DSpace at VNU: Practical nonlinear analysis of steel–concrete composite frames using fiber–hinge method

Journal of Constructional Steel Research 74 (2012) 90–97 Contents lists available at SciVerse ScienceDirect Journal of Constructional Steel Research Practical nonlinear analysis of steel–concrete composite frames using fiber–hinge method Cuong NGO-HUU a, 1, Seung-Eock KIM b,⁎ a b Faculty of Civil Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet St., Dist 10, Ho Chi Minh City, Vietnam Department of Civil and Environmental Engineering, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea a r t i c l e i n f o Article history: Received 26 July 2011 Accepted 28 February 2012 Available online 28 March 2012 Keywords: Steel–concrete composite frames Nonlinear analysis Fiber–hinge method Stability functions a b s t r a c t A fiber–hinge beam–column element considering geometric and material nonlinearities is proposed for modeling steel–concrete composite structures The second-order effects are taken into account in deriving the formulation of the element by the use of the stability functions To simulate the inelastic behavior based 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 element comprising of three segments is treated as one general element with twelve degrees of freedom The mid-length cross-section of the end fiber segment is divided into many fibers of which the uniaxial material stress–strain relationship is monitored during analysis process The proposed procedure is verified for accuracy and efficiency through comparisons to the results obtained by the ABAQUS structural analysis program and established results available from the literature and tests through a variety of numerical examples The proposed procedure proves to be a reliable and efficient tool for daily use in engineering design of steel and steel–concrete composite structures © 2012 Elsevier Ltd All rights reserved Introduction Steel–concrete composite structures comprised of steel, reinforced concrete, and steel–concrete composite members have widely used for constructing buildings and bridges due to their efficiency in structural, economic and construction aspects Therefore, extensive experimental and theoretical studies have been conducted to provide a better understanding on the behavior of the composite structure and its components under applied loading Together with the more and more application of the composite structures, there are increasing needs in having a reliable structural analysis program capable of predicting the second-order inelastic response of steel–concrete composite structures Recently, as the design profession moves towards a performance-based approach, the accurate detailed information on how a structure behaves under different levels of loads is necessary in evaluation of the expected level of performance Obviously, this requires a comprehensive analysis procedure that can consider all key factors influencing the strength of structure and produce results consistent with the current design code requirements with sufficient accuracy For daily design purpose, the nonlinear analysis program should be able to get the reliable results in a minimized time, especially in a time-consuming earthquake-resistant design The degree ⁎ Corresponding author Tel.: + 82 3408 3291; fax: + 82 3408 3332 E-mail address: sekim@sejong.ac.kr (S.-E KIM) Formerly Adjunct Researcher of Constructional Technology Institute, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul 143-747, South Korea 0143-974X/$ – see front matter © 2012 Elsevier Ltd All rights reserved doi:10.1016/j.jcsr.2012.02.018 of success in predicting the nonlinear load–displacement response of frame structures significantly depends on how the nonlinear effects to be simulated in numerical modeling The steel and concrete components can be modeled separately using plate, shell and solid elements of available commercial threedimensional nonlinear finite element packages or self-developed programs of researchers and then are assembled together by some connection or interface elements to simulate the shear connectors/ interaction between these components, as recently presented by Baskar et al [1] and Barth and Wu [2], among others This continuum method can best capture the nonlinear response of the composite structures and is usually used instead of