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Lateral torsional buckling strength of unsymmetrical plate girders with corrugated webs
Engineering Structures 81 (2014) 123–134
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Lateral torsional buckling strength of unsymmetrical plate girders with corrugated webs Sherif A. Ibrahim ⇑ Structural Engineering Department, Ain Shams University, Abbasia, Cairo, Egypt
a r t i c l e
i n f o
Article history: Received 28 April 2014 Revised 22 July 2014 Accepted 23 September 2014
Keywords: Lateral torsional buckling Shear center Warping constant Un-symmetrical section Corrugated web Plate girder
a b s t r a c t This paper deals with lateral torsional buckling of plate girders with corrugated webs (CWG). The research includes both theoretical and finite element analyses for un-symmetrical plate girders with corrugated webs subjected to uniform moment. A new warping constant for this un-symmetrical cross-section is derived taking into consideration the cross-sectional variation along the girder length. Trapezoidal corrugated web profile is taken into consideration in the derivation. The location of the shear center is determined and a closed form of the warping constant is derived. The un-symmetry of the webless beam, which agrees with the case of CWG, is considered in the calculation of the lateral torsional buckling strength. The results are verified with those obtained using the finite element technique and gave good agreement. A comparison with different specifications and codes is conducted to investigate the effect of un-symmetry on the plate girder strength. The effect of cross-section geometrical configurations on the lateral torsional buckling strength under uniform moment is investigated and discussed. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Previous researches and common practices on plate girders with corrugated webs (CWG) indicated that its weight can be much lower, up to 30%, compared with plate girders with flat webs having the same static capacity. As well, using the corrugated webs increases the out-of-plane stiffness and resistance against lateral torsional buckling without the need to add transverse stiffeners or to use thicker web plates. Researchers conducted both analytical and experimental studies on general behavior, bending strength, lateral buckling and lateral torsional buckling capacities of CWG [1–9]. Different types of web configurations and geometrical parameters were investigated. Many researchers confirmed, both experimentally and analytically, that the contribution of the corrugated web to the ultimate moment capacity can be ignored. Experimentally, the average measured stresses in the flanges, in the elastic range, were close to those obtained using simple bending theory and ignoring the contribution of the web. The longitudinal stresses in the corrugated web, almost all over the web height, were insignificant except very close to the flanges [1]. Ibrahim et al. [2] suggested also that shear deformation of CWG has a considerable contribution and should be added when calculating the vertical deflection of the girder. This is ⇑ Tel.: +20 1222455017; fax: +20 222689592. E-mail address: [email protected] http://dx.doi.org/10.1016/j.engstruct.2014.09.040 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.
basically because of the accordion effect in the corrugated web longitudinal direction, which gives it high flexibility and hence it might be ignored in the cross-section static calculations. Till recent, there is no or scarce studies about CWG having unequal top and bottom flanges. Such cross-sections can be used widely in crane tracks girders utilized in the industrial steel buildings. These crane girders are commonly used as un-symmetrical sections and using the corrugated web would improve its lateral torsional buckling capacity, especially for the un-supported compression flange. In this research, the flexure and torsional rigidities are calculated for un-symmetrical CWG cross section using some basis of the previous studies. The shear center location is determined based on ignoring the corrugated web contribution, due to its high flexibility in the longitudinal direction, as proved before both experimentally and analytically. A new expression for the warping constant of un-symmetrical CWG is derived then the elastic lateral torsional buckling capacity expression for this girder under uniform moment is suggested. A series of finite element analyses are conducted to verify the proposed equations utilized in this study. Different cross-section geometrical parameters, such as cross section un-symmetry parameter ‘‘q’’ and corrugated web parameters a, d and h as given in Fig. 1(a) are studied to figure their effect on the lateral torsional buckling capacity of the unsymmetrical CWG. The ultimate
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strength is compared with design aspects of the Eurocode and new design suggestions are proposed. 2. Theoretical background Lindner [6] proposed some empirical formulas for the warping constant of CWG based on tests results. The study confirmed that the torsional section constant may be calculated using the same method as the plate girder with flat web. He also concluded that the interaction between local flange plate buckling and global lateral torsional buckling must be taken into consideration. Sayed-Ahmed [7] concluded that the resistance of CWG to lateral torsional buckling is higher than those with flat webs, up to 37%. He calculated the lateral torsional buckling strength of the girders using the same conventional equation used for plate girders with flat webs, except using an equivalent web thickness as a function of the corrugated web geometrical configurations. Ibrahim [8] investigated the lateral torsional buckling capacity of CWG using tubular rectangular flanges. He used the same formula utilized for the elastic capacity of plate girder with flat web but with the modified warping constant suggested by Lindner [6] and using a reduced effective torsional buckling stiffness (GJr). It was concluded that this combination of corrugated webs and tubular flanges may result in a higher lateral torsional buckling strength up to 46% than conventional plate girder with flat web having the same cross-sectional area. Moon et al. [3] derived an expression, based on the force method, for the warping constant Cw of symmetrical CWG based on ignoring the web contribution. The location of the shear center of cross-section lies at a distance d outside the corrugated web. They calculated the warping constant based on average corrugation depth davr. They also used a reduced shear modulus proposed by Samanta and Mukhopadhyay [10]. The equation for the modified shear modulus of corrugated plate is as follows
Gc ¼
ða þ bÞ G ða þ cÞ
2
C wðMoonÞ ¼ C wðflatÞ þ Iw dav r
ð2a þ bÞ dmax 2ða þ bÞ
ð2Þ
(a)
ð3aÞ
where Cw(flat) is the warping constant of symmetric plate girder with flat web and is given as
C wðflatÞ ¼
Iy;c 2 h 4 w
ð3bÞ
where Iy,c is the second moment of inertia of CWG about the weak axis and ignoring the web in the calculation. Iw is the second 3
moment of inertia of web plate and is given as hw12tw . By substituting Eq. (2) into Eq. (3a), the warping constant of symmetric CWG based on Moon et al. [3] can be enhanced as a function of both geometrical cross-section and corrugated web parameters as in the following equation
C wðMoonÞ ¼ C wðflatÞ þ Iw
2a þ b 2a þ 2b
2
2
dmax
ð4Þ
They utilized the formula used to calculate the lateral torsional buckling strength of plate girder with flat web using the new warping constant derived in Eq. (4). The elastic lateral torsional buckling of CWG under uniform moment can be expressed as
Mcr ¼
ð1Þ
where G is the shear modulus of flat plates, (a + b) is the projected length of the folded plate and (a + c) is the actual length of the corrugated plate as shown in Fig. 1(a). Fig. 1(b) shows the geometrical configuration of a singly symmetric CWG and the off-center location of the corrugated web plate. They suggested an average corrugation depth davr, for simplicity, to take the change in corrugation depth into account as follows
dav r ¼
The warping constant Cw was evaluated by integrating the normalized unit warping Wn curve across the entire cross-section, and based on ignoring the web due to its accordion effect. The final value of the warping constant was to be calculated by summation of the normalized units warping, but without a final expression given for the warping constant. The final expression for the warping constant of symmetric CWG is developed in the current research by the author and is adopted in its final form as
p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Lb
EIy;c Gc J c
1 þ Wc ;
sffiffiffiffiffiffiffiffiffi EC w with W c ¼ Lb Gc J c
p
ð5Þ
where Lb is the lateral buckling length of the CWG and Jc is the pure torsional constant of CWG which is considered the same as the corresponding plate girder with flat web and for symmetrical I-shape cross-section and is given as
Jc ¼
1 2bf t3f þ hw t 3w 3
ð6Þ
Nguyen et al. [4] used the same approach as Moon et al. [3] but considered the corrugated web to be fully active and taken into consideration in the cross-section static properties calculations. The moments of inertia of symmetric CWG was given as
(b)
Fig. 1. (a) Trapezoidal corrugated web profile and (b) cross-section of un-symmetric CWG.
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S.A. Ibrahim / Engineering Structures 81 (2014) 123–134 2
3
2
hw hw t w hw þ ¼ ð6bf t f þ hw t w Þ 2 12 12 3 2 2 bf tf ð2bf t f þ bf hw tw þ 12hw tw d Þ ¼ 6ðbf t f þ hw t w Þ
Z
y2 dA;
x2 dA
Ix;cðNguyenÞ ¼ bf tf
ð7aÞ
Ix;c ¼
Iy;cðNguyenÞ
ð7bÞ
By applying these equations, the values of the second moments of inertia can be obtained as
The shear center in this case is located at a distance Xc from the outside of the corrugated web and it was given by
Xc ¼
6bf t f 6bf t f þ hw t w
d
ð8Þ
2
bf t f hw 3 2 2 ð6bf t f þ bf hw t w þ 12hw t w d Þ 24 ð6bf t f þ hw t w Þ ! 2 3 ð6bf t f Þ hw 3 hw t w 2 bf t f þ d 24 12 ð6bf t f þ hw t w Þ
C wðNguyenÞ ¼ C wðflatÞ þ Iw X c dav r
ð9bÞ
ð10Þ
It is generally noticed that Cw in the derived expressions for both Moon et al. [3] and Nguyen et al. [4] is a quadratic function of the corrugation depth d. Another approximation is made by Nguyen et al. [4], to account for the cross-section change along the plate girder length, was to consider the warping constant as the average value at d = 0 and at d = dmax. In this case, the average ffiffi . The final form of Eq. (10) depth of corrugation would be dav r ¼ dpmax 2 would be as follows
X c;max dmax 2
ð11Þ
In this part, the location of the centroid about X and Y-axes for un-symmetrical CWG as well as the second moments of inertia about both axes are determined. The web rigidity will be considered in the derivation first, and then the case of ignoring the corrugated web contribution will be applied. Fig. 1(b) shows the cross section configuration of un-symmetrical CWG where the location of the centroid is determined using the simplified geometrical analysis as follows
Qy ; A
Y¼
Qx A
ð12aÞ
where Qx and Qy are the first moments of areas about the X and Y axes, respectively. A is the cross-sectional area and is given as
A ¼ b1 t 1 þ b2 t 2 þ ðhw t f Þt w
ð12bÞ
Using Eqs. (12a) and (12b), the coordinates of the centroids are calculated as
X¼
hw t w d; A
Y¼
ð15aÞ
Iy;c
ð15bÞ
2b1 t 1 þ hw t w hw A 2
b3 t
1 1 2 2 where A1 = b1 t1, A2 = b2 t2, Aw = (hw tf)tw, Iy1 ¼ 12 and Iy2 ¼ 12 . By ignoring the web contribution in the above mentioned equations, the modified values can be given as follows
A1 hw ¼ lhw ; X ¼ 0; Y ¼ A1 þ A2 A1 A2 2 h and Iy;c ¼ ðIy1 þ Iy2 Þ Ix;c ¼ A1 þ A2 w
ð16Þ
Based on the equation proposed by Samanta and Mukhopadhyay [10], the shear modulus of the corrugated plate is defined in Eq. (1). Similar concept was suggested by Ibrahim et al. [2] where it was suggested to take the shear deformation into account when calculating the vertical deflection of beam using the corrugated web. The pure torsional constant shall be adopted in this study using the same value as in the case of plate girders with flat webs. Therefore, Jc is given by the equation
Jc ¼
1 ðb1 t 31 þ b2 t 32 þ hw t 3w Þ 3
It has been suggested by Trahair [17] and Nethercot and Trahair [18] that an easily calculated measure of the un-symmetrical I cross-section with different upper and lower flanges properties can be given as
Iy1 hw Iy1 þ Iy2
ð18Þ
where q is the unsymmetry parameter and ysc is the distance from the center of the lower flange to the shear center. This approach will be adopted herein since the web is ignored and the cross-section can be treated as the case of webless beam. It is presumed that the shear flow is distributed equally all over the web height as shown in Fig. 2(a) and is given by qw ¼ hVw , where qw is the shear flow of the corrugated web and V is the shear force acting on the cross-section. Generally, the shear flow q due to the change of the bending normal stress can be obtained using the relation
q¼
V Ix
Z
x 0
ytf dx
ð19Þ
using the above equation, the shear flow distribution across the flange can be determined as shown in Fig. 2(a). Due to the corrugation depth d, the unbalanced shear forces of the flanges FH1 and FH2 are generated. The value of these flange forces can be determined as the sum of the shear flows acting on the flanges and can be expressed as
ð13Þ
The values of the second moments of inertia Ix,c and Iy,c of the un-symmetrical CWG about the strong and weak axes, respectively, can be obtained using the equations
ð17Þ
3.2. Location of shear center of un-symmetrical CWG
Y sc ¼ qhw ¼
3. Cross-section properties of un-symmetrical CWG
X¼
2 hw 3 4A1 A2 A2w 4Aw þ A 12 ðA1 þ A2 ÞAw 2 ¼ ðIy1 þ Iy2 Þ þ d A
Ix;c ¼
3.1. Shear modulus and pure torsional constant
Substituting for Cw(flat), Iw and Xc, the warping constant Cw(Nguyen) can be rewritten as
C wðNguyenÞ ¼ C wðflatÞ þ Iw
ð14Þ
A
ð9aÞ
and this expression can be written in another form as
C wðNguyenÞ ¼
A
b3 t
This expression gives smaller value than d given by Moon et al. [3] which neglects the web contribution. Nguyen et al. [4] neglected the accordion effect between the flanges and the web when calculating the warping constant for symmetric CWG while it was considered based on the full cross-section geometrical configuration. The expression for Cw in this case is defined as
C wðNguyenÞ ¼
Iy;c ¼
Z
F H1 ¼
V V A1 ðhw YÞd and F H2 ¼ A2 Yd Ix Ix
ð20Þ
FH1 and FH2 are proportional to the corrugated web depth d. The shear center location of this cross-section can be determined using
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(a)
(b)
Fig. 2. Shear flow and shear force acting on un-symmetric CWG due to bending: (a) shear flow distribution, and (b) shear force distribution and shear center position.
