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Modified substitute structure method for analysis of existing R/C structures Yoshida, Sumio 1979

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MODIFIED SUBSTITUTE STRUCTURE METHOD FOR ANALYSIS OF EXISTING R/C STRUCTURES by S U M I O \ Y O S H I D A B. A. S c . , 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 , 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE • i n " 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 t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March, 19 79 (c) Sumio Y o s h i d a , 19 79 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 n a t p March, 1979 ABSTRACT The modified substitute structure method i s developed for the earthquake hazard evaluation of ex i s t i n g reinforced concrete buildings constructed before the most recent advances i n seismic design codes. The main c h a r a c t e r i s t i c of the proposed method i s the use of modified l i n e a r analysis for predicting the behaviour, including i n e l a s t i c response, of ex i s t i n g structures when sub-jected to a given type and i n t e n s i t y of earthquake motion, represented by a l i n e a r response spectrum. The procedure i n -volves an extention of the substitute structure method, which was o r i g i n a l l y proposed by Shibata and Sozen as a design procedure. With properties and strengths of a structure known, the modified substitute structure method computes d u c t i l i t y demand of each member v i a an e l a s t i c modal analysis, in which reduced f l e x u r a l s t i f f n e s s and substitute damping factors are used i t e r a t i v e l y . As a r e s u l t of the analysis, i t i s possible to describe, in general terms, the location and extent of damage that would occur in a structure subjected to earthquakes of d i f f e r e n t i n t e n s i t y . Several reinforced concrete structures of d i f f e r e n t sizes and strengths were tested by the proposed method and the r e s u l t s compared with a nonlinear dynamic analysis. In general, a small number of i t e r a t i o n s was required to obtain an estimate of damage r a t i o s . The method appears to work well for structures i n which y i e l d i n g i s not extensive and widespread. Furthermore, i t appears to work better for those in which y i e l d i n g occurs mainly in beams and the e f f e c t of higher modes i s not predominant. Though further research i s necessary, the modified substitute structure method can constitute an i n t e g r a l part of the r a t i o n a l r e t r o f i t procedure. i v TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES i x ACKNOWLEDGEMENTS x i i CHAPTER 1. INTRODUCTION 1.1 Background 1 1. 2 Literature Survey 4 (a) ATC Report 4 (b) Okada and Bresler 6 (c) Freeman, N i c o l e t t i , and T y r r e l l . . . 9 1.3 Purpose and Scope 11 2. SUBSTITUTE STRUCTURE METHOD 2.1 Modal Analysis 14 (a) Equation of Motion 14 (b) Periods and Mode Shapes 15 (c) Response Spectra 16 (d) Modal Forces IV (e) Combination of Forces and Displacements 18 2.2 Substitute Structure Method (a) Development 20 (b) Substitute Structure Method 2 3 (c) Computer Program 29 V CHAPTER Page 2.3 Examples and Observations (a) Frames with F l e x i b l e Beams 31 (b) Soft-Story Frame 33 (c) 2-Bay, 3-Story Frame 35 2.4 Equal-Area S t i f f n e s s Method (a) Observation 37 (b) Equal-Area S t i f f n e s s 38 (c) Examples 39 (d) Area for Further Studies 40 3. MODIFIED SUBSTITUTE STRUCTURE METHOD 3.1 Modified Substitute Structure Method 42 3.2 Computer Program 51 3.3 Convergence 54 3.4 Accelerated Convergence 6 0 4. EXAMPLES 4.1 Assumptions and Comments 65 4.2 Examples (a) 2-Bay, 2-Story Frame 69 (b) 3-Bay, 3-Story Frame 72 (c) 1-Bay, 6-Story Frame 76 (d) 3-Bay, 6-Story Frame 79 (e) Observations 83 5. FACTORS AFFECTING MODIFIED SUBSTITUTE STRUCTURE METHOD 5.1 E f f e c t of Higher Modes 86 5.2 Spectrum 91 5.3 Guidelines for Use of Method 96 V I CHAPTER Page 5.4 Further Studies 9 9 6. CONCLUSION 101 BIBLIOGRAPHY 16 7 APPENDIX A. Modification of Damage Ratio - Strain Hardening Case 169 B. Computer Program 17 3 v i i LIST OF TABLES Table Page 2.1 Natural Periods and Smeared Damping Ratios for 3-, 5-, and 10-Story Frames 103 2.2 Computed Damage Ratios for 3-, 5-, and 10-Story Frames 10 4 2.3 Comparison of Damage Ratios for 3-, 5-, and 10-Story Frames 105 2.4 Computed Natural Periods for 3-, 5-, and 10-Story Frames 106 3.1 Natural Periods for 2-Bay, 3-Story Frame A 106 3.2 Damage Ratios for 2-Bay, 3-Story Frame A 107 3.3 Natural Periods for 2-Bay, 3-Story Frame B 108 3.4 Number of Iterations - 2-Bay, 3-Story Frame B.... 108 3.5 Damage Ratios for 2-Bay, 3-Story Frame B 109 4.1 Natural Periods for 2-Bay, 2-Story Frame 109 4.2 Displacements for 2-Bay, 2-Story Frame 110 4.3 Natural Periods for 3-Bay, 3-Story Frame 110 4.4 Displacements for 3-Bay, 3-Story Frame 110 4.5 Natural Periods for 1-Bay, 6-Story Frame I l l 4.6 Displacements for 1-Bay, 6-Story Frame I l l 4.7 Natural Periods for 3-Bay, 6-Story Frame I l l 4.8 Displacements for 3-Bay, 6-Story Frame 112 5.1 Natural Periods for 3-Bay, 6-Story Frame A -Spectrum B 112 5.2 Displacements for 3-Bay, 6-Story Frame A -Spectrum B 112 v i i i Table Page 5.3 Natural Periods for 3-Bay, 6-Story Frame B -Spectrum B ^ 113 5.4 Displacements for 3-Bay, 6-Story Frame B -Spectrum B 113 5.5 Natural Periods for 3-Bay, 6-Story Frame B -Spectrum A 113 5.6 Displacements for 3-Bay, 6-Story Frame B -Spectrum A 114 5.7 Natural Periods for 3-Bay, 6-Story Frame B -E l Centro EW Spectrum and Taft S69E Spectrum 114 5.8 Displacements for 3-Bay, 6-Story Frame B -E l Centro EW Spectrum 115 5.9 Displacements for 3-Bay, 6-Story Frame B -Taft S69E Spectrum 115 5.10 Natural Periods for 3-Bay, 6-Story Frame A -E l Centro EW Spectrum and Taft S69E Spectrum 116 5.11 Displacements for 3-Bay, 6-Story Frame A -E l Centro EW Spectrum 116 5.12 Displacements for 3-Bay, 6-Story Frame A -Taft S69E Spectrum 117 i x LIST OF FIGURES Figure Page 1.1 Load-Deflection Curve for E l a s t i c and El a s t o p l a s t i c Structure 118 2.1 Idealized Hysteresis Loop for Reinforced Concrete System 118 2.2 Force-Displacement Curve - D e f i n i t i o n of Damage Ratio 119 2.3 Flow Diagram for Substitute Structure Method 12 0 2.4 Member Properties and Design Moments for 3-, 5-, 10-Story Frames 12 3 2.5 Smoothed Response Spectrum - Design Spectrum A... 124 2.6 Soft Story Frame A - Member Properties and Yie l d Moments.. 125 2.7 Soft Story Frame A - Damage Ratios for Individual Earthquakes 126 2.8 Soft Story Frame B - Member Properties and Yie l d Moments 12 7 2.9 Soft Story Frame B - Damage Ratios for Individual Earthquakes 12 8 2.10 2-Bay, 3-Story Frame - Member Properties and Yi e l d Moments.. 129 2.11 2-Bay, 3-Story Frame - Damage Ratios for Individual Earthquakes * 130 2.12 Force-Displacement Curve - D e f i n i t i o n of Equal-Area S t i f f n e s s 131 3.1 Moment-Rotation Curve - Modification of Damage Ratio... 132 X F i g u r e Page 3.2 F l o w D i a g r a m f o r M o d i f i e d S u b s t i t u t e S t r u c t u r e Method 13 3 3.3 2-Bay, 3 - S t o r y Frame A - Member P r o p e r t i e s and Y i e l d Moments 136 3.4 2-Bay, 3 - S t o r y Frame A - P l o t o f P e r i o d s v s . Number o f I t e r a t i o n s 137 3.5 2-Bay, 3 - S t o r y Frame A - P l o t o f Damage R a t i o s v s . Number o f I t e r a t i o n s 138 3.6 2-Bay, 3 - S t o r y Frame B - Member P r o p e r t i e s and Y i e l d Moments 139 3.7 2-Bay, 3 - S t o r y Frame B - P l o t o f P e r i o d s v s . Number o f I t e r a t i o n s 140 3.8 2-Bay, 3 - S t o r y Frame B - Damage R a t i o s Computed a t t h e End o f 4, 12, 20, and 200 I t e r a t i o n s 141 3.9 2-Bay, 3 - S t o r y Frame B - P l o t o f Damage R a t i o s v s . Number o f I t e r a t i o n s 142 4.1 2-Bay, 2 - S t o r y Frame - Member P r o p e r t i e s and Y i e l d Moments 14 3 4.2 2-Bay, 2 - S t o r y Frame - Damage R a t i o s 14 3 4.3 2-Bay, 2 - S t o r y Frame - Damage R a t i o s f o r I n d i v i d u a l E a r t h q u a k e s 144 4.4 3-Bay, 3 - S t o r y Frame - Member P r o p e r t i e s and Y i e l d Moments 145 4.5 3-Bay, 3 - S t o r y Frame - Damage R a t i o s 146 4.6 3-Bay, 3 - S t o r y Frame - Damage R a t i o s f o r I n d i v i d u a l E a r t h q u a k e s 14 7 x i Figure Page 4.7 1-Bay, 6-Story Frame - Member Properties and Yi e l d Moments 14 8 4.8 1-Bay, 6-Story Frame - Damage Ratios 149 4.9 1-Bay, 6-Story Frame - Damage Ratios for Individual Earthquakes 150 4.10 3-Bay, 6-Story Frame - Member Properties and Yie l d Moments 151 4.11 3-Bay, 6-Story Frame - Damage Ratios 152 4.12 3-Bay, 6-Story Frame - Damage Ratios for Individual Earthquakes 15 3 5.1 Smoothed Response Spectrum - Design Spectrum B... 154 5.2 3-Bay, 6-Story Frame A - Damage Ratios 155 5.3 3-Bay, 6-Story Frame A - Plot of Damage Ratios for Beams i n the Exterior Bay. 156 5.4 3-Bay, 6-Story Frame B - Damage Ratios 15 7 5.5 3-Bay, 6-Story Frame B - Plot of Damage Ratios for Beams i n the Exterior Bay. 158 5.6 3-Bay, 6-Story Frame B - Damage Ratios 159 5.7 3-Bay, 6-Story Frame B - Damage Ratios for Individual Earthquakes 160 5.8 E l Centro EW Spectrum and Design Spectrum.A 161 5.9 Taft S69E Spectrum and Design Spectrum A 162 5.10 3-Bay, 6-Story Frame B - Damage Ratios 16 3 5.11 3-Bay, 6-Story Frame B - Damage Ratios 16 4 5.12 3-Bay, 6-Story Frame A - Damage Ratios 16 5 5.13 3-Bay, 6-Story Frame A - Damage Ratios 166 A . l Moment-Rotation Curve I 7 2 x i i ACKNOWLEDGEMENTS The author wishes to express his sincere gratitude to his supervisors, Dr. N. D. Nathan, Dr. D. L. Anderson, and Dr. S. Cherry for t h e i r advice and guidance during the research and preparation of t h i s thesis. Thanks are also due to Mr. R. Grigg, the C i v i l Engineering Department program l i b r a r i a n , for his advice and assistance. The f i n a n c i a l support of the National Research Council of Canada i n the form of Postgraduate Scholarship i s g r a t e f u l l y acknowledged. March, 19 79 Vancouver, B r i t i s h Columbia 1 CHAPTER 1 INTRODUCTION 1.1 Background During the l a s t two decades a great deal of progress has been made in understanding the behaviour of buildings during major earthquake motions. The new knowledge r e s u l t i n g from research and observation has been incorporated i n building codes. It i s not reasonable to expect the majority of newly designed buildings to be able to survive a major earthquake motion with tolerable damage. Unfortunately, i n any large c i t y there e x i s t many buildings which were designed and constructed before the recent advances in seismic codes. The performance of these buildings are at best uncertain i f and when a sizable earthquake strikes the area. The c i t y authorities must assess the seismic r i s k s involved in such buildings from time to time. This point arises most often when an owner of an old building wishes to change the occupancy or do a s t r u c t u r a l a l t e r a t i o n . Before issuing a new building permit, the authorities must make a decision on how well i t com-p l i e s with current codes. Unless the building i s judged to be safe, they must decide on the modifications that have to be made in order to upgrade i t to a s a t i s f a c t o r y l e v e l . Upon t h e i r recommendations the owner can decide whether i t i s fea s i b l e to 2 carry on with his plan or whether i t i s more economical to replace the building with a new one. ' I t i s , therefore, necessary to develop a methodology to screen and evaluate e x i s t i n g buildings against seismic hazards. Many issues are involved here, but the most d i f f i c u l t one i s how to assess the degree of compliance with the current seismic codes. It i s appropriate here to describe b r i e f l y the ph i l o s -ophy behind the current codes, which should be borne i n mind when the evaluation of e x i s t i n g buildings i s discussed l a t e r . The current code procedure for the design of new buildings i s based on the assumption that a structure w i l l y i e l d i n a major earthquake, but that i t s ultimate displacement w i l l be approximately equal to the displacement of the same structure i f i t remained e l a s t i c during the earthquake as i l l u s t r a t e d in Fi g . 1.1. It should be noted that the s t i f f n e s s of the struc-ture i s usually predetermined by the layout and the design for gravity loads. The combination of d u c t i l i t y and strength must be chosen such that the structure reaches i t s maximum load-maximum displacement rela t i o n s h i p with only a tolerable l e v e l of damage. The code, such as the National Building Code of Canada,^ achieves th i s combination of strength and d u c t i l i t y by e s t i -mating the available d u c t i l i t y for the p a r t i c u l a r s t r u c t u r a l system selected for the design of the building, and the load l e v e l i s set accordingly. Thus a d u c t i l e system may be designed for a lower load l e v e l than a more b r i t t l e system. The code also s p e c i f i e s the detailed design requirements to ensure that t h i s d u c t i l e f a i l u r e mode occurs before the b r i t t l e f a i l u r e 3 modes associated with shear, bond or d e t a i l f a i l u r e . The code actually gives a q u a s i - s t a t i c force such that the structure i s s a t i s f a c t o r y i f i t can r e s i s t that force, provided that i t i s detailed properly to ensure the anticipated d u c t i l i t y and that i t i s also detailed c o r r e c t l y to ensure the desirable flexure f a i l u r e mode. It should now be clear that without the philosophy described above the code s t a t i c force i s meaningless. It i s not the actual force which a structure i s expected to receive during a major earthquake i f i t i s designed and detailed d i f -ferently from the current codes. The ex i s t i n g buildings were obviously designed with a d i f f e r e n t philosophy from the one implied i n the current codes, and merely applying the quasi-s t a t i c load i s a questionable approach. The best way to analyze e x i s t i n g buildings i s to subject them to a nonlinear time-step analysis. Recent advances in computer technology have made thi s approach possible. But the cost involved i n such analysis i s s t i l l p r o h i b i t i v e l y high and i t requires very accurate modelling of the entire structure. The high cost and tediousness make t h i s analysis impractical except i n very few cases. Several proposals have been made to f i n d a more p r a c t i c a l way to treat the problem of analyzing the ex i s t i n g buildings, which i s becoming known by the somewhat i n f e l i c i t i o u s term, " r e t r o f i t . " 4 1.2 Lit e r a t u r e Survey The l i t e r a t u r e survey i n th i s section i s intended to be an introduction to the approaches that must be followed i n order to i d e n t i f y the p o t e n t i a l l y hazardous buildings and to estimate an extent of hazards and an associated damage. Three papers are discussed. (a) ATC Report The Applied Technology Council i n the United States made a f i r s t attempt at a comprehensive procedure f o r the seismic 2 hazard evaluation of e x i s t i n g buildings. The relevant section of ATC I I I , the report of the council, i s b r i e f l y discussed here. ATC III points out that there are probably thousands of buildings i n the United States which are p o t e n t i a l l y earthquake hazardous. Since a thorough study of a l l buildings i s econom-i c a l l y impossible, they suggest a graduated procedure. They are, (1) Selection to i d e n t i f y p o t e n t i a l l y hazardous buildings (2) Evaluation to esta b l i s h the possible extent of hazards (3) Correction to ensure the elimination of unacceptable hazards. The f i r s t step i s to screen the p o t e n t i a l l y hazardous buildings. The seismic hazard i s related to the severity of the ground motion and the usage of buildings. The severity of the ground motion i s indicated by the Seismic Hazard Index SHI correlated with ground motion. SHI ranges from 1 to 4, with the higher number in d i c a t i n g greater severity. The usage of 5 the buildings i s indexed by the Seismic Hazard Exposure Group SHE. SHE ranges from I to I I I , with the higher number i n d i c a t i n g less usage. The buildings i n the area where the Seismic Hazard Index i s less than or equal to 3 are excluded from analysis. In the area where SHI i s 4, the newer buildings and SHE-II and SHE-III buildings with low occupancy are also exempt. The buildings with h i s t o r i c a l values are subjected to the alternate procedure. The evaluation procedure may be q u a l i t a t i v e or quanti-t a t i v e . A q u a l i t a t i v e evaluation i s required for SHE-II and - I I I groups. The procedure i s prescribed i n the report. It involves a judgement on the adequacy of the primary s t r u c t u r a l system and nonstructural elements, and i t can be c a r r i e d out very rapidly. SHE-I buildings and those judged uncertain i n the previous analysis are subject to more thorough a n a l y t i c a l studies. The aseismic design procedure for new constructions are stipulated. The procedure involves the determination of an earthquake capa-c i t y r a t i o , R , which i s a r a t i o of actual l a t e r a l seismic force capacity of an e x i s t i n g system or element to the capacity required to meet the p r e v a i l i n g seismic code provisions for the design of new buildings. The occupancy pote n t i a l are also used to assess building hazards. The t o t a l l a t e r a l seismic force i s d i s t r i b u t e d over the building height and the r e s u l t i n g applied member moment, shear, and a x i a l forces are evaluated at p a r t i c u l a r sections. The member capacities can be calculated from the known section and material properties. The earthquake capacity r a t i o i s computed by d i v i d i n g the section capacity available for earthquake loading 6 by the seismically induced load. The r a t i o s are computed for moments, shear, a x i a l forces, and d r i f t . The smallest r a t i o governs the earthquake capacity of the building. In the author's opinion, a d i s t i n c t i o n should be made in f a i l u r e modes. A f a i l u r e in bending i s much more preferable to a f a i l u r e i n shear and i t i s not proper to treat them equally i n choosing the governing earthquake capacity r a t i o . Unless the earthquake capacity r a t i o i s greater than or equal to one, there i s a hazard which i s a function of the b u i l d i n g and the occupancy po t e n t i a l . ATC sets the minimum acceptable earthquake capacity r a t i o s and those which f a i l to meet the requirements must be strengthened or demolished according to the schedule outlined i n the report. (b) Okada and Bresler Okada and Bresler i n "Strength and D u c t i l i t y Evaluation of Exis t i n g Low-Rise Reinforced Concrete Buildings - Screening 3 Method" describes a procedure for evaluating the seismic safety of low-rise reinforced concrete structures. Their method con-s i s t s of series of steps which are repeated in successive cycles with more refined modeling. Each cycle represents a "screening". Three screening cycles are proposed and the f i r s t screening cycle, the f i r s t execution of the basic procedure, i s described in d e t a i l i n t h e i r paper. It also shows how t h i s procedure can be applied to e x i s t i n g school buildings. The f i r s t screening i s based on approximate evaluation of the load-deflection c h a r a c t e r i s t i c s of the f i r s t or weakest story. The second involves a more precise estimate of o v e r a l l s t r u c t u r a l behaviour, and in the t h i r d screening nonlinearity of each member i s modeled. In describing the f i r s t screening procedure, the authors point out that the c r i t e r i a which define the permissible damage res u l t i n g from a s p e c i f i e d earthquake are the most important factors which determine s t r u c t u r a l adequacy. Two grades of earthquake motions and two corresponding degrees of building damage are chosen. Three types of f a i l u r e modes, bending, shear and shear bending are considered. The procedure consists of fi v e major steps, namely, Cl) s t r u c t u r a l modeling (2) a n a l y t i c a l modeling (3) strength safety evaluation (4) d u c t i l i t y safety evaluation (.5) synthesis evaluation of safety. The s t r u c t u r a l modeling i s i n i t i a t e d by i d e n t i f y i n g the load transmission system of the bui l d i n g from examining drawings, design calculations and f i e l d investigations. The main items to be determined are st r u c t u r a l system, load i n t e n s i t y , properties of materials, design method, and other special s t r u c t u r a l features. Several models may have to be considered. The a n a l y t i c a l modeling i s done to evaluate s t r u c t u r a l response under l a t e r a l forces. The shear cracking strength, C s c l ' u l t i m a t e shear strength, Cg u l, and bending strength, Cg ^, in terms of base shear c o e f f i c i e n t s are computed. The compar-ison of the three i d e n t i f i e s the type of f a i l u r e . The strength i s evaluated with respect to shear cracking, ultimate shear strength, and bending strength. The capacity with respect to 8 each of these three f a i l u r e modes and t h e i r r e l a t i v e values are weighed heavily i n evaluating the structure. The fundamental period and the modal p a r t i c i p a t i o n factor are computed i n an approximate manner. The strength safety evaluation determines the adequacy of l a t e r a l strength. For t h i s purpose a l i n e a r earthquake response analysis i s used with a standardized response spectrum. In c a l -culating the l i n e a r response i n terms of base shear c o e f f i c i e n t , C E, the building assumed to be a story-level-lumped-mass system with the number of stories equal to the number of degrees of freedom. Only the f i r s t mode shape i s considered. The d u c t i l i t y safety evaluation estimates the f i r s t story displacement using nonlinear displacement response spectra and modified modal p a r t i c i p a t i o n factor to i d e a l i z e the nonlinear behaviour of the building. The response d u c t i l i t y of the building, which i s modeled as the equivalent one-mass system, i s compared with the s p e c i f i e d l i m i t value. The f i n a l step i s the synthesis evaluation of safety. The assumptions and unknowns incorporated into the screening process and the need for modification of the e x i s t i n g b u ilding are care-f u l l y analyzed. Those buildings which f a i l e d to pass the f i r s t screening are c l a s s i f i e d uncertain and must go through the second and subsequent screening procedure. The procedure set fort h by Okada and Bresler represents a r a t i o n a l approach to the problem of evaluating ex i s t i n g buildings, and the present method of analysis could be f i t t e d into t h e i r screening process. 9 (c) Freeman, N i c o l e t t i , and T y r r e l l The procedure described i n "Evaluation of E x i s t i n g Buildings for Seismic Risk -- A Case Study of Puget Sound Naval Shipyard, 4 Bremerton, Washington," by Freeman et a l . i s intended to f i l l the gap between s t a t i s t i c a l procedures for large areas, and detailed s t r u c t u r a l dynamic analysis of in d i v i d u a l buildings. Its main feature i s a very rapid screening process and a simple analysis with minimum of c a l c u l a t i o n . The structure at the Puget Sound Naval Shipyard at Bremerton, Washington, was studied and the findings were reported. A t o t a l of 9 6 buildings of d i f f e r e n t siz e , age, materials, type of construction and occupancy i s eval-uated for the o v e r a l l v u l n e r a b i l i t y to earthquake damage. The study i s performed i n s i x phases, namely, (1) a v i s u a l survey of 9 6 buildings C2) investigation of two representative buildings (3) determination of the seismicity of the area (4) estimation of probable damage for 80 buildings (5) detailed investigation of f i v e c r i t i c a l buildings (6) estimation of the average annual costs of expected earthquake damage for 40 buildings. Phases C D to (3) need l i t t l e explanation. The findings i n the second phase are used for the next phases of study. In the t h i r d phase response acceleration spectra are constructed from the seismic records i n the area and are used for the phase four of the study. The fourth phase i s the most relevant to t h i s report. In analyzing the structures emphasis was placed on minimization of the man-hours spent. The l a t e r a l force strength capacities were 10 roughly approximated and the non-structural materials were also looked at. The base shear capacities were used to est a b l i s h the y i e l d l i m i t and the ultimate l i m i t . The former i s defined as the base shear represented by the force required to reach the capacity of the most r i g i d l a t e r a l f o r c e - r e s i s t i n g system. The l a t t e r i s defined as the base shear "required to cause the most f l e x i b l e l a t e r a l f o r c e - r e s i s t i n g system to y i e l d a f t e r the collapse or y i e l d of the more r i g i d ones. These were converted to spectral acceleration capacities by di v i d i n g by the weight of structure. The dynamic response c h a r a c t e r i s t i c s and the periods were estimated by approximate methods. Assumptions were made to simplify the evaluation of damage. The damage l e v e l was assumed to vary l i n e a r l y from 0% at y i e l d l i m i t to 100% at the ultimate l i m i t . In the i n e l a s t i c range nonlinear effects were taken into account by l i n e a r l y varying the damping between the two l i m i t s . The procedure used for e s t i -mating damage was based on r e c o n c i l i a t i o n of the demand spectral acceleration and the capacity of the structure i n r e l a t i o n to periods and damping. A graphical solution for estimating per-centage damage was developed. The analysis was done i n two directions and a weighted average was computed. Sets of response spectra were chosen to represent the earthquake motions with d i f f e r e n t return periods. From the damage le v e l s associated with those return periods the annual costs were computed for the 80 buildings. The authors claim that the r e s u l t of the procedure can be used to decide which buildings are most susceptible to earthquake damage and that the effects of modification on e x i s t i n g structures can be found. 11 1.3 Purpose and Scope The three papers discussed i n the previous section i l l u s -trate the type of approach that must be taken i n order to analyze a large number of e x i s t i n g buildings which are p o t e n t i a l seismic hazards. They a l l set up screening procedures to select poten-t i a l l y hazardous buildings and then subject them to seismic analysis. It i s beyond the scope of t h i s thesis to comment on the screening procedure; the s t r u c t u r a l analysis, however, deserves a few comments. 