conducting the high cost and time-consuming full-scale physical testing However, in order to model a complete structure, so many shell, plate, and solid finite elements 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 and lumped plasticity approaches based on the degree of refinement used to represent inelastic behavior The distributed method uses the highest refinement while the lumped method allows for a significant simplification The beam–column member in the former is divided into many finite elements and the cross-section of each element is further modeled by fibers of which the stress–strain relationships are monitored during the 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 spreading throughout the cross-section and along the member length The residual C 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 constant value since the fibers are sufficiently small The solution of the distributed method can be considered to be relatively accurate and easily be included the coupling effects among of axial, lateral, and torsion deformations However, it is generally recognized that this method is too computationally intensive and hence usually applicable only for research purposes (e.g., checking and calibrating the accuracy of simplified inelastic analysis methods, and establishing design charts and equations) because a very refined discretisation of the structure is necessary and the numerical integration procedure is relatively timeconsuming, especially for large-scale space structures as normally encountered in design Therefore, it is not efficient to apply them in a daily practical design The beam–column member in the lumped method is modeled by an appropriate method eliminating its further subdivision, and the plastic hinges representing the plastic interaction between axial force and the biaxial moments are assumed to be lumped at both ends of the member This plastic hinge is usually based on a specific yield surface and an approximate function to simulate the gradual yielding of the cross-section Although this method is less accurate in comparison with the distributed method, it was shown to be very simple, fast, and capable of providing results accurate enough for practical design, as presented by Porter and Powell [7], El-Tawil and Deierlein [8] and Liu et al [9] However, this method is usually applied for nonlinear analysis of frame structures composed of steel, reinforced concrete and encased composite members because the yield surface for steel and reinforced concrete composite section, especially for steel I-beam and reinforced concrete slab section, is not always available and accurate for every section Moreover, the gradual reduction in strength of the general composite section under gradual loading is not easy to model In this research, to take advantage of computational efficiency of the common lumped approach and overcome its above-mentioned weakness, a fiber–hinge beam–column element is introduced to model the steel and steel-composite composite members This is a development from the work done by Ngo-Huu and Kim [10] for nonlinear analysis of steel space structures The proposed element is divided into two end fiber–hinge segments and an interior elastic segment to simulate the inelastic behavior of the material The cross-section at mid-length of end fiber–hinge segment is divided into steel and/or concrete fibers so that the uniaxial stress–strain relationship of cross-sectional fibers can be monitored during the analysis process based on the relevant constitutive model and the flow theory of plasticity This is a good alternative for inelastic representation instead of using the specific yield surface in usual plastic hinge model Herein, the stability functions obtained from the exact buckling solution of a beam–column subjected to end forces are used to accurately capture the second-order effects The nonlinear responses of structures in a variety of numerical examples of steel and steel–concrete composite frames are compared with the existing exact solutions, the results from the experiments, and those obtained by the finite element package ABAQUS and plastic zone analyses to show the reliability and efficiency of the proposed approach in applying for practical design purpose Reductions of torsional and shear stiffnesses are not considered in the fiber–hinge