buk8-elastic limit state flexural–torsional postbuckling analysis of bars with open thin-walled...

Engineering Structures 22 (2000) 472–479www.elsevier.com/locate/engstruct

Elastic limit state flexural–torsional postbuckling analysis of barswith open thin-walled cross-sections under axial thrust

G.I. Ioannidis, D.J. Polyzois, A.N. Kounadis*

Department of Structural Analysis and Steel Bridges, National Technical University of Athens, 42 Patission Street, Athens 10682, Greece

Received 1 April 1998; received in revised form 6 July 1998; accepted 8 August 1998

Abstract

This work deals with the elastic limit state flexural–torsional postbuckling analysis of simply supported bars with open thin-walled asymmetric cross-sections under axial thrust. As it is well known stocky bars with the above type of cross-sections alwaysfail by flexural–torsional buckling in the case of asymmetric cross-sections (whose centroid does not coincide with the shear centre),while in the case of monosymmetric cross-sections the failure may occur either through flexural (Euler) buckling or flexural–torsional buckling depending on the geometric characteristics of the bars. In all the above three cases the critical state is associatedwith postbuckling strength. In this paper attention focuses on the first yielding occurring at the initial part of the post-critical pathof flexural–torsional buckling. This is, in case of bars made from ideal elastic–ideal plastic material, associated with the maximumcombined normal stress, due to axial compression, bending and warping, which, along with the nonlinear equilibrium equation,yield the maximum (ultimate) elastic load-carrying capacity. The elastic limit state postbuckling analysis given here is demonstratedwith the aid of equal-leg angles commonly used in trusses. 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Central compression; Elastic bars; Flexural–torsional buckling; Open thin-walled asymmetric cross-sections

1. Introduction

The use of light-weight and stiff structures is steadilyincreasing in modern structural design. Thus, thin-walled(closed or open) cross-sections are extensively used invarious engineering applications. However, the design ofstructures composed of thin-walled cross-sections posesparticular problems in their analysis, which becomemore severe in the case of asymmetric cross-sections,whose centroid does not coincide with the shear centre.

Instability problems of thin-walled sections have beenthe subject of extensive research. An excellent referencein this area is the early classical work presented by Vlas-sov [1]. Reviewing the present state of the art one couldrefer to the books presented by Chen and Atsutra [2] andTrahair [3], in addition to a large number of papers basedon linear analyses. Moreover, studies concerning thepostbuckling behaviour of beams and beam-columnsunder transverse loading have been presented by severalauthors [4–7]. Nevertheless, to the knowledge of the

* Corresponding author. Tel:+ 30 7723441; fax:+ 30 7723442

0141-0296/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (98)00091-1

authors, there is a lack of references in the area of thepostbuckling response of bars with asymmetric or singlysymmetric thin-walled open sections under axiallyapplied thrust [8].

In the present work the establishment of the initialpart of the postbuckling equilibrium path is necessary forthe determination of the ultimate elastic load-carryingcapacity of the bar related to the first yielding. The abovepostbuckling path is established using a simple and easyto apply technique presented by Kounadis [9].

2. Linear analysis

Consider the general case of a bar with length,, ofconstant thin-walled open cross-section, subjected to acompressive centrally applied loadP (Fig. 1). If the cen-troid of the cross-sectionC does not coincide with theshear centre, the buckling of the bar usually occursthrough a combination of bending and torsion. Ifx andy are the principal centroidal axes of the cross-section,andxo, yo the coordinates of the shear centreS, the equi-librium of the bar in a slightly deformed configuration

473G.I. Ioannidis et al. /Engineering Structures 22 (2000) 472–479

Fig. 1. Simply supported bar with a thin-walled open cross-section. Displacements of the shear centre (u, v, w) and the centroid after buckling.

resulting from the translation and rotation of the cross-section is considered. The translation is defined bydeflectionsu (along the axisx) andv (along the axisy)of the shear centreS (as well as the centroidC). Then,the shear centre moves fromS to S′ and the centroidfrom C to C′. The rotation of the cross-section about thenew position of the shear centreS′ is denoted byw andthe final position of the centroid byC″.

