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THE EFFECT OF THE PANEL ZONE DEFORMATION ON THE REALISTIC BEHAVIOR OF THE FRAMES by Ekaterina Foster Diploma in Mechanical Engineering, Russia, 1988 Bauman Moscow Technical School A thesis submitted to the Faculty of the Graduate School of the University of Colorado at Denver in partial fulfillment of the requirement for the degree of Master of Science Civil Engineering 1994
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This thesis for the Master of Science degree by Ekaterina Foster has been approved for the Department of Civil Engineering by Andreas S. Vlahinos Peter E. Jenkins
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Foster, Ekaterina (M.S., Civil Engineering) THE EFFECT OF THE PANEL ZONE DEFORMATION ON THE REALISTIC BEHAVIOR OF THE FRAMES Thesis directed by Associate Professor Andreas S. Vlahinos ABSTRACT In this project, the elastic and inelastic behavior of the semirigid connections of laterally loaded frames with and without continuity plates was presented. The modeling considerations of the rigid end assumptions are considered and compared to the more accurate shell finite element model. The deformation a.nd the stress distribution of elastic subassemblages are presented using a detailed shell finite element model. Finally, the inelastic behavior of a subassemblage with and without continuity plates was generated using bilinear kinematic hardening flow rule. This abstract accurately represents the content of the candidate's thesis. I recommend its publications. Signed Andreas Vlahinos
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v CONTENTS CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1 1-1 Historical Review ................................................. 1 1-2 Statement of the Problem ......................................... 3 II. THE EFFECT OF THE RIGID ENDS ON DEFORMATION ......... 4 11-1 Problem Description ............................................. 4 11-2 Problem Idealization ............................................. 6 11-3 Discretization .................................................... 7 11-4 Modeling Verification . . . . . . . . . . . . . . . . . . . . . . 8 11-5 Results and Conclusions .......................................... 9 III. LINEAR BEHAVIOR OF THE SUBASSEMBLAGES USING SHELL FINITE ELEMENT MODEL ....................................... 21 111-1 Problem Description ........................................... 21 111-2 Problem Idealization ........................................... 22
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Vl 111-3 Discretization .................................................. 22 111-4 Modeling Verification .......................................... 23 111-5 Results and Conclusions ........................................ 24 IV. INELASTIC BEHAVIOR OF THE SUBASSEMBLAGE WITH AND WITHOUT CONTINUITY PLATES USING A SHELL FINITE ELEMENT MODEL ........................................ 58 IV-1 Inelastic Behavior of the Model without Continuity Plates ...... 58 IV-1-1 Problem Description ...................................... 58 IV-1-2 Problem Idealization ...................................... 60 IV-1-3 Discretization ............................................. 60 IV-1-4 Results and Conclusions .................................. 61 IV -2 Inelastic Behavior of the Model with Continuity Plates .......... 62 IV-2-1 Problem Description ...................................... 62 IV -2-2 Problem Idealization ...................................... 63 IV-2-3 Discretization ............................................. 63 IV-2-4 Results, Comparison and Conclusions ..................... 63
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Vll V. CONCLUSIONS AND RECOMMENDATIONS ...................... 77 V -1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 77 V-2 Recommendations .............................................. 78 APPENDIX: A. Preprocessor Input for the Beam FE Model of Subassemblage ........ 79 B. Preprocessor Input for the Buckling Analysis of the Subassemblage ... 81 C. Preprocessor Input for the Simple Shell FE Model of a Cantilever Beam Used as Verification Problem .................................. 82 D. Preprocessor Input for the Linear Elastic Shell FE Model of the Subassemblage ...................................................... 83 E. Preprocessor Input for the Inelastic Shell FE Model of the Subassemblage without the Continuity Plates . . . . . . . . . . . . 85 F. Preprocessor Input for the Inelastic Shell FE Model of the Subassemblage with the Continuity Plates ........................... 87 REFERENCES ........................................................... 89 NOTATION .............................................................. 91
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Vlll FIGURES Figures 11-1-1 Multistory Unbraced Frames ........................................ 12 11-1-2 Connection Geometry ............................................... 13 11-1-3 Column Displacement ............................................... 14 11-4-1 Bending Moment Diagram for Actual Loading ....................... 15 11-4-2 Bending Moment Diagram for Unit Virtual Load ..................... 15 11-5-1 Deflection with and without Shear Effects . . . . . . . . . . . . . 16 11-5-2 Deflection for Various Type of Analyses .............................. 17 11-5-3 Contribution of Various Types of the Effects ......................... 18 11-5-4 Critical Loads and Corresponding Mode Shapes ...................... 19 11-5-5 Critical Loads for C = 0 and C = 1 .................................. 20 111-3-1 Typical Finite Elements of the Elastic Model ........................ 32 111-3-2 Typical Finite Elements of the Zoomed Joint ........................ 33
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IX 111-3-3 Typical Loading and Boundary Conditions of the Elastic Model 34 111-5-1 Typical Principal Stress Directions of the Elastic Model .............. 35 111-5-2 Typical Vectorial Representation of Nodal Displacements ............. 36 111-5-3 Horizontal Deformation Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 12' Case 1 ...... 37 111-5-4 Horizontal Deformation Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 18' Case 2 ...... 38 111-5-5 Horizontal Deformation Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 24' Case 3 ...... 39 111-5-6 Horizontal Deformation Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 12' Case 4 ...... 40 111-5-7 Horizontal Deformation Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 18' Case 5 ...... 41 111-5-8 Horizontal Deformation Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 24' Case 6 ...... 42
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X 111-5-9 Horizontal Deformation Distribution of the Beam-Column Subassemblage W30x132/W24x176 with Beam Span 18' Case 7 ...... 43 111-5-10 Equivalent Stress Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 12' Case 1 ..................... 44 111-5-11 Equivalent Stress Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 18' Case 2 . . . . . . . . . . 45 111-5-12 Equivalent Stress Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 24' Case 3 ..................... 46 111-5-13 Equivalent Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 12' Case 4 ..................... 47 111-5-14 Equivalent Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 18' Case 5 ..................... 48 111-5-15 Equivalent Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 24' Case 6 ..................... 49 111-5-16 Equivalent Stress Distribution of the Beam-Column Subassemblage W30x132/W24x176 with Beam Span 18' Case 7 . . . . . . . . . . 50
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Xl 111-5-17 Shear Stress Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 12' Case 1 . . . . . . . . . . 51 111-5-18 Shear Stress Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 18' Case 2 ..................... 52 111-5-19 Shear Stress Distribution of the Beam-Column Subassemblage W27x114/W18x119 with Beam Span 24' Case 3 ..................... 53 111-5-20 Shear Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 12' Case 4 ..................... 54 111-5-21 Shear Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 18' Case 5 ..................... 55 111-5-22 Shear Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 24' Case 6 ..................... 56 111-5-23 Shear Stress Distribution of the Beam-Column Subassemblage W30x132/W24x176 with Beam Span 18' Case 7 . . . . . . . . . . 57 IV-1-1 The Assumed Stress-Strain Curve .................................. 59 IV-1-2 Finite Elements of the Inelastic Model Case 7 ........................ 65
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IV-1-3 Horizontal Deformation Distribution of the Beam-Column Subassemblage W30x132/W24x176 with Beam Span 18' Xll Inelastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 IV-1-4 Equivalent Stress Distribution of the Beam-Column Subassemblage W30x132/W24x176 with Beam Span 18' F 11 = 45kips ................ 67 IV-1-5 Equivalent Stress Distribution of the Beam-Column Subassemblage W30x132/W24x176 with Beam Span 18' F 11 = 57kips ................ 68 IV-2-1 Zoomed Part of the Joint with Continuity Plates ..................... 69 IV -2-2 Load Displacement Curve of Inelastic Model with Continuity Plates . 70 IV-2-3 Von Mises and Shear Stress at the Center of the Subassemblage versus Displacement ................................................. 71 IV-2-4 Von Mises and Shear Stress at the Center of Continuity Plates versus Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 IV -2-5 Load Displacement Curves of Inelastic Models with and without Continuity Plates . . . . . . . . . . . . . . . . . . . . . . . . . 73 V-2-1 Potential Configurations for Panel Zone Reenforcement ............... 78
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Xl11 TABLES Tables 11-1 Deflection for the Various Types of Analyses ......................... 11 11-2 Critical Loads for C = 0, and C = 1 . . . . . . . . . . . . . . . . 12 111-1 Summary of Case 1 ................................................. 25 111-2 Summary of Case 2 . . . . . . . . . . . . . . . . . . . . . . . . 26 111-3 Summary of Case 3 . . . . . . . . . . . . . . . . . . . . . . . . 27 111-4 Summary of Case 4 . . . . . . . . . . . . . . . . . . . . . . . . 28 111-5 Summary of Case 5 ................................................. 29 111-6 Summary of Case 6 ................................................. 30 111-7 Summary of Case 7 ................................................. 31 IV-2-1 Equilibrium Path for the Inelastic Behavior with and without Continuity Plates . . . . . . . . . . . . . . . . . . . . . . . . . 7 4
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XIV IV-2-2 Shear and von Mises Stresses at the Center of the Subassemblage versus Deflection .................................................... 75 IV -2-3 Shear and von Mises Stresses at the Center of the Continuity Plates versus Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . 76
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XV ACKNOWLEDGEMENTS Special acknowledgements and appreciation is due to several individuals who greatly supported and contributed to this research work. To the Professor An dreas S. Vlahinos, the author's adviser at the University of Colorado at Denver who provided me with computer resources through his consulting for the large nonlinear analyses of the problems that could not be run at the school computers. Also to Dr. Charney Finley structural engineer at J .R. Harris & Company who did much of the study in the field of stability of semirigid connections of frames and provided the wonderful idea for this project. Special thanks to Doctor Yang Cheng Wang for his helpful knowledge of various computer programs. Finally, I am grateful to the Dean Peter E. Jenkins and Dr. Bruce Janson who are professors at University of Colorado at Denver for participating in the Graduate Committee.
