Post-bucklinganalysisofnon...

International Journal of Non-Linear Mechanics 40 (2005) 445–463

www.elsevier.com/locate/nlm

Post-buckling analysis of non-prismatic columns under generalbehaviour of loading

M. Zyczkowski∗

Politechnika Krakowska, Institute of Applied Mechanics,Cracow University of Technology, ul. Warszawska 24, PL-31-155 Kraków, Poland

Received 10 July 2003; received in revised form 21 April 2004; accepted 26 May 2004

Abstract

The paper shows the effects of behaviour of loading during the buckling process on the value of critical force and initialstability of post-buckling path for elastic, non-prismatic columns. Perturbation method combined with Croll’s manoeuvremakes it possible to derive general formulae for the first or the second correction of the force, and hence to analyze stabilityin the post-buckling range. The effects of behaviour of active and reactive forces may be essential: in numerous examplesthe boundary values of structural parameters separating stability and instability are evaluated. Pre-buckling geometry changesare analyzed as well.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Post-buckling analysis; Loading behaviour; Non-prismatic columns; Inverse method

1. Introduction

1.1. Post-buckling analysis of columns

Analysis of finite deflections of prismatic elasticcolumns after buckling was initiated by Euler in the18th century; he used elliptic integrals and ellipticfunctions, and the relevant deflection line was called“Euler’s elastica”. This direction was later widely de-veloped: classical results were presented in the booksby Popov[1] and Frisch-Fay[2]. In most cases the

∗ Tel./fax: +48-12-648-45-31.E-mail address:[email protected](M. Zyczkowski).

0020-7462/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2004.05.014

behaviour of force during buckling was assumed asfollows: the point of application fixed to the mat-ter, and the direction fixed in space (“dead load”).This type of behaviour is being called the “Eulerianbehaviour” of loading. In many cases, however, theloading behaviour does not conform to this scheme(path-dependent loading).

Probably the first exact analysis of finite deflectionsof columns under non-Eulerian loading behaviour isdue to Stern[3]. He considered elastica for a cantilevercolumn loaded via a rigid rod with sliding upper end(Fig. 1) and determined the regions of stable and unsta-ble post-buckling path. The direction of force loadingthe column is here no longer constant in space. Fromamong more recent results concerning non-Eulerian

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446 M. Zyczkowski / International Journal of Non-Linear Mechanics 40 (2005) 445–463

P

f

l deformable

rigida

Fig. 1. Stern’s column after buckling.

forces and based on elastica we mention Wilson andSnyder[4] and Kandakis[5] (follower forces).

Advantages of Euler’s approach are connected withexactness of the analysis and with applicability withinthe whole range of loadings and deflections. However,the results are not perspicuous and stability of post-buckling behaviour is difficult to be estimated. In 1945Koiter [6] proposed an alternative approach based onenergy criterion of stability combined with perturba-tions, namely expansions into power series of a certainsmall parameter� characterizing deformations. Thisparameter should be defined as a suitable norm forfunction or functions describing deformation, mono-tonically increasing during the buckling process (con-trol parameter). In the simplest case of a cantilever(clamped–free) column usually the deflection at thefree endf is chosen as that parameter: it may be re-garded as a Chebyshev’s norm for deflections (thoughin some more complicated problems this coincidencemay not hold true). From among other proposals defin-ing � we mention that by Riks[7] (current length ofthe equilibrium path, obviously increasing during theprocess) and by Privalova and Seyranian[8] (Gaussiannorm for the deflection angle). For the sake of gener-ality we make use of an arbitrary small parameter�; itwill be specified just in particular examples. The signs

of coefficients of expansion of the loading parameterP into power series of� determine stability of post-buckling behaviour. Such expansions can also be de-rived directly from the governing non-linear equationof the problem without energy considerations. Thenthe manoeuvre proposed by Croll[9] (orthogonaliza-tion of the expanded equation with a suitably chosenfunction and integration by parts) makes it possibleto evaluate a higher-order expansion coefficient forP

by using just a lower-order deflection function. Thisapproach will be used in the present paper.

For some applications it is important to know re-lations of the type� = �(P ), inverse with respect tothose described above. Such relations may be ob-tained for stable post-buckling behaviour. For a pris-matic column under Eulerian force it was derived byZyczkowski [10] who used inversion of the relevantpower series forP = P(�), [11]. However, conver-gence of such series is rather poor; it can be improvedby a suitable extrapolation procedure[12].

Koiter’s theory and related approaches were ex-tensively developed, mainly in connection with shellbuckling problems since initial post-buckling be-haviour of shells is in most cases unstable. Moreover,for imperfect shells it determines upper critical loading(limit point) in terms of the imperfection parameter.Many results were presented in the survey papers byHutchinson and Koiter[13], Budiansky[14], Potier-Ferry [15], and in the books by Thompson and Hunt[16], Dym [17], Huseyin [18], Bažant and Cedolin[19], Troger and Steindl[20], Nguyen [21] and byAtanackovic [22]. From among more recent papersdevoted to post-buckling behaviour of prismatic elas-tic columns we mention here Plaut[23] and Kolkka[24] (multiparameter loading), Plaut[25], Kounadis[26], Rao and Rao[27] (non-conservative loading)Haslach[28], Kounadis and Mallis[29], Szymczakand Mikulski [30] (effects of material non-linearity),Tauchert and Lu[31], Hui [32], Lee and Waas[33],Wu and Zhong[34] (columns on elastic foundation),Damil and Potier-Ferry[35] (higher-order expansionterms), Luongo and Pignataro[36] (nearly symmet-ric structures), Godoy[37] (dependence on certainparameters, e.g. responsible for deformation of across-section), Beda[38] and Wu [39] (secondarybuckling).

Post-buckling analysis of non-prismatic elas-tic columns was initiated by Frauenthal[40]: he

M. Zyczkowski / International Journal of Non-Linear Mechanics 40 (2005) 445–463 447

investigated columns determined by classical uni-modal optimization and proved that for the Eulerianbehaviour of force the initial post-buckling path isalways stable. A similar approach for columns elas-tically clamped at both ends, subject to multimodaloptimal design, is due to Privalova and Seyranian[8] who found that post-buckling behaviour of statesemanating from mixed modes is unstable. Also Lee,Wilson and Oh[41], Lee and Oh[42] investigatedpost-buckling behaviour of tapered columns designedby using parametrical optimization. Abdelkader[43] found an apparently exact elastica solution forcolumns with monotonically varying stiffness, but fora slightly changed non-linear governing equation: inone term of this equation the moment of inertiaI (s)was approximately replaced byI2(s)/I (0), whereI (0) is that moment at the free end. Also Domokosand Holmes[44] gave certain elastica solutions fornon-prismatic columns.

