Stability of thin-walled members having arbitrary flange shape and flexible web

Stability of thin-walled members having arbitrary flange shape and flexible x reb C. K. Chin, F. G. A. AI-Bermani and S. Kitipornchai

Department of Civil Engineering, The University of Queensland, St. Lucia, Queensland, Australia (Received September 1990, revised November 1990)

A finite element method is presented for analysing thin-walled structural members comprising a flexible web connected to one or two rigid flanges of arbitrary shape. A general thin-walled beam-column element is used to model the flanges while a thin plate element is used to model the web. Based on the derived total potential energy func- tional, explicit linear and geometric stiffness matrices for the two types of element are obtained. Using static condensation and appropriate transformations, the beam-column element and the plate element are combined to yield a super element with 22 degrees of freedom capable of modelling the flexural, torsional, web distortional and coupled web and flange local buckling modes of a general thin- walled member. The technique may be used to predict the elastic buckling load of members under any loading and boundary conditions. Several numerical examples are presented to demonstrate the accuracy, efficiency and versatility of the method.

Keywords: beam-column, bifurcation analysis, finite element, buckling, plate, stability, thin-walled member, web distortion

Elastic buckling analyses of general thin-walled structures do not usually consider cross-sectional distortion I-6. This assumption is valid for members with relatively compact or rigid cross-sections. In the case of fabricated girders or general thin-walled built-up sections such as those shown in Figure 1, however, where the webs are relatively slender, the buckling mode is likely to consist of lateral translation and twist accompanied by web distortion (see Figure 2a) or coupled web and flange local buckling (see Figure 2b). Web distortion is also important in heavily loaded beams where one of the flanges is partially restrained 7-9 (see Figure 2c).

Several finite element methods have been presented to study the effect of web distortion. Johnson and Will ~° discretized the flanges and web into thin plate elements thus permitting the entire cross-section to distort freely. Many degrees of freedom (d.o.f.) are needed, however, to model the member distortional buckling behaviour accurately. Akay et al. i~ used one-dimensional elements to model the flange and plate elements to model the web. The web was discretized vertically into several plate elements. Local and distortional buckling modes of the web were predicted. 0141-0296 /92 /020121 ~ 12 Q 1992 B u t t e r w o r t h - H e i n e m a n n Ltd

The finite strip method 12:3 was employed by Hancock 14 to study local, distortional, and flexural- torsional buckling of simply supported I-section members under uniform moment. Due to the restrictive

7

) (a) (b) (c)

Figure I Thin-walled sections with arbitrary flange shapes and flexible web

Eng. Struct. 1992, Vol. 14, No 2 121

Stabil i ty o f thin-wal led members: C. K. Chin et al.

!

1 T

W T WT= 0

I[ii J = I 1

W B -1 WB=OxB=O

'1 " " I ' '

WT= 0 T 0

OxB/ ii,

Figure 2 Buckling modes

(a) (b) (c)

nature of the assumed Fourier series displacement functions, the method cannot readily be applied to other loading and boundary conditions. An approximate energy method for analysing the effect of web distortion on the buckling of doubly symmetric I-beams subject to uniform moment and axial compressive loading has been presented by Hancock et al.'5. Bradford and T rahair 16.17 investigated the influence of web distortion on the buckling behaviour of general thin-walled open sections by treating the flange as an assembly of rectangular strips connected together.

This paper presents a finite element method for analysing thin-walled structural members composed of a flexible web connected to one or two rigid flanges of arbitrary shape (Figure 1). A 3-node general thin-walled beam-column element is used to model the flange. The element has 7 d.o.f, at each end and a single axial d.o.f. at the mid-length node located along the element centroidal axis. The out-of-plane bending of the web plate is modelled using a single thin plate element having 4 corner nodes with 4 d.o.f, per node. The in-plane membrane stiffness is modelled using 3 mid-height nodes with 3 d.o.f, at each end and a single d.o.f, at the centre. Using static condensation and appropriate transformations, the beam-column element and the thin plate element are combined to yield a 22 d.o.f, super element capable of modelling the elastic lateral, web distortional and coupled web and flange local buckling behaviour of general thin-walled structural members.

Problem formulation

Element reference axes and basic assumptions

The following assumptions are made for the beam- column flange element

• the element, but not necessarily the member, is prismatic and straight

• cross-sections of the flanges are rigid • Vlasov's assumption 's of negligible shear strain in

the middle surface is valid • shear deformations of cross-sections are negligible if

the element is applied to thin-walled closed sections.

For the web plate element, the following assumptions are made: Kirchhoff's plate assumptions remain valid

for the out-of-plane plate bending; and Bernoulli's assumptions are valid for the in-plane membrane stiffness. The assumptions of homogeneous, elastic and isotropic material and of conservative loading are made for all elements.

Figure 3 shows a 3-node 15 d.o.f, line element of general thin-walled open section used to model the flange. The right-hand orthogonal coordinate system x, y, z is chosen so that y and z pass through the end cross- section shear centres of the element before deformation, S and S', and are parallel to the principal y and ~ axes of the cross-section. A parallel set of axes :~, 3~,, pass through the end cross-section centroids of the element, C and C'.

