NUMERICAL PERFORMANCE EVALUATION OF BRACED FRAME SYSTEMS. Ingvar Rafn Gunnarsson

Transcription

1 NUMERICAL PERFORMANCE EVALUATION OF BRACED FRAME SYSTEMS Ingvar Rafn Gunnarsson A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering University of Washington 24 Program Authorized to Offer Degree: Department of Civil & Environmental Engineering

2 University of Washington Graduate School This is to certify that I have examined this copy of a master s thesis by Ingvar Rafn Gunnarsson and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Committee Members: Dawn E. Lehman Charles W. Roeder Greg R. Miller Date:

3 In presenting this thesis in partial fulfillment of the requirements for a master s degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this thesis is allowable only for scholarly purposes, consistent with fair use as prescribed in the U.S. Copyright Law. Any other reproduction for any purposes or by any means shall not be allowed without my written permission. Signature Date

4 University of Washington Abstract NUMERICAL PERFORMANCE EVALUATION OF BRACED FRAME SYSTEMS Ingvar Rafn Gunnarsson Chair of Supervisory Committee: Assistant Professor Dawn E. Lehman Department of Civil & Environmental Engineering The use of braced frames has become more popular in seismic design of buildings. This research focuses on the performance of Concentrically Braced Frames both Special Conventional Braced Frames (SCBF) and Buckling Restrained Braced Frames (BRBF). The objective of this research was to create a practical model for the seismic analysis of braced steel frames. The model used beam-column elements for braces and framing elements and lumped rotational springs for the connections rather than shell elements. A model of an isolated brace was developed and validated to the predicted response of design provisions and previous experimental results. The cyclic response of a single story one bay SCBF system was compared to the response of test frames. The effect of brace slenderness of SCBF systems on the response was calculated and compared to the response of a BRBF system due to seismic loading. The response of two types of three story braced frames was calculated. The resistance of the gussetplate connections was modeled using elastic springs resulting in improved global behavior of the system.

5 The developed brace model simulates the response of a single brace well compared to past experiments and theoretical predictions. When the brace is included in a frame the simulated response did not match the experimental results as well. It is believed that the discrepancy is due to frame action and nonlinear connection behavior that was seen in the experiments but not accurately simulated in the model.

6 Table of Contents Page List of Figures... iii List of Tables...vii Chapter 1: Introduction Introduction Research overview Thesis organization...3 Chapter 2: Analytical procedure Introduction Analytical software Element formulation Constitutive laws System equation solvers Convergence Connection stiffness Rotational stiffness of the frame Rotational stiffness of the brace gusset-plate connection...15 Chapter 3: Validation of inelastic models Introduction Modeling buckling behavior Convergence Comparison to the AISC buckling curve Braces with fixed ends Post buckling behavior of braces Braced frame behavior Modeling of frames Comparison between analytical and experimental results...31 Chapter 4: Cyclic response Introduction Behavior of brace Braced frame behavior UW test Description of experiments Model details Analytical results Summary...62 i

7 Chapter 5: Seismic response of braces Introduction Ground motions Response of one story model Description of models Response resulting from El Centro ground motion Response resulting from Newhall 9 ground motion Effects of slenderness...84 Chapter 6: Three story building Introduction Design loading Braced frame systems IVBF system Design Model Results XBF system Design Model Results Chapter 7: Conclusions Introduction Summary Conclusions Modeling of brace Modeling of systems Modeling of dynamic response Recommendation for future research Bibliography Appendix A: Modeling using OpenSees A.1 Model generation A.2 Analysis generation A.3 Recorder generation A.4 Analysis execution Pocket Material: CD-ROM ii

8 List of Figures Figure Number Page 2.1: Example of the cross section of W-sections : Example of cross section of HSS-section : Steel1 constitutive law : Steel2 constitutive law : Model configuration for rotational stiffness of frame and dimensions used : Model configuration for rotational stiffness of brace gusset plate connection : Layout of braces with pinned and fixed ends : Buckling load results for HSS5x5x3/8 brace with various numbers of elm : Buckling curves for HSS5x5x3/8 tube section for various imperfections : Buckling curves for W4x13 section for various imperfections : Comparison between program and AISC for HSS5x5x3/8 member : Buckling load results for HSS5x5x3/8 brace with various numbers of elm : Comparison between program and AISC for HSS5x5x3/8 member : Behavior of HSS5x5x3/8 brace under monotonic loading : The frame analyzed : Initial imperfection of Wakabayashi braced frame : Experimental results for braced frame without vertical loading : OpenSees results for braced frame without vertical loading : Experimental results for braced frame with vertical loading : OpenSees results for braced frame with vertical loading : Experimental test results of tubular strut : Details of OpenSees model : OpenSees model results of tubular strut : OpenSees model results of tubular strut using Steel2 constitutive law : Dimensions and members of test frames : Dimensions of gusset-plates for Frames HSS1 and HSS : Shear tab beam-column connections details : Measured displacement history of tests : HSS1 test results : HSS2 test results : Applied load and boundary conditions of model : Fiber details : Modeling of gusset-plate connections : Schematic of connection details of Cases A to E : Values of α relative to gusset-plates of frames HSS1 and HSS : HSS1 Case A results : HSS1 Case B results : HSS1 Case C results...51 iii