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I-1 Historical Review CHAPTER I INTRODUCTION Moment resisting frames are currently being used as preferred structural con figuration for the tall buildings. The serviceability requirements produce very large sections for the beams and columns. The contribution of the beams and columns on the total system behavior in the elastic range is well understood and can be determined with a reasonable degree of confidence. In the current design practice, the effect of the connection of the beams and columns (beam-column subassemlage) is ignored by assuming that the connection is rigid. Over the last 20 years, several research projects have questioned this assumption. Bertero, Popov and Krawinkler1 (1972) investigated the behavior of the beam-column subassemlages under repeated loadings. Two types of structural steel half scale subassemblages of the multistory unbraced frames were tested. Some of the conclusions are large shear deformation and buckling of the panel zone can occur, and a weak panel zone can lead to the development of an internal panel zone plastic hinge. For the high lateral loads, the P 6 effects must be included in the design. In evaluating o, not only flexural contribution of the columns and beams should be considered, but also the shear deformation of the panel zone.
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2 In 1975, Becker2 reported his experimental study on panel zone effect on the strength and stiffness of steel rigid frames. The results of the study concluded that the panel zone (the section of the subassemblage between the flanges of the beam and column) can be the weakest element in steel rigid frame. The strength and stiffness of the entire frame may depend on the shear capacity of the panel zone. Thus, the panel zone is especially important for rigid frames subjected to lateral load. Tsai and Popov3 (1989) studied the seismic panel zone design effect on elastic story drift in steel frames. The results are based on analytical studies for two representative steel frames with different panel zones. They recommended a simple approximate procedure for determining the elastic story drift due to the panel zone. They proposed a procedure that eliminates the need for adding additional degrees of freedom at the joints, but is based on the assumption of elastic behavior. Recently, Sibai and Frey4 (1993) introduced a new semirigid joint element for nonlinear analysis of flexibly connected frames. They also concluded that joint flexibility significantly reduces the structural strength capacity in unbraced frames. Shear flexibility has great influence on the strength and displacement of unbraced frames, but it is insignificant for braced frames. In this study, the effect of panel zone will be investigated considering in elastic large deformation analysis of several realistic subassemblages. Connections
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3 with and without continuity plates were examined. Also, the effects of axial and shear deformation are assessed. 1-2 Statement of the Problem The objective of this study is to investigate the effect of the flexibility of a re alistic moment resisting connection on the shear deformation of laterally loaded multibay, multistory frames. Connections with and without continuity plates are examined. Also, the effects of axial and shear deformation are assessed. The effect of the length of rigid end models on the total deformation is examined.
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4 CHAPTER II The EFFECT OF RIGID ENDS ON DEFORMATION 11-1 Problem Description In the model idealization of frame structures, we consider the beams and columns to be line elements at the center line of those structural elements. The subassemblage is defined as a portion of the frame which contains the connection and the portion egestion beams and columns as shown on Figure 11-1-1. The crosssections of the beams will be much larger after the beam meets the column at the connection. If the column width were negligibly small, the abrupt change of the cross-section could be neglected. Since the column width for realistic frame structures is relatively large (1/20 to 1/10 of the beam length), this abrupt change of the cross-section cannot be neglected. It has been suggested in several textbooks and papers that the portion of the beam that is located between the flanges of the column should be modeled as a rigid segment. To undimensionalize the results, a parameter C was introduced which is the length of the rigid end over the beam height. (1)
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5 where xis the rigid end length and db is the beam height. Figure II-1-2 (a): shows the connection geometry with realistic dimen sions, Figure II-1-2 (b): plots a typical beam model without the rigid ends (C = 0) and Figure II-1-2 (c): shows the beam model with the rigid ends (C = 1). In this investigation, a more precise model of the beam-column connection assemblage has been developed and as described in Chapter III. To investigate the validity of rigid end modeling, several finite models of the beam-column connection assemblage have been developed with varying lengths of the rigid end, with the maximum length of the rigid end equal to the beam height. Figure II-1-3 shows the geometry and sign convention of this connection. The deformation of the top end of the column was computed for the subassemblage identified as Case 7 and several values of the parameter C [C = 0, 0.1, ... 1]. The geometry and dimensions for this subassemblage are described in Chapter III. Since the effect of axial, shear, and flexural deformation is not a priori known, three different models were constructed. The computed deformation of the top of the column is shown at the Figure II-1-2. The first model computes the deflection 6 f considering the flexural effects only. The second model computes the deflection 6 fa considering the flexural and axial effects. The last model computes the deflection 6 fas considering the flexural, axial, and shear effects.
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6 In order to examine the importance of geometric nonlinearities, a finite element model was implemented using large deformation analysis. The same set of deformation was computed and compared to the linear results. To examine the effect of the axial loads on the bending stiffness, the buckling loads of the subassemblage were calculated and compared to the applied loads. A finite element model was developed to compute the critical loads and the corresponding mode shapes. II-2 Problem Idealization The following assumptions were made: The material considered is A36 steel with modulus of elasticity E = 29000 ksi and Poisson's ratio v = 0.3. The beams and columns are perfectly straight, and all geometric and load ing imperfections are ignored. The left end of the beam is considered 'pinned' and the right end is supported by roller. A portion of the beam and the column of length C x db at the joint is con sidered rigid. The loading is a horizontal force of F 11 = 1000 kips applied to both ends of the column in opposite directions.
