Distortional analysis of simply supported box girders with inner 1

diaphragms considering shear deformation of diaphragms using 2

initial parameter method 3

Yangzhi Ren a,b*, Wenming Cheng b, Yuanqing Wang a, Qingrong Chen b, Bin Wangc 4 5 a Key Lab of Civil Engineering Safety and Durability of China Education Ministry, Department of 6 Civil Engineering, Tsinghua University, Beijing, China, 100084 7 b Department of Mechanical Engineering, Southwest Jiaotong University, No.111, North Section 1, 8 Second Ring Road, Chengdu, Sichuan, China, 610031 9 c College of Engineering, Design and Physical Sciences, Brunel University, London, Uxbridge 10 UB8 3PH, UK. 11 12 *:Corresponding author, e-mail address: renyz66@mail.tsinghua.edu.cn 13 14 Abstract: In this paper, the distortion of simply supported girders with inner diaphragms 15 subjected to concentrated eccentric loads is investigated using initial parameter method (IPM), in 16 which the in-plane shear deformation of diaphragms is fully considered. A statically indeterminate 17 structure was modeled with inner redundant forces, where the interactions between the girder and 18 diaphragms were indicated by a distortional moment. Considering the compatibility condition 19 between the girder and diaphragms, solutions for the distortional angle, warping displacements 20 and stresses were derived and further simplified by establishing a matrix equation system. The 21 validity of IPM was intensively verified by a finite element analysis and distortional experiments. 22 Parametric studies were then performed to examine the effect of the diaphragm number on the 23 distortional angle, warping displacements and stresses under various ratios of height to span of the 24 girder and the diaphragm thicknesses. Besides, stabilities of the local web plate and mid-span 25 diaphragm were analyzed based on IPM for box girders with symmetrical three inner diaphragms. 26 Results show that the local web plate will buckle before overall yielding with the increment of the 27 eccentric loads Pj, and the mid-span diaphragm is constantly stable in the whole deformation 28 process. It shows that more attentions should be paid on the stability of the local web plate than 29 overall yielding for girders subjected to eccentric loads. 30 Keywords: simply supported girder; diaphragm; eccentric load; shear deformation; distortion; 31 initial parameter method; finite element analysis; experiment; local stability 32 1 Introduction 33

During the past several decades, box girders have been widely applied in buildings and 34 bridges due to their large bending and torsional stiffness. However, they are generally susceptible 35 to the cross-sectional distortion [1] under eccentric loads due to their quadrilateral instability. 36 Therefore, excessive distortional warping and transversal bending stresses will be produced 37 besides the torsional and bending ones in box girders. In a special case, the distortional warping 38 stresses may be significant to the torsional and bending ones. In order to control the distortion, 39 diaphragms are installed at the interior of the girder, which can increase not only the stability of 40

the local plate, but also the resistance to the warping deformations and stresses [2,3,4]. 41 Researches on the distortion of girders with inner diaphragms have been performed for many 42

decades. The distortion of box girder was initially studied by Dabrowski [5] who first formulated 43 the distortion of box girders with a symmetrical cross section. Li [6,7] proposed that the shear 44 strain of the cross section cannot be ignored when the distortional shear rigidity is significant 45 compared to the distortional warping one for box girders. Wright [8] proposed the Beam on Elastic 46 Foundation (BEF) analogy for the distortion of girders with inner diaphragms, where the 47 diaphragms are analogous to inner supports. Based on BEF, Hsu [9,10] proposed the Equivalent 48 Beam on Elastic Foundation (EBEF) analogy considering the shear strain of the cross section, and 49 found that the EBEF analogy is more versatile than BEF due to its simplicity in analyzing more 50 complex problems such as non-uniform sections and multi-span beams. 51

Interactions between the girder and diaphragms is the key issue for the distortion of girders 52 with inner diaphragms. A statically indeterminate structure [11] was modeled with redundant 53 forces acting along the junctions between the girder and diaphragms. Moreover, the force method 54 was applied to calculate redundant forces, where elements in the stiffness matrix were obtained 55 from the finite strip method [12]. The numerical results were then extended to multi-span bridges 56 [13] and long-span curved bridges [14]. An outstanding contribution was made by Suetake [15], 57 where the girder was regarded as an assembly of thin plates, and the extended trigonometric series 58 method (ETS) was applied to analyze the stresses and deformations for box girders. Comparisons 59 with FEM results show that ETS has a high accuracy. However, it is inconvenient to apply since 60 there are many simultaneous nonlinear equations to solve even for girders with few diaphragms, 61 e.g. there are up to 720 equations for a girder with two diaphragms. 62

The wall thickness of diaphragms and the number of diaphragms in a girder will make a 63 significant influence on the displacements and stresses. Park [16,17] proposed a new beam 64 element with nine degrees of freedom per node for girders. Studies showed that the distortional 65 warping and transversal bending stresses were reduced by increasing the diaphragm number. 66 Similar conclusions can be drawn for straight multi-cell box girders with diaphragms [18,19]. For 67 horizontally curved bridges, the rational spacing between adjacent diaphragms was provided [20] 68 according to the ratio between the distortional warping stress and the bending stress. Using FEM, 69 Zhang [21] found that the rational number for diaphragms is 3 to 5 when the ratio of width to 70 height of the cross section is 1.5 and the rational number is 9 when the ratio is 4.5. Li [22] 71 proposed a new finite element solution, and found that the distortional warping stress for 72 cantilever girders can be ignored when the spacing between adjacent diaphragms is less than one 73 fifth of the span; while for simply supported and fixed girders, the spacing is less than one eighth 74 of the span. 75

Initial parameter method (IPM) was proposed first by Vlasov [23] to analyze the non-uniform 76 torsion of beams. Analogous to IPM in non-uniform torsion, IPM can be extended to analyze 77 distortions of girders. Considering the effect of shear strains of the cross section, Xu [24,25] 78 developed an equation with the variable distortional angle, and established two categories of IPMs 79 of the fourth order, classified by the ratio of the distortional stiffness. Harashima [26] proposed a 80 distortional equation with a distortional warping function, and established the fifth-order IPM. 81 Both IPMs in distortion have a high efficiency compared with FEM. However, IPMs has not been 82 extended into the distortion for girders with inner diaphragms. 83

For distortions of a girder with inner diaphragms, an assumption of infinite-rigidity 84

diaphragm was generally made in most studies [16, 20, 27], where the in-plane deformation of 85 diaphragms was totally restrained and warping was free. Similar assumptions can be found in 86 distortion of curved box beams [28], where the distortional angle at the location of diaphragms is 87 set as zero. However, the infinite-rigidity assumption is just an approximation, which is not 88 applicable to thin flexible diaphragms. The main objective of this work is to investigate the 89 distortion of simply supported girders with inner flexible diaphragms under concentrated eccentric 90 loads, where the in-plane shear deformation of diaphragms is fully considered. Interactions 91 between the girder and diaphragms are indicated by a distortional moment. Based on the 92 compatibility condition between the girder and diaphragms, solutions for both the distortional 93 angle and warping function are obtained from the IPM. Taking a simply supported girder with 2, 5 94 and 9 diaphragms, respectively, as an example, the distortional solutions from IPM were obtained, 95 then verified by a FE analysis and experiments. This was followed by a parametric study, in which 96 distortional deformations and stresses were investigated in terms of the diaphragm number and 97 thickness and the height to span ratio of the girder. Based on the proposed IPM, stabilities of both 98 the local web plate and mid-span diaphragm were examined for girders with three symmetrical 99 inner diaphragms. A series of curves were obtained for the relations between the critical buckling 100 load and the position of diaphragms under various height to width ratios of the cross section. 101 2 Structural model 102

