Engineering Structures 145 (2017) 44–59

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Engineering Structures

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

Distortional analysis of simply supported box girders with innerdiaphragms considering shear deformation of diaphragms using initialparameter method

http://dx.doi.org/10.1016/j.engstruct.2017.05.0040141-0296/� 2017 The Authors. Published by Elsevier Ltd.This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

⇑ Corresponding author at: Key Lab of Civil Engineering Safety and Durability ofChina Education Ministry, Department of Civil Engineering, Tsinghua University,Beijing 100084, China.

E-mail address: renyz66@mail.tsinghua.edu.cn (Y. Ren).

Yangzhi Ren a,b,⇑, Wenming Cheng b, Yuanqing Wang a, Qingrong Chen b, Bin Wang c

aKey Lab of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, ChinabDepartment of Mechanical Engineering, Southwest Jiaotong University, No.111, North Section 1, Second Ring Road, Chengdu, Sichuan 610031, ChinacCollege of Engineering, Design and Physical Sciences, Brunel University, London, Uxbridge UB8 3PH, UK

a r t i c l e i n f o

Article history:Received 14 November 2016Revised 27 April 2017Accepted 3 May 2017Available online 11 May 2017

Keywords:Simply supported girderDiaphragmEccentric loadShear deformationDistortionInitial parameter methodFinite element analysisExperimentLocal stability

a b s t r a c t

In this paper, the distortion of simply supported girders with inner diaphragms subjected to concentratedeccentric loads is investigated using initial parameter method (IPM), in which the in-plane shear defor-mation of diaphragms is fully considered. A statically indeterminate structure was modeled with innerredundant forces, where the interactions between the girder and diaphragms were indicated by a distor-tional moment. Considering the compatibility condition between the girder and diaphragms, solutionsfor the distortional angle, warping displacements and stresses were derived and further simplified byestablishing a matrix equation system. The validity of IPM was intensively verified by a finite elementanalysis and distortional experiments. Parametric studies were then performed to examine the effectof the diaphragm number on the distortional angle, warping displacements and stresses under variousratios of height to span of the girder and the diaphragm thicknesses. Besides, stabilities of the localweb plate and mid-span diaphragm were analyzed based on IPM for box girders with symmetrical threeinner diaphragms. Results show that the local web plate will buckle before overall yielding with theincrement of the eccentric loads Pj, and the mid-span diaphragm is constantly stable in the whole defor-mation process. It shows that more attentions should be paid on the stability of the local web plate thanoverall yielding for girders subjected to eccentric loads.� 2017 The Authors. Published by Elsevier Ltd. This is an openaccess article under the CCBY license (http://

creativecommons.org/licenses/by/4.0/).

1. Introduction

During the past several decades, box girders have been widelyapplied in buildings and bridges due to their large bending and tor-sional stiffness. However, they are generally susceptible to thecross-sectional distortion [1] under eccentric loads due to theirquadrilateral instability. Therefore, excessive distortional warpingand transversal bending stresses will be produced besides thetorsional and bending ones in box girders. In a special case, the dis-tortional warping stresses may be significant to the torsional andbending ones. In order to control the distortion, diaphragms areinstalled at the interior of the girder, which can increase not onlythe stability of the local plate, but also the resistance to the warp-ing deformations and stresses [2–4].

Researches on the distortion of girders with inner diaphragmshave been performed for many decades. The distortion of box gir-der was initially studied by Dabrowski [5] who first formulated thedistortion of box girders with a symmetrical cross section. Li [6,7]proposed that the shear strain of the cross section cannot beignored when the distortional shear rigidity is significant com-pared to the distortional warping one for box girders. Wright [8]proposed the Beam on Elastic Foundation (BEF) analogy for the dis-tortion of girders with inner diaphragms, where the diaphragmsare analogous to inner supports. Based on BEF, Hsu [9,10] proposedthe Equivalent Beam on Elastic Foundation (EBEF) analogy consid-ering the shear strain of the cross section, and found that the EBEFanalogy is more versatile than BEF due to its simplicity in analyzingmore complex problems such as non-uniform sections and multi-span beams.

Interactions between the girder and diaphragms is the key issuefor the distortion of girders with inner diaphragms. A staticallyindeterminate structure [11] was modeled with redundant forcesacting along the junctions between the girder and diaphragms.

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 45

Moreover, the force method was applied to calculate redundantforces, where elements in the stiffness matrix were obtained fromthe finite strip method [12]. The numerical results were thenextended to multi-span bridges [13] and long-span curved bridges[14]. An outstanding contribution was made by Suetake [15],where the girder was regarded as an assembly of thin plates, andthe extended trigonometric series method (ETS) was applied toanalyze the stresses and deformations for box girders. Compar-isons with FEM results show that ETS has a high accuracy. How-ever, it is inconvenient to apply since there are manysimultaneous nonlinear equations to solve even for girders withfew diaphragms, e.g. there are up to 720 equations for a girder withtwo diaphragms.

The wall thickness of diaphragms and the number of dia-phragms in a girder will make a significant influence on the dis-placements and stresses. Park [16,17] proposed a new beamelement with nine degrees of freedom per node for girders. Studiesshowed that the distortional warping and transversal bendingstresses were reduced by increasing the diaphragm number. Simi-lar conclusions can be drawn for straight multi-cell box girderswith diaphragms [18,19]. For horizontally curved bridges, therational spacing between adjacent diaphragms was provided [20]according to the ratio between the distortional warping stressand the bending stress. Using FEM, Zhang [21] found that therational number for diaphragms is 3 to 5 when the ratio of widthto height of the cross section is 1.5 and the rational number is 9when the ratio is 4.5. Li [22] proposed a new finite element solu-tion, and found that the distortional warping stress for cantilevergirders can be ignored when the spacing between adjacentdiaphragms is less than one fifth of the span; while for simplysupported and fixed girders, the spacing is less than one eighthof the span.

Initial parameter method (IPM) was proposed first by Vlasov[23] to analyze the non-uniform torsion of beams. Analogous toIPM in non-uniform torsion, IPM can be extended to analyze dis-tortions of girders. Considering the effect of shear strains of thecross section, Xu [24,25] developed an equation with the variabledistortional angle, and established two categories of IPMs of thefourth order, classified by the ratio of the distortional stiffness.Harashima [26] proposed a distortional equation with a distor-tional warping function, and established the fifth-order IPM. BothIPMs in distortion have a high efficiency compared with FEM. How-ever, IPMs has not been extended into the distortion for girderswith inner diaphragms.

tp(i+1)

zpi

zp(i+1) Pj zj

P j+1zj+1

l

(a) simply supported box girder

y xz

Otpi

Inner diaphrPin joint

with inner diaphragms

Fig. 1. Girder with inner diaphragms un

For distortions of a girder with inner diaphragms, an assump-tion of infinite-rigidity diaphragm was generally made in moststudies [16,20,27], where the in-plane deformation of diaphragmswas totally restrained and warping was free. Similar assumptionscan be found in distortion of curved box beams [28], where the dis-tortional angle at the location of diaphragms is set as zero. How-ever, the infinite-rigidity assumption is just an approximation,which is not applicable to thin flexible diaphragms. The mainobjective of this work is to investigate the distortion of simply sup-ported girders with inner flexible diaphragms under concentratedeccentric loads, where the in-plane shear deformation of dia-phragms is fully considered. Interactions between the girder anddiaphragms are indicated by a distortional moment. Based on thecompatibility condition between the girder and diaphragms, solu-tions for both the distortional angle and warping function areobtained from the IPM. Taking a simply supported girder with 2,5 and 9 diaphragms, respectively, as an example, the distortionalsolutions from IPM were obtained, then verified by a FE analysisand experiments. This was followed by a parametric study, inwhich distortional deformations and stresses were investigatedin terms of the diaphragm number and thickness and the heightto span ratio of the girder. Based on the proposed IPM, stabilitiesof both the local web plate and mid-span diaphragm were exam-ined for girders with three symmetrical inner diaphragms. A seriesof curves were obtained for the relations between the critical buck-ling load and the position of diaphragms under various height towidth ratios of the cross section.