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7 11-3 Discretization The model is discretized with 44 nodes and 45 two-node elastic beam elements. This 2-D element has usual application for bending members with symmetric cross-sections. Each node has three degrees of freedom: translations in the nodal x and y directions and rotations about the nodal z axes. One end of the beam is 'pinned', another one is restrained with a roller. Appendix A shows the preprocessor input file for this finite element model. 11-4 Modeling Verification The model was verified by predicting the deflection of a column under lateral load by the method of virtual work. In general the deflection o is: (2) where o 1 is the deflection considering flexural effects, 00 is the deflection considering axial effects and o. the deflection considering shear effects. 0 = rL MLmu d rL NLnu d rL FVLVu d lo EI x + lo EA x + Jo GJ x (3) where, ML bending moment due to actual loading NL axial force due to actual loading
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8 V L shear force due to actual loading mu bending moment due to unit virtual loading nu axial force due to unit virtual loading v u shear force due to unit virtual loading E modulus of elasticity I moment of inertia A area J polar moment of inertia G shear modulus F form factor For the simple model we verified the deflection considering flexural effects only. (4) Equation (5) was obtained using the bending moment diagram for the actual load from Figure 11-4-1 and the bending moment diagram for the virtual load from Figure 11-4-2: !kb.F, b. (!f.k F11Le) b. 6 -2 2 2 1/J 2 + 2 2 2 2 3 4 1 -Elc Eh (5)
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Simplifying equation (5) into equations into (6) and (7) yields where, Lb = 18x 12(in) Lc = 12.5 X 12(in) E = 29x 106(psi) 61 = (Lclb + Lblc) 24Elclb beam span column length moment of inertia of the beam moment of inertia if the column modulus of elasticity horizontal force 6, = 2.0639 9 (6) (7) In Table 11-1 for the parameter value C = 0, the deflection 61 for linear and nonlinear analyses have values equal to those computed in this chapter. The difference between the predicted and analytical displacement was less than 0.1 %. 11-5 Numerical Results and Conclusions The subassembly considered has a beam with an area A = 38.9(in2), moment of inertia I.r = 5770(in4), length Lb = 18(ft), and a column with area A= 51.7(in2),
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10 moment of inertia Ix = 5680(in4 ), length Lb = 12,5(ft). Table 11-1 presents the deflection hat the top of the column for various values of C [C = 0, 0.1, ... 1] and various types of analysis. Figure II-5-1 plots the deflection OJ for the linear analy sis with and without shear effects. One may conclude that the effect of geometric nonlinearities for this load level and these stiffness ratios is negligible. The effect of axial deformation in both linear and nonlinear analysis is also negligible. The effect of shear deformation is significant. The shear deformation is about 41% of the total deformation and cannot be neglected. Figure 11-5-2 shows the deflection h for various values of C and three types of analyses: (1) the shell model analysis described in Chapter III, (2) the beam model with flexural effects only, and (3) the beam model with flexural and shear effects. The deflection decreases in cases II and III as C decreases. The deflection in case I is independent of C, since Cis not a parameter in this model. The most important conclusion from this figure is that the most realistic deformation closest to the shell model results appears for the value of C = 0. Thus, to consider a realistic joint flexibility, the beam models of the frame structures should not have any rigid ends. Figure 11-5-4-3 shows the contribution of the flexural, axial, and shear effects in panel zone to the total deformation. It should be noted that the shear and panel zone contributions, which are usually neglected, are more than 45%. The axial contribution has virtually no effect. The panel zone contributions is
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11 about 22%, and the significance of this contribution is extremely important, since the total frame deformation used in current practice ignores the cumulative effect of the the panel zone deformation. Appendix B shows the finite element model that computes the critical loads and corresponding mode shapes. Figure 11-5-4 and Table 11-2 shows the first five critical loads for the cases of C = 0 and C = 1. Figure 11-5-5 shows the first four modes shapes for C = 1 and their corresponding critical load. It may be concluded that the effect of C on the critical load is negligible, especially for the first mode. The first critical load is about 24 times the existing load. Thus, the effect of the axial loads on the bending stiffness of the subassemblage is also negligible. Table 11-1: Deflection for Various Types of Analyses Nonlinear Linear c or Ora Or as CASE 7 or Or as K, Kras 0 2.06 2.07 2.68 2.98 2.06 2.68 484740 372490 0.1 1.97 1.98 2.58 2.98 1.97 2.58 507287 387197 0.2 1.83 1.84 2.48 2.98 1.88 2.48 531444 402792 0.3 1.79 1.79 2.38 2.98 1.8 2.39 557093 419172 0.4 1.71 1.71 2.29 2.98 1.71 2.29 584343 436383 0.5 1.63 1.63 2.2 2.98 1.63 2.20 613316 454476 0.6 1.54 1.55 2.11 2.98 1.55 2.11 644142 473503 0.7 1.47 1.47 2.02 2.98 1.48 2.03 676964 493518 0.8 1.40 1.4 1.94 2.98 1.40 1.94 711936 514584 0.9 1.33 1.33 1.86 2.98 1.33 1.86 749227 536763 1 1.26 1.26 1.78 2.98 1.27 1.79 789020 560123
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12 Table 11-2: Critical Loads for C = 0 & C = 1 Mode# C=O C=1 1 18.69 26.26 2 35.62 47.19 3 88.27 90.52 4 94.83 100.4 5 106.5 106.7 r I 12 lee1 l 4 18 reel r l 4 I 24 reel -,-I ,.. -, I I --. --. 20 12.5 feel I I i I I i i : l a Figure 11-1-1. Multistory Unbraced Frames
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13
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Ac I c A b I b r-F F I I L b -----------J Figure Il-1-3. Column Displacement l4 Lc I I
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15 F F Lc/2 Lb F Lc/2 Lb F Figure II-4-1. Bending Moment Diagram for Actual Loading Lc/2Lb Lc/2Lb Figure II-4-2. Bending Moment Diagram for Unit Virtual Load
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c ....._... c 0 t5 3.5 -3 +-----.. ----.. -----------------------2.5 2 Q) 1.5 Q) 0 1 0.5 ----------------------1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c [--f Nonlinear ----*fas Nonlinear I Figure II-5-L Deflection with and without Shear Effects 1 ....... 0')
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3.5 3 2.5 !--------c c 2 0 (.) Q) 1.5 '+= Q) 0 1 0.5 ---0 --:: j I I I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 c Figure II-5-2. Deflection for Various Type of Analyses f Nonlinear -fa Nonlinear fas Nonlinear -f Linear fas Linear -Shell Model --J
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Panel Zone 22% Axial 0% Figure 11-5-3. Contribution of the Various Types of Effects Flexural 55% ,_ 00
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19 F = 26.3 F = 47.2 F = 90.5 F = 100.4. Figure 11-5-4. Critical Loads and Corresponding Mode Shapes
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120 100 0 -< 80 0 ..J ..J -< 60 u E-o C2 40 u 20 0 C=l CRITICAL LOADS FOR C = 0 C = 1 2 3 4 s MODE NUMBER C=O Figure 11-5-4. Critical Loads for C = 0 and C = 1 "' 0
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CHAPTER III LINEAR BEHAVIOR OF SUBASSEMBLAGES USING SHELL FINITE ELEMENT MODEL III-1 Problem Description 21 In this chapter, the linear behavior of several subassemblages widely used in current practice was investigated. For seven cases of various cross-sections of a beam and column, the stresses such as shear and von Mises, and also the deformation of the top end of the column were computed. Beam-column dimensions of the seven cross-sections analyzed were: Beam/Column Shape 12ft Beam Span 18ft Beam Span 24ft Beam Span W27x114 / W18x119 Case 1 W36x210 / W24x250 Case 4 W36x210 / W24x176 Case 2 Case 5 Case 7 Case 3 Case 6 The seventh case was selected to investigate the material nonlinear behavior of the subassemblage with and without continuity plates. This study is presented in the next chapter. For the analysis of all seven cases, the full story column height of 12.5 feet
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22 was used. Tables 111-1 through 111-7 present the dimensions and a sketch of each subassemblage accordingly. III-2 Problem Idealization The following assumptions for the linear model were made: The material is linearly elastic with modulus of elasticity E = 29000 ksi and Poisson's ratio v = 0.3. The beams and columns are perfectly straight, and all geometric and load ing imperfections are ignored. The left end of the beam is considered "pinned," and the right end is supported by a roller. The subassemblage considered geometrically symmetric over the x-z plane, and the difference between the deflection of the full and half model are negligibly small. Therefore, only half of model was utilized. The loading is a horizontal force of F = 500 kips applied horizontally to both ends of the column in opposite directions. III-3 Discretization The typical subassemblage is discretized with 1421 nodes and 1272 four-node quadrilateral shell elements. This type of element has both bending and mem-
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23 brane capabilities. Each node has six degrees of freedom: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. Estimated degrees of freedom for the model are 4062. A plot of all the elements is shown in Figure 111-3-1. For clarity, all elements have been shrunk by 20%. Customized mesh in the joint area presented in Figure 111-3-2. Smaller element sizes were created at the locations of high stress gradients. Toward the ends of the subassemblage, the size of the elements increased, creating accordingly fewer elements. This procedure minimizes the percentage of error and the use of space when the model runs. This mesh results from several iterations of mesh configurations and error estimations. Figure 111-3-3 shows all nodes with the appropriate boundary conditions and loadings. The ends of the beam have been restrained against translation in x, y, and z directions and against rotation about the y axes. All nodes lie in the y-z plane, which cuts the full model in half, have been restrained in the direction normal to the cutting plane since the geometry and the loading are symmetric about this plane. 111-4 Modeling Verification The model was verified by comparing the deflection from finite element modeling of the simply supported beam with analytical solution. Appendix D presents the input and deflection of the verified model.
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24 III-5 Results and Conclusions The elastic finite element analysis was done on CU-Denver's VAX-8800 computer and it requires about 10 minutes CPU time. The input file was set up for automated parametric analysis and is shown in the Appendix C. Seven combinations of the beam-column cross-section where analyzed lin early. Summary of each case corresponds to Tables 111-1 to 111-7. The top portion of all seven tables shows the dimensions, material properties, and sketch of the subassemblage. The bottom portion reflects the results of each analysis with the maximum values of 6, Ueqv, Tma.r, and k. Figure 111-5-1 shows principal stress directions of the typical elastic model. Vectorial representation of nodal displacement is presented in Figure 111-5-2. The horizontal deformation for the each of seven cases of the subassemblages is shown in Figures 111-5-3 through 111-5-9, respectively. Figures 111-5-10 to 111-5-16 plot the distribution of the equivalent (von Mises) stress Ueqv for the seven cases. Figures 111-5-17 to 111-5-23 present the shear stress distribution of T ma.r for the seven cases. It should be noted that for all cases, the maximum of both stresses are concentrated exactly in the joint section of the subassemblage.
