THE NONLINEAR THREE-DIMENSIONAL RESPONSE OF STRUCTURES TO EARTHQUAKE EXCITATION by BRUCE MALCOLM MASON B.Sc.(App.) Queen's Univers i ty, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of C i v i l Engineering) We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1978 jO\ Bruce Malcolm Mason, 1978 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that copy ing , o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1WS i i THE NONLINEAR THREE-DIMENSIONAL RESPONSE OF STRUCTURES TO EARTHQUAKE EXCITATION ABSTRACT The three-dimensional response of structures, comprised of members possessing nonlinear const i tut i ve re lat ionsh ips , subjected to earthquake exc i tat ion i s investigated. The input involves considera-t ion of the simultaneous horizontal t rans lat ion of the structure in two mutually perpendicular d i rect ions in addit ion to tors ional behavior; ve r t i c a l ground motion has not been included. Material non l inear i t ie s are represented in a s imp l i f i ed fashion by the use of a b i l i nea r moment-rotat ion re la t ionsh ip , subject to the kinematic hardening ru le , at each member end. Relevant concepts of structural theory are reviewed and ex-pressed in a format allowing for the use of matrix algebra. A general form of the member s t i f fnes s matrix i s presented, and a method of f o r -mulating and subsequently s t a t i c a l l y condensing the structure s t i f fnes s matrix i s also presented. The interact ion of ax ia l loads with the nonlinear material behavior i s included. For columns subjected to bending about two p r i n -c ipa l axes, the ef fect of including b iax ia l interact ion ef fects on the y i e l d surface i s considered. A method of ca lcu lat ing the unbalanced forces resu l t ing from nonlinear behavior i s out l ined. i i i An energy balance is established whereby the energy input by the earthquake into the structure i s accounted for by the various mecha-nisms with which the structure i s able to d iss ipate or store th i s input energy. The importance of analyzing the var iat ion of the various energy forms i s stressed. The theory developed i s applied to two structural examples. The problems encountered in attempting to apply th i s theory to various models of a structure intended to represent a typ ica l midsized o f f i c e building are discussed. This structure has a dual component st ructura l system in which an eccentric shear core i s designed to r e s i s t the ho r i -zontal loads and an exter ior framing system i s designed to^res i s t ver-t i c a l loads and to act as a second l i n e of defense with respect to ho r i -zontal loads. The results obtained from an earthquake analysis of a structure, comprised of exter ior f i ve storey frames, are presented. Spec i f i c aspects of these re su l t s , such as the amount of energy d i s s i -pated by various mechanisms, are analyzed in d e t a i l . The ef fect of assumptions made concerning the structural res-ponse are discussed and suggestions for future developments in th i s f i e l d of analys i s , which w i l l be aided by advancements made in computer technology, are also given. iv TABLE OF CONTENTS Paje ABSTRACT '. i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i NOTATION x i i ACKNOWLEDGEMENTS x i i i CHAPTER 1. INTRODUCTION 1 1.1 Background 1 1.2 Purpose and Scope 5 1.3 Assumptions and Limitat ions 8 2. STRUCTURAL THEORY 10 2.1 Solution of the Incremental Equi l ibr ium Equation 10 2.2 Member Properties 18 2.3 Geometric St i f fness Matrix 31 2.4 Formulation of the Reduced Frame St i f fness Matrix 39 2.5 Formulation of the Reduced Structure St i f fness Matrix 43 3. NONLINEAR MATERIAL BEHAVIOR AND ENERGY CALCULATION 46 3.1 Calculat ion of Member Deformations 46 3.2 Calculat ion of Member End Forces 49 3.3 Y ie ld C r i t e r i a 53 V CHAPTER Page 3.4 Member State Determination 59 3.5 Unbalanced Force Corrections 64 3.6 Calculat ion of Incremental Member P l a s t i c Rotations 70 3.7 Calcualt ion of Energy Quantities 72 4. MODELLING OF A SIXTEEN STOREY OFFICE BUILDING . . . . 81 4.1 Description of the Structural Model 82 4.2 Preliminary Analysis 84 4.3 Computer Modelling of the Structure 94 4.4 Summary of Computer Models 101 5. RESULTS OF A TIME STEP ANALYSIS OF A FIVE STOREY . . BUILDING 104 5.1 The Structural Model 104 5.2 The Earthquake Acceleration Record 106 5.3 Selection of the Time Step Length 108 5.4 Response Results 110 5.5 Summary of the Analysis 116 6. SUMMARY AND CONCLUSIONS 118 BIBLIOGRAPHY 122 APPENDIX A: CALCULATION OF THE EVENT FACTOR FOR A BIAXIAL COLUMN 185 APPENDIX B: CALCULATION OF THE ENERGY INPUT INTO A STRUCTURE 187 v i LIST OF TABLES Tab1e(s) Page 1 Factors used in time step integration 126 2 Possible member states 126 3 Constants used in bui ld ing the member s t i f fnes s matrix 127 4 Example design eccen t r i c i t i e s and tors ional moments 128 5 Torsional magnification factors for front frame of eccentr ic shear core model 129 6 Modal analysis of model A 130 7 Modal analysis of model B 131 v i i LIST OF FIGURES Figure(s) Page 1 Generalized co-ordinates 132 2 Plan view of typ ica l storey 132 3 B i l i nea r moment-rotation re lat ionsh ip at a member end 133 4 Typical moment-rotation re lat ionsh ip at an end of a reinforced concrete beam , . . 133 5 (a): Moment-rotation re lat ionsh ip at an end of the e l a s t i c component 134 (b): Moment-rotation re lat ionsh ip at an end of the e l a s t i c - pe r f e c t l y p l a s t i c component 134 6 Two component model 134 7 Typical hysteret ic behavior 135 8 Member degrees of freedom 135 9 Model of shear core 136 10 Beam with r i g i d end stubs ( l i n t e l beam) 136 11 Notation used in formulation of the geometric s t i f fnes s matrix 137 12 (a): Forces due to un i t ve r t i c a l displacement of j o i n t i (member state A) 137 (b) : Forces due to removal of moment at j o i n t i . . 137 (c) : Forces due to unit ve r t i ca l displacement of j o i n t i (member state B) 137 v i i i Figure(s) Page 13 (a): Forces due to unit rotat ion of j o i n t i (member state A) 138 (b) : Forces due to removal of moment at j o i n t i . . 138 (c) : Forces due to unit rotat ion of j o i n t i (member state B) 138 14 (a): Bui lding without ve r t i c a l compat ib i l i ty . . . . 139 (b): Bui lding with ve r t i ca l compat ib i l i ty 139 15 (a): Gross frame degrees of freedom. 140 (b): Reduced frame degrees of freedom 140 16 (a): Gross structure degrees of freedom 141 (b): Reduced structure degrees of freedom 141 17 Forces and deformations of a typ ica l members . . . . 142 18 (a): Ax i a l - f l e xu ra l interact ion diagram 142 (b) : B iax ia l in teract ion diagram . 143 (c) : Y ie ld surface for b iax ia l columns 143 19 (a): Unbalanced moment corresponding to member end y ie ld ing .... .-.'"3 144 (b) : Unbalanced moment corresponding to member end reversal - .^.? 144 (c) : Unbalanced moment corresponding to a change in y i e l d moment at a hinge 145 20 Floor plan of a t yp i ca l f l oo r of the core type bui lding 146 21 Frame 1 of the core bui ld ing 147 i x Figure(s) Page 22 Frames 2 and 3 of the core bui lding 148 23 Walls 8 and 9 of the core bui lding 149 24 Typical def lect ion shapes 150 25 Lateral seismic forces (kips) required by the NBC (in the q d i rect ion) 150 26 Model A . 151 27 Model B 152 28 Model C . . ' 153 29 Properties of the f i ve storey framed bui lding . . . . 154 30 Envelope function for earthquake type B 155 31 (a): Accelerat ion, ve loc i t y and displacement for earthquake B-l 156 (b): Accelerat ion, ve loc i t y and displacement for earthquake B-2 157 32 (a): Pseudo-velocity spectrum for earthquake B-l . . 158 (b): Pseudo-velocity spectrum for earthquake B-2 . . 159 33 (a): T r i p a r t i t e logarithmic p lot of spectra for earthquake B-l 160 (b): T r i p a r t i t e logarithmic p lot of spectra for earthquake B-2 161 34 (a): Acceleration ( f t . / sec 2 ) of storey 5 in the q d i rect ion 162 (b): Relative ve loc i t y (ft./sec) of storey 5 in the q d i rect ion 163 X Figure(s) Page 34 (c) : Absolute ve loc i t y (ft./sec) of storey 5 in the q d i rect ion 164 (d): Displacement ( f t . ) of storey 5 in the q d i rect ion 165 35 (a): Acceleration ( f t ./sec ) of storey 5 in the r d i rect ion 166 (b) : Relative ve loc i t y (ft./sec) of storey 5 in the r d i rect ion 167 (c) : Absolute ve loc i t y (ft./sec) of storey 5 in the r d i rec t ion 168 (d) : Displacement ( f t . ) of storey 5 in the r d i rect ion 169 36 (a): Acceleration (rad./sec ) of the rotat ional component of storey 5 170 (b) : Relative ve loc i ty (rad/sec) of the rotat ional component of storey 5 171 (c) : Absolute ve loc i t y (rad/sec) of the rotat ional component of storey 5 172 (d) : Displacement (rad) of the rotat ional component of storey 5 173 37 (a): Acceleration vs. storey level envelope . . . . 174 (b) : Relat ive ve loc i t y vs. storey level envelope . . 174 (c) : Absolute ve loc i t y vs. storey level envelope • • 175 (d) : Displacement vs. storey level envelope . . . . 175 38 (a): Exter ior beams shear envelope (kips) 176 (b): Inter ior beams shear envelope (kips) 176 x i Figure(s) Page 39 (a): Exter ior column shear envelope (kips) 177 (b): In ter io r column shear envelope (kips) . . . . . 177 40 (a): Exter ior column axia l envelope (kips) 178 (b): Inter ior column ax ia l envelope (kips) 178 41 Var iat ion of recoverable energy during the f i r s t 22 seconds 179 42 Var iat ion of to ta l energy during the f i r s t 22 seconds 180 43 Number of excursions into the p l a s t i c range at storey levels 181 44 Accumulated p l a s t i c rotat ion 182 45 Dissipated energy ( k i p - f t . ) of various components . . 183 46 Dissipated energy of the ent i re bui ld ing 184 x i i NOTATION The use and meaning of each symbol i s defined in the text at the point where i t i s introduced. Matrix notation i s used as opposed to summation convention. Vector, or one dimensional arrays, are underlined by a s ingle l i n e , e.g. xj while matrix, or two-dimensional arrays, are underlined by a double l i n e , e.g. £ . The number of dots above a var iab le, e.g. x, represent the number of times the var iable i s d i f fe rent i a ted with respect to time. x i i i ACKNOWLEDGEMENTS The author wishes to express his gratitude to his advisor, Dr. N.D. Nathan, for his valuable advice and guidance during the re -search and preparation of th i s thes i s . He also expresses his thanks to Dr. D.L. Anderson and Dr. S. Cherry for the i r suggestions and advice during the research work. The f inanc ia l support of the National Research Council in the form of a UniveBsity of B r i t i s h Columbia research ass istantship i s also g ra te fu l l y acknowledged. 1 CHAPTER 1 INTRODUCTION 1.1 Background Nonlinear behavior of structures has, unfortunately, been found to occur frequently durring moderate to major earthquakes. This behavior, in the form of members being stressed beyond the y i e l d point and hence exh ib i t ing p l a s t i c or nonlinear propert ies, precludes the use of a r e l a t i v e l y simple e l a s t i c analysis in order to describe mathemati-c a l l y the response of a structure to earthquake exc i ta t ion . The need for a nonlinear analysis has been emphasized by many authors, e.g. Clough; Benuska and Wilson [6], Degenkolb [8], Goel and Berg [11], Selna; M o r r i l l and Ersoy [26], and Wynhoven and Adams [34], [35]. During the past two decades there has been a great deal of research and considerable advance in the understanding of the response of high r i s e buildings (multi-degree of freedom systems) to earthquake exc i ta t i on . Research of th i s nature i s heavily dependent upon computer technology and has c lose ly pa ra l l e led , and hence been l imi ted by, ad-vances in computer e f f i c i ency and capacity. A good introduction to the subject of the nonlinear response of structures to earthquakes can be found in the book by Blume; Newmark and Corning [3]. This book provides a discussion of the background for i ne l a s t i c design and introduces the conceptsoof d u c t i l i t y and energy absorption for reinforced concrete structures. An intensive analysis 2 of s ingle degree of freedom systems i s presented and the theory i s ex-tended to approximate the maximum values to be obtained for a 24 storey frame. The nonlinear approximation of the const i tut ive re lat ionships of the members of a structure is an area undergoing much research. The most basic nonlinear model i s one with a b i l i nea r moment-rotation r e l a -t ionship in which the member i s assumed to be l i nea r l y e l a s t i c un t i l some point (the y i e l d moment), a f ter which the moment-curvature slope changes. A s ingle component model of th i s b i l i nea r re lat ionsh ip i s presented by Giberson [10]; however, a more convenient model i s the two component model presented by Clough; Benuska and Wilson [6]. The concept of a time step analysis of mult i degree of f r ee -dom systems i s presented by Clough; Benuska and Wilson [6] and Clough and Benuska [5] in which a r e l a t i v e l y crude analysis of a twenty storey bui lding frame subjected to several earthquake accelerat ion records i s performed. Time step analys i s , which i s now a commonly used method in nonlinear dynamic analys i s , approximates the response history of a structure by subdividing the earthquake accelerat ion record into a series of small time steps. Over each time step, the response i s as-sumed to be l i nea r , based upon the condit ion of each member of the structure at the beginning of the time step. The accuracy of the re -su lts obtained i s very sens i t ive to the length of the time step used. Discussionsooftthe length of time step to be used are presented by Heidebrecht; Lee and Fleming [16] and Walpole and Shepherd [31]. 3 In a time step analys i s , each member i s examined to see whether i t s state has changed at the end of each time step. For the b i l i n ea r moment-rotation re lat ionsh ip a change in state at a member end may take one of two forms. An end which was i n i t i a l l y e l a s t i c may be subject to a moment exceeding the y i e l d moment; the member end should have yielded and a p l a s t i c hinge i s introduced as an i n i t i a l condition for the next time step. A l te rna t i ve l y , a member end which i n i t i a l l y contained a hinge may, during a time step, have a decrease in the absolute rotat ion at the end; the hinge i s then removed for the next time step. Hence in th i s type of analysis the nonlinear response of the frame is approximated as a series of successively changing l i near time increments. Other researchers have extended the time step analysis theory to other plane frames using more sophist icated models. More notable ex-amples of research can be found in a r t i c l e s by Goel and Berg [1.1], Pekau; Green and Sherbourne [25], Spencer [27] and Walpole and Shepherd [31], [32]. The idea l i za t i on of a structure as a plane frame subjected to one component of an earthquake has the advantage of being a r e l a t i v e l y inexpensive method of analyzing the response history of the structure. There are, however, some serious drawbacks which detract from the v a l i d i t y of resu lts obtained from a plane frame approximation of the actual s t ruc -ture. For example, a column in an actual structure w i l l generally be subjected to motion about both of i t s pr inc ipa l axes. B r e s l e r [ 4 ] and Parme; Nieves and Gouwens [24] have shown that a concrete column w i l l y i e l d at a lower moment about one axis due to the presence of a moment 4 about the other pr inc ipa l ax i s . Also, the formation of a p l a s t i c hinge due to motion pr imar i ly along one axis w i l l a l t e r the behavior of the member along the other ax i s . A plane frame analysis i s unable to account for these features. In add i t ion, Newmark [23] has stated that a rotat ional compo-nent of ground motion should be considered. This component induces to r -sional behavior in the structure which cannot be accounted for in a plane frame analys i s . Torsional responses may also occur in-ia structure resu l t ing from asymmetryypf plan; layouts s t i f fnes s .onrstrength d i s t r i bu t i on . Berg and S t rat ta [2] present evidence of tors ional behavior in buildings sub-jected to the Alaska, 1964 earthquake and Hanson and Degenkolb [13] state that the Venezuela, 1967 earthquake also induced tors ional behavior. The importance of including tors ional behavior in earthquake analysis has been stressed by many authors such as Degenkolb [8] , Hart; Lew and Di J u l i o J r . [15] and Mazi lu; Sandi and Teodorescu [20]. Recently, several authors have presented resu l t s of analyses of structures including tors ional e f f ec t s . Koh; Takase and Tsugawa [18], MacKenzie [19] and Tso and Bergmann [30] present the theory of mathemati-cal models of structures which are condensed to contain three degrees of freedom at each storey l e v e l : two mutually perpendicular trans lat ions plus one rotat ional or tors ional degree of freedom. These mathematical models are tested on eccentric structures in attempts to determine the conditions under which the tors ional response becomes a s i gn i f i can t or dominant factor . MacKenzie [19] also explored the undesirable phenomenon 5 of beating. These analyses were re s t r i c ted to structures in which a l l members are assumed to remain l i nea r l y e l a s t i c during the earthquake. More recent ly, researchers have attempted mathematical models of structures including tors ional behavior and allowing for material non l i nea r i t i e s . Anagnostopoulous; Roesset and Biggs [1] , Selna; Mo r r i l l and Ersoy [26], Tso [28], Tso and Asmis [29], and Wynhoven and Adams [34], [35] have presented results obtarined with the i r nonlinear tors ional models. These authors have generally applied t he i r models to small structures and have made assumptions which reduce the cost of so lut ion by computer at the expense of accuracy. However, results obtained by these authors are valuable steps towards the accurate solut ion of the response of structures to earthquakes. One of the most important steps in at ta in ing a thorough under-standing of the nonlinear response of structures to earthquake exc i tat ion i s a study of the various ways in which energy inputtbyythe^earthquake i s stored and diss ipated by the structure. Goel and Berg [11], Hanson and Fan [14] and Pekau; Green and Sherbourne [25] have presented resu l t s showing the var iat ion of the sundry forms of energy in the structure with respeo.t-to time, and a s imi la r study i s made in th i s thes i s . 