ROSE SCHOOL PERIODS OF VIBRATION FOR DISPLACEMENT-BASED ASSESSMENT OF RC BUILDINGS

Transcription

1 I.U.S.S. Istituto Universitario di Studi Superiori Università degli Studi di Pavia EUROPEAN SCHOOL OF ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK ROSE SCHOOL PERIODS OF VIBRATION FOR DISPLACEMENT-BASED ASSESSMENT OF RC BUILDINGS A Dissertation Submitted in Partial Fulfilment of the Requirements for the Master degree in EARTHQUAKE ENGINEERING By HELEN CROWLEY Supervisor: Dr RUI PINHO July 2003

2 The dissertation entitled Periods of vibration for displacement-based assessment of RC buildings, by Helen Crowley, has been approved in partial fulfilment of the requirements for the Master Degree in Earthquake Engineering. Rui Pinho Nigel Priestley

3 Abstract ABSTRACT Simple empirical relationships are available in many design codes to relate the height of a building to its fundamental period of vibration. These relationships have been realized for force-based design and so produce conservative estimates of period such that the lateral shear force will be conservatively predicted from an acceleration spectrum. Where assessment of a structure is concerned, it is the displacement demand that gives an indication of the damage that can be expected, this displacement would be underestimated with the use of the aforementioned period height formulae. Furthermore, the period of vibration of interest in assessment is the yield period, which is calculated using the yield stiffness, also often referred to as the cracked or elastic stiffness. The derivation of a yield period height formula for use in displacement-based assessment of European buildings is thus the focus of this dissertation. Analytical fibre element models of RC frames of varying height have been developed and the yield period has been sought using eigenvalue, pushover and dynamic analyses. Keywords: displacement; assessment; period; yield; RC buildings; i

4 Acknowledgements ACKNOWLEDGEMENTS The author wishes to thank in particular Dr Rui Pinho for all of his guidance and support throughout the duration of this research project and his invaluable critique during the final stages of the dissertation. In addition, this work has greatly benefited from the comments and advice of Professor Nigel Priestley. Also, the author is grateful to Professor Shunsuke Otani for introducing the work of Professor Shunsuke Sugano during his lectures at the ROSE School, Pavia which proved most useful to the research project. Furthermore, the author would like to thank Professor Shunsuke Sugano himself, for his kind assistance in supplying data from his past work. ii

5 List of Figures TABLE OF CONTENTS Page Abstract.i Acknowledgements..ii Table of contents.iii List of Figures.vi List of Tables...ix 1 INTRODUCTION Foreword A Proposed Deformation-Based Seismic Vulnerability Assessment Procedure Dissertation outline THE RELATIONSHIP BETWEEN PERIOD AND HEIGHT Empirical period-height formulae for force-based design Empirical period-height formulae for structural assessment CASE STUDIES AND MODELLING APPROACH Introduction Description of case studies One-Storey Frame Pavia Frame Model Monterosso Building ICONS Four-Storey Frame Greek Five-Storey frames...23 iii

6 List of Figures Five and Six-Storey Romanian Frames Seven-Storey US-Japanese Frame Two and Seven-Storey Italian Frames Three, Six and Eight-Storey Italian Frames Non-linear Finite Element Package Introduction to the program Modelling parameters adopted EIGENVALUE ANALYSIS USING YIELD STIFFNESS Introduction Gross stiffness eigenvalue analysis Eigenvalue analysis in SeismoStruct Results of gross stiffness eigenvalue analyses Strength-independent stiffness reduction factors Application of stiffness reduction factors in SeismoStruct Results of eigenvalue analyses using strength-independent stiffness reduction factors Strength-dependent stiffness reduction factors Analytical study of member yield stiffness Results of eigenvalue analyses using analytically derived member yield stiffness Experimentally derived member yield stiffness Results of eigenvalue analyses using experimentally derived member yield stiffness YIELD PERIOD FROM PUSHOVER ANALYSIS Introduction Conventional and Adaptive Pushover Period calculation using a pushover curve Initial (gross) period calculation using pushover analysis Yield period calculation using a pushover curve Yield displacement of an RC frame Period calculation using yield displacement YIELD PERIOD FROM DYNAMIC TIME-HISTORY ANALYSIS Introduction Time History Records Period calculation using displacement response time-history record Results of yield period from dynamic time history analysis...80 iv

7 List of Figures 7 COMPARISON OF PERIOD V HEIGHT CURVES Summary of all period v height curves Period calculation using base shear capacity from code requirements Simplified Formula CONCLUSIONS REFERENCES...97 v

8 List of Figures LIST OF FIGURES 1.1 A deformation-based seismic vulnerability assessment procedure (Glaister and Pinho, 2003) 2.1 Results of regression analysis for RC MRF (Chopra and Goel, 1997) 3.1 Results of gross stiffness eigenvalue analyses of all case studies 3.2 One storey frame (Varum, 1996) 3.3 Beam and column sections (Varum, 1996) 3.4 Elevation view of the Pavia frame (Moratti, 2000) 3.5 Typical plan of Monterosso building (Cosenza, 2000) 3.6 Plan of fourth floor of Monterosso building (Cosenza, 2000) 3.7 RC Column sections (Cosenza, 2000) 3.8 Elevation of the ICONS Frame (Carvalho et al, 1999) 3.9 Column Reinforcement Details for the ICONS frame (Carvalho et al, 1999) 3.10 Beam reinforcement details for the ICONS Frame (Carvalho et al, 1999) 3.11 Model of Greek Frame No. 20 (Table 3.2) 3.12 Model of Greek Frame No. 23 (Table 3.2) 3.13 Model of Five-Storey Romanian Frame No 29 (Table 3.2) 3.14 Model of Six-Storey Romanian Frame No 30 (Table 3.2) 3.15 Plan and Section of the Seven-Storey RC Frame-Wall Structure (ACI, 1984) 3.16 Section reinforcement according to 81 Yugoslavian design code (ACI, 1984) 3.17 Geometry and reinforcement of two-storey Italian frame (FIB, 2003) 3.18 Geometry and reinforcement of seven-storey Italian frame (FIB, 2003) 3.19 The Three, Six and Eight-storey Italian Frames (Moratti, 2000) 3.20 Column sections, reinforcement and loading (Moratti, 2000) 3.21 Local chord reference system (SeismoStruct, 2003) 3.22 Elastic component of element stiffness matrix (SeismoStruct, 2003) 3.23 Discretistion of an RC section into fibres (SeismoStruct, 2003) 3.24 Location of integration Gauss points within an element (SeismoStruct, 2003) 3.25 Menegotto Pinto (1973) steel model used in SeismoStruct (2003) 3.26 Mander et al (1988) concrete model used in SeismoStruct (2003) vi

