Engineering Structures 32 (2010) 2157–2165

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Engineering Structures

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Elastic lateral-torsional buckling strength of I-girder with trapezoidal webcorrugations using a new warping constant under uniform momentNgoc Duong Nguyen, Sung Nam Kim, Seung-Ryong Han, Young-Jong Kang ∗School of Civil, Environmental & Architectural Engineering, Steel Structural Laboratory, Korea University, 5-1 Anam-dong, Sungbuk-ku, Seoul 136-701, South Korea

a r t i c l e i n f o

Article history:Received 21 October 2009Received in revised form10 March 2010Accepted 11 March 2010Available online 13 April 2010

Keywords:Elastic lateral-torsional buckling strengthShear centerWarping constantTrapezoidal web corrugationsI-girder

a b s t r a c t

While I-girders with trapezoidal web corrugations have been used in various structural applicationsand bridges, lateral torsion–flexure of I-girders with trapezoidal web corrugations still needs to beinvestigated, especially the general formulation for cross-section properties. This paper presents theresults of theoretical and finite element analyses of the lateral-torsional buckling of an I-girder withtrapezoidal web corrugations under uniform moment. By using the numerical equations and method,the location of shear center and warping constant are proposed. The elastic lateral-torsional bucklingstrength is then calculated via proposed cross-section properties. The results are comparedwith previousstudies in the literature. A series of finite element analyses with different corrugation profiles and lengthsare investigated and the results are compared with proposed formulas. Through comparative numericalstudies, the proposed formulas are successfully verified. The effects of corrugation profiles on elasticlateral-torsional buckling strength are also investigated and discussed.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Corrugated steel webs have been widely used in various struc-tures, especially in the bridges due to their several advantages.Firstly, they can be used to replace the stiffened steel plates of theplate girders to prevent out-of-plane displacements. Secondly, cor-rugated steel webs improve the aesthetics of structures and reducethe cost of beam fabrication. Consequently, many researchers havecarried out studies on corrugated steel webs [1–9]. The range ofstudies on the flexural and torsional behavior of the I-girder withcorrugated webs can be summarized as follows. Elgaaly et al. [1]found the fact that the contribution of the web to the ultimatemoment capacity of a beam with corrugated webs is negligibleand the ultimate moment capacity should be based on the flangeyield stress. Abbas et al. [2,3] showed that the I-girder with corru-gated webs in flexure cannot be analyzed using the conventionalbeam theory alone. Under the action of in-plane loads, a torsionalmoment is produced and an I-girder with corrugated webs twistsout-of-plane simultaneously as it deflects in-plane. The in-planebending behavior is therefore analyzed using the conventionalbeam theory, whereas the out-of-plane torsional behavior is an-alyzed as a flange transverse bending problem.Lateral-torsional buckling is one of the main design aspects

of the flexural members composed of thin-walled I-girders [4].In spite of its importance, studies on this behavior for I-girders

∗ Corresponding author. Tel.: +82 2 3290 3317; fax: +82 2 921 5166.E-mail address: yjkang@korea.ac.kr (Y.-J. Kang).

0141-0296/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2010.03.018

with trapezoidal web corrugations are scarce. Lindner [5] studiedlateral-torsional behavior of steel I-girders with trapezoidal webcorrugations. The study found that the torsional section constantJc for an I-girder with trapezoidal web corrugations does notdiffer from those of a beam with flat web and that the warpingsection constant C∗w is different. The study presented a formula tocalculate a warping constant based on the test results. The studyalso concluded that the interaction between local plate bucklingand overall lateral-torsional buckling must be taken into account.Sayed-Ahmed [6] showed that resistance to lateral torsion–flexurebuckling of I-girders with trapezoidal web corrugations is 12%to 37% higher than that of the resistance of plate girders withtraditional plane webs to lateral torsion–flexure buckling. Thus,the equations used to calculate the critical moment of girders withplanewebswould underestimate the capacity of plate girders withcorrugated webs to resist lateral torsion–flexure buckling.Most recently, based on the force method with accordion effect

between the web and flange, Moon et al. [4] suggested that thelocation of the shear center of the cross-section of the I-girderwith trapezoidal web corrugations is located at a distance of 2dfrom the center of the upper and lower flanges. Their study alsoproposed the approximated method for estimating the warpingconstant of an I-girder with trapezoidal web corrugations usingthe concept of the average corrugation depth davg. However, todate, the general formulas for the cross-section properties of an I-girderwith trapezoidal web corrugations have still not been found.Therefore, the aims of this paper are to solve these problems andto consider the effects of corrugation profiles on the elastic lateral-torsional buckling strength of an I-girder with trapezoidal webcorrugations.