1.2 Purpose and Scope The purpose of th i s thesis i s to extend the work of previous investigators concerned with the nonlinear tors ional response of s t ruc-tures to earthquake exc i t a t i on . A parametric study of the var ia t ion of 6 response values with respect to structural parameters in very large buildings was precluded by the expense involved in performing nonlinear dynamic analyses on ex i s t ing computer systems. For th i s reason, i t was the intention of the author to subject a typ ica l midsized o f f i ce bu i ld ing , modelled as accurately as poss ible, to a complete earthquake record and to examine the var iat ion with respect to time of the important response parameters. Theory concerning the response of structures to earthquakes has advanced very rapid ly in recent times, and has been accompanied by a p ro l i f e ra t i on of associated l i t e r a t u r e . As research in th i s area i s s t i l l progressing i t i s f e l t by the author that a "state of the a r t " thesis would be valuable to future researchers. For th i s reason, r e l e -vant material i s gathered from various sources and presented in th i s thesis in conjunction with theory required for the spec i f i c approach used by the author. The dynamic equi l ibr ium equation i s formulated and two tech-niques of solving the equation are discussed. The b i l i nea r member proper-t i e s and the four possible member states are examined and a general form of the member s t i f fnes s matrix i s presented. The ef fects of shear de-f l e c t i o n , rdgid end stubs (to account for f i n i t e j o i n t s ize) and geomet-r i c non l inear i t ies are reviewed and included in the member s t i f f ne s s ma-t r i x . A review of theory presented by MacKenzie [19], and Nathan and MacKenzie [21], concerning the formation and subsequent condensation of the structure s t i f fnes s matrix to contain three degrees of freedom (gen-era l i zed co-ordinates) per storey, i s summarized. 7 The interact ion of ax ia l loads on the y i e l d moment i s examined and, in the case of columns subjected to bending about two mutually per-pendicular axes, the y i e l d surface i s defined. An algorithm used in the member state determination phase i s presented. This algorithm examines a member at each end to determine whether a p l a s t i c hinge should be i n -troduced or subsequently removed. The conditions under which unbalanced forces may appear are examined and modifications to theory necessitated by the occurrence of unbalanced forces are presented. Methods used to ca lcu late various energy quantit ies are pre-sented and an energy balance i s discussed. The energyt-balance relates to the fact that the to ta l energy input by an earthquake into a structure should equal the sum of the energy quantit ies appearing in the structure. These structure energy quant it ies are: (1) the k ine t i c energy of the structure, (2) the recoverable s t ra in energy stored in the s t ructure, (3) the non-recoverable s t ra in energy diss ipated by the structure due to nonlinear, hysteret ic behavior and, (4) energy diss ipated by viscous damping. The structure which was selected to r e f l e c t a typ ica l midsized o f f i c e bui lding i s a modified version of a structure which was recently constructed in Vancouver. The sixteen storey bui ld ing has an eccentr ic elevator and s t a i r core which i s designed to r e s i s t the earthquake loads spec i f ied by the National Bui lding Code of Canada [22], in conjunction with an exter io r reinforced concrete framing system designed to indepen-dently r e s i s t twenty f i ve per cent of the spec i f ied earthquake loads. Several d i f fe rent methods of modelling th i s dual component system are 8 presented and the problems 'which were .encountered in each model are d i s -cussed. Because of the d i f f i c u l t y in modelling th i s type of s t ructure, and since the cost of solving the problem by computer was larger than o r i g i n a l l y ant ic ipated, th is model was abandoned. However a descr ipt ion of the models and the i r associated problems i s included as i t i s f e l t that th i s would be a valuable asset to others doing research in th i s area. In order tbeillustrateetheetheowyyppesenteddinnthi's--thesis, a symmetric f i ve storey framed bui lding was analyzed. This structure con-s i sted of four exter io r frames, each s imi la r to the top f i ve storeys of the frame analyzed by Clough; Benuska 0and,Wilson [6]. An eccent r i c i t y was introduced by separating the centre of mass and the centre of s t i f f -ness. The structure was subjected to earthquake records Bl and B2 which were a r t i f i c i a l l y generated by Jennings; Housner and Tsai [17]. These earthquake records were scaled to a maximum acceleration of 0.25'-,g and applied simultaneously in the two perpendicular d i rect ions . The resu l t s of th i s ana lys i s , and a discussion of the re su l t s , are presented. 1.3 Assumptions and Limitations V i r t u a l l y a l l of the l im i ta t ions governing th i s work are im-posed by the necessity of obtaining a solut ion by computer within rea-sonable economic bounds. I t i s f e l t by the author that imminent improve-ments in computer technology, coupled with the advent of micro-computers, w i l l relax most of these economic constraints and allow a more accurate analysis to be performed. 9 The theory used to assemble the structure s t i f fnes s matrix i s based upon the assemblage of plane frame s t i f fnes s matrices. For th i s reason, the structure must be such that i t may be modelled accurately as a combination of plane frames. Floors are assumed to be r i g i d with respect to in-plane deformations and of neg l ig ib le s t i f fnes s with res-pect to out of plane t rans lat ion or ro tat ion. The tors ional r i g i d i t y of indiv idual members i s neglected. The centre of mass of a l l storey levels must l i e on a s ingle ve r t i ca l l i ne in the program used in th i s research. The moment-rotation re lat ionsh ip at a member end i s assumed to be b i l i n ea r . The material i s assumed to have kinematic hardening, and s t i f fnes s degradation i:s ignored. Ax i a l - f l e xu ra l in teract ion surfaces are assumed and for columns subjected to bending about two mutually per-pendicular axes, the y i e l d surface i s assumed to be an e l l i p s e . The s ize of the problem which may be analyzed using the com-puter program developed by the author is a function of the storage ca-pac i t ies of the computer system in addit ion to constraints imposed by economy. The program was designed to require a minimum of storage while maintaining maximum e f f i c i ency . I t i s f e l t that a conventional structure of twenty to twenty f i v e storeys approaches the l i m i t of f e a s i b i l i t y of analysis imposed by the theory outl ined in th i s thesis on current com-puter systems. The author has attempted to keep the number of l im i ta t ions used in th i s analysis to a minimum. The most important l im i ta t ions are described above, however addit ional l im i t a t i on s , in the form of assump-tions necessitated by economy, are discussed as they are introduced into the relevant sections of th i s thes is . 10 CHAPTER 2 STRUCTURAL THEORY Previous work done by MacKenzie [19] in the three dimensional response of structures during earthquakes was re s t r i c ted by the assump-t ion that a l l members remain l i n ea r l y e l a s t i c throughout the response history of the structure. It i s obvious that allowing for material non-l i n e a r i t i e s w i l l s i g n i f i c an t l y a l t e r results obtained for typ ica l s t ruc-tures subject to moderate to major earthquakes. However, the general approach to the problem and much of the theory used in the formation and subsequent condensation of the s t i f fnes s matrices is s t i l l v a l i d despite the inc lus ion of material non l inear i t i e s . For these reasons the general approach to the problem used by MacKenzie [19] i s u t i l i z e d in th i s thesis and incorporated into th i s chapter for the sake of completeness. 2.1 Solution of the Incremental Equil ibrium Equation As stated by Clough and Penzien [7], "the only generally a p p l i -cable method for the analysis of a rb i t ra ry nonlinear systems i s the nu-merical step by step integration of the equation of motion. The response history i s divided into short equal time increments >At ,:and the response i s calculated during each increment for a l i nea r system having properties determined at the beginning of the i n t e r v a l . The nonlinear behavior i s approximated as a sequence of analyses of successively changing l i nea r systems". 11 The equi l ibr ium equation in incremental form (assuming viscous damping) i s the fol lowing-wel l : known second order d i f f e r e n t i a l equation. M Ax + C B Ax + Kg Ax_ = - M Ax g (2.1) in which Mr! = mass matrix Cg = damping matrix KD = (K C D - Kro) with K C D = i n i t i a l tangent s t i f f ne s s =1 = § ! ^gB Jti matn.x KGB = 9 e o m e 1 : r ' ' c s t i f fnes s matrix Mx = incremental displacement vector r e l a t i v e to the ground Ax = incremental ground accelerat ion vector. Dots represent d i f f e ren t i a t i on with respect to time. The sub-sc r i p t B represents values at the beginning of the time step or tangent values. Damping i s assumed, in (2.1), to be viscous. While th i s i s not the actual case, th is assumption i s accepted for ease of computation. The equi l ibr ium equation i s solved for the incremental displace-ment vector as described in Section 2 . 1 . i i . As solut ion of the equation involves inversion of a matrix of an order equal to the number of degrees of freedom, i t i s necessary that the number of degrees of freedom be kept as small as possible while s t i l l accurately describing the structural motion. This i s accomplished by o r i g i n a l l y including three degrees of freedom at each j o i n t : a horizontal and ve r t i ca l t rans lat ion plus a ro -tat iona l degree of freedom. The number of degrees of freedom i s then reduced by s t a t i c condensation to three per storey. The three degrees 12 of freedom per storey, or generalized co-ordinates, are horizontal t rans-lat ions in two mutually perpendicular d i rect ions q and r plus a r o ta -t ion about the centre of mass as shown in Figure 1. S ta t i c condensation reduces the order of the matrices in (2.1) to three times the number of storeys and allows for an accurate analysis of the response of the s t ruc -ture about two mutually perpendicular d i rect ions in addition to analysis of the tors ional response of the structure. 2.1.1 Description of Terms in the Equil ibrium Equation 2.1. i (.a).\. Mass Matrix.' The mass matrix i s derived by applying a unit accelerat ion i n -dependently to each generalized co-ordinate. The 1-1 term of the mass matrix generates the force in the q d i rect ion resu l t ing from a/unit accelerat ion in the q d i rec t ion at the f i r s t storey. The appl icat ion of a unit accelerat ion to any of the three generalized co-ordinates of a par t i cu la r storey level w i l l produce forces only at that storey l e v e l . This implies that the structure mass matrix may be comprised of 3 by 3 submatrices corresponding to each storey l e v e l . I f the geometric o r i g in of the structure and the centre of mass at a par t i cu la r storey level are as indicated in Figure 2, then the storey massvmatrix w i l l be as shown in (2.2) m 0 -mne " T 0 m m;e qq 1 m 12 m -mr:e nwe„ try* 13 where m = to ta l mass of the storey e , e = distance in the q, r d i rect ions from the structure q o r i g in to the centre of mass a, b = storey width in q, r d i rec t ions . The storey mass matrix shown in (2.2) assumes that the mass of a par t i cu la r storey i s evenly d i s t r ibuted about the centre of mass. The 3-3 term i s the mass polar moment of i n e r t i a for a rectangular storey. I f the structure or ig in and the centre of mass co inc ide, quan-t i t i e s e^ and e^ equal zero and the storey mass matrix i s a diagonal matrix. I f thi>s is true at every storey level then the structure mass matrix w i l l also be a diagonal matrix and may be expressed as in (2.3) M = I < m ', m 1 me >T (2.3) i dent i t y matrix vector of storey masses vector of storey mass polar moments of i n e r t i a . There are many structures for which the centre of mass of a par t i cu la r storey i s not on the ve r t i c a l l i ne jo in ing the centres of mass of other storeys, in which case the coupled form of the mass matrix shown in (2.2) must be used. However, because of the great computational saving resu l t ing from the use of a diagonal mass matrix, the program developed by the author contains th i s r e s t r i c t i o n . This implies that where = m V = 14 only structures in which the centres of mass at a l l storey levels can be approximated by points on a s ingle ve r t i ca l l i ne may be analyzed. 2.1. i (b)- St i f fness ^Ma:trtx-Derivation of the structure s t i f fnes s matrix i s presented in Section 2.5. Using the co-ordinate system shown in Figure 1, the con-densed s t i f fnes s matrix is as expressed in (2.4) K qq KB - ' V ' :•' fge KSB = i, p t t i/B ,.n> i r r ' '"0 KB K r 6 (2.4) K eq K 9 r | KR =§£ 1 where the submatrix K , for example, represents the forces in the q d i rect ion resu l t ing from unit displacements of the f loors in the r d i -rect ion , stored f l oo r by f l oo r . The superscript B indicates the v a r i -a b i l i t y of each submatrix with respect to time according to the formation of p l a s t i c hinges. 2.1. i(c) v "Geometric Stif.fnesssMa'tftlxx. The geometric s t i f fnes s matrix accounts for second order geo-metric non l inear i t ie s induced in a member by changes in geometry in the presence of a x i a l l y directed load components. Derivation of the member geometric s t i f fnes s matrix i s presented in Section 2.3. The member 15 geometric s t i f fnes s matrices may be assembled and then reduced using s t a t i c condensation s im i la r to the technique used for the l i nea r tangent s t i f fnes s matrix; resu l t ing in a matrix K^g of the same form as (2.4). A l te rnat i ve l y the member geometric s t i f fnes s matrices may be incorporated into the member l i nea r tangent s t i f fnes s matrices in which case a f te r s t a t i c condensation the overal l structure s t i f fnes s matrix including geometric non l inear i t ies w i l l be represented by (2.4). assumption that damping i s proportional to e i ther or both of the mass and s t i f fnes s d i s t r i bu t i on s , as expressed in (2?5) 2.1. i (dc): Dampj:rigc;Matrtx;-The damping matrix C R i s evaluated under the commonly used C D = a M + B K, B (2.5) o'where t , (a (2.6) B (2.7) wi th a) = natural frequency of the f i r s t l i nea r mode of v ibrat ion £ s = f ract ion of c r i t i c a l damping which i s s t i f fnes s proportional 5 = f ract ion of c r i t i c a l damping which i s mass proport ional. Assumption (2.5) i s pa r t i cu l a r l y useful in l inear e l a s t i c analysis as mode shapes and frequencies calculated for the undamped case w i l l be 16 the same when damping i s included i f th i s re la t ion i s assumed. The re -la t ion i s also useful in the nonlinear case as i t means the equi l ibr ium equations involve only two matrices: the mass and s t i f fnes s matrices. 2.1.i(e) Displacement Vectors The vector Ax , representing incremental displacements of the generalized co-ordinates r e l a t i ve to the ground, may be expressed in part i t ioned form as in (2.8) Ax = < Aq ! Ar j A6 >T (2.8) with Aq_ equal to incremental displacements in the q d i rec t ion of each storey l e v e l , and so f o r th . S imi lar relat ionships may be expressed fo r incremental ve lo-c i t i e s and accelerations of the generalized co-ordinates. The vector of incremental ground accelerations may s im i l a r l y be expressed as: Ax„ = < Aq„ A6* >T (2.9) _ g _ where each term of Aq , e t c . , i s equal to, the'incremental ground acce-1 erat ionqi nrfthe.s.tqe-?di r ec t i on. 2 .1 . i i Solution Techniques In order to solve the incremental equi l ibr ium equation (2.1) an assumption must be made concerning the var iat ion of one of the response 17 parameters Ax , Ax or Ax over the time increment. The two most com-monly used methods are (a) the Wilson 9 Method which uses the assumption that the response acceleration varies l i n ea r l y over an extended time increment. (b) the Newmark 3 Method which assumes a constant response accele-rat ion over a time increment equal to the average of values at the beginning and end of the increment. These solutionttechniques may be expressed in general form as Ax x At Ax Ax = a l t - a 5 x - a g x At (2.11) Ax = R _ 1 Af (2.12) a aa 3a,, Af = m [(•£§" + a S 5 ) x + ( a 3 + aa 6 At) x - Ax ] + 3 K g (a g x + a gx At) (2.14) where a. , i = 1, 6 are defined for each method in Table 1. Values i of x^ and x. represent response ve l oc i t i e s and accelerations at the beginning of the time step. 18 Solution of incremental displacements using (2.12) allows for incremental ve l oc i t i e s and accelerations to be solved using (2.11) and (2.10). These incremental values are then added to values at the begin-ning of the time step to give values to be used as i n i t i a l conditions for the next time step. I t should be noted that the Wilson 6 method i s made uncondi-t i o n a l l y stable by ca l cu la t i ng values over an extended time step B At where 6 > 1.37 . The incremental displacements over th i s extended time step are solved by (2.12) and then the incremental accelerations over the normal time step are found by d iv id ing (2.10))by 6,. These inc re -mental accelerations are then used to ca lcu late incremental ve l oc i t i e s and displacements over the normal time step using the l i nea r acce lera-t ion assumption. I t should also be noted that while the pseudo load vector Af w i l l have to be calculated every time step, the pseudo s t i f fnes s matrix k w i l l be a constant i f the status of each member i s unchanged. The matrix k 'Msoonly calculated (and then inverted) i f a member status changes. 