Effect of secondary warping on lateral buckling

Effect of secondary warping on lateral buckling Mario M. Attard Department of Civil Engineering, Monash University, Clayton, Victoria, Australia

Ray Lawther School of Civil Engineering, University of New South Wales, Kensington, New South Wales, Australia (Received July 1988; revised December 1988)

In the lateral buckling analysis of beams, secondary warping or thickness warping is usually ignored. For angle and tee beams where the contour warping is zero, the secondary warping is shown to have a significant effect on the lateral buckling load. Simply supported tee beams under moment gradient are investigated. Angles and tee sections are shown to be prone to 'localized' torsional buckling and a simple formula is presented to calculate this load. The Laplace equation for the warping distribution is solved by a direct application of finite element plate bending elements to obtain more precise values for the secondary warping in tee sections, and this compares well with approximate formulae available in Bleich (1952).

Keywords: lateral buckling, secondary warping, thickness warping, tee beams, angles The torsional response of most thin-walled open beams can be greatly influenced by restraining the cross-section's ability to warp. The warping of thin-walled open crosssections can be approximated by two components: the warping of the contour or section profile; and the warping across the thickness of the plates making up the cross-section (Figure 1). The latter is referred to as the thickness warping or secondary warping. For most thin-walled open cross-sections, the contribution of secondary warping to the torsional stiffness is insignificant when compared with contour warping. Hence, the secondary warping is usually ignored when considering either the static response or the stability of thin-walled open beams. One exception pointed out by Bleich ~, is when considering the torsional buckling of extremely short struts of cross-sections in which the centre lines of the plate elements intersect at the shear centre, such as an angle, cruciform or tee section. For these cross-sections, the contour warping is zero while the secondary warping leads to small but finite value for the warping constant, which can be an important consideration. This paper examines the importance of secondary warping on the lateral buckling of tee beams under moment gradient. When the flange is in tension under bending, it is demonstrated that a localized torsional buckle can be critical and that the inclusion of secondary warping in the analysis can have a significant effect. 112

Eng. Struct. 1989, Vol. 1 1, April

Approximate formula for secondary warping The Saint Venant solution for the warping distribution of prismatic elastic beams under uniform torsion is based on the assumption of infinitesimal displacements, nondeformability of the cross-section and the existence of a straight axis of twist parallel to the centroidal axis. For a cross-section described as a region R, bounded by a curve P, the warping distribution, denoted by ~o,is shown to satisfy the Laplace equation V2w = 0 in R

(1)

with the following boundary condition (Y - Ys - ao~/Oz)dy + (z - z, + &o/~y)dz = 0 on e (2) A right-handed orthogonal coordinate system has been adopted with y and z being the principal centroidal axes of the cross-section. The coordinates of the shear centre are y, and z,. The following conditions relating to axial and bending equilibrium apply to the warping distribution r o ) dR

0

(3)

fR ~ z dR = 0

(4)

~coy

(5)

dR

=0

0141-0296/89/02112-07/$03.00 © 1989 Butterworth & Co (Publishers) Ltd

Secondary warping effect on lateral buckling: M. M. Attard and R. Lawther For non-uniform torsion, further simplifying assumptions are necessary. The Vlasov z assumption of zero shear strain at the middle surface of elements forming an open, thin-walled cross-section leads to an approximate formula for the contour warping. By further assuming that each element behaves as a thin shelP '4 satisfying the Kirchoff assumption that straight lines remain normal to the middle surface during deformation, the secondary warping can be approximated by o9 = - n r(p)

L V

I

2 5 4 m m (lOin)

l

_i -I

I

19 mm (0.75 in) 3 3 2 mm

( 13.07 in)

IZmm

(0.47 in)

(6)

where n is the distance measured normal to the contour, r(p) is the perpendicular distance from the normal at the contour to the shear centre and p is the profile or contour coordinate. The secondary warping is approximated by a linear displacement across the thickness of the crosssection, being zero at the middle surface (see Figure 1). A measure of the influence of warping restraint on-the torsional stiffness is given by the warping constant, defined as I~ = f o92 dR JR

