A critical review of the r.c. frame existing building

Transcription

Istituto Universitario
di Studi Superiori
Università degli
Studi di Pavia
EUROPEAN SCHOOL FOR ADVANCED STUDIES IN
REDUCTION OF SEISMIC RISK
ROSE SCHOOL
A CRITICAL REVIEW OF THE R.C. FRAME EXISTING
BUILDING ASSESSMENT PROCEDURE ACCORDING TO
EUROCODE 8 AND ITALIAN SEISMIC CODE
A Dissertation Submitted in Partial
Fulfillment of the Requirements for the Master Degree in
EARTHQUAKE ENGINEERING
by
VASSILIS MPAMPATSIKOS
Supervisors: Dr LORENZA PETRINI, Dr ROBERTO NASCIMBENE
May, 2008
The dissertation entitled “A Critical Review of the R.C. Frame Existing Building Assessment
Procedure according to Eurocode 8 and Italian Seismic Code”, by Vassilis Mpampatsikos, has
been approved in partial fulfilment of the requirements for the Master Degree in Earthquake
Engineering.
Name of Reviewer 1 Dr. Lorenza Petrini
Name of Reviewer 2 Dr. Roberto Nascimbene
Index
ABSTRACT
In all Italian regions characterized by significant values of PGA, the assessment of the seismic
response of the existing structures is a priority, since the majority of the building heritage was
designed according to out-of-date or even non-seismic codes, possibly assuming values of PGA lower
than those considered nowadays. The uncertainties about the nonlinear behaviour of the structures may
be relevant, since the potential development and location of inelastic zone, as well as their ductility
capacity, are, in general, unknown. It is, therefore, unlikely that a force-based assessment, obtained
through an elastic analysis with the internal forces reduced by the behaviour factor, yields satisfactory
results. The direct consequence is that the nonlinear behaviour of the structure should be faced
directly, with corresponding strong increase of the complexity of the assessment process.
This issue was taken into account in this work; in particular, the assessment of R.C. frame buildings
has been performed according to all the possible approaches proposed in both Italian Seismic Code
[OPCM 3431] and Eurocode 8 [UNI EN 1998-3]. Both Codes consider the nonlinear methods of
analysis as the normal way to evaluate the seismic demand, while they limit the use of linear analyses
by strict conditions of applicability, in order to assure a uniform distribution of the nonlinearity.
Concerning the assessment of the response, both Codes require a force- (strength-) based procedure
for the brittle mechanisms (shear) and a displacement-based approach for the ductile ones (flexure).
The evaluation of both shear and deformation (chord rotation) capacities of the structural members of
a building subjected to a seismic input requires, in general, lengthy and not simple calculations.
On the base of these considerations, the aims of this work may be summarized as follows: I) checking
the importance of considering, in linear analyses, the effective secant stiffness of the structural
members instead of a fixed ratio of their gross stiffness. II) Checking the consistency of the results
obtained applying the two Codes, in order to show if the simpler approach of the Italian Seismic Code
may yield satisfactory results and testing if the two proposed formulas (empirical and theoretical) for
computing the chord rotation capacity may yield results close to each other. III) Suggesting simplified
approaches for the assessment procedure, concerning the evaluation of both seismic demand and
capacity of the structural members. IV) Checking the importance of the definition of a bidimensional
failure curve (which is not considered by the two Codes) in the assessment of the ductile mechanisms.
In order to give answers to the above mentioned goals, four public R.C. frame structures, built
according to out-of-date seismic codes, were examined. Considering that all buildings are irregular,
characterized by different structural configurations and by a wide number of structural members, with
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Index
different shapes, dimensions, lengths and reinforcement content, the conclusions valid for the four
buildings were used to draw the following rules, that may be judged as general, although influenced by
the considered numerical models.
Concerning the assessment of the demand, the chord rotation may be simply evaluated as the drift, for
the columns, and as the joint rotation, for the beams. Concerning the assessment of the ductile
response: I) assuming, in linear analyses, the stiffness of the structural members equal to a ratio
between 50% and 100% of their gross stiffness yields unconservative results compared to those
obtained considering their actual secant stiffness at yielding. II) Both Codes yield close results when
the chord rotation capacity is evaluated on the base of the empirical formula, while the Italian Seismic
Code tends to underestimate the results when the theoretical formula is considered. The theoretical
formula is very sensitive to the value of the shear span, while the empirical formula yields much stable
results. III) The procedure based on the empirical formula can be sensibly simplified without any loss
in the accuracy of the results, removing the dependency of the chord rotation capacity from the
seismic demand. IV) The definition of a bidimensional failure curve may be crucial for buildings
characterized by a significant torsional response and, in particular, for the dynamic linear analysis.
Concerning the assessment of the brittle response: I) the Italian Seismic Code yields results which
grossly underestimate those obtained through Eurocode 8 and, hence, the use of the procedure
suggested in Eurocode 8 is recommended. II) Assuming, in linear analyses, the stiffness of the
structural members equal to a ratio between 50% and 100% of their gross stiffness may be considered
a suitable choice for simplifying the assessment. III) The procedure can be sensibly simplified without
any loss in the accuracy of the results, removing the dependency of the shear capacity from the seismic
demand.
Keywords: Existing R.C. frame building assessment, chord rotation assessment, shear assessment,
Eurocode 8-3, Italian Seismic Code, consistency and sensitivity of assessment procedures
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ACKNOWLEDGEMENTS
The authors would like to thank the “Servizio Sismico della Regione Toscana” and, in particular,
Arch. Ferrini for making the data of the two considered buildings (architectural and structural
drawings, descriptive details, results of in-situ testing, etc) available for this study. The authors would
also like to acknowledge Dr. Rui Pinho for the useful discussions on the subject and the assistance in
the numerical analyses and to thank the EUCENTRE Geotechnical Staff for its precious assistance.
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Index
TABLE OF CONTENTS
Page
ABSTRACT
i
ACKNOWLEDGEMENTS
iii
TABLE OF CONTENTS
iv
LIST OF FIGURES
viii
LIST OF TABLES
xxiv
1 General Aspects of Seismic Assessment Procedure for Existing Buildings
1
1.1 Introduction and Outline of the Work
1
1.2 Limit States
2
1.3 Knowledge Levels
5
1.4 Ductile and Brittle Mechanisms
7
1.5 Ductility and Behaviour Factor in Existing Buildings
8
2 Methods of Analysis
10
2.1 General Aspects
10
2.2 Conditions for the Applicability of Linear Methods
11
2.2.1 1st Condition of Applicability
nd
12
2.2.2 2 Condition of Applicability (only Italian Seismic Code)
14
2.2.3 Further Considerations about the Applicability of Linear Methods
16
2.3 Common Problems related to Modelling
18
2.4 Modelling for Linear Elastic Methods of Analysis
21
2.5 Nonlinear Static Analysis
24
2.6 Nonlinear Dynamic Analysis
35
3 Assessment Procedure for R.C. Frame Structures
39
3.1 Ductile Mechanisms: Chord Rotation Demand
39
3.2 Ductile Mechanisms: Chord Rotation Capacity
43
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Index
3.2.1 Damage Limitation Limit State
44
3.2.2 Near Collapse Limit State: Empirical Approach
49
3.2.3 Near Collapse Limit State: Theoretical approach
52
3.3 Ductile Check: Biaxial Bending
60
3.4 Brittle Mechanisms: Shear Capacity
62
3.4.1 Shear Capacity according to the Italian Seismic Code
63
3.4.2 Shear Capacity according to Eurocode 8
66
3.5 Brittle Mechanisms: Shear Demand
4 Analyzed Buildings
4.1 Sede Comunale - Vagli Sotto (Lucca, Tuscany)
70
75
75
4.1.1 Geometry
77
4.1.2 Materials
78
4.1.3 Knowledge Level
78
4.1.4 Seismic Input
78
4.1.5 Loads
79
4.2 Scuola Elementare “Pascoli” - Barga (Lucca, Tuscany)
79
4.2.1 Geometry
80
4.2.2 Materials
81
81
4.2.4 Seismic Input
81
4.2.5 Loads
82
4.3 Scuola Media Inferiore “Puccetti” - Gallicano (Lucca, Tuscany)
82
4.3.1 Geometry
84
4.3.2 Materials
84
85
4.3.4 Seismic Input
85
4.3.5 Loads
86
4.4 Scuola Media Inferiore Don Bosco - Rapagnano (Ascoli Piceno, Marche)
86
4.4.1 Geometry
88
4.4.2 Materials
88
88
4.4.4 Seismic Input
89
4.4.5 Loads
89
5 Assessment of the Sede Comunale (Vagli Sotto)
5.1 Linear Analyses
5.1.1 Computational Model
91
91
91
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Index
5.1.2 Eigenvalue Analyses
94
5.1.3 Dynamic Linear Analysis
96
5.2 Nonlinear Analyses
109
109
5.2.2 Static Nonlinear Analysis
110
5.2.3 Dynamic Nonlinear Analysis
120
6 Assessment of the Scuola Elementare Pascoli (Barga)
6.1 Linear Analyses
128
128
128
6.1.2 Eigenvalue Analysis
130
132
144
144
146
156
7 Assessment of the Scuola Media Inferiore Puccetti (Gallicano)
7.1 Linear Analyses
164
164
164
166
168
180
180
182
193
8 Assessment of Scuola Media Inferiore Don Bosco (Rapagnano)
8.1 Linear Analyses
200
200
200
201
204
214
214
217
226
9 Summary of the Results and Conclusions
9.1 Summary of the Results of Sede Comunale (Vagli Sotto)
9.1.1 Applicability Conditions of the Linear Methods
233
233
233
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Index
9.1.2 Assessment of Ductile Mechanisms
234
9.1.3 Assessment of Brittle Mechanisms
236
9.2 Summary of the Results of Scuola Elementare Pascoli (Barga)
238
238
239
241
9.3 Summary of the Results of Scuola Media Inferiore Puccetti (Gallicano)
243
243
244
247
9.4 Summary of the Results of Scuola Media Inferiore Don Bosco (Rapagnano)
248
249
249
253
9.5 Comparisons and Conclusions about the Assessment of Ductile Mechanisms
254
9.6 Conclusions about the Assessment of Brittle Mechanisms
261
REFERENCES
264
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Index
LIST OF FIGURES
Page
Figure 1.1. Performance Levels and Limit States....................................................................................4
Figure 1.2. Knowledge Levels, allowed methods of analysis and Confidence Factors...........................6
Figure 1.3 “q-factor” determined through pushover analysis ..................................................................8
Figure 2.1. Joint bending moment equilibrium...................................................................................... 14
Figure 2.2. Equilibrium conditions of beams.........................................................................................16
Figure 2.3. In-plan deformability of slab due to large distance between stiff vertical elements ...........19
Figure 2.4. Typical M-Φ diagram of a R.C. section ..............................................................................21
Figure 2.5 Method of equality of areas..................................................................................................26
Figure 2.6. Typical σ-ε relationship of confined and unconfined concrete............................................ 29
Figure 2.7. Bi-linear idealization of σ-ε relationship of steel.................................................................30
Figure 2.8. Load control pushover analysis ...........................................................................................30
Figure 2.9. Response control pushover analysis.................................................................................... 31
Figure 3.1. Definition of shear span ......................................................................................................40
Figure 3.2. Chord rotation demand for columns....................................................................................41
Figure 3.3. Chord rotation demand for beams, neglecting the contribution of gravity loads ................ 42
Figure 3.4. Total chord rotation demand for beams...............................................................................43
Figure 3.5. Joint vertical forces equilibrium..........................................................................................51
Figure 3.6. M-Φ diagram .......................................................................................................................52
Figure 3.7. Shear capacity according to “D.M. 09-01-1996”................................................................65
Figure 3.8. Degradation of shear capacity due to development of plastic hinges..................................67
Figure 3.9. Equilibrium conditions for shear demand (Eurocode 8 and Italian Seismic Code).............72
Figure 3.10. Equilibrium conditions proposed in this work to compute shear demand.........................73
Figure 4.1. South-East view of the Sede Comunale .............................................................................. 76
Figure 4.2. South-West view of the Sede Comunale .............................................................................76
Figure 4.3. North-West view of the Sede Comunale .............................................................................76
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Index
Figure 4.4. North-East view of the Sede Comunale ..............................................................................76
Figure 4.5. Plan and front views of the Sede Comunale........................................................................77
Figure 4.6. Elastic horizontal acceleration spectra for the Sede Comunale........................................... 79
Figure 4.7. Elastic horizontal displacement spectra for the Sede Comunale .........................................79
Figure 4.8. Main entrance view of the Scuola Pascoli...........................................................................80
Figure 4.9. Rear view of the Scuola Pascoli ..........................................................................................80
Figure 4.10. Floors plan view of the Scuola Pascoli..............................................................................80
Figure 4.11. Elastic horizontal acceleration spectra for the Scuola Pascoli...........................................82
Figure 4.12. Elastic horizontal displacement spectra for the Scuola Pascoli......................................... 82
Figure 4.13. Main entrance of the Scuola Puccetti ................................................................................83
Figure 4.14. Rear view of the Scuola Puccetti.......................................................................................83
Figure 4.15. Lateral view of the Scuola Puccetti ................................................................................... 83
Figure 4.16. Lateral view of the Scuola Puccetti ................................................................................... 83
Figure 4.17. Plan views of the Scuola Puccetti......................................................................................83
Figure 4.18. Front views of the Scuola Puccetti ....................................................................................84
Figure 4.19. South view of the Scuola Don Bosco ................................................................................ 86
Figure 4.20. Frontal view of the Scuola Don Bosco.............................................................................. 86
Figure 4.21. East view of the Scuola Don Bosco ..................................................................................87
Figure 4.22. North view of the Scuola Don Bosco ................................................................................ 87
Figure 4.23. Plan views of the Scuola Don Bosco.................................................................................87
Figure 4.24. Front views of the Scuola Don Bosco ............................................................................... 87
Figure 4.25. Elastic horizontal acceleration spectra for the Scuola Don Bosco .................................... 89
Figure 4.26. Elastic horizontal displacement spectra for the Scuola Don Bosco ..................................89
Figure 5.1. Front view of Sede Comunale .............................................................................................93
Figure 5.2. Rear view of Sede Comunale ..............................................................................................93
Figure 5.3. 1st mode of interest of Sede Comunale................................................................................ 96
Figure 5.4. 2nd mode of interest of Sede Comunale ...............................................................................96
Figure 5.5. Ductile check: dynamic linear analysis, empirical form, OPCM, different EI..................100
Figure 5.6. Ductile check: dynamic linear analysis, empirical form, EC8, different EI......................100
Figure 5.7. Ductile check: dynamic linear analysis, theoretical form, OPCM, different EI................100
Figure 5.8. Ductile check: dynamic linear analysis, theoretical form, EC8, different EI ....................100
Figure 5.10. Ductile check: dynamic linear analysis, empirical form, EC8, different EI....................100
Figure 5.11. Ductile check: dynamic linear analysis, theoretical form, OPCM, different EI..............101
Figure 5.12. Ductile check: dynamic linear analysis, theoretical form, EC8, different EI ..................101
Figure 5.13. Ductile check: dynamic linear analysis, empirical form, EIeff, OPCM vs. EC8 ..............101
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Index
Figure 5.14. Ductile check: dynamic linear analysis, theoretical form, EIeff, OPCM vs. EC8 ............101
Figure 5.15. Ductile check: dynamic linear an., empirical form, EIeff, OPCM, Ls=M/V vs. Ls=L/2 ...102
Figure 5.16. Ductile check: dynamic linear an., empirical form, EIeff, EC8, Ls=M/V vs. Ls=L/2........102
Figure 5.17. Ductile check: dynamic linear an., theoretical form, EIeff, OPCM, Ls=M/V vs. Ls=L/2..102
Figure 5.18. Ductile check: dynamic linear an., theoretical form, EIeff, EC8, Ls=M/V vs. Ls=L/2......102
Figure 5.19. Ductile check: dynamic linear analysis, empirical form, EIeff, OPCM, Ggrav vs. Gseism ...103
Figure 5.20. Ductile check: dynamic linear analysis, empirical form, EIeff, EC8, Ggrav vs. Gseism .......103
Figure 5.21. Ductile check: dynamic linear an., theoretical form, EIeff, OPCM, Ggrav vs. Gseism..........103
Figure 5.22. Ductile check: dynamic linear an., theoretical form, EIeff, EC8, Ggrav vs. Gseism ..............103
Figure 5.23. Ductile check: dynamic linear analysis, theoretical form, EIeff, OPCM, different φy ......103
Figure 5.24. Ductile check: dynamic linear analysis, theoretical form, EIeff, EC8, different φy ..........103
Figure 5.25. Ductile check: dynamic linear an., emp. form, EIeff, OPCM, correct vs. simplified .......104
Figure 5.26. Ductile check: dynamic linear an., emp. form, EIeff, EC8, correct vs. simplified............104
Figure 5.27. Ductile check: dynamic linear an., theor. form, EIeff, OPCM, correct vs. simplified ......104
Figure 5.28. Ductile check: dynamic linear an., theor. form, EIeff, EC8, correct vs. simplified ..........104
Figure 5.29. Ductile check: dynamic linear an., EIeff, OPCM, theor. vs. emp., correct vs. simplified 105
Figure 5.30. Ductile check: dynamic linear an., EIeff, EC8, theor. vs. emp., correct vs. simplified.....105
Figure 5.31. Ductile check: dynamic linear an., emp. form., EIeff, OPCM, uni- vs. bi-axial bending .105
Figure 5.32. Ductile check: dynamic linear an. emp. form., EIeff, EC8, uni- vs. bi-axial bending ......105
Figure 5.33. Ductile check: dynamic linear an., theor. form., EIeff, OPCM, uni vs. bi-axial bending .105
Figure 5.34. Ductile check: dynamic linear an., theor. form., EIeff, EC8, uni- vs. bi-axial bending ....105
Figure 5.35. Brittle check: dynamic linear analysis, OPCM, different EI ...........................................106
Figure 5.36. Brittle check: dynamic linear analysis, EC8, different EI ...............................................106
Figure 5.39. Brittle check: dynamic linear analysis, OPCM, EIeff, Ggrav vs. Gseism...............................107
Figure 5.40. Brittle check: dynamic linear analysis, EC8, EIeff, Ggrav vs. Gseism ...................................107
Figure 5.41. Brittle check: dynamic linear analysis, EC8, EIeff, different φy .......................................108
Figure 5.42. Brittle check: dynamic linear analysis, EC8, EIeff, Ls=M/V vs. Ls=L/2...........................108
Figure 5.43. Brittle check: dynamic linear an., EIeff, correct OPCM vs. simplified OPCM vs. EC8...108
Figure 5.44. Brittle check: dynamic linear an., EC8, correct vs. simplified approach ........................ 108
Figure 5.45. Concrete: nonlin. confinement model ............................................................................. 109
Figure 5.46. Steel: bilinear strain hardening model ............................................................................. 109
Figure 5.47. Uniform distr., positive X, X-dir view ............................................................................ 111
Figure 5.48. Modal distr., positive X, X-dir view................................................................................111
Figure 5.49. Uniform distr., positive Y, Y-dir view ............................................................................ 111
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Index
Figure 5.50. Modal distr., positive Y, Y-dir view................................................................................111
Figure 5.51. Uniform distr., positive Y, X-dir view ............................................................................ 111
Figure 5.52. Modal distr., positive Y, X-dir view................................................................................111
Figure 5.53. Pushover Uniform X-dir, SDOF capacity curve: equivalent area method ......................112
Figure 5.54. Pushover Uniform X-dir, MDOF capacity curve: DL, SD and NC LS...........................112
Figure 5.55. Pushover Modal X-dir, SDOF capacity curve: equivalent area method .........................112
Figure 5.56. Pushover Modal X-dir, MDOF capacity curve: DL, SD and NC LS ..............................112
Figure 5.57. Pushover Uniform Y-dir, SDOF capacity curve: equivalent area method ......................113
Figure 5.58. Pushover Uniform Y-dir, MDOF capacity curve: DL, SD and NC LS...........................113
Figure 5.59. Pushover Modal Y-dir, SDOF capacity curve: equivalent area method .........................113
Figure 5.60. Pushover Modal Y-dir, MDOF capacity curve: DL, SD and NC LS ..............................113
Figure 5.61. Pushover Uniform and Modal X-dir, MDOF capacity curves: SD LS............................ 113
Figure 5.62. Pushover Uniform and Modal Y-dir, MDOF capacity curves: SD LS............................ 113
Figure 5.63. Ductile check: static nonlinear analysis, empirical form, OPCM vs. EC8......................114
Figure 5.64. Ductile check: static nonlinear an., theoretical form, OPCM vs. EC8 ............................114
Figure 5.65. Ductile check: static nonlinear analysis, theoretical form, OPCM, different φy ..............114
Figure 5.66. Ductile check: static nonlinear an., theoretical form, EC8, different φy ..........................114
Figure 5.67. Ductile check: static nonlinear analysis, empirical form, OPCM, Ls=M/V vs. Ls=L/2 ...115
Figure 5.68. Ductile check: static nonlinear an., empirical form, EC8, Ls=M/V vs. Ls=L/2 ...............115
Figure 5.69. Ductile check: static nonlinear analysis, theoretical form, OPCM, Ls=M/V vs. Ls=L/2 .115
Figure 5.70. Ductile check: static nonlinear an., theoretical form, EC8, Ls=M/V vs. Ls=L/2 .............115
Figure 5.71. Ductile check: static nonlinear an., empirical form, OPCM, NGrav vs. NSeism...................116
Figure 5.72. Ductile check: static nonlinear an., empirical form, EC8, NGrav vs. NSeism .......................116
Figure 5.73. Ductile check: static nonlinear an., theoretical form, OPCM, NGrav vs. NSeism .................116
Figure 5.74. Ductile check: static nonlinear an., theoretical form, EC8, NGrav vs. NSeism .....................116
Figure 5.75. Ductile check: static nonlinear an., empirical form, OPCM, correct vs. simplified........117
Figure 5.76. Ductile check: static nonlinear an., empirical form, EC8, correct vs. simplified ............117
Figure 5.77. Ductile check: static nonlinear an., theoretical form, OPCM, correct vs. simplified ......117
Figure 5.78. Ductile check: static nonlinear an., theoretical form, EC8, correct vs. simplified ..........117
Figure 5.79. Ductile check: static nonlinear an., OPCM, theor. vs. emp., correct vs. simplified ........118
Figure 5.80. Ductile check: static nonlinear an., EC8, theor. vs. emp., correct vs. simplified ............118
Figure 5.81. Ductile check: static nonlinear an., empirical form, OPCM, uni- vs. bi-axial bending...118
Figure 5.82. Ductile check: static nonlinear an., empirical form, EC8, uni- vs. bi-axial bending.......118
Figure 5.83. Ductile check: static nonlinear an., theor. form, OPCM, uni- vs. bi-axial bending ........118
Figure 5.84. Ductile check: static nonlinear an., theor. form, EC8, uni- vs. bi-axial bending............. 118
Figure 5.85. Brittle check: static nonlinear analysis, OPCM, Ggrav vs. Gseism ......................................119
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Index
Figure 5.86. Brittle check: static nonlinear analysis, EC8, Ggrav vs. Gseism ..........................................119
Figure 5.87. Brittle check: static nonlinear analysis, EC8, different φy ...............................................120
Figure 5.88. Brittle check: static nonlinear analysis, EC8, Ls=M/V vs. Ls=L/2 ..................................120
Figure 5.89. Brittle check: static nonlinear analysis, correct OPCM vs. simplified OPCM vs. EC8 ..120
Figure 5.90. Brittle check: static nonlinear analysis, EC8, correct vs. simplified approach................120
Figure 5.91. 5% damped response spectra of artificial accelerograms (100% intensity) ....................121
Figure 5.92. Equivalent viscous damping properties........................................................................... 121
Figure 5.93. Main accelerogram in X, X-dir view............................................................................... 122
Figure 5.94. Main accelerogram in X, Y-dir view............................................................................... 122
Figure 5.95. Main accelerogram in Y, Y-dir view............................................................................... 122
Figure 5.96. Main accelerogram in Y, X-dir view............................................................................... 122
Figure 5.97. Control node displ: main acc. in X..................................................................................122
Figure 5.98. Control node displ: main acc. in Y..................................................................................122
Figure 5.99. Ductile check: dynamic nonlinear an., empirical form, OPCM vs. EC8.........................123
Figure 5.100. Ductile check: dynamic nonlin. an., theoretical form, OPCM vs. EC8.........................123
Figure 5.101. Ductile check: dynamic nonlinear an., theoretical form, OPCM, different φ ................123
Figure 5.102. Ductile check: dynamic nonlin. an., theoretical form, EC8, different φ ........................123
Figure 5.103. Ductile check: dynamic nonlinear an., empirical form, OPCM, Ls=M/V vs. Ls=L/2....124
Figure 5.104. Ductile check: dynamic nonlin. an., empirical form, EC8, Ls=M/V vs. Ls=L/2............124
Figure 5.105. Ductile check: dynamic nonlinear an., theoretical form, OPCM, Ls=M/V vs. Ls=L/2 ..124
Figure 5.106. Ductile check: dynamic nonlin. an., theoretical form, EC8, Ls=M/V vs. Ls=L/2 ..........124
Figure 5.107. Ductile check: dynamic nonlinear an., theor. form, OPCM, correct vs. simplified ......125
Figure 5.108. Ductile check: dynamic nonlin. an., theor. form, EC8, correct vs. simplified ..............125
Figure 5.109. Ductile check: dynamic nonlinear an., OPCM, theor. vs. emp., correct vs. simplified.125
Figure 5.110. Ductile check: dynamic nonlin. an., EC8, theor. vs. emp., correct vs. simplified.........125
Figure 5.111. Ductile check: dynamic nonlinear an., empir. form, OPCM, uni- vs. bi-axial bending 125
Figure 5.112. Ductile check: dynamic nonlin. an., empir. form, EC8, uni- vs. bi-axial bending ........125
Figure 5.113. Ductile check: dynamic nonlinear an., theor. form, OPCM, uni- vs. bi-axial bending .126
Figure 5.114. Ductile check: dynamic nonlin. an., theor. form, EC8, uni- vs. bi-axial bending ......... 126
Figure 5.115. Brittle check: dynamic nonlinear an., EC8, different φy ................................................126
Figure 5.116. Brittle check: dynamic nonlinear an., EC8, Ls=M/V vs. Ls=L/2 ...................................126
Figure 5.117. Brittle check: dynamic nonlinear an., OPCM vs. EC8..................................................127
Figure 5.118. Brittle check: dynamic nonlinear an., EC8, correct vs. simplified approach ................ 127
Figure 6.1. Front view of Scuola Pascoli.............................................................................................130
Figure 6.2. Rear view of Scuola Pascoli ..............................................................................................130
Figure 6.3. 1st mode of Scuola Pascoli.................................................................................................132
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Index
Figure 6.4. 2nd mode of Scuola Pascoli................................................................................................132
Figure 6.5. 3rd mode of Scuola Pascoli, view in X............................................................................... 132
Figure 6.6. 3rd mode of Scuola Pascoli, view in Y............................................................................... 132
Figure 6.13. Ductile check: dynamic linear analysis, empirical form, OPCM, different EI................136
Figure 6.23. Ductile check: dynamic linear an., empirical form, EIeff, OPCM, Ggrav vs. Gseism ...........139
Figure 6.24. Ductile check: dynamic linear an., empirical form, EIeff, EC8, Ggrav vs. Gseism................139
Figure 6.33. Ductile check: dynamic linear an., EIeff, OPCM, theor. vs. emp., correct vs simplified . 141
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Index
Figure 6.49. Concrete: nonlinear confinement model .........................................................................145
Figure 6.50. Reinforcement steel: bilinear (strain hardening) model ..................................................145
Figure 6.69. Ductile check: static nonlinear an., empirical form, OPCM, Ls=M/V vs. Ls=L/2 ...........151
Figure 6.71. Ductile check: static nonlinear an., theoretical form, OPCM, Ls=M/V vs. Ls=L/2 .........151
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Index
Figure 6.81. Ductile check: static nonlinear an., OPCM, theor. vs. emp., correct vs. simpl. ..............153
Figure 6.82. Ductile check: static nonlinear an., EC8, theor. vs. emp., correct vs. simpl. ..................153
Figure 6.83. Ductile check: static nonlinear an., empir. form, OPCM, uni- vs. bi-axial bending .......154
Figure 6.84. Ductile check: static nonlinear an., empir. form, EC8, uni- vs. bi-axial bending............154
Figure 6.85. Ductile check: static nonlinear an., theor. form, OPCM, uni- vs. bi-axial bending ........154
Figure 6.86. Ductile check: static nonlinear an., theor. form, EC8, uni- vs. bi-axial bending............. 154
Figure 6.94.Equivalent viscous damping properties............................................................................157
Figure 6.100. Control node displ: main acc. in Y................................................................................ 158
Figure 6.101. Ductile check: dynamic nonlinear an., empirical form, OPCM vs. EC8.......................159
Figure 6.107. Ductile check: dynamic nonlinear an., theor. form, OPCM, Ls=M/V vs. Ls=L/2..........160
Figure 6.108. Ductile check: dynamic nonlin. an., theor. form, EC8, Ls=M/V vs. Ls=L/2..................160
Figure 6.111. Ductile check: dynamic nonlinear an., OPCM, theor. vs. emp., correct vs. simpl. .......161
xv
Index
Figure 6.112. Ductile check: dynamic nonlin. an., EC8, theor. vs. emp., correct vs. simpl. ............... 161
Figure 6.119, Brittle check: dynamic nonlinear an., OPCM vs. EC8..................................................163
Figure 7.1. View of Scuola Puccetti ....................................................................................................166
Figure 7.2. 1st mode of Scuola Puccetti, view in Y.............................................................................. 168
Figure 7.3. 1st mode of Scuola Puccetti, view in X.............................................................................. 168
Figure 7.4. 2nd mode of Scuola Puccetti, view in X............................................................................. 168
Figure 7.15. Ductile check: dynamic linear an., empirical form, EIeff, OPCM vs. EC8 ......................173
Figure 7.16. Ductile check: dynamic linear an., theoretical form, EIeff, OPCM vs. EC8 ....................173
xvi
Index
Figure 7.31. Ductile check: dynamic linear an., EIeff, OPCM, theor. vs. emp., correct vs. simplified 176
Figure 7.48. Steel: bilinear (strain hardening) model ..........................................................................181
Figure 7.50. Uniform distr., positive X, top view................................................................................183
Figure 7.52. Modal distr., positive X, top view ...................................................................................183
Figure 7.54. Uniform distr., positive Y, top view................................................................................183
Figure 7.56. Modal distr., positive Y, top view ...................................................................................183
xvii
Index
Figure 7.87. Ductile check: static nonlinear an., theoretical form, OPCM, uni- vs. bi-axial bending.191
Figure 7.88. Ductile check: static nonlinear an., theoretical form, EC8, uni- vs. bi-axial bending .....191
Figure 7.93. Brittle check: static nonlinear analysis, correct OPCM, simplified OPCM, EC8 ...........193
xviii
Index
Figure 7.100. Main accelerogram in Y, X-dir view............................................................................. 194
Figure 7.101. Control node displ.: main acc. in X ............................................................................... 195
Figure 7.102. Control node displ: main acc. in Y................................................................................ 195
Figure 7.103. Ductile check: dynamic nonlinear an., empirical form, OPCM vs. EC8.......................195
Figure 7.113. Ductile check: dynamic nonlinear an., OPCM, theor. vs. emp., correct vs. simpl. .......197
Figure 7.114. Ductile check: dynamic nonlin. an., EC8, theor. vs. emp., correct vs. simpl. ............... 197
Figure 8.1. View of Scuola Don Bosco ...............................................................................................201
Figure 8.2. 1st mode of Scuola Don Bosco, view in Y.........................................................................203
Figure 8.3. 2nd mode of Scuola Don Bosco view in X.........................................................................203
Figure 8.4. 3rd mode of Scuola Don Bosco, view in X ........................................................................ 203
Figure 8.5. 3rd mode of Scuola Don Bosco view in Y .........................................................................203
xix
Index
Figure 8.30. Ductile check: dynamic linear an. EIeff, OPCM, theor. vs. emp., correct vs. simplified .211
Figure 8.47. Steel: bilinear (strain hard.) model ..................................................................................215
Figure 8.48. Top view of the Scuola Don Bosco: cross braces locations ............................................ 216
xx
Index
Figure 8.83. Ductile check: static nonlinear an., theoretical form, OPCM, uni- vs. bi-axial bending.224
Figure 8.84. Ductile check: static nonlinear an., theoretical form, EC8, uni- vs. bi-axial bending .....224
xxi
Index
Figure 8.98. Control node displ: main acc. in Y..................................................................................228
Figure 8.99. Ductile check: dynamic nonlinear an., empirical form, OPCM vs. EC8.........................228
Figure 9.1. Ductile check: dynamic linear an., empirical form, EIeff, OPCM vs. EC8 ........................255
Figure 9.2. Ductile check: dynamic linear an., theoretical form, EIeff, OPCM vs. EC8 ......................255
Figure 9.3. Ductile check: static nonlinear an., empirical form, EIeff, OPCM vs. EC8........................255
Figure 9.4. Ductile check: static nonlinear an., theoretical form, EIeff, OPCM vs. EC8......................255
Figure 9.5. Ductile check: dynamic nonlinear an., empirical form, EIeff, OPCM vs. EC8 ..................255
xxii
Index
Figure 9.6. Ductile check: dynamic nonlinear an., theoretical form, EIeff, OPCM vs. EC8 ................255
Figure 9.13. Ductile check: static nonlinear an., emp. form, EIeff, EC8, correct vs. simplified...........258
Figure 9.14. Ductile check: static nonlinear an., theor. form, EIeff, EC8, correct vs. simplified..........258
Figure 9.15. Ductile check: dynamic nonlinear an., emp. form, EIeff, EC8, correct vs. simplified......258
Figure 9.16. Ductile check: dynamic nonlinear an., theor. form, EIeff, EC8, correct vs. simplified ....258
Figure 9.17. Ductile check: dynamic linear an., EIeff, EC8, theor. vs. emp., correct vs simplified......259
Figure 9.18. Ductile check: static nonlinear an., EIeff, EC8, theor. vs. emp., correct vs simplified.....259
Figure 9.19. Ductile check: dynamic nonlinear an., EIeff, EC8, theor. vs. emp., correct vs simplified 259
Figure 9.20. Ductile check: dynamic linear an., emp. form., EIeff, EC8, uni- vs. bi-axial bending .....260
Figure 9.21. Ductile check: static nonlinear an., emp. form., EIeff, EC8, uni- vs. bi-axial bending.....260
Figure 9.22. Ductile check: dynamic nonlinear an., emp. form., EIeff, EC8, uni- vs. bi-axial bending260
Figure 9.23. Brittle check: dynamic linear analysis, EIeff, OPCM vs. EC8 .........................................261
Figure 9.24. Brittle check: static nonlinear analysis, EIeff, OPCM vs. EC8.........................................261
Figure 9.25. Brittle check: dynamic nonlinear analysis, EIeff, OPCM vs. EC8....................................261
Figure 9.28. Brittle check: dynamic linear analysis, EC8, correct vs. simplified approach ................263
Figure 9.30. Brittle check: dynamic nonlinear analysis, EC8, correct vs. simplified approach ..........263
xxiii
Index
LIST OF TABLES
Page
Table 5.1. Comparison between the first three modes of interest obtained considering both the soilstructure interaction and fixed foundations...................................................................................93
Table 5.2. All modes of interest, different EI ........................................................................................94
Table 5.3. Conditions of applicability of linear methods according to both Codes...............................98
Table 5.4. Comparison between the first three modes - SeismoStruct and SAP models..................... 110
Table 6.1. All modes of interest, different EI ......................................................................................131
Table 6.2. Conditions of applicability of linear methods according to both Codes.............................134
xxiv
Chapter 1. General Aspects of Seismic Assessment Procedure for Existing Buildings
1 General Aspects of Seismic Assessment Procedure for Existing
Buildings
1.1 Introduction and Outline of the Work
In all Italian regions characterized by significant values of PGA, the assessment of the seismic
response of existing structures is a priority, since the vast majority of the building heritage
was designed according to out-of-date seismic codes or, in the worst cases, even to nonseismic codes, possibly assuming values of PGA lower than those considered nowadays. For
most of these structures, the uncertainties about the nonlinear behaviour are relevant, since,
generally, the potential development and location of inelastic zones, as well as their ductility
capacity, are not known. The uncertainties will increase if the building is not regular, since the
nonlinear behaviour will not spread uniformly throughout the structure and the inelastic zones
will not form all at the same level of intensity of the seismic input. The consequence is that it
is difficult to define general objective criteria to evaluate the global expected ductility
capacity of the structure with a satisfactory degree of approximation, without performing any
nonlinear analysis. Therefore, it is hard to define a direct correlation between the nonlinear
internal forces that develop in the system during the seismic excitation and those experienced
by an equivalent indefinitely elastic structure. For this reason, the force-based assessment
procedure, obtained using an elastic analysis and reducing the internal forces by the so-called
behaviour factor “q”, does not yield, in general, satisfactory results. Hence, the nonlinear
behaviour of the structure should be faced directly, using both force- (strength-) based and
displacement-based assessment procedures. The assessment process, therefore, turns out to be
considerably complex and lengthy.
A further source of complexity lies in the fact that both mechanical and geometrical properties
of existing buildings are, in general, not known in a satisfactory way. Anyway, even if the
knowledge of the original design data is exhaustive, there could be discrepancies between the
original drawings and the actual structure. Moreover, the mechanical properties could have
undergone degradation with respect to their original values. These considerations involve that
the “Knowledge Level” of the geometry, details and materials must be introduced as a further
variable in the assessment process.
In this work, the assessment of the seismic response of R.C. frame buildings has been
examined. All procedures suggested in both Italian Seismic Code [OPCM 3431, Attachment
2, Ch. 11, 03/05/2005] and Eurocode 8 [UNI EN 1998-3, August 2005] were critically
considered and performed.
1
In the following part of this Chapter, the theoretical concepts, which lie at the base of the
seismic assessment procedures recommended by the two considered Codes, will be discussed.
In particular, attention will be paid to those aspects which differ from the design of new
constructions, i.e. the definition of the Limite States, the introduction of the “Knowledge
Levels”, the definition of the ductile and brittle mechanisms of the structural members (beams
and columns) and the role of the behaviour factor “q” for the existing R.C. frame buildings.
In Chapter 2, all possible assessment procedures suggested by the two considered Codes will
be examined in detail, trying to identify all aspects which may lead to different possible
interpretations and to describe all cases in which the applicability of the analysis is doubtful.
Some of these problems, highlighted in the text using boldface, will be discussed in this work,
while the answers to the other raised questions will be left as subjects for possible future
researches.
In Chapter 3, the assessment procedures of the seismic response of the structural members
(beams and columns), based on all the proposed methods of analysis, will be deeply described
and commented, with the aims of clarifying those aspects which may lead to different
interpretations and suggesting possible ways to simplify the assessment procedures,
concerning the evaluation of both seismic demand and capacity of the structural members.
The main target of this work is, in fact, to try to get faster and simpler approaches to evaluate
the structural response at the member level, without loosing in accuracy of the results.
Chapter 4 consists in the detailed description of the four considered R.C. frame buildings.
They are irregular both in plan and elevation, with different structural configurations and
values of mechanical properties. Considering also the wide number of assessed structural
members, characterized by different shapes, dimensions, lengths and reinforcement content,
the conclusion based on the obtained results can be, hence, judged as satisfactory, although
influenced by the considered numerical models.
In Chapters 5, 6, 7 and 8, all the results obtained from the assessment of the four buildings
will be shown through tables and bar charts. All methods of analysis (linear and nonlinear,
static and dynamic) are considered, even in those cases in which some of them do not satisfy
applicability conditions (discussed in Chapter 2). In fact, the principal aim of this work is to
compare the different assessment procedures of Eurocode 8 and Italian Seismic Code at
structural member level, suggesting eventually simplified solutions. The evaluation of the
reliability of the different methods of analysis, in order to determine the need of retrofitting,
will be object of future research works.
In Chapter 9, the conclusions drawn in Chapters 5, 6, 7 and 8 will be, first, summarized and,
then, compared, to see which considerations are common to all the studied buildings, in order
to propose general rules for obtaining fast but also reliable assessment procedures of both
ductile and brittle mechanisms of the structural members.
1.2 Limit States
The assessment and consequent retrofitting of existing buildings are based on the idea that the
construction needs to satisfy different performance levels, depending on the intensity of the
2
seismic input. Requiring that the building remains in its elastic range independently of the
intensity of the seismic action would lead to a very expensive retrofitting. Hence, the aim of
the Codes is, instead, to minimize the expected total cost, considering both economic efforts
required for the retrofitting and costs of repairing or even reconstructing the building after an
earthquake. The parameter which plays the most important role is the relationship between the
expected rate of occurrence and the intensity of the ground motion, that is the so-called
“seismic hazard level”. Therefore, the structure should be able to:
•
resist a minor level of earthquake ground motion without damages. Namely it should
guarantee a sufficient stiffness to remain in its full elastic range;
•
resist a moderate level of earthquake ground motion without structural damage, but
possibly experience some non-structural damage. Namely it should be sufficiently
strong to guarantee that seismic demands are less than capacities in terms of forces;
•
resist a major level of earthquake ground motion without collapse, but possibly
experience wide structural as well non-structural damage. Namely, it should be
sufficiently ductile to guarantee the capacity of developing a dissipative nonlinear
behaviour.
Moreover, retrofitting a structure in order that its response remains elastic, independently of
the intensity of the seismic action, may lead to several problems besides the economic ones,
like the limited capacity of the structure to dissipate energy, since there is no hysteretic
damping and the lack of control on the possible forming of nonlinear modes under a ground
motion of larger intensity than that considered in the assessment procedure. On the base of
these considerations, the following different performance levels are identified:
•
“operational”: the structure should experience neither structural nor non-structural
damage;
•
“immediate occupancy”: the structure should experience lightly damages, limited to
non-structural elements. It should be easily and economically repaired at a later stage;
•
“life-safety”: the structure should guarantee life-safety and retain a certain lateral
stiffness and resistance, even if it undergoes heavy damage. The reparability, even if
possible, may be not economic and demolition and reconstruction may be preferable;
•
“near collapse”: the structure should remain stable and capable of carrying vertical
loads, although it could be at the verge of local collapse and, therefore, has to be
considered unsafe. Repair is not technically feasible and the demolition is necessary.
Since the vast majority of existing buildings were designed without considering capacity
design principles, there is a lack of control of the potential development of nonlinear
mechanisms. The location of potential inelastic zones and the intensity of the seismic action at
which they will form are unknown. Moreover, the structure is not prevented from the forming
of brittle modes (such as soft stories or shear failures). The consequence is that it is not
sufficient to analyze the building at the Severe Damage Limit State (which corresponds to the
3
Ultimate Limit State in the design process) in order to consider it safe, since it is possible that
the complete nonlinear mechanism of the entire structure will form in correspondence to a
stronger ground motion. There is, therefore, the need of assessing the building also at the
incipient collapse of the building, in order to assure that, even at that stage, the structure nor
becomes unstable neither presents local or even global brittle collapse and retains, however,
the capability of carrying the vertical loads. For these reasons, both Eurocode 8 and Italian
Seismic Code require the definition of three different Limit States (LS), which may be
characterized as follows (Figure 1.1):
•
LS of Damage Limitation (DL): the structure is lightly damaged, presents negligible
permanent deformations (elements prevented from significant yielding) and retains
full lateral strength and stiffness. Only non-structural components may show cracks
and damages which, however, could be economically repaired. The Codes require that
this LS should be checked under a ground motion characterized (in Italy) by a return
period of 72 years (probability of exceedance of 50% in 50 years). It should be
identify with the “immediate occupancy” performance level.
•
LS of Significant Damage (SD): the structure is significantly damaged, presents
moderate permanents deformations and some residual lateral strength and stiffness. It
is still capable of sustaining vertical loads and after-shocks of moderate intensity. The
Codes require that this LS shall be checked under a ground motion characterized by a
return period of 475 years (probability of exceedance of 10% in 50 years). It should be
identify with the “life-safety” performance level.
•
LS of Near Collapse (NC): the structure is heavily damaged, presents large permanents
deformations and low residual lateral strength and stiffness. It is still capable of
sustaining vertical loads, but it would probably not survive another earthquake. The
Codes require that this LS shall be checked under a ground motion characterized by a
return period of 2475 years (probability of exceedance of 2% in 50 years). It should be
identified with the “near collapse” performance level.
Figure 1.1. Performance Levels and Limit States
4
1.3 Knowledge Levels
The degree of knowledge of geometry, details and materials is a variable which must be
considered in the procedure of assessment of existing buildings. The idea is to penalize the
assessment of the structure in function of the lack of accuracy in the data acquisition process.
In fact, the less exhaustive the Knowledge Level (KL) is, the larger the dispersion of the
values of local and global capacity of the structure will be (in terms of both forces and
deformations). Hence, in order to guarantee a protection against this uncertainty, a parameter
called “Confidence Factor” (CF), function of the acquired KL, is introduced to penalize the
mechanical properties of the materials used in the assessment calculations. It means that the
mean values obtained from in-situ tests and from additional sources of information should be
divided by the CF to reduce the capacity or multiplied by the CF to increase the demand.
Following both Eurocode 8 and Italian Seismic Code, the KL is classified as follows:
•
KL1 : limited knowledge;
•
KL2 : normal knowledge;
•
KL3: full knowledge.
Depending on the KL, three different values of the CF are defined. Namely:
•
CFKL1 = 1.35;
•
CFKL2 = 1.2;
•
CFKL3 = 1.
The first step is to acquire the data which may be classified into three different categories:
•
geometry of structural elements (including possible eccentricities and interactions with
non-structural infills);
•
mechanical properties of the materials (with particular attention to possible local
defects due to time degradation or previous damages);
•
structural details (connections and amount of reinforcement in structural members).
This data may be obtained from original outline and detailed construction drawings, simulated
design, in-situ inspections (to verify and determine the geometrical and mechanical properties
of the structural members) and in-situ testing (to determine the mechanical properties of the
materials).
Therefore, the KL depends not only on the original documents of the design but also on the
accuracy of in-situ testing and inspections. Considering that extended and comprehensive insitu testing and inspections allow to use a low CF (even CF = 1 if KL3 is determined) and
lead to an easier check of the LS requirements, it is clear that the Codes promote the
attainment of a high KL. In fact, a more detailed acquisition of data, although may be a long
5
and laborious process, could likely lead to both considerable money saving in the retrofitting
stage and more reliable results in the assessment process.
Moreover, if only a limited KL is reached, the nonlinear analyses will be not permitted. The
reason is that any nonlinear analysis requires detailed inputs at both material level (definitions
of constitutive monotonic or cycling curves) and member level (exactly position and quantity
of reinforcements): therefore, only a good degree of knowledge of the structure allows to
define correctly a nonlinear model. In fact, a nonlinear analysis based on limited and
uncertain data, although requires a long modelling process, will likely give results which are
not more reliable than those obtained from a linear elastic analysis, which needs only simple
inputs (the Young modulus of the materials and the dimensions of structural members).
On the other hand, in lots of cases, the lack of regularity (in terms of geometry, mass, stiffness
and resistance distribution) and of redundancy do not allow to predict correctly the response
of the buildings using elastic linear methods, since distribution of internal forces and
deformations may change significantly from the elastic to the inelastic response. In these
cases, there is the need to use nonlinear methods of analysis, which, however, can be applied
only if at least KL2 is reached. Therefore, if the structure to assess and retrofit is characterized
by an irregular and not redundant structural configuration, the attainment of at least a KL2 is
prescribed (and not only promoted) by both Eurocode 8 and Italian Seismic Code.
In Figure 1.2 the relationships among KL, required data acquisition processes, corresponding
CF and allowed methods of analyses are shown.
Figure 1.2. Knowledge Levels, allowed methods of analysis and Confidence Factors
6
1.4 Ductile and Brittle Mechanisms
Regarding R.C. frame buildings, the preferred nonlinear seismic response is considered to be
the beam sway mechanism: according to this inelastic mechanism, the “plastic hinges”
(nonlinear flexural behaviour) will form at the end of the beams and at the base of the
columns, while any other part of the structure should remain elastic. This mechanism allows
to obtain high global displacement capacity of the structure with relatively low deformation
demand at a member level, avoiding any concentration of deformation, which will be
uniformly spread throughout the building. If this ductile mode will form at a lower seismic
intensity with respect to that required to activate any other possible mechanism, it will
precede and, therefore, prevent any other inelastic mode, such as flexure modes in columns,
shear modes in both beams and columns, bond failure with slippage of longitudinal
reinforcement in anchorage zones, damages in connections, nodes and at the foundation level.
All these mechanisms must be prevented, since they are:
•
classified as brittle (it is the case of shear and bond), in the sense that they are
characterized by fast degradation in both stiffness and strength with cycling, by
limited deformation capacity and by narrow and unstable hysteretic loops and,
therefore, also by little energy dissipation, or
•
characterized by an uncertain behaviour, which is difficult to model (it is the case of
nodes and connections), or
•
very important for the stability of the entire structure to undergo inelastic deformations
(it is the case of vertical elements, namely columns and walls which, therefore, should
remain elastic except for the base), or
•
very difficult to inspect and repair (foundations).
Concerning the design of new buildings, the capacity-design procedure guarantees that all
these brittle modes are limited to their elastic range, since they are designed in terms of
forces, on the base of the maximum possible demands delivered by ductile mechanisms.
Unfortunately, existing buildings are in general not guaranteed from the development of
possible non-ductile failure modes. All members, therefore, should be considered suitable for
the possible forming of both ductile and brittle mechanisms and should be checked for both.
The definition of ductile and brittle mechanisms given by both Eurocode 8 and Italian Seismic
Code is function of the type of the structure. Regarding R.C. frame buildings:
•
“ductile mechanism” is defined as flexure (with and without axial force) in beams,
columns and walls
•
“brittle mechanism” is defined as shear in beams, columns, walls and joints.
The brittle failure modes are checked using a force- (strength-) based procedure, since the aim
is to limit them to an elastic range, while the ductile mechanisms are checked following a
displacement-based procedure, that is in terms of deformations, since the aim is to check if
they are capable to develop a sufficient deformation capacity and energy dissipation; in fact,
7
once a plastic hinges has formed, the internal bending moments remain roughly constant and
equal to the yielding values, while the deformations grow with the increasing of the seismic
input level. A detailed description of these two procedures will be treated in Chapter 3.
Finally, it is important to underline that the idea of “protecting” the structure from the
development of non-ductile modes is extended also to the assessment and retrofitting
procedure of existing building by reducing, in the calculation of shear capacities of the
members, the mechanical properties of the materials not only by the CF but also by the partial
factor of the material. It means that, as regards R.C. frame buildings, the possible
development of shear modes should be checked in a more restrictive manner with respect to
the possible forming of ductile ones, not only because the former are not capable to develop
large deformations and energy dissipation but also because they may lead to sudden and
abrupt local collapses or even to the global failure of the entire building.
1.5 Ductility and Behaviour Factor in Existing Buildings
The respect of the capacity-design rules and the global ductility capacity of the structure
μ d = d u d y are strictly correlated, since, in the design process, the former is the way to
determine with a good degree of approximation the value of the latter, without the need to
implement any nonlinear pushover analysis. Instead, as regards the assessment of existing
buildings, it is impossible to determine a-priori which kinds of mechanisms (ductile or brittle)
are likely to develop, and where and in which order they will form. If the structure is
symmetric and regular, in both plan and elevation, the inelastic demand will reasonably
spread all over the building uniformly, and the nonlinear mechanisms will be all activated
roughly at the same time. However, even in this case, it is impossible to establish a-priori if
the forming of ductile mechanisms will precede that of brittle modes or vice versa. Therefore,
one can predict the ductility capacity of the structure without carrying out a nonlinear
pushover analysis only on the base of his own sensibility and experience and, therefore, this
judgment cannot be objective.
In Figure 1.3 a typical nonlinear pushover analysis, carried out to determine the ductility
capacity of the structure and the behaviour factor “q”, is shown.
Figure 1.3 “q-factor” determined through pushover analysis
8
It is important to underline that the global ductility capacity and the behaviour factor “q”,
even if determined through a nonlinear pushover analysis, could be imprecise, since the
correctness of the results depends on the accuracy of the model and, therefore, is function of
the achieved KL.
The global ductility capacity (and, hence, also the “q-factor”) of an existing structure is
normally low if compared to that of a new construction, since, in general, old buildings were
not designed to deform inelastically under the design seismic action. However, even if the
value of “q” is high (seismically designed and regular structure), the need to carry out a
nonlinear analysis to evaluate “q” in a satisfactory way makes the q-factor linear elastic
analysis too laborious and complex, since the accuracy of the results is much lower than that
obtained through a nonlinear analysis (which, unlike linear methods, is able to take into
account the change in the distribution of internal forces and deformations due to the
development of inelastic modes). On the other hand, if the value of “q” is chosen subjectively,
without carrying out any nonlinear analysis, the reliability of the results will reduce
drastically, with the serious risk of performing an unsafe assessment, if the chosen value of
“q” is larger than the actual one. It is also possible to select a very low value of “q” which
accounts only for the overstrength of the members and neglects any energy dissipation and
ductility capacity of the structure (“q” = 1.5 for R.C. frame buildings), without running any
nonlinear analysis. In this way, however, it is very likely that the seismic demands will be too
overestimated and the assessment and retrofitting will be too severe and expensive.
For all these reasons, the linear elastic analyses with the seismic input reduced by the “qfactor” (which is the normal way to design a new construction) is not recommended, although
it could be adopted also for the assessment of existing buildings. Therefore, in the following
part of this work, the assessment based on linear elastic analyses with the seismic input
reduced by the “q-factor” will be not taken into consideration.
9
Chapter 2. Methods of Analysis
2 Methods of Analysis
In this Chapter, the attention is focused on the methods of analysis and subsequent
verification of existing R.C. frame buildings. During the detailed description of all the
possible assessment procedures suggested by both Eurocode 8 Part 3 and Italian Seismic
Code [OPCM 3431], particular attention is paid to identify and clarify those aspects which
may lead to different possible interpretations and/or are treated in different ways in Eurocode
8 and in Italian Seismic Code. In particular, attempts to propose simplified approaches for the
assessment procedure, concerning the evaluation of both seismic demand and capacity of the
structural members, will be shown, in order to get faster and simpler ways to assess the
seismic response of the structure, without loosing in accuracy of the results.
2.1 General Aspects
Both Eurocode 8 and Italian Seismic Code allow to assess existing buildings on the base of
the following linear and nonlinear methods:
(i)
elastic linear static (lateral force) analysis;
(ii)
elastic linear dynamic (multi-modal response spectrum) analysis;
(iii) nonlinear static (pushover) analysis;
(iv) nonlinear dynamic (time history) analysis.
The normal methods for the assessment procedure are the nonlinear ones and, in particular,
the static nonlinear (pushover) analysis. In fact, as already expressed in Chapter 1, the
nonlinear response of an existing building may be complex and difficult to be predicted
through linear methods, especially if the structure has been designed without following any
seismic rule. In particular:
•
if the building is irregular and non symmetric concerning geometry, mass, stiffness
and resistance distribution, there may be an eccentricity between the point of
application of seismic inertia forces (centre of mass) and the point in which the
response is concentrated (which shifts from the centre of stiffness to the centre of
resistance as the structure develops an inelastic behaviour). In these cases the torsional
response is of primary importance and, consequently, displacements/deformations will
concentrate in the more flexible (less resistant) side of the building. For this reason,
10
this “weak” part of the structure is likely to develop an inelastic behaviour faster than
the “strong” side, which experiences lower displacements/deformations. Therefore, the
elastic and the inelastic response may be very different and any linear analysis would
not be able to represent the forces and displacements distributions under a strong
seismic action in a satisfactory way.
•
If there are discontinuities of the load path from the top to the base of the building,
eccentricities in the nodes and/or lack of redundancy, there may be local failures due
to the impossibility of redistributing the seismic demands among all structural
elements and, once again, a linear analysis will be not suitable to capture correctly the
response.
•
If the building is not compact and far from a rectangular shape, each floor will not
behave as a unique rigid body and the complexity and uncertainty of the results of the
analysis will grow significantly. In fact, the degrees of freedom will become from
three per each floor to three times the number of structural nodes per each floor and,
moreover, there will be the need to model the in-plan stiffness of the slabs. Even in the
case of an accurate model, however, the concentration of demands in the re-entrant
corners would lead to local plasticization that cannot be captured by an elastic
analysis.
For all these reasons, even if the linear methods are allowed in the assessment procedure, both
Eurocode 8 and Italian Seismic Code require to satisfy conditions of their applicability, in
order to avoid their use whenever they may lead to unreliable results. These applicability
conditions will be deeply discussed in Sec. 2.2.
2.2 Conditions for the Applicability of Linear Methods
Both Codes require to check that the distribution of the ratios of the bending moment
demands to the corresponding capacities is roughly constant for all structural members. The
aim is to verify that the plastic hinges will form uniformly throughout the structure and almost
at the same intensity of the seismic action. If this condition is not satisfied, the load path will
change significantly when the nonlinear mechanisms will form and, therefore, the information
furnished by a linear analysis will be too poor to be taken into consideration.
Moreover, the Italian Seismic Code requires also that, during the analysis, for each structural
member, the forming of plastic hinges at the ends of the member itself should precede the
shear failure. In fact, the brittle failure of an element would cause a sudden change in the load
path and, therefore, a sudden redistribution of internal forces and deformations which cannot
be captured by a linear analysis. This verification is considered as a criterion for the
applicability of linear methods only in the Italian Seismic Code, while in Eurocode 8 it is
considered as redundant, because all elements of the retrofitted structure will, however, fulfil
this condition, irrespective of the method of analysis.
The big limit of these criteria is that they are not an a-priori way to determine whether a linear
analysis furnishes correct results, since it is necessary to perform the analysis in order to
verify its applicability. There is, hence, the concrete possibility to model a structure, run an
11
analysis and make calculations to obtain results just to know that they are not useful to
determine correctly the response of the building. Hence, possible future investigations could
be aimed at:
•
checking the validity of the criteria of applicability of linear analyses proposed by
Eurocode 8 and Italian Seismic Code and, if required, to propose possible corrections.
•
finding a guideline to choose a-priori the best method of analysis, on the base of the
structural shape and configuration.
Both criteria for the applicability of linear methods will be shown in Section 2.2.1 and 2.2.2,
respectively.
