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CHAPTER 5

PROPOSED WARPING CONSTANT

5.1 INTRODUCTION

Generally, lateral torsional buckling is a major design aspect of

flexure members composed of thin-walled sections. When a thin walled

section is subjected to flexure about its strong axis with insufficient lateral

bracing, out- of plane bending and twisting can occur as the applied load

approaches its critical value. At this critical value, lateral torsional buckling

occurs. The equations used to calculate the critical lateral-torsional buckling

strength of the I-girder with flat webs would underestimate the capacity of the

I-girder with corrugated web

Linder (1990) proposed an empirical formula for the warping

constant of I-girder with corrugated web on the basis of test results. The

warping constant of I-girder with corrugated web is larger than that of I-girder

with flat web. Sectional warping constant Cwis determined either by

mathematical integration proposed by Galambos (1968) or by complex

formulas. Computation of warping constant for open thin walled section is

greatly simplified by recognizing the linear variation of unit warping constant

properties (w, w0, Wn, Figure 5.4) between the two consecutive intersection

points of plate elements. As a result, the sophisticated integral form for Cw is

represented by numerical expression suitable for computer coding.

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The cross-section of the beam varies along the span of the beam

due to the corrugation profile. It is difficult to find the warping constant for

varying depths of corrugation.

No design rules are available in Australian/ New Zealand Standards

(AS/NZS-4600:2005) and North American Specifications (AISI-S100:2007)

for calculating moment carrying capacity of I-section with trapezoidal

corrugated web. In order to find lateral buckling moment capacity, elastic

buckling stress for flexural buckling about the y-axis foy and elastic buckling

stress for torsional buckling foz have to be calculated.

It is found that properties such as polar radius of gyration of the

cross section about the centre, radius of gyration, Section modulus, Warping

constant etc varies along the longitudinal direction due to a change in depth of

in change of properties. Due to change in depth

y-axis also changes along the length and the geometric properties are not

constant at all sections.

In this chapter, the procedure to find the warping constant for

lipped I-beam with trapezoidal corrugated web and to locate the shear centre

have been proposed. The proposed warping constant is also validated by

using finite element analysis.

5.2 SHEAR MODULUS & TORSIONAL RIGIDITIES OF

LIPPED I-BEAM WITH CORRUGATED WEB

Generally, the shear modulus of the corrugated plates is smaller

than that of the flat plates. The shear modulus of the corrugated plates used in

this study was proposed by Samanta &Mukhopadhyay (1999). The shear

modulus Gcog of the corrugated plate is defined as

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coga bG G Ga c

(5.1)

where G is the shear modulus of the flat plates and is the ratio of the

projected length (a+b) to the actual length of the corrugated plates (a+c).

Pure torsional constant Jcog of the lipped I-beam with corrugated

web is the same as that of the lipped I-beam with flat webs. Because the pure

torsional constant of a section is equal to the sum of the pure torsional

constants of each individual element, in the case of lipped I-beam with

corrugated web, Jcog can be expressed as the sum of the pure torsional

constants of the two flanges, four lips and corrugated web. Therefore,

(5.2)

5.3 SHEAR CENTER OF LIPPED I-BEAM WITH

CORRUGATED WEB

It is presumed that the shear flow is evenly distributed over the total

depth of the web as shown in Figure 5.1 and it is given by qw=V/hw, where

qw V shear

force acting on the cross-section.

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Figure 5.1 Shear flow distribution Figure 5.2 Shear flow distribution and location of shear center

The shear flow due to the change of the bending normal stress

can be determined by using the relationship.

q = - (V*Qx)/(Ixx) (5.3)

where x is the first moment of the area about the x-axis. From Figure 5.2, it

is found that the unbalanced shear force on the flange Vf is generated due to

the corrugation depth. The magnitude of f can be determined by the sum of

the shear flows acting on the flanges and expressed as

Vf = (V/hw)*d = qw*d (5.4)

f is proportional to the corrugated depth d. For lipped I-beam with flat web

f is equal to zero. Figure 5.2 shows the shear force acting on the

cross-section of lipped I-beam with corrugated web. The location of the shear

center of this cross-section is determined by the moment equilibrium. There is

no twisting of the cross-section, when the applied load passes through the

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shear center. Therefore, the summation of the moment about shear centre is

equal to zero. The location of shear centre is obtained as Xo= - d.