Post-buckling analysis of elastic-plastic structureswas initiated by Hutchinson[45]. He consideredthe Ryder–Shanley model allowing for imperfec-tions modified by adding a transversal spring, andthen a perfect spherical shell. In a subsequent pa-per [46] Hutchinson derived general expression forpost-buckling behaviour of perfect elastic–plastic-hardening columns with symmetric cross-section.Fractional powers of the deflection were introducedwith the exponents depending on the cross-sectionalshape. Tvergaard and Needleman[47] consideredalso asymmetric sections. Their approach resembledto a certain degree the method of generalized powerseries used byZyczkowski[48,49] to describe plasticinteraction curves for combined bending with normalforce. Lewis et al.[50] considered elastic bucklingunder concentrated and distributed forces, followedby partly elastic, and partly elastic–perfectly plasticpost-buckling behaviour; appearance of plastic defor-mations usually changed a stable post-buckling char-acteristic into an unstable one. Plastic buckling andpost-buckling of prismatic columns was analyzed byLeger and Potier-Ferry[51], Chang[52] (allowing forimperfections), and by Cimetiere and Leger[53] (es-timation of maximal loading). Plastic post-bucklinganalysis of non-prismatic columns was initiated byLeger and Potier-Ferry[54].

In the case of plates and shells the post-bucklinganalysis is particularly effective if combined with the

finite-element method. In the case of columns suchan approach is also more and more developed. Thebibliography and numerous results are presented in thebook by Waszczyszyn et al.[55].

1.2. Behaviour of loading during buckling andpost-buckling

Most of the papers mentioned above consideredcolumns under Eulerian behaviour of the force. How-ever, in engineering applications this is not always thecase and other types of behaviour may appear. This factmay result not only in numerical values of the criticalforces, but may even bring essential qualitative effectslike loss of stability of columns under tension[56,57].This would be impossible under Eulerian loading un-less large strains resulting in a decrease of the cross-section are allowed for. In the post-buckling rangestability may change into instability. Many particularcases of loading behaviour were considered be Feo-dosyev[58], Rzhanitsyn[59], Panovko and Gubanova[60], Ziegler [61], Bažant and Cedolin[19].

Classification of the behaviour of loading dependson the type of support. Consider first the critical stateof a perfect column loaded axially by a concentratedforce P . The most interesting is the case of loadingat the free end: then in the adjacent position (or dur-ing small vibrations) an additional transversal com-ponentH and an additional momentM may appear.They may depend on the deflectionf and on theangle of deflection� at this end. In the descriptionof the critical state the linear homogeneous functionsH=H(f, �) andM=M(f, �) are sufficient, and hencethe behaviour of loading is described by four param-eters. General equations for this case were derived byKordas [62] both in the range of static and of ki-netic criterion of stability. Gajewski andZyczkowski[57,63] determined optimal shapes of columns in thisgeneral case with prescribed critical force as the opti-mization constraint, but under additional assumptionof conservative behaviour of loading. In the case ofa simply supported end the behaviour of loading maybe described by one functionM =M(�), and in thecase of clamped end movable both in longitudinal andtransversal directions—by one functionH=H(f ). Fi-nally, in the case of clamping admitting motion only inlongitudinal direction—the problem of the behaviourof loading does not exist.


In the post-buckling range the problem is muchmore complicated. Then the longitudinal componentof loadingV may change, and the longitudinal dis-placement of the endu must also be taken into ac-count. For a free end we describe the behaviour ofloading by three non-linear functionsH =H(f, �, u),M = M(f, �, u), andV = V (f, �, u). For a simplysupported endM =M(�, u) andV = V (�, u); for aclamped end movable in both directionsH =H(�, u),V = V (�, u); for a clamped end movable only in lon-gitudinal directionV = V (u).

Of course, the loading may also act inside theinterval of the column, then its behaviour can be de-scribed as at a free end. Moreover, some more sophis-ticated cases of loading behaviour may appear, butthey will not be discussed in the present paper. Sincethe problems of loading behaviour are most interest-ing for cantilever columns, we restrict our presentconsiderations to such type of support. The elaboratedmethod can be used for other support conditions aswell.

1.3. Aim of the paper and basic assumptions

The aim of the present paper is to analyze ini-tial post-buckling stability of elastic non-prismaticcolumns allowing for various types of behaviour ofactive and reactive forces during buckling, describedin a general manner. Two approaches will be used: adirect approach for given stiffness distribution, and aninverse approach for given deflection line in its linearapproximation—then the relevant stiffness distribu-tion is subject to evaluation. This inverse approach,introduced to buckling problems byZyczkowski[64], and called the “method of assumption of theexact equation of the deflection line”, was recentlybroadly employed by Elishakoff and his collabora-tors [65–67]. It is particularly useful in structuraloptimization problems, where the stiffness distribu-tions is in any case a priori unknown and should bedetermined. Optimal design of columns with addi-tional constraint of stable post-buckling behaviourwas initiated by Bochenek[68–70]. He discretizedthe problems, expressed them in terms of non-linearprogramming, and used the method of moving asymp-totes to obtain numerical results. The present paperis regarded as a basis for analytical approach to suchproblems.

The assumptions are as follows:

• The analysis will be restricted to cantilever columns,perfect, linearly elastic, loaded at the free end inpre-critical range by an axial forceP .

• The behaviour of loading in post-buckling rangewill be described by three functionsH=H(f, �, u),M=M(f, �, u), andV =V (f, �, u), both for activeand for reactive forces. It is assumed that these func-tions may be expanded into power series of theirarguments.

• Only initial post-buckling behaviour will be studied,but the method used makes it possible to evaluatefurther coefficients of the power series expansion.

• Axis of the column is assumed to be incompressible(like in the classical Euler’s analysis). Transverseshear effects will be neglected. Initial post-bucklinganalysis of prismatic Eulerian column allowing forshear effects was given by Huang and Kardomateas[71].

• Purely static approach to stability analysis will beused. This approach is sufficient for all cases ofconservative behaviour of loading, but also holdsfor some non-conservative cases (“conservative sys-tems of the second kind”, Leipholz[72,73]).

• Thin-walled columns with possible mode interac-tion will not be discussed.

• In principle, we assume, that there are nopre-critical geometry changes affecting loadingbehaviour. However, in the last section thecomplications due to possible geometry changeswill be discussed as well.

2. An illustrative example

2.1. A symmetric system

A general description of the behaviour of loadingin the post-buckling range is much more complicatedthan in the critical state, hence we start our consid-erations with an illustrative example. It is apparentlysimple but shows many important details.