(7)

Figure 3 Dimensions of a 345BT70 tee section

Figure 2 shows the formulae for the warping constant derived using equations (6) and (7) for cross-sections that display zero contour warping. In Appendix A, the warping distribution and warping constant for a 345BT70 tee beam (see Figure 3) are calculated by solving equations (1) and (2) using finite elements usually used for the analysis of the bending of isotropic plates. The percentage difference between the warping constant calculated using the approximate formula and the finite element analysis is about 3%. Localized torsional buckling of tee beams In the design of tee beams, lateral buckling and torsional buckling are important considerations because of the low warping stiffness and the possible reduction in torsional stiffness due to the unsymmetric nature of the bending stresses. The unsymmetric nature of the bending stresses is related to the monosymmetry of the cross-section. These considerations apply to other cross-sections of low warping stiffness such as angles. The equilibrium equations in the buckled state, assuming small displacements and ignoring the effects of initial bending curvature, are given by

Figure 1 Warping of a thin-walled open section L.. " ~

T

i

n

l

d

"

•

b

E I . WS,~ - M ~ = 0

(8)

EI~,dp,:x,,- (GJ - 2fl xM,)~b,x - M= Ws,,, = 0

(9)

in which E is the Young's modulus, G is the shear modulus, J is the torsion constant, I , is the second moment of area about the y axis, M= is the bending moment about the z axis, ~ is the angle of twist, Ws is the lateral movement of the shear centre in the z direction and//1 is a geometric quantity defined by

T

//1 = ~ Rectongle : Zw: f3d31144 b _j

b

Tee:Zw : f?b 3 + fw3h3 144 36

(yz2 + ya) dR - y,

(10)

Cruciform : .Z'w:(tl3d3 + f~b~}/144

-L

Unequal leq angle: .Z"w =f5 (bS+d 3) 56

Figure 2 Approximate formulae for warping constant for some sections having zero contour warping

where I.. is the second moment of area about the z axis. //1 reflects the degree of monosymmetry and is zero for symmetric sections. If the first term in equation (9) is insignificant, due to a low warping stiffness, then the torsional rigidity is dependent on G J - 2//IM=. This rigidity can reduce to zero if the quantity//1M= is positive. For a tee beam this can occur when the bending puts the flange in tension. Engel and Goodier s, when investigating the buckling of end loaded angle cantilevers, observed buckles that were predominately torsional buckles localized at the root of the cantilever. They postulated that the localized torsional stiffness at the root of the cantilever must be Eng. Struct. 1989, Vol. 11, April

113

Secondary warping effect on lateral buckling. M. M. Attard and R. Lawther

zero and estimated the critical moment by setting GJ 2//1M= to zero. Their estimates matched the experimental results closely. This type of buckle for angles and tee beams has largely been overlooked in the past. An estimate of the critical moment M,, for a postulated 'localized' torsional buckle confined to a very small distance is therefore Mc, =

GJ/2//I

•

\\

i i

"

2.0

/7:057

.

.

.

.

__

(11)

Note that this equation is independent of the length of the beam.

AT:I

0

"

~

_

' .

-

.

.

.

Lateral buckling under m o m e n t gradient For a simply supported monosymmetric beam of length L under uniform bending, the critical moment Mo for lateral buckling derived by Goodier 6 is given in nondimensional form by MoL = ~2 [6 + (3 2 + g 2 + 1//t2) t/2] (ElyyGJ) 1/2

(12)

- 1.0

- 0.5

0.5

0

I. 0

End moment factor,

Figure5 Modificationfactorfornon-uniformbending,

, Equation (17); - - - , i torsional buckle at left support, equation (11 ); - . -, T torsional buckle at right support, equation (11 )

where

( Elyy ~ 1/2

3 = - / / i \G----~]

(13)

is a parameter reflecting the degree of monosymmetry and

g2= EIw

(14)