The connection and bond between member and its components are perfect The panel–zone deformation of the beam-to-column joint is neglected 2.2 Beam–column element accounting for P − δ Effect To capture the effect of the axial force acting through the lateral displacement of the beam–column element relative to its chord (P − δ effect), the stability functions are used to minimize modeling and solution time Generally only one element is needed per a physical member in modeling to accurately capture the P − δ effect Similar to the formulation procedure presented by Chen and Lui [11], the incremental force– displacement relationship of the space beam–column element may be expressed as [10] ðEAÞc _ > P > L > > > > _ > > M > yA > > > > = < _ > M yB ¼6 _ > > >M > zA > > > > > _ > > >M > ; : zB > T_ 0 0 kiiy kijy 0 kijy kiiy 0 0 kiiz kijz 0 kijz kiiz 0 0 δ_ > 7> > > > > > θ_ yA > > > > 7> > > < 7 θ_ yB = 7> _ > > > > θ zA > > > θ_ > 7> > > zB > 5> ; : ðGJ Þc _ϕ L ð1Þ kiin ¼ S1n ðEI n Þc L ð2aÞ kijn ¼ S2n EI n ịc L 2bị and m X EAịc ẳ 3aị Ei Ai iẳ1 EI n ịc ẳ m X 3bị Ei ni Ai iẳ1 GJ ịc ẳ m X   2 Gi yi ỵ zi Ai 3cị iẳ1 in which m is the total number of fibers divided on the monitored crosssection; Ei and Ai are the tangent modulus of the material and the area of i th fiber, respectively; yi and zi are the coordinates of i th fiber in the cross-section; S1n and S2n (n = y, z) are the stability functions with respect to y and z axes, and are shown as S1n The following assumptions are made in the formulation of the composite beam–column element: All elements are initially straight and prismatic Plane crosssection remains plane after deformation Local buckling and lateral–torsional buckling are not considered All members are assumed to be fully compact and adequately braced Large displacements are allowed, but strains are small _ ,M _ , P_ , and T_ are incremental end moments with respect to where M nA nB _ and ϕ_ n axis (n = y, z), axial force, and torsion, respectively; θ_ nA , θ_ nB , δ, are incremental joint rotations with respect to n axis, axial displacement, and the angle of twist, respectively; Formulation 2.1 Basic assumptions 91 S2n pffiffiffiffiffiffi À pffiffiffiffiffiffiÁ À pffiffiffiffiffiffiÁ > π ρn sin π ρn −π ρn cos π ρn > > À Á > if P b p p ffiffiffiffiffi ffi ffiÁ ffiffiffiffiffi ffi < 2−2 cos π ρ −π ρ sinÀπ pffiffiffiffiffi ρn n n ¼ À Á À Á p p ffiffiffiffiffi ffi p ffiffiffiffiffi ffi ffiffiffiffiffi ffi > π2 ρn cosh π ρn −π ρn sinh π ρn > > À pffiffiffiffiffiffiÁ if P > > p : 22 cosh p n ỵ ρn sinh π ρn ð4aÞ pffiffiffiffiffiffi À pffiffiffiffiffiffiÁ π2 ρn −π ρn sin π ρn > > > À pffiffiffiffiffiffiÁ ffi À pffiffiffiffiffiffiÁ > < 2−2 cos π ρ −π pffiffiffiffiffi ρn sin π ρn n ¼ pffiffiffiffiffi pffiffiffiffiffi > π ρy sinh π ρy −π2 ρy > > > À pffiffiffiffiffiffiÁ : pffiffiffiffiffiffi À pffiffiffiffiffiffiÁ 2−2 cosh π n ỵ n sin n 4bị if P b if P > where ρn = P/(π2EIn/L2) with n = y, z and P is positive in tension 92 C NGO-HUU, S.-E KIM / Journal of Constructional Steel Research 74 (2012) 90–97 y 2.3 Beam–column element accounting for material nonlinearity To model the material nonlinearity based on the concentrated plasticity approximation, the beam–column element is modeled by two end fiber segments and a middle elastic segment as shown in Fig In order to monitor the gradual plastification throughout the element's cross-section, the cross-section located at the mid-length of the end segment is divided into many fibers to track the inelastic behavior of the section and element (Fig 2) Each fiber is represented by its area and coordinate location corresponding to its centroid The interior segment is assumed to behave elastically similar to the elastic part in the common plastic hinge For hot-rolled steel section, the residual stresses are directly assigned to fibers as the initial stresses The ECCS residual stress pattern of I-shape hot-rolled steel section is used for this study Because the hybrid element has three segments with 24 DOFs, a static condensation method developed by Wilson [12] is applied so that the element is treated as one element with 12 DOFs normally found in a general beam–column element Twelve DOFs of two interior nodes (nodes and in Fig 1) must be condensed out to leave twelve DOFs of two exterior nodes (nodes I and J, also denoted as nodes and in Fig 1) This reduces the