a thin walled beam element for curved bridge analysis

8/3/2019 A Thin Walled Beam Element for Curved Bridge Analysis

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Computers di S~rucfures Vol. 18, No. 6, pp. 103S1046, 1984 004>7949/84 $3.00 + .@IPrinted in Great Britain. Pergamon Press Ltd.

A THIN-WALLED BOX BEAM FI NITE ELEMENTFOR CURVED BRIDGE ANALYSIS

S. H. ZHANGResearch Institute of the Ministry of Communications, Chinaand

L. P. R. LYONSFinite Element Analysis Ltd. 15 Holborn Viaduct. London, ECIA ?BP, England

(Received 25 February 1983; received for publication 19 April 1983)AbstractPractical design of single and multispan curved bridges requires an analysis procedure which is easy andeconomical to use, and provides a physical insight into structural response under general loading conditions. In thework presented, the thin-walledbeam theory has been directly combined with the finite element technique toprovide a new thin-walled box beam element. The beam element includes three extra degrees-of-freedom over thenormal six degrees-of-freedom beam formulation, to take into account the warping and distortional effects as wellas shear. The beam may be curved in space and variable cross-sections may be included. The performance of thebox beam element has been compared favourably against results obtained from full 3D shell element analysis,differential equation solutions and experimental results.

1. INTRODUCTIONIt is well known that the use of thin-walled box beams isdominant in modern bridge construction having curvedalignments, as it requires appreciably great torsionalrigidity. Nevertheless, the advanced box type of struc-ture presents a major problem in the prediction of itsspecific structural response to a general loading case. Inmost cases, in addition to the conventional structuralactions, cross-sections of thin-walled box beams willwarp out of the sectional plane and distort in the sec-tional plane, as well. Figure 1 shows the linearly in-dependent distortional modes for various box beamcross-sections [16]. These effects may be significant incurved bridges, or in bridges subjected to large live loadswith relatively great eccentricity.Indeed, the analysis of such a deformable box beamhas been the focus of attention for many investigators inthe last two decades, and various theories and analyticalmethods have been developed. It is obviously not pos-sible in this paper to give a thorough review of theexisting methods. However, it is now increasingly ac-cepted that the finite element method is the most power-ful and versatile numerical tool of all the methods avail-able. One of the representative finite elements, which isparticularly suitable for the analysis of shell boxes, is theflat thin shell box element developed by Lyons [15].Jirousek et al. have presented a special macro-element,which is formed from a modified Ahmads thick shellelement and an assembly element, for practical ap-plication to prestressed curved box-girder bridges [111.Recent progress also permits a full three-dimensionalanalysis by using semiloof shell elements with doublecurvature [141.To finite element researchers it would seem that thetask of establishing an advanced numerical method forthe static analysis of box beams has already beenfulfilled, yet in practical design terms engineers have totake account of the computing costs and to deal with thevolumes of computing output which to some extentprevent a physical understanding of box beam behaviour.Indeed, in many design offices, despite the rapid progressof computer hardware and software, engineers stillprefer to use an approximate grillage analysis [7] or other

simplified methods [161 or preliminary analysis or whereother circumstances permit. In harmony with the relativedimensions of medium and long span box bridges, one ofthe logical conclusions for finite element researchers is toextend the capability of a conventional beam elementand reduce the analysis of a bridge deck to a one-dimensional subdivision, whilst retaining the main struc-tural actions.Research work has been carried out assuming a dis-crete subdivision in the longitudinal direction, i.e. a sub-division into beam elements, by several investigators.Formulations which take account of torsional warpingeffects have been used for prismatic beams with opensections [l, 5, 12, 13, 211 or beams with undeformableclosed sections [6]. Baiant and Nimeiri [2] have con-tributed a skew ended beam element for box beamscurved or straight in space taking both transverse dis-tortion and longitudinal warping into consideration.Mikkolo and Paavola [17] have conducted a somewhatsimilar approach for the analysis of a rectangular single-cell box girder with side cantilevers. It is observed thatthe methods given by the latter two papers are availableonly in the case of single-cell boxes. Since the knownfunctions describing the deformation modes of the cross-section must be chosen in advance for each type ofcross-section, difficulties exist in extending thesemethods for more complicated or more general types ofcross-section.Thin-walled beam theory applicable to box beams hasbeen established by Vlasov [19], and elaborated byDabrowski [4] and numerous other authors. The load-deflection equations under the combined vertical (qy),lateral (qz), longitudinal (qJ and twisting moment (m,)distributed loads can be written as follows

_ EJl& + GJ& _ !$ f), _ F V+ EL + GJ=,, + mR x = o

EZ,1035


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1036 S. H. ZHANG nd L. P. . YONS

Fig. 1. Modes of distortion (schematic) for various box beam cross-sections.