the moment equilibrium. There is no cross-section twisting that occurs when the applied load V passes through the shear center. The shear center location of the un-symmetrical cross-section of CWG can be obtained as
VX C ¼ F H1 ð1 qÞ hw þ F H2 q hw
ð21Þ
Substituting Eq. (20) into the above equilibrium equation, the shear center location can be determined as
V V VX C ¼ A1 ðhw YÞð1 qÞdhw þ A2 Y qdhw Ix Ix
ð22Þ
Finally, the location of the shear center can be determined by substituting for Y from Eq. (16) into Eq. (22), and this location will be positioned as shown in Fig. 2(b) to the outside of the corrugated web at a distance XC
XC ¼
2 hw A1 A2 d Ix A1 þ A2
ð23aÞ
Since the web is ignored in this case, by substituting for Ix in Eq. (16) into the above equation it will give the location of the shear center as
2
hw XC ¼ A1 A2 A1 þA2
2
hw
A1 A2 A1 þ A2
d¼d
2
h2w 12
hw 6bf tf þ hw tw
bf t f bf t f 6bf t f d¼ d 2bf t f 6bf t f þ hw t w
Since the open section is made up of thin plate elements, the computation of the cross-section warping constant may be greatly simplified by considering the linear variation of the unit warping properties. Firstly, the cross section is considered as a series of connected thin plate elements [11]. Each plate element has a length Lij and thickness tij, then the normalized unit warping at point i of any element ij can be given as
W ni ¼
n 1 X ðW oi þ W oj Þt ij Lij W oi 2A 0
ð23bÞ
ð24Þ
where Wni = qoi Lij and qoi is the distance from the shear center to the element. Fig. 3 shows the direction of paths for calculating the normalized unit warping and the warping constant of the unsymmetrical section. Using Eq. (24) and calculating the path as shown in Fig. 3, the normalized unit warping at each point can be expressed as
b2 þa 2 ¼ qhw d þ a ¼ ð1 qÞhw d þ a
W n1 ¼ qhw
ð25aÞ
W n2 W n3
ð25bÞ ð25cÞ
W n4 ¼ ð1 qÞhw
This is the same value obtained by Moon et al. [3] for the case of symmetric CWG and ignoring the web. If the contribution of the corrugated web is considered for the case of symmetric CWG, then replacing Ix in Eq. (23a) using the value of Ix,c(Nguyen) in Eq. 7(a), it will give XC as
Xc ¼
3.3. Warping constant of un-symmetrical CWG, CW,uc
W n5 W n6
b1 þa 2
b2 ¼ qhw þ a 2 b1 ¼ ð1 qÞhw þ a 2
ð25dÞ ð25eÞ ð25fÞ
ð23cÞ
This is the same result obtained before by Nguyen et al. [4]. As a result, the shear center when ignoring the web is at a distance of 2d from the centerline of the two flanges and at a distance of qhw from the center of the bottom flange as shown in Fig. 2(b). The location of the shear center will be used next to calculate the warping constant of the unsymmetrical CWG, CW,uc. From here on, for the rest of the study, the web rigidity and area will be ignored to consider for the accordion effect of the corrugated web.
Fig. 3. Direction of paths for calculating the warping constant CW,u of un-symmetric CWG.
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S.A. Ibrahim / Engineering Structures 81 (2014) 123–134
where q is the un-symmetry parameter indicated before in Eq. (18), as well another additional parameter is indicated as 2
a¼
hw t w ð1 2qÞd 2A
ð25gÞ
this parameter represents an additional term due to unsymmetry of the cross-section. This term did not show up in the derivation by both Moon et al. [3] and Nguyen et al. [4] when they tackled the same problem due to the symmetry of the cross-section in these cases. Considering q = 0.5 in the above equations will result in the same expressions developed by Moon et al. [3] for Wni values. The warping constant CW,uc can be obtained by integrating the normalized unit warping Wn across the entire cross-section, and CW,uc can be expressed by
C W;uc ¼
n 1X ðW 2ni þ W ni W nj þ W 2nj Þt ij Lij 3 0
3
2
hw t w 2 d 3
ð27aÞ
This expression maybe rewritten in another form after substituting for a as 2 ÞIy;c hw
C W;uc ¼ qð1 q
3 3hw t w h tw 2 ð2q 1Þ2 w d þ 4qð1 qÞ þ 4 A 12
ð27bÞ
To get an easier and shorter form for the same expression, it can be written as 2
2
C W;uc ¼ qð1 qÞIy;c hw þ RIw d
C W;uc:av r ¼
1 aþb
Z
a
C W;uc:max dz þ
0
1 aþb
Z
aþb
FðC W;uc Þdz
2
¼ C W;uf þ RIw d
and R ¼ 4qð1 qÞ þ wð2q 1Þ2 3hw tw where w ¼ 4 A
where C W;uc:max ¼ C W;uf þ KIw dmax
4 Z2 2 FðC W;uc Þ ¼ RIw dmax Z b b
C W;uc ¼
C W;uc ¼ C W;uf þ
Since the corrugation depth d of the CWG is varying along the length of the plate girder, it has to get an average value of the corrugation depth to simplify the calculations. For example Moon et al. [3] suggested the value of davr as given in Eq. (2), where the expression is based on averaging the area enclosed by the corrugated web, (a + 0.5b)dmax, over the projected length of the corrugated plate, a + b, under this area. Nguyen et al. [4] calculated an average value for both Iy and Cw(Nguyen) at corrugation depths equal to zero and at dmax, as given in Eqs. (9–11). These assumptions are mainly based on rational suggestions. In this part, an attempt to get an accurate value of davr is approached based on considering the cross-sectional variation along the longitudinal axis Z of the CWG. For the longitudinal fold
2 þ C W;uf þ RIw dmax
a b 1 2 2 C W;uf þ RIw dmax þ C W;uf þ RIw dmax aþb aþb 3
ð28bÞ
3.4. Average depth of corrugation in un-symmetrical CWG, davr
!
ð30Þ
where the coordinates of this function is shifted along Z-axis as shown in Fig. 4(b) and the value of Z ranges from 0 to b. Substituting of Eq. (30) into Eq. (29) would result in the final form of the warping constant of un-symmetrical CWG taking into consideration the geometrical variation of the web depth and is given by
Hence, finally
It is worth mentioning that the first part of the expression, qð1 qÞIy;c h2w , is the same value of warping constant of un-symmetrical plate girder with flat web CW,uf, and the second part is the additional contribution of the corrugated web effect to the cross-section warping constant. Eq. (28a) is written finally in the same format as Eqs. (3a) and (10), with the main change is the addition of factor R which represents the change of crosssection unsymmetry as well as the ratio of the web area to the cross-sectional area. To validate the new expression with previous studies, for the case of symmetric CWG where q equals 0.5 then R equals 1.0 and consequently will lead to the same results in Eq. (3a) by Moon et al. [3].
ð29bÞ
The value of CW,uc.max is a constant of d. Fig. 4(b), shows the variation of the quadratic function of d which represents the change of the warping constant along the inclined fold projection b, and this geometric function is derived as
ð28aÞ
ð28cÞ
ð29aÞ
a
2
ð26Þ
Substituting Eqs. (25a)–(25f), the warping constant of unsymmetrical CWG can be given as
C W;uc ¼ qð1 qÞIy;c hw a2 A þ ½1 3qð1 qÞ
with length a, the value of the warping constant is at its maximum with a maximum corrugation depth dmax. For the inclined fold with projected length b, the value of the warping constant varies from a maximum at point 1 to a minimum equal to CW,uf at point 2 then to a maximum again at point 3, as shown in Fig. 4(a). The average CW,uc for the inclined fold would be in-between the minimum value of CW,uf and the maximum value of CW,uc.max. The accurate average corrugation depth and average warping constant would result by integrating these variables along the longitudinal axis of the girder. For the warping constant along the projected length of the inclined fold b, the only variable in Eq. (28) is the depth of corrugation d. Hence, the integration would be in the form
a þ ðb=3Þ 2 RIw dmax aþb
ð31Þ
ð32Þ
and the average depth in this case would be defined as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ ðb=3Þ dmax dav r ¼ aþb
ð33Þ
Fig. 5(a) and (b) shows the previous Eq. (33) in comparison with similar expressions previously studied, where C = a and C = 0.8a in these figures, respectively. The figures cover the range of corrugation angle h from 20° up to 70°, so as to cover all practical ranges of web inclination angle. The ratio between average corrugated depth to maximum corrugated depth is considered constant with a value of 0.707 by Nguyen et al. [4], whereas it ranges from 0.758 to 0.873 and from 0.785 to 0.893 by Moon et al. [3] for the case of C = a and C = 0.8a, respectively. In the current study, these ratios range from 0.823 to 0.911 and from 0.845 to 0.926, respectively. The current study gives higher value of davr/dmax than in Moon et al. [3] by about 6% in average, but it considers the variation of the warping constant over the longitudinal axis of the plate girder. The relation between the value of CW,uc./CW,uf and 2dmax/b2 is plotted in Fig. 6, where the increase in the warping constant CW,uc. in un-symmetrical CWG can reach up to 50% over the warping constant of plate girder with flat web. With the increase of corrugation depth, the ratio CW,uc./CW,uf of increases. Fig. 7 illustrates the same ratios of CW,uc./CW,uf in a different presentation where in the case of 2dmax/b2 up to 0.45, there is a negligible effect for changing the parameter q on the ratio of CW,uc./CW,uf and with the increase of 2dmax/b2 over 0.45, the effect of q magnifies. The reason is the effect of factor R in Eq. (28) which contributes significantly in the value of CW,uc since R is a quadratic function of q.
