THE NONLINEAR THREE-DIMENSIONAL RESPONSE OF STRUCTURES TO EARTHQUAKE EXCITATION by BRUCE MALCOLM MASON B.Sc.(App.) Queen's U n i v e r s i t y , 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 t h 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 presenting an advanced the I Library further for his of this written thesis degree shall agree scholarly by this at it purposes for freely permission may representatives. thesis partial the U n i v e r s i t y make that in is financial British 2075 Wesbrook Place Vancouver, Canada V6T 1WS Columbia, British by for gain Columbia shall the that not requirements I agree r e f e r e n c e and copying t h e Head o f understood Depa r t m e n t of of for extensive permission. The U n i v e r s i t y of available be g r a n t e d It fulfilment of this be a l l o w e d that study. thesis my D e p a r t m e n t copying, or for or publication without my ii THE NONLINEAR THREE-DIMENSIONAL RESPONSE OF STRUCTURES TO EARTHQUAKE EXCITATION ABSTRACT The three-dimensional response of s t r u c t u r e s , comprised of members possessing nonlinear c o n s t i t u t i v e r e l a t i o n s h i p s , subjected to earthquake e x c i t a t i o n i s i n v e s t i g a t e d . The input involves considera- t i o n of the simultaneous horizontal t r a n s l a t i o n of the structure in two mutually perpendicular d i r e c t i o n s in a d d i t i o n to t o r s i o n a l v e r t i c a l ground motion has not been included. behavior; Material n o n l i n e a r i t i e s are represented i n a s i m p l i f i e d fashion by the use of a b i l i n e a r momentr o t a t i o n r e l a t i o n s h i p , subject to the kinematic hardening r u l e , at each member end. Relevant concepts of s t r u c t u r a l theory are reviewed and expressed i n a format allowing for the use of matrix algebra. A general form of the member s t i f f n e s 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 f n e s s matrix i s also presented. The i n t e r a c t i o n of a x i a l loads with the nonlinear material behavior i s included. For columns subjected to bending about two p r i n - c i p a l axes, the e f f e c t of including b i a x i a l i n t e r a c t i o n e f f e c t s on the y i e l d surface i s considered. A method of c a l c u l a t i n g the unbalanced forces r e s u l t i n g from nonlinear behavior i s o u t l i n e d . iii An energy balance i s established whereby the energy input by the earthquake i n t o the structure i s accounted f o r by the various mechanisms with which the structure i s able to d i s s i p a t e or store t h i s input energy. The importance of analyzing the v a r i a t i o n of the various energy forms i s stressed. The theory developed i s applied to two s t r u c t u r a l examples. The problems encountered in attempting to apply t h i s theory to various models of a structure intended to represent a t y p i c a l midsized o f f i c e building are discussed. This s t r u c t u r e has a dual component s t r u c t u r a l system in which an e c c e n t r i c shear core i s designed to r e s i s t the h o r i zontal loads and an e x t e r i o r framing system i s designed t o ^ r e s i s t v e r t i c a l loads and to act as a second l i n e of defense with respect to h o r i zontal loads. The r e s u l t s obtained from an earthquake a n a l y s i s of a s t r u c t u r e , comprised of e x t e r i o r f i v e storey frames, are presented. S p e c i f i c aspects of these r e s u 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 e f f e c t of assumptions made concerning the s t r u c t u r a l r e s ponse are discussed and suggestions f o r future developments i n t h i s f i e l d of a n a l y s i s , which w i l l be aided by advancements made in computer technology, are also given. iv TABLE OF CONTENTS Paje ABSTRACT '. ii TABLE OF CONTENTS iv LIST OF TABLES vi LIST OF FIGURES vii NOTATION xii ACKNOWLEDGEMENTS xi i i CHAPTER 1. 2. INTRODUCTION 1 1.1 Background 1 1.2 Purpose and Scope 5 1.3 Assumptions and L i m i t a t i o n s 8 STRUCTURAL THEORY 10 2.1 Solution of the Incremental E q u i l i b r i u m Equation 10 2.2 Member Properties 18 2.3 Geometric S t i f f n e s s Matrix 31 2.4 Formulation of the Reduced Frame S t i f f n e s s Matrix 39 2.5 Formulation of the Reduced Structure S t i f f n e s s Matrix 3. 43 NONLINEAR MATERIAL BEHAVIOR AND ENERGY CALCULATION 46 3.1 C a l c u l a t i o n of Member Deformations 46 3.2 C a l c u l a t i o n of Member End Forces 49 3.3 Yield Criteria 53 V CHAPTER Page 3.4 Member State Determination 59 3.5 Unbalanced Force Corrections 64 3.6 C a l c u l a t i o n of Incremental Member P l a s t i c Rotations 3.7 4. 5. 70 C a l c u a l t i o n of Energy Quantities 72 MODELLING OF A SIXTEEN STOREY OFFICE BUILDING . . . . 81 4.1 Description of the S t r u c t u r a l Model 82 4.2 Preliminary Analysis 84 4.3 Computer Modelling of the Structure 94 4.4 Summary of Computer Models RESULTS OF A TIME STEP ANALYSIS OF A FIVE STOREY 101 . . BUILDING 6. 104 5.1 The S t r u c t u r a l Model 104 5.2 The Earthquake A c c e l e r a t i o n Record 106 5.3 S e l e c t i o n o f the Time Step Length 108 5.4 Response Results 110 5.5 Summary of the Analysis 116 SUMMARY AND CONCLUSIONS BIBLIOGRAPHY APPENDIX A: 122 CALCULATION OF THE EVENT FACTOR FOR A BIAXIAL COLUMN APPENDIX B: 118 CALCULATION OF THE ENERGY INPUT INTO A STRUCTURE 185 187 vi LIST OF TABLES Tab1e(s) Page 1 Factors used i n time step i n t e g r a t i o n 126 2 Possible member states 126 3 Constants used in b u i l d i n g the member s t i f f n e s s matrix 127 4 Example design e c c e n t r i c i t i e s and t o r s i o n a l moments 128 5 Torsional magnification f a c t o r s f o r f r o n t frame of e c c e n t r i c shear core model 129 6 Modal analysis of model A 130 7 Modal analysis of model B 131 vii LIST OF FIGURES Figure(s) Page 1 Generalized co-ordinates 132 2 Plan view of t y p i c a l storey 132 3 B i l i n e a r moment-rotation r e l a t i o n s h i p at a member end 133 4 Typical moment-rotation r e l a t i o n s h i p at an end o f a r e i n f o r c e d concrete beam 5 (a): , . . Moment-rotation r e l a t i o n s h i p at an end of the e l a s t i c component (b): 133 134 Moment-rotation r e l a t i o n s h i p at an end of the e l a s t i c - p e r f e c t l y p l a s t i c component 134 6 Two component model 134 7 Typical h y s t e r e t i c 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 i n formulation of the geometric s t i f f n e s s matrix 12 (a): 137 Forces due to u n i t v e r t i c a l displacement of j o i n t i (member state A) (b) : Forces due to removal of moment at j o i n t i (c) : Forces due to unit v e r t i c a l displacement of j o i n t i (member state B) 137 . . 137 137 viii Figure(s) 13 Page (a): Forces due to u n i t r o t a t i o n 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 (c) : Forces due to u n i t r o t a t i o n of j o i n t i . . (member state B) 14 15 16 138 (a): B u i l d i n g without v e r t i c a l c o m p a t i b i l i t y . . . . 139 (b): B u i l d i n g with v e r t i c a l c o m p a t i b i l i t y 139 (a): Gross frame degrees of freedom. 140 (b): Reduced frame degrees of freedom 140 (a): Gross structure degrees of freedom 141 (b): Reduced structure degrees of freedom 141 17 Forces and deformations of a t y p i c a l members 18 (a): A x i a l - f l e x u r a l i n t e r a c t i o n diagram (b) : B i a x i a l i n t e r a c t i o n diagram (c) : Y i e l d surface f o r b i a x i a l columns (a): Unbalanced moment corresponding to member end 19 yielding (b) : . . . . .... .-.'"3 (c) : 142 . 143 143 144 144 Unbalanced moment corresponding to a change i n y i e l d moment at a hinge 145 Floor plan of a t y p i c a l f l o o r of the core type building 21 142 Unbalanced moment corresponding to member end reversal-.^.? 20 138 Frame 1 of the core b u i l d i n g 146 147 ix Figure(s) Page 22 Frames 2 and 3 of the core b u i l d i n g 148 23 Walls 8 and 9 of the core b u i l d i n g 149 24 Typical d e f l e c t i o n shapes 150 25 Lateral seismic forces (kips) required by the NBC (in the q d i r e c t i o n ) 150 26 Model A . 151 27 Model B 152 28 Model C . . ' 153 29 Properties of the f i v e storey framed b u i l d i n g . . . . 154 30 Envelope function for earthquake type B 155 31 (a): A c c e l e r a t i o n , v e l o c i t y and displacement f o r earthquake B-l (b): 156 A c c e l e r a t i o n , v e l o c i t y and displacement f o r earthquake B-2 32 33 157 (a): Pseudo-velocity spectrum f o r earthquake B-l . . 158 (b): Pseudo-velocity spectrum for earthquake B-2 . . 159 (a): T r i p a r t i t e logarithmic p l o t of spectra f o r earthquake B-l (b): 160 T r i p a r t i t e logarithmic p l o t of spectra f o r earthquake B-2 34 (a): 161 A c c e l e r a t i o n ( f t . / s e c ) of storey 5 in the q 2 direction (b): 162 Relative v e l o c i t y ( f t . / s e c ) of storey 5 in the q d i r e c t i o n 163 X Figure(s) 34 Page (c): Absolute v e l o c i t y ( f t . / s e c ) of storey 5 i n the q d i r e c t i o n 35 164 (d): Displacement ( f t . ) of storey 5 in the q d i r e c t i o n (a): A c c e l e r a t i o n ( f t . / s e c ) of storey 5 i n the r direction (b) : 166 R e l a t i v e v e l o c i t y ( f t . / s e c ) of storey 5 i n the r d i r e c t i o n (c) : 167 Absolute v e l o c i t y ( f t . / s e c ) of storey 5 i n the r d i r e c t i o n 36 168 (d) : Displacement ( f t . ) of storey 5 i n the r d i r e c t i o n (a): A c c e l e r a t i o n (rad./sec ) of the r o t a t i o n a l component of storey 5 (b) : Relative v e l o c i t y (rad/sec) of the r o t a t i o n a l 171 Absolute v e l o c i t y (rad/sec) of the r o t a t i o n a l component of storey 5 (d) : 172 Displacement (rad) of the r o t a t i o n a l component of storey 5 37 38 169 170 component of storey 5 (c) : 165 173 (a): A c c e l e r a t i o n vs. storey level envelope . . . . (b) : R e l a t i v e v e l o c i t y vs. storey l e v e l envelope . . 174 (c) : Absolute v e l o c i t y vs. storey l e v e l envelope • • 175 (d) : Displacement vs. storey l e v e l envelope 175 (a): E x t e r i o r beams shear envelope (kips) 176 (b): I n t e r i o r beams shear envelope (kips) 176 . . . . 174 xi Figure(s) 39 40 41 Page (a): E x t e r i o r column shear envelope (kips) 177 (b): I n t e r i o r column shear envelope (kips) (a): E x t e r i o r column a x i a l envelope (kips) 178 (b): I n t e r i o r column a x i a l envelope (kips) 178 . . . . . 177 V a r i a t i o n of recoverable energy during the f i r s t 22 seconds 42 V a r i a t i o n of t o t a l energy during the f i r s t 22 seconds 43 Number of excursions i n t o the p l a s t i c range at storey l e v e l s 179 180 181 44 Accumulated p l a s t i c r o t a t i o n 182 45 Dissipated energy ( k i p - f t . ) of various components . . 183 46 Dissipated energy of the e n t i r e b u i l d i n g 184 xii NOTATION The use and meaning of each symbol i s defined in the t e x t 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 i n g l e l i n e , e.g. x j while m a t r i x , or two-dimensional arrays, are underlined by a double l i n e , e.g. £ . The number of dots above a v a r i a b l e , e.g. x, represent the number of times the v a r i a b l e i s d i f f e r e n t i a t e d with respect to time. xi i i ACKNOWLEDGEMENTS The author wishes to express his gratitude to his a d v i s o r , Dr. N.D. Nathan, f o r his valuable advice and guidance during the r e search and preparation of t h i s t h e s i s . He also expresses his thanks to Dr. D.L. Anderson and Dr. S. Cherry f o r t h e i r suggestions and advice during the research work. The f i n a n c i a l support of the National Research Council in the form of a UniveBsity of B r i t i s h Columbia research a s s i s t a n t s h i p i s also g r a t e f u 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 e x h i b i t i n g p l a s t i c or nonlinear p r o p e r t i e s , precludes the use of a r e l a t i v e l y simple e l a s t i c a n a l y s i s i n order to describe mathematic a l l y the response of a structure to earthquake e x c i t a t i o n . The need f o r 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 excitation. Research of t h i s nature i s heavily dependent upon computer technology and has c l o s e l y p a r a l l e l e d , and hence been l i m i t e d by, advances in computer e f f i c i e n c y and c a p a c i t y . A good introduction to the subject of the nonlinear response of structures to earthquakes can be found i n the book by Blume; Newmark and Corning [3]. This book provides a discussion of the background f o r i n e 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 s t r u c t u r e s . An intensive analysis 2 of s i n g l e degree of freedom systems i s presented and the theory i s extended to approximate the maximum values to be obtained f o r a 24 storey frame. The nonlinear approximation of the c o n s t i t u t i v e r e l a t i o n s h i p s of the members of a s t r u c t u r e i s an area undergoing much research. The most basic nonlinear model i s one with a b i l i n e a r moment-rotation r e l a t i o n s h i p in which the member i s assumed to be l i n e a r l y e l a s t i c u n t i l some point (the y i e l d moment), a f t e r which the moment-curvature slope changes. A s i n g l e component model of t h i s b i l i n e a r r e l a t i o n s h i p 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 a n a l y s i s of m u l t i degree of f r e e dom systems i s presented by Clough; Benuska and Wilson [6] and Clough and Benuska [5] i n which a r e l a t i v e l y crude analysis of a twenty storey b u i l d i n g frame subjected to several earthquake a c c e l e r a t i o n records i s performed. Time step a n a l y s i s , which i s now a commonly used method in nonlinear dynamic a n a l y s i s , approximates the response h i s t o r y of a s t r u c t u r e by subdividing the earthquake a c c e l e r a t i o n record i n t o a s e r i e s of small time steps. Over each time step, the response i s as- sumed to be l i n e a r , based upon the c o n d i t i o n of each member of the structure at the beginning of the time step. The accuracy of the r e - s u l t s obtained i s very s e n s i t i v e 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 a n a l y s 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 e a r moment-rotation r e l a t i o n s h i p a change in s t a t e 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 y i e l d e d and a p l a s t i c hinge i s introduced as an i n i t i a l condition f o r the next time step. A l t e r n a t i v e 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 i n the absolute r o t a t i o n at the end; the hinge i s then removed f o r the next time step. Hence in t h i s type of analysis the nonlinear response of the frame i s approximated as a s e r i e s of successively changing l i n e a r time increments. Other researchers have extended the time step analysis theory to other plane frames using more s o p h i s t i c a t e d 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 i d e a l i z a t i o n 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 h i s t o r y of the s t r u c t u r e . There are, however, some serious drawbacks which d e t r a c t from the v a l i d i t y of r e s u l t s obtained from a plane frame approximation of the actual s t r u c ture. For example, a column in an actual s t r u c t u r e w i l l generally be subjected to motion about both of i t s p r i n c i p a 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 p r i n c i p a l a x i s . A l s o , the formation of a p l a s t i c hinge due to motion p r i m a r i l y along one axis w i l l a l t e r the behavior of the member along the other a x i s . A plane frame analysis i s unable to account f o r these f e a t u r e s . In a d d i t i o n , Newmark [23] has stated that a r o t a t i o n a l component of ground motion should be considered. This component induces t o r - sional behavior in the structure which cannot be accounted f o r i n a plane frame a n a l y s i s . Torsional responses may also occur in-ia structure r e s u l t i n g from asymmetryypf plan; layouts s t i f f n e s s .onrstrength d i s t r i b u t i o n . Berg and S t r a t t a [2] present evidence of t o r s i o n a l behavior in buildings sub- jected to the Alaska, 1964 earthquake and Hanson and Degenkolb [13] state that the Venezuela, 1967 earthquake also induced t o r s i o n a l behavior. importance of including t o r s i o n a l behavior in earthquake analysis The has been stressed by many authors such as Degenkolb [ 8 ] , Hart; Lew and Di J u l i o J r . [15] and M a z i l u ; Sandi and Teodorescu [20]. Recently, several authors have presented r e s u l t s of analyses structures i n c l u d i n g t o r s i o n a l e f f e c t s . of Koh; Takase and Tsugawa [18], MacKenzie [19] and Tso and Bergmann [30] present the theory of mathematical models of structures which are condensed to contain three degrees of freedom at each storey l e v e l : two mutually perpendicular t r a n s l a t i o n s plus one r o t a t i o n a l or t o r s i o n a l degree of freedom. These mathematical models are tested on e c c e n t r i c structures in attempts to determine the conditions under which the t o r s i o n a l response becomes a s i g n i f i c a n t or dominant f a c t o r . MacKenzie [19] also explored the undesirable phenomenon 5 of beating. These analyses were r e s t r i c t e d to structures in which a l l members are assumed to remain l i n e a r l y e l a s t i c during the earthquake. More r e c e n t l y , researchers have attempted mathematical models of structures including t o r s i o n a l behavior and allowing for material nonlinearities. Anagnostopoulous; Roesset and Biggs [ 1 ] , Selna; M o r r i l l and Ersoy [26], Tso [28], Tso and Asmis [29], and Wynhoven and Adams [34], [35] have presented r e s u l t s obtarined with t h e i r nonlinear t o r s i o n a l models. These authors have generally applied t h e i r models to small structures and have made assumptions which reduce the cost of s o l u t i o n by computer at the expense of accuracy. However, r e s u l t s obtained by these authors are valuable steps towards the accurate s o l u t i o n of the response of structures to earthquakes. One of the most important steps i n a t t a i n i n g a thorough understanding of the nonlinear response of structures to earthquake e x c i t a t i o n i s a study of the various ways in which energy inputtbyythe^earthquake i s stored and d i s s i p a t e d by the s t r u c t u r e . Goel and Berg [11], Hanson and Fan [14] and Pekau; Green and Sherbourne [25] have presented r e s u l t s showing the v a r i a t i o n of the sundry forms of energy in the s t r u c t u r e with respeo.t-to time, and a s i m i l a r study i s made in t h i s t h e s i s . 1.2 Purpose and Scope The purpose of t h i s thesis i s to extend the work of previous i n v e s t i g a t o r s concerned with the nonlinear t o r s i o n a l response of s t r u c tures to earthquake e x c i t a t i o n . A parametric study of the v a r i a t i o n of 6 response values with respect to s t r u c t u r a l parameters in very large buildings was precluded by the expense involved i n performing nonlinear dynamic analyses on e x i s t i n g computer systems. For t h i s reason, i t was the i n t e n t i o n of the author to subject a t y p i c a l midsized o f f i c e b u i l d i n g , modelled as accurately as p o s s i b l e , to a complete earthquake record and to examine the v a r i a t i o n with respect to time of the important response parameters. Theory concerning the response of structures to earthquakes has advanced very r a p i d l y in recent times, and has been accompanied by a p r o l i f e r a t i o n of associated l i t e r a t u r e . As research i n t h i s area i s s t i l l progressing i t i s f e l t by the author that a " s t a t e of the a r t " thesis would be valuable to future researchers. For t h i s reason, r e l e - vant material i s gathered from various sources and presented in t h i s thesis i n conjunction with theory required f o r the s p e c i f i c approach used by the author. The dynamic e q u i l i b r i u m equation i s formulated and two techniques of solving the equation are discussed. The b i l i n e a 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 f n e s s matrix i s presented. The e f f e c t s of shear de- f l e c t i o n , rdgid end stubs (to account f o r f i n i t e j o i n t s i z e ) and geometr i c n o n l i n e a r i t i e s are reviewed and included in the member s t i f f n e s s matrix. A review of theory presented by MacKenzie [19], and Nathan and MacKenzie [21], concerning the formation and subsequent condensation of the s t r u c t u r e s t i f f n e s s matrix to contain three degrees of freedom (gene r a l i z e d co-ordinates) per storey, i s summarized. 7 The i n t e r a c t i o n of a x i a 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 perpendicular axes, the y i e l d surface i s defined. An algorithm used i n 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 c a l c u l a t e various energy q u a n t i t i e s are p r e sented and an energy balance i s discussed. The energyt-balance r e l a t e s to the f a c t that the t o t a l energy input by an earthquake i n t o a structure should equal the sum of the energy q u a n t i t i e s appearing in the s t r u c t u r e . These s t r u c t u r e energy q u a n t i t i e s are: (1) the k i n e t i c energy of the s t r u c t u r e , (2) the recoverable s t r a i n energy stored in the s t r u c t u r e , (3) the non-recoverable s t r a i n energy d i s s i p a t e d by the s t r u c t u r e due to nonlinear, h y s t e r e t i c behavior and, (4) energy d i s s i p a t e d by viscous damping. The s t r u c t u r e which was selected to r e f l e c t a t y p i c a l midsized o f f i c e b u i l d i n g i s a modified version of a structure which was r e c e n t l y constructed in Vancouver. The sixteen storey b u i l d i n g has an e c c e n t r i c elevator and s t a i r core which i s designed to r e s i s t the earthquake loads s p e c i f i e d by the National B u i l d i n g Code of Canada [22], in conjunction with an e x t e r i o r reinforced concrete framing system designed to independently r e s i s t twenty f i v e per cent of the s p e c i f i e d earthquake loads. Several d i f f e r e n t methods of modelling t h 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 t h i s type of s t r u c t u r e , and since the cost of s o l v i n g the problem by computer was larger than o r i g i n a l l y a n t i c i p a t e d , t h i s model was abandoned. However a d e s c r i p t i o n of the models and t h e i r associated problems i s included as i t i s felt that t h i s would be a valuable asset to others doing research i n t h i s area. In order tbeillustrateetheetheowyyppesenteddinnthi's--thesis, a symmetric f i v e storey framed b u i l d i n g was analyzed. This s t r u c t u r e con- s i s t e d of four e x t e r i o r frames, each s i m i l a r to the top f i v e storeys of the frame analyzed by Clough; Benuska and,Wilson [ 6 ] . 0 An e c c e n t 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 s t r u c t u r e 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 a c c e l e r a t i o n of 0.25'-,g and applied simultaneously i n the two perpendicular d i r e c t i o n s . The r e s u l t s of t h i s a n a l y s i s , and a discussion of the r e s u l t s , are presented. 1.3 Assumptions and L i m i t a t i o n s V i r t u a l l y a l l of the l i m i t a t i o n s governing t h i s work are im- posed by the necessity of obtaining a s o l u t i o n by computer w i t h i n r e a 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 r e l a x most of these economic c o n s t r a i n t s and allow a more accurate analysis to be performed. 