2 The ATC-III report suggests the use of the q u a s i - s t a t i c seismic forces i n the current codes for the analysis. As was explained i n the f i r s t section of t h i s chapter, these forces are meaningless unless the structures were designed with the d u c t i l e properties and the proper d e t a i l i n g implied by the code recom-mendations. Even i f a structure can carry only a f r a c t i o n of the q u a s i - s t a t i c forces, collapse or major damage may not occur, because i n actual earthquakes the forces w i l l be r e d i s t r i b u t e d and the building w i l l respond d i f f e r e n t l y depending on i t s duc-t i l e properties. 3 Bresler's methods takes nonlinearity into account by modelling the structure as a one-mass system and through the use of nonlinear response spectra. The analysis i s intended for low-r i s e structures and, for this purpose, the assumptions and s i m p l i f i c a t i o n s that the authors made are s a t i s f a c t o r y . An extension of the method to the analysis of medium- to high-rise buildings w i l l , however, involve major modifications to t h e i r method. 12 4 Freeman's method i s at best approximate. Their approach to i n c l u s i o n of nonlinearity i n t h e i r analysis has many assump-tions and s i m p l i f i c a t i o n s . The method i s probably e f f e c t i v e for screening many one- to two-story, single-bay buildings, but the extension of t h i s method to larger buildings i s of questionable value. It i s cle a r that a procedure for analysis of e x i s t i n g buildings against seismic hazards must be developed, esp e c i a l l y for those buildings which are judged uncertain a f t e r the i n i t i a l screening process. The procedure must be capable of handling medium- to high-rise structures without major assumptions and s i m p l i f i c a t i o n s . It i s desirable that d i f f e r e n t earthquake motions can be used to obtain a good estimate of behaviour of the structure and that .the analysis should include the ef f e c t s of nonlinearity a f t e r the y i e l d of some of the members. At the same time the procedure must be simple and reasonably economical to use. Such a procedure i s developed and described in the subse-quent chapters. The modified substitute structure method i s intended to f i l l the gap between s i m p l i f i e d s t r u c t u r a l analysis and the f u l l - s c a l e , nonlinear time-step analysis. The proposed method i s suitable for reinforced concrete frame structures, but i t i s hoped that i t can be used for shear-wall type buildings and s t e e l structures. The procedure i s a modified e l a s t i c modal analysis, which i s developed from a design concept proposed by Shibata and Sozen.^ 5 . The design procedure proposed by Shibata and Sozen i s described f i r s t in order to discuss the theory and assumptions which are es s e n t i a l i n understanding the proposed method. A b r i e f discussion modal analysis i s included. Examples of the design procedure are also presented. An alternate approach i s described and the findings are discussed. Then the modified substitute structure method i s presented in the next chapter. The theory behind t h i s procedure i s d i s -cussed as well as the development of the computer program. Since i t i s an i t e r a t i v e procedure, convergence c r i t e r i a are discussed. A method to achieve faster convergence i s introduced. In order to test the v a l i d i t y of the modified substitute structure method, frames of d i f f e r e n t type and height are ana-lyzed. A comparison of re s u l t s with those of nonlinear dynamic analysis i s presented. A l l the assumptions are presented and described i n t h i s section. In the f i n a l chapter factors which a f f e c t the r e s u l t s of the analysis are discussed, and a preliminary guideline i s presented for successful applications of the method. The areas where further research i s necessary are mentioned. 14 CHAPTER 2 SUBSTITUTE STRUCTURE METHOD 2.1 Modal Analysis Modal analysis i s an approximate dynamic analysis to solve the response of a multi-degrees-of-freedom system to a given earthquake motion. Although i t i s intended for analysis of elas-t i c systems, a thorough knowledge of t h i s method i s es s e n t i a l for the discussion of the subsequent sections. Since i t i s not the intention of t h i s paper to explain the dynamics of structures subjected to the earthquake motion, the discussion i s kept very b r i e f . The subject i s covered i n Clough and Penzien. (a) Equation of Motion The basic equation of motion for a multi-degrees-of-freedom system i s given by [m] (ii) + [c] (u) + [k] (u) = -x[m] (I) (2.1) where [m] = mass matrix [c] = damping matrix [k] = s t i f f n e s s matrix (ii) , (u) , (u) = acceleration, v e l o c i t y , and displacement corresponding to each degree of freedom. 15 x = ground acceleration (I) = i d e n t i t y vector where every entry i s a unity The mass of the system i s usually lumped at the modes for si m p l i c i t y i n computation. I f such an assumption i s made, the mass matrix becomes diagonal., Discussion of the damping matrix i s beyond the scope of th i s paper. Modal analysis does not require an evaluation of this matrix, although the damping value i n each mode i s required for synthesis of the r e s u l t s . The s t i f f n e s s matrix i s formed by assembling the member s t i f f n e s s matrices. The procedure i s i d e n t i c a l to that of frame analysis. The f u l l member matrix with three degrees of freedom at each member end i s 6 x 6. If only bending deformation i s of interes t , i t s size i s reduced to 4 x 4. (b) Periods and Mode Shapes Solution of the free, undamped system yie l d s mode shapes and natural frequencies. The equation of motion becomes, [m] (ii) + [k] Cu) = CO) (2.2) The solution to thi s equation i s of the form, (u) = (A) s i n a)t (2.3) with Cu) - - c a 2 (A) sin 031 (2.4) 16 Substitute equations (2.3) and (2.4) into (2.2), -a)2 [m] (A) + [k] (A) = (0) (2.5) For a n o n t r i v i a l solution, [k] - O J 2 [m] | = 0 (2.6) This i s an eigenvalue problem of the form, [B] = X[C] (2.7) in which [B] i s a symmetric, banded matric and fC] i s a diagonal matrix. Eigenvalues associated with equation (2.6) correspond eigenvectors correspond to the mode shapes. I f n i s the rank of the mass matrix, [m], there are n natural frequencies and n mode shapes. (c) Response Spectra Given an earthquake record, i t i s r e l a t i v e l y simple to compute the response spectra. The peak acceleration, v e l o c i t y , or displacement of a single-degree-of-freedom system with a given value can be determined from the response spectra. In the modal analysis of multi-degree-of-freedom systems, with the assumption that a damping r a t i o for each mode i s known, a peak response for each mode can be read from the response spectra when natural periods are known. When a damping r a t i o i s small, to the squares of the angular frequencies, w 2. Associated 17 with l i t t l e error the peak acceleration, v e l o c i t y , and d i s -7 placement are related i n the following manner, Where S & = peak acceleration corresponding to the natural frequency, oi. S = peak v e l o c i t y v S^ = peak displacement. The choice of a damping r a t i o leaves some room for a debate. It i s generally taken to be 5 to 10% of c r i t i c a l for concrete and 2 to 5% of c r i t i c a l for s t e e l . S t r i c t l y speaking, the response spectra are v a l i d for one earthquake of known peak ground acceleration, but they can be scaled up or down depending on the peak ground acceleration which i s appropriate for a p a r t i c u l a r s i t e with certain assump-tions on magnitude and p r o b a b i l i t y of occurrence. Cd) Modal Forces Suppose that the acceleration spectrum i s given and that the damping rat i o s for a l l the modes are know or estimated; then, i t i s a r e l a t i v e l y simple to set up a force vector corresponding to each mode. Modal p a r t i c i p a t i o n factors, a, must f i r s t be computed. Let r denote the r th mode and T the transpose of a vector. The modal p a r t i c i p a t i o n factor for the r th mode can be computed as follows, 18 a = CA r) T[m] (T) (2.10) r (A r) T[m](A r) where (A ) = a vector representing the mode shape for the r th mode [m] = mass matrix (I) = i d e n t i t y vector whose elements are a l l unity. Then the force vector for the r th mode becomes (F r) = (A r)a rS^[m] (2.11) where (F ) = force vector S & = peak acceleration corresponding to r th mode natural frequency and damping. The modal displacements and response forces can be computed in the i d e n t i c a l manner to that used i n the s t i f f n e s s method i n a plane frame analysis. That i s , (F r) = [k] (A r) (2.12) where [k] = structure s t i f f n e s s matrix (A ) = modal displacements i n global coordinates, r r With (F ) known, (A ) can be computed by simply inverting the s t i f f n e s s matrix, [k]. The member forces can be calculated from the displacement vector, (A ) . (e) Combination of Forces and Displacements These forces and displacements for each mode correspond to the peak response. It i s not l i k e l y that these in d i v i d u a l maxima 19 occur at the same time; therefore, summing up the absolute values of these forces and displacements may r e s u l t i n over-estimating the response. It i s found that the root-sum-square (RSS) approach gives a more reasonable estimate. The in d i v i d u a l modal responses are combined by taking the square root of the sum of the squares of the responses. Contributions from the higher modes diminish very rapidly. For t h i s reason i t i s usually s u f f i c i e n t to take the f i r s t three or four modes for computation. For low-rise structures only the f i r s t mode i s s u f f i c i e n t f or a l l the p r a c t i c a l purposes. For high-rise structures higher modes play more dominant roles, and, hence, cannot be neglected. 20 2.2 Substitute Structure Method (a) Development Gulkan and Sozen performed a series of experiments to test the response of reinforced concrete structures to seismic g motions. The tests were r e s t r i c t e d to the single-degree-of-freedom system. They found that the basic c h a r a c t e r i s t i c s of reinforced concrete structure which determine the response to earthquakes are a change i n s t i f f n e s s and a change i n energy di s s i p a t i o n capacity, both of which are related to the maximum displacement. During strong motions the s t i f f n e s s of reinforced concrete decreases because of cracking of concrete, s p a l l i n g of concrete, and s l i p p i n g and reduction i n e f f e c t i v e modulus of s t e e l . The r e s u l t of t h i s i s that the period of the structure increases as i t undergoes i n e l a s t i c deformation. The area within a cycle of the force-displacement curve i s a measure of the energy dissipated by the system. They found that the area within the hysteretic loop increases with increase i n displace-ment into the i n e l a s t i c range of response. The e f f e c t of the hysteresis loop and the change in s t i f f -ness i s said to lead to a quantitative, r e l a t i o n s h i p between li n e a r response analysis and i n e l a s t i c analysis. A concept of substitute damping and e f f e c t i v e s t i f f n e s s are then introduced in order to interpret the i n e l a s t i c response i n terms of a li n e a r response analysis, using a spectral response curve. Consider an i d e a l i z e d symmetrical hysteresis loop as shown in Fig. 2.1. It follows Takeda 1s hysteresis loop which was 9 used as an a n a l y t i c a l model in the experiment by Takeda et a l . I t i s assumed that the structure has already undergone several cycles of i n e l a s t i c deformation. Let y be the o r i g i n a l s t i f f n e s s ; then the slope of the unloading curve BC, i s Y[—) where n i s the d u c t i l i t y and a i s a constant. The shape of the hysteresis curve i s such that i t i s approximately represented by a l i n e a r l y v i brating system with equivalent viscous damping."^ It i s assumed that the energy input i s e n t i r e l y dissipated by an imaginary viscous damper associated with the horizontal v e l o c i t y of the mass. Using t h i s idea, the substitute damping r a t i o , B g i s given by, B 2mto / ^ ( u ) 2 d t = -m/^  x u dt (2.13) s o u u where m = mass u = v e l o c i t y x = ground acceleration T = period of v i b r a t i o n to2 = measured absolute acceleration/measured o absolute displacement. The left-hand side of the equation represents the energy d i s s i -pated per cycle and the right-hand side represents the energy input per cycle. On the hysteresis loop diagram i t can be seen that ^ area EBC _ 1/2 (hysteresis loop area) (2.14) area ABF 1/2 (energy input) a i s taken as 0.5, then i t can be shown that 22 (1 - l / v ^ T ) (2.15) where n = d u c t i l i t y . 8 From the .experimental data Gulkan and Sozen gave the following It i s assumed in equation (2.16) that B g has a threshhold value of 0.02 at =1.0. The slope of the l i n e AE i s the e f f e c t i v e s t i f f n e s s and i s equal to y / n . The natural period corresponding to the e f f e c t i v e s t i f f n e s s i s T /n Gulkan and Sozen proposed a design procedure for a r e i n -forced concrete structure which can be i d e a l i z e d as a single-degree-of-freedom system. The design base shear can be c a l c u l a -ted as follows: (1) assume an admissible value of d u c t i l i t y , ri , (2) calculate the s t i f f n e s s based on the cracked section, (3) determine the natural period, T, (4) calculate the substitute damping r a t i o , B g , corresponding to the assumed value of d u c t i l i t y , n , (5) obtain base shear and maximum displacement by entering a spectral response diagram with an increased natural period of T >^T and a damping r a t i o equal to B g . Even though t h i s design procedure i s intended for a single-degree-of-freedom system, the basic concepts are d i r e c t l y transferred to the substitute structure method, which i s a design method for multi-story reinforced concrete frames. expression for the substitute damping r a t i o , B g , (2.16) 23 (b) Substitute Structure Method The substitute structure method was conceived by Shbata and 5 8 Sozen. It i s an extension of the method by Gulkan and Sozen which was described i n the previous section. The method i s intended for multi-story reinforced concrete frames and i s a design procedure to esta b l i s h the minimum strengths that the components must have so that a tolerable response displacement i s not l i k e l y to be exceeded. The main c h a r a c t e r i s t i c s of the sub-s t i t u t e structure method are the d e f i n i t i o n of a substitute frame, which i s a f i c t i t i o u s frame with i t s s t i f f n e s s and damping d i f f e r i n g from the actual frame, and the c a l c u l a t i o n of the design forces from modal analysis of the substitute frame using a l i n e a r response spectrum. These c h a r a c t e r i s t i c s are chosen such that the forces and the deformations from the analysis agree with these from the nonlinear dynamic analysis. 5 Shibata and Sozen l i s t the following conditions which must be s a t i s f i e d i n order to use the substitute structure method. (1) The system can be analyzed i n one v e r t i c a l plane. (2) There are no abrupt changes in geometry or mass along the height of the system. (3) Columns, beams and walls may be designed with d i f f e r e n t l i m i t s of i n e l a s t i c response, but the l i m i t should be the same for a l l beams i n a given bay and a l l columns on a given axis. (4) A l l st r u c t u r a l elements and jo i n t s are reinforced to avoid s i g n i f i c a n t strength decay as a r e s u l t of repeated rever-sals of the anticipated i n e l a s t i c displacements. (5) Nonstructural components do not int e r f e r e with s t r u c t u r a l response. The f i r s t condition implies that the method i s subject to the l i m i t a t i o n s of plane frame analysis. Such effects as torsion and b i a x i a l bending must be neglected. The second condition r e s t r i c t s the use of t h i s method to structures of regular shapes with uniform d i s t r i b u t i o n of mass and s t i f f n e s s . The third-con-d i t i o n deserves the most attention. The fact that the beams and columns may have d i f f e r e n t i n e l a s t i c deformation l i m i t s i s a big step forward from the conventional method in which the duc-t i l i t y of the entire structure must be chosen to be uniform. This point i s perhaps the biggest advantage i n using the substi-tute structure method. It i s usually desirable to allow the beams to y i e l d and absorb the bulk of energy while the columns remain e l a s t i c . The t h i r d condition does, however, exclude the p o s s i b i l i t y that t h i s method may be used for the design of soft story frames. The conditions (4) and (5) need l i t t l e explanation. Before the design procedure i s presented, terms p a r t i c u l a r to t h i s method must be explained. As mentioned previously, a substitute frame i s a f i c t i t i o u s frame with i t s s t i f f n e s s and damping related but not i d e n t i c a l to the actual frame. A damage r a t i o , u , i s used instead of d u c t i l i t y , n . Consider a force-displacement curve or a moment-rotation curve as i n F i g . 2.2. D u c t i l i t y i s usually defined as the r a t i o of ultimate displace-ment to y i e l d displacement, or n = ^ (2.17) The damage r a t i o on the other hand i s the r a t i o of the i n i t i a l s t i f f n e s s of the substitute frame, or 25 = slope AB H slope AC They are i d e n t i c a l for the e l a s t o - p l a s t i c case, but i f the s t i f f -ness afte r y i e l d has a p o s i t i v e slope, the damage r a t i o i s always smaller than d u c t i l i t y . Suppose s i s the r a t i o of the s t i f f n e s s a f t e r y i e l d to the i n i t i a l s t i f f n e s s ; that i s , the r a t i o of the slope of BC to the slope of AB in F i g . 2.2. Then the r e l a t i o n between the damage r a t i o and d u c t i l i t y i s 1 + (n - l ) s (2.19) where u = damage r a t i o ri = d u c t i l i t y s = r a t i o of s t i f f n e s s a f t e r y i e l d to i n i t i a l s t i f f n e s s A substitute damping r a t i o i s defined and computed i n an i d e n t i -c a l manner to that described i n the previous section. The damage r a t i o , however, i s used instead of d u c t i l i t y ; hence, 3 s = 0.2(1 - l//y") + 0.02 (2.20) where B = substitute damping r a t i o s y = damage r a t i o . The design procedure w i l l now be described . A necessary assumption i s that the preliminary member sizes of the actual structure are known from gravity loads and other functional requirements. Then the following steps are involved. (1) Assume an acceptable value of damage r a t i o , u, for each group of members. (2) Define the f l e x u r a l s t i f f n e s s of substitute-frame elements as C E I ) s i = u a i (2.21) i where (EI) . = f l e x u r a l s t i f f n e s s of i th substitute-frame s i element (EI) . = f l e x u r a l s t i f f n e s s of i th element i n the a i actual frame u. = tolerable damage r a t i o for i th element. * i (3) Compute natural periods, mode shapes and modal forces for the undamped substitute structure. (4) Compute an average or a "smeared" damping r a t i o for each mode. B s i = 0.2(1 - l / / y i ) + 0.02 (2.22) m^ =\T^ ^ s i <2-23> i i L. where P. = (M2 . + M2 . + M . R. ) (2.24) l 6 (EI) . a i D I a i r>i s i and 6 . = substitute dampinq r a t i o of i th member s i £>m = smeared substitute damping for m th mode P^ = f l e x u r a l s t r a i n energy i n i th element i n the m th mode = length of frame element i (EI) . = assumed s t i f f n e s s of substitute frame element s i M ., M, . = end moments of substitute frame element i a i b i for m th mode. (5) Repeat the modal analysis using the smeared damping r a t i o s and compute the root-sum-square (RSS) forces. (6) Compute the design forces, .v +, v , F = F rss abs i i rss 2v rss where = design force for i th element F. = root-sum-square forces for i th element 1 rss ^ v = RSS base shear rss v , = maximum value for absolute sum of any two abs J of the modal base shears. (7) To avoid the r i s k of excessive i n e l a s t i c action i n the columns increase the design moments of the columns by 20%. In the f i r s t step a designer can choose how much i n e l a s t i c deformation can be allowed i n each element group. Since the target damage r a t i o s are always greater than or equal to one, i t i s clear i n the second step that the natural periods of the sub-s t i t u t e frame are always greater than these of the actual frame. Steps 3 and 4 are necessary, because substitute damping r a t i o s may be d i f f e r e n t for each element group. A smeared damping r a t i o i s computed for each mode by assuming that each element c o n t r i -butes to modal damping i n proportion to i t s r e l a t i v e f l e x u r a l s t r a i n energy associated with the mode shape. Elements with complex s t i f f n e s s can be used to compute the smeared damping r a t i o s , but the f l e x u r a l energy approach i s easier to use and has more physical meaning. The sixth step i s an extra factor of 28 safety in case any combination of two modes produces undesirable r e s u l t s . The l a s t step i s desirable in a design procedure, because f a i l u r e i n a column p r i o r to f a i l u r e i n adjoining beams may lead to catastrophic f a i l u r e of a structure. A l i n e a r response spectrum i s used i n the analysis; the authors suggest that a smoothed spectrum be used. It i s mentioned as a c r i t i c a l feature of t h i s method that i t becomes plausible only with the understanding that the force response decreases as the structure becomes more f l e x i b l e ; therefore, the smoothed spectrum, in r e l a t i o n to the natural periods of the substitute structure, should have a shape such that the spectral acceleration response decreases with an increase i n period. Implicit assumptions and l i m i t a t i o n of the substitute structure method are now discussed. It i s i m p l i c i t l y assumed that the moment d i s t r i b u t i o n i n a l l the members are l i n e a r and that the points of i n f l e c t i o n are placed at or near the mid-points of the member spans. With these assumptions, i t becomes clear that the shape of force-displacement curve i s i d e n t i c a l to that of the moment-rotation curve. Otherwise di v i d i n g the actual f l e x u r a l s t i f f n e s s by the damage r a t i o greater than one may not be a correct approach. These assumptions are reasonable in beams which are more l i k e l y to receive i n e l a s t i c deformations but they may not be so v a l i d i n columns as shown by Blume et a l . This point, however, i s not an important factor as long as columns are designed with a target damage r a t i o of one, which i s desirable i n most p r a c t i c a l applications. In practice, unless the design moments are known, the s t i f f n e s s of a f u l l y cracked section that must be used to calculate the s t i f f n e s s of the substitute frame cannot be deter-mined. An educated guess i s required and at the end of the calculations, i t must be checked that the guess was indeed reasonable. The design moments correspond to extra moment capacities required over the capacities for the gravity loads. Two ends of a member must be capable of handling the same design moment both i n po s i t i v e and negative d i r e c t i o n s . This require-ment again i s reasonable for beams, but may not be so for columns. The authors designed the test frames using the substitute structure method. These test frames were subjected to nonlinear time-step analysis, and they state that the frames behaved well and that i n e l a s t i c deformation occurred at the prescribed loca-tions . (c) Computer Program Use of a computer i s almost as esse n t i a l i n the substitute structure method, as i t i s i n the case of regular modal analysis. A flow diagram i s shown i n F i g . 2.3. Only minor modifications are required to convert an ex i s t i n g modal analysis program to be used for the substitute structure method. A target damage r a t i o for each member must be read i n and stored when s t r u c t u r a l data are read i n . At t h i s stage i t may be advantageous to compute and store a substitute damping r a t i o for each member. When the structure s t i f f n e s s matrix i s assembled from member s t i f f n e s s matrices, f l e x u r a l components of the member s t i f f n e s s matrix must be divided by the appropriate target damage r a t i o . The structure s t i f f n e s s matrix becomes 30 that of the substitute frame, and t h i s matrix i s used to compute natural periods and associated mode shapes. Calculations of modal responses are performed twice: on the f i r s t cycle modal forces are computed for the undamped sub-s t i t u t e structure; f l e x u r a l s t r a i n energy for each member i s computed and stored for each mode. A smeared damping r a t i o for each mode i s computed according to equation (2.23). With the smeared damping known the computation of modal forces and d i s -placements are repeated. Root-sum-square forces and displace-ments are computed on the second cycle, but s t r a i n energy calculations are not required. From the modal base shears RSS base shear and the maximum value of the absolute sum of any two of the modal base shears must be computed. To compute the design forces the RSS forces are multi p l i e d by the factor i n equation (2.25). Furthermore, the column moments must be increased by 20%. If a li n e a r response spectrum i s chosen as was suggested 5 by Shibata and Sozen, only one inversion of the structure s t i f f -ness matrix i s necessary. The program i s a very e f f i c i e n t one that requires small storage and l i t t l e CPU time. If a regular plane frame analysis program i s to be converted, subroutines for setup of mass matrix, response spectrum, and computation of natural periods, mode shapes, and modal p a r t i c i p a t i o n factors must be added. 2.3 Examples and Observations (a) Frames with F l e x i b l e Beams In order to test the computer program for the substitute 5 structure method, sample frames from Shibata and Sozen's paper were chosen and the re s u l t s were compared with t h e i r s . The frames are 3-, 5-, and 10-stories high and they consist of s t i f f columns and f l e x i b l e beams. The data for the three frames are shown i n F i g . 2.4. The width i n each case was 24 feet and the story height was uniform at 11 feet with a weight of 72 kips concentrated at each story. The target damage r a t i o s were one for columns and six for beams in a l l three frames. Since the moments of i n e r t i a of the cracked sections were not known, the assumptions made by Shibata and 5 Sozen were repeated; that i s , 1/3 of moment of i n e r t i a of the gross section was used for beams and 1/2 for the columns. The 5 desxgn spectrum A i n t h e i r paper was used (Fig. 2.5). It i s an acceleration spectrum derived from l i n e a r response spectra of six earthquake motions; namely, two components of E l Centro 1940, two components of Taft 19 52, and two components of Managua 19 72. The peak ground acceleration was normalized at 0.5 g. It was assumed that the design response acceleration for any damping factor, 8, could be related to the response for B = 0.02 by using, Response acceleration,for B _ 8 ,„ Response acceleration for B = 0.02 6 + 100 B U.