Equating the internal and the corresponding externalbending and torsional moments at an arbitrary pointalong the axisz, the following system of differentialequations of equilibrium for a simply suppotred bar, canbe obtained [10]:

− EIyd2u/dz2 = P(u + yow)

− EIxd2v/dz2 = P(v − xow)

GJdw

dz− ECw

d3w

dz3 = Pyo

dudz

− Pxo

dvdz

+ sIp

dw

dz6 (1)

whereEIx and EIy are the bending rigidities about theprincipal centroidal axesx and y, respectively;GJ andECw are the torsional and warping rigidity of the cross-section respectively;s = P/A, with A being the cross-sectional area and

Ip = Ix + Iy.

The above system of equations can also be writtenas follows:

EIyd2udz2 + Pu = − Pyow

EIxd2vdz2 + Pv = Pxow

ECw

d3w

dz3 − SGJ −Io

APD dw

dz= Pxo

dvdz

− Pyo

dudz

(2)

where

Io = Ix + Iy + (x2o + y2

o)A (3)

Differentiating the last of Eq. (2) with respect toz andsubstitutingu″ andv″ from the first two of Eq. (2) onecan obtain the following differential equation

ECww″″ − SGJ −Io

APDw″ − P(k2

xx2o + k2

yy2o)w (4)

= Pk2yyou − Pk2

xxov

where the prime denotes differentiation with respect toz and

k2x = P/EIx, k2

y = P/EIy (5)

Using the shape functions

u = uosinpz,

, v = vosinpz,

, w = wosinpz,

(6)

(wherevo, uo and wo are the lateral deflections and theangle of rotation at the middle of the bar) which satisfythe boundary conditions

u(0) = v(0) = w(0) = 0, u(,) = v(,) = w(,) = 0

d2u(0)dz2 =

d2v(0)dz2 =

d2w(0)dz2 = 0,

d2u(,)dz2 =

d2v(,)dz2 =

d2w(,)dz2 6 (7)

one can obtain, for a non-trivial solution, the followinginstability equation [10]

474 G.I. Ioannidis et al. /Engineering Structures 22 (2000) 472–479

||P − Py 0 Pyo

0 P − Px − Pxo

Pyo − Pxo

Io

A(P − Pt)

||

= 0 (8)

where

Px =p2EIx

,2 , Py =p2EIy

,2 , Pt =AIoSGJ +

p2

,2 ECwD (9)

denote the critical loads of flexural buckling about thex andy axes and the critical load of torsional buckling,respectively. Eq. (8) can be expanded under the form

Ip

Io

P3 + [AIo

(Pxy2o + Pyx2

o) − (Px + Py (10)

+ Pt)]P2 + (PxPy + PxPt + PyPt)P − PxPyPt = 0

Clearly, the smallest value ofP, obtained from theabove cubic equation is the critical buckling load for thecase in which buckling occurs with combined bendingand torsion. It can be shown [10] that Eq. (10) has threepositive roots, the smallest of which (critical load) issmaller thanPx, Py andPt.

For the case of a section in which the axisy is an axisof symmetry,xo = 0, Eq. (8) reduces to:

(P − Px)[Io

A(P − Py)(P − Pt) − P2y2

o] = 0 (11)

From Eq. (11) one can get

eitherPcr = Px (12)

or Pcr =1

2a FPt + Py − √(Pt + Py)2 − 4aPtPyGwherea = 1 − Ay2

o/Io

The smallest of the above two loads related to Eq.(12) is the critical one.

3. Postbuckling analysis

In order to study the postbuckling behaviour of thebar and establish the initial part of the postbuckling equi-

librium path, a more accurate relationship for the curva-ture should be used. Thus, the first and second of Eq.(1) can be written as follows [11]

P(u + yow) = − EIyu″

(1 − u′2)1/2

P(v − xow) = − EIxv″

(1 − v′2)1/26 (13)

Note that the above formulas have been based on theapproximation sinw . w and thus the nonlinear term−w3/6 has been omitted. Instead, as was shown in previousanalyses [6] the effect of such a nonlinearity on theinitial postbuckling path is for the purposes of thepresent analysis, negligibly small.