2. Structural model

Consider a simply supported box girder with inner diaphragmsunder concentrated eccentric loads Pj (j = 1,2,. . .,m). The coordinatesystem O-xyz is illustrated in Fig. 1a with the original O at the shearcenter onone endof the girder. For analysis, thedistancesbetweenOand themid-lines ofwebs B andD aremarked by n1 and n2 in Fig. 1b,respectively. The girder ismadeof a homogeneous, isotropic and lin-early elastic material with its Young’s and shear moduli denoted byE and G, respectively. The girder span is l. The thicknesses of web Band D are t1 and t2 and their height is h, while the width of flangesA and C is b and the thickness t3. The thickness for the ith diaphragmis tpi (i = 1,2,. . .,n) and itsmid line ismarked by zpi, measured fromO.The load Pj is located on the top of web D at zj.

Fig. 2a shows that the eccentric load Pj can be decomposed intothree components [29,30] – the flexural load, the torsional and the

b

h

t2t1

t 3

y

xO

t 3

N

M

n1 n2

(b) cross section

J

K

A

B

C

D

Pj

Nodes:M,N,J,KFlanges:A,C; Webs:B,D

agms

s

s

s

s

der eccentric loads Pj (j = 1,2,. . .,m).

Fig. 2. Loading decomposition, deformations and stresses of girders.

46 Y. Ren et al. / Engineering Structures 145 (2017) 44–59

distortional load. In Fig. 2b, the cross-section rigidly rotates aroundO with a torsional angle h under torsional loads. Fig. 2c illustratesthe transversal deformations in the web and the flange under dis-tortional loads, where uM and vM are horizontal and vertical dis-placements at node M, respectively. The variation of the rightangle at node N is defined as the distortional angle v, given byv = v1 + v2. Moreover, the warping displacement wd and the stressrd, produced by the distortional moment Bd, are shown in Fig. 2d.There also exists the shear stress sd along the cross-section profile,developed by the distortional moment Md, as shown in Fig. 2e.

This article will focus on the distortional deformation and stres-ses of a simply supported box girder with inner diaphragms sub-jected to concentrated eccentric loads.

3. Distortion of girders without diaphragm

In distortion, when the shear stiffness of the cross section has asignificant value in comparison with the warping one, the influ-ence of shear strain of the cross section on deformation and stres-ses cannot be ignored [6,7]. The distortional differentiate equationcan be expressed as [7]

EItWðzÞ0000 � EIcEItGIk

WðzÞ00 þ EIcWðzÞ ¼ m0d ð1Þ

where It, Ic and Ik are distortional warping, frame and shear con-stants, respectively; W(z) is the distortional warping function; md

is the distributed distortional moment. The number of the apostro-phes of W(z) indicates the first, second and forth differentiationswith respect to the z-coordinate.

Under concentrated distortional loads, md = 0, and the solutionfor Eq. (1) is given as [26]

WðzÞ ¼X4i¼1

BiuiðzÞ ð2Þ

where Bi (i = 1, 2, 3, 4) are coefficients determined by theboundary conditions, and ui are defined as

u1 = cosh(k1z)sin(k2z), u2 = cosh(k1z)cos(k2z), u3 = sinh(k1z)cos(k2z), u4 = sinh(k1z)sin(k2z)

where ki (i = 1,2) are distortional coefficients.

k1 ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4EIcEIt

sþ EIcGIk

vuut; k2 ¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4EIcEIt

s� EIcGIk

vuutRelations between the warping function and the distortional

angle are [7]

vðzÞ ¼ � EItEIc

W 000ðzÞ;BdðzÞ ¼ �EItW0ðzÞMdðzÞ ¼ �EItW

00ðzÞ ð3Þ

Substituting Eq. (2) into Eq. (3), the matrix equation is given by

ZðzÞ ¼ UðzÞB ð4Þwhere

UðzÞ¼

� EItEIcu000

1 ðzÞ � EItEIcu000

2 ðzÞ � EItEIcu000

3 ðzÞ � EItEIcu000

4 ðzÞu1ðzÞ u2ðzÞ u3ðzÞ u4ðzÞu0

1ðzÞ u02ðzÞ u0

3ðzÞ u04ðzÞ

u001ðzÞ u00

2ðzÞ u003ðzÞ u00

4ðzÞ

26664

37775; B¼B1;B2;B3;B

T4;

ð5ÞZ(z) is the state vector in IPM,

ZðzÞ ¼ fvðzÞ; WðzÞ; �BdðzÞEIt

;MdðzÞEIt

gT

ð6Þ

The boundary conditions for a simply supported girder are

vð0Þ ¼ 0;Bdð0Þ ¼ 0; for the initial end z ¼ 0;

vðlÞ ¼ 0;BdðlÞ ¼ 0; for the ultimate end z ¼ l:

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 47

Correspondingly, the state vectors on both ends are

Zð0Þ ¼ 0; Wð0Þ; 0;Mdð0ÞEIt

� �T; ZðlÞ

¼ 0; WðlÞ; 0;MdðlÞEIt

� �T

For z = 0, Z(0) =U(0)B, we have

B ¼ ½Uð0Þ�inv � Zð0Þ ð7Þwhere [U(0)]inv is the inverse matrix of U(0).Substituting Eq. (7) into Eq. (4) yields in the state vector Z(z)

ZðzÞ ¼ PðzÞ � Zð0Þ ð8Þwhere P(z) is the transfer matrix, and P(z) =U(z)�[U(0)]inv.Eq. (8) is the standard form of IPM for the distortion of girders

without diaphragms. However, the transfer matrix P(z) is compli-cated. Based on the relations between ui(z) (i = 1,2,3,4) and theirdifferentiations (see Eq. (A1)–(A3) in Appendix A), the matrix P(z) is simplified as

PðzÞ ¼

�SC 0001 ðzÞ �KC000

2 ðzÞ �SKC0003 ðzÞ KC 000

4 ðzÞSK C1ðzÞ C2ðzÞ SC3ðzÞ �C4ðzÞSK C

01ðzÞ C 0

2ðzÞ SC03ðzÞ �C 0

4ðzÞ� S

K C001ðzÞ �C 00

2ðzÞ �SC 03ðzÞ C00

4ðzÞ

26664

37775 ð9Þ

where S ¼ 12k21þ2k22

, K ¼ EItEIc, C1ðzÞ ¼ u3ðzÞ

k1� u1ðzÞ

k2, C2ðzÞ ¼ u2ðzÞ�

k21�k222k1k2

u4ðzÞ, C3ðzÞ ¼ 3k21�k22k1

u3ðzÞ � k21�3k22k2

u1ðzÞ, C4ðzÞ ¼ u4ðzÞ2k1k2

.

