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The reponse of multi-story prestressed concrete frames to seismic loading Bannister, David Earl 1979

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THE RESPONSE OF MULTI-STORY PRESTRESSED CONCRETE FRAMES TO SEISMIC LOADING by DAVID EARL BANNISTER B . S c , C i v i l E n g i n e e r i n g , W a l l a W a l l a C o l l e g e , 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE > if THE FACULTY OF GRADUATE STUDIES Department o f C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September 1979 © Dav i d E a r l B a n n i s t e r , 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l l m e n t o f t h e requ i rement s f o r an advanced deg ree a t t he U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g ree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . * Department o f C i v i l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date / 2 / ? ? ? ABSTRACT T h i s t h e s i s i s conce rned w i t h t h e re sponse o f m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e f rames to s e i s m i c l o a d i n g , and the d u c t i l i t y demands o f t h e c o n s t i t u e n t members. In t h a t r e g a r d an i d e a l i z e d model f o r the end m o m e n t - p l a s t i c r o t a t i o n r e l a t i o n s h i p o f p r e s t r e s s e d c o n c r e t e members was d e v e l o p e d based on a p u b l i s h e d moment - cu rva tu re i d e a l i z a t i o n f o r p r e s t r e s s e d c o n c r e t e . The i d e a l i z e d m o m e n t - r o t a t i o n m o d e l , wh i ch i n c l u d e d s t i f f n e s s and s t r e n g t h d e g r a d a t i o n , was used to i n t r o d u c e a l l p o s t - e l a s t i c a c t i o n In a beam-column e l e m e n t , which c o n s i s t e d o f an e l a s t i c beam c o n n e c t i n g c o n c e n t r a t e d h i n g e s modeled as n o n l i n e a r r o t a t i o n a l s p r i n g s . The subsequent use o f t he e lement in the n o n l i n e a r a n a l y s i s o f a t y p i c a l m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e frame i n d i c a t e d t h a t both the l a t e r a l d i s p l a c e m e n t s , and the g i r d e r end r o t a t i o n a l and h i n g e c u r v a t u r e d u c t i l i t i e s wou ld be somewhat h i g h e r f o r a p r e s t r e s s e d c o n c r e t e frame than f o r a r e i n f o r c e d c o n c r e t e f rame w i t h the same i n i t i a l s t i f f n e s s and s t r e n g t h . As an e f f o r t t o l i m i t d e f l e c t i o n s , and m i n i m i z e member damage under moderate s e i s m i c l o a d i n g the use o f a h i g h e r base shea r f o r a p r e s t r e s s e d c o n c r e t e s t r u c t u r e than f o r a comparab le r e i n f o r c e d c o n c r e t e s t r u c t u r e appears w a r r a n t e d . I i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i L IST OF FIGURES v i L IST OF TABLES x i INDEX OF NOTATION x i i ACKNOWLEDGEMENT x v i i CHAPTER 1 INTRODUCTION AND SCOPE OF RESEARCH 1 1.1 I n t r o d u c t i o n 1 1.2 M o t i v a t i o n and Scope o f Research 2 1.3 Format 3 CHAPTER 2 REVIEW OF THE DYNAMIC CHARACTERISTICS OF PRESTRESSED CONCRETE 5 2. 1 I n t r o d u c t i o n 5 2.2 Damping 5 2.3 Energy A b s o r p t i o n and D i s s i p a t i o n 7 2.4 Response 12 2.5 D u c t i l e C a p a c i t y and Demand 16 2.6 B e h a v i o u r in Ea r thquake s 20 2.7 C o n c l u s i o n s 21 CHAPTER 3 DERIVATION OF MOMENT-ROTATION MODEL 23 3.1 I n t r o d u c t i o n 23 3.2 Moment -Curva ture R e l a t i o n s h i p 28 3.3 D e s c r i p t i o n o f Beam/Column E lement Used t o Model P r e s t r e s s e d C o n c r e t e Members 29 3.3.1 D e s c r i p t i o n o f E lement 29 i i i Page 3.3.2 F o r m u l a t i o n o f E lement S t i f f n e s s 33 3.4 D e r i v a t i o n o f T h e o r e t i c a l Moment -Rota t ion Curves 35 3.4.1 D e s c r i p t i o n o f Model Beam 35 3.4.2 T h e o r e t i c a l Background 38 3.4.3 D e r i v a t i o n o f End Moment-Hinge R o t a t i o n Curves 40 3.5 I d e a l i z e d Moment -Ro ta t i on Model 42 3.5.1 I d e a l i z a t i o n o f End Moment-Hinge R o t a t i o n Curves 42 3.5.2 H y s t e r e s i s Law 42 3.5.3 Numer ica l V a l ue o f Parameters 52 3.5.4 A c c u r a c y o f Moment -Ro ta t i on Curves 55 CHAPTER 4 COMPARATIVE STUDY OF THE EARTHQUAKE RESPONSE OF PRESTRESSED AND REINFORCED CONCRETE STRUCTURES 57 4.1 I n t r o d u c t i o n 57 4.2 N o n l i n e a r E q u a t i o n s o f Mo t i on 58 4.2.1 Damping 58 4.2.2 N o n l i n e a r E q u a t i o n s o f Mo t i on 63 4.2.3 S t a t e D e t e r m i n a t i o n 65 4.3 Programme o f A n a l y s i s 69 4.3.1 S t r u c t u r e A n a l y z e d 69 4.3.2 E x c i t a t i o n 71 4.3.3 Time Increment 72 4.3.4 S chedu le o f Cases S t u d i e d 73 4.3.5 System of N o t a t i o n 74 4.4 O u t l i n e o f Compara t i ve Study 75 4.4.1 Method o f P r e s e n t a t i o n 75 i v Page 4.4.2 E q u i v a l e n t P l a s t i c H inge Length 77 4.5 R e s u l t s o f Compara t i ve Study 79 4.5.1 E f f e c t o f Damping 79 4.5.2 E f f e c t o f Ea r thquake I n t e n s i t y 92 4.5.3 E f f e c t o f Ea r thquake Record 11 4 4.5.4 E f f e c t o f S t r e n g t h D i s t r i b u t i o n 128 4.5.5 Compar i son o f the Response o f P r e s t r e s s e d and R e i n f o r c e d C o n c r e t e Frames 138 4.5.6 C o n c l u s i o n s 1^ 7 CHAPTER 5 REVIEW OF CURRENT RECOMMENDATIONS FOR THE ASEISMIC DESIGN OF PRESTRESSED CONCRETE STRUCTURES 149 5.1 I n t r o d u c t i o n 149 5.2 C r i t e r i a F o r Code Recommendations 150 5.3 Code Base Shear P r o v i s i o n s 151 5.4 V e r t i c a l D i s t r i b u t i o n P r o v i s i o n s 153 5.5 C o n c l u s i o n s 154 CHAPTER 6 CONCLUSIONS 155 6.1 Summary o f R e s u l t s 155 6.2 Recommendations For F u r t h e r Study 158 APPENDIX A COMPUTER PROGRAMS 163 APPENDIX B DERIVATION OF IDEALIZED END MOMENT-PLASTIC ROTATION RELATIONSHIP 191 APPENDIX C A/20/2.2/2/6 STRUCTURAL PROPERTIES 199 APPENDIX D EARTHQUAKE PROPERTIES 207 v LIST OF FIGURES F i g u r e Page 2.1 Compara t i ve Energy D i s s i p a t i o n o f P r e s t r e s s e d and R e i n f o r c e d C o n c r e t e Members, Re f . 23 9 2.2 T h e o r e t i c a l Compar i son o f the Energy D i s s i p a t i o n o f P r e s t r e s s e d and R e i n f o r c e d C o n c r e t e , Re f . 7 10 2.3 Compar i son o f E l a s t o - P l a s t ? c , Degrad ing S t i f f n e s s , and P r e s t r e s s e d C o n c r e t e H y s t e r e t i c M o d e l s , R e f . 12 11 2.4 Compar i son o f Moment-End D e f l e c t i o n Curves f o r F u l l y P r e s t r e s s e d , p a r t i a l l y P r e s t r e s s e d , and R e i n f o r c e d C o n c r e t e Members, Re f . 8 13 2.5 Compar ison o f the Response o f Th ree V e r s i o n s o f a P o r t a l Frame, Re f . 12 15 2.6 Compar i son o f t he P e r i o d vs D i s p l a c e m e n t F a c t o r f o r Three V e r s i o n s o f a P o r t a l Frame, Re f . 12 18 2.7 S e c t i o n C u r v a t u r e Requ i rements f o r P r e s t r e s s e d C o n c r e t e Members, Ref . 12 19 3.1 E a r l y E x p e r i m e n t a l End-Moment-End R o t a t i o n Curve f o r P r e s t r e s s e d C o n c r e t e , Ref . 21 24 3.2 E a r l y I d e a l i z e d Model f o r the H y s t e r e t i c B e h a v i o u r o f P r e s t r e s s e d C o n c r e t e , Re f . 9 25 3.3 E x p e r i m e n t a l Moment -Ro ta t i on Curves f o r P r e s t r e s s e d C o n c r e t e Beams, Re f . 7 26 3.4 E x p e r i m e n t a l Moment -Rota t ion Curves f o r P r e s t r e s s e d C o n c r e t e Columns, Re f . 7 27 3-5 I d e a l i z e d Moment -Curva ture Model f o r P r e s t r e s s e d C o n c r e t e Members, Re f . 11 30 3-6 Compar i son o f E x p e r i m e n t a l , T h e o r e t i c a l , and I d e a l i z e d Moment -Curvature Curves f o r P r e s t r e s s e d C o n c r e t e Columns, Re f . 7 31 3.7 Compar i son o f E x p e r i m e n t a l , T h e o r e t i c a l , and I d e a l i z e d Moment -Curvature Curves f o r P r e s t r e s s e d C o n c r e t e Beams, Re f . 7 32 v i F i g u r e Page 3.8 E lement I d e a l i z a t i o n and De fo rmat i on s 34 3.9 Model Member 37 3.10 T h e o r e t i c a l P r o p e r t i e s o f Model Member 39 3.11 T y p i c a l D e r i v e d End M o m e n t - P l a s t i c R o t a t i o n Curves f o r S tage One 43 3.12 T y p i c a l D e r i v e d End M o m e n t - P l a s t i c R o t a t i o n Curves f o r S tage Two 44 3.13 T y p i c a l D e r i v e d End M o m e n t - P l a s t i c R o t a t i o n Curves f o r S tage Three 45 3.14 Compar ison o f D e r i v e d S tage Two End M o m e n t - P l a s t i c R o t a t i o n Curves f o r Three Cases 46 3.15 I d e a l i z e d H y s t e r e t i c Model f o r H i n g e - C a s e One 47 3.16 I d e a l i z e d H y s t e r e t i c Model f o r H i n g e - C a s e Two 48 3.17 I d e a l i z e d H y s t e r e t i c Model f o r H inge -Ca se Th ree 49 3.18 D e s c r i p t i o n o f H y s t e r e s i s Law 51 4.1 T y p i c a l End Moment-Hinge R o t a t i o n R e l a t i o n s h i p s f o r Hinges 61 4.2 T y p i c a l End Moment-Hinge R o t a t i o n R e l a t i o n s h i p s f o r H inges 62 4.3 Computat ion o f F o r c e Increment , Re f . 39 67 4.4 L i n e a r and N o n l i n e a r F o r c e Increment , Re f . 39 68 4.5 T i m e - D i s p l a c e m e n t Response f o r Top Node v e r s u s Damping. . 81 4.6 Maximum L a t e r a l S t o r y D i s p l a c e m e n t s v e r s u s Damping 82 4.7 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Damp ing . . . 83 4.8 G i r d e r H inge C u r v a t u r e D u c t i l i t y Demand v e r s u s Damping. . 84 4.9 G i r d e r Accumu la ted Hinge R o t a t i o n F a c t o r v e r s u s Damping. 85 4.10 E x t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Damping 86 v i i Fi gure Page 4.11 Internal Column End Rotational Ducti l ity Demand versus Damping 87 4.12 Hinge Status Diagrams versus Damping 88 4.13 Time-Displacement Response for Top Node versus Earthquake Intensity 94 4.14 Maximum Lateral Story Displacements versus Earthquake Intensity 95 4.15 Girder End Rotational Ducti l i ty Demand versus Earthquake Intensity 96 4.16 Girder Hinge Curvature Ducti l ity Demand versus Earthquake Intensity 97 4.17 Girder Accumulated Hinge Rotation Factor versus Earthquake Intensity 98 4.18 External Column End Rotational Ducti l ity Demand versus Earthquake Intensity 99 4.19 Internal Column End Rotational Ducti l ity Demand versus Earthquake Intensity 100 4.20 Hinge Status Diagrams versus Earthquake Intensity 101 4.21 Time-Displacement Response for Top Node versus Earthquake Intensity 102 4.22 Maximum Lateral Story Displacements versus Earthquake Intensity 103 4.23 Girder End Rotational Ducti l i ty Demand versus Earthquake Intensity 104 4.24 Girder Hinge Curvature Ducti l ity Demand versus Earthquake Intensity 105 4.25 Girder Accumulated Hinge Rotation Factor versus Earthquake Intensity 106 4.26 External Column End Rotational Ducti l ity Demand versus Earthquake Intensity 107 4.27 Internal Column End Rotational Ducti l ity Demand versus Earthquake Intensity 108 vi i i Fi gure Page 4.28 H inge S t a t u s Diagrams v e r s u s Ea r thquake I n t e n s i t y 109 4.29 T i m e - D i s p l a c e m e n t Response f o r Top Node v e r s u s Ea r thquake Record 116 4.30 Maximum L a t e r a l S t o r y D i s p l a c e m e n t s v e r s u s Ea r thquake Record 117 4.31 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Ea r thquake Record 118 4.32 G i r d e r Hinge C u r v a t u r e D u c t i l i t y Demand v e r s u s Ea r thquake Record 119 4.33 G i r d e r Accumu la ted Hinge R o t a t i o n F a c t o r v e r s u s E a r t h q u a k e Record 120 4.34 E x t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s E a r t h q u a k e Record 121 4.35 I n t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Ea r thquake Record 122 4.36 H inge S t a t u s Diagrams v e r s u s Ea r thquake Record 123 4.37 T i m e - D i s p l a c e m e n t Response f o r Top Node v e r s u s S t r e n g t h D i s t r i b u t i o n 129 4.38 Maximum L a t e r a l S t o r y D i s p l a c e m e n t s v e r s u s S t r e n g t h D i s t r i b u t i o n 130 4.39 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand v e r s u s S t r e n g t h D i s t r i b u t i o n 131 4.40 G i r d e r H inge C u r v a t u r e D u c t i l i t y Demand v e r s u s S t r e n g t h D i s t r i b u t i o n 132 4.41 G i r d e r Accumu la ted H inge R o t a t i o n F a c t o r v e r s u s S t r e n g t h D i s t r i b u t i o n 133 4.42 E x t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s S t r e n g t h D i s t r i b u t i o n 134 4.43 I n t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s S t r e n g t h D i s t r i b u t i o n 135 4.44 H inge S t a t u s Diagrams v e r s u s S t r e n g t h D i s t r i b u t i o n 136 ix Figure Page 4.45 Maximum Story Displacement versus Bui ld ing Version (Prestressed or Reinforced) 140 4.46 G i rder End Rotational D u c t i l i t y Demand versus Version (Prestressed or Reinforced) 141 4.47 External Column End Rotat ional D u c t i l i t y versus Version (Prestressed or Reinforced) 142 4.48 Internal Column End Rotat ional D u c t i l i t y versus Version (Prestressed or Reinforced) 143 B. l End-Moment-Plast ic Rotation Envelope f o r Stage One, Case One 192 B.2 End-Moment-Plast ic Rotation Envelope f o r Stage One, Case Two 193 B.3 End-Moment-Plast ic Rotation Envelope f o r Stage One, Case Three 194 D.l Accelerogram, El Centro NS 1940 208 D.2 Response Spectrum, El Centro NS 1940 209 D.3 Re la t i ve Ve loc i t y Spectrum, El Centro NS 1940 210 D.4 Accelerogram, Taft S69E 1952 21 1 D.5 Response Spectrum, Taft S69E 1952 212 D.6 Re la t i ve Ve loc i t y Spectr im, Taft S69E 1952 213 D.7 Accelerogram, P a r k f i e l d N65E, 1966 214 D.8 Response Spectrum, P a r k f i e l d N65E 1966 215 D.9 Relat ive Ve loc i t y Spectrum, P a r k f i e l d N65E I966 216 x LIST OF TABLES Table Page B.l The Determination of Equations for Point D 197 B. 2 The Determination of Equations for Points R and L 198 C l Bui 1 ding A/20/2.2/2-6 200 C. 2 Relative Member Stiffness 201 C.3 Member Sizes 202 C.4 Cracking Moments 203 C.5 Rotation at Cracking 204 C.6 Curvature @ Cracking 205 C.7 Static Loading 206 xi INDEX OF NOTATION A = n u m e r i c a l s e i s m i c c o e f f i c i e n t • A - e f f e c t i v e s h e a r a r e a A , = a r e a o f e l a s t i c c u r v a t u r e joe A , = a r e a o f p l a s t i c c u r v a t u r e Ag = g ro s s a r e a o f s e c t i o n A = s t e e l a r e a in t e n s i o n s i A = s t e e l a r e a in compre s s i on A p s = a r e a o f p r e s t r e s s i n g s t e e l b - w i d t h o f c r o s s s e c t i o n fc_] = t angent s t r u c t u r a l damping m a t r i x D = dead load d - d i s t a n c e from extreme compres s i on f i b e r to t h e c e n t r o i d o f the p r e s t r e s s i n g s t e e l o r t e n s i o n r e i n f o r c i n g s t e e l E = e a r t h q u a k e l oad E l » e f f e c t i v e f l e x u r a l s t i f f n e s s o f member E =» modulus o f e l a s t i c i t y o f c o n c r e t e c 1 E s = modulus o f e l a s t i c i t y o f s t e e l F - l a t e r a l f o r c e a t l e v e l x x F - f o u n d a t i o n f a c t o r f - s t r e s s In c o n c r e t e c i f = 2 8 day c o n c r e t e c y l i n d e r c o m p r e s s i v e s t r e n g t h x i i i j ' f j j = e l a s t i c f l e x i b i l i t y c o e f f i c i e n t s s i s pu se s t r e s s in s t e e l u l t i m a t e s t e e l s t r e s s = c a l c u l a t e d s t r e s s in p r e s t r e s s i n g s t e e l a t d e s i g n ^ S ( u l t i m a t e ) loads = u l t i m a t e s t r e n g t h in p r e s t r e s s i n g s t e e l =» e f f e c t i v e s t r e s s in p r e s t r e s s i n g s t e e l a f t e r l o s s e s = y i e l d s t r e n g t h o f n o n - p r e s t r e s s e d r e i n f o r c e m e n t 6A = e f f e c t i v e s h e a r s t i f f n e s s H = h e i g h t o f b u i l d i n g above base h *» o v e r a l l depth o f s e c t i o n h x = h e i g h t o f l e v e l x above base I » moment o f I n e r t i a o f s e c t i o n o r Importance f a c t o r I =» moment o f I n e r t i a o f c r a c k e d s e c t i o n c r 1^  = moment o f i n e r t i a o f g r o s s s e c t i o n K = b u i l d i n g t ype f a c t o r K^j = h i n g e s t i f f n e s s a t node i K^j = h i n g e s t i f f n e s s at node j [ Kj\ = t angent s t r u c t u r e s t i f f n e s s = e f f e c t i v e s t r u c t u r e s t i f f n e s s K = e l a s t i c s t i f f n e s s e Kj ,l<2 = po s t c r a c k i n g and pos t c r u s h i n g s t i f f n e s s K . . . K , . , K . . = e l a s t i c s t i f f n e s s c o e f f i c i e n t s II ' J J J xi i j L - l i v e l o a d s , l eng th o f span L = c r u s h i n g l eng th f o r model member CU 3 3 L ^ = e q u i v a l e n t p l a s t i c h i nge l eng th M = m a t e r i a l s f a c t o r , NZLC M_ r = moment a t f i r s t c r a c k i n g i M c r = pseudo c r a c k i n g moment M c u - moment a t c r u s h i n g M = e l a s t i c end moment e M . = member end moment ed M j w = loop w i d t h moment M s « moment a t a p a r t i c u l a r s e c t i o n = u l t i m a t e moment c a p a c i t y P , _ P - l o a d , inc rement o f l o ad P , A P = e f f e c t i v e l o a d , i nc rement o f e f f e c t i v e l oad r, r = d i s p l a c e m e n t , inc rement o f d i s p l a c e m e n t r , _ r = v e l o c i t y , i nc rement o f v e l o c i t y r, A r = a c c e l e r a t i o n , Increment o f a c c e l e r a t i o n r Q = v a l u e o f v e l o c i t y a t b e g i n n i n g o f t ime s t e p r Q = v a l u e o f a c c e l e r a t i o n at b e g i n n i n g o f t ime s t e p r ^ . ^ T g = a c c e l e r a t i o n , inc rement o f a c c e l e r a t i o n o f the ground s = s t r u c t u r a l type f a c t o r , NZLC s , ds = member f o r c e s x i v T = p e r i o d o f v i b r a t i o n o f s t r u c t u r e th T = p e r i o d o f v i b r a t i o n o f the n node n r t , & t = t ime and t ime increment U = u l t i m a t e s t r e n g t h d e s i g n l oad V , A V = l a t e r a l l o a d , inc rement o f l a t e r a l l o ad V = l a t e r a l f o r c e o f f i r s t c r a c k i n g in t he f rame c r 3 v , dv = member d e f o r m a t i o n s W = we igh t o f s t r u c t u r e w^ = we ight o f l e v e l x x , x =* e f f e c t i v e d i s p l a c e m e n t , i nc rement o f e f f e c t i v e d i s p l a c e m e n t ^ = damping c o e f f i c i e n t f o r mass p r o p o r t i o n a l v i s c o u s damping ft = damping c o e f f i c i e n t o f s t i f f n e s s p r o p o r t i o n a l damping ce 1 e Y = K-/K cu 2 e £ = c o n c r e t e s t r a i n c £ = s t e e l s t r a i n s 9 = r o t a t i o n 8 = end r o t a t i o n a t f i r s t c r a c k i n g c r 0 e = e l a s t i c component o f end r o t a t i o n 8^ = h i n g e r o t a t i o n 8 ^ m = maximum h i n g e r o t a t i o n 8^ = p l a s t i c component o f end r o t a t i o n UQ = member end r o t a t i o n d u c t i l i t y xv u^ - hinge curvature duct i l i ty u^ = accumulated hinge rotation factor <_ = shear f l ex ib i l i t y factor 6 = section curvature 6 = curvature at f i r s t cracking cr 3 Xj = portion of damping in the i t ' 1 mode X^. - stiffness proportional damping in the i * ^ mode \ m j =s mass proportional damping in the i*^1 mode = frequency of the n**1 node t/L- = factor defined in thesis xv i ACKNOWLEDGEMENT The a u t h o r w i shes t o e x p r e s s h i s thanks t o h i s main a d v i s o r , Dr. D. L. A n d e r s o n , f o r h i s i n v a l u a b l e a d v i c e and g u i d a n c e d u r i n g the r e s e a r c h f o r and p r e p a r a t i o n o f t h i s t h e s i s . The a d v i c e g i v e n d u r i n g the t ime t h i s work was i n p r e p a r a t i o n and the s t i m u l a t i o n o f the a u t h o r ' s i n t e r e s t in the f i e l d o f p r e s t r e s s e d c o n c r e t e by Dr . R. A . S p e n c e r i s g r a t e f u l l y acknowledged. The f i n a n c i a l s u p p o r t f o r t h i s t h e s i s was from the N a t i o n a l Research C o u n c i l o f Canada in the form o f a Research A s s i s t a n t s h l p . x v i i 1 CHAPTER 1 INTRODUCTION AND SCOPE OF RESEARCH 1.1 INTRODUCTION For many years i t has been recognized that prestressed concrete components have performed well tn buildings under gravity loading. The ab i l i ty to achieve asthetical ly pleasing architectural forms and the possible economies resulting from the use of precast components has led designers to express an Interest in prestressed concrete. However, since the increased concern regarding the seismic resistance of buildings developed, there has been considerable discussion and controversy about the performance of prestressed concrete components used as primary seismic 12 3 4 resistant elements. Early researchers ' raised questions concerning the performance of Joints and connections, the energy absorption and dissipation characteristics, the application of code provisions to pre-stressed concrete, and the degree of damage to non-structural f ixtures. A number of studies^'^'' ' ' '^'^ since that time have addressed the above and other relevant questions regarding the seismic resistance of prestressed concrete. There continues to be, however, a dearth of information in the areas of analytical data on the inelast ic behaviour of prestressed concrete members and/or structures and the performance of such structures under earthquake loading. A result of this deficiency of information is that prestressed concrete has not won wide acceptance as a lateral load resisting material. In fact, many codes discourage or forbid the use of 2 prestressed concrete members as primary s e i s m i c r e s i s t a n t elements. 1.2 MOTIVATION AND SCOPE OF RESEARCH This study was prompted by a concern over the pa u c i t y o f information on the response o f m u l t i - s t o r y prestressed concrete b u i l d i n g s and the d u c t i l i t y demands of the c o n s t i t u e n t members. Recently the FIP pub-l i s h e d general recommendations f o r the aseismic design of pr e s t r e s s e d concrete s t r u c t u r e s and the New Zealand Prestressed Concrete I n s t i t u t e ' ^ published d e t a i l e d recommendations f o r the se i s m i c design of d u c t i l e p r estressed concrete frames. However, the d e f i c i e n c y of a n a l y t i c a l data has c o n t r i b u t e d to the lack o f d e t a i l e d p r o v i s i o n s i n most codes f o r the seism i c r e s i s t a n t design of pres t r e s s e d concrete members and/or s t r u c t u r e s . In f a c t , the unique response of prestressed concrete frames Is not r e a d i l y accounted f o r i n the se i s m i c philosophy o f the t y p i c a l code which has been developed around the response of s t e e l and r e i n f o r c e d concrete structures'*. One may design a s t r u c t u r e wtth a high strength t o perform e l a s t i c a l l y , o r w i t h a high d u c t i l e c a p a c i t y to perform i n e l a s t l e a l l y . However, w h i l e p r e s t r e s s e d concrete s t r u c t u r e s may have the high r e s i s -tance a s s o c i a t e d w i t h the f i r s t c l a s s , they c h a r a c t e r i s t i c a l l y develop the large deformations a s s o c i a t e d w i t h the second'^. This i s because of both the lower percentage of c r i t i c a l viscous damping and the lower energy d i s s i p a t i o n t y p i c a l of pres t r e s s e d concrete members as compared with r e i n f o r c e d concrete members. Recent s t u d i e s by Bl a k e l e y and P a r k " ' ' 2 and Thompson and Park** have d i s p e l l e d the f e a r that p r e s t r e s s e d concrete Is a more b r i t t l e m a t e r i a l than r e i n f o r c e d concrete. Research i n d i c a t e d that c a r e f u l 3 d e t a i l i n g o f f u l l y and p a r t i a l l y p r e s t r e s s e d c o n c r e t e members r e s u l t e d in an a b i l i t y to undergo l a r g e i n e l a s t i c d e f o r m a t i o n s under bo th mono-t o n i c and h i g h i n t e n s i t y c y c l i c l o a d i n g . However, w h i l e the s t u d i e s by 9 17 Spencer ' of the d i s p l a c e m e n t s , energy d i s s i p a t i o n , and d u c t i l i t y r e q u i r e m e n t s o f s e v e r a l v e r s i o n s o f a m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e f rame and the i n v e s t i g a t i o n by B l a k e l e y ^ o f t he d i s p l a c e m e n t , energy d i s s i p a t i o n , and c u r v a t u r e r e q u i r e m e n t s o f a p o r t a l f rame were s i g n i f i -cant b e g i n n i n g s , the re sponse o f m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e s t r u c -t u r e s i s not y e t c l e a r . The r e s u l t s o f the above i n v e s t i g a t i o n s have i n d i c a t e d t h a t p r e -s t r e s s e d c o n c r e t e members may p e r f o r m w e l l under e a r t h q u a k e l o a d i n g . In f a c t , some r e s e a r c h e r s ^ * ^ have sugge s ted t h a t t he use o f p r e s t r e s -sed c o n c r e t e as a l a t e r a l l o ad r e s i s t i n g m a t e r i a l may be e x p e d i e n t . T h i s s t udy was d i r e c t e d t o a n a l y z i n g the re sponse o f m u l t i - s t o r y p r e -s t r e s s e d c o n c r e t e f rames and t h e c a p a b i l i t y o f t h e c o n s t i t u e n t members t o s e r v e as p r i m a r y s e i s m i c r e s i s t a n t e l e m e n t s . The s cope o f t he i n v e s -t i g a t i o n i n c l u d e d the d e r i v a t i o n o f a m o m e n t - r o t a t i o n model f o r p r e -s t r e s s e d c o n c r e t e members and the subsequent use o f t h e model i n the n o n l i n e a r a n a l y s i s o f a t y p i c a l m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e f r a m e . The d u c t i l i t y demands o f the members was the a r e a of s p e c i a l c o n s i d e r a -t i o n . 1.3 FORMAT The fo rmat o f t h i s t h e s i s i s as f o l l o w s : Chap te r Two i s a rev iew o f pa s t e x p e r i m e n t a l and a n a l y t i c a l s t u d i e s o f p r e s t r e s s e d c o n c r e t e wh ich were d i r e c t e d to d e t e r m i n i n g i t s dynamic c h a r a c t e r i s t i c s . The 4 d e r i v a t i o n o f an i d e a l i z e d m o m e n t - r o t a t i o n model f o r use i n the n o n -l i n e a r dynamic a n a l y s i s o f m u l t i - s t o r y frames i s p r e s e n t e d i n C h a p t e r T h r e e . In C h a p t e r Four the p r e v i o u s l y d e r i v e d model i s used t o de te rm ine the re sponse o f a t y p i c a l m u l t i - s t o r y f r a m e . S e v e r a l p r e s t r e s s e d c o n c r e t e v e r s i o n s o f the frame were s u b j e c t e d t o a number o f e a r t h q u a k e r e c o r d s w i t h v a r y i n g maximum i n t e n s i t i e s . The r e s u l t s o f a c o m p a r a t i v e s tudy a r e p r e s e n t e d . In Chap te r F i v e c u r r e n t recommendat ions f o r the a s e i s m i c d e s i g n o f p r e s t r e s s e d members a n d / o r s t r u c t u r e s a re rev iewed and a d d i t i o n -a l recommendat ions a r e p r o p o s e d . C o n c l u s i o n s a r e made and s u g g e s t i o n s f o r f u r t h e r r e s e a r c h a r e g i v e n in C h a p t e r S i x . 5 CHAPTER 2 REVIEW OF THE DYNAMIC CHARACTERISTICS OF PRESTRESSED CONCRETE 2.1 INTRODUCTION Since the f i r s t development of prestressed concrete, it has been recognized that i t is a unique material. Observations have shown that while it exhibits many of the characteristics of reinforced concrete, there are a number of signif icant differences. Histor ica l ly , invest i -gators have carried out studies, both experimental and analyt ical , to determine what these s imi lar i t ies and differences are, and how they affect the performance of prestressed concrete members and/or structures. This chapter presents a brief overview on a number of characteristics which would be expected to affect both the response of prestressed concrete structures and the capability of prestressed concrete members to serve as primary seismic resistant elements. More detailed reviews of the properties of prestressed concrete have been written by Blakeley^' 13 16 , and more recently by Hawkins 2.2 DAMPING The damping associated with the dynamic response of a structure is typical ly a combination of two dist inct types. The f i r s t , and the one considered here, is e last ic viscous damping. The second is inelastic hysteretic damping. 6 In one of the e a r l i e s t i n v e s t i g a t i o n s o f the damping of pres t r e s s e d 20 concrete members, Penzien c a r r i e d out steady s t a t e and f r e e v i b r a t i o n t e s t s on small s c a l e beams. V a r i a b l e s included the concrete compressive s t r e n g t h , grade o f p r e s t r e s s i n g bar, and the e c c e n t r i c i t y of the bars. Results showed that i n i t i a l damping in the e l a s t i c range was less than one percent o f c r i t i c a l . Previous loading h i s t o r y was found to have a s i g n i f i c a n t e f f e c t on the value obtained, e s p e c i a l l y when loading had progressed i n t o the p o s t - e l a s t i c range. Damping rose to two percent of c r i t i c a l at the f i r s t onset o f c r a c k i n g and from three percent to s i x percent o f c r i t i c a l a f t e r c r a c k i n g had developed. The l e v e l o f p r e s t r e s s and concrete compressive strength a f f e c t e d damping to the extent they a f f e c t e d the onset o f c r a c k i n g . 21 In a l a t t e r study, Spencer i n v e s t i g a t e d the s t i f f n e s s and damping p r o p e r t i e s of nine small s c a l e c e n t r a l l y p r e s t r e s s e d concrete beams. Va r i a b l e s included the l e v e l o f p r e s t r e s s , concrete s t r e n g t h , geometric c o n f i g u r a t i o n , and a p p l i c a t i o n of end r o t a t i o n s to produce e i t h e r uniform moment or uniform shear. Results were expressed i n terms of D (, the t o t a l damping energy d i s s i p a t e d by a member during one c y c l e o f steady-s t a t e l o a d i n g . I t was observed that D ( was n e g l i g i b l e before c r a c k i n g and when the amplitude of load was i n s u f f i c i e n t to re-open tension cracks. Values increased with i n c r e a s i n g and r o t a t i o n s up t o , and f o r some members, beyond f a i l u r e . Both s t i f f n e s s and damping values were found to be independent o f the frequency and the number of c y c l e s , but dependent on the type of loa d i n g . 22 Subsequently Brondum-Nielsen conducted f r e e v i b r a t i o n t e s t s on f u l l s i z e p r e s t r e s s e d concrete beams. I t was observed that damping values decreased as the l e v e l o f p r e s t r e s s increased or the load magnitude decreased. 7 B l a k e l e y conducted f r e e v i b r a t i o n t e s t s on a cracked beam-column subassembledge. The values o f two and one h a l f percent o f c r i t i c a l v i s cous damping which were obtained confirmed Penzien's e a r l i e r study. One o f the few t e s t s to date on pres t r e s s e d concrete frames has been c a r r i e d out by Nakano^. A number of o n e - t h i r d s c a l e frames were subjected to q u a s i - s t a t i c l a t e r a l l o a d i n g . Results showed that damping values rose from one percent of c r i t i c a l a t the onset of c r a c k i n g and up to seven percent o f c r i t i c a l when loading had progressed i n t o the i n -e l a s t i c range. I t may be concluded that damping values f o r t y p i c a l p r e s t r e s s e d concrete members may be s i g n i f i c a n t l y a f f e c t e d by t h e i r design. For example, as the l e v e l of p r e s t r e s s and/or concrete compressive str e n g t h Is increased, the onset of c r a c k i n g , and thus increased damping val u e s , w i l l be delayed. Since damping i n the i n i t i a l stages i s small as com-pared to r e i n f o r c e d concrete, i t would be expected that e x t r a consider-a t i o n should be given to the I n t e r a c t i o n w i t h other components o f the s t r u c t u r a l system, n o n - s t r u c t u r a l f i x t u r e s , and the supporting s o i l . 2.3 ENERGY ABSORPTION AND DISSIPATION The energy a b s o r p t i o n , and more importantly, the energy d i s s i p a t i o n c h a r a c t e r i s t i c s of the members have a s i g n i f i c a n t e f f e c t on the response of a s t r u c t u r e . H i s t o r i c a l l y , there has been considerable d i s c u s s i o n and debate on the energy p r o p e r t i e s of pr e s t r e s s e d concrete as compared w i t h r e i n f o r c e d concrete. 1 2 Some e a r l y i n v e s t i g a t o r s ' concluded that experimental and 8 t h e o r e t i c a l evidence i n d i c a t e d that the energy abs o r p t i o n c a p a c i t y f o r p r e s t r e s s e d concrete was at l e a s t equal to that of r e i n f o r c e d concrete. However, i t i s the energy d i s s i p a t i o n c h a r a c t e r i s t i c s of the m a t e r i a l that a f f e c t s the response of a s t r u c t u r e to dynamic loading. This point was i l l u s t r a t e d by Candy as shown i n F i g . 2.1. He s t a t e d that because of e l a s t i c recovery e f f e c t s , the energy d i s s i p a t e d by a pre-s t r e s s e d concrete member i s s i g n i f i c a n t l y l e ss than f o r a comparable r e i n f o r c e d concrete member. B l a k e l e y ^ compared t h e o r e t i c a l l y obtained moment-curvature 2k curves f o r a prestressed concrete member to those obtained by Kent f o r a comparable r e i n f o r c e d concrete member. Both members had equiva-le n t s t e e l areas, maximum moment c a p a c i t y , and i n i t i a l s t i f f n e s s . Results as depicted i n F i g . 2.2 showed that f o r the f i r s t p o s t - e l a s t i c c y c l e the pr e s t r e s s e d concrete s e c t i o n d i s s i p a t e d only f o r t y percent o f the energy of the r e i n f o r c e d concrete s e c t i o n . In a subsequent comparative a n a l y s i s o f a p o r t a l frame using the e l a s t o - p l a s t i c , degrading s t i f f n e s s , and pr e s t r e s s e d concrete models 12 shown i n F i g . 2.3, Park and Blake ley found t h a t , as expected, the prest r e s s e d concrete frame d i s s i p a t e d considerably l e s s energy than the e l a s t o - p l a s t l c system. However, i t was found that the energy d i s s i p a t e d i n the pr e s t r e s s e d concrete v e r s i o n by viscous damping was one and one h a l f to two times that of the e l a s t o - p l a s t i c model. This was due to the gre a t e r amplitude o f the cy c l e s t y p i c a l of the pres t r e s s e d concrete v e r s i o n . I t i s now g e n e r a l l y accepted that the energy d i s s i p a t i o n of a prestressed concrete member subjected to low amplitude c y c l i c loading w i l l be s i g n i f i c a n t l y l e s s than f o r a comparable r e i n f o r c e d concrete 9 F i g . 2.1 Compara t i ve Energy D i s s i p a t i o n o f P r e s t r e s s e d and R e i n f o r c e d C o n c r e t e Members , Ref . 23 F i g . 2.2 T h e o r e t i c a l Comparison o f t he Energy D i s s i p a t i o n o f P r e s t r e s s e d and R e i n f o r c e d C o n c r e t e , Ref . 7 '«*',' I STAGE 3 Fig. 2.3 Comparison of Elasto-Plastlc, Degrading Stiffness, and Prestressed Concrete Hysteretic Models, Ref. 12 12 member. S i g n i f i c a n t energy d i s s i p a t i o n does not occur u n t i l the y i e l d i n g o f the s t e e l o r the crushing o f the concrete i n the hinges. I t should be noted, as shown i n F i g . 2.4, that f o r large p o s t - e l a s t i c cur-vatures and/or f o r members w i t h s i g n i f i c a n t amounts of a d d i t i o n a l de-formed bar r e i n f o r c i n g , the h y s t e r e s i s curves w i l l come to resemble those t y p i c a l o f r e i n f o r c e d concrete members**' 2.4 RESPONSE It i s g e n e r a l l y agreed that the response o f a pre s t r e s s e d concrete s t r u c t u r e w i l l be somewhat g r e a t e r than that of a comparable r e i n f o r c e d concrete s t r u c t u r e . However, l i t t l e experimental o r a n a l y t i c a l work has been done i n t h i s area, e s p e c i a l l y i n regard to the response of m u l t i - s t o r y b u i l d i n g s . In one of the few other n o n l i n e a r a n a l y s i s to date which were d i r e c t l y concerned with the response o f prestressed concrete s t r u c t u r e s , 9 Spencer s t u d i e d two r e i n f o r c e d concrete and s i x pre s t r e s s e d concrete versions of a twenty s t o r y frame. The b u i l d i n g was one which had been s t u d i e d e a r l i e r by Clough and Benuska 1 0. Spencer found that f o r the prestressed concrete b u i l d i n g the l a t e r a l displacements were up to f i f t y percent greater and the i n t e r s t o r y d r i f t s were up to seventy percent g r e a t e r than those o f the r e i n f o r c e d concrete v e r s i o n s o f the b u i l d i n g . He concluded that w h i l e a p r e s t r e s s e d concrete b u i l d i n g could be designed to withstand a severe earthquake, major damage to n o n - s t r u c t u r a l f i x t u r e s might occur. In a l a t e r study of the same b u i l d i n g Spencer'^ analyzed the e f f e c t o f i n f i l l panels on c o n t r o l l i n g i n t e r s t o r y displacements. Ramberg-UNIT 1 (PRESTRESSED) 1K = 1160 psi UNIT 2 (PARTIALLY PRESTRESSED) fpc = 386 psi UNIT 3 (REINFORCED) 0 F i g . 