^b; The natural periods and smeared damping factors of the three frames are l i s t e d i n Table 2.1 along with Shibata and 5 Sozen's r e s u l t s . The design moments are shown on Fig. 2.4. The design moments for the 3-story frame agreed with those given by 5 Shibata and Sozen. The design moments for 5- and 10-story frames were not shown i n the paper. One may conclude that the program was capable of reproducing the r e s u l t s shown i n Shibata 5 and Sozen's paper. The three frames were then tested in a similar fashion to 5 that employed by Shibata and Sozen. An i n e l a s t i c dynamic program, 12 SAKE, was used to compute the response history of each frame to earthquake motions. This program was selected, because i t was written exclusively for concrete frames. Its effectiveness was 13 reported by Otani and Sozen. A record of Managua 19 72 earth-quake was not available; therefore, two components of E l Centro 19 40 and two components of Taft 19 52 were used. These accelera-ti o n records were normalized so that the peak ground acceleration was 0.5 g in a l l four records. The y i e l d moments were set at the design moments. S t i f f n e s s beyond y i e l d was taken as 3% of the i n i t i a l s t i f f n e s s . The damping was taken to be proportional to s t i f f n e s s , corresponding to 2% damping for the f i r s t mode. Com-puted damage ra t i o s of three frames are shown i n Table 2.2. Comparison of some of the results with those by Shibata and 5 Sozen i s shown i n Table 2.3. The three-story frame behaved very well . None of the c o l -umns yielded and the beam damage r a t i o s were s i x or less i n a l l four earthquakes. Thus the structure designed by the substitute structure method behaved as expected. In the f i v e - s t o r y frame, E l Centro EW record produced the worst r e s u l t . The columns 33 yielded at three locations and the damage ratios of the beams, except the f i r s t - f l o o r beam, were about seven. The frame, how-ever, behaved very well in the other three earthquake motions. The columns remained i n the e l a s t i c range and the beam damage rat i o s were less than f i v e . The ten-story frame produced the worst results of the three frames. Like the fi v e - s t o r y frame E l Centro EW motion produced the most unfavorable r e s u l t s . The columns yielded at many locations. The f i f t h story column exhibited a damage r a t i o of about seven. A l l the beams exceeded the target ^damage r a t i o of s i x and some reached a damage r a t i o of about ten. The results were much better i n the other three earth-quakes. Although the columns yielded at a few locations i n two earthquakes, i n e l a s t i c deformations were not excessive. The beam damage r a t i o s were a l l less than s i x . These results agreed q u a l i t a t i v e l y with those by Shibata and Sozen, ~* but not quantitatively (Table 2.3). The quantitative difference was the smallest for the three-story frame. The big -gest discrepancy occurred i n the ten-story frame, esp e c i a l l y in E l Centro EW motion. The difference may be due to modeling of elements i n the nonlinear dynamic program, duration of earthquake motion, or difference i n earthquake records caused by d i g i t i z a t i o n of the records or f i l t e r i n g . (b) Soft-Story Frame 5 Shxbata and Sozen r e s t r i c t e d a choice of a target damage r a t i o for each element i n order that the substitute structure method may be used successfully. They stated that columns, beams, and walls may be designed with d i f f e r e n t target damage r a t i o s f but 34 that the target damage rati o s should be the same for a l l beams in a given bay and a l l columns on a given axis- This condition implies that a soft-story frame may not be designed properly by the substitute structure method. Two examples were tested i n order to check the necessity for this r e s t r i c t i o n . Two three-story frames similar to the one used i n the pre-vious section were used. Data for the f i r s t example are shown in F i g . 2.6. The ground f l o o r was designed as a "soft story". A target damage r a t i o of two was assigned to the f i r s t - s t o r y columns and one to the rest of columns. A target damage r a t i o of six was given to the f i r s t - f l o o r beam and one to the other beams. The frame consists of one 24-foot bay with 11 foot story heights. The f l o o r weight i s 72 kips for each l e v e l . The moment of i n e r t i a of the f i r s t story columns was 3/4 of that of the columns above. The beams had constant moment of i n e r t i a . The design moments were computed by the substitute structure method and are shown i n F i g . 2.6. The design spectra shown in F i g . 2.5 were used. The frames were then subjected to four earthquake motions, using the non-12 l i n e a r dynamic analysis program, SAKE. Each earthquake record was normalized so that the maximum ground acceleration was 0.5 g. The design moments were used as the y i e l d moments. Two per cent st i f f n e s s - p r o p o r t i o n a l damping and 3% s t r a i n hardening were assumed in the nonlinear analysis. The r e s u l t s of four runs are shown in F i g . 2.7. The frame t r i e d to behave i n the way i t was designed to: the f i r s t - s t o r y columns yielded i n a l l four cases. E l Centro EW motion produced the worst r e s u l t ; the damage r a t i o reached 2.8. Taft S69E motion produced the smallest damage r a t i o , which was 1.2. The rest of the columns remained e l a s t i c . The 35 f i r s t - f l o o r beam yielded i n every case and the damage r a t i o ranged from 3.7 to 6.1. The second-floor beam remained more or less e l a s t i c except for one case. The t h i r d - f l o o r beam did not behave as well as the other beams. It yielded i n a l l four cases, but the damage ra t i o s were less than 1.5 except i n E l Centro EW motion. Although the test frame did not perform very well during E l Centro EW motion, the results from other motions seem to indicate that the substitute structure method produced a successful design of a soft story frame in t h i s example. In the second example the soft story was moved from the f i r s t story to the second story. The data are shown in F i g . 2.8. The same design spectrum was used and the substitute structure method was used to compute the design moments. Those design moments are shown i n F i g . 2.8. The frame was again subjected to the four earthquake motions i n an i d e n t i c a l manner, with the same assumptions being made i n the nonlinear dynamic analysis. The results are shown i n F i g . 2.9. They were not as good as the f i r s t example, since the second-story columns remained e l a s t i c i n a l l four cases, although they were designed to y i e l d . The other columns remained e s s e n t i a l l y e l a s t i c . The beams behaved better than the columns. The second f l o o r beam did y i e l d i n every earthquake; with the damage r a t i o ranging from 2.6 to 4.4 which i s less than the target damage r a t i o of s i x . The other beams es s e n t i a l l y remained i n the e l a s t i c range. (c) 2-Bay, 3-Story Frame The results of the soft-story frames were inconclusive. The method worked well in the f i r s t example, but only a f a i r 36 r e s u l t was obtained i n the second example. A two-bay, three-story frame was used to te s t whether the substitute structure method could be used for a frame with randomly assigned target damage r a t i o s . The data for the structure are shown i n F i g . 2.10. The design spectrum was the same one used i n the previous examples. The target damage r a t i o s were randomly assigned. The substitute structure was used to compute the design moments, but the column moments were not increased by 20%, because they could y i e l d before the beams. The nonlinear dynamic analysis was c a r r i e d out i n an ident-i c a l manner as i n the previous examples. The same four earth-quake records were used. The results of four runs are shown i n Fig. 2.11. The structure behaved quite well when the average damage rat i o s of four earthquakes are compared with the target damage r a t i o s . E l Centro EW motion produced the biggest damage while Taft motions produced the le a s t . In general, the bottom-story columns received more damage than they were expected to take, but the damage r a t i o s of the second-story columns were very close to the target damage r a t i o s . The third-story columns were damaged less severely than they were designed for. The same trend i s found i n the beams, but none of the average damage r a t i o s were higher than the target damage r a t i o s . The results of t h i s example seem to indicate that the sub-s t i t u t e structure method can be used to design a structure in which d i f f e r e n t target damage rat i o s are assigned for beams i n the same bay and for columns on the same axis. It appears that beams work better than columns. 37 2.4 Equal-Area S t i f f n e s s Method (a) Observation As was discussed i n the section 2.3(a), three frames were designed using the substitute structure method and they were sub-jected to nonlinear dynamic analysis. When the dynamic analysis program was run, time-history plots of displacements and moments were obtained as a part of the output. Upon observation of these plots i t was possible to pick up the periods of the most dominant vibration, and i t was found that these periods were peculiar to the frames, not to the earthquake motions. Furthermore, these periods were d i f f e r e n t from the natural periods of the actual frames and from those of the substitute frames. Table 2.4 l i s t s the natural periods of the actual frames and substitute frames for the f i r s t mode as well as the observed periods from the dynamic analyses. In a l l three cases the ob-served periods were longer than the natural periods of the actual frames, but shorter than the natural periods of the substitute frames. This seemed to imply that the substitute structure method did not give the correct natural periods of a structure when i t underwent i n e l a s t i c deformation. Some e f f o r t was made to f i n d a method which would give a better estimate of the natural periods of a structure which would be subjected to i n e l a s t i c deformation. This was f e l t to be important, since modal analysis was to be used, i n which the response i s read against the period. 38 (b) Equal-Area S t i f f n e s s The preceding observation supports the theory that the s t i f f n e s s of a system i s reduced when i t i s subjected to strong motions such that i t s deformations exceed the e l a s t i c l i m i t . At the same time i t seems to indicate that the s t i f f n e s s used i n the substitute structure method i s too small: true e f f e c t i v e s t i f f -ness l i e s somewhere between the e l a s t i c s t i f f n e s s and the s t i f f -ness of the substitute structure. Consider the load-deflection curve i n Fi g . 2.12. Assume that i t i s an e l a s t o - p l a s t i c case so that the damage r a t i o and d u c t i l i t y are the same. When a target damage r a t i o i s chosen, the maximum displacement i s i m p l i c i t l y selected. The system i s allowed to undergo a deformation on the loading curve up to the point C. The area under the curve i s equal to the area of the trapezoid ABCD. It i s possible to make up a f i c t i t i o u s e l a s t i c system which reaches the same ultimate displacement and has the same area under i t s l i n e a r load-deflection curve AED as the area of the b i l i n e a r curve ABCD, while both systems reach the same ultimate displacement, A , and absorb the same energy of defor-mation in doing so, the e l a s t o - p l a s t i c system has the y i e l d force, F , as maximum force and the f i c t i t i o u s e l a s t i c system reaches y F^, which i s greater than the y i e l d force. The slope of the l i n e AE i s the s t i f f n e s s of thi s e l a s t i c system, which the author c a l l s an "equal-area s t i f f n e s s " . By equating the two areas, the equal-area s t i f f n e s s can be expressed i n terms of the i n i t i a l s t i f f n e s s and the target damage r a t i o , 1 v l i z ; (2.27) 39 where k q = equal-area s t i f f n e s s k = i n i t a l s t i f f n e s s u = target damage r a t i o . The y i e l d force i s unknown, but i t i s expressed i n terms of the maximum force, F^, F = F_ (V y .} (2.28) y 1 L2y - 1J where F = y i e l d force y •* F^ = maxium force and y = target damage r a t i o . If the moment-curvature curve of an element has the same shape as that of the load-deflection durve, the f l e x u r a l s t i f f -ness of the element can be reduced according to equation (2.26). This s t i f f n e s s can be used to solve for the natural periods of the system. This approach i s , of course, very hypothetical and there i s no experimental data to support i t . The concept of substitute damping loses much of i t s meaning, because i t was derived from the s i m p l i f i e d hysteresis loop of degraded r e i n -forced concrete. But t h i s hypothesis can be tested a n a l y t i c a l l y by modifying the s t i f f n e s s part of the substitute structure program. (c) Examples The same three frames used i n section 2.2(a) were used to test the equal-area s t i f f n e s s method. The target damage rati o s were set at one for the columns and six for the beams. When the fl e x u r a l components of the member sti f f n e s s e s were assembled, they 40 were reduced according to the equation (2.27); that i s , 2p. - 1 . a . . . (EI) (EI) (2.29) ex ax where (EI) ex equal-area s t i f f n e s s of element i (EI) s t i f f n e s s of i th element of actual frame ax target damage r a t i o of i th element. The natural periods of the three frames were computed using the equal-area s t i f f n e s s . The periods corresponding to the f i r s t mode are l i s t e d on Table 2.4. Those periods agreed very well with the dominant periods observed i n the nonlinear analysis. Therefore, as far as the natural periods are concerned, t h i s approach gives a more r e a l i s t i c estimate. (d) Area for Further Studies method were used as the y i e l d moments i n the nonlinear dynamic analysis. If a method to obtain the same design forces could be developed, t h i s equal-area s t i f f n e s s method would become more at t r a c t i v e . An e f f o r t was made to f i n d a way to compute design forces that are si m i l a r to those from the substitute structure method, but i t was not possible to obtain a s a t i s f a c t o r y r e s u l t . Further studies may be worthwhile, because the agreement in periods i s too good to ignore. Any further research should be started with a single-degree-of freedom system. A theory to support t h i s hypothesis needs to be established along with The design forces computed by the substitute structure 41 experimental data. I f a l i n e a r response spectrum i s to be used, a new method of computing suitable damping properties must be developed. 42 CHAPTER 3 MODIFIED SUBSTITUTE STRUCTURE METHOD 3.1 Modified Substitute Structure Method The term, " r e t r o f i t " , i s defined in the f i r s t chapter. It describes the problem of evaluating the performance of e x i s t i n g buildings against seismic hazards. A r e t r o f i t procedure i s , then, a procedure for analyzing e x i s t i n g buildings. It i s inevitable that almost a l l the structures y i e l d and suffer i n e l a s t i c deforma-ti o n under a strong earthquake motion; such a procedure, there-fore, must perform some sort of i n e l a s t i c analysis. It must be capable of i d e n t i f y i n g the locations and extent of damage associ-ated with a p a r t i c u l a r earthquake motion. If a structure i s to f a i l , the mode of f a i l u r e must be i d e n t i f i e d . I t i s desirable that a method be f l e x i b l e enough to handle earthquakes of d i f f e r -ent nature and magnitude. At the same time i t must be reasonably economical and easy to use i n order to become a p r a c t i c a l t o o l for average engineers. The use of a computer i s probably inevitable because of the nature of the problem, but a program to run such an analysis must be easy to write and economical to operate. The modified substitute structure method f u l f i l l s the aforementioned requirements. As the name suggests, i t was devel-5 oped from the substitute structure method by Shibata and Sozen. At present i t s use i s r e s t r i c t e d to reinforced concrete structures as i s the case for the substitute structure method i t s e l f , but with proper modifications the method may be used for analysis of st e e l and other structures. I t i s a modified e l a s t i c analysis i n which the s t i f f n e s s and damping properties are changed for use with modal analysis so that the forces and deformations agree with nonlinear dynamic analysis. A l i n e a r response spectrum i s used to compute the i n e l a s t i c response. The concepts of substi-tute damping, damage r a t i o , and substitute s t i f f n e s s are borrowed from the substitute structure method. The difference between a design procedure and a r e t r o f i t procedure i s worth noting. In a seismic design procedure the i n i t i a l s t i f f n e s s of the structure i s known approximately from other requirements. A designer can choose and specify the amount of i n e l a s t i c deformation each element i s allowed to undergo in a given earthquake motion. It i s the design forces or y i e l d forces that must be determined. In the substitute structure method, the s t i f f n e s s of the actual frame i s known or i t can be estimated f a i r l y p r e c i s e l y . Target damage r a t i o s are selected for elements by a designer. Hence, the substitute s t i f f n e s s and substitute damping r a t i o s of the elements are prescribed. Natural periods, associated mode shapes, and modal p a r t i c i p a t i o n factors need to be computed only once. After computation of a smeared damping r a t i o for each mode, modal forces are calculated and combined as sp e c i f i e d . No i t e r a t i o n i s required during computation. In a r e t r o f i t procedure the i n i t i a l s t i f f n e s s and the y i e l d moments and other strength properties of a structure are known or they can be found from design calculations, drawings, and f i e l d invest-igations. What i s known i s the amount of i n e l a s t i c deformation; that i s , the damage r a t i o for each member must be computed given an earthquake motion. In the modified substitute structure method a suitable combination of modal forces must agree with the known y i e l d forces. To achieve t h i s the damage r a t i o s of a l l the elements must be estimated pr e c i s e l y so that correct substitute s t i f f n e s s and substitute damping r a t i o s can be used. This, of course, i s impossible to do; otherwise there would be no need to perform an analysis. It i s , therefore, inevitable that an i t e r a -t i v e process must be used. After each i t e r a t i o n damage r a t i o s must be modified to approach nearer to the correct values. This i s c e r t a i n l y a disadvantage, because more computations are re-quired and hence more costs. But i f the number of i t e r a t i o n s are small, i t i s s t i l l an economical alternative to f u l l - s c a l e non-li n e a r dynamic analysis. Before the procedure for the modified substitute structure method i s described i n d e t a i l , several conditions are l i s t e d . They must be s a t i s f i e d i n order to apply the modified substitute structure method properly. These conditions are: Cl) the system can be analyzed i n one v e r t i c a l plane, (2) there i s no abrupt change in geometry and preferably in mass along the height of the system, (3) reinforcement of a l l members and join t s are known such that t h e i r a b i l i t y to withstand repeated reversals of i n e l a s t i c deformation without s i g n i f i c a n t strength decay can be estimated, and (4) nonstructural components do not i n t e r f e r e with s t r u c t u r a l response. The aforementioned conditions are si m i l a r to those l i s t e d by 45 5 Shibata and Sozen i n the substitute structure method. In fact, i t should be noted that, a f t e r convergence, the f i n a l i t e r a t i o n of the modified substitute structure method i s i d e n t i c a l to the design procedure, and therefore has exactly the same r e s t r i c t i o n s and v a l i d i t y . The following i s the step-by-step description of the proce-dure for the modified substitute structure method. It must be remembered that the y i e l d force cannot be exceeded at any time. CD Perform a modal analysis on the assumption of e l a s t i c behaviour. Damping ra t i o s must be chosen so that they are appropriate for the given earthquake. Compute the root-sum-square (RSS) forces. C2) Find the members i n which RSS moments exceed the y i e l d moments. Note that the bigger of the two end moments i s used. C3) In such members modify the damage r a t i o s according to the formula that w i l l be described l a t e r on. The other members w i l l have a damage r a t i o of one. C4) Follow steps (.2) to C5) for the substitute structure method which was described on pages 26 and 27 i n Chapter 2.2(b). Compute the RSS moments. (5) Compare the RSS moments with the y i e l d moments. Modify the damage r a t i o s according to the formula to be discussed l a t e r (6) Repeat the steps (4) and (5) u n t i l a l l the computed moments, except i n those members for which the damage r a t i o s are one, are equal to the respective y i e l d moments. (7) The members i n which the damage r a t i o s are greater than one w i l l receive i n e l a s t i c deformation. Check i f each member 46 can take such deformation. If not, such a member w i l l f a i l . I t i s now possible to make an estimate of the locations and extent of damage i n the whole structure. Similar checks can be made for other components of in t e r n a l force. An ordinary e l a s t i c modal analysis i s performed i n the f i r s t i t e r a t i o n , because at thi s stage i t i s not clear i f a structure w i l l go through i n e l a s t i c deformations i n a given earh-quake. A value for damping must be chosen; a r a t i o of 10% of c r i t i c a l i s appropriate for a reinforced concrete structure sub-jected to a strong earthquake motion. Since i t i s impossible to exceed the y i e l d moments, those members i n which the computed moments are greater than t h e i r y i e l d moments w i l l y i e l d . In the t h i r d step the f i r s t estimate of damage r a t i o s i s made. Starting from the second cycle of i t e r a t i o n , the substitute structure method i s used to compute the natural periods, mode shapes, and modal forces. Damage ra t i o s calculated at the end of the previous i t e r a t i o n are used to compute the substitute s t i f f n e s s and sub-s t i t u t e damping r a t i o s . The root-sum-square moments are used throughout the i t e r a t i o n s . In the substitute structure method they were increased according to equation (2.25) and the column moments were further increased by 20% to obtain the design moments. This approach i s acceptable i n a design procedure, because i t would provide an extra margin of safety. But i n order to be on the conservative side i t i s advisable to use the root-sum-square moments and ignore the factor in equation (2.25) . Increasing the column moments by 20% here i s , of course, t o t a l l y absurd. Unless correct damage ra t i o s are obtained i n the previous i t e r a t i o n , the computed moments do not agree with the y i e l d moments except for 47 the members which remain e l a s t i c . The damage ra t i o s must be modified arid another i t e r a t i o n must be made. At some stage a l l the damage r a t i o s w i l l converge to the correct values and the i t e r a t i o n process w i l l be stopped. Then an evaluation of the performance of the structure can be carried out as outlined in the l a s t step. It must be noted that the e f f e c t of s t r a i n hardening i s ignored i n the discussion above, but i t can be included with only a s l i g h t modification. It i s now appropriate to explain a way to modify damage rat i o s at the end of each i t e r a t i o n . Consider the e l a s t o p l a s t i c case shown in F i g . 3.1. Suppose at the end of the f i r s t i t e r a -t i o n , which i s an ordinary modal analysis, the computed moment was which i s greater than the y i e l d moment, M^. Since the member was assumed to behave e l a s t i c a l l y , i t followed the l i n e OA and reached the point B with the moment, M^ , and the rotation, <J)^. Since a computed moment cannot exceed M^ , the s t i f f n e s s , k, must be reduced i n the next i t e r a t i o n . I t i s assumed that the rotation, cj>^, was correct. A point B' i s located on the p l a s t i c part of the moment-rotation curve and the slope of the l i n e OB' i s used as the s t i f f n e s s for the next i t e r a t i o n . The damage r a t i o corresponding to t h i s new s t i f f n e s s can be calculated from the geometry. The damage r a t i o at the end of the f i r s t i t e r a t i o n i s given by, M. 1 y 2 > 1 (3.1) M y where 2 damage r a t i o to be used i n the second i t e r a t i o n M. 1 computed moment i n the f i r s t i t e r a t i o n 48 M = y i e l d moment y J Suppose that at the end of the second i t e r a t i o n the computed moment, M2, s t i l l exceeded the y i e l d moment, M •; that i s , i t reached the point C on the curve. It means that the assumed s t i f f n e s s was s t i l l too big and that the damage r a t i o must be increased. This time a point C i s located on the curve and the slope of the l i n e OC1 i s used to define the new s t i f f n e s s . A new damage r a t i o corresponding to the new s t i f f n e s s can be obtained from the geometry. M2 1-U = (3.2) J M y where ^3 = damage r a t i o to be used i n the t h i r d i t e r a t i o n 1^ 2 = damage r a t i o used i n the second i t e r a t i o n = computed moment at the end of the second i t e r a -t i o n M = y i e l d moment. y It i s possible that the computed moment, M^, was less than the y i e l d moment, M . The s t i f f n e s s must now be increased; that i s , the damage r a t i o must be decreased. The new damage r a t i o can be computed from the geometry i n a si m i l a r way and the same r e l a -tion as i n equation (3.2) can be obtained. Attention must be paid t h i s time, since i f the new damage r a t i o i s less than one, i t must be set at one. In general, at the end of the n th i t e r a t i o n the new damage r a t i o can be computed by the following equation. 