shear wall 2

11
Shear lag analysis of thin-walled box girders based on a new generalized displacement Yuan-Hai Zhang , Li-Xia Lin School of Civil Engineering, Lanzhou Jiaotong Univ., 88 West Anning Rd., Lanzhou, Gansu Province 730070, People’s Republic of China article info Article history: Received 23 January 2013 Revised 8 December 2013 Accepted 26 December 2013 Available online 6 February 2014 Keywords: Thin-walled box girders Shear lag effect Generalized displacement Finite segment method Deflection Initial parameter Stress distribution abstract In many papers about the shear lag analysis of thin-walled box girders, the maximum angular rotation attributable to the in-plane shear deformation of flanges is adopted as generalized displacement. How- ever, the generalized displacement is not very simple and clear and the analytical procedure is relatively complicated. Moreover, various types of warping displacement function for shear lag were assumed which may cause some confusion. In this paper, a new method for analyzing shear lag effect in thin- walled box girders is proposed in which the additional deflection induced by shear lag effect is adopted as the generalized displacement. Based on the generalized moment defined in this paper, the shear lag deformation state is separated from the flexural deformation state of the corresponding elementary beam and analyzed as a fundamental deformation state. The quadratic parabola is demonstrated to be the rea- sonable curve of the warping displacement function in the shear lag effect analysis of a box girder and the accuracy of the degrees of the warping functions is evaluated. The so-called negative shear lag is illus- trated through the generalized moment. The governing differential equation and the boundary condition for the additional deflection are established by applying the principle of minimum potential energy, and the initial parameter solution to the differential equation is provided. A very simple and convenient for- mula of the shear lag warping stress is proposed which has the same form as that of the bending stress of elementary beam. A finite beam segment element with 8 degrees of freedom is developed to analyze the shear lag effect in complex continuous box girders with varying depth. Two plexiglass models of contin- uous box girders are analyzed and the calculated results are in agreement with the test results, which validates the analytical method and the element presented. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction Thin-walled box girders are widely used in modern bridge engi- neering due to various advantages. So far, a large number of re- search studies on the box girders have been conducted. As an important mechanical property of thin-walled beams, shear lag ef- fect of thin-walled box girders has been a popular research topic among many scholars for decades. Reissner [1] first analyzed the shear lag effect of a doubly symmetric rectangular box girder with- out side cantilever slabs by using a variation method based on the principle of minimum potential energy and a quadratic parabola was adopted as the displacement function for shear lag warping. Reissner’s method was extended later to analyze the shear lag of box girders with side cantilever slabs by many scholars, among whom Kuzmanovic and Graham [2] and Dezi and Mentrasti [3] and Chang and Yun [4] adopted quadratic and quartic parabolas, respectively; while most scholars preferred to the cubic parabola [5–11]. In addition, curves of cosine and catenary were also used by some scholars, such as Ni and Qian [12], and Gan and Zhou [13]. To denote the different warping displacements for shear lag among the slabs of a box girder with trapezoidal cross section, Dezi and Mentrasti [3] and Luo et al. [5] chose three independent gen- eralized displacements for the top, bottom, and side cantilever slabs, respectively. Reissner’s method was further applied to ana- lyze the shear lag effect in curved box girders by Luo and Li [14] and Zhang and Li [15], etc., among whom Zhang and Li [15] also considered the secondary shear deformation for restraint torsion. Lin and Zhao [16] proposed a least-work solution to the shear lag of thin-walled girders in which infinite terms of high-order polyno- mial were used to describe the uneven longitudinal displacement in the flanges, and the method was extended later to analyze the inelastic shear lag behavior in steel box beams [17]. Sa-nguanman- asak et al. [18] and Lertsima et al. [19] investigated the stress concentration in a flange due to shear lag in simply supported and continuous box girders by three-dimensional finite element method using shell elements and the effects of the way load was applied and the dependency of finite element mesh on the shear lag were carefully treated. In recent years, shear lag analysis of 0141-0296/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.12.031 Corresponding author. Tel.: +86 931 493 8689; fax: +86 931 493 8532. E-mail address: [email protected] (Y.-H. Zhang). Engineering Structures 61 (2014) 73–83 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Upload: grantherman

Post on 27-Jan-2016

234 views

Category:

Documents


2 download

DESCRIPTION

paper

TRANSCRIPT

Page 1: shear wall 2

Engineering Structures 61 (2014) 73–83

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier .com/ locate /engstruct

Shear lag analysis of thin-walled box girders based on a new generalizeddisplacement

0141-0296/$ - see front matter � 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.12.031

⇑ Corresponding author. Tel.: +86 931 493 8689; fax: +86 931 493 8532.E-mail address: [email protected] (Y.-H. Zhang).

Yuan-Hai Zhang ⇑, Li-Xia LinSchool of Civil Engineering, Lanzhou Jiaotong Univ., 88 West Anning Rd., Lanzhou, Gansu Province 730070, People’s Republic of China

a r t i c l e i n f o

Article history:Received 23 January 2013Revised 8 December 2013Accepted 26 December 2013Available online 6 February 2014

Keywords:Thin-walled box girdersShear lag effectGeneralized displacementFinite segment methodDeflectionInitial parameterStress distribution