2.2 Member Properties The force-displacement re lat ionsh ip for each member i s expressed in terms of a member s t i f fnes s matrix. The member s t i f fnes s matrices for each member of a pa r t i cu la r frame are added together, using well establ ished 19 procedures, to form the s t i f fnes s matrix for that frame. The frame s t i f f -ness matrices are then combined to form the structure s t i f fnes s matrix. Each end of each member of the structure may become subject to a stress greater than the y i e l d stress in which case y i e ld i ng w i l l occur. Y ie ld ing , or subsequent reversa l , at any member end w i l l change i t s s t i f fnes s matrix and hence the structure matrix. 2 . 2 . i Member Idea l izat ion The nonlinear behavior at a member end i s assumed to be repre-sented in a b i l i nea r fashion as shown in Figure 3. The moment-rotation re lat ionsh ip at each member end i s thus assumed to have two l i nea r slopes: a slope of k when the moment at the end i s less than the y i e l d moment (unyielded); and a slope of pk when the end i s y ie lded. This b i l i n ea r moment-rotation re lat ionsh ip i s v i r t u a l l y an exact descr ipt ion of the behavior of steel members, however for members constructed from r e i n -forced concrete a b i l i nea r moment-rotation curve i s only an approximation. The actual re lat ionsh ip for a typ ica l reinforced concrete section i s shown in Figure 4 , however for the sake of economy this complex r e l a t i o n -ship i s approximated by the b i l i n ea r re lat ionsh ip of Figure 3. I f p equals zero, Figure 3 represents the commonly used e la s -t i c - p e r f e c t l y p l a s t i c re la t ionsh ip , and i f the moment exceeds the y i e l d moment M^ , i t may be said that a p l a s t i c hinge has formed at that mem-ber end. To include the s t ra in hardening e f fec t a value of p greater 20 than zero should be assigned; a value of p in the range .03 to .05 times the o r i g ina l slope k i s most commonly used. Despite the fact that the member end w i l l not be per fect ly p l a s t i c when the moment exceeds My , the term " p l a s t i c hinge" i s s t i l l used to describe member ends in th i s state. In order to model th i s b i l i nea r moment-rotation behavior, each member i s considered to be comprised of two components in p a r a l l e l . One component remains f u l l y e l a s t i c and contributes a f ract ion p to the to ta l member. The second component i s e l a s t i c - pe r f e c t l y p l a s t i c and con-tr ibutes a f ract ion q (= 1-p) to the member. The moment-rotation curves for these components are shown in Figure 5. I t i s ea s i l y v i sua l i zed that mult ip ly ing the moment axis in Figure 5(a) by p and in Figure 5(b) by q (= 1-p) and then adding these calculated moments together w i l l produce the b i l i nea r re lat ionsh ip shown in Figure 3. A t yp i c a l l y displaced two component model i s shown in Figure 6. The tota l rotat ion of a member end i s comprised of two parts: (1) j o i n t rotat ion and (2) chord rotat ion due to j o i n t t rans la t i on . These r o ta -tions are related by (2.15) 6. = a). - y (2.15) where 6^ = tota l rotat ion of member end i co. = j o i n t i rotat ion y = chord rotat ion. 21 For the e l a s t i c - pe r f e c t l y p l a s t i c component i t can be seen from Figure 6 that (2.16) holds. 6. = cfr. + a. (2.16) where a>. = e l a s t i c rotat ion at member end i = p l a s t i c rotat ion at member end i The nonlinear, i ne l a s t i c behavior of a;typicalimember end i s shown in Figure 7. Numbers 1 through 9 describe the progression of the moment-rotation re lat ionsh ip throughout the history of a typ ica l loop. A progression of the type shown in Figure 7 i s known as kinematic harden-ing which i s characterized by the formation of a closed hysteresis loop. Numbers (1) through (5) describe the f i ve possible member end states at any par t i cu la r time. A member end in status (1), (3) or (5) w i l l exh ib i t i t s o r i g ina l s t i f fnes s and may be considered to be in an unyielded state. A member end in states (2) or (4) i s in a y ie lded s tate; that i s a p las -t i c hinge has formed at the member end. In th i s case the s t i f fnes s of the end i s derived so le ly from the e l a s t i c component; the e l a s t i c - pe r -f e c t l y p l a s t i c component has zero s t i f f ne s s . At any pa r t i cu la r time a member end w i l l e i ther have a p l a s t i c hinge or not. Hence each member w i l l be in one of four possible states as described in Table 2. The hysteret ic behavior described in Figure 7 i s not a r e a l i s -t i c descr ipt ion of the behavior of a reinforced concrete member. In reinforced concrete, the occurrence of p l a s t i c rotat ion after y i e ld ing w i l l be accompanied by s i gn i f i c an t cracking arid spa! l ing of concrete in 22 the section of the member subjected to tension. When the end condit ion reverses i t i s not r e a l i s t i c to suppose that the end w i l l regain i t s o r i -ginal s t i f f ne s s . A l so, due to cracking and spa l l i n g , i t i s not r e a l i s t i c to assume that the y i e l d moment in the opposite d i rect ion w i l l be the same as i f i t had not y ielded previously. To properly model the actual behavior of a reinforced concrete member i t i s necessary to use a de-grading s t i f f ne s s model whereby the reversed s t i f f ne s s and the reversed y i e l d moment are decreased proport ional ly to the amount of p l a s t i c r o t a -t ion in the opposite d i r ec t i on . Degrading s t i f fnes s models such as those proposed by Gulkan and Sozen [12], Hart; Lew and Di Ju l i o J r . [15], and Spencer [27] are more r e a l i s t i c models than the model used in th i s thes i s , however i t i s f e l t that use of the more r e a l i s t i c models would resu l t in a prohi bi tnvelyyexpens.ivenanalysi s. 2 .2 . i i The Member St i f fness Matrix Each of the four possible member states of Table 2 requires a separate member s t i f fnes s matrix to describe the re lat ionsh ip of moments and shears to rotat ions and trans lat ions fo r the e l a s t i c - pe r f e c t l y p las -t i c component. Formation of a p l a s t i c hinge w i l l not a f fec t the r e l a -t ion between ax ia l forces and ax ia l deformations, nor w i l l i t a f fec t the s t i f fnes s values of the e l a s t i c component of the member. The overal l member s t i f fnes s matrix may be expressed as (2.17) k = k a + p k b i + q k b 2 - k G (2.17) 23 where k = the overal l member s t i f fnes s matrix k = the axia l s t i f fnes s matrix a_ = the bending s t i f fnes s matrix of the e l a s t i c component = the bending s t i f fnes s matrix of the e l a s t i c - pe r f e c t l y p l a s t i c component vbi vba kg = the geometric s t i f fnes s matrix. The degrees of freedom of a typ ica l member are shown in Figure 8. Using thennotation of Figure 8 with 12 E I g = shear def lect ion factor = A GL A = cross-sectional area I moment of i ne r t i a E = Young's modulus G = shear modulus A y = shear area the ax ia l s t i f fnes s matrix i s described in (2.18) k = — ^ L 3 x xy o 2 -x -xy o y o -xy 2 -y o o o o o sym. 2 X xy y 2 0 0 (2.18) and the bending s t i f fnes s matrix of the e l a s t i c component i s shown in (2.19) 24 EI f f c i " |25 (l+g) •12xy 12x2 • 6yL 2 6xL 2 4L'*(l+g/4) •12y-:?yz12xy ::6yl_ 2 12y2 12xy -12x 2--. -6xL 2 -12xy - 6yL 2Sy . '6xL 2 C 2iL*(l-g/2) g6y|2 sym. 12x2 6xL 2 4L*(l+g/4) (2.19) The evaluation of the bending s t i f fnes s matrix of the e l a s t i c - pe r f e c t l y p l a s t i c component i s dependent upon the member state as shown in Table 2. For member state A b2A Bi (2.20) for k' b i described in (2.19) Ror member state B 'b2B 3y 2 3 " -3xy 03x 2 sym. EI 0 30.; 0 il'(-i'+g/4) - 3 / 33xy 0 3y 2 3xy ?3x 2 0 -3xy 3x 2 -3yL 2 33xL 2 0 3yL 2 -3xL 2 3L* (2.21) 25 For member,..state G EI L 5(l+g/4) 3 / -3xy 3x sym -3yL 2 3xL 2 • V 3xy 3yL 2 3 / 3xy -3x 2 -3xL 2 -3xy 0 0 0 0 3x' 0 (2.22) and for member state D L n = 0 = a nu l l matrix b2D = (2.23) The geometric s t i f fnes s matrix accounts for the second order non l inear i t ie s resu l t ing from ax ia l load e f fec t s . Derivation of the geometric member s t i f f ne s s matrix i s presented in Section 2.3 and the ahalyscis;.deso&ibedilindtnrist'seefcionti gnofescthe e ' f fec f fo f tax ia l 1 oads on bendsingi s t i f fness? 33 For solution by computer i t i s more convenient to express the overal l member s t i f fnes s matrix in a more general form. This i s achieved by expressing (2.17) in the form shown in (2.24) i i = k a + P k b i + kb2A + % kb2B + q s kb2C + % k b k = k + (p + a) k. + ki n + q o k. r 2D (2.24) 26 where constants P ( = P + q ) > q 2 and q 3 are defined in Table 3. The general form of the bending s t i f fnes s matrix of a typ ica l member including both components i s shown in (2.25) k b = k b , + k b 2 E]_ . 5 2 y a, -xya 1 x a * • 2 2 2 -yL a xL a, kLag. J 3 3 4 3 -y a. xya 1 yL'aa 3 y 2 22 xya -x a, -xLaa -xya J 1 1 1 3 1 - yL 2 a 2 x L 2 a 2 --.liLa^ y l / a 2 sym. 2 x^a. i 2 i h -XL a L a 2 5 (2.25) in which a. f l + f 2 a. = f a. a. = f. f + q 3 ^ 3 (2.26) a. f + q 3 M 2 a 6 = P (1 - g/2)/3 and P + q. f = P + q^ (2.27) 2P (1 + g/4)/3 27 and 6P TTT-gJ K ( i + g/4) 3% ( i + g/4) (2.28) In the current analysis i t i s assumed that the structure i s comprised of only two member types: horizontal and ve r t i ca l members. This r e s t r i c t i o n on member types allows the overal l member s t i f fnes s ma-t r i x to be s imp l i f i ed for each of the member types. For horizontal mem-bers x = L and y = 0 and for ve r t i c a l members x = 0 and y = L . Subst itut ion of these values into equations (2.25) and (2.18) allows the overal l membersstiffness matrix to be s imp l i f i ed into the forms shown in (2.29) and (2.30): For horizontal members k = M. = L 0 0 sym. 0 0 0 - 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 EI 0 a - . sym. 0 a L a L 3 it 0 0 0 0 0 -a -a 3L 0 a i 2 2 0 a L a c L 0 -a.L a_L 2 ° 2 5 (2.29) 28 and for ve r t i c a l members k = = L 0 1 sym. 0 0 0 0 0 0 0 0-1 0 0 1 0 0 0 0 0 0 EI a. 0 0 -a 3L 0 - a 1 0 -a 2L ilO a, L a L a. 3 i 0 0 0 0 a 6 L a 2 L sym. 0 a 5 L (2.30) 2 . 2 . i i i Modif icat ion of Member S t i f fness Matrix' for Effects of Rigid End Stubs One of the d i f f i c u l t i e s involved in frame analysis concerns the modelling of shear re s i s t i ng elements such as reinforced concrete cores surrounding elevator shafts and s t a i rwe l l s . The usual pract ice i s to mo-del the core as a column situated at the centroid of the cross sect ion. This procedure requires that beams framing into or between these cores be given special consideration to account for the f i n i t e width of the cores. A typ ica l plan view of a coupled elevator and s ta i rwe l l core system i s shown in Figure 9. In the true system, the l i n t e l beams jo in ing the two cores w i l l be much shorter that the beam used - i n '!4he molde^iwhi chins: assumed to span from.coltheglihtel"*beam ,to:have i n f i n i t e l y s t i f f ends ( r i g i d stubs)iof'-leng.th....ia':'jarid: ^br'c.as:sbown*irTFigure" T-0.' 29 Derivation of the member bending s t i f fnes s matrix for beams with r i g i d end stubs i s presented below. Arrays which contain the super-sc r ip t re late to the f l e x i b l e section of the beam while arrays with no superscript re la te to the tota l length of the beam to be consistent with the sign convention shown in Figure 10. The basic s t i f fnes s re lat ionships are shown in (2.31) f ' = k' A' (2.31a) f = k A (2.31b) A transformation matrix may be found to re la te the forces of the two co-ordinate systems as expressed in (2.32a) f = T f'_ (2.32a) and the p r inc ip le of contragradience may be used to re late the d i sp lace-ments of the two co-ordinate systems as expressed in (2.32b) A^ _ = t L a (2.32b) Subst itut ion of equations (2.32) into equation (2.31a) provides equation (2.33) from which the member s t i f fnes s matrix of the tota l mem-ber may be expressed in terms of the s t i f fnes s matrix of the f l e x i b l e section of the beam in equation (2.34) f = I k' I T A (2.33) 30 The transformation matrix may be found from elementary s ta t i c s and may be expressed as shown in (2.35) T = 1 0 0 0 0 0 0 1 a 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -b 1 (2.35) It may be possible that the neglect of the f i n i t e depth of r e -l a t i v e l y deep beams may have a s i gn i f i cant e f fect on the s t i f fnes s ma-t r i c e s of columns framing into these beams. However i t i s f e l t that th i s e f fect i s neg l ig ib le in most cases, hence thetef fect of r i g i d end stubs i s l imi ted to beams framing into or between shear walls in which case neglect of the f i n i t e wjd.th> of the walls could lead to a s izeable error in modelling. The member bending s t i f fnes s matrix i s found by eq. (2.34), The overal l member s t i f fnes s matrix of eq. (2.29) applies to beams with r i g i d end stubs with the factors defined in (2.26) replaced by those of (2.36) a, a. a„ a.-a. f + a a 2 1 (2.36) f + q + 2af + a a. 3 ^3 2 1 f 3 + % + 2 f ? f l + p ' ( l - g/2)/3 + a f j + 8f 2 + aB 0 4 31 in which a = a/L 3 = b/L and factors defined in (2.27) arid (2.28) are v a l i d . 2.3 Geometric S t i f fness Matrix The geometric s t i f fnes s matrix accounts for second order (geo-metric) non l inear i t ie s induced in a member subject to a x i a l l y directed (load components. The importance of including geometric nonl i nea r i t i e s has been pointed out by many researchers, e.g. Anagnostopoulous; Roesset and Biggs [1], Degenkolb [8]; Hanson and Fan [14], Pekau; Green and Sherbourne [25], and Wynhoven and Adams [34], [35]. In a rigorous analysis of geometric non l i near i t i e s , an i t e r a t i v e procedure i s required to determine the incremental displacements over a time step. The incremental displacements are found in the presence of the ax ia l loads ex i s t ing at the end of the previous time step, and from these incremental displacements the incremental member forces are ca l cu -lated. The incremental displacements are then solved again in the pre-sence of the revised ax ia l loads and a new set of member loads are ca l cu -lated. This procedure i s continued un t i l the difference between the ax ia l loads calculated in two successive i te rat ions reaches some acceptable tolerance. The rigorous i t e r a t i v e approach!'is, however, p roh ib i t i ve l y time consuming and i t i s necessary to make an assumption in order to avoid 32 performing i te rat ions over each time step. In th i s analys i s , the e f fect of geometric non l inear i t ie s was re s t r i c ted to the assumption that the ax ia l load in each member remains constant over the ent i re earthquake h istory. That i s to say, the ax ia l loads due to the earthquake are as-sumed to be small compared to those due to gravity loads. This assump-t ion causes the member s t i f fnes s matrices to be constant over a time step and pseudo-linear analysis based on superposition i s s t i l l v a l i d . The geometric s t i f fnes s matrix may be considered to be com-prised of two components: (a) terms comprising the so-cal led 1P-A e f f e c t s ' re su l t ing from the fact that ve r t i ca l loads w i l l contribute a transverse com-ponent of force to a displaced column and (b) terms resu l t ing fromttheuchange; int.the^bending s t i f fnes s properties of;?a: member subjected to an ax i a l . l oad . In the current analys i s , the columns are assumed to be sub-jected to a constant ax ia l compressive Toad resu l t ing from the dead weight of the structure, and the beams ace assumed to never be subjected to an ax ia l load. The l a t t e r assumption i s held to be j u s t i f i e d by the diaphragm action of the f l oo r s lab, which constrains the beams with res -pect to ax ia l deformation. In other words, ax ia l loads are ca r r ied , l a rge ly , by the slab. In der iv ing the geometric s t i f fnes s matrix for a column, the notation shown in Figure 11 i s used. While thennotatdon and degrees of freedom used here are d i f fe rent from those used previously, the derived 33 column s t i f fnes s matrix is eventually transformed into the previously used system shown in Figure 8. 2.3.i Member State A Geometric St i f fness Matrix A closed form solut ion to the d i f f e r e n t i a l equation for members subjected to ax ia l loads (beam columns) may be expressed in terms of s t a -b i l i t y functionseasepresented by Gere and Weaver [9] and shown in (2.37), for the degrees of freedom shown in Figure 11. •12 S i _ EI 6L-.S-5 4L2 S, V L 3 --3:2 S x -6L S 2 M 2 : 6EIS2 • 2L 2S l i sym. 12S, -6LS2 4L S 3 (2.37) For a compressive ax ia l load N , the s t a b i l i t y functions s are defined as <. _ s in \ji 1 ~ 1 2 * c , _ _ ip(sin - i> cos \j>) S 3 - 4 ^ c _ ^ (1 - cos ^) b \ - 6d,c - s in T\I) (2.38) 2*, in which <{> = 2 - 2 cos tJ; - s in \> (2.39) A!NL !EI (2.40) 34 I f sine and cosine are expanded in terms of 4> with higher order terms dropped out, the s t a b i l i t y functions of (2.37) may be rede-fined as in (2.41) 1 -1 10 l - 4 -1 30 = 1 4_ 60 1 + 1 60 (2.41) in which , _ . - _ NL " " f i " (2.42) Subst itut ion of the approximate s t a b i l i t y functions of (2.41) into the force-def lect ion re la t i on (2.37) resu lts in (2.43) V. EI_ . 