9 The theory used to assemble the structure s t i f f n e s s matrix i s based upon the assemblage of plane frame s t i f f n e s s matrices. For t h i s reason, the s t r u c t u r e 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 n e g l i g i b l e s t i f f n e s s with r e s pect to out of plane t r a n s l a t i o n or r o t a t i o n . i n d i v i d u a l members i s neglected. The t o r s i o n a l r i g i d i t y of The centre of mass of a l l storey l e v e l s must l i e on a s i n g l e v e r t i c a l l i n e i n the program used in t h i s research. The moment-rotation r e l a t i o n s h i p at a member end i s assumed to be b i l i n e a r . The material i s assumed to have kinematic hardening, and s t i f f n e s s degradation i:s ignored. A x i a l - f l e x u r a l i n t e r a c t i o n surfaces are assumed and f o r columns subjected to bending about two mutually perpendicular axes, the y i e l d surface i s assumed to be an e l l i p s e . The s i z e of the problem which may be analyzed using the computer program developed by the author i s a function of the storage c a p a c i t i e s of the computer system in a d d i t i o n to constraints imposed by economy. The program was designed to require a minimum of storage while maintaining maximum e f f i c i e n c y . I t i s f e l t that a conventional s t r u c t u r e 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 o u t l i n e d i n t h i s thesis on current computer systems. The author has attempted to keep the number of l i m i t a t i o n s used in t h i s analysis to a minimum. The most important l i m i t a t i o n s are described above, however a d d i t i o n a l l i m i t a t i o n s , in the form of assumptions necessitated by economy, are discussed as they are introduced i n t o the relevant sections of t h i s thesis. 10 CHAPTER 2 STRUCTURAL THEORY Previous work done by MacKenzie [19] in the three dimensional response of structures during earthquakes was r e s t r i c t e d by the assumpt i o n that a l l members remain l i n e a r l y e l a s t i c throughout the response h i s t o r y of the s t r u c t u r e . 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 a n t l y a l t e r r e s u l t s obtained f o r t y p i c a l s t r u c tures subject to moderate to major earthquakes. However, the general approach to the problem and much of the theory used i n the formation and subsequent condensation of the s t i f f n e s s matrices i s s t i l l v a l i d despite the i n c l u s i o n of material n o n l i n e a r 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 i n t h i s thesis and incorporated into t h i s chapter f o r the sake of completeness. 2.1 Solution of the Incremental Equilibrium Equation As stated by Clough and Penzien [ 7 ] , "the only generally a p p l i - cable method for the analysis of a r b i t r a r y nonlinear systems i s the numerical step by step i n t e g r a t i o n of the equation of motion. h i s t o r y i s divided into short equal time increments The response >At ,:and the response i s c a l c u l a t e d during each increment f o r a l i n e a r system having properties determined at the beginning of the i n t e r v a l . The nonlinear behavior is approximated as a sequence of analyses of successively changing l i n e a r systems". 11 The e q u i l i b r i u m equation i n incremental form (assuming viscous damping) i s the f o l l o w i n g - w e l l : known second order d i f f e r e n t i a l equation. M Ax + C Ax + Kg Ax_ = B in which Mr! = mass matrix Cg damping matrix K =1 = = D (K Kro) with = § ! - ^gB CD K - M Ax Jti = K CD GB = (2.1) g i n i t i a l tangent s t i f f n e s s matn 9 . x e o m e 1 : r '' c s t i f f n e s s matrix Mx = incremental displacement vector r e l a t i v e to the ground Ax = incremental ground a c c e l e r a t i o n vector. Dots represent d i f f e r e n t i a t i o n with respect to time. script values. B The sub- represents values at the beginning of the time step or tangent Damping i s assumed, i n (2.1), to be viscous. While t h i s i s not the actual case, t h i s assumption i s accepted f o r ease of computation. The e q u i l i b r i u m equation i s solved f o r the incremental d i s p l a c e ment vector as described i n Section 2 . 1 . i i . As s o l u t i o n of the equation involves i n v e r s i o n of a matrix of an order equal to the number o f 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 d e s c r i b i n g the s t r u c t u r a l motion. This i s accomplished by o r i g i n a l l y including three degrees of freedom a t each j o i n t : a horizontal and v e r t i c a l t r a n s l a t i o n plus a r o - t a t i o n a l degree o f 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 r a n s l a t i o n s in two mutually perpendicular d i r e c t i o n s q t i o n about the centre of mass as shown in Figure 1. and r plus a r o t a - S t a 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 r u c ture about two mutually perpendicular d i r e c t i o n s i n addition to analysis of the t o r s i o n a l response of the s t r u c t u r e . 2.1.1 Description of Terms in the Equilibrium Equation 2.1. i (.a).\. Mass Matrix.' The mass matrix i s derived by applying a u n i t a c c e l e r a t i o n i n - dependently to each generalized co-ordinate. matrix generates the force i n the a c c e l e r a t i o n in the q q The 1-1 term of the mass d i r e c t i o n r e s u l t i n g from a/unit d i r e c t i o n at the f i r s t storey. The a p p l i c a t i o n of a u n i t a c c e l e r a t i o n to any of the three generalized co-ordinates of a p a r t i c u l a r storey l e v e l 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 i n of the s t r u c t u r e and the centre of mass at a p a r t i c u l a r storey l e v e l are as indicated in Figure 2, then the storey massvmatrix w i l l be as shown in (2.2) m 0 0 m -mr:etry* nwe„ -mne "T m;e qq 1 m 12 m 13 where m = t o t a l mass of the storey e , e = distance i n the q, r d i r e c t i o n s from the s t r u c t u r e o r i g i n to the centre of mass = storey width in q a, b q, r d i r e c t i o n s . The storey mass matrix shown i n (2.2) assumes that the mass of a p a r t i c u l a r storey i s evenly d i s t r i b u t e d about the centre of mass. The 3-3 term i s the mass polar moment of i n e r t i a f o r a rectangular storey. I f the structure o r i g i n and the centre of mass c o i n c i d e , quantities matrix. e^ and e^ equal zero and the storey mass matrix i s a diagonal I f thi>s i s true at every storey l e v e l then the s t r u c t u r e mass matrix w i l l also be a diagonal matrix and may be expressed as in (2.3) M where = I < m ', m 1 m > T e (2.3) i d e n t i t y matrix vector of storey masses m V = = vector of storey mass polar moments of i n e r t i a . There are many structures f o r which the centre of mass of a p a r t i c u l a r storey i s not on the v e r t i c a l l i n e j o i n i n g the centres of mass of other s t o r e y s , in which case the coupled form of the mass matrix shown in (2.2) must be used. However, because of the great computational saving r e s u l t i n g from the use o f a diagonal mass matrix, the program developed by the author contains t h i s r e s t r i c t i o n . This implies that 14 only structures in which the centres of mass at a l l storey l e v e l s can be approximated by points on a single v e r t i c a l l i n e may be analyzed. 2.1. i (b)- S t i f f n e s s ^Ma:trtx- Derivation of the structure s t i f f n e s s matrix i s presented i n Section 2.5. Using the co-ordinate system shown in Figure 1, the con- densed s t i f f n e s s matrix i s as expressed in B - K SB qq i, p = t K ' i/B ,.> i V ' :•' K K 1 eq where the submatrix K n r r ''"0 t K 9r (2.4) fge K r6 (2.4) B K | K R =§£ , f o r example, represents the forces i n the d i r e c t i o n r e s u l t i n g from unit displacements of the f l o o r s in the r e c t i o n , stored f l o o r by f l o o r . The superscript B r q di- i n d i c a t e s 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 ) " G e o m e t r i c Stif.fnesssMa'tftlxx. v The geometric s t i f f n e s s matrix accounts f o r second order geometric n o n l i n e a r i t i e s induced i n a member by changes in geometry in the presence of a x i a l l y d i r e c t e d load components. Derivation of the member geometric s t i f f n e s s matrix i s presented in Section 2.3. The member 15 geometric s t i f f n e s s matrices may be assembled and then reduced using s t a t i c condensation s i m i l a r to the technique used f o r the l i n e a r tangent s t i f f n e s s m a t r i x ; r e s u l t i n g in a matrix K^g of the same form as (2.4). A l t e r n a t i v e l y the member geometric s t i f f n e s s matrices may be incorporated into the member l i n e a r tangent s t i f f n e s s matrices in which case a f t e r s t a t i c condensation the o v e r a l l structure s t i f f n e s s matrix including geometric n o n l i n e a r i t i e s w i l l be represented by (2.4). 2.1. i (dc): Dampj:rigc;Matrtx;- The damping matrix C R i s evaluated under the commonly used assumption that damping i s proportional to e i t h e r or both of the mass and s t i f f n e s s d i s t r i b u t i o n s , as expressed i n C D = (2?5) 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 n e a r mode of v i b r a t i o n £ = f r a c t i o n of c r i t i c a l damping which i s s t i f f n e s s proportional = f r a c t i o n of c r i t i c a l proportional. 5 Assumption (2.5) s damping which i s mass i s p a r t i c u l a r l y useful in l i n e a r e l a s t i c a n a l y s i s as mode shapes and frequencies c a l c u l a t e d for the undamped case w i l l be 16 the same when damping i s included i f t h i s r e l a t i o n i s assumed. The r e - l a t i o n i s also useful i n the nonlinear case as i t means the e q u i l i b r i u m equations involve only two matrices: 2.1.i(e) the mass and s t i f f n e s s matrices. Displacement Vectors The vector Ax , representing incremental displacements of the generalized co-ordinates r e l a t i v e to the ground, may be expressed i n p a r t i t i o n e d form as i n (2.8) Ax with = < Aq ! Ar j A6 > (2.8) T Aq_ equal to incremental displacements in the q d i r e c t i o n of each storey l e v e l , and so f o r t h . S i m i l a r r e l a t i o n s h i p s may be expressed f o r incremental v e l o c i t i e s and a c c e l e r a t i o n s of the generalized co-ordinates. The vector of incremental ground a c c e l e r a t i o n s may s i m i l a r l y be expressed as: where each term of < Aq„ A6* > (2.9) Ax„ = Aq , e t c . , i s equal to, the'incremental ground acce- _g_ T 1 e r a t i o n q i nrfthe.s.tqe-?di r e c t i on. 2.1.ii S o l u t i o n Techniques In order to solve the incremental e q u i l i b r i u m equation (2.1) an assumption must be made concerning the v a r i a t i o n o f 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 a c c e l e r a t i o n varies l i n e a r l y over an extended time increment. (b) the Newmark 3 Method which assumes a constant response a c c e l e r a t i o n over a time increment equal to the average of values at the beginning and end of the increment. These solutionttechniques may be expressed i n general form as Ax x At Ax Ax = a Ax = R = m [(•£§" + a S 5 5 x - a g x At (2.11) Af _1 a Af - a lt ) (2.12) aa x + (a 3 3a,, + aa 6 At) x - Ax ] + 3 K (a x + a x At) g g g (2.14) where of a. , i = 1, 6 i x^ and are defined f o r each method in Table 1. Values x. represent response v e l o c i t i e s and accelerations a t the beginning of the time step. 18 Solution of incremental displacements using (2.12) allows f o r incremental v e l o c i t i e s and a c c e l e r a t i o n s 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 f o r the next time step. I t should be noted that the Wilson 6 method i s made uncondit i o n a l l y stable by c a l c u l a t i n g values over an extended time step where 6 > 1.37 . B At The incremental displacements over t h i s extended time step are solved by (2.12) and then the incremental a c c e l e r a t i o n s over the normal time step are found by d i v i d i n g (2.10))by 6,. These i n c r e - mental a c c e l e r a t i o n s are then used to c a l c u l a t e incremental v e l o c i t i e s and displacements over the normal time step using the l i n e a r a c c e l e r a t i o n assumption. I t should also be noted that while the pseudo load vector w i l l have to k be c a l c u l a t e d every time step, the pseudo s t i f f n e s s matrix w i l l be a constant i f the status of each member i s unchanged. matrix Af The k 'Msoonly c a l c u l a t e d (and then inverted) i f a member status changes. 2.2 Member Properties The force-displacement r e l a t i o n s h i p for each member i s expressed in terms of a member s t i f f n e s s matrix. The member s t i f f n e s s matrices f o r each member of a p a r t i c u l a r frame are added together, using well e s t a b l i s h e d 19 procedures, to form the s t i f f n e s 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 f n e s 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 i n which case y i e l d i n g w i l l Y i e l d i n g , or subsequent occur. r e v e r s a l , at any member end w i l l change i t s s t i f f n e s s matrix and hence the structure matrix. 2.2.i Member I d e a l i z a t i o n The nonlinear behavior at a member end i s assumed to be repre- sented i n a b i l i n e a r fashion as shown in Figure 3. The moment-rotation r e l a t i o n s h i p at each member end i s thus assumed to have two l i n e a 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 i e l d e d . This b i l i n e a r moment-rotation r e l a t i o n s h i p i s v i r t u a l l y an exact d e s c r i p t i o n of the behavior of steel members, however f o r members constructed from r e i n forced concrete a b i l i n e a r moment-rotation curve i s only an approximation. The actual r e l a t i o n s h i p f o r a t y p i c a l reinforced concrete section i s shown in Figure 4 , however f o r the sake of economy t h i s complex r e l a t i o n ship i s approximated by the b i l i n e a r r e l a t i o n s h i p of Figure 3. If p equals zero, Figure 3 represents the commonly used e l a s - t i c - p e r f e c t l y p l a s t i c r e l a t i o n s h i p , 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 r a i n hardening e f f e c t a value of p greater 20 than zero should be assigned; a value of times the o r i g i n a l slope k p in the range .03 to .05 i s most commonly used. Despite the f a c t that the member end w i l l not be p e r f e c t l y 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 this state. In order to model t h i s b i l i n e a r moment-rotation behavior, each member i s considered to be comprised of two components i n p a r a l l e l . component remains f u l l y e l a s t i c and contributes a f r a c t i o n t o t a l member. p One to the The second component i s e l a s t i c - p e r f e c t l y p l a s t i c and con- tributes a fraction q (= 1-p) to the member. for these components are shown in Figure 5. The moment-rotation curves I t i s e a s i l y v i s u a l i z e d that m u l t i p l y i n g the moment axis in Figure 5(a) by p and in Figure 5(b) by q (= 1-p) and then adding these c a l c u l a t e d moments together w i l l produce the b i l i n e a r r e l a t i o n s h i p shown in Figure 3. A t y p i c a l l y displaced two component model i s shown in Figure 6. The t o t a l r o t a t i o n of a member end i s comprised o f two p a r t s : r o t a t i o n and (2) chord r o t a t i o n due to j o i n t t r a n s l a t i o n . (1) j o i n t These r o t a - tions are r e l a t e d by (2.15) 6. where 6^ = = t o t a l r o t a t i o n of member end co. = joint y chord r o t a t i o n . = i (2.15) a). - y rotation i 21 For the e l a s t i c - p e 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. where a>. = cfr. + a. (2.16) = e l a s t i c r o t a t i o n at member end i = p l a s t i c r o t a t i o n at member end i The nonlinear, i n e l a s t i c behavior of a;typicalimember end i s shown i n Figure 7. Numbers 1 through 9 describe the progression of the moment-rotation r e l a t i o n s h i p throughout the h i s t o r y of a t y p i c a l loop. A progression of the type shown i n Figure 7 i s known as kinematic hardening which i s characterized by the formation of a closed hysteresis loop. Numbers (1) through (5) describe the f i v e possible member end states a t any p a r t i c u l a r time. A member end i n status ( 1 ) , (3) or (5) w i l l e x h i b i t i t s o r i g i n a l s t i f f n e s s and may be considered to be i n an unyielded s t a t e . A member end i n states (2) or (4) i s i n a y i e l d e d s t a t e ; t h a t i s a p l a s t i c hinge has formed at the member end. In t h i s case the s t i f f n e s s of the end i s derived s o l e l y from the e l a s t i c component; the e l a s t i c - p e r f e c t l y p l a s t i c component has zero s t i f f n e s s . At any p a r t i c u l a r time a member end w i l l e i t h e r 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 h y s t e r e t i c behavior described i n Figure 7 i s not a r e a l i s t i c d e s c r i p t i o n of the behavior of a r e i n f o r c e d concrete member. In reinforced concrete, the occurrence of p l a s t i c r o t a t i o n a f t e r y i e l d i n g w i l l be accompanied by s i g n i f i c a n t cracking arid s p a ! l i n g of concrete i n 22 the section of the member subjected to tension. When the end c o n d i t i o n 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 n e s s . A l s o , due to cracking and s p a 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 i n the opposite d i r e c t i o n w i l l be the same as i f i t had not y i e l d e d p r e v i o u s l y . To properly model the actual behavior of a reinforced concrete member i t i s necessary to use a degrading s t i f f n e s s model whereby the reversed s t i f f n e s s and the reversed y i e l d moment are decreased p r o p o r t i o n a l l y to the amount of p l a s t i c r o t a t i o n in the opposite d i r e c t i o n . Degrading s t i f f n e s s models such as those proposed by Gulkan and Sozen [12], Hart; Lew and Di J u 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 t h i s t h e s 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 r e s u l t i n a prohi bi tnvelyyexpens.ivenanalysi s. 2.2.ii The Member S t i f f n e s s Matrix Each o f the four p o s s i b l e member states of Table 2 requires a separate member s t i f f n e s s matrix to describe the r e l a t i o n s h i p of moments and shears to r o t a t i o n s and t r a n s l a t i o n s f o r the e l a s t i c - p e r f e c t l y p l a s t i c component. Formation of a p l a s t i c hinge w i l l not a f f e c t the r e l a - t i o n between a x i a l forces and a x i a l deformations, nor w i l l i t a f f e c t the s t i f f n e s s values of the e l a s t i c component of the member. The o v e r a l l member s t i f f n e s 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 o v e r a l l member s t i f f n e s s matrix k a_ = the a x i a l s t i f f n e s s matrix = the bending s t i f f n e s s matrix of the e l a s t i c component = the bending s t i f f n e s s matrix of the e l a s t i c - p e r f e c t l y p l a s t i c component = the geometric s t i f f n e s s matrix. bi v ba v kg The degrees of freedom of a t y p i c a l member are shown in Figure 8. Using thennotation of Figure 8 with g = shear d e f l e c t i o n f a c t o r A = c r o s s - s e c t i o n a l area moment of i n e r t i a E = Young's modulus G = shear modulus = shear area I A y = 12 E I A GL the a x i a l s t i f f n e s s matrix i s described in (2.18) x k ^ = — L sym. xy y o o o -xy o 2 o xy y o 0 0 2 3 -x -xy o -y o (2.18) 2 X 2 and the bending s t i f f n e s s matrix of the e l a s t i c component i s shown in (2.19) 24 •12xy EI |2 (l+g) ffci" • 6yL 12x sym. 2 6xL 2 4L'*(l+g/4) 2 5 •12y-:?y 12xy ::6yl_ z 12xy -12x --. -6xL 2 - 6yL Sy .'6xL 2 2 12y 2 -12xy C 2iL*(l-g/2) 2 2 g6y| 12x 2 6xL 2 2 4L*(l+g/4) (2.19) The evaluation of the bending s t i f f n e s s matrix of the e l a s t i c - p e 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 for k' bi (2.20) Bi b2A described in (2.19) Ror member state B 3y EI 'b2B il'(-i'+g/4) 3 " 2 sym. -3xy 03x 0 30.; 0 -3/ 33xy 0 3y ?3x 0 -3xy 0 3yL 3xy -3yL 2 2 2 33xL 2 (2.21) 2 3x 2 2 -3xL 2 3L* 25 For member,..state G 3/ -3yL EI L (l+g/4) sym 3x -3xy 3xL 2 5 2 (2.22) 3xy • V -3x 3xy 3yL -3xL 2 -3xy 2 3x' 0 0 0 0 0 3/ 2 and f o r member state D L n b2D = 0 = = (2.23) a n u l l matrix The geometric s t i f f n e s s matrix accounts f o r the second order n o n l i n e a r i t i e s r e s u l t i n g from a x i a l load e f f e c t s . Derivation of the geometric member s t i f f n e s s matrix i s presented i n Section 2.3 and the ahalyscis;.deso&ibedilindtnrist'seefcionti gnofescthe e ' f f e c f f o f t a x i a l 1 oads on bendsingi s t i f f n e s s ? 33 For s o l u t i o n by computer i t i s more convenient to express the o v e r a l l member s t i f f n e s s matrix in a more general form. This i s achieved by expressing (2.17) in the form shown in (2.24) ii = k a + P bi k + k b2A + % k b2B + q s k b2C % + k b 2D (2.24) k = k + (p + a) k. + ki n + q o k. r 26 where constants P ( = P + q ) > q 2 and q are defined i n Table 3. 3 The general form of the bending s t i f f n e s s matrix of a t y p i c a l member i n c l u d i n g both components i s shown in (2.25) 2 y a, -xya x a 1 2 k b = k b, + k b xya xya -x a, 1 -yL a i n which a. f l + a. = f a. = f. a. f a. f a 6 = = y 2 22 1 xL a 13 2 --.liLa^ -xya x^a. 1 yl/a -XLi a 2 2 2 2 (2.26) + q ^3 M 2 P (1 - g/2)/3 P + q. P + q^ 2P (1 + g/4)/3 iL a h + q 3 and f 3 f 3 -xLaa 2 2 (2.25) 4 3 yL'aa 1 2 2 kLag. 3 -y a. J sym. 2 xL a , 3 J . 5 2 2 -yL a E]_ *• (2.27) 5 27 6P and TTT-gJ K (2.28) ( i + g/4) 3 % ( i + g/4) In the current analysis i t i s assumed that the s t r u c t u r e i s comprised of only two member types: horizontal and v e r t i c a l members. This r e s t r i c t i o n on member types allows the o v e r a l l member s t i f f n e s s mat r i x to be s i m p l i f i e d f o r each of the member types. bers x = L and y = 0 and f o r v e r t i c a l members For horizontal mem- x = 0 and y = L . S u b s t i t u t i o n of these values into equations (2.25) and (2.18) allows the o v e r a l l membersstiffness matrix to be s i m p l i f i e d into the forms shown in (2.29) and (2.30): For horizontal members k = M. = L 0 0 0 0 -10 sym. 0 EI 0 1 0 0 0 0 0 0 0 0 0 0 0 0 a - . sym. 0 a L a L (2.29) 0 0 it 0 0 -a -a L 3 3 0 0 a i 2 0 a L 2 a L c ° 2 0 -a.L a_L 2 5 28 and f o r v e r t i c a l members a. k = = L 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0-1 0 2.2.iii 0 0 sym. EI 0 -a L 3 0 a, L -a 1 0 a3L a. i 0 0 0 0 -a L 0 sym. ilO 2 aL 6 (2.30) aL 2 0 aL 5 M o d i f i c a t i o n of Member S t i f f n e s s Matrix' for E f f e c t s 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 r e s i s t i n g elements such as reinforced concrete cores surrounding e l e v a t o r shafts and s t a i r w e l l s . The usual p r a c t i c e i s to mo- del the core as a column situated at the centroid of the cross s e c t i o n . This procedure requires that beams framing into or between these cores be given special consideration to account f o r the f i n i t e width of the cores. A t y p i c a l plan view of a coupled e l e v a t o r and s t a i r w e l l core system i s shown in Figure 9. In the true system, the l i n t e l beams j o i n i n g the two cores w i l l be much shorter that the beam used - i n ' 4he molde^iwhi chins: assumed to span ! from.col<umn to' column.;! ::This ppoblem^may^be^a^ mate.. mahner*byyc"ohsidering>theglihtel"*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':' arid: ^br'c.as:sbown*irTFigure" T-0.' j approxi- 29 Derivation of the member bending s t i f f n e s s matrix for beams with r i g i d end stubs i s presented below. Arrays which contain the super- s c r i p t r e l a t e to the f l e x i b l e section of the beam while arrays with no superscript r e l a t e to the t o t a l length of the beam to be consistent with the sign convention shown i n Figure 10. The basic s t i f f n e s s r e l a t i o n s h i p s are shown in (2.31) f' = k' A' (2.31a) f = kA (2.31b) A transformation matrix may be found to r e l a t e the forces of the two co-ordinate systems as expressed in (2.32a) f = T f'_ (2.32a) and the p r i n c i p l e of contragradience may be used to r e l a t e the d i s p l a c e ments of the two co-ordinate systems as expressed i n (2.32b) A^_ = t L a (2.32b) S u b s t i t u t i o n of equations (2.32) into equation (2.31a) provides equation (2.33) from which the member s t i f f n e s s matrix of the t o t a l member may be expressed in terms of the s t i f f n e s s matrix of the f l e x i b l e section o f the beam in equation (2.34) f = I k' I T A (2.33) 30 The transformation matrix may be found from elementary s t a t i c s and may be expressed as shown i n (2.35) T = 1 0 0 0 0 0 0 1 0 0 0 0 0 a 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 -b 1 (2.35) I t 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 g n i f i c a n t e f f e c t on the s t i f f n e s s mat r i c e s of columns framing into these beams. However i t i s f e l t that t h i s e f f e c t i s n e g l i g i b l e in most cases, hence t h e t e f f e c t of r i g i d end stubs i s l i m i t e d 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 i z e a b l e e r r o r in modelling. The member bending s t i f f n e s s matrix i s found by eq. (2.34), The o v e r a l l member s t i f f n e s 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„ f a.