journal of constructional steel researchcuong ngo-huu a,1, seung-eock kim b,⁎ a faculty of civil...

Journal of Constructional Steel Research 74 (2012) 90–97

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Journal of Constructional Steel Research

Practical nonlinear analysis of steel–concrete composite frames usingfiber–hinge method

Cuong NGO-HUU a,1, Seung-Eock KIM b,⁎a Faculty of Civil Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet St., Dist. 10, Ho Chi Minh City, Vietnamb Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea

⁎ Corresponding author. Tel.: +82 2 3408 3291; fax:E-mail address: [email protected] (S.-E. KIM).

1 Formerly Adjunct Researcher of Constructional Tecversity, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, So

0143-974X/$ – see front matter © 2012 Elsevier Ltd. Aldoi:10.1016/j.jcsr.2012.02.018

a b s t r a c t
a r t i c l e i n f o
Article history:Received 26 July 2011Accepted 28 February 2012Available online 28 March 2012

Keywords:Steel–concrete composite framesNonlinear analysisFiber–hinge methodStability functions

A fiber–hinge beam–column element considering geometric and material nonlinearities is proposed formodeling steel–concrete composite structures. The second-order effects are taken into account in derivingthe formulation of the element by the use of the stability functions. To simulate the inelastic behaviorbased on the concentrated plasticity approximation, the proposed element is divided into two end fiber–hinge segments and an interior elastic segment. The static condensation method is applied so that the ele-ment comprising of three segments is treated as one general element with twelve degrees of freedom. Themid-length cross-section of the end fiber segment is divided into many fibers of which the uniaxial materialstress–strain relationship is monitored during analysis process. The proposed procedure is verified for accu-racy and efficiency through comparisons to the results obtained by the ABAQUS structural analysis programand established results available from the literature and tests through a variety of numerical examples. Theproposed procedure proves to be a reliable and efficient tool for daily use in engineering design of steeland steel–concrete composite structures.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Steel–concrete composite structures comprised of steel, reinforcedconcrete, and steel–concrete composite members have widely usedfor constructing buildings and bridges due to their efficiency in struc-tural, economic and construction aspects. Therefore, extensive exper-imental and theoretical studies have been conducted to provide abetter understanding on the behavior of the composite structureand its components under applied loading. Together with the moreand more application of the composite structures, there are increas-ing needs in having a reliable structural analysis program capable ofpredicting the second-order inelastic response of steel–concrete com-posite structures. Recently, as the design profession moves towards aperformance-based approach, the accurate detailed information onhow a structure behaves under different levels of loads is necessaryin evaluation of the expected level of performance. Obviously, this re-quires a comprehensive analysis procedure that can consider all keyfactors influencing the strength of structure and produce results con-sistent with the current design code requirements with sufficientaccuracy. For daily design purpose, the nonlinear analysis programshould be able to get the reliable results in a minimized time, espe-cially in a time-consuming earthquake-resistant design. The degree

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of success in predicting the nonlinear load–displacement responseof frame structures significantly depends on how the nonlinear ef-fects to be simulated in numerical modeling.

The steel and concrete components can be modeled separatelyusing plate, shell and solid elements of available commercial three-dimensional nonlinear finite element packages or self-developed pro-grams of researchers and then are assembled together by some con-nection or interface elements to simulate the shear connectors/interaction between these components, as recently presented byBaskar et al. [1] and Barth and Wu [2], among others. This continuummethod can best capture the nonlinear response of the compositestructures and is usually used instead of conducting the high costand time-consuming full-scale physical testing. However, in order tomodel a complete structure, so many shell, plate, and solid finiteelements must be used and, as a result, it is too time-consuming.