2.k Comparison of Moment-End Def lect ion Curves f o r Fu l l y Prestressed, P a r t i a l l y Prestressed, and Reinforced Concrete Members, Ref. 8 14 Osgood f u n c t i o n s were used to model the h y s t e r e t i c a c t i o n o f the e l e -ments. I t was assumed that the load-deformation p r o p e r t i e s o f the panels were frequency independent. Results showed that n o n - s t r u c t u r a l : i n t e r -s t o r y elements could be very e f f e c t i v e i n reducing i n t e r s t o r y d r i f t . I n t e r s t o r y displacements were reduced one h a l f to one t h i r d those o f the same frame without i n f i l l panels. It was found that n o n y i e l d i n g elements w i t h r e l a t i v e l y low energy d i s s i p a t i o n were more e f f e c t i v e i n c o n t r o l l i n g d r i f t s than were y i e l d i n g elements w i t h higher energy d i s -s i p a t i o n . Spencer concluded that there could be severe consequences, however, i f f a i l u r e s occured in the n o n - s t r u c t u r a l elements during s e i s m i c loading. 12 Park and Blaketey i n the same comparative study r e f e r r e d to e a r l i e r concluded that under earthquake loading the maximum d i s p l a c e -ment o f a pres t r e s s e d concrete frame would be about 1.4 times that of a comparable r e i n f o r c e d concrete s t r u c t u r e . Shown i n F i g . 2.5 are the displacement-time response curves f o r the three versions o f the p o r t a l frame s t u d i e d . The curves show the higher amplitude and longer period of v i b r a t i o n t y p i c a l o f pres t r e s s e d concrete s t r u c t u r e s as compared with other types o f b u i l d i n g s , jo Butcher has reported on recent research on the response of prestressed concrete framed s t r u c t u r e s subjected to such earthquake records as the Jennings A2 a r t i f i c i a l r e c o r d . Results showed that the maximum displacement o f a f u l l y p r e s t r e s s e d concrete frame was on the average, 1.3 times that o f a r e i n f o r c e d concrete frame w i t h the same i n i t i a l s t r e n g t h , s t i f f n e s s , and damping. I t was noted, however, that there was s i g n i f i c a n t v a r i a t i o n i n the response from the average value w i t h the range being from 0.7 to 2.4. Otsplacement (inchn) 01 r» 3 or a a 50 Si 16 It may be concluded that a prestressed concrete frame may perform sat is factor i ly in an earthquake. Indications are, however, that severe 25 damage may occur to non-structural f ixtures. A recent AIJ publication expressed concern over the possible incompatibilities of prestressed concrete framing with many of the standard curtain finishes and/or other lateral load resisting mechanisms. Should prestressed concrete members come into wider acceptance as lateral load resisting elements, extreme care must be taken to ensure compatibility with other components of the structural system, non-structural f ixtures, and architectural elements. More research is needed in this area. 2.5 DUCTILE CAPACITY AND DEMAND While the curvature and member duct i l i ty demands for reinforced concrete members are generally assumed to be well known, this is not the case for prestressed concrete. While the lateral displacements of a prestressed concrete frame may be greater than those of a compar-able reinforced concrete frame, it is not clear that the duct i l i ty demands wi l l be proportionately higher. 9 In the nonlinear study referred to previously, Spencer found that the maximum duct i l i ty demands for member end rotations were forty per-cent less, and for curvatures were seventy percent less for the pre-stressed concrete structure as compared with the reinforced concrete structure. Spencer explained that this somewhat surprising result could be attributed, in part, to the longer hinge length assumed for the prestressed concrete members. 26 In a later investigation Inomata made a comparative study of 17 prestressed and re in forced concrete beams. Those beams were subjected to reversed q u a s i - s t a t i c loading. Results were presented in the form of d u c t i l i t y factors which inomata def ined as the r a t i o o f the measured d e f l e c t i o n under maximum load to that under the ca l cu la ted load. The l im i t load was ca l cu la ted in accordance with the current FIP-CEB recom-27 mendations . Those recommendations proposed that the ult imate f l e x u r a l l im i t strength should be ca l cu l a ted on the basis of the design strength of the mater ia l s and the l i m i t i n g of the poss ib le s tee l elongation to one percent or less . Inomata s tated that values of 3.8 to 5.7 obtained fo r prestressed concrete members were acceptable when compared with the values of 2.5 to 2.8 obtained f o r re in forced concrete members. 12 ' ; ' ' ' *' In t h e i r comparative study, Park and Blakeley have presented some of the most informative resu l t s to date. F i g . 2.6 shows p lots o f i the displacement d u c t i l i t y vs period fo r the three load-deformation models. It may be noted that there Is a general trend fo decreasing displacement d u c t i l i t y demand as the per iod increases. This f ac to r favors the more f l e x i b l e construct ion t yp i ca l of prestressed concrete s t ruc tures . As part o f the study an ana lys i s o f sect ion curvature requirements fo r prestressed concrete members was undertaken. Results were presented in the form of the table and nomograms shown in F i g . 2.7. It was concluded that adequate d u c t i l i t y o f members could be ensured provid ing s tee l areas and ax i a l load leve l s are kept sma l l . It was fu r ther noted that s i g n i f i c a n t add i t iona l d u c t i l i t y may be ava i l ab l e subsequent to crushing of the concrete in the hinges. As such, catas -t roph ic f a i l u r e of members should not occur f o r the values of curvatures shown in the t ab le . Further research is necessary to determine the app l i c a t i on of the curvature resu l t s to mu l t i - s to ry prestressed concrete s t ruc tu re s . • ON •n T I n -i o» O 9» O 3 3 r j "O n O o> • - i n 3D -ft Ifl n o o 3 ° — J -h » r» » 3* (0 < » "0 -I » o o 3 Q . O 5 -ti o 0) — -1 IV cr o OJ n — 3 a 3 Dltplacrmtnt ductility factor, U 8- 6 81 19 Section curvature ratios Displacement Column-sidesway mechanism Beam-sidesway mechanism ductility Column<urvature ratio Beam-cunature ratio Column-curxature ratio factor M <P /<P ~ maxc ' crc ' maxb T crb ' maxc T crc H = 1 0 D H = 1 6 D H = 8 D H = 1 4 D H = 8 D H = 14D 4 13 1 9 13 2 1 9 15 6 2 1 3 2 2 2 3 7 15 2 5 8 2 9 4 4 3 2 5 3 2 2 3 6 10 3 6 5 6 4 1 6 9 2 8 4 7 1 2 4 4 6 8 5 0 8 5 3 4 5 7 14 5 2 8 1 5 9 101 4 0 6 8 16 6 0 9 3 6 8 1 1 6 5 1 7 8 Fig. 2.7 Section Curvature Requirements for Prestressed Concrete Members, Ref. 12 20 g Park and Thompson found that the duct i l i ty of prestressed concrete beams was enhanced by the presence of deformed bar reinforcing in the compression zone of the members. It was further found that transverse steel was necessary in the plast ic hinge zones and in the beam-column joint cores to ensure ductile behaviour and prevent diagonal tension fa i lure. The repair and subsequent retesting of a damaged unit demon-strated that damaged prestressed concrete members may feasibly be repaired and most of the original strength restored. 2.6 BEHAVIOUR IN EARTHQUAKES 28 29 30 A number of reports ' ' have been presented on the behaviour under seismic loading of structures incorporating significant amounts prestressing. There is, however, relatively l i t t l e information on the performance of fu l ly framed prestressed concrete structures under major earthquake loading. Most structures which were reported on had prestressing for functions other than lateral load resisting elements. Many incorporated prestressing or post-tens toning for f loor slabs or roof systems. In spite of this, valuable information has been gained on several c r i t i c a l areas of design. In general, prestressed concrete members and/or structures have performed well in major earthquakes. In many cases prestressed concrete members suffered no more than minor cracking, even in the face of 32 severe foundation settlement . Where failures have occured it has usually been due to an Init ia l fa i lure of the supporting system and/ 31 or connections. Berg , however, reported on one case where major dam-age to a precast prestressed concrete structure occured due to bending 21 about the weak axis of the supporting columns. 2.7 CONCLUSIONS From the results of experimental testing several points may be noted. F i r s t , the percentage of c r i t i c a l viscous damping is less for the prestressed concrete members as compared with comparable reinforced concrete members. This is especially true in the e last ic and early portion of the cracked-elastic stages. Secondly, monatonic and high intensity cyc l ic loading tests have shown that prestressed concrete elements possess a large ductile capacity. Thirdly, while the i n i t i a l hysteretic energy dissipation is small, once crushing of the concrete in the hinges has commenced, significant dissipation capacity is avalIable. An early nonlinear study of a multi-story prestressed concrete frame supported the intuit ive view that a prestressed concrete structure would have greater lateral displacements than a comparable reinforced concrete structure. It was found, however, that the rotational and curvature duct i l i ty demands were less for the prestressed concrete frame. A subsequent study of a portal frame showed that the displacement duct i l i ty demands for a prestressed concrete structure would be higher than those of a comparable reinforced concrete structure. It was, however, concluded that prestressed concrete members are capable of achieving the high curvatures necessary to prevent fa i lure in a severe earthquake. It has been found that structures containing prestressed concrete 22 elements have performed well in major earthquakes. Fully framed pre-stressed concrete structures have yet to be tested under severe seismic loading, however. An analysis of the l i terature shows research is needed in a number of areas, including nonlinear analyses of multi-story prestressed concrete and mixed structures, additional quasi-static and dynamic testing of multi-story prestressed concrete frames, and further analysis of the damage to non-structural elements and fixtures due to seismic loading. 2 3 CHAPTER 3 DERIVATION OF MOMENT-ROTATION MODEL 3.1 INTRODUCTION The v a l i d i t y o f any n o n l i n e a r dynamic a n a l y s i s depends on the use of a h y s t e r e t i c model which r e f l e c t s as c l o s e l y as p o s s i b l e observed c h a r a c t e r i s t i c s of r e a l members. It has g e n e r a l l y been agreed that the h y s t e r e t i c response of t y p i c a l p r e s t r e s s e d concrete members i s markedly d i f f e r e n t from that of comparable r e i n f o r c e d concrete members. This i s because of both the tower energy d i s s i p a t i o n and e l a s t i c recovery 21 e f f e c t s a s s o c i a t e d w i t h prestressed concrete. H i s t o r i c a l l y , i t has been thought that the h y s t e r e t i c response f o r p r e s t r e s s e d concrete members w i t h t i t t l e o r no a d d i t i o n a l deformed bar r e i n f o r c i n g would be s i m l l l a r to that shown on F i g . 3.1. q Spencer , in the e a r l i e s t n o n l i n e a r study of a p r e s t r e s s e d concrete frame, used the b i - l i n e a r end moment-end r o t a t i o n h y s t e r e t i c model shown i n F i g . 3.2. That model, which c o n s i s t e d of e l a s t i c and cracked-e l a s t i c p o r t i o n s r e f l e c t e d experimental observation of the c y c l i c t e s t i n g of prestressed concrete members. However, i t d i d not account f o r the s t i f f n e s s and strength degradation which might be s u s t a i n e d under severe 7 8 s e i s m i c loading. Recent s t u d i e s ' , both experimental and a n a l y t i c a l , have demonstrated the a b i l i t y of p r e s t r e s s e d concrete members to s u s t a i n large post-crushing deformations. Some moment-rotation curves obtained by Btaketey^ are shown i n F i g s . 3.3 and 3.4. In a d d i t i o n , i t has been 2k FJg. 3.1 Early Experimental End-Moment-End Rotation Curve for Prestressed Concrete, Ref. 21 25 Fig. 3.2 Early Idealized Model for the Hysteretic Behaviour of Prestressed Concrete, Ref. 9 26 % dia stirrup 2-12w/0.276 "dia cables Fig. 3.3 Experimental Moment-Rotation Curves for Prestressed Concrete Beams, Ref. 7 27 " F i g . 3.4 E x p e r i m e n t a l Moment -Rota t ion Curves f o r P r e s t r e s s e d C o n c r e t e Co lumns, Re f . 7 28 shown that while the hysteretic energy dissipation in the e las t ic and cracked-elastic range in small, once crushing of the concrete in the hinges has commenced, significant energy dissipation capacity exists. The work described in this chapter generalizes existing moment-rotation hysteretic models for prestressed concrete members in that it allows for post-crushing deformations and includes stiffness and strength degradation. 3.2 MOMENT-CURVATURE RELATIONSHIP Any hysteretic moment-rotation model is based on an assumed moment-curvature relationship. That moment-curvature relationship is characteristic of the material, and it has long been recognized that the relationship for prestressed concrete is d ist inct ly different from that of steel or reinforced concrete. Early experimental studies aimed at determining monotonic or cyc l ic load behaviour for prestressed concrete members did not account for the extreme deformations which might be sustained under a severe 5 21 26 earthquake. Cyclic tests ' ' were performed at load levels below maximum strength. Likewise, theoretical investigations neglected the post-crushing range. Priestly^ and Sherbourne and Parameswar^ deter-mined the moment-curvature relationship for a prestressed concrete member under monotonic load only up to a theoretical maximum concrete compressive 34 strain of 0.004. Paranagama and Edwards studied the cyc l ic load behaviour of prestressed concrete, however they considered repeated loading in one direction only. The apparent result of these and other studies was that post-crushing curvatures were not accounted for In the 29 b l - H n e a r moment-curvature re l a t i on sh ip which was genera l ly assumed. A recent inves t i ga t ion by Blakeley^ extended previous work to account f o r the large deformations and high curvatures which could be expected under extreme seismic loading. He c a r r i e d out both e x p e r i -mental and theore t i ca l s tudies on the monotonic and c y c l i c load behaviour of prestressed concrete members. Experimental resu l t s were obtained from the c y c l i c te s t ing of f u l l sca le beam-column subassemblages. The columns were prestressed and the beams were post-tensioned and grouted. A mortar j o i n t connected the column and beam with only the tendons extending across i t . Theoret i ca l resu l t s were obtained a n a l y t i c a l l y from a computer program designed to ca l cu l a te the moment-curvature c h a r a c t e r i s t i c s o f prestressed concrete sect ions based on i dea l i zed s t r e s s - s t r a i n re la t ionsh ips fo r the concrete and pres t ress ing s t e e l . As one resu l t of th i s inves t i ga t ion he proposed the genera l ized i d e a l -ized moment-curvature re l a t i on sh ip shown in F i g . 3.5. That model cons i s t s o f e l a s t i c , c r a c k e d - e l a s t i c , and i n e l a s t i c or post -crushing por t ions . The c lose c o r r e l a t i o n between experimental r e s u l t s , theoret -i c a l de r i va t i ons , and i dea l i zed moment-curvature curves is shown in F i g s . 3.6 and 3.7. This model was used as a basis f o r the moment-rotat ion model descr ibed in subsequent sec t ions . 3.3 DESCRIPTION OF BEAM/COLUMN ELEMENT USED TO MODEL PRESTRESSED CONCRETE MEMBERS 3.3.1 Descr ipt ion of Element The most accurate method of determining the theo re t i c a l d u c t i l i t y requirements would be to monitor the curvature at every sect ion along M STAGE 2 F i g . 3.5 Ideal ized Moment-Curvature Model fo r Prestressed Concrete Members, Ref. 11 o 31 ( O t p E A L i z E D C O R V E S g. 3.6 Comparison of Experimental, Theoretical, and Idealized Moment-Curvature Curves for Prestressed Concrete Columns, Ref. 7 32 Fig. 3.7 Comparison of Experimental. Theoretical and Idealized Moment-Curvature Curves for Prestressed Concrete Beams, Ref. 7 33 the hinge length of a member during a nonl inear dynamic ana ly s i s . However, the computational e f f o r t required makes th i s method unsuitable f o r p r a c t i c a l a p p l i c a t i o n . As a r e s u l t , most researchers use some form of moment-rotation model f o r nonl inear s tud ies . These models genera l ly use e i t h e r concentrated or d i s t r i bu ted hinges to account f o r Q the i n e l a s t i c a c t i o n . In h i s nonl inear study Spencer used a f i n i t e hinge length to account f o r the i n e l a s t i c a c t i o n . The element used in the model descr ibed In th i s study Is formulated to account f o r the f l exu ra l s t i f f n e s s and strength degradation which c h a r a c t e r i s t i c a l l y occur when a prestressed concrete member is subjected to c y c l i c loading. The element cons i s t s o f a l i nea r e l a s t i c beam e l e -ment connecting concentrated hinges as shown In F i g . 3.8. A l l post -e l a s t i c a c t i o n , inc luding degrading f l exura l s t i f f n e s s , is introduced by means of an end moment-hinge ro ta t ion re l a t i on sh ip fo r the hinges which are modeled as nonl inear ro ta t iona l spr ings . 3.3.2 Formulation o f Element S t i f f ne s s The complete element s t i f f n e s s matrix includes contr ibut ions from both the e l a s t i c beam and the concentrated hinges. The ax ia l s t i f f n e s s f o r the element is constant and is given by • EA . 1™ L 1 The f l e x u r a l s t i f f n e s s f o r the e l a s t i c beam is given by ids . Ej_ L k k II Ij k i j k j j dv, dv. where El Is the e f f e c t i v e f l e x u r a l s t i f f n e s s . The factors k j j , k- j_ 34 F i g . 3.8 Element I d e a l i z a t i o n and Deformations 35 and k.. may be m o d i f i e d t o i n c l u d e the e f f e c t s o f s hea r d e f o r m a t i o n and / J J o r non p r i s m a t i c beam e f f e c t s . To fo rm t h e comp le te e lement f l e x u r a l s t i f f n e s s m a t r i x , the e l a s t i c beam f l e x u r a l s t i f f n e s s m a t r i x may be i n v e r t e d to f o rm the c o r r e s p o n d i n g _ i f l e x i b i l i t y m a t r i x . The c u r r e n t f l e x i b i l i t y v a l u e s f o r the h i n g e s , K ^ _ i and K a re then added to g i v e the e lement f l e x i b i l i t y m a t r i x a s : dv. i i 'h? f . . • J f . . + J J 1 The e lement f l e x i b i l i t y m a t r i x may then be i n v e r t e d to y i e l d the comp le te s t i f f n e s s m a t r i x . The g l o b a l s t i f f n e s s m a t r i x may then be formed by the s t a n d a r d p r o c e d u r e . 3.4 DERIVATION OF THEORETICAL MOMENT-ROTATION CURVES 3.4.1 D e s c r i p t i o n o f Model Beam It w i l l be a p p r e c i a t e d t h a t the b e h a v i o u r o f a r e a l p r e s t r e s s e d c o n c r e t e member s u b j e c t e d t o c y c l i c l o a d i n g i s complex. Depending on a number o f f a c t o r s , i n c l u d i n g g e o m e t r i c c o n f i g u r a t i o n , end moment p a t t e r n s , and magni tude o f l o a d i n g , the p l a s t i c c u r v a t u r e s may not be l i m i t e d to 7 8 35 the e n d s , but may ex tend o v e r a s i g n i f i c a n t p o r t i o n o f the member ' ' 7 8 A d d i t i o n a l l y , o b s e r v a t i o n o f members t e s t e d e x p e r i m e n t a l l y ' has shown t h a t under h i g h I n t e n s i t y c y c l i c l o a d i n g a p o r t i o n o f t he member at each end may s u s t a i n c u r v a t u r e s in exce s s o f t h o s e r e q u i r e d to cause y i e l d i n g o f t h e s t e e l a n d / o r c r u s h i n g o f the c o n c r e t e . As s u c h , i n the d e r i v a t i o n o f the i d e a l i z e d model f o r the h y s t e r e t i c a c t i o n o f a p r e s t r e s s e d c o n c r e t e 36 member proposed i n t h i s study, the n o n l i n e a r curvatures were i n t e g r a t e d over the e n t i r e length o f the member to form t h e o r e t i c a l end moment-p l a s t i c r o t a t i o n curves. These were i d e a l i z e d and used i n conjunction w i t h the concentrated hinges o f the pres t r e s s e d concrete beam-column element described e a r l i e r . In the d e r i v a t i o n of the t h e o r e t i c a l moment-rotation curves, the complex a c t i o n o f a r e a l member was represented by the model beam shown i n F i g . 3.9. I t w i l l be noted that the model member c o n s i s t s of a c e n t r a l p o r t i o n where e l a s t i c and/or p o s t - e l a s t i c a c t i o n may occur, and at each end, a crushing zone, L c y , where a l l post-crushing a c t i o n was assumed to occur. This crushing length was defined as that length of the member i n which the curvature a s s o c i a t e d w i t h the onset the crushing o f the concrete i s reached before strength degradation occurs. The ne c e s s i t y to define a crushing length arose because, i n a re a l member, not!cable strength degradation does not occur when an i n f i n i t e s m a l s e c t i o n a t the member end reaches the u l t i m a t e moment and crushing o f the concrete ensues. In a re a l member, the crushing a c t i o n would be spread over a f i n i t e length. In view o f t h i s , the crushing zone of the model beam was defined as f i v e percent of the length o f the member. I t was f e l t that f o r most members th t s would be i n keeping with experimental r e s u l t s , and acceptable in view o f curr e n t r e c o m m e n d a t i o n s ^ ' ^ ' ^ ' f o r equivalent p l a s t i c hinge lengths. In c o n s t r a s t to the assumption made above, i n a r e a l member, strength degradation would occur before the e n t i r e length _ c u had reached crushing. The assumption was made, however, f o r computational ease. I t is f e l t that f o r most members the assumption w i l l not m a t e r i a l l y a f f e c t the response. However, f o r members which have a high l e v e l of Fig. 3.9 Model Member 38 p r e s t r e s s and l i t t l e a d d i t i o n a l deformed b a r r e i n f o r c i n g , s t r e n g t h d e g r a d a t i o n w i l l be u n d e r e s t i m a t e d and energy d i s s i p a t i o n w i l l be o v e r -e s t i m a t e d . 3.4.2 T h e o r e t i c a l Background C o n s i d e r a member l oaded w i t h equa l and o p p o s i t e end moments as shown i n F i g . 3.10a. If t he moment - cu rva tu re r e l a t i o n s h i p shown i n F i g . 3.5 i s a c c e p t e d , then f o r t he model beam d e s c r i b e d e a r l i e r , t he M/E I , o r c u r v a t u r e d iagram wou ld be as shown in F i g . 3.10b. A p o r t i o n o f the member remains e l a s t i c , a p o r t i o n i s in t he c r a c k e d - e l a s t i c r ange , and a p o r t i o n has p r o g r e s s e d i n t o the i n e l a s t i c , o r p o s t - c r u s h i n g r a n g e . F i g . 3.10c shows the d e f l e c t e d shape o f t h e member. It w i l l be n o t e d t h a t the end r o t a t i o n , Bj, i s composed o f two components . The e l a s t i c r o t a t i o n i s r e p r e s e n t e d by 9^ , w h i l e t h e p l a s t i c r o t a t i o n i s shown as 8 . P A c o n v e n i e n t method f o r d e t e r m i n i n g t h e end r o t a t i o n s o f a member s u b j e c t e d t o some moment l o a d i n g p a t t e r n i s the use o f c o n j u g a t e beam a n a l y s i s . R e c a l l t h a t the s h e a r and moment o f the c o n j u g a t e beam l o a d e d w i t h t he H/EI d i ag ram c o r r e s p o n d t o the s l o p e and d e f l e c t i o n o f the r e a l beam. Note t h a t t he a r e a o f t he c u r v a t u r e d i ag ram may be d i v i d e d i n t o two r e g i o n s as shown i n F i g . 3.10b. The f i r s t i s a t r i -a n g u l a r r e g i o n o f e l a s t i c c u r v a t u r e , w h i l e the second Is the r e g i o n o f p l a s t i c c u r v a t u r e . If o n l y t he p l a s t i c p o r t i o n o f t h e end r o t a t i o n Is d e s i r e d , o n l y the p l a s t i c r e g i o n o f the c u r v a t u r e d i ag ram i s used to c a l c u l a t e the end s h e a r s o f t he c o n j u g a t e beam. Fig. 3.10 Theoretical Properties of Model Member 3.4.3 Derivation of End Moment-Hinge Rotation Curves While the moment-curvature relationship for a given material is unique, each end moment pattern determines a different end moment-rotation relationship. The usual procedure is to produce some moment-rotation relationship based on a particular end moment configuration and use that model in a nonlinear analysis. Typically, the relationship corresponding to equal end moments producing reversed curvature is used. In fact, the end moment pattern may vary widely from that on which the model is based. As a result, it was decided to include Idealizations for three basic end moment patterns in the model. Those cases were the usual equal end moment producing reversed curvature, and In addition moment at one end only and uniform moment producing continuous curvature. It would be expected that, based on the r e l -ative stiffnesses of the surrounding columns and girders, one could choose the idealization most l ikely to represent the behaviour of a particular member. A computer program, MREL, was written to determine the end moment-hinge rotation relationship for the various end moment config-urations. For the purpose of analysis the model beam described ear l ier was represented as a uniform member with non-dimensionalized properties. That member was divided into an arbitrary number of increments. The stiffness of each increment, or section, was monitored through the use of a hysteresis routine based on Blakeley's^ moment-curvature model. The program operated by incrementing the end moments to produce a monotonic or cyc l ic loading pattern. In the analysis the end moments could continue to Increase, as shown in Fig. 3.3, until the section a length L c u from the member end had reached the crushing curvature. No section, however, was permitted to exceed M . When the entire crushing r cu 3 length had sustained the curvature associated with the onset of crushing strength degradation was in i t iated. For any loading pattern the analysis was carried out by repetition of the procedure described below: 1. Input end moment Increment, A M g^ 2. Calculate the Incremental moments AM for each of the sections. 5 3. For a given section determine i f the add!tional AM has resulted in a change of st i f fness. If a change of stiffness has occured determine the proportion of M necessary to cause the change. 4. Determine the Incremental section curvature, A 5. If no change of stiffness has occured add f u l l Increment of _M and_JZf to M$ and 0^. If a change has occured add the proportion of AM g and resultantA# s required to cause change. Then calculate the additional ASS with the new stiffness due to the balance of AM and update moment and curvature. 6. Repeat steps 2 to 3 for each section. 7. Using the conjugate beam principle determine the end rotation. Subtract the e las t ic portion to leave the plast ic component of end rotation. 8. Repeat steps 1 to 7 for as many increments of end moment as required to produce desired loading pattern. A l i s t ing of MREL is contained in Appendix A. Each of the three cases considered consisted of three stages. Those stages, as with Blakeley's^ moment-curvature Idealization, correspond respectively to loading following cracking of the concrete, loading following crushing of the member in one direction only, and loading subsequent to crushing of the concrete In both directions. For each of the three cases a number of computer runs were made to generate a data base for idealization. Various incremental end moment patterns were 42 input to produce h y s t e r e t i c curves f o r a range of reversa l patterns f o r each of the three stages. Typ ica l examples of derived end moment-hinge rotat ion curves are shown In F ig s . 3.11 - 3.13. F i g . 3.14 shows the c lose correspondence between the curves f o r the f i r s t two cases. Deta i l s are given In Appendix B. 3.5 IDEALIZED MOMENT ROTATION MODEL 3.5.1 Idea l i za t ion of End Moment-Hinge Rotation Curves Any nonl inear ana lys i s uses some model to monitor the in te rac t ion o f force and d e f l e c t i o n o r moment and ro ta t i on . The funct ion of the model is to monitor the s t i f f n e s s of an element and/or hinge based on present resu l t s and ce r ta in parameters. Before the resu l t s o f the der ived hys teres i s curves could be app l ied to a nonl inear ana l y s i s , i t was necessary to formulate an i dea l i zed model. S t a t i s t i c a l ana ly s i s were made f o r each stage of the three cases to determine the curves of best f i t f o r a number of para-meters which contro l the h y s t e r e t i c curves. Deta i l s are given in Appendix B. F i g s . 3.15 - 3.17 show the i dea l i zed curves f o r each o f the three cases. With the i dea l i z a t i on s as a bas i s , a group of sub-routines compatible with DRAIN-2D, a dynamic ana lys i s program was wr i t t en . A l i s t i n g o f these subroutines appears in Appendix A. 3.5.2 Hysteres is Law While there is some v a r i a t i o n between the three cases the F i g . 3.11 Typica l Derived End Moment-Plastic Rotation Curves f o r Stage One 03 rv Fig. 3.12 Typical Derived End Moment-Plastic Rotation Curves for Stage Two F f g . 3.13 T y p i c a l D e r i v e d End M o m e n t - P l a s t i c R o t a t i o n Curves f o r S tage Three 46 Fig. 3.14 Comparison of Derived Stage Two End Moment-Plast ic Rotation Curves for Three Cases M M STAGE THREE F i g . 3.15 I d e a l i z e d H y s t e r e t i c Model f o r H inge -Case One F i g . 3.16 I d e a l i z e d H y s t e r e t i c Model f o r H inge -Case Two M STAGE THREE F i g . 3.17 I d e a l i z e d H y s t e r e t i c Model f o r Hinge-Case Three 50 h y s t e r e s i s law f o r each Is e s s e n t i a l l y as described below: 1. Stage One On I n i t i a l loading the hinge i s assumed to have i n f i n i t e s t i f f n e s s up to the pseudo cracking moment, M' , at points + A, as shown i n F i g . 3.18. At that point the hinge s t i f f n e s s i s changed to K j , and loading continues along l i n e A-B. Should a r e v e r s a l occur at point C during loading along l i n e A-B, the l o c a t i o n of poi n t D i s , based on s t a t i s t i c a l data, c a l c u l a t e d and the hinge s t i f f n e s s Is modified based on the value o f moment and hinge r o t a t i o n at point C. Unloading then commences along l i n e C-D and continues along l i n e D-A to complete un-loading and s t a r t negative loading. Reversal and subsequent reloading at any point between C and -A w i l l r e s u l t i n r e t u r n i n g along D-A and/or C-D to the point of o r i g i n a l r e v e r s a l . Loading would then continue along A-B. 2. Stage Two If during loading the ul t i m a t e moment, M c u, at po i n t s + B i s reached, the hinge s t i f f n e s s Is changed to zero and remains at that value u n t i l y i e l d i n g has progressed to poi n t E. At that point f u r t h e r loading causes the hinge s t i f f n e s s to change to K^, a negative s t i f f n e s s , and commencement of strength degradation. F i g . 3.18 shows stage two where i n i t i a l post-crushing loading occurs in the p o s i t i v e d i r e c t i o n . Should a r e v e r s a l at point F occur during loading along B-E or beyond E, the l o c a t i o n o f points G and L i s c a l c u l a t e d , as described i n the f o l l o w i n g s e c t i o n , and the hinge s t i f f n e s s i s modified based on the value o f moment and hinge r o t a t i o n at the point of r e v e r s a l . Unloading then commences along l i n e F-G and continues along l i n e G-A to complete unloading and s t a r t negative loading. Reversal between F and G causes STAGE TWO 8 f "1 STAGE ONE STAGE THREE F i g . 3.18 D e s c r i p t i o n o f H y s t e r e s i s Law 52 reloading along F-G. I f a rev e r s a l occurs during negative loading along A-B, say at 1, unloading moves along I-H towards point H rather than Point 0 as In stage one. Further unloading continues along l i n e s H-L and L-F to complete negative unloading and s t a r t p o s i t i v e l oading. Reversal during unloading along G-A, say at G', causes loading along G'-L. Reversal during loading along H-L, say at H', causes un-loading along H'-A. 3. Stage Three A f t e r moments grea t e r than M £ u have been sustained In both the p o s i t i v e and negative d i r e c t i o n s the behaviour is as shown f o r stage three In F i g . 3.18. Unloading a f t e r F-G continues along G-L and L-F as opposed to stage two where i t continued along l i n e G-A. Reversal during unloading along G-L causes loading along G'-L. Reversal during loading along L-F causes unloading along L'-G. 3.5.3 Numerical Value o f Parameters. The numerical values f o r the parameters obtained from the a n a l y s i s described above are l i s t e d below. For a l l cases the r a t i o of M /M i s taken as 1.8. A d d i t i o n a l l y , Kj Is the slope of the c r a c k e d - e l a s t i c p o r t i o n and i s the slope o f the degrading s t r e n g t h p o r t i o n . I. Case One: Equal and Opposite End Moments K * " i e L where El Is the e f f e c t i v e f l e x u a l s t i f f n e s s o f the member Y ce - 0.222 Y - K„/K -z e -0.010 (beams) cu • -0.031 (columns) M A " M ' , * 1.2 M (See Appendix B) /» c r cr M D - 0.6 M'r e D « S 8 C [ 6 . 6 6 - 13.40 + M s f ^ * ] MG = ° ' 3 3 M c u Mti -' 0.55 M I cu 8 c h - 5.0 e cu 8 L G - 9 f [ ° - 2 7 + oWfc;V °-00,7m] I f 9- > 20 0 f ' cu »CU/ » CUV 9 L G - 0 . 7 0 9 f M, - 0.40 M lw c r 2. Case Two: Moment a t one End Only K - I f L e L where El Is the e f f e c t i v e f l e x u r a l s t i f f n e s s o f the member Y - K./K • 0.222 ce 1 e Y - K,/K » -0.018 (beams) cu l e -0.054 (columns) M = M' - 1.2 M A c r c r M_ « 0.6 M' D c r e D « 6 c [ 6 . 6 6 - .3.40 6.95 ( V ] cu cu M G - 0 . 3 3 M c u M L » 0 . 5 O M c u 9 c h - 2 ' 5 9 c u f o r e < e, < io e cu ^ f ^ cu 2. • « [O.M • 0.12 ( ^ - o - o o . f y ] if e. > 10 e f / cu e L G-o . 7 o ef M. - 0.40 M 1w c r 3. Case T h r e e : U n i f o r m Moment e L where El i s the e f f e c t i v e f l e x u r a l s t i f f n e s s o f the member Y » K./K » 0.178 ce 1 e Y ^. . = V K _ " -0.003 (beams) cu z e • -0.009 (columns) M' - M cr cr M. » 0.60 M D c r • D - » c t5-78 " ,2-79 (ir) * 7-'3 (ir-) ] \ c u ; » c u ' -* 55 M G - 0 . 3 3 M c u M L » 0 . 5 0 M c u e . - e ch cu for e c u < ef < 10 e c u e L G» ef [ 0 . 2 3 • 0.05 /JtV \ cu' J i f ef > 10 e c u e L G - o . 7 o ef M. » 0.30 M 1w cr One would expect the moment-rotation curves for the case of uniform moment to be similar to the moment-curvature relationship on which they are based. As a result, some of the parameters for this case are the same as those for Biakeley's^ moment-curvature idealization. 