Pushover Analysis of an Existing- Thesis

PUSHOVER ANALYSIS OF AN EXISTING REINFORCED CONCRETE STRUCTURE Jón Örvar Bjarnason

Byggingarverkfræði MSc.

Leiðbeinendur: Eyþór Rafn Þórhallsson Eysteinn Einarsson

Tækni- og verkfræðideild Vor 2008

Tækni- og verkfræðideild Heiti verkefnis: Pushover Analysis of an Existing Reinforced Concrete Structure Námsbraut: Byggingarverkfræði MSc.

Tegund verkefnis: Jarðskjálftagreining burðarvirkis

Önn: Vor 2008

Ágrip:

Námskeið: T899-MEIS

Höfundur: Jón Örvar Bjarnason

Umsjónarkennari: Eyþór Rafn Þórhallsson

Leiðbeinendur: Eysteinn Einarsson Eyþór Rafn Þórhallsson

Viðfangsefni þessa verkefnis er jarðskjálftagreining þriggja hæða steinsteyptrar byggingar sem skemmdist í skjálftunum. Gerðar eru tvennskonar greiningar á byggingunni. Fyrst er gerð sveifluog svörunarrófsgreining, þar sem þrívítt módel af byggingunni er gert í smáeininga forriti sem miðast við línulega eiginleika byggingarinnar. Niðurstöður þeirrar greiningar eru síðan nýttar í ólínulega greiningu sem gerð er á einum vegg byggingarinnar, þar sem tekið er tillit til ólínulegra efniseiginleika mannvirkisins. Ólínulega greiningin er notuð til að meta hvernig veggurinn stendur af sér jarðskjálftaálag sem reiknað er skv. hönnunarstaðli (EC8). Gerður er samanburður á raunverulegum skemmdum hússins og þeim skemmdum sem smáeininga módel af veggnum gefur til kynna við áætlað raunálag í jarðskjálftunum. Loks var lagt mat á tillögur sem gerðar hafa verið að styrkingu veggjarins. Helstu niðurstöður greiningar eru að veggurinn þolir jarðskjálftaálag sem reiknað er skv. EC8, en stenst hinsvegar ekki kröfur staðalsins um styrk byggingareininga í brotmarkaástandi. Raunverulegum skemmdum veggjarins bar ágætlega saman við niðurstöður útreikninga. Loks sýndu niðurstöður að tillaga að styrkingu veggjarins eykur jarðskjálftaþol umtalsvert og mun minni skemmdir koma fram.

Prófdómari: Bjarni Bessason

Dagsetning: 2008-05-29

Dreifing: opin

Í Júní árið 2000 riðu tveir stórir jarðskjálftar yfir Suðurland. Þrátt fyrir umtalsverðar skemmdir mannvirkja urðu engin alvarleg slys á fólki og engin hús hrundu.

Lykilorð íslensk: Jarðskjálfti Steinsteypa Ólínuleg greining

lokuð

Lykilorð ensk: Earthquake Concrete Nonlinear Analysis

til:

Háskólinn í Reykjavík ♦ Ofanleiti 2, 103 Reykjavík ♦ Höfðabakka 9, 110 Reykjavík ♦ sími: 599 6200 www.ru.is

ABSTRACT

The main purpose of this study is to analyze the seismic response of a reinforced concrete structure which suffered damage in the South Iceland earthquakes in June 2000. Two different methods are used to analyse the structure. First, a 3D linear elastic response spectrum analysis (RSA) is carried out in a finite element program, using response spectra within the European seismic design code (EC8). The response spectrum analysis is followed with a pushover analysis of one of the walls which suffered the most severe damage. A new finite element model is constructed where the nonlinear properties of the reinforced concrete are implemented. The pushover analysis is combined to the response spectra approach through the comprehensive N2 method, which is a novelty in EC8. Results of the pushover analysis are used to estimate the performance of the structure under the design earthquake load. Deformation of the structure, stress strain and evaluation of cracks in the reinforced concrete are reviewed. Damage obtained in the analysis and the actual reported damages are then compared. Finally, the effect of strengthening is estimated. The study showed that despite serious damage, the structural wall is able to withstand the earthquake design load. However, the structure does by no means meet the design code requirements. Observed damage was fairly consistent with that actually reported, but according to the analysis the N2 procedure tends to be conservative. As shown in the study, the pushover analysis based on the N2 method is a desirable procedure in order to evaluate the seismic resistance of existing structures, especially if there is doubt that response spectrum analysis provides sufficient information.

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PREFACE The financial support by Íbúðalánasjóður and the excellent study facility provided by VST Consulting Engineers Ltd. is gratefully acknowledged. I would like to thank my supervisors Eysteinn Einarsson and Eyþór Rafn Þórhallsson for their excellent guidance and inspiration and valuable comments, through the entire thesis work. I would like to thank Ingunn Sæmundsdóttir for her great support during this work. I would like to thank Guðbrandur Steinþórsson and Ari Guðmundsson for their comments and for reading and correcting the manuscript and Indriði Sævar Rikharðsson for his assistance according to the finite element programming. I also would like to thank Fjóla Sigtryggsdóttir and Brian Atkin for their valuable comments and support. Finally, I would like to thank my fiancé, Valgerður Júlíusdóttir for her incessant patience and encouragement through my entire BSc. and MSc. study.

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TABLE OF CONTENTS Abstract ..................................................................................................................... i Preface ..................................................................................................................... ii Table of Contents ....................................................................................................iii List of Figures .......................................................................................................... v List of Tables ........................................................................................................viii 1 Introduction ......................................................................................................... 1 1.1 Background and Problem Statement ............................................................ 1 1.2 Aim of the Study .......................................................................................... 3 1.3 Objectives..................................................................................................... 3 1.4 Scope of this Work ....................................................................................... 4 2 Research Methodology ....................................................................................... 5 2.1 Introduction .................................................................................................. 5 2.2 Models and Numerical Simulation .............................................................. 6 2.3 Experimental Work ...................................................................................... 6 3 Literature Review ................................................................................................ 8 3.1 Introduction .................................................................................................. 8 3.2 Nonlinear Analysis of Reinforced Concrete ................................................ 8 3.3 Analytical Modelling of Reinforced Concrete ........................................... 13 3.4 Concrete Model Calibration ....................................................................... 26 3.5 Elastic Response Spectrum Analysis ......................................................... 30 3.6 Pushover Analysis ...................................................................................... 35 4 Case Study......................................................................................................... 48 4.1 Introduction ................................................................................................ 48 4.2 Previous Study of the Building .................................................................. 55 4.3 Response Spectrum Analysis of the Building ............................................ 56 4.4 Pushover Analysis of the Back Side Wall ................................................. 61 4.5 Results of the Pushover Analysis ............................................................... 65 5 Discussion ......................................................................................................... 76 iii

5.1 Level of Ductility ....................................................................................... 76 5.2 Comparison of Reported and Computed Damage ..................................... 77 5.3 Retrofit Proposals....................................................................................... 80 6 Conclusions ....................................................................................................... 86 6.1 Final Conclusions....................................................................................... 86 6.2 Prognosis .................................................................................................... 87 6.3 Further Research ........................................................................................ 87 References .............................................................................................................. 89

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LIST OF FIGURES Figure 1-1 Horizontal acceleration, E-W directional component recorded at Hella in June 17th 2000 [2] ....................................................................................... 1 Figure 1-2 Left: Two storey residential houses damaged in June 2000 earthquakes. Right: Crack in pumice inner wall [5] ............................................................. 2 Figure 3-1 Load-deflection relationship of reinforced concrete [11] ...................... 8 Figure 3-2 Uniaxial stress-strain relationship for concrete [14] ............................ 10 Figure 3-3 Strength envelope for concrete under biaxial load [15] ....................... 11 Figure 3-4 Failure criterion presented in the three dimensional principal stress [16] ................................................................................................................. 12 Figure 3-5 Stress-stain diagram for reinforcement steel [17] ................................ 12 Figure 3-6 Load-displacement relationship for typical reinforced concrete element [11] ................................................................................................................. 14 Figure 3-7 Failure surface for concrete [18] .......................................................... 15 Figure 3-8 Yield surface and the kinematic hardening rule [18] ........................... 16 Figure 3-9 The solid65 concrete element in ANSYS [18]..................................... 18 Figure 3-10 Stress-strain curves for concrete and steel in the ANSYS model ...... 20 Figure 3-11 Two load steps of Newton-Raphson iteration solution [18] .............. 22 Figure 3-12 Definition of the ductility factor [16] ................................................. 23 Figure 3-13 Stress-strain diagrams for concrete subjected to various types of confinement [16] ............................................................................................ 24 Figure 3-14 Layout of poorly and well confined concrete columns [26] .............. 24 Figure 3-15 Concept of the elasto-perfectly plastic structural behaviour .............. 25 Figure 3-16 Relationship between strength and ductility [13] .............................. 26 Figure 3-17 Model of the Bresler-Scordelis A-1 beam in ANSYS ....................... 27 Figure 3-18 Reinforcement layout of the A-1 beam and corresponding model in ANSYS .......................................................................................................... 27 Figure 3-19 Load-displacement relationship from ANSYS and the BreslerScordelis test .................................................................................................. 29 Figure 3-20 Distribution of cracks in the A-1 beam at ultimate load (467 kN) from ANSYS and the Bresler-Scordelis test [30] ................................................... 30 Figure 3-21 Response spectra for the June 17th recorded ground acceleration in Hella (5% damping) [2] ................................................................................. 32 Figure 3-22 Elastic response spectra for subsoil class A ....................................... 34 Figure 3-23 Demand spectra for constant ductility in AD format, normalized to 1,0g peak ground acceleration, for subsoil class A........................................ 40 v

Figure 3-24 Elastic-perfectly plastic idealization of an equivalent SDOF system [6] ................................................................................................................... 43 Figure 3-25 Determination of the target displacement for of an equivalent SDOF system [6] ....................................................................................................... 46 Figure 4-1 Front side and N-W-gable of the analyzed structure [4] ...................... 49 Figure 4-2 Back side and S-E-gable of the analyzed structure [4] ........................ 49 Figure 4-3 Side views of the structure [4] ............................................................. 50 Figure 4-4 Plan layout of the basement [4]............................................................ 51 Figure 4-5 Location of the main structural damage in the June 2000 earthquakes [48] ................................................................................................................. 53 Figure 4-6 Shear Crack in back-side basement column [48] ................................. 54 Figure 4-7 Shear crack in stairwell wall in the basement [48] .............................. 54 Figure 4-8 Finite element model of the structure in Sap2000 ............................... 57 Figure 4-9 Response spectra for the RSA of the structure .................................... 58 Figure 4-10 Mode shape 1 with 0,153 sec vibration period and 65% mass participation ................................................................................................... 59 Figure 4-11 Mode shape No. 51 with 0,038 sec vibration period and 9% mass participation. .................................................................................................. 60 Figure 4-12 Model of the back side wall in ANSYS ............................................. 61 Figure 4-13 Mesh of the Solid65 concrete elements ............................................. 62 Figure 4-14 Pushover curves of the wall for uniform and modal load patterns (MDOF system) ............................................................................................. 65 Figure 4-15 Ultimate compression strain in minor column in the basement ......... 66 Figure 4-16 Elastic-perfectly plastic idealization of the capacity curve (uniform load pattern) ................................................................................................... 67 Figure 4-17 Elastic-perfectly plastic idealization of the capacity curve for modal load pattern .................................................................................................... 68 Figure 4-18 Cracks at target displacement for the uniform load pattern ............... 69 Figure 4-19 Cracks at target displacement for the modal load pattern .................. 70 Figure 4-20 Formation of first, second and third cracks and cracks at target displacement, in main column on 1st floor at 190 kN, 340 kN, 856 kN and 1.101kN base shear, respectively (uniform load pattern) .............................. 70 Figure 4-21 First, second and third cracks and cracks at target displacement in main column in the basement, at 196 kN, 489 kN, 987 kN and 1.101 kN respectively (uniform load pattern)................................................................ 71 Figure 4-22 First, second and third cracks and cracks at target displacement, in main column on 1st floor at 163 kN, 210 kN, 585 kN and 721kN base shear, respectively (modal load pattern) .................................................................. 71

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Figure 4-23 First and second cracks and cracks at target displacement in main columns in the basement, at 210 kN, 512 kN and 721 kN respectively (modal load pattern) ................................................................................................... 72 Figure 4-24 Location of elements with maximum steel stress .............................. 72 Figure 4-25 Tensile stress in element A in the 1st floor column ............................ 73 Figure 4-26 Compression stress in element B ....................................................... 74 Figure 4-27 Stress in confinement reinforcement in element C ............................ 74 Figure 5-1 Definition of the maximum ductility for the uniform load pattern ...... 76 Figure 5-2 Elastic response spectra from EC8 and Hella 2000 in June 17th in E-W direction with 5% damping ratio [2] .............................................................. 77 Figure 5-3 Formation of the diagonal shear crack in the model compared to the actual damage (uniform load pattern to the left)............................................ 78 Figure 5-4 Cracks in 1st floor columns at the formation of shear crack in the basement column (uniform load pattern to the left) ...................................... 79 Figure 5-5 Retrofit proposal of the back side wall [48] ......................................... 80 Figure 5-6 Model of the proposed strengthening in ANSYS ................................ 81 Figure 5-7 Pushover curves for the proposed strengthening of the wall ............... 82 Figure 5-8 Elastic-perfectly plastic idealization of the capacity curve for the strengthening proposal with, uniform load pattern ........................................ 82 Figure 5-9 Elastic-perfectly plastic idealization of the capacity curve for the strengthening proposal, with modal load pattern ........................................... 83 Figure 5-10 Cracks at target displacement for the proposed retrofit (uniform load pattern) ........................................................................................................... 84 Figure 5-11 Cracks at target displacement for the proposed retrofit (modal load pattern) ........................................................................................................... 84 Figure 5-12 Cracks in 1st floor columns at target displacement (uniform load pattern to the left) ........................................................................................... 85

