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Torsional response of multi-storey buildings using 3-D inelastic dynamic analysis Zaghloul, Hassan Mohamed 1989

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TORSIONAL RESPONSE OF MULTI-STOREY BUILDINGS USING 3-D INELASTIC DYNAMIC ANALYSIS By Hassan Mohamed Zaghloul B. Sc. (Civil Engineering) University Of Alexandria, Egypt 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1989 © Hassan Mohamed Zaghloul, 1989 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of CIV\U ErVfralK/EERj/vt^ The University of British Columbia Vancouver, Canada Date .TnUj Xh Wf¥\  DE-6 (2/88) Abstract This thesis investigates the static code provisions as they pertain to torsion of the 1985 edition of the National Building Code Of Canada (NBCC 85) for eccentric multistory buildings. This is done by calculating the displacements and ductility demand of several practical five storey eccentric buildings designed according to the 1985 code, and comparing the response to similar non eccentric buildings. The analysis is carried out using a modified version of the computer program PITSA, which carries out a pseudo elastic dynamic analysis to model the inelastic response. A • modification to the program, developed in this thesis, accounts for the effect of gravity forces on the ductility demand. A number of parameters are considered, namely the type of eccentricity, the aspect ratio, the gravity loads, gravity load distribution, and the initial eccentricity ratio. The effect of the design on each parameter is investigated. The following factors are seen to largely affect the reponse, but are not recognized in the code: 1. The static eccentricity specified in the code is not stated whether it is a result of an eccentric center of mass (CM) or an eccentric center of rigidity (CR) building. This study shows that the behavior of the CR buildings are different from CM buildings in that the bigger frames are more damaged in CR buildings but the smaller frames are more damaged in the CM buildings. 2. Gravity loads have a potentially large impact on the response. For beams carrying no gravity loads, the ductility demand in the upper floors is about 15, while if u the gravity loads are considered to be eccentrically distributed, the ductility de-mand ranges from 2 to 5 with the bigger frames underdesigned and smaller frames overdesigned. When gravity loads are uniformly distributed, the code provisions are about right. 3. The ±50% increase in the nominal torsion specified in the code can be changed without a significant change in the ductility demand of the longitudinal frames as the torsional moments are essentially carried by the transverse frames. 4. The increase in the building dimension in the direction parallel to the earthquake results in an increase in the dynamic amplification, and the torsional provisions can generally be said to cover the highest possible dynamic amplification, as the design is generally acceptable for these buildings. The result of that is an overdesign in buildings with small aspect ratios, or alternatively, small dynamic amplification. 5. The torsional provisions tend to overdesign the bigger frames in CM buildings and overdesign the smaller frames in CR buildings for large eccentricity ratios. Other findings pertinent to this study show the following: 1. The code-specified period used in the calculation of the design base shear is a con-servative estimate. This period should be established using the structural properties and deformation characteristics of the resisting elements in a properly substantiated analysis. 2. The Modified Substitute Structure Method can now model a building with earth-quake as well as static loads. 3. PITS A is a reliable tool in the evaluation of the damage in a three-dimensional frame buildings. iii 4. The torsional moments are essentially carried by the transverse frames, and the longitudinal frames resist lateral loads for an earthquake applied in the longitudinal direction. iv Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgement viii 1 INTRODUCTION 1 1.1 Background 1 1.2 Objective . 3 1.3 Survey of the Literature 3 1.4 Scope 6 1.5 General Approach 7 1.6 Sequence of Presentation 8 2 BACKGROUND ON DUCTILITY 9 2.1 General 9 2.2 Ductility Types 9 2.3 Calculation of the Demand Ductility Factor 10 2.3.1 Displacement Ductility Factor 12 2.3.2 Curvature Ductility Factor 13 2.3.3 Rotational Ductility Factor 13 2.4 Disadvantages of the Ductility Ratio 14 v DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 16 3.1 General 16 3.2 Two Approaches to Dynamic Analysis 16 3.3 The Modified Substitute-Structure Method (MSSM) 17 3.3.1 General 17 3.3.2 Evolution of the Method 18 3.3.3 Terms Relevant to MSSM 19 3.3.4 Details of the MSSM 23 3.3.5 Idealization of Three-Dimensional Model 26 3.4 Three Dimensional Time-Step Analysis 27 3.4.1 General 27 3.4.2 Hysteretic Model 28 3.4.3 Response Parameters 28 GROUND MOTION AND DESIGN OF MODEL STRUCTURES 30 4.1 Introduction 30 4.2 Base Motions For Earthquake Simulations . . 31 4.3 Mathematical Modelling 35 4.4 Description of the Models 36 4.5 Principal Design Steps 39 4.5.1 Determination of Design Base Shear 39 4.5.2 Distribution of Base Shear 41 4.5.3 Determination of the Structural Eccentricity 42 4.5.4 Calculation of Torsional Moments 45 4.5.5 Analysis 46 4.5.6 Moment Redistribution 47 vi 4.5.7 Load and Strength Reduction Factors 47 4.5.8 Determination of the Yield Moment 49 4.5.9 Strong Column - Weak Girder Criterion 49 5 ALTERATION TO MSSM 51 5.1 Introduction 51 5.2 Effect of Gravity 52 5.2.1 Testing of DRAINTABS 52 5.2.2 Testing the Effect of Gravity Load Moments on the Ductility Ratio 54 5.3 Alteration to MSSM 56 5.3.1 First Method 56 5.3.2 Second Method 57 5.4 Program Testing 58 5.4.1 General • 58 5.4.2 Results 59 5.5 Execution Cost 60 5.6 Discussion 61 6 PARAMETRIC STUDY 67 6.1 General Scope 67 6.2 Parameters Considered 68 6.3 Model Codes 69 6.4 Effect of the Movement of Center of Resistance 71 6.5 Numerical Results 72 6.5.1 Eccentricity Type 73 6.5.2 Aspect Ratio 74 6.5.3 Presence of Gravity Loads 75 vii 6.5.4 Gravity Load Distribution 78 6.5.5 Initial Eccentricity Ratio 78 6.6 Base Shear 85 6.7 Effect of Period of Vibration 87 6.7.1 On The Design 87 6.7.2 On The Analysis 88 6.8 Discussion ; 89 7 SUMMARY AND CONCLUSIONS 92 7.1 Summary 92 7.2 Conclusions 95 Bibliography 97 Appendices 106 A BUILDING PROPERTIES 106 vni List of Tables 1.1 Design Eccentricities from Various Building Codes 2 4.1 Diaphragm Data in Different Sets 38 4.2 Calculation of the Lateral Forces and Shears in the Case of an Aspect Ratio=1.5 for an Earthquake in the Longitudinal Direction 43 4.3 Calculation of the Lateral Forces and Shears in the Case of an Aspect Ratio=3 for an Earthquake in the Longitudinal Direction 43 5.1 Comparison of Execution Costs 61 A . l Summary of Model Codes 107 A.2 Stifness Values EI in kips.ft2 used for the Girders in Different Models . 107 A.3 Stifness Values EI in kips.ft2 used for Each Column Line in Different Models with Aspect Ratio=1.5 108 A.4 Areas in Inch2 used for Each Column Line in Different Models 108 A.5 Stifness Values EI in kips.ft2 used for Each Column Line in Different Models with an Aspect Ratio=3 . 108 A.6 Yield Moments My in kips.ft in the Girders of Model CMxxzl 109 A.7 Yield Moments My in kips.ft in the Girders of Model CMxx31 and CRxxzlllO A.8 Yield Moments My in kips.ft in the Girders of Models CRxxzl I l l A.9 Yield Moments My in kips.ft in the Girders of Models CRxxz2 112 A.10 Yield Moments My in kips, ft in the Girders of Models CMxxl2 and CMxx22113 A.11 Yield Moments My in kips.ft in the Girders of Models CMxx32 115 IX A. 12 Initial Elastic Periods in Seconds in the Different Models A.13 Final Periods in Seconds in the Different Models . . . . . . List of Figures 2.1 Resistance-Displacement Relationship 11 3.1 Damage Ratio Definition 21 3.2 Relationship between the Damage Ratio and the Ductility Ratio for Dif-ferent Strain Hardening Values 21 4.1 Shibata and Sozen Spectrum Versus Normalized 3% Damped Response Spectra and their Average 33 4.2 Comparison of Scaled Shibata and Sozen Response Spectrum, Scaled Av-eraged Normalized Response Spectrum, and NBCC 80 Response Spectrum. 34 4.3 Layout of the Models Under Consideration. Top: Elevation of the Frames in the Longitudinal Direction; Middle: Plan of the Model with Aspect Ratio=1.5; Bottom: Plan of the Model with Aspect Ratio=3 37 4.4 Strong Column - Weak Girder Criterion 50 5.1 Checking the Static Load Solution 53 5.2 Layout and Properties of Tarn's Building (taken from Tarn 1985) 55 5.3 Ductility Demand in Members with and without Gravity Loads with Max-imum Horizontal Ground Acceleration of 0.25g 63 5.4 Ductility Demand in Members with and without Gravity Loads with Max-imum Horizontal Ground Acceleration of 0.50g 63 5.5 First Method to Incorporate Gravity Loads to MSSM: Moment Rotation Relationship at the Member Ends 64 XI 5.6 Peak Ductility Demand Versus Damage Ratio for Frame 1 of Tarn's Build-ing. Left: Peak Ductility Demand from Four Records Versus Modified PITSA, Right: Average from the Four Records Versus Modified and Un-modified PITSA 65 5.7 Peak Ductility Demand Versus Damage Ratio for Frame 4 of Tarn's Build-ing. Left: Peak Ductility Demand from Four Records Versus Modified PITSA, Right: Average from the Four Records Versus Modified and Un-modified PITSA 65 5.8 Peak Ductility Demand Versus Damage Ratio in a 30% Eccentric Building. Left: Frame 1, Right: Frame 3. 66 6.1 Layout and Element Numbering: (a) Elevation, (b) Plan of CM Build-ing with Different Aspect Ratios, (c) Plan of CR Building with Different Aspect Ratios 70 6.2 Target Damage Ratio for the Evaluation of the Design 72 6.3 Effect of the Eccentricity Type 74 6.4 Effect of the Aspect Ratio, (a) Dynamic Amplification of Floor Rotations, (b) Dynamic Amplification of Floor Displacements, (c) Damage Ratio in CM2011, (d) Damage Ratio in CM2012 76 6.5 Effect of Gravity Load Presence 77 6.6 Effect of the Distribution of Gravity Loads 79 6.7 Effect of the Eccentricity Ratio on CM Buildings with Aspect Ratio=1.5 and Different Distribution of Gravity 81 6.8 Effect of the Eccentricity Ratio on CM Buildings with Aspect Ratio=3 and Different Distribution of Gravity . . . 82 xn 6.9 Effect of the Eccentricity Ratio on CR Buildings with Aspect Ratio=1.5 with and without Gravity 83 6.10 Effect of the Eccentricity Ratio on CR buildings with and without Gravity for Aspect Ratio=3 : 84 6.11 Distribution of Base Shears from the Dynamic Analysis Under Three-Dimensional Action 86 6.12 Effect of Initial Period on the Ductility Demand 88 xm Acknowledgement Dr. D. L. Anderson proposed this project. I am deeply indebted to him for giving his support and invaluable assisatnce and ideas throughout the course of the study, and for his conscientious proofreading. Gratitude is also due to Dr. Andre Filiatrault for his help during this work and for proofreading this thesis and his valuable suggestions to improve the presentation. Special thanks are extended to Dr. N. D. Nathan. His willingness to help when I needed it is greatly appreciated. The research assistantship from the Department Of Civil Engineering is gratefully ac-knowledged. Finally, I want to express my appreciation to my fellow graduate students, Gerard Can-isius, J. D. Dolan, and Steven Kuan. xiv To my mother, Kariman, and the late memory of my father, who have always believed in me. xv Chapter 1 INTRODUCTION 1.1 Background One important aspect in multistory building design is to ensure that the building will perform in a satisfactory manner under lateral forces caused by seismic ground motions. If asymmetry exists in the geometry, stiffness, or mass distribution in the plan of the building, lateral forces lead to torsional, as well as lateral, displacements. Even in nom-inally symmetric structures, torsion may arise because of rotational ground motions or if the horizontal ground motion is not uniform over the base. During the Alaskan earth-quake in 1964, Green (1987) mentioned that the J. C. Penney building in Anchorage was so severely damaged by torsion that it was necessary to demolish the structure. From a design viewpoint, it is necessary to know the magnitude of such torsional effects so that the required strengths and stiffnesses of the different lateral resisting ele-ments can be estimated. Special torsional provisions have been made by different codes to allow for this effect. The most common form of torsional provision is the requirement that buildings be designed for the application of lateral loads and torsional moments at each storey. The torsional moment at each storey is obtained by multiplying the storey shear by a quantity called 'design eccentricity ex\ In most seismic codes, the design ec-centricity expression consists of two parts. The first part is termed dynamic eccentricity and is a function of the structural eccentricity e, the distance between the center of mass 1 Chapter 1. INTRODUCTION 2 and center of rigidity. This dynamic eccentricity takes into account the difference in stiff-ness and mass distributions of the building in plan as well as the dynamic amplification of the torsional response. The second part, commonly referred to as accidental eccentricity, accounts for other factors which can not be quantitatively predetermined. These may include the actual (as opposed to the assumed) mass and stiffness distribution, and the rotational component of ground motion about a vertical axis. The accidental eccentricity is a function of the plan dimension of the building. The design eccentricity for most seismic codes can be written in the form ex=ed + pDn (1.1) where e<£ and f3Dn are the dynamic and accidental eccentricities respectively; Dn is a specified plan dimension at the level under consideration; and is a coefficient, taken to be 0.05 or 0.1, depending on the building code. The design eccentricities stipulated in five seismic codes are listed in Table 1.1. Table 1.1: Design Eccentricities from Various Building Codes Design Eccentricity, ex Reference Canada Mexico New Zealand Turkey U.S.A. 1.5e + 0.1D„ 0.5e- 0.1Dn 1.5e + 0.1Dn e - 0.1Dn 1.7e- -£+0.1Z> n e - 0.1Dn e + 0.05Dn e + 0.05Dn e - 0.05£ n National Building Code 1985 Rosenblueth (1979) Standard Association of New Zealand (1976) Ministry of Reconstruction (1975) ATC3 - 06, Applied Technical Council (1978) Column 1 2 3 It is evident from studies by Newmark and Rosenblueth (1971) and Elms (1977) that the Chapter 1. INTRODUCTION 3 code provisions for torsion are, to a large extent, based on the findings of elastic studies obtained by analyzing monosymmetric single-story building models, i.e. eccentric in one direction only. While these are undoubtedly helpful, there must always remain the question of the influence of inelastic behavior. 1.2 Objective The primary objective of the investigation reported in this thesis is to examine the accuracy and applicability of the torsional provisions recommended in the 1985 edition of the NBCC for multi-storey buildings. This is done using a three-dimensional inelastic dynamic analysis. 1.3 Survey of the Literature The torsional phenomenon and the validity of code provisions have been investigated by several researchers (Humar 1984; Awad and Humar 1982; Tso and Dempsey 1980; Tsicnias and Hutchinson 1981; Tso and Dempsey 1980). These studies used the single storey monosymmetric building model and an elastic analysis to investigate the NBCC 80 torsional provisions, which can be summarized as follows: 1. Torsional moments in the horizontal plane of the building shall be computed in each storey using the following formula: 2. The design eccentricity, ex, in Equation 1.2 shall be computed by one of the fol-lowing equations, whichever provides the greater stresses: X (1.2) ex = 1.5e + 0.05Pn,or (1.3) Chapter 1. INTRODUCTION 4 0.50e - 0.051>n (1.4) 3. The structural eccentricity, e, is defined by Fe-(1.5) F-where F{ = lateral force applied at level i, eix = distance between the center of mass at floor i and the center of rigidity at floor x. 4. When the maximum design eccentricity exceeds 0.25Dn, either a dynamic analysis shall be made or the adverse effects of torsion as computed from the code shall be doubled. Tso and Dempsey (1980) compared the magnitude of eccentricities obtained from a dy-namic analysis with those specified by the NBCC 80. On the basis of this comparison, they concluded that the code provisions underestimate the torsional moments when the eccentricity of the building is small and the frequency ratio, defined as the ratio of the uncoupled torsional frequency to the uncoupled translational frequency, is close to unity. Tsicnias and Hutchinson (1981) arrived at a similar conclusion for buildings with small eccentricities and/or values of frequency ratio close to unity. For large eccentricities and/or high values of the frequency ratio, they mentioned that the building code provide reasonable estimates of the earthquake response. They also found that a decrease in the transverse building dimension, i.e. perpendicular to the eccentricity, results in a decrease in the dynamic amplification of the eccentricity. A critical evaluation of the NBCC 80 torsional provisions was given by Tso and Meng (1982). It was shown that for regular, asymmetrical buildings, i.e., buildings having cen-ters of mass and centers of rigidity of the floors lying along two vertical lines, the NBCC 80 torsional provisions are generally applicable. Inaccuracies of the code provisions arise in two circumstances. The code is nonconservative when the structural eccentricity is Chapter 1. INTRODUCTION 5 small and when the torsional and lateral frequencies are close to one another (i.e., modal coupling occurs). The code is overly conservative when the structural eccentricity is large because the torsional and lateral frequency ratios are unlikely to be close and the code requires the doubling of the torsional effect for design. Another finding of the study concerns the use of the torsional provisions in NBCC 80 in the design of buildings with setbacks. It was found that the torsional moments in the setback portions were grossly underestimated. As a result of the evaluation studies carried out, Tso (1983) proposed four recom-mendations to improve some of the shortcomings of the seismic torsional provisions in the NBCC 80. The first two recommendations aim at providing static torsional moment estimates that are more compatible with the dynamic torsional moments for regular ec-centric buildings. The last "two recommendations aim at clarifying the limitations of the static code provisions. These recommendation can be stated as follows: 1. The accidental eccentricity component should be increased from 0.05 to 0.10D. This reduces some of the unconservatism when the design eccentricity is small and when modal coupling occurs. 2. The requirement to double the torsional effect if the design eccentricity exceeds 0.25.D should be deleted, as the study by Dempsey and Tso (1982) indicated that the torsional effect is adequately estimated when the building eccentricity is large, and further doubling the torsional effect is thus conservative. 