Stiffened plates and cylindrical shells under interactive buckling

*Corresponding author.�Former doctoral student.

Finite Elements in Analysis and Design 38 (2001) 155}178

Sti!ened plates and cylindrical shells under interactive buckling

Srinivasan Sridharan*, Madjid Zeggane�

Department of Civil Engineering, Washington University in St. Louis, Campus Box 1130, St. Louis, MO 63130, USA

Abstract

The interaction of local and overall buckling in sti!ened plates and cylindrical shells has been analyzedusing a novel "nite elements in which local buckling deformation has been embedded. Amplitude modula-tion, a key feature of the interactive buckling has been incorporated in the element formulation. The modelhas the following additional features: (i) the inclusion of a key secondary local mode where the cross-sectionhas complete or approximate double symmetry; and (ii) the introduction of a simple approach for capturinglocalization of local buckling; this involves incorporating a single local buckling mode in the analysis, butletting the amplitude modulation function to be di!erent for di!erent elements. Numerical examples of plateand shell structures are presented to throw light on these aspects of the methodology as well as todemonstrate the accuracy and e$ciency of the model. � 2001 Elsevier Science B.V. All rights reserved.

Keywords: Finite elements; Buckling; Plate structures; Cylindrical shells; Modal interaction; Local buckling; Amplitudemodulation; Localization of buckling; Imperfection sensitivity

1. Introduction

Cylindrical shell and panels are often reinforced with stringers to enhance their sti!ness inresisting axial compression. The sti!ening elements not only enhance the buckling resistance butalso reduce the imperfection-sensitivity of the shells. Because of the resistance o!ered by thesti!eners to radial movement, &local' buckling modes whose nodal lines do not coincide with thelocation of the sti!eners are simply eliminated. This has the e!ect of minimizing the nonlinearmodal interactions which are the source of imperfection-sensitivity in unsti!ened shells.

0168-874X/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 8 7 4 X ( 0 1 ) 0 0 0 5 6 - 7

However there remain principally two distinctive modes of buckling which dominate thebehavior of sti!ened shells:(i) the short-wave local mode(s) in which the sti!ener-skin junction remains essentially straight, i.e.

the shell-skin buckles between the sti!eners.(ii) the overall long-wave mode in which the cross-sections of sti!eners undergo signi"cant

translations in the direction normal to the shell, i.e. shell-skin bends carrying the sti!eners withit. The optimum design of the shells often leads to a con"guration for which the critical stressesare close to each other. Thus, a study of nonlinear modal interaction of local and overallinstabilities is of considerable signi"cance in the context of optimal design of such shells. Theproblem of interaction of an Euler buckling with local plate buckling was studied in the 1960sand 1970s by several investigators (see for example, [1}3]). Tvergaard presented a detailedanalysis of sti!ened plates under interactive buckling [4]. The problemwas studied Byskov andHutchinson [5] using an asymptotic approach and solutions in the classical mold. Koiter andPignatrao [6,7] authored two papers of fundamental importance to the interaction of local andoverall buckling in sti!ened panels and sti!ened shells. They introduced the technique ofamplitudemodulation of the local buckling mode and simply neglected the mixed second order"eld arising out of the interaction of local and overall buckling modes.

The 1980s saw further studies on interactive buckling, most notably by Sridharan and hisco-workers. A "nite strip approach was used [8] to extract the buckling modes and the secondorder "elds to simplify the analysis. Next, a special beam element was developed to deal with theinteraction of local and overall buckling . This is an Euler}Bernoulli beam element in which thelocal buckling deformation was embedded. The local buckling mode(s) and the periodic part ofsecond order "elds are determined a priori by the asymptotic procedure. These e!ects are built intoa beam "nite element in terms of two additional degrees of freedom, for each local buckling modeconsidered, thus allowing automatically for the amplitude modulation [9,10]. The theoretical basisof such an approach has been discussed by Sridharan and Peng [11]. It is shown that the amplitudemodulation is the key factor in the interaction; it performs the function of capturing the contribu-tions of several neighboring modes of the same longitudinal description as the fundamental localmode, but with di!ering circumferential wave numbers. For doubly symmetric cross-sections it isvitally important to consider a companion secondary local mode which is triggered by theinteraction of the overall buckling mode and the primary local mode. The use of amplitudemodulated local modes and the induction of key secondary modes as full participants in theinteraction pulls in all the additional patterns of deformation generated in the interaction of anoverall mode with the primary local mode. So much so, it is super#uous to consider the mixedsecond order "eld arising by the interaction of these two modes. Use of a beam element in theformulation restricts, however, the applicability of this approach to beam-like structures.