3.5.4 Accuracy of Moment-Rotation Curves It would be expected that the value of the parameters given for the three cases would be affected by member properties. For example, as the amount of deformed bar reinforcing is increased, i t would be t expected that the hysteretic curves would come to resemble those for ordinary reinforced concrete. Additionally, one would expect the amount of transverse reinforcing, level of axial load, and degree of prestressing to be significant factors. However, for typical prestressed concrete members with additional lateral and transverse reinforcing, 56 t h e pa rameter s g i v e n s h o u l d p r o v i d e adequa te s o l u t i o n a c c u r a c y . It s h o u l d be kept In mind t h a t the model i s d e s i g n e d to a c c o u n t f o r t h e energy d i s s i p a t i o n , and the s t i f f n e s s and s t r e n g t h d e g r a d a t i o n p r o p e r -t i e s o f p r e s t r e s s e d c o n c r e t e members as opposed to p r e d i c t i n g the e x a c t r e sponse o f a p a r t i c u l a r b u i l d i n g . 5 7 CHAPTER 4 COMPARATIVE STUDY OF THE EARTHQUAKE RESPONSE OF PRESTRESSED AND REINFORCED CONCRETE STRUCTURES 4.1 INTRODUCTION Modern seismic design codes require a building to be designed with a high strength if it is to perform elast ica l ly or with a high ductile capacity i f it is to perform inelast leal ly. If ductile behav-iour is assumed the structure wi l l be loaded, perhaps s ignif icant ly, into the post-elastic range when subjected to a strong earthquake. Histor ica l ly , there has been concern about the possible b r i t t l e nature and premature fa i lure of prestressed concrete members used as seismic 7 8 resisting elements. A number of recent experimental studies ' have demonstrated the ab i l i ty of carefully detailed prestressed concrete members to sustain large post-crushing curvatures. These studies have shown that prestressed concrete members may be designed to possess substantial ductile capacity. However, there continues to be a dearth of quantitative values for the duct i l i ty demands of members of pre-stressed concrete framed structures subjected to seismic loading. Apart 9 17 from the studies by Spencer ' , l i t t l e work has been done on the response of multi-story prestressed concrete frames, and the duct i l i ty demands and energy dissipation properties of the constituent members. This chapter presents the results of a comparative study of prestres-sed and reinforced concrete versions of a typical multi-story framed 58 s t r u c t u r e . P r e v i o u s work Is e x t e n d e d in t h a t t he p r e s t r e s s e d c o n c r e t e h y s t e r e t i c i d e a l i z a t i o n i n c l u d e s s t i f f n e s s and s t r e n g t h d e g r a d a t i o n , and a number o f e a r t h q u a k e r e c o r d s a t v a r y i n g i n t e n s i t y a r e u sed . 4.2 NONLINEAR EQUATIONS OF MOTION 4.2.1 Damping Most n o n l i n e a r t ime s t e p a n a l y s e s use some f o r m o f v i s c o u s damping in a d d i t i o n to the h y s t e r e t i c damping o f the members. Because o f t h e c o m p l e x i t y and l a c k o f u n d e r s t a n d i n g o f the damping mechanism in the r e a l s t r u c t u r e , the s i m p l i f i c a t i o n p roposed by R a l e i g h i s g e n e r a l l y u sed . That assumes t h a t the v i s c o u s damping m a t r i x may be w r i t t e n a s : [c]= -.[M] • / S [ K ] That i s , the v i s c o u s damping m a t r i x may be assumed t o be p r o p o r t i o n a l t o the mass and s t i f f n e s s m a t r i c e s o f t he s t r u c t u r e . Fo r mass p r o p o r -t i o n a l damping, the f r a c t i o n o f c r i t i c a l v i s c o u s damping In the ? ^ m o d e , A m i , i s g i v e n by A - . F o r s t i f f n e s s p r o p o r t i o n a l damping, the f r a c t i o n o f c r i t i c a l v i s c o u s damping In the ? mode, A ^ j , i s g i v e n by A K i = Tj Thus i t i s a p p a r e n t t h a t in the h i g h e r modes w i t h s m a l l T the e f f e c t i v e n e s s o f mass p r o p o r t i o n a l damping d e c r e a s e s w h i l e t h a t o f s t i f f n e s s p r o p o r t i o n a l damping i n c r e a s e s . D e t a i l s a r e g i v e n in most 36 37 dynamics o f s t r u c t u r e s t e x t s , i e . C lough , B i ggs 59 There appears to be no accepted value fo r the viscous damping r a t i o to be app l ied in the nonl inear ana lys i s of a prest ressed concrete framed 9 7 s t r u c t u r e . However, both Spencer , and Blakeley , found that the percentage of v iscous damping assumed had a s i g n i f i c a n t e f f e c t on the response of the s t ructures they analyzed. As such, a cor rect representat ion of the v iscous damping r a t i o would seem fundamental to the accurate p red ic t ion of the d u c t i l i t y demands f o r prest ressed concrete members. In an e l a s t i c ana l ys i s where h y s t e r e t i c damping is Ignored and a l l energy d i s s i p a t i o n is assumed occur through v iscous damping, the equivalent v iscous damping r a t i o f o r re in forced concrete is genera l ly taken to be from f i v e to ten percent , depending on s t ruc tu re 7 8^ and/or member damage, with an average value of eight percent ' . It 4 20 21 w i l l be r e c a l l e d from Chapter 2 that experimental s tudies ' ' of the dynamic proper t ies of prest ressed concrete have suggested that the damping r a t i o Is lower in prest ressed concrete members than in compar-ib le re in forced concrete members, perhaps only o n e - h a l f as much. This would imply a damping r a t i o on the order of four percent of c r i t i c a l fo r the e l a s t i c ana lys i s of a prest ressed concrete frame subjected to r e l a t i v e l y Intense se ismic loading . In a nonl inear a n a l y s i s , however, where the v iscous damping is in add i t ion to the h y s t e r e t i c damping the v iscous damping r a t i o assumed depends on several f a c t o r s . These include the in tens i t y of loading and resu l tant s t ruc tu re and member damage, the in te rac t ion of the s t r u c t u r a l system and the nonstructura l components and f i x t u r e s , and the accuracy of the h y s t e r e t i c model In representing small amplitude behaviour. In the cons iderat ion of the relevance of the above f a c t o r s to 60 this investigation several points were noted. F i r s t , as described in section 3.5.2 and shown in Fig. 3.18 and Figs. 4.1 and 4.2, once crushing has occured, the idealized model for the hysteretic behaviour of a prestressed concrete member accounts for some energy dissipation during small amplitude cycling. This is in contrast to the elasto-plast ic model generally used for reinforced concrete. The second factor considered was the effect of the nonstructural components and fixtures on damping. It was conservatively estimated that a l l resistance to lateral deformation would be provided by the structural frame and that there would be negligible interaction between the nonstructual curtain walls and other components, and the frame. As such, nonstructural elements would not have a signif icant effect on the damping. Final ly, q in his nonlinear study, Spencer found that the assumption of a damping ratio of even one percent mass proportional and two percent stiffness proportional viscous damping resulted in s ixty-six percent of the total energy being dissipated by viscous damping. In the same study, other analyses with higher damping ratios attributed s ignif icantly higher percentages of the total energy dissipated to viscous damping. As stated 9 by Spencer in his study , it is f e l t that to have such a high percentage of the energy dissipated by viscous damping Is unreal ist ic for the non-linear analysis of an open framed structure. In this study the damping ratio was limited to one percent of c r i t i c a l in the fundamental mode. It is thought that while a viscous damping ratio on the order of four percent of c r i t i c a l may be reasonable for an e last ic analysis of a prestressed concrete frame, it could not be jus t i f ied for the nonlinear analyses in this study. The value of one percent of c r i t i c a l was chosen for the viscous damping ratio because it o ON F i g . 4.1 Typical End Moment Hinge Rotation ~* Relat ionships f o r Hinges F i g . k.2 T y p i c a l End Moment H inge R o t a t i o n R e l a t i o n s h i p s f o r Hinges ON 63 r e f l e c t s experimental observation of the damping values of prestressed concrete members and frames in the e l a s t i c stage up t o the onset of cr a c k i n g . I t i s thought that the value chosen w i l l g i ve a reasonable upper bound f o r the d u c t i l i t y demands of a m u l t i - s t o r y moment r e s i s t i n g p restressed concrete frame. 4.2.2 Nonlinear Equations o f Motion The n o n l i n e a r equations of motion were solved using the time step 39 i n t e g r a t i o n procedure contained in the DRAIN-2D base subroutines . The program uses the constant a c c e l e r a t i o n i n t e g r a t i o n method because i t has the q u a l i t i e s o f being s t a b l e f o r a l l time increments and per i o d s , and not introducing numerical damping into the system. A d e t a i l e d d e s c r i p t i o n o f the constant a c c e l e r a t i o n method may be found in most 36 37 dynamics o f s t r u c t u r e s t e x t s , i e . Clough Briggs . The b a s i c equation of dynamic e q u i l i b r i u m f o r any f i n i t e time step may be w r i t t e n as: fM]{Vr) + (c T]{_r) + [ K _ ] { A r) - U P) ( 4 . 1 ) where (Ar\, {4 r\, ( i i r ] , and {_ pj are the f i n i t e increments of a c c e l e r a t i o n , v e l o c i t y , displacement, and load r e s p e c t i v e l y , [ M ] is the mass matrix while [C-] and [K_] are the tangent damping and s t i f f n e s s matrices f o r the s t r u c t u r e at the beginning o f the time step. Results o f the constant a c c e l e r a t i o n method allow {_ r ] and (&r] to be w r i t t e n as 2 o ~ ' A t and (&r) - - 2 ' r ' - + i r l (4.3) O O 41 ^ t 2 64 where r Q and r Q are the velocity and acceleration values respectively at the beginning of the time step and 4 t is the length of the time step. Upon substitution of equations (4.2) and (4.3), and the assumed form of the damping matrix equation (4 .1) becomes: [ ( ! L . 2 + & ) „ • ,) ^ ( L R ) . { „ ) L 1 A t &tJ v A t ' , J + M ( 2 > r o + n'ro + 2«ro) +^Mi2'ro) Because i t is not convenient in the computational scheme to evaluate the term /$[Kt] «^  2 r ^ , the transformation procedure proposed 40 by Wilson is introduced. Rearranging equation (4.4) gives: ' 4_ A t ' • & pl * (Ml{ 2R0 + AVO + W0 < (i,-5) It is expedient to define an 'e f fect ive ' displacement vector (4 x] as: M -fir • 'KM - ififo) ( " - 6 ) Substituting equation (4.6) into (4.5) gives: M + ["]{ 2 r 0 • / t r o * 2<0 (1,-7) Solving equation (4.6) for (d r^  and substituting the result into equation (4.7) y ields: [ a i M • KT] fax] - (AP) + [M] {2r"o + 4 1 o o o 65 Equation (4.8) may be expressed more simply as [K_1 {&X} - (A P) (4.10) where [K-| Is the 'e f fect ive ' stiffness and Is given by [R-] » [uXM + K-1 and p\ IS the effective Increment of load and Is given by {AP) - M • [Ml {2 r Q • ^ ' r o • UrQ - iJlr^] (4.11) In the determination of the earthquake response of a structure, {& P\ is replaced by the term -[M*]{_ r'g^ w n e r e & r g \ ' s t n e Incremental ground acceleration and (A r\ represents the motion of the structure relative to the ground. It follows that once (jx^ has been determined, equation (4.6) may be solved for r^. The incremental velocit ies and accelerations may then be determined from equations (4.2) and (4.3) . 4.2.3 State Determination As a result of the solution of Equation (4.8) the incremental hodel displacements and resultant incremental member deformations at the end of any time step may be determined. To complete the analysis of the incremental response, it is additionally necessary to determine the Increments of member forces corresponding to the computed incremental deformations, taking into account any change of stiffness of the hinges at the member ends. This procedure is typical ly termed the 'state determination'. The hysteretic relationship used to model the nonlinear action of a prestressed concrete member was the prestressed concrete idealization described in Chapter Three. That idealization Is piece wise linear 66 and thus a change o f h i nge s t i f f n e s s i s assumed t o o c c u r s u d d e n l y r a t h e r than p r o g r e s s i v e l y as in a r e a l member. T h i s change o f s t i f f n e s s i s termed an " e v e n t " . If no e v e n t s o c c u r i n a t ime s t e p , the member f o r c e i nc rement w i l l be l i n e a r as shown by the d o t t e d l i n e in F i g . 4.4. However, i f one o r more e v e n t s o c c u r , the c a l c u l a t e d i n c r e m e n t a l d i s p l a c e m e n t s and d e f o r -mat ions w i l l not be s t r i c t l y c o r r e c t because they have been computed on the b a s i s o f l i n e a r b e h a v i o u r . It may be assumed, however , t h a t e s s e n -t i a l l y the same d i s p l a c e m e n t s and d e f o r m a t i o n s wou ld r e s u l t had the n o n l i n e a r i t y been taken i n t o a c c o u n t . T h i s i s r e a s o n a b l e because the t ime s t e p i s assumed to be sma l l r e l a t i v e t o the p e r i o d and t h e re sponse o f t he s t r u c t u r e i s assumed to r e s u l t p r i m a r i l y f rom damping and i n e r t i a e f f e c t s 3 9 . The p r o c e d u r e f o r computa t i on and a d d i t i o n o f f o r c e i nc rement s in any t ime s t e p i s i l l u s t r a t e d i n F i g . 4.3. The i n i t i a l l i n e a r f o r c e inc rement i s A S ^ . At t he o c c u r a n c e o f an e v e n t , the p e r c e n t a g e n e c e s -s a r y to reach the e v e n t , «*Jj A S J , IS added t o the t o t a l member f o r c e s . S u b s e q u e n t l y , the e lement s t i f f n e s s m a t r i x i s m o d i f i e d , the new member f o r c e I n c r e m e n t , A S ^ , i s c a l c u l a t e d , and the p e r c e n t a g e needed to reach the next event o r comp le te the d e f o r m a t i o n i n c r e m e n t , fli^ & S^, i s d e t e r m i n e d . As shown in F i g . 4.4, t h i s w i l l r e s u l t , in g e n e r a l , i n the a d d i t i o n o f a n o n l i n e a r f o r c e i n c r e m e n t , AS... , f o r the t ime s t e p . Because one o r N L more even t s have o c c u r e d , t h e r e w i l l be an e q u i l i b r i u m unba l ance f o r c e o f A S ^ f o r the member, and thus dynamic e q u i l i b r i u m w i l l no t be s a t i s f i e d . E q u i l i b r i u m unba lances may a d d i t i o n a l l y r e s u l t f rom s t i f f n e s s p r o p o r -t i o n a l damping f o r c e s due to changes in the e lement s t i f f n e s s m a t r i x a t 67 F i g . 4.3 Computat ion o f F o r c e Increment , Re f . 39 FORCE, S ^ DEFORMATION, v F i g . U.k L i n e a r and N o n l i n e a r F o r c e Increment , Ref 69 the occurance of an event. If these unbalances are uncorrected, s igni -ficant errors may accumulate and severely affect the accuracy of the computed response. As a result, temporary corrective loads are applied at the beginning of the time step following the occurance of one or more events to eliminate the error caused by the unbalances. The 39 procedure is explained in greater detail in the DRAIN-2D users manual . A.3 PROGRAMME OF ANALYSIS 4.3.1 Structure Analyzed The building which is considered In this study is a moment resisting frame structure or ig inal ly designed by Clough and Bensuka and used in their studies for the Federal Housing Administration'^. Since that time 41 9 17 it has been studied by GIberson and Spencer ' . The structure does not represent the best or most ef f ic ient design for a prestressed concrete frame. However, an advantage in using this particular frame is that both reinforced and prestressed concrete versions have been studied ear l ier . As such, as basis for veri f icat ion of the computer program and comparison of results exists. The properties of the structure are given in Appendix C. The frame was i n i t i a l l y designed using approximate procedures for s tat ic and gravity loads, and lateral loads as specified by the 1964 Uniform Building Code. Then using the relative stiffness values shown in Table C.2, a computer analysis was made to determine the "design" forces and moments in the members. The frame is designated as A/20/2.2/2/6 with the symbols having the following significance: 70 A: the s t i f f n e s s p r o p e r t i e s o f t he frame v a r y w i t h h e i g h t 20: the number o f s t o r i e s above g round l e v e l 2 . 2 : the p e r i o d o f the fundamenta l mode 2: the r a t i o o f c r a c k i n g ( y i e l d moments f o r r e i n f o r c e d c o n c r e t e v e r s i o n ) moments t o the d e s i g n moments f o r the beams 6: t he r a t i o o f t he c r a c k i n g moments t o the d e s i g n moments f o r the co lumns . To d e t e r m i n e the l e v e l o f e a r t h q u a k e Imp l i ed In the b u i l d i n g d e s i g n , a s t a t i c a n a l y s i s w i t h the g r a v i t y loads shown i n T a b l e C.7 and the l a t e r a l loads c a l c u l a t e d in a c c o r d a n c e w i t h the 1964 Un i f o rm B u i l d i n g Code was made f o r t h i s I n v e s t i g a t i o n . The base s h e a r , V, as s p e c i f i e d by tha t code f o r a moment r e s i s t i n g frame in zone t h r e e i s a p p r o x i m a t e l y 0 .03G. Because the b u i l d i n g has a h e i g h t t o depth r a t i o o f l e s s than f i v e , a l l o f the base s h e a r was a p p l i e d In the u sua l t r i a n g u l a r d i s t r i b u t i o n o v e r the h e i g h t o f the f r ame . The moments In the g i r d e r s r eached about t w o - t h i r d s o f t he c r a c k i n g moments o v e r most o f t he h e i g h t o f t h e b u i l d i n g w i t h s l i g h t l y lower v a l u e s in the top s e v e r a l s t o r i e s . T h i s wou ld imply i n c i p i e n t y i e l d i n g o f most o f the g i r d e r s f o r a base s h e a r o f a p p r o x i m a t e l y 0 .04G. If a s y s tem d u c t i l i t y o f f o u r i s assumed t h i s would imply a base s h e a r o f about 0.16G f o r an e q u i v a l e n t e l a s t i c s y s tem. F o r a p e r i o d o f 2.2 seconds and a damping r a t i o o f f i v e p e r c e n t , the dynamic a m p l i f i c a t i o n f a c t o r i s about 0 .46 . T h i s i m p l i e s the b u i l d i n g was d e s i g n e d f o r a maximum ground a c c e l e r a t i o n o f about t h i r t y - f o u r p e r c e n t o f g r a v i t y . In t he o p e r a t i o n o f the computer program and t h e a n a l y s i s o f t h e b u i l d i n g d e s c r i b e d the f o l l o w i n g a s sumpt ions were made: 1. The f o u n d a t i o n was i n f i n i t e l y r i g i d and t o r s i o n was n e g l e c t e d . 71 2. The structure was an open moment resisting frame. Al l resistance to lateral displacements was assumed to be provided by the beams and columns. In a real building, nonstructural fixtures and i n f i l l panels may provide addi-tional resistance to lateral movement. 3. A separate stat ic analysis was made of the structure with the loading pattern shown In Table C.7. The resulting forces and moments were applied as i n i t i a l loads for the nonlinear dynamic analyses. 4. Nonlinear effects were introduced by means of hinges at each end of a beam or column. 5. Shear deflection in the beams and columns was included. 6. The P-A effect was modeled approximately by the inclusion of a geometric st i f fness. 7. There was no interaction between axial load and column moment. 8. A l l joints at the same f loor level were constrained to move lateral ly together. That Is, there was no extension of the beams. 9. Vertical degrees of freedom were not considered. That Is, there was no column extension. 10. A l l mass was assumed to be concentrated at the nodes of a floor level. Only horizontal movement of the mass was cons i dered. 11. While the program included the hysteresis laws for three end moment patterns, only that corresponding to equal and opposite end moments producing reversed curvature was used. 4.3.2 Excitation Three earthquake records were used in this study, those being El Centro NS 1940, Taft S69E 1952, and Parkfield N65E 1966. The f i r s t two records are sinusoidal in nature while the third is of the impulsive type. El Centro has a peak acceleration of 35 percent of gravity which occurs 2.12 seconds after the start of the record. Taft has a peak acceleration of 18 percent of gravity which occurs 6.40 72 seconds after the start of the record. Parkfield has a peak accelera-tion of 49 percent of gravity which occurs 3.74 seconds after the start of the record. El Centro was used at both its normal intensity and scaled to give a peak acceleration of 50 percent of gravity. The Taft and Parkfield records were both scaled to 50 percent of gravity. Details concerning the response spectra and earthquake records are given in Appendix D. For a l l the cases studied the f i r s t twenty seconds of the earth-quake record was used. That length was chosen to ensure that the peak ground acceleration, velocity and displacement, maximum structure response, and the greatest proportion of the spectral response pract i -cal in relation to computer costs, would occur. 4.3.3 Time Increment The value of the time increment used for a l l the nonlinear analyses made in this investigation was 0.01 second. A study was i n i t i a l l y made of the A/20/2.2/2/6 frame with 0.005, 0.01, and 0.05 second time steps. The responses of the 0.01 second and 0.005 second analyses were found to be almost identical, while that of the 0.05 second analysis was found to diverge considerably. As such, to minimize computer costs, the 0.01 second increment was used. Using the 0.01 second interval, the time taken to compute the response for twenty seconds of earth-quake record was approximately 350 seconds of CPU time on an AMDAHL 470 V/6 Model II computer. 73 4.3.4 S c h e d u l e o f Cases S t u d i e d In the s tudy o f t h e re sponse to s e i s m i c l o a d i n g o f t h e m u l t i -s t o r y p r e s t r e s s e d c o n c r e t e f r amed b u i l d i n g d e s c r i b e d e a r l i e r , and o f the r e s u l t i n g d u c t i l i t y demands o f the members, the e f f e c t o f s e v e r a l v a r i a b l e s was i n v e s t i g a t e d . Those v a r i a b l e s were the magn i tude and type o f v i s c o u s damping, the e a r t h q u a k e r e c o r d and i n t e n s i t y , and the s t r e n g t h d i s t r i b u t i o n o f the f rame. In t h i s r e g a r d t he f o l l o w i n g n o n -l i n e a r dynamic t ime s t e p a n a l y s e s were made: B u i l d i n g A / 2 0 / 2 . 2 / 2 / 6 - p r e s t r e s s e d c o n c r e t e v e r s i o n A. E a r t h q u a k e r e c o r d - E l C e n t r o NS 1940 @ 0.35G 1. AAECNS35 - damping r a t i o - 1 . 0 % mass p r o p o r t i o n a l v i s c o u s damping. 2. ABECNS35 - damping r a t i o - 0 . 3 7 % mass p r o p o r t i o n a l and 0.63% s t i f f n e s s p r o p o r t i o n a l v i s c o u s damping. 3. ADECNS35 - damping r a t i o - 1 . 0 % s t i f f n e s s p r o p o r t i o n a l v i s c o u s damping. B. 1.0% mass p r o p o r t i o n a l v i s c o u s damping 4. AAECNS50 - e a r t h q u a k e r e c o r d - E l C e n t r o NS 1940 @ 0 .50G. 5. AAZASE50 - e a r t h q u a k e r e c o r d - T a f t S69E 1952 § 0 .50G. 6. AAPKNE50 - e a r t h q u a k e r e c o r d - P a r k f i e l d N65E 1966 § 0 .50G. C. 1.0% s t i f f n e s s p r o p o r t i o n a l v i s c o u s damping 7. ADECNS50 - e a r t h q u a k e r e c o r d - E l C e n t r o NS 1940 § 0 .50G. B u i l d i n g B / 2 0 / 2 . 2 / V / 6 - p r e s t r e s s e d c o n c r e t e v e r s i o n D. 1.0% mass p r o p o r t i o n a l v i s c o u s damping 8. BAECNS35 - e a r t h q u a k e r e c o r d - E l C e n t r o NS 1940 § 0.35G The B has the f o l l o w i n g s i g n i f i c a n c e : The d e s i g n s t r e n g t h t o c r a c k i n g moment r a t i o v a r i e s l i n e a r l y f rom f o u r a t t he top o f t he b u i l d i n g to two at the bottom. In a l l other respects the B/20/2.2/V/6 frame Is identical to the A/20/2.2/2/6 frame described ear l ier and In Appendix C. In addition to the analyses l isted above, the results of several previous studies of the A/20/2.2/2/6 frame were considered. A. Building A/20/2.2/2/6 - reinforced concrete version 1. CCECNS32 - damping ratio - 10.0% mass proportional viscous damping, ref. 10 2. CSECNS32 - damping ratio 10.0% mass proportional viscous damping, ref. 9 B. Building A/20/2.2/2/6 - prestressed concrete version 3. ASECNS32 - damping ratio - 1.0% mass proportional and 2.0% st iffness proportional viscous damping, structure 5, ref. 9. 9 Spencer's results were or ig inal ly calculated using a different rotational duct i l i ty relationship than that contained in this study. To provide a basis for comparison, his results were recalculated on the basis of the definition of rotational duct i l i ty contained in this thesis. The prestressed concrete version of the frame studied by Spencer and considered in this investigation was, as noted refered to structure f ive in his report. That version was chosen because it had the min-imum value of damping used and maximum values for the response parameters. 4.3.5 System of Notation For convenience In identifying the principle parameters considered in this investigation, an eight symbol reference code was developed. As may be noted in the previous schedule of cases studied, the f i r s t letter refers to the version of the A/20/2.2/2/6 frame studied where 75 A: refers to the basic prestressed concrete version B: refers to the prestressed concrete version with a varied design strength to cracking moment ratio for the girders, that is the B/20/2.2/V/6 frame C: refers to the reinforced concrete version The second letter describes the damping values or makes reference to another study where A: refers to 1.0 percent mass proportional damping in the fundamental mode B: refers to 0.37 percent mass proportional and 0.67 st i f fness proportional damping in the fundamental mode D: refers to 1.0 percent st i ffness proportional damping In the fundamental mode C: refers to the FHA study'*' by Cloush and Benuska g S: refers to the nonlinear study by Spencer The third and fourth letters identify the earthquake record. The f i f t h and sixth letters describe the component of the record. The last two numbers refer to the scaled value, in terms of percent of gravity, of the maximum horizontal ground acceleration for that non-1inear analysis. k.h OUTLINE OF COMPARATIVE STUDY k.k.1 Method of Presentation Al l results presented are the maximum values, either positive or negative, recorded during the twenty seconds of earthquake excitation, of a number of parameters which indicate the capacity of the structure to withstand seismic loading. Those parameters are the maximum story lateral displacements, the member end rotational and hinge curvature 76 d u c t i l i t i e s , the t o t a l a c c u m u l a t e d h i n g e r o t a t i o n f a c t o r s , and the h i n g e s t a t u s d i a g rams . W h i l e no one p a r a m e t e r g i v e s a comp le te p i c t u r e o f t h e re sponse o f t he s t r u c t u r e t o e a r t h q u a k e l o a d i n g , when t aken t o g e t h e r , a r e a s o n a b l y a c c u r a t e i n t e r p r e t a t i o n may be made. The member end r o t a t i o n a l d u c t i l i t i e s were d e f i n e d i n the f o l l o w i n g manner: 9 + 8. c r hm u8 " 9 c r - I + e hm e c r where 6 H ' c r L (2 + 3>) C R " T T ET " and _ - ^y1-, ( s h e a r f l e x i b i l i t y f a c t o r ) L A G The term 0 r e f e r s to the r o t a t i o n a t c r a c k i n g , the term M' was c r c r d e f i n e d as the ' e f f e c t i v e * c r a c k i n g moment, and term 8 ^ r e f e r s t o t h e maximum h i n g e r o t a t i o n . 0 ^ i s shown as 0^ on F i g s . 3.15 - 3.18. For t h e c a s e where a member does not reach c r a c k i n g the member end r o t a t i o n a l d u c t i l i t y f a c t o r was d e f i n e d a s : 0 , M . ed ed u f t "* 0 ™ » 9 c r M c r where 8 .and M . r e f e r t o the maximum e l a s t i c member end r o t a t i o n and ed ed moment r e s p e c t i v e l y . H inge c u r v a t u r e d u c t i l i t i e s were d e f i n e d a s : 8. ^cr ' -ph where & « ^ c r E l 77 The term d r e f e r s t o the s e c t i o n c u r v a t u r e a t c r a c k i n g w h i l e the term L ^ r e f e r s t o the e q u i v a l e n t p l a s t i c h i n g e l e n g t h . T o t a i a c c u m u l a t e d h i n g e r o t a t i o n f a c t o r s were d e f i n e d in the f o l l o w i n g manner: The h i n g e s t a t u s d iagrams shown the s t a t e o f t h e h i n g e s , c r a c k e d o r c r u s h e d , a t t he end o f twenty seconds o f e a r t h q u a k e e x c i t a t i o n . The member end r o t a t i o n a l d u c t i l i t y f a c t o r r e p r e s e n t s t h e r a t i o o f the maximum member end r o t a t i o n , e l a s t i c p l u s p l a s t i c , to the c r a c k i n g r o t a t i o n . The h i n g e c u r v a t u r e d u c t i l i t y f a c t o r r e p r e s e n t s the r a t i o o f the maximum s e c t i o n c u r v a t u r e , t o the c r a c k i n g c u r v a t u r e . These d u c t i -l i t y f a c t o r s p r o v i d e an i n d i c a t i o n o f the maximum a m p l i t u d e o f n o n -l i n e a r member b e h a v i o u r . The t o t a l a c c u m u l a t e d h i n g e r o t a t i o n f a c t o r p r o v i d e s an i n d i c a t i o n o f the t ime h i s t o r y o f t he member n o n l i n e a r i t y . The h i n g e s t a t u s d iagram p r o v i d e s a v i s u a l i n d i c a t i o n o f t he damage t o the members as a r e s u l t o f s e i s m i c l o a d i n g . E q u i v a l e n t P l a s t i c H inge Length In the c a l c u l a t i o n o f t he c u r v a t u r e d u c t i l i t i e s , i t was n e c e s s a r y t o d e f i n e an e q u i v a l e n t p l a s t i c h i n g e l e n g t h , L . . To da te t h e r e i s p h no a c c u r a t e method o f p r e d i c t i n g h i n g e l e n g t h s f o r p r e s t r e s s e d c o n c r e t e members, a l t h o u g h a number o f recommendat ions have been made. It wou ld a p p e a r , however , t h a t the h i n g e l e n g t h assumed has a p ronounced e f f e c t 9 on the r e s u l t s o f t h e a n a l y s i s . Spencer p a r t i a l l y e x p l a i n e d the law c u r v a t u r e d u c t i l i t i e s he o b t a i n e d as due to the g r e a t e r h i n g e l e n g t h he u t e c r 78 assumed f o r p r e s t r e s s e d c o n c r e t e members. He f e l t t h a t s i n c e the p r e s e n c e o f p r e s t r e s s i n g d e l a y s t he deve lopment o f c u r v a t u r e c o n c e n -t r a t i o n due to y i e l d i n g o f t he s t e e l o r c r u s h i n g o f t h e c o n c r e t e , a l o n g e r h i n g e l e n g t h f o r p r e s t r e s s e d c o n c r e t e members was r e a l i s t i c . 35 In the ab sence o f any a c c e p t e d e q u a t i o n , the p r o p o s a l s o f C o r l e y , 7 8 B l a k e l e y , and Thompson were c o n s i d e r e d . C o r l e y I n v e s t i g a t e d the r o t a t i o n a l c a p a c i t y o f r e i n f o r c e d c o n c r e t e members. As one r e s u l t he p roposed an e q u a t i o n t o d e t e r m i n e the s p r e a d o f p l a s t i c i t y a l o n g a member beyond a d i s t a n c e o f d/2 f rom the p o i n t o f maximum moment. He c o n c l u d e d t h a t f o r r e i n f o r c e d c o n c r e t e members the e x t e n t t o wh i ch p l a s t i c i t y ex tends a l o n g a beam is governed by s e v e r a l g e o m e t r i c a l f a c t o r s . These i n c l u d e the e f f e c t i v e depth d , the degree o f f l e x u r a l r e i n f o r c e m e n t , and the d i s t a n c e a l o n g a member from the s e c t i o n o f m a x i -mum moment t o the a d j a c e n t s e c t i o n o f z e r o moment . However, i n c o n -s i d e r i n g the a p p l i c a t i o n o f C o r l e y ' s p r o p o s a l s t o p r e s t r e s s e d c o n c r e t e members, B l a k e l e y ^ found t h a t they p r e d i c t e d l o n g e r e q u i v a l e n t p l a s t i c h i n g e l e n g t h s than h i s e x p e r i m e n t a l r e s u l t s i n d i c a t e d , and as s u c h , f e l t they would be n o n - c o n s e r v a t i v e . B l a k e l e y s t a t e d t h a t In h i s t e s t i n g programme o f p r e s t r e s s e d c o n c r e t e beams, no d i a g o n a l t e n s i o n c r a c k s were o b s e r v e d . Whereas in r e i n f o r c e d c o n c r e t e members t h e s e c r a c k s would a l l o w f o r the s p r e a d o f p l a s t i c i t y . B l a k e l e y p r o p o s e d t h a t an a p p r o x -ima t i on f o r the e q u i v a l e n t p l a s t i c h i n g e l eng th c o u l d be taken as o n e -h a l f the o v e r a l l depth o f t he member I f the s u r f a c e c o n c r e t e s t r a i n exceeded 0.0035. F o r a maximum c o n c r e t e s t r a i n on the o r d e r o f 0.002, he p roposed t h a t the l e n g t h o f t he e q u i v a l e n t p l a s t i c h i n g e be taken as t h r e e q u a r t e r s o f the o v e r a l l depth o f the the member. R e c e n t l y g Thompson t e s t e d a number o f f u l l y and p a r t i a l l y p r e s t r e s s e d beam -79 column a s s e m b l i e s . H i s r e s u l t s s u p p o r t e d B l a k e l e y ' s p r o p o s a l t h a t t he e q u i v a l e n t p l a s t i c h i n g e l e n g t h be t aken as o n e - h a l f o f t h e o v e r a l l depth o f t h e member. In t h i s s t udy t h e e q u i v a l e n t p l a s t i c h i nge l e n g t h was, as recom-mended by B l a k e l e y and Thompson, t aken as o n e - h a l f o f t he o v e r a l l depth o f t h e member. It was f e l t t h a t the use o f t h i s v a l u e wou ld g i v e an upper bound on the c u r v a t u r e d u c t i l i t i e s o b t a i n e d , e s p e c i a l l y f o r t ho se members where c r u s h i n g o f t h e c o n c r e t e had not o c c u r e d . 4.5 RESULTS OF COMPARATIVE STUDY 4.5.1 E f f e c t o f Damping 9 10 41 E a r l i e r s t u d i e s ' ' o f both r e i n f o r c e d and p r e s t r e s s e d c o n c r e t e v e r s i o n s o f t h e A / 2 0 / 2 . 2 / 2 / 6 frame have shown t h a t t he magni tude and t ype o f v i s c o u s damping used in t he n o n l i n e a r a n a l y s e s had a s i g n i f i -c an t e f f e c t on the v a l u e s o b t a i n e d f o r a number o f t he re sponse p a r a -m e t e r s . The re sponse o f any s t r u c t u r e wou ld be e x p e c t e d to be a f f e c t e d by the v a l u e o f t he damping r a t i o s assumed, however , the frame c o n s i d e r e d appears to demons t ra te a p a r t i c u l a r s e n s i t i v i t y to t h i s v a r i a b l e . The reason l i e s p r i m a r i l y in t h e somewhat e x c e s s i v e t a p e r o f t h e s t i f f n e s s p r o p e r t i e s o f t h e f r ame. P r e v i o u s s t u d i e s o f the r e i n f o r c e d c o n c r e t e v e r s i o n o f t he b u i l d i n g have found t h a t the r e l a t i v e l y weak d e s i g n o f i t s upper s t o r i e s leads to a s i g n i f i c a n t c o n t r i b u t i o n by the h i g h e r modes in c e r t a i n o f i t s re sponse p a r a m e t e r s ' 0 ' ' * ' . W h i l e a modal a n a l y s i s was not i n c l u d e d in t h i s i n v e s t i g a t i o n the r e s u l t s o f an e a r l i e r n o n -l i n e a r s tudy o f the p r e s t r e s s e d c o n c r e t e v e r s i o n have i n d i c a t e d tha t 80 the modal contribution pattern would be similar to that of the rein-forced concrete version. This combined with the more f lexible nature characteristic of prestressed concrete structures, the lower level of hysteretic energy dissipation than for a comparable reinforced concrete structure, and the degrading stiffness and strength properties of the hinge moment-rotation idealization used in this study suggests that the damping values assumed might be Important in the determination of the response of the frame. For this Investigation, three analyses of the frame were considered in studying the effect of varying the viscous damping ratios on the lateral displacements and the rotational and curvature duct i l i ty demands. Those were the AAECNS35, ABECNS35, and ADECNS35 analyses described In section 4.3.4. The damping ratios for the ABECNS35 analysis were chosen to provide approximately one-third of the percentage of c r i t i c a l viscous damping as those for the ASECNS32 analysis. The results obtained are presented in Figs. 4.5 to 4.12. Shown In Figs. 4.5 and 4.6 are plots of the time - displacement response for the top . of the frame and the maximum lateral story dis-placements respectively for the three analyses. Examination of the figures shows that a change in the magnitude and type of viscous damping assumed had only a moderate effect on the absolute values of these response parameters. This may be explained by the fact that for the reinforced concrete version of the A/20/2.2/2/6 frame, the lateral displacements were found to be primarily a function of the fundamental 41 mode, although the second and third modes made some contribution Observation of Fig. 4.6 shows that this is essentially the case for the prestressed concrete version of the frame. While there is a sl ight decrease In the lateral story displacements in the upper portion of 81 ABECNS35 AAECN-S35 ADECNS35 F i g . k.T Time - D i sp l a cement Response f o r Top Node v e r s u s Damping F F g . 