9 4.19: HSS1 Case D results : HSS1 Case E results : HSS1 Case C results with double shear tab connections rotational stiffness : HSS1 Case C results with Steel2 constitutive law : HSS1 Case C results with Steel2 constitutive law and α = : HSS1 Case E results with Steel2 constitutive law : HSS2 Case A results : HSS2 Case B results : HSS2 Case C results : HSS2 Case D results : HSS2 Case C results with Steel2 constitutive law and α = : El Centro acceleration normalized to gravity : Newhall 9 acceleration normalized to gravity : Acceleration response spectra for El Centro and Newhall : One story frame : Response of X1-A-1 frame due to El Centro : Deflected shape of X1-A-1 frame for positive story drift : Deflected shape of X1-A-1 frame for negative story drift : Force displacement response of the X1-A-1 braces : Response of X1-B frame due to El Centro : Force displacement response of the X1-B braces : Response of X1-C frame due to El Centro : Story drift of the frame for the three cases due to El Centro : Response of X1-A-1 frame due to Newhall : Deflected shape of the X1-A-1 frame for positive story drift : Deflected shape of the X1-A-1 frame for negative story drift : Force displacement response of the X1-A-1 braces : Response of X1-B frame due to Newhall : Force displacement response of the X1-B braces : Response of X1-C frame due to Newhall : Story drift of the frame for the three cases due to Newhall : Response for braces with various slenderness parameters, due to El Centro : Response for braces with various slenderness parameters, due to Newhall : Floor plans and elevations for 3 story SAC building : Analyzed frames : Forces in braces due to equivalent lateral earthquake forces : Forces acting on columns 1 through : Forces acting on beams 1 through : Initial imperfection of IVBF frame : Forces and masses acting on the model : Boundary conditions : Lateral roof displacement of IVBF due to El Centro : Deformed shape at positive locations of IVBF due to El Centro : Deformed shape at negative locations of IVBF due to El Centro...14 iv

10 6.12: Force-displacements plots for the braces of IVBF due to El Centro : Force-displacements plots for the beams of IVBF due to El Centro : Force-displacements plots for the columns of IVBF due to El Centro : Story drifts for the IVBF system due to El Centro : Lateral roof displacement of IVBF - BRBF due to El Centro : Deformed shape at positive locations of IVBF - BRBF due to El Centro : Deformed shape at negative locations of IVBF - BRBF due to El Centro : Force-displacements plots for the braces of IVBF - BRBF due to El Centro : Force-displacements plots for the beams of IVBF - BRBF due to El Centro : Force-displacements plots for the columns of IVBF BRBF due to El Centro : Story drifts of IVBF - BRBF due to El Centro : Lateral roof displacement of IVBF due to Newhall : Deformed shape at positive locations of IVBF due to Newhall : Deformed shape at negative locations of IVBF due to Newhall : Force-displacements plots for the braces of IVBF due to Newhall : Force-displacements plots for the beams of IVBF due to Newhall : Force-displacements plots for the columns of IVBF due to Newhall : Story drifts of IVBF due to Newhall : Lateral roof displacement of IVBF BRBF due to Newhall : Deformed shape at positive locations of IVBF BRBF due to Newhall : Deformed shape at negative locations of IVBF-BRBF due to Newhall : Force-displacements plots for the braces of IVBF BRBF due to Newhall : Force-displacements plots for the beams of IVBF BRBF due to Newhall : Force-displacements plots for the columns of IVBF BRBF due to Newhall : Story drifts for the of IVBF BRBF due to Newhall : Forces in braces due to equivalent lateral force : Initial imperfection : Forces and masses acting on the model : Boundary conditions : Lateral roof displacement of XBF due to El Centro : Deformed shape at positive locations of XBF due to El Centro : Deformed shape at negative locations of XBF due to El Centro : Force-displacements plots for the braces of XBF due to El Centro : Force-displacements plots for the beams of XBF due to El Centro : Force-displacements plots for the columns of XBF due to El Centro : Story drifts of XBF due to El Centro : Lateral roof displacement of XBF BRBF due to El Centro : Deformed shape at positive locations of XBF BRBF due to El Centro : Deformed shape at negative locations of XBF BRBF due to El Centro : Force-displacements plots for the braces of XBF - BRBF due to El Centro : Force-displacements plots for the beams of XBF BRBF due to El Centro : Force-displacements plots for the columns of XBF BRBF due to El Centro : Story drifts of XBF - BRBF due to El Centro : Lateral roof displacement of XBF due Newhall v

11 6.56: Deformed shape at positive locations of XBF due Newhall : Deformed shape at negative locations of XBF due Newhall : Force-displacements plots for the braces of XBF due Newhall : Force-displacements plots for the beams of XBF due Newhall : Force-displacements plots for the columns of XBF due Newhall : Story drifts of XBF due Newhall : Lateral roof displacement of XBF BRBF due Newhall : Deformed shape at positive locations of XBF BRBF due Newhall : Deformed shape at negative locations of XBF BRBF due Newhall : Force-displacements plots for the braces of XBF BRBF due Newhall : Force-displacements plots for the beams of XBF BRBF due Newhall : Force-displacements plots for the columns of XBF BRBF due Newhall : Story drifts of XBF - BRBF due Newhall A.1: Coordinates of points A.2: Time increments for 1 second vi