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25 CASE Shape d(in) lw(in) br(in) tr(in) L(rt) Beam W27Xll4 33.5 4090 159 27.29 0.57 10.07 0.93 12 Column Wl8Xll9 35.1 2190 253 18.97 0.655 11.265 1.06 12.5 Material Properties: E = 29000(ksi), v = 0.3, u = 36000(psi), E, = 29000(psi) Continuity Plates: ( ) Yes, (X) No. l f d -l. L ., _: -u,__j 6( in) 4.836 Table III-1. Summary of Case 1
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Shape Beam W27X114 Column W18X119 Material Properties: Continuity Plates: ( d l. --b,___j 6(in) 5.445 26 CASE 2 A(in2 ) lxx(in4 I (in4 ) d(in) tw(in) br(in) tr(in) L(ft) 33.5 4090 159 27.29 0.57 10.07 0.93 18 35.1 2190 253 18.97 0.655 11.265 1.06 12.5 E = 29000(ksi), v = O.J, 0' = 36000(psi), E, = 29000(psi) ) Yes, (X) No. 837.1 Table III-2. Summary of Case 2
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Shape Deam W27X114 Material Properties: Continuity Plates: ( -l. L_ ,. I --b, -------l 6(in) 6.014 L r CASE 3 I (in4 ) d(in) tw(in) 159 27.29 0.57 253 0.655 v = 0.3, No. Table III-3. Summary of Case 3 27 llr(io) tr(in) L(ft) 10.07 0.93 24 11.265 1.06 12.5 E, = 29000 psi 860.7
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28 CASE 4 Shape d(in) lw(in) br(in) tr(in) L(ft) Beam W36X210 36.69 0.83 12.18 1.36 12 Colu W24X250 26.34 1.04 13.85 1.89 12.5 lerial Properties: E = 29000(ksi), "= 0.3, tT = 36000( psi), E, = 29000(psi) Continuity Plates: ( ) Yes, (X) No. L, l. 6(in) Tmu(ksi) 1.34 231.9 Table III-4. Summary of Case 4
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CASE 5 Shape (in4 ) d(in) lw(in) Beam W36X210 36.69 0.83 Column W24X250 724 26.34 1.04 terial Properties: E = 29000(ksi), v = 0.3, u = 36000(psi), Continuity Plates: ( ) Yes, (X) No. d -l. _. b,____j 6(in) 1.526 lr Table III-5. Summary of Case 5 29 br(in) lr(in) L(ft) 12.18 1.36 18 13.85 1.89 12.5 E, = 29000(psi) 253.5
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30 CASE 6 Shape A(in1 ) lu(in4 ) (in4 ) d(in) lw(in) br(in) tr(in) L(ft) Beam W36X210 61.8 13200 411 36.69 0.83 12.18 1.36 24 Column W24X250 73.5 8490 724 26.34 1.04 13.85 1.89 12.5 terial Properties: E = 29000(ksi), v = 0.3, q = 36000(psi), E, = 29000(psi) Continuity Plates: ) Yes, (X) No. L, rl -l. 6(in) Tmu(ksi) 1.7 264.4 Table 111-6. Summary of Case 6
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31 CASE 7 Shape I (in4 ) d(in) l,..(in) br(in) tr(in) L(ft) Beam W30X132 196 30.31 0.615 10.545 1.00 18 Column W24X176 51.7 5680 479 25.24 0.750 12.890 1.340 12.5 Material Properties: E = 29000 ksi II=: 0.3, E, = 29000(psi) Conlinwty Plates: (X) Yes, ( ) No. lr l. _. -b,__j 6(in) 2.978 495.5 Table 111-7. Summary of Case 7
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32 Figure 111-3-1. Typical Finite Elements of the Elastic Model
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33 Figure 111-3-2. Typical Finite Elements of the Zoomed Joint
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34 : .. ... ... -:. ... .. .. ... .. Figure 111-3-3. Typical Loading and Boundary Conditions of the Elastic Model
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--...--..... .... .... ,. "' " I ,/' "' .., +. .;< __ ..... ,, *. \ ., \ ... ,., :< .(( X X X X X ... "' "' X: / .,.. .... .... !Xi '""' ""' -'X. ..... ..... II ... I .. I Figure III-5-1. Typical Principal Stress Directions of the Elastic Model 35 *
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36 I :mil .. T I T T 1 = Figure 111-5-2. Typical Vectorial Representation of Nodal Displacements
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DISPLACEMENT STEP"'1 SUB =1 TIME=l RSYS"'O DMX "'4.836 SEPC .. 25.081 DSCA"'2.088 XV El YV "'-1 zv -=1 DISTE100.96 37 YF .816 A-ZS-60 CENTROID HIDDEN Figure III-5-3. Horizontal Deformation Distribution of the Beam-Column Subassemblage W27xll4/Wl8xl19 with Beam Span 12' Case 1
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DISPLACEMENT STEP=1 SUB -1 TIME-1 RSYS"'O DMX 5. 4 4 5 SEPC=25.628 DSCA"'2.151 XV YV -1 zv -1 DIST-117.127 38 YF .816 A-ZS-60 CENTROID HIDDEN Figure III-5-4. Horizontal Deformation Distribution of the Beam-Column Subassemblage W27xll4/Wl8xl19 with Beam Span 18' Case 2
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DISPLACEMENT STEP-=1 SUB =1 TIME=1 RSYS=O DMX =6.014 SEPC=25.964 DSCA"'2.217 XV YV -1 zv -1 DIST=133.294 39 YF -=2.816 A-ZS-60 CENTROID HIDDEN Figure III-5-5. Horizontal Deformation Distribution of the Beam-Column Subassemblage W27xll4/Wl8xll9 with Beam Span 24' Case 3
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DISPLACEMENT STEP=1 SUB =1 TIME=1 RSYS=O DMX =1.34 SEPC.,23.262 DSCA.554 XV -1 YV -1 zv DIST.25 40 YF -3.462 A-ZS-60 CENTROID HIDDEN Figure III-5-6. Horizontal Deformation Distribution of the Beam-Column Subassemblage W36x210/W24.x250 with Beam Span 12' Case 4
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I I I I I I I I I : I I I I I I I I I 1 ........... DISPLACEMENT SUB TIMEE1 RSYS .. O DMX SEPCE23 .638 DSCA .694 XV YV -1 zv -1 DISTl17.417 YF .462 A-ZS-60 CENTROID HIDDEN Figure III-5 7 Horizontal Deformation Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 18' Case 5 41
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DISPLACEMENT STEP .. 1 SUB "'1 TIME .. 1 RSYSO DMX -=1.7 SEPC .. 23.843 DSCA"'7.858 XV YV ,._1 zv .. 1 DIST .. l33.585 42 YF .. 3.462 A-ZS-60 CENTROID HIDDEN Figure-5-8. Horizontal Deformation Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 24' Case 6
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DISPLACEMENT SUB TIME=1 DMX =2.967 DSCA.954 XV YV zv -1 DIST-117.309 YF .222 Figure III-5-9. Horizontal Deformation Distribution of the Beam-Column Subassemblage W30xl32/W24xl76 with Beam Span 18' Case 7 43
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NODAL SOLUTION STEP"'l SUB "'1 TIME=l SEQV TOP (AVG) DMX .836 SMN SMX SMXB.115E+07 l;:::;:;;;::=:;::;=;:;:j 4 .. 160406 237358 314310 391262 468214 545166 622118 699070 Figure III-5-10. Equivalent Stre ss Distributio n of the Beam-Column Subassemblage W27xll4/Wl8xll9 with Beam Span 12' Case 1 44
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NODAL SOLUTION STEP=l SUB TIME"'l SEOV TOP (AVG) DMX "'5. 44 5 SMN E760.486 SMX .. 734047 SMXB .. 0.115E+07 (::::::x::i::;:::::l W!tiJ}l .. -760.486 82237 163713 245189 326666 408142 489618 571094 652571 734047 Figure lll-5-ll. Equivalent Stress Distribution of the Beam-Column Subassemblage W27xll4/Wl8xll9 with Beam Span 18' Case 2 45
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NODAL SOLUTION STEPcl SUB =1 TIMR=l SEQV (AVG) TOP DMX =6.014 SMN SMX .,752362 SMXB.115E+07 r::w:::::':::'J ; 3 .. -170391 253530 336669 419807 502946 586084 669223 752362 Figure III-5-12. Equivalent Stress Distribution of the Beam-Column Subassemblage W27xl14/W18xll9 with Beam Span 24' Case 3 4G
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NODAL SOLUTION STEP SUB THiF.l SEQV TOP (AVG) DMX .34 SMN SMX SMXB-374000 I:'}Wi;:::;:;:j 3 -46235 68316 90398 112480 134561 156643 178724 200806 Figure III-5-13. Equivalent Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 12' Case 4 47
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NODAL SOLUTION STEP"'1 SUB TIME SEOV TOP (AVG) DMX .526 SMN SMX SMXB 7 -49915 74143 98371 122599 146827 171055 195283 219511 Figure III-5-14. Equivalent Stress Distribution of the 13eam-Columu Subassemblage W36x210/W24x250 with Beam Span 18' Case 5 48
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NODAL SOLUTION STEP=1 SUB =1 TTME=1 SEOV (AVG) TOP DMX "'1.7 SMN .434 SMX "'228935 SMXB 6 : 3 4 -51514 76860 102206 127552 152898 178244 203590 228935 Figure III-5-15. Equivalent Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 24' Case 6 49
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NODAL SOLUTION STEP=1 SUB =1 TIME=l SEQV TOP (AVG) DMX -=2.967 SMN -=5176 SMX -=438183 SMXB t::::::;?::':d ; ; 8 !W@ 101400 ltm1l 149512 197624 245736 293848 341960 390071 -438183 Figure III-5-16. Equivalent Stress Distribution of the Beam-Column Subassemblage W30xl32/W24xl76 with Beam Span 18' Case 7 50
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NODAL SOLUTION STEP"'1 SUB =1 TIME"'1 SINT TOP (AVG) DMX "'4.836 SMN SMX "'798080 SMXB.123E+07 ; : 9 183191 271032 .. 358873 -446715 -534556 -622398 710239 798080 51 Figure III-5-17 Shear Stress Distribution of the Beam-Column Subassernblage W27x114/Wl8xll9 with Beam Span 12' Case 1
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NODAL SOLUTION SUB "'1 TIME"'l SINT (AVG) TOP DMX SMN .133 SMX .. 837080 SMXB.123E+07 : 8 3 3 HW#PJ 1111111111 186701 279612 372523 465435 558346 651258 744169 837080 52 Figure III-5-18. Shear Stress Distribution of the Beam-Column Subasscmblage W27xll4/Wl8x119 with Beam Span 18' Case 2