+I, tGJ,R e;+q,=O

Details of the notation are given later. Slope-defiection[3.9] and finite difference [10,20] mathematical tech-niques have been developed by Neins et al. to solve theabove equations.A strategy which directly combines the thin-walledbeam theory represented by eqns (l)-(4), and the finiteelement technique, has been adopted in the presenttheoretical work to give a new thin-walled box beamelement. In addition to the usual six degrees-of-freedomat each beam node, three extra degrees-of-freedom havebeen incorporated in the finite element formulation totake account of longitudinal warping and transverse dis-tortion. These additional degrees-of-freedom are the rateof change of twisting angle &, distortional angle of thecross-section yd, and the rate of change of distortionalangle -rd.The thin-walled box beam element, which canbe used effectively to cope with the static analysis ofsingle or multicell box beams curved in space and sub-jected to general loading conditions, may well be regar-ded as the general representation of a conventional beamelement.

2.BASlCASSUMFTIONSThe usual assumptions associated with linear elasticsmall displacement theory have been adopted. Theseassumptions can be generally stated as being that thestructural material is homogeneous, isotropic and linearlyelastic, and that the actual deformations are small com-pared with the structural dimensions. Additionalassumptions, which are mainly related to thin-walledstructural behaviour, have been considered in this ap-proach, and are as follows:(i) The dimensions of the cross-sections are

significantly less than the span lengths and less than theradii of curvature in the case of curved members. Thelength/width ratios of the individual component platesfrom which the boxes are assembled should not be lessthan 3.(ii) The thickness of the wails are small compared withthe dimensions of the cross-section.(iii) For bending action, plane sections remain plane,but not necessarily normal to the beam axis, thus allow-ing for shear deformation.(iv) For warping torsion analysis, cross-sections areassumed to remain undeformed in their own plane, butmay rotate about the flexural axis (locus of the shearcentres) and be subject to longitudinal warping.(v) The in-plane longitudinal bending action of an in-dividual component plate is represented by elementarybeam theory and the shear deformation caused by dis-tortion is neglected.(vi) The bending action of an individual componentplate normal to its plane is represented by the flexuralbehaviour of an equivalent transverse frame.(vii) Diaphragms are considered to be infinitely orfinitely stiff in their own plane, but perfectly flexible inthe direction normal to the plane.

3. INITEEL~NTFO~LATION3.1 ~~nifjon of element geometryConsider the thin-walled box beam element with vari-able cross-sections shown in Fig. 2. The element iscurved in space but the cross-sections are generated bystraight lines. A cross-section is assumed to have avertical axis of symmetry to simplify the analysis ofdistortion, however, for bending and torsion analysis thisassumption is unnecessary.The element axis is defined as the locus.of the cen-troids which may be eccentric from but parallel to theflexural axis. The element has two end nodes and amidpoint node situated on the axis.A local rectangular coordinate system (x, y, z) alongthe axis curve is used in the element formulation. Theorigin of the Cartesian coordinate system is located atthe centroid of the cross-section, and the orientation of


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A thin-walled box beam finite element for curved bridge analysis 1037

Fig.2. Thin-walled box beam element with three nodes.the local axes yz is assumed to coincide with the prin-cipal axes of the cross-section. The local x axis istangential to the element axis in the direction of node 1towards node 3. The local y axis usually represents thevertical axis of symmetry; the local z axis being definedby a right handed orthogonal system.The gIoba1 Cartesian Coordinates are in terms of anatural coordinate & as well, which varies between - 1and t 1 on the respective faces of the element. Letr=X.i+Y.jtZrkbethepositionvectorofapointPon the element axis, then a unit tangent vector along thex direction is

where i , j and k are unit vectors in the global X, Y, Zdirections respectively, and the Lame factor or theJacobian factor referring to the Jacobian matrix is

To fix the orientation of the local y axis, the coordinatesof the reference nodes 4-6, which in most cases can bethe mid-points of the top flanges, are required. The localz axis can then be determined by vector multiplication

e, = e, X ey3.2 Stress-strain relationship

(7)The generalised displacements in the local coordinatesystem are given by

6 = lu, 0, w, %, @yt , @,, d, ydlT (8)where u, u and w are the translations along the local x, y,z axes respectively, 0, is the angle of twist, 8, is the rateof twist, Or and 0, are average rotations about the y and2 axes respectively, yd is the distortional angle, yd is therate of distortion.The corresponding dispIacements in the global systemare given by

