An investigation on stability of composite i girder bridge made of steel for bridge high performance structure SBHS

An investigation on stability of composite i girder bridge made of steel for bridge high performance structure (SBHS) An investigation on stability of composite i girder bridge made of steel for bridge high performance structure (SBHS)

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

NGUYEN XUAN BACH

AN INVESTIGATION ON STABILITY OF COMPOSITE I GIRDER BRIDGE MADE

OF STEEL FOR BRIDGE HIGH PERFORMANCE STRUCTURE (SBHS)

MASTER’S THESIS

Hanoi, 2020

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

NGUYEN XUAN BACH

AN INVESTIGATION ON STABILITY OF COMPOSITE I GIRDER BRIDGE MADE

OF STEEL FOR BRIDGE HIGH PERFORMANCE STRUCTURE (SBHS)

MAJOR: INFRASTRUCTURE ENGINEERING

CODE: 8900201.04QTD

RESEARCH SUPERVISOR:

Dr DANG VIET DUC

Ha Noi, 2020

ACKNOWLEDGEMENTS

I would like to give sincere thanks to my supervisor, Dr Dang Viet Duc, for his guidance and supports throughout the time I study on this thesis By his advice and inspiration, Dr Duc has given me a passion for structural engineering and the motivation to excel in all my efforts I am honored to have conducted this research under his guidance

I would also like to thank Prof Hironori Kato, Prof Nguyen Dinh Duc, Dr Phan Le Binh, Dr Nguyen Tien Dung, and the teachers of Master Infrastructure Engineering (MIE) Program Their instruction throughout my studying career in Viet Nam Japan University is greatly appreciated Also, their recommendation and question is really invaluable for me, which make my knowledge better

I would like to thank all staff of Japan University, both past and present, for all their support and help throughout two years, I have studied in Viet Nam Japan University I will never forget my experiences here with them

I am indebted to my parents for their endless support Without them, none of this would have been possible They have given me as much as possible They are my inspiration

I would especially like to thank my girlfriend, Hoang Ngan, for being my best friend and my unending support I would like to extend my thanks to her family for giving me support and confidence throughout over the time

ABSTRACT

The steel-concrete composite I-girder is one of the most popular structural types for highway and railway bridges They are design to a trend using larger twin girders and simple crossbeam instead of system more two girders and complex k-frame As a result, the slender ratio system girder becomes larger which makes the lateral buckling can occur in the system Current American bridge design specifications (AASHTO 2012) simplify the flexural design of I-girder sections by treating local and global inelastic stability phenomenon independently According to specifications, if a section is compact and sufficiently braced against lateral instability, the member will achieve or exceed its theoretical plastic moment capacity Treating local and global buckling has been proven by past experience to be suitable when formulating flexural design provisions for lower strength steels However, new researches are proving that these provisions have recently been inappropriate to apply to the design of High Performance Steel Besides, application of bridge high performance steels, SBHS500 girders is expected to be an economical solution for composite girder bridges Steels SBHS500 with yield strengths of 500, have been standardized in 2008 in Japanese Industrial Standards (JIS) They present the advantage of high yield strength, good weldability However, if compared to conventional (normal) steels they show different inelastic behavior, such as almost no yield plateau, smaller ductility, and a greater yield-to-tensile strength ratio Consequently, the main objective of the current study is to investigate the effect of crossbeam spacing on Ultimate Flexural Resistance of composite I-girder systems which are SM490Y and SBHS500 systems Finite element modes of SBHS500 and SM490Y systems, employing nonlinear analysis, are used to determine crossbeam spacing and Ultimate Flexural Resistance, precisely After indicating that limit of crossbeam spacing in conventional design standards are over conservative for composite SBHS500 steel girders