GJL 2

is a measure of the relative importance of warping in a flexural-torsional buckle. An alternative parameter to equation (13) to measure the degree of monosymmetry for tee beams has been proposed by Kitipornchai and Trahair 7, and is

~h ( Elyy ~1/2 K = -~- ~,G---J-~)

(15)

in which h is the distance between the centroid of the flange and the end of the web. The typical range suggested for K was 0.1-2.5. In this paper the range of J~ examined is 0.5-2.5. The lower value of R is associated with long beams and/or compact sections while the higher value of 2.5 is associated with short beams and/or slender sections. When a simply supported monosymmetric beam is subjected to non-uniform bending, it has been suggested (see the discussion in Kitipornchai and Wang s) that the critical buckling moment can be approximated using a modification factor m applied to Mo, that is Me, = m Mo

(16)

where m = 1.75 + 1.05//+ 0.3//2 = 2.56

(17)

In equation (17),//is the end moment ratio as shown in Figure 4. Kitipornchai and Wang 8 demonstrated that equations (16) and (17) give erroneous results for tee

beams when the web is subjected to significant compression. They did not report that this was due to localized torsional buckling at the support, as will be demonstrated below. They also assumed a zero warping constant and hence ignored the contribution of secondary warping. In Figure 5 the modification factor based on equation (17) is plotted along with derived relationships using equation (11). Assuming equation (11) is correct, it can be seen that localized torsional buckling at the supports will be the critical mode for a large range of beams, especially when the beam is placed in double curvature under a positive end-moment ratio.

Finite elemeat model The finite element formulation of Attard 9 was employed to study the lateral and torsional buckling of tee beams. The tee section chosen for this study, a M5BT70, has the following geometric and material properties: E = 200000 MPa G = 80000 MPa Iyy = 2.6 x 107 mm 4 I== = 9.59 x 107 mm 4 J = 7.662 x 105 mm 4 I,, = 2.53 x 109 mm 6 //x = 120.9 mm (flange bottom, negative ff flange top) y, = 75.68 mm The warping constant was calculated using the approximate formula of Figure 2. The values of ~ adopted are shown in Table 1 with the corresponding beam lengths and parameters defined in equations (13) and (14). In the work of Kitipornchai and Wan8 s energy solutions were determined with varying numbers of terms in the Fourier sine series. Although convergence was Table I

M

Jfrflr

/~

0.5

1.0"

2.5

L(m)

10 8.25E- 5 0.112

5 3.30E- 4 0.223

2 2.06E- 3 0.557

EIw/GJL 2 6

Figure 4 Simply supported beam under non-uniform bending

114

Eng. Struct. 1989, Vol. 11, April

Secondary warping effect on lateral buckling: M. M. Attard and R. Lawther achieved for the buckling loads, the buckled shape for the angle of twist for some problems displayed a curious wavy shape that possibly relates to a lack of convergence for the buckled shape. Various beam meshes (with I,, set to zero) were tried, to examine the effect on the twist of the calculated eigenmode. Initially 10 equally spaced elements were used. For a tee beam orientated with the flange at the bottom, p = 1.0 and L= 2.0 m, this model produced the twist buckled shapes shown in Figure 6. A wavy shape is displayed similar to that determined by Kitipornchai and Wang. By examining Figure 5, it can be postulated that a localized torsional buckle at the left support is likely and hence the equally spaced element model would not be an appropriate model. It was therefore decided to use a model with a finer mesh near the support.

a

T0rsionol buckleat support~ _

1.0"

Secoe
x O

~ " ~

%%J ]

E

-o

0 L/2

0

L

b 1.0-

Secondory warping

a

wl.O-

LI2

L

Figure 8

Eigenmode solution for (a) twist and (b) lateral deflection, for a tee beam with its flange at the top of the web,/] = 0.25 and L = 2 m, using the refined mesh 0

L,~/2

\ a

\

b

I.O"