49 M _ . . . n yn+l y n >1 -C3.31 M y where u ,. = damage r a t i o to be used i n the n+l th n+l J i t e r a t i o n y n = damage r a t i o used i n n th i t e r a t i o n Mn = computed moment i n n th i t e r a t i o n M = y i e l d moment. y If Vn+2_ equals y n for a l l the members, the i t e r a t i o n process i s complete. When the moment-rotation curve a f t e r y i e l d exhibits s t r a i n hardening, the si t u a t i o n i s a l i t t l e more complex. If such i s the case, the y i e l d moment i s not the absolute l i m i t . The com-puted moment can be and w i l l be greater than the y i e l d moment provided that the damage r a t i o i s greater than one. Derivation of the formula for the new damage rati o s i s shown i n Appendix A. It i s , yn+l Mn' yn >1 (3.4) M (1 - s) + s.y .M y n n where ^n+l ~ modified damage r a t i o to be used i n n+l th i t e r a t i o n = damage r a t i o used i n n th i t e r a t i o n M = computed moment i n n th i t e r a t i o n M = y i e l d moment y s = r a t i o of s t i f f n e s s a f t e r y i e l d to i n i t i a l s t i f f n e s s . 50 Inherent lim i t a t i o n s of the modified substitute structure method are now discussed. The moment-rotation curve of each member must be such that i t can be approximated by a b i l i n e a r curve. Furthermore, i t must have the same shape as that of the load-deflection curve. I f l i n e a r l y d i s t r i b u t e d moment with a point of i n f l e c t i o n i n the mid-span of a member i s assumed, t h i s condition i s s a t i s f i e d . The moment capacity of each member i s assumed to be the same for both ends and for both posi t i v e and negative moments. If the computed moment at one end of the member i s greater than at the other end, the bigger moment i s chosen to compute the damage r a t i o . 51 3.2 Computer Program The use of a computer i s esse n t i a l for p r a c t i c a l applica-tions of the modified substitute structure method. The i t e r a t i v e process that i s required i n the method can be incorporated i n the program quite e a s i l y . The program i t s e l f can be written with l i t t l e d i f f i c u l t y . If an e l a s t i c modal analysis program i s available, r e l a t i v e l y few modifications are necessary. The flow diagram of the modified substitute structure pro-gram i s shown i n Fig . 3.2. Data for s t r u c t u r a l d e f i n i t i o n , member properties, and j o i n t locations are read i n and stored in the f i r s t part of the program. The damage r a t i o s of a l l the members should be i n i t i a l i z e d at one. Then the mass matrix should be set up; i t remains unchanged throughout the i t e r a t i o n process. The structure s t i f f n e s s matrix i s assembled from member matrices. The f l e x u r a l part of the member s t i f f n e s s i s modified according to the damage r a t i o using equation (2.21). Since a l l the damage ra t i o s are set at one i n the f i r s t i t e r a t i o n the struc-ture s t i f f n e s s matrix i s the same as i n the e l a s t i c analysis. This matrix and the mass matrix are used to solve for natural periods, associated mode shapes, and modal p a r t i c i p a t i o n factors. Since i t involves a regular eigenvalue problem, a l i b r a r y sub-routine i s usually available. I n i t i a l l y a suitable set of damping r a t i o s should be given by the user. Ten per cent damping for a l l the modes was used by the author. A spectrum subroutine i s c a l l e d and a peak ground acceleration i s returned. Then a load vector i s set up and the s t i f f n e s s matrix i s inverted to solve for deflections. Modal forces can be computed i n the usual 52 manner. This process i s repeated for a l l the modes and RSS forces and displacements are computed. At the end RSS moments are com-pared with respective y i e l d moments. If the y i e l d moment of any member i s exceeded, i t e r a t i o n i s necessary. The damage r a t i o of such a member i s modified according to equation (3.3) or (3.4). From the second i t e r a t i o n the substitute structure method i s used to compute modal forces and displacements. The structure s t i f f n e s s matrix i s reassembled using the new set of damage r a t i o s and the computation of natural periods, mode shapes, and modal p a r t i c i p a t i o n factors i s repeated. Substitute damping r a t i o s of a l l the members should be calculated at t h i s stage using equation (2.20). Modal forces are calculated twice. Forces for the undamped case are computed f i r s t to calculate the f l e x u r a l s t r a i n energy stored i n each member. Smeared damping r a t i o s for a l l the modes are computed using equations (2.23) and (2.24). They are used to get the peak ground accelerations from the spectrum. Modal forces and displacements are recomputed and RSS forces and displacements are obtained at the end. Equation (3.3) or (3.4) i s used to modify the damage r a t i o s . Further i t e r a t i o n s are necessary u n t i l a l l the damage ra t i o s stop changing. In practice, very many i t e r a t i o n s are necessary to achieve t h i s and more prac-t i c a l convergence c r i t e r i a must be used to keep the number of ite r a t i o n s at a reasonable l e v e l . The program used by the author i s l i s t e d in Appendix B. The cost of running the program depends d i r e c t l y on the number of i t e r a t i o n s . If the convergence can be accelerated, the saving i n CPU time and hence cost can be substantial. An attempt was made to achieve accelerated convergence and a method i s described i n a subsequent section of t h i s chapter. Obviously the proposed method i s more costly than an ordinary modal analysis because of the amount of computation involved, but the storage requirement i s roughly the same and the CPU time required for t h i s method i s s t i l l a f r a c t i o n of that for the f u l l - s c a l e non-l i n e a r dynamic analysis. Therefore, o v e r a l l cost of running th i s analysis i s s t i l l small compared to the cost of running the nonlinear dynamic analysis. Coupled with ease of data setup, the advantage of the modified substitute structure method over the nonlinear dynamic analysis i s substantial. 54 3.3 Convergence In order to test whether the modified substitute structure method actually works, test frames are required. The damage rat i o s of a l l the members i n such frames must be known for a given l i n e a r response spectrum. Since the method u t i l i z e s the substitute structure method, i t i s possible to design a frame by the substitute structure method and then subject i t to analysis by the modified substitute structure method. When the i t e r a t i o n procedure i s complete the computed damage r a t i o s should be equal to the target damage ra t i o s assigned in the design method. Since the RSS forces are used as the computed forces, the design forces in the substitute structure method must also be the RSS forces, not the forces which are increased by the factor i n equation (2.25). Two frames were tested t h i s way. The f i r s t test frame i s a 2-bay, 3-story frame. The data are shown on F i g . 3.3. The substitute structure method was used to compute the y i e l d moments and natural periods which are also shown on F i g . 3.3. RSS moments were taken as the design moments. The target damage r a t i o s were one for the columns and six for the beams. The same response spectrum as i n the previous examples was used (Fig. 2.5). This frame was then subjected to the modi-f i e d substitute structure analysis to test the convergence of periods and damage r a t i o s . The i t e r a t i o n was carr i e d out 20 times and the natural periods and damage ra t i o s were printed at the end of each i t e r a t i o n . The damping r a t i o s for a l l three modes were taken as 10% of the c r i t i c a l i n the f i r s t cylce of i t e r a t i o n . The three natural periods computed i n each i t e r a t i o n are l i s t e d on Table 3.1. To i l l u s t r a t e the speed of convergence, each period i s normalized to that computed i n the substitute structure method and the plot of the normalized periods versus the number of i t e r a t i o n s i s shown on F i g . 3.4. As can be seen from the plot, the natural period for the f i r s t mode converged very rapidly. It took only f i v e i t e r a t i o n s for the f i r s t mode periods to be within 1% of the correct period. The convergence of the second mod period and the t h i r d mode period were slower; they were within 1% of the correct periods a f t e r 13 i t e r a t i o n s . The second mode periods approached the correct value more rapidly during the f i r s t few i t e r a t i o n s than the t h i r d mode period. The damage ra t i o s of selected columns and beams are l i s t e d i n Table 3.2 and the plot i s shown in F i g . 3.5. The damage ra t i o s of column 1 and beam 1 converged very rapidly. Only 6 it e r a t i o n s were necessary before they were within 1% of th e i r respective target damage r a t i o s . Convergence of damage r a t i o i n beam 2 was slower and i t took 15 it e r a t i o n s to be within 1% of the target damage r a t i o . Column 2 had the slowest convergence of the four members. Its damage r a t i o was within 1% of the target damage r a t i o at the end of 20 i t e r a t i o n s . As can be seen from the two plots, the periods converged faster than the damage r a t i o s . Among the natural periods, the lowest mode period converged at the fastest rate, and the highest mode the slowest. As far as the convergence of the damage rati o s i s concerned those of the members in the lower story converged faster than in the upper story. This i s l o g i c a l , because the response of the members i n the lower story i s governed by the lower mode and the convergence of the natural periods and hence 56 the mode shapes i s faster for the lower mode. The same 2-bay, 3-story frame was used i n the second example. The member properties were the same as i n the f i r s t frame, the target damage r a t i o s were changed. The columns had target damage rati o s of two, one, and three. The same damage ratio s were assigned to a l l columns on the same axis. The t a r -get damage ra t i o s for beams were s i x i n one bay and two i n the other bay. The substitute structure method was used to compute the y i e l d moments, which were RSS moments. Those forces and the natural periods are shown i n F i g . 3.6. The frame was then anal-yzed by the modified substitute structure method. The natural periods computed i n the f i r s t 20 i t e r a t i o n s are tabulated i n Table 3.3. The plot of normalized periods are shown in Fig. 3.7. The convergence of the f i r s t two periods was again very rapid. The period for the t h i r d mode, however, was r e l a -t i v e l y slow. It was within 4.3% of the correct value a f t e r 20 it e r a t i o n s . The damage ra t i o s converged very slowly. F i g . 3.8 shows the damage r a t i o s at the end of selected numbers of i t e r a -tions. At the end of 20 i t e r a t i o n s the damage ra t i o s of the third-story columns and beams were s t i l l quite d i f f e r e n t from the target ones. The i t e r a t i o n was carr i e d out 200 times and by then they did converge to the correct values. The pl o t of damage rat i o s against the number of i t e r a t i o n s i s shown i n F i g . 3.9. The rate of convergence of damage r a t i o s were much slower i n the second example than i n the f i r s t example. F i g . 3.9 shows that about 100 i t e r a t i o n s were necessary to achieve reasonable estimate of damage r a t i o s . The same conclusion i n the previous example applies i n the second example. 57 The r e s u l t s of these two examples showed the following. (1) The natural periods converge at a faster rate than the damage r a t i o s . (2) The natural periods for the lower modes converge faster than those i n the higher modes. C3) In general, the damage r a t i o s of the upper story columns and beams converge at a slower rate than those of the lower s t o r i e s . (4) Both the damage r a t i o s and the natural periods do not con-verge monotonically. This point i s p a r t i c u l a r l y true in the f i r s t few cycles of i t e r a t i o n s . (5) The rate of convergence slows down as the number of i t e r a -tions increases. The most rapid changes occur during the f i r s t few cycles of i t e r a t i o n . These observations were confirmed i n the other examples that w i l l be shown l a t e r on. It i s , i n practice, impossible to carry out the i t e r a t i o n process u n t i l a l l the damage ra t i o s cease to fluctuate. As soon as a good estimate of damage ratios i s obtained, the i t e r a t i o n procedure should be stopped. Some c r i t e r i o n must be established for t h i s purpose. It i s possible, but not p r a c t i c a l , to keep track of every damage r a t i o at the end of each i t e r a t i o n . I t i s also impractical to set the l i m i t on the number of i t e r a t i o n s at a certain number. The two examples i n thi s section i l l u s t r a t e d t h i s point very c l e a r l y . If the number of i t e r a t i o n s i s set at 30, say, then the l a s t 10 to 15 i t e r a t i o n s i s t o t a l l y unnecessary in the f i r s t example. On the other hand, inaccurate estimate of damage ratios results i n the second example. Two approaches 58 seem possible as suitable convergence c r i t e r i a . One approach i s to compare the values of the damage r a t i o of each member at the end of the i t e r a t i o n with that of the previous i t e r a t i o n . The following formula may be used. ( V n - l C y i } n - 1 < 6 (3.5) where ^ i ^ n = ^ a m a < ? e r a t i o of i th element at the end of n th i t e r a t i o n ( y ^ ) n _ 1 = damage r a t i o of the same element at the end of n-1 th i t e r a t i o n 6 = constant If t h i s i s true for a l l the elements in the structure, the i t e r a -t i o n i s complete and the forces, displacements, and damage r a t i o s can be printed. An alternative approach i s to compare the com-puted moments with the y i e l d moments. The following formula i s suitable for thi s purpose. (M. ) - (M . ) i n y i (M .) y i < e (3.6) where ^ M i ^ n = c o m P u t e d RSS moment in i th element during n th i t e r a t i o n (M .) = y i e l d moment for i th element y l J £ = constant. If t h i s inequality i s s a t i s f i e d for a l l the elements with damage ratio s greater than one, no more i t e r a t i o n i s necessary. A d e f i n i t e advantage of the f i r s t method i s that i t i s a di r e c t comparison of the damage ra t i o s computed i n the l a t e s t two i t e r a t i o n s . The second method i s an i n d i r e c t comparison of the damage r a t i o s . I t i s not clear how much change i s made on damage r a t i o s . The f i r s t approach has a d e f i n i t e disadvantage, because the denominator changes at every i t e r a t i o n . Because of t h i s reason the second approach was adopted by the author. It i s hoped that t h i s c r i t e r i o n produces a more uniform r e s u l t for d i f f e r e n t types of structures. With a l i t t l e experience a s u i t -able value for e can be spec i f i e d . In running a computer program i t i s desirable to set the l i m i t on the number of i t e r a t i o n s , because no output would be obtained i f a value for e was too small and CPU time exceeded the l i m i t set by the user. In spite of the foregoing discussion, i t should be noted that, i n practice, because of the inaccuracies i n modeling the structure, i n predicting the earthquake, and in c o r r e l a t i n g damage r a t i o with actual damage, the results are not s i g n i f i c a n t to a high degree of prec i s i o n . 60 3.4 Ac eele rated C onverge nce The cost of running the modified substitute structure method i s roughly proportional to the number of i t e r a t i o n s that i s necessary to meet the convergence c r i t e r i o n . If there i s a way to accelerate the convergence, the method becomes a more powerful t o o l . An e f f o r t was made to achieve t h i s goal and the following procedure was developed. It was observed in the two examples i n the l a s t section that the most rapid changes in the damage r a t i o occurred during the f i r s t several cycles of the i t e r a t i o n process and then the damage rati o s gradually approached the f i n a l values. The damage ra t i o s are modified at the end of each i t e r a t i o n by the use of equation (3.3) or (3.4). It appeared possible to make over-corrections on the damage rati o s i n order to speed up the con-vergence. It i s easy to keep track of the difference between the new damage r a t i o of an element, and the damage r a t i o of the same element i n the previous i t e r a t i o n . The following formula was proposed for overcorrection of damage r a t i o s . (y.)' = (y.) + a (y.) - (y.) . (3.7) p i n i n i n l n-1 overcorrected damage r a t i o of i th element used for n th i t e r a t i o n damage r a t i o of i th element computed at the end of n-1 th i t e r a t i o n using equation (3.3) or (3.4) = damage r a t i o of i th element used i n n-1 where (y • ) ' i n ( u - ) i n -61 th i t e r a t i o n a = posit i v e constant. What i s proposed i n equation (3.7) i s that some f r a c t i o n of the difference between the modified damage r a t i o and the previous damage r a t i o be added to the modified damage r a t i o . Since a i s a posit i v e constant, the overcorrected damage r a t i o i s smaller than the modified one when the damage r a t i o i s altered to have a lower value than the previous one, but overcorrected damage rat i o s cannot be less than one. I t was found that applying t h i s overcorrection from the beginning could lead to an unexpected re s u l t , because the damage r a t i o s change quite rapidly during the f i r s t stage of the i t e r a t i o n procedure. The damage ra t i o s may fluctuate up and down v i o l e n t l y from one i t e r a t i o n to another. It i s strongly advisable that the constant, a , be set to zero during the f i r s t f i v e to ten i t e r a t i o n s , so that the overcorrec-ti o n procedure i s applied when the damage ra t i o s change at a small rate. I f such a precaution i s taken, the value of a may be set at as high as one to achieve faster but s t i l l smooth convergence. The following example i l l u s t r a t e s the usefulness of the procedure. I t also shows how cl o s e l y the damage r a t i o s approach the exact values when d i f f e r e n t l i m i t s are used as convergence c r i t e r i a . The second example i n the previous section was used. A l l the relevant information i s shown in Fig . 3.6. Recall the convergence c r i t e r i o n proposed in the previous section. It was CM..). . - CM .) v l n v y i (M .) y i < e C3.6) where e i s a constant. e was set at 10 , 10 , and 10 . When the r e l a t i o n i n (3.6) was s a t i s f i e d for a l l the members, the i t e r a t i o n procedure was stopped. Six runs were made i n t o t a l . In the f i r s t three runs no overcorrection was made and the numbers of i t e r a t i o n s required to achieve the three convergence c r i t e r i a were recorded. In the next three runs the same three -2 -3 convergence c r i t e r i a were used; that i s , e was set at 10 ,10 , -4 and 10 , but the overcorrection of damage r a t i o s was applied, a was set at 1.0 at the end of the f i f t h i t e r a t i o n and the number of i t e r a t i o n s required was recorded for each run. The results are given in Table 3.4. -2 When e was set at 10 , i t took 29 i t e r a t i o n s to s a t i s f y t h i s c r i t e r i o n without overcorrection of damage r a t i o s . When the damage rati o s were overcorrected, the number of i t e r a t i o n s was -3 reduced to 18 for a saving of 11 i t e r a t i o n s . At £ = 10 158 i t e r a t i o n s were required without overcorrection technique. With i t , the number was reduced to only 81 for a saving of 77 -4 i t e r a t i o n s . At e = 10 the convergence c r i t e r i o n was not met aft e r 200 i t e r a t i o n s when overcorrections were not made, but i t was met af t e r 12 4 i t e r a t i o n s when they were made. Clearly t h i s technique accelerated the convergence of the damage r a t i o s . The number of i t e r a t i o n s was reduced by one t h i r d to almost one ha l f . The saving i n computation i s substantial when the convergence i s slow i n a case such as the example used here. The gain i s not so s i g n i f i c a n t when the convergence i s fast, as i t i s i n the f i r s t example in the previous section. Since i t i s impossible to pre-d i c t the rate of convergence beforehand, t h i s technique should be used a l l the time. On rare occasions the method produced bad results i n which the damage ra t i o s o s c i l l a t e d . In order to avoid t h i s p o s s i b i l i t y , a may be set at a constant less than one or the application of the technique may be delayed u n t i l more than 10 ite r a t i o n s are completed. Table 3.5 shows how cl o s e l y the damage ra t i o s of a l l the members approached the exact value when d i f f e r e n t e values were spe c i f i e d . Overcorrections were made i n a l l cases. The same -2 applies to the natural periods. When an e of 10 was reached, some of the damage ra t i o s were s t i l l quite far from the exact ones; the third-story columns and beams f a l l i n th i s category. The natural period for the t h i r d mode d i f f e r s the most from the exact one but the difference i s less than three percent. At -3 e = 10 almost a l l the damage rati o s are very close to the exact values. The natural periods are even closer to the exact -4 values than the damage r a t i o s . At e = 10 both the damage ra t i o s -2 and the natural periods are p r a c t i c a l l y exact. e set at 10 i s -3 probably too coarse. e should be set at somewhere between 10 -2 -3 and 10 . It was found from other runs that e set at 10 pro-duced s a t i s f a c t o r y r e s u l t s . However, i f the modified substitute structure method i s used to obtain a rough estimate e may be set -2 at a value a l i t t l e smaller than 10 ; and t h i s generally i s a l l that i s warranted i n practice. It may be possible to incorporate the overcorrection of damage rati o s into the formula for modifying the damage ra t i o s at the end of the i t e r a t i o n . When equations (3.3) and (3.4) were derived, i t was assumed that the same rotation would be obtained in the next i t e r a t i o n . The substitute s t i f f n e s s and hence the damage r a t i o was increased or decreased accordingly to s a t i s f y t h i s assumption. But t h i s assumption i s not absolutely necessary. Another assumption i s possible and with such an assumption a new formula may be derived to achieve faster convergence. Further study i s possible i n this area. As a f i n a l remark i n this chapter i t i s worth noting that the two examples i n the previous section, even though they were i d e n t i c a l frames, except for the y i e l d moments, lay on the two extreme sides as far as the rate of convergence was concerned. It was very rare that the damage r a t i o s of a structure converged at a faster rate than they did i n the f i r s t example, or at a slower rate than i n the second example. Even when the size of a structure i n the f i r s t example was considerably greater than the structure i n the second example, fewer i t e r a t i o n s were required to s a t i s f y the same convergence c r i t e r i o n . In general, less than 20 i t e r a t i o n s are necessary to obtain a good estimate on damage r a t i o s for most of the structures i n practice. 65 CHAPTER 4 EXAMPLES 4.1 Assumptions and Comments The goal of the modified substitute structure analysis i s to predict the behaviour of an exis t i n g reinforced concrete struc-ture under a given earthquake motion. Tests must be performed to fi n d out whether the method f u l f i l l s t h i s intent. It i s almost impossible to do an actual experiment. The test must be done a n a l y t i c a l l y . Among many a n a l y t i c a l methods, a nonlinear dynamic analysis produces the most accurate prediction of the behaviour of a structure which i s subjected to an earthquake motion. It i s , therefore, essential that the modified substitute structure method produce a r e s u l t which i s comparable to that obtained from the nonlinear dynamic analysis. A series of test frames were analyzed by the modified sub-s t i t u t e structure method. The same frames were also subjected to analyses using the nonlinear dynamic analysis program. The results from the two analyses were compared. The extent of damage represented by damage r a t i o s , locations of damage and the d i s -placements are the quantities of in t e r e s t . Before the results are described i n d e t a i l , a l l the relevant information and assump-tions w i l l be discussed i n t h i s section. A t o t a l of four frames were tested. They were not modeled from actual e x i s t i n g buildings, but they were intended to repre-sent small- to medium-sized reinforced concrete structures. A test on a larger structure was not possible mainly due to the l i m i t a t i o n s of the nonlinear dynamic analysis program. The high cost involved i n the analysis was another reason to l i m i t the size of a test frame. In order to s a t i s f y the second condition l i s t e d in section 3.2, they were a l l regular frames with no abrupt change i n geometry. The dimensions of a frame were determined so that they would represent an actual building of comparable size. Member sizes and properties were chosen somewhat a r b i t r a r i l y and are not necessarily completely r e a l i s t i c . Since the method would be used i n practice for analysis of buildings that may not have been designed to r e s i s t earthquakes, the member properties were deliberately chosen i n an a r b i t r a r y fashion. It was f e l t that i f the modified substitute structure method worked for these test frames, i t would work for more r e a l i s t i c structures. Since the test was a n a l y t i c a l , there was no r e s t r i c t i o n on the choice of these parameters. The following assumptions were made i n the modeling of frames for use with the modified substitute structure program. Beams and columns were modeled as l i n e members. Their a x i a l deformations were ignored. The bottom columns were assumed to be fixed at ground l e v e l . A j o i n t was modeled as a point. Moment capacities at the two ends of a member were taken to be equal. P -A e f f e c t i n the columns were not included. Upon running a program overcorrection of damage r a t i o s was applied a f t e r the f i r s t ten cycles of i t e r a t i o n was over. a i n the equation (3.7) was set at 0.95. Equation (3.6) was used as a convergence c r i t e r i o n and e was set at 10 . Iteration was stopped as soon as t h i s convergence c r i t e r i o n was s a t i s f i e d . 