9 List of Figures 3.27 Bilinear stress-strain steel model with strain hardening (SeismoStruct, 2003) 4.1 Gross stiffness eigenvalue analysis of inner and outer frames 4.2 Gross stiffness eigenvalue analysis of inner frames 4.3 Gross stiffness eigenvalue analysis of older frames 4.4 Comparison of gross stiffness eigenvalue period height curve with best-fit and upper bound curves from Chopra and Goel (1997) study 4.5 Period v height graph for Paulay and Priestley (1992) reduced stiffness eigenvalue analyses 4.6 Comparison of Paulay and Priestley (1992) reduced stiffness period - height curve and gross stiffness period height curve 4.7 Moment-curvature relationship under design assumption, i.e. constant stiffness (Priestley, 2003) 4.8 Moment-curvature relationship under realistic conditions, i.e. constant yield curvature (Priestley, 2003) 4.9 Moment-curvature curves for a rectangular column (Priestley, 2003) 4.10 Effective stiffness of large rectangular columns (Priestley, 2003) 4.11 Period v height graph for Priestley (2003) reduced stiffness eigenvalue analyses 4.12 Comparison of the period height curves using the yield stiffness according to Priestley (2003) and using traditional stiffness reduction factors (Paulay and Priestley, 1992) 4.13 Calculated and observed stiffness reduction factor at yielding (Sugano, 1970) 4.14 Curves of equation (4.10), ratio of secant to initial stiffness plotted against axial load ratio for 1% reinforcement ratio 4.15 Period v height graph for Sugano reduced stiffness eigenvalue analyses 4.16 Comparison of the period height curves using the yield stiffness according to Sugano (1970) with that according to Priestley (2003) 5.1 SDOF Substitute structure 5.2 Period v height results using the initial stiffness of pushover curves 5.3 Comparison of best-fit curves using the initial stiffness of pushover curves and gross stiffness eigenvalue analysis 5.4 Elasto-plastic representation of a pushover curve 5.5 Period v height results using yield stiffness at 75% ultimate strength of pushover curve 5.6 Comparison of best-fit curves using eigenvalue analysis with yield stiffness according to Sugano (1970) and pushover analysis using the yield stiffness at 75% ultimate strength 5.7 Typical beam-column sub-assemblage (Priestley, 2003) 5.8 Period v height results using yield stiffness at yield displacement of pushover curve 5.9 Comparison of best-fit curves using yield stiffness at 75% ultimate strength of pushover curve and at yield displacement 6.1 Acceleration time-history records for (a) Sakaria and (b) Emeryville 6.2 Acceleration spectra for Sakaria and Emeryville time-history records vii

10 List of Figures 6.3 Displacement spectra for Sakaria and Emeryville time-history records 6.4 Fourier amplitude spectrum showing definite peak amplitude 6.5 Fourier amplitude spectrum with numerous amplitude peaks 6.6 Displacement response time-history at maximum displacement 6.7 Period v height results from dynamic time-history analysis using Sakaria record 6.8 Period v height results from dynamic time-history analysis using Emeryville record 6.9 Comparison of best-fit curves found using eigenvalue, pushover, and dynamic time-history analysis 7.1 Comparison of all period height curves both developed and discussed in Chapters 2 to Comparison of yield period height curves 7.3 Proposed yield period height curve and formula for displacement-based assessment of RC frames 7.4 Period v height results using equation (7.6) 7.5 Comparison of best-fit curves using codified base shear coefficient and pushover base shear coefficient in equation (7.6) 7.6 Comparison of proposed yield period height curve with a simplified linear curve viii

11 List of Tables LIST OF TABLES 3.1 Summary table of all case studies 3.2 Summary Table of older frames used in the study 3.3 Added mass to each floor of the Pavia frame 3.4 Table of columns in Monterosso building (Cosenza, 2000) 3.5 Height and mass of each floor of the Monterosso building 3.6 Material properties ( Monterosso Building) 3.7 Height and mass of each floor of the ICONS frame 3.8 Material Properties ( ICONS frame) 3.9 Height and mass of each floor of the Greek frames 3.10 Height and mass of each floor of the US/Japanese frame 3.11 Height and mass of each floor of the two-storey frame 3.12 Height and mass of each floor of the seven-storey Italian frame 3.13 Material properties (two and seven-storey Italian frames) 4.1 Effective member moment of inertia (Paulay and Priestley, 1992) 4.2 Effective stiffness of members in FEMA 356 (ASCE, 2000) 7.1 Base shear coefficients obtained from both design codes and pushover analyses and respective yield period data calculated using equation (7.6) ix

12 Chapter 1. Introduction 1 INTRODUCTION 1.1 Foreword The determination of the natural period of vibration of an RC structure is an essential procedure in earthquake design and assessment. An improved understanding of the global demands on a structure under a design earthquake can be obtained from this single characteristic. This property is dependent on the mass, strength and stiffness of the structure and is thus affected by many factors such as the structural regularity, the number of storeys and bays and the section properties including dimensions and extent of cracking. Cracking of RC members is a phenomenon often ignored in period calculation however it generally occurs under gravity loading and after moderate seismic action. The stiffness of RC members significantly decreases after cracking and so this stiffness reduction should be adequately modelled in analyses to determine an expected period of vibration. Simple empirical relationships are available in many design codes to relate the height of a building to its fundamental period of vibration. However these relationships have been realized for force-based design and so produce conservative estimates of period such that the base shear force will be conservatively predicted. Where assessment of a structure is concerned it is the displacement that gives an indication of the damage that can be expected (Priestley, 1997). Hence the displacement of a structure needs to be conservatively estimated; however with a conservative period-height relationship the displacements will be underpredicted. Therefore a relationship is sought that will give more accurate predictions of the period of vibration for use in displacement-based assessment of RC structures. In particular, 1