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

tp(i+1)

zpi

zp(i+1) Pj zj

P j+1zj+1

l

(a) simply supported box girder with inner diaphragms

y xz

Otpi

Inner diaphragmsPin joint

s

s

s

s

103 Fig.1 Girder with inner diaphragms under eccentric loads Pj (j=1,2,…,m) 104

Consider a simply supported box girder with inner diaphragms under concentrated eccentric 105 loads Pj (j=1,2,…,m). The coordinate system O-xyz is illustrated in Fig.1a with the original O at the 106 shear center on one end of the girder. For analysis, the distances between O and the mid-lines of 107 webs B and D are marked by n1 and n2 in Fig.1b, respectively. The girder is made of a homogeneous, 108 isotropic and linearly elastic material with its Young’s and shear moduli denoted by E and G, 109 respectively. The girder span is l. The thicknesses of web B and D are t1 and t2 and their height is h, 110 while the width of flanges A and C is b and the thickness t3. The thickness for the ith diaphragm is 111 tpi (i=1,2,…,n) and its mid line is marked by zpi, measured from O. The load Pj is located on the top 112 of web D at zj. 113

Fig.2a shows that the eccentric load Pj can be decomposed into three components [29, 30] – 114 the flexural load, the torsional and the distortional load. In Fig.2b, the cross-section rigidly rotates 115 around O with a torsional angle θ under torsional loads. Fig.2c illustrates the transversal 116 deformations in the web and the flange under distortional loads, where uM and vM are horizontal 117

and vertical displacements at node M, respectively. The variation of the right angle at node N is 118 defined as the distortional angle χ, given by χ=χ1+χ2. Moreover, the warping displacement wd and 119 the stress σd, produced by the distortional moment Bd, are shown in Fig.2d. There also exists the 120 shear stress τd along the cross-section profile, developed by the distortional moment Md, as shown 121 in Fig.2e. 122

This article will focus on the distortional deformation and stresses of a simply supported box 123 girder with inner diaphragms subjected to concentrated eccentric loads. 124

Pj

Pjn2

b

+

x

y

Flexural loads

x

y

x

y

b

Pjn1b

Pjn1

bPjn1b

Eccentric load Pj

x

y

Pjn1

2bPjn1

2b

Pjn1

2h

Pjn1

2h

x

y

Pjn1

2bPjn1

2b

Pjn1

2h

Pjn1

2h

+

Torsional loads Distortional loads

deformedoriginal

(a) loading decomposition

(b) torsional deformation

(c) distortional frame deformation

n1 n2

h

χ1

χ2

(d) distortional warping

σdN

(e) distortional shearing stress

N

MσdM

Bd

Md

wdN

wdM

τd

θ

v MM

uM

N

J

K

σdJ

wdJ

σdK

wdK

125 Fig.2 Loading decomposition, deformations and stresses of girders 126

3 Distortion of girders without diaphragm 127 In distortion, when the shear stiffness of the cross section has a significant value in 128

comparison with the warping one, the influence of shear strain of the cross section on deformation 129 and stresses cannot be ignored [6,7]. The distortional differentiate equation can be expressed as [7] 130

d( ) ( ) ( )tt c c

k

EIEI W z '''' EI W z '' EI W z m 'GI

− + = (1) 131

where It, Ic and Ik are distortional warping, frame and shear constants, respectively; W(z) is the 132 distortional warping function; md is the distributed distortional moment. The number of the 133 apostrophes of W(z) indicates the first, second and forth differentiations with respect to the 134 z-coordinate. 135

Under concentrated distortional loads, md=0, and the solution for Eq.(1) is given as [26] 136

4

1( ) ( )i i

iW z B zϕ

=

=∑ (2) 137

where Bi (i=1, 2, 3, 4) are coefficients determined by the boundary conditions, and φi are defined as 138 φ1=cosh(λ1z)sin(λ2z), φ2=cosh(λ1z)cos(λ2z), φ3=sinh(λ1z)cos(λ2z), φ4=sinh(λ1z)sin(λ2z) 139 where λi (i=1,2) are distortional coefficients. 140

141

2c c

t k

EI EIEI GI

λ = + , 241

2c c

t k

EI EIEI GI

λ = − 141

Relations between the warping function and the distortional angle are [7] 142

( ) ( )t

c

EIz W zEI

χ ′′′= − , d ( ) ( )tB z EI W z′= − , d ( ) ( )tM z EI W z′′= − (3) 143

Substituting Eq.(2) into Eq.(3), the matrix equation is given by 144 Z(z)=Φ(z)B (4) 145 where 146

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

t t t t

c c c c

EI EI EI EIz z z zEI EI EI EI

z z z zzz z z

''' ''' ''' '''

' ' ' ''' '' '' ''

zz z z z

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕϕ ϕ ϕ ϕϕ ϕ ϕ ϕ

− − − − =

Φ , B={B1, B2, B3, B4}T, 147

Z(z) is the state vector in IPM, 148

T

d d( ) ( )( ) ( ), ( ), ,t t

B z M zz z W zEI EI

χ

= −

Z (5) 149

The boundary conditions for a simply supported girder are 150 χ(0)=0, Bd(0)=0, for the initial end z=0; 151 χ(l)=0, Bd(l)=0, for the ultimate end z=l. 152

Correspondingly, the state vectors on both ends are 153

T

d (0)(0) 0, (0), 0,t

MWEI

=

Z , T

d ( )( ) 0, ( ), 0,t

M ll W lEI

=

Z (6) 154

For z=0, Z(0)=Φ(0)B, we have 155 B=[Φ(0)]inv·Z(0) (7) 156 where [Φ(0)]inv is the inverse matrix of Φ(0). 157

Substituting Eq.(7) into Eq.(4) yields in the state vector Z(z) 158 Z(z)=P(z)·Z(0) (8) 159

where P(z) is the transfer matrix, and P(z)= Φ(z)·[Φ(0)]inv. 160

Eq.(8) is the standard form of IPM for the distortion of girders without diaphragms. However, 161 the transfer matrix P(z) is complicated. Based on the relations between φi(z) (i=1,2,3,4) and their 162 differentiations (see Eq.(A1)~Eq.(A3) in Appendix I), the matrix P(z) is simplified as 163

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

SC ''' z KC ''' z SKC ''' z KC ''' zS C z C z SC z C zK

z S C ' z C ' z SC ' z C ' zKS C '' z C '' z SC '' z C '' zK

− − − − =

− − − −

P (9) 164

where 2 21 2

12 2

Sλ λ

=+

, t

c

EIKEI

= , 3 11

1 2

( ) ( )( ) z zC z ϕ ϕλ λ

= − , 2 2

1 22 2 4

1 2

( ) ( ) ( )2

C z z zλ λϕ ϕλ λ−

= − , 165

2 2 2 21 2 1 2

3 3 11 2

3 3( ) ( ) ( )C z z zλ λ λ λϕ ϕλ λ− −

= − , 44

1 2

( )( )2

zC z ϕλ λ

= . 166

The jth distortional load in IPM is indicated by a vector Zj, given by 167

T

0, 0, 0, jj

t

MEI

=

Z (10) 168

where Mj is the distortional moment for the jth distortional load, and Mj =Pj·n1/2 [26]; n1 is the 169 distance between the web D and point the original O (see Fig.1b). 170 4 Distortion of box girders with inner diaphragms 171 4.1 IPM solution 172

For analysis, a statically indeterminate structure is modeled with inner redundant forces acting 173 along the junctions between the girder and the diaphragms. The entire model is shown in Fig.3a 174 and the horizontal and vertical redundant forces Hej and Vej are illustrated in the zoomed picture, 175 where the subscript e indicates the webs and flanges and e =A,B,C,D (see Fig.1b). The small 176 circles indicate the joints between the girder and diaphragms where redundant forces are located. 177

(b) boundary on ith diaphragm(a) a statically indeterminate system

yx

zO

Inner diaphragmsPin joint

HAVA

HD

VDHC

VC

HB

VB

A

B

C

D

HDr HA2

VDr VA2

HAr

VAr

HD2VD2

HCr

VCr

HC2

VC2

HBr

VBr

HB2

VB2

Zoomed picture

mpi

τd =mpibh

yx

Otpi

178 Fig.3 A statically indeterminate structure and the equivalent boundaries for the ith diaphragm 179

In order to analyze the interactions between the girder and diaphragms, two assumptions are 180 made: 181