3. The expression used to define the structural eccentricity should be deleted. The rea-son is that it conveys the impression that this expression can be applied to buildings having different values of eccentricities at different floors, while the study by Tso and Meng (1982) showed that this expression does not lead to results comparable to dynamic analysis in these cases. Chapter 1. INTRODUCTION 6 4. A statement should be added to clarify the limitation of the torsional provisions given in the code. The reason is that for irregular, asymmetrical buildings, there are currently no suitable formulae to give realistic estimates of the torsional effects, and dynamic analysis is the most reliable approach to estimate the force distribution within such structures. Humar (1984) discussed the first recommendation proposed by Tso (1983), and judged the design eccentricity expression to be unnecessarily conservative. He recommended an 'effective' design eccentricity that would closely match the results of the dynamic analysis. This 'effective' design eccentricity is based on the study be Dempsey and Tso (1982), and is given by ex = e + 0.10Dn (1.6) In the reply by Tso (1984), Equation 1.6 was said to be adequate for a class of buildings with uniform mass distribution in plan, but it would underestimate the torsional effect for structures with mass distribution more concentrated within the core, like buildings with columns and/or walls forming clusters near the core, or for buildings with lateral load resisting elements located near the perimeter of the building. The four recommendations proposed by Tso (1983) found their way in the NBCC 85. 1.4 Scope The scope of the research reported here is limited as follows: 1. The analysis and design of the models in this study is limited to buildings of regular shape, as it is stated explicitly in the code that the torsional provisions are only applicable to regular buildings, or buildings with the centers of mass and centers of rigidity lying on approximately two vertical lines. Chapter 1. INTRODUCTION 7 2. Torsion in the building is assumed to arise only from lateral forces acting on ec-centricities in the mass and stiffness distributions at various floor levels in the structures, and not from torsional ground motions. 3. The second part of the dynamic eccentricity expression, namely the accidental eccentricity, is ignored in the design as Blume et al. (1970) suggested that it is essentially included in the design eccentricity expression to account for torsional ground motions. The design is thus based on the first part of the design eccentricity expression, namely 1.5e or 0.5e. 4. Buildings are assumed to be of reinforced concrete only. 1.5 General Approach In order to evaluate the torsional provisions, a comparison of the member ductility de-mands in different parts of the structure is done. To simulate realistic conditions, build-ings have been designed to satisfy the code provisions for a location such as Vancouver where the peak ground acceleration (PHA) and peak ground velocity (PHV) are both 0.2 (units of g and m/s respectively). The stiffnesses of the beams and columns are assumed to represent a practical situation, and the application of the quasi-static forces specified in NBCC 85 results in the yield moments of the beams and columns. The response of the buildings when subjected to a ground excitation with the type and intensity that may be expected in Vancouver is then calculated. The 1985 edition of NBCC does not provide a spectrum for use in dynamic analysis, but the 1980 edition of NBCC does and this is used in the analysis. The dynamic analyses are carried out using a modified version of the program PITSA which uses the 'Modified Substitute-Structure Method' to carry out a pseudo elastic dynamic analysis to model the inelastic response. A modification to the program, developed in this thesis, accounts for the effect of the gravity forces on the Chapter 1. INTRODUCTION 8 ductility demand. The ductility demand in the different eccentric buildings are obtained from PITSA and compared with the ductility demand for a symmetric building. The accuracy of PITSA to predict ductility demands and lateral deflections is determined by comparison with averaged response from the time step analysis program DRAINTABS. 1.6 Sequence of Presentation The torsional effect on building structures is examined in this study by comparing duc-tility requirement of different frames. Chapter 2 provides the reader with a general background on ductility. Chapter 3 presents an overview of the two computer programs used in the design and analysis of the models under consideration. In Chapter 4, the ground motion and the design steps used in this study with particular reference to the earthquake-resistant design provisions in the NBCC 85 are discussed. Chapter 5 introduces the modification to the "Modified Substitute-Structure Method" to incorporate the effect of gravity on the first order analysis of buildings subjected to earthquakes. In Chapter 6, numerical results followed by a discussion of these results are presented. A summary and conclusions are presented in Chapter 7. Chapter 2 BACKGROUND ON DUCTILITY 2.1 General For most building structures situated in areas of high seismic risk, it is economically infeasible to design the structure to remain elastic under the forces induced by intense ground shaking. Therefore, inelastic structural response during severe but rare seismic events is tolerated. Preventation of collapse under severe earthquakes is closely related to the ductility de-mand imposed by extreme ground motions, and also on the ability of the structure and its components to sustain at a number of instances the associated plastic deformations without significant loss of lateral load resistance. Without utilizing base isolation tech-niques or building in energy absorbing features, the only alternative to requiring ductility is to provide so much resistance that members would not exceed their elastic limit. 2.2 Ductility Types Ductility is usually expressed as a ductility factor, which is defined as the ratio between the maximum response to the yield response, of a structural member or system. Response here could mean curvatures, rotations or displacements. Some common types of ductility factors are : 1. Rotational Ductility of a joint in a flexural member. 2. Curvature Ductility Factor. 9 Chapter 2. BACKGROUND ON DUCTILITY 10 3. Relative Displacement Ductility of a storey in a building. 4. Overall Displacement Ductility of a building. The storey ductility factor is essentially denned by use of a relationship in which the dis-placement is the relative storey deflection between the floor above and the floor beneath. The overall system ductility factor is in general some weighted average of the storey duc-tility factors. The member ductility factor may be considerably higher than the storey ductility factor, which in turn may be somewhat higher than the overall ductility factor. Newmark (1982) suggested that for an overall ductility factor of 3 to 5 to be developed in a structure, the storey ductility factor may have to vary between 3 to 8 or 10, and the individual member ductility factors will probably he in the range of 5 to 15 or even more. In this regard the ductility factor as defined by Newmark (1982) is given by the ratio of maximum deformation to the deformation at the effective yield deformation, rather than the ratio of the maximum deflection to the elastic limit deflection or displacement as shown in Figure 2.1. In this figure, the elasto-plastic approximation to the actual curvilinear resistance-displacement curve is drawn so that the areas under both curves are the same at the "effective" elastic limit displacement uy and at the selected value of maximum displacement um. 2.3 Calculation of the Demand Ductility Factor For a structure to successfully withstand a severe earthquake, the members must un-dergo substantial inelastic deformation without failure. The inelastic deformation or the ductility demand should be measured in some manner and compared to an allowable value. The best measure is the maximum strain attained, but in practice, ductility is represented in terms of either a displacement, a curvature, or a spring rotation. Chapter 2. BACKGROUND ON DUCTILITY 11 Figure 2.1: Resistance-Displacement Relationship Chapter 2. BACKGROUND ON DUCTILITY 12 In general the displacement ductility factors are used in assessing or attempting to pre-dict overall building response, such as estimating base shear or overturning moment. The curvature or rotational ductility ratios are used in assessing individual member damage or capacity. For assessing member ductility the curvature ductility ratio is the most useful in that it enables one to estimate the maximum strains in the member. The yield curvature is usually well defined and is a function of the section property only, independent of the length, load or the rest of the structure. However, it is very difficult to calculate or mea-sure the curvature at a section once it moves into the post elastic range. Most analysis programs work with nodal rotations at the ends of the members and to translate these plastic rotations to curvatures requires making some assumption on how the curvatures are distributed along some 'plastic hinge length'. The rotation ductility factor is another measure to assess the member ductility. It pro-vides a means for estimating the real damage in the structure by using the maximum plastic hinge rotation, which is calculated by most analysis programs. The difficulty in calculating the rotation ductility lies in the determination of the yield rotation, as it is dependent on the loading on the member. However, in earthquake engineering, the yield rotation for a beam with equal end rotations due to equal and opposite end moments is generally used. 2.3.1 Displacement Ductility Factor Displacement Ductility Factor is defined as: A, ' m a x > 1 (2.1) where Chapter 2. BACKGROUND ON DUCTILITY 13 A m 0 x is the maximum lateral deflection in the postelastic range, and Ay is the lateral deflection when yield is first reached. When a number of load cycles are involved, A y is taken as the lateral deflection when yield is first reached in the first load excursion into the postelastic range. For a multi-story building, the displacements A v and A u giving the displacement ductility factor are measured at a suitable position (e.g., at the roof level). In a multistory frame, plastic hinges tend to develop at critical sections throughout the structure, but may not develop at the same load. Hence the lateral load-deflection relationship will not be bilinear but will tend to be curved because the stiffness decreases gradually as the plastic hinges de-velop at various load levels. To assess the displacement ductility factor in such a case, Fintel (1982) suggested an approximate bilinear lateral load-deflection curve, taking the deflection at first yield as the deflection due to the static design load applied to the frame behaving elastically. 2.3.2 Curvature Ductility Factor The Curvature Ductility Factor can be expressed as follows: where ipmax is the maximum curvature in the postelastic range, and tpy is the curvature at first yield. 2.3.3 Rotational Ductility Factor The Rotation Ductility Factor Tjg, based on an assumed antisymmetric deformation mode, is used to provide a comparative representation of the maximum fibre strain developed (2.2) Chapter 2. BACKGROUND ON DUCTILITY 14 in a beam. The rotational ductility ratio is defined at each end of a beam. When the end moment is plastic, the rotation ductility ratio is denned as one plus the ratio of the absolute maximum plastic hinge rotation developed during the excursion into the plastic range to the yield rotation for a beam with equal end rotations, and no transverse end displacements. This can be expressed as Vs = 1 + ^ (2.3) where: \8P\ is the absolute maximum plastic hinge rotation, My is the yield moment of the section, and 6y is the yield rotation, and can be calculated as where I is the length of the member, EI is the elastic beam bending stiffness. For the cases considered here it is taken as —sYil- for the beams, and Q.^,EIgTOS8 for the columns. 2.4 Disadvantages of the Ductility Ratio 1. In experimental investigations designed to study the possibility of fracture or un-stable response, the curvature ductility ratio is very difficult to measure. 2. The maximum ductility factor is only a measure of the maximum deformations or strain. By itself, it does not include the effect of many cycles of inelastic deforma-tions. Hence, by itself it is not a complete measure of the fracture or deterioration of the material. Chapter 2. BACKGROUND ON DUCTILITY 15 3. For general loading conditions, with torsional moments, shear, axial forces, and bending moments acting at a section, it is difficult to get one factor that measures the behavior of all the different loading conditions. Chapter 3 DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 3.1 General The determination of the rotational ductility factor requires the maximum absolute plas-tic hinge rotation in the members, which requires a nonlinear dynamic analysis. Two non-linear approaches are used in this study, the program DRAINTABS which is a con-ventional time-step analysis or the so-called "Modified Substitute-Structure Method" which is a pseudo-elastic modal analysis. 3.2 Two Approaches to Dynamic Analysis The term dynamic analysis, particularly in earthquake engineering, is associated with two different analytical approaches. In the first approach, referred to as the mode superposi-tion method, and which uses the response spectrum approach, approximate values of the maximum response are determined by combining the response from the different modes of vibration behaving as an independent Single Degree-of-Freedom (SDOF) systems. The second approach to dynamic analysis, generally termed the time step method, employs a direct numerical integration of the equations of motion for a particular earthquake input. For a linearly elastic system with viscous damping, which allows the uncoupling of the equations of motion, modal analysis permits a significant saving in computing time, since the determination of the modes of vibration and the associated frequencies is carried out only once. For an elastic plastic system, the equations are nonlinear and so can not 16 Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 17 be uncoupled into separate modes. Thus the mode superposition method, in its usual form, can not be used for calculating the inelastic response. In the time step method, the set of simultaneous differential equations in incremental form can be linearized and transformed into a set of algebraic equations, and then solved. The time step method, because it can be linearized over a time increment, can be used to solve for the inelastic response. The time step method applied to the analysis of a typical multi-storey structure, gen-erally requires the use of a large computer, and can be time-consuming and expensive, particularly when inelastic response is considered. Its use may be justifiable only for a few important projects. It does, however, provide more reliable values than mode super-position using response spectra, and for inelastic response it is the only rigorous means of determining the deformability requirements of members corresponding to a particular design and earthquake input motion. For design purposes, the use of a single earthquake record as input may not provide sufficient assurance of the behavior of a proposed structure under future earthquakes. For this purpose, the use of a set of earthquake records, possibly including artificially generated accelerograms, is advisable. 3.3 The Modified Substitute-Structure Method (MSSM) 3.3.1 General Before the recent advances in seismic codes, many buildings" were designed and con-structed. The performance of such buildings will be at best uncertain if and when a sizeable earthquake strikes the area, and it is important to analyze these buildings to determine whether they can withstand an expected earthquake. The best way to ana-lyze these buildings is to subject them to a nonlinear time-step analysis, but the cost and Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 18 tediousness of using this kind of analysis is prohibitively high. The "Modified Substitute-Structure Method" (MSSM) is an alternative method of analysis that provides the means of locating the plastic hinges and determining approximately the ductility demands in different parts of the structure with much less computational effort than that required in a time step analysis. 3.3.2 Evolution of the Method Gulkan and Sozen (1974) developed a design procedure for determining the yield forces in a simple reinforced concrete frame structure for a desired level of ductility in the mem-bers. This method was later extended to Multi Degree Of Freedom (MDOF) reinforced concrete systems by Shibata and Sozen (1976), who called it "The Substitute-Structure Method". The method uses the linear mode superposition approach utilizing the elastic spectrum for the earthquake input. The substitute structure used in the analysis uses reduced member stiffnesses, the amount of reduction depending on the member ductility, as well as increased damping to account for the inelastic energy dissipation. Yoshida et al. (1979) extended the "Substitute-Structure Method" towards the analysis of existing reinforced concrete buildings. The new procedure was given the name "The Modified Substitute-Structure Method". In this method, the initial stiffness and yield moments and other strength properties of a structure are known or can be found from design calculations, drawings and/or field investigations. What is unknown is the amount of inelastic deformation expected in each member for a given earthquake motion. Since the inelastic deformation, expressed here as a ductility ratio, is not known apriori, an it-erative procedure is used whereby the ductility ratio in each member is changed until the calculated member moments agree with the actual yield moments. This requires several iterations but the method remains much more economical than the non-linear time step method of analysis. Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 19 The MSSM was later extended by Metten (1981) to the analysis of coupled shear walls. He also added a convergence speeding routine to increase its efficiency. Hui (1984) later studied the problem of convergence, and proposed a new formula for calculating the damping ratio. Hui also modified the program to include members with plastic hinges at each end. This eliminated the need to assume equal yield deformations at the two ends of a member that was present in the original formulation. The extension of the Modified Substitute Structure Method to three dimensional struc-tures was carried out by Tarn (1985), and his version was coded PITSA, which stands for Pseudo Inelastic Torsional Seismic Analysis. The overall structure stiffness matrix is obtained from the frame stiffness matrices and is condensed to retain three degrees of freedom, two translations and one rotation at each floor. 