.h. ‘ '~.- U -.'-\ ~ "- “ v .«r 13’; ”‘3‘ k ‘2‘ .. 3M ‘ . '3 I Q x)»:- . v '. I, k " . ») mx“ - _yh‘ '> " '1‘ "5f ‘v‘u . 3w nu 9‘1 5‘4“ V‘( 4 I 2 51"?" $54 '1‘ . .. . 5"“> ‘1‘ (9.3) n \- V \ \‘I fiat.“ ... .w; 1' /"‘ . I.“ .‘ m V a ,. ! '5 “E“ Q . 1:31-14": ) I I ‘IQ‘I . 3‘. ‘g‘l‘ll‘QT‘. , fiat/4 1." ‘ «no !‘3’21‘ L .4 l “War-‘11: "In . I ' '1 ‘ {AK 4‘" “~" "4'1? _' ~35. A?! d! n 'r.» - A. I F ‘53"? '( '~‘ A; " ‘1'! J: ‘ 1' vi“. vi’.é .‘(1 I '34,, 1“ fl; ‘1‘“ 1 “ ”1" A u ‘ 7, 13>. J. I lit . lllmllllllullllllfllll 2 I (*1 H <6 7 ‘7 31293 LIBRARY Michigan State University This is to certify that the dissertation entitled INELASTIC-BUCKLING BEHAVIOR OF STEEL STRUTS: HYSTERETIC MODELING AND APPLICATIONS TO NONLINEAR "POST-FAILURE" ANALYSIS OF SPACE TRUSSES 0 presented by Mohamad-Samer Alawa has been accepted towards fulfillment of the requirements for Ph.D. Civil Engineering degree in i 03—. S &b ’ t 4 3' Major professor Date 11—6-1987 MS U is an Affirmative Action/Equal Opportunity Institution ‘ ‘ 7‘ - 0-12771 MSU LlBRARlES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. INELASTIC-BUCKLING BEHAVIOR OF STEEL STRUTS: HYSTERETIC MODELING AND APPLICATIONS TO NONLINEAR “POST-FAILURE" ANALYSIS OF SPACE TRUSSES BY Mohamad-Samer Alawa A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSPHY Department of Civil and Environmantal Engineering 1988 555 4547 ABSTRACT INELASTIC-BUCKLING BEHAVIOR OF STEEL STRUTS: HYSTERETIC MODELING AND APPLICATIONS TO NONLINEAR 'POST-FAILDRE' ANALYSIS OF SPACE TRUSSES By Mohamad-Samer Alawa A computationally efficient. element model was developed for ac- curately predicting the inelastic-buckling behavior of steel struts under generalized axial excitations. The model was based on theoretical concepts and the physics of the steel strut behavior, with limited reliance on experimental data. It is capable of simulating the effects of the element initial imperfection and end eccentricities, gradual plastification, deterioration under cyclic loads, and residual stresses on its response characteristics. The developed model of steel struts was verified against a wide variety of test results, and it was used for numerical studies leading to recommendations for selection of variables in the design of steel struts. The element model was incorporated into a computer program for non- linear analysis of space trusses. This computer program is capable of? predicting the inelastic force redistributions within the space truss systems following the yielding and buckling of some critical elements, with due consideration to the complex nonlinear behavior of the X—braces and the in-plane redundants. The structural analysis program is shown to predict the experimental nonlinear behavior, failure mechanism, and "post-failure" residual strength of different space truss systems with a reliable accuracy. The advantages of this program in.the design of structures with ductile failure modes and also in nonlinear analysis of the complex multiple-structure systems have been discussed. ACKNOWLEDGEMENT I would like to express my sincere gratitude to Dr. Parviz Soroushian under whose guidance this dissertation was written. His sug- gestions and criticism from the very beginning of the study up to the preparation of the final manuscript were of great value. I want also to thank Dr. R. K. Wen, Dr. J. L. Segerlind, and Dr. W. E. Kuan for their help as members of my guidance committee. This study was supported.by Michigan Energy and Resource Researdh Association (MERRA) and Michigan Utilities. This support is sincerely appreciated. Financial assistance from the Department of Civil and Environmental Engineering, Michigan State University, is gratefully ac- knowledged. Finally, thanks are extended to the Albert H. Case Center for Computer Aided Design in the Engineering College for providing the necessary computing and plotting facilities. iv TABLE OF CONTENTS LIST OF TABLES ................................................. LIST OF FIGURES ................................................ LIST OF NOTATIONS .............................................. 1 INTRODUCTION ................................................ 2 .A LITERATURE REVIEW ON THE INELASTIC CYCLIC BEHAVIOR OF STEEL STRUTS AND THEIR CONNECTION ........................... 2.1 INTRODUCTION ......................................... 2.2 PHYSICAL BEHAVIOR OF STEEL STRUTS UNDER GENERALIZED LOADS ................................................ 2.3 EXPERIMENTAL RESULTS ON STEEL STRUTS ................. 2.3.1 Formation of Plastic Hinges ................. 2.3.2 Effects of Slenderness Ratio ................ 2.3.3 Effects of Local Buckling ................... 2.3.4 Effects of Initial Imperfection and Residual Stresses .................................... 2.3.5 Effective Length Concept ..... ................ 2.4 ANALYTICAL MODELING OF STEEL STRUTS UNDER GENERALIZED 2.4.2 Phenomenological Methods .................... 2.4.3 Physical Theory Models ...................... 2.5 STATE-OF-THE-ART IN PHYSICAL THEORY MODELING OF STRUTS 2.5.1 Gugerli Model ............................... 2.5.2 Ikeda Model ................................. 2.5.3 Zayas Model ................................. PHYSICAL BEHAVIOR OF STEEL STRUT CONNECTIONS ......... EXPERIMENTAL RESULTS ON STRUT CONNECTIONS ............ CONNECTION MODELING AND DESIGN ....................... SUMMARY AND CONCLUSIONS .............................. NNNN \OCDNCh ix 10 10 10 ‘ 13 l3 13 15 l7 l7 18 20 25 29 32 34 38 3 THE DEVELOPED ELEMENT MODEL ................................. 41 3. INTRODUCTION ......................................... 41 3.2 THE DEVELOPED FORMULATION ............................ 42 3.3 REFINEMENTS OF THE BASIC FORMULATION ................. 49 3.4 ADVANTAGES OF THE PROPOSED MODEL ..................... 58 3.5 SUMMARY AND CONCLUSIONS .............................. 60 4 VERIFICATION OF THE ELEMENT MODEL AND PARAMETRIC STUDIES .... 61 4.1 INTRODUCTION ......................................... 61 4.2 EMPIRICAL DERIVATION OF THE MODEL VARIABLES .......... 62 4.3 COMPARISONS WITH MONOTONIC EXPERIMENTAL RESULTS ...... 64 4.4 COMPARISONS WITH CYCLIC EXPERIMENTAL RESULTS ......... 71 4.5 NUMERICAL STUDIES AND DESIGN RECOMMENDATIONS FOR STEEL STRUTS ............................................... 89 4.5.1 The Influence of Yield Strength ............. 89 4.5.2 The Influence of End Connection Restraints 1. 93 4.5.3 The Influence of Cross-Sectional Shape ...... 93 4.5.4 The Influence of Initial Imperfection ....... 95 4.5.5 The Influence of End Eccentricity ........... 98 4.6 SUMMRY AND CONCLUSIONS ............................... 100 5 INELASTIC ANALYSIS OF SPACE TRUSSES ......................... 103 5.1 INTRODUCTION ......................................... 103 5.2 X-BRACING AND IN-PLANE REDUNDANTS : A REVIEW OF THE BEHAVIOR ................................ _ ............. 105 5.2.1 X-Bracing ................................... 106 5.2.2 In-Plane Redundants ......................... 107 5.3 A BACKGROUND ON THE INCREMENTAL NONLIEAR STRUCTURAL ANALYSIS OF SPACE TRUSSES ............................ 110 5.4 THE ADOPTED STRUCTURAL MODELING APPROACH ............. 114 5.4.1 X-Bracing Systems ........................... 115 5.4.2 In-Plane Redundants ......................... 117 5.5 ADOPTED STRUCTURAL ANALYSIS PROCEDURES AND NUMERICAL TECHNIQUES ........................................... 119 5.6 COMPARISONS WITH STRUCTURAL TEST RESULTS ............. 120 5.6.1 Double Layer Space Truss System ............. 121 5.6.2 Isolated X-Braced System .................... 131 5.6.3 Lattice Transmission Tower .................. 134 vi 5.7 SUMMARY AND CONCLUSIONS .............................. 6 SUNNART'AND CONCLUSIONS ..................................... LIST OF REFERENCES ooooooooooooooooooooooooooooooooooooooooooooo vii LIST OF TABLES 3 Cyclically Loaded Steel Struts .......................... 63 The Empirical Values of the Element Model Variables ..... 63 Material and Geometric Properties of L 3.54in X 3.54in X 0.27in (L 90mm X 90mm X 7mm) Tested in Monotonic Compression45 ............................................ 66 Properties of the Steel Struts Tested Under Monotonic Loads‘e"7 ............................................... 69 Properties of the Steel Struts Tested Under Cyclic Loadsl’3 74 Properties of Tower Elements ............................ 136 viii M NNNNNM .10 .ll .12 .13 2.14 NNNN .15 .16 .17 .18 .19 .20 .21 .22 .23 LIST OF FIGURES Steel Struts and Their Connections in Braced Frames and Space Trusses ........................................... Cyclic Behavior of Steel Struts ........................ Detailed Behavior of Plastic Hinges (TS 4in x 4in x 1/2 in, k1/r-80)1- ........................ Effects of Slenderness Ratio on the Strut Hysteretic Performance (W6x20)3 .................................... Effects of Cross-sectional Shape (partially caused by local buckling) on the Strut Hysteretic Performance10 .......... Effects of Initial Imperfection and Residual Stresses on the Initial Buckling load of Steel Strutsfi’s’7 ............ Inelastic Buckling Shape Compared with the Elastic Curve Earlier Strut Models2 .................................... Finite Element Model .................................... Examples of Phenomenological Models ..................... Typical Element Geometry in Physical Theory Models ...... Definition of the Location of the Instantaneous Neutral Axis ynsa .............................................. i.. Axial Load-Displacement Curve Used in the Gegerli Model Comparison Between Experimental and Gugerli Axial Load- Displacement Curves1 .................................... Empirical Relationships Used in Formulation of the Physical Theory Strut Model in Reference 1 ....................... Theoretical and Experimental Hysteresis Loops ........... The Physical Theory Strut Model of Reference 2 .......... Possible Failure Modes of Bolted Connections56 ........... Bending Moment in Connections Caused by Bowing of Steel Struts .................................................. Formation of End Plastic Hinges Due to The Flexural Yield- ing of Connections ...................................... Slippage at Bolted Connections .......................... Rotational Restraint of Gusset Plates in In-Plane and Out- Plane Bending ........................................... Effects of The Bolted Connection Slippage on The Steel Strut ix ll 11 12 12 14 14 16 16 21 21 22 24 26 26 31 31 31 33 33 .24 2.25 2.26 2.27 2.28 43 Behavior Under Load .................................... Idealization of The Steel Struts End Connections by Linear and Rotational Springs .................................. Strut Concentric Axial Force Used in Connection Design Axial-Flexural Forces Usually Induced in The Strut End . Connections ............................................. Cantilever Beam Simulation of Gusset Plates ............. 42 Effective Width of Gusset Plates ....................... Pinned-End Physical Theory Model ........................ Geometry and Deformation of Half-Strut .................. Effects of Partial Plastification on Axial Force-Plastic ninge Moment Relationship (W8X20, kl/r-120) ............. Effects of Partial Plastification on the Axial Force- Plasic Hinge Rotation Relationship1 ..................... Proposed Idealization of Partial Plastification at the Plastic Hinge ........................................... Proposed Model of Tangent Modulus of Elasticity ......... Idealized Axial Load-Displacement Upon Buckling of Straig- ht Struts ............................................... Sensitivity of the Developed Hysteretic Model to the Variations of the Empirical Variables (W6X20, k1/r-80) Experimental and Analytical Maximum Compressive Strength 45 of Angular Struts ...................................... Experimental and Analytical Axial Load-Deformation Relati- 46 47 onships of Concentrically Loaded Angles ............. Experimental and Analytical Axial Load-Deformation Relati- 46 47 ’ onships of Eccentrically Loaded Angles .............. Experimental and Analytical Axial Load-Deformation Relati- 46 47 ’ onships of Biaxially Loaded Angles .................. Experimental and Analytical Comparisons of Strut 1, Table 1 Experimental and Analytical Comparisons of Strut 2, Table 1 4.5 ..................................................... Experimental and Analytical Comparisons of Strut 3, Table 1 4.5 ..................................................... Experimental and Analytical Comparisons of Strut 4, Table 35 35 35 37 37 37 41 44 51 51 52 55 58 65 67 7O 72 73 76 78 80 UIU'IUWU'ILD UlbU-JNH U'IU1U1U! .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 0“ .10 .11 1 4.5 ..................................................... Experimental and Analytical Axial Load—Displacement Relationships of Strut 5 (Table 4.5) .................... Experimental and Analytical Axial Load-Displacement Relationships of Strut 6 (Table 4.5)3 .................... Experimental and Analytical Axial Load-Displacement Relationships of Strut 7 (Table 4.5)3 .................... Experimental and Analytical Axial Load-Displacement Relationships of Strut 8 (Table 4.5) .................... Experimental and Analytical Axial Load-Displacement Relationships of Strut 9 (Table 4.5)3 .................... Effects of Different Yield Strengths on the Axial Load- Displacement Relationships ....... i ....................... Effects of Variations in Yield Strength on the Axial Load- Displacement Relationships .............................. Effects of Rotational End Fixities on the Axial Load- Displacement Relationships .............................. Effects of Cross-Sectional Shapes on the Axial Load- Displacement Relationships .............................. Effects of Initial Imperfection on the Axial Load- Displacement Relationships .............................. Effects of End Eccentricities on the Axial Load- Displacement Relationships .............................. Analytical vs. Experimental Axial Forces in Truss Elements44 X—Bracing and In-Plane Redundants ....................... X-Bracing Test Configaration and Typical Results ........ Effects of In-Plane Redundants on X-Braced Systems ..... Stiffness Analysis Simulation in Incremental Nonlinear Analysis ................................................ Convergence Criteria in Incremental Nonlinear Analysis Typical X-Braced Panels with Similar or Opposing Senses of Forces in Crossing Diagonal Braces ...................... Partial / Complete Buckling Over the Element Length ..... The Selected Structural Systems ......................... The Geometry of the Double Layer Space Truss ............ Overall Analytical and Experimental Comparisons of the 57 Double Layer Space Truss ............................... xi 82 84 85 86 87 88 90 92 94 96 97 99 104 105 106 109 '112 112 116 118 122 123 124 5.12 U1 U1UlU'IU'I .13 .14 .15 .16 .17 57 Picture of the Buckled Structure ........................ Detailed Analytical Performance of the Double Layer Space Truss ................................................... Detailed Behavior of an Isolated X-Braced System ........ Tower Geometry .......................................... Tower Perspective ....................................... Analytical Behavior of Some Key Members in the Tower xii 124 126 132 135 136 140 K 5: Hi m tn rt Oa‘< n L) mi'p in t: r* R: 0 LIST OF NOTATION cross-sectional area incremental midspan moment incremental axial force incremental midspan lateral deflection incremental end axial displacement incremental plastic hinge axial displacement incremental end rotational displacement incremental plastic hinge rotational displacement incremental and displacement vector incremental plastic hinge displacement vector initial elastic modulus tangent modulus of elasticity yield stress axial tangent stiffness matrix of strut 2x2 tangent stiffness matrix of strut strut length midspan moment axial force strut buckling load axial yield force reduction factor accounts for local buckling as specified by AISC manual17 indicative of the level of the axial-flexural force combination indicative of the "first-yield" interaction diagram midspan lateral deflection initial imperfection elastic axial displacement geometric shortening plastic hinge displacement accumulated plastic hinge displacement tensile yield displacement axial strain, equals to 6/L axial strain when the P-PC xiii fully plastic interaction function , - partial derivative of Q with respect to M - partial derivative of Q with respect to P -u::< weighting factor plastic deformation scalar Qty-2000 end rotational displacement. Subscripts e - elastic p - plastic xiv CHAPTER 1 INTRODUCTION Steel struts are the main load carrying elements in space trusses, and also dominate the response of braced structural systems to certain loading conditions. While generally considered as pure axial elements in structural systems, the actual behavior of steel struts might involve substantial bending, caused by factors such as end eccentricities, initial imperfections, and post-buckling bowing of the element. Under cyclic loads, additional factors such as degradation of the strength and tangent modulus at critical regions further complicate the axial- flexural behavior of steel struts. Gradual plastification is another peculiarity of the strut performance, which can have tangible effects on its response to both monotonic and cyclic loads. Improvements in inelastic-buckling modeling of steel struts are pre-requisites to the development of reliable nonlinear structural analysis programs capable of simulating the inelastic force redistribu- tion in space trusses and braced structural systems under generalized loading conditions. Improved inelastic structural analysis techniques based on the refined element models will be effective tools in predict- ing the failure mechanism and "post-failure" behavior of structural systems, thereby providing information on the ductility of structural failure leading to efficient design modifications for improving this ductility. The "post-failure" behavior is of special significance in situations like complete transmission line systems, where a specific structure (e.g., a transmission line tower) reacts like a component of a larger system. Such structures will redistribute their forces to the adjacent components, if they reach the failure conditions. This redistribution phenomenon is strongly dependent on the "post-failure" behavior and residual strength of the collapsing components. Development of nonlinear analysis techniques for complete transmission line systems, capable of simulating the inelastic force redistributions following the failure of certain line structures is one of the major concerns of the electric power industry?4 The Research project reported herein has been concerned with the development of a reliable and computationally efficient model of steel struts, and the application of this element model for a more accurate prediction.of the inelastic behavior of space trusses, without a detailed treatment of the complexities in the connection nonlinear be- havior. This investigation has been performed in two phases. Phase one concentrated.on the element modeling, while the structural analysis ac- tivities were conducted in phase two. Chapters 2,3, and 4 report on different.stages of phase one, and Chapter 5 is devoted to the descrip- tion of phase two. 5 Chapter 2 presents the physical performance of steel struts ob- served in tests under monotonic and cyclic loads. This chapter also critically reviews the available element modeling methodologies for predicting the steel strut performance. A brief discussion on the ex- perimental behavior and analytical modeling of the common connections of steel struts is also presented in Chapter 2. Chapter 3 describes the development of the new strut model which is distinguished from the previous ones by its computational efficiency, accuracy in predicting the test results performed on a wide variety of steel struts, and a heavy reliance on the theoretical concepts governing the steel performance (with minimum dependence on specific test data). The developed element model has been verified in Chapter 4, using runnerous test data reported in the literature. The results of an analytical numerical study with the developed element model, aimed at generating some practical recommendations for design of steel struts, are also discussed in Chapter 4. Chapter 5 presents an illustration of some peculiarities of the space truss nonlinear behavior, and briefly discusses the analytical techniques popularly used in nonlinear analysis of structural systems. The advantages and shortcomings of these techniques in application to space trusses are also stated. The process of incorporating the developed element model into a structural analysis program for nonlinear analysis of the space truss systems is illustrated in Chapter 5. This Chapter makes a comparison between the experimental nonlinear behavior of three typical space trusses and the predictions of the developed analytical techniques. Detailed discussions on the inelastic force redistributions and failure mechanisms of these truss systems, as predicted by the analytical models, are also presented, and comments are made on the advantages of a reliable nonlinear structural analysis program in optimizing the structural design and in securing a ductile mode of failure. CHAPTER2 A LITERATURE REVIEW ON THE INELASTIC CYCLIC BEHAVIOR OF STEEL STRUTS AND THEIR CONNECTIONS 2.1 INTRODUCTION Steel struts are commonly used in braced frames (Figure 2.1a) and truss structural systems (Figure 2.1b). Their inelastic-buckling be- havior tends to dominate the nonlinear response characteristics of SUChL structures under static and dynamic loads. The steel strut performance is influenced not only by the material and geometric properties of the element itself, but also by the performance of their and connections. The end connections can modify the steel strut behavior by providing rotational restraint, generating end eccentricities, causing slippage (in the case of bolted connections), and weakening the element in the . vicinity of the connections. This chapter presents a review of the literature on the experimen: tal behavior of steel struts and the modeling techniques adopted for predicting their hysteretic behavior. The chapter also deals briefly with the experimental behavior and analytical modeling of the strut con- nections. 2.2 PHYSICAL BEHAVIOR OF STEEL STRUTS UNDER GENERALIZED LOADS The inelastic-buckling behavior of steel struts under cyclic loads, Figure 2.2a, involves several complex physical phenomena including: a) Braced Frame b) Steel Trusses Figure 2.1 Steel Struts and Their Connections in Braced Frames and Space Trusses buckling under compression and yielding in tension (Figure 2.2b), non- linear behavior prior to buckling in compression (Figure 2.2c), post- buckling loss of the compressive resistance (Figure 2.2d), deterioration of the buckling load in the subsequent inelastic cycles (Figure 2.2a), progressive degradation of tangent moduli during cycles in compression (Figure 2.2f) and in tension (Figure 2.2g), and plastic growth in the brace length (Figure 2.2h). The accuracy of the computed inelastic response of steel trusses and braced frames depends in large part on the accuracy of the strut 1 2 4 models. An ideal strut model is a theoretically-based (and thus generally applicable) one, capable of properly describing the axial force-deformation characteristics of the strut ( accounting for the physical phenomena shown in Figure 2.2). Moreover, the model should be computationally efficient for analysis of large structural systems. A considerable amount of information is now available on the initial buckling load of steel colt..unns.5’6’7 The understanding. of the inelastic behavior of struts under generalized loading conditions is, however, just emerging. A number of proposals have been made for mathe- matical modeling of the strut inelastic-buckling behavior under cyclic 1 2 3 4 a 9 I 1 7 loads, but the state of the art in this area has not yet developed to the point where all of the influential factors can be ac- counted for analytically. The key problems in this area seem to be associated with : a) Development of a computationally efficient and generally applicable model for predicting the strut hysteretic behavior;1’2 b) Effect of local plate buckling on the overall hysteretic behavior 134510 of struts; ’ ’ ’ ’ c) Degradation of the steel strut capacity and tangent modulus under repeated inelastic load reversals (resulting from the Bauschinger % 7U ' 53:7": . ’V' a) General Behavior b) Yielding and Buckling behavior P - \ so \ ‘ C) Nonlinearity Prior to d) Post—Buckling Loss of Initial Buckling Resistance I ’ . I ' "c I ' O O .I I '0‘. .’ 4 ‘ / O l o ”A 6 - ‘ a a 7 7AA I! ' i- 6 4; / ' l: o ‘l 0 FL. .; ’¢;~ ' fi‘ \ Compressive Tangent Modulus Degradation P f) Nag-d— ‘\ Q’s \ ‘ g) Tensile Tangent MOdUIUS h) Plastic Growth in Length Degradation Figure 2.2 Cyclic Behavior of Steel Struts 3 effect and gradual plastification of the highly stressed regions) ; and d) Nonlinearities prior to buckling of the virgin element loaded in compression (due to initial imperfection and residual stresses), 2513 and the consequent reduction of the initial buckling load; ’ ’ 2.3 EXPERIMENTAL RESULTS ON STEEL STRUTS Experimental studies have helped to identify the following impor- tant aspects of the strut behavior under cyclic loads 2.3.1 Formation of Plastic Hinges : Inelastic deformations concentrate in the critical regions of steel struts (plastic hinges) ,2’3’1"15’16 and repeated inelastic load reversals tend to pronounce this concentration.3 Figure 2.3a shows a typical plastic hinge axial force- moment relationship obtained in cyclic tests on steel struts.1 The theoretical first-yield and fully-plastic interaction diagrams are also presented. The actual plastic hinge behavior shown in this figure tends to deviate from the commonly assumed elastic-fully plastic type of response. The gradual plastification of the hinge seems to be a major factor influencing the behavior. This is confirmed in Figure 2.3b, while compares an empirical axial force-plastic hinge rotation relation- ship with a theoretical curve that has been based on an assumed rigid- fully plastic behavior of the hinge. It should be emphasized that plastic hinges deform both rotation- ally and.axially. The plastic hinge axial deformations have been found to significantly influence the hysteretic performance of steel struts 2 especially those with low and intermediate Slenderness ratios. NORMALIZED AXIAL FORCE. P/P. NORMALIZED PLASTIC HINGE MOMENT, HIM, a) Axial Force vs. Plastic. Hinge Momenst l.0 . TEST &' t THEORY ————— I «5 _ U 0.5 ’lrz/ // g ’5'.” / 1 LL. ————— tfl-T /’ . _J ________ , 1 _,- < O I I I § L ————— ——J ____ 41-:L.I I Y Q ‘“ u.) N :3 -O.5 - < 2'. c: 0 Z >‘I.O' 1 L 1 1 0.5 0.4 0.3 0.2 0.! O PLASTIC HINGE ROTATION. 0 (RADIANS) b) Axial Force vs. Plastic Hinge Rotation Figure 2.3 Detailed Behavior of Plastic Hinges 1 (TS 4in x 4in x 1/21n, Kl/r 80) 10 2.3.2 Effects of Slenderness Ratio : The strut effective Slenderness ratio has a dominant effect on its inelastic-buckling behavior.2’3 Struts with lower Slenderness ratios exhibit a more stable hysteretic performance. This is typically shown in Figure 2.4 which compares the experimental hysteretic envelopes of two struts with similar cross sec- tions but with different Slenderness ratios ( hysteretic envelopes a enclose all the hysteretic loops). 2.3.3 Effects of Local Buckling : Local plate buckling is generally recognized as a major factor causing the differences in the inelastic- buckling performance of struts with similar Slenderness ratios but with different cross-sectional sizes and shapes.3’10A T-section with unstif— fened plate elements having relatively large width-to-thickness ratios is shown in Figure 2.5 to have inferior buckling load, post-buckling behavior and energy dissipation capacity, when compared with a pipe made from a relatively thick plate.10 Test results have also indicated that the local buckling distortions tend to grow with increasing compressive strains, and they generally disappear at tension yielding.10 Even "compact" sections are observed to suffer local plate buckling at large inelastic compressive strains.10 The adverse effects of local plate buckling tend to grow under repeated inelastic load reversals (which cause deterioration at the plate buckling locations). 2.3.4 Effects of Initial Imperfection and Residual Stresses : These factors mainly reduce the initial compressive stiffness and strength of the virgin elements. Typical effects of the variations in initial im- perfection and residual stresses of struts on their response to monotonic compressive loads are shown in Figures 2.6a and 2.6b, respec- tively. ll ‘0 ——-K(h'|20 ———KIIIOUO O u O NORMALIZED AXIAL LOAD P/ P, 6 u. ——.-— p 401‘ _‘ _‘ ' ‘ NORMALIZED AXIAL DISPLACEMENT 8/87 Figur 2.4 Effects of Slenderness Ratio on the Strut 3 Hysteretic Performance (W6X20) '05 -— I we 1 20 —--¢ 4.0.337 ---04.41v4 ah ‘i 05— 0 <1 0 .1 .1 S. x 0 <1 0 Lu '3 .1 <1 2 -05- a: O 2 ll: I00 PINNED-PINNED _‘o J l J _ _ J l -4 -2 o 2 4 NORMALIZED AXIAL DISPLACEMENT 8/8, Figure 2.5 Effects of Cross-sectional Shape (partially caused by 10 local buckling) on the Strut Hysteretic Performance Load Axial Coup- Increasing 12 Imperfection buckling Load Residual Stress Zero 0 Low 0 High Axial Displ. a) Initial Imperfection Slenderness Ratio b) Residual Stersses Figure 2.6 Effects of Initial Imperfection and Residual Stresses on the Initial Buckling load of NORMALIZEO LAT. DISPL. ( A/Ammt) 5 6 7 teel Struts.’ ’ W6 x20 -— ELASTIC THEORY l/r =40 --- CYCLE 3 -'— CYCLE 5 .iii in I 1 1 05 NORMALJZED LENGTH(X/I) Figure 2.7 Inelastic Buckling Shape Compared with 3 the Elastic Curve 13 2.3.5 Effective Length Concept : Figure 2.7 compares, at different stages of a cyclic loading history, the buckled shapes of a steel strut fixed at one end and hinged at the other.3 Although the curvature tends to concentrate more in the plastic hinge region in later cycles, the inflection points coincide with the elastic predictions. The buckled shape is also found in Ref. 3 to be independent of the strut Slenderness ratio. Based on these observations, it seems reasonable to adhere to the effective length concept, according to which the portion of strut between the inflection points acts as a hinged-hinged element, even un- der inelastic load cycles. 2.4 ANALYTICAL MODELING OF STEEL STRUTS UNDER GENERALIZED LOADS Examples of the earlier steel strut hysteretic models used for non- linear dynamic analysis of space trusses and braced structures are: "tension-only" without compression resistance, Figure 2.8a, and "yield- elastic buckling", Figure 2.8b.2 These models can not realistically represent the actual hysteretic performance of struts shown in Figure 2.2a. More recently, three different modeling techniques have been used to predict the inelastic-buckling behavior of struts : finite element, phenomenological, and physical theory. 2.4.1 Finite Element Models : In this approach,1’2 the strut is divided longitudinally into a series of segments, Figure 2.9, which are further subdivided into a number of layers, and a large displacement analysis is performed for the complete system. The finite element technique is generally applicable to many types of problems. In this approach, only the member geometry and material properties need to be defined. The finite element technique in application to steel struts is, however, 14 AXIAL II AXIAL I LOAD TENSION LOAD ' TENSION AXIAL AXIAL DISPL. DISPL. COMPRESSION COMPRESSION . - 2 a) Ten51°n °nly b) Yield-Elastic Bukling 2 Figure 2.8 Earlier Strut Models . ‘- C—u— n” -§ M P P ’ Figure 2.9 Finite Element Model 15 computationally expensive, and given the large number of degrees of freedom in each member and the large storage requirements for the ele- ment property variables, the method is generally considered to be impractical for nonlinear analysis of large structures.1’2'3 Another disadvantage of the finite element modeling technique in application to some structural systems like lattice transmission towers with bolted connections is the inconsistency in the accuracies achievable at dif- ferent modeling, and analysis steps. Simulation of the behavior of a bolted connection and selection of the element material and geometric characteristics, residual stresses and initial imperfections for input to the finite element model are currently done at degrees of accuracy which are far below that of the finite element modeling technique. Hence, it is questionable if the accuracy of the final analytical results is any where close to the accuracy generally expected from the time- and storage-intensive finite element technique. The finite ele- ment models can, however, be useful tools in analytical studies on small subcomponents of structures and their elements. They can substitute experimental techniques commonly used to verify the simpler and more practical element models. 2.4.2 Phenomenological Models : The phenomenological models are cur- rently the most common strut models used for nonlinear analysis of steel trusses and braced structures. 1 Phenomenological models are based on simplified rules that only mimic the experimental axial force- displacement relationships, Figure 2.10. These models have only one local degree of freedom, axial deformation, and they are computationally efficient. Their users, however, should usually specify numerous em- pirical input parameters for each strut. Such parameters can be AXIAL LOAD I YIELD"/-. 16 AXIAL LOAD YIELD AXIAL DISPL. "BUCKLING 19 a) Nilforoushan, AXIAL LOAD I YIELD ‘/—7 I I I I I II AXIAL 01551. BUCKLING 20 b) Singh AXIAL‘ YIELD /~/ . 10 d) Jain AXIAL DISPL. ‘* 0BUCKLING AXIAL LOAD l YIELD ‘77 / I ‘IBUCKLING v AXIAL ofSpL 4 c) Maison LOAD _ AXIAL DISPL. 2 e) Ikeda IBUCKLING 1 Figure 2.10 Examples of Phenomenological Models PL ASTIC HINGE ’ P —>o” \ELASTIC Figure 2.11 Typical Element Geometry in Physical Theory Models 17 selected properly only if test results on struts similar to the ones under consideration become available. 2.4.3 Physical Theory Models : These models incorporate simplified theoretical formulations based on some assumptions on the physical be- havior of struts. The main assumption of these models is that inelastic deformations are concentrated in dimensionless plastic hinges at criti- cal locations of struts, Figure 2.11. Input parameters for physical theory strut models are simply the member geometric and material properties. They also have a small number of degrees of freedom. Their computational efficiency, however, has been damaged by the fact that the available formulations of physical theory strut models are based on the force method of analysis.1’2 This deficiency has been overcome ,111 this study by successfully applying a formulation based on the displacement method of analysis to these models. With this improvement, the physical theory modeling technique seems to combine the realism of the finite element approach with the computational simplicity of the phenomenologi- cal modeling technique, and to provide a promising method for simulating the inelastic-buckling behavior of steel struts in large structures. 2.5 STATE—OF-THE-ART IN PHYSICAL THEORY MODELING OF STRUTS Several physical theory models have been developed for simulating 1 a 9 10 16 18 22 23 24 the inelastic-buckling behavior of steel struts. ’ ’ ’ ’ ’ ’ ’ ’ These models usually have a plastic hinge at midspan which connects two elastic segments of the element, Figure 2.11. The axial component of the plastic hinge deformation was neglected in the early physical theory strut models in favor of the rotational component; however, both components have been included in the later 18 models. Most of the available physical theory models have been based on the following set of assumptions 1) Material properties are elastic-perfectly plastic; 2) The plastic state of the hinge is described by an interaction curve relating the fully plastic values of moment and axial force; 3) Partial plastification within the hinge and along the element length are disregarded; 4) The element bends uniaxially; 5) There is no possibility of local or torsional-flexural buckling; 6) Plane sections remain plane after bending; 7) Shear deformations are negligible; and 8) The effective length concept is valid even under inelastic load cycles. In the following, first a simple (basic) physical theory model is introduced, and then two recent models that represent the state-of the- art in this area are critically reviewed. 2.5.1 Gugerli Model : The axial displacement 6 of the physical theory element model proposed in Reference 38 consists of five components, 6 - 6 + 6 + 6 + 6 + 6 (2 l) e g P P0 CY where : 6 elastic axial displacement; on I geometric shortening; On I plastic hinge displacement; 0-. I accumulated plastic hinge displacement; and 6ty - tensile yield displacement. The elastic axial displacement 6e is expressed as 19 PL 6 - EA (2.2) e where A is the cross-sectional area. The expression for 6g is 6 h (k) 02 L 2 3 g " 1 p ( . ) where h1(k) = if P>O 2 k2 _ |P| L El 6p- plastic hinge rotation; and L - member length. The axial displacement, 6p, associated with the plastic hinge deformations takes the form : 7% P d6 (P ) * 5 =1 —P—— dP (2.4) P Po dP where Po is the axial force value at which the plastic hinge initially becomes plastic. Equation 2.4 takes into account the effects of the plastic hinge axial and rotational deformations on the strut length. A plastic rule 39 based on Druker’s postulate was also used by Gugerli to arrive at an expression for the incremental axial plastic hinge deformation d6 20 d6 - - d9 2.5 p yn p ( > where yn denotes the location of instantaneous neutral axis, Figure 2.12 The plastic hinge rotation 0p can be determined by solving the com- patibility and equilibrium equations of the elastic segments of the element model, with due consideration to the boundary conditions EI M - k — 0 2.6 g< ) L p ( ) where : M - plastic hinge moment; -—1'(— tan —k— if PO Five different zones shown in Figure 2.13 are used in formulating cyclic inelastic-buckling behavior of steel struts based on the above equations. The Gugerli model distinguishes between the elastic and plastic states of the plastic hinge behavior, and it accounts for the yielding of the strut when it straightens under tension. A typical com- parison of the Gugerli model with test results is presented in Figure 2.14. A major limitation of the Gugerli model is its failure to simu- late the drastic deterioration in the buckling load with cycles. 2.5.2 Ikeda Model : In this physical theory element model presented in Reference 1, the axial displacement of the strut is assumed to consist of seven components 6=6 +6 +6 +6 +6 +6 +6 (2.7) e g p m 21 118,, d8dy) i 89(y) d8p(Y) Figure 2.12 Definition of the Location of the Instantaneous 3 Neutral Axis yn3 . AXUUJFORCE,P AXIAL DISPLACEMENT, 6 Figure 2.13 Axial load-Displacement Curve Used 1 in the Gugerli Model 22 8 UN.) -2 -l 0 A I 2 I V . ‘ 1 V V Y V r Y T 1 ‘ 1500 300 EXPERIMENT - I000 200 b '00 '- d M P P (KIPsm % mm 400 ~ _ -500 -200 b l J J -* 4000 -50 -25 o 25 so 8 (mm) a UN.) -2 -l . . o 1 g - 1 g ‘ ‘ ‘ ‘ ‘ :500 30° * GUGERLI MODEL — nooo 200 ~ I, /’ a 500 '00 ‘- All p p “09510 0(KN) "00 t a -500 '200 P L , 7 1 1 — -sOOO -50 -25 o 25 so 8 (mm) Figure 2.14 Comparison Between Experimental and Gugerli 1 Axial Load-Displacement Curves 23 where : 6 , 6 , 6 , 6 and 6 were defined in the previous section e P P0 ty (see Equation 2.1) 6m - corrective displacement to remove errors resulting from sudden change of tangent modulus upon load reversal; and 6mo a residual corrective displacement. The elastic axial and geometric displacements, 6e£nu16 g are derived in Reference 1 using the equilibrium and compatibility equa- tions, with due consideration to the boundary conditions. The plastic hinge contributions to the element axial displacement, 6 and 6pc, are derived in terms of axial load using the fully-plastic interaction diagram concept together with the outward normal plastic flow rule. The physical theory strut model developed in Reference 1 also incorporates a number of empirical refinements illustrated below : a) Corrective factors are applied to the theoretical fully-plastic interaction diagram, Figure 2.15a; b) The material tangent modulus, used in calculating 6e and 6g, is expressed as a function of axial load, Figure 2.15b; and c) An axial force-plastic hinge rotation relationship is used to ac- count for the gradual plastification of the hinge, Figure 2.15c. Figure 2.16 presents a typical comparison between an experimental cyclic axial load-displacement relationship and the theoretical predic- tion of the model of Reference 1. This comparison is made for a 21/2 in X 0.049 in. pipe having a yield strength of 27.4 ksi and an effective length of 50 in. The model is observed to overestimate the initial and consequent buckling loads, and the maximum tensile loads and tangent stiffnesses of the strut in later cycles. 24 M-Lfll‘lo, wmucu r M INTERACTION cuuvc \ m rensum u - Lam THEORETICAL P-M INTER ACTION CURVU \ M - LII P) c, \ EMPIRICAL P-M INTERACTION CURVE IN COMPRESSION a) Fully-Plastic Interaction Diagram I use“ IDEALIZAI’ION cunvc wucu AXIAL ( I? 8 x l 8 l FORCE oecuuse _, I / I 5 l / I < o 4: i I a I e 4 J : use“ IDEALIZATION g I // ' /" CURVE WHEN AXIAL < l / I roucs menace z I '/ l / I S_' y' 1 Z 0 e. 82 e 3 NORMALIZED TANGENT MODULUS. ElE (E - 29000 ksi) b) Tangent Modulus vs. Axial Load c) Plastic Hinge Rotation vs. Axial Load Figure 2.15 Empirical Relationships Used in Formulation of the Physical Theory Strut Model in Reference 1 25 The discrepancies between the experimental results and theoretical predictions observed in Figure 2.16 are possibly caused by the following shortcomings of the model developed in Reference 1 a) Effects of the residual stresses and initial imperfection on the behavior of the virgin element in compression are neglected; b) The incorporated modifications in the fully-plastjx: interaction diagram and axial force-plastic hinge rotation relationship are purely empirical and based on limited test data. Their general applicability is thus questionable; c) Modification of the tangent modulus in this model is based on the assumption that the element is under a uniform stress condition along its length, disregarding the concentration of high axial- flexural stresses near the plastic hinge; and d) The possibility of local plate buckling is disregarded in the model. V There are a relatively large number of empirical parameters that should be input to this model. Moreover, the formulation of Reference 1 expresses the strut axial displacement in terms of axial force axui thus it is computationally inefficient for incorporation into the conven- tional computer programs for nonlinear analysis of large structures, that generally follow the displacement method of analysis. 2.5.3 Zayas Model : In order to overcome the computational inefficiency of the physical theory strut models, Reference 2 has introduced, but not fully developed or verified, a theoretical basis for development of an efficient formulation for these models, which is capable of illustrating axial force in terms of axial displacement. In this formulation, one half of the strut, that is assumed to be symmetric, is treated as a two- degree-of-freedom system (Figure 2.17), and the incremental forces , dP 26 20 fn‘ EXPERlHENT E 5 0.. Lu' U I: 0 LL. ._1 S x < 0.50 AXIAL DISPLACEMENT. 6 (IN) 20 5; Harman MODEL. & 5 Io - c- / 8 o / Z a: O u. ..I . .5 «o X < .90 I I -o.so 4125 o on 0.50 AXIAL DISPLACEMENT, 8 (IN) . 