12 List of Tables Table Number Page 3.1: Name and properties of braces : Name and properties of HSS5x5x3/8 braces : Name and properties of W4x13 braces : Capacity of brace : Wakabayashi test member dimension : Material properties of the test frames : Dimensions of members : Rotational stiffness for HSS1 and HSS : Connection types of Cases A to E : Summary of HSS1 results for Cases A through E : Summary of HSS1 results for refined Case C model : Summary of HSS2 results : Details of models : Cross-sectional properties of braces : Natural periods of the systems : Loading on 3-story frame : Seismic mass : Vertical distribution of seismic forces : Load on columns : Moment load on beams : Required strength of columns : Summarization of member selection : Dimensions of members used : Summary of IVBF system response due to El Centro ground motion : Summary of IVBF system response due to Newhall 9 ground motion : Summarization of member selection : Dimensions of members used : Summary of XBF system response due to El Centro ground motion : Summary of XBF system response due to Newhall ground motion vii

13 Acknowledgements The author would like to gratefully acknowledge the Valle Scholarship and Scandinavian Exchange Program for their great support during the study at the University of Washington. Professors Charles Roeder and Dawn Lehman are highly acknowledged for their input and knowledge. Professor Greg Miller is also thanked for serving on the defense and committee reviewing this thesis. Faculty and staff at the Department of Civil and Environmental Engineering are also acknowledged for making the study at the University of Washington great and very memorable. viii

14 1 Chapter 1: Introduction 1.1 Introduction Braced frames are a popular system to resist seismic loading. Because the design and fabrication of braced frames is easier than of moment resistant frames, and more complex and more stringent guidelines have been introduced to the design of Special Moment Resisting Frames (SMRF) after the 1994 Northridge earthquake. The research in this report focuses on the response of Concentrically Braced Frames both Special Concentrically Braced Frames (SCBFs) and Buckling Restrained Braced Frames (BRBFs). The inelastic behavior of SCBF systems is dominated by brace buckling, yielding and the post buckling behavior. Inelastic performance of the brace is nonsymmetrical, because of difference in the tensile and compressive strength of the brace, and the deterioration in the resistance after buckling. The braces of BRBF systems are encased in either tube or other structural element to prevent brace buckling, which results in equal tensile and compression resistances and symmetric response. However braces are commonly considered to be trusses in design. Actual connections do not respond as pinned joints; the braces are connected to the beams and columns using gusset-plates. The use of gusset-plates results in increase in the flexural stiffness of the connection and a reduction in the effective length of the brace. Both reducing the effective length and increasing the connections stiffness increases the buckling capacity of the brace. It also influences the behavior of the system and both truss and frame action must be considered.

15 1.2 Research overview The objective of this research was to create a practical model for the seismic analysis of a steel braced frame to accurately predict the nonlinear response. Models were developed using beam-column elements for the braces and frame members, rather than using more detailed model consisting of shell elements and three dimensional meshes. Nonlinear analytical models were developed to model the behavior of the SCBF system under both monotonic and cyclic loading. To model the buckling response, the brace was subdivided into multiple nonlinear beam-column elements and given an initial imperfection. The number of elements and the magnitude of the initial imperfection were optimized. Nonlinear constitutive law was used to represent the steel and nonlinear geometric effects were included at the element level. The models were verified using previous experimental results. The model simulated both postbuckling and cyclic behavior. Beam-column elements were used for the framing elements without an initial imperfection. The stiffness of the gusset-plate connections was modeled using a combination of elastic rotational springs and rigid end zones. The flexibility of shear tab connections was simulated using elastic rotational springs. Previous experimental results and the AISC design provisions were used for the verification of brace response. The cyclic response of a single story braced frames were verified using experimental results. No experimental data was available to verify the response of the three story braced frames. 2

16 1.3 Thesis organization This thesis is divided into 4 main parts, development of analytical procedure, monotonic response, cyclic response and seismic response. Chapter 2 discusses the selection of analytical software, elements, constitutive law, solvers and connections stiffness. Chapter 3 focuses on simulating the response of braces subjected to monotonic loading and a single story braced frame. Chapter 4 covers the development of models for a simple brace and single story braced frame where the connections details are modeled using provisions discussed in Chapter 2. The cyclic response is compared to experimental results. Chapters 5 and 6 cover the modeling of braced frames subjected to seismic loading. Chapter 5 covers the response of a single story frame with various brace properties and Chapter 6 covers the response of three story braced frames. Conclusions are summarized in Chapter 7 as well as recommendations for further research are listed. Appendix A discusses the modeling using OpenSees. OpenSees scripts and Matlab code is provided as pocket material on a CD-ROM. 3