3 30L 12 6L 4L sym. •t,112 -6L 12 6L 2L 2 -6L 4L 2 336 3L 4L sym. --36 -3L 36 3L -L 2 -3L 4L 2 (2.43) which may be expressed as 1 = CKS - K G ] 6 (2.44) 35 where Kg = member bending s t i f fnes s matrix in absence of ax ia l loads Kg = member geometric s t i f fnes s matrix I t should also be noted that the geometric s t i f fnes s matrix in (2.43) may also be derived in a "consistent f i n i t e element" sense. This approach i s used by Clough and Penzien [7] and i s based on the re la t ion K G1j * N ^,(x)/ 9U)L ; ) (30-13(p) -(30-13cp) 0 (30-13) (30-3*) sym. •(30-3) (30-3cp)L' (2.48) from which the geometric s t i f fnes s matrix may be expressed as the sum of terms accounting for the change in bending s t i f fnes s matrix plus terms accounting for the P-A effects as shown in (2.49) N (30-cf))L 6+<() 0 0 sym. -(6+cj)) 0 6+cj) 6L 0 -6L 6L" *1 1 0 0 sym. •1 0 1 0 0 0 0 (2.49) 2.3 . i i i Member State C Geometric St i f fness Matrix The geometric s t i f fnes s matrix for a member in state C may be derived in a s imi la r manner to that used for members in state B. The resu l t ing geometric s t i f fnes s matrix for the e l a s t i c - pe r f e c t l y p l a s t i c component of a member in state C i s as shown in (2.50) _G (30-cj))L (6+cp) 6L 6L 2 sym. •(6+(J)) -6L (6+cj>) 0 0". 0 0 1 0 0 sym. -1 0 1 0 0 0 0 (2.50) 38 2.3.iv Member State D, Geometric St i f fness Matrix The e f fec t of ax ia l loads on the e l a s t i c - pe r f e c t l y p l a s t i c com-ponent of state D members w i l l only be in the form of P-A e f fec t s . This may be accounted for by the geometric s t i f fnes s matrix shown in (2.51) II 1 0 0 sym. -1 0 1 0 0 0 (2.51) 2.3.V General Form of Member Geometric St i f fness Matrix As stated previously, the assumption has been made that only columns w i l l be subject to ax ia l loads. Using the notation of Table 2 and (2.'42), the geometric s t i f fnes s matrix for a column subjected to a compressive ax ia l load N may be expressed for a l l states as shown in (2.52) N L a. 0 0 sym. 0 K I 0 u i a' a? i a,-a i CO P. 0 0 0 4 P, i a' 0 0,2 J ' Li. 1 0 0 sym. 0 0 0 - 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 (2.52) 39 in which a. = (6 + 4>) (q 2 + q 3 ) a ( % + 6 q 2 ) L a. a. (iO + 6q 3)L (2.53) a a. P_ 30 p e 2 30 - * % 30 - cp (2.54) An approximation for the geometric s t i f fnes s matrix which i s commonly used in nonlinear dynamics i s to neglect the f i r s t matrix in (2.52). In th i s case only the second matrix, which represents P-A e f -fect s , i s used to approximate geometric non l inear i t ie s . This approxima-t ion i s accepted ninto the computer program developed by the author. 2.4 Formation of the Reduced Frame St i f fness Matrix The member state matrices derived in Section 2.2 are based upon an i dea l i za t i on containing three degrees of freedom at each,member end. 40 Inclusion of a l l member matrices w i l l re su l t in a structure s t i f fnes s matrix containing three degrees of freedom at each j o i n t in the structure. However i t i s desirable to reduce the number of degrees of freedom in the structure s t i f fnes s matrix as the e f f o r t required for the solut ion of the incremental-displacements varies accordingly. As pointed out in Section 2.1, the structure s t i f fnes s matrix i s reduced to contain only three de-grees of freedom per storey and a discussion of the methods used to re -duce these degrees of freedom i s now presented. The structure i s viewed to be an assemblage of plane frames which allows for the s t i f fnes s matrix of each plane frame to be obtained independently and then added into the structure s t i f fnes s matrix. Several degrees of freedom may be eliminated i f ax ia l deforma-tions of the beams are neglected in each frame. This assumption implies that the f loors at each storey level are r i g i d with respect to in-plane deformation which i s reasonable for a f l oo r system of reasonable th i ck -ness continuously connected to the beams. Additional degrees of freedom may be eliminated i f a shear-beam model i s used, i . e . the columns are assumed to be inextensible allowing for reduction of a l l ve r t i ca l degrees of freedom. However th i s assump-t ion has-been shown, by Weaver and Nelsonri[33L]]tdolead 'to- a s izeable overestimation of s t i f f ne s s , espec ia l l y in t a l l or slender buildings which have a high aspect r a t i o . In addit ion i t i s desirable to enforce ve r t i ca l compat ib i l i ty at corner columns common to two frames, e spec ia l l y for structures whose predominant displacement mode i s of a bending, rather than shear type. I f th i s ve r t i ca l compat ib i l i ty i s not included 41 the undesirable re su l t shown in Figure 14(a) resu lts in which frames perpendicular to a par t i cu la r motion do not contribute to resistance of the motion. Inclusion of the ve r t i ca l compat ib i l i ty condition causes the more r e a l i s t i c case shown in Figure 14(b) to re su l t . For these rea-sons i t i s necessary to include the ve r t i ca l degrees of freedom at j o i n t s on columns common to more than one frame. These degrees of f ree -domTjwiiflI1 hereafter be referred to as ' c ompat ib i l i t y ' degrees of freedom. of freedom shown in Figure 15a and then each frame s t i f fnes s matrix i s s t a t i c a l l y condensed to contain the reduced number of degrees of freedom shown in Figure 15b. matrix must be added into the frame matrix in a unique manner. The frame matrix must be part i t ioned in terms of the fol lowing three d i s t i n c t types of degrees of freedom: (1) the horizontal t rans lat ions of the storey levels j i (2) the compat ib i l i ty degrees of freedom x_ (3) other degrees of freedom §_ which are independant of a l l Thus the frames are i n i t i a l l y assembled to contain the degrees In order to condense the frame s t i f fnes s matrix each member other frames ( 'other ' degrees of freedom). The part i t ioned frame matrix re lat ionsh ip i s shown in (2.55) H X x (2.55) 0 K K 6 2 1 22 42 Forces corresponding to S_ w i l l be 0_ since in th i s dynamic analysis only horizontal i n e r t i a forces are considered. I t should be noted that in Section 3.5 th i s procedure w i l l . be repeated for the case where unbalanced forces A ex i s t in which case a d i f fe rent condensation procedure i s required. However, in the absence of unbalanced loads the fol lowing procedure i s v a l i d . As matrix K 2 2 may be stored in banded form i t i s su itable for decomposition into the product of a lower t r iang le matrix times i t s transpose using Choleski 's method K = L c lJc (2.56) 22 F F Operations on the lower pa r t i t i on of (2.55) w i l l re su l t in the re lat ionsh ip x >T (2.57) which can be used in conjunction with the upper pa r t i t i on to give the re lat ionsh ip ij} = [ K l x - ( C \ 2 l ) J ( L - 1 K 2 1 ) ] F {^ } (2.58) in which case the reduced frame s t i f fnes s matrix Kc may be express in terms of only tworrmatrices K and (L K 2 1 ) in a form ea s i l y soluble by computer and requir ing a minimum of storage: 6 = e L J / ^ l / Y ^ p < h 43 K F = K n , (L " K 2 1 ) _ (L K 2 l ) ( (2.59) 2.5 Formation of the Reduced Structure St i f fness Matrix The reduced frame s t i f fnes s matrices must be added into the structure s t i f fnes s matrix in a unique manner. The reduced structure T degrees of freedom are denoted by < g_ r 9 > with corresponding genera-l i z ed forces < C[ R_ e_ >^ as shown in Figure 16b. For th i s system q_ and £ are vectors representing horizontal t rans lat ions in two mutually perpendicular d i rect ions (one each per storey) and 9_ i s a vector of rotations about the structure o r i g i n . For a par t i cu la r frame the fol lowing quantit ies may be defined I = d i rect ion cosine of frame in q d i rect ion q I = d i rec t ion cosine of frame in r d i rec t ion If l = perpendicular distance from the structure o r i g i n to the frame. The horizontal trans lat ions of a frame may be expressed in terms of the structure co-ordinates q , r and 9 from the ea s i l y v i s -ual ized re lat ionsh ip shown in (2.60) r q + I r + I„ 9 = < I q a. r — 9 — c i „ > i £- r (2.60) and the horizontal forces at each storey of a frame may be expressed in 44 terms of the structural generalized forces Q , R , 9 by applying the p r inc ip le of v i r t ua l displacements. H a H < R > . (2.61) H > 0 > I f the reduced frame force-def lect ion re lat ionsh ip of (2.58) i s expressed in part i t ioned form as shown in (2.62) H / y = ' — x_ Khh ', Khx Kxh ' Kxx (2.62) then the introduction of equations (2.60) and (2.61) into the part i t ioned frame s t i f fnes s re lat ionsh ip w i l l allow i t , with appropriate manipula-t ions , to be expAessedsfn :;!temsro€-!:4heT.s-t-K!uetuFal ,co'-ot?dinates.'.- q » r ahd-.;"6 nas shown in (2.63) 4 I2 q sym. q hx 1.' R- C S h I K, r hx £ .It !G lr S h 2 T e Khh x_ \ in r xh 0 xh (2.63) When the contributions of a l l frames have been added into (2.63) the unreduced structure force-def lect ion equation i s obtained in 45 terms of the degrees of freedom shown in Figure 16(a). I t i s now poss i -ble to use s t a t i c condensation to reduce the compat ib i l i ty degrees of freedom. The generalized forces x_ must equal fJ as ve r t i c a l i n e r t i a forces are ignored. This allows the structure s t i f fnes s matrix to be s t a t i c a l l y condensed using the same procedure as for frame matrices. The f i na l structure s t i f fnes s matrix i s shown in the structure force-dis-placement equation (2.64) Q R = K s -(2.64) in which K s = [K^ - (L"\ 2 1) T (L \ ^ ) \ (2.65) 46 CHAPTER 3 NONLINEAR MATERIAL BEHAVIOR AND ENERGY CALCULATION One of the major problems associated with a nonlinear analysis concerns the member state determination phase of the analys i s . In an analysis in which the members are assumed to remain l i n e a r l y e l a s t i c the structure s t i f fnes s matrix need only be calculated once and member forces calculated as desired. However, in a nonlinear analysis member forces must be calculated every time step i n order to check whether any members have yielded or subsequently reversed. This phase of the analys is i s known as the member state determination phase. I f a member state changes during a time step, the structure s t i f fnes s matrix must be recalculated and unbalanced forces resu l t ing from the nonl inear i ty must be accounted fo r . This chapter out l ines the methods used to perform the above ca lcu lat ions . 3.1 Calculat ion of Member Deformations The b i l i near member properties assumed in therfimodel require that member deformations be calculated incrementally and that the to ta l member deformations at any time be calculated by superposing the inc re -mental deformations which have occurred up to that time. Solution of the equi l ibr ium equation (2.1) y ie lds a vector of incremental displacements of the generalized co-ordinates < A£ Ar A6 >T . 47 This vector must be expanded to account for incremental member deforma-t ions. This expansion i s done in two steps: f i r s t l y the incremental displacements of the compat ib i l i ty degrees of freedom are calculated and secondly the incremental displacements of degrees of freedom relevant only to a par t i cu la r frame are calculated for each frame in turn. These ca lcu lat ions y i e l d a vector of incremental j o i n t displacements from which the incremental member deformations may be selected for each frame. The incremental displacements of the compat ib i l i ty degrees of freedom are calculated from the structure force-displacement re lat ionsh ip (2.63) in which the forces corresponding to the compat ib i l i ty degrees of freedom are set equal to fJ as in (3.1) i . e . AF < ; > = , T K c 21S _ < Af 0 1 K 2 2S Ax in which AF, Af = incremental generalized forces, displacements Ax = incremental displacements of the compat ib i l i ty degrees of freedom and the subscript s refers to submatrices of the structure s t i f fnes s matrix in order to avoid confusion with those of the frame s t i f fnes s matrices. Using the Choleski decomposition technique described in Section 2.4 the incremental compat ib i l i ty displacements may be expressed in the fol lowing form: Ax = - LJ"1 [L"1 K_ 2 1 ] s Af (3.2) 48 The vector of incremental s t ructura l displacements < Af Ax > can be transformed into a vector of incremental displacements for frame i : < Ah-. Ax.j >T in which for frame i Ahj^ = I q i Ag_ + I r i AT + I Q i A0 (3.3) and Ax. are the compat ib i l i ty degrees of freedom corresponding to frame i . Solution of the incremental displacements of a l l other degrees of freedom for frame i , A6- 9 i s achieved by usage of (2.57) written in terms of incremental values as = " i a (L"1 K 2 1) FJ < ^ i ^ j . > T ( 3 , 4 ) where the subscript Fi refers to matrices for frame i . Equations (3.3) and (3.4) are applied separately to each frame to produce a vector of incremental displacements < Ah Ax A6 > , which describes the incremental displacement of each j o i n t , fo r each frame. The incremental member deformations, in terms of three degrees of f ree -dom at each member end, may be selected from the incremental j o i n t d i s -placements; and from these incremental deformations the incremental mem-ber end forces may be ca lcu lated. 49 3.2 Calculat ion of Member End Forces I t i s convenient to introduce a new co-ordinate system and no-tat ion as shown in Figure 17 in order to express the incremental member forces in as concise a manner as possible. The to ta l incremental member end forces are comprised of the sum of forces in the e l a s t i c component and those in the e l a s t i c - pe r f e c t l y p l a s t i c component and these two compo-nents are solved separately. In Section 2.3 the geometric s t i f fnes s matrix i s derived and expressed as two separate components: terms accounting for P-A ef fects and terms accounting for the change in bending s t i f fnes s properties of members subject to ax ia l loads. In the current analysis only the c o n t r i -bution of P-A ef fects i s considered in formation of the geometric s t i f f -ness matrix, and for s imp l i c i t y the derivat ion of column force r e l a t i o n -ships i s also re s t r i c ted to geometric non l inear i t ies in the form of P-A e f fec t s . The incremental ax ia l force i s calculated for columns only, as beams are constrained with respect to ax ia l deformation. I f the inc re -mental ax ia l shortening of a member i s 6 then the incremental ax ia l load in a member, AN , i s expressed in (3.5) AN = ~ 6 (3.5) in which A = column cross-sectional area E = Young's modulus L = column length 50 Incremental moments and shears in amember may be expressed in terms of incremental member end rotat ions. The member end rotat ion 9 i s expressed as the j o i n t rotat ion co minus the chord rotat ion y as shown in (2.15). For a beam with r i g i d end stubs as shown in Figure 10 i t i s desirable to calculate the member end forces at the ends of the f l e x i b l e section of the beam. In th i s case, the chord rotat ion must be modified by including the t rans lat ion of the member ends caused by rotat ion of the r i g i d stubs. Using the notation shown in Figure 10 the chord rotat ion of the f l e x i b l e beam may be expressed as in (3.6) and a l l further member properties refer to the f l e x i b l e section of the beam. 6.-6. . j i a b to c\ Y = - ^ - j ; [j " i " r; w j ( 3 ' 6 ) in which 6., 6. = ve r t i c a l displacements at ends i , j of the 1 J to ta l beam y = chord rotat ion of the f l e x i b l e section of the beam. In ca lcu lat ion of incremental member forces i t i s also desira-ble to define the quantit ies shown in (3.7) K a L (1 + g) k = 2Ej,.(] - 9/2) K b L (1 + g) in which I = member moment of i n e r t i a g = member shear def lect ion factor (3.7) 51 Incremental moments in the e l a s t i c component are expressed in terms of incremental member end rotat ions in (3.8) A M IEL A M 2EL A e. A e. (3.8) in which p equals a factor describing the contr ibution of the e l a s t i c component to the tota l member, i . e . the y ie lded slope divided by the i n i t i a l slope of the assumed b i l i nea r moment-rotation re la t ionsh ip , and the subscript EL indicates forces in the e l a s t i c component. The incremental moments in the e l a s t i c - pe r f e c t l y p l a s t i c com-ponent must be expressed separately for each possible member s tate. The possible member states are described in Table 2. For member state A, incremental forces in the e l a s t i c - pe r f e c t l y p l a s t i c component are solved by (3.9) A M l EP A M EP A k k. A 0 a b l (1 - p) > k. k A 0 b a 2 (3.9) in which the subscript EP indicates forces in the e l a s t i c - pe r f e c t l y p l a s t i c component. 52 For member state B, A M IEP A M 2EP J B (1 - P) a k a A 6 \ (3.10) A 6, For member state C A M iEP y = ( i - p) ' 2 E P j k a " k b / k a \ (3.11) A 8. and for member state D A M iEP A M 2EP 0 0 (1 - p) •< > 0 0 A6 2 V. J (3.12) These incremental moments are calculated assuming that the mem-ber state at the end of the previous time step remains unchanged during the ent i re current time step. However i t i s possible that the member state may actua l l y change during the time step. For instance the addi -t ion of the incremental moments calculated over a time step to the to ta l moment ex i s t ing at the beginning of the time step for a member o r i g i n a l l y in state A may produce a moment greater that the y i e l d moment at an end. 53 In th i s case the member end actua l ly y ie lds during the time step and the incremental moments in the e l a s t i c - pe r f e c t l y p l a s t i c component are i n -correct. The methods used in compensating for th i s error are outl ined in the fol lowing sections. 