supper-TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION 1

1.1 Introduction of composite girder bridge 1

1.2 Issue of design for composite twin bridges 4

1.2.1 Thicker steel plates and new steel grades 4

1.2.2 Assumption of lateral-torsional buckling 6

1.3 SECTION CLASSIFICATION 7

1.3.1 Behavior of bending beam 7

1.3.2 Requirement Unbraced Length of compact section 9

1.3.3 Requirement of Wed Plate 11

1.4 Objectives 12

1.5 Overview of Thesis Organization 12

CHAPTER 2 LITERATURE REVIEW 13

CHAPTER 3 METHODOLOGY AND MODELING 16

3.1 Methodology 16

3.1.1 Nonlinear finite element analysis 16

3.1.1 Material modeling 17

3.1.2 Initial Geometrical Imperfection 18

3.2 Finite element model 20

3.2.1 The simplified twin I girder composite structure 20

3.2.2 Steel and concrete material model 23

3.2.3 Meshing of composite girder model 25

3.2.4 Initial geometrical imperfection 25

3.2.5 Phases structural analysis 26

CHAPTER 4 RESULT AND DISCUSSION 28

4.1 Effect of crossbeam spacing on the ultimate flexural resistance 28

4.2 Effect of stiffener spacing on the ultimate flexural resistance 30

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 32

5.1 Conclusion 32

5.2 Recommendations for future research 33

REFERENCE 34

APPENDIX 36

Appendix 1 Properties of composite section 36

Appendix 2 Checking dimension of section by provision of AASHTO 2010 41

Appendix 3 Results and bending moment and curvature plots 42

LIST OF FIGURES

Fig 1.1 The composite twin I-girder bridges in France (Le Viaduc de l'Hyrôme,2011) 2 Fig 1.2 Global buckling mode in I-girder under loading of wet concrete and simplified

outstanding plate 3

Fig 1.3 The actual test stress-strain relation of conventional and high strength SBHS steel grades (Dang Viet Duc, 2013) 4

Fig 1.4 Nagata Bridge made of SBHS 500 steel grade with truss system (Source: https://www.pinterest.com/pin/297659856597529870) 5

Fig 1.5 Tokyo Gate made of SBHS 500 steel grade with full-welded truss girder (Source https://photos.com/featured/tokyo-gate-bridge-i-kadek-wismalana) 5

Fig 1.6 Twin I-girder system (Yoseph Yura, 2008) 7

Fig 1.7 Beam Behavior (Yura, Galambos, and Ravindra 1978) 8

Fig 1.8 Classification of beding section referred by AASHTO-2010 9

Fig 1.9 Basic form of all I-section flexural resistance equations follow unbraced length referred by AASHTO 2010 10

Fig 1.10 Assumption of tress distribution in composite steel section referred by AASHTO 2010 11

Fig 2.1 Finite element model of Twin-I girder bridge 13

Fig 2.2 Steel material comparison for different girder depths 14

Fig 2.3 Stability comparison for different stiffener spacing 14

Fig 3.1 Constitutive Law-True Stress versus True Strain (abaqus-docs.mit.edu) 17

Fig 3.2 Dominant buckling modes of I-girder systems 19

Fig 3.3 Vertical view of the model 20

Fig 3.4 Plan view and condition boundary 20

Fig 3.5 Horizontal view of the model 21

Fig 3.6 Bending moment produced by displacement control method 22

Fig 3.7 Idealized stress-strain relations for steel material model (Dang Viet Duc, 2013) 23

Fig 3.8 Idealized stress-strain relations for concrete material model (Material Laboratory of Ha Noi University of Transport and Communication) 23

Fig 3.9 Cross-section and type of elements 25

Fig 3.10 Initial deflection for the beam supported simply as the composite girder models on plan view (a) and 3D model (b) 26Fig 3.11 Different construction phases 27Fig 4.1 Curvature relationship between bending moment and rotation of SM 490Y structure with varying crossbeam spacing 28Fig 4.2 Curvature relationship between bending moment and rotation of SBHS500 structure with varying crossbeam spacing 29Fig 4.3 Curvature relationship between bending moment and rotation of SM 490Y structure with varying stiffener spacing 30Fig 4.4 Curvature relationship between bending moment and rotation of SBHS500 structure with varying stiffener spacing 31