a b s t r a c t

In many papers about the shear lag analysis of thin-walled box girders, the maximum angular rotationattributable to the in-plane shear deformation of flanges is adopted as generalized displacement. How-ever, the generalized displacement is not very simple and clear and the analytical procedure is relativelycomplicated. Moreover, various types of warping displacement function for shear lag were assumedwhich may cause some confusion. In this paper, a new method for analyzing shear lag effect in thin-walled box girders is proposed in which the additional deflection induced by shear lag effect is adoptedas the generalized displacement. Based on the generalized moment defined in this paper, the shear lagdeformation state is separated from the flexural deformation state of the corresponding elementary beamand analyzed as a fundamental deformation state. The quadratic parabola is demonstrated to be the rea-sonable curve of the warping displacement function in the shear lag effect analysis of a box girder and theaccuracy of the degrees of the warping functions is evaluated. The so-called negative shear lag is illus-trated through the generalized moment. The governing differential equation and the boundary conditionfor the additional deflection are established by applying the principle of minimum potential energy, andthe initial parameter solution to the differential equation is provided. A very simple and convenient for-mula of the shear lag warping stress is proposed which has the same form as that of the bending stress ofelementary beam. A finite beam segment element with 8 degrees of freedom is developed to analyze theshear lag effect in complex continuous box girders with varying depth. Two plexiglass models of contin-uous box girders are analyzed and the calculated results are in agreement with the test results, whichvalidates the analytical method and the element presented.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Thin-walled box girders are widely used in modern bridge engi-neering due to various advantages. So far, a large number of re-search studies on the box girders have been conducted. As animportant mechanical property of thin-walled beams, shear lag ef-fect of thin-walled box girders has been a popular research topicamong many scholars for decades. Reissner [1] first analyzed theshear lag effect of a doubly symmetric rectangular box girder with-out side cantilever slabs by using a variation method based on theprinciple of minimum potential energy and a quadratic parabolawas adopted as the displacement function for shear lag warping.Reissner’s method was extended later to analyze the shear lag ofbox girders with side cantilever slabs by many scholars, amongwhom Kuzmanovic and Graham [2] and Dezi and Mentrasti [3]and Chang and Yun [4] adopted quadratic and quartic parabolas,respectively; while most scholars preferred to the cubic parabola[5–11]. In addition, curves of cosine and catenary were also used

by some scholars, such as Ni and Qian [12], and Gan and Zhou[13]. To denote the different warping displacements for shear lagamong the slabs of a box girder with trapezoidal cross section, Deziand Mentrasti [3] and Luo et al. [5] chose three independent gen-eralized displacements for the top, bottom, and side cantileverslabs, respectively. Reissner’s method was further applied to ana-lyze the shear lag effect in curved box girders by Luo and Li [14]and Zhang and Li [15], etc., among whom Zhang and Li [15] alsoconsidered the secondary shear deformation for restraint torsion.Lin and Zhao [16] proposed a least-work solution to the shear lagof thin-walled girders in which infinite terms of high-order polyno-mial were used to describe the uneven longitudinal displacementin the flanges, and the method was extended later to analyze theinelastic shear lag behavior in steel box beams [17]. Sa-nguanman-asak et al. [18] and Lertsima et al. [19] investigated the stressconcentration in a flange due to shear lag in simply supportedand continuous box girders by three-dimensional finite elementmethod using shell elements and the effects of the way load wasapplied and the dependency of finite element mesh on the shearlag were carefully treated. In recent years, shear lag analysis of

Page 2: shear wall 2

Nomenclature

The following symbols are used in this paper:A area of whole cross section of box girderAt, Ab, Ac area of top slab, bottom slab, and the two cantilever

slabs, respectivelyAf area of cross section for shear lagb1, b2 half width of top and bottom slabs, respectivelyb3 width of side cantilever slabCi constant of integration (i = 1–4)E, G Young’s and shear modulus, respectivelyf additional deflection of box girder induced by shear lag

effectF element nodal force vectorPp element equivalent nodal force vector for shear lagIx moment of inertia of whole cross section about centroi-

dal x-axisIf, Iyf moment and product of inertia of whole cross section

for shear lag, respectivelyIx generalized moment of inertia of whole cross section for

shear lagk Reissner’s parameterki,j element in element stiffness matrixl span of box girder or length of elementM bending momentMx generalized moment for shear lagNf element shape function matrix for shear lagP concentrated load

q distributed load or shear flowQ shear forceQx generalized shear force for shear lags contour coordinate measured along the central line of

thin wallt slab thicknessu longitudinal displacement at any point on cross sectionw deflection of box girder corresponding to elementary

beamx, y, z coordinatesys, yx distances between centroid of cross section and mid-

plane of top and bottom slabs, respectivelyc in-plane shear deformation of slabd element nodal displacement vectorg modification factorh additional angular displacement due to shear lagk shear lag coefficientl Poisson’s ratioP total potential energyr total normal stress considering shear lag effectrx shear lag warping stresss shear stressu angular displacement of elementary beamx generalized warping displacement function for shear lagxf warping displacement function for shear lag

(a)

(b)Fig. 1. Box girder with trapezoidal cross section. (a) Coordinate system and load. (b)Cross section.

74 Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83

more complex box girders such as skewed continuous box girderswas conducted by Zhang et al. [20–22].

However, in most of the existing related papers, the maximumangular rotation attributable to the in-plane shear deformation offlanges is adopted as the generalized displacement of a box girder,which is not very simple and clear and also not easy to be under-stood especially for practical engineers. Moreover, in the existinganalytical methods, the shear lag deformation state is analyzed to-gether with the flexural deformation state of the correspondingelementary beam, which complicates the shear lag analysis proce-dure. And it is also inconvenient to distinguish between the shearlag warping stress and the flexural stress of the corresponding ele-mentary beam. In addition, various types of warping displacementfunction for shear lag were assumed in these papers, therefore, it isevident that a relatively acceptable displacement function forshear lag warping is still lacking. Zhang [23] proposed a generalstress formula expressed through the generalized internal forces(shear lag moment and bending moment) and relevant geometricalproperties of cross section. If the shear lag effect is ignored, thestress formula will be reduced to the well-known formula in theelementary beam theory. However, the stress formula is still notsimple enough.

In this paper, the quadratic parabola is proved to be thereasonable curve of the shear lag warping displacement functionand the additional deflection induced by shear lag effect isadopted as the generalized displacement to describe the shearlag deformation state of a box girder, and the shear lagdeformation state is analyzed as an independent fundamentaldeformation state. A very simple formula is proposed to calculatethe shear lag warping stress. The governing differential equationand the corresponding boundary condition on the additionaldeflection are established by applying an energy variation methodand the initial parameter solution to the differential equation isgiven. A finite beam segment element with 8 degrees of freedomis developed to analyze the shear lag effect in complex

continuous box girder with varying depth and the correspondingnumerical examples are provided.

With the development of modern bridge engineering, single-cell box girders with long cantilever plates and large spacing ofwebs have been widely used, therefore, the paper mainly focuseson the shear lag analysis of single-cell box girders. However, thepresented method is generally applicable to the shear lag analysisof any other thin-walled girders with wide flanges.