The jth distortional load in IPM is indicated by a vector Zj, givenby

Zj ¼ 0; 0; 0;Mj

EIt

� �T

ð10Þ

where Mj is the distortional moment for the jth distortional load,and Mj = Pj�n1/2 [26]; n1 is the distance between the web D andpoint the original O (see Fig. 1b).

4. Distortion of box girders with inner diaphragms

4.1. IPM solution

For analysis, a statically indeterminate structure is modeledwith inner redundant forces acting along the junctions betweenthe girder and the diaphragms. The entire model is shown inFig. 3a and the horizontal and vertical redundant forces Hej andVej are illustrated in the zoomed picture, where the subscript eindicates the webs and flanges and e = A,B,C,D (see Fig. 1b). The

Fig. 3. A statically indeterminate structure and the

small circles indicate the joints between the girder and diaphragmswhere redundant forces are located.

In order to analyze the interactions between the girder and dia-phragms, two assumptions are made:

(1) Self balance of the in-plane forces of diaphragms

Under distortional loads, the summation of all vertical and hor-izontal redundant forces and moments are zero for diaphragms.That is

Xe¼A;B;C;D

Xrj¼2

Hej ¼ 0;X

e¼A;B;C;D

Xrj¼2

Vej ¼ 0X

e¼A;B;C;D

Xrj¼2

Hejyej þX

e¼A;B;C;D

Xrj¼2

Vejxej ¼ 0

ð11Þwhere xej and yej are the distances between the redundant forces Vej,Hej and the original O.

Based on the self balance assumption, only the distortionalcomponent for redundant forces is reserved. Referred to the forma-tion of external momentMj [26,27], the distortional components ofthe redundant forces are gathered and indicated by a concentrateddistortional moment Mpi for the ith diaphragm. Therefore, theinteractions between the girder and diaphragms can be indicatedby a moment Mpi (i = 1,2,. . .,n). Only Mpi,, opposite to Mj, will resistthe distortional deformation and stresses. In IPM, Mpi is indicatedby a vector Zpi, given by

Zpi ¼ 0; 0; 0;Mpi

EIttpi

� �T

ð12Þ

(2) Compatibility condition between the girder and diaphragms

In-plane shear strains of diaphragms are considered, given bycpi =Mpi /(Gbhtpi). The compatibility condition is that the distor-tional angle at the mid line of diaphragms is opposite to the in-plane shear strain of diaphragms. That is v(zpi) = �cpi (0 5 i 5 n).This is a key aspect for the distortion of girders with innerdiaphragms.

Combining Eq. (8) and Eq. (12), the state vector Z(z) can beexpressed as

ZðzÞ ¼ PðzÞZð0Þ �XRi¼1

Z zpiþtpi=2

zpi�tpi=2Pðz� ziÞZpidzi �

XSj¼1

Pðz� zjÞZj

ð13Þwhere R and S are the numbers of diaphragms and moments Mj

before point z, respectively. The transfer matrices P(z � zi) and P

Mpi

Mpi

equivalent boundaries for the ith diaphragm.

48 Y. Ren et al. / Engineering Structures 145 (2017) 44–59

(z � zj) are obtained from P(z) by substituting the variable z by‘z � zi’ and ‘z � zj’.

For z = l, Eq. (13) changes into

ZðlÞ ¼ PðlÞZð0Þ �Xni¼1

Z zpiþtpi=2

zpi�tpi=2Pðl� ziÞZpidzi �

Xmj¼1

Pðl� zjÞZj ð14Þ

where the vectors Z(l) and Z(0) are referred to Eq. (6); W(0) andMd(0) in vector Z(0) can be calculated from the first and third simul-taneous equations in Eq. (14). Then, substituting W(0) and Md(0)into Eq. (13), the distortional angle and the warping function canbe obtained as

vðzÞ ¼Pn

i¼1R1giðzÞMpi þ

Pmj¼1S

1e24ðz; zjÞMj

2k1k2EIcU24ðl; lÞ ð15Þ

(a) n=2

(c) n=9

yx

zO

yO

yx

zO

yO

Fig. 4. Meshing grid, ending boundaries a

(b) D(a) Distortional warping displacement

Fig. 5. 3D contours of a girder with two diaphragms u

(b) D(a) Distortional warping displacement

Fig. 6. 3D contours of a girder with five diaphragms un

WðzÞ ¼Pn

i¼1R2giðzÞMpi þ

Pmj¼1S

2e24ðz; zjÞMj

2k1k2EItU24ðl; lÞ ð16Þ

where

R1giðzÞ ¼

2tpi

n0031 l� zpi;tpi2

� �U33

42ðz; lÞ þ n0231 l� zpi;tpi2

� �U13

42ðl; zÞ� �

� H Rþ 12� i

� �2U24ðl; lÞ

tpin0231 z� zpi;

tpi2

� �

R2giðzÞ ¼

2tpi

n0031 l� zpi;tpi2

� �U30

42ðl; zÞ � n0231 l� zpi;tpi2

� �U10

42ðl; zÞ� �

þ H Rþ 12� i

� �2U24ðl; lÞ

tpin0ð�1Þ24 z� zpi;

tpi2

� �

(b) n=5

conditionsloading(d)

xz

Ph

Ph

Pv

Pv

Ph

Ph

Pv

Pv

at section z1=0.45l at section z2=0.55l

xz

nd loading conditions in FEA model.

istortional warping stress (c) Frame deformation

nder distortional loads (tp = 0.01 m, amp = 3000).

istortional warping stress (c) Frame deformation

der distortional loads (tp = 0.01 m, amp = 10,000).

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 49

S1e24ðz; zjÞ ¼ K24ðz; zjÞ � H Sþ 1

2� j

� �u000

4 ðz� zjÞU24ðl; lÞ

S2e24ðz; zjÞ ¼ K24ðz; zjÞ þ H Sþ 1

2� j

� �u4ðz� zjÞU24ðl; lÞ

H(x) is the unit step function. H(x) = 1 for x > 0 and H(x) = 0 forx < 0

Uijðx; yÞ ¼u000

i ðxÞ u000j ðyÞ

u0iðxÞ u0

jðyÞ

; Kijðx; yÞ ¼

uiðxÞ U4iðl� y; lÞujðxÞ U4jðl� y; lÞ

;

Kijðx; yÞ ¼ d3Kijðx;yÞdx3

(b) D(a) Distortional warping displacement

Fig. 7. 3D contours of a girder with nine diaphragms u

0 0.5-2

-1

0

1

2 x 10-6

0 0.5 1-5

0

5

10

15

x 10-5

IPM-tp/t=0.5IPM-tp/t=1IPM-tp/t=2FEA-tp/t=0.5FEA-tp/t=1FEA-tp/t=2(z

)

z/l z/l

wdN

( z)/m

(a) distortional angle (z) (b) warping displ

IPM-tp/t=0.IPM-tp/t=1IPM-tp/t=2

Fig. 8. The distortional angle, warping displacements and stresses between IPM andthicknesses.

