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NONLINEAR RESPONSE OF COMPOSITE STEEL-CONCRETE BOX GIRDER BRIDGESSanjay TiwariIndian Institute of Technology(IIT Roorkee)Roorkee (India)SummaryCellular steel section composite with a concrete deck is one of the most suitablesuperstructures in resisting torsional and warping effects induced by highwayloading. This type of structure has inherently created new design problems forengineers in estimating its load distribution when subjected to moving vehicles.Current composite steel and concrete bridges are designed using full interactiontheory assuming there is no any relative displacement or slip at interface of concreteand steel. However, in the assessment of existing composite bridges thissimplification may not be warranted as it is often necessary to extract the greatestcapacity and endurance from the structure. This may only be achieved using partial-interaction theory which truly reflects the behaviour of the structure. This paperpresents a non linear three dimensional finite element model incorporating the slip ofshear connectors using commercially available software ANSYS, capable ofanalyzing composite box girder bridges of various geometries. The proposed modelhas been validated against the published results from literature. A distribution factorapproach has been suggested as a simplified design method for the preliminaryproportioning of such bridges.Introduction:The use of composite bridges in interchanges of modern highway systems has becomeincreasingly popular for functional, economic as well as aesthetic considerations. Thistype of construction leads to an efficient transverse load distribution due to excellenttorsional stiffness of the section. Further utilities and services can be readily providedwithin the cells. Among the refined methods, the FEM is the most general andcomprehensive technique of analysis capturing all aspects affecting the structuralresponse. But it is too involved and time consuming to be used for routine designpurpose. Practical requirement in the design process necessitate a need for a simplerdesign method. This paper presents a three dimensional non linear model using theFinite element method in which two and four lane bridges of 20m span has beenanalyzed using a commercially available package ANSYS. The effect of cross bracingshas been presented in detail. Results from published literature are used to substantiate theanalytical modeling. Based on the parametric study, load distribution factors are deducedfor such bridges subjected to IRC loadings. Concept of Distribution Factor: The concept of the distribution factor allows the design engineer to considerthe transverse effect of wheel loads in determining the shear and moments of girdersunder the longitudinal as well transverse placement of live loads, thus simplifying theanalysis and design of bridges. According to the approach of the load distribution,maximum shear and moments in bridge are obtained first as if the wheel loads areapplied directly to bridge as a beam. These values are then multiplied by the
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appropriate live-load distribution factors to obtain critical live-load shear andmoments in the different girders in a bridge. M MDF = max ----------- (1) M The above relationship used for calculation of moment distribution factor,MDF, carried by each girder of the bridge, the maximum moment, M, was calculatedin a simply supported girder subjected to one train of IRC (Indian Road Congress:Standard Specifications and Code of Practice for Road Bridges) Class 70R wheelloads. The longitudinal moment carried by each girder of the prototype bridge, Mmax,was calculated by integrating the normal stresses at midspan, determined from thefinite-element analysis. Bridge ModelingA four-node shell element ‘shell43’ with six degrees of freedom at each node wasused to model the concrete deck, steel webs, steel bottom flange, and end diaphragms.A three dimensional two-node beam element ‘beam188’ was adopted to model thesteel top flanges, cross bracings and top-chords. The modeling of shear connector isdone using three mutually perpendicular nonlinear springs which constitutes for thestiffness in three directions