computational time when assembling the total stiffness matrix and solving the system of linear equations In addition, this twelve DOF element is adaptive with the existing beam–column program so that the coding time can be reduced However, it also requires that a reverse condensation needs be performed in order to compute the deformations at the ends of the segments to evaluate the degree of yielding of the fiber–hinge In a general nonlinear analysis, the element stiffness matrices of the current step are evaluated based on the state of the system determined at the end of previous step Once the incremental fiber strain of the cross section is evaluated, the flow theory of plasticity is applied to determine the incremental fiber stress based on the relevant uniaxial material stress–strain relationship represented in the following section The strain hardening of the steel material that causes an increase in member strength is considered These let the proposed approach simulate a more realistic behavior of structure than the common plastic hinge method does bs Concrete fiber ds z d Steel fiber bf Fig The partition of monitored cross-section into fibers 2.5 Element stiffness matrix accounting for P − Δ effect The P − Δ effect is the effect of axial load P acting through the relative transverse displacement of the member ends Δ The end forces and displacements used in Eq (1) are shown in Fig 4(a) The sign convention for the positive directions of element end forces and displacements of a finite element is shown in Fig 4(b) 2.4 Constitutive model of material The constitutive material models in explicit functions of strain for steel and concrete recommended by Eurocode-2 [13] are used in this research as shown in Fig The tensile strength of concrete is neglected The concrete stress–strain relation in compression is described as following expressions   ! ε n σ ¼ fc′ 1− 1− for ≤ ε ≤ ε0 ε0 ð5aÞ ẳ fc for b u 5bị (a) Steel where f′c is the concrete compressive cylinder strength; n is the exponent; ε0 is the concrete strain at maximum stress and εu is the ultimate strain For fc′ ≤ 50 MPa, n = and ε0 = 0.002 and these values are used for all relevant examples presented in this research (b) Concrete Fig Beam–column element comprising of three segments Fig Constitution models of material C NGO-HUU, S.-E KIM / Journal of Constructional Steel Research 74 (2012) 90–97 93 Eq (9) can be partitioned as ½K n 1212 ẳ ẵK n ẵK n T2 ẵK n ẵK n ! 10ị where a 60 60 ẵK n ẳ 60 40 (a) Forces 6 ½K n Š2 ¼ 6 (b) Displacements By comparing the two figures, the equilibrium and kinematic relationships can be expressed in symbolic form as ð6aÞ where n o n o d_ e ẳ ẵT 612 d_ L 6bị a¼ n o n o where f_n and d_ L are the incremental end force and displacement vectors of an element and are expressed as n oT f_n ¼ f r n1 r n2 r n3 r n4 r n5 r n6 rn7 r n8 r n9 r n10 r n11 r n12 g ð6aÞ d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 g ð6bÞ n o n o and f_e and d_ e are the incremental end force and displacement vectors in Eq (1) [T]6 × 12 is a transformation matrix written as ½T Š6Â12 6 6 6 ¼6 6 6 −1 0 0 1 − L 0 − L 0 L 0 L 0 0 0 0 0 0 0 0 L L 0 1 − 0 L 0 − 0 L 0 0 −1 0 07 7 07 7 07 7 15 T 0 0 −d 0 −f e 0 ð11aÞ 0 −e i c7 07 07 05 j ð11bÞ −c 7 7 7 n ð11cÞ ð12Þ Eq (8) is used to enforce no side-sway in the beam–column member If the beam–column member is permitted to sway, additional axial and shear forces will be induced in the member These additional axial and shear forces due to a member sway to the member end displacements can be related as n o n o f_ s ¼ ½K s Š d_ L ð13Þ n on o where f_ s , d_ L , and [Ks] are incremental end force vector, end displacement vector, and the element stiffness matrix They may be written as r s2 r s3 r s4 r s5 r s6 r s7 r s8 r s9 rs10 r s11 rs12 g ð14aÞ ð7Þ n oT d_ L ¼ f d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 g ð14bÞ 8ị [Kn] is the element stiffness matrix expressed as ẵK n 1212 ẳ ẵT 612 ẵK e 66 ẵT 612 c7 07 07 05 h 0 e g C iiz ỵ 2C ijz ỵ C jjz C iiz ỵ C ijz Et A bẳ cẳ L L L2 C iiy ỵ 2C ijy ỵ C jjy C iiy ỵ C ijy GJ f ¼ e ¼ d¼ L L L2 g ¼ C iiy h ¼ C iiz i ¼ C ijy j ¼ C ijz m ¼ C jjy n ¼ C jjz n oT f_ s ¼ f r s1 Using the transformation matrix by equilibrium and kinematic relations, the force–displacement relationship of an element may be written as n o n o f_n ẳ ẵK n d_ L −a 0 −b 0 0 0 −c 0 f 0 a 0 0 60 b 0 60 d e ẵ K n ẳ 60 0 f 40 e m c 0 n o n o T f_n ẳ ẵT 612 f_e d2 0 d −e Fig Element end force and displacement notations n oT d_ L ¼ f d1 b 0 c 9ị ẵK s 1212 ẳ ½K s Š −½K s ŠT −½K s Š ½K s Š ! ð14cÞ where a 6 b ẵK s ẳ 6 0 a −b c 0 c 0 0 0 0 0 0 0 0 0 07 07 07 05 ð15Þ 94 C NGO-HUU, S.