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Minimum seismic requirements for masonry walls Chung, Rufus Wai-Kwong 1978

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MINIMUM SEISMIC REQUIREMENTS FOR MASONRY WALLS by RUFUS WAI-KWONG CHUNG B.A.Sc. (1975) The U n i v e r s i t y of B r i t i s h Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s t h e s i s as conforming to the req u i r e d standard The U n i v e r s i t y of B r i t i s h Columbia A p r i l 1978 © Rufus Wai-Kwong Chung, 1978 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it fr e e l y 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 f i n a n c i a l gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date A P r i l 28 1978 ABSTRACT A study by computer a n a l y s i s was conducted to e s t a b l i s h the minimum percentage of s t e e l required i n the seismic design of masonry walls with respect to transverse f o r c e s . Buildings of masonry load-bearing shear wall and masonry i n f i l l - f r a m e construction were studied. The b u i l d i n g s were modelled as plane frame structures. Linear e l a s t i c s p e c t r a l analyses were performed by using the modified computer program DYNAMIC. By examining the response of the models to a peak ground a c c e l e r a t i o n of 1.0 g, a c o r r e l a t i o n between the fundamental period of a masonry wa l l structure and i t s response a c c e l e r a t i o n s at d i f f e r e n t height i s found, i . e . , envelopes of dynamic amplication f a c t o r are developed. A new empirical formula for estimating the fundamental period of a structure with masonry walls i s recommended. The minimum s t e e l i s established i n t y p i c a l w a l l panels designed for the i n e r t i a forces imposed by a major earthquake i n seismic Zone 3. i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS . . .. i i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS , .. .. x CHAPTER 1. INTRODUCTION 1 1.1 Background 1 1.2 Purpose and Scope 3 2. BACKGROUND ON DYNAMIC ANALYSIS .. 5 2.1 Introduction 5 2.2 Spectral Analysis 6 2.3 Method of Structural Modelling 13 2.4 Wide Column Model for Shear Walls 14 2.5 Effective Width of Slabs 15 2.6 I n f i l l Walls 16 3. MODELLING OF BUILDINGS FOR DYNAMIC ANALYSIS 18 3.1 Introduction 18 3.2 Load-Bearing Shear Wall Building i n Vancouver.. 19 3.3 Park Lane Towers 23 3.4 Infill-Frame Building 25 i v Page 4. DISCUSSION OF THE ANALYTICAL EFFORT 28 4.1 Computer Program 28 4.2 Design Spectrum 29 4.3 Dynamic An a l y s i s and Results 31 4.3.1 Load-Bearing Shear Wall Buildings . . . . 35 4.3.2 I n f i l l - F r a m e B u i l d i n g 45 5. RESULTS-ANALYSIS AND RECOMMENDATIONS 55 5.1 Observations 55 5.2 L e v e l C o e f f i c i e n t 56 5.3 Fundamental Period of Buildings .. .. 64 6. MINIMUM STEEL REQUIREMENTS 68 6.1 Introduction .. .. 68 6.1.1 Racking Forces 68 6.1.2 Transverse Forces 69 6.2 Design Earthquake 70 6.3 Design Formula for Masonry Walls i n Bending .. 73 6.4 Wall Panel Designing 74 6.4.1 Load-Bearing Shear Wall 75 6.4.2 I n f i l l Wall 77 7. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 79 7.1 Summary and Conclusions 79 7.2 Recommendations f o r Future Research 81 V Page REFERENCES 83 APPENDIX A BEAM STIFFNESS MATRIX FOR WIDE .., COLUMN FRAME 86 APPENDIX B MODELLING OF INFILL WALLS 93 APPENDIX C COMPUTER LISTING OF SUBPROGRAM SPECTR . . . . 96 APPENDIX D INPUT AND RESULT DATA OF MODEL ANALYSES .. 98 LIST OF TABLES Table Page 1 COMPARISON OF FUNDAMENTAL PERIODS 66 2 MEMBER PROPERTIES OF STRUCTURE 1A 100 3 MEMBER PROPERTIES OF STRUCTURE IB 102 4 MEMBER PROPERTIES OF STRUCTURE 2 104 5 DYNAMIC ANALYSIS RESULTS OF STRUCTURE 1A .... 105 6 DYNAMIC ANALYSIS RESULTS OF STRUCTURE IB ..... 109 7 DYNAMIC ANALYSIS RESULTS OF STRUCTURE 2 .... 112 v i i LIST OF FIGURES Number — 1 Single-degree-of-freedom system 6 2 Response spectra for elastic systems, 1940 EL Centro earthquake (from Ref. 3) 10 3 Wide column frame model 15 4 Typical floor plan of load-bearing shear wall building i n Vancouver 20 5 Typical floor plan of Park Lane Towers 24 6 Typical floor plan of infill-frame building .... 26 7 Peak ground motion bounds and elastic average response spectrum for 1.0 g max. ground acceleration (from Ref. 30) .. 30 8 Torsional induced acceleration 34 9a Structure 1A: R-S-S storey displacements ....... 37 9b Structure 1A: R-S-S storey displacements 38 10a Structure 1A: Storey accelerations of 5-storey model 39 10b Structure 1A: Storey accelerations of 10-storey model 40 10c Structure 1A: Storey accelerations of 15-storey model 41 lOd Structure 1A: Storey accelerations of 20-storey model 42 lOe Structure 1A: Storey accelerations of 25-storey model Li v i i i F i g u r e Page Number ___s— 11 Variation of response acceleration with fundamental period at levels h/H = 0.2, 0.4, 0.6, 0.8, 1.0 of Load-Bearing Shear Wall Buildings .. 44 12a Structure 2: R-S-S storey displacements 47 12b Structure 2: R-S-S storey displacements . . . . . . 48 13a Structure 2: Storey accelerations of 5-storey model 49 13b Structure 2: Storey accelerations of 8-storey model 50 13c Structure 2: Storey accelerations of 10-storey model 51 13d Structure 2: Storey accelerations of 15-storey model 52 13e Structure 2: Storey accelerations of 20-storey model 53 14 Variation of response acceleration with fundamental period at levels h/H = 0.2, 0.4, 0.6, 0.8, 1.0 of Infill-Frame Buildings 54 15a Variation of response acceleration with fundamental period at h/H = 0.2 57 15b Variation of response acceleration with fundamental period at h/H =0.4 58 15c Variation of response acceleration with fundamental period at h/H = 0.6 59 15d Variation of response acceleration with fundamental period at h/H = 0.8 60 15e Variation of response acceleration with fundamental period at h/H = 1.0 61 16 Comparison of fundamental periods by computer and by Code empirical formula 67 ix F i g u r e Page Number — ° — 17 Effect of u and X on the design load 72 18 Models for (i) shear wall panel and ( i i ) i n f i l l panel 76 19 Beam element for wide column frame 87 20 Equivalent strut model of i n f i l l wall 94 21 Plane frame model of Structure 1A 99 22 Plane frame model of Structure IB 101 23 Plane frume model of Structure 2 103 ACKNOWLEDGEMENTS I would l i k e to express my gratitude to Dr. N.D. Nathan who served as my major adviser f o r t h i s study. In p a r t i c u l a r h i s advice and suggestions during the preparation of my th e s i s are very much appreciated. I would also l i k e to thank Dr. S. Cherry f o r h i s advice and suggestions and Dr. D. L. Anderson f o r many h e l p f u l discussions. Appreciation i s due to National Research Council and Portland Cement A s s o c i a t i o n f o r permissions to r e p r i n t copyright f i g u r e s . F i n a l l y , I l i k e to thank my wife, Yvonne, f o r typing t h i s t h e s i s . Also her patience and understanding at times of my study are deeply appreciated. CHAPTER 1 INTRODUCTION 1.1 Background Masonry has been used as a b u i l d i n g material since the e a r l i e s t days of man. T r a d i t i o n a l l y masonry structures were designed and b u i l t with no s t e e l reinforcement and supported only g r a v i t y loads. These structures had t h i c k walls and heavy f l o o r s and were u s u a l l y one- or two-story b u i l d i n g s . The heavy g r a v i t y load tended to s t a b i l i z e the structure against l a t e r a l forces due to minor wind or seismic a c t i o n . These structures were not capable of r e s i s t i n g large l a t e r a l forces because of the low t e n s i l e strength of masonry. They were e s p e c i a l l y r i s k y i n earthquake Zone 3 where major earthquakes would most l i k e l y occur. In the l a s t few decades, r e i n f o r c e d masonry has been introduced. S t e e l reinforcement i n masonry allows a bet t e r use of the masonry m a t e r i a l . M u l t i s t o r y r e i n f o r c e d masonry b u i l d i n g s have been b u i l t . These b u i l d i n g s are found to provide a better re s i s t a n c e to l a t e r a l forces and to behave very well during earthquakes. Reinforced masonry i s now recognized as another h i g h - r i s e b u i l d i n g material and i s very popularly used. Reinforced masonry i s a r e l a t i v e l y new material to which very l i m i t e d research has been devoted. Present designers often simply follow the b u i l d i n g codes, but many clauses i n the b u i l d i n g codes are not f u l l y j u s t i f i e d . Only one of these w i l l be mentioned i n the following. The National B u i l d i n g Code of Canada 1 9 7 5 ^ " ^ s p e c i f i e s : "4.4.3.30. (1) Reinforced masonry walls shall be reinforced horizontally and vertically with steel having a total cross-sectional area not less than 0.002 times the gross cross-sectional area of the wall so that not less than 1/3 of the recruired steel is either vertical or horizontal." These minima appear to have t h e i r o r i g i n s i n t r a d i t i o n rather than to be based on experimental evidence. They have considerable economic impact on the use of masonry, and cause considerable controversy among designers and contractors. The presumed purpose of t h i s s t e e l i s twofold: 1. To prevent the masonry from f a l l i n g out due to transverse forces normal to the plane of the w a l l . 2. To lend some d u c t i l i t y to the wa l l with respect to racking deformations corresponding to i n -plane loading. 