viz., stiffness parallel to stud longitudinal axis andstiffness perpendicular to longitudinal axis (one parallel to the bridge axis and oneperpendicular to bridge axis). The shell elements in the top flange were connectedusing ‘combin39’, type nonlinear spring elements, with the elements (flanges) of theweb to ensure load-slip relation. Because of their insignificant flexural and torsionalstiffness, cross-bracing and top-chord members are considered as axial membersloaded in tension and compression. Two different constraints were used in themodeling, namely, the roller support at one end of the bridge, constraining bothvertical and lateral displacements at the lower end nodes of each web, and the hingesupport at the other end of the bridge, restricting all possible translations at the lowerend nodes of each web.Modeling of Shear Connector:- Kuan-Chen Fu and Feng Lu (Kuan-Chen 2003) suggested that the shear studcan be modeled by a bar element, which can be seen as two independent linear springswith a stiffness K N parallel to the longitudinal axis of the bar and K T perpendicularto the axis. Note that EsAs KN = hs -------- (2)Where, Es = elastic modulus; As = area of cross section; and hs = height of stud.Along the tangent surface, the constitutive behaviour is defined by a typical load-slipfunction proposed by Yam and Chapman (Yam 1972), which is P = a (1 − e −by ) --------- (3)Where P = load; a and b = constants; and y = interface slip.By choosing two points on the function such that the relationship y2=2y1 ismaintained, the constants a and b can be determined as P×Pa= 2 P 2 − P1 ---------- (4) 1 P1 b = log y1 P 2 − P1 ---------- (5)
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Therefore, the stiffness in the tangential direction is dP KT = = abe − by dy ------------ (6)In the present problem modeling of shear connector is done using three mutuallyperpendicular nonlinear springs which constitutes for the stiffness in three directionsviz., stiffness parallel to stud longitudinal axis and stiffness perpendicular tolongitudinal axis (one parallel to the bridge axis and one perpendicular to bridge axis).Each bar element provides a dimensionless link between the concrete deck elementand neighboring top flange element of the girder.Material Modelling Material nonlinearity is incorporated in the analysis using nonlinear materialmodel available in ANSYS software. For concrete Drucker-Prager failure criterion isused while for steel bilinear isotropic hardening is used as yielding criterion.Concrete modeling:-The non-linear response of concrete is caused by four major material effects: crackingof the concrete; aggregate interlock; and time dependent effects such as creep,shrinkage, temperature, and load history.In spite of its obvious shortcomings the linear theory of elasticity combined withcriteria defining “failure” of concrete is most commonly used material law forconcrete in reinforced concrete analysis. The linear elastic modeling can besignificantly improved by using the non-linear theory of elasticity.The Drucker-Prager (DP) option available in ANSYS is applicable to granular(frictional) material such as soils, rock, and concrete, and uses the outer coneapproximation to the Mohr-Coulomb law. This option uses the Drucker-Prager yieldcriterion with either an associated or non-associated flow rule. The yield surface doesnot change with progressive yielding, hence there is no hardening rule and thematerial is elastic- perfectly plastic.The equivalent stress for Drucker-Prager is 1 1 T 2σ e = 3βσ m + { S } [ M ]{ S } 2 ------------ (7)Where,σ m = (σ x + σ y + σ z ) 1 3 = mean or hydrostatic stress. ------------ (8){ s} = deviatoric stress[ M ] = plastic compliance matrix. 2 sin φ =β = material constant 3 ( 3 − sin φ ) ------------ (9)Where, φ = angle of internal friction.The material yield parameter is defined as 6c × cos φσy = 3 ( 3 − sin φ ) ------------ (10)Where, c = cohesion value.The yield criterion is then