17 4 Chapter 2: Analytical procedure 2.1 Introduction This chapter will discuss the analytical procedure of the research, the first step was to find and select an analysis program that would meet the requirements of the research program. This program would have to be widely available, be able to perform cyclic nonlinear analysis using the Finite Element Method (FEM) and include elements capable of modeling nonlinear geometric effects. When the program had been selected the next step was to research the various components of the modeling, including the appropriate elements, material properties, geometric formulation, convergence criteria and connection details. This chapter will first discuss the selection of the program, then in the following order, the elements, constitutive law, available solvers, convergence criteria and finally the modeling of the connections. 2.2 Analytical software After some discussion and research it was decided to use OpenSees a software framework for developing applications to simulate the performance of structural and geotechnical systems subjected to earthquakes [Website, OpenSees]. OpenSees is intended to serve as a computational platform for research in performance-based earthquake engineering at PEER (Pacific Earthquake Engineering Research center). OpenSees has also been selected as the official program of the National Earthquake

18 5 Engineering Simulation (NEES) program, and so it is expected to have increasing acceptance and development. OpenSees has built in nonlinear beam-column elements capable of modeling nonlinear behavior (Chapter 2.3). OpenSees has a couple of available nonlinear analysis algorithms including: Newton algorithm, modified Newton algorithm, Newton line search algorithm and Krylov-Newton algorithm (Chapter 2.5). OpenSees is under constant development, sponsored by PEER, which goal is to improve the modeling and computational simulation through open-source development. The idea behind open-source is to allow programmers on the internet to read, redistribute and modify the source code of the software [Website, Open Source]. This both speeds the development of the software and makes it better than traditional closed model. Because OpenSees is an open-source program future research of braced frame would have the possibility to develop and implement appropriate elements and constitutive laws for application to braced frame system. Although the capabilities discussed above are attractive, OpenSees is not a perfect program. The capabilities of OpenSees are constantly growing, current capabilities can be broken down to modeling, analysis and structural reliability [Website, OpenSees]. In the author s opinion the main disadvantages are the lack of graphical user interface (GUI), difficulties in obtaining system equation to converge and lack of documentation. According to the OpenSees website some GUI is under development. One GUI is available on the website but no responsibility is taken for this accessory modifying

19 6 certain operating system files, thus the author was unable to test this GUI. Matlab code was developed to plot the various results, including the deformed shape of the frame and force-displacements plots [Program, Matlab]. Most of the time it was possible to overcome the convergence issue by using smaller increment for the force or the time. Sometimes it was impossible to overcome the convergence problems without either changing the model or making some assumptions to simplify it. The lack of documentation about the various components of OpenSees was also a considerable problem while doing the research. It is expected that that future developments of the program will produce more and better documentation. 2.3 Element formulation The modeling of the frame required the appropriate choice of elements. For the beams and columns OpenSees has a selection of three beam-column elements to model nonlinear beam-column behavior. One model is based on displacement/stiffness method with distributed plasticity, and two others are based force/flexibility method that is one with distributed plasticity and one with concentrated plasticity with elastic interior. It was decided to use force/flexibility based nonlinear beam-column elements with 4 integration points along the length of each element. This element is based on the non-iterative (or iterative) force formulation, and considers the spread of plasticity along the element [Website, OpenSees]. Studies have shown that these elements, which are based on force interpolation functions inside the elements, are better suited to

20 7 describe the nonlinear behavior of structural members. This is particularly true when the element is under conditions of strain softening [Spacone, 1996]. Irrespective of the geometry and the constitutive law of the beam element, these interpolation functions represent the exact solution to the governing equations [Neuenhofer, 1997]. It is stated that the integration along the element is based on Gauss-Lobatto quadrature rule. Quadrature is a name used for numerical integration; it operates by evaluating the function at specific points, multiplying the resulting number by an appropriate weighting factor and adding results [Cook, 22]. When Gauss quadrature is applied to integrate stiffness matrices, each coefficient of the matrix is a function that has to be integrated over the element [Cook, 22]. Gauss quadrature locates sampling points and assigns weights so as to minimize integration error making it possible to use fewer sampling points to acquire some level of accuracy [Cook, 22]. Gauss-Lobatto quadrature rule is a modification of the Gauss quadrature rule that uses the ends of the element as sampling points. As the BRBF braces are designed not to buckle, ideally it is possible to model them using a single truss element. Truss elements only transfer axial loads, and thus no moment can be induced from them to the nonlinear beam-column elements used to model the beams and columns. The cross section of all the members could be created either by defining the force deformation quantities (Force-deflection, Moment-curvature) or by using fiber sections. It was decided to create the sections using fiber sections, which enabled the creation of the various steel cross sections, i.e. W-sections and HSS-sections. The fiber

21 8 sections enable the program to calculate the various force deformation quantities based on the geometry of the cross-section, the constitutive law and the strain distribution in the members by making the assumption that plane sections remain plane. The W-section was formed from three quadrilateral regions, i.e. both flanges and the web. Then these regions were divided into fibers. This was done relative to the expected direction of buckling/bending, i.e. more fibers were used in the expected direction of buckling/bending, see Figure 2.1. By dividing for example each region of a W16x45 into 1x2 fibers if it is anticipated to buckle/bend in the strong axis direction each fiber in the flange will be 3.52 in by.565 in and each fiber in the web will be 1.61 in by.173 in. Regions Strong Axis Weak Axis Figure 2.1: Example of the cross section of W-sections Similarly the HSS-section was formed from eight quadrilateral regions. This was done to have continuity between all the regions. These regions were also divided into fibers relative to the direction of buckling/bending, see Figure 2.2.

