# Panel zone behaviour in steel moment resisting frames

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Panel zone behaviour in steel moment resisting frames

Istituto Universitario di Studi Superiori Università degli Studi di Pavia EUROPEAN SCHOOL FOR ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK ROSE SCHOOL PANEL ZONE BEHAVIOUR IN STEEL MOMENT RESISTING FRAMES A Dissertation Submitted in Partial Fulfilment of the Requirements for the Master Degree in EARTHQUAKE ENGINEERING AND ENGINEERING SEISMOLOGY by FRANCISCO JOSE DAVILA-ARBONA Supervisors: Dr MIGUEL CASTRO, Dr AHMED ELGHAZOULI May, 2007 The dissertation entitled “Panel Zone Behaviour in Steel Moment Resisting Frames”, by Francisco Jose Davila-Arbona, has been approved in partial fulfilment of the requirements for the Master Degree in Earthquake Engineering and Engineering Seismology. Dr. Miguel Castro …… … ……_______… Dr. Ahmed Elghazouli……_____… … …… 3 Abstract ABSTRACT The response of a moment-resisting frame depends on the characteristics of its main components, namely the columns, the beams and the connections. For the connection type considered in this study the response is mainly governed by the panel zone. This component is defined as the column web portion delimited by the beam continuity plates and the column flanges. The panel is described to be an element mainly subjected to shear stresses and therefore its failure mode is governed by shear yielding. Tests demonstrated that the shear failure mode was stable and ductile under cyclic loading. This dissertation aims to provide a better understanding of the panel zone role in the seismic behaviour of steel moment-resisting frames. The influence of this component on the global and local seismic demands is examined through numerical studies carried out using idealized systems. The ultimate goal of the work is to suggest a design criterion for the panel zone that leads to a more optimized performance of the various components of a structure. The numerical studies undertaken clearly illustrated the importance of this component and the need for its consideration in both the analysis and design stages. Keywords: panel pone; distortions, shear yielding; plastic rotations, curvatures. i Acknowledgements ACKNOWLEDGEMENTS To God To my supervisors: Dr. Miguel Castro and Dr. Ahmed Elghazouli To my parents and family To my lovely wife To my friends To the European Commission, Rose School staff and Imperial College staff. ii Index TABLE OF CONTENTS Page ABSTRACT ............................................................................................................................................i ACKNOWLEDGEMENTS....................................................................................................................ii TABLE OF CONTENTS ......................................................................................................................iii LIST OF FIGURES ................................................................................................................................v LIST OF TABLES...............................................................................................................................viii 1. INTRODUCTION .............................................................................................................................9 1.1 General.......................................................................................................................................9 1.2 Objective ..................................................................................................................................10 1.3 Dissertation Outline .................................................................................................................10 2. LITERATURE REVIEW ................................................................................................................12 2.1 Introduction..............................................................................................................................12 2.2 Experimental studies to understand the Panel Zone role .........................................................12 2.3 Strength and Energy Dissipating Capacity of the Panel Zone .................................................13 2.4 Stiffness and deformability of the Panel Zone.........................................................................14 2.5 Numerical Studies....................................................................................................................15 2.6 Recent Developments ..............................................................................................................16 2.7 Recommendations to Design Guidelines and Code Provisions ...............................................16 2.8 Concluding Remarks................................................................................................................17 3. PANEL ZONE BEHAVIOUR AND DESIGN ...............................................................................18 3.1 Introduction..............................................................................................................................18 3.2 Panel Zone under Shear ...........................................................................................................19 3.3 Representing Analytically Panel Zone Strength and Stiffness ................................................19 3.3.1 Elastic Range .................................................................................................................20 3.3.2 Post-elastic range ...........................................................................................................21 iii Index 3.4 Modelling the Panel Zone ........................................................................................................23 3.5 Design Guidelines and Code Provisions..................................................................................23 3.5.1 FEMA 350 Seismic Guidelines......................................................................................24 3.5.2 AISC Seismic Provisions ...............................................................................................25 3.5.3 Eurocode ........................................................................................................................27 4. NUMERICAL INVESTIGATION..................................................................................................28 4.1 Introduction..............................................................................................................................28 4.2 Cruciform Sub-Assemblage.....................................................................................................29 4.2.1 Description.....................................................................................................................29 4.2.2 Analytical Demands in the Panel Zone..........................................................................30 4.2.3 Design Criterion for the Panel Zone ..............................................................................31 4.3 Numerical Modelling ...............................................................................................................33 4.3.1 OpenSees Finite Element Program for Non-linear Analysis .........................................33 4.3.2 OpenSees Model representation.....................................................................................33 4.4 Analysis Procedures and Response Parameters .......................................................................35 4.4.1 Analysis Procedures.......................................................................................................35 4.4.2 Response Parameters .....................................................................................................35 4.5 Discussion of Results...............................................................................................................37 4.6 Concluding Remarks................................................................................................................42 5. PARAMETRIC STUDIES ..............................................................................................................43 5.1 Introduction..............................................................................................................................43 5.2 Parameters Considered.............................................................................................................43 5.3 Influence of the Panel Zone to Beam capacity ratio ................................................................44 5.4 Influence of the Beam Span .....................................................................................................53 5.5 Influence of the Beam Depth ...................................................................................................56 5.6 Influence of the Panel Zone Second Yield Distortion .............................................................60 5.7 Influence of Steel Strain Hardening........................................................................................63 5.8 Influence of the Gravity Load Level........................................................................................65 6. CONCLUSIONS AND FUTURE RESEARCH .............................................................................74 6.1 Summary and Conclusions ......................................................................................................74 6.2 Recommendations for Future Research ...................................................................................75 REFERENCES .....................................................................................................................................76 APPENDIX A.........................................................................................................................................1 A.1 OpenSees Input File ...........................................................................................................2 iv Index LIST OF FIGURES Page Figure 1.1. Panel Zone ................................................................................................................9 Figure 2.1. Timeline: panel zone development.........................................................................17 Figure 3.1.Moment distributions at joints.................................................................................18 Figure 3.2.Moment converted into Shear .................................................................................19 Figure 3.3.Analytical Models ...................................................................................................20 Figure 3.4.Geometrical parameters...........................................................................................21 Figure 3.5.Strength of the panel zone .......................................................................................22 Figure 3.6.Joint Models of the Panel Zone for Moment Resisting Frames ..............................23 Figure 4.1. Cruciform sub-assemblage. ....................................................................................28 Figure 4.2. Sub-assemblage geometrical properties and boundary conditions.........................29 Figure 4.3. Forces in the Panel Zone under lateral loading conditions.....................................30 Figure 4.4. Design criterion for the panel zone ........................................................................31 Figure 4.5. Design criterion for the panel zone. (a) Strong panel (b) Weak panel ...................32 Figure 4.6. Model Representation in OpenSees........................................................................33 Figure 4.7. Uniaxial Hardening Material..................................................................................34 Figure 4.8. Panel Representation. a) Beam Column Joint Element (Lowes et al., 2004) b) Trilinear model adopted for the shear panel ..........................................................................35 Figure 4.9. Plastic Hinge Concept ............................................................................................37 Figure 4.10. Pushover Curve ....................................................................................................38 Figure 4.11. Deformed Shape 4% Drift ...................................................................................38 Figure 4.12. Contribution from Beam, Panel and Column to the drift .....................................39 Figure 4.13. Beam Elastic and Plastic Contribution to the drift ..............................................39 Figure 4.14. Plastic Hinge Rotation Beams ..............................................................................40 v Index Figure 4.15. Normalized Plastic Hinge Length ........................................................................41 Figure 4.16. Curvature Ductility...............................................................................................41 Figure 4.17. Panel Distortion ....................................................................................................42 Figure 4.18. Panel Distortion Ductility.....................................................................................42 Figure 5.1. Influence of the Panel Zone to Beam capacity ratio in the Pushover Curve..........45 Figure 5.2. Contribution to Deformation: Panel Zone to Beam capacity ratio.........................46 Figure 5.3. Beam Contribution to Deformation: Panel Zone to Beam capacity ratio ..............47 Figure 5.4. Beam Plastic Contribution to Deformation: Panel Zone to Beam capacity ratio...48 Figure 5.5. Panel Contribution to Deformation: Panel Zone to Beam capacity ratio ...............49 Figure 5.6. Column Contribution to Deformation: Panel Zone to Beam capacity ratio ...........49 Figure 5.7. Beam Plastic Hinge Rotations: Panel Zone to Beam capacity ratio.......................50 Figure 5.8. Plastic Hinge Length: Panel Zone to Beam capacity ratio .....................................50 Figure 5.9. Curvature Ductility: Panel Zone to Beam capacity ratio........................................51 Figure 5.10. Panel Distortion: Panel Zone to Beam capacity ratio...........................................52 Figure 5.11. Panel Distortion Ductility: Panel Zone to Beam capacity ratio ...........................53 Figure 5.12. Pushover Curve: Beams Span ..............................................................................54 Figure 5.13. Plastic Hinge Rotation: Span to Beam Depth ratio ..............................................54 Figure 5.14. Curvature Ductility: Span to Beam Depth ratio ...................................................55 Figure 5.15. Panel Zone Distortion Ductility: Span to Beam Depth ratio ................................56 Figure 5.16. Pushover Curve: Beam Depth ..............................................................................57 Figure 5.17. Plastic Hinge Rotation: Beam Depth....................................................................58 Figure 5.18. Plastic Hinge Length: Beam Depth ......................................................................58 Figure 5.19. Curvatures: Beam Depth ......................................................................................59 Figure 5.20. Curvature Ductility: Beam Depth.........................................................................59 Figure 5.21. Panel Zone Distortion Ductility: Beam Depth .....................................................60 Figure 5.22. Pushover Curve: Second Yield Distortion ...........................................................61 Figure 5.23. Plastic Hinge Rotation: Second Yield Distortion .................................................61 Figure 5.24. Curvature Ductility: Second Yield Distortion......................................................62 Figure 5.25 Panel Zone Distortion Ductility: Second Yield Distortion....................................62 Figure 5.26. Pushover Curve: Strain Hardening.......................................................................63 Figure 5.27. Plastic Hinge Rotation: Strain Hardening ............................................................64 Figure 5.28. Curvature Ductility: Strain Hardening .................................................................64 Figure 5.29. Panel Zone Distortion Ductility: Strain Hardening ..............................................65 vi Index Figure 5.30. Multi-bay Structure ..............................................................................................65 Figure 5.31. Multi-bay Structure ..............................................................................................66 Figure 5.32. Pushover Curve: Gravity Load.............................................................................67 