3.3 Y ie ld C r i t e r i a I t i s necessary that the conditions for which a member end may be considered to have yielded or subsequently unyielded be accurately defined. There are three d i f fe rent y i e l d c r i t e r i a which must be used in th i s analys i s . Beam members are constrained against axial deformation hence the member may be considered unyielded un t i l the tota l moment sur-passes the yieflld moment at the member end. Columns, however, are sub-jected to ax ia l loads which necessitates consideration of a x i a l - f l e xu r a l interact ion in determining whether the end has y ie lded. In add i t ion, columns which are common to more than one frame w i l l have moments about both pr inc ipa l axes and the interact ion of the two components of moment plus ax ia l loads must be considered. The three y i e l d c r i t e r i a are thus related to the fol lowing three cases: -(1) beam members, (2) columns subject to bending about one axis only (uniaxial columns), (3) columns subject to bending about two axes (b iax ia l columns). 54 3.3.i Beam Members In general there w i l l be four d i f f e ren t y i e l d moments for each beam member: a pos i t ive and negative y i e l d moment at each member end. For a prismatic and symmetric steel beam these y i e l d moments are a l l i n i -t i a l l y equal, but for a reinforced concrete beam the re in forc ing steel near the top of the beam w i l l generally not be the same as that near the bottom of the beam at a par t i cu la r member end. In add i t ion, quantit ies of steel at one endsof"cthe~beam are commonly d i f fe rent from those at the other end of the beam. Hence for beams constructed from reinforced con-crete i t i s necessary to consider four d i f fe rent values for the y i e l d moments of the beam. It i s also necessary to consider the i n i t i a l moments induced in the beams by the dead weight of the f l oo r s labs. The member end mo-ments calculated during the earthquake are those caused so le ly by the earthquake, in absence of i n i t i a l moments caused by dead loads. Hence the i n i t i a l moments must be subtracted from the actual y i e l d moments to express the e f fec t i ve y i e l d moment in terms of the moment, in addit ion to the dead load moment, which w i l l cause y i e l d . For example consider a member end which w i l l y i e l d when subjected to a pos i t ive or negative moment of 100 k ip - feet . If the dead load moment is 40 kip feet in the negative d i rect ion then i t may be said that the pos i t ive y i e l d moment i s 140 kip feet while the negative y i e l d moment is 60 kip feet. 55 3 .3 . i i Uniaxial Columns The y i e l d moment for uniaxial columns i s a function of the cur-rent value of the axia l load in the member. It i s common pract ice in reinforced concrete construction to use e i ther c i r c u l a r or square columns with ve r t i c a l reinforcement located symmetrically about two axes. The pos i t ive and negative y i e l d moments w i l l be equal for these columns, so th i s r e s t r i c t i o n , although not v i t a l , i s accepted in the current analys i s . A typ ica l a x i a l - f l e xu ra l interact ion curve i s shown in Figure 18 fo r co-lumns subject to the above r e s t r i c t i o n . which are required to express the pos i t ive or negative y i e l d moment in terms of the ax ia l load It may be seen from Figure 18 that there are three equations for N > 0 (tension) M. (3.13) for N < 0 and N > N (compression) B (3.14) and for N < N g (compressfori) 'B (3.15) in which N ax ia l load (negative for compression) ultimate compressive load (negative) N t ultimate tens i le load (pos i t ive) N B balanced load (negative) y i e l d moment in presence of ax ia l load N 56 M = y i e l d moment with 0 ax ia l load y Mg = balanced moment. The actual interact ion surface for a reinforced concrete co-lumn i s generally a smooth curve, hence the interact ion curve shown in Figure 18a consist ing of s t ra ight l i n e approximations of the curve i s not exact. Interaction curves for other materials may s im i l a r l y be ap-proximated by appropriately def ining the points described above. 3 . 3 . i i i B iax ia l Columns Columns which are common to more +than one frame, and hence which are subject to two perpendicular components of bending, have a much more complex y i e l d surface. The y i e l d moment about one axis of the column .is a function of both the moment about the other axis of the co-lumn and the ax ia l load. For b iax ia l columns, the y i e l d surface i s a three dimensional surface ex i s t ing in a space defined by the moment about one ax i s , the moment about the other axis and the ax ia l load. A view of th is interact ion surface perpendicular to the plane in which the moment in one d i rect ion equals zero w i l l provide the in teract ion diagram shown in Figure 18a. The same holds for a view of the interact ion sur-face perpendicular to the plane in which the other moment equals zero, as expected. The shape of the y i e l d surface in a plane defined by a constant ax ia l load i s represented in a so-cal led b iax ia l interact ion diagram. In a plane frame analysis the e f fec t of bending out of plane, i . e . b iax ia l 57 e f f e c t s , are ignored. However i t has been stated by Degenkolb [8] that these b iax ia l ef fects are important and research by Selna; Mo r r i l l and Ersoy [26] on the f a i l u r e of the Ol ive View Psych iatr ic Day C l i n i c sup-ports th i s conclusion. The inclus ion of b iax ia l e f fects causes a column to y i e l d in a given d i rect ion at a lower moment due to the presence of a moment in the other d i r ec t i on . Much research has been done concerning the shape of the b iax ia l interact ion diagrams for a var iety of column sections. The shape of the curve for reinforced concrete columns is a function of many variables such as re inforc ing steel arrangement, the steel y i e l d strength, the con-crete compressive strength, the dimensions of the column cross sect ion, etc . In an attempt to describe the b iax ia l interact ion diagram, Bresler [4] proposed the fol lowing formula (3.16) y i e l d moments about the x, y axes yieldnmoment about the x axis with o moment about the y axis y i e l d moment about the y axis with o moment about the x axis constants determined empi r ica l l y and depending upon column properties. where M . M x y M = xo M = yo m , n = Parme; Nieves and Gouwens.[24] proposed that the fol lowing formula be used to define the b iax ia l interact ion diagram for rectangular 58 reinforced concrete columns log .5 log .5 / M \ log 3 / M \ log 3 in which the factor 3 was determined ana l y t i c a l l y as a function of several column properties. In order to reduce the number of variables required to define the b iax ia l interact ion diagram i t i s necessary to make an assumption. In th i s analys i s , i t has been assumed that the b iax ia l columns are such that values of m and n equal to 2 in the formula proposed by Bresler [4] and 3 equal to — - in the formula proposed by Panne; Nieves and rr Gouwens^[24-]]are appropriate. This assumption makes the-b iax ia l i n t e r -action diagram an e l l i p s e as shown in Figure 18b and expressed in (3.18) (3.18) The y i e l d surface for b iax ia l columns subject to the above as-sumption i s shown in Figure 18c. While th is y i e l d surface i s not general i t i s f e l t to be representative of the y i e l d surface of reinforced con-crete columns and results obtained using th i s y i e l d surface should resem-ble the actual b iax ia l behavior of these columns. 59 3.4 Member State Determination This phase of the analysis involves checking each member to see whether i t s state changes during a time step, and updating member properties. Because of the d i f fe rent y i e l d c r i t e r i a used for beam mem-bers; columns which are subject to bending about one axis (uniaxial co-lumns); and for b iax ia l columns, a d i f fe rent technique i s used for these three types of members. 3 . 4 . i Beam Members The method used in the member state determination phase i n -volves ca lcu lat ing the f ract ion of the time step for which the i n i t i a l member state remains unchanged. I f th i s f ract ion i s equal to one the i n i t i a l member state existed for the ent i re time step and ca lcu lat ions for th i s member are correct. However, i f th i s f ract ion i s less than one, the i n i t i a l member state must have changed during the time step, and ca lcu lat ions for th i s member are incorrect as a resu l t of th i s change in state. A change in state w i l l hereinafter be referred to as an ' event ' . I t i s convenient to introduce an assumption in order to correct the calculated.incremental forces. The calculated incremental d i sp lace-ment; is'assuriied to jbe correct;-''.whether an event occurred or not. While this'a'ssumptio'n.is"not•.rigorous.?oif the time-step is kept' small errors ihtroducedfbythi ' s assumption-will be s l i g h t . For a member end which i s e l a s t i c at the beginning of a time step, the f ract ion of the time step un t i l an event (hereinafter referred 60 to as the event factor) i s calculated according to (3.19) M. y = -1 (3.19) in which y = the event factor for the member end under consideration I f a value of y greater than 1.0 i s calculated by (3.19), an event did not occur and y i s set equal to 1.0. For a member end which i s p l a s t i c at the beginning of the time step, a c r i t e r i o n must be established to determine whether the p l a s t i c hinge w i l l reverse (and hence be removed) or remain. The c r i t e r i o n used in th i s analysis i s to calculate the energy diss ipated in the p l a s t i c hinge and to remove the hinge i f the energy diss ipated during a time step i s calculated to be less than zero. The energy diss ipated at a p l a s t i c hinge i s calculated according to (3.20) = the y i e l d moment of the e l a s t i c - pe r f e c t l y p l a s t i c component at the member end = ( 1 - P ) M y = the y i e l d moment of the member end =;:. the i n i t i a l moment in the e l a s t i c - pe r f e c t l y p l a s t i c component at the member end = the incremental moment in the e l a s t i c - pe r f e c t l y p l a s t i c component at the member end. A E D * M E P A a (3.20) in which AE D incremental energy diss ipated at the hinge M, 'EP the moment in the e l a s t i c - pe r f e c t l y p l a s t i c component at the hinge Aa the incremental p l a s t i c rotat ion at the hinge. 61 Equations used to ca lcu late the incremental p l a s t i c rotat ion are presented in Section 3.6. I f the energy diss ipated in the.ihinge over a time step is less than zero a value of u = 0 i s assigned, while i f the incremental energy diss ipated i s greater than zero, \i i s set equal to 1. Hence the member forces and state are determined in an i t e r a -t i ve manner in which the member state may change one or more times per time step. I f the event factor at e i ther end of a member i s less than one, the state of the c r i t i c a l member end i s changed and the member event factor i s set equal to the event factor at thateerid. A new-member event factor i s then calculated for the revised member state. This procedure i s repeated un t i l the sum of member event factors equals one. 3 .4 . i i Uniaxial Columns For members subject to ax ia l loads, the member state determina-t ion phase i s performed based upon the y i e l d moment in the presence of ax ia l loads ex i s t ing at the beginning of the time step. However the ax ia l loads, and hence the y i e l d moments, w i l l change during the time step. This problem i s solved by rechecking the member state again at the end of the time step. For a member end which i s e l a s t i c at the end of a time step, a p l a s t i c hinge is inserted i f the magnitude of the mo-ment in the e l a s t i c - pe r f e c t l y p l a s t i c component i s greater than the y i e l d moment.,calculated at the end of the time step. On the other hand, the p l a s t i c hinge i s removed from a member end which i s p l a s t i c at the 62 end of a time step i f the moment in the e l a s t i c - pe r f e c t l y p l a s t i c compo-nent i s less than the revised y i e l d moment. 3 . 4 . i i i B iax ia l Columns For columns subject to bending about two axes, introduction of a hinge at a column end w i l l a f fect the behavior of the column in each d i r ec t i on . That i s , a hinge cannot ex i s t in the column with res-pect to bending about one axis without the inc lus ion of a hinge with res-pect to bending about the other ax i s . Determination of the state of b i -ax ia l columns must include current values describing the member in both d i rect ions . The b iax ia l interact ion diagram of Figure 18b i s assumed to be in the shape of an e l l i p s e . For th i s e l l i p t i c a l y i e l d re la t ionsh ip , the event factor for an e l a s t i c end i s expressed in (3.21) = a a - a a + / a - (a a - a a ) i 1 2 3 4 / 5 v 1 4 2 3 ' / 0 0 , x y = : (3.21) a 5 in which a a a 1 " M y i A M E P i 2 " V . M EPj 3 M y j a - A " E p J a 63 2 a = a + a 5 AM, EP' M, 'EP = the incremental, to ta l moments in the e l a s t i c -perfect ly p l a s t i c component M, y = the y i e l d moment in the e l a s t i c - pe r f e c t l y p l a s t i c component and the subscripts i and j indicate two perpendicular axes. Derivation of (3.21) i s given in Appendix A. The event factor for a p l a s t i c end i s found in a manner s im i l a r to that used for e l a s t i c ends. The energy diss ipated at the hinge i s calculated and i f the energy i s less than zero the hinge i s removed and an event factor of zero i s assigned. I f the energy diss ipated i s greater than zero the hinge remains and an event factor of one i s assigned. The tota l energy diss ipated at a p l a s t i c hinge i s the sum of the energies diss ipated in the two perpendicular d i rec t ions , as shown in (3.22) in which the subscripts i and j indicate values in two perpendicular d i rect ions . The i t e r a t i v e procedure used for b iax ia l columns i s ident ica l to that used for uniaxial members; that i s the procedure i s continued un t i l the sum of event factors for a time step equals one. In b iax ia l columns the ax ia l load generally varies over a time step, hence i t i s necessary to recheck the member state based on the y i e l d moment at the end of the time step. I f the check shown in (3.23) i s s a t i s f i e d , an e l a s t i c end w i l l remain e l a s t i c , however a p l a s t i c end AE D = M E p i Aou + M E p j Aa. (3.22) 64 w i l l become e l a s t i c . Likewise, i f the check i s not s a t i s f i ed an e l a s t i c end w i l l become p l a s t i c while a p l a s t i c end w i l l remain p l a s t i c . (3.23) 3.5 Unbalanced Force Corrections When an event occurs, member forces at the end of a time step w i l l d i f f e r from those calculatedc'based on i n i t i a l member states. Figure 19a shows the unbalanced moment, ,*?which results when a member end y ie lds and Figure 19b shows the unbalanced moment resu l t ing from the r e -versal of a p l a s t i c hinge during a time step. In both of these cases the member i s assumed to be in pos it ion 1 at the beginning of the time step. The member end rotates an amount A9 which places the end at po-s i t i o n 2 based upon the i n i t i a l state of the end. However as movement from posit ion 1 to posit ion 2 d ictates the occurrence of an event, the member end i s revised to be in pos it ion 3. The difference in moment be-tween posit ions 2 and 3 i s the unbalanced moment at the j o i n t which must be accounted for in order to maintain equi l ibr ium. Unbalanced forces also re su l t at an end when an event occurs at the other end. In general, the unbalanced moments for a member equal the difference between the moments calculated based upon i n i t i a l condi-tions and the moment calculated by the i t e r a t i v e procedure described in Section 3.4 and shown in (3.24) 65 n M E P i + 1 = M E P i + * ^ E P j < 3- 2 4' in which Mrp,- = moment in e l a s t i c - pe r f e c t l y p l a s t i c component at the beginning of the time step M F P - j + = moment in e l a s t i c - pe r f e c t l y p l a s t i c component at 1 the end of the time step y. = event factor for i t e ra t i on number j ANLp. = incremental moment in the e l a s t i c - pe r f e c t l y p l a s t i c J component for i t e r a t i on j n = number of i te rat ions occurring for the member over the time step. Unbalanced forces may also resu l t at the p l a s t i c hinges of mem-bers subject to ax ia l loads. These forces re su l t from the fact that a change in ax ia l load over a time step may cause an excursion outside of the y i e l d surface. Calculat ion of unbalanced forces resu l t ing from th i s e f fec t i s shown in Figure 19(c). The change'.in the axiaTCforce in the member moves i t from posit ion 1 to posit ion 2 which i s outside the y i e l d surface. In order to bring the member end back onto the y i e l d surface i t i s necessary to apply the unbalanced moment Mu to thermember end which w i l l bring i t to the admissable pos i t ion 3. Unbalanced shear forces w i l l also re su l t for each member in which unbalanced moments occur. Unbalanced shear forces are calculated from simple equi l ibr ium as shown in (3.25) K V - U = E P u i L E P " J (3.25) in which L = member length 66 and the subscript u indicates unbalanced forces and subscripts i and j indicate values at ends i and j . In a s t a t i c nonlinear analysis unbalanced forces are corrected by using an i t e r a t i v e scheme such as the Newton-Raphson method. In th i s method the unbalanced forces are reapplied to the revised structure and incremental displacements and member forces caused by the appl icat ion of the unbalanced forces are added to those caused by the i n i t i a l loads. Another set of unbalanced forces may resu l t from appl icat ion of the f i r s t set of unbalanced forces, and th i s set i s then appl ied. This i t e r -at ive method i s continued unt i l the calculated set of unbalanced forces become neg l i g ib le . In theory the Newton-Raphson method could also be applied in th i s dynamic case, however i t s use would require an i t e ra t i on for each time step in which an event occurs which i s not economically feas ib le . An approximate method of t reat ing these unbalanced forces i s to apply them in the fol lowing time step. Hence while these unbalanced forces w i l l be permitted to act for one time step, i f the time step i s kept small the error introduced by th i s assumption should be neg l i g ib le . 