12 A nonlinear dynamic analysis program for frames, SAKE, was used to compute the response history of each frame. The s t i f f -ness after y i e l d was taken as 2% of the i n i t i a l s t i f f n e s s . The analysis was made with viscous damping proportional to s t i f f n e s s , corresponding to a damping r a t i o of 2% for the f i r s t mode. Joints were modeled as i n f i n i t e l y r i g i d beams, with sizes proportioned according to the member sizes. A time step corresponding to 1/30 to 1/50 of the smallest period was used for numerical i n t e r a t i o n . Response calculations were done at every f i v e to ten time steps. Choosing a proper response spectrum i s beyond the scope of 5 t h i s thesis. The design spectrum A i n Shibata and Sozen's paper was used for the modified substitute structure analysis. As mentioned in section 2.3, i t was derived from response spectra of six earthquake motions (Fig. 2.5). Equation (2.26) was used to compute the response acceleration when the damping r a t i o was d i f -ferent from 2%. The maxium ground acceleration was 0.5 g. Four of the six earthquake records, from which the design spectrum was made, were used to compute the response h i s t o r i e s . They were E l Centro EW, E l Centro NS, Taft S69E, and Taft N21E. Each record was normalized to give a peak ground acceleration of 0.5 g. The duration of each earthquake record was chosen such that each frame received the maximum damage during that duration. Unless otherwise noted, the f i r s t 15 seconds of each earthquake record were used for computation. CPU time for running the two programs i s given to i l l u s t r a t e the difference i n cost, but i t should be 68 noted that the cost for storage was much higher for the nonlinear dynamic analysis program, because i t required more memory. The damage ra t i o s and displacements were used for compari-son of the two analyses. Since the design spectrum was the average spectrum of the six earthquakes, the re s u l t s of the mod-i f i e d substitute structure analysis should be viewed as "average" of the four nonlinear dynamic analyses. 4.2 Examples 69 (a) 2-Bay, 2-Story Frame The two-bay, two-story frame of F i g . 4.1 was used as a test frame. The widths of both bays were 30 feet. The ground story was 12 feet i n height and the second story was 10 feet high. The f l o o r weights for the f i r s t and second story were 120 kips and 1Q0 kips respectively. The exterior columns were bigger than the i n t e r i o r columns. Their cracked transformed moments of i n e r t i a were taken as approximately one-half of the gross section. The moments of i n e r t i a for beams were about one-third of the gross section. An e l a s t i c analysis was run to compute the natural periods. As shown i n Table 4.1, the periods for the two modes were 0.50 sec. and 0.13 sec. respectively, representing a short period structure. The y i e l d moments were assigned randomly such that each member was expected to receive a d i f f e r e n t amount of i n e l a s t i c deformation. In the modified substitute structure analysis i t took 24 i t e r a t i o n s to s a t i s f y the convergence c r i t e r i o n . The CPU time on the Amdahl V/6-II computer was 0.91 sec. The natural periods of the substitute frame computed i n the l a s t i t e r a t i o n were 0.76 sec. for the f i r s t mode and 0.18 sec. for the second mode (See Table 4.1). The f l o o r displacements were computed as the root-sum-square of the modal displacements and are shown i n Table 4.2. The displacement of the f i r s t f l o o r was 1.8 i n . and that of the second f l o o r was 3.8 i n . The d i s t r i b u t i o n of damage r a t i o s was quite random as expected (See F i g . 4.2). A l l the f i r s t - s t o r y columns yielded. The damage r a t i o s for those columns were 4.2, 70 2.6, and 1.4 respectively. The second-story columns did not y i e l d very much. One of the exterior columns remained e l a s t i c . A l l the four beams yielded. The f i r s t - f l o o r beam i n the l e f t bay had the biggest damage r a t i o at 4.8. Response h i s t o r i e s of the test frame to four earthquake 12 motions were computed by the computer program, SAKE. The f i r s t 15 seconds of earthquake records were used for response computa-ti o n . 0.003 sec. was chosen as the time step for numerical i n t e -gration. CPU time was 12.9 sec. for E l Centro EW motion, 12.2 sec. for E l Centro NS, 11.8 sec. for Taft S69E, and 11.4 sec. for Taft N21E. Results of the nonlinear analyses are shown i n F i g . 4.3 and Table 4.2. The two components of the E l Centro earthquake resulted i n more damage to the test frame than the two components of Taft earthquake. The displacements and damage r a t i o s i n Fi g . 4.3. and Table 4.2 were the maximum values recorded in the re-sponse h i s t o r i e s . The displacement of the f i r s t - f l o o r ranged from 1.3 i n . to 2.8 i n . for d i f f e r e n t motions. The mean maximum d i s -placement was 2.1 i n . The second f l o o r displacement ranged from 2.7 i n . to 5.3 i n . with a mean of 4.2 i n . The damage r a t i o s in Fig . 4.3 correspond to the bigger of the two damage ra t i o s for each member. In the E l Centro EW motion a l l of the f i r s t - s t o r y columns suffered extensive damage with damage rati o s ranging from 3.3 to 9.6. On the other hand, none of the columns on the second story yielded. A recorded damage r a t i o less than one i n F i g . 4.3 im-p l i e s that the maximum computed moment was that f r a c t i o n of the y i e l d moment. The l e f t exterior column had the lea s t damage. A l l four beams yielded with t h e i r damage ra t i o s ranging from 2.8 to 71 6.4. In the E l Centro NS motion the f i r s t - s t o r y columns suffered approximately the same amount of damage as i n the previous case. The damage rati o s for the beams were also approximately the same as those i n the E l Centro EW motion. Two of the second-story columns, however, yielded with damage ra t i o s of 1.3. The Taft S69E motion produced the least damage. As i n the two previous cases the f i r s t story columns yielded, but the damage ra t i o s were roughly a half of those with E l Centro. The same applies to the beam damage r a t i o s . The three columns on the second story remained e l a s t i c , but the maximum moments were com-parable to those found i n the E l Centro EW motion. The Taft N21E motion was more severe, but i t was not strong enough for the second-story columns to y i e l d . The damage r a t i o s for the other columns ranged from 2.0 to 6.5 and those for the beams from 2.0 to 4.6. The members which remained e l a s t i c reached roughly the same maximum moments in a l l four motions, but those which yielded suffered d i f f e r e n t amounts of damage i n each motion. When the average f l o o r displacements from the nonlinear dynamic analyses are compared with those from the modified sub-s t i t u t e structure analysis as i n Table 4.2, i t i s found that the l a t t e r predicted smaller displacements i n both s t o r i e s . The d i f -ference was greater for the f i r s t - f l o o r displacement which was about 20% o f f than for the second-floor displacement which was about 10% o f f . Nevertheless the estimate was reasonable. Fig. 4.2 shows the comparison of the average damage ra t i o s of the four motions with the predicted values. In a q u a l i t a t i v e sense the modified substitute structure analysis c o r r e c t l y predicted that the columns on the f i r s t story would y i e l d and that the extent of damage would be greatest for the l e f t exterior column and least for the r i g h t exterior column. But the predicted damage ra t i o s were about 60% of the average damage r a t i o s of the four motions. The prediction for the second-story columns was good except for the r i g h t exterior column. The modified substi-tute structure method predicted that t h i s column would y i e l d s l i g h t l y , but i t did not happen. It only reached 60% of i t s moment capacity. The prediction of damage r a t i o s for the beams was quite good. Although they were s l i g h t l y underestimated, they were a l l within 20% of the average values. The cost of running one nonlinear analysis was about 13 times that of the modified substitute structure analysis in t h i s example. (b) 3-bay, 3-story Frame The three-bay, three-story frame was tested i n the second example. Data i s shown in F i g . 4.4. The width of bays was 30 feet for the exterior bays and 2 0 feet for the i n t e r i o r bay. The f i r s t story was 15 feet high and the second and t h i r d s t o r i e s were 12 feet high. The f l o o r weights were 240 kips for the f i r s t f l o o r , 200 kips f o r the second, and 180 kips for the t h i r d . As i n the l a s t example, exterior columns were bigger than i n t e r -i o r columns. In each group of columns the f i r s t - s t o r y columns and the second-story columns were given the same dimension. The third-story columns were made smaller than the others. The beam sizes were reduced at higher f l o o r s . One half of the moment of i n e r t i a l of the gross section was used for the cracked trans-formed section. The r i g h t exterior column on the second story had much smaller moment of i n e r t i a than i t s counterpart. The moment 73 of i n e r t i a of each beam was taken as one t h i r d of that of the gross section. The y i e l d moments of the columns were set at high values, es p e c i a l l y i n the f i r s t story, so that the columns would not y i e l d too much. The beam y i e l d moments were smaller i n the l e f t bay than i n the other bays. An e l a s t i c analysis was performed p r i o r to the test to compute the natural periods of the e l a s t i c frame. As shown i n Table 4.3, they were 0.94 s e c , 0.30 s e c , and 0.14 sec. These periods were much longer than those i n the previous example. The re s u l t s of the modified substitute structure analysis are shown i n F i g . 4.5, Table 4.3, and Table 4.4. Only 14 i t e r a -tions were necessary to s a t i s f y the convergence c r i t e r i o n . I t took 0.92 sec. of CPU time to do a l l the necessary computation. The three natural periods of the substitute frame were 1.22 s e c , 0.36 s e c , and 0.16 sec. The increase i n natural periods over those from the e l a s t i c analysis was less for t h i s frame than the previous frame. The horizontal displacements of the three f l o o r s were 2.2 i n . for the f i r s t , 5.0 i n . for the second, and 8.0 i n . for the t h i r d , i ndicating a f a i r l y uniform pattern of displace-ments (See Table 4.4). Predicted damage r a t i o s are shown i n Fig. 4.5. A damage r a t i o less than one i s equivalent to the r a t i o of the computed moment to the y i e l d moment. If the two end moments were d i f f e r e n t in a member, the bigger of the two was used. A l l the columns on the f i r s t story had damage r a t i o s of 1.1, indicating that t h e i r y i e l d moment capacities were s l i g h t l y exceeded. One of the columns on the second story yielded to a damage r a t i o of 1.5, but the other three remained i n the e l a s t i c range. The moment capacities of the third-story columns were f u l l y u t i l i z e d , as the i r damage rati o s were almost 1.0 exactly. Two beams i n the right exterior bay remained e s s e n t i a l l y e l a s t i c ; others had damage rati o s ranging from 2.0 to 5.2. Nonlinear dynamic analyses were run to compute the response h i s t o r i e s of the frame in the four earthquake motions. The f i r s t 15 seconds of the records were used. Since t h i s was a bigger frame than the previous one, a considerably longer time on the Amdahl V/6-II computer was required for computation. The average CPU time of one run was about 28 seconds, double the time required i n the previous example. A time increment of 0.003 sec. was selected for numerical integration. The results of four runs are shown i n Fi g . 4.6. Displace-ments are shown in Table 4.4. They exhibited a large v a r i a t i o n from one earthquake to another. E l Centro EW component produced the biggest displacements, twice as big as those i n Taft N21E component. E l Centro NS produced the second biggest displace-ment and Taft S69E motion followed. The average displacements were 2.2 i n . for the f i r s t f l o o r , 4.7 i n . for the second, and 7.5 i n . for the t h i r d . The t h i r d - f l o o r displacement, for example, ranged from 5.2 i n . i n Taft N31E to 10.6 i n . in E l Centro EW. The same trend was found in damage r a t i o s . The damage ra t i o s were the highest i n the E l Centro EW motion. A l l the columns on the f i r s t story had damage ra t i o s around 1.8. The ri g h t i n t e r i o r columns on the second and t h i r d s t o r i e s yielded as well, but the rest of the columns remained e l a s t i c . The two exterior columns on the t h i r d story had the lowest computed moments. A l l the beams yielded with damage 75 ra t i o s ranging from 1.5 to 6.4. The beams i n the ri g h t bay experienced the least damage. Ine l a s t i c deformations occurred in the same beams and columns i n the E l Centro NS motion. The damage r a t i o s of these members, however, were lower i n thi s motion than in EW motion. The moment capacities were reached i n the two columns on the second story, but they did not y i e l d . In the Taft S69E motion only one column underwent i n e l a s t i c deformation, the ri g h t i n t e r i o r column on the second story, with, damage r a t i o of 1.2. A l l the columns on the f i r s t story and two on the t h i r d story had computed moments equal to or a l i t t l e less than t h e i r respective y i e l d moments. Two of the beams did not y i e l d , although t h e i r damage r a t i o s were almost one. Damage ratio s of the other beams ranged from 1.8 to 4.6 which were much lower than the values found i n the E l Centro EW motion. The damage ra t i o s were the lowest i n the Taft N21E motion. A l l of the columns and two beams remained e l a s t i c and those which yielded had damage r a t i o s ranging from 1.5 to 4.0. Average damage ra t i o s are shown in F i g . 4.5. When these values are compared with those predicted by the modified sub-s t i t u t e structure analysis, there i s a remarkable agreement. The prediction of beam damage ra t i o s i s excellent without exception. Even the worst one was o f f by only 15%. The prediction of column damage ra t i o s was a l i t t l e worse than for the beams. Only the damage ra t i o s of the exterior columns on the t h i r d story were s l i g h t l y o f f , but others were i n good agreement. The average displacements also agreed very well with those predicted by the modified substitute structure analysis, as shown i n Table 4.4. At least for t h i s example i t i s safe to to say that the modified 76 substitute structure analysis c o r r e c t l y predicted the res u l t s of the nonlinear dynamic analysis. This i s a remarkable achievement when the difference i n CPU time i s concerned. (c) 1-Bay, 6-Story Frame Fig. 4.7 shows the data for the one-bay, six-story frame that was used as the t h i r d t e s t frame. The width of the frame was 35 feet, and the story height was constant at 13 feet for an o v e r a l l height of 78 feet. The f l o o r weight was constant up to the f i f t h story at 100 kips, but at the top story i t was reduced to 90 kips. The column sizes were decreased at every second story. Beam size was constant up to the f i f t h f l o o r , with a smaller beam being used at the top f l o o r . The moment of i n e r t i a of a l l the members were taken as approximately one-half of the values based on gross section. The y i e l d moments of the columns were reduced progressively up the height of the frame. The y i e l d moments of the beams were large, except at the top, compared to those of the columns. I t was hoped that columns would receive a f a i r amount of damage. E l a s t i c periods were computed for a l l six modes and the values are shown in Table 4.5. The natural periods for the f i r s t two modes were 1.1 sec. and 0.37 sec. respectively. The smallest period was 0.0 8 sec. The modified substitute structure analysis was c a r r i e d out in the usual manner. It was necessary to perform 9 6 i t e r a t i o n s to achieve the convergence. CPU time was 2.3 sec. The natural periods of the substitute frame, as shown i n Table 4.5, were considerably longer than the periods of the actual frame. The period for the f i r s t mode was 1.85 sec. and the second mode period was 0.84 sec. Usually the natural period for the highest mode of the substitute structure i s not much d i f f e r e n t from that of the actual frame, but they were quite d i f f e r e n t i n t h i s example. The former was 0.13 sec. and the l a t t e r was 0.0 8 sec. The displace-ment pattern was also quite unique (See Table 4.6). The displace-ment of the second fl o o r was much greater than the f i r s t f l o o r . There was a big difference between the fourth-floor displacement and the f i f t h - f l o o r displacement. The damage r a t i o s are shown in F i g . 4.8. The analysis pre-dicted that the damage ra t i o s would vary widely among the members. The column i n the f i r s t three stories were 2.5, 6.6, and 2.9 respectively. The large damage r a t i o for the second-story column i s the reason for the big jump i n displacement between the second and t h i r d f l o o r . Two columns, the one i n the fourth story and the one i n the si x t h story, did not y i e l d . A large i n e l a s t i c deformation was predicted i n the f i f t h - s t o r y column with a damage r a t i o of 16.6. The f i r s t - f l o o r beam had a damage r a t i o of 1.5, ind i c a t i n g a small amount of i n e l a s t i c deformation. The beams on the next three f l o o r s did not y i e l d . The beams on the f i f t h and sixth f l o o r s were given large damage ra t i o s of 9.5 and 6.8 respectively. Response h i s t o r i e s of the frame were computed by the non-li n e a r dynamic analysis program, using the f i r s t 20 sec. of the four earthquake records. The time increment for numerical in t e -gration was set at 0.004 sec. The average CPU time was 42.6 sec. The damage r a t i o s for in d i v i d u a l earthquake motions are shown i n Fig . 4.9. E l Centro EW motion produced by far the worst r e s u l t . 78 The damage due to other motions were si m i l a r to each other i n magnitude. In E l Centro EW motion a l l the members except the top-story column suffered severe damage and the f l o o r displacements were large, as shown i n Table 4,6. Damage rati o s of the columns i n the f i r s t f i v e s t o r i e s ranged from 6.3 to 14.4 i n the f i r s t story. The thir d - s t o r y column was also damaged badly with a damage r a t i o of over 10. A l l the beams experienced a large amount of i n e l a s t i c deformation, with damage r a t i o s ranging from 6.3 to 10.8, with the highest value in the f i f t h - f l o o r beam. In E l Centro NS motion a l l s i x columns yielded. The smallest damage r a t i o was 1.3 and the highest was 5.2. The columns on the t h i r d , fourth and f i f t h s t o r i e s were damaged more than the other three. A l l the beams also yielded. The damage ra t i o s increased up the height of the building except at the f i f t h f l o o r where the damage r a t i o of the beam was the highest at 8.1. The displacements were small compared to those found i n E l Centro EW motion. The d i s -placement of the f i r s t f l o o r was p a r t i c u l a r l y small (See Table 4.6). In Taft S69E motion every member of the frame yielded. Among the columns those i n the f i r s t four stories received the most damage, with damage r a t i o s about six. The damage r a t i o s of the beams on the f i r s t three f l o o r s were approximately the same at about 3.5. The other three were damaged to a greater extent. The damage r a t i o of the f i f t h - f l o o r beam was the highest at 9.5, while the other two beams had damage ratios of about seven. The increase i n displacements was quite uniform i n the f i r s t four f l o o r s . Taft N21E motion resulted i n quite a d i f f e r e n t pattern 79 of damage r a t i o s . Most of the damage in the columns was concen-trated in the second-story column and the third-story column with damage rati o s of 7.9 and 6.6 respectively. The damage r a t i o s of the other four columns were small. The damage in the beams was concentrated i n the f i r s t two f l o o r beams, where the damage r a t i o s were 5.5 and 3.2. The other four beams escaped with minor damage. The displacements above the t h i r d f l o o r did not increase s i g n i -f i c a n t l y . The average displacements are shown in Table 4.6 and average damage ra t i o s i n F i g . 4.8. They were very d i f f e r e n t from the figures computed by the modified substitute structure analysis. The displacement patterns were quite d i f f e r e n t . The prediction by the modified substitute structure analysis resulted i n an underestimate of the displacements of the f i r s t four f l o o r s . The prediction of damage ra t i o s was also poor. The damage was not concentrated i n a p a r t i c u l a r column or a beam, but was spread over the whole structure. The modified substitute structure method f a i l e d i n t h i s test frame. (d) 3-Bay, 6-Story Frame The 3-bay, 6-story frame shown i n F i g . 4.10 was used as the fourth test frame. Each bay was 2 4 f t . wide and story height was constant at 11 f t . A weight of 200 kips was concentrated at each story. Members sizes were uniform along the height. They were 24 i n . by 24 i n . for columns and 18 i n . by 30 i n . for beams. One half of the moment of i n e r t i a of the gross section was used to compute the i n i t i a l s t i f f n e s s of columns, and one t h i r d for beams. In t h i s example a l l the columns were intended to remain e l a s t i c . 80 For t h i s purpose the substitute structure method was used to compute design moments. These moments were used as a guide to est a b l i s h the y i e l d moments. The periods are summarized i n Table 4.7. The computed periods for the f i r s t two modes were 1.1 sec. and 0.34 sec. respectively, while the period for the highest mode was 0.0 75 sec. These e l a s t i c periods were comparable to those of the l a s t test frame. The modified substitute structure analysis was carr i e d out in the usual manner; 16 i t e r a t i o n s were required to s a t i s f y the convergence c r i t e r i o n . The CPU time was 2.30 sec. on the Amdahl V/6-II computer. As shown i n Table 4.7, the natural periods for the f i r s t two modes were 1.66 sec. and 0.48 sec. respectively, while the shortest period was 0.0 76 sec. The f i r s t two periods of the substitute frame were much longer than the corresponding periods of the e l a s t i c frame, but the other periods were r e l a -t i v e l y unchanged. As far as the displacements, shown i n Table 4.8, were concerned, the second-floor displacement was quite large compared to the f i r s t - f l o o r displacement. The increase in displacement from the f i f t h f l o o r to the sixth f l o o r was small. The r e l a t i v e displacement was quite uniform for the other f l o o r s , the top deflection being 8.8 i n . Damage ra t i o s are shown i n Fi g . 4.11. Those of the columns were roughly constant at around 0.8; that i s , the computed moments of a l l the columns were about 80% of the y i e l d moments. A l l the y i e l d i n g took place i n the beams. The beams i n the exterior bays had higher damage rati o s than those i n the i n t e r i o r bay. In both bays the bottom beams had the highest damage r a t i o s . 81 They decreased at an increasing rate with height i n the frame. For the beams i n the exterior bays the damage r a t i o s ranged from 3.5 to 4.5. For those i n the i n t e r i o r bay they ranged from 2.0 to 2.7. Response h i s t o r i e s of the frame to the four earthquake motions were computed. The f i r s t 15 sec. of records were used i n each run. Numerical integration was performed every 0.002 sec. and the response c a l c u l a t i o n was done afte r every f i v e time steps. Each nonlinear dynamic analysis was expensive, as i t required, on the average, 120 sec. of CPU time. A summary of results i s shown i n Fi g . 4.12. In E l Centro EW motion three of the exterior columns yielded. They were the third.-, fourth- and f i f t h - s t o r y columns and t h e i r damage ra t i o s were about 1.5. None of the i n t e r i o r columns yielded, but the maximum moments of the three columns were equal to or just below the y i e l d moments. A l l the beams yielded to some extent. The second-floor beams received the highest damage r a t i o s . The f i r s t - f l o o r beams and the t h i r d -f l o o r beams were damaged to the same extent as the second-floor beams. Damage ra t i o s decreased rapidly with height above the t h i r d story. The top beam i n the i n t e r i o r bay almost remained e l a s t i c . The top defle c t i o n was 9.8 i n . Response of the frame to E l Centro NS motion was moderate. None of the columns yielded with t h e i r damage ra t i o s ranging from 0.5 8 to 0.96. In both the i n t e r i o r bay and the exterior bays the highest damage r a t i o was found i n the f i r s t - f l o o r beams. It was 3.2 for the i n t e r i o r bay and 5.0 for the exterior bay. The damage rati o s decreased steadily with height. The top beam i n 82 the i n t e r i o r bay did not y i e l d . The top deflection was 6.3 i n . The f l o o r displacement did not increase much above the t h i r d -story. Taft S69E motion was more severe than E l Centro NS motion. The columns on the f i f t h story yielded. The damage r a t i o of the exterior column was 1.5 and the i n t e r i o r column 1.1. The maxi-mum moments of several columns were very close to the y i e l d moments. In the exterior bay the maximum damage r a t i o was 5.5 at the bottom beam. The damage rati o s of the beams on the next three 'floors were about the same at 4.8. The top beam had the lowest damage r a t i o at 2.8. The same trend was found i n the beams i n the i n t e r i o r bay, but the damage r a t i o s were smaller. The highest damage r a t i o was 3.5 and the lowest was 1.6. The displacement at the top was 7.3 i n . Taft N21E motion produced si m i l a r results to those i n E l Centro NS motion. A l l the columns remained i n the e l a s t i c range. The damage ra t i o s ranged from 0.58 to 0.89. The f i f t h - s t o r y columns were quite far from y i e l d i n g . Damage r a t i o s of the beams decreased with height i n each bay. In the exterior bay they were 4.4 at the bottom and 1.0 at the top. In the i n t e r i o r bay they were 2.8 and 0.61 respectively. The displacement of the top f l o o r was 5.4 i n . which was the smallest for the four records. The average damage ra t i o s and displacements are shown i n Fi g . 