2158 N.D. Nguyen et al. / Engineering Structures 32 (2010) 2157–2165

Nomenclaturea Length of flat webb Projection length of inclined panelc Length of inclined paneld Depth of corrugationdmax Maximum depth of corrugationdavg Average depth of corrugation from result of Moon

et al. [4]Lo Corrugation wavelengthbf Width of flangetf Thickness of flangetw Thickness of webteq Web equivalent thickness from result of Sayed-

Ahmed [6]hw Height of corrugated websE Young’s modulus of elasticityν Poisson’s ratioG Shear modulus of elasticity of flat platesGc Shear modulus of elasticity of corrugated platesJc Pure torsional constant of I-girder with trapezoidal

web corrugations(xc, y′c) Coordinates of the centroid C of I-girder with

trapezoidal web corrugations in the x–y′ axes(xic, y′ic) Coordinates of the centroid of ith element of I-girder

with trapezoidal web corrugations in the x–y′ axes(xi, yi), (xj, yj) Coordinates of the points i and j in the x–y

axesIx,c, Iy,c Moments of inertia of I-girder with trapezoidal web

corrugations about strong axis (x-axis) and weakaxis (y-axis), respectively

Ixy,c Product of inertia of I-girder with trapezoidal webcorrugations

Ai Area of ith elementA Total area of whole cross-section(Xo, Yo) Coordinates of the shear center S in the x–y axes(Iwx,c, Iwy,c) Geometric properties of I-girder with trape-

zoidal web corrugationsρij Distance from the centroid C to the elementρoi Distance from the shear center S to the elementtij Thickness of plate element (i− j)Lij Length of plate element (i− j)Xs Distance from the shear center S to the center of

upper and lower flanges OWni Normalized unit warping at point iCw,c Warping constant of I-girder with trapezoidal web

corrugationsCw,co Warping constant of I-girder with trapezoidal web

corrugations from result of Moon et al. [4]C∗w Warping constant of I-girder with trapezoidal web

corrugations from result of Lindner [5]Cw,flat Warping constant of I-girder with flat webs [12]Mocr Elastic lateral-torsional buckling strength of I-girder

with trapezoidal web corrugationsM∗ocr Elastic lateral-torsional buckling strength of I-girder

with trapezoidal web corrugations from result ofLindner [5]

M∗∗ocr Elastic lateral-torsional buckling strength of I-girderwith trapezoidal web corrugations from result ofSayed-Ahmed [6]

M∗∗∗ocr Elastic lateral-torsional buckling strength of I-girderwith trapezoidal web corrugations from result ofMoon et al. [4]

Mocr,flat Elastic lateral-torsional buckling strength of I-girderwith flat webs [12]

Mocr,FEM Elastic lateral-torsional buckling strength of I-girderwith trapezoidal web corrugations from result ofFEM

(U1,U2,U3) Displacements about directions 1, 2, 3, respec-tively

(R1, R2, R3) Rotations about directions 1, 2, 3, respectively

a b

Fig. 1. (a) Dimensions of cross-section, (b) Dimensions of corrugation profiles.

In this study, using the numerical equations and numericalmethod [10,11], formulas are proposed to calculate the locationof centroid, moments of inertia and product of inertia of an I-girder with trapezoidal web corrugations. The location of shearcenter and warping constant are then derived. The elastic lateral-torsional buckling strength under uniform moment is determinedby taking the average elastic lateral-torsional buckling strengthat d = 0 and d = dmax using the proposed formulas. Thecritical moment used in this study is the same simple formulaof lateral-torsional buckling strength of an I-girder with flatwebs [12]. A series of finite element analyses are performed. Theproposed values obtained from this study are verified throughcomparisons with finite element analyses and other numericalresults in the literature. Through comparative studies, the elasticlateral-torsional buckling strength of an I-girder with trapezoidalweb corrugations with proposed cross-section properties underuniform moment is successfully verified. Finally, the effects ofcorrugation profiles on elastic lateral-torsional buckling strengthare investigated and compared with other studies and finiteelement analyses.