Let v and w denote the displacements of the shear centre of the flange element in the y and z directions, Ox the angle of twist about the shear centre axis, u the axial displacement along the centroidal axis CC', and 0y and 0 z the rotations about ~ and ~ axes of the element. Let Um be the axial displacement at the mid-length node along the centroidal axis of the element. Seven actions (Fx, Fy, Fz, M~, My, M z and M=) with corresponding

M y a , O y a

I End , Y- .Y !Node a ~ ~ , u a

z I I/ ) $~-..--F Centroidal / / / / i x . 8~. "~L ' ' 4 ~ \ za.Wa

, ; / / - ' End , . ^ . ~ ~ m / ,/fM=a.Oxa

- - t / Mzb.ezb Y / Y ~Shear Centre , / / Axe,/ F x b , U b / ~ R'/Fzh,Wh ~// / ,~..,,._.. zb b /

, / /. p/ Mwb,exb

X

Figure 3 General thin-walled beam-column element

122 Eng. Struct. 1992, Vol. 14, No 2

y5

iyT2,0yT2 t W ~-

ixT2 FzT2,0 ~ " " ~ ' W T 2 x T 2 j ~ ~ ~

Mzi,Ozi~ t l _

M 11 0.~ 1 i / U k Z _ _

x E3 2 / / . ~

Fyi,Vi

Figure 4 Rectangular thin plate element

displacement components (u, v, w, 0~, Oy, 0 z and O0~/Ox) can be applied at each end of the element as shown in Figure 3.

Figure 4 shows a 23 d.o.f, rectangular thin plate element used to model the web. The right-hand orthogonal coordinate system x, y, z is chosen so that x and y pass through the web middle surface along the sides of the rectangular plate element. A parallel set of axes £, y, ~ is chosen so that the • axis passes through the web mid-height nodes as shown in Figure 4. Four actions (Fz, Mx, My and M~) with corresponding displacement components (w, Ox, 0 r and OOx/OX) can be applied at each corner node of the thin plate element. In addition, three actions (Fx, Fy and Mz) with corresponding displacement components (u, v and 0~) can be applied at the two end nodes of the web mid-height. A displacement, uk, can be applied at the central mid-height node along the $ axis.

Total potential energy

The total potential energy of a continuum can be expressed as

11 = U - V (1)

in which U is the strain energy stored in the continuum and V is the potential of the external loading. For a continuum, the strain energy, U, is given by

U = ~ ({eLlr[D] {eLl +2{olr{eu})dv v

(2)

in which {o} are the cartesian components of the Cauchy stress tensor in the deformed state, [D] is the material matrix, v is the volume of the continuum, and {eLI and [eN] are the linear and nonlinear components of the strain tensor, respectively.

The general expression for the strain-displacement relation in a continuum can be written in indicial form 19

e 0 = %,j + eN,, (3)

Stability of thin-walled members: C. K. Chin et al.

in which

eL,, = I/2(ui.j + uj.i) (4a)

eNi j = l /2(Uk, i Uk , j ) (4b)

Stiffness matrices for flange element

Figure 3 shows a general thin-walled beam-column element used to model the flange. Since the element is basically a line element, the components e,.,., ez= and 3'yz of the strain tensor are zero and the matei:ial matrix is a diagonal matrix, i.e.

[ D ] = [ E G GI (5)

in which E and G are the Young's modulus and the shear modulus of the material, respectively.

For small but finite deformations, the displacements a, ~ and ~, of an arbitrary point on the cross-section may be expressed in terms of the axial displacement of the centroid, u, the shear centre displacements, v and w, and the angle of twist, Ox, about the shear centre axis, thus

~ = V - - OxZ

~ = w + O x y

av aw a = u - Y - - - -

in which

(6a)

(6b)

(6c) 0x

y = y - yo (7a)

= z - Zo (7b)

Yo, Zo are coordinates of the centroid relative to the shear centre of the element and o~s is the sectorial coordinate.

The direct and the shear stresses acting at any point of the flange element in generalized principal coordinates 3~ and ~ are

+ (-1%o, + My O2)

y M,~w, (8a)

7xy -- Aff (8b)

gzb 7-xz = - - AI (8c)

in which

x p l = l - -

L (9a)

Eng. Struct. 1992, Vol. 14, No 2 123

Stabi l i ty of th in-wal led members: C. K. Chin et al.

X P2 = - (9b)

L

L is the length of the element, A s the cross-sectional area of the flange element, Iri and Izy the second moments of area about the principal )~ and Z axes of the flange element, respectively, given by

l ' f= f~ g2 dA (10a) 1

/ .4= jA y2 dA (10b) !

and I~ is the warping section constant, given by

IAj 2dA (11) I~ = ~ s

The subscripts a and b refer to the far and near nodes of the flange element, respectively, (see Figure 3).

