Implementation and verification of a masonry panel

Transcription

Istituto Universitario
di Studi Superiori
Università degli
Studi di Pavia
EUROPEAN SCHOOL FOR ADVANCED STUDIES IN
REDUCTION OF SEISMIC RISK
ROSE SCHOOL
IMPLEMENTATION AND VERIFICATION OF A MASONRY
PANEL MODEL FOR NONLINEAR DYNAMIC ANALYSIS OF
INFILLED RC FRAMES
A Dissertation Submitted in Partial
Fulfilment of the Requirements for the Master Degree in
EARTHQUAKE ENGINEERING
by
ELENI SMYROU
Supervisor: Dr RUI PINHO
June, 2006
The dissertation entitled “Implementation and verification of a masonry panel model for
nonlinear dynamic analysis of infilled RC frames”, by Eleni Smyrou, has been approved in
partial fulfilment of the requirements for the Master Degree in Earthquake Engineering.
Rui Pinho …… …
Helen Crowley………… …
………
……
Abstract
ABSTRACT
The effect of infill panels on the response of RC frames subjected to seismic action is widely
recognised and has been subject of numerous experimental investigations, while several attempts to
model it analytically have been reported. In this work, the implementation, within a fibre-based Finite
Elements program, of an advanced double-strut nonlinear cyclic model for masonry panels is
described. The accuracy of the model is first assessed through comparison with experimental results
obtained from pseudo-dynamic tests of large or full-scale frame models. This is followed by a
sensitivity study whereby the relative importance of each parameter necessary to calibrate the model is
evaluated, so that guidance on the general employment of the latter can be given. Furthermore, a
representative range of values for the geometrical and material properties of the infill panels has been
also defined. Finally, the assessment of the behaviour of the infill panel model is completed by testing
the model under fully nonlinear dynamic conditions.
Keywords: infill panels; nonlinear pseudo-dynamic and dynamic analysis; earthquake response; model
calibration; sensitivity analysis
i
Acknowledgements
ACKNOWLEDGEMENTS
I would like to sincerely thank my supervisor Dr. Rui Pinho for proposing me such an interesting
research topic and for being always willing and available to help. Thanks also to Dr. Helen Crowley
for her efforts to facilitate my work and to Carlos Blandon for his helpful hints. The contribution of
Dr. Francisco Crisafulli and Dr. Humberto Varum, whose work has been extensively referred, as well
as the technical support from Dr. Stelios Antoniou are gratefully acknowledged. Finally, I would like
to thank Ihsan Engin Bal for his continuous support throughout this work.
ii
Index
TABLE OF CONTENTS
Page
ABSTRACT ............................................................................................................................................i
ACKNOWLEDGEMENTS....................................................................................................................ii
TABLE OF CONTENTS ......................................................................................................................iii
LIST OF FIGURES ...............................................................................................................................vi
LIST OF TABLES.................................................................................................................................xi
1. INTRODUCTION .............................................................................................................................1
2. DESCRIPTION OF THE INFILL PANEL MODEL........................................................................4
2.1 Introduction................................................................................................................................4
2.2 Description of the Constitutive Material Model ........................................................................4
2.2.1 Envelope Curve in Compression......................................................................................5
2.2.2 Unloading and Reloading ................................................................................................5
2.2.3 Tensile Behaviour ............................................................................................................8
2.2.4 Small Cycle Hysteresis ....................................................................................................8
2.2.5 Local Contact Effects of Cracked Material on the Hysteresis Response .........................9
2.2.6 Cyclic Shear Behaviour .................................................................................................10
2.2.7 Mann and Müller Theory ...............................................................................................11
2.3 Formulation of the Element Model..........................................................................................13
2.3.1 Equivalent Strut Approach.............................................................................................13
2.3.2 Description of the Model ...............................................................................................14
2.3.3 Modelling Aspects .........................................................................................................15
2.3.4 Fibre-based Programme .................................................................................................16
2.3.5 Parameters of the model.................................................................................................17
2.4 Mechanical Parameters ............................................................................................................17
2.4.1 Compressive strength.....................................................................................................17
iii
Index
2.4.2 Elastic Modulus .............................................................................................................18
2.4.3 Tensile Strength .............................................................................................................19
2.4.4 Bond Shear Strength, Coefficient of Friction and Maximum Shear Stress ...................19
2.4.5 Strains ............................................................................................................................20
2.5 Geometrical Parameters ...........................................................................................................20
2.5.1 Vertical Separation between struts.................................................................................20
2.5.2 Area of Strut...................................................................................................................21
2.6 Empirical Parameters ...............................................................................................................24
2.7 Openings ..................................................................................................................................25
3. PRESENTATION OF THE EXPERIMENTAL CASE STUDY....................................................29
3.1 Difficulties in Experimental Tests of Infilled Frames..............................................................29
3.2 The ICONS Frame ...................................................................................................................29
3.2.1 Introduction....................................................................................................................29
3.2.2 Geometry and Reinforcement Detailing ........................................................................30
3.2.3 Material Properties.........................................................................................................35
3.2.4 Mass ...............................................................................................................................38
3.3 Testing Procedure ....................................................................................................................38
3.3.1 Input Motion ..................................................................................................................38
3.3.2 Pseudo-dynamic Test Method........................................................................................41
3.3.3 Modelling the Case Study Frame...................................................................................42
3.4 Calculation and Selection of the Model Parameters for the Case Study Frame ......................42
3.4.1 Compressive strength.....................................................................................................43
3.4.2 Elastic Modulus .............................................................................................................43
3.4.3 Tensile Strength .............................................................................................................43
3.4.4 Bond Shear Strength, Coefficient of Friction and Maximum Shear Stress ...................44
3.4.5 Strains ............................................................................................................................44
3.4.6 Horizontal and Vertical Offset.......................................................................................44
3.4.7 Vertical Separation between Struts and Thickness of Infill Panel .................................44
3.4.8 Area of the Strut.............................................................................................................45
3.4.9 Empirical Parameters .....................................................................................................46
4. VERIFICATION OF THE NUMERICAL MODEL ......................................................................47
4.1 Preliminary Verification ..........................................................................................................47
4.1.1 Modal Analysis of the Bare Frame ................................................................................47
4.1.2 Static Time-history Analysis of the Bare Frame............................................................48
4.2 Sensitivity Analysis .................................................................................................................51
4.3 Pseudo-dynamic Analyses Results for the Infilled Frame .......................................................52
iv
Index
4.3.1 Peak Values of Base Shear.............................................................................................52
4.3.2 Time-history of Base Shear............................................................................................53
4.3.3 Cumulative of Base Shear..............................................................................................57
4.4 Dynamic Analyses Results for the Infilled Frame ...................................................................60
5. CONCLUSIONS AND FUTURE DEVELOPMENTS ..................................................................78
5.1 Conclusions..............................................................................................................................78
5.2 Future Developments ...............................................................................................................79
REFERENCES .....................................................................................................................................81
APPENDIX A.........................................................................................................................................1
v
List of Figures
LIST OF FIGURES
Page
Figure 2. 1. Envelope curve in compression [Crisafulli, 1997] ...............................................................5
Figure 2. 2. Proposed curve for unloading and reloading [Crisafulli, 1997] ...........................................6
Figure 2. 3. Stress-strain curves for unloading branch [Crisafulli, 1997]................................................6
Figure 2. 4. a) Reloading curve and associated parameters b) Definition of the change point for the
unloading curve [Crisafulli, 1997] ..................................................................................................7
Figure 2. 5. Model assumed for the tensile behaviour of masonry ..........................................................8
Figure 2. 6. Typical cyclic response with small cycle hysteresis [Crisafulli, 1997]................................9
Figure 2. 7. Local contact effects for the cracked masonry [Crisafulli, 1997] ......................................10
Figure 2. 8. Analytical response for cyclic shear response of mortar joints ..........................................11
Figure 2. 9. State stress in the masonry panel and stress distribution in the brick based on Mann and
Müller’s assumptions [Crisafulli, 1997] .......................................................................................12
Figure 2. 10. Envelope curve for masonry shear strength by Mann and Müller [Crisafulli, 1997].......13
Figure 2. 11. Linear normal stress distribution acting on a brick [Crisafulli, 1997]..............................13
Figure 2. 12. Modified strut models [Crisafulli, 1997]..........................................................................14
Figure 2. 13. Infill panel element configuration ....................................................................................15
Figure 2. 14. Shear spring modelling.....................................................................................................15
Figure 2. 15. Discretization of RC cross-section in a fibre-based model [SeismoSoft, 2006] ..............17
Figure 2. 16. Stress state considered to evaluate the strength of masonry [Crisafulli, 1997] ................18
Figure 2. 17. Configuration with the geometrical properties of infill panel ..........................................21
Figure 2. 18. Variation of the ratio bw/dw as a function of the parameter hּλ [Decanini and Fantin,
1986] .............................................................................................................................................22
Figure 2. 19. Variation of the ratio bw/dw as function of the parameter hּλ...........................................23
Figure 2. 20. Variation of the area of the masonry strut as function of the axial strain.........................23
Figure 2. 21. Formulation of struts around openings: (a) position of openings, (b) struts for monolithic
infill panels, (c) struts for separating infill panels [Thiruvengadam, 1985]..................................27
vi
List of Figures
Figure 2. 22. Equivalent struts in infill panels with openings [Hamburg, 1993]...................................28
Figure 3. 1. The ICONS frames .............................................................................................................30
Figure 3. 2. Plan and elevation views of RC frame with infill panels [Carvalho et al., 1999] .............31
Figure 3. 3. Slab reinforcement [Carvalho et al., 1999] ........................................................................32
Figure 3. 4. Beam reinforcement details [Carvalho et al., 1999]...........................................................33
Figure 3. 5. Column reinforcement details: cross sections and lap-splice [Carvalho et al., 1999]........34
Figure 3. 6. Elevation view of the infilled frame – Location and dimensions of openings [Carvalho et
al., 1999] .......................................................................................................................................35
Figure 3. 7. Best-fit of steel constitutive law [Carvalho et al., 1999]....................................................37
Figure 3. 8. Displacement spectrum of input motion.............................................................................39
Figure 3. 9. Acceleration spectrum of input motion ..............................................................................39
Figure 3. 10. Ground motion accelerations for 475, 975 and 2000yrp ..................................................40
Figure 4. 1. Comparison of the base shear for the bare frame (475yrp record) .....................................49
Figure 4. 2. Comparison of the base shear for the bare frame (975yrp record) .....................................51
Figure 4. 3. Sensitivity analysis results of infill model parameters in terms of deviation from the base
model.............................................................................................................................................51
Figure 4. 4. Comparison of the base shear of the infilled frame for pseudo-dynamic analysis (475yrp
record) ...........................................................................................................................................51
record) ...........................................................................................................................................51
record) ...........................................................................................................................................51
Figure 4. 7. Normalised cumulative absolute values of base shear versus time for the 475yrp seismic
input record ...................................................................................................................................58
input record ...................................................................................................................................59
input record ...................................................................................................................................59
Figure 4. 10. Comparison of the 1st floor displacement of the infilled frame for dynamic analysis
(475yrp record) .............................................................................................................................63
Figure 4. 11. Comparison of the 2nd floor displacement of the infilled frame for dynamic analysis
(475yrp record) .............................................................................................................................64
Figure 4. 12. Comparison of the 3rd floor displacement of the infilled frame for dynamic analysis
(475yrp record) .............................................................................................................................65
vii
List of Figures
Figure 4. 13. Comparison of the 4th floor displacement of the infilled frame for dynamic analysis
(475yrp record) .............................................................................................................................66
Figure 4. 14. Comparison of the base shear of the infilled frame for dynamic analysis (475yrp record)
.......................................................................................................................................................67
(975yrp record) .............................................................................................................................68
(975yrp record) .............................................................................................................................69
(975yrp record) .............................................................................................................................70
(975yrp record) .............................................................................................................................71
.......................................................................................................................................................72
model.............................................................................................................................................51
record) ...........................................................................................................................................51
record) ...........................................................................................................................................51
record) ...........................................................................................................................................51
input record ...................................................................................................................................58
input record ...................................................................................................................................59
input record ...................................................................................................................................59
(475yrp record) .............................................................................................................................63
(475yrp record) .............................................................................................................................64
(475yrp record) .............................................................................................................................65
(475yrp record) .............................................................................................................................66
viii
List of Figures
.......................................................................................................................................................67
(975yrp record) .............................................................................................................................68
(975yrp record) .............................................................................................................................69
(975yrp record) .............................................................................................................................70
(975yrp record) .............................................................................................................................71
.......................................................................................................................................................72
(2000yrp record) ...........................................................................................................................73
(2000yrp record) ...........................................................................................................................74
(2000yrp record) ...........................................................................................................................75
(2000yrp record) ...........................................................................................................................76
.......................................................................................................................................................77
Figure A. 1. Normalised cumulative absolute values of displacement of 1st floor versus time for the
475yrp seismic input record ............................................................................................................1
Figure A. 2. Normalised cumulative absolute values of displacement of 2nd floor versus time for the
Figure A. 3. Normalised cumulative absolute values of displacement of 3rd floor versus time for the
Figure A. 4. Normalised cumulative absolute values of displacement of 4th floor versus time for the
Figure A. 5. Normalised cumulative absolute values of base shear versus time for the 475yrp seismic
input record .....................................................................................................................................3
ix
List of Figures
input record .....................................................................................................................................6
2000yrp seismic input record ..........................................................................................................6
x
List of Tables
LIST OF TABLES
Page
Table 2.1. Suggested and limit values for the empirical parameters. ...................................................25
Table 3. 1. Concrete average compressive strength after tests on specimens........................................36
Table 3. 2. Mean mechanical properties of steel specimens after tests. ................................................37
Table 3. 3. Material properties of infill panels.......................................................................................38
Table 3. 4. Mass of the infilled frame....................................................................................................38
Table 3. 5. Empirical Parameters...........................................................................................................46
Table 4. 1. Estimated natural periods of the bare frame model. ............................................................48
Table 4. 2. Comparison of base shear peak values. ...............................................................................52
Table 4. 3. Base shear peak values at the instant of the experimental peak value.................................53
Table 4. 4. Base shear peak values at the instant of the numerical peak value......................................53
Table 4. 5. Final cumulative values of base shear. ................................................................................57
Table 4. 6. Displacement peak values for the 475yrp record.................................................................61
Table 4. 7. Displacement peak values for the 975yrp record.................................................................61
Table 4. 8. Displacement peak values for the 2000yrp record...............................................................62
Table 4. 9. Base shear peak values. .......................................................................................................62
xi
Chapter 1. INTRODUCTION
1. INTRODUCTION
Infill panels are widely used as interior partitions and external walls in buildings, but they are
usually treated as non-structural elements and not included in the design. As recent studies
have shown, a properly designed infilled frame can be superior to a bare frame in terms of
stiffness, strength and energy dissipation. However, modern earthquake codes deter the
designers from reducing the design seismic action effects or from relying on the beneficial
presence of infill walls. So the latter is not included in the design and the infill walls
constitute a second line of defence and a source of significant overstrength. This code
approach is enforced mainly due to the inherent uncertainty associated to the numerous
parameters on which the behaviour of the infill panels depends.
Specifically, the variability of the mechanical properties of infill panels, depending on both
the mechanical properties of their materials and the construction details, introduces difficulty
in predicting the behaviour of infill panels. Additionally, the overall geometry of the structure
i.e. number of bays and stories, aspect ratio of infill panels, and the detailing of the reinforced
concrete members are aspects that should be considered. The location and the dimensions of
openings play also an important role in the evaluation of the strength and stiffness of the infill
panels.
Finally, important source of uncertainty is the type of interaction between the infill and the
frame, which strongly influences the behaviour of the infilled frame by altering the loadresisting mechanisms of its individual components. It has become clear after experimental
tests that the lateral resistance of an infilled frame is not equal to the sum of the resistances of
the infill and the surrounding frame. The infilled frame works as system, as some sort of
composite material in fact, at least at low-load levels, while increase of load leads to partial
separation of the infill panel from the frame.