1.2 Purpose and Scope The purpose of t h i s t h e s i s i s to e s t a b l i s h the minimum percentages of s t e e l f o r the seismic design of masonry wa l l s . These minima w i l l be established with respect to the tranverse force (out-of-plane bending); the e f f e c t of the racking force ( i n -plane shearing) i s al s o recognized and w i l l be discussed b r i e f l y . Two common types of bu i l d i n g s with masonry w a l l con-s t r u c t i o n w i l l be i n v e s t i g a t e d . (a) Load- bearing shear w a l l b u i l d i n g s , and (b) I n f i l l e d frame b u i l d i n g s . T y p i c a l b u i l d i n g s of each type w i l l be modelled as plane frame structures and then subjected to dynamic an a l y s i s by the response spectrum approach. Response w i l l be obtained f o r a range of buil d i n g s with d i f f e r e n t numbers of s t o r i e s so that the r e l a t i o n s h i p between the fundamental period and the storey a c c e l e r a t i o n at each l e v e l of the b u i l d i n g can be determined. That i s to say, envelopes of the dynamic a m p l i f i c a t i o n f a c t o r w i l l be sought f o r . An i n d i v i d u a l w a l l panel w i l l then be designed to prevent i t s f a l l i n g out under the i n e r t i a f o r c e a r i s i n g from i t s own mass. Code-type recommendations w i l l also be developed. In the dynamic a n a l y s i s , the structures w i l l be assumed to behave as l i n e a r systems. The e l a s t i c modal response of the f i r s t three modes of v i b r a t i o n w i l l be obtained and superposed. I t i s recognized that i n p r a c t i c e , nonlinear behaviour of the structures w i l l occur i n severe earthquakes. Thus, the true dynamic behaviour of the structures can be determined only i f allowance i s made f o r n o n l i n e a r i t y i n the a n a l y s i s . Nevertheless, p l a s t i c deformations of i n e l a s t i c structures absorb large amounts of energy from earthquake-generated motions and s t i f f n e s s degradations have the e f f e c t of reducing the response of the systems. Therefore, the maximum response accelerations obtained from the l i n e a r e l a s t i c dynamic a n a l y s i s w i l l be conservative and can be considered as the upper l i m i t s . CHAPTER 2 BACKGROUND ON DYNAMIC ANALYSIS 2.1 Introduction In recent years, there has been a growing acceptance of the dynamic an a l y s i s of structures subjected to earthquake loading. The response of a structure can be determined by eit h e r the time-history ground motion approach or the response spectrum approach. In the time-history ground motion approach, the response of a structure i s obtained by numerical step-by-step integrations i n the time domain. An accurate response, within the mathematical model, can be obtained f o r a p a r t i c u l a r ground motion. This method i s often proven to be un s a t i s f a c t o r y because a d i f f e r e n t ground motion w i l l cause a d i f f e r e n t response of a structure, and the response i s also dependent on the time step one has chosen. I t i s extremely d i f f i c u l t to p r e d i c t the d e t a i l e d c h a r a c t e r i s t i c s of an earthquake f o r a given s i t e and a choice of a short time step w i l l make the an a l y s i s very c o s t l y . In most analyses, where only the maximum response of a structure i s required, i t i s often found simpler to use the response spectrum approach. However, i t provides only approximate r e s u l t s f o r structures with multi-degrees-of-freedom, and i s applic a b l e only to l i n e a r structures with damping that s a t i s f i e s the uncoupled modal response equations. 2.2 Spectral Analysis x mass m columns, s t i f f n e s s k 1 / / / / / / / / J44 1 x y g Figure 1 Single-degree-of-freedom system Figure 1 shows a linear one-storey single-degree-of-freedom system with mass m, column stiffness k and damping coefficient c. The displacements x , x and y are related by the expression: x = x g + y (2.1) where x = absolute displacement of the mass, y = mass displacement relative to the ground, X g = ground displacement. The d i f f e r e n t i a l equation of motion for the system may be written as: my + cy + ky = -mxo (2.2) where the dots denote differentiations with respect to time. The solution for Equation (2.2) may be obtained by the application of the Duhamel integral. By neglecting the slight difference between the damped and undamped frequencies, the response of a . (5) damped structure at any time t i s : y ( t ) = - - x (T ) exp [ -5u)(t-T)] s i n [ t o ( t - T ) ] dx (2.3a) 0) O o . y(t) = X g ( T ) e x p [ - c > ( t - T ) ] cos[to(t-r)] dx - fc>y(t) (2.3b) x(t) = - o)2y(t) - 2?wy(t) (2.3c) where co = undamped natural circular frequency of vibration of the structure = v^ kTm £ = fraction of c r i t i c a l damping - c 2v^k In most practical problems where only the maximum numerical values of the response are of interest, i t is convenient to introduce the spectral velocity: S v = | Qf X g ( x ) exp[-£u)(t-r)] s i n [ c o ( t - T ) ] dx 1 ^ (2.4) For systems with small damping (£<0.2) and short period (T<1 s e c ) , as are normally encountered in building structures, i t can be shown that the maximum quantities of the relative displacement, relative velocity and absolute acceleration can be expressed respectively as y(t) = S, = - S (2.5a) J max d to v . \ / X(t) = S = coS (2.5c) max a v \*..-»w where S d i s the s p e c t r a l displacement and S the s p e c t r a l i ^- d*) ac c e l e r a t i o n From Equation (2.4), i t i s apparent that the s p e c t r a l v e l o c i t y i s a function of 1) the c h a r a c t e r i s t i c s of the ground motion and 2) the c h a r a c t e r i s t i c s of the structure defined by i t s n a t u r a l period of v i b r a t i o n and damping. By studying the recorded accelerations of the past earthquakes, Newmark^^ i d e a l i z e d the ground motion of an earthquake by a polygon made up of three upper bounds as shown i n Figure 2. The average e l a s t i c response spectrum can be obtained by m u l t i p l y i n g the peak ground motion bounds by a m p l i f i c a t i o n f a c t o r s which are functions of the period and damping of a str u c t u r e . With the peak ground a c c e l e r a t i o n , damping and period of v i b r a t i o n given, the maximum response of a structure S,, S and S can be determined. r d' v a The National B u i l d i n g C o d e ^ ^ has adopted t h i s procedure f o r dynamic a n a l y s i s . 10 Undamped natural period T, sec. Figure 2 Response spectra f o r e l a s t i c systems, 1940 EI Centro earthquake, (from Ref.3) 11 The dynamic analysis of a multi-degree-of-freedom system is more complex. However, for linear structures with (9) small damping, the normal mode theory can be applied to simplify the problem. According to the normal mode concept, the principal modes of vibration of a multi-degree-of-freedom system are uncoupled. Each mode responds to the base excitation as an independent single-degree-of-freedom system. The modal maxima of a structure can be determined by the spectral analysis as was discussed for single-degree-of-f reedom systems. For the i * " * 1 mode, the modal maximum response of the 3^ mass can be found by the following equations: y i j max = * i j ^ 1 S d i ( 2 - 6 a ) y. . = <J>. . y. S . (2.6b) i ] max 1 j 1 v i \ • / X. . =<)>.. y. S . (2.6c) 1 3 max 1 j 1 a i v ' th where <)>. . = relative displacement of the j mass in 1 3 the i * " ^ 1 mode Y^ = modal participation factor of the i t n mode E m. <)>. . = ] . . . 2 J = 1 n 2 m. (j>. . J 1 J 12 The actual response of the structure can then be obtained by superposing the modal maxima. Since the modal maxima probably would not occur simultaneously, the root-sum-square method w i l l normally give a better estimate of the actual response. The root-sum-square method consists of taking the square root of the sum of the modal maxima squared. Since the effect of higher modes on a building i s usually insignificant, only the f i r s t few modes need to be included. From Equation (2.6c), i t i s apparent that the maximum absolute acceleration i s directly proportional to the relative displacement, of a structure. It i s clear that the relative displacement i s zero at the base of a structure. This in turn w i l l give zero absolute acceleration at the base by Equation (2.6c) which contradicts the fact that the base, which i s attached to the ground must have the same acceleration as the ground. In order to rectify this situation, a separate response mode of constant acceleration with a magnitude equal to the peak ground acceleration w i l l be superposed with the other normal modes to obtain the actual acceleration by the root-sum-square method, i.e.: 2 2 2 2 X. = / x * + X, / + X_ . + x» . + j max / g max 1 j max 2 j max 3 j max (2.7) 1 3 This w i l l normalize the a c c e l e r a t i o n at the base of a structure to the ground a c c e l e r a t i o n . The response accelerations so obtained w i l l be r e f e r r e d to as "Adjusted Ac c e l e r a t i o n s " i n the future. 2 . 3 Method of S t r u c t u r a l Modelling When analyzing a b u i l d i n g f o r earthquake loading, i t i s a common p r a c t i c e to s i m p l i f y the an a l y s i s by assuming the structure to be represented by a set of plane frames perpendicular to each other i n the d i r e c t i o n s of the p r i n c i p a l axes of the b u i l d i n g . Assuming that each component of the earthquake forces i s r e s i s t e d only by frames i n that d i r e c t i o n , independent a n a l y s i s i s required for each p r i n c i p a l d i r e c t i o n . I t i s assumed that t h i s approach w i l l provide adequate r e s i s t a n c e to h o r i z o n t a l a c c e l e r a t i o n s i n any other d i r e c t i o n . In case of a symmetrical structure where the t o r s i o n a l e f f e c t i s i n s i g n i f i c a n t , a two-dimensional computer program can be used f o r the a n a l y s i s . I t i s us u a l l y assumed that the f l o o r s act as r i g i d diaphragms, so that the b u i l d i n g can be modelled as a s e r i e s of two-dimensional frames connected by a x i a l l y r i g i d s t r u t s which enforce equal l a t e r a l d e f l e c t i o n s at each story l e v e l . I A Each frame can be represented by a planar assemblage of beams and columns. The degrees of freedom considered i n the plane of each frame are the v e r t i c a l and r o t a t i o n a l displacements of each beam-column j o i n t , and a s i n g l e h o r i z o n t a l displacement f o r each f l o o r l e v e l . In the buildings considered herein, which have r i g i d basement structures, columns are assumed to be f i x e d at the ground l e v e l . 2.A Wide Column Model f or Shear Walls In the wide column frame analogy, i t i s recognized that the e f f e c t of the f i n i t e j o i n t widths due to column depths w i l l contribute s i g n i f i c a n t l y to the s t i f f n e s s of a b u i l d i n g . A common device to account f o r the f i n i t e widths of the shear walls or columns i n frame analyses i s to consider that p o r t i o n of a member from the face to the n e u t r a l axis of the w a l l as a segment with i n f i n i t e f l e x u r a l and a x i a l s t i f f n e s s e s . Thus a beam i n the frame a n a l y s i s can be represented by a member with three sections. The middle s e c t i o n i s the o r i g i n a l beam length taken as the c l e a r distance between walls and columns. There i s one i n f i n i t e l y r i g i d s e c t i o n attached to each end of the middle s e c t i o n , representing the half-widths of the walls or columns. The walls can be represented by l i n e members at t h e i r n e u t r a l axes since the beams are u s u a l l y shallow and t h e i r depths can be neglected. The member s t i f f n e s s ACTUAL STRUCTURE I = oo I = oo A = oo A = °° Figure 3 Wide column frame model. matrix f o r a three-sectioned beam member can be formulated by the s t i f f n e s s method. The d e r i v a t i o n i s presented i n Appendix A. The e f f e c t of shear d e f l e c t i o n s can be included i n a l l members; such e f f e c t i s e s p e c i a l l y important f o r shear w a l l s . 