2. Shear lag warping stress

Fig. 1 shows a thin-walled box girder with trapezoidal cross sec-tion, subjected to flexure in the y–z plane under distributed load

Page 3: shear wall 2

Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83 75

p(z). The z axis is taken to coincide with the centroidal axis and x, yare taken as the principal inertia directions. For a box girder, shearlag effect reduces the in-plane stiffness of flanges, thus increasesinevitably the box girder deflection. In this paper, the additionaldeflection induced by shear lag effect is adopted as the generalizeddisplacement to describe the shear lag deformation state of a boxgirder. So, the longitudinal displacement at any point of the crosssection of the box girder shown in Fig. 1 can be expressed as

uðx; y; zÞ ¼ �y w0ðzÞ þ f 0ðzÞ½ � þ gxfðx; yÞf 0ðzÞ¼ �yw0ðzÞ �xðx; yÞf 0ðzÞ ð1Þ

where u(x, y, z) is the longitudinal displacement at any point of thecross section of the box girder; w(z) is the deflection of the box gir-der corresponding to the elementary beam; f(z) is the additionaldeflection induced by shear lag effect; xf(x, y) is the warping dis-placement function for shear lag; g is a modification factor consid-ering the self-equilibrium condition of shear lag warping stresses;and x(x, y) is the generalized warping displacement function forshear lag. x(x, y) is given by

xðx; yÞ ¼ y� gxfðx; yÞ ð2Þ

The normal stress r(x, y, z) at any point of the cross sectionfollows

rðx; y; zÞ ¼ E@u@z¼ �Eyw00ðzÞ � Exðx; yÞf 00ðzÞ ð3Þ

where E is the Young’s modulus.It is quite evident that the first item at the right side of the sec-

ond equal-sign in Eq. (3) is the bending stress rb of the elementarybeam theory and the second is the shear lag warping stress rx, i.e.,

rx ¼ �Exf 00 ð4Þ

The bending moment M of a box girder is the bending stressresultant on the cross section, while the shear lag warping stressesmust lead to no axial force and no bending moment, thus,Z

ArxdA ¼ 0 ð5Þ

ZArxydA ¼ 0 ð6Þ

Substituting Eq. (4) into Eqs. (5) and (6) and noting the relation(2), we obtainZ

AxfdA ¼ 0 ð7Þ

g ¼ Ix

Iyfð8Þ

where Ix is the well-known moment of inertia of the cross sectionabout the centroidal axis x; and Iyf is the product of inertia of thecross section corresponding to shear lag, i.e., Iyf =

RAyxfdA.

The generalized moment corresponding to the shear lag warp-ing stress is defined by

Mx ¼Z

ArxxdA ð9Þ

Substituting Eq. (4) into Eq. (9) gives

Mx ¼ �EIxf 00 ð10Þ

where Ix ¼R

A x2dA, and Ix can be called the generalized moment ofinertia for shear lag warping. According to the relations of Eqs. (2)and (8), Ix can be expressed as

Ix ¼ g2If � Ix ð11Þ

in which If is the moment of inertia for shear lag warping, i.e.,If ¼

RA x2

f dA.

Using Eq. (4) and (10), a very simple formula of shear lag warp-ing stress can be obtained, i.e.,

rx ¼Mx

Ixx ð12Þ

The total normal stress at any point of the cross section of a boxgirder is the sum of the normal stress in elementary beam theoryand the shear lag warping stress, i.e.,

r ¼ MIx

yþMx

Ixx ð13Þ

3. Displacement function for shear lag warping

The displacement function for shear lag warping should be se-lected cautiously for it reflects directly the shear lag warping stressdistribution. The shear lag warping stress in the cross section of abox girder must meet the axial equilibrium condition (7) when avertical load acts on the box girder.

Shear lag effect of a box girder is caused by the in-plane sheardeformation of flange plates. Therefore, the displacement functionfor shear lag warping can be investigated from the in-plane sheardeformation. For the box section with a vertical symmetrical axisshown in Fig. 1(b), the flexural shear flow at any point of the topslab (between two webs) can be expressed as

qðsÞ ¼ �Q y

Ix

Z s

0ytds ¼

Qy

Ixysts ð14Þ

where Qy is the vertical shear on the cross section; ys is the distancebetween centroid and the middle surface of the top slab; t is theslab thickness; and s is the contour coordinate measured alongthe central line of wall and its origin is at the intersection point be-tween the symmetrical axis and the middle surface of top slab.

The in-plane shear deformation of the top slab can be approxi-mately expressed as

cðsÞ ¼ @u@s¼

Q yys

GIxs ð15Þ

where u is the longitudinal displacement of the top slab; and G isthe shear modulus. In Eq. (15), the effect of the lateral displacementof the top slab on the shear deformation is assumed to be negligible.

Integrating Eq. (15) with respect to s gives

u� u0 ¼Q yys

2GIxs2 ð16Þ

in which u0 is the longitudinal displacement at the origin of s.Eq. (16) shows that the longitudinal displacement of the top

slab of a box girder subjected to flexure is distributed transverselyas a quadratic parabola. The same conclusion is also suitable for thebottom and cantilever slabs. Thus, the displacement function forshear lag warping is assumed as

xf ¼

�ys 1� x2

b21

h iþ d for top slab

�ys 1� ðb1þb3�xÞ2

b23

h ib3b1

� �2þ d for cantilever slab

yx 1� x2

b22

h ib2b1

� �2yxys

� �þ d for bottom slab

d for web slab

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

ð17Þ

the symbols of which are illustrated in Fig. 1(b).As indicated in Eq. (17), the constant d is added to the entire

cross section, through which the displacement function xf can sat-isfy the equilibrium condition (7). In addition, to consider the dif-ferent warping among the top, bottom, and cantilever slabs, the xf

for the cantilever slab is modified in proportion to the square of its

Page 4: shear wall 2

76 Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83

width, and for the bottom slab, the xf is also modified in propor-tion to the distance away from the centroidal axis x.

Substituting Eq. (17) into Eq. (7) yields

d ¼ 2ys

3AAt þ Ac

b3

b1

� �2

� Abb2

b1

� �2 yx

ys

� �2" #

ð18Þ

in which A is area of the entire cross section; and At, Ac, Ab is the areaof the top slab, cantilever slabs at both sides, and bottom slab of thebox girder, respectively.

From Eq. (18) d is readily seen to be equal to zero for a doublysymmetric rectangular box section without cantilever slabs. For arectangular box girder with side cantilever slabs used widely inpractice, only if the dimensions of the cross section conform tothe following condition (i.e., d = 0)

At þ Acb3

b1

� �2

¼ Abyx

ys

� �2

ð19Þ

will the shear lag warping stresses satisfy automatically axial equi-librium condition.