Figure 5.33. Curvatures at a drift level of 4%...........................................................................68 Figure 5.34. Plastic Hinge Rotations IntL: Gravity Load .........................................................69 Figure 5.35. Plastic Hinge Rotations IntR: Gravity Load .........................................................69 Figure 5.36. Curvature Ductility L2: Gravity Load (thick cref) ...............................................70 Figure 5.37. Curvature Ductility R2: Gravity Load..................................................................71 Figure 5.38. Distortion Ductility PZ1: Gravity Load ...............................................................71 Figure 5.39. Distortion Ductility PZ2-PZ3: Gravity Load .......................................................72 Figure 5.40. Distortion Ductility PZ4: Gravity Load ...............................................................73 vii Index LIST OF TABLES Page Table 4.1. Sub-assemblage Geometric Properties. ...................................................................30 Table 4.3. Panel Zone thickness for the control case................................................................33 Table 4.4. Output Parameters from the Analysis......................................................................35 Table 4.5. Response Parameters ...............................................................................................36 Table 5.1. Parameters considered in the parametric studies .....................................................43 Table 5.2. Summary Control Case Sub-assemblage.................................................................44 Table 5.3. Cases for Panel Zone to Beam Capacity ratio .........................................................44 Table 5.4. Cases for Span to Beam Depth ratio ........................................................................53 Table 5.5. Cases for Beam Depth .............................................................................................56 Table 5.6. Cases for Panel Second Yield Distortion.................................................................60 Table 5.7. Cases for Strain Hardening of Steel.........................................................................63 Table 5.8. Cases for Gravity Load level ...................................................................................66 viii Chapter 1. Introduction 1. INTRODUCTION 1.1 General The response of a moment-resisting frame depends on the characteristics of its main components, namely the columns, the beams and the connections. There are several types of connection configurations such as flush end plate, extended end plate, welded flanges-bolted web or welded flanges-welded webs connections, among others. In the present study the connection type considered is welded flanges and welded webs in which the response is mainly governed by the panel zone. This component is defined as the column web portion delimited by the beam continuity plates and the column flanges, as shown in Figure 1.1. Figure 1.1. Panel Zone In the beginnings of the 1970’s, the importance of the panel zone in the response of the frames was identified and, experimental and analytical studies were carried out to characterize this component behaviour. The panel was described to be an element mainly subjected to shear stresses and therefore its failure mode was governed by shear yielding. Tests demonstrated that the shear failure mode was stable and ductile under cyclic loading. These attractive features were taken into account in design regulations by the end of the 1980’s which allowed the panel zone to be considered as a dissipative component. 9 Chapter 1. Introduction However, the 1994 Northridge earthquake resulted in severe damage in the connections of steel moment-resisting frames which had not been observed before. Extensive research was triggered which detected that excessive distortions in the panel zone played an important role in the development of weld failures in the connections. Since then research has been carried out to define new panel zone design criteria that can lead to both effective and reliable performance of panel zones during frame response. Different approaches have been proposed in recent seismic guidelines but until now a consistent approach has not been fully established and validated. 1.2 Objective This dissertation aims to provide a better understanding of the panel zone role in the seismic behaviour of steel moment-resisting frames. The influence of this component on the global and local seismic demands is examined through numerical studies carried out using idealized systems. The ultimate goal of the work is to suggest a design criterion for the panel zone that leads to a more optimized performance of the various components of a structure. 1.3 Dissertation Outline The dissertation is composed of seven chapters. In Chapter 2 a literature review on previous research done on panel zone is provided. The key experimental and numerical studies are described. The panel zone behaviour under unbalanced loading is discussed in Chapter 3. The existing approaches to evaluate the stiffness and capacity of this component are introduced along with various modelling techniques available to represent this component in a numerical analysis. The chapter terminates with a review of design guidelines and code provisions for the design of the panel zone. Chapter 4 describes a numerical investigation aiming to investigate the influence of the panel zone on the lateral response of an idealised cruciform system based on a “balanced design” for the panel zone. The nonlinear finite element program OpenSees, which is used to model the structure, is described as well as the procedures adopted in the analysis. The performance of the structure is then discussed based on several response parameters. A parametric investigation to investigate the influence of several geometric, design and modelling on the performance of moment frames is presented Chapter 5. The selected parameters and the cases considered are firstly described. The results obtained are then compared with those obtained for the structure analysed in Chapter 4 which is used as a control case. The chapter ends with a discussion and some design considerations are made. In Chapter 6 an application example is presented in order to validate the conclusions drawn in the previous chapters. The structure consists of a five-storey three-bay moment frame designed according Eurocode 3 but adopting different design criteria for the panel zone. The global performance of the structure and the local response of the various components is presented and a discussion of the results obtained is provided. 10 Chapter 1. Introduction Finally, Chapter 7 presents the conclusions of the work along with proposals for future research. 11 Chapter 2. Literature Review 2. LITERATURE REVIEW 2.1 Introduction The following sections describe the main issues regarding panel zone performance in moment-resisting frames. An early stage into investigation of the topic and understanding of the mechanism is outlined. The experimental research is presented first, followed by a description of the main aspects of the behaviour of the panel zone in terms of resistance and energy dissipating capacity. The influence of lateral displacements caused by panel zone distortions is presented and analytical studies together with modelling proposals also described. Finally a suggestion made by researchers to design guidelines and code provisions ends this chapter. 2.2 Experimental studies to understand the Panel Zone role In the early 1970’s recognition was made of the importance of beam to column connections in the behaviour of moment-resisting steel frames. The basic requirements for buildings were divided in two main topics: 1) to perform under serviceability conditions and 2) to minimize the possibility of failure of a structure when submitted to a low probability of occurrence of a seismic event. To accomplish this, it was necessary to rely on the capacity of energy dissipation of the structure. In other words, the building was not expected to resist forces in the elastic range, since this would mean a high economic demand on the structural configuration. On the other hand using the inelastic range of the structure required a thorough understanding of the performance and hence motivated further research. In the beginning of the 1970’s, research was conducted to comprehend the inelastic behaviour of joints in moment-resisting frames [Fielding and Huang, 1971; Bertero et al., 1972]. In order to understand different loading regimes, several loading conditions were simulated on the tests, whereby gravity and cyclic seismic loads were applied to different sub-assemblages. In addition, the effect of axial loads on the performance of connections subjected to shear was also carried out. Several years later, a number of test were performed by Popov [1985] in order to verify the extreme loading conditions on joints and to study the cyclic behaviour of large beam assemblies. Ten years after, Tsai et al. [1995] carried out further testing on similar joint subassemblies. The main objective was to study the performance of seismic steel beam-column 12 Chapter 2. Literature Review moment joints. The research concluded about the significant influence of the panel zone in the joint behaviour and it has shown that the inelastic deformation capacity of the joint can be enhance if the panel zone is correctly proportioned. Accordingly, among the parameters studied in both researches was the design criteria used for the panel zone. The studies recommended involving the panel as a dissipative component by developing a yield mechanism simultaneously with the beam flexural yield mechanism. The Northridge-California earthquake in 1994 triggered a large amount of research activity in the United States. Failure modes were studied in depth and design criteria were analysed under the light of the results. Failure of joint welds and premature fracture were observed and associated to excessive distortions in the panel zone. Several documents published in the following years [FEMA-267, FEMA-353, FEMA-355D], discussed the reasons for the poor behaviour of the connections and the factors affecting their performance. The concepts of seismic design in typical connections were considered as additional research subjects. Additional testing was aimed to further understand the balance of energy dissipation between the panel zone and the beam by varying the panel zone capacity in terms of the moment capacity of the beam. Participation of the panel zone to the inelastic response contributed to the reduction of the demands on the beams in terms of deformation [Lee C.H. et al., 2005]. Thorough investigations to address the influence of different details for stiffening the column in steel moment-resisting connections under cyclic loading were also performed. For example, effectiveness of the doubler plate connection detail was addressed by Lee D. et al. [2005]. Analogous research was carried out in Europe by Dubina et al. [2001] and Ciutina and Dubina [2006] to understand the cyclic performance of beam-to-column joints. The panel zone was documented as a ductile component, capable of dissipating energy by allowing stable hysteretic loops. The studies intended to clarify the performance of moment-resisting connections, influenced by different reinforcing solutions to the panel zone region. Analytical studies have been based on the experimental work presented in this section. In addition, the principal considerations regarding panel zone resistance, stiffness and analysis have been confronted against these experimental results, as presented in the subsequent sections. 2.3 Strength and Energy Dissipating Capacity of the Panel Zone Theoretical analysis of the yield condition of the web panel were carried out and verified against experimental tests. The typical load response from the panel was characterised by three stages. First, elastic shear response followed by yielding, according to the Von Mises criterion. Second, reserve in strength attributed to the surrounding elements of the panel. Finally, a post yield strength characterised by strain hardening of the steel. The actions in the panel are summarized in terms of simplified models of constant shear distribution in the column web; this is done by assuming that the flexural moment of the beam is transmitted through the beam flanges mainly as a couple of forces, as presented by Fielding and Huang [1971] and Bertero et al. [1972]. The post yield range was considered to be stable and to sustain considerable load above the yield capacity. The panel was recognized as a 13 Chapter 2. Literature Review dissipating energy component. Considerations of the elastic and inelastic range of the panel was done by an in depth experimental and clear analytical understanding of the load deformation behaviour of joints and the associated strain stress regime. Initial suggestions were made by the researchers cited above about shear stiffening of the panel based on different criteria. Fielding and Huang [1971] proposed to base the stiffness according to the required rigidity of the connection. Bertero et al. [1972] suggested not making the panel very weak because the overall energy absorption capacity of connection could be reduced by not allowing other components to develop their strength. On the other hand, suggested that the panel zone should not be too strong in order to avoid loosing its inelastic dissipative capacity. A discussion on the strength, stiffness and energy dissipation of joints was presented by Krawinkler [1978]. Importance was given to the capability of the joint to undergo severe inelastic strain reversal without loosing strength. An upper limit for the maximum strength and stiffness of a frame was established when all joints were designed for the maximum demands imposed by the elements framing into them. A discussion of the need for design criteria to balance out these parameters was hence given. A tri-linear model to represent this component in the analysis was proposed and suggestions about considering the second strength capacity attained by the surrounding elements to allow the joint to participate in energy dissipation through the inelastic range were stated. This publication strongly influenced subsequent guidelines in the topic as described in the following sections. Panel zone design procedures started to be questioned, as a result of which an opinion paper by Englekirk [1999] was presented where the behaviour objectives of steel moment resisting frames compared to experimental results from eight subassemblies were discussed. The promotion of panel zone yielding was considered to result in poor frame performance. As a minimum objective the author proposed to support a criterion where the panel zone and framing beams yield simultaneously. Therefore, the capacity of the panel was suggested to be related to yield rather than to contributions from the surrounding elements or strain hardening; this hence, promoted reinforced panel zones. Induced kinks in the columns and problems in the weld of the beam flanges were suggested to be minimised by controlling excessive panel zone distortions through stiffening of the panel. 2.4 Stiffness and deformability of the Panel Zone From experimental test, Bertero et al. [1972] identified that the contribution to the top displacement of the sub-assemblage was highly influenced by the panel zone distortion. In the mid 1970’s further analytical studies were carried out by Krawinkler et al. [1975] pointing out the importance of considering the influence of joint deformations in frames in terms of stiffness and energy absorption. Adequate control of shear deformations was promoted to attained stable hysteretic behaviour into the inelastic range. CHECK! Further analytical studies stated the necessity of including the deformability of the joint in frame analysis. Suggestions about designing the panel to sustain the maximum strength developed in the beams, to avoid early deterioration of the lateral stiffness of the joint and the entire frame, were presented. To reduced the demands on the plastic hinge rotations of the 14 Chapter 2. Literature Review beams, controlled deformations were allowed in the joint as long as the stiffness was not the principal design consideration of the joint. An important parameter identified was that large deformations of the panel zone could lead to local kinking in the column flanges, as already mentioned before. Similarly, in an analytical study Popov [1987] critically reviewed experimental results to determine different contributions from the joint elements to the flexibility of the structure. The need to avoid considering the joints as rigid members was stated. A discussion of the flexibility of the joint when subjected to seismic loads was also carried out and an analytical modelling proposal presented. Moreover, necessity for further experimentation on large joints with different member geometries was suggested. Analogously, after the Northridge earthquake, an analytical study was carried out by Schneider and Amidi [1998] to account for the effect of panel distortions on the behaviour of moment resisting frames. Results suggested that the base shear can be overestimated by 30% and the drift underestimated by 10% if the contributions from the panel distortions are not to be considered. The lateral strength of the frame can be also overestimated if the flexibility due to the panel zone is not incorporated. Recommendations about avoiding rigid eccentricities at the beam and column ends, to account for the finite dimension of the panel zone, were stated. 