2.2.1 1st Condition of Applicability
According to both Eurocode 8 and Italian Seismic Code, considering each i-th end section of
each structural member, denoting by ρi = Di/Ci the ratio of the bending moment demand Di
obtained directly from the linear elastic analysis under the seismic load combination to the
corresponding capacity Ci, and by ρmax and ρmin the maximum and minimum values of ρi
respectively, the ratio ρmax/ρmin should not exceed a maximum acceptable value, beyond that
the inelastic response of the building is considered to differ significantly from the elastic one.
This limit should be judged, hence, as the threshold for the acceptability of the results of the
analysis.
First, it is important to emphasize that both demand Di and capacity Ci should be obtained
considering the mean values of the mechanical properties of the materials, as determined
during the data acquisition process, regardless any reductive coefficient, like the CF and/or
the partial factor of the material. In fact, the aim is not to evaluate whether the retrofitting is
needed, but it is to state the validity of the analysis and, therefore, the most probable values of
demands and capacities should be considered.
Concerning the calculation of the bending moment capacities of columns, both Codes suggest
to use the axial load, N, obtained applying only the gravity loads of the seismic combination,
neglecting, therefore, the change in axial load due to the overturning moment (which affects
particularly the perimetral columns). The seismic action, in fact, causes an increment (or
decrement) ΔN with respect to the axial force due to gravity loads; when the seismic input
reverses, the effect of the overturning moment on columns axial load reverses too (from
compression to tension or vice versa) and the incremental axial force becomes roughly -ΔN.
Hence, N due only to gravity loads (neglecting the contribution of the overturning moment)
may be considered approximately as the mean N that the structure may undergo under a
seismic excitation. Therefore, accounting for N due only to gravity loads may be regarded as
an enough precise approximation for evaluating the possibility of accepting the results of
linear methods.
In order to compute ρmax and ρmin, the Italian Seismic Code suggests to consider all the
locations (ends of beams and columns) where ρi ≥ 2 and to limit the ratio ρmax/ρmin to 2.5. This
procedure is quite simple and fast, since it is sufficient to calculate the bending moment
12
capacity at each end of each structural member and, then, to compare it to the demand
obtained from the analysis.
Eurocode 8, instead, proposes a much more complex way to evaluate to ratio ρmax/ρmin. First, it
suggests to consider the locations where ρi ≥ 1 and to limit the ratio ρmax/ρmin to a
recommended value between 2 and 3 (2.5 in Italy). Not each end section of structural
members where ρi ≥ 1 should be considered, but only those that can develop plastic hinges. In
fact, only the weaker between the columns and the beams can develop plastic hinges because,
once the plastic hinges formed either in columns or in beams, the value of the internal forces
(bending moments and shears) at that joint is locked (any increment of flexure at the end
sections which are still in their elastic range could not be balanced, since in plastic hinge
regions the bending moments have reached their capacity values). The comparison between
the sum of flexural capacities of the beams and columns that frame into a joint allows to
understand where hinging is feasible. Therefore, denoting by MRc and MRb the flexural
capacities of columns and beams, respectively, if:
•
∑ M Rc > ∑ M Rb , then plastic hinges will likely develop in beams and, consequently,
only the beams should be considered for the evaluation of ρmax and ρmin;
•
∑ M Rc < ∑ M Rb , then plastic hinges will likely develop in columns and, thereby, only
the columns should be considered for the evaluation of ρmax and ρmin.
In the above-mentioned joint flexural equilibrium, the columns are considered to develop
opposite bending moments, as well as beams. This hypothesis is a normal assumption in
frame structures, since the stiffness of columns and beams are comparable, while it would be
not reasonable in case of wall structures. In this latter case, in fact, it is very likely that the
bending moment in structural walls does not reverse at floor levels. However, since in this
work only frame buildings are analyzed, this assumption can be considered as correct.
The above-mentioned joint equilibrium condition should be checked separately in each of the
two principal directions of the building. In fact, although the seismic action is actually threedimensional, taking into account the interaction of the two orthogonal bending moments in
columns seems to be too laborious and lengthy in this stage.
For each principal direction, the check has to be performed considering the bending moments
acting both clockwise and counter clockwise. Therefore, the following 2 different equilibrium
conditions should be checked (Figure 2.1):
(i) both column flexural capacities acting on the joint clockwise, while both beam
flexural capacities acting on the joint counter clockwise;
(ii) both column flexural capacities acting on the joint counter clockwise, while both beam
flexural capacities acting on the joint clockwise.
13
Figure 2.1. Joint bending moment equilibrium
In order to assure that a plastic hinge will not form, a member end should remain elastic
considering both equilibrium conditions.
The Italian Seismic Code neglects the equilibrium at the joint into which the considered
element frames. As a result, the maximum obtainable bending moments and shears could be
overestimated, since, actually, they could be locked to inferior values as a consequence of the
possible plasticization of end sections of other structural members which frame into the
considered joint. However, the assumption of neglecting joint equilibrium conditions leads to
an estimation of the applicability of linear methods which lies on the safe side.
On the other hand, although this procedure suggested by Eurocode 8 is rigorous and
theoretically more precise than the solution adopted by the Italian Seismic Code, it is not
immediate and requires lengthy calculations. Therefore, since the procedure proposed by the
Italian Seismic Code is much faster and simpler, an aim of this work is to check if, adopting
Eurocode 8 procedure, the improvement in the results is enough remarkable to justify the
complexity of this method.
2.2.2 2nd Condition of Applicability (only Italian Seismic Code)
Considering each i-th end section of each structural member, denoting by Ci the shear (brittle
mechanism) capacity evaluated using the mean values of material properties, Ci should be
larger than the corresponding demand Di. Denoting by ρi the ratio of the bending moment
demand obtained from the analysis to the corresponding capacity, if ρi < 1, Di will be obtained
directly from the analysis, while, if ρi ≥ 1, Di will be computed through the equilibrium of the
considered structural member, loaded by the capacity bending moments acting at the two
member ends and, in the case of beams, also by the gravity loads acting along the member. In
this way it is possible to take into consideration the fact that the shear demand does not
increase indefinitely with the external forces (as it happens in a linear elastic analysis) but it is
limited by the forming of plastic hinges at member ends.
14
Concerning columns, this procedure leads to simple calculations. In fact, since typically no
transverse loads act on columns, shears are constant along the height and, hence, it is
sufficient to:
•
consider the values of bending moments which derive from the analysis at the ends of
the member;
•
compare them to the corresponding capacities and take the bigger ones;
•
compute shear demands through equilibrium. Denoting by 1 and 2 the end sections of
the column and considering as positive the bending moments and shears which act
clockwise:
V1+
= V2+
M 1− + M 2−
=
L
(2.1)
M 1+ + M 2+
L
(2.2)
V1− = V2− =
The maximum value between Eqs. (2.1) and (2.2) is the maximum shear demand that the
column may undergo under the seismic combination.
The procedure is more complicated in the case of beams, since it is necessary to take into
account the contribution of gravity loads along the member. Considering the same
conventions for shear and moment used for Eqs. (2.1) - (2.2), the equilibrium may be obtained
by the superposition of the following two systems (Figure 2.2):
•
beam considered simply supported, loaded by gravity loads q;
•
beam considered simply supported, loaded only by end bending moment demands
(both acting clockwise and counter clockwise) limited to their corresponding
capacities.
Since the gravity loads generate reactions which are upwards directed, the maximum demands
of shear force at end sections 1 and 2 may be computed, respectively, as:
V1+
= V1, MAX = V1,q
M 1− + M 2−
+
L
(2.3)
M 1+ + M 2+
L
(2.4)
V2− = V2, MAX = V2,q +
15
Figure 2.2. Equilibrium conditions of beams
2.2.3 Further Considerations about the Applicability of Linear Methods
A deepening on some aspects is required. First, it should be noted that the linear static
(equivalent lateral force) analysis can be applied in very few cases, since it is difficult that an
existing building satisfies both conditions of regularity in elevation and those required to
accept the result of linear methods. Moreover, even when the linear static procedure can be
used, the linear dynamic analysis should be preferred, since it takes into account the effect of
higher modes. As a consequence of adopting the linear dynamic analysis, the results are given
in envelope form. Therefore, at a first sight, it could appear impossible to check correctly the
conditions of applicability of linear methods, since the use of equilibrium considerations is
required. In fact, the maximum/minimum values of bending moment at the two ends of a
structural member do not necessarily correspond to the same load condition, and, hence,
computing the shear demands as the maximum/minimum shear values obtained through all
possible equilibrium of the maximum/minimum bending moments may lead to overestimated
values. Actually, the maximum/minimum shear forces obtained through equilibrium
conditions of the maximum/minimum values of elastic bending moments (bending moments
obtained directly from the analysis) correspond exactly to the maximum/minimum shear
demands obtained directly from the analysis. It means that the maximum/minimum shear
values can be reasonably and correctly found from the equilibrium of bending moments, even
if they are given in envelope form. Therefore, the conditions for the applicability of linear
methods can be checked correctly also if a linear dynamic analysis is performed.
Concerning the evaluation of shear demand needed to check the second condition of
acceptability of linear methods, the Italian Seismic Code divides the procedure into two cases:
ρi < 1, and, ρi ≥ 1. As a matter of fact, however, three different situations should be
considered:
(i) ρi < 1 at both member ends. The shear demand should be evaluated either directly
from the analysis or through equilibrium, since both procedures furnish the same
result.
(ii) ρi ≥ 1 at both member ends. The shear demand should be evaluated through
equilibrium conditions, considering the flexural capacities as the bending moment
demands acting at the two member ends.
(iii)ρi < 1 at one end and ρi ≥ 1 at the other one. In this case the member end where ρi ≥ 1
develops a plastic hinge, while the other one (ρi < 1) remains elastic. Hence, the shear
16
demand should be evaluated considering the bending moments obtained from the
analysis, if ρi < 1, and the respective capacities (positive/negative), if ρi ≥ 1.
Therefore, all these three cases may be evaluated using the same procedure, that is through
equilibrium conditions.
The Italian Seismic Code is very restrictive regarding the second condition of applicability of
linear analysis. In fact, it is sufficient that one structural member does not meet this condition
to reject all the results of the analysis. The idea is that if one brittle mechanism develops,
suddenly a different load path will be activated. However, if the structure is sufficiently
redundant and the brittle mechanisms sufficiently rare, the change in the distribution of forces
will be evident only in proximity of the failed member, while the load path will tend to match
the original one far from the location of the brittle mechanism. Further future investigation are
required in order to evaluate if, even in the case of a limited percentage of members which
undergo a brittle failure, the results of the linear analysis diverge significantly from those
obtained using nonlinear methods and, consequently, should be invalidated, as stated by the
Italian Seismic Code.
According to both Eurocode 8 and Italian Seismic Code, the reference analysis methods for
the assessment of existing buildings are the nonlinear ones and, in particular, the nonlinear
static analysis. The linear methods are permitted only in few cases, because the conditions for
their applicability are very restrictive. Nevertheless, some doubts may arise about the fact that
the results of the nonlinear analyses will be always more reliable than those obtained through
a linear analysis, if the conditions for its applicability are not satisfied. In fact, it should be
considered that some structures are fairly regular but not enough to satisfy the first condition
of applicability of linear methods and/or do not present spread brittle mechanisms but only
rare shear failures which are, however, sufficient to reject the results of the analysis,
according to the Italian Seismic Code. In these cases, if the structure is irregular as regards the
shape (neither compact nor approximately symmetric), it should be investigated the
possibility that the results obtained from a linear dynamic analysis may match better the
maximum values of the response of a nonlinear dynamic analysis than those obtained from a
nonlinear static analysis based on conventional pushovers, characterized by unidirectional
seismic lateral forces. In fact, despite the lack of accuracy of the linear dynamic analysis due
to the impossibility of catching the change in load path, a nonlinear static analysis shows a
considerable loss of information caused by ignoring both torsion and higher modes
contribution (that, instead, can not be neglected if the structure is neither symmetric nor
compact).
Further future investigations are, hence, needed to determine for which kinds of structural
configuration a linear dynamic analysis should be preferred to a nonlinear static, and for
which cases, instead, neither a linear dynamic nor a nonlinear static analysis may yield
acceptable results and, therefore, a nonlinear dynamic analysis is required. The choice to
determine the most suitable method should not be based only on the reliability of the results.
In fact, unless a poor KL of the properties of the structures has been acquired, the information
given by a nonlinear dynamic analysis is always the most realistic one. It should be
considered, on the other hand, that the nonlinear dynamic analysis requires long modelling
17
and running time of analysis and also a complex procedure for both management of the results
and data processing.
2.3 Common Problems related to Modelling
In this section, the most important problems related to the modelling of both linear (dynamic)
and nonlinear (static and dynamic) methods are developed. All types of analysis have in
common the following aspects.
As regards the definition of masses, three different approaches are feasible:
(i) concentrating the whole floor mass in the centre of mass of each floor. The advantages
are the velocity of modelling and the simplicity of the eigenvalue analysis, since each
floor is characterized only by three degrees of freedom (two horizontal displacements
and one rotation around the vertical axis). The main limits of this choice are that:
•
the floor must be considered as a rigid body, which is not always true, as it will
be shown afterward;
•
not only the translational mass, but also the rotational mass should be defined;
•
the local effects due to concentration of mass in particular zones of the
buildings cannot be captured.
(ii) concentrating the mass in the structural joints. The advantages are that:
•
the in-plan deformations of slabs can be taken into account;
•
only the definition of translational masses is required, while the rotational
inertia and, therefore, the torsional effects are implicitly defined.
The weakness of this approach with respect to the previous one is that three degrees of
freedom are associated to each structural joint and, hence, the model is less simple and
the understanding of the global behaviour of the structure less clear, since local modes
may be activated. On the other hand, the possibility of taking into account local effects
due to irregular mass distribution may lead to a better assessment of the seismic
response of the structure and, therefore, it may be seen as a further benefit of this
approach.
(iii)distributing the mass along the structural elements. It requires to define a translational
mass per unit of length for each member. The advantages are the same of choosing to
concentrate the masses at structural joints and also the possibility to account for the
activation of local modes at a member level. The main shortcoming is that, in this
way, the eigenvalue problem shifts from a discrete one (finite number of degrees of
freedom) to a continuous one and it leads to more complex computations.
18
On the base of these considerations, the choice to lump the whole floor mass in the centre of
mass of each floor seems to be inadequate, except for those structures which are very regular
in shape, masses, stiffness and resistance distribution.
The second approach (concentrating the masses in the structural joints) is suitable for
nonlinear static analysis, since, as it will be shown in the following part of this Chapter, the
evaluation of the mode shapes based on lumped masses is consistent with the choice of
distributing the lateral pushover forces in the structural joints.
The third approach (distributing the mass along the structural elements) is suitable for linear
and nonlinear dynamic analyses, since the modelling is faster than lamping the masses in the
structural joints and, moreover, the results are more accurate.
Modelling the slabs as rigid diaphragms is feasible only if they are enough stiff to make the
in-plan deformations negligible. According to this assumption, the inertia forces caused by
seismic motion are distributed to the vertical elements, proportionally to their stiffness (in
elastic range) or to their resistance (in inelastic range). The clearest advantage is the gain in
simplicity and the reduction of uncertainties of the response (number of degrees of freedom
reduced to three per each floor). Unfortunately, in the case of existing buildings, the in-plan
stiffness of slabs needs to be investigated and they cannot be modelled a-priori as rigid.
Eurocode 8 states that a diaphragm may be taken as rigid only if its horizontal displacements,
obtained modelling its actual in-plan flexibility, nowhere exceed those resulting from the rigid
diaphragm assumption by more than 10%. Therefore, in order to establish whether the floor
slabs may be considered as rigid in their plan, it is necessary to perform two analyses and to
model the in-plan flexibility of the diaphragms.
Possible conditions that may invalidate the rigid diaphragm assumption are the following:
•
not compact configuration and plan view far from rectangular (T, U, H, L or even
more complicated shapes). In this case, significant in-plan stresses can develop at reentrant corners of such non-rectangular diaphragms, causing early cracking;
•
large openings in floor slabs, due to internal patios or stairways. In this case, the
continuity of the force path can disrupt;
•
large distance between strong and stiff vertical elements compared to the transverse
dimension of the diaphragm. The slab develops significant deflections and flexural
stresses within its plane (Figure 2.3).
Figure 2.3. In-plan deformability of slab due to large distance between stiff vertical elements
19
As a result of these considerations, concerning the assessment of existing R.C. frame
buildings, floor slabs should be included in the model as non-rigid elements of appropriate inplan stiffness. In this work two different possibilities of modelling are proposed:
(i) using shell Finite Elements (F.E.);
(ii) substituting the slab panels with X-diagonal braces.
The first method is used for linear elastic analysis. A single shell F.E. is adopted to model
each slab. The in-plan stiffness of the slab is obtained through the definition of the Young’s
modulus of an equivalent material and the thickness of the shell.
The second approach consists in substituting the slab panel with X-diagonal braces,
characterized by an elastic axial stiffness equal to the in-plan elastic stiffness of the slab
panel. For a fairly rectangular slab panel of dimension Lx ≈ Ly ≈ L, defining as:
•
J the moment of inertia of the section (Jx ≈ Jy ≈ J);
•
As the shear area;
•
Ec and Gc the Young’s modulus and the shear modulus of concrete, respectively,
the in-plan elastic stiffness of the slab panel, Kslab, is evaluated using the following
approximated formula:
K slab =
1
3
L
L
+
12 EC J AS GC
(2.5)
Since the elastic axial stiffness of the diagonal brace, Kb, is:
Kb =
Eb Ab
Lb
(2.6)
where:
•
Eb is the Young’s modulus of the material chosen for the diagonal brace;
•
Ab is the area of the section of the diagonal brace;
•
Lb is the length of the diagonal brace,
once chosen the material for the diagonal braces and set Kslab = Kb, it is possible to evaluate
the area Ab and to model the diagonal braces. The limit of this approach is that it provides a
reliable approximation of the in-plan stiffness only if Lx ≈ Ly for each slabs panel. This
method was used in this work for modelling the slabs in nonlinear analyses.
20
2.4 Modelling for Linear Elastic Methods of Analysis
As already underlined in the previous sections, the linear dynamic analysis should always be
preferred to the linear static analysis, since it is more accurate (it takes into account higher
modes effects), while the time required for modelling and performing the two analyses is
roughly the same. Therefore, in this work only the linear dynamic analysis is treated.
The main problem of modelling a linear elastic analysis is the correct definition of stiffness of
concrete members. Assuming a bi-linear (elastic – perfectly plastic) shape of the momentcurvature diagram, the elastic stiffness used in analysis should correspond to the secant
stiffness at yielding (Figure 2.4).
The Italian Seismic Code allows to consider as cracked stiffness a value between 50% and
100% of corresponding gross stiffness. In general, this assumption leads to values which
overestimate the real secant stiffness at yielding and, therefore, the structure seems to be more
rigid with respect to how it is actually. The direct effect is that the natural periods are
underestimated and, consequently, the internal forces are overestimated, while displacements
and deformations are underestimated. Therefore, within the force- (strength-) based approach,
using a high estimate of effective cracked stiffness is a conservative choice.
Unfortunately, the assessment of existing buildings is a force- (strength-) based procedure
regarding the brittle mechanisms but also a displacement-based method concerning the ductile
mechanisms. Consequently, adopting as cracked stiffness a value equal or larger than 50% of
corresponding gross stiffness leads to an assessment of the ductile modes which may not be
on the safe side.
Moreover, the effective stiffness depends strongly on the depth of the neutral axis which is
function of the axial load. Therefore, in general, the ratio of actual secant stiffness at yielding
to the gross stiffness is higher in columns than in beams and tends to reduce from the base to
the top of the building. Considering an equal value of that ratio for all structural members of
the building may yield imprecise results.
Figure 2.4. Typical M-Φ diagram of a R.C. section
In the case of assessment of existing building, however, not only the dimensions of the
structural members but also the position and quantity of the reinforcement are known. It is
21
therefore possible to evaluate, for each structural member, the mean value of the actual secant
stiffness at yielding (effective stiffness) EI eff . Defined
•
the chord rotation, θ, as the angle between the chord connecting the end section of the
member to the section at which M = 0 and the tangent to the member axis at the end
section;
•
the shear span length, Ls, as the distance between the end of the member and the
section where M = 0 (i.e. the point of contraflexure of the member),
EI eff may be obtained as:
EI eff
+
M y−,1 −
M y+, 2 +
M y−, 2 − ⎞
1 ⎛⎜ M y ,1 +
Ls ,1 + − Ls ,1 + + Ls , 2 + − Ls , 2 ⎟
=
⎟
4 ⎜⎝ 3θ y+,1
3θ y ,1
3θ y , 2
3θ y , 2
⎠
(2.7)
where θy is the yielding chord rotation capacity (defined in details in Sec. 3.2.1), the indexes 1
and 2 refer to the two ends of the considered member, the apexes + and – refer to positive and
negative bending, respectively.
Taking Ls = L/2 (where L is the member length), Eq. (2.7) can be simplified as:
EI eff =
L 4 M y ,i
∑
24 i =1 θ y ,i
(2.8)
This formula corresponds to consider each structural member as made up by two cantilevers
of length Ls = L/2, fixed at the two respective member ends.
Eq. (2.8) requires a further consideration concerning columns, since the yielding moment is
function of the axial load N, which changes during the seismic excitation (particularly in
perimetral columns). However, since N due to vertical loads is roughly equal to the mean N
that a member may experience during the seismic excitation, it seems to be a good choice to
evaluate EIeff considering N due to the gravity loads of the seismic combination.
In this work, three different possible ways to evaluate EIeff,i of each i-th structural member
were developed for each of the four considered buildings. In particular, in growing order of
precision:
(i) EI eff ,i = 100% EI gross ,i , for all structural members;
(ii) EI eff ,i = 50% EI gross ,i , for all structural members;
(iii) EI eff ,i computed through Eq. (2.8).
Considering that the purpose of using a linear analysis is to get a fast method to assess the
structural response, assigning the value of EIeff,i computed through Eq. (2.8) to each i-th
structural member, separately, may be regarded as a too lengthy procedure. Hence, in this
work, three different solutions were adopted. In growing order of precision:
22
n
∑
(iii-a) EI eff ,i =
j =1 EI gross , j
n
∑
(iii-b) EI eff ,i =
n
EI eff , j
j =1 EI gross , j
n
∑
(iii-c) EI eff ,i =
EI eff , j
n
EI eff , j
EI gross ,i , where n refers to all structural members;
EI gross ,i , where n refers to all beams and columns, separately;
j =1 EI gross , j
EI gross ,i , where n refers separately to T-beams, rectangular
n
beams and columns of each floor.
An aim of this work is to test the improvements in the results which may be obtained using
EIeff computed through Eq. (2.8) instead of applying a value between 50% and 100% EIgross.
Since the evaluation of EIgross does not require any My-θy calculation, the use of EIeff according
to Eq. (2.8) will be suggested only if the results obtained considering a value between 50%
and 100% EIgross seriously underestimate the actual response.
A further aim is to establish the improvement in the results obtained considering the most
precise application of Eq. (2.8) (method iii-c) with respect to the methods iii-a and iii-b.
Although the assessment of EIeff according to Eq. (2.8) allows to obtain an accurate estimation
of the member actual secant stiffness at yielding, it could happen that applying Eq. (2.8) to all
structural members will lead to an underestimation of the actual secant stiffness at yielding of
the whole structure. In fact, the yielding of the structure (idealized as the corner point of the
bilinear approximation of the force-displacement curve of the structure) does not correspond,
in general, to the local yielding of all structural members. The majority of the members could
be still in their elastic range, resulting in a higher value of EIeff with respect to that obtained
through Eq. (2.8). It is, therefore, possible that the structural stiffness will be underestimated
with the consequence that the displacements/deformations will be overestimated and the
internal forces underestimated. A better approach could be to compute EIeff through an
equation equal to Eq. (2.8) in the form, but where the member yielding capacities (My,i and
θy,i) are substituted by the respective demand values ( M D ,i and θ D ,i ), obtained applying a
seismic action characterized by the intensity required to reach exactly the yielding point of the
structure. This procedure, even if theoretically more correct, was not taken into account in this
work since it was considered too complex and lengthy and since further future investigations
are needed to test its validity.
Finally, it is important to dwell upon the following consideration as regards the application of
linear dynamic analysis. If all relevant modal responses can be regarded as independent of
each other, both Eurocode 8 and Italian Seismic Code will allow to combine their effects
using the SRSS rule, which neglects any correlation among the modes. Actually, for a
computer, the additional complexity due to the use of the CQC rule (which instead accounts
for the correlation among the modes) is not a problem. Therefore, there is no reason to
implement a SRSS instead of a CQC which is more general, accurate and always acceptable.
23
2.5 Nonlinear Static Analysis
According to both Eurocode 8 and Italian Seismic Code, the nonlinear static analysis is the
reference method in assessment practice of existing buildings. It is based on pushover
analyses carried out under constant gravity loads and increasing lateral forces, applied at the
location of the masses to simulate the inertia forces induced by the seismic action. As the
model may account for both geometrical and mechanical nonlinearity, this method can
describe the evolution of the expected plastic mechanisms and of structural damage, with the
limit that the seismic input can simulate only a single horizontal component of the seismic
motion and does not reverse.
Pushover analysis was initially developed for 2-D analyses. On the other hand, the irregularity
in plan of lots of existing buildings does not allow to consider separate 2-D frames and a 3-D
analysis is required.
The lateral forces Fi applied to masses mi (the best choice, as already shown, is to lump the
masses at structural nodes, in order to account automatically for torsional inertia and for local
effects) remain proportional to a certain pattern of horizontal displacement Φi:
Fi = λ (miΦ i )
(2.9)
According to both Eurocode 8 and Italian Seismic Code, two different lateral load patterns
should be used:
(i) a “uniform pattern”, Φi = 1, which attempts to simulate an inelastic response
dominated by a soft-storey mechanism (development of plastic hinges at both top and
bottom ends of all columns of a storey, in general the ground floor, which is subjected
to highest lateral forces). The lateral drifts concentrate, hence, at the soft-storey and
this causes the storeys above to move with roughly the same lateral displacement.
(ii) a “modal pattern”, which should follow the fundamental elastic translational mode
shape. The “modal pattern” tries to simulate the response up to global yielding, or
even beyond that point, if a beam-sway mechanism (“strong” columns which remain
elastic except for the base and “weak” beams which develop plastic mechanisms at
their ends) governs the inelastic response.
Since the inelastic mechanisms which are likely to develop in existing buildings are, in
general, unknown, the results obtained using the two standard lateral force patterns should be
considered as an envelope of the actual response, which should lie between the two capacity
curves. Therefore, the most unfavourable results of the two pushover analyses should be
adopted. On the other hand, since in general the existing buildings are not regular structures,
their responses are more complex than soft-storey or beam-sway mechanisms. Therefore, the
statement that the actual behaviour of the structures should be included between the two
capacity curves needs to be confirmed by further future investigations.
Denoting as X and Y the two principal horizontal orthogonal directions of the structure, eight
different pushover analyses should be performed, four in X-direction (“modal” towards
24
positive X, “modal” towards negative X, “uniform” towards positive X, “uniform” towards
negative X) and four in Y-direction (“modal” towards positive Y, “modal” towards negative
Y, “uniform” towards positive Y, “uniform” towards negative Y).
Each pushover analysis leads to determine a capacity curve, that is a relationship between the
total base shear Fb and the horizontal displacement dn of a representative point of the
structure, termed “control node”. According to Eurocode 8, it should be taken in
correspondence to the centre of mass of the roof of the building. The demand at the
considered LS is determined separately, using the displacement response spectrum. It allows
to compute the point on the capacity curve which corresponds to the so-called “target
displacement” of the “control node” and to the total base shear demand. Therefore, for each
curve, three “target displacements” and three corresponding total base shears are defined, in
order to identify the seismic demand at the DL, DS and NC LS, respectively. All the internal
forces and deformations, attained when the “control node” displacement reaches the “target
displacement”, are taken as the demand values of the structural members.
Since the “target displacements” are defined in terms of spectral quantities, it is necessary to
build the equivalent single-degree-of-freedom (SDOF) system of the structure. Both Eurocode
8 and Italian Seismic Code define the equivalent SDOF system on the base of the following
procedure, developed by Fajfar for 2-D pushover analysis.
The horizontal displacements Φi are, first, normalized so that at the control node, n, Φn = 1.
The total mass, the total base shear and the control displacement of the equivalent SDOF
system are, then, defined, respectively, as
m ∗ = ∑ miΦ i
F∗ =
Fb
d∗ =
dn
Γ
Γ
(2.10)
(2.11)
(2.12)
where Γ, called “transformation factor”, is given by:
Γ =
m∗
∑ miΦ i
=
2
2
∑ miΦ i
∑ miΦ i
(2.13)
In the “modal pattern”, Φi emulates the shape of the first elastic translational mode in each
horizontal principal direction of the building. Therefore, the “transformation factor” is the
participation factor of that mode in the direction of application of lateral forces.
In the case of “uniform pattern”, instead, since Φi = 1 for each point in which the masses and
the equivalent lateral forces are applied, Γ = 1 and, hence, the MDOF and the SDOF merge
into the same system.
25
Once the SDOF capacity curve has been defined, an estimation of the elastic period T* of the
equivalent SDOF system is required for the determination of the “target displacement” from
the displacement response spectrum. The capacity curve of the SDOF is idealized as an
elastic-perfectly plastic curve with:
•
yielding force F y∗ taken as the peak base shear of the SDOF system;
•
the slope of the first branch of the bi-linear curve (that is the elastic stiffness of the
SDOF system) determined so that the areas under the actual capacity curve and its
elastic-perfectly plastic idealization up the peak of the capacity curve are the same
(Figure 2.5).
These two conditions lead to define the yield displacement d ∗y of the bi-linear SDOF system
as:
d ∗y
⎛ ∗
E ∗peak
⎜
= 2 d peak −
⎜
Fy∗
⎝
⎞
⎟
⎟
⎠
(2.14)
where d ∗peak is the displacement at the peak point of the equivalent SDOF system and E ∗peak
is the area subtended by the “capacity curve” up to d ∗peak . E ∗peak represents, hence, the
deformation energy stored in the actual SDOF curve up to that peak point.
Figure 2.5 Method of equality of areas
The equivalent SDOF elastic stiffness is, then, defined as:
K ∗ = F y∗ d ∗y
(2.15)
while the elastic period of the equivalent SDOF system is estimated as:
T ∗ = 2π
m∗
K∗
(2.16)
The elastic displacement d e*,t is, then, determined directly from the 5%-damped elastic
displacement response spectrum.
26
The “target displacement” d t* of the equivalent SDOF system is obtained from the elastic
displacement d e*,t on the base of the so-called “equal displacement rule”, modified for short
period (i.e. rigid) structures. According to this approach:
•
if T* is longer than the corner period of pseudo-acceleration spectrum, Tc, the target
displacement d t* of the equivalent SDOF system will be equal to the elastic target
displacement d e*,t :
d t*
•
=
d e*,t
2
( )
⎡T ∗ ⎤
= S e T * ⎢ ⎥ , if T* ≥ Tc
⎣ 2π ⎦
∗
(2.17)
if T* < Tc, the target displacement will be corrected as follows:
d t*
d e*,t ⎛
T ⎞
=
⎜1 + (qu − 1) c∗ ⎟ ≥ d e*,t , if T* < Tc
qu ⎝
T ⎠
where qu =
m ∗ S e (T ∗ )
Fy∗
(2.18)
(2.19)
Finally, the “target displacement” of the MDOF system at the “control node” is obtained as:
d t = Γ ⋅ d t∗
(2.20)
In nonlinear analyses, there are two different approaches to account for the mechanical non
linearity:
(i) concentrating it only in particular zones;
(ii) considering it widespread throughout the whole model.
According to the first approach, the structure is modelled as linear elastic, except for the zones
in which a nonlinear behaviour is expected to develop. Therefore, there is the need to identify
these zones in terms of location, extension and properties of inelastic behaviour, defined
through the M-θ curve. This approach is, hence, characterized by the use of concentrated
plastic hinges. The advantages are that:
•
the direct introduction in the model of the amount of longitudinal reinforcement is not
required.
•
It is possible to take into consideration, through an appropriate definition of the M-θ
relationship, not only the flexural behaviour, but also the influence of different aspects
that may be important to better define the nonlinear response. In particular, there are
models calibrated on experimental data which allow to account also for shear
deformation, slippage of longitudinal bars, node deformations and interactions
between structural members and infills.
27
This methodology is, therefore, suitable for the verification of new seismically designed
buildings, for which it is easy to identify the location of plastic hinges thanks to the respect of
capacity design procedures. Moreover, regarding the structural members of new buildings,
there is an exhausting literature of possible M-θ models. Unfortunately, concerning the
assessment of existing buildings, it is hard to understand where the plastic hinges can develop
and to define correctly not only their location but also their extension and behaviour (i.e. the
characteristic curve M-θ). A great experience is, therefore, needed in order to model in a
correct way the structure using concentrated plastic hinges.
According to the second approach, the inelastic behaviour is defined at a material level and
the whole structure is modelled as nonlinear. Since the stresses and strains are, in general, not
constant on the sections of structural members and the σ-ε relationship is nonlinear, each
section is subdivided in a grid of small areas, each one characterized by constant stress and
strain values. The forces and deformations are then computed through integration. It requires
to subdivide each structural member into a certain number of fibres. Therefore, this kind of
approach is called “nonlinear fibre element” modelling. The advantages are that:
•
the nonlinearity is defined only at a material level (M-θ curves are not required).
•
There is no need to recognize location and extension of nonlinear zones because they
are automatically identified.
The shortcomings are that:
•
the amount of reinforcement must be modelled. Hence, the accuracy of the results
depends strongly on the KL of the steel content.
•
Only the flexural behaviour can be modelled. Nevertheless, neglecting the shear
deformation and the slippage of longitudinal bars may yield inaccurate results, in
particular if the structural members are characterized by small values of Ls h and if
the steel bars are not ripped, respectively.
•
The integration procedure requires long time, especially if a huge number of fibres is
needed to achieve the wished precision.
Despite these shortcomings, the “nonlinear fibre element” approach is the most suitable in the
case of the assessment of existing R.C. buildings, since it furnishes as output not only the
forces and the structural displacements, but also information about position, extension, and, if
a time history analysis is performed, chronological forming order of the plastic hinges.
In the following, important choices concerning the modelling of a pushover analysis are
shown.
The first choice concerns the definition of the σ-ε relationship of the materials (concrete and
reinforcement steel for R.C. buildings).
28
Regarding concrete, it is important to take into account the confinement due to transversal
reinforcement, in particular for the seismically designed structures, detailed with close spaced
stirrups. The confinement effect can be taken into consideration through the evaluation of the
so-called confinement factor kc, defined as:
kc =
f cc
fc
(2.21)
where fc is the unconfined concrete strength and fcc is the confined one.
kc is function of stirrups spacing, restraining effect of longitudinal bars, transversal steel
yielding tension and concrete strength in compression (a more detailed description of the
concrete model adopted in this work will be developed in Chapter 3). The most important
effect of confinement, however, does not lie in the increment of the peak compressive
concrete strength which is in general low, but in flatter descending branch of constitutive
curve and, mainly, in the increase of deformation capacity of the material, which leads to the
increase of rotation capacity of the plastic hinges. Therefore, the effect of confinement could
not be ignored, since it allows to account for the residual inelastic curvature capacity of the
sections, after spalling of unconfined concrete cover (Figure 2.6).
Figure 2.6. Typical σ-ε relationship of confined and unconfined concrete
Experimental data show that the mean tensile strength of concrete is roughly 10% of the
compressive strength. However, considering an existing building, it is very likely that the
members have already developed tensile cracks due to shrinkage and temperature effects
and/or to their load history. Therefore, it seems to be a correct choice to neglect the
contribution of both tensile strength of concrete and Poisson modulus.
Regarding steel, a simple bi-linear constitutive curve seems to be suitable, since more
complex models are characterized by parameters which may be difficult to determine, in
particular in the case of existing buildings. In order to determine correctly the bi-linear model
of steel, it is important to define the post-yielding tangent stiffness (that is the slope of the
second branch of the curve), which allows to account for the overstrength due to the strainhardening effect (a more detailed description of the steel model adopted in this work will be
developed in Chapter 3) (Figure 2.7).
29
Figure 2.7. Bi-linear idealization of σ-ε relationship of steel
Another important choice regards the definition of the type of control. The pushover analysis
is based on the application of horizontal incremental lateral forces Fi = λ (miΦ i ) . Determined
the values of mi and Φi, which remain fixed during the analysis, there are three different
possibilities of controlling the increment of load through the load factor λ.
(i) Load control. The lateral loads Fi increase step by step so that, considering the generic
i-th and (i+1)-th step:
λi +1 − λi = Δλ = constant
In this way it is impossible to evaluate the descending branch of the capacity curve,
since the applied lateral load can only increase (Figure 2.8). This could be a very
heavy limit, because, in general, the “target displacements” corresponding to the SD
and NC LS lie in the descending branch of the F-Δ curve.
Figure 2.8. Load control pushover analysis
(ii) Displacement control. The nodal displacements increase step by step through a
constant multiplicative factor, which is the same for each point of the structure. The
lateral forces needed to obtain those displacements are then evaluated. Although this
method allows to define the descending branch of the capacity curve of the structure, it
should not be considered a good solution, since the displacement pattern does not
change during the analysis while, actually, the inelastic deformed configuration may
differ significantly from the elastic one.
30
(iii)Displacement response control. The lateral loads Fi increase step by step through a
multiplicative factor λi, determined so that:
u ci +1 − u ci = Δu = constant
where u ci and u ci +1 are the “control node” displacements at the generic i-th and (i+1)-th
step.
This is the best way to control the factor λ, since it is possible to determine the
descending branch of the F-Δ curve and also to take into account the change in the
displacement pattern due to the spread of the inelastic behaviour (Figure 2.9).
Figure 2.9. Response control pushover analysis
In this work the displacement response control was adopted for the pushover analyses.
The definition of the “modal pattern” of lateral forces can be not easy, especially if the
structural configuration is neither symmetric nor compact. In this case, in fact, it is probable
that the percentage of modal mass associated to the rotation around the vertical axis will be
large and, hence, the first mode will not be purely translational. Therefore, the displacements
Φi will not be unidirectional but will have also a component in the orthogonal direction. It
means that also the lateral forces Fi should not be identified only by their amplitude in the
considered direction, but by a vector in the horizontal plane. The main problem is that the
Fajfar’s procedure was though for 2-D systems and, therefore:
•
the capacity curve is unidirectional;
•
the evaluation of the “transformation factor” Γ is based only on the normalized
displacements in the direction considered for the evaluation of the capacity curve.
On the other hand, neglecting the orthogonal component of lateral forces Fi would lead to an
uncorrected assessment of the torsional effects. A possible future development may be,
therefore, to look for a procedure suitable for the 3-D pushover of asymmetric and noncompact structures, for which the torsional component is important as well as the translational
one. Since in this work the Fajfar’s procedure was applied (as indicated by both Eurocode 8
31
and Italian Seismic Code), the “modal pattern” of lateral forces was taken as unidirectional,
proportional to the displacements of the first mode in the considered principal direction of the
building.
The choice of the “control node” is another problem typical of asymmetric and non-compact
structures. According to both Eurocode 8 and Italian Seismic Code, the “control node” should
be taken in correspondence to the centre of mass of the top floor. If the structure configuration
is compact and roughly symmetric with respect to both principal directions of the building,
the centre of mass will lie next to both centres of stiffness and resistance. In this case,
therefore, the displacement of the “control node” is mainly directed towards the pushing
direction, while the contribution of the orthogonal component is negligible. If, instead, the
structure is neither compact nor symmetric, there may be a considerable eccentricity between
the centre of mass and the centre of stiffness, in elastic range, and between the centre of mass
and the centre of resistance, in inelastic range. In this case the following two problems may
arise.
(i) The “control node” may rotate around a point which shifts from the centre of stiffness
to the centre of resistance, besides translating in the pushing direction. This rotation
leads to an additional contribution to the “control node” displacement. The projection
of this “rotational displacement” in the pushing direction may oppose to the
“translational displacement”. In this case, if the “rotational displacement” is large
enough, at a certain step of the analysis it will be impossible to find a load factor λ to
obtain an increment Δu of the displacement of the “control node”. In fact, each
positive value λ will lead to a decrement of the “control node” displacement and,
consequently, the analysis will stop. If the step at which the analysis stops corresponds
to a value of the “control node” displacement lower than the “target displacement” of
the considered LS, it will be impossible to assess the building. Although choosing
different “control nodes” for different patterns of lateral forces could seem to be a
satisfactory way to avoid this shortcoming, actually this choice would not allow to
compare the different capacity curves and to combine the results obtained from the
pushovers performed in the two principal directions of the building. This problem,
therefore, needs further future investigations.
(ii) Choosing the “control node” in correspondence to the centre of mass of the top floor
could give rise to a value of the “transformation factor” Γ (Eq. (2.13)) lower than 1. It
means that the capacity curve of the SDOF system could be amplified with respect to
that of the corresponding MDOF system. The reason why it is possible that Γ < 1 may
be explained as follows. According to both Eurocode 8 and Italian Seismic Code, the
horizontal displacements Φi are normalized so that at the “control node” Φn = 1.
Therefore, if the torsional modal mass associated to the first mode is relevant, the
more flexible part of the structure will undergo displacements which could be (much)
larger than the “control node” displacement. Moreover, it will be possible that some
node displacements will be negative. The denominator of Eq. (2.13) can be, hence,
larger than the numerator, leading to Γ < 1.
32
In this work the “control node” was chosen in correspondence to the centre of mass of the top
floor, according to both considered Codes.
In order to avoid problems of convergence in the assessment of the capacity curves, the
“control node” could be chosen in correspondence to the centre of stiffness (in elastic range)
or of resistance (in inelastic range) of the top floor, because it is the point where the resultant
of the response can be concentrated. Considering that, in general, the “target displacement” at
the DL LS belongs to the elastic branch of the capacity curve, the “control node” may be
chosen in correspondence to the centre of stiffness. If, instead, the response at the SD or at the
NC LS is assessed, the “control node” may be chosen in correspondence to the centre of
resistance, since, in this case, the “target displacement” can be reasonably supposed to belong
to the inelastic branch of the capacity curve. This procedure could allow to bypass the
problem of the lack of convergence of pushover analysis, because the “control node” would
experience only translation towards the direction of the capacity curve. The problem that Γ
could be lower than 1, however, would remain unsolved.
Another possible choice could be to choose the “control node” in correspondence to the point
which undergoes the largest displacement in the direction of the capacity curve, according to
the 1st mode deformed shape in the same direction. In this way Γ would be always larger than
1 but, on the other hand, it is very likely that the horizontal displacement of the “control node”
will be affected by torsional response with the resulting problems of convergence in the
evaluation of the capacity curve. Moreover, since in general the point which experiences the
largest displacement according to the 1st mode deformed shape in X direction is different from
that in Y direction, it will be impossible to combine the results obtained through a pushover in
X with those given by a pushover in Y.
It is clear that the problem of the best choice of the “control node”, as a function of the
structural configuration, needs further future investigations.
Another topic which needs a deepening concerns how to assess the structure on the base of
pushover analyses, as the actual seismic action is bidirectional in the horizontal plane while
the pushover forces are unidirectional. Since, in general, the regularity conditions are not
satisfied, a 3-D model of the building is required and, hence, eight conventional pushover
analyses should be performed (four in each of the two principal directions).
The Italian Seismic Code suggests to treat separately the results of the eight pushover
analyses. For each i-th end section of each structural member, hence, the effects Ei (both
internal forces and deformations) should be evaluated eight times, corresponding to the eight
performed pushover analyses. The highest value of Ei governs the assessment. This procedure
is based on the assumption that the building will simply translate toward the pushing
direction. Therefore, all the effects Ei will develop only in the pushing direction too. This
assumption can be considered as a reasonable approximation only if a compact, fairly
rectangular, symmetrical and regular (concerning mass, stiffness and resistance distribution)
structure is considered. In this case, in fact, Ei in the direction orthogonal to the pushing
direction will be negligible. Therefore, the seismic actions acting in the two principal
directions may be considered as uncoupled, as suggested by the Italian Seismic Code.
33
Unfortunately, the behaviour of an irregular building is much more complicated than a simply
translation toward the pushing direction, since the torsional response may yield a considerable
rotation of the structure. Hence, significant Ei will develop also in the orthogonal direction. In
this case, the assumption of considering the two orthogonal responses as uncoupled needs to
be investigated, as it could lead to unreliable results.
Eurocode 8, instead, suggests to consider each effect Ei as given by the combinations of the
outcomes of two orthogonal pushover analyses. Denoting by Exi and Eyi the effects (both
forces and deformations) on each i-th end of each structural member due to pushovers in Xdirection and in Y-direction, respectively, Eurocode 8 advises to use one of the two following
combination rules:
Ei = E xi2 + E yi2
(2.22)
Ei ,1 = E xi + 0.3 ⋅ E yi
Ei , 2 = 0.3 ⋅ E xi + E yi
(2.23)
The SRSS combination rule (Eq. (2.22)) may overestimate the value of Ei. In fact, summing
the square of the effects implies that any reduction in the value of Ei due to opposite signs of
Exi and Eyi is replaced by an increase of Ei. In order to avoid this possible overestimation of Ei,
in this work, the second approach (Eq. (2.23)) was considered.
Theoretically, for each i-th end sections of each structural member, Ei should be evaluated 32
times (16 Ei,1 and 16 Ei,2), according to all the possible combinations (four pushover in X and
four pushover in Y). Actually, in this work, the possibility of combining the effects obtained
through the application of the “uniform” pattern with the effects obtained according to the
“modal” pattern is excluded. Therefore, only 16 combinations (eight Ei,1 and eight Ei,2) were
considered.
If, on one hand, the procedure suggested by the Italian Seismic Code seems to be too
simplified, on the other hand, the idea of combining the effects of two pushovers proposed by
Eurocode 8 may yield inaccurate results and is, however, theoretically incorrect. In fact, the
procedure of combining the effects will be correct only if an elastic analysis is performed. It
may be more reasonable combining the pushover forces, and then evaluating the capacity of
the structure due to bidirectional forces. Anyway, the Fajfar’s procedure has been developed
for 2-D systems and it leads to 1-D capacity curves. Since the demand is bidirectional in the
horizontal plane, it would be more useful evaluating a “capacity surface” instead of two
separated capacity curves in the two horizontal principal directions. Unfortunately, at the
present-day, there are no procedures that allow to evaluate the demand points corresponding
to the considered LS on a capacity surface. Therefore, even if it is theoretically incorrect,
nowadays the only way to consider the interaction of the pushover analyses in X and Y
directions is to combine their results, as suggested by Eurocode 8.
In this work, the results of the assessment performed according to the Italian Seismic Code
and to Eurocode 8 were compared, in order to check if they yield close results, although the
34
Italian Seismic Code suggests to neglect the combination of the results in the two horizontal
principal directions of the building.
Since this subject is strictly correlated to the structural configuration (regularity of both
building shape and distribution of mass, stiffness and resistance), the assessment of several
other buildings of different characteristics is required to get more general conclusions.
2.6 Nonlinear Dynamic Analysis
The nonlinear dynamic analysis allows to represent in the most accurate way what really
happens when the structure experiences a ground motion. Unlike nonlinear static analysis,
which requires a complex procedure to estimate the seismic demand, the nonlinear dynamic
analysis enables to determine directly the demand quantities. Moreover, it allows to establish
the correct chronological order of forming of inelastic mechanisms. This last result cannot be
reached through a pushover analysis, since the loads are unidirectional (while in a 3-D time
history the input ground motion is in general bi- or tri- directional) and do not reverse.
Although this analysis is very sophisticated, it requires very deep information concerning both
nonlinear geometrical and mechanical properties of the building. In fact, since the assessment
of the seismic demand is very sensitive to the data input, if the KL is poor, the nonlinear
dynamic analysis will yield uncertain results, even less reliable than those obtained through
linear methods. Moreover, the time required to perform a time history analysis is much longer
than the time needed to carry out any other type of analysis. Considering that both Codes
prescribe to perform at least seven time history analyses in order to account for the mean
values of the results, it is clear that this method is very expensive and should be used only if
all other techniques furnish unacceptable results and, at the same time, the acquired KL is not
poor.
The response may be heavily influenced by:
•
the characteristics of the applied accelerograms;
•
the choice of the computational software;
•
the characteristic curves of the materials;
•
the definition of the damping model.
Concerning the choice of the accelerograms, Eurocode 8 and the Italian Seismic Code allow
to use both artificial and natural ground motions.
Artificial time series are obtained directly from the response pseudo-acceleration spectrum by
spectral matching. It involves modifying the frequency content of the time series to match the
considered spectrum at all periods. The main advantage of this approach is that it allows to
reduce the number of series that need to be run, leading to a significant cost saving. Moreover,
it is generally difficult to select suitable recorded natural accelerograms, since they should
reflect the frequency content, duration and distribution of large amplitudes along the duration,
which are peculiar to the considered site. On the other hand, it should be noted that spectrum
35
compatible time series (that is artificial ground motions) are not “realistic”, since the response
pseudo-acceleration spectrum from which they are obtained is an envelope of multiple
earthquakes and, therefore, a time series that matches the entire spectrum is representing more
than one earthquake occurring at once. For this reason such time series seems to overestimate
the structural response (too conservative). Moreover, artificial time series tend to smooth the
large peaks typical of actual recorded accelerograms.
Natural accelerograms are obtained through records and need only to be scaled in order to
match the response pseudo-acceleration spectrum at a specified spectral period. The main
advantage is that this approach leads to realistic ground motions which are feasible and,
therefore, the actual response of the structure is not overestimated. On the other hand, the
disadvantage is that several ground motions are needed to match entirely the considered
pseudo-acceleration spectrum. Moreover, generally, it is difficult to have natural ground
motion recorded at the site of interest or at locations with the wished characteristics.
Therefore, in lots of cases the criteria for selecting initial time series to scale are too stringent.
This argument is only superficially mentioned in this work and need of further deepening,
since this is not a focal point of this dissertation.
In this work, for each considered building, artificial ground motions were selected, such that
their frequency content matches the same pseudo-acceleration spectrum used to perform both
linear dynamic and nonlinear static analyses at the SD LS.
The use of a “fibre elements” computational software, which accounts for mechanical
nonlinearity directly at a material level, is the most suitable option for assessing the response
of existing buildings (for the same reasons already explained concerning the modelling of
pushover analysis).
The cyclic nature of the nonlinear dynamic analysis requires to define the hysteretic behaviour
of the materials (unlike the pushover analysis which requires only the definition of monotonic
curves σ-ε). The most important requirement is to represent correctly the energy dissipation
due to post-elastic unloading and reloading cycles. Simple hysteresis models, which require
the definition of few and clear parameters, should be preferred for two reasons:
(i) they are less likely to lead to numerical problems than elaborate and complex models.
In fact, a very important attribute of the model is the numerical robustness of the
algorithm, since local numerical problems could develop into lack of convergence and
global instability of the response.
(ii) It may happen that the values to assign to some parameters of complex models are
difficult to be determined and, hence, default values are used. Unfortunately, in
general, the default values were calibrated on the properties of new constructions and,
therefore, they may be unsuitable for the assessment of existing buildings.
In this work, concerning concrete, the uniaxial nonlinear constant confinement model that
follows the constitutive relationship proposed by Mander et al. [1988] and the cyclic rules
proposed by Martinez-Rueda and Elnashai [1997] was chosen.
36
Regarding steel, instead, the uniaxial bilinear stress-strain model with kinematic strain
hardening was used. This simple model is characterized by easily definable parameters
(Young’s modulus, yield strength and strain hardening stiffness) and by its computational
efficiency.
The damping effect is normally obtained combining an equivalent viscous damping with the
contribution of the hysteresis of the materials. The hysteretic damping is taken into account
directly through the definition of the cyclic characteristic curve σ-ε of the materials. Hence,
only the definition of the equivalent viscous damping is required. If the ground motion
intensity is low enough to maintain the response of the structure in its elastic range or even if
the structural behaviour is inelastic but not characterized by wide and fat hysteretic cycles, it
will be crucial to give a correct evaluation of the equivalent viscous damping in order to
obtain reliable results. Increasing the ground motion intensity, the importance of the hysteretic
damping will increase too, while a correct assessment of the equivalent viscous damping will
be less significant. Therefore, concerning the assessment of the response at the DL LS, a
correct evaluation of the equivalent viscous damping is fundamental, while the hysteretic
properties of the materials are less important, and vice versa regarding the assessment at the
SD and NC LS. Actually, the introduction of even a very small quantity of equivalent viscous
damping, compared to the hysteretic one, might turn out to be very beneficial in terms of the
numerical stability of highly inelastic dynamic analyses, given that the viscous damping
matrix will have a "stabilizing" effect in the system of equations. As such, its use is generally
recommended, albeit with small values.
The equivalent viscous damping takes into account the effects of the following phenomena:
•
radiation of kinematic energy through foundations;
•
kinetic friction between structural and non structural members;
•
viscosity in materials.
Since it is hard to represent in a satisfactory way each of these physical phenomena, the
equivalent viscous damping should be characterized by a simple expression, which allows to
furnish a good approximation of their actual values. In order to simplify the dynamic problem,
the equivalent viscous damping should be selected so that the damping matrix will be
diagonal in principal (mode shapes) coordinates. In fact, this property will allow to treat the
MDOF problem as a superposition of SDOF systems. In order to gain this goal, the most
common choice is to use the Rayleigh damping matrix. Denoting by C, M and K the damping,
the mass and the stiffness matrices, respectively, the Rayleigh damping matrix is such that:
[C ] = a0 [M ] + a1 [K ]
4πξ
T1 + T2
T1T2ξ
a1 =
π (T1 + T2 )
a0 =
(2.24)
37
where T1 and T2 represent, respectively, the period of the fundamental mode of vibration of
the structure, and the period of the highest mode of interest. For irregular structures, selecting
T2 could be not trivial and, anyway, it is a subjective decision that requires experience, in
order not to obtain an unrealistic assessment of the equivalent viscous damping.
Other simplified approaches can be obtained from the Rayleigh damping matrix by taking:
•
a0 = 4πξ T1 and a1 = 0 (C proportional only to M). The parameter a0 decreases with
increasing T. In this way higher modes response may be more easily excited (weakly
damped).
•
a0 = 0 and a1 = T1ξ π (C proportional only to K). The parameter a1 increases with
increasing T. Therefore, higher modes response may be suppressed (strongly damped).
Both solutions (C proportional only to M or only to K) allow to bypass the problem of
selecting T2.
The way to evaluate K has to be defined, in order to compute C when a1 ≠ 0 . K can be chosen
as:
•
constant and equal to initial elastic stiffness. The main advantage of this approach is
that the analysis is much faster and also more stable. On the other hand, it should be
underlined that a constant K tends to increase effective damping factors when an
inelastic response develop, because, in that case, the actual stiffness is (much) lower
than the initial elastic stiffness. Therefore, this approach could underestimate the
actual response.
•
Instantaneous tangent stiffness. K should be computed at each step (or even at each
iteration), leading to a heavy algorithm and, consequently, to longer analyses.
However, this choice seems to be important in order to avoid an overestimation of the
equivalent viscous damping in inelastic range.
On the base of all these considerations, in this work the damping matrix C was assumed
proportional only to instantaneous tangent stiffness K, updated at each step but not at every
iteration, since the latter choice would give rise to higher numerical instability and to very
long run times.
38
Chapter 3. Assessment Procedure for R.C. Frame Structures
3 Assessment Procedure for R.C. Frame Structures
The assessment procedure for existing buildings is displacement-based concerning ductile
mechanisms and force- (strength-) based regarding brittle mechanisms. In particular, in the
case of R.C. structures, the ductile modes should be checked in terms of chord rotation
(defined in Sec. 2.4), while the brittle modes should be assessed in terms of shear.
Regarding frame buildings, the structural members (beams and columns) are, in most of the
cases, slender elements. The shear forces are, therefore, low compared to the bending
moments and, consequently, flexural deformations dominate the behaviour. The controlling
factor is the so-called “shear span ratio”, Ls h , where Ls is the shear span length (defined in
Sec. 2.4) and h is the height of the cross-section. The lower is the shear span ratio, the more
important will be the shear stresses with respect to the normal stresses. In the case of beams
and columns, in general, the shear span ratio is high and, therefore:
•
the force transfer mechanisms by flexure and shear can be considered uncoupled and
independent. They practically act in series and this is the reason why the ductile and
brittle assessment procedures are separated. This assumption is not true for squat
columns and short beams since, in this case, both force transfer mechanisms of merge
into one;
•
the Navier-Bernoulli plane section hypothesis can be applied for the calculations of
the member deformations, without particular loss in precision;
•
the Mörsch-Ritter truss model can be used for the evaluation of shear capacities.
In the following sections, the assessment procedures for ductile and brittle mechanisms are
shown.
3.1 Ductile Mechanisms: Chord Rotation Demand
The ductile mechanisms are assessed at a member level, through the evaluation of the chord
rotation demand and the correspondent capacity at the ends of each structural element (beams
and columns). The demand is evaluated in correspondence with the mean values of concrete
strength, fcm, longitudinal steel yielding strength, fylm, and transversal steel yielding strength,
fywm. The capacity is computed considering the mean values of the material properties (fcm, fylm
and fywm) reduced by the determined CF.
39
As already explained in Sec. 2.4, the chord rotation is defined as the angle between
•
the chord connecting the centroid of the end section of the member and the centroid of
the section at which M = 0,
•
the tangent to the member axis at the end section.
Since M is proportional to the curvature Φ, the section where M = 0 corresponds to the point
of contraflexure of the member. Therefore, each structural member is considered as formed by
two cantilevers, fixed at the member ends and characterized by a length equal to the shear
span length, Ls (Figure 3.1):
Ls =
M
V
(3.1)
where M and V are the bending moment and shear demands at the considered member end,
respectively.
Figure 3.1. Definition of shear span
The chord rotation at member ends is the most important and convenient deformation measure
for concrete members, because:
•
unlike curvatures, which are difficult to measure experimentally, deflections at the end
of the shear span can be reliably measured;
•
most published experimental results have been obtained as tip deflections on simple or
double cantilever specimens.
If the nodal rotation of the considered member end is low with respect to the drift of the
equivalent cantilever, then the chord rotation, θ , may be reasonably defined as:
θ=
Δ
Ls
(3.2)
where Δ is the tip deflection at the shear span end.