Using the approximations

u″(1 − u′2)1/2 = u″S1 +

12

u′2D andv″

(1 − v′2)1/2

= v″S1 +12

v′2DEq. (13) can be further simplified as follows

u″ + k2yu = − k2

yyow −12

u″u′2 (14)

v″ + k2xv = k2

xxow −12

v″v′2

Following the approximate analytical technique men-tioned in the indroduction [9] for solving nonlinearboundary-value problems, the shape functions Eq. (6)are introduced in the second member of Eq. (14)which become

u″ + k2yu = − k2

yyowosinpj +u3

op4

2cos2pj·sinpj (15)

v″ + k2xv = k2

xxowosinpj +v3

op4

2cos2pj·sinpj

where

j = z/,, u = u/,, v = v/,, yo = yo/,, xo = xo/, (16)

k2y = k2

y,2, k2

x = k2x,

2, uo = uo/,, vo = vo/,

Taking into account the boundary conditions

u(o) = v(o) = u(1) = v(1) = 0 (17)

the general integrals of Eq. (15) are


u(j) = − Fk2yyowo − (p4/8)u3

o

k2y − p2 Gsinpj

+(p4/8)u3

o

(k2y − 9p2)

sin3pj,(k2yÞp2,9p2) (18)

v(j) = Fk2xxowo + (p4/8)v3

o

k2x − p2 Gsinpj

+(p4/8)v3

o

(k2x − 9p2)

sin3pj,(k2xÞp2,9p2)

Integrating once the third of Eq. (2) and substitutingu and v from Eq. (18) into the second member the fol-lowing differential equation is obtained:

w″(j) + k2t w(j) = b2[(A1yo + A2xo)sinpj (19)

+ (A4xo − A3yo)sin3pj]

where

k2t = k2

t ,2, k2t = SIo

AP − GJD/ECw, b2 = P,4/ECw

and

A1 =k2

yyowo − (p4/8)u3o

k2y − p2 , A2 =

k2xxowo + (p4/8)v3

o

k2x − p2

A3 =(p4/8)u3

o

(k2y − 9p2)

, A4 =(p4/8)v3

o

(k2x − 9p2)

6 (20)

Eq. (19) can be solved with the aid of the boundaryconditionsw(o) = w(1) = 0, resulting in the followingfunction for the angle of rotation

w(j) = b2FSA1yo + A2xo

(k2t − p2) Dsinpj + SA4xo − A3yo

(k2t − 9p2) Dsin3pjG

(21)

Application of Eqs. (18) and (21) atj = 0.5 results inthe following system of nonlinear equations inu0, v0 andw0

uo(k2y − p2)(k2

y − 9p2) = − k2yyowo(k2

y − 9p2) − p6u3o

vo(k2x − p2)(k2

x − 9p2) = k2xxowo(k2

x − 9p2) − p6v3o (22)

wo(k2t − p2)(k2

t − 9p2) = b2hyo[A1(k2t − 9p2)

+ A3(k2t − p2)] + xo[A2(k2

t − 9p2) − A4(k2t − p2)]j

Setting

rx =Cw

Ix,2 , ry =

Cw

Iy,2 , m =

Io

A,2 , l =GJ,2

ECw

(23)

Eq. (22) can be transformed as follows

uo(b2ry − p2)(b2ry − 9p2) + p6u3o =

− b2ryyo(b2ry − 9p2)wo

vo(b2rx − p2)(b2rx − 9p2) + p6v3o =

b2rxxo(b2rx − 9p2)wo (24)

wo(mb2 − l − p2)(mb2 − l − 9p2)

= b2hyo[A1(mb2 − l − 9p2) + A3(mb2 − l − p2)]