2.5 Numerical Studies Numerical studies were carried out by El-Tawil et al. [1999] in order to understand the effect of the inelastic behaviour of the panel zone on the possibility of fracture of the welds in steel connections. This was addressed by three-dimensional, nonlinear, finite-element models of different subassemblies based on the design criteria specified before the Northridge earthquake. A parametric study was presented where the principal geometric parameters were varied to estimate the participation to the overall performance of the connection. Results showed that although weak panel zones controlled the beam plastic rotations, this could lead to high stress concentrations in the welds of the flanges resulting in brittle failures. According to the authors, the panel zone models capable of dissipating energy and contributing to the ductility of the connection. These results confirmed the experimental observations but it was suggested that special attention should be paid to control excessive deformations. As part of a global study on buildings to enhance the elastic models utilised to design steel moment resisting frames, Foutch and Yun [2001] performed analysis on different 9-storey and 20-storey buildings designed accordingly to 1997 NEHRP provisions. Nonlinear behaviour was considered together with detailed modelling representations. The investigation involved the study of the centreline dimensions parameter and its influence to stiffness. Nonlinear springs for the beam connections and for the panel zones were used to simulate the inelastic response of the elements. The fracture of beam connections to simulate the preNorthridge type of connection was also taken into account. Static pushover and dynamic analyses were carried out. 15 Chapter 2. Literature Review The work has shown that the models that used centreline dimensions were more flexible and hence weaker than other models. This was considered to be conservative for the design of new buildings. The inclusion of connection fracture in the models resulted in a reduced drift capacity and significantly higher drift demands compared to the framing having ductile connections. 2.6 Recent Developments Motivated by the fact that the panel zone can control the performance of a structure, the need for realistic panel zone models to represent accurately the overall behaviour of frames was addressed by Kim and Engelhardt [2001],. Improvements to previous spring models were made to obtain better correspondence with experimental results. The monotonic loading used in the proposed model was based on a quadri-linear response, and it also included shear and bending deformations. The same authors proposed a cyclic model, based on Dafalia’s bounding surface theory. The models do not predict the ultimate state of the panel zone, neither buckling nor fracture at the corners. Recommendations about further study for composite behaviour were suggested and extensive comparisons with experimental data were presented. Importance to the modelling of the panel zone in steel and composite moment frames was addressed by Castro et al. [2005]. After reviewing existing analytical models to represent the panel zone a new approach was proposed to model composite joints. Validation of the proposal was done against detailed finite element models as well as against experimental data available. The work has shown that force distributions vary from composite joints to plain steel joints. The study was carried out for stress distributions which capture appropriately the distribution of plasticity. An important conclusion from the same work was that the extent of the second range of the behaviour is directly related to the thickness of the column flange and differs from the initial proposal from Krawinkler [1978] who suggested the second yield occurring at four times the yield distortion. The aforementioned study highlighted the force regimes involved in steel and composite joints. The study also discussed the implementation in analytical frame modelling analyses. 2.7 Recommendations to Design Guidelines and Code Provisions To conclude the chapter, Figure 2.1 includes a timeline with relevant events for the evolution of the panel zone development. From experimental and analytical studies, suggestions have been made to regulations over the last decades. When the post yielding range started to be considered stable and beneficial at the beginnings of the 1970’s, the AISC design formula was considered to be conservative by Fielding and Huang [1971]. In the late 1970’s Krawinkler [1978] presented a revision for the AISC design criteria in the light of the experimental studies available. In the late 1980’s and beginning of the 1990’s, an important change in the design provisions was implemented for moment-resisting frames. As documented by Tsai and Popov [1990], the 16 Chapter 2. Literature Review new provisions motivated that yielding in the panel zones took place before or at the same time as beam yielding. This was encouraged by the investigations carried to-date showing the adequate behaviour of the panel zone under inelastic demands. The AISC Specification 1990 and the Uniform Building Code 1988 allowed the use of thinner doubler plates and in some cases neglected the in need, permitting more economic designs. However, this consideration implied the necessity of controlling frame deflections. Consideration for the panel zone contribution to the total drift of a structure was proposed by means of a simplified adjustment to the results from standard elastic analysis. Different approaches have been proposed for the design of the panel zone; a diverse approach is presented in FEMA-350 [2000] seismic provisions. The main objective of the new criteria is to make the beams and the panels to yield simultaneously. The provisions were evaluated and discussed by Jun Jin and El-Tawil [2005], based on the experimental data available and the results from dynamic analyses. According to the limited results the authors considered, that the provisions may not be totally adequate since low levels of panel zone participation were identified. Dubina et al. (2001) European test for the cyclic performance of moment resisting joints. Bertero et al. (1972) Experimental investigations with gravity and cyclic loads. Panel recognized as an energy dissipating component. Popov (1985) Tests to verify the extreme loading conditions on joints. Krawinkler et al. (1975) Influence of joint deformation in frames in terms of stiffness and energy dissipation. 1970 1980 Krawinkler (1978) Proposal of analytical tri-linear model. Fielding and Huang (1971) Theoretical analysis for yield condition verified against test. Failure mode due to yielding. Northridge earthquake (1994) Excessive distortions in panel zones. Damage in welds. FEMA-350 (2000) Guidelines for seismic design of moment-resisting frames 1990 UBC (1988) Panel Zone Strength increased 2000 AISC (2000) Supplement 2. Shear strength demand no longer estimated from load combinations. Lee C.H. et al. (2005) Tests to study the effects of the panel zone and RBS in moment-resisting joints. 2007 Ciutina and Dubina et al. (2006) European test to clarify the influence of different reinforcing solutions to the Lee D. et al. (2005) panel zone in moment-resisting Tests to study the influence connections. of different details to reinforce moment-resisting joints. Effect of the panel zone and doubler plates studied. Figure 2.1. Timeline: panel zone development 2.8 Concluding Remarks A general overview of previous research in the panel zone was presented in this chapter. The main experimental studies were presented. Subsequently, studies in the relevance of the stiffness of the panel zone in the overall lateral performance of frames were described. The main numerical studies were provided, along with recommended modelling and recent developments. Finally, the evolution of suggestions to design guidelines and code provisions were presented. 17 Chapter 3. Panel Zone Behaviour and Design 3. PANEL ZONE BEHAVIOUR AND DESIGN 3.1 Introduction In a moment-resisting frame the stiffness and resistance to lateral loads are developed through the transfer of bending moments between beams and columns, either by semi-rigid or rigid beam to column connections. The characteristics of the joint behaviour are therefore related to the column, beam and connection properties. As part of the connection the panel zone plays an important role in terms of strength, stiffness and ductility, for the overall behaviour of the frame. The beam-column joints can fail for different causes according to Krawinkler et al. [1975]; among them column web crippling, column web buckling, column flange distortion, and shear yielding and buckling of the panel zone. In the following sections particular attention will be given to the shear failure mode. The panel zone is firstly described and, after that the methodologies used to represent analytically the strength and stiffness of the panel zone are discussed, in addition to the procedures to model the panel as part of a structural frame. Finally, a review of design guidelines and code specifications for the design of panel zones is conducted. Vertical Loading Unbalanced Moment Vertical Loading Balanced Moment W W W Vcol Lateral Loading Unbalanced Moment EXTERNAL JOINT Vcol Lateral Loading Unbalanced Moment INTERNAL JOINT Figure 3.1.Moment distributions at joints. 18 Chapter 3. Panel Zone Behaviour and Design 3.2 Panel Zone under Shear The forces acting in the panel zone result from the force distribution applied to the joint. There are different types of forces caused by different loading conditions. Under vertical loading the moments in external joints are unbalanced, whereas the moments for internal joints are balanced, as it can be observed in Figure 3.1. When lateral loading is considered the moment distribution is unbalanced for both external and internal joints. Unbalanced moments at the joints are transmitted trough the connection components and equilibrated by the column. An unbalanced moment produces a shear stress distribution in the panel zone where the higher stresses concentrate at the middle of the panel and reduce moderately towards the corners, according to Krawinkler et al. [1975]. A simplified methodology for the shear stress distribution is to assume a constant shear stress throughout the panel zone, implying that the bending moment is transmitted into the joint through a couple of forces concentrated at the centroid of the flanges of the beams, as presented in Figure 3.2. Although axial stresses and bending stresses are also present, the principal stress regime that develops in the panel is due to the shear imposed by the force pair. Experimental tests performed on moment-resisting connections provided important information regarding the panel behaviour, [Fielding et al. 1971; Krawinkler et al., 1975; Popov, 1985; Tsai et al., 1995; Dubina et al. 2001]. Accordingly, the behaviour is typically characterised by three stages. In the elastic range, shear strains are predominant. Subsequently, yielding occurs and a significant reduction in stiffness is observed. In the second stage, additional strength in the panel develops due to the contribution of surrounding elements, i.e. column flanges and continuity plates. Finally in the third stage a considerable decrease in stiffness is observed accompanied by high strains. Inelastic deformations take place in a strain hardening regime before local buckling or other mode of failure occurs. Δ M/d pz ΔM d pz=d b - t bf Δ M/d pz Figure 3.2.Moment converted into Shear 3.3 Representing Analytically Panel Zone Strength and Stiffness A representation of the experimental moment-distortion relation is presented in Figure 3.3, as well as different proposals for the analytical representation of the panel zone behaviour. Proposals regarding how the observed experimental behaviour should be modelled were 19 Chapter 3. Panel Zone Behaviour and Design suggested by several authors. Fielding and Huang [1971] proposed a bi-linear model. Few years later Krawinkler [1978] proposed a tri-linear model and, as a more recent suggestion, Kim and Engelhardt [2001] proposed a quadri-linear model. The models mainly agree for the elastic range, the main difference being in terms of the inelastic range. The model most widely used throughout the literature is the tri-linear model proposed by Krawinkler. In following paragraphs a description of the elastic and post-elastic ranges is provided. Experimental Response Panel Zone KEL Bi-Linear Model Fielding and Huang (1971) a) KEL b) Tri-linear Model Krawinkler (1978) Cuadri-linear Model KEL Kim and Engelhardt (2001) c) d) Figure 3.3.Analytical Models 3.3.1 Elastic Range For the elastic range the commonly adopted expression can be derived from mechanics. Equation 3.1 expresses the shear stress τ as the product between the shear modulus GS of the steel material and the web panel distortion γ. Equation 3.2 expresses the shear stress τ as function of the moment M, the panel zone height dpz and the shear area Av. Therefore the moment resistance of the panel can be expressed as in Equation 3.3. τ = GS .γ (3.1) M d pz V = Av Av (3.2) M = GS . Av .d pz .γ (3.3) τ= The differences in the previous expression regard the definition of the shear area Av and the distance between the forces dpz. Fielding and Huang (1971) proposed Av= dc.twc, while Krawinkler at al. (1975) proposed Av=(dc-tcf)tcw. Figure 3.4 presents the convention and the two different alternatives for the shear area. Only when the thickness of the column flange is significant as in the case of deep columns, the differences between the two expressions 20 Chapter 3. Panel Zone Behaviour and Design become relevant. The distance between the equivalent forces is sometimes assumed to be dpz, however considerable differences are only expected for deep beams having thick flanges. In order to find the moment in the panel that will cause yielding the Von Mises yield criterion is usually applied. The yield shear stress can be expressed as in Equation 3.4 when the effects of axial loads need to be accounted; Fy is the yield stress of the steel and Py the axial capacity of the column. The yield moment is presented in Equation 3.5. ⎛P τy = 1− ⎜ ⎜P 3 ⎝ y Fy ⎞ ⎟ ⎟ ⎠ 2 (3.4) M y , pz = Av .d pz .τ y (3.5) Comparisons of experimental and analytical predictions of the yield moment of the panel are adequate despite of the small differences mention above. The elastic rotational stiffness can then be derived as shown below: M = K EL .γ (3.6) K EL = GS . Av .db (3.7) B tbf r dpz =d b -t bf A db A wpz= d c -t cf r t cw t cf t bw B bcf bbf Section B-B Av = dc.tcw Fielding and Huang (1971) A v = (dc-tfc)t cw Krawinkler et al. (1975) dc Section A-A Figure 3.4.Geometrical parameters 3.3.2 Post-elastic range The panel zone presents an important reserve of strength after yielding; this can be attributed mainly to the elements surrounding the panel. The column flanges provide strength due to their bending resistance and the beam webs together with the continuity plates provide in21 Chapter 3. Panel Zone Behaviour and Design plane stiffness. From experimental results Krawinkler (1978) proposed an analytical expression for the post-elastic stiffness KP-EL. An estimation of the post yield strength based assuming inelastic distortions of four times the yield distortion was also suggested. Figure 3.5 presents a graphical interpretation. K P − EL = 4.E S .bcf .t cf 10 2 = 1.04.GS .bcf .t cf 2 (3.8) To estimate the post yield strength the yield strength plus the additional strength must be calculated. The final expression for the post-elastic yield strength is presented in Equation 3.9. Once the post-elastic strength is reached and yielding has occurred, strain hardening of the material is considered with a KSH stiffness presented in Equation 3.10, where μ is a strain hardening parameter. ⎛ 3.12.b fc .t fc 2 ⎞ ⎟ M y , P − EL = Av .db.