2O

r

\

E e-

o

\

J

o

L;'2

\ \\

L

\

Figure 6

First and second eigenmode solution for a tee beam with its flange at the bottom of the web, p = 1.0 and L = 2 m, using 10 equally spaced elements

/ A/,Torlioilil lxJckleat support

i.o]-

.--

I

a

-I.0

l

l

l

l

i

-0.5

Secondary

i

t

I

l

I

I

0 End moment factor,

I

I

I

I

I

I

0.5

1.0

0.5

1.0

b OI

,

,

0

LI2

L

o

Lia

L

Cd

-~ 5

ff 0 Figure 7

Eigenmode solution for (a) twist and (b) lateral deflection, for a tee beam with its flange at the bottom of the web, ~ = 1.0 and L = 2 m, using the refined rmmh

,

- 1.0

.015

0 End momentfactor, 13

Figure 9 Comparison of FEM results (Iw = 0) with equation (11 ). • FEM results (no warping); - - - , torsional buckling at support, equation (11 )

Eng. Struct. 1989, Vol. 11, April

115

Secondary warping effect on lateral buckling." M. M. Attard and R. Lawther The beam was subdivided into a symmetric mesh with 34 elements. The first 1/14th of the beam length at both supports was subdivided into 8 elements. The next 1/14th of the beam length was subdivided into 4 elements while the remaining section of the beam was subdivided using 10 elements. Two typical resulting eigenmode solutions are shown in Figures 7 and 8. A pronounced localized torsional buckle is displayed at the supports where the flange is in tension, when the warping constant is taken as zero. The rate of twist is almost infinite at these supports. The twist eigenmode has a smoother profile when secondary warping is considered. Figure 9 shows the comparison of the finite element results with 1w set to zero, and the approximate critical moment calculated using equation (11). It is seen that in the region where localized torsional buckle is critical, equation (11) provides a good approximation to the critical load if secondary warping is ignored.

a

.~

.j"

1

I

i

K~

;° l

l

I

-0.5

-l.O

I

I

I

I

I

I

~

0 End moment foctor, I~

b

Effect of secondary warping

//

0.5

1.0

~

/

A comparison of the FEM results for the critical moment w i t h and without secondary warping included, is shown in Figures lOa and lOb. Figures l l a and l l b compare the modification factor of equation (17) with FEM results. The critical moment is shown to be higher when secondary warping is included. The percentage difference,

z.o t

/

~=o.5

./

t.0

I

/~ =2.5

T

E

2.5 I

I

I

-I.0

I

I

I

l

-0.5

I

i

I

I

i

I

0

I

I

I

I

I

0.5

I.O

End moment lector, ~ f

Figure 11 Effect o f secondary w a r p i n g on modification factor. - N o w a r p i n g (Iw = ~)); - - - , e q u a t i o n (17)

-I.0

-0.5

0 End moment foctor,

0.5

t.0

secondary w a r p i n g

included; - . - ,

shown in Figures 12a and 12b, can be of the order of 100% for short beams with the flange at the top of the web and/~ < 0.5.

Conclusions oJ

b

v

\

~4 / /7= 0.5 /

,..=..

t

K =1.0

~_-2.5 ! ,I -I.0

I

I

I

I

-0.5

I

I

I

I

I

I

l

t

0 End moment footer, [3

I

t

I

I

0.5

Figure 10 Effect o f secondary w a r p i n g o n critical moment. - N o Warping (Iw = ~ ) ; - - - ,

116

secondary w a r p i n g included

Eng. Struct. 1989, Vol. 11, April

I

0

Tee beams and angle sections, because of their low torsional stiffness coupled with the possible reduction of torsional stiffness due to the unsymmetric nature of their bending stresses, are prone to a localized torsional buckle. A simple equation, Me, = GJ/2~I, is proposed that can be used to check if localized torsional buckling is critical. The inclusion of secondary warping in the finite element analysis has been shown to result in differences of up to 100%. Secondary warping is therefore an important factor in the lateral buckling analysis of tee beams. The effect of this factor could equally be significant for other monosymmetric sections of low warping stiffness. Only the fundamental mode of buckling has been considered. For higher modes the effect of this factor can be much more marked. It is shown in Appendix A that the approximate formula for the secondary warping constant for tee beams

Secondary warping effect on lateral buckling: M. M. Attard and R. Lawther 13 Edwardes, RJ. and Kabaila, A.P. Plate buckling in nonconservative structural systems, Proc. 3rd Int. Conf. in Aast. on FEM Universityof New South Wales, Sydney, 1979

,®- -]=

13=0 . t ~

0o%/y_.