Consider post-buckling behaviour of an elastic can-tilever column of the lengthl, joined with a transverserigid beam of the length 2b, and additionally supportedby two symmetric, oblique linear springs with equalstiffnessC (Fig. 2). In order to simplify the problem


bb

rigid

l

deformable

C C

P

Fig. 2. Column with a transverse rigid beam and springs.

we assume Eulerian behaviour of the external forceP and focus our attention on the behaviour of reac-tive forces in the springs. The column after bucklingis shown inFig. 3. The elongations of the springs Iand II equal

�lI=√l2I + b2

I −√l2 + b2,

�lII=√l2II + b2

II −√l2 + b2,

where

lI = l − u− b sin �, bI = b cos� + f,lII = l − u+ b sin �, bII = b cos� − f. (2)

The system of coordinates adopted is shown inFig. 4. The symbolsf, �, and u(0) = u, denote, inturn, deflection, angle of deflection, and axial dis-placement at the free end. We look for the functionw = w(s), wheres is measured along the deformedaxis assumed to be inextensible, 0�s� l.

As it is usual in post-buckling analysis we employthe perturbation method and expand the functionw(s)

in a power series of a certain representative small pa-rameter�. Since we confined here our considerations

b bf

P

α

II I

II

l

l

I

u(0) = u

R I

IV

IHR IIIIV

IIH

Fig. 3. Column after buckling.

w

u(0) = u

α

f

x, u

z, w

s

x

Fig. 4. System of coordinates adopted.

just to symmetric systems,w is an odd function of�,and we may write

w =∞∑i=1

w2i−1(s)�2i−1 = w1(s)� + w3(s)�3 + . . .

(3)


The remaining characteristic generalized displace-ments are related tow by the formulae

u= l −∫ l

0

√1 − w′2 ds = 1

2�2

∫ l

0w′

12 ds

+(

1

8

∫ l

0w′

14 ds +

∫ l

0w′

1w′3 ds

)�4 + . . . (4)

�= arcsinw′(0)= w′1(0)� +

[w′

3(0)

+1

6w′

13(0)

]�3 + . . . . (5)

In what follows, we use the notation

f = f1� + f3�3 + . . . , � = �1� + �3�3 + . . . ,u= u2�2 + u4�4 + . . . . (6)

It is seen thatf1, �1, andu2 are fully determinedby w1, whereasf3, �3 and u4 needw3, and so on.Similarly, in post-buckling range of a symmetric sys-tem we use the expansion

P = Pcr + P2�2 + P4�4 + . . . (7)

and our aim is to determineP2 the sign of which deter-mines stability of initial post-buckling behaviour ([16]use (7) with slightly different notation of coefficientsPi).

The reactive forcesRI andRII are directly propor-tional to�lI and�lII , (1). For our purposes more im-portant are the componentsH andV , and namely

HI = − bI√l2I + b2

I

RI = −CbI

1 −

√l2 + b2√l2I + b2

I

,

VI = lI√l2I + b2

I

RI = ClI

1 −

√l2 + b2√l2I + b2

I

,

HII = bII√l2II + b2

II

RII = CbII

1 −

√l2 + b2√l2II + b2

II

,

VII = − lII√l2II + b2

II

RII = ClII

1 −

√l2 + b2√l2II + b2

II

.

(8)

The column after buckling is loaded by the longitu-dinal forceV=P+VI +VII , transversal forceH=HI +HII , and the momentM = (HII −HI)b sin � + (VI −VII )b cos�. Expanding the roots in (8) into power se-ries and taking into account the orders of magnitudeof f, �, andu, we first obtain

V=P + C{

1

(l2 + b2)[(l − u)(u2 − 2lu+ f 2)

−2b2 sin � (f cos� − l sin � + u sin �)]− 3b2l

(l2 + b2)2(l2 sin2 � + f 2 cos2 �

−2f l sin � cos� + . . .)+ . . .}, (9)

H=C{

1

(l2 + b2)[f (−u2 + 2lu− f 2)

+2b2 cos�(l sin � − f cos� − u sin �)]+ 3b2

(l2 + b2)2[f (l2sin2� + f 2cos2�

−2f l sin � cos� + . . .)+ cos�(2l2u sin� − 2f lu cos� − f 2l sin �

+ f 3 cos� + . . .)] + . . .}, (10)

M=Cb2{

2

l2 + b2 [f sin � + (l − u) cos�]×[f cos� − (l − u) sin �]− 3l cos�

(l2 + b2)2(2l2u sin � − 2f lu cos�

− f 2l sin � + f 3 cos� + . . .)}. (11)

Finally, expanding trigonometric functions intopower series and making use of (6) and (7) weobtain the following power series of the smallparameter�:

V=(Pcr + P2�2 + . . .)+ C{

1

(l2 + b2)2

×[(l2 − 2b2)lf 21 + 2b2(2l2 − b2)f1�1

+b2l(−l2 + 2b2)�21

− 2l2(l2 + b2)u2]�2 + . . .}, (12)


H=C{

2b2

l2 + b2 (−f1 + l�1)�

+ 1

(l2 + b2)2[(−l2 + 5b2)f 3

1 − 9b2lf 21�1

+(5l2 + 2b2)b2f1�21

−43(l

2 + b2)b2l�31 + 2(l2 − 2b2)lf 1u2

+2(2l2 − b2)b2�1u2

+ 2(l2 + b2)b2l(−f3 + l�3)]�3 + . . .}

(13)

M=Cb2{

2l

l2 + b2 (f1 − l�1)�

+ 1

(l2 + b2)2[−3lf 3

1 + (5l2 + 2b2)f 21 �1

−4(l2 + b2)lf 1�21 + 4

3(l2 + b2)l2�3

1

+2(2l2 − b2)f1u2 + 2(−l2 + 2b2)�1u2

+ 2(l2 + b2)l(f3 − l�3)]�3 + . . .}. (14)

We do not continue this particular example sepa-rately, but use the results obtained here to formulate ageneral approach.

2.2. An asymmetric system

It is sufficient to leave just one spring in the aboveexample to obtain an asymmetric system. ThenV =P + VI , H = HI , M = MI , and many terms subjectpreviously to reduction remain: the series are neitherodd, nor even, but all terms are present. Instead of(3)–(6) we have to write

w=w1(s)� + w2(s)�2 + w3(s)�3

+ . . .=∞∑t=1

wt(s)�t , (15)

�=w′1(0)� + w′

2(0)�2

+[w′

3(0)+1

6w′

13(0)

]�3 + . . .=

∞∑t=1

�t�t , (16)

u=1

2�2

∫ l

0w′

12 ds + �3

∫ l

0w′

1w′2 ds

+ . . .=∞∑t=2

ut�t , (17)

P

M

V

H

Fig. 5. Loading of the column after buckling.