- f + a a 2 + q 3 a. f 3 (2.36) 1 + 2af ^3 + % + a a. 2 + 2 f ? f l 1 + p ' ( l - g/2)/3 + a f j + 8 f 2 + aB 0 4 31 in which a = a/L 3 = b/L and f a c t o r s defined i n (2.27) arid (2.28) are v a l i d . 2.3 Geometric S t i f f n e s s Matrix The geometric s t i f f n e s s matrix accounts for second order (geo- metric) n o n l i n e a r i t i e s induced in a member subject to a x i a l l y d i r e c t e d (load components. The importance of including geometric nonl i n e a 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 a n a l y s i s of geometric n o n l i n e a r 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 i n the presence of the a x i a l loads e x i s t i n g at the end of the previous time step, and from these incremental displacements the incremental member forces are c a l c u lated. The incremental displacements are then solved again i n the pre- sence of the revised a x i a l loads and a new set of member loads are c a l c u lated. This procedure i s continued u n t i l the d i f f e r e n c e between the a x i a l loads c a l c u l a t e d i n two successive i t e r a t i o n s reaches some acceptable tolerance. The rigorous i t e r a t i v e approach!'is, however, p r o h i b i t i v e l y time consuming and i t i s necessary to make an assumption in order to avoid 32 performing i t e r a t i o n s over each time step. In t h i s a n a l y s i s , the e f f e c t of geometric n o n l i n e a r i t i e s was r e s t r i c t e d to the assumption that the a x i a l load in each member remains constant over the e n t i r e earthquake history. That i s to say, the a x i a l loads due to the earthquake are as- sumed to be small compared to those due to g r a v i t y loads. This assump- t i o n causes the member s t i f f n e s s matrices to be constant over a time step and pseudo-linear a n a l y s i s based on superposition i s s t i l l valid. The geometric s t i f f n e s s matrix may be considered to be comprised of two components: (a) terms comprising the s o - c a l l e d P-A e f f e c t s ' r e s u l t i n g from 1 the f a c t that v e r t i c a l loads w i l l contribute a transverse component of force to a displaced column and (b) terms r e s u l t i n g fromttheuchange; int.the^bending s t i f f n e s s properties of;?a: member subjected to an a x i a l . l o a d . In the current a n a l y s i s , the columns are assumed to be sub- jected to a constant a x i a l compressive Toad r e s u l t i n g from the dead weight of the s t r u c t u r e , and the beams ace assumed to never be subjected to an a x i a 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 o o r s l a b , which constrains the beams with r e s pect to a x i a l deformation. In other words, a x i a l loads are c a r r i e d , l a r g e l y , by the s l a b . In d e r i v i n g the geometric s t i f f n e s s matrix f o r a column, the notation shown i n Figure 11 i s used. While thennotatdon and degrees of freedom used here are d i f f e r e n t from those used previously, the derived 33 column s t i f f n e s s matrix i s eventually transformed i n t o the previously used system shown i n Figure 8. 2.3.i Member State A Geometric S t i f f n e s s Matrix A closed form s o l u t i o n to the d i f f e r e n t i a l equation f o r members subjected to a x i a 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 i n Figure 11. •12 S _ M 3 --3:2 S x 6EIS 2 : 4L S, sym. -6L S 12S, 2 6L-.S-5 EI L V i • 2L S 2 2 2 li (2.37) -6LS 4L S 2 For a compressive a x i a l load 3 N , the s t a b i l i t y functions s are defined as <. _ sin c \ \ji b 1 _ S 3 in which ~ 1 2 *c, _ - ip(sin _ - ^ (1 - cos ^) 6d, - sin 2*, - i> cos \j>) 4^ <{> = 2 - 2 cos A!NL !EI c tJ; - s i n \> (2.38) T\I) (2.39) (2.40) 34 I f sine and cosine are expanded i n 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 redefined as i n (2.41) 1 - = 10 1 1 + 1 1 , 4_ 60 (2.41) l - 430 in which 1 _ <f> " .- " _ 60 NL fi" (2.42) S u b s t i t u t i o n of the approximate s t a b i l i t y functions of (2.41) into the f o r c e - d e f l e c t i o n r e l a t i o n (2.37) r e s u l t s in (2.43) 12 V. 6L 4L •t,112 -6L 6L 2L EI_ sym. . 3 12 2 -6L 4L 2 336 30L 3L 4L sym. --36 -3L 36 -L -3L 3L 2 4L 2 (2.43) which may be expressed as 1 = CK S - K ] 6 G (2.44) 35 where K g Kg = member bending s t i f f n e s s matrix in absence of a x i a l = member geometric s t i f f n e s s matrix loads I t should also be noted that the geometric s t i f f n e s s matrix i n (2.43) may also be derived in a " c o n s i s t e n t f i n i t e element" sense. This approach i s used by Clough and Penzien [7] and i s based on the r e l a t i o n K where ijj(x) G1j * ^,(x)/ <R(x) ;dx N (2.45) are the cubic hermitian i n t e r p o l a t i o n functions expressed in (2.46) ^(x)L L ; 4> U)L - 3x L + 2x 3 2 2 ; 3 3 3x L - 2x 9 2 <Mx)L xL 2 3 - 2x L + x 3 x (2.46) 2. - x L Thus an approximate method of expressing the member geometric s t i f f n e s s matrix i s found from (2.43) and may be expressed as the sum of two component contributions as shown in (2.47) 6 i/ - " G 30L K .1 3L 4L -6 -3L 3L -L sym. iO 0 sym. 6 -1 0 1 0 0 0 2 2 -3L 4L (2.47) 0 36 As the second component matrix represents the geometric s t i f f ness matrix commonly used to account f o r P-A e f f e c t s , the f i r s t component matrix may be considered to represent terms r e l a t i n g to the change i n bending s t i f f n e s s r e s u l t i n g from the compressive a x i a l load N. I t should be noted that the e f f e c t of shear d e f l e c t i o n s is^no^" included i n the geometric s t i f f n e s s matrix. 2.3.ii Member State B Geometric S t i f f n e s s Matrix Derivation of the geometric s t i f f n e s s matrix f o r members i d e a l ized to have a hinge at one end i s based upon the superposition of d i s placement cases. This procedure isoonj'y v a l i d i f the a x i a l load i s a constant independant of displacement due to bending. The superposition of the cases shown in Figures 12a and 12b r e s u l t in the case shown in Figure 12c wm«i:ch shows the member forces corresponding to the f i r s t c o l umn of the member state B geometric s t i f f n e s s matrix. S i m i l a r l y cases shown in Figures 13a and 13b may be superposed to produce Figure 13c which comprises the fourth column of Kg . The other terms of the geo- metric s t i f f n e s s matrix may be determined from symmetry. This d e r i v a t i o n i s based on the f a c t that higher order terms in tj) (with accompanying higher denominators) may be neglected. In view of t h i s assumption, t h i s d e r i v a t i o n should only be considered v a l i d f o r (j) < 1 ; however, t h i s i s generally the case i n p r a c t i c e . The f o r c e - d e f l e c t i o n r e l a t i o n f o r the e l a s t i c - p e r f e c t l y p l a s t i c component of a member in state B may be expressed as shown in (2.48) 37 (30-13(p) sym. 3EI L (30-d>) (2.48) 3 M„ -(30-13cp) 0 (30-13<J>) •(30-3<j>) (30-3*) (30-3cp)L' from which the geometric s t i f f n e s s matrix may be expressed as the sum of terms accounting f o r the change i n bending s t i f f n e s s matrix plus terms accounting f o r the P-A e f f e c t s as shown in (2.49) 1 6+<() N 0 sym. 0 0 *1 (30-cf))L -(6+cj)) 6L 2.3.iii 0 6+cj) 0 -6L 6L" sym. 0 (2.49) •1 0 1 0 0 0 0 Member State C Geometric S t i f f n e s s Matrix The geometric s t i f f n e s s matrix f o r a member in state C may be derived i n a s i m i l a r manner to that used f o r members i n state B. The r e s u l t i n g geometric s t i f f n e s s matrix f o r the e l a s t i c - p e 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 i n (2.50) 1 (6+cp) 6L _G 6L 2 sym. 0 (6+cj>) -1 (30-cj))L •(6+(J)) -6L 0 0". 0 0 0 0 sym. 0 1 0 0 0 (2.50) 38 2.3.iv Member State D, Geometric S t i f f n e s s Matrix The e f f e c t of a x i a l loads on the e l a s t i c - p e r f e c t l y p l a s t i c component of state D members w i l l only be in the form of P-A e f f e c t s . This may be accounted f o r by the geometric s t i f f n e s s matrix shown in (2.51) 1 II 0 0 sym. -1 0 1 0 0 2.3.V (2.51) 0 General Form of Member Geometric S t i f f n e s s Matrix As stated previously, the assumption has been made that only columns w i l l be subject to a x i a l loads. Using the notation of Table 2 and (2.'42), the geometric s t i f f n e s s matrix for a column subjected to a compressive a x i a l load N may be expressed f o r a l l states as shown in (2.52) 1 a. 0 0 0 N L I 0 K i a' a? CO u sym. P. 0 4 P, i i ' Li. a,i a 0 0 0 0 sym. 0 (2.52) -10 0 1 0 0 0 0 0 0 0 a' 0,2 0 0 0 0 0 0 J 0 39 in which a. = (6 + 4>) ( q a (%+6q )L a. (iO + 6q )L 2 + q) 3 2 3 (2.53) a. a a. P_ 30 p e 2 30 - * (2.54) % 30 - cp An approximation for the geometric s t i f f n e s s matrix which i s commonly used i n nonlinear dynamics i s to neglect the f i r s t matrix in (2.52). In t h i s case only the second matrix, which represents P-A e f - f e c t s , i s used to approximate geometric n o n l i n e a r i t i e s . This approxima- t i o n i s accepted ninto the computer program developed by the author. 2.4 Formation of the Reduced Frame S t i f f n e s s Matrix The member state matrices derived i n Section 2.2 are based upon an i d e a l i z a t i o n containing three degrees of freedom at each,member end. 40 Inclusion of a l l member matrices w i l l r e s u l t in a s t r u c t u r e s t i f f n e s s matrix containing three degrees of freedom at each j o i n t in the s t r u c t u r e . However i t i s d e s i r a b l e to reduce the number of degrees of freedom i n the structure s t i f f n e s s matrix as the e f f o r t required f o r the s o l u t i o n of the incremental-displacements varies accordingly. As pointed out i n Section 2.1, the s t r u c t u r e s t i f f n e s s matrix i s reduced to contain only three degrees of freedom per storey and a discussion of the methods used to r e duce these degrees of freedom i s now presented. The s t r u c t u r e i s viewed to be an assemblage of plane frames which allows f o r the s t i f f n e s s matrix of each plane frame to be obtained independently and then added into the structure s t i f f n e s s matrix. Several degrees of freedom may be eliminated i f a x i a l deformations of the beams are neglected in each frame. This assumption implies that the f l o o r s at each storey l e v e l are r i g i d with respect to in-plane deformation which i s reasonable f o r a f l o o r system of reasonable t h i c k 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 i n e x t e n s i b l e allowing for reduction of a l l v e r t i c a l degrees of freedom. However t h i s assump- t i o n has-been shown, by Weaver and Nelsonri[33L]]tdolead 'to- a s i z e a b l e overestimation of s t i f f n e s s , e s p e c i a l l y i n t a l l or slender buildings which have a high aspect r a t i o . In a d d i t i o n i t i s d e s i r a b l e to enforce v e r t i c a l c o m p a t i b i l i t y at corner columns common to two frames, e s p e c i a l l y for structures whose predominant displacement mode i s of a bending, rather than shear type. I f t h i s v e r t i c a l c o m p a t i b i l i t y i s not included 41 the undesirable r e s u l t shown in Figure 14(a) r e s u l t s i n which frames perpendicular to a p a r t i c u l a r motion do not contribute to resistance of the motion. Inclusion of the v e r t i c a l c o m p a t i b i l i t y condition causes the more r e a l i s t i c case shown in Figure 14(b) to r e s u l t . For these r e a - sons i t i s necessary to include the v e r t i c a l degrees of freedom at j o i n t s on columns common to more than one frame. These degrees of f r e e - domTjwiiflI hereafter be referred to as ' c o m p a t i b i l i t y ' degrees of freedom. 1 Thus the frames are i n i t i a l l y assembled to contain the degrees of freedom shown in Figure 15a and then each frame s t i f f n e s s matrix i s s t a t i c a l l y condensed to contain the reduced number of degrees o f freedom shown i n Figure 15b. In order to condense the frame s t i f f n e s s matrix each member matrix must be added i n t o the frame matrix in a unique manner. The frame matrix must be p a r t i t i o n e d in terms of the f o l l o w i n g three d i s t i n c t types of degrees of freedom: (1) the horizontal t r a n s l a t i o n s of the storey l e v e l s (2) the c o m p a t i b i l i t y degrees o f freedom x_ (3) other degrees of freedom ji §_ which are independant of a l l other frames ( ' o t h e r ' degrees of freedom). The p a r t i t i o n e d frame matrix r e l a t i o n s h i p i s shown in (2.55) H x X 0 K 21 K 22 6 (2.55) 42 Forces corresponding to S_ w i l l be 0_ since in t h i s dynamic a n a l y s i s only horizontal i n e r t i a forces are considered. I t should be noted that i n Section 3.5 t h i s procedure w i l l . b e repeated f o r the case where unbalanced forces procedure i s required. e x i s t i n which case a d i f f e r e n t condensation A However, in the absence of unbalanced loads the f o l l o w i n g procedure i s v a l i d . As matrix K 22 may be stored i n banded form i t i s s u i t a b l e for decomposition i n t o the product o f a lower t r i a n g l e matrix times i t s transpose using Choleski 's method K = L l F F (2.56) J c 22 c Operations on the lower p a r t i t i o n of (2.55) w i l l r e s u l t i n the r e l a t i o n s h i p 6 = eLJ/^l/Y^p < h x > (2.57) T which can be used in conjunction with the upper p a r t i t i o n to give the relationship ij} = [K l x - (C\ 2 l ) J (L - 1 K 2 1 )] {^} F in which case the reduced frame s t i f f n e s s matrix terms of only tworrmatrices K and (L K ) 21 by computer and r e q u i r i n g a minimum of storage: (2.58) K c may be express in in a form e a s i l y soluble 43 K 2.5 F = K n, K ( " L 2 1 )_ (L K 2 l ) (2.59) ( Formation of the Reduced Structure S t i f f n e s s Matrix The reduced frame s t i f f n e s s matrices must be added into the structure s t i f f n e s s matrix i n a unique manner. T degrees of freedom are denoted by l i z e d forces and £ < C[ R_ e_ >^ < g_ r 9 > The reduced structure with corresponding as shown in Figure 16b. genera- For t h i s system q_ are vectors representing horizontal t r a n s l a t i o n s in two mutually perpendicular d i r e c t i o n s (one each per storey) and 9_ i s a vector of r o t a t i o n s about the structure o r i g i n . For a p a r t i c u l a r frame the f o l l o w i n g q u a n t i t i e s may be defined I I I q fl = d i r e c t i o n cosine of frame in q direction = = d i r e c t i o n cosine of frame i n r d i r e c t i o n perpendicular distance from the structure o r i g i n to the frame. The horizontal t r a n s l a t i o n s of a frame may be expressed in terms of the structure co-ordinates q , r and 9 from the e a s i l y v i s - ualized r e l a t i o n s h i p shown in (2.60) r q a. q + I r r + I„ 9 9 — — = < cI i„ > i £- r (2.60) and the horizontal forces at each storey of a frame may be expressed in 44 terms of the s t r u c t u r a l generalized forces p r i n c i p l e of v i r t u a l Q , R , 9 by applying the displacements. H a H < H R . (2.61) > 0 > > I f the reduced frame f o r c e - d e f l e c t i o n r e l a t i o n s h i p of (2.58) i s expressed in p a r t i t i o n e d form as shown in (2.62) H / y ' — K hh ', hx K xh ' xx K (2.62) = x_ K then the i n t r o d u c t i o n of equations (2.60) and (2.61) i n t o the p a r t i t i o n e d frame s t i f f n e s s r e l a t i o n s h i p w i l l allow i t , with appropriate manipulat i o n s , to be expAessedsfn ;!temsro€-!:4heT.s-t-K uetuFal ,co'-ot?dinates.'.- q » r : ! ahd-.;"6 nas shown i n (2.63) 4 I 2 sym. q q 1.'' R- £ x_ I C Sh 2 .It \ in ! G r l r Sh xh T e 0 K r hx K, hx (2.63) hh xh When the contributions of a l l frames have been added i n t o (2.63) the unreduced structure f o r c e - d e f l e c t i o n equation i s obtained i n 45 terms of the degrees of freedom shown in Figure 16(a). I t i s now p o s s i - ble to use s t a t i c condensation to reduce the c o m p a t i b i l i t y degrees of freedom. The generalized forces forces are ignored. x_ must equal fJ as v e r t i c a l inertia This allows the structure s t i f f n e s s matrix to be s t a t i c a l l y condensed using the same procedure as f o r frame matrices. The f i n a l structure s t i f f n e s s matrix i s shown in the s t r u c t u r e f o r c e - d i s placement equation (2.64) Q R in which = K K s s (2.64) - = [K^ - (L"\ ) (L \ ^ ) \ T 21 (2.65) 46 CHAPTER 3 NONLINEAR MATERIAL BEHAVIOR AND ENERGY CALCULATION One of the major problems associated with a nonlinear concerns the member state determination phase of the a n a l y s i s . analysis 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 s t r u c t u r e s t i f f n e s s matrix need only be c a l c u l a t e d once and member forces c a l c u l a t e d as desired. However, in a nonlinear a n a l y s i s member forces must be c a l c u l a t e d every time step i n order to check whether any members have y i e l d e d or subsequently reversed. This phase of the a n a l y s i s is known as the member state determination phase. I f a member state changes during a time step, the structure s t i f f n e s s matrix must be r e c a l c u l a t e d and unbalanced forces r e s u l t i n g from the n o n l i n e a r i t y must be accounted f o r . This chapter o u t l i n e s the methods used to perform the above c a l c u l a t i o n s . 3.1 C a l c u l a t i o n of Member Deformations The b i l i n e a r member properties assumed i n the fimodel require r that member deformations be c a l c u l a t e d incrementally and that the t o t a l member deformations at any time be c a l c u l a t e d by superposing the i n c r e mental deformations which have occurred up to that time. Solution of the e q u i l i b r i u m equation (2.1) y i e l d s 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 deformations. This expansion i s done in two steps: f i r s t l y the incremental displacements of the c o m p a t i b i l i t y degrees of freedom are c a l c u l a t e d and secondly the incremental displacements of degrees of freedom relevant only to a p a r t i c u l a r frame are c a l c u l a t e d for each frame i n t u r n . These c a l c u l a t i o n s y i e l d a vector of incremental j o i n t displacements from which the incremental member deformations may be selected f o r each frame. The incremental displacements of the c o m p a t i b i l i t y degrees of freedom are c a l c u l a t e d from the structure force-displacement r e l a t i o n s h i p (2.63) i n which the forces corresponding to the c o m p a t i b i l i t y degrees of freedom are set equal to fJ as in (3.1) i.e. , AF < ; > Af 21Sc _ = < 0 in which T K 1 K 2 2S Ax AF, Af = incremental generalized f o r c e s , displacements Ax = incremental displacements of the c o m p a t i b i l i t y degrees of freedom and the subscript s r e f e r s to submatrices of the structure stiffness matrix in order to avoid confusion with those of the frame s t i f f n e s s matrices. Using the Choleski decomposition technique described in Section 2.4 the incremental c o m p a t i b i l i t y displacements may be expressed in the f o l l o w i n g form: Ax = - LJ" 1 [L" 1 K _ ] Af 21 s (3.2) 48 The vector of incremental s t r u c t u r a l displacements < Af Ax > can be transformed into a vector of incremental displacements f o r frame i : < Ah-. Ax.j > in which f o r frame T Ahj^ = and frame Ax. I Ag_ qi + i I r i AT + I A0 Qi (3.3) are the c o m p a t i b i l i t y degrees of freedom corresponding to i . Solution of the incremental displacements of a l l other degrees of freedom f o r frame i , A6- i s achieved by usage of (2.57) w r i t t e n in 9 terms of incremental values as = " i a where the subscript Fi (L" 1 K 21 )J F < ^ i ^ j . > T ( 3 , 4 ) r e f e r s 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 , f o r each frame. The incremental member deformations, in terms of three degrees of f r e e 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 member end forces may be c a l c u l a t e d . 49 3.2 C a l c u l a t i o n of Member End Forces I t i s convenient to introduce a new co-ordinate system and no- t a t i o n as shown i n Figure 17 i n order to express the incremental member forces i n as concise a manner as p o s s i b l e . The t o t a l incremental member end forces are comprised of the sum of forces in the e l a s t i c component and those i n the e l a s t i c - p e r f e c t l y p l a s t i c component and these two components are solved separately. In Section 2.3 the geometric s t i f f n e s s matrix i s derived and expressed as two separate components: terms accounting f o r P-A e f f e c t s and terms accounting f o r the change in bending s t i f f n e s s properties of members subject to a x i a l loads. In the current analysis only the c o n t r i - bution of P-A e f f e c t s i s considered in formation of the geometric s t i f f ness matrix, and for s i m p l i c i t y the d e r i v a t i o n of column force r e l a t i o n ships i s also r e s t r i c t e d to geometric n o n l i n e a r i t i e s in the form of P-A effects. The incremental a x i a l force i s c a l c u l a t e d f o r columns only, as beams are constrained with respect to a x i a l deformation. mental a x i a l shortening of a member i s load in a member, then the incremental a x i a l AN , i s expressed in (3.5) AN = in which 6 I f the i n c r e - ~ 6 A = column c r o s s - s e c t i o n a l area E = Young's modulus L = column length (3.5) 50 Incremental moments and shears in amember may be expressed in terms of incremental member end r o t a t i o n s . 9 i s expressed as the j o i n t r o t a t i o n as shown in (2.15). The member end r o t a t i o n co minus the chord r o t a t i o n y For a beam with r i g i d end stubs as shown in Figure 10 i t i s desirable to c a l c u l a t e the member end forces at the ends of the f l e x i b l e section of the beam. In t h i s case, the chord r o t a t i o n must be modified by i n c l u d i n g the t r a n s l a t i o n of the member ends caused by r o t a t i o n of the r i g i d stubs. Using the notation shown in Figure 10 the chord r o t a t i o n of the f l e x i b l e beam may be expressed as in (3.6) and a l l f u r t h e r member properties r e f e r to the f l e x i b l e section of the beam. 6.-6. = Y in which 6., 6. 1 J y j -^-j; i a [j " . b " r; j to w i ( 3 i , j ' c\ 6 ) = v e r t i c a l displacements at ends t o t a l beam of the = chord r o t a t i o n of the f l e x i b l e section of the beam. In c a l c u l a t i o n of incremental member forces i t i s also desirable to define the q u a n t i t i e s shown in (3.7) K a L (1 + g) (3.7) k K in which b = 2Ej,.(] - 9/2) L (1 + g) I = member moment of i n e r t i a g = member shear d e f l e c t i o n f a c t o r 51 Incremental moments in the e l a s t i c component are expressed i n terms of incremental member end r o t a t i o n s in A A in which p M M (3.8) A e. IEL (3.8) A 2EL e. equals a f a c t o r describing the c o n t r i b u t i o n of the e l a s t i c component to the t o t a l member, i . e . the y i e l d e d slope divided by the i n i t i a l slope of the assumed b i l i n e a r moment-rotation r e l a t i o n s h i p , and the s u b s c r i p t EL i n d i c a t e s forces in the e l a s t i c component. The incremental moments i n the e l a s t i c - p e r f e c t l y p l a s t i c component must be expressed separately f o r each possible member s t a t e . The possible member states are described in Table 2. For member state A, incremental forces i n the e l a s t i c - p e r f e c t l y p l a s t i c component are solved by (3.9) A M k lEP (1 - p) A M in which the subscript p l a s t i c component. a k. b A 0 l > k. b k A 0 (3.9) EP A EP i n d i c a t e s forces i n the e l a s t i c - p e r f e c t l y a 2 52 For member state B, A6 A MIEP \ (1 - P) A M2EP a J B k (3.10) A 6, a For member state C A M iEP k y = a " k b a / k \ (3.11) (i - p) A ' EPj 2 8. and f o r member state D 0 A MiEP 0 (1 - p) A M2EP > •< 0 A6 0 V. 2 (3.12) J These incremental moments are c a l c u l a t e d assuming that the member state at the end of the previous time step remains unchanged during the e n t i r e current time step. However i t i s possible that the member s t a t e may a c t u a l l y change during the time step. For instance the a d d i - t i o n of the incremental moments c a l c u l a t e d over a time step to the t o t a l moment e x i s t i n g at the beginning of the time step f o r 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 t h i s case the member end a c t u a l l y y i e l d s during the time step and the incremental moments i n the e l a s t i c - p e r f e c t l y p l a s t i c component are i n correct. The methods used in compensating f o r t h i s e r r o r are o u t l i n e d in the f o l l o w i n g sections. 