1 Figure 2.16 Theoretical and Experimental Hysteresis Loops Figure 2.17 The Physical Theory Strut Model of Reference 2 27 and dM, are related to the incremental displacements, d6 and d0, through a tangent stiffness matrix, kt : dP d6 [dM] - kt [d0] (2.8) The total incremental strut end deformations are assumed to be the sum of the elastic and plastic parts. d6 d6 d6 [d6] ' [doe] + [clap] (2'9) e P where the subscripts e and p represent the elastic and plastic contribu- tions, respectively. Zayas developed the following incremental equation, where the in- cremental axial force dP and moment dM at the plastic hinge are related to the elastic strut end deformations by means of the elastic force- deformation matrix B. dP d6 [w] .. B [(102] (2.10) The plastic portion of end displacements are related to the plastic hinge deformations by means of a geometric transformation matrix. d6 cosfi Lsinfl dL [clap] " [ o 1 ] [app] (2.11) P P where Lp is the axial plastic hinge deformation. . 39 The outward normal flow rule can be applied to the plastic hinge deformations. 28 dL ' O, [M] - [,9] . (2.12) P where : A - plastic deformation scalar; ¢ formula for the interaction curve; é’P - derivative of é with respect to P; and 0,M - derivative of 0 with respect to M. Combining Equations 2.11 and 2.12, the flow rule for the strut end deformations can be written as follows [22p] - A J (2.13) p [c050 é’P + L51n6 é’M] Q’M where : J - Employing the condition that the interaction function remains constant, one can arrive at the following equations dP d6 [1.] - B [.9] -J. (2.1.) d6 ’ A — a [do] .<2.15> where G - [Q,: B J]'1 [¢,T B] ‘1’. is ' [If] Finally, Equations 2.14 and 2.15 can be combined and solved for the tangent stiffness matrix Kt' Kt = (B - B.J.G) (2.16) 29 Static condensation is then used to reduce the element into a single-degree-of-freedom system : dP - K6 . d6 (2.17) K + x ( P F1- Ktzl tll t12 Kt22 - P.F2 where : K6 - K . A 1-( _ ) Kt22 P.F2 The formulation introduced in Reference 22 employs the outward nor- mal plastic flow rule and the fully-plastic interaction diagram concepts to simulate the plastic hinge behavior. This formulation, however, does not incorporate any of the refinements employed in Reference 1 (e.g. variation of tangent modulus as a function of force level and gradual plastification of the hinge). Some corrections and modifications were found to be necessary, and were incorporated in this research, for im- proving the above formulation. The approach, as introduced in Reference 2 and then modified.in.this study, provides an alternative implementa- tion strategy for physical theory brace models. 2.6 PHYSICAL BEHAVIOR OF STEEL STRUT CONNECTIONS Connections in space trusses can have detrimental effects on their overall behavior under load. The ability of connections to transfer moment and shear between elements, the rotational restraint they provide at element ends, their flexural capacity or capacity under different load combinations, slippage of bolted connections, and the weakening of elements at connections are among the factors related to connection be- havior that tend to modify the element and structural response characteristics under generalized loads. 30 Connections of steel trusses are generally designed for monotonic axial forces of elements. Figure 2.18 presents the possible failure modes of a typical bolted connection with gusset plate under this load- ing condition. The connections of struts in space trusses might also be subjected to large bending moments in addition to axial loads. The bending moments are usually caused by the bowing of elements under axial loads, Figure 2.19, and by the frame action of trusses (which depends on the rotational rigidity of connections). The connections capable of providing substantial rotational restraints at element ends, which are also the ones developing bending moments, might yield in flexure. The plastic hinges will thus form in- side the connections at element ends (Figure 2.20), causing a reduction in the end rotational restraint. The flexural yielding moment of con- nections depends on the direction of bending, the level of axial load, the history of loading, and also the details of connection design. Another critical aspect of the bolted connection behavior is the relative movement, slippage, of the connected elements and plates. Slippage occurs initially when the friction between the connected plates is overcome, Figure 2.21a, noticing that the bolts will not in general bear against the hole periphery before some slippage takes place. Eventually, after some slippage, the bolt will come in touch with the ‘hole periphery and will induce stress concentration and yielding at the contact region, leading to the enlargement of the hole, Figure 2.21b, and continuation of slippage at higher loads. Upon the load reversal, a more extensive frictional slippage (with no bearing on the hole) will take place before the contact is made in the other direction (inside the enlarged hole). As a result of this action, the slippage will tend to become more dominant under cyclic load applications. I HTD g L a 1 RM q: J J \l l_J Inn Shear lailur‘: OI bolt (bl Shear failure of plate Q; cfiwq \ _. K J \I _} (cl Bearing failure (d) Bearing failure of plate ol bolt ' E #:j» U ,\ 'hx l r 4 (cl Tensile Iailure (0 Bending fauure (g) Tensile failure ol bolts of bolts of plate 56 Figure 2.18 Possible Failure Modes of Bolted Connections .(Trc Figure 2.19 Bending Moment in Connections Caused by Bowing of Steel Struts ”( 4A 175% p P Figure 2.20 Formation of End Plastic Hinges Due to The Flexural Yielding of Connections 32 The element ends are generally weakened by the connecting action (e.g., drilling holes or welding). Welding might induce large residual stresses and the drilling of holes reduces the net cross sectional area. At both the welded and bolted connections, significant stress concentra- tion might be produced, leading to premature yielding and failure. 2.7 EXPERIMENTAL RESULTS ON STRUT CONNECTIONS Experimental studies on steel struts have generally been performed on struts having hinged or fixed supports. Test results on steel struts with realistic end connections are scarce, 'and very few comprehensive tests on truss connections are available in the literature. References 40,41 and 42 have reported results of monotonic tests performed during 1950's on truss connections with gusset plates. The results indicate that a complex state of stress is induced in the gusset plates, leading to major material and geometric nonlinearities. Test results summarized in Reference 15 indicate that the rotational restraint provided by the gusset plates influences the buckling strength of steel struts under monotonic loading. The restraining effects of the gusset plate connections strongly depend on the direction of element buckling. If the buckling causes bending of the gusset plates in their planes, the plates are capable of providing large rotational restraints (close to fixity), and their large in-plane flexural capacity forces the plastic hinges to form inside the element adjacent to the connections, Figure 2.22a. If the element buckling causes bending of the gusset plates out of their plane, the rotational restraint at the element ends are relatively small, and plastic hinges tend to occur inside the gusset plates, Figure 2.22b. Increasing the thickness (e.g., the rotational stiffness) of gusset plates is found to be especially effective in.ine creasing the element buckling load. The buckling load is not influenced 33 a) With No Bearing Against Hole Periphery b) Enlarging of Hole by Bearing Figure 2,21 Slippage at Bolted Connections a) In-Plane Bending b) Out-Plane Bending Figure 2.22 Rotational Restraint of Gusset Plates in In—Plane and Out-Plane Bending 34 much by the increase in the gusset plate yield strength, (e.g., moment capacity). The behavior of steel struts might be adversely influenced by a brittle failure associated with the rotation of the gusset plate connec- tions (as is true for their brittle axial failure modes). The test results presented in Reference 43 have indicated that gusset plates tend to fracture in a brittle manner at relatively small out-of-plane rota- tions, if sufficient free length is not provided in the plate for the formation of a plastic hinge. As mentioned earlier, the slippage of bolted connection (Figure 2.21) is an important factor influencing the connection and element be- havior under load. Slippage causes a sudden jump in the strut axial deformations at a constant axial load . As can be seen in Figure 2.23, slippage may occur at relatively low axial loads, and can be followed by an almost elastic behavior of the strut. At higher loads, however, the local stress concentration at the bolt contact regions leads to yield- ing, hole enlargement and major nonlinearities. The increased slippage under load reversal is an indication of the enlargement of the bolt holes during the earlier load applications. This phenomenon would be pronounced as the load cycles are repeated. The considerable "pinching" of the hysteretic loops resulting from the slippage of bolted connec- tions damages the strut behavior by reducing its hysteretic energy absorption capacity and axial stiffness. 2 . 8 CONNECTION MODELING AND DESIGN Limited analytical studies have been conducted on the steel strut 1 connections. Some investigators have simply substituted the strut end connections with linear and/or rotational springs, Figure 2.24. 35 I: . _____ 0' o I'— - l I on I I , _/ I I l I I l 04 - I I I 0.2-»! I She I g . / . l -. -; ' i i o I 3. SI; 03$ SICTIOI 2133:1ka m: J -¢{_ Figure 2.213 Effects of The Bolted Connection Slippage ‘3 on The Steel Strut Behavior Under Load ‘éCMNflSV/I5i:fz/”'_“‘\\\\\\5§\fisy~w:éw nlu u Figure 2.24 Idealization of The Steel Struts End Connections by Linear and Rotational Springs Figure 2.25 Strut Concentric Axial Force Used in Connection Design 36 However, no refined approaches have been reported for deriving the con- stitutive models (e.g., characteristic force and deformation values and 'hysteretic rules) of these idealized springs as functions of the physi- cal properties of typical strut connections. Some simple simulations, like those which treat the connection gusset plate as a cantilever beam:3 do not seem to realistically reproduce the actual behavior (especially with bolted connections) described above. Inadequate analytical simulation studies have also led to the adop- tion of simplistic design approaches for the steel strut connections. The current practice in design of steel strut connections is to propor- tion the connection to resist a constant tension force acting through the centroid of the member, Figure 2.25. The results of existing tests?3 however, indicate that the pure axial loading condition seldom exist in reality. The post-buckling behavior of steel struts induces large bending moments in addition to axial forces in connections. Hence, the design of strut end connections for the combined effects of bending moment and axial force, Figure 2.26, seems to be more ap- propriate. The plasticity condition of the connection under axial- flexural forces may be assumed to define the maximum force limits in designing the connections. In spite of the complex stress distribution existing in the steel 40 41 42 44 45 ’ ’ ’ ’ has been strut connections, the beam theory (Figure 2.27) found to satisfactorily predict the maximum of stresses in gusset plates under monotonic loading. In addition, Reference 42 proposes that only an effective width of the gusset plate in bolted connections is fully functional in resisting the applied forces. The maximum stresses should be, according to Reference 42, based on this effective width which can be Obtained (see Figure 2.28) by drawing 30 degree lines from the outer 37 Figure 2.26 Axial—Flexural Forces Usually Induced in The Strut End Connections Figure 2.27 Cantilever Beam Simulation of Gusset Plates 42 Figure 2.28 Effective Width of Gusset Plates 38 fastener in the first row to intersect with a line perpendicular to the line of force action passing through the bottom row of the fasteners. Recent studies, have indicated that some connection details might have detrimental effects on the ductility of failure. Reference 43, for example, suggests that a free length of the gusset plate equal to two times its thickness should be provided beyond the element ends. This free length enables the end plastic hinges, forming in the gusset plates bending out of their plane, to go through large rotations without restraints that cause brittle fracture of the plates in bending. 2.9 SUMMARY AND CONCLUSIONS The inelastic-buckling behavior of steel struts and their connec- tions plays a detrimental role in the response of steel trusses and braced frames to static and dynamic loads. This chapter has summarized the results of experimental studies on the inelastic, monotonic and cyclic, behavior of steel struts and their connections, and has criti- cally reviewed the suggested analytical techniques for predicting the response characteristics of the strut elements and connections under generalized loading conditions. The experimental results on steel struts have indicated that the element behavior under cyclic loads is strongly influenced by : a) concentration of inelastic deformations in the critical regions (plastic hinges); b) Slenderness ratio of the struts; c) local plate buckling; and d) initial imperfections and residual stresses. Inelastic cyclic test data have also indicated that the effective length concept, commonly applied to struts under monotonic loads, is also applicable under cyclic loads. Connections of the steel struts can significantly modify the strut behavior under generalized loads. They influence the element behavior 1- 39 through : a) their ability to transfer moment and shear between adjacent elements; b) rotational restraint of the element ends; c) weakening the element by the fastening action (drilling holes and welding); d) yield- ing and failure of the connections under axial-flexural forces; and e) slippage at the bolted connections. The more recently developed analytical models for predicting the steel strut load-deformation behavior under generalized loads can be categorized as the finite element, phenomenological, and physical theory models. Among these, the physical theory simulation techniques seem to combine the realism and generality of the finite element approach with the computational simplicity of the phenomenological modeling technique, and to provide a promising method for simulating the inelastic¥buckling behavior of steel struts in large structures. The physical theory models incorporate simplified theoretical formulations based on some assumptions on the physical behavior of struts. From a comprehensive review of the available physical theory models of steel struts, it may be concluded that improvements are needed in : a) their computational efficiency in analysis of large structures; b) consideration of the j gradual plastification of the element; c) accounting for the initial imperfections and residual stresses which cause nonlinearities prior to 'buckling of the virgin element loaded in compression; and d) ec- centricities of the axial load at element ends. There are also other aspects of the available physical theory strut models which require im- provements, but have been outside the scope of this investigation. These include : a) simulation of local plate buckling, b) the end rota- tional restraint provided at connections, and c) the possibility of torsional-flexural buckling. Limited analytical studies have been reported on steel strut con- nections, especially under cyclic loads. The suggested simplistic 40 simulations of the connection physical performance do not seem to provide realistic means of reproducing the actual behavior, especially in the case of bolted connections. The current practice in design of steel strut connections, based on simple connection models and loading conditions, also fails to account for the combined effects of the trans- ferred axial-flexural forces and the complex stress distribution existing inside connections. 