4.6 Maximum L a t e r a l S t o r y D i sp l a cement s ve r sus Damping • F i g . k.7 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand ver sus Damping ' F i g . 4.8 Gi r d e r H inge C u r v a t u r e D u c t i l i t y Demand ver sus Damping G I R D E R A C C U M U L A T E D H I N G E R O T A T I O N F A C T O R S T O R Y E L C E N T R O NS 1 9 4 0 NUM A A E C N S 3 5 A 6 E C N S 3 5 A D E C N S 3 5 2 0 417-06 6 2 . 4 6 3 2 . 9 1 19 1 9 6 . 9 7 78.22 50.76 18 144.82 78.99 57.05 17 1 1 0 . 7 1 7 0 . 2 2 54.83 16 1 4 2 . 2 3 84.81 65.93 15 90 .58 6 9 . 6 5 58.21 14 68.05 47.12 4 3 - 4 3 13 5 6 . 0 2 4 2 . 3 4 3 6 . 9 4 12 42.49 33.93 31-50 11 2 8 . 5 7 2 4 . 8 8 2 5 - 1 1 10 2 5 . 4 2 20.75 20.33 9 2 4 . 9 9 1 9 - 4 0 18.39 6 2 8 . 8 9 2 1 . 1 5 20.09 7 2 9 . 0 7 25-"15 22.63 6 2 7 . 2 2 26.95 2 6 . 7 7 5 2 4 . 2 2 2 5 - ' t 3 26.81 4 2 6 . 6 5 2 6 . 5 5 28.58 3 24.67 2 4 . 3 7 2 6 . 0 0 2 1 9 . 8 1 20.74 2 1 . 6 6 1 16.75 17.67 18.16 \ i 1 / // / // / / // \ I (WEOtSSS \ = l . O M H RBECNS35 X = 0 . 3 7 ; i » , 0 . 6 7 J K A0ECN535 >> - 1.001K 1 1 1 1 1 0 0 5 0 - 0 100.0 1 5 0 . 0 2 0 0 . 0 2 5 0 . 0 3 0 0 . 0 3 5 0 . 0 400 GIRDER HCCUM HINGE ROTATION FACTOR F i g . k.S G i r d e r Accumu la ted Hinge R o t a t i o n F a c t o r v e r s u s Damping E X T E R N A L C O L U M N END R O T A T I O N A L D U C T I L I T Y S T O R Y E L C E N T R O NS 191(0 NUM A A E C N S 3 5 A B E C N S 3 5 A D E C N S 3 5 2 0 1 . 1 8 1 .08 1 .07 19 1.34 0 . 9 5 0.88 18 0.86 0.78 0.79 17 0.79 0.73 0 . 7 2 16 0.78 0 . 6 9 0.67 15 0.83 0.73 0.66 14 0.65 0 . 6 2 0.59 13 0.76 0 . 6 2 0.56 12 0 . 7 2 0.66 0.63 11 0.68 0.65 0 . 6 4 10 0.58 0.58 0.59 9 0.68 0.59 0.56 8 0 . 6 6 0.65 0 . 6 4 7 0 . 6 4 0.66 0.65 6 0.65 0 . 6 0 0.62 5 0.62 0.62 0.62 l i 0 . 6 4 0.63 0.63 3 0.68 0.65 0.66 2 0.67 0.67 0.67 1 0.77 0 . 7 7 0.76 HRECN535 >. = l .OOJH RBCCNS35 X = 0.37JI4.0.611K RQECHS35 X = l .OOJK 1 1 1 . 5 2 . 0 2 5 3 . 0 EXT COLUMN END ROT DUCTILITY I— 3 . 5 4 . 0 OO ON ' F t g . 4.10 E x t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand ver sus Damping INTERNAL COLUMN END ROTATIONAL DUCTILITY STORY EL CENTRO NS 1940 NUM AAECNS35 ABECNS35 A0ECNS35 20 8.35 3.03 2.36 19 2.35 1 72 1.37 18 1.22 0.98 1.00 17 0.97 0.90 0.89 16 0.91 0.82 0.78 15 0.94 0.83 0.81 14 0.79 0.75 0.71 13 0.88 0.72 0.66 12 0.81 0.75 0.71 11 0.75 0.73 0.71 10 0.62 0.63 0.65 9 0.74 0.64 0.61 8 0.73 0.72 0.70 7 0.64 0.72 0.71 6 0.70 0.67 0.68 5 0.69 0.70 0.70 4 0.70 0.70 0.71 3 0.74 0.71 0.72 2 0.76 0.75 0.75 1 0.78 0.77 0.77 O k ^ _ — . — ii \ K V \\/ W \\ -n » r « K « V = l nniH R8ECNS35 X = 0 . 3 1 J N . 0 . 6 3 J K R0ECNS35 X = 1.Q0JK i i I I I ' 0 . 0 0 . 5 1 .0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 INT COLUMN ENO ROT DUCTILITY F i g . 4.11 I n t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Damping ? 1 ft 4 J 4 I • •< ' 4 4 * * * * * * * * * * • * f * * A A * 7 "77/7 7T TSST/ / . ' i > J > > 1 1 > i • 11 • > 11 > 11). 21 19 18 17 tri>n> n >/t/>i > j i / > r i III ' i /11 i /1} i' 2 0 19 IB 17 16 15 l « 13 12 I I 10 9 8 7 6 5 l | 3 2 I - A - A m . A 7 7; 4 4 'm 4 4 A ' *4 4 • • . 7 A ? • • A * A * A 2 * * * * 1 1 > > 1 1 111 1 1 1 1 1 1 1 1 1 1 1 . AAECNS35 ABECNS35 ADECNS35 oo co F i g . 4.12 Hinge S t a t u s Diagrams ver sus Damping 89 the frame with Increasing percentages of stiffness proportional damping, the effect is not significant over most of the height of the building. In fact, because of the reduced member damage implied In the lower duct i l i ty demands shown in Figs. 4.7 and 4.8 for the ABECNS35 and ADECNS35 analyses, a more effective transfer of the shear forces down the frame results In the s l ight ly higher lateral displacements shown for the middle portion of the building with Increased stiffness propor-tional damping. By contrast, the lateral displacement configuration predicted for the structure by the AAECNS35 analysis is the result of an inabi l i ty to effectively transfer the shear forces down the frame. A natural reduction in the relative stiffness of the columns and girders as shown in Table C.2, amplified by the increased member damage implied in suddenly higher gradient of girder duct i l i ty demands and accumulated hinge rotation factors, both of which occur about mid-height, results in an abrupt decrease in structure st i f fness. The end effect Is that the upper portion of the frame suffers relatively more for the AAECNS35 analysis than for the other two analyses, principal ly as a result of Its substantially secondary osc i l l a t ion. The significant affect the degree of stiffness proportional viscous damping assumed has on the girder end rotational and girder hinge curvature duct i l i ty demands for members in the upper stories may be attributed primarily to the importance of the higher modes in deter-mining the interfloor displacements. In a nonlinear modal analysis 41 with almost no damping, Giberson found that for the upper portion of the reinforced concrete version of the frame up to eight modes were signif icant. The results presented in this study show that, even for the small values used, as the percentage of stiffness proportional 90 damping i s i n c r e a s e d , the a c t i o n o f t he h i g h e r modes i s s u b s t a n t i a l l y d i m i n i s h e d , and the g i r d e r r o t a t i o n a l and c u r v a t u r e d u c t i l i t y demands a r e c o r r e s p o n d i n g l y r e d u c e d . It i s a l s o impor t an t t o no te t h a t , as wou ld be e x p e c t e d , in the m i d d l e and lower p o r t i o n s o f t h e f rame where the impor tance o f the h i g h e r modes in d e t e r m i n i n g the i n t e r f l o o r d i s p l a c e m e n t s i s r e d u c e d , the deg ree o f s t i f f n e s s p r o p o r t i o n a l damping used has o n l y a moderate e f f e c t on the v a l u e s o b t a i n e d f o r t h e s e r e -sponse p a r a m e t e r s . The p o s i t i v e e f f e c t the i n c r e a s e d s t i f f n e s s p r o p o r -t i o n a l damping had on the g i r d e r a c c u m u l a t e d h i n g e r o t a t i o n f a c t o r s i s shown i n F i g . 4.9. The lower v a l u e s o b t a i n e d f o r t h e f a c t o r s in t h e upper p o r t i o n o f t he frame f o r t he ABECNS35 and ADECNS35 a n a l y s e s , combined w i t h the a p p r o x i m a t e l y same l a t e r a l d i s p l a c e m e n t s and lower d u c t i l i t y demands as compared w i t h t h e AAECNS35 a n a l y s i s , s u gge s t s t h a t somewhat l e s s f a t i g u e wou ld o c c u r i n the h i n g e r e g i o n s o f the g i r d e r s i n t h e upper s t o r i e s f o r t ho se c a s e s . In F i g s . 4.10 and 4.11 i t i s seen t h a t v a r y i n g the damping r a t i o had l i t t l e i n f l u e n c e on the column b e h a v i o u r o v e r most o f t h e h e i g h t o f the b u i l d i n g . W i th the e x c e p t i o n o f the top s e v e r a l f l o o r s , the columns remain e l a s t i c and t h e i r r o t a t i o n a l d u c t i l i t y demands remain e s s e n t i a l l y u n i f o r m . The r e d u c t i o n in re sponse i s most n o t i c e a b l e f o r the i n t e r i o r co lumns , however, t h i s i s s i g n i f i c a n t o n l y in the upper s t o r i e s o f t h e f r ame. The r e s u l t s demons t ra te the v a l u e o f a s t r o n g column - weak beam c o n f i g u r a t i o n . The p r e f e r e n t i a l y i e l d i n g o f the g i r d e r s r e s u l t s in the co lumns , f o r t h e most p a r t , r ema in ing e l a s t i c and the ma in tenance o f t he v e r t i c a l i n t e g r i t y o f t he s t r u c t u r e . A v i s u a l i n d i c a t i o n o f the damage t o the members o f t h e f rame as a r e s u l t o f the t h r e e a n a l y s e s i s p r e s e n t e d i n F i g . 4.12. A s e m i -91 c i r c l e r e p r e s e n t s c r a c k i n g and a t r i a n g l e r e p r e s e n t s c r u s h i n g o f t h e h i n g e in the d i r e c t i o n o f the s ymbo l . The d iagrams show t h a t , as p r e d i c t e d by the o t h e r re sponse p a r a m e t e r s , most o f t he r e d u c t i o n in re sponse w i t h i n c r e a s e d s t i f f n e s s p r o p o r t i o n a l v i s c o u s damping o c c u r s in t he t o p s e v e r a l s t o r i e s . It s h o u l d be n o t e d , however , t h a t a h i n g e c o u l d have j u s t r e a c h e d c r u s h i n g o r , a l t e r n a t i v e l y , c o u l d have s u s t a i n e d c o n s i d e r a b l e s t r e n g t h and s t i f f n e s s d e g r a d a t i o n and s t i l l be shown as a t r i a n g l e . The h i n g e s t a t u s d iagrams a r e most m e a n i n g f u l when v i ewed w i t h the r o t a t i o n a l and c u r v a t u r e d u c t i l i t i e s i n m ind . The s u b s t a n t i a l v a r i a t i o n , e s p e c i a l l y in the upper s t o r i e s o f t he f r ame , o f some o f t he re sponse pa rameter s between t h e t h r e e a n a l y s e s r a i s e s the q u e s t i o n o f a p p l i c a t i o n t o a r e a l s t r u c t u r e . E a r l i e r s t u d i e s by bo th G i b e r s o n ' * ' and C lough and B e n u s k a ' " have found t h a t as the t a p e r o f the s t i f f n e s s p r o p e r t i e s o f a f rame i s r e d u c e d , the c o n t r i b u t i o n o f the h i g h e r modes becomes l e s s i m p o r t a n t . As such i t wou ld be e x p e c t e d t h a t the c h o i c e o f the type o f v i s c o u s damping wou ld not have as g r e a t an i n f l u e n c e on the r e s p o n s e . The v a l i d i t y o f t h a t s t a tement i s shown In the r e s u l t s o b t a i n e d f o r t he m i d d l e and lower p o r t i o n o f the f rame c o n s i d e r e d . T h e r e , where the c o n t r i b u t i o n o f t he h i g h e r modes i s no t as s i g n i f i c a n t , the v a l u e s o b t a i n e d f o r the g i r d e r and column d u c t i l i t y demands a r e a p p r o x i m a t e l y Independent o f the a n a l y s i s . I f , on the o t h e r hand , the h i g h e r modes a r e Important i n d e t e r m i n i n g the r e s p o n s e , the use o f s t i f f n e s s p r o p o r t i o n a l damping may be u n c o n s e r v a t i v e . The r e s u l t s o b t a i n e d f o r t h e upper p o r t i o n o f t he f rame i n d i c a t e t h a t even a sma l l p e r c e n t a g e o f s t i f f n e s s p r o p o r t i o n a l v i s c o u s damping can m a t e r i -a l l y reduce the v a l u e o f some re sponse p a r a m e t e r s . 21 k2 I t s h o u l d be kept in mind t h a t a number o f o t h e r s t u d i e s ' have 92 s u g g e s t e d t h a t most o f t he damping In s t r u c t u r e s Is e s s e n t i a l l y f r e q u e n c y Independent a n d , as s u c h , Is not c o r r e c t l y r e p r e s e n t e d by v i s c o u s damping. The use o f o t h e r forms o f damping, f o r example Columb type damping may be more r e a l i s t i c . Spencer has s u g g e s t e d t h a t a damping m a t r i x based on d i s p l a c e m e n t s r a t h e r than v e l o c i t i e s may g i v e a more a c c u r a t e r e p r e s e n t a t i o n o f t he t r u e damping mechanism. Where v i s c o u s damping i s u s e d , however , the r e s u l t s o f t h i s s e c t i o n i n d i c a t e t h a t t he use o f mass p r o p o r t i o n a l damping i s the more c o r r e c t c h o i c e . In c o n t r a s t t o mass p r o p o r t i o n a l damping, s t i f f n e s s p r o p o r -t i o n a l damping Is unbounded, and as s u c h , v e r y l a r g e damping r a t i o s may o c c u r In the h i g h e r modes. T h i s may l e a d to t h e s i t u a t i o n where the damping f o r c e s exceed the i n e r t i a f o r c e s and the energy d i s s i p a t e d by v i s c o u s damping i s g r e a t e r than t h a t d i s s i p a t e d by h y s t e r e t i c damp-i n g . The r e s u l t s o b t a i n e d f o r the member d u c t i l i t i e s under t h o s e c o n -d i t i o n s may not r e a l i s t i c a l l y r e f l e c t t he demands i n a r e a l f r ame . It i s impor t an t t o n o t e the upper f l o o r s o f the frame c o n s i d e r e d a r e somewhat weaker than they wou ld be had they been d e s i g n e d a c c o r d i n g to c u r r e n t s e i s m i c recommendat ions . As s u c h , t h e r e s u l t s o f t he AAECNS35 a n a l y s i s f o r the top s e v e r a l s t o r i e s a r e no t n e c e s s a r i l y r e p r e s e n t a t i v e o f t h o s e wh i ch might o c c u r In a m u l t i - s t o r y moment r e s i s t i n g p r e s t r e s s e d c o n c r e t e f r ame . It Is t h o u g h t , however , t h a t in g e n e r a l , t he v a l u e s o b t a i n e d a r e r e p r e s e n t a t i v e o f t he d u c t i l i t y demands and member damage l i k e l y t o o c c u r in an open f ramed s t r u c t u r e . 4.5.2 E f f e c t o f Ea r thquake I n t e n s i t y An impor tan t c o n s i d e r a t i o n in t h e a s e s m i c d e s i g n o f a s t r u c t u r e 93 Is t he l e v e l o f e a r t h q u a k e i t i s d e s i g n e d to r e s i s t . In a c c o r d a n c e w i t h the c u r r e n t s e i s m i c d e s i g n p h i l o s o p h y , i t i s d e s i r a b l e f o r a b u i l d i n g t o s u s t a i n o n l y m i n o r damage as a r e s u l t o f a moderate e a r t h -quake , and to r e s i s t s e v e r e s e i s m i c l o a d i n g w i t h o u t c o l l a p s e . The lower l e v e l s o f h y s t e r e t i c and v i s c o u s damping, combined w i t h i t s more f l e x i b l e re sponse and d e g r a d i n g s t i f f n e s s and s t r e n g t h p r o p e r t i e s , r a i s e s t he q u e s t i o n o f t he a b i l i t y o f a p r e s t r e s s e d c o n c r e t e s t r u c t u r e to s u r v i v e ma jo r e a r t h q u a k e l o a d i n g . It i s c o n c e i v a b l e tha t w h i l e a p r e s t r e s s e d c o n c r e t e f r amed s t r u c t u r e may p r e f o r m s a t i s f a c t o r i l y under moderate s e i s m i c l o a d i n g , i t may s u f f e r c a t a s t r o p h i c f a i l u r e when s u b j e c t e d to a more s e v e r e e a r t h q u a k e . To e v a l u a t e the e f f e c t o f v a r y i n g the e a r t h q u a k e a c c e l e r a t i o n i n t e n s i t y on the re sponse p a r a -m e t e r s , two l e v e l s o f maximum a c c e l e r a t i o n were c o n s i d e r e d . Those were E l C e n t r o NS 19^0 a t an u n s e a l e d i n t e n s i t y o f a maximum a c c e l e r a t i o n o f t h i r t y - f i v e p e r c e n t o f g r a v i t y , wh i ch i s a p p r o x i m a t e l y t he ma jo r e v e n t f o r wh ich the b u i l d i n g was d e s i g n e d , and E l C e n t r o NS 1940 s c a l e d t o g i v e a maximum a c c e l e r a t i o n o f f i f t y p e r c e n t o f g r a v i t y . The r e s u l t s o f f o u r a n a l y s e s a r e d i s c u s s e d i n t h i s s e c t i o n . Those a r e the AAECNS35 and ADECNS35 a n a l y s e s r e f e r r e d to i n the p r e v i o u s s e c t i o n , and a d d i -t i o n a l l y , the AAECNS50 and ADECNS50 a n a l y s e s . The r e s u l t s o b t a i n e d a r e p r e s e n t e d in F i g s . A.13 to 4.28. Shown i n F i g . 4.13 a r e the t ime - d i s p l a c e m e n t p l o t s o f t he top o f the frame f o r the AAECNS35 and AAECNS50 a n a l y s e s . O b s e r v a t i o n o f the F i g u r e shows t h a t as e x p e c t e d , t h e i n c r e a s e d i n t e n s i t y o f t he a c c e l e r a t i o n r e c o r d , r e s u l t e d In an i n c r e a s e in t he re sponse o f the A / 2 0 / 2 . 2 / 2 / 6 f rame. However, F i g . 4.14 shows t h a t most o f the i n c r e a s e in t he l a t e r a l d i s p l a c e m e n t s o c c u r e d In t he top s e v e r a l s t o r i e s o f the AAECNS35 AAECNS50 R g . 4.13 Time - D i s p l a c e m e n t Response f o r Top Node v e r s u s Ea r thquake I n t e n s i t y F i g . 4.14 Maximum L a t e r a l S t o r y D i sp l acement s v e r s u s Ear thquake I n t e n s i t y F i g . 4.15 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Ear thquake I n t e n s i t y F i g . 4.16 G i r d e r Hinge C u r v a t u r e D u c t i l i t y Demand v e r s u s Ear thquake I n t e n s i t y F i g . 4 .17 G i r d e r Accumula ted v e r s u s Ear thquake Hinge R o t a t i o n F a c t o r I n t e n s i t y EXTERIOR COLUMN END ROTATIONAL DUCTILITY STORY EL CENTRO NS 1940 NUM AAECNS35 AAECNS50 Z INCREASE 20 1.18 2.23 89.0 19 1-34 2.02 50.7 18 0.86 1.26 46.5 17 0.79 1.37 73.4 16 0.73 0.92 17-9 15 0.G3 0.78 -6.0 14 0.65 0.71 9.2 13 0.76 0.75 -1.3 12 0 .72 0.72 0.0 11 0.68 0.69 1.5 10 0.58 0.68 17-2 9 0.68 0.72 5-9 3 0.66 0.58 -12.1 7 0.64 0.61 -4.7 6 0.65 0.66 1.5 5 0 .62 0.63 1.6 4 0.64 0.66 3.1 3 0.68 0.70 2.9 2 0.67 0.62 -7.5 1 0.77 0.75 -2.6 / \ RRECN33S X * l . O J H RNECNSSO \ - l . M N -I 1 1 1 1 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 EXT COLUMN END ROT DUCTILITY 4.0 F i g . 4.18 E x t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand ver sus Ea r thquake I n t e n s i t y INTERIOR COLUMN END ROTATIONAL DUCTILITY STORY EL CENTRO NS 1940 NUH AAECNS35 AAECNS50 Z INCREASE 20 8.35 22.74 172 19 2.35 3.47 47.7 18 1 . 2 2 1.91 56.6 17 0.97 2 . 0 6 112 16 0.91 1.17 28.6 15 0-94 0.90 - 4.3 14 0.79 0.35 7.6 13 0.88 0.86 -2.3 12 0.81 0.37 7-4 11 0.75 0.77 2.7 10 0 . 6 2 0.79 2 7 . 4 9 0.74 0 . 8 0 8.1 8 0.73 0.67 -8 .2 7 0.64 0.66 3.1 6 0.70 0.71 1 . 4 5 0.69 0.69 0 . 0 4 0.70 0 . 7 2 2.9 3 0.74 0.75 1 . 4 2 0.76 0 . 7 0 -7-9 1 0.78 0.83 6.4 u / // I A /1 \ i \ i y i \ MECN5S0 X = 1.0 JM l.S 2.0 2.5 3.0 JNT COLUMN END ROT DUCTILITY F i g . 4.19 I n t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand ve r su s Ear thquake I n t e n s i t y AAECNS35 AAECNS50 F i g . 4.20 Hinge S t a t u s Diagrams v e r s u s Earthquake I n t e n s i t y ADECNS35 ADECNS50 F i g . 4.21 Time - D i sp l a cement Response f o r Top Node v e r s u s Ea r thquake I n t e n s i t y F i g . 4.22 Maximum L a t e r a l S t o r y D i sp l a cement s v e r s u s Ear thquake I n t e n s i t y F f g . 4.23 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Ear thquake I n t e n s i t y F i g . 4.24 G i r d e r v e r s u s Hinge C u r v a t u r e D u c t i l i t y Demand Ear thquake I n t e n s i t y 3-0 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 GIRDER RCCUM HINGE ROTATION FACTOR O ON - F i g . 4 .25 G i r d e r Accumulated H inge R o t a t i o n F a c t o r v e r s u s Ear thquake I n t e n s i t y EXTLRIOR COLUMN ENO ROTATIONAL DUCTILITY STORY EL CENTRO NS 1940 NUM ADECNS35 ADECNSbO Z INCREASE 20 1.07 1 .15 7.5 19 0 .88 1.47 6 7 . 0 18 0 .79 1 .09 38.0 17 0 .72 0.78 8.3 16 0 . 6 7 0 .93 38.8 15 0 .66 0 .82 24.2 14 0 .59 0 .73 23.7 13 0 .56 0.75 3 3 9 12 0.63 0.69 9.5 11 0.64 0.70 9-4 10 0 .59 0 .70 18.6 9 0 .56 0.72 28.6 8 0 .64 0.67 4 .7 7 0 .65 0.63 -3.1 6 0.62 0.66 6 .5 5 0.62 0.64 3 2 4 0 .63 0.66 4 .8 3 0 .66 0.70 6.1 2 0 . 6 7 0 .77 14.9 1 0.76 0 .86 13.2 L t i o 5* o / / 1/ \\ 1 \ > 1 \ 1 1 / \ \j j \ RDECMS35 > - 1.0IK RDECHSSO V = ].0« OS 1.0 _ IS 2.0 2.5 3.0 EXT COLUMN END ROT DUCTILITY o Fig. 4.26 External Column End Rotational Ductil ity Demand versus Earthquake Intensity Fig. 4.27 Internal Column End Rotational Ducti l ity Demand versus Earthquake Intensity ADECNS35 ADECNS50 F i g . A.28- H inge S t a t u s Diagrams v e r s u s Ear thquake I n t e n s i t y no f rame. In f a c t , e x a m i n a t i o n o f F i g s . 4.14 t o 4.20 shows t h a t , w h i l e t h e r e i s some v a r i a t i o n i n the v a l u e o f t he re sponse p a r a m e t e r s , v a r y i n g t h e I n t e n s i t y o f the a c c e l e r a t i o n r e c o r d appeared to have l i t t l e e f f e c t o v e r most o f the h e i g h t o f t he b u i l d i n g . On ly above the s i x -t e e n t h f l o o r do the v a l u e s o f t h e pa rameter s b e g i n to I n c rea se s i g n i -f i c a n t l y f o r the AAECNS50 a n a l y s i s . It i s e v i d e n t t h a t the r e s u l t s show no l i n e a r o r p r o p o r t i o n a l v a r i a t i o n i n the v a l u e s o f t he response pa rameter s c o r r e s p o n d i n g to the v a r i a t i o n o f the a c c e l e r a t i o n r e c o r d I n t e n s i t y . The r e s u l t s r e p r e s e n t the re sponse l i k e l y t o o c c u r f o r a m u l t i -s t o r y p r e s t r e s s e d c o n c r e t e f rame w i t h s u b s t a n t i a l t a p e r o f the s t r u c t u r a l p r o p e r t i e s . The h i g h degree o f t a p e r o f t he s t i f f n e s s p r o p e r t i e s f o r the f rame c o n s i d e r e d r e s u l t s In the c o n c e n t r a t i o n o f the n o n l i n e a r i t y and r e s u l t a n t member damage In the upper l e v e l s o f t he b u i l d i n g s . The p a t t e r n o f a s u d d e n l y h i g h e r g r a d i e n t f o r the v a l u e s o f t he re sponse pa rameter s f o r the AAECNS50 a n a l y s i s i s , as was the c a s e f o r t he AAECNS35 a n a l y s i s , the r e s u l t o f an ab rup t d e c r e a s e i n the s t r u c t u r e s t i f f n e s s . O b s e r v a t i o n o f T a b l e C.2 shows t h a t the s h a r p e s t d e c r e a s e in t h e r e l a t i v e s t i f f n e s s o f bo th columns and beams o c c u r s about the s i x t e e n t h s t o r y . T h i s combined w i t h the i n c r e a s e d member damage due to r e l a t i v e l y h i g h e r d e f o r m a t i o n s In the upper l e v e l s and a c c e n t u a t e d by the subsequent s t i f f n e s s and s t r e n g t h d e g r a d a t i o n o f the g i r d e r h i n g e s r e s u l t s i n t he h i g h e r re sponse o f the top s e v e r a l f l o o r s . The lower v a l u e s f o r many o f t he re sponse pa rameter s f o r the AAECNS50 a n a l y s i s as compared w i t h the AAECNS35 a n a l y s i s a r e the r e s u l t o f an i n a b i l i t y o f t he weakened f rame t o t r a n s f e r the s h e a r f o r c e s o f t h e upper p o r t i o n down the f r ame. It w i l l be n o t e d however , In s p i t e o f t he lower I l l o r a p p r o s l m a t e l y same v a l u e s f o r many o f the response p a t t e r n s , the g i r d e r a ccumu la ted h i n g e r o t a t i o n f a c t o r s show a s i g n i f i c a n t i n c r e a s e o v e r the h e i g h t o f the b u i l d i n g . T h i s sugge s t s a h i g h e r deg ree o f o s c i l l a t i o n and i m p l i e s t h a t i n c r e a s e d h i n g e f a t i g u e and r e s u l t a n t member damage a r e l i k e l y as a r e s u l t o f the i n c r e a s e d e a r t h q u a k e i n t e n -s i t y . The end e f f e c t o f the re sponse p a t t e r n , however, i s to i n d i c a t e t h a t the member damage wou ld be m i n i m i z e d o v e r the m a j o r i t y o f the frame a t the expense o f the upper s t o r i e s . In f a c t , i t i s q u e s t i o n a b l e whether r e a l members c o u l d accommodate w i t h o u t f a i l u r e the d u c t i l i t y demands p r e d i c t e d f o r some members a t the top o f t he f rame by the AAECNS50 a n a l y s i s . It i s impor tan t t o no te t h a t even w i t h a f o r t y - t h r e e p e r c e n t i n c r e a s e In e a r t h q u a k e i n t e n s i t y , the columns remained e l a s t i c o v e r most o f the h e i g h t o f the b u i l d i n g . On ly In the top s e v e r a l s t o r i e s o f the f rame d i d s e v e r e n o n l i n e a r i t y o c c u r . The r e s u l t s demons t r a te tha t as long as the g i r d e r s a r e a b l e t o p r o v i d e the r e q u i r e d energy d i s s i p a t i o n c a p a c i t y , the columns w i l l t end to remain e l a s t i c . The r e s u l t s shown in F i g s . 4.13 to 4.20 a r e l i k e l y t o be t y p i c a l o f an open f r ame where , because o f weak d e s i g n a n d / o r p ronounced t a p e r -ing o f t h e s t i f f n e s s p r o p e r t i e s , the h i g h e r modes c o n t r i b u t e a s i g n i f i -can t and perhaps p redomina te p o r t i o n o f c e r t a i n o f t he r e sponse p a r a -m e t e r s . By c o m p a r i s o n , F i g s . 4.21 t o 4.28 p r e s e n t r e s u l t s f o r the ca se where the e f f e c t o f the h i g h e r modes has been reduced through the use o f s t i f f n e s s p r o p o r t i o n a l damping. F i g u r e s 4.21 and 4.22 show t h a t the use o f s t i f f n e s s p r o p o r t i o n a l damping d i d no t reduce the maximum l a t e r a l s t o r y d i s p l a c e m e n t s a t the top o f t he f r a m e . T h i s would be e x p e c t e d as the l a t e r a l d i s p l a c e m e n t s a r e p r i m a r i l y a f u n c t i o n o f t he f i r s t s e v e r a l 112 modes. It w i l l be n o t e d however , t h a t in c o n t r a s t t o the r e s u l t s o f t he AAECNS50 a n a l y s i s , t h e r e i s a g e n e r a l i n c r e a s e in most o f t h e r e s -ponse pa rameter s in t he m i d d l e and lower s e c t i o n s o f t he frame f o r the ADECNS50 a n a l y s i s w i t h an i n c r e a s e In e a r t h q u a k e I n t e n s i t y . The lower i n c r e a s e i n the d u c t i l i t y demands in the upper p o r t i o n o f t he f rame f o r the ADECNS50 a n a l y s i s as shown In F i g s . 4.23 t o 4 .25, and the lower mem-b e r damage I m p l i e d , r e s u l t s i n a l a r g e r t r a n s f e r o f t he s h e a r f o r c e s down the f r a m e . The r e s u l t i s a h i g h e r p o r t i o n o f t he n o n l i n e a r i t y o c c u r i n g in t he m i d d l e and lower s e c t i o n s o f the b u i l d i n g f o r the ADECNS50 a n a l y s i s than f o r t he AAECNS50 a n a l y s i s . However, the m a j o r i t y o f t h e i n c r e a s e in re sponse f o r bo th a n a l y s e s o c c u r s In the upper p o r t i o n o f the s t r u c t u r e . It wou ld be e x p e c t e d t h a t a more u n i f o r m d i s t r i b u t i o n o f the n o n l i n e a r i t y wou ld r e s u l t f o r a f rame w i t h reduced t a p e r o f the s t i f f n e s s p r o p e r t i e s . I t Is impor tan t to n o t e t h a t because o f t h e h i g h l e v e l o f damping f o r t he A0ECNS50 a n a l y s i s , the re sponse i s not l i k e l y t o be r e p r e s e n t a t i v e o f t h a t o f the a c t u a l s t r u c t u r e . The v a l u e o f the a n a l y s i s l i e s p r i m a r i l y In an o b s e r v a t i o n o f t he r e a c t i o n o f t he b u i l d i n g to a l a r g e r t r a n s f e r o f s h e a r down t h e f r a m e . The r e s u l t s o f t he a n a l y s e s d i s c u s s e d In t h i s s e c t i o n demons t r a te the c a p a b i l i t y o f a m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e f rame t o w i t h s t a n d s e v e r e s e i s m i c l o a d i n g . Over most o f t h e h e i g h t o f the b u i l d i n g the d u c t i l i t y demands a r e w e l l w i t h i n the d u c t i l e c a p a c i t y o f c a r e f u l l y d e -t a i l e d p r e s t r e s s e d c o n c r e t e members. The p a r t i c u l a r f rame s t u d i e d s u s -t a i n s e x c e s s i v e d u c t i l i t y demands in t he top s e v e r a l s t o r i e s . T h i s , and the re sponse p a t t e r n s emphas ize the d e t r i m e n t a l a s p e c t s o f p ronounced t a p -e r o f t he s t i f f n e s s p r o p e r t i e s o f a s t r u c t u r e and p a r t i c u l a r l l y the 111 e f f e c t s o f a b r u p t d e c r e a s e s In the r e l a t i v e s t i f f n e s s o f members. 113 Because o f lower h y s t e r e t i c and v i s c o u s damping , and a more f l e x i b l e re sponse than f o r r e i n f o r c e d c o n c r e t e , p r e s t r e s s e d c o n c r e t e f rames appear t o be e s p e c i a l l y s u s c e p t i b l e to the "wh ip l a s h " e f f e c t , and t h i s i s a c c e n t u a t e d by sudden changes in s t r u c t u r e s t i f f n e s s . A b a s i c p r i n -c i p l e o f a s e i s m i c d e s i g n o f s t r u c t u r e s i s the a v o i d a n c e o f any ab rup t changes i n s t i f f n e s s o r s t r e n g t h p r o p e r t i e s . As s u c h , where the s t i f f -ness p r o p e r t i e s o f a p r e s t r e s s e d c o n c r e t e f r ame a r e t a p e r e d , they s h o u l d be v a r i e d c o n t i n u o u s l y r a t h e r than In b l o c k s as f o r the frame c o n s l d e r e d . An e a r t h q u a k e , r e g a r d l e s s o f i n t e n s i t y , i n p u t s energy i n t o a b u i l d i n g as a r e s u l t o f g round m o t i o n . Some o f t h i s energy may be d i s s i p a t e d by d e s i g n e d damping mechanisms. A d d i t i o n a l energy may be d i s s i p a t e d by n o n s t r u c t u r a l components such as w a l l s and i n f i l l p a n e l s . However, in most c a s e s , as t he i n t e n s i t y o f t he s e i s m i c l o a d i n g i n -c r e a s e s , the s t r u c t u r e d sys tem i s c a l l e d upon to p r o v i d e the n e c e s s a r y energy d i s s i p a t i o n c a p a c i t y . F o r a moderate to s e v e r e e a r t h q u a k e It would be e x p e c t e d t h a t the m a j o r i t y o f the energy d i s s i p a t i o n f o r a f rame s t r u c t u r e wou ld be p r o v i d e d by h y s t e r e t i c damping. As s u c h , t he maximum d u c t i l i t y demands, w i l l be dependent on the d i s t r i b u t i o n o f t he energy d i s s i p a t i o n th rough the f r a m e . A l t h o u g h t h e r e a r e l i k e l y t o be s p e c i a l c a s e s , i n g e n e r a l an e f f i c i e n t d e s i g n w i l l a t tempt t o s p r e a d the n e c e s s a r y n o n l i n e a r i t y as u n i f o r m l y as p o s s i b l e th roughout t h e s t r u c t u r a l s y s t e m , and in p a r t i c u l a r , the g i r d e r s . It wou ld be e x p e c t e d t h a t t h i s wou ld r e s u l t i n s l i g h t l y h i g h e r g i r d e r d u c t i l i t y demands i n the lower and m i d d l e s e c t i o n s and somewhat lower member d u c t i l i t i e s in the upper p o r t i o n o f t he frame c o n s i d e r e d . 114 4.5.3 Effect of Earthquake Record Most other nonlinear s t u d i e s 9 * ' 0 ' ' ^ have used the acceleration record of El Centro NS 1940 to provide the dynamic excitation. However, to obtain an envelope of values for the response parameters it is necessary to subject the structure under consideration to a variety of ground motions with a range of record characteristics. In addition to intensity, as represented by peak ground acceleration, other significant ground motion characteristics affecting the seismic response of a bui ld-ing are the impulsive or sinusoidal nature and the frequency content of an accelerogram. The impulsive or sinusoidal nature of a particular ground motion refers to the rate at which it supplies energy to a structural system, a factor Bertero suggested is important In the class-i f icat ion of earthquakes. The frequency characteristics of a given acceleration record refer to the relative strength of the various com-ponent waves which compose that ground motion. The importance of know-ing the frequency content of a record lies in the possible occurance of a quasi-reasonance condition when a dominant frequency of the ground motion approachs a natural frequency of the structure. One method of measuring the frequency content of a record is through the use of the 42 relative velocity spectrum for that accelerogram. Flntel and Ghosh have proposed that earthquake records may be c lass i f ied as "peaking" or "broad banded" depending on whether the relative velocity spectrum exhibits dominant frequencies over a well defined range or whether It remains approximately f lat within the period range at and Immediately beyond the fundamental period. A further subgroup Is the "ascending" broad banded spectrum" which they defined as one for which the relative 115 velocity spectrum increased with increasing periods in the range of interest. In this section, three representative ground motions were considered. Those were El Centro NS 1940, Taft S69E 1952, and Park-f ie ld N65E 1966. Observation of the accelerograms contained In Appendix D shows that Taft and El Centro are sinusoidal in nature, while Park-f ie ld is of the impulsive type. Examination of the relative velocity spectra suggests that El Centro may be c lass i f ied as broad banded ascending while Taft and Parkfleld are broad banded descending in the period range of interest. To provide a basis for comparison a l l three earthquake records were scaled to give a maximum acceleration of f i f t y percent of gravity. It should be noted that this does not imply uni-form spectral Intensities. The results obtained are presented in Figs. 4.29 to 4.36. Shown in Figs. 4.29 and 4.30 are the time-displacement response for the top of the A/20/2.2/2/6 frame and the maximum lateral story displacements respectively for the AAECNS50, AATASE50, and AAPKNE50 analyses. These results, together with the girder duct i l i ty demands shown In Figs. 4.31 and 4.32 demonstrate the sensit iv ity of structural response to the ground motion type. The particular configuration of these response parameters may be explained In the context of the earth-quake properties described ear l ier . For example, It wi l l be observed that the maximum lateral displacements for the Taft analysts are sub-stantial ly less than those for the El Centro or Parkfleld analyses over most of the height of the building. Examination of the response spectra In Appendix D for the three earthquakes shows that for a fundamental period of 2.2 seconds, a damping ratio of two percent, and a maximum acceleration of f i f t y percent of gravity, the dynamic amplification AAPKNE50 F i g . 4.29 Time-Displacement Response fo r Top Node versus Earthquake Record 4.0 F ig . 4.30 Maximum Lateral Story Displacements versus Earthquake Record F i g . 4.31 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand v e r s u s Ear thquake Record F i g . 4.32 G i r d e r Hinge C u r v a t u r e D u c t i l i t y Demand ve r su s Ear thquake Record GIRDER ACCUMULATED HINGE ROTAT 1 ON FACTOR STORY EL CENTRO NS 1940 TAFT S69E 1952 PARKFIELD N65E 1966 NUM AAECNS50 AATASE50 AAPKNE50 20 1268.43 1147.89 388.68 19 563.36 338.41 229.45 18 3 2 6 . 4 0 2 3 2 . 4 5 188.63 17 215.33 154.01 102.46 16 147.40 9 0 . 2 3 83-12 15 98.20 60.51 57.03 73.86 64.87 4 0 . 4 2 13 72 .48 57.92 38.89 12 55.05 41.44 42.59 II 42 .21 35-80 42-31 10 48.90 34.69 4 8 . 1 6 9 4 8 . 7 2 30.33 42.88 8 4 0 . 3 7 28.31 46.00 7 35.80 23-88 44.11 6 3 2 . 4 8 21.95 4 1 . 2 2 5 3 2 . 2 6 23.69 38.77 <t 32-74 25-68 40.66 3 2 7 . 2 8 25-64 36-73 2 27 .10 24.38 33-11 1 2 3 . 8 4 2 2 . 1 6 14.00 C Q Q / -' ' — — - V A / w 4 / " / i ( i \ t \ i \ i \ ' 1 t y / / / / i j i i 11 j I i i i i i A i • i RACLN5S0 X = l.DUft RRTRSE50 X " i .OQJM RRPMC90 X • l.OOiH 1 T • 1 — 100.0 150.0 200.0 250.0 . — -GIRDER ACCUM H I N G E ROTATION FACTOR 400 F i g . 4.33 G i r d e r Accumula ted Hinge R o t a t i o n F a c t o r ve r su s Ear thquake Record EXTERIOR COLUMN ROTATIONAL DUCTILITY STORY EL CENTRO NS 1940 TAFT S69E 1952 PARKFIELD N65E 1966 NUM AAECNS50 AATASE50 AAPKNE50 20 2.23 2.37 159 19 2.02 1.55 2.07 18 1.26 1.25 1.37 17 1.37 0.95 0.95 16 0.92 0.74 0.82 15 0.78 0.73 0.72 1*4 0-71 0.69 0.69 13 0.75 0.69 0.87 . 12 0.72 0.67 0.89 11 0.69 0.68 0.71 10 0.68 0.63 0.82 9 0.72 0.62 0.86 8 0.58 0.59 0.83 7 0.61 0.56 0.80 6 0.66 0.57 0.78 5 0.63 0.53 0.79 it 0.66 0.67 0.82 3 0.70 0.61 0.82 2 0.62 0.73 0.90 1 0.75 0.80 1.32 fa J.** V h / i i I \ / \ i ( \ ( ^ v. mxusa x - O.OIH MTASE50 X = 0.01H MPWE50 X = 0.01H o.s 1.5 2.0 2.5 3 . 0 EXT COLUMN END ROT D U C T I L I T Y Fig. 4.34 External Column End Rotational Ductil ity Demand versus Earthquake Record INTERIOR COLUMN ROTATIONAL DUCTILITY STORY EL CENTRO NS 1940 TAFT S69E 1952 PARKFIELD N65E 1966 HUH A A E C N S 5 0 A A T A S E 5 0 AAPKNE50 20 22.74 8 . 8 5 7-97 19 3.47 2 . 8 8 3.38 18 1.91 1.92 1.96 17 2 . 0 6 1.50 1.35 16 1.17 0.86 0 . 9 6 15 0 . 9 0 0 . 8 7 0 . 