In order to extend the scope of this approach Sridharan et al. [12] developed a shell element inwhich local buckling deformation was embedded. However, only relatively simple examples wereconsidered for illustration, such as sti!ened panels studied previously by Tvergaard. The methodwas validated by comparisons with detailed nonlinear "nite element analysis using a commercial"nite element code , viz., ABAQUS [13, version 5.8].

On date there exist few solutions to the problem of interactive buckling accomplished by meansof nonlinear "nite element analysis. Such an analysis would obviously entail considerable

156 S. Sridharan, M. Zeggane / Finite Elements in Analysis and Design 38 (2001) 155}178

computational resources as it would require the use of a su$ciently "ne mesh to capture the localdeformation (such as sinusoidal ripples of small wavelength) in a structure long and broad enoughfor the overall instability to be of importance.

From the point of view of modeling interactive buckling using a modal approach , prismaticstructural components can be divided into two broad categories: (i) Structures whose cross-sectionexhibits strong asymmetry with respect to the axis of bending or overall buckling. Typical examplesof these would be slender plates attached to stocky sti!eners, e.g. Tvergaard panels [4]. (ii)Structures which exhibit near or complete symmetry with respect to axis of bending or overallbuckling. Typical examples of these are rectangular box- or I-section columns.

Consider case (i): In its simplest form, the structure has only two signi"cant modes of buckling,viz., Euler buckling mode and the plate buckling mode with scaling parameters �

�and �

�respec-

tively. The potential energy function governing the interaction can be written as:

�"��a

�!� b

��

�#�

��a

�!� b

��

�#c

� � ��#2higher order terms, (1)

where the key term of interaction is the c� � �

term. This term involves the energy of interaction ofoverall bending associated with Euler buckling with the midsurface stretching associated with localbuckling. Such a potential energy function was employed by Tvergaard [4] and subsequentlythoroughly investigated by Hunt [14].

Next consider the case (ii). In this case, the key term c� � �

vanishes because of symmetry of thecross-section with respect to the axis of bending: the overall buckling stress/strain is antisymmetricwith respect to the axis of bending while the mid-surface stretch associated with local buckling issymmetric with respect to the same axis. In order to compute a higher order term involving both��and �

�, one has to compute the mixed second order "eld which arises by an interaction of the

two modes. Such a procedure will yield a biquadratic term, viz., ��[10]. The inherent numerical

di$culties of such a procedure have been discussed in literature [9] and will not be repeated here.A robust approach to this problem is into incorporate an additional key local mode (associatedwith the scaling parameter �

�) as a principal mode in the analysis from the outset. This mode will

be antisymmetric with respect to the axis of bending if the primary local mode is symmetric (Fig. 1)and vice versa and gives rise to a key trilinear term as shown below:

�"��a

�!� b

��

�#�

��a

�!� b

��

�#�

��a

�!� b

��

�#c

� � ��

#2higher order terms. (2)

Note that the incorporation of the secondary mode is essential in modeling the localized deforma-tion in the compression #ange of the square box column, i.e. the local deformation in the #angewhich su!ers additional compression due to overall buckling is further accentuated while that inthe opposite tension #ange is alleviated (Fig. 1). In order to ensure accuracy, post-local bucklinge!ects must be accounted for in the analysis. Thus not only the secondary local mode, but also thesecond order "eld associated with it and the mixed second order "eld that arises by the interactionof two local modes, have to be included. Thus we have two local modes and three second order"elds. This has been done by Sridharan and Ali [9,15] and Ali and Sridharan [16].