LIST OF TABLES

Table 1.1 Definition and web slenderness limits in AASHTO 2010 12

Table 2.1 Cross section dimension based on same section area 13

Table 2.2 Stress and displacement reponse for different cross-beam spacing 15

Table 2.3 Stress and displacement response for different cross-beam depth 15

Table 3.1 Steel girder systems investigated in the study 21

Table 3.2 Characteristic of steel in the study 24

Table 3.3 Characteristic of concrete in the study 24

Table A.1 Result of plastic moment 39

Table A.3 Result of internal moment of sections 42

LIST OF ABBREVIATION UFR Ultimate Flexural Resistance

LTB Lateral-torsional buckling

CHAPTER 1 INTRODUCTION

1.1 Introduction of composite girder bridge

Steel-concrete composite girders have experienced a veritable domination in both highway and Railway bridges structures The girders are highly competitive in area of medium structures (45-90m) Nowadays, in common usage composite girder

is normally taken to mean steel and concrete, which tasks a concrete slap part to total compression and a steel girder to in-plane flexural force and tension Regarding the service characteristics, the concrete slap work well to build asphalt surface layers Modern composite bridges built in compliance with seismic codes are highly resistant to earthquakes Moreover, the kind of bridges are lighter and more slender than equivalent span and width concrete bridges, this reduces the loads sustained by their supports in the event of an earthquake Steel-concrete composite structures are sustainable development, which is the durable construction, robust structures, sparing in material and guarantee reducing environmental and human health impacts at an acceptable economic cost

Nowadays, with excellent industry, a steel production has been developed considerably and thus resulted in the decreasing cost of steel material However, in

a lot of countries, the shortage of workforce is steeply becoming serious As a result, the increase of labor cost has changed the total cost of bridge construction Comparing to the effect of material saving, the effect of labor cost growth on the total cost is dominated Consequently, to reduce consuming workforce in bridge construction, steel-concrete girders have been designed in the trend of simpler structures, such as thicker web plates and fewer number of stiffeners and cross beam So that, the time of erection and fabrication will be reduced

frames which supply torsion stiffener (shown in Fig 1.1-a) Nevertheless, since

1995 the great majority of girder composite was called twin girder cross-beam

structures (presented in Fig 1.1-b) This structure has two main I-girders connected

to the concreted slap, usually simply reinforced, and interlinked by secondary beams called crossbeam Typically, the bottom flange is larger because it provides the necessary tensile forces caused by positive bending When the deck is hardened and completed, it will be permanently joined the top flange, and they will act together in bending Therefore, the twin girders with simple interconnection could

be very economical

However, the spacing of cross beams plays important roles because it transfers force from one girder to another and provides global stability of whole system If the crossbeams are placed too closely, the number of cross beams increases and

results in the rising cost of material and labor On the other hand, if the spacing between crossbeams is larger, the system of girder unable to achieve global stability (i.e., global instability described by next section) Additionally, in the twin structures, the top flanges of girder are not connected to the concrete slap during casting of concrete which not yet supports system to lateral buckling As a result, during construction, the girders are at very high risk of buckling to carry their self-weight load, concrete self-weight load, construction loads, etc The two techniques used to increase the girder’s capacity preventing buckling instability during construction (before the concrete slap casted) One of popular techniques using temporary support is called shored construction method Another technique is un-shored construction method without temporary support According to AASHTO

“the shored construction method generally is expected more economical” In shored construction, before the concrete slap is harden, the smaller top flange is more vulnerable to lateral buckling because it is the main contributor to lateral buckling strength in positive bending The global buckling can appear which makes girders system unstable or collapsed unless the system is suitably supported by

un-crossbeams (as shown by Fig 1.2)