To reduce the modeling and computational expense, “line element”method has been proposed and it can be classified into distributed andlumpedplasticity approaches based on the degree of refinement used torepresent inelastic behavior. The distributedmethod uses the highest re-finement while the lumped method allows for a significant simplifica-tion. The beam–column member in the former is divided into manyfinite elements and the cross-section of each element is furthermodeledby fibers of which the stress–strain relationships are monitored duringthe analysis process, as recently presented by Ayoub and Filippou [3],Salari and Spacone [4], Pi et al. [5], McKenna et al. [6], among others.Therefore, this method is able to model the plastification spreadingthroughout the cross-section and along themember length. The residual

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91C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97

stress in each fiber of the steel section can directly be assigned as con-stant value since the fibers are sufficiently small. The solution of the dis-tributed method can be considered to be relatively accurate and easilybe included the coupling effects among of axial, lateral, and torsion de-formations. However, it is generally recognized that this method is toocomputationally intensive and hence usually applicable only for re-search purposes (e.g., checking and calibrating the accuracy of simpli-fied inelastic analysis methods, and establishing design charts andequations) because a very refined discretisation of the structure isnecessary and the numerical integration procedure is relatively time-consuming, especially for large-scale space structures as normallyencountered in design. Therefore, it is not efficient to apply them in adaily practical design.

The beam–column member in the lumped method is modeled byan appropriate method eliminating its further subdivision, and theplastic hinges representing the plastic interaction between axialforce and the biaxial moments are assumed to be lumped at bothends of the member. This plastic hinge is usually based on a specificyield surface and an approximate function to simulate the gradualyielding of the cross-section. Although this method is less accuratein comparison with the distributed method, it was shown to be verysimple, fast, and capable of providing results accurate enough forpractical design, as presented by Porter and Powell [7], El-Tawil andDeierlein [8] and Liu et al. [9]. However, this method is usually ap-plied for nonlinear analysis of frame structures composed of steel,reinforced concrete and encased composite members because theyield surface for steel and reinforced concrete composite section, es-pecially for steel I-beam and reinforced concrete slab section, is notalways available and accurate for every section. Moreover, the gradu-al reduction in strength of the general composite section undergradual loading is not easy to model.

In this research, to take advantage of computational efficiency of thecommon lumped approach and overcome its above-mentioned weak-ness, a fiber–hinge beam–column element is introduced to model thesteel and steel-composite composite members. This is a developmentfrom the work done by Ngo-Huu and Kim [10] for nonlinear analysisof steel space structures. The proposed element is divided into twoend fiber–hinge segments and an interior elastic segment to simulatethe inelastic behavior of the material. The cross-section at mid-lengthof end fiber–hinge segment is divided into steel and/or concrete fibersso that the uniaxial stress–strain relationship of cross-sectional fiberscan be monitored during the analysis process based on the relevantconstitutivemodel and the flow theory of plasticity. This is a good alter-native for inelastic representation instead of using the specific yieldsurface in usual plastic hinge model. Herein, the stability functionsobtained from the exact buckling solution of a beam–column subjectedto end forces are used to accurately capture the second-order effects.The nonlinear responses of structures in a variety of numerical exam-ples of steel and steel–concrete composite frames are compared withthe existing exact solutions, the results from the experiments, andthose obtained by the finite element package ABAQUS and plasticzone analyses to show the reliability and efficiency of the proposedapproach in applying for practical design purpose.

2. Formulation

2.1. Basic assumptions

The following assumptions are made in the formulation of thecomposite beam–column element:

1. All elements are initially straight and prismatic. Plane cross-section remains plane after deformation.

2. Local buckling and lateral–torsional buckling are not considered. Allmembers are assumed to be fully compact and adequately braced.

3. Large displacements are allowed, but strains are small.

4. Reductions of torsional and shear stiffnesses are not considered inthe fiber–hinge.

5. The connection and bond between member and its componentsare perfect. The panel–zone deformation of the beam-to-columnjoint is neglected.