0 0.5 1-1

0

1

2

3

4

5 x 10-5

0 0.5-5

0

5 x 10-7

χ(z)

z/l z/l

wdN

(z)/m

(a) distortional angle χ(z) (b) warping displa

IPM-tp/t=0.5IPM-tp/t=1IPM-tp/t=2

IPM-tp/t=0.5IPM-tp/t=1IPM-tp/t=2

FEA-tp/t=0.5FEA-tp/t=1FEA-tp/t=2

Fig. 9. The distortional angle, warping displacements and stresses between IPM and FEAthicknesses.

Uijmnðx; yÞ ¼

uðiÞm ðxÞ uðiÞ

n ðxÞuðjÞ

m ðyÞ uðjÞn ðyÞ

; nijmnðx; yÞ ¼

uðiÞm ðxÞ �uðiÞ

n ðxÞuðjÞ

m ðyÞ uðjÞn ðyÞ

where um(i)(x) and un

(i)(y) are the ith differentiation of functionsum(x) and un(y). un

(�1)(y) is the integral of function un(y), given by

uð�1Þ2 ðyÞ ¼ k1u3ðyÞ þ k2u1ðyÞ

k21 þ k22; uð�1Þ

4 ðyÞ ¼ k1u1ðyÞ � k2u3ðyÞk21 þ k22

When the calculated point z is located in the thickness of (R + 1)th diaphragm (zp(R+1)–tp(R+1)/2 5 z 5 zp(R+1)+tp(R+1)/2), an additionalangle vadd and function Wadd should be involved,

istortional warping stress (c) Frame deformation

nder distortional loads (tp = 0.01 m, amp = 20,000).

0 0.5 1-3

-2

-1

0

1

2

3

4 x 106

1

acement wdN(z)

5

z/l

dN(z

)/Pa

(c) warping stress dN(z)

FEA-tp/t=0.5FEA-tp/t=1FEA-tp/t=2

IPM-tp/t=0.5IPM-tp/t=1IPM-tp/t=2FEA-tp/t=0.5

FEA-tp/t=2FEA-tp/t=1

FEA for a simply supported girder with two diaphragms of different diaphragm

1 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

2 x 106

cement wdN(z)z/l

σ dN(z

)/Pa

(c) warping stress σdN(z)

FEA-tp/t=0.5FEA-tp/t=1FEA-tp/t=2

IPM-tp/t=0.5IPM-tp/t=1IPM-tp/t=2FEA-tp/t=0.5

FEA-tp/t=2FEA-tp/t=1

for simply supported girders with five diaphragms varying with three diaphragm

50 Y. Ren et al. / Engineering Structures 145 (2017) 44–59

vadd ¼ MpðRþ1Þ½2k1k2 �u004ðz� zpðRþ1Þ þ tpðRþ1Þ=2Þ�

2k1k2EIctpðRþ1Þð17Þ

Wadd ¼ MpðRþ1Þuð�1Þ4 ðz� zpðRþ1Þ þ tpðRþ1Þ=2Þ2k1k2EIttpðRþ1Þ

ð18Þ

where zp(R+1), tp(R+1) and Mp(R+1) are the mid-line location, the thick-ness and the distortional moment of the (R + 1)th diaphragm,respectively.

Based on Eqs. (15)–(18), both the angle v(z) and the function W(z) are related to Mj and Mpi. Since Mj has been given in Eq. (10),solutions rest in Mpi.

4.2. Derivation of Mpi

The compatibility condition gives the equation

Xni¼1

T1giðzpTÞMpi þ

Xmj¼1

T1e24ðzpT ; zjÞMj

�U24ðl; lÞtpT

u004

tpT2

� �� 2k1k2 1þ EIc

Gbh

� �� �MpT ¼ 0 ð19Þ

for the Tth diaphragm (T = 1,2,. . .,n), where

0 0.5 10

0.5

1

1.5

2

2.5

x 10-5

0 0-4

-2

0

2

4 x 10-7

χ(z)

z/l z

w dN(z

)/m

(a) distortional angle χ(z) (b) warping disp

IPM-tp/t=0.5IPM-tp/t=1IPM-tp/t=2

IPM-tp/t=0IPM-tp/t=1IPM-tp/t=2

FEA-tp/t=0.5FEA-tp/t=1FEA-tp/t=2

Fig. 10. The distortional angle, warping displacements and stresses between IPM and FEthicknesses.

n=2, with shear strain of the cross section

0 0.5 1-2

0

2

4

6

8

10 x 10-5

0 0-1.5

-1

-0.5

0

0.5

1

1.5 x 10-6n=2, without shear strain of the cross section

χ

z/l

wd

z

14.9%13.3%

17.8%

(a) (b

Fig. 11. Comparison of girders with and without the shear strain of cross section for (a) thgirders with 2 and 5 diaphragms.

T1giðzpTÞ ¼ 2

tpin0031 l� zpi;

tpi2

�U33

42ðzpT ; lÞ þ n0231 l� zpi;tpi2

�U13

42ðl; zpTÞh i

�H T � 12 � i

� 2U24ðl;lÞtpi

n0231 zpT � zpi;tpi2

�T1e24ðzpT ; zjÞ ¼ K24ðzpT ; zjÞ � H kT þ 1

2 � j�

U24ðl; lÞu0004 ðzpT � zjÞ

kT is the number of distortional loads before the Tth diaphragm.The matrix equation system for Eq. (19) is

g �Mp þ e ¼ 0 ð20Þwhere

Mp ¼ fMp1;Mp2; . . . ;MpngT; g¼

R11 T1g2ðzp1Þ T

1gnðzp1ÞT1g1ðzp2Þ R22 T

1gnðzp2Þ. . . . . . . . . . . .

T1g1ðzpnÞ T

1g2ðzpnÞ Rnn

26664

37775

e ¼Xmj¼1

T1e24ðzp1; zjÞMj;

Xmj¼1

T1e24ðzp2; zjÞMj; :::;

Xmj¼1

T1e24ðzpn; zjÞMj

" #T

and the diagonal elements in matrix g is

Rii ¼ T1giðzpiÞ �

U24ðl; lÞtpi

u004

tpi2

� �� 2k1k2 1þ EIc

Gbh

� �� �

.5 1 0 0.5 1-5

0

5

10

15 x 105

/llacement wdN(z)

z/l

σ dN(z

)/Pa

(c) warping stress σdN(z)

.5IPM-tp/t=0.5IPM-tp/t=1IPM-tp/t=2

FEA-tp/t=0.5FEA-tp/t=1FEA-tp/t=2

FEA-tp/t=0.5FEA-tp/t=1FEA-tp/t=2

A for simply supported girders with nine diaphragms varying with three diaphragm

.5 1 0 0.5 1-2

-1

0

1

2

3

4 x 106

/l

σ d

z/l) (c)

n=5, with shear strain of the cross sectionn=5, without shear strain of the cross section

e distortional angle, (b) warping displacements and (c) stresses of simply supported

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 51

Consequently, Mpi can be obtained by

Mpi ¼ �Xmj¼1

QijMj ð21Þ

according to the Cramer rule. Qij = |gi|/|g|, where the |g| indicatesthe determinant of g, and gi is the same of g except for the ith col-umn [T

1e24ðzp1; zjÞ, T1e24ðzp2; zjÞ; . . . ; T1e24ðzpn; zjÞ]T

4.3. Simplification of v(z) and W(z)

Substituting Eq. (21) into Eq. (15), and the angle v(z) changesinto

vðzÞ ¼Pn

i¼1

Pmj¼1

S1e24ðz;zjÞ

n � R1giðzÞQij

�Mj

2k1k2EIcU24ðl; lÞ ð22Þ

where n and m are the total numbers of diaphragms and distor-tional loads, respectively.