3.3.3 Terms Relevant to MSSM Damage Ratio Figure 3.1 represents the relation between the applied moment, M, and the end rotation, 6, caused by flexural deformation within the span of a member. The ductility ratio is defined as the ratio of maximum rotation to yield rotation . The damage ratio on the other hand is the ratio of the initial stiffness to the secant stiffness at the maximum moment, and is denoted here by /i. Qualitatively, the damage and ductility ratios are both a measure of the inelastic rotation in the member, but quantitatively they are identical only for elastic perfectly plastic response. The damage ratio can be expressed in terms of the ductility ratio by " = I T ( ^ <"> in which Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 20 p, is the damage ratio, 77 is the ductility ratio, and s is the strain hardening ratio Figure 3.2 shows the relation between the damage and the ductility ratios for different strain hardening values. It is seen that there is not much difference between the two for the small practical values of s of about 0.01. Convergence Criteria Because the MSSM is an iterative procedure, criteria are needed to assess the desired level of accuracy. Yoshida (1979) used two criteria, both involving individual members, that must be satisfied simultaneously: 1. The computed bending moment and the yield moment for every member must be within 5% of each other, i.e. 2. The change in damage ratio between successive iterations be limited to one per cent if the damage ratio is greater than five. When the damage ratio is less than five, the absolute difference of the damage ratios between successive iteration be less than 0.1, i.e Mn-My My < 0.05 if p > 1 | < 0.01 if p > 5 p,n - / i n _ i | < 0.1 if p < 5 Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 21 y max Rotation, 0 Figure 3.1: Damage Ratio Definition. 15 E s=D.O : i s = 0:001: • _• _ s = 6.01: "s=p.1 ; - A' • \f i i i i i i i i L 0 2 4 6 8 1 0 1 2 1 4 1 6 Ductility Ratio Figure 3.2: Relationship between the Damage Ratio and the Ductility Ratio for Different Strain Hardening Values Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 22 Modal Combination In the mode superposition method the problem of how to combine the modal contri-butions to best represent the maximum response has long been a problem. Methods currently used are the root sum square (RSS) method, sum of absolute values method, and the complete quadratic combination (CQC) method. Wilson et al. (1981) found the RSS method to considerably underestimate the response of undamped or lightly damped systems, whereas it might yield conservative results for highly damped systems. It also provides a nonconservative estimate in the case where two or more modes have natural frequencies close to each other. The CQC method is a weighted form of the RSS method. It was initially developed by Rosenblueth and Elordury (1969), and later simplified by Wilson et al. (1981). It overcomes many of the problems associated with the RSS method but is more complex to use. The form of the CQC method is given as: (3.2) in which Pij = (3.3) (1 - r 2 ) 2 + 4/3i/?ir(l + r-2) + 4(/3? + ffy where Qi is the maximum contribution if the ith mode for the response of interest, Pij is the cross-modal coefficient, /3i is the damping ratio in the ith mode, and r is the ratio of modal periods Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 23 It should be noted that Equation 3.3 is a complete quadratic form including all cross-modal terms. When natural periods are well separated, the cross-modal coefficients are small and the CQC method degenerates into the RSS method. The cross-modal terms become significant when the natural periods are close to one another. 3.3.4 Details of the MSSM The MSSM is an elastic spectral analysis in which the stiffness and damping properties are altered until the calculated moments are all equal to or less than the yield moments. A step-by-step description of the method is now presented. 1. Perform a modal analysis on the assumption of elastic behavior using 2 or 3% critical damping. 2. Calculate Eigenvalues and Eigenvectors to determine the natural period and mode shapes of the structure. 3. Compute the maximum displacements and forces using the CQC method. 4. Find the members in which the maximum moments exceed the yield moments. 5. In these members, calculate the damage ratio at each end according to the following formula: where fi[2^ is the damage ratio in the second iteration for the ith member, Af/1^ is the combined modal moment from the first iteration in the ith member, and Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 24 Myi is the yield moment at the appropriate location. The damage ratio in the members that do not yield is kept at one, because those members still have the initial stiffness. 6. Define the flexural stiffness of the substitute frame elements as: (EJU = (3.5) Pi where (EI)si,(EI)ai is the flexural stiffness of the ith element in the substitute and actual frames, respectively. Hi is the damage ratio for the ith element for the next iteration. 7. Compute an average or a "smeared" damping ratio for each mode P.i = 0 .2 (1 - -^z) + 0.2 (3.6) fln=£^Vfti (3-7) where Pi = e j i T k + M " + M * M " ) <3-8) and /3,i is the substitute damping ratio of the ith member, /3m is the smeared substitute damping for mth mode, Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 25 Pi is the flexural strain energy of the ith element in the mth mode, Li is the length of frame element i, and Mai,Mbi are the end moments of substitute frame element i for mth mode. 8. Repeat the modal analysis using the new smeared damping ratios and stiffnesses and compute the new member forces. 9. Compare the new moments with the yield moments. Modify the damage ratios according to the formula: > + 1 ) _ M^p<r Myi(l-s) + spr'M'i in which /4 n + 1 ) = „ „ \ " , n , „ ( B ) (3-9) ^ ( n + i ) . g ^ e m o ( j j f j e ( j d a m a g e ratio of the ith member to be used in the [n + l)th iteration, p\n^ is the damage ratio of the ith member used in the nth iteration, is the computed moment in the nth iteration, Myi is the yield moment in the ith member, and s is the ratio of stiffness after yield to initial stiffness. Equation 3.9 was derived by Yoshida (1979) on the basis that the rotation 6i for the next iteration will not change much. 10. Repeat steps 6 to 9 until all the convergence criteria are satisfied. The final values of pi are then taken to be the damage ratio for the structure, which provides an estimate of the locations and extent of damage in the whole structure. Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 26 3.3.5 Idealization of Three-Dimensional Model The structural idealization considered in PITSA can be summarized as follows: 1. The structure is idealized as a series of arbitrarily oriented plane frames connected by horizontal rigid diaphragms, thus three degrees of freedom at each floor describe the longitudinal displacements of the plane frames. 2. The mass is assumed to all be attached to the rigid diaphragms. Individual frame members are assumed massless enabling static condensation of many of the frame degrees of freedom. 3. Torsional stiffnesses of individual members are ignored. 4. The compatibility of vertical displacements for columns common to two or more frames is enforced, but no rotational compatibility is enforced for common columns. This introduces some errors when the frames are not mutually perpendicular. 5. No axial-flexural interaction is assumed for columns. The stiffness matrix of each plane frame is obtained using standard techniques and stat-ically condensed using the Choleski method to retain only one horizontal translation at each floor level, as well as those additional degrees of freedom required for vertical compatibility at columns common to adjacent frames. Assembling the structure stiffness matrix requires the transformation of the condensed frame stiffness matrices from the local to the global coordinates. The global coordinates system consists of three degrees of freedom, two translations and one rotation. The structure stiffness matrix is further condensed in the same manner as the frame stiffness matrices to retain only two translations and one rotation at the center of mass of each floor. Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 27 3.4 Three Dimensional Time-Step Analysis 3.4.1 General For nonlinear analysis, the most general approach is to consider the response of a nonlin-ear system as being essentially linear over a small interval of time, and to incrementally add these small time steps to get the entire response over time. Israel and Powell (1977) developed the computer program DRAINTABS at the University of California, Berkeley to calculate the inelastic dynamic response of three-dimensional buildings of essentially arbitrary configuration. The program combines the features of two computer programs, TABS and DRAIN-2D. The former performs the elastic analysis of three-dimensional buildings and the latter performs inelastic seismic analyses of two-dimensional struc-tures. The building is idealized in DRAINTABS as a series of independent plane substructures interconnected by horizontal rigid diaphragms. The global coordinate system consists of three degrees of freedom, two translations and one rotation at the center of rigidity of each diaphragm. A major limitation in this program is that the coupling of the substruc-tures through common columns is not fully taken into account, so that the idealization is not suitable for tube-type buildings. DRAINTABS can generate a variety of useful information such as maximum deformation and corresponding forces at all significant locations, and time history records of deformations at particular points in the structure. In spite of the limitations of the method, associated mainly with the uncertainties of the stiffness and damping to be used in the model, nonlinear dynamic analysis represents a powerful tool in the study of the earthquake response of structures. Its use provides insight into the basic mechanism of earthquake resistance of framed structures as well as a measure of the magnitude of the structural response to earthquakes. Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES, 28 3.4.2 Hysteretic Model The degrading stiffness beam bending element is used to model the reinforced concrete beams, which characteristically exhibit degrading flexural stiffness properties when sub-jected to cyclic loads. The element is assumed to consist of a linear elastic beam element with nonlinear rotational springs at each end. All plastic deformation effects, including the effects of degrading stiffness, are introduced by means of the moment-rotation rela-tionships at the springs. Israel and Powell (1977) mentioned that the moment-rotation relationship used in DRAINTABS is an extended version of Takeda's model, which in-cludes stiffness changes at flexural cracking and yielding, and strain hardening character-istics. Unloading stiffness is reduced by an exponential function of the previous maximum deformation. There is a set of rules for load reversals at a displacement amplitude less than the previous peak amplitudes. Flexural deformation, connection deformation and bar slip of longitudinal reinforcement within the connection are considered in defining the backbone curve. Israel and Powell (1977) extended the Takeda model in DRAINTABS to reflect the ob-served behavior in practice as follows: 1. A reduction of the unloading stiffness. The amount of this reduction depends on the largest previous hinge rotations. 2. Incorporation of a variable reloading stiffness, which is larger than that of the original Takeda model and also depends on the past rotation history. 3.4.3 Response Parameters The principal objective of the nonlinear analysis program is the evaluation of the maxi-mum inelastic flexural deformation produced in each member of the structure during the course of the earthquake. This deformation is presented by the maximum plastic hinge Chapter 3. DYNAMIC STRUCTURAL ANALYSIS PROCEDURES 29 rotation 8P, which is evaluated for each end of each member at each time increment. The maximum value of 6P is a measure of the ductility requirement imposed on the member by the earthquake. In order that this ductility requirement may be interpreted readily, the angle 6p is com-pared with the maximum elastic rotation angle 9y which the member may develop. It is also useful to know how near the maximum moment is to the yield moment in the members which do not yield. The ductility ratio is less than unity in these cases, and is the ratio of the maximum moment to the yield moment. Chapter 4 GROUND MOTION AND DESIGN OF MODEL STRUCTURES 4.1 Introduction The National Building Code Of Canada provides design recommendations for the seismic design of structures. The following philosophy governing the earthquake performance of building-type structures is implied: • The structure should be designed to remain elastic during earthquakes of moderate intensities which may occur several times during the life of the structure. • For the most intense earthquake expected during the life of the structure inelastic deformation is permitted, and the structure should be designed to survive the excitation without collapse. However, the code only gives recommendations for the latter case and the current edition (1985) of the NBCC specifies an elastic design procedure using reduced design forces to reflect inelastic behavior. For the purpose of this study, the cross sections of the beams and columns in the models are chosen to represent realistic conditions, and the yield moments are the values to be calculated. In doing so, the provisions specified in subsection 4.1.9 - 1985 and commentary J of the National Building Code of Canada, are used. The buildings are assumed to be located in a region with seismicity as given for Vancouver. The design base shear is first determined and distributed in the manner specified in the code. Then, the 30 Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 31 torsional moments are computed after determining the locations of the centers of mass and rigidity. Having the lateral forces and torsional moments as well as the gravity loads, a linear static analysis is performed to yield member design forces. The dead and live load design moments are redistributed, and combined with the earthquake moments using the load combination factors of the code to give the member design yield moments. These moments are fixed as the yield moments for the beams. For the columns, interaction diagrams are used to obtain the appropriate yield moments. Finally, the assigned yield moments are checked so that the sum of the flexural strength of the columns meeting at a joint is equal to or greater than the sum of the framing beams in the same plane, producing a strong column - weak beam design. 4.2 Base Motions For Earthquake Simulations In determining the response of a structure to future unknown earthquakes, the choice of a suitable ground motion, or motions, is one of the most important parts of the analysis. If the mode superposition method is used, then the use of a smoothed response spectrum provides input that accounts for the different frequency content of many different earth-quake ground motions. If the time step method is employed, ground motion records are needed and it is very difficult to get a single record that has a frequency content covering all realistic values. Therefore, it is preferable to consider a number of representative input motions when determining the likely maximum response of a particular structure. Shibata and Sozen (1976) proposed a smoothed response spectrum to fit the average re-sponse spectrum for the following four earthquake ground motions records used in their study: 1. El Centro 1940 NS. 2. El Centro 1940 EW. Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 32 3. Taft 1952 S69E. 4. Taft 1952 N21E. The four earthquake ground motions are scaled to give a maximum horizontal ground acceleration of 0.2g. The normalized 3% damped response spectra for the measured base motions are compared with the smoothed spectrum in Figure 4.1. In investigating how well the program PITSA predicts the ductility demand or displace-ments, Tam (1985) compared the results using PITSA and the Shibata and Sozen re-sponse spectrum, with the averaged response from DRAINTABS using the four scaled earthquakes forming this spectrum. The inelastic spectral accelerations are accounted for by the increase in the initial damping assumed at the beginning of the iterations. The results using PITSA agreed very well with the averaged response from DRAINTABS, and the same method is used in this study when comparisons are made between PITSA and DRAINTABS. Because this study is mainly directed towards studying the NBCC 85 recommenda-tions, it would be consistent to use the response spectrum implied in this code, which is taken to be the spectrum applied in NBCC 80. Therefore, to fit the NBCC 80 response spectrum, the spectral accelerations in Shibata and Sozen response spectrum are scaled by a factor of 1.3. Figure 4.2 shows the comparison between the scaled Shibata and Sozen spectrum, the scaled average spectrum of the four earthquakes that form Shibata and Sozen response spectrum, and the NBCC 80 response spectrum. It is clear that the results obtained using the scaled average response spectrum for the four earthquakes, or the smoothed response spectrum, will compare fairly well with NBCC 80 response spectrum, especially for periods greater than 0.5 seconds. Therefore, Shibata and Sozen response spectrum is used with a maximum horizontal ground acceleration of 0.26g to represent a maximum horizontal ground acceleration of 0.2g in the NBCC 80 response Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 33 1.2 1 -0.8 CO c" o to (0 0.6 CD CL CO 0.4 0.2 -.01 .25 .5 .75 1. 1.25 1.5 1.75 2. Period in seconds 2.25 Figure 4.1: Shibata and Sozen Spectrum Versus Normalized 3% Damped Response Spec-tra and their Average. Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 34 1.2 Figure 4.2: Comparison of Scaled Shibata and Sozen Response Spectrum, Scaled Aver-aged Normalized Response Spectrum, and NBCC 80 Response Spectrum. Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 35 spectrum. 