2.2. Beam–column element accounting for P−δ Effect

To capture the effect of the axial force acting through the lateraldisplacement of the beam–column element relative to its chord (P−δeffect), the stability functions are used to minimize modeling and solu-tion time. Generally only one element is needed per a physical memberin modeling to accurately capture the P−δ effect. Similar to the formu-lation procedure presented by Chen and Lui [11], the incremental force–displacement relationship of the space beam–column element may beexpressed as [10]

_P_MyA_MyB_MzA_MzB_T

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

¼

EAð ÞcL

0 0 0 0 0

0 kiiy kijy 0 0 00 kijy kiiy 0 0 00 0 0 kiiz kijz 00 0 0 kijz kiiz 0

0 0 0 0 0GJð ÞcL

26666666664

37777777775

_δ_θyA_θyB_θzA_θzB_ϕ

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

ð1Þ

where _MnA, _MnB, _P , and _T are incremental endmoments with respect ton axis (n=y, z), axial force, and torsion, respectively; _θnA, _θnB, _δ, and _ϕare incremental joint rotations with respect to n axis, axial displace-ment, and the angle of twist, respectively;

kiin ¼ S1nEInð ÞcL

ð2aÞ

kijn ¼ S2nEInð ÞcL

ð2bÞ

and

EAð Þc ¼Xmi¼1

EiAi ð3aÞ

EInð Þc ¼Xmi¼1

Ein2i Ai ð3bÞ

GJð Þc ¼Xmi¼1

Gi y2i þ z2i� �

Ai ð3cÞ

inwhichm is the total number of fibers divided on themonitored cross-section; Ei and Ai are the tangentmodulus of thematerial and the area ofith fiber, respectively; yi and zi are the coordinates of ith fiber in thecross-section; S1n and S2n (n=y, z) are the stability functions withrespect to y and z axes, and are shown as

S1n ¼π

ffiffiffiffiffiffiρn

psin π


p� �−π2ρn cos π


p� �2−2 cos π


p� �−π


psin π


p� � if P b 0

π2ρn cosh πffiffiffiffiffiffiρn

p� �−π


psinh π


p� �2−2 cosh π


p� �þ πffiffiffiffiffiffiρn

psinh π


p� � if P > 0

8>>>><>>>>:

ð4aÞ

S2n ¼

π2ρn−πffiffiffiffiffiffiρn

psin π


p� �2−2 cos π


p� �−π


psin π


p� � if P b 0

πffiffiffiffiffiρy

psinh π

ffiffiffiffiffiρy

p� �−π2ρy

2−2 cosh πffiffiffiffiffiffiρn

p� �þ πffiffiffiffiffiffiρn

psin π


p� � if P > 0

8>>>><>>>>:

ð4bÞ

where ρn=P/(π2EIn/L2) with n=y, z and P is positive in tension.

bf

Concrete fiber

z

d

bs

ds

Steel fiber

y

Fig. 2. The partition of monitored cross-section into fibers.

(a) Steel

92 C. NGO-HUU, S.-E. KIM / Journal of Constructional Steel Research 74 (2012) 90–97

2.3. Beam–column element accounting for material nonlinearity

To model the material nonlinearity based on the concentratedplasticity approximation, the beam–column element is modeled bytwo end fiber segments and a middle elastic segment as shown inFig. 1. In order to monitor the gradual plastification throughout theelement's cross-section, the cross-section located at the mid-lengthof the end segment is divided into many fibers to track the inelasticbehavior of the section and element (Fig. 2). Each fiber is representedby its area and coordinate location corresponding to its centroid. Theinterior segment is assumed to behave elastically similar to the elasticpart in the common plastic hinge. For hot-rolled steel section, theresidual stresses are directly assigned to fibers as the initial stresses.The ECCS residual stress pattern of I-shape hot-rolled steel section isused for this study.