40
If, instead, the tip deflection is low with respect to the nodal rotation of the considered end
section, then θ may be defined simply as the nodal rotation of the considered member end.
The former case is typical of columns under seismic input. In general, in fact, since the
building is pushed laterally by the ground motion, the lateral drifts at shear span ends are
much larger than the nodal rotations at columns ends. The values obtained under the
hypothesis of neglecting the end nodal rotations of the columns will be very precise if the
flexural stiffness of the beams is much larger than those of the columns. This is typical of
structures designed for gravity loads only, without following capacity design procedures.
However, even if at a structural joint the sum of the stiffness of the beams is smaller than that
of columns, it is reasonable to assume that the nodal rotations of columns are negligible with
respect to the drifts of the equivalent cantilevers.
Denoting as θ1 and θ2 the nodal rotation and the drift at the shear span end, respectively
(Figure 3.2):
θ = θ 2 − θ1 ≅ θ 2
(3.3)
Figure 3.2. Chord rotation demand for columns
Moreover, since the sign of θ1 is opposite to that of θ2, the assumption that θ = θ2 leads to a
small overestimation of the chord rotation demand and, hence, to a safe side assessment.
If the beams flexural stiffness is larger than the columns stiffness (very common in old R.C.
frame buildings), the bending moments at the two ends of each column will be characterized
by similar values and opposite signs. If the columns stiffness is larger than the beams
stiffness, instead, the bending moments at the two ends of each column can be very different
and it is even possible that they will be characterized by same sign and/or will not reverse at
floor levels. Therefore, increasing the beams stiffness with respect to the columns stiffness,
the values of bending moments at the two column ends will tend to become more similar to
each other. In this case (beams stiffer than columns), also the shear span length at both
columns ends can be reasonably considered equal to half columns length, regardless of the
seismic demand. This hypothesis would be exact if the model were built up by shear type
frames (beams infinitely stiff with respect to columns).
41
In available specimens, in general, the shear span ratio is maintained constant during testing,
since there are no distributed loads. Therefore, concerning columns, the experimental results
reflect accurately the actual situation.
Concerning beams, instead, the chord rotation may be simply defined as the nodal rotation of
the considered member end. This assumption will be precise if the presence of gravity loads is
neglected. In this case, the chord connecting the member end with the point of contraflexure
will be roughly horizontal and, hence, the nodal rotation will be the only contribution to the
chord rotation. Moreover, if the frames are made up by equal beams (in both geometry of the
cross-sections and length) and equal columns, the bending moments at the two ends of a beam
will be characterized by same value and opposite sign. In this case (Figure 3.3):
•
the chord connecting the member end with the point of contraflexure will be perfectly
horizontal.
•
The nodal rotations at both beam ends will be characterized by same value and
opposite sign. Consequently, the chord rotations computed at both ends will be equal
to each other.
•
The shear span length will be equal to half beam length.
Figure 3.3. Chord rotation demand for beams, neglecting the contribution of gravity loads
Unfortunately, the actual situation is much more complicated, since gravity loads act on
beams. Considering the actual response of a beam as the superposition of the following two
systems:
(i) beam unloaded and end sections undergoing the nodal rotations due to the seismic
input;
(ii) beam fixed at both ends, loaded by gravity loads,
the overall chord rotation may be seen as the sum of the nodal rotation (system 1) and the drift
at the point of contraflexure due to gravity loads (system 2). Denoting as θ1 and θ2 the chord
rotation due to system 1 and 2, respectively (Figure 3.4):
θ = θ1 + θ 2
(3.4)
42
Figure 3.4. Total chord rotation demand for beams
As the seismic action reverses, θ1 changes sign, while θ2 remains the same. Therefore, the
overall chord rotation and also the shear span length depend on the sign of the seismic action.
The shear span length is minimum and the chord rotation demand is maximum when both
moment and shear due to lateral loads are characterized by the same sign as the moment and
shear due to gravity loads (top flange in tension). As a consequence, the gravity loads are a
further source of unsymmetrical behaviour (in addition to the lack of symmetry of the crosssection and of the reinforcement). If the shear span length is minimum (the top flange is in
tension), the effect of shear on response will be more significant than when the bottom of the
beam is in tension.
Although the gravity loads influence the ductile response of a beam, it should be noted that, in
general, the nodal rotations due to lateral loads are larger than the drift due to gravity loads.
Increasing the ground motion intensity, the contribution of gravity loads to the chord rotation
demand will decrease. Therefore, in general (and in particular in the case of assessment at the
SD and NC LS), there will be no remarkable lack of accuracy if θ2 is neglected and the chord
rotation demand is assumed to be equal to θ1, i.e. to the nodal rotation of the considered beam
end.
Since in most of the available specimens the shear span is maintained constant during testing,
the experimental results do not match accurately the beam actual response.
3.2 Ductile Mechanisms: Chord Rotation Capacity
The chord rotation capacity depends on both geometrical and mechanical properties of the
shear span and on the seismic input. In fact, the shear span length is defined as the ratio of
bending moment demand to shear demand at the considered member end and, moreover, the
curvature capacity is influenced by the amount of axial load. The chord rotation capacity,
hence, may not be defined as an inner property of a member, since the same member may
develop different values of capacity as the seismic action changes.
According to both Eurocode 8 and Italian Seismic Code, the chord rotation capacity is
evaluated only at DL and NC LS, while the value at SD LS is taken, conventionally, as 3/4 the
value determined at NC LS.
43
3.2.1 Damage Limitation Limit State
The chord rotation capacity corresponds to the yield point of the considered member end. In
Eurocode 8 the following expression is proposed:
(
)
⎛
Ls + α s d − d '
h
θ y = φy
+ 0.0013⎜⎜1 + 1,5
Ls
3
⎝
d f
CF
⎞
⎟⎟ + 0.13φ y b ylm
f cm CF
⎠
(3.5)
where:
•
φ y is the yield curvature of the end section.
•
α s is a 0-1 variable which takes into account the tension shift of the bending moment
due to diagonal cracking. Such an increase would not take place unless flexural
yielding is preceded by diagonal cracking. So, if the shear force that causes diagonal
cracking, Rc, [Eurocode 2, EN 1992-1-1:2004, 6.2.2(1)] is smaller than the shear force
developed at flexural yielding of the end section, Vy = My/Ls, then α s = 1 , otherwise
αs = 0.
•
d b is the mean diameter of the tension reinforcement.
In the Italian Seismic Code, instead, the following formula is suggested:
θ y = φy
⎛
Ls
h
+ 0.0013⎜⎜1 + 1,5
Ls
3
⎝
d f
CF
⎞
⎟⎟ + 0.13φ y b ylm
f cm CF
⎠
(3.6)
The possible shift of bending moment due to diagonal cracking is, therefore, not taking into
consideration. This assumption simplifies a lot the expression, since the evaluation of α s is,
in general, long and not trivial.
A part from this discrepancy, the way to compute the chord rotation capacity of structural
members at DL LS is the same for both considered Codes.
Eq. (3.6) (and, similarly, Eq. (3.5)) is made up of three terms:
(i) φ y Ls 3 accounts for the flexural deformation. It is theoretically computed, supposing
a triangular distribution of the curvature along the shear span length. The effect of
gravity loads on the curvature is, hence, neglected.
(ii) 0.0013(1 + 1,5 h Ls ) accounts for the contribution of shear deformation and is purely
empirical. It may be noted that if the shear span ratio Ls/h decreases, the shear effects
will increase.
(iii) 0.13φ y (d b f ylm CF ) f cm CF accounts for the fixed-end rotation due to slippage of
longitudinal steel bars from their anchorage zone located in column-beam joint. This
fixed-end rotation is due to the elongation of the tensile bars between the end of their
44
embedment length and the end section of the member. This elongation will increase
significantly if bar yielding penetrates into the anchorage zone.
A deepening on the application of Eqs. (3.5) – (3.6) seems to be opportune.
In order to compute the chord rotation capacity, the yielding curvature has to be assessed. It
may be evaluated theoretically, on the base of the following assumptions.
•
The Navier-Bernoulli hypothesis is applied. The normal strain ε can be, hence,
computed, as ε = φy.
•
The concrete is considered indefinitely linear elastic in compression, with elastic
modulus that may be taken as [Fib Bulletin 24 – Appendix 4.A]:
1
⎛ f ⎞3
E c = 0.85 ⋅ 2.15 ⋅ 10 4 ⎜ cm ⎟ , where fcm is in MPa.
⎝ 10 ⎠
(3.7)
This formula, used in this work, differs from those suggested by both Codes, since
both Eurocode 8 and Italian Seismic Code refer to expressions valid for new
constrictions that could be improper for the assessment of existing buildings.
•
The concrete does not react to tension, since both beams and columns are considered
already cracked, due to their load history and/or to the constraint of both shrinkage and
temperature strains. Therefore, the M-φ diagram can be taken as linear up to yielding.
•
The steel is considered linear in both tension and compression up to the yielding
force, f ylm CF . The slope of the linear elastic branch was assumed, in this work, as Es
= 200000 MPa.
•
Perfect bond between the steel bars and the surrounding concrete is supposed. The
steel strain is, hence, equal to the strain of the surrounding concrete.
It should be noted that the first and the last assumptions lead to neglect the shear deformation
and the rigid-body rotation due to slip of tensile steel, respectively. This is the reason why
these two contributions need to be added separately to Eqs. (3.5) – (3.6).
The condition of yielding of a reinforced concrete section is commonly considered to
correspond to the yielding of the tensile longitudinal reinforcement, the compressive concrete
being considered as linear elastic. However, if the axial load value is high, it may happen that,
when the tensile steel yields, the compressive concrete has already developed a strongly
nonlinear behaviour. Hence, in this case, the evaluation of an elastic strain limit of the
compressive concrete is needed in order to define an “apparent yielding” of the section. The
available test results on yielding of members with high axial load [Panagiotakos and Fardis,
2001] suggest to consider the following value as the elastic strain limit of the compressive
concrete [Fib Bulletin 24, Appendix 4.A]:
45
ε c ,max =
1.8 f cm
E c (CF )
(3.8)
The elastic strain limit of the compressive concrete is, hence, assumed equal to 90% of the
strain at peak stress, εco = 0.002.
Therefore, denoting as xy the depth of the neutral axis and as d the distance between the
tensile steel and the extreme compressed fibre, the following two distinct possible values of
theoretic yielding curvature should be assessed for each end section, in both positive and
negative bending.
(i) Tensile steel yielding curvature:
φy =
(
f ylm
)
(3.9)
E s d − x y ⋅ CF
(ii) Compressive concrete “apparent yielding” curvature:
φy =
1.8 f cm
E c x y ⋅ CF
(3.10)
The lower of the values furnished by Eq. (3.9) and Eq. (3.10) is considered as the correct yield
curvature.
Available experimental results show that the yielding curvature is sensitive mainly to the
height of the section, h and to the yielding strain of the longitudinal tensile steel, εsyl.
It means that other quantities such as the amount of longitudinal reinforcement, the σ-ε law of
concrete and the amount of axial load do not influence much the value of yielding curvature,
but affect significantly the value of yielding moment and, therefore, the secant stiffness at
yielding.
In this work, the yielding curvature was computed, for each end section, in both positive and
negative flexure, in four different ways:
(i) according to the theoretical approach shown previously and considering the axial load
N due only to the gravity loads of the seismic combination.
(ii) According to the theoretical approach and considering N due to the seismic load
combination. If a linear dynamic analysis is performed, this procedure will lead to
calculate the yield curvature for both maximum and minimum N.
(iii)According to the following empirical expressions [M.J.N. Priestley, 1993]:
for rectangular beams: φ y =
1.87ε syl
h
(3.11)
46
for T-section beams: φ y =
for columns: φ y =
1.7ε syl
2.1ε syl
h
h
(3.12)
(3.13)
(iv) According to the following empirical expression valid for both beams and columns
[D.E. Biskinis, 2006]:
φy =
1.75ε syl
h
(3.14)
The theoretical approach with N obtained from the seismic combination is considered to be
the correct way of evaluating the yielding curvature. On the other hand, this procedure is long
and laborious. An aim of this work is, hence, to check if this approach is necessary or if it
could be replaced by faster calculations (according to empirical expressions), without any
sensible loss of precision. In particular, the reason of assessing the yielding curvature with
these four different procedures is double.
(i) To show if the two empirical formulations of yielding curvature (Eqs. (3.11) – (3.14))
can furnish a correct approximation of the theoretical curvature also in the case of
existing buildings. The available empirical formulas were, in fact, calibrated on
members representative of new, seismically designed, constructions. The accuracy of
the values obtained from the proposed empirical expressions, hence, should be
checked for existing buildings, since they could be characterized by non seismic
details and by degraded mechanical properties of the materials.
(ii) To establish if the theoretical yielding curvature, computed on the base of N due to
gravity loads only, is consistent with that calculated considering N due to the seismic
combination. At a first sight, the former procedure could appear useless, since it could
seem to be comparable to the latter in terms of both complexity and time required for
the assessment. Actually, it will be true only if a static analysis is carried out. If,
instead, a dynamic (either linear or nonlinear) analysis is performed, the results will be
given in envelope form. The time required for performing the assessment on the base
of N due to the seismic combination will be, therefore, doubled. Moreover, the
complexity of the procedure will increase too, because, in order to furnish a correct
estimation of the actual member capacity, M and V values which correspond to both
maximum and minimum N should be considered. In fact, taking the
maximum/minimum envelope values of V and M acting together with the
maximum/minimum envelope values of N may lead to a significant overestimation (or
underestimation) of the actual member capacity. This non trivial problem may be
bypassed assessing the yielding curvature on the base of N due to gravity loads only.
The shear span length has to be evaluated in order to determine both flexural and shear
contributions of the yielding chord rotation capacity. The correct definition of shear span
length is the ratio of the bending moment demand, M, to the corresponding shear demand, V
47
(Eq. (3.1)). If a nonlinear static analysis is performed, single values of M and V will be
obtained for each LS. Hence, the application of Eq. (3.1) will be simple.
If, instead, a linear dynamic analysis is carried out, the evaluation of shear span length will be
much more complicated, for both following reasons:
(i) as already explained previously, both M and V are furnished in envelope form;
(ii) both M and V obtained through an elastic linear analysis grow indefinitely,
proportionally to the external forces, while, instead, they should be limited to their
corresponding capacity values.
In this work, the following procedure has been considered to evaluate the shear span length
when a linear dynamic analysis is performed.
The maximum positive and negative bending moments, M, at each member end, are got
directly from the analysis. Each M is then compared to the corresponding capacity and the
smaller value is taken as demand value. For each structural member, therefore, four M
demands (two per each end) are evaluated.
Shear forces, V, are computed through equilibrium, considering, in the case of beams, also the
contribution of the gravity loads. This procedure is the same as that considered for the
evaluation of the second condition of applicability of linear methods of analysis (Sec. 2.2.2).
The possible equilibrium conditions are four. Actually, in this work, only two of them were
considered. In fact, the maximum positive M at one member end was supposed to act together
with the maximum negative bending moment at the other end and vice versa. This
assumption, which may be seen as arbitrary since it may lead to an overestimation of
maximum and minimum V, is actually correct. In fact, assuming the elastic M (obtained
directly from the analysis) as demand values, the application of the two considered
equilibrium conditions allowed to obtain, for all beams and columns of all the considered
structures (1370 structural members) exactly the V furnished by the analysis itself.
Finally, for each end of each structural member, two values of shear span length were
calculated, corresponding to maximum and minimum M, respectively.
This rigorous procedure is complex and lengthy, since:
•
the shear span length is not an inner property of the member but has to be recomputed
every time the analysis is carried out, because it is function of the results of the
analysis itself (M and V).
•
the evaluation of bending moment capacity is needed, in both positive and negative
flexure.
In this work, the results obtained from the correct definition of shear span length (Eq. (3.1))
were compared to those found by simply assuming Ls = L/2. The aim is to check if this
48
simplified procedure can be accepted and, hence, adopted as a possible alternative to the
correct definition of shear span length.
Summarizing, the procedure suggested by both Eurocode 8 and Italian Seismic Code leads to
complex and lengthy calculations (in particular if a linear dynamic linear analysis is
performed), since the chord rotation capacity is function of both cross-section properties and
seismic demand. If the cross-section deformation capacity (yielding curvature) could be
evaluated by simple formulas (i.e. according to the empirical expressions proposed by
Priestley and Biskinis) and if the shear span length could be taken as half the member length,
independently from the results of the analysis, the gain in simplicity and velocity would be
huge. In this work, the reliability of the values obtained applying the simplified procedure will
be checked.
3.2.2 Near Collapse Limit State: Empirical Approach
Two different approaches, one based on theoretical assumptions and the other one based on
experimental results, are proposed by both Eurocode 8 and Italian Seismic Code for the
evaluation of the chord rotation capacity at NC LS.
The empirical expression for chord rotation capacity at flexural failure proposed by both
Codes is calibrated on cycling load conditions and developed on the base of statistical
methods. According to both Eurocode 8 and Italian Seismic Code, the following formula is
proposed:
θu =
1
γ el
⎡ max(0.01; ω ') f cm ⎤
0.016 ⋅ 0.3ν ⋅ ⎢
⋅
⎥
⎣ max(0.01; ω ) CF ⎦
( )
0.225
⎛ Ls ⎞
⎜ ⎟
⎝ h ⎠
0.35
25
f
⎛
⎜ αρ sx ywm
⎜
f cm
⎝
⎞
⎟
⎟
⎠
(3.15)
where:
•
γ el is equal to 1.5 for seismic structural members.
•
ν = ( N Ac ) ⋅ (CF f cm ) , N positive for compression.
•
ρ sx = Asx bw s h , Asx being the transverse steel area parallel to the X-direction of
loading;
•
(
α = (1 − s h 2b0 ) ⋅ (1 − s h 2h0 ) ⋅ 1 − ∑ bi2 6h0 b0
)
is the confinement effectiveness
factor, where h0 and b0 are the dimensions of confined concrete core, delimited by
hoops centreline, while bi is the i-th spacing between two adjacent longitudinal bars
laterally restrained by a stirrup corner or a cross-tie.
Since in this work, for all the assessed structures, the longitudinal bars are laterally
restrained only at stirrups corners (typical of R.C. frame existing buildings), the
confinement effectiveness factor becomes:
49
⎛
s ⎞⎛
s
α = ⎜⎜1 − h ⎟⎟⎜⎜1 − h
⎝ 2b0 ⎠⎝ 2h0
2
2
⎞⎛⎜ (b0 − (d w + d l ,c )) + (h0 − (d w + d l ,c ))
⎟⎟ 1 −
3h0 b0
⎠⎜⎝
⎞
⎟
⎟
⎠
(3.16)
where dw and dl,c are the diameters of the stirrups and of the corner longitudinal bars,
respectively.
Concerning the assessment of R.C. frame members not detailed for earthquake resistance,
both Codes require to multiply the value obtained from Eq. (3.15) times a reduction factor
(0.825 for Eurocode 8, 0.85 for Italian Seismic Code). Since all the constructions considered
in this work were built without seismic details, the expressions proposed by Eurocode 8 and
Italian Seismic Code become, respectively:
( )
0.016
⎛f ⎞
θu =
⋅ 0.3ν ⋅ ⎜ cm ⎟
1.2375
⎝ CF ⎠
θu =
( )
0.016
⎛f ⎞
⋅ 0.3ν ⋅ ⎜ cm ⎟
1.275
⎝ CF ⎠
0.225
0.225
⎛ Ls ⎞
⎜ ⎟
⎝ h ⎠
0.016 ⎡ max(0.01; ω ') f cm ⎤
θu =
⋅
1.2375 ⎢⎣ max(0.01; ω ) CF ⎥⎦
θu =
0.016 ⎡ max(0.01; ω ') f cm ⎤
⋅
1.275 ⎢⎣ max(0.01; ω ) CF ⎥⎦
⎛ Ls ⎞
⎜ ⎟
⎝ h ⎠
0.225
0.225
0.35
25
0.35
25
⎛ Ls ⎞
⎜ ⎟
⎝ h ⎠
⎛ Ls ⎞
⎜ ⎟
⎝ h ⎠
f
⎛
⎜ αρ sx ywm
⎜
f cm
⎝
f
⎛
⎜ αρ sx ywm
⎜
f cm
⎝
0.35
25
0.35
25
⎞
⎟
⎟
⎠
⎞⎫
⎟
⎟
⎠⎪
⎪
⎬ , for columns
⎪
⎪
⎭
f
⎛
⎜ αρ sx ywm
⎜
f cm
⎝
f
⎛
⎜ αρ sx ywm
⎜
f cm
⎝
⎞
⎟
⎟
⎠
(3.17)
⎞⎫
⎟
⎟
⎠⎪
⎪
⎬ , for beams
⎪
⎪
⎭
(3.18)
The only difference between the two Codes in the evaluation of θu consists, therefore, in the
value proposed for this reduction factor.
Two aspects of Eq. (3.15) need a deepening, namely the way to choose the axial load N and to
compute the shear span length Ls.
The correct N should be the value obtained from the analysis under the seismic load
combination. However, as it was already underlined concerning the evaluation of yielding
chord rotation, if a linear dynamic analysis is performed, a double assessment procedure will
be required, since the capacity should be evaluated in correspondence of both Nmax and Nmin.
Therefore, if it were possible to consider N due only to gravity loads (which is roughly the
mean N that the columns undergo during a seismic input), the number of required operation
would be halved. In this work, the consistence of the results obtained using N due to gravity
loads with those found considering N from the seismic load combination is tested.
Moreover, choosing N due to gravity loads allows to bypass the problem to define how to
calculate Nmax and Nmin in the case of linear dynamic analysis. In this work, Nmax and Nmin
were taken directly from the analysis, but this assumption may lead to an overestimation of
the actual ΔN, axial load that develops (particularly in perimetral columns) to equilibrate the
overturning moment. In fact, since the analysis is elastic, ΔN induced in the columns by the
lateral forces may be unrealistically large, particularly when the seismic action is enough
50
strong to cause the development of nonlinear mechanisms throughout the structure. Since a
linear analysis is not able to detect the development of nonlinear mechanisms, the moment
demands, MDi, can be larger than the corresponding capacities, MCi. Therefore, in order to
respect the equilibrium at the joints into which the structural members frame, also the shear
demands, VDi, and the axial load demands due to seismic action, ΔN, may be larger (in
absolute value) than the actual ones. While both Eurocode 8 and Italian Seismic Code suggest
to limit MDi to MCi, if MDi > MCi, and to compute the shear demand VDi on the base of
equilibrium considerations, they do not explain how to limit ΔN. In order to satisfy the
vertical translational equilibrium at the structural joints, the following condition should be
applied (Eq. (3.19)), that limits ΔN on the base of the maximum shear demands attainable at
the end sections of the beams that frame into the considered joint.
∑ VDb = N Dc 2 − N Dc1 = ΔN
(3.19)
Eq. (3.19) corresponds to the two equilibrium conditions shown in Figure 3.5.
Figure 3.5. Joint vertical forces equilibrium
The possibility of simplifying the assessment procedure considering N due to gravity loads
does not dependent only on the properties of the structural members (shape, dimensions and
reinforcement content) but also on the structural configuration of the building and, in
particular, on the slenderness of the building, defined as H/B, where H and B are the total
building height and the total building width in the direction of seismic loads, respectively.
Increasing the building slenderness, in fact, the variation of N due to seismic action in the
perimetral columns increases too, with the consequence that the ductility capacity of
perimetral columns reduces. This reduction may affect seriously the assessment. Therefore, a
wide range of values of building slenderness should be analyzed in order to draw general
conclusions about this topic.
The problem related to the choice of how to compute the value of shear span length has been
already discussed (Sec. 3.2.1).
51
Finally, it must be considered that Eq. (3.15) is fully empirical and calibrated on structural
members typical of new constructions. Therefore, the reliability of the results, in the case of
the assessment of existing buildings, may be questionable, in particular if the materials have
experienced a sensible degradation of their mechanical properties and/or if the reinforcement
is heavily under-dimensioned. In these cases, in fact, the values of fcm, ρsx and ν may be out of
the range of the values considered to calibrate Eq. (3.15).
3.2.3 Near Collapse Limit State: Theoretical approach
The second approach to compute the chord rotation capacity at NC LS is based on theoretical
considerations. Assuming a bi-linear idealization of the moment curvature diagram (Figure
3.6), the curvature grows linearly from the point of contraflexure (free end of equivalent
cantilever) to the yielding section. From that yielding section to the fixed end of the member,
the curvature still grows linearly but the slope becomes much higher (due to the reduction of
tangent stiffness). The length of this last part of the member, where the nonlinear behaviour
develops, is called length of plasticization, lpl.
Figure 3.6. M-Φ diagram
A more simplified approach is proposed for the calculation of the curvature at ultimate
conditions. The overall curvature is divided into two contributions:
(i) elastic curvature, which grows linearly from zero at the free end of the equivalent
cantilever to φy at the fixed end of the equivalent cantilever;
(ii) plastic curvature (φu - φy), constant over a length Lpl, called “plastic hinge length”. Lpl
is, therefore, shorter than the real length of plasticization lpl.
As a consequence, also the ultimate chord rotation capacity is evaluated as the sum of an
elastic part, due to the elastic curvature, plus a plastic part due to the constant plastic
curvature developed over the plastic hinge length.
The elastic part may be taken equal to the yielding chord rotation, θy, already computed for
the assessment at the DL LS.
The plastic part of the chord rotation, θ u , pl , may be calculated as:
52
⎛
θ u , pl = (φu − φ y ) ⋅ L pl ⋅ ⎜⎜1 −
⎝
L pl ⎞
⎟
2 Ls ⎟⎠
(3.20)
It may seem that Eq. (3.20) does not account for any other effect beside the flexural one.
Actually, both shear and bond-slip contributions to the ultimate chord rotation capacity are
dealt with indirectly through the plastic hinge length Lpl, which is not a physical quantity, but
a conventional one. The Italian seismic Code proposes the following formula, calibrated for
cycling loading conditions:
L pl = 0.1Ls + 0.17 h + 0.24
d bl f ylm CF
f cm CF
(3.21)
where dbl is the mean diameter of the tension reinforcement, while both fcm and fylm are in
MPa.
Two different possible expressions of Lpl are suggested in Eurocode 8. One of these two
formulas is equal to Eq. (3.21). Hence, in order to be consistent with the Italian Seismic Code,
Eq. (3.21) is used also for the assessment based on Eurocode 8.
Finally, the overall ultimate chord rotation capacity is obtained as:
θu =
L pl
⎛
1 ⎡
⎢θ y + φu − φ y L pl ⎜⎜1 −
γ el ⎢⎣
⎝ 2 Ls
(
)
⎞⎤
⎟⎥
⎟
⎠⎥⎦
(3.22)
According to both Codes, no reduction factor should be applied to Eq. (3.22) to deal with the
lack of seismic details, since the evaluation of φy and φu are already based on the actual
geometrical and mechanical properties of the cross-sections.
The factor γ el is assumed equal to 1.5 in the Italian Seismic Code and to 2 in Eurocode 8, for
seismic structural members. The assessment performed according to Eurocode 8 is, therefore,
more conservative than that obtained considering the Italian Seismic Code.
The time required for the calculations is much longer in the case of Eq. (3.22), since a correct
evaluation of the theoretical ultimate curvature φu is complex, as it will be shown in the
following part of this chapter.
Unlike Eqs. (3.17) and (3.18), which are purely empirical, Eq. (3.22) is a mixed approach,
since it is based on theoretical considerations concerning the computation of both φy and φu,
but it is empirical too with reference to the evaluation of Lpl.
A deepening is required to establish how to assess the ultimate curvature φu . Section failure is
conventionally considered to take place when the moment capacity drops at 80% of its peak
value. On the base of this consideration:
•
if the spalling of concrete cover, which is not confined by the stirrups, causes a sudden
drop of the resisting moment larger than 20%, then the spalling of concrete cover may
53
be regarded as the failure of the section and, therefore, the curvature at spalling may
be considered as the ultimate curvature. This condition is common for columns with
low confinement ratio (typical of existing buildings without seismic details) and/or
high axial load ratio.
•
if the resisting moment, computed considering only the confined core of the section, is
larger than 80% of the resisting moment of the unspalled section, then the ultimate
curvature will be attained at the failure of the spalled section, due to crashing of
concrete core (if the axial load ratio is high) or to buckling of compression bars
(especially of corners ones) or to rupture of one or more tensile bars (if the member is
lightly reinforced).
In order to calculate both curvature and resisting moment at spalling of concrete cover, the
following assumptions were considered:
•
Navier-Bernoulli hypothesis (plan sections remain plan once deformed).
•
Perfect bond (no slip) between longitudinal bars and surrounding concrete.
•
Cross-sections already cracked and, hence, no concrete resistance to tension.
•
Bi-linear characteristic curve of longitudinal reinforcement steel. It was considered
elastic linear (Es = 200000 MPa) until the yielding point, f ylm CF , and then perfectly
plastic until the ultimate strain value, εsul. Therefore, the strain-hardening effect was
not taken into account in the calculation of the flexural capacity of the sections. From
experimental data on nude steel bars under monotonic loading conditions
ε su = 10 − 12% was determined, depending on the steel properties. However, it should
be considered that εsul of the tensile reinforcement under cycling loading conditions is
only a small fraction of the experimental εsu obtained on the nude bar under monotonic
loading conditions. In this work ε sul = 0.04 was assumed (according to the Italian
Seismic Code). The linear elastic-perfectly plastic curve of bar steel was considered to
be symmetric (same behaviour in compression and in tension).
•
Compressive unconfined concrete σ-ε law rising parabolically up to the peak point
( ε co = 0.002 ) and than staying constant up to the ultimate strain, εcu. The peak stress
was evaluated simply as f cm CF . Any reduction factor due to long term effects which
is, normally, incorporated in the design tools, in fact, should not be applied. It means
that the compressive peak concrete stress should not be multiplied times 0.85
(reduction factor according to Eurocode 2, Eurocode 8 and Italian Codes for R.C.
structures). In this work ε cu = 0.004 was assumed.
The calculation of curvature and bending moment capacity at spalling of concrete cover is not
trivial, since both equilibrium and compatibility conditions have to be verified. In this work,
the following procedure was considered.
(i)
Before any calculation, the strain distribution at the ultimate conditions of the
unspalled section was assumed. Failures of either extreme compressive concrete layer
54
or tensile reinforcement were both considered. Moreover, for each of these two
conditions, the compressive reinforcement was supposed either already yielded or still
in its elastic range.
(ii)
The equilibrium of the section was, then, computed on the base of the assumptions
made in step (i) and the value of the neutral axis depth, xu,unsp, was obtained.
(iii) Steps (i) and (ii) were repeated for each possible assumption on strain distribution (all
the possible values of xu,unsp were, hence, calculated).
(iv) For each xu,unsp, the assumptions on strain distribution were, then, checked through the
application of compatibility conditions (assuming both Navier-Bernoulli and perfect
bond hypotheses).
(v)
The correct curvature and bending moment capacities at spalling of concrete cover
were, finally, evaluated in correspondence to the only value of xu,unsp that satisfies both
equilibrium and compatibility conditions.
In particular, in this work, this procedure was developed for T-sections. In fact, the
rectangular shape may be seen as a particular case of the T-shape, characterized by equal
bottom and top width. Therefore, in the following, only the procedure developed for Tsections will be shown.
If xu,unsp is smaller than the flange thickness, the section may be considered as it were
rectangular, characterized by a constant width equal to the top (flange) width. Therefore, each
assumption on strain distribution should be considered twice, first supposing that xu,unsp lies in
the flange and than assuming that it lies in the web.
Since in old buildings the degradation of mechanical properties of the materials may be
remarkable and strong ground motions may lead to very high compression and tension axial
load ratios in perimetral columns, the strain distributions corresponding to all the possible
section failures (from pure tension to pure compression) should be considered.
In this work, the following possible deformed configurations of the cross-section were
supposed in order to compute xu,unsp at spalling of concrete cover.
1. Pure tensile failure. The applied N is larger than the maximum tensile force that the
section can bear (i.e. the sum of the areas of all longitudinal bars times their yielding
stress). This rare condition may happen in perimetral columns, when a tensile N acts
on the section and, in the meanwhile, the section is under-reinforced.
2. Tensile failure. The neutral axis lies outside the section, which reacts only thanks to
the reinforcement. Like the previous case, it is possible only in columns subjected to
tensile N. Since, in general, the columns are symmetrically reinforced, the ultimate
conditions are obtained when the bars farther from the neutral axis fail in tension (εsl =
εsul), while the bars closer to the neutral axis are still in their elastic range (εsl’ < εsyl).
55
3. Tensile failure of the reinforcement farther from the neutral axis (εsl = εsul),
reinforcement closer to the neutral axis in tension and still in its elastic range (εsl’ <
εsyl), neutral axis in the cover of the section and concrete σ-ε law considered linear
(compressive concrete strain and stress are likely to be very small). Like the previous
two cases, it is feasible only in columns subjected to tensile N.
4. Failure of the tensile reinforcement (εsl = εsul), compressive reinforcement still in its
elastic range (εsl’ < εsyl), concrete σ-ε law considered parabolic-constant, rectangular
section or T-section with neutral axis in the flange (equivalent rectangular section).
5. Failure of the tensile reinforcement (εsl = εsul), compressive reinforcement still in its
elastic range (εsl’ < εsyl), concrete σ-ε low considered parabolic-constant, T-section
with neutral axis in the web.
6. Failure of the tensile reinforcement (εsl = εsul), compressive reinforcement already
yielded (εsl’ > εsyl), concrete σ-ε law considered parabolic-constant, rectangular section
or T-section with neutral axis in the flange (equivalent rectangular section).
7. Failure of the tensile reinforcement (εsl = εsul), compressive reinforcement already
yielded (εsl’ > εsyl), concrete σ-ε law considered parabolic-constant, T-section with
neutral axis in the web.
8. Failure at the extreme fibre of the compressive concrete (εc = εcu), compressive
reinforcement still in its elastic range (εsl’ < εsyl), tensile reinforcement already yielded
(εs > εsyl), rectangular section or T-section with neutral axis in the flange (equivalent
rectangular section).
reinforcement still in its elastic range (εsl’ < εsyl), tensile reinforcement already yielded
(εs > εsyl), T-section with neutral axis in the web.
reinforcement already yielded (εsl’ > εsyl), tensile reinforcement already yielded (εs >
εsyl), rectangular section or T-section with neutral axis in the flange (equivalent
reinforcement already yielded (εsl’ > εsyl), tensile reinforcement already yielded (εs >
εsyl), T-section with neutral axis in the web.
reinforcement still in its elastic range (εsl’ < εsyl), tensile reinforcement still in its
elastic range (εs < εsyl), rectangular section or T-section with neutral axis in the flange
(equivalent rectangular section).
56
reinforcement still in its elastic range (εsl’ < εsyl), tensile reinforcement still in its
elastic range (εs < εsyl), T-section with neutral axis in the web.
reinforcement already yielded (εsl’ > εsyl), tensile reinforcement still in its elastic range
(εs < εsyl), rectangular section or T-section with neutral axis in the flange (equivalent
reinforcement already yielded (εsl’ > εsyl), tensile reinforcement still in its elastic range
(εs < εsyl), T-section with neutral axis in the web.
16. Failure at the extreme fibre of the compressive concrete (εc = εcu), reinforcement
farther from the neutral axis already yielded (εsl’ > εsyl), reinforcement closer to the
neutral axis in compression and still in its elastic range (εs < εsyl), neutral axis in the
cover of the section.
neutral axis in compression and still in its elastic range (εs < εsyl), neutral axis outside
the section which is all compressive.
neutral axis in compression and already yielded (εs > εsyl), neutral axis outside the
section which is all compressive.
19. Pure compressive failure. The external N is larger than the maximum axial force that
the section can bear (i.e. the sum of the concrete area times the peak concrete
resistance plus the total longitudinal steel area times its yielding stress).
The last four configurations are feasible only in perimetral columns, when they are
characterized by a huge compressive axial load ratio. Therefore, since all the columns
considered in this work are rectangular (or square), the computation was developed only for
rectangular sections.
These 19 possible deformed configurations were considered for both positive and negative
flexure.
In the case of columns, since the sections are symmetrically reinforced, the values of both
positive and negative curvature coincide. Hence, only 13 cases were analyzed (those
corresponding to rectangular sections).
In the case of beams, the first three and the last four cases are unfeasible, since no axial load is
considered to act on beams. Therefore, 12 cases were considered for rectangular beams (six in
positive and six in negative bending). For T-shape beams, instead, the equilibrium was
57
computed on the base of 24 different strain distributions (12 in positive and 12 in negative
bending).
In the first and last strain configurations, both curvature and bending moment capacities are
null.
In cases 2, 3, 4, 5, 6, 7 the failure of the section is due to steel rupture. The curvature at
spalling of concrete cover is, hence, equal to:
φu ,unsp =
ε sul
d − xu ,unsp
(3.23)
In case 2 the value of the neutral axis is negative.
In cases 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 the failure of the section is due compressive
concrete failure. Therefore, curvature at spalling of concrete cover is equal to:
φu ,unsp =
ε cu
xu ,unsp
(3.24)
In cases 17 and 18 the neutral axis is larger than the overall height of the section.
In order to compute both curvature and resisting moment at ultimate conditions of the
confined concrete core, a confining model of the compressive concrete must be selected and
considered in calculations. Eurocode 8 suggests two different confinement models. According
to both, the σ-ε law rises parabolically until the peak point (εcco; fccm) and then stays constant
until the ultimate strain εccu. The values of fccm, εcco, εccu are determined respectively as
follows:
0.86
⎡
⎛ αρ sx f ywm ⎞ ⎤ ⎫
⎟ ⎥⎪
f ccm = f cm ⎢1 + 3.7⎜⎜
⎟ ⎥⎪
f
⎢
cm
⎝
⎠ ⎦
⎣
⎪
⎡
⎛ f ccm
⎞⎤
⎪ st
ε cco = 0.002⎢1 + 5⎜⎜
− 1⎟⎟⎥
⎬1 Model
⎝ f cm
⎠⎦⎥
⎪
⎣⎢
αρ sx f ywm
⎪
ε ccu = 0.004 + 0.5
⎪
f ccm
⎪
⎭
(3.25)
where α and ρ sx are defined in Eq. (3.15).
58
⎫
⎧ ⎛
αρ sx f ywm ⎞ if
⎟ ⎯⎯→ αρ sx f ywm ≤ 0.05 f cm
⎪
⎪ f cm ⎜⎜1 + 5
f cm ⎟⎠
⎪ ⎝
⎪
f ccm = ⎨
⎪⎪ nd
f
αρ
if
⎪ f ⎛⎜1.125 + 2.5 sx ywm ⎞⎟ ⎯⎯→
αρ
f
0
.
05
f
>
cm ⎜
sx
ywm
cm
⎬2 Model
⎪⎩ ⎝
f cm ⎟⎠
⎪
2
⎪
αρ sx f ywm
⎛f ⎞
ε cco = 0.002 ⋅ ⎜⎜ ccm ⎟⎟ ; ε ccu = 0.004 + 0.2
⎪
f cm
⎝ f cm ⎠
⎭⎪
(
)
(
)
(3.26)
Eq. (3.25), proposed by Fardis and Panagiotakos [2001], is the modified model proposed by
Mander et al. [1988]. It is more accurate than Eq. (3.26), suggested in Eurocode 2. On the
other hand, Eurocode 8 recommends to use Eq. (3.26) when Eq. (3.21) is considered for the
evaluation of the plastic hinge length Lpl. Therefore, in this work, Eq. (3.21) and Eq. (3.26)
were selected to assess the plastic hinge length and the properties of the confined concrete
core, respectively.
The procedure to compute the curvature at ultimate conditions of the confined concrete core is
analogous to that already considered for the calculations at spalling of concrete cover. The
only differences are that the confined σ-ε law of compressive concrete should be used instead
of unconfined one (fcm, εco, εcu should be replaced by fccm, εcco, εccu) and that the centrelines of
the closed stirrups should be regarded as the border of the resisting section. Accordingly, only
the strain configurations of rectangular sections should be considered, since the two lateral
parts of T-beam flange are regarded as unconfined. In particular, in this work, only the
following strain configurations were considered, in order to compute the neutral axis depth at
ultimate conditions of the confined concrete core, xu,sp.
•
In the case of beams, the configurations 4, 6, 10, 14 were considered for both positive
and negative flexure. Hence, eight cases were analyzed.
•
In the case of columns, the configurations 1, 2, 4, 6, 10, 14, 17, 18, 19 were
considered. Since the sections are symmetrically reinforced, the values computed for
positive and negative flexure coincide. Hence, nine cases were analyzed.
Once computed the correct value of xu,sp, both curvature and resisting moment at ultimate
conditions of the confined concrete core were assessed.
For each end section of each structural member, for both positive and negative bending,
considering N from both gravity loads and seismic combination (Nmin and Nmax if a dynamic
analysis is performed), the bending moment capacities at both spalling of concrete cover and
ultimate conditions of confined core were computed and, then, compared. When the ratio of
the resisting bending moment at ultimate conditions of the confined core to the resisting
bending moment at spalling of concrete cover was less than 0.8, the spalling of the cover was
judged as the section failure. Otherwise, the confined core of the section was assumed to
recover from the loss of the cover and, hence, the section failure was considered to correspond
to the ultimate conditions of the confined concrete core.
59
Taking into account the residual capacity of concrete core after spalling of concrete cover
leads to a significant increase in the curvature capacity. In fact, the ultimate strain of confined
compressive concrete, εccu, is (much) larger than the unconfined one, εcu. Moreover, after the
spalling of cover, the effective depth of the section decreases and, therefore, the curvature
capacity increases.
On the other hand, in general, the maximum value of resisting moment is obtained in
correspondence to the capacity of the unspalled section. The reason why this happens may be
explained as follows. The moment capacity can be computed as the force resisted by the
compressive concrete times the length of the inner lever arm. The values of the force resisted
by the compressive concrete of both unconfined gross section and confined core are similar to
each other (because fccm is only slightly larger than fcm and the compressive concrete area
reduces with spalling of cover), while the length of the inner lever arm decreases after
spalling. Therefore, the resisting bending moment computed on the unspalled section is, in
general, larger that the moment capacity computed on the confined concrete core. If the
neutral axis depth is large (i.e. when a large compressive N acts on the section), the bending
moment capacity will be very sensitive to the reduction of the length of the inner lever arm. In
this case it is likely that the reduction of the length of the inner lever arm will lead to a
sensible reduction of the moment capacity, which will be possibly lower than 80% of that
computed at spalling conditions. It is probable, hence, that perimetral columns subjected to
large compressive N will fail at spalling of concrete cover and will be characterized, therefore,
by a low deformation (curvature) capacity.
Eq. (3.15) and Eq. (3.22) are expected to yield close results, since both Codes allow the use of
one indifferently to the other. Actually, the results obtained through the theoretical approach
(Eq. (3.22)) are very sensitive to how the shear span Ls is evaluated. In fact, it may happen
that, if Ls is computed according to its correct definition (Eq. (3.1)) and if the bending
moment demand at the considered member end is particularly low, the value of Ls will be low
too, possibly shorter than the plastic hinge length Lpl. In particular, if Lpl > 2Ls, than the
term 1 − L pl (2 Ls ) of Eq. (3.22) will be negative and θu will be lower than θy or even
negative, if L pl >> 2 Ls .
(
)
An aim of this work is to test the reliability of the assessment of ductile mechanisms based on
Eq. (3.22), considering both correct (Eq. (3.1)) and simplified (Ls = L/2) definition of the
shear span length Ls. In order to achieve this goal, the results obtained using Eq. (3.22) are
compared to those obtained applying Eq. (3.15) (which is less sensitive to Ls).
3.3 Ductile Check: Biaxial Bending
Biaxial bending may be crucial in the assessment of ductile mechanisms. In fact, in general,
the results of seismic analyses show that columns may undergo biaxial bending. In particular:
•
if a linear dynamic analysis is performed, the building will be excited by an input
which takes into account the contribution of all modes of interest. As a result, the
magnitudes of the two orthogonal columns bending moments will be comparable, as
well as the interstorey drifts.
60
•
If a building characterized by a strong torsional response is analyzed through a
pushover analysis, the perimetral columns (particularly those located in the flexible
side of the structure) will undergo bending moments and drifts not only in the pushing
direction, but also in the orthogonal one, leading to a bi-axial bending excitation.
•
If a time history analysis is considered, since the input ground motion is applied in
both horizontal principal directions simultaneously, the columns will be subjected to
bi-axial bending.
Although this topic may be crucial for the assessment of a building subjected to seismic
excitation, the available experimental data are very limited and, therefore, the present-day
knowledge of the inelastic behaviour of R.C. columns under biaxial cycling moments is
neither deep nor accurate, well behind the understanding of the behaviour in uniaxial cycling
flexure. Nevertheless, it is clear from the few available results [Bousias et al., 1992, 1995,
2002] that there is a strong coupling between the behaviours in the two principal transverse
directions. In fact, the moment-curvature response in one principal direction is affected by
magnitude and history of the moment-curvature in the orthogonal one, in the following way:
•
stiffness and resistance in each individual direction reduce with respect to those in
uniaxial bending;
•
deformation (curvature) capacity in each principal direction reduces with respect to
that in uniaxial bending.
Both Codes do not explain how to consider the problem of the evaluation of the chord
capacity in biaxial bending. In fact:
•
Eq. (3.15) is purely empirical and calibrated on experimental data obtained under
uniaxial bending. Therefore, it is not suitable for the evaluation of biaxial chord
rotation capacity.
•
Eq. (3.22) is function of the shear span ratio, Ls/h, which is defined only for uniaxial
bending. Hence, also Eq. (3.22) is suitable only for the assessment of uniaxial chord
rotation capacity.
For these reasons, in this work, the interaction between the two orthogonal column chord
rotations was taken into consideration, even if not suggested by both Codes. The following bidimensional failure curve [Bousias et al., 2002] was used:
⎛ θ Dx
⎜
⎜θ
⎝ Cx ,uni
2
⎞ ⎛ θ Dy
⎟ +⎜
⎟ ⎜θ
⎠ ⎝ Cy ,uni
2
⎞
⎟ =1
⎟
⎠
(3.27)
According to this expression, the chord rotation demands in the two principal directions, θDx
and θDy, are normalized to the corresponding uniaxial capacities, θCx,uni and θCy,uni, in order to
define the threshold of the bi-dimensional safe domain.
61
The approach based on two separate chord rotation checks in the two principal directions was
compared with that based on one single check which considers the coupling of the chord
rotations in the two principal directions through Eq. (3.27).
An aim of this work is, therefore, to test if the definition of a bi-dimensional failure curve
leads to a substantial improvement in the accuracy of the results with respect to the
conventional uniaxial check, performed separately in the two principal directions.
Since this subject is strictly correlated to the structural configuration (regularity of building’s
shape, distribution of mass, stiffness and resistance), the assessment of several other buildings
of different characteristics is required to get more general conclusions.
3.4 Brittle Mechanisms: Shear Capacity
The brittle mechanisms are assessed at a section level, through the comparison of shear
demand and corresponding capacity at both ends of each structural member. Unlike the
verification of ductile mechanisms, the assessment is required only at the most severe (i.e.
NC) LS.
The approach to assess the shear capacity of R.C. structural members proposed in the Italian
Seismic Code differs from that recommended in Eurocode 8. In fact, the Italian Seismic Code
suggests to treat the problem of computing the shear capacity as it was a non-seismic assess,
with the only restriction of limiting the contribution of the concrete to the value of shear
resistance computed for the same section considered without lateral reinforcement. Hence, the
procedure proposed in the Italian Seismic Code is very quick to apply. On the other hand, the
response induced by the seismic action is more complicated and it is affected by lots of
phenomena which are neglected by the approach proposed in the Italian Seismic Code. These
phenomena, due to both cyclic nature of the seismic excitation and possible development of
nonlinear mechanisms at the member ends, lead to a degradation of the shear resistance of
R.C. members. In particular, the cyclic nature of the seismic excitation induces the following
effects:
•
reduction of aggregate interlock along diagonal cracks, as their interfaces become
smoother;
•
degradation of dowel action;
•
development of flexural cracks throughout the depth of the member and reduction of
the contribution of the compressive zone.
The degradation of shear resistance is larger within flexural plastic hinges at member ends,
because:
•
flexural cracks are larger and intersect diagonal cracks;
•
the compressive zone is more damaged and its size is reduced;
62
•
longitudinal bars develop inelastic strains or even buckle and reduce their
effectiveness in dowel action;
•
the compressive zone has to resist also the effects of the diagonal strut of the truss
mechanism.
3.4.1 Shear Capacity according to the Italian Seismic Code
According to the indications furnished in the Italian Seismic Code, the shear resistance is
computed as in a non-seismic design. In this work, the approach proposed in “Decreto
Ministeriale (D.M.) 09-01-1996” was considered. According to “D.M. 09-01-1996”, the
Mörsch-Ritter truss is assumed as the resisting model, the inclination of the compressive
concrete struts is considered constant and equal to 45°. The shear resistance, VR, is, therefore,
computed as the minimum between the value that causes the transverse reinforcement to yield
in tension and the value that leads to the failure in diagonal compression of the concrete web:
V R = min(Vc + Vw ;Vc ,max )
(3.28)
The (Vc + Vw ) term is the shear that causes the transverse reinforcement to yield in tension.
Vc is the concrete contribution to shear resistance. “D.M. 09-01-1996” prescribes to assume
the minimum value between that relative to the actual reinforced section and that computed
considering the same section without transversal reinforcement. In the former case, Vc takes
into account the contribution of tensile strength of the concrete. Vc must be considered since
Vw accounts only for the resistance offered by the tensile steel and the compressive concrete
struts. In the latter case, Vc is a purely empirical term, due to the aggregate interlock along
diagonal cracks, the dowel action of the longitudinal bars and the tensile strength of concrete
between diagonal cracks. The following expression was assumed for Vc:
Vc =
⎡
⎤
⎛
⎛ A
⎞⎞⎞
f ctm
d (mm) ⎞ ⎛⎜
⎛
bw dδ ⎢min⎜ 0.25 max⎜1.6 −
;1⎟ ⋅ ⎜1 + 50 min⎜⎜ sl ;0.02 ⎟⎟ ⎟⎟ ⎟;0.6⎥
⎜
1000 ⎠ ⎝
γ c CF
⎝
⎢⎣
⎥⎦
⎝ bw d
⎠ ⎠ ⎟⎠
⎝
(3.29)
where:
•
f ctm is the mean value of tensile strength of the concrete, evaluated as
f ctm = 0.25 ⋅ ( f cm )2 3
•
Asl is the total area of tensile longitudinal reinforcement;
•
δ is a term which takes into account the contribution of axial load, N. For beams
(N=0), δ = 1 . For columns in tension, δ = 0 . For columns in compression,
δ = 1 + M dec M A,max , where:
o Mdec is the bending moment that causes the neutral axis to lie exactly in
correspondence to the extreme fibre of the section, which, therefore, is all in
compression, with a triangular distribution of stresses.
63
o MA,max is the maximum moment that acts on the considered section, at least equal
to Mdec. In a nonlinear analysis MA,max = max(MA, Mdec), where MA is the moment
obtained directly from the analysis. Instead, if a linear elastic analysis is
performed, MA,max = max(min(MA, MR), Mdec), where MR is the bending moment
capacity, evaluated by means of the mean values of the material properties divided
by both partial factor of the concrete (γc = 1.6 in Italy) and the acquired CF. Hence,
if a column subjected to compressive N is considered:
M dec
, for nonlinear analyses
max(M A , M dec )
(3.30)
M dec
, for linear analyses.
max(min(M A , M R ), M dec )
(3.31)
δ = 1+
δ = 1+
Vw is the resisting force due to the transversal reinforcement:
Vw = Asw ⋅
f ywm
γ s CF
⋅
(d − d ')
s
(3.32)
where:
•
Asw is the total area of transversal reinforcement in the direction of the acting shear;
•
s is the spacing in the direction of the member axis between the centreline of two
adjacent transversal bars.
The shear that causes the failure in diagonal compression of the concrete web is computed,
according to the Mörsch-Ritter truss model, as:
Vc,max =
0.6 f cm
bw d
2γ c CF
(3.33)
In a diagonal strut the maximum compressive stress is less than its uniaxial value, due to the
presence of tensile stresses and strains in the orthogonal direction. According to Eq. (3.33),
the maximum compressive stress is taken equal to 0.6 fcm.
In the case of beams and columns subjected to tension VR is constant and can be determined
on the base of geometrical and mechanical properties of the structural members. If, instead, a
column subjected to compression is considered, VR will be sensitive also to the seismic
demand through the factor δ (Eqs. (3.30)-(3.31)), which is function of both N and M demands.
If a linear dynamic analysis is performed, it will not be trivial to determine N corresponding to
Mmax and Mmin. A possibility is to compute δ for both Nmax and Nmin, for both Mmax and Mmin,
and then choose the most unfavourable (smallest) value of δ. On the other hand, this
procedure may be too conservative and, however, requires to assess twice the value of Vc for
Mmax and twice for Mmin. If N due only to gravity loads (which is roughly the mean N that the
64
columns undergo during a seismic input) is considered for the assessment, both Mdec and MR
will be constant, independent on the seismic action and, therefore, δ will depend on the
demand only through M. Hence, choosing N from gravity loads will allow to bypass the
problem of determine the correspondence N-M and also to halve the number of required
operations. In this work, the consistency of the results obtained using N due to gravity loads
with those found considering N from the seismic load combination is tested. As already
remarked concerning the assessment of ductile mechanisms, the possibility of simplifying the
assessment procedure considering N due to gravity loads depends strictly on the slenderness
of the building, defined as H/B, where H and B are the total building height and the total
building width in the direction of seismic loads, respectively. Increasing the building
slenderness, in fact, the variation of N due to seismic action in the perimetral columns
increases too and this may affect seriously the bending moment capacity. Therefore, a wide
range of values of building slenderness should be analyzed in order to draw general
conclusions.
When a plastic hinge forms, the bending moment demand, MA,max, will be locked to a constant
value equal to its corresponding capacity, MR. Assuming constant N, δ will be constant
(constant Mdec and MR) and, hence, VR will be constant too, regardless of the intensity of
seismic loads. It shows that Eq. (3.28) is not sensitive to the development of plastic
mechanisms at the member ends, as the degradation of shear capacity in inelastic regions is
neglected. In a nonlinear static analysis, if the point representative of the structural demand
belongs to the descending branch of the capacity curve, evaluating the local shear demands at
that point would be not consistent with Eq. (3.28). It is the reason why the Italian Seismic
Code proposes to evaluate the local shear demands at the peak point of the capacity curve
(maximum base shear). This choice is consistent with the non-seismic way of assessing VR.
According to Eq. (3.29), Vc is subjected to an instantaneous drop when N passes from
compression to tension. The code, in fact, proposes to neglect the contribution of concrete
when a tensile N is applied to the considered member (Figure 3.7).
Figure 3.7. Shear capacity according to “D.M. 09-01-1996”
In a R.C. frame building subjected to a seismic excitation, if the additional contribution to N
due to the overturning moment is larger than N obtained from the gravity loads in some of the
perimetral columns, those columns will pass several times from compression to tension and
65
vice versa, with the consequence that their assessment will be penalized. It must be
considered, moreover, that it is common for an existing building to be under-reinforced, in
particular concerning the amount of stirrups close to the structural joints. In this case, Vc
becomes the principal source of VR. Hence, if a column subjected to a very low compressive N
and the same column subjected to a very low tensile N are considered, it will be possible that
in the former case VR will be much larger than in the latter, while, actually, the two values
should be similar.
An aim of this work is to show if, neglecting Vc, the assessment of tensile columns will be
too severe, compared to the more accurate procedure suggested in Eurocode 8.
3.4.2 Shear Capacity according to Eurocode 8
Eurocode 8 takes into account the effects due to both cycling nature of seismic load and
possible development of nonlinear mechanisms in member ends, by decreasing VR with
increase of cycling inelastic deformations. Since the chord rotation is considered the most
meaningful deformation quantity of the structural members, VR is assumed to be function of
the plastic part of the chord rotation ductility demand, μθpl,dem :
μθpl,dem = μθ ,dem − 1 =
θ dem − θ y
θy
(3.34)
where θ y can be computed through Eq. (3.5).
The following expression is proposed for VR:
VR =
[
⎛
⎞
f
1 ⎧⎪ h − x dem
min⎜⎜ N ;0.55 Ac cm ⎟⎟ + 1 − 0.05 min 5; μθpl,dem
⎨
1.15 ⎪⎩ 2 Ls
γ c CF ⎠
⎝
⎡
⎛
⎛ Ls
⎢0.16 max(0.5;100 ρ tot )⎜⎜1 − 0.16 min⎜ 5;
⎝ h
⎢⎣
⎝
(
)]
⎤ ⎫⎪
⎞ ⎞ f cm
Ac + Vw ⎥ ⎬
⎟ ⎟⎟
⎠ ⎠ γ c CF
⎥⎦ ⎪⎭
(3.35)
where:
•
xdem is the neutral axis depth, determined in correspondence of N and M demands that
act on the section when it experiences the considered value of V demand;
•
N is positive for compression, taken equal to zero for tension;
•
ρ tot = Asl (bd ) is the total longitudinal reinforcement ratio;
•
Vw = ( Asw s ) (d − d') f ywm (γ s CF ) is the contribution of transverse reinforcement.
Eq. (3.35) can be applied to beams and slender columns and refers only to diagonal tension
failure.
66
Although the shear failure should be regarded as brittle mechanism since V-γ loops are
narrow, inverted S-shaped and provide little energy dissipation, the idea that lies at the basis
of Eq. (3.35) is that two distinct behaviours should be considered.
(i) “Brittle shear”. Ultimate shear failure occurs before flexural yielding, at relatively low
deformations and is characterized by a large drop in lateral load resistance. It means
that, computed the shear corresponding to the flexural yielding, Vy = My/Ls, and the
elastic shear resistance, VR,0, (according to Eq. (3.35), where μθpl,dem = 0 and xdem = xy),
VR,0 < Vy.
(ii) “Ductile shear”. Concrete members subjected to cycling loading may first yield in
flexure but at the end fail showing strong and clear shear effects, like wide inclined
cracks, while the peak force resistance experiences a gradual drop with cycling. It
means that Vy is smaller than VR,0 but larger then VR at ultimate conditions (according
to Eq. (3.35), where the demand quantities ( μθpl,dem , xdem and Ls) are those referring to
the actual demand at ultimate conditions). Since VR decreases linearly with μθpl,dem ,
there is a plastic deformation level which corresponds to this “relative ductile” shear
failure (Figure 3.8).