The shear lag warping displacement function xf and thegeneralized displacement function x are graphically shown inFig. 2.

The formulas of the geometrical properties for shear lag can beeasily derived as follows

If ¼ A � d2 þ Atys8

15ys �

43

d� �

þ Acys8

15ys

b3

b1

� �2

� 43

d

" #b3

b1

� �2

þ Ab8

15y2

x

ys

� �b2

b1

� �2

þ 43

d

" #y2

x

ys

� �b2

b1

� �2

ð20Þ

(a)

(b)Fig. 2. Diagrams of displacement functions in slabs of box girder. (a) xf. (b) x.

Iyf ¼23

y2s At þ Ac

b3

b1

� �2

þ Abb2

b1

� �2 yx

ys

� �3" #

ð21Þ

For multi-cell cross sections of box girders, the warping dis-placement function can be adopted in a similar manner and Eqs.(20) and (21) are still suitable.

If cubic parabola is chosen as the shear lag warping displace-ment function, 2/3 in Eqs. (18) and (21) transforms to 3/4, and 8/15 and 4/3 in Eq. (20) become 9/14 and 3/2, respectively. Quarticparabola is used, then 2/3, 8/15 and 4/3 above transform to 4/5,32/45 and 8/5, respectively.

4. Governing differential equation and its solution

The total potential energy of the box girder shown in Fig. 1 canbe expressed as follows:

P ¼ 12

Zl

ZA

r2

Eþ s2

G

� �dAdz�

Zl

pðwþ f Þdz

¼ 12

Z l

0EIxw002 þ EIxf 002 þ g2GAff 02�

dz�Z l

0pðwþ f Þdz ð22Þ

where Af ¼R

A @xf=@xð Þ2dA, and it is called the area of cross sectionfor shear lag warping and can be calculated by using the followingformula

Af ¼43

Atys

b1

� �2

þ Acb3

b1

� �4 ys

b3

� �2

þ Abyx

b2

� �2 yx

ys

� �2 b2

b1

� �4" #

ð23Þ

The first-order variation for the total potential energy can beobtained easily, i.e.,

dP ¼Z l

0EIxw

0000 � p�

dwdzþZ l

0EIxf

0000 � g2GAff 00 � p�

dfdz

þ EIxw00dw0jl0 � EIxw000dwjl0 þ EIxf 00df 0jl0þ g2GAff 0 � EIxf 000�

df jl0 ð24Þ

Thus, according to the principle of minimum potential energy,setting the first-order variation equal to zero, i.e., dP = 0, the gov-erning differential equations of the box girder can be given asfollows:

EIxw0000 � p ¼ 0 ð25Þ

EIxf0000 � g2GAff 00 � p ¼ 0 ð26Þ

It is obvious that Eq. (25) is the well-known differential equation ofthe elementary beam, and Eq. (26) is the governing differentialequation for shear lag. Only one independent displacement is in-volved in Eqs. (25) and (26), respectively. Eq. (26) can be solvedfor the additional deflection f, and the generalized moment Mx

can be obtained only by using Eq. (10). It is thus clear that the shearlag deformation state of the box girder can be analyzed indepen-dently as a fundamental deformation state.

Eq. (26) reduces to

f0000 � k2f 00 ¼ p

EIxð27Þ

where k is Reissner’s parameter, that is,

k ¼ g

ffiffiffiffiffiffiffiffiGAf

EIx

s

The boundary items in the first-order variation expression (24)show that the internal forces corresponding to the deflection w andthe angular rotation �w0 in elementary beam theory are shearforce �EIxw000 and bending moment �EIxw00, respectively, and thegeneralized internal force corresponding to the additional angular

Page 5: shear wall 2

Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83 77

rotation �f 0 induced by shear lag effect is the generalized moment�EIxf 00, which is completely the same with that defined in Eq. (10).The generalized force corresponding to the additional deflection fcan be called the generalized shear force and denoted by Qx, thatis,

Qx ¼ g2GAff 0 � EIxf 000 ð28Þ

The boundary conditions in solving the differential Eq. (27) canbe summarized as following:

for a fixed end: f = 0, f 0 = 0;for a simply supported end: f = 0, f 00 ¼ 0; andfor a free end: f 00 ¼ 0, f 000 � k2f 0 ¼ 0.

The differential Eq. (27) is a fourth order inhomogeneous differ-ential equation with constant coefficient and the general solutionto the corresponding homogeneous equation is

f ¼ C1 þ C2zþ C3 sinh kzþ C4 cosh kz ð29Þ

therefore, the first derivative of the additional deflection, the gener-alized moment and the generalized shear force can be easily ob-tained as follows:

f 0 ¼ C2 þ k C3 cosh kzþ C4 sinh kzð Þ ð30Þ

Mx ¼ �EIxf 00 ¼ �EIxk2 C3 sinh kzþ C4 cosh kzð Þ ð31Þ

Qx ¼ g2GAff 0 � EIxf 000 ¼ EIxk2C2 ð32Þ

Four initial parameters are denoted by f0, f 00, Mx0 and Qx0 which arethe relevant generalized displacements and internal forces at theleft end of the box girder. Setting z = 0 in Eqs. (29)–(32), we canestablish the relations between the initial parameters and the inte-gral constants Ci(i = 1–4). This leads to the following initial param-eter solutions:

f ¼ f0 þ f 00sinh kz

kþ Mx0

k2EIxð1� cosh kzÞ þ Qx0

k3EIxðkz� sinh kzÞ

ð33Þ

f 0 ¼ f 00 cosh kz� Mx0

kEIxsinh kzþ Qx0

k2EIxð1� cosh kzÞ ð34Þ

Mx ¼ �f 00kEIx sinh kzþMx0 cosh kzþ Qx0sinh kz

kð35Þ

Qx ¼ Qx0 ð36Þ

Eqs. (33)–(36) are restricted to the case where there are no loadsapplied to the span interior of the box girder. For the box girder sub-jected various kinds of loads shown in Fig. 3, Eqs. (33) and (35), forexample, become

ξ ξ

Fig. 3. External loads on box girder.

f ¼ f0 þ f 00sinh kz

kþ Mx0

k2EIxð1� cosh kzÞ þ Qx0

k3EIxðkz� sinh kzÞ

�����

a

P

k3EIxkðz� aÞ � sinh kðz� aÞ½ ��

����b

L

k2EIx1� cosh kðz� bÞ½ �

�����

c

Z z

c

pðnÞk3EIx

kðz� nÞ � sinh kðz� nÞ½ �dn ð37Þ

Mx ¼ �f 00KEI sinh KZ þMx0 cosh KZ þ Qx0sinh KZ

k�����

a

Pk

� sinh KZðz� aÞ �����

b

L cosh kðz� bÞ�����

c

Z z

c

pðnÞk

� sinh kðz� nÞdn ð38Þ

where the items with symbols ka are suitable for the segment ofz > a, and the rest symbols are similar in meaning. The upper limitz of the integrals should be changed into d when z > d.