Fig 1.2 Global buckling mode in I-girder under loading of wet concrete and

simplified outstanding plate

1.2 Issue of design for composite twin bridges

1.2.1 Thicker steel plates and new steel grades

For new design of twin bridges in Japan or France, the thicker steel plates with thickness up to 100 mm have been made which make girders less slender However, the thicker steel plates cause the lager self-weight and hence necessary to study on these kinds of girder

Additionally, high strength steel plates are widely using in structures of bride which has been completely believed by consultants and contractor in developing the industrial building The steel for bridge high performance structure (SBHS) was standardized by Japanese Industrial Standards (JIS 2008) in 2008, including SBHS

500 and SBHS 700 with design yield strength are 500 MPa and 700 MPa, respectively The steels have weld-ability and good steelwork, but also their inelastic behavior is different compared to conventional steel because of no yield

area and yield-to-tensile strength ratio (presented in Fig 1.3) Specially, the design

yield strength would not reduce if the thickness of steel plates changes

Fig 1.3 The actual test stress-strain relation of conventional and high strength

SBHS steel grades (Dang Viet Duc, 2013) Nowadays, the SBHS 500 was employed to build in several bridges, for example Nagata Bridge and Tokyo Gate Bridge The Nagata Bridges could be seen

as steel-concrete composite girder bridge but it is space pipe truss system instead of

the conventional I-shaped or box girder system (as shown Fig 1.4) Thickness of

steel pipes in this bridge project is up to 67 mm The pipe truss system was fabricated with cold formability and on-site welding

Fig 1.4 Nagata Bridge made of SBHS 500 steel grade with truss system

(Source: https://www.pinterest.com/pin/297659856597529870)

The Tokyo Gate Bridge can be seen as a Mega Structure and World’s scale fully welded continuous truss bridge The bridge has 4 traffic lanes, whole length of 2933 m, the longest span of 440 m, and marine area of 1,618 m, the ship tolerance of 300m x 54.6m, the maximum height from water level of 87.8 m The bridge importantly connects the central breakwater landfill and Koto-City Wakasu, straddling the Tokyo Port Third Seaway The general view of Tokyo Gate Bridge is

largest-shown in Fig 1.5

Fig 1.5 Tokyo Gate made of SBHS 500 steel grade with full-welded truss girder

(Source https://photos.com/featured/tokyo-gate-bridge-i-kadek-wismalana)

However, these bridges are steel truss or pipe system which mainly suffers tension or compression Besides, few studies on local buckling of SBHS steel using for structures which investigate just stability in a plates or a single girder So that, the use of SBHS steel in composite twin I-girder bridge, which is necessary to conduct precise studies of global buckling in this kind of structure

1.2.2 Assumption of lateral-torsional buckling

Lateral-torsional buckling (LTB) of a girder is failure mode that involves lateral movement and twist of the girder cross section Thus, this failure also is seen Global Lateral Buckling (GLB) The classic solution for LTB of a simple supported girder (Ti-moshenko and Gere 1991) bent about the strong axis by a uniform moment, Mcr, is

√

Where, Lb= distance between points along the length where twist is prevented

E= modulus, G= shear modulus (usually taken as E/2.6)

Iy= weak axis (lateral moment of inertia), J=torsional constant

Cw= warping constant, For Twin I girder Cw=Iy(ho/2)2

It should be noted that Eq 1.3.2.1 was derived using only the no twist condition at each end of the unbraced length The design for LTB of twin beam in bridge design specification is based on Eq 1.3.2.1 In many applications, brace and crossbeam are used to reduce Lb thus increasing M0 so that yielding, not buckling, controls the strength of the girder Current design specifications only consider the

lateral buckling of individual girder (as shown Fig 1.6-a) The force in compression

flange causes lateral moment of individual girder to occur between braced points The design for torsional braced point is given in AAHSTO 2010 Bridge Design Specification (AASHTO 2010) Although the common design practice is to evaluate

the lateral buckling capacity of the girder between braced points, but a global system buckling mode can occur over the span Lg represented by the dotted lines in