4.3 Mathematical Modelling Mathematical modelling is an important part of the process for analyzing structures for response to earthquake-like ground motions. Assumptions made in constructing the mathematical model, concerning cracked, uncracked, or transformed concrete sections; slab participation; and participation of nonstructural elements, can greatly affect the outcome of the analysis, especially in relationships to periods, lateral displacements, and response accelerations. The proper mathematical model for a particular situation de-pends on the magnitude of the lateral motion. For example, very small (e.g. ambient) motion may cause all elements of rigidity, structural and nonstructural, to participate. With large motions, many of these elements, which are brittle and have low strength ca-pacities, will crack and will participate little if at all in the lateral stiffness of the building. Therefore, for small motions, the mathematical model would include transformed con-crete section properties of all the structural elements, including the floor slabs; it would also include the participation of such nonstructural elements as interior partitions, exte-rior walls and stairs. For large motions, the mathematical model may include cracked section properties of highly stressed members; it may use gross concrete section properties; and it may neglect or reduce the participation of the nonstructural elements. In addition to stiffness characteristics, the weight and the distribution of gravity loads must be considered. In this study, the models are subjected to a major earthquake, and the gross concrete sections properties are used. To make allowances for cracking, CSA Standards CAN 3-A23.3-M84 assume member stiffnesses to be 0.5 of the gross EI for all beams and 0.8 of the gross EI for all columns. In addition to that, the following may be Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 36 noted: 1. Each beam is represented by a single element, so that the plastic hinges can only form at the ends. 2. Element lengths from center of joints have been assumed throughout. 3. P — M interaction is not taken into account for columns and beams. 4. The P — A effect is not considered as a previous study by Goel (1969) showed that it does not materially alter the response for low or medium-rise structures. 5. A three percent critical damping is assumed as initial damping. 6. The slope of yielding branch in the moment-rotation relationship is taken to be horizontal, i.e elastic perfectly plastic. 4.4 Description of the Models The reinforced concrete buildings under consideration are rectangular in plan, with mo-ment resisting frames in both directions. Two aspect ratios are considered. The buildings in the first set have an aspect ratio of 1.5, and the buildings in the second set have an aspect ratio of 3. The typical plan and elevation of the buildings are shown in Figure 4.3. The models have the same elevation in the longitudinal direction, which is taken as the direction of earthquake ground motion, but the plan dimensions are different. They are all based on a 23 feet column spacing. The first set of models is 69 feet by 46 feet in plan, has a floor height of 12 feet and is five storeys tall, giving three frames in the longitudinal direction and four frames in the transverse direction. The second set of models is 69 feet by 23 feet in plan, with two frames in the longitudinal direction, and four frames in the transverse direction. The columns have constant cross sections throughout the height of 2 !=r *-< • CD OQ C n O O CD !-*> t) e-t- 03 !=r e-K CO 0 Mod dinal Layo O 1 *-« • n 0 ecti >-*» <rt-on; ni cn T3 on; CD 0 o e-t-& EI del t« & cn P> <r+ p f t—• • o II nd II CD CO 1-1 O CD O o cn cL n n p> o 1 H CD O P> P? Cn ttf o e-K O 3 CD P> X 10 FRAME 3 CO © 10 CO m — cn FRAME 4 ro CD u "S FRAME 5 *. oo m 10 ro ro ' _ i - • CO CO • FRAME 4 (n m FRAME 5 FRAME 2 Z FRAME 1 3 FRAME 6 FRAME 7 CJI © O o c 3 n rr. CP -n 3J > CO CO > o 5 ro CO 3J O C CO 3J O O -» ro 5 ffi. Z! o o - 1 CO o •2 to o § to o o § to 03 • - I O o O to ta CO H to Q to Co CO Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 38 Table 4.1: Diaphragm Data in Different Sets Aspect Ratio 1.5 3 Weight (kips) 400 200 Rotary moment of inertia (kip — ft2) 229250 88170 Column 1 2 3 the building, the bases of the lowest storey segments being assumed fixed. The beams and slabs have the same dimensions at all floor levels. The element dimensions are chosen to be within the practical range, and are tabulated in Appendix A. The characteristics of the multistory models used in this study are the following: 1. All floors of each model have the same plan and the same locations for columns. Thus, all floors have the same radius of gyration about the vertical axis through the centroid of the diaphragm. The diaphragm data for both models are given in Table 4.1. 2. The models are assumed to be eccentric in only the transverse or y-direction. 3. The principal axes of resistance for all stories are identically oriented long the x and y-axes. 4. The ratio of the beam stiffness to the column stiffness is the same for each frame, i.e all the frames should deform in the same shape for a given distribution of loading, the only difference being the magnitude of the displacements. For this reason the center of stiffness and the shear center should coincide and should be in the same location at each floor, and should thus all he along one straight line. Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 39 4.5 Principal Design Steps The application of the earthquake-resistant design provisions of NBCC 85 to the design of typical members in the five storey moment-resisting frame building are listed below. These design steps result in a yield moment for each beam and column to be used in the non-linear dynamic analysis. Heidebrecht and Tso (1983) noted that the torsional provisions for multistory structures are strictly applicable only to eccentric buildings where the centers of mass and centers of rigidity he on approximately vertical lines. For eccentric buildings that do not satisfy such conditions, the code provision may or may not be valid to in estimating the torsional effect induced by seismic ground motions. The code suggests the use of the more reliable approach of dynamic analysis to estimate the torsional effect under such circumstances. 4.5.1 Determination of Design Base Shear In seismic design, one of the fundamental parameters for estimating the earthquake effect on a structure is the Design Base Shear. This parameter represents the total horizontal seismic force that may be assumed to act parallel to the principal axis of the structure considered. NBCC 85 also specifies that independent design about the other principal axis shall be considered to provide adequate resistance in the structure for earthquake forces applied in any direction. This procedure is commonly known as the "quasi-static approach". The total lateral force, or Base Shear V, is given by: V=vIKSFW (4.1) where: v is a zonal velocity ratio whose value depends on the seismic zone in which the structure is located, as determined from the seismic risk maps. The risk level corresponds to Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 40 a return period of approximately 470 years (Heidebrecht and Tso, 1983), I is the occupancy importance factor, reflecting the greater conservation in design desir-able for buildings housing areas of public assembly or facilities that must remain operational after a major earthquake. NBCC 85 assigns a value of 1.3 for all post-disaster buildings and schools, and 1.0 for all other buildings, i f is a factor depending on the structural system used. This factor is intended to account for differences in the available ductilities or energy-dissipation capacities of various structural elements. The values of K assigned to different structural systems have been greatly influenced by past observations. Values of K range from 0.7 for a ductile moment-resisting space frame designed to resist the entire lateral load to 3.0 for elevated tanks, S is a coefficient related to flexibility of the structure, F is the foundation factor, designed to account for possible magnification of dynamic structural response due to adverse soil conditions, and W is the total dead load plus the following: 1. 25 per cent of design the snow load. 2. 60 per cent of the storage load for areas used for storage, the full contents of any tanks. To calculate the base shear V using Equation 4.1, the importance factor J, and the foundation factor JP, are assigned a value of unity. The period of the structure based on the code suggested value of 0.1 times the number of storeys in the longitudinal direction is estimated as 0.5 seconds. A value of K=0.7 is used because the structure consists of moment-resisting frames. For the calculation of W, the loads are assumed to be uniformly Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 41 distributed, with the dead load taken to be 110 psf, the snow load taken to be 25% of 70 psf, and 25% of 70 psf taken as live load which is assumed to represent 60% of the storage load. The base shear in the transverse direction is the same as the longitudinal direction because the structure consists of moment-resisting frames in both directions. 4.5.2 Distribution of Base Shear For structures having regular shapes or framing systems, the total lateral force or base shears, V, is to be divided into two groups: a concentrated load, Ft, applied at the top of the structure (to simulate the whiplash effect due to the higher modes of vibration) and the balance, (V - Ft) to be distributed over the entire height of the building, generally as concentrated loads at the floor levels. The latter is to be distributed in a triangular manner, a distribution corresponding essentially to the fundamental mode of vibration, increasing from zero at the base to a maximum at the top. The top load, Ft, is given by: Ft 0.004T/(^)2 < 0.15V 0.00 if ( ^ ) < 3.0 or Ta < 1.0 seconds in which: hn is the total height, and D„ is the dimension of the lateral force resisting system in a direction parallel to the applied forces. The magnitude of the distributed forces, Fx, making up the balance of the total lateral force, that is (V - Ft), is given by: Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 42 in which: Fx is the lateral force at level x above the base, wx, Wi is that portion of the total weight W, located at or assigned to level x or i respectively, hx, hi is the height above the base to level x ot i respectively, and n is the uppermost level of the structure. At each level x, the force Fx is to be applied through the center of mass. The distribution of the base shear for the models under consideration are tabulated in Tables 4.2 and 4.3. 4.5.3 Determination of the Structural Eccentricity In order to calculate the floor torque using code provisions, it is necessary to establish the structural eccentricity at the different floors of the building. Assuming the line of action of the resultant lateral loads at each floor is known, the problem of determining the structural eccentricity then reduces to locating the center of rigidity at each floor. The concept of the center of rigidity arose from considering the behavior of single-storey buildings with rigid roof diaphragm (Lin 1951). It is defined as the point through which the resultant of the lateral load passes without causing any rotation of the di-aphragm. For a single-storey building, this point can be located by the requirement that the first moment of the stiffness of the lateral resisting elements about the center of rigidity should be zero. This gives rise to the interchanging use of the term "center of rigidity" and "center of stiffness". The extension of this concept to eccentric multistory buildings is not trivial. In one inter-pretation (Humar 1984), the center of rigidity at a floor is defined as the point through Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 43 Table 4.2: Calculation of the Lateral Forces and Shears in the Case of an Aspect Ra-t io^.5 for an Earthquake in the Longitudinal Direction. Seismic Force Floor Storey Height Storey Level Level K Weight Wxhx Lateral Force Storey Shear ft kip kip. ft kip kip 5 5 60 400 24000 30 30 4 4 48 400 19200 24 54 3 3 36 400 14400 18 72 2 2 24 400 9600 12 84 1 1 12 400 4800 6 90 2000 72000 90 Col. 1 2 3 4 5 6 7 Table 4.3: Calculation of the Lateral Forces and Shears in the Case of an Aspect Ratio=3 for an Earthquake in the Longitudinal Direction.  Seismic Force Floor Level Storey Level Height K Storey Weight Wxhx Lateral Force Storey Shear ft kip kip. ft kip kip 5 5 60 200 12000 15 15 4 4 48 200 9600 12 27 3 3 36 200 7200 9 36 2 2 24 200 4800 6 42 1 1 12 200 2400 3 45 1000 36000 45 Col. 1 2 3 4 5 6 7 Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 44 which the resultant lateral force at that floor can pass and that floor will not undergo any rotation. The other floors may or may not rotate. Riddell and Vasquez (1984) inter-preted the concept of eccentricity in the dynamic sense, and concluded that the centers of rigidity exist only for a very special class of multistory buildings, namely buildings with proportional framing. Cheung and Tso (1986) defined the centers of rigidity of a multistory building as the set of points located at floor levels such that when the given distribution of lateral loading passes through them, no rotational movement of the building about a vertical axis will occur. This definition is used in this study. Tso (1989) showed that the static eccentricity 'e' as used in NBCC 85 should be referred to the shear center instead of the center of rigidity at the level under consideration. In his study, the eccentricity is defined as the horizontal distance between the location of the resultant of all forces at and above the level being considered and the shear center at the level being considered. Having defined the center of rigidity, it is necessary to develop a procedure to de-termine the locations of these centers for the models under consideration The following steps are used to do so. 1. For framed type structures, with uniform cross sections through the height, the centers of rigidity of all floors will he on a straight line. 2. Apply a set of horizontal forces through any point on the diaphragms and calcu-late the diaphragm rotations using any three-dimensional space frame program. DRAINTABS is used for this calculation in this study. 3. Repeat the analysis with the same set of forces through a different point in the diaphragms, and calculate another another set of diaphragm rotations. Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 45 4. Since these analyses are elastic, linear interpolation is used to find the exact location where the forces would be applied to give zero diaphragm rotations. This defines the center of rigidity. 4.5 .4 Calculation of Torsional Moments NBCC 85 includes a formula for the computation of the torsional moments which are to be applied simultaneously with the lateral forces: i = l where: Mtx is the torsional moment at level x, Ft is the top load, Fi is the lateral force applied at level i, and ex is the design eccentricity at level x. The torsional moments calculated according to such a formula are called static torques. They are essentially an extension of the static lateral forces used in computing storey shears. The design eccentricity is computed by whichever of the following expressions provides the greater stresses: ex = 1.5e (4.4) ex = 0.5e (4.5) where (4.3) Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 46 e is the structural eccentricity, the distance between the location of the resultant of all forces at and above the level being considered and the center of rigidity at the level being considered, and As mentioned before, the accidental eccentricity part in the design eccentricity expression is ignored in the design. 4.5.5 Analysis A linear analysis of the structure is made for each of the dead loads, live loads, and lateral earthquake loads, using DRAINTABS. This step yields a dead load moment MJJ, and a live load moment ML at each end of each member. The earthquake moment MQ is the largest moment from the following analyses: (a) Lateral loads and torsional moments using a design eccentricity^ 1.5e in the longitudinal direction. (b) Lateral loads and torsional moments using a design eccentricity= 0.5e in the longitudinal direction. (c) Lateral loads only in the transverse direction, as the building is not eccentric in this direction. In a normal design procedure, these design forces would serve to proportion the reinforcing of the various members. For the purpose of the present investigation, however, no detail design is required; it is merely sufficient to establish a yield moment for each member to serve as input for the non-linear dynamic analysis. The yield moments in these members are obtained for the most unfavorable combination of graAnty and lateral loads. For the beams, the yield moment is taken as the largest moment from the load combinations. The yield condition of the columns presented a more difficult problem due to the interaction Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 47 effect of the axial forces. The strengths of these columns are determined using the interaction diagrams for axial load and moment resistance for rectangular columns with an equal number of bars on all four faces. 4.5.6 Moment Redistribution Full utilization of the plastic capacity of reinforced concrete beams and frames requires an extensive analysis of all possible mechanisms and an investigation of rotation require-ments and capacities at all proposed hinge locations. The increase of design time may not be justified by the limited gains obtained. On the other hand, a restricted amount of redistribution of elastic moments can safely be made without complete analysis, yet may be sufficient to obtain most of the advantages of limit analysis. A limited amount of redistribution is permitted under the NBCC 85. The beams of a ductile moment resisting frame are designed and detailed to ensure that they possess considerable ductility. Hence, for these members, it is conservative to utilize the maxi-mum redistribution permitted by Clause 8.4 (i.e. 20 per cent). While it is possible to redistribute the earthquake moments, care must be exercised to ensure that the total column shears in any one storey remain unchanged after redistri-bution. A simpler approach is to redistribute only the dead and live load moments. This modification of moments does not mean a reduction in safety factor below that implied in code safety provisions; rather, it means a reduction of the excess strength which would otherwise be present in the structure because of the actual redistribution of moments that would occur before failure. 4.5.7 Load and Strength Reduction Factors NBCC 85 separates the strength provision for structural safety into two parts, load factors and strength factors. Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 48 The code requires that the ultimate capacity of a structure and its component elements be at least equal to or greater than the required ultimate strength. The required strength is determined by a combination of load factors, that is, specified service loads multiplied by appropriate load factors. Load factors are intended to ensure adequate safety against an increase in service loads beyond loads specified in design so that failure is extremely unlikely. Different load factors are used for different types of loads. Before combining these loads to obtain a design ultimate load, reduced load factors are specified to allow for the lesser possibility of certain types of loads occurring simultaneously. The design moments are determined from the larger of (a) 1.25 x MD + 1.5 X ML, or (b) 1.25 xMD + 1.5 x M g . o r (c) 1.25 x MD + 0.7 (1.5 x ML + 1.5 x MQ). The capacity of the structure is obtained by applying a strength reduction factor, <j>, to the nominal resistance of the element determined from basic mechanics. The factor cb is intended to allow for approximations in the calculations and variations in the material strengths, workmanship, and dimensions. Each of these may be within tolerable limits, but in combinations they may result in undercapacity. The basic strength equation for a section may be said to give the ideal strength, assuming that the equation is scientifically correct, that the materials are as strong as specified, and that sizes are as shown in the drawings. The dependable or reliable strength of the section to be used in the design calculations is taken as the ideal strength multiplied by c/> which depends on the importance of the variable quantities. For reinforced concrete members in flexure without axial load, <6 is taken as 0.90 and for members with spiral reinforcement, cj> is taken as 0.75. The ideal strength is calculated using the specified strengths of the concrete and steel. Because these strength values are normally exceeded in a real structure, an additional reserve of strength is available. Chapter 4. GROUND MOTION AND DESIGN OF MODEL STRUCTURES 49 4.5.8 Determination of the Yield Moment The final step is the determination of the yield moments in each member. In general, there are four different yield moments for beam members, a positive and a negative yield moment at each member end. For a prismatic and symmetric beam, these yield moments are all initially equal, but for a reinforced concrete beam the top reinforcing steel is generally not the same as the bottom steel. In addition, quantities of steel at one end of the beam are commonly different from those at the other end of the beam. Hence, for beams constructed from reinforced concrete, it is necessary to consider four different values for the yield moments. However, for this study, the positive and negative yield moments at an end are assumed to be equal because of the limitation imposed in PITSA that only a single value of yield moment can be used at a member end. Tong (1988) mentioned that the difference in yield moments between two ends of an element are insignificant. Therefore, a single value for the yield moment is assumed for each member. Furthermore, the yield moments for adjacent beams framing into a column must be equal because the same amount of steel is available for each beam. For these reasons the yield moment at each floor is taken to be the same in each bay of a frame. 4.5.9 Strong Column - Weak Girder Criterion Because of the relatively large inelastic deformations a building designed by the present code may undergo during a strong earthquake, proper provisions must be made to ensure that the structure has adequate ductile flexural behaviour (with ability to undergo large inelastic reversible deformations ) for individual members such as beams and columns, and preventing other non-ductile types of failure. The code thus prescribes the so-called strong column-weak girder design with the intent of confining yielding to the beams while the columns remain elastic throughout their seismic response. This is accomplished by Figure 4.4: Strong Column - Weak Girder Criterion requiring that the sum of the flexural strengths of the columns meeting at a joint, under the design axial loads, be equal to or greater than | times the sum of the moment strengths of the beams framing into the same joint ( Figure 4.4 ); that is ( M a + M 3 ) > 1(M2 + M 4 ) (4.6) Anderson and Bertero (1969) compared the dynamic response of frames designed accord-ing to this philosophy with that of frames designed using allowable stress methods with yielding allowed in the columns, and also of frames designed to achieve minimum weight (hence minimum stiffness). Their study showed that the present code philosophy in this regard results in better structural performance under earthquake loading than the other design methods. Appendix A contains the yield moments in the different buildings. Chapter 5 ALTERATION TO MSSM 5.1 Introduction In the early stages of this study, the program DRAINTABS was used to investigate the effect of the gravity loads on the ductility ratio. Gravity loads can influence the response in two ways; by the so-called P-delta effect which is ignored in this study, and by the creation of primary moments which can cause premature yielding in some beams. When the gravity loads were incorporated into DRAINTABS, by specifying the fixed end moments on each beam, the calculated maximum plastic hinge rotation (or ductility ratio) increased. This raised two questions: 1. Does specifying the fixed end moments on each beam in DRAINTABS lead to the correct value of the ductility ratio ? 2. How could PITSA be altered to account for the gravity load moments ? To answer the first question, the maximum plastic hinge rotation of a one storey building was obtained in two ways and the results compared. To incorporate the effect of gravity loads in PITSA, two different methods were tried. Testing of the modified versions is carried out by comparison from DRAINTABS. 51 Chapter 5. ALTERATION TO MSSM 52 5.2 Effect of Gravity 5.2.1 Testing of DRAINTABS This section examines whether DRAINTABS' analysis using the fixed end forces gives the correct plastic hinge rotations. To investigate this a one storey framed building is analyzed in two ways and the results are compared. The first analysis uses six elements to model each beam and the gravity loads are treated as concentrated loads at the interior nodes. This analysis should give the correct result but requires many more degrees of freedom. In the second case the building is modelled using one element for each beam using the fixed end forces to account for the gravity loads. This is the desired approach for large structures as it minimizes the number of degrees of freedom. The one story framed building is shown in Figure 5.1. The frames in the direction of the applied ground motion consist of one 60-ft wide bay, and a 40-ft wide bay frame in the transverse direction. The diaphragm weighs 520 kips. The yield moments of the columns and beams are 600 and 240 k-ft respectively. 30 inch square columns are used, and 18 by 36 inch beams are used throughout. In the first analysis the static loads are applied at the nodes as shown in Figure 5.1 for model 1. In the second analysis, the fixed end forces indicated in model 2 are used. The structure is subjected to El Centro 1940 NS ground motion, with a maximum horizontal ground acceleration of 0.20g. A time step of 0.01 second is used. The resulting maximum plastic hinge rotations in the two models at the member ends are 0.003245 and 0.003444 radians respectively. For the cross section under consideration, this plastic hinge rotation represents a ductility factor of 1.85. This value of the ductility factor is small and the good comparison may be due to the small values of the plastic hinge rotations. To examine whether the plastic hinge rotations would be the same if their absolute values are higher, the sizes are decreased to 18 inch Chapter 5. ALTERATION TO MSSM 53 5.0 k 5.0 k 5.0 k 5.0 k 5.0 k Model 1 146 k-ft 12.5 12.5 146 k-ft Model 2 12' /77m77777777777777777777777777777777777777777777^ FRAME 4 Ground Motion I • CM CM >)< CR LU FRAM • • FRAME 1 60' 40 ' Figure 5.1: Checking the Static Load Solution Chapter 5. ALTERATION TO MSSM 54 square columns, and 14 by 20 inch beams. The yield moments are also changed to 180 kip-ft in all beams, and the maximum horizontal ground acceleration is raised to 0.25g. The resulting maximum plastic hinge rotations are 0.005835 radians in both models when gravity loads are not used, and 0.042104 and 0.038553 radians for models 1 and 2 respectively when gravity loads are introduced. This represents a ductility factor of about 3.27 for the first model, and 3.12 for the second model. Although the results are not identical, the two analyses produce estimates of the ductility ratio that are quite close, and offer proof that the inclusion of the gravity loads as fixed end moments satisfactorily accounts for the increase of the ductility ratio. 5.2.2 Testing the Effect of Gravity Load Moments on the Ductility Ratio The five story frame building used by Tarn (1985) is used to test the effect of gravity load moments on the ductility ratio. This building is shown in Figure 5.2. The frames in the direction of the applied ground motion consist of three 20-feet wide bays. The story height is 12 feet, giving a total building, height of 60 feet. The diaphragm weights for each floor are 520 kips, and the building is symmetric. The yield moments of the columns and beams are 2000 and 333 k-ft respectively everywhere. No strain hardening after yield is assumed. Two values of maximum horizontal ground acceleration are used, namely 0.25g and 0.50g for the first 10 seconds of El Centro 1940 NS earthquake record. A time step of 0.01 seconds is used for numerical integration. To arrive at the effect of gravity on the peak ductility demand, two cases are studied. The structure is assumed to have no gravity loads in one case, and gravity loads are incorporated as fixed end forces in DRAINTABS in the other case. Gravity loads are as-sumed to be uniformly distributed, with the dead load taken to be 330 psf. Consequently, the load vector is modified in the solution to include these initial conditions. Figures 5.3 and 5.4 show the peak ductility demand in the members of a longitudinal frame for a Chapter 5. ALTERATION TO MSSM Columns: section 30"x30" Beams: section 18"x36" Storey Weight = 520 kips 520 kips 520 kips 520 kips 520 kips Element Numbering / 25 30 35 24 29 34 23 2B 33 22 27 32 21 28 31 /77777777777777777777777777m7777777777m 5@12' FRAME 4 Ground Motion m w 2@20' Figure 5.2: Layout and Properties of Tarn's Building (taken from Tarn 1985) Chapter 5. ALTERATION TO MSSM 56 maximum horizontal ground acceleration of 0.25g and 0.50g respectively. The inspection of these charts shows the increase in the maximum ductility demand when gravity loads are considered. The grey bars indicated in Figures 5.3 and 5.4 by "Ratio", represent a quantity equal to 1.5 times the ratio of the gravity moment to the yield moment for the member under consideration. The results show that on average the effect of primary moments on the peak ductility demand is to increase the ductility factor in the beams by a value approximately equal to 1.5 times the applied primary moment over the yield moment at a member end. The analyses also show that the columns do not yield, i.e. the ductility demand in the columns are less than one, and are unchanged when gravity loads are included. Similarly, there is no change in the lateral floor displacements because the static solution shows that gravity loads do not cause appreciable lateral displacement in the building. 5.3 Alteration to MSSM Two different methods are used in trying to incorporate the effect of gravity loads in the MSSM analysis. 5.3.1 First Method The moments induced at the member ends due to an earthquake will generally be equal and of opposite signs if gravity loads are ignored. When gravity is considered, its effect is to add hogging moments at each end of the member. During earthquake motion, these hogging moments will tend to increase the moments at one end and decrease it at the other end of the member. In previous versions of MSSM, the yield moments were assumed to be from earthquake only. Chapter 5. ALTERATION TO MSSM 57 In an attempt to modify MSSM for the gravity loads, the yield moment at one end of the beam was decreased by the gravity load moments and increased at the other end. Figure 5.5 schematically shows such an arrangement for the end where the yield moment has been decreased. The results from PITSA will be the damage ratio p\ calculated as if the moment rotation curve had its origin at 01. However, the definition of the ductility ratio is one plus the plastic hinge rotation divided by the yield rotation. This would be the same as the value of the damage ratio p2 drawn with reference to the original moment rotation curve in Figure 5.5. This is given by: where R is the ratio of the gravity load moment to the yield moment at the end under consideration. Similarly, at the other end of the member, the yield moment to be used in the analysis is the sum of. the yield moment and the gravity moment at that end, and the new damage ratio is calculated from the old damage ratio as follows: As will be shown in section 5.4.2, this method does not produce good agreement with DRAINTABS results. 5.3.2 Second Method The second method is based on the results obtained in section 5.2.2. The average increase in the beam ductility factors due to the presence of gravity loads is found to be 1.5 times the ratio of the gravity moment to the yield moment at a member end for the cases investigated here. Since the ductility ratio ignoring gravity compares well with the damage ratios from MSSM by several researchers (Yoshida 1979, Metten 1981, Hui P2 =pi(l-R) + R (5.1) p2 = p1(l + R) + R (5.2) Chapter 5. ALTERATION TO MSSM 58 1984, Tam 1985), adding the ratio of gravity moment to yield moment to the damage ratio calculated in PITSA would correspond to the ductility ratio obtained from DRAINTABS with gravity loads included. Therefore, the definition of the damage ratio is changed to take into account the effect of the gravity loads as follows: MDR is the "modified" damage ratio due to the combined effect of gravity and earth-quake loads, DR is the damage ratio calculated by MSSM ignoring gravity loads, and a is a function of the structure and the earthquake considered. In words, the modified damage ratio is equal to the damage ratio at the end of the iteration plus a factor that depends on the structure and the earthquake characteristics times the ratio of gravity (primary) moments to yield moment of the member under consideration. It should be noted that the calculated displacements are not altered by this modification, which agrees with the results from DRAINTABS output. Both methods are incorporated into the computer program PITSA, and the results obtained from the comparison with DRAINTABS results are shown in the following section. 5.4 Program Testing 5.4.1 General To use the modified computer program with confidence, it should be checked against an established procedure. The time history computer program DRAINTABS is chosen for this purpose. MDR = DR + ct (5.3) Chapter 5. ALTERATION TO MSSM 59 The prime response parameter to be compared is the peak ductility demand for each member in the frame under consideration. In doing so, the damage ratios are shown for the modified PITSA versus the ductility ratios obtained from each of the four earthquake records, followed by a comparison of the averaged results from DRAINTABS versus modified and original PITSA results. The frames at the periphery of the building parallel to the ground motion direction are used in the comparison. 5.4.2 Results The building used by Tarn (1985) is used again in this section to test the magnitude of the modified damage ratio with the maximum ductility demand from DRAINTABS. The reason for using this building is that the results obtained without using gravity in both programs compared very well. Therefore, the success or failure of using the modified version will be attributed only to the modification due to gravity loads, and not to other reasons that may cause inaccuracies in the MSSM. In this model, the centers of mass in all floor levels are displaced four feet from the center of the structure in the direction perpendicular to the earthquake motion, as shown in Figure 5.2. This represents a uni-form eccentricity of 10%. The maximum horizontal ground acceleration is 0.25g. Figures 5.6 and 5.7 contain charts of the peak ductility demand calculated from DRAINTABS output for the four earthquake records forming the Shibata and Sozen response spectrum versus the modified damage ratio using each of the two methods. The average value of the DRAINTABS results is then compared with the original and the modified damage ratio found by the two methods. These charts shows the comparison for frames 1 and 4 respectively. Because frame 4 is closer to the center of mass, it is subjected to higher forces, and since the yield moments are equal in both frames, the damage ratios in this frame are higher than frame 1. The average value of damage ratio in frame 1 from DRAINTABS is 2.39, compared to an average value of 1.95 from the first method, and Chapter 5. ALTERATION TO MSSM 60 an average value of 2.29 from the second method. Original PITSA results in an average value of 1.42. Similarly, frame 4 has an average damage ratio of 2.73 from DRAINTABS, while "modified" PITSA results in a value of 2.15 from the first method, and 2.68 from the second method. Original PITSA results in an average value of 1.78. This comparison shows the improvement in the calculation of the damage ratio especially using the sec-ond method which will be used in the work that follows. The second method is used to examine building CM3011 that is designed according to the requirements of NBCC 85. It has an eccentricity of 30% in the center of mass. In Figure 5.8, DRAINTABS results show that frame 1 has an average damage ratio of 3.2, and that the modified PITSA using method 2 predicts 2.8. The comparison also shows that PITSA predicts that frame 3 is very close to the elastic range with a damage ratio of about 1.3, which is essentially the same damage predicted by DRAINTABS. From these results it is concluded that PITSA is a reliable tool in predicting the damage pattern and magnitude of the ductility demand. 5.5 Execution Cost The execution cost of PITSA depends on a number of factors, such as the number of modes, number of degrees of freedom, half-band width of the stiffness matrix and the number of iterations required to converge. On the other hand, the number of time steps required to go through the complete duration of the record affects the computing cost in DRAINTABS. In other words, a longer record costs more, if the step size is held constant. Also, as more members yield, the cost in running DRAINTABS increases, because the change in the yield status requires a new frame stiffness matrix to be constructed. Table 5.1 compares the execution cost of the two computer programs using the main frame Amdahl V/8 computer at the University of British Columbia with a commercial account Chapter 5. ALTERATION TO MSSM 61 Table 5.1: Comparison of Execution Costs PITSA DRAINTABS Model CM3011 CM3011 Maximum Acceleration (g) 0.2 0.2 Number of Modes 6 N / A Number of DOF in structure 45 45 Number of Iterations 5 N / A Number of Time steps N / A 1000 Execution Time in second (CPU) 8.973 66.012 Computing Cost in $ 2.36 14.512 Column 1 2 3 and at normal priority. 5.6 Discussion Based on the results obtained, the following summarizes the observations: 1. Although only one example was considered, it appears that using the fixed end forces from the gravity loads in DRAINTABS produces the correct increase in the ductility ratio. 2. For those beams subjected to gravity loads, adding to the damage ratio after the end of the iterations a value equals to a times the ratio of primary moments to the member yield moment produces reasonable predictions of the ductility ratio for the MSSM analysis. The factor a depends on the structure and the earthquake characteristics and is to be investigated in another study. Thus, it is now possible to use this method to evaluate existing buildings due to earthquake and gravity loads combined. Chapter 5. ALTERATION TO MSSM 62 3. PITSA is a reliable tool in predicting the damage pattern of multistory framed structures. 