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FLEXURAL TORSIONAL BUCKLING OF THREE DIMENSIONAL

THIN WALLED ELASTIC BEAMS

Katy Saadé1, Bernard Espion2 (Member ASCE) and Guy Warzée3

ABSTRACT

This paper presents an advanced finite element formulation of the torsional behavior of thin walledbeams with arbitrary asymmetrical open or closed cross sections. The warping includes two terms todescribe the warping along the midline (global warping) and across the thickness (local warping) of thethin walls and is able to model the effects of non-uniform torsion. An updated corotational Lagrangianformulation is presented in order to study the flexural-torsional buckling problem. To solve the nonlinearproblem, an incremental-iterative technique combines the Newton Raphson method with the constant arclength of incremental displacements. Numerical examples are presented to show the performance, accuracyand efficiency of the proposed warping function for Eulerian stabilit y of thin walled beams and columns.

Keywords: Thin-walled beams, Finite element method, Flexural-Torsional Buckling.

INTRODUCTIONThe carrying capacity of thin-walled beams and columns is often governed by instability.

Thin walled beams may fail in a flexural or/and torsional buckling mode: the beam suddenlydeflects laterally or twists out of the plane of loading. In beam geometrical nonlinearities, strainsremain small and the large movements are mostly due to rigid body motions. In this paper, anupdated corotational lagrangian formulation is thus used. The reference is the last knownconfiguration. The corotational formulation eliminates the rigid body rotations from theincremental solution. High order terms of nodal parameters in the element beam model areneglected and rigid body motions are separated from local deformations. The largedisplacement/rotation problem is solved step by step with moderate rotations. Results presentedin this paper are limited to Eulerian stability analysis. The criterion to determine the bucklingstate is the singularity of the tangent stiffness matrix of the structure. Geometric buckling isanalyzed for thin walled beams with asymmetrical cross sections, where centroïd and shearcenter do not coincide. The study, deriving from Proki G work (Proki HJILKNM2MPO�Q , uses a singlewarping function which is valid for either open or closed cross sections.

1 Department of Continuum Mechanics, University of Brussels (ULB), C.P. 194/5, Av. F. D. Roosevelt 50, 1050

Brussels. Email : ksaade@smc.ulb.ac.be2 Department of Civil Engineering, University of Brussels (ULB). Email : bespion@ulb.ac.be3 Department of Continuum Mechanics, University of Brussels (ULB). Email : gwarzee@smc.ulb.ac.be

2

KINEMATIC MODELThe warping displacement is described by two terms. The first term describes the local

warping and depends on the derivative of the torsional rotation angle �

x,x , on the distance to theshear center hn and on the distance to the midline e (Fig.1).

x,xlocwarpingu ����� where Z(y,z)=hn(s).e (1)

The second term describes the global warping and is measured by a linear combination ofdisplacement parameters at selected nodes (ui) of the cross section.

¦� �� n

1ii

iglobwarping )x(u)z,y(u (2)

Figure 1. Cross-section of a thin walled beam

The functions �

i describe the mode of deformation between adjacent transversal nodes of thecross section.

The displacements at a point q are:

°°°°

¿

°°°°

¾

½

°°°°

¯

°°°°

®

­

T�

T��

:�ZT�

�

°°°°

¿

°°°°

¾

½

°°°°

¯

°°°°

®

­T�

�

°°°°

¿

°°°°

¾

½

°°°°

¯

°°°°

®

­T

�

°°°°

¿

°°°°

¾

½

°°°°

¯

°°°°

®

­

°°°°

¿

°°°°

¾

½

°°°°

¯

°°°°

®

­¦

xC

xC

n

1ii

ix,xzy0

q

q

q

)yy(

)zz(

)x(u)z,y(

0

v

y

w

0

z

0

0

u

w

v

u

(3)

The additional parameters (ui) are only related to the out of plane displacement due to thetorsional behavior. The cross section remains plane when subjected to normal force or bending.The displacements (ui) are not independent; they cannot take arbitrary values. They represent thewarping but they must induce no global elongation of the cross section neither global rotation ofthe section along flexural axis y and z. So, from the 6+n degrees of freedom, only 6+n-3 are

1

2

3 4

: (x,y)=1

z

Gy

qhn

e

s

C

3

independent: three translations (u,v,w), three rotations (�

x,�

y,�

z) and n-3 relative longitudinaldisplacements, where n is the number of transversal nodes of a cross section.