Apart from the complexity in their behaviour, infill walls have been blamed for structural
failures inducing brittle shear failures of the reinforced columns, as well as short-column
phenomena. Moreover, interruption of the infill walls in height causes over-strengthening of
the other floors and introduces a soft-storey, which is highly undesirable from the earthquake
resistance point of view, since the inelastic deformation demands are concentrated in a part of
the building. Finally, asymmetric distribution of the infill panels in plan may generate
torsional response to the translational horizontal components of the seismic action.
1
Despite the aforementioned cases of undesired structural behaviour, field experience,
analytical and experimental research have demonstrated that the beneficial contribution of the
infill walls to the overall seismic performance of the building, especially when the latter
exhibits limited engineering seismic resistance. In fact, infill panels through their in-plane
horizontal stiffness and strength decrease the storey drift demands and increase the storey
lateral force resistance respectively, while their contribution to the global energy dissipation
capacity is significant, always under the assumption that they are effectively confined by the
surrounding frame.
The prediction of the failure mechanism developed before collapse becomes rather difficult
since factors such as the relative stiffness of the frame and infill panel, the strength of its
components and the dimensions of the structure are involved. In reality, the collapse of the
system usually results from a combination of simple types of failure, commonly referred to as
modes of failure. After a detailed review of the literature, the principle modes of failure
observed in experimental tests can be summarised in three: shear cracking, compression
failure and flexural cracking. Their occurrence depends on the material properties and the
stress state in the panel.
Several attempts have been made to describe realistically the behaviour of the infill panels,
including the potential modes of failure. Analytical models have been developed in an effort
to overcome the difficulties and uncertainties in the numerical simulation of the behaviour of
the infilled frames and account for most of the phenomena associated with it. The different
techniques proposed for idealizing the infilled frames can be divided into two different
categories in terms of simulation approach: micro-models (fundamental) and macro-models
(simplified).
The first group of models is based on a finite element representation of each infill panel, using
discrete elements and appropriate constitutive relations for the reinforced frame, the infill
panel and the interface between frame and panel. The second group includes simplified
considerations based on a physical understanding of the behaviour of an infill panel, which is
treated as part of system that consists of the frame and the infill panel itself. Although the
micro-models can represent the behaviour of the masonry for large displacements, rotation
and sliding between blocks and can detect new contacts automatically during the calculation,
their use is limited because of their complexity and the large amount of information
demanded. On the other hand, the macro-models, characterized by their advantageous
simplicity, can describe the overall response but often do not capture the local phenomena
appearing between the surrounding frame and the infill panel.
In order to facilitate the understanding and follow the evolution of the modelling attempts of
infill panels’ behaviour, an extensive number of references is given in chronological order.
The reader is referred to the work of Mallick and Severn [1967], Goodman et al. [1968],
Mallick and Garg [1971], Kost et al. [1974], Riddington and Stafford Smith [1977], King and
Pandey [1978], Liauw and Kwan [1984], Rivero and Walker [1984], Dhanasekar et al.
[1985], Shing et al. [1992], Chrysostomou [1991], Syrmakezis and Asteris [2001] for further
information about the micro-models developed based on finite element approach. For a
detailed literature review of the macro-models, the work of Polyakov [1956], Holmes [1961],
2
Klingner and Bertero [1976], Liauw and Lee [1977], Thiruvengadam [1985], Doudoumis and
Mitsopoulou [1986], Syrmakezis and Vratsanou [1986], D’Asdia et al. [1990], Panagiotakos
and Fardis [1994] and Crisafulli [1997] is recommended.
3
Chapter 2. DESCRIPTION OF THE INFILL PANEL MODEL
2. DESCRIPTION OF THE INFILL PANEL MODEL
2.1 Introduction
A refined model to represent the overall response of the masonry-frame system as well as the
interaction between the masonry and the reinforced surrounding frame accurately is required.
Different analytical models have been used to describe the behaviour of infilled frames.
Detailed literature review is included in Chapter 1, however, the reader is referred again to the
work of Klingner and Bertero [1978], Panagiotakos and Fardis [1994] and Crisafulli [1997].
Crisafulli [1997] proposed an advanced nonlinear cyclic model for masonry panels that
exhibits significant advantages in its use and implementation. Specifically, the suggested
analytical model offers the possibility to model the material with different levels of accuracy
according to the information availability. It also takes account of the local phenomena caused
by the interaction between infill panel and surrounding frame and due to its assumptions
allows its easier implementation.
It is rather difficult to develop a general model that can describe all the potential types of
failure. The main modes of failure reported are due to shear cracking, the compressive failure
and failure because of flexural cracking. In order to define which failure mode may take
place, it is necessary to estimate the masonry strength. A large set of equations have been
proposed by several authors [Mann and Müller, 1982; Paulay and Priestley, 1992; Mehrabi et
al., 1996; Decanini et al., 1987; Biondi et al., 2000].
Crisafulli [1997], who adopted the equivalent strut approach for the proposed model,
introduced different configurations of struts for the principle modes of failure for masonry.
Due to the appreciable level of complexity and uncertainty involved, the model finally
presented in the following sections is intended to describe only the shear failure of the
masonry panel, which is the most common mode of failure.
2.2 Description of the Constitutive Material Model
The cyclic compressive behaviour of the masonry in the model is represented by several
hysteresis rules to consider different behaviours for loading, unloading or reloading. The
relationship between stress and strain, at a given state, depends on the actual strain and some
parameters related to the previous stress-strain history. The model also considers the local
contact effects of the cracked material, the effect of the small inner cycles and the tensile
behaviour of masonry, offering thus generality.
4
2.2.1 Envelope Curve in Compression
The strain-stress curves found in literature for unreinforced masonry are valid up the
maximum compression stress and do not consider the particular characteristics of masonry
behaviour, therefore, Crisafulli [1997] assumed that the expression proposed by Sargin et al.
[1971] originally for concrete can approximately represent the envelope curve for masonry.
The consideration of the softening branch can be really important in the modelling of infilled
frames. Failure of the infill panel usually occurs at small lateral displacements, before the
frame reaches its strength but the system of frame – infill panel is able to resist increasing
lateral loads. So this beneficial effect of the frame, that restrains the cracked panel, leads to
smoother decrease of the resistance of the infill panel. The descending branch of the strength
envelope can be alternatively described by a parabolic curve as it is shown in Figure 2.1. This
consideration is intended to allow a better control of the descending branch of the envelope
when the cyclic model is applied to the representation of masonry struts.
Figure 2. 1. Envelope curve in compression [Crisafulli, 1997]
It is assumed that the envelope curve is independent of the loading history and coincides
approximately with the stress-strain curve obtained under monotonic loading. Combescure et
al. [1996] suggested that the compressive strength of masonry should be reduced as a result of
the cyclic loading. However, due to lack of experimental information it is difficult to quantify
this reduction.
2.2.2 Unloading and Reloading
Unloading and reloading is a complex phenomenon that is very difficult to be modelled
accurately. Generally, the approach adopted by Crisafulli is based on an analytical model that
uses a curve which passes through two predefined points, where the slope of the curve is
known (Figure 2.2). A nonlinear continuous expression is proposed to represent the
unloading-reloading curves, the main advantage of which is that the slope of the curve can be
imposed at both ends.
Experimental results indicate that the unloading curves exhibit a simple curvature and have
shapes dependent on the level of unloading strain [Naraine and Sinha, 1989]. The unloading
curve (rule 2), as it is shown in Figure 2.3, starts from the envelope curve (εun, fun – rule 1) and
5
finishes with a residual or plastic deformation εpl, which seems to be the most important
parameter in determining the unloading curve. For the prediction of the value of εpl, empirical
expressions have been proposed [Subramaniam and Sinha, 1995] but with limited validity.
Crisafulli [1997] expanded the general approach suggested by Mander et al. [1988]
introducing an empirical constant in the calculation of εpl.
fm
f2
2
E2
Es
E1
f1
1
ε1
εm
ε2
Figure 2. 2. Proposed curve for unloading and reloading [Crisafulli, 1997]
fm
(εun, fun)
1
Eun
2
εp
Epl,u
εm
Emo
(εa, fa)
Figure 2. 3. Stress-strain curves for unloading branch [Crisafulli, 1997]
The tangent moduli corresponding to the beginning and to the plastic strain of the unloading
curve are given respectively in proportion and as a function of the initial modulus. The shape
of the curve can be controlled by changing the initial and final modulus.
6
The unloading curve starts when the compressive strain εm reaches the plastic strain εpl. Figure
2.4Figure 2. 4 shows the path and the parameters that define the curve. After that point the
compression stress increases following a path different from the one corresponding to
unloading. The shape of the reloading curve is complex, showing double curvature with mild
concavity in the low stress region and a sharp reversal in curvature near the envelope [Otter
and Naaman, 1989].
The reloading curve consists of two curves. The first one (rule 4) goes from the point
reloading (εpl, 0) to an intermediate point (εch, fch). Then the second curve (rule 5) continues
until the envelope curve is reached. The modulus used as final for rule 4 is used as initial for
rule 5, assuring continuity. The resultant curve and its derivative are continuous, representing
thus successfully the changes of curvature observed in tests of masonry.
Mander et al. [1988] uses a linear reloading curve combined with parabolic transition, while
the strain εre is computed using an empirical expression. Otter and Naaman [1989] suggest a
linear expression to approximate the behaviour in reloading, pointing out however that this
expression may not work well for large values of strain, while modification is required for
small strains. The method by Yankelevsky and Reinhardt [1987], according to which the
reloading strain is computed as the intersection of a predefined line with the envelope curve,
exhibits problems in the range of low and medium strain levels.
fm
5
Εre
fm
(εun, fun)
(εre, fre)
Εch
(εre, fre)
(εch, fch)
(εch, fch)
4
5
Εpl,r
εm
εpl
4
εpl εb
(a)
εm
Ech
εun – fun/Eun
(b)
Figure 2. 4. a) Reloading curve and associated parameters b) Definition of the change point for the
unloading curve [Crisafulli, 1997]
For the evaluation of εre the model assumes that the reloading strain is proportional to the
difference of the unloading strain and the strain at the beginning of reloading, with
differentiation for the case of small cycles. The gained generality of the model compensates
for the increased complexity of the model in the calculation of the reloading. Finally, the
tangent modulus is defined at both ends of the curve, intended to limit abrupt changes of
stiffness.
7
It is also noted that the model accounts also for the cases in which no stress develops in the
masonry. Such a situation occurs when the masonry is under tensile strains or when the
ultimate compressive strain has been exceeded or after the tensile strength has been reached.
2.2.3 Tensile Behaviour
The behaviour of masonry in tension is scarcely investigated, so the model proposed here is
based mainly on experimental data corresponding to concrete. It is assumed that when no
previous compression has taken place the stress-strain relationship in tension (rule 6) is
linearly elastic until the tensile strength is reached. At that point a brittle failure occurs no
tensile strength can be resisted in further cycles. The elastic modulus is taken as the initial
modulus in compression. Tension softening, a local material softening observed during strain
controlled experiments, is also considered in the model. In case of previous compressive
strain, degradation of the tensile strength and the elastic modulus has been reported [Mander
et al., 1988]. The tensile strength thus is assumed zero when the plastic strain exceeds the
magnitude of the strain at maximum compressive strength.
An example of the tensile behaviour that the model follows is given with Figure 2.5. Initially,
a compressive cycle applied affects the subsequent tensile behaviour. With decrease of the
strain the tensile loading occurs. When the strain reverses before the reduced tensile strength
is reached, the linear unloading happens following the same line. In a second cycle of
compression applied, tensile loading takes place again but the tensile strength and the elastic
modulus are reduced due to the increase of the plastic strain. Augment of the tensile stress
continues until ft, when failure occurs.
fm
B
F
f’t/Emo
H
D
E
ε’
εmm
εpl
εpl
O
A
G
C
I
ft
f’t
Figure 2. 5. Model assumed for the tensile behaviour of masonry
2.2.4 Small Cycle Hysteresis
The rules previously described define the loops that start from and return to the envelope
curve with only one reversal after complete unloading. But reversals can happen at any place
during the loading history. For the sake of completeness, the model proposed by Crisafulli
includes the effect of the inner loops.
Because of the complexity of the behaviour and of lack of data, Crisafulli conducted tests on
standard concrete cylinders with different combinations of complete and inner loops. The
8
conclusions drawn were that the successive inner loops increase the reloading strain, do not
affect the plastic deformation and remain inside the cycle defined for the complete unloading
and reloading curves. The former can exhibit change in direction of its concavity depending
on the starting point of the loading curve, while the latter show no inflection point. A typical
cyclic response with small cycle hysteresis is presented in Figure 2.6Figure 2. 6.
fm
εm
Figure 2. 6. Typical cyclic response with small cycle hysteresis [Crisafulli, 1997]
2.2.5 Local Contact Effects of Cracked Material on the Hysteresis Response
The usual assumption is that the cracked masonry cannot bear any compressive stress until the
complete closure of the cracks. However, experimental results [Bolong et al., 1980; Stevens et
al., 1991; Xinrong, 1995] indicate that the compressive stress starts to augment when the
strain is reversed, following a soft response. The contact effects are more important as the
width of the cracks becomes larger [Bolong et al., 1980]. This observation can be explained
due to the presence of small particles that flake off during cracking and remain in the cracks,
as well as due to the misalignment of the crack surface, that causes progressive contact and
gradual transfer of compression across the cracks.
Crisafulli introduced some specific conditions to the hysteretic model in order to simulate the
fact that the material can carry compressive stresses before the cracks are completely closed.
The consideration of contact effects produces wider hysteresis loops and gradual increase of
the compressive stress in the reloading process.
An example of the cyclic response including contact effects is illustrated in Figure 2.7. After
the first cycle, reloading starts when the strain is equal to εcl. The response is soft at the
beginning and becomes stiffer until the normal reloading curve is reached. Second reloading
begins immediately after reversal of strain because the strain is smaller than εcl.
9
Figure 2. 7. Local contact effects for the cracked masonry [Crisafulli, 1997]
2.2.6 Cyclic Shear Behaviour
The adopted model is capable of representing the shear behaviour when bond failure happens
along the mortar joints. It is assumed that the behaviour of the latter is linear elastic while the
shear strength is not reached. Unloading and reloading are also in the elastic range. Thus, the
shear stress τ is equal to the shear deformation γ times the shear modulus Gm.
The model consists of two simple rules and includes the axial load in the masonry as a
variable in the shear strength. The shear strength is evaluated following a bond-friction
mechanism, consisting of a frictional component and the bond strength τo (elastic responserule 1). The former depends on the coefficient of friction µ and the compressive stress
perpendicular to the mortar joints, fn (Equation 2.1).
'
τ m = τ o + µ f n ≤ τ max
if fn < 0
(2.1a)
τm =τo
if fn ≥ 0
(2.1b)
In Figure 2.8, where the analytical response for cyclic shear response of mortar joints is
shown, τmax represents an upper limit for the shear strength according to analytical and
experimental data, which show that for medium to high values of the compressive strength, fn,
the previous Equation is not valid. The values of µ and τo should be such as to reflect the real
strength of the masonry.
When the shear strength is reached, the bond between mortar and brick is destroyed and
cracks appear in the affected region. In this phase, one part of the infill panels slides, with
respect to the other part and only the frictional mechanism remains (sliding-rule 2).
Consequently the shear strength is given by Equation (2.2), where µr is the residual coefficient
of friction.
'
τ m = µ f n ≤ τ max
if fn < 0
(2.2a)
10
τm = 0
if fn ≥ 0
(2.2b)
It is assumed that the unloading and reloading after the bond failure follows a linear
relationship. This process can be represented by rule 1, using Equation 2.1. The reloading line
increases the shear stress until the shear strength is reached and sliding starts again (Figure
2.8).
It has to be stressed that the normal stress fn controls the shear stress which can be resisted by
the mortar joints. This aspect is included in the model but not considered in Figure 2.8, where
it is assumed that fn remains constant. In masonry panels subjected to cyclic or dynamic
loading, the normal stress usually changes as the panel deforms in shear. As a result, the shape
of the hysteresis loops can be different from that illustrated in the figure [Blandon, 2005].