It is found that this shear center is located at a distance of 2d from

, the center of upper and lower flange (Figure 5.2).

5.4 WARPING CONSTANT OF LIPPED I-BEAM WITH

CORRUGATED WEB

The warping constant is determined either by integration forms or

by numerical forms. The open section is made up of thin plate elements.

Warping constant is determined by numerical forms, considering the section

is composed of a series of inter-connected plate elements. If a plate element of

length Lij and thickness tijis considered, then the normalized unit warping at

points i and j of any element (i-j) is given by

Wni=[( ) (woi+woj)tijLij]-woi (5.6a)

Wnj=[( ) (woi+woj)tijLij]-woj (5.6b)

woj=woi oij * Lij (5.6c)

where, oij is the distance between the shear center to the tangent of element ij

(Figure 5.3), wo is the unit warping with respect to shear center and

woiandwojare thecorresponding values of wo at the ends of element i and j

(Figure 5.4)

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Figure 5.3 Coordinates and tangential Figure 5.4 Distribution of Won distances element in an a plate open thin-walled section

Figure 5.5 Direction for path for calculating warping constant of lipped I-beam with corrugated web

Figure 5.5 shows the direction of the path for calculating the

warping constant of the lipped I-beam with corrugated web. Using the

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equation (5.6) and calculating the path as shown in Figure 5.5, the simplified

form Wni of the lipped I-beam with corrugatedweb can be expressed as

(5.7a)

(5.7b)

(5.7c)

(5.7d)

(5.7e)

(5.7f)

(5.7g)

(5.7h)

(5.7i)

(5.7j)

The general formula of Cw of any arbitrary section composed of thin plate is

given by

(5.8)

Cw,cogis obtained by using equation (5.8) and Wni as described in equation

(5.7). It is found that Wni varies along the longitudinal direction due to a

change in d, which results in a change in Cw,cog.The average corrugation depth

davg suggested in this study for considering the change in d. davg is given by

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davg=[(2a+b)*dmax] / [2*(a+b)] (5.9)

Procedure for calculating Cw,cog is as follows

a) Calculation of average corrugation depth davg byusing

equation (5.9)

b) Evaluation of the normalized unit warping at point i and Wni

of the lipped I-beam with corrugated webs by using equation

(5.7 ) with davg in Step (a)

c) Determination of the warping constant of the lipped I-beam

with corrugated webs Cw,cog by using equation (5.8) with Wni

obtained in Step (b)

5.5 LATERAL-TORSIONAL BUCKLING STRENGTH OF

LIPPED I-BEAM WITH CORRUGATED WEB

If a uniformly distributed load or any other transverse load acts on

the I-beam, shear force induced is taken up by web. In the case of the lipped I-

beam with corrugated web, the attachment of corrugated web to the flanges is

not along the center line of the beam. It is attached eccentrically, as the profile

of web varies along the span. Due to this, force derived from shear in the

corrugated web causes out-of-plane transverse bending of the upper and lower

flanges (Figure 5.6).

In this study, uniform bending is adopted to investigate the lateral-

torsional buckling strength. The boundary condition used in this study is

simple support in flexure and torsion.

It is assumed that the formula of the lateral-torsional buckling

strength of the lipped I-beam with flat web can be applied to lipped I-beam

with corrugated webs under uniform bending. The beam under uniform

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bending deflects in-plane without any torsional behavior, even if the shear

center and the center of beam do not coincide.