Fig 1.6-b The global buckling of the girder system has traditionally not been a

design consideration because of assuming braced point not to displace laterally.(Yoseph Yura, 2008)

Fig 1.6 Twin I-girder system (Yoseph Yura, 2008) 1.3 SECTION CLASSIFICATION

1.3.1 Behavior of bending beam

The resistance of steel beams in flexure is dependent on the stability effects both locally and globally If the beam is able to remain stable at high loads, then beam can develop a bending resistance beyond the first yield moment My and perhaps even attaining its full plastic moment resistance, Mp If the beam’s stability

is limited by either local or global buckling then the bending resistance may be less than Mp, and it the buckling is severe, less than My In order to prevent local or global buckling prior to the attainment of Mp, the beam must be compact and adequately braced (Salmon and Johson 1996) The term “compact” refers to adequate proportioning of the cross-sectional plate elements and also crossbeam spacing (i.e unbraced length) of the girder such that Mp can be achieved and maintained through a finite cross-sectional rotation A compact beam should have adequate rotation capacity for inelastic force redistribution to take place (Yura et al 1978)

Fig 1.7 Beam Behavior (Yura, Galambos, and Ravindra 1978)

The behavior of a beam bent about the strong axis is shown in Fig 1.7 The

beam will ultimately fail with the occurrence of either local buckling of compression flange, local buckling of the web, or globally lateral buckling of the beam Due to the sufficient ductility of the grades of AASHTO20010 allow for flexural application involving moment redistribution, failure by tensile rupture will not occur prior to a buckling type failure associated with compression (Yuar et Al 1978)

The beam behavior in Fig 3.1 can be classified into 4 categories:

1 The plastic range where the cross section is able to reach the plastic moment, Mp, and maintain this strength through sufficient rotation capacity in order to permit moment redistribution in indeterminate structures (Yuara et Al 1978)

2 Inelastic response where plastic moment Mp is achieved but little rotation capacity is exhibited, as a result of inadequate stiffener of the flange and /or web

to resist local buckling, or inadequate lateral support to resist lateral-torsional buckling, while the flange is inelastic

3 The inelastic response where the moment strength Mr, the moment above which residual stress cause inelastic behavior to begin, is reached or exceeded,

however, local buckling of the flange or web, or global buckling prevent achieving the plastic moment Mp

4 Elastic behavior with moment strength, Mcr, being controlled by elastic buckling, any or all of, local flange buckling, local web buckling, global-torsional buckling

When a beam is adequately braced against lateral-torsional buckling, local buckling of the flange and or web will control the achievement of the beam’s plastic moment capacity, Mp (Salmon and Johnson 1996) Local buckling of the beam plate elements or global buckling of whole beam can cause premature failure of the entire section, or at least it will cause stress to become nonuniform and reduce the overall strength of the beam Thus, current design provisions require that the plates elements and unbraced length to be adequately proportioned, or “compact” in applications requiring the attainment of Mp AASHTO 2012 states that compact sections are capable of developing a fully plastic stress distribution and can possess

a rotational capacity of approximately 3 before the onset of local or global buckling

(Yura et al 1978) Classification of section is shown as in Fig 3.2

Fig 1.8 Classification of beding section referred by AASHTO-2010

1.3.2 Requirement Unbraced Length of compact section

As shown in section 1.2.2, all of the I-section flexural resistance equation in the AASHTO 2010, are based on logic of identifying the two anchor points (shown

in Fig 3.2-1) Anchor point 1 is located at the length Lb=Lp for lateral-torsional

buckling (LTB) to development of the maximum potential flexural resistance as compact section Anchor point 2 is located at the length Lr for which inelastic and elastic LTB are the same For Lb>Lr, LTB resistance are governed by elastic buckling