Figure 3.8. Degradation of shear capacity due to development of plastic hinges
According to Eq. (3.35), beyond a limit value of μθpl,dem = 5 , assuming xdem constant (xdem does
not change significantly once plastic moment developed), VR is constant at its lowest value
VR,min. For beams (N=0), when μθpl,dem ≥ 5 , V R ,min = 0.75V R ,0 . Hence:
•
if V R ,0 < V y , the element will undergo a “brittle shear” failure;
•
if VR ,min < V y < VR ,0 , the element will undergo a “ductile shear” failure;
•
if V y < VR ,min , the element will not fail in shear.
The axial load demand affects only the first part of the Vc term of Eq. (3.35), which becomes,
for beams and columns subjected to tension:
67
{[
)]
(
1
1 − 0.05 min 5; μθpl,dem ⋅
1.15
⎡
⎛
⎛ Ls
⎢0.16 max(0.5;100 ρ tot )⎜⎜1 − 0.16 min⎜ 5;
⎝ h
⎝
⎣⎢
VR =
⎤ ⎫⎪
⎞ ⎞ f cm
Ac + Vw ⎥ ⎬
⎟ ⎟⎟
⎠ ⎠ γ c CF
⎦⎥ ⎪⎭
(3.36)
Therefore, unlike Eq. (3.29), Vc will not vanish if the member is subjected to tensile N but it
will decrease smoothly when N passes from compression to tension.
Eq. (3.35) accounts only for the shear failure by diagonal tension of structural members. This
kind of failure mechanism is typical of members which are not axially loaded (beams) and
those which can be regarded as slender elements, although subjected to compression
(columns). A measure of the slenderness of a R.C. member is the shear span ratio Ls/h. If in a
R.C. column Ls/h > 2, than the column may be regarded as slender and the failure in diagonal
tension according to the Mörsch-Ritter truss may be considered as the only possible shear
failure mode. If, instead, Ls/h < 2, then the column should be regarded as squat and the failure
by web crushing along the diagonal of the column should be considered instead of Eq. (3.35),
since Eq. (3.35) does not represent the actual shear failure mechanism of squat columns.
For R.C. columns characterized by Ls/h < 2, Eurocode 8 suggests the following empirical
expression to evaluate VR under cycling loading (with units: MN and meters):
[
(
)]
4
1 − 0.02 min 5; μθpl,dem ⋅
8.05
⎛
⎞
⎛
Nγ c CF ⎞
f
⎜⎜1 + 1.35
⎟⎟[1 + 0.45(100 ρ tot )] min⎜⎜ 40; cm ⎟⎟bw (d − d ') sin (2δ )
Ac f cm ⎠
γ c CF ⎠
⎝
⎝
V R ,max =
(3.37)
where δ = arctan(h 2 Ls ) is the angle between the diagonal strut and the axis of the column.
Eq. (3.37) shows that also the shear failure of squat columns due to diagonal compression is
affected by the development of inelastic behaviour at member ends. VR,max, in fact, reduces
linearly with increasing in μθpl,dem and stabilizes for μθpl,dem ≥ 5 at a value equal to 90% of that
at μθpl,dem = 0 .
Unlike the failure in diagonal tension (which can be considered relatively ductile, if it is
preceded by flexural yielding), the shear failure of squat columns must be regarded as brittle,
even if the member experiences a flexural inelastic behaviour before diagonal compressive
failure of the concrete.
Both Eqs. (3.35) and (3.37) are fully empirical and calibrated on structural members typical of
new constructions. Therefore, the reliability of the results in the case of the assessment of
existing buildings, may be questionable, in particular if the materials have experienced a
sensible degradation of their mechanical properties and/or if the reinforcement is heavily
under-dimensioned. In these cases, in fact, the values of fcm and ρtot may be out of the range of
the values considered to calibrate Eqs. (3.35) and (3.37).
Both Eqs. (3.35) and (3.37) depend on:
68
•
N (directly and Eq. (3.37) also through xdem)
•
Ls (directly and Eq. (3.39) also through δ)
•
φy (needed to define θy, required to define μθpl,dem ).
The correct evaluation of VR requires to assume N from the seismic load combination, Ls from
Eq. (3.1) and φy computed theoretically, according to Eqs. (3.9) – (3.10). Hence, the
procedure may be long and not trivial. In fact, assuming N due to the seismic load
combination, if a dynamic (both linear and nonlinear) analysis is performed, the results will
be given in envelope form. The time required for performing the assessment will be,
therefore, doubled with respect to that required if N due to gravity loads is assumed.
Moreover, in order to compute xdem, M and N values corresponding to both Vmax and Vmin
should be considered. It could be possible to consider the maximum/minimum envelope
values of M and N acting together with the maximum/minimum envelope values of V, but this
procedure may yield unrealistic values of the actual member capacity. This problem may be
bypassed assuming N due to gravity loads. In this way, in fact, xdem will depend only on M,
and, as already explained in Sec. 2.2.3, M values corresponding to Vmax and Vmin can be
correctly found from equilibrium considerations, taking into account also the contribution of
the gravity loads, in the case of beams.
An aim of this work is, hence, to test the consistency of the results obtained using N due to
gravity loads with those found considering N from the seismic load combination, in order to
establish if it is possible to simplify the assessment of VR (Eqs. (3.35) – (3.37)).
As already stated in Sec. 3.2.2, the possibility of simplifying the assessment procedure
considering N due to gravity loads does not dependent only on the properties of the structural
members but also on the slenderness of the building, H/B, where H and B are the total
building height and the total building width in the direction of seismic loads, respectively.
Therefore, a wide range of values of building slenderness should be analyzed in order to draw
general conclusions about this topic.
Evaluating Ls according to its correct definition (Eq. (3.1)) may be complicated if a linear
dynamic analysis is carried out, since both M and V are furnished in envelope form and grow
indefinitely, proportionally to the external forces, instead of being limited to their
corresponding capacity values. The correct way to compute M and V in the case of a linear
dynamic analysis is shown in Sec. 3.2.1. This correct procedure is complex and long, since:
•
Ls is not an inner property of the member, but is function of the results of the analysis
itself (M and V) and, hence, needs to be recomputed every time the analysis is carried
out.
•
The evaluation of bending moment capacity is needed, in both positive and negative
flexure.
In this work, the results obtained from the correct definition of Ls (Eq. (3.1)) were compared
to those found by simply assuming Ls = L/2. The aim is to check if this simplified procedure
69
can be accepted and, hence, adopted as a possible alternative to the correct definition of
shear span length.
Evaluating φy according to the theoretical approach (Eqs. (3.9) – (3.10)) requires long and
laborious calculations. On the other hand, simple empirical formulas are available in
literature. In this work, the expressions proposed by Priestley [1993] (Eqs (3.11) – (3.12) –
(3.13)) and Biskinis [2006] (Eqs (3.14)) were considered. According to these formulas, the
value of φy is sensitive only to the section height, h, and the yielding strength of longitudinal
steel, εsyl. These empirical expressions were calibrated on members representative of new,
seismically designed, constructions and, therefore, their validity needs to be checked for
members typical of existing buildings, that could be characterized by non-seismic details and
degraded mechanical properties of the materials. An aim of this work is, hence, to check if it
is possible to assess the value of φy using the considered empirical expressions (Eqs. (3.11)
– (3.12) – (3.13)) – (3.14)) without any sensible loss of accuracy, also in the case of the
shear capacity assessment (Eqs. (3.35) – (3.37)) of structural members of existing buildings.
3.5 Brittle Mechanisms: Shear Demand
If a nonlinear analysis is carried out, the values of internal forces at each step will correctly
represent the actual distribution of the demand in structural members. For this reason, the
shear demand is assumed to be equal to the values obtained directly from the analysis, the
structure being modelled using the mean values of material properties, as defined in the data
acquisition process.
Concerning nonlinear dynamic analysis, denoting as Di and Ci the shear demand and capacity
of the i-th structural member, the value Di that governs the assessment of brittle modes is not
necessarily the maximum absolute value obtained from the analysis. Theoretically, in fact, the
ratio Ri,j = Di,j/Ci,j should be evaluated at each j-th time step of the analysis. Di,j corresponding
to the highest value of Ri,j will govern the assessment process. Although these considerations
are correct, the direct consequence of their application is that the procedure will be long and
expensive.
On the other hand, as already shown in Sec. 3.4, Ci depends on the value of axial load
demand, Ni and, according to Eurocode 8, also on the chord rotation ductility demand, μθi,dem
= θi,dem/θyi. Neglecting the variation of Ci due to the variation of Ni, if the considered member
undergoes plastic deformations, Ci will be minimum when μθi,dem is maximum. Since it is
likely that the maximum values of both μθi,dem and Di are in correspondence to the highest
peak of the global response, the maximum Ri can be reasonably assumed to be obtained in
correspondence of the maximum absolute value of Di. These considerations do not take into
account the influence of the variation of Ni on the assessment of Ci. Considering only Ni as
parameter (i.e. fixing the value of μθi,dem), Ci is minimum when Ni is maximum (minimum
compression or, if the considered section experiences tensile response, maximum tension). In
general, maximum Ni and maximum absolute values of Di are obtained at different time steps.
Therefore, in order to maximize Ri, also Di corresponding to the maximum Ni should be
considered.
70
As a consequence of these considerations, in this work, for each i-th member end, Ri is
evaluated in correspondence to:
•
maximum Di and corresponding Ci;
•
minimum Di and corresponding Ci;
•
minimum Ci, obtained in correspondence to the maximum tensile (or the minimum
compressive) Ni, and corresponding Di.
It is very probable that, following this procedure, the maximum absolute value of Ri will be
captured.
Concerning nonlinear static analysis, Eurocode 8 suggests to consider Di obtained in
correspondence to the “control node” displacement representative of the considered LS. The
Italian Seismic Code, instead, advises to consider Di obtained in correspondence to the
“control node” displacement representative of the considered LS only if the “control node”
displacement is smaller than the displacement obtained at the peak point of the capacity curve.
Otherwise Di should be those corresponding to the peak point of the capacity curve. The
reason of this difference between the two Codes dwells in the way in which Ci is evaluated.
According to the Italian Seismic Code, since Ci remain constant (fixed Ni) for all points on the
capacity curve while Di are largest at the peak point, the choice of not considering the
descending branch of the pushover curve for the evaluation of Di seems to be a suitable
solution, in order to get a safe side assessment. These considerations cannot be extended to
Eurocode 8, because the reduction of the magnitude of Di in the descending branch of the
capacity curve goes with the degradation of corresponding Ci. Therefore, the choice of
considering the point representative of the considered LS for the evaluation of Di is, in this
case, appropriate.
If a linear elastic analysis is performed, Di obtained directly from the analysis may
overestimate the response, since they increase linearly, while actually they should be limited
because of the development of nonlinear mechanisms. In fact, as already shown in Sec. 2.2
concerning the conditions of applicability of linear methods of analysis, the forming of plastic
hinges at one or both ends of the considered structural member locks the values of bending
moment demand to the corresponding capacities and, therefore, also Di are locked to values
that may be computed through equilibrium. It is important to remind that the assessment
based on a linear analysis will be allowed only if the conditions of applicability are satisfied.
These conditions assure that the actual distribution of internal forces does not differ
significantly from the linear one when plastic mechanisms develop. Limiting the bending
moments to their capacities is sufficient, therefore, to correctly assess Di through equilibrium
considerations.
According to both Codes, considering each i-th end section of each structural member, the
value of bending moment obtained from the analysis, MA,i, should be compared to the
corresponding capacity MR,i,mean, evaluated using the mean properties of the materials, as
determined through the data acquisition process. If MA,i < MR,i,mean, then MA,i will be taken as
the bending moment demand, Mi. If, instead, MA,i > MR,i,mean, the bending moment demand
71
will be evaluated as Mi = MR,i,CF, where MR,i,CF is the corresponding bending moment capacity
computed by means of the mean material properties multiplied by the attained CF. The reason
why MR,i,CF is considered instead of MR,i,mean is to penalize the assessment of brittle
mechanisms, since they dissipate a limited amount of energy and may lead to abrupt local
collapses or even to the failure of the structure.
It is interesting to note that, if MR,i,mean < MA,i < MR,i,CF, the assessed demand Mi will be larger
than the elastic demand MA,i, since Mi = MR,i,CF and MR,i,CF > MA,i. (Figure 3.9). The theoretical
concept that lies at the base of this procedure should be, instead, to limit the bending moment
values derived from the analysis to their capacities. In fact, once ensured that the distribution
of internal forces does not change substantially from an elastic to an inelastic response, MA,i
obtained from a linear elastic analysis should be an upper bound of the actual bending
moments Mi, which will be equal to MA,i, if the response is elastic, and smaller, if some plastic
mechanisms develop. It is, unlikely, to obtain a response in which the actual bending moment
demands Mi are larger that MA,i, computed through a linear elastic analysis, as may happen
following the procedure suggested by both Codes.
Figure 3.9. Equilibrium conditions for shear demand (Eurocode 8 and Italian Seismic Code)
It may be a better choice to consider the bending moment capacities computed by means of
the mean material properties amplified by the acquired CF since the beginning of the
procedure for the evaluation of shear demand. Following this assumption, the shear demand is
evaluated through equilibrium, by considering the bending moments at the member ends
equal to the minimum between the values obtained from the analysis and the corresponding
capacities computed using the mean material properties amplified by the acquired CF (Figure
3.10; Eq. (3.38)):
M i = min (M A,i ; M R ,i ,CF )
(3.38)
Following this approach, the maximum attainable bending moment demand is simply shifted
to a larger value. Hence, any possible jump of the value obtained from the linear elastic
analysis to a larger one will be avoided.
72
Figure 3.10. Equilibrium conditions proposed in this work to compute shear demand
Except for the values to assign to the mechanical properties of the materials involved in the
calculation of bending moment capacity, the procedure for evaluating the shear demand to use
in the assessment of brittle modes is the same as that already shown concerning the
verification of the second condition of applicability of linear methods of analysis (Sec. 2.2.2).
However, it seems opportune to remark the following aspects.
Since it is possible that at one member end MA,i > MR,i,CF while, at the opposite end, MA,i <
MR,i,CF, it seems a good solution, in terms of simplicity and velocity of the assessment
procedure, to derive the shear demands Di always through the equilibrium of the member,
considering at each member end MA,i, if MA,i < MR,i,CF, and MR,i,CF, if MA,i > MR,i,CF. If MA,i <
MR,i,CF at both member ends, Di computed through equilibrium will be equal to those obtained
from the analysis.
If a linear dynamic analysis is performed, the results will be obtained in envelope form.
Although the possible equilibrium conditions are four for each structural member, only two of
them should be taken into account, namely those that consider the maximum positive bending
moment at one end acting together with the maximum negative bending moment at the other
end and vice versa (bending moment considered positive in correspondence to tensile inferior
fibres).
Concerning the assessment of beams, Di are obtained from the equilibrium of both bending
moments applied at the two end sections of the member and gravity loads, which, in general,
cannot be neglected, especially for slender beams and low intensity of seismic action.
According to the Italian Seismic Code, the equilibrium conditions should be applied only on
the considered member. Eurocode 8, instead, suggests to consider also the influence of the
adjacent structural members through the moment equilibrium of the joint in which the
member ends frame. The maximum bending moment demand at the i-th end section, Mi, is,
therefore, the minimum among MA,i, MR,i,CF and the maximum bending moment that may be
delivered by means of equilibrium considerations on the joint into which the considered
member end frames, Mi,joint (Eq. (3.39)).
73
(
M i = min M A,i ; M R ,i ,CF ; M i , jo int
)
(3.39)
Since the development of plastic hinges at some member ends that frame into the considered
joint locks the bending moment values of the other member ends, it is possible that Mi,joint <
MR,i,CF and, therefore, that Mi < MR,i,CF. This procedure suggested by Eurocode 8, although is
more precise than that proposed by Italian Seismic Code, is much more expensive from a
computational point of view. In this work, thanks to the obtained results, it will be possible
to give a measure on the improvement that can be reached following Eurocode’s procedure
and to judge the convenience of its application.
74
Chapter 4. Analyzed Buildings
4 Analyzed Buildings
In this section a detailed description of the four buildings analyzed in this work is presented.
All considered buildings are real constructions, designed and build between 1962 and 1978,
according to different regulations, which nowadays may be judged as out-out-date. Therefore,
the seismic response of all considered cases need to be assessed, in order to check the need of
retrofitting. All these structures are public buildings, irregular in plan and elevation,
characterized by different material properties (concerning the mean strength of both concrete
and steel), plan shapes (both compact and not compact, symmetric and not symmetric),
resisting systems (mono-directional or bi-directional frames), interstorey heights, beam
lengths, sections shapes and dimensions and reinforcement content.
None of the considered buildings is a tall structure. In fact the overall heights H are between 8
m and 12 m. The lack of tall and slender structures could not allow to draw general
conclusions about the possibility of simplifying the assessment procedure considering N from
gravity loads instead of N from the seismic combination, since the results are strictly
correlated to the structural slenderness, defined as the ratio H/B (where B is the width of the
structure in the lateral loads direction). On the other hand, it should be considered that the vast
majority of R.C. frame structures built in Italy before 1980 in high seismic zones is
characterized by a small number of storeys, while tall and slender R.C. frame buildings are
rare. Therefore, considering buildings not taller than 12 m may be judged not to be a very
severe limitation.
Except for the possibility of simplifying the assessment procedure assuming N from gravity
loads, the four considered buildings allowed to draw general conclusions concerning all the
other aims explained in Chapters 2 and 3.
4.1 Sede Comunale - Vagli Sotto (Lucca, Tuscany)
The Sede Comunale, located in Vagli Sotto (Lucca, Tuscany) is a two-storey R.C. frame
structure with masonry infills, prefabricated floor slabs, a small basement floor that covers
about 25% of the plan view and sloping roofs made of inclined R.C. slabs (Figure 4.1 - Figure
4.2 - Figure 4.3 - Figure 4.4). The building was designed and constructed in 1965, according
to “R.D. n. 2229/39”.
The shape of the building is compact and approximately rectangular in plan, with an
expansion on the south-east side (Figure 4.1). The maximum dimensions in plan are 27.25 m
75
and 13.60 m. The clear interstorey heights are 2.05 m for the basement, 3.20 m for the ground
floor and 3.50 m per the first floor.
The R.C. frames are mono-directional and oriented parallel to the short sides of the building
(east-west direction). The frames are connected with secondary beams in correspondence with
the interstorey slabs.
At foundation level, a mat slab of dimensions 11 m by 3.7 m is located in the central part of
the building, in correspondence with the basement. This mat slab supports R.C. walls which
in turn support some of the columns of the first floor. In the remaining part of the building,
the R.C. columns are directly supported by footings.
The floor and roof slabs are made of unidirectional R.C. ribs with interposed brick blocks and
a 40 mm topping slab of cement conglomerate. The roof inclined slabs are supported by R.C.
columns of minor dimensions (Figure 4.5).
Figure 4.1. South-East view of the Sede Comunale
Figure 4.2. South-West view of the Sede Comunale
Figure 4.3. North-West view of the Sede Comunale
Figure 4.4. North-East view of the Sede Comunale
76
PLAN VIEW
GROUND FLOOR
FIRST FLOOR
220
BASEMENT
FRONT VIEW
+90
+380
EAST VIEW
SOUTH VIEW
15
WEST VIEW
NORTH VIEW
Figure 4.5. Plan and front views of the Sede Comunale
4.1.1 Geometry
The foundation of the structure is composed of footings, connected with R.C. beams 700 mm
deep and 800 mm wide in longitudinal direction, 700 mm deep and 500 mm wide in
transversal direction.
The dimensions of perimetral beams section at the floor level vary from 250 mm by 500 mm
at the supports to 50 mm by 500 mm at the middle of the member. The diameter of
longitudinal bars is 16 mm, while the transversal reinforcement is constituted by stirrups of 6
mm or 8 mm diameter at a spacing of 200 mm. Primary beams dimensions are 250 mm by
500 mm, with longitudinal reinforcement bars of 12 mm or 16 mm diameter and transversal
reinforcement constituted by stirrups of 6 mm diameter at a spacing of 200 mm. Other beams
of minor length are 160 mm by 400 mm, with longitudinal reinforcement bars of 16 mm
diameter and transversal reinforcement constituted by stirrups of 6mm diameter at a spacing
of 200 mm. All stirrup hooks are at 90°.
All R.C. columns are rectangular-shaped. Their dimensions are 250 mm by 400 mm. Their
longitudinal reinforcement consists in four bars of 16 mm diameter, while their transversal
reinforcement is constituted by stirrups of 6 mm diameter at a spacing varying from 150 mm
to 200 mm.
The sloped roof slabs are of composite construction: R.C. ribs 160 mm deep with interposed
brick blocks and a 40 mm topping slab of cement conglomerate. They are supported by beams
160 mm by 250 mm, with longitudinal reinforcement bars of 12 mm diameter, which in turn
are supported by columns 160 mm by 250 mm, with longitudinal reinforcement consisting in
four bars of 12 mm diameter.
77
4.1.2 Materials
Information on material properties were acquired from two main sources, namely, the design
report and in-situ material quality tests. Mean value of the compressive strength of concrete,
fcm, has been based on the results of the in-situ, destructive and non-destructive investigations
to test the quality of concrete carried out by the Regione Toscana, “Programma delle attività
d’indagini su edifice pubblici in cemento armato zona sismica”, in 2002 and 2003. The results
of destructive and non-destructive tests showed a population of very dispersive values,
between 4.00 MPa and 12.61 MPa. fcm = 8.3 MPa has been considered for the structural
assessment.
Laboratory tests have not been performed to determine the quality of longitudinal and
transversal reinforcement steel. fylm = 440 MPa (Feb44k) and fywm = 440 MPa (Feb44k) were
adopted as mean yield strength of longitudinal and transversal reinforcement steel,
respectively, based on data from the structural drawings and the design calculation report.
The geometry of the building is available from original design drawings verified by recent site
survey. The structural details are available from original blue prints, verified by a recent
limited site survey of the main structural members. Nominal values of material properties
have been obtained from the original design and limited in-situ destructive and nondestructive testing. The lack of information about the present condition of the reinforcement
steel should force to place the building in KL1. On the other hand, it should be considered that
the variability of steel properties is generally lesser than concrete. Hence, considering also the
quality and adequacy of the remaining information available, the building has been
categorized in KL2 (Adequate Knowledge Level). Therefore, all methods of analysis are
permitted.
4.1.4 Seismic Input
According to the Italian Seismic Code, the elastic response spectrum for the Ultimate Limit
State (ULS) was assumed as the response spectrum for the SD LS. The response spectrum for
the DL LS was obtained by reducing the response spectrum corresponding to the SD LS by a
factor of 2.5. The response spectrum for the NC LS was obtained by scaling the response
spectrum corresponding to the SD LS by a factor of 1.5.
According to the Seismic Classification of Italian Municipalities, “OPCM 3271, Attached 1”,
the building was categorized as belonging to “Seismic Zone 2”. Therefore, the peak horizontal
ground acceleration (ag) on rock (“Soil Type A”) at the SD LS was assumed equal to 0.25 g.
Since a value of Vs,30 (average shear waves velocity in the top 30 m) between 603 m/s
(seismic refraction test) and 627 m/s (seismic down-hole test) was determined, the soil was
classified as “Soil Type B” (S = 1.25; TB = 0.15; TC = 0.50; TD = 2.0).
According to both considered Codes, the Importance Factor (γI) of the structure was assumed
equal to 1.4 (buildings of primary importance for civil protection during earthquakes).
78
Both horizontal acceleration response spectra and horizontal displacement response spectra,
corresponding to “Soil Type B”, γI = 1.2, for all considered LS are shown in Figure 4.6 Figure 4.7.
0.45
1.6
Horiz. Spectral Displacement [m]
H oriz. Spectral A cceleration [g]
1.8
LD LS
SD LS
NC LS
1.4
1.2
1
0.8
0.6
0.4
0.2
0
LD LS
0.4
SD LS
0.35
NC LS
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
Structural Period [s]
3
3.5
4
Figure 4.6. Elastic horizontal acceleration spectra
for the Sede Comunale
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure 4.7. Elastic horizontal displacement spectra
for the Sede Comunale
4.1.5 Loads
The values of dead and live loads were obtained directly from the original design calculation
documents of the building.
The load due to snow was computed in accordance with the regulation “Decreto Ministeriale
16/01/1996 – Carichi e sovraccarichi”. Since the building is located in Toscana (“Snow zone
2”), at 600 m above the see level, q s = 1.75kN / m 2 was determined as the snow load.
4.2 Scuola Elementare “Pascoli” - Barga (Lucca, Tuscany)
The Scuola Elementare “Pascoli”, located in Barga (Lucca, Tuscany) is a two-storey R.C.
frame structure with masonry infills, prefabricated floor slabs, a small basement floor that
covers about 25% of the plan view and sloping roofs made of inclined R.C. slabs (Figure 4.8 Figure 4.9). The building was designed and constructed in 1978, according to “Legge n° 1086
5/11/1971, Legge n° 64 2/2/1974 and D.M. 3/3/1975”.
The shape of the building is approximately square and symmetric with respect to the
longitudinal axis. The maximum dimensions in plan are 41.3 m and 35.6 m. The structure
rises to a height of 8 m above the ground level and the foundation extends to a depth of 1.2 m
below the ground level. The clear height of the ground and first floor is 3.4 m, while the clear
height of the basement room is 2.6 m. The basement room is constituted by R.C. columns and
R.C. walls.
The building has two internal staircases located symmetrically on either sides of a common
landing area facing the main entrance on the ground floor. The rear central portion of the
building is single-storey. The common landing area of the stairs on the first floor opens onto a
terrace (Figure 4.10).
79
Figure 4.8. Main entrance view of the Scuola
Pascoli
Figure 4.9. Rear view of the Scuola Pascoli
Figure 4.10. Floors plan view of the Scuola Pascoli
4.2.1 Geometry
The foundation of the structure is composed of continuous, inverted R.C. T-beams 1200 mm
deep in both principal directions. The widths of the web and flange of the T-beam are 400 mm
and 1000 mm, respectively. In few locations these T-beams are interconnected by smaller
beams of square cross-section (400 mm). Longitudinal reinforcement bars of diameters 8 mm,
14 mm, 16 mm and 20 mm and stirrups of diameter 8 mm at a spacing of 200 mm or 300 mm
have been used in the foundation beams.
The structure is composed of bi-directional R.C. frames. Most of the beams in the structure
are 800 mm deep and 400 mm wide, 800 mm deep and 300 mm wide or 600 mm deep and
400 mm wide. Longitudinal reinforcement bars of diameters 12 mm, 14 mm, 16 mm and 20
mm have been used in the beams of the superstructure. The stirrups are of diameter 10 mm at
spacing of 100 mm, or of diameter 8 mm at a spacing of 200 mm.
The R.C. columns are square-shaped of dimension 400 mm. The columns have 4, 6 or 8
longitudinal reinforcement bars of diameters 16 mm or 20 mm. Since no information on the
transverse reinforcement was available, stirrups of diameter 8 mm at a spacing of 300 mm
were assumed for the columns.
80
The floor and roof slabs are of composite construction: prefabricated R.C. ribs with interposed
brick blocks and a 40 mm topping slab of cement conglomerate. The composite height of the
slab is 200 mm + 40 mm.
4.2.2 Materials
Information on material properties were acquired from two main sources, namely, the design
report and in-situ material quality tests. Mean value of the compressive strength of concrete,
fcm, has been based on the results of the in-situ, destructive and non-destructive investigations
to test the quality of concrete carried out by the Regione Toscana, “Programma delle attività
d’indagini su edifice pubblici in cemento armato zona sismica”, in 2002 and 2003. The results
of destructive and non-destructive tests showed good quality of concrete in the structure. fcm =
30 MPa has been considered for the structural assessment.
Laboratory tests have not been performed to determine the quality of longitudinal and
transversal reinforcement steel. fylm = 440 MPa (Feb44k) and fywm = 440 MPa (Feb44k) were
adopted as mean yield strength of longitudinal and transversal reinforcement steel,
respectively, based on data from the structural drawings and the design calculation report.
survey. The structural details are available from original blue prints verified by a recent
limited site survey of the main structural members. The structural details available from the
drawings are not complete. Nominal values of material properties have been obtained from
the original design and limited in-situ destructive and non-destructive testing. The lack of
information about the present condition of the reinforcement steel should force to place the
building in KL1. On the other hand, it should be considered that the variability of steel
properties is generally lesser than concrete. Hence, considering also the quality and adequacy
of the remaining information available, the building has been categorized in KL2 (Adequate
Knowledge Level). Therefore, all methods of analysis are permitted.
4.2.4 Seismic Input
Since a value of Vs,30 (average shear waves velocity in the top 30 m) equal to 754 m/s was
determined through the seismic refraction test, the soil was classified as “Soil Type B” (S =
1.25; TB = 0.15; TC = 0.50; TD = 2.0).
81
equal to 1.2 (the building houses a school).
0.4
1.6
Displacement Elastic Spectrum [m]
Horizontal Spectral A cceleration [g]
corresponding to “Soil Type B”, γI = 1.2, for all considered LS are shown in Figure 4.11 Figure 4.12.
LD LS
1.4
SD LS
1.2
NC LS
1
0.8
0.6
0.4
0.2
0
SL-DL
SL-DS
SL-CO
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
for the Scuola Pascoli
0
0.5
1
1.5
2
2.5
3
3.5
4
for the Scuola Pascoli
4.2.5 Loads
Dead and live loads were determined from the original design calculations of the building.
The volumetric weight of concrete was assumed as 25 kN/m3. The slab weights used for the
first floor were 4.2 kN/m2, 4.1 kN/m2 3.9 kN/m2 or 3.8 kN/m2, depending on the location. The
slab weights used for the roof were 4.7 kN/m2, 4.2 kN/m2 or 4.1 kN/m2. The dead loads of
external and internal walls were 8.54 kN/m and 5.59 kN/m, respectively. The dead and live
loads of the rain water pipes and parapets were 5.15 kN/m and 2.45 kN/m, respectively. The
live loads used for floor slabs of the building were 3 kN/m2 and 5 kN/m2, whereas 2.68 kN/m2
was the live load used for the terrace.
4.3 Scuola Media Inferiore “Puccetti” - Gallicano (Lucca, Tuscany)
The Scuola Media Inferiore “Puccetti” located in Gallicano (Lucca, Tuscany) is a two-storey
R.C. frame structure with masonry infills, prefabricated floor slabs, a small basement floor
and sloping roofs of varying height (Figure 4.13 - Figure 4.14 - Figure 4.15 -Figure 4.16). The
structure is roofed by inclined R.C. slabs supported by short columns rising above the second
storey. The building has a C-shaped plan form with re-entrant corners; hence, the structural
configuration is not compact. The building is also not symmetrical with respect to both
principal directions, as one arm of the ‘C’ is longer than the other (Figure 4.15).
The original structure was constructed between 1962 and 1963, in accordance with the
regulations provided for the seismic zone category II, “R.D. n.2229/39”.
Originally the main entrance to the structure led to a double-height space supported by a
massive portal frame on one end. In 1980, this space was altered to accommodate a new
82
classroom on the first floor, thereby transforming the double-height space into two vertically
partitioned spaces. The R.C. frame of the new classroom was constructed within the original
structure of the school but isolated right from the foundation level. A considerably large
portion of the roof was also repaired in 1980. Both the addition and the repair works were
carried out in accordance with the regulation: “Norme tecniche di edilizia–Legge 2/2/1974
n.64” (Figure 4.17 - Figure 4.18).
Figure 4.13. Main entrance of the Scuola Puccetti
Figure 4.14. Rear view of the Scuola Puccetti
Figure 4.15. Lateral view of the Scuola Puccetti
Figure 4.16. Lateral view of the Scuola Puccetti
Figure 4.17. Plan views of the Scuola Puccetti
83
Figure 4.18. Front views of the Scuola Puccetti
4.3.1 Geometry
The foundation of the original structure is composed of continuous inverted mono-directional
beams 980 mm deep. The alignment of these primary foundation beams varies with their
location and they are connected by smaller square beams of depth 380 mm in the
perpendicular direction.
The structure is composed of mono-directional frames with few of these interconnected by
beams. Most of the beams in the structure are 360 mm wide and 460 mm deep and few are
360 mm wide and 540 mm deep. The beam of the entrance portal is 1150 mm deep.
Longitudinal reinforcement bars of diameters 14 mm, 16 mm and 20 mm have been used in
the beams of the superstructure and 8 mm bars have been used along with 16 mm bars in the
foundation beams. Transverse reinforcement is of 6 mm diameter stirrups at a spacing of
either 200 mm or 250 mm. Stirrups in the foundation beams are of 8 mm diameter with 275
mm spacing.
The columns are square, with side length of 380 mm. Perimeter columns have eight
longitudinal reinforcement bars of 16 mm diameter, whereas interior columns have six bars of
16 mm diameter. Transverse reinforcement bars are of 8 mm diameter at 160 mm spacing.
Columns of the entrance portal are 400 mm by 600 mm and increase to 400 mm by 900 mm
at the beam column joint with four longitudinal bars of 20 mm diameter and 6 mm diameter
transverse reinforcement bars at spacing of 200 mm. The stirrup hooks are at 90°.
The floor and roof slabs are of composite construction: prefabricated R.C. ribs with interposed
brick blocks and a 40 mm topping slab of cement conglomerate. Two types of slabs have been
used in this building: “monotrave” and “bitrave” with composite heights of 29 cm (25+4) and
24 cm (20+4), respectively.
4.3.2 Materials
In-situ destructive and non-destructive investigations were carried out in 2002 by the Regione
Toscana, “Programma delle attività d’indagini su edifici pubblici in cemento armato, zona
sismica”, to test the quality of concrete. The results of the non-destructive tests on concrete
showed the heterogeneity of the tested material and consequently variability of resistance. The
84
results of the destructive tests showed that the compressive strength of concrete, fcm, was
between 15.2 MPa and 20 MPa. A mean value of 18 MPa has been considered for the
structural assessment.
Tests have not been performed to determine the quality of reinforcement steel in the structure.
The mean value of yield strength of both longitudinal (fylm) and transversal (fywm)
reinforcement steel has been adopted as 440 MPa (Feb44k).
the original design and limited in-situ destructive and non-destructive testing. The quality of
concrete was determined testing about 14 cores from ground and first floor columns
(approximately 25% of the total number of columns). The present condition of the
reinforcement steel is unknown. This should force to place the building in KL1. On the other
hand, it should be considered that the variability of steel properties is generally lesser than
concrete. Hence, considering the quality and adequacy of the remaining information available,
the building has been categorized in KL2 (Adequate Knowledge Level). Therefore, all
methods of analysis are permitted.
4.3.4 Seismic Input
The following values of Vs,30 (average shear waves velocity in the top 30 m) were determined
through the seismic refraction test, corresponding to three different locations: 613 m/s, 573
m/s and 747 m/s. Since all values of Vs,30 are between 360 m/s and 800 m/s, the soil was
classified as “Soil Type B” (S = 1.25; TB = 0.15; TC = 0.50; TD = 2.0).
Both horizontal acceleration response spectra and horizontal displacement response spectra
are equal to those already shown in Figure 4.11 and Figure 4.12 for the Scuola Elementare
“Pascoli”, located in Barga (“Soil Type B”, γI = 1.2, all considered LS).
85
4.3.5 Loads
Due to the absence of original design calculation documents of the building, standard values
have been used to calculate the dead loads and live loads. The volumetric weight of concrete
has been assumed as 25 kN/m3. Weights of the external and internal walls have been assumed
as 3 kN/m2 and 2 kN/m2, respectively. The live load has been assumed as 3 kN/m2. The slab
weights have been assumed as 5.58 kN/m2 (“bitrave” slab) and 3.82 kN/m2 (“monotrave”
slab). The live load on the roof (inaccessible roof) has been assumed to be 0.5 kN/m2.
4.4 Scuola Media Inferiore Don Bosco - Rapagnano (Ascoli Piceno, Marche)
The Scuola Media Inferiore “Don Bosco”, located in Rapagnano (Ascoli Piceno, Marche), is a
three-storey R.C. frame structure, with masonry infills and a flat roof. The building was
designed in 1958 and constructed between 1958 and 1962, according to “R.D. n. 2229/39”.
The shape of the building is compact and approximately rectangular in plan, not symmetric
with respect to the two principal directions. The maximum dimensions in plan are 23.84 m
along X-direction and 14.44 m along Y-direction. The clear interstorey heights are 3.20 m for
both ground and first floors, 3.45 m for the second floor (Figure 4.19 - Figure 4.20 - Figure
4.21 - Figure 4.22).
The R.C. frames are mono-directional and oriented parallel to the short sides of the building
(east-west direction). The frames are connected with secondary beams in correspondence with
the interstorey slabs. The building is supported by deep, indirect foundations, made of piles,
connected by foundation beams, which support directly the columns of the superstructure.
The floor and roof slabs are made of unidirectional R.C. ribs with interposed brick blocks and
a topping slab of cement conglomerate.
In 1981 a gym was designed and built in 1983. This is a double-height space, located in the
South side of the building, at the first floor (Figure 4.19). The works to build the gym were
carried out in accordance with the regulation “Norme tecniche di edilizia–Legge 2/2/1974
n.64”.
Figure 4.19. South view of the Scuola Don Bosco
Figure 4.20. Frontal view of the Scuola Don Bosco
86
Figure 4.21. East view of the Scuola Don Bosco
Figure 4.22. North view of the Scuola Don Bosco
The plan views and the directions of the slab are shown in Figure 4.23, while the front views
are illustrated in Figure 4.24.
Figure 4.23. Plan views of the Scuola Don Bosco
Figure 4.24. Front views of the Scuola Don Bosco
87
4.4.1 Geometry
The structure is composed of mono-directional frames connected with secondary beams in
correspondence with the interstorey slabs. Most of the beams in the structure are 300 mm
wide and 400 mm deep, 240 mm wide and 300 mm deep, or 130 mm wide and 650 mm deep.
Sections of many other different dimensions are present, although less common than the three
sections above mentioned. For the structural assessment, part of the beams is considered as
rectangular, while others are treated as T-beam (a portion of the topping slab of cement
conglomerate is considered to collaborate with the beam to resist the bending moment).
Longitudinal reinforcement bars of diameters 6 mm, 8 mm, 10 mm, 12 mm, 14 mm, 16 mm
and 18 mm have been used in the beams, while their transverse reinforcement is made of
stirrups of 6 mm or 8 mm diameter at a spacing of 100 mm or 120 mm or 150 mm. The
stirrup hooks are at 90°.
Some of the columns are square, with side length equal to 300 mm or 400 mm. Other columns
are rectangular-shaped, with dimensions 300 mm by 350 mm, 300 mm by 400 mm or 400 mm
by 500 mm. Their longitudinal reinforcement consists in four bars of 16 mm diameter
(columns 300x300 mm and 300x350 mm), six bars of 16 mm diameter (columns 400x500
mm), or four bars of 18 mm diameter (columns 300x350 mm and 300x400 mm). Their
transversal reinforcement is made of stirrups of 6 mm diameter at spacing of 150 mm
(columns 300x300 mm, 300x350 mm and 300x400 mm) or of 200 mm (columns 400x400
mm and 400x500 mm). The stirrup hooks are at 90°.
The floor and roof slabs are of composite construction: prefabricated R.C. ribs, with
interposed brick blocks and a 30 mm topping slab of cement conglomerate. The type of slab is
“2000”, with composite heights of 28.5 cm (25.5+3) for the floors and 35 cm (32+3), for the
roof. Figure 4.23 shows the direction of the floor and roof slabs. At the second floor there is a
R.C. balcony, with thickness varying from 6.5 cm at the free end to 17.5 cm at the fixed end.
4.4.2 Materials
Since a limited number of in-situ investigations were performed, the mean values of
compressive strength of concrete, fcm, yield strength of longitudinal reinforcement steel, fylm,
and yield strength of transversal reinforcement steel, fywm, were determined from the original
structural drawings. They are, respectively: fcm = 16.6 MPa, fylm = 215 MPa (Feb22k) and fywm
= 215 MPa (Feb22k).
the original design and limited in-situ testing. The lack of information about the present
condition of the reinforcement steel and the limited information about the present condition of
the concrete should force to place the building in KL1. Nevertheless, considering also the
quality and adequacy of the remaining information available, the building has been
88
categorized in KL2 (Adequate Knowledge Level), in order to permit the use of both linear and
nonlinear methods of analysis.
4.4.4 Seismic Input
Since the values of Vs,30 (average shear waves velocity in the top 30 m) determined for the
foundation soil are between 180 m/s and 360 m/s, the soil was classified as “Soil Type C” (S
= 1.25; TB = 0.15; TC = 0.50; TD = 2.0).
0.4
1.6
Displacement Elastic Spectrum [m]
Horizontal Spectral A cceleration [g]
corresponding to “Soil Type C”, γI = 1.2, for all considered LS are shown in Figure 4.25 Figure 4.26.
LD LS
1.4
SD LS
1.2
NC LS
1
0.8
0.6
0.4
0.2
0
SL-DL
SL-DS
SL-CO
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
3.5
4
for the Scuola Don Bosco
0
0.5
1
1.5
2
2.5
3
3.5
4
for the Scuola Don Bosco
4.4.5 Loads
The values of dead and live loads were obtained directly from the original design calculation
documents of the building. The load due to snow was computed in accordance with the
regulation “Decreto Ministeriale 16/01/1996 – Carichi e sovraccarichi”. Since the building is
located in Marche (“Snow zone 1”), at 314 m above the see level, and since the roof is flat,
q s = 1.55kN / m 2 was determined as the snow load.
89
In the following Chapters 5, 6, 7 and 8, for each building, the results obtained for each
assessment procedure will be shown through tables and bar charts. On the base of the detailed
description of the results, indications will be given about which rules should be followed in
order to obtain a fast but also reliable assessment procedure. In Chapter 9, all results will be,
then, compared, to see which considerations are common to all the considered buildings, in
order to propose general rules for the assessment of both ductile and brittle mechanisms of the
structural members.
90
Chapter 5. Assessment of Sede Comunale (Vagli Sotto)
5 Assessment of the Sede Comunale (Vagli Sotto)
The assessment of the seismic behaviour of the Sede Comunale located in Vagli Sotto has
been performed according to all methods of analysis proposed by the two considered Codes
(except for the linear static analysis, since the structure is regular neither in plan nor in
elevation). All analyses were based on 3-D models, as the lack of regularity did not allow to
consider two planar separated models in the two principal directions.
The linear analyses (eigenvalue and dynamic linear analyses, performed assumed all the ways
to determine the members stiffness EI shown in Sec. 2.4) were modelled through the software
“SAP2000 Advanced I 10.0.7”. The nonlinear analyses (eight pushover and eight time history
analyses) were performed through the software “SeismoStruct, Version 4.0.3, Built 30”, a
nonlinear fibre model software which considers both material nonlinearity (uni-axial
nonlinear constitutive models are available to describe concrete and steel cyclic behaviour)
and geometrical nonlinearity (second order effects).
5.1 Linear Analyses
The following assumptions characterize the SAP2000 model.
Ec = 17174 MPa, computed through Eq. (3.7), where fcm = 8.3 MPa.
Beams and columns were modelled using 3-D beam elements with six degrees of freedom.
The beam-column joints were not modelled as rigid. In fact, considering the beam-column
joints as rigid seemed to be unconservative, as it would lead to neglect the shear deformation
of the joint.
The R.C. walls at the basement level were modelled with shell finite elements, characterized
by EI = EIgross = 17174 Igross Nmm2 (uncracked sections) and thickness = 250 mm.
The floor and roof slabs were incorporated in the model using shell finite elements, as the
Codes do not allow to model a-priori the in-plan flexibility of the slabs as infinite. Since the
slabs consist of R.C. ribs with interposed brick blocks and a topping slab of cement
conglomerate, an equivalent homogeneous section was estimated. The equivalent thickness of
the slab was calculated as t = (Vc + mVs ) A , where m is the moduli ratio, m = G s Gc ; Gs and
91
Gc are the shear moduli of steel and concrete, respectively; Vc is the volume of concrete in the
slab and Vs is the volume of steel in the slab.
Assuming Ec = 17174 MPa, Es = 200000 MPa, νc (Poisson modulus of concrete) = 0.2 and νs
(Poisson modulus of steel) = 0.3:
Gc = E c 2(1 + ν ) = 7156MPa
G s = E s 2(1 + ν ) = 76923MPa
m = G s Gc = 10.75
For a slab characterized by L = 4250 mm (length in the direction of the seismic action) and L’
= 4400 mm (length in the direction perpendicular to the seismic action):
Vc = 41
.25
⋅ 42
.44
⋅ 04
.3
04 + 10(0.08 ⋅ 4.25 ⋅ 0.12) = 1.156m 3
44
14442444
3
topping
[
(
ribs
)
(
) ]
Vs = 10 ⋅ 2 ⋅ 10 2 π 4 ⋅ 4.25 + 10 ⋅ 1.25 ⋅ 10 2 π 4 ⋅ 2 ⋅ 10 −6 = 8.64 ⋅ 10 −3 m 3 .
Finally, the thickness of the equivalent homogeneous section of the slab was obtained as:
t = (Vc + m ⋅ Vs ) A = (1.156 + 10.75 ⋅ 0.00864) (4.25 ⋅ 4.4) = 67mm
This calculation was repeated for all slabs of the building.
The masses were distributed along the structural members (both beams and columns).
In order to model the building with members stiffness equal to 100% EIgross, EI = EIgross =
17174 Igross Nmm2 was assumed.
In order to model the building with members stiffness equal to 50% EIgross, EI = 0.5 EIgross =
In order to model the building with members stiffness equal to the mean actual secant
stiffness at yielding (effective stiffness), EI eff , computed through Eq. (2.8), it was assumed
that:
N
∑
EI =
EI eff ,i
i =1 EI gross ,i
N
EI gross
⎧i = All _ members ⇒ 36.6% EI
gross = 6284 ⋅ I gross
⎪
⎪i = All _ beams ⇒ 31.8% EI gross = 5460 ⋅ I gross
⎪i = Re c _ beams ⇒ 36.1% EI gross = 6207 ⋅ I gross
⎪i = T − beams ⇒ 26.8% EI gross = 4601 ⋅ I gross
⎪
= ⎨i = All _ columns ⇒ 43.5% EI gross = 7469 ⋅ I gross
⎪i = Ground _ floor _ columns ⇒ 55.3% EI gross = 9494 ⋅ I gross
⎪i = 1st floor _ columns ⇒ 49.6% EI
⎪
nd
⎪i = 2 floor _ columns ⇒ 37.9% EI gross = 6511 ⋅ I gross
⎪⎩i = Roof _ floor _ columns ⇒ 36.0% EI gross = 6191 ⋅ I gross
92
Figure 5.1 and Figure 5.2 show the model of the Sede Comunale used for linear analyses.
Figure 5.1. Front view of Sede Comunale
Figure 5.2. Rear view of Sede Comunale
In order to establish if the effect of the soil–structure interaction needs to be taken into
consideration, two eigenvalue analyses were performed, one taking into account the soilstructure dynamic interaction and the other one considering fixed foundations. Both analysis
were performed assuming EI = 50% EIgross.
The procedure suggested by Gazetas [1991] was used to define the elements of the stiffness
matrix that models the soil-structure dynamic interaction. This method furnishes elements of
the stiffness matrix characterized by complex values, which depend on the frequency content
of the seismic input. The stiffness matrix of the soil-structure interaction was initially
computed for the static case (null frequency). Then it was computed also for the dynamic
case, considering the natural frequency of the first mode of the structure as the input
frequency. Since the frequency content of the seismic input is function of the stiffness matrix,
in general an iterative process is required to reach the convergence. For the considered
structure, instead, the convergence was reached at the first iteration. It means that the dynamic
effect has not any influence on the soil-structure interaction.
Table 5.1 shows the comparison between the values of the natural periods of the first three
modes of interest (modal mass > 5%) obtained considering both soil-structure interaction and
fixed foundations. The values of natural periods obtained from the two different models are
very similar (differences less than 2%). It means that the flexibility of the system soilfoundation is negligible with respect to the flexibility of the superstructure. Therefore, in
order to simplify the model, fixed foundations were assumed for the Sede Comunale.
Table 5.1. Comparison between the first three modes of interest obtained considering both the soilstructure interaction and fixed foundations
Periods [s]
I Mode
II Mode
III Mode
Soil-structure interaction
1.075
0.784
0.614
Fixed foundations
1.066
0.774
0.603
93
5.1.2 Eigenvalue Analyses
Table 5.2 illustrates the numerical results of the eigenvalue analyses performed according to
all the ways to determine the stiffness EI of the structural member described in Sec. 2.4. The
tables show all the modes of interest (modal mass > 5%) needed to get a cumulative modal
mass at least equal to 90% for both translations in the two horizontal principal directions and
rotation around the vertical axis.
Table 5.2. All modes of interest, different EI
Mode
#
Period
[s]
1
3
4
6
8
44
51
126
0.774
0.559
0.431
0.280
0.175
0.065
0.061
0.033
Mode
#
Period
[s]
2
3
4
6
8
56
62
82
165
1.066
0.774
0.603
0.393
0.245
0.075
0.072
0.063
0.033
% Modal
Mass
Transl. in
X
0.60
0.00
0.00
0.08
0.00
0.02
0.12
0.00
% Modal
Mass
Transl. in
X
0.60
0.00
0.00
0.07
0.00
0.11
0.05
0.07
0.00
EI = 100% EIgross
%
% Modal % Modal
Cumulative
Mass
Mass
Transl. in Rotation Modal Mass
around Z Transl. in X
Y
0.00
0.08
0.60
0.11
0.03
0.60
0.60
0.49
0.47
0.00
0.01
0.68
0.11
0.09
0.68
0.05
0.11
0.78
0.02
0.00
0.95
0.00
0.00
0.98
EI = 50% EIgross
% Modal % Modal
%
Mass
Mass
Cumulative
Transl. in Rotation Modal Mass
Y
around Z Transl. in X
0.00
0.08
0.60
0.11
0.02
0.60
0.60
0.48
0.46
0.00
0.01
0.67
0.10
0.09
0.67
0.02
0.10
0.80
0.07
0.07
0.86
0.00
0.01
0.94
0.02
0.01
0.97
% Cum.
%
Cumulative Modal Mass
Rotation
Modal Mass
around Z
Transl. in Y
0.00
0.08
0.11
0.10
0.60
0.58
0.60
0.59
0.71
0.69
0.82
0.85
0.85
0.87
0.90
0.91
%
% Cum.
Cumulative Modal Mass
Modal Mass
Rotation
Transl. in Y
around Z
0.00
0.08
0.11
0.10
0.59
0.57
0.59
0.58
0.69
0.68
0.76
0.78
0.84
0.85
0.84
0.86
0.91
0.92
EI = mean EIeff (all members)
Mode
#
Period
[s]
2
3
4
6
8
62
75
93
181
1.230
0.896
0.701
0.457
0.285
0.083
0.076
0.067
0.034
% Modal
Mass
Transl. in
X
0.60
0.00
0.00
0.07
0.00
0.10
0.00
0.12
0.00
% Modal
Mass
Transl. in
Y
0.00
0.11
0.48
0.00
0.10
0.00
0.08
0.00
0.01
% Modal
Mass
Rotation
around Z
0.08
0.02
0.46
0.01
0.09
0.02
0.15
0.01
0.01
% Cum.
%
%
Cumulative Cumulative Modal Mass
Rotation
Modal Mass Modal Mass
around Z
Transl. in X Transl. in Y
0.60
0.00
0.08
0.60
0.11
0.10
0.60
0.58
0.56
0.67
0.58
0.57
0.67
0.69
0.67
0.79
0.75
0.69
0.79
0.83
0.84
0.93
0.84
0.86
0.97
0.90
0.91
94
EI = mean EIeff (beams - columns)
Mode
#
2
3
4
6
8
67
72
94
189
Mode
#
2
3
4
6
9
73
75
191
% Cum.
%
%
% Modal % Modal % Modal
Mass
Mass
Mass
Rotation
Transl. in Transl. in Rotation Modal Mass Modal Mass
around
Z
around Z Transl. in X Transl. in Y
Y
X
0.00
0.08
0.60
0.00
0.08
1.175
0.60
0.865
0.00
0.11
0.02
0.60
0.11
0.10
0.00
0.60
0.59
0.57
0.680
0.48
0.46
0.430
0.08
0.00
0.01
0.67
0.59
0.58
0.269
0.00
0.10
0.09
0.67
0.69
0.67
0.079
0.12
0.00
0.05
0.81
0.75
0.73
0.075
0.01
0.08
0.11
0.82
0.83
0.84
0.066
0.06
0.00
0.01
0.92
0.83
0.86
0.035
0.00
0.02
0.01
0.97
0.91
0.91
EI = mean EIeff (rectangular and T beams - columns per floor)
% Cum.
%
%
Period
Mass
Mass
Mass
[s]
Rotation
Transl. in Transl. in Rotation Modal Mass Modal Mass
around Z
around Z Transl. in X Transl. in Y
Y
X
0.00
0.07
0.58
0.00
0.07
1.188
0.60
0.875
0.00
0.11
0.02
0.58
0.11
0.10
0.00
0.58
0.57
0.54
0.704
0.46
0.45
0.441
0.10
0.00
0.01
0.67
0.57
0.56
0.276
0.00
0.11
0.10
0.68
0.69
0.67
0.076
0.10
0.02
0.12
0.78
0.77
0.80
0.074
0.09
0.06
0.04
0.87
0.83
0.84
0.035
0.00
0.01
0.01
0.97
0.90
0.91
Period
[s]
The number of modes required to get at least 90% of the modal mass is large and varies from
126 for EI = 100% EIgross to 191 for EI = EIeff (computed separately for rectangular beams, Tbeams and columns of each floor). These huge numbers of required modes may be justified
considering that the slabs are quite flexible, since both value of Ec and thickness of the
homogeneous equivalent section of the slabs (67 mm) are small. Moreover, the presence of an
inclined roof increases the number of local modes, making the eigenvalue problem more
complex.
Two modes govern the dynamic problem: a pure translational mode in X that involves 60% of
the modal mass and a mode characterized by both translation in Y (modal mass = 46-49%)
and rotation around Z (modal mass = 45-47%). Therefore, although the structural
configuration is compact and roughly rectangular, the first modes in X and Y are not
characterized by a very large amount of modal mass. The reason is the lack in regularity of
the structural frames, regarding both geometry and mechanical properties. Moreover, while
the first mode in X is a purely translational mode, the first mode in Y and the first torsional
mode merge into a single mode. Although it is not a problem for the dynamic analyses, it
becomes a handicap for the assessment according to the static nonlinear analysis, since
pushover analyses were developed for 2-D systems that experience only translation toward
the pushing direction (see Sec. 2.5).
95
Changing the stiffness of the structural members EI, the natural periods change too, but the
increase of natural periods is not linear with the decrease of EI. In fact, considering the first
mode:
T1, EI gross
T1,0.5 EI gross
T1, EI gross
T1, EI eff
=
=
0.5EI gross
0.774
= 0.5
= 0.726 , while
EI gross
1.066
EI eff
0.774
= 0.366 .
= 0.629 − 0.658 , while
EI gross
1.175 − 1.230
Figure 5.3 and Figure 5.4 show the first mode in X (pure translation) and the first mode in Y
and Z (both translation and torsion), respectively.
Figure 5.3. 1st mode of interest of Sede Comunale
Figure 5.4. 2nd mode of interest of Sede Comunale
The dynamic linear analysis consists in a multi-modal response spectrum analysis, performed
on a linear elastic model. As already stated in Sec. 1.4, the analyses were carried out applying
elastic response spectra, while inelastic response spectra were not considered, since the
determination of the “q-factor” is subjective and difficult, in particular for complex structures.
Since, for the Sede Comunale, accounting for vertical seismic input is not required, only the
horizontal elastic 5%-damped pseudo-acceleration response spectrum defined in Sec. 4.1.4
was applied in both horizontal principal directions. The modal superposition was performed
applying the CQC rule, considering enough modes to obtain a cumulative modal mass at least
equal to 90% of the total mass for both translation toward the two principal directions and
First, the conditions of applicability of linear methods proposed by the two Codes (Sec. 2.2)
were checked. Concerning the first condition of applicability (Sec. 2.2.1), Table 5.3 shows
that the value of ρmax/ρmin reduces when the members stiffness EI decreases. This reduction is
not linear with the reduction of EI. In fact:
( ρ max / ρ min ) 0.5 EI gross
( ρ max / ρ min ) EI gross
( ρ max / ρ min ) EI eff
= 0.758 , while
0.5EI gross
EI gross
= 0.713 − 0.733 , while
= 0.5
EI eff
EI gross
= 0.366 .
96
The reason of the nonlinear reduction of the ratio ρmax/ρmin can be explained as follows.
Reducing the stiffness EI of structural members, the global stiffness of the structure reduces
too and, therefore, the natural periods increase. For the Sede Comunale, assuming EI = 100%
EIgross, the periods of the first and second modes of interest are, respectively, T1 = 0.744 s and
T2 = 0.431 s. T1 belongs to the descending branch of the elastic response spectrum (T1 > TC),
while T2 lies in the plateau of the spectrum (TB < T2 < TC). Decreasing the members stiffness,
T1 and T2 shift, respectively, to T1 = 1.066 s and T2 = 0.603 s, in the case of EI = 50% EIgross,
and to T1 = 1.175 - 1.230 s and T2 = 0.680 - 0.704 s, in the case of EI = EIeff computed through
Eq. (2.8). Therefore, decreasing EI, the spectral ordinates corresponding to the first two
modes of interest decrease too, yielding smaller seismic forces. Since the value of ρmin is fixed
to 1 in Eurocode 8 and to 2 in the Italian Seismic Code, ρmax is the only variable in ρmax/ρmin.
ρmax increases with increasing the intensity of the input forces and vice versa; hence, ρmax/ρmin
decreases with decreasing EI.
This conclusion is not general, but depends on both stiffness of the structure and shape of
elastic response spectrum (in particular on the value of corner period, TC). The Sede
Comunale is characterized by a very low value of fcm (8.3 MPa) and, hence, by a low value of
Ec which yields large values of natural periods. For very stiff structures (characterized by a
large value of fcm, small interstorey heights and low number of floors), instead, it could be
even possible that the building experiences an increase of ρmax/ρmin with the decrease of EI.
This may happen if the periods of the principal modes (even T1, for extremely stiff structures)
belong to the ascending branch of the elastic response spectrum.