Now, consider how such a scenario can be modeled with plate/shell elements carrying localbuckling deformation embedded in them. It would again, seem necessary to incorporate degrees offreedom to represent the secondary local mode, together with the relevant second order "elds. This

S. Sridharan, M. Zeggane / Finite Elements in Analysis and Design 38 (2001) 155}178 157

Fig. 1. The local modes of buckling in a square box column.

has been successfully attempted in the present study. More importantly, a simpler alternative ofachieving the same result is also presented. This, is considering just a single local mode but lettingthe amplitude modulation function be di!erent for di!erent plate elements. Thus the cross-sectional pattern of local buckling deformation of the structure evolves freely under the in#uence ofoverall buckling deformation.

Another issue that is relevant is the relative importantance of the short wave contribution of thesecond order "elds for obtaining an accurate estimate of the maximum load. In this study anattempt is made to answer this question with some numerical examples.

We now summarize the objectives of the present paper:(i) Demonstrate the applicability of the concept `locally buckleda elements to shell and plate

structures, especially those with double symmetry. In these cases, there is a possibility oflocalization of deformation, con"ned typically to one of the elements.

(ii) Examine the e!ects, respectively, of the second local mode and the second order "elds in thecomputations.

(iii) Finally, the accuracy of the model by comparison with detailed FE analysis and earliersimpli"ed formulations.

In order to make the paper complete, we present a brief outline of the formulation of &locallybuckled' plate/shell elements. This is followed by several worked examples and commentarythereof. The paper ends with a summary of the conclusions.

2. Theory

In this section, the theory and formulation of the present "nite element model is outlined.


Fig. 2. Coordinate axes of a cylindrical shell.

2.1. Displacement, strain and stress vectors

The displacement variables are

�u��"�u, v,w, �,��, (3)

where u, v and w are the displacement components in the axial (x-), transverse (y-) and outwardnormal (z-) directions, respectively, at any point on the middle surface shell or sti!ener (Fig. 2) and� and � are the rotations of the normal in the xz and yz planes respectively.

The generic strain vector �3� may now be de"ned as in Reissner}Mindlin theory:

��"��, �

�,

��,

�,

�,

��,

��,

��. (4)

Of these, �� "��, �

�,

�� are the inplane strain components, ��"�

�,

�,

�� are the curvature

components, and ��"��,

�� are the transverse shearing strain components.

The following strain}displacement relations are assumed for the shell/sti!ener:

��"

�u�x

#

12��

�v�x�

�#�

�w�x�

�

�, (5a)

��"

�v�y

#

wR

#

12��

�u�y�

�#�

�w�y �

�

�, (5b)

��

"

�u�y

#

�v�x

#

�w�x

�w�y

, (5c)

�"

��x

, (5d)

�"

��y

, (5e)

��

"

��y

#

��x

, (5f)


��

"�#

�w�x

, (5g)

��

"�#

�w�y

. (5h)

These may be viewed as Donnell's strain}displacement relations modi"ed to account for transverseshear deformation and the large in-plane motions of sti!eners such as would occur under overallbuckling. Thus, the expression �

�is augmented to include a nonlinear term in v*a key term which

enables the modeling of overall bending/buckling phenomena. The nonlinear terms in u and v canbe neglected in dealing with a purely local buckling problem. The Eqs. 5(a)}5(h) can be written inan abbreviated form

��"¸

��(u

�)#�

�¸

��(u

�) (6)

with i"1,2 , 8 and j"1,2 5.To correspond with ��, a generic stress vector � � is de"ned. This consists of force-resultants

�N�"�N�N

�N

��, moment resultants �M�"�M

�M

�M

��, and transverse shear forces

�Q�"�Q�Q

��.