(1) Self balance of the in-plane forces of diaphragms 182 Under distortional loads, the summation of all vertical and horizontal redundant forces and 183

moments are zero for diaphragms. That is 184

A,B,C,D 2

0r

eje j

H= =

=∑ ∑ , A,B,C,D 2

0r

eje j

V= =

=∑ ∑ , A,B,C,D 2 A,B,C,D 2

0r r

ej ej ej eje j e j

H y V x= = = =

+ =∑ ∑ ∑ ∑ (11) 185

where xej and yej are the distances between the redundant forces Vej, Hej and the original O. 186 Based on the self balance assumption, only the distortional component for redundant forces is 187

reserved. Referred to the formation of external moment Mj [26,27], the distortional components of 188 the redundant forces are gathered and indicated by a concentrated distortional moment Mpi for the 189 ith diaphragm. The interactions between the girder and diaphragms can be indicated by a moment 190 Mpi (i=1,2,…,n). Only Mpi,, opposite to Mj, will resist the distortional deformation and stresses. In 191 IPM, Mpi is indicated by a vector Zpi, given by 192

T

0, 0, 0, pii

t

mEI

=

pZ (12) 193

(2) Compatibility condition between the girder and diaphragms 194 In-plane shear strains of diaphragms are considered, given by γpi = Mpi /(Gbhtpi). The 195

compatibility condition is that the distortional angle at the mid line of diaphragms is opposite to 196 the in-plane shear strain of diaphragms. That is χ(zpi)= –γpi (0≦i≦n). This is a key aspect for the 197 distortion of girders with inner diaphragms. 198

Combining Eq.(8) and Eq.(12), the state vector Z(z) can be expressed as 199

/2

1 1/2

( ) ( ) (0) ( ) d ( )pi pi

pi pi

z tR S

i i i j ji jz t

z z z z z z z+

= =−

= − − − −∑ ∑∫ pZ P Z P Z P Z (13) 200

where R and S are the numbers of diaphragms and moments Mj before point z, respectively. The 201 transfer matrices P(z–zi) and P(z–zj) are obtained from P(z) by substituting the variable z by ‘z–zi’ 202 and ‘z–zj’. 203

For z=l, Eq.(13) changes into 204

/2

1 1/2

( ) ( ) (0) ( ) d ( )pi pi

pi pi

z tn m

i i i j ji jz t

l l l z z l z+

= =−

= − − − −∑ ∑∫ pZ P Z P Z P Z (14) 205

where the vectors Z(l) and Z(0) are referred to Eq.(6); W(0) and Md(0) in vector Z(0) can be 206 calculated from the first and third simultaneous equations in Eq.(14). Then, substituting W(0) and 207 Md(0) into Eq.(13), the distortional angle and the warping function can be obtained as 208

( )1 1

241 1

1 2 24

( ) ( , )

2 ( , )

n m

R i pi S j ji j

c

z M z z Mz

EI Φ l l

η εχ

λ λ= =

+=∑ ∑

(15) 209

( )2 2

241 1

1 2 24

( ) ( , )

2 ( , )

n m

R i pi S j ji j

t

z M z z MW z

EIΦ l l

η ε

λ λ= =

+=∑ ∑

(16) 210

where 211

1 00 33 02 1331 42 31 42

022431

2( ) , ( , ) , ( , )2 2

2 ( , )1 ,2 2

pi piR i pi pi

pi

pipi

pi

t tz l z Φ z l l z Φ l z

t

tΦ l lH R i z zt

η ξ ξ

ξ

= − + −

− + − −

, 212

2 00 30 02 1031 42 31 42

0( 1)2424

2( ) , ( , ) , ( , )2 2

2 ( , )1 ,2 2

pi piR i pi pi

pi

pipi

pi

t tz l z Φ l z l z Φ l z

t

tΦ l lH R i z zt

η ξ ξ

ξ −

= − − −

+ + − −

, 213

124 24 4 24

1( , ) ( , ) ( ) ( , )2S j j jz z Λ z z H S j ''' z z Φ l lε ϕ = − + − −

, 214

( )224 24 4 24

1( , ) ( , ) ( , )2S j j jz z Λ z z H S j z z Φ l lε ϕ = + + − −

, 215

H(x) is the unit step function. H(x)=1 for x>0 and H(x)=0 for x<0 216

( ) ( )

( , )( ) ( )

i jij

i j

''' x ''' yΦ x y

' x ' yϕ ϕϕ ϕ

= , 4

4

( ) ( , )( , )

( ) ( , )i i

ijj j

x Φ l y lΛ x y

x Φ l y lϕϕ

−=

−,

3

3

d ( , )( , )

dij

ij

Λ x yΛ x y

x= 217

( ) ( )

( ) ( )

( ) ( )( , )

( ) ( )

i iij m nmn j j

m n

x xΦ x y

y yϕ ϕϕ ϕ

= , ( ) ( ) ( )( ) ( )

( ) ( )

( ) ( ),i i

m nijmn j j

m n

x xx y

y yϕ ϕ

ξϕ ϕ

−= , 218

where φm(i)(x) and φn(i)(y) are the ith differentiation of functions φm(x) and φn(y). φn(-1)(y) is the 219 integral of function φn(y), given by 220

( 1) 1 3 2 12 2 2

1 2

( ) ( )( ) y yy λϕ λ ϕϕλ λ

− +=

+, ( 1) 1 1 2 3

4 2 21 2

( ) ( )( ) y yy λϕ λ ϕϕλ λ

− −=

+. 221

When the calculated point z is located in the thickness of (R+1)th diaphragm (zp(R+1)–tp(R+1)/2222

≦z≦zp(R+1)+tp(R+1)/2), an additional angle χadd and function Wadd should be involved, 223

( 1) 1 2 4 ( 1) ( 1)add

1 2 ( 1)

2 ( 2)2

p R p R p R

c p R

M '' z z tEI t

λ λ ϕχ

λ λ+ + +

+

− − + = (17) 224

( )( 1)

( 1) 4 ( 1) ( 1)add

1 2 ( 1)

22

p R p R p R

t p R

M z z tW

EI tϕ

λ λ

−+ + +

+

− += (18) 225

where zp(R+1), tp(R+1) and Mp(R+1) are the mid-line location, the thickness and the distortional 226 moment of the (R+1)th diaphragm, respectively. 227

Based on Eqs.(15)~(18), both the angle χ(z) and the function W(z) are related to Mj and Mpi. 228 Since Mj has been given in Eq.(10), solutions rest in Mpi. 229 4.2 Derivation of Mpi 230

The compatibility condition gives the equation 231

1 1 2424 4 1 2

1 1

( , )( ) ( , ) 2 1 02

n mpT c

T i pT pi T p pTj T

T ji p

j Mt

t EIΦ l lz MG

z z M 'bh

'η ε ϕ λ λ= =

+ + − − =

∑ ∑ (19) 232

for the Tth diaphragm (T=1,2,…,n), where 233

1 00 33 02 1331 42 31 42

022431

2( ) , ( , ) , ( , )2 2

2 ( , )1 ,2 2

pi piT i pT pi pT pi pT

pi

pipT pi

pi

t tz l z Φ z l l z Φ l z

t

tΦ l lH T i z zt

η ξ ξ

ξ

= − + −

− − − −

234

124 24 24 4

1( , ) ( , ) ( , ) ( )2T pT j pT j T pT jz z Λ z z H k j Φ l l ''' z zε ϕ = − + − −

, 235

kT is the number of distortional loads before the Tth diaphragm. 236 The matrix equation system for Eq.(19) is 237

η·Mp+ε=0 (20) 238

where { }T

1 2, ,...,p p p pnM M M=M ;