8 5 14 0.85 0.83 0.83 13 0.86 0.76 1.01 12 0.87 0.73 1.03 II 0.77 0.76 0.82 10 0.79 0.68 0.93 9 0 . 8 0 0 . 6 7 0.97 8 0.67 0 . 6 5 0.96 7 0.66 0 . 5 9 0 . 7 9 6 0-71 0.61 0 . 8 8 5 0 . 6 9 0 . 6 4 0.91 4 0 . 7 2 0 . 8 0 0 . 9 4 3 0.75 0.71 0.93 2 0 . 7 0 0 . 8 2 1.11 1 0.83 0 . 6 0 1.34 U J ¥' >~ cc o ^ ^ ^ ^ / / 1 / I / 1 1 \ 1 I 1 \J // \ 1 1 1 \ i { 1 V / i ( \ 1 i \ y ( 1 1 I \ \ oarriiwi \. = n AIM RRTRSESfJ X « Q.01H . . . . RHPKNE50 X * 0.01M 0.0 o.s 1.5 2.0 2.5 3.0 I N T COLUMN END ROT D U C T I L I T Y 3.5 4.0 F i g . 4.35 Internal Column End Rotational D u c t i l i t y Demand versus Earthquake Record f * *J 4 ¥ 1 A A" f* < '4 4" • * — "4 4 1 • ? * 4' f f i '4 4 1 * f V f ' 4 4 ' • V • < • 4 ' 4 4 f f A 4 • • ? A A • f A A A A • f • • A A A A • f A A • V A A A A • • A A • • A A V V . A V • M A W~ ^ V A A ? • A A t A A ? • * A ¥ • -• • # V # • # 9 ^ 0 • • # A A 0 A,, A A A A A A V V ^ • # 7 "tftt tt >r>j>> -# r r r v y > f t * -0 #>-I AAECNSSO AAPKNE50 AATASE50 F i g . 4.36 Hinge S t a t u s Diagrams v e r s u s Ear thquake Record 124 f a c t o r f o r T a t t i s about 0.38 as compared w i t h a p p r o x i m a t e l y 0.62 f o r E l C e n t r o and P a r k f i e l d . It i s i n t e r e s t i n g t o no te t h a t the r a t i o o f the maximum l a t e r a l s t o r y d i s p l a c e m e n t s a t the top o f t he f rame f o r the T a f t and P a r k f i e l d a n a l y s e s i s a lmos t e x a c t l y the same as the r a t i o o f t he dynamic a m p l i f i c a t i o n f a c t o r s f o r those two a n a l y s e s . A d d i t i o n a l l y , e x a m i n a t i o n o f t h e r e l a t i v e v e l o c i t y s p e c t r a f o r t he t h r e e e a r t h q u a k e s , w i t h the same f a c t o r s as a b o v e , shows t h a t E l C e n t r o and P a r k f i e l d have r e l a t i v e v e l o c i t i e s ten and t h i r t y - t h r e e p e r c e n t h i g h e r r e s p e c t i v e l y in the fundamenta l mode than t h a t o f T a f t . I t i s f u r t h e r Importnat t o no te t h a t the r e l a t i v e v e l o c i t y o f t h e second mode f o r El C e n t r o Is f o u r t e e n and e i g h t y - t h r e e p e r c e n t h i g h e r r e s p e c t i v e l y than tho se f o r T a f t and P a r k f i e l d . W h i l e the s p e c t r a a r e based on the re sponse o f e l a s t i c s i n g l e deg ree o f f reedom s y s t e m s , they p r o v i d e an i n d i c a t i o n o f t h e r e l a t i v e s t r e n g t h a t g round mot ions and t h e i r c a p a c i t y to e x i s t the v a r i o u s modes a t a s t r u c t u r a l r e s p o n s e . As wou ld be e x p e c t e d , t h e n , the l a t e r a l d i s p l a c e m e n t c o n f i g u r a t i o n f o r the T a f t a n a l y s i s i s seen to be p r i m a r i l y a f u n c i ton o f t h e fundamenta l mode in t he lower and m i d d l e p o r t i o n s o f t he b u i l d i n g w h i l e t he second mode makes a n o t i c e -a b l e c o n t r i b u t i o n in the upper s e c t i o n o f t h e f r ame. The maximum l a t e r a l s t o r y d i s p l a c e m e n t c o n f i g u r a t i o n f o r the P a r k f i e l d a n a l y s i s i s , as s ugge s ted by the r e l a t i v e s p e c t r u m , seen to be m a i n l y a f i r s t mode r e s p o n s e . By c o n t r a s t , the l a r g e r c o n t e n t o f h i g h e r f r e q u e n c i e s in the E l C e n t r o r e c o r d , p a r t i c u l a r l y in the a r e a o f t he second n a t u r a l f r e q u e n c y o f the s t r u c t u r e c o n s i d e r e d , combined w i t h the a s c e n d -ing n a t u r e o f the r e l a t i v e v e l o c i t y spec t rum and the weak d e s i g n o f the upper s t o r i e s s e r v e s to p roduce the l a t e r a l d i s p l a c e m e n t re sponse shown f o r the AAECNS50 a n a l y s i s . 125 As w i t h the l a t e r a l d i s p l a c e m e n t s , t he g i r d e r r o t a t i o n a l and c u r v a t u r e d u c t i l i t y demands o f the f rame c o n s i d e r e d were found to be s i g n i f i c a n t l y i n f l u e n c e d by t h e e a r t h q u a k e p r o p e r t i e s . F o r examp le , the lower v a l u e s f o r the g i r d e r d u c t i l i t i e s in the bot tom h a l f o f the frame f o r t he E l C e n t r o and T a f t a n a l y s e s as compared w i t h t h e P a r k -f i e l d a n a l y s i s wou ld be e x p e c t e d f o r s i n u s o i d a l r e c o r d s as compared w i t h those o f the i m p u l s i v e t y p e . The ab rup t i nput o f r e l a t i v e l y l a r g e r amounts o f energy i n t o t h e s t r u c t u r a l s y s tem g e n e r a l l y r e s u l t s , depending o f c o u r s e on the r e l a t i v e f r e q u e n c y and energy c o n t e n t , i n an i n c r e a s e d re sponse f o r a b u i l d i n g s u b j e c t e d to an i m p u l s i v e r e c o r d as compared w i t h a s i n u s o i d a l a c c e l e r o g r a m . The predominance o f t he f u n d a -menta l mode due to the p a r t i c u l a r f r e q u e n c y c o n t e n t o f the P a r k f i e l d e a r t h q u a k e sugge s t s a r e d u c e d c o n t r i b u t i o n by the h i g h e r modes in d e t e r m i n i n g the i n t e r s t o r y d i s p l a c e m e n t s . The r e s u l t , as shown i n F i g s . 4.31 and 4 . 3 2 , Is an a v o i d a n c e o f the c o n c e n t r a t i o n o f member damage and a s p r e a d o f n o n l i n e a r i t y th roughout the f r ame. By c o n t r a s t , w h i l e the energy input o f t he E l C e n t r o r e c o r d i s more u n i f o r m o v e r t i m e , the h i g h e r modes a r e more s i g n i f i c a n t in d e t e r m i n i n g the g i r d e r d u c t i l i t i e s f o r the AAECNS50 a n a l y s i s . As s t a t e d e a r l i e r , the r e s u l t s i n d i c a t e t h a t a l a r g e r c o n t e n t o f h i g h e r f r e q u e n c i e s In the E l C e n t r o a c c e l e r o g r a m leads to a r e l a t i v e l y l a r g e r e x c i t a t i o n o f t he h i g h e r modes in d e t e r m i n i n g the i n t e r s t o r y d i s p l a c e m e n t s . T h i s combined w i t h the weak d e s i g n wh ich i n c r e a s e s t h e impor tance o f t h e h i g h e r modes u l t i m a t e l y l eads to the c o n c e n t r a t i o n o f n o n l i n e a r i t y , as d e s c r i b e d In p r e v i o u s s e c t i o n s , in the top o f t he f r a m e . The a c t i o n i s f u r t h e r a g g r a v a t e d by the " a s c e n d i n g " n a t u r e o f t he E l C e n t r o r e l a t i v e v e l -o c i t y spec t rum in the range o f t he fundamenta l p e r i o d o f the frame 126 c o n s i d e r e d . As a s t r u c t u r e d i s s i p a t e s energy th rough n o n l i n e a r i t y the n a t u r a l p e r i o d s t y p i c a l l y i n c r e a s e . The I n c r e a s i n g r e l a t i v e v e l o c i t y spec t rum f o r the E l C e n t r o e a r t h q u a k e in the p e r i o d range o f i n t e r e s t i m p l i e s an i n c r e a s e d re sponse w i t h i n c r e a s i n g s t r u c t u r a l damage. On the o t h e r hand , the " d e s c e n d i n g " n a t u r e o f t he r e l a t i v e v e l o c i t y s p e c t r a f o r the T a f t and P a r k f l e l d r e c o r d s In the p e r i o d range o f i n t e r e s t i m p l i e s a r e d u c t i o n i n re sponse w i t h an I nc rea se In s t r u c t u r e n o n l i n e a r -i t y . T h i s f a c t o r a i d s i n l i m i t i n g t h e v a l u e o f t h e g i r d e r r o t a t i o n a l and c u r v a t u r e d u c t i l i t y demands f o r t he AATASE50 and AAPKNE50 a n a l y s e s . The e f f e c t o f v a r y i n g the g round mot ion t y p e on the column d u c t i l i t y demands i s shown in F i g s . 4.34 t o 4 .36 . It w i l l be o b s e r v e d t h a t the re sponse Is most a f f e c t e d by the s i n u s o i d a l o r i m p u l s i v e n a t u r e o f the a c c e l e r o g r a m . The abrupt i n t r o d u c t i o n o f l a r g e amounts o f energy Into the s t r u c t u r a l s y s tem by the P a r k f l e l d r e c o r d and the c o r r e s p o n d i n g i n c r e a s e in s t r u c t u r a l re sponse r e s u l t s In the somewhat h i g h e r d u c t i l i -t i e s shown f o r the AAPKNE50 a n a l y s i s . However, i t Is Important t o note t h a t In s p i t e o f t he I n t e n s i t y o f the r e c o r d s , f o r a l l the a n a l y s e s s e r i o u s n o n l i n e a r i t y o c c u r e d o n l y i n the top s e v e r a l s t o r i e s . The r e s u l t s I n d i c a t e t h a t t h e r e s h o u l d be no danger o f t he c o l l a p s e due t o column f a i l u r e , even In a s e v e r e e a r t h q u a k e , o f a m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e framed s t r u c t u r e d e s i g n e d a c c o r d i n g t o c u r r e n t s e i s m i c c o d e s . 41 In an e a r l i e r n o n l i n e a r s t udy G i b e r s o n i n v e s t i g a t e d the e f f e c t o f E l C e n t r o and seven J e n n i n g s a r t i f i c i a l e a r t h q u a k e r e c o r d s on the re sponse o f a r e i n f o r c e d c o n c r e t e v e r s i o n o f t he A / 2 0 / 2 . 2 / 2 / 6 f r ame. As one r e s u l t o f t h o s e a n a l y s e s he c o n c l u d e d t h a t the p a t t e r n o f modal c o n t r i b u t i o n t o the s t r u c t u r a l r e sponse and the c o n f i g u r a t i o n o f t h e d u c t i l i t y demands ove r the h e i g h t o f t h e frame were more a f u n c t i o n o f 127 s t r u c t u r a l p r o p e r t i e s than o f t h e ground mot ion e x c i t a t i o n . The p r e -s t r e s s e d c o n c r e t e v e r s i o n o f the frame appears t o be somewhat more s e n -s i t i v e t o the e a r t h q u a k e r e c o r d u s e d . It w i l l be r e c a l l e d , however , t h a t as the t a p e r o f t h e s t r u c t u r a l , and p a r t i c u l a r l y the s t i f f n e s s p r o p e r -t i e s o f a f rame i s r e d u c e d , the Importance o f t he h i g h e r modes In d e t e r -m i n i n g t h e re sponse w i l l be c o r r e s p o n d i n g l y r e d u c e d . As s u c h , i t wou ld be e x p e c t e d tha t the h i g h e r f r e q u e n c i e s c o n t a i n e d In an a c c e l e r o g r a m would be somewhat l e s s s i g n i f i c a n t In d e t e r m i n i n g the re sponse o f a f rame de s i gned a c c o r d i n g t o p r e s e n t e a r t h q u a k e c o d e s . The " p e a k e d " , " a s c e n d -i n g " , o r " d e s c e n d i n g " n a t u r e o f r e l a t i v e v e l o c i t y spec t rum o f a ground mot ion in the a r e a o f t he fundamenta l p e r i o d o f a p r e s t r e s s e d c o n c r e t e f rame Is l i k e l y t o be o f c o n s i d e r a b l e i m p o r t a n c e , however, In t he d e t e r -m i n a t i o n o f I ts r e s p o n s e . S i m i l a r l y , the r e s u l t s i n d i c a t e t h a t t he s i n u s o i d a l o r Impu l s i ve n a t u r e o f an a c c e l e r a t i o n r e c o r d w i l l have a s i g n i f i c a n t a f f e c t on the re sponse o f a p r e s t r e s s e d c o n c r e t e f r a m e . T h i s , and the v a r i a t i o n i n the v a l u e o f the re sponse pa rameter s u n d e r l i n e s the Importance o f d e t e r m i n i n g t h e c o r r e c t ground mot ion f o r a s i t e when the n o n l i n e a r a n a l y s i s o f a s t r u c t u r e Is b e i n g c o n t e m p l a t e d . In rega rd s t o t he r e s u l t s p r e s e n t e d in t h i s s e c t i o n , i t ts thought t h a t the a c c e l e r a t i o n r e c o r d s used r e p r e s e n t a s u f f i c i e n t l y b road s p e c -trum o f g round mot ion types such t h a t t he v a l u e s o b t a i n e d i n d i c a t e the l i k e l y range o f t h e d u c t i l i t y demands o f a p r e s t r e s s e d c o n c r e t e frame s u b j e c t e d to a major e a r t h q u a k e . It i s Important to n o t e t h a t t he d u c t i l -i t y demands o f a p a r t i c u l a r b u i l d i n g w i l l be s t r o n g l y dependent on the s t r u c t u r a l s t i f f n e s s and s t r e n g t h p r o p e r t i e s , as w e l l as g round mot ion t y p e . However, t h e r e s u l t s p r e s e n t e d r e i n f o r c e e a r l i e r s t a tement s c o n -c e r n i n g the c a p a b i l i t y o f a m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e frame to 128 w i t h s t a n d s e v e r e s e i s m i c l o a d i n g . Over most o f t he h e i g h t o f the b u i l d i n g t he d u c t i l i t y demands a r e w e l l w i t h i n the d u c t i l e c a p a c i t y o f c a r e f u l l y d e t a i l e d p r e s t r e s s e d c o n c r e t e members. 4.5.4 E f f e c t o f S t r e n g t h D i s t r i b u t i o n In p r e v i o u s s e c t i o n s the e f f e c t o f the s e v e r e t a p e r o f t h e s t r u c t u r a l p r o p e r t i e s on the re sponse o f t h e p r e s t r e s s e d c o n c r e t e f rame has been emphas i zed . It has a l s o been p o i n t e d ou t t h a t t he a p p l i c a t i o n o f the l a t e r a l f o r c e d i s t r i b u t i o n s p e c i f i e d by c u r r e n t s e i s m i c codes would r e s u l t In a c o m b i n a t i o n o f reduced t a p e r o f s t i f f -ness p r o p e r t i e s and i n c r e a s e d member s t r e n g t h i n the upper p o r t i o n o f t h e f r ame . In t h i s s e c t i o n , o n l y the e f f e c t o f t he s t r e n g t h d i s t r i b u -t i o n In t he g i r d e r s on the re sponse pa rameter s was c o n s i d e r e d . It wou ld be e x p e c t e d , o f c o u r s e , t h a t an I n c r e a s e In g i r d e r s t r e n g t h wou ld r e s u l t In reduced n o n l i n e a r i t y . The o b j e c t i v e o f t h i s s e c t i o n was t o d e t e r m i n e t he magn i tude o f t h e r e d u c t i o n and the e f f e c t on the column d u c t i l i t y demands. In t h a t r e ga rd t h e re sponse o f t h e B/20/2.2/ V/6 frame d e s c r i b e d In s e c t i o n 4.3.4 Is compared w i t h t h a t o f the s t a n -d a r d A/20/2.2/2/6 f r ame . The r e s u l t s o b t a i n e d a re p r e s e n t e d In F i g s . 4.37 t o 4.44. Shown In F i g . 4.37 a r e the t ime - d i s p l a c e m e n t p l o t s o f t h e top o f the f rame f o r the AAECNS35 and BAECNS35 a n a l y s e s . O b s e r v a t i o n o f t h e F i g u r e shows t h a t , as e x p e c t e d , an I n c r e a s e In g i r d e r s t r e n g t h r e s u l t s In a d e c r e a s e In re sponse and a s m a l l e r I n c rea se i n p e r i o d due to s t r u c t u r e damage. The l a r g e r l a t e r a l d i s p l a c e m e n t s f o r t h e BAECNS35 a n a l y s i s In the lower p o r t i o n o f the b u i l d i n g as shown In F i g . 4.38 a r e AAECNS35 BAECNS 35 F i g . 4.37 Time - D i s p l a c e m e n t Response f o r Top Node v e r s u s S t r e n g t h D i s t r i b u t i o n F i g . 4.38 Maximum L a t e r a l S t o r y D i sp l acement s v e r s u s S t r e n g t h D i s t r i b u t i o n F i g . 4.39 G i r d e r v e r s u s End R o t a t i o n a l D u c t i l i t y Demand S t r e n g t h D i s t r i b u t i o n F i g . 4.40 G i r d e r v e r s us Hinge C u r v a t u r e D u c t i l i t y Demand S t r e n g t h D i s t r i b u t i o n 0.0 30.0 60.0 90.0 120.0 [ 50.0 180.0 210.0 24) GIRDER ACCUM H I N G E ROTATION FACTOR F i g . 4.1»1 G i r d e r Accumula ted Hinge R o t a t i o n F a c t o r v e r s u s S t r e n g t h D i s t r i b u t i o n F i g . k.kZ E x t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s S t r e n g t h D i s t r i b u t i o n INTERNAL COLUMN END ROTATIONAL DUCTILITY STORY NUM EL CENTRO NS 1940 AAECNS35 BAECNS35 % INCREASE 20 8.35 5.34 -36 19 2.35 3.46 47 18 1.22 2.24 84 17 0.97 1.43 47 16 0.91 1.23 35 15 0.94 1.30 38 14 0.79 0.99 25 13 0.88 0.92 4.5 12 0 . 8 1 0.83 2 . 5 11 0.75 0 . 8 0 6.7 10 0 . 6 2 0 . 7 2 16 9 0.74 0.75 1.4 8 0.73 0.81 11 7 0 . 6 4 0.86 34 6 0.70 0.79 13 5 0.69 0.77 12 4 0.70 0.79 13 3 0.74 0.76 2.7 2 0.76 0.85 12 1 0.78 0.91 17 03 o / / \ / v 1 \ \ v -1 /) 1 (( M |\ \ 1 1 1 j ' 1 ; \ \ \ 1 aorru's-K \ = 1 n » 8RECHS35 \ = 1.0/N 1 5 2.0 2.5 3.0 INT COLUMN END ROT D U C T I L I T Y 3.5 F i g . 4.43 I n t e r n a l Column End R o t a t i o n a l D u c t i l i t y Demand v e r s u s S t r e n g t h D i s t r i b u t i o n ' * A 1 •< 1 * m{ » • • ( >• •< A 1 • •< *' 1 • • ' • • ' • •< 1 • • • •< 20 19 16 15 U 13 12 II 10 9 1 3 CRD / I f J / J t t f J J J / / f i l l I i ' > ' ) ! > ) / / > ) Ii AAECNS35 BAECNS35 ON F i g . k.kk H inge S t a t u s Diagrams v e r s u s S t r e n g t h D i s t r i b u t i o n 137 e v i d e n c e o f a more e f f e c t i v e t r a n s f e r o f the s h e a r f o r c e s down the f r ame . The e f f e c t o f v a r y i n g the s t r e n g t h d i s t r i b u t i o n on the re sponse pa rameter s i s most e v i d e n t in the g i r d e r d u c t i l i t y demands and g i r d e r a ccumu la ted h i n g e r o t a t i o n f a c t o r s shown i n F i g s . 4.39 t o 4.41 r e s -p e c t i v e l y . R e s u l t s show t h a t f o r a l l t h r e e pa rameter s t h e r e l a t i o n s h i p between the p e r c e n t a g e d e c r e a s e i n re sponse and t h e c o r r e s p o n d i n g s t o r y number i s a p p r o x i m a t e l y u n i f o r m o v e r the h e i g h t o f t he f r a m e . The r e -l a t i o n s h i p between the p e r c e n t a g e d e c r e a s e i n re sponse and t h e p e r -cen tage I n c r e a s e in t he c r a c k i n g moments f o r t he B/20 f rame as compared t o the A/20 f rame was f o u n d to be a p p r o x i m a t e l y l i n e a r o v e r the h e i g h t o f the b u i l d i n g f o r the t h r e e p a r a m e t e r s . The v a l u e s o b t a i n e d demon-s t r a t e t h a t the d u c t i l i t y demands i n an open p r e s t r e s s e d c o n c r e t e frame can be r e a s o n a b l y c o n t r o l l e d . The r a t i o o f d e s i g n to c r a c k i n g moments f o r the B/20 frame i s p r o b a b l y t oo h i g h to be p r a c t i c a l In a r e a l b u i l d -i n g , however , the d u c t i l i t y demands f o r the g i r d e r s a r e perhaps lower than might be a c c e p t a b l e f o r t h i s magn i tude o f e a r t h q u a k e . It i s a l s o impor t an t t o no te t h a t f o r a f rame d e s i g n e d w i t h the l a t e r a l f o r c e d i s -t r i b u t i o n s p e c i f i e d by c u r r e n t s e i s m i c codes the r e l a t i v e s t i f f n e s s and d e s i g n moments o f t h e members, p a r t i c u l a r l y In the upper s t o r i e s , wou ld l i k e l y be somewhat h i g h e r than tho se o f t h e A / 2 0 / 2 . 2 / 2 / 6 f r a m e . T h i s would a l l o w a lower d e s i g n s t r e n g t h to c r a c k i n g moment r a t i o f o r the g i r d e r s w h i l e s t i l l l i m i t i n g the d u c t i l i t y r e q u i r e m e n t s . The e f f e c t o f v a r y i n g the s t r e n g t h d i s t r i b u t i o n on the column d u c t i l i t y demands i s shown in F i g s . 4.42 t o 4 .44. As would be e x p e c t e d , a s t r e n g t h e n i n g o f t h e g i r d e r s r e s u l t s in l e s s energy b e i n g d i s s i p a t e d by those members and a c o r r e s p o n d i n g i n c r e a s e In column d u c t i l i t y demands. 138 The e f f e c t i s most evident in the upper p o r t i o n of the frame where the reduction in g i r d e r d u c t i l i t y requirements was most dramatic. However, had the strength of the columns been increased to maintain the same r a t i o to the g i r d e r strength as i n the A/20 frame, the increase in column d u c t i l i t y demands would have been minimized. The r e s u l t s of t h i s s e c t i o n i n d i c a t e that the response of a frame designed according to current seismic codes and subjected to an El Centro i n t e n s i t y earthquake would be somewhere between that of the B/20 and A/20 frames. The increased s t i f f n e s s of the members in the upper p o r t i o n would f u r t h e r reduce the maximum displacement at the top of the frame and produce a greater f i r s t mode c o n t r i b u t i o n in the l a t e r a l displacement c o n f i g u r a t i o n . It would be expected t h a t , w h i l e there would l i k e l y be some increase in g i r d e r d u c t i l i t y demands over the height of the b u i l d i n g , they would be considerably less than f o r the A/20 frame. F i n a l l y the values obtained i n d i c a t e that f o r the design strength to c r a c k i n g moment r a t i o f o r the A/20 frame the columns would remain e l a s t i c except p o s s i b l y in the top s e v e r a l f l o o r s . 4.5.5 Comparison o f the Response of Prestressed and Reinforced Concrete Frames 7 9 19 A number of other s t u d i e s have confirmed the i n t u i t i v e b e l i e f that the l a t e r a l displacements of a prestressed concrete frame w i l l be somewhat greater than those of a r e i n f o r c e d concrete s t r u c t u r e w i t h the same i n i t i a l s t i f f n e s s and strength p r o p e r t i e s . This may be a t t r i b u t e d to both the lower energy d i s s i p a t i o n c apacity c h a r a c t e r i s t i c of pre-s t r e s s e d concrete and the lower percentage of viscous damping assumed 139 f o r a p r e s t r e s s e d c o n c r e t e frame as compared w i t h a r e i n f o r c e d c o n c r e t e sy s tem^. However, t h e r e l a t i o n s h i p between the d u c t i l i t y demands f o r members o f comparab le p r e s t r e s s e d and r e i n f o r c e d c o n c r e t e s t r u c t u r e s has not been as c o n c l u s i v e l y e s t a b l i s h e d . To d a t e , the o n l y o t h e r p u b -l i s h e d s tudy wh ich has a d d r e s s e d the q u e s t i o n o f t he d u c t i l i t y demands f o r members o f m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e framed b u i l d i n g s was a t h a t by Spencer . As a r e s u l t o f t h a t s t udy Spencer c o n c l u d e d t h a t t he d u c t i l i t y r equ i rement s f o r members o f a p r e s t r e s s e d c o n c r e t e f rame wou ld be somewhat l e s s than those f o r a r e i n f o r c e d c o n c r e t e frame w i t h the same i n i t i a l s t i f f n e s s and s t r e n g t h . A compar i son o f t he r e s u l t s o b t a i n -ed in t h i s and e a r l i e r s t u d i e s o f both p r e s t r e s s e d and r e i n f o r c e d c o n -c r e t e v e r s i o n s o f t he A/20/2.2/2/6 f rame i s d i s c u s s e d i n t h i s s e c t i o n . F i g u r e s 4.45 t o 4.48 p r e s e n t t he v a l u e s wh i ch were o b t a i n e d f o r the r e l e v a n t r e sponse p a r a m e t e r s . 7 9 19 P r e v i o u s s t u d i e s have sugge s ted t h a t the l a t e r a l d i s p l a c e m e n t s o f a p r e s t r e s s e d c o n c r e t e frame wou ld be , on the a v e r a g e , t h i r t y to f i f t y p e r c e n t g r e a t e r than tho se f o r a comparab le r e i n f o r c e d c o n c r e t e f r a m e , a l t h o u g h a wide range o f v a l u e s has been o b t a i n e d . The h y s t e r e t i c and s t r u c t u r a l p r o p e r t i e s o f a b u i l d i n g , t o g e t h e r w i t h t h o s e o f the e x c i t i n g g round mot ion a r e o f c o n s i d e r a b l e impor tance in de te rm lng a p a r t i c u l a r re sponse r e l a t i o n s h i p . F i g u r e 4 .45 shows t h a t t he maximum l a t e r a l d i s p l a c e m e n t o f t he top o f the frame f o r t he AAECNS35 a n a l y s i s was about f i f t y - f i v e p e r c e n t g r e a t e r than tha t f o r the r e i n f o r c e d c o n -c r e t e v e r s i o n , s l i g h t l y more than t h a t o b t a i n e d in the ASECNS32 a n a l y s i s by S p e n c e r . By c o m p a r i s o n , t h e maximum l a t e r a l s t o r y d i s p l a c e m e n t a t t he top o f the b u i l d i n g f o r the BAECNS35 a n a l y s i s d i s c u s s e d in t he p r e -v i o u s s e c t i o n was about f o r t y - s i x p e r c e n t g r e a t e r than tha t f o r t he r e -Fi„g. 4.45 Maximum S t o r y D i s p l a c e m e n t v e r s u s B u i l d i n g V e r s i o n ( P r e s t r e s s e d o r R e i n f o r c e d ) GIRDER END ROTATIONAL DUCTILITY STORY NUM AAECNS35 ASECNS32 DUCTILITY * OF R/C DUCTILITY % OF R/C 20 18.95 370 1 .00 19.5 19 11.60 333 1.88 62 .0 13 9 .45 176 3.02 56.3 17 9.16 190 2.67 55.4 16 11.61 288 2.90 72.0 15 8.76 289 3.55 117 lit 7.63 302 4.16 164 13 6 .96 321 4.51 208 12 5.27 298 4.89 276 11 3-50 192 4 .70 258 10 3.00 150 5.52 276 9 2.71 136 5.30 265 a 3.42 162 4 .89 232 ' 7 3.64 141 4 .70 182 6 3.65 135 4.51 166 5 3.42 123 4.33 156 i. 3 -59 116 4 .70 152 3 3.44 114 4.70 166 2 3.14 III 4.00 141 1 2.77 131 3-69 175 CC LO QQO CC \ s Y\ V \A K ^ fi \ / i\ \ 1 \ \ / / 1 / / / \ \ j ! 1 1 / / / / (WEQIS3S X = l.OJH RSECHS32 >.= 1.0IH,2.0JK CSCCMS3Z X ~ 10.(W cceaisx A > tootn 0.0 3.0 6.0 9.0 12.0 IS. 0 18.0 G I R D E R E N D ROT DUCTILIir F\g. kA6 G i r d e r End R o t a t i o n a l D u c t i l i t y Demand ve r su s V e r s i o n ( P r e s t r e s s e d o r R e i n f o r c e d ) 142 / / / / 1 1 1 If l\ \ \ I \ 1 \ I 1 \ \ \ 1 1 1 1 / / \ \ 1 w nnmwas X = l niM CXECNS32 X = 10.0JH "1 1 1 1 1 1 1 1 1 Q.O 0.5 1.0 l.S 2.0 2.S 3.0 3.5 4.0 E X T C O L U M N E N D R O T ( D U C T I L I T Y F i g * 4.47 E x t e r n a l Column End R o t a t i o n a l D u c t i l i t y v e r s u s V e r s i o n ( P r e s t r e s s e d o r R e i n f o r c e d ) 143 1 I 1 -1 \ / ' ( ' \ \ / -I 1 1 / / \ / \ \ - ){ / 1 \ \ \ 1 1 — 1 - -o o r r u w X = I n»M CCECNS32 X = 10.0ZH | v r T " 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 I N T C O L U M N E N D R O T D U C T I L I T Y F i g . 4.48 I n t e r n a l Column End R o t a t i o n a l D u c t i l i t y v e r s u s V e r s i o n ( P r e s t r e s s e d o r R e i n f o r c e d ) 144 i n f o r c e d c o n c r e t e v e r s i o n o f the f r ame . It s h o u l d be no ted t h a t the h y s t e r e t i c r e l a t i o n s h i p f o r the r e i n f o r c e d c o n c r e t e v e r s i o n s was the e l a s t o - p l a s t i c mode l . Had a d e g r a d i n g s t i f f n e s s model been u s e d , i t wou ld be e x p e c t e d t h a t the l a t e r a l d i s p l a c e m e n t s o f t h e r e i n f o r c e d c o n c r e t e v e r s i o n s would be somewhat g r e a t e r , and t h u s , the r a t i o the maximum l a t e r a l d i s p l a c e m e n t s f o r t he p r e s t r e s s e d and r e i n f o r c e d c o n -c r e t e v e r s i o n s o f t h e frame would be s l i g h t l y l ower . The r a t i o o f t h e l a t e r a l d i s p l a c e m e n t s was a d d i t i o n a l l y dependent on the degree o f v i s c o u s damping assumed. The n o n l i n e a r a n a l y s i s o f the r e i n f o r c e d 9 10 c o n c r e t e v e r s i o n s were pe r fo rmed w i t h a v i s c o u s damping r a t i o o f ten p e r c e n t . As d e s c r i b e d in s e c t i o n 4 . 2 , t h i s magn i tude o f v i s c o u s damping i s somewhat e x c e s s i v e f o r an i n e l a s t i c a n a l y s i s . A r e d u c t i o n In the p e r c e n t a g e o f v i s c o u s damping f o r t he r e i n f o r c e d c o n c r e t e v e r -s i o n wou ld f u r t h e r s e r v e to reduce the r a t i o o f the maximum l a t e r a l d i s p l a c e m e n t s between t h a t and the p r e s t r e s s e d c o n c r e t e v e r s i o n . The f a c t t h a t the l a t e r a l d i s p l a c e m e n t s o f a p r e s t r e s s e d c o n c r e t e f rame w i l l be somewhat g r e a t e r than tho se o f a comparab le r e i n f o r c e d c o n c r e t e s t r u c t u r e s h o u l d no t n e c e s s a r i l y be v iewed as a m a t e r i a l weak-n e s s . Non s t r u c t u r a l components can be d e s i g n e d t o accommodate the I n c r e a s e d d e f o r m a t i o n s . A l t e r n a t e l y , S p e n c e r ' ^ f o u n d t h a t t he use o f r i g i d i n f i l l p a n e l s was v e r y e f f e c t i v e in r e d u c i n g bo th i n t e r s t o r y and l a t e r a l d i s p l a c e m e n t s . It Is c l e a r , however , t h a t If s i m i l a r l a t e r a l d i s p l a c e m e n t s a r e d e s i r e d f o r open r e i n f o r c e d and p r e s t r e s s e d c o n c r e t e f r a m e s , the l a t t e r w i l l r e q u i r e a somewhat h i g h e r i n i t i a l s t i f f n e s s and s t r e n g t h than i t s r e i n f o r c e d c o n c r e t e c o u n t e r p a r t . Shown in F i g . 4.46 a r e the g i r d e r end r o t a t i o n a l d u c t i l i t y f a c t o r s f o r both p r e s t r e s s e d and r e i n f o r c e d c o n c r e t e v e r s i o n s o f the frame c o n -145 s i d e r e d . It i s impor t an t t o no te t h a t , bo th the r e s u l t s o f t h i s i n -v e s t i g a t i o n , and in g e n e r a l , the r e c a l c u l a t e d r e s u l t s f o r s t r u c t u r e f i v e o f S p e n c e r ' s s t udy p r e d i c t h i g h e r r o t a t i o n a l d u c t i l i t y demands f o r a p r e s t r e s s e d c o n c r e t e f rame than f o r a comparab le r e i n f o r c e d c o n c r e t e f r ame. The lower v a l u e s o b t a i n e d f o r the upper s t o r i e s o f the b u i l d i n g f o r the ASECNS35 a n a l y s i s a r e the r e s u l t o f t he magn i tude and t y p e o f damping u s e d . I t may be r e c a l l e d t h a t in the same non -l i n e a r s t udy S p e n c e r found t h a t t he I n t e r s t o r y d i s p l a c e m e n t s o f a p r e s t r e s s e d c o n c r e t e frame were up t o s e v e n t y p e r c e n t g r e a t e r than those f o r the r e i n f o r c e d c o n c r e t e v e r s i o n o f t h e f r ame . S i n c e the magn i tude o f t h e I n t e r s t o r y d i s p l a c e m e n t s i s a pr ime f a c t o r In d e t e r m i n i n g the d u c t i l i t y r e q u i r e m e n t s , i t wou ld be e x p e c t e d t h a t the demands f o r a p r e s t r e s s e d c o n c r e t e frame wou ld be somewhat g r e a t e r than those o f a r e i n f o r c e d c o n c r e t e f rame w i t h the same I n i t i a l s t i f f n e s s and s t r e n g t h . The r e s u l t s p r e s e n t e d In F i g u r e 4.46 show t h a t f o r the p r e s t r e s s e d c o n c r e t e v e r s i o n o f t he A / 2 0 / 2 . 2 / 2 / 6 frame s u b j e c t e d to E l C e n t r o NS 1940 a t a maximum a c c e l e r a t i o n o f t h i r t y - f i v e p e r c e n t o f g r a v i t y , the g i r d e r d u c t i l i t y demands range from a low o f one hundred e l e v e n p e r c e n t t o a h i g h o f t h r e e hundred e i g h t y - t h r e e p e r c e n t o f t ho se f o r the r e i n -f o r c e d c o n c r e t e v e r s i o n . It has been shown in p r e v i o u s s e c t i o n s , however, t h a t the re sponse o f an open p r e s t r e s s e d c o n c r e t e f rame d e s i g n e d a c c o r d -ing t o c u r r e n t e a r t h q u a k e codes would be somewhat l e s s than t h a t o f t h e AAECNS35 a n a l y s e s , and as s u c h , wou ld not s u f f e r the d r a m a t i c i n c r e a s e in g i r d e r d u c t i l i t y r e q u i r e m e n t s In the upper s t o r i e s . In the FHA'° s tudy i t was s u g g e s t e d t h a t an open m u l t i - s t o r y r e i n f o r c e d c o n c r e t e frame d e s i g n e d a c c o r d i n g to the code l a t e r a l f o r c e r e q u i r e m e n t s and s u b j e c t e d to an E l C e n t r o i n t e n s i t y e a r t h q u a k e wou ld r e q u i r e g i r d e r end 146 rotational duct i l i t ies on the order of f ive. The results of this study suggest that to resist the same magnitude earthquake, a comparable open multi-story prestressed concrete frame would require girder end rota-tional duct i l i t ies on the order of six to eight. Shown in Figs. 4.47 and 4.48 are the column duct i l i ty demands for the prestressed and reinforced concrete versions of the frame considered. In both cases the columns remained e last ic over most of the height of the building although the demands for the prestressed concrete version are considerably closer to cracking. This is the result of both higher lateral and Interstory displacements. It is important to note that for both the reinforced and prestressed concrete versions of the frame, the ratio of the design strength of the columns to the cracking, or yeildlng moments was six. The results Indicate that for both, and particularly the prestressed concrete version this ratio could not be materially reduced without s ignif icantly increasing the column duct i l i ty require-ments and possibly endangering the vert ical Integrity of the structure. The results further demonstrate the need for the ductile detailing of the columns of multl-story frames. There has been some suggestion^ that mixed frames should be sub-jected to nonlinear dynamic time step analyses. The results of this study Indicate that for such a frame with the same in i t i a l stiffness and strength properties, the response Is not l ikely to be considerably different from that of a corresponding prestressed concrete frame as long as the columns remain e las t ic , although the higher viscous damping ratio typical of reinforced concrete may produce some reduction In response. In the case where severe nonlinearity occurs In the columns, however, the larger hysteretic energy dissipation capacity of the 147 reinforced concrete columns wi l l be more effective In l imiting the response than would the prestressed concrete columns. The primary advantages in using prestressed concrete columns l ie in the possible ease and economy of construction and limitation of permanent deformations as a result of nonlinear action. 4.5.6 Conclusions While the nonlinear analyses described in the previous sections were limited, they serve to indicate the l ikely trends in the seismic response of open multi-story prestressed concrete frames. The follow-ing is a summary of the conclusions reached as a result of the inves-tigations in this chapter. 1. The results support those of other studies which found that the lateral displacements of an open prestressed concrete frame would be, on the average, thirty to f i f t y percent greater than those of a comparable reinforced concrete frame. 2. Where viscous damping Is used In addition to hysteretic damping the use of mass proportional viscous damping is suggested. 3. The results indicate that prestressed concrete frames appear to be more sensitive to the taper of the s t i f f -ness properties than are s iml l lar reinforced concrete frames. Severe taper of the stiffness properties leads to large Interstory displacements and correspondingly excessive duct i l i ty demands in the upper portion of a frame. 4. Due to the weak design of the upper portion of the frame considered and the ground motion used, the Increase In earthquake Intensity resulted In increased nonlinearity primarily In the top several stories. The results indicate, however, that for a frame designed according to current seismic codes the increase in response and part iculart ly in the girder duct i l i ty demands would occur over the height of the frame. 148 As a r e s u l t o f lower energy d i s s i p a t i o n , i n c r e a s e d f l e x i b i 1 i t y , a n d d e g r a d i n g s t i f f n e s s and s t r e n g t h t y p i c a l o f b o t h , p r e s t r e s s e d c o n c r e t e f rames a p p e a r to be more s e n s i t i v e to d i f f e r e n t a c c e l e r a t i o n r e c o r d s than do comparab le r e i n f o r c e d c o n c r e t e f r a m e s . The shape o f the r e l a t i v e v e l o c i t y spec t rum in the range o f the f u n d a -menta l p e r i o d and the s i n u s o i d a l o r i m p u l s i v e n a t u r e o f t he g round mot ion a r e l i k e l y t o be o f c o n s i d e r a b l e Im-p o r t a n c e in d e t e r m i n i n g t h e re sponse o f an open m u l t i -s t o r y p r e s t r e s s e d c o n c r e t e f r ame. As the t a p e r o f t h e s t r u c t u r a l , and p a r t i c u l a r l l y the s t i f f n e s s p r o p e r t i e s o f a f rame i s r e d u c e d , the impor tance o f t h e h i g h e r modes in de te rm lng the re sponse i s reduced and the e f f e c t o f t h e h i g h e r f r e q u e n c i e s In an a c c e l e r o g r a m i s c o r r e s -p o n d i n g l y m i n i m i z e d . The r e s u l t s i n d i c a t e t h a t the g i r d e r d u c t i l i t y demands o f an open m u l t l - s t o r y p r e s t r e s s e d c o n c r e t e f rame d e s i g n e d w i t h the l a t e r a l l oads s p e c i f i e d by c u r r e n t s e i s m i c codes and s u b j e c t e d t o an E l C e n t r o i n t e n s i t y e a r t h q u a k e w i l l be on the o r d e r of s i x t o e i g h t . F o r a weak beam - s t r o n g column c o n f i g u r a t i o n the n o n l i n e a r a c t i o n w i l l o c c u r p r i m a r i l y i n the g i r d e r s w i t h the columns rema in ing e l a s t i c e x c e p t p o s s i b l y In the top s e v e r a l s t o r i e s . The r e s p o n s e o f a weak beam - s t r o n g column t ype mixed frame d e s i g n e d a c c o r d i n g t o c u r r e n t e a r t h q u a k e codes i s l i k e l y to be a lmos t I d e n t i c a l t o t h a t o f a comparab le p r e s t r e s s e d c o n c r e t e frame u n l e s s s e v e r e n o n l i n e a r i t y o c c u r s in the co lumns . CHAPTER 5 REVIEW OF CURRENT RECOMMENDATIONS FOR THE ASEISMIC DESIGN OF PRESTRESSED CONCRETE STRUCTURES 5.1 INTRODUCTION W h i l e p r o v i s i o n s r e l a t i n g t o the s e i s m i c d e s i g n o f s t e e l and r e i n f o r c e d c o n c r e t e s t r u c t u r e s have been embodied in the major b u i l d i n g codes f o r somet ime, i t i s o n l y i n the l a s t few y e a r s t h a t s i g n i f i c a n t p r o g r e s s has been made in d e v e l o p i n g s i m i l a r recommendat ions f o r the a s e i s m i c d e s i g n o f p r e s t r e s s e d c o n c r e t e s t r u c t u r e s . The h i s t o r i c a l r e s i s t a n c e o f many code a u t h o r s to a c c e p t p r e s t r e s s e d c o n c r e t e as a l a t e r a l l oad r e s i s t i n g m a t e r i a l has been spawned p r i m a r i l y by bo th a l a c k o f p r a c t i c a l e x p e r i e n c e c o n c e r n i n g the p re fo rmance o f p r e s t r e s s e d c o n c r e t e s t r u c t u r e s under e a r t h q u a k e l o a d i n g , and the c o n t i n u e d p a u c i t y o f e x p e r i m e n t a l and a n a l y t i c a l s t u d i e s p r o v i d i n g q u a n t i t a t i v e v a l u e s f o r the re sponse o f m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e b u i l d i n g s i n g e n -e r a l , and the d u c t i l i t y demands o f the c o n s t i t u e n t members i n p a r t i c u l a r . 