13 Chapter 1. Introduction an assessment procedure for urban areas has recently been proposed (Glaister and Pinho, 2003) wherein a relationship between period and height at yield is a requirement of the methodology. 1.2 A Proposed Deformation-Based Seismic Vulnerability Assessment Procedure The work of Glaister and Pinho (2003) on a simplified deformation-based method for seismic vulnerability assessment has shown how capacity curves of displacement v period can be constructed for buildings at a given limit state using analytical relationships between displacement capacity and height, and empirical relationships between height and elastic period. This is shown conceptually in Figure 1.1. displacement LS3 LS2 LS1 η LS1 η LS2 P LS1 cumulative frequency P LSi percentage of buildings failing LSi η LS3 P LS2 Demand Spectra P LS3 T LS3 T LS2 T LS1 effective period 0 H LS3 H LS2 H LS1 Height H LSi = f (T Lsi, LSi) FIGURE 1.1 A deformation-based seismic vulnerability assessment procedure (Glaister and Pinho, 2003) The displacement capacity-height curves are produced by theoretical considerations of failure mode, member dimensions and material properties for pre-yield and post-yield limit states. An empirical period-height relationship dependent on the limit state is substituted into the equations to produce displacement capacity-period curves. These can then be plotted on the same graph as the displacement demand spectra shown in Figure 1.1 which need to be scaled by a damping factor which is related to the ductility demand and so, in turn, the limit state under consideration. The intersections between the displacement capacity curve and the displacement spectrum, for a given limit state, provide the boundaries where demand and capacity coincide. The range of periods over which the capacity curve is found to be below the 2

14 Chapter 1. Introduction demand spectrum needs to be transformed to a range of equivalent heights, using an empirical period-height relationship. This range can then be used with a cumulative distribution function of buildings with height to find the proportion of the building stock failing the given limit state. At the core of the proposed methodology lays a relationship between period and height that must be valid throughout the entire displacement range for all limit states. For yield limit states where the ductility is less than or equal to one, the period of the equivalent SDOF system is equal to that of the actual building before or at yielding. For post-yield limit states the period of the substitute structure can be obtained by the secant stiffness to the point of maximum deflection on a force-displacement curve. Hence the period is highly dependent on the ductility, and so a relationship between ductility and period is required. Assuming an elasto-plastic force-displacement relationship, the secant stiffness to the point of maximum deflection can be shown to be a geometric function of the elastic stiffness and ductility. Since the elastic period is also a function of elastic stiffness, it can be assumed that the effective period (T LSi ) of the inelastic structure is a function of elastic period and ductility alone: T = (1.1) LSi TLSy µ LSi Hence the realisation of an elastic period-height relationship which represents periods of vibration at yield is seen to be a pivotal formula in the proposed assessment methodology. 1.3 Dissertation outline The second chapter of this dissertation is concerned with recent empirical relationships that have been proposed between period and height and their validity for use in displacement-based structural assessment. The third chapter presents the finite element analysis program, SeismoStruct, which is used in this dissertation to model a range of buildings to investigate the relationship between yield period and height. Some modelling issues are discussed here and the RC frames which have been modelled are presented. 3

15 Chapter 1. Introduction The fourth chapter provides a discussion of yield period calculation using eigenvalue analysis. The representation of the yield stiffness of RC members using stiffness reduction factors and more accurate strength-dependent factors is discussed. The results of eigenvalue analyses using each of the yield stiffness representations are presented. The fifth chapter introduces the method of period calculation by representing the structure as a single degree of freedom substitute structure and subsequently calculating the structural stiffness from its pushover curve. The yield stiffness of the pushover curve is predicted using both the recommendations of Park (1988) and Priestley (2003). In the sixth chapter, the use of dynamic time history analyses to find the yield period is discussed and the results are presented. Two time-history records have been used and the results serve to substantiate the methods of yield period calculation developed in Chapters 4 and 5. The penultimate chapter summarises and compares the curves of period v height obtained in Chapters 4, 5 and 6 and concludes with a discussion of the recommended yield period height formula to be used in displacement-based assessment of RC buildings. A method of period calculation using the base shear coefficient prescribed in codes is also developed and discussed and comparisons are made with the previously derived period-height curves. In the final chapter, the conclusions from this study are assembled and some suggestions for future work are provided. 4

16 Chapter 2. The relationship between period and height 2 THE RELATIONSHIP BETWEEN PERIOD AND HEIGHT 2.1 Empirical period-height formulae for force-based design Empirical formulae have traditionally been obtained by fitting curves through regression analysis to the periods of vibration of buildings measured from their motion during earthquakes. The most useful period data comes from structures which respond elastically to the ground motion, however this is often the most difficult to obtain. Accumulation of such data is slow due to the relatively few number of buildings equipped with accelerographs and the infrequency of earthquakes causing the required motion (Chopra and Goel, 1997). Generally, the empirical formula is given as a function of height alone because although other factors such as horizontal dimensions have been used in regression analyses, the height has been found to play the most important role in fundamental period prediction (Hong and Hwang, 2000). The first empirical formulae employed in seismic design codes, such as the U.S. building code ATC3-06 (ATC 1978), had the form: 0.75 T = C t H (2.1) where Ct was taken to be 0.03 for reinforced concrete moment-resisting frames (RC MRF) and H the height in feet. This particular form was theoretically derived using Rayleigh s method with the following assumptions: the equivalent static lateral forces are distributed linearly over the height of the building, the seismic base shear is proportional to 1/T 2/3 and the distribution of stiffness with height produces a uniform storey drift under the linearly distributed lateral 5

17 Chapter 2. The relationship between period and height forces. The numerical value of the constant C t was obtained from the measured periods of buildings during the 1971 San Fernando earthquake. Other simple formulae have been recommended such as in the NEHRP-94 (NEHRP 1994) provisions where the period can be taken as T = 0.1N, where N is the number of storeys; however this formula is restricted to buildings of less than 12 storeys with minimum storey height of 10ft (3.05m). In the European earthquake design code, EC8 (CEN, 2002), the following period-height relationship is specified for force-based design of moment-resisting frames: 0.75 T = 0.075H (2.2) As mentioned, this form of period-height relationship can be obtained from Rayleigh s method under certain assumptions. The height is measured in metres in Europe, thus the C t coefficient has been adapted from feet to metres (0.03x = 0.073), and has also been rounded up for simplicity. This formula should ideally be updated for use in EC8 because, as shown by Chopra and Goel (1997), the period-height formula has found to be better represented by a different exponent. Also, due to differences in design and construction practice between Europe and California, it might be the case that a distinct period-height formula is required in EC8. In Taiwan, Hong and Hwang (2000) have shown that differences in construction practices between California and Taiwan do indeed lead to different period height formulae. The data obtained during moderate earthquakes from vibration-measuring devices located in Taiwanese RC moment resisting frames has been processed by Hong and Hwang (2000). Although the authors derived a relationship between the measured fundamental period and height using the same methodology as the U.S. building code formula (represented in equation 2.1), they found that the Taiwanese buildings were much stiffer than the Californian and so produced lower predictions of the period of vibration for a given height. The results of this Taiwanese study therefore provide evidence for the need for a distinct formula between the period and height of a building, depending on its design and construction background. In order to update the aforementioned U.S. empirical formula, Chopra and Goel (1997) collected additional data from eight Californian earthquakes, starting with the 1971 San Fernando earthquake and ending with the 1994 Northridge earthquake, from 42 Steel Moment 6