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NODAL SOLUTION STEPa1 SUB z1 TIM.E=1 s rN'J" TOP (AVG) DMX =6 .014 SMN "'4751 SMX SMXB.125E+07 1 mmrt.n 194 97 2 290083 385193 480304 575415 670525 765636 860747 53 Figure III-5-19. Shear Stress Distribution of the Beam-Column Subasscmblagc W27xll4/W 18xll9 with Beam Span 24' Case 3
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NODAL SOLUTION STEP SUB .. 1 Tl.ME .. l SINT (AVG) TOP DMX .34 SMN SMX SMXB-400457 ; ; 0 W@ 53387 11\\W 78885 -104383 129880 -155378 180876 206373 -231871 54 Figure fii-5-20 Shear Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 12' Case 4
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NODAL SOLUTION STEP-1 SUB "'1 TIME .. 1 SINT TOP (AVG) DMX .526 SMN SMX SMXB-397587 1 :::::::'::::;:::;:;::'1 1 mwm 57637 85613 113589 ... 141565 -169541 -197518 -225494 -253470 5 5 Figure lii5-21 Shear Stress Distribution of the Beam-Column Subassemblage W36x210/W24x250 with Beam Span 18' Case 5
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NODAL SOLUTION STEP-= 1 SUB -=1 TIME-=1 SINT TOP (AVG) DMX -=1.7 SMN .665 SMX -=264352 SMXB l:':':::::::::::;::;q 1111111 -949.665 30217 59483 88750 118017 147284 176551 205818 235085 264352 56 Figure III-5-22. Shear Stress Distribution of the Beam-Column Subasscmblagc W36x210/W24x250 with Beam Span 24' Case 6
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NODAL SOLUTION STEP=l SUB =1 SINT TOP (AVG) DMX =2.967 SMN =5977 SMX =495456 r:::::::::::::::::::l 4 twm 111111111 114750 169137 223523 277910 332296 386683 441069 495456 57 Figure III-5-23. Shear Stress Distribution of the Bcam-Coluillll SuGasscmblage W30xl32/W24xl76 with Beam Span 18' Case 7
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58 CHAPTER IV INELASTIC BEHAVIOR OF THE SUBASSEMBLAGE WITH AND WITHOUT CONTINUITY PLATES USING A SHELL FINITE ELEMENT MODEL IV-1 Inelastic Behavior of the Model without Continuity Plates IV-1-1 Problem Description In the linear model of the previous chapter, the stress levels obtained were much greater than the yield stress. In order to obtain more realistic results, inelastic analysis was necessary. For this, the seventh case with the shape beam and column dimensions of W30x132 and W24x176, and a beam span length of 18 feet was studied using inelastic analysis. In this case, the analysis with nonlinear material properties and bilinear kinematic hardening behavior was performed. The assumed stress strain curve is shown in Figure IV-1-1. The results of the previous chapter show the stress distribution at the joint. One may notice that in several regions, the stress levels are several times greater than the yield stress u11 For example, the maximum equivalent stress t7eqv in Case 7 is 438 ksi that is 12 times greater than u11 This indicates the
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59 ( IOU I) .5 I .5 2.5 3.5 4.5 EPS Figure IV-1-1. Assumed Stress-Strain Curve. need for inelastic analysis. It might also be noticed that the region that exceeds u11 is located near the connection, which is why this specific part of the subassemblage has been remodeled with nonlinear analyses. These changes are described later in this chapter in Section IV-1-3 on Discretization.
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60 IV-1-2 Problem Idealization The assumptions from Chapter III for the linear model were also used in the nonlinear analysis with several changes: The material properties of the joint follow the bilinear kinematic hardening How rule. The assumed strength values for steel A36 are: modulus of elasticity E = 29000 ksi, Poisson's ratio 11 = 0.3, stress yield at cr11 = 36000 psi, tangent modulus Et = 29000 psi. The lateral load is applied in the same directions as the linear model but with several substeps. The value of the force was increasing from 45 kips through 60 kips. IV -1-3 Discretization To reduce the computational burden on the VAX-8800 system, but still obtain meaningful results, the whole model was not analyzed for nonlinear behavior. Only part of the subassemblage, where the stresses were higher than 0.8u11, was analyzed nonlinearly, assuming that differences between linear and nonlinear be havior towards the ends of the cross-section are negligibly small. Therefore, a finite element model much like that described in Chapter III was used with 1421 nodes and 1272 shell elements. The same element type as in the previous model was used except for the joint in the subassemblage. To model the joint, new type element with nonlinear capabilities was introduced. Figure
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61 IV-1-2 shows the locations of both types of elements in the model. Each node has six degrees of freedom as was described in Chapter III. The boundary conditions, mesh, and direction of loading were unchanged from the fully linear model. The values of the horizontal forces have been in creased from 45 kips through 60 kips. IV -1-4 Results and Conclusions Appendix D shows the input file for the nonlinear model of case 7 that was set up parametrically. The maximum number of iterations was required to be less than 400. A hundred loading sub-steps were permitted. The data of the load displacement for inelastic behavior is presented in Table IV-2-1 and Figure IV-2-5. Horizontal deformation is displayed in Figure IV-1-3. The distribution of the equivalent (von Mises) stress under different lateral loads F 11 = 45 kips and F 11 = 57.315 kips is presented in Figures IV-1-4 and IV1-5. It should be noted that at load step F 11 = 45 kips, the equivalent stress is slightly higher that the yield stress ( CT eqv = 36005 psi). Thus the plastic stage of the process has been slowly taken place and deflection is an almost invisible c = 0.29". The maximum deflection of c = 1. 75" was reached at the load step F 11 = 57.315 kips. When the equivalent stress reached the critical value of CTeqv = 36764 psi, the plastified joint part of the subassemblage collapsed.
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62 IV -2 Inelastic Behavior of the model with the Continuity Plates IV-2-1 Problem Description The previous model has been modified to investigate the significance of the strength contribution of the continuity plates added to the subassemblage. The stresses, such as shear and von Mises and also the deformation of the top end of the column, were computed and compared to the model without continuity plates. IV-2-2 Problem Idealization Similar assumptions as for the nonlinear model without continuity plates were made. They were described in this chapter earlier. Several corrections have appeared: The loading is a horizontal force of F 11 = 45 kips through 70 kips applied with several substeps as described above for the model without the plates. The geometric and welding imperfections of the continuity plates were ig nored. The flow rule considered bilinear kinematic hardening. IV -2-3 Discretization The model for Case 7 with nonlinear behavior was taken for the foundation of
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63 this model with continuity plates. The model was discretized with 1509 nodes and 1368 elements. Two types of elements were operated which were described previously. As a result of adding the continuity plates, number of elements and nodes in the model has increased. Figure IV-2-1 shows the elements and mesh of the attached continuity plates. The boundary conditions and direction of loading were unchanged from the model without continuity plates. The horizontal force has been increased from 45 kips to 70 kips that have been applied in several substeps. IV-2-4 Results, Comparison and Conclusions To differentiate the results with continuity plates the previous model has been utilized. Appendix E shows the input file for the nonlinear model of case 7 with the continuity plates. The resulting load displacement with continuity plates is presented in Table IV-2-1. Figure IV-2-2 shows the load displacement curve. Increases in the loading force through stepsizes of 48, 52, 58, 60, and 62 kips indicates the expanding changes in the stress distribution in the structure. In other words, the dark portion indicates the section of the panel that exceeds the yield of u11 = 36100 psi, and the light section indicates that stress levels are in the elastic range ( u 11 j 36000 psi). Figure IV-2-3 and Table IV-2-2 present the von Mises and shear stresses in the element at the center of the subassemblage versus deflection.