The reduction of ρmax/ρmin with the decrease of EI has an interesting consequence, as the
choice of EI may be crucial to determine whether the results furnished by linear analyses can
be accepted. In fact, choosing a high value of EI (e.g. EI = EIgross), the chord rotation demands
Di will be small and, therefore, it will be easier to satisfy Di < Ci, but, on the other hand, large
values of ρmax/ρmin will be obtained and, hence, it will be more difficult to satisfy the
conditions of applicability of linear analyses.
Although Eurocode 8 suggests to take into account both member and joint equilibrium to
determine ρmax/ρmin, the results in terms of ρmax are identical to those achieved applying the
recommendations of the Italian Seismic Code. In fact, it is very probable that the member end
which experiences ρmax according to the Italian Seismic Code (i.e. on the base of the member
equilibrium alone) is weaker than the other members which frame into the joint and,
therefore, it will experience ρmax also according to Eurocode 8. In the light of these
considerations, for the Sede Comunale, there is no reason to consider the joint equilibrium in
order to determine the value of ρmax/ρmin and the simpler and faster procedure suggested in the
Italian Seismic Code is recommended.
Considering that:
•
both Codes fix the maximum allowable value of ρmax/ρmin to 2.5,
•
ρmax values obtained applying both Codes are almost equal to each other,
•
Eurocode 8 suggests ρmin = 1, while, the Italian Seismic Code proposes ρmin = 2,
97
the condition of applicability of linear analyses suggested in Eurocode 8 is twice more
conservative with respect to that proposed in the Italian Seismic Code.
The second condition of applicability of linear analyses is considered only in the Italian
Seismic Code (Sec. 2.2.2). The results of the check (Table 5.3) show a percentage between
10-11% (for EI = EIeff) and 13% (for EI = 100% EIgross) of structural members which do not
satisfy the recommendations. Therefore, the second condition of applicability of linear
analyses is less sensitive to the choice of EI than the first condition. In analogy with the first
condition, also the second condition will become more restrictive if large values of EI are
assumed, as the shear demands will increase with increasing EI (except for very stiff
structures).
Table 5.3. Conditions of applicability of linear methods according to both Codes
CONDITIONS OF APPLICABILITY OF LINEAR ANALYSES
FIRST CONDITION
SECOND CONDITION
EI = 100% EIgross
OPCM 3431
EC8
OPCM 3431
8.05
8.05
Number of Sections
1154
ρmax
ρmax
2.01
1.00
Not Verified
153
ρmin
ρmin
ρmax/ρmin
4.00
ρmax/ρmin
8.03
% Not Verified
13
EI = 50% EIgross
OPCM 3431
EC8
OPCM 3431
6.10
6.10
Number of Sections
1154
ρmax
ρmax
2.00
1.02
Not Verified
127
ρmin
ρmin
ρmax/ρmin
3.05
ρmax/ρmin
6.01
% Not Verified
11
EI = mean EIeff (all elements)
OPCM 3431
EC8
OPCM 3431
5.90
5.90
Number of Sections
1154
ρmax
ρmax
2.00
1.00
Not Verified
118
ρmin
ρmin
ρmax/ρmin
2.95
ρmax/ρmin
5.89
% Not Verified
10
OPCM 3431
EC8
OPCM 3431
5.76
5.76
Number of Sections
1154
ρmax
ρmax
2.01
1.01
Not Verified
124
ρmin
ρmin
ρmax/ρmin
2.87
ρmax/ρmin
5.71
% Not Verified
11
OPCM 3431
EC8
OPCM 3431
5.74
5.74
Number of Sections
1154
ρmax
ρmax
2.00
1.00
Not Verified
123
ρmin
ρmin
ρmax/ρmin
2.87
ρmax/ρmin
5.74
% Not Verified
11
For the Sede Comunale, all conditions of applicability of linear methods (first condition
according to both Codes and second condition according to the Italian Seismic Code) are not
satisfied. It means that the results furnished by linear analyses should be judged as not
reliable, as the load path is likely to change noticeably when nonlinear mechanisms develop.
Anyway, it must be considered that, if most members experience very similar ρ values and
only a limited number of members is characterized by sensibly larger ρ values, ρmax/ρmin will
be large, although the overall behaviour of the building is quite regular. Therefore,
considering only ρmax/ρmin to evaluate the possibility of accepting linear analyses may not be
98
judged as an exhaustive method. The same problem affects the second condition of
applicability: if only a limited number of members do not satisfy the check, it will be possible
that the change in the load path will be not so remarkable to justify the choice of rejecting the
possibility of using linear analyses. Therefore, other studies are required to determine if the
conclusion to reject linear methods is unavoidable or if the dynamic linear analysis may be
considered useful at least to catch the global seismic response of the building and to express a
judgment about the need of retrofitting.
Although the conditions of applicability are not satisfied, the dynamic linear analysis was
performed, since the principal aim of this work is to compare the different assessment
procedures of Eurocode 8 and Italian Seismic Code at structural member level, suggesting
eventually simplified solutions. The evaluation of the reliability of the different methods of
analysis, in order to determine the need of retrofitting, will be object of future research works.
The results of the assessment of ductile mechanisms will be shown in the following charts.
The assessment was performed according to both Codes and both empirical (Eq. (3.15)) and
theoretical (Eq. (3.22)) θu, with the following aims:
•
checking if the two Codes yield close results.
•
Checking if the two approaches (empirical and theoretical) yield close results.
•
Showing if the choice of members stiffness EI affects strongly the results.
•
Testing if possible simplifications in the procedures can be applied without any
sensible loss in accuracy of the results. In particular, N from gravity loads (gravity N)
instead of N from the seismic combination (seismic N), Ls = L/2 instead of Ls from Eq.
(3.1) and φy from empirical expressions (Eqs. (3.11) to (3.14)) instead of φy from
theoretical assumptions (Eqs. (3.9) – (3.10)) were assumed in order to check the
possibility of removing the dependency of the chord rotation capacity from the seismic
demand and proposing faster and less complex calculations.
•
Understanding if the definition of a bi-dimensional failure curve (Eq. (3.27)) is
required or if the conventional uniaxial check allows to obtain satisfactory results.
All results refer to the percentages of structural members which do not satisfy the verification.
The percentage was computed for all structural members, for beam, for columns and, also, for
columns of each floor separately, in order to check the possible development of soft-storey
mechanisms.
The sensitivity of the assessment procedure to EI is shown from Figure 5.5 to Figure 5.12. All
charts refer to the percentage of members which do not satisfy the verification according to
both Codes and both empirical (Eq. (3.15)) and theoretical (Eq. (3.22)) θu. In particular,
Figure 5.5 - Figure 5.6 - Figure 5.7 - Figure 5.8 show that, decreasing EI, the percentage of
members which do not satisfy the verification increases, for both beams and columns. This
tendency will be more evident if Eq. (3.15) is used to assess θu. Therefore, assuming EIeff
computed through Eq. (2.8) as the most accurate choice of EI, it is clear that, for the Sede
99
Comunale, both 100% EIgross and 50% EIgross yield unconservative results. Figure 5.9 - Figure
5.10 - Figure 5.11 - Figure 5.12 show that close results are obtained considering the mean EIeff
computed i) for all members, ii) separately for beams and columns, or iii) separately for T and
rectangular beams and for columns of each floor. In particular, the results are very close to
each other if Eq. (3.22) is used to assess θu, while a difference can be appreciated if Eq. (3.15)
is assumed. However, this difference is small, except for the columns of the roof floor.
DUCTILE CHECK - DYNAMIC LIN. - EMP. FORM. - OPCM
DUCTILE CHECK - DYNAMIC LIN. - EMP. FORM. - EC8
100
90
100
100%EI
90
50%EI
80
50%EI
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
All Columns 1st Floor Col 2nd Floor Col 3rd Floor Col
Roof Col
Figure 5.5. Ductile check: dynamic linear analysis,
empirical form, OPCM, different EI
All Elements
Roof Col
DUCTILE CHECK - DYNAMIC LIN. - THEOR. FORM. - EC8
100%EI
90
100%EI
50%EI
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff
0
All Elements
All Beams
Roof Col
theoretical form, OPCM, different EI
All Elements
All Beams
Roof Col
theoretical form, EC8, different EI
100
100
EI_eff_uniform
90
EI_eff_beam_col
80
100
50%EI
90
All Beams
empirical form, EC8, different EI
DUCTILE CHECK - DYNAMIC LIN. - THEOR. FORM. - OPCM
100
80
EI_eff
0
All Elements
90
100%EI
EI_eff_uniform
EI_eff_beam_col
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_precise
0
All Elements
All Beams
Roof Col
All Elements
All Beams
Roof Col
Figure 5.10. Ductile check: dynamic linear
analysis, empirical form, EC8, different EI
100
100
100
EI_eff_uniform
90
90
EI_eff_beam_col
80
EI_eff_uniform
EI_eff_beam_col
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_precise
0
All Elements
All Beams
Roof Col
All Elements
All Beams
Roof Col
analysis, theoretical form, EC8, different EI
Figure 5.13 and Figure 5.14 compare the percentages of members which do not satisfy the
verification according to the Italian Seismic Code and Eurocode 8. Although the formulas
proposed in both Codes are very close to each other, Eurocode 8 suggests more complex
calculations. In fact, the flexural deformation at yielding (Eqs. (3.5) – (3.6)) is computed as
φ y Ls + α s d − d ' 3 in Eurocode 8, and as φ y Ls 3 in the Italian Seismic Code. Moreover,
the Italian Seismic Code proposes to evaluate M and V demands through the member
equilibrium, while Eurocode 8 suggests to consider also the equilibrium of the joint into
which the considered member frames. The results obtained from the two Codes are close to
each other. Therefore, for the Sede Comunale, the use of the simpler approach proposed in the
Italian Seismic Code is advised.
[
(
)]
DUCTILE CHECK - DYNAMIC LIN. - EMP. FORM. - EI_eff
DUCTILE CHECK - DYNAMIC LIN. - THEOR. FORM. - EI_eff
100
100
90
OPCM
90
OPCM
80
EC8
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
empirical form, EIeff, OPCM vs. EC8
All Elements
All Beams
Roof Col
analysis, theoretical form, EIeff, OPCM vs. EC8
The comparison between the results obtained using the correct definition of Ls (Eq. (3.1)) and
Ls = L/2 is shown from Figure 5.15 to Figure 5.18. The charts refer to the percentages of
members which do not satisfy the verification according to both Codes and both empirical
(Eq. (3.15)) and theoretical (Eq. (3.22)) θu. Concerning the empirical θu, the use of Ls = L/2
yields a very small decrease in the percentage of members which do not satisfy the
verification with respect to Eq. (3.1). Concerning the theoretical θu, the decrease in the
percentage of members which do not satisfy the verification obtained using Ls = L/2 is larger
(but less than 20%). Therefore, it suggests that the procedure may be simplified, particularly if
the assessment is performed according to Eq. (3.15).
101
DUCTILE CHECK - DYN. LIN. - EMP. FORM. - OPCM - EI_eff
100
90
80
DUCTILE CHECK - DYN. LIN. - EMP. FORM. - EC8 - EI_eff
100
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
Roof Col
Figure 5.15. Ductile check: dynamic linear an.,
empirical form, EIeff, OPCM, Ls=M/V vs. Ls=L/2
All Elements
All Beams
Roof Col
empirical form, EIeff, EC8, Ls=M/V vs. Ls=L/2
DUCTILE CHECK - DYN. LIN - THEOR. FORM - OPCM - EI_eff
100
80
Ls=L/2
0
All Elements
90
Ls=M/V
DUCTILE CHECK - DYN. LIN. - THEOR. FORM. - EC8 - EI_eff
100
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
Roof Col
theoretical form, EIeff, OPCM, Ls=M/V vs. Ls=L/2
All Elements
All Beams
Roof Col
theoretical form, EIeff, EC8, Ls=M/V vs. Ls=L/2
The comparison between the results obtained using the gravity N and the seismic N is shown
from Figure 5.19 to Figure 5.22. The charts refer to the percentages of members which do not
satisfy the verification according to both Codes and both empirical (Eq. (3.15)) and theoretical
(Eq. (3.22)) θu. Concerning the empirical θu, the seismic N yields a very small increase in the
percentage of members which do not satisfy the verification with respect to the gravity N.
Concerning the theoretical θu, the increase in the percentage of members which do not satisfy
the verification obtained using the seismic N is larger (up to 20%, considering all columns).
However, it must be remarked that the seismic ΔN considered in this case is related with the
hypothesis of indefinitely elastic behaviour and, hence, is larger than ΔN obtained when the
development of nonlinear mechanisms is accounted for (Sec. 3.2.2). Therefore, at least
concerning the assessment performed according to Eq. (3.15), the procedure may be
simplified.
102
100
90
80
100
N_Grav
90
N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
Roof Col
empirical form, EIeff, OPCM, Ggrav vs. Gseism
All Elements
All Beams
Roof Col
analysis, empirical form, EIeff, EC8, Ggrav vs. Gseism
100
80
N_Seism
0
All Elements
90
N_Grav
100
N_Grav
90
N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
N_Grav
N_Seism
0
All Elements
All Beams
Roof Col
theoretical form, EIeff, OPCM, Ggrav vs. Gseism
All Elements
All Beams
Roof Col
theoretical form, EIeff, EC8, Ggrav vs. Gseism
The comparison between the results obtained considering φy from theoretical assumptions
(Eqs. (3.9) – (3.10)) and from empirical expressions (Eqs. (3.11) to (3.14)) is shown in Figure
5.23 and Figure 5.24. The charts refer to the percentages of members which do not satisfy the
verification according to the theoretical θu (Eq. (3.22)) and to both Codes. Empirical and
theoretical φy yield very close results. Hence, the empirical formulas calibrated for new
seismically designed structures may be applied without any loss of accuracy.
100
90
100
FI_Theoric
90
FI_Fardis
80
FI_Theoric
FI_Fardis
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
FI_Priestley
0
All Elements
All Beams
Roof Col
theoretical form, EIeff, OPCM, different φy
All Elements
All Beams
Roof Col
analysis, theoretical form, EIeff, EC8, different φy
103
Figure 5.25 to Figure 5.28 show the comparison between the results obtained using the correct
procedure (seismic N; Ls from Eq. (3.1); theoretical φy) and the simplified procedure (gravity
N; Ls = L/2; empirical φy). The charts refer to the percentages of members which do not satisfy
the verification according to both Codes and both empirical (Eq. (3.15)) and theoretical (Eq.
(3.22)) θu. Concerning the empirical θu, the simplified approach yields a small decrease in the
percentage of members which do not satisfy the verification with respect to the correct
approach. Concerning the theoretical θu, instead, the decrease in the percentage of members
which do not satisfy the verification obtained using the simplified approach is noticeable.
Comparing the results obtained considering both correct and simplified approach and both
empirical and theoretical θu (Figure 5.29 - Figure 5.30), it appears that empirical and
theoretical θu yield close results when the correct approach is considered, while, concerning
the simplified procedure, the theoretical θu yields unconservative results. Therefore, for the
Sede Comunale, the assessment of ductile mechanisms based on the dynamic linear analysis
can be performed according to the correct approach of either Eq. (3.15) or Eq. (3.22), or to the
simplified approach of Eq. (3.15). In fact, the simplified approach of Eq. (3.15) allows a quite
accurate and very quick assessment of the ductile response of the structure and, hence, can be
accepted even if it is slightly unconservative.
100
100
90
Ls=L/2; N_Grav
90
Ls=L/2; N_Grav
80
Ls=M/V; N_Seism
80
Ls=M/V; N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
Figure 5.25. Ductile check: dynamic linear an., emp.
form, EIeff, OPCM, correct vs. simplified
All Elements
90
80
Roof Col
emp. form, EIeff, EC8, correct vs. simplified
100
All Beams
100
FI_Fardis; Ls=L/2; N_Grav
90
FI_Theoric; Ls=M/V; N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
theor. form, EIeff, OPCM, correct vs. simplified
All Elements
All Beams
Roof Col
theor. form, EIeff, EC8, correct vs. simplified
104
DUCTILE CHECK - DYNAMIC LIN. - OPCM - EI_eff
100
90
80
DUCTILE CHECK - DYNAMIC LIN. - EC8 - EI_eff
100
EMP. FORM; Ls=L/2; N_Grav
EMP. FORM; Ls=M/V; N_Seism
TH. FORM; FI_Fardis; Ls=L/2; N_Grav
TH. FORM; FI_Theoric; Ls=M/V; N_Seism
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
Figure 5.29. Ductile check: dynamic linear an., EIeff,
OPCM, theor. vs. emp., correct vs. simplified
All Elements
All Beams
Roof Col
EIeff, EC8, theor. vs. emp., correct vs. simplified
The comparison between the results obtained using the conventional uniaxial check and those
found considering the definition of a bidimensional failure curve (Eq. (3.27)) is shown from
Figure 5.31 to Figure 5.34. The charts refer to the percentages of members which do not
(Eq. (3.22)) θu. All charts show that, for the considered building, the definition of a
bidimensional failure curve does not improve on the accuracy of the results. Therefore, the
conventional uniaxial check is recommended for the assessment of the Sede Comunale, as it
yields enough accurate results and is much faster that the bidimensional check.
100
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
form., EIeff, OPCM, uni- vs. bi-axial bending
All Elements
All Beams
Roof Col
Figure 5.32. Ductile check: dynamic linear an.
emp. form., EIeff, EC8, uni- vs. bi-axial bending
100
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
theor. form., EIeff, OPCM, uni vs. bi-axial bending
All Elements
All Beams
Roof Col
theor. form., EIeff, EC8, uni- vs. bi-axial bending
105
In the following charts the results of the assessment of brittle mechanisms will be shown,
according to both Codes, with the following aims:
•
checking if the two Codes yield close results. In particular, considering that Eurocode
8 suggests a much more complicated procedure, the target is too see if the indications
furnished by the Italian Seismic Code allow to obtain accurate results.
•
•
Testing if possible simplifications in the procedures can be applied without any
sensible loss in accuracy of the results. In particular, gravity N instead of seismic N, Ls
= L/2 instead of Ls from Eq. (3.1), empirical φy (Eqs. (3.11) to (3.14)) instead of
theoretical φy (Eqs. (3.9) – (3.10)) and a fraction of EIgross (between 50 and 100%)
instead of EIeff (Eq. (2.8)) were assumed in order to check the possibility of removing
the dependency of the shear capacity from the seismic demand and proposing faster
and less complex calculations.
The sensitivity of the assessment procedure to EI is shown from Figure 5.35 to Figure 5.38.
All charts refer to the percentages of members which do not satisfy the verification according
to both Codes. In particular, Figure 5.35 and Figure 5.36 show that, according to both Codes,
decreasing EI the percentage of members which do not satisfy the verification decreases too.
Therefore, assessing the brittle mechanisms using a fixed fraction of EIgross (between 50 and
100%) yields conservative results. Moreover, the difference in percentage of members which
do not satisfy the verification assuming 100% EIgross and mean EIeff (Eq. (2.8)) is less than
10%. Hence, for the Sede Comunale and within the limits of brittle assessment, assuming a
fixed fraction of EIgross between 50% and 100% may be considered a suitable choice, since the
assessment of EIeff according to Eq. (2.8) is much longer and yields small improvements in
the results.
Figure 5.37 and Figure 5.38 show that very close results are obtained considering the mean
EIeff computed i) for all members, ii) separately for beams and columns, or iii) separately for
T and rectangular beams and for columns of each floor.
BRITTLE CHECK - DYNAMIC LIN. - OPCM
BRITTLE CHECK - DYNAMIC LIN. - EC8
100
90
100
100%EI
100%EI
90
50%EI
50%EI
80
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff
0
All Elements
All Beams
Roof Col
Figure 5.35. Brittle check: dynamic linear analysis,
OPCM, different EI
All Elements
All Beams
Roof Col
EC8, different EI
106
100
100
EI_eff_uniform
90
80
EI_eff_uniform
90
EI_eff_beam-col
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_beam-col
EI_eff_precise
0
All Elements
All Beams
Roof Col
OPCM, different EI
All Elements
All Beams
Roof Col
EC8, different EI
in Figure 5.39 and Figure 5.40. The charts refer to the percentages of members which do not
satisfy the verification according to both Codes. Concerning Eurocode 8, the seismic N yields
a very small increase in the percentage of members which do not satisfy the verification with
respect to the gravity N. Therefore the procedure may be simplified. Concerning the Italian
Seismic Code, instead, the increase in the percentage of members which do not satisfy the
verification obtained using the seismic N is much larger (up to 40% considering all columns).
Although the seismic ΔN used in this case is related with the hypothesis of indefinitely elastic
behaviour and, therefore, is larger than ΔN obtained when the development of nonlinear
mechanisms is considered, such strong decreases in the percentage of members which do not
satisfy the verification do not allow to simplify the procedure. This huge difference is due to
the fact that the concrete shear resistance of columns will become null if the columns
experience a tensile N (see Fig. 3.20).
BRITTLE CHECK - DYNAMIC LIN. - OPCM - EI_eff
BRITTLE CHECK - DYNAMIC LIN. - EC8 - EI_eff
100
100
N_Grav
N_Grav
90
90
N_Seism
N_Seism
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
OPCM, EIeff, Ggrav vs. Gseism
All Elements
All Beams
Roof Col
EC8, EIeff, Ggrav vs. Gseism
The comparison between the results obtained considering the theoretical φy (Eqs. (3.9) –
(3.10)) and the empirical φy (Eqs. (3.11) to (3.14)) is shown in Figure 5.41. The chart refers to
the percentages of members which do not satisfy the verification according only to Eurocode
8 (the expression suggested in the Italian Seismic Code does not depend on φy). Empirical and
theoretical φy yield very close (practically identical) results. Therefore, for the Sede
Comunale, the empirical formulas may be applied without any loss of accuracy.
107
Ls = L/2 is shown in Figure 5.42. The chart refers to the percentages of members which do not
satisfy the verification according only to Eurocode 8 (the expression suggested in the Italian
Seismic Code does not depend on Ls). The use of Ls = L/2 yields a very small increase in the
percentage of members which do not satisfy the verification with respect to Eq. (3.1), leading
to a safe side assessment. Hence, for the Sede Comunale, the procedure may be simplified.
100
100
FI_Theoric
Ls=M/V
90
FI_Fardis
90
80
FI_Priestley
80
Ls=L/2
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
EC8, EIeff, different φy
All Elements
All Beams
Roof Col
EC8, EIeff, Ls=M/V vs. Ls=L/2
Figure 5.43 shows the comparison between the results obtained according to the Italian
Seismic Code (both seismic and gravity N) and to Eurocode 8. Considering the more complex
approach proposed in Eurocode 8 as the most accurate, it is clear that assessing the brittle
mechanisms according to the Italian Seismic Code with gravity N yields results which grossly
underestimate the percentage of members which do not satisfy the requirements. Moreover,
also the correct approach (seismic N) of the Italian Seismic Code yields inaccurate results (the
percentage is 20% less than for Eurocode 8). Therefore, for the Sede Comunale, the procedure
suggested in Eurocode 8 is recommended. Figure 5.44 shows the comparison between the
results obtained using the correct approach (seismic N; Ls from Eq. (3.1); theoretical φy; mean
EIeff) and the simplified approach (gravity N; Ls = L/2; empirical φy; 50% EIgross) of the
assessment procedure suggested in Eurocode 8. The simplified approach yields a small
increase in the percentage of members which do not satisfy the verification with respect to the
correct approach, leading to a safe side assessment. Hence, for the Sede Comunale, the
procedure may be simplified.
BRITTLE CHECK - DYNAMIC LIN. - EI_eff
100
100
OPCM; N_Grav
N_Grav; Ls=L/2; FI_Fardis; 50%EI
90
OPCM; N_Seism
90
80
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
N_Seism; Ls=M/V; FI_Theoric; EI_eff
0
All Elements
All Beams
Roof Col
Figure 5.43. Brittle check: dynamic linear an., EIeff,
correct OPCM vs. simplified OPCM vs. EC8
All Elements
All Beams
Roof Col
Figure 5.44. Brittle check: dynamic linear an.,
EC8, correct vs. simplified approach
108
The following assumptions characterize the SeismoStruct model.
A uniaxial nonlinear constant confinement model was assumed. The confinement effects
provided by the transversal reinforcement were incorporated through the rules proposed by
Mander et al. [1988], later modified by Martinez-Rueda and Elnashai [1997].
The following four parameters were defined in order to describe the mechanical
characteristics of the material (Figure 5.45):
•
concrete compressive strength, fc = 8.3 MPa.
•
Tensile strength, ft ≈ 0 MPa (concrete sections considered already cracked).
•
Strain at peak stress, εco = 0.002.
•
Confinement factor, kc (Eq. (2.21)). kc was taken equal to 1 for concrete cover (no
confinement) and between 1.02 and 1.07 for concrete core, depending on both core
dimensions and transversal steel content.
The uniaxial bilinear stress-strain model with kinematic strain hardening was assumed for the
longitudinal steel bars. Three parameters were defined in order to describe the mechanical
•
modulus of elasticity, Es = 200000 MPa.
•
Yield strength, fy = 440 MPa.
•
Strain hardening parameter, μ = E sp E s = 0.005 , where Esp is the post-yield stiffness.
Figure 5.45. Concrete: nonlin. confinement model
Figure 5.46. Steel: bilinear strain hardening model
The definition of the ultimate properties (ultimate concrete strain εcu and ultimate steel strain
εsu) were not required, since SeismoStruct does not consider the failure of members but, once
reached the ultimate conditions, R.C. members continue to have a residual strength.
Beams and columns were modelled using 3-D inelastic beam elements. Every member was
subdivided into four elements. This subdivision allowed to take into account the change in the
reinforcement content (both longitudinal and transversal) between the ends and the middle
part of the member, leading to a more accurate assessment of the inelastic behaviour.
109
The basement structural walls were modelled using frame elements rigidly connected to the
corner joints.
Since the software SeismoStruct does not allow to model shell elements, the flexibility of the
slabs was modelled using the simplified method described in Sec. 2.3 (Eqs. (2.5) – (2.6)).
According to this procedure, the slab was substituted by two cross braces, connected to the
corners of the slab through hinges, in order to avoid any moment transfer.
In the following, the procedure adopted for estimating the dimensions of the brace is shown
for the same slab already considered in Sec. 5.1.1 (L’ = 4250 mm, L = 4400 mm). The
thickness of the equivalent slab computed for the SAP model was determined as 67 mm (Sec.
5.1.1). Therefore:
(
)
I = 4400 3 ⋅ 67 12 = 4.756 ⋅ 1011 mm 4
As = 5 6 A = 5 6 ⋅ (4400 ⋅ 67) = 245667 mm 2
Gc = 7156 MPa
[
(
]
)
K b = 4250 3 12 ⋅ 17174 ⋅ 4.756 ⋅ 1011 + 4250 (245667 ⋅ 7156 )
−1
= 312429 N / mm
Once known Kb and l = L' 2 + L2 = 4250 2 + 4400 2 = 6117 mm , the area of the brace was
computed as: Ab = 312429 ⋅ 6117 / 17174 = 111288mm 2 . Considering a circular section, D =
376 mm was determined.
The correctness of this approach was tested comparing the natural periods obtained using the
SAP model and those found using SeismoStruct, with beams and columns modelled as linear
elastic (E = 17174 MPa). The results are shown in Table 5.4. The differences are in the order
of 1% for the first mode of interest, 0% for the second and 7% for the third. Since the
differences are very small, the SeismoStruct model was judged to be consistent with the SAP
model.
Table 5.4. Comparison between the first three modes - SeismoStruct and SAP models.
Periods [s]
I Mode
II Mode
III Mode
SeismoStruct
0.783
0.559
0.400
SAP
0.774
0.559
0.431
The nonlinear static analysis of 3-D models is based on eight pushover analyses (“modal” and
“uniform” pattern of lateral forces, applied in positive and negative X and Y directions). In
order to perform the analysis:
•
only the definition of the monotonic stress-strain model of the materials (Figure 5.45
and Figure 5.46) was required.
110
•
The masses were lumped in the structural joints. This solution is consistent with the
choice of applying the later forces in the structural joints (Sec. 2.3).
•
The displacement response control was chosen, in order to evaluate also the
descending branch of the capacity curves (Sec. 2.5; Fig. 2.9).
The deformed shapes according to both “modal” and “uniform” pattern of lateral forces,
applied in both X and Y are shown from Figure 5.47 to Figure 5.52. In these figures, the cross
braces used to model the in-plan slabs stiffness are omitted, in order to make the deformed
shapes clearer. All deformed shapes refer to the SD LS.
Considering both “modal” and “uniform” pattern of lateral forces applied in X, the deformed
shapes denote a pure translation toward X, in agreement with the deformed shape of the first
mode. Considering both “modal” and “uniform” pattern of lateral forces applied in Y, the
deformed shapes denote both translation toward Y and rotation around Z, in agreement with
the deformed shape of the second mode.
Figure 5.47. Uniform distr., positive X, X-dir view
Figure 5.48. Modal distr., positive X, X-dir view
Figure 5.49. Uniform distr., positive Y, Y-dir view
Figure 5.50. Modal distr., positive Y, Y-dir view
Figure 5.51. Uniform distr., positive Y, X-dir view
Figure 5.52. Modal distr., positive Y, X-dir view
111
The definition of the demand quantities (both forces and deformations) corresponding to the
considered LS were determined following the procedure shown in Sec. 2.5.
For the “uniform” pattern of lateral forces in X (Figure 5.53 and Figure 5.54):
•
Γx = 1 (hence, the SDOF and the MDOF merge into the same system).
•
d *y = 72mm (Eq. (2.14)); K * = 23392 MPa (Eq. (2.15)); T * = 0.951s (Eq. (2.16)).
•
For the SD LS: d t* = d t = 129mm (Eqs. (2.17) – (2.20)); Vb = 1681kN .
For the “modal” pattern of lateral forces in X (Figure 5.55 and Figure 5.56):
•
Γx = 1.229 (Eq. (2.13)).
•
d *y = 66mm (Eq. (2.14)); K * = 13253MPa (Eq. (2.15)); T * = 1.263s (Eq. (2.16)).
•
For the SD LS: d t = 211mm (Eqs. (2.17) – (2.20)); Vb = 895kN .
CAPACITY CURVE - SDOF (=MDOF)
CAPACITY CURVE - MDOF (=SDOF)
1800
1800
1400
1400
Vb (kN)
1200
Vb (kN)
Vb,SD
1600
1600
1000
800
Vb,NC
1200 V
b,LD
1000
800
600
600
400
400
200
200
Δ
0
0
*
Δ
y
50
*
100
150
Δ LD
0
m
200
250
300
350
0
400
50
Δ NC
Δ SD
100
150
200
250
300
350
400
Δ control node (mm)
Figure 5.53. Pushover Uniform X-dir, SDOF
capacity curve: equivalent area method
Figure 5.54. Pushover Uniform X-dir, MDOF
capacity curve: DL, SD and NC LS
CAPACITY CURVE - SDOF
CAPACITY CURVE - MDOF
1200
900
800
Vb,LD
1000
700
800
Vb (kN)
600
Vb (kN)
Vb,SD
500
400
Vb,NC
600
400
300
200
200
100
Δ
0
0
50
*
y
Δ
Δ LD
*
m
100
0
150
200
250
300
Figure 5.55. Pushover Modal X-dir, SDOF capacity
curve: equivalent area method
0
50
100
Δ NC
Δ SD
150
200
250
300
350
400
Figure 5.56. Pushover Modal X-dir, MDOF
For the “uniform” pattern of lateral forces in Y (Figure 5.57 and Figure 5.58):
•
Γy = 1 (hence, the SDOF and the MDOF merge into the same system).
•
112
For the SD LS: d t* = d t = 57 mm (Eqs. (2.17) – (2.20)); Vb = 3103kN .
•
For the “modal” pattern of lateral forces in Y (Figure 5.59 and Figure 5.60):
•
Γy = 1.147 (Eq. (2.13)).
•
•
3500
3500
3000
3000
Vb,NC
2500
Vb (kN)
2500
Vb (kN)
Vb,SD
2000
1500
Vb,LD
2000
1500
1000
1000
500
500
Δ
0
*
Δ
y
0
*
50
Δ LD
0
m
100
150
200
250
0
300
Δ NC
Δ SD
50
100
Figure 5.57. Pushover Uniform Y-dir, SDOF
200
250
300
Figure 5.58. Pushover Uniform Y-dir, MDOF
1800
2000
1600
1800
Vb,SD
Vb,NC
1400
1600
Vb,LD
1400
Vb (kN)
1200
Vb (kN)
150
1000
800
600
1200
1000
800
600
400
400
200
Δ
0
*
0
y
Δ
200
*
m
50
Δ LD
0
100
150
200
250
0
300
50
Δ SD
100
Δ NC
150
200
250
300
Figure 5.59. Pushover Modal Y-dir, SDOF capacity
Figure 5.60. Pushover Modal Y-dir, MDOF
Figure 5.61 and Figure 5.62 illustrate the MDOF curves and the points representative of the
SD LS for the pushover in X and Y, respectively.
CAPACITY CURVES Y - MDOF
CAPACITY CURVES X - MDOF
3500
1800
VSD,UNIF
1600
UNIFORM
UNIFORM
VSD,UNIF
MODAL
2500
1200
1000
Vb (kN)
Vb (kN)
3000
MODAL
1400
VSD,MOD
800
600
2000
VSD,MOD
1500
1000
400
200
500
ΔSD,MOD
ΔSD,UNIF
ΔSD,UNIF
0
ΔSD,MOD
0
0
50
100
150
200
250
300
350
400
Figure 5.61. Pushover Uniform and Modal X-dir,
MDOF capacity curves: SD LS
0
50
100
150
200
250
300
Figure 5.62. Pushover Uniform and Modal Y-dir,
113
theoretical (Eq. (3.22)) θu, with the same aims (except for the sensitivity to EI) already
described for the assessment based on the dynamic linear analysis (Sec. 5.1.3).
DUCTILE CHECK - STATIC NONLIN. - THEOR. FORM.
DUCTILE CHECK - STATIC NONLIN. - EMP. FORM.
100
100
90
OPCM
90
OPCM
80
EC8
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
Figure 5.63. Ductile check: static nonlinear analysis,
empirical form, OPCM vs. EC8
All Elements
Roof Col
100
FI_Theoric
90
FI_Theoric
FI_Fardis
FI_Fardis
80
DUCTILE CHECK - STATIC NONLIN. - THEOR. FORM. - EC8
DUCTILE CHECK - STATIC NONLIN - THEOR. FORM - OPCM
100
90
All Beams
Figure 5.64. Ductile check: static nonlinear an.,
theoretical form, OPCM vs. EC8
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
FI_Priestley
0
All Elements
All Beams
Roof Col
theoretical form, OPCM, different φy
All Elements
All Beams
Roof Col
theoretical form, EC8, different φy
proposed in both Codes are very close to each other, Eurocode 8 suggests more complex
calculations (Sec. 5.1.3). Moreover, also the way to compute the demand is different. In fact,
the Italian Seismic Code proposes to consider separately the results obtained from each
pushover, while Eurocode 8 suggests to consider each demand quantity as the outcome of the
combination of the values furnished by a pushover in X and a pushover in Y (Eq. (2.23)).
Concerning the empirical θu (Eq. (3.15)), the results obtained from Eurocode 8 are slightly
conservative and very close to those obtained from the Italian Seismic Code. Therefore, for
the Sede Comunale, there is no need to use the procedure suggested in Eurocode 8 and, hence,
the simpler procedure proposed in the Italian Seismic Code is recommended. Concerning the
theoretical θu, instead, the percentage of members which do not satisfy the verification
obtained from Eurocode 8 is larger than that obtained from the Italian Seismic Code. This
difference is due in particular to the way of computing the seismic demand, since, as already
shown for the assessment based on the dynamic linear analysis, the more complex way to
assess the capacity proposed in Eurocode 8 yields close results to those obtained applying the
114
recommendations of the Italian Seismic Code. Further future investigations are needed to
determine whether Eq. (2.23) yields more reliable demands values than those obtained
considered each pushover separately.
DUCTILE CHECK - STATIC NONLIN. - EMP. FORM. - EC8
DUCTILE CHECK - STATIC NONLIN. - EMP. FORM. - OPCM
100
100
90
80
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
Roof Col
empirical form, OPCM, Ls=M/V vs. Ls=L/2
All Elements
All Beams
Roof Col
empirical form, EC8, Ls=M/V vs. Ls=L/2
100
100
80
Ls=L/2
0
All Elements
90
Ls=M/V
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
Roof Col
theoretical form, OPCM, Ls=M/V vs. Ls=L/2
All Elements
All Beams
Roof Col
theoretical form, EC8, Ls=M/V vs. Ls=L/2
(3.10)) and the empirical φy (Eqs. (3.11) to (3.14)) is shown in Figure 5.65 and Figure 5.66.
The charts refer to the percentages of members which do not satisfy the verification according
to the theoretical θu (Eq. (3.22)) and to both Codes. Empirical and theoretical φy yield very
close results. Therefore, for the Sede Comunale, the empirical formulas calibrated for new
(Eq. (3.15)) and theoretical (Eq. (3.22)) θu. Concerning the empirical θu, the use of Ls = L/2
yields results very close to those obtained considering Eq. (3.1). Hence, the procedure may be
simplified. Concerning the theoretical θu, instead, the decrease in the percentage of members
which do not satisfy the verification obtained using Ls = L/2 is huge (even larger than 100%).
This shows that Eq. (3.22) is very sensitive to the value of Ls. This result is much more
evident than in the linear analyses, since the inelastic Ls are likely to change with respect to
the elastic Ls, possibly resulting in very small values, leading to θu = 0.
115
100
90
80
100
N_Grav
90
N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
Roof Col
empirical form, OPCM, NGrav vs. NSeism
All Elements
All Beams
Roof Col
empirical form, EC8, NGrav vs. NSeism
100
80
N_Seism
0
All Elements
90
N_Grav
DUCTILE CHECK - STATIC NONLIN - THEOR. FORM - EC8
100
N_Grav
90
N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
N_Grav
N_Seism
0
All Elements
All Beams
Roof Col
theoretical form, OPCM, NGrav vs. NSeism
All Elements
All Beams
Roof Col
theoretical form, EC8, NGrav vs. NSeism
(Eq. (3.22)) θu. All charts show that gravity N and seismic N yield very close results. Hence,
concerning both empirical and theoretical formulas of θu of both Codes, the procedure may be
simplified. This conclusion clashes with the results based on the dynamic linear analysis. This
can be justified considering that the seismic ΔN experienced in a linear analysis is, in general,
(much) larger than ΔN obtained through a nonlinear analysis.
The comparison between the results obtained using the correct procedure (seismic N; Ls from
Eq. (3.1); theoretical φy) and the simplified procedure (gravity N; Ls = L/2; empirical φy) is
shown from Figure 5.75 to Figure 5.78. The charts refer to the percentages of members which
do not satisfy the verification according to both Codes and to both empirical (Eq. (3.15)) and
theoretical (Eq. (3.22)) θu. Concerning the empirical θu, the simplified and the correct
approaches yield very close results. Therefore, for the Sede Comunale, the assessment
procedure may be simplified. Concerning the theoretical θu, instead, the decrease in the
percentage of members which do not satisfy the verification obtained using the simplified
approach is noticeable and can be ascribed mainly to the fact that Eq. (3.22) is very sensitive
to the value of Ls.
116
empirical and theoretical θu (Figure 5.79 and Figure 5.80), it appears that the results obtained
applying both approaches of the empirical θu and the simplified approach of the theoretical θu
are close to each other, but very different from those obtained considering the correct
approach of the theoretical θu. The deformed shapes (in particular Figure 5.47 and Figure
5.48) show that a soft-storey mechanism is likely to develop at the second storey, while the
first storey (the basement) remains practically undeformed. This behaviour was correctly
captured using both correct and simplified approaches of the empirical θu and the simplified
approach of the theoretical θu, while it was not detected using the theoretical θu with the
correct definition of Ls (Eq. (3.1)). Hence, for the Sede Comunale, concerning the theoretical
evaluation of θu, there is the need of considering Ls = L/2, since the use of Eq. (3.1) yields
results too sensitive to the values of Ls.
yields enough accurate results and is much faster than the bidimensional check.
100
100
90
Ls=L/2; N_Grav
90
Ls=L/2; N_Grav
80
Ls=M/V; N_Seism
80
Ls=M/V; N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
empirical form, OPCM, correct vs. simplified
All Elements
All Beams
Roof Col
empirical form, EC8, correct vs. simplified
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
theoretical form, OPCM, correct vs. simplified
All Elements
All Beams
Roof Col
theoretical form, EC8, correct vs. simplified
117
DUCTILE CHECK - STATIC NONLIN. - OPCM
100
90
80
DUCTILE CHECK - STATIC NONLIN. - EC8
100
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
All Elements
100
All Beams
Roof Col
EC8, theor. vs. emp., correct vs. simplified
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
empirical form, OPCM, uni- vs. bi-axial bending
All Elements
Roof Col
empirical form, EC8, uni- vs. bi-axial bending
100
All Beams
100
90
MONO-AXIAL BENDING
90
80
BI-AXIAL BENDING
80
MONO-AXIAL BENDING
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
theor. form, OPCM, uni- vs. bi-axial bending
All Elements
All Beams
Roof Col
theor. form, EC8, uni- vs. bi-axial bending
The results of the assessment of brittle mechanisms will be shown in the following charts. The
assessment was performed according to both Codes, with the same aims (except for the
sensitivity to EI) already described for the dynamic linear analysis (Sec. 5.1.3).
Concerning the assessment of the brittle mechanisms according to the Italian Seismic Code, it
may be useful to remind that, if the point representative of the LS belongs to the descending
branch of the capacity curve, the demand quantities will be those corresponding to the peak
point of the capacity curve.
118
satisfy the verification according to both Codes. Concerning Eurocode 8, gravity and seismic
N yield nearly identical results. Regarding the Italian Seismic Code, the seismic N yields a
very small (practically negligible) increase in the percentage of members which do not satisfy
the verification with respect to the gravity N. For the Sede Comunale, hence, the procedures
suggested by the two Codes may be simplified. This conclusion clashes with the results based
on the dynamic linear analysis. The fact that the results obtained using the static nonlinear
analysis are less sensitive to the choice of N can be justified considering that the seismic ΔN
experienced by the structure in a linear analysis is, in general, (much) larger than ΔN obtained
through a nonlinear analysis.
BRITTLE CHECK - STATIC NONLIN. - OPCM
BRITTLE CHECK - STATIC NONLIN. - EC8
100
100
N_Grav
N_Grav
90
90
N_Seism
N_Seism
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
Figure 5.85. Brittle check: static nonlinear analysis,
OPCM, Ggrav vs. Gseism
All Elements
All Beams
Roof Col
Figure 5.86. Brittle check: static nonlinear
analysis, EC8, Ggrav vs. Gseism
Figure 5.87 illustrates the comparison between the results obtained considering the theoretical
φy (Eqs. (3.9) – (3.10)) and the empirical φy (Eqs. (3.11) to (3.14)). The chart refers to the
percentages of members which do not satisfy the verification according only to Eurocode 8
(the expression suggested in the Italian Seismic Code does not depend on φy). Empirical and
theoretical φy yield very close (practically identical) results. Therefore, for the Sede
Comunale, the empirical formulas may be applied without any loss of accuracy.
Figure 5.88 shows the comparison between the results obtained using the correct definition of
Ls (Eq. (3.1)) and Ls = L/2. The chart refers to the percentages of members which do not
Seismic Code does not depend on Ls). The use of Ls = L/2 yields a very small (practically
negligible) increase in the percentage of members which do not satisfy the verification with
respect to the percentage obtained considering Eq. (3.1). Hence, for the Sede Comunale, the
Figure 5.89 shows the comparison between the results obtained from the Italian Seismic Code
(using both seismic and gravity N) and from Eurocode 8. Considering the more complex
mechanisms according to the recommendations of the Italian Seismic Code (considering both
seismic and gravity N) yields results which grossly underestimate the percentage of members
119
which do not satisfy the requirements. Therefore, for the Sede Comunale, the procedure
suggested in Eurocode 8 is recommended.
Figure 5.90 shows the comparison between the results obtained using the correct approach
(seismic N; Ls from Eq. (3.1); theoretical φy) and the simplified approach (gravity N; Ls = L/2;
empirical φy) of the assessment procedure suggested in Eurocode 8. The simplified approach
yields a small increase in the percentage of members which do not satisfy the verification with
respect to the correct approach, leading to a safe side assessment. Hence, for the Sede
Comunale, the procedure may be simplified.
100
100
FI_Theoric
Ls=M/V
90
FI_Fardis
90
80
FI_Priestley
80
Ls=L/2
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
EC8, different φy
All Elements
All Beams
Roof Col
EC8, Ls=M/V vs. Ls=L/2
BRITTLE CHECK - STATIC NONLIN.
100
100
OPCM; N_Grav
90
90
N_Grav; Ls=L/2; FI_Fardis
80
N_Seism; Ls=M/V; FI_Theoric
OPCM; N_Seism
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
All Elements
All Beams
Roof Col
The nonlinear dynamic analysis of the Sede Comunale was carried out on the base of eight
time history analyses, each one preformed with an accelerogram acting in one horizontal
principal direction and the same accelerogram, but with the intensity reduced to 30%, applied
in the orthogonal principal direction. Each accelerogram, therefore, allowed to perform two
time history analysis. Hence, in this work, four accelerograms were selected. These
accelerograms were artificially generated, according to the procedure and the attenuation law
proposed by Sabetta and Pugliese [1996]. The magnitude considered to select the
accelerograms varies from 6.0 to 7.0. The generated time histories were, then, modified to
120
better match the shape of the Code elastic spectrum at SD LS (Figure 5.91). The vertical
acceleration was not applied, as it was not required by the Codes.
It may be useful to remind that the reason why eight time history analyses were performed is
that, according to both Codes, carrying out at least seven time history analyses allows to
assess the structural members on the base of the average (instead of the maximum) peak
response quantities. However, since the capacity is not a fixed value but depends on the
demand, in this work the assessment was based on the average of the maximum values of
|Di/Ci| obtained from each time history analysis.
In order to perform the nonlinear dynamic analysis:
•
the masses were distributed along the structural members.
•
The equivalent viscous damping matrix C was assumed proportional only to
instantaneous tangent stiffness K, updated at each step but not at every iteration
(Figure 5.92):
[C ] = a1 [K ] where a1 = T1ξ
π.
Assuming T1 ≈ 0.8s and ξ = 0.05 ⇒ a1 = T1ξ π = (0.8 ⋅ 0.05) π = 0.0127 .
SPEUDO-ACCELERATION 5% DAMPED SPECTRA (100%)
1.3
1.2
Mag=6.0
Mag=6.5
Mag=6.5_long
Mag=7.0
Mean
Target
PSEUDO-ACCELERATION [g]
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
T [s]
2
2.5
Figure 5.91. 5% damped response spectra of
artificial accelerograms (100% intensity)
3
Figure 5.92. Equivalent viscous damping
properties
The deformed shapes corresponding to the main accelerogram (100% of the intensity) applied
in X and Y are illustrated in Figure 5.93 - Figure 5.94 and in Figure 5.95 - Figure 5.96,
respectively.
In analogy with both eigenvalue and static nonlinear analyses, the deformed shapes denote a
pure translation toward X when the main accelerogram is applied in X, while both translation
toward Y and rotation around Z are observed when the main accelerogram is applied in Y.
This behaviour is confirmed by the time history of the displacement of the control node
(chosen at the centre of mass of the top floor). In fact, applying the accelerogram with its full
intensity in X and scaled to 30% in Y, the maximum absolute values of the control node
displacement are 147 mm in X and 21 mm in Y (Figure 5.97). This implies nearly a pure
translation towards X. Applying the accelerogram with its full intensity in Y and scaled to
30% in X, instead, the maximum absolute values of the control node displacement are 58 mm
121
in X and 67 mm in Y (Figure 5.98). This confirms that the structure experiences both
translation towards Y and rotation around Z.
Figure 5.93. Main accelerogram in X, X-dir view
Figure 5.94. Main accelerogram in X, Y-dir view
Figure 5.95. Main accelerogram in Y, Y-dir view
Figure 5.96. Main accelerogram in Y, X-dir view
CONTROL NODE DISPLACEMENT
80
120
DISPL. IN X
90
DISPL. IN Y
DISPL. IN X
60
DISPL. IN Y
40
60
DISPL [mm]
DISPL [mm]
150
30
0
-30
-60
20
0
-20
-40
-90
-60
-120
-150
-80
0
2
4
6
8
10
12
14
16
T [s]
Figure 5.97. Control node displ: main acc. in X
0
2
4
6
8
10
12
14
16
T [s]
Figure 5.98. Control node displ: main acc. in Y
theoretical (Eq. (3.22)) θu, with the same aims described for the assessment based on the static
nonlinear analysis (Sec. 5.2.2).
Figure 5.99 and Figure 5.100 show the comparison between the percentages of members
which do not satisfy the verification according to both Codes. Although the formulas
proposed in the Italian Seismic Code and in Eurocode 8 are very similar, Eurocode 8 suggests
more complex calculations (Sec. 5.1.3). Concerning both empirical and theoretical θu, the
results obtained from Eurocode 8 are very close to those obtained from the Italian Seismic
Code. Hence, for the Sede Comunale, there is no need to use the procedure suggested in
122
Eurocode 8 and the use of the simpler procedure proposed in the Italian Seismic Code is
recommended.
DUCTILE CHECK - DYNAMIC NONLIN. - EMP. FORM.
DUCTILE CHECK - DYNAMIC NONLIN. - THEOR. FORM.
100
100
90
OPCM
90
OPCM
80
EC8
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
Figure 5.99. Ductile check: dynamic nonlinear an.,
All Elements
All Beams
Roof Col
Figure 5.100. Ductile check: dynamic nonlin. an.,
to the theoretical θu (Eq. (3.22)) and to both Codes. The theoretical φy yields slightly
conservative results with respect to the empirical φy. The results are practically identical
concerning beams, while a difference (in the order of 10%) can be seen for columns. Since the
differences among the results obtained through the different approaches are small, for the
Sede Comunale, the empirical formulas calibrated for new seismically designed structures
may be applied without any loss of accuracy.
DUCTILE CHECK - DYN. NONLIN. - THEOR. FORM. - EC8
DUCTILE CHECK - DYN. NONLIN. - THEOR. FORM. - OPCM
100
100
90
FI_Theoric
90
FI_Theoric
FI_Fardis
FI_Fardis
80
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
FI_Priestley
0
All Elements
All Beams
Roof Col
theoretical form, OPCM, different φ
All Elements
All Beams
Roof Col
theoretical form, EC8, different φ
(Eq. (3.15)) and theoretical (Eq. (3.22)) θu. Concerning the empirical θu, Ls = L/2 yields
results very close to Eq. (3.1). Therefore, the procedure may be simplified. Concerning the
theoretical θu, instead, the decrease in the percentage of members which do not satisfy the
123
verification obtained using Ls = L/2 is huge (even larger than 100%). This shows that Eq.
(3.22) is very sensitive to the value of Ls.
DUCTILE CHECK - DYN. NONLIN. - EMP. FORM. - EC8
DUCTILE CHECK - DYN. NONLIN. - EMP. FORM. - OPCM
100
100
90
80
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
Roof Col
All Elements
All Beams
Roof Col
100
100
80
Ls=L/2
0
All Elements
90
Ls=M/V
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
Roof Col
All Elements
All Beams
Roof Col
Figure 5.107 and Figure 5.108 show the comparison between the results obtained using the
correct procedure (seismic N; Ls from Eq. (3.1); theoretical φy) and the simplified procedure
(gravity N; Ls = L/2; empirical φy). The charts refer to the percentages of members which do
not satisfy the verification according to the theoretical θu (Eq. (3.22)) and to both Codes. The
decrease in percentage of members which do not satisfy the verification obtained using the
simplified approach is noticeable and can be ascribed mainly to the fact that Eq. (3.22) is very
sensitive to the value of Ls. Figure 5.109 and Figure 5.110 show that the results obtained
applying both correct and simplified approaches of the empirical θu and the simplified
approach of the theoretical θu are close to each other, but very different from those obtained
considering the correct approach of the theoretical θu. The deformed shape of Figure 5.93
shows that a soft-storey mechanism is likely to form at the second storey, while the first
storey remains practically undeformed. This behaviour was correctly captured using both
approaches (correct and simplified) of the empirical θu and the simplified approach of the
theoretical θu, while it was not detected using the correct approach of the theoretical θu.
Hence, for the Sede Comunale, concerning the theoretical evaluation of θu, there is the need
of considering Ls = L/2, since Eq. (3.1) yields results too sensitive to the values of Ls.
124
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
theor. form, OPCM, correct vs. simplified
All Elements
DUCTILE CHECK - DYNAMIC NONLIN. - OPCM
100
90
80
All Beams
Roof Col
theor. form, EC8, correct vs. simplified
DUCTILE CHECK - DYNAMIC NONLIN. - EC8
100
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
All Elements
All Beams
Roof Col
yields enough accurate results and is much faster that the bidimensional check.
100
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
empir. form, OPCM, uni- vs. bi-axial bending
All Elements
All Beams
Roof Col
empir. form, EC8, uni- vs. bi-axial bending
125
100
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
All Elements
All Beams
Roof Col
assessment was performed according to both Codes, with the same aims already described for
the static nonlinear analysis (Sec. 5.2.2).
Figure 5.115 illustrates the comparison between the results obtained considering the
theoretical φy (Eqs. (3.9) – (3.10)) and the empirical φy (Eqs. (3.11) to (3.14)). The chart refers
to the percentages of members which do not satisfy the verification according only to
Eurocode 8 (the expression suggested in the Italian Seismic Code does not depend on φy).
Empirical and theoretical φy yield very close results. Therefore, for the Sede Comunale, the
empirical formulas may be applied without any loss of accuracy.
Figure 5.116 shows the comparison between the results obtained using the correct definition
of Ls (Eq. (3.1)) and Ls = L/2. The chart refers to the percentages of members which do not
Seismic Code does not depend on Ls). The use of Ls = L/2 results in a very small (practically
respect to the percentage obtained considering Eq. (3.1). Hence, for the Sede Comunale, the
BRITTLE CHECK - DYNAMIC NONLIN. - EC8
100
100
90
FI_Theoric
90
80
FI_Fardis
80
70
FI_Priestley
Ls=M/V
Ls=L/2
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
Figure 5.115. Brittle check: dynamic nonlinear an.,
EC8, different φy
All Elements
All Beams
Roof Col
Figure 5.117 shows the comparison between the results obtained from the Italian Seismic
Code and Eurocode 8. Considering the approach proposed in Eurocode 8 as more accurate, it
126
is clear that assessing the brittle mechanisms according to the Italian Seismic Code yields
results which grossly underestimate the percentage of members which do not satisfy the
requirements. Hence, for the Sede Comunale, the procedure suggested in Eurocode 8 is
recommended.
yields slightly conservative results (small increase in percentage of members which do not
satisfy the verification with respect to the correct approach). Hence, for the Sede Comunale,
the procedure may be simplified.
BRITTLE CHECK - DYNAMIC NONLIN.
100
100
90
OPCM
90
80
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Roof Col
OPCM vs. EC8
All Elements
All Beams
Roof Col
127
Chapter 6. Assessment of Scuola Elementare Pascoli (Barga)
6 Assessment of the Scuola Elementare Pascoli (Barga)
The assessment of the seismic behaviour of the Scuola Elementare Pascoli located in Barga
has been performed according to all methods of analysis proposed by the two considered
Codes (except for the linear static analysis, since the structure is regular neither in plan nor in
elevation). All analyses were based on 3-D models, as the lack of regularity did not allow to
consider two planar separated models in the two principal directions.
analyses) were performed through the software “SeismoStruct, Version 4.0.3, Built 30”.
6.1 Linear Analyses
Ec = 26357 MPa, computed through Eq. (3.7), where fcm = 30 MPa.
joints as rigid seemed to be unconservative, since it would lead to neglect the shear
deformation of the joint.
The R.C. walls at the basement level were modelled with shell finite elements, characterized
by EI = EIgross (uncracked sections) and thickness = 200 mm.
the slab was calculated as t = (Vc + mVs ) A , where m is the moduli ratio, m = Gs Gc ; Gs and
128
Gc = Ec 2(1 + ν ) = 10982 MPa
G s = E s 2(1 + ν ) = 76923MPa
m = G s Gc = 7.00
For a square slab, L = 6800 mm:
Vc = 6800
40 + 0.002(250 ⋅ 40) = 1088000mm 3
1
42⋅4
3
1442443
[
topping
(
)
ribs
(
)
(
)
(
)]
Vs = 2 ⋅ 6800 6 π 4 + 3400 8 2 π 4 + (3400 + 4533) 12 2 π 4 + 3400 14 2 π 4 ⋅ 0.002 = 3952mm 3
2
t = (Vc + m ⋅ Vs ) A = (1088000 + 7.00 ⋅ 3952) 6800 = 164mm
that:
N
∑
EI =
EI eff ,i
i =1 EI gross ,i
N
EI gross
⎧i = All _ members ⇒ 17.2% EI gross = 4532 ⋅ I gross
⎪⎪i = T − beams ⇒ 12.8% EI
= 3386 ⋅ I gross
= ⎨i = All _ columns ⇒ 23.8%gross
EI gross = 6261 ⋅ I gross
⎪i = Ground _ floor _ columns ⇒ 26.9% EI
⎪
st
⎪⎩i = 1 floor _ columns ⇒ 19.9% EI gross = 5256 ⋅ I gross
For the Scuola Pascoli the values of EIeff /EIgross are much smaller than for the Sede Comunale
(Sec. 5.1.1). The reason of this difference lies in the difference between the values of fcm.
Concerning the Sede Comunale, in fact, since fcm is very small (8.3 MPa), the neutral axis
depths are, in general, large, leading to a large amount of concrete resisting area and, hence, to
large values of EIeff. The opposite happens in the case of the Scuola Pascoli: fcm is large (30
MPa) and the neutral axis depths are small, leading to a small amount of concrete resisting
area and, hence, to small values of EIeff.
129
Figure 6.1 and Figure 6.2 show the model of the Scuola Pascoli realized with SAP2000.
Figure 6.1. Front view of Scuola Pascoli
Figure 6.2. Rear view of Scuola Pascoli
The soil properties which characterize the Scuola Pascoli are very similar to those which
characterize the Sede Comunale. Since for the Sede Comunale the flexibility of the system
soil-foundation was considered negligible with respect to the flexibility of the superstructure,
the soil–structure interaction was not taken into account and fixed foundations were assumed
for the Scuola Pascoli.