P = Pcr + P1� + P2�2 + . . .=∞∑t=0

Pt�t (18)

and the analysis of such systems is more complicated.

3. General approach to post-buckling behaviour

3.1. The governing non-linear equation

For the general behaviour of loading, characterizedby theH , M andV (Fig. 5), the exact equation ofbuckling may be written in the form

EI(s)w′′√

1 − w′2+ Vw +M +Hx = 0, (19)

where

x =∫ s

0

√1 − w′2(s)ds (20)

and s denotes the variable of integration.

3.2. Expressions for the generalized forces

Making use of the expression derived in Section2 for the behaviour of reactive forces and assumingsimilar behaviour of active (external) force, we nowpresentV , H andM by the series of homogeneous


polynomials of thenth order of magnitude (not nec-essarily of thenth degree, sinceu is proportional to�2). For the general, asymmetric case

V=P +∞∑n=1

n∑i=0

n∑j=0

q∑k=0

(PV ijk + Cvijk)

×f i�j uk , (21)

H=∞∑n=1

n∑i=0

n∑j=0

q∑k=0

(PHijk + Chijk)

× f i�j uk , (22)

M=∞∑n=1

n∑i=0

n∑j=0

q∑k=0

(PMijk + Cmijk)

× f i�j uk , (23)

under the conditioni + j + 2k = n. In these expres-sionsq = n/2 for evenn andq = (n − 1)/2 for oddn. In the symmetric case in (21) just even terms re-main,n = 2,4,6 . . ., and in (22) and (23)—just oddterms, n = 1,3,5 . . . . The coefficientsVijk, Hijk,andMijk describe the behaviour of the external (ac-tive) force, vijk, hijk, and mijk—the behaviour ofreactive forces (like in (9)—(11) after expansion oftrigonometric functions), andC denotes the stiffnessof reactive springs. For several springs with variousstiffnesses an appropriate sum should be taken.

Moreover, we introduce the expansion of the forceP : (18) in the general case or (7) for symmetric struc-tures. Making use of expansions (16) and (17) we mayintroduce a shorter notation

V=Pcr + (P1 + PcrV1 + Cv1)�+(P2 + P1V1 + PcrV2 + Cv2)�2 + . . . ,

H=(PcrH1 + Ch1)� + (P1H1 + PcrH2 + Ch2)

×�2 + . . . ,M=(PcrM1 + Cm1)� + (P1M1 + PcrM2 + Cm2)�2

+ . . . ,(24)

where

V1=V100f1 + V010�1,

V2=V200f21 + V110f1�1 + V020�2

1

+V001u2 + V100f2 + V010�2,

H1=H100f1 +H010�1,

H2=H200f21 +H110f1�1 +H020�2

1

+H001u2 +H100f2 +H010�2, (25)

and so on. For symmetric cases some terms vanish andinstead of (24) we obtain

V=Pcr + (P2 + PcrV2 + Cv2)�2 + . . . ,H=(PcrH1 + Ch1)�

+(P2H1 + PcrH3 + Ch3)�3 + . . . ,M=(PcrM1 + Cm1)�

+(P2M1 + PcrM3 + Cm3)�3 + . . . , (26)

whereH1 andM1 retain their form (25)whereas

V2=V200f21 + V110f1�1 + V020�2

1 + V001u2,

H3=H300f31 +H210f

21 �1 +H120f1�2

1 +H030�31

+H101f1u2 +H011�1u2

+H100f3 +H010�3,

M3=M300f31 +M210f

21 �1 +M120f1�2

1 +M030�31

+M101f1u2 +M011�1u2

+M100f3 +M010�3. (27)

Our aim is to findP1 in the asymmetric case andP2 in the symmetric case without solving the equa-tions forw2(s) orw3(s), respectively. From this pointof view it is convenient to change slightly the abovenotation. Namely, in the asymmetric case all terms ofH2, h2,M2, andm2 are known ifw1 is known, exceptthe last terms containingf2 and�2. Similarly, in thesymmetric caseH3, h3,M3 andm3 are known exceptthe terms containingf3 and�3. The formulae forP1or P2 will be effective iff2, �2 or f3, �3 can be elimi-nated. Hence we change notation (24) and write, moreexplicitly,

H=(PcrH1 + Ch1)� + [P1H1 + PcrH2 + Ch2

+(PcrH100 + Ch100)f2

+(PcrH010 + Ch010)�2]�2 + . . . (28)

and similarly forM in the asymmetric case, as well as

H=(PcrH1 + Ch1)� + [P2H1 + PcrH3 + Ch3

+(PcrH100 + Ch100)f3

+(PcrH010 + Ch010)�3]�3 + . . . (29)


and similarly forM in the symmetric case. The sym-bolsH2, h2, H3, h3 and similar forM denote here theknown parts of the coefficients of�2 or �3, determinedby (25) or (27) with the last terms omitted.

3.3. The linear approximation, critical force

Substituting (24) and (28) into (19) and expandingthis equation into power series of� we obtain for theasymmetric case

EI(s)(w′′1� + w′′

2�2 + . . .)(1 + 1

2w′1

2�2 + . . .)

+[Pcr + (P1 + PcrV1 + Cv1)� + . . .]×(w1� + w2�2 + . . .)+ (PcrM1 + Cm1)�+[P1M1 + PcrM2 + Cm2 + (PcrM100

+Cm100)f2 + (PcrM010 + Cm010)�2]�2

+ · · · +{(PcrH1 + Ch1)� + [P1H1

+PcrH2 + Ch2 + (PcrH100 + Ch100)f2

+(PcrH010 + Ch010)�2]�2 + . . .} [s

− �2

2

∫ s

0w′

12(s)ds + . . .

]= 0 (30)

and similarly for the symmetric case

EI(s)(w′′1� + w′′

3�3 + . . .)(1 + 1

2w′1

2�2 + . . .)

+[Pcr + (P2 + PcrV2 + Cv2)�2 + . . .]×(w1� + w3�3 + . . .)+ (PcrM1 + Cm1)�+[P2M1 + PcrM3 + Cm3 + (PcrM100

+Cm100)f3 + (PcrM010 + Cm010)�3]�3

+ · · · +{(PcrH1 + Ch1)� + [P2H1

+PcrH3 + Ch3 + (PcrH100 + Ch100)f3

+(PcrH010 + Ch010)�3]�3 + . . .} [s

− �2

2

∫ s

0w′

12(s)ds + . . .