Because the hybrid element has three segments with 24 DOFs, astatic condensation method developed by Wilson [12] is applied sothat the element is treated as one element with 12 DOFs normallyfound in a general beam–column element. Twelve DOFs of two inte-rior nodes (nodes 3 and 4 in Fig. 1) must be condensed out to leavetwelve DOFs of two exterior nodes (nodes I and J, also denoted asnodes 1 and 2 in Fig. 1). This reduces the computational time whenassembling the total stiffness matrix and solving the system of linearequations. In addition, this twelve DOF element is adaptive with theexisting beam–column program so that the coding time can be re-duced. However, it also requires that a reverse condensation needsbe performed in order to compute the deformations at the ends ofthe segments to evaluate the degree of yielding of the fiber–hinge.In a general nonlinear analysis, the element stiffness matrices of thecurrent step are evaluated based on the state of the system deter-mined at the end of previous step. Once the incremental fiber strainof the cross section is evaluated, the flow theory of plasticity is ap-plied to determine the incremental fiber stress based on the relevantuniaxial material stress–strain relationship represented in the follow-ing section. The strain hardening of the steel material that causes anincrease in member strength is considered. These let the proposedapproach simulate a more realistic behavior of structure than thecommon plastic hinge method does.

2.4. Constitutive model of material

The constitutive material models in explicit functions of strain forsteel and concrete recommended by Eurocode-2 [13] are used in thisresearch as shown in Fig. 3. The tensile strength of concrete isneglected. The concrete stress–strain relation in compression is de-scribed as following expressions

σ ¼ fc′ 1− 1− εε0

� �n for 0≤ ε≤ ε0 ð5aÞ

σ ¼ fc′ for ε0b ε≤ εu ð5bÞ

where fc′ is the concrete compressive cylinder strength; n is the expo-nent; ε0 is the concrete strain at maximum stress and εu is the ulti-mate strain. For fc′≤50 MPa, n=2 and ε0=0.002 and these valuesare used for all relevant examples presented in this research.

Fig. 1. Beam–column element comprising of three segments.

2.5. Element stiffness matrix accounting for P−Δ effect

The P−Δ effect is the effect of axial load P acting through the rel-ative transverse displacement of the member ends Δ. The end forcesand displacements used in Eq. (1) are shown in Fig. 4(a). The signconvention for the positive directions of element end forces and dis-placements of a finite element is shown in Fig. 4(b).

(b) Concrete

Fig. 3. Constitution models of material.

(a) Forces

(b) Displacements

Fig. 4. Element end force and displacement notations.


By comparing the two figures, the equilibrium and kinematic rela-tionships can be expressed in symbolic form as

_fnn o

¼ T½ �T6�12_fe

n oð6aÞ

_den o

¼ T½ �6�12_dL

n oð6bÞ

where _fnn o

and _dL

n oare the incremental end force and displacement

vectors of an element and are expressed as

_fnn oT ¼ rn1 rn2 rn3 rn4 rn5 rn6 rn7 rn8 rn9 rn10 rn11 rn12f g

ð6aÞ

_dL

n oT ¼ d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12f gð6bÞ

and _fen o

and _de

n oare the incremental end force and displacement

vectors in Eq. (1). [T]6×12 is a transformation matrix written as

T½ �6�12 ¼

−1 0 0 0 0 0 1 0 0 0 0 00 0 −1

L0 1 0 0 0

1L

0 0 0

0 0 −1L

0 0 0 0 01L

0 1 0

01L

0 0 0 1 0 −1L

0 0 0 0

01L

0 0 0 0 0 −1L

0 0 0 10 0 0 1 0 0 0 0 0 −1 0 0

26666666666664

37777777777775

ð7Þ

Using the transformation matrix by equilibrium and kinematic re-lations, the force–displacement relationship of an element may bewritten as