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LIST OF TABLES Table 3-1 Input parameters for material properties of concrete ............................ 19 Table 3-2 Reinforcement in the A-1 beam ............................................................ 27 Table 3-3 Input parameters for the A-1 beam test ................................................. 28 Table 3-4 Importance categories according to EC8............................................... 33 Table 3-5 Values of the parameters describing the elastic response spectrum ...... 33 Table 4-1 Geometry of structural elements ........................................................... 52 Table 4-2 Model properties for the RSA in Sap2000 ........................................... 57 Table 4-3 Dead and imposed load in the RSA in Sap2000 [kN/m2] ..................... 58 Table 4-4 Load cases in the RSA........................................................................... 58 Table 4-5 Horizontal displacements of the structure in the fundamental mode in the RSA .......................................................................................................... 60 Table 4-6 Input parameters for S200 concrete (material model 1) ........................ 63 Table 4-7 Input parameters for S250 concrete (material model 3) ........................ 63 Table 4-8 Input parameters for S400 and St37 steel (material models 2 and 4).... 64 Table 4-9 Lateral load patterns according the N2 method..................................... 64 Table 4-10 Results of the pushover analysis ......................................................... 69 Table 4-11 Stress condition in reinforcement in element A, B and C at target displacement [MPa] ....................................................................................... 73 Table 5-1 Target displacement and base shear for the assumed seismic load on the structure in June 17th ...................................................................................... 79 Table 5-2 Results of the pushover analysis for the proposed retrofit .................... 83

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Chapter 1 Introduction

1 1.1

INTRODUCTION Background and Problem Statement

1.1.1 South Iceland Earthquakes in June 2000 During the last centuries, earthquakes have several times caused considerable damage in the South part of Iceland. In June 2000, after 88 years of rest, this area was struck by two major earthquakes of magnitude 6.6 (Ms). A considerable number of houses suffered serious damage and several houses were deemed to be not repairable, but despite intensive surface fissuring and serious damage of a considerable number of structures, the earthquakes caused no serious injuries or structural collapse [1].

Acceleration component (g)

The earthquake acceleration was recorded in several places in the area, for three directional components. Figure 1-1 shows the E-W directional components of the recorded acceleration in the village Hella, in June 17th, where the peak acceleration reached 0,47g. Similar peak ground acceleration in the N-S direction was 0,17g [2]. 0,4 0,3 0,2 0,1 0 -0,1 0 -0,2 -0,3 -0,4 -0,5 -0,6

5

10

15

Time [sec]

Figure 1-1 Horizontal acceleration, E-W directional component recorded at Hella in June 17th 2000 [2]

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Subsequently, extensive damage evaluation was carried out, based on visual observation and some measurements, but did not generally include any seismic analysis of the damaged structures. Most of the buildings in the area are one or two storey concrete or wood constructions. Generally, the damaged structures were old, had poor foundation systems, with external walls constructed of pumice or concrete with low reinforcement ratio [3, 4].

Figure 1-2 Left: Two storey residential houses damaged in June 2000 earthquakes. Right: Crack in pumice inner wall [5] 1.1.2 Seismic Analysis of Reinforced Concrete Structures So far the linear elastic method along with the response spectra is the most used approach in the field of seismic analysis and practical design of structures, in other words, the response spectrum analysis (RSA). The analysis is not able to either predict the fracture mechanism or account for redistribution of forces during progressive yielding of the material. Instead it assumes linear elastic behaviour and inelastic material properties are taken into account through modification factors, which are mainly based on empirical research. Several mathematical tools are available for such analysis, mainly based on the finite element method [6]. The best way to assess the performance and to predict the demand on a structure subjected to seismic action would in many cases be the nonlinear time-history analysis. However, it requires considerable judgment and experience to perform. Many parameters, like characteristic ground motion that may affect the area must be clearly defined [7]. The nonlinear time-history analysis is also still very time consuming and requires powerful analytical tools. Thus a more simple, but yet reliable method of structural analysis is desirable. During the last 10-15 years, the nonlinear static analysis (pushover analysis) has been gaining ground within the seismic engineering field as an alternative mean of 2


analysis. The main purpose of the pushover analysis is to assess the structural performance by estimating the strength and deformation capacity using a nonlinear static analysis and compare these capacities with the demands at the corresponding performance level. The method accounts for material inelasticity and geometrical nonlinearity as well as the redistribution of internal forces. Hence, the pushover analysis is able to provide crucial information on response parameters that cannot be obtained by conventional RSA. In this study, the pushover analysis is connected to the seismic design code, EC8, through the relatively new N2 method. The simplicity of the pushover analysis makes it a more attractive approach than the nonlinear time-history analysis. However, the method has no robust theoretical background and is not exempt from some limitation such as the inability to consider higher mode effects [8]. Only a few three storey reinforced concrete buildings are located in Hella, one of those, which suffered damage in June 2000, was chosen to be analysed. The building was constructed in 1975, one year before issue of the first seismic design code in Iceland [4].

1.2

Aim of the Study

The main aim of this study is to increase knowledge and proficiency in seismic analysis of existing reinforced concrete structures. Emphasis is set on the N2 method which connects the pushover analysis to the demands in EC8. The expectation is that the results of the research will provide information on how the pushover analysis, along with the N2 method, can be used to obtain the seismic response of existing reinforced concrete structures, which might be vulnerable to structural damage under earthquake action.

1.3

Objectives

1. Construct two different finite element computer models to predict the seismic response of the reinforced concrete structure. First a 3D RSA is carried out in a finite element program (Sap2000) in order to obtain the fundamental vibration and mode shape of the structure. The modal analysis is followed with a pushover analysis of one of the structure’s wall 3


performed in another finite element programme (ANSYS), where the nonlinear behaviour of the reinforced concrete is taken into account. 2. Results of the pushover analysis are used to estimate the performance of the structure under design earthquake load and code requirements are discussed. 3. Compare the results from the pushover analysis to the actual damage and the actual seismic load. 4. Re-evaluate the seismic response of the wall using proposed retrofit of the building.

1.4

Scope of this Work

This research is organized in six chapters. The main chapters of this work are those outlined below: Second chapter: Brief and general discussion of the appropriate research methodology is carried out. The focus is set on modelling and numerical simulation as well as the experimental nature of this study. Third chapter: An overview of the most important literature relevant to the seismic response of the structure is given. The discussion is based on the properties of reinforced concrete, structural ductility and finite element modelling of the structure with emphasis on the nonlinear behaviour. Fourth chapter: A three storey reinforced concrete structure is analysed, first a RSA is carried out and then a pushover analysis is performed of one of the walls of the structure, under the design seismic load. Fifth chapter: Performance of the structure under the design load is discussed and the obtained damage in the analysis is compared to the actual reported damage. Finally, the effect of the proposed strengthening of the structure is estimated. Sixth chapter: The most important aspects of research are summarized. A thorough presentation of the main findings and conclusion and the identification of the possible project for future research are also outlined. 4

Chapter 2 Research Methodology

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RESEARCH METHODOLOGY

2.1

Introduction

In this section the research methodology in seismic engineering is briefly discussed, focusing on the most important aspect, appropriate for this study. “The scientific method refers to the procedure of selecting the appropriate techniques for a research project; in other words, evaluating alternative courses of scientific action. The scientific method is the “instrument” that scientists use to find the answers to questions. It is the process of thinking through the possible solutions to a problem and testing each possibility to find the best solution [9]. Typical seismic analysis of a structure is based on both theoretical and empirical research methods. The basic steps of theoretical research method are [10]: 1. Create formal models based on mathematics (logic). 2. Define concepts within these. 3. Prove properties of the concepts. 4. Abstraction, hide details to make the whole more understandable and to make it possible to prove properties of it. An example of the theoretical research method is evaluation of differential equations to describe the motion of structural systems. Such equations are based on pure mathematics and assumed physical properties and therefore relatively easy to account for. The basic steps of the empirical research method are as follows [10]: 1. Perform experiments. 2. Obtain results and draw conclusions.

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Empirical related experiment is for example the development of the behaviour factor, used in linear analysis of structures to account for the ability of inelastic energy dissipation. Developing such a factor is too complicated to be carried out using theoretical methods only.

2.2

Models and Numerical Simulation

Within the seismic engineering field, the numerical simulation is used to predict the actual seismic load acting during a seismic event and subsequently to evaluate the corresponding response of a given structure. A typical numerical simulation process in the field of seismic engineering is listed below [10]: 1. A formal model of the structure is constructed. Here geometry and material properties of the structure are modelled. Subsequently, the model is constrained and the load is attached. The model must be validated faithfully, in order to capture the relevant aspects of the modelled system. 2. Experiments (artificial experiments) are carried out in a computer. Here, for example the effect of different material properties and different reinforcement layout in structural elements is evaluated. 3. Collect statistics and draw conclusions. In other words, sufficient numerical simulation may be used to, generalize the results to the real system.

2.3

Experimental Work

In this study the research method is experimental in its nature where a mathematical model is used to predict the nonlinear behaviour of reinforced concrete. The model is then used to evaluate the seismic response of an existing damaged structural wall, which may be considered as a special case study. Laboratory tests, which are more similar to pure experiment in nature, would give results that are more precise in order to examine the nonlinear behaviour of a reinforced concrete element compared to mathematical modelling, but only a few laboratories (none in Iceland) would be able to perform such tests and the expense would generally be enormous. Results for numerous laboratory tests for beams, columns and walls exist in textbooks and reports. Precisely documented previous 6


tests can therefore be used to verify the accuracy of different finite element models. Furthermore, models can be used on existing structures in order to ascertain their seismic performance and, if needed, used to provide necessary structural remedial measures to buildings. Considering the South Iceland earthquakes in June 2000, data analysis of all structural damage can give an indication of how a particular structure or part of a structure would perform. A team of engineers collected data of damage in the June 2000 earthquakes. The work was carried out on the behalf of Iceland Catastrophic Insurance. By analysing the data the engineers were able to compare damage to building types, building material and so on. However, such data analysis can not evaluate the seismic response of a specific damaged structure in order to propose a retrofit, especially not in case of complex structures.

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Chapter 3 Literature Review

3 3.1

LITERATURE REVIEW Introduction

This chapter gives a brief overview of the most important topics, which are essential for the seismic analysis of the three storey reinforced concrete structure. The emphasis is set on modelling of the nonlinear behaviour of reinforced concrete, which is a complex subject and the N2 pushover analysis procedure, which is a novelty in the European seismic design code, EC8.

3.2

Nonlinear Analysis of Reinforced Concrete

Figure 3-1 illustrates the typical load-displacement relationship for a reinforced concrete element. The highly nonlinear relationship can be divided in three intervals, uncracked elastic stage interval where the deflection increases linearly with increased load, cracking propagation interval and the plastic stage interval where yield of reinforcement and crushing of concrete occurs. Load

Yielding of steel Crushing of concrete Cracking Elastic

Deflection

Figure 3-1 Load-deflection relationship of reinforced concrete [11] Three time-independent components of the nonlinear behaviour of reinforced concrete are considered in this study, cracking of the concrete, plasticity of concrete in compression and plasticity of the reinforcement bars. Time8


independent components, not considered here, are bond between concrete and the reinforcement, aggregate interlock of a cracked concrete and dowel action of the reinforcement bars. Time-dependent components of the nonlinear behaviour like creep, shrinkage and temperature change are not relevant for this study [11, 12]. 3.2.1 Basic Material Properties The characteristic compressive strength of concrete (fck) is defined in EC2 (3.1.2.2) as the value of strength below which 5% of all possible strength test results for the specified concrete may be expected to fall. When estimating the resistance of a structural member subjected to earthquake load, it should not be assumed that concrete has any significant tensile strength. The main reason for this is that microcracks, e.g. due to shrinkage, can appreciably affect the tensile strength, which thus is not easy to assess in each instance. On the other hand, it may be necessary to assume some tensile strength when evaluating the load at which crack formation starts in reinforced concrete [13]. In EC2 (3.1.2.3) the mean tensile strength fctm is given by: (3.1)

fctm = 0,3fck2/3

The Modulus of elasticity Ec, which depends on strength class of the concrete and the actual properties of the aggregates, is generally found from secant measurement under applied compression load up to 0,4fck [13, 14]. Figure 3-2 shows stress-strain diagram of plain concrete and the definition of Modulus of elasticity from EC2.

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fc fck

0,4fck Ec εc1

εult

Figure 3-2 Uniaxial stress-strain relationship for concrete [14] The relation between the characteristic compressive cylinder strength fck and Modulus of elasticity is given by: (3.2)

Ecm = 9500(fck + 8)1/3

where Ecm and fck are in MPa. According to the Icelandic National Application document to EC2, the elastic modulus of concrete should be multiplied with 0,6 for porous aggregate, but 0,9 otherwise, see EC2 national annex 3.1.2.5.2. Poisson ratio for elastic strain is the ratio of transverse strain to longitudinal strain in the direction of the uniaxial compressive force. EC2 (3.1.2.5.3) recommends the value of 0,2 for concrete. 3.2.2 Concrete Response to Multiaxial Loading Study of response of plain concrete to uniaxial loading is generally insufficient with regard to understand the behaviour of confined concrete. Moreover, response in elements like structural walls and deep beams is rather described by considering biaxial loading. Equation (3.3) shows the fundamental matrix equation of plain stress for an isotropic element:

0 ⎡σx ⎤ ⎡1 ν ⎤ ⎡ εx ⎤ 0 ⎢ σ y ⎥ = E /(1 − ν 2 ) ⎢ν 1 ⎥ ⋅ ⎢ εy ⎥ ⎢⎣τ xy ⎥⎦ ⎢⎣0 0 (1 − ν) / 2⎥⎦ ⎢⎣ γ xy ⎥⎦

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(3.3)


where E is the elastic modulus and ν is the Poisson’s ratio. Figure 3-3 shows the strength envelope for a 200 x 200 x 50 mm concrete plate under biaxial loading. The graph was obtained by testing [15].