Substituting equations (3) and (5)-(8) into (2) and neglecting the higher-order and less significant terms, the expression for strain energy U~ of the flange element may be obtained as follows

/O:v \Z ' +

us=2 £[ "taxi

[OzO'"x2 Gj(OO~) z] dx

0r(0 v CwV T JoL\~/ + \OxJ

+ Ow Ov 00, 2yo - 2Zo ax g. £ + r~ OxxJJ

aoq Ox J

l'o OOx ( av & + M~aO,~ ~ - 7 -

l ~ OOx(~ ~,, - MybP2OXX ~ X - 2 0

.lo oo (ow + M~oo,~ £ +7

Ox}

ax ) dx

f - M z b o z - - dx o 2 ax/

+t I'o ( lo( °v au av d x - Fzb Ox

Ox

/02w\2

dx

OW

Ox

on o;) + O--x

(12)

in which J is the St. Venant torsion constant and r~ is the polar radius of gyration about the shear centre,

respectively

r~= I. J'Ar (Y2 + Z2)dA (13)

The terms/~y, /~z and/3~ are given by

13y = ~ (g3 + gy2) dA + 2Zo (14a)

/3z=/~4.i' A' (23+yg2) dA+2yo (14b)

I IA~ /3~ = ~ w,(372 + gz) dA (14c)

To solve equation (12), the following interpolation functions to approximate the displacement fields are adopted: a quadratic interpolation polynomial (f2) for axial displacement u and a Hermifian interpolation polynomial (f3) for the transverse displacements v, w and the angle of twist Ox. The displacement fields can be expressed

/ut[ }/,u.,, v (f3> Ivel

w <z> Iw, l I

ox ~3) {Ox<l]

(15) in which

and

{u<} =(Ua Um Ub) r (16a)

{v<} =(va 0za Vb Ozb> r (16b)

{%1 = (w, -Oy. w b --Oyb) r (16C)

O0x~ 0x~ 00x~\r [0xe} = 0~. cgx Oxxl (16d)

<f2) = (PI( 1 - 2p2) 4pip2 - P2(Pi - 02)) (17a)

( f s ) = ((3 - 2p,) p~p~p2£ (3 - 2p2)p~ - p,p~L)

(l'Ib) Substituting equation (15) into (12) and integrating,

the expression for the strain energy Uy of the flange element may be written as

Us= % lrsi r[gLs + goi] Irsi (18)

in which I t s and K~s are the linear and geometric stiffness matrices of the flange element, and {r;I is the nodal displacement vector for the flange element

=(ua u,, us Vo O~a Vb Ozb W~ O.v~ I rs} /

.00xbk ) r wb 0,, Ox~ 00~. O~b (19)

" ~ " ~I

124 Eng. Struct. 1992, Vol. 14, No 2

The expression for the total potential energy for the flange element according to equation (1) can thus be expressed as

H = V=lr/]r[K£:+ Kcyllr/] - [r/lrlF:] (20)

in which {Fy] is the nodal force vector

My~ F~b Myb M~. M~ M~b M,~b) r

(21)

Invoking the equilibrium condition, one obtains

[K£f+ K~f] Irf} = {F/I (22)

in which the linear and geometric stiffness matrices Kq and Kc/can be derived explicitly.

Stiffness matrices for web plate element

Figure 4 shows a fiat plate element for modelling the web. Assuming no coupling between the membrane and the out-of-plane bending stiffnesses, the strain energy of the element can be represented by adding the membrane strain energy to the out-of-plane bending strain energy 2°, i.e.

U~ = U~ + Up (23)

Membrane stiffness. Following Bernoulli's assumptions, the displacements of an arbitrary point on the cross- section of the thin plate element in the :~ and y directions (see Figure 4) because of in-plane membrane stretching can be written in terms of the in-plane displacements of the plate centroidal axis, Uw and v~, i,e.

_ Ovw a = Uw - Y - - (24a)

Ox

= v., (24b)

~, = 0 (24c)

The direct and shear stresses acting on the plate can be expressed as:

o,= = F~ + (Mup ' _ M~/p2) y--- (25a) .,'14,

%' = Fyj (25b) Aw

in which

lz~ = dla,. 372 dA (26)

A~ is the cross-sectional area of the web. The subscripts i and j refer to the far and near nodes along the :~ axis (see Figure 4).

Stabi l i ty of th in-wal led members: C. K. Chin et al.

Because the strain components %y, e=, %~ and "ty~ are taken to be zero, the material matrix is a diagonal matrix, i.e.

[D] = [ E G] (27)

Substituting equations (3) and (24)- (27) into (2), and neglecting higher-order terms, the expression for the membrane strain energy Um may be obtained

:,,wVl Um= e< \ ax / + el '\ o-Z; ) J

+ f rn, ?Vw Ou. Ov 4 30L 2 \ ax I - Fyj Ox ax J dx

(28)

Using a quadratic interpolation function (f~) for the axial displacement, Uw, and a Hermitian interpolation polynomial function (f3) for the transverse displacement, v~, the displacement field can be expressed as

IvUWl=[ (f2) (f3)] f{uel](tvel) (29)

in which

{Ue} = ( U i U k Uj) T (30a)

{ve} =(vi Ou vj 0v) r (30b)

Substituting equation (29) into (28) and integrating, the equilibrium equation for the membrane actions can be written as

IFml = [Kt + [rml (31)

in which

IFml =(Fx, Fxk Fy, Mz; F,j T (32)

{r,~] =(ui Uk Uj V i Ozi Vj Oz, j ) T (33)

The linear and geometric stiffness matrices, K~ and Ko,, for the in-plane plate membrane stiffness can be derived explicitly.

Out-of-plane bending stiffness. Following Kirchhoff's hypothesis of negligible transverse shear deformation, a line normal to the plate undeformed middle surface is assumed to remain normal to the deformed middle surface 21. Therefore, the displacements of an arbitrary point on the cross-section of the thin plate element (see Figure 4) in the x, y and z directions due to out-of-plane plate bending can be written in terms of the middle surface out-of-plane displacement, w~, i.e.