3 . 5 . i Reduction of the Unbalanced Force Vector Addition of the unbalanced member forces into a vector for each frame follows standard procedures based on the frame code numbers. This w i l l r e su l t in a load vector containing the unbalanced forces at each degree of freedom for each frame. The unbalanced load vector may 67 be part i t ioned to the form < H ! A ! X > T in which IH A unbalanced forces in the horizontal d i rect ion unbalanced forces at the compat ib i l i ty degrees of freedom X = unbalanced forces at a l l other degrees of freedom From equation (2.57) i t was shown that (2.57) 6 = L^" 1 [L"V 2 J_] F < h j x > T which expresses displacements of the "other" degrees of freedom in terms of the displacements of the horizontal and compat ib i l i ty degrees of freedom. Subscript F indicates matrices for the par t i cu la r frame. V i r tua l work equations can then be used to show that r ~\ .U - JJL> A c (Lf-) Fi£ ] * (3.26) in which a superscript 1 indicates unbalanced forces resu l t ing a f te r forces _X have been s t a t i c a l l y condensed. i Forces H_. can be expressed in terms of the generalized forces Q , R and e_. from equation (2.61) "a - H -i - T r - ' I CD| ^ 6 (2.61) 68 Hence for each frame a reduced vector of unbalanced forces < Q.R 8 A ' >T can be obtained. When the contributions of a l l frames have been added into th i s vector, the transformation to generalized co-ordinates may be performed as shown in (3.27) 4 R - l j - i 1 E(L K 2 1 ) ^ L s ] A (3.27) 3.5. i i Modif icat ion of Equi l ibr ium Equation Unbalanced forces must be accounted for in order to maintain equi l ibr ium. The e f fec t of unbalanced forces occurring during a pa r t i cu -l a r time step should be accounted for by performing an i t e r a t i v e proce-dure within the time step. In the current analysis unbalanced forces are allowed to ex i s t for the time step during which they occur but are accounted for in the fol lowing time step. The incremental displacements in the time step fol lowing the occurrence of an event are solved by using the equi l ibr ium equation shown in (2.1) modified by adding the reduced unbalanced load vector F £ to the pseudo-load vector as fol lows. M Ax + C B Ax + k B Ax = M A x„ + F (3.28) 3 . 5 . i i i Modif ication of Equations Used to Expand the Incremental Displacement Vector The theory presented in Section 3.1 used to expand the i nc re -mental displacement vector from generalized co-ordinates to gross frame 69 degrees of freedom must be modified due to the existence of unbalanced forces. Previously presented theory i s based upon the fact that forces ex i s t i ng at compat ib i l i ty and ' o ther ' frame degrees of freedom equal zero. However, unbalanced forces may resu l t at these degrees of freedom d i c ta t i ng modif ication of the equations. The incremental s t ructura l force-displacement re lat ionsh ip shown in (3.1) and modified to include unbalanced forces i s shown in (3.29) AF AF. uc K : KT u s 21S K . K 21S 22S Af Ax (3.29) in which AF i s the vector of unbalanced forces occurring at the com-uc 3 p a t i b i l i t y degrees of freedom. Equation (3.29) may be used to express incremental d i sp lace-ments of the compat ib i l i ty degrees of freedom in terms of the incremen-ta l generalized displacements in the presence of unbalanced forces at the compat ib i l i ty degrees of freedom in a form shown in (3.30) ^ • . CLS L;I FUC T - i c L i ] s - (3.30) S im i l a r l y the incremental frame force-displacement r e l a t i o n -ship must be modified for frames in which unbalanced forces ex i s t at 'other ' degrees of freedom. The incremental displacements of these 'o ther ' degrees of freedom may be expressed in terms of incremental 70 displacements of the frame horizontal and compat ib i l i ty degrees of free-dom as shown in (3.4) i f no unbalanced forces ex i s t and as shown in (3.31) i f unbalanced forces at the other frame degrees of freedom F .. do e x i s t . A 6 i = A-F1 L F i ] _ l F u o i " L F i 1 « L ' 1 | C . i V l < A A Ax. >T (3.31) in which the subscripts i and Fi indicate values for vectors and ma-t r i ce s for frame i respect ively. Hence the expansion of the incremental displacements i s per-formed according to equations (3.2) and (3.4) for time steps in which no unbalanced forces occur and (3.30) and (3.31) for time steps in which unbalanced forces act. 3.6 Calculat ion of Incremental Member P l a s t i c Rotations The most important parameter in the analysis of s t ructura l re s -ponse to earthquakes i s the amount of p l a s t i c rotat ion resu l t ing in each member end. P l a s t i c ro tat ion , which may be converted into the commonly used d u c t i l i t y factor , i s c r i t i c a l for reinforced concrete members and provides an ind icat ion of s t ructura l damage or f a i l u re resu l t ing from an earthquake. The amount of p l a s t i c rotat ion occurring at a p l a s t i c hinge depends on the state of the other member end. For members with a hinge at each end, the tota l rotat ion occurring during a time step at each 71 hinge i s p l a s t i c i . e . for member state D Aa = Aco - Ay (3.32) Aa„ • = Aw - Ay 2 2 1 in which A a 1 , Aa 2 = incremental p l a s t i c rotations at ends 1, 2 respect ively Aw , Aco2 = incremental j o i n t rotat ions at ends 1 and 2 Ay = incremental chord rotat ion. For members with only one hinge, the incremental p l a s t i c rota-t ion at the hinge over a time step i s shown in (3.33) Aa. = (Aa). - Ay) + c(Au>. - Ay) (3.33) Aa, = 0 J in which c = the carry-over factor c - } ] (3.35) in which At = length of time step • F6' F i = ^ a s e s n e a r f ° r c e a t the beginning, end of the time step a , a = ground acceleration at the beginning, end of the 1 time step v Q = ground ve loc i t y at the beginning of the time step. This ca lcu lat ion must be performed for each component of the earthquake. In the current analys is , the rotat ional component of the earthquake i s not included hence energy i s input only by the two mutually perpendicular t rans lat iona l components. I t should be noted that the incremental energy input may be e i ther pos i t ive or negative. A negative value indicates that the s t ruc -ture i s feeding energy back into the ground. 74 3 .7 . i i K inet ic Energy of the Structure The tota l k inet i c energy of the structure i s found from (3.36) E k = 1"*A.= *A ( 3 ' 3 6 ) in which = absolute ve l oc i t i e s of the generalized co-ordinates. Solution of (2.11) gives the incremental r e l a t i ve ve l o c i t i e s of the generalized co-ordinates; the tota l r e l a t i ve ve loc i t y vector at any time is found by summing incremental ve l oc i t i e s obtained for a l l of the time steps up to the time in question. The r e l a t i ve ve loc i t y vector re lates to the ve loc i t i e s of the generalized co-ordinates with respect to the base of the structure. To be converted to absolute ve l oc i t i e s the to ta l ground ve loc i t y in the d i rect ion of each co-ordinate must be added. For the three d i rect ions q , r , 9 the absolute ve l oc i t i e s may be expressed as r i •1 • q >- * • + < V . 0 . in which the subscript A represents absolute values, the subscript R represents r e l a t i ve values, and the subscript g represents values at ground l e v e l . 75 3 . 7 . i i i Recoverable Strain Energy Stored in the Structure The s t ra in energy stored in a structure may be subdivided into three components: (1) ax ia l energy (2) bending energy of the e l a s t i c component (3) bending energy of the e l a s t i c - pe r f e c t l y p l a s t i c component. The ax ia l energy i s calculated incrementally for each member over each time step. The incremental axial energy stored in a member is shown in (3.38) AEft = (N. + M - ) A6 (3.38) A o 2 in which N q = ax ia l load at the beginning of the time step AN = incremental ax ia l load A6 = incremental ax ia l deformation. The bending energy stored in the e l a s t i c component is also calculated incrementally for each member over each time step, as shown in (3.39) AM , AM. , AE e l = (M e l + - ^ i ) i (AO.) + (M e l + - T ± ) i . ( A e j ) (3.39) o o in which M 1 = moment in the ax ia l component at the beginning of e o of the time step AMg-j = incremental moment in the e l a s t i c component A6 = incremental rotat ion of the member end 76 and the subscripts i and j indicate the two member ends. Calculat ion of the bending energy stored in the e l a s t i c - pe r -f e c t l y p l a s t i c component must be performed separately for each member state. For a member with no hinges the incremental bending energy i s calculated in a manner s imi la r to .that used for the e l a s t i c component, that is AM AM A E E p = ( M E p + - J ^ . ) . (A6.) + (MEp + -fZ-). (ABj) (3.40) in which the subscript EP replaces the subscript EL in (3.39) i n d i -cat ing the e l a s t i c - pe r f e c t l y p l a s t i c component. For members with a hinge at one end, careful attention must be paid to ca lcu lat ion of incremental bending energy. In the analysis presented in th i s thes i s , i t was assumed that there would be no change in the recoverable bending energy stored at a p l a s t i c hinge; the tota l change in energy at that j o i n t would be due to hysteret ic d i s s ipa t ion . Hence the incremental bending energy was assumed to occur only at the e l a s t i c end of the member as shown in (3.41) AM p A E E p = ( M E p + - f - ) . A6. (3.41) o in which the subscript i refers to the e l a s t i c member end. After the analysis was performed i t has come to the author 's attention that th i s analysis w i l l not conserve energy as the energy levels computed did not balance completely. Further invest igat ion 77 revealed that the f a i l u r e of the energy balance can be explained in part by examination of equations (2.16) and (3.33) 9, = . + a. = to. - y (2.16) and (2.15) J J J J Aa, = (Aw, - Ay) + c (Aw. -Ay) (3.33) in which $. = tota l e l a s t i c rotat ion of the e l a s t i c - pe r f e c t l y p l a s t i c J component at end j a, = tota l p l a s t i c rotat ion at end j w. = j o i n t rotat ion at end j Y = chord rotat ion c = carry-over factor and equation 3.33 i s fo r members with a hinge at end j but e l a s t i c at end i . In order to s a t i s f y these equations i t i s necessary that eq. (3.42) should hold: c (Aw. - Ay) (3.42) which, in e f f ec t , states that the e l a s t i c rotat ion at a member end con-ta in ing a p l a s t i c hinge w i l l change over a time step i f the other (elas-t i c ) member end experiences ro ta t ion . Hence the incremental bending energy at th i s p l a s t i c member end w i l l also change and i t i s necessary to include th i s e f fec t into eq. (3.41). The revised eq. (3.43) super-sedes eq. (3.41) and must be used to s a t i s f y the energy balance. AMFp AE E p = (M E p + -f^-). A6i - (H£p). c A 6. (3.43) 78 In the case of a member with p l a s t i c hinges at both ends i t i s c lear that no recoverable incremental bending energy w i l l be stored in the e l a s t i c - pe r f e c t l y p l a s t i c component. 3.7.iv Energy Dissipated due to Hysteretic Behavior The occurrence of p l a s t i c rotat ion at a p l a s t i c hinge w i l l cause energy to be d iss ipated. This energy d i s s ipat ion i s an important parameter in studying the response of structures to earthquake e x c i t a -t i on . As the amount of energy diss ipated by viscous damping in a typ ica l structure w i l l be a r e l a t i v e l y small quantity and the k inet i c energy of the structure w i l l be a small quantity at the maximum c y c l i c d i sp lace-ment of the structure, excess energy quantit ies must be in the form of recoverable s t ra in energy and energy diss ipated due to nonlinear (hyste-r e t i c ) behavior. Hence there are two approaches which are used in the design of a structure to absorb the energy input by an earthquake: (1) designing a s t i f f and strong structure in which most of the ' energy may be absorbed in recoverable s t ra in energy and which w i l l exh ib i t very l i t t l e nonlinear behavior (2) designing a less s t i f f and/or strong structure but with each member end detai led to as to be capable of withstanding large p l a s t i c deformations. This i s the basis of a duct i l e design which w i l l d iss ipate large quantit ies of energy in the from of nonlinear rotat ions. 79 While these two design philosophies are eas i l y supported by heur i s t i c arguments, there i s a de f i n i t e lack of empirical or ana lyt ica l methods of determining the r e l a t i ve merits of these methods quant i ta t i ve ly for multi-degree of freedom structures. The author believes that a more complete understanding of the behavior of multi-degree of freedom st ruc-tures in earthquakes may be obtained by examing the various forms of energy absorbed by d i f fe rent types of structures. The energy diss ipated in one nonlinear cycle of a member end i s equal to the area enclosed by the hysteret ic loop on the moment-rota-t ion curve. In one time step, the energy diss ipated at a p l a s t i c hinge i s calculated according to eq. (3.20) A EDH = MEP A a (3.20) arid the tota l " eneijgy diss ipated in a structure during a time step i s found by summing these incremental energies for each p l a s t i c hinge. 3.7.v Energy Dissipated due to Viscous Damping The incremental energy diss ipated due to viscous damping i s found by assuming that a force, equal to the average of the viscous forces acting at the beginning and end of the time step, acted.through-out the i n t e r v a l . Viscous forces at the beginning and end of a time step are calculated in eqs. (3.44a) and (3.44b) respect ively F b = (°fflL + *b (3.44a) 80 F e = (am + BkJ x g (3.44b) in which the notation of Section 2.1 i s used. Hence the average force vector, F ^ , i s assumed to act through the i n -cremental displacements of the structure Ax , d i s s ipat ing the incremen-ta l energy quantity shown in eq. 3.46 e b FA V = ^ + 2 ( 3 ' 4 5 ) A E D V = AX (3.46) 3.7.vi Energy Balance At any par t i cu la r time the energy input by the earthquake into the structure should equal the various forms of energy appearing in the structure. Af ter n time steps have been performed, the fol lowing b a l -anceeshould ex i s t E A E I N = E k + < A EAi + A E E L i + A E W + £ A EDH + ^ 81 CHAPTER 4 MODELLING OF A SIXTEEN STOREY OFFICE BUILDING The importance of using a structural model containing three dimensional response parameters in order to accurately describe the non-l inear response of a structure to earthquake exc i tat ion has already been establ ished. While results obtained using a three-dimensional model w i l l be more accurate, even for symmetric structures, than those from a plane frame analys is , the difference becomes most pronounced for structures in which torsion i s more apt to occur. Torsional response becomes an important factor in core type structures in which the primary s t i f f en ing elements are located near the centre of r i g i d i t y . This class of structures w i l l have a much smaller tors ional resistance than structures in which the primary s t i f f en ing e l e -ments are on the periphery of the structure, with the largest possible moment arm about the centre of r i g i d i t y . Large tors ional o s c i l l a t i o n s may also resu l t in i r r egu la r l y shaped structures such as the L-shaped bui ld ing and other eccentr ic structures in which the centre of r i g i d i t y and the centre of mass do not coincide. A bui ld ing layout which combines these two undesirable tors ional s i tuat ions by having a core displaced from the centre of mass i s a s i tuat ion which should be avoided. The current trend in the design of medium sized high r i se buildings i s towards structures in which shear walls comprise the main l a te ra l load re s i s t i ng elements. In areas of r e l a t i v e l y high earthquake 82 r i s k , a common bui lding type i s one with a dual-component l a te ra l load re s i s t i ng system comprised of a duct i le moment-resisting space frame in conjunction with shear wal ls . The shear walls are designed so that they are capable, when acting alone, of carrying the tota l s t a t i c l a te ra l forces spec i f ied by the code. The frame i s desngned so that i t i s able to carry, as a system separate from the shear wa l l s , the ve r t i ca l loads in addit ion to 25% of the abovementioned l a te ra l load. The shear walls are commonly s ituated around the elevator shaft, hence the name "core-type" bui lding may be applied to th i s type of structure. In many cases, arch i tects specify that the core i s to be s ituated away from the geomet-r i c centre of the bui lding in order to provide for a larger area of open space. This causes torsion to be an important parameter. Since the dual component bui lding i s becoming so commonly used, i t i s f e l t by the author that i t i s a bui ld ing type which should undergo some accurate analysis to determine i t s adequacy when subjected to ear th -quake exc i ta t i on . The bui lding which was selected for analysis herein i s a var iat ion of a recently constructed bui ld ing located in downtown Vancouver. The intent was to examine the ef fect of tors ion on th i s bu i l d -ing and to determine whether the manner in which tors ion i s treated in current bui ld ing codes i s adequate. 