1 111 2 1 1

1 11 2 22 2

1 11 2

( ) ( )( ) ( )

...... ... ...( ) ( )

T p T n p

T p T n p

T pn T pn nn

R z zz R z

z z R

η ηη η

η η

=

η ; 239

T

1 1 124 1 24 2 24

1 1 1( , ) , ( , ) ,..., ( , )

m m m

T p j j T p j j T pn j jj j j

z z M z z M z z Mε ε ε= = =

= ∑ ∑ ∑ε ; 240

241 and the diagonal elements in matrix η is 242

1 244 1 2

( , )( ) 2 12pi c

ii T i pipi

t EIΦ l lRt Gbh

z ''η ϕ λ λ = − − +

. 243

Consequently, Mpi can be obtained by 244

1

m

pi ij jj

M Q M=

= −∑ (21) 245

according to the Cramer rule. Qij =|ηi|/|η|, where the |η| indicates the determinant of η, and ηi is the 246

same of η except for the ith column [ 124 1( , )T p jz zε , 1

24 2( , )T p jz zε ,…, 124 ( , )T pn jz zε ]T. 247

4.3 Simplification of χ(z) and W(z) 248 Substituting Eq.(21) into Eq.(15), and the angle χ(z) changes into 249

124 1

1 1

1 2 24

( , )( )

( )2 ( , )

n mS j

R i ij ji j

c

z zz Q M

nz

EI Φ l l

εη

χλ λ

= =

−

=∑∑

(22) 250

where n and m are the total numbers of diaphragms and distortional loads, respectively. 251

The number of calculation steps is m×n in Eq.(22) and it would be time-consuming for 252 girders with many diaphragms under many distortional loads. For solution, a matrix equation is 253 established, given by 254 η·ψ = α (23) 255 where 256

11 12 1

21 22 2

1 2

...... ... ...

n

n

n n nn

ψ ψ ψψ ψ ψ

ψ ψ ψ

=

ψ ;

11 12 1

21 22 2

1 2

...... ... ...

n

n

n n nn

α α αα α α

α α α

=

α and the elements in matrix α are 257

1 1 1 124 24

1

1 1 124 24

1

( , ) ( ) ( ) ( , ) ( )

( , ) ( ) ( , ) ( )

m

S j T k pg R k T pg j jj

gk m

S j gg R g T pg j jj

z z z n z z z M g kα

z z R n z z z M g k

ε η η ε

ε η ε

=

=

− ≠ = − =

∑

∑. 258

In this approach, the distortional angle χ(z) can be simplified as 259

( )11 2 24

12 ( , )

n

iiic

z ψEI Φ l l

χλ λ =

= ∑ . (24) 260

For the warping function W(z), a matrix equation can also be established similar to Eq.(23), 261

and the elements αgk (g, k =1,2,…n) in matrix α are 262

2 1 2 124 24

1

2 2 124 24

1

( , ) ( ) ( ) ( , ) ( )

( , ) ( ) ( , ) ( )

m

S j T k pg R k T pg j jj

gk m

S j gg R k T pg j jj

z z z n z z z M g kα

z z R n z z z M g k

ε η η ε

ε η ε

=

=

− ≠ = − =

∑

∑ 263

Therefore, the function W(z) can be simplified as 264

11 2 24

1( )2 ( , )

n

iiit

W z ψEIΦ l lλ λ =

= ∑ . (25) 265

Taking the node N (see Fig.1b) of the cross section as an example, the distortional warping 266 displacement wdN and the stress σdN are given by 267

ddN

d

( )( )4( 1)

W z bhw z ββ

= −+

, ddN

d

( )( )4( 1)

EW z bhz βσβ′

= −+

(26) 268

where βd is the ratio of the distortional warping stresses between nodes J and N, and 269

3 1d

3 2

33

bt htbt ht

β +=

+. 270

5 Verifications of IPM 271 5.1 Verifications with FEA 272

To verify the proposed IPM, simply supported girders with 2, 5 and 9 diaphragms are 273 investigated for three diaphragm thicknesses, respectively by FEA using a commercial code 274 ANSYS. All girders are modeled with the Young’s modulus E=2.1×1011Pa, the Poisson’s ratio 275 υ=0.3, the span l=1m, the width b=0.1m, the height h=0.2m and the flanges and webs thicknesses 276 t=0.01m. Diaphragms are evenly distanced along the span with the thicknesses tp= 0.005m, 0.01m 277 and 0.02m, respectively. 278

Figs.4a, b and c show the meshing grids of the girders and diaphragms using Shell63 element 279 in FEA model, where translations in the x- and y- axial directions and rotations about the y- and 280 z-axes on both ends are restrained. A convergent test shows that 1650 to 2026 elements are 281 adequate in terms of the diaphragm number. Fig.4d shows the loading conditions, where two 282 concentrated distortional loads are applied along the flange and web in cross sections z1=0.45l and 283 z2=0.55l, and Ph=1.25kN and Pv=2.5kN, respectively. 284

(a) n=2 (b) n=5

(c) n=9 (d) loading conditions

yx

zO

yx

zO

yx

zO

Ph

Ph

Pv

Pv

Ph

Ph

Pv

Pv

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

yx

zO

285 Fig.4 Meshing grid, ending boundaries and loading conditions in FEA model 286

Fig.5 to Fig.7 show 3D contours of the distortional warping deformations and stresses for 287

simply supported girders with 2, 5 and 9 diaphragms, respectively. The ‘amp’ indicates the 288

amplified factor for deformations. It is seen that the largest displacement and stress occur at the 289

junction between webs and flanges at the loading sections. With the increment of diaphragm 290

number, the largest stress reduces from 5.51MPa to 1.55MPa and the displacement from 1.61μm to 291

0.268μm, and frame deformations at the loading sections clearly become small. 292

(b) Distortional warping stress(a) Distortional warping displacement (c) Frame deformation

293

Fig.5 3D contours of a girder with two diaphragms under distortional loads (tp=0.01m, amp=3000) 294

(b) Distortional warping stress(a) Distortional warping displacement (c) Frame deformation

295

Fig.6 3D contours of a girder with five diaphragms under distortional loads (tp=0.01m, amp=10000) 296

(b) Distortional warping stress(a) Distortional warping displacement (c) Frame deformation

297

Fig.7 3D contours of a girder with nine diaphragms under distortional loads (tp=0.01m, amp=20000) 298 Fig.8 to Fig.10 show the distortional angle, warping displacements and stresses at node N 299

from IPM and FEA for simply supported girders with 2, 5 and 9 diaphragms, respectively, in the 300

relative diaphragm thickness tp /t=0.5, 1 and 2. The distortional angle in FEA results can be 301

calculated by 302

N M N JUX UX UY UY( )zh b

χ − −= + (27) 303

where UXN and UXM are x-axial displacements at nodes N and M (see Fig.1b), respectively; UYJ 304

and UYN are y-axial displacements at nodes J and N, respectively. 305

0 0.5 1-3

-2

-1

0

1

2

3

4 x 106

0 0.5 1-2

-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

χ(z)

z/l z/l

wdN

(z)/m

(a) distortional angle χ(z) (b) warping displacement wdN(z)

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

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

306 Fig.8 The distortional angle, warping displacements and stresses between IPM and FEA for a simply supported 307

girder with two diaphragms of different diaphragm thicknesses 308 Good agreements are observed between IPM and FEA in Fig.8 to Fig.10 for the distortional 309

angle, warping displacements and stresses for simply supported girders with inner diaphragms. 310

Compared the girders braced by 2 diaphragms with those by 5 and 9 diaphragms, it’s worth noting 311

that the mid-span diaphragm effectively restrains the transversal deformation. 312

For the distortional angle, the largest error between IPM and FEA occurs at the loading 313

sections, where the FEA result is 23.68% higher than the IPM one for girders with two diaphragms, 314

and reduces to 13.86% for those with five diaphragms and 10.18% for those with nine diaphragms. 315