τ y ⎜1 + ⎜ ⎟ A . db v ⎝ ⎠ (3.9) K SH = μ.K EL (3.9) The main differences from the previous tri-linear model and the quadri-linear model proposed by Kim and Engelhardt (2001) regard the inclusion of the shear and bending deformations modes. Participation of the column flanges thickness influencing the panel resistance in the elastic range and inelastic range was taking into account. When thick column flanges were considered, good agreement between experimental results and the model proposed were observed in contrast with other proposals. Cyclic conditions were also considered by the same authors as part of the model. M K SH M y, P-EL KP-EL My Tri-linear Model Krawinkler KEL 3γy γy 4γy γ Figure 3.5.Strength of the panel zone 22 Chapter 3. Panel Zone Behaviour and Design 3.4 Modelling the Panel Zone The flexural strength of beam to column moment-resisting joints and its continuity is what provides lateral load capacity to a frame. Generally, in the analysis of frames the joint is considered to be rigid. No relative change in angle of rotation between the beam and the column centrelines is assumed to occur. There are some structural models that consider centreline dimensions for the elements and can be refined by introducing eccentricities to the joint; this can over predict the strength and stiffness of the structural frame. For a reliable lateral load analysis, joint behaviour must be taken into account as the overall behaviour of the frame can be significantly influenced by the joint deformations. When subjected to earthquakes a structural frame can observe a reduction in stiffness if a proper joint design is not carried out. In order to state the elastic and inelastic behaviour of the panel, zone the stress-strain distribution and the load-deformation behaviour of joints must be understood and should be included in the structural model. Different options for representation of the panel zone in moment- resisting frames have been proposed, namely, the scissors model and the frame model. The scissors model incorporates a rotational spring between the column and the beams, as shown in Figure 3.6(b). The rotational spring accounts for the relative deformation between the elements. Normally, rigid links are considered in the joint as extensions of the column and the beams. The frame model which is presented in Figure 3.6(c), has evolved since the initial conception by Krawinkler et al (1975). Initially rigid links constituted the frame and rotational springs were located at each of the joints. At present, the model is typically considered as rigid links with a translational spring that accounts for the relative rotation between the members and vertical translation between the beams. The properties of the springs in the models can be easily derived from the expressions provided before in this chapter. Kγ a) b) KT c) Figure 3.6.Joint Models of the Panel Zone for Moment Resisting Frames 3.5 Design Guidelines and Code Provisions The design philosophy of the main national guidelines and regulations for structural performance under seismic loading is to allow structures to deform and dissipate energy into the inelastic range. Therefore, structures are expected to observe large deformations in a ductile manner avoiding collapse. Yield mechanisms having stable deformation cycles capable of dissipating energy avoiding fracture and significant decrease in strength are 23 Chapter 3. Panel Zone Behaviour and Design desired. On the other hand, brittle failure modes which induce sudden reductions in strength, reduction in rotation capacity and cause fractures, should therefore be avoided. At a given joint, the yield mechanism can develop in the column, the beam or the connection. To avoid collapse due to excessive story drifts, local ductility demands and a development of a soft storey mechanism, the column is aimed to remain elastic, thus invoking the weak beamstrong column mechanism as part of the capacity design criteria. Thus, the beam and connection yield mechanisms are the remaining dissipative components of interest in the seismic response of steel frames. Sharing of plasticity between the two elements is proposed in the recent literature. As outlined in Chapter 2, the main research concerning joint behaviour under lateral loading was carried out at the beginning of the 1970’s. Design regulations were based on the results obtained from experimental investigations and remained unaltered during the 1970’s and great part of the 1980’s. The main objective was to ensure that the panel zone would remain elastic under a given set of loading conditions, consequently not as a dissipative component. Only on the 1988 Uniform Building Code Regulation [UBC,1988] the shear strength of the panel zone was increased. This was motivated by the outstanding ductility performance and the post yield strength observed in the component. The main expressions established in national guidelines and regulations are presented below. 3.5.1 FEMA 350 Seismic Guidelines The 1994 Northridge earthquake in California triggered a number of experimental and analytical studies to further understand the behaviour of steel moment-resisting frame buildings. It was thought before the event that this type of structural configuration was considered among the most efficient earthquake-resistant type of structures. However, the 1994 earthquake evidenced the contrary, major connection failures were reported mostly by cracking of the beam bottom flange welds. As a result, different entities constituted the SAC Joint Venture to organize a systematic methodology for research on the subject. Among the objectives were the development of reliable guidelines and standards for the repair of damaged steel buildings, the design of new buildings and retrofit of existing steel buildings at risk. The SAC Joint Venture then was sponsored by the Federal Emergency Management Agency (FEMA), in a cooperative agreement to perform problem focused studies of the seismic performance of steel moment frame buildings and connections of various configurations. The objective was to develop recommendations for professional practice and criteria for steel construction. The FEMA-350 Guidelines is one of the compiled publications aimed to provide recommended criteria for design; it is a document for organizations occupied in the development of building codes and standards for regulations in the design and construction of steel moment-frame structures that may be subject to the effects of earthquake ground shaking. The recommended criteria for new steel moment-resisting buildings in FEMA-350, defines the strength of the panel zone such that yielding of the panel is initiated almost 24 Chapter 3. Panel Zone Behaviour and Design simultaneously as the flexural yielding of the beams, thus generating two yield mechanisms at the same time. The required web panel thickness is given by: h − db h t= (0.9)0.6 Fyc R yc d c (d b − t bf CyM c ) (3.10) Where t is the equivalent thickness of the panel zone including doubler plates, CyMc accounts for the action including overstrength from the beams, h is the average story height, db is the beam depth, dc is the column depth, tfb is the beam flange thickness, Fyc is the yield stress of the material for the column and Ryc account for steel overstrength in the column. 3.5.2 AISC Seismic Provisions The American Institute of Steel Construction (AISC), is responsible for the American National Standard: Specification for Structural Steel Buildings and Seismic Provisions for Structural Steel Buildings. The objective of the Specification is to provide design criteria in a standardized document in agreement with practice development in design of steel buildings and other structures. The objective of the Seismic Provisions is to provide design and construction rules for structural steel and composite systems under high seismic demands. The nominal shear strength Rn as well as the way to estimate the required shear strength Ru have been established by the specification and are related by the following expression Ru=φvRn, where φv is the resistance factor for the panel zone strength. The following expressions for panel zone nominal shear strength design are specified by AISC 360-05. When the effect of panel zone deformation on frame stability is not considered in the analysis: For Pr≤0.40Pc Rn = 0.6 Fy d c t w (3.11) For Pr>0.40Pc ⎛ P ⎞ Rn = 0.6 Fy d c t w ⎜⎜1.4 − r ⎟⎟ Pc ⎠ ⎝ (3.12) When frame stability, including plastic panel zone deformation is considered in the analysis: For Pr≤0.75Pc ⎛ 3.bcf t cf 2 ⎞ ⎟ Rn = 0.6 Fy d c t w ⎜1 + ⎜ ⎟ d d t b c w ⎠ ⎝ (3.13) For Pr>0.75Pc 25 Chapter 3. Panel Zone Behaviour and Design ⎛ 3.bcf t cf 2 Rn = 0.6 Fy d c t w ⎜1 + ⎜ db dctw ⎝ ⎞⎛ ⎟⎜1.9 − 1.2 Pr ⎟⎜⎝ Pc ⎠ ⎞ ⎟⎟ ⎠ (3.14) Where Pr is the axial design capacity, Pc is equal to 0.6Py (Py is the axial yield strength of the column), Fyc is the column minimum specified yield stress, bcf is the column flange width, tcf the column flange thickness, db the beam depth, dc the column depth and tw the panel zone thickness. Equations 3.11-3.14 are based on Krawinkler (1978) proposal. Although slight modifications have occurred over time no significant modifications have been made. On the other hand, the required shear strength demand of the panel zone under seismic loading specified by the AISC has evolved over the years. Two main approaches have been proposed, depending either on load combinations or on flexural strength at yielding of the connecting beams. The Specification is continuously under improvements and several editions have been published over the years. Recalling the Specifications of 1992, a concise revision of the main evolutions of the code referring to the panel zone required strength, follows. (a) AISC 1992 Seismic Provisions. The provisions in 1992 attempted to be consistent with the UBC 1991, by determining the shear demand from the analysis using the appropriate load combinations. The resistance factor φv for the panel zone design strength was equal to 0.75. However, there was a limiting value based on the flexural capacity the framing beams could develop, equal to 0.9ΣφbMp; where φb is 0.9 and Mp is the beam plastic moment. This limiting value is referred as a capping value and its intention was to avoid exceeding realistic loading combination values. (b) AISC 1997 Seismic Provisions. The provisions in 1997 were kept the same for estimating the shear demand in the panel. However, a structural overstrength magnification factor Ω0 was incorporated in the calculation of the earthquake demands. Moreover, the capped shear generated from the framing beams was modified to 0.8ΣRyMp, where Ry accounts for yield strength above the minimum value. The intention with this capping was to account for the favourable effects of gravity loads in internal joints. In 1999 a supplement was added to the provisions, where the main change was related with the capped shear demand of 0.8ΣM*pb, where ΣM*pb= Σ(1.1RyMp+Mv). The additional moment Mv, resulted from the projection of the plastic hinge moments away from the column to the column face. A significant change was made in a supplement published in 2000, where the required strength was no longer estimated from load combinations. A minimum required shear strength Ru value was established from the summation of the moments at the column faces, determined by projecting the expected moments at the plastic hinges to the column faces, but removing the 0.8 factor. In addition, to relate beam and panel zone yielding, the resistance factor φv was change from 0.75 to 1.0. In this Supplement the thickness of the panel is established from the methodology adopted for proportioning prequalified connections. 26 Chapter 3. Panel Zone Behaviour and Design (c) AISC 2002 Seismic Provisions. The provisions in 2002 remained on the same line as AISC 2000, and the shear demand and the design shear strength remain unchanged. Nevertheless, a thickness condition was added to the panel zone as a requirement against local buckling of the column web. t ≥ (d z + wz ) 90 (3.15) Where t is the thickness of the column web or the doubler plates, dz is the panel zone depth between continuity plates and wz is the panel zone width between column flanges. (d) AISC 2005 Seismic Provisions. The provisions in 2005 intended to merge the load and resistance factor design (LRFD) and the allowable stress design (ASD) design philosophies in one document, although no major change was considered for the panel zone compared to previous provision. The φv factor for available strength of the panel zone remained 1.0. The required shear strength and the design shear strength remained as in AISC 2002 with minor modifications to notation. 3.5.3 Eurocode The Eurocode is a compilation of different building code regulations developed by the European Committee for Standardization. The Eurocode is organised in 10 different sections in which the objective of Eurocode 8: Design of Structures for Earthquake Resistance, is the design and construction of buildings and civil engineering works in seismic regions. The shear resistance of column web panels should satisfy the following expression: Vwp,Ed/Vwp,Rd≤1.0, where Vwp,Ed is the shear force in the panel due to action effects taking into account the plastic resistance of the adjacent dissipative zones in beams or connections, and Vwp,Rd is the shear resistance in accordance to Eurocode 3 1-8 expressed as: Vwp , Rd = 0.9 f y ,wc Avc (3.16) 3.γ M 0 Where fy,wc is the yield stress of the column web, Avc is the shear area of the column and γM0 is usually assumed as 1.0 but is dependent on the national annex. If the column has web stiffeners or continuity plates, the codes allows the strength of the panel to be increased by: Vwp ,add , Rd = 4M pl , fc , Rd ds but Vwp ,add , Rd ≤ 2M pl , fc , Rd + 2M pl , st , Rd ds (3.17) Where Mpl,fc,Rd is the design plastic moment resistance of a column flange, ds is the distance between centrelines of the stiffeners and Mpl,st,Rd is the design plastic moment resistance of a stiffener. 27 Chapter 4. Numerical Investigation 4. NUMERICAL INVESTIGATION 4.1 Introduction In this chapter an investigation is carried out to evaluate the influence of the panel zone design in structural frame response. This is achieved through the analysis of a cruciform subassemblage, which is representative of the behaviour of a moment-resisting frame. The properties of the structure are firstly described as well as the design criterion adopted for the panel zone. Following that, the numerical model of the structure is introduced and the procedures adopted in the analysis along with the response parameters are established. The results obtained are then presented and a discussion is made regarding the performance of the structure with particular emphasis on the influence of the panel zone. B tbf r dpz =d b -t bf A db A wpz= d c -t cf r t cw t bw B bbf Section B-B bcf t cf dc Section A-A Figure 4.1. Cruciform sub-assemblage. 28 Chapter 4. Numerical Investigation 4.2 Cruciform Sub-Assemblage 4.2.1 Description In order to understand the panel zone role in the response of a structural frame, a simplified sub-assemblage is studied (Figure 4.2). The beam span (L) is assumed as 8.0m and the storey height (H) is assumed as 3.5m. The columns of the sub-assemblage span from the mid-height of the storeys and the beams from mid-length of the spans, where the contra-flexure points are generally located under lateral loading. The boundary conditions can then be established. The ends of the beams are assumed as vertically restrained, but able to rotate and to move horizontally. The bottom of the column is assumed vertically and horizontally restrained, but able to rotate. The top of the column is assumed free to displace. A summary of the simplified structure with the corresponding geometrical parameters and the boundary conditions are shown in Figure 4.2. The static equilibrium of the sub-assemblage is accomplished in terms of the external shear force in the column (Vcol) and the vertical reactions at the beam ends. Vcol H/2 H/2-dpz /2 L/2 L/2-wpz /2 L/2 L/2-wpz /2 wpz dpz RL= Vcol H L RR= Vcol H L H/2 H/2-dpz /2 Vcol Figure 4.2. Sub-assemblage geometrical properties and boundary conditions. For the design of the frame members, conventional procedures are used to determine the size of the columns and of the beams. The members are designed to sustain the demands from vertical and lateral loading, and to provide sufficient rigidity to the structure in order to fulfil drift limits. The typically adopted weak beam-strong column concept is employed for the column design To achieve this desirable energy dissipation mechanism, the column size was determined to be at least 1.3 times stronger than the plastic moment of the beams. The steel sections determined from the design with the respective dimensions and plastic moment relations, are presented in Table 4.1. Having defined the beams and column sizes, the focus is now on the design of the connection, more precisely the web panel. In the following sections, the demands and criteria adopted in the design of the panel zone are described. 