Figure 5.6 Deformed shape of I-

beam with corrugated

webs under uniformly

distributed loadMoon

J et al. (2009)

Figure 5.7 Deformed shape of I-

beam with corrugated

webs under uniform

bendingMoon J et al.

(2009)

The elastic lateral torsional buckling strength of the beam is expressed as

2( )( )4cr y cog cog TM EI G J I W (5.10a)

,( / )T w cog cog cogW EC G JL

(5.10b)

where L is the length of the beam and WT represents the effect of the warping

torsional stiffness.

5.6 VERIFICATION OF THE PROPOSED EQUATIONS

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In this section, the proposed warping constant of the lipped I-beam

with corrugated webs is verified with finite element analysis.

The warping constant from the FEA Cw,FEM for the lipped I-beam

with corrugated web is calculated by

22 2

, 2 2

( **( * *

cog cogCrFEMw FEM

yy

G JM L LCE I E

(5.11)

where Mcr,FEMis the elastic lateral torsional buckling strength from FEA.

Linder(1990) suggested the following empirical formula for the

warping constant of I-section with corrugated web

Cwlinder = Cw,flat + (Cw*L2 2) (5.12a)

Cw = [(2*dmax)*hw]/[8*ux*(a+b)] (5.12b)

2 3

2

*( ) *( )2* * * 600* * * *

w xx yywx

w xx yy

h a b I IhUG a t a E I I

(5.12c)

where Cw,flat is the warping constant of the I-section with flat web. Equation

(5.12) suggested by Linder(1990) without lip in flanges is compared with the

proposed warping constant in line with Moon (2009).

An Eigen-value analysis is performed by using ANSYS.12 to

evaluate the lateral-torsional buckling strength of the lipped I-beam with

corrugated web. Four node Shell-181 element is used. Figures 5.8, 5.9 and

5.10 show a typical loading and boundary condition of the FE model. The end

moments are applied in the form of compression and tension on the top and

bottom flanges, respectively. The beams are considered to be simply

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supported in the flexure and torsion and the following boundary conditions

are implemented.

Figure 5.8 Loading and Boundary Condition of the FE Model

Figure 5.9 Loading and Boundary Condition of the FE Model at the left end

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Figure 5.10 Loading and Boundary Condition of the FE Model at the right end

Displacements about directions x, y and z(ux, uy and uz) and the

rotation about direction Z ( z) at point A are restrained. Displacement about

directions x and y( ux , uy) and the rotation about direction Z ( z) at point B are

restrained. ux at the line a and b are restrained and uy at the line c and d are

also restrained. To verify the finite element model used in this study, the

lipped I-beam with flat web is modeled and the result is compared with the

theoretical lateral-torsional buckling strength. The dimensions and result of

the analysis are shown in Table 5.1.

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Table 5.1 Dimensions and Result of FE and Theory Moment for flat web

tw

mm hw

mm bf

mm tf

mm L

mm bl

mm tl

mm FE Moment Nmm ×106

Theory Moment Nmm × 106

Error %

1.2 250 100 2 3000 15 2 12.669 13.087 3.29

An appropriate mesh size of 20mm × 20mm is chosen after a mesh

sensitivity analysis in order to get accurate results.

Table 5.2 Dimensions of models with corrugated web

Model No.

a mm

b mm

c mm

dmax mm

tw

mm hw

mm bf

mm tf

mm bl

mm tl

mm L

mm

Aspect Ratio

TCIAE1 60 57.95 60 7.75 15 1.2 300 100 2 15 2 3000 1

TCIAE2 60 51.90 60 15 30 1.2 300 100 2 15 2 3000 1

TCIAE3 60 42.42 60 21.21 45 1.2 300 100 2 15 2 3000 1

TCIAE4 60 30.00 60 25.95 60 1.2 300 100 2 15 2 3000 1

TCIAR1 60 20 28.3 10 45 1.2 300 100 2 15 2 3000 0.47

TCIAR2 60 40 56.6 20 45 1.2 300 100 2 15 2 3000 0.94

TCIAR3 60 60 84.85 30 45 1.2 300 100 2 15 2 3000 1.41

TCIAR4 60 80 113.1 40 45 1.2 300 100 2 15 2 3000 1.89

Table 5.2 shows the dimensions of the models with corrugated web.