The number of modes required to get at least 90% of the modal mass varies from three to
eight. The reason why a so small number of modes is required may be ascribed to the high
stiffness of the slabs (equivalent depth = 163 mm and Ec = 26357 MPa). Hence, each floor
behaves like a rigid body and this simplifies the eigenvalue problem.
Three modes govern the dynamic problem: a pure translational mode in X (modal mass = 4752 %), a mode characterized mainly by translation in Y (modal mass = 87-91 %) but also by
rotation around Z (modal mass = 38-42 %) and a third mode characterized mainly by rotation
around Z (modal mass = 49-50 %) but also by translation in X (modal mass = 40-41 %).
While the first mode in X is a purely translational mode, the second and the third mode are
characterized by both translation and rotation. Although it is not a problem for the dynamic
analyses, it becomes a handicap for the assessment according to the static nonlinear analysis,
since pushover analyses were developed for 2-D systems that experience only translation
toward the pushing direction (Sec. 2.5).
130
EI = 100% EIgross
% Modal % Modal
Mode Period
Mass
Mass
#
[s]
Transl. in Transl. in
X
Y
1
2
3
0.396
0.372
0.360
0.52
0.00
0.40
0.00
0.91
0.00
% Modal
Mass
Rotation
around Z
% Cum.
Modal Mass
Transl. in X
% Cum.
Modal Mass
Transl. in Y
% Cum.
Modal Mass
Rotation
around Z
0.00
0.41
0.50
0.52
0.52
0.92
0.00
0.91
0.91
0.00
0.41
0.91
% Cumul.
Modal Mass
Transl. in X
% Cumul.
Modal Mass
Transl. in Y
% Cumul.
Modal Mass
Rotation
around Z
0.00
0.91
0.91
0.00
0.41
0.91
EI = 50% EIgross
% Modal % Modal
Mode Period
Mass
Mass
#
[s]
Y
X
% Modal
Mass
Rotation
around Z
0.00
0.00
0.51
0.52
0.91
0.41
0.00
0.92
0.50
% Cumul.
Mode Period
Mass
Mass
Mass
Modal Mass
#
[s]
Transl. in Transl. in Rotation
Transl. in X
around Z
Y
X
0.00
0.00
0.52
1
0.934
0.52
0.00
0.52
2
0.881
0.91
0.42
0.00
0.92
3
0.852
0.40
0.50
1
2
3
0.556
0.523
0.506
0.51
0.00
0.40
% Cumul.
Modal Mass
Transl. in Y
0.00
0.91
0.91
% Cumul.
Modal Mass
Rotation
around Z
0.00
0.42
0.91
% Modal % Modal
Mode Period
Mass
Mass
#
[s]
Y
X
0.00
1
0.825
0.50
0.00
2
0.780
0.90
0.00
3
0.756
0.41
% Modal
Mass
Rotation
around Z
0.00
0.40
0.50
% Cumul.
Modal Mass
Transl. in X
% Cumul.
Modal Mass
Transl. in Y
0.50
0.50
0.91
0.00
0.90
0.90
% Cumul.
Modal Mass
Rotation
around Z
0.00
0.41
0.91
% Modal % Modal
Mode Period
Mass
Mass
#
[s]
Y
X
0.00
1
0.814
0.47
0.00
2
0.772
0.87
0.00
3
0.747
0.41
7
0.327
0.09
0.00
0.317
0.00
0.11
8
% Modal
Mass
Rotation
around Z
0.00
0.38
0.49
0.01
0.05
% Cumul.
Modal Mass
Transl. in X
% Cumul.
Modal Mass
Transl. in Y
0.47
0.47
0.88
0.97
0.97
0.00
0.87
0.87
0.88
0.99
% Cumul.
Modal Mass
Rotation
around Z
0.00
0.38
0.87
0.88
0.94
131
mode:
T1, EI gross
T1,0.5 EI gross
T1, EI gross
T1, EI eff
=
=
0.5 EI gross
0.396
= 0.5
= 0.712 , while
EI gross
0.556
EI eff
0.396
= 0.172 .
= 0.424 − 0.486 , while
EI gross
0.814 − 0.934
The first mode in X (mode 1) is shown in Figure 6.3, the first mode in Y (mode 2) in Figure
6.4 and the first rotational mode (mode 3) in Figure 6.5 and Figure 6.6.
Figure 6.3. 1st mode of Scuola Pascoli
Figure 6.4. 2nd mode of Scuola Pascoli
Figure 6.5. 3rd mode of Scuola Pascoli, view in X
Figure 6.6. 3rd mode of Scuola Pascoli, view in Y
Since, for the Scuola Pascoli, accounting for vertical seismic input is not required, only the
132
were checked.
Concerning the first condition of applicability (Sec. 2.2.1), Table 6.2 shows that the value of
ρmax/ρmin reduces when the members stiffness EI decreases. This reduction is not linear with
the reduction of EI. In fact:
= 0.923 , while
0.5 EI gross
EI gross
= 0.495 − 0.519 , while
= 0.5
EI eff
EI gross
= 0.172 .
The reason why ρmax/ρmin undergoes a nonlinear reduction with the reduction of EI has been
already explained in Sec. 5.1.3, concerning the assessment of the Sede Comunale. For the
Scuola Pascoli, assuming EI = 100% EIgross, the periods of the first three modes of interest
are, respectively, T1 = 0.396 s, T2 = 0.372 s and T3 = 0.360 s. All three periods lie in the
plateau of the response spectrum (TC = 0.5 s). Decreasing the members stiffness, the periods
shift, respectively, to T1 = 0.556 s, T2 = 0.523 s, T3 = 0.506 s in the case of EI = 50% EIgross,
and to T1 = 0.814-0-934 s, T2 = 0.772-0.881 s, T3 = 0.747-0.852 s, in the case of EI = EIeff
computed through Eq. (2.8). Therefore, decreasing EI, the spectral ordinates corresponding to
the first three modes of interest decrease too, yielding smaller seismic forces. This reduction
is very limited (practically negligible) for EI = 50% EIgross since all three periods are ≈ TC,
while it is evident for EI = EIeff. A consequence of this reduction is that, assuming EI = EIeff,
the first condition of applicability of linear methods will become less restrictive than the
verification on the base of EI = 50-100% EIgross.
therefore, it will experience ρmax also according to Eurocode 8. Therefore, for the Scuola
Pascoli, there is no reason to consider the joint equilibrium in order to determine the value of
ρmax/ρmin and the simpler and faster procedure suggested in the Italian Seismic Code is
recommended.
Considering that:
•
•
•
133
0% (for EI = EIeff) and 1% (for EI = 100% EIgross) of structural members which do not satisfy
the recommendations. In analogy with the first condition, also the second condition will
become less restrictive if EI = EIeff is assumed, as the shear demands will decrease with
respect to EI = 50-100% EIgross.
FIRST CONDITION
OPCM 3431
11.92
ρmax
2.00
ρmin
ρmax/ρmin
5.96
SECOND CONDITION
EI = 100% EIgross
EC8
11.92
ρmax
1.18
ρmin
ρmax/ρmin
10.14
OPCM 3431
Number of Sections
Not Verified
% Not Verified
926
6
1
EI = 50% EIgross
OPCM 3431
11.00
ρmax
2.00
ρmin
ρmax/ρmin
5.49
OPCM 3431
ρmax
ρmin
ρmax/ρmin
5.90
2.01
2.94
EC8
OPCM 3431
11.00
Number of Sections
ρmax
1.05
Not Verified
ρmin
ρmax/ρmin
10.52
% Not Verified
EC8
OPCM 3431
5.90
Number of Sections
ρmax
1.01
Not Verified
ρmin
ρmax/ρmin
5.82
% Not Verified
926
6
1
926
1
0
OPCM 3431
ρmax
ρmin
ρmax/ρmin
6.19
2.00
3.10
EC8
ρmax
ρmin
ρmax/ρmin
6.19
1.10
5.61
OPCM 3431
Number of Sections
Not Verified
% Not Verified
926
4
0
OPCM 3431
ρmax
ρmin
ρmax/ρmin
6.17
2.00
3.08
EC8
ρmax
ρmin
ρmax/ρmin
6.17
1.00
5.80
OPCM 3431
ρmax
Not Verified
% Not Verified
926
4
0
For the Scuola Pascoli all conditions of applicability of linear methods (first condition
according to both Codes and second condition according to the Italian Seismic Code) were
not satisfied. Therefore, according to the Codes, the results furnished by linear analyses
should be judged as not reliable. Anyway, as already explained for the Sede Comunale (Sec.
5.1.3):
•
concerning the first condition of applicability, considering only ρmax/ρmin to evaluate
the possibility of accepting linear analyses may not be judged as an exhaustive
method.
134
•
Concerning the second condition of applicability, it is very likely that the change in
the load path will be not so remarkable to justify the choice of rejecting the possibility
of using linear analyses since only a very limited number of members (0-1%) do not
satisfy the check.
Therefore, other studies are required to determine if the conclusion to reject linear methods is
unavoidable or if the dynamic linear analysis may be considered useful at least to catch the
global seismic response of the building and to express a judgment about the need of
retrofitting. Although the conditions of applicability are not satisfied, the dynamic linear
analysis was performed, since the principal aim of this work is to compare the different
assessment procedures of Eurocode 8 and Italian Seismic Code at structural member level,
suggesting eventually simplified solutions. The evaluation of the reliability of the different
methods of analysis, in order to determine the need of retrofitting, will be object of future
research works.
theoretical (Eq. (3.22)) θu, with the same aims described for the Sede Comunale (Sec. 5.1.3).
The sensitivity of the assessment procedure to EI is shown in Figure 6.7 and Figure 6.8, in
terms of percentage of members which do not satisfy the verification. The results reflect the
high stiffness of the structure (T1 = 0.396 s, assuming EI = EIgross), due to the fact that fcm is
large (30 MPa), the columns are stocky (L = 400 mm) and the total height of the building is
only 8 m. The consequence is that only a very limited percentage of structural members do
not satisfy the chord rotation check. Hence, showing the results in terms of percentages of
members which do not satisfy the verification is, for the ductile assessment of the Scuola
Pascoli, not particularly meaningful in order to compare different approaches of assessment.
For this reason, all charts will be shown in terms of percentages of the mean values of
Di Ci , where Di and Ci are the chord rotation demand and corresponding capacity of each ith structural member, respectively. Figure 6.9 - Figure 6.10 - Figure 6.11 and Figure 6.12
show that, decreasing EI, the mean values of Di Ci increase sensibly, for both beams and
columns. Therefore, assuming EIeff computed though Eq. (2.8) as the most accurate choice of
EI, it is clear that, for the Scuola Pascoli, both 100% EIgross and 50% EIgross yield
unconservative results.
100
90
100
100%EI
90
50%EI
80
100%EI
50%EI
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
135
80
90
100%EI
100%EI
70
80
50%EI
50%EI
70
60
EI_eff
EI_eff
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Columns
1st Floor Col
2nd Floor Col
70
All Beams
140
100%EI
60
100%EI
120
50%EI
EI_eff
50%EI
EI_eff
50
100
40
80
30
60
20
40
10
20
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
analysis, theoretical form, OPCM, different EI
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
Figure 6.13 - Figure 6.14 - Figure 6.15 and Figure 6.16 show that results close to each other
were obtained considering the mean EIeff computed i) for all members, ii) separately for
beams and columns, or iii) separately for T and rectangular beams and for columns of each
floor. A difference can be appreciated for the columns of the first storey. In fact, computing
the mean EIeff separately for the columns of each floor allows to take into account that the
columns of the first storey are stiffer (and, hence, deform less) because they bear a larger
compressive N than the upper storey. However, since this difference is limited to the first
storey columns while, considering all elements, the results are very close to each other, for the
Scuola Pascoli, using the mean EIeff computed for all members yields quite accurate results.
100
90
EI_eff_uniform
EI_eff_uniform
EI_eff_beam_col
80
120
100
EI_eff_precise
EI_eff_beam_col
EI_eff_precise
70
80
60
50
60
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
136
90
180
EI_eff_uniform
EI_eff_uniform
80
160
EI_eff_beam_col
70
EI_eff_beam_col
140
EI_eff_precise
60
120
50
100
40
80
30
60
20
40
10
20
0
EI_eff_precise
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
Figure 6.17 and Figure 6.18 compare the mean values of Di Ci , computed according to the
Italian Seismic Code and Eurocode 8. Although the formulas proposed in both Codes are very
similar to each other, Eurocode 8 suggests more complex calculations (Sec. 5.1.3).
Concerning the empirical formula of θu (Eqs. (3.15)), since the results obtained from the two
Codes are quite close to each other, there is no need to follow the procedure suggested in
Eurocode 8 and, hence, the simpler approach proposed in the Italian Seismic Code is
recommended. Concerning the theoretical formula of θu (Eq. (3.22)), instead, the mean values
of Di Ci given by Eurocode 8 are larger (i.e. more conservative). Hence, assuming the
procedure suggested in Eurocode 8 as correct, the simpler approach of the Italian Seismic
Code yields quite inaccurate results.
140
90
80
OPCM
70
EC8
120
OPCM
EC8
100
60
50
80
40
60
30
40
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
analysis, theoretical form, EIeff, OPCM vs. EC8
Ls = L/2 is shown from Figure 6.19 to Figure 6.22. The charts refer to the mean values of
Di Ci , computed according to both Codes and both empirical (Eq. (3.15)) and theoretical
(Eq. (3.22)) θu. All charts show that Ls = L/2 yields nearly the same values of Di Ci as Eq.
(3.1). Concerning the assessment performed according to Eq. (3.15), since the differences are
less than 10%, the procedure may be simplified. Concerning the assessment based on (Eq.
(3.22)), instead, it must be considered that for 11 beams θu = 0, since Lpl > Ls. The
contribution of these beams can not be taken into account in the evaluation of the mean values
of Di Ci , since when Ci = 0 , Di Ci → ∞ . Hence, the results obtained using the correct
137
definition of Ls (Eq. (3.1)) and Ls = L/2 (Figure 6.21 and Figure 6.22) are more similar to each
other than if they were shown in terms of percentages of members which do not satisfy the
verification. Anyway, since only 11 members are characterized by theoretical θu = 0, the
contribution of these members may be judged as negligible. Hence, the procedure may be
simplified also if the assessment is performed according to Eq. (3.22).
80
90
Ls=M/V
Ls=M/V
80
70
Ls=L/2
Ls=L/2
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
1st Floor Col
2nd Floor Col
140
Ls=M/V
60
All Columns
70
All Beams
Ls=M/V
120
Ls=L/2
50
100
40
80
30
60
20
40
10
20
0
Ls=L/2
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
The comparison between the results obtained using the gravity and the seismic N is shown
from Figure 6.23 to Figure 6.26. The charts refer to the mean values of Di Ci , computed
according to both Codes and both empirical (Eq. (3.15)) and theoretical (Eq. (3.22)) θu.
Concerning the empirical θu, the gravity and the seismic N yield very close results.
Concerning the theoretical θu, following the recommendations of Eurocode 8, the mean values
of Di Ci obtained assuming the seismic N are larger than assuming the gravity N
(differences up to 25% for the columns). However, it must be remarked that the seismic ΔN
considered in this case is related with the hypothesis of indefinitely elastic behaviour and,
hence, is larger than ΔN obtained when the development of nonlinear mechanisms is
accounted for (Sec. 3.2.2). Therefore, the procedure may be simplified, particularly
concerning the assessment performed according to Eq. (3.15) and also to Eq. (3.22), if the
recommendations of the Italian Seismic Code are followed.
138
80
90
N_Grav
N_Grav
80
70
N_Seism
N_Seism
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
1st Floor Col
2nd Floor Col
140
N_Grav
60
All Columns
empirical form, EIeff, EC8, Ggrav vs. Gseism
70
All Beams
N_Grav
120
N_Seism
50
100
40
80
30
60
20
40
10
20
0
N_Seism
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
(3.10)) and the empirical φy (Eqs. (3.11) to (3.14)) is shown in Figure 6.27 - Figure 6.28. The
charts refer to the mean values of Di Ci , computed according to the theoretical θu (Eq.
(3.22)) and to both Codes. Empirical and theoretical φy yield close results. Therefore, for the
Scuola Pascoli, the empirical formulas calibrated for new seismically designed structures may
be applied without any loss of accuracy.
70
160
FI_Theoric
60
FI_Theoric
140
FI_Fardis
FI_Priestley
120
FI_Fardis
FI_Priestley
50
100
40
80
30
60
20
40
10
20
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
analysis, theoretical form, EIeff, EC8, different φy
139
N; Ls = L/2; empirical φy). The charts refer to the mean values of Di Ci , computed
Concerning the empirical θu, the simplified approach yields a small decrease in the mean
values of Di Ci with respect to the correct approach. Concerning the theoretical θu, instead,
the decrease in the mean values of Di Ci obtained using the simplified approach is
noticeable, particularly if the recommendations of Eurocode 8 are followed. Comparing the
results obtained considering both correct and simplified approach of both empirical and
theoretical θu (Figure 6.33 - Figure 6.34), all approaches of the Italian Seismic Code yield
close results while, considering Eurocode 8, the correct approach of the theoretical θu yields
very conservative results with respect to the other procedures. Therefore, for the Scuola
Pascoli, the assessment of ductile mechanisms based on Eq. (3.15) may be simplified without
any remarkable loss in the accuracy of the results. Concerning Eq. (3.22), instead, the large
values obtained applying the correct approach may be due to the use of ΔN related with the
hypothesis of indefinitely elastic behaviour instead of ΔN obtained accounting for the
development of nonlinear mechanisms (Sec. 3.2.2). The fact that the results obtained from the
simplified approach of Eq. (3.22) are very close to those obtained from both correct and
simplified approaches of Eq. (3.15) may lead to state that the correct approach of Eq. (3.22)
yield too conservative results. Further future investigations, considering ΔN obtained
accounting for the development of nonlinear mechanisms instead of ΔN related with the
hypothesis of indefinitely elastic behaviour, are required to confirm this conclusion.
70
90
80
Ls=L/2; N_Grav
Ls=M/V; N_Seism
60
80
Ls=L/2; N_Grav
70
Ls=M/V; N_Seism
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
All Elements
2nd Floor Col
form, EIeff, OPCM, correct vs. simplified
70
All Columns
1st Floor Col
2nd Floor Col
140
60
All Beams
120
50
100
40
80
30
60
20
40
10
20
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
140
90
80
140
120
70
100
60
50
80
40
60
30
40
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
Figure 6.33. Ductile check: dynamic linear an., EIeff,
OPCM, theor. vs. emp., correct vs simplified
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
Figure 6.35 to Figure 6.38. The charts refer to the mean values of Di Ci , computed
according to both Codes and both empirical (Eq. (3.15)) and theoretical (Eq. (3.22)) θu. All
charts show that the use of the bidimensional failure curve yield an increase in the mean
values of Di Ci of columns (for the beams the verification is only uniaxial) between 26 and
28%. This difference is due to the lack of compactness of the Scuola Pascoli. Therefore, the
use of the bidimensional failure curve improves on the accuracy of the results and, hence, is
recommended.
100
120
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
MONO-AXIAL BENDING
100
70
BI-AXIAL BENDING
80
60
50
60
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
80
70
All Columns
1st Floor Col
2nd Floor Col
90
All Beams
180
MONO-AXIAL BENDING
160
BI-AXIAL BENDING
140
60
120
50
100
40
80
30
60
20
40
10
20
0
MONO-AXIAL BENDING
BI-AXIAL BENDING
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
141
according to both Codes, with the same aims already described for the Sede Comunale (Sec.
5.1.3).
to both Codes. In particular, Figure 6.39 and Figure 6.40 show that, according to both Codes,
the percentage of members which do not satisfy the verification decreases with decreasing EI.
Therefore, assessing the brittle mechanisms using a fixed fraction of EIgross (between 50 and
100%) yields conservative results. Moreover, concerning Eurocode 8, the difference in
percentage of members which do not satisfy the verification assuming 100% EIgross and mean
EIeff (Eq. (2.8)) is less than 10%. Hence, for the Scuola Pascoli, if the recommendations of
Eurocode 8 are followed, assuming a fixed fraction of EIgross between 50% and 100% may be
considered a suitable choice, since the assessment of EIeff according to Eq. (2.8) is much
longer and yields small improvements in the results.
Figure 6.41 and Figure 6.42 show that very similar results were obtained considering the
mean EIeff computed i) for all members, ii) separately for beams and columns, or iii)
separately for T and rectangular beams and for columns of each floor.
100
90
100
100%EI
90
80
100%EI
50%EI
50%EI
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
OPCM, different EI
All Elements
All Columns
1st Floor Col
2nd Floor Col
EC8, different EI
100
100
EI_eff_uniform
EI_eff_uniform
90
90
EI_eff_beam-col
80
All Beams
EI_eff_beam-col
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_precise
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
OPCM, different EI
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
EC8, different EI
142
satisfy the verification according to both Codes. Concerning Eurocode 8, the seismic N yields
a very small (practically negligible) increase in the percentage of members which do not
satisfy the verification with respect to the gravity N. Hence, the procedure may be simplified.
Concerning the Italian Seismic Code, instead, the increase in the percentage of members
which do not satisfy the verification obtained using the seismic N is much larger and can not
be neglected. Although the seismic ΔN used in this case is related with the hypothesis of
indefinitely elastic behaviour and, therefore, is larger than ΔN obtained when the development
of nonlinear mechanisms is considered, such huge difference between the results obtained
considering the seismic N and the gravity N do not allow to simplify the procedure. This
difference is due to the fact that the concrete shear resistance of columns will become null if
the columns experience a tensile N (see Fig. 3.20).
100
100
90
N_Grav
N_Grav
90
N_Seism
N_Seism
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
theoretical φy yield very close (practically identical) results. Therefore, for the Scuola Pascoli,
the empirical formulas may be applied without any loss of accuracy.
100
100
FI_Theoric
90
80
Ls=M/V
90
FI_Fardis
Ls=L/2
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
143
Seismic Code does not depend on Ls). The use of Ls = L/2 yields a small increase (about 15%)
in the percentage of members which do not satisfy the verification with respect to the
percentage obtained considering Eq. (3.1), leading to a safe side assessment. Hence, for the
Scuola Pascoli, the procedure may be simplified.
Figure 6.47 shows the comparison between the results obtained following the procedure
suggested in the Italian Seismic Code (both seismic and gravity N) and the formula proposed
in Eurocode 8. Considering the more complex approach proposed in Eurocode 8 as the most
accurate, it is clear that assessing the brittle mechanisms according to the Italian Seismic Code
with both gravity and seismic N yields results which grossly underestimate the percentage of
members which do not satisfy the requirements. Therefore, for the Scuola Pascoli, the use of
the procedure suggested in Eurocode 8 is recommended.
(seismic N; Ls from Eq. (3.1); theoretical φy; mean EIeff) and the simplified approach (gravity
N; Ls = L/2; empirical φy; 50% EIgross) of the assessment procedure suggested in Eurocode 8.
The simplified approach yields a small increase (about 15%) in the percentage of members
which do not satisfy the verification with respect to the correct approach, leading to a safe
side assessment. Hence, for the Scuola Pascoli, the procedure may be simplified.
90
80
100
OPCM; N_Grav
100
OPCM; N_Seism
90
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
Figure 6.48. Brittle check: dynamic linear an.,
144
•
concrete compressive strength, fc = 30 MPa.
•
•
•
•
•
•
Figure 6.49. Concrete: nonlinear confinement
model
Figure 6.50. Reinforcement steel: bilinear (strain
hardening) model
The basement structural walls were modelled using frame elements rigidly connected to the
corner joints.
145
In the following, the procedure adopted for estimating the dimensions of the brace is shown
for the same slab already considered in Sec. 6.1.1 (square slab, L = L’ = 6800 mm).
The thickness of the equivalent slab computed for the SAP model was determined as 164 mm
(Sec. 6.1.1). Therefore:
(
)
I = 6800 3 ⋅ 164 12 = 4.299 ⋅ 1012 mm 4
As = 5 6 A = 5 6 ⋅ (6800 ⋅ 164) = 929735mm 2
Gc = 10982 MPa
[
(
]
)
K b = 6800 3 12 ⋅ 26357 ⋅ 4.299 ⋅ 1012 + 6800 (929735 ⋅ 10982 )
−1
= 1114545 N / mm
Once known Kb and l = L' 2 + L2 = 6800 2 + 6800 2 = 9617 mm , the area of the brace was
computed as: Ab = 1114545 ⋅ 9617 / 26357 = 406653mm 2 .
Considering a circular section, D = 719 mm was determined.
elastic (EI = 50% EIgross = 13179 MPa). The results are shown in Table 6.3. The differences
are totally negligible. Hence, the SeismoStruct model was judged to be consistent with the
SAP model.
Periods [s]
I Mode
II Mode
III Mode
SeismoStruct
0.556
0.523
0.500
SAP
0.556
0.523
0.494
The nonlinear static analysis was based on eight pushover analyses (“modal” and “uniform”
pattern of lateral forces, applied in positive and negative X and Y directions). In order to
perform the analysis:
•
•
•
146
For the “uniform” pattern of lateral forces in X (Figure 6.55 - Figure 6.56):
•
•
d *y = 42mm (Eq. (2.14)); K * = 171445MPa (Eq. (2.15)); T * = 0.612 s (Eq. (2.16)).
•
•
Γx = 0.865 (Eq. (2.13)).
•
•
For the “uniform” pattern of lateral forces in Y (Figure 6.59 - Figure 6.60):
•
•
147
•
•
Γy = 1.217 (Eq. (2.13)).
•
d *y = 43mm (Eq. (2.14)); K * = 135126 MPa (Eq. (2.15)); T * = 0.682 s (Eq. (2.16)).
•
8000
8000
7000
7000
Vb,NC
6000
Vb (kN)
6000
Vb (kN)
Vb,SD
5000
4000
3000
5000
Vb,LD
4000
3000
2000
2000
1000
1000
Δ
0
*
Δ
y
0
50
*
Δ LD
0
m
100
150
200
250
300
Δ NC
Δ SD
0
50
100
250
300
8000
7000
7000
6000
6000
5000
Vb,SD
Vb,NC
5000
Vb (kN)
Vb (kN)
200
4000
3000
Vb,LD
4000
3000
2000
2000
1000
1000
Δ
0
0
Δ
*
y
50
Δ LD
*
m
0
100
150
200
250
0
300
Δ NC
Δ SD
50
100
150
200
250
300
8000
8000
7000
7000
Vb,SD
Vb,NC
6000
Vb (kN)
6000
Vb (kN)
150
5000
4000
3000
5000
Vb,LD
4000
3000
2000
2000
1000
1000
Δ
0
0
20
Δ
*
y
40
60
80
*
Δ LD
0
m
100
120
140
160
180
200
0
Δ SD
50
Δ NC
100
150
200
148
8000
6000
Vb,NC
7000
5000
Vb,SD
6000
Vb (kN)
Vb (kN)
4000
3000
2000
Vb,LD
5000
4000
3000
2000
1000
1000
Δ
0
0
*
Δ
y
50
Δ LD
*
0
m
100
150
0
200
50
Δ SD
100
Δ NC
150
200
250
CAPACITY CURVES Y - MDOF
8000
8000
VSD,UNIF
7000
7000
MODAL
6000
VSD,UNIF
VSD,MOD
5000
Vb (kN)
Vb (kN)
6000
VSD,MOD
UNIFORM
4000
3000
2000
5000
4000
UNIFORM
3000
MODAL
2000
1000
ΔSD,MOD
1000
ΔSD,UNIF
0
ΔSD,UNIF
0
0
50
100
150
200
250
300
350
0
50
ΔSD,MOD
100
150
200
250
The Scuola Pascoli is roughly symmetric but not compact. The centre of mass is located in the
central part of the structure, which is stiffer than the lateral arms. Therefore, it is likely that
the joints located in the perimetral arms of the structure experience larger displacements than
the centre of mass. This is the reason why the “transformation factor” Γx is smaller than 1 (Γx
= 0.865). Hence, for the Scuola Pascoli, the choice to locate the control node at the centre of
mass of the top floor may be questionable. Further future investigations are needed to
recognize which is the best control node location in order to assess the response in the most
reliable way and to understand also if the results of the assessment are sensitive to the choice
of the control node.
Although for the Scuola Pascoli the correctness of the results of the static nonlinear analysis is
doubtful, in this work the procedures suggested by both Codes were followed and the
assessment based on the static nonlinear analysis was performed.
described for the assessment based on the dynamic linear analysis (Sec. 5.1.3). Since the
structure is very stiff, only a limited percentage of structural members do not satisfy the chord
rotation check. Hence, showing the results in terms of percentages of members which do not
149
satisfy the verification is not particularly meaningful in order to compare different approaches
of assessment. For this reason, all charts will be shown in terms of percentages of the mean
values of Di Ci , where Di and Ci are the chord rotation demand and corresponding capacity
of each i-th structural member, respectively.
Figure 6.65 - Figure 6.66 compare the mean values of Di Ci , computed according to the
similar to each other, Eurocode 8 suggests more complex calculations (Sec. 5.1.3). Moreover,
also the way to compute the demand is different. In fact, the Italian Seismic Code proposes to
consider separately the results obtained from each pushover, while Eurocode 8 suggests to
consider each demand quantity as the outcome of the combination of the values furnished by a
pushover in X and a pushover in Y (Eq. (2.23)). Concerning the empirical θu (Eq. (3.15)), the
results obtained from Eurocode 8 are slightly conservative and very close to those obtained
from the Italian Seismic Code. Therefore, for the Scuola Pascoli, there is no need to use the
procedure suggested in Eurocode 8 and, hence, the simpler procedure proposed in the Italian
Seismic Code is recommended. Concerning the theoretical θu, instead, the mean values of
Di Ci obtained from Eurocode 8 are larger than those obtained from the Italian Seismic
Code. This difference is due, in particular, to the way of computing the seismic demand,
since, as already shown for the assessment based on the dynamic linear analysis, the more
complex way to assess the capacity proposed in Eurocode 8 yields results close to those
obtained applying the recommendations of the Italian Seismic Code. Further future
investigations are needed to determine whether Eq. (2.23) yields more reliable demands
values than those obtained considered each pushover separately.
100
120
90
OPCM
80
EC8
OPCM
100
EC8
70
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
Figure 6.65. Ductile check: static nonlinear
analysis, empirical form, OPCM vs. EC8
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
The charts refer to the mean values of Di Ci , computed according to the theoretical θu (Eq.
(3.22)) and to both Codes. Empirical and theoretical φy yield close results (a difference < 15%
can be appreciated if the recommendations of Eurocode 8 are followed). Hence, for the
Scuola Pascoli, the empirical formulas calibrated for new seismically designed structures may
be applied without any sensible loss of accuracy.
150
(Eq. (3.22)) θu. Concerning the empirical θu (Figure 6.69 - Figure 6.70), the use of Ls = L/2
yields results close to those obtained considering Eq. (3.1). Therefore, the procedure may be
simplified. Concerning the theoretical θu (Figure 6.71 - Figure 6.72), instead, the difference is
huge (using Ls = L/2 there is an underestimation > 50%). This difference is due to the fact that
Eq. (3.22) is very sensitive to the value of Ls. This result is more evident than for the linear
analyses because the inelastic Ls are likely to change with respect to the elastic Ls, possibly
resulting in very small values, leading to θu = 0.
120
FI_Theoric
FI_Theoric
100
100
90
FI_Fardis
80
FI_Priestley
FI_Fardis
FI_Priestley
70
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
analysis, theoretical form, OPCM, different φy
All Elements
100
All Beams
All Columns
1st Floor Col
2nd Floor Col
80
90
Ls=M/V
80
Ls=L/2
Ls=M/V
70
Ls=L/2
60
70
50
60
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
100
All Columns
1st Floor Col
2nd Floor Col
120
All Beams
100
Ls=M/V
90
Ls=M/V
Ls=L/2
80
Ls=L/2
70
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
151
charts show that gravity and seismic N yield very close results. Hence, concerning both
empirical and theoretical formulas of θu of both Codes, the procedure may be simplified. This
conclusion clashes with the results based on the dynamic linear analysis. This can be justified
considering that the seismic ΔN experienced by the structure in a linear analysis is, in general,
100
80
90
N_Grav
80
N_Seism
N_Grav
70
N_Seism
60
70
50
60
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
120
100
All Beams
All Columns
1st Floor Col
2nd Floor Col
100
N_Grav
90
N_Grav
N_Seism
80
N_Seism
70
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
shown from Figure 6.77 to Figure 6.82. The charts refer to the mean values of Di Ci ,
computed according to both Codes and both empirical (Eq. (3.15)) and theoretical (Eq. (3.22))
θu. Concerning the empirical θu, the simplified and the correct approaches yield close results.
Therefore, for the Scuola Pascoli, the assessment procedure may be simplified. Concerning
the theoretical θu, instead, the simplified approach leads to an underestimation of the mean
values of Di Ci larger than 50%. This difference is due mainly to the fact that Eq. (3.22) is
very sensitive to the value of Ls. Comparing the results obtained considering both correct and
simplified approach and both empirical and theoretical θu (Figure 6.81 - Figure 6.82), it
appears that the results obtained applying both approaches of the empirical formula and the
simplified approach of the theoretical formula are close to each other, but different from those
152
obtained considering the correct approach of the theoretical formula. The difference is huge
particularly for beams and if the recommendations of Eurocode 8 are followed. Figure 6.81
and Figure 6.82 confirm that the correct approach of Eq. (3.22) is too sensitive to the value of
Ls. Therefore, for the Scuola Pascoli, concerning the theoretical evaluation of θu, there is the
need of considering Ls = L/2 instead of Eq. (3.1).
100
80
90
Ls=L/2; N_Grav
80
Ls=M/V; N_Seism
Ls=L/2; N_Grav
70
Ls=M/V; N_Seism
60
70
50
60
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
100
120
90
80
100
70
80
60
50
60
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
All Elements
2nd Floor Col
100
All Columns
1st Floor Col
2nd Floor Col
DUCTILE CHECK - STATIC NONLIN. - EC8
DUCTILE CHECK - STATIC NONLIN. - OPCM
120
All Beams
100
90
80
70
80
60
50
60
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
OPCM, theor. vs. emp., correct vs. simpl.
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
EC8, theor. vs. emp., correct vs. simpl.
The comparison between the results obtained using the conventional uniaxial check and
assuming a bidimensional failure curve (Eq. (3.27)) is shown from Figure 6.83 to Figure 6.86.
153
The charts refer to the mean values of Di Ci , computed according to both Codes and both
empirical (Eq. (3.15)) and theoretical (Eq. (3.22)) θu. All charts show that the definition of a
bidimensional failure curve does not improve on the accuracy of the results. This conclusion
clashes with the results based on the dynamic linear analysis. The reason of this difference
may be explained as follows. A pushover analysis is performed with unidirectional lateral
forces and, hence, it is likely that the degradation of stiffness in the pushing direction will be
faster than in the orthogonal direction. Therefore, the displacements/deformations in the
direction orthogonal to that of the lateral forces will be, in general, small. In a linear analysis,
instead, the stiffness is constant and, hence, the difference between the uniaxial and the
biaxial check is more evident.
100
80
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
MONO-AXIAL BENDING
70
BI-AXIAL BENDING
60
70
50
60
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
120
All Columns
1st Floor Col
2nd Floor Col
120
MONO-AXIAL BENDING
MONO-AXIAL BENDING
100
All Beams
100
BI-AXIAL BENDING
80
80
60
60
40
40
20
20
0
BI-AXIAL BENDING
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
sensitivity to EI) already described for the dynamic linear analysis in Sec. 5.1.3.
154
satisfy the verification according to both Codes. Concerning Eurocode 8, gravity and seismic
N yield nearly identical results. Regarding the Italian Seismic Code, the seismic N yields a
small increase in the percentage of members which do not satisfy the verification with respect
to the gravity N (differences < 10%). For the Scuola Pascoli, hence, the procedures suggested
by the two Codes may be simplified. This conclusion clashes with the results based on the
dynamic linear analysis. The fact that the results obtained using the static nonlinear analysis
are less sensitive to the choice of N can be justified considering that the seismic ΔN
experienced by the structure in a linear analysis is, in general, (much) larger than ΔN obtained
through a nonlinear analysis.
100
90
100
N_Grav
90
N_Seism
N_Grav
N_Seism
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
EC8, Ggrav vs. Gseism
φy (Eqs. (3.9) – (3.10)) and the empirical φy (Eqs. (3.11) to (3.14)). The chart refer to the
theoretical φy yield very close (practically identical) results. Therefore, for the Scuola Pascoli,
the empirical formulas may be applied without any loss of accuracy.
Ls (Eq. (3.1)) and Ls = L/2. The chart refer to the percentages of members which do not satisfy
the verification according only to Eurocode 8 (the expression suggested in the Italian Seismic
Code does not depend on Ls). The use of Ls = L/2 yields a small increase (< 10%) in the
percentage of members which do not satisfy the verification with respect to Eq. (3.1). Hence,
for the Scuola Pascoli, the procedure may be simplified.
which do not satisfy the requirements. Therefore, for the Scuola Pascoli, the procedure
155
respect to the correct approach, leading to a safe side assessment. Hence, for the Scuola
Pascoli, the procedure may be simplified.
100
100
FI_Theoric
90
80
90
FI_Fardis
Ls=M/V
Ls=L/2
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
EC8, different φy
All Elements
1st Floor Col
2nd Floor Col
100
OPCM; N_Grav
90
OPCM; N_Seism
80
All Columns
100
90
All Beams
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
The nonlinear dynamic analysis of the Scuola Pascoli was carried out on the base of eight
156
•
•
(Figure 6.94):
[C ] = a1 [K ] where a1 = T1ξ
π.
Assuming T1 = 0.396s and ξ = 0.05 ⇒ a1 = T1ξ π = (0.396 ⋅ 0.05) π = 0.0063 .
1.1
1
Mag=6.0
Mag=6.5
Mag=6.5_long
Mag=7.0
Mean
Target
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
T [s]
2
2.5
3
Figure 6.94.Equivalent viscous damping properties
respectively.
In analogy with both eigenvalue and static nonlinear analyses, the deformed shape is mainly
characterized by a translation toward X when the main accelerogram is applied in X, and by a
translation toward Y when the main accelerogram is applied in Y. This behaviour is
confirmed by the time history of the displacement of the control node (chosen at the centre of
mass of the top floor). In fact, applying the accelerogram with its full intensity in X and scaled
to 30% in Y, the maximum absolute values of the control node displacement are 70 mm in X
and 21 mm in Y (Figure 6.99). Applying the accelerogram with its full intensity in Y and
scaled to 30% in X, instead, the maximum absolute values of the control node displacement
are 19 mm in X and 73 mm in Y (Figure 6.100).
157
80
80
DISPL [mm]
40
DISPL. IN X
60
DISPL. IN Y
40
DISPL [mm]
60
20
0
-20
DISPL. IN X
DISPL. IN Y
20
0
-20
-40
-40
-60
-60
-80
-80
0
2
4
6
8
10
12
14
16
T [s]
0
2
4
6
8
10
12
14
16
T [s]
nonlinear analysis (Sec. 5.2.2). Since the structure is very stiff, only a limited percentage of
structural members do not satisfy the chord rotation check. Hence, showing the results in
terms of percentages of members which do not satisfy the verification is, for the ductile
assessment of the Scuola Pascoli, not particularly meaningful in order to compare different
approaches of assessment. For this reason, all charts will be shown in terms of percentages of
the mean values of Di Ci , where Di and Ci are the chord rotation demand and corresponding
capacity of each i-th structural member, respectively.
Figure 6.101 and Figure 6.102 show the comparison between the mean values of Di Ci ,
computed according to both Codes. Although the formulas proposed in the Italian Seismic
Code and in Eurocode 8 are very similar, Eurocode 8 suggests more complex calculations
(Sec. 5.1.3). Concerning the empirical θu, the results obtained from Eurocode 8 are close to
those obtained from the Italian Seismic Code. Hence, the use of the simpler procedure
proposed in the Italian Seismic Code is recommended. Concerning the theoretical θu, instead,
Eurocode 8 yields conservative results and therefore its use is suggested.
(3.22)) and to both Codes. The theoretical and empirical φy yield very close results. Hence, for
the Scuola Pascoli, the empirical formulas calibrated for new seismically designed structures
158
60
70
OPCM
60
50
EC8
OPCM
EC8
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
1st Floor Col
2nd Floor Col
70
FI_Theoric
50
All Columns
60
All Beams
FI_Theoric
60
FI_Fardis
FI_Priestley
FI_Fardis
FI_Priestley
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
(Eq. (3.22)) θu. Concerning the empirical θu, Ls = L/2 yields results very close to Eq. (3.1).
Hence, the procedure may be simplified. Concerning the theoretical θu, instead, the difference
is large (using Ls = L/2 there is an underestimation ≈ 30%, if the recommendations of the
Italian Seismic Code are followed). This difference is due to the fact that Eq. (3.22) is very
sensitive to the value of Ls. This difference is larger than for the linear analyses but smaller
than for the nonlinear static analysis.
60
60
Ls=M/V
50
Ls=M/V
50
Ls=L/2
40
40
30
30
20
20
10
10
0
Ls=L/2
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
159
70
60
Ls=M/V
50
Ls=M/V
60
Ls=L/2
Ls=L/2
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
All Elements
2nd Floor Col
theor. form, OPCM, Ls=M/V vs. Ls=L/2
All Beams
All Columns
1st Floor Col
2nd Floor Col
theor. form, EC8, Ls=M/V vs. Ls=L/2
(gravity N; Ls = L/2; empirical φy). The charts refer to the mean values of Di Ci , computed
according to the theoretical θu (Eq. (3.22)) and to both Codes. The simplified approach leads
to underestimate the results obtained with the correct approach (the difference is ≈ 30%, if the
recommendations of the Italian Seismic Code are followed). This difference (much smaller
than for the static nonlinear analysis) is due mainly to the fact that Eq. (3.22) is very sensitive
to the value of Ls.
empirical and theoretical θu (Figure 6.111 - Figure 6.112), it appears that, concerning the
assessment of the columns response, all results are quite close to each other (the differences
are < 15%). Concerning beams, instead, both approaches of the empirical θu and the
simplified approach of the theoretical θu yield results close to each other, but different from
those obtained considering the correct approach of the theoretical θu, which overestimate the
results (difference ≈ 30% for both Eurocode 8 and Italian Seismic Code). It confirms that the
correct approach of Eq. (3.22) is too sensitive to the value of Ls. Therefore, for the Scuola
Pascoli, concerning the theoretical evaluation of θu, there is the need of considering Ls = L/2
instead of Eq. (3.1).
60
70
50
60
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
160
70
60
80
70
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
charts show that, for the considered building, the definition of a bidimensional failure curve
does not improve on the accuracy of the results. Therefore, the conventional uniaxial check is
recommended for the assessment of the Scuola Pascoli, as it yields enough accurate results
and is much faster that the bidimensional check.
60
MONO-AXIAL BENDING
50
70
60
BI-AXIAL BENDING
MONO-AXIAL BENDING
BI-AXIAL BENDING
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Columns
1st Floor Col
2nd Floor Col
70
60
MONO-AXIAL BENDING
50
All Beams
60
BI-AXIAL BENDING
MONO-AXIAL BENDING
BI-AXIAL BENDING
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
161
the static nonlinear analysis in Sec. 5.2.2.
Empirical and theoretical φy yield very close results. Therefore, for the Scuola Pascoli, the
Seismic Code does not depend on Ls). The use of Ls = L/2 results in a small increase (≈ 10%)
in the percentage of members which do not satisfy the verification with respect to Eq. (3.1),
leading to a safe side assessment. Hence, for the Scuola Pascoli, the procedure may be
simplified.
100
90
100
90
FI_Theoric
Ls=M/V
Ls=L/2
80
FI_Fardis
80
70
FI_Priestley
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
EC8, different φy
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
requirements. Hence, for the Scuola Pascoli, the procedure suggested in Eurocode 8 is
recommended.
yields slightly conservative results (small increase, ≈ 10%, in percentage of members which
do not satisfy the verification with respect to the correct approach). Hence, for the Scuola
Pascoli, the procedure may be simplified.
162
100
100
90
OPCM
90
80
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
Figure 6.119, Brittle check: dynamic nonlinear an.,
OPCM vs. EC8
All Elements
All Beams
All Columns
1st Floor Col
2nd Floor Col
163
Chapter 7. Assessment of Scuola Media Inferiore Puccetti (Gallicano)
7 Assessment of the Scuola Media Inferiore Puccetti (Gallicano)
The assessment of the seismic behaviour of the Scuola Media Inferiore Puccetti located in
Gallicano has been performed according to all methods of analysis proposed by the two
considered Codes (except for the linear static analysis, since the structure is regular neither in
plan nor in elevation). All analyses were based on 3-D models, as the lack of regularity did
not allow to consider two planar separated models in the two principal directions.
7.1 Linear Analyses
Ec = 22230 MPa, computed through Eq. (3.7), where fcm = 18 MPa.
Gc = E c 2(1 + ν ) = 9263MPa
164
G s = E s 2(1 + ν ) = 76923MPa
m = G s Gc = 8.30
For a square slab, L = 3500 mm:
Vc = 3500
⋅ 3500
⋅3
40 + 5(3500 ⋅ 22544) = 884520000mm 3
144
244
1442443
[
topping
(
)
ribs
]
Vs = 14 ⋅ 2 ⋅ 2500 ⋅ 6 π 4 + 2 ⋅ 108 ⋅ 3500 = 2735203mm 3 .
2
t = (Vc + m ⋅ Vs ) A = (884520000 + 8.30 ⋅ 2735203) 3500 2 = 74mm
that:
N
∑
EI =
EI eff ,i
i =1 EI gross ,i
N
EI gross
⎧i = All _ members ⇒ 23.1% EI
⎪
⎪i = Re c _ beams ⇒ 22.7% EI gross = 5042 ⋅ I gross
⎪i = T − beams ⇒ 18.8% EI gross = 4171 ⋅ I gross
⎪
= ⎨i = All _ columns ⇒ 27.7% EI gross = 6151 ⋅ I gross
⎪i = Ground _ floor _ columns ⇒ 31.8% EI gross = 7080 ⋅ I gross
⎪i = 1st floor _ columns ⇒ 29.9% EI
⎪
nd
For the Scuola Puccetti the values of EI eff EI gross are smaller than for the Sede Comunale
(Sec. 5.1.1) and larger than for the Scuola Pascoli (Sec. 6.1.1). The reason of these differences
lies in the difference between the values of fcm and was already explained in Sec. 6.1.1.
Figure 7.1 shows the model of the Scuola Puccetti realized with SAP2000.
165
Figure 7.1. View of Scuola Puccetti
The soil properties which characterize the Scuola Puccetti are very similar to those which
characterize both Sede Comunale and Scuola Pascoli. Therefore, also for the Scuola Puccetti,
the flexibility of the system soil-foundation was considered negligible with respect to the
flexibility of the superstructure and, hence, fixed foundations were assumed.
Five modes are required to get at least 90 % of the modal mass, regardless of EI. Two modes
govern the dynamic problem: the first mode, characterized by both translation in Y (modal
mass = 65-70 %) and rotation around Z (modal mass = 76-82 %) and the second mode,
characterized mainly by translation in X (modal mass = 80-84 %). Hence, the first
translational mode in Y and the first torsional mode merge into a single mode. Although it is
not a problem for the dynamic analyses, it becomes a handicap for the assessment according
to the static nonlinear analysis, since pushover analyses were developed for 2-D systems that
experience only translation toward the pushing direction (Sec. 2.5).
mode:
T1, EI gross
T1,0.5 EI gross
T1, EI gross
T1, EI eff
=
=
0.5 EI gross
0.433
= 0.747 , while
= 0.5
EI gross
0.580
EI eff
0.433
= 0.538 − 0.585 , while
= 0.231 .
EI gross
0.740 − 0.804
The first mode (translation in Y and rotation around Z) is shown in Figure 7.2 and Figure
7.3,while the first translational mode in Y is shown in Figure 7.4.
166
Mode Period
#
[s]
1
2
3
4
5
0.433
0.402
0.364
0.143
0.137
Mode Period
#
[s]
1
2
3
4
5
0.580
0.552
0.503
0.198
0.191
Mode Period
#
[s]
1
2
3
4
5
0.804
0.785
0.720
0.287
0.277
Mode Period
#
[s]
1
2
3
4
5
0.753
0.731
0.673
0.264
0.255
Mode Period
#
[s]
1
2
3
4
5
0.740
0.718
0.660
0.257
0.248
EI = 100% EIgross
% Cum.
% Cum.
Mass
Mass
Mass
around Z
Y
X
0.02
0.02
0.65
0.65
0.76
0.04
0.11
0.86
0.69
0.84
0.03
0.18
0.02
0.89
0.86
0.09
0.00
0.07
0.97
0.87
0.01
0.13
0.03
0.98
1.00
EI = 50% EIgross
% Cum.
% Cum.
Mass
Mass
Mass
around Z
Y
X
0.03
0.03
0.67
0.67
0.78
0.05
0.09
0.86
0.71
0.84
0.03
0.16
0.02
0.89
0.87
0.09
0.00
0.05
0.98
0.87
0.00
0.13
0.04
0.98
1.00
% Cum.
% Cum.
Mass
Mass
Mass
around Z
Y
X
0.07
0.07
0.65
0.65
0.82
0.09
0.04
0.87
0.74
0.80
0.02
0.14
0.03
0.90
0.88
0.09
0.00
0.04
0.98
0.88
0.00
0.12
0.04
0.98
1.00
% Cum.
% Cum.
Mass
Mass
Mass
X
Y
around Z
0.04
0.04
0.70
0.70
0.78
0.06
0.08
0.86
0.76
0.82
0.03
0.11
0.04
0.89
0.87
0.09
0.00
0.05
0.98
0.87
0.00
0.12
0.04
0.98
1.00
% Cum.
% Cum.
Mass
Mass
Mass
X
Y
around Z
0.04
0.04
0.69
0.69
0.77
0.06
0.08
0.85
0.75
0.82
0.03
0.12
0.04
0.88
0.86
0.10
0.00
0.06
0.98
0.86
0.00
0.13
0.05
0.98
1.00
% Cum.
Modal Mass
Rotation
around Z
0.76
0.87
0.89
0.95
0.99
% Cum.
Modal Mass
Rotation
around Z
0.78
0.87
0.89
0.95
0.99
% Cum.
Modal Mass
Rotation
around Z
0.82
0.87
0.90
0.94
0.99
% Cum.
Modal Mass
Rotation
around Z
0.78
0.85
0.90
0.94
0.99
% Cum.
Modal Mass
Rotation
around Z
0.77
0.85
0.89
0.94
0.99
167
Figure 7.2. 1st mode of Scuola Puccetti, view in Y
Figure 7.3. 1st mode of Scuola Puccetti, view in X
Figure 7.4. 2nd mode of Scuola Puccetti, view in X
Since, for the Scuola Puccetti, accounting for vertical seismic input is not required, only the
were checked. Concerning the first condition of applicability (Sec. 2.2.1), Table 7.2 shows
that the value of ρmax/ρmin reduces when the members stiffness EI decreases. This reduction is
not linear with the reduction of EI. In fact:
= 0.781 , while
0.5 EI gross
EI gross
= 0.439 − 0.445 , while
= 0.5
EI eff
EI gross
= 0.231 .
168
Scuola Puccetti, assuming EI = 100% EIgross, the periods of the first two modes are,
respectively, T1 = 0.433 s and T2 = 0.402 s. Both periods lie in the plateau of the response
spectrum (TC = 0.5 s). Decreasing the members stiffness, the periods shift, respectively, to T1
= 0.580 s, T2 = 0.551 s, in the case of EI = 50% EIgross, and to T1 = 0.740-0-804 s, T2 = 0.7180.785 s, in the case of EI = EIeff computed through Eq. (2.8). Therefore, decreasing EI, the
spectral ordinates corresponding to the first two modes decrease too, yielding smaller seismic
forces. A consequence of this reduction is that, assuming EI = EIeff, the first condition of
applicability of linear methods will become less restrictive than the verification on the base of
EI = 50-100% EIgross. In particular, for the Scuola Puccetti, the choice of EI is crucial to
determine whether the results furnished by linear analyses can be accepted. In fact, choosing
EI = 50-100% EIgross, ρmax/ρmin will be larger than 2.5 (limit for the applicability of linear
analyses) while, according to the Italian Seismic Code, if EI = EIeff, ρmax/ρmin will be smaller
than 2.5 and, hence, the condition for the applicability of linear analyses is satisfied.
determine ρmax/ρmin, the results in terms of ρmax are very similar to those achieved applying the
therefore, it will experience ρmax also according to Eurocode 8. Therefore, for the Scuola
Puccetti, there is no reason to consider the joint equilibrium in order to determine the value of
recommended.
Considering that Eurocode 8 suggests to assume ρmin = 1 while the Italian Seismic Code
proposes ρmin = 2, the condition of applicability of linear analyses suggested in Eurocode 8 is
twice more conservative with respect to that proposed in the Italian Seismic Code. This is the
reason why, assuming EI = EIeff, the condition will be satisfied only if the recommendations
of Eurocode 8 are followed.
1% (for EI = EIeff) and 2% (for EI = 50-100% EIgross) of structural members which do not
satisfy the recommendations. In analogy with the first condition, also the second condition
will become less restrictive if EI = EIeff is assumed, as the shear demands will decrease with
respect to EI = 50-100% EIgross.
For the Scuola Puccetti, even if the recommendations of the Italian Seismic Code and EI =
EIeff are considered, the results furnished by linear analyses should be judged as not reliable,
since the second condition of applicability is not satisfied. Anyway, as already explained for
the Sede Comunale (Sec. 5.1.3):
169
•
concerning the first condition of applicability, considering only ρmax/ρmin to evaluate
the possibility of accepting linear analyses may not be judged as an exhaustive
method.
•
Concerning the second condition of applicability, it is very likely that the change in
the load path will be not so remarkable to justify the choice of rejecting the possibility
of using linear analyses since only a very limited number of members (1-2 %) do not
satisfy the check.
Therefore, other studies are required to determine if the conclusion to reject linear methods is
unavoidable or if the dynamic linear analysis may be considered useful at least to catch the
global seismic response of the building and to express a judgment about the need of
retrofitting.
FIRST CONDITION
SECOND CONDITION
EI = 100% EIgross
OPCM 3431
EC8
OPCM 3431
10.79
10.65
Number of Sections
946
ρmax
ρmax
2.00
1.01
Not Verified
21
ρmin
ρmin
ρmax/ρmin
5.39
ρmax/ρmin
10.58
% Not Verified
2
EI = 50% EIgross
OPCM 3431
EC8
OPCM 3431
8.43
8.10
Number of Sections
946
ρmax
ρmax
2.00
1.02
Not Verified
20
ρmin
ρmin
ρmax/ρmin
4.21
ρmax/ρmin
7.97
% Not Verified
2
OPCM 3431
EC8
OPCM 3431
4.76
4.76
Number of Sections
946
ρmax
ρmax
2.01
1.00
Not Verified
6
ρmin
ρmin
ρmax/ρmin
2.38
ρmax/ρmin
4.75
% Not Verified
1
OPCM 3431
EC8
OPCM 3431
4.74
4.55
Number of Sections
946
ρmax
ρmax
2.00
1.00
Not Verified
8
ρmin
ρmin
ρmax/ρmin
2.37
ρmax/ρmin
4.53
% Not Verified
1
OPCM 3431
EC8
OPCM 3431
4.80
4.59
ρmax
946
ρmax
ρmax
4.59
1.00
Not Verified
7
ρmin
ρmin
ρmax/ρmin
2.40
ρmax/ρmin
4.59
% Not Verified
1
170
theoretical (Eq. (3.22)) θu, with the same aims already described for the Sede Comunale in
Sec. 5.1.3.
The sensitivity of the assessment procedure to EI is shown in Figure 7.5 and Figure 7.6, in
terms of percentage of members which do not satisfy the verification. Since only a very
limited percentage of structural members do not satisfy the chord rotation check (in particular,
for EI = 100% EIgross, all members satisfy the check), showing the results in terms of
percentages of members which do not satisfy the verification is, for the ductile assessment of
the Scuola Puccetti, not particularly meaningful in order to compare different approaches of
assessment. For this reason, all charts will be shown in terms of percentages of the mean
values of Di Ci , where Di and Ci are the chord rotation demand and corresponding capacity
of each i-th structural member, respectively. Figure 7.7 - Figure 7.8 - Figure 7.9 - Figure 7.10
show that, decreasing EI, the mean values of Di Ci increase sensibly, for both beams and
columns. Therefore, assuming EIeff computed though Eq. (2.8) as the most accurate choice of
EI, it is clear that, for the Scuola Puccetti, both 100% EIgross and 50% EIgross yield inaccurate
and unconservative results.
100
90
100
100%EI
90
50%EI
80
50%EI
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
All Columns
Ground
Floor Col
1st Floor Col 2nd Floor Col Roof Floor
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
120
100
100%EI
100%EI
50%EI
80
EI_eff
0
All Elements
90
100%EI
100
EI_eff
50%EI
EI_eff
70
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
171
160
120
100%EI
100
100%EI
140
50%EI
EI_eff
120
50%EI
EI_eff
80
100
80
60
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
All Beams
All Columns
Ground
Floor Col
Col
Figure 7.11 - Figure 7.12 - Figure 7.13 - Figure 7.14 show that similar results were obtained
considering the mean EIeff computed i) for all members, ii) separately for beams and columns,
or iii) separately for T and rectangular beams and for columns of each floor. A difference can
be appreciated for the columns of the first storey. In fact, computing the mean EIeff separately
for the columns of each floor allows to take into account that the columns of the first storey
are stiffer (and, hence, deform less) because they bear a larger compressive N. However, since
this difference is small (≤ 10%) and limited to the first storey columns, for the Scuola
Puccetti, using the mean EIeff computed for all members yields quite accurate results.
120
120
EI_eff_uniform
100
EI_eff_uniform
EI_eff_beam_col
100
EI_eff_precise
EI_eff_beam_col
EI_eff_precise
80
80
60
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
180
120
EI_eff_uniform
EI_eff_uniform
160
100
EI_eff_beam_col
EI_eff_beam_col
140
EI_eff_precise
80
EI_eff_precise
120
100
60
80
40
60
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
172
Figure 7.15 and Figure 7.16 compare the mean values of Di Ci , computed according to the
similar to each other, Eurocode 8 suggests more complex calculations (Sec. 5.1.3).