_f nn o

¼ Kn½ � _dL

n oð8Þ

[Kn] is the element stiffness matrix expressed as

Kn½ �12�12 ¼ T½ �T6�12 Ke½ �6�6 T½ �6�12 ð9Þ

Eq. (9) can be partitioned as

Kn½ �12�12 ¼ Kn½ �1 Kn½ �2Kn½ �T2 Kn½ �3

ð10Þ

where

Kn½ �1 ¼

a 0 0 0 0 00 b 0 0 0 c0 0 d 0 −e 00 0 0 f 0 00 0 −e 0 g 00 c 0 0 0 h

26666664

37777775

ð11aÞ

Kn½ �2 ¼

−a 0 0 0 0 00 −b 0 0 0 c0 0 −d 0 −e 00 0 0 −f 0 00 0 e 0 i 00 −c 0 0 0 j

26666664

37777775

ð11bÞ

Kn½ �3 ¼

a 0 0 0 0 00 b 0 0 0 −c0 0 d 0 e 00 0 0 f 0 00 0 e 0 m 00 c 0 0 0 n

26666664

37777775

ð11cÞ

where

a ¼ EtAL

b ¼ Ciiz þ 2Cijz þ Cjjz

L2c ¼ Ciiz þ Cijz

L

d ¼ Ciiy þ 2Cijy þ Cjjy

L2e ¼ Ciiy þ Cijy

Lf ¼ GJ

Lg ¼ Ciiy h ¼ Ciiz i ¼ Cijy j ¼ Cijz m ¼ Cjjy n ¼ Cjjz

ð12Þ

Eq. (8) is used to enforce no side-sway in the beam–columnmem-ber. If the beam–column member is permitted to sway, additionalaxial and shear forces will be induced in the member. These addition-al axial and shear forces due to a member sway to the member enddisplacements can be related as

_f sn o

¼ Ks½ � _dL

n oð13Þ

where _f sn o

, _dL

n o, and [Ks] are incremental end force vector, end dis-

placement vector, and the element stiffness matrix. They may bewritten as

_f sn oT ¼ rs1 rs2 rs3 rs4 rs5 rs6 rs7 rs8 rs9 rs10 rs11 rs12f g

ð14aÞ

_dL

n oT ¼ d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12f gð14bÞ

Ks½ �12�12 ¼ Ks½ � − Ks½ �− Ks½ �T Ks½ �

ð14cÞ

where

Ks½ � ¼

0 a −b 0 0 0a c 0 0 0 0−b 0 c 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

26666664

37777775

ð15Þ

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0

Euler's theoretical solution

Fiber hinge element (proposed)

CRC curve

Fiber hinge element (proposed)residual stress included

P/P

y

λcy

Fig. 6. Strength curve of pinned-ended steel column.


and

a ¼ MzA þMzB

L2; b ¼ MyA þMyB

L2; c ¼ P

Lð16Þ

By combining Eqs. (8) and (13), we obtain the general beam–columnelement force–displacement relationship as

_fLn o

¼ K½ �local _dL

n oð17Þ

where

_fLn o

¼ _fnn o

þ _f sn o

ð18Þ

K½ �local ¼ Kn½ � þ Ks½ � ð19Þ

3. Numerical Examples

An analysis program developed based on the above-mentionedformulation is verified for accuracy and efficiency by the comparisonsof its predictions with the experimental test results, available exactsolution, and the results obtained by the use of the commercial finiteelement package ABAQUS [14] and the spread-of-plasticity methods.In the numerical modeling created by the proposed program, eachframe member is modeled as one or two beam–column elementsusing the proposed fiber plastic hinge element. The *CONCRETEDAMAGED PLASTICITY option in ABAQUS is used for modelingconcrete. The *CONCRETE COMPRESSION HARDENING option is usedto defined the stress–strain behavior of concrete in uniaxial compres-sion outside the elastic range following the nonlinear curve ofEq. (28). Compressive stress data are provided as a tabular function ofinelastic strain.

3.1. Steel Pinned-Ended Column

The nonlinear analyses are performed for axially compressed steelpinned-ended column as shown in Fig. 5 to verify the accuracy of theprogram in capturing second-order, inelastic, and residual stress ef-fects. The section isW8×31 and the yield stress and Young's modulusof the material are E=200 GPa and σy=250MPa, respectively. Theradius of gyration about weak-axis of the section is ry=51.2mm.Only one proposed element is used to model the column. Two casesof excluding and including residual stresses are surveyed.