4.1 Description of the Structural Model The actual bui ld ing on which the analysis i s based i s a 14 storey reinforced concrete bu i ld ing. While i t was desirable to model th i s bui ld ing as accurately as poss ib le, i t was also necessary to make 8 3 var iat ions to the bui lding to enable an accurate yet economic analys is to be made subject to the assumptions inherent in the computer program. The economic constraints meant that the re lat ionsh ip between the s t ruc-ture which was studied and the actual bui ld ing was not a close one, but the author believes that the structure that was analyzed represents a typ ica l midsized o f f i c e bui lding designed under current techniques. The plan of a typ ica l f l oo r of the modified bui lding i s shown in Fig. 20. The core i s located at the back face of the bui ld ing which causes the undesirable s i tuat ion in which torsion i s l i a b l e to become an important response parameter. Aumique feature of th i s core i s that the back wall i s s p l i t down the middle throughout the ent i re height of the bu i ld ing. This causes the centre of r i g i d i t y to move closer to the cen-treoof mass reducing the design eccen t r i c i t y . However, cutt ing the back wall reduces the s t i f fnes s of the bui lding with respect to horizontal t rans lat ion as well as ro tat ion. An interest ing s i de l i gh t to the analy-s i s of th i s bui lding i s to examine whether cutt ing the back wall causes lower stresses to develop when the bui ld ing i s subject to a design earth-quake. The modified structure was a rectangular bui lding 100 f t . in the q d i rect ion by 75 f t . in the r d i rec t i on . The f i r s t storey was 16 ft., above ground f o i l owed, by 15 storeys 'of~12 f t . ^ f l o o r to f l oo r height giving a tota l bui lding height of 196 f t . Future sections of th i s chapter w i l l describe various methods used to model th i s bui ld ing in an attempt to accurately incorporate the behavior of the core into the frame analys is . In each of these models 84 frames 1, 2 and 3 as numbered in F ig. 20 are i d e n t i c a l , and these frames are described in Figs. 21 and 22. In add i t ion, walls numbered 8 and 9 remain unchanged throughout the various models. These walls run in the q d i rect ion and the i r s t i f fnes s in the r d i rect ion is neglected. The structural parameters of walls 8 and 9 are described in F ig. 23. The methods in which the^other components of the bui lding are included are described as they are encountered in the indiv idual descriptions of the models. 4.2 Preliminary Analysis The purpose of th i s analysis i s to determine whether the ap-proximate earthquake analysis as proposed by bui lding codes w i l l provide a s u f f i c i e n t l y accurate estimate of the forces resu l t ing from a probable earthquake. In t h i s respect, the techniques prescribed by the National Bui lding Code of Canada (NBC) (22) are examined with reference to the model described previously. The analysis i s r e s t r i c ted to s t a t i c shears in the q d i rect ion which causes tors ion to occur because of the eccen-t r i c i t y of the model. 4.2.1 Calculat ion of the Lateral Seismic Base Shear The base shear force, V , i s calculated as the product of the quantit ies shown in (4.1) V = ASKIFW (4.1) in which A = the assigned horizontal ground accelerat ion ra t i o which 85 equals 0.08 for seismic zones 3 in which the ci.ty of Vancouver is. s i t u -ated. The quantity S relates the seismic response to the f i r s t natu-ral frequency of the structure, as shown in (4.2) S = (4.2) in which T = the fundamental period of the bu i ld ing. The fundamental period of the structure was l a te r calculated from an eigenvalue analys i s , but, for consistency, an approximate empirical value of T i s calculated according to (4.3) for use with the code s t a t i c ana-l y s i s : 0.05 h f = 0- (4.3) / I T with the bui lding height h n = 196 f t . and the bui ld ing width, D = 100 f t . , a resu l t ing fundamental period T of 0.98 sec. is ca l cu -la ted , resu l t ing in a value of S = 0.503. The quantity K accounts for var iat ions in the structure sys-tem and i s evaluated in part with respect to the measured performance of structures in actual earthquakes. For the structure under considerat ion, which i s designed to contain a complete duct i le moment re s i s t i ng space frame (capable of re s i s t i ng 25% of the prescribed s t a t i c shear when acting alone) in conjunction with shear walls (capable of r e s i s t i ng the tota l shear load alone), a value of K = 0.8 may be assigned. 86 A value of 1.0 may be assigned to the importance factor , I , and the s o i l conditions at the s i t e of the model are assumed to be such that a foundation factor , F , of 1.0 may be used. For the defined values of the various coef f i c ient s the base shear force V may be expressed in terms of the tota l weight of the bui ld ing (plus 25% of the design snow load), W , in terms of (4.4) V = 0.032 U (4.4) The dead load of the bui lding is calculated by representing i t as a uniform load of 175 pounds per square foot acting over each f loor giving a tota l weight of 1310 kips per storey causing a tota l seis-mic base shear force of 670 kips for th is structure. 4.2.2 D i s t r ibut ion of the Lateral Seismic Shear Force A port ion, F^ . , of the tota l base shear force, V , i s concen-trated at the top of the bui ld ing to account for upper mode v ibrat ions . The NBC defines th is quantity as shown in (4.5) h 2 F t = 0.004 V (=2-) (4.5) 1 u s in which h n equals the height of the bui ld ing and D s equals the dimension of the l a te ra l fo rce - res i s t ing system in a d i rect ion pa ra l l e l to the applied force. The quantity Dg i s d i f f i c u l t to evaluate for a dual component system such as the model concerned. However, i t i s f e l t that the intent ion 87 of the code i s to define the quantity according to the primary l a t e r a l force re s i s t i ng element which i s the eccentr ic shear core (D g = 27 f t . ) . In the l i g h t of th i s assumption, the upper-mode shear force is c a l -culated to be 0.21 V. However the code st ipulates that force F^ "need not exceed 0.15 V" from which i t i s inferred that th is l im i t i n g value (100 kips) should be applied in th i s case. The remaining l a te ra l shear is d i s t r ibuted over the bui lding height according to the formula (4.6) •' W.- h F x = ( V _ F t ) - f ^ - (4.6) in which F = the l a te ra l force to be applied at storey x X W = the weight of storey x X h = the height of storey x X n = the number of storeys. Appl icat ion of formula (4.6) produces the shear d i s t r i bu t i on shown in F ig. 25. 4.2.3 Calculat ion of Torsional Moments The NBC states that " tors iona l moments in the horizontal plane of the bui ld ing shal l be computed in each storey" based on a formula in which the tota l l a te ra l shear force at and above a pa r t i cu la r level i s mu l t ip l ied by the design eccent r i c i t y of the storey to produce the ap-pl ied tors ional moment. The design eccen t r i c i t y , e , i s computed by A one of the fol lowing equations? whichever provides the greater stresses. 88 e v = 1.5 e + 0.05 Dn (4.7a) e = 0.5 e - 0.05 Dn (4.7b) in which Dn equals the plan width of the bu i ld ing. The quantity 0.05 Dn represents tors ional influences due to accidental e c cen t r i c i t i e s which may resu l t from such things as the addi -t ion of wall panels or pa r t i t i on s . The f i r s t term of (4.7) equals the eccen t r i c i t y ; the computed distance between the centre of mass and the centre of r i g i d i t y at the par t i cu la r storey, increased or decreased by 50 per cent. The 50 per cent factor i s e s sent ia l l y an ignorance factor explained by the complex nature of torsion as i t occurs during an earth-quake. One point concerning the determination of eccen t r i c i t y deserves discussion. This point concerns the fact that the centre of r i g i d i t y could be defined in several ways. For example, i f the f loors above and below that under consideration were held f ixed and a s ingle load was ap-p l ied to the f l oo r being considered, there would be one point in the f l oo r such that any force applied at that point would cause t rans lat ion in i t s own l i ne of action with no rotat ion of the f l o o r . This point could be defined as the centre of r i g i d i t y of the f l oo r . A l t e rna t i ve l y , suppose that no constraints were appl ied, but that the single force was applied to the f l oo r under consideration. Again there would be one point such that loads applied there would cause no rotat ion of that f l o o r , a l -though other f loors might rotate. Again, that point could be defined as 89 the centre of r i g i d i t y of that f l oo r . Th i rd ly , suppose that loads were applied to a l l f loors in some pattern (say a unit load at each f l o o r , or in the pattern of the base shear d i s t r i bu t i on discussed above). The loads could be placed in a single ve r t i ca l plane and the pos it ion of that plane which caused no rotat ion of a given f loor would define i t s centre of r i g i d i t y ; or the load could be placed in a d i f fe rent locat ion on each f l oo r so that no f l oo r rotated. This would define the centre of r i g i d i -t i e s of a l l f loors at once. The code i s not c lear as to whether one of these or some other de f i n i t i on i s envisaged. The author believes that an eas i l y calculated method of deter-mining storey eccen t r i c i t i e s can be i l l u s t r a t e d with reference to the model under consideration. The s t i f fnes s contributionoof the indiv idual components (frames or walls) of the bui lding may be approximated by use of equation (4.8) l / X -kj - - j p - L - (4.8) T. l/X, i -'• in which k. = the f ract ion of the tota l s t i f fnes s contributed by J component j X. = the displacement of component j J n = the number of components in the d i rect ion under consideration. where displacements X are obtained for a par t i cu la r load case. As the s t i f fnes s coe f f i c i en t s k vary with the type of load case as-sumed, i t i s f e l t that a t r iangular load d i s t r i b u t i on , s im i l a r to the l a te ra l seismic load d i s t r i bu t i on s t ipu lated by the codes would be the 90 most appropriate load case to apply in order to maintain consistency. Appl icat ion of th i s t r iangu lar load case to each component of the bui ld ing in turn w i l l enable factor k to be evaluated for each component. This factor k may be calculated at each storey. Ca lcu la -t ion of factors k enables the eccent r i c i t y of each storey to be eva lu-ated according to (4.9) n 1*1 ^ 6 I eo = V £*.9> E k i=l 1 in which e- = the eccent r i c i t y of storey j J e- = the eccent r i c i t y (distance from the centre of mass to the centroid of the component) of component i Calculat ion of storey eccen t r i c i t i e s according to th i s method w i l l produce the result that the eccent r i c i t y decreases with respect to height for a bui ld ing of th i s type. This var iat ion of eccent r i c i t y may be explained by examinjnghther.defleetedhshape'ofthe'iitwoimain components of t h i s bu i ld ing, as shown in F ig. 24. The deflected shape of the frame components i s more or less l i nea r while the deflected shape of the shear wall elements i s of a higher order when each of these components i s ex-amined independently. This indicates that the percentage of the tota l s t i f fnes s contributed by the shear wall elements decreases with respect to height with an accompanying increase in the percentage s t i f f ne s s of the frame components. For the structure under examination, the eccen-91 t r i c i t y of the bui lding is governed by the eccent r i c i t y of the shear core in the lower s to r ie s , while th i s eccent r i c i t y reduces with respect to height as the moderating influence of the frame components assumes a more s i gn i f i can t ro le . This method of ca lcu lat ing eccen t r i c i t y was performed on a var ia t ion of the bui lding model and the resu lts are presented in Table 4. Results shown are for a 15 storey bui ld ing with 9 foot storey heights and hence they w i l l d i f f e r for the bui lding described previously. How-ever as values shown in Table 4 are based on a buiilding with ident ica l plan dimensions and l a te ra l force re s i s t i ng elements, they are f e l t to be analogous to those obtained for the bui ld ing under consideration. In the lower two s to r ie s , the calculated e c cen t r i c i t y , when increased by 50 per cent, exceeds 25 per cent of the plan width (75 f t ) in which case values shown for the design eccent r i c i t y have been doubled as required by the NBC. While th i s method of ca lcu lat ing eccent r i c i t y i s not rigorous, i t i s f e l t by the-.author that the method of establ i sh ing the centre of r i g i d i t y of a storey and ca lcu lat ing eccent r i c i t y i s not defined e x p l i c -i t l y enough by the NBC. I t i s believed that a method, such as that des-cribed in th i s sect ion, should be required by the code, as the e f fect of tors ion in a structure may vary s i g n i f i c an t l y i f an a l ternat ive method i s fol lowed. Calculat ion of the tors ional moment at storey x, Mt , i s performed according to equation (4.10) 92 x (4.10) The torsional, moments'to be applied to each storey of the bui lding under consideration are shown in Table 4. I t should be noted that there are two values of the tors ional moment to be applied at each storey. This re lates to the two values of design eccent r i c i t y as calculated by equa-t ion (4.7) and the tors ional moment which causes the greatest stress in a par t i cu la r component i s to be the design moment for that component. 4.2.4 Summary of the Preliminary Analysis method in which tors ional moments shal l be included in the analys i s . As i t i s f e l t thatrthe code appreciates the.fact that analyses more, compli-cated than plane frame analysis are rare ly performed in a design o f f i c e , i t appears that the intended method of including tors ional moments i n -volves representing the tors ional moment as a set of *addit ional shear loads in the various components. The addit ional shear load at storey x for component i , . , i s calculated according to the fol lowing f o r -mula (4.11) The NBC i s also not spec i f i c in i t s requirements governing the F. t x i M. tx (4.11) in which fil the tota l number of components of the bui ld ing y distance from the centre of r i g i d i t y to the centroid of the component 93 The e f fect of tors ion may be considered in terms of tors ional magnification factors which indicate the amount by which the design l a t -eral seismic forces are increased as a resu l t of the computed e c c e n t r i c i -ty . These magnification factors are l i s t e d at each storey of the front frame (frame number 1) of th i s model in Table 5. S imi lar ca lcu lat ions may be made on the other components. In order to examine the manner in which tors ional response i s treated by the code, the magnification fac -tors may be compared with those obtained by using the nonlinear time step analysis method presented previously. The time step analysis would be performed twice; once on the model containing the eccentr ic shear core as shown in F ig. 20 and then on a var iat ion of the model in which the shear core is relocated so that the centre of r i g i d i t y and centre of mass coincide. In order to compare the dynamic analysis with the NBC s t a t i c analysis i t would be necessary to se lect a par t i cu la r response parameter to compare. The use of magnification factors to compare the influence of tors ion i s not an exact technique as the NBC s t a t i c analysis i s based upon e l a s t i c analysis while the response of the structure as calculated by the time step analysis i s based on nonlinear theory. However i t can be shown that a member end w i l l rotate approximately the same whether i t undergoes p l a s t i c rotat ion or not. In the l i g h t of th i s assumption, the tors ional magnif ication factors may be expressed in terms of the ra t i o of member end rotations (or d u c t i l i t y factors) obtained in the eccentr ic and symmetric cases. Any s i gn i f i can t difference in the tors ional magni-f i c a t i on factors obtained by the NBC s t a t i c analysis and the time step analysis may be an ind icat ion that the manner in which tors ional e f fects due to eccen t r i c i t y are accounted for by the NBC i s unsat isfactory. 94 4.3 Computer Modelling of the Structure As stated previously, modelling of th i s dual component shear wall-frame structure posed a d i f f i c u l t problem as a resu l t of r e s t r i c -t ions imposed by theory. As the computer program requires that the structure be idea l i zed as a combination of plane frames, i t i s necessary to make several approximations in combining the shear core with the framing system. The eventual resu l t of the attempts at modelling th i s pa r t i cu la r bui ld ing system i s that analysis based upon the theory des-cribed in th i s thesis would not provide s u f f i c i en t accuracy. This sec-t ion describes the various attempts at modelling the bui lding and explains the sources of d i f f i c u l t y . As explained previously there appears to be l i t t l e doubt that modelling of frames 1, 2 and 3 and walls 8 and 9 as numbered on F ig. 20 may be done in a straightforward manner and with s u f f i c i e n t accuracy. The various models presented herein represent attempts to accurately i n -clude the shear core in terms of the remaining plane components shown in Fig. 20. 