OW w a = - z - - (34a)

8x

aWw ~7 = - z - - (34b)

ay

~v = Ww (34c)

Eng. Struct. 1992, Vol. 14, No 2 125

Stabi l i ty of th in-wal led members: C. K, Chin et al.

The direct and shear stresses acting on the plate can be expressed as in equation (25), taking %y = trzz = r,~ = ry~ = 0. The material matrix can be written as 0[

[D ] = 1 - / / 2 1 0 (35) 0 1 - v

2

in which v is Poisson's ratio. Substituting equations (3), (25), (34) and (35) into (2)

and neglecting higher-order terms, the expression for the thin plate bending strain energy Up can be obtained

+ 2v 02ww o2Ww "q- 2(1 -|---|[I/02Ww\2q 'x' 'y2 ")\'xayf J dx dy

+,w + (Mz:, -,,.,., q( 4' 2 aojo L\Aw Iz~/\~x/

2Fy, OWw OWw] + dx dy (36)

A w ax ayJ

in which

Dw - Eta (37) 12(1 - v 2)

tw, L and hw are the thickness, length and height of the thin plate element, respectively, (see Figure 4).

To solve equation (36), Hermitian interpolation polynomial functions (f3) and (N3) are used to approximate the displacement fields in the x and y directions, respectively, i.e.

Ww = (N3) ~ )

h ~{O~Bel ( ) [Wre} ((38)

L IO. e )

in which

IWBel = (Wsl -OyB~ ws2 -Oym) r (39a)

{wre} = (wn -0yn wr2 -Oyr2) r (39b)

OOxal 0~ (39c) [0xsel= 0xa~ 0x ax /

( a0xr2~ r 00xrl 0x72 (39d) [ O~re l = Oxn ax ax /

(N3) = ((3 - 2 ( i )~ ~ 2 h w (3 - 2~2)~ - ~l ~2h~ ) (40)

in which

hw (41a)

~2 - y (41b) hw

and (f3) is given in equation (17b). The subscripts T, B, 1 and 2 refer to the t_op, bottom, far and near ends, respectively.

Substituting equation (38) into (36) and integrating, the equilibrium of the out-of-plane actions can be expressed as

IG} = [K,. + K~.] Ir.I

in which

(42)

{Fp} =(FzB I My m Fz~ 2 Mys2 Mxs , M=B ,

M~ M~ r~. M:~ F: M:~ Mxn M~n Mxr2 M~r2 ) r (43)

Ire} =(win OyBI WB2 ORB'2 oo xn l Ox

OxB2 ax wn Oyn w~ Oyr2 Oxr~

aO~n 0~ O0xn~ r Ox o-X-x~ (44)

The linear and geometric stiffness matrices, Kt~ and Kop, for the out-of-plane plate bending stiffness can be derived explicitly.

Method of solution The stiffness matrices for the flange and the web elements can be assembled into a super element to model

Yc

MyT2,0y ~ ~ ~ ~ ~ ~ - ~ " ~ T I

~- Centroidal Axis I ~c..1~Fx2,U2 of Super E l e m e n ~

M=a2,exs2

Figure 5 Super element

126 Eng. Struct. 1992, Vol. 14, No 2

the behaviour of a general thin-walled member having a flexible web through appropriate transformations. The generalized nodal displacements in the local principal axes, Ir:l (equation (19)) are first transformed to corresponding displacements, {~:l, in the axes parallel to the local axes of the super element (see Figure 5), i.e.

{ rf] = [ TR] { ?:} (45)

The flange element generalized nodal displacements, If:l , can also be expressed in terms of the super element generalized nodal displacements, {r~}, using the following transformation

I~:1 = [ ~ ] {r,I

in which {r,} is given by

(46)

[rs} =(ul vl O: wnl Oxnl OyBI OOxBIox

WTI OxTI OyTl OOxT1 U2 12 2 OZ2 Ox

w~ 0~ Oyn2 O0~ wr2 O~r2 Oyn Ox

OOxn )r OX U3 (47)

The web membrane stiffness is derived relative to the centroidal axes of the web and must be transformed to the centroidal axes of the super element, via

{r,,} = [T,,] {rsl (48)

in which [r~} is given by equation (33). The transformation matrices [TR], [T:] and [T~] are given in the Appendix.

Assembling the flange and web stiffness matrices accordingly and employing static condensation to con- dense out the mid-length nodal degree of freedom, u3, a 22 × 22 super element stiffness matrix combining the flange and the web elements can be obtained

1F122 × i = [KL + gG]superlr}22 x | (49)

in which

[r] =(ul vl Ozl wnl O,B, eyBi OOxBlox

O0~rl WTI OxTI Oyrl OX U2 V2 Oz2

WB2 OxB2 OyB2 OX WT2 0x72 OyT2

O0~r2 \ r

Ox/ (50)

A mid-length node is added to the flange element and the web membrane element to avoid incompatibility of


axial displacement when joining the flange to the web element 22. If the section contains only one flange, then one of the flange stiffnesses is ignored.