Fig. 6 presents a comparison of buckling loads obtained by theproposed program's analysis, the above-mentioned Euler's theoretical

P

W8x

31

L

Fig. 5. Pined-ended column under axially compressed load.

exact solution, and CRC column curve (Chen and Lui [11]) with a largerange of the column length. Since the proposed element is based onthe stability functions derived from the governing differential equa-tion of beam–column, it is capable of predicting the exact bucklingload of the column by the use of only one element per member inmodeling. Whereas, as stated by Liew et al. [15], the cubic elementin the common finite element method over-predicts the bucklingloads by about 20% if the pinned-ended column is modeled by oneelement. The strength curves corresponding to the slendernessparameter about weak axis λcy of all cases are shown in Fig. 6. It canbe seen that the curves are almost identical for both cases with themaximum error of about 2%. This example demonstrates the accuracyand efficiency of the proposed element in predicting the bucklingloads of the column.

3.2. Continuous composite beam tested by Ansourian

Six continuous steel–concrete composite beams tested byAnsourian[16] are often used as benchmark tests by other researchers. In thisstudy, the two-span continuous beam indicated as CTB1 by Ansourianis used to verify the accuracy of the present method. Pi et al. [5] usedthe distributed plasticity finite element method to model this beam.This beam has two spans 4 m and 5 m long and is loaded by a con-centrated load at the mid-length of the shorter span as shown inFig. 7. The cross-section of CTB1 consisted of an IPE200 steel section(flanges 8.5 mm×100 mm, web 183 mm×6.5 mm) and a concreteslab 100 mm×800 mm. The shear connection consisted of 66 welded

P = 200 kN

A

A

L = 4 m

A - A

IPE 200

800 mm

100

mm

L = 5 m

v

Fig. 7. Continuous composite beam CTB1.

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60

Deflection, v (mm)

Load

fact

or

Experiment, Ansourian (1981)

Plastic zone method, Pi et al. (2006)


Fig. 8. Load–displacement curves of composite beam CTB1.

0.0

0.3

0.6

0.9

1.2

1.5

0 30 60 90 120 150 180

Displacement, u (mm)

Load

fact

or

Shell and solid elements, ABAQUS


Fig. 10. Load–displacement curves of composite portal frame.


studs 19 mm×75mm, resulting in a connection strength of 150% of therequired strength in positive bending and 160% in negative bending.Therefore, as mentioned by Ansourian, the effects of slip from thetest were relatively small. Therefore, the interaction between the steelI-beam and concrete slab can be considered to be fully restrained. Thecontribution of rebars into the strength of concrete slab is assumedto be negligible in this study. The compressive strength of concreteis fc′=30 MPa, the yield strength of steel fy=277 MPa, and the elasticmodulus of steel E=2×105 MPa.

Two proposed elements and four finite elements of Pi et al. [5] areused to model each member in the continuous beam. As shown inFig. 8, the load–displacement curves obtained from the proposedmethod achieves a good approximation of the distributed analysisof Pi et al. [5] while those of both numerical analyses are slightlydifferent to the experimental curve.

3.3. Composite portal frame

Fig. 9 shows a steel–concrete composite portal frame comprising of asteel–concrete composite beam rigidly connected to two steel columns.

A - A

W12x27

1219 mm

102

mm

P

P = 150 kN

W12

x50

H =

5 m

W12

x 50

A

A

u

L = 8 m

4 m 4 m

Fig. 9. Steel–concrete composite portal frame.

The compressive cylinder strength of concrete is fc′=16MPa and theultimate strain isεu=0.00806. For steel material, the yield strength isfy=252.4MPa, the elastic modulus E=2×105 MPa, and the strainhardening modulus ES=6×103 MPa. The beam-section consists of aW12×27 steel section and a concrete slab 102 mm×1219mm. TheW12×50 section is used for the columns. A concentrated loadP=150 kN is applied at the mid-length of the beam while another lat-eral concentrated force with the same value is applied into the top ofthe left column. The vertical and lateral loads are proportionally appliedto the structure until the structure is collapsed. To predict the nonlinearbehavior of the structure, the column and beam aremodeled by the useof one and two proposed elements, respectively. For ABAQUSmodelingserved for verification purpose, the bare steel frame ismodeled by using5852 quadrilateral shell elements S4R and the concrete slab is modeledby using 5376 hexahedral solid elements C3D8R. The top flange area ofthe steel beam and the corresponding concrete slab area is fully con-strained by using the *TIE function of ABAQUS to simulate a fully com-posite interaction between two components.