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64 Figure IV-2-4 and Table IV-2-3 shows the von Mises and shear stresses at the center of the edge of the continuity plate (node A) and at the center of the beam next to the continuity plate (node B). Two load displacement curves with and without continuity plates are shown in Figure IV-2-5. In both curves, inelastic behavior was considered and the load limit for the assemblage with continuity plates is about 20% larger than without continuity plates. It can also be observed that after the load limit has been reached, the subassemblage does not provide any more resistance and act as a "plastic hints." The significant increase of the strength in the structure with continuity plates was observed.
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65 Figure IV-1-2 Finite Elements of the Inelastic Model Case 7
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66 Figure IY-1-3. Horizontal Deformation Distribution of the Beam-Column Subassemblage W30xl32/W24x176 with Beam Span 18' (Inelastic Model Case 7)
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NODAL SOLUTIOt\ STEP SUB -8 TIME=1 SEQV TOP (AVG) DMX SMN SMX r::=:::''''''''''""l B1\'l -.285717 -555.445 -36005 555.445 4494 8433 12372 16311 20250 24189 Figure IV-1-4. Equivalent Stress Distribution of the Beam-Column Subassemblagc W30xl32/W24xl76 with Beam Span 18' F v = 15kips 67
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SUB m999999 SEQV TOP (AVG) DMX "'1.774 SMN .815 SMX .. 400.815 4441 84 82 12522 16562 20603 24643 28683 32724 36764 Figure IV 1-5 Equivalent Stress Distribution of the Beam-Column Subasscmblage W30x132/W24xl76 with Beam Span 18' = 57kips 68
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69 Figure IY-2-1. Zoomed Part of the Joint with Continuity Plates
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70 I k 1 ps l 30 20 F 10 6 I 1 n l 0.0 2.0 4.0 6.0 Figure IV-2-2. Load Displacement Curve of Inelastic Model with Continuity Plates
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Slress ( ks i l 40 30 20 10 0.0 Shear Stress Von Mises Stress Element A 2.0 4.0 6.0 8.0 b (in l Figure IV -2-3. Yon Mises and Shear Stress at the Center of the Subassemblage versus Displacement -..J -
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Sl ress psi l 12000 9000 6000 3000 I I 0 0.0 Node A 2.0 Node B Max Shear Von Mises \ \ -cr -cr --'J ..cr-.......... .cr --o--,, .t: :: ---o -----o ---o ----o --o ---{) <( ,a' Node A 4.0 6.0 Node 8 8 (in) 8.0 Figure IV-2-4. Von Mises and Shear Stress at the Center of Continuity Plates versus Displacement -..J t-.:1
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F
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74 Table IV-2-1: Equilibrium Path for Inelastic Behaviour with & without Continuity Plates Without Plate With Plate Load 6 Load 6 0.45 0.003 0.45 0.003 0.9 0.006 28.8 0.17 1.8 0.01 45.0 0.26 3.6 0.02 45.25 0.26 7.2 0.04 45.5 0.27 14.4 0.09 46.0 0.27 28.8 0.18 47.0 0.27 45.0 0.28 48.0 0.28 45.15 0.28 50.0 0.29 45.3 0.28 52.0 0.30 45.6 0.29 54.0 0.33 46.2 0.29 58.0 0.45 46.8 0.31 60.0 0.61 48.0 0.31 62.0 0.87 50.4 0.35 64.0 1.40 52.8 0.44 66.0 3.23 54.0 0.52 66.125 3.39 55.2 0.65 66.25 3.56 56.4 1.07 66.375 3.73 57.0 1.74 66.625 4.08 57.3 1.74 66.875 4.45 57.315 1.74 67.125 4.82 67.375 5.21 67.5 5.40 67.625 5.60 67.875 6.00 68.0 6.20 68.125 6.40 68.25 6.60 68.375 6.81 68.5 7.01 68.625 7.22 68.75 7.42 68.875 7.63 69.0 7.83 69.125 8.04 69.25 8.25 69.375 8.46 69.5 8.66 69.625 8.1 69.75 9.08 69.875 9.29 70.0 9.50
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75 I Table IV-2-2: Shear and von Mises stresses at the I Center of the Assemblage versus Deflection Load 6 Tma .. O"cqv 0.45 0.003 408.377 353.7 0.9 0.005 816.754 707.3 1.8 O.Dl 1633.51 1415 3.6 0.02 3267.02 2829 7.2 0.04 6534.03 5659 14.4 0.08 13068.1 11317 28.8 0.17 26136.1 22635 45 0. 26 40837.7 35367 45.25 0.26 41018.6 35523 45.5 0.27 41199.4 35680 46 0.27 41561 35993 47 0.27 41571.8 36002 48 0.28 41574.1 36004 50 0.29 41579.8 36009 52 0.31 41585.3 36014 54 0.33 41595.6 36023 58 0.46 41643.6 36065 60 0.61 41702.7 36116 62 0.87 41799.2 36199 64 1.4 41999.1 36372 66 3.21 42679.3 36961 66.125 3.39 42738.7 37013 66.25 3.56 42799.7 37066 66.375 3.73 42862.4 37120 66.625 4.08 42993.2 37233 66.875 4.45 43128.5 37350 67.125 4.82 43267.1 37471 67.375 5.21 43409.9 37594 67.5 5.40 43482.2 37657 67.625 5.6 43555.5 37720 67.875 6.0 43702.9 37848 68 6.2 43777.4 37912 68.125 6.40 43852.1 37977 68.25 6.60 43926.7 38042 68.375 6.81 44002.5 38107 68.5 7.01 44078.1 38173 68.625 7.22 44153.4 38238 68.75 7.42 44230 38304 68.875 7.63 44305. 38370 69 7.83 44382.2 38436 69.125 8.04 44458.8 38503 69.25 8.25 44535.82 38569 69.375 8.46 44612.7 38636 69.5 8.66 44689.8 38703 69.625 8.873 44767 38769 69.75 9.08 44844.3 38836 69.875 9.29 44921.8 38904 70 9.5 44999.4 38971
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76 Table IV-2-3: Shear and von Mises Stresses at the Center of the Continuity Plate versus Deflection Load 6 Tmax Tmaxc Ucqv Ucqvc 45 0.26 8669 1862 1810 7593 45.3 0.26 8706 1869 1816 7625 45.6 0.27 8742 1875 1822 7656 46.2 0.27 8815 1887 1834 7720 47.4 0.27 9009 929 1875 7891 48.6 0.28 9204 1971 1915 8063 51 0.29 9717 2106 2041 8522 53.4 0.30 10310 2279 2201 9050 55.8 0.33 10903 2500 2394 9573 60.6 0.45 11474 2593 2388 10038 63 0.61 11140 2114 1873 9737 65.4 0.87 10701 2389 2186 9347 67.8 1.40 10102 5227 4550 8810 70.2 3.23 9842 8765 7591 8568 70.4 3.39 9836 8960 7760 8562 70.5 3.56 9830 9149 7924 8556 70.7 3.73 9826 9332 8082 8551 71 4.08 9820 9670 8375 8545 71.3 4.45 9817 9986 8649 8542 71.6 4.82 9820 10275 8900 8543 71.9 5.21 9828 10535 9125 8549 72 5.40 9833 10658 9233 8554 72.2 5.60 9840 10775 9334 8559 72.5 6.00 9855 10997 9527 8571 72.6 6.20 9864 11105 9620 8579 72.8 6.40 9873 11208 9710 8586 72.9 6.60 9882 11309 9798 8594 73.1 6.81 11403 9879 8603 73.2 7.01 9904 11494 9959 8612 73.4 7.22 9915 11586 10039 8622 73.5 7.42 9928 11669 10111 8633 73.7 7.63 9940 11755 10185 8643 73.8 7.83 9953 11835 10255 8654 74 8.04 9970 11900 10300 8670 74.1 8.25 9981 11987 10387 8678 74.3 8.46 9996 12061 10451 8690 74.4 8.66 10010 12132 10512 8703 74.6 8.87 10026 12200 10572 8716 74.7 9.08 10041 12267 10630 8729 74.9 9.29 10057 12330 10685 8742 75 9.50 10073 12393 10739 8756
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CHAPTER V CONCLUSIONS AND RECOMMENDATIONS V-1 Conclusions 77 Based on the analysis of this study, the conclusions can be summarized as follows: The shear and panel zone contribution to the total deformation of the sub assemblage is more than 45% and cannot be neglected. The axial contri bution has virtually no effect. The panel zone contribution is about 22%. The significance of this contribution is extremely important since the total frame deformation used in current practice ignores this cumulative effect of panel zone deformation. To consider a realistic joint flexibility, the beam models of the frame struc tures should not have any rigid ends. The stress distribution in the subassemblage under lateral load is concentrated at the joint. Thus, in analyzing the model, more attention should be given to the cross-section. The increase of the load limit for the subassemblage using the continuity plates is about 20%.