The stress}strain relations take the form

�N

M�"�A B

B D��

�, (7a)

�Q�"kGM t��, (7b)

where k is the shear correction factor (taken as 5/6), GM is the averaged transverse shear modulus andt is the thickness of the shell. The stress}strain relations may be written in the abbreviated form

�"H

��. (8)

2.2. Determination of the local buckling xelds

2.2.1. NotationThe following notation will be employed in the sequel: A single superscript over a displacement,

stress or strain will indicate a "rst order quantity and a double supercript likewise will refer toa second order quantity; the superscript (o) is reserved for the overall buckling mode, while (1) and(2) will refer to the local modes considered in the ascending order of the corresponding criticalstresses; double superscripts (11) and (22) indicate the second order "elds associated with theprimary local mode and secondary local mode, respectively, and (12) stands for the mixed secondorder "eld (m.s.o.f.) arising by the interaction of local modes 1 and 2. The local buckling problem issolved following the standard "nite strip procedure [17]. The salient features thereof will now bementioned.


2.2.2. The linear stability (eigenvalue) problemThe potential energy function corresponding to the nth local mode (u��) can be written in the

form

� ��"��[H

��¸��

(u��) ) ¸

�� l(u��l )# H

�) ¸

�� (u��

�)], (9)

where ��represents the stress in the unbuckled state, and a dot operation indicates multiplication

and integration over the volume of the structure and the * stands for stress in the unbuckled state.For a stringer-sti!ened cylindrical shell composed of a specially orthotropic material, the

displacement functions that satisfy the di!erential equations are of the form

�u��, ��"�u��, ��

��

�(y) sin�

m�x¸ �,

�v��,w��,��"�v��,w��

�, ��

��

�(y) cos�

m�x¸ �. (10)

Here u�, v

�, 2 are the degrees of freedom (d.o.f.) and �

�are appropriately chosen polynomial shape

functions. In the present work, the functions ��are chosen in a hierarchical form as explained later.

For su$ciently large m, the boundary conditions at the end are deemed not to in#uence the localbuckling process.

Designating the d.o.f.'s of the local buckling problem as q��, the potential energy function

(Eq. (9)) may be expressed in the form

� ��"��a��

��!�b��

�� q��

�q��

(11)

where � is the load parameter and i, j range over all the d.o.f.'s considered in the buckling problem.The equilibrium equation is

�a��

!� b��

� q��o ��

"0. (12)

Solution of the linear eigenvalue problem in Eq. (12) gives the critical stress for local bucklingand the eigenmode, q��

�.

2.2.3. Second order xeld problemThe potential energy function governing the second order "eld problem is given by

� ��"��[H

��¸

��(u��

�) ) ¸

��l(u��l )# H

�) ¸

��(u��

�)

#H��¸

��(u��

�)� ) ¸

��l(u��l )#2¸

��(u��

�) ) ¸

��l(u��l , u��l )�], (13)

where �u�� refers to the second order "eld sought.The displacement functions must be chosen keeping in view the solution to the di!erential

equations of the second order "eld problem. The right hand side vector �r� of the di!erential


equations consists of three sets of terms in general:

�r�"r�(y)#r�(y) cos�2m�x

¸ �#r�(y) sin�2m�x

¸ �, (14)

The last two terms are rapidly varying trigonometric functions (m�1) in x, while the "rst term isindependent of x. The second order "eld problem may, then, be viewed as that of a cylindrical shellsubjected to two loads which vary sinusoidally in the x-direction, together with a third term whichremains constant in the x-direction.

The solution takes the form

�u��"�u��#�u�� cos�2m�x

¸ �#�u�� sin�2m�x

¸ �, (15)

where �u�� is a function of x and y, whereas �u�� and �u�� are functions of y only and can be viewedas the particular solution of the di!erential equation. �u�� is a slowly varying function with respectto x and contains additional terms needed to enforce the end boundary conditions. Note that thesolution cannot contain a component in the form of the buckling mode in the asymptoticprocedure [18]. Also because of the slowly varying nature of �u��, it is decoupled in the solutionprocess from �u�� and �u��.