2.5 E f f e c t i v e Width of Slabs In recent years, slab-shear w a l l b u i l d i n g s have become very popular i n North America. This system consists of continuous 16 slabs spanning between load-bearing shear walls. In the analysis of such structural systems subjected to la t e r a l loads, i t i s often convenient to assume an effective width of the slab to act as a beam, coupling the shear walls. A study of the effective width of coupling slabs has been carried out by many p e o p l e ^ * ^ ^ ^ , but their studies have been restricted to some simple plan con-figurations. For many other configurations, useful results are not available. In a typical frame consisting of beams and columns, with the floor slab monolithic with the beams, the effective width of the slab working as a T-beam flange i s specified by the ACI (12) Standard 318-71 . The same slab effective width i s assumed for slab-shear wall buildings in this analysis. It i s found, however, that inaccuracy i n estimating the effective widths of the coupling slabs w i l l generally not affect the overall behaviour of the structure significantly. 2.6 I n f i l l Walls In many frame type structures, walls and partitions are made of masonry units. Although such elements are often considered to be non-structural, they contribute significantly to the la t e r a l stiffness of the structure. When the effect of 17 i n f i l l s i s included in an analysis, the frequency of vibration of the structure w i l l be increased and i t s mode of response w i l l be changed. The f i r s t major investigation of i n f i l l e d frames sub-jected to racking loads was started by Polyakov i n 1948. Polyakov found that i n a l l his tests, the i n f i l l s f i r s t separated from the frames, except near the compression areas, by cracking around the perimeters. He also found that the usual mode of failu r e of the complete i n f i l l following the separation was by cracking of the mortar jointing along the compression diagonal. This led to Polyakov's proposal for analysing i n f i l l s as diagonal bracing struts. In the present analysis, due to the two way action in earthquake loading, i t i s appropriate to represent each i n f i l l by two diagonal bracing struts crossing each other. The diagonal struts are assumed to have the same elastic properties as the beams and columns in the frame. The sectional area of the struts i s determined by equating the lat e r a l displacements for the i n f i l l e d frame and the braced model for the same applied lateral force, a (31) procedure which i s similar to Hrennikoff's framework method The detail derivation i s shown in Appendix B. CHAPTER 3 MODELLING OF BUILDINGS FOR DYNAMIC ANALYSIS 3.1 Introduction Masonry walls are commonly used i n two types of construction load-bearing shear w a l l s , and i n f i l l w a l l s for frame b u i l d i n g s . A t y p i c a l b u i l d i n g of each type w i l l be analyzed by dynamic a n a l y s i s to determine t h e i r d i f f e r e n c e i n response when subjected to earthquake loadings. The b u i l d i n g s are modelled as two-dimensional frame structures. Shear and bending deformations are considered f o r a l l elements. One t y p i c a l storey i s studied f o r each b u i l d i n g ; the s e c t i o n a l area and moment of i n e r t i a are obtained f o r each beam and column i n that storey based on t h e i r uncracked sections. Members i n other s t o r i e s are assumed to be the same as those i n the t y p i c a l storey which i s an appropriate assumption f o r reinfor c e d concrete and masonry b u i l d i n g s . For the purpose of the dynamic a n a l y s i s , the mass of the b u i l d i n g i s assumed to be concentrated at the f l o o r l e v e l s . Only the dead load i s used f o r d e r i v i n g the storey masses. L i v e load, since i t i s l i k e l y to be present only i n part during an earthquake and i s generally not p o s i t i v e l y attached to the b u i l d i n g , w i l l not contribute to the i n e r t i a forces on the s t r u c t u r e . T h i s assumption i s based on the seismic design method i n the N a t i o n a l B u i l d i n g Code, which implies that only the dead load should be used i n obtaining l a t e r a l f orces. E x i s t i n g b u i l d i n g s w i l l be used as our object of study. The b u i l d i n g s w i l l be modelled as c l o s e l y to the r e a l ones as p o s s i b l e . However, i t should be c l a r i f i e d that i t i s not our i n t e n t i o n to a c c u r a t e l y analyze a s p e c i f i c b u i l d i n g , so changes and assumptions w i l l be made to s i m p l i f y our analysis and meet our requirements under the c r i t e r i o n that i t s t i l l remains a r e a l i s t i c b u i l d i n g . 3.2 Load-Bearing Shear Wall B u i l d i n g i n Vancouver A t y p i c a l apartment b u i l d i n g with rein f o r c e d masonry load-bearing shear w a l l s , located i n Vancouver, B.C., i s studied. The b u i l d i n g has a mixed beams and f l a t p l a t e construction 20 Figure 4 T y p i c a l f l o o r plan of load-bearing shear wall b u i l d i n g i n Vancouver supported by shear walls. The shear walls are constructed with reinforced and grouted 8" thick concrete blocks with f = 3,000 m psi. High tensile reinforcing steel of 60,000 psi yield strength is used throughout. Floor to floor heights are 8' - 8" and a typical floor area is about 10,000 sq. f t . The dead load of one storey i s calculated to be about 1,680,000 lbs. which averages out to 168 lbs. per sq. f t . of floor area. The elastic and shear moduli of the reinforced masonry are assumed to be 3,000,000 psi and 1,200,000 psi respectively based on the formulae: E = 1000 f (3.1) m G = 400 f m (3.2) given by the Commentaries on Part 4 of the National Building Code 1975 . They agree with the values used by Kenneth Medearis. Associates i n their analysis of a masonry load-bearing shear wall building, the Park Lane Towers, located i n Denver, Colorado (20) In dynamic a n a l y s i s , the l a t e r a l s t i f f n e s s c o n t r i b u t i o n of walls perpendicular to the response d i r e c t i o n can be neglected and walls p a r a l l e l to the response can be grouped i n t o frames. The shear w a l l s are represented by columns with f i n i t e width. A n a l y s i s i s c a r r i e d out only f o r the N-S d i r e c t i o n i n which the walls are grouped i n t o seven frames. Due to symmetry, only the west h a l f of the b u i l d i n g , as shown i n Figure 4, needs to be analyzed. Frame 1 c o n s i s t s of seven w a l l s . Two slender ones are neglected since t h e i r c o n t r i b u t i o n to the l a t e r a l s t i f f n e s s of the b u i l d i n g w i l l be n e g l i g i b l e . Frame 2 c o n s i s t s of s i x w a l l s . Frame 3 i s represented by four columns. The elevator shaft and the w a l l s surrounding the s t a i r c a s e s are considered to act as one l a r g e column. It i s assumed to be connected to i t s adjacent walls by f i c t i c i o u s beams with n e g l i g i b l e bending s t i f f n e s s i n order to complete a regular g r i d . In the cases where one w a l l i s p a r a l l e l and i d e n t i c a l to the other, they are . combined i n t o one column so as to reduce the degrees of freedom. Frame 4 c o n s i s t s of two long walls and i s shared by both halves of the b u i l d i n g . Thus the member s t i f f n e s s e s i n t h i s frame are halved. 3.3 Park Lane Towers Another masonry load-bearing shear wall building, the Park Lane Towers, with a typical floor plan shown in Figure 5, is analyzed so as to compare with the results of the previous building. The towers had been dynamically analyzed by Kenneth Medearis Associates, and a comparison of the two studies can be used to check our accuracy in building modelling. The Park Lane Towers complex, located in Denver, Colorado, consists of three essentially identical high-rise towers. Each of the towers is about 206' high, having twenty main levels above the ground plus a two-level penthouse. The storey heights are 9' - 2". Each storey has a floor area of approximately 7,800 sq. f t . The towers are of reinforced masonry load-bearing shear wall construction. The masonry walls are 10" thick. Both reinforced grouted brick and block masonry are used. The floor and roof are constructed with 12" precast, prestressed concrete double tees with 2h" poured-in-place con-crete topping. The elastic and shear moduli of the walls are taken as 3,000,000 psi and 1,200,000 psi respectively. Analysis is performed only for the E-W direction. Only 24 Figure 5 T y p i c a l f l o o r plan of Park Lane Towers the north half w i l l be analyzed due to symmetry. Again only those walls aligning with the response are assumed effective in resisting la t e r a l loads. The walls are grouped into three frames in the model. Frame 1 consists of five walls, frame 2 consists of two and frame 3 has one. The effect of the f i n i t e width of the shear walls i s accounted for i n the analysis. 3.4 Infill-Frame Building Great d i f f i c u l t y was encountered in finding a high-rise building with pure frames and i n f i l l s construction. Most frame buildings designed nowadays consist of some shear walls or elevator shafts which act as the principal lateral load resisting members. Dynamic response of this type of structure i s expected to be very similar to that of a load-bearing shear wall building. A pure frame-infill building i s desirable so that a comparison can be made. The design of a reinforced concrete office building located in Vancouver, B.C. is altered to a r e a l i s t i c frame building with masonry i n f i l l s . Each storey is 12' high and has a floor area of approximately 7,000 sq. f t . The framing, as shown in Figure 6, i s 30' i n the E-W and 25' i n the N-S directions. N CN LU UJ LL. Figure 6 T y p i c a l f l o o r plan of i n f i l l - f r a m e b u i l d i n g The floors are 5" thick. The corner and internal columns are 24" x 24". The external or face columns are 24" x 30". The beam sizes are 19" wide and 22" deep (including the thickness of the 5" slab). The columns, beams and floor slabs are of reinforced concrete with an average f^ of 4,000 psi . The corresponding elastic and shear moduli are 3,600,000 psi and 1,440,000 psi respectively. The i n f i l l walls are of masonry material. A shear modulus of 500,000 psi i s adopted based on Benjamin's (21) tests on unreinforced masonry panels and supported by the (22) test results obtained by Blume Only the N-S direction w i l l be analyzed. There is a total of four frames aligning with the response in this direction. Each frame consists of four columns. Due to symmetry, only the two frames on the east side, representing one-half of the building, w i l l be analysed. The outside frame i s f i l l e d com-pletely with wall panels. The other has panels only in one bay. The masonry i n f i l l walls are represented by diagonal struts. The depth effect of the members in the frame (i.e. the f i n i