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(a)
(b)
Fig. 4. (a) Corrugated web depth profile and warping constant variation and (b) integration function along the inclined fold projection.
4. Lateral torsional buckling capacity of un-symmetrical CWG The combination of un-symmetrical steel cross-section in conjugate with the corrugated web is quite sophisticated especially for the case where the girder is subjected to lateral torsional buckling. In this study the girders are investigated under uniform moment and the boundary conditions are both simply supported in flexure and torsion. The expression used for calculating the elastic lateral torsional buckling for the un-symmetrical CWG, Mcr,cw, will be considered the same as in calculating the elastic lateral torsional buckling strength for un-symmetrical plate girder with flat web by Galambos [12] and Chen and Lui [16], with the proposed new warping constant formulas and ignoring the corrugated web contribution in the moment of inertia calculations
(a)
Mcr;cw ¼
p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Lb
EIy;c Gc J c
1 þ W1 þ W2 þ W2 ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi EC W;uc ; with W 1 ¼ Lb Gc J c
p
sffiffiffiffiffiffiffiffiffi
W2 ¼
pbx EIy;c 2Lb
Gc Jc
ð34Þ
where Lb is the length of the un-symmetrical CWG and W1 represents the warping torsional stiffness term and W2 represents the unsymmetrical parameter where the parameter bx is another coefficient presenting the degree of cross-section un-symmetry. The general expression for bx is defined as
bx ¼
(b) Fig. 5. Comparison between average depth of corrugated web and corrugation angle for: (a) c = a, (b) c = 0.8a.
Fig. 6. Comparison between CW,uc./CW,uf and 2dmax/b2 for different q parameter.
1 Ix
Z
yðx2 þ y2 ÞdA yo
ð35aÞ
A
where yo = (q l) hw as shown in Fig. 8. Kitipornchai and Trahair [13] studied the case of mono-symmetric webless I-beams, which matches the case of CWG when ignoring the web contribution. They derived an equation for the unsymmetry property bx which is given by
bx Iy ¼ ð2q 1Þ þ ðl qÞ hw Ix
ð35bÞ
Fig. 7. Variation of CW,uc./CW,uf and q parameter for different 2dmax/b2 values.
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Fig. 9. Loading and boundary conditions of finite element models. Fig. 8. Webless I-section suggested by Kitipornchai and Trahair [13].
This equation will be utilized in the current research for evaluating the elastic lateral torsional buckling strength for unsymmetrical CWG.
4.1. Verification of lateral torsional buckling capacity of un-symmetrical CWG with proposed equation using finite element analysis In this section, the lateral torsional buckling strength of unsymmetrical CWG, using the new warping constant developed equation, is verified and compared with the results obtained using the finite element method for the same models. Twelve models are considered for the analysis using different geometrical parameters as indicated in Table 1. The finite element analysis is performed using the commercial structural analysis program ‘‘ANSYS V.11’’ [15]. The used element from the program library is the four nodes thin shell element ‘‘Thin shell93’’ which has the capability of both geometrical and material non-linearities. The typical loading and boundary conditions of all models are considered as shown in Fig. 9. The girders are considered same as in the mathematical model which is simply supported both in bending and torsion. Displacements about directions X, Y and Z and rotation about Z-axis are restrained for point (A), whereas for point (B) the displacements about directions X and Y and rotation about Z-axis are only restrained. Displacements in Y-direction for both lines (1) and (2) are restrained and for lines (3) and (4) the displacements in X-direction are the only restrained. The end moments are applied as an equivalent tension and compression forces applied on the bottom and top flanges, respectively. Fig. 10 shows the typical finite element geometry and mesh used in all models, where both the top and bottom flanges are modeled using 10 elements per flange width. Twenty elements are used through the web height in all verification models. The horizontal fold width is modeled using 5 elements whereas the
inclined fold width is modeled using 6 elements. The aspect ratio of any shell element did not exceed 2 so as to obtain accurate results. The simple span of the girder is kept constant in all the verification models at L = 13,600 mm. The geometrical imperfection is considered as L/1000 out-of-plane in the middle of the girder span. The material model of the finite element analysis is considered as bi-linearly elastic perfectly plastic with steel yield stress fy = 355 MPa and ultimate stress fu = 510 MPa. The automatic Arc-length control technique of the program is adopted for all models in conjunction with the Modified Newton–Raphson solution technique. 4.2. Comparison between proposed equation and finite element analysis results Table 2. demonstrates the results of the critical buckling moment from the finite element analysis, MFEA, compared with the critical elastic buckling moment capacity, Mcr,cw, obtained from Eq. (34) and utilizing the derived warping constant CW,uc for unsymmetrical CWG using Eq. (32). The ratios illustrated are relative to the plastic moment capacity MP of the un-symmetrical CWG with web neglected, where the value of MP is expressed as
MP ¼ f y
2A1 A2 hw A1 þ A2
ð36Þ
The ratio of MFEA/ Mcr,cw ranges from 0.9 to 1.055 with an average value of 0.983 as illustrated in Table 2, which is considered as a good agreement between the two methods. Fig. 11 shows the deformed shape at failure for one of the finite element models, CWG-1, where the failure mode is the elastic lateral torsional buckling of the compression flange. This type of failure is typical for all models considered herein. The models are designed geometrically to keep the failure mode within the elastic limit. Figs. 12(a) and 12(b) show the applied end moment relation against top lateral
Table 1 Geometrical configuration of verification and analytical finite element models. Model No.
hw (mm)
tw (mm)
b1 (mm)
t1 (mm)
b2 (mm)
t2 (mm)
dmax (mm)
a (mm)
b (mm)
h (°)
L (mm)
CWG-1 CWG-2 CWG-3 CWG-4 CWG-5 CWG-6 CWG-7 CWG-8 CWG-9 CWG-10 CWG-11 CWG-12
900 900 900 900 900 1200 1500 600 900 900 900 900
6 6 6 6 6 6 6 6 6 6 6 6
300 300 300 300 300 300 300 300 200 240 360 420
15 15 15 15 15 15 15 15 15 15 15 15
200 200 200 200 200 200 200 200 200 200 200 200
15 15 15 15 15 15 15 15 15 15 15 15
60 30 90 60 60 60 60 60 60 60 60 60
180 180 180 240 300 180 180 180 180 180 180 180
160 160 160 160 160 160 160 160 160 160 160 160
36.9 20.6 48.4 36.9 36.9 36.9 36.9 36.9 36.9 36.9 36.9 36.9
13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600
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Fig. 10. Finite element model and elements mesh arrangement.
Table 2 Comparison between proposed equations and finite element analysis results. Model No.
Mcr,cw (kN m)
MP (kN m)
MFEA (kN m)
M cr;cw MP
M FEA MP
M FEA M cr;cw
CWG-1 CWG-2 CWG-3 CWG-4 CWG-5 CWG-6 CWG-7 CWG-8 CWG-9 CWG-10 CWG-11 CWG-12
395.8 387.3 413.2 398.3 400.3 507.1 625.1 290.2 144.1 218.5 628 966.4
1150.2 1150.2 1150.2 1150.2 1150.2 1533.6 1917 766.8 958.5 1045.6 1232.4 1298.6
392.9 375 404.6 406.7 409.5 470.8 602.3 297.5 129.6 199.7 647.6 1020
0.344 0.337 0.359 0.346 0.348 0.331 0.326 0.378 0.150 0.209 0.510 0.744
0.341 0.326 0.352 0.354 0.356 0.307 0.314 0.388 0.130 0.191 0.525 0.785
0.993 0.968 0.979 1.021 1.023 0.928 0.964 1.025 0.90 0.914 1.031 1.055
Average results = 0.983.
displacement and vertical displacement at mid-span of the model CWG-1 compression flange, respectively. Excessive lateral displacement of the compression flange is clear in the figure due to the elastic twisting of the girder cross-section without any gain in the moment capacity after reaching the critical buckling load. 4.3. Effect of cross-section and corrugated web geometrical configuration on lateral torsional buckling capacity of un-symmetrical CWG The effect of different geometrical parameters such as corrugation depth, dmax, horizontal fold length, a, web height hw, and the un-symmetry parameter, q, on the critical buckling moment will be discussed. The values of these variables are indicated in the models from CWG-1 to CWG-12 listed in Table 1. The variation in non-dimensional Mcr,cw/MP with the change of 2dmax/b2 is shown in Fig. 13. The results of both the finite element analysis and the proposed equation are plotted in the same figure. It indicates that
Fig. 11. Lateral torsional buckling deformed shape of Finite element model CWG-1.
S.A. Ibrahim / Engineering Structures 81 (2014) 123–134
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Fig. 13. Variation of Mcr,cw./MP of proposed equation and FEA for different 2dmax/b2 values.
Fig. 12a. Girder end moments versus top lateral displacement of compression flange Ux for model. CWG-1.
Fig. 14. Variation of Mcr,cw./MP of proposed equation and FEA for different a/twvalues.
Fig. 12b. Girder end moments versus vertical displacement of compression flange Uy for model CWG-1.
the increase of elastic lateral torsional buckling capacity due to the increase of web corrugation depth is about 7% and this is mainly due to the increase in the warping constant value CW,uc. Same result is confirmed by the finite element ratios in the same plot. The effect of changing the ratio a/tw from 30 to 50 results only in a minute increase in the Mcr,cw/MP ratio by only 1% as shown in Fig. 14. The variation of web slenderness ratio hw/tw with critical buckling moment capacity ratio Mcr,cw/MP is shown in Fig. 15 where it is found that with the increase of hw/tw ratio from 100 to 250, the ratio of Mcr,cw/MP decreases by 14%. Referring back to Eq. (36) where the value of MP is proportionally linear with hw/tw, whereas the increase in the moment capacity of the un-symmetrical CWG is not increasing in value by the same linear rate according to Eq. (34). The influence of the un-symmetry parameter, q, is clearly shown in Fig. 16, the ratio of Mcr,cw/MP increases from 0.15 at q = 0.5 to 0.744 at q = 0.9, due to the large increase of Iy which contributes significantly in the elastic lateral torsional buckling capacity of un-symmetrical CWG. The results of the finite element analysis also confirmed the results as shown in the same figure.