Appendix A: Calculation on warping displacements and stiffness

~ 5o

B 0

0.5

1.0

1.5

2.0

2.5

J7 50

When a bar is acted on by torsion, the cross-section will not remain plane (except for the case where the crosssection is circular). This appendix gives details of calculations of the warping displacements and stiffness of the T-section used in this paper. This is done to assess the thin-walled approximations used. The out-of-plane movement, or warping displacement can be calculated using equations (1) to (5). The shear centre is located by the further condition f

40

~-om \

~ ao I0

0.5

1.0

1.5

2.0

25

Percentagedifference in critical moment when secondary warping is considered.

Figure 12

presented by Bleich ~ compares well with a more detailed solution of the Laplace equation for warping. A finite element procedure using plate bending elements has been presented to solve the Laplace equation numerically.

References 1 Bleich,F. Buckling Strength and Metal Structures McGraw-Hill, New York, 1952 2 Vlasov, V.S. Thin-Walled Elastic Beams (translated from 2nd Russian edition). Israel Program for ScientificTranslation, Jerusalem, 1961 3 Gjelsvik,A..The Theory of Thin WalledBars John Wileyand Sons, New York, 1981 4 Attard, M.M. Nonlinear theory of non-uniform torsion of thinwalled open beams, Thin-Walled Struct. 1986,4, 101-134 5 Engel,H.L. and Goodier, J.N. Measurementsof torsional stiffness changes and instabilitydue to tension,compressionand bending,J. Appl. Mech., Trans. ASME 1953, 553-560. 6 Goodier,J.N. Torsionaland flexuralbucklingofbars ofthin-waUed open sectionunder compressiveand bendingloads, J. Appl. Mech., Trans. ASME 1942,9, A103-107 7 Kitipornchai, S. and Trahair, N.S. Buckling properties of monosymmetric I-beams, J. Struct. Div. ASCE 1980, 106, 941 8 Kitipornchai, S. and Wang, C.M. Lateral buckling of tee beams undermoment gradient, Computersand Structures 1986,23, 69-76 9 Attard, M.M. Lateral buckling analysis of beams by the FEM, Computers and Structures 1986, 23, 217-231 10 Timoshenko, S. and Goodier, J.N. Theory of Elasticity McGraw-Hill, New York, 1951 11 Reissner,E. On torsion with variable twist, Osterreich. Ing. Arch. 1955, 9, 218-224 12 Bognor, F.K., Fox, R.L. and Schmit, L.A. The generation of interelement-compatible stiffness and mass matrices by the use of interpolation formulas, AFFDL-TR-66-80,Wright-Patterson Air Force Base, 1986

oJ2 dR = minimum

(A1)

This minimum value of the integral gives the warping stiffness Iw. The coordinates of the shear centre and the warping stiffness can be found by starting with an origin at the centroid and then applying a parallel axis calculation, in the same way that the centroid and second moments of area can be found by starting with an arbitrary origin. The derivation of the above equations can be found in Timosbenko and Goodier 1° and the parallel axis equations are given by Reissner 11 The differential problem is solved approximately using finite elements to calculate the least-squares error in the differential equation, while satisfying the boundary conditions precisely (except at corner points, where contradictions occur, as explained later). If we consider the general Poisson equation V2~o = f ( z , y)

(A2)

then an approximate solution to= has an error e = V2t~= - f

(A3)