+ xo[A2(mb2 − l − 9p2) − A4(mb2 − l − p2)]j

In the case in which the critical load of the bar isdetermined according to the second of Eq. (12), the barloses its stability developing simultaneously lateraldeflectionsu and angles of twistw. For each value ofthe dimensionless loadb2, the corresponding values ofuo, wo, can be readily obtained for given values of theparameters involved by solving the nonlinear equilib-rium Eq. (24) with respect touo andwo. These valuescan then be used to establish the respective equilibriumpaths (i.e. the plotsb2 versusuo andb2 versuswo). It isevident that the trivial solutionuo = 0, wo = 0 which rep-resents the fundamental equilibrium paths satisfies Eq.(24). The intersection of the fundamental path with thenonlinear postbuckling path, given by Eq. (24), corre-sponds to the critical bifurcation state. Clearly, if theshear centre coincides with the centoidxo = yo = 0 andthen wo = 0 (Euler flexural buckling).

4. Computation of stresses

Since previous analyses have shown that the lateral-torsional behaviour of the above bars is associated withpostbuckling strength [8] the present work deals with theonset of first yielding occurring at the initial postbuck-ling path.

Clearly, first yielding occurs when the maximum nor-mal stress in the cross-section becomes equal to the yieldstress of the material of the bar. This stress is given by

smax = sy = s0 + sby + sw (25)

wheres0 = P/A is the uniform stress due to axial com-pression;sby = M/Zy = Pu0/Zy is the maximum bendingstress (Zy the elastic section modulus about they axis)and sw is the maximum normal stress due to warping.

The normal warping stresssw in Eq. (25) is definedas [12]

sw(z) = Ew″(z)(w1 − w1) (26)

where


w1 = Es

0

rsds+ nrn (27)

The distancesrs and rn in Eq. (27) are shown in Fig.2, representing an open thin-walled cross-section, whereS is the shear centre andA (being the intersection of theaxess and h) is an arbitrary point of the mean line (ofthe cross-section) of lengths, w1 is the mean value ofw1 and n the distance from the mean line to any pointon the cross-section. For thin-walled open sections themaximum value ofn is t/2.

5. Application to an equal leg angle

5.1. Linear analysis

Consider the case of a pin-ended bar, with an equalleg angle cross-section, subjected to a centrally appliedload P (Fig. 3), which loses its stability through a flex-ural–torsional mode of buckling. The critical load canbe calculated according to Eq. (11). As shown in Fig. 3,x, y are the principal axes of the angle cross-section,jandh are the geometric axes,C the centroid,S the shearcentre,b the width of each leg andt is the thickness.

Fig. 2. Distancesrs and rn related to the warping constant of a thin-walled cross-section.

Fig. 3. Simply supported bar with an equal-leg angle cross-sectionunder an axial load.

Assuming thatt , b, the cross-sectional properties canbe approximated as follows:

A = 2bt (28)

(cross-sectional area),

jo = ho =b4

, yo = −b4

√2 (29)

(position of shear centre) and

Iy =13

tb3, Ix =112

tb3,

Ip = Ix + Iy =512

tb3

Io = Ip + Ay2o =

23

tb3

6 (30)

(moments of inertia). Finally, the torsional constantJand the warping constantCw are given by

J =23

bt3, Cw =118

t3b3 (31)

The buckling loads corresponding to Eq. (9) can bedetermined using Eqs. (28), (30) and (31) as follows

Px =p2

12Etb3

,2 ,

Py =p2

3Etb3

,2

Pt =3Et3

b S 13.9

+p2

18b2

,2D6 (32)

With the aid of the second and the third of Eq. (32)the critical loadPcr of Eq. (12) can be expressed as a


function of the geometric properties of the bar. In Eq.(12) the coefficienta is: a = 1 − Ay2

o/Io = 5/8.Using the critical stressscr = Pcr/A, after some elabor-

ation, the following expression for the dimensionlesscritical stress is obtained

scr

E=

215

p2ht2A + b2 − [( t2B + b2)2 − 2.50Bt2b2]0.5j

(33)

where

b = b/,, t = t/b (34)

and

B =3

1.3p2 +12

b2 (35)