4.11 and Table 4.8. The prediction by the modified sub-s t i t u t e structure method was compared with the average values of the four nonlinear analysis r e s u l t s . Column damage ra t i o s were predicted reasonably well. Those of the three exterior columns 83 were s l i g h t l y underestimated, but they were not bad. The damage ratio s of the beams on the f i r s t two floo r s were overestimated. Those on the top three floors were underestimated. In more general terms, the modified substitute structure analysis pre-dicted f a i r l y uniform damage ra t i o s i n the beams up the height of the frame with a small decrease towards the top f l o o r , but the average damage ra t i o s were higher at the bottom and decreased quite rapidly with height. The prediction was s t i l l reasonable, esp e c i a l l y when the two top beams were excluded. As far as displacements were concerned, the two methods agreed very well up to the t h i r d f l o o r . The modified substitute structure analysis overestimated the displacements above the t h i r d f l o o r , but the difference was not substantial. In t h i s example the modified substitute structure method worked reasonably well . (e) Observations Four test frames were analyzed by the modified substitute structure method. The res u l t s were compared with those by the nonlinear dynamic analysis. The method worked very well i n the three-bay, three-story frame. Average damage ra t i o s and d i s -placements of the four earthquake motions agreed with those pre-dicted i n the modified substitute structure analysis. The method was less successful i n the two examples, the two-bay, two-story frame and the three-bay, six-story frame. But i t was s t i l l possible to obtain good estimates of damage ra t i o s and displace-ments. For these three frames damage ra t i o s converged very rapidly. The difference i n the CPU time was enormous, esp e c i a l l y i n the three-bay, six-story frame. When th i s point i s taken into 84 consideration, i t i s reasonable to c l a s s i f y the results of these three examples as success. The method did not work well for the single-bay, six-story frame. The damage rat i o s predicted by the method were quite d i f f e r e n t from those computed i n the nonlinear dynamic analysis. It should be pointed out that the frame was badly designed and that excessive y i e l d i n g took place in every member in the frame. The modified substitute structure method does not seem to work i n such badly designed structures. But at least the method was able to predict that the frame would behave very poorly. In practice i t w i l l be rare that such a structure exists i n an area where a strong earthquake i s l i k e l y to occur. Most importantly, however, i t must be observed that the actual behaviour of t h i s structure, as determined by the f u l l dynamic analysis, was t r u l y unpredict-able. That i s to say, i t behaved d i f f e r e n t l y in d i f f e r e n t earth-quakes, so i t i s not surprising that the modified substitute structure method was unable to predict the behaviour s a t i s f a c t o r -i l y . It i s suspected that a structure in which there i s wide-spread and extensive y i e l d i n g may exhibit t h i s type of behaviour and should, therefore, be considered unsafe, even i f damage rat i o s would be otherwise acceptable. The results found in the two-bay, two-story frame and the three-bay, six-story frame may be considered as t y p i c a l of the modified substitute structure method. Considering that these frame were highly hypothetical and that no p a r t i c u l a r e f f o r t was made to control the behaviour of the structure, the method would be l i k e l y to work at least as well i n a r e a l structure, the behaviour of which i s l i k e l y to be more controlled. 85 Since the modified substitute structure analysis i s so much cheaper to run than the nonlinear dynamic analysis, i t can be used repeatedly to see the e f f e c t of modifications. From the res u l t s of such analyses a recommendation can be made on what steps can be taken to upgrade the performance of a building to a s a t i s -factory l e v e l . 86 CHAPTER 5 FACTORS AFFECTING MODIFIED SUBSTITUTE STRUCTURE METHOD 5.1 E f f e c t of Higher Modes Design Spectrum A i n Ref. 5 was used as a smoothed response spectrum i n the previous chapter. Spectrum B from the same references was derived from the 8244 Orion, San Fernando 19 71 record and i s shown i n F i g . 5.1. Among the four test frames, the 3-bay, 6-story frame shown i n F i g . 4.10 was selected and the frame was analyzed by the modified substitute structure method in the same manner as before, except that Design Spectrum B was used as a smoothed response spectrum. The modified substitute method had worked reasonably well for t h i s frame when Design Spectrum A i n Fig. 2.5 was used. The purpose of t h i s analysis was to see i f the method could work equally well for a d i f f e r e n t type of earthquake motion, represented by a d i f f e r e n t response spectrum. The properties of the test frame were unchanged and the analysis was carried out with the same assumptions as i n Chap. 4. The maximum ground accelerations was taken as 0.5 g. With the new response spectrum i t took 27 i t e r a t i o n s to s a t i s f y the con-vergence c r i t e r i o n set i n Chap. 4. Natural periods for the actual frame and the substitute frame are shown i n Table 5.1 and 87 displacements i n Table 5.2. Damage ra t i o s are shown i n F i g . 5.2. Most of the i n e l a s t i c deformations occurred i n the beams, a l l of which yielded. Damage ra t i o s in the beams in a given bay increased with height up the frame. Only the second-story columns yielded. A nonlinear dynamic analysis was done, using the f i r s t 20 seconds of the 8244 Orion 19 71 record to compute the response history of the frame. The maximum ground acceleration was norm-a l i z e d at 0.5 g as before. Maximum displacements are shown i n Table 5.2 and damage r a t i o s i n F i g . 5.2. The results of the non-li n e a r analysis were quite d i f f e r e n t from those of the modified substitute structure analysis. A plot of damage r a t i o s for beams in the exterior bay i s shown i n F i g . 5.3. It i s clear that the modified substitute structure method grossly overestimated the damage rati o s of upper-story beams. A si m i l a r trend was seen i n the previous example, though i t was much less noticeable. Although t h i s finding was very disappointing, an e f f o r t was made to f i n d out the reason why the method f a i l e d to work for the test frame with t h i s response spectrum. The f l o o r weights of the test frame were changed to see i f the natural periods of the frame had any e f f e c t . They were reduced from 2 00 kips per f l o o r to 130 kips per f l o o r to decrease the natural periods. The rest of the properties were the same as shown in F i g . 4.10. The modi-f i e d substitute structure analysis and the nonlinear dynamic analysis were carried out i n an i d e n t i c a l manner. Natural periods of the actual frame and the substitute frame are l i s t e d i n Table 5.3. Displacements in the two analyses are shown i n Table 5.4 and damage ra t i o s i n F i g . 5.4. The results of the two analyses 88 agreed very well t h i s time. The displacements computed i n the modified substitute structure analysis were almost i d e n t i c a l at a l l l e v e l s to those i n the nonlinear dynamic analysis. Beam damage ra t i o s agreed very well as shown in F i g . 5.5 in which the damage ra t i o s for beams in the exterior bay are plotted. Columns yielded s l i g h t l y at three locations i n the nonlinear dynamic analysis, though the modified substitute structure method pre-dicted that a l l the columns would remain i n the e l a s t i c range. Nevertheless, column damage r a t i o s agreed very well i n general. Thus natural periods did a f f e c t the accuracy of the modified substitute structure analysis. There are two ways to explain why the analysis of two frames, i d e n t i c a l except for the f l o o r weights, resulted i n f a i l u r e i n one case and success i n another. One possible explan-ation i s that an actual response spectrum i s very rugged with many peaks and troughs. When a smoothed spectrum i s used, the response acceleration at a certain period may be overestimated, while that at another period may be underestimated. The natural periods of the substitute frame i n the f i r s t case were such that correct response accelerations were not obtained from a smoothed response spectrum. The other explanation i s based on the e f f e c t of higher modes. The shape of a response spectrum i s such that responses due to higher modes play a more prominent role for a structure with longer periods. For a t y p i c a l structure the longest period, and possible the second longest period, may cor-respond to the downward sloping part of the response spectrum. As the natural periods of a substitute frame increase, response accelerations for the lower modes become smaller and less 89 s i g n i f i c a n t compared to those for the higher modes. Since the modal damping r a t i o s of substitute structures decrease for higher modes, higher modes a f f e c t response calculations even more. Therefore, i t i s possible that the substitute structure method overestimates the e f f e c t of higher modes and that this point shows up more c l e a r l y in a structure with longer periods. In order to see which explanation was more l i k e l y to be true, a test frame with shorter periods was analyzed using the o r i g i n a l smoothed response spectrum; that i s , Design Spectrum A in Fig. 2.5 was used. The analysis procedure i n Section 4.2(d) was repeated. The f l o o r weight of the 3-bay, 6-story frame was reduced to 130 kips at a l l l e v e l s , but the rest of the properties were as shown i n F i g . 4.10. The response h i s t o r i e s of the frame to four earthquake motions were also computed by the nonlinear dynamic analysis program. Natural periods of the actual frame and the substitute frame are i n Table 5.5 and displacements are l i s t e d i n Table 5.6. Damage ra t i o s are shown in F i g . 5.6 and F i g . 5.7. Average damage r a t i o s from the four nonlinear analyses agreed well with those i n the modified substitute structure analy-s i s . Better agreement was observed i n the response of upper sto r i e s for t h i s frame than the test frame used i n the l a s t chapter. Thus, although the difference was less apparent i n the case of Design Spectrum A, the frame with shorter periods again worked better. This seems to support the second explanation. Although the difference between a smoothed response spectrum and an actual response spectrum may a f f e c t response computations i n the modi-f i e d substitute structure method, the results described i n t h i s 90 section favours the argument that the modified substitute struc-ture method works better for a structure with shorter periods. Or conversely, the method overestimates the contribution from higher modes. The substitute damping r a t i o i s calculated i n accordance with equation (2.20) i n Chapter 2, and modal damping ra t i o s are computed on the assumption that each element c o n t r i -butes to the modal damping i n proportion to the s t r a i n energy associated with i t i n each mode shape. This has the e f f e c t of making modal damping r a t i o s higher i n the lower modes. In terms of energy i t implies that lower modes dissipate more energy. This i s probably true, but when response calculations are made, th i s works against the o r i g i n a l intention. Since response accelerations i n higher modes with smaller damping r a t i o s are much greater, responses in higher modes are probably given more weight than they should have. When the response acceleration is calculated from the design spectra i n Ref. 5, lower damping ra t i o s do in fact have a proportionally greater e f f e c t ; t h i s should have the e f f e c t of s l i g h t l y de-emphasizing higher modes, which tend to have lower damping, but the evidence here indicates that t h i s e f f e c t should be increased to de-emphasize them s t i l l further. 5.2 Spectrum 91 A smoothed response spectrum deviates from a r e a l response spectrum at many places. Although the curve i s drawn i n such a way that the difference i n the two spectra i s minimized, a s i z -able difference may occur at certain periods. This point arises often when a smoothed response spectrum i s derived from more than one response spectrum. Recall that in the examples i n Chapter 4 the damage due to E l Centro EW motion was consistently higher than that anticipated i n the modified substitute structure analysis. On the other hand, the response h i s t o r i e s of test frames to Taft S69E motion agreed reasonably well with the modified substitute structure analysis. These results may p a r t l y be caused by the discrepancy between a smoothed response spectrum and an actual response spectrum. A computer program was used to generate the response spectra 14 for E l Centro EW motion and Taft S69E motion. The difference between a smoothed spectrum and E l Centro EW spectrum i s i l l u s -trated in F i g . 5.8. The two spectra are reasonably similar in. shape and magnitude at 2% damping r a t i o except at a few places where peaks i n the actual spectrum are considerably above the smoothed spectrum. At 10% damping r a t i o , however, E l Centro EW spectrum i s consistently above the smoothed spectrum for a period greater than 0.4 sec. The response acceleration from the actual spectrum i s 50% to 100% greater than the smoothed spectrum. It appears that the big difference i n the two spectra at high damping r a t i o s explains i n a q u a l i t a t i v e manner the discrepancy in the re s u l t s of the modified substitute structure analysis and the nonlinear dynamic analysis. The smoothed response spectrum and Taft S69E spectrum are plotted i n F i g . 5.9. For Both damping rati o s the smoothed spectrum i s reasonably close to the actual spectrum. This seems to explain q u a l i t a t i v e l y why the results of the two analysis were not very far apart. From these observations i t seemed possible that a better estimate of damage ra t i o s and displacement could be obtained i f an actual response spectrum was used instead of the smoothed spectrum. Response accelerations were computed at a short incre-ment of periods for several damping r a t i o s ranging from 3 = 0.0 to 8 = 0.20 from E l Centro EW record and Taft S69E record. In t o t a l 50 periods for both cases were chosen to complete a table of response spectra. The maximum ground acceleration was norma-l i z e d at 0.5 g. The modified substitute structure analysis was performed in the same way as before except for the following change. The spectral acceleration was read d i r e c t l y or i n t e r -polated from the table. The period was interpolated f i r s t and then the damping. Suppose that the period, T, and the damping r a t i o , 8, were known and that the spectral acceleration corres-ponding to t h i s period and damping was to be computed. Two periods, and 1^> were located i n the table such that T lay between T^ and 1^• Then two damping r a t i o s , 8^ and were found from the table such that 3 was between 3-^  and Q^- Using a li n e a r i nterpolation, spectral accelerations corresponding to T at 3 j a r*d T at were calculated. A l i n e a r i n t e r p o l a t i o n was again performed to compute the acceleration at 6. Several frames were analyzed by the modified substitute structure method, using E l Centro EW spectrum and Taft S69E spectrum. Although the response spectra were no longer smooth, the damage r a t i o converged. In other v/ords, i t was possible to fi n d a substitute structure such that the computed moments were equal to the y i e l d moments for a l l the members which yielded. The number of it e r a t i o n s increased i n many cases. It was found that the overcorrections of damage ra t i o s resulted, i n some cases, in unstable behaviour; the damage r a t i o s o s c i l l a t e d from one i t e r a t i o n to another. The 3-bay, 6-story frame i n F i g . 4.10, with f l o o r weights taken as 130 kips at a l l l e v e l s , was analyzed by the modified substitute structure method, using E l Centro EW spectrum. Twenty i t e r a t i o n s were necessary to achieve convergence. With a smoothed response spectrum i t took 13 i t e r a t i o n s to s a t i s f y the i d e n t i c a l convergence c r i t e r i o n . The results from t h i s analysis were com-pared with those from the previous analyses. The natural periods are summarized i n Table 5.7, the displacements in Table 5.8, and the damage ra t i o s i n F i g . 5.10. The natural periods of the sub-s t i t u t e frame were longer with E l Centro EW spectrum than with the smoothed spectrum. The displacements agreed a l i t t l e better with those from the nonlinear dynamic analysis. The damage ra t i o s were higher with the r e a l response spectrum than with the smoothed response spectrum. They were closer to the damage rati o s found in the nonlinear dynamic analysis, but the modified substitute structure method s t i l l underestimated the damage r a t i o s at lower level s and overestimated those at upper l e v e l s . The results con-firmed the observation that the smoothed spectrum was unconserva-t i v e for E l Centro EW motion. The analysis was repeated, using Taft S69E spectrum. It took 14 i t e r a t i o n s for the damage rati o s to converge, while 13 ite r a t i o n s were required with the smooth response spectrum. The comparison of natural periods i s shown in Table 5.7, the d i s -placements i n Table 5.9, and damage ra t i o s i n F i g . 5.11. The same trend observed i n the analysis with E l Centro EW spectrum was present, but the two modified substitute structure analyses did not d i f f e r s i g n i f i c a n t l y . I t indicates that the smoothed response spectrum represented Taft S69E motion well. Compared with the nonlinear dynamic analysis, the r e a l response spectrum produced s l i g h t l y better results than the smoothed spectrum, but the improvement was marginal. The same frame, except that the f l o o r weight was set at 200 kips, was next tested. This i s the i d e n t i c a l frame used in Section 4.2(d). The analysis with E l Centro EW spectrum was done i n the same manner. The periods, displacements, and damage ra t i o s are shown i n Table 5.10, Table 5.11, and F i g . 5.12 respec-t i v e l y . Thirty-three i t e r a t i o n s were required, while i t took 16 it e r a t i o n s with the smoothed spectrum. The res u l t s were very disappointing. The modified substitute structure method with E l Centro EW spectrum badly overestimated the displacements and the damage r a t i o s , e s p e c i a l l y for the beams. The displacements were too large at a l l ; l e v e l s , but the deviation from the non-l i n e a r dynamic analysis r e s u l t s became progressively larger at upper l e v e l s . Some y i e l d i n g i n the columns was observed, but those columns did not y i e l d in the nonlinear analysis while others did. The beam damage rati o s increased with height when E l Centro EW spectrum was used. But i n the nonlinear analysis the opposite trend was observed. 95 The analysis was repeated with Taft S69E spectrum. The number of i t e r a t i o n s was 23, an increase of 7 i t e r a t i o n s over the analysis with the smoothed spectrum. The results are summarized in Table 5.10, Table 5.12, and F i g . 5.13. They compared more fav-orably t h i s time with those from the nonlinear analysis. The mod-i f i e d substitute structure method with Taft S6 9E spectrum, again, overestimated the displacements and damage r a t i o s , but not as badly as in the l a s t example. The results for the two test frames indicate that using a r e a l response spectrum does not guarantee a better estimate of damage r a t i o s and displacements. This observation was confirmed in the analyses of other frames. A marginal improvement was achieved with the use of a r e a l response spectrum while a bad est-imate of damage r a t i o s resulted i n some cases. The improvement, i f any, was so small and the increase i n computation so big that i t would not be p r a c t i c a l to employ t h i s approach. I t i s more useful to make a smoothed response spectrum closer to the r e a l response spectrum and perform the modified substitute structure analysis with the smoothed spectrum. The difference i n r e s u l t between th i s analysis and the nonlinear dynamic analysis should be regarded as an inherent error due to the approximate nature of th i s analysis. I t must, of course , also be borne i n mind that the future earthquake w i l l not have a record i d e n t i c a l to those of the past, either. Thus the smoothed spectrum represents the future earth-quake just as well as does the " r e a l " spectrum from a past earth-quake. However, the foregoing discussion does indicate that one source of "error" i n the modified substitute structure method lay in the smoothing and averaging of the spectrum. 96 5.3 Guidelines for Use of Method As was i l l u s t r a t e d i n the example i n Chapter 4, the modi-f i e d substitute structure works very well for some structures, while i t works poorly for others. An e f f o r t was made to estab-l i s h the conditions which must be s a t i s f i e d i n order to apply the method successfully for analysis of e x i s t i n g buildings. The author, however, has so far been unable to set firm guidelines. More research i s necessary to achieve t h i s goal; therefore, the following comments should be interpreted with caution. The modified substitute structure method i s an extention of the substitute structure method. Therefore, the success of the former depends greatly on the success of the l a t t e r . As described i n Chapter 2, certain conditions must be s a t i s f i e d in order for the method to work. They are also applicable to the modified substitute structure method with the exception of one condition. The preliminary r e s u l t s indicate that the damage ra t i o s of beams in a given bay or the damage ra t i o s of columns on a given axis need not be the same. The modified substitute structure method works well for small structures. The 2-bay, 2-story frame and the 3-bay, 3-story frame i n the l a s t chapter can be used to support t h i s argu-ment. Although t h e i r member properties and strengths were not very uniform, the results agreed very well with those from the nonlinear dynamic analysis. It appears that any structure up to four-story high can be analyzed by the modified substitute struc-ture method quite sucessfully. Some caution i s necessary to interpret the results for medium-rise structures. Although the method works reasonably well for most of the structure, there are instances when i t pro-duces erroneous r e s u l t s . When a structure i s badly underdesigned for a given ground motion and y i e l d i n g takes place i n almost a l l the members, the modified substitute structure method may work very poorly. The 6-story frame i n Chapter 4 i s a good example. Though the method can show q u a l i t a t i v e l y that a structure i s behaving poorly, the damage rati o s and displacements may be quite d i f f e r e n t from the nonlinear dynamic analysis. I n t u i t i o n should be used to judge whether the res u l t s are reasonable. In th i s par-t i c u l a r 6-story frame, however, i t was noted that the "actual" behaviour was e r r a t i c : the dynamic analysis led to a very d i f -ferent answer from the d i f f e r e n t earthquake records. Thus one reason why the modified substitute structure method was unable to give a good answer was that there was no " r e a l " answer. One may conclude that when there are few load paths and extensive y i e l d -ing the behaviour of the structure i n future earthquakes i s e s s e n t i a l l y unpredicable, and the modified substitute structure method w i l l , of course, f a i l . As long as the damage rati o s are not very high, say, less than f i v e i n any member, the res u l t s can be received with c o n f i -dence. The method seems to work better when y i e l d i n g i s concen-trated in beams. The method may overestimate the damage rati o s for upper-story beams, but they are usually not very far from those i n the nonlinear analysis. A l l of the 3-bay, 6-story frames can be used as evidence for th i s argument. A multi-bay structure seems to work better with the method. High-rise structures, greater than 10 s t o r i e s , say, have 98 not been tested. They can be analyzed by the modified substitute structure method at a r e l a t i v e l y small cost. The damage ra t i o s converge quite rapidly, but t h e i r accuracy has not been compared with the nonlinear dynamic analysis, mainly because of high cost involved i n such an analysis. It i s hoped that the method works as well for high-rise structures as i t does for medium-rise structures. 99 5.4 Further Studies The modified substitute structure method was proposed for analysis of e x i s t i n g reinforced concrete structures. The empha-s i s of the research by the author was placed on the development of the procedure for the proposed method. Although a series of test frames were analyzed and the r e s u l t s were compared with the nonlinear dynamic analysis, the findings are s t i l l preliminary. More researches are needed to establish the true effectiveness and the l i m i t a t i o n s of the modified substitute structure method. Some of the areas for further studies are discussed in t h i s section. A multi-bay, high-rise structure has not been tested, and the performance of the method for such a frame i s not known pre-c i s e l y . The results from the modified substitute structure analysis should be compared with the nonlinear dynamic analysis. Though the cost f o r the nonlinear analysis w i l l be undoubtedly high, the careful choice of an earthquake record may help keep i t at a reasonable l e v e l . More r e a l i s t i c structures should also be tested. Actual reinforced concrete structures may be used as test frames for t h i s purpose. The r e s u l t s of such analysis w i l l help set up better guidelines for a p p l i c a b i l i t y of the method as i t stands at the present time. An attempt should also be made to improve the present pro-cedure. The method becomes more f l e x i b l e and, hence, more p r a c t i c a l i f some of the r e s t r i c t i o n s are removed. For example, at present a single value for the y i e l d moment i s assigned to each member. If the moment capacities of' the two end of a member 100 d i f f e r e n t , the method cannot be applied c o r r e c t l y without a s u i t -able s i m p l i f i c a t i o n i n the modeling of such a member. The current procedure should be modified to handle t h i s case. It i s also desirable to include the e f f e c t of a x i a l forces i n the analysis. Behaviour of columns can be estimated more p r e c i s e l y i f such modi-f i c a t i o n s are made. As was discussed b r i e f l y i n the f i r s t section of t h i s chap-ter, the present method for computation of "smeared" or average modal damping rat i o s may not be the best way: i t appears that the e f f e c t of higher modes are overemphasized. Perhaps a new way to combine the damping r a t i o for each member can be developed to give more r e a l i s t i c modal damping r a t i o s . So far only reinforced concrete frame structures were tested. In practice, i t i s very rare to f i n d reinforced concrete struc-tures without shear walls. The a p p l i c a b i l i t y of the modified sub-s t i t u t e structure method to shear walls should be investigated. If the present method did not work well with shear walls, a d i f f e r e n t way of modifying s t i f f n e s s and damping r a t i o s would have to be developed. It i s possible that the modified substitute structure method can be altered to handle structures made of other materials, such as s t e e l . If suitable rules to modify s t i f f n e s s and damping ra t i o s are developed for s t e e l structures, the method can be used i n a similar manner for analysis of e x i s t i n g s t e e l buildings. It probably i s not very d i f f i c u l t to study the hysteresis loop of a steel structure a f t e r several cycles of i n e l a s t i c deformation. The s t i f f n e s s and damping properties may be determined i n a g s i m i l a r manner to that used by Gulkan and Sozen. 