The following two examples illustrate the application of the ini-tial parameter method to a cantilever box girder and a two-spancontinuous box girder, respectively. For the cantilever box girdersubjected to vertical uniform load shown in Fig. 4, noting thatf0 = 0, f 00 ¼ 0;Qx0 ¼ pl, we can get the additional deflection fromEq. (37) as follows

f ¼ Mx0

k2EIxð1� cosh kzÞ þ pl

k3EIxðkz� sinh kzÞ �

Z z

0

� p

k3EIxkðz� nÞ � sinh kðz� nÞ½ �dn

¼ Mx0

k2EIxð1� cosh kzÞ þ pl

k3EIxðkz� sinh kzÞ

� p

2k4EIxk2z2 þ 2� 2 cosh kz� �

ð39Þ

where the initial parameter Mx0 can be determined from theboundary condition at the tip end (f 00 ¼ 0 at z = l). Thus, the expres-sion of the additional deflection of the cantilever box girder is

f ¼ p

k4EIx cosh klk2z l� z

2

� �cosh klþ kl sinh ðkl� kzÞ

h� kl sinh klþ cosh kz� 1

ið40Þ

Letting z = l in Eq. (40) we obtain the additional deflection at thetip end as following

fl ¼p

2k4EIx cosh klk2l2 cosh kl� 2kl sinh klþ 2 cosh kl� 2h i

Fig. 5(a) shows a two-span continuous box girder subjected to avertical concentrated load P at each midspan. Due to the symmetryof the structure and the load, the box girder can be analyzed as asingle span shown in Fig. 5(b). It is obvious that f0 = 0 andMx0 = 0, thus we get

f ¼ f 00sinh kz

kþ Qx0

k3EIxðkz� sinh kzÞ�

����l2

P

k3EIxk z� l

2

� ��sinh k z� l

2

� �� �

Fig. 4. Cantilever box girder under uniformly-distributed load.

Page 6: shear wall 2

(a)

(b)Fig. 5. Two-span continuous box girder simulated as single span. (a) Two-spancontinuous box girder. (b) Single span box girder.

Fig. 6. Generalized moment and additional deflection of cantilever box girder.

Fig. 7. Generalized moment and additional deflection of two-span continuous boxgirder.

iθ jθif jf

iw jw

iϕ jϕ

Fig. 8. Nodal displacements of box beam element.

78 Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83

where the initial parameters f 00 and Qx0 are determined from theboundary conditions, i.e., f = f0 = 0 at z = l. Finally, the expressionsof the additional deflection and the generalized moment can belisted as follows

For z 6 l/2

f ¼ P

k3EIxkz�

kl2 cosh kl� sinh kl

2

klcosh kl� sinh klkz�

klcosh kl2� sinh kl

2� kl2

klcosh kl� sinh klsinh kz

!

ð41Þ

Mx ¼Pk

kl cosh kl2 � sinh kl

2 � kl2

kl cosh kl� sinh klsinh kz ð42Þ

For z > l/2

f ¼ P

k3EIx

kl2þ sinh k z� l

2

� ��

kl2 cosh kl� sinh kl

2

kl cosh kl� sinh klkz

"

�klcosh kl

2 � sinh kl2 � kl

2

kl cosh kl� sinh klsinh kz

#ð43Þ

Mx ¼Pk

kl cosh kl2 � sinh kl

2 � kl2

kl cosh kl� sinh klsinh kz� sinh k z� l

2

� �" #ð44Þ

The generalized moment Mx and additional deflection f of the can-tilever and continuous box girders are graphically shown in Figs. 6and 7, respectively.

Compared with the shear lag analysis method in existing liter-atures, the proposed analysis method has the following advanta-ges: (1) Adopting the additional deflection as the generalizeddisplacement for shear lag analysis has very clear physical signifi-cance meaning and is easy to be accepted especially by engineersin practice. (2) In the proposed method, shear lag is uncoupledwith the flexure of elementary beam. Therefore, the shear lagdeformation state of box girders can be easily analyzed as a funda-mental deformation state. (3) Based on the generalized momentand the relevant geometrical properties defined, the shear lagwarping stress can be easily calculated by applying a very simpleformula as that in elementary beam theory.

5. Finite segment element

Based on the above analytic method, a box beam element withtwo nodes is presented to analyze the shear lag effect of complexbox girders, such as continuous box girders with varying depth.The nodal displacements of the box beam element are shown inFig. 8. The element nodal displacement vector can be expressedas

d ¼ wi ui fi hi wj uj fj hj� T ð45Þ

where wi and wj are the corresponding elementary beam deflectionsat nodes i and j, respectively; ui and uj are the corresponding angu-lar rotations at nodes i and j, respectively; fi and fj are the additionaldeflections at nodes i and j, respectively; and hi and hj are the addi-tional angular rotations at nodes i and j, respectively.

Page 7: shear wall 2

Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83 79

Corresponding to the element nodal displacement vector, theelement nodal force vector can be expressed as

F ¼ Qi Mi Qxi Mxi Q j Mj Qxj Mxj� T ð46Þ

where Qi and Qj are the shear forces of corresponding elementarybeam at nodes i and j, respectively; Mi and Mj are the correspondingbending moments at nodes i and j, respectively; Qxi and Qxj are thegeneralized shear forces for shear lag at nodes i and j, respectively;and Mxi and Mxj are the generalized moments for shear lag atnodes i and j, respectively.

The element stiffness matrix and equivalent nodal force vectorcan be derived easily. In fact, in the stiffness matrix and the equiv-alent nodal force vector, the elements corresponding to the degreesof freedom of the elementary beam can be quoted directly in anybooks on matrix analysis of member system structure, while theelements corresponding to the degrees of freedom for shear lagcan be derived by employing the initial parameter solutions estab-lished above.