18 Chapter 2. The relationship between period and height Resisting Frames, 27 Reinforced-Concrete and 16 RC Shear Frames. The periods were measured in the two orthogonal lateral directions and the two data points are shown in Figure 2.1 as circles, connected by a vertical line. Although not shown on the graph in Figure 2.1, the authors also compared the data with the previously mentioned empirical formula employed in the U.S. building codes (T = 0.03H 3/4 ). They found that the code formula was close to the lower bound of measured periods for buildings up to 160ft (approximately 16 storeys) but predicted periods significantly shorter than the measured periods for buildings in the height range of ft. Therefore a more accurate best-fit curve has been found by Chopra and Goel (1997) from regression analysis to determine the coefficients α and β in the following chosen form: T = αh β (2.3) FIGURE 2.1 Results of regression analysis for RC MRF (Chopra and Goel, 1997) Various regression analyses were undertaken by Chopra and Goel (1997) with and without constraints on the variable β. The scatter was obviously found to be greater for the constrained regression analyses, but as the unconstrained β was found to be 0.92, a constrained analysis 7

19 Chapter 2. The relationship between period and height with β = 0.9 did not cause a significant increase in scatter, hence the latter power was chosen due to its numerical simplicity. Figure 2.1 shows the best-fit, best-fit + 1σ and the best-fit 1σ curves (where σ represents standard deviation) for the RC MRF. Also shown in Figure 2.1 is the distinction between the results obtained from ground motion where ü g < 0.15g and those where ü g 0.15g. It can be observed that the periods obtained from stronger shaking are much higher than those from less intensive shaking. This is due to the increased cracking of RC members leading to reduced stiffness, and so the periods observed during earthquakes of strong shaking are much higher than those predicted with code formulas. Such code formulas are calibrated to intentionally underestimate the period by approximately 10-20% at first yield of the building (Chopra and Goel, 1997). This is a conservative assumption when applying a force-based design method to a structure as the underestimation of the period produces a conservative estimation of the base shear force. Hence for conservatism the authors have recommended the lower bound curve of best fit 1σ to be used in codes for estimating the fundamental period of vibration (Chopra and Goel, 1997). The equation of this lower bound curve is given below in two forms depending on whether the height is measured in feet or metres: 0.9 TL = 0.016H (feet) (2.4) 0.9 TL = H (metres) (2.5) 2.2 Empirical period-height formulae for structural assessment When assessing structures to a given limit state it is known that displacements and/or drifts give a better indication of damage than forces (Priestley, 1997). As design displacement spectra generally increase with period, within the range considered applicable for RC frame buildings, underestimating the period would lead to underestimations of the seismic displacements: an un-conservative outcome where assessment is concerned. The upper bound curve of best-fit + 1σ of the previously discussed study (Chopra and Goel, 1997) has been suggested as that which should be used to predict building periods when 8

20 Chapter 2. The relationship between period and height seismic displacements need to be estimated from displacement spectra (Chopra and Goel, 2000). The equation of this upper bound curve is given below in two forms depending on whether the height is measured in feet or metres: 0.9 TU = 0.023H (feet) (2.6) 0.9 TU = 0.067H (metres) (2.7) Although this formula provides an improved method of period estimation for structural assessment, it should be noted that the relationship has been obtained by considering just Californian buildings, and so it should be applied with caution in other parts of the world with different construction practices. Also it does not specifically define a yield period as desired for the displacement-based assessment method described in section 1.2, but rather an upper bound to the best-fit periods of vibration obtained during strong shaking, as defined by Chopra and Goel (1997). This strong shaking was purposely taken as that which does not cause the yield limit of the structures to be reached; this was specified because the motivation for producing such equations was for force-based design applications. Hence although formulae do exist to relate the period of vibration of a building to its height, an empirical yield period-height relationship for European RC moment-resisting frames for use in displacement-based assessment has not yet been established and is thus the focus of this dissertation. 9

21 Chapter 3. Case studies and modelling approach 3 CASE STUDIES AND MODELLING APPROACH 3.1 Introduction As previously mentioned, the objective of this dissertation is to find an empirical relationship between fundamental period and height at yield using analytical methods rather than the measured periods of vibration during earthquakes. This decision was taken due to both the lack of European buildings equipped with frequency measuring devices and the added flexibility available when using analytical methods. Therefore many buildings have been modelled using a non-linear finite element package, Seismostruct (2003), and the required information has been obtained from various publications. Firstly, the buildings used in the study are presented, focusing on the important data of country, year, number of storeys and bays and mass. The older frames are defined as those designed to codes with no capacity design principles considered; generally this can be interpreted as all frames designed before the 1980 s. It will be shown that it is these older frames which are of interest in this study and so only the information for these frames will be fully presented. The finite element program SeismoStruct (2003) will be described in some detail and the modelling parameters adopted in the program for all frames will be summarised. 3.2 Description of case studies In order to model frames in SeismoStruct, detailed design data of the frames is required and in this study these have been obtained from various theses and publications. The frames that have initially been chosen cover both recent capacity-designed bare frames as well as older, more 10

22 Chapter 3. Case studies and modelling approach flexible, column-sway bare frames. Figure 3.1 summarises the frames that have been modelled. Aside from the geometrical properties of the structure such as beam and column dimensions, the important information required from the publications is the reinforcement detailing and material properties. It soon became apparent during the study that the older frames are of more interest and should be considered separately from the more recently designed buildings. The scope of this study is to produce a period-height formula to be used for displacement-based assessment of RC buildings. It can be considered that more recently designed buildings, designed following capacity design principles, are not likely to be included in an urban assessment scheme. This would mean that only older buildings, not designed to capacity design principles need to be included in this study. The need to consider the two categories of buildings separately has been further emphasized by the results of the eigenvalue analyses, which will be fully described and presently in Chapter 4. The fundamental period taken from eigenvalue analyses of both recently designed and older (pre-1980) frames are shown together in Figure 3.1, with the former shown as solid black dots and the latter as black rings. The recently designed buildings are shown to be much stiffer and so the fundamental period is lower; this can be attributed to capacity design principles leading to large columns being required. Hence only the older frames were used in further analyses, as listed in Table old frames new frames 1.5 Period (s) Height (m) FIGURE 3.1 Results of gross stiffness eigenvalue analyses of all case studies 11