Figure 3-3 Strength envelope for concrete under biaxial load [15] Figure 3-3 indicates that the strength is increased by approximately 25% when the ratio α = σ1/ σ2 ≈ 0,5 and it increases by 16% when the ratio is 1,0. The biaxial tensile strength is not very dependent upon α and it may be said to be similar to that under uniaxial loading [16]. In reality, most structural elements are subjected not only to biaxial loading, but rather to triaxial loading. This applies to e.g. containment vessels, prestressed concrete reactor vessels and offshore platforms [16]. Confined concrete is subjected to triaxial loading and its response under those conditions can thus be of importance regarding earthquake loads. The behaviour of concrete under triaxial loading is complex and is outside the scope of this thesis, and therefore only a brief overview shall be given here. A more thorough treatment may be found in textbooks like in [11]. Usually, the strength surface is described by the meridians and the shape functions in the deviatoric plane. The deviatoric plane is the plane that is perpendicular to

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the hyrdrostatic axis, namely σ1 = σ2 = σ3 where σ1, σ2, σ3 are the principal stresses [16].

Figure 3-4 Failure criterion presented in the three dimensional principal stress [16] 3.2.3 Steel Reinforcement Figure 3-5 shows a typical stress-strain diagram for reinforcement steel. It is well known that the ultimate strain decreases with increasing strength. This is a tendency not very different from that exhibited by plain concrete. fs

Strain predicted by linear relationship

ftk fy

0,002

Es

εs

Figure 3-5 Stress-stain diagram for reinforcement steel [17]

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When steel is subjected to cyclic loading, the elastic modulus, as a rule, does not change considerably, if stress reversal does not occur. Under earthquake conditions, though, stress reversal may occur, although large compressive strains are uncommon. In this case, it is important to investigate the behaviour of steel under continuously repeated loading. A decrease in stiffness occurs due to stress reversals. This non-linear behaviour takes place below the yield limit of the steel and is commonly referred to as the Bauschinger-effect. On the other hand, the stress reversals become almost linear upon repetition [16].

3.3

Analytical Modelling of Reinforced Concrete

The accuracy of the prediction of the behaviour of a structure depends on the development of reliable analytical models of the critical regions. Analytical model must reflect properties of concrete, both in tension and compression, response of the steel bars and synergy of the two materials. In order to build model of the structure with adequate precision, a stress-strain relationship for both concrete and steel bars and a failure theory, must be available. Different approaches are used to implement the stress-strain relationship under various stress states, namely, nonlinear elastic, plasticity based model, plastic fracturing theory etc. [11]. By using the finite element method and performing a nonlinear analysis with appropriate constitutive relations, deformational and failure characteristics, it is possible to predict the actual behaviour of reinforced concrete.

13


Figure 3-6 Load-displacement relationship for typical reinforced concrete element [11] Figure 3-6 shows Load displacement relationship for typical concrete element from cracking in tension to crushing in compression. The relationship is linear in the tension zone and from the origin to the proportional (yield) point A. The AC interval represents the elasto-plastic response of the element and CD represents the perfectly plastic interval. The plasticity is characterized by the irreversible straining εp that occurs in a material once a certain level of stress is reached in other words it provides a mathematical relationship to characterise the elastoplastic response. The theory is based on three components: The yield criterion, flow rule and the hardening rule. Detailed discussion of the plasticity theory is for an example in [11, 18]. 3.3.1 Failure Criteria for Concrete The load carrying capacity of concrete under multiaxial stress condition is a function of the state of the stress and can be predicted by using failure criteria. Most failure criteria are given as a hypothesis and whose application to different material should be evaluated from tests [12]. Several failure criteria have been formulated for concrete under general stress state using mathematical models with one to five parameters. Initiation of cracking, yielding and ultimate carrying capacity has been used in formulating the failure criteria [11]. The multiaxial stress condition for isotropic material failure criteria can be expressed as an invariant function of the stress condition as follows: 14


(3.4)

f (σ1 , σ 2 , σ 3 ) = 0

Where σ1, σ2, σ3 are the principal stresses. Equation (3.4) can be replaced in terms of three principal stress invariants: (3.5)

f ( I1 , J 2 , J 3 ) = 0

where I1 is the first invariant of the stress tensor and J2, J3 are the second and third invariants of the deviatoric stress tensor. In this study, the William and Warnke failure criteria is used [19]. It is three dimensional failure criteria, which considers a five-parameter failure surface model for concrete in compression and tension. The model has been widely used for analytical models of concrete. The model, presented by three dimensional graph, has a conical shape with curved meridians (second order parabolas) and a deviatoric cross section, as shown in figure 3-4. More simple description of the three-dimensional failure surface is shown in Figure 3-7.

σyp f c´

Cracking

fr Cracking σxp fr

Cracking

σzp > 0 (Cracking) σzp = 0 (Crushing)

f c´

σzp < 0 (Crushing)

Figure 3-7 Failure surface for concrete [18]

15


The most significant nonzero principal stresses are in the x and y directions, represented by σxp and σyp respectively. Three failure surfaces are shown as projection on the σxp and σyp plane. The mode failure is a function of the sign of σzp (principal stress in z-direction). Hence, if σxp and σyp are negative (compression) and σzp is positive (tensile), cracking would be predicted. However, if σzp zero or negative, the material is assumed to crush [18, 19]. Cracking occurs when the principal stress in any direction lies outside the failure surface and the elastic modulus of the concrete becomes zero in the direction parallel to the principal tensile stress direction. On the other hand, crushing occurs when all principal stresses are compressive and lies outside the failure surface and the elastic modulus becomes zero in all principal directions and the element becomes inactive. The hardening rule reflects the changing of the yield surface with progressive yielding, so that the stress states for subsequent yielding can be established. Two hardening rules are most commonly used: work hardening and kinematic hardening. Kinematic hardening is used in this study and it assumes that the yield surface remains constant in size and the surface translates in the stress space with progressive yielding, as shown in figure 3-8 [18]. σ2 σ3

Initial yield surface σ1 = σ2= σ3

Subsequent yield surface

σ1

σ1

σ2

Figure 3-8 Yield surface and the kinematic hardening rule [18] The constitutive equations for the nonlinear kinematic hardening are based on the elastic von Mises yield function and the associated flow rule. The model can simulate the Bauschinger-effect (explained before) and the monotonic hardening [18].

16


3.3.2 Concrete Modelling ANSYS 10, a nonlinear three-dimensional finite element program is used in this study to simulate the response of the reinforced concrete. ANSYS provides a dedicated three dimensional eight-node solid isoparametric element, Solid65 to model the response. The model only considers monotonic loading cases and does not predict the cyclic behaviour of the material. Generally the basic steps, involved in the finite element method analysis in this study, consist of the following [20]: 1. Create and discretize the solution domain into finite elements i.e. to subdivide the problem into nodes and elements (Figure 3-9). 2. Assume a shape function to represent the physical behaviour of an element, i.e. a continuous function is assumed to represent the approximate solution of an element. In this study beam and shell elements are used in the elastic analysis (Sap2000) and the Solid65 octahedron concrete element in the pushover analysis (ANSYS). 3. Develop equations for an element. 4. Assemble the elements to present the entire problem, constructing the global stiffness matrix. 5. Apply boundary conditions, initial conditions and loading. 6. Solve a set of linear or nonlinear algebraic equations simultaneously to obtain nodal results such as displacements.

17


Figure 3-9 The solid65 concrete element in ANSYS [18] The Solid65 Element is capable of plastic deformation, cracking in three orthogonal directions and crushing. Solid65 has one solid element and up to three rebar materials. Rebar specifications are input as real constants, including material properties, volume ratio and the orientations. The steel bars are capable of tension, compression but no shear. The following assumptions and restrictions are given by the ANSYS documentation for the Solid65 element [18]: 1. Cracking is permitted in three orthogonal directions at each integration point. 2. If cracking occurs at an integration point, the cracking is modelled through an adjustment of material properties, which effectively treats the cracking as a “smeared band” of cracks, rather than discrete cracks. 3. The concrete material is assumed to be initially isotropic. 4. Whenever the reinforcement capability of the element is used, the reinforcement is assumed to be “smeared” throughout the element. 5. The concrete material data are input as a series of nine constants for the Solid65 elements.

18


Table 3-1 Input parameters for material properties of concrete No. 1 2 3 4 5 6 7 8 9

Description Shear transfer coefficient for an open crack Shear transfer coefficient for a closed crack Uniaxial tensile cracking stress Uniaxial crushing stress Biaxial crushing stress

Label βt βc ft fc fcb

Ambient hydrostatic stress, for use with constant 7 and 8

σah

Biaxial crushing stress under the ambient of hydrostatic f1 stress state Uniaxial crushing stress under the ambient of hydrostatic f2 stress state Stiffness multiplier for cracked tensile conditions Tc

The William and Warnke criteria use the five input parameters, ft, fc, fcb, f1 and f2. The failure surface can be specified with a minimum of two constants, ft and fc and the other three constants default to William and Warnke model [19]: fcb = 1,20 fc f1 = 1,45 fc f2 = 1,725 fc The shear transfer coefficient for cracks β varies form zero to one. Zero represents a smooth crack with no shear transfer and one represents rough crack with full shear transfer. Previous studies have shown that convergence is difficult to achieve if the shear transfer coefficient for an open crack is below 0,2 [21, 22]. Hence, the value of the shear transfer coefficient for an open crack βt is set to 0,2 in this study. Stiffness multiplier for cracked tensile conditions, Tc is a multiplier for tensile relaxation. Tc Varies between zero and one where zero represents complete loss of tensile stress at cracking [18]. When a principal stress in the Solid65 concrete element exceeds the ultimate tensile strength according to figure 3-7, a cracking sign represented by a circle appears. The cracking sign appears perpendicular to the direction of the principal stress. The crack may occur in all three principal directions [18].

19


Linear isotropic properties with Ec as the elastic modulus and bilinear kinematic hardening properties with EcT as the secant modulus of plasticity are used to predict the stress-strain relationship of the concrete. Similar stress-strain relationship is introduced for the reinforcement steel. Figure 3-10 shows the idealized stress-strain relationship for concrete and steel, used in ANSYS.

fc EcT 0,8fc

Ec εt

6εt

Tc ft

ε

σ EsT fy Es -εy

εy ε

-fy -σ

Figure 3-10 Stress-strain curves for concrete and steel in the ANSYS model

3.3.3 Finite Element Discretization One of the most important step of the finite element modelling is the meshing of the model, in other words, choosing the element size. Convergence of the solution is among other based on the mesh density. Sufficient density is practically achieved when an increase in the mesh density no longer has an appreciable effect 20


on the results. Coarse mesh is not able to obtain good results in case of stress concentration, for example in window corners. Studies have shown that the size of the Solid65 elements should be about two to three times greater than the maximum aggregate size of the concrete, in order to correctly and realistically model the actual cracks [21]. 3.3.4 Nonlinear Solution In the nonlinear static analysis, the total applied load is divided into a series of load steps (load increment). At the end of each load step solution, the stiffness matrix of the model is adjusted to reflect the nonlinear changes in the structural stiffness before proceeding to the next load step. The equation of equilibrium can be defined as: (3.6)

[K]{Δu} = {ΔP}

where K is the stiffness matrix u is the displacement and P is the applied force, for a small load increment during which the behaviour is assumed to be linear elastic. For more convenient computational purpose, the equation may be written as: (3.7)

[Kt]{Δu} + {Rt} = {P}

where Kt is the tangent stiffness matrix for the current load increment and Rt is the restoring force at the beginning of the load increment [18, 23]. In ANSYS, this updating process of the stiffness is based on the Newton-Raphson equilibrium iteration [18]. It provides convergence at the end of each load step within given tolerance limits. Figure 3-11 shows the use of the Newton-Raphson approach in a single-degree-of-freedom system (SDOF) nonlinear analysis.

21


Load

Converged solution Displacement

Figure 3-11 Two load steps of Newton-Raphson iteration solution [18] Before each solution, the Newton-Raphson approach estimates the out-of-balance load vector which is the difference between the loads corresponding to the element stress and the applied load [24]. ANSYS subsequently obtains a linear solution, using the out-of-balance loads and checks for convergence. This process is re-evaluated until the convergence criteria are satisfied. In this study the convergence criteria are based on load and displacement. Several studies have shown that convergence of solution for the Solid65 concrete models are difficult to achieve if the default criteria in ANSYS are used [21, 25]. Hence the default tolerance is increased to a maximum of five times the default limit, up to 0,5% for force checking and 5% for displacement checking. 3.3.5 Ductility Ductility is a basic property of reinforced concrete subjected to loads from earthquake. Ductility of a structural element of reinforced concrete is a description of its capacity to withstand nonlinear deformations without considerable loss of strength, as well as the capacity to dissipate earthquake energy through hysteresis loops. Ductility is a desirable property in reinforced concrete as it induces redistribution of stresses and can give a warning of impending failure. By ensuring that a structural system or any part thereof possesses adequate ductility it may be assumed that the structure will withstand earthquake loads considerably in excess of that indicated by its strength as calculated by linear methods. Designing a structure with the provision of ductility in the structure or parts thereof may result in a considerable economic gain. Despite the concept of 22


ductility being fairly well known it’s direct application to the evaluation of the material properties of reinforced concrete is rather complex. In order to assess the ductility of a structural element one uses the factor of ductility. The factor is defined as the ratio of the ultimate deformation to the deformation at the yielding: μ=

Δu Δy

(3.8)

The ultimate deformation Δu is usually defined as that value where the strength of the structural element or the material has decreased slightly, for reinforced concrete this loss is often set at 15%. Figure 3-12 shows the definition of the ductility factor. F

Fy

Δu

Δ

Δy

Figure 3-12 Definition of the ductility factor [16] The higher the factor of ductility possessed by a structure, the higher it’s safety against failure or collapse in the event of an earthquake. In order to ensure the ductility of a structural element it is required that the materials, concrete and reinforcement, possess ductility. Thus the mechanical properties of reinforced concrete are the deciding factor as to the ductility of any structural element and its capacity to withstand cyclic loading [16]. A desirable source of inelastic deformation of a structure or structural elements is rotation in potential plastic hinges. It is convenient to relate section rotations (curvature) to causative bending moments. The maximum curvature ductility is expressed as: 23


μφ =

φm φy

(3.9)

where φ m is the maximum curvature expected to be attained or relied on and φ y is the yield curvature, which is normally defined as the first yielding of the tensile reinforcement [13]. In order to gain ductility in form of increased ultimate strain in reinforced concrete columns, the reinforcement is crucial. Properly confined columns provide significantly more ductility because deformability is increased due to the triaxial state of stress (figure 3-13 and figure 3-14). Lateral swelling is prevented in form of closed ties (hoops) or spirals [13, 16].