4. In terms of computing cost, PITSA is considerably cheaper to use than DRAINTABS. Chapter 5. ALTERATION TO MSSM 63 Ductility Ratio Element Number Figure 5.3: Ductility Demand in Members with and without Gravity Loads with Maxi-mum Horizontal Ground Acceleration of 0.25g Ductility Ratio 21 22 23 24 25 26 27 28 29 30 Element Number Figure 5.4: Ductility Demand in Members with and without Gravity Loads with Maxi-mum Horizontal Ground Acceleration of 0.50g Chapter 5. ALTERATION TO MSSM 64 Figure 5.5: First Method to Incorporate Gravity Loads to MSSM: Moment Rotation Relationship at the Member Ends Chapter 5. ALTERATION TO MSSM 65 Ductility Ratio 4 Ductility Ratio 3 -1 -/ / / / / / / / / / / / / / / C3Av»r»g» 12 Original PITSA • M»thod 1 • Mtthod 2 23 24 25 Element Number Element Number Figure 5.6: Peak Ductility Demand Versus Damage Ratio for Frame 1 of Tarn's Building. Left: Peak Ductility Demand from Four Records Versus Modified PITSA, Right: Average from the Four Records Versus Modified and Unmodified PITSA Ductility Ratio Ductility Ratio 21 22 23 24 25 21 22 23 24 25 Element Number Element Number Figure 5.7: Peak Ductility Demand Versus Damage Ratio for Frame 4 of Tarn's Building. Left: Peak Ductility Demand from Four Records Versus Modified PITSA, Right: Average from the Four Records Versus Modified and Unmodified PITSA Chapter 5. ALTERATION TO MSSM 66 Damage Ratio Damage Ratio Element Number Element Number Figure 5.8: Peak Ductility Demand Versus Damage Ratio in a 30% Eccentric Building. Left: Frame 1, Right: Frame 3. Chapter 6 PARAMETRIC STUDY 6.1 General Scope The research work reported herein is an evaluation of the torsional provisions of the 1985 edition of the NBCC. It is intended to demonstrate the order of magnitude of the flexural ductility that may be required for the girders in typical frame buildings during an earthquake. Five storey rectangular frame buildings designed according to the requirements of the NBCC 85 are employed; the structural variation consists only of changes in the member stiffnesses, yield moments, and eccentricity ratio in each set. These buildings are subjected to ground motion excitation represented by the response spectrum developed by Shibata and Sozen, which has been multiplied by an appropriate factor such that it is close to the NBCC 80 response spectrum. The buildings are analyzed for their inelastic response to the above mentioned ground motion using the computer program PITSA which utilizes the Modified Substitute-Structure Method. The program computes the maximum forces and damage ratios developed in each column and girder for each frame in the building, as well as the maximum storey displacements and rotations. The prime response parameter of interest is the damage ratio in the members of the eccentric buildings, which is to be compared with the damage ratio for similar concentric buildings. 67 Chapter 6. PARAMETRIC STUDY 68 6.2 Parameters Considered The parameters considered in this study are: 1. Type of Eccentricity This parameter refers to whether the mass or the stiffness center is eccentric with respect to the building center. A building is called mass eccentric (CM) if the center of stiffness is located at the center of the building, but the mass center is not. Similarly, the building is termed stiffness eccentric (CR) if the mass center is at the center of the building but the stiffness center is not. The two cases are shown in Figure 6.1. 2. Aspect Ratio The ratio of the building dimension in the direction parallel to the ground motion to the dimension in the perpendicular direction or as shown in Figure 6.1. Tsicnias (1981) emphasized a study by Tso (1980) showing that an increase in the aspect ratio results in an increase in the dynamic amplification of the torsional response, even for a nonlinear system. Therefore, two values of the aspect ratio are considered to represent the cases with moderate and severe dynamic amplification, namely 1.5 and 3. 3. Gravity Loads For stiffness eccentric building two different assumptions are made as to how the floors are supported on the beams. In one case the floors are assumed to be two way slabs and so the floor loads are equally divided in each direction, interior beams get-ting twice the load of exterior beams. Another case simulates the situation where intermediate beams carry essentially all the load to the transverse girders and so the beams in the longitudinal frames are assumed to carry no gravity loads. Chapter 6. PARAMETRIC STUDY 69 A similar situation arises for the mass eccentric buildings but here a further com-plication is that the floor load may vary in the transverse direction, and so the gravity loads are not distributed equally in the frames of the longitudinal direction. 4. Eccentricity Ratio This parameter represents the distance between the center of mass and the center of stiffness e, to the transverse plan dimension D as shown in Figure 6.1. The values of this parameter are taken to range from zero for a symmetric building to an exceptionally large eccentricity of 38 percent. 6.3 Model Codes For the ease of tracking the different cases studied, a code consisting of two letters followed by four digits are assigned to each model. The first two letters show the eccentricity type, in which CM means an eccentric center of mass, and CR means an eccentric center of stiffness. The first two digits show the percentage of the eccentricity The third digit indicates the type of gravity load, as follows: 3 r d digit=l : Uniform distribution of gravity loads. 3 r d digit=2 : Non uniform distribution of gravity loads. 3 r d digit=3 : All gravity loads are assumed to be carried by the transverse frames. The fourth digit is a code for the value of the aspect ratio, where: 4th digit=l : Aspect ratio =1.5 with three frames in the longitudinal direction, and four frames in the transverse direction. 4th digit=2 : Aspect ratio = 3 with two frames in the longitudinal direction. FRAME 3 FRAME 4 3 *J t _ i . • <r+- era &•* p 1-1 q n Si P ST o CD ^ o P> &-CD" 3 CD P> 3 o cr i -S CD #cra S. 'pT Bt 5 ' w era 3 P> 5-- sr. M > » *T3 CD O pa pj S E o era FRAME4 FRAME5 FRAME 1 -n > m ro FRAME5 n < n CD FRAMES FRAMES — D — FRAME7 FRAME 3 FRAME4 . 0 0 . FRAME4 O -X-FRAME5 FRAME 1 -n 31 > m ro FRAME 5 2 S • CO FRAMES 8 CD o CO § o FRAMES FRAME 7 Chapter 6. PARAMETRIC STUDY 71 4th digit=3 : Aspect ratio = ~ with two frames in the longitudinal direction. For example, the model CR3012 represents an eccentric center of rigidity building with an eccentricity ratio of 30 percent. The design of this model is based on a uniform distribution of gravity loads to the frames, and it has an aspect ratio of 3. 6.4 Effect of the Movement of Center of Resistance The increase of damage ratios in one frame means that this frame is subjected to an increasing amount of inelastic behavior, and so a reduction in effective stiffness. De-pending on the frame involved this could lead to an increased eccentricity. Tong (1988) showed that the increase of damage ratios in one side of a building relative to the other side would mean the movement of the center of resistance away from the damaged side towards the other side that has little damage. Further excitation may lead to further damage in this frame and consequently the movement of the center of resistance further away from it. Eventually, this could lead to the collapse of the structure. This movement of the center of resistance leads to an important observation regarding the difference between the CR and CM buildings. From the design, it is noted that frames 2 or 3, for aspect ratios 3 and 1.5 respectively, are always stronger than frame 1. Therefore in a CR building, if frames 2 or 3 is damaged more than frame 1, the center of resistance will move away from frames 2 or 3 towards frame 1 and so the eccentricity will decrease. For a C M building, if frame 1 has more damage than frames 2 or 3, the center of resistance will move towards frames 2 or 3 causing an increase in the eccentricity, thus a decrease in the torsional moments for further excitation. The ductility demand in the members of the longitudinal frames at the periphery of the building are only used because they represent the maximum and minimum ductility de-mand in the longitudinal direction. The transverse frames will always be elastic because Chapter 6. PARAMETRIC STUDY Damage Ratio 72 24 26 26 27 28 29 30 31 Element Number 32 33 34 35 Figure 6.2: Target Damage Ratio for the Evaluation of the Design the earthquake is assumed to be in the longitudinal direction only. 6.5 Numerical Results The distribution of the ductility demand in the beams of the longitudinal frames at the periphery of the building are calculated. To arrive at a target value for the local ductility demand, or alternatively the damage ratio, a dynamic analysis is performed on a symmetric structure subjected to the same ground motion used in this study. This building has an initial period of 0.95 seconds. Figure 6.2 shows the damage ratios in the members of a frame in the longitudinal direction. The element numbers refer to the beams in the frames in the direction of the earthquake and are shown in Figure 6.1. A damage ratio of 3.5 is the average value for a symmetric building with any aspect ratio, and thus can be considered as a reference value seeing that the damage is essentially uniform throughout the building. If the torsional provisions of the code produce a building with uniform damage ratio of about 3.5, then the torsional provisions would be considered adequate. During the design of the buildings it was noted that for buildings where the mass is Chapter 6. PARAMETRIC STUDY 73 eccentric, i.e. CM type buildings, the frame closest to the center of mass, frame 3 in Figure 6.1, is governed by the design eccentricity ex — 1.5e provision in the code, and that frame 1 is governed by the design eccentricity ex — 0.5e provision. For the CR type of buildings where the mass is considered concentric but the rigidity is eccentric, the design of the stiffer frame, frames 2 or 3 which is closest to the center of rigidity, is always governed by the design eccentricity ex — 0.5e, while for frame 1 the design is generally governed by ex = 1.5e. It is also noted that frame 3 is stronger than frame 1 in the CR as well as CM buildings. Thus frames 2 or 3 can be named the "bigger" frame, and frame 1 the "smaller" frame. 6.5.1 Eccentricity Type In order to evaluate the effect of eccentricity type on the design, two models are con-sidered, namely CM3012 and CR3012. The eccentricity ratio is equal to 30% in both models, and the design is based on an uniform distribution of gravity loads. The initial fundamental periods in CR3012 and CM3012 are 0.954 and 0.923 seconds respectively, and the final periods are 1.35 and 1.32 seconds. Figure 6.3 shows the damage ratio in frames 1 and 2 in both models. The charts show that the CR building has large damage in the bigger frame, frame 2, and low damage in the smaller frame, frame 1. This means that the center of resistance moved away from frame 2 towards frame 1, and thus the eccentricity decreased. On the other hand, frame 2 is the bigger frame in CM3012. The damage in this frame can be seen to be less than frame 1, thus resulting in the movement of the center of resistance towards frame 2 and so the eccentricity decreases, although in this case the damage ratio is very much the same in the two frames. Therefore the behavior is not the same in both models as the bigger frame has more damage in the CR building but less damage in the C M building. However, the movement of the center of resistance is such that the eccentricity decreases in both buildings, but the decrease in Chapter 6. PARAMETRIC STUDY 74 Damage Ratio s 22 23 24 Element Number • FRAME 1 m FRAME 2 Damage Ratio s • FRAME 1 SFRAME 2 22 23 24 Element Number Figure 6.3: Effect of the Eccentricity Type CR3012 will be much more than CM3012. In both cases the frames design governed by the 0.5e eccentricity had the most damage. 6.5.2 Aspect Ratio In order to examine the effect of aspect ratio on the response, the damage ratios in different parts of the structure are examined. Two models are considered for the purpose of this investigation, namely CM2011 and CM2012. Both models are eccentric with respect to the center of mass, with an eccentricity ratio of 20%. The design is based on a uniform distribution of gravity loads. The initial periods are 1.016 and 0.95 seconds and the final periods are 1.497 and 1.313 in CM2011 and CM2012 respectively. The only difference between these models is the aspect ratio. CM2011 has an aspect ratio of 1.5 while CM2012 has an aspect ratio of 3. To obtain an estimate of the dynamic amplification in the buildings with different aspect ratios, the ratio of the dynamic response to the static response at a floor level is used, where response here could mean floor rotations or floor lateral displacements. The static Chapter 6. PARAMETRIC STUDY 75 response is obtained in each building using the code-specified forces as lateral loads and an eccentricity equal to the static eccentricity 'e' to calculate the torsional moments which are applied with the lateral loads. The dynamic amplification of three models, namely CM2011, CM2012 and CM2013 are shown in Figure 6.4, which also shows the damage ratios in the members of CM2011 and CM2012. From Figure 6.4 (a) and (b), it is evident that the increase in the aspect ratio results in an increase in the dynamic amplification in the building. This result agrees with Tso (1980) and Tsicnias (1981). In addition to that, the damage ratio in frame 1 shows that the damage decreases with the increase in the aspect ratio in the lower three floors. For frame 3 in CM2011 or frame 2 in CM2012, the increase in the aspect ratio results in an increase in the damage ratio, especially in the lower floors which results in a uniform distribution of damage in the two frames at the periphery of the building. This can be considered as a satisfactory performance, as the average damage is about 3 which is just less than the target damage. Therefore, for small aspect ratios, the performance of the models suggests that the smaller frames are underdesigned and the bigger ones are overdesigned. For the high aspect ratio, the design of both frames is generally acceptable. 6.5.3 Presence of Gravity Loads The increase in the yield moments due to gravity loads greatly affects the damage pattern in the structure. If gravity loads are considered and distributed uniformly, the damage pattern will tend to be relatively uniform in the building, whereas if the member do not carry gravity loads and the member yield strength is based only on the earthquake loads, the ductility ratio goes much higher in some members. This is shown in Figure 6.5 by comparing the behavior of models CM2012 and CM2032. CM2012 assumes a uniform distribution of gravity loads to the beams while CM2032 assumes all gravity loads are carried by the transverse beams. The increase of the ductility demand in the beams of Chapter 6. PARAMETRIC STUDY 76 Dynamic Amplification 3 2.5 1.5 0.5 Aspect Ratio-1,5 Aspect Ratlo=3 AspBct.Ratlo-1/3 2 3 4 Floor Level Dynamic Amplification 14 12 10 Aspect Ratio° 1.5 Aspect. Ratlo=3 Aspect Ratio-1/3 2 3 4 Floor Level (a) (b) Damage Ratio & Damage Ratio 6 22 23 24 Element Number 21 22 23 24 25 Element Number (c) (d) Figure 6.4: Effect of the Aspect Ratio, (a) Dynamic Amplification of Floor Rotations, (b) Dynamic Amplification of Floor Displacements, (c) Damage Ratio in CM2011, (d) Damage Ratio in CM2012 Chapter 6. PARAMETRIC STUDY Damage Ratio Damage Ratio • FRAME 1 El FRAME 2 21 22 23 24 25 21 22 23 24 25 Element Number Element Number Damage Ratio 8] CR3012 s -i i i i i l 21 22 23 24 25 Element Number • FRAME 1 • FRAME 2 Damage Ratio s • FRAME 1 • FRAME 2 22 23 24 25 Element Number Figure 6.5: Effect of Gravity Load Presence Chapter 6. PARAMETRIC STUDY 78 the upper floors seen in CM2032 was also observed by Tam (1985) and attributed to the relatively longer period of vibration, which resulted from the more flexible structure that occurs when the damage ratio increases. To examine whether gravity presence will lead to the same behavior in the eccentric center of stiffness cases, buildings CR3012 and CR3032 are also shown in Figure 6.5. As can be seen, the behavior is essentially the same although not quite as pronounced when gravity loads are ignored. 6.5.4 Gravity Load Distribution This parameter is present only in the cases of eccentric center of mass. The behavior of model CM2011 versus CM2021 and CM3011 versus CM3021 is shown in Figure 6.6. Structures CM2011 and CM3011 have the floor mass distributed uniformly, i.e. the beams in frames 1 and 3 have equal gravity loads. CM2021 and CM3021 have larger gravity loads on frame 3 than frame 1, reflecting the higher mass on that side of the building. The results are consistent with those shown in Figure 6.5 for CM2012 and CM2032 where it is seen that in frames with higher gravity loads and higher yield moments, there is a lower ductility demand. Similarly frames with lower yield moments have a higher ductility factor. From these results, it is clear that the ratio of the gravity load moments to the yield moments have a potentially large impact on the ductility demand and its distribution through the height of the structure. 