122 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 C w is 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, w 0 , W n , Figure 5.4) between the two consecutive intersection points of plate elements. As a result, the sophisticated integral form for C w is represented by numerical expression suitable for computer coding.
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122
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