Fig 1.9 Basic form of all I-section flexural resistance equations follow unbraced

length referred by AASHTO 2010

In the section 6.10.8.2.3 in AASHTO 2010, for unbraced lengths in which the member is prismatic, the lateral buckling resistance of the twin I girders shall be taken as:

Lb ≤ Lp Where: Lp= limiting unbraced length to achieve the nominal flexural resistance

under uniform bending (m)

= 1.0rt√

Lr=limiting unbraced length to achieve the onset of nominal yielding in either flange under uniform bending with consideration of compression flange residual stress effects (m)

= πrt√

rt effective radius of gyration for lateral torsional buckling (m)

(3.2.1)

=

√

Fyr=compression-flange stress at the onset of nominal yielding within the cross-section, including residual stress effect, but not including

compression-flange lateral bending taken as the smaller of 0.7Fyc and Fyw, but not less than 0.5Fyc

1.3.3 Requirement of Wed Plate

For steel composite twin I girder is suffered positive flexure, the influence of local buckling in compressive flange on flexural strength is eliminated Web slenderness maintain an important influence on flexural strength of composite girder and ways to classify section based on web slenderness limits in current AASHTO 2010 is discussed in this section

Fig 1.10 Assumption of tress distribution in composite steel section referred by

AASHTO 2010 AASHTO toke the quantities of plastic moment Mp and yield moment My to define section class The section can achieve full plastic stress distribution as shown

Fig 1.10 The yield moment My obtained when yield stress attaining at extreme

fiber of either top or bottom flange (presented in Fig 1.10) AASHTO specification

considers web slenderness as ratio of twice the compression depth of web to web

thickness 2Dcp/tw or 2Dc/tw where Dcp and Dc, described in Fig 1.10, are compression depths corresponding to plastic and yield moment stress distributions, respectively The comparison of web slenderness limits proposed by the design

code is presented in Table 1.1

Table 1.1 Definition and web slenderness limits in AASHTO 2010

Design code Section class Definition Web slenderness limit

1.4 Objectives

Ultimate Flexural Resistance: to investigate the effect of crossbeam spacing

on Ultimate Flexural Resistance of composite twin I girders in un-shored construction The investigation is conducted with two types of structure that are made of (A) conventional steel structure (SM 490Y) and (B) high performance steel

structure (SBHS500)

Section Classification Based on Ultimate Flexural Resistance: according to

AASHTO standard, either the system can reach moment plastic (ultimate flexural resistance) or not, so it is compact or non-compact It makes design of Twin I-

girder bridges more suitable for applying new steel grade (SBHS500)

1.5 Overview of Thesis Organization

This thesis is organized as follows In Chapter 2, introduction of articles related

to research issues and explanation for necessity of study is reviewed Chapter 3 describes the FEM model that will be analyzed by ABAQUS software In Chapter

4, the results of the study are presented and discussed Conclusions from this study are contained in Chapter 5

CHAPTER 2 LITERATURE REVIEW

Study of Haiying Ma and Xuefei Shi (2016)

Haiying Ma published the paper (Study on Behavior Twin I-girder Bridge

System with Cross-beams) addressing the effects of depth girder, flange-thickness

ration, web depth-thickness ratio and also crossbeam spacing on behavior of the system The study was developed based on a twin I-girder with span of 35m (as

shown in Fig 2.1) and processed the finite element method (FEM) numerical

models in elastic range These models are four types of girder depths, including

1.25, 1.5, 1.75 and 2.0m, the detail section in the Table 2.1

Fig 2.1 Finite element model of Twin-I girder bridge

Table 2.1 Cross section dimension based on same section area

In study of Haiying Ma, from the section properties and response of different girder depths, girder section with higher depth has smaller stress and displacement response than lower girder depth under the same condition and material is used In other way, with the same stress requirement that means load required Girder with

higher depth may save material which is more economical (as shown by Fig 2.2)