The number of calculation steps is m � n in Eq. (22) and itwould be time-consuming for girders with many diaphragmsunder many distortional loads. For solution, a matrix equation isestablished, given by

g � w ¼ a ð23Þ

Fig. 12. anti-symmetri

20

1500

150

20

1500

350

(a) stiff unit on flanges (mm)

(b) stiff unit on webs (mm)

Welded field

Welded field

St

Fig. 13. Stif

where w ¼w11 w12 w1nw21 w22 w2n. . . . . . . . . . . .wn1 wn2 wnn

0BB@

1CCA; a ¼

a11 a12 a1n

a21 a22 a2n

. . . . . . . . . . . .an1 an2 ann

0BB@

1CCA and

the elements in matrix a are

agk ¼Xm

j¼1½S1e24ðz;zjÞT1gkðzpgÞ=n� R

1gkðzÞT1e24ðzpg ;zjÞ�Mj ðg–kÞXm

j¼1½S1e24ðz;zjÞRgg=n� R

1ggðzÞT1e24ðzpg ;zjÞ�Mj ðg¼ kÞ

8<:

In this approach, the distortional angle v(z) can be simplified as

vðzÞ ¼ 12k1k2EIcU24ðl; lÞ

Xni¼1

wii ð24Þ

For the warping function W(z), a matrix equation can also beestablished similar to Eq. (23), and the elements agk (g, k = 1,2,. . .n) in matrix a are

agk ¼Xm

j¼1½S2e24ðz;zjÞT1gkðzpgÞ=n� R

2gkðzÞT1e24ðzpg ;zjÞ�Mj ðg–kÞXm

j¼1½S2e24ðz;zjÞRgg=n� R

2gkðzÞT1e24ðzpg ;zjÞ�Mj ðg¼ kÞ

8<:

Therefore, the function W(z) can be simplified as

WðzÞ ¼ 12k1k2EItU24ðl; lÞ

Xni¼1

wii ð25Þ

cal loading design.

(c) actual picture of stiff unit

Welded field

Stiff unit on flangesiff unit on webs

f units.

52 Y. Ren et al. / Engineering Structures 145 (2017) 44–59

Taking the node N (see Fig. 1b) of the cross section as an exam-ple, the distortional warping displacement wdN and the stress rdN

are given by

wdNðzÞ ¼ �WðzÞbdbh4ðbd þ 1Þ ;rdNðzÞ ¼ � EW0ðzÞbdbh

4ðbd þ 1Þ ð26Þ

where bd is the ratio of the distortional warping stresses betweennodes J and N, and bd ¼ 3bt3þht1

3bt3þht2

5. Verifications of IPM

5.1. Verifications with FEA

To verify the proposed IPM, simply supported girders with 2, 5and 9 diaphragms are investigated for three diaphragm thick-nesses, respectively by FEA using a commercial code ANSYS. Allgirders are modeled with the Young’s modulus E = 2.1 � 1011 Pa,the Poisson’s ratio t = 0.3, the span l = 1 m, the width b = 0.1 m,the height h = 0.2 m and the flanges and webs thicknessest = 0.01 m. Diaphragms are evenly distanced along the span withthe thicknesses tp = 0.005 m, 0.01 m and 0.02 m, respectively.

Fig.4a, b and c show the meshing grids of the girders and dia-phragms using Shell63 element in FEA model, where translationsin the x- and y- axial directions and rotations about the y- andz-axes on both ends are restrained. A convergent test shows that1650 to 2026 elements are adequate in terms of the diaphragmnumber. Fig. 4d shows the loading conditions, where two concen-trated distortional loads are applied along the flange and web in

Hydraulic cylinder

Hydraulic jack

Loading head Support Connect

FCS hloa

Tested girders

(a) entire scheme

Stiff unitWheel units

Fig. 14. Entire exper

A1B1

A2B2

A3B3

A4B4

A5B5

A6B6

A7B7

A8B8

A9B9

A10B10

A11B11

N

MJ

K

Rectangular tu

(a) scheme of tested points

Dial gag

y

xzO

Magnetic ba

Fig. 15. The scheme

cross sections z1 = 0.45l and z2 = 0.55l, and Ph = 1.25 kN andPv = 2.5 kN, respectively.

Figs. 5–7 show 3D contours of the distortional warping defor-mations and stresses for simply supported girders with 2, 5 and9 diaphragms, respectively. The ‘amp’ indicates the amplified fac-tor for deformations. It is seen that the largest displacement andstress occur at the junction between webs and flanges at the load-ing sections. With the increment of diaphragm number, the largeststress reduces from 5.51 MPa to 1.55 MPa and the displacementfrom 1.61 lm to 0.268 lm, and frame deformations at the loadingsections clearly become small.

Figs. 8–10 show the distortional angle, warping displacementsand stresses at node N from IPM and FEA for simply supportedgirders with 2, 5 and 9 diaphragms, respectively, in the relativediaphragm thickness tp /t = 0.5, 1 and 2. The distortional angle inFEA results can be calculated byFig. 9

vðzÞ ¼ UXN � UXM

hþ UYN � UYJ

bð27Þ

where UXN and UXM are x-axial displacements at nodes N and M(see Fig. 1b), respectively; UYJ and UYN are y-axial displacementsat nodes J and N, respectively.

Good agreements are observed between IPM and FEA inFigs. 8–10 for the distortional angle, warping displacements andstresses for simply supported girders with inner diaphragms. Com-pared the girders braced by 2 diaphragms with those by 5 and 9diaphragms, it’s worth noting that the mid-span diaphragm effec-tively restrains the transversal deformation.

ion beam

ydraulic servo ding system

(b) loading equipment

imental scheme.

be

(b) actual picture

es

se

of tested points.

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 53

For the distortional angle, the largest error between IPM andFEA occurs at the loading sections, where the FEA result is23.68% higher than the IPM one for girders with two diaphragms,and reduces to 13.86% for those with five diaphragms and 10.18%for those with nine diaphragms. Since there is no diaphragms orstiffeners at the loading sections, the error between IPM and FEAcan be attributed to the local stress concentration. So the distor-tional angle obtained from IPM is susceptible to the influence ofstress concentration.

In addition, the influence of shear strains of the cross section ondistortional deformations and stresses are examined in Fig. 11 forsimply supported girders with 2 and 5 diaphragms, where thecompatibility condition between the girder and diaphragms is con-sidered. It is seen that the shear strain of the cross section makeslittle effect on warping displacements and stresses, but a largeinfluence on the distortional angle. The largest difference occursat the loading sections z = 0.45l and z = 0.55l, which is 14.9% forgirders with 2 diaphragms and 17.8% with 5 diaphragms. Also,the error at the mid span for girders with 2 diaphragms is 13.3%.Thus shear strains of the cross section cannot be ignored whenthe transversal deformation of girders is considered.