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78 V -2 Recommendations To continue this research, more work is necessary in the following areas: To investigate the inelastic behavior of the subassemblage with continuity plates with a presence of axial load on the column. To investigate the existence of local buckling on vanous components of the subassemblage, a buckling analysis of the shell model is important to conduct. To develop a joint with a one degree of freedom element to reflect the panel zone deformation in terms of its geometric characteristic such as tw, bh til To optimize the shape, the dimensions and locations of the continuity plates to enforce the subassemblage strength capacity need to be examined. Figure V-2-1 represents several options of using "continuity plates." I il j II -" ---000 000 000 -r --= II II II Figure V-2-1. Potential Configurations for Panel Zone Reenforcement.
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APPENDIX A Preprocessor Input fot the Beam FE Model of the subassemblage /prep7 c=0.3 Ah=38.9 lxh=5770 dh=30.31 lh=12*18 Ac=51.7 Ixc=5680 dc=25.24 lc=12*12.5 twh=0.615 twc=0.75 fb=Ah/( db-twb )/twb fc=Ac/( dc-twc)/twc ppp=0.8* Ac*36000 mp,ex,1,29e6 et,1,3 r,l,lOOOO* Ab,Ixb,db r,2,10000* Ac,Ixc,dc r,3,10000* Ab,lOOOO*Ixb,db k,3 k,l,-lb/2 k,2,-c*dc/2 k,4,c*dc/2 k,5,lb/2 k,6,c*db/2 k,7,1c/2 k,8,-c*db/2 k,9,-1c/2 1,1,2 1,2,3 1,3,4 1,4,5 1,3,6 1,6,7 1,3,8, 1,8,9 LESIZE,l,lO LESIZE,4,10 LESIZE,6,10 79
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LESIZE,2,,1 LESIZE,3,,1 LESIZE,5,1 LESIZE,7,1 real,1 lmesh,1 lmesh,4 real,2 lmesh,6 lmesh,S real,3 lmesh,2 lmesh,3 lmesh,S lmesh,7 fini /solu NLGEO,ON SSTIF,ON AUTOTS,ON LNSRCH,ON NEQIT,30 NSUBST,10,30,2 OUTRES,NSOL,ALL dk,l,ux,,uy dk,S,uy fk, 7 ,fy,-ppp FK,9,Fy,ppp fk, 7 ,fx,lOOOOOO FK,9,FX,-1000000 SOLVE save fini fpostl set,last get,deltf,node,24,u,x kf=lOOOOOO/deltf fini /prep7 r,l,Ab,Ixb,db r ,2,Ac,Ixc,dc fini /solu NLGEO,ON SSTIF,ON AUTOTS,ON LNSRCH,ON NEQIT,30 NSUBST,10,30,2 OUTRES,NSOL,ALL dk,1,ux,,uy dk,S,uy fk, 7 ,fy,-ppp FK,9,Fy,ppp fk, 7 ,fx, 1000000 FK,9,FX,-1000000 SOLVE fini fpostl set,last get ,del tfa,node,24, u ,x kfa= 1000000/ deltfa fini fprep7 r,1,Ab,Ixb,db,fb r ,2,Ac,Ixc,dc,fc fini /solu NLGEO,ON SSTIF,ON AUTOTS,ON LNSRCH,ON NEQIT,30 NSUBST ,10,30,2 OUTRES,NSOL,ALL dk,1,ux,,uy dk,5,uy fk,7,fy,-ppp FK,9,Fy,ppp fk,7,fx,1000000 fk,9,fx,-1000000 80
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81 APPENDIX B Preprocessor Input su bassemblage for the FE Model for Buckling Analysis of the /prep7 c=1 Ab=38.9 lxb=5770 db=30.31 lb=12*18 Ac=51.7 lxc=5680 dc=25.24 lc=12*12.5 twb=0.615 twc=0.75 fb=Ab/( db-twb )/twb fc=Ac/( dc-twc)/twc ppp=0.8* Ac*36000 mp,ex,1,29e6 et,1,3 r,1,Ab,Ixb,db,fb r ,2,Ac,Ixc,dc,fc r,3,10000* Ab,10000*Ixb,db r ,3,A b,Ixb,dc,df k,3 k,1,-1b/2 k,2,-c*dc/2 k,4,c*dc/2 k,5,1b/2 k,6,c*db/2 k,7,lc/2 k,8,-c*db/2 k,9,-1c/2 1,1,2 1,2,3 1,3,4 1,4,5 1,3,6 1,6,7 1,3,8 1,8,9 LESIZE,1,10 LESIZE,4,10 LESIZE,6,10 LESIZE,8,10 LESIZE,2,1 LESIZE,3,1 LESIZE,5,1 LESIZE, 7,1 real,1 1mesh,1 lmesh,4 real,2 lmesh,6 1mesh,8 real,3 1mesh,2 lmesh,3 lmesh,5 lmesh,7 fini /solu pstres,on dk,1,ux,,uy dk,S,uy fk, 7 ,fy,-ppp FK,9,Fy,ppp fk, 7 ,fx, 1000000 fk,9 ,fx,-1 000000 solve fini /solu antype,buck bucopt,subsp,5 save & solve & fini /solu expass,on m.xpand,5 outres,nsol,all & solve & fini
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82 APPENDIX C Preprocessor Input for the Simple Shell FE Model of a Cantilever Beam Used as Verification Problem /PREP7 MP,EX,1,29e6 ET,1,63 lb=lO tb=O.l r,l,tb wb=l k,l k,2,lb k,3,lb,wb k,4,wb a,1,2,3,4, elsi,0.5 amesh,l fini /solve nsel,loc,x d,all,all nsel,loc,x,lb f,22,fz,-0.5 ,23,fz,-2 f,2,fz,-0.5 nsel,all solve fini
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83 APPENDIX D Preprocessor Input for the Linear Elastic Shell FE Model of the Subassemblage Case 4 hfb=12.18 tfb=1.36 twh=0.83/2 dh=36.69-tfb 1h=12*12 bfc=13.85 tfc=1.89 twc=1.04/2 1c=12*12.5 dc=26.34-tfc fprep7 et,1,63 mp,ex,1,29e6 mp,nuxy,1,0.3 r,1,tfb r,2,twb r,3,tfc r,4,twc k,1,dc/2 k,2,dc/2,dh/2 k,3,dc/2,hfb/2,dh/2 k,4,1h/2 k,5 k,6,dc/2,hfb/2 k,7,dc/2,hfc/2 k,8,dc/2,1c/2 1,1,2 1,2,3 1,1,4 1,1,5 1,2,8 1,1,6 1,6,7 ad rag, 1 ,2, ,3 adrag,4,,,1,5 adrag,6, 7 ,,1 ,5 numm,all
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ESHAPE,2 LESIZE,8,,6 LESIZE,1,,6,1 I 5 LESIZE,21 ,,6 LESIZE,15,6,1/5 LESIZE,16,6 LESIZE,13,6,5 LESIZE,4,,6,5 LESIZE,27,1 LESIZE,22,1 LESIZE,7,1 LESIZE,10,12,10 LESIZE,12,,12,10 LESIZE,3,,12,10 LESIZE,5,12,10 LESIZE,26,12,10 LESIZE,28,12,10 LESIZE,18,12,10 LESIZE,24,4,5 LESIZE,2,4,5 LESIZE,6,4,5 real,4 ames,3 real,2 ames,! real,! ames,2 real,4 ames,4 real,3 amesh,5,8,1 ARSYM,X,ALL numm,all ARSYM,Z,ALL numm,all NSEL,S,LOC,X,lh/2 CERIG,62,ALL,roty,uz nsel,all nsel,s,loc,x,-lb /2 CERIG,380,ALL,roty,uz nsel,all n,all,uy d,all,uy d,all,rotx d,all,rotz da,l,symm da,4,symm da,3,symm da,ll,symm da,12,symm da,28,symm da,20,symm da,9,symm da,25,symm da,27,symm da,17,symm da,l9,symm fini /solu DK,4,UX,,UY,UZ DK,16,UZ,,UY FK,lS,FX,l000000/2 FK,36,FX,-1000000/2 save solve fini 84