]= 0. (31)

Considering the terms with�1 we obtain in bothcases the following linear equation:

EI(s)w′′1 + Pcrw1 + PcrM1 + Cm1

+(PcrH1 + Ch1)s = 0 (32)

with the boundary conditions

w1(0)= 0, w′1(0)= �1, w1(l)= f1,

w′1(l)= 0. (33)

The second-order equation (32) needs two boundaryconditions, hence the remaining two make it possibleto find �1 in terms off1 and the critical forcePcr.

If we denote byF1(s;Pcr) andF2(s;Pcr) two lin-early independent functions satisfying the homoge-neous part of (32), then the general integral of (32)takes the form

w1=C1F1(s;Pcr)+ C2F2(s;Pcr)

−(M1 + C

Pcrm1

)−

(H1 + C

Pcrh1

)s. (34)

The boundary conditions (33) yield four equations; allof them are linear with respect toC1, C2, f1, and�1[hidden also inM1, m1, H1, andh1, (25)], and non-linear with respect toPcr. EliminatingC1, C2, and�1we obtain an equation forPcr. The relevant formulaeare lengthy and will not be quoted in their generalform.

For a prismatic column (34) takes the form

w1=C1 sin ks + C2 cosks −(M1 + C

Pcrm1

)

−(H1 + C

Pcrh1

)s, (35)

with k =√Pcr/EI . Elimination ofC1 andC2 makes

it possible to evaluate�1,

�1 = (PcrH100 + Ch100) (1 − coskl)+ (PcrM100 + Cm100) k sin kl

Pcr coskl − (PcrH010 + Ch010) (1 − coskl)− (PcrM010 + Cm010) k sin klf1 (36)


and to derive the following transcendental equation forPcr

[(PcrH100 + Ch100) (PcrM010 + Cm010)

−(PcrM100 + Cm100) (PcrH010 + Ch010)] k(2−2 coskl − kl sin kl)+ [(PcrH010 + Ch010)

+(PcrM100 + Cm100)]Pcrk(coskl − 1)

−(PcrM010 + Cm010)Pcrk2 sin kl

−(PcrH100 + Ch100)Pcr(sin kl − kl coskl)

+P 2crk coskl = 0. (37)

Eq. (37) gives a generalization of that derived byKordas[62] for the caseC = 0 (in a slightly differentnotation). On the other hand, Kordas allowed also forelastic clamping of the column, and this effect was notconsidered here.

Multiplying (32) byw′′1, integrating over the whole

interval 0�s� l, performing integration by parts, andtaking the boundary conditions (33) into account, wearrive at the formula

Pcr =∫ l

0 EI(s)w′′1

2 ds − C(m1�1 + h1f1)∫ l0 w

′1

2 ds +M1�1 +H1f1

(38)

This generalization of the well-known Rayleigh–Ritzformula determinesPcr in an implicit form, since theright-hand side depends also on the critical force viaw1, f1, and�1. However, it may serve for approximateevaluation ofPcr.

3.4. The second-order terms for the asymmetric case

Considering in (30) the terms with�2 we obtain thefollowing linear equation

EI(s)w′′2 + Pcrw2 + (P1 + PcrV1 + Cv1)w1

+(P1M1 + PcrM2 + Cm2

+(PcrM100 + Cm100)f2 + (PcrM010 + Cm010)�2

+[P1H1 + PcrH2 + Ch2

+(PcrH100 + Ch100)f2

+(PcrH010 + Ch010)�2]s = 0, (39)

with the boundary conditions

w2(0)= 0, w′2(0)= �2, w2(l)= f2,

w′2(l)= 0. (40)

Elimination ofw2 will be achieved by the manoeu-vre proposed by Croll[9]. We multiply (39) byw′′

1,integrate the second term by parts and making use ofboundary conditions obtain

∫ l

0w2w

′′1 ds =

∫ l

0w′′

2w1 ds, (41)

hence

∫ l

0[EI(s)w′′

1 + Pcrw1]w′′2 ds

+∫ l

0{(P1 + PcrV1 + Cv1)w1w

′′1

+[P1M1 + PcrM2 + Cm2 + (PcrM100

+Cm100)f2 + (PcrM010 + Cm010)�2]w′′1

+[P1H1 + PcrH2 + Ch2 + (PcrH100

+Ch100)f2

+(PcrH010 + Ch010)�2]sw′′1} ds = 0 (42)

Now it is seen that the bracket in the first integralmay be eliminated by using (32) and we obtain onlysimple integrals which may be evaluated by integrationby parts and using the boundary conditions. The re-sulting equation is linear with respect toP1 and makesit possible to find it without calculation ofw2, but theresult is effective only iff2 and�2 vanish, sincef2and �2 cannot be determined without calculation ofw2. Vanishing of the coefficients off2 and�2 takesplace if

H010 =M100, and h010 =m100. (43)

It will be shown in Section 5 that (43) is satisfiedif the loading is conservative. We restrict here ourconsiderations to conservative loadings, assume (43)and then

P1=− 1∫ l0 w

′1

2 ds +M1�1 +H1f1

×[(PcrV1 + Cv1)

∫ l

0w′

12 ds

+ (PcrM2 + Cm2)�1 + (PcrH2 + Ch2)f1

].

(44)


It is seen thatP1 = 0 for symmetric systems, sincethen V1 = v1 = M2 = m2 = H2 = h2 = 0. It mayalso vanish for asymmetric systems, for example ifthe spring stiffnessC is chosen in such a way as tomake zero the expression in square bracket.C is alsohidden inw1, f1 and in�1, for example, (35), (36),and hence the relevant equation forC is not as simpleas it may seem. The conditionP1 = 0 is necessary forstable post-buckling behaviour; sufficient conditionsmay be formulated just by analysis of the higher-orderterms.

3.5. The third-order terms for the symmetric case

Considering in (31) the terms with�3 we obtain theequation similar to (39), but with additional effects ofgeometrical non-linearity

EI(s)w′′3 + Pcrw3 + 1

2EI(s)w′1

2w′′

1

+(P2 + PcrV2 + Cv2)w1

+[P2M1 + PcrM3 + Cm3 + (PcrM100 + Cm100)f3

+(PcrM010 + Cm010)�3]+[P2H1 + PcrH3 + Ch3 + (PcrH100 + Ch100)f3

+(PcrH010 + Ch010)�3]s−1

2(PcrH1 + Ch1)

∫ s

0w′

12(s)ds = 0 (45)

and the relevant boundary conditions are

w3(0)= 0, w′3(0)= �3 − 1

6�3

1, w3(l)= f3,

w′3(l)= 0. (46)

Orthogonalization withw′′1 is in this case similar as

before, and the double integral appearing in the lastterm of (45) may be reduced to a single integral bychanging the order of integration:

∫ l

0

[w′′

1(s)

∫ s

0w′

12(s)ds

]ds

=∫ l

0

[w′

12(s)

∫ l

s

w′′1(s)ds

]ds = −

∫ l

0w′

13 ds.