In the TCIAE series models, the corrugation angle is varied from 15º to 60º

with increase in dmax keeping aspect ratio (a:c) as one. In the TCIAR series

models, the aspect ratio is varied by keeping a corrugation angle of 45º with

an increase of dmax. Figures 5.11 and 5.12 show the lateral torsional buckling

shape obtained from FEA for TCIAE4 and TCIAR2 models respectively.

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Table.5.3 shows the comparison of warping constant of Cw,cog,

Cw,linderand Cw,FEM. Figures 5.13 and 5.14 show the comparison of results for

varying aspect ratio and corrugation angle respectively. It is found that Cw,cog

is in good agreement with Cw FEM, while Cw,linder generally overestimates the

warping constant of the lipped I-beam with corrugated webs. The difference

between Cw FEMand Cw,linder

(Figure 5.14) and also the difference between Cw FEMand Cw,linder increases

with increasing aspect ratio(Figure 5.13).

Table 5.3 Comparison of warping constant of proposed method, FEA and Linder

Model No.

Cw,cog

mm6× 109 Cw,linder

mm6× 109 Cw,FEM

mm6× 109 Cw,FEM/Cw,cog Cw,FEM/Cw,linder

TCIAE1 14.118 18.766 14.144 1.002 0.754

TCIAE2 14.169 29.953 14.353 1.013 0.479

TCIAE3 14.551 47.835 14.883 1.022 0.311

TCIAE4 15.175 71.196 16.087 1.060 0.226

TCIAR1 14.098 24.431 13.920 0.929 0.570

TCIAR2 14.471 44.926 14.882 1.028 0.331

TCIAR3 15.317 70.639 16.060 1.049 0.227

TCIAR4 16.545 98.756 17.576 1.062 0.177

Mean 1.028 0.385

Standard Deviation 0.027 0.200

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Figure 5.11 Lateral-torsional buckling shape from finite element analyses for TCIAE4

Figure 5.12 Lateral-torsional buckling shape from finite element analyses for TCIAR2

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0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.5 1 1.5 2

C

Aspect Ratio (a:c)

Cw,FEM/Cw,cog Cw,FEM/Cw,linder

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

15 30 45 60

Cw

,FE

M/C

w,c

og o

r C

w,F

EM/C

w,li

nder

Corrugation Angle

Cw,FEM/Cw,cog Cw,FEM/Cw,linder

Figure 5.13 Comparison of the warping constant of corrugated web with respect to aspect ratio

Figure 5 14 Comparison of the warping constant of corrugated web with respect to corrugation angle

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Linder(1990) proposed an equation for warping constant under a

transverse loading. Lateral displacement occurs along the flange locations of

the corrugated web lipped I-beam under such transverse loading, because the

shear forces in the corrugated web in inclined panel should be divided into a

lateral component in the out-of-plane direction due to the corrugated shape.

(1990) work includes the effect of lateral

displacement component. Therefore, Linder(1990) results do not match with

proposed warping constant result under uniform bending condition.

5.7 CONCLUSION

A simplified method for estimating the warping constant of the

lipped I-beam with corrugated web is evaluated. In this study, the depth of

corrugation varies along the span of the beam for which the average depth is

taken for calculating the warping constant.

from the middle of the

upper and lower flanges.

Numerical validation has been carried out to verify the

appropriateness of the warping constant proposed in this study by using the

FEA software ANSYS. 12 and it is found that the numerical results are quite

closer to the proposed method.