22 9 Regions Tube Figure 2.2: Example of cross section of HSS-section A geometric transformation is necessary to transform beam element stiffness and resisting force from the local coordinate system of the elements to the global coordinate system of the frame. OpenSees has a selection of three geometric transformations: linear, P-delta and corotational. For beams and columns P-Delta Coordinate Transformation formulation was used. P-delta geometric transformation considers the second order P-delta effects when transferring from the local coordinate system to the local coordinate system. A Corotational Coordinate Transformation formulation was used to model the geometric-transformation of the braces. When using corotational coordinate transformation a local coordinate system is attached to each element, and that coordinate system rotates and translates with the element in the deformation process while the global coordinate system remains fixed [Cook, 22]. Element deformation is decomposed into rigid-body component and straining component, this enables the components to be addressed independently [Cook, 22]. This transformation includes large displacement relationships which are necessary to model the post-buckling

23 behavior. This formulation also made it possible to model the brace under cyclic loading Constitutive laws OpenSees has two types of uniaxial constitutive laws to model the behavior of steel under cyclic loading. Steel1 models the behavior of the steel using bilinear stress-strain curves, and Steel2 is based on Giuffré-Menegotto-Pinto model [Fedeas]. Steel1 model was used as the default constitutive law as it is the most common constitutive law to model the nonlinear behavior of steel. For most of the analysis the uniaxial bilinear steel constitutive law with kinematic strain hardening was used to model the material behavior. Figure 2.3 shows the material parameters of this constitutive law where f y is the yield stress, E is the modulus of elasticity and b is the strain hardening ratio. Figure 2.3 also shows the hysteric behavior of this constitutive law. σ f y 1 be σ 1 E ε f y /E ε Figure 2.3: Steel1 constitutive law

24 Because this constitutive law is bilinear there is no limit to the tension or compression strength or the ductility. In reality steel has limited ductility at which it fractures. Future research might be able to develop better constitutive law to improve the model of the braces. The Giuffré-Menegotto-Pinto constitutive law is similar to the bilinear constitutive law except the curve is rounded were the yielding occurs, see Figure 2.4. The material parameters of this constitutive law are the yield stress, f y, the modulus of elasticity, E, the strain hardening ratio, b, an exponent that controls the transition between elastic and hardening branch, R, and parameters for the change of R with cyclic loading history, c 1 and c σ f y 1 be σ bigger R 1 E ε f /E ε Figure 2.4: Steel2 constitutive law As for the Steel1 constitutive law, Steel2 has no limit to the tension or compression strength.

25 2.5 System equation solvers OpenSees has several available solvers to solve the nonlinear system equation including: Newton algorithm, Newton line search algorithm, modified Newton algorithm and Krylov-Newton algorithm. The Newton algorithm uses Newton-Raphson method to advance to the next step, by updating the tangent stiffness each time. The modified Newton method works almost the same except the tangent stiffness is only updated once for every load/time increment. The Newton line search method is based on the Newton-Raphson method and uses line search to advance to the next time step. It was decided to use the Krylov-Newton algorithm to solve the system equation because it was the currently fastest and most reliable algorithm available in the OpenSees framework. This algorithm is based on the modified Newton-Raphson algorithm and uses Krylov subspace acceleration to advance to the next time step [Website, OpenSees; Carlson, 1998] Convergence OpenSees has three available convergence tests, Norm Unbalance, Norm Displacement Increment and Energy Increment. It was decided to use the Energy Increment test which tests positive force convergence using the following check: where T.5( U R) < tol U is the displacement increment, R is the unbalance of the system equation and tol is the specified tolerance [Website, OpenSees]. The tolerance was set as

26 and if the system equation solver did not satisfy the convergence test in 2 increments the convergence test returned failure to converge. If the convergence failed the time/load increment was minimized. Some convergence problems needed simplification of the model for example it was impossible to use pinned connection for models subjected to seismic loading. Some numbered errors would be returned, but no list of these error values or solutions to them was acquired Connection stiffness Another research study was under way at the University of Washington which purpose is to quantify the joint stiffness and to evaluate the interaction between the braces and beam-column connections for braced frames [Yoo, 24]. When complete this will provide rotational stiffness of the frame and both axial and rotational stiffness of the brace gusset-plate connection. This was done by developing an elastic finite element model, from which a parametric study was done to approximate the influence of several dimensional parameters on the stiffness. This finite element model was developed using shell elements and three dimensional meshes, which takes more time both to create and solve than the type of modeling done in this research Rotational stiffness of the frame The rotational stiffness of the frame due to the effects of the gusset plate was calculated by applying moment to the beam and column as can be seen on Figure 2.5.