In this paper, Eq. (33) is chosen as the additional deflectionfunction of the box beam element. Thus, the additional angularrotation function of the element is

hðzÞ ¼ �f 00 cosh kzþ Mx0

kEIxsinh kz� Qx0

k2EIxð1� cosh kzÞ ð47Þ

According to the arrangement of the degrees of freedom in Eq.(45) we know that, if the end i of the box beam element fixed attwo ends takes place an unit additional deflection fi = 1 while therest displacements for shear lag equal zero (hi = hj = fj = 0), the cor-responding nodal forces at two ends give the elements in the thirdcolumn of the stiffness matrix. Thus, setting f0 = 1, f 00 ¼ �hi ¼ 0, andemploying the displacement condition fj = hj = 0 at end j of the ele-ment, we can obtain the initial parameters Qx0 and Mx0, thenchanging the signs of Qx0 and Mx0 gives the third column elementsk3,3 and k4,3 in the stiffness matrix. Substituting Qx0 and Mx0 intoEqs. (35), (36) and setting z = l, then k8,3 and k7,3 are obtained. Inthe same way, all the rest elements corresponding to the degreesof freedom for shear lag can be determined. The upper triangularelements in the stiffness matrix for shear lag can be listed asfollows

k3;3 ¼ �k3;7 ¼ k7;7 ¼k3EIx

Dsinh kl

k3;4 ¼ k3;8 ¼ �k4;7 ¼ �k7;8 ¼ �k2EIx

Dðcosh kl� 1Þ

k4;4 ¼ k8;8 ¼kEIx

Dkl cosh kl� sinh klð Þ

k4;8 ¼kEIx

Dðsinh kl� klÞ

where D = kl sinh kl + 2–2 cosh kl.The shear lag displacement function of the box beam element

can be expressed through the element nodal displacement vector,i.e.

f ðzÞ ¼ N f d ð48Þ

where Nf is the element shape function matrix for shear lag, and canbe expressed as

Nf ¼ 0 0 N1ðzÞ N2ðzÞ 0 0 N3ðzÞ N4ðzÞ½ �

in which

N1ðzÞ ¼1D

sinh kl sinh kzþ kl� kzð Þ þ 1� cosh klð Þð1þ cosh kzÞ½ �

N2ðzÞ ¼1

kDkl cosh kl� sinh klð Þðcosh kz� 1Þ þ ðcosh kl� 1Þkz½

þ cosh kl� 1� kl sinh klð Þ sinh kz�

N3ðzÞ ¼1D

sinh klðkz� sinh kzÞ þ ð1� cosh klÞð1� cosh kzÞ½ �

N4ðzÞ¼1

kDðsinh kl�klÞðcosh kz�1Þþðcosh kl�1Þðkz�sinh kzÞ½ �

Based on the Eq. (48), the equivalent nodal force vector corre-sponding to shear lag can be easily determined by applying theprinciple of virtual work. For the vertical uniformly-distributedload p on the element, the equivalent nodal force vector Pp forshear lag can be obtained as follows

Pp ¼ pZ l

0NT

f dz ¼ p2

0 0 l �f 0 0 l f½ �T ð49Þ

where

f ¼ 1

k2DðklÞ2ð1þ cosh klÞ þ 4 cosh kl� kl sinh kl� 1ð Þh i

:

6. Numerical examples

6.1. Example 1

In this example we examine the shear lag effect of a cantileverbox girder with a span length of 470 mm, subjected to uniformly-distributed load p = 1 kN/m over the total length (Fig. 9a). TheYoung’s modulus of the material is E = 2900 MPa and the Poisson’sratio is l = 0.4. Fig. 9(b) shows the detail dimensions of the crosssection. Fig. 9(c) shows the curves of the shear lag coefficient k (de-fined as the ratio between the normal stress allowing for shear lagand that obtained from the elementary beam theory) along theflange at two cross sections of the box girder (marked A and B inFig. 9a). The first section A is located at the support where shearlag is maximal. The second section is 300 mm from support wherethe so-called negative shear lag is significant.

It can be seen from Fig. 9(c) that shear lag coefficients obtainedby the proposed method are symmetrically distributed to the webslab, since the in-plane shear flows in the top and cantilever slabsare identical according to the flexural theory of thin-walled boxgirders. However, finite shell element analysis shows that the realnormal stresses and the shear lag coefficients in the top and canti-lever slabs are quite different, and the normal stress at the edge ofcantilever slab is much smaller than that at the middle of the topslab due to different boundary conditions. For the details, seeexample 2 (Fig. 11).

Fig. 10 shows the diagrams of bending moment M and shear lagmoment Mx of the cantilever box girder subjected to uniform loadp = 1 kN/m. In fact, On the basis of the shear lag moment Mx

defined in the paper, the so-called negative shear lag can be illus-trated clearly through comparing the diagram of bending momentM with that of the shear lag moment Mx. Negative shear lag occursin the region where the shear lag moment and bending momentare of contrary sign.

6.2. Example 2

In this example we examine the accuracy of different degrees ofthe warping displacement functions for different shear lag intensi-ties in two-span continuous box girders with the same crosssection as that shown in Fig. 9(b). The span arrangement of thetwo-span continuous box girders is 340 mm + 340 mm and750 mm + 750 mm, respectively. Each box girder is subjected to

Page 8: shear wall 2

(a)

(b)

(c)Fig. 9. Numerical example of a cantilever box girder subjected to uniform load.

4702350

-200

-160

-120

-80

-40

0

40

Inte

rnal

mom

ent (

N·m

)

Longitudinal coordinate z (mm)

M10Mω

Negative shear lag

Fig. 10. Bending moment and generalized moment of the cantilever box girderexample.

(a)

(b)

(c)

(d)Fig. 11. Shear lag coefficients of two-span continuous box girders. (a) Mid-spancross section (l = 430 mm, b1/l = 0.22). (b) Internal support cross section(l = 430 mm, b1/l = 0.22). (c) Mid-span cross section (l = 750 mm, b1/l = 0.1). (d)Internal support cross section (l = 750 mm, b1/l = 0.1).