2. Shear modulus and pure torsional constant

Generally, the shearmodulus of corrugatedwebs is smaller thanthat of flat plates [4]. In this study, the formula for calculatingshear modulus of corrugated webs proposed by Samanta andMukhopadhyay [13] is adopted as follows

Gc =a+ ba+ c

G (1)

where G is the shear modulus of the flat plates and (a + b) is theprojection length of the actual length (a+ c).Fig. 1 shows the geometry data of cross-section and corrugation

profiles.Using the study [5], the pure torsional constant Jc of a steel I-

girder with trapezoidal web corrugations does not differ from abeam with a flat web. Therefore, Jc is given by

Jc =2bf t3f + hwt

3w

3. (2)

3. Lateral-torsional bucking of I-girder with trapezoidal webcorrugations

Using the numerical equations and numerical method de-scribed below, the location of centroid (xc, y′c), moments of inertia(Ix,c, Iy,c) and product of inertia (Ixy,c) of the I-girder with trape-zoidal web corrugations are derived. Next, the location of shearcenter is studied. The warping constant of the I-girder with trape-zoidal web corrugations is then determined. Finally, the elasticlateral-torsional buckling strength is obtained via section proper-ties.

N.D. Nguyen et al. / Engineering Structures 32 (2010) 2157–2165 2159

Fig. 2. The location of the centroid C in x–y′ axes.

3.1. The location of centroid of I-girder with trapezoidal webcorrugations

The location of centroid C of the I-girder with trapezoidal webcorrugations is determined by considering a section comprisedof a series of interconnected plate elements of areas Ai [10].The centroid C is defined as a point in the x–y′ plane that hascoordinates in the x–y′ axes as shown in Fig. 2.

xc =Qy′A

(3a)

y′c =QxA

(3b)

A =n∑i=1

Ai (3c)

Qx =∑(∫

Aiy′dA

)=

∑(Qx)i =

∑Aiy′ic (3d)

Qy′ =∑(∫

AixdA

)=

∑(Qy′)i =

∑Aixic (3e)

where Qx and Qy′ are the first moments of area about the x and y′-axes, respectively. Ai is the area of ith element and A is total area ofwhole cross-section.Using Eqs. (3a)–(3e), the coordinates of centroid C are given by

xc =twhwd

2bf tf + twhw(4)

y′c = 0 (5)

where (xc, y′c) are the coordinates of the centroid C of the I-girderwith trapezoidal web corrugations in the x–y′ axes.

3.2. Moments of inertia and products of inertia of I-girder withtrapezoidal web corrugations

The determination of moments of inertia (Ix,c, Iy,c) and productof inertia (Ixy,c) can be obtained in a similar way by considering thesection comprised of a series of interconnected plate elements [11].Considering a given element ij of thickness tij and length Lij, thecross-sectional properties Ix,c , Iy,c , and Ixy,c are given by

Ix,c =13

b∑0

(y2i + yiyj + y2j )tijLij (6a)

Iy,c =13

b∑0

(x2i + xixj + x2j )tijLij (6b)

Ixy,c =13

b∑0

(xiyi + xjyj)tijLij +16

b∑0

(xiyj + xjyi)tijLij. (6c)

Using Eqs. (6a)–(6c), the cross-sectional properties Ix,c, Iy,c , andIxy,c are obtained

Ix,c =12h2wtf bf +

112h3wtw (7)

Iy,c =tf bf (2tf b3f + twhwb

2f + 12d

2twhw)

6(2tf bf + twhw)(8)

Ixy,c = 0. (9)Iy,c is a quadratic function of d and varies when d changes from

d = 0 to d = dmax. Thus, to simplify the calculation, Iy,c is given bythe average value of Iy,c with d = 0 and Iy,c with d = dmax.