29 Chapter 4. Numerical Investigation Table 4.1. Sub-assemblage Geometric Properties. d bf tf tw Mpl mm mm mm mm kN.m 400 180 13.5 8.6 340 330 300 16.5 9.5 484 Steel Beam IPE400 Steel Column HEA340 4.2.2 Analytical Demands in the Panel Zone The demand in the panel zone is function of the moments in the beams and the shear in the column. The moment and shear force diagrams of the structure under lateral loading are illustrated in Figure 4.3(a). For the symmetric structure Mb,L=Mb,R and hence the moments acting at the boundaries of the panel zone can be expressed as follows. M b , L = M b , R = Vcol H ( L 2 − wpz 2) L (4.1) Vcol P Vcol RL RR M b,L/d pz M b,R/d pz Vcol d pz V b,L M b,R M b,L V b,R Vcol M b,L/d pz M b,R/d pz Vcol RL RR P Vcol wpz a) b) Figure 4.3. Forces in the Panel Zone under lateral loading conditions The forces at the boundary of the panel zone are summarized in Figure 4.3(b). The behaviour of the panel zone is mainly influenced by shear stresses. As shown by the shear force diagram, the shear force in the column is opposite in sign to the shear forces transferred through the beam flanges, therefore it reduces the demands. Thus the shear acting in the panel can be expressed as shown. VPZ = M b,L d pz + M b,R d pz − Vcol (4.2) 30 Chapter 4. Numerical Investigation The aim is to calculate the shear observed by the panel as function of the moments developed at the beams. Expressing the moment in the beams as Mb, the shear in the panel zone can be estimated as follows. VPZ = Mb Mb Mb ⎛L⎞ + − ⎜ ⎟ d pz d pz ( L 2 − wpz 2) ⎝ H ⎠ ⎡ 1 1 ⎛L − VPZ = 2M b ⎢ ⎜ ⎢⎣ d pz ( L − wpz ) ⎝ H (4.3) ⎞⎤ ⎟⎥ ⎠ ⎥⎦ (4.4) 4.2.3 Design Criterion for the Panel Zone Having defined the demands in the panel zone, a design criterion for the component needs to be established. For the structure under analysis, the panel is going to be proportioned such that yielding of this component initiates at the same load level that develops the plastic capacity of the connecting beams, as shown in Figure 4.4. However, the panel zone strength is defined at the first yield Vy,pz as marked by a circle in Figure 4.4(a) and not at the second yield as proposed by several design codes. This design criterion is referred hereafter as a “balanced design”. Capacity Demand V α =1 ⎡ 1 1 ⎛L VPL , PZ = 2 M b , PL ⎢ − ⎜ − ( ) d L w pz ⎝ H ⎣⎢ pz K SH V y, P-EL VPZ ⎞⎤ ⎟⎥ ⎠ ⎦⎥ K P-EL Vy, pz M b =Mb,PL KEL V y, pz =A v .τ y VPZ =f(Mb ) VPZ =f(Mb,PL ) 3γy γy γ 4γy (a) γ (b) Figure 4.4. Design criterion for the panel zone The design criterion can be generalized by defining a capacity to demand ratio α as follows: α= Vy , pz VPL , PZ α>1 strong panel; α<1 weak panel (4.5) An α factor equal to one corresponds to a “balanced design”. On the other hand, an α factor exceeding one implies a strong panel whereas an α factor below indicates a weak panel design. The strong and weak panel zone design scenarios are illustrated in Figure 4.5. For the strong panel zone case, α>1, yielding in the beams occurs before the panel yields. Therefore 31 Chapter 4. Numerical Investigation the panel remains in the elastic range as presented by the circles and the dotted line in Figure 4.5(a). For the weak panel zone, α<1, yielding in the panel occurs before the beams reach the plastic moment. Capacity V V y, P-EL Demand VPZ K SH STRONG PANEL α >1 K P-EL Vy, pz KEL M b =Mb,PL τ V pz=A v . ELASTIC RANGE 3γy γy γ γ 4γy (a) Capacity V VPZ Demand WEAK PANEL α <1 K SH V y, P-EL KP-EL Vy, pz 3γy γy M b =Mb,PL ⎛ 3.12.b fc .t fc 2 ⎞ VP− EL = Av .τ ⎜1 + ⎟ ⎜ Av .db ⎟⎠ ⎝ KEL POST-ELASTIC RANGE 4γy γ γ (b) Figure 4.5. Design criterion for the panel zone. (a) Strong panel (b) Weak panel The controlling parameter for the strength of the panel zone is the shear area Av, defined as the panel width wpz times the thickness tpz. Since the width is fixed by the column size, the area is directly proportional to the panel zone thickness tpz. Hence the criterion to define different strength values will be function of this parameter. From Equation 4.5 the following expression for the panel zone thickness is derived. t pz = α 2M b, PL ⎛ 1 ⎜ wpz ⎜⎝ τ y ⎞⎡ 1 1 ⎛L − ⎜ ⎟⎟ ⎢⎢ d ⎠ ⎣ pz ( L − wpz ) ⎝ H ⎞⎤ ⎟⎥ ⎠ ⎦⎥ (4.6) The panel zone thickness is defined as tpz= tcw + tdp, where tcw is the thickness of the column web and tdp is the additional thickness provided by the doubler plates. The thickness of the panel calculated from Equation 4.6 is presented in Table 4.3 32 Chapter 4. Numerical Investigation Table 4.1. Panel Zone thickness for the control case Parameter Panel thickness (tpz) Column web thickness (tcw) Doubler plates thickness (tdp) Value 31.3 mm 9.5 mm 21.8 mm In the next section a description of the numerical model of the structure is presented. 4.3 Numerical Modelling 4.3.1 OpenSees Finite Element Program for Non-linear Analysis The finite element program used in the numerical studies is Open System for Earthquake Engineering Simulation, OpenSees, developed by the University of California at Berkeley (http://www.berkeley.edu/OpenSees). The program was created for the sole purpose to be used for research projects. OpenSees is an object-oriented finite element analysis framework used to simulate structural and geotechnical scenarios in earthquake engineering. 4.3.2 OpenSees Model representation The sub-assemblage is represented in OpenSees using two different types of finite elements. The column and beams are represented by non-linear beam-column elements and the panel by a beam-column joint element, as shown in Figure 4.6. Displacement Control PANEL ZONE Beam column joint element Rigid springs except shear panel H/2 See Figure 4.7 L/2 COLUMN Non-linear beam column element B i-node j-node i-node Integration Point B A A BEAM Non-linear beam column element Fiber Sections Section A-A j-node E SH = μ. E S Section B-B ES Fiber Sections Bi-Linear Model Strain Hardening Steel Figure 4.6. Model Representation in OpenSees (a) Nonlinear Beam Column element. This is a force-based element that considers the spread of plasticity across the section and along the member. The number of integration points is defined by the user and for this study five integration points chosen. Each integration point is characterised by a cross section and each section is defined by one or more uniaxial materials. A description of the Fibre Section Model is presented next. 33 Chapter 4. Numerical Investigation • Fibre Section Model. A fibre section enables to discretize a cross section in subregions denominated patches. The fibre section object is composed by different patch objects of simple geometrical shapes. For example, the flanges and the web of the Isections are defined as rectangular patches. Each patch is described by a general geometric configuration and can be subdivided using the Fibre Command. The fibre command allows creating uniaxial fibre objects to be added to the section. A uniaxial material is then assigned to each fibre. Figure 4.6 presents the discretized fibres for the flanges and the webs of the column and the beam sections. A sub-routine was created to generate and discretize each steel section based on its geometrical properties. The material model adopted for the steel in columns and beams is the Uniaxial Hardening Material. The properties considered are shown in Figure 4.7. The selected uniaxial material combines linear kinematic and isotropic hardening. The isotropic hardening is neglected and only kinematic hardening is considered. In this study the strain hardening parameter μ is assumed equal to 1%. Stress E SH Fy ES Hardening Material E S = 210000 MPa F y = 275 MPa ESH = Strain Es ES (H iso+H kin ) ES +Hiso+Hkin H iso = 0 ES .μ 1- μ H kin = μ = strain hardening parameter Fy Figure 4.7. Uniaxial Hardening Material (b) Beam Column Joint Element. This is used to create a joint element between the beams and the columns. As shown in Figure 4.8(a), the element is composed of 13 different components. As discussed, the main interest of the connection in the present research is to represent the panel zone behaviour, hence all the 12 translational springs are assumed as rigid. The parameter that will control the behaviour is the shear panel. The properties of the shear panel can be specified by a user defined moment-distortion relationship. In order to represent a tri-linear behaviour, a sub-routine combining two bilinear relationships in parallel was programmed. The properties assigned to the element are shown in Figure 4.8(b). 34 Chapter 4. Numerical Investigation M KSH = μ . K EL My,P-EL KP-EL = M y,pz 4 E S.bcf .tcf2 10 μ = strain hardening parameter K EL = GS .Av .dpz M y, pz = Av .τy .d pz 3.12.b cf .t cf2 ) MP-EL = Av .τy .d pz (1+ A v .d pz γy a) 4γ y b) γ Figure 4.8. Panel Representation. a) Beam Column Joint Element (Lowes et al., 2004) b) Tri-linear model adopted for the shear panel 4.4 Analysis Procedures and Response Parameters 4.4.1 Analysis Procedures The structure is analysed in OpenSees by performing a nonlinear static (or pushover) analysis under displacement control. The control node is the top node, for which successive horizontal displacements of the order of one millimetre are applied to simulate the drift observed by the frame during a lateral loading condition. The target displacement for the sub-assemblage is calculated based on a storey drift of 4%. The OpenSees software enables to record response parameters of interest; these recordings are stored in text files. The following response parameters are recorded for each step of the analysis: reactions at the boundary conditions, load factor, nodal displacements, strains at the extreme fibre of the I-sections and panel zone moments and distortions. Table 4.4 summarizes the recorded parameters. Table 4.3. Output Parameters from the Analysis Load factor Reactions Nodal displacements Strains at the Extreme Fibre I-section Distortion of the panel zone λ RL, RR, RX δx, δy ε γ To be able to post-process this data the software Matlab is used. After loading the results these are manipulated and stored in an Excel spreadsheet and subsequently processed. 4.4.2 Response Parameters Having obtained the output parameters from the analysis, post-processing of the results is carried out to obtain additional response parameters for the sub-assemblage. These can be grouped into parameters describing the overall structure response, and parameters which are 35 Chapter 4. Numerical Investigation related to the local response of the beams, columns and panel zones. A summary of all the response parameters is presented in Table 4.5 and the expressions adopted for the beam response are provided in Figure 4.9. For the panel zone, the contribution to the top displacement is function of the distortion and the geometrical configuration of the subassemblage as presented below. ⎡ 4(L 2 − w pz 2)(H 2 − d pz 2) − w pz .d pz ⎤ Δ PZ = γ ⎢ ⎥ 2(L 2 − w pz 2) + w pz ⎢⎣ ⎥⎦ (4.7) Table 4.2. Response Parameters Component STRUCTURE Parameter Units Description Δ =ΔΒΕΑΜ+ΔPZ+Δ COL [m] Top displacement Drift=Δ/H [%] Top displacement over storey height Vcol [kN] Column Shear force μ = Δ/Δy θ pl = L ∫ Structure ductility φ .dx − φ y .Lp [mrad] Plastic hinge rotation L − Lp BEAM Lph/L’ Normalized plastic hinge length μ φ= φ/φy Curvature ductility μ θ= θ/θy Rotation ductility EL Cont=Δ Δ [%] Elastic Contribution to top displacement PL Cont=ΔBEAM-PLiv/Δ [%] Plastic Contribution to top displacement BEAM Cont=Δ [%] Beam Contribution to top displacement = EL+PL iii BEAM-EL / v BEAM / Δ γ [mrad] Distortion μ γ= γ/γy PANEL Distortion ductility ΔPZ [m] Panel Zone top displacement PZ Cont=ΔPZvi/Δ [%] Panel Zone Contribution to top displacement Vcol ⋅ (H 2 − d pz 2) 3 Δ COL = 2 COLUMN 3.E S .IxC COL Cont=ΔCOL/Δ Column top displacement [%] Column Contribution to top displacement 36 Chapter 4. Numerical Investigation R L ph L' M pl : plastic moment M y : yield moment Moment Diagram M pl My φ Curvatures φ y =My /EI x φy Deflections θ pl Δ BEAM-PL L' Δ BEAM = ∫ φ .x.dx 0 L ' − L ph Δ BEAM − EL = θ pl ∫ _ φ .x.dx + φ y . x .L ph 0 Δ BEAM-EL Δ Δ BEAM − PL = L' ∫ _ φ .x.dx − φ y . x .Lph L ' − L ph Figure 4.9. Plastic Hinge Concept 4.5 Discussion of Results The lateral performance of the sub-assemblage is now examined with the emphasis on the influence of the panel zone in the overall response of the structure and based on the response parameters described in the previous section. The global response of the structure can be evaluated based on the pushover curve presented in Figure 4.10. The load factor applied to the structure to reach the objective displacement is referred in terms of Vcol. Three main points can be identified, namely, the initial yield of the beams, the point where the panel yields and the beams reach the plastic moment and finally, the point where the panel reaches the second yield. The load factor and the corresponding drift for each of these points are also provided in the plot. 37 Chapter 4. Numerical Investigation 250 200 [1.20%; 181kN] Beam first yield 150 Panel zone yield and beam plastic moment [1.45%; 202kN] Vcol[kN] [2.65%; 216kN] Panel zone second yield 100 50 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 4.10. Pushover Curve Attention is now driven to the deformation of the structure. The scaled deformed shape for a 4% drift is presented in Figure 4.11. The contribution from the various components (beams, column and panel zone) to the flexibility of the structure is illustrated in Figure 4.12 for increasing levels of deformation of the system. 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Figure 4.11. Deformed Shape 4% Drift Before yielding takes place, the contribution to displacement is shared by the components as follows: the beams contribute 67.7%, followed by the column with 19.7% and then the panel with 12.6%. When yielding in the beams initiates, an increase in the contribution from these members occurs. However, when yielding of the panel takes place, the contribution from this component to the global deformation observes a gradual increase up to 30% (for 4% drift). This effect is obviously reflected in a reduction in terms of the contribution from the column 38 Chapter 4. Numerical Investigation and the beams. It becomes clear that panel yielding leads to an attenuation of the demand imposed on the beam. This will be further discussed and illustrated later in this section. 100% Contribution to Deformation 100% 75% Beam 50% 25% Panel Column 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 4.12. Contribution from Beam, Panel and Column to the drift As presented in the previous section, the participation of the beams to the deformation of the structure can be divided into two terms namely, the elastic the plastic contributions which are illustrated in Figure 4.13. It is clear from the figure that the plastic contribution from the beam initiates when yielding takes place. The elastic contribution of the beam becomes less relevant as the plastic hinge develops in the beams. At approximately 3% drift the plastic contribution becomes more important then the elastic contribution. At the final displacement equivalent to 4% drift, the plastic and elastic contribution are 61% and 39%, respectively of the total contribution of the beam. 100% Contribution to Deformation 100% 75% Beam 50% Beam EL Beam PL 25% 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 4.13. Beam Elastic and Plastic Contribution to the drift It is worth noting that the total deformation of the structure at 4% drift results from 31% and 69% of elastic and plastic deformation of the components, respectively. 39 Chapter 4. Numerical Investigation Having discussed the global response of the structure, the focus hereafter will be on the local response of the components. Regarding the beams, the plastic hinge rotation against drift is presented in Figure 4.14. As expected from the plastic contribution of the beams, the plastic hinge rotation increases as the deformation of the structure increases. Consistently with Figure 4.12, three different ranges can be observed. In the first range, the rate of increase in the rotations is larger compared to the rate after the panel yields. Similarly, the rate at which rotations increase after second yield of the panel is lower than the previous range. It is therefore clear the attenuating effect in terms of plastic demand that panel yielding provides to the beams. 20 15 θ pl [m rad] 10 θpl θpl 5 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 4.14. Plastic Hinge Rotation Beams The maximum plastic rotation of the beam at 4% drift is 16mrad. If plastic behaviour was only concentrated on the beams, plastic rotations of about 28mrad would be expected to develop in the beams for this level of drift. However, due to the development of inelasticity in the panel, the plastic beam rotations are clearly lower leading to a more desirable element response. Another local response parameter involving the beam is the spread of plasticity along the member which is typically represented by the plastic hinge length Lph. Obviously, the plastic hinge length does not increase in steps, it rather spreads as a continuous function. However, the calculations are performed between integration points and hence the jumps in plastic hinge length visible in Figure 4.15. When the plastic moment in the beams is reached, the plastic hinge length is equal to 10% L’, which is around 400mm, approximately the beam depth. For the final displacement the plastic hinge reaches a value around 1.5 times the beam depth. 