Figure 3-13 Stress-strain diagrams for concrete subjected to various types of confinement [16]

Poorly confined

Figure 3-14 Layout of poorly and well confined concrete columns [26] More convenient and visible quantity to evaluate either the ductility imposed on a structure caused by an earthquake or the structure’s capacity to develop ductility is displacement. Based on the ductility, a rather simple empirical observation may be used to estimate the equivalent displacement of a structure. Figure 3-15 shows the concept of the elasto-perfectly plastic structural behaviour, where Δeq is the expected

24


maximum structural displacement whether the structure behaves elastically or inelastically [27].

Feq Ky

Fy Keq Δy

Δeq

Figure 3-15 Concept of the elasto-perfectly plastic structural behaviour Based on this assumption the ductility factor is: μ=

Feq Fy

=

Δ eq

(3.10)

Δy

According to Paulay and Priestley [13] structures may be classified in relation to the level of ductility permitted of the structure. Class A, includes fully elastic structures with ductility factor equal to one (path OAA´) with corresponding elastic strength SE.

25


Figure 3-16 Relationship between strength and ductility [13] In class B the response is ductile, where path OBB´ represents restrained ductility, for example structures with large areas of structural walls, which inherent possesses significant strength with respect to lateral forces. Finally, path OCC´ represents fully ductile response where structures are designed to possess the maximum ductility potential that can reasonably be achieved at carefully detailed inelastic regions.

3.4

Concrete Model Calibration

3.4.1 Introduction Numerous studies have shown that the Solid65 concrete element in ANSYS is capable of predict the actual nonlinear behaviour with fairly acceptable accuracy [21, 28, 29]. In most of the studies, the ANSYS models are compared to laboratory tests, where the load-displacement relationship and evaluation of cracks are compared. In 1961, Bresler and Scordelis [30] tested twelve beams of different size with different reinforcement layout, in a laboratory. This classical beam test has been widely use by engineers, for example to verify analytical computer models. The test was re-examined in 2004 [31] and despite of some minor difference in materials properties and loading arrangements, the Bresler-Scordelis test were reproducible with reasonably good accuracy. In this section, one of the beams (A1) is modelled in ANSYS and the results are compared to verify the accuracy of the analytical results. 26


The finite element model consists of 1.976 nodes and 1.332 Solid65 elements. Density of the mesh and layout of the elements is chosen in order to obtain correct moment capacity and convergence of the results.

P

3.660 mm

Figure 3-17 Model of the Bresler-Scordelis A-1 beam in ANSYS Table 3-2 Reinforcement in the A-1 beam Reinforcement Tension Compression Hoops

Number of bars 4 2 c/c 210

Diameter [mm] 28,7 12,7 6,4

Area [mm2] 2.600 1,0 2,2 MPa

Figure 3-18 Reinforcement layout of the A-1 beam and corresponding model in ANSYS

27


The material constants, used in the ANSYS model of the beam are listed in the table below. Table 3-3 Input parameters for the A-1 beam test No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Description Shear transfer coefficient for an open crack (βt) Shear transfer coefficient for a closed crack (βc) Uniaxial tensile cracking strength (ft) Uniaxial ultimate compression strength (fc) Yield strength of concrete Stiffness multipl. for cracked tensile conditions (Tc) Modulus of elasticity for concrete (Ec) Secant modulus of plasticity for con. (ET) Poisson ratio for concrete (νc) Ultimate concrete strain (εc) Weight of concrete (γc) Yield strength of tension steel Yield strength of compression steel Modulus of elasticity compr. steel and hoops (Es1) Modulus of elasticity for tension steel (Es2) Mod. of plasticity for compr. steel and hoops (ET1) Modulus of plasticity for tension steel (ET2) Ultimate strain for compr. steel and hoops (εs1) Ultimate strain for tension steel (εs2) Weight of steel (γc)

0,2 1,0 3,0 MPa 24,1 MPa 19,28 MPa 0,6 27.000 MPa 1.723 MPa 0,2 0,35% 25 kN/m3 555 MPa 345 MPa 201.000 MPa 206.000 MPa 1.002 MPa 9.000 MPa 1,6% 1,3% 77 kN/m3

The force P is attached to the nodes in the middle of the beam and increased in 200 steps, until the ultimate load (467 kN) in the Bresler-Scordelis test is reached.

28


ANSYS

Bresler & Scordelis

500 450 400 350

P [kN]

300 250 200 150 100 50 0 0

2

4

6

8

10

12

14

Displacement [mm]

Figure 3-19 Load-displacement relationship from ANSYS and the BreslerScordelis test The correlation between the load-displacement relationship obtained from ANSYS and from the Bresler-Scordelis test is fairly good (figure 3-19). Cracks start to form at about 40 kN in both cases and displacement at the ultimate load is the same. The failure of the beam in the Bresler-Scordelis test was in form of shearcompression failure. The behaviour could be characterized as shear-flexural in nature and it exhibited several diagonal-tension cracks during the last load stage [31]. Crack patterns obtained from ANSYS and from the Bresler-Scordelis test are similar and location of diagonal tension cracks is also similar.

29


Flexural crack

Diagonal tensile crack

Figure 3-20 Distribution of cracks in the A-1 beam at ultimate load (467 kN) from ANSYS and the Bresler-Scordelis test [30]

3.5

Elastic Response Spectrum Analysis

3.5.1 Theoretical Background RSA is the most commonly used method in the field of seismic design. Numerous textbooks provides a detail study of the subject, see for example [32, 33]. The equation of motion for undamped free vibration of multi-degree-of-freedomsystem (MDOF) can be written in matrix form as [23]: (3.11)

M ü+K u = 0

where M and K are the mass and stiffness matrices of the structure, respectively. The motions of a system in free vibration are harmonic, hence, the displacement vector can be represented as: (3.12)

u = ū sin(ωt)

where ū is the displacement vector, ω is the angular frequency and t is the time. Differentiating twice with respect to time gives: (3.13)

ü = -ω2 u

and substituting equation (3.13) into equation (3.11) gives the eigenvalue equation: 30


(3.14)

(K - ω2M) u = 0

The solution of the classical equation above is based on the fact that in order for a set of homogenous equilibrium equations to have a nontrivial solution, the determinants of the coefficient matrix must be zero: (3.15)

det(K - ω2M) = 0

By expanding the determinant by minors, the polynomial of degree N is obtained which is the frequency equation. The N roots of the polynomial represent the frequencies of the N modes of vibration. Then the frequencies can be substituted into equation (3.14) which can be solved for the relative amplitude of motion for each of the displacement components in the particular mode of vibration [23]. The fundamental mode, which is particularly important in this study, is the one with the longest vibration time. Eigenvalue problems for large structures, as analyzed in this study, can only be solved by computers, in most cases through special structural finite element programs. Generally, the procedure of the elastic analysis can be described in the following five steps: 1. The mode shapes and vibration periods of the structure are determined (as described above). 2. The spectral acceleration for each mode are obtained, using a response spectrum. 3. For each mode, the effective modal masses are determined and from these the maximum inertia forces. 4. For the maximum inertia forces of the modes, the maximum values of the response are determined (moments, shears displacement and so on). 5. The quantities of the modes under consideration are superimposed, usually by using the SRSS (square root of the sum of their squares) or the CQC (complete quadratic combination) methods.

31


A response spectrum is a plot of the maximum acceleration response for SDOF system for various vibration periods subjected to an earthquake ground motion [6]. It is influenced by different frequency contents and ground acceleration. It is practical for many structural applications to present the maxima of the response parameters rather than their values at each instance during the response. A family of curves may be computed for any given excitation in order to show the effect of various structural damping (for example 1%, 5%, 10% and 20%) in terms of equivalent linear viscous damping, ξ [6]. Response spectra can be computed either from natural or artificial seismic acceleration diagrams, by implementing a numerical algorithm to integrate the equation of motion for a SDOF system using a given measured action-deformation relationship and damping ratios [32]. Figure 3-21 shows a response spectra computed for the June 17th recorded ground acceleration in Hella in EW direction [2].

1,8 1,6 1,4

Se/g

1,2 1 0,8 0,6 0,4 0,2 0 0,0

0,5

1,0

1,5

2,0

2,5

3,0

T [sec]

Figure 3-21 Response spectra for the June 17th recorded ground acceleration in Hella (5% damping) [2] In EC8 the earthquake motion for computation of the design seismic action is generally represented by a smooth elastic ground acceleration response spectrum. To form the elastic response spectrum for a specific site, it is necessary to know the effective peak ground acceleration. The reference horizontal peak ground acceleration agR is derived from zonation maps in the National Annexes Documents to EC8, where the national territories are divided into seismic zones 32


depending on the local seismic hazard. The horizontal peak ground acceleration corresponds to a reference return period of 475 years which correspond to 90% probability of not being exceeded in 50 years, which meet the criteria of the ultimate limit state, according to EC8 [34]. The design ground acceleration is given by: a g = γ I ⋅ a gR

(3.16)

where γI is the importance factor and agR is the reference peak ground acceleration for given seismic zone. Buildings are classified into four importance classes, mainly depending on the importance for public safety and probability of human losses in case of collapse. Table 3-4 Importance categories according to EC8 Category Buildings γI factor I Buildings of minor importance for public safety, e.g. 0,8 agricultural buildings II Buildings of intermediate size and normal use, e.g. 1,0 apartment house and office buildings III Buildings whose seismic resistance is of importance in view of the consequences associated with a collapse, e.g. 1,2 schools and assembly halls IV Buildings whose integrity during earthquakes is of vital 1,4 importance for civil protection, e.g. hospitals The seismic action is influenced by local subsoil conditions. Seven subsoil classes (ground types) are given in EC8. In Iceland, due to general geological conditions, ground type A is usually the most appropriate one. Ground type A represents conditions when the structure is founded on rock or rock-like geological formation including at most 5 m of weaker material at the surface (EC8 clause 3.1.2). Parameters based on subsoil class A are given in the table below. Table 3-5 Values of the parameters describing the elastic response spectrum Constant S TB TC TD

Description Soil factor Lower limit of the constant spectral acceleration Upper limit of the constant spectral acceleration Beginning of the constant displacement response range 33

Value 1,0 0,15 s 0,40 s 2,00 s


The following expressions define the response spectra in EC8:

⎡ T ⎤ 0 ≤ T ≤ TB : Se (T ) = a g ⋅ S ⋅ ⎢1 + (η ⋅ 2,5 − 1)⎥ ⎣ TB ⎦

(3.17)

TB ≤ T ≤ TC : Se (T ) = a g ⋅ S ⋅ η ⋅ 2,5

(3.18)

⎡T ⎤ TC ≤ T ≤ TD : Se (T ) = a g ⋅ S ⋅ η ⋅ 2,5 ⋅ ⎢ C ⎥ ⎣T⎦

(3.19)

⎡T ⋅ T ⎤ TD ≤ T ≤ 4s : Se (T ) = a g ⋅ S ⋅ η ⋅ 2,5 ⋅ ⎢ C 2 D ⎥ ⎣ T ⎦

(3.20)

where η is the damping correction factor, which may be taken as 1,0 for 5% viscous damping. Figure 3-22 shows the elastic response spectra for subsoil class A, where Se(T) is the ordinate of the elastic response spectrum and T is the vibration period of the linear SDOF system. 3,0 2,5

Se(T)

2,0 1,5 1,0 0,5 0,0 0,0

0,5

1,0

1,5

2,0 T [sec]

2,5

3,0

Figure 3-22 Elastic response spectra for subsoil class A

34

3,5

4,0


3.6

Pushover Analysis

3.6.1 Background to Pushover Analysis The pushover analysis (also named nonlinear static analysis) was introduced back in 1970’s and for the last 10-15 years it has been noticed as a powerful engineering tool. Initially, the discussion on the pushover analysis was on the range of applicability of the method and its advantage and disadvantage compared to elastic or dynamic nonlinear methods. Compared to nonlinear dynamic analysis, the pushover analysis is relatively simple and much less time consuming. On the other hand, pushover analysis is more realistic and more comprehensive than one based on a RSA. Pushover analysis is able to consider the inelastic response characteristics and therefore provide information of performance of a structure in a seismic event, which the linear approach is not capable of [35]. A comprehensive discussion of the pushover analysis is given in [8, 35, 36]. The main purpose of the pushover analysis is to compare the strength and deformation capacity with the demands at the corresponding performance level, by using a static nonlinear analysis algorithm. The analysis considers geometrical non-linearity and material inelasticity, as well as the internal force redistribution. It is carried out under constant gravity loads and monotonically increased lateral forces, applied at the location of the masses in the structural model, to simulate the inertia forces. The method is able to describe the evaluation of plastic mechanism and structural damage as a function of the lateral forces since they are increased monotonically [6]. The pushover analysis may be described as an extension of the lateral force method of linear analysis in to the nonlinear regime. However, the method is based on many assumptions and may in some cases provide misleading results, as explained in the end of this section. Pushover analysis may be provided if there is a doubt that simple analysis provides sufficient information of the structural seismic resistance. The pushover analysis provides more relevant information and response characteristics that can not be obtained from a RSA. Pushover analysis is also feasible for seismic analysis of existing structures and design of retrofit schemes. The following examples of the information given by the pushover analysis are as follows [35]:

• Force demands on potential brittle elements, such as axial force demands in columns, force demands on braced connections, moment demands on beams

35


to columns connections, shear force demands in deep reinforced concrete spandrel beams and in unreinforced masonry wall piers, etc. • Estimates of deformation demands for elements that have to deform inelastically in order to dissipate the energy imparted to the structure by ground motion. • Consequence of strength deterioration of individual elements on the behaviour of the structural system. • Identification of critical regions in which the deformation demands are expected to be high and that have to become the focus of thorough detailing. • Identification of strength discontinuities in plan or elevation than will lead to changes in dynamic characteristics in the inelastic range. • Estimating of the interstory drifts, which account for strength or stiffness discontinuities and may be used to control damage and evaluate P-delta effects. • Estimates of global drifts, which may be used to assess the potential for pounding. The advantage of the response characteristics given above comes at the cost of additional analysis effort, associated with incorporating all important elements and modelling their inelastic properties. Three dimensional analytical model of a structure would be the most preferable one, but at this time, few adequate analytical tools are available for that purpose. However, the capability of computers is growing fast and for the last few years, sophisticated finite element computer programs for instance Sap2000 [37] and ETABS [38], have introduced Pushover analysis of steel and concrete frame structures as an addition to their programmes. In Sap2000 and ETABS, the nonlinear properties of the elements are implemented in the form of yield hinges, chosen and defined by the structural designer. Other finite element programs, for instance ANSYS (see chapter 3.3.2) and Cosmos/M, can perform pushover analysis where the nonlinear material properties are considered. However a three dimensional model of a typical structure would be cumbersome and with few exceptions too time consuming for a typical design process in the consulting engineering field.

36


The pushover analysis has no robust theoretical foundation. The basic assumption is that the response of a MDOF (multi-degree-of-freedom) structural system can be related to the response of an equivalent SDOF system. This implies that the response is controlled by a single mode and that the shape of the mode is constant throughout the time history [35]. It is clear that both these assumptions are not correct. However, several pilot studies have indicate that these assumptions result in fairly good prediction of the maximum seismic response of MDOF structures, as long as the response is dominated by a single mode. Several studies have shown that results of experimental tests and nonlinear dynamic analysis are similar to those obtained from the pushover analysis [39, 40]. 3.6.2 EC8 and the N2 Pushover Procedure For the last year, the control of structural damage during seismic events has been gaining more attention. One of the ways to achieve the damage control is through the pushover analysis, which has recently been introduced into the seismic design methodology [40]. In other words, to combine the pushover analysis and the response spectrum approach. Examples of this are the capacity spectrum method applied in ATC-40 [41], FEMA-273 [7] and the N2 method introduced in EC8. Studies have shown that the FEMA-273 procedure and the N2 procedure can yield exactly the same results if the same displacement shape and lateral load distribution are assumed. The difference lies in the visualisation provided in the N2 method [42]. In EC8, the primary use of nonlinear analysis is to evaluate the seismic performance of new designs or to assess existing or retrofitted buildings. On the assessment and retrofitting of buildings in EC8, the reference analysis is the nonlinear one. EC8 specifies two additional uses for the pushover analysis [6]: 1. To verify or revise the value of the factor αu/α1 incorporated in the basic or reference value q0 of the behaviour factor of concrete, steel or composite buildings, to account for over-strength due to redundancy of the structural system. For more details, see EC8 clause 5.2.2.2 2. To design buildings based on a nonlinear static analysis and deformationbased verification of its ductile members, instead of force-based design with linear elastic analysis and the design spectrum that incorporates the behaviour factor q. In this case, the seismic action is defined in terms of the target displacement, derived from the elastic spectrum with 5% damping ratio, instead of the design response spectrum. 37


The N2 procedure was developed at the University of Ljubljana in Slovenia in mid-eighties [43, 44]. It is a novelty in EC8. The method is based on the following steps: 1. Compute the lateral force pattern. 2. Perform the pushover analysis. 3. Compute the equivalent SDOF capacity curve. 4. Idealize the capacity curve as an elastic-perfectly plastic. 5. Compute the seismic demand according to EC8. 6. Compute the target displacement of the MDOF system.

Seismic Demand in AD Format Before following the six steps above, the seismic demand may be represented graphically in an AD (acceleration-deformation) format for constant ductility. It should be noted that the construction of the demand spectra is not necessary for the computational procedure. It is shown here for better visualisation of the procedure [40]. For an elastic SDOF system, the following relationship applies:

S de =

T2 S ae 4π 2

(3.21)

where Sae and Sde are the values on the elastic acceleration and displacement spectrum respectively, corresponding to the period T and a fixed viscous damping. For an inelastic spectrum, the acceleration Sa and the displacement Sd can be determined as:

Sa =

S ae Rμ

(3.22)

38


Sd =

μ Sde Rμ

(3.23)

where μ is the ductility factor, defined as the ratio between ultimate displacement and the yield displacement of the SDOF system [45]. Rμ is the reduction factor due to ductility, i.e. due to hysteretic energy dissipation of ductile structures [40, 46]. Several proposals for the reduction factor exist; see for example [46]. The N2 method makes use of the following bilinear spectrum for the reduction factor Rμ proposed by Vidic T. [45]: R μ = (μ − 1)

T +1 Tc

Rμ = μ

if T < Tc

(3.24)

if T ≥ Tc

(3.25)

where Tc is the characteristic period of the ground motion, usually defined as the transition period where the constant acceleration segment of the response spectrum (short period range) passes the constant velocity segment of the spectrum (medium period range). The N2 method suggests that the equal displacement rule applies in the medium and long period ranges. In other words, that the displacement of the inelastic system is equal to the displacement of the corresponding elastic system with the same period [40]. Figure 3-23 shows the demand spectra for different value of constant ductility, normalized to 1,0g peak ground acceleration, subsoil class A and 5% damping. The vibrations period is represented as radial lines.

39


3 μ=1 2,5

μ = 1,5

Sa (g)

2

μ=2 μ=3

1,5

μ=4 1

T = 0,15 T = 0,4

0,5

T=2

0 0,00

0,01

0,02

0,03

0,04

0,05

Sd [m]

Figure 3-23 Demand spectra for constant ductility in AD format, normalized to 1,0g peak ground acceleration, for subsoil class A Lateral Force Pattern In the N2 method, the vector of the lateral load F used in the pushover analysis is given by: (3.26)

F=pMΦ

where M is the mass matrix, p controls the lateral load magnitude and Φ is the displacement shape vector. According to EC8, the lateral forces Fi applied to storey masses mi are taken to remain proportional to a certain pattern of horizontal displacements Φi: (3.27)

Fi = p mi Φi

The pushover analysis should be performed using both of the following lateral load patterns (EC8 4.3.3.4.2.2): A uniform pattern, corresponding to uniform unidirectional lateral acceleration which yields in Φi = 1, and the lateral forces becomes:

40


Fi =

mi

∑

n

mj j=1

Fb =

mi

∑

n

mj j=1

p

(3.28)

where Fb is the base shear force and mi and mj are the storey masses A modal pattern, which depends on the type of linear analysis applicable of the particular structure. The two types are: 1. If the building satisfies the conditions for the application of lateral force analysis, an inverted triangular similar to the one used in that method, which yields in Φi = Zi (EC8 4.3.3.2.1). 2. If the building does not meet the conditions of the lateral load method, a pattern that simulates the peak inertia forces of the fundamental mode shape in the horizontal directions should be adopted. The fundamental mode shape Φi is derived from the RSA (section 3.5) and the lateral forces become: Fi =

miΦi

∑ j=1 m jΦ i n

Fb

(3.29)

The most unfavourable results from the two lateral load patterns above (uniform and modal) should be applied in the design process. A practical possibility (not required by EC8) would be to use both the load patterns and envelope the results. Equivalent SDOF Capacity Curve In order to model the MDOF system as an SDOF system, the starting point is the equation of motion of a planar MDOF model that explicitly includes only lateral translational degrees of freedom [23, 42]: (3.30)

Mü+R=Mla

where u and R are vectors representing displacements and internal forces, l is a unit vector and a is the ground acceleration as a function of time. Here, damping is not included in the derivation. It is included in the response spectrum. Assuming that the displacement shape Φ is constant during the structural response to ground motion, the displacement vector is: 41


(3.31)

u = Φ dn

where dn is the roof top displacement and Φ is normalised in such way that the component at the top displacement is equal to one. From statics, the internal forces are equal to the statically applied external load: (3.32)

F=R

Inserting equation (3.26), (3.31) and (3.32) into equation (3.30) and multiplying from the left with ΦT the equation of motions becomes: ..

ΦT M Φ d n + ΦT M Φ p = - ΦT M l a

(3.33)

and by multiplying and dividing left hand side with ΦT, M and l the equation of motion of the SDOF system thus becomes: ..

(3.34)

m* d * + F* = -m* a where m* is the mass of the equivalent SDOF system given by: m* = ΦT M l =

∑m Φ i

(3.35)

i

The force-displacement characteristics F* and d* can be determined from the MDOF capacity curve, by employing the equations below: F* =

Fb Γ

(3.36)

d* =

dn Γ

(3.37)

Fb =

∑ F = ΦT M i

(3.38)

l p = p m*

where Fb is the base shear force of the MDOF model, and dn is the displacement of the control node. The constant Γ represents the modal participation factor of the

42


normal mode in the direction of application of the lateral force. In other words a transformation factor for the MDOF system to the SDOF system:

Γ=

m* ∑ m i Φ i2

(3.39)

Elastic-Perfectly Plastic Idealization of the Capacity Curve The capacity curve of a MDOF system shows the nonlinear relationship between the lateral displacement and the base shear force of the structure. The displacement is usually taken at a certain node (control node) at the roof level of the structure. The equivalent SDOF curve is essential in order to determine the seismic demand. Figure 3-24 shows an elastic-perfectly plastic idealization of a SDOF system capacity curve.

Plastic mechanism

Figure 3-24 Elastic-perfectly plastic idealization of an equivalent SDOF system [6] The yield force Fy* of the elastic-perfectly plastic curve in figure 3-24 is taken as the ultimate strength of the SDOF system which is equal to the value of F* at the formation of the complete plastic mechanism. In order to determine the elastic stiffness of the elastic-perfectly plastic curve, it is assumed that the area under the SDOF curve and the idealized curve up to formation of the plastic mechanism are equal [6]. This gives the value of the yield displacement of the elastic-perfectly plastic SDOF curve:

43


⎛ E* ⎞ d *y = 2⎜ d *m − m* ⎟ ⎜ Fy ⎟⎠ ⎝

(3.40)

where d*m is the displacement of the equivalent SDOF system at the formation of the plastic mechanism and E *m is the deformation energy under the SDOF capacity curve up to that point. The vibration period of the equivalent SDOF system is:

T = 2π *

m * d *y

(3.41)

Fy*

If the structure remains elastic up to the yield point Fy* , the vibration period is identical to the one obtained in the RSA of the structure, e.g. in an elastic modal analysis or from the Rayleigh quotient given by EC8 [6]. Target Displacement and Seismic Demand The structural response quantities to a given seismic load can not be applied directly from the capacity curve of the pushover analysis. The target displacement must be estimated as the displacement demands for the corresponding equivalent SDOF system transformed to the SDOF domain using a shape vector and the reference SDOF displacement d*. According to EC8, the target displacement of the equivalent SDOF structure is determined from the 5% damped elastic response spectrum. The reduction factor Rμ introduced above may also be represented as the ratio between the acceleration in the structure with unlimited elastic behaviour Se(T*) and in the structure with limited strength F*y/m*: Rμ =

S e (T * ) Fy* / m *

(3.42)

If the elastic period T* is equal or larger than the corner period Tc, the equal displacement rule applies (the displacement of the inelastic system is equal to the displacement of the corresponding elastic system with the same vibration period) and the target displacement of the SDOF system is equal to: 44


⎛ T* ⎞ d = d = Sae (T )⎜⎜ ⎟⎟ ⎝ 2π ⎠ * t

* et

2

T* ≥ Tc

*

(3.43)

If the period T* is less than Tc, the target displacement of the SDOF is modified on the basis of the q-μ-T relationship [6, 42, 45]. The ductility demand can now be computed by rearrange equation (3.24). μ = ( R μ − 1)

Tc +1 T*

T* < Tc

(3.44)

If F*y/m* ≥ Se(T*), the response is elastic and thus: T* < Tc

d *t = d *et

(3.45)

If F*y/m* < Se(T*), the response is nonlinear and thus:

d *t =

d *et ⎛ T ⎞ ⎜ 1 + R μ − 1 c* ⎟ ≥ d *et Rμ ⎝ T ⎠

(

)

T* < Tc

(3.46)

The elastic response spectra and demand spectra can now be plotted as a spectral acceleration against spectral displacement, where the vibration period T* is represented by radial lines. Figure 3-25 shows the determination of the target displacement of an equal SDOF system with (a) vibration period T* longer than Tc and (b) period shorter than Tc.