3.3 Yield Criteria I t i s necessary that the conditions for which a member end may be considered to have y i e l d e d or subsequently unyielded be accurately defined. There are three d i f f e r e n t y i e l d c r i t e r i a which must be used i n this analysis. Beam members are constrained against a x i a l deformation hence the member may be considered unyielded u n t i l the t o t a l moment surpasses the yieflld moment at the member end. Columns, however, are sub- jected to a x i a l loads which necessitates consideration of a x i a l - f l e x u r a l i n t e r a c t i o n i n determining whether the end has y i e l d e d . In a d d i t i o n , columns which are common to more than one frame w i l l have moments about both p r i n c i p a l axes and the i n t e r a c t i o n of the two components of moment plus a x i a l loads must be considered. r e l a t e d to the f o l l o w i n g three cases: The three y i e l d c r i t e r i a are thus - (1) beam members, (2) columns subject to bending about one axis only ( u n i a x i a l columns), (3) columns subject to bending about two axes ( b i a x i a l columns). 54 3.3.i Beam Members In general there w i l l be four d i f f e r e n t y i e l d moments f o r each beam member: a p o s i t i v e 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 ini- t i a l l y equal, but f o r a reinforced concrete beam the r e i n f o r c i n g s t e e l 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 p a r t i c u l a r member end. In a d d i t i o n , q u a n t i t i e s of steel at one endsof" the~beam are commonly d i f f e r e n t from those at the c other end of the beam. Hence f o r beams constructed from reinforced con- crete i t i s necessary to consider four d i f f e r e n t values f o r 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 o o r s l a b s . The member end mo- ments c a l c u l a t e d during the earthquake are those caused s o l e l y by the earthquake, in absence of i n i t i a l moments caused by dead loads. the i n i t i a l Hence moments must be subtracted from the actual y i e l d moments to express the e f f e c t i v e y i e l d moment i n terms of the moment, in a d d i t i o n 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 p o s i t i v e or negative moment of 100 k i p - f e e t . If the dead load moment i s 40 kip feet in the negative d i r e c t i o n then i t may be said that the p o s i t i v e y i e l d moment i s 140 kip feet while the negative y i e l d moment i s 60 kip f e e t . 55 3.3.ii Uniaxial Columns The y i e l d moment for u n i a x i a l columns i s a function of the c u r rent value of the a x i a l load in the member. I t i s common p r a c t i c e i n reinforced concrete construction to use e i t h e r c i r c u l a r or square columns with v e r t i c a l reinforcement located symmetrically about two axes. The p o s i t i v e and negative y i e l d moments w i l l be equal for these columns, so t h i s r e s t r i c t i o n , although not v i t a l , i s accepted i n the current a n a l y s i s . A t y p i c a l a x i a l - f l e x u r a l i n t e r a c t i o n curve i s shown in Figure 18 f o r columns subject to the above r e s t r i c t i o n . I t may be seen from Figure 18 that there are three equations which are required to express the p o s i t i v e or negative y i e l d moment in terms of the a x i a l load and for N > 0 (tension) for N < 0 and (compression) for N < N (3.13) M. N > N g (compressfori) in which N (3.14) B (3.15) 'B a x i a l load (negative f o r compression) ultimate compressive load (negative) N t ultimate t e n s i l e load ( p o s i t i v e ) N B balanced load (negative) y i e l d moment in presence of a x i a l load N 56 M y Mg = y i e l d moment with = balanced moment. 0 a x i a l load The actual i n t e r a c t i o n surface for a reinforced concrete c o lumn i s generally a smooth curve, hence the i n t e r a c t i o n curve shown in Figure 18a c o n s i s t i n g of s t r a i g h t l i n e approximations of the curve i s not exact. Interaction curves f o r other materials may s i m i l a r l y be ap- proximated by appropriately d e f i n i n g the points described above. 3.3.iii B i a x i a 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 column and the a x i a l load. For b i a x i a l columns, the y i e l d surface i s a three dimensional surface e x i s t i n g i n a space defined by the moment about one a x i s , the moment about the other axis and the a x i a l load. A view of t h i s i n t e r a c t i o n surface perpendicular to the plane in which the moment i n one d i r e c t i o n equals zero w i l l provide the i n t e r a c t i o n diagram shown i n Figure 18a. The same holds for a view of the i n t e r a c t i o n s u r - face perpendicular to the plane i n 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 a x i a l load i s represented in a s o - c a l l e d b i a x i a l i n t e r a c t i o n diagram. In a plane frame analysis the e f f e c t of bending out of plane, i . e . b i a x i a l 57 e f f e c t s , are ignored. However i t has been stated by Degenkolb [8] that these b i a x i a l e f f e c t s are important and research by Selna; M o r r i l l and Ersoy [26] on the f a i l u r e of the O l i v e View P s y c h i a t r i c Day C l i n i c supports t h i s conclusion. The i n c l u s i o n of b i a x i a l e f f e c t s causes a column to y i e l d in a given d i r e c t i o n at a lower moment due to the presence of a moment in the other d i r e c t i o n . Much research has been done concerning the shape of the b i a x i a l i n t e r a c t i o n diagrams f o r a v a r i e t y of column s e c t i o n s . The shape of the curve for r e i n f o r c e d concrete columns i s a function of many v a r i a b l e s such as r e i n f o r c i n g steel arrangement, the s t e e l y i e l d strength, the concrete compressive strength, the dimensions of the column cross s e c t i o n , etc. In an attempt to describe the b i a x i a l i n t e r a c t i o n diagram, B r e s l e r [4] proposed the f o l l o w i n g formula (3.16) where M . M x y y i e l d moments about the x, y axes M = xo yieldnmoment about the x axis with the y axis o moment about M = yo y i e l d moment about the y axis with the x axis o moment about m , n = constants determined e m p i r i c a l l y and depending upon column p r o p e r t i e s . Parme; Nieves and Gouwens.[24] proposed that the f o l l o w i n g formula be used to define the b i a x i a l i n t e r a c t i o n diagram f o r rectangular 58 reinforced concrete columns log .5 / M /M \ log 3 in which the f a c t o r log .5 \ log 3 was determined a n a l y t i c a l l y as a function of 3 several column p r o p e r t i e s . In order to reduce the number of variables required to define the b i a x i a l i n t e r a c t i o n diagram i t i s necessary to make an assumption. In t h i s a n a l y s i s , i t has been assumed that the b i a x i a l columns are such that values of [4] and 3 m and equal to n —- equal to 2 in the formula proposed by Bresler in the formula proposed by Panne; Nieves and rr Gouwens^[24-]]are appropriate. This assumption makes t h e - b i a x i a 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 f o r b i a x i a l columns subject to the above assumption i s shown in Figure 18c. While t h i s 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 r e i n f o r c e d concrete columns and r e s u l t s obtained using t h i s y i e l d surface should resemble the actual b i a x i a l behavior of these columns. 59 3.4 Member State Determination This phase of the a n a l y s i s 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 f e r e n t y i e l d c r i t e r i a used f o r beam mem- bers; columns which are subject to bending about one axis ( u n i a x i a l columns); and for b i a x i a l columns, a d i f f e r e n t technique i s used f o r these three types of members. 3.4.i Beam Members The method used in the member state determination phase i n - volves c a l c u l a t i n g the f r a c t i o n of the time step f o r which the i n i t i a l member state remains unchanged. I f t h i s f r a c t i o n i s equal to one the i n i t i a l member state existed f o r the e n t i r e time step and c a l c u l a t i o n s for t h i s member are c o r r e c t . However, i f t h i s f r a c t i o n i s less than one, the i n i t i a l member state must have changed during the time step, and c a l c u l a t i o n s f o r t h i s member are i n c o r r e c t as a r e s u l t of t h i s change in state. A change in state w i l l h e r e i n a f t e r be r e f e r r e d to as an ' e v e n t ' . I t i s convenient to introduce an assumption i n order to c o r r e c t the calculated.incremental forces. The c a l c u l a t e d incremental d i s p l a c e - ment; is'assuriied to jbe correct;-''.whether an event occurred or not. this'a'ssumptio'n.is"not•.rigorous.?oif While the time-step i s kept' small errors i h t r o d u c e d f b y t h i ' 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 r a c t i o n of the time step u n t i l an event (hereinafter referred 60 to as the event f a c t o r ) i s c a l c u l a t e d according to (3.19) M. y in which y = - (3.19) 1 = the event f a c t o r f o r the member end under consideration = the y i e l d moment of the e l a s t i c - p e r f e c t l y p l a s t i c component at the member end = (1-P) = the y i e l d moment of the member end M y =;:. the i n i t i a l moment i n the e l a s t i c - p e 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 - p e r f e c t l y p l a s t i c component at the member end. I f a value of greater than 1.0 i s c a l c u l a t e d by (3.19), y 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 t h i s analysis i s to c a l c u l a t e the energy d i s s i p a t e d in the p l a s t i c hinge and to remove the hinge i f the energy d i s s i p a t e d during a time step i s c a l c u l a t e d to be less than zero. The energy d i s s i p a t e d at a p l a s t i c hinge i s c a l c u l a t e d according to (3.20) A E in which AE D D * M EP A a (3.20) incremental energy d i s s i p a t e d at the hinge M'EP , the moment in the e l a s t i c - p e 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 r o t a t i o n at the hinge. 61 Equations used to c a l c u l a t e the incremental p l a s t i c r o t a t i o n are presented in Section 3.6. I f the energy d i s s i p a t e d i n the.ihinge over a time step i s less than zero a value of u = 0 i s assigned, while i f the incremental energy d i s s i p a t e d i s greater than zero, \i i s set equal to 1. Hence the member forces and state are determined i n an i t e r a t i v e manner in which the member state may change one or more times per time step. I f the event f a c t o r at e i t h e r 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 f a c t o r at thateerid. A new-member event f a c t o r i s then c a l c u l a t e d f o r the revised member s t a t e . This procedure i s repeated u n t i l the sum of member event factors equals one. 3.4.ii Uniaxial Columns For members subject to a x i a l loads, the member state determinat i o n phase i s performed based upon the y i e l d moment i n the presence of a x i a l loads e x i s t i n g at the beginning of the time step. However the a x i a 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 i s inserted i f the magnitude of the moment in the e l a s t i c - p e 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 - p e r f e c t l y p l a s t i c component i s less than the revised y i e l d moment. 3.4.iii B i a x i a l Columns For columns subject to bending about two axes, i n t r o d u c t i o n of a hinge at a column end w i l l a f f e c t the behavior of the column in each d i r e c t i o n . That i s , a hinge cannot e x i s t in the column with r e s - pect to bending about one axis without the i n c l u s i o n of a hinge with respect to bending about the other a x i s . Determination of the state of b i - a x i a l columns must include current values d e s c r i b i n g the member in both directions. The b i a x i a l i n t e r a c t i o n diagram of Figure 18b i s assumed to be in the shape of an e l l i p s e . For t h i s e l l i p t i c a l y i e l d r e l a t i o n s h i p , the event f a c t o r f o r an e l a s t i c end i s expressed in (3.21) = a i y in which a - a a 3 1 2 = a 1 " yi M A M a 2 3 a a V . a EPi " M M - EPj yj A " E p + / a 4 J / a: - (a 5 5 v 1 a 4 - a a) 2 3 ' / 0 0 , x (3.21) 63 a5 AM, , EP' M'EP M, y and the subscripts a = the incremental, t o t a l moments in the e l a s t i c p e r f e c t l y p l a s t i c component = the y i e l d moment in the e l a s t i c - p e r f e c t l y p l a s t i c component i + a 2 = and j i n d i c a t e two perpendicular axes. Derivation of (3.21) i s given i n Appendix A. The event f a c t o r f o r a p l a s t i c end i s found in a manner s i m i l a r to that used f o r e l a s t i c ends. The energy d i s s i p a t e d at the hinge i s c a l c u l a t e d and i f the energy i s less than zero the hinge i s removed and an event f a c t o r of zero i s assigned. I f the energy d i s s i p a t e d i s greater than zero the hinge remains and an event f a c t o r of one i s assigned. The t o t a l energy d i s s i p a t e d at a p l a s t i c hinge i s the sum of the energies d i s s i p a t e d in the two perpendicular d i r e c t i o n s , as shown in (3.22) AE D in which the subscripts = M i and E p i Aou + M j E p j Aa. (3.22) i n d i c a t e values in two perpendicular directions. The i t e r a t i v e procedure used f o r b i a x i a l columns i s i d e n t i c a l to that used f o r u n i a x i a l members; that i s the procedure i s continued u n t i l the sum o f event f a c t o r s f o r a time step equals one. In b i a x i a l columns the a x i a 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 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 e d 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 s t a t e s . 19a shows the unbalanced moment, Figure ,*?which r e s u l t s when a member end y i e l d s and Figure 19b shows the unbalanced moment r e s u l t i n g 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 p o s i t i o n 1 at the beginning of the time step. The member end rotates an amount s i t i o n 2 based upon the i n i t i a l A9 which places the end a t po- state of the end. However as movement from p o s i t i o n 1 to p o s i t i o n 2 d i c t a t e s the occurrence of an event, the member end i s revised to be in p o s i t i o n 3. The d i f f e r e n c e in moment be- tween positions 2 and 3 i s the unbalanced moment at the j o i n t which must be accounted f o r i n order to maintain e q u i l i b r i u m . Unbalanced forces also r e s u l t at an end when an event occurs at the other end. In general, the unbalanced moments f o r a member equal the d i f f e r e n c e between the moments c a l c u l a t e d based upon i n i t i a l condi- tions and the moment c a l c u l a t e d 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 in which Mrp,M -j F P * ^EPj < - ' 3 2 4 = moment i n e l a s t i c - p e r f e c t l y p l a s t i c component at the end of the time step = event f a c t o r f o r i t e r a t i o n number = incremental moment i n the e l a s t i c - p e r f e c t l y p l a s t i c component f o r i t e r a t i o n j = number of i t e r a t i o n s occurring f o r the member over the time step. J n EPi + moment i n e l a s t i c - p e r f e c t l y p l a s t i c component at the beginning of the time step 1 ANLp. + 1 M = + y. EPi = j Unbalanced forces may also r e s u l t at the p l a s t i c hinges of members subject to a x i a l loads. These forces r e s u l t from the f a c t that a change i n a x i a l load over a time step may cause an excursion outside of the y i e l d surface. C a l c u l a t i o n of unbalanced forces r e s u l t i n g from t h i s e f f e c t i s shown i n Figure 19(c). The change'.in the axiaTCforce i n the member moves i t from p o s i t i o n 1 to p o s i t i o n 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 M u to thermember end which w i l l bring i t to the admissable p o s i t i o n 3. Unbalanced shear forces w i l l also r e s u l t f o r each member i n which unbalanced moments occur. Unbalanced shear forces are c a l c u l a t e d from simple e q u i l i b r i u m as shown i n (3.25) K V in which L = U = E P u i member length E P L "J (3.25) 66 and the subscript j u indicates unbalanced forces and subscripts i n d i c a t e values at ends i and 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 t h i s method the unbalanced forces are reapplied to the revised structure and incremental displacements and member forces caused by the a p p l i c a t i o n 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 r e s u l t from a p p l i c a t i o n of the f i r s t set of unbalanced f o r c e s , and t h i s set i s then a p p l i e d . This i t e r - a t i v e method i s continued u n t i l the c a l c u l a t e d set of unbalanced forces become n e g l i g i b l e . In theory the Newton-Raphson method could also be applied in t h i s dynamic case, however i t s use would require an i t e r a t i o n f o r each time step in which an event occurs which i s not economically f e a s i b l e . An approximate method of t r e a t i n g these unbalanced forces i s to apply them in the following 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 e r r o r introduced by t h i s assumption should be n e g l i g i b l e . 3.5.i Reduction of the Unbalanced Force Vector Addition of the unbalanced member forces i n t o a vector f o r each frame follows standard procedures based on the frame code numbers. This w i l l r e s u l t in a load vector containing the unbalanced forces at each degree of freedom for each frame. The unbalanced load vector may 67 T be p a r t i t i o n e d to the form in which < H ! A ! X > IH unbalanced forces in the horizontal d i r e c t i o n A unbalanced forces at the c o m p a t i b i l i t y degrees of freedom X = unbalanced forces a t a l l other degrees of freedom From equation (2.57) i t was shown that 6 L^" [L"V _] 1 = 2J F <hjx> T (2.57) which expresses displacements of the "other" degrees of freedom in terms of the displacements of the horizontal and c o m p a t i b i l i t y degrees of freedom. Subscript F indicates matrices for the p a r t i c u l a r frame. V i r t u a l work equations can then be used to show that ~\ r - JJL> .U c( forces 1 ] F A in which a s u p e r s c r i p t Lf-) i£ * (3.26) i n d i c a t e s unbalanced forces r e s u l t i n g a f t e r _X have been s t a t i c a l l y condensed. i Forces forces Q , R and H_. can be expressed in terms of the generalized e_. from equation (2.61) "ai - H T r -' CD| ^6 I (2.61) 68 Hence f o r 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 i n t o t h i s vector, the transformation to generalized coordinates may be performed as shown in (3.27) 4 3.5.ii j - l R E(L K )^ L 2 1 - i s 1 ] A (3.27) M o d i f i c a t i o n of E q u i l i b r i u m Equation Unbalanced forces must be accounted for in order to maintain equilibrium. The e f f e c t of unbalanced forces occurring during a p a r t i c u - l a r time step should be accounted f o r by performing an i t e r a t i v e procedure within the time step. In the current analysis unbalanced forces are allowed to e x i s t f o r the time step during which they occur but are accounted f o r in the f o l l o w i n g time step. The incremental displacements in the time step f o l l o w i n g the occurrence of an event are solved by using the e q u i l i b r i u m equation shown in (2.1) modified by adding the reduced unbalanced load vector F £ M Ax + C 3.5.iii to the pseudo-load vector as f o l l o w s . B Ax + k B Ax = M A x„ + F (3.28) M o d i f i c a t i o n of Equations Used to Expand the Incremental Displacement Vector The theory presented in Section 3.1 used to expand the i n c r e 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 f a c t that forces e x i s t i n g at c o m p a t i b i l i t y and ' o t h e r ' frame degrees of freedom equal zero. However, unbalanced forces may r e s u l t at these degrees of freedom d i c t a t i n g m o d i f i c a t i o n of the equations. The incremental s t r u c t u r a l force-displacement r e l a t i o n s h i p shown in (3.1) and modified to include unbalanced forces i s shown in (3.29) K AF K AF. uc in which AF : us uc K Af T 21S (3.29) . K 21S Ax 22S i s the vector of unbalanced forces occurring at the com3 p a t i b i l i t y degrees of freedom. Equation (3.29) may be used to express incremental d i s p l a c e ments of the c o m p a t i b i l i t y degrees of freedom in terms of the increment a l generalized displacements in the presence of unbalanced forces at the c o m p a t i b i l i t y degrees of freedom in a form shown in (3.30) ^ • . CL L;I F S T-i UC c Li s] (3.30) S i m 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 e x i s t a t ' o t h e r ' degrees of freedom. The incremental displacements of these ' o t h e r ' degrees of freedom may be expressed in terms of incremental 70 displacements of the frame h o r i z o n t a l and c o m p a t i b i l i t y degrees of freedom as shown i n (3.4) i f no unbalanced forces e x 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 in which the subscripts t r i c e s f o r frame i Fi ] _ l F i u o i " Fi « ' L and Fi 1 L 1 | C .iVl < A A Ax. > T (3.31) i n d i c a t e values f o r vectors and ma- respectively. Hence the expansion of the incremental displacements i s performed according to equations (3.2) and (3.4) f o r time steps in which no unbalanced forces occur and (3.30) and (3.31) f o r time steps in which unbalanced forces a c t . 3.6 C a l c u l a t i o n of Incremental Member P l a s t i c Rotations The most important parameter in the a n a l y s i s of s t r u c t u r a l r e s - ponse to earthquakes i s the amount of p l a s t i c r o t a t i o n r e s u l t i n g i n each member end. P l a s t i c r o t a t i o n , which may be converted i n t o the commonly used d u c t i l i t y f a c t o r , i s c r i t i c a l f o r r e i n f o r c e d concrete members and provides an i n d i c a t i o n of s t r u c t u r a l damage or f a i l u r e r e s u l t i n g from an earthquake. The amount of p l a s t i c r o t a t i o n occurring at a p l a s t i c hinge depends on the s t a t e of the other member end. For members with a hinge at each end, the t o t a l r o t a t i o n 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„ • = 2 in which A a , Aa Aw 2 - Ay 1 2 = incremental p l a s t i c r o t a t i o n s at ends 1, 2 respectively Aw , Aco = incremental j o i n t r o t a t i o n s at ends 1 and 2 = incremental chord r o t a t i o n . 1 2 Ay For members with only one hinge, the incremental p l a s t i c rotat i o n at the hinge over a time step i s shown in (3.33) Aa. = (Aa). - Ay) + c(Au>. - Ay) (3.33) Aa, J in which c = = 0 the carry-over f a c t o r c and the subscript hinge and subscript i - <K$ (3-34) i n d i c a t e s the member end containing the p l a s t i c j indicates the e l a s t i c member end. The carry-over f a c t o r shown in (3.34) i s c a l c u l a t e d ignoring the change in bending s t i f f n e s s caused by a x i a l loads, hence i t i s v a l i d only f o r an a n a l y s i s in which geometric n o n l i n e a r i t i e s may be approximated accurately by j u s t P-A e f f e c t s . 