5.2. Verifications with experiments

For further verification of the IPM, a series of experiments wereperformed using four groups of samples – girders with no dia-phragms, one, two and three diaphragms, subjected to distortionalloads. Diaphragms are equally distanced along the span. Both gird-ers and diaphragms were fabricated from carbon structural steelplates (yield strength 235 MPa) of 8 mm thickness. All girders are

-3

-2

-1

0

1

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10A11

IPMQuadratic fittingExperimentsSingularity point

(a) x-axial displacementTested points

UX

N/m

m

0.5

1

1.5

2

2.5

B1 B2 B3 B4 B5

IPMQuadExpeSing

(b) y-axial dTested

UY

N/m

m

Fig. 16. Comparison of transversal displacements and an

Fig. 17. Strain me

3 meters long, with height h = 0.6 m and width b = 0.346 m, givinga 30� angle between the diagonal and the web. All girders aresealed by a steel plate of 6 mm thickness on both ends.

To simultaneously produce two concentrated distortional loads,as in Fig. 4d, two steps were taken as follows

(1) For distortional loads, as shown in Fig. 12, units weredesigned with two groups of wheels anti-symmetricallyabout the shear center O of the cross section. This is todecompose the horizontal power force Psource produced byFCS hydraulic servo system into two orthogonal loadingcomponents Ph and Pv.

(2) For concentrated loads, as shown in Fig. 13, four stiff units of1.5 m in length and 0.15 m and 0.350 m in width, respec-tively, were welded onto flanges and webs symmetricallyover the mid span, and the loading sections were hencelocated at z1 = 0.75 m and z2 = 2.25 m, respectively.

During the experiments, compressive loadings were appliedalong the diagonal of the cross section by two anti-symmetricalwheel units, as shown in Fig. 12a. The entire experimental setupis shown in Fig. 14a, including two FCS hydraulic servo loading sys-tems, two connection beams, two wheel units, four stiff units, thetested girder and two supports. The maximum loading was set to5t with a loading speed of 2 mm/s, controlled by a loading equip-ment in Fig. 14b.

Fig. 15a shows measured points on the girder, with twelveequally distanced points A1–A11 on webs and B1–B11 on flanges.The transversal deformations on measured points were measuredby dial gages, magnetically attached to a rectangular tube of

B6 B7 B8 B9 B10B11

ratic fittingrimentsularity point

isplacement points

0.5 1 1.5 2 2.5

-5

0

5

10

15

x 10-3

(c) distortional anglez/m

χ/ra

d

IPMQuadratic fittingExperimentsSingularity point

gle between IPM and experiments under 5t loading.

asurements.

54 Y. Ren et al. / Engineering Structures 145 (2017) 44–59

3.5 m length, as shown in Fig. 15b. Measurements at each pointgave the x- and y- axial deformations UXN and UYN (N = 1,. . ., 11),and the distortional angle is calculated as

vðzÞ ¼ 2UXN

hþ 2UYN

bð28Þ

Compared with those from IPM, the transversal displacementsand distortional angle are depicted in Fig. 16 for girders without

Fig. 18. Comparisons of distortional warping stresses between IP

1 2 4 6 8 90

0.2

0.4

0.6

0.8

1

1.2

n(a) varying with h/l

d/d0

h/l=0.1h/l=0.2h/l=0.3h/l=0.4h/l=0.5

Fig. 19. The non-dimensiona

diaphragms under the source loading of 5t. There exist some errorsbetween the IPM and experimental results at the singularity pointsA1 for x-axial displacement and B10 and B11 for y-axial displace-ment, which are mainly caused by the residual strains of weldingand manufacturing. Eliminating the influences induced by singu-larity points, fitting lines were obtained from experimental resultsby applying the quadratic fitting method provided in the softwareMATLAB. Fig. 16 shows reasonable agreements between two

M and experiments (n being the total diaphragm number).

1 2 4 6 8 90

0.2

0.4

0.6

0.8

n(b) varying with tp/t

d/d0

tp/t=0.25tp/t=0.5tp/t=1tp/t=2tp/t=4

l warping stress rd/rd0.

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 55

results.For distortional warping stresses, strains (ed�0� ,ed�45� anded�90� ) in three directions were measured at points A1–A6 usingstrain gauges shown in Fig. 17a and recorded by the static strainequipment in Fig. 17b. The warping stress is calculated as

rd ¼ E1� t2

ðed�90� þ ted�0� Þ ð29Þ

where E = 2.1 � 1011Pa and t = 0.3.Fig. 18 shows calculated distortional warping stresses on the

tested points A1–A6 for girders with (a) no diaphragm, (b, c) onediaphragm, (d, e) two diaphragms and (f) three diaphragms,respectively. It is seen that the warping stresses obtained fromthe IPM and experiments have the same up-and-down trend andwith apexes at A3 for girders with diaphragms. The errors mainlycome from the drop of dh (Fig. 12a) between two horizontal sourceforces, which are mainly caused by the residual strain during man-ufacturing and welding. And this will produce torsional warpingstress besides the distortional one.

6. Parametric study

In this section, the effects of the ratio of height to span of thegirder h/l, the diaphragm thickness tp and number n are examinedon the distortion of simply supported girders with equally dis-tanced inner diaphragms, where the loading and boundary condi-tions are referred to those in Section 5.1.

1 2 4 6 8 90

0.1

0.2

0.3

0.4

0.5

0.6

/0

n(a) varying with h/l

h/l=0.1h/l=0.2h/l=0.3h/l=0.4h/l=0.5

Fig. 20. The non-dimensiona

1 2 4 6 8 90

0.5

1

1.5h/l=0.1h/l=0.2h/l=0.3h/l=0.4h/l=0.5

wd/ w

d0

n(a) varying with h/l

Fig. 21. The non-dimensional wa

Fig. 19 shows the non-dimensional warping stress rd/rd0 fornode N at the loading section z = 0.45l varying with the ratios h/land tp/t, where rd0 is the warping stress of girders without dia-phragms. It is seen that except for the girder with two diaphragmsunder h/l = 0.1, the non-dimensional stress rd/rd0 remains lessthan 1 and becomes smaller with the increment of n, tp/t and h/l,respectively. Besides, the rd/rd0 decreases substantially when thenumber of diaphragm reduces from 2 to 1, or increases from 2 to3, clearly indicating that the mid-span diaphragm plays a signifi-cant role in the reduction of distortional warping stress.

Fig. 20 shows the non-dimensional distortional angle v/v0 atthe loading section z = 0.45l varying with h/l and tp/t, where v0 isthe distortional angle for girders without diaphragms. It is seenthat except for the girder with two diaphragms under h/l = 0.1,the non-dimensional distortional angle v/v0 remains less than0.25 and becomes smaller with the increment of n, tp/t and h/l,respectively. This implies that the inner diaphragms are capableof restraining the transversal deformation of the cross section,especially when n > 3. Similarly, compared with the distortionalangle at n = 2, those at n = 1 and 3 decrease significantly due tothe restraint by the mid-span diaphragm.