Fig. 15. Variation of Mcr,cw./MP of proposed equation and FEA for different hw/tw values.
Fig. 16. Variation of Mcr,cw./MP of proposed equation and FEA for different q parameter values.
4.4. Comparison of elastic lateral torsional buckling capacity between un-symmetrical CWG and corresponding plate girder with flat web Figs. 17–20 show the ratio of non-dimensional ratio Mcr,cw/MFlat against the same geometrical parameters mentioned before, 2dmax/b2, a/tw, hw/tw and q, respectively. MFlat is the elastic lateral
torsional buckling strength of un-symmetrical plate girder with flat web. It can be seen from Fig. 17 that the ratio Mcr,cw/MFlat increases with the increase of dmax where the gain in the moment capacity due to using the corrugated web can reach 11% more than
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S.A. Ibrahim / Engineering Structures 81 (2014) 123–134
the corresponding plate girder with flat web. It is worth mentioning that the maximum value of dmax is chosen as 0.9 of the narrower lower flange outstand, where it is restrained by the maximum flange width as well as by the local buckling of the compression flange. In Fig. 18, the change of a/tw does not exhibit a major change in Mcr,cw/MFlat, only a gain of 1% is obtained in the lateral torsional buckling capacity, the same value as the result obtained previously in Fig. 14. The increase of Mcr,cw/MFlat ranges from 2.7% to 9.2% with the change of hw/tw ratio from 100 to 250, as shown in Fig. 19, respectively. The increase of hw/tw ratio contributes in the increase of web plate moment of inertia and consequently to the increase of warping constant CW,uc as indicated in Eq. (32). Fig. 20 shows the ratio between Mcr,cw/MFlat and the un-symmetry parameter q where this ratio tends to increase and ranges from 1.024 to 1.092. The increase in q value will results in an increase of the warping constant value based on Eq. (28).
Fig. 17. Comparison between Mcr,cw. and Mflat versus different 2dmax/b2 values.
5. Ultimate moment resistance of un-symmetrical CWG according to Eurocode 3 (EN 1993-1-5:2006) Annex D Eurocode EC-3, Part 1-5 [14]: Plated structural elements – Annex D, covers design rules for both symmetrical and unsymmetrical plate girders with both trapezoidal and sinusoidal corrugation profiles. The conventional assumption to ignore the web moment resistance of the girder is considered in the rules of this annex. For a simply supported girder, the lateral torsional buckling is based on the smallest axial resistance of each of the flanges times the distance between the flanges centroids. It is based on the minimum value of the corresponding moment due to tension flange yielding, compression flange yielding or compression flange lateral torsional buckling. Herein, the controlling failure mode is the lateral torsional buckling of the girder and the moment resistance MRd for fully effective flanges is given according to the EC-3 – Annex D as
M Rd ¼
b1 t 1
cM1
vLT f y hw þ
t1þ t 2 2
ð37aÞ
where cM1 is a partial factor representing resistance of member for instability = 1.0 and
vLT ¼
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 uLT þ u2LT bk2LT
1:0 1=k2LT
ð37cÞ
kLT
ð37dÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffi MP ¼ Mcr;cw
where vLT is the reduction factor due to out-of-plane buckling, aLT is an imperfection factor which is considered in this study as 0.76 for buckling curve (d). For all models considered in this research, the value of hw/bf > 2.0; hence buckling curve (d) is utilized. b and kLT,0 are parameters depending on the type of beam section. In one method of design, the EC3 uses conservative values of b = 1.0 and kLT,0 = 0.2 whereas the other less conservative method uses the values of b = 0.75 and kLT,0 = 0.4. These values are also considered for the case of plate girder subjected to pure moment. The effective area of the compression flange is considered based on the flange slenderness parameter kP and should satisfy the following condition to have the flange with a full effective width
b1 =2t 1 pffiffiffiffiffiffi 0:748 28:4e K r
where e ¼
ð38aÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 235=f y and Kr is the buckling factor of the compression
flange outstand plate and is expressed as
Fig. 19. Comparison between Mcr,cw. and Mflat versus different hw/tw values.
ð37bÞ
uLT ¼ 0:5½1 þ aLT ðkLT kLT;0 Þ þ bk2LT
kP ¼
Fig. 18. Comparison between Mcr,cw. and Mflat versus different a/tw values.
Fig. 20. Comparison between Mcr,cw. and Mflat versus different q parameter values.
K r ¼ 0:43 þ
b1 2
þd a þ 2b
!2 P 0:6
ð38bÞ
In this study, all the cross sections investigated are considered to be with fully effective compression flanges thus to depict its behavior due to lateral torsional buckling manifestation without interference with other failure modes. All models, from CWG-1 to CWG-12, are chosen to have elastic lateral torsional buckling by considering the range of buckling parameter kLT between the values 1.16 and 1.71.
S.A. Ibrahim / Engineering Structures 81 (2014) 123–134
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Finite element analysis is performed for these models and the results are shown in Fig. 22. The values of the non-dimensional ratio Mcr,cw/MP are plotted against kLT using the proposed method, finite element analysis and EC3-Annex D, considering different values for parameters b and kLT,0, respectively. The results show that using the EC3 buckling curve with values of b = 0.75 and kLT,0 = 0.4 is more consistent with both the finite element results and the proposed method than using the factors b = 1.0 and kLT,0 = 0.2. The ratio of MFEA/Mcr,cw ranges from 0.88 to 1.05. The ratio between EC3 with the suggested parameters and the proposed method ranges from 0.78 to 1.03. 6. Conclusions Fig. 21. Variation of Mcr,cw./MP of proposed equation and EC3 with kLT (results of elastic buckling analysis).
Fig. 22. Variation of Mcr,cw./MP of proposed equation and EC3 with kLT (results of inelastic buckling analysis).
In Fig. 21, the values of the non-dimensional ratio Mcr,cw/MP are drawn against kLT for all models. Mcr,cw values are evaluated using the proposed method for calculating the elastic lateral torsional buckling moment capacity of un-symmetrical CWG, as well as using (EC3-Annex D) methodology. Both values of parameters b = 1.0, 0.75 and kLT,0 = 0.2, 0.4 are used, respectively. It is clearly shown that the results of Mcr,cw/MP have better agreement between the proposed method and the EC3 design method which utilizes the parameters b = 0.75 and kLT,0 = 0.4 than for the conservative method which uses the conventional parameters of b = 1.0 and kLT,0 = 0.2. It is recommended thus for the case of un-symmetrical CWG design against lateral torsional buckling to use (EC3-Annex D) with the design parameters b = 0.75 and kLT,0 = 0.4, respectively. 5.1. Inelastic lateral torsional buckling capacity of un-symmetrical CWG By using Eqs. (37a–37d), the inelastic lateral torsional buckling strength of CWG can be depicted as well as the elastic lateral torsional buckling capacity mentioned before. To validate using these equations for the inelastic region of un-symmetrical CWG, both of the proposed method, using Eqs. (32) and (34), as well as the finite element analysis are used. The geometrical properties of model CWG-1 cross-section are used in this part of the analysis. The value of the non-dimensional buckling parameter kLT is selected in the range of 0.53–1.71 by varying the length of the girder L from 4000 mm to 13,600 mm, so as to cover both elastic and inelastic ranges.
In this study, the lateral torsional buckling strength of unsymmetrical plate girder with corrugated web (CWG) subjected to uniform pure flexure moment is investigated. Pure torsional constant and shear modulus are presented based on previous work. Different geometrical properties of the un-symmetrical crosssection such as moments of inertia and location of the shear center are determined. Modified forms of warping constants, for the case of symmetric cross-section, and derived by others were modified to a final form for comparison. Based on a proposed equation for the shear center location, the warping constant is derived using numerical methodology. The average depth of the variable trapezoidal corrugated web is determined by integrating the warping constant all over the beam length to obtain the warping constant in its final form. The formula used for calculating the elastic lateral torsional buckling strength of un-symmetrical plate girder with conventional flat web is utilized in this research to calculate the same failure strength for un-symmetrical CWG. The concept of ignoring the corrugated web in the flexure resistance calculation is adopted herein. Finite element analysis is performed taking into consideration different geometrical cross-section and corrugated web parameters. The comparison between the finite element results and the proposed formulas showed a good agreement. The depth of corrugation proved to have a major influence on the critical lateral torsional buckling strength of the CWG. It is found that the un-symmetrical CWG has an elastic lateral torsional buckling capacity up to 11% more than the corresponding plate girder with conventional flat web. Finally, a comparative study is made with the EC3 approach for both the elastic and ultimate strengths of CWG and less conservative design factors in the Eurocode is suggested to be used when using the EC3- Annex D in the design of un-symmetrical CWG. References [1] Elgaaly M, Seshadri A, Hamilton RW. Bending strength of steel beams with corrugated webs. J Struct Eng ASCE 1997;123(6):772–82. [2] Ibrahim Sherif A, El-Dakhakhni Wael W, Elgaaly Mohamed. Behavior of bridge girders with corrugated webs under monotonic and cyclic loading. Eng Struct 2006;28:1941–55. [3] Moon JiHo, Yi Jong-Won, Choi Byung H, Lee Hak-Eun. Lateral-torsional buckling of I-girder with corrugated webs under uniform bending. ThinWalled Struct 2009;47:21–30. [4] Nguyen Ngoc Duong, Kim Sung Nam, Seung-Ryong Han, Kang Young-Jong. Elastic lateral-torsional buckling strength of I-girder with trapezoidal web corrugations using a new warping constant under uniform moment. Eng Struct 2010;32:2157–65. [5] Abbas HH, Sause R, Driver RG. Behavior of corrugated web I-girders under inplane loading. J Eng Mech, ASCE 2006;132(8):806–14. [6] Lindner J. Lateral torsional buckling of beams with trapezoidally corrugated webs. Stab Steel Struct Budapest, Hungary 1990:305–10. [7] Sayed-Ahmed EY. Lateral torsion-flexure buckling of corrugated web steel girders. Proc Inst Civil Eng Struct Build 2005;158(1):53–69. [8] Ibrahim, Sherif A. Comparative study on lateral torsional buckling of tubular plate girders using different web systems. In: Proc 11th ICSGE colloquium, Cairo, Egypt; May 2005.
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[9] Gil H, Lee S, Lee HE, Yoon T. Study on elastic buckling strength of corrugated web. J Struct Eng KSCE 2004;24(1A):192–202 [in Korean]. [10] Samanta A, Mukhopadhyay M. Finite element static and dynamic analyses of folded plates. Eng Struct 1999;21:227–87. [11] Heins CP. Bending and torsional design in structural members. Lexington books. MA, USA: D.C. Heath and Company; 1975. [12] Galambos TV. Guide to stability design criteria for metal structures. 5th ed. New York, N.Y.: John Wiley & Sons Inc.; 1998. [13] Kitipornchai S, Trahair NS. Buckling properties of mono-symmetric I-beams. University of Queensland, Research Report No. CE4; May 1979.