72 3.7 C a l c u l a t i o n of Energy Quantities One of the most important methods of gaining an understanding of the nonlinear response of structures to earthquakes i s to study the v a r i a t i o n of the sundry forms of energy in the structure with respect to time. While t h i s valuable study i s conspicuously absent in the l i t e r - ature, several researchers such as Goel and Berg [11], Hanson and Fan [14] and Pekau; Green and Sherbourne [25] have included an energy analy- s i s on nonlinear frames subject to earthquake e x c i t a t i o n . The c a l c u l a t i o n of energies can also be used to check on the accuracy with which the model i s being analyzed. This check i s in the form of an energy balance in which the t o t a l energy input by the e a r t h quake into the structure should equal the sum of the f o l l o w i n g q u a n t i t i e s : (1) k i n e t i c energy of the structure (2) recoverable s t r a i n energy stored in the structure (3) t o t a l energy d i s s i p a t e d by viscous (4) t o t a l energy d i s s i p a t e d by h y s t e r e t i c e f f e c t s damping (non-recoverable s t r a i n energy) and t h i s energy balance i s performed at the end of every time step. Note that "energy input" as defined herein may be p o s i t i v e or negative, i . e . feedback to the ground i s included in ttois term. 73 3.7.i Energy Input into Structure by Earthquake The energy input during a time step i s found by i n t e g r a t i n g the product of the base shear and the ground v e l o c i t y with respect to time. In order to perform t h i s i n t e g r a t i o n i t i s necessary to make the assumption that the base shear force and the ground a c c e l e r a t i o n both vary 1 i n e a r l y over a time step. The d e r i v a t i o n of (3.35) which i s used to c a l c u l a t e the energy input in a time step by one component of the earthquake i s given i n Appendix B. 3a A E IN T = [ F o { 3 v o + A t + a 4 ( 1 ) } 5a + F i'« { 3 v o + A t ( 4 + 3a > }] (3.35) in which At • 6' i F F a , a = ^ step = a s e Q s n e a r f° r c e a t the beginning, end of the time = ground a c c e l e r a t i o n at the beginning, end of the time step = ground v e l o c i t y at the beginning of the time step. 1 v length of time step This c a l c u l a t i o n must be performed for each component of the earthquake. In the current a n a l y s i s , the r o t a t i o n a l component of the earthquake i s not included hence energy i s input only by the two mutually perpendicular t r a n s l a t i o n a l components. I t should be noted that the incremental energy input may be e i t h e r p o s i t i v e o r negative. A negative value i n d i c a t e s that the s t r u c - ture i s feeding energy back into the ground. 74 3.7.ii K i n e t i c Energy of the Structure The t o t a l k i n e t i c energy of the structure i s found from (3.36) E in which = k 1"*A.= *A = ( 3 ' 3 6 ) absolute v e l o c i t i e s of the generalized co-ordinates. Solution of (2.11) gives the incremental r e l a t i v e v e l o c i t i e s of the generalized co-ordinates; the t o t a l r e l a t i v e v e l o c i t y vector at any time i s found by summing incremental v e l o c i t i e s obtained f o r a l l of the time steps up to the time in question. The r e l a t i v e v e l o c i t y vector r e l a t e s to the v e l o c i t i e s of the generalized co-ordinates with respect to the base of the s t r u c t u r e . To be converted to absolute v e l o c i t i e s the t o t a l ground v e l o c i t y in the d i r e c t i o n of each co-ordinate must be added. For the three d i r e c t i o n s q , r , 9 the absolute v e l o c i t i e s may be expressed as r • •1 q i >- * • + < V . 0 . in which the subscript A represents absolute values, the subscript R represents r e l a t i v e values, the subscript g represents values at ground l e v e l . and 75 3.7.iii Recoverable S t r a i n Energy Stored in the Structure The s t r a i n energy stored in a structure may be subdivided into three components: (1) a x i a 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 - p e r f e c t l y p l a s t i c component. The a x i a l energy i s c a l c u l a t e d incrementally for each member over each time step. The incremental a x i a l energy stored in a member i s shown in (3.38) AE A ft in which = (N. + M - ) A6 o 2 (3.38) = a x i a l load at the beginning of the time step AN = incremental a x i a l load A6 = incremental a x i a l deformation. N q The bending energy stored in the e l a s t i c component i s also c a l c u l a t e d incrementally f o r each member over each time step, as shown i n (3.39) AE el = (M AM , +-^ ) i el o in which M = 1 e o i (AO.) + ( M + el AM. , -T ) .(Ae ) ± i j (3.39) o moment in the a x i a l component at the beginning of of the time step AMg-j = incremental moment in the e l a s t i c component A6 = incremental r o t a t i o n of the member end 76 and the subscripts i and j i n d i c a t e the two member ends. C a l c u l a t i o n of the bending energy stored in the e l a s t i c - p e r f e c t l y p l a s t i c component must be performed separately f o r each member state. For a member with no hinges the incremental bending energy i s c a l c u l a t e d in a manner s i m i l a r to .that used f o r the e l a s t i c component, that i s AM AE E p = (M E p AM + -fZ-). + - J ^ . ) . (A6.) + (M Ep in which the s u b s c r i p t EP replaces the s u b s c r i p t (ABj) (3.40) EL in (3.39) i n d i - c a t i n g the e l a s t i c - p e r f e c t l y p l a s t i c component. For members with a hinge at one end, careful a t t e n t i o n must be paid to c a l c u l a t i o n of incremental bending energy. In the a n a l y s i s presented in t h i s t h e s 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 t o t a l change in energy at that j o i n t would be due to h y s t e r e t i c d i s s i p a t i o n . 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 AE = E p (M E p p + - f - ) . A6. (3.41) o in which the s u b s c r i p t i r e f e r s to the e l a s t i c member end. A f t e r the analysis was performed i t has come to the author's attention that t h i s a n a l y s i s w i l l not conserve energy as the energy l e v e l s computed did not balance completely. Further i n v e s t i g a t i o n 77 revealed that the f a i l u r e of the energy balance can be explained i n part by examination of equations 9, = <j>. + a . Aa, = (Aw, - Ay) + c (Aw. -Ay) J in which $. (2.16) and (3.33) J = to. - y J (2.16) and (2.15) J = t o t a l e l a s t i c r o t a t i o n of the e l a s t i c - p e r f e c t l y p l a s t i c component at end j a, = t o t a l p l a s t i c r o t a t i o n at end w. = j o i n t r o t a t i o n at end Y = chord r o t a t i o n c = carry-over J j j factor and equation 3.33 i s f o r members with a hinge at end end i . (3.33) j but e l a s t i c at 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 e c t , states that the e l a s t i c r o t a t i o n at a member end cont a i n i n g a p l a s t i c hinge w i l l change over a time step i f the other t i c ) member end experiences r o t a t i o n . Hence the incremental (elas- bending energy at t h 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 t h i s e f f e c 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. AM AE E p = (M Ep Fp + -f^-). A6 i - (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 l e a r that no recoverable incremental bending energy w i l l be stored in the e l a s t i c - p e 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 r o t a t i o n at a p l a s t i c hinge w i l l cause energy to be d i s s i p a t e d . This energy d i s s i p a t i o n i s an important parameter in studying the response of structures to earthquake e x c i t a tion. As the amount of energy d i s s i p a t e d by viscous damping in a t y p i c a l structure w i l l be a r e l a t i v e l y small quantity and the k i n e t 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 s p l a c e ment of the s t r u c t u r e , excess energy q u a n t i t i e s must be i n the form of recoverable s t r a i n energy and energy d i s s i p a t e d due to nonlinear r e t i c ) behavior. (hyste- 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 i n which most of the ' energy may be absorbed i n recoverable s t r a i n energy and which w i l l e x h i b 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 d e t a i l e d to as to be capable of withstanding large p l a s t i c deformations. This i s the basis of a d u c t i l e design which w i l l d i s s i p a t e large q u a n t i t i e s of energy in the from of nonlinear r o t a t i o n s . 79 While these two design philosophies are e a s i l y supported by h e u r i s t i c arguments, there i s a d e f i n i t e lack of empirical or a n a l y t i c a l methods of determining the r e l a t i v e merits of these methods q u a n t i t a t i v e l y for multi-degree of freedom s t r u c t u r e s . The author believes that a more complete understanding of the behavior of multi-degree of freedom s t r u c tures in earthquakes may be obtained by examing the various forms of energy absorbed by d i f f e r e n t types of s t r u c t u r e s . The energy d i s s i p a t e d in one nonlinear cycle of a member end i s equal to the area enclosed by the h y s t e r e t i c loop on the moment-rotat i o n curve. In one time step, the energy d i s s i p a t e d at a p l a s t i c hinge i s c a l c u l a t e d according to eq. (3.20) AE DH = M EP (3.20) A a arid the t o t a l " eneijgy d i s s i p a t e d 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 d i s s i p a t e d due to viscous damping i s found by assuming that a f o r c e , equal to the average of the viscous forces acting at the beginning and end of the time step, acted.throughout the i n t e r v a l . Viscous forces at the beginning and end of a time step are c a l c u l a t e d in eqs. (3.44a) and (3.44b) r e s p e c t i v e l y F b = (°fflL + *b (3.44a) 80 F = e (am + BkJ x (3.44b) g in which the notation of Section 2.1 i s used. Hence the average force vector, F ^ , i s assumed to a c t through the i n cremental displacements of the s t r u c t u r e Ax , d i s s i p a t i n g the incremen- t a l energy quantity shown i n eq. 3.46 e F A AE 3.7.vi ^ = V D = V b 2 + ( 3 ' 4 5 ) (3.46) AX Energy Balance At any p a r t i c u l a r time the energy input by the earthquake i n t o the s t r u c t u r e should equal the various forms of energy appearing i n the structure. After n time steps have been performed, the f o l l o w i n g b a l - anceeshould e x i s t E A E IN = E k + < Ai AE + A E ELi + A E W + £ AE DH + ^ 81 CHAPTER 4 MODELLING OF A SIXTEEN STOREY OFFICE BUILDING The importance of using a s t r u c t u r a l model containing three dimensional response parameters in order to accurately describe the nonl i n e a r response of a structure to earthquake e x c i t a t i o n has already been established. While r e s u l t s obtained using a three-dimensional model w i l l be more accurate, even f o r symmetric s t r u c t u r e s , than those from a plane frame a n a l y s i s , the difference becomes most pronounced f o r structures in which t o r s i o n i s more apt to occur. Torsional response becomes an important f a c t o r in core type structures in which the primary s t i f f e n i n g elements are located near the centre of r i g i d i t y . This c l a s s of structures w i l l have a much smaller t o r s i o n a l resistance than structures in which the primary s t i f f e n i n g e l e ments are on the periphery of the s t r u c t u r e , with the l a r g e s t moment arm about the centre of r i g i d i t y . possible Large t o r s i o n a l o s c i l l a t i o n s may also r e s u l t in i r r e g u l a r l y shaped structures such as the L-shaped b u i l d i n g and other e c c e n t r i c structures in which the centre of r i g i d i t y and the centre of mass do not c o i n c i d e . A b u i l d i n g layout which combines these two undesirable t o r s i o n a l s i t u a t i o n s by having a core displaced from the centre of mass i s a s i t u a t i o n which should be avoided. The current trend in the design of medium sized high r i s e buildings i s towards structures in which shear walls comprise the main l a t e r a l load r e s i s t i n g elements. In areas of r e l a t i v e l y high earthquake 82 r i s k , a common b u i l d i n g type i s one with a dual-component l a t e r a l load r e s i s t i n g system comprised of a d u c t i l e moment-resisting space frame in conjunction with shear w a l l s . The shear walls are designed so that they are capable, when a c t i n g alone, of carrying the t o t a l s t a t i c l a t e r a l forces s p e c i f i e d by the code. The frame i s desngned so that i t i s able to c a r r y , as a system separate from the shear w a l l s , the v e r t i c a l in addition to 25% of the abovementioned l a t e r a l load. loads The shear walls are commonly situated around the elevator s h a f t , hence the name " c o r e type" b u i l d i n g may be applied to t h i s type o f s t r u c t u r e . In many cases, a r c h i t e c t s s p e c i f y that the core i s to be s i t u a t e d away from the geometr i c centre of the b u i l d i n g in order to provide for a l a r g e r area of open space. This causes t o r s i o n to be an important parameter. Since the dual component b u i l d i n g i s becoming so commonly used, i t i s f e l t by the author that i t i s a b u i l d i n g type which should undergo some accurate analysis to determine i t s adequacy when subjected to e a r t h quake e x c i t a t i o n . The b u i l d i n g which was selected f o r a n a l y s i s herein i s a v a r i a t i o n of a r e c e n t l y constructed b u i l d i n g located i n downtown Vancouver. The i n t e n t was to examine the e f f e c t of t o r s i o n on t h i s b u i l d - ing and to determine whether the manner in which t o r s i o n i s treated in current b u i l d i n g codes i s adequate. 4.1 Description of the S t r u c t u r a l Model The actual b u i l d i n g on which the analysis i s based i s a 14 storey r e i n f o r c e d concrete b u i l d i n g . While i t was d e s i r a b l e to model t h i s b u i l d i n g as accurately as p o s s i b l e , i t was also necessary to make 83 v a r i a t i o n s to the b u i l d i n g to enable an accurate yet economic a n a l y s i s to be made subject to the assumptions inherent i n the computer program. The economic constraints meant that the r e l a t i o n s h i p between the s t r u c ture which was studied and the actual b u i l d i n g was not a close one, but the author believes that the structure that was analyzed represents a t y p i c a l midsized o f f i c e b u i l d i n g designed under current techniques. The plan of a t y p i c a l f l o o r of the modified b u i l d i n g i s shown i n Fig. 20. The core i s located at the back face of the b u i l d i n g which causes the undesirable s i t u a t i o n in which torsion i s l i a b l e to become an important response parameter. Aumique feature of t h i s core i s that the back wall i s s p l i t down the middle throughout the e n t i r e height of the building. This causes the centre of r i g i d i t y to move c l o s e r to the cen- treoof mass reducing the design e c c e n t r i c i t y . However, c u t t i n g the back wall reduces the s t i f f n e s s of the b u i l d i n g with respect to horizontal t r a n s l a t i o n as well as r o t a t i o n . An i n t e r e s t i n g s i d e l i g h t to the analy- s i s of t h i s b u i l d i n g i s to examine whether c u t t i n g the back wall causes lower stresses to develop when the b u i l d i n g i s subject to a design e a r t h quake. The modified structure was a rectangular b u i l d i n g 100 f t . i n the q d i r e c t i o n by 75 f t . in the r direction. 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 o o r height giving a t o t a l b u i l d i n g height of 196 f t . Future sections of t h i s chapter w i l l describe various methods used to model t h i s b u i l d i n g in an attempt to accurately incorporate the behavior of the core into the frame a n a l y s i s . In each of these models 84 frames 1, 2 and 3 as numbered i n F i g . 20 are i d e n t i c a l , and these frames are described in Figs. 21 and 22. In a d d i t i o n , walls numbered 8 and 9 remain unchanged throughout the various models. q d i r e c t i o n and t h e i r s t i f f n e s s i n the r These walls run i n the d i r e c t i o n i s neglected. s t r u c t u r a l parameters of walls 8 and 9 are described i n F i g . 23. The The methods in which the^other components of the b u i l d i n g are included are described as they are encountered i n the i n d i v i d u a l d e s c r i p t i o n s of the models. 4.2 Preliminary Analysis The purpose of t h i s a n a l y s i s i s to determine whether the ap- proximate earthquake a n a l y s i s as proposed by b u i l d i n g codes w i l l provide a s u f f i c i e n t l y accurate estimate of the forces r e s u l t i n g from a probable earthquake. In t h i s respect, the techniques prescribed by the National Building Code of Canada (NBC) (22) are examined with reference to the model described previously. in the q The analysis i s r e s t r i c t e d to s t a t i c shears d i r e c t i o n which causes t o r s i o n to occur because of the eccen- t r i c i t y of the model. 4.2.1 C a l c u l a t i o n of the Lateral Seismic Base Shear The base shear f o r c e , V , i s c a l c u l a t e d as the product of the q u a n t i t i e s shown i n (4.1) V in which A = = ASKIFW (4.1) the assigned horizontal ground a c c e l e r a t i o n r a t i o which 85 equals 0.08 f o r seismic zones 3 in which the ci.ty of Vancouver is. s i t u ated. The quantity S r e l a t e s the seismic response to the f i r s t natu- ral frequency of the s t r u c t u r e , as shown in = S in which T = (4.2) (4.2) the fundamental period of the b u i l d i n g . The fundamental period of the structure was l a t e r c a l c u l a t e d from an eigenvalue a n a l y s i s , but, f o r consistency, an approximate empirical value of T i s c a l c u l a t e d according to (4.3) for use with the code s t a t i c ana- lysis: f with the b u i l d i n g height D = 0.05 h = h n /IT = 0- (4.3) 196 f t . and the b u i l d i n g w i d t h , 100 f t . , a r e s u l t i n g fundamental period l a t e d , r e s u l t i n g i n a value of The quantity K S = T of 0.98 sec. is c a l c u - 0.503. accounts f o r v a r i a t i o n s in the structure sys- tem and i s evaluated i n part with respect to the measured performance of structures in actual earthquakes. For the structure under c o n s i d e r a t i o n , which i s designed to contain a complete d u c t i l e moment r e s i s t i n g space frame (capable of r e s i s t i n g 25% of the prescribed s t a t i c shear when acting alone) i n conjunction with shear walls (capable of r e s i s t i n g the t o t a 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 f a c t o r , 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 f a c t o r , F , of 1.0 may be used. For the defined values of the various c o e f f i c i e n t s the base shear force V may be expressed in terms of the t o t a l weight of the b u i l d i n g (plus 25% of the design snow l o a d ) , W , in terms of V = 0.032 U (4.4) (4.4) The dead load of the b u i l d i n g i s c a l c u l a t e d by representing i t as a uniform load of 175 pounds per square foot acting over each f l o o r g i v i n g a t o t a l weight of 1310 kips per storey causing a t o t a l seis- mic base shear force of 670 kips f o r t h i s s t r u c t u r e . 4.2.2 D i s t r i b u t i o n of the Lateral Seismic Shear Force A p o r t i o n , F^. , of the t o t a l base shear f o r c e , V , i s concen- trated at the top of the b u i l d i n g to account for upper mode v i b r a t i o n s . The NBC defines t h i s quantity as shown i n F t h 0.004 V (=2-) s = 1 (4.5) 2 (4.5) u in which h n equals the height of the b u i l d i n g and D s equals the dimension of the l a t e r a l f o r c e - r e s i s t i n g system in a d i r e c t i o n p a r a l l e l to the applied f o r c e . The quantity D g i s d i f f i c u l t to evaluate f o r a dual component system such as the model concerned. However, i t i s f e l t that the i n t e n t i o n 87 of the code i s to define the quantity according to the primary l a t e r a l force r e s i s t i n g element which i s the e c c e n t r i c shear core (D g = 27 f t . ) . In the l i g h t of t h i s assumption, the upper-mode shear force culated to be 0.21 V. not exceed 0.15 V" is However the code s t i p u l a t e s that force F^ cal"need from which i t i s i n f e r r e d that t h i s l i m i t i n g value (100 kips) should be applied i n t h i s case. The remaining l a t e r a l shear i s d i s t r i b u t e d over the b u i l d i n g height according to the formula (4.6) •' W.- h F in which F x = ( V _ F t ) - f ^ - (4.6) = the l a t e r a l force to be applied at storey = the weight of storey x = the height of storey x = the number of storeys. x X W X h X n A p p l i c a t i o n of formula (4.6) produces the shear d i s t r i b u t i o n shown in F i g . 25. 4.2.3 C a l c u l a t i o n of Torsional Moments The NBC states that " t o r s i o n a l moments i n the horizontal plane of the b u i l d i n g s h a l l be computed in each storey" based on a formula in which the t o t a l l a t e r a l shear force at and above a p a r t i c u l a r l e v e l is m u l t i p l i e d by the design e c c e n t r i c i t y of the storey to produce the app l i e d t o r s i o n a l moment. The design e c c e n t r i c i t y , e , i s computed by A one of the f o l l o w i n g equations? whichever provides the greater stresses. 88 e e in which D n v = 1.5 e + 0.05 D = 0.5 e - 0.05 D (4.7a) n (4.7b) n equals the plan width of the b u i l d i n g . The quantity 0.05 D n represents t o r s i o n a l influences due to accidental e c c e n t r i c i t i e s which may r e s u l t from such things as the a d d i tion of wall panels or p a r t i t i o n s . The f i r s t term of (4.7) equals the e c c e n 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 p a r t i c u l a r storey, increased or decreased by 50 per cent. The 50 per cent f a c t o r i s e s s e n t i a l l y an ignorance f a c t o r explained by the complex nature of torsion as i t occurs during an e a r t h quake. One point concerning the determination of e c c e n t r i c i t y deserves discussion. This point concerns the f a c t that the centre of r i g i d i t y could be defined i n several ways. For example, i f the f l o o r s above and below that under consideration were held f i x e d and a single load was app l i e d to the f l o o r being considered, there would be one point i n the f l o o r such that any force applied at that point would cause t r a n s l a t i o n in i t s own l i n e of action with no r o t a t i o n 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 o o r . Alternatively, suppose that no constraints were a p p l i e d , but that the single force was applied to the f l o o r under consideration. Again there would be one point such that loads applied there would cause no r o t a t i o n of that f l o o r , a l though other f l o o r s might r o t a t e . Again, that point could be defined as 89 the centre of r i g i d i t y of that f l o o r . T h i r d l y , suppose that loads were applied to a l l f l o o r s 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 b u t i o n discussed above). The loads could be placed i n a single v e r t i c a l plane and the p o s i t i o n of that plane which caused no r o t a t i o n of a given f l o o r 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 f e r e n t l o c a t i o n on each f l o o r so that no f l o o r r o t a t e d . t i e s of a l l f l o o r s at once. This would define the centre of r i g i d i - The code i s not c l e a r as to whether one of these or some other d e f i n i t i o n i s envisaged. The author believes that an e a s i l y c a l c u l a t e d method of d e t e r mining storey e c c e n 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 f n e s s contributionoof the i n d i v i d u a l components (frames or walls) of the b u i l d i n g may be approximated by use of equation (4.8) l/X- kj - -jp-LT. (4.8) l/X, i -'• in which k. = the f r a c t i o n o f the t o t a l s t i f f n e s s contributed by component j X. = the displacement of component n = the number of components i n the d i r e c t i o n under consideration. J J where displacements j X are obtained f o r a p a r t i c u l a r load case. As the s t i f f n e s s c o e f f i c i e n t s k vary with the type of load case as- sumed, i t i s f e l t that a t r i a n g u l a r load d i s t r i b u t i o n , s i m i l a r to the l a t e r a l seismic load d i s t r i b u t i o n s t i p u l a t e d by the codes would be the 90 most appropriate load case to apply in order to maintain consistency. A p p l i c a t i o n of t h i s t r i a n g u l a r load case to each component of the b u i l d i n g in turn w i l l enable f a c t o r component. This f a c t o r t i o n of f a c t o r s k k k to be evaluated f o r each may be c a l c u l a t e d at each storey. Calcula- enables the e c c e n t r i c i t y o f each storey to be e v a l u - ated according to (4.9) n 1*1 ^ o e = £*.9> V E i=l in which 6I k 1 e- = the e c c e n t r i c i t y of storey e- = the e c c e n t r i c i t y (distance from the centre of mass to the centroid of the component) of component i J j C a l c u l a t i o n of storey e c c e n t r i c i t i e s according to t h i s method w i l l produce the r e s u l t that the e c c e n t r i c i t y decreases with respect to height f o r a b u i l d i n g of t h i s type. This v a r i a t i o n of e c c e n t r i c i t y may be explained by examinjnghther.defleetedhshape'ofthe'iitwoimain components of t h i s b u i l d i n g , as shown in F i g . 