Since there is no diaphragms or stiffeners at the loading sections, the error between IPM and FEA 316

can be attributed to the local stress concentration. So the distortional angle obtained from IPM is 317

susceptible to the influence of stress concentration. 318

0 0.5 1-1

0

1

2

3

4

5 x 10-5

0 0.5 1-5

0

5 x 10-7

0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

2 x 106

χ(z)

z/l z/l

wdN

(z)/m

(a) distortional angle χ(z) (b) warping displacement wdN(z)z/l

σ dN(z

)/Pa

(c) warping stress σdN(z)

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

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

319 Fig.9 The distortional angle, warping displacements and stresses between IPM and FEA for simply supported 320

girders with five diaphragms varying with three diaphragm thicknesses 321

0 0.5 10

0.5

1

1.5

2

2.5

x 10-5

0 0.5 1-4

-2

0

2

4 x 10-7

0 0.5 1-5

0

5

10

15 x 105

χ(z)

z/l z/l

wdN

(z)/m

(a) distortional angle χ(z) (b) warping displacement wdN(z)z/l

σ dN(z

)/Pa

(c) warping stress σdN(z)

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

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

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

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

322 Fig.10 The distortional angle, warping displacements and stresses between IPM and FEA for simply supported 323

girders with nine diaphragms varying with three diaphragm thicknesses 324 n=2, with shear strain of the cross section

0 0.5 1-2

0

2

4

6

8

10 x 10-5

0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5 x 10-6

0 0.5 1-2

-1

0

1

2

3

4 x 106

n=2, without shear strain of the cross section

χ

z/l

wd

z/l

σ d

z/l

14.9%13.3%

17.8%

(a) (b) (c)

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

325

Fig.11 Comparison of girders with and without the shear strain of cross section for (a) the distortional angle , (b) 326 warping displacements and (c) stresses of simply supported girders with 2 and 5 diaphragms 327

In addition, the influence of shear strains of the cross section on distortional deformations 328 and stresses are examined in Fig.11 for simply supported girders with 2 and 5 diaphragms, where 329 the compatibility condition between the girder and diaphragms is considered. It is seen that the 330 shear strain of the cross section makes little effect on warping displacements and stresses, but a 331 large influence on the distortional angle. The largest difference occurs at the loading sections 332 z=0.45l and z=0.55l, which is 14.9% for girders with 2 diaphragms and 17.8% with 5 diaphragms. 333 Also, the error at the mid span for girders with 2 diaphragms is 13.3%. Thus shear strains of the 334 cross section cannot be ignored when the transversal deformation of girders is considered. 335 5.2 Verifications with experiments 336

For further verification of the IPM, a series of experiments were performed using four groups 337 of samples – girders with no diaphragms, one, two and three diaphragms, subjected to distortional 338 loads. Diaphragms are equally distanced along the span. Both girders and diaphragms were 339 fabricated from carbon structural steel plates (yield strength 235MPa) of 8mm thickness. All 340 girders are 3 meters long, with height h=0.6m and width b=0.346m, giving a 30° angle between 341 the diagonal and the web. All girders are sealed by a steel plate of 6mm thickness on both ends. 342

To simultaneously produce two concentrated distortional loads, as in Fig.4d, two steps were 343 taken as follows 344

(1) For distortional loads, as shown in Fig.12, units were designed with two groups of wheels 345 anti-symmetrically about the shear center O of the cross section. This is to decompose the 346 horizontal power force Psource produced by FCS hydraulic servo system into two orthogonal 347 loading components Ph and Pv. 348

(2) For concentrated loads, as shown in Fig.13, four stiff units of 1.5m in length and 0.15m 349 and 0.350m in width, respectively, were welded onto flanges and webs symmetrically over the 350 mid span, and the loading sections were hence located at z1=0.75m and z2=2.25m, respectively. 351

Wheel units Stiff units Girders

(a) anti-symmetrical design (b) actual picture of wheel unit

Ph

Pv

Psource

Ph

Pv

Psource δh

Wheels

352 Fig.12 Anti-symmetrical loading design 353

During the experiments, compressive loadings were applied along the diagonal of the cross 354 section by two anti-symmetrical wheel units, as shown in Fig.12a. The entire experimental setup is 355 shown in Fig.14a, including two FCS hydraulic servo loading systems, two connection beams, two 356 wheel units, four stiff units, the tested girder and two supports. The maximum loading was set to 357 5t with a loading speed of 2mm/s, controlled by a loading equipment in Fig.14b. 358

20

1500

150

20

1500

350

(a) stiff unit on flanges (mm)

(b) stiff unit on webs (mm) (c) actual picture of stiff unit

Welded field

Welded field

Welded field

Stiff unit on flangesStiff unit on webs

359 Fig.13 Stiff units 360

Hydraulic cylinder

Hydraulic jack

Loading head Support Connection beam

FCS hydraulic servo loading system

Tested girders

(a) entire scheme

Stiff unitWheel units

(b) loading equipment 361 Fig.14 Entire experimental scheme 362

Fig.15a shows measured points on the girder, with twelve equally distanced points A1~A11 on 363 webs and B1~B11 on flanges. The transversal deformations on measured points were measured by 364 dial gages, magnetically attached to a rectangular tube of 3.5m length, as shown in Fig.15b. 365 Measurements at each point gave the x- and y- axial deformations UXN and UYN (N=1,..., 11), and 366 the distortional angle is calculated as 367

N N2UX 2UY( )zh b

χ = + (28) 368

A1B1

A2B2

A3B3

A4B4

A5B5

A6B6

A7B7

A8B8

A9B9

A10B10

A11B11

N

MJ

K

Rectangular tube

(a) scheme of tested points (b) actual picture

Dial gages

y

xzO

Magnetic base

369

Fig.15 The scheme of tested points 370 Compared with those from IPM, the transversal displacements and distortional angle are 371

depicted in Fig.16 for girders without diaphragms under the source loading of 5t. There exist some 372 errors between the IPM and experimental results at the singularity points A1 for x-axial 373 displacement and B10 and B11 for y-axial displacement, which are mainly caused by the residual 374

strains of welding and manufacturing. Eliminating the influences induced by singularity points, 375 fitting lines were obtained from experimental results by applying the quadratic fitting method 376 provided in the software MATLAB. Fig.16 shows reasonable agreements between two results. 377

-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 B6 B7 B8 B9 B10B11

IPMQuadratic fittingExperimentsSingularity point

(b) y-axial displacementTested points

UY

N/m

m

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

378 Fig.16 Comparison of transversal displacements and angle between IPM and experiments under 5t loading 379

For distortional warping stresses, strains ( d-0ε ° , d-45ε ° and d-90ε ° ) in three directions were 380

measured at points A1~A6 using strain gauges shown in Fig.17a and recorded by the static strain 381 equipment in Fig.17b. The warping stress is calculated as 382

( )d d-90 d-021Eσ ε υευ ° °= +

− (29) 383

where E=2.1×1011Pa and υ=0.3. 384

385 Fig.17 Strain measurements 386

Fig.18 shows calculated distortional warping stresses on the tested points A1~A6 for girders 387 with (a) no diaphragm, (b, c) one diaphragm, (d, e) two diaphragms and (f) three diaphragms, 388 respectively. It is seen that the warping stresses obtained from the IPM and experiments have the 389 same up-and-down trend and with apexes at A3 for girders with diaphragms. The errors mainly 390 come from the drop of δh (Fig.12a) between two horizontal source forces, which are mainly 391 caused by the residual strain during manufacturing and welding. And this will produce torsional 392 warping stress besides the distortional one. 393

0

10

20

30

40

50

-15

-10

-5

0

5

10

15

20

-30

-20

-10

0

10

20

30

σ d/M

pa

(a) n=0σ d

/Mpa

(b) n=1

σ d/M

pa

(c) n=1

A1 A2 A3 A4 A5 A6 A1 A2 A3 A4 A5 A6 A1 A2 A3 A4 A5 A6Tested points Tested points Tested points