The load–displacement curves obtained by ABAQUS and the pro-posed programs are shown in Fig. 10. It can be seen that the curvescorrelate well. With using the same Intel Pentium 2.21 GHz, 3.00 GBof RAM computer, the computational times of the ABAQUS and pro-posed programs for this problem are 48 min and 20 s, respectively.This result proves the high computational efficiency of the proposedcomputer program.

Fig. 11. Geometry and dimension of steel arch bridge.

W21x101

800 mm

200

mm

Fig. 12. Composite section of tie beams. 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.000 0.001 0.002 0.003 0.004

Mid-span displacement ratio, v/L

Load

fact

or (

λ)

Bare steel bridge

Bridge considering concrete slab

1.123

1.198

Limit Loads

Fig. 14. Load–displacement curves of steel arch bridge.


3.4. Steel arch bridge with concrete slab

Fig. 11 shows a steel arch bridge which is 7.32 m (24 ft) wide and61.0 m (200 ft) long. The elastic modulus of E=2×105 MPa and yieldstress of fy=248MPa are used for steel material. For the concretematerial, the compressive cylinder strength is fc′=27.58MPa and the ul-timate strain is εu=0.00467. To include the effect of the concrete slab,the steel–concrete composite section comprised of the steel beamW21×101 and reinforced concrete slab 200 mm×800 mm as shownin Fig. 12 is used for the bridge tie beams. The steel square box sectionof 24×24×1/2 is used for the arch ribs while the wide flange sectionof W8×10 and W10×22 are used for the vertical truss members andthe lateral braces of the bridge, respectively. Fig. 13 shows the designloads applied to the structure as concentrated vertical and lateral loads.

Each member of the bridge is modeled by one proposed fiber–hinge element. In order to evaluate the increase in strength of thebridge when the composite action of concrete slab is consideredinto the load-carrying capacity of the steel tie, two analysis casesare performed: (1) the bare steel bridge; (2) the bridge with compos-ite section of tie.

The load–displacement curves of the analyses at the mid-span ofmiddle tie beam in two above-mentioned cases are shown inFig. 14. The bare steel arch bridge encountered ultimate state whenthe applied load ratio reached 1.123. The system resistance factor of0.95 is used since the bare steel bridge collapsed by tension yieldingat the vertical truss member. Since the ultimate load ratio λ resultsin 1.07 (=1.123×0.95) which is greater than 1.0, the member sizesof the bridge are adequate for strength requirement. It can be seenthat when the concrete slab is considered in the modeling, the stiff-ness of the bridge is significantly increased. However, the ultimateload factor of the bridge considering concrete slab slightly increases6.3% compared to that of the bare steel bridge.

(a) Vertical load

(b) Lateral load

30λ

167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ

(Units : kN)

7.32

m

30λ 30λ 30λ 30λ 30λ 30λ 30λ 30λ

Fig. 13. Load conditions of steel arch bridge.