Fig 2.2 Steel material comparison for different girder depths

Additionally, stiffeners and cross beam spacing also considered in the study They indicate that stiffener spacing are smaller than girder depth, the stability of system is affected considerably But when the spacing is larger than 3m, the stability varies slightly Therefore, the spacing is not needed to be small In term of position of crossbeam in section, they are nearer to the bottom flange, the transverse

displacement and relative displacement decrease (presented by Fig 2.3) And thus,

the crossbeam is suggested to arrange near girder bottom

Fig 2.3 Stability comparison for different stiffener spacing

Table 2.2 Stress and displacement reponse for different cross-beam spacing

Crossbeam

spacing

(mm)

Maximum transverse displacement (mm)

Relative transverse displacement(mm)

Stress at mid-span (Mpa)

Stress at support (Mpa)

Relative transverse displacement(mm)

Stress at mid-span (Mpa)

Stress at support (Mpa)

of stability in twin I girders In addition, the study is only investigated to the effects

of crossbeam spacing on stress at middle point of girder, but not considered to effects on ultimate flexural strength

CHAPTER 3 METHODOLOGY AND MODELING

3.1 Methodology

3.1.1 Nonlinear finite element analysis

FEM is the most useful numerical technique for analyzing structures This method is a relatively inexpensive and efficient time Successful use of FEM in many studies, involving compound structures or interactions of structural members has been one of the motivations for applying the method in this study

A three-dimensional FE model is developed in ABAQUS (2018), a purpose finite element program to perform elasto-plastic large displacement analysis of composite girders In consideration of twin I girder composite bridges, the number of cross beam and also stiffener is reduced but sufficiently restricted the girder to lateral-torsional buckling

general-The objective of the current study is to investigate the effect of crossbeam spacing (unbraced length) has on flexural ductility of girder This includes the attainment of the cross-section’s plastic moment and its ability to maintain this capacity through some amount of rotation capacity In order to investigate this behavior need to draw a full plot, both loading and unloading, of the twin I girders

is relationship between internal moment and versus rotation response as shown in

Fig 1.7 Thus, this plot would achieve a nonlinear displacement based finite

element analysis is performed A nonlinear analysis is used rather than a linear analysis, since the girder will undoubtedly show nonlinear behavior prior to reaching its ultimate flexural resistance

In the structure, there are two important nonlinearities that are material nonlinearity and geometric nonlinearity These nonlinearities are produced by finite deformation coupled with changes in material stiffener under applied loading Both nonlinearities are proper to the current study in which they are combined with the

inelastic global (Earls 1999) Geometric nonlinearity controls in a situation where it

is inappropriate to formulate equilibrium in terms of the undeformed state of the structure Material nonlinearity means that the material has just exceeded the yield strength as undergoing plastic deformation

3.1.1 Material modeling

Many constitutive models have been developed in the scope of the FE method

to reproduce the nonlinear behavior of construction materials, especially concrete and steel The FE analysis requires a constitutive model that relates the stress within the material to an applied strain The constitutive models capable of modeling the behavior of the concrete and steel are presented in the following sections

Implementation of metal plasticity in Abaqus

True-stress versus true-strain characteristics of the material are used in nonlinear finite element analysis since nonlinear element formulations permit the consideration of updated structural configurations Engineering stress-strain response does not give a true indication of the deformation characteristics of a structural steel because it is based entirely on the original dimensions of a given specimen Ductile materials, such as steel, exhibit localized geometric changes and therefore, the relevant stress and strain measures are different from the measured

engineering stress and strain values Fig 1.9 illustrates the true stress-true strain

curve Specific values for the various steels used in the current study will be discussed in section 3.3.2

Fig 3.1 Constitutive Law-True Stress versus True Strain (abaqus-docs.mit.edu)