6311211113 THE DEVEIDPED ELEMENT MODEL 3.1 INTRODUCTION The physical theory models of steel struts incorporate simplified theoretical formulations based on some assumptions on the physical be- ‘havior of'struts?"’5’9 The main assumption of these models is that the inelastic deformations at axial loads below the yield limit are con- centrated in dimensionless plastic hinges at critical locations of struts (Figure 3.1). The theoretical basis of the physical theory strut models improves their reliability and general applicability. PLASTIC HINGE ,dfif// ”‘ “ ~ ” ‘ Figure 3.1 Pinned-End Physical Theory Model The input parameters to physical theory strut models are simply the basic geometric and material properties of the element. These models also involve a small number of degrees of freedom. They are thus con- 'venient to define, and structural analysis employing these models would require relative small amounts of computer time and memory. The formulations presented in the literature for physical theory strut models, however, are computationally inefficient in the sense that 41 42 they use the axial force of the element as the input variable defining the external effects and the output is the axial displacement of the element. Hence, in application of these models to nonlinear analysis of structures using the conventional displacement method of analysis, a time-consuming iterative solution is necessary to obtain the axial force when the axial displacement is determined for each element. Other shortcomings of many of the available physical theory strut models result from their assumption of a rigid-perfectly plastic behavior in the plastic hinge and a fully elastic behavior in the element outside the plastic hinge. In the actual behavior, the plastic hinge plastifies partially prior to full yielding, and the partial plastification might also penetrate into the elastic segments outside the plastic region. The above shortcomings of the available physical theory strut models have been overcome in this study through: a) development of a computationally efficient formulation which defines the strut axial force in terms of axial displacement; and b) refining the conventional physical theory strut model in order to practically simulate the partial plastification and degradation in the plastic hinge region and along the element length. This chapter describes the proposed efficient formulation.of'the physical theory strut models and the refinements incorporated for simulating the partial plastification of the element. The accuracy of the developed model in predicting the experimental monotonic and cyclic performance of a variety of steel struts will be demonstrated in the next chapter. 3.2 THE DEVEIDP- FORMULATION The major assumptions on the basis of which the new formulation for physical theory strut model has been developed are 43 l)'The effective length concept is assumed to be valid even under in— elastic cyclic loads. Test results2’3 support the validity of this concept (which implies that a strut with any support conditions can be simulated as a simply supported element spanning between the strut inflection points). 2) In the basic formulation, the inelastic axial-flexural deformations (except for'axial yield in tension) are assumed to be concentrated in a dimensionless plastic hinge at the center of the effective length. It is also assumed that the plastic hinge behaves in a rigid-fully plastic manner. Later refinements of the basic for- mulation, however, will account approximately for the partial plastification outside the plastic hinge and along the element length, and the gradual plastification prior to full yielding of the plastic hinge. 3) Shear deformations are neglected. 4) Plane sections are assumed to remain plane after bending. 5) The element is assumed to bend and buckle uniaxially. Based on the above assumptions, an incremental solution procedure is employed where the incremental forces are related to the incremental deformations by means of a tangent stiffness matrix. For the purpose of developing this stiffness matrix, the strut geometry (Figure 3.1) may be simplified as shown inFigure 3.2 using the symmetry about the mid-span txf the effective length. This figure also shows the element forces and deformations. In the development of the basic formulamdxnn, a key con: sideration for improving the computational efficiency is to express the element forces in terms of its displacements. When the plastic hinge is activated (i.e. when the axial-flexural forces at the plastic hinge location reach the fully-plastic interaction 44 P dA Figure 3.2 Geometry and Deformations of Half-Strut diagram), the total incremental end deformations of the strut, d6 and d6 are the sums of the elastic, dSe and dfle, and plastic, dSP and dop dv - dv i dv (3.1) p e . _ |d5| . _ I“ l . _ I“ l where . du [Idal , due Idozl , and dup Idogl Note: n this ormulation whereve 4' i used. L+) corresponds to tensile axial forceI and the (~) to compressive axial force. The midspan lateral deflection, dA, is also assumed to be the sum total of the elastic and plastic parts. |dA| - |dAp| i ldAel (3.2) The elastic midspan lateral deflection can be expressed in terms of the elastic end deformations: 45 (D I H- O p O N f—_—| 0. § 0 L_:: -+ - C . due (3.3) The plastic midspan lateral deflection may be kinematicaly related to the plastic hinge deformations, dip and dip: dAp - i [_s;n9 -% c050] dip (3.4) where : d; - ldip' P Idopl The plastic end deformations are related to the plastic hinge deformations by means of a transformation matrix: d6p coso -L sin0 dip do ' o 1 d? (3'5) P P Combining Equations 3.4 and 3.5 one can express the plastic lateral deflection of midspan in terms of the plastic end deformations: -1 _ _§iflfl _L cosfi -L sine |d6 | “p i" 2 2m“ [ 0 1 ] [magi] _ .5221 ____L__ |d5 | i I 2 2 c050] [|d0§|] - i D - d . up (3 6) The incremental axial force and plastic hinge moment, (11’ and dM, can be related to the elastic end deformations of the strut and the plastic lateral midspan deflection by means of the elastic force- 46 deformation matrix, B: dS - i B due i R dAp (3.7) where : dS - [:3]; and R - [g] Combining Equations 3.6 and 3.7, one gets: dS - i B due i Q dup (3.8) where : Q - [P21 P32] The outward normal flow rule can be applied to the plastic hinge deformations; dup - A 0 4?,5 (3.9) . <5. . where . é’s [Q P] , O - fully plastic interaction function; é’P - partial derivative of ¢ with respect to P; 0,“ - partial derivative of o with respect to M; and A - plastic deformation scalar. The fully plastic interaction function, Q, is being derived with due consideration to static equilibrium between the possible forces at the plastic hinge location (P and M) and the resultants of the internal stresses across the plastic hinge (assuming all the fibers in the plas- tic hinge reached their yield stress in tension or in compression). Combining Equations 3.5 and 3.9, the flow rule for the strut end 47 deformations can be obtained : dl/ - X o J (3.10) P . where : J _ [¢,P coso - 0,“ L 51nd] Q," The condition that the interaction function remains constant yields that the incremental forces are tangent to the interaction surface: O - 0 d¢ - 0 o,: . as - o (3.11) combining Equations 3.1 and 3.8, one gets: dS - B dv + (i Q -B) dup (3.12) Equations 3.10, 3.11, and 3.12 can now be solved for the plastic scalar, A A - G o dv (3.13) where : c - -[oTs (iQ-B) J1'1[¢?S B] Equations 3.10, 3.12, and 3.13 can be solved for the tangent stiff- ness matrix, Kt : dS - i Kt du (3.14) where : Kt - [B + (iQ-B) J G] 48 Equation 3.14 can be used to solve for the incremental forces whenL the ixuzremental end displacements are defined. The incremental midspan lateral deflection can also be expressed in terms of the incremental end displacements using Equations 3.2, 3.3, and 3.6 : dA - i F dv (3.15) where: F - [C + (D-C) J G] The tangent stiffness matrix, kt’ as well as the matrix relating the midspan lateral deflection to end deformations, F, were derived in the above formulation assuming that the plastic hinge has reached yield conditions, and is loaded inelastically. e t oa‘d , dvp and A are zero and thus matrices kt and F can be simplified as fol- lows k - B (3.16) F - C (3.17) The tangent stiffness matrix, kt’ is a 2x2 matrix. For use in com- puter structural analysis it is more convenient to represent the brace in terms of the single degree of freedom, d6. The tangent stiffness, kt' can be reduced to a scalar by imposing the conditions of equi- librium: M - p . A (3.18) or dM.- dP . A + P . d A (3.19) 49 Combining Equations 3.14, 3.15, and 3.19, the single degree of freedom stiffness equations can be obtained : dP - K5 - d6 (3.20) do - z - d6 (3.21) :11 :12 x - 9.? :22 2 where k - , and 5 K .A t12 1 ’ ( K - P F ) t22 ° 2 z _ A -x + P'Fl Kt21 K:22 ' P"“12 5 K:22 ’ P F2 After each incremental step the element geometry is internally up- dated and the matrices k6 and F are computed according to the above formulations for use in the next step. In each step, the incremental value of axial displacement (d6) is the input. The proposed element model is computationally efficient in application. to complete struc- tures, because in step by step nonlinear structural analysis, following the conventional displacement method, the input to the element model is the incremental displacement. This is compatible with the proposed ele- ment formulation, and thus iterative solutions at the element level can be avoided. 3.3 REFINEHENTS OF THE BASIC FORMULATION The basic formulation for physical theory modeling of steel struts presented above needs to be modified in order to account for: a) the effects of the partial plastification in the plastic hinge region and 50 along the element length; b) full tensile yielding; and c) the pos- sibility of buckling of straight element (the element might be straight prior to loading or be straightened by tensile yielding during the loading). A.general discussion on these refinements and the procedures followed for incorporating them into the basic formulation are described below. The experimental data on the cyclic behavior of the plastic hinge in steel struts indicate a partial plastification of the hinge before its full yielding. Figure 3.3a presents a typical experimental relationship between the axial load and bending moment at the plastic hinge region of a cyclically loaded steel struts. Figure 3.3b shows the same relationship produced analytically, assuming an elastic-perfectly plastic hinge behavior. The experimental plastic hinge behavior tends to deviate from an elastic performance before the fully plastic condi- tion is reached. The gradual plastification process is indicative of some partial yielding across the critical sections under the axial- flexural loading. Another indication of the partial plastification in plastic hinges is shown in Figure 3.4. This figure compares an ex- perimental axial load-plastic hinge rotation relationship with the corresponding analytical relationship developed assuming an elastic- perfectly plastic behavior. The analytical curve can not predict the gradual transition from an elastic to a perfectly plastic behavior. In order to simulate the partial plastification of hinges in physi- cal theory modeling of steel struts, it is proposed that the overall performance of the struts beyond a "first—yield" level of the axial- flexural load combinations can be presented as a weighted average of the elastic and fully plastic types of performance, with the fully plastic one becoming dominant as the fully plastic interaction diagram is ap- proached. 51 ‘M “ 1 j a) Experimental :00 .§ I x INA 4 g o " I g PIN-3 5 4 -m'I -wEJaoV-zvoof-{oo' 0 W150 1 :50 V no MOMENT, 06p: - In 300 ‘ A 1 tical ”0‘ b) as y .3 I x Iflh AXIAL LW. # o -‘w‘ 1 -100. 4 -3021“ v -foo f-"oo V b V I?” - 250 V 500 HOUENI, Kips - in Figure 3.3 Effects of Partial Plastification on Axial Force- Plastic Hinge Moment Relationship (W8X20, Kl/r-lZO) 0: it |.0 — GUGERLI MODEL uj ---- EXPERIMENT U P “o‘ 05 . LL . '<' ' '17:: 52 O ’ < LN 8 b :3 ~05.» .4 < )- 5 -‘O a 1 n 1 g 0.5 0.4 0.3 0.2 0.! 0 PLASTIC HINGE ROTATION, 9 (RADIANS) Figure 3. 4 Effects of Partial Plastification on Axial Force- Plastic Hinge Rotation Relationship 52 Fully-Plastic (1:1) Plastic 1_Elastic P/PY 1.... \\‘\. First-Yield /,__ \ / \ 0 #0 BC . 