27 Then the stiffness was calculated as the applied moment divided by the rotation calculated from movement of point Rf relative to R see Figure Rf b θ b R a Figure 2.5: Model configuration for rotational stiffness of frame and dimensions used The stiffness was calculated for various configurations of the gusset-plate by changing for example the size of the gusset-plate, the angle of the brace and the member sizes of the beam and column. From those values a quantitative relationship was developed using non-linear data fitting method. From which the stiffness, K Rot. Frame, can be calculated as: K I I Rot. Frame b 13 c = 2.599( θb ) K abt + abt a a b b t t where K is the stiffness of the frame without the gusset plate, θ b is the angle of the brace in units of radians, I b is the moment of inertia of the beam, I c is the moment of

28 inertia of the column, a and b are the dimensions of the gusset-plate and t is the thickness of the gusset-plate Rotational stiffness of the brace gusset-plate connection The rotational stiffness of the brace gusset-plate connection was calculated by applying moment on the brace first in-plane and second out-of-plane as seen on Figure 2.6. Then the stiffness was calculated as the applied moment divided by the rotation calculated from movement of point Rb relative to R see Figure 2.6. out-of-plane Rb in-plane L b b θ b R a Figure 2.6: Model configuration for rotational stiffness of brace gusset plate connection and dimensions used As for the rotational stiffness of the frame the stiffness was calculated for number of configuration from which a quantitative relationship was developed using non-linear data fitting method. This relationship is as follows for the in-plane rotational stiffness:

29 16 K I I EtA abt abt gusset in plane.64 9 b 13 c = 2.224( θb ) a a b b t t and for the out-of-plane rotational stiffness: K gusset out of plane.5 Lb Lb Lb = ( θb ) Et t a b a a b b t t where E is modulus of elasticity, t is the thickness of the gusset-plate, A is the area of the gusset-plate, θ b is the angle of the brace in units of radians, I b is the moment of inertia of the beam, I c is the moment of inertia of the column, a and b are the dimensions of the gusset-plate, t is the thickness of the gusset-plate and L b is the length of the brace welded to the gusset-plate.

30 17 Chapter 3: Validation of inelastic models 3.1 Introduction This chapter will include description of the modeling of isolated braces and braced frame systems to calculate the response due to monotonic loading. The objective of these models is to validate the predicted response by comparing it to design provisions and previous experimental results. The model was validated by developing a model of an isolated brace. The convergence of the brace was analyzed, and the element mesh needed to obtain a convergent solution while providing reasonable economy was established. Then the buckling load of the model was approximated to the AISC provision by varying the initial imperfection. When the variables of the isolated brace model had been validated to the AISC provision, the response of a single story cross braced frame was modeled and compared to previous experimental results. 3.2 Modeling buckling behavior As mentioned before the first step was to model the behavior of a brace using nonlinear beam-column elements with fiber section and uniaxial bilinear constitutive law. It was decided to start with a model of a simple brace, i.e. with both ends pinned and thus effective length equal to the true length. The load, P, was put on one end in the direction of the brace, see Figure 3.1 below. The material properties of the steel

31 were as follows: yield stress: f = 5ksi, modulus of elasticity: E = 29ksi and strain hardening: b = 3%. y 18 v-axis L P u-axis Pinned ends v-axis L P u-axis Fixed ends Figure 3.1: Layout of braces with pinned and fixed ends Initial imperfection was needed for the brace to buckle, without it the model would predict that the brace remains straight until it yielded. It was decided that the π brace should have an initial crookedness v sin u L with the maximum imperfection, v, in the middle of the brace, since similar imperfections have been evaluated in past theoretical work [Batterman, 1967]. For the convergence test it was decided to use an initial imperfection of v = L/ 5. The buckling load was calculated by using load control to increase the applied load until the brace buckled. Relatively small load steps were necessary to capture the buckling behavior, for the analysis a load increment of.5kips was used.

32 3.2.1 Convergence HSS5x5x3/8 brace was modeled with various numbers of elements as can be seen in Table 3.1 which also includes the names given to the braces. The buckling load was calculated for various lengths of the braces from which the slenderness parameter was calculated as: 19 kl λc = rπ f y E where k = 1 is the effective length and r = 1.89in is the radius of gyration. Table 3.1: Name and properties of braces Brace name Number of elements Initial imperfection HSS5-M4-5 4 L/5 HSS5-M6-5 6 L/5 HSS5-M1-5 1 L/5 HSS5-M L/5 HSS5-M2-5 2 L/5 Figure 3.2 shows that increasing the number of segments from 1 to 14 or 2 had virtually no impact on the predicted buckling load for stocky columns ( λ c less than 1.5). The increased number of segments adds a moderate effect on more slender columns, but these more slender columns are not greatly affected by inelastic buckling. As a result, the 1 segment model was selected for this analysis program.