101 CHAPTER 6 CONCLUSION The modified substitute structure method has been presented for determining damage ra t i o s i n an exis t i n g reinforced concrete building. These values are required for establishing the location and extent of damage which would occur i n an earthquake. It i s obvious that they cannot be predicted precisely for uncertain future seismic events; thus, i n spite of i t s imprecision, the method may constitute a useful part of the r a t i o n a l r e t r o f i t procedure. At present i t i s not always possible to predict the accur-acy of the modified substitute structure analysis, but the method appears to work well for structures in which y i e l d i n g i s not extensive and widespread. In addition the preliminary findings indicate that i t works better i f y i e l d i n g occurs mainly i n beams. There i s an indicati o n that the e f f e c t of higher modes i s over-emphasized. It i s hoped that further research would c l a r i f y requirements for successful application of the method. Although not perfected, the modified substitute structure method of f e r s a cheap and e f f e c t i v e way of estimating damage rat i o s or d u c t i l i t y demands under one or more l e v e l of seismic a c t i v i t y . Though less precise, i t i s much cheaper than a f u l l -scale nonlinear dynamic analysis and, as an additional advantage, an analysis can be done on a smaller sized computer. Its 102 advantage over a l i n e a r e l a s t i c analysis i s that i t takes account of the r e d i s t r i b u t i o n of forces as members begin to y i e l d . A s l i g h t l y higher cost of computation i s amply rewarded with t h i s additional information on i n e l a s t i c behaviour of a structure, which cannot be obtained by a conventional modal analysis. 103 Natural Periods i n sec Smeared Damping Ratios Mode Computed S & S* Computed S & S* 10-Story 1 3.1807 3.18 0.1061 0.106 Frame 2 0.8763 0.87 0.0805 0.081 3 0.3945 0 . 39 0.0525 0.053 4 0.2172 0.22 0.0383 0.038 5 0.1358 0 .14 0.0312 0.032 6 0.0930 0.093 0.0272 0.027 7 0.0681 0.068 0.0244 0.024 8 0.0531 0.053 0.0224 0.022 9 0.0442 0.044 0.0211 0.021 10 0.0397 0.040 0.0204 0.020 5-Story 1 1.5868 1.58 0.0991 0.099 Frame 2 0.4101 0.41 0.0680 0.068 3 0.1751 0. 18 0.0409 0.041 4 0.0967 0.097 0.0283 0.028 5 0.0670 0 . 067 0.0218 0.022 3-Story 1 0.8525 0 .85 0.0852 0 .086 Frame 2 0.1883 0 .19 0.0454 0.045 3 0.0784 0.078 0.0245 0.025 * Shibata and Sozen Table 2.1 Natural Periods and Smeared Damping Ratios for 3-, 5-, and 10-Story Frames 104 Damage Ratios E l Centro EW E l Centro NS Taft S69E Taft N21E Average Frame 1 2 3 co 4 6 b 3 6 0 7 u 8 9 10 2.0 4.4 4.8 2.5 6.9 1.0 1.1 1.8 0.96 4.0 0.85 0.90 0.97 0.91 0.94 0.81 0.72 0.90 0.89 0.92 0.98 1.1 0.90 0.88 0.97 0.95 0.98 0.95 0.90 0.90 0.95 1.4 1.1 0.92 1.0 1.1 1.2 1.7 0 . 96 0.92 1.2 2.0 1.9 1.3 2.5 0.98 0. 99 1. 35 0.93 1.7 10-Story 1 2 3 4 «* 6 CD D m 7 8 9 10 6.5 7.6 8.3 8.1 8.6 9.3 9.8 9.9 9.9 9.9 4.1 4.3 4.6 4.7 4.7 4.8 4.7 4.4 4.1 3.9 5.0 5.0 4.9 4.5 4.1 4.2 4.1 3.9 3.6 3.4 4.9 5.0 5.1 5.1 5.0 4.8 4.6 4.2 4.0 4.0 5.1 5.5 5.7 5.6 5.6 5.8 5.8 5.6 5.4 5.3 Frame 1 § 2 1 3 4 O 5 1.1 3.9 0.97 1.1 1.0 0.98 0.86 0.97 1.1 0.89 0.87 0.84 0.89 0.93 0.87 0.84 0.70 0.78 0.88 0.85 0.95 1.6 0.90 1.0 0.90 5-Story 1 co 2 e 3 cu 4 0 0 5 5.4 7.1 7.1 6.7 6.7 4.8 4.6 4.7 4.4 4.2 4.4 4.4 4.1 4.1 3.9 3.7 3.3 3.2 2.8 2.4 4.6 4.8 4.8 4.5 4.3 u fe > i . 1 , H 2 0 z u 3 0.95 0.89 0.91 0.90 0.94 0. 89 0.64 0.61 0.84 0.77 0.65 0.86 0.82 0.73 0.88 3-Stor , 1 £ 2 ffl 3 6 . 3 6.1 6.0 5.8 6 . 0 6.3 4.0 3.3 2.7 4.9 4.5 4.1 5.3 5.0 4.8 Table 2.2 Computed Damage Ratios for 3-, 5-, and 10-Story Frames 105 Damage Ratios E l Centro EW Taft S69E Computed S & S* Computed S & S* 10-Story Column 1 2.0 0.95 0.98 0.58 Frame 2 4.4 1.2 1.1 0.80 3 4.8 1.0 0.90 0.70 4 2.5 0.98 0.88 0.80 5 6.9 2.8 0.97 0.90 6 1.0 1.2 0.95 0.80 7 1.1 0.96 0.98 0.80 8 1.8 0.98 0.95 0.85 9 0.96 0. 85 0.90 0 . 80 10 4.0 1.7 0.90 0 . 80 Beam 1 6.5 6.9 5.0 5.5 2 7.6 7.2 5.0 5 . 5 3 8.3 7.5 4.9 5.0 4 8.1 7.8 4.5 4.9 5 8.6 7.5 4 .1 4.8 6 9 . 3 8.8 4.2 4.6 7 9.8 9.6 4.1 4.8 8 9.9 9.9 3.9 3.8 9 9.9 9.8 3.6 3.0 10 9.9 10.0 3.4 2.2 5-Story Column 1 1.1 0.90 0.87 0.70 Frame 2 3.9 2.2 0.84 0 . 70 3 0.97 0.94 0.89 0.80 4 1.1 2.3 0.93 0.80 5 1.0 0.96 0.87 0. 90 Beam 1 5.4 7.0 4.4 4.4 2 7.1 8.3 4.4 4.3 3 7.1 8.4 4.1 3.6 4 6 . 7 7.3 4.1 2.5 5 6.7 6.9 3.9 1.5 3-Story Column 1 0.95 0.97 0.64 0.65 Frame 2 0. 89 0.90 0.61 0.61 3 0.91 0.90 0.84 0.90 Beam 1 6 . 3 6 . 8 4.0 4.5 2 6.1 6 . 3 3.3 3.7 3 6.0 6.0 2.7 3.0 * Shibata and Sozen' Table 2.3 Comparison of Damage Ratios for 3-, 5-, 10-Story Frame's 106 Natural Periods for the F i r s t Modes i n sec I n i t i a l E l a s t i c Substitute Structure Nonlinear Analysis Average Equal-Area S t i f f n e s s 3-Story Frame 0.50 0.85 0.65 0.72 5-Story Frame 0. 85 1.58 1.20 1.29 10-Story Frame 1.58 3.18 2 . 50 2.55 Table 2.4 Computed Natural Periods for 3-, 5-, and 10-Story Frames Natural Periods i n sec No. of Iterations Mode 1 Mode 2 Mode 3 1 1.0679 0 . 3233 0.1804 2 1.3701 0. 3632 0.1917 3 1.7655 0.4484 0.2231 4 1.7810 0.4486 0.2129 5 1.7945 0.4513 0.2074 6 1.8004 0.4505 0.2033 7 1.8066 0.4496 0 .2009 8 1.8076 0.4476 0.1990 9 1.8073 0.4455 0.1975 10 1.8069 0.4439 0.1964 11 1.8067 0.4431 0.1960 12 1.8060 0.4423 0.1956 13 1.8052 0.4414 0.1952 14 1.8046 0.4405 0.1948 15 1.8046 0.4397 0.1944 16 1.8041 0.4390 0.1940 17 1.8036 0.4386 0.1937 18 1.8035 0.4383 0.1936 19 1.8036 0.4381 0.1934 20 1.8036 0.4380 0.1933 Su b s t . ( a ) 1.8036 0.4377 0.1932 (a) Natural periods computed i n the substitute structure analysis Table 3.1 Natural Periods for 2-Bay, 3-Story Frame A 107 Damage Ratios No. of Iterations Column 1 Column 2 Beam 1 Beam 2 1 1.000 1.155 2.853 1. 344 2 1.205 1.848 6.084 2.538 3 1.079 1.964 6 . 382 3.281 4 1.000 2.030 6 . 281 4.119 5 1.000 1.986 6 .116 4.758 6 1.000 1.881 6.021 5.195 7 1.000 1.749 5.982 5 . 453 8 1.000 1.621 5.975 5.612 9 1.000 1.508 5.981 5 .716 10 1.000 1.409 5.988 5. 785 11 1.000 1. 324 5.992 5.827 12 1.000 1.250 5.996 5. 857 13 1.000 1.188 5.999 5.882 14 1.000 1.134 6 .002 5.905 15 1.000 1.087 6.004 5.926 16 1.000 1.057 6 .006 5.945 17 1.000 1.038 6.006 5.961 18 1.000 1.025 6.006 5.973 19 1.000 1. 017 6.006 5.988 20 1.000 1.011 6.006 5.992 S u b s t . ( a ) 1.000 1.000 6.000 6 . 000 (a) Target damage ra t i o s i n the substitute structure analysis Table 3.2 Damage Ratios for 2-Bay, 3-Story Frame A 108 Natural Periods i n sec _ _ J-NO. Of Iterations Mode 1 Mode 2 Mode 3 1 1.0674 0.3233 0.1804 2 1.2606 0.3694 0 .2062 3 1.6682 0.4758 0.2666 4 1.6338 0.4609 0.2579 5 1.6371 0.4601 0.2568 6 1.6379 0.4605 0.2563 7 1.6382 0.4620 0.2560 8 1.6375 0.4636 0.2556 9 1.6366 0.4650 0.2552 10 1.6360 0.4663 0,2546 12 1.6350 0.4682 0.2534 14 1.6339 0.4692 0.2518 16 1.6331 0,4697 0.2503 18 1.6325 0 .4699 0 .2489 20 1.6320 0.4700 0.2476 S u b s t . ( a ) 1.6307 0.4633 0.2 37 5 (a) Natural periods computed i n the substitute structure analysis Table 3.3 Natural Periods for 2-Bay, 3-Story Frame B Number of Iterations t =10"2 t = i o " 3 -4 i =10 0.0 29 158 200 1.0 18 81 124 D i f f 11 77 76 Table 3.4 Number of Iterations - 2-Bay, 3-Story Frame B 109 Damage Ratios Member _2 £ =10 =10 3 £=10 4 After 100 Exact Iterations Col. 1 1.969 1.998 2.001 2.000 2. 2 1.489 2.002 2 .003 2.003 2 . 3 3. 476 2.017 1.996 2.002 2. Col. 4 l.ooo 1.000 1.000 1.000 1. 5 1.000 1.000 1.000 1.000 1. 6 1.496 1.036 1.005 1.013 1. Col. 7 2.973 2.999 3.002 3.001 3. 8 3.143 3.013 3.003 3.005 3. 9 3.582 3.049 3.003 3.019 3. Beam 1 6 . 016 5.993 5.995 5.995 6 . 2 6.160 6.000 5.999 5.999 6 . 3 4.675 5.956 5.991 5.981 6 . Beam 4 1.992 2.001 2 .002 2 .001 2. 5 1.968 1.999 2 .001 2.000 2. 6 1.496 1.964 1.995 1.987 2. No. of It e r - 18 81 124 100 ations i Table 3. 5 Damage Ratios for 2-Bay, 3-Story Frame B Natural Periods i n sec Mode I n i t i a l Substitute E l a s t i c 1 0.50 0.76 2 0.13 0.18 Table 4.1 Natural Periods for 2-Bay, 2-Story Frame 110 Displacements i n inches Level Centro Centro Taft Taft Average Subst. EW NS S69E N21E 1 2.8 2.6 1.3 1.9 2.1 1.8 2 5.3 5.1 2.7 3.6 4.2 3.8 Table 4.2 Displacements for 2-Bay, 2-Story Frame Natural Periods i n sec Mode I n i t i a l Substitute Nonlinear E l a s t i c Average 1 0.94 1.22 2 0.30 0. 36 1.04 3 0.14 0.16 Table 4.3 Natural Periods for 3-Bay, 3-Story Frame Displacements i n inches Level Centro Centro Taft Taft Average Subst EW NS S69E N21E 1 3.0 2.4 1.8 1.6 2.2 2.2 2 6.7 5.2 3.8 3.0 4.7 5.0 3 10 .6 7.9 • 6.2 5.2 7.5 8.0 Table 4.4 Displacements for 3-Bay, 3-Story Frame I l l Natural Periods i n sec Mode I n i t i a l Substitute Nonlinear E l a s t i c Average 1 1.08 1.85 2 0. 37 0.84 3 0.21 0. 38 1.65 4 0.15 0.28 5 0 .10 0.17 6 0.077 0 .13 Table 4.5 Natural Periods for 1-Bay, 6-Story Frame Displacements i n inches Level Centro Centro Taft Taft Average Sub st EW NS S69E N21E 1 3.7 0.74 1.4 2.4 2.1 0. 71 2 8.2 1.7 3.3 4.8 4.5 2.1 3 12.0 3.0 4.8 6 . 1 6.5 2.9 4 14.5 4.5 6.7 6.6 8.1 3 . 3 5 17.0 6.5 9.4 6.9 10.0 6.8 6 19. 3 8.4 11.6 7.2 11.6 8.6 Table 4.6 Displacements for 1-Bay, 6-Story Frame Natural Periods i n sec Mode I n i t i a l Substitute Nonlinear E l a s t i c Average 1 1.07 1.66 2 0.34 0.48 3 0.19 0.24 1.25 4 0.12 0.14 5 0.090 0 .096 6 0.075 0.076 Table 4.7 Natural Periods for 3-Bay, 6-Story Frame 112 Displacements in inches Level Centro Centro Taft . Taft Average Subst EW NS S69E N21E 1 1.3 1.1 1.3 0.98 1.2 1.1 2 3.5 2.9 3.1 2.5 3.0 3.0 3 5.9 4.5 4.5 3.7 4.7 5.0 4 7.9 5.5 5.8 4.6 6.0 6 . 7 5 9.2 6 .1 6.6 5.1 6 . 8 7.9 6 9.8 6.3 7.3 5.4 7.2 8.8 Table 4.8 Displacements for 3-Bay, 6-Story Frame Natural Periods i n sec Mode I n i t i a l Substitute E l a s t i c 1 1.07 2.24 2 0.34 0.63 3 0.19 0.29 4 0.12 0 .16 5 0.090 0 .11 6 0 .075 0.078 Table 5.1 Natural Periods for 3-Bay, 6-Story Frame A -Spectrum B Displacements i n inches Level Substitute Nonlinear 1 1.4 1.6 2 5.2 4.5 3 9.4 8.0 4 13. 3 10.9 5 16.8 13.1 6 19.7 14.0 Table 5.2 Displacements for 3-Bay, 6-Story Frame A -Spectrum B 113 Natural Periods i n sec Mode I n i t i a l Substitute E l a s t i c 1 0 . 86 1.20 2 0.27 0 .34 3 0.15 0 .17 4 0.099 0 .11 5 0.073 0.076 6 0.060 0.061 Table 5.3 Natural Periods for 3-Bay, 6-Story Frame B -Spectrum B Displacements i n inches Level Substitute Nonlinear 1 1.1 1.2 2 3.3 3.3 3 5.5 5.5 4 7.4 7.4 5 8.7 8.5 6 9.4 9.1 Table 5.4 Displacements for 3-Bay, 6-Story Frame B -Spectrum B Natural Periods i n sec Mode I n i t i a l Substitute E l a s t i c " 1 0.86 1.20 2 0.27 0. 34 3 0.15 0 .17 4 0.099 0.11 5 0.073 0 .076 6 0.060 0.061 Table 5.5 Natural Periods for 3-Bay, 6-Story Frame B - Spectrum A 114 Displacements i n inches Level Centro Centro Taft Taft Average Subst EW NS S69E N21E 1 1. 3 1.1 1.1 0.95 1.1 0.94 2 3.4 2.8 2.9 2.6 2.9 2.6 3 5.4 4.3 4.5 4.1 4.6 4.3 4 6.9 5.4 5.6 5.2 5.8 5.5 5 7.7 6.0 6.2 5.9 6.4 6.4 6 8.0 6 . 3 6.4 6.1 6 . 7 6.8 Table 5.6 Displacements for 3-Bay, 6-Story Frame B -Spectrum A Natural Periods i n sec I n i t i a l Modified Subst. Str . Analysis Mode E l a s t i c Smooth Centro EW Taft S69E Spectrum Spectrum Spectrum 1 0.86 1.20 1.32 1.23 2 0.27 0.34 0 . 36 0.34 3 0.15 0.17 0.18 0.18 4 0.099 0.11 0.11 0.11 5 0.073 0.076 0.076 0.076 6 0.060 0 . 061 0.061 0.061 Table 5.7 Natural Periods for 3-Bay, 6-Story Frame B -E l Centro EW Spectrum and Taft S69E Spectrum 115 Displacements i n inches Substitute Structure Nonlinear Level Smooth E l Centro EW E l Centro EW Spectrum Spectrum 1 0.94 1.1 1.3 2 2.6 3.2 3.4 3 4 . 3 5.4 5.4 4 5.5 7.2 6.9 5 6.4 8.4 7.7 6 6 . 8 9.0 8.0 Table 5.8 Displacements for 3-Bay, 6-Story Frame B -E l Centro EW Spectrum Displacements i n inches Substitute Structure Nonlinear Level Smooth Taft S69E .Taft S69E Spectrum Spectrum 1 0.94 0.98 1.1 2 2.6 2.8 2.8 3 4.3 4.5 4.3 4 5.5 5.8 5.4 5 6.4 6.8 6.0 6 6 . 8 7.3 6 . 3 Table 5.9 Displacements for 3-Bay, 6-Story Frame B -Taft S69E Spectrum 116 Natural Periods i n sec Modified Subst. Str.. Analysis I n i t i a l Smooth Centro EW Taft S69E Mode E l a s t i c Spectrum Spectrum Spectrum 1 1.07 1.66 2.04 1.82 2 0. 34 0.48 0.58 0.52 3 0.19 0.24 0.28 0.25 4 0.12 0.14 0.16 0.15 5 0.090 0.096 0.11 0.098 6 0.075 0.076 0.082 0.076 Table 5.10 Natural Periods for 3-Bay, 6-Story Frame A -E l Centro EW Spectrum and Taft S69E Spectrum Displacements i n inches Level Substitute Structure Nonlinear E l Centro EW Smooth . i Spectrum E l Centro EW Spectrum 1 1.1 1.5 1.3 2 3.0 4.8 3.5 3 5.0 8.4 5.9 4 6.7 11.6 7.9 5 7.9 14.3 9.2 6 8.8 16 .4 9.8 Table 5.11 Displacements for 3-Bay, 6-Story Frame A -E l Centro EW Spectrum 117 Displacements i n inches Level Substitute Structure Nonlinear Taft S69E Smooth Spectrum Taft S69E Spectrum 1 1.1 1.3 1.3 2 3.0 3.9 3.1 3 5.0 6.7 4.5 4 6.7 9.2 5.8 5 7.9 11.2 6.6 6 8.8 12.4 7.3 Table 5.12 Displacements for 3-Bay, 6-Story Frame A -Taft S69E Spectrum 118 Max. load-deflection r e l a t i o n s h i p for hypothetical structure which remains e l a s t i c Max. load-deflection r e l a t i o n s h i p for actual structure which y i e l d s Deflection F i g . 2.1 Idealized Hysteresis Loop for Reinforced Concrete System 119 120 Start Read: 1. str u c t u r a l information 2. j o i n t information 3. member information including target damage ra t i o s Compute: 1. number of unknowns 2. half bandwidth 3. member substitute damping r a t i o s • Assemble the mass matrix 1 1. Compute member s t i f f n e s s matrices. Modify the f l e x u r a l part of st i f f n e s s e s according to the target damage r a t i o s . 2. Assemble the s t r u c t u r a l s t i f f n e s s matrix. i = = 1 n = = 0 © g. 2.3 Flow Diagram for Substitute Structure Method 0 121 No Yes Set (3 = 0 for a l l the modes Compute the response acceleration for n th mode Set up the load vector Compute the f l e x u r a l s t r a i n energy stored i n each member Recall the smeared damping ra t i o s N = number of modes Yfes Compute the f l e x u r a l s t r a i n energy stored in each member © g. 2 .3 Flow Diagram for Substitute Structure Method 122 Compute the smeared damping r a t i o for n th mode n = n + 1 N= Number of modes Yes i = 2 1 Compute RSS displacements ' and RSS forces Compute the design forces jp _ r liabs +^rss design rss 2 V r s s Increase the column moments by 2 0% • Stop Fig. 2.3 Flow Diagram for Substitute Structure Method 123 M =173 k - f t J£ 216 186 366 19 5 404 216 366 404 Beams Size 3-Story Frame 18"x 30" 13,500 i n 5-Story Frame 18"x 30" 13,500 in^ 10-Story Frame 18"x 30" 13,500 i n Columns 3-Story Frame 24"x 24" 13,824 in' 5-Story Frame 24"x 24" 13,824 i n ' 10-Story Frame 30"x 30" 33,750 in 4 206 448 219 428 228 401 228 446 217 496 19 7 695 128 1165 448 428 401 446 496 695 1165 E = 3,600 k s i Floor weight i s 72 kips at a l l levels My=199 k - f t 239 226 313 237 328 233 417 183 831 239 313 328 417 • h 831 M =212 k - f t _,y_ 254 231 407 189 850 254 407 850 24 ' 24 ' 24 ' F i g . 2.4 Member Properties and Design Moments for 3-, 5-, 10-Story Frames 124 0.0 0.5 1.0 1.5 2.0 2.5 3.0 12 10 8 6 4 2 Period i n sec. Frequency i n hertz Response Acceleration for _ 8 Response Acceleration for (i=0.02 6 + 100p Fig. 2.5 Smoothed Response Spectrum - Design Spectrum A 125 /*• = 1 1 /A = 1 /A -1 1 yU = 1 /A = 6 /*- 2 fA = 2 24 W = 72 kips W = 72 kips W = 72 kips yU = Target damage r a t i o M =325 k - f t y E = 3600 k s i 390 669 390 736 239 736 Yie l d Moments 728 728 Size I 1st Story- 24" x 24" 10,368 Columns 2nd Story 24" x 24" 13,824 3rd Story 24" x 24" 13,824 Beams 18" x 30" 13,500 F i g . 2.6 Soft Story Frame A - Member Properties and Yi e l d Moments 3.1 0.84 1.9 0.88 6.1 2.8 126 1.5 0.79 1.1 0.80 4.4 1.8 1.2 0.79 0.93 0.70 3.7 1.2 E l Centro EW E l Centro NS Taft S69E 1.3 0.79 0.96 0.72 3.9 1.3 1.8 0.81 1.2 0.77 4.5 Taft N21E Average F i g . 2.7 Soft Story Frame A - Damage Ratios for Individual Earthquakes 127 /*= 1 yu= 1 /U = 6 /U = 2 >U = 1 = 1 = 1 = 1 W = 72 kips W = 72 kips W = 72 kips y U = Target Damage Ratio 24 M =583 k - f t y 699 238 610 707 902 699 610 902 Size Columns 1 24"x24" 2 24"x24" 3 24"x24" 13,824 i n 10,368 in1 13,824 in^ Beams 18"x30" 13,500 i n Design Moments Fi g . 2.8 Soft Story Frame B - Member Properties and Yie l d Moments 0.96 0 .78 4.2 0.79 1.2 1.2 E l Centro EW 128 0.96 0 .78 4.1 0.82 1.1 0.96 E l Centro NS 1.0 0.82 4.4 0 .84 1.2 1.1 Taft S69E 0.60 0.48 2.6 0.53 0.70 0.66 Taft N21E 0 .89 0 .71 3.8 0 .74 1.0 0.97 Average F i g . 2.9 Soft Story Frame B - Damage Ratios for Individual Earthquakes 129 /i= 6 }K= 2 CM y(A= 2 //= 2 yU = 4 yU = 6 fK= 1 yU= 2 yU = 2 yU = 2 50 50 W = 600 kips fk= 3 W = 600 kips JJi= 3 W = 600 kips / A -  3 E = 3,600 k s i M =339 k - f t 745 y 478 1061 962 1069 610 1288 524 1323 1127 2513 614 823 938 Y i e l d Moments Columns Beams Size 21" x 21' 20" x 36' 16,000 i n 40,000 i n 4 F i g . 2.10 2-Bay, 3-Story Frame - Member Properties and Y i e l d Moments 130 4.2 1.5 0.70 2.0 0.92 2.0 6.7 7.2 3.2 3.2 4.4 E l Ce 1.9 rttro EW 4.4 1.6 1.7 1.8 0 .90 1.9 1.7 4.2 0.90 1.6 1.8 0.77 2.8 4.7 5.6 3.0 0 .90 2.6 Taft S69E 3.8 1.3 0 .69 1.7 3.7 5.1 2.6 0.89 1.8 1.7 3.3 1.1 0.70 0.87 1.6 1.6 4.2 1.8 5.0 2.2 2.7 1.0 E l Centro NS 3.1 0.96 0.67 0.85 1.3 1.5 2.3 0.95 4.0 1.6 1.8 0.76 Taft N21E 2.2 2.5 Average 1.7 2.8 3.6 1.2 1.3 2.6 2.1 1.1 3.6 Fi g . 2.11 2-Bay, 3-Story Frame - Damage Ratios for Individual Earthquakes 131 F i g . 2.12 Force-Displacement Curve - D e f i n i t i o n of Equal-Area S t i f f n e s s 132 i g . 3.1 Moment-Rotation Curve - Modification of Damage Ratio 133 Start Read: 1. st r u c t u r a l information 2. j o i n t information 3. member information. Compute: 1. number of unknowns 2. half bandwidth. "  Set the damage r a t i o s at one V Assemble the mass matrix f  k = 1 Compute member s t i f f n e s s matrices. Modify the f l e x u r a l s t i f f n e s s e s according to the damage r a t i o s . Assemble the s t r u c t u r a l s t i f f n e s s matrix. Yes = 1 ^ > No Compute member substitute damping r a t i o s . } Compute: 1. natural periods 2. mode shapes 3. modal p a r t i c i p a t i o n factors. F i g . 3.2 Flow Diagram for Modified Substitute Structure Method 134 1 i = 1 and n = 0 Yes Set (3=0 for a l l modes s Recall smeared damping r a t i o s Compute the response acceleration for n th mode Set up the load vector Compute modal displacements and modal forces No Compute the f l e x u r a l s t r a i n energy stored i n each member. Compute the smeared damping r a t i o for n th mode Set (b at appropriate values g. 3.2 Flow Diagram for Modified Substitute Structure Method i = 2 Compute RSS displacements and RSS forces Write: 1. RSS displacements 2. RSS forces 3. damage r a t i o s Y Stop Flow Diagram for Modified Substitute Structure Method 136 C N /A = 6 /A = 1 A= 6 yU = 1 /U= 6 / A = 6 yU= 1 M= 6 yU = 1 /A= 6 /A: 50 M =461 k - f t Y 50 461 1612 1701 W = 600 kips yU= 1 W = 600 kips yU= 1 /U = Target Damage Ratio JU= i 722 501 917 501 746 443 921 443 722 W = 600 kips E = 3,600 k s i Columns Beams Size 21" x 21" 20" x 36" I 16,000 i n 4 40,000 i n 4 Natural Periods Mode 1 1.804 sec. Mode 2 0.4 38 sec. Mode 3 0.19 3 sec. 746 1612 Y i e l d Moments Fig . 3.3 2-Bay, 3-Story Frame A - Member Properties and Y i e l d Moments Number of Iterations Fig. 3.4 2-Bay, 3-Story Frame A - Plot of Periods vs. Number of Iterations Fig. 3.5 2-Bay, 3-Story Frame - Plot of Damage Ratios vs. Number of Iterations 139 /A. = 6 /U = 2 CN rH yU = 2 jX = 1 / A = 3 /A = 6 JU. = 2 CN rH i JU= 2 jU = 1 / A = 3 / A = 6 / A = 2 CN rH }k = 2 / A = 1 yU= 3 1 r * W = 600 kips W = 600 kips W = 600 kips Target Damage Ratio 50 ' 50 E = 3,600 k s i Size My=365 k - f t 8 6 6 Columns 21"x21" 16,000 in Beams 20"x36" 40,000 in' 368 1212 751 487 1299 Natural Periods 517 1389 761 Mode 1 1.6307 sec. Mode 2 0.4633 sec. 525 1349 Mode 3 0.2375 sec. 1171 2513 967 Y i e l d Moments Fi g . 3.6 2-Bay, 3-Story Frame B - Member Properties and Y i e l d Moments Period 3 Period 2 -Period 2 iod I _ 1 | ! ! 1 I 1 I 1 1 1 • 1 ' 1 1 1  5 10 15 20 Number of Iterations 2-Bay, 3-Story Frame B - Plot of Periods vs. Number of Iterations 141 3.48 1.41 3.35 5.47 1.34 1.76 2.50 6.17 1.45 2.03 2.13 1.09 After 4 i t e r a t i o n s 3.48 1.41 2.17 3.63 1.34 5. 47 1. 76 3.45 2.11 1.45 6 . ]7 2. 03 3.20 1.94 1.09 < 2.17 3.45 3.20 After 12 i t e r a t i o n s 4.28 1.42 5.98 1.99 3.63 6.07 1.47 1.92 1.84 6.01 1.15 1.98 1.95 1.00 3.49 3.43 2.97 2.00 6.00 1.01 2.00 2.00 6.00 1.00 . 2.00 2.00 1.00 3.02 3.01 3.00 After 20 i t e r a t i o n s After 200 i t e r a t i o n s F i g . 3.8 2-Bay, 3-Story Frame B - Damage Ratios Computed at the End of 4, 12, 20, and 200 Iterations 143 M =115 k - f t 60 Y F i g . 4.1 140 90 19 5 200 210 200 30 ' Exterior Columns Interior Columns Beams 1st Story 2nd Story 110 600 30 ' Size 21" x 21" 18" x 18" 18" x 21" 15" x 18" W = 100 kips W = 120 kips E = 4,320 k s i 8,100 i n 4,375 i n ' 4,630 i n ' 2,430 in' 2-Bay, 2-Story Frame - Member Properties and Yie l d Moments 2.3 4.2 0.61 2.3 Modified Substitute Structure Analysis Average of 4 Nonlinear Dynamic Analysis F i g . 4.2 2-Bay, 2-Story Frame - Damage Ratios 144 0 . 59 3.3 0 .74 1.6 3.0 2.0 3.7 0.82 0.94 0.56 0.78 0.94 3.6 1.7 4.6 2.1 4.4 2.5 1.1 6.5 3.7 -to s^v •** Taft S69E 0 .56 2.0 •«* *« *^  Taft N21E F i g . 4.3 2-Bay, 2-Story Frame - Damage Ratios for Individual Earthquakes 145 M =60 k - f t 90 250 2 225 110 385 170 780 30 205 305 410 240 530 20 1125 240 430 480 600 30 W = 180 kips 355 W = 200 kips 325 W = 240 kips 895 E = 3,600 k s i 1 Exterior Columns 2 3 1 Interior Columns 2 3 1 Beams 2 3 (a) I for the r i i s 4,375 i n 4 Size I 24" X 24" 13,800 . 4 in 24" X 24" 13,800 . 4 in 21" X 21" 8,100 . 4 in 21" X 21" 8,100 . 4 in 21" X 21" 8,100 . 4 in 18" X 18" 4,375 . 4 in 18" X 24" 6,910 . 4 i n 18" X 21" 4,630 . 4 in 18" X 18" 2,920 . 4 in ght-hand-side column Fi g . 4.4 3-Bay, 3-Story Frame - Member Properties and Y i e l d Moments 146 5.2 4.5 1.1 1.0 4.9 1.0 2.1 1.1 2.0 0.70 3.6 0 . 86 3.3 1.5 1.0 1.1 1.1 1.1 Modified Substitute Structure  Analysis 0.62 4 . 3 0.94 2.4 1.2 2.1 0.64 0.68 1.3 0.80 3.8 0.91 3.7 1.6 1.2 1.3 1.2 1.2 * •< Average of 4 Nonlinear Dynamic Analysis F i g . 4.5 3-Bay, 3-Story Frame - Damage Ratios 147 6.4 6.2 1.5 5.5 5.3 1.3 0.54 0.86 1.8 0.68 0 .56 0.94 1.2 5.7 3.3 2.8 4.5 2.5 2.2 0.81 0.95 2.1 0.80 0.99 1.0 1.8 5.0 5.2 1.6 4.1 4.2 1.3 1.9 1.7 1.8 1.9 1.4 1.3 1.4 E l Centro EW E l Centro NS 0.67 0.78 1.4 4.6 0 . 79 3.7 0.73 3.1 1.0 4.4 1.0 2.0 0.84 3.1 0.98 0.97 Taft S69E 0.94 1.8 1.2 0.97 0.95 0.64 0.61 0.99 4.0 0.60 3.2 0.66 2.4 0.91 3.7 0.93 1.7 0.84 2.4 0.89 0.84 0 .84 1.5 1.0 0.78 0.85 0.55 0.53 0.84 Taft N21E Fi g . 4.6 3-Bay, 3-Story Frame - Damage Ratios for Individual Earthquakes 148 M - 110 k - f t V  130 235 180 363 235 458 262 518 296 487 422 130 180 235 262 296 422 Size (in) I (in 4) 1 21x30 24 ,000 2 21x30 24,000 3 21x27 19,200 4 21x27 19,200 5 21x21 9 ,400 6 21x21 9 ,400 1 15x36 33,700 2 15x36 33,700 3 15x36 33,700 4 15x36 33,700 5 15x36 33,700 6 15x31 11,600 E = 3,600 k s i 35 1-Bay, 6-Story Frame - Member Properties and Yi e l d Moments 149 6.8 5.7 0.85 1.1 9.5 7.5 16 .6 3.7 1.0 5.0 0.97 5.5 0.94 3.8 2.9 7.2 1.0 3.9 6.6 6.2 1.5 4.7 2.5 6.9 Modified Substitute Structure Analysis Average of 4 Nonlinear Dynamic Analysis F i g . 4.8 1-Bay, 6-Story Frame - Damage Ratios 150 8.4 6.1 0.96 6.3 10.8 7.1 1.3 4.3 8.1 4.7 8.1 10.4 .5 6.3 7.4 8.2 5.2 5.2 3.3 3.7 2.2 1.7 14.4 3.7 E l Centro EW E l Centro NS 7.2 1.1 9.5 3.1 6.8 7.0 3.8 6.5 3.0 5.1 3.4 6.7 Taft S69E 1.1 0.84 1.7 1.1 1.3 1.7 1.3 6.6 3.2 7.9 5.5 2.8 Taft N21E Fi g . 4.9 1-Bay, 6-Story Frame - Damage Ratios for Individual Earthquakes 151 My=153 k - f t 228 153 r H r H 183 19 8 309 237 330 267 330 276 416 228 1020 447 317 471 376 550 423 550 438 661 356 1020 447 198 471 237 550 267 550 276 661 228 1020 183 309 330 330 416 1020 24 24 24 • Size I Columns 24" x 24" 13,824 in' Beams 18" x 30" 13,500 i n 4 E = 3,600 k s i Floor weight i s 200 kips at a l l leve l s F i g . 4.10 3-Bay, 6-Story frame - Member Properties and Yie l d Moments 152 3.5 2.0 0.84 3.9 0.83 2.3 , 0.81 4.2 0.83 2.5 0 . 78 4.3 s 0 .80 2.6 0 .80 4.5 1 0.83 2.7 . 0.77 4.5 0.79 2.7 0.72 1.7 i 0 . 78 0.95 , 0.86 2.7 0.70 1.6 1.1 3.6 . 0.91 2.2 1.0 4.3 0.88 2.4 1.1 4.7 •• s 0.93 3.0 i 0.70 5.2 0.75 3.3 0.81 1 0.88 Modified Substitute Structure Analysis Average of 4 Nonlinear Dynamic Analyses F i g . 4.11 3-Bay, 6-Story Frame - Damage Ratios 153 2.0 1.0 0 .89 3.2 i 0.81 2.0 , 1.5 4.7 0.99 3.0 , 1.4 5.7 s 0.99 3.7 1.5 6.0 1.0 3.8 , 0.86 5.9 0.85 3.8 f 0.86 E l Cenl J 0.93 bro EW 2.8 1.6 , 0.90 3.6 0.81 2.3 1 1.5 4.6 ) 1.1 2.9 1.0 4.9 0.91 3.1 0.94 4.8 0.89 3.0 0.71 5.5 0.76 3.5 , 0.91 0.98 Taft S69E 1.0 0.59 . 0.82 2.0 ) ) 0.58 1.1 0.70 2.6 s 0.76 1.5 0 . 86 3.5 > 0.83 2.2 0.96 4.5 5 0.91 2.8 , 0.59 5.0 > 0 .69 3.2 0.76 E l Cen 0.82 tro NS 1.0 0.61 . 0.82 1.9 0.58 1.1 < 0.69 2.5 ) 0 .77 1.5 0.77 3.1 i 0.79 1.9 0.88 3.7 i 0 .89 2.3 t 0.63 4.4 i 0.68 2.8 0 .71 l 0.78 Taft N21E F i g . 4.12 3-Bay, 6-Story Frame - Damage Ratios for Individual Earthquakes 154 2.5 -Frequency i n Hertz Response Acceleration for ft _ 8 Response Acceleration for p=0.02 6 + 100p> F i g . 5.1 Smoothed Response Spectrum - Design Spectrum B 155 12.4 8.0 0.97 11.0 0.84 6.7 , 0.96 10.3 > 0.91 6.4 1.0 9.9 0.90 6.1 0.92 9.5 0.91 5.8 3.1 6.3 > 2.3 3.9 0.90 4.2 i 0.96 2.5 0.94 4.6 s 0.85 3.0 { 5.6 7.5 \ 3.7 5.0 1.9 7.8 —4 1.3 5.1 f 3.6 8.0 J2.4 5.2 2.7 7.1 1.9 4.6 f 1.1 i 1.2 Modified Substitute Structure Analysis ( Spectrum B ) W = 200 k i p s / f l o o r Nonlinear Dynamic Analysis ( 8244 Orion 1971 ) Fi g . 5.2 3-Bay, 6-Story Frame A - Damage Ratios F i g . 5.3 3-Bay, 6-Story Frame A - Plot of Damage Ratios for Beams i n the Exterior Bay 157 2.1 0.85 3.1 0.91 4.1 0.87 4.6 0.82 4.8 0.84 4.7 0.75 1.1 0.82 1.9 0.91 2.4 0 .88 2.8 0.87 2.9 0.85 2.9 0 .82 Modified Substitute Structure Analysis ( Spectrum B ) W = 130 k i p s / f l o o r 1.7 0.90 2.9 1.5 4.6 1.2 5.4 0.92 5.6 0.78 5.6 0.80 0.84 0.73 1, 1.1 2.9 0.95 3.4 0 .89 3.5 0.80 3.6 0.81 Nonlinear Dynamic Analysis ( 8244 Orion 1971 ) F i g . 5.4 3-Bay, 6-Story Frame B - Damage Ratios 158 * — M o d i f i e d Subst. Str. M e t h o d ( S p e c t r u m B ) Nonlinear A n a l y s i s 18244 Orion) 3 N \ \ \\ \\ \ \ \ \ i \ • \ \ V • 1 i i 1 i (i i 1 1 1 I O.O 2.0 4.0 6.0 e R a t i o s 3-Bay, 6-Story Frame B - Plot of Damage Ratios for Beams i n the Exterior Bay 1.3 159 0.71 0.84 2.1 i 0.67 1.2 i 0.75 2.8 J 0.83 1.7 ( 0 .75 3.2 1 0.79 0.76 3.6 0.83 2.1 0.67 3.8 l 0.74 2.3 { 0 .64 0.87 i 0.71 0 .52 0.73 1.8 \ 0.51 1.0 { 0.78 3.0 \0.83 1.8 f 0 .94 3.9 \ 0.83 2.4 0.90 4.6 \ 0.87 2.9 0.63 5.0 i 0 .70 3-2 , 0.75 •TO ' 0.82 M o d i f i e d S u b s t i t u t e S t r u c t u r e A n a l y s i s ( S p e c t r u m A ) W = 130 k i p s / f l o o r A v e r a g e o f 4 N o n l i n e a r Dynamic A n a l y s e s F i g . 