[14] European committee for standardization. Eurocode3. Design of steel structure; 2006. [15] ANSYS User’s Manual (version 11). Houston: Swanson Analysis Systems Inc.; 2006. [16] Chen WF, Lui EM. Structural stability, theory and implementation. Elsevier Science Publishing Co., Inc.; 1987. [17] Trahair NS. The behavior and design of steel structures. London: Chapman and Hall; 1977. [18] Nethercot DA, Trahair NS. Lateral buckling approximations for elastic beams. Struct Eng 1976;54:197–204.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Lateral torsional buckling strength of unsymmetrical plate girders with corrugated webs Sherif A. Ibrahim ⇑ Structural Engineering Department, Ain Shams University, Abbasia, Cairo, Egypt
a r t i c l e
i n f o
Article history: Received 28 April 2014 Revised 22 July 2014 Accepted 23 September 2014
Keywords: Lateral torsional buckling Shear center Warping constant Un-symmetrical section Corrugated web Plate girder
a b s t r a c t This paper deals with lateral torsional buckling of plate girders with corrugated webs (CWG). The research includes both theoretical and finite element analyses for un-symmetrical plate girders with corrugated webs subjected to uniform moment. A new warping constant for this un-symmetrical cross-section is derived taking into consideration the cross-sectional variation along the girder length. Trapezoidal corrugated web profile is taken into consideration in the derivation. The location of the shear center is determined and a closed form of the warping constant is derived. The un-symmetry of the webless beam, which agrees with the case of CWG, is considered in the calculation of the lateral torsional buckling strength. The results are verified with those obtained using the finite element technique and gave good agreement. A comparison with different specifications and codes is conducted to investigate the effect of un-symmetry on the plate girder strength. The effect of cross-section geometrical configurations on the lateral torsional buckling strength under uniform moment is investigated and discussed. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Previous researches and common practices on plate girders with corrugated webs (CWG) indicated that its weight can be much lower, up to 30%, compared with plate girders with flat webs having the same static capacity. As well, using the corrugated webs increases the out-of-plane stiffness and resistance against lateral torsional buckling without the need to add transverse stiffeners or to use thicker web plates. Researchers conducted both analytical and experimental studies on general behavior, bending strength, lateral buckling and lateral torsional buckling capacities of CWG [1–9]. Different types of web configurations and geometrical parameters were investigated. Many researchers confirmed, both experimentally and analytically, that the contribution of the corrugated web to the ultimate moment capacity can be ignored. Experimentally, the average measured stresses in the flanges, in the elastic range, were close to those obtained using simple bending theory and ignoring the contribution of the web. The longitudinal stresses in the corrugated web, almost all over the web height, were insignificant except very close to the flanges [1]. Ibrahim et al. [2] suggested also that shear deformation of CWG has a considerable contribution and should be added when calculating the vertical deflection of the girder. This is ⇑ Tel.: +20 1222455017; fax: +20 222689592. E-mail address: [email protected] http://dx.doi.org/10.1016/j.engstruct.2014.09.040 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.
basically because of the accordion effect in the corrugated web longitudinal direction, which gives it high flexibility and hence it might be ignored in the cross-section static calculations. Till recent, there is no or scarce studies about CWG having unequal top and bottom flanges. Such cross-sections can be used widely in crane tracks girders utilized in the industrial steel buildings. These crane girders are commonly used as un-symmetrical sections and using the corrugated web would improve its lateral torsional buckling capacity, especially for the un-supported compression flange. In this research, the flexure and torsional rigidities are calculated for un-symmetrical CWG cross section using some basis of the previous studies. The shear center location is determined based on ignoring the corrugated web contribution, due to its high flexibility in the longitudinal direction, as proved before both experimentally and analytically. A new expression for the warping constant of un-symmetrical CWG is derived then the elastic lateral torsional buckling capacity expression for this girder under uniform moment is suggested. A series of finite element analyses are conducted to verify the proposed equations utilized in this study. Different cross-section geometrical parameters, such as cross section un-symmetry parameter ‘‘q’’ and corrugated web parameters a, d and h as given in Fig. 1(a) are studied to figure their effect on the lateral torsional buckling capacity of the unsymmetrical CWG. The ultimate
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S.A. Ibrahim / Engineering Structures 81 (2014) 123–134
strength is compared with design aspects of the Eurocode and new design suggestions are proposed. 2. Theoretical background Lindner [6] proposed some empirical formulas for the warping constant of CWG based on tests results. The study confirmed that the torsional section constant may be calculated using the same method as the plate girder with flat web. He also concluded that the interaction between local flange plate buckling and global lateral torsional buckling must be taken into consideration. Sayed-Ahmed [7] concluded that the resistance of CWG to lateral torsional buckling is higher than those with flat webs, up to 37%. He calculated the lateral torsional buckling strength of the girders using the same conventional equation used for plate girders with flat webs, except using an equivalent web thickness as a function of the corrugated web geometrical configurations. Ibrahim [8] investigated the lateral torsional buckling capacity of CWG using tubular rectangular flanges. He used the same formula utilized for the elastic capacity of plate girder with flat web but with the modified warping constant suggested by Lindner [6] and using a reduced effective torsional buckling stiffness (GJr). It was concluded that this combination of corrugated webs and tubular flanges may result in a higher lateral torsional buckling strength up to 46% than conventional plate girder with flat web having the same cross-sectional area. Moon et al. [3] derived an expression, based on the force method, for the warping constant Cw of symmetrical CWG based on ignoring the web contribution. The location of the shear center of cross-section lies at a distance d outside the corrugated web. They calculated the warping constant based on average corrugation depth davr. They also used a reduced shear modulus proposed by Samanta and Mukhopadhyay [10]. The equation for the modified shear modulus of corrugated plate is as follows
Gc ¼
ða þ bÞ G ða þ cÞ
2
C wðMoonÞ ¼ C wðflatÞ þ Iw dav r
ð2a þ bÞ dmax 2ða þ bÞ
ð2Þ
(a)
ð3aÞ
where Cw(flat) is the warping constant of symmetric plate girder with flat web and is given as
C wðflatÞ ¼
Iy;c 2 h 4 w
ð3bÞ
where Iy,c is the second moment of inertia of CWG about the weak axis and ignoring the web in the calculation. Iw is the second 3
moment of inertia of web plate and is given as hw12tw . By substituting Eq. (2) into Eq. (3a), the warping constant of symmetric CWG based on Moon et al. [3] can be enhanced as a function of both geometrical cross-section and corrugated web parameters as in the following equation
C wðMoonÞ ¼ C wðflatÞ þ Iw
2a þ b 2a þ 2b
2
2
dmax
ð4Þ
They utilized the formula used to calculate the lateral torsional buckling strength of plate girder with flat web using the new warping constant derived in Eq. (4). The elastic lateral torsional buckling of CWG under uniform moment can be expressed as
Mcr ¼
ð1Þ
where G is the shear modulus of flat plates, (a + b) is the projected length of the folded plate and (a + c) is the actual length of the corrugated plate as shown in Fig. 1(a). Fig. 1(b) shows the geometrical configuration of a singly symmetric CWG and the off-center location of the corrugated web plate. They suggested an average corrugation depth davr, for simplicity, to take the change in corrugation depth into account as follows
dav r ¼
The warping constant Cw was evaluated by integrating the normalized unit warping Wn curve across the entire cross-section, and based on ignoring the web due to its accordion effect. The final value of the warping constant was to be calculated by summation of the normalized units warping, but without a final expression given for the warping constant. The final expression for the warping constant of symmetric CWG is developed in the current research by the author and is adopted in its final form as
p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Lb
EIy;c Gc J c
1 þ Wc ;
sffiffiffiffiffiffiffiffiffi EC w with W c ¼ Lb Gc J c
p
ð5Þ
where Lb is the lateral buckling length of the CWG and Jc is the pure torsional constant of CWG which is considered the same as the corresponding plate girder with flat web and for symmetrical I-shape cross-section and is given as
Jc ¼
1 2bf t3f þ hw t 3w 3
ð6Þ
Nguyen et al. [4] used the same approach as Moon et al. [3] but considered the corrugated web to be fully active and taken into consideration in the cross-section static properties calculations. The moments of inertia of symmetric CWG was given as
(b)
Fig. 1. (a) Trapezoidal corrugated web profile and (b) cross-section of un-symmetric CWG.