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1 1 T 2F = 3βσ m + { S } [ M ]{ S } − σ y = 0 2 ------------ (11)This yield surface is cone with material parameters chosen such that it corresponds tothe outer aspices of the hexagonal Mohr-Coulomb yield surface.Fig.1 Mohr-Coulomb and Drucker-Prager yield surfaces.Steel modeling:-Plasticity theory provides a mathematical relationship that characterizes the elasto-plastic response of materials. The yield criterion determines the stress level at whichyielding is initiated. For multi-component stresses, this is represented as a function ofthe individual components, f ({σ}), which can be interpreted as an equivalent stressσe. The material will develop plastic strains. If σe is less than σy, the material is elasticand the stresses will develop according to the elastic stress-strain relations. Theequivalent stress can never exceed the material yield since in this case plastic strainswould develop instantaneously, thereby reducing the stress to the material yield.Fig.2 Stress strain relationship for bilinear isotropic hardening.This option (bilinear isotropic hardening) uses Vonmises yield criterion withassociated flow rule and isotropic (work) hardening.The equivalent stress is,
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1 3 2σe = {S }T [ M ]{S } ------------ (12) 2 And the yield criterion is, 1 3 2F = { S } [ M ]{ S } −σk = 0 ------------ (13) T 2 For work hardening σk is a function of the amount of plastic work done. For the caseof isotropic plasticity assumed here, σk can be directly determined from equivalentplastic strain.Inputs required for the Software for Material modeling:-Concrete: - For specifying Drucker-Prager failure criterion only three inputs arerequired in ANSYS software 1) angle of internal friction = 45˚; 2) cohesion= 3kN/m2;3) flow angle 0˚ [Diganta Goswami, 2003] additionally inputs required are modulusof elasticity E(27000MPa); Poisson’s ratio µ (0.2).Steel: - For specifying Yield criterion for steel Bilinear Isotropic hardening option isused inputs required are modulus of elasticity of steel E (200000MPa); Poisson’s ratioµ (0.3); yield strength of steel fy(250MPa) and tangent modulus (0) i.e. perfectlyelasto-plastic behaviour.Input for COMBINE39: -The element requires force Vs deflection relationship as input for accounting itstransverse stiffness. In the present study load slip relation has been taken frompublished literature (Dennis 2005).Validation of Proposed ModelThe results from an experimental study on a beam model [L.C.P.Yam 1968] are usedto validate the modeling adopted for current bridge. The details of the above referredexperiment are as below:A number of simply supported and continuous composite beams were tested atImperial College. One of these specimens is analyzed to validate the models. Thesimply supported beam selected from the test series which is loaded at midspan. Thebeam consist of as 152mm thick concrete slab and I-section steel girder,304×152mm×0.196kN/m, connected by 100 uniformly distributed head studs,19×100mm. The geometric configuration of beam is as shown in figure 3 below. Thematerial properties are steel: Es=2×105MPa, µ=0.3, Concrete: ES=3×104MPa, µ=0.2.
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a) Elevation and cross section b) Finite Element Idealization.Fig.3 Modeling of composite beam for validation .
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Only one quarter of beam is considered in analysis taking advantage of doublesymmetry of the specimen. The finite element mesh is as shown in figure 5.3. Theinterface slip values are compared in the following Table 1 for a load of 448kN. Andvery good coincidences with experimental values are observed. Table 1 Comparison of results for interface slip with literature review. Results from Interface slip (mm) At 2.0m from left hand At midspan At support support Test 0 0.139 0.508 ANSYS 0 0.151 0.436Loading Conditions: The live load considered is the IRC Class 70R wheeled vehicle loading. Theseloads were first applied on a simply supported girder, with a span equal to that of thebridge prototype, to determine which case produced maximum moment at midspan orthe maximum shear force at the support. Subsequently two loading cases wereconsidered for each bridge prototype including central and eccentrically placed IRCClass 70R wheeled loading, and the bridge dead load. The live load was considered asstatic patch loads of appropriate contact dimensions as per IRC: 6-2000 in theanalysis. The