24. The deflected shape of the frame components i s more or l e s s l i n e a r while the deflected shape of the shear wall elements i s of a higher order when each o f these components i s examined independently. This i n d i c a t e s that the percentage of the t o t a l s t i f f n e s s contributed by the shear wall elements decreases with respect to height with an accompanying increase in the percentage s t i f f n e s s of the frame components. For the structure under examination, the eccen- 91 t r i c i t y of the b u i l d i n g i s governed by the e c c e n t r i c i t y of the shear core i n the lower s t o r i e s , while t h i s e c c e n t r i c i t y reduces with respect to height as the moderating influence of the frame components assumes a more s i g n i f i c a n t r o l e . This method of c a l c u l a t i n g e c c e n t r i c i t y was performed on a v a r i a t i o n of the b u i l d i n g model and the r e s u l t s are presented in Table 4. Results shown are f o r a 15 storey b u i l d i n g with 9 foot storey heights and hence they w i l l d i f f e r f o r the b u i l d i n g described p r e v i o u s l y . How- ever as values shown in Table 4 are based on a buiilding with i d e n t i c a l plan dimensions and l a t e r a l force r e s i s t i n g elements, they are f e l t to be analogous to those obtained f o r the b u i l d i n g under consideration. In the lower two s t o r i e s , the c a l c u l a t e d e c c e n 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 e c c e n t r i c i t y have been doubled as required by the NBC. While t h i s method o f c a l c u l a t i n g e c c e n t 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 e s t a b l i s h i n g the centre of r i g i d i t y of a storey and c a l c u l a t i n g e c c e n t 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 t h i s s e c t i o n , should be required by the code, as the e f f e c t of t o r s i o n i n a structure may vary s i g n i f i c a n t l y i f an a l t e r n a t i v e method i s followed. C a l c u l a t i o n of the t o r s i o n a l moment at storey performed according to equation (4.10) x, M t , is 92 x (4.10) The torsional, moments'to be applied to each storey of the b u i l d i n g under consideration are shown in Table 4. I t should be noted that there are two values of the t o r s i o n a l moment to be applied at each storey. This r e l a t e s to the two values o f design e c c e n t r i c i t y as c a l c u l a t e d by equat i o n (4.7) and the t o r s i o n a l moment which causes the greatest stress in a p a r t i c u l a r component i s to be the design moment f o r that component. 4.2.4 Summary of the Preliminary Analysis The NBC i s also not s p e c i f i c in i t s requirements governing the method i n which t o r s i o n a l moments s h a l l be included i n the a n a l y s i s . As i t i s f e l t t h a t r t h e code appreciates t h e . f a c t that analyses more, c o m p l i cated than plane frame analysis are r a r e l y performed i n a design o f f i c e , i t appears that the intended method of i n c l u d i n g t o r s i o n a l moments i n volves representing the t o r s i o n a l moment as a set o f * a d d i t i o n a l shear loads in the various components. for component i , The a d d i t i o n a l shear load at storey . , i s c a l c u l a t e d according to the f o l l o w i n g f o r - mula (4.11) F. txi in which fil y M.tx (4.11) the t o t a l number of components of the b u i l d i n g distance from the centre of r i g i d i t y to the centroid of the component x 93 The e f f e c t of t o r s i o n may be considered i n terms of t o r s i o n a l magnification factors which i n d i c a t e the amount by which the design l a t eral seismic forces are increased as a r e s u 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 f r o n t frame (frame number 1) of t h i s model in Table 5. may be made on the other components. Similar calculations In order to examine the manner i n which t o r s i o n a l response i s treated by the code, the magnification f a c tors may be compared with those obtained by using the nonlinear time step analysis method presented previously. The time step a n a l y s i s would be performed twice; once on the model containing the e c c e n t r i c shear core as shown in F i g . 20 and then on a v a r i a t i o n of the model in which the shear core i s 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 s e l e c t a p a r t i c u l a r response parameter to compare. The use of magnification f a c t o r s to compare the influence of t o r s i o n 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 c a l c u l a t e d 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 r o t a t i o n or not. In the l i g h t of t h i s assumption, the t o r s i o n a l magnification f a c t o r s may be expressed i n terms of the r a t i o of member end r o t a t i o n s (or d u c t i l i t y factors) obtained in the e c c e n t r i c and symmetric cases. Any s i g n i f i c a n t d i f f e r e n c e in the t o r s i o n a l magni- f i c a t i o n factors obtained by the NBC s t a t i c analysis and the time step analysis may be an i n d i c a t i o n that the manner in which t o r s i o n a l e f f e c t s due to e c c e n t r i c i t y are accounted f o r by the NBC i s unsatisfactory. 94 4.3 Computer Modelling of the Structure As stated p r e v i o u s l y , modelling of t h i s dual component shear wall-frame structure posed a d i f f i c u l t problem as a r e s u l t of r e s t r i c tions imposed by theory. As the computer program requires that the structure be i d e a l i z e d 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 r e s u l t of the attempts at modelling t h i s p a r t i c u l a r b u i l d i n g system i s that analysis based upon the theory described in t h i s thesis would not provide s u f f i c i e n t accuracy. This sec- t i o n describes the various attempts at modelling the b u i l d i n g 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 i g . 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 i n terms of the remaining plane components shown in F i g . 20. 4.3.1 Model A The f i r s t attempt at i d e a l i z i n g the s t r u c t u r a l system i s shown i n F i g . 26. This model i s expected to accurately model the behavior of the shear core i n the r d i r e c t i o n i f the s t r u c t u r a l parameters assigned to the columns of frames 5 and 6 with respect to motion in the r direc- t i o n 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 t h i s shear core involves assigning s t r u c t u r a l values to the members of frame 7 in order to accurately analyze the response of the core to motions in the i n t e r i o r section of the core i s of an H q direction. As the shape c h a r a c t e r i s t i c of a wide flange s e c t i o n , the walls of t h i s section which run in the r direction may be considered analogous to the flanges and the wall running in the q d i r e c t i o n thought of as the web of a wide flange s e c t i o n . of t h i s section could be accurately modelled f o r motion i n the Behavior q direc- t i o n by i d e a l i s i n g the section as a column, of s t r u c t u r a l properties defined by the wide flange s e c t i o n , placed at the centroid of the s e c t i o n . This technique would not enable the b i a x i a l e f f e c t s of the core to be considered as the core i s modelled by d i f f e r e n t components in the and r q directions. An a l t e r n a t i v e method of i d e a l i z i n g the core with respect to motion i n the q to form a frame. d i r e c t i o n i s to use 3 columns connected by r i g i d beams The two external columns of t h i s frame 7 would coincide with the i n t e r n a l columns of frames 5 and 6 to allow the b i a x i a l e f f e c t s to be accounted f o r . Each of these external columns would represent a flange of the cross s e c t i o n . The rimterior column would represent the web of the wide flange and the response of t h i s i n t e r i o r column would be independent of motion i n the r direction. To approximate the behavior of the core i t was f e l t that the e x t e r i o r 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 c r o s s - s e c t i o n a l area equal to zero. The beams j o i n i n g the columns would be assigned a r t i f i c i a l l y high values of moment of i n e 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 t h i s model frame revealed that i t w i l l not provide an accurate d e s c r i p t i o n of the actual core. The deflected shape of the core should resemble a c a n t i l e v e r d e f l e c t e d shape as shown in F i g . 24b. Modelling of the core as described w i l l r e s u l t i n a d e f l e c t e d shape c h a r a c t e r i s t i c of a frame as shown in F i g . 24a. This important d i f f e r e n c e in the s t r u c t u r a l i d e a l i z a t i o n prohibited use of t h i s model. A v a r i a t i o n of t h i s technique could be developed to accurately describe the behavior of the core. In t h i s v a r i a t i o n , the i n t e r i o r co- lumn would be assigned the s t r u c t u r a l properties of the core and the ext e r i o r columns and the beams would be assigned properties set equal to 0. A f t e r incremental displacements of the building have been solved, the a x i a l force induced i n t o the e x t e r i o r 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 r e s u l t i n g from a x i a l forces in the e x t e r i o r columns. This technique would accu- r a t e l y describe the motion of the core in a l l d i r e c t i o n s as well as enab l i n g the b i a x i a l e f f e c t s of the core to be analyzed. However, t h i s v a r i a t i o n would require s i g n i f i c a n t a l t e r n a t i o n s 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 g e n e r a l i t y of the program. 4.3.2 Model B From examination of model type A i t was r e a l i z e d t h a t , except for the proposed v a r i a t i o n of the model, i t i s not possible to accurately describe motion in the q d i r e c t i o n i f the core i s i d e a l i z e d to have 2 components with respect to motion in the contains only one component in the as shown in F i g . 27. r r direction. Hence, model B d i r e c t i o n to describe the c o r e , This component consists of 2 columns connected by the r i g i d end stub l i n t e l beam s i m i l a r to frame 5 or frame 6 of model A. As f o r model A, i f appropriate s t r u c t u r a l properties are assigned to the columns and beams, i t i s f e l t that an accurate d e s c r i p t i o n of the b u i l d ing with respect to motion in the r d i r e c t i o n w i l l be obtained. One inherent f a u l t of t h i s model i s the f a c t that i t p a r t i a l l y neglects the t o r s i o n a l s t i f f n e s s c o n t r i b u t i o n of the shear core by c o n s i dering i t to e x i s t in one plane. Before a n a l y s i s of the structure was performed an i n v e s t i g a t i o n of t h i s reduction in the t o r s i o n a l r i g i d i t y was c a r r i e d out to determine whether t h i s would introduce a s i g n i f i c a n t error. The e f f e c t of i d e a l i z i n g the core in a s i n g l e plane may be examined e a s i l y with respect to the s t a t i c l a t e r a l force method of the NBC. The design e c c e n t r i c i t y arid r e s u l t a n t t o r s i o n a l moments w i l l not be a f fected as the t o t a l l a t e r a l s t i f f n e s s with respect to motion i n the d i r e c t i o n i s unchanged. q However the t o r s i o n a l moments w i l l be d i s t r i b u t e d 98 in a d i f f e r e n t manner upon a p p l i c a t i o n of equation (4.11). may be c a l c u l a t e d by reducing the value of n This e f f e c t by 2 and e l i m i n a t i n g the c o n t r i b u t i o n to the t o r s i o n a l r i g i d i t y made by components 5 and 6. The revised magnification f a c t o r s , calculated.n'n-.this manner, w i l l be a p p l i c a b l e ! <feotilM.s imodeil;.?..- Comparn'sonnoifrtheamagniffeationrifa'etors-relating to t h i s model and the s t a t i c model i n d i c a t e that the error introduced by t h i s approximation i s not s i g n i f i c a n t with respect to the p r e c i s i o n r e q u i r e d , f o r frame number 1. The e f f e c t of t h i s r e d u c t i o n i i n t o r s i o n a l r i g i d i t y may also be examined on the mathematical model of the b u i l d i n g used in the computer program. The computer program developed by Mackenzie [19] i s based upon the same theory as that described in t h i s thesis i n the l i n e a r range. Models of the structure corresponding to the accurate l i n e a r model A were compared with model B with respect to t o r s i o n a l behavior in the l i n e a r range using t h i s program. A t o r s i o n a l moment was applied at a height of 2/3 of the t o t a l b u i l d i n g height and the r o t a t i o n s obtained at the v a r i ous s t o r e l y l e v e l s were compared. When subject to t h i s moment, the d i f - ference i n the storey r o t a t i o n s was small (no greater than 10%) hence i t may be concluded that model B provides an adequate d e s c r i p t i o n of the t o r s i o n a l c h a r a c t e r i s t i c s of the b u i l d i n g . It may be argued that a possible nonlinear state may be reached, i . e . i f a s i g n i f i c a n t part of the e x t e r i o r frames are in a p l a s t i c state while the core walls are in an e l a s t i c s t a t e , in which case the t o r s i o n a l r i g i d i t y of the core becomes a much more important c o n t r i b u t i o n to the t o t a l t o r s i o n a l r i g i d i t y of the b u i l d i n g . In t h i s case model B may be 99 expected to provide a s i g n i f i c a n t l y d i f f e r e n t t o r s i o n a l response than would r e s u l t i n the actual b u i l d i n g . Tables 6 and 7 i l l u s t r a t e r e s u l t s of a modal comparison of the two models'. motion i n the While the periods for the f i r s t three modes representing r d i r e c t i o n (with no motion i n the q d i r e c t i o n ) are equal f o r each model, the periods and mode shapes corresponding to coupled q d e f l e c t i o n and r o t a t i o n d i f f e r . This may r e s u l t from another possible source of e r r o r concerning the revised model of the back wall (frame 4 ) . By using j u s t one column to describe the shear core the beams connecting the core to the corner ( e x t e r i o r ) columns w i l l be made longer. As the 2 s t i f f n e s s c o n t r i b u t i o n of these beams i s a function of ( I / L ) , (I/L (I/L ) and ) the moment of i n e r t i a (I) may be adjusted to give the proper r e - s u l t f o r only one of these three types. With the sources of e r r o r kept i n mind an attempt to analyze the s t r u c t u r a l response was made. This analysis was continued u n t i l a state was reached in which continuation of the analysis was f e l t f u t i l e . This s t a t e occurred when the a x i a l loads occurring in the columns r e p r e senting the shear core reached a c r i t i c a l p o i n t . The a x i a l load N^. , shown on F i g . 18a, represents the ultimate t e n s i l e strength of a column. When t h i s load i s reached i t i s necessary to introduce an a x i a l s l i p j o i n t (analogous to the r o t a t i o n a l hinge inserted when the y i e l d moment i s exceeded) to adequately describe the r e s u l t s . As the theory incorpo- rated in t h i s analys-is did not include the p o s s i b i l i t y of a x i a l y i e l d i n g o c c u r r i n g , i t was impossible to provide an accurate d e s c r i p t i o n of the model once a x i a l loads exceeding N + were reached. 100 The theory used i n the computer program was intended to be appropriate f o r framed structures in which the width of the frames i s such that a x i a l y i e l d i n g w i l l generally not occur. I f t h i s theory i s applied to very slender components, such as shear cores, i t i s apparent that the e f f e c t of a x i a l y i e l d i n g must be included. Because of the s i g - n i f i c a n t amount of r e v i s i o n which would be required to include a x i a l y i e l d i n g i n t o the theory, an a l t e r n a t i v e method of i d e a l i z i n g the s t r u c ture was examined. 4.3.3 Model C From the d i f f i c u l t i e s encountered in attempting to model t h i s b u i l d i n g as described i n the previous two sections of thcis t h e s i s , i t was f e l t that a f e a s i b l e a l t e r n a t i v e model was to consider the e n t i r e shear core as a s i n g l e column as shown in F i g . 28. This model w i l l t o - t a l l y neglect the t o r s i o n a l r i g i d i t y of the shear core. In a d d i t i o n , t h i s model w i l l not permit consideration of the coupled shear wall e f f e c t caused by the doorway openings. However, i f the l i n t e l beams are r e i n - forced with heavy diagonal steel the e f f e c t w i l l be s i m i l a r to a cross bracing system used on trusses. with respect to motion i n the In t h i s case the response of the core r d i r e c t i o n w i l l be s i m i l a r to that of a s i n g l e wall as opposed to the coupled shear walls used p r e v i o u s l y . Analysis of t h i s model was performed u n t i l an undesirable trend developed. As expected, the column representing the shear core was by f a r the most s i g n i f i c a n t force r e s i s t i n g element, and the magnitude of forces r e s u l t i n g in t h i s column was very large. During the 101 earthquake analysis a time step of 0.01 second was used. was considered to be the economic l i m i t . This time step Unbalanced forces r e s u l t i n g from excursions i n t o the nonlinear range must be corrected f o r in an approximate manner as described in Section 3. In t h i s case the unbalanced forces r e s u l t i n g from nonlinear behavior of the shear core were of a very large magnitude and the approximate method of accounting f o r these unbalanced forces broke down. This problem may be a l l e v i a t e d by reducing the s i z e of the time step which w i l l generally reduce the magnitude of the unbalanced f o r c e s , however th.is approach was economically u n f e a s i b l e . I t i s f e l t by the author that the most rigorous s o l u t i o n to t h i s problem would be to use a Newton-Raphson i t e r a t i o n scheme w i t h i n each time step to properly account f o r these large unbalanced forces. It is possible that j u s t two or three i t e r a t i o n s of the Newton-Raphson method would provide s u f f i c i e n t accuracy. 4.4 Summary of Computer Models The d i f f i c u l t y in modelling t h i s b u i l d i n g has led the author to several conclusions regarding analysis of shear wall-framed structures using theory developed f o r r i g i d frames. In addition to the problems encountered and described above there are a couple of points which also deserve mention. The method of accounting for p l a s t i c or nonlinear behavior in framed s t r u c t u r e computer programs i s to i n s e r t a p l a s t i c hinge of i n f i n i t e l y small width at a member end. However, in actual f a c t the p l a s t i c 102 zone at a member end w i l l not be r e s t r i c t e d to a p a r t i c u l a r point but w i l l act over a length of the element proportional to i t s depth. Consi- dering a hinge to be a point may be adequate for a member whose length i s large compared to i t s depth, however f o r members i n which the reverse i s t r u e , the f i n i t e length of hinge d i s t r e s s should be considered. Hence f o r the shear core of t h i s model i t does not appear j u s t i f i e d to speak of a member with a hinge at one end but not at the other end. The zone of p l a s t i c action in a shear core of these dimensions may extend over two or three s t o r i e s . Closer care should be taken in applying frame theory to the nonlinear behavior of shear w a l l s . Another point which deserves c l o s e r a t t e n t i o n concerns the c r i t e r i o n f o r which a hinge may be considered to reverse i n a b i a x i a l column. I t was found i n t h i s analysis that the c r i t e r i o n of negative energy d i s s i p a t i o n , as described i n Section 3 . 4 does not provide an adequate r u l e for hinge r e v e r s a l . Removal of a hinge according to t h i s c r i t e r i o n causes an event f a c t o r of zero to be assigned and the member to be reanalyzed. However, i f the hinge should a c t u a l l y have remained, an event f a c t o r of zero w i l l again be assigned to t h i s revised member, and the hinge i s r e i n s e r t e d . This behavior causes the undesirable f e a - ture of an i n f i n i t e loop to occur. In order to avoid t h i s problem an assumption was made which w i l l a f f e c t the accuracy of r e s u l t s obtained. I t was assumed that only one event occurred during a time step f o r each member, i . e . a f t e r the i n i t i a l event f a c t o r i s determined, the remaining portion of the time step i s assumed to not incur f u r t h e r member s t a t e changes. 103 In summary the assumptions and techniques inherent in the theory presented in t h i s t h e s i s did not y i e l d accurate a n a l y s i s of b u i l d i n g s containing slender shear cores, using the computer program developed by the author. I t i s f e l t that the cost of obtaining an accurate analysis i s such that work of t h i s type w i l l nbttbe wor'thv'Whiilie u n t i l the imminent improvements in computer technology s i g n i f i c a n t l y reduce the cost of solution. 104 CHAPTER 5 RESULTS OF A TIME STEP ANALYSIS OF A FIVE STOREY BUILDING The o r i g i n a l i n t e n t of t h i s research was to examine the r e s ponse to an earthquake of a building c h a r a c t e r i s t i c of current b u i l d i n g techniques. However, i n Chapter 4 i t was explained that the common method of incorporating shear walls with a framing system provides a model which i s d i f f i c u l t to analyze using the theory presented. Hence to obtain some meaningful r e s u l t s i t was necessary to r e s t r i c t the a n a l y s i s to a framed s t r u c t u r e . While i t i s r e a l i z e d that the b u i l d i n g under con- s i d e r a t i o n i s not representative of a common b u i l d i n g technique, i t i s f e l t that a presentation of the r e s u l t s obtained would provide a valuable step towards obtaining a better understanding of the nonlinear response of s t r u c t u r e s . 5.1 The S t r u c t u r a l Model As the r e s t r i c t i o n s imposed on the building type preclude the use of a complex model r e q u i r i n g great expense, a s i m p l i f i e d b u i l d i n g as shown i n F i g . 