5.2. Nonlinear analysis

The postbuckling equilibrium equations can beobtained from Eq. (24) by taking into account thatx0 = 0 and by introducing the expressions forA1 andA3

given in Eq. (20). Thus,

(b2ry − p2)(b2ry − 9p2)uo + p6u3o =

− b2ryyo(b2ry − 9p2)wo

(b2rx − p2)(b2rx − 9p2) + p6v2o = 0 (36)

wo(mb2 − l − p2)(mb2 − l − 9p2)

= b2yoFSb2ryyowo −p4

8u3

oD mb2 − l − 9p2

b2ry − p2

+p4

8u3

o

mb2 − l − p2

b2ry − 9p2 GSubstituting the expression forwo, given by the first

of Eq. (36), into the third of Eq. (36), yields

p6h(mb2 − l − 9p2)f(b2) + b4ryy2o[b2(m + ry)

− l − 10p2]ju2o = − (mb2 − l − 9p2) (37)

(ryb2 − p2)(ryb

2 − 9p2)f(b2)

where

f (b2) = ry(m − y2o)b4 − [ry(p2 + l) + mp2]b2 (38)

+ p2(p2 + l)

and

rx =23

t2b2,

ry =16

t2b2,

m =13

b2,

l = 4.6154/b2

yo = − b√2/4

(39)

The dimensionless deformationu0 corresponding to acertain value of the dimensionless compressive loadb2,can now be computed using Eq. (37).

With the aid of Eqs. (37) and (38) the postbucklingequilibrium path (b2 versusu0) can be established forequal leg angles. It has been shown [13] that the criticalload is associated with a stable symmetric bifurcationpoint. Nevertheless, the postbuckling equilibrium path isquite shallow and thus the bar cannot exhibit significantpostbuckling strength. An example of such an equilib-rium path, corresponding tob = t = 0.10, is shown inFig. 4.

5.3. Elastic limit state

The normal stressess0 and sby used in Eq. (25) canbe determined with the aid of Eqs. (28), (30), (31) and(34) and Fig. 3, as follows:

s0 = P/A =136

Et2b4b2

sby = My/Zy = Pu0

b√22

/Iy =√212

Et2b3u0b26 (40)

where u0 is obtained from Eq. (37).Referring to Eqs. (26) and (27), for an equal-leg angle

cross-section,rs = 0, w1 = 0 and, therefore, the normalstress due to warping can be written as

sw = Ew″(z)w1 = Ew″(z)nrn = Ew″(z)bt/2 (41)

At mid-height of the bar, according to Eq. (6),w″(1/2)= w0p

2/,2. Hence, the maximum normal stress due towarping is

sw =12

bt Ew0

p2

,2 =p2

2Etb2w0 (42)

Using Eqs. (40) and (42) Eq. (25), the condition corre-sponding to the first yielding of the bar, can beexpressed as

Esy

tb2

2 H 118

tbSb + 3√2u0Db2 + p2w0J = 1 (43)


Fig. 4. Initial postbuckling pathb2 versusuo of a simply supported bar with equal leg cross-section, under simultaneous bending and torsion dueto axial load. First yielding point.

With the aid of this equation the level of the externalloading b2, related to the onset of first yielding of thebar, can be determined taking into account the geometricproperties of the cross-section and the material proper-ties (E and sy). The associated deformationsu0 andw0

can be computed from Eqs. (36) and (37).The above analysis can be easily extended to other

boundary conditions. For instance, for a column withbuilt-in ends we have the following boundary conditions

u(0) = v(0) = w(0) = 0, u(,) = v(,) = w(,) = 0 (44)

u′(0) = v′(0) = w′(0) = 0, u′(,) = v′(,) = w′(,) = 0

The third of Eq. (2) or Eq. (4) (independent from theboundary conditions) is still valid, while the two first ofEq. (2) become

EIyu″ + Pu = − Pyow + EIyu″(0)

EIxv″ + Pv = Pxow + EIxv″(0)J (45)

These three equations and the boundary conditions Eq.(44) are satisfied by taking a solution in the form

u =uo

2 S1 − cos2pz

, D, v =vo

2 S1 − cos2pz

, D, (46)

w =wo

2 S1 − cos2pz

, DInserting these expressions into the aforementioned

three equations we obtain again Eq. (8), wherep2/,2 inPx, Py andPt given in Eq. (9) must be replaced by 4p2/,2.Regarding the postbuckling analysis, Eq. (15) is stillvalid after adding in their RHS the term 2up2 (in thefirst of Eq. (15)) and 2vp2 (in the second of Eq. (15)).The subsequent procedure follows exactly the steps ofthe analysis outlined in Section 3.