-E KIM / Journal of Constructional Steel Research 74 (2012) 9097 and Euler's theoretical solution 1.0 bẳ MyA ỵ MyB L2 ; P c¼ L 0.8 By combining Eqs (8) and (13), we obtain the general beam–column element force–displacement relationship as n o n o f_L ẳ ẵK local d_ L Fiber hinge element (proposed) ð16Þ ð17Þ CRC curve Fiber hinge element (proposed) residual stress included 0.6 P/Py M þM a ¼ zA zB ; L 0.4 where 0.2 n o n o n o f_L ¼ f_n ỵ f_ s 18ị ẵK local ẳ ẵK n ỵ ẵK s 19ị 0.0 0.0 1.0 2.0 cy 3.0 4.0 5.0 Fig Strength curve of pinned-ended steel column Numerical Examples An analysis program developed based on the above-mentioned formulation is verified for accuracy and efficiency by the comparisons of its predictions with the experimental test results, available exact solution, and the results obtained by the use of the commercial finite element package ABAQUS [14] and the spread-of-plasticity methods In the numerical modeling created by the proposed program, each frame member is modeled as one or two beam–column elements using the proposed fiber plastic hinge element The *CONCRETE DAMAGED PLASTICITY option in ABAQUS is used for modeling concrete The *CONCRETE COMPRESSION HARDENING option is used to defined the stress–strain behavior of concrete in uniaxial compression outside the elastic range following the nonlinear curve of Eq (28) Compressive stress data are provided as a tabular function of inelastic strain exact solution, and CRC column curve (Chen and Lui [11]) with a large range of the column length Since the proposed element is based on the stability functions derived from the governing differential equation of beam–column, it is capable of predicting the exact buckling load of the column by the use of only one element per member in modeling Whereas, as stated by Liew et al [15], the cubic element in the common finite element method over-predicts the buckling loads by about 20% if the pinned-ended column is modeled by one element The strength curves corresponding to the slenderness parameter about weak axis λcy of all cases are shown in Fig It can be seen that the curves are almost identical for both cases with the maximum error of about 2% This example demonstrates the accuracy and efficiency of the proposed element in predicting the buckling loads of the column 3.1 Steel Pinned-Ended Column 3.2 Continuous composite beam tested by Ansourian The nonlinear analyses are performed for axially compressed steel pinned-ended column as shown in Fig to verify the accuracy of the program in capturing second-order, inelastic, and residual stress effects The section is W8 × 31 and the yield stress and Young's modulus of the material are E = 200 GPa and σy = 250 MPa, respectively The radius of gyration about weak-axis of the section is ry = 51.2 mm Only one proposed element is used to model the column Two cases of excluding and including residual stresses are surveyed Fig presents a comparison of buckling loads obtained by the proposed program's analysis, the above-mentioned Euler's theoretical Six continuous steel–concrete composite beams tested by Ansourian [16] are often used as benchmark tests by other researchers In this study, the two-span continuous beam indicated as CTB1 by Ansourian is used to verify the accuracy of the present method Pi et al [5] used the distributed plasticity finite element method to model this beam This beam has two spans m and m long and is loaded by a concentrated load at the mid-length of the shorter span as shown in Fig 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 concrete slab 100 mm× 800 mm The shear connection consisted of 66 welded P P = 200 kN A v A L=5m 800 mm IPE 200 100 mm L W8x31 L=4m A-A Fig Pined-ended column under axially compressed load Fig Continuous composite beam CTB1 1.0 1.5 0.8 1.2 Load factor Load factor C NGO-HUU, S.