40 Chapter 4. Numerical Investigation 20% Lph/L' 15% 10% 5% 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 4.15. Normalized Plastic Hinge Length In terms of cross section response, the curvature ductility of the beams is represented in Figure 4.16. The cross section analyzed corresponds to the integration point next to the column. As discussed before for the plastic rotation, the slope of the curvature ductility is characterized by three ranges. The slope of each range decreases as the panel yields. The curvature ductility increases up to 12.8 times for a level of drift equal to 4%. 16 12 μφ 8 4 μ φ=1 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 4.16. Curvature Ductility After describing the local response of the beams, the discussion will now focus on the panel zone. In Figure 4.17 the panel distortion is presented, where the trilinear behaviour specified for the component can be easily observed. It is clear that at 1.45% the first yield of the panel occurs. Beyond this point, high distortions develop in the panel up to15mrad when 4% of drift is applied to the structure. 41 Chapter 4. Numerical Investigation 16 12 γ [m rad] 8 90−γ 90+γ 4 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 3.0% 3.5% 4.0% Drift Figure 4.17. Panel Distortion 8 6 μγ 4 2 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 4.0% Drift Figure 4.18. Panel Distortion Ductility The panel ductility is presented in Figure 4.18. In the figure the limits for the elastic and postelastic range are marked by dotted lines. The final panel distortion is 7.4 times the yield distortion, which is well within the strain hardening range of the panel. 4.6 Concluding Remarks In this chapter a numerical study aiming to investigate the influence of the panel zone in the overall response of a cruciform sub-assemblage was carried out. The criterion of a “balance design” for the panel zone was introduced, whereby the panel zone thickness is established in such a way that yielding in the panel initiates when the beams reach the plastic moment. The results obtained have clearly shown a reduction of the plastic beam demand due to yielding of the panel zone component. The structure analysed is used as reference case in the next chapter in which a parametric study is carried out in order to assess the influence of different geometrical and design parameters on the response of MR systems. 42 Chapter 5. Parametric Studies 5. PARAMETRIC STUDIES 5.1 Introduction The following chapter intends to identify the principal parameters that have an influence in the panel zone response by carrying out a parametric study. The effect of different factors as: design considerations for the panel, the influence of geometrical properties of the structure and the effect of different modelling criteria are identified and presented in the following sections. In the previous chapter, the description of a cruciform sub-assemblage and the main modelling criteria was presented. The results from this structure are used as a reference case for the parametric studies. The intention is to study the effect of varying a specific parameter by comparing the results with the control case. The global response of the structure is analyzed and the differences in the local response of the beams and the panel zone are compared. 5.2 Parameters Considered Several parameters can influence the behaviour of a frame in relation to the panel zone. A selection of parameters considered to be the more relevant regarding the panel zone, are presented in Table 5.1. Table 5.1. Parameters considered in the parametric studies Parameter Range Unstrengthen, 80%, 90%, 100%, 110%, Rigid 6.4m, 7.2m, 8.0m, 8.8m, 9.6m 300mm, 360mm, 400mm, 450mm, 500mm 300mm, 400mm, 500mm, 600mm Property Panel Zone to Beam Capacity Ratio α Beam Span L Beam depth L=8m db Beam depth with L/db=20 db Panel Zone Second Yield Distortion n.γy 3γy , 4γy , 5γy , 6γy Modelling Strain Hardening of the Steel μ β 0%, 1%, 2% Modelling None, 25%, 50%, 75% Modelling/Design Gravity load level Design Geometrical Geometrical Geometrical 43 Chapter 5. Parametric Studies The summary of the control case is presented in Table 5.2 to bear in mind the main changes in each of the parameters analyzed. For each of the parameters studied different values have being assigned to generate different cases. The evaluation of the results is always compared against the control case. Table 5.2. Summary Control Case Sub-assemblage Parameter Span Storey height Steel Beam IPE400 Steel Column HEA340 Panel Zone α=100% Case Ref.: CC L 8.0m H 3.5m db 400mm dc 330mm tpz 31.3mm 386.5mm dpz wpz 313.5mm Second yield distortion n.γy Strain Hardening Steel μ β 4γy 1% Gravity Load Level None The following sections present the different cases studied and relevant results. 5.3 Influence of the Panel Zone to Beam capacity ratio To understand the influence of the panel zone to beam capacity ratio, six cases are considered as shown in Table 5.3. Three cases are below the balance design and are considered to have weak panel zones, and two cases are above the balance design and are considered to have strong panel zones. The main difference for each case is in terms of the panel zone thickness as it can be observed in the table. The ratio between tpz and tcw is presented as an indication of how many times the column web thickness has to be added to reach the required design thickness of the panel. Table 5.3. Cases for Panel Zone to Beam Capacity ratio Case α PZUNSTR PZ80 PZ90 CREF PZ110 PZRIGID Unstrengthen 80% 90% 100% 110% Rigid tpz [mm] tpz /tcw 9.5 25.1 28.2 31.3 34.5 - 1.00 2.64 2.97 3.29 3.63 - First, attention is on the global response of the sub-assemblage by examining the differences between the pushover curves illustrated in Figure 5.1. The PZUNSTR and PZRIGID cases are considered boundaries for the design of the panel zone and therefore bound lines delimit the behaviour of the sub-assemblage. The unstrengthen panel considers a column web where no 44 Chapter 5. Parametric Studies additional doubler plates are added. The rigid panel on the other hand neglects any contribution from the joint to the flexibility of the structure and it can be assumed as a very strong panel zone design. The discussion for the pushover curves is developed on three main aspects: stiffness, strength and deformation, for which relevant conclusions are presented below. The stiffness of the structures is observed to be influence by the design of the panel zone, as shown in Figure 5.1. By comparing the range in which the elastic stiffness of the structure varies, relevant differences can be observed from the PZUNSTR and PZRIGID cases; however for the range between 80% and 110%, the variations in stiffness are less significant. Regarding the strength, clear differences can be observed when yielding initiates. For the three cases below the balance design (PSUNSTR, PZ80, PZ90), considered to have weak panels, yielding in the panel is expected to occur before yielding in the beams and therefore the strength of the structure is governed by the yield strength of the panel zone. For the two cases above the balance design (PZ110, PZRIGID), considered to have strong panels, yielding in the beams are expected to occur before yielding in the panel and therefore the strength of the structure is governed by the yield strength of the beams. This can be observed from the marks inserted in the figure to identify first and second yield of the panel. For a final drift level of 4%, significant differences can be observed between the balance design case and the unstrenthen case, where strength reduces from 223kN to 85kN, equivalent to a 62% reduction. Conversely, the strength for balance design and the rigid case, increase from 223kN to 234kN, equivalent to a 5% increase. Therefore the final strength of the structure is controlled by the design of the panel zone mainly in the weak panel zone range. 250 100% 110% 200 Rigid 80% 90% 150 Panel zone first yield Vcol [kN] Panel zone second yield 100 Unstrengthen 50 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.1. Influence of the Panel Zone to Beam capacity ratio in the Pushover Curve Concerning the deformations in the sub-assemblages different yield displacements are observed for each case. The deformations are directly proportional to the stiffness and strength, since this two where discussed it can be observed that for the unstrengthen case the 45 Chapter 5. Parametric Studies drift level of 0.5% at which the panel yields is related to low stiffness and low strength of this case. The drift level range for the remaining cases is between 1.0% and 1.3%. Finally, it can be stated that assumptions of rigid joints in structural models are unrealistic if the panel zone is not strengthened. The contribution to deformation of the structure according to the different panel zone to beam capacity ratios are shown in Figure 5.2, discriminating the participation of the different components (column, beams and panel). Although individual comparisons are carried out below for each component and each case, the simultaneous comparison allows extracting the following. The participation from the panel compared to the beam can be appreciated. As the panel becomes stronger its contribution becomes less relevant; consistently the beam contribution becomes more significant. Also, the plastic participation of the beams is observed to gain more participation as the panel becomes stronger. The importance in participation in comparison with the other components can easily be notice. 100% 100% 80% Unstr Panel 75% Contribution [%] Contribution [%] 75% 50% Beam 25% Panel Beam 50% Beam EL 25% Column Column 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 0% 0.0% Dri ft Top Node [%] Beam PL 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Top Node [%] 100% 100% 100% 90% 75% Beam Contribution [%] Contribution [%] 75% 50% Panel Beam EL 1.0% 1.5% 2.0% Beam PL Panel Beam PL Column 0.5% Beam EL 50% 25% 25% 0% 0.0% Beam 2.5% 3.0% 3.5% 0% 0.0% 4.0% 0.5% Column 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 100% 100% 110% Beam Rigid Beam 75% Contribution [%] 75% Contribution [%] 4.0% Drift Top Node [%] Drift Top Node [%] Beam PL Beam EL 50% 25% Beam PL Beam EL 50% 25% Column Column Panel 0% 0.0% 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Top Node [%] Drift Top Node [%] Figure 5.2. Contribution to Deformation: Panel Zone to Beam capacity ratio As mentioned before, the participation from each component is analyzed separately. By extracting the beam contribution to deformation from each case, Figure 5.3 is generated. It can be observed that weak panel zones significantly reduce the beam participation to the overall deformation of the structure. While strong panel zones require more participation from the beams for the total deformation of the structure. Looking at the PZ110 case, it is known that yielding in the beams occur first, however when the panel zone reaches its firtst yield at a drift level of 2.8%, an attenuation in the demands on the beams can be observed. 46 Chapter 5. Parametric Studies 100% Rigid Contribution to Deformation 110% 75% 100% 50% 90% 80% 25% Unstrengthen 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.3. Beam Contribution to Deformation: Panel Zone to Beam capacity ratio From examining separately the plastic contribution to deformation from the beams, Figure 5.4 is extracted. The following can be concluded: • The rigid bound clearly establishes the maximum contribution to deformation that can be expected from the beams. Below this bound participation can only be observed until the 80% case. For panel zone to beam capacity ratio below 80%, the beams remain elastic and all the plastic deformations are concentrated in the panel. • For the 110% case the plastic contribution starts simultaneously with the balance case. However, clear differences are observed when the panel yields for the control case and shares the contribution to deformation. Only one change in slope is observed for the 110% case, when the panel reaches the first yield at a drift level of 2.8%. • Although panel zone thickness variations from 80% to 110% are within a narrow band, differences in plastic participation from the beam are completely different and vary significantly the response of the structure. 47 Chapter 5. Parametric Studies Contribution to Deformation 100% 75% Rigid 110% 50% 100% 25% 90% 0% 0.0% 80% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.4. Beam Plastic Contribution to Deformation: Panel Zone to Beam capacity ratio The different cases for panel zone contribution to deformation are illustrated in Figure 5.6. The following can be concluded: • A large range for participation of the panel zone is observed. Variations in thickness are decisive for the response of the panel. • For weak panel zones, transition from first yield to second yield occur within a level of 0.6% drift; beyond this, strain hardening of the material governs the response. This comment not very important • For the strong panel (110%) the contribution to deformation reduces as the beam plastic contribution is develop. However as soon as the first yield strength in the panel is reach due to strain hardening, the participation increases. • The balance case show to be an optimize design in terms of panel zone contribution from every range (elastic, post-elastic and strain hardening). Not clear 48 Chapter 5. Parametric Studies 100% Contribution to Deformation Unstrengthen 75% 80% 90% 50% 100% 25% 110% 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.6. Panel Contribution to Deformation: Panel Zone to Beam capacity ratio For the column contribution to deformation Figure 5.7 is presented. As expected, the range of variation is very narrow since this component remains elastic throughout the deformation of the sub-assemblage. As the panel becomes weaker, the participation from the column reduces. Contribution to Deformation 100% 75% 50% Rigid 25% Unstrengthen 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.7. Column Contribution to Deformation: Panel Zone to Beam capacity ratio Focusing now in the local response of the beams, the plastic hinge rotation is illustrated in Figure 5.8. The following can be concluded: • Once again it can be observed that the plastic rotations are beneficially reduced by yielding of the panel zone for the balanced and strong panel. The contribution of the panel is always beneficial • Sub-assemblages with weak panels required very large deformations to imposed significant rotations on the beams. Not relevant 49 Chapter 5. Parametric Studies • For panel zone designs below 80% of the balance thickness, plastic deformations will be controlled exclusively by the panel. No plastic rotations for panel that are 80% or less of the capacity of the beam. 40 30 θ pl [m rad] Rigid 110% 20 100% 10 90% 0 0.0% 80% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.8. Beam Plastic Hinge Rotations: Panel Zone to Beam capacity ratio The plastic hinge length is shown in Figure 5.9: • Consistently with the plastic contribution from the beams, the plastic hinge length reduces as the panel zone is weaker. • Basically yielding gradually spreads along the member according to the plastic participation from the beams. For panel zones below the balance design, the plastic demands are less and hence the plasticity spreads along a shorter length. 