14 R e c e n t l y , the FIP p u b l i s h e d g e n e r a l recommendat ions f o r the a s e i s m i c d e s i g n o f p r e s t r e s s e d c o n c r e t e s t r u c t u r e s , and the S e i s m i c Committee o f t h e New Z e a l a n d P r e s t r e s s e d C o n c r e t e I n s t i t u t e p u b l i s h e d d u c t i l e 19,^ ,45,46 d e t a i l e d r e c o m m e n d a t i o n s f o r the d e s i g n and d e t a i l i n g o f t i l e p r e s t r e s s e d c o n c r e t e f r a m e s . The New Z e a l a n d Load ing s Code r e p r e s e n t s the most s i g n i f i c a n t p r o g r e s s t o da te in the a c c e p t a n c e o f p r e s t r e s s e d c o n c r e t e as a l a t e r a l l oad r e s i s t i n g m a t e r i a l . Most c o d e s , 150 however, continue to e x p l i c i t l y or i m p l i c i t l y discourage or p roh ib i t the use of p res t ress ing to develop the d u c t i l e moment capacity required to r e s i s t l a t e r a l loadings. This chapter presents a b r i e f c r i t i q u e of current recommendations and code prov is ions fo r the aseismic design of m u l t i - s t o r y prest ressed concrete s t ructures in view of the resu l t s presented In the previous chapter . Several proposals r e l a t i n g to the current National Bu i ld ing Code o f Canada are made. The s p e c i f i c recommendations for the d u c t i l e 7 8 design of prest ressed concrete members have been discussed by others ' 14 15 ' and w i l l not be dealt with in th is chapter . 5.2 CRITERIA FOR CODE RECOMMENDATIONS The resu l t s presented in the previous chapter , together with 9 17 7 8 those of other s t u d i e s , i e . Spencer ' , Blakeley , and Thompson , suggest that the acceptance of prest ressed concrete as a l a t e r a l load r e s i s t i n g mater ia l may be j u s t i f i e d . The values obtained in th is study for the l a t e r a l displacements of the frame, and the d u c t i l i t y demands of the members were somewhat higher than those of a re in forced concrete frame with the same i n i t i a l s t i f f n e s s and s t rength . However, e x p e r l -7 8 mental resu l ts suggest that the d u c t i l i t y demands predicted fo r members of a prestressed concrete frame would be wi th in the d u c t i l e capacity of members designed according to the recommendations of the New Zealand Prest ressed Concrete I n s t i t u t e ' ^ . As such, the area of primary concern would appear to be not the question of the c a p a b i l i t y of an adequately designed prest ressed concrete frame to r e s i s t without f a i l u r e the severe se ismic loading that may occur once during the 151 b u i l d i n g s l i f e t i m e , but r a t h e r the q u e s t i o n o f s e r v i c e a b i l i t y o f t he s t r u c t u r e as a r e s u l t o f a moderate e a r t h q u a k e . Perhaps the p r i n c i p l e advantage o f the use o f p r e s t r e s s e d c o n c r e t e as a l a t e r a l l oad r e s i s t i n g m a t e r i a l i s the l i m i t a t i o n o f permanent d e f o r m a t i o n s due to e l a s t i c r e c o v e r y e f f e c t s . To a c h i e v e the c o n t i n u e d b e n e f i t s o f t h i s c h a r a c t e r i s t i c i t i s d e s i r e a b l e to l i m i t member damage under moderate s e i s m i c l o a d i n g to c r a c k i n g . The s i g n i f i c a n t l o s s o f g p r e s t r e s s s h o u l d c e r t a i n l y be a v o i d e d . Park and Thompson demons t ra ted the f e a s i b i l i t y o f r e p a i r i n g damaged beam-column suba s semb lage s , however, the r e p a i r o f damaged members o f p r e s t r e s s e d c o n c r e t e frames i s l i k e l y to be bo th c o s t l y and l e s s e f f e c t i v e . A f u r t h e r f a c t o r in regards t o the s e r v i c e a b i l i t y c r i t e r i a i s the p o s s i b l e damage to n o n s t r u c t u r a l w a l l s , i n f i l l p a n e l s , and o t h e r f i x t u r e s t h a t would r e s u l t f rom the h i g h e r d e f o r m a t i o n s t y p i c a l o f a p r e s t r e s s e d 25 c o n c r e t e f r a m e . In f a c t , a r e c e n t A I J p u b l i c a t i o n has e x p r e s s e d conce rn o v e r the p o s s i b l e i n c o m p a t a b i 1 i t i e s o f p r e s t r e s s e d c o n c r e t e f r a m i n g w i t h many o f the s t a n d a r d c u r t a i n f i n i s h e s a n d / o r o t h e r l a t e r a l load r e s i s t i n g mechanisms. 19 In v iew o f t h e s e and o t h e r f a c t o r s the s p e c i f i c a t i o n o f a h i g h e r base shea r f o r p r e s t r e s s e d c o n c r e t e s t r u c t u r e s appears w a r r a n t e d . 5.3 CODE BASE SHEAR PROVISIONS The NBC a l l o w s the use o f dynamic t ime s t e p a n a l y s e s i n the d e t e r m i n a t i o n o f the s e i s m i c d e s i g n loads p r o v i d i n g t h a t t h e h o r i z o n t a l g round a c c e l e r a t i o n used as i npu t Is not l e s s than t h a t s p e c i f i e d by the NBC. However, because o f a number o f f a c t o r s , i n c l u d i n g c o s t , h i g h e r 152 degree o f c o m p l e x i t y , and u n c e r t a i n t y in t he models f o r members and s t r u c t u r e s , l e v e l and t y p e o f damping, and a p p l i c a b l e ground mot ion f o r s i t e , e x c e p t f o r impor tan t b u i l d i n g s , most d e s i g n e r s choose t o use the e q u i v a l e n t s t a t i c a n a l y s i s a p p r o a c h . F o r tha t method the NBC e q u a t i o n f o r the base s h e a r r e s u l t i n g f r om s e i s m i c l o a d i n g i s e x p r e s s e d as V - ASKIFW where A » f a c t o r wh ich r e f l e c t s t h e h o r i z o n t a l g round a c c e l e r a t i o n f o r a p a r t i c u l a r a r e a . S • n u m e r i c a l c o e f f i c i e n t based on the fundamenta l n a t u r a l p e r i o d . K => a f a c t o r wh ich r e f l e c t s d i f f e r e n t s t r u c t u r a l sys tems and m a t e r i a l t y p e s . I = a c o e f f i c i e n t wh ich r e f l e c t s t h e impor tance o f a s t r u c t u r e . F = f o u n d a t i o n f a c t o r . W h i l e t h e r e i s some v a r i a t i o n In t h e f a c t o r s , the p h i l o s o p h y o f the UBC and NZLC base s h e a r s e i s m i c e q u a t i o n s i s the same as t h a t o f the NBC. However, an impor tan t and s i g n i f i c a n t d i f f e r e n c e in the NZLC as opposed t o the UBC o r NBC Is t he d i v i s i o n o f t h e K f a c t o r In to a s t r u c t u r a l t ype f a c t o r and a m a t e r i a l s f a c t o r . T h i s f a c i l i t a t e s r e -c o g n i t i o n o f the un ique c h a r a c t e r i s t i c s o f a m a t e r i a l , f o r example e l a s t i c and h y s t e r e t i c damping p r o p e r t i e s , i ndependent o f c o n s t r u c t i o n t y p e . The s t r u c t u r a l t ype f a c t o r in the NZLC v a r i e s f rom 0.8 f o r d u c t i l e f rames and the d u c t i l e c o u p l e d s h e a r w a l l s t o 2.5 f o r d i a g o n a l t e n s i o n b r a c i n g ' . The m a t e r i a l s f a c t o r v a r i e s f rom 0 .8 f o r s t r u c -t u r a l s t e e l t o 1.0 f o r r e i n f o r c e d c o n c r e t e and 1.2 f o r p r e s t r e s s e d 19 45 46 c o n c r e t e and masonry ' . The c h o i c e o f a m a t e r i a l s f a c t o r i s complex and i s based on bo th m a t e r i a l c h a r a c t e r i s t i c s and p re fo rmance 153 r e c o r d s under s e i s m i c l o a d i n g . In rega rd s t o t he m a t e r i a l s f a c t o r f o r p r e s t r e s s e d c o n c r e t e the NZLC s t a t e s : The v a l u e o f M ( m a t e r i a l s f a c t o r ) = 1.2 when used in p r e s t r e s s e d c o n c r e t e d u c t i l e f rames s h o u l d a t t h i s s t a g e be rega rded as t e n t a t i v e and s u b j e c t t o rev iew when s u f f i c i e n t re sponse a n a l y s e s o f m u l t i - s t o r y s t r u c t u r e s s u b j e c t e d t o e a r t h -quake mot ions have been made; the i n c r e a s e o f 20% above the v a l u e f o r r e i n f o r c e d c o n c r e t e i s i n t e n d e d to a l l o w f o r t he I n c rea sed re sponse o f p r e s t r e s s e d c o n c r e t e s t r u c t u r e s ' ^ * 1 * . The r e s u l t s o f t he n o n l i n e a r a n a l y s e s pe r f o rmed f o r t h i s s tudy sugges t t h a t the c h o i c e o f a m a t e r i a l s f a c t o r on the o r d e r o f 1.2 i s w a r r a n t e d . 5.4 VERTICAL DISTRIBUTION PROVISIONS The frame c o n s i d e r e d in t h i s s t udy was o r i g i n a l l y d e s i g n e d w i t h the l a t e r a l l oads a p p l i e d in a s t r a i g h t t r i a n g u l a r d i s t r i b u t i o n . The v a l u e s o b t a i n e d showed t h a t t he impor tance o f the h i g h e r modes due t o the s e v e r e t a p e r o f t he s t i f f n e s s p r o p e r t i e s r e s u l t e d i n the upper s t o r i e s o f t h e frame b e i n g p a r t i c u l a r l y s e n s i t i v e t o the " w h i p l a s h " a f f e c t , more so than the comparab le r e i n f o r c e d c o n c r e t e v e r s i o n . Modern s e i s m i c codes s p e c i f y t h e a p p l i c a t i o n o f a p o r t i o n o f the base s h e a r as a c o n c e n t r a t e d f o r c e a t t he top o f t h e s t r u c t u r e f o r s l e n d e r b u i l d i n g s . T h i s would reduce t o a s i g n i f i c a n t degree the h i g h d u c t i l i t y demands and e x c e s s i v e d e f o r m a t i o n s In the upper p o r t i o n o f the f rame c o n s i d e r e d . The h i g h e r m a t e r i a l s f a c t o r f o r p r e s t r e s s e d c o n c r e t e s h o u l d a c c o u n t f o r the I n c r e a s e d tendancy t o s u f f e r f rom w h i p -l a s h e f f e c t s . 154 5.5 CONCLUSIONS On the b a s i s o f t h i s s t u d y , and bo th s t u d i e s by o t h e r s 7 ' * * ' 9 , 1 7 15 19 44 45 46 and in v iew o f o t h e r code p r o v i s i o n s , s e v e r a l recom-mendat ions in the r e g a r d t o the c u r r e n t NBC a r e made. 1. That p r e s t r e s s i n g be e x p l i c i t l y a c c e p t e d as a means o f d e v e l o p i n g the d u c t i l e moment c a p a c i t y o f members o f bo th d u c t i l e moment r e s i s t i n g f rame - s hea r w a l l systems (K = 0.8) where the frame i s d e s i g n e d to c a r r y t w e n t y - f i v e p e r c e n t o f the l a t e r a l l o a d , and d u c t i l e moment r e s i s t i n g f rames where the frame p r o v i d e s most o r a l l o f t h e r e s i s t a n c e t o l a t e r a l loads (K - 0.8). 2. That the most e f f i c i e n t method o f a c c o u n t i n g f o r the un ique c h a r a c t e r i s t i c s o f p r e s t r e s s e d c o n c r e t e i n p a r t i c u l a r , and a d d i t i o n a l l y , r e i n f o r c e d c o n c r e t e and s t r u c t u r a l s t e e l wou ld be t o r e p l a c e the c u r r e n t K f a c t o r w i t h a m a t e r i a l s f a c t o r and a s t r u c t u r a l t ype f a c t o r s i m i l a r t o the NZLC. 3. That as a means o f l i m i t i n g d e f l e c t i o n s and r e d u c i n g both member and n o n s t r u c t u r a l damage, the s p e c i f i c a t i o n o f a h i g h e r m a t e r i a l s f a c t o r t o a ccoun t f o r the i n c r e a s e d re sponse appear s w a r r a n t e d . A v a l u e on the o r d e r o f 1.2 wou ld seem r e a s o n a b l e . 155 CHAPTER 6 CONCLUSIONS 6.1 SUMMARY OF RESULTS The c o n c l u s i o n s reached as a r e s u l t o f t h i s s tudy a r e d e t a i l e d a t the end o f each c h a p t e r . A summary Is p r e s e n t e d be low: An i d e a l i z e d model f o r the end m o m e n t - p l a s t i c r o t a t i o n r e l a t i o n -s h i p o f p r e s t r e s s e d c o n c r e t e members was d e v e l o p e d based on a pub-l i s h e d 7 ' ^ moment - curva tu re i d e a l i z a t i o n f o r p r e s t r e s s e d c o n c r e t e . The i d e a l i z e d m o m e n t - r o t a t i o n m o d e l , wh ich i n c l u d e d s t i f f n e s s and s t r e n g t h d e g r a d a t i o n , was used to i n t r o d u c e a l l p o s t - e l a s t i c a c t i o n i n a beam-column e l e m e n t , wh ich c o n s i s t e d o f an e l a s t i c beam c o n n e c t i n g c o n c e n t r a t e d h inge s modeled as n o n l i n e a r r o t a t i o n a l s p r i n g s . It would be e x p e c t e d tha t the a c c u r a c y o f the i d e a l i z e d m o m e n t - r o t a t i o n model wou ld be a f f e c t e d by member p r o p e r t i e s . For examp le , one would e x p e c t the amount o f l o n g i t u d i n a l and t r a n s v e r s e deformed b a r r e i n -f o r c i n g , l e v e l of a x i a l l o a d , and degree o f p r e s t r e s s i n g t o be s i g n i -f i c a n t f a c t o r s in d e t e r m i n g the h y s t e r e t i c re sponse o f a member. However, f o r members d e t a i l e d a c c o r d i n g t o the c u r r e n t recommenda t i on s 7 8,14,15 f o r d u c t i l e d e s i g n o f p r e s t r e s s e d c o n c r e t e , the model s h o u l d p r o v i d e adequate s o l u t i o n a c c u r a c y . It s h o u l d be kept In mind tha t the model i s d e s i g n e d to a c c o u n t f o r the energy d i s s i p a t i o n , and d e g r a d i n g s t i f f n e s s and s t r e n g t h p r o p e r t i e s o f p r e s t r e s s e d c o n c r e t e 156 members as opposed t o p r e d i c t i n g the exac t re sponse o f a p a r t i c u l a r b u i I d i n g . The p r e s t r e s s e d c o n c r e t e beam-column e lement was s u b s e q u e n t l y used in the n o n l i n e a r a n a l y s i s o f a t y p i c a l m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e f rame. It was found t h a t the l a t e r a l d i s p l a c e m e n t o f t h e top o f the frame f o r the p r e s t r e s s e d c o n c r e t e v e r s i o n were up t o f i f t y - f i v e p e r c e n t g r e a t e r than t h o s e o f a r e i n f o r c e d c o n c r e t e v e r s i o n w i t h the same i n i t i a l s t i f f n e s s and s t r e n g t h . In p r a c t i c e , a p r e -s t r e s s e d c o n c r e t e frame i s l i k e l y t o have s m a l l e r members, and thus be more f l e x i b l e than a r e i n f o r c e d c o n c r e t e frame d e s i g n e d f o r the same l o a d i n g . T h i s e f f e c t was p a r t i a l l y a c c o u n t e d f o r in the lower v i s c o u s damping r a t i o a s s i g n e d the p r e s t r e s s e d c o n c r e t e v e r s i o n . The r e s u l t s o f t h i s s t udy i n d i c a t e tha t t he g i r d e r end r o t a t i o n a l d u c t i l i t y demands f o r members o f a p r e s t r e s s e d c o n c r e t e f rame d e s i g n e d a c c o r d i n g to c u r r e n t NBC o r UBC s e i s m i c codes and s u b j e c t e d t o an E l C e n t r o i n t e n s i t y e a r t h q u a k e w i l l be on the o r d e r o f s i x t o e i g h t . T h i s i s somewhat h i g h e r than the v a l u e o f f i v e wh i ch an e a r l i e r s t udy by C lough and B e n u s k a ' 0 s ugge s ted would be the g i r d e r d u c t i l i t y r e q u i r e -ments f o r a r e i n f o r c e d c o n c r e t e f rame s u b j e c t e d t o the same l o a d i n g . The r e s u l t s f u r t h e r show t h a t f o r a weak beam-s t rong column c o n f i g u r a -t i o n , the columns o f an a d e q u a t e l y d e s i g n e d p r e s t r e s s e d c o n c r e t e frame s h o u l d remain e l a s t i c , e x c e p t p o s s i b l y in the top s e v e r a l s t o r i e s . The e f f e c t o f s e v e r a l pa rameter s on the re sponse o f t he p r e s t r e s s e d c o n c r e t e f rame c o n s i d e r e d was i n v e s t i g a t e d . R e s u l t s i n d i c a t e d t h a t b e -cause o f the lower p e r c e n t a g e o f v i s c o u s damping, and the lower h y s t e r e -t i c energy d i s s i p a t i o n c h a r a c t e r i s t i c o f p r e s t r e s s e d c o n c r e t e , a p r e s t r e s -sed c o n c r e t e f rame w i l l be more s e n s i t i v e to the t a p e r o f t h e s t i f f n e s s 157 p r o p e r t i e s than a comparab le r e i n f o r c e d c o n c r e t e f rame. Severe t a p e r o f the s t i f f n e s s p r o p e r t i e s o f a f rame leads to a s i g n i f i c a n t i n c r e a s e in the impor tance o f the h i g h e r modes in d e t e r m i n i n g the r e s p o n s e . T h i s i s l i k e l y t o lead to both l a r g e i n t e r - s t o r y and l a t e r a l d e f o r m a t i o n s , and e x c e s s i v e g i r d e r d u c t i l i t y demands f o r m u l t i - s t o r y p r e s t r e s s e d c o n -c r e t e f r a m e s . Because o f the unbounded n a t u r e o f s t i f f n e s s p r o p o r t i o n a l damping, i t s use may l ead t o u n r e a l i s t i c r e d u c t i o n s i n r e sponse when t h e h i g h e r modes a r e s i g n i f i c a n t . As s u c h , the use o f mass p r o p o r t i o n a l damping i s recommended. The r e s u l t s o b t a i n e d sugges t t h a t the s i n u s o i d a l o r i m p u l s i v e n a t u r e o f an a c c e l e r o g r a m , and the shape o f r e l a t i v e v e l o -c i t y spec t rum in the p e r i o d range at and immed ia te l y beyond the fundamen-t a l p e r i o d a r e l i k e l y to be o f c o n s i d e r a b l e impor tance i n d e t e r m i n i n g the re sponse o f a d u c t i l e m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e moment r e s i s -t i n g f r ame. As the Importance o f the h i g h e r modes i s r e d u c e d , the e f f e c t o f the f r e q u e n c y c o n t e n t o f a g i v e n ground mot ion was f ound to be c o r r e s p o n d i n g l y r e d u c e d . On the b a s i s o f t h i s s t u d y , and s t u d i e s by o t h e r s 7 ' * * ' 9 , ' 7 i t was c o n c l u d e d t h a t a m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e frame d e s i g n e d a c c o r d -ing to c u r r e n t NBC o r UBC s e i s m i c codes c o u l d r e s i s t w i t h o u t f a i l u r e the s e v e r e s e i s m i c l o a d i n g t h a t may o c c u r once d u r i n g the b u i l d i n g s l i f e t i m e . However, as a method o f l i m i t i n g e x c e s s i v e l a t e r a l and i n t e r -s t o r y d e f o r m a t i o n s , and member damage under moderate s e i s m i c l o a d i n g i t was sugge s ted t h a t the s p e c i f i c a t i o n o f a h i g h e r base s h e a r f o r p r e s t r e s -sed c o n c r e t e appear s w a r r a n t e d . 158 6.2 RECOMMENDATIONS FOR FURTHER STUDY In the c o u r s e o f t h i s i n v e s t i g a t i o n i t has been e v i d e n t t h a t f u r t h e r r e s e a r c h Is needed in s e v e r a l a reas r e g a r d i n g the s e i s m i c r e s i s t a n c e o f p r e s t r e s s e d c o n c r e t e . These i n c l u d e : 1. A d d i t i o n a l n o n l i n e a r dynamic t ime s t e p a n a l y s e s o f m u l t i -s t o r y p r e s t r e s s e d c o n c r e t e frames u s i n g b u i l d i n g s d e s i g n e d in a c c o r d a n c e w i t h the c u r r e n t and recommended code p r o -v i s i o n s . To e s t a b l i s h the s e n s i t i v i t y o f p r e s t r e s s e d c o n -c r e t e f rames to g round mot ion t y p e and i n t e n s i t y i t i s recommended t h a t a c c e l e r o g r a m s o t h e r than those used i n t h i s s tudy be i n c l u d e d as i n p u t . 2. E x p e r i m e n t a l a n d / o r a n a l y t i c a l i n v e s t i g a t i o n s to de te rm ine a p p r o p r i a t e e x p r e s s i o n s f o r the e q u i v a l e n t p l a s t i c h i n g e l eng th s f o r p r e s t r e s s e d c o n c r e t e members. 3. E x p e r i m e n t a l and a n a l y t i c a l s t u d i e s on the e l a s t i c damping and energy d i s s i p a t i o n c h a r a c t e r i s t i c s o f p r e s t r e s s e d c o n -c r e t e members. A. Shak ing t a b l e t e s t s on m u l t i - s t o r y p r e s t r e s s e d c o n c r e t e f rames to d e t e r m i n e t h e i r p re fo rmance under e a r t h q u a k e l o a d i n g . 159 L IST OF REFERENCES 1. De spey roux , J . , " T h e Use o f P r e s t r e s s e d C o n c r e t e In E a r t h q u a k e - R e s i s t a n t D e s i g n , " P r o c e e d i n g s , 3WCEE, New Z e a l a n d , Jan I965, V o l . I l l , pp. IV-203-215. 2. Guyon, Y . , " E n e r g y A b s o r p t i o n o f P r e s t r e s s e d C o n c r e t e Beams, " P r o c e e d i n g s , 3WCEE, New Z e a l a n d , Jan 1965, V o l . I l l , pp. IV-216-223. 3. L i n , T . Y . , " D e s i g n o f P r e s t r e s s e d C o n c r e t e B u i l d i n g s f o r Ea r thquake R e s i s t a n c e , " J o u r n a l o f t he S t r u c t u r a l D i v i s i o n , ASCE, V o l . 91, No. ST5, O c t , 1965, pp. 1-17-4. R o s e n b l u e t h , E . , " D i s c u s s i o n o f R e f e r e n c e 3; J o u r n a l o f the S t r u c t u r a l D i v i s i o n , ASCE, V o l . 92, No. ST1. 5. Nakano, K., " E x p e r i m e n t s on B e h a v i o r o f P r e s t r e s s e d C o n c r e t e F o u r - s t o r y Model S t r u c t u r e on L a t e r a l F o r c e , " P r o c e e d i n g s , 3WCEE, New Z e a l a n d , Jan 1965, V o l . I l l , pp. IV-572-590. 6. P r i e s t l e y , M . J . N . , "Moment R e d i s t r i b u t i o n i n P r e s t r e s s e d C o n c r e t e Cont inuous Beams, " Phd T h e s i s , U n i v e r s i t y o f C a n t e r b u r y , New Z e a l a n d , I966, I89 pp. 7. B l a k e l e y , R.W.G., " D u c t i l i t y o f P r e s t r e s s e d C o n c r e t e Frames Under S e i s m i c L o a d i n g , " Phd T h e s i s , U n i v e r s i t y o f C a n t e r b u r y , New Z e a l a n d , 1971, 230 pp . 8. Thompson, Kev in J . , and P a r k , R., " C y c l i c Load T e s t s on P r e s t r e s s e d and P a r t i a l l y P r e s t r e s s e d Beam-Column J o i n t s , " J o u r n a l o f the PC I , V o l . 22, No. 5, S e p t / O c t 1972, pp. 84-110. 9. S p e n c e r , R .A . , " T h e N o n - l i n e a r Response o f a M u l t i - s t o r y P r e s t r e s s e d C o n c r e t e S t r u c t u r e t o Ea r thquake L o a d i n g , " P r o c e e d i n g s , 4WCEE, C h i l e , Jan I969, V o l . 11, pp. A4-139-154. 10. C l o u g h , R.W., Benuska , K . L . , and T . y . L i n and A s s o c . , "FHA Study o f S e i s m i c Des ign C r i t e r i a f o r High R i s e B u i l d i n g s , " U.S. Dept . o f Hous ing and Urban Deve lopment , F e d e r a l Hous ing A d m i n i s t r a t i o n , W a s h i n g t o n , D . C , Aug I966, 347 pp. 11. B l a k e l e y , R.W.G., and P a r k , R., " P r e s t r e s s e d C o n c r e t e S e c t i o n s w i t h C y c l i c F l e x u r e , " J o u r n a l o f the S t r u c t u r a l D i v i s i o n , ASCE, V o l . 99, No. ST8, Aug 1973, pp. 1717-1742. 12. B l a k e l e y , R.W.G., and Pa rk , R., "Response o f P r e s t r e s s e d C o n c r e t e S t r u c t u r e s to Ea r thquake M o t i o n s , " New Z e a l a n d E n g i n e e r i n g , New Z e a l a n d , Feb 1973, pp 42-54. 160 13. B l a k e l e y , R.W.G., P a r k , R., and S h e p a r d , R., "A Review o f the S e i s m i c R e s i s t a n c e o f P r e s t r e s s e d C o n c r e t e , " B u l l e t i n , New Z e a l a n d S o c i e t y f o r Ea r thquake E n g i n e e r i n g , V o l . 3, No. 1, March 1970, pp 3-23. ]k. "Recommendat ions f o r the Des i gn o f A s e i s m i c P r e s t r e s s e d C o n c r e t e S t r u c t u r e s , " F e d e r a t i o n I n t e r n a t i o n a l e de l a P r e c o n t r a i n t e , Nov 1977. 15. S e i s m i c Committee o f t he New Z e a l a n d P r e s t r e s s e d C o n c r e t e I n s t i t u t e , "Recommendations f o r the Des i gn and D e t a i l i n g o f D u c t i l e P r e s t r e s s e d C o n c r e t e Frames f o r S e i s m i c L o a d i n g , " B u l l e t i n o f the New Z e a l a n d N a t i o n a l S o c i e t y f o r Ea r thquake E n g i n e e r i n g , V o l . 9, No. 2, June 1976, pp 89-96. 16. Hawkins , N e i l M . , " S e i s m i c R e s i s t a n c e o f P r e s t r e s s e d and P r e c a s t C o n c r e t e S t r u c t u r e s , " PCI J o u r n a l , V o l . 22, No. 6, Nov/Dec 1977. pp 80-110. 17. S p e n c e r , R .A . , "The Ea r thquake Response o f P r e s t r e s s e d C o n c r e t e S t r u c t u r e s w i t h N o n - s t r u c t u r a l I n t e r f l o o r E l e m e n t s , " P r o c e e d i n g s o f the F i r s t Canad ian C o n f e r e n c e on Ea r thquake E n g i n e e r i n g , Canada, May 1971, pp 354-366. 18. Z a v r i e v , K .S . , "Dynamics and Ea r thquake R e s i s t a n c e o f P r e s t r e s s e d R e i n f o r c e d C o n c r e t e S t r u c t u r e s , " P r o c e e d i n g s , 3WCEE, New Z e a l a n d 1965, V o l . 11, pp IV-645. 19- B u t c h e r , G.W., " T h e New Zea l and Load ings Code and I t s A p p l i c a t i o n t o the Des i gn o f S e i s m i c R e s i s t a n t P r e s t r e s s e d C o n c r e t e S t r u c t u r e s , " B u l l e t i n o f the New Z e a l a n d N a t i o n a l S o c i e t y f o r Ea r thquake E n g i n e e r i n g , V o l . 9, No.3, Sept 1976, pp 162-165. 20. P e n z l e n , J . , "Damping C h a r a c t e r i s t i c s o f P r e s t r e s s e d C o n c r e t e , " J o u r n a l o f the A . C . I . , V o l . 61, No. 9, Sept 1964, pp 1125-1147. 21. S p e n c e r , R .A. , " S t i f f n e s s and Damping o f N ine C y c l i c a l l y Loaded P r e s t r e s s e d Members , " J o u r n a l o f the P . C . I . , V o l . 14, No. 3, June I969, PP 39-52. 22. B r o n d u m - N l e l s e n , T . , " E f f e c t o f P r e s t r e s s on the Damping o f C o n c r e t e , " R e s i s t a n c e and U l t i m a t e Deformabi11ty of S t r u c t u r e s A c t e d on by W e l l D e f i n e d Loads , IABSE, L i s b o n , 1973. 23. Candy, C . F . , D i s c u s s i o n o f R e f e r e n c e I; P r o c e e d i n g s , 3WCEE, New Z e a l a n d , Jan 1965, V o l . 3. 24. K e n t , D . C , " I n e l a s t i c B e h a v i o u r o f R e i n f o r c e d C o n c r e t e Members w i t h C y c l i c L o a d i n g , " Phd T h e s i s , U n i v e r s i t y o f C a n t e r b u r y , New Z e a l a n d , 1967> 246 pp. 161 25. A r c h i t e c t u r a l I n s t i t u t e o f J a p a n , " D e s i g n E s s e n t i a l s i n Ear thquake R e s i s t a n t B u i l d i n g s , " E l s e v i e r Company, 1970, 295 pp. 26. Inomata, E . S . , " C o m p a r a t i v e S tudy on B e h a v i o u r o f P r e s t r e s s e d and R e i n f o r c e d C o n c r e t e Beams S u b j e c t t o Load ing R e v e r s a l s , " PCI J o u r n a l , V o l . 16, No. 1, J a n / F e b , 1971, pp 21-37. 27. F . I . P . - C . E . B . , "Recommendations P r a t i q u e s Pour l e C a l c u l e t L ' E x e c u t i o n des Ouvrages en Beton P r e c o n t r a i n t , " ( P r a c t i c a l recommendat ions f o r the d e s i g n and c o n s t r u c t i o n o f p r e -s t r e s s e d c o n c r e t e s t r u c t u r e ) , Comite Europeen du B e t o n , B u l l e t i n D * I n f o r m a t i o n , No. 54, March 1966 ( i n F r e n c h ) . 28. Kunze, W.E., S b a r o u n i s , J . A . , and A m r h e i n , J . E . , " B e h a v i o u r o f P r e s t r e s s e d C o n c r e t e S t r u c t u r e s Dur ing the A l a s k a n E a r t h q u a k e , " PCI J o u r n a l , V o l . 10, No. 2, A p r i l I965, pp 80-91. 29. Inomata, E . S . , "A Report on the B e h a v i o u r o f S t r u c t u r e s Employ ing P r e s t r e s s e d C o n c r e t e D u r i n g N l i g a t a E a r t h q u a k e , " J o u r n a l o f Japan P r e s t r e s s e d C o n c r e t e E n g i n e e r i n g A s s o c i a t i o n , V o l . 6, No. 5, Oct 1964, pp 38-42. 30. F i n t e l , M . , " B e h a v i o u r o f S t r u c t u r e s in t he Caraca s E a r t h -q u a k e , " C i v i l E n g i n e e r i n g , ASCE, Feb I968, pp 42-46. 31. B e a r g , C . V . , " T h e S k o p j e , Y u g o s l a v i a E a r t h q u a k e , " Amer ican Iron and S t e e l I n s t i t u t e , 1964, pp 78. 32. S u t h e r l a n d , W.M., " P r e s t r e s s e d C o n c r e t e Ea r thquake R e s i s t a n t S t r u c t u r e s - Deve lopment , P e r f o r m a n c e , and C u r r e n t R e s e a r c h , " P r o c e e d i n g s , 3WCEE, New Z e a l a n d , Jan 1965, V o l . I l l , pp IV-463-494. 33. S h e r b o u r n e , A . N . , and Parameswar, H . C . , " L i m i t A n a l y s i s o f Cont inuous P r e s t r e s s e d Beams, " J o u r n a l o f t he S t r u c t u r a l D i v i s i o n , ASCE, V o l . 94, No. ST1, Jan I968, pp 19-40. 34. Paranagma, D .D.O. , and Edwards, A . D . , "Moment D e f o r m a t i o n C h a r a c t e r i s t i c s o f P r e s t r e s s e d C o n c r e t e Beams S u b j e c t t o F l u c t u a t i n g L o a d s , " : PCI J o u r n a l , V o l . 14, No. 4, Aug I969, pp 62-74. 35. C o r l e y , W.G., " R o t a t i o n a l C a p a c i t y o f R e i n f o r c e d C o n c r e t e Beams, " J o u r n a l o f the S t r u c t u r a l D i v i s i o n , ASCE, V o l . 92, No. ST3, Oct 1963, PP 121-146. 36. C l o u g h , R.W., and P e n z i e n , S . , "Dynamics o f S t r u c t u r e s , " M c G r a w - H i l l , 1975, 634 pp . 37. B i g g s , J o h n , " I n t r o d u c t i o n t o S t r u c t u r a l D y n a m i c s , " M c G r a w - H i l l . 162 38. B lume, John A . , Newmark, N.M., and C o r n i n g , L . H . , " D e s i g n o f M u l t i - s t o r y R e i n f o r c e d C o n c r e t e B u i l d i n g s f o r Ea r thquake M o t i o n s , " PCA, 1961 , 318 pp. 39- Kanaan, Amin E . , and P o w e l l , Graham H . , "DRAIN-2D U s e r ' s Guide and S u p p l e m e n t , " Report No. EERC 73-6 and EERC 73-22, Ear thquake E n g i n e e r i n g Research C e n t r e , C o l l e g e o f E n g i n e e r -i n g , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y . 40. W i l s o n , E . L . , "Method o f A n a l y s i s f o r the E v a l u a t i o n o f F o u n d a t i o n - S t r u c t u r e I n t e r a c t i o n , " P r o c e e d i n g s , 4WCEE, C h i l e , Jan I969, pp A-6 , 87-99. 41. G i b e r s o n , M . F . , " The Response o f N o n l i n e a r M u l t i - s t o r y S t r u c t u r e s S u b j e c t e d to Ea r thquake E x c i t a t i o n , " Phd T h e s i s , C a l i f o r n i a I n s t i t u t e o f T e c h n o l o g y , C a l i f o r n i a , 1967, 232 pp. 42. F i n t e l , Mark and Ghosh, S .K . , " E f f e c t o f W a l l S t r e n g t h on the Dynamic I n e l a s t i c S e i s m i c Response o f Y i e l d i n g W a l l - E l a s t i c Frame I n t e r a c t i v e S y s t e m s , " P r o c e e d i n g s , T h i r d Canad ian C o n f e r e n c e on Ea r thquake E n g i n e e r i n g , M o n t r e a l , June 1979, V o l . 2, pp 949-968. 43. Inomata, S . , " A s p e c t s o f t h e FIP Recommendations f o r Des i gn o f A s e i s m i c P r e s t r e s s e d C o n c r e t e S t r u c t u r e s , " B u l l e t i n o f t he New Z e a l a n d N a t i o n a l S o c i e t y f o r Ea r thquake E n g i n e e r i n g , Sept 1976, V o l . 9, No. 3, pp 159-161. 44. Andrews, A . L . , " D e s i g n o f S e i s m i c R e s i s t a n t P r e s t r e s s e d C o n c r e t e S t r u c t u r e s , " B u l l e t i n o f the New Z e a l a n d N a t i o n a l S o c i e t y f o r Ea r thquake E n g i n e e r i n g , S e p t . 1976, V o l . 9, No. 3, PP 175-180. 45. K o l s t o n , D., " N o t e s in the New Z e a l a n d Ea r thquake Load ing P r o v i s i o n s , " B u l l e t i n o f t he New Z e a l a n d N a t i o n a l S o c i e t y f o r Ea r thquake E n g i n e e r i n g , June 1975, V o l . 8, No. 2, pp 102-113. 46. "Code o f P r a c t i c e f o r Genera l S t r u c t u r a l Des i gn and Des i gn Load ings f o r B u i l d i n g s , " NZS 4203, S t andard s A s s o c i a t i o n o f New Z e a l a n d , Feb 1976. APPENDIX A COMPUTER PROGRAMS A. I COMPUTER FACILITIES During the time th is thes is was in preparat ion the main academic computer operated by the U .B .C . Computer Centre was an AMDAHL 470 V/6 Model II computer which ran under the Michigan Terminal System. Core equipment included a Central Processing Unit with four m i l l i o n bytes of s torage, two PDP-8 computers fo r equipment c o n t r o l , and three PDP-11 which operated as f ront end communications computers. Per ipheral storage capacity included a f i xed head storage u n i t , twenty d isk dr ives capable of s t o r i n g one hundred megabytes each, and seven 9 - t r a c k tape d r i v e s . P l o t t i n g f a c i l i t i e s cons is ted of two PDP-8 c o n t r o l l e d Calcomp 563 29 inch p l o t t e r s . A .2 PROGRAM MREL As descr ibed in Chapter Three the program MREL was wr i t ten to determine t h e o r e t i c a l end moment-hinge ro ta t ion curves fo r p r e -s t ressed concrete members subjected to any of three end moment pat -terns . The ana l ys i s was based on B l a k e l e y ' s ^ i d e a l i z e d moment curvature model. The three end moment patterns considered were 164 e q u a l end moments p r o d u c i n g r e v e r s e d c u r v a t u r e , moment at one end o n l y , and u n i f o r m moment. The input r e q u i r e d i s the i n c r e m e n t a l end moments t o p roduce the d e s i r e d , i e , mono ton i c o r c y c l i c , l o a d i n g p a t t e r n . A l i s t i n g i s g i v e n be low. I M P L I C I T R E A L ' 8 (A-H,0-7.) D I M E N S I O N KODY ( J 0 0 ) , B N T T ( 3 0 0 ) . C I I T T ( 3 0 0 ) , B M T ( 3 0 0 ) , ROT (JOO) DI MENS I ON B H I C ( 3 0 0 ) , C U I C ( 3 0 0) , EKCO ( J 0 0 ) , E K I C ( 3 0 0 ) , B K K P ( 3 ) , D U M ( 3 0 0 ) DI R E N S I O N K F A C ( 3 0 0 ) , TDK ( 3 0 0 ) , IND ( 3 0 0 ) , B T N ( 3 0 0 ) , DOT ( 3 0 0 ) , CUT ( 3 0 0 ) D I M E N S I O N O f l T I ( 3 0 0 ) , U H T 2 ( 3 0 0 ) , B H C R ( 2 ) , C U C R ( 2 ) , B H C I I ( 2 ) ,CUCU(2) D I N E N S I O N BRTD ( 3 0 0 ) , C 0 T D (3 0 0 ) , C O I P ( 3 0 0 ) , E K P ; ( JO 0) , EK I H ( JO 0 ) DI HENS I O N B H U L T ( 3 0 0 , 2) , E K H J ( 3 0 0 ) , E K J P ( 3 0 0 ) , EKGA ( 3 0 0 ) , E K 3 L ( J 0 O | D I M E N S I O N 3 U E L ( 3 0 0 ) , D U B A I ( 2 ) C Z I N I T I A L I Z E P A B A N E T E B S C EKH= 1 . 00 B N C R ( 1) = 1 . 0 0 B B C R ( 2 ) DO C U C B (1) = 1 . 0 0 C U C B (2) = - 1 . DO B M C U ( 1 ) = 1 . 8 D 0 BMCU (2) = - 1 . 8 D 8 C U C U ( I ) = 6 . 3 3 3 3 3 3 3 C U C U ( 2 ) = - 6 . 3 3 J 3 3 3 3 B N H A X I 1) * 1 . 8 D 0 BHHAX ( 2 ) = - 1 . 8 D 0 E K E P ( 1 ) =EKH E K B P ( 2 ) = 0 . 1 5 D 0 * E K H M B I T E < 8 , 6 ) 6 P O B N A T ( 2 9 H B E A R (1) OB COL DUN ( 0 ) E L E M E N T ) C A L L P B E A D C G O S E B ' , M I : ' , N E 1 T ) I P ( N E L T . E Q . O ) J O TO 7 E K E P ( J ) = - 0 . 0 0 3 D O * E K H GO TO 8 7 B K E P ( 3 ) = - 0 . 0 0 9 D 0 * E K H 8 C O N T I R U e M R I T E ( 8 . 4 | 4 P O R B A T ( 5 I , 5 H N S T E P , 4 I , l ( H N I H C , 4 X , « H N B C R , 4 X , U H N N H L ) C A L L F B E A D ( ' G t J S B B ' , ' 4 1 : ' , N S T E P , N I N C . N H C B . N N H L ) HB I T E ( 8 , 9 ) 9 P O B H A T ( / / 1 2 X , 2 9 H MOMENT END C O N D I T I O N S . T Y P E : / / 1 4 4 H EQDAL END 110(1 E N T S - R E V E R S E D C U H V A T O K E = 1 / 2 4 4 H B 3 R E N T AT ONE END E Q U A L S Z E R O = 2 / J 4 4 H EQUAL END BON E N T S - C O N T I N U O U S C U R V A T U R E S ) C A L L P R R A D ( ' G U S E R ' , • 1 1 : • , K « O B ) C C R E A D I V I H C H E R B H T A L MOMENTS C DO 10 I = 1 , | | S T E P B E A D ( 5 , 1 1 ) D B H ( I ) 11 P O B H A T ( P 1 0 . J) 10 C O N T I N U E C C I N T I A L I Z E A R R A Y S C DO 2 3 1 = 1 , N S T E P B N T ( I ) = 0 . 