(47)

Final formula forP2 is effective if conditions (43)are satisfied, since thenf3 and �3 are eliminated.Under these conditions we obtain for conservativeloading

P2= 1∫ l0 w

′1

2 ds +M1�1 +H1f1

×[

1

2

∫ l

0EI(s)w′

12w′′

12 ds − (PcrV2 + Cv2)

×∫ l

0w′

12 ds + 1

2(PcrH1 + Ch1)

×∫ l

0w′

13 ds − (PcrM3 + Cm3)�1

−1

6(PcrM1 + Cm1)�3

1 − (PcrH3 + Ch3)f1

].

(48)

This formulae gives the main result of thepresent paper, sinceP2>0 determines locally sta-ble, and P2<0—locally unstable post-bucklingbehaviour.

4. Inverse approach

4.1. General formulae for the symmetric case

The inverse approach described in Section 1.3, con-sists here in assumption of the functionw1 = w1(s)

and calculation of the relevant stiffness distributionEI = EI(s). From (32) we obtain

EI(s)=− 1

w′′1[Pcr(w1 +M1 +H1s)

+C(m1 + h1s)]. (49)

Considering the points = l (clamped end) and sub-stitutingw1(l) = f1 we evaluate the relevant criticalforce

Pcr = −EI(l)w′′1(l)+ C(m1 + h1l)

f1 +M1 +H1l. (50)


Further, substituting (49) into (48) and performing in-tegration by parts we obtain for this approach

P2= 1∫ l0 w

′1

2 ds +M1w′1(0)+H1w1(l)

×[

1

6Pcr

∫ l

0w′

14 ds+2

3(PcrH1+Ch1)

∫ l

0w′

13 ds

−(PcrV2 + Cv2)

∫ l

0w′

12 ds

−(PcrM3 + Cm3)w′1(0)

− (PcrH3 + Ch3)w1(l)

]. (51)

It should be noted that some authors consideringEulerian behaviour of the force, e.g.[17,40], derivefor P2 a formula different from (48) or (51). However,their formula is in full agreement with those derivedhere, namely it may be obtained by the schemeP2 =4P2 −3P2, where 4P2 is determined by (48), and 3P2by (51).

4.2. A simple example of the deflection line

The simplest example of the deflection line is theassumption of a linear function for the derivativew′

1=w′

1(s), namely

w′1 = B(l − s). (52)

Integrating (52) and making use of the boundary con-ditions (33) we findB = 2f1/l

2 or B = �1/l, hence

w1 =(

2s

l− s2

l2

)f1 or w1 =

(s − s2

2l

)�1. (53)

Assumption (52) means constant curvature of the de-flection line in the linear approximation. Moreover,Gajewski andZyczkowski [57,63] proved that (53)gives optimal solution for out-of-taper-plane bucklingof plane-tapered columns under general conservativebehaviour of loading if the critical force is assumedas the design objective.

From (49) and (50) we obtain here

EI(s)=−Pcr

2s2 +

[Pcr

(l + H1

B

)+ Ch1

B

]s

+Pcr

BM1 + C

Bm1, (54)

Pcr = BEI(l)− C(m1 + h1l)

f1 +M1 +H1l. (55)

In view of the linear function determiningw′1 the

evaluation of the integrals in (51) is particularly simplein this case and one obtains

P2= 1

2Bl2 + 3H1l + 6M1

×[Pcr

(B3l4

5+ B2H1l

3 − 2BV 2l2

−3H3l − 6M3)

+C(B2h1l3 − 2Bv2l

2 − 3h3l − 6m3)

](56)

These formulae expressEI(s), Pcr andP2 in termsof f1 or �1. For Eulerian behaviour of the force andlack of reactive forces (56) is reduced to

P2

Pcr= B2l2

10= 2f 2

1

5l2= �2

1

10. (57)

5. On conservative behaviour of forces

In the general case of post-buckling behaviour theincremental work is determined by

dW =H df +M d� + V du, (58)

whereas considering just buckling we neglect the lastterm as a higher-order term. Hence, in the linearizedcase we have just one condition of conservative be-haviour

�H��

= �M�f

(59)

or, making use of (25)

H010 =M100, h010 =m100. (60)

Exactly these conditions were necessary to eliminatef2 and�2 or f3 and�3 from the expressions forP1 orP2, (43). Hence the relevant formulae (44) and (48)are valid only for conservative loadings.

For arbitrarily largef, �, andu we have three con-ditions ensuring conservative behaviour of forces; be-sides (59) they take the form

�H�u

= �V�f,

�V��

= �M�u. (61)


Making use of (25) and (27) we obtain a sequenceof equations likeV100 = 2H200, H210 = 3M300, andso on. It may be checked that the forces (9), (10), and(11) in example 2.1 satisfy all these equations, hencethey are conservative.

6. Numerical examples

6.1. Column supported by two oblique springs

Return now to the column discussed in Section2.1, Fig. 2. We have hereH1 = H3 =M1 =M3 = 0,whereas the values ofhijk, mijk, and vijk, may beread from Eqs. (12) to (14). The calculations for aprismatic column making use of the function (35) andof (36) and (37) are cumbersome here and will not bequoted. On the other hand, using an inverse approachand assuming the deflection line (53) we obtain forthe respective non-prismatic columns relatively sim-ple formulae. It is convenient to assume here�= f , itmeansf1 = 1, fi = 0 for i >1. Since Eqs. (13) and(14) result inm1 = −h1l we obtain from (50) simplyPcr = 2EI(l)/ l2 like for a column without springs.The conclusion may be misleading here: the criticalforce remains the same, but the shape of the columnchanges according to (54). For the columns of pre-scribed shape the critical force depends onm1 andh1,for example for prismatic columns the valuekl= �/2does not satisfy the transcendental equation (37).

Returning to the columns (54) we obtain from (51)

P2= 2

5l2Pcr + C

3l(l2 + b2)2

×(l4 − 5b2l2 − 240b4). (62)

For small spring stiffnessC and small ratiob/l theinitial post-buckling behaviour is stable, but for largerC andb/l it turns into unstable.