t e size of the joints) i s ignored. CHAPTER 4 DISCUSSION OF THE ANALYTICAL EFFORT 4.1 Computer Program The e x i s t i n g computer program DYNAMIC i s used f o r analyzing the b u i l d i n g s . DYNAMIC i s the concatenation of programs wr i t t e n by several people at the U n i v e r s i t y of B r i t i s h Columbia. It performs l i n e a r e l a s t i c small d e f l e c t i o n dynamic a n a l y s i s of structure problems; pi n - p i n and f i x - f i x members may be used. There are three major parts to the program DYNAMIC STRDYN, FREQ and DYNAM. STRDYN assembles the global s t i f f n e s s and mass matrices by the s t i f f n e s s method. FREQ finds the natural frequencies and mode shapes of the stru c t u r e . DYNAM fi n d s the response of the structure to earthquake ground ac c e l e r a t i o n s . DYNAMIC has been modified to account f o r the s t i f f n e s s c o n t r i b u t i o n of member depths. A l l members are assumed to be three sectioned with the middle s e c t i o n representing the c l e a r 29 l e n g t h of the member and the r i g i d s e c t i o n s at each end r e p r e s e n t i n g the h a l f - w i d t h s of the j o i n t s . The above model i s necessary f o r a n a l y z i n g shear w a l l b u i l d i n g s . I f i t i s d e s i r e d t o n e g l e c t a j o i n t w i d t h , the l e n g t h of that r i g i d s e c t i o n can simply be set equal to zero. E f f e c t s of shear d e f l e c t i o n s as w e l l as f l e x u r a l d e f l e c t i o n s are i n c l u d e d i n f o r m u l a t i n g the member s t i f f n e s s m a t r i x (Appendix A ) . The o r i g i n a l program uses the Newmark Spectrum f o r s p e c t r a l a n a l y s i s . A d i f f e r e n t spectrum i s used f o r the present a n a l y s i s . Thus the subroutine SPECTR i s r e w r i t t e n and a l i s t i n g i s attached i n Appendix G. 4.2 Design Spectrum The dynamic a n a l y s i s procedure described i n the N a t i o n a l B u i l d i n g Code 1975 makes use of a design spectrum i n which the h o r i z o n t a l peak ground motion bounds are given by three i n t e r s e c t i n g s t r a i g h t l i n e s . However, the response spectrum i s found t o be u n r e a l i s t i c . The 1975 Code has ignored the f a c t that the response of a s t r u c t u r e w i l l be equal to the ground motion a t h i g h frequencies (or low p e r i o d s ) . This w i l l i n c r e a s e s i g n i f i c a n t l y the c o n t r i b u t i o n of higher modes to the a c t u a l response of a s t r u c t u r e . 30 PERIOD T. SEC 'Figure 7 Peak ground motion bounds and elastic average response spectrum for 1.0 g max. ground acceleration (from Ref. 30) 31 A more r e a l i s t i c elastic average response spectrum, which appears in the proposed revision to the National Building Code 1975 i s used in this study. The response is assumed to be the same as the acceleration ground motion bound for periods lower than 0.03 sec. For periods within the range of 0.03 sec. and 0.1 sec.j the response i s assumed to vary linearly on the logarithmic plot. The peak ground motion bounds and the elastic response spectrum for 1.0 g maximum ground acceleration are shown in Figure 7. 4.3 Dynamic Analysis and Results The three buildings, which have been modelled as plane frame structures, w i l l be subjected to dynamic analysis. Linear elastic behaviour i s assumed. For convenience, the load-bearing shear wall building i n Vancouver and the Park Lane Towers w i l l be referred to as Structures 1A and IB respectively and the i n f i l l -frame building as Structure 2 in future discussions. The response of the buildings i s calculated by the structural analysis program DYNAMIC. The buildings w i l l be subjected to a horizontal peak ground acceleration of 1.0 g. Storey accelerations and relative displacements due to the f i r s t three response modes and their 32 root-sum-square values w i l l be found. The adjusted accelerations, as discussed i n section 2.2, w i l l also be obtained so as to rationalize the response acceleration at the base of a structure. Due to the linear relation between the response and the ground acceleration, the response can later be scaled linearly for any value of peak ground acceleration desirable. The multipliers for the spectral bounds, that are used in the spectral analysis, are based on 10% of c r i t i c a l damping as recommended by the National Building Code for masonry construction and concrete frame with masonry walls structures. The effects of. v e r t i c a l ground accelerations are not considered since they are usually smaller than the horizontal accelerations. Besides, since the v e r t i c a l vibration frequencies of a building are generally higher than the horizontal ones, coupling of the v e r t i c a l and horizontal modes of vibration w i l l be unlikely. For buildings with small eccentricites, the torsional modes of vibration can also be assumed uncoupled from the horizontal ones for similar reasons. Because of the symmetrical design in the stiffness and mass of the buildings, computed eccentricity between the center of mass and the center of r i g i d i t y at each storey level i s n e g l i g i b l e . Nevertheless, the National B u i l d i n g Code requires a minimum design e c c e n t r i c i t y , e, of 0.05 times the plan dimension, D , to account f o r the a c c i d e n t a l t o r s i o n , which i s intended to n ' account f o r the p o s s i b l e a d d i t i o n a l t o r s i o n a r i s i n g from v a r i a t i o n s i n the estimates of the r e l a t i v e r i g i d i t i e s , uncertain estimates of dead and l i v e loads at the f l o o r l e v e l s , a d d i t i o n of w a l l panels and p a r t i t i o n s a f t e r completion of the b u i l d i n g , v a r i a t i o n of the s t i f f n e s s with time, and i n e l a s t i c or p l a s t i c a c t i o n . T h e a c c i d e n t a l t o r s i o n a l moment tends to increase the i n e r t i a forces or a c c e l e r a t i o n s on one side of the b u i l d i n g and decrease the i n e r t i a forces or a c c e l e r a t i o n s on the other side. The i n e r t i a a c c e l e r a t i o n induced by the moment can be approximated by assuming the f l o o r diaphragm at each storey as a r i g i d rectangular plate with uniform storey mass and thickness, and by studying the k i n e t i c s of the p l a t e . For a square p l a t e with dimension D n and storey mass m, i t s mass moment of i n e r t i a about the v e r t i c a l c e n t r o i d a l axis i s : (4.1) I t i s shown i n Figure 8 that the t o r s i o n a l a c c e l e r a t i o n caused by the minimum design t o r s i o n a l moment, v a r i e s l i n e a r l y across the width, D n, i n the d i r e c t i o n perpendicular to the earthquake loading. This t o r s i o n a l a c c e l e r a t i o n i s maximum at the two ends with magnitude 34 Figure 8 T o r s i o n a l induced a c c e l e r a t i o n x which is found to be 0.15 times the translational inertia a acceleration, x. To account for the effect of accidental torsion, an adjustment factor of 1.15 can be used to raise the inertia forces or response accelerations obtained from the translational dynamic analysis by 15% to arrive at an upper bound value. In order to study the correlation between the response and the fundamental period of a structure, buildings of different heights are analyzed. This i s achieved by adding to or reducing the number of storeys of the building. Since a l l structure stiffnesses and storey masses (except the roof) are assumed the same for each storey, only the mass of the top storey needs to be changed. For ease of comparison, the storey level in each model w i l l be represented by the dimensionless ratio, h/H, in which h i s the height of the n*"*1 storey and H i s the height of the entire building. 4.3.1 Load-Bearing Shear Wall Buildings For Structure 1A, 5 models are analyzed. Computer runs are made for models with 5, 10, 15, 20 and 25 storeys. In each model, the mass of the top storey i s assumed to be half of the others. The f i r s t three natural periods are as follows: 36 No. of Storey T - C s e c ) T 9(sec) T„(sec) 5 0.115 0.034 0.018 10 0.313 0.084 0.041 15 0.592 0.150 0.071 20 0.960 0.230 0.106 25 1.417 0.321 0.145 A l l results are plotted for observation. The R-S-S relative displacements are plotted in Figures 9a - 9b. The R-S-S along with the adjusted accelerations are plotted in Figures 10a -lOe. The adjusted accelerations are plotted against the fundamental periods at h/H = 0.2, 0.4, 0.6, 0.8, 1.0 respectively in Figure 11. For Structure IB, the original tower i s analyzed as a 20-storey structure, with the two-level penthouse included in the top storey. Results are compared with those obtained by Kenneth Medearis Associates (KMA). KMA had analyzed the tower experimentally by taking measurements of the tower micro vibrations at various levels and theoretically by model analysis. Both the KMA and the present results are tabulated below for comparison. 37 1.0 0.8 ^ 0.6 H—t LU X _ l < r-o U_ ° 0.4 o M r-O 0.2 0.0 A 5-STOREY MODEL • 10-STOREY MODEL 0.0 0.1 0.2 0.3 DISPLACEMENT ( f t ) Figure 9a Structure 1A: R-S-S storey displacements DISPLACEMENT ( f t ) Figure 9b Structure 1A: R-S-S storey displacements 39 40 41 0.0 2.0 4.0 ACCELERATION (g) Figure 10c Structure 1A: Storey accelerations of 15-storey model 42 43 Figure lOe Structure 1A: Storey accelerations of 25-storey model Figure 11 Variation of response acceleration with fundamental period at levels h/H = 0.2, 0.4, 0.6,0.8,1.0 of Load-Bearing Shear Wall Buildings 45 T^sec) T 2(sec) T 3(sec) KMA. (experimental) 0. 83 0.27 0.14 KMA(theoretical) 0. 95 0.25 0.12 Present study 1. 12 0.28 0.13 Despite the limited information on the Park Lane Towers, which meant that several details were asssumed, our results compare very closely with those of KMA. One may conclude that our procedure for modelling the building was reasonably similar to that used by KMA. Computer runs are also made for models with 15, 10 and 5 storeys. This time the mass of the top storey i s l e f t to be the same as the others. The adjusted accelerations versus the fundamental periods are also plotted in Figure 11. 