The global stiffness matrix for the structure may be obtained by a standard assembly procedure in the finite element method as follows

[KL + Ko]gloOat = ~ [C] T[KL + Ko]super[C] n

(51)

in which n is the number of elements, [KL + Ka] is the super element stiffness matrix, and [ C] is the transformation matrix relating the local coordinates of the super element to the global coordinates of the structure.

The bifurcation load can be obtained by solving the following eigenvalue equation

I KL + XcrKo I~tob,t = 0 (52)

Several methods for solving the above equation are available (e.g. Reference 23). In the present work the bisection method is used to obtain the critical load factor ~ c r •

N u m e r i c a l e x a m p l e s

Based on the described formulation, a number of studies are presented to demonstrate the efficiency, versatility and accuracy of the proposed finite element method. Most of the numerical examples chosen have been investigated previously using different techniques. In general, it is found that 5 to 10 elements are needed for convergence, depending on the type of problem.

Doubly symmetric I-beams and beam-columns under uniform moment

Hancock et al. 15 studied the effect of web distortion on simply supported doubly symmetric 1-beams under combined axial load and uniform moment using an approximate energy approach. The proposed finite element method has been applied to the same problem, using 5 elements. The derived buckling interaction curves for axial load and moment are compared with those given by Hancock et al. ~5, and with the classical rigid web solution in Figure 6. It is seen that the proposed finite element method gives a slightly higher buckling load when the axial compressive force is small.

Recently, van Erp 24 used the finite strip method to study web distortional buckling of plate girders. He compared the finite strip solutions with the Hancock et al. ~Senergy solutions and concluded that the Hancock et al. solutions are slightly more conservative for stocky beams (i.e. beams with high web depth to length ratio). For the purpose of this paper, the same problem of a simply supported doubly symmetric plate girder 1-beam under uniform moment is analysed by the proposed finite element method. The beam dimensions (beam A) are given in Table 1. The present authors' results using 6 elements are compared with those of van Erp, and the Hancock et al. solutions in Table 2. The finite element

Eng. Struct. 1992, Vol. 14, No 2 127

Stability of thin-walled members: C, K. Chin et al.

1.0 I I I I

0.8

a. o

0.6

n- 10

0 J 0.4

X <

0.2

-~Tf \ ~ \ \ \

tw "k "k, \

Me Mc

i i

Bf /h = 0.4 - Tf / tw = 8

L/h = 30

I 1 1 0 0.2 0.4 0.6

\ \

\ \

\ \ \ \

\ \ \ \ \ \1 0.8 1.0

Buckl ing Moment Ratio, Mc/M o

Figure 6 Comparison of results wi th Hancock et a l? 5 for simply supported doubly symmetr ic I-section beam-columns under uniform moment. ( ) This paper, (5 elements); (-- --), Han- cock et a/.15; ( . . . . . . ) rigid web

solutions are in excellent agreement with those obtained by v a n Erp 24.

Monosymmetric 1-beams under uniform moment

Bradford and Waters :5 presented an approximate energy method to study the web distortional buckling of simply supported monosymmetric I-beams under uniform moment. The buckling solution was reduced to solving the eigenvalue of a 4 × 4 matrix. In order to

Table 1 Dimensions (mm) of beams in examples (E = 2 1 0 G P a , = 0.3)

B e a m

Flange Flange Web Web width, thickness, depth, thickness, Bf Tf h w t

A 100.0 10.0 440.0 2.0 X 1 300.0 35.0 900.0 18.5 X2 268.0 25.4 718.8 15.6 X3 153.5 18.9 427.3 10.7 X4 133.4 7.8 187.6 5.8

Table 2 Buckling moments (kN-m) for a plate girder under uni form bending

Length Rigid Hancock van Erp 24 This paper

3 97.8 76.4 88.7 89.5 5 41.5 37.4 37.8 37.8

10 15.8 15.0 15.0 15.0 20 7.0 7.0 7.0 7.0

demonstrate the accuracy of the proposed finite element method, one of the examples analysed by Bradford and Waters 25 is selected. The present authors' results using 5 elements are compared with those of Bradford and Waters in Figure 7. The present finite element solutions are in excellent agreement with those derived by Brad- ford and Waters.

Beams with tension flange restraints

The stability of doubly symmetric I-beams with tension flange restraints (see Figure 2c) was studied by Williams and Jemah 8. They calculated the buckling stress for four beams (beams X1 to X4, see Table 1) using the VIPASA computer program 7~. Goltermann and Svensson 9 used an approximate analytical method of solving the differential equilibrium equation to obtain solutions and found good agreement with those reported by Williams and Jemah 8. The proposed finite element method has been applied to these four beams. In the analysis, all the degrees-of-freedoms of the nodes at the tension flange were restrained. The present authors' results, using 8 elements are compared in Table 3. In general, the proposed finite element solutions are in reasonable agreement with those obtained by Williams and Jemah 8 and by Goltermann and Svensson 9, except for the beams with thicker flanges. This is because the proposed finite element method assumes that the junction line between the tension flange and the web is restrained, while in References 8 and 9, the restraint was assumed to be at the middle fibre of the tension flange. Moreover, accurate modelling of the web has been made

UJ u_ o

L..