τ
τmax
2
Bond failure
τo
2
1
Gm
Gm
1
γ
1
1
2
−τmax
Figure 2. 8. Analytical response for cyclic shear response of mortar joints
2.2.7 Mann and Müller Theory
Mann and Müller [1982] developed a failure theory that explains the behaviour of
unreinforced masonry subjected to shear and compressive stresses, based on equilibrium
considerations. Their theory is based on basic concepts and thus allows the estimation of the
strength of the masonry in a better and more general manner compared to empirical
formulations. The main assumptions of the theory are:
•
The stress acting on the direction parallel to the bed joint is small enough to be
neglected.
•
The shear stress and the axial compressive stress are uniform in the masonry panel.
Therefore, they represent an average value of the stresses.
•
No shear stresses can be transferred by the head joints.
11
Shear stresses in the bed joints produce a torque in each individual brick which must be
equilibrated by a vertical couple (Figure 2.9Figure 2. 9). This couple modifies the vertical
stress distribution and it is assumed that one brick half is subjected to a stress fn1 (Equation
2.3) and the other half to a smaller stress fn2 (Equation 2.4). According to the convention used,
a negative value of fn1 or fn2 represents tensile stress.
f n1 = f n +
2b
τ
d
(2.3)
f n2 = f n −
2b
τ
d
(2.4)
+
τ
b
d
fn
+
2bτ
d
Figure 2. 9. State stress in the masonry panel and stress distribution in the brick based on Mann and
Müller’s assumptions [Crisafulli, 1997]
The combined state of stress produces failure in different ways, depending on the relative
values of the axial and shear stresses. Three distinct cases are considered: shear-friction
failure, diagonal tension failure and compressive failure. The envelope curve for the masonry
shear strength, which shows the range of stress for each type of failure, is presented in Figure
2.10. Mann and Müller proposed expressions corresponding to the three types of failure,
managing to describe the phenomena adequately.
Mann and Müller’s theory was modified by Crisafulli, suggesting a different distribution of
stresses. Specifically, the consideration of uniform stresses seems rather improbable so
Crisafulli proposed a linear distribution as shown in Figure 2.11.
12
τm
τ∗o
f’m
Shear friction
failure
Diagonal Tension
failure
Compressive
failure
fn
Figure 2. 10. Envelope curve for masonry shear strength by Mann and Müller [Crisafulli, 1997]
3b
τ
d
τ
3b
τ
d
Figure 2. 11. Linear normal stress distribution acting on a brick [Crisafulli, 1997]
2.3
Formulation of the Element Model
2.3.1 Equivalent Strut Approach
In the adopted model the equivalent strut approach is adopted, considering a multi-strut
formulation in order to achieve a better representation of the effect of the masonry panel on
the surrounding frame.
Crisafullli investigated the limitations of single strut model, which is the simplest rational
representation used for the analysis of infilled frames. At the same time he studied the
influence of different multi-strut models on the structural response of infilled frames, focusing
on the stiffness of the structure and the actions induced in the surrounding frame.
Numerical results obtained for the three strut models shown in Figure 2.12 were compared to
those corresponding to an equivalent finite element model. In the analyses the area of the strut
was kept constant, static lateral load was imposed and linear elastic behaviour was assumed,
13
Ams
a) Single-Strut Model
Ams / 2
hz = z/2
hz = z/3
but for the finite element models in which nonlinear effects were considered to describe the
separation of the infill panel-frame interface.
b) Double-Strut Model
c)Triple-Strut Model
Figure 2. 12. Modified strut models [Crisafulli, 1997]
The results indicated that the stiffness of the infilled frame is similar in the cases considered,
slightly decreased for the double- and triple-strut models. Especially for the triple-strut model
the stiffness may vary significantly depending on the distance between the struts hz. Increase
of the distance hz, which can be evaluated as a fraction of the contact length, causes reduction
of the stiffness, that is mainly controlled by the mechanical properties of the columns.
Furthermore, the single-strut model underestimated the bending moments, the double-strut
model led to larger values, while the triple-strut model constituted a better approximation
despite some differences at the ends of the columns. Similar conclusions were drawn for the
shear forces too. Finally, the maximum axial forces in the concrete members were
approximately equal in all models.
The results demonstrated that the single-strut model, despite its simplicity, offers adequate
estimation of the stiffness of the infilled frame and the axial forces induced in the frame
members by lateral loads. Nevertheless, a more refined model is necessary for obtaining
realistic values for the bending moments and the shear forces of the surrounding frame.
Although the single-strut model constitutes a sufficient tool for the prediction of the overall
response and the triple-strut model outbalances in precision, Crisafulli adopted the doublestrut model approach, enough accurate and less complicated compared to the aforementioned
models. However, in the proposed model the struts are not explicitly connected to the frame.
Detailed description and configuration of the model are given in the next section.
2.3.2 Description of the Model
The proposed model consists of four-node masonry panel elements, designed to represent the
behaviour of infill panels in framed structures. Each panel (Figure 2.13) is represented by five
strut members, two parallel struts in each diagonal direction and a single strut acting across
two opposite diagonal corners to carry the shear from the top to the bottom of the panel
(Figure 2.14). This last strut acts across the diagonal which can be on compression and so
connects different top and bottom corners depending on the deformation of the panel.
14
Xoi
V
Yoi
φ
U
Internal
Dummy
hz
dm
Figure 2. 13. Infill panel element configuration
The first four struts use the masonry strut hysteresis model, developed by Crisafulli et al.
[2000]. As already discussed the model consists of rules that take into account the possibility
of different stress paths, while the shear strut uses a bilinear hysteresis rule. The shear
modelling with a shear spring in both directions of loading is shown in Figure 2.14.
Yoi
Xoi
Figure 2. 14. Shear spring modelling
2.3.3 Modelling Aspects
As can be observed in the detailed configuration of the model given in Figure 2.7, the
introduction of 4 dummy nodes is intended to represent the contact length between the frame
and the infill panel, allowing to somehow approximately take account of the effect of local
phenomena, while the 4 internal nodes stand for the frame-infill contact at the exterior part of
the column and the beam, considering thus the reduction of the infill panel’s dimensions due
to the depth of the reinforced concrete members.
15
The total element stiffness is distributed in a given proportion to the shear spring (Ks) and to
the struts (KA) as given by the following equations:
Ks = γ s
Ams E m
cos 2 θ
dm
K A = (1 − γ s )
Ams E m
2 ⋅ dm
(2.5)
(2.6)
where γs is the percentage of total stiffness received by the shear spring, Em is the elastic
modulus of the masonry, Ams is the area of the struts, dm and θ are the length and the
inclination respectively of the diagonal of the panel.
The stiffness matrix and coordinates transformation procedure is obtained from equilibrium
and compatibility of forces and displacements. All the internal forces are transformed to the
exterior 4 nodes where the element is connected to the frame. The obtained displacements and
forces in the dummy nodes have to be transferred to the adjacent internal nodes and next the
displacements in the internal nodes are transformed into displacements in the external nodes.
The transformation of the displacements and forces coming from the shear spring is simpler,
given that only the step from internal to external nodes has to be carried out. The direction of
the shear spring depends on the displacement direction.
A limitation that is worth noting is that the model does not have the capacity of modelling
potential plastic hinges developing in the columns, even though the model take into account
the effect due to eccentricity of the struts. If the formation of plastic hinges in the length of
the column is important a different strut configuration should be implemented.
2.3.4 Fibre-based Programme
The adopted model was implemented in SeismoStruct [SeismoSoft, 2006]. SeismoStruct is an
internet-downloadable fibre-based Finite Element package capable of predicting the large
displacement behaviour of space frames under static or dynamic loading, considering both
geometric nonlinearities and material inelasticity. Specifically, the sectional stress-strain state
of beam-column elements is obtained through the integration of the nonlinear uniaxial
material response of the individual fibres, in which the section has been subdivided, thus fully
accounting for the spread of inelasticity along the member length and across the section depth.
The discretization of a typical reinforced concrete cross-section is depicted, as an example, in
Figure 2.15. If a sufficient number of fibres (100-300 in spatial analysis) is employed, the
distribution of material nonlinearity across the section area is accurately modelled, even in the
highly inelastic range. In this was, in a fibre model the gradual spreading of inelasticity over
the cross section and the element height leads to a smoother transition between elastic and
inelastic element behaviour compared to a lumped plasticity model.
16
Figure 2. 15. Discretization of RC cross-section in a fibre-based model [SeismoSoft, 2006]
2.3.5 Parameters of the model
The model consists of numerous parameters that can be distinguished in mechanical,
geometrical and empirical parameters. The mechanical and geometrical parameters are
required to define the behaviour of the masonry struts. All variables needed as input data are
presented in the following sections as well as with recommendations for the selection or
calculation of their values and the values that are finally implemented. The empirical
parameters involved in the model are necessary for the calculation of different parameters
associated with the cyclic behaviour. Crisafulli proposed for these parameters a range of
recommended values, which have been obtained after experimental tests.
2.4
Mechanical Parameters
2.4.1 Compressive strength
The compressive strength fmθ is the parameter that mainly controls the resistance of the strut
and has to be distinguished from the standard compressive strength of the masonry by taking
into account the inclination of the compression principal stresses and the mode of failure
expected in the infill panel.
Specifically, the failure theory proposed by Mann and Müller [1982] and modified by
Crisafulli [1997] has been developed considering the shear and normal stresses in the bed
joint and assuming that the axial stress parallel to the bed joints can be neglected. Based on
equilibrium considerations the following equation was obtained:
f n = f 1 sin 2 θ
(2.7)
In case of having an estimate or experimental results from tests for the compressive strength
of masonry perpendicular to the bed joints, it is possible to evaluate the principal stress f1 by
transforming Equation (2.7), which in fact coincides with the compressive strength fmθ in the
direction of the strut (Figure 2.16).
17
Figure 2. 16. Stress state considered to evaluate the strength of masonry [Crisafulli, 1997]
2.4.2 Elastic Modulus
The elastic modulus Em represents the initial slope of the strain-stress curve and its values
exhibit a large variation. Different approaches can be found in the literature for the calculation
of Em. Since masonry is a composite material consisting of bricks and mortar which have
distinct properties, several researchers [Ameny et al., 1983; Binda et al., 1988; Drysdale et
al., 1994; Sahlin, 1971] assumed linear elastic behaviour for both materials and that the sum
of deformation of the bricks and mortar joints is equal to the compressive deformation of
masonry, concluding in the same equation.
Other researchers related the modulus of elasticity of masonry walls with the compressive
strength of the material. These empirical equations result in their majority in a range of values
between 400fmθ<Em<1000fmθ [Crisafulli, 1997]. Specifically, Paulay and Priestley [1992] and
Sahlin [1971] give the following expression
E m = 750 f m
(2.8)
while San Bartolomé [1990] proposes that
E m = 500 f m
(2.9)
Sinha and Pedreschi [1983] after tests on masonry prisms made with different bricks and
mortars resulted in
E m = 1180 f m0.83
(MPa)
(2.10)
E m = 2116 f m0.50
(MPa)
(2.11)
while Hendry [1990] reports that
18
Due to this last dispersion of values, several researchers [Paulay and Priestley, 1992; Sahlin,
1971] concluded in the same expression:
E m = 1000 f m
(2.12)
2.4.3 Tensile Strength
The tensile strength ft represents the tensile strength of the masonry or the bond-strength of
the interface between frame and infill panel. Its presence offers generality in the model but it
can even be assumed zero since it is much smaller than the compressive strength with
insignificant effect on the overall response.
2.4.4 Bond Shear Strength, Coefficient of Friction and Maximum Shear Stress
The shear strength results as the combination of two mechanisms, namely, bond strength and
the friction resistance between the mortar joints and the bricks. As already mentioned in
section 2.2.6, the shear strength can be expressed as the sum of the initial shear bond strength
τo and the coefficient of friction µ, multiplying the latter with the absolute value of the normal
compressive strength in the direction perpendicular to the bed joints. This approach to
describe shear is commonly adopted by the design codes independently of the mechanism of
failure. The parameters τo and µ can be evaluated by direct shear tests or design specifications,
though the former usually lead to overestimated values [Wan and Yi, 1988; Riddington and
Ghazali, 1988], while the latter are usually conservative. Mann and Müller [1982] proposed
expressions for reducing the usually overestimated values from shear tests.
An ample sample of shear bond strengths, τo, has been measured by different researchers.
Hendry [1990] presented experimental results obtained from tests using diverse materials and
τo varied from 0.3 to 0.6MPa. Paulay and Priestley [1992] indicated that typical values range
from 0.1 to 1.5MPa, while Shrive [1991] limits this range between 0.1 to 0.7MPa. Similar
values were reported by Stöckl and Hofmann [1988] for clay and sand-lime masonry units
and Atkinson et al. [1989] for a wide range of materials. Several empirical expressions have
been proposed, dependent on different parameters, but their use must be cautious considering
the numerous variables that affect the shear bond strength.
Due to lack of clear knowledge about the factors that affect the coefficient of friction
contradictory results have been reported. The measured values of µ range from 0.1 to 1.2
according to several researchers [Sahlin, 1971; Stöckl and Hofmann, 1988; Hendry, 1990;
Paulay and Priestley, 1992]. Atkinson et al. [1989] observed that the coefficient of friction
ranges between 0.7 and 0.85 for a wide variety of materials, thus they recommend a value of
0.7 as a reliable lower bound for estimating µ. Paulay and Priestley [1992] suggest a value of
0.3 for design purposes.
Finally, the maximum shear stress τmax is the maximum permissible shear stress in the infill
panel and can be estimated using the expressions suggested in the modified Mann and
Müller’s theory [Crisafulli, 1997] according to the expected mode of failure (Figure 2.10).
19
2.4.5 Strains
(a) Strain at max stress έm. It represents the strain at maximum strength and influences via
the modification of the secant stiffness of the ascending branch of the stress-strain curve. This
parameter may vary from 0.002 to 0.005 according to Crisafulli [1997].
(b) Ultimate strain εu. It is used to control the descending branch of the stress-strain curve,
which is modelled with a parabola to obtain better control of the strut response. For larger
values such as 20έm, the decrease of the compressive strength becomes smoother.
(c) Closing strain εcl. This parameter defines the strain after which the cracks partially close
allowing compression stresses to develop. For large values its effect is not considered in the
analysis. Its suggested values range between 0 and 0.003.
2.5 Geometrical Parameters
The geometrical parameters of the model are the horizontal and vertical offset, the thickness
of the panel, the vertical separation between the struts and the area of the strut. Explanations
for the first two parameters are given below, while the last two are presented in detail in the
following sections, accompanied by a thorough literature review.
The Horizontal and Vertical offset, xoi and yoi represent the reduction of the infill panel’s
dimensions due to the depth of the frame members, so they can be easily computed. In the
model these two parameters define the coordinates of the dummy nodes.
The thickness tw stands for the thickness of the panel.
2.5.1 Vertical Separation between struts
The Vertical separation between struts hz leads to reasonable results for values of 1/3 to 1/2 of
the contact length. The contact length z, as defined by Stafford Smith [1966], who introduced
the dimensionless relative stiffness parameter λ, is given by
π
2λ
(2.13)
E m t w sin( 2θ )
4 E c I c hw
(2.14)
z=
where
λ=4
in which EcIc is the bending stiffness of the columns, while the other parameters are explained
in Figure 2.17.
20
d
w
bw
hw
h
z
lw
Figure 2. 17. Configuration with the geometrical properties of infill panel
2.5.2 Area of Strut
The Area of strut Am is defined as the product of the panel thickness and the equivalent width
of the strut bw, which normally varies between 10% and 25% of the diagonal of the infill
panel, as Stafford Smith [1962] concluded based on experimental data and analytical results.
There are also numerous empirical expressions by different authors for the evaluation of the
equivalent width, presented hereinafter.