-E KIM / Journal of Constructional Steel Research 74 (2012) 90–97 0.6 0.4 95 0.9 0.6 Experiment, Ansourian (1981) 0.2 Plastic zone method, Pi et al (2006) Shell and solid elements, ABAQUS 0.3 Fiber hinge element (proposed) Fiber hinge element (proposed) 0.0 0.0 10 20 30 40 50 60 Deflection, v (mm) 30 60 90 120 150 180 Displacement, u (mm) Fig Load–displacement curves of composite beam CTB1 Fig 10 Load–displacement curves of composite portal frame studs 19 mm× 75 mm, resulting in a connection strength of 150% of the required strength in positive bending and 160% in negative bending Therefore, as mentioned by Ansourian, the effects of slip from the test were relatively small Therefore, the interaction between the steel I-beam and concrete slab can be considered to be fully restrained The contribution of rebars into the strength of concrete slab is assumed to be negligible in this study The compressive strength of concrete is fc′ = 30 MPa, the yield strength of steel fy = 277 MPa, and the elastic modulus of steel E = × 10 MPa Two proposed elements and four finite elements of Pi et al [5] are used to model each member in the continuous beam As shown in Fig 8, the load–displacement curves obtained from the proposed method achieves a good approximation of the distributed analysis of Pi et al [5] while those of both numerical analyses are slightly different to the experimental curve The compressive cylinder strength of concrete is f′c = 16 MPa and the ultimate strain isεu = 0.00806 For steel material, the yield strength is fy = 252.4 MPa, the elastic modulus E = × 10 MPa, and the strain hardening modulus ES = × 103 MPa The beam-section consists of a W12 × 27 steel section and a concrete slab 102 mm× 1219 mm The W12 × 50 section is used for the columns A concentrated load P = 150 kN is applied at the mid-length of the beam while another lateral concentrated force with the same value is applied into the top of the left column The vertical and lateral loads are proportionally applied to the structure until the structure is collapsed To predict the nonlinear behavior of the structure, the column and beam are modeled by the use of one and two proposed elements, respectively For ABAQUS modeling served for verification purpose, the bare steel frame is modeled by using 5852 quadrilateral shell elements S4R and the concrete slab is modeled by using 5376 hexahedral solid elements C3D8R The top flange area of the steel beam and the corresponding concrete slab area is fully constrained by using the *TIE function of ABAQUS to simulate a fully composite interaction between two components The load–displacement curves obtained by ABAQUS and the proposed programs are shown in Fig 10 It can be seen that the curves correlate well With using the same Intel Pentium 2.21 GHz, 3.00 GB of RAM computer, the computational times of the ABAQUS and proposed programs for this problem are 48 and 20 s, respectively This result proves the high computational efficiency of the proposed computer program 3.3 Composite portal frame Fig shows a steel–concrete composite portal frame comprising of a steel–concrete composite beam rigidly connected to two steel columns P = 150 kN A u A 4m W12x50 W12x50 4m H=5m P L=8m A-A W12x27 102 mm 1219 mm Fig Steel–concrete composite portal frame Fig 11 Geometry and dimension of steel arch bridge 96 C NGO-HUU, S.-E KIM / Journal of Constructional Steel Research 74 (2012) 90–97 Limit Loads 1.198 800 mm 1.2 1.123 200 mm W21x101 Load factor (λ) 1.0 0.8 0.6 0.4 Bare steel bridge 0.2 Bridge considering concrete slab Fig 12 Composite section of tie beams 0.0 0.000 0.001 0.002 0.003 0.004 Mid-span displacement ratio, v/L 3.4 Steel arch bridge with concrete slab Fig 14 Load–displacement curves of steel arch bridge Fig 11 shows a steel arch bridge which is 7.32 m (24 ft) wide and 61.0 m (200 ft) long The elastic modulus of E = × 10 MPa and yield stress of fy = 248 MPa are used for steel material For the concrete material, the compressive cylinder strength is f′c = 27.58 MPa and the ultimate strain is εu = 00467 To include the effect of the concrete slab, the steel–concrete composite section comprised of the steel beam W21 × 101 and reinforced concrete slab 200 mm× 800 mm as shown in Fig 12 is used for the bridge tie beams The steel square box section of 24 × 24 × 1/2 is used for the arch ribs while the wide flange section of W8 × 10 and W10× 22 are used for the vertical truss members and the lateral braces of the bridge, respectively Fig 13 shows the design loads 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 the bridge when the