09

t-

O

(0

m t- O .m o9 c"

E_ o

-2 10 \ i Mc i Mc \

\ l u 1 \ \\ BTXTT " ~ - ]

Local - -_ \ ii I h 15 3 Buckl ing ~ \ t w - - ~ - I -

---// ~ \ ' \ \ BB×TB ~ J

~ , \ BT/h = 0 .063 ~ ( \ TT/tw = 2.5

~ \ \ BT/BB =0.5 ~ \ TT/TB = 1.0

1(~4 _ ' ~ \ \ h / tw= 200 -

1(~0 .5 1 10 100 Dimensionless Length, L/h

Figure 7 Comparison of results wi th Bradford and Waters 25 for s imply supported monosymmetr ic I-beams under uniform moment . ( a, ) This paper (5 elements), ( - - - ), rigid web; ( ) energy method Bradford and Waters 25

128 Eng. Struct. 1992, Vol. 14, No 2

Table 3 Elastic critical stress (MPa) in middle f ibre of compres- sion f lange

Length Will iams Goltermann Beam (mm) and Jemah 8 and Svensson 9 This paper

X1 7399 455 467 477 X1 12333 489 506 507 X1 30830 454 465 477

X2 6010 525 534 548 X2 10016 567 581 586 X2 25040 524 532 548

X3 3322 591 606 623 X3 5537 635 658 660 X3 13842 591 604 623

X4 2042 1128 1137 1142 X4 3404 1225 1242 1225 X4 8510 1124 1133 1140

for the finite element method whereas Williams and Jemah 8 and Ooltermann and Svensson 9 -assumed the web height to be between the flanges' centroids.

Hollow flanges I-beams under moment gradient

The proposed finite element method has been used to analyse the distortional buckling of a simply supported doubly symmetric hollow (circular) flanged I-beam (see Figure 8) subject to unequal end moments, M and/3M. In this example the web is relatively flexible (2 mm thick) to highlight the distortional effect. Buckling solu-

400

350

A E I 300

z

o 250

a) 200 E O

c~ 150 t--

O

m 100

50

I ~ i i i I

13=+1 \

\ \ \

\ \ L L _1

p=o\ \ . \ \

\ x \ \ 2 - ~ _ ~ 25 \ \

r =-I \ \ \ \ \ (mm)

~=+1 . \ \

I I I I I 0 2 4 6 8 10 12

Beam Length, L (m)

Figure 8 Influence of web distort ion on buckling of simply supported hollow flanged beams under moment gradient. ( ), Flexible web; ( - - - ) , rigid web ( 5 - 1 0 elements used)


tions for flexible and rigid web cases are compared in Figure 8 for end moment ratios, ~ = - 1 (uniform moment), 0 (one end moment) and + 1 (double cur- vature). In general, 5 elements are sufficient for uniform moment, but for the case of high moment gradient, 10 elements are needed. It is seen that the effect of web distortion is very significant for this beam.

Two-span continuous 1-beams under concentrated loads

The elastic flexural-torsional buckling of continuous elastic symmetric I-beams had been investigated theoretically and experimentally by Trahair 27. The significance of buckling interaction between adjacent beam spans has ben reported. In the study the effect of web distortion was not considered. The proposed finite element method can be applied to this problem to incor- porate the effect of web distortion. Methods such as the finite strip method 13 and the approximate energy method 25 cannot easily be applied.

A two-span continuous aluminium I-beam subject to mid-span concentrated loads P1 and P2 acting at the top flange level is shown in Figure 9. This beam was tested by Trahair 27. To study the web distortional effect, the thickness of the web was reduced arbitrarily by 50% (0.5tw) and 80% (0.2tw) respectively. The authors' finite element solutions, using 8 elements, are compared with Trahair's test results 27 in Figure 9. Also shown are the buckling curves obtained ignoring the effect of web distortion, using Chan and Kitipornchai's beam- column finite element program 5. It is seen that the buckling interaction curve derived for the beam with the

500

400

A O0 .Q

o j 3 0 0

0 --1

20O

100

L~ads at Top Flange 91 92

1 I /k A

L 60" _L 6o" , i

_ 4 - - - 2

1 . 2 4 2 " ~ j.. /

,TTT I

3 E = 9400ksi -

tw G=3870ks i

AA

,I

I I k

0 i I 100 2O0 300

Load, P1 (Ibs)

Figure 9 cont inuous I-beams. ( zx ), Trahair 's tests27; ( elements); ( - - - ) , rigid web s

400

Influence of web distortion on .buckling of two-span ), this. paper (8

Eng. Struct. 1992, Vol. 14, No 2 129


1.4 l J i ]

1.2 o

~o 1.0

6

tr 0.8 E E o 0.6

e- .~ 0.4

03

0.2

Lateral Buckling

/ ~ Coupled

10~_~-h=400 \ w e b & Flange

Me Mc (} L = 32001

1 I I I I 0.2 0.4 0.6 0.8 1.0 1.2

Flange Width to W e b D e p t h Ratio, B,/h

Figure 10 Inf luence of coupled web and f lange local buckl ing on simply-supported doubly symmetric I -beams under uni form moment . ( ), Robert and Jhita2e; ( 0 ), Azizian2S; ( z~ ) this paper (8 elements)

original web thickness case agrees very well with test results and with the rigid web solution. The influence of web distortion, however, becomes more significant as the web thickness is reduced. In such cases, ignoring the effect Of the web distortion would result in an overestimation of the buckling capacity of the beam.