23 Chapter 3. Case studies and modelling approach TABLE 3.1 Summary table of all case studies No. Country Decade Design Code Heig No. of No. Mass Comments ht storeys of (t) (m) bays 1 Portugal 1 80 s RSA Regular, internal 2 Portugal 90 s DBD2%CD Regular, internal 3 Portugal 80 s RSA Regular, internal 4 Portugal 90 s DBD1%CD Regular, internal 5 Portugal 2 90 s EC Regular, external 6 Portugal 90 s EC Regular, external 7 Portugal 90 s EC Regular, external 8 Portugal 90 s EC Regular, external 9 Portugal 90 s EC Pilotis, external 10 Portugal 90 s EC Pilotis, external 11 Portugal 90 s EC Pilotis, external 12 Portugal 90 s EC Pilotis, external 13 Portugal 90 s EC Horiz. irregular, int. 14 Portugal 3 90 s EC Regular, internal 15 Portugal 90 s EC Regular, internal 16 Portugal 90 s EC Regular, internal 17 Italy 4 70 s Italian, 70 s Horiz. irreg. internal 18 Italy 70 s Italian, 70 s Regular, internal, central penthouse 19 Italy 5 70 s Italian, 70 s /3 scale model, horiz. irregular 20 Greece 6 60 s Greek, Regular, internal 21 Greece 60 s Greek, Regular, internal 22 Greece 60 s Greek, Pilotis, internal 23 Greece 60 s Greek, Pilotis, internal 24 Italy 7 60 s Italian, Horiz. Irreg, internal 25 Italy 60 s Italian, 60 s Horiz. Irreg, internal 26 Italy s Italian, 50-60s Regular, Internal 27 Italy s Italian, 50-60s Regular, Internal 28 Romania s Romanian, 20-40s Regular, Internal 29 Romania s Romanian, 20-40s Regular, Internal 30 Portugal s Portuguese, Horiz. Irreg, internal s 31 Yugoslavia s Yugoslavian Regular, Internal References: 1) Romäo (2002), 2) Arêde (1997), 3) Varum (1996), 4) Cosenza (2000), 5) Moratti (2000), 6) Zeris et al (2002), 7) FIB (2003), 8) Bostenaru (2003) 9) Carvalho et al (1999) 10) ACI (1984) 12

24 Chapter 3. Case studies and modelling approach TABLE 3.2 Summary Table of older frames used in the study No. Country Decade Design Code Heig No. of No. Mass Comments ht storeys of (t) (m) bays 14 Portugal 90 s EC Regular, internal 17 Italy 70 s Italian, 70 s Horiz. irreg. internal 18 Italy 70 s Italian, 70 s Regular, internal, central penthouse 19 Italy 70 s Italian, 70 s /3 scale model, horiz. irregular 20 Greece 60 s Greek, Regular, internal 21 Greece 60 s Greek, Regular, internal 22 Greece 60 s Greek, Pilotis, internal 23 Greece 60 s Greek, Pilotis, internal 24 Italy 60 s Italian, Horiz. Irreg, internal 25 Italy 60 s Italian, 60 s Horiz. Irreg, internal 26 Italy s Italian, 50-60s Regular, Internal 27 Italy s Italian, 50-60s Regular, Internal 28 Romania s Romanian, 20-40s Regular, Internal 29 Romania s Romanian, 20-40s Regular, Internal 30 Portugal s Portug., s Horiz. Irreg, internal 31 Yugoslavia s Yugoslavian Regular, Internal All of the 16 frames shown in Table 3.2 will now be described in some detail; the reader should refer to the references given for detailed information of the buildings not presented herein One-Storey Frame A one storey frame (number 14 in Table 3.2) has been designed by Varum (1996) to EC8 and is shown in Figure 3.2. The beam and columns have section size and reinforcement as shown in Figure

25 Chapter 3. Case studies and modelling approach FIGURE 3.2 One storey frame (Varum, 1996) FIGURE 3.3 Beam and column sections (Varum, 1996) The material properties were provided by Varum through written communication; the compressive strength of concrete f c was taken as 24.6MPa and the tensile strength as 2.2MPa, the yield strength of the steel was given as 460MPa with a yield strain of 0.235% and a strain hardening parameter (see section 3.3.2) of Although this frame has been designed to EC8, a code which considers capacity design principles, it has been included in the study of older frames as it is currently the only one storey frame which has been obtained. It is assumed that the design of such a low-rise frame would not differ greatly from a design to an older code. This assumption appears to be valid when one considers the results shown in Figure 3.1 where the low-rise older and newer frames produce similar gross stiffness periods of vibration, whereas the difference becomes greater as the height increases. 14

26 Chapter 3. Case studies and modelling approach Pavia Frame Model Moratti (2000) describes the design of a frame to just gravity loads according to the Italian specifications of the 1970 s. It has three-storeys and three bays and has been built to 2/3 scale at the Department of Structural Mechanics at the University of Pavia, Italy. Figure 3.4 shows the geometry of the frame (number 19 in Table 3.2) in an elevation view (dimensions are in cm). The columns have square cross-section with each side equal to 200 mm, while the beams have dimensions of 200/330 mm. The columns were reinforced with 6Φ8 mm bars and the beams with 2Φ8 mm bars top and bottom and 2Φ12mm bars top and bottom. The material properties were obtained from testing carried out at the laboratory in Pavia, and the concrete was found to have a compressive strength of 17MPa and a tensile strength of 3MPa. The steel reinforcement was found to have yield strength of 370MPa and an ultimate strength of 470MPa. FIGURE 3.4 Elevation view of the Pavia frame (Moratti, 2000) During testing RC concrete blocks were placed on each floor to simulate slab and live loading. Table 3.3 shows the additional mass placed at each floor: 15

27 Chapter 3. Case studies and modelling approach TABLE 3.3 Added mass to each floor of the Pavia frame Floor Added Mass (tonnes) Monterosso Building In the GNDT report edited by Cosenza (2000), a four story RC building designed to just gravity loading in the 1970 s in Catania, Italy is presented (referred to as the Monterosso building). The building has a transversal axis of symmetry and a typical plan of the first three storeys is shown in Figure 3.5. The height of each storey varies between 2.75 and 3.25m, giving a total height of the building of 11.8m (Table 3.5). The fourth storey, being only a cover to the central staircase, covers a smaller area as shown in Figure 3.6. The column sections are tabulated in Table 3.4 and the detailing of the longitudinal reinforcement is shown in Figure 3.7. Two frames have been taken from this building, a transverse frame (columns 3, 17, 30, 38 in Figure 3.5) and a longitudinal frame (columns in Figure 3.5). These frames have previously been summarized in Table 3.2 as numbers 17 and 18 respectively. 16