3.3. The location of shear center of I-girder with trapezoidal webcorrugations

Using the same procedure as that used in Section 3.2 [11], thelocation of shear center S is computed by

Xo =Ixy,c Iwx,c − Iy,c Iwy,cI2xy,c − Ix,c Iy,c

(10a)

Yo =Ix,c Iwx,c − Ixy,c Iwy,cI2xy,c − Ix,c Iy,c

(10b)

Iwx,c =13

b∑0

(wixi + wjxj)tijLij +16

b∑0

(wixj + wjxi)tijLij (10c)

Iwy,c =13

b∑0

(wiyi + wjyj)tijLij +16

b∑0

(wiyj + wjyi)tijLij (10d)

wk =

j=k∑i=1

ρijLij (10e)

where the summation∑b0 represents the sum of each element in

the cross-section and ρij is the distance from the centroid C to theelement. Fig. 3(a) shows the direction of the path for calculatingthe location of shear center.Using Eqs. (10a)–(10e) and calculating the path as shown in

Fig. 3(a), the values of Iwx,c, Iwy,c and the location of shear centerS of the I-girder with trapezoidal web corrugations are obtainedIwx,c = 0 (11)

Iwy,c =2h2wtf bf d(twhw + 3tf bf )3(2tf bf + twhw)

(12)

Xo =8tf bf d(twhw + 3tf bf )

(2tf bf + twhw)(6tf bf + twhw)(13)

Yo = 0 (14)where Iwx,c and Iwy,c are geometric properties of the I-girder withtrapezoidal webs corrugations. (Xo, Yo) are the coordinates of theshear center S in the x–y axes.The distance from the shear center S to the center of the upper

and lower flanges O is determined asXs = Xo + xc

=(12tf bf + twhw)d6tf bf + twhw

= d+6tf bf

6tf bf + twhwd. (15)

2160 N.D. Nguyen et al. / Engineering Structures 32 (2010) 2157–2165

ab

Fig. 3. (a) Direction of paths for calculating the location of shear center S and warping constant Cw,c of the I-girder with trapezoidal web corrugations; (b) Geometry ofrectangular elements.

Eq. (15) gives a smaller value than the value of 2d obtained fromthe forcemethod [4], since the numerical equations and numericalmethod did not account for the assumption of accordion effectbetween the flange and web.

3.4. The warping constant of I-girder with trapezoidal web corruga-tions

Using the same procedure as that used in Section 3.2 [11], thewarping constant can be obtained. Firstly, the normalized unitwarping at point i for any element ij is given by

Wni =12A

n∑0

(woi + woj)tijLij − woi (16a)

A =n∑0

tijLij (16b)

woi = ρoiLij (16c)

where the term 12A

∑n0(woi + woj)tijLij is the sum effect of all

elements across the section and ρoi is the distance from shearcenter to the element.Using Eqs. (16a)–(16c) and calculating the path as shown in

Fig. 3(a), the values ofWni are obtained as

Wn1 = −hw(−6tf b2f − bf twhw + 2twhwd)

4(6tf bf + twhw)(17a)

Wn2 =3dtf bf hw6tf bf + twhw

(17b)

Wn3 = −3dtf bf hw6tf bf + twhw

= −Wn2 (17c)

Wn4 =hw(−6tf b2f − bf twhw + 2twhwd)

4(6tf bf + twhw)= −Wn1 (17d)

Wn5 = −hw(6tf b2f + bf twhw + 2twhwd)

4(6tf bf + twhw)(17e)

Wn6 =hw(6tf b2f + bf twhw + 2twhwd)

4(6tf bf + twhw)= −Wn5. (17f)

Next, the warping constant Cw,c can now be evaluated byintegrating the Wn curve across the entire cross-section. For theentire cross-section, Cw,c can be expressed by

Cw,c =13

n∑0

(W 2ni +WnjWni +W2nj)tijLij. (18)

Substituting Eqs. (17a)–(17f) into Eq. (18), Cw,c is given by

Cw,c =h2wtf bf (6tf b

3f + twhwb

2f + 12d

2twhw)

24(6tf bf + twhw). (19)

Cw,c is a quadratic function of d and varies when d changes from0 to dmax. Therefore, to simplify the calculation, Cw,c is given by theaverage value of Cw,c with d = 0 and Cw,c with d = dmax.