Coupled web and flange local buckling of 1-beams

Roberts and Jhita 2s used an energy method while Azizian 29 used the finite element method to study the coupled web and flange local buckling of a relatively stocky doubly symmetric I-beam (see Figure 10) under uniform moment. The variation of the buckling moment ratio Mc/l~lc with respect to the ratio B//h is shown in Figure 10. The value for ~r c represents the buckling moment of a similar beam with cross-sectional distortion suppressed. It is seen that for low By/h ratios, the beam buckles laterally without web distortion. As the flange width to web depth ratio increases, however, the buckled mode changes from lateral to coupled web and flange local buckling. The present finite element solutions using 8 dements are in excellent agreement with those derived by Roberts and Jhita 2s and Azizian 29.

Conclusions

A finite element technique for predicting the elastic web distortional and coupled web and flange local buckling of general thin-walled members comprising arbitrary flange shape and flexible web has been presented. The flange, which is assumed to be rigid in its own plane, is modelled as a general thin-walled beam-column element, while the web part is modelled as a thin plate

element. The flanges and the web stiffnesses are assembled into a super element with 22 d.o.f, capable of predicting the lateral, web distortional and coupled web and flange local buckling behaviour of thin-walled members. Several numerical examples have been presented to demonstrate the accuracy and efficiency of the method. In all cases, the proposed finite element technique predicts results which are in close agreement with independent published solutions. The method is versatile and can be applied to general thin-wall members of the prescribed shape, subject to any loading, restraint and boundary conditions.

Acknowledgments

The work in this paper forms part of the project 'Stability of beams and beam-columns' supported by the Australian Research Council (ARC) under Project grant no. 834 and by Tube Technology Pty Ltd (Palmer Tube Group).

The authors wish to thank Dr S.L. Chan of the Department of Civil and Structural Engineering, Hong Kong Polytechnic and Dr C.M, Wang, Department of Civil Engineering, National University of Singapore for some useful suggestions and Mr Warren H. Traves, Gutteridge Haskins and Davey Pty Ltd for proof-reading the manuscript.

References

1 Pekoz, T.B. and Winter, G. Torsional-flexura| buckling of thin- walled sections under eccentric load, J. Struct. Div., ASCE 1969, 95(ST5), 941-961

2 Barsoum, R.S. and Gallagher, R.H. Finite element analysis of torsional and torsional-flexural stability problems, Int. J. Num. Methods Eng. 1970, 2, 335-352

3 Yang, Y.B. and McGuire, W. Stiffness matrix for geometric nonlinear analysis, J. Struct. Eng. ASCE 1986, 112(4), 853-877.

4 Kitipornchai, S. and Chan, S.L. Nonlinear finite element analysis of angle and tee beam-columns, J. Struct. Eng., ASCE 1987, 113(4), 721-739

5 Chan, S.L. and Kitipornchai, S. Geometric nonlinear analysis of asymmetric thin-walled beam-columns, Eng. Struct. 1987, 9, 243 -254

6 AI-Bermani, F.G.A. and Kitipornchai, S. Nonlinear analysis of thin- walled structures using least element/member, J. Struct. Eng., ASCE 1990, 116(1), 215-234

7 Bradford, M.A. and Trahair. N.S. Lateral stability on beams of seats, J. Struct. Eng., ASCE 1983, 109(ST9), 2212-2215

8 Williams, F.W. and Jemah, A.K. Buckling curves for beams, J. Const. Steel Res. 1987, 7(2), 133-147

9 Goltermann, P. and Svensson, S.E. Lateral distortional buckling: predicting elastic critical stress, J. Struct. Eng., ASCE 1988, 114(7), 1606-1625

10 Johnson, C.P. and Will, K.M. Beam buckling by finite element procedure, J. Struct. Div. Div., ASCE 1974, 100(ST3), 669-685

11 Akay, H.U., Johnson, C.P. and Will, K.M. Lateral and local buckling of beams of frames, J. Struct. Div., ASCE 1977, 103(ST9), 1821 - 1832

12 Plank, R.J. and Wittrick, W.H. Buckling under combined loading of thin flat-walled structures by a complex finite strip method, Int. J. Num. Methods Eng. 1974, 8, 323-339

13 Cheung, Y.K. Finite strip method in structural analysis Pergamon Press, New York, 1976

14 Hancock, G.J. Local, ,distortional and lateral buckling of I-beams, J. Strcut. Div. , ASCE 1978, 10,I(STll), 1787-1798

15 Hancock, G.J., Bradford, M.A. and Trahair, N.S. Web distortion and flexural-torsional buckling, J. Struct. Div. ASCE 1980, 106(ST7), 1557-1571

16 Bradford, M.A. and Trahair, N.S. Distortional buckling of I-beams J. Struct. Div., ASCE 1981, 107(ST2), 355-370

1:30 Eng. Struct . 1 9 9 2 , Vol . 14, No 2

17 Bradford, M.A. and Trahair, N.S. Distortional buckling of thin-web beam-columns Eng. Struct. 1982, 4, 2 - 1 0

18 Vlasov, V.Z. Thin-walled elastic beams (2nd edn), National Science Foundations, Washington D.C., 1961

19 Washizu, K. Variational methods in elasticity and plasticity (2rid edn) Pergamon Press, New York, 1975

20 Chu, T.C. and Schnobrich, W.C. Finite element analysis of transla- tional shells, Comp. Struct. 1972, 2, 197-222

21 Cook, R.D. Concepts and applications of finite element analysis, (2nd edn), John Wiley, New York, 1981

22 Miller, R.E. Reduc2ion of the error in eccentric beam modelling Int. J. Num. Methods Eng. 1980, 15, 575-582

23 Bathe, K.J. and Wilson, E.L. Numerical methods in finite element analysis, Prentice-Hall, New York, 1976

24 van Erp, G.M. Advanced buckling analysis of beams with arbitrary

A p p e n d i x

Transformation matrices [ T R] , [ Tf] and [Tm ]

C C

- S S

C

- S

C

S

C

in which C = cos ¢, S = sin ¢~ and ~ = angular displacement between the local principal axes of the flange element and the local axes of the super element.