Experiment, Psource=2tIPM, Psource=2tExperiment, Psource=3tIPM, Psource=3t

Psource=2.3t ExperimentIPM

Psource=3.4t ExperimentIPM

394

-5

0

5

10

15

-5

0

5

10

15

20

A1 A2 A3 A4 A5 A6 A1 A2 A3 A4 A5 A6 A1 A2 A3 A4 A5 A6-3

-2

-1

0

1

2

3

4

σ d/M

pa

(d) n=2

σ d/M

pa

(e) n=2

σ d/ M

pa

(f) n=3Tested points Tested points Tested points

ExperimentIPM

ExperimentIPM

ExperimentIPM

Psource=2t Psource=3t Psource=3.4t

395 Fig.18 Comparisons of distortional warping stresses between IPM and experiments (n being the total diaphragm 396

number) 397 6 Parametric study 398

In this section, the effects of the ratio of height to span of the girder h/l, the diaphragm 399

thickness tp and number n are examined on the distortion of simply supported girders with equally 400

distanced inner diaphragms, where the loading and boundary conditions are referred to those in 401

Section 5.1. 402

Fig.19 shows the non-dimensional warping stress σd/σd0 for node N at the loading section 403

z=0.45l varying with the ratios h/l and tp/t, where σd0 is the warping stress of girders without 404

diaphragms. It is seen that except for the girder with two diaphragms under h/l=0.1, the 405

non-dimensional stress σd/σd0 remains less than 1 and becomes smaller with the increment of n, tp/t 406

and h/l, respectively. Besides, the σd/σd0 decreases substantially when the number of diaphragm 407

reduces from 2 to 1, or increases from 2 to 3, clearly indicating that the mid-span diaphragm plays 408

a significant role in the reduction of distortional warping stress. 409

Fig.20 shows the non-dimensional distortional angle χ/χ0 at the loading section z=0.45l 410

varying with h/l and tp/t, where χ0 is the distortional angle for girders without diaphragms. It is 411

seen that except for the girder with two diaphragms under h/l=0.1, the non-dimensional 412

distortional angle χ/χ0 remains less than 0.25 and becomes smaller with the increment of n, tp/t and 413

h/l, respectively. This implies that the inner diaphragms are capable of restraining the transversal 414

deformation of the cross section, especially when n>3. Similarly, compared with the distortional 415

angle at n=2, those at n=1 and 3 decrease significantly due to the restraint by the mid-span 416

diaphragm. 417

1 2 4 6 8 90

0.2

0.4

0.6

0.8

1

1.2

1 2 4 6 8 90

0.2

0.4

0.6

0.8

n n(a) varying with h/l (b) varying with tp/t

σ d/σ

d0

σ d/σ

d0

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

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

418 Fig.19 The non-dimensional warping stress σd/σd0 419

1 2 4 6 8 90

0.1

0.2

0.3

0.4

0.5

0.6

1 2 4 6 8 90

0.05

0.1

0.15

0.2

0.25

χ/χ 0

n n(a) varying with h/l (b) varying with tp/t

χ/χ 0

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

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

420 Fig.20 The non-dimensional distortional angle χ/χ0 421

1 2 4 6 8 90

0.5

1

1.5

1 2 4 6 8 90

0.2

0.4

0.6

0.8

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

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

wd/w

d0

n n(a) varying with h/l (b) varying with tp/t

wd/w

d0

422 Fig.21 The non-dimensional warping displacement wd/wd0 423

Fig.21 shows the non-dimensional warping displacement wd/wd0 for node N at the loading 424

section z=0.45l varying with the ratios h/l and tp/t, where wd0 is the warping displacement for 425

girders without diaphragms. It is seen that except for the girder with two diaphragms under h/l=0.1, 426

the non-dimensional displacement wd/wd0 in Fig.21a remains less than 1.0 and converges to a fixed 427

value between 0.1 and 0.2 for large diaphragm numbers. 428

7 Buckling of both the local web plate and the mid-span diaphragm 429 Taking a simply supported girder with uniform three inner diaphragms as an example, Fig.22 430

shows 3D contours for the warping displacements and stresses, where the measurement, loading 431

and boundary conditions are referred to those in Section 5.1. It is seen from Fig.22 that the 432 maximum displacement occurs at the local web plate at the loading sections, and the maximum 433 stress occurs at the edge of the mid-span diaphragm, which may result in local buckling. It is thus 434 necessary to check the stabilities for both local web plate and mid-span diaphragm. 435

In this section, the proposed IPM was carried to analyze the critical buckling values Pcr1 and 436 Pcr2 for the eccentric load Pj according to the stabilities of the local web plate and the mid-span 437 diaphragm, respectively. Inner diaphragms include one fixed (diaphragm II) at the mid span and 438 two (diaphragms I and III) symmetrical to the mid-span one. The loading and boundary conditions 439 are referred to those in Section 5.1. 440

Diaphragm I Diaphragm IIIMid diaphragm(diaphragm II)

(a) 3D contour of distortional warping displacement (b) deformation of mid-span diaphragm

(c) 3D contour of distortional warping stress (d) Stress of mid-span diaphragm

Original frame

Deformed frame

Maximum warping displacement

Maximum warping stress

441

Fig.22 3D contours of warping displacement and stress of girders with three uniform inner diaphragms (tp=5mm) 442 Fig.23 shows the mixed stress boundary condition for the local web plate between the 443

loading section and the mid span, in which a = 0.05l. Under distortional loads, the warping stress 444

σd varies linearly from σd1 at the loading section to σd2 (=βσd1) at the mid span and from σd1(2) on 445

the top to –σd1(2) at the bottom. There also exists a constant shear stress τd1 (=ησd1) on all four sides 446

of the cross-section, and a parabola τd2 on both lateral sides. Both shear stresses can be obtained 447

from [26] 448

d d

dd d

d

( ) ( )d( )( , ) ( )

( )dF

tF

S s s sM zs z S s

I t s s

ρτ

ρ

= − −

∫

∫

(30) 449

where Md(z) is the distortional moment in Fig.2; It is the distortional warping constant in Eq.(1); t is 450

the thickness of local web plate, given by t=0.01m; F is the cross-sectional area; ρd(s) is the 451

distance between the distortional center [6] and the mid line of the cross-sectional profile; Sd(s) is 452

the second moment of area, given by Sd(s)=∫ŵds, ŵ is the distortional sectorial coordinate [7], s is 453

the circumferential coordinate around the cross-sectional profile. 454

Under the mixed stress boundary condition, the out-of-plane deformation wp for local web 455

plate can be indicated by the combination of two orthogonal sinusoidal functions in z- and y-axial 456

directions, given by 457

p1 1

sin sinuvu v

u z v yw Aa hπ π∞ ∞

= =

= ∑∑ (31) 458

where u and v are the numbers of half waves in z- and y-axial directions; Auv is the undetermined 459

coefficient, which is the maximum of the combination of two sinusoidal functions. 460

y xz

O

Loading sectionPin joint Mid-span sectionLocal web plate

σd1 σd2=βσd1

β:(–) β:(+)

τd2

z

y

τd1=ησd1

τd1 σd2=βσd1

σd1

σd σd σd

a

h

σd2

(a) local web plate (b) stress boundaries around local web plate

–σd1 –σd2

τd2

461 Fig.23 Mixed stress boundaries of local web plate 462

For analysis, the distortional warping stress σd and the shear stress τd for the local web plate 463

can be presumed as [31] 464

d d12 11 1y zh a

βσ σ − = − −

(32) 465

2

d d1 d2 d1 d11 yy

a hβτ τ τ σ η σ −

= + = + −

(33) 466

where β is the ratio of the warping stresses σd1 to the σd2; η is the ratio of the warping stress σd1 to 467

the constant shear stress τd1. Both ratios are calculated from IPM. 468

To obtain the critical buckling stress (σd1)cr for the local web plate, Galerkin equation [32] is 469 applied, given by (p, q=1,2,…,∞) 470