6.5.5 Initial Eccentricity Ratio Several combinations are used to examine the effect of eccentricity on the ductility de-mand. For each eccentricity type, both aspect ratios are considered. For each aspect Chapter 6. PARAMETRIC STUDY 79 Damage Ratio Damage Ratio 6 22 23 24 25 Element Number 22 23 24 25 Element Number Damage Ratio 7 2t 22 23 24 25 Element Number Damage Ratio 7 22 23 24 Element Number Figure 6.6: Effect of the Distribution of Gravity Loads Chapter 6. PARAMETRIC STUDY 80 ratio, gravity loads are assumed to be uniform or eccentric in the longitudinal direction, or totally carried by the transverse direction. In each case, two values of eccentricity ratio are used. Figure 6.7 shows the effect of increasing the eccentricity ratio on the damage ratio in a C M building with an aspect ratio equal to 1.5 and different distribution of gravity loads. The inspection of these charts shows that the increase in the eccentricity ratio results in lower damage ratios in frame 3, while frame 1 in general has higher damage ratios. Figure 6.8 shows the increase in the eccentricity ratio in CM0612 and CM3012 for a uniform distribution of gravity and in CM1322 and CM2522 for an eccentric distribution of gravity. CM1332 and CM2532 represent the case with no gravity loads. The decrease in the damage ratio of frame 2 with the increase in the eccentricity ratio is evident from these charts. Figure 6.9 shows the comparison for CR buildings in which gravity loads are assumed uniform in CR1611 and CR3811 and are ignored in CR1631 and CR3831. The increase in the eccentricity ratio in these cases results in a decrease in the damage of frame 1 and a slight increase in the damage of frame 3, the stronger frame, which is the reverse of the CM cases. The level of damage seen in frame 1 is generally low and thus the design may be considered severe for the smaller frames in the CR buildings. For a higher aspect ratio, Figure 6.10 compares CR1012 with CR3012 when gravity loads are considered, and CR1032 with CR3032 when gravity is ignored. The same trend of damage observed in the smaller aspect ratio is evident in the higher aspect ratio. This shows that the increase in the eccentricity ratio results in a decrease in the damage of the smaller frames and a slight increase in the damage of the bigger frames. Chapter 6. PARAMETRIC STUDY 81 Damage Ratio 5 Damage Ratio 5 22 23 24 Element Number 22 23 24 Element Number Damage Ratio 7 Damage Ratio 7 22 23 24 Element Number 22 23 24 Element Number Damage Ratio 20 21 22 23 24 25 Element Number • FRAME 1 0 FRAME 3 Damage Ratio 20 21 22 23 24 25 Element Number • FRAME 1 • FRAME 3 Figure 6.7: Effect of the Eccentricity Ratio on CM Buildings with Aspect Ratio=1.5 and Different Distribution of Gravity. Chapter 6. PARAMETRIC STUDY 82 Damage Ratio 4 Damage Ratio 22 23 2+ Element Number Damage Ratio s 22 23 24 Element Number Damage Ratio 18 21 22 23 24 25 Element Number Damage Ratio 6 21 22 23 24 25 Element Number Damage Ratio 22 23 24 Element Number 21 22 23 24 25 Element Number Figure 6.8: Effect of the Eccentricity Ratio on CM Buildings with Aspect Ratio=3 and Different Distribution of Gravity Chapter 6. PARAMETRIC STUDY 83 Damage Ratio s Damage Ratio 6 CR3811 I FRAME 1 £3 FRAME 3 21 22 23 24 Element Number II III 21 22 23 24 25 Element Number Figure 6.9: Effect of the Eccentricity Ratio on CR Buildings with Aspect Ratio=1.5 with and without Gravity Chapter 6. PARAMETRIC STUDY 84 Damage Ratio 5 Damage Ratio 5 22 23 24 25 Element Number 22 23 24 25 Element Number Damage Ratio 20 22 23 24 25 Element Number Damage Ratio 20 15 10 CR3032 21 22 23 24 25 Element Number I FRAME 1 ED FRAME 2 Figure 6.10: Effect of the Eccentricity Ratio on CR buildings with and without Gravity for Aspect Ratio=3 Chapter 6. PARAMETRIC STUDY 85 6.6 Base Shear In an effort to find another method for determining the design eccentricity that would achieve an even distribution of damage for higher initial eccentricity ratios, the design eccentricity formulae specified in different codes were examined, and some calculations were carried out using the ATC provisions. The ATC does not amplify the calculated eccentricity, and has been used to redesign the models CM1011, CM2011, CM3011. The resulting damage ratios were compared with those using the NBCC 85 provisions, and surprisingly, there was hardly any change in the damage ratio of frames 1 and 3 using the new design, and the transverse frames were still in the elastic range. To investigate this result, the base shears for the new designs were obtained from the dynamic analysis and plotted in Figure 6.11. This Figure shows that the forces in the longitudinal frames are quite uniform, and that there is hardly any change in base shears in the longitudinal frames 1, 2 and 3 as the eccentricity increases, but that the base shears in the transverse frames 4 and 5. The reason for that may be due to the yielding of the longitudinal frames, which reduces the stiffness of these frames relative to the transverse frames, that still have their initial stiffness because they do not suffer any damage for an earthquake applied in the longitudinal direction. The static analysis was carried out in two steps for CM3011, first with the lateral forces alone, and then with the torsional moments alone. These analyses show clearly that the torsional moments are essentially resisted by frames 4, 5, 6, and 7 and that frames 1, 2 and 3 resist the lateral forces only. Therefore, changing the torsional provisions does not lead to much of a change in the distribution of the forces in the longitudinal frames, and specifying another design eccentricity does not help improve the design of the frames 1, 2 and 3 as the increase or decrease in the torsional moments primarily affects frames 4, 5, 6 and 7 in the transverse direction. Chapter 6. PARAMETRIC STUDY 86 Dynamic Base Shear (kips) • 10% Eccentricity Ratio • 20% Eccentricity Ratio • 30%Eecantricity Ratio Frame Number Figure 6.11: Distribution of Base Shears from the Dynamic Analysis Unci-." Three-Dimensional Action Additional analyses were done where the earthquake was applied with the same intensity in the transverse direction only and then in both directions. In the first case, because the building is symmetric in this direction no base shears were present in the longitudinal direction, and only the transverse frames carried the base shears. The damage ratio in the transverse direction was about 3.5, which is the same magnitude as the longitudinal symmetric case. When earthquakes were applied simultaneously in both directions, the resulting base shears in frames 1, 2 and 3 were changed very slightly from their value when the earthquake was only applied in the longitudinal direction, and the increase in the base shear was essentially only in the transverse frames. The damage ratios in the transverse direction were found to be uniform, with an average value of 4.2 in the beams. However, the application of the earthquakes simultaneously and with the full intensity in both directions is severe, and a realistic situation would be to apply two components Chapter 6. PARAMETRIC STUDY 87 of the earthquake with an intensity of 0.18g for each component, giving a total intensity of 0.26g applied at 45 degrees with the building axes. The result of this analysis showed that the damage in the transverse frames was about 3. The longitudinal frames had less damage than if the earthquake is only applied in the longitudinal direction but with the same distribution, i.e. frame 1 has more damage than frame 3. This shows that applying the earthquake in the longitudinal direction only produces the highest damage in the longitudinal frames, and the effect of having two components of the earthquake with full intensity will mainly be to increase the damage in the transverse frames. 6.7 Effect of Period of Vibration 6.7.1 On The Design The calculation of the design base shear requires the fundamental period T of the build-ing. In the absence of a precise determination of T, the code suggests that periods in seconds be taken as 0.1 times the number of stories. For the five storey building under consideration, their estimated period is 0.5 seconds. The elastic analysis of the symmetric building CM0012 shows that the fundamental period is 0.95 seconds. If another design is done using a fundamental period of 0.95 seconds, the seismic response factor S will change from 0.311 to 0.226, which will result in a decrease of 40% in the design base shear. The effect of the decrease in the design base shear on the damage ratios of the symmetric case can be estimated by increasing the spectral acceleration in the analysis by the same ratio 40%. This results in an increase of the average damage ratio from 3.5 for the build-ing with an design estimated period of 0.5 seconds to 4.5 for the building with period 0.95 seconds. Therefore, the code-specified period provides a conservative estimate of Chapter 6. PARAMETRIC STUDY 88 Damage Ratio s C R 3 4 1 2 inn 21 22 23 2* 25 Element Number • FRAME 1 • FRAME 2 Damage Ratio 5 21 22 23 24 25 Element Number Figure 6.12: Effect of Initial Period on the Ductility Demand the fundamental period of a building in the sense that the design will be based on higher forces than if the "real" period is used. 6.7.2 On The Analysis The stiffnesses of the CM buildings are chosen to simulate a practical situation. For the CR buildings, the eccentricity is introduced by assigning different stiffnesses to the different parts in the building. If the chosen overall stiffnesses are assumed to be very different from the CM case, the resulting buildings will have different initial periods and the ductility demand cannot be compared directly to the CM cases. To illustrate this point, a comparison is made between two buildings both designed for T = 0.5 seconds but having different periods. CR3412 has an initial period of vibration of 1.46 seconds, while CR3012 has an initial period of 0.95 seconds. The results of the dynamic analyses shown in Figure 6.12, show that the ductility demand in CR3012, which has a lower period, is higher than the other case. The reason is due to the higher Chapter 6. PARAMETRIC STUDY 89 spectral acceleration for CR3012 due to its lower period. The difference in the calculated spectral acceleration, which is about 50%, results in an increase in the average damage ratio of the stiffer frame of about 45%. Therefore, for eccentric buildings with an initial period equal to the initial period of the symmetric building, the damage ratios can be compared. For buildings with different periods, the change in the damage ratio magnitudes has to be taken into consideration. 6.8 Discussion A three-dimensional inelastic torsional seismic analysis is performed on a total of twenty seven models. The structural damage of the frames at the periphery of the buildings is compared using the damage ratio in each member. The results show the following: 1. For buildings with uniform distribution of gravity loads: (a) The design of CM structures is about right, and the torsional provisions can be said to be adequate for high aspect ratios. For small aspect ratios, the bigger frame is overdesigned, and the smaller frame is slightly underdesigned. (b) CR structures appear to be overdesigned for frame 1, the weaker frame, for all eccentricity ratios. The design of frame 3, the stronger frame, is underde-signed. 2. For non-uniform distribution of gravity loads: (a) Frame 1, the weaker frame, is underdesigned. (b) Frame 3, the stronger frame, is overdesigned. The trend of frame 1 being underdesigned and frame 3 being overdesigned, increases with the increase in the initial eccentricity ratio. Chapter 6. PARAMETRIC STUDY 90 3. For no gravity loads: (a) Higher floors are very much underdesigned, especially in CM buildings. (b) Lower floors are generally acceptable in the CM buildings. In the CR buildings, the smaller frame is always overdesigned, especially for high eccentricity ratios. 4. The three-dimensional dynamic and static analyses show that the increase in the eccentricity ratio results in an increase in the forces in the transverse direction although the ground motion is only applied in the longitudinal direction. This may be due to the yielding in the longitudinal frames, which reduces the stiffnesses of these frames, and thus the relative stiffness between the longitudinal and transverse direction changes. The result is the transfer of the loads to the frames that do not yield, i.e. the transverse frames. 5. The application of the earthquake in the longitudinal direction results in moments in the transverse frames approaching yield. If another earthquake is applied simul-taneously with the same intensity in the transverse direction, which is concentric, a small change in the longitudinal base shear accompanied by a large increase in the transverse base shear will result. This results in damage in the transverse frames that exceeds that found for the transverse loading alone. If the intensity of both earthquakes is multiplied by ^ , the resulting damage will decrease in all members and the damage in the transverse frames will be less than for the symmetric case. Therefore, for buildings with the same physical properties, the following may be con-cluded: 1. The code provides a conservative estimate of the period that is used in the calcu-lation of the base shear. Chapter 6. PARAMETRIC STUDY 91 2. The behavior of the models with eccentric center of mass is different from those with an eccentric center of rigidity because the bigger frame has more damage in the CR buildings while the higher damage occurs in the smaller frame in the CM building. 3. When gravity loads are not considered in the design, the yield moments in the members of the upper stories should be increased because of the effect of higher modes on the response. When the gravity load distribution is eccentric, the in-crease in the eccentricity ratio results in an overdesign in the bigger frames, and an underdesign in the smaller frames. In the cases of eccentric center of mass, and if the distribution of gravity loads is uniform, the code provisions is about right. For CR buildings, the smaller frames are overdesigned. 4. The trend of the torsional moments to overdesign the bigger frames in CM buildings, and overdesign the smaller frames in CR buildings increases with the increase in the eccentricity ratio. 5. The torsional moments are essentially distributed to the transverse frames, and the longitudinal frames resist lateral loads only. Thus any change in the torsional provisions does not result in a significant change in the ductility demand in the longitudinal direction. Chapter 7 SUMMARY AND CONCLUSIONS 7.1 Summary The torsional provisions of the NBCC 85 specify that structures should be designed for the worst condition that results from increasing the nominal torsion by ±50% as well as ±0.1 times the transverse dimension of the building (0.1D). The ±50% increase is thought to account for increased dynamic amplification of the torsional motion as well as changes in the eccentricity due to changing stiffness of the structure components. The ±0.1D increase is to account for errors in determining the center of mass and difficulty in determining the center of rigidity (or shear center), especially for multistory structures. In this study only the ±50% increase in eccentricity has been studied and the±0.1D terms have been neglected entirely. In order to investigate the accuracy and applicability of the torsional provisions of the NBCC 85, the ductility demand of several practical five storey eccentric buildings de-signed according to the requirements of NBCC 85 are compared to the ductility demand in a similar concentric building. The buildings are assumed to be in a location such as Vancouver where the peak horizontal acceleration and peak horizontal velocity are 0.2 (units of g and m/s respectively). The structural variations consist of changes in the stiffnesses and yield moments of the members. The choice of the buildings are such that they simulate a practical combination of different parameters, namely: 1. Eccentricity Type. 92 Chapter 7. SUMMARY AND CONCLUSIONS 93 2. Aspect Ratio. 3. Gravity Loads. 4. Eccentricity Ratio. The buildings are then analyzed for their inelastic response when subjected to a ground excitation with the type and intensity that may be expected in Vancouver. This is done using the program PITSA, which is a program that carries out a pseudo elastic dynamic analysis to model the inelastic response. Modifications to the program, developed in this thesis, accounts for the effect of gravity loads on the ductility demand. The calculated ductility demand on different parts of each building are then compared to the ductility demand in a similar concentric building, and the effect of each parameter on the ductility demand is examined. The results from the analyses show the following: 1. The code-specified period used in the calculation of the design base shear pro-vides a conservative estimate of the period of a structure. This period should be established using the structural properties and deformation characteristics of the resisting elements in a properly substantiated analysis. 2. The behavior of the buildings with eccentric center of mass is different from those with eccentric center of rigidity. The bigger frame has more damage in stiffness eccentric buildings while the smaller frame is more damaged in mass eccentric buildings. 3. Gravity loads have a potentially large impact on the ductility demand and its distribution through the height of the building. When the members are not designed to carry gravity loads, the higher modes of vibration result in a large underdesign in the upper floor level beams, which leads to a member rotational ductility demand Chapter 7. SUMMARY AND CONCLUSIONS 94 of about 15. For an eccentric distribution of gravity loads, the bigger frames are always overdesigned with a ductility demand of about 2 and the smaller frames are always underdesigned with a ductility demand of about 5. The code provisions do not recognize such a distribution. For a uniform distribution of gravity loads, the code provisions are about right, as the damage is not excessive in any of the members. 4. The torsional provisions tend to overdesign the bigger frames in mass eccentric buildings for large eccentricity ratio, and overdesign the smaller frames in stiffness eccentric buildings. 5. For an earthquake applied in the longitudinal direction, the torsional moments are essentially carried by the transverse frames, and the longitudinal frames resist the lateral forces only. If the full value of the earthquake is applied in each direction, the increase in base shears will only be in the transverse frames if the building is concentric in this direction, and the damage in this direction will be higher than for a symmetric case. When one earthquake is applied at an angle making 45 degrees with the building axes, the resulting damage in the transverse frames will be less than the symmetric case. 6. Modifications made to MSSM allow it to include the effect of gravity loads in the calculated damage ratios. 7. The computer program PITSA is a reliable tool in the evaluation of the damage in a three-dimensional building structure. Most notably, the damage of different parts of the building relative to each other can be evaluated, which is important in " the evaluation of the behavior of torsionally unbalanced buildings. Chapter 7. SUMMARY AND CONCLUSIONS 95 7.2 Conclusions The study shows that the following points are not recognized in the code but are seen to have a potential impact on the response: 1. The static eccentricity specified in the code is not stated whether it is a result of an eccentric center of mass (CM) or an eccentric center of rigidity (CR) building. This study shows that the behavior of the CR buildings are different from CM buildings. 2. Gravity loads have a potentially large impact on the ductility demand and its distribution through the height of the building. When the members are not designed to carry gravity loads, the upper floor level beams will be underdesigned. For an eccentric distribution of gravity loads, the bigger frames are always overdesigned and the smaller frames are always underdesigned. For a uniform distribution of gravity loads, the code provisions are about right. 3. The increase in the aspect ratio results in an increase in the dynamic amplification, and the torsional provisions appear to cover the higher dynamic amplification cases as the design is generally acceptable for these buildings. The result of that is an overdesign in buildings with small aspect ratios, or alternatively, small dynamic amplification. 4? The increase in the eccentricity ratio results in an overdesign in the bigger frames in CM buildings and an overdesign in the smaller frames in CR buildings. 