6 = [U, v, w 4x, 4,3 4z, 8x,YdtYdl (9)where U, V and W are translations along the XYZglobal axes respectively, c&., (p, and #Pi are rotationsabout the same axes respectiveiy, whilst the rate of twist6, and distortional variables -v,+.Y,+emain in locals. Th e1 1 ,..

thin-walled box element has, therefore, nine degrees-of-freedom at each node.With reference to the principal local system defined,the relation between the generalised stress resultants andthe generalised strain components has been normalised,and can be expressed from the Timoshenko beam theoryand the thin-walled beam theory as (Fig. 3)

a=Dewhere the generalised stress vector is

(10)

Tu =

[N,, Q,, Qz. M,,, My, Mz,-!- Br, MA B,,

ELI 7(11)

in which N, is the axial force, Q, QL are the shearforces, MS, is the pure torsional moment, M,, Mz are theprimary bending moments, BI is the torsional warpingbimoment, Md is the distortional moment and B,, is thedistortional warping bimoment. The total torsionalmoment is the sum of the pure and warping torsionalmoments,i&=il&,th& (I?)

where the warping torsional moment M, can be obtainedfrom the first differential of the torsional bimoment,

Mw= Br. (13)A similar relationship exists between the distortionalmoment and the distortional bimoment,

Md = Br,. 04)It should be emphasised that the shear forces and thetorsional moment are referred to the shear centre of thecross-section.The generalised strain vector is

axial strain au6x = zshear strain in y-direction

yx +aU=aU_,;a x ay a x


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1038 S. H. ZHANGand L. P. R. LYONS

Fig. 3. Generalised orces and displacements.shear strain in z-direction

torsional strain

flexural strain about y-axis

flexural strain about z-axis

torsional warping strain

distortional strainCL = /d

distortional warping strain.

The radius of curvature R introduced in the torsionalwarping strain component can be expressed as1?=[($)2+($)l+($)2]-~ (16)

where X, Y and Z are the global coordinates of thepoints on the beam axis curve. It can be seen that amodification has been made to the torsional warpingstrain to take into account the effect caused by the initialcurvature. It states that the torsional warping displace-ment can be assumed as not only being proportional tothe rate of twist but also to the bending rotation.Finally, as a result of the orthogonalisation the rigiditymatrix can be shown to be

(17)where: A is cross-sectional area; A,,, A,, are effectiveshear areas in the y and the z-directions respectively; Jr istorsional moment of inertia; I,, I, are primary bendingmoments of inertia about the y and z axes respectively; J,is torsional warping moment of inertia; p, is warping shearparameter; Ja is distortional second moment of area: andJr1 is distortional warping moment of inertia.It should be noted that the bending moments of inertiashould be calculated on the basis of an effective flangebreadth replacing the actual width to account for theeffect of shear lag [18].The material properties are characterised by theYoungs modulus of elasticity E and the shear modulusG which is expressed as

G-E--2(1t v) (18)


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A thin-walled box beam finite element for curved bridge analysis 1039where v is the Poissons ratio. The conversion modulus The strain matrix B is then obtained by combiningof elasticity is eqns (22) and (24), hence

3.3 Shape functi ons to define the displacement fieldOnly Co continuity is required for the extensional-flexural effects, and the following quadratic shape func-tions are usedNi=~([2t&,) for i=land3Ni = (1 - .$*) for i = 2 (20)

where &,= &. The quadratic shape functions are alsoused for mapping the element geometry.In contrast to the extensional-flexural effects, it isnecessary to satisfy C, continuity for warping torsionand distortion. A series of special fifth order polynomialshape functions have been derived and can be expressesas

Ni,=$(4t5[o-2[2-3[d)for i = 1 and 3 (21)

Ni2 = $5 (1 [,,)(1 - 5)Ni, = (l- 5) for i = 2Ni, = .Ji[(1 - [*)

where &,= & and Ji is the Jacobian factor with respectto nodal coordinates.3.4 Formulation of displacements and strainsThe generalised displacements in the local coordinatesystem are

8 = NS = [N1N2N&2, 6 (22)where 6 are the nodal values of the global displace-ments and

Ni =

B=LN (25)and L = BS. (26)3.5 The element sti$ness matrix and the equivalent nodalforce vectorThe element stiffness matrix which is of order 27 x 27is expressed by the usual relationship

+(1/z) flk = I B=DBdx= I JBT DB d[. (27)-(I/Z) -1The exact integration of eqn (27) may be achieved byusing a three-point Gauss quadrature for the axial andbending contributions and a six-point Gauss quadraturefor the torsional and distortional contributions. Sinceshear deformation has been included in the formulationfor which rotations due to bending are interpreted asshear strains, an excess of shear strain energy is storedby the element. This problem can be overcome by usinga reduced integration scheme. Thus, the two-point in-tegration procedure, which exactly integrates the bendingcontribution, tut underintegrates the shear contribution isused with the six-point integration.The nodal force vector equivalent to internal andexternal forces is written as