Holmes [1961] suggested that
bw = dw /3
(2.15)
Mainstone [1971] obtained a set of equations for different levels of performance, to which
Equation (2.16) belongs.
bw = 0.16 λh-0.3dw
(2.16)
Klingner and Bertero [1978] adopted the Equation (2.17), which has been previously
proposed by Mainstone and Weeks [1970] and was included in FEMA 274 [1997] for the
analysis and rehabilitation of buildings.
bw = 0.175(λ ⋅ h) −0.4 d w
(2.17)
Liauw and Kwan [1984] presented Equation (2.18), taking θ equal to 25o and 50o in order to
represent the commonest cases in practical engineering.
bw =
0.95hw cos θ
λh
(2.18)
21
0.9
0.8
Decanini
(Uncracked))
0.7
Decanini
(cracked)
0.6
bw/dw
0.5
0.4
0.3
0.2
0.1
0
0
2
4
λh
6
8
10
Figure 2. 18. Variation of the ratio bw/dw as a function of the parameter hּλ [Decanini and Fantin, 1986]
Decanini and Fantin [1986], based on tests on framed masonry under lateral loading, proposed
a two sets of equations for different states of masonry. The variation of the strut width versus
parameter h·λ is plotted in Figure 2.18.
Finally, Paulay and Priestley [1992] give a conservative value for the estimation of bw, useful
for design purposes.
bw = dw /4
(2.19)
All the aforementioned expressions by several researchers are plotted and compared in Figure
2.19.
22
Holmes
0.8
Paulay & Priestley
Mainstone
Liauw & Kwan (50' )
bw / dw
0.6
Liauw & Kwan (25' )
Decanini (uncracked)
Decanini (cracked)
0.4
0.2
0
0
2
4
h-λ
6
8
10
Figure 2. 19. Variation of the ratio bw/dw as function of the parameter hּλ
Due to cracking of the infill panel, the contact length between the frame and the infill
decreases as the lateral and consequently the axial displacement increases, affecting thus the
area of equivalent strut. In order to gain generality and achieve control of the variation of the
stiffness and the axial strength of the strut, the value of the residual area is inserted in the
model as percentage of the initial area. It is assumed that the area varies linearly as function of
the axial strain (Figure 2.20), so the two strains between which this variation takes place are
required. There is insufficient information to estimate the practical values of the parameters
describing this reduction, as Crisafulli [1997] mentions. Details on which values were
implemented in the model are given in section 3.4.8.
Ams
Am1
Am2
ε2
ε1
ε
Figure 2. 20. Variation of the area of the masonry strut as function of the axial strain
23
2.6 Empirical Parameters
Furthermore, a number of empirical parameters are involved in the calculation of different
parameters associated with the cyclic behaviour. Crisafulli proposes for these parameters a
range of recommended values, which have been obtained after experimental results. A short
explanation about their meaning is given below:
γun: it defines the unloading modulus in proportion to the initial modulus Em and modifies the
internal cycles, not the envelope.
αre: it is used to predict the strain at which the loop reaches the envelope after unloading.
αch: it predicts the strain at which the reloading curve has an inflexion point, controlling the
loops’ fatness. Specifically, increase of the parameter incurs fatter loops.
βa: it defines the auxiliary point used to define the plastic deformation after complete
unloading.
βch: it defines the stress at which the reloading curve exhibits an inflection point.
γplu: it defines the modulus of the hysteretic curve at zero stress after complete unloading in
proportion to Em.
γplr: it defines the modulus of the reloading curve after total unloading.
ex1: it controls the influence of εun in the degradation stiffness.
ex2: it increases the strain at which the envelope curve is reached after unloading and
represents cumulative damage inside repeated cycles, important when there are repeated
consecutive cycles inside same inner loops.
γs: it represents the proportion of the panel stiffness assigned to the shear spring.
αs: the reduction shear factor represents the ratio of the maximum shear stress to the average
stress in the panel.
24
Table 2.1. Suggested and limit values for
the empirical parameters.
Parameters
γun
αre
αch
βa
βch
γplu
γplr
ex1
ex2
γs
αs
Suggested
values
1.5 – 2.5
0.2 – 0.4
0.3 – 0.6
1.5 – 2.0
0.6 – 0.7
0.5 – 0.7
1.1 – 1.5
1.5 – 2.0
1.0 – 1.5
0.5– 0.75
1.4 – 1.65
Limit
values
≥1
≥0
0.1 – 0.7
≥0
0.5 – 0.9
0 – 1.0
≥1
≥0
≥0
Crisafulli [1997] defined the limit values for which each parameter has meaning and proposed
a suggested range of values, in which he concluded after experiments and for which he had
realistic results. These limit and suggested values of the empirical parameters are presented in
Table 2.1. A set of values for the empirical parameters needs to be selected, concluding thus
in a base model, which will be used in the analyses for the verification of the accuracy of the
infill panel model. Therefore, a sensitivity analysis is required before in order to assess the
relative importance of each empirical parameter and its effect on the behaviour of the model.
2.7 Openings
The presence of openings in infilled panels constitutes an important uncertainty in the
evaluation of the behaviour of infilled frames. Several researches have investigated the
influence of the openings on strength and stiffness, the prediction of which becomes rather
difficult due to the variability in the location and the dimensions of openings.
According to Sortis et al. [1999], the presence of openings modifies the structural behaviour
of the infill panels by reducing the strength and stiffness. Moreover, the openings decrease the
loading corresponding to the initial cracking stage with premature development of cracks due
to the stress concentration in the aperture corners and the energy dissipation capacity.
Benjamin and Williams [1958] measured 50% reduction of the ultimate strength in infilled
frames for a centred opening with dimensions equal to 1/3 of the infill panel’s dimensions. At
the same time for the same infilled frame, they noticed during the loading process up to the
50% of the ultimate load that the presence of the opening led to slight reduction of the
stiffness, but when the load increased further the stiffness sharply decreased. Similar results
for the strength reduction were obtained by Gostič and Žarnič [1999] after tests on a scaled
two-storey, two-bay infilled frame including windows and doors.
25
Mallick and Garg [1971] investigated the effect of possible positions of openings on the
lateral stiffness of infilled frames. Their tests indicated that small centred openings had no
significant effect, while considerable decrease was recorded in cases of openings located close
the loaded ends of the compressed diagonal [CEB, 1996].
On the other hand, Liauw and Lee [1977] reported that the openings’ presence did not
influence the stiffness of the infilled frames significantly. Moreover, Dawe and Young [1985]
did not observe any significant reduction of the ultimate strength of the infilled frames with
openings. According to the same authors, it seems that the negative effect of a door on the
stiffness of the infilled frame is more pronounced for a symmetrically located opening.
Nevertheless, in case of large amplitude cyclic actions, where the eccentricity of an opening is
favourable during the opposite direction, it seems that the symmetrical location is preferable
[CEB, 1996].
Fiorato et al. [1970] have found that the reduction of the load resistance of an infilled frame is
not proportional to the reduction of the cross-sectional area on the infill due to openings. In
their tests, openings that decreased the horizontal cross-sectional area of an infill panel by
50% led to a strength reduction of about 20-28%.
Mosalam et al. [1997] confirmed that observation by additional tests on frames with
symmetric openings. They reported that the infilled frames including openings show lower
initial strength and more ductile behaviour than the frame with solid infill panels. For a
decrease of 17% of the cross-sectional area, the maximum load resistance of the frame with
the symmetric window was almost the same as that without openings, while the presence of a
door led to reduced load resistance by 20%. Finally, they observed change in the crack
patterns. The cracks tended to form at the corners of the openings and propagate towards the
loaded corners.
Bertoldi et al. [1994] proposed a set of expressions for the calculation of the reduction
coefficient rac. The parameters that they used for characterizing the strength and stiffness
reduction of the infill panel due to apertures are the ratio between aperture area and panel area
(Aa), the ratio between the aperture width and panel width (Ac) and the existence and type of
strengthening in the aperture. An infill panel with opening can be considered effective in
terms of influence in the structural behaviour if the following conditions yield [Sortis et al.,
1999]:
Aa (%) ≤ 25%
(2.20)
Ac (%) ≤ 40%
(2.21)
In the case of non-strengthened opening, rac is given by Equation (2.22).
rac = 0.78e −0.322 ln Aa + 0.93e −0.762 ln Ac ≤ 1
(2.22)
26
Bertoldi et al. [1999] using finite elements models and experimental tests obtained values of
the diagonal strut width for walls with and without openings.
In the model proposed by Mallick and Garg [1971] using plane rectangular elements, they
tried to study the design problem of openings in the infill panels, obtaining under static
monotonic loading satisfactory numerical results compared to experimental results.
Utku [1980] investigated the effect of openings parameters such as aspect ratio, position and
area, on the stiffness and strength of single-storey walls with single openings under
earthquake loading, analyzing the walls as plane stress problems assuming linearly elastic
isotropic material and small deformations.
In a different approach, Thiruvengadam [1985] proposed the use of several diagonal struts in
each direction to simulate the effect of an infill panel, rendering thus the presence of openings
easier to be considered (Figure 2.21). Similarly, Hamburg [1993] proposed a multi-strut
configuration that can account for the openings also, but the evaluation of the characteristics
of the struts is rather complicated (Figure 2.22).
(a)
(b)
(c)
Figure 2. 21. Formulation of struts around openings: (a) position of openings, (b) struts for monolithic
infill panels, (c) struts for separating infill panels [Thiruvengadam, 1985]
27
Figure 2. 22. Equivalent struts in infill panels with openings [Hamburg, 1993]
Papia [1988], using elastic ‘beam’ finite elements to discretize the surrounding frame and
modelling the infill panel as a shell in tension, tries to estimate the loss in stiffness due to
centred openings keeping the same ratio of the opening’s dimensions with respect to the
dimensions of the infill panel itself. He proposes an approximately linear reduction of
stiffness for a wide range of ratios between the infill panel and the opening dimensions.
A representative analytical work on the influence of the openings on the elastic stiffness of
infill panels has been presented by Giannakas et al. [1987]. The approach used to model the
infill walls is a finite element method, under the assumption of homogeneous, elastic and
isotropic material for the masonry.
Asteris [2003] proposes stiffness reduction factors for different opening percentages after
analyzing a number of single-storey, single-bay infilled frames with different configurations.
The problem is studied in the linear range, using a new finite element technique based on
contact points between frame and infill panel [Asteris, 1996] to model the behaviour of
infilled frames under lateral loads.
The distinct lack of recommendations or an integrated way to quantify the effect of the
openings’ presence, especially in the nonlinear range, emerged after the extensive literature
review. Therefore, as described in section 3.4.8, empirical reduction factors were used to
decrease the strut area, and thus account for the influence of the openings.
28
Chapter 3. PRESENTATION OF THE EXPERIMENTAL CASE STUDY
3. PRESENTATION OF THE EXPERIMENTAL CASE STUDY
3.1 Difficulties in Experimental Tests of Infilled Frames
The behaviour of infilled frames has been subject of numerous experimental investigations.
The results obtained from these experimental attempts improved the knowledge and
understanding of the phenomena related to the behaviour of infilled frames. Nevertheless, the
difficulties faced during the experimental tests cannot be underestimated.
Specifically, the variability of the materials used in infilling the RC frames and the sensitivity
of these materials to scale effects constitute main difficulties. Small-scale models should be
considered as rather unreliable, but on the other hand specimens of large dimensions are
associated with high cost. So the limited number of affordable tests comes in contrast with the
numerous influencing parameters that have to be investigated. Moreover, the large dimensions
and weight of specimens demand special test facilities, while the distributed mass of infilled
frames renders the application of pseudo-dynamic techniques complicated.
The difficulties in testing lead to problems in the systematic evaluation of the experimental
results. The cost of tests prohibits the separate investigation of the influential parameters,
although they are numerous, and leads to overexploitation of the tests. So the influence of
each parameter cannot be distinguished with certainty, since the changing parameters during a
test are many. The high cost does not allow the repetition of identical tests, making
impossible the evaluation of scatter, which is inherent in the investigated phenomena.
Moreover, it is rather rare to find directly comparable tests in literature, so no quantitative
conclusions can be drawn. Many of the results cannot be considered reliable, since they were
obtained from tests on specimens where the reinforced concrete members were scaled down,
while for the infill panels, materials available in the market were used. Finally, most of the
test specimens are single-bay, single-storey plane frames, although the aim is to study the
behaviour of the structure as a whole.
3.2 The ICONS Frame
3.2.1 Introduction
In an effort to study the behaviour of the infill panels, pseudo-dynamic tests on a full-scale
frame model were carried out at the ELSA reaction-wall laboratory within the framework of
the ICONS research programme [Pinto et al., 1999]. Two frames, identical in geometry,
construction and detailing (Figure 3.1), were constructed and tested, one was bare and the
29
other infilled. The full-scale, four-storey, three-bay reinforced concrete frame considered in
this work was infilled with brick walls that included openings of different dimensions.
Figure 3. 1. The ICONS frames
3.2.2 Geometry and Reinforcement Detailing
The tested RC frame can be regarded as representative of the design and construction practice
of 40-50 years ago in countries such as Italy, Greece and Portugal. The frame was thus not
expected to meet the modern code seismic design requirements, for which reason no specific
detailing provisions were considered, no preferential inelastic dissipation mechanisms were
assumed and no specific ductility or strength was provided [Pinto et al., 1999].
The frame is designed to withstand only vertical loads, while its resistance to horizontal loads,
as far as the ultimate limit state is concerned, is approximately 8 % of its weight. Similarly,
the lateral resistance in terms of allowable stresses is 5 % of its weight reflecting the common
practice in those decades.
30
Figure 3. 2. Plan and elevation views of RC frame with infill panels [Carvalho et al., 1999]
The four-storey frame consists of two bays of 5.0m span and one bay of 2.5m span (Figure
3.2Figure 3. 2). The inter-storey height is 2.7m, the slab thickness is 0.15m (Figure 3.3) with
a width of 4.0m.
31
Figure 3. 3. Slab reinforcement [Carvalho et al., 1999]
All beams have equal geometry in all floors: 0.25×0.50m for the longitudinal beams,
0.20×0.50m for the transversal ones (Figure 3.4). The columns have the same geometrical
characteristics along the height, i.e. 0.20×0.40m and 0.20×0.30m, apart from the second
‘strong’ column with dimensions 0.60×0.25m, that is the only one that works in its stronger
axis and plays dominant role in the structural response of the frame. The section of the
stronger column decreases to 0.50×0.50m on the third and fourth storey (Figure 3.5).
Duplication of the longitudinal reinforcement, which consists of smooth round bars
commonly used in the past years, occurs at the bottom of the columns of the 1st and 3rd floor
due to lap-splice for 0.70m (Figure 3.5). Stirrups are provided every 0.15m. The column
reinforcement splicing, joints and stirrup detailing are characterized by a distinct lack of
confinement, characteristic in the non-ductile reinforced concrete structures constructed till
the late 70’s.
Specifically, inadequate transversal reinforcement, maximum distance between longitudinal
bars, inexistence of transversal reinforcement in the joints, inadequate bends of the stirrups
and no specific mechanism for energy dissipation were identified as main seismic design
deficiencies. The frame does not satisfy most of the current requirements in terms of detailing
and global deformation mechanisms, therefore poor seismic performance is expected with
development of premature storey mechanisms.
32
Transversal Beams
Figure 3. 4. Beam reinforcement details [Carvalho et al., 1999]
33
Figure 3. 5. Column reinforcement details: cross sections and lap-splice [Carvalho et al., 1999]
34
The infill walls are non-load bearing. The long external bay infill contains a window opening
(1.2 m × 1.0 m) at each of the four levels. The central bay contains a doorway
(2.0 m × 1.75 m) at ground level and window openings (2.0 m × 1.0 m) in each of the upper
three levels of the building. The external short (2.5 m span) bay contains solid infill panels,
i.e. without openings. The general layout of the location and the dimensions of the infill
panels are given in Figure 3.6.
Figure 3. 6. Elevation view of the infilled frame – Location and dimensions of openings [Carvalho et al.,
1999]
Further information about the case study frame as well as the tests conducted in ELSA can be
found in Pinto et al. [1999], Carvalho et al. [1999], Pinho and Elnashai [2000] and Varum
[2003].