Concerning the empirical θu (Eq. (3.15)), since the results obtained from the two Codes are
very close to each other, there is no need to follow the procedure suggested in Eurocode 8
and, hence, the simpler approach proposed in the Italian Seismic Code is recommended.
Concerning the theoretical θu (Eq. (3.22)), instead, the mean values of Di Ci are larger
(particularly for columns) when the recommendations of Eurocode 8 are followed. Hence,
assuming the procedure suggested in Eurocode 8 as correct, the simpler approach of the
Italian Seismic Code yields quite inaccurate results.
120
160
OPCM
140
OPCM
100
EC8
120
EC8
80
100
60
80
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
80
All Columns
Ground
Floor Col
Col
theoretical form, EIeff, OPCM vs. EC8
120
100
90
All Beams
Ls=M/V
Ls=M/V
100
Ls=L/2
70
Ls=L/2
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
All Beams
All Columns
Ground
Floor Col
Col
160
120
Ls=M/V
Ls=M/V
140
100
Ls=L/2
Ls=L/2
120
80
100
80
60
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
173
(Eq. (3.22)) θu. All charts show that Ls = L/2 and Eq. (3.1) yield very close values of Di Ci .
Concerning the assessment performed according to Eq. (3.15), the procedure may be
simplified. Concerning the assessment based on (Eq. (3.22)), instead, it must be considered
that for 17 beams and 28 columns θu = 0, since Lpl > Ls. The contribution of these beams can
not be taken into account in the evaluation of the mean values of Di Ci , since when Ci = 0 ,
Di Ci → ∞ . Hence, the results obtained using the correct definition of Ls (Eq. (3.1)) and Ls
= L/2 (Figure 7.19 and Figure 7.20) are more similar to each other than if they were shown in
terms of percentage of members which do not satisfy the verification. Anyway, since only 45
members (7% of the total number of members) are characterized by theoretical θu = 0, the
contribution of these members may be judged as negligible. Hence, the procedure may be
simplified also if the assessment is performed according to Eq. (3.22).
90
80
120
100
N_Grav
N_Grav
100
N_Seism
70
N_Seism
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
All Columns
Ground
Floor Col
Col
120
All Beams
160
N_Grav
N_Grav
140
100
N_Seism
N_Seism
120
80
100
60
80
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
Concerning the empirical θu, the gravity and the seismic N yield very close results and, hence,
the procedure may be simplified. Concerning the theoretical θu, the mean values of Di Ci
174
obtained assuming the seismic N are larger than assuming the gravity N (differences up to
40% for the columns). However, it must be remarked that the seismic ΔN considered in this
case is related with the hypothesis of indefinitely elastic behaviour and, hence, is larger than
ΔN obtained when the development of nonlinear mechanisms is accounted for (Sec. 3.2.2).
Hence, further future investigations are needed to understand if the procedure can be
simplified if Eq. (3.22) is assumed.
The comparison between the results obtained considering φy from theoretical assumptions
(Eqs. (3.9) – (3.10)) and from empirical expressions (Eqs. (3.11) to (3.14)) is shown in Figure
7.25 - Figure 7.26. The charts refer to the mean values of Di Ci , computed according to the
theoretical θu (Eq. (3.22)) and to both Codes. Empirical and theoretical φy yield close results.
Therefore, for the Scuola Puccetti, the empirical formulas calibrated for new seismically
designed structures may be applied without any loss of accuracy.
120
160
FI_Theoric
100
FI_Theoric
140
FI_Fardis
FI_Priestley
120
FI_Fardis
FI_Priestley
80
100
60
80
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
theoretical form, EIeff, EC8, different φy
N; Ls = L/2; empirical φy). The charts refer to the mean values of Di Ci , computed
Concerning the empirical θu, the simplified approach yields a small decrease in the mean
values of Di Ci with respect to the correct approach. Concerning the theoretical θu, instead,
the decrease in the mean values of Di Ci obtained using the simplified approach is
noticeable (≈ 40%). Comparing the results obtained considering both correct and simplified
approach of both empirical and theoretical θu (Figure 7.31 - Figure 7.32), according to the
Italian Seismic Code, both correct approaches and the simplified approach of Eq. (3.15) yield
close results while the simplified approach of Eq. (3.22) underestimates the mean values of
Di Ci . According to Eurocode 8, the tendency of underestimate the results assuming the
simplified approach of Eq. (3.22) is less evident, while the correct approach of Eq. (3.22)
overestimates the values obtained considering both approaches of Eq. (3.15). Therefore, for
the Scuola Puccetti, the assessment of ductile mechanisms based on Eq. (3.15) may be
simplified without any remarkable loss of accuracy. Concerning Eq. (3.22), instead, further
future investigations are required to understand if the large difference between the correct and
the simplified approach is due to the use of ΔN related with the hypothesis of indefinitely
175
elastic behaviour, instead of ΔN obtained accounting for the development of nonlinear
mechanisms (Sec. 3.2.2).
120
100
90
Ls=L/2; N_Grav
80
Ls=M/V; N_Seism
Ls=L/2; N_Grav
100
70
Ls=M/V; N_Seism
80
60
60
50
40
40
30
20
20
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
emp. form, EIeff, OPCM, correct vs. simplified
All Columns
Ground
Floor Col
Col
120
All Beams
160
140
100
120
80
100
60
80
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
100
Ground
Floor Col
Col
160
All Columns
120
All Beams
140
120
80
100
60
80
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
EIeff, OPCM, theor. vs. emp., correct vs. simplified
All Elements
All Beams
All Columns
Ground
Floor Col
Col
176
charts show that the use of the bidimensional failure curve yield an increase in the mean
values of Di Ci of columns ≈ 25% (for the beams the verification is only uniaxial). This
difference is due to the lack of compactness and to the particular shape (asymmetric C) of the
Scuola Puccetti. Therefore, the use of the bidimensional failure curve improves on the
accuracy of the results and, hence, is recommended.
140
120
140
MONO-AXIAL BENDING
120
BI-AXIAL BENDING
BI-AXIAL BENDING
100
100
80
80
60
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
200
160
140
MONO-AXIAL BENDING
MONO-AXIAL BENDING
180
MONO-AXIAL BENDING
BI-AXIAL BENDING
160
BI-AXIAL BENDING
120
140
100
120
100
80
80
60
60
40
40
20
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
according to both Codes, with the same aims already described for the Sede Comunale (Sec.
5.1.3).
to both Codes. In particular, Figure 7.37 and Figure 7.38 show that the percentage of
members which do not satisfy the verification decreases with decreasing EI. Therefore,
assessing the brittle mechanisms using a fixed fraction of EIgross between 50 and 100% yields
conservative results. In particular, according to Eurocode 8, the increase in the percentages of
members which do not satisfy the verification is only between 15% and 20%. Hence, for the
Scuola Puccetti, if the recommendations of Eurocode 8 are followed, assuming a fixed
fraction of EIgross between 50% and 100% may be considered a suitable choice, since the
177
assessment of EIeff according to Eq. (2.8) is much longer and yields small improvements in
the results.
Figure 7.39 and Figure 7.40 show that results very close to each other were obtained
considering the mean EIeff computed i) for all members, ii) separately for beams and columns,
or iii) separately for T and rectangular beams and for columns of each floor.
100
90
100
100%EI
100%EI
90
50%EI
80
50%EI
80
EI_eff
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
OPCM, different EI
All Elements
Ground
Floor Col
Col
100
EI_eff_uniform
EI_eff_uniform
90
90
EI_eff_beam-col
80
All Columns
EC8, different EI
100
All Beams
EI_eff_beam-col
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_precise
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
OPCM, different EI
All Elements
All Beams
All Columns
Ground
Floor Col
Col
EC8, different EI
satisfy the verification according to both Codes. Concerning Eurocode 8, the difference in
percentage of members which do not satisfy the verification assuming the gravity and the
seismic N is large (≈ 40%, considering all columns). Further future investigations are required
to understand if this difference is due to the use of the seismic ΔN related with the hypothesis
of indefinitely elastic behaviour instead of ΔN obtained accounting for the development of
nonlinear mechanisms (Sec. 3.2.2). Concerning the Italian Seismic Code, the difference in
percentage of members which do not satisfy the verification assuming the gravity and the
seismic N is much larger (> 100%) and can not be neglected. This difference is due to the fact
that the concrete shear resistance of columns will become null if the columns experience a
tensile N (see Fig. 3.20).
178
theoretical φy yield very close (practically identical) results. Therefore, for the Scuola
Puccetti, the empirical formulas may be applied without any loss of accuracy.
Seismic Code does not depend on Ls). The use of Ls = L/2 yields a small increase (< 15%) in
the percentage of members which do not satisfy the verification with respect to Eq. (3.1),
leading to a safe side assessment. Hence, for the Scuola Puccetti, the procedure may be
simplified.
100
90
100
N_Grav
90
N_Seism
N_Grav
N_Seism
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
90
80
All Columns
Ground
Floor Col
Col
100
All Beams
100
FI_Theoric
90
FI_Fardis
Ls=M/V
Ls=L/2
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
179
members which do not satisfy the requirements. Therefore, for the Scuola Puccetti, the use of
the procedure suggested in Eurocode 8 is recommended.
If all members are considered, the simplified and the correct approaches yield very close
results. On the other hand, it must be underline that, considering only the beams, the
simplified approach overestimates the correct approach (difference ≈ 25%), while,
considering only the columns, the simplified approach underestimates the correct approach
(difference > 50%). Further future investigations are required to understand if this difference
will reduce if ΔN obtained accounting for the development of nonlinear mechanisms is used
instead of ΔN related with the hypothesis of indefinitely elastic behaviour (Sec. 3.2.2).
100
OPCM; N_Grav
90
90
OPCM; N_Seism
80
100
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
Figure 7.46. Brittle check: dynamic linear an., EC8,
correct vs. simplified approach
•
concrete compressive strength, fc = 18 MPa.
•
•
180
•
•
•
•
Figure 7.48. Steel: bilinear (strain hardening) model
In the following, the procedure adopted for estimating the brace’s dimensions is shown for the
same slab already considered in Sec. 7.1.1 (square slab, L = L’ = 3500 mm). The thickness of
the equivalent slab computed for the SAP model was determined as 74 mm. Hence:
(
)
I = 3500 3 ⋅ 74 12 = 2.646 ⋅ 1011 mm 4
As = 5 6 A = 5 6 ⋅ (3500 ⋅ 74) = 216008mm 2
Gc = 9263 MPa
[
(
)
]
K b = 3500 3 12 ⋅ 22230 ⋅ 2.646 ⋅ 1011 + 3500 (216008 ⋅ 9263)
−1
= 424326 N / mm
181
Once known Kb and l = L' 2 + L2 = 3500 2 + 3500 2 = 4950mm , the area of the brace was
347 mm was determined. This calculation was repeated for all slabs of the building.
are in the order of 1%, 13%, and 14% for the first, the second and the third mode,
respectively. The difference is, hence, negligible concerning the first mode, while it is small
but not insignificant for the other two modes. This difference may be justified considering
that the structure is not compact and therefore, the way to model the in-plan stiffness of the
slabs can affect the results of the model, particularly at the re-entrant corners, which may
undergo concentrations of stresses and deformations. Anyway, since the differences are not
large, the SeismoStruct model was judged to be consistent with the SAP model.
Periods [s]
SeismoStruct
SAP
I Mode
0.585
0.580
II Mode
0.485
0.552
III Mode
0.438
0.503
pattern of lateral forces, applied in positive and negative X and Y directions). In order to
perform the analysis:
•
•
•
182
Figure 7.50. Uniform distr., positive X, top view
Figure 7.52. Modal distr., positive X, top view
Figure 7.54. Uniform distr., positive Y, top view
Figure 7.56. Modal distr., positive Y, top view
183
•
•
•
•
Γx = 0.118 (Eq. (2.13)).
•
•
5000
5000
4500
4500
4000
4000
3500
Vb (kN)
Vb (kN)
3500
3000
2500
2000
Vb,SD
Vb,LD
Vb,NC
3000
2500
2000
1500
1500
1000
1000
500
500
Δ
0
0
Δ
*
y
50
*
Δ LD
0
m
100
150
200
250
0
300
Δ NC
Δ SD
50
100
150
200
250
300
3000
25000
Vb,NC
2500
20000
Vb,SD
Vb (kN)
Vb (kN)
2000
15000
10000
1500
Vb,LD
1000
5000
500
Δ
Δ
*
y
*
0
0
0
50
100
Δ LD
m
150
200
250
0
Δ SD
10
Δ NC
20
30
40
•
•
184
•
•
Γy = 0.778 (Eq. (2.13)).
•
•
2000
3000
Vb,SD
Vb,NC
Vb,LD
1800
2500
1600
1400
Vb (kN)
Vb (kN)
2000
1500
1000
1200
1000
800
600
400
500
Δ
0
Δ
*
y
0
200
*
m
Δ LD
0
50
100
150
Δ NC
Δ SD
0
200
50
100
150
200
4000
4000
Vb,SD
3500
3500
3000
Vb (kN)
Vb (kN)
3000
2500
2000
1500
2500
Vb,LD
2000
1500
1000
1000
500
500
Δ
0
0
20
40
Δ
*
y
60
*
80
Δ LD
0 V
b,NC
0
m
100
20
Δ SD
40
60
80
100
CACAPITY CURVES Y - MDOF
4000
5000
VSD,UNIF
4500
VU,UNIF
3500
4000
UNIFORM
MODAL
3000
3000
2500
UNIFORM
VSD,MOD
2000
Vb (kN)
Vb (kN)
3500
MODAL
1500
2500
2000
VU,MOD
1500
1000
1000
500
500
ΔSD,UNIF
ΔSD,MOD
0
0
50
100
150
ΔSD,MOD
0
200
250
0
25
50
ΔSD,UNIF
75
100
125
150
175
200
185
For the Scuola Puccetti the use of the static nonlinear analysis is questionable. In fact,
according to what explained in detail in Sec. 2.5, many problems arose due to the fact that the
structural configuration is neither compact nor symmetric. First, when lateral unidirectional
forces are applied, the structure undergoes a strong torsional response, leading to important
displacements also in the orthogonal direction. This effect, which is clear from the images
(Figure 7.49 to Figure 7.56), can not be captured by the analysis, since it was developed for 2D systems. Further future investigations are needed to understand how significant the change
in the results will be if the contribution of forces/displacements in the orthogonal direction is
considered too. The figures show also that one side is less stiff than the rest of the structure
and deforms much more that the centre of mass, while the torsional response causes some
joints to experience displacements opposite to the direction of loading (in particular when the
“modal” distribution of forces is applied). These are the reason why the “transformation
factor” Γ is smaller than 1 (Γx = 0.118 and Γy = 0.778). Hence, for the Scuola Puccetti, the
choice to locate the control node at the centre of mass of the top floor may be judged as
inappropriate. As already explained in Sec. 2.5, further future investigations are needed:
•
to know if the use of the “modal” distribution of forces may yield reliable results also
in the case of non-compact and asymmetric structures;
•
to recognize which is the best control node location in order to assess the response in
the most reliable way;
•
to understand if the results of the assessment are sensitive to the choice of the control
node.
Although for the Scuola Puccetti the correctness of the results of the static nonlinear analysis
is doubtful, in this work the procedures suggested by both Codes were followed and the
assessment based on the static nonlinear analysis was, hence, performed
described in Sec. 5.1.3 for the assessment based on the dynamic linear analysis.
Since only a limited percentage of structural members do not satisfy the chord rotation check,
showing the results in terms of percentages of members which do not satisfy the verification
is, for the ductile assessment of the Scuola Puccetti, not particularly meaningful in order to
compare different approaches of assessment. For this reason, all charts will be shown in terms
of percentages of the mean values of Di Ci , where Di and Ci are the chord rotation demand
and corresponding capacity of each i-th structural member, respectively.
Figure 7.67 - Figure 7.68 compare the mean values of Di Ci , computed according to the
similar to each other, Eurocode 8 suggests more complex calculations (Sec. 5.1.3). Moreover,
also the way to compute the demand is different. In fact, the Italian Seismic Code proposes to
consider separately the results obtained from each pushover, while Eurocode 8 suggests to
consider each demand quantity as the outcome of the combination of the values furnished by a
186
pushover in X and a pushover in Y (Eq. (2.23)). Concerning both empirical and theoretical θu,
the mean values of Di Ci obtained from Eurocode 8 are larger than those obtained from the
Italian Seismic Code. The difference is more evident when the theoretical θu is assumed. This
difference is due, in particular, to the way of computing the seismic demand. Further future
investigations are needed to determine whether Eq. (2.23) yields more reliable demands
values than those obtained considered each pushover separately.
250
300
OPCM
OPCM
250
200
EC8
EC8
200
150
150
100
100
50
50
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
charts refer to the mean values of Di Ci , computed according to the theoretical θu (Eq.
(3.22)) and to both Codes. Empirical and theoretical φy yield close results. Hence, for the
Scuola Puccetti, the empirical formulas calibrated for new seismically designed structures
may be applied without any sensible loss of accuracy.
(Eq. (3.22)) θu. Concerning the empirical θu (Figure 7.71 - Figure 7.72), assuming the Italian
Seismic Code, Ls = L/2 and Eq. (3.1) yield results close to each other. In particular, Ls = L/2
yields slightly unconservative results (difference in the order of 15%). Although a larger
difference can be appreciated for the columns of the lower storeys, the procedure may be
simplified. Assuming Eurocode 8, instead, Ls = L/2 yields unconservative results (difference
in the order of 25%). Hence, simplifying the procedure would yield inaccurate results.
Concerning the theoretical θu (Figure 7.73 - Figure 7.74), the difference between the correct
and the simplified approaches is huge for both Codes (using Ls = L/2 there is an
underestimation > 30%). This difference is due to the fact that Eq. (3.22) is very sensitive to
the value of Ls. This result is more evident than for the linear analyses because the inelastic Ls
are likely to change with respect to the elastic Ls, possibly resulting in very small values,
leading to θu = 0.
187
180
250
FI_Theoric
FI_Theoric
160
FI_Fardis
140
FI_Fardis
200
FI_Priestley
FI_Priestley
120
150
100
80
100
60
40
50
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
Ground
Floor Col
Col
300
Ls=M/V
200
All Columns
250
All Beams
Ls=M/V
250
Ls=L/2
Ls=L/2
200
150
150
100
100
50
50
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
All Columns
Ground
Floor Col
Col
300
250
Ls=M/V
Ls=M/V
200
All Beams
250
Ls=L/2
Ls=L/2
200
150
150
100
100
50
50
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
Assuming the empirical θu, the charts show that gravity and seismic N yield very close results.
Assuming the theoretical θu, the charts show that gravity N yields slightly unconservative
results (differences in the order of 10%). The difference between the two approaches is
evident only for the ground floor while it is negligible for the upper floor levels. In fact the
seismic ΔN is null at the top of the building and maximum at the base. Hence, the procedure
188
may be simplified, even if particular attention should be paid to the assessment of the ground
floor columns. This conclusion clashes with the results based on the dynamic linear analysis.
This can be justified considering that the seismic ΔN experienced by the structure in a linear
analysis is, in general, (much) larger than ΔN obtained through a nonlinear analysis.
300
250
N_Grav
200
N_Grav
250
N_Seism
N_Seism
200
150
150
100
100
50
50
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
160
140
All Columns
Ground
Floor Col
Col
180
All Beams
250
N_Grav
N_Grav
200
N_Seism
N_Seism
120
150
100
80
100
60
40
50
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
shown from Figure 7.79 to Figure 7.84. The charts refer to the mean values of Di Ci ,
computed according to both Codes and both empirical (Eq. (3.15)) and theoretical (Eq. (3.22))
θu. Concerning the empirical θu, assuming the recommendations of the Italian Seismic Code,
the simplified and the correct approaches yield results close to each other. In particular, the
simplified approach yields slightly unconservative results (difference ≈ 15%). Although a
larger difference can be appreciated for the columns of the lower storeys, the procedure may
be simplified. Assuming the recommendations of Eurocode 8, instead, the simplified
approach yields unconservative results (difference ≈ 25%). Hence, simplifying the procedure
would yield inaccurate results. Concerning the theoretical θu, the simplified approach leads to
a serious underestimation of the mean values of Di Ci (difference > 50%). This difference
is due mainly to the fact that Eq. (3.22) is very sensitive to the value of Ls. Hence, simplifying
the procedure would yield inaccurate results.
189
300
250
Ls=L/2; N_Grav
Ls=L/2; N_Grav
200
250
Ls=M/V; N_Seism
Ls=M/V; N_Seism
200
150
150
100
100
50
50
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
160
140
All Columns
Ground
Floor Col
Col
180
All Beams
250
200
120
150
100
80
100
60
40
50
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
DUCTILE CHECK STATIC NONLIN. - OPCM
DUCTILE CHECK STATIC NONLIN. - EC8
300
250
250
200
200
150
150
100
100
50
50
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
2nd Floor Col Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
2nd Floor Col Roof Floor Col
recommended for the assessment of the Scuola Puccetti, as it yields enough accurate results
and is much faster that the bidimensional check. This conclusion clashes with the results
190
based on the dynamic linear analysis. The reason of this difference may be explained as
follows. A pushover analysis is performed with unidirectional lateral forces and, hence, it is
likely that the degradation of stiffness in the pushing direction will be faster than in the
orthogonal direction. Therefore, the displacements/deformations in the direction orthogonal to
that of the lateral forces will be, in general, small. In a linear analysis, instead, the stiffness is
constant and, hence, the difference between the uniaxial and the biaxial check is more evident.
300
250
MONO-AXIAL BENDING
200
MONO-AXIAL BENDING
250
BI-AXIAL BENDING
BI-AXIAL BENDING
200
150
150
100
100
50
50
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
All Elements
Col
160
140
All Columns
Ground
Floor Col
Col
180
All Beams
250
MONO-AXIAL BENDING
MONO-AXIAL BENDING
BI-AXIAL BENDING
200
BI-AXIAL BENDING
120
150
100
80
100
60
40
50
20
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
theoretical form, OPCM, uni- vs. bi-axial bending
All Elements
All Beams
All Columns
Ground
Floor Col
Col
theoretical form, EC8, uni- vs. bi-axial bending
sensitivity to EI) already described for the dynamic linear analysis (Sec. 5.1.3).
satisfy the verification according to both Codes. The charts show that gravity and seismic N
yield nearly identical results. For the Scuola Puccetti, hence, the procedures suggested by the
two Codes may be simplified. This conclusion clashes with the results based on the dynamic
191
linear analysis. The fact that the results obtained using the static nonlinear analysis are less
sensitive to the choice of N can be justified considering that ΔN experienced by the structure
in a linear analysis is, in general, (much) larger than ΔN obtained through a nonlinear
analysis.
100
100
90
N_Grav
90
N_Grav
80
N_Seism
80
N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
All Beams
All Columns
Ground
Floor Col
Col
φy (Eqs. (3.9) – (3.10)) and the empirical φy (Eqs. (3.11) to (3.14)). The chart refer to the
theoretical φy yield very close (practically identical) results. Therefore, for the Scuola
Puccetti, the empirical formulas may be applied without any loss of accuracy.
Ls (Eq. (3.1)) and Ls = L/2. The chart refer to the percentages of members which do not satisfy
the verification according only to Eurocode 8 (the expression suggested in the Italian Seismic
Code does not depend on Ls). The use of Ls = L/2 yields a small increase (< 10%) in the
percentage of members which do not satisfy the verification with respect to Eq. (3.1). Hence,
for the Scuola Puccetti, the procedure may be simplified.
which do not satisfy the requirements. Therefore, for the Scuola Puccetti, the procedure
yields a small increase (< 10%) in the percentage of members which do not satisfy the
verification with respect to the correct approach, leading to a safe side assessment. Hence, for
the Scuola Puccetti, the procedure may be simplified.
192
100
100
FI_Theoric
90
FI_Fardis
80
FI_Priestley
90
Ls=M/V
80
Ls=L/2
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
All Elements
EC8, different φy
All Beams
All Columns
Ground
Floor Col
analysis, EC8, Ls=M/V vs. Ls=L/2
100
100
OPCM; N_Grav
90
OPCM; N_Seism
80
Col
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
90
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
correct OPCM, simplified OPCM, EC8
All Elements
All Beams
All Columns
Ground
Floor Col
Col
analysis, EC8, correct vs. simplified approach
The nonlinear dynamic analysis of the Scuola Puccetti was carried out on the base of eight
time history analyses, each one performed with an accelerogram acting in one horizontal
•
•
(Figure 7.96):
193
[C ] = a1 [K ] where a1 = T1ξ
π.
Assuming T1 = 0.433s and ξ = 0.05 ⇒ a1 = T1ξ π = (0.433 ⋅ 0.05) π = 0.0069 .
1.1
1
Mag=6.0
Mag=6.5
Mag=6.5_long
Mag=7.0
Mean
Target
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
T [s]
2
2.5
3
Figure 7.96. Equivalent viscous damping properties
respectively. In analogy with both eigenvalue and static nonlinear analyses, the deformed
shape is mainly characterized by a translation toward X when the main accelerogram is
applied in X, and by both a translation toward Y and a rotation around Z when the main
accelerogram is applied in Y. This behaviour is confirmed by the time history of the
displacement of the control node, chosen at the centre of mass of the top floor (Figure 7.101 Figure 7.102).
194
50
50
40
DISPL. IN X
DISPL. IN Y
DISPL [mm]
DISPL [mm]
30
20
10
0
-10
40
DISPL. IN X
30
DISPL. IN Y
20
10
0
-10
-20
-20
-30
-30
-40
-40
-50
-50
0
2
4
6
8
10
12
14
0
16
2
4
6
8
10
12
14
16
T [s]
T [s]
Figure 7.101. Control node displ.: main acc. in X
nonlinear analysis (Sec. 5.2.2). Since only a limited percentage of structural members do not
satisfy the chord rotation check, showing the results in terms of percentages of members
which do not satisfy the verification is, for the ductile assessment of the Scuola Puccetti, not
particularly meaningful in order to compare different approaches of assessment. For this
reason, all charts will be shown in terms of percentages of the mean values of Di Ci , where
Di and Ci are the chord rotation demand and corresponding capacity of each i-th structural
member, respectively.
Figure 7.103 and Figure 7.104 show the comparison between the mean values of Di Ci ,
computed according to both Codes. Although the formulas proposed in the Italian Seismic
Code and in Eurocode 8 are very similar, Eurocode 8 suggests more complex calculations
(Sec. 5.1.3). Concerning both empirical and theoretical θu, the results obtained following the
recommendations of the Italian Seismic Code and Eurocode 8 are close to each other. Hence,
the use of the simpler procedure proposed in the Italian Seismic Code is recommended.
90
70
OPCM
60
EC8
80
OPCM
70
EC8
50
60
40
50
30
40
30
20
20
10
10
0
0
All Elements
All Beams
All Columns Ground Floor 1st Floor Col 2nd Floor Col
Col
Roof Floor
Col
All Elements
All Beams
Col
Roof Floor
Col
(3.22)) and to both Codes. The theoretical and empirical φy yield very close results. Hence, for
195
the Scuola Puccetti, the empirical formulas calibrated for new seismically designed structures
90
90
FI_Theoric
80
80
FI_Fardis
70
FI_Fardis
70
FI_Priestley
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
Col
Roof Floor
Col
All Elements
All Beams
Col
Roof Floor
Col
70
70
Ls=M/V
Ls=M/V
60
Ls=L/2
50
50
40
40
30
30
20
20
10
10
0
Ls=L/2
0
All Elements
All Beams
Col
Roof Floor
Col
All Elements
All Beams
Col
Roof Floor
Col
90
80
FI_Priestley
0
All Elements
60
FI_Theoric
90
Ls=M/V
80
Ls=L/2
Ls=M/V
Ls=L/2
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Col
Roof Floor
Col
All Elements
All Beams
Col
Roof Floor
Col
Ls=L/2 is shown from Figure 7.107 to Figure 7.110. The charts refer to the mean values of
(Eq. (3.22)) θu. Concerning the empirical θu, Ls = L/2 and Eq. (3.1) yield results close to each
other. In particular, the results obtained considering Ls = L/2 are slightly unconservative
196
(difference in the order of 15%). Hence, the procedure may be simplified. Concerning the
theoretical θu, instead, the difference is much larger (using Ls = L/2 there is an
underestimation > 50%). This difference is due to the fact that Eq. (3.22) is very sensitive to
the value of Ls.
(gravity N; Ls = L/2; empirical φy). The charts refer to the mean values of Di Ci , computed
according to the theoretical θu (Eq. (3.22)) and to both Codes. The simplified approach leads
to underestimate the results obtained with the correct approach (difference > 50%). This
difference is due mainly to the fact that Eq. (3.22) is very sensitive to the value of Ls.
empirical and theoretical θu (Figure 7.113 - Figure 7.114), it appears that both approaches of
the empirical θu and the simplified approach of the theoretical θu yield results close to each
other (differences ≤ 20%), but different from those obtained considering the correct approach
of the theoretical θu, which overestimates the results (difference > 50%). It confirms that the
correct approach of Eq. (3.22) is too sensitive to the value of Ls. Therefore, for the Scuola
Puccetti, concerning the theoretical evaluation of θu, there is the need of considering Ls = L/2
instead of Eq. (3.1).
90
80
70
90
80
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Col
Roof Floor
Col
All Elements
80
70
Col
Roof Floor
Col
90
All Beams
90
80
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
Col
Roof Floor
Col
All Elements
All Beams
Col
Roof Floor
Col
197
recommended for the assessment of the Scuola Puccetti, as it yields enough accurate results
and is much faster that the bidimensional check.
70
60
70
MONO-AXIAL BENDING
60
BI-AXIAL BENDING
BI-AXIAL BENDING
50
50
40
40
30
30
20
20
10
10
0
MONO-AXIAL BENDING
0
All Elements
All Beams
All Columns Ground Floor 1st Floor Col 2nd Floor Col Roof Floor
Col
Col
All Elements
Col
Col
90
All Beams
90
MONO-AXIAL BENDING
MONO-AXIAL BENDING
80
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
BI-AXIAL BENDING
0
All Elements
All Beams
Col
Col
All Elements
All Beams
Col
Col
the static nonlinear analysis in Sec. 5.2.2.
Empirical and theoretical φy yield very close results. Therefore, for the Scuola Puccetti, the
198
satisfy the verification according to Eurocode 8. The use of Ls = L/2 results in a small increase
(< 10%) of percentage of members which do not satisfy the verification with respect to Eq.
(3.1), leading to a safe side assessment. Hence, for the Scuola Puccetti, the procedure may be
simplified.
100
100
90
FI_Theoric
90
Ls=M/V
80
FI_Fardis
80
Ls=L/2
70
FI_Priestley
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
EC8, different φy
All Elements
All Beams
All Columns
Ground
Floor Col
Col
requirements. Hence, for the Scuola Puccetti, the procedure suggested in Eurocode 8 is
recommended.
(seismic N; Ls from Eq. (3.1); theoretical φy) and the simplified approach (gravity N; Ls=L/2;
yields slightly conservative results (small increase, <10%, in percentage of members which do
not satisfy the verification with respect to the correct approach). Hence, for the Scuola
Puccetti, the procedure may be simplified.
100
100
90
OPCM
90
80
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground
Floor Col
Col
OPCM vs. EC8
All Elements
All Beams
All Columns
Ground
Floor Col
Col
199
Chapter 8. Assessment of Scuola Media Inferiore Don Bosco (Rapagnano)
8 Assessment of Scuola Media Inferiore Don Bosco (Rapagnano)
The assessment of the seismic behaviour of the Scuola Media Inferiore Don Bosco located in
Rapagnano has been performed according to all methods of analysis proposed by the two
considered Codes (except for the linear static analysis, since the structure is regular neither in
plan nor in elevation). All analyses were based on 3-D models, as the lack of regularity did
not allow to consider two planar separated models in the two principal directions.
8.1 Linear Analyses
Ec = 21638 MPa, computed through Eq. (3.7), where fcm = 16.6 MPa.
(Poisson modulus of steel) = 0.3, the following thicknesses of the equivalent homogeneous
section of the slab were obtained:
200
•
t = 75mm , for the slabs of the first and second floor;
•
t = 108mm , for the slab of the roof floor;
•
t = 120mm , for the slab of the balcony.
stiffness at yielding, EI eff , computed through Eq. (2.8), it was assumed that:
N
∑
EI =
EI eff ,i
i =1 EI gross ,i
N
EI gross
⎧i = All _ members ⇒ 22.3% EI gross = 4825 ⋅ I gross
⎪i = Re c _ beams ⇒ 14.2% EI
⎪
=
−
⇒
= 4263 ⋅ I gross
19
.
7
%
i
T
beams
EI
⎪
= ⎨i = All _ columns ⇒ 31.0%gross
EI gross = 6708 ⋅ I gross
⎪i = Ground _ floor _ columns ⇒
35.1% EI gross = 7595 ⋅ I gross
⎪
st
Figure 8.1 shows the model of the Scuola Don Bosco realized with SAP2000.
Figure 8.1. View of Scuola Don Bosco
For the Scuola Don Bosco, since the flexibility of the system soil-foundation was considered
to be negligible with respect to the flexibility of the superstructure, the soil–structure
interaction was not taken into account and fixed foundations were assumed.
201
EI = 100% EIgross
% Modal
Mode Period
Mass
#
[s]
Transl. in
X
0.00
1
0.587
2
0.539
0.73
0.13
3
0.501
8
0.213
0.00
0.194
0.06
10
% Modal
Mass
Transl. in
Y
0.81
0.01
0.03
0.07
0.00
% Modal
Mode Period
Mass
#
[s]
Transl. in
X
0.00
1
0.824
2
0.758
0.73
0.13
3
0.706
8
0.299
0.00
0.274
0.06
10
% Modal
Mass
Transl. in
Y
0.81
0.01
0.03
0.07
0.00
% Modal
% Cum.
% Cum.
Mass
Rotation
around Z
0.00
0.81
0.32
0.01
0.73
0.82
0.87
0.85
0.54
0.03
0.87
0.94
0.00
0.93
0.94
EI = 50% EIgross
% Modal
Mass
Rotation
around Z
0.32
0.01
0.54
0.03
0.00
% Cum.
% Cum.
0.00
0.74
0.87
0.87
0.93
0.81
0.82
0.85
0.94
0.94
% Cum.
Modal Mass
Rotation
around Z
0.32
0.33
0.87
0.91
0.92
% Cum.
Modal Mass
Rotation
around Z
0.32
0.33
0.87
0.91
0.92
% Modal
Mode Period
Mass
#
[s]
Transl. in
X
0.00
1
1.216
2
1.124
0.74
0.13
3
1.050
8
0.445
0.00
0.406
0.05
10
% Modal
Mode Period
Mass
#
[s]
Transl. in
X
0.00
1
1.170
2
1.067
0.71
0.14
3
0.990
0.363
0.09
15
% Modal % Modal
% Cum.
Mass
Mass
Modal Mass
Transl. in Rotation
Transl. in X
Y
around Z
0.00
0.80
0.31
0.02
0.01
0.74
0.03
0.87
0.55
0.06
0.02
0.87
0.00
0.00
0.92
% Modal % Modal
% Cum.
Mass
Mass
Modal Mass
Transl. in Rotation
Transl. in X
around Z
Y
0.00
0.77
0.31
0.02
0.01
0.71
0.03
0.85
0.53
0.00
0.01
0.95
% Cum.
Modal Mass
Transl. in Y
0.80
0.82
0.85
0.94
0.94
% Cum.
Modal Mass
Transl. in Y
0.77
0.79
0.81
0.92
% Cum.
Modal Mass
Rotation
around Z
0.31
0.33
0.88
0.91
0.92
% Cum.
Modal Mass
Rotation
around Z
0.31
0.32
0.85
0.91
% Modal
Mode Period
Mass
#
[s]
Transl. in
X
0.00
1
1.155
2
1.038
0.64
0.19
3
0.969
0.364
0.09
15
% Modal
Mass
Transl. in
Y
0.75
0.01
0.02
0.00
% Modal
Mass
Rotation
around Z
0.30
0.00
0.52
0.01
% Cum.
% Cum.
0.00
0.64
0.84
0.94
0.75
0.77
0.79
0.91
% Cum.
Modal Mass
Rotation
around Z
0.30
0.30
0.83
0.90
202
The number of modes required to get at least 90% of the modal mass is not small and varies
from 10 to 15. This can be justified considering that the slabs of the Scuola Don Bosco are
characterized by very different areas and, therefore, by very different stiffness. Moreover, the
thickness of the equivalent homogeneous section of the floor slabs is only 75 mm. Hence, the
slabs (and, in particular, the slab of the gym, which is very wide) are likely to be quite flexible
in their plane and this makes the eigenvalue problem more complex. Three modes govern the
dynamic problem. The first mode is characterized mainly by translation in Y (modal mass =
75-81 %) but also by rotation around Z (modal mass = 30-32 %). The second mode is a pure
translational mode in X (modal mass = 64-73 %), while the third mode is characterized
mainly by rotation around Z (modal mass = 52-55 %) but also by translation in X (modal
mass = 13-19 %). Concerning the first and the third modes, translation and rotation are
coupled. Although it is not a problem for the dynamic analyses, it becomes a handicap for the
assessment according to the static nonlinear analysis, since pushover analyses were developed
for 2-D systems that experience only translation toward the pushing direction (Sec. 2.5).
increase of natural periods is not linear with the decrease of EI. In fact:
T1, EI gross
T1,0.5 EI gross
T1, EI gross
T1, EI eff
=
=
0.5 EI gross
0.587
= 0.5
= 0.712 , while
EI gross
0.824
EI eff
0.587
= 0.223 .
= 0.483 − 0.508 , while
EI gross
1.155 − 1.216
The first, the second and the third modes are shown in Figure 8.2, Figure 8.3, Figure 8.4 and
Figure 8.5, respectively.
Figure 8.2. 1st mode of Scuola Don Bosco, view in Y
Figure 8.3. 2nd mode of Scuola Don Bosco view in X
Figure 8.4. 3rd mode of Scuola Don Bosco, view in X
Figure 8.5. 3rd mode of Scuola Don Bosco view in Y
203
Since, for the Scuola Don Bosco, accounting for vertical seismic input is not required, only
the horizontal elastic 5%-damped pseudo-acceleration response spectrum defined in Sec. 4.4.4
were checked.
Concerning the first condition of applicability (Sec. 2.2.1), Table 8.2 shows that the value of
ρmax/ρmin reduces when the members stiffness EI decreases. This reduction is not linear with
the reduction of EI. In fact:
= 0.694 , while
0.5 EI gross
EI gross
= 0.368 − 0.444 , while
= 0.5
EI eff
EI gross
= 0.223 .
Scuola Don Bosco, assuming EI = 100% EIgross, the periods of the first three modes of interest
are, respectively, T1 = 0.587 s, T2 = 0.539 s and T3 = 0.501 s. All three periods lie in the
descending branch of the response spectrum (TC = 0.5 s). Decreasing the members stiffness,
the periods shift, respectively, to T1 = 0.824 s, T2 = 0.758 s, T3 = 0.706 s in the case of EI =
50% EIgross, and to T1 = 1.155-1.216 s, T2 = 1.038-1.124 s, T3 = 0.969-1.050 s, in the case of
EI = EIeff computed through Eq. (2.8). Therefore, decreasing EI, the spectral ordinates
corresponding to the first three modes of interest decrease too, yielding smaller seismic
forces. A consequence of this reduction is that, assuming EI = EIeff, the first condition of
applicability of linear methods will become less restrictive than the verification on the base of
EI = 50-100% EIgross.
therefore, it will experience ρmax also according to Eurocode 8. Therefore, for the Scuola Don
Bosco, there is no reason to consider the joint equilibrium in order to determine the value of
204
recommended.
Considering that:
•
•
•
Seismic Code (Sec. 2.2.2). The results of the check (Table 8.2) show that the shear capacity of
every structural member (evaluated through the mean values of the material properties) is
larger that the respective seismic demand. Hence, for the Scuola Don Bosco, the second
condition of applicability of linear methods is satisfied.
FIRST CONDITION
SECOND CONDITION
EI = 100% EIgross
OPCM 3431
EC8
OPCM 3431
56.84
56.84
Number of Sections
910
ρmax
ρmax
2.03
1.00
Not Verified
0
ρmin
ρmin
ρmax/ρmin
27.95
ρmax/ρmin
56.61
% Not Verified
0
EI = 50% EIgross
OPCM 3431
EC8
OPCM 3431
39.44
39.44
Number of Sections
910
ρmax
ρmax
2.00
1.00
Not Verified
0
ρmin
ρmin
ρmax/ρmin
19.70
ρmax/ρmin
39.29
% Not Verified
0
OPCM 3431
EC8
OPCM 3431
25.26
25.26
Number of Sections
910
ρmax
ρmax
2.00
1.01
Not Verified
0
ρmin
ρmin
ρmax/ρmin
12.61
ρmax/ρmin
24.96
% Not Verified
0
OPCM 3431
EC8
OPCM 3431
20.90
20.90
Number of Sections
910
ρmax
ρmax
2.01
1.00
Not Verified
0
ρmin
ρmin
ρmax/ρmin
10.40
ρmax/ρmin
20.88
% Not Verified
0
OPCM 3431
EC8
OPCM 3431
24.13
24.13
ρmax
910
ρmax
ρmax
2.00
1.00
Not Verified
0
ρmin
ρmin
ρmax/ρmin
12.06
ρmax/ρmin
24.13
% Not Verified
0
205
On the other hand, since the values of ρmax/ρmin are much larger than the limits of the Codes,
the first condition of applicability was not satisfied. Hence, according to both Codes, the
results furnished by linear analyses should be judged as not reliable. Anyway, as already
explained in Sec. 5.1.3, considering only ρmax/ρmin to evaluate the possibility of accepting the
results of linear analyses may not be judged as an exhaustive method, since it is possible that
few structural members, much weaker than the others, will invalidate the analysis. Other
studies are required to determine if the conclusion to reject linear methods is unavoidable or if
the dynamic linear analysis may be considered useful at least to catch the global seismic
response of the building and to express a judgment about the need of retrofitting. Although
the conditions of applicability are not satisfied, the dynamic linear analysis was performed,
since the principal aim of this work is to compare the different assessment procedures at
structural member level, suggesting eventually simplified solutions. The evaluation of the
reliability of the different methods of analysis will be object of future research works.
theoretical (Eq. (3.22)) θu, with the same aims already described in Sec. 5.1.3. All results refer
to the percentages of structural members which do not satisfy the verification. The percentage
was computed for all structural members, for beam, for columns and, also, for columns of
each floor separately, in order to check the possible development of soft-storey mechanisms.
The sensitivity of the assessment procedure to EI is shown from Figure 8.6 to Figure 8.13. All
charts refer to the percentages of members which do not satisfy the verification according to
both Codes and both empirical (Eq. (3.15)) and theoretical (Eq. (3.22)) θu. Figure 8.6 - Figure
8.7 - Figure 8.8 - Figure 8.9 show that, decreasing EI, the percentage of members which do
not satisfy the verification increase sensibly, for both beams and columns. This tendency is
more evident if Eq. (3.15) is used to assess θu. Therefore, assuming EIeff computed though Eq.
(2.8) as the most accurate choice of EI, it is clear that, for the Scuola Don Bosco, both 100%
EIgross and 50% EIgross yield unconservative results. Figure 8.10 - Figure 8.11 - Figure 8.12 Figure 8.13 show that results close to each other were obtained considering the mean EIeff
rectangular beams and for columns of each floor. In particular, the results are very close to
each other if Eq. (3.22) is used to assess θu, while a difference can be appreciated if Eq. (3.15)
is assumed. However, this difference is small, except for the columns of the ground floor.
100
90
100
100%EI
90
50%EI
80
100%EI
50%EI
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
206
100
90
100
100%EI
90
50%EI
80
50%EI
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
Ground Floor
Col
1st Floor Col
Roof Floor Col
EI_eff_uniform
90
EI_eff_uniform
EI_eff_beam_col
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_precise
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
100
EI_eff_uniform
90
EI_eff_beam_col
80
All Columns
100
EI_eff_beam_col
90
All Beams
100
80
EI_eff
0
All Elements
90
100%EI
EI_eff_uniform
EI_eff_beam_col
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_precise
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
proposed in both Codes are very similar to each other, Eurocode 8 suggests more complex
calculations (Sec. 5.1.3). Concerning the empirical θu (Eq. (3.15)), since the results obtained
from the two Codes are very close to each other, there is no need to follow the procedure
suggested in Eurocode 8 and, hence, the simpler approach proposed in the Italian Seismic
Code is recommended. Concerning the theoretical θu (Eq. (3.22)), instead, the percentages of
members which do not satisfy the verification according to Eurocode 8 are larger (difference
207
≈ 25%). Hence, assuming the procedure of Eurocode 8 as correct, the simpler approach of the
Italian Seismic Code yields quite inaccurate results.
100
100
90
OPCM
90
OPCM
80
EC8
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
(Eq. (3.15)) and theoretical (Eq. (3.22)) θu. Concerning both empirical and theoretical θu, the
use of Ls = L/2 yields a small decrease (< 10%) in the percentage of not verified members
with respect to Eq. (3.1). Hence, the procedure may be simplified.
100
100
90
Ls=M/V
80
Ls=L/2
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
All Elements
Roof Floor Col
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
80
Ls=L/2
0
0
90
Ls=M/V
100
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
208
The comparison between the results obtained using seismic and gravity N is shown from
(Eq. (3.22)) θu. Concerning both empirical and theoretical θu, the gravity N yields a decrease
in the percentage members which do not satisfy the verification with respect to the seismic N.
The difference between the results of the two approaches is ≈ 20% for the Italian Seismic
Code and 15% for Eurocode 8. In particular, the decrease is larger for the columns of the
ground floor, while it is negligible for the columns of the roof floor. In fact, the seismic ΔN is
maximum at the base and becomes null at the top of the building. Although the difference
between the results obtained assuming the gravity and the seismic N is not negligible, it must
be remarked that the seismic ΔN considered in this case is related with the hypothesis of
indefinitely elastic behaviour and, hence, is larger than ΔN obtained when the development of
nonlinear mechanisms is accounted for (Sec. 3.2.2). Hence, further future investigations are
needed to understand if the procedure can be simplified.
100
100
90
N_Grav
90
N_Grav
80
N_Seism
80
N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
All Beams
100
90
N_Grav
90
N_Grav
80
N_Seism
80
N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
charts refer to the percentages of members which do not satisfy the verification according to
the theoretical θu (Eq. (3.22)) and to both Codes. Empirical and theoretical φy yield very close
results. Therefore, for the Scuola Don Bosco, the empirical formulas calibrated for new
209
100
90
100
FI_Theoric
90
FI_Fardis
80
FI_Theoric
FI_Fardis
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
FI_Priestley
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
theoretical form, EIeff, EC8, different φy
do not satisfy the verification according to both Codes and both empirical (Eq. (3.15)) and
theoretical (Eq. (3.22)) θu. Concerning both empirical and theoretical θu, the simplified
approach yields a decrease in the percentage of members which do not satisfy the verification.
According to the Italian Seismic Code, the difference between the results of the two
approaches is ≈ 30-35 %, while, following Eurocode 8, the difference is ≈ 20-25 %. Further
future investigations are required to understand if the difference between the results of the
correct and the simplified approaches is due to the use of ΔN related with the hypothesis of
indefinitely elastic behaviour, instead of ΔN obtained accounting for the development of
nonlinear mechanisms (Sec. 3.2.2).
(Eq. (3.22)) θu. All charts show that the use of the bidimensional failure curve yields an
increase in the percentage of members which do not satisfy the verification. The increase is
large if Eq. (3.15) is assumed (≈ 25%, considering all columns), while, if Eq. (3.22) is used,
the increase is smaller (≈ 15%, considering all columns). Therefore, at least if the assessment
is performed according to Eq. (3.15), the use of the bidimensional failure curve is advised.
100
100
90
Ls=L/2; N_Grav
90
Ls=L/2; N_Grav
80
Ls=M/V; N_Seism
80
Ls=M/V; N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
emp. form, EIeff, OPCM, correct vs. simplified
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
210
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
90
80
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
All Beams
100
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
Figure 8.30. Ductile check: dynamic linear an. EIeff,
All Elements
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
All Beams
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
emp. form., EIeff, OPCM, uni- vs. bi-axial bending
All Elements
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
Figure 8.33. Ductile check: dynamic linear an. emp.
form., EIeff, EC8, uni- vs. bi-axial bending
100
All Beams
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
211
In the following charts, the results of the assessment of brittle mechanisms will be shown,
according to both Codes, with the same aims already described for the Sede Comunale in Sec.
5.1.3.
to both Codes. Figure 8.36 and Figure 8.37 show that the percentage of members which do
not satisfy the verification decreases with decreasing EI. Therefore, assessing the brittle
mechanisms using a fixed fraction of EIgross between 50 and 100% yields conservative results.
In particular, according to Eurocode 8, the increase in the percentages of members which do
not satisfy the verification is < 10%. Hence, for the Scuola Don Bosco, if the
recommendations of Eurocode 8 are followed, assuming a fixed fraction of EIgross between
50% and 100% may be considered a suitable choice, since the assessment of EIeff according to
Eq. (2.8) is much longer and yields small improvements in the results. Figure 8.38 and Figure
8.39 show that results very close to each other were obtained considering the mean EIeff
rectangular beams and for columns of each floor.
100
90
100
100%EI
100%EI
90
50%EI
50%EI
80
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
OPCM, different EI
All Elements
All Beams
All Columns
Ground Floor
Col
Roof Floor Col
EC8, different EI
100
100
EI_eff_uniform
EI_eff_uniform
90
90
EI_eff_beam-col
80
EI_eff_precise
EI_eff_beam-col
80
1st Floor Col
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
OPCM, different EI
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
EC8, different EI
The comparison between the results obtained using gravity and seismic N is shown in Figure
8.40 and Figure 8.41. The charts refer to the percentages of members which do not satisfy the
212
verification according to both Codes. Concerning Eurocode 8, the gravity N yields slightly
unconservative results (the difference between the percentages of members which do not
satisfy the verification obtained assuming the gravity and the seismic N is < 20%, considering
all columns). Further future investigations are required to understand if this difference is due
to the use of ΔN related with the hypothesis of indefinitely elastic behaviour, instead of ΔN
obtained accounting for the development of nonlinear mechanisms (Sec. 3.2.2). Concerning
the Italian Seismic Code, the difference in the percentage of members which do not satisfy the
verification assuming the gravity and the seismic N is much larger (> 100%) and cannot be
neglected. This difference is due to the fact that the concrete shear resistance of columns will
become null if the columns experience a tensile N (see Fig. 3.20).
100
100
N_Grav
90
N_Grav
80
80
N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
90
N_Seism
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
theoretical φy yield very close (practically identical) results. Therefore, for the Scuola Don
Bosco, the empirical formulas may be applied without any loss of accuracy.
Seismic Code does not depend on Ls). Ls = L/2 and Eq. (3.1) yield results close to each other.
In particular, Ls = L/2 yields a small increase (≈ 5%) in the percentage of beams and a small
decrease (≈ 15%) in the percentage of columns that do not satisfy the verification. Hence, for
the Scuola Don Bosco, the procedure may be simplified.
members which do not satisfy the requirements. Therefore, for the Scuola Don Bosco, the use
of the procedure suggested in Eurocode 8 is recommended.
213
The chart shows that the simplified and the correct approaches yield results very close to each
other. Hence, for the Scuola Don Bosco, the procedure may be simplified.
100
FI_Theoric
100
90
FI_Fardis
90
Ls=M/V
80
FI_Priestley
80
Ls=L/2
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Roof Floor Col
100
OPCM; N_Grav
90
OPCM; N_Seism
80
1st Floor Col
100
90
Ground Floor
Col
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
Figure 8.45. Brittle check: dynamic linear an., EC8,
correct vs. simplified approach
Mander et al. [1988], later modified by Martinez-Rueda and Elnashai [1997]. The following
four parameters were defined in order to describe the mechanical characteristics of the
material (Figure 8.46):
•
concrete compressive strength, fc = 16.6 MPa.
214
•
•
•
•
•
•
Figure 8.47. Steel: bilinear (strain hard.) model
subdivided into four elements. This subdivision allows to take into account the change in the
Since the software SeismoStruct does not allow to model 2D elements, the flexibility of the
According to this procedure, each slab should be substituted by two cross braces, connected to
the corners of the slab through hinges, in order to avoid any moment transfer. On the other
hand, since most slabs of the Scuola Don Bosco are characterized by columns located not
only at the corners but also along each side, modelling the slab stiffness using only two cross
braces was not judged as a suitable choice. In fact, in this way, the ends of the columns where
no braces are located would not be restrained horizontally, as it would happen if the slab were
modelled with a 2-D element. Hence, in order to bypass this shortcoming, the slabs which
present columns not only at the corners but also along each side were subdivided into smaller
areas. For each area the equivalent braces were dimensioned. According to this procedure,
215
each end of each column is restrained, as it happens actually, leading to avoid local spurious
modes and to improve the accuracy of the results (Figure 8.48).
Figure 8.48. Top view of the Scuola Don Bosco: cross braces locations
The procedure adopted for estimating the dimensions of the brace is shown for a slab (L =
2140 mm; L’ = 6360 mm) of the Scuola Don Bosco. The thickness of the equivalent slab used
in the SAP model is 75 mm (Sec. 8.1.1). Therefore:
(
)
I = 2140 3 ⋅ 75 12 = 6.125 ⋅ 1010 mm 4
As = 5 6 A = 5 6 ⋅ (2140 ⋅ 75) = 133750mm 2
Gc = 9016 MPa
[
(
]
)
K b = 2140 3 12 ⋅ 21638 ⋅ 6.125 ⋅ 1010 + 2140 (133750 ⋅ 9016 )
−1
= 418270 N / mm
Once known Kb and l = L' 2 + L2 = 2140 2 + 6360 2 = 6710mm , the area of the brace was
406 mm was determined.
are negligible. Hence, the SeismoStruct model was judged consistent with the SAP model.
Periods [s]
I Mode
II Mode
III Mode
SeismoStruct
0.826
0.760
0.703
SAP
0.824
0.758
0.706
216
pattern of lateral forces, applied in positive and negative X and Y directions).
In order to perform the analysis:
•
•
•
braces used to model the in-plan stiffness are omitted, in order to make the deformed shapes
clearer. All deformed shapes refer to the SD LS.
•
217
•
d *y = 49mm (Eq. (2.14)); K * = 30898MPa (Eq. (2.15)); T * = 0.806 s (Eq. (2.16)).
•
•
Γx = 1.232 (Eq. (2.13)).
•
•
1600 Vb,SD
1400
1400
1200
1200 Vb,LD
1000
1000
Vb (kN)
Vb (kN)
1600
800
600
Vb,NC
800
600
400
400
200
200
*
d
0
0
50
m
100
0
150
200
250
0
300
Δ SD
Δ LD
*
d
y
50
100
200
250
300
1600
1200
1400
1000
1200
Vb (kN)
800
Vb (kN)
Δ NC
150
600
400
Vb,SD
Vb,NC
Vb,LD
1000
800
600
400
200
200
*
d
0
0
50
100
m
150
200
250
300
350
400
Δ SD
Δ LD
0
*
d
y
0
50
100
150
Δ NC
200
250
300
350
400
•
•
d *y = 61mm (Eq. (2.14)); K * = 19770 MPa (Eq. (2.15)); T * = 1.010 s (Eq. (2.16)).
•
For the SD LS: d t* = d t = 137 mm (Eqs. (2.17) – (2.20)); Vb = 1193kN .
•
Γy = 1.282 (Eq. (2.13)).
218
•
•
For the SD LS: d t = 194mm (Eqs. (2.17) – (2.20)); Vb = 1068kN
1400
1200
1200
1000
1000
Vb (kN)
Vb (kN)
1400
800
600
Vb,SD
Vb,NC
Vb,LD
800
600
400
400
200
200
0
Δ LD
*
*
d
0
d
y
50
100
m
150
0
200
250
300
0
350
50
150
200
250
300
350
400
1200
900
800
Vb,SD
Vb,NC
Vb,LD
1000
700
800
Vb (kN)
600
Vb (kN)
Δ NC
Δ SD
100
500
400
600
400
300
200
200
100
*
d
0
0
d
50
100
Δ LD
0
*
y
m
150
200
250
300
0
350
50
100
Δ NC
Δ SD
150
200
250
300
350
400
Figure 8.60. Pushover Modal Y-dir, MDOF capacity
curve: DL, SD and NC LS
SD LS for the pushover in X and in Y, respectively
CAPACITY CURVES Y -MDOF
1600 V
b,SD,UNIF.
1400
1400
1200
Vb,SD,MOD.
1200
Vb,SD,MOD
1000
1000
UNIFORM
Vb (kN)
Vb (kN)
Vb,SD,UNIF
MODAL
800
800
UNIFORM
600
600
MODAL
400
400
200
200
0
50
100
ΔSD,UNIF
ΔSD,MOD.
ΔSD,UNIF.
0
150
ΔSD,MOD
0
200
250
300
0
50
100
150
200
250
300
350
400
described for the assessment based on the dynamic linear analysis in Sec. 5.1.3.
219
proposed in both Codes are very similar to each other, Eurocode 8 suggests more complex
calculations (Sec. 5.1.3). Moreover, also the way to compute the demand is different. In fact,
the Italian Seismic Code proposes to consider separately the results obtained from each
pushover, while Eurocode 8 suggests to consider each demand quantity as the outcome of the
combination of the values furnished by a pushover in X and a pushover in Y (Eq. (2.23)).
Concerning the empirical θu (Eq. (3.15)), the results obtained from Eurocode 8 are slightly
conservative and very close to those obtained from the Italian Seismic Code. Hence, for the
Scuola Don Bosco, there is no need to use the procedure suggested in Eurocode 8 and the
simpler procedure proposed in the Italian Seismic Code is recommended. Concerning the
theoretical θu, instead, the percentage of members which do not satisfy the verification
obtained from Eurocode 8 is larger than that obtained from the Italian Seismic Code
(difference > 30%). This difference is larger than that experienced by assessing the building
on the base of the dynamic linear analysis and it is due mainly to the way of computing the
seismic demand. Further future investigations are needed to determine whether Eq. (2.23)
yields more reliable demands values than those obtained considered each pushover separately.
100
100
90
OPCM
90
OPCM
80
EC8
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
to the theoretical θu (Eq. (3.22)) and to both Codes. Empirical and theoretical φy yield very
close results. Therefore, for the Scuola Don Bosco, the empirical formulas calibrated for new
(Eq. (3.15)) and theoretical (Eq. (3.22)) θu. Concerning the empirical θu, Ls = L/2 and Eq.