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3
2
hw hw t w hw þ ¼ ð6bf t f þ hw t w Þ 2 12 12 3 2 2 bf tf ð2bf t f þ bf hw tw þ 12hw tw d Þ ¼ 6ðbf t f þ hw t w Þ
Z
y2 dA;
x2 dA
Ix;cðNguyenÞ ¼ bf tf
ð7aÞ
Ix;c ¼
Iy;cðNguyenÞ
ð7bÞ
By applying these equations, the values of the second moments of inertia can be obtained as
The shear center in this case is located at a distance Xc from the outside of the corrugated web and it was given by
Xc ¼
6bf t f 6bf t f þ hw t w
d
ð8Þ
2
bf t f hw 3 2 2 ð6bf t f þ bf hw t w þ 12hw t w d Þ 24 ð6bf t f þ hw t w Þ ! 2 3 ð6bf t f Þ hw 3 hw t w 2 bf t f þ d 24 12 ð6bf t f þ hw t w Þ
C wðNguyenÞ ¼ C wðflatÞ þ Iw X c dav r
ð9bÞ
ð10Þ
It is generally noticed that Cw in the derived expressions for both Moon et al. [3] and Nguyen et al. [4] is a quadratic function of the corrugation depth d. Another approximation is made by Nguyen et al. [4], to account for the cross-section change along the plate girder length, was to consider the warping constant as the average value at d = 0 and at d = dmax. In this case, the average ffiffi . The final form of Eq. (10) depth of corrugation would be dav r ¼ dpmax 2 would be as follows
X c;max dmax 2
ð11Þ
In this part, the location of the centroid about X and Y-axes for un-symmetrical CWG as well as the second moments of inertia about both axes are determined. The web rigidity will be considered in the derivation first, and then the case of ignoring the corrugated web contribution will be applied. Fig. 1(b) shows the cross section configuration of un-symmetrical CWG where the location of the centroid is determined using the simplified geometrical analysis as follows
Qy ; A
Y¼
Qx A
ð12aÞ
where Qx and Qy are the first moments of areas about the X and Y axes, respectively. A is the cross-sectional area and is given as
A ¼ b1 t 1 þ b2 t 2 þ ðhw t f Þt w
ð12bÞ
Using Eqs. (12a) and (12b), the coordinates of the centroids are calculated as
X¼
hw t w d; A
Y¼
ð15aÞ
Iy;c
ð15bÞ
2b1 t 1 þ hw t w hw A 2
b3 t
1 1 2 2 where A1 = b1 t1, A2 = b2 t2, Aw = (hw tf)tw, Iy1 ¼ 12 and Iy2 ¼ 12 . By ignoring the web contribution in the above mentioned equations, the modified values can be given as follows
A1 hw ¼ lhw ; X ¼ 0; Y ¼ A1 þ A2 A1 A2 2 h and Iy;c ¼ ðIy1 þ Iy2 Þ Ix;c ¼ A1 þ A2 w
ð16Þ
Based on the equation proposed by Samanta and Mukhopadhyay [10], the shear modulus of the corrugated plate is defined in Eq. (1). Similar concept was suggested by Ibrahim et al. [2] where it was suggested to take the shear deformation into account when calculating the vertical deflection of beam using the corrugated web. The pure torsional constant shall be adopted in this study using the same value as in the case of plate girders with flat webs. Therefore, Jc is given by the equation
Jc ¼
1 ðb1 t 31 þ b2 t 32 þ hw t 3w Þ 3
It has been suggested by Trahair [17] and Nethercot and Trahair [18] that an easily calculated measure of the un-symmetrical I cross-section with different upper and lower flanges properties can be given as
Iy1 hw Iy1 þ Iy2
ð18Þ
where q is the unsymmetry parameter and ysc is the distance from the center of the lower flange to the shear center. This approach will be adopted herein since the web is ignored and the cross-section can be treated as the case of webless beam. It is presumed that the shear flow is distributed equally all over the web height as shown in Fig. 2(a) and is given by qw ¼ hVw , where qw is the shear flow of the corrugated web and V is the shear force acting on the cross-section. Generally, the shear flow q due to the change of the bending normal stress can be obtained using the relation
q¼
V Ix
Z
x 0
ytf dx
ð19Þ
using the above equation, the shear flow distribution across the flange can be determined as shown in Fig. 2(a). Due to the corrugation depth d, the unbalanced shear forces of the flanges FH1 and FH2 are generated. The value of these flange forces can be determined as the sum of the shear flows acting on the flanges and can be expressed as
ð13Þ
The values of the second moments of inertia Ix,c and Iy,c of the un-symmetrical CWG about the strong and weak axes, respectively, can be obtained using the equations
ð17Þ
3.2. Location of shear center of un-symmetrical CWG
Y sc ¼ qhw ¼
3. Cross-section properties of un-symmetrical CWG
X¼
2 hw 3 4A1 A2 A2w 4Aw þ A 12 ðA1 þ A2 ÞAw 2 ¼ ðIy1 þ Iy2 Þ þ d A
Ix;c ¼
3.1. Shear modulus and pure torsional constant
Substituting for Cw(flat), Iw and Xc, the warping constant Cw(Nguyen) can be rewritten as
C wðNguyenÞ ¼ C wðflatÞ þ Iw
ð14Þ
A
ð9aÞ
and this expression can be written in another form as
C wðNguyenÞ ¼
A
b3 t
This expression gives smaller value than d given by Moon et al. [3] which neglects the web contribution. Nguyen et al. [4] neglected the accordion effect between the flanges and the web when calculating the warping constant for symmetric CWG while it was considered based on the full cross-section geometrical configuration. The expression for Cw in this case is defined as
C wðNguyenÞ ¼
Iy;c ¼
Z
F H1 ¼
V V A1 ðhw YÞd and F H2 ¼ A2 Yd Ix Ix
ð20Þ
FH1 and FH2 are proportional to the corrugated web depth d. The shear center location of this cross-section can be determined using
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(a)
(b)
Fig. 2. Shear flow and shear force acting on un-symmetric CWG due to bending: (a) shear flow distribution, and (b) shear force distribution and shear center position.
the moment equilibrium. There is no cross-section twisting that occurs when the applied load V passes through the shear center. The shear center location of the un-symmetrical cross-section of CWG can be obtained as
VX C ¼ F H1 ð1 qÞ hw þ F H2 q hw
ð21Þ
Substituting Eq. (20) into the above equilibrium equation, the shear center location can be determined as
V V VX C ¼ A1 ðhw YÞð1 qÞdhw þ A2 Y qdhw Ix Ix
ð22Þ
Finally, the location of the shear center can be determined by substituting for Y from Eq. (16) into Eq. (22), and this location will be positioned as shown in Fig. 2(b) to the outside of the corrugated web at a distance XC
XC ¼
2 hw A1 A2 d Ix A1 þ A2
ð23aÞ
Since the web is ignored in this case, by substituting for Ix in Eq. (16) into the above equation it will give the location of the shear center as
2
hw XC ¼ A1 A2 A1 þA2
2
hw
A1 A2 A1 þ A2
d¼d
2
h2w 12
hw 6bf tf þ hw tw
bf t f bf t f 6bf t f d¼ d 2bf t f 6bf t f þ hw t w
Since the open section is made up of thin plate elements, the computation of the cross-section warping constant may be greatly simplified by considering the linear variation of the unit warping properties. Firstly, the cross section is considered as a series of connected thin plate elements [11]. Each plate element has a length Lij and thickness tij, then the normalized unit warping at point i of any element ij can be given as
W ni ¼
n 1 X ðW oi þ W oj Þt ij Lij W oi 2A 0
ð23bÞ
ð24Þ
where Wni = qoi Lij and qoi is the distance from the shear center to the element. Fig. 3 shows the direction of paths for calculating the normalized unit warping and the warping constant of the unsymmetrical section. Using Eq. (24) and calculating the path as shown in Fig. 3, the normalized unit warping at each point can be expressed as
b2 þa 2 ¼ qhw d þ a ¼ ð1 qÞhw d þ a
W n1 ¼ qhw
ð25aÞ
W n2 W n3
ð25bÞ ð25cÞ
W n4 ¼ ð1 qÞhw
This is the same value obtained by Moon et al. [3] for the case of symmetric CWG and ignoring the web. If the contribution of the corrugated web is considered for the case of symmetric CWG, then replacing Ix in Eq. (23a) using the value of Ix,c(Nguyen) in Eq. 7(a), it will give XC as
Xc ¼
3.3. Warping constant of un-symmetrical CWG, CW,uc
W n5 W n6
b1 þa 2
b2 ¼ qhw þ a 2 b1 ¼ ð1 qÞhw þ a 2
ð25dÞ ð25eÞ ð25fÞ
ð23cÞ
This is the same result obtained before by Nguyen et al. [4]. As a result, the shear center when ignoring the web is at a distance of 2d from the centerline of the two flanges and at a distance of qhw from the center of the bottom flange as shown in Fig. 2(b). The location of the shear center will be used next to calculate the warping constant of the unsymmetrical CWG, CW,uc. From here on, for the rest of the study, the web rigidity and area will be ignored to consider for the accordion effect of the corrugated web.
Fig. 3. Direction of paths for calculating the warping constant CW,u of un-symmetric CWG.
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where q is the un-symmetry parameter indicated before in Eq. (18), as well another additional parameter is indicated as 2
a¼
hw t w ð1 2qÞd 2A
ð25gÞ
this parameter represents an additional term due to unsymmetry of the cross-section. This term did not show up in the derivation by both Moon et al. [3] and Nguyen et al. [4] when they tackled the same problem due to the symmetry of the cross-section in these cases. Considering q = 0.5 in the above equations will result in the same expressions developed by Moon et al. [3] for Wni values. The warping constant CW,uc can be obtained by integrating the normalized unit warping Wn across the entire cross-section, and CW,uc can be expressed by
C W;uc ¼
n 1X ðW 2ni þ W ni W nj þ W 2nj Þt ij Lij 3 0
3
2
hw t w 2 d 3
ð27aÞ
This expression maybe rewritten in another form after substituting for a as 2 ÞIy;c hw
C W;uc ¼ qð1 q
3 3hw t w h tw 2 ð2q 1Þ2 w d þ 4qð1 qÞ þ 4 A 12
ð27bÞ
To get an easier and shorter form for the same expression, it can be written as 2
2
C W;uc ¼ qð1 qÞIy;c hw þ RIw d
C W;uc:av r ¼
1 aþb
Z
a
C W;uc:max dz þ
0
1 aþb
Z
aþb
FðC W;uc Þdz
2
¼ C W;uf þ RIw d
and R ¼ 4qð1 qÞ þ wð2q 1Þ2 3hw tw where w ¼ 4 A
where C W;uc:max ¼ C W;uf þ KIw dmax
4 Z2 2 FðC W;uc Þ ¼ RIw dmax Z b b
C W;uc ¼
C W;uc ¼ C W;uf þ
Since the corrugation depth d of the CWG is varying along the length of the plate girder, it has to get an average value of the corrugation depth to simplify the calculations. For example Moon et al. [3] suggested the value of davr as given in Eq. (2), where the expression is based on averaging the area enclosed by the corrugated web, (a + 0.5b)dmax, over the projected length of the corrugated plate, a + b, under this area. Nguyen et al. [4] calculated an average value for both Iy and Cw(Nguyen) at corrugation depths equal to zero and at dmax, as given in Eqs. (9–11). These assumptions are mainly based on rational suggestions. In this part, an attempt to get an accurate value of davr is approached based on considering the cross-sectional variation along the longitudinal axis Z of the CWG. For the longitudinal fold
2 þ C W;uf þ RIw dmax
a b 1 2 2 C W;uf þ RIw dmax þ C W;uf þ RIw dmax aþb aþb 3
ð28bÞ
3.4. Average depth of corrugation in un-symmetrical CWG, davr
!
ð30Þ
where the coordinates of this function is shifted along Z-axis as shown in Fig. 4(b) and the value of Z ranges from 0 to b. Substituting of Eq. (30) into Eq. (29) would result in the final form of the warping constant of un-symmetrical CWG taking into consideration the geometrical variation of the web depth and is given by
Hence, finally
It is worth mentioning that the first part of the expression, qð1 qÞIy;c h2w , is the same value of warping constant of un-symmetrical plate girder with flat web CW,uf, and the second part is the additional contribution of the corrugated web effect to the cross-section warping constant. Eq. (28a) is written finally in the same format as Eqs. (3a) and (10), with the main change is the addition of factor R which represents the change of crosssection unsymmetry as well as the ratio of the web area to the cross-sectional area. To validate the new expression with previous studies, for the case of symmetric CWG where q equals 0.5 then R equals 1.0 and consequently will lead to the same results in Eq. (3a) by Moon et al. [3].