loads are so placed in accordance with IRC: 6-2000, section: II-Loadsand Stresses.Description of Bridge Prototypes:A parametric study has been carried out using the proposed finite element model inwhich two and four lane bridges of various geometries has been analyzed.Theparameters considered are number of cells, number of lanes.For this study, 8 simply supported single-span bridges of different configurationswere used. The basic cross-sectional configurations for the bridges studied arepresented in Table 2. The symbols used in the first column in Table 2 representdesignations of the bridge types considered: l stands for lane, c stands for cell, and thenumber at the end of the designation represents the span length in meters. Forexample, 2l-3c-20 denotes a simply supported bridge of two-lane, three-cell and of 20m span. The cross sectional symbols used in Table 2 are shown in Fig. 4. The numberof lanes was taken as 2 and 4. Number of cells ranged from 1 to 4 for two-lanebridges and 4 to 7 for 4 lane bridges. When changing the number of cells for the same bridge width, the thicknessesof the top steel flanges, webs, and bottom flanges were altered to maintainapproximately the same overall flexural stiffness as well as shear rigidity of the cross
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section. The bridge width was taken as 8.5 m for two lane bridges and 16.7m for fourlane bridges. Fig. 4: Geometric details of model Table 2: Geometries of Prototype Bridges in Parametric StudyBridge Cross Sectional Dimensions (mm)type A B C D F t1 t2 t3 t42l-1c-20 8500 4250 300 800 1050 22 20 16 2502l-2c-20 8500 2835 300 800 1050 22 14 12 2502l-3c-20 8500 2125 300 800 1050 16 10 10 2502l-4c-20 8500 1700 300 800 1050 16 8 10 2504l-4c-20 16700 3340 300 800 1050 18 16 12 2504l-5c-20 16700 2785 300 800 1050 18 14 10 2504l-6c-20 16700 2385 300 800 1050 18 12 10 2504l-7c-20 16700 2085 300 800 1050 16 10 10 250 The moduli of elasticity of concrete and steel were taken as 27 and 200 GPa,respectively. Poisson’s ratio was assumed as 0.2 for concrete and 0.3 for steel. Enddiaphragms were provided at the supports with minimum thickness and the crossbracings were provided at some interval along the span. The material for the enddiaphragms and the cross bracings were taken to be the same as those for the webs.Determination of Load Distribution Factors:To determine the distribution factors, the bridge deck was loaded with wheel loadspositioned along the longitudinal direction of the bridge that produced the maximummoment. The wheels were then moved transversely across the width of the bridge forthe maximum response per girder. The maximum interior and exterior girder momentsand web shears were calculated in each loading case. (a) Cross-section symbols for five-cell bridge
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(b) Idealized cellular bridge for moment distribution Fig. 5 Cross-section of five cell bridge prototype The cellular cross section was divided into I-beam shaped girders as shown in Fig. 5(b). Each idealized girder consisted of the web, steel top flange, concrete deck slab, and steel bottom flange. Results and Discussions: Effects of Cross-Bracing systems: The torsional stiffness of a box girder results from three components: the Saint- Venant rigidity, the warping rigidity, and the distortional rigidity. Increasing the flexibility of any of these components reduces the rigidity of the box girder. Adding bracings between support lines is generally required for stability purposes at the construction phase. Tables (2 & 3) show the effect of bracings on the moment distribution between idealized girders for central lane loadings and eccentric lane loadings respectively. Table 3: Effect of cross bracings on moment distribution factors for central lane loadingBridge Number Outer type of cross girder Interior Girders Outer bracings girder between I II III IV V VI supports2l 3c 20 0 0.12 0.26 0.26 - - - - 0.122l 3c 20 1 0.17 0.21 0.21 - - - - 0.172l 3c 20 2 0.14 0.24 0.24 - - - - 0.142l 3c 20 3 0.16 0.22 0.22 - - - - 0.162l 3c 20 5 0.16 0.22 0.22 - - - - 0.164l 7c 20 0 0.05 0.17 0.25 0.29 0.29 0.25 0.17 0.054l 7c 20 1 0.14 0.22 0.2 0.2 0.2 0.2 0.22 0.144l 7c 20 2 0.08 0.19 0.23 0.26 0.26 0.22 0.19 0.084l 7c 20 3 0.12 0.22 0.22 0.22 0.22 0.22 0.22 0.14l 7c 20 5 0.14 0.2 0.21 0.21 0.21 0.21 0.2 0.14