Unlike in the asymptotic procedure for the initial postbuckling analysis [17], it is not necessaryto compute �u�� at this stage. Rather we shall let it arise, by the interaction of L

�(u��) terms with the

degrees of freedom associated with the "nite element mesh to be introduced later. Because of itsslowly varying character, a relatively coarse "nite element mesh must be able to pick up thedeformation associated with �u��. So, we need to compute only the solution vectors associated withthe trigonometric terms at this stage.

For a specially orthotropic material, the displacement "elds to be computed take the simpli"edform

�u��,��"�u��

, ��

��(y) sin�

2m�x¸ �,

�v��,w��

, ��

�"�v��

,w��

, ��

��(y) cos�

2m�x¸ �, (16)

where u��

,2 are the d.o.f.'s second order "eld.The potential energy function (Eq. (10)) can now be expressed in terms of the d.o.f.'s q��

�and

q��

de"ning the "rst and second order "elds, respectively, and takes the form

� ��"��(a��

��!�b��

��)q��

�q��

#c��

q��

q��

q��

(r, s"1,2 , n�); (i, j"1,2,2 , n

�), (17)

where n�stands for the d.o.f.'s of the trigonometric part of the second order "eld. The equations of

equilibrium takes the form

(a��

!� b��

) q��

"!c��

q��

q��, (18)

� is set equal to ��

in the calculations.


2.3. Mixed second order xeld

The potential energy function governing this problem can be expressed in the form

� ��"��[H

��¸��

(u��

) ) ¸��(u��

)# H

�) ¸

��(u��

�)

#H��2¸

��(u��

�) ) ¸

��(u��

, u��

)#2¸

��(u��

�) ) ¸

��(u��

, u��

)

#2¸��(u��

�) ) ¸

��(u��

, u��

)�]. (19)

Here we consider the second order "eld arising out of interaction of two local modes of the samewave length. As discussed in a later section, signi"cant interaction occurs only between the overallmode and two local modes of the same (or nearly the same) wavelength. Thus, we assume the twolocal modes have the same number of half-waves, m. This would lead to a di!erential equationwhich has the same structure as Eq. (15) and results in a solution of the type given in Eq. (16). Onceagain we need to compute only the periodic part of the solution.

The local buckling displacements in a problem where two local modes are active can be writtenin the form

ul"u��#u��

�#u��

�#u��

��#u��

�, (20)

where u"�u v w � � ��; ��and �

�are the scaling parameters of the two local buckling modes,

respectively; the single subscripted u's represent the local buckling "elds as given by Eq. (10) and thedouble subscripted u's, viz., u��, u�� , and u�� , stand for the "rst , mixed and second order "elds,respectively.

2.4. Amplitude modulation in the x-direction

However, as already mentioned the amplitudes of the local modes vary in the longitudinaldirection as the local buckling deformation comes under the in#uence of overall buckling of thestructure. Thus, the local buckling amplitudes are `modulateda to enable the model to depict theaccentuation of local buckling in regions where compressive stresses develop due to overallbuckling. Apart from this, because of its `slowly varyinga character, the amplitude modulationperforms the role of accounting for the neighboring local modes of the same transverse descriptionas the associated local mode but of slightly di!ering half-wavelength, which are liable to betriggered by the interaction [11,12].

Thus we let the amplitude vary with respect to x- according to a `slowly varyinga modulatingfunction. The local buckling deformation is thus represented in the form

u"�u��

#u��

��(x)#u��

��

#u��

��

#u��

��

��(x)�

�(x), (21)

�&s represent the degrees of freedom associated with amplitude modulating functions �&s.

2.5. The wavelength of the secondary local mode

It has been assumed in the foregoing treatment that the two local modes which together describethe local buckling deformation have the same number of half-wavelengths, m. This requires some


elucidation. The most signi"cant term in the interaction of two local modes with the overall modeis the trilinear term c

��. This term arises by the product of stress associated with the overall

buckling with the bilinear coupled term representing the midsurface strain, viz., L��

(u��, u��)(i.e. (�w

�/�x) �w

�/�x) and integrating the same over the structure. In so far as the overall buckling

displacements are `slowly varyinga functions of x, and their derivatives may be treated as nearlyconstant. Thus, the said integral tends to vanish unless the longitudinal wavelengths (or number ofhalf-waves, m) are the same. Thus only the interaction of local modes of the same wavelength withthe overall mode is of importance.