25% Rigid 20% 110% 100% 15% 90% Lph/L' 10% 80% 5% 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.9. Plastic Hinge Length: Panel Zone to Beam capacity ratio 50 Chapter 5. Parametric Studies Looking at the cross section of the beam next to the column the curvature ductility is illustrated in Figure 5.10. The following can be observed: • The curvature ductility for the rigid, 110%, 100%, 90% and 80% case are 17.9, 16.3, 12.8, 7.2 and 1.6, respectively. • The maximum demand observed by the beams is the one corresponding to the rigid case. However, this case is unlikely (not say unlikely) to be designed and implemented in a connection. Check… because. • For panel designs below the 80% case, plasticity in the beams is not developed; therefore the ductility in the beam will depend exclusively on the panel zone design. • The range of weak panel zone (80% to 100%) is characterized by significant differences in curvature ductility. It can be observed that this range has an increment of 5.6 units of ductility for each 10% increment in thickness. (Do not quantify, identify trends) • On the other hand, passing from 100% to 110% gives an increment of 3.5 units. Remove 20 Rigid 15 μφ 110% 100% 10 90% 5 0 0.0% 80% Unstregnthen 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.10. Curvature Ductility: Panel Zone to Beam capacity ratio Now looking at the local response of the panel zone, the distortions for the component are shown in Figure 5.11. The following can be observed: • For a drift level of 4% the distortions for the unstrengthen, 80%, 90%, 100% and 110% case are 41.6mrad, 33.3mrad, 26.5mrad, 15.0mrad and 5.0mrad, respectively. (Don’t quantify) 51 Chapter 5. Parametric Studies • A panel zone that is not strengthen can observed high distortions up to (41.6mrad). Excessive distortions can triggered as discussed before - undesirable failure modes other mechanism failure which are not desired, hence a target distortion must be identified and the corresponding panel zone thickness should be designed accordingly. Clearly important that if going to participate is determined • For 3% drift , typical limit Design Guidelines, ductility close to the second yield distortion 50 40 Unstrengthen γ [mrad] 30 80% 90% 20 100% 10 110% 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.11. Panel Distortion: Panel Zone to Beam capacity ratio The panel zone ductility is presented in Figure 5.12. The following can be observed: • The first yield (μγ=1) and second yield (μγ=4) are marked by dotted lines and the post-elastic range is highlighted. Only the 110% case remains in the post-elastic range. • The corresponding distortion ductility for the unstrengthen, 80%, 90%, 100% and 110% case are 21.0, 16.9, 13.5, 7.6 and 2.6, respectively. 52 Chapter 5. Parametric Studies 24 18 Unstrengthen 80% μγ 90% 12 100% 6 μγ=4 110% μγ=1 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.12. Panel Distortion Ductility: Panel Zone to Beam capacity ratio 5.4 Influence of the Beam Span To study the influence of the beams span, the cases shown in Table 5.5 are studied. Introduction based on a balance design. Table 5.5. Cases for Span to Beam Depth ratio Case LDB16 LDB18 CREF LDB22 LDB24 L [m] 6.4 7.2 8.0 8.8 9.6 L/ db 16 18 20 22 24 The pushover curve is presented in Figure 5.13. The following can be concluded: • The span is inversely proportional to the stiffness of the sub-assemblage. • Since all the cases have the same beams, the final strength of the sub-assemblages is the same but is reached at different drift levels. They are not exactly the same • Yielding initiates first for shorter beams. Because the yield drift is inversely proportional to the span, 53 Chapter 5. Parametric Studies 250 200 L/db =16 L/db =18 L/db =20 L/db =22 150 L/db =24 Vcol[kN] 100 50 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.13. Pushover Curve: Beams Span Looking at the local response of the beams, the plastic hinge rotations are presented in Figure 5.14. The following can be observed: • As yielding initiates first for the shorter beams, for levels of drift up to 3%, the plastic rotations are (inversely proportional) to the span. (more general) • Not relevant particularly for high drift levels However, regardless of the beam span, the plastic rotations achieved beyond a drift level of 3% are similar for all the cases. 20 15 θ pl [m rad] 10 L/db =16 L/db =18 5 0 0.0% 0.5% 1.0% 1.5% L/db =20 L/db =22 L/db =24 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.14. Plastic Hinge Rotation: Span to Beam Depth ratio Nonetheless as observed in Figure 5.15, the curvature ductility is different for each case along the deformation of the sub-assemblage. The following can be concluded: 54 Chapter 5. Parametric Studies • For a defined cross section, the slope of the linear bending moment distribution under lateral loading varies according to the span. The gradient of the bending moment distribution for a short beam is higher than for a long beam, therefore the plastic hinge length is directly proportional to the beam span. • Since the plastic rotations are the integration of the curvatures, for given θpl a short span with shorter yield length requires higher curvatures to achieve a given rotation level compared to long spans with longer yield lengths. 16 12 μφ 8 L/db =16 L/db =18 L/db =20 L/db =22 L/db =24 4 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.15. Curvature Ductility: Span to Beam Depth ratio Focusing in the panel zone local response the distortion ductility is presented in Figure 5.16. The following can be observed: • The strength demands observed by the panel for a given level of drift are higher for shorter spans, hence yielding of the panel initiates first. 55 Chapter 5. Parametric Studies 10 8 L/db =16 6 L/db =18 μγ μγ=4 4 L/db =20 L/db =22 L/db =24 2 0 0.0% μγ=1 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.16. Panel Zone Distortion Ductility: Span to Beam Depth ratio • The response of the panel is same for all the deformation but offset according to initiation of yielding. • Justification.. interest to note that the capacity is (Look sheet) 5.5 Influence of the Beam Depth A geometrical parameter that has an influence in the panel zone behaviour is the beam depth. The cases considered to study the variation of this parameter are presented in Table 5.4. To be consistent with the numerical model presented in the previous chapter, the ratio L/db=20 is kept constant. Also the columns are selected to satisfy capacity design criteria of weak beamstrong column. All the panels satisfy a balance design (α=100%). Since the column dimensions are different for all the cases, the balanced thickness and the panel zone thickness to column web thickness ratio are presented. As shown in the table, deep beams required more shear area Av given that the demands observed by the panel are higher. However, since larger column sections are considered the panel zone width is increased and the ratio tpz /tcw reduces with increasing column sections. Table 5.4. Cases for Beam Depth Case BD300 CREF BD500 BD600 db [mm] 300 400 500 600 Beam IPE300 IPE400 IPE500 IPE600 Column HEA260 HEA340 HEA450 HEA550 L [m] 6.0 8.0 10.0 12.0 Av [mm2] 6579 9813 12905 16667 tp z [mm] 27.7 31.3 30.8 32.3 tpz /tcw 3.70 3.30 2.68 2.58 For the overall behaviour of the structure, the different pushover curves are presented in Figure 5.17. The following can be concluded: 56 Chapter 5. Parametric Studies • Because the sub-assemblages have different structural configurations the curves are not comparable. • However the differences in strength and stiffness can be observed. • Deep beams yielding before shallow beams and induce yielding in the panel at an earlier stage. 800 Beam first yield Panel zone first yield IPE600 600 Panel zone second yield Vcol [kN] IPE500 400 IPE400 [CREF] 200 IPE300 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.17. Pushover Curve: Beam Depth After looking at the global response of the sub-assemblage, attention is drawn to the local response of the beams. In Figure 5.18 the plastic hinge rotation of the beams is presented, for which the following conclusions can be extracted: • Before discussing the figure, from mechanics it can be deduce that smaller plastic rotations are expected for deep beams if the beam span is kept constant. This is related to the curvatures of the beam, defined as the difference between the strains divided by the beam depth. The strain is property of the material and flange geometry and therefore the maximum plastic curvature for a deep beam is smaller that for a shallow beam. Since the plastic rotation is the integral of the plastic curvatures for equal spans smaller rotations are expected. • However, higher plastic rotations are observed for deep beams. This is explained because higher beams are related to longer spans, for which larger yield length are expected and hence the plastic rotation integral is larger. 57 Chapter 5. Parametric Studies 20 15 θ pl [mrad] 10 5 IPE600 IPE500 IPE400 IPE300 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.18. Plastic Hinge Rotation: Beam Depth Accordingly the plastic hinge length is presented in Figure 5.19. The following can be concluded: • Since the plastic hinge length is normalized and also the ratio L/db is kept constant, similar values are expected for the different curves. The curves have similar trend but yielding initiates at different stages for the each depth. 25% 20% IPE500 IPE600 15% Lph/L' 10% IPE300 IPE400 5% 0% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.19. Plastic Hinge Length: Beam Depth Looking at the cross section of the beams, the curvatures are presented in Figure 5.20. The following can be observed: • As expected shallow beams observed higher curvatures compared to deep beams. 58 Chapter 5. Parametric Studies 0.12 0.1 IPE300 0.08 IPE500 φ [rad] 0.06 IPE400 [CREF] 0.04 IPE600 0.02 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.20. Curvatures: Beam Depth • However when the curvatures are normalized by the yield curvature, as in Figure 5.21, the curvature ductility plots follow a similar trend apart from the different initiations of yielding. • The final curvatures reached by the different curves is around 12 times the respective yield curvature. 16 12 μφ 8 IPE600 IPE500 4 IPE400 IPE300 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.21. Curvature Ductility: Beam Depth Interest is now focused on the panel zone response, for which the distortion ductility is shown in Figure 5.22. The following can be concluded: • Beams with greater depths induce yielding in the panel before shallow beams. • Final distortions in the panel are higher for deep beams than for shallow beams. 59 Chapter 5. Parametric Studies 10 8 6 IPE600 μγ IPE500 μγ=4 4 IPE400 IPE300 2 μγ=1 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.22. Panel Zone Distortion Ductility: Beam Depth 5.6 Influence of the Panel Zone Second Yield Distortion The panel zone model as discussed previously is defined in terms of stiffness and strength. The first yield point is considered to be well established, although the second strength capacity attributable to the surrounding elements is not clearly specified in terms of distortion. The following analysis intend (pretends) to illustrate the effect of this parameter in the response of the sub-assemblage. Table 5.3. 5.7. Cases for Panel Second Yield Distortion Case n. γy NG3 CREF NG5 3 γy 4 γy 5 γy In term of global response, the second yield distortion is irrelevant as observed in the pushover curve presented in Figure 5.23. 60 Chapter 5. Parametric Studies 250 200 150 Vcol [kN] 100 50 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.23. Pushover Curve: Second Yield Distortion For this parameter more attention is given to the local response of the beams and panel. In Figure 5.24 the plastic hinge rotation of the beams is presented, for which the following conclusions can be extracted: • As soon as the second yield is reached in the panel a relieve in the plastic rotations of the beams is observed. • Try to put the figures together 20 15 NG5 θ pl [mrad] CREF 10 NG3 5 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.24. Plastic Hinge Rotation: Second Yield Distortion Looking at the cross section of the beams, the curvature ductility is presented in Figure 5.25. The following can be observed: • As the panel reaches second yielding the curvature ductility of the beams reduce. 61 Chapter 5. Parametric Studies 16 NG5 12 CREF NG3 μφ 8 4 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.25. Curvature Ductility: Second Yield Distortion Looking specifically the panel zone response, Figure 5.26 presentes the distortion ductility. The following can be concluded: • As the second yield in the panel extends the final distortion ductility reduces. • Quantify the difference around 10% for 3 and 5 cases 10 8 NG3 6 μγ CREF NG5 4 2 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.26 Panel Zone Distortion Ductility: Second Yield Distortion 62 Chapter 5. Parametric Studies 5.7 Influence of Steel Strain Hardening Several parameters can influence the behaviour of a frame in relation to the panel zone. A Table 5.8. Cases for Strain Hardening of Steel Case SH0.5 CREF SH2.0 Strain Hardening μ 0.5% 1.0% 2.0% The global response of the structure for different strain hardening factors is observed in Figure 5.27. The pushover curve present slight variations for the final load factor for different strain hardening levels. 250 μ=2.0% μ=1.0% 200 μ=0.5% 150 Vcol [kN] 100 50 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.27. Pushover Curve: Strain Hardening In Figure 5.18 the plastic hinge rotation of the beams is presented, for which the following conclusions can be extracted: • As the strain hardening factor is lower, the plastic rotations in the beams increase. • The strain hardening influence significantly, the rotaions in the beamsSignificant differences can be expected for strain hardening values below 0.5%, for which the numerical analysis start to become unstable and plastic rotations of the beams tend to indetermined rotations (disproportionate numerical) values. 63 Chapter 5. Parametric Studies 20 μ=0.5% 15 θ pl [mrad] μ=1.0% 10 μ=2.0% 5 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.28. Plastic Hinge Rotation: Strain Hardening Looking at the cross section of the beams, the curvature ductility is presented in Figure 5.29. The following can be observed: • Lower strain hardening values, impose higher curvatures to the beam as a result of a low resistance to the load factor in terms of deformation. 20 15 μ=0.5% μφ μ=1.0% 10 μ=2.0% 5 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.29. Curvature Ductility: Strain Hardening In Figure 5.30 the distortion ductility for the panel zone is presented. The following can be concluded: • Contrary to the beams the panel observes higher distortions for higher strain hardening levels. This is a result from the demands imposed by the beams. 64 Chapter 5. Parametric Studies • Beams develop more plastic rotations and the panels will reduce the plastic distortions. • Higher strain hardening values develop a certain load factor at an earlier stage than lower strain hardening values and hence the panel is submitted to the demands at a later stage where the final distortions are lower. (Not relevant) 10 8 μ=2.0% 6 μγ μ=1.0% μ=0.5% 4 2 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.30. Panel Zone Distortion Ductility: Strain Hardening 5.8 Influence of the Gravity Load Level In order to consider vertical loads in the performance of the joint, the sub-structure was not suitable for the vertical loading purpose. Assuming the contra flexure points is a good approximation for lateral loading condition but not accurate for vertical and lateral loading. Moreover, the bending moment diagram under distributed vertical loads can only be correctly reproduced if the entire beam is modelled. For this reason an extension of the substructure is made to accurately account for vertical loads. The extension of the sub-structure will be denominated the multi-bay model, this model consist of three bays as shown in Figure 5.31. The contra flexure points are imposed only at the mid-height of the columns. The geometrical properties and the analysis procedures are consistent with the sub-assemblage study until now. Vcol Vcol Vcol W Vcol Vcol W Vcol W Vcol Vcol Figure 5.31. Multi-bay Structure 65 Chapter 5. Parametric Studies Different load levels are considered for this case, based on the plastic moment capacity of the beam characterized by the equation shown below. β= w( L − wpz ) 2 12 (5.1) M b , PL Three different values of distributed load w where applied to obtain the cases shown below. Table 5.7. Cases for Gravity Load level Gravity Level β None 25% 50% 75% Case CREF GL25 GL50 GL75 Vertical Loading Balanced W W Vcol Lateral Loading Un-balanced Vertical + Lateral Loading INTERNAL JOINT Figure 5.32. Multi-bay Structure The global response of the structure is represented by the pushover curve in Figure 5.33, from which the following conclusions can be extracted: • As shown in Figure 5.32, the gravity loads at the face of the columns produce negative moments and when lateral loads are applied positive and negative moments are generated. This means that at one side of the joint the moments are added and at the other side the moments are subtracted. Therefore yielding of the beams occurs at different instants. 66 Chapter 5. Parametric Studies • The fact that the beams yield at different load levels is the reason to for the change in slope observed observe the change in gradient of the pushover curves. For higher levels of gravity loads the difference between the two yields increases. • Reduction in the capacity, vertical loads are taking part of the cpacity of the structure. • The stiffness of the multi-bay varies significantly when a mechanism is developed. Higher gravity loads levels induce earlier the plastic hinge in the beams and hence the deterioration of stiffness is observed. 