29 was modelled. The b u i l d i n g was comprised of four i d e n - t i c a l r e i n f o r c e d concrete frames, each i d e n t i c a l to the top f i v e storeys of the frame analyzed by Clough; Benuska and Wilson [ 6 ] , arranged symmetrically. The weight of each storey was c a l c u l a t e d by increasing the weights used by Clough et a l for the 25 f t . frame spacing according to 105 the 60 f t . width used in t h i s analysis. The e f f e c t of t o r s i o n was induced by separating the centre of mass from the centre of r i g i d i t y (geometric centre of the b u i l d i n g ) . The centre of mass was assumed to be located 5 per cent of the b u i l d i n g width, or 3 f e e t , from the centre of r i g i d i t y in the r direction. Determination of the column properties required to define the a x i a l force - moment i n t e r a c t i o n diagram posed a problem as no d e t a i l s of the r e i n f o r c i n g steel are presented in the d e s c r i p t i o n of the plane frame a n a l y s i s . In t h i s case a quantity of t e n s i l e steel required to r e s i s t the l i m i t i n g moment, which i f exceeded would require the use of compression s t e e l , was assumed to e x i s t i n each member. For the beams, a s i m i l a r procedure was used to determine the ultimate moment which was then assigned to the beam ends with respect to both p o s t i t i v e and negat i v e moments. The p o s i t i v e ultimate moments were then increased and the negative moments decreased (in absolute sense) by 17 per cent to a p p r o x i mate the e f f e c t s of the i n i t i a l moment induced i n t o the beams by dead loads. For beam ends framing into corner columns, the ultimate moments were a r b i t r a r i l y set at 50 per cent of those c a l c u l a t e d f o r the beam, to approximate the usual proportioning of r e i n f o r c i n g steel i n framed structures. The i n i t i a l a x i a l loads induced i n t o the columns by dead loads were assumed to be one-ninth of the t o t a l storey load f o r each of the eight i n t e r i o r columns and o n e - t h i r t y - s i x t h for each of the four e x t e r i o r or corner columns. 106 Damping was assumed to be mass proportional with no c o n t r i b u t i o n made by s t i f f n e s s proportional damping. This assumption allows f o r s i g n i f i c a n t s i m p l i f i c a t i o n s to be made in s o l u t i o n of the equations of motion. In a d d i t i o n , i t i s uncertain whether the incremental s t i f f n e s s proportional damping should be based on the o r i g i n a l structure s t i f f n e s s or the s t i f f n e s s at time of s o l u t i o n . A r a t i o of c r i t i c a l damping of 0.02 was assumed f o r t h i s b u i l d i n g . 5.2 The Earthquake A c c e l e r a t i o n Record The most serious problem i n r e l a t i n g r e s u l t s obtained from a time step a n a l y s i s with the actual r e s u l t s expected from the occurrence of an earthquake concern the s e l e c t i o n of an appropriate accelerogram record to be input into the a n a l y s i s . As the strong-motion shaking of major earthquakes occurs in a l o c a l i z e d area at very infrequent time i n t e r v a l s , there i s a d e f i n i t e lack of recorded strong-motion accelerograms. El Centro, C a l i f o r n i a , 1940 and T a f t , C a l i f o r n i a , 1952 are the two most commonly used records measured near the epicentre of strong earthquakes. However the v a r i a t i o n of i n t e n s i t y , duration and frequency content occ u r r i n g in these earthquakes and comparable records obtained i n South America r e s u l t s in s i g n i f i c a n t s t a t i s t i c a l f l u c t u a t i o n s . I t appears that many more records must be obtained before the c h a r a c t e r i s t i c s of strong motion earthquakes may be accurately s t a t i s t i c a l l y defined. A recent trend in the determination of appropriate a c c e l e r o gram records involves a geophysical examination of the conditions near 107 a site. By examination of the c h a r a c t e r i s t i c s of the f a u l t l i n e , i n con- j u n c t i o n with the s o i l and rock conditions between the f a u l t and the s i t e , the.properties of the ground shaking which would occur at a s i t e due to a c t i v i t y along the f a u l t l i n e may be i n f e r r e d . The i m p l i c a t i o n of t h i s work i s that motion which has been recorded at a p a r t i c u l a r s i t e may bear l i t t l e r e l a t i o n to the motion to be expected at another s i t e . However, geophysical work in t h i s f i e l d i s c u r r e n t l y in a developing t h e o r e t i c a l state and use of an accelerogram to be expected f o r a p a r t i c u l a r s i t e does not seem j u s t i f i e d at the present time. By the same token, i t appears that the widespread use of the El Centro and T a f t a c c e l e r o grams i s also not j u s t i f i e d . As i t w i l l be a s i g n i f i c a n t length of time u n t i l an adequate quantity of recorded accelerograms are obtained, or u n t i l a d e f i n i t e d e s c r i p t i o n of the accelerogram to be expected at a p a r t i c u l a r s i t e has been d e f i n e d , many researchers have s t a t i s t i c a l l y developed ensembles of simulated earthquake records. These records are based upon a s t a t i s - t i c a l evaluation of recorded earthquake accelerogram p r o p e r t i e s , the most important of which are duration, i n t e n s i t y and frequency content. Currently among the more commonly used simulated accelerograms are those developed by Jennings; Housner and Tsai [17]. These earthquake records are based upon a Gaussian random process developed previously by the f i r s t two authors to model the c e n t r a l portion of strong earthquakes, m u l t i p l i e d by a time dependant envelope f u n c t i o n , r e s u l t i n g in a nonstationary random process. These records have been created in pairs of d i f f e r e n t earthquake types and f o r each 108 type i t i s intended that the two accelerograms may serve to represent the two components of the same earthquake. This feature i s p a r t i c u l a r l y d e s i r a b l e f o r the current a n a l y s i s i n which the three-dimensional motion of the structure i s included. Earthquake type B was selected f o r the current a n a l y s i s . This earthquake has a t o t a l duration of 50 seconds of which 11 seconds of the motion are"irit'the:strongTmbtionppoHcion. TTheeenvMppe^function for this earthquake type i s shown in F i g . 30 and the a c c e l e r a t i o n , v e l o c i t y and displacement records are shown f o r the two components, B-l and B-2, in Figs. 31a and 31b. The pseudo-velocity spectra are shown i n Figs. 32a and 32b and are r e p l o t t e d on the standard t r i p a r t i t e logarithmic paper in Figs. 33a and 33b. Earthquake type B i s designed to represent the . strong gr-ound.shak.ing i n a magnitude 7^0^ gijeaterrearthquake^with the* same spectrum i n t e n s i t y as the El Centro 1940 earthquake. I t has been shown by Newmark [23] that a r o t a t i o n a l component of ground motion may also occur during an earthquake. This component may have a s i g n i f i c a n t e f f e c t on the t o r s i o n a l response of the s t r u c t u r e , but i t was neglected i n t h i s 5.3 analysis. Selection of the Time Step Length The frame described i n Section 5.1 was subjected to component B-l i n the q d i r e c t i o n and component B-2 in the r direction. In order to s e l e c t a length of time step which would provide accurate r e s u l t s i t i s necessary to determine the natural periods of the b u i l d i n g . 109 The National B u i l d i n g Code of Canada [22] states that f o r framed structures the fundamental period may be estimated as one-tenth of the number of storeys which implies a natural period of 0.5 for t h i s b u i l d i n g . seconds A more accurate eigenvalue analysis r e s u l t e d in a period of 0.91 seconds f o r motion i n the q d i r e c t i o n (with coupled t o r s i o n a l motion due to the e c c e n t r i c i t y of the centre of mass), seconds f o r motion i n the r 0.91 d i r e c t i o n and 0.52 seconds for motion p r e - dominantly t o r s i o n a l with coupled motion in the q direction. As the period s p e c i f i e d by the NBC i s a r e s u l t of measurements of actual b u i l d ings in which a d d i t i o n a l s t i f f e n i n g elements such as walls and p a r t i t i o n s cause the periods to be lower than f o r the framing system t r e a t e d alone, i t i s f e l t that the r e s u l t s of the eigenvalue analysis are appropriate. Hence in determining the time step to be used and the damping f a c t o r a value of 0.91 seconds was selected f o r the period of the f i r s t mode of v i b r a t i o n . I t was stated by Walpole and Shepherd [31] that a time step equal to |g- of the f i r s t period would provide accurate r e s u l t s . Hence for t h i s structure a time step less than 0.023 seconds should be used by t h i s c r i t e r i o n . Olough and Penzien [7] state that the time step should be no more than of the period of the highest mode shape which i s desired to be included i n the a n a l y s i s . For t h i s model the lowest period corresponds to the r o t a t i o n a l mode shape of the 5th mode which i s 0.05 seconds. be used. From t h i s c r i t e r i o n a time step of 0.005 seconds should Because the accelerograms f o r the input earthquake are defined at every 0.25 seconds, a time step of 0.005 seconds would allow the r e c ords to be accurately read i n , hence t h i s value was chosen. no 5.4 Response Results Use of the 0.005 second time step r e s u l t e d in 200 sets of r e s - ponse c a l c u l a t i o n s to be made f o r each second of the earthquake record and 10,000 sets of c a l c u l a t i o n s required f o r the e n t i r e record. each set of c a l c u l a t i o n s i t was d e s i r a b l e to monitor several For variables such as the displacements and rotations of the generalized co-ordinates, the member end f o r c e s , e l a s t i c and p l a s t i c deformations and energy d i s sipated. Hence a n a l y s i s of t h i s s i m p l i f i e d b u i l d i n g produced m i l l i o n s of pieces of data which posed a great problem to overcome i n order to analyze the output. In t h i s p a r t i c u l a r case the most e f f f i c i e n t manner of handling t h i s data was to output a l l data onto a magnetic tape and then use a u x i l l i a r y computer programs which could read the required data from the tape and p l o t the v a r i a t i o n of the response parameters and tabulate envelope values. Typical p l o t s of response values are shown in Figs., 34 to 36. These plots r e l a t e to the response of the f i f t h storey of the b u i l d i n g and Figs. 34, 35 and 36 correspond to motion in the (torsion) d i r e c t i o n s r e s p e c t i v e l y . a maximum value of 24 f t / s e c q , r and 6 F i g . 34a shows the a c c e l e r a t i o n with (75 per cent of g). Fig. 34b shows the r e l a t i v e v e l o c i t y which has a maximum of 2.1 f t . per second. The abso- lute v e l o c i t y , shown in F i g . 34c, i s computed by adding the ground v e l o c i t y to the r e l a t i v e v e l o c i t y and i s the important parameter in computing k i n e t i c energy. The peak value of absolute v e l o c i t y i s s l i g h t l y greater than for r e l a t i v e v e l o c i t y and equals 2.3 f t . per second. The expected lower displacement frequency i s shown rim F i g . 34d with a maximum of Ill roughly 0.3 f t . When these maximum values are compared with those shown for the response spectra of earthquake B-l in Figs. 32a and 33a i t can be seen that these response values are reasonable f o r a structure of fundamental period equal to 0.91 second and a damping r a t i o of 0.02. A more accurate comparison with these response spectra cannot be made as the analysis i s based on nonlinear behavior while the spectra a r e , of course, based on l i n e a r e l a s t i c theory. The envelope response values are shown in F i g . 37 and show the maximum values, in both a p o s i t i v e and negative sense, of the accel e r a t i o n , absolute and r e l a t i v e v e l o c i t i e s , and displacement at each storey l e v e l . Acceleration and v e l o c i t y curves are more or less symmet- r i c about the zero axis implying that the maximum values of these v a r i a bles were approximately equal i n both the p o s i t i v e and negative d i r e c tions of the axes. The displacement curves, however, are not symmetric, i n d i c a t i n g a permanent o f f s e t of the b u i l d i n g . This o f f s e t may also be v i s u a l i z e d in F i g . 35d. Envelope values for member forces were also c a l c u l a t e d and are shown in Figs. 38, 39 aridd40. Figs. 38a and 38b show the envelope of shear forces for the e x t e r i o r and i n t e r i o r beams r e s p e c t i v e l y . be noted that the shears i n frames 1 and 4, which run in the It q should direc- t i o n , are v i r t u a l l y i d e n t i c a l ; as are the shears for frames 2 and 3 which run in the r direction. This may be explained by the f a c t that shears in the e l a s t i c - p e r f e c t l y p l a s t i c component cannot exceed those c o r r e s ponding to the y i e l d moment. Thus, the occurrence of t o r s i o n w i l l not increase the shear forces s i g n i f i c a n t l y more than i f t o r s i o n did not 112 occur. Values for the shear envelope of the e x t e r i o r columns and i n - t e r i o r columns are shown i n Figs. 39a and 39b r e s p e c t i v e l y . Values f o r the column a x i a l force maximums for e x t e r i o r and i n t e r i o r columns are shown in Figs. 40a and 40b r e s p e c t i v e l y . The i n t e r i o r column a x i a l forces are v i r t u a l l y i d e n t i c a l i n each frame and i t should be noted that these columns do not experience t e n s i l e forces during the earthquake record. E x t e r i o r columns, however, do experience s i g n i f i c a n t a x i a l t e n - s i l e forces. Because the y i e l d moment of a column i s g r e a t l y decreased when the column undergoes tension, as shown i n F i g . 18, and as the y i e l d moment i s further reduced because of the b i a x i a l e f f e c t s of these corner columns, i t appears that i n c l u s i o n of these e f f e c t s i s necessary in the nonlinear analysis of t h i s b u i l d i n g . I t has been stated previously that a study of the f l u c t u a t i o n s in the various l e v e l s of energy in a structure during an earthquake may be a valuable tool i n developing a b e t t e r understanding of nonlinear r e s ponse. For t h i s reason, energy values have been c a l c u l a t e d at the end of each time step. Fig. 41 shows the v a r i a t i o n of recoverable s t r a i n energy occurring during the strong motion section of the earthquake. Curve C defines the v a r i a t i o n of k i n e t i c energy of the b u i l d i n g comprised of the a r i t h m e t i c sum of the energies in the three component d i r e c t i o n s . While the troughs of t h i s curve are generally not equal to zero there are a few times (e.g. at about 10.2 seconds) at which the t o t a l k i n e t i c energy of the b u i l d i n g becomes very close to zero. Curve A represents the sum of the k i n e t i c energy plus the recoverable s t r a i n energy stored in the b u i l d i n g as computed in the a n a l y s i s . I t was noted i n Section 113 3 . 7 . i i i that an e r r o r i n c a l c u l a t i o n of recoverable s t r a i n energy e x i s t e d in the computer program. This error occurs when a member has a p l a s t i c hinge at one end and i s i n an e l a s t i c state at the other end and undergoes further p l a s t i c r o t a t i o n at the hinge. In t h i s case an a d d i t i o n a l amount of recoverable s t r a i n energy i s accumulated each time t h i s event occurs. While a corrected run was not performed, an approximate corrected curve, as shown by curve B, may be devised by s e t t i n g the s t r a i n energy of the structure to a small quantity when a peak i n the k i n e t i c energy curve occurs. Figure 42 shows the r e l a t i o n s h i p between the energy d i s s i p a t e d by the structure and the recoverable energy of the s t r u c t u r e . Curve C of F i g . 42 i s the same as curve B of Fig. 41; the corrected recoverable energy curve. Curve B of F i g . 42 adds the e f f e c t of energy d i s s i p a t e d by viscous damping. Curve A adds the e f f e c t of energy d i s s i p a t e d by h y s t e r e t i c behavior or non recoverable s t r a i n energy. This curve com- pares favourably with the t o t a l energy input into the structure as c a l culated according to (3.35). The small difference in these values may be explained in part by the various assumptions used i n the c a l c u l a t i o n of the energy q u a n t i t i e s . For instance, the incremental energy d i s s i - pated at the p l a s t i c hinge of a column i s c a l c u l a t e d using the y i e l d moment in presence of the a x i a l load at the end of the time step. As the a x i a l load a c t u a l l y changes during the time step, an e r r o r in the c a l c u l a t i o n of d i s s i p a t e d energy w i l l occur. Another source of e r r o r r e l a t e s to the c r i t e r i o n f o r removing a hinge in a b i a x i a l column, as described iin Section 4.4. The assumption 114 that only one event occurs in a member during a time step may cause the c a l c u l a t i o n of h y s t e r e t i c energy d i s s i p a t e d to be i n c o r r e c t . : Nonetheless, i t i s f e l t that the t o t a l energy input i n t o the s t r u c t u r e , as shown by curve A of F i g . 42, and the sum of recoverable energy plus energy d i s s i pated due to damping, as shown by curve B, are accurate values f o r t h i s building. Hence the distance between the two curves represents energy d i s s i p a t e d due to h y s t e r e t i c behavior and the s l i g h t d i f f e r e n c e between these values and those c a l c u l a t e d by the program represent the e f f e c t of these sources of e r r o r . Figure 43 shows the number of times that the various elements make excursions into the p l a s t i c range. The e x t e r i o r beams show a large number of excursions, ranging up to 30, at each of the lower four storey levels. This large number of excursions may be explained by the low value of the y i e l d moment for the beams framing into the corner columns. On the other hand, the i n t e r i o r beams undergo few p l a s t i c excursions, ranging up to a maximum of two. The columns undergo the greatest number of p l a s t i c stages at the ground l e v e l where a maximum value of 35 i s reached. One s i g n i f i c a n t feature which can be seen from these plots i s that the level of p l a s t i c a c t i v i t y decreases r a p i d l y at the f i r s t storey level where a maximum of 6 excursions was experienced. This level then increased to a maximum of 14 at the second storey l e v e l and then decreased above t h i s l e v e l . This feature may be explained by the change to a "smaller column s i z e at the second storey l e v e l . I t sho.usiadbesnOitedd.thatttheEefl.ements- of_frame 4 generally experience a greater number of excursion into the p l a s t i c 115 range than the other frame running in the q Frames two and t h r e e , which run i n the d i r e c t i o n , show approximately r d i r e c t i o n , frame number 1. the same number of excursions. Figure 44 shows the v a r i a t i o n of the maximum accumulated p l a s t i c r o t a t i o n s of the various elements with respect to storey height. Research into nonlinear s t r u c t u r a l behavior commonly reports l e v e l s of p l a s t i c a c t i v i t y in terms of a d u c t i l i t y f a c t o r equal to the maximum p l a s t i c r o t a t i o n divided by the y i e l d r o t a t i o n plus one. This f a c t o r i s , however, a d i f f i c u l t concept to v i s u a l i z e f o r components subjected to a x i a l loads in which the y i e l d moment changes with respect to the level of a x i a l load. For t h i s reason, i t i s f e l t by the author that the amount of p l a s t i c r o t a t i o n i s a more reasonable quantity to report. Figure 44 shows s i m i l a r features as F i g . 43 i n which the l e v e l of p l a s t i c a c t i v i t y i s generally greater i n frame 4 than f o r frame 1 while frames 2 and 3 experience s i m i l a r l e v e l s . A l s o , the l a r g e s t l e v e l of p l a s t i c r o t a t i o n i s observed i n the columns at the ground l e v e l with a large decrease at the f i r s t storey l e v e l followed by an increase at the second storey l e v e l . The t o t a l p l a s t i c r o t a t i o n s experienced by the i n - t e r i o r columns i s roughly the same as f o r the e x t e r i o r columns while those of the e x t e r i o r beams i s s i g n i f i c a n t l y greater than f o r the i n t e r i o r beams. Figure 45 shows the amount of energy d i s s i p a t e d by h y s t e r e t i c behavior p l o t t e d at each storey l e v e l f o r each component of each frame. In general the beams d i s s i p a t e d s l i g h t l y under one-half of the t o t a l energy d i s s i p a t e d with the remaining amount s p l i t between the i n t e r n a l 116 and external columns i n a r a t i o of between 3 and 4 to one. While Figs. 43 and 44 showed that the number of excursions i n t o the p l a s t i c range and the accumulated p l a s t i c r o t a t i o n s of the i n t e r i o r and e x t e r i o r c o l umns was roughly the same, the f a c t that the i n t e r i o r columns d i s s i p a t e d a s i g n i f i c a n t l y greater amount of energy than the e x t e r i o r columns may be explained by the f a c t that the e x t e r i o r columns become p l a s t i c at a lower moment level due to the higher a x i a l forces in these members. Figure 45 also shows that approximately one-half of the t o t a l energy was d i s s i p a t e d by frames running in each d i r e c t i o n . However frames 2 and 3 d i s s i p a t e d roughly the same amount of energy while the level d i s s i p a t e d in frame 4 was roughly twice as much as that f o r frame 1. The d i s p a r i t y in energy l e v e l s d i s s i p a t e d by the two frames running in the q d i r e c t i o n i n d i c a t e s that a plane frame a n a l y s i s , which would d i v i d e these q u a n t i t i e s evenly between the two frames, would r e s u l t i n a s i g n i f i c a n t misrepresentation of these energy l e v e l s . It appears that the i n c l u s i o n of three dimensional behavior i n c l u d i n g t o r s i o n a l e f f e c t s has a s i g n i f i c a n t e f f e c t on t h i s a n a l y s i s . Figure 46 shows the data of F i g . 45 combined to represent the t o t a l values f o r the s t r u c t u r e . 5.5 Summary of the Analysis I t i s apparent that the nonlinear analysis of even a simple framed b u i l d i n g such as that described in t h i s chapter w i l l provide a multitude of data, a small portion of which may be presented as shown 117 in t h i s t h e s i s . Of course no general conclusions can be i n f e r r e d from a s i n g l e a n a l y s i s , but i t i s f e l t that a comparative analysis of r e s u l t s of t h i s type, performed on further s t r u c t u r a l models, w i l l provide a better understanding of the response of buildings to strong motion earthquakes. 118 CHAPTER 6 SUMMARY AND CONCLUSIONS The t h e o r e t i c a l basis and p r a c t i c a l a p p l i c a t i o n s of a mathematical model f o r analyzing the nonlinear three-dimensional response of structures to earthquake e x c i t a t i o n has been presented. The nonlinear behavior of the model i s represented by the use of a b i l i n e a r moment-rotation curve. In the l i g h t of t h i s r e s t r i c t i o n a general form of the member s t i f f n e s s matrix, i n c l u d i n g shear d e f l e c t i o n s , geometric n o n l i n e a r i t i e s and r i g i d end stubs was defined. Methods used i n the formation and subsequent reduction of the structure s t i f f ness matrix, to allow f o r e f f i c i e n t s o l u t i o n of the incremental d i f f e r e n t i a l equation, were presented. The e f f e c t s of a x i a l force i n t e r a c t i o n and b i a x i a l bending on the nonlinear behavior were included using s p e c i f fccassumptions on t h e i r i n f l u e n c e . A technique of s o l v i n g the incremen- t a l equation and including the nonlinear e f f e c t s in t h i s equation were presented. The equations used to c a l c u l a t e the l e v e l of energy input i n t o a structure and in the various forms which the energy i s dispersed throughout the structure were discussed. The theory developed was applied to various models of a s t r u c ture intended to represent a t y p i c a l b u i l d i n g designed under current techniques. Various problems were encountered i n the modelling of t h i s dual component shear wall-frame b u i l d i n g . The accurate modelling of a shear wall using frame elements was found to be complicated, e s p e c i a l l y i f b i a x i a l e f f e c t s are to be included. In a d d i t i o n , as shear cores are 119 i n v a r i a b l y slender elements, sizeable a x i a l loads develop and the i n t e r action of these a x i a l forces on the nonlinear behavior becomes an important f a c t o r . In the model under c o n s i d e r a t i o n , i t was found that a x i a l forces reached a c r i t i c a l point at which, i t was necessary to include axial y i e l d i n g . Other f a c t o r s reader the nonlinear, three dimensional of shear wall and core elements a questionable endeavour. analysis Idealization of the p l a s t i c zone of a wall as an i n f i n i t e s i m a l hinge does not seem justified. In a d d i t i o n , the magnitude of loads developed plus the low natural periods of these components require that a small time step be used i n order that the errors introduced by the approximations required to economically include nonlinear behavior not dominate the a n a l y s i s . Having abandoned the shear core b u i l d i n g , a model comprised of four e x t e r i o r f i v e storey frames was analyzed. While no general con- clusions can be made from t h i s s i n g l e example, a few points deserve d i s cussion. The most noteworthy observation which can be made i s that the r e s u l t s of t h i s analysis d i f f e r s i g n i f i c a n t l y from those which would be obtained by use of a plane frame a n a l y s i s . The i n c l u s i o n of three-dimen- sional e f f e c t s i n conjunction with t o r s i o n cause response parameters of two i d e n t i c a l and p a r a l l e l frames to show s i g n i f i c a n t l y d i f f e r e n t values. In a d d i t i o n , s o l u t i o n of t h i s r e l a t i v e l y small problem y i e l d e d l i t e r a l l y m i l l i o n s of pieces of data. I n t e r p r e t a t i o n of these r e s u l t s presented a considerable problem and comparison with a l t e r n a t i v e examples would appear to be an undertaking of immense proportion. 