4.3.2 Infill-Frame Building For Structure 2, models with 5, 8, 10, 15 and 20 storeys are analyzed. As for the type 1 structures, in each model the mass of the top storey i s assumed to be half of the others. The f i r s t three natural periods of the models are as follows: Storey T^sec) T 0(sec) T 3(sec 5 0.210 0.070 0.042 8 0.380 0.119 0.065 10 0.519 0.156 0.082 15 0.960 0.260 0.130 20 1.537 0.381 0.184 The R-S-S relative displacements are plotted in Figures 12a - 12b. The accelerations are plotted in Figures 13a - 13e. The adjusted accelerations versus the fundamental periods at h/H = 0.2, 0.4, 0.6, 0.8, 1.0 are plotted in Figure 14. DISPLACEMENT (ft) Figure 12a Structure 2: R-S-S storey displacements 48 1.0 0.8 h 0.6 x X o LU X _J < c5 0.4 h-u_ o z : o i—i u 0.2 / •——IB u » A i | * / / D £ O / / / / m l e / / / / // • A • 'A' / / I- • / • /// h B A 0 h80 0 • 10-STOREY MODEL A 15-STOREY MODEL • 20-STOREY MODEL 0.0 « 1 —I 1 I 0.0 1.0 2.0 DISPLACEMENT ( f t ) F i g u r e 12b S t r u c t u r e 2: R-S-S s t o r e y displacements 49 Figure 13a Structure 2: Storey accelerations of 5-storey model 0 .0 2.0 4.0 ACCELERATION (g) Figure 13b Structure 2: Storey accelerations of 8-storey model 51 0.0 2.0 4.0 ' ACCELERATION (g) Figure 13c Structure 2: Storey accelerations bf 10-storey model 52 53 0.0 2.0 4.0 ACCELERATION (g) Figure 13e Structure 2: Storey accelerations of 20-storey model Figure 14 Variation of response acceleration with fundamental period at levels h/H = 0.2, 0.4, 0.6, 0.8, 1.0 of Infill-Frame Buildings CHAPTER 5 RESULTS-ANALYSIS AND RECOMMENDATIONS 5.1 Observations From the storey displacements plots, i t i s observed that both the shear wall buildings and the infill-frame buildings have a combined shear beam and cantilever behaviour. Cantilever action is seen to predominate; especially at the lower levels of the buildings. Some portal action i s observed at the upper levels. As for the infill-frame buildings, this indicates that the i n f i l l walls contribute significantly to the latera l stiffness of a frame building. The frames are changed from a supposedly shear beam behaviour to a predominantly cantilever behaviour due to the i n f i l l stiffnesses. It i s also observed that the higher modes become more significant in buildings of long periods. Their effect i s more easily recognized at the upper levels, h/H =0.6 and 0.8, from the R-S-S storey acceleration plots. From the a c c e l e r a t i o n versus period p l o t s , i t i s seen that the response a c c e l e r a t i o n of a structure i s period dependent. For a l l three types of structures that were analyzed, there seems to be a s i m i l a r c o r r e l a t i o n between the response a c c e l e r a t i o n at given l e v e l s and the period. Separate a c c e l e r a t i o n at d i f f e r e n t l e v e l s versus period curves are p l o t t e d i n Figures 15a - 15e with r e s u l t s from a l l three Structures p l o t t e d on the same graph. I t i s seen that a l l data f a l l roughly on the same curve. An upper bound i s drawn f o r each curve to include a l l data points. These curves can be i n t e r p r e t e d as the envelopes of dynamic a m p l i f i c a t i o i f a c t o r f o r masonry b u i l d i n g s . 5,2 L e v e l C o e f f i c i e n t The National B u i l d i n g Code recommends the e q u a t i o n . V = ASKIFW (5.1) f o r the c a l c u l a t i o n of the base shear of a s t r u c t u r e , where A i s the assigned h o r i z o n t a l design ground a c c e l e r a t i o n given i n the Table of C l i m a t i c Data i n Part 2 of the Code. The base shear i s then d i s t r i b u t e d along the height of the b u i l d i n g by assuming a l a t e r a l i n e r t i a f o r c e d i s t r i b u t i o n that i s approximately t r i a n g u l a r 1 . 0 0.1 Figure 15b V a r i a t i o n of °-6 0 . 8 1.0 PERIOD (sec) response a c c e l e r a t i o n with fundamental 2.0 3.0 period at h/H =0.4 CO 6.0 4.0 « 3.0 o LU Lu 2.0 CJ < Q LU 00 1.0 0.1 T r i s r J I I L i — i — i r o STRUCTURE IA Q STRUCTURE IB A STRUCTURE 2 T r J i « ' » _L J L 2.0 0,2 0.3 0,4 0.6 0.8 1.0 PERIOD (sec) Figure 15c Variation of response acceleration with fundamental period at h/H =0.6 3.0 6.0 60 2 LU _! LU (_) U < Q LU r-LO ID "2 Q < 4.0 3.0 2.0 • 0 -0.1 ® STRUCTURE 1A . • STRUCTURE IB A STRUCTURE 2 °'2 °'3 0.4- ' 0.6 0.8 1.0 PERIOD (sec) Fig u r e 15d V a r i a t i o n of response a c c e l e r a t i o n w i t h fundamental 2.0 p e r i o d a t h/H =0.8 3.0 O 6.0 60 4.0 3.0 © 1 1 ~T i r i — j — i r © STRUCTURE IA e STRUCTURE IB A STRUCTURE 2 T r LU _J LU U o < Q LU H-00 2.0 1.0 J I _L ± - — I I I I L J L 0.1 0.2 0.3 0.4 0.6 0.8 1.0 PERIOD (sec) 2.0 3.0 Figure 15e V a r i a t i o n of response a c c e l e r a t i o n with fundamental period at h/H = 1.0 62 in shape with the apex at the base for stubby structures with fundamental periods less than about 1 second. For more slender structures, the Code suggests a redistribution of forces by applying part of the base shear as a concentrated force at the top of the structure. These procedures are intended for developing a shear force envelope, but the inertia forces so obtained give, in effect, forces at any floor which can be used for the design of masonry walls. For the design of parts of buildings, the Code gives a lateral force, V ( 1 0 ) • • P V = AS W (5.2) P P P ' The values of are given as 10.0 for cantilever walls and 2.0 for other exterior or interior walls. The design force given by Equation (5.2) appears to be a b i t crude. It should be graded, at least p a r t i a l l y , for height as implied by the Code suggested shear force envelope. With the available response results on hand, la t e r a l forces at any floor for use in design of masonry walls can be obtained by the introduction of a level coefficient (or level dynamic ampli-fication factor), a. Similarly, a more accurate distribution of 63 shear or inertia forces for masonry buildings can also be determined. The inertia acceleration at the i t h floor can b e expressed a. = a. A S K I F ( 5 . 3 ) where a ± = level coefficient at level i . With the inertia acceleration known, the inertia forces at the .th x rloor can be obtained by: V = a i Wp (5.4) F. - a. W. t r c \ i x i (->.5) where V = later a l force on a part of the structure (eg. walls) at level i , W^  = weight of a part of a structure, F. = inertia force at level i , x ' W^  = the portion of dead load which i s located at level i . For simplicity, a building can be divided into five equal sections along i t s height. The values of a at h/H = 0.2, 0.4, 0.6, 0.8, 64 1.0 can be obtained from Figures 15a - 15e. These values should be increased by 15% to include the e f f e c t of a c c i d e n t a l t o r s i o n s . For intermediate h/H values, the a can be assumed constant and equal to that corresponding to the next larger h/H r a t i o . 5.3 Fundamental Period of Buildings In l i e u of more accurate estimates, the Code has recommended . . , f , (10) the empxrxcal formula : T (5.6) f o r the determination of the fundamental period f o r b u i l d i n g s . It i s assumed i n the formula that the period v a r i e s l i n e a r l y with the t o t a l height of a b u i l d i n g for the same dimension D. This assumption may be true for s t e e l b u i l d i n g s , i t i s d e f i n i t e l y untrue f o r r e i n f o r c e d concrete or masonry load-bearing shear w a l l and i n f i l l - f r a m e b u i l d i n g s . The periods obtained by the empirical formula, T , are compared to those obtained by computer analyses, T , for both Structures IA and 2 (see Table 1). Both the load-bearing shear w a l l and i n f i l l panel systems are found to be s t i f f e r than expected from the empirical formula. The.empirical period exceeds the computed value by as much as .140% in the worst prediction; the error is greatest with the lowest structures. The ratio T /T is c e plot t e d against T £ in Figure 16. The following formula i s derived to predict the fundamental period of reinforced concrete or masonry load-bearing shear wall and i n f i l l frame buildings: T = (0.35 + 0.5T )T~ e e 2 or T = 0.0175 — + 0.00125 ~ (5.7) fi D where D is the dimension of the building in feet in a direction parallel to the applied forces. • Table 1 COMPARISON OF FUNDAMENTAL PERIODS No. of Stories Height of (H - f t . Bldg. ) Period by Computer (T c - sec) Period by Formula ,rp 0.05H IT - sec) / D Percentage Error in T e(%) T Ratio -~ T e Structure 1A: D = 60 f t . 5 10 15 20 25 43.3 86.7 130.0 173.3 216.7 0.115 0.313 0.593 0.959 1.417 0.280 0.559 0.839 1.119 1.399 + 143 + 79 + 41 + 17 1 0.41 0.56 0.71 0.86 1.01 Structure 2: D = 75 f t . 5 8 10 15 20 60.0 96.0 120.0 156.0 192.0 0.210 0.380 0.519 0.961 1.537 0.346 0.554 0.693 1.039 1.386 + 65 + 46 + 34 + 8 - 10 0.61 0.69 0.75 0.92 1.11 CHAPTER 6 MINIMUM STEEL REQUIREMENTS 6.1 Introduction A major earthquake i s the most severe loading to which a b u i l d i n g might ever be subjected. This i s e s p e c i a l l y true f o r b u i l d i n g s located i n seismic Zone 3 where major earthquakes w i l l most l i k e l y occur. For t a l l b u i l d i n g s with masonry w a l l s , s t e e l reinforcement i s required to s a f e l y guard the walls from f a i l u r e . The amount of s t e e l required depends on the earthquake forces the w a l l has to r e s i s t . There are two major types of forces which have to be considered: 1. Racking forces (In-plane shearing), and 2. Transverse forces (Out-of-plane bending). 6.1.1 Racking Forces i I t i s r e a l i z e d that w a l l s , e i t h e r load-bearing shear walls or i n f i l l panels, are the dominant l a t e r a l load r e s i s t i n g 69 elements in a building. Unreinforced masonry subjected to i n -plane shear forces is presumed to f a i l in a b r i t t l e manner, and the minimum steel as required by the Code is intended to give rise to a ductile behaviour. In the minimum earthquake force design formula, Equation (5.1), the coefficient K reflects the damping and expected d u c t i l i t y of a structural system. Types of construction that are recognized to have performed well in earthquakes are assigned lower values of K. For reinforced masonry shear wall buildings and frame buildings with masonry i n f i l l s , the assigned K value i s 1.3 which can be shown to correspond to an expected d u c t i l i t y of about 2. S o the key to masonry seismic design requirements for in-plane shear forces i s to design a wall panel which has a ductile be-haviour consistent with the specified K coefficient. In order to achieve this, experimental effort is required by testing wall panels with various amounts of steel reinforcement. Only then can the amount of steel necessary to provide the required degree of d u c t i l i t y for a particular seismic zone be determined. Such effort is not attempted in this thesis. j 6.1.2 Transverse Forces thquake-induced response accelerations of a building 70 produce i n e r t i a forces i n elements, a r i s i n g from t h e i r own masses. The i n e r t i a force on a wall panel can be analyzed as a uniform load a c t i n g normal to the face of the w a l l by assuming that the whole w a l l has the same a c c e l e r a t i o n . With the maximum dynamic response already obtained f o r the masonry w a l l b u i l d i n g s , the i n e r t i a forces acting on the walls can be determined by Equation (5.4) given the input peak ground a c c e l e r a t i o n appropriate to s p e c i f i c zones and s i t e s . These i n e r t i a forces w i l l cause out-of-plane bending i n the w a l l s . Since masonry has a f a i r l y low t e n s i l e strength, s t e e l reinforcement i s needed.in the high tension areas. The behaviour of r e i n f o r c e d masonry i s quite s i m i l a r to that of r e i n f o r c e d concrete. By assuming that the masonry walls are coherent with bonded reinforcement and that the edge conditions provide adequate support, the same p r i n c i p l e s a p p l i c a b l e to reinfo r c e d concrete may be employed f o r meeting the minimum s t e e l requirements f o r out-of-plane bending i n masonry wa l l s . 