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85 APPENDIX E Preprocessor Input for the Inelastic Shell FE Model of the Subassemblage without Continuity Plates hfb=12.18 tfb=1.36 twh=0.83/2 dh=36.69-tfb 1h=12*12 bfc=13.85 tfc=1.89 twc=1.04/2 1c=12*12.5 dc=26 .34-tfc /prep7 e=29e6 et,1,63 mp,ex,1,e mp,nuxy,1,0.3 fy=36000 et=e/1000 et,2,43 mp,ex,2,e mp,nuxy,2,0.3 tb,bkin,2,1 t bdata,1 ,fy,et r,1,tfb r,2,twb r,3,tfc r,4,twc k,1,dc/2 k,2,dc/2,db/2 k,3,dc/2,bfb/2,db/2 k,4,1b/2 k,5 k,6,dc/2,bfb/2 k, 7 ,dc/2,bfc/2 k,8,dc/2,lc/2 1,1,2 1,2,3 1,1,4 1,1,5
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1,2,8 1,1,6 1,6,7 adrag,1,2,,3 adrag,4,,,1 ,5 adrag,6, 7 ,,1,5 numm,all ESHAPE,2 LESIZE,8,6 LESIZE,1,6,1/5 LESIZE,21,,6 LESIZE,15,6,1/5 LESIZE,16,6 LESIZE,13,6,5 LESIZE,4,6,5 LESIZE,27 ,1 LESIZE,22,1 LESIZE, 7,1 LESIZE,10,12,10 LESIZE,12,12,10 LESIZE,3,12,10 LESIZE,5,,12,10 LESIZE,26,12,10 LESIZE,28,12,10 LESIZE,18,,12,10 LESIZE,24,,,4,5 LESIZE,2,4,5 LESIZE,6,4,5 real,4 ames,3 real,2 ames,1 real,1 ames,2 real,4 ames,4 real,3 amesh,5,8,1 ARSYM,X,ALL numm,all ARSYM,Z,ALL numm,all nse1,s,loc,x,-dc,dc nse1,r ,1oc,z,-db,db esln,s emodif,all,mat ,2 emodif,all,type,2, esel,al1 nse1,all NSEL,S,LOC,X,lb/2 CERIG,62,ALL,roty,uz nsel,all nsel,s,loc,x,-lb /2 CERIG,289,ALL,roty,uz nsel,all n,all,uy da,1,symm da,4,symm da,3,symm da,ll,symm da,12,symm da,28,symm da,20,symm da,9,symm da,25,symm da,27,symm da,17,symm da,19,symm NUMM,ALL & save & fini /solu NLGEO,OFF SSTIF,OFF autots,on lnsrch,on neqit,400 nsubst,100,1000,2 86 outres,all,all DK,4,UX,,UY,UZ DK,16,UZ,UY FK,15,FX,90000/2 FK,36,FX,-90000/2 FK,15,FX,120000/2 FK,36,FX,-140000/2 & lswrite solve & fini
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87 APPENDIX F Preprocessor Input for the Inelastic Shell FE Model of the Subassemblage with the Continuity Plates hfb=12.18 tfb=1.36 twh=0.83/2 dh=36.69-tfb lh=12*12 bfc=13.85 tfc=1.89 twc=1.04/2 lc=12*12.5 dc=26.34-tfc /prep7 e=29e6 et,1,63 mp,ex,1,e mp,nuxy,1,0.3 fy=36000 et=e/1000 et,2,43 mp,ex,2,e mp,nuxy,2,0.3 tb,bkin,2,1 tbdata,1,fy,et r,1,tfb r,2,twb r ,3,tfc r,4,twc k,1,dc/2 k,2,dc/2,db /2 k,3,dc/2,hfb/2,dh/2 k,4,lh/2 k,5 k,6,dc/2,bfb /2 k,7,dc/2,bfc/2 k,8,dc/2,lc/2 1,1,2 1,2,3 1,1,4 1,1,5 1,2,8 1,1,6 1,6,7 adrag,l ,2,,3 adrag,4,,,1,5 adrag,6, 7 ,,1,5 numm,all ESHAPE,2 LESIZE,8,6 LESIZE,1,6,1/5 LESIZE,21,6 LESIZE,15,,6,1/ 5 LESIZE,16,6 LESIZE,13,6,5 LESIZE,4,6,5 LESIZE,27 ,,1 LESIZE,22,1 LESIZE, 7,1 LESIZE,10,,12,10 LESIZE,12,,12,10 LESIZE,3,,12,10 LESIZE,5,12,10 LESIZE,26,12,10 LESIZE,28,12,10 LESIZE,18,12,10 LESIZE,24,4,5 LESIZE,2,,4,5 LESIZE,6 ,4,5 real,4 ames,3 real,2 ames,l real,l ames,2 real,4 ames,4
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rea.l,3 amesh,5,8,1 ARSYM,X,A11 numm,a.ll ARSYM,Z,A11 numm,a.ll nsel,s,loc,x,-dc,dc nsel,r ,loc,z,-db,db esln,s emodif,a.ll,mat ,2 emodif,a.ll, type,2, esel,a.ll nsel,a.ll NSE1,S,10C,X,lb/2 CERIG,62,A11,roty,uz nsel,a.ll nsel,s,loc,x,-lb /2 CERIG,289,A11,roty,uz nsel,a.ll lnsrch,on n,a.ll,uy da,l,symm da,4,symm da,3,symm da,ll,symm da,12,symm da,28,symm da,20,symm da,9,symm da,25,symm da,27,symm da,17,symm da,19,symm NUMM,A11 1,17,3 1,47,32 1DIV ,46,2 1DIV ,50,2 A,17,12,13,22,4 A,22,13,2,3,4 A,4 7 ,44,35,31,4 A,35,23,32,31,4 1ESIZE,68,4,5 1ESIZE, 72,,4,5 1ESIZE,58,6,1/5 1ESIZE,50,,,6,5 LESIZE,46,6,5 1ESIZE,56,6,1/5 ASE1,33,34,35,36 REA1,1 TYPE,2 AMESH,A11 ASE1,A11 numm,all save fini /solu N1GEO,OFF SSTIF,OFF autots,on lnsrch,on neqit,400 nsubst,100,1000,2 outres,all,all DK,4,UX,,UY,UZ DK,16,UZ,UY FK,15,FX,90000/2 FK,36,FX,-90000/2 lswrite solve FK,15,FX,120000/2 FK,36,FX,-140000/2 lswrite solve fini 88
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NOTATIONS Ab Shear area of the beam Ac Shear area of the column db Beam depth de Column depth E : Modulus of elasticity Et : Tangent modulus of elasticity f : Form factor FEA Finite element analysis Fu : Lateral load k : Subassemblage lateral stiffness lxb : Beam moment of inertia lxc : Column moment of inertia 4 :Beam bay lc : Column span tfb : Beam flange thickness tfc : Column :flange thickness twb : Beam web thickness twc Column web thickness v Poisson's ratio 5 : Deflection of the subassemblage due to lateral load 89
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6 J Deflection considering :flexural effects only 6 fa Deflection considering flexural and axial effects 6 faa Deflection considering flexural, axial and shear effects CTeqv Von Mises equivalent stress CT 11 : Yield stress T : Shear stress 90
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91 REFERENCES 1. Bertero V. Vitelmo, Egor P. Popov and Helmut Krawinkler. "Beam-Column Subassemblages under Repeated Loading," Structural Division, pp. 1137-1159; 1972. 2. Beker Roy. "Panel Zone Effect on the Strength and Stiffness of Steel Rigid Frames," Engineering Journal, pp. 19-29; 1975. 3. Tsai Keh-Chyuan, Egor P. Popov. "Seismic Panel Zone Design Effect on Elastic Story Drift in Steel and Frames," Journal of Structural Engineering, Vol. 116, pp. 3285-3301; 1989. 4. Sibai W. Atamaz & F. Frey. "New Semi-Rigid Joint Elements for Non-linear Analysis of Flexibly Connected Frames," Journal Constructural Steel Re search, Vol. 25, pp. 185-199; 1992. 5. Vlahinos, A. S., C. V. Smith & G. J. Simitses. "A Nonlinear Solution Scheme for Multistory, Multibay Plane Frames," International Journal Com puter Structures, Vol. 22, pp. 1035-1045; 1986.
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