FINITE ELEMENT MODELThe beam finite element has three longitudinal nodes (Fig. 2). For each end node, there are

6+n degrees of freedom (u,v,w,�

x,�

y,�

z,ui,…). For the central node, 5 degrees of freedom(v,w,

�x,�

y,�

z…) exist. The beam displacements are expressed in terms of the nodal displacementsby using two functions. The transverse displacements (v,w) and the rotations (

�x,

�y,

�z) are

interpolated by a quadratic function N.

�����

���

�������

� ���)21(

)1(4231

N

2T avec

lx��

(4)

For the longitudinal displacements (u0, u

i…), a linear interpolation function N � is used.

� ��

�� � ����� 1

N T (5)

Fig 2. Finite element model

BUCKLINGUsing the updated lagrangian formulation, the incremental virtual work is given by:

³³ ����� "!!v

ii

v

ijijijklijkl dVdfudV)*eddd( (6)

where the volume V corresponds to the last known configuration which is taken as reference. dij kl

O,C

z,w

y,v

x,u

Tz

Ty

Tx

u 01T x1

v T y1

w T z1

u 11u 21…u n1

1 1

1

2

# x2

v T y2

w T z2

2

2

u 03#

x3

v T y3

w T z3

u 13u 23 …u n3

3 3

3

4

is an element of the Hooke matrix; �ij is the linear part of the Green-Lagrange strain tensor; e*ij is

the non linear part of it. �ij is the Cauchy stress tensor; fi is the external force vector. The equation

(6) can be developed and transformed into a matrix formulation:

[KT] {q} = {F} where [KT] =[KL ] + [KNL] (7)

An incremental iterative method is used to solve the nonlinear system of equations (7). Itassumes that the solution is known at the initial discrete step (t), and iterations are performed tocalculate the (t+1) equilibrium configuration by considering the equilibrium between the exteriorload forces and the nodal interior forces (equivalent to stresses in the element).

Critical loads are calculated by taking into consideration that the structure, already inequilibrium, reaches instability if there is more than one equilibrium position for the sameloading level. The criterion to determine this buckling state is the singularity of the tangentstiffness matrix [KT] of the structure.

EXAMPLESThe above-described finite element model has been implemented into a software. In the

following examples, critical loads for columns, beams and frames are compared with analyticalvalues obtained with other kinematic formulations.

Example 1: Plane frame buckling

The first example illustrates the plane flexural buckling of a portal frame. The columns andthe beam of the frame are identical and the closed cross section is given below (Fig. 3).E=210GPa, G=80GPa.

The difference between the finite element results and the solution given by Timoshenko(1961) is shown in Fig.3. The numerical value of the first critical load given by the softwareconverges to the value given the analytical solution of Timoshenko (Pcr=652N).

Fig.3 Stability analysis of a frame consisting of members with closed crosssection

Example 2: Flexural torsional buckling of a column

A column with open monosymmetric cross section is submitted to an axial load (Fig.4),

t=0.001m

0.04m

0.04m

PP P

5m

5m

Difference with reference solution for the first critical load

0.1%

1.0%

10.0%

100.0%

1 10 100

Number of elements

5

L=20m, E=210GPa, G=80GPa. The non coincidence of the centroïd(G) and the shear center(C)leads to a coupling between the flexural and the torsional buckling.

Fig. 4 Stability analysis of a column with open monosymmetric cross section

Two analytical studies are used to compare the results of the software. The first is based onVlassov’s theory and the second is based on the warping function described in this paper. Thebuckling non linear equations are developed from a state of combined torsion, bending and axialcompression. They are obtained from general equilibrium equations written for the deformedbeam or column. The solution is given by taking into consideration the boundary conditions.

The difference between the numerical value of the first critical load obtained by the softwareand the reference value based on Vlassov’s theory (Pcr=102223N) is illustrated in Fig.4. Theanalytical value based on the above described warping function converge to this value with adifference of less than 0.0001%.

Fig. 5 Critical loads for the centrally loaded column (Fig. 4)

P

L

Difference with reference solution for the first critical load

0.1%

1.0%

10.0%

100.0%

1 10 100Number of elements

0.1m

0.2m

tfs=0.02m

tw=0.01m

G

C

0.25m

Tfi=0.014m

0

5

10

15

20

25

30

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06 1.E+07

Critical loads [N]

Num

ber

of e

lem

ents

6

In a general flexural torsional buckling of a beam-column, the (xy) bending modes, the (xz)bending modes and the twisting modes are coupled. The first-order theory gives threehomogeneous equations and represents an eigenvalue problem. When, the shear center (C) andthe centroïd (G) coincide, the equations are uncoupled and the solution gives a discrete set ofbuckling modes. The lowest critical load is, in general, of practical significance. When C and Gdo not coincide, buckling involves simultaneously torsion and bending, and the critical load islower than if torsional effects are prevented.