3.2.3 Material Properties
The materials were chosen to have properties as similar as possible to those used in the
construction in the late ‘70s in the southern European countries. The concrete used
corresponds to a normal weight low strength concrete, class C16/20, but since compressive
strength tests on concrete specimens were carried out, the average values were obtained for
each type of element (beams, columns, slabs) of all floors. The concrete specimens were
cubes of 150mm side and four cubes have been tested for each casting phase.
35
It is noteworthy that the values obtained and presented in Table 3.1 were confirmed by
extensive non-destructive tests on the structure. Furthermore, in order to confirm the real
properties of the concrete further tests were performed using concrete core specimens from
the non-damaged regions after the completion of the tests. The results from the compressive
strength tests on cylindrical specimens were in good agreement with those of the cubic
specimens.
A small variance was found for each casting phase, whilst large differences corresponded to
the various casting phases, something that should be taken in consideration in the numerical
analyses. The concrete compression tests on the specimens led to an average strength of
16MPa indicating the poor quality of concrete and the low construction standards. However,
this was expected. The quality control in the construction of the models was intentionally low
so that the outcome would be a structure that would represent worst case scenarios of typical
reinforced concrete construction of forty or more years ago.
Table 3. 1. Concrete average compressive
strength after tests on specimens.
Specimen group
(casting phase)
Compressive cubic
ultimate strength
(MPa)
Base foundation
31.75
Columns 1st floor
16.66
st
Slab/Beams 1 floor
13.24
Columns 2nd floor
13.78
nd
Slab/Beams 2 floor
18.10
Columns 3rd floor
16.50
rd
Slab/Beams 3 floor
21.63
Columns 4th floor
13.58
th
Slab/Beams 4 floor
16.98
The steel used was class Fe B22k according to Italian standards with nominal values of yield
stress, ultimate strength and ultimate strain equal to 215MPa, 335MPa and 24% respectively.
Tensile strength tests on steel bar specimens have been carried out and from a best-fit of
experimental diagrams for the reinforcing bars, the mean mechanical properties were
estimated. The best-fit was based on a linear regression for the elastic initial branch and on a
non-linear regression using the Mander [Mander et al., 1988] model for the hardening branch
as shown in Figure 3.7. The values obtained are summarised in Table 3.2.
36
Stress (MPa)
ICONS Structure
Best-fit of steel constitutive law
600
Mander
f = 451,54 - 108,18 . [(22,91 - ε) / 19,88]
500
5,06
Linear
f = 2044,78 ε
400
300
200
100
0,168 %
3,027 %
0
0
2
4
6
8
10
12
14
16
18
20
Strain
Figure 3. 7. Best-fit of steel constitutive law [Carvalho et al., 1999]
Table 3. 2. Mean mechanical properties of steel
specimens after tests.
Mechanical properties
Value
Young modulus – E m
204.5 GPa
Yield stress – f sym
343.6 MPa
Hardening strain – ε shm
3.03 %
Tangent modulus
at beginning of hardening – E sh
2.8 GPa
Ultimate strength – f sum
451.5 MPa
Ultimate strain – ε sum
22.9 %
The mechanical properties of steel obtained from tests on specimen bars differ considerably
from the nominal values. In fact the steel tested and used in the construction of the frame has
significantly higher strength. The explanation for such difference lies in the fact that only
minimum strength requirements were included in the old codes, leading thus to steel strength
much higher than the nominal values.
The non-load bearing infill panels were constructed after the frame, using hollow bricks,
horizontally perforated with dimensions 0.120×0.245×0.245m. Plaster of 15mm was applied
on both sides of the wall, causing an evident increase in stiffness and strength. Wallets
representative of the masonry infill panels were tested in the horizontal, vertical and diagonal
direction. The average values for the compression, shear and tensile strength of the masonry
obtained are presented in Table 3.3.
37
Table 3. 3. Material properties of infill panels.
Masonry
Wallet
Strength
Average
Values
(MPa)
Compressive
1.10
Tensile
Shear
0.575
0.575
Type of Test
Compressive
perpendicular to bed joints
Diagonal Compression
Diagonal Compression
3.2.4 Mass
The overall mass is calculated as 179.23 tonne. This mass corresponds to the self-weight of
the frame, the live load, the finishings, other self-loads and certainly to the infill panels. For
the infill panels, a specific weight by m2 of wall equal to 0.785kN/m2 was considered in the
calculations. The mass for each node of the infilled frame is presented in detail in Table 3.4.
Table 3. 4. Mass of the infilled frame.
Mass (tonne)
Floor
Column 1
Column 2
Column 3
Column 4
1
9.943
17.371
13.679
6.255
2
9.943
17.371
13.679
6.255
3
9.943
17.371
13.679
6.255
4
7.752
14.230
10.991
4.513
3.3 Testing Procedure
3.3.1 Input Motion
The input seismic motions were chosen to be representative of a moderate-high European
hazard scenario. A probabilistic seismic hazard analysis was carried out. Consistent with the
hazard, acceleration time-histories were artificially generated [Campos-Costa and Pinto,
1999] yielding a set of twelve uniform hazard response spectra for increasing return periods.
Finally, three acceleration time-histories of increasing return periods of 475, 975 and 2000
years were used for the experiment. The return periods for the input motions were chosen so
as to test the structure under the different seismic hazard levels specified in the FEMA-273
documents [FEMA-273, 1997]. They correspond to "Rare" (475 years) and "Very Rare" (975
and 2000 years) events, under which a structure has to meet the "Life Safety" and "Collapse
Prevention" performance levels, respectively.
The displacement and acceleration linear elastic spectra for 5% damping corresponding to
475, 975 and 2000yrp are presented respectively in Figures 3.8 and 3.Figure 3. 99. The time
38
series of acceleration for 475, 975 and 2000yrp with peak accelerations of 0.22g, 0.29g and
0.38g, respectively, are plotted in Figure 3.10.
Figure 3. 8. Displacement spectrum of input motion
Figure 3. 9. Acceleration spectrum of input motion
39
Figure 3. 10. Ground motion accelerations for 475, 975 and 2000yrp
40
3.3.2 Pseudo-dynamic Test Method
The experimental seismic response was obtained by applying pseudo-dynamic test method.
This method uses a direct step-by-step integration technique to compute the displacement
response of the specimen subjected to numerically specified seismic excitation record,
utilizing the non-linear restoring forces actually developed by the specimen during the test.
In a pseudo-dynamic test the test structure is idealized as a discrete-parameter system having
a finite number of dynamic degrees of freedom, such as the governing equation of motion can
be simplified to a system of second-order differential equations, which can be expressed in
matrix form as
Ma + Cv + r (d ) = p( t )
(3.1)
in which M and C are the mass and viscous damping matrices of the system, v and a are the
nodal velocity and acceleration vectors, first- and second order time derivatives of the nodal
displacement vector d, r is the structural restoring force vector, which is non linear function of
d, and p is the external force excitation applied to the system. Equation (3.2) can be
conveniently solved by means of a direct step-by-step integration method to obtain the
displacement response d to any arbitrary external excitation p.
The pseudo-dynamic test proceeds in a stepwise manner under a step-by-step integration
procedure. In each step, the computed displacements d are quasi-statically imposed on the
specimen by means of computer controlled actuators. The restoring forces r developed by
structural deformations and measured at the end of a step are used to compute the
displacement response in the next step, based on the prescribed values of M and C as well as
on a numerically specified excitation history p. This process is repeated until the entire
response history is obtained [Shing and Mahin, 1985].
In the pseudo-dynamic tests, the matrices of the inertia and viscous damping forces may be
calculated from the preliminary dynamic identification tests performed on the structure (e.g.
free vibration and stiffness tests). Alternatively, these matrices are computed by the static
condensation of the matrices corresponding to the complete structure to the degrees of
freedom of interest. The numerically modelled inertia and viscous damping forces are a
relatively straightforward matter compared to the non-linear structural restoring forces, which
are measured experimentally because of the difficulty in modelling them accurately. The
process automatically accounts for the hysteretic damping due to inelastic deformation and
damage of the structural materials, which is the major source of energy dissipation. Typically,
the viscous damping matrix C is considered null in a pseudo-dynamic test [Pinto et al., 1996].
For the case study frame three seismic records as external excitation were used as already
explained in previous section. One degree of freedom corresponding to a longitudinal
horizontal displacement was considered per floor, where displacements were applied to the
structure by means of actuators.
41
3.3.3 Modelling the Case Study Frame
The case study frame was modelled in SeismoStruct. Each structural member was subdivided
into four inelastic beam-column elements with smaller length at the member ends so as to
ensure the accurate modelling of expected plastic-hinge zones. Beams and columns are
modelled as extending from the centre of one beam-column joint to the centre of the next. The
mass is modelled as lumped mass, while the vertical loads defined in the model are applied in
the beam-column nodes and simulate the self-weight of the frame, including the weight of the
finishings and the masonry infill panels, as well as the live load. Inelastic frame elements
were employed, divided in 200 fibres.
For concrete, the Mander et al. [1997] nonlinear model with constant confinement is used,
while for steel the Menegotto-Pinto [1973] model with Filippou hardening isotropic rule
[1983] is chosen. The material properties were modelled in detail by using all the available
information obtained from experiments on specimens during the construction of the case
study frame (see section 3.2.3). The reinforced concrete and steel properties used are given
respectively in Tables 3.1 and 3.2. The choice not to use a uniform value for the material
properties of the concrete members but distinct values for each type of elements (i.e. columns,
beams) increased the modelling effort, which was compensated by limiting the uncertainties
concerning the construction materials, and thus obtaining a more realistic representation of the
frame.
Furthermore, attention was paid to the accurate modelling of the reinforcement of the several
regions of the concrete members. This resulted in an increased number of sections
corresponding to the beams ends and the beam mid-spans, with different reinforcement
distributions. Finally, the lap-splice in the columns of the first and third floor was taken under
consideration.
The effective width of the slab was taken equal to 1.0m and 0.6m for long and short spans
respectively. The slab participation was measured during the tests by placing transducers at
the top and bottom of the first floor slab in critical zones in such a way that was possible to
contrast the long-span bays versus the short-span bay. The values used in the modelling were
obtained after measurements and indicate smaller slab collaboration width than the one
estimated by the code provisions.
The boundary conditions for ground columns were defined as fixed supports, representing
thus the strong foundation, consisting of a thick continuous slab and high foundation beams,
which was provided in the laboratory with the aim of fixing the structure to the laboratory
strong floor, avoiding sliding and overturning during testing. In the rest nodes the out-ofplane degrees of freedom were restrained since the analysis conducted was two-dimensional.
3.4 Calculation and Selection of the Model Parameters for the Case Study Frame
After an extensive literature review concerning the parameters involved in the model,
recommendations for the calculation or selection of them were given in detail in section 2.4.
Hereinafter, the values of the mechanical, geometrical and empirical parameters of the model
that were finally implemented are presented, as well as the steps of computing or choosing
them.
42
3.4.1 Compressive strength
Five infill masonry specimens (1.0m ×1.0m wallets without plaster) were constructed and
tested in compression for perpendicular and parallel to the bed joints direction at the
University of Pavia. The specimens were constructed using the same materials (block units
and mortar) and with the same geometry (joints thickness, fabric) as the ones used for the
infill panels of the case study frame. Further details about the series of tests conducted in
Pavia can be found in Pinto et al. [2001].
From the results of the tests performed on the specimens, the average value for the strength of
masonry in the direction perpendicular to the bed joints was estimated to be 1.11MPa. It is
noted that the values obtained from the test results exhibit high coefficient of variance,
however, the information derived from them constitutes the closest estimate of the infill
panel’s properties.
The value of 1.1MPa was used to compute the compressive strength in the direction of the
strut with the help of Equation (2.7). According to the aspect ratio of the infill panel the value
of the compressive strength of the strut fmθ changes. The long- and the short-bay panels have
net dimensions 2.2×4.6m and 2.2×2.3m respectively, so the value of fmθ for the long- and the
short-bay panels was estimated as 5.0MPa and 3.5MPa respectively.
3.4.2 Elastic Modulus
As demonstrated in section 2.4.2 the elastic modulus exhibits large variation. In the
implementation of the model Equation (2.12) in which several researchers resulted was
employed. In accordance with Equation (2.12) and the already calculated values of fmθ, the Em
values for the long- and short-bay panels are estimated as 5000MPa and 3500MPa
respectively.
3.4.3 Tensile Strength
As already mentioned, the value of tensile strength has no significant effect on the overall
response since it is much smaller than the compressive strength. That is why it can even be
assumed zero.
However, diagonal compression tests on masonry wallets have been conducted with the aim
to evaluate the conventional tensile strength, which can be related to the shear strength of the
masonry walls. According to the standards (RILEM recommendations, ASTM standards), the
tensile strength is typically estimated from the load at failure of the specimen assuming that
the material is elastic, isotropic and homogeneous. A square masonry panel is subjected to a
compressive force applied at two opposite corners along a diagonal until the panel cracks.
The wallets had nominal dimensions 1.0m ×1.0m and were constructed at the same time,
using the same materials (block units and mortar) and with the same geometrical requirements
in terms of joint and plaster thickness as the infill walls [Varum, 2003]. The specimens were
constructed at the ELSA laboratory and tested at the laboratory of University of Pavia. It has
to be noted that among the twelve specimens tested, four of them were with plaster on both
faces. The tensile strength of these specimens was evaluated to be 0.575MPa, which is the
value used in the implementation of the model.
43
3.4.4 Bond Shear Strength, Coefficient of Friction and Maximum Shear Stress
In accordance with section 2.4.4, the implemented value of bond shear strength was 0.3MPa,
as an average value that agrees with the recommendations of the several researchers.
Similarly, the coefficient of friction was selected equal to 0.7, while a reasonable estimation
for the maximum shear stress was 1MPa.
3.4.5 Strains
The value of 0.0012 for strain at max stress έm provided the best results, so it was
implemented for all the infill panels of the model.
In sequence, the value of 0.024 was used for the ultimate strain εu. This large value that is
20έm was chosen in order to achieve a smoother decrease of the compressive strength.
Finally, the value for the closing strain εcl was taken as 0.003.
3.4.6 Horizontal and Vertical Offset
The horizontal and vertical offset was calculated based on the dimensions of beams and
columns as well as the net dimensions of the infill panels. Beams and columns are modelled
as extending from the centre of one beam-column joint to the centre of the next. In the centre
of beam-column joint the external nodes are located. So the vertical offset yoi for all panels is
equal to half the beam depth divided by the net height of the infill panel, i.e. 12%.
In case of horizontal offset, the value of xoi varies since the columns have different
dimensions in their strong axis, as well as the bays do not have equal length. It is reminded
that the columns have the same geometrical characteristics along the height, i.e. 0.20×0.40m
and 0.20×0.30m, apart from the second ‘strong’ column between the two long bays with
dimensions 0.60×0.25m. The full infill panel has 2.3m net length, while the other two longer
infill panels have net length equal to 4.6m. Consequently, the horizontal offset for the full
infill panel is equal to 0.10/2.3, i.e. 4.3%. Since one value for the offset is inserted in the
model, in the case of the longer bays it was assumed that the stronger column dominates so
the xoi was computed as 0.3/4.6 that gives a value of 6.5%.
3.4.7 Vertical Separation between Struts and Thickness of Infill Panel
The vertical separation between struts is percentage of the contact length. The definition and
the relative expressions for the estimation of the contact length and the relative stiffness
parameter were presented in section 2.5.1. Based on the geometry of the case study frame, the
angle of the struts for the long- and short-bay infill panels and the moment of inertia of the
columns were computed.
Having in mind that the vertical separation hz fluctuates between 1/3 and 1/2 of the contact
length, average values and approximations in the calculation of the contact length and
consequently of hz are justified. The values finally implemented for hz are 23% and 17% for
the long- and the short-bay infill panels respectively.
It has to be noted that the contact length is different for each side of the infill panels due to the
different dimensions of the columns. In this case an average value for z was assumed.