(28)where: b is body force vector; q is distributed forcevector including patch loads; P is concentrated forcevector; l is initial strain vector including temperatureeffects; and u0 is initial stress vector.3.6 Boundary conditionsThe application of boundary conditions is generallyself-evident. For some conventional support conditionsused in bridge construction the following holds true:

Nii . e, Nij . e, Nik . e,Nii.eY Nij.e, Nikse,Nii. e, Nij . e, Nik. e,0 0 00 0 00 0 00 0 00 0 00 0 0

000Nili *e,Nii . eYNii. e,aNi,-iiee,ax00

000Nii *e,Nij *e,Nij .e,

aNi,.axJ.e00

0 0 0 00 0 0 00 0 0 0Nilk.e, Ni, 0 0Nik.e, 0 0 0Nik.e, 0 0 0%k.e, !!!!$ 0 00 0 Ni, Ni20 a2Nil aNi20 -~ax ax

(23)

_Iwhere i . e,, j . e, . . . ., k 9e, are the direction cosines.With the displacements known within the element thestrains are obtained from

E=LS (24)where L is a linear operator and can be written directly._.. ~.

(a) If the beam is fixed at the support, no deformationarises in the support cross-section. and therefore thefollowing may be writtenu=n=w=O

0, = ey = 8, = 0I I$U=V=W=O& = 9, = 4, = 0 (29)

from the detinition of the generahsed strain vector. 0, = yd = yd = 0.


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1040 S. H. ZHANCnd L. P. R. LYONS(b) If the support cross-section is connected by apinned support, braced by a rigid diaphragm, and yet isfree to warp, then the independent boundary conditionsare

0, = yd = 0 (30)(c) If the cross-section is supported by a linear rollerwhich is orientated perpendicularly to the longitudinalbeam axis, and also is braced by a rigid diaphragm, but isfree to warp, the boundary conditions may be adjusted to

r=w=O8, = Yd= 0 (31)

(d) If an interior diaphragm is introduced to resisttransverse deformation only, thus inhibiting distortionalbehaviour at that location, then-yd= 0. (32)

From the above conditions, it is obvious in some casesthe imposed restrictions at the boundaries are in the localxyz directions. Thus, the necessary transformation of thestructural stiffness equations should be made to deal withthe boundary conditions.3.1 Interaction between bending, torsion and distortionfor curved box beams

Although in the finite element formulation the trans-verse distortional effect is treated as an equivalentstraight box beam with a span equal to the developedlength of the axis curve, interacting effects betweenbending, torsion and distortion need to be taken intoconsideration in the case of a curved box beam.Firstly, it can be seen from the differential eqn (4) thatadditional distortional forces occur in a curved box beamdue to the radial components, uR. of the longitudinalbending stresses. The system of radial forces can bereplaced by a force acting through the shear centre alongthe local z-direction and a twisting moment. The ad-ditional distortional moment per unit length can thereforebe expressed asM,m,,,=p- R (33)

in which p is the distortional factor due to initial cur-vature and can be calculated by

~ =++j- (34)z f Awhere b, and b, are the widths of the top and bottomflanges between the side webs respectively. Since I, isusually obtained with reference to an effective flangebreadth, eqn (34) can be approximated as

(35)where 4 is designated as the shear lag factor and isexpressed as

(36)

in which L is the bending moment of inertia with respectto the entire cross-section.Unfortunately, the primary bending moment M,required for the distortional analysis is generally notknown in advance. Thus, following the primary analysis,in which M, is determined, an iterative procedure isnecessary to compute the secondary distortional forcesmd.R.Secondly, in the case of a curved box beam, theinteraction of structural effects caused by the defor-mation of the cross-section can be accounted for within asecond iteration. During this analytical stage the angularrate of distortion obtained, can be treated as an initialbending strain, (yd/R). This numerical technique impliesthe effect of distortion is to reduce the effective rigidityof the cross-section. Normally this effect is notsignificant in practical curved bridges and can then beignored.

4. INTERNALTRESSYSTEMThe normal stresses at the points on the mid-line of theflanges can be obtained by the following expression,

Nx M. Mz B, B,,ax=-tfz-~-vt--_rt-co,,A 1, L J, Jrr (37)where o, and orr are the torsional warping and thedistortional warping functions respectively. The factor 5is expressed as

5=4 $ +(3&i-l)( ,) 7-+@)21or

for parts between webs, (38a)

i= ; *+Np[l_(p)]0

a thin walled beam element for curved bridge analysis

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