45


Figure 3-25 Determination of the target displacement for of an equivalent SDOF system [6] The expected target displacement of the MDOF system can be estimated by multiplying the displacement demand for the SDOF model with the transformation factor:

D t = d *tΓ

(3.47)

The target displacement Dt represents a mean value for the applied seismic load. Now the pushover analysis is employed again, the structure is pushed to the target displacement, and all relevant quantities of the structure are obtained, for example local demands in terms of joint rotations and storey drifts. Distribution of deformation of the structure from the pushover analysis should after this, correspond to that which would be obtained in a dynamic analysis [39]. Limitation of Pushover analysis It is crucial to consider the limitation of the pushover analysis. Some important issues are given below:

46


•

The pushover analysis is static and cannot predict the dynamic behaviour of the structure with large accuracy.

•

The pushover analysis could underestimate affects of modes that may occur in a structure subjected to severe seismic events and exaggerate others. This applies in case of higher modes i.e. in tall buildings. Hence, the pushover analysis becomes inaccurate if higher mode effects are important. However pushover analysis procedure where effects of higher modes are considered has been available for the last few years [47].

•

The load pattern affects the results dramatically. Each load pattern is likely to favour certain deformation mode. Therefore more than one load pattern should always be considered in the pushover analysis [35].

•

Incorporation of torsion effects due to mass, stiffness and strength irregularities could affect the results and also 3-D problems like orthogonality effects, direction of loading and semi rigid diaphragms.

47

Chapter 4 Case Study

4 4.1

CASE STUDY Introduction

Hella is a small town in South Iceland, about 100 km east of Reykjavík. Hella is relatively close to the epicentre of the June 2000 earthquakes and several buildings suffered damage there. Even though most of the damaged buildings were old concrete and masonry (pumice) shear wall buildings with poor, little or no reinforcement, some more recent constructions were damaged. The case study is an analysis of one such building, where the focus is set on the pushover analysis, which is combined to EC8 through the N2 method, in order to obtain the seismic response and the performance of the building under the design seismic load. Both RSA and pushover analysis of the structure are carried out. In the RSA analysis, the whole structure is modelled in a 3D finite element program, in order to obtain the fundamental mode shape, vibration period and the base shear. The RSA is followed by a nonlinear pushover analysis of the back side wall of the structure, by using a 3D finite element model in ANSYS. Nonlinear properties of the reinforced concrete are considered in order to predict the response of structure to a static lateral load, in a more realistic way. The building is a three storey, cast in place, reinforced concrete structure, constructed in 1975. Health care, drugstore and offices are located in the building. The first seismic design code was issued in 1976 in Iceland. Hence the structure was not designed according to a seismic code [48].

48


Figure 4-1 Front side and N-W-gable of the analyzed structure [4]

Figure 4-2 Back side and S-E-gable of the analyzed structure [4] 4.1.1 Structural System The structural system consists of in-situ reinforced concrete walls, columns and slabs. The plan geometry is rather simple. The concrete roof slabs and the floors are carried by the floor columns (400 mm x 400 mm) and the structural walls. In the west corner of the building is a stairwell, made of structural walls, between the basement floor and the roof slab. The floor plan is about 14,5 m wide and 25 m long (figure 4-4). The basement walls have no openings in the S-W-side and in the gables, but there are windows and doors openings in the back side of the basement. The gables 49


have few openings and may be assumed to be relatively stiff with a shear wall action, but the long sides have large openings and therefore less stiffness and lateral load resistance (figure 4-3). The lateral load resistance in the longitudinal direction is provided by frame action of the column-slab and column-deep-beam (column-wall) system. Thin steel sheets supported by a wooden substructure make the roof [48].

Front side (S-W)

S-E-gable

Back side (N-E)

N-W-gable

Figure 4-3 Side views of the structure [4]

50


Figure 4-4 Plan layout of the basement [4]

4.1.2 Material Properties The material properties are derived from the original drawings. Only one drawing of the reinforcement exist today, including description of the specified properties of the concrete and the reinforcement steel. Walls and slabs are made of K200 concrete. According to EC2, the characteristic compression strength is comparable to C16 concrete with fck = 16 MPa and mean compression strength fcm = 16 + 8 = 24 MPa. Columns are made of K250 concrete, comparable to C20 with fck = 20 MPa and mean compression strength fcm = 20 + 8 = 28 MPa. According to EC2, with equation (3.2) multiplied with 0,9, the characteristic modulus of elasticity is 24.750 MPa for the S200 concrete and 26.100 MPa for the S250 concrete. Poisson ratio is assumed to be 0,2. The structure is reinforced with two types of steel. Reinforcement in walls and longitudinal reinforcement in columns is made of S400 steel but hoops in main columns are made of St37 steel. The S400 steel is ribbed with 400 MPa yield strength and the St37 steel has 235 MPa yield strength. The ultimate strain for both the S400 and the St37 steel is assumed to be 1,5%.

51


4.1.3 Reinforcement Layout Main wall columns in the front and back side walls are 200 mm x 430 mm with eight φ16 mm longitudinal bars and confined with φ8 hoops spaced 200 mm. The minor columns are 200 mm x 180 mm and have four φ10 mm longitudinal bars, but no hoops i.e. with out confinement reinforcement. Structural walls have φ8 mm bars spaced 250 mm, in both horizontal and vertical directions. Two φ10mm bars are placed around windows and doors. The main slab columns are 400 mm x 400 mm with eight φ16 mm longitudinal bars and confined with φ8 hoops spaced 250 mm. The roof slab is 180 mm thick and floor slabs are 200 mm thick. Table 4-1 Geometry of structural elements Parameter Wall thickness 1st and 2nd floor slab thickness Roof slab thickness Main columns in front and back sides Minor columns in front and back sides Main floor columns

Value 180 mm 200 mm 180 mm 200 mm x 430 mm 200 mm x 180 mm 400 mm x 400 mm

4.1.4 Structural Damage Location of the main damage is shown on figure 4-5. Most of the structural damage occurred in the basement. Three out of five columns in the basement failed in shear, where diagonal cracks occurred (figure 4-6). Shear cracks occurred also in the stairwell walls. The following damages were reported by the Icelandic Catastrophic Insurance [48]: 1. Basement, shear failure of main wall columns (200 mm x 430 mm) no. 3, 4, and 5 (see Figure 4-6). Shear failure of all minor columns (180 mm x 200 mm). 2. Shear cracks in longitudinal back wall. 3. Cracks in basement slabs and permanent offset close to columns. 4. Cracks in the 2nd floor slab spreading from the corners of the building. 5. Shear cracks in stairwell walls, see figure 4-7. 52


6. Cracks in unreinforced part of columns at ground level on the SW longitudinal wall.

Figure 4-5 Location of the main structural damage in the June 2000 earthquakes [48] Main columns are covered with insulation and cement mortar on the inside of the structure. The cover was only removed from the columns in the basement, hence damage of columns on the 1st and 2nd floor is not known with good accuracy at this moment. However the columns are visible from the outside, hence, any major damage in the 1st and 2nd floor columns should have been detected.

53


Shear crack

Figure 4-6 Shear Crack in back-side basement column [48]

Shear crack

Figure 4-7 Shear crack in stairwell wall in the basement [48]

54


4.2

Previous Study of the Building

In 2003, F. Sigtryggsdóttir [48] studied the structure. The work was carried out for the Iceland Catastrophic Insurance. A RSA was carried out, using a finite element program. The main purpose of the study was to compare the analytical results of a conventional seismic design process, using EC8, to the actual reported damage of the structure. Some simplified methods were used to take nonlinearities into account e.g. by removing structural elements from the finite element model if exceeding their ultimate capacity and replace fixed supports with springs. The aim of the study was also to propose structural remedial measures to the building. The seismic response of the structure was both computed with design response spectrum analysis according to EC8 and by using a time history analysis. The design response spectrum was computed with ground acceleration ag = 0,5g and structural behaviour factor q = 1,5 and subsoil class A. A time history analysis was carried out with a 50 Hz sample of the ground acceleration, measured in Hella in June 17th. The first 70 eigenfrequencies of the structure were computed to accumulate at least mass participation factor (MPF) of 90% which is required for the RSA according to EC8. The first vibration period was in the longitudinal direction, T = 0,155 sec. with MPF = 63%. The second vibration period was in the transverse direction, T = 0,09 sec. and MPF = 62%. The total base shear obtained, was 4.928 kN and 4.800 kN for the first and second vibration periods, respectively. According to Sigtryggsdóttir analysis, member forces exceeded their estimated capacity in several places. Shear forces in main columns and minor columns in the basement back side, exceeded their estimated shear capacity. Furthermore structural walls in the stairwell were subjected to high shear stress, likely to produce shear cracks. For those elements, the analytical results were quite consistent with the actual damage recorded. However, the analytical results indicated more damage in other areas than the actual damage recorded, for example on first and second floor. This should not be surprising since the RSA is elastic and nonlinear behaviour of the structure is only taken in to account trough the behaviour factor q, which estimates the ductility and the energy dissipation of the structure according to guidelines in EC8 [48]. For details of the q factor, see EC8 clause 5.2.2.2.

55


F. Sigtryggsdóttir proposed strengthening of the structure, mainly by closing windows with concrete infill, in order to obtain more lateral stiffness and to redistribute member forces more evenly. Strengthening proposals given by Sigtryggsdóttir are discussed further in section 5.3.

4.3

Response Spectrum Analysis of the Building

In this section a dynamic elastic response spectrum analysis (RSA) of the structure is carried out with the finite element program Sap2000. Results of the RSA are the eigenfrequencies (vibration periods) and associate mode shapes of the structure (see section 3.5). Mass participation of each mode shape for both Xand Y-directions is computed. Torsion effects of the structure may be obtained by the dynamic analysis, through the mode shapes. The main purpose of the RSA in this study is to compute the base shear of the structure (especially of the back-side wall) and the fundamental mode shape of the structure in the longitudinal direction, by using the RSA design process in EC8. It is also important to find out if the structure is vulnerable to torsion effects or if higher modes are dominating in the response. 4.3.1 3D finite Element Model The structure is modelled in three dimensions with the X-, Y- and Z-axes representing the longitudinal, transverse and the vertical direction, respectively. The model consists of 11.888 nodes, 331 beam elements and 11.588 shell elements. The floor diaphragms and the walls are modelled with four-node shell element and the columns are modelled with frame elements. The model is constrained in X-, Y-, and Z-directions at the supports.

56


Figure 4-8 Finite element model of the structure in Sap2000 4.3.2 Elastic Model Properties The table below shows the material properties used in the RSA. Table 4-2 Model properties for the RSA in Sap2000 Constant E1 E2 ν γc ag

Description Elastic modulus of concrete walls Elastic modulus of concrete columns Poisson’s ratio of concrete Weight of reinforced concrete Design ground acceleration

Value 24.750 MPa 26.100 MPa 0,2 25 kN/m3 0,4g

4.3.3 Structural Load The horizontal peak ground acceleration is γIagR = 0,4g, determined from the National Application Document to EC8, with γI = 1,0 for importance class II (clause 4.2.5). The response spectrum for the RSA, for subsoil class A, is shown on the figure below.

57


1,2 1,0

Se/g

0,8 0,6 0,4 0,2 0,0 0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

T [sec]

Figure 4-9 Response spectra for the RSA of the structure According to EC8, the modal mass consists of the dead load and 30% of the imposed load of the structure. The dead load on 1st and 2nd floor, consist of concrete and a 40 mm layer of cement mortar. Weight of the wooden roof structure is estimated as 0,5 kN/m2. The table below shows dead and imposed load on the structure. Table 4-3 Dead and imposed load in the RSA in Sap2000 [kN/m2] Floor 1st floor 2nd floor Roof

Dead load 5,0 5,0 5,0

Imposed load 1,5 (30%) 1,5 (30%) 0,0

Total 6,5 6,5 5,0

The following three load cases were computed: Table 4-4 Load cases in the RSA Description Load case Dead and imposed load (as used for the modal mass analysis) G Seismic load in longitudinal direction of the structure EX Seismic load in the transverse direction of the structure EY

58


In order to obtain the seismic base shear force in the longitudinal direction (Xdirection) according EC8, clause 4.3.3.5.1, the load cases are combined as follows: (4.1)

G “+” EX “+” 0,3 EY

4.3.4 Results of the RSA The main results from the RSA are similar to what were obtained by Sigtryggsdóttir F. (see section 4.2). 57 and 56 modes must be computed to cumulate 90% mass participation in the X- and Y-direction, respectively. The fundamental mode has a vibration period of 0,153 sec, in the X-direction with 65% mass participation. The second mode has a vibration period of 0,09 sec, in the Y-direction, with 64% mass participation.

Figure 4-10 Mode shape 1 with 0,153 sec vibration period and 65% mass participation Modes 3 to 20 mainly consist of vertical deformation of the floor slabs. Torsion occurs in mode shape 21 and 22, but with insignificant mass participation. Only mode shapes 1, 51 and 56 of the total 70 modes computed have more than 2% modal mass participation in the X-direction. Mode shapes 51 and 56 have about 9% mass participation and the floor slabs moves in opposite directions (figure 4-11). From the above, it can be concluded that torsion effects and higher modes do not contribute significantly in the seismic structural response. This fact is of great importance for the following pushover analysis (see section 3.6.1).

59


Figure 4-11 Mode shape No. 51 with 0,038 sec vibration period and 9% mass participation. Total base shear of the structure in the X-direction for the first 57 modes, combined with the CQC method, is 6.650 kN. The base shear of the back side wall of the structure, which will be analysed further in next section, is 905 kN which is about 14% of the total base shear. The modal load pattern for the pushover analysis is based on the horizontal displacement of the first (fundamental) mode of the structure (figure 4-10). Table 4-5 Horizontal displacements of the structure in the fundamental mode in the RSA Floor Roof 2nd 1st

Displacement 7,8 mm 5,6 mm 1,1 mm

60


4.4

Pushover Analysis of the Back Side Wall

As mentioned in section 4.1.4, most serious damage occurred in the columns in the back-side of the structure. Therefore, the back side wall is chosen for the pushover analysis. 4.4.1 3D Model of the Back Side Wall The wall is modelled in three dimensions where the X-axis represents the longitudinal direction of the wall and the direction of the lateral load and Y-axis represents the vertical direction.