28 Chapter 3. Case studies and modelling approach FIGURE 3.5 Typical plan of Monterosso building (Cosenza, 2000) 17

29 Chapter 3. Case studies and modelling approach FIGURE 3.6 Plan of fourth floor of Monterosso building (Cosenza, 2000) TABLE 3.4 Table of columns in Monterosso building (Cosenza, 2000) 18

30 Chapter 3. Case studies and modelling approach FIGURE 3.7 RC Column sections (Cosenza, 2000) TABLE 3.5 Height and mass of each floor of the Monterosso building Floor Height (m) Mass (tonnes) The permanent loads and 30% live loads lead to a mass of each storey as shown in Table 3.5. The beams dimensions are shown in Figure 3.5 and the reinforcement is generally 2Φ10 top and bottom and 2Φ12 bars at the bottom with an additional Φ12 bar at each end of the beam. The slabs have a width of (150+40) mm, and span in the transversal direction. The material properties as assumed in Cosenza (2000) are shown in Table 3.6: 19

31 Chapter 3. Case studies and modelling approach TABLE 3.6 Material Properties ( Monterosso building) Concrete Compressive strength, f c MPa Tensile strength, f t 2.0 MPa Ultimate strain, ε cu Steel Yield strength, f y 380 MPa Ultimate strength, f ult 475 MPa Yield strain, ε y 0.18% Ultimate strain, ε ult 14% ICONS Four-Storey Frame Figure 3.8 shows an elevation of a four-storey, three-bay reinforced concrete bare frame designed and built at Ispra, Italy for pseudo-dynamic testing by the European Commission, undertaken by the ECOEST II / ICONS 1 collaboration (Carvalho et al, 1999). The frame was designed essentially for gravity loads and a nominal lateral load of 8% of its weight. The reinforcement details attempted to mirror the practices used in southern European countries in the 1950 s; the columns were non-ductile, smooth reinforcing bars were used, capacity design principles were ignored and lap splicing occurred in critical regions. Figures 3.9 and 3.10 show column and beam reinforcement details respectively; the column stirrup detail should be noted in particular. The height to each floor of the frame and the mass of each floor are given in Table ECOEST II is the European Consortium of Earthquake Shaking Tables ICONS Innovative Concepts for Seismic Design of New and Existing Structures 20

32 Chapter 3. Case studies and modelling approach FIGURE 3.8 Elevation of the ICONS Frame (Carvalho et al, 1999) FIGURE 3.9 Column Reinforcement Details for the ICONS frame (Carvalho et al, 1999) 21

33 Chapter 3. Case studies and modelling approach FIGURE 3.10 Beam reinforcement details for the ICONS Frame (Carvalho et al, 1999) 22

34 Chapter 3. Case studies and modelling approach TABLE Height and mass of each floor of the ICONS frame Floor Height (m) Mass (tonnes) The material properties were obtained during testing of the frame. The concrete was defined as C16/C20 and had the properties as shown in Table 3.8. The longitudinal steel reinforcement consisted of smooth round bars (FeB22k) with the properties as given in Table 3.8. TABLE 3.8 Material Properties ( ICONS frame) Concrete Compressive strength, f c Tensile strength, f t 16.3 MPa 1.9 MPa Steel Yield strength, f y 343 MPa Ultimate strength, f ult 452 MPa Yield strain, ε y % Ultimate strain, ε ult 23% Greek Five-Storey frames Zeris et al (2002) describe various Greek frames designed to Greek codes from 1959 and From this paper, along with correspondence with the authors, information was obtained such that four frames could be modelled. Two orthogonal frames (numbers 20 and 21 in Table 3.2) were taken from a five storey building designed to the 1959 Greek code. These frames have a constant storey height of 3m whilst one frame has four bays and the other has six bays, the model of the six bay frame is shown in Figure

35 Chapter 3. Case studies and modelling approach FIGURE 3.11 Model of Greek Frame No. 20 (Table 3.2) The design loads were taken equal to: 1.5 kn/m 2 surcharge and 2.0 kn/m 2 live load, interior light partition weight 1.0 kn/m 2 uniform over the entire floor plan and exterior infills 3.6 kn/m 2 of façade area. The frames were modelled, as mentioned previously, with the total dead load plus 30% of the live load. The height to each floor and the mass of each floor has thus been calculated as given in Table 3.9. TABLE Height and mass of each floor of the Greek frames Floor Height (m) Mass (tonnes) 4 bay Mass (tonnes) 6 bay Both frames have 350/350 mm square columns at the first floor, reduced to 300/300 mm square at the second floor from where they are further reduced to 250/250 mm up to the roof. Column reinforcement is typically governed by code minima (0.8 to 1.0% gross section steel ratio) however the third storey columns are reinforced with 2% gross section steel ratio, leading to a large strength discontinuity between the second and third storeys. Beams are 24

36 Chapter 3. Case studies and modelling approach 200/500 mm with light reinforcement (2Φ10 top and 4/5Φ10 bars bottom usually). The slab is 120 mm thick on all floors. The frames were designed with concrete compressive strength of 16MPa and steel reinforcement with yield strength of 220MPa, and so these were the properties used in the model. A four bay frame and a six bay frame were also taken from a five storey building similar to that described above, however the first storey height is 5m and so would probably produce a soft storey. The model of the four bay frame is shown in Figure Column dimensions and reinforcement are similar to those in the previous frames; however the columns of the ground floor are now 400/400 mm. The beams also have the same dimensions and similar reinforcement. FIGURE 3.12 Model of Greek Frame No. 23 (Table 3.2) Five and Six-Storey Romanian Frames Bostenaru (2003) describes the retrofit of many interwar Romanian RC frames and written communication with the author has lead to the modelling in this study of the original frames of both a five- and six-storey regular frame (numbers 28 and 29 respectively in Table 3.2). The five-storey frame has a storey height of 3m and five bays, as shown in Figure

37 Chapter 3. Case studies and modelling approach FIGURE 3.13 Model of Five-Storey Romanian Frame No 29 (Table 3.2) The six-storey frame also has a storey height of 3m with four equal 4.5m bays, as shown in Figure FIGURE 3.14 Model of Six-Storey Romanian Frame No 30 (Table 3.2) The columns of both frames are 500 by 500 mm square and the beams are 500 deep in both frames with a width of 250mm in the five-storey frame and 300mm in the six-storey frame. Reinforcement of these members follows the practice during the interwar period in Romania. 26