5.6 3-Bay, 6 - S t o r y Frame B - Damage R a t i o s 160 0.85 0.52 , 0.72 1.8 0.50 1.0 0.84 3.3 > 0.87 2.0 1.2 4.7 > 0.93 2.9 1 0.95 5.4 0.91 3.5 4 0.72 5.8 > 0.77 3.7 0.83 •*> i 1 0.90 E l Centro EW 0.69 1.7 <, 0.49 0.97 t 0.77 2.8 \ 0.82 1.7 { 0.85 3.6 t 0.83 2.2 ( 0 .89 4.5 1 0.87 2.8 1 0 .59 5.0 i 0.68 3.2 0.75 0.82 0.94 0.56 , 0 .79 2.0 0.55 1.1 0.77 3.0 j 0 .83 1.8 , 0.87 3.7 ) 0 .86 2.3 . 0.91 4.3 0.87 2.7 0.60 4.9 0.67 3.1 0.74 E l Cen s 0.81 v* tro NS 0.85 0.51 , 0 . 70 1.7 0 .49 0.96 * 0.74 2.7 0.78 1.6 0.89 3.6 i 0.87 2.2 0 .84 4.2 > 0.83 2.6 0.60 4.4 0.69 2.8 0.68 > 0 .74 Taft S69E Taft N21E F i g . 5.7 3-Bay, 6-Story Frame B - Damage Ratios for Individual Earthquakes Fig. 5.8 E l Centro EW Spectrum and Design Spectrum A I I I 1 I I 1 I I I ! I 1 I 1 I • 1 1 1 L O.O 0.5 1.0 1.5 2.0 Period i n S e c Response Acceleration in CJ o in o —i— — i — to © o o o - o n> —s O o b 4 H-vQ a -3 fD Cu cn hh H- rt 3 c n c n c n v o T j M ( D o c n rj fD C O 3 rt > S Z9T 16 3 t 0.85 2.7 a r - ^ u . S 0.76 1.6 s 0.89 3.7 0.91 2 ' 3 t 0.88 4.3 i 0.89 2 7 0.82 4.5 0.88 2.8 1 0.81 4.5 J 0.85 2.8 ) 0.74 0.81 0.85 0.52 , 0 .72 1.8 0.50 1.0 , 0.84 3.3 > 0.87 2.0 . 1.2 4.7 1 0.93 2.9 i 0.95 5.4 J 0.91 3.5 0.72 5.8 1 0.64 3.7 { 0.83 } 0.90 MSSA^1) (El Centro EW Spectrum) NDA^J (El Centro EW Motion) (2) 1.3 0.71 0.84 2.1 > 0.67 1.2 0.75 2.8 S 0.83 1.7 f 0.75 3.2 \ 0.75 1.9 0.76 3.6 0.83 2.1 0.67 3.8 i 0.74 2.3 f 0.64 1 0.71 W = 130 k i p s / f l o o r (1) Modified Substitute Structure Analysis (2) Nonlinear Dynamic Analysis MSSA(1> (Smooth Spectrum) F i g . 5.10 3-Bay, 6-Story Frame B - Damage Ratios 16 4 1.5 0.79 , 0.85 2.3 0.71 1.4 ( 0.78 3.0 ) 0.85 1.8 i 0.77 3.4 s 0.81 2.1 0.78 3.7 i 0.84 2.3 0.70 3.8 > 0 .76 2.4 f 0.67 \ 0.74 MSSA ( 1 ) (Taft S69E Spectrum) 1.3 0.71 0.84 2.1 0.75 2.8 0.75 3.2 0.76 3.6 0.67 3.8 0.64 (1) 0.67 1.2 0.83 1.7 0.83 1.9 0.83 2.1 0.74 2.3 0.71 MSSAV (Smooth Spectrum) 0.85 0.70 1.7 0 .74 2.7 0 .89 3.6 0 .84 4.2 0 .60 4.4 0.68 (2) 0.51 0 .49 0.96 0.78 1.6 0.87 2.2 0.83 2.6 0.69 2.8 0 .74 NDA '(Taft S69E Motion) W = 130 k i p s / f l o o r (1) Modified Substitute Structure Analysis (2) Nonlinear Dynamic Analy F i g . 5.11 3-Bay, 6-Story Frame B - Damage Ratios 165 7.7 5.1 0.97 7.4 0.94 4.8 { 1.0 7.3 I 0.97 4.7 < 1.0 7.1 0.93 4.6 t 1.0 7.1 0.97 4 - 5 1 2.2 5.8 1.7-3.8 0.94 > 1.0 MSSA^ (El Centro EW Spectrum) 3.5 2.0 0.84 3.9 0.81 4.2 0.78 4.3 0.80 4.5 0.77 4.5 0.72 -5 0.83 2.3 0.83 2.5 0.80 2.6 0.83 2.7 0.79 2.7 0.78 (1) MSSA' (Smooth Spectrum) 2.0 0.89 3.2 1.5 4.7 1.4 5.7 1.5 6.0 0.86 5.9 0.86 (2) 1.0 0.81 2.0 0.99 3.0 0.99 3.7 1.0 3.8 0.85 3.8 0.93 NDA ' ( E l Centro EW Motion) W = 200 ki p s / f l o o r (1) Modified Substitute Structure Analysis (2) Nonlinear Dynamic Analysis F i g . 5.12 3-Bay, 6-Story Frame A - Damage Ratios 166 0.90 4.7 5.0 0.97 5.4 0.95 5.6 0.87 5.6 1.0 5.3 0.86 2.9 0 .88 3.1 0.95 3.4 0.92 3.6 0.89 3.6 0.96 3.4 0.93 (1) MSSAV ; ( T a f t S69E Spectrum) 2.8 1.6 0.90 3.6 > 0.81 2.3 1.5 4.6 i 1.1 2.9 1.0 4.9 l 0.91 3.1 0.94 4.8 > 0 .89 3.0 , 0.71 5.5 i 0.76 3.5 0.91 j 0.98 3.5 0.84 3.9 0.81 4.2 0.78 4.3 0.80 4.5 0.77 4.5 0.72 (1) 2.0 0.83 2.3 0.83 2.5 0.80 2.6 0.83 2.7 0.79 2.7 0.78 MSSA (Smooth Spectrum) W = 200 k i p s / f l o o r (1) Modified Substitute Structure Analysis (2) Nonlinear Dynamic Analysis Fig 5.13 3-Bay, 6-Story Frame A - Damage Ratios 167 BIBLIOGRAPHY 1. Associate Committee on the National Building Code 19 75, Supplement No. 4 to the National Building Code of Canada, National Research Council of Canada, Ottawa, Ontario, 1975. 2. Applied Technology Council, "Tentative Provisions for the Development of Seismic Regulations for Buildings", ATC 3-06 NSF 78-8, Applied Technology Council, Palo Alto, C a l i f o r n i a , June, 1978. 3. Okada, T. and Bresler, B., "Strength and D u c t i l i t y Evaluation of E x i s t i n g Low-Rise Reinforced Concrete Buildings - Screening Method", EERC 76-1, Earthquake Engineering Research Center, University of C a l i f o r n i a , Berkeley, C a l i f o r n i a , February, 1976. 4. Freeman, S. A., N i c o l e t t i , J. P., and T y r r e l l , J. V., "Evaluation of E x i s t i n g Buildings for Seismic Risk — A Case Study of Puget Sound Naval Shipyard, Bremerton, Washington", Proceedings of the U. S. National Conference on Earthquake Engineering, Ann Arbor, I l l i n o i s , June, 1975, pp. 113-122. 5. Shibata, A. and Sozen, M. A., "Substitute-Structure Method for Seismic Design i n R/C", Journal of the Structural Division, ASCE, Vol. 10 2, No. STl, January, 19 76, pp.1-18. 6. Clough, R. W. and Penzien, J., Dynamics of Structures, McGraw-Hill, New York, 1975, pp. 545-610. 7. Hudson, D. E., "Some Problems i n the Application of Spectrum Technique to Strong-Motion Earthquake Analysis", B u l l e t i n 16 8 of the Seismological Society of America, Vol. 52, No. 2, A p r i l , 1962, pp. 417-430. 8. Gulkan, P. and Sozen, M. A., "In e l a s t i c Response of Reinforced Concrete Structures to Earthquake Motions", Journal of the American Concrete I n s t i t u t e , Vol. 71, No. 12, December, 1974, pp. 604-610. 9. Takeda, T., Sozen, M. A., and Nielsen, N. N., "Reinforced Concrete Response to Simulated Earthquakes", Journal of Structural Division, ASCE, Vol. 96, No. ST12, December, 1970, pp. 2557-2573. 10. Jenning, P. C , "Equivalent Viscous Damping for Yielding Structures", Journal of the Engineering Mechanics D i v i s i o n , ASCE, Vol. 94, No. EMl, February, 1968, pp. 103-116. 11. Blume, J. A., Newmark, N. M., and Corning, L. H., Design of Multistory Reinforced Concrete Buildings for  Earthquake Motions, Portland Cement Association, Chicago, 1961, pp. 73-86. 12. Otani, S., "SAKE. A Computer Program for In e l a s t i c Response of R/C Frames to Earthquakes", Structural Research Series No. 413, C i v i l Engineering Studies, University of I l l i n o i s , Urbana, I l l i n o i s , November, 1974. 13. Otani, S. and Sozen, M. A., "Behaviour of Multistory Reinforced Concrete Frames during Earthquakes", Structural Research Series No. 39 2, C i v i l Engineering Studies, University of I l l i n o i s , Urbana, I l l i n o i s , November, 1972. 14. Nigam, N. C. and Jennings, P. C , " D i g i t a l Calculation of Response Spectra from Strong Motion Earthquake Records", Earthquake Engineering Research Laboratory, C a l i f o r n i a I n s t i t u t e of Technology, Pasadena, C a l i f o r n i a , June, 1968. 169 Appendix A Modification of Damage Ratio - Strain Hardening Case Consider the b i l i n e a r moment-rotation curve shown i n Fi g . A . l . Let k = i n i t i a l s t i f f n e s s , s = r a t i o of s t i f f n e s s a f t e r y i e l d to i n i t i a l s t i f f n e s s , = damage r a t i o used i n n th i t e r a t i o n , j^n+-^ = damage r a t i o to be used i n n+1 th i t e r a t i o n , M = y i e l d moment, y Mn = computed moment i n n th i t e r a t i o n , <p - y i e l d rotation, and c£n = rotation corresponding to M n on l i n e OC. M' and 6 ' are the moment and rotation at B, which i s an i n t e r -n n section of li n e s OC and AC'. Assume that the damage r a t i o , j x ^ , used i n the n th i t e r a t i o n was too small; point C i s o f f the b i l i n e a r curve. Therefore, the damage r a t i o must be increased i n the next i t e r a t i o n . I t i s assumed that the rotation, 0 n , i s correct and that the slope of l i n e OC i s used as the new s t i f f n e s s . The new damage r a t i o , JJ[n+-^r i s derived i n a following manner. k M Slope of l i n e OC*: _ n+1 (A.l) M / M Slope of l i n e OC: n _ . I n_ (A.2) From (A.2), <pn = yWn { A ' 3 ) 170 S u b s t i t u t e e q u a t i o n (A.3) i n t o ( A . l ) . M n + l A1 n+l V k k ' Mn+1 ^ M n / Z^n+l /*n M n \ M n + 1 (A.4) S l o v e f o r M , , i n terms o f M and U , n + l y rn M , = M 1 + s • k • ( d> - d> • ) n + l n 'n " n M = M 1 + s-k n n M' n k/yW n k / / A n = M' + S M U n n / ln - s M' n/^n M . , = M' ( 1 n + l n - s • U ) + M1 • s • U Pn n f*n (A. 5) Now s o l v e f o r M 1, n M 1 = M + s- k • ( <£> 1 - - i ) n y r n " y M' M + s • k y n k / / X n M \ y M ( l - S ) + M ' - S y W y n / n 171 1 - s M1 n = M (A.6) V 1 " s M n Substitute equation (A.6) into (A.5) 1 - s M , , = M n+l y 1 - s ( 1 " s/*n ) + S ' /VMn Mn+1 = My ( 1 - s ) + s.^n-Mn (A.7) Substitute equation (A.7) into (A.4). / n n n+l (3.4) My ( 1 - s ) + s.^ n M n 172 Fi g . A . l Moment-Rotation Curve 173 Appendix B Computer Program The FORTRAN IV program for the modified substitute structure method i s l i s t e d i n the following i n th i s appendix. The subroutine, MOD3, i s written for an e l a s t o - p l a s t i c case. Important variables are explained i n each subroutine. 174 DIMENSION KL (50) , KG (50) , ABEA (50) ,CBMOM (50) ,BMCAP(100) , 1 DA L I B AT (50) ,ND(3,50) , HP (6,50) ,XM (50) ,¥M (50) ,DM (50) ,S (500) 2F(100) ,TITLE(20) ,SDAMP(5Q) ,AV (50) DIMENSION 11 (300) DIMENSION AMASS (50) ,EVAL(2Q) ,EVEC(50,20) C C SAMPLE MAIN PROGRAM C IUNIT=7 CALL CONTBL (TITLE,N1J, NBH, E, G, 7) CALL SETUP (NRJ,NRtt,E,G,XM,YM,DM,ND,NP,ABEA,CBMOM,DAMEAT, AV,KL,KG, 1NU,NB,SDAHP,BHCAP,IUNIT,0) NMODES=10 ICOUNT=0 AMAX=.5 IFLAG=0 CALL MASS (NU,ND,AMASS,IUNIT,NBJ) IUNIT=6 IMAX=200 IM=IMAX-1 1=0 BETA=0. EBRQB=1.E-3 10 CONTINUE 1=1+ 1 CALL BUILD (NU,NB,XM,YM,DH,NP,AREA,CEMOM,A7,E,G,DAMRAT,KL , KG, NRM,S, 1500) CALL EIGEN (NU, NB ,S,500,AMASS,EVAL,EVEC,NMODES,IUNIT) IF (I .GE. .10) BETA=. 95 CALL MOD3 (ICOONT,2,NBJ,NRM,NU,NB,NMODES,S,500,ND,NP,XH,Y 3,DS,AREA, 1CBMOM,DAMBAT,KL,KG,SDAMP,BMCAP,E,AMASS,EVEC,EVAL,AMAX, IS IGN, 21 UNIT, BETA,ERROR , 1) I1(I)=ISIGN WRITE (8,201) (DAMRAT (II) ,11=1, NBM) 201 FORM AT (• »,15F8. 3) IF (IPLAG. EQ. 1 . AND. I. EQ.IMAX) GO TO 40 IF(IFLAG.EQ. 1) GO TO 20 IF(I.EQ. 1 .AND. ISIGN.EQ.O) GO TO 46 IF(I.EQ.IM .OB. ISIGN.EQ.O) GO TO 35 GO TO 10 35 CONTINUE IFLAG=1 IUNIT=7 GO TO 10 20 CONTINUE WRITE(IUNIT,30) I 30 FOBS AT(*- *,5X,* NO. OF ITERATIONS =»,I5///) GO TO 50 40 CONTINUE WRITE (IUNIT,45) I 45 FOBMAT(*-•,5X,'DOES NOT CONVERGE AFTER*,15,* ITERATION 175 s v / / ) GO TO 50 46 CONTINUE ICOUNT=0 IFLAG=1 IUNIT=7 WHITE (IBMIT,48) 18 FORK AT ('-* ,5X, 'MEMBERS DO NOT YIELD •///) GO TO 10 50 CONTINUE WRITE(IUNIT,60) BETA,ERROR 60 FORMAT('-* ,5X, 'BETA =»,F5.3,///5X,•ERROR = «,F8.6///) JJJ=I-1 WRITE (7,200) (11 CU) ,IJ=1*JJJ) 200 FORM AT (• »,2016) STOP END 176 SUBROUTINE CONTRL(TITLE, NBJ,NRM, E,G,IUNIT) DIMENSION TITLE(20) C C BEAD IN TITLE C BEAD (5, 1) {TITLE (I) ,1=1,2 0) C C BEAD IN NEJ,NBM,E,G C NBJ = NUMBEB OF JOINTS C NRM = NUMBEB OF MEMBERS C E ELASTIC MODULUS IN KSI C G SHEAfi MODULUS IN KSI C BEAD (5,2) NBJ, NRM, E, G HBITE (IUNIT,3) (TITLE(I) ,1=1,20) WRITE (IUNIT,4) E, G WRITE (I0NIT,5) WBITE (IUNIT,6) NBJ, NRM RETURN 1 FOBMAT(20A4) 2 FORMAT(2I5,2F10. 0) 3 FOBMAT(» 1 * ,20'A4) 4 FOBHAT(»-»,5X,«E =» , F8.3, 5X , • G =* ,F8. 3) 5 FOBM AT 10 {»*»)) 6 FOBM AT (*-* , ' NO* OF JOINTS*, • = » , 15, 10X, • NO. OF MEMBERS = »,I5) END 177 SUBROUTINE SETUP (NRJ,NRM,£,G,XM,YM,DM,ND,NP,AREA,CRMOM,D AMRAT,A?, 1 KL,KG,NU,NB,SDAMP,B MCAP,IUNIT,IFL AG) C C SET UP TBE FRAME DATA FOR MODIFIED SUBSTITUTE C STRUCTURE METHOD C DIMENSION KL (NRM) , KG (NRM), AREA (NRM) , CRMOM (NRM) , SDAMP (NRM) , 1 DAMRAT (NRM) , AV (NRM) , ND(3,NRJ), NP(6,NRM), XH (NRM), 2 YM (NRM), DM(NRM) DIMENSION X(100), Y(100), JNL(IOO), JNG(100), BMCAP (NRM) C C JN JOINT NUMBER C ND(1,JN) JOINT DEGREE OF FREEDOM IN X-•DIRECTION C ND(2,JN) = JOINT DEGREE OF FREEDOM IN Y-•DIRECTION C ND (3, JN) JOINT DEGREE OF FREEDOM IN ROTATION c X(JN) = X-COORDINATE OF JN IN FEET c Y(JN) Y-COOEDINATE OF JN IN FEET c MN = MEMBER NUMBER c JNL(MN) = LESSER JOINT NUMBER c JNG (MN) GREATER JOINT NUMBER c KL (MN) = MEMBER TYPE AT LESSER JOINT c KG(MN) •= MEMBER TYPE AT GREATER JOINT c AREA (MN) = AREA IN IN**2 c CRMOM (MN) = MOMENT OF INERTIA IN IN**4 c DAMRAT (MN) DAMAGE RATIO FOR MN c AV(MN) - SHEAR AREA IN IN**2 c BMCAP (MN) = YIELD MOMENT IN K-FT c XM (MN) MEMBER LENGTH IN X-DIRECTION c YM (MN) - MEMBER LENGTH IN Y-DIRECTION c DM(MN) MEMBER LENGTH c SDAMP (MN) SUBSTITUTE DAMPING RATIO FOR MN c NP(I,MN) = MEMBER DEGREE OF FREEDOM c NU NUMBER OF UNKNOWNS c c NB = HALF BANDWIDTH WRITE (IUNIT, 1) WRITE (IUNIT, 2) C C READ IN JOINT DATA AND COMPUTE NO- OF DEGREES OF FREEDOM C NU=1 C DO 50 1=1,NRJ READ (5,3) JN, ND(1,I), HD(2,I), ND(3,I), X(I), Y(I ) C DO 40 -8=1,3 IF(ND(K,I)-1) 30,10,20 10 ND(K,I)=NU NU=NU+1 GO TO 40 20 JNN=ND(K,I) 178 ND (K,I)=ND(K, JNN) GO TO 40 30 CONTINUE ND (K,I) =0 40 CONTINUE C C PHINT JOINT DATA C WRITE (IUNIT,4) I, X (I) , 1(1)-, ND(1,I), ND(2,I), ND (3,1) 50 CONTINUE C NU=NU-1 WRITE (I0NIT,5) WRITE (IUNIT,6) WRITE (IUNIT,7) C C READ IN MEMBER DATA AND COMPUTE THE HALF BANDWIDTH C NB=0 C DO 180 1=1,NRM READ (5,8) MN, JNL(I) , JNG (I) , KL(I), KG(I), AREA (I ) , 1 CBMOH(I), DAMBAT (I) , AV (I) , BMCAP(I) IF (IFLAG. NE. 1) GO TO 70 IF(DAHRAT (I).NE.Q.) GO TO 60 DAMR AT (I) = 1-60 GO TO 80 70 DAMBAT (I) = 1. 80 CONTINUE JX.=JNL(I) JG=JNG(I) XM (I)=X(JG)-X(JL) YH(I)=Y(JG)-Y<JL) DM (I)=SQBT ( (XM (I) ) **2 + (YM (I) ) **2) DAMAGE=DAMBAT(I) ROOT=SQBT(DAMAGE) SDAMP (I) =0. 0 2*0- 2* ( 1 1 . /BOOT) NP(1,I)=ND(1,JL) NP(2,I)=ND(2,JL) HP(3,I)=ND(3,JL) NP(4,I)=ND(1,JG) NP(5,I)=ND(2,JG) NP{6/I)=HD(3,JG) MAX=0 C DO 110 K=1,6 IF (NP (K ,1) -MAX) 100,100,90 90 MAX=NP(K,I) 100 CONTINUE 110 CONTINUE C MIN=1Q0Q C DO 150 K=1,6 179 IF(NP(K,I)) 140,140,120 120 IF (NP (K,I)-HIN) 130,140, 140 130 MIN=NP(K,I) 140 CONTINUE 150 CONTINUE C NBB=MAX-MIN•1 IF (NBB-NB) 170,170,160 160 NB=NBB 170 CONTINUE C C PEINT MEMBER DATA C WRITE (IUNIT,9) I, JNL(I), JNG(I), DM (I), XM(I), YM CD , 1 NP(1,I), NP(2,I), NP(3,I), NP(4,I), NP(5,I), NP(6,I) , 2 AREA (I), CRMOH(I), DAMBAT (I) , AV(I) , BMCAP(I) , KL{I), 3 KG (I) C CHANGE THE LENGTHS FBOM FEET TO INCHES XM (I)=XH(I) *12. YM (I)=YM(I) *12. DH(I)=DM(I) *12. 180 CONTINUE C C PRINT THE NO. OF DEGREES OF FREEDOM AND THE HALF BANDWID TH C WRITE (IUNIT,11) NU WRITE (IUNIT,12) NB RETURN 1 FORM AT (*~ * , * JOINT DATA') 2 FOBMAT(* 0 *,7X, ,JN ,,3X,'X(FEET)»,3X,» Y(FEET)•,4X,» NDX«,2X ,1NDY', 1 2X, ,NDB«) 3 FORMAT(4I5,2F10.5) 4 FOBMAT (* »,5X,I4,2F10.3,2X,3I5) 5 FOBMAT{»-•,«MEMBER DATA') 6 FORMAT( #0» ,7X,»MN JNL JNG LENGTH XM (FT) YH (FT) NP1 NP 2 NP3 NP4 1HP5 NP6 AREA I (CRACKED) DAMAGE A? * ,4X, * MOMENT *, 2 4X,»KL«,3X,»KG») 7 FORM AT (* *,19X,» (FEET) • #«1X^« (SQ.IN) *,2X,« (IN**4) •,6X,«R ATIO», 1 2X, f(SQ.IN)« CAPACITY*) 8 FOBMAT(5I5,5F10.5) 9 FOBS AT (* »,5X,3I4,3F8.2,6I4,F8. 1,F12. 1 ,2F8. 3 ,F10. 2, 215) 11 FOBM AT(*- *,•NO.OF DEGREES OF FREEDOM OF STRUCTURE =«,I5) 12 FOBMAT( ,0 i,»HALF BANDWIDTH OF STIFFNESS MATRIX =',I5) END 180 SUBROUTINE MASS(NU,ND,AMASS,IUNIT,NRJ) C C THIS SUBROUTINE SETS UP THE MASS MATRIX C C ND(J,I)=DEGBEES OF FREEDOM OF I TH JOINT C WTX,WTY,8TR=X-MASS,Y-MASS,ROT.MASS IN FORCE UNITS(KIPS 0 R IN-KIPS) C AMASS (I) =MASS MATRIX C NMASS=NO.OF MASS POINTS C C MASSES ARE LUMPED AT NODES.. THE MASS MATRIX IS DIAGONAL IZED. C DIMENSION ND(3,NRJ), AMASS (NU) C C READ IN NO. OF NODES WITH MASS C READ (5,1) NMASS WRITE (IDNIT,2) WRITE (IUNIT,3) NMASS WRITE (IUNIT,4) WRITE (ION IT, 5) C C ZERO MASS MATRIX C DO 10 1=1,NU AMASS (I)=0. 10 CONTINUE C C READ IN X-HASS,Y-MASS AND SOT. MASS (IN UNITS OF WEIGHT ) C DO 50 1=1,NMASS READ (5,6) JN, WTX, HTY, WTR WRITE (IUNIT,7) JN, WTX, WTY, WTR N1=ND(1, JN) N2=ND(2,JN) N3=ND(3,JN) IF(N1.EQ. 0) GO TO 20 AMASS (N 1) =AH ASS (N 1) + (WTX/386. 4) 20 IF(N2.EQ. 0) GO TO 30 AMASS (N2) =A8ASS (N2) * (WTY/386.4) 30 IF(N3.EQ. 0) GO TO 40 AMASS (N3) =AMASS (H3) > (WTR/386.4) 40 CONTINUE 50 CONTINUE C RETURN 1 FORM AT (15) 2 FORMAT (///110 («*»)) 3 FORM AT (' — *,• NO. OF NODES WITH MASS*,* = ',I5) 4 FORMAT(,0«,7X,'JN*,3X,,X-MASS*,'*X, "Y-MASS* ,2X, ' ROT. MASS ' ) 5 FORMAT (• «,12X,' (KIPS) *,4X,» (KIPS) • ,2X,» (IN-KIPS) ') 6 FORM AT (15, 3F 10.0) 7 FORM AT (• ' ,5X,I4,3F10.3) 181 END 182 SUBROUTINE BUILD (NU,NB,XM,YM,DM,NP, AREA,CRMOM, AV,E,G,DAM BAT, 1 KL,KG,NBM,S,IDIM) C C THIS SUBROUTINE CALCULATES THE STIFFNESS MATBIX OF EACH C MEMBER AND ADDS IT INTO THE STRUCTURE STIFFNESS MATRIX. C FLEXURAL STIFFNESSES OF MEMBERS ABE MODIFIED C ACCORDING TO THE DAMAGE RATIOS. C THE FINAL STIFFNESS MATBIX S IS RETURNED. C C DAMRAT (I) = DAMAGE RATIO FOR I TH MEMBER C S(I) = STRUCTURE STIFFNESS MATRIX C SM(I) = MEMBER STIFFNESS MATRIX C ) , NRM) DIMENSION XM (NRM) , YM (NRM) , DM (NRM) , NP(6,NRM), ABEA (NRM 1 CRMOM(NRM), AV(NRM) , DAMRAT (NRM) , KL (NRM), KG{ DIMENSION S(IDIM), SM(21) C C ZERO STRUCTURE STIFFNESS MATRIX C DO 10 1=1,IDIM S(I)=0. 10 CONTINUE C C BEGIN MEMBER LOOP C DO 200 1=1,NRM C C ZERO MEMBER STIFFNESS NATRIX C DO 20 J=1,21 SM (J)=0. 20 CONTINUE C DM2= DM (I) *DM (I) XM2=XM(I) *XH (I) YM2= YM (I) *YM (I) XMYM=XH (I) *YM(I) F=AREA (I) *E/(DM (I) *DM2) H=0. IF(AV(I).EQ.O. .OB.G.EQ.O.) GO TO 30 H=12-*E*CRMOM(I)/(AV (I) *G*DM2) 30 XM2F=XM2*F ¥M2F=YM2*F XMYMF=XM¥M*F C C FILL IN PIN-PIN SECTION OF MEMBER STIFFNESS MATRIX C SM (1)=XM2F SM (2)=XHYMF SM(1)=-XM2F SM (5)=-XMYMF SM(7)=YM2F SH(9)=-XMYHF 183 SM (10)=-YM2F SM (16)=XM2F SM (17)=XMYMF SH(19)=YM2F IF (KL (I) +KG (I) -1) 100,40,50 40 F=3. *E*CRHOM (I) /(DM2*DM2*DM (I) * ( 1- «/H/4. ) ) /DAMRAT (I) GO TO 60 50 F=12.*E*CBMGM(I)/(DM2*DM2*DM(I)*(1.*H))/DAMBAT(I) C C FILL IN TEBMS WHICH ABE COBaON TO PIN-FIX,FIX-PIN,AND C FIX-FIX a EMBERS C 60 XH2F=XM2*F YM2F=YM2*F XMYMF=XMYM*F DM2F=DH2*F SH(1)=SM(1) *YM2F SH (2) =SM (2)-XMYMF SM (4)=SM (4)-YM2F SM(5)=SM(5) +XHYMF SM (7)=SM (7) +XM2F SB (9)=SB(9) +XMYHF Sa(10)=SM (10)-XM2F SM(16)=SM(16) +YM2F SM (17) =SH (17)-XMYMF SM(19) = SH(19) + XH2F IF(KL(I) -KG (I)) 70,80,90 C C FILL IN REMAINING PIN-FIX TERMS C 70 SM (6) =-YH (I) *DM2F SM (11)=XM (I) *DM2F SH(18)=-SM(6) SM (20) =-SN (11) SM(21) = DH2*DM2F GO TO 100 C C FILL IN REMAINING FIX-FIX TEEMS C 80 SM (3) =- YM (I) *DM2F*. 5 SM (6)*SM{3) SM(8)=XH(I)*DM2F*-5 SM(11)=SM<8) SM(12)=DH2*DM2F*(4.+H)/12. SH (13)=-SM (3) SH (14) =-SM (8) SM(15)=DM2*DM2F*(2.-H)/12-SM (18)=-SM (3) SM(20)=-SM (8) SM (21)=SM (12) GO TO 100 C C FILL IN REMAINING FIX-PIN TERMS C 90 SH (3)=-YM (I) *DM2F SM (8)=XM(I) *DM2F 184 SM (12)=DM2*DM2F SM (13)=-SM (3) SM (14)=-SM{8) 100 CONTINUE C C ADD THE MEMBER STIFFNESS MATRIX SM INTO THE STROCTORE C STIFFNESS MATRIX S-C NB1=NB-1 C DO 190 3=1,6 IF(NP(J,I)) 190,190,110 110 J1= (J-1)*(12-J)/2 C DO 180 L=J,6 IF(NP(L,I)) 180, 180,120 120 IF(NP(J,I)-NP(L,I)) 150,130, 160 130 IF(L-J) 140,150, 140 140 K=(NP (L,I)-1) •NBUNP (J,I) H-J1+L S (K)=S(K) *2.*SM (N) GO TO 180 150 K= ( H P < J , I) -1) * NB 1 + NP ( L , I) GO TO 170 160 K=(NP{L,I)-1)*NBH-NP(J,I) 170 N=J1*1 S(K)=S (K) *SM(N) 180 CONTINUE C 190 CONTINUE C 200 CONTINUE C RETURN END 185 SUBROUTINE EIGEN (NU,NB,S,IDIM,AHASS,EVAL,EVEC,NMODES,IUN IT) C C THIS SUBROUTINE COMPUTES A SPECIFIED NO- OF NATURAL FSEQ UENCIES C AND ASSOCIATED MODE SHAPES C C NU=NQ. OF DEGREES OF FREEDOM C NB=HALF BANDWIDTH C NMODES=NO. OF MODE SHAPES TO BE COMPUTED C AMASS(I)=MASS MATRIX M=RANK OF MASS MATRIX C S(I)=STIFFNESS MATRIX STORED BY COLUMNS C EVAL (I)—NATURAL FREQUENCIES C EVEC (I,J)=MODE SHAPES C DIMENSION S(IDIM), AMASS (NU) , EVAL(NMODES), EVEC (50,20) , 1 SCR (900) DIMENSION CMASS(100), SS(500) C C COMPUTE THE RANK OF MASS MATRIX C M=0 C DO 10 1=1,NU CMASS (I) =AMASS (I) IF (AMASS (I).EQ.O. ) GO TO 10 M=M*1 10 CONTINUE C IF(NMODES.GT.M) NMODES=M IF(NMODES.EQ.0) NMODES-M WRITE (IUNIT,1) NMODES C C CALL RVPOW TO COMPUTE EIGENVALUES AND EIGENVECTORS C DO 20 1=1,500 SS(I)=S(I) 20 CONTINUE C EPS=0. EPSV=0. CALL RVPOWR(SS,CMASS#NU,NB,EVEC#50#EVAL,NMODES,EPS,EPSV, 100, 1 SCR,M) C C PRINT EIGENVALUES AND EIGENVECTORS (MODE SHAPES) C WRITE (IUNIT, 2) WRITE (IUNIT, 3) WRITE (IUNIT, H) C DO 30 1=1,NMODES EVAL1=EVAL(I) EVAL (I) =SQRT (EVAL 1) FEEQ=EVAL (I)/6. 283185308 PERIOD=6.283185308/£VAL(I) 186 WRITE (IUNIT,5) I, EVAL1, EVAL (I), FREQ, PERIOD 30 CONTINUE C WRITE (IDNIT,6) NMODES WRITE (IUNIT,7) (1,1=1,NMODES) C DO 40 1=1,M WRITE (IONIT,8) (EVEC (I, J) ,J= 1 , NMODES) 40 CONTINUE C RETORN 1 FGRHAT{*-*,*NQ. OF MODES TO BE ANALIZED = » ,I5///11 0 {» *• ) ///) 2 FORM AT (///110 (»*•)) 3 FORMAT(*Q*,5X,•MODES',4X,» EIGENVALUES*,6X,* NATURAL FREQU ENCIES* , 1 13X,*PERIODS*) 4 FORMAT {• •,30X,• (RAD/SEC) •,5X,*(CYCS/SEC) *,8X,* (SECS) •) 5 FORM AT {' *,5X,I5,4F15.4) 6 FORMAT{*0«,5X,*MODE SHAPES CORRESPONDING TO FIRST*,15,* FREQUENCIE IS') 7 FORMAT{*0•,10112) 8 FORMAT {* *,10F12. 6) END 187 SUBROUTINE MOD3(ICOUNT,ISPEC,NRJ,NRM,NU,NB,NMODES,S,IDIM ,ND,NP,XM, 1 YM,DH, AREA,CRMOM,DAMRAT,KL,KG# SDAMP,BMCAP,E,& MASS,EVEC, 2 EVAL , AM AX, I SIGN, I UNIT, BET A , ERROR, XBASE) C C MODIFIED SUBSTITUTE STRUCTURE METHOD C THIS SUBROUTINE COMPUTES JOINT DISPLACEMENTS AND MEMBER FORCES C NEW DAMAGE RATIOS WILL BE CALCULATED AND RETURNED. C C ICOUNT = 0 IF DAMPING IS SET AT 10% AND ELASTIC C ANALYSIS IS TO BE CARRIED OUT C ISPEC = 1 FOR SPECTRUM A, 2 FOR B, AND 3 FOR C C IDIM = DIMENSION OF S(I) C ISIGN = NUMBER OF MEMBERS FOR WHICH DAMAGE RATIOS C ARE MODIFIED C IUNIT = OUTPUT DEVICE UNIT C BETA = CONSTANT FOR ACCELERATED CONVERGENCE C ERROR = CONSTANT FOR CONVERGENCE CRITERION C IBASE = 1 IF BASE SHEAR IS TO BE PRINTED C c DIMENSION ND(3,NRJ), NP(6,NRH), XM (NRM) , YM(NRM), DM (NRM ) . 1 AREA (NRM), CRMOM (NRM) , DAMRAT (NRM), KL (NRM) , K G (NRM) , 2 AMASS (NRM) , EVEC (50, 20), EVAL (NMODES) , S (IDIM) 3 SDAMP (NRM), ZETA(10), PI (100) DIMENSION BMASS(40), IDOF(100), ALPHA (20), RMS (7, 100), 1 F(100) , D(6) DIMENSION BMCAP (1) C C CALCULATE THE MODAL PARTICIPATION FACTOR C JJ=1 C DO 10 1=1,NU IF (AMASS (I) . EQ-O. ) 30 TO 10 BMASS (JJ) =AMASS (I) IDOF (JJ)=I JJ=JJ*1 10 CONTINUE C C JJ=JJ-1 DO 3 0 1=1,NMODES AMT=0. AMB=0. ALPHA (I) =0. DO 20 J=1,JJ AMT=AMT*BMASS (I) *EVEC (J , I) AM B=AMB*BMASS(I)*EVEC(J,I)**2 20 CONTINUE 188 C ALPH &(I) = AMT/AMB 30 CONTINUE C WRITE (IUNIT,1) C DO 4 0 1=1,NMODES WRITE (IUNIT,2) I, ALPHA (I) 40 CONTINUE C C WHEN KK=1, MODAL FORCES FOR UNDAMPED SUBSTITUTE STRUCTUR E ARE C COMPUTED. THEY ARE USED TO COMPUTE * SMEARED * DAMPING VA LUES, C WHICH ARE USED TO CALCULATE THE ACTUAL RESPONSE OF THE S UBSTITUTE C STRUCTURE C INDEX=1 C DO 420 KK=1,2 IF(ICOUNT-I) 400,70,50 50 CONTINUE IF (KK.NE. 1) GO TO 70 C DO 60 K=1,NMODES ZETA<K)=0. 60 CONTINUE C 70 CONTINUE SHRMS=0. C C ZERO ABSO(J,I) AND RMS (J,I) C DO 90 1=1,100 C DO 80 J=1,7 RMS (J,I)=0. 80 CONTINUE C 90 CONTINUE C C CALCULATE THE MODAL DISPLACEMENT VECTOR C DO 290 K=1,NMODES C C CALCULATE NATURAL PERIOD AND CALL SPECTA C WN = 6. 2831 85308/EVAL (K) DAMP=ZETA (K) CALL SPECTS(ISPEC,DAMP,WN,AMAX,SA) C C ZERO LOAD VECTOR C DO 100 J=1,NU F(J)=0. 189 100 CONTINUE C FF=0. C C COMPUTE LOAD VECTOR C FAC=SA*ALPHA(K)*386.4 C DO 110 J=1,JJ I1=IDOF(J) F (I1)=EVEC (J,K) *FAC*AMASS (11) FF=FF+F(I1) 110 CONTINUE C C CALCULATE THE BASE SHEAR C IF(KK.NE.2) GO TO 120 SHRMS=SHRHS+FF**2 IF(K.LT.NMODES) GO TO 120 SHRMS=SQBT (SHRMS) 120 CONTINUE C C COMPUTE DEFLECTIONS C C C CALL SUBROUTINE FBAND C RATIO=1.E-7 CALL FBAND(S,F,NU,NB,INDEX,RATIO,DET,JEXP,0,0,0.) INDEX=INDEX+1 C DO 160 1=1,NRJ DX=0-DY=0. DR=0. N1=ND(1,I) N2=ND(2,I) N3=ND(3,I) IF(NI-EQ-O) GO TO 130 DX=F(N1) BBS(1,1)=RMS(1,I)*DX**2 130 CONTINUE IF(N2.EQ.0) GO TO 140 DY=F(N2) RMS (2,1)-BBS(2,1)*DY**2 140 CONTINUE IF(N3.EQ.O) GO TO 150 DR=F (N3) RMS(3,1)-BBS(3,1)+DR**2 150 CONTINUE 160 CONTINUE C C COMPUTE MEMBER FORCES C SIGPI=0. 190 DO 260 1=1,NBM DO 200 J=1,6 N1 = NP(J,I) IF{N1) 180,180,170 170 D(J)=F(N1) GO TO 190 180 D(J)=0. 190 CONTINUE 200 CONTINUE L-D{2) *YL) XL=XM(I) ¥L=YM (I) DL=DM (I) &XIAL= (ABEA (I) *E/DL**2) * (D (4) *XL + D (5) *¥L-D ( 1)*X IF (KL (I) .EQ. 0 . AND. KG (I) .EQ.O) GO TO 230 DV= (D (2) *XL—D (1) *YL+D (4) *YL-D (5) *XL)/DL AK=CBMOM (I) *E/DL/DAMBAT (I) BHL=-AK*(6.*DV/DL*4.*D(3) +2. *D(6) )/12. SHEAB=AK*6. * (2- *DV/DL*D (3) *D (6) ) /DL BMG=BHL+SHEAB*DL/12. IF (KL (I) -KG (I)) 210,240,220 210 BHG=BMG+BML*.5 SHEAB=SHEAB•1.5*BHL/(DL/12.) BML=0. GO TO 240 220 BML=BML*BMG*.5 SHEAB=SHEAB-1.5*BMG/(DL/12.) BHG=0. GO TO 240 230 BMG=0. BEL=0., SHEAB=0. 240 CONTINUE C C COMPUTE THE RELATIVE FLEXU8AL STBAIN ENEBGY C IF(KK.NE.1) GO TO 250 PI (I) = (BML**2*BMG**2*BML*BMG) /6./AK SIGPI=SIGPI*PI(I) 250 CONTINUE C C ACCUMULATE ABSOLUTE SUM AND BMS SUM C BMS (4,1) =BMS(4,I) *AXIAL**2 BMS(5,I)=BHS(5,1)+SHEAB**2 BMS (6,I)=BMS(6,I) +BSL**2 BMS (7,1)=BMS(7,1)*BMG**2 260 CONTINUE C C COMPUTE THE SMEABED DAMPING FOB EACH MODE C IF(KK.NE.I) GO TO 280 C DO 270 1=1,NBM 191 ZETA (K) = ZETA (K) + PI(I) *SDAMP(I) 270 CONTINUE C ZETA (K) =ZETA (K) /SIGPI 280 CONTINUE 290 CONTINUE C IF (KK. EQ. 1) GO TO 420 C C PRINT RMS DISPLACEMENTS AND FORCES C WRITE (IUNIT,4) WRITE (IUNIT,5) WRITE (IUNIT#3) C DO 310 1=1,NBJ C DO 300 J=1,3 SCRAT=RaS{J,I) RMS (J, I) =SQRT{SCRAT) 300 CONTINUE C WRITE (IUNIT,6) I, (BMS (J,I) , J= 1,3) 310 CONTINUE C C MODIFY DAMAGE RATIOS C WRITE (10NIT,7) IF(IBASE.NE. 1) GO TO 320 WRITE (IUNIT,8) SHBHS 320 CONTINUE WRITE (IUNIT,9) ISIGN=0 C DO 390 1=1,NRM IF (RMS (6,I)-RMS (7,1) ) 330,330,340 330 BIG=RMS(7,I) GO TO 350 340 BIG=RMS(6,I) 350 CONTINUE BM=SQBT (BIG) DAM0LD=DAMBAT (I) DAMRAT (I) =BM/BMCAP (I) *DAMRAT (I) DAMBAT (I) =DAMRAT (I) +BETA* (DAMRAT (I) —D AMOLD) IF (DAMBAT (I) . LT. 1.0) GO TO 360 CHECK=ABS (BM-BMCAP (I) ) /BMCAP (I) IF (CHECK. GT.ERROR) ISIGN=ISIGN+1 . GO TO 370 360 CONTINUE DAMRAT (I) = 1. 370 CONTINUE SDAMP (I)=0.02+0.2*(1.-1-/SQBT(DAMRAT(I))) C DO 380 J=4,7 RMS (J,I) =SQBT (BMS (J,I) ) 380 CONTINUE 192 C WRITE (I0NIT,11) I, (RHS(J,I) ,J=4,7) , BMCAP(I), D AMRAT (I) 390 CONTINUE C GO TO 420 400 CONTINUE C DO 4 10 1=1,NMODES ZETA (I) =. 1 410 CONTINUE C ICOUNT=ICOUNT*1 WRITE {IHNIT,12) 420 CONTINUE C ICOUNT=ICOUNT+1 RETURN 1 FORM AT {•-',* MODAL PARTICIPATION FACTOR*,/) 2 FORM AT(* »,5X,•MODE»,I5,5X,F10.5) 3 FORMAT (*-' ,7X,*JOINT NO. »,10X, 'X-DISP(IN) ' , 10X, • Y-DISP (I N)',7X, 1 'ROTATION (RAD)') 4 FORMAT (* —• , 110 (**') ) 5 FORM AT('-',*ROOT MEAN SQUARE DISPLACEMENTS *) 6 FORMAT {* • ,6X,I10,3F20.4) 7 FORMAT(*-*,* ROOT MEAN SQUARE FORCES') 8 FORM AT(1H0,7X,»RSS BASE SHEAR = *,F10.3) 9 FORMAT(* — * ,8X,* MN *,1QX,* AXIAL*,10X,•SHEAR*,11X,*BML*,12X ,* BMG', 1 9X,'MOMENT*,1 OX,'DAMAGE*/21X,•KIPS»,12X,»KIPS',2( 9X, » (K—FT) 2*), 8X, * CAPACITY' ,9X, * RATIO *) 11 FORMAT {* • ,5X,I5,6F15.3) 12 FORMAT(*~* ,110 (•*•)) END 193 SUBROUTINE SPECTR (ISPEC,DAMP,WN,AMAX,SA) C C ISPEC=1 IF SPECTRUM A IS USED C =2 IF SPECTRUM B IS USED C =3 IF SPECTRUM C IS USED C DAMP=DAMPING FACTOR (FRACTION OF CRITICAL DAMPING) C WN =NATURAL PERIOD IN SECONDS C AMAX=MAXIMUH GROUND ACCELERATION (FRACTION OF G) C SA =RESPONSE ACCELERATION (FRACTION OF G) C IF (ISPEC. EQ. 2) GO TO 10 IF (ISPEC. EQ. 3) GO TO 60 C C SPECTRUM A C IF (WN.LT. 0. 15) SA=25.*AMAX*WN IF (WN.GE-0. 15 .AND. WN.LT.0.4) SA=3.75*AMAX IF(WN.GT-0-4) SA=1.5*AMAX/WN GO TO 90 C C SPECTRUM B C 10 CONTINUE IF (WN.LT.0.1875) GO TO 20 IF (WN.LT. 0-53333333) GO TO 30 IF (WN.LT. 1. 6666667) GO TO 40 IF(WN.LT.1.81666667) GO TO 50 SA=2.*AMAX/(WN-0.75) GO TO 90 20 SA=20.*AMAX*WN GO TO 90 30 SA=3.75*AMAX GO TO 90 40 SA=2.*AMAX/WN GO TO 90 50 SA=1.875*AMAX GO TO 90 C C SPECTRUM C C 60 CONTINUE IF(WN.LT.O. 15) GO TO 70 IF (WN.LT. 0.38333333) GO TO 80 SA=0. 5*AHAX/ (WN-0. 25) GO TO 90 70 SA=25.*AMAX*WN GO TO 90 80 SA=3.75*AMAX 90 CONTINUE SA=SA*8./(6.*100.*DAMP) RETURN END 

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