4.3.1 Model A The f i r s t attempt at i dea l i z i ng the structural system i s shown in F ig. 26. This model i s expected to accurately model the behavior of the shear core in the r d i rect ion i f the structural parameters assigned to the columns of frames 5 and 6 with respect to motion in the r d i r e c -t ion are representative of the corresponding properties of the actual structure. The beams of frames 5 and 6 are l i n t e l beams spanning the 95 doorway opening in the wall and to accurately model these beams a beam with r i g i d end stubs must be used. The problem in modelling th is shear core involves assigning s t ructura l values to the members of frame 7 in order to accurately ana-lyze the response of the core to motions in the q d i rec t i on . As the i n t e r i o r section of the core i s of an H shape charac te r i s t i c of a wide flange sect ion, the walls of th i s section which run in the r d i rec t ion may be considered analogous to the flanges and the wall running in the q d i rect ion thought of as the web of a wide flange sect ion. Behavior of th i s section could be accurately modelled for motion in the q d i r e c -t ion by i dea l i s i ng the section as a column, of structural properties de-fined by the wide flange sect ion, placed at the centroid of the sect ion. This technique would not enable the b iax ia l e f fect s of the core to be considered as the core i s modelled by d i f f e ren t components in the q and r d i rect ions . An a l te rnat i ve method of i dea l i z i ng the core with respect to motion in the q d i rect ion i s to use 3 columns connected by r i g i d beams to form a frame. The two external columns of th i s frame 7 would coincide with the internal columns of frames 5 and 6 to allow the b iax ia l ef fects to be accounted fo r . Each of these external columns would represent a flange of the cross sect ion. The rimterior column would represent the web of the wide flange and the response of th i s i n t e r i o r column would be independent of motion in the r d i r ec t i on . To approximate the behavior of the core i t was f e l t that the exter ior columns would be assigned cross sectional areas equal to that of the flange with moments of i n e r t i a set 96 equal to zero. The i n t e r i o r column would have a moment of i n e r t i a equal to that of the web with a cross-sectional area equal to zero. The beams jo in ing the columns would be assigned a r t i f i c i a l l y high values of moment of i ne r t i a and y i e l d moment to ensure that the beams w i l l behave as r i g i d components. Closer examination of the behavior of th i s model frame revealed that i t w i l l not provide an accurate descr ipt ion of the actual core. The deflected shape of the core should resemble a cant i lever deflected shape as shown in F ig. 24b. Modelling of the core as described w i l l re su l t in a deflected shape charac te r i s t i c of a frame as shown in F ig. 24a. This important difference in the structural i dea l i z a t i on prohibited use of th i s model. A var iat ion of th i s technique could be developed to accurately describe the behavior of the core. In th i s va r i a t i on , the i n t e r i o r co-lumn would be assigned the s t ructura l properties of the core and the ex-t e r i o r columns and the beams would be assigned properties set equal to 0. After incremental displacements of the building have been solved, the ax ia l force induced into the exter ior columns could be computed based upon the end rotations of the i n t e r i o r column. The moments in the i n -t e r i o r column would then be adjusted to include the couple resu l t ing from ax ia l forces in the exter ior columns. This technique would accu-rate ly describe the motion of the core in a l l d i rect ions as well as ena-bl ing the b iax ia l ef fects of the core to be analyzed. However, th is var iat ion would require s i gn i f i can t a lternat ions to the computer program, and i t was preferred that a modelling technique be used which would 97 adequately analyze the response while maintaining the general i ty of the program. 4.3.2 Model B From examination of model type A i t was rea l i zed that, except for the proposed var iat ion of the model, i t i s not possible to accurately describe motion in the q d i rect ion i f the core i s ideal ized to have 2 components with respect to motion in the r d i rec t i on . Hence, model B contains only one component in the r d i rec t ion to describe the core, as shown in F ig. 27. This component consists of 2 columns connected by the r i g i d end stub l i n t e l beam s im i l a r to frame 5 or frame 6 of model A. As for model A, i f appropriate st ructura l properties are assigned to the columns and beams, i t i s f e l t that an accurate descr ipt ion of the bu i l d -ing with respect to motion in the r d i rect ion w i l l be obtained. One inherent f au l t of th i s model i s the fact that i t p a r t i a l l y neglects the tors ional s t i f fnes s contr ibut ion of the shear core by cons i -dering i t to ex i s t in one plane. Before analysis of the structure was performed an invest igat ion of th i s reduction in the tors ional r i g i d i t y was carr ied out to determine whether th i s would introduce a s i gn i f i can t error. The e f fect of i dea l i z i ng the core in a s ingle plane may be ex-amined eas i l y with respect to the s t a t i c l a te ra l force method of the NBC. The design eccent r i c i t y arid resultant tors ional moments w i l l not be a f -fected as the tota l l a te ra l s t i f fnes s with respect to motion in the q d i rect ion i s unchanged. However the tors ional moments w i l l be d i s t r ibuted 98 in a d i f fe rent manner upon appl icat ion of equation (4.11). This e f fect may be calculated by reducing the value of n by 2 and e l iminat ing the contr ibut ion to the tors ional r i g i d i t y made by components 5 and 6. The revised magnification factors , calculated.n'n-.this manner, w i l l be app l i ca -ble! § f (6-0) i i 1 _ y 10 ; L2 (12-6-£) fl 5 L 3 (6-& ) II 10 L 2 FIG. 12(a): FORCES DUE TO UNIT VERTICAL DISPLACEMENT OF JOINT i (MEMBER STATE A) (270-9 (0) El 30-0 fj (30"V) f l r27O-90v Ei V 30-0 ; L 3 FIG. 12(b): FORCES DUE TO REMOVAL OF MOMENT AT JOINT H i f 30-130') j, L 3 ^ T T J ^ T ; 1 2 30-0 FIG. 12(c): FORCES DUE TO UNIT VERTICAL DISPLACEMENT OF JOINT i (MEMBER STATE B) 138 FIG. 1 3 ( a ) : FORCES DUE TO UNIT ROTATION OF JOINT j (MEMBER STATE A) F i G . 1 3 ( b ) : FORCES DUE TO REMOVAL OF MOMENT AT JOINT i 3_LL f 30-3 4K 2 vin_/+\ / 30-0 3EI ,30-3 L2 ^30-0 ; /3O-3 0V ^30-0 ; FIG. 1 3 ( c ) : FORCES DUE TO UNIT ROTATION OF JOINT j (MEMBER STATE B) 139 FIG. 14 (b): BUILDING WITH VERTICAL COMPATIBILITY 140 e i t 3 -CORNER COLUMN e 4 ^3 1 3 0 11 a JJ/\Wt /I) VA ) 18 5 12 t ^ 6 CORNER COLUMN FIG. 15(a): GROSS FRAME DEGREES OF FREEDOM .1 FIG. 15(b): REDUCED FRAME DEGREES OF FREEDOM 141 FIG. 16(b) REDUCED STRUCTURE DEGREES OF FREEDOM 142 FIG. 17: FORCES AND DEFORMATION OF A TYPICAL MEMBER COMPRESSIVE (NEGATIVE) AXIAL LOAD TENSILE (POSITIVE) AXIAL LOAD MOMENT FIG. 1 8 ( a ) : AX IAL-FLEXURAL INTERACTION DIAGRAM 143 F I G . 1 8 ( c ) YIELD SURFACE FOR BIAXIAL COLUMNS MOMENT M 2 >> Mu My — ' M, -/ i /, 3 L 8 2 = 8 3 ROTATION FIG. 1 9 ( a ) : UNBALANCED MOMENT CORRESPONDING TO MEMBER END YIELDING MOMENT FIG. 1 9 ( b ) : UNBALANCED MOMENT CORRESPONDING TO MEMBER END REVERSAL 145 FIG 1 9 ( c ) : UNBALANCED MOMENT CORRESPONDING TO A CHANGE IN YIELD MOMENT AT A HINGE 146 FIG..20: FLOOR PLAN OF A TYPICAL FLOOR OF THE CORE TYPE BUILDING 147 AXIAL LOADS PER STOREY 20k 75k 100k 1 75k 20k 19" x 40" DP. (TYP. BEAM) (TYP. EXT. COLUMN) 30" x 24" OP. (TYP. INT. COLUMN) 2 3 4 x 25' 100' MOMENTS (KIP-FT.) AND AXIAL FORCES (KIPS) TO DEFINE INTERACTION DIAGRAM COLUMN LINE 1 COLUMN LINE 2 CO .UMN L l NE 3 STOREY My Mb Pt Pt My Mb Pb Pt Pc My Mb Pb Pt Pc 7-16 1-7 0-1 135 180 240 500 780 945 675 1060 1195 210 290 380 1800 3050 3720 180 300 625 635 1050 1510 840 1320 1490 290 480 1310 2260 3880 5240 240 400 625 695 1150 1510 840 1320 1490 380 600 1310 2290 3950 5240 ULTIMATE MOMEN EXTERIOR BEAMS INTERIOR BEAMS STOREY LINE 1 LINE 2 LINE 1 LINE 2 POS. NEG. POS. NEG. POS. NEG. POS. NEG. 2-16 350 -175 350 -505 495 -360 495 -435 1 355 -180 355 -595 870 -450 580 -530 S (KIP-FT.) FIG. 21: FRAME 1 OF THE CORE BUILDING AXIAL LOADS PER STOREY 20k 60k 60k 2 2 3 x 25' 19" x 22" DP. (TYP. BEAM) (TYP. EXT. COLUMN) 30" x 24" DP. (TYP. INT. BEAM) MOMENTS (KIP-FT.) AND AXIAL FORCES (KIPS) TO DEFINE INTERACTION DIAGRAM STOREY COLUMN LINE 1 COLUMN LINE 2 My M|j p b P t p c M y M b P b P t p c 7-16 135 500 675 210 1800 240 695 840 380 2290 1-7 180 780 1060 290 3050 400 1150 1320 600 3950 0-1 240 945 1195 380 3720 625 1510 1490 1310 5240 ULTIMATE MOMENTS (KIP-FT.) EXTERIOR BEAMS INTERIOR BEAMS STOREY LINE 1 LINE 2 LINE 1 LINE 2 POS. NEG. POS. NEG. POS. NEG. POS. NEG. 2-16 250 -100 250 -255 355 -255 355 -255 1 295 -100 295 -380 405 -380 405 -380 FIG. 22: FRAMES 2 AND 3 OF THE CORE BUILDING 149 WALL 8 i a1 -6' MOMENTS (KIP-FT) AND AXIAL FORCES (KIPS) TO DEFINE AXIAL INTERACTION DIAGRAM yy x x - 7.29 x 10s in6 I. A = 1780 in 2 = 0.25 E = 3.6 x 10s psi STOREY AXIAL U y Mb P c 15-16 50 1 , 540 10,540 1,810 170 5, 420 14-15 100 13-14 150 12-13 200 11-12 250 . 1 ' t \ 10-11 300 2,700 10,870 1,810 260 5,450 9-10 350 8-9 400 7-8 450 < < ' 6-7 500 3, 380 17,700 3, 020 300 9, 130 5-6 550 4-5 600 3-4 650 2-3 700 1-2 750 0-1 800 \ r i t i i WALL 9 IB'-B". r j z c 6" MOMENTS (KIP-FT) AND AXIAL FORCES (KIPS) TO DEFINE AXIAL INTERACTION DIAGRAM STOREY AXIAL M y V P b p c 15-16 30 47 5 6, 355 1 , 410 100 3,620 14-15 60 13-14 90 12-13 120 11-12 150 10-1 1 180 9-10 210 8-9 240 7-8 270 6-7 300 • \ > t -> 5-6 330 47 5 10 46 5 2, 350 100 6, 035 4-5 360 3-4 390 2-3 420 i-2 450 0-1 480 • r v - 1 > \ - 3.88 x I06 in6 X X A = 1190 in 2 I = 0 yy JI = 0-25 E = 3.6 x 106 psi FIG. 23: WALLS 8 AND 9 OF THE CORE BUILDING (a) FRAME (b) WALL FIG. 24: TYPICAL DEFLECTION SHAPES FIG. 25: LATERAL SEISMIC FORCES (KIPS) REQUIRED BY THE NBC (IN THE q DIRECTION) Cl L> O- -o- -0- © BEAM 1 (in4) Av (in2) B1 8,000 320 B2 10,560 440 B3 125,000 360 COLUMN DIRECTION I (in4) A (in2) Av (in2) Cl q 27,650 288 240 C2 q 6.65 x 106 3455 1390 C2 r 8.1 x 10 6 3455 2140 C3 q 54,000 720 600 C4 . q 0 27 40 0 C4 r 13.4 x 10 6 5450 2620 C5 q 1.78 x 108 10,900 5000 FIG. 26: MODEL A C2 38C ©I »BI(9) B3 C4 CI 0 a el -o- -a- -o- -d BEAM I (in4) Av (in2) B1 8,000 320 B2 10,560 440 B3 250,000 720 Av = SHEAR AREA COLUMN DIRECTION 1 (in4) A (in 2) Av (in2) CI q 27,650 288 240 C2 q 1 .33 x 107 6910 2780 C2 r 1.62 x 107 6910 4280 C3 q 54,000 720 600 C4 q 1.78 x 10 8 10,900 5000 C4 r 2.68 x 107 10,900 5240 FIG 27: MODEL B CI B1 C3 B2 0 -TJ-C2 "HI BJ CD f I C4 M -Q- © -D-C3 BEAM 1 (in4) Av (in2) B1 8,000 320 B2 10,560 440 COLUMN DIRECTION I (in4) A (in 2) Av (in2) C1 q 27,650 288 240 C2 q 1.33 x 107 6910 2780 C3 q 54,000 720 600 C4 q 1.78 x 108 10,900 6000 C4 r 3.11 x 108 17,800 11 ,040 FIG. 28: MODEL C 154 \K 2 ° ' » |« 2 0 ' >I- 2 0 ' BEAM LEVEL - * - 0" STOREY POLAR STOREY WT.MOMENT OF WT. INERTIA (k> ( k - f ' t 2 ) 655 1 EXT-(393000) 612 (370000) 612 (370000) 612 (370000) 370 (222000) SQUARE COLUMNS BEAMS 18° 20' INT.WIDE 22 " 13" 24" 15" DEEP 20" 22 " COLUMN LINE PLAN VIEW OF TYPICAL STOREY ELEVATION VIEW OF TYPICAL FRAME E = 5000 KSI ULTIMATE MOMENTS IN BEAMS ( K I P - F T . ) LEVELS LINE 1 LINE 2 POS NEG POS NEG 1-2 340 -240 680 -480 3-5 230 -165 460 -325 MOMENTS ( K I P - F T ) AND AXIAL FORCES ( K I P S ) TO DEFINE INTERACTION DIAGRAM BEAM COLUMN LINE 1 COLUMN LINE 2 LEVELS My Mb Pb Pt Pc My Mb Pb Pt Pc 1-2 290 570 610 580 1870 440 930 900 720 2640 3-5 210 420 500 480 1520 320 700 750 580 2210 FIG. 29: PROPERTIES OF THE FIVE STOREY FRAMED BUILDING EARTHQUAKE B ^t(sec) 50 tn 156 rl5 ( FT/SEC 2) Ho , |3Q 140 ,50 TIME ( SEC ) EARTHQUAKE B-l Figure 31(a) Acceleration, velocity and displacement for earthquake B-l 160 EAR HQUAKE B-l ,0* .OS ,0 8 .1 ~ ,2 .4 .6 .8 i PERIOD (sees) & 6 id Figure 33(a) T r i p a r t i t e logarithmic p l o t of spectra for earthquake B-l 161 EARTHQUAKE B-2 •^f 1 " ' ,02 ,0 4 .06 .08 .1 TI ~* ^ I 2 4 PERIOD (sees) Figure 33(b) Tripartite logarithmic plot of spectra for earthquake B-2 162 -i r 659'9T SZO-M NOUUc)313338 8t- KtS'K-Fi'gure 34(a). Acceleration (ft./sec ) of storey 5 in the q direction Figure 34(b) Relative velocity (ft./sec) of storey 5 in the q direction - 1 — i 'O-5 » ' D 0-C UI30T3A S9H J.3M0IS Figure 34(c) solute velocity (ft./sec) of storey 5 in the q direction Figure 34(d) Displacement (ft.) of storey 5 in the q direction Figure 35(a) 2 Acceleration ( f t . / s e c ) of storey 5 i n the r d i r e c t i o n Figure 35(b) Relative velocity (ft./sec) of storey 5 in the r direction 168 Figure 35(d) Displacement (ft.) 'of storey 5 in the r direction 170 Figure 36(a) 2 Acceleration (rad./sec ) of the rotational component of storey 5 Figure 36(b) Relative velocity (ft./sec) of the rotational component of storey 5 IO ROT nro o-o eo'o- sso'o- 6sro- «.ro-( i-OIX) UI3BTJA S8B JGittliS Figure 36(c) Absolute velocity (ft./sec) of the rotational component of storey 5 173 Figure 36(d) Displacement (ft.) of the rotational component of storey 5 174 -3.0 - 2 .0 -1.0 1.0 2.0 3.0 - 3 . 0 -2.0 -1.0 1.0 2.0 3.0 -1 x 1 0 ~ 2 1 x 1 0 " Q (FT / SEC 2 ) R (FT / SEC 2 ) 8 (RAD/SEC) FIG.37(D): RELATIVE VELOCITY Vs. STOREY LEVEL ENVELOPE 175 Q ( F T . / S E C . ) R C F T . / S E C . ) 8 ( R A D / S E C ) FIG-37(c) ABSOLUTE VELOCITY Vs. STOREY LEVEL ENVELOPE Q ( F T . ) R ( F T . ) 8 ( R A O . ) FIG.37(d): DISPLACEMENT Vs. STOREY LEVEL ENVELOPE 176 FIG.38 (D) : INTERIOR BEAMS SHEAR ENVELOPE ( K I P S ) 4 FIG.39(D): INTERIOR COLUMN SHEAR ENVELOPE (KIPS) FIG.40(D): INTERIOR COLUMN AXIAL ENVELOPE 179 120 4-14 . 15 16 17 18 19 '. 20 21. 22 TIME (SECONDS) Figure 41 Variation of recoverable energy during the f i r s t 22 seconds 180 Figure 42 Variation of total energy during the f i r s t 22 seconds 181 FIG.43 NUMBER OF EXCURSIONS INTO THE PLASTIC RANGE AT STOREY LEVELS 182 A. EXTERIOR BEAMS B. INTERIOR BEAMS __i 1 1 1 1 i 1 1 1 v i 1 1 1 1 i 1 -.04 - .03 -.02 -.01 .01 .02 .03 .04 .05 - .003 -.002 -.001 .001 .002 .003 C. EXTERIOR COLUMNS 0. INTERIOR COLUMNS FIG.44 ACCUMULATED PLASTIC ROTATION 183 FRAME 1 FRAME 4 TOTALS 106.2 K IP -FT . TOTAL — 220 K IP -FT . (15 .7% OF TOTAL (32 .5% OF TOTAL ENERGY DISSIPATED) ENERGY DISSIPATED) FRAME 2 FRAME 3 TOTAL— 175.0 K IP -FT. TOTAL— 176.4 K IP -FT. (25.8'/, OF TOTAL (26% OF TOTAL ENERGY DISSIPATED) ENERGY DISS IPATED) FIG.45 DISSIPATED ENERGY (KIP-FT.) OF VARIOUS COMPONENT 184 TOTAL 677 .6 K IP -FT . FIG.46 DISSIPATED ENERGY OF THE ENTIRE BUILDING 185 APPENDIX A CALCULATION OF THE EVENT FACTOR FOR A BIAXIAL COLUMN The f ract ion of a time step for which an e l a s t i c end of a b i -ax ia l column remains unyielded, ca l led the event factor y 3 , can be c a l -culated from the information shown in F ig. A . l . , Mu ( t dM, ^ \ M Figure A. l At ifehe beginning of the time step the member end was, in the e l a s t i c state 1 with moments about the two pr inc ip le axes of M . and M . . At the end of the time step the member end i s in a p l a s t i c state and has experienced incremental moments about the two p r i nc ip le axes of dM and dM . Assuming a l inear var iat ion of these moments with res-1 2 3 pect to time, the condition at y i e l d may be expressed as shown in (A. l) /M . + p d M \ 2 / M . + y d M \ * 186 Introducing the factors a 1 to a 5 shown in (A.2) M • d M M . a - _JJL a = L a = 21 1 M E P l 2 M E P l 3 M EP 2 d K 2 2 2 , 2 a = rj a = a + a t M R N 5 2 4 Er2 (A.2) equation (A.l) may be rearranged as a quadratic function of the event factor y as shown in (A.3) a 5 y 2 + ( Z a ^ + ^ a J ^ ^ .0 (A.3) which may be solved for y as shown in the fol lowing steps: •2a a--. - 2a;a -± / 4 ' a f a 2 + 8a Saa a * 4a 2 a 2 4.i4a-: (a/-+ a 2 - 1)' 1 2 • -3 34 i 1 2 1 2 3 4 3 4 5 v 1 3 ' U = - - 2a 5 (A .4) /T . 2 2 - , 2 2 I -a a - a a ± / 2a a aaaa --a^an 7 a a + a °1 2 3 4 V " l 2 -3 1-lti, 223 ,'. 1 it 5 y = (A.5) C 5 a s -a a 1 2 a a 3 % y = (a a v 1 4 a a ) 2 3 ' (A.6) It may be shown by examination of th i s equation in each of the four quadrants that the pos i t ive square root w i l l always give the proper event factor . 187 APPENDIX B CALCULATION OF THE ENERGY INPUT INTO A STRUCTURE The energy input during a s ingle time increment i s found by integrat ing the product of the base shear force and the ground ve loc i ty with respect to time, as shown in (B.l) A E i n F(t) v(t) dt (B. l ) in which A E- n = incremental energy input F(t) = base shear force v(t) == gground ve loc i ty T = length of time increment As the base shear force i s only calculated at the beginning and end of the time increment, the assumption of l i nea r var ia t ion of base shear force over the time increment i s assumed as shown in (B.2) (B.2) in which F q = base shear force at the beginning of the time increment F = base shear force at the end of the time increment The earthquake ground accelerat ion record i s d i g i t i z e d with the intent ion that i t i s to vary l i n ea r l y between the points spec i f ied 188 on a record. Hence th i s l i nea r var iat ion of ground acceleration may be integrated to give the parabolic var iat ion of ground ve loc i ty as shown in (B.3) a - a 2 v(t) = v + a t + ( 1 T °) t (B.3) o o T in which v = ground ve loc i ty at the beginning of the time 0 increment a = ground accelerat ion at the beginning of the time 0 increment a i = ground accelerat ion at the end of the time increment With these spec i f ied var iat ions of base shear force and ground ve loc i ty the quantity F(t) v(t) may be expressed as (B.4) F (a T - v ) + f v t F(t) v(t) = F v +-2 2 ^ L_£_ t -v ' o o T t F a - 3F a + 2F a 2 (F - F ) (a - a ) 3 , 0 1 0 0 1 0 . . 1 O' ^ 1 O' . ^ + == * + 5 1 t L 1 2T: (B.4) Integration of (B.4) according to (B. l ) results in (B.5) A 2 E. = F v T + [F (a T - v ) + F v ] I + (F a - 3F a + 2F a ) in 0 0 0 v 0 0 1 o J 2 v 0 1 0 0 1 0' 6 T 2 + (F, - F ) (a - a ) -5— v 1 0 1 0 8 189 which may be rearranged to be in the form shown in (B.6) 3a + a 5a + 3a (B.6)