(3.1) yield results close to each other. Hence, the procedure may be simplified. Concerning
the theoretical θu, instead, the decrease in the percentage of members which do not satisfy the
verification obtained using Ls = L/2 is huge (> 50%). It is a proof that Eq. (3.22) is very
sensitive to the value of Ls. This result was not evident in the linear analysis and this can be
220
justified considering that the inelastic values of Ls are likely to change with respect to their
elastic values, possibly resulting in very small values, leading to θu = 0.
100
90
100
FI_Theoric
90
FI_Fardis
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
Ground Floor
Col
1st Floor Col
Roof Floor Col
Ls=M/V
90
Ls=L/2
80
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
80
All Columns
100
70
90
All Beams
100
80
FI_Priestley
0
All Elements
90
FI_Theoric
FI_Fardis
80
100
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
(Eq. (3.22)) θu. All charts show that gravity and seismic N yield close results. In particular,
assuming Eq. (3.15), the results are almost identical, while considering Eq. (3.22), the gravity
221
N yields slightly unconservative results (difference ≤ 20%, considering all columns). Hence,
for both empirical and theoretical θu of both Codes, the procedure may be simplified. This
conclusion clashes with the results based on the dynamic linear analysis. This can be justified
100
100
90
80
N_Grav
90
N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
100
80
N_Seism
0
All Elements
90
N_Grav
N_Grav
90
N_Grav
N_Seism
80
N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
do not satisfy the verification according to both Codes and both empirical (Eq. (3.15)) and
theoretical (Eq. (3.22)) θu. Concerning the empirical θu, the simplified and the correct
approaches yield very close results. Hence, for the Scuola Don Bosco, the assessment
procedure may be simplified. Concerning the theoretical θu, instead, the decrease in the
percentage of members which do not satisfy the verification obtained using the simplified
approach is noticeable (even larger than 50%). Figure 8.79 and Figure 8.80 illustrate the
results obtained considering all approaches of both θu, following the recommendations of the
Italian Seismic Code and of Eurocode 8, respectively. The charts show that, concerning
beams, the results obtained applying both approaches of the empirical θu and the simplified
approach of the theoretical θu are very close to each other, while the correct approach of the
theoretical θu overestimates the results. Concerning columns, instead, the results obtained
applying both approaches of the empirical θu and the correct approach of the theoretical θu are
222
close to each other, while the simplified approach of the theoretical θu underestimates the
results. Globally, it appears that, according to the Italian Seismic Code, the simplified
approach of the theoretical θu underestimates the results with respect to the other approaches,
while, according to Eurocode 8, the correct approach of the theoretical θu overestimates the
results. Therefore, for the Scuola Don Bosco, it may be concluded that the assessment
according to the empirical θu can be simplified, while, concerning the theoretical θu, the charts
confirm that Eq. (3.22) is too sensitive to the value of Ls.
conventional uniaxial check is recommended for the assessment of the Scuola Don Bosco, as
it yields enough accurate results and is much faster that the bidimensional check.
100
90
80
100
Ls=L/2; N_Grav
90
Ls=L/2; N_Grav
Ls=M/V; N_Seism
80
Ls=M/V; N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
All Beams
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
223
DUCTILE CHECK STATIC NONLIN. - OPCM
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
All Beams
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
100
90
MONO-AXIAL BENDING
90
80
BI-AXIAL BENDING
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
MONO-AXIAL BENDING
BI-AXIAL BENDING
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
theoretical form, OPCM, uni- vs. bi-axial bending
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
theoretical form, EC8, uni- vs. bi-axial bending
sensitivity to EI) already described for the dynamic linear analysis in Sec. 5.1.3.
224
satisfy the verification according to both Codes. The charts show that, considering the Italian
Seismic Code, the gravity N yields quite unconservative results. According to Eurocode 8,
instead, gravity and seismic N yield results very close to each other. Hence, for the Scuola
Don Bosco, the procedure suggested by Eurocode 8 may be simplified. This conclusion
clashes with the results based on the dynamic linear analysis. The fact that the results obtained
using the static nonlinear analysis are less sensitive to the choice of N can be justified
100
100
N_Grav
90
N_Grav
80
80
N_Seism
70
70
60
60
50
50
40
40
30
30
20
20
10
10
90
N_Seism
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
φy (Eqs. (3.9) – (3.10)) and the empirical φy (Eqs. (3.11) to (3.14)). The chart refers to the
theoretical φy yield very close (practically identical) results. Therefore, for the Scuola Don
Bosco, the empirical formulas may be applied without any loss of accuracy.
Ls (Eq. (3.1)) and Ls = L/2. The chart refers to the percentages of members which do not
Seismic Code does not depend on Ls). The use of Ls = L/2 yields a very small (practically
respect to Eq. (3.1). Hence, for the Scuola Don Bosco, the procedure may be simplified.
which do not satisfy the requirements. Therefore, for the Scuola Don Bosco, the procedure
225
respect to the correct approach, leading to a safe side assessment. Hence, for the Scuola Don
Bosco, the procedure may be simplified.
100
90
FI_Theoric
FI_Fardis
FI_Priestley
100
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
EC8, different φy
90
Ls=L/2
0
All Elements
100
Ls=M/V
90
80
OPCM; N_Grav
OPCM; N_Seism
EC8
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
The nonlinear dynamic analysis of the Scuola Don Bosco was carried out on the base of eight
226
The equivalent viscous damping matrix C was assumed proportional only to instantaneous
tangent stiffness K, updated at each step but not at every iteration (Figure 8.92):
[C ] = a1 [K ] where a1 = T1ξ π .
Assuming T1 = 0.587 s and ξ = 0.05 ⇒ a1 = T1ξ π = (0.587 ⋅ 0.05) π = 0.00934 .
1.1
1
Mag=6.0
Mag=6.5
Mag=6.5_long
Mag=7.0
Mean
Target
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
T [s]
2
2.5
3
Figure 8.92. Equivalent viscous damping properties
respectively. The time history of the displacement of the control node (chosen at the centre of
mass of the top floor) are shown in Figure 8.97 and Figure 8.98. Applying the accelerogram
with its full intensity in X and scaled to 30% in Y, the maximum absolute values of the
control node displacement are 109 mm in X and 42 mm in Y, while applying the
accelerogram with its full intensity in Y and scaled to 30 % in X, the maximum absolute
values of the control node displacement are 26 mm in X and 129 mm in Y.
227
120
100
DISPL. IN X
DISPL. IN Y
DISPL [mm]
DISPL [mm]
80
60
40
20
0
-20
-40
-60
-80
-100
-120
0
2
4
6
8
10
12
14
140
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
-120
-140
DISPL. IN X
DISPL. IN Y
0
16
2
4
6
8
10
12
14
16
T [s]
T [s]
The following charts show the results of the assessment of ductile mechanisms. The
assessment was performed according to both Codes and both empirical (Eq. (3.15)) and
nonlinear analysis in Sec. 5.2.2.
Figure 8.99 and Figure 8.100 show the comparison between the percentages of members
which do not satisfy the verification according to both Codes. Although the formulas
proposed in the Italian Seismic Code and in Eurocode 8 are very similar, Eurocode 8 suggests
more complex calculations (Sec. 5.1.3). Concerning the empirical θu, the results obtained
from Eurocode 8 are very close to those obtained from the Italian Seismic Code. Hence, there
is no need to use the procedure suggested in Eurocode 8 and the simpler procedure proposed
in the Italian Seismic Code is recommended. Concerning the theoretical θu, instead, the results
obtained from Eurocode 8 are conservative (difference ≈ 30%).
100
100
90
OPCM
90
OPCM
80
EC8
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
to the theoretical θu (Eq. (3.22)) and to both Codes. The theoretical and the empirical φy yield
nearly identical results. Hence, for the Scuola Don Bosco, the empirical formulas calibrated
for new seismically designed structures may be applied without any loss of accuracy.
228
(Eq. (3.15)) and theoretical (Eq. (3.22)) θu. Considering both empirical and theoretical θu, Ls =
L/2 underestimates the results with respect to Eq. (3.1). Concerning the theoretical θu, the
decrease in the percentage of members which do not satisfy the verification obtained using Ls
= L/2 shows that Eq. (3.22) is very sensitive to the value of Ls.
100
90
100
FI_Theoric
90
FI_Fardis
80
FI_Fardis
80
FI_Priestley
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
Ground Floor
Col
1st Floor Col
Roof Floor Col
Ls=M/V
90
Ls=L/2
80
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
80
All Columns
100
70
90
All Beams
100
80
FI_Priestley
0
All Elements
90
FI_Theoric
100
Ls=M/V
90
Ls=L/2
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Ls=M/V
Ls=L/2
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
229
(gravity N; Ls = L/2; empirical φy). The charts refer to the percentages of members which do
not satisfy the verification according to the theoretical θu (Eq. (3.22)) and to both Codes. The
decrease in the percentage of members which do not satisfy the verification obtained using the
simplified approach is noticeable (> 40% for the Italian Seismic Code and ≈ 30% for
Eurocode 8) and can be ascribed mainly to the fact that Eq. (3.22) is very sensitive to the
value of Ls.
conventional uniaxial check is recommended for the assessment of the Scuola Don Bosco, as
it yields enough accurate results and is much faster that the bidimensional check.
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
All Beams
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
230
100
100
90
MONO-AXIAL BENDING
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
the static nonlinear analysis (Sec. 5.2.2).
Empirical and theoretical φy yield very close results. Therefore, for the Scuola Don Bosco, the
Seismic Code does not depend on Ls). The use of Ls = L/2 results in a small increase (< 10%)
in the percentage of members which do not satisfy the verification with respect to Eq. (3.1),
leading to a safe side assessment. Hence, for the Scuola Don Bosco, the procedure may be
simplified.
requirements. Hence, for the Scuola Don Bosco, the procedure suggested in Eurocode 8 is
recommended.
yields slightly conservative results (small increase, < 10%, in percentage of members which
do not satisfy the verification with respect to the correct approach). Hence, for the Scuola Don
Bosco, the procedure may be simplified.
231
100
FI_Theoric
FI_Fardis
90
100
90
Ls=M/V
80
80
Ls=L/2
70
70
60
60
50
50
40
40
30
30
20
20
10
10
FI_Priestley
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
All Elements
Roof Floor Col
EC8, different φy
Ground Floor
Col
1st Floor Col
Roof Floor Col
100
80
All Columns
100
90
All Beams
OPCM
90
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
OPCM vs. EC8
All Elements
All Beams
All Columns
Ground Floor
Col
1st Floor Col
Roof Floor Col
232
Chapter 9. Summary of the Results and Conclusions
9 Summary of the Results and Conclusions
In this Chapter, the results obtained from the assessment of each of the four considered
buildings will be, first, summarized. The results will be, then, compared, in order to see if the
conclusions drawn for each building separately can be generalized, with the purpose of giving
an answer to all the questions introduced in Chapters 2 and 3.
9.1 Summary of the Results of Sede Comunale (Vagli Sotto)
The Sede Comunale located in Vagli Sotto (Tuscany) is a compact and roughly rectangular
building, irregular in plan and in elevation. It is characterized by low values of fcm (8.3 MPa)
and Ec (17174 MPa), resulting in a rather flexible structure (T1 varying from 0.77 s,
considering EIgross, to 1.18-1.23 s, considering EIeff).
All the methods of analysis (linear dynamic, nonlinear static and dynamic) proposed in the
Italian Seismic Code [OPCM 3431] and in Eurocode 8 were considered in order to assess
both ductile and brittle responses of the structural members (beams and columns).
First, the applicability conditions of the linear methods suggested by both Codes were
checked, considering three different solutions to evaluate the members stiffness EI (100%
EIgross, 50% EIgross and EIeff computed according to Eq. (2.8)), in order to evaluate if the
choice of EI affects strongly the results.
The check on the first condition shows that:
•
for both Codes, for all the considered choices of EI, ρmax/ρmin exceeds the threshold for
the applicability of linear methods.
•
The value of ρmax/ρmin reduces when the members stiffness EI decreases (Table 5.3).
This reduction is not linear with the reduction of EI and depends on both structural
stiffness and shape of the elastic response spectrum (in particular, on the value of
corner period, TC). Since the Sede Comunale is rather flexible (T1 >> Tc), the reduction
of ρmax/ρmin obtained considering EIeff instead of EIgross is large.
•
Although Eurocode 8 suggests to take into account both member and joint equilibrium
to determine ρmax/ρmin, the results in terms of ρmax are almost identical to those
achieved applying the recommendations of the Italian Seismic Code. It suggests not to
233
consider the joint equilibrium in order to determine the value of ρmax/ρmin and to adopt
the simpler and faster procedure suggested in the Italian Seismic Code.
The check on the second condition (considered only in the Italian Seismic Code) shows a
percentage between 10-11% (for EI = EIeff) and 13% (for EI = EIgross) of structural members
which do not satisfy the recommendations (Table 5.3). Hence, also the second condition of
applicability of linear analyses is sensitive to the choice of EI, but less than the first condition.
In analogy with the first condition, this second condition will become more restrictive too, if
large values of EI are assumed.
In this Section, the results obtained assessing the ductile response of the structural members
are shown, with particular reference to the following aims, already explained in Chapters 2
and 3:
(i)
checking if the two considered Codes yield close results. Although the formulas
proposed in both Codes are very similar to each other, Eurocode 8 suggests more
complex calculations, in both computation of the flexural deformation at yielding and
evaluation of Ls.
Concerning the empirical θu (Eq. (3.15)), for all methods of analysis, the results
obtained from the two Codes are close to each other (Figure 5.13, Figure 5.63, Figure
5.99).
Concerning the theoretical θu (Eq. (3.22)), according to both linear and nonlinear
dynamic analyses, the results obtained from Eurocode 8 are close to those obtained
from the Italian Seismic Code (Figure 5.14, Figure 5.100) while, according to the
nonlinear static analysis, the percentage of members which do not satisfy the
verification obtained from Eurocode 8 is larger than that obtained from the Italian
Seismic Code (Figure 5.64). This difference is due, in particular, to the way of
computing the seismic demand (the Italian Seismic Code considers separately the
results of each pushover, while Eurocode 8 considers each demand quantity as the
outcome of the combination of the values furnished by a pushover in X and a
pushover in Y).
(ii)
Decreasing EI, the percentage of members which do not satisfy the verification
increases. This tendency will be more evident if the empirical θu is considered. Hence,
234
assuming EIeff computed through Eq. (2.8) as the most accurate choice of EI, both
100% EIgross and 50% EIgross yield unconservative results (Figure 5.5 to Figure 5.8).
Results close to each other are obtained considering the mean EIeff computed i) for all
members, ii) separately for beams and columns, or iii) separately for T and rectangular
beams and for columns of each floor (Figure 5.9 to Figure 5.12).
(iii) Testing if possible simplifications in the procedures can be applied without any
Regarding the possibility of using the gravity N instead of the seismic N:
•
concerning the empirical θu, for all methods of analysis, gravity and seismic N
yield very close results (Figure 5.19 - Figure 5.20, Figure 5.71 - Figure 5.72).
•
Concerning the theoretical θu, according to both static and dynamic nonlinear
analyses, gravity and seismic N yield very close results (Figure 5.73 - Figure
5.74). According to the dynamic linear analysis, instead, the decrease in the
percentage of members which do not satisfy the verification obtained using the
gravity N is up to 20% (Figure 5.21 - Figure 5.22). This can be justified
considering that the seismic ΔN experienced in a linear analysis is, in general,
(much) larger than the actual ΔN obtained accounting for the development of
nonlinear mechanisms.
Regarding the possibility of using Ls = L/2 instead of Ls from Eq. (3.1):
•
concerning the empirical θu, according to both static and dynamic nonlinear
analyses, the use of Ls = L/2 yields results very close to those obtained
considering Eq. (3.1) (Figure 5.67 - Figure 5.68, Figure 5.103 - Figure 5.104).
According to the dynamic linear analysis, the use of Ls = L/2 yields a small
decrease in the percentage of members which do not satisfy the verification
with respect to Eq. (3.1) (Figure 5.15 - Figure 5.16).
•
analyses, the decrease in the percentage of members which do not satisfy the
verification obtained using Ls = L/2 is huge, even larger than 100% (Figure
5.69 - Figure 5.70, Figure 5.105 - Figure 5.106). This shows that Eq. (3.22) is
very sensitive to the value of Ls. According to the dynamic linear analysis,
instead, a small decrease (less than 20%) in the percentage of members which
do not satisfy the verification obtained using Ls = L/2 can be appreciated
(Figure 5.17 - Figure 5.18).
235
Empirical and theoretical φy yield very close results, for all methods of analysis
(Figure 5.23 - Figure 5.24, Figure 5.65 - Figure 5.66, Figure 5.101 - Figure 5.102).
Considering the correct procedure (seismic N; Ls from Eq. (3.1); theoretical φy) and the
simplified procedure (gravity N; Ls = L/2; empirical φy):
•
concerning the empirical θu, for all methods of analysis, the simplified and the
correct approaches yield close results (Figure 5.25 - Figure 5.26, Figure 5.75 Figure 5.76).
•
concerning the theoretical θu, for all methods of analysis, the decrease in the
simplified approach is noticeable and can be ascribed mainly to the fact that
Eq. (3.22) is very sensitive to the value of Ls (Figure 5.27 - Figure 5.28, Figure
5.77 - Figure 5.78, Figure 5.107 - Figure 5.108).
(iv) Checking if the two different formulas of θu (empirical, Eq. (3.15) and theoretical, Eq.
(3.22)) yield close results.
If the dynamic linear analysis is performed, empirical and theoretical θu yield close
results when the correct approach is considered, while, concerning the simplified
procedure, the theoretical θu yields unconservative results (Figure 5.29 - Figure 5.30).
For both nonlinear static and dynamic analyses, the results obtained applying both
approaches of the empirical θu and the simplified approach of the theoretical θu are
close to each other, but very different from those obtained considering the correct
approach of the theoretical θu (Figure 5.79 - Figure 5.80, Figure 5.109 - Figure 5.110).
The deformed shapes show a soft-storey mechanism that was correctly captured using
both correct and simplified approaches of the empirical θu and the simplified approach
of the theoretical θu, while it was not detected using the correct approach of the
theoretical θu. It suggest that, concerning the theoretical evaluation of θu, there should
be the need of considering Ls = L/2, since Eq. (3.1) yields results too sensitive to the
values of Ls.
(v)
Understanding if the definition of a bidimensional failure curve is required.
For both empirical and theoretical θu, for all methods of analysis, the definition of a
bidimensional failure curve does not improve on the accuracy of the results (Figure
5.31 to Figure 5.34, Figure 5.81 to Figure 5.84, Figure 5.111 to Figure 5.114).
In this Section, the results obtained assessing the brittle response of the structural members
and 3:
236
(i)
checking if the two Codes yield close results. Considering that Eurocode 8 suggests a
more complex procedure, the target is to understand if the recommendations of the
Italian Seismic Code allow to obtain accurate results.
For all methods of analysis, assessing the brittle mechanisms according to the
recommendations of the Italian Seismic Code yields results which grossly
underestimate the percentage of members which do not satisfy the requirements
(Figure 5.43, Figure 5.89, Figure 5.117).
In the light of these considerations, only the results obtained according to Eurocode 8
will be shown in the following part of this Section.
(ii)
decreases too (Figure 5.36). Hence, assessing the brittle mechanisms using a fixed
fraction of EIgross (50 - 100%) yields conservative results. Moreover, the difference in
percentage of members which do not satisfy the verification assuming 100% EIgross
and mean EIeff (Eq. (2.8)) is less than 10%.
Results very close to each other are obtained considering the mean EIeff computed i)
for all members, ii) separately for beams and columns, or iii) separately for T and
rectangular beams and for columns of each floor (Figure 5.38).
theoretical φy (Eqs. (3.9) – (3.10)) and a fraction of EIgross between 50 and 100%
instead of EIeff (only for the dynamic linear analysis) were assumed in order to check
the possibility of removing the dependency of the shear capacity from the seismic
•
concerning the dynamic linear analysis, the gravity N yields a very small
with respect to the seismic N. (Figure 5.40).
•
Concerning the nonlinear analyses, gravity and seismic N yield practically
identical results (Figure 5.86).
Ls = L/2 yields a very small (practically negligible) increase in the percentage of
members which do not satisfy the verification with respect to Eq. (3.1), for all
methods of analysis (Figure 5.42, Figure 5.88, Figure 5.116).
237
Finally, the results of the assessment based on all methods of analysis show that the
simplified approach (gravity N, Ls = L/2, empirical φy and 50% EIgross) yields a small
increase in the percentage of members which do not satisfy the verification with
respect to the correct approach (seismic N, Ls from Eq. (3.1), theoretical φy, mean
EIeff), leading to a safe side assessment (Figure 5.44, Figure 5.90, Figure 5.118).
9.2 Summary of the Results of Scuola Elementare Pascoli (Barga)
The Scuola Pascoli, located in Barga (Tuscany) is a non-compact and roughly square
building, irregular in plan and in elevation. It is characterized by high values of fcm (30 MPa)
and Ec (26357 MPa), resulting in a rather rigid structure (T1 varying from 0.40 s, considering
EIgross, to 0.81-0.93 s, considering EIeff).
•
for both Codes, for all the considered choices of EI, ρmax/ρmin exceeds the threshold for
the applicability of linear methods.
•
corner period, TC). Since the Scuola Pascoli is rather rigid (T1 < Tc, if EIgross is
assumed), the reduction of ρmax/ρmin considering 50% EIgross instead of 100% EIgross is
very small, while it is evident assuming EI = EIeff.
•
percentage between 0% (for EI = EIeff) and 1% (for EI = EIgross) of structural members which
do not satisfy the recommendations (Table 6.2). Hence, also the second condition of
applicability of linear analyses is sensitive to the choice of EI. In analogy with the first
condition, this second condition will become more restrictive too, if large values of EI are
assumed.
238
and 3:
(i)
evaluation of Ls.
obtained from Eurocode 8 are slightly conservative and close to those obtained from
the Italian Seismic Code (Figure 6.17, Figure 6.65, Figure 6.101).
Concerning the theoretical θu (Eq. (3.22)), for all methods of analysis, the results
obtained from Eurocode 8 are conservative and the mean values of |Di/Ci| much larger
than those obtained from the Italian Seismic Code (differences ≈ 20% for the dynamic
nonlinear, ≈ 30% for the static nonlinear and > 30% for the dynamic linear analysis)
(Figure 6.18, Figure 6.66, Figure 6.102). Hence, assuming the procedure suggested by
Eurocode 8 as correct, the simpler approach of the Italian Seismic Code yields quite
inaccurate results.
(ii)
Decreasing EI, the mean values of |Di/Ci| increases sensibly, for both empirical and
theoretical θu. Hence, assuming EIeff computed through Eq. (2.8) as the most accurate
choice of EI, both 100% EIgross and 50% EIgross yield unconservative results (Figure
6.9 to Figure 6.12).
beams and for columns of each floor. A difference can be appreciated for the columns
of the first storey. In fact, computing the mean EIeff separately for the columns of each
floor allows to take into account that the columns of the first storey are stiffer (and,
hence, deform less), because they bear the largest compressive N. However, since this
difference is limited to the first storey columns, using the mean EIeff computed for all
members yields quite accurate results (Figure 6.13 to Figure 6.16).
239
•
yield very close results (Figure 6.23 - Figure 6.24, Figure 6.73 - Figure 6.74).
•
Concerning the theoretical θu, according to the nonlinear methods of analysis,
gravity and seismic N yield very close results (Figure 6.75 - Figure 6.76).
According to the dynamic linear analysis, instead, following the
recommendations of Eurocode 8, the mean values of |Di/Ci| obtained assuming
the seismic N are larger than assuming the gravity N (differences up to 25% for
the columns) (Figure 6.25 - Figure 6.26). This can be justified considering that
the seismic ΔN experienced in a linear analysis is, in general, (much) larger
than the actual ΔN obtained accounting for the development of nonlinear
mechanisms.
•
concerning the empirical θu, for all methods of analysis, the use of Ls = L/2
yields results very close to those obtained considering Eq. (3.1) (Figure 6.19 Figure 6.20, Figure 6.69 - Figure 6.70, Figure 6.105 - Figure 6.106).
•
analyses, the decrease in the mean values of |Di/Ci| obtained using Ls = L/2 is
huge (≈ 30%, for the dynamic nonlinear, and even > 50%, for the static
nonlinear analysis) (Figure 6.71 - Figure 6.72, Figure 6.107 - Figure 6.108).
This shows that Eq. (3.22) is very sensitive to the value of Ls. According to the
dynamic linear analysis, instead, a small decrease (< 15%) in the mean values
of |Di/Ci| obtained using Ls = L/2 can be appreciated (Figure 6.21 - Figure
6.22).
•
concerning the empirical θu, for all methods of analysis, the simplified and the
correct approaches yield close results (Figure 6.29 - Figure 6.30, Figure 6.77 Figure 6.78).
•
concerning the theoretical θu, for all methods of analysis, the decrease in the
mean values of |Di/Ci| obtained using the simplified approach is noticeable, in
240
particular for the nonlinear analyses, and can be ascribed mainly to the fact that
Eq. (3.22) is very sensitive to the value of Ls (Figure 6.31 - Figure 6.32, Figure
6.79 - Figure 6.80, Figure 6.109 - Figure 6.110).
For all methods of analysis, the results obtained applying both approaches of the
empirical θu and the simplified approach of the theoretical θu are close to each other,
but different from those obtained considering the correct approach of the theoretical
θu. This difference is huge in particular for the static nonlinear analysis and it is larger
if the procedure of Eurocode 8 is followed, while, if the recommendations of the
Italian Seismic Code are considered and the dynamic linear analysis is used, the
results of the correct approach of the theoretical θu are close to all the other results
These considerations suggest that, concerning the theoretical θu, there should be the
need of considering Ls = L/2, since Eq. (3.1) yields results too sensitive to the values
of Ls.
(v)
For the dynamic linear analysis, concerning both empirical and theoretical θu, the use
of the bidimensional failure curve yields an increase in the mean values of |Di/Ci| of
columns (for the beams the verification is only uniaxial) between 26 and 28%. This
difference is due to the lack of compactness of the Scuola Pascoli (Figure 6.35 to
Figure 6.38).
For the nonlinear analyses (both static and dynamic), instead, concerning both
empirical and theoretical θu, the definition of a bidimensional failure curve does not
improve on the accuracy of the results (Figure 6.83 to Figure 6.86, Figure 6.113 to
Figure 6.116).
The difference between the results based on linear and nonlinear methods of analysis
is due to the fact that, in a nonlinear analysis, it is likely that the degradation of
stiffness in the pushing direction (for the static analysis) or in the direction where the
main accelerogram acts (for the dynamic analysis) will be faster than in the orthogonal
direction. Therefore, the displacements/deformations in the orthogonal direction will
be, in general, small. In a linear analysis, instead, the stiffness is constant and, hence,
the difference between the uniaxial and the biaxial check is more evident.
and 3:
241
(i)
(ii)
The gravity N yields results very close to those obtained assuming the seismic N, for
all methods of analysis (Figure 6.44Figure 6.88).
Ls = L/2 yields a slightly increase in the percentage of members which do not satisfy
the verification with respect to Eq. (3.1), for all methods of analysis (Figure 6.46,
Figure 6.90, Figure 6.118).
respect to the correct approach (seismic N, Ls from Eq. (3.1), theoretical φy and mean
242
9.3 Summary of the Results of Scuola Media Inferiore Puccetti (Gallicano)
The Scuola Puccetti, located in Gallicano (Tuscany) is neither compact nor symmetric, since
it has a C-shaped plan form, with one arm of the ‘C’ longer than the other one. It is irregular
in plan and in elevation and characterized by fcm = 18 MPa and Ec = 22230 MPa, resulting in a
rather rigid structure (T1 varying from 0.43 s, considering EIgross, to 0.74-0.80 s, considering
EIeff).
•
the value of ρmax/ρmin reduces when the members stiffness EI decreases (Table 7.2).
corner period, TC). Since the Scuola Puccetti is rather rigid (T1 ≈ Tc, if EIgross is
assumed), the reduction of ρmax/ρmin considering 50% EIgross instead of 100% EIgross is
small (≈ 20%), while it is much larger (> 50%) assuming EI = EIeff.
•
According to Eurocode 8, for all the considered choices of EI, ρmax/ρmin exceeds the
threshold for the applicability of linear methods. According to the Italian Seismic
Code, instead, the choice of EI is crucial to determine whether the results furnished by
linear analyses can be accepted. In fact, choosing EI = 50-100% EIgross, ρmax/ρmin will
be larger than 2.5 (limit for the applicability of linear analyses), while, if EI = EIeff,
ρmax/ρmin will be smaller than 2.5 and, hence, the condition for the applicability of
linear analyses is satisfied.
•
percentage between 1% (for EI = EIeff) and 2% (for EI = EIgross) of structural members which
do not satisfy the recommendations (Table 7.2). Hence, also the second condition of
applicability of linear analyses is sensitive to the choice of EI. In analogy with the first
condition, this second condition will become more restrictive too, if large values of EI are
assumed.
243
Although the conditions of applicability are not satisfied (even considering the Italian Seismic
Code and assuming EI = EIeff, in fact, the results of the linear analyses should be rejected, as
the second condition of applicability is not satisfied), the dynamic linear analysis was
and 3:
(i)
proposed in both Codes are similar to each other, Eurocode 8 suggests more complex
calculations, in both computation of the flexural deformation at yielding and
evaluation of Ls.
Concerning the empirical θu (Eq. (3.15)), according to both linear and nonlinear
dynamic analyses, the results obtained from Eurocode 8 are close to those obtained
from the Italian Seismic Code (Figure 7.15, Figure 7.103). According to the static
nonlinear analysis, instead, the results obtained from Eurocode 8 are slightly
conservative, since the mean values of |Di/Ci| are larger than those obtained from the
Italian Seismic Code (differences ≈ 15%) (Figure 7.67).
Concerning the theoretical θu (Eq. (3.22)), according to the nonlinear dynamic
analysis, the results obtained from Eurocode 8 are close to those obtained from the
Italian Seismic Code (Figure 7.16) while, according to both linear dynamic and
nonlinear static analyses, the results obtained from Eurocode 8 are conservative and
the mean values of |Di/Ci| much larger than those obtained from the Italian Seismic
Code (differences ≈ 20% for the dynamic linear and ≈ 30% for the static nonlinear
analysis) (Figure 7.68, Figure 7.104). Hence, assuming the procedure suggested by
Eurocode 8 as correct, the simpler approach of the Italian Seismic Code yields quite
inaccurate results.
(ii)
Decreasing EI, the mean values of |Di/Ci| increase sensibly, for both empirical and
theoretical θu. Hence, assuming EIeff computed through Eq. (2.8) as the most accurate
choice of EI, both 100% EIgross and 50% EIgross yield unconservative results (Figure
7.7 to Figure 7.10).
244
hence, deform less), because they bear the largest compressive N. However, since this
difference is small and limited to the first storey columns, using the mean EIeff
computed for all members yields quite accurate results (Figure 7.11 to Figure 7.14).
•
yield close results (Figure 7.21 - Figure 7.22, Figure 7.75 - Figure 7.76).
•
Concerning the theoretical θu, according to the nonlinear methods of analysis,
the seismic N yields slightly conservative results. The difference between the
results obtained assuming the gravity and the seismic N is evident only for the
columns of the ground floor, as the seismic ΔN is null at the top of the building
and maximum at the base (Figure 7.77 - Figure 7.78). According to the
dynamic linear analysis, instead, the mean values of Di Ci obtained
assuming the seismic N are larger than assuming the gravity N (differences up
to 40% for the columns) (Figure 7.23 - Figure 7.24). This can be justified
•
concerning the empirical θu, according to the dynamic linear analysis, Ls = L/2
yields results very close to those obtained considering Eq. (3.1) (Figure 7.17 Figure 7.18). According to the nonlinear methods of analysis, instead, Ls = L/2
yields slightly unconservative results. In particular, the differences between the
results obtained assuming Ls = L/2 and Eq. (3.1) are ≈ 15% for the dynamic
nonlinear and for the static nonlinear analysis if the Italian Seismic Code is
used, and ≈ 25% for the static nonlinear analysis according to Eurocode 8
(Figure 7.71 - Figure 7.72, Figure 7.107 - Figure 7.108).
•
analyses, the decrease in the mean values of |Di/Ci| obtained using Ls = L/2 is
huge, even > 100% for the dynamic nonlinear analysis (Figure 7.73 - Figure
7.74, Figure 7.109 - Figure 7.110). This shows that Eq. (3.22) is very sensitive
to the value of Ls. According to the dynamic linear analysis, instead, a small
245
decrease (< 15%) in the mean values of |Di/Ci| obtained using Ls = L/2 can be
appreciated (Figure 7.19 - Figure 7.20).
•
concerning the empirical θu, according to the dynamic linear analysis, the
simplified approach yields a small decrease in the mean values of Di Ci with
respect to the correct approach (Figure 7.27 - Figure 7.28). According to the
nonlinear methods of analysis, instead, the simplified approach yields slightly
unconservative results. In particular, assuming the Italian Seismic Code, the
results are close to those obtained considering the correct approach (difference
≈ 15%), while, following the recommendations of Eurocode 8, the difference
increases up to 25% (Figure 7.79 - Figure 7.80).
•
Concerning the theoretical θu, for all methods of analyses, the decrease in the
mean values of |Di/Ci| obtained using the simplified approach is noticeable (≈
40%, for the linear dynamic, > 50% for the nonlinear analyses). This
difference between the two approaches is due mainly to the fact that Eq. (3.22)
is very sensitive to the value of Ls. (Figure 7.29 - Figure 7.30, Figure 7.81 Figure 7.82, Figure 7.111 - Figure 7.112).
According to both linear dynamic and nonlinear static analyses, all results are close to
each other. In particular, assuming the Italian Seismic Code, the simplified approach
of the theoretical θu tends to underestimate the values obtained considering the other
approaches, while, following the recommendations of Eurocode 8, the correct
approach of the theoretical θu tends to overestimate the values obtained considering
the other approaches (Figure 7.31 - Figure 7.32, Figure 7.83 - Figure 7.84).
According to the dynamic nonlinear analysis, the results obtained applying both
approaches of the empirical θu and the simplified approach of the theoretical θu are
close to each other, but very different from those obtained considering the correct
approach of the theoretical θu, which overestimates sensibly the values obtained
considering the other approaches (differences > 50%) (Figure 7.113 - Figure 7.114).
These considerations suggest that, concerning the theoretical evaluation of θu, there
should be the need of considering Ls = L/2, since Eq. (3.1) yields results too sensitive
to the values of Ls.
(v)
246
of the bidimensional failure curve yields an increase in the mean values of |Di/Ci| of
columns ≈ 25% (for the beams the verification is only uniaxial). This difference is due
to the lack of compactness and to the particular shape (asymmetric “C”) of the Scuola
Puccetti (Figure 7.33 to Figure 7.36).
Figure 7.118).
and 3:
(i)
(ii)
247
•
according to the dynamic linear analysis, the difference in percentage of
members which do not satisfy the verification assuming the gravity and the
seismic N is large (≈ 40%, considering all columns) (Figure 7.42). This
difference can be justified considering that the seismic ΔN experienced in a
linear analysis is, in general, (much) larger than the actual ΔN obtained
accounting for the development of nonlinear mechanisms.
•
According to the nonlinear analyses, instead, the gravity N yields results very
close to those obtained assuming the seismic N (Figure 7.90).
Ls = L/2 yields a slightly increase in the percentage of members which do not satisfy
the verification with respect to Eq. (3.1), for all methods of analysis (Figure 7.44,
Figure 7.92, Figure 7.120).
(Figure 7.43 - Figure 7.91 - Figure 7.119).
Finally, the results of the assessment based on the nonlinear analyses show that the
simplified approach (gravity N, Ls = L/2, empirical φy and 50% EIgross) yields a very
small increase in the percentage of members which do not satisfy the verification with
EIeff), leading to a safe side assessment (Figure 7.94, Figure 7.122). According to the
dynamic linear analysis, instead, if all members are considered, the simplified and the
correct approaches yield very close results. On the other hand, considering only the
beams, the simplified approach overestimates the correct approach (difference ≈ 25%),
while, considering only the columns, the simplified approach underestimates the
correct approach (difference > 50%) (Figure 7.46). This difference is mainly due to
the fact that the seismic ΔN experienced in a linear analysis is, in general, (much)
larger than the actual ΔN obtained accounting for the development of nonlinear
mechanisms.
9.4 Summary of the Results of Scuola Media Inferiore Don Bosco (Rapagnano)
The Scuola Don Bosco located in Rapagnano (Marche) is a compact and roughly rectangular
building, not symmetric with respect to the two principal directions, irregular in plan and in
elevation. It is characterized by fcm = 16.6 MPa, Ec = 21638 MPa, T1 varying from 0.59 s,
considering EIgross, to 1.16-1.22 s, considering EIeff.
248
•
for both Codes, for all the considered choices of EI, ρmax/ρmin are much larger than the
threshold for the applicability of linear methods.
•
corner period, TC). Since, for the Scuola Don Bosco, T1 > Tc, the reduction of ρmax/ρmin
obtained considering EIeff instead of EIgross is rather large.
•
The check on the second condition (considered only in the Italian Seismic Code) shows that
the shear capacity of every structural member is larger than the respective demand (Table
8.2). Hence, the second condition of applicability of linear analyses is satisfied.
Although the first condition of applicability is not satisfied, the dynamic linear analysis was
and 3:
(i)
evaluation of Ls.
249
obtained from the two Codes are close to each other (Figure 8.14, Figure 8.63, Figure
8.99).
Concerning the theoretical θu (Eq. (3.22)), for all methods of analysis, the percentage
of members which do not satisfy the verification obtained from Eurocode 8 is larger
than that obtained from the Italian Seismic Code. In particular, the differences are ≈
25% for the linear dynamic, ≈ 30% for the nonlinear dynamic and > 30% for the
nonlinear static analysis (Figure 8.15, Figure 8.64, Figure 8.100).
(ii)
increases. This tendency will be more evident if the empirical θu is considered. Hence,
assuming EIeff computed through Eq. (2.8) as the most accurate choice of EI, both
100% EIgross and 50% EIgross yield unconservative results (Figure 8.6 to Figure 8.9).
hence, deform less), because they bear the largest compressive N (Figure 8.10 to
Figure 8.13). However, since this difference is limited to the first storey columns,
using the mean EIeff computed for all members yields quite accurate results.
•
concerning the empirical θu, according to the nonlinear analyses, the results
obtained from Eurocode 8 are very close to those obtained from the Italian
Seismic Code. (Figure 8.20 - Figure 8.21, Figure 8.71 - Figure 8.72).
According to the linear dynamic analysis, the gravity N yields a decrease in the
percentage members which do not satisfy the verification with respect to the
seismic N. The difference is ≈ 20% for the Italian Seismic Code and 15% for
Eurocode 8. In particular, the decrease is larger for the columns of the ground
floor, as the seismic ΔN is maximum at the base and becomes null at the top of
the building. This difference may be justified considering that the seismic ΔN
experienced in a linear analysis is, in general, (much) larger than the actual ΔN
obtained accounting for the development of nonlinear mechanisms.
250
•
Concerning the theoretical θu, for all methods of analysis, the gravity N yields
a decrease in the percentage of members which do not satisfy the verification
with respect to the seismic N. In particular, the difference is larger for the
dynamic linear analysis (≈ 15-20%) than for the nonlinear analyses (< 15%)
(Figure 8.22 - Figure 8.23, Figure 8.73 - Figure 8.74). This can be justified
•
concerning the empirical θu, according to both linear dynamic and nonlinear
static analyses, the use of Ls = L/2 yields results close to those obtained
considering Eq. (3.1) (Figure 8.16 - Figure 8.17, Figure 8.67 - Figure 8.68).
According to the nonlinear dynamic analysis, the very small percentages of
members which do not satisfy the verification do not allow to draw significant
considerations (Figure 8.103 - Figure 8.104).
•
analyses, the decrease in the percentage of members which do not satisfy the
verification obtained using Ls = L/2 is huge (> 30% for the nonlinear dynamic
and > 50% for the nonlinear static analysis) (Figure 8.69 - Figure 8.70, Figure
8.105 - Figure 8.106). This shows that Eq. (3.22) is very sensitive to the value
of Ls. According to the dynamic linear analysis, instead, the use of Ls = L/2
yields results close to those obtained considering Eq. (3.1) (Figure 8.18 Figure 8.19).
•
concerning the empirical θu, according to the dynamic linear analysis, the
simplified approach yields a decrease in the percentage of members which do
not satisfy the verification with respect to the correct approach (Figure 8.26 Figure 8.27). The difference is ≈ 30% for the Italian Seismic Code and 20% for
Eurocode 8. This difference may be justified considering that the seismic ΔN
experienced in a linear analysis is, in general, (much) larger than the actual ΔN
obtained accounting for the development of nonlinear mechanisms. According
to the nonlinear methods of analysis, instead, the correct and the simplified the
approaches yield close results (Figure 8.75 - Figure 8.76).
•
Concerning the theoretical θu, for all methods of analysis, the decrease in the
simplified approach is noticeable (≈ 20-30% for the linear dynamic, ≈ 30-40%
for the nonlinear dynamic and > 50% for the nonlinear static analysis) and may
251
be ascribed mainly to the fact that Eq. (3.22) is very sensitive to the value of Ls
(Figure 8.28 - Figure 8.29, Figure 8.77 - Figure 8.78, Figure 8.107 - Figure
8.108).
If the dynamic linear analysis is performed, empirical and theoretical θu yield close
results. In particular, following the recommendations of the Italian Seismic Code, the
empirical θu yields slightly larger percentages of members which do not satisfy the
verification with respect to the theoretical θu, while, assuming Eurocode 8, the
theoretical θu yields slightly larger percentages of members which do not satisfy the
verification with respect to the empirical θu (Figure 8.30 - Figure 8.31).
Concerning the nonlinear analyses, following the recommendations of the Italian
Seismic Code, the results obtained applying both approaches of the empirical θu and
the correct approach of the theoretical θu are close to each other, but very different
from those obtained considering the simplified approach of the theoretical θu, which
tends to underestimate the percentage of members which do not satisfy the
verification. Following the recommendations of Eurocode 8, instead, the results
obtained applying both approaches of the empirical θu and the simplified approach of
the theoretical θu are close to each other, but very different from those obtained
considering the correct approach of the theoretical θu, which tends to overestimate the
percentage of members which do not satisfy the verification (Figure 8.79 - Figure
8.80).
(v)
of the bidimensional failure curve yields an increase in the percentage of members
which do not satisfy the verification. The increase is large if the empirical θu is
assumed (≈ 25%, considering all columns), while, if the theoretical θu is used, the
increase is smaller (≈ 15%, considering all columns) (Figure 8.32 to Figure 8.35).
Figure 8.112).
252
and 3:
(i)
(ii)
decreases too, even if the difference in percentage of members which do not satisfy the
verification assuming 100% EIgross and mean EIeff (Eq. (2.8)) is less than 10% (Figure
8.37).
•
concerning the dynamic linear analysis, the gravity N yields a very small
with respect to the seismic N (Figure 8.41).
•
Concerning the nonlinear analyses, gravity and seismic N yield practically
identical results (Figure 8.86).
253
Ls = L/2 yields a very small (practically negligible) increase in the percentage of
members which do not satisfy the verification with respect to Eq. (3.1), for all
methods of analysis (Figure 8.43, Figure 8.88, Figure 8.114).
9.5 Comparisons and Conclusions about the Assessment of Ductile Mechanisms
The comparison of the most significant results, obtained from the assessment of the ductile
mechanisms of the four considered buildings, will be shown in the following charts. In the
previous Chapters, for two buildings (Sede Comunale and Scuola Don Bosco), the results
were shown in terms of percentage of structural members which do not satisfy the
verification, while, for the other two buildings (Scuola Pascoli and Scuola Puccetti), the
results were shown in terms of percentage of the mean values of Di Ci . Hence, in order to
compare the two different kinds of results, in the following charts, the results obtained for
each of the four considered buildings were normalized. It means that the highest value in
terms of percentage of either structural members which do not satisfy the verification or mean
values of Di Ci , obtained for each building, for each analysis, is assumed equal to 100%
and all the other percentages are scaled consequently. On the other hand, it is important to
remind that the aim of this work is to compare different approaches of assessment and not to
quantify the members which do not satisfy the Codes’ requirements. Hence, showing the
normalized results will not lead to any loss of information. For each chart, the normalized
results refer only to all structural members. The results in terms of beams, columns and
columns of each floor, computed separately, can be observed in Chapters 5, 6, 7 and 8, for the
Sede Comunale, the Scuola Pascoli, the Scuola Puccetti and the Scuola Don Bosco,
respectively.
Concerning the empirical θu (Eq. (3.15)), for all considered buildings, for all methods of
analysis, the results obtained from Eurocode 8 are slightly conservative and close to those
obtained from the Italian Seismic Code (Figure 9.1 - Figure 9.3 - Figure 9.5). Hence,
considering that the procedures proposed in Eurocode 8 to compute both flexural deformation
at yielding and Ls are more complex and lengthy than the approaches suggested in the Italian
Seismic Code, the use of the Italian Seismic Code is recommended.
Concerning the theoretical θu (Eq. (3.22)), for all considered buildings, for all methods of
analysis, the percentages obtained from Eurocode 8 are, in general, larger than those obtained
from the Italian Seismic Code (Figure 9.2 - Figure 9.4 - Figure 9.6). In particular, only in the
case of the dynamic nonlinear analysis, for both Sede Comunale and Scuola Puccetti, the
differences between the results obtained from the two Codes are negligible, while, for all the
other cases, the differences vary from less than 10% (Sede Comunale, dynamic linear
254
analysis) to more than 30% (Scuola Pascoli, dynamic linear analysis). Hence, assuming the
procedure suggested by Eurocode 8 as correct, the simpler approach of the Italian Seismic
Code yields quite inaccurate results.
100
100
90
OPCM
90
80
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
OPCM
EC8
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Puccetti
Scuola Don Bosco
100
Scuola Pascoli
OPCM
100
EC8
90
OPCM
90
80
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Puccetti
Scuola Don Bosco
DUCTILE CHECK - DYNAMIC NONLIN.
THEOR. FORM.
100
100
90
OPCM
90
80
EC8
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
Scuola Pascoli
OPCM
EC8
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
The sensitivity of the assessment procedure to EI is shown in Figure 9.6 and Figure 9.7. All
charts refer to the results obtained considering Eurocode 8, for both empirical (Eq. (3.15)) and
theoretical (Eq. (3.22)) θu. Figure 9.6 shows that, decreasing EI, the percentages increase
sensibly, for all buildings. Therefore, assuming EIeff computed through Eq. (2.8) as the most
255
accurate choice of EI, it is clear that, both 100% EIgross and 50% EIgross yield unconservative
results. Figure 9.7 shows that close results are obtained considering the mean EIeff computed
i) for all members, ii) separately for beams and columns, or iii) separately for T and
rectangular beams and for columns of each floor. The same considerations extend to the
results obtained assuming the Italian Seismic Code.
100
100
100%EI
90
100%EI
50%EI
90
EI_eff
50%EI
80
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
100
Scuola Pascoli
EI_eff_uniform
EI_eff_uniform
EI_eff_beam_col
90
EI_eff_precise
EI_eff_beam_col
80
Scuola Don Bosco
100
90
Scuola Puccetti
80
EI_eff_precise
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Figure 9.11 to Figure 9.16 show the results of the attempt to simplify the assessment
procedure assuming N from gravity loads, Ls = L/2 and φy from empirical expressions (Eqs.
(3.11) to (3.14)) instead of N from the seismic combination, Ls from Eq. (3.1) and φy from
theoretical assumptions (Eqs. (3.9) – (3.10)), respectively. All charts refer to the results
obtained considering Eurocode 8, for both empirical (Eq. (3.15)) and theoretical (Eq. (3.22))
θu.
Concerning the empirical θu, the results obtained assuming the simplified approach slightly
overestimate the values obtained considering the correct approach (Figure 9.11 - Figure 9.13 Figure 9.15). This is a general trend, confirmed by the assessment of all considered structures,
according to all methods of analysis. Two exceptions are the Scuola Don Bosco, assessed
according to the dynamic linear analysis, and the Scuola Puccetti, assessed according to the
static nonlinear analysis. In both cases, the simplified approach grossly underestimates the
results of the correct approach (differences in the order of 30%). On the other hand, it should
be considered that the assessments of both Scuola Don Bosco, based on the dynamic linear
256
analysis, and Scuola Pascoli, based on the static nonlinear analysis, may yield unreliable
results. In fact, concerning the Scuola Don Bosco, the values of ρmax/ρmin exceed widely the
Codes’ threshold of 2.5 (ρmax/ρmin varying from 12.1 to 56.6, depending on both choice of EI
and considered Code), while, concerning the Scuola Pascoli, unusual values of the
“transformation factor” (Γx = 0.118 and Γy = 0.778) and questionable location of the control
node (in the centre of mass) contribute to yield uncertain results. With the exception of these
two cases, the differences between the results of the two approaches are in the order of 1015% for the dynamic linear analysis and < 10% for the nonlinear analyses (both static and
dynamic). The difference between the results of linear and nonlinear analyses may be justified
considering that the seismic ΔN experienced in a linear analysis is, in general, (much) larger
than the actual ΔN obtained accounting for the development of nonlinear mechanisms.
Concerning the theoretical θu, for all considered buildings, for all methods of analysis, the
results obtained assuming the simplified approach grossly overestimate the values obtained
considering the correct approach (Figure 9.12 - Figure 9.14 - Figure 9.16). The differences
between the results of the two approaches vary from about 20% (Scuola Pascoli, dynamic
nonlinear analysis) to values larger than 100% (Sede Comunale, both static and dynamic
nonlinear analyses) and may be ascribed mainly to the fact that Eq. (3.22) is very sensitive to
the value of Ls.
Hence, the possibility of removing the dependency of the chord rotation capacity from the
seismic demand and proposing faster and less complex calculations is limited to the
empirical θu. The same considerations extend to the results obtained assuming the Italian
Seismic Code.
100
100
90
Ls=L/2; N_Grav
90
80
Ls=M/V; N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
257
100
100
90
Ls=L/2; N_Grav
90
80
Ls=M/V; N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Puccetti
Scuola Don Bosco
DUCTILE CHECK - DYN. NONLIN.
THEOR. FORM. - EC8
100
Scuola Pascoli
100
90
Ls=L/2; N_Grav
90
80
Ls=M/V; N_Seism
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
The comparison between the results obtained using the empirical and the theoretical θu,
according to both correct and simplified approach is shown in Figure 9.17 - Figure 9.18 Figure 9.19. All charts refer to the results obtained considering Eurocode 8.
The results obtained assuming both simplified and correct approaches of the empirical θu and
the simplified approach of the theoretical θu are close to each other, while, the correct
approach of the theoretical θu grossly overestimates the results obtained with all the other
procedures, proving that Eq. (3.22) is very sensitive to the value of Ls. This is a general trend,
confirmed by most considered cases. In particular, the difference between the results of the
correct approach of the theoretical θu and all the other approaches is huge for the nonlinear
analyses, as the inelastic Ls are likely to change with respect to the elastic Ls, possibly
resulting in very small values, leading to θu = 0. The only exceptions to these considerations
are the Scuola Puccetti, assessed according to the static nonlinear analysis, the Scuola Don
Bosco, assessed according to the dynamic linear analysis (both already mentioned concerning
the comment of Figure 9.11 to Figure 9.16), the Sede Comunale, assessed according to the
dynamic linear analysis and the Scuola Don Bosco, assessed according to the dynamic
nonlinear analysis. For these last two cases, the results obtained from the simplified approach
of the theoretical θu diverge from those obtained considering both approaches of the empirical
θu. The general conclusion that can be drawn is that the theoretical θu is too sensitive to the
value of Ls. The empirical θu, instead, yields much stable results and, hence, its use is
258
recommended. The same considerations extend to the results obtained assuming the Italian
Seismic Code.
100
90
100
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
EIeff, EC8, theor. vs. emp., correct vs simplified
DUCTILE CHECK
DYNAMIC NONLIN. - EC8
100
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
EIeff, EC8, theor. vs. emp., correct vs simplified
90
80
70
60
50
40
30
20
10
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Figure 9.19. Ductile check: dynamic nonlinear an., EIeff, EC8, theor. vs. emp., correct vs simplified
found considering the definition of a bidimensional failure curve (Eq. (3.27)) is shown in
Figure 9.17 - Figure 9.18 - Figure 9.19. All charts refer to the results obtained considering
Eurocode 8, according only to the empirical θu, since the theoretical θu has been judges as too
sensitive to Ls. Unlike all the other comparisons shown in the previous charts, the structural
configuration plays a very important role to understand if the definition of a bidimensional
failure curve improves the reliability of the results. Both Sede Comunale and Scuola Don
Bosco are characterized by a compact, roughly symmetrical (rectangular) plan configuration;
the Scuola Pascoli is not compact and not symmetrical, although its plan view is roughly
square; the Scuola Puccetti is the most irregular case, as it is characterized by a C-shaped plan
form, with one arm of the ‘C’ longer than the other one. Although the Scuola Don Bosco
could seem to be rather regular, it should be noted that in a part of the building there is a
double-height space which houses a gym. This part of the building is, hence, more flexible
and this induces an irregular (torsional) response.
These considerations reflect in the results obtained from the assessment based on the dynamic
linear analysis. In fact, Figure 9.20 shows that, for the Sede Comunale, the definition of a
bidimensional failure curve does not improve on the accuracy of the results, while, for all the
259
other buildings, the use of the conventional uniaxial check underestimates the results obtained
considering the bidimensional failure curve. The difference between the two approaches is in
the order of 15-20%. It is useful to remind that for the beams the verification is only uniaxial
and, hence, considering only columns, the difference increases to 20-25%.
For the nonlinear analyses (both static and dynamic), instead, for all considered buildings, the
definition of a bidimensional failure curve does not improve on the accuracy of the results
(Figure 9.21 - Figure 9.22). This can be justified considering that, in a static analysis, the
lateral forces are unidirectional and, in a dynamic analysis, the intensity of the accelerogram
in one principal direction is reduced to 30%. Hence, it is likely that the degradation of
accelerogram acts at its full intensity (for the dynamic analysis) will be faster than in the
orthogonal direction. Therefore, the displacements/deformations in the orthogonal direction
will be, in general, small. In a linear analysis, instead, the stiffness is constant and, hence, the
difference between the uniaxial and the biaxial check is more evident.
The same considerations extend to the results obtained assuming the Italian Seismic Code.
100
100
MONO-AXIAL BENDING
90
90
MONO-AXIAL BENDING
80
80
BI-AXIAL BENDING
70
70
60
60
50
50
40
40
30
30
20
20
10
10
BI-AXIAL BENDING
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
100
90
MONO-AXIAL BENDING
80
BI-AXIAL BENDING
70
60
50
40
30
20
10
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Figure 9.22. Ductile check: dynamic nonlinear an., emp. form., EIeff, EC8, uni- vs. bi-axial bending
260
9.6 Conclusions about the Assessment of Brittle Mechanisms
The comparison of the most significant results, obtained from the assessment of the brittle
mechanisms of the four considered buildings, will be shown in the following charts. All
results refer to the percentages of structural members which do not satisfy the verification. For
each chart, the percentages were computed only for all structural members, while the results
in terms of beams, columns and, columns of each floor, computed separately, can be observed
in Chapters 5, 6, 7 and 8, for the Sede Comunale, the Scuola Pascoli, the Scuola Puccetti and
the Scuola Don Bosco, respectively.
For all methods of analysis, for all considered buildings, assessing the brittle mechanisms
according to the Italian Seismic Code yields results which grossly underestimate the
percentages of members which do not satisfy the requirements obtained through Eurocode 8
(Figure 9.23 - Figure 9.24 - Figure 9.25). In particular, the differences between the results
obtained following the two Codes vary from about 30% (Sede Comunale, dynamic linear
analysis) to values larger than 100% for most cases. Hence, the use of the procedure
In the light of these considerations, only the results obtained according to Eurocode 8 will be
shown in the following part of this Section.
100
100
90
OPCM
90
OPCM
80
EC8
80
EC8
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
EIeff, OPCM vs. EC8
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
EIeff, OPCM vs. EC8
100
90
OPCM
80
EC8
70
60
50
40
30
20
10
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Figure 9.25. Brittle check: dynamic nonlinear analysis, EIeff, OPCM vs. EC8
261
The sensitivity of the assessment procedure to EI is shown in Figure 9.26 and Figure 9.27. For
all considered buildings, decreasing EI, the percentages of members which do not satisfy the
verification decrease too, even if the difference between assuming 100% EIgross and mean EIeff
(Eq. (2.8)) is very small (included between less than 5%, for both Sede Comunale and Scuola
Don Bosco, and about 15%, for the Scuola Puccetti) (Figure 9.26). Hence, within the limits of
brittle assessment, assuming a fixed fraction of EIgross between 50% and 100% may be
considered a suitable choice, since the assessment of EIeff according to Eq. (2.8) is much
longer and yields small improvements in the results.
Results very close to each other are obtained considering the mean EIeff computed i) for all
members, ii) separately for beams and columns, or iii) separately for T and rectangular beams
and for columns of each floor (Figure 9.27).
100
100%EI
100
90
50%EI
90
EI_eff_beam-col
80
EI_eff_precise
80
EI_eff
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
EI_eff_uniform
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
EC8, different EI
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
EC8, different EI
Figure 9.28 - Figure 9.29 - Figure 9.30 show the results of the attempt to simplify the
assessment procedure assuming N from gravity loads, Ls = L/2, φy from empirical expressions
and EI = 50% EIgross instead of N from the seismic combination, Ls from Eq. (3.1), φy from
theoretical assumptions and EI = mean EIeff, respectively. The results of the assessment based
on all methods of analysis show that, for all considered buildings, the simplified approach
yields results very close to those obtained considering the correct approach. In particular, the
differences in the results vary from less than 5% to 15%. In most cases, the simplified
approach yields a small increase in the percentage of members which do not satisfy the
verification with respect to the correct approach, leading to a safe side assessment. Hence, the
procedure may be simplified without any loss in the accuracy of the results.
262
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
100
90
80
70
60
50
40
30
20
10
0
Sede Comunale
Scuola Pascoli
Scuola Puccetti
Scuola Don Bosco
Figure 9.30. Brittle check: dynamic nonlinear analysis, EC8, correct vs. simplified approach
263
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A critical review of the r.c. frame existing building

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