6.2 Design Earthquake It i s generally recognized that a major earthquake w i l l generate a much greater input than the Code s p e c i f i e s . An express-ion f o r the peak ground ac c e l e r a t i o n s that are a c t u a l l y implied by 71 the q u a s i - s t a t i c procedure of the National B u i l d i n g Code has been (23) developed by Anderson, Nathan and Cherry . Their i n t e r p r e t a t i o n of the Code procedure can be summarized as follo w i n g : Working l e v e l load = ASKIFW Y i e l d or ultimate l e v e l load = XASKIFW E l a s t i c load for major earthquake = uXASKIFW Average s t r u c t u r a l response a c c e l e r a t i o n as a f r a c t i o n of g = uXASKIF , average a c c e l e r a t i o n Spectral a c c e l e r a t i o n = 2 — — •— p yXASKIF „ . ' s p e c t r a l a c c e l e r a t i o n Ground a c c e l e r a t i o n = — c • uXASKIF BD (6.1) where X = load f a c t o r , u = d u c t i l i t y r a t i o corresponding to the K c o e f f i c i e n t , 6 = 0.6 for a uniform c a n t i l e v e r i n the f i r s t mode, D = dynamic a m p l i f i c a t i o n f a c t o r i n the f i r s t mode, E l a s t i c Load = i n e r t i a force on hypot h e t i c a l b u i l d i n g with same i n i t i a l s t i f f n e s s which remains ' e l a s t i c . By using the developed expression the ground a c c e l e r a t i o n implied by the quasi-static procedure was found to vary with the funda-mental period of a building and the type of structure. The implied ground acceleration tends to increase with the period of a structure, T (since the ratio S/D in Equation (6.1) increases with T). The Code appears to be conservative for structures with long periods. By applying Equation (6.1) to the seismic Zone 3 where A = 0.08 with y = 2, X = 1.8, K= 1.3, I =1, F = 1, 8 = 0.6, S = 0.5 and D = 1.1, i t can be shown that .the ground acceleration is equal to 0.28 of gravity. The values of S and D correspond to a damping ratio of 10% and a period of 1 second, which represent a typical medium high-rise masonry construction. LOAD ELASTIC LOAD = yXV ULTIMATE LOAD = XV / I 7i"~1 WORKING LOAD = V = ASKIFW / I I D I S P L A C E M E N T Figure 17 Effect of y and X on the design load 6-3 Design Formula for Masonry Walls in Bending Face loading perpendicular to the masonry surface has (24) received much research attention for unreinforced walls Combined bending and compression loading of unreinforced walls (25) has also been researched . Scrivener, on the other hand, has conducted a series of face loading tests on reinforced brick Of.\ walls . In the tests, i t was found that the ultimate load could be predicted to within a few percent by considering the brick wall as a l i g h t l y reinforced wide beam and applying the ultimate moment theory as for reinforced concrete. The stress-strain curve for the brick masonry was assumed to be the same as that for concrete in order that the concrete constant 0.59 could be used. Thus the Whitney Formula: \ = A s f y [ d " ° - 5 9 ( A s f y / f c b ) ] ( 6 - 2 ) where M = ultimate moment, u ' A = cross-sectional area of steel, s f = yield stress of steel, y d = depth to center of gravity of steel, b = beam width, f^ = concrete compressive strength. I 74 with replaced by the masonry compressive strength, f^, can be used for the design of masonry walls. The Whitney Formula i s satisfactory for very l i g h t l y reinforced masonry because the second term within the bracket, which contains 0.59 and f , c affects the results by only a small percentage. 6.4 Wall Panel Designing The Code-implied ground acceleration corresponding to A - 0.08 (Zone 3) for masonry buildings with fundamental period ol 1 second has been found to be 0.28 g. With an amplication factor obtained from Figure 15e at T = 1 second, the maximum response acceleration (including the accidental torison effect) i s calculated to be 0.9 g at the top storey of a building. The corresponding uniform face load, q , on a wall panel can be assessed with the wall density known. By assuming isotropic properties and appropriate boundary conditions of the walls, the maximum bending moment due to the face load can be obtained. By assuming that the steel reinforcement i s placed along the center, the wall can then be designed for this maximum bending moment according to the ACI code by using Equation (6.2), and applying the appropriate load and capacity reduction factors. 75 By applying the Code-given formula, Equation (5.2), in the design of parts of buildings for lateral forces, the design inertia accelerations corresponding to A = 0.08 are computed to be 0.8 g and 0.16 g for cantilever walls and other interior or exterior walls respectively. The value for cantilever walls compares closely with the maximum 0.9 g from the present analysis but the 0.16 g for exterior or interior walls i s very much below our present value and seems to be unfit for design application. 6.4.1 Load-Bearing Shear Wall A continuous shear wall can be assumed fixed to the slabs at floor levels and attached at some points to perpendicular walls. In general, each shear wall panel can be modelled as a plate with three edges fixed and the fourth edge free as shown in Figure 18(i). A wall panel of dimension a x b is assumed to be 8" thick and has a density of 90 psf. From Reference (28), the bending moment i s found to be maximum when a/b = 0.75, where: IM I = 0.107 q a 2 (6.3) 1 y 'max o This bending moment i s compared to those for walls with other possible boundary conditions and is proved to be the highest. The t ( i ) FIXED / / / / / / / > • / / / / / / / / / thickness t UJ LU LL FIXED b to oo ( i i ) S.S. thickness t S.S. oo 00 Figure 18 Models for ( i ) shear w a l l panel and ( i i ) i n f i l l panel 77 panel i s then designed to the maximum moment. The steel area required i s found to be dependent on the wall dimensions. It i s approximately proportional to the square of the height to thickness ratio, a/t, of the wall. The vertical steel area is calculated to be 0.0006 bt for a/t = 12, where bt i s the gross cross-sectional area of the wall. The ver t i c a l compressive load on the wall i s ignored in the calculation because compressive loads w i l l increase the ultimate bending capacity of a wall as only failures i n i t i a t e d by yielding of the tension steel are of concern. The steel area thus obtained i s on the conservative side. The maximum bending moment in the other direction for the same panel i s : |M I = 0.05 q a 2 (6.4) ' x 'max no which gives a design horizontal steel area of 0.0002 bt for a/t = 12. The total steel area adds up to 0.0008 bt. For an a/t ratio of 18, the total area of steel required i s 0.0018 bt. 6.4.2 I n f i l l Wall An i n f i l l wall panel of height a and width b can be modelled as a plate with four sides simply supported. A wall panel is assumed to be 8" thick and has a density of 60 psf. For a square plate (a/b = 1.0), the maximum bending moment i s : l M I = IM I = 0.046 q a 2 Ks ' x 'max 1 y 'max Ho (6.5) The corresponding design steel is 0.0004 bt in both the ver t i c a l and horizontal directions which gives a total steel area of 0.0008 bt for a/t = 18. For a/b = 0.2, which i s the same as a one-way slab, Reference (29) gives: |M I = 0.125 q a 2 (6.6) y 'max o^ |M [ = 0.0375 q a 2 / F I . 7 N x 'max Mo V . O . / ; The v e r t i c a l steel is calculated to be 0.001 bt and the horizontal steel 0.0001 bt which give a total of 0.0011 bt for a/t = 18. CHAPTER 7 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 7.1 Summary and Conclusions In order to e s t a b l i s h the minimum percentages of s t e e l for the seismic design of masonry wa l l s , the racking and t r a n -sverse forces caused by a major earthquake on the walls have to be considered. The minimum s t e e l with respect to racking forces i s decided by the d u c t i l i t y requirements as implied by the Buil d i n g Code. The reinforcement with respect to transverse forces i s decided by the wall's a b i l i t y to r e s i s t i n e r t i a forces due to i t s own mass. A study has been c a r r i e d out to e s t a b l i s h the minimum s t e e l required i n masonry walls for a major earthquake with respect to the transverse forces. Buildings with masonry load-bearing shear walls and frame b u i l d i n g s with masonry i n f i l l panels were modelled as plane frame structures. An e x i s t i n g computer program DYNAMIC with some modifications was used f o r the l i n e a r e l a s t i c dynamic analysis. The response of the structure was found by the response spectrum approach. It i s assumed that the response accelerations so obtained provide the upper limits and are con-servative. Findings based on the results of the dynamic analysis are summarized as followings: (a) The base response acceleration of a structure obtained by the spectral analysis is zero which i s obviously unrealistic. Equation (2.7) is recommended to adjust the accelerations so as to normalize the base acceleration to the ground acceleration. ( b ) A l l models analyzed, including the infill-frames, are observed to behave predominantly lik e cantilevers with some portal action at the top of the buildings. The i n f i l l panels contribute significantly to the stiffness of a frame building. (c) The results of the Park Lane Towers are compared with those reported by Kenneth Medearis Associates. They agree very closely with each other. (d) The buildings analyzed are found to be s t i f f e r than expected from Equation (5.6) given by the Code. A more appropriate empirical formula, Equation (5.7), i s recommended for estimating the fundamental periods of masonry buildings. 81 (e) A correlation between the response acceleration and the fundamental period of vibration of a structure i s developed for levels h/H = 0.2, 0.4, 0.6, 0.8, 1.0. (f) The la t e r a l load design formula, Equation (5.2), for parts of buildings, given by the National Building Code, appears to be very unconservative. A level coefficient i s recommended for finding the inertia acceleration envelope. (g) In order to account for the accidental torsion, the response forces or accelerations obtained should be increased by 15%. (h) The maximum response acceleration of the top storey of a typical masonry high-rise building in Zone 3 i s found. The minimum steel required in the masonry walls to resist the inertia forces can be determined by applying the Whitney Formula. For a height to thickness ratio a/t = 18, the total steel cross-sectional area required i s computed to be 0.0018 and 0.0011 times the gross cross-sectional area of the wall respectively for the load-bearing shear wall and the i n f i l l wall. The amount of steel required i n other levels or zones can be established similarly. 