44
Furthermore, the contact length varies along the height of the frame for the infill panels that
are in contact with the strong column, however, this variation is insignificant. An estimation
of 30000MPa for the elastic modulus of concrete was used, while for the elastic modulus of
masonry the values already estimated were inserted in the expression of λ.
Finally, the thickness tw of the infill panel was taken as 0.15m, considering that the thickness
of bricks is 0.12m and both sides of the panel are covered with plaster of 15mm width.
3.4.8 Area of the Strut
The significance of evaluating correctly the area of the equivalent strut consists in the fact that
the stiffness of the infill panels and consequently of the structure is directly affected. In cyclic
loading, stiffness degradation is observed after the cracking of infill panels. Moreover, the
presence of openings reduces the initial stiffness of the panels. So the effect of cyclic loading
and openings can be accounted for by varying the area of the struts.
Knowing that the area of the strut varies between 10% and 25% of the diagonal of the infill
panel, a first approximation for the equivalent strut area can be made. However, the area was
calculated in more detail by utilising the relations for estimating the width of the equivalent
strut, since the thickness of the strut is already known.
First, the values of λ for each infill panel have already been calculated for the needs of the
contact length. The width of the equivalent strut is computed for all the approaches proposed
by several researchers and presented in section 2.5.2. The most satisfactory results were
obtained by using the expression by Holmes [1961] as that is given by Equation (2.15). It is
pointed out that the values according to Equation (2.15) gave the highest estimation for bw and
consequently for the initial area of the struts.
The areas of struts for the long- and the short-bay infill panel were initially found equal to
0.2550m2 and 0.1591m2 respectively. However, consideration is required for the presence of
openings, as well as for the effect of cyclic loading on the strut area. The decrease in stiffness
can be realised by reducing the elastic modulus or/and the equivalent strut area since stiffness
of the strut is proportional to both Em and Am as clearly seen by Equation (2.5). In the
implemented hysteresis model the elastic modulus does not remain constant, which inherently
implies stiffness reduction.
In the case study frame there are openings of different dimensions and position (Figure 3.6).
Specifically, the left-side infill panels include a large window that is not centred, the panel in
the mid-bay of the ground floor has a door, while the rest infill panels of the mid-bays of the
upper floors have a small centred window. This configuration entails different percentage of
reduction will be assumed for each opening. That is why, in the model the strut areas for the
infill with the small window, the large window and the door are estimated as 70%, 60% and
50% respectively of the equivalent strut area initially calculated before considering the
presence of openings. These percentages are comparable with the ones proposed by Pinho and
Elnashai [2000]. However, it has to be stressed that the lack of recommendations in the
literature about the effect of the openings was remarkable and reflects the fact that no specific
way of quantifying the area reduction due to openings has been found.
45
As already explained the model offers the option to consider the degradation of stiffness due
to cyclic loading by reducing the initial area after a certain level of strain. The model gains
thus generality, though there is insufficient information to estimate the practical values of the
parameters describing this reduction, as Crisafulli [1997] mentions. This option of the model
was employed, so the residual area is assumed 10% of the area calculated after the reduction
due to openings. The displacements between which this reduction takes place were defined as
0.0003 and 0.0006 for the small infill panel and 0.0005 and 0.0009 for the large infill panels.
Finally, no strength reduction due to openings was considered since in the literature this point
is not fully clarified.
3.4.9 Empirical Parameters
The finally selected and implemented values for the empirical parameters are presented in
Table 3.5. In the same Table, the suggested and the limit values, already given in Table 2.1,
are included too in order to facilitate the comparison. It is reminded that the suggested and the
limit values were obtained by Crisafulli [1997] after calibration of experimental data.
However, out-of-range values were used for four of the parameters since this led to a better
match with the experimental results.
Table 3. 5. Empirical Parameters.
Parameters
Suggested
values
Limit
values
γun
1.5 – 2.5
≥1
Used
value
1.7
αre
0.2 – 0.4
≥0
0.2
αch
0.3 – 0.6
0.1 – 0.7
0.7
βa
1.5 – 2.0
≥0
2.0
βch
0.6 – 0.7
0.5 – 0.9
0.9
γplu
0.5 – 0.7
0 – 1.0
1.0
γplr
1.1 – 1.5
≥1
1.1
ex1
1.5 – 2.0
≥0
3.0
ex2
1.0 – 1.5
≥0
1.0
γs
0.5– 0.75
0.7
αs
1.4 – 1.65
1.5
46
Chapter 4. VERIFICATION OF THE NUMERICAL MODEL
4. VERIFICATION OF THE NUMERICAL MODEL
4.1 Preliminary Verification
Before proceeding with the analyses of the infilled frame, there was need to verify the
accuracy of the numerical model of the bare frame. The aim was to ensure that the numerical
model before inserting the infill panel elements represented satisfactorily the bare frame.
Checking the modelling of the bare frame is prerequisite for the validity of the analysis that
will follow. By limiting the other parameters of uncertainty, the infill panel model is rendered
in fact the only variable, the effect of which should be examined.
4.1.1 Modal Analysis of the Bare Frame
Initially, a modal analysis was undertaken to provide a first insight into the structure. The
values of the periods of the 1st, 2nd, 3rd and 4th mode are computed and presented in Table 4.1,
which also contains the experimental estimates of the natural frequencies. In order to provide
data for modal identification of the bare frame, a very low intensity earthquake (nondestructive test) was applied to the structure before the pseudo-dynamic test takes place,
allowing the excitation of all modes. For further and detailed information about these methods
which are based on time-domain linear models and are extensively applied to the results of
pseudo-dynamic tests at ELSA laboratory, refer to Molina et al. [1999]. It should be
mentioned that before the pseudo-dynamic tests some modal dynamic tests were also carried
out in order to assess the natural frequencies of the bare frame. The tests were conducted by
exciting the structure with an instrumented impact hammer of 5Kgr mass. A good agreement
was verified between the natural frequencies estimated with the non-destructed tests and those
experimentally measured.
The first observation is that the contribution of the fundamental period is dominant. The
fundamental period can be characterised long for a four-storey reinforced building, reflecting
the flexibility of the structure mainly due to the inexistence of infill panels. As far as the value
of the first period is concerned, the fundamental period of the numerical model is 0.679sec,
exceeding only by 6.6% the estimated first period of the test frame, which was found equal to
0.637sec. Generally, the satisfactory agreement between the numerical and experimental
values of periods is remarked.
47
Table 4. 1. Estimated natural periods of the bare frame
model.
Period
(sec)
1st
2nd
3rd
4th
Experimental
Numerical
0.64
0.22
0.13
0.09
0.68
0.25
0.15
0.10
4.1.2 Static Time-history Analysis of the Bare Frame
As a second step, the numerical model for the frame was subjected to static-time history. In
static time-history analysis, the applied loads (displacements, forces or a combination of both)
can vary independently in the pseudo-time domain, according to a prescribed load pattern.
The applied load in a nodal position is given by a function of a time-dependent load factor and
the nominal load. This type of analysis is typically used to model static testing of structures
under various force or displacement patterns (e.g. cyclic loading).
The displacements obtained during the pseudo-dynamic test were imposed on the numerical
model of the bare frame by employing static time-history analysis. The values of base shear
were numerically computed and compared with the equivalent experimental results. It is noted
that the bare frame was tested only for the 475yrp and 975yrp records. In fact the latter was
interrupted to avoid collapse of the frame that had experienced severe deformations. In
Figures 4.1 and 4.2 the base shear time-histories from the numerical analyses and the pseudodynamic tests are plotted, offering an immediate comparison of the response of the bare
frame. The numerical results match satisfactorily with the experimental ones, despite the
former tend to underestimate the shear in several cycles. The good coincidence between the
numerical and experimental results is evident, confirming thus that the numerical model
constitutes an adequate representation of the tested bare frame. Consequently, the conclusion
that the only variable in the verification of the infilled frame is the infill panel model can be
drawn.
48
Figure 4. 1. Comparison of the base shear for the bare frame (475yrp record)
49
Figure 4. 2. Comparison of the base shear for the bare frame (975yrp record)
50
4.2 Sensitivity Analysis
For the definition of the numerical model a set of mechanical, geometrical and empirical
parameters is required. Using this set of values for all parameters, the accuracy of the
numerical model remains to be checked and verified. In order to conclude in a base model, a
sensitivity study has been conducted in order to evaluate the relative importance of the less
intuitive and harder to calibrate empirical parameters included in the model. The sensitivity
analysis was essentially based on varying each empirical parameter within the limit range of
values proposed by Crisafulli [1997] and presented in Table 2.1. The measure of the influence
of each parameter has been the difference between the final cumulative absolute value of base
shear for the modified model and the equivalent cumulative value of the base model,
normalised by the latter. The cumulative procedure, used as measure of the accuracy of the
model, is presented in more detail in section 4.3.3. The results of the sensitivity study are
presented in Figure 4.3. Those empirical parameters that did not exhibit a significant influence
on the behaviour of the base model (less than 1%) are not included
As easily noted in the results of Figure 4.3, the most influential parameters are γun, αch and ex1.
The parameter γun has the most significant influence on the model’s behaviour, exhibiting a
wide range from -8% to 21% in the case of 975yrp record or from -7% to 22% in the case of
2000yrp record. Considerable effect is induced also by αch with comparable values for all
records, while ex1 affects mainly in the range of large displacements (2000yrp record). Since
γun determines the shape of the loop by controlling the tangent modulus at the beginning of the
unloading curve and ex1 controls the tangent modulus corresponding to the plastic strain of the
unloading curve again, while αch defines the fatness of the loops, all parameters in fact decide
the amount of dissipated energy.
un 1
γung1.0
2.3
γung un2.3
22
-2
21
5
-7
-8
re 0.3
αrea 0.3
-1
αcha ch0.10.1
8
12
8
-1
-1
b ch0.5
0.5
βch
-2
γplu 0.5
-1
g plu 0.5
-2
1
γplr
1
g plr
1.5
1.5
2.02
εx1 ex1
ex1 1.5
εx1 1.5
-20
1
1
-10
0
3
475
975
2000
5
6
9
10
20
30
40
50
% variation from the base model
model
51
It is reminded that the final values of empirical parameters implemented in the model are
presented in Table 3.5.
4.3 Pseudo-dynamic Analyses Results for the Infilled Frame
After the preliminary verification of the numerical model for the bare frame, the infill panel
elements were implemented in the model. The calculation or selection of their properties was
done in accordance with the recommendations included in Chapter 2. The values used for all
the parameters are presented in section 3.4. Then, static time-history analyses were carried out
in the infilled frame with the aim to simulate the pseudo-dynamic tests conducted in the
laboratory. That is why the displacements obtained from the experiment on the infilled frame
were imposed on the numerical model. Furthermore, unified input file was created that
contained the three records of displacements in sequence and including intervals between
them that contained null values. Thus, the numerical model was subjected to the
displacements of the records of increasing intensity consecutively, accounting in this way for
the residual displacements and the accumulated damage caused on the frame.
The accuracy of the numerical model is evaluated by comparing the results of the analyses
with the experimental results. The comparison was done in terms of time-histories,
cumulative and maximum absolute values of base shear. It has to be noted that the case study
frame did not manage to complete the test under the 2000yrp record, which was interrupted as
the frame approached imminent collapse.
4.3.1 Peak Values of Base Shear
In an attempt to quantify the accuracy of the model, the peak values of base shear as absolute
values and as percentage of the equivalent experimental ones are reported in Table 4.2.
Table 4. 2. Comparison of base shear peak values.
Input motion
475yrp
975yrp
2000yrp
Numerical
Experimental
%
Max abs values Max abs values
(kN)
(kN)
763
793
1.04
868
814
0.94
535
521
0.97
However, the inherent limitation of the way the aforementioned results are presented is that
they do not elucidate whether the peak values coincide in time or occur at different instants. In
case the latter happens, it is interesting to examine whether the experimental or numerical
peak of base shear exhibits a relatively large difference from the equivalent numerical or
experimental value that takes place at the same time. In Tables 4.3 and 4.4 the experimental
and numerical maximum absolute values for the three records of input motion are compared
respectively to the numerical and experimental absolute values of base shear happening at the
same time-step. It is noted that in the case of 975yrp record the peak experimental and
numerical base shear occur simultaneously.
52
Table 4. 3. Base shear peak values at the instant of the experimental peak
value.
Input motion
475yrp
975yrp
2000yrp
Experimental
Equivalent
Time
%
max abs values numerical values
(kN)
(kN)
(sec)
763
745
11.170 0.98
868
814
6.725 0.94
535
506
4.550 0.95
Table 4. 4. Base shear peak values at the instant of the numerical peak value.
Input motion
475yrp
975yrp
2000yrp
Time %
Equivalent
Numerical
max abs values experimental values
(sec)
(kN)
(kN)
793
727
6.556 1.09
814
868
6.725 0.94
521
510
4.380 1.02
It is reminded that the percentages given in all the three Tables above correspond always to
the ratio of the numerically computed peak values over the equivalent experimentally
obtained values. These ratios are such that it can be inferred that the model achieves to predict
with satisfactory precision the peak value of base shear for all three seismic input records.
4.3.2 Time-history of Base Shear
From the analyses conducted for the three records of different return period, the time-histories
of base shear of the infilled frame for each record were obtained and plotted in Figures 4.4,
4.5 and 4.6, offering an immediate graphical comparison. Considering the instantaneous
nature of the records, peak values of base shear are a first and useful means of comparison
since good match of the peak values constitutes an indication and necessary prerequisite of
the accuracy of the model, without though giving an overall picture of the precision of the
numerical model. The comparison in terms of time-histories is considered to be more
representative because thus the success of the numerical model to predict the response of the
infilled frame can be examined for the whole duration of the input records.
A first overall observation is that the analytical results demonstrate a good match with those
of the experiment. Some differences are identified in several parts of the time-histories. For
example, at the beginning of the time-history corresponding to the 475yrp record (Figure 4.4)
the model tends to underestimate the base shear, while at the end of the same record a few
differences occurring at the peaks of a limited number of cycles are observed. On the other
hand, the base shear for the 975yrp record coincides perfectly with the experimental base
shear in most parts of the graph. Equally remarkable is the match between the numerical and
53
Figure 4. 4. Comparison of the base shear of the infilled frame for pseudo-dynamic analysis (475yrp record)
54
55
56
experimental results for the 2000yrp record, though its duration is short since the frame failed
to finish it.
4.3.3 Cumulative of Base Shear
Additionally to the peak absolute values and the time-histories of base shear, a cumulative
evaluation process has been used in order to compare the results numerically. Specifically, the
cumulative-absolute-base-shear values of both experimental and analytical analyses are
normalized with the total-absolute-base-shear of the experimental tests, as described by
Equation (4.1).
cumulative ( t ) =
base shear ( t ) numerical
∑ base shear
(4.1)
exp erimental
The graphical results of this cumulative evaluation process are given in Figures 4.7, 4.8 and
4.9. The numerical results, i.e. the final cumulative values of base shear for each input motion
record, are included in Table 4.5, showing the deviation from the unit value of cumulative
corresponding to the experimental data.
Table 4. 5. Final cumulative values of
base shear.
Input motion Final cumulative
value
475yrp
0.87
975yrp
0.94
2000yrp
1.04
The cumulative of base shear was selected as means of comparison because it provides a
representative picture of the cumulative error and thus of the deviation from the experimental
values, as well as of the time intervals that the latter takes place. Specifically, the cumulative
error for the 475yrp record exhibits the largest deviation, though this is still within an
acceptable range. The cumulative corresponding to the numerical model starts to deviate early
but after the first five seconds the difference between the two lines stabilises till the end of the
record. This picture is in agreement with the time-history for the 475yrp record. In fact the
cumulative depicts the differences observed at the first cycles of the base shear time-history
(Figure 4.4), reflecting the tendency of the numerical model to underestimate the response of
the infilled frame.