80 Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83

uniformly-distributed load p = 1 kN/m over the total length. TheYoung’s modulus of the material is E = 2900 MPa and the Poisson’sratio is l = 0.4. The box girders are analyzed by applying the pre-sented method (the warping displacement function is chosen asquadratic, cubic, and quartic parabola, respectively) and shell ele-ment of ANSYS. In the numerical analysis by ANSYS, the340 mm + 340 mm box girder is divided into 5168 shell elementsand 7600 shell elements are used for the 750 mm + 750 mm boxgirder. The shear lag coefficient at the mid-span cross sectionand internal support cross section of each box girder is calculatedand shown in Fig. 11. It can be seen from Fig. 11 that, the shear lag

Page 9: shear wall 2

(a)

(b)Fig. 12. Two-span continuous box girder model (Unit: mm). (a) Span arrangementand load. (b) Cross section.

(a)

(b)

(c)Fig. 13. Three-span continuous box girder model (unit: mm). (a) Span arrangementand testing cross section. (b) Cross sections. (c) Location of the measuring points.

Fig. 14. Distribution of generalized moment for shear lag.

Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83 81

effect of the box girder with span 340 mm (the corresponding ratioof width to length b1/l = 0.22) is much more significant than thatwith span 750 mm (ratio of width to length b1/l = 0.1); the shearlag effect at the internal support cross section is more significantthan that at the middle of span for each box girder; the shear lageffect at the mid-span cross section of the box girder with span750 mm is not significant and can be ignored. Fig. 11 also showsthat the distribution of the shear lag coefficient obtained by thepresent method is close to that by the shell element of ANSYS forboth box girders and the shear lag coefficient based on the warpingfunction of quadratic parabola is closest to that from ANSYS forboth box girders.

6.3. Example 3

In this example, a two-span continuous box girder model withconstant depth provided by Wei et al. [11] is analyzed by applyingthe analytical solution based on the initial parameter method. Thedetailed dimensions of the box girder model are shown in Fig. 12.The Young’s modulus of the material is E = 2800 MPa and the Pois-son’s ratio is l = 0.37. The concentrated load is P = 20 N. The max-imum stresses calculated and tested in the top slab of the two-spancontinuous box girder are listed in Table 1 and the numerical re-sults based on the FEM (ANSYS) are also listed in Table 1. 5168shell elements are used in the finite element analysis. It can beseen from Table 1 that the calculated values are in agreement withthe test values on the whole.

6.4. Example 4

In this example we examine the effectiveness of the presentedelement through a three-span continuous box girder model withvarying depth provided by Luo et al. [24]. The details of the modelare shown in Fig. 13. The span arrangement of the box girder modelis 460 mm + 860 mm + 460 mm, and the depth varies in a qua-dratic parabola form, being 80 mm at the two internal supportsand 40 mm at the middle of the main span. The Young’s modulusof the material is E = 2600 MPa and the Poisson’s ratio is l = 0.4.

Table 1Maximum stresses in the top slab of the two-span continuous box girder.

Location of cross sections Present method (kPa) FEM (kPa)(1) (2) (3)

Middle span �45.81 �51.83Interior support 53.31 51.99

The normal stresses at three cross sections numbered I, II and IIIwere tested, and the locations of the measuring points (numberedr–z) are shown in Fig. 13(c). Loads were applied in two cases: (1)A vertical concentrated force (P = 137.33 N) was applied to themodel at the middle of the main span; (2) vertical uniform load(p = 0.5 N/mm) is applied to the whole length. For the detail loadcase and other information, see reference [24].

The three-span continuous box girder is analyzed by using thecomputer program developed by the authors. The box girder is di-vided into 120 segment elements with uniform cross section. Thegeometrical properties of cross section of each element are chosento the average values of those at two ends. The calculated stresses

Experiment (kPa) |(2)–(4)|/|(4)| (%) |(3)–(4)|/|(4)| (%)(4) (5) (6)

�50.95 10.09 1.7360.09 11.28 13.48

Page 10: shear wall 2

Table 2Stress comparison at cross section I–I under uniformly distributed load.

Measuring points Present element (MPa) FEM (MPa) Experiment (MPa) |(2)–(4)|/|(4)| (%) |(3)–(4)|/|(4)| (%)(1) (2) (3) (4) (5) (6)

r �0.1679 �0.1609 �0.1476 13.75 9.01s �0.1753 �0.1944 �0.1680 4.34 15.71t �0.1953 �0.2361 �0.2135 8.52 10.59u �0.1930 �0.2285 �0.2147 10.11 6.43v �0.1606 �0.1932 �0.1671 3.89 15.62w �0.1477 �0.1749 �0.1467 0.68 19.22x �0.1930 �0.2285 �0.2138 9.73 6.88y �0.1953 �0.2361 �0.2126 8.14 11.05z �0.1679 �0.1609 �0.1755 4.33 8.32

08710980-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Def

lect

ion

(mm

)

Longitudinal coordinate z (mm)

wfw+f

Fig. 15. Deflection curves under concentrated load.

08710980-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Def

lect

ion

(mm

)

Longitudinal coordinate z (mm)

wfw+f

Fig. 16. Deflection curves under uniform load.

Table 3Mid-span deflection increase of the central span.

Concentrated load Uniform load

w (mm) f (mm) v (mm) f/w w (mm) f (mm) v (mm) f/w

0.1926 0.0430 0.2601 22.3% 0.2283 0.0545 0.2940 23.9%

82 Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83

at Section I–I by using shell element and the presented box beamelement as well as those from test provided by Luo et al. [24] arelisted in Table 2. Because of the limited paper length, the resultsof the other sections are not listed. It can be seen from Table 2 thatthe calculated values are in agreement with the test values and theshell element results on the whole.

Fig. 14 shows the diagrams of the generalized moment for shearlag in the three-span continuous box girder model. It can be seenfrom Fig. 14 that, for the concentrated load case, the maximumgeneralized moment occurs at the middle of the main span, andthose at the two internal supports are also relatively large (byabsolute values). However, they attenuate quickly with the dis-tance away from the mid-span or support location. For the uniformload case, the maximum generalized moment (by absolute values)occurs at the two internal supports and also attenuates quickly.

Figs. 15 and 16 show the deflection distribution of the continu-ous box girder under the concentrated and uniformly-distributedloads, respectively. That the shear lag effect reduces the in-planestiffness of the flange slabs results in the increase of the deflectionof the box girder. It can be seen from Fig. 15 that, for the concen-trated load case, the increase of the deflection due to shear lag oc-curs mainly near the middle of the main span, while the increase inthe side spans is negligible. However, from Fig. 16 we can see that,for the uniformly-distributed load case, the increase of the deflec-tion at the middle of the main span is more remarkable. In addi-tion, the influence of shear lag on the deflection in side spans isalso obvious.