6.2. General formulae for Stern’s column

In 1979 Stern[3] considered finite deflections ofa prismatic column shown inFig. 1. Stern gave an

exact analysis using elliptic integrals and found thatfor a <0.6l the initial post-buckling behaviour is un-stable. Now we derive the formulae forP2 for Stern’snon-prismatic columns.

It is seen that in the case under considerationV =Pand

H = P f√a2 − f 2

= P fa

+ 1

2Pf 3

a3 + . . . . (63)

Assuming� = f we obtainH1 = 1/a, H3 = H3 =1/2a3, . . .M1 =M3 = . . . V2 =V4 = . . .=0. Eqs. (48)and (51) yield here

P2= 1∫ l0 w

′1

2 ds + 1a

(1

2

∫ l

0EI(s)w′

12w′′

12 ds

+Pcr

2a

∫ l

0w′

13 ds − Pcr

2a3

), (64)

P2= 1∫ l0 w

′1

2 ds + 1a

(1

6

∫ l

0w

′41 ds

+ 2

3a

∫ l

0w′

13 ds − 1

2a3

)Pcr. (65)

6.3. Prismatic Stern’s column

For prismatic columns we may use (35) and (37),thenw1 is determined by

w1 = sin ks

ak coskl− s

a(66)

and the critical force by the transcendental equation

tgkl = 1 + ��

kl, (67)

where�=l/a; the case� → 0 corresponds to Eulerianbehaviour of the force. Substituting (66) into (64) weobtain

P2

Pcr= �2[kl(1 − 8 cos2 kl − 8 cos4 kl)+ sin kl coskl(1 + 14 cos2 kl)] − 8�kl cos4 kl

8l2 cos2 kl(kl + 2kl cos2 kl − 3 sin kl coskl). (68)

It would be desirable to eliminate from (68) thecritical force hidden ink and to obtain this function interms of� only. This elimination cannot be done in ananalytical form since (67) is a transcendental equation


with respect tok. However, we may use an inverseprocedure and express from (67)� in terms ofk, then

P2

Pcr= k2 kl(1 − 8 cos2 kl)+ sin kl coskl(1 + 6 cos2 kl)

8(sin kl − kl coskl)2(kl + 2kl cos2 kl − 3 sin kl coskl). (69)

Equating the numerator in (69) to zero we can findthe boundary parameters separating initially stablefrom initially unstable post-buckling behaviour of aprismatic Stern’s column. Numerical solution of therespective transcendental equation gives the smallestpositive root (kl)b = 1.03409, and then from (67)�b = 1.59904, (a/ l)b = 1/�b = 0.62538. Stern[3]estimated this boundary value as lying inside theinterval 0.6<(a/l)b<0.7, and the above result con-firms his estimation. For the Eulerian behaviour of theforce� → 0 we obtainkl = �/2 and the well-knownformulaP2/Pcr = �2/32l2 = 0.3084/l2.

6.4. Inverse approach for a non-prismatic Stern’scolumn

We consider now non-prismatic columns corre-sponding to the deflection line (53). From (54)–(56)we obtain

EI(s)= EI(l)

1 + �

[(2 + �)

s

l− s2

l2

], (70)

Pcr = 2EI(l)

l2(1 + �), (71)

P2

Pcr= 1

l2

16+ 40� − 15�3

40+ 30�. (72)

Equating the numerator in (72) to zero weobtain a cubic equation with the positive root�b = 1.8049, (a/ l)b = 1/�b = 0.55405. The intervalof stable initial post-buckling behaviour is in this caselarger than for a prismatic column. This fact may beexplained by an attribute of optimality of the function(53) mentioned in Section 4.2. In the Eulerian case� → 0 we obtainP2/Pcr = 0.4/l2, and this value isalso higher than that for a prismatic column.

6.5. Model of a Stern’s column

Just for the sake of comparison we give here a briefanalysis of a one-degree-of-freedom model of a Stern’scolumn shown inFig. 6. It consists of two rigid bars;

ϕ

l rigid

K

f

P

rigida

Fig. 6. Model of Stern’s column.

the lower bar of the lengthl is supported by an elastichinge with the rotational spring stiffnessK. Hence,it corresponds to a non-prismatic elastic column witha very small stiffness in the vicinity of clamping andlarge stiffness elsewhere. Of course, such a shapeis far from the optimal shape and it will be inter-esting to compare the results with those obtainedabove.

The equation of equilibrium of such a model leadsto

P = K�√

1 − �2 sin2 �

l sin �(√

1 − �2 sin � + � cos�) , (73)

where � is defined as above. Expanding (73) intopower series of� we obtain

P = K

(1 + �)l

[1 + 1

6(1 + 3� − 3�2)�2 + . . .

].

(74)


f

l deformable

P

H

C

Fig. 7. Column supported by a transverse spring.

The termP2 determining stability of post-bucklingbehaviour is equal to zero for�=�b =1.2638, hence(a/ l)b = 0.79129. Indeed, the range of unstable be-haviour(a/ l)<0.79129 is in this case larger than inboth cases discussed above.

6.6. Column supported by a transversal spring

Consider a column under the forceP of Eulerian be-haviour, additionally supported by a transversal springof stiffnessC (Fig. 7). It is assumed that the right-hand end of the spring may slide in axial direction soas to ensure constant direction of that spring in space.

At a first glance the system may look asymmetric,but, in fact, it is symmetric since both senses of buck-ling are equally privileged. Assuming� = f , and thesigns shown inFig. 5we haveh100=−1 and all othercoefficientshijk, mijk, andvijk vanish. From (48) and(51) we obtain

P2= 1

2∫ l

0 w′1

2 ds

×(∫ l

0EIw′

12w′′

12 ds − C

∫ l

0w′

13 ds

), (75)

P2= 1

6∫ l

0 w′1

2 ds

×(Pcr

∫ l

0w′

14 ds − 4C

∫ l

0w′

13 ds

). (76)

It is seen that for small values ofC the post-bucklingbehaviour is stable, whereas beyond a certain bound-ary valueCb it becomes unstable. If we use the inverseapproach assumingw1 =w1(s), then (76) is effectivein its explicit form, and we may write

Ccrb = l

∫ l0 w

′41 ds

4∫ l

0 w′1

3 ds(77)

where the dimensionless spring stiffness is related tothe critical forcePcr, namely defined byCcr=Cl/Pcr.For the deflection line (53) we obtainCcr

b = 0.4.On the other hand, if the functionEl = EI(s) is

given, then the right-hand side of (75) equated to zerogives just an implicit formula forCb, sinceC is thenhidden also inw1. For example, for a prismatic columnwe obtain from (35)

w = C

Pcr

(− sin ks

k coskl+ s

)(78)

and from (37)

C(sin kl − kl coskl)+ Pcrk coskl = 0. (79)

Substituting (78) and (79) into (75) with an addi-tional conditionP2=0 we arrive at the following tran-scendental equation

kl(1 − 8 cos2 kl − 8 cos4 kl)

+ sin kl coskl(1 + 14 cos2 kl)= 0 (80)

with the smallest positive rootkl = 1.8574. Fi-nally, substituting this value into (79) we obtainCcr

b = 0.35376, hence the interval of stable initialpost-buckling behaviour of the prismatic column0<Ccr<Ccr

b is here smaller than for the non-prismatic column discussed above.