ð29bÞ
The value of CW,uc.max is a constant of d. Fig. 4(b), shows the variation of the quadratic function of d which represents the change of the warping constant along the inclined fold projection b, and this geometric function is derived as
ð28aÞ
ð28cÞ
ð29aÞ
a
2
ð26Þ
Substituting Eqs. (25a)–(25f), the warping constant of unsymmetrical CWG can be given as
C W;uc ¼ qð1 qÞIy;c hw a2 A þ ½1 3qð1 qÞ
with length a, the value of the warping constant is at its maximum with a maximum corrugation depth dmax. For the inclined fold with projected length b, the value of the warping constant varies from a maximum at point 1 to a minimum equal to CW,uf at point 2 then to a maximum again at point 3, as shown in Fig. 4(a). The average CW,uc for the inclined fold would be in-between the minimum value of CW,uf and the maximum value of CW,uc.max. The accurate average corrugation depth and average warping constant would result by integrating these variables along the longitudinal axis of the girder. For the warping constant along the projected length of the inclined fold b, the only variable in Eq. (28) is the depth of corrugation d. Hence, the integration would be in the form
a þ ðb=3Þ 2 RIw dmax aþb
ð31Þ
ð32Þ
and the average depth in this case would be defined as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ ðb=3Þ dmax dav r ¼ aþb
ð33Þ
Fig. 5(a) and (b) shows the previous Eq. (33) in comparison with similar expressions previously studied, where C = a and C = 0.8a in these figures, respectively. The figures cover the range of corrugation angle h from 20° up to 70°, so as to cover all practical ranges of web inclination angle. The ratio between average corrugated depth to maximum corrugated depth is considered constant with a value of 0.707 by Nguyen et al. [4], whereas it ranges from 0.758 to 0.873 and from 0.785 to 0.893 by Moon et al. [3] for the case of C = a and C = 0.8a, respectively. In the current study, these ratios range from 0.823 to 0.911 and from 0.845 to 0.926, respectively. The current study gives higher value of davr/dmax than in Moon et al. [3] by about 6% in average, but it considers the variation of the warping constant over the longitudinal axis of the plate girder. The relation between the value of CW,uc./CW,uf and 2dmax/b2 is plotted in Fig. 6, where the increase in the warping constant CW,uc. in un-symmetrical CWG can reach up to 50% over the warping constant of plate girder with flat web. With the increase of corrugation depth, the ratio CW,uc./CW,uf of increases. Fig. 7 illustrates the same ratios of CW,uc./CW,uf in a different presentation where in the case of 2dmax/b2 up to 0.45, there is a negligible effect for changing the parameter q on the ratio of CW,uc./CW,uf and with the increase of 2dmax/b2 over 0.45, the effect of q magnifies. The reason is the effect of factor R in Eq. (28) which contributes significantly in the value of CW,uc since R is a quadratic function of q.
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(a)
(b)
Fig. 4. (a) Corrugated web depth profile and warping constant variation and (b) integration function along the inclined fold projection.
4. Lateral torsional buckling capacity of un-symmetrical CWG The combination of un-symmetrical steel cross-section in conjugate with the corrugated web is quite sophisticated especially for the case where the girder is subjected to lateral torsional buckling. In this study the girders are investigated under uniform moment and the boundary conditions are both simply supported in flexure and torsion. The expression used for calculating the elastic lateral torsional buckling for the un-symmetrical CWG, Mcr,cw, will be considered the same as in calculating the elastic lateral torsional buckling strength for un-symmetrical plate girder with flat web by Galambos [12] and Chen and Lui [16], with the proposed new warping constant formulas and ignoring the corrugated web contribution in the moment of inertia calculations
(a)
Mcr;cw ¼
p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Lb
EIy;c Gc J c
1 þ W1 þ W2 þ W2 ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi EC W;uc ; with W 1 ¼ Lb Gc J c
p
sffiffiffiffiffiffiffiffiffi
W2 ¼
pbx EIy;c 2Lb
Gc Jc
ð34Þ
where Lb is the length of the un-symmetrical CWG and W1 represents the warping torsional stiffness term and W2 represents the unsymmetrical parameter where the parameter bx is another coefficient presenting the degree of cross-section un-symmetry. The general expression for bx is defined as
bx ¼
(b) Fig. 5. Comparison between average depth of corrugated web and corrugation angle for: (a) c = a, (b) c = 0.8a.
Fig. 6. Comparison between CW,uc./CW,uf and 2dmax/b2 for different q parameter.
1 Ix
Z
yðx2 þ y2 ÞdA yo
ð35aÞ
A
where yo = (q l) hw as shown in Fig. 8. Kitipornchai and Trahair [13] studied the case of mono-symmetric webless I-beams, which matches the case of CWG when ignoring the web contribution. They derived an equation for the unsymmetry property bx which is given by
bx Iy ¼ ð2q 1Þ þ ðl qÞ hw Ix
ð35bÞ
Fig. 7. Variation of CW,uc./CW,uf and q parameter for different 2dmax/b2 values.
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Fig. 9. Loading and boundary conditions of finite element models. Fig. 8. Webless I-section suggested by Kitipornchai and Trahair [13].
This equation will be utilized in the current research for evaluating the elastic lateral torsional buckling strength for unsymmetrical CWG.
4.1. Verification of lateral torsional buckling capacity of un-symmetrical CWG with proposed equation using finite element analysis In this section, the lateral torsional buckling strength of unsymmetrical CWG, using the new warping constant developed equation, is verified and compared with the results obtained using the finite element method for the same models. Twelve models are considered for the analysis using different geometrical parameters as indicated in Table 1. The finite element analysis is performed using the commercial structural analysis program ‘‘ANSYS V.11’’ [15]. The used element from the program library is the four nodes thin shell element ‘‘Thin shell93’’ which has the capability of both geometrical and material non-linearities. The typical loading and boundary conditions of all models are considered as shown in Fig. 9. The girders are considered same as in the mathematical model which is simply supported both in bending and torsion. Displacements about directions X, Y and Z and rotation about Z-axis are restrained for point (A), whereas for point (B) the displacements about directions X and Y and rotation about Z-axis are only restrained. Displacements in Y-direction for both lines (1) and (2) are restrained and for lines (3) and (4) the displacements in X-direction are the only restrained. The end moments are applied as an equivalent tension and compression forces applied on the bottom and top flanges, respectively. Fig. 10 shows the typical finite element geometry and mesh used in all models, where both the top and bottom flanges are modeled using 10 elements per flange width. Twenty elements are used through the web height in all verification models. The horizontal fold width is modeled using 5 elements whereas the
inclined fold width is modeled using 6 elements. The aspect ratio of any shell element did not exceed 2 so as to obtain accurate results. The simple span of the girder is kept constant in all the verification models at L = 13,600 mm. The geometrical imperfection is considered as L/1000 out-of-plane in the middle of the girder span. The material model of the finite element analysis is considered as bi-linearly elastic perfectly plastic with steel yield stress fy = 355 MPa and ultimate stress fu = 510 MPa. The automatic Arc-length control technique of the program is adopted for all models in conjunction with the Modified Newton–Raphson solution technique. 4.2. Comparison between proposed equation and finite element analysis results Table 2. demonstrates the results of the critical buckling moment from the finite element analysis, MFEA, compared with the critical elastic buckling moment capacity, Mcr,cw, obtained from Eq. (34) and utilizing the derived warping constant CW,uc for unsymmetrical CWG using Eq. (32). The ratios illustrated are relative to the plastic moment capacity MP of the un-symmetrical CWG with web neglected, where the value of MP is expressed as
MP ¼ f y
2A1 A2 hw A1 þ A2
ð36Þ
The ratio of MFEA/ Mcr,cw ranges from 0.9 to 1.055 with an average value of 0.983 as illustrated in Table 2, which is considered as a good agreement between the two methods. Fig. 11 shows the deformed shape at failure for one of the finite element models, CWG-1, where the failure mode is the elastic lateral torsional buckling of the compression flange. This type of failure is typical for all models considered herein. The models are designed geometrically to keep the failure mode within the elastic limit. Figs. 12(a) and 12(b) show the applied end moment relation against top lateral
Table 1 Geometrical configuration of verification and analytical finite element models. Model No.
hw (mm)
tw (mm)
b1 (mm)
t1 (mm)
b2 (mm)
t2 (mm)
dmax (mm)
a (mm)
b (mm)
h (°)
L (mm)
CWG-1 CWG-2 CWG-3 CWG-4 CWG-5 CWG-6 CWG-7 CWG-8 CWG-9 CWG-10 CWG-11 CWG-12
900 900 900 900 900 1200 1500 600 900 900 900 900
6 6 6 6 6 6 6 6 6 6 6 6
300 300 300 300 300 300 300 300 200 240 360 420
15 15 15 15 15 15 15 15 15 15 15 15
200 200 200 200 200 200 200 200 200 200 200 200
15 15 15 15 15 15 15 15 15 15 15 15
60 30 90 60 60 60 60 60 60 60 60 60
180 180 180 240 300 180 180 180 180 180 180 180
160 160 160 160 160 160 160 160 160 160 160 160
36.9 20.6 48.4 36.9 36.9 36.9 36.9 36.9 36.9 36.9 36.9 36.9
13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600 13,600
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Fig. 10. Finite element model and elements mesh arrangement.
Table 2 Comparison between proposed equations and finite element analysis results. Model No.
Mcr,cw (kN m)
MP (kN m)
MFEA (kN m)
M cr;cw MP
M FEA MP
M FEA M cr;cw
CWG-1 CWG-2 CWG-3 CWG-4 CWG-5 CWG-6 CWG-7 CWG-8 CWG-9 CWG-10 CWG-11 CWG-12
395.8 387.3 413.2 398.3 400.3 507.1 625.1 290.2 144.1 218.5 628 966.4
1150.2 1150.2 1150.2 1150.2 1150.2 1533.6 1917 766.8 958.5 1045.6 1232.4 1298.6
392.9 375 404.6 406.7 409.5 470.8 602.3 297.5 129.6 199.7 647.6 1020
0.344 0.337 0.359 0.346 0.348 0.331 0.326 0.378 0.150 0.209 0.510 0.744
0.341 0.326 0.352 0.354 0.356 0.307 0.314 0.388 0.130 0.191 0.525 0.785
0.993 0.968 0.979 1.021 1.023 0.928 0.964 1.025 0.90 0.914 1.031 1.055
Average results = 0.983.
displacement and vertical displacement at mid-span of the model CWG-1 compression flange, respectively. Excessive lateral displacement of the compression flange is clear in the figure due to the elastic twisting of the girder cross-section without any gain in the moment capacity after reaching the critical buckling load. 4.3. Effect of cross-section and corrugated web geometrical configuration on lateral torsional buckling capacity of un-symmetrical CWG The effect of different geometrical parameters such as corrugation depth, dmax, horizontal fold length, a, web height hw, and the un-symmetry parameter, q, on the critical buckling moment will be discussed. The values of these variables are indicated in the models from CWG-1 to CWG-12 listed in Table 1. The variation in non-dimensional Mcr,cw/MP with the change of 2dmax/b2 is shown in Fig. 13. The results of both the finite element analysis and the proposed equation are plotted in the same figure. It indicates that
Fig. 11. Lateral torsional buckling deformed shape of Finite element model CWG-1.
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Fig. 13. Variation of Mcr,cw./MP of proposed equation and FEA for different 2dmax/b2 values.
Fig. 12a. Girder end moments versus top lateral displacement of compression flange Ux for model. CWG-1.
Fig. 14. Variation of Mcr,cw./MP of proposed equation and FEA for different a/twvalues.