On the contrary, the cumulative corresponding to the 975yrp record demonstrates excellent
match within the duration of the record. Some differences occurring at the peaks of a limited
number of cycles between seconds ten and twelve lead to a limited deviation of the
cumulative noticed at the end of the record. Finally, as far as the 2000yrp record is concerned,
it has been found that the numerical cumulative plot matches perfectly with the equivalent
57
experimental one. Though, this result is accepted with reservation because the test under the
2000yrp record was interrupted, since the frame approached imminent collapse, and thus it is
perhaps to be expected that the cumulative error is lower due to the reduced length of this
record.
Figure 4. 7. Normalised cumulative absolute values of base shear versus time for the 475yrp seismic input
record
58
record
record
59
4.4 Dynamic Analyses Results for the Infilled Frame
The numerical results obtained from the static-time history exhibited satisfactory match with
the experimental ones, verifying thus the accuracy of the infill panel model. Furthermore,
calibration of the model has been completed by conducting sensitivity analysis, which
determined the relative importance of the participating empirical parameters.
The facilities in the ELSA laboratory include a reaction-wall and not a shaking table, that is
why the tests conducted were not dynamic but pseudo-dynamic. The pseudo-dynamic test
conditions were fully simulated by the static time-history analyses, the results of which were
presented in the previous sections. The excellent behaviour of the model encouraged to try the
model under fully dynamic conditions. The model is tested with dynamic time-history
analysis in order to prove its robustness.
A unified input file, containing the three acceleration time-histories and including intervals
between them that contained null values, was created and applied at the base of the numerical
model. Thus, the numerical model was subjected to accelerations records of increasing
intensity consecutively, accounting in this way for the residual displacements and the
accumulated damage caused on the frame.
The displacements of each floor obtained for each seismic record were compared to the
displacements measured during the tests. Furthermore, the experimentally measured base
shear was compared to the base shear from the numerical analyses. The dynamic time-history
gave satisfactory results, which were drastically improved after few modifications in the
initial parameters. These changes are summarized in the increase of both the initial and the
residual strut area. As shown in section 2.5.2, there are different approaches for the
calculation of the strut area, leading to a variation of the area. Moreover, the reduction due to
the presence of openings introduces another uncertainty in the estimation of the area. That is
why an increase of the order of 5% in the initial strut area, after having accounted for the
openings effect, is fully justified. Similar uncertainty is inherent in the estimation of the
residual area. Concluding, in the numerical model for the dynamic time-history analyses the
only changed parameters were the initial strut area, that was increased 5%, and the residual
strut area, that was increased from 10% to 20%.
It is underlined that none of the empirical parameters was modified, maintaining the set of
values of the base model obtained in the previous sections. Finally, it is noted that the
damping model used in the analyses for the 475yrp record was Rayleigh damping with 5%
and 7% damping in the 1st and 3rd mode respectively, while for the analyses for the 975yrp
and 2000yrp records the damping was in proportion to the stiffness employing the option of
tangent stiffness and with stiffness parameter equal to 0.00123 approximately. As the
intensity of the input motion increases, the frame-infill panel interface progressively suffers
more damage, i.e. the gap increases, and hence the contribution of this effect to the equivalent
viscous damping diminishes. This fact justifies the choice of different damping model for the
475yrp record. By using Rayleigh damping, which is as known proportional to both mass and
stiffness, higher damping is obtained for the same percentage of damping.
60
The numerical model exhibits satisfactory behaviour as shown by the match with the
experimental results. It manages to describe successfully the frequency content and reach the
peak values in most cycles. The most differences are observed in the case of the 475yrp
record (Figures 4.10 to 4.14), where the model underestimates the behaviour having though
some peaks that exceed the experimental values. However, this mainly tends to occur in the
1st floor displacement (Figure 4.10) and is limited in the upper floors displacements (Figures
4.11 to 4.13). The 4th floor displacement time-history (Figure 4.13), which exhibits the most
interest and should give the most reliable values, has a satisfactory match. The behaviour of
the model under the 975yrp record (Figures 4.15 to 4.19) can be characterized as excellent
since the numerical results coincide with the experimental. The same is valid for the 2000yrp
record (Figures 4.20 to 4.24), despite the small shift that appears at the end of the timehistories. The reliability of the values at around 5sec may be characterized as doubtful since
the structure has suffered extensive damage, reason that led to the interruption of the test.
The peak values of floor displacements and base shear are given in the following Tables,
while the displacement and the base shear time-histories for the three seismic records obtained
from the dynamic analyses are presented hereinafter. Additionally, in Appendix A the
cumulative graphs of the displacements and base shear for each seismic record are can be
found for having a more complete picture of the behaviour of the numerical model.
Table 4. 6. Displacement peak values for the 475yrp record.
Numerical
Input motion Experimental
475yrp
(mm)
(mm)
st
1 floor
3.26
3.90
nd
2 floor
6.44
5.78
3rd floor
8.49
7.69
4th floor
10.17
8.34
%
1.19
0.90
0.91
0.82
Numerical
975yrp
(mm)
(mm)
st
1 floor
11.48
11.57
nd
2 floor
16.49
14.73
rd
3 floor
19.67
18.00
th
4 floor
22.25
20.32
%
1.01
0.89
0.91
0.91
61
Numerical
2000yrp
(mm)
(mm)
st
1 floor
34.75
25.79
nd
2 floor
38.74
30.37
rd
3 floor
39.81
35.21
th
4 floor
40.59
37.17
%
0.74
0.78
0.88
0.92
Table 4. 9. Base shear peak values.
Input motion
475yrp
975yrp
2000yrp
Experimental
Numerical
%
(kN)
(kN)
754
655
0.87
847
716
0.85
529
674
1.27
The assessment of the infill panel model is concluded with the results from the dynamic timehistory, which give a definitely satisfactory picture of the behaviour of the model. Some
discrepancies or deficiencies observed can be attributed to the fact that the experimental
results are derived from pseudo-dynamic tests, while the numerical results come from
dynamic analyses. However, the model succeeds in describing the response of the infilled
frame under pseudo-dynamic and dynamic conditions for the imposed seismic records of
different intensity. The results allow no doubt for the robustness of the model and fully verify
its accuracy.
62
Figure 4. 10. Comparison of the 1st floor displacement for the infilled frame for dynamic analysis (475yrp record)
63
Figure 4. 11. Comparison of the 2nd floor displacement for the infilled frame for dynamic analysis (475yrp record)
64
Figure 4. 12. Comparison of the 3rd floor displacement for the infilled frame for dynamic analysis (475yrp record)
65
Figure 4. 13. Comparison of the 4th floor displacement for the infilled frame for dynamic analysis (475yrp record)
66
Figure 4. 14. Comparison of the base shear for the infilled frame for dynamic analysis (475yrp record)
67
68
69
70
71
72
73
74
75
76
77
Chapter 5. CONCLUSIONS AND FUTURE DEVELOPMENTS
5. CONCLUSIONS AND FUTURE DEVELOPMENTS
5.1 Conclusions
The effect of infill panels on the response of RC frames subjected to seismic action is widely
recognised and has been subject of numerous experimental investigations, while several
attempts to model it analytically have been reported.
In this work, a double-strut cyclic nonlinear model for the masonry panels was implemented
in a fibre-based Finite Element program. The masonry panel model consists of different rules
for loading, unloading and reloading and considers also the tensile behaviour of the masonry,
the local contact effects of the cracked material and the effect of the small inner cycles. The
implemented four-node masonry panel element is represented by five strut members, two
parallel struts in each diagonal direction and a single strut acting across two opposite diagonal
corners to carry the shear from the top to the bottom of the panel. The first four struts use the
aforementioned rules, while the shear strut uses a bilinear hysteresis rule.
The masonry panel model was employed to reproduce the behaviour of the full-scale frame
model used as a case study for the needs of this work. The full-scale, four-storey, three-bay
reinforced concrete frame considered was pseudo-dynamically tested at the ELSA reactionwall laboratory within the framework of the ICONS research programme. Two identical in
geometry, construction and detailing frames were constructed and tested, one was bare and
the other infilled. The case study frame was infilled with brick walls that included openings of
different dimensions.
The experimental seismic response was obtained by applying pseudo-dynamic test method,
i.e. a step by step integration technique to compute the displacement response of the frame
that was subjected to three different, numerically specified seismic records, utilizing the nonlinear restoring forces actually developed during the test. The input seismic motions were
chosen to be representative of a moderate-high European hazard scenario. Consistent with the
hazard, acceleration time-histories were artificially generated and finally three of increasing
return periods of 475, 975 and 2000 years were used for the experiment.
The case study frame was modelled in the fibre-base programme. Initially, the experimental
values of base shear from the pseudo-dynamic tests on the bare frame were compared to the
results obtained for the numerical model, exhibiting a satisfactory match. Moreover, good
agreement was observed between the fundamental period of the numerical model and the
estimated first period of the test frame. Confirmation of the good match between the
78
behaviours of the bare frame numerical model and the experimental frame has been carried
out to ensure that the only variable in the verification of the infilled frame model was the infill
panel model and thus other sources of uncertainty did not have to be considered.
The infill panel elements were inserted in the model. The presence of the openings was
considered too. The calibration of the model entailed careful selection or calculation of the
parameters involved, which are distinguished in mechanical, geometrical and empirical. The
variables needed as input data were presented in detail, as well as with recommendations for
the selection or calculation of their values and the values that are finally implemented,
offering thus guidance on general employment of the infill panel model.
The numerical results obtained by static time-history have been compared to the experimental
ones in terms of time-histories, cumulative and maximum absolute values of base shear,
exhibiting a satisfactory match. The model succeeded in predicting accurately the response of
the infilled frame in all terms. Thus, the accuracy of the numerical model has been verified.
The calibration of the model was completed by conducting sensitivity analysis, in which the
relative importance of the empirical parameters was evaluated, concluding that the parameters
related to energy dissipation are dominant. The sensitivity analysis offered a set of values for
the empirical parameters that can be generally used.
The excellent behaviour of the model during the pseudo-dynamic analyses encouraged to try
the model under fully dynamic conditions. The model was tested with dynamic time-history
analysis in order to complete the assessment of the model and prove its robustness. The
results are definitely characterized as satisfactory, despite few differences from the
experimental ones, something that can be explained by considering that the experimental
results are derived from pseudo-dynamic tests, while the numerical results come from
dynamic analyses.
Undoubtedly, the model succeeded in describing the response of the infilled frame under
pseudo-dynamic and fully dynamic conditions with seismic records of different intensities,
predicting the frequent content of the excitation and the peak values of the response. It is
considered that the accuracy and reliability of the masonry panel model was verified.
5.2 Future Developments
The accuracy of the masonry panel model was verified after the results of the pseudo-dynamic
and fully dynamic analyses and their satisfactory match with the experimental results. These
findings are a good starting point for future research that will cover uncertainties traced with
this work. However, it would be useful if these results were confirmed with other case studies,
i.e. with full-scale multi-storey structures that include infill panels and have been tested
experimentally.
Specifically, the adverse local effects that the infill panels may cause due to their interaction
with the surrounding frame should be investigated. When the infilled frame is laterally loaded,
high shear forces develop at the contact points between the infill panels and the concrete
members of the surrounding frame, usually at the columns of the frame. If these shear forces
are excessive, the result may be a brittle failure. That is why these local effects because of the
79
interaction between infill panel and reinforced frame should be considered and therefore it
should be examined whether the model manages to capture them.
Finally, the lack of recommendations in the literature about the effect of the openings was
remarkable and reflects the fact that no specific way of quantifying the area reduction due to
openings has been found. The effect of openings either has been ignored or considered in an
empirical way, which definitely is not adequate. That is why the investigation of the effect of
openings is essential, considering different locations and aspect ratios of openings for frames
of several dimensions.
80
References
REFERENCES
Ameny, P., Loov, R.E. and Shrive, N.G. [1983] “Prediction of Elastic Behaviour of Masonry,”
International Journal of Masonry Construction, Vol. 3, No. 1, pp. 1-9.
Asteris, P.G. [1996] “A Method for the Modelling of Infilled Frames (Method of Contact Points),”
Proceedings of the 11th World Conference on Earthquake Engineering, Acapulco, Mexico.
Asteris, P.G. [2003] “Lateral Stiffness of Brick Masonry Infilled Plane Frames,” ASCE Journal of
Structural Engineering, Vol 129, No. 8, pp. 1071-1079.
Atkinson, R.H., Noland, J.L. and Abrams, D.P. [1985] “A Deformation Failure Theory for Stack-Bond
Brick Masonry Prism in Compression,” Proceedings of the 7th International Brick Masonry
Conference, Melbourne, Australia.
Atkinson, R.H., Amadei, B.P., Saeb, S. and Sture, S. [1989] “Response of Masonry Bed Joints in
Direct Shear,” ASCE Journal of Structural Division, Vol. 115, No. 9, pp. 2276-2296.
Benjamin, S.T. and Williams, H.A. [1958] “The Behaviour of One-storey Brick Shear Walls,” ASCE
Journal of Structural Division, Vol. 84, No.ST4, pp. 30.
Bertoldi S.H., Decanini, L.D., Santini S. and Via G. [1994] “Analytical models in infilled frames,”
Proceedings of the 10th European Conference on Earthquake Engineering, Vienna, Austria.
Bertoldi , S.H., Decanini, L.D. and Gavarini, C. [1999] “Telai Tamponati Soggeti ad Azioni Sismiche,
un Modello Semplificato: Confronto Sperimentale e Numerico,” (in Italian), Atti del 60 Convegno
Nazionale ANIDIS, 2, Perugia, Italy.
Binda, L., Fontana, A. and Frigerio, G. [1988] “Mechanical Behaviour of Brick Masonries Derived
from Unit and Mortar Charachteristics,” Proceedings of the 8th International Brick and Block
Masonry Conference, Dublin, Ireland.
Biondi, S. Colangelo, F. and Nuti, C. [2000] La Risposta Sismica dei Telai con Tamponature Muraie,
(in Italian), Gruppo Nazionale per la Difesa dai Terremoti, Rome, Italy.
Blandon-Uribes, C.A. [2005] “Implementation of an Infill Masonry Model for Seismic Assessment of
Existing Buildings,” Individual Study, European School for Advanced Studies in Reduction of
Seismic Risk (ROSE School), University of Pavia, Italy.
Bolong, Z., Mingshun, W. and Kunlian, W. [1980] “A Study of Hysteretic Curve of Reinforced
Concrete Members under Cyclic Loading,” Proceedings of the 7th World Conference on
Earthquake Engineering, Istanbul, Turkey.
81
References
Carvalho, E.C., Coelho, E. and Campos-Costa, A. [1999] “Preparation of the Full-Scale Tests on
Reinforced Concrete Frames. Characteristics of the Test Specimens, Materials and Testing
Conditions,” ICONS Report, Innovative Seismic Design Concepts for New and Existing Structures,
European TMR Network, LNEC.
Campos-Costa, A. and Pinto, A.V. [1999] “European Seismic Hazard Scenarios – An Approach to the
Definition of Input Motions for Testing and Reliability Assessment of Civil Engineering
Structures,” JRC Special Pubblication, No. X.99.XX, ELSA, JRC – Ispra, Italy.
CEN - Comité Euro-International du Béton [1996] RC Frames under Earthquake Loading: State of the
Art Report, Thomas Telford, London, England.
Chrysostomou, C.Z. [1991] “Effects of Degrading Infill Walls on the Non-linear Seismic Response of
Two-dimensional Steel Frames,” PhD Thesis, Cornell, United States.
Combescure, D., Pires, F., Cerqueira, P. and Pegon, P. [1996] “Tests on Masonry Infilled R/C Frames
and its Numerical Implementation”, Proceedings of the 11th World Conference on Earthquake
Engineering, Acapulco, Mexico.
Crisafulli, F.J. [1997] “Seismic Behaviour of Reinforced Concrete Structures with Masonry Infills,”
PhD Thesis, University of Canterbury, New Zealand.
Crisafulli, F.J., Carr, A.J. and Park, R. [2000], “Analytical Modelling of Infilled Frame Structures – A
General Overview,” Bulletin of the New Zealand Society for Earthquake Engineering, Vol. 33, No
1.