5.4. Numerical example

The analysis given above is applied to the case of anequal-leg angle cross-section withb = t = 0.10 fabri-cated from a steel having a yield stresssy = 900 N/mm2

and a modulus of elasticityE = 210 000 N/mm2.The critical buckling load corresponds to a flexural–

torsional configuration and is equal tob2 = 128150.972.Combining Eq. (37), the first of Eqs. (36) and (43), weestablish the first yielding, occurring atb2 = 128150.988.The corresponding displacements areu0 = 0.00111 andw0 = 0.113.

The position of the first yielding on the post-bucklingequilibrium path is shown in Fig. 4.

6. Conclusions

The most important conclusions of this study arethe following:

1. Using a simple and comprehensive postbucklinganalysis, the initial part of the flexural–torsional post-buckling path of bars with open thin-walled cross-sections under axial compressive force can be read-ily established.

2. Considering bars made from ideal elastic–ideal plas-tic material, their elastic limit state associated withthe first yielding can also be determined.

3. First yielding is related to the maximum normal stressin the middle cross-section of the simply supportedbar, being equal to the yield stress of the bar’smaterial. The above maximum normal stress isdetermined as a function of the uniform stress due toaxial compression of the maximum bending stressand of the maximum normal stress due to warping.One should notice the importance of the normal stressdue to warping, which can reach an appreciable per-centage of the total normal stress.


4. Since the postbuckling path, for the model chosen,is shallow, the margins of postbuckling strength arelimited and thus the load-carrying capacity is takenequal to that corresponding to the critical load.

5. Due to the shallowness of the postbuckling path firstyielding takes place near the critical state.

References

[1] Vlassov BZ. Pieces longues en voiles minces. Eyrolles, Paris,1962.

[2] Chen WF, Atsutra T. Theory of beam-columns. McGraw–Hill,NY. 1977.

[3] Trahair NS. Flexural–torsional buckling of structures. E.N. Spon,London, 1993.

[4] Woolcock ST, Trahair NS. Postbuckling behaviour of determi-nate beams. J. Eng. Mech. Div., ASCE 1974;100(EM2):151–71.

[5] Woolcock ST, Trahair NS. Postbuckling of redundant I-beams.J. Eng. Mech. Div., ASCE 1976;102(EM2):293–312.

[6] Kounadis AN, Ioannidis GI. Lateral postbuckling analysis ofbeam-columns. J. Eng. Mech. Div., ASCE 1994;120(4):695–706.

[7] Ioannidis GI, Kounadis AN. Lateral postbuckling analysis ofmonosymmetric I-beams under uniform bending. J. Constr. SteelRes. 1994;30:1–12.

[8] Kounadis AN. Postbuckling analysis of bars with thin-walledcross-sections under simultaneous bending and torsion due tocentral thrust. J. Constr. Steel Res. 1998;45:17–37.

[9] Kounadis AN. An efficient and simple approximate technique forsolving nonlinear initial and boundary-value problems. Comp.Mech. 1992;9:221–31.

[10] Timoshenko S, Gere J. Theory of elastic stability. McGraw–Hill,NY. 1961.

[11] Thompson GM, Hunt GV. General theory of elastic stability.John Wiley and Sons, London, 1973.

[12] Oden JT. Mechanics of elastic structures. McGraw–Hill, NY.1967, pp. 186–193.

[13] Ioannidis GI, Kounadis AN. Flexural–torsional postbucklinganalysis of centrally compressed bars with open thin-walledcross-section. Eng. Struct. 1999;21:55–61.

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