Fig. 12b. Girder end moments versus vertical displacement of compression flange Uy for model CWG-1.
the increase of elastic lateral torsional buckling capacity due to the increase of web corrugation depth is about 7% and this is mainly due to the increase in the warping constant value CW,uc. Same result is confirmed by the finite element ratios in the same plot. The effect of changing the ratio a/tw from 30 to 50 results only in a minute increase in the Mcr,cw/MP ratio by only 1% as shown in Fig. 14. The variation of web slenderness ratio hw/tw with critical buckling moment capacity ratio Mcr,cw/MP is shown in Fig. 15 where it is found that with the increase of hw/tw ratio from 100 to 250, the ratio of Mcr,cw/MP decreases by 14%. Referring back to Eq. (36) where the value of MP is proportionally linear with hw/tw, whereas the increase in the moment capacity of the un-symmetrical CWG is not increasing in value by the same linear rate according to Eq. (34). The influence of the un-symmetry parameter, q, is clearly shown in Fig. 16, the ratio of Mcr,cw/MP increases from 0.15 at q = 0.5 to 0.744 at q = 0.9, due to the large increase of Iy which contributes significantly in the elastic lateral torsional buckling capacity of un-symmetrical CWG. The results of the finite element analysis also confirmed the results as shown in the same figure.
Fig. 15. Variation of Mcr,cw./MP of proposed equation and FEA for different hw/tw values.
Fig. 16. Variation of Mcr,cw./MP of proposed equation and FEA for different q parameter values.
4.4. Comparison of elastic lateral torsional buckling capacity between un-symmetrical CWG and corresponding plate girder with flat web Figs. 17–20 show the ratio of non-dimensional ratio Mcr,cw/MFlat against the same geometrical parameters mentioned before, 2dmax/b2, a/tw, hw/tw and q, respectively. MFlat is the elastic lateral
torsional buckling strength of un-symmetrical plate girder with flat web. It can be seen from Fig. 17 that the ratio Mcr,cw/MFlat increases with the increase of dmax where the gain in the moment capacity due to using the corrugated web can reach 11% more than
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the corresponding plate girder with flat web. It is worth mentioning that the maximum value of dmax is chosen as 0.9 of the narrower lower flange outstand, where it is restrained by the maximum flange width as well as by the local buckling of the compression flange. In Fig. 18, the change of a/tw does not exhibit a major change in Mcr,cw/MFlat, only a gain of 1% is obtained in the lateral torsional buckling capacity, the same value as the result obtained previously in Fig. 14. The increase of Mcr,cw/MFlat ranges from 2.7% to 9.2% with the change of hw/tw ratio from 100 to 250, as shown in Fig. 19, respectively. The increase of hw/tw ratio contributes in the increase of web plate moment of inertia and consequently to the increase of warping constant CW,uc as indicated in Eq. (32). Fig. 20 shows the ratio between Mcr,cw/MFlat and the un-symmetry parameter q where this ratio tends to increase and ranges from 1.024 to 1.092. The increase in q value will results in an increase of the warping constant value based on Eq. (28).
Fig. 17. Comparison between Mcr,cw. and Mflat versus different 2dmax/b2 values.
5. Ultimate moment resistance of un-symmetrical CWG according to Eurocode 3 (EN 1993-1-5:2006) Annex D Eurocode EC-3, Part 1-5 [14]: Plated structural elements – Annex D, covers design rules for both symmetrical and unsymmetrical plate girders with both trapezoidal and sinusoidal corrugation profiles. The conventional assumption to ignore the web moment resistance of the girder is considered in the rules of this annex. For a simply supported girder, the lateral torsional buckling is based on the smallest axial resistance of each of the flanges times the distance between the flanges centroids. It is based on the minimum value of the corresponding moment due to tension flange yielding, compression flange yielding or compression flange lateral torsional buckling. Herein, the controlling failure mode is the lateral torsional buckling of the girder and the moment resistance MRd for fully effective flanges is given according to the EC-3 – Annex D as
M Rd ¼
b1 t 1
cM1
vLT f y hw þ
t1þ t 2 2
ð37aÞ
where cM1 is a partial factor representing resistance of member for instability = 1.0 and
vLT ¼
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 uLT þ u2LT bk2LT
1:0 1=k2LT
ð37cÞ
kLT
ð37dÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffi MP ¼ Mcr;cw
where vLT is the reduction factor due to out-of-plane buckling, aLT is an imperfection factor which is considered in this study as 0.76 for buckling curve (d). For all models considered in this research, the value of hw/bf > 2.0; hence buckling curve (d) is utilized. b and kLT,0 are parameters depending on the type of beam section. In one method of design, the EC3 uses conservative values of b = 1.0 and kLT,0 = 0.2 whereas the other less conservative method uses the values of b = 0.75 and kLT,0 = 0.4. These values are also considered for the case of plate girder subjected to pure moment. The effective area of the compression flange is considered based on the flange slenderness parameter kP and should satisfy the following condition to have the flange with a full effective width
b1 =2t 1 pffiffiffiffiffiffi 0:748 28:4e K r
where e ¼
ð38aÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 235=f y and Kr is the buckling factor of the compression
flange outstand plate and is expressed as
Fig. 19. Comparison between Mcr,cw. and Mflat versus different hw/tw values.
ð37bÞ
uLT ¼ 0:5½1 þ aLT ðkLT kLT;0 Þ þ bk2LT
kP ¼
Fig. 18. Comparison between Mcr,cw. and Mflat versus different a/tw values.
Fig. 20. Comparison between Mcr,cw. and Mflat versus different q parameter values.
K r ¼ 0:43 þ
b1 2
þd a þ 2b
!2 P 0:6
ð38bÞ
In this study, all the cross sections investigated are considered to be with fully effective compression flanges thus to depict its behavior due to lateral torsional buckling manifestation without interference with other failure modes. All models, from CWG-1 to CWG-12, are chosen to have elastic lateral torsional buckling by considering the range of buckling parameter kLT between the values 1.16 and 1.71.
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Finite element analysis is performed for these models and the results are shown in Fig. 22. The values of the non-dimensional ratio Mcr,cw/MP are plotted against kLT using the proposed method, finite element analysis and EC3-Annex D, considering different values for parameters b and kLT,0, respectively. The results show that using the EC3 buckling curve with values of b = 0.75 and kLT,0 = 0.4 is more consistent with both the finite element results and the proposed method than using the factors b = 1.0 and kLT,0 = 0.2. The ratio of MFEA/Mcr,cw ranges from 0.88 to 1.05. The ratio between EC3 with the suggested parameters and the proposed method ranges from 0.78 to 1.03. 6. Conclusions Fig. 21. Variation of Mcr,cw./MP of proposed equation and EC3 with kLT (results of elastic buckling analysis).
Fig. 22. Variation of Mcr,cw./MP of proposed equation and EC3 with kLT (results of inelastic buckling analysis).
In Fig. 21, the values of the non-dimensional ratio Mcr,cw/MP are drawn against kLT for all models. Mcr,cw values are evaluated using the proposed method for calculating the elastic lateral torsional buckling moment capacity of un-symmetrical CWG, as well as using (EC3-Annex D) methodology. Both values of parameters b = 1.0, 0.75 and kLT,0 = 0.2, 0.4 are used, respectively. It is clearly shown that the results of Mcr,cw/MP have better agreement between the proposed method and the EC3 design method which utilizes the parameters b = 0.75 and kLT,0 = 0.4 than for the conservative method which uses the conventional parameters of b = 1.0 and kLT,0 = 0.2. It is recommended thus for the case of un-symmetrical CWG design against lateral torsional buckling to use (EC3-Annex D) with the design parameters b = 0.75 and kLT,0 = 0.4, respectively. 5.1. Inelastic lateral torsional buckling capacity of un-symmetrical CWG By using Eqs. (37a–37d), the inelastic lateral torsional buckling strength of CWG can be depicted as well as the elastic lateral torsional buckling capacity mentioned before. To validate using these equations for the inelastic region of un-symmetrical CWG, both of the proposed method, using Eqs. (32) and (34), as well as the finite element analysis are used. The geometrical properties of model CWG-1 cross-section are used in this part of the analysis. The value of the non-dimensional buckling parameter kLT is selected in the range of 0.53–1.71 by varying the length of the girder L from 4000 mm to 13,600 mm, so as to cover both elastic and inelastic ranges.
In this study, the lateral torsional buckling strength of unsymmetrical plate girder with corrugated web (CWG) subjected to uniform pure flexure moment is investigated. Pure torsional constant and shear modulus are presented based on previous work. Different geometrical properties of the un-symmetrical crosssection such as moments of inertia and location of the shear center are determined. Modified forms of warping constants, for the case of symmetric cross-section, and derived by others were modified to a final form for comparison. Based on a proposed equation for the shear center location, the warping constant is derived using numerical methodology. The average depth of the variable trapezoidal corrugated web is determined by integrating the warping constant all over the beam length to obtain the warping constant in its final form. The formula used for calculating the elastic lateral torsional buckling strength of un-symmetrical plate girder with conventional flat web is utilized in this research to calculate the same failure strength for un-symmetrical CWG. The concept of ignoring the corrugated web in the flexure resistance calculation is adopted herein. Finite element analysis is performed taking into consideration different geometrical cross-section and corrugated web parameters. The comparison between the finite element results and the proposed formulas showed a good agreement. The depth of corrugation proved to have a major influence on the critical lateral torsional buckling strength of the CWG. It is found that the un-symmetrical CWG has an elastic lateral torsional buckling capacity up to 11% more than the corresponding plate girder with conventional flat web. Finally, a comparative study is made with the EC3 approach for both the elastic and ultimate strengths of CWG and less conservative design factors in the Eurocode is suggested to be used when using the EC3- Annex D in the design of un-symmetrical CWG. References [1] Elgaaly M, Seshadri A, Hamilton RW. Bending strength of steel beams with corrugated webs. J Struct Eng ASCE 1997;123(6):772–82. [2] Ibrahim Sherif A, El-Dakhakhni Wael W, Elgaaly Mohamed. Behavior of bridge girders with corrugated webs under monotonic and cyclic loading. Eng Struct 2006;28:1941–55. [3] Moon JiHo, Yi Jong-Won, Choi Byung H, Lee Hak-Eun. Lateral-torsional buckling of I-girder with corrugated webs under uniform bending. ThinWalled Struct 2009;47:21–30. [4] Nguyen Ngoc Duong, Kim Sung Nam, Seung-Ryong Han, Kang Young-Jong. Elastic lateral-torsional buckling strength of I-girder with trapezoidal web corrugations using a new warping constant under uniform moment. Eng Struct 2010;32:2157–65. [5] Abbas HH, Sause R, Driver RG. Behavior of corrugated web I-girders under inplane loading. J Eng Mech, ASCE 2006;132(8):806–14. [6] Lindner J. Lateral torsional buckling of beams with trapezoidally corrugated webs. Stab Steel Struct Budapest, Hungary 1990:305–10. [7] Sayed-Ahmed EY. Lateral torsion-flexure buckling of corrugated web steel girders. Proc Inst Civil Eng Struct Build 2005;158(1):53–69. [8] Ibrahim, Sherif A. Comparative study on lateral torsional buckling of tubular plate girders using different web systems. In: Proc 11th ICSGE colloquium, Cairo, Egypt; May 2005.
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