Fig. 21 shows the non-dimensional warping displacement wd/wd0 for node N at the loading section z = 0.45l varying with theratios h/l and tp/t, where wd0 is the warping displacement for gird-ers without diaphragms. It is seen that except for the girder withtwo diaphragms under h/l = 0.1, the non-dimensional displace-

1 2 4 6 8 90

0.05

0.1

0.15

0.2

0.25

n(b) varying with tp/t

/0

tp/t=0.25tp/t=0.5tp/t=1tp/t=2tp/t=4

l distortional angle v/v0.

1 2 4 6 8 90

0.2

0.4

0.6

0.8tp/t=0.25tp/t=0.5tp/t=1tp/t=2tp/t=4

n(b) varying with tp/t

wd/w

d0

rping displacement wd/wd0.

56 Y. Ren et al. / Engineering Structures 145 (2017) 44–59

ment wd/wd0 in Fig. 21a remains less than 1.0 and converges to afixed value between 0.1 and 0.2 for large diaphragm numbers.

7. Buckling of both the local web plate and the mid-spandiaphragm

Taking a simply supported girder with uniform three inner dia-phragms as an example, Fig. 22 shows 3D contours for the warpingdisplacements and stresses, where the measurement, loading andboundary conditions are referred to those in Section 5.1. It is seenfrom Fig. 22 that the maximum displacement occurs at the localweb plate at the loading sections, and the maximum stress occursat the edge of the mid-span diaphragm, which may result in localbuckling. It is thus necessary to check the stabilities for both localweb plate and mid-span diaphragm.

In this section, the proposed IPM was carried to analyze thecritical buckling values Pcr1 and Pcr2 for the eccentric load Pj accord-ing to the stabilities of the local web plate and the mid-spandiaphragm, respectively. Inner diaphragms include one fixed

Diaphragm I Diaphragm IIIMid diaphragm(diaphragm II)

(a) 3D contour of distortional warping displaceme

(c) 3D contour of distortional warping stress

Maximum warping

Fig. 22. 3D contours of warping displacement and stress of g

Fig. 23. Mixed stress bound

(diaphragm II) at the mid span and two (diaphragms I and III) sym-metrical to the mid-span one. The loading and boundary conditionsare referred to those in Section 5.1.

Fig. 23 shows the mixed stress boundary condition for the localweb plate between the loading section and the mid span, in whicha = 0.05l. Under distortional loads, the warping stress rd varies lin-early from rd1 at the loading section to rd2 (=brd1) at the mid spanand from rd1(2) on the top to –rd1(2) at the bottom. There also existsa constant shear stress sd1 (=grd1) on all four sides of the cross-section, and a parabola sd2 on both lateral sides. Both shear stressescan be obtained from [26]

sdðs; zÞ ¼ �MdðzÞItt

ðSdðsÞ �HF SdðsÞqdðsÞdsH

F qdðsÞdsÞ ð30Þ

where Md(z) is the distortional moment in Fig. 2; It is the distor-tional warping constant in Eq. (1); t is the thickness of local webplate, given by t = 0.01 m; F is the cross-sectional area; qd(s) is thedistance between the distortional center [6] and the mid line ofthe cross-sectional profile; Sd(s) is the second moment of area, given

nt (b) deformation of mid-span diaphragm

(d) Stress of mid-span diaphragm

Original frame

Deformed frame

displacement

Maximum warping stress

irders with three uniform inner diaphragms (tp = 5 mm).

aries of local web plate.

Table 1Several definite integral items used in Eq. (35).

Definite integrals u = p u– p and u + p :even u– p and u + p :oddR a0 sin upz

a sin ppza dz a

2 0 0R a0 cos upz

a sin ppza dz 0 0 �2ap

pðu2�p2ÞR a0 z sin upz

a sin ppza dz a2

40 �4a2up

p2ðu2�p2Þ2R a0 y cos upy

a sin ppya dy � a2

4uppa2

pðu2�p2Þ�pa2

pðu2�p2ÞR a0 y2 cos upy

a sin ppya dy � a3

4uppa3

uðm2�p2Þ4pa3ð3u2þp2Þp3ðu2�p2Þ3 � pa3

pðu2�p2Þ

0 0.1 0.2 0.3 0.40.5

1

1.5

2

2.5

Dimension: 2×2

time=12s

Dimension: 4×4

time=51220s

time=508s

Dimension: 3×3

zp1/l

(d1

) cr /M

Pa

Fig. 24. Convergence of critical distortional warping stress (rd1)cr.

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 57

by Sd(s)=Rwds, w is the distortional sectorial coordinate [7], s is the

circumferential coordinate around the cross-sectional profile.Under the mixed stress boundary condition, the out-of-plane

deformation wp for local web plate can be indicated by the combi-nation of two orthogonal sinusoidal functions in z- and y-axialdirections, given by

wp ¼X1u¼1

X1v¼1

Auv sinupza

sinvpyh

ð31Þ

where u and v are the numbers of half waves in z- and y-axial direc-tions; Auv is the undetermined coefficient, which is the maximum ofthe combination of two sinusoidal functions.

For analysis, the distortional warping stress rd and the shearstress sd for the local web plate can be presumed as [31]

rd ¼ rd1 1� 2hy

� �1� 1� b

az

� �ð32Þ

sd ¼ sd1 þ sd2 ¼ rd1gþ rd11� ba

y� y2

h

� �ð33Þ

where b is the ratio of the warping stresses rd1 to the rd2; g is theratio of the warping stress rd1 to the constant shear stress sd1. Bothratios are calculated from IPM.

To obtain the critical buckling stress (rd1)cr for the local webplate, Galerkin equation [32] is applied, given by (p, q = 1,2,. . .,1)

Z a

0

Z h

0

@4wp

@z4þ 2

@4wp

@z2@y2þ @4wp

@y4þ trd

@2wp

@z2þ 2tsd

@2wp

@z@y

!

� sinppza

sinqpyh

dzdy ¼ 0 ð34Þ

Substitute Eqs. (31)–(33) into Eq. (34), the latter is changed into

X1u¼1

X1v¼1

Auv

upa

� 2þ vph

� 2 �2 R a0

R h0 sinupz

a sinvpyh sinppz

a sinqpyh dzdy

�trd1upa

� 2 R a0

R h0 1�2

hy�

1�1�ba z

� sinupz

a sinvpyh sinppz

a sinqpyh dzdy

þtrd12uvp2

ah

R a0

R h0 gþ 1�b

a y� y2

h

�h icosupz

a cosv npyh sinppz

a sinqpyh dzdy

8>>>><>>>>:

9>>>>=>>>>;

¼0

ð35Þwhere the definite integrals on variables u and p are given inTable 1; and the integrals on v and q are obtained by replacing uand p.

Eq. (35) can be translated into a matrix equation set and thedeterminant of the coefficient matrix should be zero as the coeffi-cient Auv cannot be zero. Therefore, the critical stress (rd1)cr can beobtained.