4 4 4 2 2

p p p p pd d4 2 2 4 2

0 0

2 2 sin sin d d 0a h w w w w w p z q yt t z y

z z y y z z y a hπ πσ τ

∂ ∂ ∂ ∂ ∂+ + + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂

∫ ∫ (34) 471

Substitute Eqs.(31)~ (33) into Eq.(34), the latter is changed into 472

22 2

0 0

2

d10 0

2 2

d1

sin sin sin sin dzd

2 11 1 sin sin sin sin dzd

2 1 cos cos si

a h

a h

uv

u v u z v y p z q y ya h a h a h

u u z v y p z q yA t y z ya h a a h a h

uv y u z n yt y vah a h a h

π π π π π π

π β π π π πσ

π β π πσ η

+

− − − −

−+ + −

∫ ∫

∫ ∫1 1

0 0

0

n sin dzd

u v

a h p z q y ya hπ π

∞ ∞

= =

=

∑∑

∫ ∫

473

(35) 474 where the definite integrals on variables u and p are given in Tab.1; and the integrals on v and q 475 are obtained by replacing u and p. 476

Eq.(35) can be translated into a matrix equation set and the determinant of the coefficient 477 matrix should be zero as the coefficient Auv cannot be zero. Therefore, the critical stress (σd1)cr can 478 be obtained. 479

In addition, since the dimension of the coefficient matrix affects the calculation accuracy, the 480 critical stresses (σd1)cr were examined first in Fig.24 under different dimensions of the coefficient 481

matrix. In the calculation, the span l=1m, the height h=0.25m, the width b=0.1m, the thicknesses 482 for webs and flanges t=0.01m, and the diaphragm thickness tp=0.005m. It is seen that the critical 483 stress (σd1)cr tends to a converged value as the dimension of the coefficient matrix increases from 484 2×2 to 4×4. However, calculation time increases significantly from 12s to 51220s. Considering 485 both the accuracy and time, the 3×3 coefficient matrix is regarded most appropriate to calculate 486 the critical stress (σd1)cr. 487

Tab.1 Several definite integral items used in Eq.(35) 488 Definite integrals u=p u≠p and u+p :even u≠p and u+p :odd

0

sin sin dza u z p z

a aπ π

∫ 2a

0 0

0

cos sin dza u z p z

a aπ π

∫ 0 0 ( )2 2

2apu pπ−−

0

sin sin dza u z p zz

a aπ π

∫ 2

4a

0 ( )

2

22 2 2

4a up

u pπ

−

−

0

cos sin da u y p yy y

a aπ π

∫ 2

4auπ

− ( )2

2 2

pau pπ −

( )

2

2 2

pau pπ−

−

2

0

cos sin da u y p yy y

a aπ π

∫ 3

4auπ

− ( )3

2 2

pau m p−

( )( ) ( )3 2 2 3

3 2 23 2 2

4 3pa u p pau pu p ππ

+−

−−

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

(σd1

) cr /M

Pa

489 Fig.24 Convergence of critical distortional warping stress (σd1)cr 490

In order to obtain Pcr1, a linear relationship is established between the critical moment Mcr1 491 and the critical stress (σd1)cr. 492

cr1 d1 cr

e d1 e

( )( )

MM

σσ

= (36) 493

where Mcr1 is the critical value for external moment Mj based on the stability of the local web plate, 494 given by Mcr1=Pcr1·n1/2. Me is the unit external moment, i.e. Me=1Nm. (σd1)e is the corresponding 495 warping stress on top of the web at the loading section z=0.45l, produced by the unit moment Me. 496

Fig.25 gives the critical load Pcr1 varying with the location zp1 of diaphragm I for various 497

ratios h/b, based on the stability of the local web plate. It is seen that the critical load Pcr1 increases 498

remarkably when diaphragm I is located close to the loading section z1=0.45l, implying that 499

installation of a diaphragm at the loading section will enhance effectively the stability of the local 500

web plate. 501

0 0.2 0.4

10

20

30

40

50

P cr1

/kN

zp1/l

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

0.3 0.410

15

Zoomed picture

502 Fig.25 The critical load Pcr1 obtained from the stability of local web plate 503

While for the mid-span diaphragm (diaphragm II) subjected to pure shear boundary 504

conditions, the critical value (Mp2)cr for moment Mp2 is given by [33,34] 505

4 3

22 cr 2( )

(1 )p

p

E tM

πξ υ

=−

, (h/b≤2) 506

2 3

22 cr 2

89( )200 3 1

pp

E th bMb h

πυ

= + − , (h/b>2) (37) 507

where ( )

2 22 2 2 2 2 2

2 2 2 2 22 2

384 706 81 81625 25 9 25 99

b h h b h bh b h bh b

ξ + +

= + + + + +; υ is the Poisson’s ratio and 508

equals to 0.3; tp2 is the thickness of mid-span diaphragm and tp2=0.005m. 509 Based on Eq.(21), the critical moment Mcr2 for external moment Mj can be calculated by 510

2 crcr2

21 22

( )pMM

Q Q= −

+ (38) 511

where Q21 and Q22 are defined in Eq.(21). 512 Furthermore, based on equation Mj=Pj·n1/2 [26], the critical load Pcr2 can be finally obtained 513

from Eq.(38). For the stability of the mid-span diaphragm, Fig.26 gives the critical load Pcr2 514 varying with the location zp1 of diaphragm I for various values of h/b. It is seen that the critical 515 load Pcr2 increases remarkably when diaphragm I is located close to the loading section z1=0.45l, 516 indicating increased resistance to buckling. 517

Back to the girder with equally-distanced three inner diaphragms in Fig.22, the girder will 518 reach its yield strength limit of 235MPa when the eccentric load Pj is 650kN, which is much 519 smaller than the critical buckling load Pcr2 of 8743kN at zp1/l=0.25 for the ratio h/b=2 according to 520 the stability of the mid-span diaphragm. Simultaneously, the plastic yield load of 650kN is much 521 larger than the critical load Pcr1 of 12.2kN at zp1/l=0.25 according to the stability of the local web 522 plate. This implies that buckling of the local web plate will be the primary failure mode with the 523 increment of eccentric loads Pj. Hence attentions should be paid on the stability of the local web 524 plate for the design of girders subjected to eccentric loads. 525

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

526

Fig.26 The critical load Pcr2 obtained from the stability of the mid diaphragm 527 Conclusions 528

In this paper, the initial parameter method (IPM) is applied to investigate the distortion of 529 simply supported girders with inner diaphragms subjected to concentrated eccentric loads, where 530 in-plane shear deformation of diaphragms is considered. The main conclusions can be drawn as 531 follows 532

(1) Compared with results from FEA and experiments, accurate analyses can be obtained by 533 IPM for the distortional angle, warping displacements and stresses for simply supported girders 534 with inner diaphragms. Both the in-plane shear deformation of diaphragms and the compatibility 535 condition between the girder and diaphragms are taken into account in IPM. Besides, comparison 536 of results between considering the shear strain of the cross section and not shows that the shear 537 strain of the cross section cannot be ignored when calculating the distortional angle. 538

(2) Both the distortional angle and warping stresses decrease with the increment of the ratio 539 of height to span, the number of diaphragms and their thickness. The warping displacement 540 converges to a fixed value between 0.1 and 0.2 for large diaphragm numbers. And the mid-span 541 diaphragm plays a key role in reducing the distortional deformations and stresses for girders under 542 symmetrical loads. 543

(3) Stabilities of the local web plate and the mid-span diaphragm were both investigated for 544 box girders with symmetrical three inner diaphragms. Results show that both the local web plate 545 and the mid-span diaphragm increase their resistance to buckling when the diaphragm I is located 546 close to the loading sections. Moreover, the local web plate will buckle first as the primary failure 547 mode. Therefore, attentions are needed on the stabilities of the local web plate for simply 548 supported girders under eccentric loads. 549