700 600 CREF GL25 500 GL50 400 Vcol [kN] GL75 300 200 100 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.33. Pushover Curve: Gravity Load To understand the way the hinges develops in the beams the curvatures for the different load levels are presented for a drift level of 4% in Figure 5.34. The following can be pointed out: • For the reference case where no vertical load is applied, the hinges form simultaneously and equal curvatures can be observed at both sides of the internal joint. • For the GL25 (all cases) case the beneficial effect from gravity loading can be observed in the left hand side, where curvatures are lower and spread in a wider range. However on the right side, the gravity loads increase. This is consistent with the slop of the bending moment diagram for this region. • For more than 50%, For the GL50 case the effect from gravity loads is more relevant and the plastic hinge shifts from the face of the column and the curvatures are much higher at the opposite side. • For the GL75 the gravity loads influences even more and the plastic hinge develops away from the column face. This has an important effect, since the panel only is subjected (bending moment vertical is dominant and the )to one plastic 67 Chapter 5. Parametric Studies moment but not to two simultaneously. Therefore, the strength for which the panel was design is not reached and the panel does not yield concentrating plasticity in the beams. 10.00 10.00 15.00 15.00 GL25 CREF 10.00 15.00 10.00 GL50 15.00 GL75 Figure 5.34. Curvatures at a drift level of 4% The local response of the beams are analized for an internal joint discriminating between the left and right side of the connection (IntL-IntR). The plastic rotations for the IntL are presented in Figure 5.35. The following can be observed: • The effect from gravity loading significantly increases the plastic rotations in the beams. The final values for a drift level of 4% for CREF, GL25, GL50 and GL75 are 21mrad, 27mrad, 36mrad and 45mrad, respectively. • Contribution from the panel can be observed for the first three cases, where the gradient of the curves decrease. 68 Chapter 5. Parametric Studies 50 40 GL75 θ pl [mrad] 30 GL50 GL25 20 CREF 10 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.35. Plastic Hinge Rotations IntL: Gravity Load Observing the plastic rotations for the right side of the internal joint (IntR) in Figure 5.36, the following is observed: • As mentioned before yielding at the face of the column occur for cases CREF and GL25, and apart from the column face for cases GL50 and GL75. For the first two cases the beneficial effect from the gravity loadin can be observed since GL25 observes lower plastic rotations than CREF. • A different effect is observed for the other two cases, where the higher gravity loads GL75 exhibit larger plastic rotations than GL50. This is cause because the plastic hinge length for the GL75 case is larger than GL50 and hence the integration of the curvatures is calculated in a wider range. 20 15 θ pl [mrad] GL75 10 GL50 CREF 5 GL25 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.34. Plastic Hinge Rotations IntR: Gravity Load 69 Chapter 5. Parametric Studies Observing now the curvature ductility of the beams for the left side in Figure 5.35, the following can be observed: • As expected the curvature ductility are higher for the higher gravity load levels. Significant variations can be observed when the bending moments from vertical loading are considerable for the cross section moment capacity. 40 GL75 30 GL50 μ φ 20 GL25 10 0 0.0% CREF 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.35. Curvature Ductility L2: Gravity Load (thick cref) For the right side of the joint the curvature ductility is presented in Figure 5.36. The following is observed: • Two factors are relevant for the reductions of the curvature ductility. First, the reduction in the bending moment produced by the vertical loading causes that yielding initiates at a later stage, hence lower demands are imposed. Second, is that the moment diagram gradient is lower and allows spread of plasticity in a larger length, therefore curvature demands reduce to achieve certain plastic rotations. 70 Chapter 5. Parametric Studies 16 12 CREF μφ 8 GL25 4 GL75 GL50 0 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.36. Curvature Ductility R2: Gravity Load Attention is now focused on the panel zone distortion ductility. Three different cases are examined the external joints refer as PZ1 and PZ4 and the internal joints refer as PZ2-PZ3. Figure 5.37 presents the distortion ductility for PZ1. The following can be observed: • Only cases CREF and GL25 reach the panel zone strength, however at a later stage and the panel does not reach the second yield strength. • The other two cases GL50 and GL75 do not develop the yield strength of the panel, since the plastic hinge is not the column face. 10 8 6 μγ 4 μγ=4 CREF 2 0 0.0% μγ=1 GL25 GL50 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% GL75 3.5% 4.0% Drift Figure 5.37. Distortion Ductility PZ1: Gravity Load Observing the distortion ductility for the internal joint in Figure 5.38, the following can be extracted: 71 Chapter 5. Parametric Studies • The GL75 does not develop plastic moments at both boundaries of the panel zone, therefore yielding in the panel does not occur. • For the CREF and GL25, the response is similar, given that the panel observes the plastic moment from the beams, even though the plastic distributions are different, • For the GL50 case, yielding of the panel only occurs for a drift level of 2.8% and therefore second yield strength is not reached. 10 8 6 μγ 4 μγ=4 GL25 CREF 2 0 0.0% GL50 μγ=1 0.5% GL75 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Figure 5.38. Distortion Ductility PZ2-PZ3: Gravity Load Finally the external panel PZ4 is revise, for which the distortions ductility is presented in Figure 5.39. The following can be observed: • Since the plastic moments at the column face are develop completely, the panel yields for all the cases. • Only cases GL50 and GL75 reach strain hardening range. Cases CREF and GL25 remain in the post elastic range. • Higher distortions are observed for higher levels of gravity loads. 72 Chapter 5. Parametric Studies 10 8 GL75 6 GL50 μγ 4 2 0 0.0% μγ=4 GL25 CREF μγ=1 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% Drift Top Node [%] Figure 5.34. Distortion Ductility PZ4: Gravity Load 73 Chapter 6. Conclusions and Future Research 6. CONCLUSIONS AND FUTURE RESEARCH 6.1 Summary and Conclusions The research carried out and presented in this dissertation aimed at investigating the influence of the web panel zone on the seismic behaviour of steel moment-resisting frames. The numerical studies undertaken clearly illustrated the importance of this component and the need for its consideration in both the analysis and design stages. From the results obtained, the following conclusions can be drawn: • The panel zone can have a significant influence on the stiffness and capacity of a frame, particularly when its yield strength is well below the plastic capacity of the connecting beams; • Panel zones designed to achieve simultaneous yielding with the adjacent beams (‘balanced design’) can lead to important reductions on the beam plastic rotational demands; • For the cases studied, panel zones with a capacity below 80% of the connecting beams, resulted in elastic response of the beams; • The strain-hardening level of the steel can have a significant influence on the ductility demands of panels and beams when simultaneous yielding is considered. Low values of strain-hardening (around 0.5%) increase significantly the concentration of plasticity in the beams and consequently reduce the ductility demand imposed on the panels; • When gravity loads are considered in the analysis, the ductility demands imposed on the beams can be significantly higher comparing to those observed when these loads are not included. Adequately designed panel zones can attenuate these extreme demands; • High levels of vertical loads can even shift the plastic hinges away from the column face. As a result, the concept of simultaneous yielding of panel zones and beams may be more difficult to be achieved using simplified design criteria; • However, the ‘balanced design’ criterion does not seem to be accurate for panel zones in realistic moment frames. The shift of contra-flexure points in the beams and 74 Chapter 6. Conclusions and Future Research columns leads to a reduction of the shear demand on internal panels and to an increase in the external ones. This effect needs to be accounted for in future design criteria; • Panel zones which are weaker comparing to the capacity of the adjacent beams can observe extremely high distortional demands. As identified in previous studies these deformations are likely to cause failure in other components of the connection, particularly the welds. This indicates that any criterion for panel zone design should ensure and/or impose control of the maximum panel distortion; 6.2 Recommendations for Future Research The research conducted provided a clear insight into the behaviour of the panel zone in moment-resisting frames and indicated the need to carry out additional research in the topic. Future research may focus on the following topics: • Establish a more accurate design criteria for the panel zone that considers near ‘simultaneous’ yielding between panel zones and beams but that also incorporates criteria to limit the ductility demand in the panel; • Additional testing of welded beam-to-column connections to identify reliable limits for panel zone distortions; • Perform extensive parametric studies to examine the structural parameters that have an influence on panel zone shear demands. 75 References REFERENCES Bertero VV, Popov EP, Krawinkler H. [1972] “Beam-Column Subassemblages under Repeated Loading”, Journal of the Structural Division; ST5: 1137-1159. Castro, J.M., Elghazouli, A. Y., Izzuddin, B.A. [2005] “Modelling of the panel zone in steel and composite moment frames”, Journal of Engineering Structures 27; 129-144. Ciutina AL, Dubina D. [2006] “Seismic behaviour of steel beam-to-column joints with column web stiffening”, Steel and Composite Structures; Vol. 6, No 6: 493-512. Dubina D, Ciutina A, Stratan A. [2001] “Cyclic Test of Double-Sided Beam-To-Column Joints”, Journal of Structural Engineering, ASCE; 127(2):129–136. El-Tawil S, Vidarsson E, Mikesell T, Kunnath SK. [1999] “Inelastic behavior and design of steel panel zones”, Journal of Structural Engineering, ASCE;125(2):183–193. El-Tawil S. [2000] “Panel Zone Yielding in Steel Moment Connections”, Engineering Journal, AISC; Third quarter:120–131. Englekirk RE. [1999] “Extant Panel Zone Design Procedures for Steel Frames are Questioned”, Earthquake Spectra; Vol. 15, No. 2: 361-369. Fielding DJ, Huang JS [1971] “Shear in steel beam-to-column connections”, Welding Journal, 50(7):S313–26 [research supplement]. Foutch DA, Yun SY. [2002] “Modelling of steel moment frames for seismic loads”, Journal of Constructional Steel Research; 58(5–8):529–64. Jun Jin, El-Tawil S. [2005] “Evaluation of FEMA-350 seismic provisions for steel panel zones”, Journal of Structural Engineering, ASCE; 131(2):250–58. Kim KD, Engelhardt MD. [2002] “Monotonic and cyclic loading models for panel zones in steel moment frames”, Journal of Constructional Steel Research; 58(5–8):605–35. Krawinkler H. [1971] “Inelastic Behavior of Steel Beam-to-Column Subassemblages”, Earthquake Engineering Research Center Report No. EERC 71-7. Krawinkler H. [1978] “Shear in beam-column joints in seismic design of steel frames”, Engineering Journal, AISC; 15(3), 82-91. Krawinkler H, Bertero V. V. and Popov E. P. [1975]. “Shear Behavior of Steel Frame Joints”, Journal of the Structural Division, ASCE, Vol. 101, No. ST11. 76 References Lee CH, Jeon SW, Kim JH, Uang CH. [2005] “Effects of Panel Zone Strength and Beam Web Connection Method on Seismic Performance of Reduced Beam Section Steel Moment Connections”, Journal of Structural Engineering, ASCE; 131(12):1854–1865. Lee D, Cotton SC, Hajjar JF, Dexter RJ, Ye Y. [2005] “Cyclic Behavior of Steel Moment-Resisting Connections Reinforced by Alternative Column Stiffener Details I. Connection Performance and Continuity Plate Detailing”, Engineering Journal; Vol. 42 No. 4 :189–214. Lee D, Cotton SC, Hajjar JF, Dexter RJ, Ye Y. [2005] “Cyclic Behavior of Steel Moment-Resisting Connections Reinforced by Alternative Column Stiffener Details II. Panel Zone Behavior and Doubler Plate Detailing”, Engineering Journal, AISC; Vol. 42 No. 4 :215–238. Schneider SP, Amidi A. [1998] “Seismic behavior of steel frames with deformable panel zones”, Journal of Structural Engineering, ASCE; 124(1):35–42. Castro, J.M.F. [2006] “Seismic Behaviour of Composite Moment-Resisting Frames”, PhD Thesis, Imperial College London, University of London, United Kingdom. Bruneau, M., Uang, C.M., Whittaker, A. [1998] Ductile Design of Steel Structures, McGraw-Hill, New York, USA. 77 Appendix A APPENDIX A A1 Appendix A A.1 OpenSees Input File The software is composed of several tools. Following a description of a typical input file shown to describe the different features of the program used in the study. The descriptions are extracted from the OpenSees Command Language Manual (2006). (a) Basic model builder. The model builder constructs as in any finite element analysis, the analyst's first step is to subdivide the body being studied into elements and nodes, to define loads acting on the elements and nodes, and to define constraints acting on the nodes. The model builder is the object in the program responsible for building the following objects in the model and adding them to the domain: nodes, masses, materials, sections, elements, load patterns, time series, transformations, blocks and constraints. (b) Constraints. A constraint either prescribes the value of a DOF (as in imposing a support condition) or prescribes a relationship among DOF. In common terminology, a single-point constraint sets a single DOF to a known value (often zero) and a multi-point constraint imposes a relationship between two or more DOF. For example, support conditions on a three-bar truss invoke single-point constraints, while rigid links and rigid elements each invoke a multi-point constraint. (c) Plain pattern. This command is used to construct an ordinary load pattern object in the domain. (d) Recorder. The recorder command is used to define the analysis output. The node recorder will be used to output the horizontal and vertical displacements at node 3 into a file named Node3.out. The element recorder will be used to output the element forces. Element forces for element 1 will be output into file Element1.out. (i) Plain Constraints. This command creates a plain handler which is only capable of enforcing homogeneous single-point constraints. If other types of constraints exist in the domain, a different constraint handler must be specified. The command is used to enforce homogeneous single-point constraints, such as the case of homogeneous boundary conditions, where all boundary conditions are fixity, using single-point constraints (e) Numberer Command. This command is used to construct a plain numberer object. The plain numberer assigns degrees-of-freedom to the nodes based on how the nodes are stored in the domain. Currently, the user has no control over how nodes are stored. This command is used to construct the DOF_Numberer object. The DOF_Numberer object determines the mapping between equation numbers and degrees-of-freedom -- how degreesof-freedom are numbered. A2 Appendix A Plain nodes are assigned degrees-of-freedom arbitrarily, based on the input file. This method is recommended for small problems or when sparse solvers are used, as they do their own internal DOF numbering. RCM -- nodes are assigned degrees-of-freedom using the Reverse Cuthill-McKee algorithm. This algorithm optimizes node numbering to reduce bandwidth using a numbering graph. This method will output a warning when the structure is disconnected. (f) Band SPD. This command is used to construct a symmetric positive definite banded system of equations object which will be factored and solved during the analysis using the Lapack band spd solver. (gl) Norm Displacement Increment Test. This command is used to construct a CTestNormDispIncr object which tests positive force convergence if the 2-norm of the x vector (the displacement increment) in the LinearSOE object is less than the specified tolerance. (k) Newton Algorithm. This command is used to construct a NewtonRaphson algorithm object which uses the Newton-Raphson method to advance to the next time step (m) Displacement Control. This command is used to construct a StaticIntegrator object of the type DisplacementControl The displacement increment at iterations i, dU(i), is related to the displacement increment at (i-1), dU(i-1), and the number of iterations at (i-1), J(i-1), by the following: dU(i) = dU(i1)*Jd/J(i-1) A3