The least-squares functional is E = f~ e2 dR

(A4)

and the integrand, when expanded is

where the term f 2 has been omitted since it is independent of the approximation and plays no part in the minimization of E. In structural analysis terminology, the first term in the integrand leads to the stiffness matrix and the second to the load vector. It is the first that determines the suitability or otherwise of an element approximation (which here is that V2m= must be squareintegrable or, equivalently, that o>= must be C ~ continuous). The second term is evaluated as 'equivalent nodal loads'. If the equation is a Laplace equation, as here, t h e 'load vector' contributions are all zero--nonzero terms in the right-hand side all come from prescribed boundary values--' support settlements' in structural parlance. Finite elements for the analysis of bending of isotropic plates are based on minimization of functionals with an

Eng. Struct. 1989, Vol. 11, April

117

Secondary warping effect on lateral buckling." M. M. Attard and R. Lawther B Ic

AIi D

I

III

± -

b

a

C

1 I

F" E Figure A 1 Figure A 3

Table A 1 11"

TrT

Mesh

O'

Position of shear centre (ram)

/w (mm e)

2.7 3.4 4.5 0

2.49e9 2.48e9 2.45e9 2.53e9

I a b c Bleich 1

0

Convex corner

Concave corner

Figure A 2

integrand containing the dominant term [co,zz co,, 2CO,jD

(A6)

This integrand term is identical to the one arising from the Poisson equation if both the factor D and the Poisson's ratio, #, are 1. We can therefore use a platebending program to solve the Poisson equation by simply setting Poisson's ratio to 1 (we must also set the plate rigidity to 1 separately, as this becomes infinite at a Poisson's ratio of 1, and we need to evaluate some integrals such as S co2 dR). The element used here is the fully compatible rectangle of Bognor et aP 2 The T-beam is modelled using anti-symmetry, as shown in Figure A]. The boundary conditions are co = 0 on AF (from antisymmetry) ~co/az = y on vertical edges BC and DE dco/dy = - z on horizontal edges AB, DC and FE

These boundary conditions imply O2co/dzfly = + 1 on vertical edges and c 3 2 c o / d y d z = - 1 on horizontal edges, and we therefore get a contradiction at all comers. The element used requires these 'twist' second derivatives to be given, but even if this were not so, the contradiction would remain. This problem has been handled differently depending on whether the comer is convex or concave. For convex comers (B,C and E) the twist at the comer has been left unspecified. The boundary conditions are

118

Eng. Struct. 1989, Vol. 11, April

satisfied at the node points of the comer element (a,b and C in Figure A2), but are not otherwise enforced along the boundary aCb. Since no other element connects to this corner node, there is no relaxation of the C t continuity. For a concave comer (D), the twists of elements I and III have been specified, but that of element lI has been left unspecified and unconnected. The derivative aco/az is matched between elements I and II, and/ka/dy is matched between II and III. The boundary conditions have been satisfied, but the C 1 continuity has been relaxed to C o for the transverse slopes across the edges aD and bD. No attempt has been made to include these relaxations in the least squares functional The following finite element meshes were used. Meshes (b) and (c) in Fioure A3 were graded using the method shown by Edwardes and Kahaila 13. Results of these analyses are given in Table A 1. The position of the shear centre is the distance below the middle line of the flange. The calculated values of the warping stiffness are surprisingly consistent with each other, and with the approximate value given by Bleich I, which was used in the calculations in the body of this paper. What is not so consistent is the calculated position of the shear centre. Many writers report that:',,:~e Shear centre for such a section is very close to the fi~id-height of the flange, and this seems to be physically reasmmbl¢ (Bleich's approximate warping displacexaents imply a position 0.6 mm below the flange mid-height). The above analyses are consistently predicting a shear centre that is somewhat removed from this position, although there is not much agreement on precisely where. These results are presented without further comment, except to note that if the lw values are nmalculated on the basis of the shear oentm being at the r a g e mid-height, the agreement between the values is nowhere near that shown in the above Table A1.