4. Conclusions

In an effort of reducing computational expense and supplying struc-tural engineers with a reliable and efficient tool in daily engineering de-sign, a practical nonlinear analysis program for predicting nonlinearbehavior of steel–concrete composite frame structures is proposed.Based on the lumped plasticity concept, the fiber–hinge element com-prised of two exterior fiber segments and one interior elastic segmentis systematically developed so that the gradual reduction in strengthof section and element of steel and steel–concrete composite membersis reasonably captured. The stability functions are used for middle elas-tic segment to capture the second-order effect of themember. Using thestatic condensation algorithm, the elementwith three segments is trea-ted as general twelve degree-of-freedom beam–column element. Thishelps the proposed element is easily adaptive to the existing programin order to shorten the coding time and, more importantly, reducesthe number of degrees of freedom of the total structure matrix for stor-age and computational efficiency. The proposed fiber–hinge elementcan be considered as the hybrid elementwhich integrates the dominantcharacteristics of both common plastic hinge and finite elementmethods. As shown in a variety of numerical examples, the proposedmethod using only one or two elements per member is capable of con-ducting a relatively accurate result compared with time-consumingcontinuum and distributed plasticity methods that need to modelmany elements for one member. For large-scale structures with manyelements and degrees of freedom in numerical modeling as normallyencountered in a real design, this efficiency is really significant. The pro-posed analysis program can be used to evaluate the strength and stabil-ity of steel–concrete composite structures as an integrated systemrather than a group of individual members. This is very efficient to de-sign the required structure with uniform safe factor and then bringsabout economic efficiency. It can be concluded that the proposed nu-merical procedure is simple but efficient for use in practical design.

Acknowledgements

This work was supported by the National Research Foundation ofKorea (NRF) grant funded by the Korea government (MEST) (No.2011-0030847), and the Human Resources Development of the KoreaInstitute of Energy Technology Evaluation and Planning (KETEP) grantfunded by the Korea government Ministry of Knowledge Economy(No. 20104010100520).

References

[1] Baskar K, Shanmugam NE, Thevendran V. Finite-element analysis of steel-concrete composite plate girder. J Struct Eng 2002;128(9):1158–68.

[2] Barth KE, Wu H. Efficient nonlinear finite element modeling of slab on steel stringerbridges. Finite Elem Anal Des 2006;42(14–15):1304–13.

image of Fig.�12


[3] Ayoub A, Filippou FC. Mixed formulation of nonlinear steel-concrete compositebeam element. J Struct Eng 2000;126(3):371–81.

[4] Salari MR, Spacone E. Finite element formulations of one-dimensional elementswith bond-slip. Eng Struct 2001;23(7):815–26.

[5] Pi YL, Bradford MA, Uy B. Second-order nonlinear inelastic analysis of compositesteel-concrete members. Part II. J Struct Eng 2006;132(5):751–71.

[6] McKenna F, Fenves GL, Filippou FC, Scott MH. Open system for earthquake engi-neering simulation (OpenSees). Berkeley (CA): Pacific Earthquake EngineeringResearch Center, University of California; 2006. http://opensees.berkeley.edu.

[7] Porter FL, Powell GH. Static and dynamic analysis of inelastic frame structures.Report No. EERC 71–3. Berkeley (CA: University of California; 1971.

[8] El-Tawil S, Deierlein GG. Nonlinear analysis ofmixed steel-concrete frames. I: Elementformulation. J Struct Eng 2001;127(6):647-55.c.

[9] Liu S-W, Lui Y-P, Chan S-L. Advanced analysis of hybrid steel and concrete frames.Part 1: Cross-section analysis technique and second-order analysis. J Constr SteelRes 2012;70:326–36.

[10] Ngo-Huu C, Kim S-E. Practical advanced analysis of space steel frames using fiberhinge method. Thin Walled Struct 2009;47(4):421–30.

[11] Chen WF, Lui EM. Structural Stability – Theory and Implementation. New York:Elsevier; 1987.

[12] Wilson EL. The static condensation algorithm. Int J Numer Methods Eng 1974;8:198–203.

[13] Eurocode-2. Design of concrete structures – Part 1–1: General rules and rules forbuildings. Brussels (Belgium): European Committee for Standardization (CEN),Eurocode-2; 1992.

[14] ABAQUS. ABAQUS/Standard, Version 6.5. User's Manual. Providence (Rhode Island):ABAQUS Inc.; 2004.

[15] Liew JYR, Chen H, Shanmugam NE, Chen WF. Improved nonlinear plastic hingeanalysis of space frame structures. Eng Struct 2000;22(10):1324–38.

[16] Ansourian P. Experiments on continuous composite beams. Proc Inst Civ Eng1981;71(2):25–51.

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