For the sake of comparison we quote also the resultsfor a relevant one-degree-of-freedom model shown inFig. 8. ThenPcr=(K/l)+Cl, whereK is the rotationalspring stiffness, andP2 = (K/6l3) − (C/2l). In thiscase we obtainCcr

b =0.25 and the range of stable post-critical behaviour is the smallest in comparison withthe two examples discussed previously.

The dimensionless spring stiffnessCcr definedabove has a certain deficiency, namelyPcr dependsalso onC. To avoid this deficiency we may use an-other definition, relatingC to the relevant Eulerianforce (calculated forC = 0), namelyCE = Cl/PE.


P

H

f

ϕ

l rigid

K

C

Fig. 8. Model of the column supported by a transverse spring.

For the inverse approach based on (53) we obtainfrom (38)Pcr = PE + (3C/4l), hencePE = (7/4)ClandCE

b = 0.57143. For a prismatic column we obtainCE

b = 0.49463, and for the modelCEb = 0.33333. The

arrangement ofCEb for particular cases under discus-

sion is the same as ofCcrb .

7. Effects of pre-buckling geometry changes

In some cases pre-buckling geometry changes mayaffect the behaviour of loading during buckling. Thenthe formulae derived above are no longer valid andsome generalizations must be introduced.

We present here two examples of such changes:Stern’s column loaded via a compressible rod (Fig. 9)and a column loaded by an attached vessel filled witha heavy liquid with specific weight� (Fig. 10). Thediscussion of the second example is particularly inter-esting and the effects of possible geometry changeswill clearly be seen.

The vessel has the form of a symmetric triangularparellellepiped characterized by the dihedral centralangle 2� and the depthb (Fig. 10). A rectangular par-allellepiped or a cylindrical vessel seem simpler, butthis is not the case: one has to consider two rangesnamely the bottom fully covered or partly covered withliquid, and in a triangular vessel such a problem doesnot appear. The column is loaded by the concentrated

a −

∆a

l

rodelasticin compression

P

columnelasticin bending

al

Fig. 9. Stern’s column loaded via an elastic rod.

l

rigid

vessel

liquid

lβ β

h lβ

β

α

vesselrigid

liquid

elastic column

Fig. 10. Column loaded by liquid in an attached triangular vessel.

axial force

P = �Vl = �bh2l tg�, (81)

whereVl denotes the volume of liquid andhl describesthe level of liquid in the undeflected position. Afterbuckling the column is also loaded by the concentratedmoment

M = �bh3l sin � sin 2�

3 cos2 �√

cos2 � − sin2 �(82)


depending on the angle of deflection�, �<(�/2)−�.This angle will be chosen here as a small parameter,� = �, hence�1 = 1, �i = 0 for i >1.

Loading of the column may be controlled by any oftwo parameters: specific weight� or by the volume ofliquid Vl . In the first case (rather theoretical, but pos-sible e.g. in a magnetic field) we have no pre-bucklinggeometry changes and the formulae derived in the pre-vious sections hold. Expressing� in terms ofP from(81) we present the formula (82) in the form

M= Phl sin 2�

3 cos�√

cos2 � − sin2 �= 2Phl

3 cos3 �

×[� +

(1

2 cos2 �− 2

3

)�3 + . . .

]. (83)

This result coincides with (25)–(27), namely

M010= 2hl3 cos3 �

,

M030= 2hl3 cos3 �

(1

2 cos2 �− 2

3

), . . . . (84)

A quite different situation is obtained if the loadingprocess is controlled by the volume of the liquidVl , asit takes place in typical applications. Then we expresshl in terms ofP and present (82) in the form

M = P 3/2 sin 2�

3√

�b sin � cos�(cos2 � − sin2 �)(85)

and using the expansion ofP , (7) we obtain

M= 2P 3/2cr

3√

�b sin � cos3 �

{� +

[3

2

P2

Pcr

+(

1

2 cos2 �− 2

3

)]�3 + . . .

}. (86)

The series (86) may be written in the followinggeneral form, resembling (26):

M=PcrM010(Pcr)� + [P2M∗010(Pcr)

+PcrM030(Pcr)]�3 + . . . (87)

with

M010 = 2P 1/2cr

3√

�b sin � cos3 �,

M∗010 = P

1/2cr√

�b sin � cos3 �,

M030 = 2P 1/2cr

3√

�b sin � cos3 �

(1

2 cos2 �− 2

3

).

. . . . . . . . .

. . . . . . . . . (88)

Comparing (87) and (88) with (25), (26), and (27)we find two essential differences:

(1) All the coefficients in (88) depend on the criticalforcePcr, whereas in (25) and (27) they are indepen-dent ofPcr [like in (84)];

(2) The coefficientsM010 andM∗010 differ from each

other (hence the additional star in the second expres-sion), whereas in (26) they are equal to each other.

In conclusion, in the case of pre-buckling geome-try changes the method elaborated in the present pa-per may also be used, but the results are much morecomplicated.

8. Conclusions

Post-buckling analysis of non-prismatic columnssubject to general behaviour of active and reactiveforces was carried out via the perturbation method.General formulae for the first or the second correc-tion of the force,P1 or P2, were derived by usingjust the linear approximation of the non-linear gov-erning equation. They are effective in the case ofconservative loadings, whereas for non-conservativeloadings the higher-order terms of the deflection lineare necessary. Direct method for the given stiffnessdistribution as well as an inverse method based on theassumption of the equation of the deflection line wereelaborated. The effects of the behaviour of loading onstability may be essential—they were shown on sev-eral numerical examples. The inverse approach maybe regarded as a background to formulate and solvethe problems of optimal design with respect to sta-ble post-buckling behaviour. Effects of pre-bucklinggeometry changes may also be allowed for, but theycomplicate final formulae.


Acknowledgements

This paper was partly supported by the State Com-mittee for Scientific Research (KBN) under Grant No.PB 5 T07A 003 24.

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