1 NORMALIZED SENDING MOMENT, M/MP NORMALIZED AXIAL LOAD. 3" a“ / / \ Figure 3.5 Proposed Idealization of Partial Plastification at the Plastic Hinge A so called "first-yield interaction diagram" was defined as the limit for a pure elastic performance (Figure 3.5). Partial plastifica- tion is assumed to be initiated when the load combination at the plastic hinge location exceeds this "first—yield" interaction diagram. In the partial plastification region, the incremental values of the plastic hinge axial force and bending moment (dP & dM, respectively) are calcu- lated as weighted averages of the pure elastic and perfectly plastic ones (Figure 3.5) dP dP dP [dM] a 7 [dMe] + (1-7) [de] (3.22) e P where : dPe’ dMe - The axial load and bending moment increments assum- ing a pure elastic behavior; 53 de, de - The axial load and bending moment increments assum: ing a perfectly plastic behavior of the plastic hinge, with the fully plastic interaction diagram scaled down such that.the current load combination is located on it; and 7 - The weighting factor which ranges from 1.0 on the first-yield interaction diagram (pure elastic 'behavior) to 0.0 on the fully plastic interactive diagram (perfectly plastic behavior of the plastic hinge). The "first-yield" interaction diagram was assumed in this study to be a scaled down version (by a factor of 80) of the fully plastic one. When the strut is originally loaded, 50 is a function of the residual stresses (which decide the initiation of the partial plastification). The value of 80 is expected to decrease in the subsequent inelastic load cycles due to the Bauschinger effect and plastic hinge degradation. In order to consider this phenomena, 50 was expressed as a function of the inelastic load history : l pmax (3.23) ’Bo-azlfl |+b where : Elapmaxl - The sum total of the absolute values of the plastic hinge rotations at the load reversal points; a, b - Empirical coefficients (a-10.0, b-0.7 derived in Chapter 4 using 18 cyclic strut test results). The factor 7'in Equation 3.22 varies from 1.0 to 0.0 as the axial- flexural load combination moves from the elastic region to the fully 54 plastic condition. The following expression is suggested to be used for deriving 7 : 1 - 18 t m . .. —— 3.24 7 I l _ 30 I ( ) where : fit - an indicative of the level of the axial-flexural force combination (Figure 3.5); m - empirical coefficient 2.0 for P<0 (compression) ' 2 2.0 [“fi'] for P>0 (tension) P Under tension, m is expressed as a function of M/Mp to.account for the fact that as the member straightens (and the plastic hinge bending moment approaches zero), the member tends to yield under pure tension along its length. In this condition, m approaches zero (and 1 ap- proaches 1.0), in order to simulate the gradual spread of the inelastic deformations (axial yielding) along the element length and the less con- centration of inelasticities at the plastic hinge region. Another critical aspect of the steel strut hystereticlnflmyior which is not included in the basic formulation presented earlier, is the partial plastification and full-yielding along the element length. Due to the Bauschinger effect, the steel modulus of elasticity in the sub- sequent inelastic cycles tends to decrease under increasing levels of axial load. In the first cycle also, due to the presence of residual stresses, there is partial yielding along the element length at axial loads below the yield level. Another factor contributing to partial plastification along the element length is the spread of yielding from 55 0.0-i- -0‘u0 Normalized Axial Load P db CF/py -I.0 Normalized Tangent Modulus Figure 3.6 Proposed Model of Tangent Modulus of Elasticity the plastic hinge to the neighboring regions which are generally more critically stressed than locations further away from the plastic hinge. In order to approximately simulate of the partial plastification along the element length, the modulus of elasticity is expressed as a decreasing function of axial force, varying from a maximum value equal to the elastic modulus to a minimum value of the strain hardening modulus as the axial force increases from zero to the yield force of the element. The general configuration of the model for expressing the tan- gent modulus of elasticity as a decreasing function of axial force is shown in Figure 3.6. The value of a in this model was derived empiri- cally as a function of the sum total of the absolute values of the plastic hinge rotation at the load reversal points. 56 l a ' c 2|0 l + a (3.25) pmax where : c, d - Empirical coefficients (c-5.0 d-0.70, derived in Chapter 4 using 18 cyclic steel strut test results) The proposed formulation makes the partial plastification along the element length a function of the plastic hinge rotations. This reflects the fact that regions near the plastic hinge are more critically stressed and thus are more exposed to partial yielding. An important consequence of using the proposed empirical formulation of the tangent modulus of elasticity (as a decreasing function of the axial load) is the simulation of axial yielding of the strut. As the axial load in tension (or in compression) approaches the element yield strength, the steel modulus of elasticity in the element model gradually lowered to the level of the strain hardening modulus. Beyond the yield load, the modulus of elasticity stays constant at the strain hardening level. This accounts for the axial yielding of the element and the post-yield strain hardening of steel (assuming a bilinear stress-strain relationship). The possibility of sudden buckling of the straight strut was also considered in the proposed formulation of the physical theory strut modeljy. The element might be straight prior to loading, or it could be straightened during loading by yielding in tension. Buckling of the straight element is assumed to occur when the compressive axial load reaches the buckling load given by the AISC manual17 (without safety factors) 57 2 ' Q3[ 1 - -Lkl§£l 1 P for k1/rcC (RI) 2 2 n E 1/2 where : C = c Q F S Y Py - Axial yield strength; kl/r - Effective Slenderness ratio; E - Tangent modulus of elasticity; and QS - Reduction factor, accounts for local buckling as specified 17 by the AISC manual. Upon buckling of the straight element, the axial compressive load is assumed to remain constant as the lateral midspan deflection and con- sequently the end axial deformation increase to a level where the 'plastic hinge interaction diagram is reached (Figure 3.7). During this process, from the occurrence of the straight element buckling to the formation of the plastic hinge, the relationship between the axial and the midspan lateral displacements is expressed by the empirical 4 function where : A Lateral midspan strut deflection; L - Strut length; 6 - Axial strain, equals to 6/L; and £0 a Axial strain when the load equals Pc' 58 Axial load Axial Displ. — v . AISC Buckling Load . Yield Load Figure 3.7 Idealized Axial Load-Displacement Upon Buckling of Straight Struts 3.4 ADVANTAGES OF THE PROPOSED MODEL The suggested formulation for predicting the hysteretic behavior of steel struts is distinguished from the previous strut models by the fol- lowing advantages 1) It is practical (economical) for nonlinear analysis of complete structural systems. It involves a limited number of degrees of freedom and limited input information on the material and geometric characteristics of the element, when compared with the finite element and phenomenological models of steel struts. Relatively small com- puter storage and user times requirements are thus required for work. with element model. The developed formulation is also computation- ally efficient for structural analysis, in the sense that it defines 59 the element incremental forces in terms of the incremental displace- ments, and can be efficiently incorporated into the conventional computer programs for nonlinear analysis of complete structures by the incremental stiffness method of analysis. 2) In spite of some limited reliance on empirical refinements, the model is dominantly based on theoretical concepts, and thus, as confirmed in the nexmzchapter, it is generally applicable to steel struts with different geometric and material characteristics. 3) The proposed strut model can be used in structural analysis without requiring a large displacement analysis. 4) The sound.theoretical basis of the model makes it possible to incor- porate further refinements into the formulations with no conceptual problems. 5) The model approximately accounts for partial plastification within the plastic hinge and along the element length, following semi- empirical procedures which are based on the physics of strut behavior under cyclic loads. 6) The effects of the end eccentricity and initial imperfection of the element on the steel strut inelastic-buckling behavior under general- ized excitations have been accounted for in the proposed formulation. With the above advantages, as shown in the next chapter, the proposed model is capable of accurately predicting the hysteretic be- havior of steel struts with relatively small computer time and memory. 60 3.5 SUMMARY AND CONCLUSIONS The developed physical theory model of steel struts incorporates simplified theoretical formulations based on some assumptions on the physical behavior of steel struts. This model involves a limited number of degrees of freedom and requires limited computer storage for specify- ing the geometric and material characteristics of the element. A computationally efficient formulation has been developed for predicting the inelastic-buckling behavior of steel struts under generalized cyclic loading conditions. This formulation expresses the element incremental forces in terms of the incremental axial displace- ment. It can thus be efficiently incorporated into a conventional nonlinear structural analysis program which follows the incremental stiffness method of analysis. The basic formulation developed in this investigation is also capable of accounting for the effects of the initial imperfection and end eccentricities. I The basic physical theory strut models assume a rigid-perfectly plastic behavior of the plastic hinge and a pure elastic behavior along the element length outside the hinge. This basic model has been refined to consider, in a practical manner, the effects of partial plastifica- tion and degradation of the plastic hinge under cyclic loads and the softening, partial plastification and axial yielding along the element length outside the plastic hinge. These refinements were based on semi- empirical procedures, with due consideration to the physics of the strut inelastic-buckling behavior observed in tests. CHAPTER4 VERIFICATION OF THE ELEMENT MODEL AND PARAMETRIC STUDIES 4.1 INTRODUCTION The constant coefficients of the proposed physical theory brace model are derived empirically in this chapter using the results of tests on a variety of steel struts subjected to generalized axial excitations. The final version of the model is then verified by comparing its predic- tions with the experimental monotonic and cyclic axial load-deformation relationships of steel struts having different material and geometric properties, subjected to concentric and eccentric loads. Many of the tested struts used in this verification have not been used in deriving the empirical coefficients of the model. Following the full development and verification of the proposed model of steel struts, it has been used in a parametric study on the effects of different design variables on the monotonic and cyclic axial loadedeformation relationships of steel struts. In this parametric study, the effects of the material yield strength, end support rota- tional fixity (effective length), cross sectional shape, initial imperfection, and end eccentricity on the strut axial load-deformation characteristics have been evaluated. The results can help designers in optimizing their strut designs through the proper selection of design variables. 6l 62 4.2 EMPIRICAL DERIVATION OF THE MODEL VARIABLES The basic formulation of the proposed element model has been based. on some theoretical concepts and certain assumptions which comply with the physical performance of steel struts observed.iJ1 tests. 'Phe proposed refinements (for simulating the partial plastification and degradation at the plastic hinge and along the element length) were also based on some physically meaningful criteria. Partial plastification, however, is a complex phenomenon, and the proposed simple approach to the modeling of this phenomenon involves some degree of empiricism. Eighteen cyclic strut test results, reported in Reference 3, were used.tu: empirically derive the variables a & b (used for simulating the plastic hinge partial yielding and degradation in Equation 3.23 of Chapter 3), and c & d (used for idealizing the softening and partial or full yielding along the element length in Equation 3.25 of Chapter 3). Table 4.1 summarizes the material and geometric properties of these hinged-hinged steel struts which were subjected to severe cyclic loading histories. All the struts were concentrically loaded and their material yield strengths were measured using coupon tests. The variables a,b,c and d were selected to give desirable>>>rbIb bILr>bplh>~>s?>.kl>L 0.0 4 v f wngu 4 1 uuouumdec< iiiii . deucoauueaxu v Ila; mwxfl ROCHE 0&0 masons owuucooom An EX? .océ mmmzmmozflm as. on. map 00. nu on an o LIFhlPhhbbbbptLPhthbbbbbbbbLbehlL o-o Tngu sqoausaqc< ....... . aeucoauueaxu # o.- Ad/Od 'ovm ounxona daznvwaou Ad/Od ‘dvm 9Nn>lona 03ZI1VH80N maxs uofiefi on» Undone eunucooom A0 taxi .océ mmmzmmozmam nn_ ow_ nu. 00. we .om. ..n~ ..... +1.50 imau meouuxaec< IIIIII Aqueous-nun II o; oauuaoocoo Am Ehx .opé mmmzmwozmsm mm. on_ nu? 00. mm on mm o >P>>~P>PrrP>>prbyppPpp>P>>>>bn II 0.0 1mAu I v 703332 I I l I-.. #0., deucoauuennu muduum umaswc¢ mo camcouum m>wmmouaaoo adafixmz Hooauhaec< can aeuCoEHuonxm N.¢ ouswam v, Ad/Od 'dvm ounxone daznvwaou Ad/Od ‘ovm 9Nn>13na daznvnaou 68 struts with uniaxial eccentricity around the minor axis. These modifications were based on the physical behavior of steel struts ob- served in tests. It was assumed that prior to the formation of the plastic hinge, the axial deformations of the strut consist of three com- ponents related to axial shortening, bowing around the minor axis, euui ‘bowing around the major axis. After formation of the plastic hinge, in the post-peak region, the member is assumed to bow uniaxially around its minor axis. This assumption agrees with the observed behavior of biaxially loaded steel struts in tests. Upon the formation of the plas- tic hinge, it is assumed that the midspan lateral deformation around the minor axis increases to a value equal to the resultant of the minor and major axis lateral deformations, with the major axis lateral deforma- tions disappearing. The plastic hinge is formed when the axial-flexural force combination (assuming that the resultant lateral deformation oc- curs around the minor axis) reaches the fully plastic interaction diagram of the element in bending around the minor axis. In the post- peak region, additional bowing of the element is assumed to take place only around the minor axis, and the effects of the major axis ec- centricity are neglected. The shift in the direction of bending of the biaxially loaded struts is a phenomenon which has been observed in ex- periments. It is stimulated by some twisting of the element. The monotonic experimental axial load-deformation relationships of some angular elements were also compared with the theoretical ones. Table 4.4 presents the geometric and material characteristics as well as the end support conditions and eccentricities of the monotonically loaded angular struts used in these comparisonsfe"7 Figures 4.3a through 4.3d compare the experimental and analytical axial load-deformation relationships of the concentrically loaded an- gular elements No.1 through No.4 in Table 4.4. 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I . ’ . v f IIIIII \ in 703332 IIIIII neuceeauedxu . v Aq.¢ oHAeHv H usuum Au 5 .oz_zu._.mOIm ._<_x< 0.0 0.0 #6 N6 p r V r P F » p I p p n / IIIIIIIIIII u to.