3.5. Elastic lateral-torsional buckling strength of I-girder withtrapezoidal web corrugations

Lateral-torsional buckling of a steel I-girder with corrugatedwebs is a topic still to be investigated. In this study, the elasticlateral-torsional buckling strength of an I-girder with trapezoidalweb corrugations under uniform moment is investigated. Theboundary conditions are simply supported in flexure and torsion. Itis also assumed that the formula for calculating the elastic lateral-torsional buckling strength of an I-girder with flat web [12] isused in calculating the elastic lateral-torsional buckling strengthof an I-girderwith trapezoidalweb corrugationswith the proposedformulas for cross-section properties.

Mocr =π

L

√EIy,cGc Jc

√1+W 2c (20)

where L is the length of the I-girder with trapezoidal web

corrugations andWc = πL

√ECw,cGc Jc

represents the effect of warpingtorsional stiffness.To reduce the numerical error in calculation procedures,

Eq. (19) will be used directly in calculating Mocr. Therefore, Mocris also a function of d and varies when d changes from 0 to dmax. Todetermine the values ofMocr for design,Mocr is given by the averagevalue ofMocr with d = 0 andMocr with d = dmax.

4. Verification of proposed formulas

4.1. Finite element model and convergence study

Three groups of I-girders with trapezoidal web corrugationswith different corrugation profiles and lengths are modeled inABAQUS software [14]. The geometry data are shown in Table 1.The material properties are Young’s modulus of elasticity E =200,000 N/mm2, shear modulus of elasticity G = E

2(1+ν) , andPoisson’s ratio ν = 0.3. The models are built using four-nodethin shell elements (S4R5). Fig. 4 shows the results of convergenceanalysis of the model C3 in group B with the detailed geometrydata shown in Table 1. When each flange of the beam in Fig. 5is divided into fourteen elements in the direction 1 axis, thedifference between the elastic lateral-torsional buckling strength

N.D. Nguyen et al. / Engineering Structures 32 (2010) 2157–2165 2161

Table 1Geometry data of three groups analysis models.

Group Model No. a (mm) b (mm) dmax (mm) c (mm) tw (mm) hw (mm) bf (mm) tf (mm) L (mm)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

A

C1 180 140 15 143.18 8 1500 300 25 14080C2 180 140 25 148.66 8 1500 300 25 14080C3 180 140 50 172.05 8 1500 300 25 14080C4 180 140 100 244.13 8 1500 300 25 14080C5 180 140 125 286.53 8 1500 300 25 14080

B

C1 330 270 25 274.59 12 2000 500 40 16800C2 330 270 50 287.92 12 2000 500 40 16800C3 330 270 100 336.01 12 2000 500 40 16800C4 330 270 125 367.97 12 2000 500 40 16800C5 330 270 175 442.04 12 2000 500 40 16800

C

C1 350 275 30 281.47 12 2000 500 40 17500C2 350 275 60 300.04 12 2000 500 40 17500C3 350 275 120 365.00 12 2000 500 40 17500C4 350 275 150 406.97 12 2000 500 40 17500C5 350 275 180 453.02 12 2000 500 40 17500

Fig. 4. Convergence test.

Fig. 5. Loading and boundary condition of the analysis model.

using Eq. (20) with proposed cross-section properties and theelastic lateral-torsional buckling strength obtained from finiteelement analysis is only 0.01%. The corrugated webs have thesimilar mesh refinement as the flanges. That is, the corrugatedwebs are divided into 60 elements for all models of group A and40 elements for all models of group B and C in the direction 2 axisof Fig. 5. Therefore, in this study, this type of mesh refinement offinite element models is reasonable to obtain accurate numericalresults.The applied moment at each end is expressed as the tension

and compression forces at the bottom flange and the top flange,respectively. The girders are considered to be simply supported in

flexure and torsion. Point 1 is the hinged end where the displace-ments in directions 1, 2, 3 and the rotation about the direction 3are restrained. Point 2 is a roller end where the displacements indirections 1, 2 and the rotation about the direction 3 are restrained.Edges a and b are restrained in direction 2. The displacements indirection 1 for edges c and d are restrained. The typical loadingand the boundary condition of the analysis models are illustratedin Fig. 5.