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Table 4: Effect of cross bracings on moment distribution factors for eccentric lane loadingBridge Number Outertype of cross girder Interior Girders Outer bracings girder between I II III IV V VI supports2l 3c 20 0 0.24 0.35 0.22 - - - - 0.072l 3c 20 1 0.2 0.25 0.24 - - - - 0.192l 3c 20 2 0.2 0.28 0.25 - - - - 0.152l 3c 20 3 0.2 0.25 0.25 - - - - 0.182l 3c 20 5 0.2 0.26 0.25 - - - - 0.174l 7c 20 0 0.25 0.39 0.37 0.31 0.23 0.15 0.07 -0.014l 7c 20 1 0.23 0.29 0.26 0.24 0.23 0.21 0.19 0.094l 7c 20 2 0.2 0.33 0.32 0.29 0.23 0.17 0.15 0.074l 7c 20 3 0.21 0.31 0.29 0.25 0.23 0.2 0.17 0.094l 7c 20 5 0.24 0.3 0.28 0.25 0.22 0.19 0.17 0.11 It can be observed that with increase in number of cross bracing system the bending moment increases in the outer girder and decreases in the central girder for central lane loading while in the case of bridges with eccentric lane loading (Table 3) the maximum moment carried by the loaded outer girder is considerably reduced. As an example, when using five cross- bracing systems the bending moment increases up to a maximum of 81% in the outer girder and decreases by a maximum of 23% in the central girder for the bridge type 4l-7c-20, while in the case of bridges with eccentric lane loading (Table 3) the maximum moment carried by the loaded first intermediate girder is reduced by more than 21%. Thus adding cross bracings improves the ability of the cross section to transfer loads from one girder to the adjacent ones. For the span length of 20m it is revealed that adding more than three cross bracing systems has an insignificant effect on the distribution factors substantiate the fact that for the span length of 20m three number of cross bracing systems gives reasonable load distribution. Conclusion Most composite bridges are designed assuming full-interaction between the concrete deck and steel box girder interface because of the complexities of partial- interaction analysis techniques. However, in the assessment of existing composite bridges this simplification may not be warranted as it is often necessary to extract the greatest capacity and endurance from the structure. This may only be achieved using partial-interaction theory which truly reflects the behaviour of the structure. The proposed model can be efficiently utilized incorporating the realistic behavior of shear connectors. References 1. Cook, R.D., Malkus, D.S., Plesha, M.E and Witt, R.J.,(2002).“Concepts and Applications of Finite Element Analysis, 4ed”John Willey and Sons, New york, 97-99.
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2. Dennis, L., and EI-Lobody, E. (2005). “Behaviour of Headed Stud Shear Connectors in Composite Beam.” ASCE 12(1), 96-1073. Diganta Goswami, (2003), “Ground subsidence due to shallow tunneling in soft ground.,” Ph.D. Dissertation, Indian Institute of Technology Roorkee, at Roorkee.4. IRC 22-1986, “Standard Specifications and Code of Practice for Road Bridges”, Section 4 – Composite Construction, The Indian Road Congress, New Delhi.5. IRC: 5-1998, Standard Specifications and Code of Practice for Road Bridges, Section I, General Features of Design (Seventh Revision).6. IRC: 6-2000, Standard Specifications and Code of Practice for Road Bridges, Section II, Loads and Stresses (Fourth Revision).7. Kuan-Chen, F., and Feng, L. (2003). “Nonlinear Finite-Element Analysis for Highway Bridge Superstructures.” ASCE Journal of Bridge Engineering, 173-1798. L. C. P. Yam and J. C. Chapman, The inelastic behaviour of simply supported composite beams of steel and concrete. Proc. Inst. Civil Engng 41, 651-683 (1968).9. Sennah, K. M., and Kennedy, J.B. (1999a) “Load Distribution Factors for Composite Multicell Box Girder Bridges.” Journal of Bridge Engineering, 4(1), 71-7810. Seracino, R., Oehlers, D.J., and Yeo., M.F. (2001). “Partial-interaction flexural stresses in composite steel and concrete bridge beams.” Engineering Structures, 23, 1186-119311. Upadyay, A. and Klayanaraman, V., (2003), “Simplified analysis of FRP box- girders.”, Composite Structures, 59, 217-225.
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