120 Perhaps the most important conclusion which can be made from t h i s work concerns i t s place i n the r a p i d l y evolving sphere of computer analyses of s t r u c t u r a l problems. Twenty-five years ago when the com- puter f i r s t became used f o r s t r u c t u r a l problems, the s t a t i c analysis of frames r e s t r i c t e d to a small number of degrees of freedom posed a large undertaking of considerable economic consequence. Vastly more e f f i c i e n t f o l l o w i n g generations of computers, in conjunction with s o l u t i o n t e c h niques devised to reduce computational costs, allowed f o r greater f r e e dom i n the s e l e c t i o n of s t a t i c problems to be solved. As the e f f i c i e n c y of computer solutions increased i t became possible for more complex problems such as nonlinear f i n i t e element analysis and dynamic analysis to be made. In the twenty-five years of computer solutions to s t r u c t u r a l problems, the cost of solutionhhas reduced by several orders of magnitude. I t i s f e l t by the author that nonlinear dynamic a n a l y s i s using the current generation of computers i s a problem of s i m i l a r proportion to that of s t a t i c analysis twenty-five years ago. Solution may be reached w i t h i n reasonable economic l i m i t s provided the s i z e of the problem i s greatly r e s t r i c t e d . However analysis of these problems must be r e s - t r i c t e d to a minimal number of degrees of freedom and must incorporate several r e l a t i v e l y crude approximations in order to be economically justified. Many computer experts believe that in the next ten years the cost of performing a given analysis w i l l be two or three orders of magnitude less than current c o s t s . For t h i s reason the author r e a d i l y ad- mits the f a c t that theory presented i n t h i s thesis w i l l be e a s i l y 121 surpassed i n the coming years. In p a r t i c u l a r , member i d e a l i z a t i o n using a more accurate degrading s t i f f n e s s r u l e should be made. Solution of the equations of motion should be coupled with an i t e r a t i v e technique, such as the Newton-Raphson method, to accurately include nonlinear e f fects. More f l e x i b l e i d e a l i z a t i o n of the e f f e c t s of a x i a l - f o r c e i n t e r - action and b i a x i a l behavior should be allowed. These l i m i t a t i o n s , im- posed on analysis made on e x i s t i n g computer systems, w i l l i n e v i t a b l y be removed from future analyses. I t i s hoped thaththis work w i l l serve as a stepping stone f o r future research concerned with the accurate s o l u t i o n of the nonlinear, threervd.tmehs.ionaiLresponse ;r:jfieJadjn.gstors.ionailisbehavior^of framed - buildings to earthquake . e x c i t a t i o n . 122 BIBLIOGRAPHY 1. Anagnostopoulous, S.A.; Roesset, J.M. and Biggs, J.M. 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"The Torsional Response of High Rise B u i l d i n g s " , Master's Thesis, U n i v e r s i t y of B r i t i s h Columbia, Vancouver, 1974. 20. M a z i l u , P.; Sandi, H. and Teodorescu, D. "Analysis of Torsional O s c i l l a t i o n s " , Proceedings of the F i f t h World Conference on Earthquake Engineering, V o l . 1, Rome, 1973, pp. 153-162. 21. Nathan, N.D. and MacKenzie, J.R. "Torsional Analysis of Framed S t r u c t u r e s " , Internal Report. 22. National B u i l d i n g Code of Canada. Subsection 23. Newmark, N.M. (1977). " E f f e c t s of Earthquakes", 4.1.9. "Torsion in Symmetric B u i l d i n g s " , Proceedings of the Fourth World Conference on Earthquake Engineering, V o l . I l l , Santiago,f-'T969, pp. -A3: "19-32. 24. Parme, A.L.; Nieves, J.M. and Gouwens, A. "Capacity of Reinforced Rectangular Columns Subject to B i a x i a l Bending", Journal of the American Concrete I n s t i t u t e , September, 1966, pp. 911-923. 25. 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Tso, W.K. and Asmis, K.G. "Torsional V i b r a t i o n of Symmetrical S t r u c t u r e s " , Proceedings of the F i r s t Canadian Conference on Earthquake Engineering, Vancouver, 1971, pp. 178-186. 30. Tso, W.K. and Bergmann, R. "Dynamic Analysis of an Unsymmetrical High Rise B u i l d i n g " , Canadian Journal of Cavil Engineering, V o l . 3, March, 1976, pp. 107-118. 31. Walpole, W.R. and Shepherd, R. " E l a s t o - P l a s t i c Seismic Response of Reinforced Concrete Frames", American Society of C i v i l Engineers, S t r u c t u r a l D i v i s i o n , October, 1969, pp. 2031-2055. 32. Walpole, W.R. and Shepherd, R. "The I n e l a s t i c Response of a Steel Frame", Proceedings of the Fourth World Conference on Earthquake Engineering, V o l . II, Santiago, 1969, pp. A4: 195-204. 33. Weaver, W. and Nelson, M.F. "Three Dimensional Analysis of T i e r B u i l d i n g s " , American Society of C i v i l Engineering, S t r u c t u r a l D i v i s i o n , December, 1966, pp. 385-404. 34. Wynhoven, J.H. and Adams, P.F. "Analysis of Three Dimensional S t r u c t u r e s " , American Society of C i v i l Engineers, S t r u c t u r a l D i v i s i o n , January, 1972, pp. 233-248. 35. Wynhoven, J.H. and Adams, P.F. "Behavior of Structures under Loads Causing T o r s i o n " , American Society of C i v i l Engineers, S t r u c t u r a l D i v i s i o n , J u l y , 1972, pp. 1361-1376. TABLE 1: FACTORS USED IN TIME STEP INTEGRATION FACTOR NEWMARK 6 METHOD a 4 6 4 6 2 3 a, 2 3 a 5 2 3 c 0 0.5 a a I 2 WILSON 6 METHOD b TAR/ - • STATE • •' TABLE 2: <v TTWT ~~£" TW,-jrpDA'"';3' T r A POSSIBLE MEMBER STATES JOINT i JOINT j A NO HINGE NO HINGE B HINGE NO HINGE C NO HINGE HINGE D HINGE HINGE TABLE 3: CONSTANTS USED IN BUILDING THE MEMBER STIFFNESS MATRIX r"cir CASE P % A 1 0 0 B P 1-p 0 C P 0 1-p D P 0 0 ro 128 TABLE 4: STOREY EXAMPLE DESIGN ECCENTRICITIES AND TORSIONAL MOMENTS CALCULATED ECCENTRICITY DESIGN ECCENTRICITY (FT.) MAXIMUM MINIMUM TORSIONAL MOMENT (K-FT) MAXIMUM MINIMUM 1 12.42 44.77 2.46 226.7 12.45 2 10.79 39.87 1.65 403.7 16.71 3 9.67 18.27 1.09 277.5 16.55 4 8.77 16.91 0.64 342.4 13.00 5 8.00 15.74 0.25 398.5 6.33 6 7.31 14.72 -0.01 447.2 -0.3 7 6.69 13.78 -0.41 488.3 -14.5 8 6.10 12.90 -0.70 522.5 -28.4 9 5.51 12.01 -1.00 547.5 -45.6 10 5.00 11.25 -1.25 569.6 -63.3 11 4.44 10.41 -1.53 579.9 -85.2 12 3.91 9.62 -1.80 584.4 -109.4 13 3.37 8.81 -2.07 579.6 -136.2 14 2.81 7.97 -2.35 564.8 -166.6 15 2.23 7.10 -2.64 538.8 -200.5 129 TABLE 5: TORSIONAL MAGNIFICATION FACTORS FOR FRONT FRAME OF ECCENTRIC SHEAR CORE MODEL (1) SOSTcOREY DESIGN SHEAR (KIPS) (4) TORSIONAL SYMMETRIC (3) ECCENTRIC "MAG. FACTOR 1 0.31 2.14 6.90 2 0.77 4.48 5.82 3 1.37 4.09 2.99 4 2.10 5.57 2.65 5 2.96 7.09 2.40 6 3.90 8.59 2.20 7 4.94 10.11 2.05 8 6.07 11.64 1.92 9 7.33 13.10 1.80 10 8.62 14.71 1.71 11 IOC-De- 16.27 1.62 12 l l ' ^ * 17.83 1.54 13 13?20: 19.40 1.47 14 14.97 21.00 1.40 15 16.88 22.60 1.34 TABLE 6: | Q D E MODAL ANALYSIS OF MODEL A PERIOD ( s e c ' ) " i . : NUMBER'OFCCROSSOVERS (+1) q~ r 9~ 1 3.22 0 1 0 2 2.29 1 0 1 •n PC-' 3 0.94 1 0 2 4 0.67 2 0 2 5 0.64 0 2 0 6 0.34 3 0 3 7 0.25 0 3 0 8 0.20 4 0 4 9 0.16 2 0 5 1 131 TABLE 7: MODE PERIOD MODAL ANALYSIS OF MODEL B NUMBER OF CROSSOVERS (+1) (sec.) 1 3.22 0 1 0 2 2.46 2 0 1 3 1.68 1 0 2 4 0.73 2 0 2 5 0.64 0 2 0 6 0.40 2 0 3 7 0.32 2 0 3 8 0.25 0 3 0 9 0.25 4 0 4 8 ^ V A * CENTRE OF MASS i 1 STRUCTURE ORIGIN FIG. 2 : PLAN VIEW OF TYPICAL STOREY 133 My + pk 8y ROTATION FIG. 3 : BILINEAR MOMENT-ROTATION RELATIONSHIP AT A MEMBER END 1 k. ROTATION FIG. 4 : TYPICAL MOMENT-ROTATION RELATIONSHIP AT AN END OF A REINFORCED CONCRETE BEAM s 134 My 4- ROTATION 5(a): MOMENT-ROTATION RELATIONSHIP AT AN END OF THE ELASTIC COMPONENT ROTATION FIG. 5(b): MOMENT-ROTATION RELATIONSHIP AT AN END OF THE ELASTIC-PERFECTLY PLASTIC COMPONENT ELASTIC COMPONENT FIG. 6 : TWO COMPONENT MODEL FIG. 8 : MEMBER DEGREES OF FREEDOM 136 STAIRS CENTROID OF STAIR CORE COLUMN WALL LINTEL BEAM LINTEL BEAM ELEVATOR SHAFT FIG. CENTROID OF ELEVATOR CORE MODEL 9 : MODEL OF SHEAR CORE RIGID STUB FLEXIBLE SECTION OF THE BEAM .,2' e l k FIG. 10 : BEAM WITH RIGID END STUBS ( L I N T E L BEAM) 137 V,. 1 M ,B 2 2 K FIG. 11 : NOTATION USED IN FORMULATION OF THE GEOMETRIC STIFFNESS MATRIX -V>§ f_ y(6-0) ii 10 L (,2 1 ; 2 (12--£) fl 5 L 6 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 27O-90v Ei 30-0 L r V ; 3 FIG. 12(b): FORCES DUE TO REMOVAL OF MOMENT AT JOINT H i f 30-130') j, L 12 3 ^TTJ^T 30-0 FIG. 12(c): FORCES DUE TO UNIT VERTICAL DISPLACEMENT OF JOINT i (MEMBER STATE B) ; 138 FIG. FiG. 13(a): 13(b): FORCES DUE TO UNIT ROTATION OF JOINT j (MEMBER STATE A ) FORCES DUE TO REMOVAL OF MOMENT AT JOINT i 3_LL f 30-3 4K 2 vin_/+\ 30-0 / 3EI ,30-3 L ^30-0 2 FIG. ; 13(c): /3O-3 0V ^30-0 ; FORCES DUE TO UNIT ROTATION OF JOINT j (MEMBER STATE B) 139 FIG. 14(b): BUILDING WITH VERTICAL COMPATIBILITY 140 a 18 e e it ^3 3 4 0 11 3 1 5 12 t^6 -CORNER COLUMN CORNER COLUMN JJ/\Wt /I) VA ) 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 T Y P I C A L MEMBER COMPRESSIVE (NEGATIVE) AXIAL LOAD MOMENT TENSILE (POSITIVE) AXIAL LOAD FIG. 1 8 ( a ) : AX I A L - F L E X U R A L INTERACTION DIAGRAM 143 FIG.18(c) YIELD SURFACE FOR B I A X I A L COLUMNS MOMENT M 2 i /, >> Mu My — ' M, - 3 / FIG. 19(a): L 8 2 = 8 3 ROTATION UNBALANCED MOMENT CORRESPONDING TO MEMBER END YIELDING MOMENT FIG. 19(b): UNBALANCED MOMENT CORRESPONDING TO MEMBER END REVERSAL 145 FIG 19(c): UNBALANCED MOMENT CORRESPONDING TO A CHANGE IN Y I E L D 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 1 75k 20k 100k 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 7-16 CO .UMN L l NE 3 COLUMN LINE 2 COLUMN LINE 1 STOREY My Mb 135 500 675 210 1800 Pt Pt Pc My Mb Pb 290 2260 240 695 840 My Mb Pb Pt 180 635 840 Pt 380 Pc 2290 1-7 180 780 1060 290 3050 300 1050 1320 480 3880 400 1150 1320 600 3950 0-1 240 945 1195 380 3720 625 1510 1490 1310 5240 625 1510 1490 1310 5240 ULTIMATE MOMEN S (KIP-FT.) EXTERIOR BEAMS STOREY LINE 1 POS. INTERIOR BEAMS LINE 2 NEG. POS. NEG. LINE 1 POS. NEG. LINE 2 POS. NEG. 2-16 350 -175 350 -505 495 -360 495 -435 1 355 -180 355 -595 870 -450 580 -530 FIG. 21: FRAME 1 OF THE CORE BUILDING AXIAL LOADS PER STOREY 60k 20k 60k 19" x 22" DP. (TYP. BEAM) (TYP. EXT. COLUMN) 30" x 24" DP. (TYP. INT. BEAM) 2 2 3 x 25' MOMENTS (KIP-FT.) AND AXIAL FORCES (KIPS) TO DEFINE INTERACTION DIAGRAM STOREY COLUMN LINE 2 COLUMN LINE 1 My M 240 695 840 My M|j p 7-16 135 500 675 210 1800 1-7 180 780 1060 290 3050 400 1150 3720 625 1510 240 0-1 b 1195 945 P t 380 p c b P b P t INTERIOR STOREY LINE 1 POS. NEG. LINE 2 LINE 1 POS. NEG. POS. NEG. c 380 2290 1320 600 3950 1490 1310 5240 ULTIMATE MOMENTS (KIP-FT.) EXTERIOR BEAMS p BEAMS LINE 2 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 ia 1 -6' - 7.29 x 10 in s x x A = 1780 in = 0.25 I. 6 yy 2 E = 3.6 x 10 psi s MOMENTS (KIP-FT) AND AXIAL FORCES (KIPS) TO DEFINE AXIAL INTERACTION DIAGRAM STOREY AXIAL 15-16 14-15 13-14 12-13 11-12 10-11 9-10 8-9 7-8 6-7 5-6 4-5 3-4 2-3 1-2 0-1 U y Pc M b 50 1 , 540 10,540 1,810 170 100 150 200 't 250 . 1 300 2,700 10,870 1,810 260 350 400 < < 450 500 3, 380 17,700 3, 020 300 550 600 650 700 750 i it \r 800 5, 420 \ 5,450 ' 9, 130 i WALL 9 IB'-B". r - 3.88 x I0 in 6 XX jzc A = 1190 in 6" JI MOMENTS (KIP-FT) AND AXIAL FORCES (KIPS) TO DEFINE AXIAL INTERACTION DIAGRAM V Pb STOREY AXIAL M y 15-16 30 47 5 6, 355 1 410 , 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 5-6 330 47 5 10 46 5 2,350 4-5 360 3-4 390 2-3 420 i-2 450 v•r 0-1 480 1 p = 0-25 c 100 3,620 t -> 100 6, 035 > \ FIG. 2 3 : WALLS 8 AND 9 OF THE CORE BUILDING 6 I yy = 0 2 E = 3.6 x 10 psi 6 ( a ) FRAME ( b ) WALL FIG. 24: TYPICAL DEFLECTION SHAPES FIG. 25: LATERAL SEISMIC FORCES ( K I P S ) REQUIRED BY THE NBC ( I N THE q DIRECTION) Cl L> O- -o- BEAM B1 1 (in ) 4 Av (in ) 2 8,000 320 B2 10,560 440 B3 125,000 360 COLUMN Cl © -0- DIRECTION A (in ) I (in ) 2 4 27,650 q C2 q 6.65 x 10 6 C2 r 8.1 x 10 6 C3 C4 54,000 q . 0 q r 13.4 x 10 6 C5 q 1.78 x 10 8 2 240 3455 1390 3455 2140 720 27 40 C4 FIG. 26: MODEL A 288 Av (in ) 600 0 5450 2620 10,900 5000 C2 CI 38C »BI(9) ©I B3 C4 el 0 a -a- -oBEAM I (in ) 4 Av (in ) 2 B1 8,000 320 B2 10,560 440 B3 250,000 720 COLUMN CI C2 C2 DIRECTION Av = SHEAR AREA 1 (in ) 27,650 1 .33 1.62 C3 q C4 q 1.78 C4 r 2.68 x 10 x 10 7 54,000 FIG x x 10 8 10 7 27: MODEL B A (in ) Av (in ) 288 240 2 4 q q r -d -o- 7 2 6910 2780 6910 4280 720 600 10,900 5000 10,900 5240 CI C2 B1 "HI BJ C3 CD f B2 I C3 C4 M 0 © -Q- -TJ- 1 (in ) 4 Av (in ) B1 8,000 320 B2 10,560 440 BEAM COLUMN C1 C2 C3 DIRECTION 2 I (in ) A (in ) Av (in ) 27,650 4 q q -D- 2 288 240 7 6910 2780 54,000 720 600 10,900 6000 17,800 1.33 x 1 0 q 2 C4 q 1.78 x 10 8 C4 r 3.11 x 1 0 8 FIG. 28: MODEL C 11 ,040 154 BEAM LEVEL STOREY WT. STOREY POLAR WT.MOMENT OF INERTIA (k> (k-f't2) 655 (393000) 1 SQUARE COLUMNS - BEAMS INT.WIDE DEEP EXT 612 (370000) 612 18° 2 2 " 13" 20" 20' 15" 22" (370000) 612 (370000) 370 24" (222000) -*- \K 2 ° ' »|« 2 0 ' >I- 2 0 0" COLUMN L I N E ' PLAN VIEW OF T Y P I C A L STOREY ELEVATION VIEW OF T Y P I C A L FRAME E = 5000 KSI ULTIMATE MOMENTS IN BEAMS (KIP-FT.) LINE 2 LINE 1 LEVELS POS POS NEG NEG 1-2 340 -240 680 -480 3-5 230 -165 460 -325 MOMENTS ( K I P - F T ) AND A X I A L FORCES ( K I P S ) TO DEFINE BEAM LEVELS INTERACTION DIAGRAM COLUMN LINE 2 COLUMN LINE 1 My M Pb Pt Pc My M Pb Pt 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. b 2 9 : PROPERTIES OF THE F I V E b STOREY FRAMED B U I L D I N G P c EARTHQUAKE B 50 ^t(sec) tn 156 rl5 ( FT/SEC ) 2 Ho , TIME ( SEC ) EARTHQUAKE |3Q 140 B-l Figure 31(a) Acceleration, v e l o c i t y and displacement for earthquake B - l ,50 160 EAR ,0* .OS HQUAKE ,0 8 .1 ~ ,2 B-l .4 .6 .8 i & 6 PERIOD (sees) F i g u r e 33(a) T r i p a r t i t e l o g a r i t h m i c p l o t of s p e c t r a f o r earthquake B - l id 161 EARTHQUAKE B-2 TI ~* ^ I 2 4 PERIOD (sees) Figure 33(b) T r i p a r t i t e logarithmic plot of spectra for earthquake •^f 1 " ' ,02 ,0 4 .06 .08 .1 B-2 162 -i 659'9T r SZO M - NOUUc)313338 8t- KtS'K- Fi'gure 34(a). Acceleration (ft./sec ) of storey 5 i n the q d i r e c t i o n Figure 34(b) Relative v e l o c i t y (ft./sec) of storey 5 i n the q d i r e c t i o n 5»'D 0-C UI30T3A S9H Figure J.3M0IS - 1 — i'O- 34(c) solute v e l o c i t y (ft./sec) of storey 5 i n the q d i r e c t i o n Figure 34(d) Displacement ( f t . ) of storey 5 i n the q d i r e c t i o n Figure Acceleration 35(a) 2 ( f t . / s e c ) of s t o r e y 5 i n the r direction Figure 35(b) Relative v e l o c i t y (ft./sec) of storey 5 i n the r d i r e c t i o n 168 Figure 35(d) Displacement ( f t . ) 'of storey 5 i n the r d i r e c t i o n 170 Figure 36(a) 2 Acceleration (rad./sec ) of the r o t a t i o n a l component of storey 5 Figure 36(b) Relative v e l o c i t y (ft./sec) of the r o t a t i o n a l component of storey 5 IO ROT nro o-o eo'osso'o( i-OIX) U I 3 B T J A S8B JGittliS Figure 6sro- 36(c) Absolute v e l o c i t y (ft./sec) of the r o t a t i o n a l component of storey 5 «.ro- 173 Figure 36(d) Displacement ( f t . ) of the r o t a t i o n a l component of storey 5 174 -3.0 - 2 . 0 -1.0 1.0 2.0 3.0 Q(FT/SEC ) 2 FIG.37(D): - 3 . 0 -2.0 -1.0 1.0 2.0 R(FT/SEC ) 2 3.0 -1 x10~ 1x10" 2 8 (RAD/SEC) RELATIVE VELOCITY Vs. STOREY LEVEL ENVELOPE 175 Q ( F T . /SEC. ) FIG-37(c) Q(FT.) FIG.37(d): RCFT./SEC.) 8 (RAD/SEC) ABSOLUTE VELOCITY Vs. STOREY LEVEL ENVELOPE R(FT.) DISPLACEMENT Vs. STOREY LEVEL ENVELOPE 8 (RAO.) 176 FIG.38(D): INTERIOR BEAMS SHEAR ENVELOPE (KIPS) 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. TIME (SECONDS) Figure 41 V a r i a t i o n of recoverable energy during the f i r s t 22 seconds 22 180 Figure 42 V a r i a t i o n of t o t a l 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 __i -.04 C. 1 1 -.03 BEAMS -.02 1 -.01 B. 1 i 1 .01 .02 1 .03 v 1 .04 .05 i 1 -.003 0. EXTERIOR COLUMNS FIG.44 INTERIOR BEAMS -.002 1 -.001 INTERIOR ACCUMULATED PLASTIC ROTATION 1 1 i .001 .002 COLUMNS 1 .003 183 FRAME 1 TOTALS 106.2 KIP-FT. ( 1 5 . 7 % OF TOTAL ENERGY D I S S I P A T E D ) FRAME 2 TOTAL— 1 7 5 . 0 K I P - F T . (25.8'/, OF TOTAL ENERGY D I S S I P A T E D ) FIG.45 FRAME 4 TOTAL — 220 KIP-FT. ( 3 2 . 5 % OF TOTAL ENERGY DISSIPATED) FRAME 3 TOTAL— 1 7 6 . 4 KIP-FT. (26% OF TOTAL ENERGY DISSIPATED) DISSIPATED ENERGY (KIP-FT.) OF VARIOUS COMPONENT 184 TOTAL FIG.46 677.6 KIP-FT. DISSIPATED ENERGY OF THE ENTIRE BUILDING 185 APPENDIX A CALCULATION OF THE EVENT FACTOR FOR A BIAXIAL COLUMN The f r a c t i o n of a time step f o r which an e l a s t i c end o f a b i a x i a l column remains unyielded, c a l l e d the event f a c t o r y , can be c a l 3 culated from the information shown in F i g . A . l . , Mu ( t dM, ^ \ M Figure A . l At ifehe beginning of the time step the member end was, i n the e l a s t i c state 1 with moments about the two p r i n c i p l e axes of M . . M . and 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 n c i p l e axes of dM and 1 dM . 2 Assuming a l i n e a r v a r i a t i o n of these moments with r e s 3 pect to time, the condition at y i e l d may be expressed as shown in /M . + p d M \ 2 /M . + y d M\* (A.l) 186 Introducing the f a c t o r s M • a 1 E a = M P l E K = t shown in 5 (A.2) = 21 M P l EP 2 2 , 2 2 a rj M a 3 2 a a M . L 2 d to 1 d M _JJL M a Er2 R N 5 = a + a 2 4 (A.2) equation ( A . l ) may be rearranged as a quadratic function of the event factor as shown in y a 5 y (A.3) + ( Z a ^ + ^ a J ^ ^ 2 .0 (A.3) which may be solved f o r y as shown in the f o l l o w i n g steps: •2a a--. - 2a;a -± / 4 ' a f a U = 1 2 • - -3 34 - i + 8a Saa a 2 1 2 12 3 * 4a a 2 4 2a 4.i4a-: (a/-+ 2 3 4 5 v 1 a 2 3 - 1)' ' 5 (A.4) -a a °1 y /T . 2 2 -, 2 2 ± / 2a a aaaa --a^an 7 a a + a - a a 2 3 4 V " 2 2 3 ,'. a a -a a = 1 it 5 (A.5) = C y l 2 -3 1-lti, I 3 12 a 5 (a 1a 4 v % s a a ) 2 3 ' (A.6) I t may be shown by examination of t h i s equation i n each of the four quadrants that the p o s i t i v e square root w i l l always give the proper event f a c t o r . 187 APPENDIX B CALCULATION OF THE ENERGY INPUT INTO A STRUCTURE The energy input during a s i n g l e time increment i s found by i n t e g r a t i n g the product of the base shear force and the ground v e l o c i t y with respect to time, as shown i n A in which E F(t) v(t) dt in A E- (B.l) n F(t) v(t) = incremental energy input = base shear force (B.l) == gground v e l o c i t y T = length of time increment As the base shear force i s only c a l c u l a t e d at the beginning and end o f the time increment, the assumption of l i n e a r v a r i a t i o n of base shear force over the time increment i s assumed as shown i n (B.2) (B.2) in which F F q = base shear force at the beginning of the time increment = base shear force at the end of the time increment The earthquake ground a c c e l e r a t i o n record i s d i g i t i z e d with the i n t e n t i o n that i t i s to vary l i n e a r l y between the points specified 188 on a record. Hence t h i s l i n e a r v a r i a t i o n of ground a c c e l e r a t i o n may be integrated to give the parabolic v a r i a t i o n of ground v e l o c i t y as shown in (B.3) v(t) = v + a o in which v i T o - a °) t T 2 (B.3) ground v e l o c i t y at the beginning of the time increment = ground a c c e l e r a t i o n at the beginning of the time increment = ground a c c e l e r a t i o n at the end of the time increment 0 a 1 = 0 a t + ( a With these s p e c i f i e d v a r i a t i o n s of base shear force and ground v e l o c i t y the quantity F(t) v ( t ) F F(t) v(t) = F ' v v o o F a , 0 1 + may be expressed as (a T - v ) + f v +-2 2 ^ (B.4) t L_£_ t T - 3F a + 2F a 2 (F - F ) (a - a ) 00 1 0 . . 1 O' ^ 1 O' == * 51 2T: (B.4) t + 3 .^ t L 1 Integration of (B.4) according to ( B . l ) r e s u l t s i n A E. in = I+ F v T + [F (a T - v ) + F v ] 0 0 0 0 0 1 o 2 v J T + (F, - F ) (a 1 0 1 v 2 - a ) -5— 0 8 v (B.5) 2 (F a 0 1 - 3F a 00 + 2F a ) 1 0' 6 189 which may be rearranged to be i n the form shown in 3a + a (B.6) 5a + 3a (B.6)
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The nonlinear three-dimensional response of structures to earthquake excitation Mason, Bruce Malcolm 1978
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Title | The nonlinear three-dimensional response of structures to earthquake excitation |
Creator |
Mason, Bruce Malcolm |
Date Issued | 1978 |
Description | The three-dimensional response of structures, comprised of members possessing nonlinear constitutive relationships, subjected to earthquake excitation is investigated. The input involves consideration of the simultaneous horizontal translation of the structure in two mutually perpendicular directions in addition to torsional behavior; vertical ground motion has not been included. Material nonlinearities are represented in a simplified fashion by the use of a bilinear moment-rotation relationship, subject to the kinematic hardening rule, at each member end. Relevant concepts of structural theory are reviewed and expressed in a format allowing for the use of matrix algebra. A general form of the member stiffness matrix is presented, and a method of formulating and subsequently statically condensing the structure stiffness matrix is also presented. The interaction of axial loads with the nonlinear material behavior is included. For columns subjected to bending about two principal axes, the effect of including biaxial interaction effects on the yield surface is considered. A method of calculating the unbalanced forces resulting from nonlinear behavior is outlined. An energy balance is established whereby the energy input by the earthquake into the structure is accounted for by the various mechanisms with which the structure is able to dissipate or store this input energy. The importance of analyzing the variation of the various energy forms is stressed. The theory developed is applied to two structural examples. The problems encountered in attempting to apply this theory to various models of a structure intended to represent a typical midsized office building are discussed. This structure has a dual component structural system in which an eccentric shear core is designed to resist the horizontal loads and an exterior framing system is designed to resist vertical loads and to act as a second line of defense with respect to horizontal loads. The results obtained from an earthquake analysis of a structure, comprised of exterior five storey frames, are presented. Specific aspects of these results, such as the amount of energy dissipated by various mechanisms, are analyzed in detail. The effect of assumptions made concerning the structural response are discussed and suggestions for future developments in this field of analysis, which will be aided by advancements made in computer technology, are also given. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062861 |
URI | http://hdl.handle.net/2429/21161 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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