Based on the IPM, it is possible to improve the warping displacements and stresses of simply 550 supported girders through optimizing the positions and wall thickness of diaphragms. Future work 551 are needed for (1) optimization of the distortion of girders with diaphragms; (2) mechanical 552 properties of girders with perforated diaphragms. 553 Acknowledgements 554

This work was supported by the National Natural Science Foundation of China (NSFC) 555 [grant number: 51175442 and 51675450]. 556

Appendix I 557 The relationships between φi(z) (i=1,2,3,4) and their differentiations is 558

1 1 4 2 2'ϕ λϕ λ ϕ= + , 2 1 3 2 1'ϕ λϕ λ ϕ= − , 3 1 2 2 4'ϕ λϕ λ ϕ= − , 4 1 1 2 3'ϕ λϕ λ ϕ= + ; (A1) 559

( )2 21 1 2 1 1 2 32''ϕ λ λ ϕ λ λ ϕ= − + , ( )2 2

2 1 2 2 1 2 42''ϕ λ λ ϕ λ λ ϕ= − − , 560

( )2 23 1 2 3 1 2 12''ϕ λ λ ϕ λ λ ϕ= − − , ( )2 2

4 1 2 4 1 2 22''ϕ λ λ ϕ λ λ ϕ= − + ; (A2) 561

( ) ( )3 2 2 31 1 1 2 4 1 2 2 23 3'''ϕ λ λ λ ϕ λ λ λ ϕ= − + − , ( ) ( )3 2 2 3

2 1 1 2 3 1 2 2 13 3'''ϕ λ λ λ ϕ λ λ λ ϕ= − − − , 562

( ) ( )3 2 2 33 1 1 2 2 1 2 2 43 3'''ϕ λ λ λ ϕ λ λ λ ϕ= − − − , ( ) ( )3 2 2 3

4 1 1 2 1 1 2 2 33 3'''ϕ λ λ λ ϕ λ λ λ ϕ= − + − . (A3) 563

Reference 564 [1] Pezeshky P, Mohareb M. Distortional theory for the analysis of wide flange steel beam. Eng 565 Struct 2014; 75: 181-96. 566 [2] Senjanovic I, Fan Y. On torsional and warping stiffness of thin-walled girders. Thin-Walled 567 Struct 1991; 11: 233-76. 568 [3] Alghamdi SA. Static and modal analysis of twin-cell box girder structures. AIAA J 2001; 39: 569 1406-10. 570 [4] Zhang H, Desroches R, Yang ZJ, et al. Experimental and analytical studies on a streamlined steel 571 box girder. J Constr Steel Res 2010; 66: 906-14. 572 [5] Dabrowski R. Curved thin-walled girders theory and analysis. Cement and Concrete 573 Association, 1968. 574 [6] Boswell LF, Li Q. Consideration of the relationships between torsion, distortion and warping of 575 thin-walled beams. Thin-Walled Struct 1995; 21: 147-61. 576 [7] Li Q. Investigation of the warping functions of thin walled bars with closed cross section. J 577 Southwest Jiaotong Univ 1993; 6: 1-6. 578 [8] Wright RN, Abdel-Samad SR, Robinson AR. BEF analogy for analysis of box-girders. J Struct 579 Div 1968; 94: 1719-43. 580 [9] Hsu YT, Schelling DR. EBEF method for distortional analysis of steel box-girder bridges. J 581 Struct Eng 1995; 121: 557-66. 582 [10] Hsu YT, Fu CC. Application of EBEF method for the distortional analysis of steel box girder 583 bridge superstructures during construction. Adv Struct Eng 2002; 5: 211-21. 584 [11] Zhao ZM, Fang ZZ, Guo JQ. Analysis of continuous box girders with diaphragms by finite strip 585 method. Bridge Constr 1993; 4:35-53. 586 [12] Cheung YK. Finite strip method in structural analysis. Oxford: Pergamon Press; 1976. 587 [13] Zhao ZM. The calculating analysis for multiple span continuous curved box girder by the finite 588 strip method. J Fuzhou Univ (N Sci) 1997; 25: 90-4. 589 [14] Zhao ZM. Analysis of continuous curved box-girder bridge with flexible transverse 590 diaphragms by finite strip method. Comp Struct Mech Appl 1993; 10: 473-84. 591 [15] Suetake Y, Hirashima M. Extended trigonometric series analysis of box girders with 592 diaphragms. J Eng Mech. 1997; 123: 293-301. 593 [16] Park N, Lim N, Kang Y. A consideration on intermediate diaphragm spacing in steel box girder 594 bridges with a doubly symmetric section. Eng Struct 2003; 25: 1665-74. 595

[17] Park N, Choi Y, Yi G, et al. Distortional analysis of steel box girders. Steel Structures 2002; 2: 596 51-58. 597 [18] Park N, Choi S, Kang Y. Exact distortional behavior and practical distortional analysis of 598 multicell box girders using an expanded method. Comp & Struct 2005; 83: 1607-26. 599 [19] Park N, Kang YJ, Kim HJ. An independent distortional analysis method of thin-walled 600 multicell box girders. Struct Eng Mech 2005; 21: 275-93. 601 [20] Park N, Choi Y, Kang Y. Spacing of intermediate diaphragms in horizontally curved steel box 602 girder bridges. Finite Elem Anal Des 2005; 41: 925-43. 603 [21] Zhang L. Influences of diaphragm plate and geometric characteristics on distortion effect of 604 steel box girder. J Railway Eng Soc 2013; 8: 68-73. 605 [22] Li HF, Luo YF. Application of stiffness matrix of a beam element considering section distortion 606 effect. J Southeast Univ 2010; 26: 431-35. 607 [23] Vlasov VZ. Thin walled elastic beams, 2nd ed. Jerusalem, Israel. Available from Office of 608 Technical Services, US Department of Commerce,Washington, D.C.: Israel Program for Scientific 609 Translations; 1961. 610 [24] Xu X, Qiang SZ. Research on distortion analysis theory of thin-walled box girder. Eng Mech 611 2013; 30: 192-201. 612 [25] Xu X, Ye HW, Qiang SZ. Distortional analysis of thin-walled box girder taking account of 613 shear deformation. Chin J Comp Mech 2013; 30: 860-66. 614 [26] Harashima M, Usugi K. Distortional analysis of random box girders considering the shear 615 deformation. Foreign Bridge 1980; 1: 1-29. 616 [27] Sakai F, Okumura T. Influence of diaphragms on behaviour of box girders with deformable 617 cross section. Intl Assoc Bridge Struct Eng 1972;9:285-98. 618 [28] Oleinik JC, Heins CP. Diaphragms for curved box beam bridges. J Struct Div, ASCE 619 1975;101:2161–78. 620 [29] Park N, Kang YJ. Distortional analysis of multicell box girders using 3-dimensional shell 621 elements-I. proposal and application of an expanded method. J Korean Soc Civ Eng A 2004; 24: 622 557-63. 623 [30] Park N, Kang YJ. Distortional analysis of multicell box girders using 3-dimensional shell 624 elements-II. distortional analysis based on the expanded method. J Korean Soc Civ Eng A 2004; 24: 625 565-72. 626 [31] Lu YZ. Buckling of simply-supported rectangular plates under combined actions of shear and 627 bending stresses varying linearly in longitudinal direction. J Tongji Univ 1991; 19: 481-87. 628 [32] Fletcher CA. Computational Galerkin Methods. Berlin: Springer-Verlag Berlin Heidelberg; 629 1984. 630 [33] Liu HW, Lin JX, Cao ML. The Theory of Plates and Shells. China: Zhejiang University Press; 631 1987. 632 [34] Xiao MX. The Stability of Plates. China: Sichuan Science and Technology Press; 1993. 633