0 0 faOT ( I ) = 0 . DO 2 3 C O I I T I N O E 0 0 2<4 I = 1,NINC B M I C ( I ) = 0 . 0 0 C U I C ( I ) = 0 . DO C U E L ( I | = 0 . 0 0 B B T T ( I ) = 0 . DO C U T T ( I ) = 0 . DO BHTD ( I ) = 0 . D 0 C U T D ( I ) - 0 . DO KODY ( I ) - 0 E K I C ( I ) = 0 . DO E K C D ( I ) = 0 . 0 0 I N D ( I | = 1 B H U L T ( I , 1| =BBCU (1) B R U L T ( 1 , 2 ) =BHCU (2) U H T 1 ( I ) = 0 . 0 0 UHT2 ( I ) = 0 . DO K F A C ( I ) =0 24 C O K T I N O E C P C B P = 0 . DO P A C A C = 0 . D 0 K B A L = 0 . DO GO T O ( 1 4 , 1 5 , 1 5 ) , K H O B 14 01 M C = O . 5 D 0 / B I B C GO TO 16 15 D I H C = 1 . 0 D 0 / H I H C 16 C O V T I N D E C C T B A C E OOT N O N L I N E A R P A T H C 2 7 DO 4 3 0 K = 1 , N S T E P B O I = 0 . D O DO 4 2 5 I = 1 , M I N C D I N C L = D I N C * ( 1 - 1 ) I P ( I N D ( I ) . E Q . 2 . 0 R . B M I C ( I ) . L T . O . ) SO TO 2 8 I F ( O B H ( K ) . L T . O . . A N D . KODY ( I ) . E Q . 2) 3 0 TO 2 9 K F A C ( I ) = 0 . GO TO 31 2 9 K P A C ( I ) = 1 . 31 I F (DBH (K) . L T . D . DO. A N D . K O D I ( N R H L ) . E Q . 2 ) P C B F = 0 . D 0 DHB=DBH (K) I F (DBH (K) . G T . 1 . DO. A N D . P C B P . B Q . 1 . DO) D 8 B = - D B H (K) GO TO J 4 28 I F (DBB (K) . G T . D . DO. A N D . KODI ( I ) . E Q . 2) GO TO 3 2 K F A C ( I ) =0 GO TO 3 3 3 2 K P A C ( I ) = 1 33 C O N T I N U E I F (DBB (K) . 3 T . 0 . D 0 . A B D . KODY ( N N H L ) . E Q . 2 ) P C B F = 0 . DO DHB=DBN (K) I F (DBB (K) . L T . O . DO. A N D . P C B P . E Q . 1 . DO) D B B - - D B H ( K ) J 4 I F ( K . E Q . 1) GO TO 3 5 B H L = B H T ( K - 1 ) GO TO 3 6 J 5 B R L = 0 . 3 6 GO TO ( 4 1 , 4 2 , 4 3 ) , K N O H 4 1 B H I C ( I ) = ( B « l * ( O I N C L » 0 . 5 D 0 * D I N C ) ) / O . 5 D 0 D B I C = ( D N B * ( D I N C L » 0 . 5 D 0 » D I N C ) ) / 0 . 5D0 GO TO 4 4 4 2 B R I C ( I ) = ( B H L * ( D I N C L * 0 . 5 D 0 * D I N C ) ) D B I C = ( 0 R B * ( D I N C L t O . 5 D 0 « D I N C ) ) GO TO 4 4 4 1 B B I C ( I ) - B N L O B I C = D R B 4 4 CONTINUE 30 PACTOR=0.99999999-FACAC IF(DBIC.EQ.O.DO) SO T O 415 IF (KPAC (I) . EQ. f | G O T3 170 C C S E T REVERSAL HONENT IMDICATOR C REMIHD= 1. IP (IND (I) . EQ. 2) BEHIND=-1. C C RULE ONE. ELASTIC STAGE. GET FACTOR F O R CHANGE C IF (KODY (I| . N E . 0) J O TO 50 IF(DBIC. GT.O. ) G O T O 40 FAC=(BHCR(2)-BNIC (I))/DBIC IF (FAC. G E . FACTOR) GO TO 115 C FACTOR= FAC BH=BflCR(2) CUB=C0CR(2) KODE=1 EK = BKEP (2) IDK(I| =2 G O T O 115 C 40 PAC = (BBCB(1)-B1IIC(I))/DBIC IF (PAC. GE. FACTOR) GO TO 415 C FACTOB=PAC Bn=BncB(D C O R = C O C B ( 1 ) KODE=1 E K = EKEP (2) IDK(I) =1 G O TO 4 I S KOD=KODY (I) G O TO(63,70,80,90, 100, 150, 170,200,220,240,260,280,300,320,340,36 •380,400,405,410),KOD RULE THO. INELASTIC STAGE. GET FACTOR FOR CHANGE. ( A - C ) IP(REHIND'DBIC.LT.0.) G O TO 80 IF (DBIC.GT.O.) GO TO 65 FAC=(BHCU (2) -BHIC(I) ) /DBIC IF (FAC. GE. FACTOR) GOTO 415 FACTOR=FAC Bn = BliC0(2) CUR = CUCU (2) KODB=2 E K = EKEP (3) IDMD -2 UHT2 (I) =2. DO G O TO 415 C 65 PAC= (BHCU(1 ) -B« IC ( I ) ) /DB IC IP (FAC. GE. FACTOR) GO TO 415 C FACTOR = PAC BH=BHCU (1) CUR=C0CD(1| KODE-2 EK=EKEP (3) U8T1 (I) = 1. DO IDK(I) =1 GO TO 415 C C RULE THREE. LOADING ON THE DEGRADING COBVE B-E. C 70 CONTINUE GO TO 415 C C BULE POOR. UNLOAOING ALONG C-D. C 80 IP (IND(I). EQ. 1. AND. OUT 2 (I) . EQ. 2. DO) GO TO 260 IP (IND(I) .EQ.2.AND.UHT1 (I) .EQ. 1.D0| GO TO 260 IP (KODY (I) . EQ. 3) GO TO 82 IF (KODY (I) . EQ. 4) 50 TO 82 BHTT(I) =BNIC(I) CDTT(I)=OIIC(I) 81 BHCP=0. 522D0*BHCR (I ND (I) ) EK= (BHIC(I) -BHCP) / (COIC (I) -COCR (IND (I) ) ) EKCD(I)=EK ERIC (I) =EK KODI (I) =3 82 IF(HEHIND*DBIC.GT.O.) GO TO 90 FAC= (BNCP-BNIC(I))/DBIC IP(PAC.GE.FACTOR) 50 TO 415 C FACTOH=FAC BB=BMCP C0R=CUCB (IND (I) ) EK= (B(1CP»BNCB(IHD(I)) )/(COCB (IHD(I) ) •CDCB (IND (I)) ) KODE=5 IF (IND (I). EQ. 2) GO TO 83 IDK(I| =2 GO TO 84 83 TDK (I) = 1 84 GO TO 415 C C RULE FIVE. LOADING ALONG D-C C 90 IF (REHIND*DBIC.LT.O.) GOTO 80 KODY (I) =4 PAC=(BUTT(I)-BHIC(I))/DBIC IP (FAC. GE. PACTOH) GO TO 415 C FACTOR3 FAC BH=BHTT(I) CUR=CUTT (I) K0DE=1 EK = EKEP (2) IDK(I) = IND (I) ON ON GO TO 1 1 5 C C R O L E S I I . U N L 3 A D I N 3 ALONG A - 0 C 1 0 0 K 0 D Y ( I ) = 5 I P ( 8 E H I N D * D 8 I C . L T . 0 . | G O T O 150 1 0 5 I P ( D B I C . G T . 0 . | GO TO 1 1 0 F A C = ( B B C B ( 2 ) - B H I C ( I ) | / D B I C I P ( F A C . G E . P A C T O B ) GO TO 1 1 5 C F A C T O R = F A C BM= BWCR (2) C U R = C O C B ( 2 | I D K ( I ) = 2 GO TO 1 1 5 C 110 PAC= ( B N C f i ( 1 ) - B H I C ( I ) ) / O B I C I F ( F A C . G E . F A C T O B ) GO TO 1 1 5 C F A C T O R 3 F A C Bfl=BHCR ( 1) C U H = C U C B < 1 ) I D K ( I ) =1 1 1 5 EK= E K E P ( 2 ) KODE=1 GO TO 4 1 5 C C B U L B S B V E M . L O A D I N G ALONG A - D C 1 5 0 I P ( B E H I N D « D B I C . G T . O . ) G O T O 100 KODY ( I ) =6 FAC= ( B B C P - B N I C ( I ) » / D B I C I F ( F A C . G E . F A C T O B ) GO TO 4 1 5 C F A C T O B = P A C B B = B N C P C U R = C U C B ( I N D ( I ) ) K O D E = 3 EK= EKCD ( I ) I F ( I N D ( I ) . E Q . 2 ) GO TO 1 5 3 rDK ( I ) = 2 GO TO 1 5 1 1 5 3 I D K ( I ) = 1 1 5 4 GO TO 4 1 5 C C B O L E E I G H T . O N L O A D I N G FROM P O I N T F . C 1 7 0 I F ( B E N I N D * D B I ^ . G T . 0 . . A N 0 . K O D Y ( I ) . E Q . 8 | GO TO 2 0 0 I F (KODY ( I ) . E Q . 7 ) GO TO 1 7 5 I F (KODY ( I ) . E Q . 8 ) 3 0 TO 1 7 5 B H T D ( I ) = B N I C ( I ) C U T D ( I ) = C O I C ( I ) I P ( I N D ( I ) . E Q . 2 ) GO TO 177 BHDLT ( I , 1 ) = B B r D | I ) B H ( 1 A I ( 1) = B H L I F (DHT2 ( I ) . ME. 2 . DO) G O T O 1 7 4 B » 0 L T ( I , 2 ) = - B H T D ( I ) BHBAX ( 2 ) = - B B L GO TO 1 7 4 177 B H U L T ( I , 2 ) = B B T D ( I ) BflHAY ( 2 ) = B N L I F ( U H T I ( I ) . M E . 1 . D 0 ) GO TO 1 7 4 B r t U L T ( I , 1 ) = - B « r D ( I | B H N A I ( 1) = - B M L 174 B H L P = 0 . 5 D 0 * BNCO ( I N D ( I) ) B H I P = 0 . 6 1 2 D 0 * B H C B ( I M D ( I ) ) I P ( D A B S ( C U T D ( I ) | . G T . DABS ( ( 1 0 » C ( J C U ( I N D ( I ) ) ) ) ) GO TO 1 8 0 CO I P ( I ) = C 0 T D ( I ) » ( 0 . 2 3 » ( O . O S » C U T D ( I ) | / C B C U ( I B D ( I 1 ) ) GO TO 1 8 5 1 8 0 C U I P ( I | = 0 . 7 * C 0 T D ( I ) 1 8 5 C O N T I N U E EK= ( B M T D ( I ) - B H I P ) / ( C U T D ( I ) - C O I P ( I ) ) E K P G ( I ) =EK E K I C ( I ) = E K K O D Y ( I ) =7 C 1 7 5 F A C = ( B N I P - B B I C ( I ) ) / D B I C I P ( F A C . G E . F A C T O B ) GO TO 4 1 5 I P ( I N D ( I ) . E Q . 2 . A N D . UHT1 ( I ) . E Q . 1 . DO) G O T O 1 9 5 I F ( I N D ( I ) . E Q . 1 . AND. UNT2 ( I ) . B Q . 2 . D 0 ) GO TO 195 C F A C T O B = F A C B B = B B I P C U B = C 0 I P ( I ) I P ( D B I C . L T . O - ) GO TO 1 9 0 EK= ( B H I P - B n C R ( l ) ) / ( C U I P ( I ) - C 0 C B ( 1) ) I D K ( I ) =2 GO TO 1 9 1 1 9 0 EK= ( B f l I P - B H C B ( 2 ) ) / ( C U I P ( I ) - C U C B ( 2 ) ) I D K ( I ) =1 191 KODE=9 EKGA ( I ) =EK GO TO 4 1 5 C 1 9 5 F A C T O R = P A C B B = B B I P C U B = C 0 I P ( I ) B N L P = 0 . 5 D 0 * B H ; 0 ( I N D ( I ) ) EK= ( B B L P » B B I P ) / ( 2 * C U I P ( I ) ) E K G L ( I ) = EK I D K ( I ) = I N 0 ( I ) KODE=17 GO TO 4 1 5 C C B O L E N I N E . L O A D I N S ALONG F - G . C 2 0 0 I F ( B E B I N D * D B I C . L T . O . | GO TO 1 7 0 K O D Y ( I ) =8 F A C = ( B H T D ( I ) - B B I C ( I ) ) / D B I C I F ( F A C . G E . F A C T O B ) GO TO 4 1 5 C F A C T O R = F A C BB=BHTD ( I ) C U B = C U T D { I ) KODE=3 EK = E K B P (3) I D K ( I ) = I N D ( I ) GO TO 4 15 c C RULE TEN. UNLOADING ALONG G-A. C 220 IP (DBIC*REMND. GT.O.) GO TO 2»0 IP(DBIC.LT.0.) 30 TO 225 KODT(I) =9 FAC= (BHCR(I)-BNIC(I)) / DBIC IP(PAC.GE.PACTOR) GO TO 415 C PACTOR=PAC BH=BHCR (1) C0R=C0CR(1) KODE=1 EK=EKEP (2) IDK(I) = 1 GO TO 415 C 225 FAC=(BHCR(2)-BHIC(I))/DBIC IP (FAC. GE. FACTOR) GO TO 015 C FACTOR*PAC BH-BHCR (2) CUR=CUCR(2) KODE=1 EK=BKEP (2) IDK(I) =2 GO TO 415 C C RULE ELEVEN. LOADING ALONG G-A. C 240 IP(RENIKD*DBIC.LT.O.) GO TO 220 KODY (I) =10 C FAC= (BBIP-BHIC(I))/DBIC IF(PAC.GE.PACTOR) GO TO 415 C FACTOR=FAC BH=BHIP CUR=CUIP(I) KODE=8 EK=EKF3 (I) IP(DBIC.LT.O.) GO TO 245 IDK(I) = 1 GO TO 250 245 IDK(I)=2 250 GO TO 415 C C RULE TWELVE. UNLOADING ALONG I-H. C 260 IF (KODT (I) .EQ. 11) GO TO 265 IF (KODT (I). EQ. 12) GO TO 265 BHTT(I) = BHIC(I) CUTT(I)=CUIC(I) BHHP=0.712D0*BNCR (IND (I) ) EK = (BHIC(I) -BNHP) / (CUIC (I) -CUCR (IND (I) ) ) EKIH (I) =EK EKIC (I) =EK KODT (I) = 11 265 IF (BEMIND»DBi:.GT.O.) GOTO 280 PAC=(B«HP-BHIC(I)) /DBIC IF (FAC. GE. FACTOR) GO TO 415 C FACTOR= FAC BH=BBHP CUR=C0CR (IND(I) ) EK=(BHHP-B(1LP) / (CUCB (I ND (I) )-CUI P (I) ) EKRJ (I) = EK 271 KODE=13 IDK (I) = IND(I) GO TO 415 C C ROLE THIRTEEN. LOADING ALONG I-H. C 280 IP(REHIND*DBi:.LT.O.) GO TO 260 KODT(I) =12 FAC= (BUTT (I)-BHIC(I)) /DBIC IP(FAC.GE.FACTOR) GO TO 415 C FACTOR=FAC BH=BMTT (I) CUR=CUTT(I| KODE=1 EK=EKEP (2) IDK(I)=IND(I) GO TO 415 C C RULE FOURTEEN. LOADING ALONG H-J. C 300 IP(REHIHD*DBi:.GT.O.) GO TO 320 KODI(I) =13 FAC= (BHLP-BNIC (I)) /DBIC IPJPAC.GE.FACTOR) GO TO 415 C FACTOR=PAC BH= BBLP CUR=CUIP(I) KODE=15 EK = (BHTD (I) -BHLP) /(CDTD (I) -CDIP (I)) EKJF(I) = EK IP (IND (I) . EQ. 1) GO TO 315 IDK (I) = 1 GO TO 316 315 IDK(I) = 2 316 CONTINUE GO TO 415 C C RULE PIPTEEN. UNLOADING ON H-J. C 320 IF (BEHIND»DBIC. LT.O.) GOTO 300 KODY (I) =14 FAC= (BNIP-BMIC (I) )/DBIC —• IP (FAC. GE. FACTOR) GO TO 415 g> FACTOR=FAC BH=BHIP CUR=CUCR (IND(I) ) KODE=12 EK= EKIH (I) IDK (I)=TND (I) GO TO HI 5 C C ROLE SIXTEEN. LOADING ON J-P C 340 IP (RENIND*DBIC.LT.O.) GOTO 360 KODY (I) =15 FAC= (BMTD(I) -BNIC(I) ) /DBIC IP (PAC. GE. FACTOR) GO TO 415 C FACTOR* PAC BH=BHTD (I) CUR=C0TD(I) KODE=2 EK= EKEP (3) IDK (I) = IND(I) GO TO 415 C C RULE SEVENTEEN. UNLOADING ON J-F C 360 IP (REHIND*OBIC. GT.O.) GOTO 415 KODT (I) = 16 PAC= (BHLP-BNIC(I)) /DBIC IP (PAC. GE. FACTOR) GO TO 415 C PACTOH=PAC BH=BNLP CUB=CDIP(I) KODE=14 EK= EKHJ (I) IF(IND(I) .EQ.1) SO TO 365 IDK(I) = 1 GO TO 366 365 IDK(I)=2 366 CONTINUE GO TO 415 C C RULE EGIHTTEEN. UNLOADING ALONG G-L. C 380 IF (REHIND'DBIC.GT.O.) GOTO 400 KODY (I) =17 FAC= (-BHLP-BHIC (I) ) /DBIC IF (PAC. GE. PACTOR) GO TO 415 C FACTOR=PAC BH=-BHLP CUB=-CUIP(I) KODB=19 EK= (-BHTD(I)»BI1LP)/(-CUTD(I) •CUIP(I)) IF(IND(I) .EQ.2) GO TO 382 IDK(I)=2 GO TO 385 382 IDK(I) = 1 385 CONTINUE GO TO 415 C C RULE NINETEEN. LOADING ALONG G-L. C 400 IF (BEPII ND'DBIC. LT.O.) GOTO 360 KODY (I) =18 FAC=(BNLP-BtlIC (I)) /DBIC IF (PAC. GE. FACTOR) GO TO 415 C FACTOR*FAC BR=BHLP CUR=CUIP(I) KODE=7 EK= EKPG (I) IDK (I) = IND (I) GO TO 4 15 C C RULE T WENT I. LOADING ON L-P. C 405 IP (REHIND'DBIC. LT.O. DO) GO TO 410 KODI (I) =19 PAC=(-B«TD(I) -BHIC(I))/DBIC IP (PAC. GE. PACTOR) SO TO 415 C 406 PACTOB=PAC B«=-BHTD(I) COR=-CDTD(I) KDDE=2 IDK(I) = IND(I) EK = EKEP (3) 60 TO 415 C C RULE THENTI-ONE. UNLOADING ON L-P. C 410 IP (BEHIND*DBIC. GT.O. DO) GO TO 405 KODT (I) =20 FAC= (-BNLP-BMC (I) ) /DBIC IP (PAC. SB. FACTOR) GO TO 415 C PACTOB=PAC BN=-BMLP CUB=-CUIP(I) KODE=18 IDK(I) = IND(I) EK=EKGL (I) GO TO 415 C C 415 CONTINUE C C CHECK POR COMPLETION OF CYCLE C FACAC=FACAC»FACTOR IP (FACAC. GT.O. 99999900) GO TO 418 C C UPDATE MOUENTS AND CURVATURES C CALCULATE HINGE ROTATION C BfllC(I) =Brt CUIC(I)=C!IF KODY(I)=KODE I NO (I ) = I DK (I) EKIC (I| =EK IF (I. NE. NNCR) CO TO 417 CUT (K + KBAL) =CUIC (NNCR) BUT (K» KflAL) =BHIC(NHCR| KBAL=KBAL*1 417 GO TO 30 418 FACAC=O.D0 rp (KODT (I) -GT. 0) GO T3 12 1 EKIC(I) =BKH 421 CONTINUE BNIC(I) =BHIC(I) *DABS (FACTOR) *DDIC BHI=SNGL(BHIC(I) ) BNU=SNGL (BB1ILT (I,IND(l|)) IF (DABS (BUI) .LE.DABS (BHU) ) GO T3 428 BHIC (I) = BHULT (I, INO(I) ) GO TO 423 426 IF (PCRF. EQ. 1. DO. AND. BHI. EQ. BHU) GO T3 423 IF (IND (I) . EQ. 2) GO TO 429 IP (BHI. EQ. BHD. AND. DHB.LT.O. DO) G3 TO 423 GO TO 426 429 IF (BHI. EQ. BHU. AND. DBB. GT. 0. D0| G3 T3 423 426 CUIC(I) =CUIC(I)»DABS{PACTOB)*DBIC/EKIC(I| CUEL(I) = CUEL(I) *DABS (FACTOB) • DBIC/EKH IF (BHL. LE. 1.8) 30 TO 423 423 IF(I.NE.NNCR) 30 TO 422 COT (KtKBAL) = C(JIC(NHCH| BUT(K»KBAL) =BBIC(NHCR) 422 GO TO(431, 432,432) , KHOH 431 BOI=ROI*(C0IC(I)-COEL(I)) *DINC» (DINCL»0.5D0*DINC»0. 5DO) * (CUIC(I) -CO EL (I) ) *DMC» (0. 5DO- (DIHCL »0. 5D0*DINC)) GO TO 434 432 BOI=BOI»(C0IC(I)-C0EL(I>)»DIHC» (DINCL»0.5*DINC) 434 KCHAN=0 BHUC=BHHAI (IND (I) ) 425 CONTINUE C IF (KODY (NNHL). EQ. 2) PCBF= 1. DO BOT(K) = BOI IF (K. EQ. 1) GO TO 446 BHT(K) = BHT(K-1) »DHB GO TO 447 446 BHT (K) = DHB 447 KSTBP=K*KOAL BTH(K) = BHT (K) IF (DABS (BHT (K)) . GT. DABS (BHUC) ) BTH(K) = BSUC 430 COMTINOE C C MRITE RESULTS IN PILE C DO 450 1=1,NSTEP HRITE(6,455) ROT (I) , BTH (I) 455 FORBAT(2P10.4) 450 CONTINUE C DO 460 I=1,KSTEP BRITE(7,465) :DT (I) ,BUT (I) 465 FORHAT (2F10.4) 460 CONTINOE C CALL PLOTHR(B3T,BTH,NSTEP,BUCK,BHCU) CALL PL0T(15.0,0.0,-3| CALL PLOTH: (CUT,B0T,KSTEP,BBCB,DBCU) CALL PLOTND STOP END C C SUBROUTINE TO PLOT HOB ENT BOTATI3V CUB YES C SUBBOUTINE PL3T HR(ROT,BTH,NSTEP,BBCB,BHCU) IHPLICIT REAL*8(A-H,0-Z) DI HENS ION Z (2) , BHP (300) , ROP (300) , ROT (300) , BTH (300) DIMENSION BBCB (2), BHCU (2) C DO 2 1=1,NSTEP ROP(I| =SNGL (R3T (I) ) BHP (I) =SNGL (BTH (I)) 2 CONTINUE PLH=7. PLR=7 PLH2= PLH/2. PLR2=PLR/2. DBAX=0. DO 5 1=1,NSTEP 5 IF (DABS (BHP (I) ) . GT. DHAX) DHAI=DABS(BHP (I)) Z(2)=0-2(1) = 0HAX CALL SCALE(Z,2,PLH2,XHIH,DIX, 1) IHINN=-0XX*PLH2 BHAX=0. DO 10 1=1,NSTEP 10 IF (DABS (BOP(I) ) . GT. RHAX) BH AX = DABS(ROP (I)) Z(2)=0. Z(1)=-BHAX CALL SCALE(Z,2,PLB2,RHIH,BDX, 1) BHINN=-RDX»PLH2 CALL AXIS (0. ,PLB2,40H B3TATION,-4 0 C, PLR,0. , RBI NN, RDX) CALL AIIS(PLH2,0.,35H HOHEBf,35 C,PLn,90.0,IHINN,DXX) CALL PLOT(PLR2,PLH2,3) DO 20 1=1,NSTEP CALL PLOT(BOP(I)/RDI»PLR2,BHP(I)/'DII«PLH2,2) 20 CONTINUE CALL PLOT(0.,0.,-3) RETURN END C C SUBROUTINE TO PLOT HOHENT COBYATURB CURVES C SUBROUTINE PL3TNC(CUT,BUT,KSTEP,BBCB,BHCU) IHPLICIT REAL*8(A-H.O-Z) DIHENSION Z(2) ,BUP(300) ,CUP(300) ,COT(300| ,BOT(300| DI HENSION 8HCB(2) ,BHCH(2) C DO 2 1=1,KSTEP CUP (I) = SNGL (CUT (I) ) BUP(I) =SN3L (BtIT(I) ) 2 CONTINUE PLH=7. PLC=7. PLM2=PLH/2. PLC2=PLC/2. DNAX=0. O DO 5 I = 1,KSTEP IK(DAUS(DUP(I) ) .ST. DHAX) DMA X = DA DS (DU P (I)| Z(2)=0. Z(1)=DNAX CALL SCALE(Z,2>PLn2,XHIN,DIX ,1) X(IINN=-DXX»PL«2 CNAX=0. DO 10 I=1,KSTEP IF(DABS(CUP(I| | .GT.CNAX) CflAX=DADS (CUP(I|) Z(2) =0. Z(1)=-CNAX CALL SCALE(Z,2,PLC2,CHIN,CDX,1) C8INN=-CDX*PLC2 CALL AIIS(0.,PLC2,41H C,PLC,0.,CNINN,CDI) CALL AXIS (PLN2.0.,35H C.PLB.90.0,XBINN.DXX) CALL PLOT(PLC2,PLH2, 3| DO 20 I=1,KSTBP CALL PLOT (CBP(I)/CDX*PLC2,BUP(I| /DXX»PLN2,2) COHTINOE CALL PLOT(0. ,0. ,-3) RETURN END CURV ATUHE, HOHENT,35 172 A.3 PROGRAM: PRESTRESSED CONCRETE BEAM/COLUMN ELEMENT (DRAIN-2D) The method presented in the DRAIN-2D users manual for the addi t ion of elements was used in w r i t i n g a group of seven sub-routines compatible with a modif ied DRAIN-2D program. The sub-routines operated to monitor the e l a s t i c , or poss ib le nonl inear response of the elements and the h y s t e r e t i c act ion of the hinges which would occur in the l a t t e r case. A desc r ip t ion of the pre -s t ressed concrete beam/column element used is given in Chapter 39 Three of th is t h e s i s . See the DRAIN-2D users manual f o r a des-c r i p t i o n of the base program and the other sect ions referenced in the input requirements l i s t e d below: Element E ight : Prestressed Concrete Beam/Column Elements -Required Data Input Number of words of information per element is 189. E8(a) CONTROL INFORMATION FOR GROUP (815) - ONE CARD Columns 5: Punch 8 to ind icate that group cons is ts of prest ressed beam/column elements. 6 - 10: Number of elements in group. 11 - 15: Number of d i f f e r e n t element s t i f f n e s s types (max is 40) . See sect ion E8(b). 16 - 20: Number of d i f f e r e n t end e c c e n t r i c i t y types, (max is 15). See sect ion E2(c) . 21 - 25: Number of d i f f e r e n t cracking moment values for cross sect ions (max is 40) . See sect ion E2(d). 26 - 30: Number of d i f f e r e n t f i x e d end force patterns (max Is 34). See sec t ion E2(e) . 31 - 35: Number of d i f f e r e n t i n i t i a l element force patterns (max is 30) . See sect ion E2 ( f ) . 173 36 - 40: Number of d i f f e r e n t moment-rotation types (max is 40). See sect ion E8(b). 46 - 50 Code f o r energy balance. Leave blank or punch zero i f energy balance is not to be preformed. E 8 ( b ) STIFFNESS TYPES - TWO CARDS FOR EACH TYPE CARD 1: ELEMENT PROPERTIES CARD (15 , 3F10.0.3F5.0) 5 Columns 6 16 26 36 41 46 15 25 35 40 45 50 S t i f f n e s s type number in sequence beginn ing wi th 1. E l , e f f e c t i v e f l e x u r a l s t i f f n e s s . EA, e f f e c t i v e ax ia l s t i f f n e s s . GA' , e f f e c t i v e shear s t i f f n e s s . If blank or zero , shear deformations are neglected . F lexural s t i f f n e s s factor K . . . i i Flexural s t i f f n e s s f a c t o r K . . . J J Flexural s t i f f n e s s f a c t o r K . . . 'J *Note - A l l of S t i f f n e s s Types (card I) are entered in sequence fo l lowed by S t i f f n e s s Types (card 2) in sequence. CARD 2: HINGE MOMENT-ROTATION PROPERTIES CARD (15.6FI0.3). Columns 5: Hinge type number in sequence beginning with 1 and corresponding to the appropr iate s t i f f n e s s type number. 15: E l a s t i c s t i f f n e s s at node 1. Factors K 16 26 36 46 56 and are appl ied to th is va lue . Actual hinge s t i f f n e s s up to the cracking moment is i n f i n i t e . 25: E l a s t i c s t i f f n e s s at node j . 35: Value o f K^/K at node i . 45: Value of K./K at node j . 1 e J 55: Value of K 2 /K g at node i . 65: Value of *Z/Ke a t n o d e J* 174 E8 (c) through E 8 ( f ) : These sect ions are i d e n t i c a l to sect ions E2(c) through E 2 ( f ) . E8(g) ELEMENT GENERATION COMMANDS ( 1 4 1 4 . 2 F 5 . 0 , 15, F5 .0 ) Elements must be s p e c i f i e d in increasing order . Prov is ion is made f o r element generat ion . See note at end of sect ion*. Columns 4: Element number or number of f i r s t elements in a sequent ia l l y numbered se r ies of elements to be generated by th is command. 5 - 8: Node number at end i . 9 - 12: Node number at end j . 13 - 16: Node number increment for element genera-t i o n . See note at end of sec t ion* . 17 - 20: S t i f f n e s s type number. 21 - 24: Hinge moment-rotation type number. 25 - 28 : End moment pattern number: 1 - equal and opposite end moment pat tern . 2 - moment at one end on l y . 3 - uniform moment. Note - Appropriate f l e x u r a l s t i f f n e s s factors must be s p e c i f i e d . 29 - 32: End e c c e n t r i c i t y type number. Leave blank or type zero i f there is no end e c c e n t r i c i t y . 33 - 36: Cracking moment at element end i . 37 - 40: Cracking moment at element end j . 41 - 44 : Code fo r inc lud ing geometric s t i f f n e s s . Leave blank or punch zero i f geometric s t i f f n e s s is not to be inc luded. 45 - 48: Code f o r time h is to ry output . If a time h is to ry f o r element is not required leave blank or punch zero. If a time h i s t o r y , at in te rva l s p e c i f i e d on card D l , is requ l red , punch 1. 49 - 52: Fixed end force pattern number for s t a t i c dead loads on element. Leave blank or punch zero i f there are no dead loads. 175 Columns 53 - 56: F i x e d end f o r c e s p a t t e r n number f o r s t a t i c l i v e l oads on e l e m e n t . Leave b l ank o r punch z e r o i f t h e r e a re no l i v e l o a d s . 5 7 - 6 1 : S c a l e f a c t o r to be a p p l i e d to f i x e d end f o r c e s due to s t a t i c dead l o a d . 62 - 66 : S c a l e f a c t o r to be a p p l i e d to f i x e d end f o r c e s due to s t a t i c l i v e l o a d . 67 - 71: I n i t i a l f o r c e p a t t e r n number. Leave b l ank o r punch z e r o i f t h e r e a re no i n i t i a l f o r c e s . 72 - 76: S c a l e f a c t o r t o be a p p l i e d to i n i t i a l e lement f o r c e s . *Note - Node number increment f o r e lement g e n e r a t i o n must be s p e c i f i e d as z e r o i f g e n e r a t i o n r o u t i n e i s not u sed . A l i s t i n g o f the s u b r o u t i n e s f o l l o w s : SUBROUTINE INulU (KCONT, e'COST, M 03 t.A IN f t , r J, i , \ , NN) IMPLICIT REAL'S (A-H.O-Z) COBBON/GEWINF/IIDUM (30) , N IMF ( 10) , NDO?F ( 10| , J JDU H ( 10) COMBON/PASS/IGR.KKDUH (f.) COMMON/INPEL/ IMEK,IHRrtD,KSr,KSrD,Lfl(6| ,100(6) , KU E0.1 , K>i tiJ MO , PL,COSA,SINA,EC(4) ,A (2 , b ) , BFU, UFJJ, BPIJ, EAL, EKHIP, EKHJP, EKH (2) ,K0l)YX(2) , NO (4) .KODY (2) ,2KEP (2,b) ,IND (2) ,1NDUM(2) , uEVPT (2) , BNCR (2,2), B1CR I (2,2) ,UNCU (2 , 2) , ROCU (2, 2) , HOCH (2, 2 BMTOT (2) ,SFTOT (2) ,FTUT(2) ,SOTOT (2) ,ROPLS 12) , KOELS(2) ,ROPMX(2) ,P.ONMi(2) ,ROTPN(2) , HOT.'.'(2) , SEN P (10) , SENN (10) , TENP(1J),TENh(10) , Si) ACT (3) .NODI 4 NOOJ.KOUTDT, KJUfl (3) ,EK1 (2) , I MO£D, 11MOED, K3 EC (2, 2) » UMT1 (2) ,UHT2 (2) , 13 MTT (2) , HOTT (2) , BNTD (2) , rtJTD (2) , 5 BHIP(2) , ROIP(2) , BMCD(2) , R3CD (2) ,UMCP (2) ,RJCP (2) , • DNLc1 (2) , BBMW (2) ,EKCD (2) , EKFG (2) ,EKJF(2) , EKGL (2), 6 EKGA(2) , EKIH(2), BEST (11) COM HUH /HOUR/ SPP(8) ,SS1'F(8) ,DD(6) ,GA(6,6) ,FFEF( 6) ,FF(6) ,KSF (2) 1 FTYP (63,6) , PEP (SO, 7) ,K DEEP (50) , FIN IT (45,6) , » HM8T(50,7), 2 £CC(35, 4) , BNCK (70, 2) , EEX PN (60) ,ASTN (60) , U (1 8) COBBON/THIST/ITHOUT (10) ,THOUT (20) ,ITH P, IS AVE, NELTtl, NSTH, HF7, iSii C DIMENSION KCONT (1) ,ID(NN,1),X(1),Y(1) ,COH(1) ,AST(2) , YES NO (2) DIMENSION I NT (60) EQUIVALENCE (IHEM.COM ( 1) ) DATA AST/28 ,2H »/ DATA YES NO/4 H YES.4H NO / C C DATA INPUT, P. S. BEAM ELEMENTS C NDOF=6 NINFC=189 NDOPF(IGR)=tfDOP NINF(IGR) =NINPC KCCM= KCONT (1) NHEB=KCONT ( 2) NBBT=KCONT(3) NECC=KCONT(4) »SURP=KCONT (5) NF EF=KCONT(6) Nl NT= KCONT (7) NHRR=KCONT (8) PBINT 10, (KCONT (I) , 1=2, 8) . KCONT ( 10) 10 FORMAT (32H P.S. DEAM/COL ELEMENTS (TYPE ri) //// 1 3411 NO. Of ELEMENTS = 14/ 2 34H NO. OF STIPFNE33 TYPES = 14/ 3 3411 NO. OF ECCENTRICITY TYP25 = 14/ 4 34H NO. OF CRACKING MOB2N1' TYPES = 11/ 5 3411 NO. OF FIXED END POilCE PATTEliilS = 14/ 6 34H NO. OF INITIAL FORCR PATT2P.Nr> = 14/ 7 34H NO. OF ilOBL'NT-ROTATION DELATION S= 14/ 8 14H CODE FOK ENEUGY BALANCE = 14) C C INPUT STIFFNESS Pau PcKT I LS PRINT 20 20 POBBAT(////29ll ELEMENT FLEXURAL PROPERTIES// 1 2X,5H TYPE, 7X,28IIBEFEIiENCE SECTION PROPERTIES,5X, 2 26HFLEXURAL STIFFNESS FACTORS / J 2X.5H NO. , 8X,2HEI, 10X, 2HEA, 10X, 2HGA, 8X, 2HII, UX,2UJJ, 8X, 4 2HIJ/) DO 60 N= 1, NMBT READ 30,1, (FTYP (N,J) ,J=1,6) 30 POHHAT(I5,3F10.0,3F5.0) 40 PRINT 50,1, (FTYP (»,J) ,J = 1,6) 50 FORMAT (2X,14 ,2X , 3E12. 4, 2X, 3P7.0) 60 CONTINUE PBINT 70 70 PORMATJ/33H MOMENT-ROTATION CURVE PROPERTIES// 1 6H TYPE, 61, 17HINITIAL STIFFNESS, 10X.9HSTIFFNBSS, 2 11H PAHABETKBS/36I,5HK1/KE,14X,5HK2/KE/bH NO., 3 8X,1HI,9X,1HJ,9X.1HI,9X,1HJ.9X,1HI,9X,1UJ/) DO 60 N=1,WHBR READ 85,1, (HHRT (N,J) ,J=1,b) 85 PORBAT(1I5,6F10.3) 80 PRINT 90,1, (HBRT (N,J) ,J=1, 6) 90 FOBBAT(2I,I4,3X,E10.3,E10.3,F10.3,P10.3,F10.3,F10.3) C C INPOT END ECCENTRICITIES C IF (NECC.EQ.O) GO TO 140 PRINT 100 100 PORBAT(////23H END BCCEMTRICITY TYPES// 1 SH TYPE,6X,25HHOBIZONTAL ECCBNTBICITIES, 5X, 2 25H VERTICAL ECCENTRICITIES / 3 5H NO. ,4X,25H END I END J .5X, 4 25H END I END J /) DO 110 N=1,NECC READ 120, I, (ECC(N.J) ,J=1,4) 110 PRINT 130. B, (ECC(N.J) ,J=1, 4) 120 FORHAT(I5,4F10.0| 130 FORBAT (14,2 (Fl 5. 2,PI 4. 2,1X)) C C CRACKING MOMENT TYPES C 140 PRINT 150 150 FORMAT(////22H CRACKING MOMENT TYPES// 1 5H TYPE,5X,0HPOSITIVE,5X,8HNEGATIVE/ 2 5H NO.,5X,8ll MOMENT ,5X,8H MOMENT /) DO 180 N=1,NSUttF READ 160, I,BNCK(N, 1) ,BMCK(N,2) PRINT 170, N.BSCK (N. 1) , UBCK (N,2) 160 FOBBAT(I5,5X,2F10.0) 170 FORBAT(I4,2F1l. 2) 180 BMCK(N,2) =-DADS (JMCK (N ,2) ) C C FIXED END FOKCE PATTERNS C IF (NFEF-EQ.C) GO TO 2J0 PRINT 190 190 FORMAT |////25!1 FIXED END FOHCS PATTERNS// 1 BH PATTERN, IX, 4 H AX IS , 7 X, 5H AX IAL , 7X , 5HSH tAK , 6 X , 6IIBO BENT, 2 7X, 5HAXIAL,7X,5HSUEAH,6X,6hHOH2NT,5X,8tlLL. RED. / 3 8H NO. , JX.KHCOOf, 7X, 5HAT I,7X,5HAT I,6X,6H AT I , 4 7 X . 5 H A T J.7K . S I I A T J , 6 X.1.:| A f J , 5 X . H H F A C T O R / ) DO 200 N=1,NFEP READ 2)0, I ,KDFF.F ( N ) , (FE.'fN.J) , J = 1, 7) 200 PR INT 2 2 0 , N.KDFEF (N) , ( F f c F (N, J) , J = 1 ,7) 210 FOBHAT (2I5.7F10.0) 220 PORHAT(I5,I9,F13 .2,5P12 .2.F12. J) C C INITIAL FORCE PATTERNS C 230 IF ( N I N T .EQ* 0) ISO TO 2 U 0 PRINT 2<»0 240 FORMAT(////28I1 INITIAL END FOiiCE P A T T E R N S / / 1 OH ?ATrERN,7X,5IIAXIAL, 7 X , 5 I I S HEAR , bi , 6HHOI1 E N T , 7 X , 5 i ( A X I A L , 2 7X,5HS1IEAR,6X,6HH0HENT/ 3 8H N O . ,7X,5HAT I ,7X,5IIAT I,6X , 6 I ! A T I , 7 X , 5 i l A I J , 14 7X.5HAT J,6X,6II AT J /) DO 250 N=1,NINT BEAD 260, I,(PINIT(N,J),J=1,6) 250 PRINT 270, N, (FINIT (N, J) , J= 1, 6) 260 FORHAT(I5,6F10.0) 270 F0BHAT(I5,3X,SF12.2) C C ELEMENT SPECIFICATION C 280 PRINT 290 290 FORHAT(////22H ELEMENT SPECIFICATION// C 1 1X,4HELEn,2X,4HNODE, IX, 4HNODE, 2X, 4HNODE, 21, 4HSTIt', 2X,411 tlX 2 ,2X,4HENDM,2X,4HECCX, 2X, 13HCHKNG MOM ENTS, 2 X, 4HGL'GI1 ,2X, 3 4HTIHE,2X,12HFEF PATTERNS,2X,14HFEF SCALE FACT.2X, 4 16H INITIAL FORCES / 5 IX,4H N0..2X.3U 1.2X.33 J,3X,41IDIFF,2X,4HTYPE, 21 ,4HTYPE 6 ,2X,411TXPE,2X,4HTTPE, 2X, 1 3HEND I EN0 J ,2X,4HSTIi' ,2X, 7 4HHIST,2X,I2H DL LL ,2X,14H D L LL ,2X, 8 17H NO. SCALE FAC./) DO 300 J=77,139 300 CON (J) =0. K o o r x < i ) = o KODYX (2) =0 KODY (1) =0 KODY (2) =0 KST=0 C INEH=1 J10 READ 320, INEL , INOUI , I NODJ , I l i l C , IINDT, 1IUHHT , 1 1 . 1 0 E D , II E C C , IKS PI, IIKSFJ.IKGN.IKDT, IKFDL, IKFLL, FFOL, FFLL, UNIT, FFlNIT 320 FORHAT(14I4,2F5.0,I5,F5.0) C IF(IINC.E3.0) 3 0 TO 330 IF (INEL.GT. IHEN) GO TO 360 330 NODI=INODI NODJ=INODJ INC=IINC IMBT=IIMBT IECC=IIECC IH BRT=IIHNRT IHOED=IIHOED KSPI=IKSPI KSFJ=IKSPJ KG ECtt=l KGN KOUTDT=lKDT YNG=YE3 NO (2) IF (KGEOH.NE.C) YNG = YESNO (1) YNT=YESNO ( 2 ) IP (KO0TDT.NE.0) YNT=YESNO (1) KPDL=IKFDL KPLL=IKPLL FDL=FF0L FLLM=FFLL PLLP=1-IF (KFLL.EQ.Cl GO TO J40 FLLF=FEF(IKFLL,7) IF (FLLF.EQ.O.) FLLP=1.E-6 340 INIT=IINIT P1NT=FFINIT ASTT=AST (1) C IF (INEL.NE.NREH) 310,360 IP (INC. EQ.O) 30 TO 360 IF (INEL.NE.NNEN) GO TO J10 GO TO 360 C 350 NODI=NODI»INC NODJ=NODJ+INC ASTT=AST(2) C 360 JHEH=IHEN IF(INC.EQ.O) J R EH = I NEL PRINT 370, ASTT.JHEH, NODI, NODJ, INC, IHBT, IHHBT,IHOED,IECC,KSFI,KSF J i 1 YNG,YNT,KFDL,KFLL, FDL, FLLN.INIT, PINT 370 FOBHAT(A1,I4, 315,416,217,5X, A4.2 X,A4 ,216 ,2F 10. 2,16, F10. 2) C C COUNT NUHBER 3F ELEMENT TIRE HISTORIES C IF (KOUTDT.NE. 0) NELTH=NELTH* 1 C C LOCATION HATEIX C DO 380 1=1,3 LH (I)=ID(NODI, I) 380 L(1 (1*3) = ID(NODJ,I) CALL BAND C C ELEMENT PROPERTIES C XL=X(NODJ)-X (NODI) IL = Y (NODJ) -Y(NODI) IF (IECC. EQ. 0) GO TO 400 DO 390 1=1,4 390 EC (I) = ECC (I ECC, I) XL = XL-EC( 1) »EC (2) — YL=YL-EC(3) »EC(4) " v l C 400 FL = SQRT(XL«*2«-YL**2) > > J 400 PL = USQRT(XL**2»YL**2) COSA=XL/FL SINA=YL/FL EI L = FTYP (IHBT, 1) /PL EAL=FTYP(IMBT,2)/FL FACL=FTXP(IMBT,4) fACR=FTYP(IMBr, ci) FACLB=FTYP(IMBT.6) DET=(FACL«PACR-FACLR»»2) *EI L BFI I=PACR/DET BPJJ=FACL/DET Ul'IJ=-FACLR/DEr SHI 1= HNRT (IHMKT, 3) SHI2 = HMRT (IHMRT ,5) SHJ 1 = HMRT(IHMRT,4) SHJ2= HHRT (IHHHT , 6 ) C C AOD SHEARING QEFORfl AT IOll C IF (FTYP(IHBT,3). EJ.O.) GOTO 420 SHFAC=1./(PL*PTYP (IH3T,3)) BFII=BFII*SHFAC BFJJ = BF.1J»SHFAC BFIJ=BPIJ»SHFAC C C CALCULATE HINGE STRAIN HARDENING C 420 EKH (1) = HMRT (IHMRT, 1) EKEP( 1, 1| =EKH{ 1) EKEP(1,2) = EKH(1) *SHI1 EK EP ( 1, 3) =EKH (1) *SH 12 EKH (2) =HHRT (IHNRT, 2) EKEP(2,1) = EKH(2) EKEP (2, 2) =EKH(2) *SHJ1 EKEP(2,3) = EKH(2) *SHJ2 EKH(1) = EKH(1)» (10.**15) EKH (2) = EKH(2) * (10. **1S) EKHIP=EKH (1) EKHJP=EKH(2) EK 1 (1J = EKH (1) EK 1 (2) =EKH(2) I N D ( l ) - ' IND (2) = 1 C C CALCULATION OF CRACKING MOMENTS C UHCR (1,1) =BMCK (KSPI , 1) BMC B (1 ,2) = BMCK (KSFI, 2) BMCR(2, 1) = BMCK (KSFJ,1) BHCB (2,2) = BMCK (KSFJ, 2) RREC(1, 1) =0.DC BREC(1,2)=0. DO liREC(2, 1) =0.00 RREC(2,2)=0.D0 C C CALCULATION OF EVRNT POINTS C DO 421 IEND=1,2 DO 421 J=1, 2 BMCU (I END, J) = 1. 800 D0»BnCS ( 1 EN U, J) 421 CONTINUE C GO TO(424, 428,4.12) , IKOED 424 DO 425 IEND=1,2 DO 425 J = 1, 2 BMCRI (IEN5,J)= 1. 2J0»B^CK(IESo, J) ROCU (IEND,J) = (BMCU (IENU,J) -B1CRI (1END, J) ) /EK EP (IEN D , 2) ROCH(IEN0, J| =S.00»ROCU (IEN0,J) 425 CONTINUE GO TO 440 428 DO 429 IEND=1,2 UO 429 J=1,2 UHCRI (I END, J) = 1. 2D0*DMCB (I END, J) ROCU (I END, J) = (BMCU(IEND.J) -BMCRI (IEND, J) ) /E K EP ( IEND, 2) ROCH (1 END, J) =2. 500»BOCU (IEND,J) 429 CONTINUE GO TO 440 432 DO 433 IEND=1,2 DO 433 J=1,2 UHChI (IEND,J) = 1.0D0»BMCB(IEND,J) ROCU (IEND.J) = (BMCU (IEND, J ) - B H C a i (IEND, J) )/EK £P (IEND, 2) ROCU (IEND.J) = 1.C0»BOCU.(IEND,J) 433 CONTINUE C C DISPLACEMENT TR ANS FORM ATION C 440 A(1,1) = -5INA/PL A (1,2) =COSA/FL A(1,3) = 1. 4(1,4)=-* (1,1) A(1,5) = -A(1,2) *( 1.6) =0. A(2,1) = A(1,1) »(2,2)=A(1,2) A(2,3)=0. A(2,4) =A(1,4) A(2,5) = A(1,5) A(2, 6) =1. IF (IECC. EQ.O) GO TO 442 A (2, 3) =(SINA»EC (3) *COSA»EC (1) ) /FL A(1,3) = 1.»A(2,3) 4 (1,6)= (-SINA*EC(4)-C0SA*EC (2)) /PL 4(2,6) =1.»A(1.6) GO TO 445 442 EC( 1) =1.23456E10 445 CONTINUE . C C L04DS DUE TO FIXED END FORCES C DO 450 1=1,6 SPF(I) =0. 450 SSFF(I)=0. IF (KFDL*KFLL. EQ.O) GO TO 580 DO 460 1=1 ,6 DO 460 J = 1,fa 460 GA (I,J) =0. GA ( 1, 1) =L'OSA GA (1,2)=SINA GA (2, 1) =-SINA GA (2,2) =COSA GA (3, 3) =1. GA (4 ,4)=CUSA GA ( 4 , 5) =SINA GA (5,4) =-SI NA GA (5, 5) =COSA GA ( b , 6 ) = 1. FF (3) = 3 S F F (2) FF (4) =SSFK ('j) FF (5) = SSFP ( 1) FF (6) =SSFF (4) DO 630 1=1.6 IF (FP(I) . LT.3. ) GO TO 620 SENP(I) =FP (I) SENN (I) =0. GO TO 6 30 620 SENN(I) =FF (I) SENP(I) =0. 630 CONTINUE C CALL FINISH C C GENERATE HISSING ELEMENTS C IF (IHEM. EQ. NHEM) RETURN IHEH=IHEH*1 IF (INC. EQ.O) GO TO 310 IF (IHEH.EQ.INEL) GO TO 330 GO TO 350 C END SUBROUTINE STIP8 (MSTEP.NDOF,HINPC.COMS,PK,DFAC,KCONT) IMPLICIT BEAL*8(A-H.O-Z) C COMNON/INFEL/ I MEM, INEH 0, KST. KSTD, LH (6) , LND (6) , KGEOM .KG.JOMD , * FL,C0SA,SINA,EC(4| ,A(2,6) , 1 BPII,BFJJ,BPIJ,EAL,EKHIP,EKHJP.EKU (2) ,KODYX(2) , * ND(4) ,KODX (2) , £K EP (2, 6) ,IND(2) ,IN0QB(2) ,BEVPT(2) , 2 BNCB (2,2) ,BNCBI (2,2) , BHCU (2,2) ,ROCU (2, 2) , KOCH (2, 2 ) . * UHTOT(2) ,SFTOT(2) ,FTOT(2) ,BOTOT(2) ,BOPLS(2) , 3 BOELS(2),BOPHX(2| ,HONMX(2) ,BOTPN(2) ,BOTPP(2) , * SENP(IO) ,SENH(10) ,TENP(10) ,TEHN (10) ,S DACT (3) , NO0I li NODJ,KOUTDT, KDUH (3) ,EK1 (2) , I HOED, IIBCED, RBEC (2, 2} « UMT1(2) ,UHT2 (2) ,BMTT (2) , ROTT (2) ,BMTD (2) , aOTD (2) , 5 UHIP(2) , KOIP(2) , DHCD(2) , ROC0(2) ,BHCP (2) ,HJCP (2) , * BHLP(2) , BHH V (2) ,EKCD (2) , EKFG (2) , EK JF (2) , EK GL (2) , 6 EKGA (2) , EKIH (2) , BEST (1 I) COHHON /WORK/ ST (2, 2) , STT (2, 2) , AT K (6, 2) , AA (2, 6) ,PFL , AXK ,. AC , 1 FFK(6,6) ,H(1929) C DIMENSION COH(1) ,COHS(1) ,FK (6,6) , KCONT(U) EQUIVALENCE(IHEH,COa(1) ) C C STIFFNESS FORMULATION, P.S. BEAH ELEMENTS C DO 10 J=3,40 10 COM (J) =COHS (J| C C CURRENT FLEXURAL STIFFNESS C CALL FSTFU (ST.EKIl(l) ,EK1I(2)) C C PREVIOUS STIFFNBSS C C It (KPDL. EJ.O) GO TO 500 DO 170 1=1,6 470 PFEP (I) =PEF (KPDL, I) *FDL IF (KDFEF(KPDl).EQ.O) GO TO 4U0 CALL HULT (GA, FFEF, SFF.6,6, 1) GO TO 500 480 DO 490 1=1,6 490 SFF (I) = FFEP (I) C 500 IF (KFLL. EQ.O) GO TO 540 DO 510 1=1,6 FLL=FLLF*FLLH IF (I.EQ. J.OB.I.EQ.6) 1LL=FLLH 510 FFEF (I) = FEF (KFLL,I) »FLL IF (KDFEF (KFLL) .EQ. 0) GO TO 520 CALL MULT (GA, FFEF, SSFF,6,6 ,1) GO TO 540 520 DO 530 1=1,6 530 SSFF (I) =FFEF (I) C 540 CONTINUE DO 550 1=1,6 550 FF (I)=SFF (I)*5SFF(l) C CALL HULTT (GA,FF, DD,6, 6, 1) IF (IECC.EQ.OI GO TO 560 DD (3) = DD (3) -DD(1) «EC(3) •DD(2) »EC (1) DD (6)=DD(6)-DD (4) »EC (4) »DD (5) *EC (2) 560 CALL SFORCE (DD) C C HODIFY TO GET INITIAL ELEMENT FORCES C DO 570 1=1,6 PLL= 1. /FLLF IF (I.EQ.3.OH.I.EQ.6) FLL= 1 • 570 SFF(I(=3FF(I)»SSFF(I) *FLL C C INITIAL FORCES C 580 DO 590 1=1,6 590 SSFF(I) =0. IF (I NIT. EQ.O) GO TO 6 10 DO 600 1=1,6 SSFF (I) =FI NIT (1 NIT, I) * PINT 600 SFF (I) =SFF (I) »S5FF (I) C C INITIALIZE ELEMENT FORCES C 610 FTOT(1) =3FF(1) F10T (2) =SFF (U) SFTOTJ 1) =SFF(2) SPT0T(2)=SFF(5) DM TOT ( 1) =SPF (3) DM TOT (2) =SFF (6) C C INITIALIZE ARRAYS C FF (1) =Sf.Fi'' (3) FF (2) =SSFF (b) IF (MSTEP.LT.2) i J TO JO CALL FSTF8 ( STT , EK IUP , EKH JP) C C STIFFNESS DIFP EliE.NCii C DO 20 1=1,4 20 ST (1.1) =ST(I,1)-STT (I, 1) CALL MULTST (A,ST,ATK,FK,6,2) COHS (3J) = EKH (1) COHS(34)=EKH(2) RETURN C C ORIGINAL STIFFNESS AT STEP ZEEU, BKTA-0 COREN AT STEP 1 C JO FAC=1. IF (MSTEP.EQ.O) GO TO 50 FAC=DFAC DO UO 1=1, 4 40 ST(I,1)=ST(I,1) »FAC 50 CALL HULTST (A , ST, ATK, FK,6 , 2) IP (FAC. EQ.O.) GO TO 80 EAL=EAL*FAC AXK=EAL»COSA**2 FK (1,1) = FK (1,1) *AXK FK (1,4) =FK (1,4)-AXK FK (4,4) = FK (4,4) »AXK AXK=EAL»SIBA**2 FK (2,2) =FK (2,2) *AXK FK (2,5)=FK (2,5)-AXK FK(b, 5) =FK(5, 5) »AXK AXK=EAL*SINA*COSA FK (1,2) =FK (1,2) »AXK FK (1,5) =PK (1 ,5) -AXK FK (2,4) =FK (2,4) -AXK PK (4,5) =PK (4,5) »AXK IF (EC ( 1) .EQ. 1. 234 56E10) GO TO 60 EC3 = COSA»EC (3) -SINA*EC (1) EC4=SINA*EC (2) -COSA*EC (4) AXK=COSA*EC3*EAL FK (1,3)=FK(1,3) -AXK FK (3,4) =FK (3,4) +AXK AXK=SINA*EC3»EAL FK (2,3)=FK (2, 3)-AXK FK (3,5) =PK(3,5) »AXK FK (3,3) =PK (3,3) »EAL*EC3*»2 AXK=COSA»EC4»EAL FK (1.6)=FK (1,6|-AXK FK (4,6) =FK (4,6) »AXK AXK=SINA*EC4*EAL FK (2,6) =FK (2,b)-AXK FK (5,6) =FK (5,6) »AXK FK (J,6) = FK(3,6) »EC3*EC4*EAL FK (6,6) =FK (6,6) »EC4»»2»EAL EAL=EAL/FAC 60 DO 70 1=1,6 DO 70 J=I,6 70 FK(J.I) =FK(I ,J) C C ADD GEOMETRIC STIFFNESS C 80 IF (HSTEP. E Q . O . O H . K . ; E O M . EQ.O) GO TO 130 P F L = ( C O H S(8 2) - c o r . s (Hi)) /(2.»FL> DO 90 1=1,4 90 ST(I, 1) =PFL DO 100 1=1,12 100 AA (I, 1) =0. AA (1,1)=-SINA AA ( 1,2) =COSA AA (2,4)=SINA AA (2, 5) =-COSA IF (EC (1). EQ. 1. 2J456E10) GOTO 110 AA (1,3) =SINA«EC (3) »COSA*EC (1) AA (2,6) =-3INA»EC(4) -COSA»EC(2) C 110 CALL HULTST (AA.ST, ATK, FFK, 6, 2) DO 120 1=1,36 120 FK (1,1) =FK (1,1) »FPK (I, 1) C 130 IF (KCONT (10).EQ.O) GOTO 140 IF(HSTEP.GT.O) (iO TO 140 WRITE (7) (FK (1,1),1=1,36) C 140 RETURN C END SUBROUTINE RESP8 (NOOF , NINPC, KBAL.KPR,COHS , DDISH, DD, TIHE, VELH, DP AC 1, DELTA, KCONT) IHELICIT REAL'S(A-R,0-Z) C COHNON/INPEL/ IHEH, IHEHD, KST,KSTD,Lfl (6) ,LSQ (6) .KGEOH.KGEOHO, • FL,COSA,SINA,EC(4) ,A (2,6) , 1 BPII.BFJJ,BPIJ,EAL,EKHIP,EKHJP,EKIi (2) ,KODXX(2) , • NO(4) .KODI (2) , EKEP(2,6) ,IND(2) ,IN0UH(2) ,H2VPT(2) , 2 BHCB (2,2) .BHCRI (2,2) ,BHCU (2,2) ,ROCU (2, 2) , aoCH (2, 2 ) . • BNTOT (2) ,SFTOT(2) ,FTOT(2) ,BOTOT (2) ,ROPLS(2) , J BOELS(2),BOPHX(2| ,RONNX(2) ,ROTPN(2) ,ROTPP(2) , • SENP(IO) ,SENN (10) ,TENP(10) ,TENN(10) ,S CACT ( J) , NOD I 4 NODJ,KOUTDT,KDUH(3) ,EK1 (2) , I HOED, IIHOED, KiEC (2, 2) * UHT1(2) ,0HT2(2),B.1TT(2),ROTT(2),BHTD(2),i1JTJ(2), 5 BHIP(2) , ROIP(2) , BHCDJ2) ,R3CD(2) ,BHCP(2) ,RJC? (2) , * BrtLP(2) ,0HHH(2) ,EKCD (2) ,EKFG (2) , EK JF (2) , EK GL (2) , 6 EKGA (2) , EKIH (2) , REST (11) COMMON /WORK/ D Vtll , DVR J , BHDT4 ( 2) , DBH ( 2) , ST ( 2, 2) , N ( 1 990) COHMON /TIIIST/ ITHOUT(IO) ,THOUT(20) ,ITH P.IS AV t , NtLTH, NSTU, NF7,IS E C DIMENSION COM(1) , COHS ( 1) , DDISH ( 1) , DD (1) , VELH (1) ,NOD (2) ,KuONT(1) DIHEHSION REMIND (2) ,DBO (2) ,BETO(2) ,UHINC(2) DI HENSION BH(2) ,RT(2) ,IDK (2) ,KODE(2) ,EK (2) .FACTOB (2) ,FACAC(2) C EQOI VALENCE (IHSH,CON(1)), (NODI , NOD (1) ) C C STATE DETERMINATION, P.S. BEAM ELEHENT C DO 5 J=1,NINFC 5 COM (J) =COHS (J) KODYX (1) =KUDY (1) KODYX(2) = KODY(2) IF (I "EN. EC 1) IliED^O DEFORMATION INCKENENT5 fOR EXTERNAL MDQCS IF (EC (1). EQ. 1. 23456E10) U O T O 10 DDISM( 1) =D0ISB ( 1) - EC (3) *DDISM (3| 00 ISA (2) = 0JISfl (2| »EC( 1) *DDISfl ( 1 ) DDISB (4) =UDISN ( I t ) - EC (4) *DDISB (b) DDI SB (S) =DDISB (5) *EC(2) *DDISM (b) DVAX = COSA* (DDISB (4) - DDISH( 1) ) »SIN A* (DDISil (5) -DDISI1 (2) ) ROT* (SIN A* (DDIS* (4) -DDI SB ( 1) ) »COS A* (DDIS B ( 2) -DUISM(5) ) ) /F L DVRI=DDISM(3) »HOT OVRJ=DOISH (6) *HOT AXIAL FORCE INCREMENT DP AX= EAL*i)V AX PT0T(1)=FT0T(1)-3VAX FTOT(2) =FTOT(2) *DVAX ROTATIONAL I NCR EHENTS DRO(1) = DVRI DRO(2) = DY8J LINEAR HOBENT INCREMENTS CALL BHCAL8 BBTOT1 = 6BTOT (1) BHTOT2=BMTOT(2| BBINC(1) = BBTOr (1) • DBH(1) BBINC(2) =BBTOT (2) »DBH(2) INTIALIZE ARRAYS IP (TIB E.GT- 0. 000) GO T O 15 DO 16 1=1,2 ROTOT(I) =0.D0 ROTPP(I)=0. DO kOTPN(I) *0.D0 HOPLS (I)=0.D0 ROELS(I) =0.D0 ROPBX (I) =0. DO RONBX(I) =0.D0 BBTT (I) =0. DO ROTT(I) =0.D0 BBTD(I)=0. DO ROTD(l) =0.D0 BBIP(I) =0. DO ROIP(I) =0.D0 CONTINUE TRACE OUT NONLINEAR PATH DO IB I END*1.2 FACTOR (IEND) =0. DO F AC AC (I END) =0. DO CONTINUE KUAL=0 KFAC=0 KC YC=0 SFAC=0.D0 C 30 CONTINUE DO 450 IEND=1,2 PACTOR (IEND) =1.D0-FACAC (IEND) IF (FACTOR (IEND). EQ. 0. DO) GO TO 450 C C SET ROTATION REVERSAL INDICATOR C RE BI ND (I END) = 1. 