4.2. Verification elastic lateral-torsional buckling strength of I-girderwith trapezoidal web corrugations with proposed cross-sectionproperties

Herein, the elastic lateral-torsional buckling strength of themodel, shown in Fig. 5, is carried out in order to assess the proposedformulas. The values Mocr,FEM are calculated from an eigenvaluesanalysis conducted using ABAQUS software [14]. The values Mocrusing Eq. (20) are compared with the results of studies [5], [6] and[4], respectively.Lindner [5] applied the formula used to calculate the elastic

lateral-torsional buckling strength for flat webs to the elasticlateral-torsional buckling strength of an I-girder with trapezoidalweb corrugations (M∗ocr)with the empirical formula of thewarpingconstant C∗w .

C∗w = Cw,flat +cwL2

Eπ2(21a)

cw =(2dmax)2h2w8ux(a+ b)

(21b)

ux =hw2Gatw

+h2w(a+ b)

3(Ix,c + Iy,c)600a2EIx,c Iy,c

(21c)

where Cw,flat is the warping constant of the I-girder with flat webs.The values ofM∗ocr obtained from his research are compared in thisstudy.Sayed-Ahmed [6] assumed that the formula used to calculate

the critical moment of an I-girder with trapezoidal web corruga-tions (M∗∗ocr) is the same as the I-girder with flat webs having webequivalent thickness which is given by

teq =a+ ca+ b

tw. (22)

There is no theoretical evidence for Eq. (22) as it is merely aproposal by the author for lateral stability calculations. The valuesofM∗∗ocr obtained from his research are also compared in this study.Moon et al. [4] suggested a formula to calculate the elastic

lateral-torsional buckling strength (M∗∗∗ocr ) (using Eq. (12) [4]) with

2162 N.D. Nguyen et al. / Engineering Structures 32 (2010) 2157–2165

Fig. 6. Comparison of the elastic lateral-torsional buckling strength of the I-girderwith trapezoidal web corrugations of group A.

Fig. 7. Comparison of the elastic lateral-torsional buckling strength of the I-girderwith trapezoidal web corrugations of group B.

their proposed formulas of cross-section properties. The valuesof (M∗∗∗ocr ) obtained from their research are also compared in thisstudy.Variations of Mocr,FEMMocr

, Mocr,FEMM∗ocr,Mocr,FEMM∗∗ocr

, and Mocr,FEMM∗∗∗ocrwith increas-

ing dmax are plotted in Figs. 6–8. When dmax increases, the differ-ences between Mocr, M∗ocr,M

∗∗ocr,M

∗∗∗ocr , and Mocr,FEM also increase

and the errors between Mocr and Mocr,FEM are the smallest. On theother hand,Mocr gives the best agreement withMocr,FEM.

4.3. Comparison effects of corrugation profiles on elastic lateral-torsional buckling strength of I-girder with trapezoidal web corruga-tions

Figs. 9–11 show the ratios, Cw,cCw,flatand Cw,coCw,flat

(Cw,co is the warpingconstant of an I-girder with trapezoidal web corrugations [4] andCw,flat is the warping constant of an I-girder with flat webs [12]). Itcan be seen that themaximum differences between Cw,c and Cw,flatare 22%, 12%, and 13% respectively,while themaximumdifferencesbetween Cw,co and Cw,flat are 34%, 18%, and 19% respectively.This means that Cw,c gives smaller values than Cw,co when dmaxincreases and thewarping constant of the I-girderwith trapezoidalweb corrugations is larger than that of the I-girder with flat webs.In this study, the same shear modulus is used as that used

in Moon et al. [4]. According to [4], the ratio GcG decreases

with increasing dmax and the shear modulus of the I-girder withtrapezoidal web corrugations is smaller than that of the I-girderwith flat webs. Figs. 9–11 also demonstrate this.

Fig. 8. Comparison of the elastic lateral-torsional buckling strength of the I-girderwith trapezoidal web corrugations of group C.

Fig. 9. Comparison between Cw,c , Cw,co and Cw,flat of group A.

Fig. 10. Comparison between Cw,c , Cw,co and Cw,flat of group B.

The elastic lateral-torsional buckling strength is not only af-fected by the warping constant and shear modulus but also by themoment of inertia about the minor axis Iy,c . The comparisons be-tween Iy,c, Iy,co and Iy,flat (Iy,co is themoment of inertia about themi-nor axis of the I-girder with trapezoidal web corrugations [4] andIy,flat is the moment of inertia about the minor axis of the I-girderwith flat webs [12]) are plotted in Figs. 12–14. It is found that Iy,co isapproximated as equivalent to Iy,flat and its values are not changedwhen d increases, while Iy,c increases and its values are larger than

N.D. Nguyen et al. / Engineering Structures 32 (2010) 2157–2165 2163

Fig. 11. Comparison between Cw,c , Cw,co and Cw,flat of group C.