7.2 Recommendations for Future Research The minimum seismic requirement for masonry walls 82 established i n t h i s t h e s i s considers only transverse forces. A c c r.plete study should include racking forces. Therefore, the l o g i c a l next step i n the development of the minimum seismic s t e e l i s to test w a l l panel specimens with various amount of reinforcement subjected to racking loads and s i n g l e out the one that meets the d u c t i l i t y requirements. Preliminary t e s t s should be undertaken with s t a t i c loadings, but i t i s envisaged that a very s i m i l a r set and number of specimens would be tested under a dynamic c y c l i c loading, simulating earthquake response. Further t e s t s on a few f u l l scale r e i n f o r c e d walls w i l l be h e l p f u l to confirm the hypotheses from the specimen t e s t i n g . Furthermore, tin; arrangement of s t e e l such as the maximum spacing, the necessary amount i n the d i r e c t i o n normal to the span, and so f o r t h , have to be established experimentally. 83 REFERENCES 1. Sahlin, S. Structural Masonry. Prentice-Hall, Inc., Englewood C l i f f s , N.J., 1971. 2. Amrhein, J.E. Reinforced Masonry Engineering Handbook. 2nd Ed., Masonry Institute of America, Los Angeles, California, 1973. 3. Blume, J.A., Newmark, N.M. and Corning, L.H. Design of Multi-story Reinforced Concrete Buildings for Earthquake Motions. Portland Cement Association, U.S.A., 1961. 4. Wiegel, R.L., Ed., Earthquake Engineering. Prentice-Hall, Inc., Englewood C l i f f s , N.J., 1970. 5- Newmark, N.M. and Rosenblueth, E. Fundamentals of Earthquake Engineering. Prentice-Hall, Inc., Englewood C l i f f s , N.J., 1971. 6. Fi n t e l , M., Ed., Response of Multistory Concrete Structures to Lateral Forces. Publication SP-36, American Concrete Institute, Detroit, 1973. 7. Beaufait, F.W., Ed., Proceedings of the Symposium on T a l l Buildings - Planning, Design & Construction. Nashville, Tennessee, November 14-15, 1974. 8. Coull, A. and Smith, B.S., Ed., T a l l Buildings. Oxford, N.Y., Symposium Publications Division, Pergamon Press, 1966. 9. Timoshenko, S., Young, D.H. and Weaver, W. Jr. Vibration Problems in Engineering. 4th Ed., John Wiley & Sons, Inc., U.S.A., 1974. 10. National Building Code of Canada 1975. National Research Council of Canada. Associate Committee on the National Building Code, Ottawa, NRCC No.13982. 11. Commentaries on Part 4 of the National Building Code of Canada 1975. National Research Council of Canada. Associate Committee on the National Building Code, Ottawa, NRCC No. 13989. 12. Building Code Requirements for Reinforced Concrete (ACI 318-71). American Concrete Institute, Detroit, Michigan. 84 13. Cherry, S. "Basic Principles of Response of Linear Structures to Earthquake Ground Motions", Proceedings of the Symposium on Earthquake Engineering, University of Br i t i s h Columbia, September 8-11, 1965. 2-. Hudson, D.E. "Some Problems in the Application of Spectrum Techniques to Strong Motion Earthquake Analysis" Bui. Seismological Society of America, Vol. 52, No.2, 1962. 15. Newmark, N.M. and Hall, W.J. "Seismic Design C r i t e r i a for Nuclear Reactor F a c i l i t i e s " Proceedings of the Fourth World Conference on Earthquake Engineering, Vol. 2, Santiago De Chile, January 13-18, 1969. 16. Pecknold, D.A. "Slab Effective Width for Equivalent Frame Analysis". ACI Journal, April 1975, pp. 135-137. 17. Tso, W.K. and Mahmond, A.A. "Effective Width of Coupling Slabs in Shear Wall Buildings". Journal of Structural Division, ASCE, March 1977, pp. 573-586. 18. Polyakov, S.V. Masonry in Frame Buildings: An Investigation into the Strength and Stiffness of Masonry I n f i l l i n g . (English Translation) Moscow, 1957. 19. Polyakov, S.V. "On the Interaction Between Masonry F i l l e r Walls and Enclosing Frame when loaded in the Plane of the Walls". Translations in Earthquake Engineering, Earthquake Engineering Research Institute, San Francisco, 1960. 20. Kenneth Medearis Associates "An Investigation of the Dynamic Response of the Park Lane Towers to Earthquake Loadings". Report to Colorado Masonry Institute, November 1973. 21. Benjamin, J.R. and Williams, H.A. "The Behaviour of One-Story Brick Shear Walls". Proceedings of ASCE, Structural Division, N.Y., July 1958. 22. Blume, J. and Associates "Shear in Grouted Brick Masonry Wall Elements". Report to Western States Clay Products Association, San Francisco, California, 1968. 23. Anderson, D.L., Nathan, N.D. and Cherry S. "Correlation of Static and Dynamic Earthquake Analysis of the National Building Code of Canada 1975". Department of C i v i l Engineering, University of British Columbia, 1977. 85 24. Cox, F.W. and Ennenga, J.L. "Transverse Strength of Concrete Block Walls". ACI Journal, May 1958 Proc. 54, pp. 951-960. 25. Yokel, F.Y., Mathey, R.G. and Dikkers, R.D. "Strength of Masonry Walls under Compressive and Transverse Loads". National Bureau of Standards Report, U.S. Dept. of Commerce, Washington D.C, B u i l d i n g Science Series 34, March 1971 2 c . Scrivener, J.C. "Face Load Tests on Reinforced Hollow-brick Non-load-bearing Walls" New Zealand Engineering, J u l y 1969, p p . 215-220. 27. Scrivener, J.C. "Reinforced Masonry - Seismic Behaviour and Design". B u l l e t i n of N.Z. Society f o r Earthquake Engineering, V o l . 5, No.4, December 1972, pp. 143-155. 28. Bares, R. Tables f o r the Analysis of P l a t e s , Slabs and Diaphragms Based on the E l a s t i c Theory. Translated by Carel van Amerongen, Bauverlag Gmbh., Wiesbaden und B e r l i n g (Germany), 1969. 29. Timoshenko, S. and Woinowski - Krieger, S. Theory of Plates and S h e l l s , 2nd E d i t i o n , McGraw - H i l l Book Company, U.S.A., 1959. 30. Commentaries on Part 4 of the National B u i l d i n g Code of Canada 1977. Associate Committee on the National B u i l d i n g Code, Ottawa, NRCC No. 15558. 31. Hrennikoff, A. "Solution of Problems of E l a s t i c i t y by the Framework Method", Journal of Applied Mechanics, ASME, December 1941, p.A160. APPENDIX A BEAM STIFFNESS MATRIX FOR WIDE COLUMN FRAME Fi g u r e 1 9 ( i ) shows a beam spanning between two shear w a l l s (columns). P o i n t s A and B are on the c e n t r o i d a l axes of the w a l l s at beam l e v e l and are the node p o i n t s . The beam element can be modelled as shown i n Fi g u r e 1 9 ( i i ) . Normally the member s t i f f n e s s f o r a beam element i s w i i t t e n i n terms of the deformations at i t s ends. ( i ) A . .B b ( i i ) Figure 19 Beam element for wide column frame For convenience, the s t i f f n e s s matrix can be broken up into two parts, the a x i a l terms and the bending terms (K^)• 88 K = K a + (A.2) For an ordinary beam of length £' with both ends f i x e d , '^ N_6 7 F = K A l i s s t i f f n e s s matrices with shear deformations included are: K AE 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 EI £3(l+g) where g 0 0 0 0 0 0 14.4EI 2 AG I 0 12 61 0 -12 61 0 61 £2(4+g) 0 -61 £2(2-g) 0 0 0 0 0 0 0 -12 -61 0 12 -61 0 61 * 2(2-g) 0 -6£ £ 2(4+g) 89 Since the equilibrium equations of the structure are wri t t e n i n terms of the forces and deformations at the node points A and B, a transformation i s necessary. 2 ? i — \ t 1,2,3 4,5,6 A w M '1 .(— L K ' _ 31 F = K A For .small deformations, the forces at the ends of the beam, f, are re l a t e d to the nodal forces, f, by the expression: where T = 1 F = 1 F 1 0 0 0 0 0 0 1 a 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 -b 0 0 0 0 0 0 (A. 3) It can be shown that: K K 1 (A.4) where = transpose of 90 In general, the structure co-ordinates do not coincide with the member co-ordinates. It i s necessary to define a s t i f f n e s s n a t r i x i n terms of the nodal forces and deformations i n the structure c o - c r d i n a t e s (F, A). F = KA The r e l a t i o n between the nodal forces i n structure co-ordinates and those i n member co-ordinates are: T F (A. 5) where T = I t can be shown that X - y 0 0 0 0 y X 0 0 0 0 0 0 L 0 0 0 0 0 0 X - y 0 0 0 0 y X 0 0 0 0 0 0 L K = = * T K T (A.6) where T = transpose of T S u b s t i t u t i n g (A.4) into (A.6) gives K = T ?1 K T .J* T* (A.7) or K = T T. K T, T a 1 a 1 By c a r r y i n g out the matrix m u l t i p l i c a t i o n s , K& and are found as f o l l o w i n g : K AE L 2 I EI 3 2 L £ (1+g) 2 X xy 0 2 -x -xy 0 xy 2 y 0 -xy 2 -y 0 0 0 0 .0 0 0 2 -x -xy 0 2 X xy 0 -xy 2 -y 0 xy 2 y 0 0 0 0 0 0 0 12y2 -12xy -W 12xy -k 2y -12xy 12x2 k^x 12xy -12x2 k 2x -\y k^x k3 -k^x k4 -W 12xy k l Y W -12xy k 2 y 12xy -12x2 -k^x -12xy 2 12x -k 2x -k 2y k 2x k4 -k 2x k5 92 where 1^ = 6L(2a + £) k 2 = 6L(2b + £) k 3 = 12L 2a(a + £) + L 2£ 2(4 + g) k^ = 12L2ab + 6L2£(a + b) + L 2£ 2(2 - g) k 5 = 12L2b(b + £) + L 2£ 2(4 + g) The complete beam stiffness matrix, K , for the wide column frame, in the structure co-ordinates can be obtained by adding K q and V APPENDIX B MODELLING OF INFILL WALLS An i n f i l l w a l l which i s assumed to r e s i s t only shear forces can be represented by two diagonal s t r u t s . A w a l l panel oi: dimension a x b and thickness t i s shownin Figure 20(i). The panel has a shear modulus G. The equivalent s t r u t model represent-ing the w a l l panel i s shown i n Figure 20(ii). Each s t r u t has a c r o s s - s e c t i o n a l area A and an e l a s t i c modulus E. A constant shear s t r e s s , T, i s assumed ac t i n g on the wall panel. I t i s the same as having concentrated forces acting at the four corners of the panel as shown i n Figure 20(i). The same forces and shear d e f l e c t i o n are assumed i n the s t r u t model. From Figure 20(i), T = G - (B.l) a From Figure 2 0 ( i i ) , s t r u t elongation 6 ' = <5 (B.2) s t r a i n f o r c e F = M 61 L 94 Figure 20 Equivalent s t r u t model of i n f i l l w a l l From the free body at point 0 in Figure 20(111), rtb T~ = 0 (B.4) By substituting (B.l) a n d ( B. 3) into (B.4), - • M b G tb L L L 6 ~ a 6 2 ~ = 0 A = £ tL^ E 2ab (B.5) The equivalent strut area i s calculated for a masonry wall with a = 12 f t . b = 25 f t . t = 4 in. G = 500,000 psi The t = 4" corresponds to a wall thickness of 8" since the net cross-sectional area of a hollow masonry wall i s approximately 50% of the gross area. The strut i s assumed to have an elastic modulus E = 3,600,000 psi By substituting the above values into equation (B.5), the cross-sectional area of a strut i s computed to be 1.645 f t . 96 APPEND I X C COMPUTER L I S T I N G OF SUBPROGRAM SPECTR SUBROUTINE SPECTR( ITYPE,MODE,DAMP,SD,SA,AE,WN, IV.DAF) C C Mf.DE IS THE MODE NUMBER C Sn IS THE VECTOR OF SPECTRAL DISPLACEMENTS C SA I S THE VECTOR OF SPECTRAL ACCELERATIONS C IS THE VECTOR OF MODAL NATURAL FREQUENCIES (RACS/SEC) C !