Minor columns 200x180 mm Main columns 200x430 mm

5 4 3 2 1

Figure 4-12 Model of the back side wall in ANSYS The model consists of 52.889 nodes and meshed into 32.780 Solid65 elements. It is important to select sufficient density of the mesh in order to obtain a convergence of results as discussed in section 3.3.4. Size of elements in main and 61


minor columns is not only chosen on the basis of convergence, but also with location of the reinforcement bars in mind, in such way that the moment capacity of the columns can be predicted (figure 4-13).

100x100x90 mm

67x67x90 mm 67x67x125 mm

Figure 4-13 Mesh of the Solid65 concrete elements 4.4.2 Model Input Parameters Generally, the proposed procedure is to use the most probable values for material strengths (the mean concrete and steel strengths, fcm and fym) when estimating inelastic deformations and corresponding drifts, and then introduce appropriate safety factors in defining the permissible values of these deformations and drifts. For member design against flexure and shear, the standard code format is retained, and the EC8 and EC2 material safety factors would be used [49]. The ANSYS finite element model in this study consists of four material models. Model 1 is S200 (C16) concrete, for minor columns and walls, model 2 is for S400 steel reinforcement, model 3 is for S250 (C20) concrete in main columns and model 4 is for St37 reinforcement hoops in main columns. Properties of the four material models are given in the tables below.

62


Table 4-6 Input parameters for S200 concrete (material model 1) No. 1 2 3 4 5 6 7 8 9 10 11

Description Shear transfer coefficient for an open crack (βt) Shear transfer coefficient for a closed crack (βc) Uniaxial tensile cracking strength (ft) Uniaxial ultimate compression strength (fcm) Yield strength of concrete Stiffness multiplier for cracked tensile conditions (Tc) Modulus of elasticity for concrete (Ec) Secant modulus of plasticity for concrete (ET) Poisson ratio for concrete (νc) Ultimate concrete strain (εc) Weight of concrete (γc)

S200 0,2 1,0 1,9 MPa 24 MPa 19,2 MPa 0,6 24.750 MPa 1.762 MPa 0,2 0,35% 25 kN/m3

Table 4-7 Input parameters for S250 concrete (material model 3) No. 1 2 3 4 5 6 7 8 9 10 11

Description Shear transfer coefficient for an open crack (βt) Shear transfer coefficient for a closed crack (βc) Uniaxial tensile cracking strength (ft) Uniaxial ultimate compression strength (fcm) Yield strength of concrete Stiffness multiplier for cracked tensile conditions (Tc) Modulus of elasticity for concrete (Ec) Secant modulus of plasticity for concrete (ET) Poisson ratio for concrete (νc) Ultimate concrete strain (εc) Weight of concrete (γc)

S250 0,2 1,0 2,2 MPa 28 MPa 22,4 MPa 0,6 26.100 MPa 2.203 MPa 0,2 0,35% 25 kN/m3

As discussed in section 3.3.5, confinement reinforcement in columns is essential in order to gain high ductility. Maximum strain of properly confined concrete is significantly larger than that of unconfined concrete. The ultimate strain of the concrete in the main columns was computed in order to estimate ultimate strain input parameter εc in ANSYS. A method introduced by T. Paulay and M. Priestley [13] was used. It was found out, that the confinement reinforcement of the main columns is not sufficient in order to increase the ultimate concrete strain. Hence, the ultimate concrete strain is assumed εc = 0,35%.

63


Ratio and orientation of the reinforcement is implemented as real constants sets. Five sets of real constants are used to implement the reinforcement layout, where values can be entered for material model number, volume ratio, and orientation angles. The material number refers to the type of material model for the reinforcement. The volume ratio refers to the ratio of steel to concrete in the element and the orientation angles refer to the orientation of the reinforcement in the smeared model. Table 4-8 Input parameters for S400 and St37 steel (material models 2 and 4) No. 1 2 3 4 5 6

Description Yield stress of reinforcement steel (fy) Modulus of elasticity for steel (Es) Secant modulus of plasticity for steel (EP) Poisson ratio of steel (νs) Ultimate steel strain (εs) Weight of steel (γs)

S400/St37 400/235 MPa 200.000 MPa 1.035 MPa 0,3 1,5% 77 kN/m3

4.4.3 Loads and Boundary Conditions Two load patterns are used for the pushover analysis (see section 3.6.2), uniform load pattern and modal load pattern. Table 4-9 shows the normalized mode shape vectors and the force vectors for both the uniform and the modal load patterns. Table 4-9 Lateral load patterns according the N2 method Floor

Roof 2nd 1st

mi kg

Φiu uniform

27.523 35.780 35.750

1,0 1,0 1,0

Piu Uniform kN 0,769 1,000 1,000

Φim modal 1,000 0,781 0,141

Pim modal kN 1,000 0,933 0,183

The lateral load on each floor is evenly distributed (along the X-direction), attached to the element nodes. The model is pushed according to the force vector, up to about 1.200 kN in base shear, which is about 135% of the base shear obtained in the RSA in section 4.3. The lateral load is increased stepwise in 50 load steps, which is about 24 kN in each step. Initially the load was applied in 200 steps, but due to computational run time, it was decided to reduce the number of load steps down to 50. Comparison 64


of analysis with 200 load steps and 50 loads steps, showed insignificant difference in the load-displacement relationship. Displacement boundary conditions are used to constrain the model at the supports. All nodes at the base are constrained in the X-, Y-, and Z-direction, i.e. the displacement UX, UY and UZ is set to zero. The model is also constrained in the Z-direction, at the roof, to prevent unrealistic deformations.

4.5

Results of the Pushover Analysis

4.5.1 Force-Displacement Relationship The key outcome of the pushover analysis is the force-displacement relationship. Figure 4-14 shows the base shear as a function of the roof-displacement. The model behaves completely elastically up to 190 kN for the uniform load and up to 163 kN for the modal load. Uniform

modal

Linear elastic

1400 1200

Base shear [kN]

1000 800 600 400 200 0 0

10

20

30

40

50

60

Roof-displacement [mm]

Figure 4-14 Pushover curves of the wall for uniform and modal load patterns (MDOF system)

65


If the elastic part of the pushover curve for the modal load pattern is extended with a trend line, the roof-displacement at 905 kN base shear is about 7,5 mm. Table 4-5 shows lateral displacement from the RSA where the roof-displacement is 7,8 mm at 905 kN base shear. Hence a good correlation is in the loaddisplacement relationship between ANSYS and Sap2000 in the elastic range. Figure 4-14 shows that the relative roof-displacement is significantly larger for the modal load pattern. The roof-displacement for the modal load is almost two times larger than that for the uniform load at the terminal point. This can mainly be explained by the fact that the lateral force of the modal load pattern is applied to the structure at relatively more height (table 4-9).

4.5.2 N2 Method and the Target Displacement The end point of the pushover curve usually needs an engineering judgement. According to the N2 method, the structure should be pushed until a complete plastic mechanism has formed in the structure. However it is not essential to identify this point with high accuracy since it is only used for the estimation of the elastic stiffness of the equivalent SDOF system as explained in section 3.6.2. If a complete plastic mechanism does not develop between target displacement and the end point (terminal point) the SDOF capacity curve may be determined on the basis of the latter point [6].

Figure 4-15 Ultimate compression strain in minor column in the basement

66


In this study, the terminal point is defined as the point where maximum strain of the concrete or the reinforcement steel is reached (see discussion in [49]). For the uniform load pattern, the ultimate concrete strain (0,35%) is reached in a minor column in the basement at 1.174 kN base shear and 24 mm roof-displacement (figure 4-15). For the modal load pattern, the ultimate strain is reached at the same location at 1.024 kN base shear and 48 mm roof-displacement. The N2 method is based on the assumption that the response of a MDOF structural system can be related to the response of an equivalent SDOF system, as explained in section 3.6.2. Figure 4-16 and figure 4-17 shows the elastic-perfectly plastic idealization of the SDOF system capacity curve for the uniform and the modal load pattern respectively. The yield displacement of the SDOF system for uniform and modal load pattern is 17 mm and 25 mm respectively and the corresponding base shear is 1.174 kN and 821 kN respectively. 1200 1000

F* [kN]

800 600 400 200 0 0

5

10

15

20

25

30

35

40

d* [mm]

Figure 4-16 Elastic-perfectly plastic idealization of the capacity curve (uniform load pattern)

67


1200 1000

F* [kN]

800 600 400 200 0 0

5

10

15

20 d* [mm]

25

30

35

40

Figure 4-17 Elastic-perfectly plastic idealization of the capacity curve for modal load pattern Results of the N2 pushover method are listed in the table below, where the key outcome is the target displacement of the MDOF system obtained by the use of the equations in section 3.6.2. The target displacement is 22 mm for both load patterns.

68


Table 4-10 Results of the pushover analysis Description m* Γ F*y T* Sae Sa Em* d*m d*y Rμ μ d*t Dt Ft

Equivalent mass of SDOF system Participation factor Maximum base shear of the SDOF Period of the SDOF system Seismic coefficient for T* (EC8) Seismic coefficient F*y/m* (g) Deformation energy Displacement of terminal point Yield displacement of SDOF system Reduction factor based on SDOF Ductility demand based on SDOF Target displacement of the SDOF Target displacement of the MDOF Base shear at target displacement

Uniform load 139.898 kg 1,0 1.174 kN 0,29 sec 1,00 0,86 18.015 Nm 24 mm 17 mm 1,17 1,24 22 mm 22 mm 1.101 kN

Modal load 58.257 kg 1,25 821 kN 0,27 sec 1,00 1,43 21.495 Nm 39 mm 25 mm Not used Not used 18 mm 22 mm 721 kN

4.5.3 Evaluation of Cracks In accordance with the results, the focus is set on two critical parts of the backside wall i.e. main column in the basement and main column at the 1st floor (Figure 4-18).

Figure 4-18 Cracks at target displacement for the uniform load pattern

69


Figure 4-19 Cracks at target displacement for the modal load pattern According to figure 4-18 and figure 4-19, most of the cracks occur in the columns or in the walls/deep-beams around the columns. The figures also indicate that more damages are obtained at the target displacement for the uniform load. For the uniform load, first, second and third cracks occur in the main column on 1st floor when the base shear is 190 kN, 340 kN and 856 kN respectively. Corresponding base shear at target displacement for uniform load pattern is 1.101 kN. Figure 4-20 shows the evaluation of cracks for the uniform load in the 1st floor column and figure 4-21 shows the evaluation of cracks in the basement column at the same base shear.

Figure 4-20 Formation of first, second and third cracks and cracks at target displacement, in main column on 1st floor at 190 kN, 340 kN, 856 kN and 1.101kN base shear, respectively (uniform load pattern)

70


Figure 4-21 First, second and third cracks and cracks at target displacement in main column in the basement, at 196 kN, 489 kN, 987 kN and 1.101 kN respectively (uniform load pattern) Significant damage at the target displacement for the uniform load pattern is obtained. Cracks in all principal directions occur in most of the elements in the basement column. Similar figures below show evaluation of cracks for the modal load pattern. Figure 4-22 shows similar damage in the 1st floor column for both load patterns but figure 4-23 indicates less damage in the basement column for the modal load pattern, where the third cracks do not occur.

Figure 4-22 First, second and third cracks and cracks at target displacement, in main column on 1st floor at 163 kN, 210 kN, 585 kN and 721kN base shear, respectively (modal load pattern)

71


Figure 4-23 First and second cracks and cracks at target displacement in main columns in the basement, at 210 kN, 512 kN and 721 kN respectively (modal load pattern) 4.5.4 Stress in Steel Reinforcement Figure 4-24 shows elements A and B (column on the 1st floor) where maximum tension and compression stress is obtained, and element C (column in the basement) where maximum stress in the hoops is obtained.

Element A

Element B

Element C

Figure 4-24 Location of elements with maximum steel stress The longitudinal reinforcement (Y-direction) in element A yields, for both uniform and modal load patterns, before the target displacement is reached (figure 4-25). Compression stress in element B is 138 MPa and 95 MPa at the target displacement for uniform load pattern and modal load pattern respectively. The hoops in the basement main column, element C, yields due to shear stress before the target displacement is reached, for the uniform load but the stress is 125 MPa for the modal load pattern.

72


Table 4-11 Stress condition in reinforcement in element A, B and C at target displacement [MPa] Element A (tension) B (compression) C (hoops)

Uniform Yielded 138 Yielded

Modal Yielded 97 155

Yield stress in reinforcement in element A and B is 400 MPa and 235 MPa in element C. Total strain in element A at the target displacement is 0,4% for uniform load pattern and 0,37% for modal load pattern. Total strain in hoops, in element C for uniform load pattern is 0,28%. The ultimate strain capacity of the steel, for both St37 and S400, is assumed 1,5%, hence the reinforcement at target displacement will not fail. Uniform

450

Modal

400 350

Stress [MPa]

300 250 200 150 100 50 0 0

200

400

600

800

1000

Base shear [kN]

Figure 4-25 Tensile stress in element A in the 1st floor column

73

1200


Uniform

180

Modal

160 140

Stress [MPa]

120 100 80 60 40 20 0 0

200

400

600

800

1000

1200

Base shear [kN]

Figure 4-26 Compression stress in element B

Uniform

Modal

250

Stress [MPa]

200

150

100

50

0 0

200

400

600

800

1000

Base shear [kN]

Figure 4-27 Stress in confinement reinforcement in element C

74

1200


Based on evaluation of cracks and stress and strain in the reinforced concrete, discussed above, the wall will resist the seismic design load according the EC8. However, it will not fulfil the code requirements of structural resistance. Reinforcement in the wall is generally below the minimum requirements, for example, no confinement reinforcement exists in minor columns and reinforcement ratio in walls/deep-beams is far too low. At the ultimate limit state of EC8, the design values (fcd, fyd) for strength of materials are used in the structural member design. In EC2 and EC8 the material safety factors for concrete and steel are γc = 1,5 and γs = 1,15 at ultimate limit state, respectively. Section forces and capacity of element sections are not discussed in this study, but for example, the design shear capacity of the main columns, according to EC2 (clause 4.3.2) , is about 54 kN whereas the shear force in the basement main columns is at least 70 kN.