33 2 1.1 buckling load for brace with x elements / buckling load for brace with 1 elements [-] HSS5-M4-5/HSS5-M1-5 HSS5-M6-5/HSS5-M1-5 HSS5-M14-5/HSS5-M1-5 HSS5-M2-5/HSS5-M Slenderness parameter λ c [-] Figure 3.2: Buckling load results for HSS5x5x3/8 brace with various numbers of elements with respect to buckling load with 1 elements with pinned end conditions Comparison to the AISC buckling curve Once the analytical algorithm was established, a subsequent goal was to match the AISC buckling curve, since this curve is partially based upon inelastic buckling calculation. The AISC curve is defined as follows: F cr 2 λc Fy λc.658 if 1.5 =.877 F if y λc > λc where 23]. F y is the yield stress and λ kl rπ F E y c = is the slenderness parameter [AISC,

34 Various maximum imperfections, v, were used to model the brace and then the buckling curve was created by calculating the buckling load using various lengths of the braces. The buckling load was calculated both for HSS5x5x3/8 tube and W4x13 section with initial imperfection from L /15 to L / 25. As before the buckling load was calculated by increasing the load until the brace would buckle. Since the HSS5x5x3/8 tube has the same moment of inertia in both directions it has the same buckling capacity in both directions, the W4x13 section does not have the same buckling capacity in both directions and therefore the buckling was modeled in both the weak and the strong axis direction. Table 3.2 and Table 3.3 below show the names and properties of the braces analyzed. 21 Table 3.2: Name and properties of HSS5x5x3/8 braces Brace name Number of elements Initial imperfection HSS5-M L/25 HSS5-M1-5 1 L/5 HSS5-M1-1 1 L/1 HSS5-M L/15 Table 3.3: Name and properties of W4x13 braces Brace name Number of elements Initial imperfection W4-M L/25 W4-M1-5 1 L/5 W4-M1-1 1 L/1 W4-M L/15

35 Fcr [ksi] 3. Less imperfection 2. Euler AISC HSS5-M HSS5-M1-15 HSS5-M1-5 HSS5-M λ c [-] Figure 3.3: Buckling curves for HSS5x5x3/8 tube section for various imperfections Fcr [ksi] 3. Fcr [ksi] Euler Euler AISC AISC W4-M1-1 W4-M W4-M1-15 W4-M1-15 W4-M1-5 W4-M1-5 W4-M1-25 W4-M λc [-] λc [-] Strong axis Weak axis Figure 3.4: Buckling curves for W4x13 section for various imperfections From Figure 3.3 and Figure 3.4 it can be seen that maximum imperfection of v = L/ 5 provides a close approximation to the design strength of the column. The AISC buckling curve is based on a reasonable conversion of research data into design equations [AISC, 23]. According to the Commentary on the Load and Resistance

36 23 Factor Design Specification for Structural Steel Buildings this curve is essentially the same curve as column-strength curve 2P of the Structural Stability Research Council [AISC, 23]. That buckling curve was created by using imperfection in the form of half sine with the maximum imperfection equal to L/15 plus using the residual stresses of the cross-section [Bjorhovde, 1988]. Using fiber elements it was hard to model these residual stresses, plus using maximum imperfection of L/5 provided relatively good results. After deciding to use maximum imperfection of v = L/ 5 and 1 elements to model the brace, the difference between the calculated buckling load and the AISC buckling load was calculated. This was done for an HSS5x5x3/8 sections with pinned ends, Figure 3.5 shows this comparison. 1.1 Buckling load calculate using OpenSees / Buckling load calculated according to AISC [-] Slenderness parameter λ c [-] Figure 3.5: Comparison between program and AISC for HSS5x5x3/8 member with pinned ends

37 24 It is interesting to notice that the results compare relatively well and that for slenderness parameter equal to.3, 1. and 1.8 both methods give the same results. For slenderness parameter smaller than.3 and between 1. and 1.8 the program predicts a buckling load smaller than the AISC prediction. For slenderness parameter between.3 and 1. and higher than 1.8 the program predicts a larger buckling strength than the AISC buckling curve Braces with fixed ends It was also necessary to check whether the same convergence criteria applied to braces with fixed end conditions. Therefore an HSS5x5x3/8 brace with fixed ends was analyzed using OpenSees and compared to the AISC buckling curve for a brace with effective length equal to half of the true length. First the brace was analyzed using 4, 6, 1, 14 and 2 elements to check whether the same convergence applied as for the pinned brace. Figure 3.6 is comparable to Figure 3.2 on page 2.

38 buckling load for brace with x elements / buckling load for brace with 1 elements [-] HSS5-M4-5/HSS5-M1-5 HSS5-M6-5/HSS5-M1-5 HSS5-M14-5/HSS5-M1-5 HSS5-M2-5/HSS5-M Slenderness parameter λ c [-] Figure 3.6: Buckling load results for HSS5x5x3/8 brace with various numbers of elements with respect to buckling load with 1 elements with fixed end conditions From the above plot it is clear that the convergence is good for braces with fixed end conditions with 1 or more elements as it is for braces with pinned end conditions with 1 or more elements. Following this analyzes the buckling load for brace with 1 elements was compared to the AISC buckling curve for a brace with effective length equal to half of the brace length, see Figure 3.7.