00 IF (IND (IEND) .EQ.2) BEBI ND (I END) =-1. DO IF (DBB(IENO) . EQ.O. DO. AND. EKH(IEND) . N E. 0. DO) GO TO 450 IF (DRO (IEND) .EJ.O.DO.AND.KODY(IEND) .E2* 2) GO TO 450 C C RULE ONE. ELASTIC STAGE. GET PACTOR POR CHANGE. O-A. C IF(KODY (IEND) .NE.O.DO) GO TO 50 35 KO0Y(IEND)=0 IND(IEND) * 1 IF (DBB(IEND).GT.O. DO) GO TO 40 FAC=DABS( (BBCRI (IEND, 2) -BBTOT (IEND) ) /DBA (I END) ) IF (FAC. GE. FACTOR (I END) ) GO TO 450 C FACTOR (I END) *F AC BB (IEND) =BBCRI (IEND,2) RT (IEN0) = 0. DO KODE(IEND) * 1 EK (IEND) = EKEP (I END, 2) IDK (IEND) =2 GO TO 450 C 40 FAC=OAas ( (BBCRI (IEND, 1) -BBTOT (IEND) ) /DBH (IEND)) IF (FAC. GE.PACTOR (IEND) ) GO TO 450 C FACTOR (IEND) =FAC BB (I END) = BBCRI (IEND, 1) RT (IEND) =0.00 KODE(IBND) = 1 EK (IEND) =EK£P(IEUD,2) IDK(IEND) = 1 GO TO 4 50 C 50 KO D=KODY(TEND) GO TO (60,70,85 ,90, 100, 120, 140,160,185,200,220,230,24 0,260,280,30 0. • 310,320,340,350,360,380,390, 400,410,420| ,KOD C C RULE THO. CRACKING STAGE. GET FACTOU FOB CHANGE. A-B. C 60 IF (RilNINDIIKND) *DBB (IEND) .LT.O. DO) GO TO 90 c IF (DBH (IEND) .iT.O.DJ) GO TO 65 FAC=DABS( (BMCU (I CN D, 2) - DBTOT (IEND) )/D3M (IEND) ) IP (FAC. GE. FACTOR (IEND) ) GO TO 450 C FACTOR(IEND)*PAC BH ( IEN0) =BBCU(IE«D, 2) RT (IENU) = ROCU(IEND,2) KODE(ISND) -2 EK (IEND) -0. DO IDK (IEND) -2 UMT2 (I END) =2.30 GO TO 450 C 65 FAC=DADS ( (BMCU (IEND, 1) -3MTOT ( I EN D) ) /DBM (IEND) ) IF (FAC.GE.FACTOn (IEND) ) GO TO 450 C FACTOR (IEND) =FAC BH(ItNU) = BMCU{IENlJ, 1) RT (IEND) =ROCU(IEND, 1) KOD£(IEND)=2 EK (IEND) =0.D0 UBT1 (IEND) = 1.DU IDK (IEND) =1 30 TO 450 C C RULE THREE. CRUSHING OF HINGE. B - il. C 70 IF (BEHIND (IEND) »DRO (IEND) . LT. 0. DO) GO TO 160 KODY (IEND) =2 7 4 FAC=DABS( (ROCU (IEND, IN 0(1 END) ) -ROPLS(I END) )/DRO (IEND) ) IF (FAC. G£. FACTOR(IEND) ) GO TO 450 C FACTOR (I END) = FAC BH (IEND) =BHCU(IEND,IND (IEND) ) RT (I END) = ROCH (I END, IN D (I END) ) EK (IEND) =EKEP(IEND,3) IDK (IEND)=IND(IEND) KODE (IEND) = 3 GO TO 450 C C RULE FOUH. DEGRADING STAGE. E->. C 85 KODY (I END) = 3 IF (DABS(BMTOT(IEND) ) .GT.O.DO) GO TO 87 BHCU (I END, IND (I END) ) =0. DO EKH(IEND) = 0.D0 87 IF (EKH (IEND) . EQ.O. DO) GO TO *li IF (IND(IEND) .Eg. 1. AND. DRO (IEND) . LT.O. DO) CO TO 1L0 IF (IND(IEND) . iiQ. 2. AND. DRO (IEN D) . GT. 0. DO) GO TO 160 88 GO TO 450 C C RULE FIVE. UNLOADING PRCH POINT C. C-D. C 90 IF (IND (IEND) . EQ. 1. AND. UMT2 (IEND) . EQ. 2. DO) GO TO 210 IF (IND (1EIID) .EQ. 2. AND. UHT1 (IEND) . EQ. 1. DO) GOTO 240 IP ( KODY (I END) . EQ. 4) GO TO 95 IF (KODY (IEND). EQ. 5) GO TO 95 BNTT (IEND) =BHTOf (IEND) BOTT (I END) = ROPLS (TEND) IF(POTT(IEND) .EQ.O.DO) SO TO 35 BHCD (I END) =0. 6 D0«BHCRI (IZNO, IND( i END) ) BTCU= (BUTT (IEND)/DHCU (I END, IND (IEND) ) ) GO TO(92,92,93) , IMOED 92 RTCD=6.6686-13. 390* BTCII tb. 9 429»BTCU •» 2 BOCD(IRND) =nTCD«HOTT(IEND) GO TO 94 93 HTCD=5.7786-12.787»BTCD*7.1324«DTCU*«2 ROCD(IEND) = RTC D* ROTT ( I EN D) 94 EK (IENU)= (UMTUT (IEND) - BMC J( IEND) ) / (ROPLS (I END) - ROC ) (IEND) ) EK H (IEND) = EK (IEND) EKCD(I END) = EK (I END) KODY (IEND) =4 FACTOR (I END) =0. DO GO TO 4 50 C 95 K0DY(IEND)=4 IF (BEHIND (I END) *DBM (IEND) . GT.O. DO) GO TO 100 FAC = DABS ( (DHCU (IEND ) - B HTOT (I END) ) /DBS (I END) ) IF(FAC. GE. FACTOH(IEND) ) GO TO 450 C FACTOR (IEND) = FAC BH (IEND) = BMCD (IEND) RT(IEND)=ROCD(IEHD) EK (IEND) = (BHCD (IEND) •DNCRI (IEND,I ND (I END) ) ) / (BOCD (I END) ) KODE(IEND) =6 IDK (IEND) = IND (IEND) IF (PACTOR (IENO) .EQ.O.DO) GO TO 440 GO TO 450 C C RULE SIX. LOADING ALONG D-C. C 100 IF (REHIND(I END) •DBH(IEND) .LT.O.DO) SO TO 90 KODY (IEND) =5 FAC = DABS( (BMTT (IEND) -BHTOT(IEND) ) /DBH (IEND) ) IP (FAC. GE. FACTOR (IEND) ) GO TO 450 C FACTOR (IEND) =FAC DM (IEND) = BMTT(I£ND) RT (IEND) - ROTT (IEND) KOUE(IEND) =1 £K(I£N0) = EKEP(IEND,2) IDK (IEND) =IND(IEND) IF (PACTOR (IEND) . EQ.O. DO) GO TO 440 GO TO 450 C C RULE SEVEN. UNLOADING ALONi D-A. C 120 KODY(IEND)=6 IF (REMIND (I END) *DDH (IEND) . GT.O. DO) GO TO 140 IF (DBH (IEND) -3T. 0. DO) GO TO 125 FAC=DABS( (OMCRI (IEN3.2) —BMTOT (IEND) J/DBM (IEND) ) IF (FAC. GE. FACTOR (IEND) ) GO TO 450 C FACTOR (IEN3)=FAC BH (IEND) =DHCr.I (IEND, 2) RT (IEND) =0. DO IDK (IEND) =2 GO TO 130 C 125 FAC = DABS ((BMCRI (IEND, 1)-BMTOT (IEND) ) /DBH (IEND)) IF (FAC. Gf. FACTOR (IENO) ) GO TO 450 C FACTOR(IEND)=FAC BH (IEND) = UMCRI (I PHD, 1) RT ( IEND) =0.D0 IDK (IEND) = 1 130 EK(IEND) =EKEP(IEND,2) CO M KODE(I END) = 1 IF (FACTOR (IEND). EQ. 0. DO) CO TO 440 GO TO 450 C C RULE EIGHT. LOADING ALONG A-D. C 140 IF(REHINU(IEND) *0D« (IEN0) .LT.O.DO) GOTO 12C KODY (I END) = 7 FAC=DABS ( (UNCD (IEND) - BHTOT ( I E ND) ) /DBH (I END) ) IF(FAC.GE.FACTOB(IEND) ) GO TO 450 C FACTOR (IEND) =FAC 6M(IEND) = BHCD(IEND) RT (IEND) =HOCD(IEND) KODE (I END) =5 EK (IEND) =EKCD(I£ND) IUK(IEND) = INU(EEND) IP (FACTOR (IEND) .EQ. 0.00) GO TO 440 GO TO 450 C C RULE NINE. UNLOADING FROM PT. F. C 160 IF (REMIND (I ESD) * DBB (I EN 0) . GT. 0. DO . AN D. KODY (I END) . EQ.O) •GO TO 185 IP (KODY (I END). EQ. 8) GO TO 165 IF(KODY(IEND).EQ.9) GOTO 165 BHTD(IEND) =BHTOT (I END) ROTD (I END) = ROPL5 (IEND) HTCU=BOTD(IEND) /ROC U (I END, I ND (IEND) ) GO TO(161,162, 163) , INOED 161 BH IP (IEND) =0.33333333*BHCU(IENO,INO(IEND)| BH LP (I END) =0.55555555* BHCU (I END, IND (IENO)) BTCD=0.2734800*0.054357D0*HTCU-0.0017017D0*RTCU*»2 IF (DABS (BOTD (IEND) ) .GT. DABS ( (20. 0D0»BOCU (I END,I ND (IEND))) )) *RTCU=0.7D0 BO IP (IEND) =RTCD*ROTD (I END) GO TO 164 162 B8IP (IEND) =0. J3333333*Di1CU (IEND,I ND (IEND) ) BHLP(IEND) =0. 55555555*BHCU (IEND, IND (IEN D) ) RTCD=0.19252D0*O.11902D0*RTCU-O.68746D0*RTCU**2 IP (DABS (ROTD (I END) ) .GT. DABS ( ( 10. 0D0*BOCU (I END, I ND (I END) ) ) ) ) *BTCD=0.7DO ROIP(IEND) =RCTD*ROTD (IEND) GO TO 164 163 BH IP (IEND) =0. 333333 33*DHCU (IEND, I ND (IEND) ) bHLP (I END) =0. 500O00O0*Bi1C0 (I EU J, I N D (I EN D) ) RTCD=0.23000*0.050D0»RTCU IP (DABS(POTD(IEND) | .GT. DABS ( 110. 0D0*KUCU (I END, 1 ND (1 END) J ) ) ) *HTCD=0.7U0 ROIP(IEND) =nTCD*l(OTU(IEND) GO TO 164 164 CONTINUE IF (DABS (B.1T0 (IEND) ) .GT. DABS (BHLP ( IEND) ) ) UO TO 166 BHLP (I END) =0HTD(I£NJ) 8 HI V (I END) = BHL P(1EU D) -0. 4 DO* UNCI: ( I cIN J, IN i) (I EN D) ) 166 EK (IEND) = ( 1 1 H T D (IEND) - Bill V (I END) ) / (BOTD (IEND) •-ROIP(IEND) ) EK FG (UNO) -EK (Ic.'lJ) EKH (IEND) = EK (IENO) KODY (I END) = fl FACTOR(IEND)=0.00 GO TO 45C C 165 IF (DDil (IENJ) . EQ.O. DO) GOTO 450 KODY (IEND) =8 FAC=DABS( (BHIP (IEND) -BHTOT (I END) )/D BH (IEND) ) IF (FAC. GE. FACTOR (IEND) ) GO TO 450 C FACTO R(IEND) = FAC OH (IEND) =0NIP(IEN0) RT (IEN0) = ROIP(IEND) IF(IND(IEND) .EQ.2.AND.UHT1 (IEND).EQ.1.DO) GOTO 176 IF(IND(IEND) .EQ.1. AN D. U HT2 (I END) . EQ. 2. DO) GOTO 176 169 EK (IEND) = (BHIP (IEND) »BHCRI (IEND, I ND (I END)) )/(R3IP (IEND) ) IDK (IEND) =IND(IEND) KODE (IEND) = 10 EKGA (IEND) =EK (I END) GO TO 450 C 176 KODE (I END) =21 IDK(IEND) =IND(IEND) EK (IEND) = (BHIP (IEND) *BHLP (I EN D) ) / (RO IP (IEND) ••ROIP(IEND) ) EKGL(I END) =EK(IEND) GO TO 4 50 C 177 DBH (IEND) =0. DO FACTOB (IEND) = 1.DO GO TO 450 C C RULE TEN. LOADING ALONG F-G. C 185 IF (REHIND(IENU)*DBH(IEND).LT.O.DO) GO TO 160 IF (KODY* (IEND) .EQ. 2) GO TO 177 KODY (I END) =9 FAC=DABS ((ONTO (IEND) -BHTOT (IEND) ) /DON (IEND) ) IF(FAC.GE.FACTOR(IEND)) GO TO 450 C FACTOR (IEND) =FAC BH (IEND) = BHTD(IEND) BT ( IEND) =ROTD ( I END) IF (DABS (ROTD(IEND)). GT. DADS (ROCH(I£ND, IND (IEND))) ) GOTO 136 KODE (IEND) =2 EK (IEND) =0. DO GO TO 187 186 KODE (IEND) =3 EK (IENO) = EKEP(IENO, 3) 187 IDK(IEND) =IND(I£ND) IF (FACTOR (I END) . EQ. 0. DO) GO TO 180 GO TO 450 188 DBH (IEND) = 0 . DO FACTOR (IEND) =1.00 IF (JRO(IEND) *n3NIND(lEND) .LT.O. DO) GO TO 139 C DRO (IEND) =0.00 GO TO 440 189 OHO(IEND) = 0 . DO EK ( IEND) =EKil (I END) KODE (I END) =8 GO TO 440 C RULE ELEVEN. UNLOADING A LO N'.J G - A . C 200 IP(REMIND(IEN0) •DUN (IEND) .GT.O.DO) 30 TO 220 KODY(I END) =10 IF (DBH (IEND) . LT. 0. DO) GOTO 210 204 PAC=DABS((BHCRI (IEND, 1)-3.1T0T (I E ND) )/D BB (IEND)) IF (FAC.GE. FACTOR(IEND) ) GO TO 450 C FACTOR (IEND) = FAC BH (IEND) =BHCRI (IEND, 1) RT(IENO)=0. DO KOOE(IENO) = 1 EK(IEND)=EKEP(IEND, 2) IDK (IEND) = 1 IP (FACTOR (IEND) . EQ.O. DO) GO TO 440 GO TO 450 C 210 PAC=DABS( (UHCRI (I END, 2) -3HTOT (IEND) )/D B.I (I £N D) | IF (FAC.GE. FACTOB(IEND) ) GO TO 450 C FACTOR (IEN0) = PAC BH (IEND) =BBCBI (IEND,2) RT (IEND) =0. DO KODE (IEND) =1 EK (1END) = EKEP(IEND,2) IDK (IEND) = 2 IP (FACTOR (I END) . EQ. 0. DO) GO TO 440 GO TO 450 C C RULE TBELVE. LOADING ALONG G'-L. C 220 IF(BEHINO(IEN0) *O0H (IEND) .LT.O.DO) GO TO 230 IP (KODY (IEND) . EQ. 11) GOTO 225 IF (KODY (IEND). EQ. 12) GO TO 225 BBTT (I END) =BBTOT (I END) ROTT (I END) = ROPLS (IEND) RTT=ROIP (IEND) - ROTT (IEND) IF (ETT. EQ. 0 . DO) GO TO 223 EK ( IEND) =(DHLP(IEND) -BHTT (I 2ND) ) / ( ROIP (IEND) -HOTT (I END) ) GO TO 224 223 EK (IEND) = EK 1 (IEND) 224 EKH (IEND) - EK (IEND) KODY (IEND) =11 PACTOR (IEND)=3. DO GO TO 4 50 C 225 FAC=DABS( (BHLP(IEND) -UMTOT (I END) ) /DBM(ISND) ) KODY (I END) = 11 IF (FAC. GE. PACTOR (IEND) ) GO TO 450 C FACTOR(IEND)=PAC BH(IEND) =BHLP(IENU) RT(IEND) = ROIP(IEND) KODE (IEND) =18 EK (IEND)= (BHTD (I UNO) -BflLP (I i>N J) ) / ( i lOT. ) ( I ZU L)J -i< J Ic> (I END) ) EKJF (IEND) =EK (IEND) IDK (I END) =IND(I END) IP (FACTOR (IEND) . EQ. 0 . DO) GO TO 440 GO TO 4 50 C C RULE THIRTEEN. UNLOADING ALONG G»-L. C 230 IF (BEHIND (IEND) *DBN (IEND) .GT.O.DC) GO TO 220 KODY (I END) =12 C FAC=DABS ( (B.1TT (IEND) -BBTOT (TEND) ( /DBM (IEND) ) IF (FAC.GE. FACTOR (IEND) ) GO TO 4S0 C FACTOR (IEND) =FAC BB (IEND) = BHTT (IEND) RT (IEND) =R0TT( IEND) KODE (I END) =10 EK (IEND) =EKGA (I END) IDK (IEND) =IND(I£ND) IF (FACTOR (IEND) .EQ.O.DO) GO TO 440 GO TO 450 C C RULE FOURTEEN. UNLOADING ALONG I-H. C 240 IF (REBIND(IEND) *D BB (I EN D) . GT. 0. DO) GO TO 260 IF (KODY (IEND) . EQ.13) GO TO 250 IP (KODY (I 2ND) . EQ. 14) GO TO 250 BBTT (IEND) = BBTOT (IEND) ROTT(IEND) =ROPLS(IEND) GO TO(241,244,248),IBOED 241 BHLN=.4DC»BBCB (IEND,IND (IEND) ) GO TO 249 244 BBLB=0. 400*BB3B (IEND, IND (IEND)) GO TO 249 248 BHL8=0.300*OH:R (IEND,IND (IEND)) 249 BMCP(I END) = BBCRI (I END, IND (IEND)) -BBLU EK (IEND)= (BBTT (IENO) -BBC? (I END)) /HOTT (IEND) EK IH (IEND) =EK (IEND) EKH (IEND) = EK (I END) KODY (IEND) =13 FACTOR (IEND)=0. DO GO TO 450 C 250 KODY (IEND) =13 FAC=DABS ((BBCP(IEND) -BBTOT (I END) | /DBM (IEND) ) IF (FAC.GE.FACTOR (IEND) ) GO TO 450 C FACTOR (IEND) =FAC BM(IEND) = BBCP(IEND) RT (IEND) = 0.D0 EK (IEND) = EKGA(IEND) KODE (IEND) =15 IDK (IEND)=IND(IEND) IF (FACTOR (IEND) .EQ.O.DO) GO TO 440 259 GO TO 450 C C RULE FIFTEEN. LOADING ALONG I-H. C 260 IF (REMIND(IEND) *DBM(1END).LT.O.DO) GO TO 240 KODY (IENO) =14 F AC = DAB3 ( (BBTT (I END) -BMTOT (I END) ) /DBM (I END) ) IF(FAC.GE.FAcrOR(IEND| ) GO TO 450 C FACTOR (IEND) = FAC BB ( IEN D) =BMTT (I END) UT (IEND) = HOTT (I END) KODE (IEND) = 1 EK(I£ND)=EKEP(I-ND,2) IDK(IEND)=IND(IENO) IF (FACTOR (IEND) .EQ.O.DC) GO TO 4 4 0 GO TO 4 50 C C RULE SIXTEEN. LOADING ALONG H-J. C 280 IF (REMIND (I END) *DBH (IEND) . GT.O. DO) GO T3 300 KODY (IEND) =15 PAC = DABS((BMLP(IEND) -DMTOT (I END) )/DBM(IEND) ) IF (FAC. GE. FACT OH (t END) ) GO TO 450 C FACTOR (I END) = FAC 8N (IEND) =BMLP(IEN0) DT (IEND) = ROIP(IEND) KODE(ILND) =18 EK (IEND)= (BHTD (IEND) -BHLP( IEND) ) / ( ROT D (I EN D) -*ROIP (IEND) ) EK JF (IEND) = EK(IEND) IF (IND (IEND). EQ. 1) GO TO 285 IDK (IEND) =1 GO TO 286 285 IDK(IENO)=2 286 IF (FACTOR (IEND) . EQ.O. DO) GO TO 440 GO TO 450 C C RULE SEVENTEEN. UNLOADING ALONJ 11 • -A. C 300 IF(HEHIND(IBN3) *DBH (IEND) .LT.O.DO) GO TO 310 IF (KODY (IENO) . EQ. 16) GOTO 305 IF (KODT (IEND). EQ. 17) GOTO 30 5 BHTT (I END) =BHTOT (I END) ROTT (I END) = ROPLS (I END) RTT=BOTT(IEND) IF (RTT. EQ. 0. DO) GO TO 30 3 EK (IEND) = (BHTT (I END) -BHCRI (IEND, I NO (I END) ) ) / (ROTl' (IEND) ) GO TO 304 103 EK (IEND) = EK 1 (IEND) 304 EKH (IEND)=EK(IEND) KODY (IEND) =16 PACTOR (I END) =0. DO GO TO 450 C 305 PAC=DABS( (BflCRI (I EN J, I N D (I END) ) -BHTOT (I END) )/CLH(IEND)| KODY (I END) = 16 IF (FAC.GE. FACTOR (IF.ND) ) GO TO 450 C FACTOR (IEND) =FAC BH (TEND) = BHCRI (I EN D, I N D (I EN D) ) RT (IEND) =0. DO KODE (IEND) =1 EK (IEND) = EK£P(IL'ND, 2) IDK (IEND) =IUb (IEND) IF (FACTOR (IEND) . EQ.O. DO) GO 7 0 440 GO TO 450 C C hULE EIGHTEEN. LOADIflC* ALONJ ;i'-A. C 310 IF (BEHIND (ILN3) *UP.t1 (IEND) .GT.O. DO) GO TO 300 KODY (I END) =17 FAC = DABS ( (BMTT (I END) -BHTOT ( I END) ) /DBH (IEND) ) IF(FAC.GE.FACT01((IEND) ) GO TO 450 C FACTOR (IEND) =FAC BH(IEND) = B.1TT(IEND) RT (IEND) =ROTT (IEND) KODE(IEND) = 15 EK (IEND) =EKGA (I END) IDK (IEND) =IND(IEND) IF (FACTOn (IEND) .EQ.O.DO) GO TO 440 GO TO 450 C C RULE NINETEEN. LOADING ON L-F. C 320 IF (REMIND (I END) *DUH (IEND) . LT.O. DO) GO TO 34U KODY (IEND) =18 FAC=0ABS ( (BHTD (I END) - BHTOT (I END) ) /DBH (I END) ) IF (FAC.GE.PACrOB(IEND)J GO TO 450 C FACTOR (IEND) = FAC BH (TEND) =BNTD(IEND) BT (IEND) = ROTD(IEND) IF(DABS(ROTD (IEND) ) .ST.DABS (ROCH (IEND,IND(IEND))) ) GOTO 324 KODE (I END) =2 EK (IEND) = 0. DO GO TO 325 324 KODE(IEND)=3 EK (IEND) =EKEP(IEND,3) 325 IDK (IEND) = IND(IEND) IF (PACTOR (IEND) .EQ. 0.DO) GO TO 440 GO TO 450 C C RULE TWENTY. UNLOADING ALONG L' -G. C 340 IF (REHIND(IEND) *DBM (IEND) . GT.O. DO) GO TO 350 IF (KODY (IEND).EQ. 19) GO TO 345 IF (KODY (IEND) . EQ. 20) GO TO 345 BHTT (IEND) =BMTOT (IEND) ROTT (I END) =UOPLS(IEND) RTT=HOTT (I END) -ROIP(IEND) IP(RTT.LQ.O.DO) GO TO 343 EK (IEND)= (BHTT ( IEND) - BHIP (I EN D) ) / (ROTT (I BN D)-BO IP (I END) ) GO TO 344 343 EK (IEND) = EK1 (IEND) 344 EKH(IEND) =EK (IEND) KODY (I END) = 19 FACTOR (IEND) =0.00 GO TO 450 C 34 5 FAC=DABS (<0NIP( IEND) -UMTOT (IEND) ) /3B.1 (I EN D) ) IF (FAC.GE.FACTOK (IEND) ) GO TO 450 C FACTOR(IEND)= *AC BM (IEND) = BMI?(IEND) RT(IEND) =HOIP(IEND) KODE(IEND) = 10 EK (IEND) = E K G A ( IEIIJ) IDK (I END) = IND(I END) CO 186 o o H o _^, o cu O M. O oa o — M> M •~ M. <N 3. o o * o «^ o 9 Z cu z z a z C J M o cu —- o u z o w O M t M M M l w a • J w w * E i Q n *3 a q ° a a O e CD z C5 03 z O z i 3 n o3 . j a cu u a \ O \ o M — . M O o ta o -»© • w o — . o a- o o Q if* —» * z CL* O A m z ?T Q li-I Q n o « a =r 1 •J • a •» < a z Z Q O -1 O z o -4 K O z O o o z O O cu O z f- -< • cu o A • cu o CC t- c -1 CU z — M M IH M -« M O ta ta o O CO o ~i — o q * — 13 z • H q ta CO « e- o CO 1 CO q * J t O a . M ) o 3 < _ l o o < ^ , u z -4 — . Q Q *« Z a Q n a Ml a a . 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Z —• M CO • a a cu a E 4 u as z z w z i CU Q 4 O cu cu ^ cu ca II CM f " M W U H w u — U — — i CM •« CM I H CM M CM co ca ^ cu — M Q o »u a-ta z o C - J i n «c O rn ta CJ z o ^  t - o £ 3 n •e, Q Q Q o Q O o Z Z Z Z f j -a ^ CU CU W M CU « Q M M M M M ^ O l w cu U Q Q Cb CU Oi • • kU M H H J M M O cc E O E E O CU X k! CO a O Q °4 • z " 1 1 1 1 1 — cu . II 11 II II II II a cN M X . ,— z II II w H ca ca o o Q ea to tO as o z z z z z Z z i-t ca a o m cu CU CU CU W CU CU z z M M M M M M Ca cu cu u H » ' W W W M * W W X t-t O H < O CM Q a Qi Qi 0* l-t " — w CM H ^ H e- > J M M — be be w -4 MS E O E E O CM Q O a CM O a u a o s o c o c a H H U H t-t i a s a in uo « x -n E-* <"> i"0 z o o M f t U ^ . — ^ . z z — — C 3 4 ( D ( N r j M M { -Q . , w w C * O O f * W « * u U O U I O • i b CL *—• Z — . E O a w a a co a a • «J M Z Z | | II w o a — M y O M M Q Z — w SC M »x >• C U C Q Q H W Q O w o> be be t-< w w w H 6., CM CM C H H H (O Q a o z z n cu cu • M M o — ' z z cu ^ SC fM ^ U II C U II A O* II 9 il o cu _ c o A o z « Q m o z u E- Z Z C U M H al O U H -a t-t H M >* W W w - B Q CM be o be be O i-t cu ca u cu be o o u < o b* ta IF (KODY (IEND). EQ. 25) SO TO 413 IF (KODY (IEND) .FQ.26) 30 TO 41b BHTT (I END) = BKTOT (IEND) KOTT(IEND) = ROPLS(IEND) RTT=ROTT (I END) -BOIP(IEND) IF(HTT.EQ.O.DO) CO TO 41] EK (IEND)= (BHTT (IEND) -DMIP ( I EN J) ) / (ROTT (I EN D)- l«3 IP (liiND) ) GO TO 4 14 413 EK(IEND)=EK1 (IEND) 414 EKH(IEND) =EK(IEND) KODY (IEND) =25 FACTOR (IEND) =0.D0 GO TO 450 C 415 FAC=DABS ( (DHIP(IEND) -BHTOT (IEND) ) /DBH (I END) ) KODY (IEND) = 25 IF (FAC.GE. FACTOR (I END) ) GO TO 450 C FACTOB (I END) =?AC BH (IEND)=DHIP(IEND) hT (I END) =ROIP(IEND) K0DE(IEND) = 21 EK (IEND) =EKGL(IEND) IDK (TEND) = XND(IEND) IF(FACTOR(IEND).EQ.0.DO) GO TO 440 GO TO 450 C C RULE TWENTY-SEVEN. LOADING ALONG L'-G. C 420 IF (BEHIND(IEND) *DBH(IEND) . LT.O.DO) GO TO 410 KODY (IEND) =26 FAC=DABS( (BHTT (IEND)-BHTOT(IEND) ) /DBH (IEND) ) IF (FAC. GE. FACTOR (IEND) ) GO TO 450 C FACTOB(IEND)=PAC BH (IEND) =BHTT(IEND) RT (IEND) =ROTT (IEND) KODE(IEND) =24 EK (IEND) =EKJF (I END) IDK (IEND) =IND( IEND) IF (FACTOB (IEND). EQ.O. DO) CO TO 440 GO TO 450 C c 440 EKH (IEND) = EK (I END) KO OY (IEND) =KODE (IEND) BHTOT (I END) = BH (IEND) BOPLS(IEND) =RT (IEND) IND(I£ND)=IDK(IEND) C c 450 CONTINUE C IF (FACTOF ( 1) .RQ.FACTOtl (2) ) GO TO 4b0 IF (FACTOR (1) . GT. FACTOR (2) ) GO T3 460 SFAC=I ACTOR (1) GO TO 470 460 SFAC=FACTOB (2) 470 CONTINUE C c C UPDATE KOHEHTS,ROTATIONS AND EVENT DATA C DO 500 IEND=1,2 IF (FACTOR (I END) • EQ. 0. DO) GO TO 500 IF( (FACAC(IEND) »FACTOB (IEND)) .3T.0.99999) GO TO 510 BHTOT (IEND) = BH ( I EN D) ROPLS (IEND) =RT (IEND) KODY (I END) =KO0E(IEND) EKH (IEND) = EK (I END) IN D (IEND) =IDK (IEND) GO TO 530 510 IF (KODY (IEND) .EQ.O) GO TO 525 515 IF(EKH(IEND). E3.0. DO) GO TO 520 IF(DBH(IEND) .EQ.O.DO) GO TO 500 ROPLS (I EN D)= ROPLS (I END) •SFAC»DBN (IEND) /EKH (IEND) GO TO 525 520 ROPLS (I END) = ROPLS (I END) *SFAC* DBO (IEND) 525 BNTOT(IEND)=BNTOT(IEND) »SPAC*DBH(IEND) 530 BOTOT (I END) =POTOT(IEND) tSFAC»OBO (IEND) IF (HOPLS (IEND) . EQ.0. DO) GO TO 500 IP (ROPLS (IEND| .LT.O.) GO TO 540 IF (ROPLS (I END) . GT. ROPHX (IEND)) R3PHX (IEND) =ROPLS(IEND) GO TO 550 540 IF (SOPLS (IEND). LT. RONNX (IEND)) RONNX (IEND) =ROPLS(IEND) 550 IF (KODY (IENO). EQ.O) GO TO 500 IP (EKH(IEND) .EQ.O.DO) GO TO 554 DBOT=SF AC*DBA (IEND)/EKH (IEND) GO TO 556 554 DBOT=SFAC*DRO(IEND) 556 IP (DROT.LT.0.D0) GO TO 560 ROTPP(IEND) =BOTPP(IEND) tDBOT GO TO 500 560 BOTPN (IEND) =R3T?N (IEND) *DHOT 500 CONTINUE C DO 580 IEND= 1, 2 580 FACTOR ( IEND) =SFAC C C CHECK FOR COHPLETIOM OP CYCLE C DO 1000 IEN0=1,2 PAC AC (I ENO) =FACAC(I END) »FACTOH (I EN D) IF(FACAC(IEND)•3T.0.999999) GOTO 1000 KFAC=1 1000 CONTINUE C IF (KFAC.NE. 1) 30 TO 1100 KBAL= 1 KFAC=0 SFAC=0. DO CALL BHCAL8 GO TO 30 ^ C -~J 1100 CONTINUE C C TOTAL SHEAR FORCES C DSF= (BHTOT (1) - BHTOT 1 •BHTOT ( 2) -BHTOT 2) /FL SFTOT( 1) =SFTOT( 1) »DSF SFT0T(2) =SFTOr (2) -OSF C C UNBALANCED LOADS DU i i TO EVEN': C POUB=0. DO IF(KUAL.EQ.O) 30 TO B20 BNIUB=RMINC (1) -BHTOT (1) BMJUB=HH1NC (2| -D;iT0T(2) CO TO 830 820 BMIUB=0.DO BBJU0=0.DO C C DEFORMATION BATES FOR DAMPING C BET0(1) =0. DO DET0(2) = 0.D0 830 IF (DFAC. EQ.O. 0. AND. DELTA. E2. 0.0) GO T3 840 IF(TIHE.EQ.O.O) GO TO 905 KBAL=1 IF (EC (1) .EQ. 1. 123456E1 0) GO TO 860 VELB(1|=VELB(1)-EC(3) *V ELM ( 3) VELM (2) = VELB (2) *-EC (1) *VELM (3) VELH(<4) =VELB(4) - EC ( 4) * VELM (6) VELM (5) =V ELM (5) »EC(2) *V ELM (6) 860 DVAX=COSA*(VELB (4) - VELM (1) ) *SINA* (VELfl (5)-VELB (2) ) ROT=(SINA*(VELH(U) -VELM(1) ) tCOSA* (VELM (2)-VELM (5) ) ) /PL DVRI=VELH (3)*ROT DVRJ=VELfl (6) »BOT c C BETA -0 DABPING C IF(DFAC.EQ.O.O) GO TO 880 PAC=DrAC CALL FSTF8(ST,EKEP(1,1),EKEP(2,1)) BETO (1) = (ST (1, 1) *DV RI*ST (1, 2) *DVRJ) *FAC BBIUB=BBIUB»DETO (1) BETO (2) = (ST( 1, 2) »DVHI»ST(2, 2) »DVRJ) »FAC bflJUB=BBJUB*BET0(2) FO0B=EAL»DVAX*DPAC 880 CONTINUE C C SET OP UNBALANCED LOAD VECTOR C 840 IF (KBAL. EQ. 0) GO TO 905 845 SFUB=(BBIUU*BMJUB) /FL DD (1)=-SFUB*SINA-FOUB»COSA DD (2) =SFUB*C0SA-F0U3*S1NA DD (3) = BBIUB DD (4)=-3D (1) DD(5) = -DD(2) DD (6) = DMJUD IF (EC( 1) .EQ. 1. 2345oE 10) ,io TO 'JOO DD (3) = DD (3)-DD(1) * EC (3) »UD( 2) « EC ( 1) DD (6) = DD (6) - DD (4) *EC (4) »DD (5) *KC ( 2) C C EXTRACT ENVELJP2S C 905 DO 910 1=1, 1C S=BHTOT (I) If (S.LE.SENP(I) ) iO T3 907 SENP(I) = 5 TENP (I) =TIME 907 IF (S.GE. SENN (I) ) 30 TO 910 SE NN (I) =S TENN (I) =TIME 910 CONTINUE C C PRINT TIME HISTORY C C ISAVE=0 IP (KPR. LT. 0) GO TO 600 IF (KPR.EQ.O.OR.KOUTDT.EQ.O) GO TO 660 IF (ITHP.GT. 1) GO TO 640 600 IF (KCONT( 10) .EQ. 1) 30 TO 635 IP (IHED. NE. 0) GO TO 620 KK PR = IABS (KPR) PRINT 610, KKPR.TIBE 610 F0RHAT(///18II RESULTS FOB GROUP,13, 1 2811, P. S. BEAM ELEMENTS, TIN E =, FB. 3/// 2 5H ELEB,3X,4HHODE,3X.4HUYST,3X,4HENDB,4X,7HBENDING,3X, 3 5HSHEAR,4X,5HAXIAL, 4X, 8HCUR END,2X,8HCUR PLS,3X, 4 151IAC PL HINGE H0T,8X, 16HBAX PL HINGE LOT/ 5 5H NO. , 3X,4H NO. ,3X, 4HC0DE, 3X, 4UTYPE. 4X,7H BOBENT.3X, 6 511FORCE,4X,5HFOBCE, 4 X, 811 ROTATION, 2 X.6HROT ATION, 3X, 7 4HPOS. ,7X,4HNEG. ,9X, 4HPOS. ,7X,4HNEG./) IHED=1 620 PRINT 630, IBKB, (NOD (I) ,'KODY (I) , IN3ED, BBTOT (I) , SFTOT (I) ,F TOT(I) , IROTOT(I) ,ROPLS (I) ,ROTPP(I) , BOTPN (I) , ROPBX(I) ,£OUflX(I) ,1 = 1,2) 6 30 FOBSAT(I4,3I7, 3F10. 2, 2P10. 5.2F10. 5.6X, 2P10. 5/ 1 4X.3I7, 3F10. 2, 2P10. 5, 2P10. 5,6X, 2F10. 5) C GO TO 640 C C C WRITE ENERGY BALANCE DATA C 635 WR ITE (7) IBEB, (NOD (I) .BBTOT (I) ,ROPLS (I) , DRO(l) , BETO (I) ,1= 1, 2) URITE(7) (DDISM(I) ,1=1,6) C C C SET TIBE HISTORY IN /THIST/ C 640 IP (1THP.LT. 1.OH.KOUTDT.EQ.0) GO TO 660 KKPR=I ABS (KPR) ITHOUT( 1) =KKPR ITHOUT (2)=0 ITHOUT( 3) = IHEM ITHOUT (4) = KODY (1) ITHOUT(5) = K03Y (2) ITHOUT (f.) = NODI IT110UT(7) = NODJ c» DO 650 1=1,16 CO 650 THUUT (I) =UMTOT (I) TH OUT (17)=TIH3 ISAVE=1 C C SET INDICATOR FOR STIFFNESS CI1AN3E C 660 KST=0 If (KOOYX ( 1) • NE. KODY (1) . UK. KODKX (2)-NE. K3DY (2) ) KST=1 UPDATE INFORMATION IN CONS DO b70 J=35,189 670 COHS(J) =COM (J) COBS (2) = COH (2) RETURN END SUBROUTINE OUra (COBS.NINPC) IHPLICIT REAL*3(A-H,0-Z)' COBBON/INP2L/ IBEB,IBEBD,KST,KSTD,LB(b) ,LMD(6) , KG FOB , KGiO B D, « FL,COSA,SINA, EC(4) ,A (2,6), 1 BFII,BFJJ,nFIJ,EAL,EKHIP,EKIIJP,EKH (2) ,K3JYX(2) , * ND(4) .KODY (2) , EKEP (2,6) ,IND (2) ,1NDUB(2) , BEVPT(2) , 2 BBCB(2,2),BNCRI(2,2) ,3BCU(2,2) ,ROCU(2,2) ,ROCH|2,2 * BHTOT (2) , SFTOT (2) , FTOT ( 2) ,ROTOT(2) .ROPLS (2) , 3 ROELS(2) .BOPHX (2) , RONBX (2) , HOTPN (2) , ROTPP(2) , * SENP(10) ,SENN(10) ,TENP(10) ,TENN(10) ,SDA:T(3) ,HODI 4 NODJ, KOUTDT, KDU1 (3) , EK 1 ( 2) , IBOLD,IIHOED.dJEC (2,2) » HBT J (2) ,UBT2(2) , B1TT(2| , ROTT (2) ,BHTD (2) ,SJTD (2) , 5 BBIP (2) ,BOIP (2) ,BBCD (2) , ROCD (2) ,OBCP (2) , H JCP (2) , * BHLP(2) , BBHR (2) , EKCD(2| , EKFG(2) , EK JF (2) ,EKGL(2) , 6 EKGA (2) ,EKIH (2) ,REST (11) DIMENSION COM ( 1) ,COMS ( 1) EQUIVALENCE(1MEB,C0H(1) ) FINAL ENVELOPE OUTPUT, P.S. UEAB ELEMENTS DO 10 J=1,NINPC 10 COB (J) =COBS(J) IF (IMEH.EQ. 1) PRINT 20 20 FORMAT (32U P.5. BEAB/COL ELEMENTS (TYPE 8) /// 1 5(1 BLEM, 3X,4HN0DE,7X,7llBENDI NS. 1 IX, 5HSHEAR, 1 2 X , 5ilA XI AL , 2 14X,3HEND,13X,BHAC HI NGE, 5 X.8HPL 1IIUGE/ 3 5H NO. ,3X,<tH N0..7X.7H HOBENT, 3X, 4IITIME, 4 X, SHFOrtJ E , 3X , 4 4HTIBE,5I,5HPOBCE.3X,4HTIHE,5X,0HROTATION,2X,4liri.1E, 5 4X,8HROT ATION,5X,8HROTAT ION,3X,4HTIME/) PRINT 30, IBEB, NODI, (S ENP (I) , TENP (I) , I = 1, 7, 2) , HOTPP (1) ,3^NP(J) ,T 1P(9) , (SENN (I) ,TENN (I) , 1=1, 7, 2) , 33TPN ll) , SEN N (9) ,TENN (9) ,.IODJ, (JE 2 (I) ,TENP (I) ,1=2,3,2) ,IOTPP( 2) , 3 EN ? (10) , TEN ? (10) , ( SENN (I) , TENN (I) 31=2,8,2) ,BOTPN (2) , S ENN (10) , TENN (10) 30 t'0RHAT(I4, 17, JX, 3HPOS , 3 (P 10. 2 ,F7 . 3) , Fl 2. b , F7. 3 , F10. 4 , 2X , 2F 1 J. b/ 1 14X,3HNEG,3(P10.2. F7.3),P12.6,P7.3,Fl0.4,2X,2F1J.u/ 2 7X.I4, 3X,3ll?OS, 3 (F 10. 2, F7. J) , F12. I,, Fl. J , F10 . 4 , 2X.2F 1 3. o / 3 14X, 3HNE3, 3 (F 10. 2.F7..1) ,F1 2.fa ,F7 . 3 , F1C.4 ,2X,^F1J. u/) RETURN END SUBROUTINE TliPRfl (NS) IMPLICIT REAL'S (A-H.O-Z) C COHBON /THIST/ ITHOUT(10),THOUT(20) ,ITHP,ISAVi,N2LTH,NSTJ,NF7,1S E C C REORGANIZED TIME HISTORY OUTPUT, P.S. BEAH ELEMENTS C IP (NS. GT. 1) 33 TO 20 C PRINT 10, ITHOUT(I) ,ITHOUT(3) 10 PORHAT (18H1 RESULTS FOR CROUP, 13, 1 3711, P.S. BEAB/COL ELEMENTS, ELEBENT NO. .I4///5X, 2 5H TIBE,3X,4HNODE,3X,4HHYST,4X,7HBENDING,4X,5HSHEAB, 3 5X,5HAXIAL,3I,9HC0R HINGE,3X, 4 15HAC PL HINGE ROT/12X, 5 4H NO.,3X,4HCODE,4X,7H BOBENT,3X.5HFORCE, 6 5X,5HF0BCE, 3X,9HROT*TIONS, 3X, 7 4HPOS.,7X,4HNEG,7X,4HPOS. ,7X,4HNE3./) C 20 PRINT 30, TH0UT(47) .ITHOUT (6) ,ITHOUT(4) .(TIIOUT(I) ,1=1,15,2) ,ITUO UT 1(7) ,ITHOUT(5) , (THOUT(I) ,1=2,16,2) 30 FORMAT (1H0,F8. 3,18,17,3P10.2,Fl1.6,4F10.6, I IB, I7,3F10.2,F11.6,4P10.6) IP (ISE. EQ.O) GO TO 40 WRITE (NP7) THOUT (4 7) , ITHOUT (6) , ITHOUT (4) , (TUOUT (I) , 1= 1, 1S, 2) , IT HO 1UT(7) , ITHOUT (5) , (THOUT (I) ,1=2,16,2) 40 CONTINUE RETURN END SUBROUTINE PSTF8 (ST, EK HI, EKHJ) IMPLICIT REAL*8(A-H,0-Z) C C FORM 2*2 FLEXURAL STIFFNESS C C COBBON/INFEL/ IHEH. IBEBD, KST, KSTD,LB (6) ,LB D (6) ,KGEOB,KGEOBD, * FL,COSA,SINA,EC(4) ,A(2,6) , 1 BFII,BPJ.1,BPIJ,EAL,EKHIP,EKHJP,EKU (2) ,KODYX(2) , * ND(4) .KODY (2) , EKEP (2,6) , IND {2) , INDUH (2) ,HBVPT{2) , 2 BBCIt(2.2) ,BBCRI(2,2) , BHCU (2 , 2) . ROCU (2, 2) » BOCH (2, 2 ' ' » BHTOT(2) , SFTOT (2) ,FTOT(2),HOTOT(2) ,ROPLS (2) , 3 ROEL3(2) ,RO?HX(2) , RONBX (2) , ROTPN (2) ,ROTPP(2) , * SENP (10) .SENN (10) , TENP(10) ,TENN (10) .SDACT (3) .NODI M NODJ, KOUTDT, KDUB(3) ,EK1 (2), I BOED.I IBOED, KdEC (2, 2| * UMT1 (2) ,UMT2 (2) ,DMTT (2) ,ROTT (2) , BHTD (2) ,ROTD(2) , 5 BMIP(2) ,ROIP(2),BBCD(2) ,R3CD(2) ,BHCP(2) ,H3CP(2) , CO * UMLP(2) ,BBBW(2| ,EKCD (2) ,EKFG (2) ,EKJF(2) ,EKGL(2) , U> 6 EKGA(2) , EKIH(2),BEST{11) C DI BENSIOll ST (2,2) C IF (EKHI. E'J. 0. DO) GO TO 20 FII=1./EKIII»BFII GO TO .10 20 FII=0.D0 JO IF <EKHJ. EQ.O. 00) GO TO 50 FJJ=1./EKIIJ»BPJJ GO TO 60 50 PJJ=0.D0 60 IF (EKH J. EQ. 0. DO. AN 0. EKH I. EQ. 0. DO) GO TO 90 IP(EKHI.EQ.O.DO) GO TO 80 IP (EKHJ. EQ. 0. DO) GO TO 70 DET=PII»FJJ-nPIJ*»2 ST(1,1)=FJJ/DET ST (2,2)=PII/DET ST(1 #2) =-BFIJ/DET ST(2,1) = ST(1,2) GO TO 110 70 ST (1 ,1) = 1/FII ST(1.2) =0.D0 ST (2,2) =0. DO GO TO 100 80 ST (1 ,1) =0. DO ST ( 1,2) =0.D0 ST (2,2) = 1/FJJ GO TO 100 90 ST(1,1)=0. DO ST (1,2) =0.D0 ST(2,2)=0. DO 100 ST (2, 1)=ST(1,2) 110 CONTINUE C RETURN END SUB8O0TINP. BMC A L0 IMPLICIT BEAL*8(A-H,0-Z) C COMMON/INPEL/ IMEM,IHEMD,KST, KSTD ,LM(6) ,LMD(6) , KGEOH, KGEC MD, • FL.OOSA.SINA, EC(U) , A (2,6) , 1 BFII,BPJJ,BPIJ,EAL,EKHIP,EKHJP,KKH(2) ,KUDIX(2) , * ND(4) .KODT (2) ,EKEP (2,6) ,IND (2) ,1HDUH (2) , REVPT(2) , 2 3HCB (2, 2) , BHCBI(2, 2) , BMCU (2, 2) ,UOCU(2,2) ,KOCH(2,2 ) , * BHrOT(2),SPTOT(2) , FTOT (2) ,BOTOT(2) ,BOPLS(2) , 3 KOELS (2) ,ROPHX(2) ,RONHX(2) ,ROTPN (2) ,ROTPP(2) , • SENP (10) ,SENN ( 10) ,TENP( 10) , TENN ( 10) ,SDA=T(3) ,N0Dl it NODJ, KOUTDT, KDOM (3) , EK 1 (2) , IHOED, 1IMOED, R2EC (2,2) • UHT 1 (2) , UHT 2 (2) , 3MTT (2) ,R3TT(2) , BHTD (2) ,RJTD (2) , 5 BMIP(2) , ROIP (2) , BHCD (2) ,ROCD (2) , BMCP (2) , ROCP (2) , • BHLP(2) , BMHH ( 2) ,EKCD(2) ,EKFG(2) , EKJF (2) ,EKGL(2) , 6 EKGA (2) ,EKIH (2) , RE ST (11) COMMON/WORK/ DVRI , DVR J , BM INC (2| , DBH (2) ,K(199<t) C DI HENSION ST(2,2) C CALL FSTPO (ST, EKH(1) , EKH(2) ) C DBM (1) = DVnl«Sr (1, 1) »DV RJ»ST (1, 2) DBM (2) =DVRI*.ST (2, 1) »DVU J*ST (2,2) C RETURN O APPENDIX B DERIVATION OF IDEALIZED END MOMENT-PLASTIC ROTATION RELATIONSHIP B.1 STAGE ONE It wi l l be recalled that in Chapter Three a pseudo cracking moment, M was defined to be 1.2 times the theoretical cracking cr 3 moment, M In a nonlinear analysis a beam-column element remained cr of the e last ic s t i f fness , K , until one or both of the end moments ' e had reached M . At that point the subsequent nonlinear action was introduced in the appropriate hinge(s). Figures B.I - B.3 show the envelopes for stage one of the three end moment configurations considered. It wi l l be seen that for case one (equal end moments producing reversed curvature), and case two (moment at one end only) the effective member stiffness changes progressively, rather than suddenly as for case three (uniform moment). This i s , of course, because the cracking, at M , of a single section at the end does not cr J signif icantly reduce the effective member st i f fness . However, as the end moment M , continues to increase, and more sections reach the ed ' cracking moment, the effective member stiffness approaches the cracking st i f fness , Kj. It is not generally pract ica l , however, in an analytical hysteretic model to allow for a progressive change in member and/or hinge st i f fness . As such, for computational ease it was fe l t that F i g . B . l E n d - M o m e n t - P l a s t i c R o t a t i o n Enve lope f o r S tage One, Case One 193 F i g . B.2 End-Moment-Plastic Rotation Envelope f o r Stage One, Case Two F i g . B.3 End-Moment-Plast ic Rotation Envelope f o r Stage One, Case Three 195 based on the moment-rotatIon envelopes shown, the ac t ion of the i d e a l -ized end moment-hinge rotat ion model would be s i m i l a r to that of the the t h e o r e t i c a l end moment-plast ic rotat ion curves i f the nonl inear act ion was introduced when the end moment had reached a s l i g h t l y higher "pseudo" cracking moment, rather than the e f f e c t i v e t h e o r e t i c a l c rack-i ing moment. The value of 1.2 for M was chosen such that cracking s t i f f n e s s , K j , of the idea l i zed moment-rotation model would be s i m i l a r to the average cracking s t i f f n e s s of the t h e o r e t i c a l curves. It w i l l be noted that th is resu l ts in the energy d i s s i p a t i o n being s l i g h t l y overest imated. As the energy d i s s i p a t i o n in the cracked e l a s t i c stage is s m a l l , there should be a n e g l i g i b l e e f f e c t on the s t r u c t u r a l response. The equations for the locat ion of point D (see F i g . 3.18) based on the locat ion of the reversal p o i n t , C, are based on a s t a t i s t i c a l analy -s i s of the derived theore t i ca l curves. While a higher order of equation could have been used, i t is thought that the second order equations given s u f f i c i e n t l y r e f l e c t the accuracy of the h y s t e r e t i c model. The values used in the determination of the equations are given in Table B . l . B.2 STAGES TWO AND THREE As with stage one, the equations fo r the locat ion of points G and L, given the locat ion of the reversal p o i n t , F, were determined based on the derived t h e o r e t i c a l curves. Again the equations were l im i ted to second order . The values used in t h e i r de r i va t ion are presented in Table B.2. The rules for small amplitude c y c l i n g are based on i n t u i t i v e under-standing of the c h a r a c t e r i s t i c s of prest ressed concrete and log i ca l 196 o p e r a t i o n o f the h y s t e r e s i s c u r v e s , r a t h e r than t h e o r e t i c a l o r e x p e r i m e n t a l r e s u l t s . 197 TABLE B1 THE DETERMINATION OF EQUATION FOR POINT D END MOMENT CASE ONE, TWO M c 1.2 1.3 1.4 1.5 1.6 1.7 1.8 END MOMENT CASE THREE M M /M 8 / 9 C C CU C O 1.1 0.61 0.69 1.2 0 .67 0.35 1.3 0.72 0.23 1.4 0.78 0.15 1.5 0.83 0.13 1.6 O.89 0.12 1.7 0.94 0.10 1.8 1.00 0.09 M /M c cu 0.67 0.72 0.78 O.83 O.89 0.94 1.00 e /e c o 0.85 0.56 0.43 0.35 0.27 0.23 0.20 TABLE B2 DETERMINATION OF EQUATIONS FOR POINTS G AND L END MOMENT CASE ONE 6 F / e C H 0 F / 9 L G 1 .0 0.30 3.0 0.42 5.0 0.55 7.0 0.60 9 .0 0.63 13.0 0.67 16.0 0.69 20.0 0.70 END MOMENT CASE TWO V e C H * F / \ G 1.0 0.30 2.5 0.45 4.5 0.60 5.5 0.64 6.5 0.67 7.5 0.69 10.0 0.70 APPENDIX C A/20/2.2/2/6 STRUCTURAL PROPERTIES 200 STANDARD BUILDING Story Number 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 p 3 a 20 - 60 f t * j Story Weight 176K 198K 198K 198K 226K 226K 226K 234K 234K US r £ > 234K l>J -ft 284K n 284K to CO 284K -n 288K 288K 288K 314K 314K 314K — . 320K //////// /s///jf////;//s/Jv//s/////////s -; Table C.l B u i l d i n g A/20/2.2/2-6 201 E l / E l ( E l - 133500 K i p - f t 2 ) o o S t o r y Square Columns Gl r d e r s Number Ext Int 20 1.0 2.0 4.0 19 1.0 2.0 4.0 18 1.0 2.0 4 .0 17 1.5 3.0 4.0 16 1.5 3.0 6.0 15 1.5 3.0 6.0 14 3.0 6.0 6.0 13 3.0 6.0 6.0 12 3.0 6.0 6.0 11 4.5 9.0 6.0 10 4.5 9.0 8.0 9 4.5 9.0 8.0 8 6.0 12.0 8.0 7 6.0 12.0 8.0 6 6.0 12.0 8.0 5 10.0 20 .0 8.0 4 10.0 20 .0 10.0 3 10.0 20.0 10.0 2 12.0 24 .0 10.0 1 12.0 24.0 10.0 T a b l e C.2 R e l a t i v e Member S t i f f n e s s 2 0 2 Member Sizes (In) Story Number Square Columns Girders Ext Int Width Depth 20 18 22 16 31 19 18 22 16 31 18 18 22 16 31 17 20 24 16 31 16 20 24 17 35 15 20 24 17 35 14 24 29 17 35 13 24 29 17 35 12 24 29 17 35 11 27 32 17 35 10 27 32 18 37 9 27 32 18 37 8 29 34 18 37 7 29 34 18 37 6 29 34 18 37 5 32 39 18 37 4 32 39 20 39 3 32 39 20 39 2 34 41 20 39 1 34 41 20 39 Table C.3 Member Sizes 203 C r a c k i n g Moments ( k i p - f t ) S t o r y Number Square Columns . G i r d e r s Ext Int Ext Int 20 133 148 90 82 19 194 268 152 139 18 251 385 192 180 17 315 492 225 214 16 373 620 279 268 15 428 734 316 303 14 523 850 352 340 13 558 950 382 373 12 600 1033 413 404 11 68) 1158 432 423 10 708 1217 498 488 9 753 1308 518 508 8 811 1392 543 534 7 835 1458 565 557 6 851 1492 585 576 5 951 1650 585 578 4 917 1592 642 635 3 924 1617 638 631 2 1005 1717 635 628 1 1467 2850 603 594 T a b l e C.4 C r a c k i n g Moments 20k Rot § Cracking (RAD x 103) Story Number Square Columns Girders Ext Int Ext Int 20 2.4305 1.3589 0.6836 0.6228 19 3.5452 2.4668 1.1545 1.0558 18 4.5868 3.5437 1.4584 1.3672 17 3.8508 3.0280 1.7090 1.6254 16 4.5598 3.8158 1.4197 1.3638 15 5.2322 4.5173 1.6080 1.5419 14 3.2188 2.6410 1.7912 1.7302 13 3.4342 2.9518 1.9439 I.8982 12 3.6926 3.2098 2.1016 2.0558 11 2.8078 2.4184 2.1983 2.1526 10 2.9190 2.5416 1.9081 1.8698 9 3.1045 2.7316 1.9847 1.9464 8 2.5199 2.1986 2.0806 2.0460 7 2.5944 2.3022 2.1648 2.1341 6 2.6442 2.3560 2.2414 2.2069 5 1.7917 1.5847 2.2414 2.2146 4 1.4480 1.5290 1.9698 1.9483 3 1.4590 1.5530 1.9576 1.9360 2 1.5869 1.3813 1.9483 1.9268 1 2.8420 2.7953 1.8502 1.8226 Table C.5 Rotation at Cracking 205 Curvature @ Cracking (RAD/ft x 101*) Story Number Square Columns Gi rders Ext Int Ext Int 20 2.0225 1.8427 19 3.4157 3.1236 18 4.3146 4.0450 17 5.0562 4.8090 16 4.1797 4.0150 15 4.7341 4.5394 14 5.2734 5.0936 13 5.7228 5.5880 12 6.1873 6.0524 11 6.4720 6.3371 10 5.5955 5.4832 9 5.8202 5.7074 8 6.1012 6.0000 7 6.3484 6.2585 6 6.5730 6.4720 5 6.5730 6.4944 4 5.7708 5.7072 3 5.7348 5.6719 2 5.7079 5.6449 1 5.4202 5.3393 Table C.6 Curvature @ Cracking 206 S t a t i c Load ing ( K i p s ) S t o r y Square Columns G l r d e r s Number Ext Int 20 4.0 6.0 3 . 0 / f t 19 4.0 6.0 3 . 0 / f t 18 4.0 6.0 3 . 0 / f t 17 5.0 7.8 3 . 0 / f t 16 5.0 7.8 3 . 3 3 / f t 15 5.0 7.8 3 . 3 3 / f t 14 7.5 10.0 3 . 3 3 / f t 13 7.5 10.0 3 . 3 3 / f t 12 7.5 10.0 3 . 3 3 / f t 11 9.0 13.0 3 . 3 3 / f t 10 9.0 13.0 4 . 0 / f t 9 9.0 13.0 4 . 0 / f t 8 10.0 15.0 4 . 0 / f t 7 10.0 15.0 4 . 0 / f t 6 10.0 15.0 4 . 0 / f t 5 14.0 19.0 4 . 0 / f t 4 14.0 19.0 4 . 2 / f t 3 14.0 19.0 4 . 2 / f t 2 15.0 21.0 4 . 2 / f t 1 15.0 21 .0 4 . 2 / f t T a b l e C.7 S t a t i c L o a d i n g APPENDIX D EARTHQUAKE PROPERTIES -500 r .( 1MPERIRL VALLEY ERRTHQURKE MRY 18, 1940 - 2037 P5T liftOOl MO.001.0 |EL C£NTRQ(SITE IMPERIAL VALLEY IRRIGATION DISTRICT COMP SOOE o PEAK VALUES : ACCEL = 3M1.7 CM/SEC/SEC VELOCITY = 33.4 CM/SEC OISPL = 10.9 CM N> O OO TIME 30 SECONDS F i g . DI 2 0 9 RESPONSE SPECTRUM IMPERIAL VALLEY EARTHQUAKE MAY 18. 1940 - 2037 PST IIIA001 U O . 001 .0 E L C E N T R O S I T E I M P E R I A L V A L L E Y I R R I G A T I O N D I S T R I C T C O M P S O O E O A M P I N G V A L U E S A R E 0 . 2 . 5. 10 A N O 20 P E R C E N T OF C R I T I C A L PERIOD (sees) F i g . 02 RELATIVE VELOCITY RESPONSE SPECTRUM IMPERIAL VALLEY EARTHQUAKE MAY 18. 1940 - 2037 PST IIIA001 UO.001.0 EL CENTRO SITE IMPERIAL VALLEY IRRIGATION DISTRICT DAMPING VALUES ARE 0, 2, 5, 10 AND 20 PERCENT OF CRITICAL 110 COMP SOOE u U J i n CJ o U J > UJ U J cc T T T T T i 1 i i i | i i i | i i i sv FS 1 I I I 1 I I I I 1 I I M O 11 15 PERIOD - SECONDS Fig. D3 RESPONSE SPECTRUM 2 1 2 KERN COUNTY, CALIFORNIA EARTHQUAKE JULY 2 1 , 1952 - 0453 PDT IllfiOOU 52.002.0 TAFT LINCOLN SCHOOL TUNNEL COMP S69E DAMPING VALUES ARE 0. 2. 5. 10 AND 20 PERCENT OF CRITICAL PERIOD (sees) F i g . 05 RELATIVE VELOCITY RESPONSE SPECTRUM KERN COUNTY, CALIFORNIA EARTHQUAKE JULY 21. 1952 - 0U53 PDT IIIA004 52.002.0 TAFT LINCOLN SCHOOL TUNNEL COMP S69E DAMPING VALUES ARE 0, 2, 5. 10 AND 20 PERCENT OF CRITICAL 1 1 1 1 1 1 1 1 1 1 1 1 I I I | I I I | I I I sv PERIOD - SECONDS F i g . 06 PARKFIELD. CALIFORNIA EARTHQUAKE JUNE 27 . 1966 - 2026 P5T IIB033 66.001.0 CHOLAME.SHANDON. CALIFORNIA ARRAY NO. 2 COMP N65E o PEAK VALUES : ACCEL = -1179.6 CM/SEC/SEC VELOCITY = -77.9 CM/SEC DISPL = 26.3 CM 0 5 I.C 15 20 25 30 35 UO 45 TIME - SECONDS F i g . D7 215 RESPONSE SPECTRUM PRRKFIELD, CRLIFORNIR ERRTHQURKE JUNE 27 . 1966 - 2026 P 5 T IIIBD33 65.001.0 CHOLRME ShANDON. CRLIFORNIA flHRRY NO. 2 COMP N5SE DAMPING VALUES RRE 0 . 2. 5. 10 AND 20 PERCENT OF CRITICAL 4 0 0 0 2 0 0 0 PERIOD (sees) F f g . D8 RELATIVE VELOCITY RESPONSE SPECTRUM PARKF1ELO. CALIFORNIA EARTHQUAKE JUNE 27. 1966 - 2026 PST II1B033 66.001.0 CHOLAME.SHRNOON. CALIFORNIA ARRAY NO. 2 COMP N65E DAMPING VALUES ARE 0, 2, 5. 10 AND 20 PERCENT OF CRITICAL -1 1 1 1 1 1 1 1 1 1 1 1 I I I | I I I | I I I PERIOD - SECONDS F i g . D9 

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