38 Chapter 3. Case studies and modelling approach The columns were reinforced with 8 Φ 12mm smooth bars in the five-storey frame and 8 Φ 16mm smooth bars in the six-storey frame. The beams were reinforced with gross section steel ratios between 1-2%. The concrete compressive strength in these buildings is known to be around 15MPa, whilst the steel has been assumed with a yield strength of 500MPa (limited information available) Seven-Storey US-Japanese Frame A full-scale seven storey RC building has been designed and built through the US-Japan Cooperative earthquake Engineering Research Program (ACI, 1984). Four designs were made to various codes and the design chosen to be modelled for this study was to the Yugoslavian code of 1981, hence it can be considered a European building designed for an area of high seismicity and is thus applicable for this study. Figure 3.15 shows the plan and section of the RC frame-wall structure. The frame used in this study was taken as one of the outer frames, however, as there is no wall present in these frames. FIGURE 3.15 Plan and Section of the Seven-Storey RC Frame-Wall Structure (ACI, 1984) 27

39 Chapter 3. Case studies and modelling approach Figure 3.16 shows the beam and column section reinforcement obtained from the Yugoslavian design code. The end region of the outer columns is composed of section 1 and the middle region of section 2; the middle columns are composed of section 5. The end region of the beams is composed of section 3 and the middle region of section 4. The low reinforcement content, and use of stirrups rather than hoops should be noted, as this follows European practice up to years ago. FIGURE 3.16 Section reinforcement according to the 81 Yugoslavian design code (ACI, 1984) The height to each storey and mass of each floor is given in Table TABLE Height and mass of each floor of the US/Japanese frame Floor Height (m) Mass (tonnes) The material properties were taken as 29MPa compressive strength and 2.42MPa tensile strength for the concrete and 375MPa yield strength for the longitudinal steel, as suggested in ACI (1984). 28

40 Chapter 3. Case studies and modelling approach Two and Seven-Storey Italian Frames Chapter 7 of the state-of-the-art report by FIB (2003) includes the design information of two typical Italian frames designed and built according to the Italian Seismic Code of 1960 in a region of high seismic hazard. Design was made by employing a permissible working stress approach; this implies that the maximum design stress in the reinforcement is about half the yield stress. In addition, columns were not dimensioned for shear. The two planar frames have been taken from a low-rise and a high-rise multi-storey building, the former has bays and two floors whilst the latter has two bays and seven floors. Figure 3.17 shows the geometry and reinforcement of the low-rise frame (number 25 in Table 3.2) and Figure 3.18 shows the geometry and reinforcement of the high-rise frame (number 24 in Table 3.2). FIGURE Geometry and reinforcement of two-storey Italian frame (FIB, 2003) 29

41 Chapter 3. Case studies and modelling approach FIGURE 3.18 Geometry and reinforcement of seven-storey Italian frame (FIB, 2003) 30

42 Chapter 3. Case studies and modelling approach The height to each floor of the two-storey frame and the total mass assumed to load the frame at each floor are shown in Table 3.11, whilst this data for the seven-storey frame is given in Table 3.12: TABLE 3.11 Height and mass of each floor of the two-storey frame Floor Height (m) Mass (tonnes) TABLE Height and mass of each floor of the seven-storey Italian frame Floor Height (m) Mass (tonnes) The material properties for both frames have been used as presented in the report by FIB (2003), as outlined in Table 3.13: TABLE 3.13 Material properties (two and seven-storey Italian frames) Concrete Compressive strength, f c Tensile strength, f t 22 MPa 2.0 MPa Steel Yield strength, f y 230 MPa Ultimate strain, ε ult 10% 31

43 Chapter 3. Case studies and modelling approach Three, Six and Eight-Storey Italian Frames Three frames of three, six and eight- storeys were designed for gravity loading only according to the Italian code provisions available in he 1950 s and 1960 s (Moratti, 2000). The three frames have been taken from the design of the eight storey structure as shown in Figure The storey height is 3m throughout and the bay lengths are 4.5m, 2m and 4.5m. 6 PIANI 8 PIANI 3 PIANI FIGURE 3.19 The Three, Six and Eight-storey Italian Frames (Moratti, 2000) The six and the eight storey frames have been modelled in this study. The dimensions of all beams are 300mm by 500mm with 2Φ16mm bars at the top and 4Φ16mm bars at the bottom with two of the Φ16mm bottom bars bent up at the ends of the beam. The column dimensions, reinforcement and the loads to each column are shown in Figure The material properties were found from laboratory testing; the concrete was found to have a compressive strength of 19.62MPa and the longitudinal reinforcing steel a yield strength of 373MPa. 32

44 Chapter 3. Case studies and modelling approach N=5073 kg N=7313 kg 25x25 4 φ14 St φ 10 cm 25x25 4 φ14 St φ 10 cm kg kg 25x25 4 φ14 St φ 10 cm 25x25 4 φ14 St φ 10 cm kg kg 25x25 4 φ14 St φ 10 cm 25x25 4 φ14 St φ 10 cm kg kg 25x25 4 φ14 St φ 10 cm 30x30 4 φ16 St φ 10 cm kg kg 25x25 4 φ14 St φ 10 cm 30x30 4 φ16 St φ 15 cm kg kg 30x30 4 φ16 St φ 15 cm 35x35 4 φ18 St φ 15 cm kg kg 30x30 4 φ16 St φ 15 cm 35x35 4 φ18 St φ 15 cm kg 68755kg 35x35 4 φ18 St φ 15 cm 40x40 4 φ18 St φ 15 cm FIGURE 3.20 Column sections, reinforcement and loading (Moratti, 2000) 33

45 Chapter 3. Case studies and modelling approach 3.3 Non-linear Finite Element Package The non-linear finite element program SeismoStruct (2003) has been chosen to model the frames and subsequently calculate their fundamental period. This package carries out distributed inelasticity fibre analysis as opposed to a concentrated plasticity approach present in plastic hinge modelling programs. The program is capable of predicting the large displacement behaviour of space frames under static or dynamic loading, taking into account both local and global geometric nonlinearities and material inelasticity Introduction to the program In SeismoStruct, both local (beam-column effect) and global (large displacements/rotations effect) sources of geometric nonlinearity are automatically taken into account. Modelling of the latter is carried out through the employment of a co-rotational formulation (e.g. Izzuddin, 2001), whereby local element displacements and resulting internal forces are defined with regard to a moving local chord system. In this local system six basic degrees-of-freedom are employed (θ 2(A), θ 3(A), θ 2(B), θ 3(B),, θ T ), as shown in Figure Exact transformation of element internal forces (M 2(A), M 3(A), M 2(B), F, M T ) and the stiffness matrix, obtained in the local chord system, into the global system of coordinates allows for large displacements/rotations to be accounted for (e.g. Izzuddin, 1991). Figure 3.21 Local chord reference system (SeismoStruct, 2003) 34