39 Buckling load calculate using OpenSees / Buckling load calculated according to AISC [-] Slenderness parameter λ c [-] Figure 3.7: Comparison between program and AISC for HSS5x5x3/8 member with fixed ends Again it is interesting to notice that the results compare relatively well and that for slenderness parameter equal to.3, 1.1 and 1.5 both methods give the same results. For slenderness parameter smaller than.3 and between 1.1 and 1.5 the program predicts a buckling strength that is lower than the AISC prediction. For slenderness parameter between.3 and 1.1 and higher than 1.5 the program predicts a larger buckling strength than the AISC buckling curve Post buckling behavior of braces The next step was to model the behavior of the brace after buckling. In order to model the post buckling behavior the model was analyzed using displacement control, i.e. by applying displacement to the end of the column and calculating the resisting

40 27 force. Three types of braces were modeled: a) SCBF brace with pinned ends, b) SCBF brace with fixed ends and c) BRBF brace. For the brace an HSS5x5x3/8 section was used of length 2 inch. First the buckling strength of the brace is calculated according to AISC. The area of the brace was calculated as follows: ( ) 2 A= = in, the moment of inertia is: I ( ) = = 24.9in from that the governing radius of gyration is calculated as: I r = = 1.894in. A In order to calculate the buckling load according to AISC the effective length of the column has to be decided. For case a) the effective length is k = 1. and thus the slenderness parameter is: kl Fy λc = = rπ E and the load capacity is: 2 λ ( ) c P = A.658 F = kips. n y Similarly for case b) the effective length is k =.5 and from that the load capacity is calculated as:

41 28 Pn = kips. For case c) the yield capacity is calculated as: P = A F = kips. y y Using OpenSees the behavior of the braces was calculated for changes in length up to 2in. Figure 3.8 shows the force-displacement plot for all the braces. The buckling load was calculated for cases a) and b) as follows: case a) P = kips, case b) P = kips and case c) P = 347.3kips. Table 3.4 summarizes the AISC n capacity and the capacity calculated using OpenSees. y n Table 3.4: Capacity of brace Case OpenSees AISC Error kips kips % a b c Load [kips] BRBF SCBF - Fixed ends 1 5 SCBF - Pinned ends Delta length [in] Figure 3.8: Behavior of HSS5x5x3/8 brace under monotonic loading

42 It is interesting to notice that the post-buckling strength for the fixed SCBF brace is much higher than for the pinned SCBF brace as expected Braced frame behavior The behavior of braced frame under monotonic pushover loading was compared to prior experimental results. A one story frame was modeled for monotonic and cyclic load and compared to experimental results [Wakabayashi, 1974] Modeling of frames The report used cm as the length unit and metric tons as the force unit, and the comparison was made using these same units. The width of the frame was 5 cm center to center and the height was 26 cm from the bottom to the middle of the beam, see Figure 3.9. The beam, columns and braces had the shape of the W-section and the dimension can be seen in Table 3.5 below. 7tons 5cm 7tons 26cm Figure 3.9: The frame analyzed [Wakabayashi, 1974]

43 3 Table 3.5: Wakabayashi test member dimension [Wakabayashi, 1974] Depth Width T web T flange mm mm mm mm Columns Beam Braces The OpenSees results were compared to four experiments done by Wakabayashi s group. First a braced frame without vertical load under monotonic loading (Test BM ) and under cyclic loading (Test BC ). Second a braced frame with 7 ton vertical load on each column under monotonic loading (Test BM 5) and under cyclic loading (Test BC 5). The measured material properties of the members were given in the report and these same properties were used to model the frame. A modulus of Elasticity of E 24 tons / cm 2 = was used (not stated in report). The yield stress, y f, and strain hardening factor, b, for the tests can be seen in Table 3.6 below. Table 3.6: Material properties of the test frames [Wakabayashi, 1974] Beam Column Brace f y f u b f y f u b f y f u b t/cm 2 t/cm 2 - t/cm 2 t/cm 2 - t/cm 2 t/cm 2 - BM BM BC BC For the model, each column and the beam were divided into 1 elements, while each brace was divided into 2 elements. Each flange of the members was divided into 1x2 fibers and similarly each web was divided into 1x2 fibers. This was done relative to the direction of buckling/bending, i.e. beam and columns bended in the

44 31 direction of the strong axis and the braces buckled in the direction of the weak axis (Figure 2.1). The beam and columns were not given initial imperfection while the braces had maximum initial imperfection of L/5 in the middle of each unsupported brace were L is the unsupported length of the brace. Figure 3.1 below shows this imperfection scaled by a factor of 2. No out-of-plane initial imperfection was applied. The braces were modeled as pinned at the ends but the middle connection of the braces was modeled as fixed. As Figure 3.9 implies the bottom of each column was modeled as fixed and then the top of the columns was restrained against out-of-plane movement. L = 281.8cm Imperfection Imperfection Imperfection Imperfection Figure 3.1: Initial imperfection of Wakabayashi braced frame The vertical loading was applied to both upper corner nodes Comparison between analytical and experimental results Below is the comparison between experiments and OpenSees for braced frames without (Figure 3.11 and Figure 3.12) and with (Figure 3.13 and Figure 3.14) vertical loading. The experimental results are a slight higher than the OpenSees. The results could possible be improved by modeling the behavior of the brace connections, but the dimensions of the gusset-plates were not given in the report.