For the top flange element:

=

"1 0 -37~r, 0 0 0 0 0 0

0 3Yc7 Y~r 0 0 0 0 3QT 0 --QT 2L 4 2L 4

0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 --Z,T 0

0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 ~T 0

0 0 0 0 0 0 0 0 0 I

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0


cross-sections, Ph.D. Thesis, 1989, University of Eindhoven, The Netherlands

25 Bradford, M.A. and Waters, S.W. Distortional instability of fabricated monosymmetrie I-beams, Comp. Struct. 1988, 29(4), 715-724

26 Wittrick, W.H. and Williams, F.W. Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings, Int. J. Mech. Sci. 1974, 16, 209-239

27 Trahair, N.S. Elastic stability of continuous beams, J. Struct. Div., ASCE 1969, 95(ST6), 1295-1312

28 Roberts, T.M. and Jhita, P.S. Lateral, local and distortional buckling of 1-beams, Thin-Walled Struct. 1983, 1,289-308

29 Azizian, Z.G. Instability of beams and plate girders, Report, Dept. Civ. and Struct. Eng., Univ. College, Cardiff, 1982

- S

C

S

C

- S -

C I

I

(AI)

Z.cr 0 0 0 0 0 0 0 0 0 0 0 0 O"

0 0 -3Ycr YeT 0 0 0 0 -3~'~r 0 -Z~r 0 1 2L ' 4 2L 4

0 1 0 --YcT 0 0 0 0 0 0 ZcT 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 o 1 0 0 o 0 o o --Zsr 0 0 0 I

0 0 0 I 0 0 0 0 0 0 0 O0

0 0 0 0 0 0 0 0 0 0 0 O0

0 0 0 0 0 0 0 0 0 0 0 O0

0 0 0 0 0 0 0 0 1 Y~T 0 0 0

0 0 0 0 0 0 0 0 0 0 1 O 0

0 0 0 0 0 0 0 0 0 0 0 O 0

I 0 0 0 0 0 0 0 0 0 0 O0

0 0 0 0 0 0 0 0 0 I 0 O0

0 0 0 0 0 0 0 0 0 0 0 1 0

(A2)

Eng. Struct. 1992, Vol. 14, No 2 131

Stabil i ty o f thin-wal led members: C. K. Chin et al.

in which 2for, z~r = coordinates of the centroid of the top flange element relative to that of the super element, and Y,r, Z,r = coordinates of the shear centre of the top flange element relative to the local axes at the junction line between the top flange and the web of the super element.

For the bottom flange element

[ ~ 1 =

1 0 -37cB 0 0 z,-8 0 0 0 0 0 0 0 0

0 32cB YcB 3~B 0 - z J 0 0 0 0 0 0 -3YcB 2~B 2L 4 2L 4 2L 4

0 0 0 0 0 0 0 0 0 0 0 1 0 - ~ 8

0 1 0 0 -z~80 0 0 0 0 0 0 0 0

0 0 I 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 000000 1 0

0 0 0 0 0 0 000000 0 1

0 0 0 1 y~B 0 000000 0 0

0 0 0 0 0 1 000000 0 0

0 0 0 0 0 0 000000 0 0

0 0 0 0 0 0 000000 0 0

0 0 0 0 1 0 000000 0 0

0 0 0 0 0 0 I00000 0 0

0 0 0 0 0 0 000000 0 0

0 0 0 0 0 0 000000 0 0

in which ~P~B, ~B = coordinates of the centroid of the bottom flange element relative to that of the super element, and YsB, zsB = coordinates of the shear centre of the bottom flange element relative to the local axes at the junction line between the bottom flange and the web of the super element.

[ T ~ ] =

"1 0 -Yc., 0 0 0 0 0 0 0 0 0

0 3Yc~ Y¢~ 0 0 0 0 0 0 0 0 0 2L 4

0 0 0 0 0 0 0 0 0 0 0 1

0 I 0 0 0 0 0 0 0 0 0 0

0 0 I 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

in which ~9~w = coordinate of the mid-height node of the web relative to the centroidal axes of the super element.

0 0

33~cw -3~¢w

2L 4

0 -Y~w

0 0

0 0

1 0

0 1

0 0 0 0 0 0 0 0 0 '

0 0 0 0 0 0 1 2L 4

0 0 zT, cB 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 --ZsB 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

1 y,~ 0 0 0 0 0 0 0

0 0 I 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 I 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 O"

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

(A3)

(A4)

132 Eng. Struct. 1992, Vol. 14, No 2

Stability of thin-walled members having arbitrary flange shape and flexible web

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