Table 3 lists the mid-span deflections of the central span calcu-lated by using the present element and shell element (denoted byv). It can be seen that shear lag effect increases the deflection by22.3% for the concentrated load, while 23.9% for the uniform load.

Therefore, the influence of shear lag effect on the deflection of thebox girder is remarkable on the whole, and should be consideredseriously in engineering practice.

7. Conclusions

This paper presents a new method for analyzing shear lag effectin thin-walled box girders. The additional deflection induced byshear lag effect is adopted as generalized displacement to describethe shear lag deformation state. Based on the generalized momentfor shear lag defined in this paper, the shear lag deformation stateis separated from the flexural deformation state of the correspond-ing elementary beam and analyzed as a fundamental deformationstate. The governing differential equation and boundary conditionfor the additional deflection are established and the initial param-eter solution to the differential equation is given and applied toanalyze a numerical example of two-span continuous box girdermodel. A very simple and convenient formula of shear lag warpingstress is proposed which has the same form as that of the bendingstress of elementary beam. The quadratic parabola is proved anddemonstrated to be the reasonable mode of shear lag warping dis-placement function. The so-called negative shear lag can be easilydiscriminated by the generalized moment defined in the paper.

Page 11: shear wall 2

Y.-H. Zhang, L.-X. Lin / Engineering Structures 61 (2014) 73–83 83

Negative shear lag occurs in the region where the shear lag mo-ment and bending moment are of contrary sign. A finite beam seg-ment element with 8 degrees of freedom is developed and used toanalyze a three-span continuous box girder model with varyingdepth, and the calculated results are in agreement with the testresults and shell element results, validating the method and theelement presented. The increase of mid-span deflection for thethree-span continuous box girder model due to shear lag reaches22.3% and 23.9% under concentrated and uniformly-distributedloads, respectively, which should be treated seriously in engineer-ing practice.

Acknowledgments

The authors would like to gratefully acknowledge the financialsupport from the National Natural Science Foundation of China(Grant Nos. 51268029 and 51068018). The present research is alsosupported by the Program for Changjiang Scholars and InnovativeResearch Team in University of Ministry of Education of China(IRT1139).

References

[1] Reissner E. Analysis of shear lag in box beams by the principle of the minimumpotential energy. Q Appl Math 1946;4(3):268–78.

[2] Kuzmanovic BO, Graham HJ. Shear lag in box girders. J Struct Div1981;107(9):1701–12.

[3] Dezi L, Mentrasti L. Nonuniform bending-stress distribution (shear lag). JStruct Eng 1985;111(12):2675–90.

[4] Chang ST, Yun D. Shear lag effect in box girder with varying depth. J Struct Eng1988;114(10):2280–92.

[5] Luo QZ, Wu YM, Li QS, Tang J, Liu GD. A finite segment model for shear laganalysis. Eng Struct 2004;26(14):2113–24.

[6] Luo QZ, Li QS, Tang J. Shear lag in box girder bridges. J Bridge Eng2002;7(5):308–13.

[7] Luo QZ, Tang J, Li QS. Finite segment method for shear lag analysis of cable-stayed bridges. J Struct Eng 2002;128(12):1617–22.

[8] Wu YP, Liu SZ, Zhu YL, Lai YM. Matrix analysis of shear lag and sheardeformation in thin-walled box beams. J Eng Mech 2003;129(8):944–50.

[9] Wu YP, Zhu YL, Lai YM, Pan WD. Analysis of shear lag and shear deformationeffects in laminated composite box beams under bending loads. Compos Struct2002;55(2):147–56.

[10] Zhou SJ. Finite beam element considering shear-lag effect in box girder. J EngMech 2010;136(9):1115–22.

[11] Wei CL, Li B, Zen QY. Transfer matrix method considering both shear lag andshear deformation effects in non-uniform continuous box girder. Eng Mech2008;25(9):111–7 [in Chinese].

[12] Ni YZ, Qian YQ. Elastic analysis of thin-walled beam bridges. Beijing: ChinaCommunication press; 2000. p. 77–98 [in Chinese].

[13] Gan YN, Zhou GC. An approach for precision selection of longitudinal shear lagwarping displacement function of thin-walled box girders. Eng Mech2008;25(6):100–6 [in Chinese].

[14] Luo QZ, Li QS. Shear lag of thin-walled curved box girder bridges. J Eng Mech2000;126(10):1111–4.

[15] Zhang YH, Li Q. Flexural-torsional analysis of thin-walled curved box girderswith shear lag and secondary shear deformation in restraint torsion, China. CivEng J 2009;42(3):93–8 [in Chinese].

[16] Lin ZB, Zhao J. Least-work solutions of flange normal stresses in thin-walledflexural members with high-order polynomial. Eng Struct2011;33(10):2754–61.

[17] Lin ZB, Zhao J. Modeling inelastic shear lag in steel box beams. Eng Struct2012;41(8):90–7.

[18] Sa-nguanmanasak J, Chaisomphob T, Yamaguchi E. Stress concentration due toshear lag in continuous box girders. Eng Struct 2007;29(7):1414–21.

[19] Lertsima C, Chaisomphob T, Yamaguchi E. Stress concentration due to shearlag in simply supported box girders. Eng Struct 2004;26(8):1093–101.

[20] Zhang YH, Wang LL, Li Q. One-dimensional finite element method and itsapplication for the analysis of shear lag effect in box girders, China. Civ Eng J2010;43(8):44–50 [in Chinese].

[21] Zhang YH, Lin LX. A method considering shear lag effect for flexural–torsionalanalysis of skewly supported continuous box girder. Eng Mech2012;29(2):94–100 [in Chinese].

[22] Zhang YH, Su YD, Lin LX. Finite beam element analysis on shear lag effect ofskewly supported continuous box girder. J China Railway Soc2012;34(10):85–90 [in Chinese].

[23] Zhang YH. Improved finite-segment method for analyzing shear lag effect inthin-walled box girders. J Struct Eng 2012;138(10):1279–84.

[24] Luo QZ, Wu YM, Tang J, Li QS. Experimental studies on shear lag of box girders.Eng Struct 2002;24(4):469–77.