46 Chapter 3. Case studies and modelling approach The interaction between axial force and transverse deformation of the element (beam-column effect), on the other hand, is implicitly incorporated in the element cubic formulation suggested by Izzuddin (1991), whereby the strain states within the element are completely defined by the generalised axial strain and curvature along the element reference axis (x), whilst a cubic shape function is employed to calculate the transverse displacement as a function of the end rotations of the element: u θ + θ 2θ + θ = 2 A2 (3.1) L L A2 B2 3 A2 B2 2 ( x) x x θ x 2 + u θ + θ 2θ + θ = 2 A3 (3.2) L L A3 B3 3 A3 B3 2 ( x) x x θ x 3 + The resulting elastic component of the stiffness matrix of the element, as defined in the local chord system (Izzuddin, 2001) is: Figure 3.22 Elastic component of element stiffness matrix (SeismoStruct, 2003) In Figure 3.22, E denotes the modulus of elasticity, A is the cross-sectional area and I 2 and I 3 are the moments of inertia about the local axes (2) and (3). The torsional constant is denoted by J, whilst G stands for the modulus of rigidity, obtained as G = E/(2(1+ν)), where ν is the Poisson s ratio. Since a constant generalised axial strain shape function ( (x) = ) is assumed in the adopted cubic formulation, it results that its application is only fully valid to model the nonlinear response of relatively short members (Izzuddin, 1991) and hence a number of elements (3-4 per structural member) is required for the accurate modelling of structural frame members. 35

47 Chapter 3. Case studies and modelling approach It should be noted that shear strains across the element cross-section are not modelled, thus the strain state of a section is fully represented by the curvature and centroidal strains alone (Izzuddin, 1991). Also warping strains and warping effects (cross-section distortion) are not considered. The spread of inelasticity along the member length and across the section depth is explicitly modelled in SeismoStruct following a fibre modelling approach, thus allowing for an accurate estimation of damage distribution. The sectional stress-strain state of inelastic beam-column frame elements is obtained through the integration of the nonlinear uniaxial stress-strain response of the individual fibres into which the section has been subdivided. The discretisation of a typical reinforced concrete section is illustrated in Figure The user is required to define a sufficient number of fibres (about 200 is recommended for spatial analysis) and then the distribution of material nonlinearity across the section area is accurately modelled, even in the highly inelastic range. Figure 3.23 Discretisation of an RC section into fibres (SeismoStruct, 2003) The spread of inelasticity along the member length arises as a result of the inelastic cubic formulation suggested by Izzuddin (1991), on which the beam-column elements within SeismoStruct are based. Two integration Gauss points per element are used for the numerical integration of the governing equations of the cubic formulation, as shown in Figure If a sufficient number of elements is used (5-6 per structural member), then the plastic hinge length of structural members subjected to high levels of material inelasticity can accurately estimated. It is evident that if the plastic hinges are to be accurately modelled, more elements should be defined where hinges are expected to form. The division of the member into shorter elements 36

48 Chapter 3. Case studies and modelling approach also renders valid the use of the cubic formulation to model nonlinear response, as mentioned previously. Figure 3.24 Location of integration Gauss points within an element (SeismoStruct, 2003) There are seven material models of steel and concrete available within the program library such that the user can define the material behaviour to the required degree of accuracy, however it should be noted that more calibration is required for increasingly complex models. Steel models include a bilinear stress-strain model with strain hardening, a Menegotto Pinto (1973) model which utilises a damage modulus to represent more accurately the unloading stiffness under loading reversals (Figure 3.25) and a Monti Nuti (1992) model able to describe the post-elastic buckling behaviour of reinforcing bars. Figure 3.25 Menegotto - Pinto (1973) steel model used in SeismoStruct (2003) 37

49 Chapter 3. Case studies and modelling approach Concrete can be modelled most simply by a simple trilinear model with no tensile resistance. More accurate models for normal strength concrete are available considering either constant confinement following the Mander et al (1988) model (see Figure 3.26) or variable confinement as proposed by Madas and Elnashai (1992). A high-strength concrete model is also available as proposed by Kappos and Konstantinidis (1999) which allows for constant confinement modelling. Figure 3.26 Mander et al (1988) concrete model in SeismoStruct (2003) Modelling parameters adopted The following parameters have been applied to all frames that have been modelled: (1) The Mander et al (1988) constant confinement material model (see Figure 3.26) was used for concrete. This requires the input of concrete compressive strength (f c ), tensile strength (f t ), strain at peak stress and a confinement factor. The confinement factor is defined as the ratio between the confined and unconfined compressive stress of the concrete, and is used to scale up the stress-strain relationship throughout the entire strain range. This material model was used for both confined and unconfined concrete, with the confinement factor for the latter being taken as 1.0. When information regarding the tensile strength and strain at peak stress was not available, the former has been calculated as 0.5 f c and the latter taken as The bi-linear stress strain with strain hardening model was used for steel (see Figure 3.27). This model requires the definition of yield strength (f y ), modulus of elasticity (E), and a strain 38

50 Chapter 3. Case studies and modelling approach hardening parameter. The strain hardening parameter is the ratio between the post-yield stiffness (E sp ) of the material and the initial elastic stiffness (E s ). The former is defined as E sp = (f ult -f y )/(ε ult -f y /E s ), where f ult and ε ult represent the ultimate or maximum stress and strain capacity of the material, respectively. When information was not available for the modulus of elasticity and strain hardening parameter, the former was assumed to be 2E10 5 MPa and the latter to be FIGURE 3.27 Bilinear stress-strain steel model with strain hardening (SeismoStruct, 2003) (2) 4-5 elements, with smaller elements at member ends, were used to model beams and columns to ensure inelasticity could be accurately modelled; (3) The inertia was taken as the dead load plus approximately 30% of the live load; (4) Beams and columns were modelled as extending from the centre of one beam-column joint to the centre of the next. SeismoStruct cannot model shear deformations however they need to be accounted for in both joints and members so that frame deformations can be accurately predicted. Within joints, alongside the shear deformation, there is also increased flexibility due to yield penetration and bar slip. A large proportion of the current building stock in Europe is considered to have been constructed with smooth bars, following the general building practice up until about years ago. Therefore it is proposed that the inclusion of bar slip in the determination of yield stiffness is justified. 39