5. The ±50% increase in the nominal torsion specified in the code can be changed without a significant effect on the ductility demand of the longitudinal frames, as the torsional moments are essentially carried by the transverse frames. Chapter 7. SUMMARY AND CONCLUSIONS 96 6. The maximum damage in the longitudinal frames occurs by applying an earthquake in the longitudinal direction. If another earthquake with the same intensity is applied simultaneously in the transverse direction, and the building is concentric in this direction, the damage in the transverse frames will exceed that for a symmetric case. Bibliography [1] Arnold, C. and Robert Reitherman. "Building Configuration and Seismic Design," New York: John Wiley & Sons, 1982. [2] Awad, A.M. and J.L. Humar "Dynamic Response of Buildings to ground Motions," Seventh European Conference on Earthquake Engineering, Athens, Greece (1982). [3] Blume, J., N. M . Newmark and L. Corning. "Design of Multistory Reinforced Concrete Buildings for Earthquake Motions," Port-land Cement Association, Skokie, 111, 1970. [4] Clough, R.W. and Sterling B. Johnston. "Effect of Stiffness Degradation on Earthquake Ductility Requirements, " Proceeding Japan Earthquake Engineering Symposium, Tokyo , pp 227 -232 (1966). [5] Clough, R.W. and J. Penzien. "Dynamics of Structures," Mcgraw Hill Book Company, New-York, 1975. [6] Cowan, H.J. and Forrest Wilson. "Structural Systems," Van Nostrand Reinhold Company, 1981. [7] Cheung, V.W. and W.K. Tso. " Eccentricity in Irregular Multistory Buildings," Canadian Journal of Civil Engi-neering, Volume 13, pp 46-52 (1986). [8] CSA Standards. CAN 3 - A23.4 - M84. Concrete Design Handbook, CPCA (1984). 97 Bibliography 98 [9] Derecho, A.T., Donald M, Schultz and Mark Fintel. "Analysis and Design of Small Reinforced Concrete Building for Earthquake Force", Engineering Bulletin, portland cement Association, PCA, 1978. [10] Elms, D.G. "Seismic Torsional Coupling in Structures," Proceeding 6th Australian Conference on the Mech. of Structures and Materials, University of Canterbury, New Zealand, pp 189-193 (1977). [11] Fintel, M. "Handbook of Concrete Engineering," 2nd Edition, Van Nostrand reinhold Com-pany, New York, 1985. [12] Gluck, J. et al. " Dynamic Torsional Coupling in Tall Building Structures," Proceedings Institution of Civil Engineers, Part 2, Volume 67, pp 411-424 (1979). [13] Green, N.B. "Earthquake Resistant Building Design and Construction," 3rd edition, 1987. [14] Goel, S.C. "P-Delta and Axial Column Deformations on Aseismic Frames," Journal of the Structural Division, ASCE, Volume 95, No. ST8, pp 1063-1711 (August 1969). [15] Gulkan, P. and Mete A. Sozen. " Inelastic Responses of Reinforced Concrete Structures to Earthquake Motions," Journal of the American Concrete Institute, Title No. 71-41, pp 604-610 (1974). [16] Heidebrecht, A.C. and Wai-Keung Tso. "Proposed Seismic Loading Provision National Building Code of Canada 1985," ^th Bibliography 99 Canadian Conference on Earthquake Engineering, June 1983, pp kl9 - k31. [17] Heidebrecht, A.C. "Earthquake Codes and Design in Canada", 3rd Canadian Conference on Earth-quake Engineering, Volume 1, June 1979, pp 575. [18] Hoerner, J.B. "Modal Coupling and Earthquake Response of Tall Buildings," California Institute Of Technology, Pasadena, Report EERL 71-07 (1971) [19] Humar, J.L. and A. M. Awad. "Design for Seismic Torsional Forces," ^th Canadian Conference on Earthquake En-gineering, June 1983, pp 251-260. [20] Humar, J.L. "A Proposal to Improve the Static Torsional Provisions for the National Building Code of Canada: Discussion," Canadian Journal Of Civil Engineers, Volume 11, pp 668-669 (1984) [21] Humar, J.L. "Seismic Design of Buildings using a Time-history Method," 3rd Canadian Confer-ence on Earthquake Engineering, Volume 2, pp 745-777 (June 1979). [22] Humar, J.L. " Design for Seismic Torsional Forces," J^th Canadian Conference on Earthquake Engineering, Vancouver, B.C., Volume 12, No. 2, pp 150-163 (1984). [23] Hui, L. " Pseudo Non-linear Seismic Analysis," Master's thesis, U.B.C. (October 1984). Bibliography 100 [24] Husid, R. "The Effect of Gravity on the Collapse of Yielding Structures with Earthquake Excitation," Proceedings of the J^th World Conference on Earthquake Engineering, Volume 2, Santiago de Chile, pp A4-31 (January 1964). [25] Irvine, N.M. and G.E. Kountouris. "Inelastic Seismic Response of a Torsionally Unbalanced Single-storey Building Model," U.S. Department Of Commerce, Seismic behavior and design of buildings, Report Number 2, Massachusetts Institute Of Technology, Cambridge. Prepared for National Science Foundation, Washington DC, Publication Number R79-31 order No. 654. [26] Israel, R.G. and Graham H. Powell. DRAIN-TABS: A Computer Program For Inelastic Earthquake Response of Three-Dimensional Buildings, Report No. UCB/EERC - 77/08, Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley, Califor-nia (March 1977). [27] Leet, Kenneth. "Reinforced Concrete Design," McGraw-Hill Book Company. [28] Lin, T.Y. "Lateral Force Distribution in a Concrete Building Story." Journal of the American Concrete Institute, Volume 48 (12), pp 282-296 (1951). [29] Kan. C.L. and Anil K. Chopra. " Elastic Earthquake Analysis of a Class of Torsionally Coupled Buildings," Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, Volume 103, No.ST4, April 1977, pp 821-838. Bibliography 101 [30] Kan, C.L. and Anil K. Chopra. " Torsional Coupling and Earthquake Response of Simple Elastic and Inelastic Sys-tems," Journal of the Structural Division, Proceedings of the American Society of Civil Engineers, Volume 107, No. ST8, pp 1569-1587 (1981). [31] Kanaan, A.E. and Graham H. Powell. DRAIN-2D: A General Purpose Computer Program for Dynamic Analysis of Inelas-tic Plane Structures, Report No. EERC 73-6 and EERC 73-22. College of Engineer-ing, University Of California, Berkeley. (1975). [32] National Building Code Of Canada, 1980. [33] National Building Code Of Canada, 1985. [34] Metten, Andrew W.F. "The Modified Substitute-Structure Method as a Design Aid for Seismic Resistant Coupled Structural Walls," Master's thesis, U.B.C., March 1981. [35] Newmark, N.M. and W. H. Hall. "Earthquake Spectra and Design," Earthquake Engineering Research Institute, EERI, 1982. [36] Newmark, N.M. "Torsion in Symmetrical Buildings," Proceedings of the J^th World Conference on Earthquake Engineering Volume 2, Santiago de Chile, (January 1969). [37] Newmark, N.M. and E. Rosenblueth. "Fundamentals of Earthquake Engineering," Prentice-Hall, Englewood Cliffs, N.J., 1971. Bibliography 102 [38] Nilson, A.H. and George Winter. "Design of Concrete Structures", tenth edition, McGraw-Hill Book Company. [39] Pekau, O.A. and H. A. Gordon. "Buildings Susceptible to Torsional-Translational Coupling," 3rd Canadian Confer-ence on Earthquake Engineering, Volume 2, pp 1063-1090 (June 1979) [40] Pekau, O.A., R. Green and A. N. Sherbourne. "Inelastic Behavior of Frame Structures Under Static and Earthquake Forces," Pro-ceedings 1st Canadian Conference on Earthquake Engineering Vancouver, Canada, pp 303-320 (1971). [41] Park, R. and T. Paulay. "Reinforced Concrete Structures," John Wiley & Sons, 1975. [42] Pillai, S.U. and D. W. Kirk. "Reinforced Concrete Design, " 2nd edition, Mcgraw-Hill Ryerson Limited. [43] Rega, G. and F. Vsetroni and A. Vulcano. "On the Relationship Between Overall and Local Ductility Demand for Plane Frames Subjected to Earthquakes," 8th world Conference on Earthquake Engineering, San Fransisco, Volume 4, pp 575-582 (1984). [44] Riddell, R. and J. Vasquez. "Existence of Centers of Resistance and Torsional Uncoupling of Earthquake Re-sponse of Building, " Proceedings Of 8th World Conference On Earthquake Engi-neering, San Fransisco, California, Volume 4, pp 187 - 194 (1984). [45] Rutenberg, A. and 0. A. pekau. "Earthquake Response of Asymmetric Buildings, a Parametric Study", J^th Canadian Bibliography 103 Conference On Earthquake Engineering, pp 271-280 (June 1983). [46] Rosenblueth, E. "Design of Earthquake Resistant Structures", Pentech Press - London :Plymouth. [47] Rosenblueth, E. and J. Elodury. "Response of Linear Systems to Certain Transient Disturbance," Proceeding 4th World Conference on Earthquake Engineering, Santiago, Part A l , Volume 1, pp 185-196 (1969). [48] Shibata, A. and M. A. Sozen. "Substitute-Structure Method for Seismic Design in Reinforced Concrete," Journal of The Structural Division, Volume 102 (ST1), pp 1-18 (January 1976). [49] Thomas, G.R. and Nicholas Petalas. "Seismic Response of Shear Wall-frame Systems," 3rd Canadian Conference on Earthquake Engineering, Volume 2, June 1979, pp 989-1008. [50] Tarn, K.S.K. "Pseudo Inelastic Torsional Seismic Analysis Utilizing the Modified Substitute-Structure Method," Master's thesis, U.B.C., 1985. [51] Tong, Raymond K.W. "The Dynamic Behavior of the Center of Stiffness of R/C Eccentric Structures under Seismic Excitation," Master's thesis, U.B.C., 1988. [52] Tsicnias, T.G. and G.L. Hutchinson. " Evaluation of Code Requirements for the Earthquake Resistant Design of Torsion-ally Coupled Buildings," Proceedings Institutions of Civil Engineers, Part 2, Volume 71, pp 821-843 (1981). Bibliography 104 [53] Tso, W.K. "Seismic Torsional Moment Estimation for Multistorey buildings," Canadian Society For Civil Engineering Annual Conference, St. Johns, Newfoundland, June 8-10 (1989). [54] Tso, W.K. and A . C Heidebrecht and S.Cherry. "Canadian Seismic Code Provision Beyond 1985," ^th Canadian Conference on Earthquake Engineering, pp34 (1983). [55] Tso, W.K. and K. G.Asmis. "Torsional Vibration of Symmetrical Structures," Proceedings of the 1st Canadian Conference on Earthquake Engineering, Vancouver, Canada, 1971, pp 178-186. [56] Tso, W.K. and K .M. Dempsey. "Seismic Torsional Provisions for Dynamic Eccentricity," Earthquake Engineering and Structural Dynamics, Volume 8, pp 275-289 (1980). [57] Tso, W.K. "A Proposal to Improve the Static Torsional Provisions for the NBCC," Canadian Journal of Civil Engineering, Volume 10, pp 561-566 (1983). [58] Tso, W.K. "A Proposal to Improve the Static Torsional Provisions for the NBCC: Reply," Canadian Journal of Civil Engineering, Volume 10, pp 669-670 (1984). [59] Tso, W.K. "Induced Torsional Oscillations in Symmetric Structures," Earthquake Engineering and Structural Dynamics, Volume 3, pp 337-346 (1975) Bibliography 105 [60] Tso, W.K. and A. W. Sadek. "Inelastic Response of Eccentric Structures," Jfth Canadian Conference on Earth-quake Engineering, June 1983, pp261-270. [61] Tso, W.K. and V. Meng. "Torsional Provisions in Building Codes," 3rd Canadian Conference on Earthquake Engineering, Volume 1, June 1977, pp 663. [62] Wakabayashi, M. "Design of Earthquake-Resistant Buildings," Mcgraw-Hill book Company, 1986. [63] Wilson, E.L., A. Der kiureghian and E.P. Bayo. "A Replacement for the SRSS Method in Seismic Analysis," Earthquake Engineering And Structural Dynamics, Volume 9, pp 187-194 (1981). [64] Wilson, E.L. and H. H. Dovey. "TABS - Static and Earthquake Analysis of Three-Dimensional Frame and Shear-wall Buildings," Report No. EERC 72-1, Earthquake Engineering Research Center, University of California, Berkeley, May 1972. [65] Yoshida, S. " Modified Substitute-Structure Method for Analysis of Existing Reinforced Con-crete Structures," Master's thesis, U.B.C., March 1979. [66] Yoshida, S., N. D. Nathan , S. Cherry, and D. L. Anderson. "Modified Substitute Structure Method for Analysis of Existing Buildings," 3rd Canadian Conference on Earthquake Engineering, Volume 2 (1979). Appendix A BUILDING PROPERTIES This appendix shows the stifnesses, yield moments and the initial and final periods in the different models under consideration. Figure 4.3 shows the layout of the models under consideration. A single value for the yield moment is assumed for each beam. In each frame, the beams in each floor are assumed to have the same yield moment. Note that: xx means any initial eccentricity ratio, and z means any distribution of gravity loads. Frames 4 and 7 have the same properties and frames 5 and 6 also do. 106 Appendix A. BUILDING PROPERTIES 107 Table A . l : Summary o f Model Codes MODEL Gravity Aspect Ratio CMxxl l Uniform 1.5 CMxx21 Eccentric 1.5 CMxx31 N / A 1.5 CMxxl2 Uniform 3 CMxx22 Eccentric 3 CMxx32 N / A 3 CRxxl l Uniform 1.5 CRxx31 N / A 1.5 CRxxl2 Uniform 3 CRxx32 N / A 3 Column 1 2 3 Table A.2: Stifness Values EI in kips.ft2 used for the Girders in Different Models MODEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 FRAME 5 CMxxzl 192000 192000 192000 192000 192000 CR16zl 94500 94500 224000 192000 192000 CR38zl 32000 32000 362000 192000 192000 CMxxz2 172537 172537 111111 mm CR10z2 172500 253200 192000 192000 CR30z2 58000 290000 111111 mm CR34z2 20000 111111 111111 mm Col. 1 2 3 4 5 6 Appendix A. BUILDING PROPERTIES 108 Table A.3: Stifness Values EI in kips.ft2 used for Each Column Line in Different Models with Aspect Ratio=1.5 jMODEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 FRAME 5 Col. line—• 1,2,3,4 5,6,7,8 9,10,11,12 1,5 9 2,6 10 CMxxzl 296296 296296 296296 296296 296296 296296 296296 CR16zl 194400 194400 433800 194400 433800 194400 433800 CR38zl 71140 71140 723400 71140 723400 71140 723400 Col. 1 2 3 4 5 6 7 8 Table A.4: Areas in Inch2 used for Each Column Line in Different Models | MODEL Column Line 1,2,3,4 5,6,7,8 9,10,11,12 CMxxzl 400 400 400 CR16zl 324 324 484 CR38zl 196 196 625 CMxxz2 484 484 CR10z2 440 530 CR30z2 324 625 CR34z2 400 144 Col. 1 2 3 4 Table A.5: Stifness Values EI in kips.ft2 used for Each Column Line in Different Models with an Aspect Ratio=3  MODEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 Column line 1,2,3,4 5,6,7,8 1 5 2 6 CMxxz2 433807 433807 433807 433807 433807 433807 . CR10z2 360000 518200 360000 518200 360000 518200 CR30z2 71140 723400 71140 723400 71140 723400 CR34z2 38400 296296 38400 296296 38400 296296 Col. 1 2 3 4 5 6 7 Appendix A. BUILDING PROPERTIES 109 Table A.6: Yield Moments My in kips.ft in the Girders of Model CMxxzl MODEL LEVEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 FRAME 5 1 85 135 130 100 140 2 90 140 135 110 135 CM1011 3 85 130 110 100 135 4 70 115 85 75 120 5 50 100 60 60 105 1 75 135 170 100 140 2 80 140 165 110 135 CM2011 3 75 130 100 100 135 4 70 115 95 75 120 5 55 95 65 60 105 1 65 135 200 100 140 2 70 140 200 110 140 CM3011 3 70 130 100 125 135 4 65 115 105 75 120 5 55 95 65 60 105 1 80 135 140 100 140 2 90 140 140 110 140 CM1021 3 80 130 110 100 135 4 65 115 90 75 120 5 50 100 70 65 105 1 65 135 180 100 140 2 70 140 180 110 140 CM2021 3 65 130 100 100 135 4 55 115 105 75 120 5 40 95 85 65 105 1 45 135 220 100 140 2 55 140 215 110 140 CM3021 3 55 130 100 125 135 4 45 115 120 75 120 5 30 95 95 65 105 Col. 1 2 3 4 5 6 7 Appendix A. BUILDING PROPERTIES 110 Table A.7: Yield Moments My in kips.ft in the Girders of Model CMxx31 and CRxxzl MODEL LEVEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 FRAME 5 1 60 70 105 75 75 2 65 75 105 85 85 CM1031 3 55 65 85 70 70 4 40 • 45 55 50 50 5 20 25 30 25 25 1 50 70 145 75 75 2 55 75 140 85 85 CM2031 3 50 65 105 70 70 4 40 45 65 50 50 5 20 25 35 25 25 1 35 70 180 75 75 2 45 75 170 85 85 CM3031 3 45 65 125 70 70 4 30 45 80 50 50 5 20 25 40 25 25 Col. 1 2 3 4 5 6 7 Appendix A. BUILDING PROPERTIES 111 Table A.8: Yield Moments My in kips.ft in the Girders of Models CRxxzl MODEL LEVEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 FRAME 5 1 125 130 115 100 140 2 125 135 125 110 135 CR1611 3 105 125 115 100 130 4 75 115 90 75 115 5 60 . 105 70 60 85 1 100 60 85 75 75 2 100 60 100 85 85 CR1632 3 80 50 85 70 70 4 50 35 65 50 50 5 30 20 35 25 25 1 120 125 155 85 125 2 110 125 175 95 140 CR3811 3 100 115 160 90 135 4 70 105 120 80 125 5 60 100 85 75 115 1 90 50 130 75 75 2 95 50 150 85 85 CR3831 3 70 40 130 70 70 4 45 25 95 50 50 5 25 15 55 25 25 Col. 1 2 3 4 5 6 7 endix A. BUILDING PROPERTIES Table A.9: Yield Moments My in kips.ft in the Girders of Models CRxxz2 MODEL LEVEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 1 85 85 85 130 2 90 90 95 135 CR1012 3 85 80 90 130 4 70 70 80 115 5 60 60 75 85 1 55 55 50 50 2 60 65 65 65 CR1032 3 50 55 60 60 4 40 40 50 50 5 25 25 35 35 1 75 100 85 125 2 75 110 95 140 CR3012 3 70 110 90 135 4 65 100 80 125 5 50 80 75 115 1 35 70 50 50 2 40 80 65 65 CR3032 3 35 80 60 60 4 25 70 50 50 5 15 50 35 35 1 105 70 140 130 2 115 70 135 135 CR3412 3 105 65 100 130 4 85 60 80 115 5 75 60 45 85 Col. 1 2 3 4 5 6 Appendix A. BUILDING PROPERTIES 113 Table A.10: Yield Moments My in kips, ft in the Girders of Models CMxxl2 and CMxx22 MODEL LEVEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 1 80 80 90 135 2 85 85 105 145 CM0012 3 80 85 95 140 4 70 70 80 125 5 65 65 70 115 1 70 80 90 135 2 75 85 105 145 CM1322 3 70 85 95 140 4 60 75 80 125 5 50 75 70 110 1 80 95 90 135 2 85 100 105 145 CM2012 3 80 90 95 140 4 70 75 80 125 5 60 65 70 115 1 55 120 90 135 2 65 125 105 145 CM2522 3 55 115 95 140 4 45 105 80 125 5 40 90 70 115 1 75 100 90 145 2 80 110 105 145 CM3012 3 75 95 95 130 4 70 80 80 125 5 60 70 70 115 Col. 1 2 3 4 5 6 Appendix A. BUILDING PROPERTIES 114 Note that for a symmetric building, modes 1 and 4 are related to the direction of the earthquake, modes 2 and 5 are related to the transverse direction, and modes 3 and 6 are related to the rotation. Appendix A. BUILDING PROPERTIES Table A . l l : Yield Moments My in kips.ft in the Girders of Models CMxx32 MODEL LEVEL FRAME 1 FRAME 2 FRAME 3 FRAME 4 1 45 35 60 60 2 50 45 75 75 CM1332 3 45 40 65 65 4 30 30 45 45 5 20 15 30 30 1 40 65 60 60 2 45 70 75 75 CM2032 3 45 60 65 65 4 30 40 50 50 5 20 25 45 45 1 40 70 65 65 2 45 75 65 65 CM2532 3 40 60 45 45 4 30 40 30 30 5 20 25 15 15 1 40 75 60 60 2 45 80 75 75 CM3032 3 40 65 65 65 4 30 45 50 50 5 20 25 45 45 Col. 1 2 3 4 5 6 Appendix A. BUILDING PROPERTIES Table A. 12: Initial Elastic Periods in Seconds in the Different Models !Model Mode -» 1 2 3 4 5 6 CMxxzl 1.016 1.00 0.73 0.31 0.31 0.23 CM0012 0.95 0.83 0.68 0.27 0.25 0.20 CM3012 0.95 0.92 0.64 0.27 0.26 0.19 CM1322 0.95, 0.84 0.67 0.27 0.25 0.20 CM25z2 0.95 0.88 0.64 0.27 0.26 0.19 CR16zl 1.25 1.17 0.84 0.37 0.35 0.25 CR38zl 1.32 1.23 0.84 0.48 0.40 0.25 CR10z2 0.94 0.77 0.66 0.27 0.23 0.19 CR30z2 0.95 0.91 0.64 0.27 0.27 0.19 CR3412 1.46 1.31 0.94 0.44 0.40 0.28 Col. 1 2 3 4 5 6 7 Appendix A. BUILDING PROPERTIES 117 Table A. 13: Final Periods in Seconds in the Different Models iModel Mode 1 2 3 4 5 6 CM1011 1.507 1.016 0.848 0.421 0.310 0.253 CM2011 1.497 1.016 0.826 0.424 0.310 0.243 CM3011 1.517 1.157 0.873 0.450 0.347 0.242 CM1021 1.502 1.016 0.849 0.417 0.310 0.253 CM2021 1.498 1.016 0.838 0.424 0.310 0.248 CM3021 1.515 1.017 0.821 0.436 0.311 0.240 CM1031 1.867 1.016 0.865 0.534 0.310 0.263 CM2031 1.811 1.019 0.861 0.532 0.315 0.265 CM3031 1.792 1.041 0.888 0.543 0.338 0.286 CM0012 1.324 0.945 0.731 0.345 0.272 0.209 CM3012 1.32 0.945 0.714 0.349 0.272 0.206 CM1322 1.369 0.945 0.733 0.355 0.272 0.209 CM2522 1.323 0.945 0.731 0.349 0.272 0.208 CM2532 1.676 0.945 0.739 0.437 0.272 0.211 CR1611 1.53 1.26 1.00 0.44 0.37 0.29 CR1632 1.9 1.26 1.02 0.54 0.37 0.31 CR3811 1.86 1.41 1.3 0.54 0.42 0.36 CR3831 2.06 1.43 1.37 0.62 0.43 0.39 CR1012 1.28 0.944 0.73 0.34 0.27 0.21 CR1032 1.69 0.94 0.74 0.43 0.27 0.22 CR3012 1.3529 0.954 0.739 0.349 0.273 0.209 CR3032 1.621 0.95 0.75 0.42 0.27 0.21 CR3412 1.91 1.311 1.017 0.529 0.395 0.303 Col. 1 2 3 4 5 6 7 

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