In addition, since the dimension of the coefficient matrix affectsthe calculation accuracy, the critical stresses (rd1)cr were examinedfirst in Fig. 24 under different dimensions of the coefficient matrix.In the calculation, the span l = 1 m, the height h = 0.25 m, the widthb = 0.1 m, the thicknesses for webs and flanges t = 0.01 m, and thediaphragm thickness tp = 0.005 m. It is seen that the critical stress(rd1)cr tends to a converged value as the dimension of the coeffi-cient matrix increases from 2 � 2 to 4 � 4. However, calculationtime increases significantly from 12 s to 51220 s. Considering boththe accuracy and time, the 3 � 3 coefficient matrix is regardedmost appropriate to calculate the critical stress (rd1)cr.

In order to obtain Pcr1, a linear relationship is establishedbetween the critical moment Mcr1 and the critical stress (rd1)cr.

Mcr1

Me¼ ðrd1Þcr

ðrd1Þeð36Þ

whereMcr1 is the critical value for external momentMj based on thestability of the local web plate, given by Mcr1 = Pcr1�n1/2. Me is theunit external moment, i.e. Me = 1Nm. (rd1)e is the corresponding

warping stress on top of the web at the loading section z = 0.45l,produced by the unit moment Me.

Fig. 25 gives the critical load Pcr1 varying with the location zp1 ofdiaphragm I for various ratios h/b, based on the stability of the localweb plate. It is seen that the critical load Pcr1 increases remarkablywhen diaphragm I is located close to the loading section z1 = 0.45l,implying that installation of a diaphragm at the loading sectionwill enhance effectively the stability of the local web plate.

While for the mid-span diaphragm (diaphragm II) subjected topure shear boundary conditions, the critical value (Mp2)cr formoment Mp2 is given by [33,34]

ðMp2 Þcr ¼Ep4t3p2

nð1� t2Þ ; ðh=b 6 2Þ

ðMp2 Þcr ¼89h200b

þ b3h

� �Ep2t3p21� t2

; ðh=b > 2Þ ð37Þ

where n ¼ 384b2h2

9ðh2þb2Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi706625 þ 81

25h2þb2

h2þ9b2

�2þ 81

25h2þb2

9h2þb2

�2r; t is the Pois-

son’s ratio and equals to 0.3; tp2 is the thickness of mid-span dia-phragm and tp2 = 0.005 m.

Based on Eq. (21), the critical momentMcr2 for external momentMj can be calculated by

Mcr2 ¼ � ðMp2ÞcrQ21 þ Q22

ð38Þ

where Q21 and Q22 are defined in Eq. (21).Furthermore, based on equation Mj = Pj�n1/2 [26], the critical

load Pcr2 can be finally obtained from Eq. (38). For the stability ofthe mid-span diaphragm, Fig. 26 gives the critical load Pcr2 varying

0 0.2 0.4

10

20

30

40

50P c

r1/k

N

zp1/l

h/b=2.5h/b=2h/b=1.5h/b=1

0.3 0.410

15

Zoomed picture

Fig. 25. The critical load Pcr1 obtained from the stability of local web plate.

0 0.1 0.2 0.3 0.4 0.50.5

1

1.5

2

2.5 x 104

P cr2

/kN

zp1/l

h/b=2.5h/b=2h/b=1.5h/b=1

Fig. 26. The critical load Pcr2 obtained from the stability of the mid diaphragm.

58 Y. Ren et al. / Engineering Structures 145 (2017) 44–59

with the location zp1 of diaphragm I for various values of h/b. It isseen that the critical load Pcr2 increases remarkably when dia-phragm I is located close to the loading section z1 = 0.45l, indicat-ing increased resistance to buckling.

Back to the girder with equally-distanced three inner dia-phragms in Fig. 22, the girder will reach its yield strength limitof 235 MPa when the eccentric load Pj is 650kN, which is muchsmaller than the critical buckling load Pcr2 of 8743kN atzp1/l = 0.25 for the ratio h/b = 2 according to the stability of themid-span diaphragm. Simultaneously, the plastic yield load of650kN is much larger than the critical load Pcr1 of 12.2kN atzp1/l = 0.25 according to the stability of the local web plate. Thisimplies that buckling of the local web plate will be the primaryfailure mode with the increment of eccentric loads Pj. Hence atten-tions should be paid on the stability of the local web plate for thedesign of girders subjected to eccentric loads.

8. Conclusions

In this paper, the initial parameter method (IPM) is applied toinvestigate the distortion of simply supported girders with innerdiaphragms subjected to concentrated eccentric loads, where in-plane shear deformation of diaphragms is considered. The mainconclusions can be drawn as follows

(1) Compared with results from FEA and experiments, accurateanalyses can be obtained by IPM for the distortional angle,warping displacements and stresses for simply supportedgirders with inner diaphragms. Both the in-plane sheardeformation of diaphragms and the compatibility conditionbetween the girder and diaphragms are taken into accountin IPM. Besides, comparison of results between consideringthe shear strain of the cross section and not shows thatthe shear strain of the cross section cannot be ignored whencalculating the distortional angle.

(2) Both the distortional angle and warping stresses decreasewith the increment of the ratio of height to span, the numberof diaphragms and their thickness. The warping displace-ment converges to a fixed value between 0.1 and 0.2 forlarge diaphragm numbers. And the mid-span diaphragmplays a key role in reducing the distortional deformationsand stresses for girders under symmetrical loads.

(3) Stabilities of the local web plate and the mid-spandiaphragm were both investigated for box girders with sym-metrical three inner diaphragms. Results show that both thelocal web plate and the mid-span diaphragm increase theirresistance to buckling when the diaphragm I is located closeto the loading sections. Moreover, the local web plate willbuckle first as the primary failure mode. Therefore, atten-tions are needed on the stabilities of the local web platefor simply supported girders under eccentric loads.

Based on the IPM, it is possible to improve the warpingdisplacements and stresses of simply supported girders throughoptimizing the positions and wall thickness of diaphragms. Futurework are needed for (1) optimization of the distortion of girderswith diaphragms; (2) mechanical properties of girders withperforated diaphragms.

Acknowledgements

This work was supported by the National Natural Science Foun-dation of China (NSFC) [grant number: 51175442 and 51675450].

Appendix A

The relationships between ui(z) (i = 1,2,3,4) and their differenti-ations is

u01 ¼ k1u4 þ k2u2;u

02 ¼ k1u3 � k2u1u

03 ¼ k1u2 � k2u4;u

04

¼ k1u1 þ k2u3; ðA1Þ

u001 ¼ ðk21 � k22Þu1 þ 2k1k2u3; u00

2 ¼ ðk21 � k22Þu2 � 2k1k2u4;

u003 ¼ ðk21 � k22Þu3 � 2k1k2u1; u00

4 ¼ ðk21 � k22Þu4 þ 2k1k2u2;

ðA2Þ

u0001 ¼ðk31�3k1k22Þu4þð3k21k2�k32Þu2; u000

2 ¼ðk31�3k1k22Þu3�ð3k21k2�k32Þu1;

u0003 ¼ðk31�3k1k22Þu2�ð3k21k2�k32Þu4; u000

4 ¼ðk31�3k1k22Þu1þð3k21k2�k32Þu3:

ðA3Þ

Y. Ren et al. / Engineering Structures 145 (2017) 44–59 59

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