75

Chapter 5 Discussion

5 5.1

DISCUSSION Level of Ductility

As discussed in section 3.3.5, ductility is an essential property of reinforced concrete. The ductility demand for the SDOF system at the target displacement is given in table 4-10. For the uniform load pattern the demand is μ = 1,24 but for the modal load pattern, ductility above μ = 1,0 is not required. Using the yield displacement and the formation of the plastic mechanism of the SDOF system, the maximum ductility μd = d*m/d*y of the wall may be estimated, based on the elasticperfectly plastic behaviour (see also figure 3-15 and figure 3-16). 1400 1200

F* [kN]

1000 800 600 400 200 0 0

10

d* y

20 d* [mm]

d* m

30

40

Figure 5-1 Definition of the maximum ductility for the uniform load pattern For the uniform load and the modal load, μd = 1,41 and μd = 1,56 is obtained, respectively. This indicates low ductile behaviour of the wall which in general means vulnerability or fragility in case of excessive ground motion [27]. It should be pointed out that the terminal point of the pushover curves is based on an engineering judgement, which could affect the results somewhat.

76


5.2

Comparison of Reported and Computed Damage

In order to compare the observed damage in the N2 pushover analysis to the reported damage, the actual seismic load on the structure in the June 2000 earthquake must be estimated. Figure 5-2 shows the 5% damped elastic response spectra for the June 17th earthquake in E-W direction and the smooth response spectra from EC8, for peak ground acceleration equal to 0,4g. Since the seismic load in the N2 method is based on the elastic response spectra, the elastic response spectra from Hella are the most appropriate ones to use. At the fundamental elastic vibration period (0,153 sec), the seismic coefficient of the June 17th earthquake is about 65% of the seismic coefficient obtained from the EC8 spectra. Similar, in the E-W direction, the seismic coefficient is about 50% of the one obtained from EC8 [2]. Response spectra, EC8

Response spectra, Hella 2000

1,8 1,6 1,4 1,2

Se/g

1,0 0,8 0,6 0,4 0,2 0,0 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

T [sec]

Figure 5-2 Elastic response spectra from EC8 and Hella 2000 in June 17th in E-W direction with 5% damping ratio [2] The positive X-direction of the analysed wall is the N-W direction. Hence neither of the seismic coefficients (65% and 50%) directly reflects the seismic response of the wall. In general, the two horizontal components should be considered as acting simultaneously. 77


New RSA analysis is carried out, where the direction of the structure and the response spectra for EW and NS direction, recorded in June 17th in Hella, are taken into account. The base shear in the longitudinal direction of the wall is reevaluated and compared to the results from the previous RSA. The base shear in the longitudinal direction of the wall in the RSA now becomes 501 kN (56% of the previous computed base shear). Hence, it may be assumed that the actual seismic load is significantly lower than obtained by the EC8 smooth spectra. Since the cement mortar cover was only removed from the basement columns but not from the 1st and 2nd floor, it is appropriate to compare obtained calculated damage to the actual damage reported there. Figure 4-27 shows that stress in hoops in the main basement column (element C), increases rapidly between 600 kN and 800 kN in base shear, indicating that the diagonal cracks occur during that interval. This holds true for both uniform and modal load patterns. The base shear at the formation of the diagonal shear crack is 612 kN and 652 kN for the uniform and the modal load respectively (Figure 5-3).

Figure 5-3 Formation of the diagonal shear crack in the model compared to the actual damage (uniform load pattern to the left) The pushover analysis is carried out again, where the seismic coefficient is reduced to Sae = 0,56g to represent the assumed seismic load in June 17th in Hella. The table below shows the obtained target displacement and corresponding base shear of the MDOF system.

78


Table 5-1 Target displacement and base shear for the assumed seismic load on the structure in June 17th Load pattern Uniform Modal

Sae 0,56 0,56

Dt 14 mm 12 mm

Base shear 783 kN 509 kN

According to table 5-1, higher base shear is obtained in the analysis for the uniform load pattern, than required to form the basement column diagonal shear crack (figure 5-3). On the other hand, the modal load pattern yields lower base shear than required to form the basement column shear crack, which indicates underestimation of actual damage in the basement columns at the assumed seismic load in June 17th in Hella. It should be pointed out that material properties are based on information from the original drawings of the structure. Hence, the material strength is not known with high accuracy. Figure 5-4 shows cracks in the 1st floor column at the formation of the shear crack in the basement (at the same displacement and base shear). Significant damages are obtained, especially for the modal load pattern. As mentioned in section 4.1.4, the cement mortar and the insulation were not removed from the inside of the 1st and 2nd floor columns. However, any significant damage should have been visible from the outside.

Figure 5-4 Cracks in 1st floor columns at the formation of shear crack in the basement column (uniform load pattern to the left)

79


Comparison of figure 5-3 and figure 5-4 shows that the minimum base shear (or displacement) which is required to form the diagonal shear crack in the basement columns (652 kN) for the modal load pattern, yields in significant damage of the 1st floor column. This indicates that the modal load pattern exaggerates the damage in the 1st floor column compared to the actual damage reported. Similar comparison for the uniform load pattern shows less damage in the 1st floor column at the minimum required base shear forming the crack in the basement column.

5.3

Retrofit Proposals

Previous discussion about damage, ductility and especially code requirements emphasizes the importance of improving the earthquake resistance of the structure in order to enhance its performance in future earthquakes. F. Sigtryggsdóttir [48] proposed structural remedial measures to the building in her study.

Figure 5-5 Retrofit proposal of the back side wall [48] The aim of the strengthening was to generally increase stiffness and reduce member forces in main columns. Closing of windows with concrete infill and in cast concrete shear wall/column (250 mm x 1200 mm) was recommended, with vertical and horizontal reinforcement about 1,47% and 0,27% respectively. The wall/column is located in the east corner, in order to increase plan symmetry. The utility of the building is obviously reduced by the concrete infill of the windows but it was considered acceptable. Based on F. Sigtryggsdóttir study, new model of the wall is constructed where geometry and proposed reinforcement layout is considered. It should be pointed out that the proposed strengthening is not followed in details. Same material properties are used as for the S250 concrete and S400 steel, above. 80


Shear wall/column 250 x 1200 x 9400 mm

Window infill

Figure 5-6 Model of the proposed strengthening in ANSYS The pushover curves below, for the retrofitted wall, shows that the stiffness has increased significantly. For the uniform load, the curve is cut off at base shear equal to 1.535 kN when ultimate strain occurs in horizontal steel reinforcement in the upper left corner of 1st floor window (figure 5-10). For the modal load pattern, the curve is cut off at base shear equal to 1.075 kN due to the same reason. Hence the pushover curves are now cut off due to strain in the left upper corner of the 1st floor window, instead of concrete strain in minor columns in the basement. The figures below shows the pushover curves and the corresponding elastic-perfectly plastic idealization of the capacity curve for the proposed retrofit.

81


Retro-Uniform

1600

Retro-Modal

1400

Base shear [kN]

1200 1000 800 600 400 200 0 0

5

10

15

20

25

30

35

Dislpacement [mm]

Figure 5-7 Pushover curves for the proposed strengthening of the wall

1800 1600 1400

F* [kN]

1200 1000 800 600 400 200 0 0

5

10

15 d* [mm]

20

25

30

Figure 5-8 Elastic-perfectly plastic idealization of the capacity curve for the strengthening proposal with, uniform load pattern

82


1000 900 800

F* [kN]

700 600 500 400 300 200 100 0 0

5

10

15

20

25

d* [mm]

Figure 5-9 Elastic-perfectly plastic idealization of the capacity curve for the strengthening proposal, with modal load pattern The results of the pushover analysis are listed in the table below. The target displacement for uniform and modal load pattern is now 15 mm and 13 mm for the uniform and modal load pattern respectively. Table 5-2 Results of the pushover analysis for the proposed retrofit Description Equivalent mass of SDOF system Participation factor Maximum base shear of the SDOF system Deformation energy Displacement of terminal point Yield displacement of SDOF system Period of the SDOF system Reduction factor based on the ductility Target displacement of the SDOF system Target displacement of the MDOF system

83

m* Γ F*y Em* d*m d*y T* Rμ d*t Dt

Uniform load 139.898 kg 1,0 1.535 kN 27.703 Nm 27 mm 18 mm 0,25 sec Not used 15 mm 15 mm

Modal load 58.257 kg 1,25 861 kN 13.763 Nm 24 mm 15 mm 0,20 sec Not used 10 mm 13 mm


The most important effect of the retrofit is obviously that no damage occurs in the basement columns and significantly less damage occurs in the 1st floor columns. Stress in vertical reinforcement in the 1st floor columns is now below yielding, 320 MPa and 270 MPa for the uniform and modal load pattern respectively. On the other hand flexural tension cracks occur in the middle of the strengthening column (area I on figure 5-10) and in the upper left corner in the 1st floor window (area II).

II

I

Figure 5-10 Cracks at target displacement for the proposed retrofit (uniform load pattern)

Figure 5-11 Cracks at target displacement for the proposed retrofit (modal load pattern) Furthermore, vertical reinforcement in area I has yielded where the strain is 0,52% and 0,4% for the uniform and modal load patterns respectively. Horizontal reinforcement in the upper left corner of the 1st floor window has also yielded, where the strain is 1,0% and 0,8% respectively.

84


Despite the cracks and post yielding of the reinforcement around the 1st floor window, the performance of the structure at the design earthquake load is generally much better than before the strengthening. This is mainly due to the fact that the columns generally suffer significantly less damage.

Figure 5-12 Cracks in 1st floor columns at target displacement (uniform load pattern to the left) It is though clear that more stiffness comes at the cost of the ductility and the energy is dissipated through higher base shear and less displacement.

85

Chapter 6 Conclusions

6 6.1

CONCLUSIONS Final Conclusions

In this study, the seismic resistance of a three storey reinforced concrete building built in a seismic area, but designed without seismic provisions has been estimated. The evaluation of the seismic resistance has been addressed by means of a 3D RSA followed by a pushover analysis of the back side wall of the structure. The 3D RSA was carried out in order to obtain the fundamental vibration period, the first mode shape and the base shear. The fundamental elastic vibration period of the structure is 0,153 sec. and the maximum elastic roof-displacement is 7,8 mm where the first 70 mode shapes are combined with the CQC method. Total base shear of the structure is 6.650 kN and thereof 905 kN in the back side wall. Before the pushover analysis was carried out, the ability of the computer programme ANSYS to predict the nonlinear behaviour of reinforced concrete was tested against experimental data form a classical laboratory beam test. One of the tested beams was modelled in ANSYS and good correlation of the loaddisplacement relationship and evaluation of cracks was obtained between the model and the experimental data. A detailed finite element model of the building’s back side wall was constructed where the nonlinear behaviour of the reinforced concrete was considered. Size and configuration of elements was chosen in such way that good prediction of moment and shear capacity of structural elements was secured. Convergence of the nonlinear results was also considered. A pushover analysis according to the N2 method was carried out. Both a modal load pattern, corresponding to the fundamental load shape and a uniform load pattern were used. The analysis indicated that the backside wall is able to withstand the design seismic load according to EC8. However the damage will be significant and the performance of the wall does not meet the code requirements of resistance of the structural elements at the ultimate limit state. 86


When the structure is pushed to the target displacement Dt = 22 mm, serious damage occurs. Diagonal shear cracks occurs in the basement columns and both flexural and shear cracks occurs in the 1st floor columns. Significantly fewer damages are obtained in the 2nd floor columns. Damages are not considered to be critical in the deep beams (walls between columns) despite of the cracks there. The performance of the backside wall was re-evaluated for a proposed retrofit. The structural stiffness was increased and the target displacement was lowered. Generally the structural retrofit improved the seismic resistance of the structure and was considered to prevent the risk of structural collapse under the design load with much more confidence.

6.2

Prognosis

This study shows how the pushover analysis may be used in order to estimate the seismic resistance of existing structures. The study shows also how the linear analysis may be followed by a detailed nonlinear analysis of part of the structure. One of the most important advantages of nonlinear analysis beyond the RSA is the opportunity to evaluate damage. Hence, the pushover analysis can give valuable information about performance of building in the future seismic events. The pushover analysis is not confined to the ultimate limit state given by the seismic design codes. It may be used to estimate the performance according to different criteria, for example in significantly important structures like power plants or hospitals, where the utility must not be significantly reduced in a seismic event.

6.3

Further Research

A 3D pushover analysis or a 3D nonlinear dynamic analysis of the whole structure in both the directions would cover the post yielding force distribution better and also include the possible effect of rotation. However, the RSA is supposed to give a reasonable idea of this behaviour. In this case the RSA indicated that rotation does not affect the result of the analyzed wall significantly. Further more, a 3D model of the whole structure would be substantially cumbersome and time consuming. 87


The N2 method is relatively new and has not been widely used in the practical field, compared to the RSA. Furthermore, the formation of a complete plastic mechanism (terminal point) in the pushover procedure is not clearly defined. Hence similar studies, as provided here, on buildings with different structural systems are recommended. Different method of structural remedial measure to the building, based on the pushover analysis could be studied, for example, strengthening of the columns instead of windows concrete infill. More ductile behaviour of the structure could be obtained in that way. Detailed guidelines for seismic rehabilitations for building guidelines may be found in FEMA-273 [7]. It is the author’s opinion that the pushover analysis is a good compromise between simplicity and reality and it provides an excellent alternative to the most commonly used analysis, namely the linear one.

88

References

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