•« I S THE NIJVBER CF yCDES C f»,',i TS THE DYNAMIC AMPLIFICATION FACTOR C M i C T R READS WHATEVER DATA I T NEEDS ON THE F I R S T C A L L , C WHICH I S N'ADF WITH NCDE=G C I' MODE.NF.0 THEN SO, SA, ANO DA F ARE RETURNED C DIMENSION UNI I I , SC( It , SAl 1 ) I F {MODE.NE.0 ) GO TO 101 READ!5,100) ITY DE,C1» C? »C3 »C4,PGA 100 FORMAT!I 8,5P8.4) F0G=1.25664 F1G=9.66644 GA=PGA*C1 V=GA/FIG D=V/FOG A=GA*C2 V=V*C3 D=D*C4 FO=V/D F1=A/V <=2 = 6 2 . 832 F 3 = 2 0 9 . 4 4 D2=A/F2/F? 03=GA/F3/i= 3 WRITE(6,107) GA,C1,C2,C3,C4,A,V,D 97 107 P ORM AT(/•NBC'S IDEALIZED SPFCTRUM APPLIED'// *' PEAK GROUND ACC =»,F8.4/« SOIL COEFF= ' ,F8.4 f• ACC CCEFF=«, *F8.4,» VEL COEF F = ' » F8 » 4» ' DISP COEFF='» FB.4/' ACC LIMIT = *F8.4,« VEL LIMIT =»,F8.4,» DISP LIMIT =»,F8.4/) ~-TURN i : : R EO^MMCDF) l'i FREQ.GE.FC) GO Tf) 201 SCI MODE) = D " V Tn 205 20! I-(FP.EO.GF.Fl) GO TO 202 ST ! ••'CCF )=V/ PREQ GO ?05 202 IF(FRE0.GE.P25 GO TO 203 SC(MODE)=A/FREQ/FREQ GO T O 205 203 IPC CRE0.GF.F3) GO TO 204 SD( MODE ) -D?*EXP ( AL0G(D3/D2)*AL0G( FP.E0/F2 )/ALOG { F3/F2 ) ) GC TO ?05 204 SD( T O E » = GA/FREO/FREO 205 5A{M CDE 5 = SD(MODE) #WN{MODE)*WN(MODE ) C A P =•' S A (MODE) /GA-AF = 0 .0 R F T U r< N FND APPENDIX D INPUT AND RESULT DATA OF MODEL ANALYSES FRAME 1 FRAME 2 FRAMI *, FRAME H-23.61 13.8' 10.8' 23.8' A3.2' ,19.4' ,16.9' 14.5' r10.8' 13.9' 14.r 13.0' 32.0' 10 11 12 13 14 15 16 17 Figure 21 Plane frame model of Structure 1A Table 2 MEMBER PROPER ' I Ii IA Columns I ( f t 4 ) A ( f t 2 ) Beams I ( f t 4 ) 1 25.07 5.11 2 28.44 5.33 1-2 0.1 3 1.50 2.00 2-3 0.1 4 16.49 4.45 3-4 0.1 5 10.13 3.78 4-5 0.1 6 3.56 2.67 7.18 7 227.56 10.67 6-7 8 202.11 10.25 7-8 0.26 9 96.00 8.00 8-9 0.26 10 73.94 7.33 9-10 0.26 11 3.56 2.67 10-11 7.18 12 371.51 12.56 13 3.00 4.00 12-13 0.26 14 1220.00 46.00 13-14 ;'c co 15 162.03 15.12 14-15 •k co 16 571.41 9.13 0.38 17 571.41 9.13 16-17 ( f t 2 ) 2.67 2.67 2.67 2.67 5.78 7.11 7.11 7.11 5.78 7.11 4.11 a ( f t ) b ( f t ) 3.84 4.00 1.50 3.34 2.00 8.00 7.92 6.00 5.50 9.42 ( p i n - p i n ) ( p i n - p i n ) 13.70 4.00 1.50 3.34 2.84 8.00 7.92 6.00 5.50 2.00 1.50 13.70 F i c t i c i o u s Beam, pinned-pinned endsj I = A = FRAME 1 n O N >-- 2V F R A M h : 7 FRAME 3 , 15.50' , 17.85' 14.85' , 17.50' • 18.50' Figure 22 Plane frame model of Structure IB MEMBER PROPERTIES OF STRUCTURE IB Columns I ( f t 4 ) A ( f t 2 ) Beams I ( f t 4 ) A ( f t 2 ) a (ft) b (ft) 1 203.07 11.92 2 85.07 8.92 1-2 1.50 3.00 7.15 5.35 3 23.82 5.83 2-3 1.50 3.00 5.35 3.50 4 220.59 12.25 3-4 1.50 3.00 3.50 7.35 5 203.07 11.92 4-5 1.50 3.00 7.35 7.15 6 347.24 14.25 7 156.12 10.92 6-7 1.50 3.00 8.55 6.65 8 41.16 7.00 FRAME 1 FRAME 2 Figure 23 Plane frame model of Structure 2 MEMBER PROPERTIES OF STRUCTURE 2 Columns I ( f t 4 ) A ( f t 2 ) Beams I ( f t 4 ) A ( f t 2 ) 1 1.33 4.00 2 1.67 5.00 1-2 1.21 3.94 3 1.67 5.00 2-3 1.21 3.94 4 1.33 4.00 3-4 1.21 3.94 5 2.60 5.00 6 1.33 4.00 5-6 1.57 5.68 7 1.33 4.00 6-7 1.57 5.68 8 2.60 4.00 7-8 1.57 5.68 Bracing Strut: A = 1.65 f t 2 Table 5 DYNAMIC ANALYSIS RESULTS OF STRUCTURE 1A (a^ 5-Storey T l = 0.115 sec i2 = 0.034 sec T3 = 0.018 sec Storey Mass (lbs) Rel. X-Displ. ( f t ) X-Accel. (g) Adjusted Accel, (g) 0.2 0.4 0.6 0.8 1.0 840,000 840,000 840,000 840,000 420,000 0.005 0.012 0.021 0.028 0.034 0.652 1.289 - 1.949 2.605 3.253 1.194 1.632 2.191 2.790 3.403 (b) 10-Storey T l = 0.313 sec T2 = 0.084 sec T3 = 0.041 sec h/H Storey Mass (lbs) R e l . X-Displ. ( f t ) X-Accel. (g) Adjusted A c c e l , (g) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 840,000 420,000 0.011 0.032 0.058 0.087 0.118 0.149 0.179 0.208 0.234 0.258 0.398 0.880 1.280 1.571 1.804 2.029 2.284 2.626 3.070 3.550 1.076 1.332 1.624 1.862 2.063 2.262 2.493 2.810 3.228 3.689 Table 5 (Continued) (c) 15-Storey Ti = 0.592 sec T 2 = 0.150 sec T3 = 0.071 sec h/H Storey Mass Rel. X-Displ. X-Accel. Adjusted (lbs) (ft) - (g) Accel, (g) 0.067 840,000 0.016 0.267 1.035 0.1 n 840,000 0.047 0.638 1.186 o.:>. 840,000 0.088 0.997 1.412 0.21.7 840,000 0.136 1.281 1.625 0..TI3 840,000 0.191 1.473 1.780 0 . / , 840,000 0.250 1.593 1.881 0.4b/' 840,000 0.312 1.678 1.954 o . ' ; ; 840,000 0.375 1.754 2.019 0.6 840,000 0.439 1.829 2.085 0.667 840,000 0.503 1.910 2.156 0.733 840,000 0.567 2.029 2.262 0.8 840,000 0.628 2.229 2.443 0.867 840,000 0.688 2.531 2.721 0.933 840,000 0.745 2.904 3.071 1 . 0 420,000 0.800 3.297 3.446 107 Table 5 (Continued) h/H Storey Mass Rel. X-Displ. X-Accel. Adjusted (lbs) (ft) (g) Accel, (g) 0.05 840,000 0.015 0.194 1.019 0.1 840,000 0.044 0.482 1.110 0.15 840,000 0.084 0.789 1.274 0.2 840,000 0.133 1.062 1.459 0.25 840,000 0.189 1.270 1.617 0.3 840,000 0.250 1.403 1.723 0.35 840,000 0.317 1.467 1.776 0.4 840,000 0.387 1.483 1.789 0.45 840,000 0.460 1.478 1.784 0.5 840,000 0.536 1.469 1.777 0.55 840,000 0.614 1.459 1.769 0.6 840,000 0.693 1.438 1.752 0.65 840,000 0.773 1.399 1.719 0.7 840,000 0.853 1.351 1.681 0.75 840,000 0.933 1.336 1.669 0.8 840,000 1.011 1.408 1.727 0.85 840,000 1.089 1.599 1.886 0.9 840,000 1.165 1.886 2.135 0.95 840,000 1.240 2.222 2.437 1.0 420,000 1.313 2.567 2.755 23-Storey ~l = 0.960 sec ~2 = 0.230 sec T q = 0.106 sec 108 Table 5 (Continued) ie) 25-Storey 7i = 1.417 sec 12 = 0.321 sec T 3 = 0.145 sec h/H Storey Mass Rel. X-Displ. X-Accel. Adjusted (lbs) ( f t ) (g) A c c e l , (g) 0.04 840,000 0.015 * 0.137 1.009 0.08 840,000 0.043 0.354 1.061 0.] ' 840,000 0.083 0.600 1.166 0.16 840,000 0.132 0.844 1.309 0.2 840,000 0.189 1.060 1.458 0. ^ 1 840,000 0.252 1.234 1.588 0. 840,000 0.321 1.357 1.685 0. .! 840,000 0.395 1.429 1.744 0.3( 840,000 0.473 1.459 1.769 0.4 840,000 0.555 1.458 1.768 0.44 840,000 0.640 1.440 1.753 0.48 840,000 0.728 1.413 1.731 0.52 840,000 0.818 1.383 1.707 0.56 840,000 0.910 1.346 1.677 0.6 840,000 1.004 1.293 1.634 0.64 840,000 1.099 1.218 1.576 0.68 840,000 1.195 1.124 1.505 0.72 840,000 1.292 1.028 1.434 0.76 840,000 1.389 0.967 1.391 0.8 840,000 1.483 0.994 1.410 0.84 840,000 .1.581 1.134 1.512 0.88 840,000 1.677 1.368 1.695 0.92 840,000 1.771 1.655 1.934 0.96 840,000 1.864 1.960 2.201 1.0 420,000 1.957 2.263 2.475 109 Table 6 DYNAMIC ANALYSIS RESULTS OF STRUCTURE I B (a N 5-Storey l ! = 0.153 sec 72 = 0.045 sec 7 q = 0.024 sec h/H Storey Mass (lbs) Rel.X-Displ. (ft) X-Accel. (g) Adjusted Accel, (g) 0.2 660,000 0.008 0.675 1.206 0.4 660,000 0.021 - 1.298 1.638 0.6 660,000 0.034 1.861 2.113 0.8 660,000 0.046 2.424 2.622 1.0 660,000 0.056 3.019 3.180 (b) 10-Storey T i = 0.378 sec T 2 = 0.105 sec T3 = 0.052 sec h/H Storey Mass Rel.X-Displ. X-Accel. Adjusted (lbs) (ft) (g) Accel, (g) 0.1 660,000 0.018 0.453 1.098 0.2 660,000 0.047 0.976 1.397 0.3 660,000 0.083 1.385 1.708 0.4 660,000 0.123 1.657 1.936 0.5 660,000 0.166 1.852 2.105 0.6 660,000 0.208 2.027 2.260 0.7 660,000 0.251 2.224 2.439 0.8 660,000 0.291 2.508 2.700 0.9 660,000 0.328 2.916 3.083 1.0 660,000 0.363 3.392 3.536 110 Table 6 (Continued) (c) I5-Storey Ti = 0.677 sec T 2 = 0.179 sec T3 = 0.086 sec h/H Storey Mass Rel. X-Displ. X-Accel, Adjusted (lbs) (ft) . (8) Accel, (g) 0.067 660,000 0.021 0.301 1.044 0.1 M 660,000 0.057 0.689 1.215 0.2 660,000 0.104 1.039 1.442 0.2 iv/ 660,000 0.159 1.295 1.636 0. : r n 660,000 0.220 1.447 1.759 0.4 660,000 0.285 1.527 1.826 O.'n ; 660,000 0.353 1.580 1.870 0.511 660,000 0.422 1.632 1.914 0.6 660,000 0.492 1.684 1.959 0.667 660,000 0.562 1.727 1.995 0.733 660,000 0.630 1.781 2.042 0.8 660,000 0.697 1.901 2.148 0.867 660,000 0.761 2.132 2.355 0.933 660,000 0.823 2.461 2.656 1.0 660,000 0.882 2.828 3.000 Table 6 (Continued) (c'i 20-Storey TT = 1.121 sec T 2 = 0.280 sec T3 = 0.131 sec h/H Storey Mass (lbs) Rel. X-Displ. (ft) * X-Accel. (g) Adjusted Accel, (g) 0.05 660,000 0.020 0.204 1.021 0.1 660,000 0.055 0.490 1.114 0.15 660,000 0.101 0.783 1.270 0.2 660,000 0.156 1.040 1.443 0.25 660,000 0.219 1.236 1.590 0.3 660,000 0.287 1.363 • 1.691 0.35 660,000 0.360 1.425 1.741 0.4 660,000 0.437 1.441 1.754 0.45 660,000 0.517 1.431 1.746 0.5 660,000 0.599 1.417 1.734 0.55 660,000 0.683 1.404 1.723 0.6 660,000 0.768 1.383 1.707 0.65 660,000 0.854 1.343 1.674 0.7 660,000 0.940 1.277 1.622 0.75 660,000 1.025 1.205 1.566 0.8 660,000 1.110 1.173 1.542 0.85 660,000 1.193 1.240 1.593 0.9 660,000 1.275 1.430 1.745 0.95 660,000 1.356 1.711 1.982 1.0 1,200,000 1.435 2.031 2.264 Table 7 DYNAMIC ANALYSIS RESULTS OF STRUCTURE 2 (a) 5-Storey Tl = 0.210 sec T 2 = 0.070 sec 1< = 0.042 sec h/H Storey Mass (lbs) Rel. X-Displ. ( f t ) X-Accel. (g) Adjusted A c c e l , (g) 0.2 600,000 0.024 * 0.947 1.377 0.4 600,000 0.051 1.631 1.913 0.6 600,000 0.075 2.152 2.373 o.» 600,000 0.095 2.669 2.851 1.0 300,000 0.107 3.119 3.275 (b) ii-Storey T i = 0.380 sec T 2 = 0.119 sec T 3 = 0.065 sec h/H Storey Mass (lbs) Rel. X-Displ. ( f t ) X-Accel. (g) Adjusted A c c e l , (g) 0.125 600,000 0.040 0.748 1.249 0.25 600,000 0.089 1.345 1.676 0.375 600,000 0.142 1.686 1.961 0.5 600,000 0.195 1.934 2.177 0.625 600,000 0.247 2.188 2.405 0.75 600,000 0.294 2.518 2.709 0.875 600,000 0.335 2.974 3.138 1.0 300,000 0.368 3.413 3.556 113 Table 7 (Continued) (c) 10-Storey T i = 0.519 sec 7 2 = 0.156 sec 7, = 0.082 sec h/H Storey Mass (lbs) Rel. X-Displ. (ft) X-Accel. (g) Adjusted Accel, (g) 0.1 600,000 0.049 0.633 1.183 0.2 600,000 0.112 1.181 1.547 0.3 600,000 0.183 ' 1.513 1.814 0.4 600,000 0.259 1.709 1.980 0.5 600,000 0.337 1.880 2.130 0 . 6 600,000 0.415 2.063 2.293 0.7 600,000 0.490 2.277 2.487 0.8 600,000 0.560 2.595 2.781 0.9 600,000 0.624 3.024 3.185 1.0 300,000 0.679 3.437 3.580 (d) 15-Storey Ti = 0.960 sec T 2 = 0.260 sec T 3 = 0.130 sec h/H Storey Mass (lbs) Rel. X-Displ. (ft) X-Accel. (g) Adjusted Accel, (g) 0.067 600,000 0.043 0.434 1.090 0.133 600,000 0.102 0.863 1.321 0.2 600,000 0.172 1.187 1.552 0.267 600,000 0.251 1.382 1.706 0.333 600,000 0.337 1.462 1.771 | 0.4 600,000 0.429 1.471 1.779 j 0.467 600,000 0.525 1.457 1.767 0.533 600,000 0.625 1.435 1.749 0.6 600,000 0.725 1.393 1.715 0.667 600,000 0.827 1.331 1.665 0.733 600,000 0.927 1.310 1.648 0.8 600,000 1.026 1.430 1.745 0.867 600,000 1.121 1.716 1.986 0.933 600,000 1.213 2.083 2.310 1.0 300,000 1.299 2.428 2.626 Table 7 (Continued) ( e ) 20-Storey T 1 = 1.537 sec T 2 = 0.381 sec T3 = 0.184 sec h/H Storey Mass (lbs) Rel. X-Displ. ( f t ) X-Accel. (g) Adjusted Accel, (g) o.ns 600,000 0.042 0.312 1.047 0.1 600,000 0.101 0.647 1.191 0. 1 '. 600,000 0.171 0.945 1.376 o.:' 600,000 0.252 1.182 1.548 0. 600,000 0.341 1.344 1.676 0. • 600,000 0.437 1.432 1.747 0. v, 600,000 0.540 1.460 1.769 0. . 600,000 0.649 1.448 1.759 0.45 600,000 0.762 1.416 1.734 0.5 600,000 0.879 1.375 1.700 0.55 600,000 1.000 1.319 1.655 0.6 600,000 1.123 1.235 1.589 0.65 600,000 1.248 1.117 1.499 0.7 600,000 1.374 0.981 1.401 0.75 600,000 1.500 0.890 1.339 0.8 600,000 1.627 0.938 1.371 0.85 600,000 1.752 1.154 1.527 0.9 600,000 1.875 1.472 1.779 0.95 600,000 1.995 1.813 2.070 1.0 300,000 2.113 2.127 2.351 

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