— ~eouu31a . .30» I . pupa. . . .Q.O. . » pNnO Do . 0.0 Heouubg I IIIIII anaconda-E 'NOISSHHdWOO 'IVIXV 5d!» 9d!» ‘NOIssaaawoo 'lVIXV 33 3me a sauna Au 5 .oz_Zu.EOIm .._<_x< 0.0 o6 To Nd I P s p P r I r P P u I p > p ’I’ 10:32.! illllll- anaconda-mud Ae.e oHAaHV m azuum Au 5 .oz_zw.EOIm ._<_x< o... 0.0 0.0 v.0 N6 06 IIIP_.>|II_>>pIPPIP>_>Lb>Doc IQN T 1 4 we» f r r wnfi acouuxnflz lllllll Hod» gauge-Eons.“— h .u&, wax 'NOISSBHdNOO wuxv ‘le 'NOISSEEdWOO 'MXV 73 -l 4 Experimental ” v ' - s ‘ O— - c. _ -- ,9- IO-I , ‘\ Analytical 3: i’ Q m Lu 0: 0.. 2 O O 1 c 1 V W Y I 1* I V T 7 Y I 1 T 1 V V V Y I Y ‘I l V 0.0 0.2 0.4 0.6 0.8 1.0 AXIAL SHORTENING, In a) Strut 9 (Table 4.4) .30 ‘ —Experinental m ‘ .. --..- - Analytical .9- 3 x d i’ Q In LIJ m G. 2 O Q X < 1 c . -.. ,T. .1, .s. .,- ... ,.se.. 0.0 0.2 0.4 0.6 0.8 1.0 AXIAL SHORTENING, In b) Strut 10 (Table 4.4) 40 : Experimental .3. : --_---- Analytical X i" 9 U) LIJ a: CL 2 O O x ---- < O ....,4...,-...,..--,..- 0.0 0.2 0.4 0.8 0.8 1.0 AXIAL SHORTENING, In c) Strut 11 (Table 4.4) Figure 4.5 Experimental and Analytical Axial Load-Deformation 4e 47 Relationships of Biaxially Loaded Angles ’ 74 the plastic hinge axial force-bending moment relationships. these analytical results were compared with the corresponding experimental ones. Some comparative studies were also made between the proposed analytical approach and some other available analytical models,-in order to demonstrate the advantages of the proposed formulation in efficiently and reliably predicting the experimental results. Comparisons were also made between the cyclic experimental and analytical behavior of the steel struts shown in Table 4.5. The wide selection of the steel strut cross-sections, Slenderness ratios, and material yield strengths (see Table 4.5) provide data for a comprehen- sive verification of the proposed physical theory steel strut model. Figures 4.6a through 4.14a present the experimental axial force- deformation relationships of 'the steel struts introduced in Table 4.5. The corresponding analytical relationships are presented in Figures 4.6b through 4.14b. The proposed analytical model is found to be capable of accurately predicting the cyclic axial load-deformation relationships for the wide selection of steel struts. Phenomena like the Table 4.5 Properties of the Steel Struts Tested Under Cyclic Loads ' Strut Cross- Slenderness Yield Strength End Connections No. Section Ratio, Kl/r Ksi Type I W 8 X 20 120 40.4 hinged-hinged 2 W 6 X 20 80 40.2 hinged-hinged 3 W 6 X 15.5 40 50.0 hinged-hinged 4 W 6 X 20 40 40.2 hinged-fixed 5 W 6 X 20 80 40.2 hinged-hinged 6 W 6 X 16 120 44.7 hinged-hinged 7 WT 8 X 22.5 80 41.8 hinged-hinged 8 Pipe 4 Std. 80 47.5 hinged-hinged 9 TS 4 X 4 X 1/4 80 59.0 hinged-hinged 75 deteriorations of the buckling load, axial stiffness and energy absorp- tion capacity under repeated inelastic load cycles are closely simulated by the proposed model. The analytical cyclic axial load-deformation relationships of steel struts No.1 through 4 derived using the refined physical theory strut model of Reference 1 (described in Chapter 2) are also presented in Figures 4.6c through 4.9c. The refined model of Reference 1, which in: volves a large number of empirical coefficients and is computationally inefficient (i.e., presents axial displacement in terms of axial load), is clearly inferior to the proposed formulation in predicting the deteriorating hysteretic behavior of steel struts under cyclic loads. The comparisons between the experimental and analytical plastic hinge axial force-bending moment relationships are presented in Figures 4.6(d,e,f) through 4.9(d,e,f). The analytical results have been derived using the proposed physical theory formulation and also the refined model of Reference 1. These comparisons are also indicative of the high degree of accuracy and the realism of the proposed formulation. they .also indicate the superiority of the proposed formulation over the com- plex physical theory model of Reference 1. The proposed model is observed to satisfactorily simulate the partial plastification phenomenon in the plastic hinge, which leads to a gradual transition from the elastic axial force-bending moment relationship to the fully plastic one. Based on the comparisons between the experimental and analytical performance characteristics of steel struts, it may be concluded that the proposed model provides a practical (economical) tool for accurate prediction of the steel strut inelastic-buckling behavior under general- ized loading conditions. The proposed techniques for the simulation of partial plastification also seem to produce a realistic idealization of 76 um I 200- V) a 4 .2 . 100- '8 I O ._I 0‘ :§ X < -uma -zoo . . . , .4 .4.4,4. . . , . . . , - . .4 -3 -2 —1 o I 2 Axial Displacement , in a) P-6 relationship; Experiment 0) .9 X 3 .J _J g X < -200 . - 4..- 4.4.4.4.. . . r4. - . , . -3 -2 -1 o 1 ; AXIAL DISPLACEMENT, in b) P-6 relationship; Developed Model 300 I 4 1 200m .1 m 4 .9 4 x 1 . iod- IMW O " - I a : MM 3‘ o; MIN/y,” _‘ I, / —100: my 3 I -200‘ - . . -,4. - ... . . . . . - .. . .. . —3 -2 -1 o 1 2 AXIAL DISPLACEMENT, in c) P-6 relationship; Ikeda Model Figure 4.6 Experimental and Analytical Comparisons of Strut 1, 1 Table 4.5 I Normalized Axial Load .1 P/PY' Normaliied Moment , M/MP d) M-P relationship; Experiment NORMALIZED AXIAL LOAD, P/PY NORMAUZED MOMENT, M/ MP e) M-P relationship; Developed Model NORMAUZED AXIAL LOAD, P/PY O I -i 0 I NORMAUZED MOMENT, M/MP f) M-P relationship; Ikeda Model I ‘ - Figure 4.6 Experimental and Analytical Comparisons of Strut l, 1 Table 4.5 (cont' d) 78 ZOOj kips 100d -IOO-I Axial Load . -2004 —JOO . , . - . , - - ‘ -2 -1 0 I 4! Axial Displacement . in a) P-6 relationship; Experiment 300 200- Kips lOO~ -100~ AXIAL LOAD. o -2004 —300 *r ‘r v f v v f fir j 1 T fi l -2 -1 O AXIAL DISPLACEMENT. in b) P-5 relationship; Developed Model 300 200d In a 22 100« o 1 8 a O .J 1 § -um~ X .( 4 -200-1 l -300f-.,-4- f+-,-.- -2 —I O I 2 Axial Displacement . in c) P-6 relationship; Ikeda Model Figure 4.7 ExperimenEal and Analytical Comparisons of Strut 2, Table 4.5 79 >' I a. 1‘ \ a 1 W 13 I O o 1 _I . T2 1 31‘ ° 1 U 1 o . N ~_-_: 1 O 4 E . ‘- 4 o 4 z ‘ s « a -‘ W v v ‘ v v v v v v v‘vg g*7 T v v ‘ v v v v -I .5 -I .0 -O.5 0.0 0.5 I.O' Normalized Moment , M/MP d) M-P relationship; Experiment >.. a 1‘ 3y I , 1 v i 8 4 .J .I “ — 1 | .g . , < O 1 ‘o l l a.) . I .t‘ . T: . E . ‘ i o 4 Z -I -1....,-..-..... ....r...- - l .5 -1.0 -0.5 0.0 0.5 1.0 Normalized Moment , M/MP e) M-P relationship; Developed Model >. a \ O. 1 P- B o "' ...l E ff 0 g r .5 _ 3 g . g -1 A 1 1 I a j I 1 --1.5 —1.0 -o.5 0.0 0.5 1.0 Normalized Moment , M/MP f) M-P relationship; Ikeda Model Figure 4.7 Experimental and Analytical Comparisons of Strut 2, Table 4.5 (cont' a)1 80 sec 2004 n 1 Q 2: 100~ . 4 : '0 c l’ll’."" 8 llfllyfll .J ‘ :§ .t'h f§ -4004 * X < i "2°07 J -300 4* - - *r”*’ r .. 1 . - ~ -2 -1 0 1 Axial Displacement . in a) P-6 relationship; Experiment 300 2004 MW m . O. 32 100« _ J c: 3 c z; / // _. ‘ / E: -400— X < . -200-4 W -300 re . . 1 . .. r i . 4r, - — -1 0 i AXIAL DISPLACEMENT. in b) P-6 relationship; Developed Model 400 , - l 2004 n ‘ . O. :2 100- 'B 0 O .J 4 f§ ~100a {5 ‘ _200d 4 -500 , f , r , f , , a - -2 -l 0 1 Axial Displacement , in c) P-6 relationship; Ikeda Model Figure 4.8 Experimental and Analytical Comparisons of Strut 3, Table 4.5 1 P/PY Normalized Axial Load . -1 fl Y V ‘r V V Y iii V ‘7 ‘Tfi f V Normalized Moment , M/MP d) M-P relationship; Experiment >- 0.. \\ O. U C O .J '6 Y ‘1 ‘O Q) a '6 E L 0 Z -1fi-w---.f:Ar.'+..rr... -L0 00 10 Normalized Moment . M/MP e) M-P relationship; Developed Model 1 >- O. \\ O. . p ‘O O O .J B 0 '§ < 'O Q) .E — F O E \- 0 Z -1 l J —L0 on :0 Normalized Moment . M/MP f) M-P relationship; Ikeda Model Figure 4.8 Experimental and Analytical Comparisons of Strut 3, Table 4.5 (cont' d)1 82 um 2°°‘ test flgfifig - 1 445%! 3' "we ;5§:=&5“’ ‘E 0‘ ”ij’]l 3 . ‘1‘?“ 1 -4 l/ '6 -1004 ‘=‘§EE:—" i 'i I. , < J :!=!; -200e V l -Mm - - l C Axial Displacement . in a) P-6 relationship; Experiment kips Axial Load , Axial Displacement . in b) P-6 relationship; Developed.Mode1 J00 Ikeda& Mahl'n kips Axial Load . -300 - e . 1 l . . -i O 1 Axial Displacement . in c) P-6 relationship; Ikeda Model Figure 4.9 Experimental and Analytical Comparisons of Strut 4, 1 Table 4.5 Figure 4.9 A A A A A A A Normalized Axial Load , P/PY o ' 7 fl V v v v v f ' ' ' 1' V ' v I I ' r ' ' l ' -l .5 - 1.0 -0.5 0.0 0.5 l.0 Normalized Moment . M/MP d) M-P relationship; Experiment P/PY \ ‘---‘---- Normalized Axial Load , -1 - —l.5 ' f' I ' V ' ' l r ‘ l ' ' ' ' I ' ' ' -l.0 -0.5 0.0 0.5 1.0 Normalized Moment , M/MP V e) M-P relationship; Developed Model ). Q. ~\ a 1 1°! 1- O .1 :§ ,__ g o 'D 0 .g 1- E O z-l 4 - I A ' -L5 -L0 ~as 00 as to Normalized Moment . M/MP f) M-P relationship; Ikeda Model Experimental and Analytical Comparisons of Strut 4, 1 Table 4.5 (cont' d) 84 Figure 4.10 Experimental and Analytical Axial Load-displacement .500 I Experimental 200- I/// m C. .4 . 100-J I ‘ /' ,4 .8 4 24);!“ O ‘1 _J O 1 '§ 4 <1 -100~ -I ‘200 . T l . T . TT‘T T T 1 TT T 1 —3 --2 -i O 1 Axial Displacement , in 300 4 Analytical 200- I .3 1 I’M e W 1004 I ”I . // g T 1.25.41 _I ' 2 ° — / “J —- =§55~73W~ >< ~ fi‘§’§:~~-‘ < ‘0 -100— ‘200 T T T i T T TTTr T TTT 1‘—T‘T‘TTT -3 -2 -1 0 1 in AXIAL DISPLACEMENT, Relationships of Strut 5 (Table 4.5)3 kips Axial Load , ikips Axial Load , 85 500 2004 100- Experimental -1OOd -200 500 Axial Displacement , in 2004 100- Analytical —100~ -ZOO Figure 4.11 Experimental and Analytical Axial Load-displacement Axial Displacement , in 3 Relationships of Strut 6 (Table 4.5) kips Axial Load , Kips AXIAL LOAD.- Figure 4.12 Experimental and Analytical Axial Load-displacement $00 86 200- 100— -IOOd -2004 —300 Experimental / / //:.::l;/I_‘--u-:~- 1 / _ ~ ‘: :5 EFL-”5:; fit r T T T j T I T T I I I I I j dI-i —2 - -1 0 Axial Displacement , in 300 200- 100— -100- -200— -300 r r I I I I I I ‘l’ I I j -2 -1 0 1 AXIAL DISPLACEMENT, in 3 Relationships of Strut 7 (Table 4.5) kips Axial Load , Kips AXIAL LOAD, 87 200- 100« Experimental -iOO- —2004 Axial Displacement , in Analytical I I 1 I fi T 1 I I T l j’fifi ' —2 31 o 1 2 AXIAL DISPLACEMENT, in Figure 4.13 Experimental and Analytical Axial Load-displacement Relationships of Strut 8 (Table 4.5) 3 88 Figure 4.14 Experimental and Analytical Axial Load-displacement in AXIAL DISPLACEMENT, 3 Relationships of Strut 9 (Table 4.5) 600 4 Experimental 2004 (D .9- T x . 1004 .0 -l a o —’ o .15 x i <( -100- .1 —200 .ee , . - , . Te. e. . . , . -3 —2 -1 0 1 2 Axial Displacement, in 300 1 Analytical 200- (D .9- i x I f - 100a // <0: 4]] -1 I I 9 0 14’” 2 34.1?! A 32 . E§§545337533‘- < I” —100- I I —200 . . , . . ,e.s ,. a.s sa,fi - .e,a -3 -2 -1 0 1 2 89 the actual hinge behavior. 4.5 NUMERICAL STUDIES AND DESIGN RECOMMENDATIONS FOR STEEL STRUTS Development of reliable models and analysis procedures can reduce the reliance on costly and time-consuming experimental investigations for assessing the effects of different design variables on the perfor- mance characteristics of structural systems, and for optimizing the design. The developed analytical model of steel struts can be used for numerical (instead of experimental) parametric studies, leading to the selection of the optimum material and geometric properties of the struts for achieving superior performance characteristics at minimum cost. The model can also be used to numerically evaluate the effects of the varia- tions in the initial imperfection and end eccentricity of steel struts on their performance under load. The results of a numerical study aimed at generating the information needed for optimum design of steel struts are presented below. 4.5.1 Influence Of The Yield Strength : Figures 4.15a and 4.15b present the effects of yield strength on the axial load-deformation relationship of steel struts with effective Slenderness ratios of 60 and 200, respec- tively. Each figure presents the monotonic axial load-deformation diagrams as well as the envelopes of the cyclic diagrams. The envelope curves are chosen to represent the overall hysteretic performance characteristics of the struts. The steel strut under study is a Bin x Bin x l/4in (76mm x 76mm x 6.3mm) angle, and all the force values in Figure 4.15 have been normalized by the yield force of a strut with 36 Ksi (248 Mpa) yield strength. The angular section introduced above, and the described axial force 9O -eeoa Heax< one be com 1 u\Hs Ab FzmszSumE .253 DMN3<2mOz a. n a n1 a? i p r t 1 h ndl aeooao>nu ouueuouuh: 106 1nd r . 10.F - ‘ .11. as. on i E i... as. on It i n; .Azmzuofiama 4<_x< omstamoz o N1 i o1 mi .1 i p n P r b L r h F p b b n.0l aucouoco: as. an i .C ...... . Us on i .C 1N.O| [F.Oal o.o 'OVOW 'lVlXV GEZI'IVNMON Ad/d 'OVO‘I 'IVIXV OBZI'lVl‘lBDN Ad/d mmwnmcowuaaom unoaooaanmao ON anumcauum eHaH» uceuawmao mo abobmmm mH.a aaswam ob i b\Hx Am .Fzm2wojmm5 4<_x< DMNDfizmoz or o O—l Owl » L r r l P . NI naoaao>cm ouuououu»: I?’ a 10 l iiiiiiiiiiiiii i a... o... .. b-1111 . «as on i hmilllll. N .FZMZMQSQBQ 4<_x< OMN3<§mOz o m1 91 m7. 81 n r P r h NI cacouocoz «as on .. bill . «as on i .Cil 'OVO'I 'IVIXV OBZHVNMON Ad/d 'GVOW 'lVIXV OEZI‘IVWMON Ad/d 9l normalization process are typical of the ones adopted in the other stages of this parametric study. From Figure 4.lSa it may be concluded that an increase in yield strength from 36 Ksi (248 Mpa) to 50 Ksi (34S Mpa) enhances practically' all aspects of the monotonic and cyclic axial load-deformation charac- teristics of the steel strut with an effective Slenderness ratio of 60. The ultimate compressive and tensile strengths, the post-buckling resis- tance and the hysteretic energy absorption capacity of the strut all improve noticeably as the yield strength increases from 36 Ksi (248 Mpa) to 50 Ksi (345 Mpa). A comparison of Figures 4.15a and 4.15.b indicates that the more slender strut of Figure 4.le is less sensitive to the variations in yield strength, when the loading is in compression. Under tension, however, the variations in yield strength significantly influence the strut performance. It seems that designers should be careful in balanc- ing the cost versus the performance improvements corresponding to the use of higher strength materials for steel struts, especially at higher Slenderness ratios. The variations in the actual material yield strength, compared to timespecified one, for steel struts is another important concern in design. Figures 4.16a and 4.l6b present the effects of such variations on the axial load-deformation characteristics of steel struts with ef- fective Slenderness ratios of 60 and 200, respectively. The steel strut is a 3in x 3in x l/4in (76mm x 76mm x 6.3mm) angle, and the1diagrams in Figure 4.16 are produced for steel struts having the specified yield strength of 36 Ksi (248 Mpa), an increase in yield strength by 20%, auui a decrease iJijield strength by 10% . The discussion made in the pre- vious paragraph on the effects of yield strength is also valid for the results presented in Figure 4.16. It may be concluded from this figure 92 maanmaofiuaaom uaoaoomaamwa -paoq Hmwx€ can so mfiuwcauum papa? SH mcofiumHum> mo muoammm mH.