D’Asdia, P. et al. [1990] “On the Seismic Behaviour of Infilled Frames,” Proceedings of the 9th
European Conference on Earthquake Engineering, Moscow, Russia.
Dawe, J.I. and Young, T.C. [1985] “An Investigation of Factors Influencing the Behaviour of Masonry
Infills in Steel Frames Subjected to In-plane Shear,” Proceedings of the 7th International Brick
Masonry Conference, Melbourne, Australia.
Decanini, L.D. and Fantin, G.E. [1987] “Modelos Simplificados de la Mamposteria Incluida en
Porticos. Caractreisticas de Rigidez y Resistencia Lateral en Estado Limite,” (in Spanish),
Jornadas Argentinas de Ingeniería Estructural, Buenos Aires, Argentina, Vol. 2, pp. 817-836.
Dhanasekar, M. et al. [1985] “Behaviour of Brick Masonry under Biaxial Stress with Particular
Reference to Infilled Frames,” Proceedings of the 7th International Brick Masonry Conference,
Melbourne, Australia.
Doudoumis, I.N. and Mitsopoulou, E.N. [1986] “Non-linear Analysis of Multistorey Infilled Frames
for Unilateral Contact Conditions,” Proceedings of the 8th European Conference on Earthquake
Engineering, Lisbon, Portugal.
Drysdale, R.G., and Khattab, M.M. [1995] “In-plane Behaviour of Grouted Concrete Masonry under
Biaxial Tension-Compression,” American Concrete Institute Journal, Vol. 92, No. 6, pp. 653-664.
FEMA-273 [1997] NHERP Guidelines for the Seismic Rehabilitation of Buildings, Federal Emergency
Management Agency, Applied Technology Council, Washington DC,United States.
FEMA-274 [1997] NHERP Commentary on the Guidelines for the Seismic Rehabilitation of
Buildings, Federal Emergency Management Agency, Applied Technology Council, Washington
DC,United States.
82
References
Filippou, F.C., Popov, E.P. and Bertero V.V. [1983] “Modelling of R/C Joints under Cyclic
Excitations,” ASCE Journal of Structural Engineering, Vol. 109, No. 11, pp. 2666-2684.
Fiorato, A.E., Sozen, M.A. and Gamble, W.L. [1970] “An Investigation of the Interaction of
Reinforced Concrete Frames with Masonry Filler Walls,” Report UILU-ENG-70-100, Department
of Civil Engineering, University of Illinois, Urbana-Champaign IL, USA.
Giannakas, A., Patronis, D. and Fardis, M. [1987] “The Influence of the Position and the Size of
Openings to the Elastic Rigidity of Infill Walls,” (in Greek), Proceedings of 8th Hellenic Concrete
Conference, Xanthi-Kavala, Greece.
Goodman, R.E. et al. [1967] “A Model for the Mechanics of Jointed Rock,” American Society of Civil
Engineers, Journal of Soil Mechanics, Vol. 94, pp. 637-659.
Gostič, S. and Žarnič, R. [1985] “Cyclic Lateral Response of Masonry Infilled R/C Frames and
Confined Masonry Walls,” Proceedings of the 8th North American Masonry Conference, Austin,
Texas, United States.
Hamburger, R.O. [1993] “Methodology for Seismic Capacity Evaluation of Steel-frame Buildings
with Infill Unreinforced Masonry,” Proceedings of the National Conference, Earthquake
Consortium, Memphis, Tennesse, United States.
Hendry, A.W. [1990] Structural Masonry, Macmillan Education Ltd, London, England.
Holmes, M. [1961] “Steel Frames with Brickwork and Concrete Infilling,” Proceedings of the
Institution of Civil Engineers, Vol. 19, pp. 473-478.
King, G.J.W. and Pandey, P.C. [1978] “The Analysis of Infilled Frames using Finite Elements”
Proceedings of the Institution of Civil Engineers, Vol. 65, pp. 749-760.
Klingner, R.E. and Bertero, V.V. [1976] “Infilled Frames in Earthquake Resistant Construction”
Earthquake Engineering Research Center, University of California, Berkeley, EERC 76-32.
Klingner, R.E. and Bertero, V.V. [1978] “Earthquake Resistance of Infilled Frames” ASCE Journal of
the Structural Division, Vol. 104, No. ST6, pp. 973-989.
Kost, E.G. et al. [1974] “Non-linear Dynamic Analysis of Frames with Filler Panels,” ASCE Journal
of the Structural Division, Vol. 100, pp. 743-757.
Liauw, T.C. and Lee, S.W. [1977] “On the Behaviour and the Analysis of Multi-storey Infilled Frames
Subjected to Lateral Loading,” Proceedings of Institution of Civil Engineers, Part 2, Vol. 63, pp.
641-656.
Liauw, T.C. and Kwan, K.H. [1984] “Nonlinear Behaviour of Non-Integral Infilled Frames,”
Computer and Structures, Vol. 18, No. 3, pp. 551-560.
Mainstone, R.J. and Weeks, G.A. [1970] “The influence of Bounding Frame on the Racking Stiffness
and Strength of Brick Walls,” Proceedings of the 2nd International Brick Masonry Conference,
Stoke-on-Trent, United Kingdom.
Mainstone, R.J. [1971] “On the Stiffnesses and Strength of Infilled Panels,” Proceedings of the
Institution of Civil Engineers, Supplement IV, pp. 57-90.
Mallick, D.V. and Garg, R.P. [1971] “Effect of Openings on the Lateral Stiffness of Infilled Frames,”
Proceedings of the Institution of Civil Engineering, Vol. 49, pp. 193-209.
83
References
Mallick, D.V. and Severn, R.T. [1967] “The Behaviour of Infilled Frames under Static Loading,”
Proceedings of the Institution of Civil Engineering, Vol. 39, pp. 639-656.
Mander, J.B., Priestley, M.J.N. and Park, R. [1988] “Theoretical Stress-Strain Model for Confined
Concrete,” ASCE Journal of Structural Engineering, Vol. 114, No. 8, pp. 1804-1826.
Mann, W. and Müller, H. [1982] “Failure of Shear-Stresses Masonry - An Enlarged Theory, Tests and
Application to Shear Walls,” Proceedings of the British Ceramic Society, Vol. 30, pp. 139-149.
Mehrabi, A.B., Shing, P.B., Schuller, M.P. and Noland, J.L. [1996] “Experimental Evaluation of
Masonry-Infilled RC Frames,” ASCE Journal of Structural Engineering, Vol. 122, No. 3, pp. 228237.
Menegotto, M. and Pinto, P.E. [1973] “Method of Analysis for Cyclically Loaded R.C. Plane Frames
Including Changes in Geometry and Non-elastic Behaviour of Elements under Combined Normal
Force and Bending,” Symposium on the Resistance and Ultimate Deformability of Structures Acted
on by Well Defined Repeated Loads, Zurich, Switzerland.
Molina, F.J., Pegon, P. and Verzeletti, G. [1999] “Time-domain identification from Seismic Pseudodynamic Test Results on Civil Engineering Specimens,” 2nd International Conference on
Identification in Engineering Systems, Wales, United Kingdom.
Mosalam, K.M., White, R.N., and Gergely, P. [1997] “Static Response of Infilled Frames using Quasistatic Experimentation,” ASCE Journal of Structural Engineering, Vol. 123, No. 11, pp. 14621469.
Naraine, K. and Sinha, S. [1989] “Behaviour of Brick Masonry under Cyclic Compressive Loading,”
ASCE Journal of Structural Engineering, Vol. 115, No. 2, pp. 1432-1445.
Otter, D.E. and Naaman, A.E. [1989] “Model for Response of Concrete to Random Compressive
Loads,” ASCE Journal of Structural Engineering, Vol. 115, No. 11, pp. 2794-2809.
Panagiotakos, T.B. and Fardis, M.N. [2994] “Proposed Nonlinear Strut Models for Infill Panels,” 1st
Year Progress Report of HCM-PREC8 Project, University of Patras.
Papia, M. [1988] “Analysis of Infilled Frames Using a Coupled Finite Element and Boundary Element
Solution Scheme,” International Journal for Numerical Methods in Engineering, Vol. 26, pp. 731742.
Paulay, T. and Priestley, M.J.N. [1992] Seismic Design of Reinforced Concrete and Masonry
Buildings, John Wiley & Sons, New York, United States.
Pinho, R. and Elnashai, A.S. [2000] “Dynamic Collapse Testing of a Full-Scale Four Storey RC
Frame,” ISET Journal of Earthquake Engineering, Special Issue on Experimental Techniques, Vol.
37, No. 4, pp. 143-164.
Pinto, A., Verzeletti, G., Pegon, P., Magonette, G., Negro, P. and Guedes, J. [1996] “Pseudo-dynamic
Testing of Large-scale RC Bridges,” EUR Report, Joint Research Centre, Ispra, Italy.
Pinto, A., Verzeletti, G., Molina, F.J., Varum, H., Pinho, R. and Coelho, E. [1999] “Pseudo-Dynamic
Tests on Non-Seismic Resisting RC Frames (Bare and Selective Retrofit Frames),” EUR Report,
Joint Research Centre, Ispra, Italy.
84
References
Pinto, A., Verzeletti, G., Molina, F.J., Varum, H., Carvalho, E.C. and Coelho, E. [1999] “PseudoDynamic Tests on Non-Seismic Resisting RC Frames (Infilled Frame and Infill Strengthened
Frame Tests),” EU Special Publication, Joint Research Centre, Ispra, Italy.
Polyakov, S.V. [1956] “Masonry in Framed Buildings,” Gosudarstvennoe Izdatel Stvo Literatury Po
Stroitel Stvui Arkitektuze, Moscow (translated by Cairns G.L.).
Riddington, J.R. and Ghazali, M.Z. [1988] “Shear Strength of Masonry Walls,” Proceedings of the 8th
International Brick and Block Masonry Conference, Dublin, Ireland
Riddington, J. and Stafford Smith, B. [1977] “Analysis of Infilled Frames subject to Racking with
Design Recommendations,” ASCE Journal of Structural Engineering, Vol. 52, No. 6, pp. 263-268.
RILEM Draft Recommendations TC 76-LUM [1988] “General Recommendations for Methods of
Testing Load-Bearing Unit Masonry,” Materials and Structures, Vol. 21, No. 123, pp. 229-231.
Rivero, C.E. and Walker, W.H. [1984] “An Analytical Study of the Interaction of Frames and Infill
Masonry Walls,” Proceedings of the 8th World Conference on Earthquake Engineering, San
Francisco, United States.
Sahlin, S. [1971] Structural Masonry, Prentice-Hall Inc., New Jersey, United States.
San Bartolomé, A.[1990] Colleción del Ingeniero Civil, (in Spanish), Libro No. 4, Colegio de
Ingenierios del Peru.
Sargin, M., Ghosh, S.K. and Handa, V.K. [1971] “Effects of Lateral Reinforcement upon the Strength
and Deformation Properties of Concrete,” Magazine of Concrete Research, Vol. 23, No. 75-76, pp.
99-110.
SeismoSoft [2006] “SeismoStruct – A Computer Program for Static and Dynamic Nonlinear Analysis
of Framed Structures,”(online), available from URL: http://www.seismosoft.com.
Shing, P.B. et al. [1992] “Finite Elements Analysis of Shear Resistance of Masonry Wall Panels with
and without Confining Frames,” Proceedings of the 10th World Conference on Earthquake
Engineering, Madrid, Spain.
Shing, P.B. and Mahin, S.A. [1985] “Computational Aspects of a Seismic Performance Test Method
Using On-line Computer Control,” Journal of Earthquake Engineering and Structural Dynamics,
Vol. 13, pp. 507-526.
Shrive, N.G. [1991] “Materials and Material Properties,” Reinforced and Prestressed Masonry,
A.W.Hendry, Longman Scientific and Technical, London, England.
Sinha, B.P. and Pedreschi, R. [1983] “Compressive Strength and Some Elastic Properties of
Brickwork,” International Journal of Masonry Construction, Vol. 3, No. 1, pp. 19-27.
Sortis, A., Pasquale, G. and Nasini, U. [1999] Criteri di Calcolo per la Progettazione degli Interventi
– Terremoto in Umbria e Marche del 1997, (in Italian), Servizio Sismico Nazionale, Editrice
Sallustiana, Rome, Italy.
Stafford Smith, B. [1966] “Behaviour of Square Infilled Frames,” ASCE Journal of Structural
Division, Vol. 92, No. ST1, pp. 381-403.
85
References
Stevens, N.J., Uzumeri, S.M. and Collins, M.P. [1991] “Reinforced Concrete Subjected to Reversed
Cyclic Shear – Experiments and Constitutive Model,” American Concrete Institute Journal, Vol.
88, No. 2, pp. 135-146.
Stöckl, S. and Hofmann, P. [1988] “Tests on the Shear Bond Behaviour in the Bed-joints of Masonry,”
Proceedings of the 8th International Brick and Block Masonry Conference, Dublin, Ireland.
Subramaniam, K.V.L. and Sinha, S.N. [1995] “Analytical Model for Cyclic Compressive Behaviour
of Brick Masonry,” American Concrete Institute Journal, Vol. 13, pp. 401-419.
Syrmakezis, C.A. and Asteris, P.G. [2001] “Masonry Failure Criterion under Biaxial Stress State,”
ASCE Journal of Materials in Civil Engineering, Vol. 13, No. 1, pp. 58-64.
Syrmakezis, C.A. and Vratsanou, V.Y. [1986] “Influence of Infill Walls to RC Frames Response,”
Proceedings of the 8th European Conference on Earthquake Engineering, Lisbon, Portugal.
Thiruvengadam, H [1980] “On the Natural Frequencies of Infilled Frames,” Journal of Earthquake
Engineering and Structural Dynamics, Vol. 13, pp. 507-526.
Utku, B. [1980] “Stress Magnifications in Walls with Openings,” Proceedings of the 7th World
Conference on Earthquake Engineering, Istanbul, Turkey.
Varum, H.S.A. [2003] “Seismic Assessment, Strengthening and Repair of Existing Buildings,” PhD
Thesis, University of Aveiro, Portugal.
Wan, Q. and Yi, W. [1986] “The Shear Strength of Masonry Wall under Combined Stresses,”
Proceedings of the 4th Canadian Masonry Symposium, University of New Brunswick, Canada.
Xinrong, L. [1995] “Reinforced Concrete Columns under Seismic Lateral Force and Varying Axial
Load,” PhD Thesis, University of Canterbury, New Zealand.
Yankelevsky, D.Z. and Reinhardt, H.W. [1987] “Model for Cyclic Compressive Behaviour of
Concrete,” ASCE Journal of Structural Division, Vol. 113, No. 2, pp. 228-240.
86
Appendix A
APPENDIX A
Hereinafter, the cumulative graphs of floor displacements and base shear are presented for the
three seismic records. The cumulative of displacement is similarly defined as the cumulative
of base shear (see section 4.3.3).
Figure A. 1. Normalised cumulative absolute values of displacement of 1st floor versus time for the 475yrp
seismic input record
A1
Appendix A
Figure A. 2. Normalised cumulative absolute values of displacement of 2nd floor versus time for the 475yrp
Figure A. 3. Normalised cumulative absolute values of displacement of 3rd floor versus time for the 475yrp
A2
Appendix A
Figure A. 4. Normalised cumulative absolute values of displacement of 4th floor versus time for the 475yrp
Figure A. 5. Normalised cumulative absolute values of base shear versus time for the 475yrp seismic input
record
A3
Appendix A
Figure A. 6. Normalised cumulative absolute values of displacement of 1st floor versus time for the 975yrp
Figure A. 7. Normalised cumulative absolute values of displacement of 2nd floor versus time for the 975yrp
A4
Appendix A
Figure A. 8. Normalised cumulative absolute values of displacement of 3rd floor versus time for the 975yrp
Figure A. 9. Normalised cumulative absolute values of displacement of 4th floor versus time for the 975yrp
A5
Appendix A
input record
2000yrp seismic input record
A6
Appendix A
A7
Appendix A
A8
Appendix A
A9

Implementation and verification of a masonry panel

Transcription

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