composite action of concrete slab is considered into the load-carrying capacity of the steel tie, two analysis cases are performed: (1) the bare steel bridge; (2) the bridge with composite section of tie The load–displacement curves of the analyses at the mid-span of middle tie beam in two above-mentioned cases are shown in Fig 14 The bare steel arch bridge encountered ultimate state when the applied load ratio reached 1.123 The system resistance factor of 0.95 is used since the bare steel bridge collapsed by tension yielding at the vertical truss member Since the ultimate load ratio λ results in 1.07 (=1.123 × 0.95) which is greater than 1.0, the member sizes of the bridge are adequate for strength requirement It can be seen that when the concrete slab is considered in the modeling, the stiffness of the bridge is significantly increased However, the ultimate load factor of the bridge considering concrete slab slightly increases 6.3% compared to that of the bare steel bridge (Units : kN) Conclusions In an effort of reducing computational expense and supplying structural engineers with a reliable and efficient tool in daily engineering design, a practical nonlinear analysis program for predicting nonlinear behavior of steel–concrete composite frame structures is proposed Based on the lumped plasticity concept, the fiber–hinge element comprised of two exterior fiber segments and one interior elastic segment is systematically developed so that the gradual reduction in strength of section and element of steel and steel–concrete composite members is reasonably captured The stability functions are used for middle elastic segment to capture the second-order effect of the member Using the static condensation algorithm, the element with three segments is treated as general twelve degree-of-freedom beam–column element This helps the proposed element is easily adaptive to the existing program in order to shorten the coding time and, more importantly, reduces the number of degrees of freedom of the total structure matrix for storage and computational efficiency The proposed fiber–hinge element can be considered as the hybrid element which integrates the dominant characteristics of both common plastic hinge and finite element methods As shown in a variety of numerical examples, the proposed method using only one or two elements per member is capable of conducting a relatively accurate result compared with time-consuming continuum and distributed plasticity methods that need to model many elements for one member For large-scale structures with many elements and degrees of freedom in numerical modeling as normally encountered in a real design, this efficiency is really significant The proposed analysis program can be used to evaluate the strength and stability of steel–concrete composite structures as an integrated system rather than a group of individual members This is very efficient to design the required structure with uniform safe factor and then brings about economic efficiency It can be concluded that the proposed numerical procedure is simple but efficient for use in practical design Acknowledgements 167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ 167λ 7.32 m (a) Vertical load 30λ 30λ 30λ 30λ 30λ 30λ 30λ 30λ 30λ (b) Lateral load Fig 13 Load conditions of steel arch bridge This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No 2011-0030847), and the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No 20104010100520) References [1] Baskar K, Shanmugam NE, Thevendran V Finite-element analysis of steelconcrete 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 stringer bridges Finite Elem Anal Des 2006;42(14–15):1304–13 C NGO-HUU, S.-E KIM / Journal of Constructional Steel Research 74 (2012) 90–97 [3] Ayoub A, Filippou FC Mixed formulation of nonlinear 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nonlinear analysis, the element stiffness matrices of the current step are evaluated... forces are used to accurately capture the second-order effects The nonlinear responses of structures in a variety of numerical examples of steel and steel–concrete composite frames are compared with... proves the high computational efficiency of the proposed computer program 3.3 Composite portal frame Fig shows a steel–concrete composite portal frame comprising of a steel–concrete composite beam rigidly