Numerically, when the number of finite elements increases, the number of detected criticalloads increases. The number of detected buckling modes increases because it depends on thediscretization and on the interpolation functions. Fig. 5 shows the critical values obtained for thesame example (fig. 4) up to 10MN. The squares in fig. 5 represented along the horizontal axis arethe reference values of the buckling loads (based on Vlassov’s theory). The other sets of valuescorresponds to buckling loads detected by finite element analyses with increasing number ofelements (1, 2, 5, 10, 16 and 20 elements). For one element, there are only three critical valueswhen the applied load P increases from zero to 10MN. For two elements, the numerical values ofbuckling loads are improved and other critical loads appear and so on... So, each critical valueconverges to the reference solution with the increasing number of elements.

Example 3: Lateral torsional beam buckling

An I beam is loaded by two couples at its ends and is therefore subjected to uniform bending(Fig.6) L=20m, E=300GPa, G=99.5GPa. The critical value corresponding to the lateral torsionalbuckling is Mcr=6262.26Nm. Fig. 6 shows how the numerical solution converges to the referenceanalytical solution (Timoshenko, 1961).

Fig. 6 Lateral torsional buckling of an I beam

Example 4: Torsional buckling of a column

The last example illustrates a case of pure torsional buckling. A column with a cruciformsection submitted to an axial load is considered (Fig. 7). The thickness of the walls is t=4mm.L=1m, G=80.8GPa, E=210GPa. According to Vlassov’s theory, this kind of cross section doesnot warp. The theoretical torsional Eulerian buckling load is 258 398N. To initiate the torsionalbuckling of the column, a perturbation is needed in the finite element analysis; this is introducedby applying a small torsional moment Mx at mid height of the column. Fig. 7 gives the

M M

0.08m

0.08m

0.38m

tf=0.01m

tw=0.00375m

Difference with reference solution for the first critical load

0%

1%

10%

100%

1 10 100Number of elements

7

relationship between the axial load and the angle of twist at mid height for increasing values of Pand Mx. The horizontal line

� is the critical load 258398N. The curves represent the geometrical

non-linear variation of the angle of twist at mid height for different values of the ratio Mx/P:Mx/P=1.5 10-7 m for the curve � , 5 10-7 for � , 1.5 10-6 for � , 1.5 10-5 for � and 1.5 10-4 for � .The load-angle of twist relationship are obviously influenced by the magnitude of the appliedtorsional perturbation, but all curves reach asymptotically the level of the elastic buckling load.

Fig. 7 Torsional buckling of a column with cruciform cross section

SUMMARY AND CONCLUSIONSA finite element formulation has been used to study the flexural torsional buckling of elastic

structures composed of arbitrary thin walled cross sections. Some numerical examples for openand closed sections are presented. The numerical results are in excellent agreement with existinganalytical solutions. This model, which can be easily applied to an assembly of beams andcolumns with all kind of profiles, is currently used to study the flexural torsional buckling ofstructures combining beams with different cross sections (open or/and closed).

0.E+0

1.E+5

2.E+5

3.E+5

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007�[rad]

P[N

]

�

� �

� �

�

8cm

8cm

8cm

8cm

P

L

8

REFERENCESCriesfield M. A. (1997), "Non-linear Finite Element Analysis of Solids and Structures", Wiley,

Chichester, England.De Ville De Goyet V. (1989), "L'analyse statique non linéaire par la méthode des éléments finis

des poutres à section non symétrique", Ph.D. thesis, Université de Liège, Belgium (in French).Murray N. (1986), "Introduction to the theory of thin-walled structures", Oxford University

Press, New York, USA.Proki

� A. (1990), "Thin walled beams with open and closed cross section", Ph.D. thesis, Univ. of

Belgrade, Yugoslavia (in Serbian).Shakourzadeh H., Guo Y.Ch. and Batoz J. L. (1996), "On the large displacement and instability

of 3D elasto-plastic thin walled beam structure", Wiley, Eccomas 1996, Volume 2.Timoshenko S., Gere J. (1961), “Theory of elastic stability” , McGraw-Hil l, New York, USA.Trahair N. S. (1993), “Flexural-Torsional Buckling of Structures” , E & FN SPON, London.Bleich F. (1952), “Buckling strength of metal structures ” , McGraw-Hil l, New York, USA.