Fig. 12. Comparison between Iy,c , Iy,co and Iy,flat of group A.

Fig. 13. Comparison between Iy,c , Iy,co and Iy,flat of group B.

Iy,flat. The ratiosIy,cIy,coand WcWco (Wco is the effect of warping torsional

stiffness of the I-girder with trapezoidal web corrugations [4]) areshown in Figs. 15–17. It can be seen that Wc is smaller than Wcoand Iy,c is larger than Iy,co. However, the differences between Iy,cand Iy,co are larger than that between Wc and Wco. This explainswhyMocr in this study is larger thanM∗∗∗ocr [4].Figs. 18–20 show the ratios Mocr

Mocr,flat,M∗ocrMocr,flat

,M∗∗ocrMocr,flat

,M∗∗∗ocrMocr,flat

andMocr,FEMMocr,flat

. It can be seen that the values ofMocr,M∗ocr,M∗∗ocr,M

∗∗∗ocr and

Fig. 14. Comparison between Iy,c , Iy,co and Iy,flat of group C.

Fig. 15. Variation of Iy,cIy,co andWcWcowith dmaxLo of group A.

Fig. 16. Variation of Iy,cIy,co andWcWcowith dmaxLo of group B.

Mocr,FEM are higher than Mocr,flat. The values of the ratio MocrMocr,flat

are found to most closely agree with Mocr,FEMMocr,flat

. The maximumdifferences between Mocr and Mocr,flat are 29%, 20%, and 21% forgroups A, B, and C respectively, while the maximum differencesbetweenM∗∗∗ocr andMocr,flat are 11%, 9%, and 9% for groups A, B, andC respectively.

2164 N.D. Nguyen et al. / Engineering Structures 32 (2010) 2157–2165

Fig. 17. Variation of Iy,cIy,co andWcWcowith dmaxLo of group C.

Fig. 18. Comparison between the elastic lateral-torsional buckling strengths ofthe I-girder with trapezoidal web corrugations and the I-girder with flat webs ofgroup A.

Fig. 19. Comparison between the elastic lateral-torsional buckling strengths ofthe I-girder with trapezoidal web corrugations and the I-girder with flat webs ofgroup B.

5. Concluding remarks

This study reported the elastic lateral-torsional bucklingstrength and general cross-section of an I-girder with trapezoidalweb corrugations under uniform moment. Following previousstudies, the shear modulus and pure torsional constant are

Fig. 20. Comparison between the elastic lateral-torsional buckling strengths ofthe I-girder with trapezoidal web corrugations and the I-girder with flat webs ofgroup C.

presented. Next, using the numerical equations and numericalmethod, the location of the centroid is determined. The momentsof inertia, product of inertia and the location of shear centerare then derived. The distance from the shear center to thecenter of the upper and lower flanges is lower than 2d, whichis suggested by Moon et al. [4]. The warping constant is alsoderived using the numerical equations and numerical methodbased on the proposed formula for the location of shear center.With the proposed cross-section formulas, the elastic lateral-torsional buckling strength of the I-girder with trapezoidal webcorrugations is obtained by using the same formula as that usedfor the I-girder with flat webs [12] and by taking the averagevalue of Mocr with d = 0 and Mocr with d = dmax. A seriesof finite element analyses with different corrugation profiles andlengths as well as results from previous studies’ are used to verifythe proposed formulas. Through comparative numerical studies,the proposed formulas are successfully verified. Finally, the elasticlateral-torsional buckling strength of an I-girder with trapezoidalweb corrugations is not only affected by a warping constant andshear modulus [4] but also by the moment of inertia about theminor axis Iy,c . The value of the elastic lateral-torsional bucklingstrength of the I-girder with trapezoidal web corrugations is about21% to 29% larger than the I-girder with flat webs.

Acknowledgement

This work was supported by the Korea Science and EngineeringFoundation (KOSEF) Grant funded by the Korea government(MEST) (No. R0A-2005-000-10119-0).

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