¢ mudwfih cow 1 u\Hx Ab .Fzmfiwojmmfi 4<_x< DMNfifizmoz Ad/d ‘ovm WIXV aaznvwaorx o. n a n1 can r P p — b b b )0 uoi aeno~e>=u oauououua: r 190 woo ‘ \ a 10. ~\\ a3.inniha ....... as. «.3 i blil . as. can i sci 0F .pzmsmofiamfi ._<_x< ou~3<§moz Nhl 3| 0| Ql b h D I ’i D b I I D I b I ”'0' ouaouocoz «9—iunixa 111111 assassixuiiiii .txoanixwlllli [Neel IP-OII 0.0 'OVO‘I WIXV OEZIWWMON Ad/d ob i “\Hx Ab szZmoSn—mfi .._<_x< QMNQSKOZ . a . t . . m.Fl manage/cu Ugandan»: 1 \\/ lo._.l \ u ImoOII 10.0 And I'll ‘I l I, l ‘\ \ IOoP «as ¢.~n W hm iiiiiii . as. «.3 i bill in; «3— 0.2” I bill “EN .szimoajumfi ._<_x< DwNflafmoz 3 nl owl n—l Owl - P p n i b “oFll o couaco « x «as ¢.Nn I biiiiii aa_finei.uiilll «3.9eniaaillll IOoF' 'm.0' 4 “Ho 'OVO'I 'IVIXV CBZI'IVWUON Ad/d 'OV01 'IVIXV OBZI'IVWHON Ad/d 93 that the effects of a 10% reduction in the material yield strength on the monotonic and cyclic axial load-deformation characteristics of steel struts is relatively small. 4.5.2 The Influence Of The End Connection Restraints : Figures 4.17a and 4.l7b present the effects of changing the end rotational fixities of .steel struts on axial loadodeformation characteristics, for struts with slenderness ratios (calculated as the full length divided by the minimum radius of gyration, disregarding the specified end fixities) of 60 and 200, respectively. The three curves on each diagram of Figure 4.17 cor- respond to the hinged-hinged, hinged-fixed, and fixed-fixed rotational end conditions. Lateral end deformations were restrained in all cases. The corresponding effective length factors are thus 1.0, 0.7 and 0.5, for the three end conditions. A comparison of Figures 4.17a and 4.l7b indicates that the provi- sion of and rotational restraints is highly effective in enhancing the monotonic and cyclic axial load-defamation characteristics when, loading is in compression, in steel struts with higher slenderness ratios (Figure4.l7b). The behavior of the more slender struts under tension and that of the shorter strut under tension and compression (except for the post-buckling resistance of shorter struts) are not much influenced by the changes in the end rotational restraints. This information can be very helpful to designers in making the decision on the details of the end connections. 4.5.3 The Influence 0f Cross-Sectional Shape: In order to compare the performance of steel struts with different cross-sectional shapes, typi- cal struts having different shapes but similar lengths and cross- 94 O— maazmaoaumHam Damsaomaamfia -eaoa Haaxa are so mafiuaxaa ecu HacoAbauam mo aboammm AH.¢ magmas com 1 0\H An .hzmznmojama 4<_X< DwN_i. p m- L Q C.F.| oqcouocoz n6..xi ..... Ad..xll1l. oznix .J — .. . ~ . a u , 1001 r 06 'OVO'I 'WIXV OEZIWVNMON Ad/d 'CIVOW WVIXV OBZI‘IVWHON Ad/d cm i 0\H Ab szzwojuma 4<_x< DwN_._<§moz ON Op 0 o—I DNl hi p nonoaegu uauououua: \. 2 Dél m; .tzmzmofiama ._<_x< omwsszmoz 0 ml opl r » W.Fi uuGOuocor n.O I x llllll Ad..xillil.. o; l x I .Pi ,o,, o //,,,,, // I--- //// 1,1- III/ Imdi 0.0 'OV01 'IVIXV OEZI'IVV‘IMON Ad/d 'OVO'l "IVIXV GBZI‘IVWHON Ad/ d 95 sectional areas were analyzed under monotonic and cyclic loads. For the 100 in. and 200 in. long steel struts, Figures 4.l8a and 4.l8b, respec— tively, show the axial load-deformation relationships (both monotonic and cyclic) of struts with Angle, WT, Pipe, and Square Tube cross- sections. It should be noted that the steel struts in either Figure 4.18a or Figure 4.l8b have equal length but not similar slenderness ratios. They were chosen to have comparable cross-sectional areas and consequently material costs. In this approach we will be comparing the performance characteristics of steel struts with comparable prices but different cross-sectional shapes. From Figure 4.18, it may be concluded that the cross-sectional shape effect on the inelastic-buckling behavior of steel struts is more pronounced for the longer struts when the axial load is in compression. The Square Tube followed closely by Pipe are superior to the other cross-sectional shapes under both monotonic and cyclic loads. It is worth mentioning that in optimizing the cross-sectional shape, attention should also be paid to the cost of connections which are usually used together with certain cross-sectional shapes. 4.5.4 The Influence Of Initial Imperfection : Out-of-straightness is commonly observed in steel struts at the job sites. It is important for the designers and constructors to understand the effects of different levels of initial imperfections on the axial load-deformation relation- ships of steel struts. Figures 4.l9a and 4.l9b present the effects of the variations in initial imperfection on compressive axial load- deformation relationship of the 3in x 3in x l/4in (76mm x 76mm x 6.3mm) angular struts with effective slenderness ratios of 60 and 200, respec- tively. The three levels of initial imperfection presented in these 96 mnwnmcoauaaom uGanoaHQmwa -pmoA Hawx< mnu co mammnm aaaowuoom-mmouo mo muoammm mH.q muswwm Cw com I A A9 .hzwzuojama 4<_x< owNfi b b kl i > > P r b b i i i o.F| 3:30:02 a 2.2.55 3 iiiii 3n .4 3.2 III: . I .. ., «:53 t. .lli . \W/ 22.53 a . / x \ / ../ imdi / f, / /- a / // / //l I--- . / I], / Y / III I 0.0 'OVOT 'MXV OEZIWWHON Ad/d Ad/d 'OVO'I WIXV OBZHVWHON .cs coal 4 Ab .tzmzwofiama a<_x< omwflszmoz o. n a on o... A . p . . 0.—| won—32:.» 03332:. n. 4 /.. imdi rod de 22.5.8 ma 111111 . .5“ a 2.2 III: 10.. 9.53 5 ll 1| Stuns a . m; .Azmzuofiama I.<_x< ommjszmoz o n1 o... p i i r r t r i l O._.l 7,. 3:33:02 4 //,,. 2.33.3 3.1-1-1 e ,. .3“ e .3. II I //,., “in." .5 ill . / 3:53 a / II / / X. / 11:: imdl / xx / /// JIIII / / I I / / I'll ll! 0.0 'OVO'I 'iVIXV OBZI'IVWMON Ad/d Ad/d 'ovm 111va aaznvwaON 97 ' A0 " 0.0 1.0— - a, - L/lOO _________ A0 - L/lOOO .0 <3 .1 1 fi fi r 0 I V ' i' 5 1 I 10 1 I is 1 I 20 NORMALIZED AXIAL SHORTENING, NORMALIZED AXIAL COMPRESSION, P/PY a) kl/r - 60 A0 - 0.0 -——-————-—- A0 - L/lOO C124 _________ A0 - L/lOOO 0.0 1 I I I j— ‘I I I # T I I I I fir NORMALIZED AXIAL COMPRESSION, P/PY NORMALIZED AXIAL SHORTENING, b) kl/r - 200 Figure 4.19 Effects of Initial Imperfection on the Axial Load- Displacement Relationships 98 figures correspond to the maximum out-of-straightnesses of 0.0% , 0.1% , and 1.0% of the length. Figure 4.19 indicates that the increase in initial imperfection, especially from 0.1% to 1.0% of the length, sig- nificantly reduces the ultimate compressive strength as well as the strut pre-peak tangent stiffness. The post-buckling behavior of the struts, however, is not much influenced by the presence of the initial imperfection. 4.5.5 The Influence Of End Eccentricity : Figures 4.20a and 4.20b present the effects of the variations in end eccentricities around the minor and major axes, respectively, on the compressive axial load- deformation relationship of a 3in x 3in x l/4in (76mm x 76mm x 6.3mm) angular struts with an effective slenderness ratio of 60. Figures 4.20c and 4.20d present similar results for the strut with an effective slenderness ratio of 200. The four curves in each figure correspond to end eccentricities of 0.0, 0.3in (7.6 mm), lin (25.4mm), and 3in (76mm). The increase in end eccentricity around minor axis is observed to sig- nificantly reduce the ultimate compressive strength, pre-peak tangent stiffness and post-peak compressive resistance of the struts with an effective slenderness ratio of 200 (Figure 4.20c). Similar effects of the minor axis eccentricity are observed in Figure 4.20a for the strut with an effective slenderness ratio of 60, except that its pre-peak tan- gent stiffness is not much influenced by the increase in minor axis eccentricity. In the strut with an effective slenderness ratio of 60, the adverse effects of the major axis eccentricity are comparable to those of the minor axis eccentricity (compare Figures 4.20a and 4.20b). For the more slender struts with an effective slenderness of 200, however, the major 99 maanmcoauaaam uaosoomHamHa -pmoA Hawxd ecu Go moHuHoHNDCaoom new no muoommm o~.¢ ouawfim Aoou I H\Hxv muxa “ohms ofiu pedou< AU Aoo~ I H\Hxv maxd Honda ecu pcsou< Ao 02.55.05 .232 834502 oz_zquxm ._<_x< 834252 o e N a r L b L b b _ bl L 1! 00° ill. 'I I I I! I 'l’-’ " an o.m I a iiiiiiii Cu o.m I 01:151- 1N6 5 o4 I oIII :« OJ I eIII candle I a.“ n.0le I 0.0 I a 0.0 I e nd nd Ad/d ‘ovm NOISSBHdWOO oaznvwaON Ad/d ‘avm NOISsaadwoo uaznvwaon Ace I H\Hxv mafia Mamas an» pcsou< An now I u\axv maxa HocHa onu paaou< Am 02.2whmOIm 4439‘ CNN—442102 UZ.ZMHKOIW .._<_x< OwN3<2mOZ ON mp or n O t P I L I — F P > L h b 00 as o." i a ......... as o.n o .------. aces...1:1|: . i Iiliii 1 in-“ 1 og.i.illlli i 9o... a o.— Ad/d ‘avm Nassaadwoo 03ZI'IVV‘IHON Ad/d 'avm NOISsaadwoo aaznvwaoru 100 axis eccentricity reduces the ultimate strength and the pre-peak tangent stiffness of the struts far less than the minor axis eccentricity (compare Figures 4.20c and 4.20d). 4.6 SUMMARY AND CONCLUSION : The coefficients in the proposed analytical model of steel struts were derived empirically, and the final model was positively verified using monotonic and cyclic test results on steel struts with wide ranges of'nmterial and geometric properties and cross-sectional shapes. The fully developed and verified element model was then used for a ‘numerical study on the effects of the material yield strength, end sup- port rotational fixity, cross-sectional shape, initial imperfection, and end eccentricity of steel struts on their monotonic and cyclic axial load-deformation relationships. From the results of the numerical study it may be concluded that a) For the less slender steel struts with an effective slenderness ratio of 60, the ultimate compressive and tensile strengths, the post-buckling compressive resistance, and the hysteretic energy absorption capacity of the strut all substantially increase with increasing yield strength. The more slender struts, with an effective slenderness ratio of 200, are less sensitive to the yield strength variations when loaded in compres- sion. Under tension, however, the effects of variations in yield strength on the performance of the more slender strut are still sig- nificant. b) End rotational restraints are highly effective in enhancing the monotonic and cyclic axial load-deformation characteristics of steel struts wdlflilnigher slenderness ratios under compression. The behavior of the more slender strut under tension and that of the shorter strut 101 under tension or compression (except for its post-buckling resistance) are not much influenced by providing end rotational restraints. c) The cross-sectional shape effects on the strut performance are more pronounced for the longer struts. At the same cross-sectional areas, among the four cross-sectional shapes considered (Square Tube, Circular Pipe,VflL and Angle), the Square Tube followed closely by the Pipe- section are the more superior ones under monotonic and cyclic loads. The WT comes next followed by the Angle. d) The increase in initial imperfection significantly reduces the ul- timate compressive strength as well as the pre-peak stiffness of the steel struts. The post-buckling behavior, .however, is not much in- fluenced by the increase in initial imperfection. e) The increase in end eccentricity around the minor axis significantly reduces the ultimate compressive strength, the pre-peak tangent stiff- ‘ness, and.the post-peak compressive strength of the more slender struts with effective slenderness ratios of about 200. The effects of minor axis eccentricity on the less slender struts are also similar, except that the pre-peak tangent stiffness is not much sensitive to the minor axis eccentricity. The major axis eccentricity effects 0 0:2 j 1 I 014 I ‘ 1 0:6 j T r70:8 CENTRAL VERTICAL DEFLECTION, in O C) Figure 5.11 Overall Analytical and Experimental Comparisons 51 of the Double Layer Space Truss 51 Figure 5.12 Picture of the Buckled Structure 125 Major inelasticities and collapse of the structural system were in- itiated by the buckling of some compression members at the upper layer of the truss near the applied load locations (see Figure 5.12). In the analytical model, major inelasticities were initiated by the buckling of elements 3 and 3'in Figure 5.13a simultaneously with the buckling of the corresponding elements in the other three quarters of the structure. This was followed by the buckling of the elements 7 and 7'and a major force redistribution within the system. In spite of all these, the structure could still preserve major fractions of its ul- timate load resistance at large deformations. Figures 5.13b,c, and d show typical relationships between the axial forces of the steel struts in the upper layer, web and lower layer, respectively, of the structural system and the vertical displacement at the center node of the lower layer. From Figure 5.13b through 5.13d, it may be concluded that the peak resistance of the complete structure is reached following the buck- ling of elements 3 and 3', when the descending branch of the compressive load-deformation relationship in these elements is initiated. Figure 5.13t>:is also indicative of major internal force redistributions within the compressive elements of the upper layer following the buckling of elements 3,3' ,7 and 7'. The key aspect of this phenomenon is the un- loading of those upper layer elements which are connected to the buckling ones (elements 1,1' ,2,2',8,8',9, and 9' in Figures 5.13a,b), another significant aspect of the force redistribution within the upper layer elements is the accelerated rate of compressive force increases in elements 4,4',5,5',6, and 6' at the boundary of the upper layer. These phenomena indicate that following the buckling of elements 3,3' ,7, and 7, the post-peak load carrying capacity of the structure has been upheld by the transfer of the compression forces to the elements at the ex- terior of the upper layer. 126 a) One Quarter ’Figure 5.13 Detailed Analytical Performance of the Double Layer. Space Truss 127 . Ac .uCoOV mmsHH oommm uohmq mansoo 0:» mo mocmauomuom Hwofiuzamc< poaamuoo mH.m oudwwm E .zo:om._.imo ._ .Zwfizmo md one no No 0.0 F P L L #1 _ F . .Vorlu mucoEon “chad woman An , \P H r p P L 1.1 I I D! ._ I I co 5’ T T If <0. C? T11 I I <1I 0| T I I 9! 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