2.6. Modixcation of local buckling deformation in the transverse direction

So far our description of local buckling consists of two amplitude-modulated local modes oforthogonal transverse description but of the same wave length and all the associated second ordere!ects. To this we add yet another feature in the present study : The amplitude modulatingfunctions are given freedom to di!er from element to element, thus giving the the local bucklingdeformation freedom to vary across the section. This tends to make the secondary local modesomewhat redundant*an issue which will be considered later.

2.7. Neglection of mixes second order xelds (m.s.o.f.), u��

The m.s.o.f u�� arising of out the interaction of a typical local mode (n"1,2) with the overallmode, gives, in principle, the additional patterns of deformation (orthogonal to all the "rst order"elds already accounted for) that are generated during the interaction. In an earlier paper theauthors have [12] have discussed the pitfalls in the evaluation of the m.s.o.f. arising from theinteraction of local and overall modes. In brief, accuracy is lost due to the fact interaction in a longprismatic structure gives rise to a number of local modes with eigenvalues close to that of theprincipal local mode at which the m.s.o.f. is evaluated. On the other hand, the m.s.o.f. can helpidentify the local modes that are triggered in the interaction [11,15]. Once these modes areidenti"ed, they must be included in the analysis as principal (`mastera) modes in the analysis.

The inclusion of a key secondary local mode when found to be necessary, the amplitudemodulation of the local modes and the freedom given for the local buckling deformation toundergo modi"cation freely in the transverse direction*all these fully account for any change ofthe local buckling deformation. Further, we have a "nite element mesh that would automaticallyaccount for any changes in the overall deformation under the in#uence of the local modes. Alsonote that these e!ects are given the status of a "rst order "eld*`mastera rather than `slavea "elds.

The m.s.o.f. arising from the local and overall buckling mode interaction must be evaluatedrequiring it to be orthogonal with respect to the totality of the "rst order "eld (which includes thefundamental local modes and their respective neighbors implied in the amplitude modulation).Such a calculation results in a m.s.o.f. with all its destabilizing contents &squeezed out'. It cantherefore be conveniently neglected. Note that a typical m.s.o.f. strain takes the form

��

"¸�(u��)#¸

��(u��

�,u��

�), (22)

where the "rst term represents the contribution of mixed second order displacement "eld and thesecond term arises by the mixing of the two fundamental modes. If only the "rst term on R.H.S. is


neglected, Eq. (22) is clearly inconsistent and therefore the second term which acts as the driver ofthe m.s.o.f. must also be neglected. Thus, we set all the m.s.o.f stresses, strains and displacements areset to zero.

2.8. Finite element formulation

2.8.1. Choice of shape functionsA p-version "nite element approach is adopted. Thus, a set of hierarchical polynomial functions

are selected to represent the displacements. For a su$ciently high &p' ( polynomial degree), theproblems of shear and membrane locking associated with lower order elements become incon-sequential. A relatively small number of elements (in comparison to the h-version approach) wouldbe su$cient to accurately model the structure and this results in considerable savings of e!ort indata input. The type of polynomials �

�chosen in the present work are integrals of the Legendre

functions, advocated by Szabo and Babuska [19]. These have been successfully used in previouswork by the authors (see for example [17]).

2.8.2. Strain-displacement matrixAs stated earlier, a cylindrical shell element based on Donnell's theory admitting shear-

deformation via Reissner-Mindlin theory (Eq. 5(a)}5(h)) is employed. The displacement functionstake the form

�u��"�u��, v

��,w

��, �

��, �

��

�(x)�

�(y). (23)

As a "rst step in the formulation we set up the B-matrix which relates the incremental strains to theincremental degrees of freedom. To this end, each strain component is expressed in terms ofdisplacement variables of the local and overall "elds. Thus the mid-plane strain �

�take the form

��"u

��

�#�

�(w

��w

�l#v

��v�l)��

��

��

��l

#��

��

��#��

��

��

#��

��

��

��

�#��

��

��

��

�#��

��

��

��

�, (24)

where the last two lines give the local buckling contributions from two local modes. For example,the contributions from the nth local mode are

��"u��

��

�(y)�

�cos (�

�x), (25a)

��

"

��4

w��

w��

��(y)�

�(y)#�2��u��

��

�(y)#

��4

w��

w��

��(y)�

�(y)�cos (2��x), (25b)

where ��

"m�/¸, and a prime denotes di!erentiation w.r.t. x.Expressions similar to Eq. (25) are written down for all the strain components. The incremental

strain vector ��"��

��

��

��

��

��

��

��

� can be related to the incremental


degrees of freedom. The full set of d.o.f's �q� can be divided into two categories: those that depictoverall action �q�� and others that control the amplitude modulating functions associated withlocal buckling �ql�, i.e. �q��"��q��ql��. The increments of strains are expressed in terms of theq's with the aid of B-matrices as follows

��l�"B�

��q�#B�

��q�q�, (26a)

��l�"

��

�[B��] cos(n�

�x)#[B�

��] sin(n�

�x)� ql

�

#

��

�[B��

] cos (n��x)#[B�

��] sin(n�

�x)� ql

�ql

�. (26b)

The elements of the matrix [B�] can be grouped under two distinct categories (i) those that arewhich describe the variations of the overall strain quantities and (ii) those which describe thevariations of the strain associated with local buckling via amplitude modulating function. (Thelatter comes from the quadratic L

�(w��) and L

��(w��, w��) terms.) Both these variations are treated

as `slowa compared to cos (��x) or sin (�

�x) associated with local buckling.

Current stresses �� too are arranged in a similar form and these must be available for everyintegration point on the surface of the structure.

� �"� ��

��

[� �� cos (i�

�x)#� �

�� sin (i�

�x)] (27)

The tangential sti!ness matrix [K] is a sum of two matrices [K�] and [K

�] whose elements are

derived from:

[K��]"�

�B��

H��

B��

�dx dy#�

�B��

��dx dy (28a)

[Kl

��]"

12 �

�

��

��

B��

H��

B��

#B��

H��

B��dx dy

#

12�

�

��

B��

��

#B��

��dxdy. (28b)

Note that [K�] houses overall, local and interactive terms whereas [K

] contains only local

buckling terms. The simpli"ed form of Eq. (28) is due to the `slowly varyinga nature of the overalldisplacements and the amplitude modulating functions.

The vector of unbalanced forces at any stage in the analysis are given by

� f �"� f�

�!�

[B]�� dx dy, (29)


where f�

is the externally applied force. From this point on, standard procedures of nonlinearanalysis are used to trace the load de#ection relationship. Appropriate imperfections are used tode"ne the initial geometry corresponding to zero stress.

3. Numerical examples

3.1. Finite element discretization

Each longitudinal shell segment/plate member is either represented by a single element ordivided longitudinally into two elements. A su$ciently high polynomial ( up to p"5) is used in thex-direction. Quadratic functions are employed for the representation of the amplitude modulatingfunction. Thus, the number of elements are of the same order as the longitudinal plate stripsconstituting the structure.

3.2. Objectives

In this section the results of a numerical study are presented with the following broad objectivesin view:

1. Compare the results obtained by the present simpli"ed model with that produced by a generalpurpose commercial program to illustrate the accuracy and e$ciency of the former. Theexample of a 5-bay cylindrical structure is selected for this.

2. Study the performance of the present model in cases where two local buckling modes must beconsidered in a modal interactive buckling analysis. Examples of box and I-section columns areconsidered for this purpose.

3. To examine the roles respectively of the secondary local mode and the periodic part of thesecond order "elds in the determination of the maximum load carrying capacity under interac-tive buckling.

3.3. Stringer-stiwened shell structure

Fig. 3 shows one half of a 5-bay stringer-sti!ened shell structure studied. The geometry of thecross-section is given by the parameters: b ( sti!ener spacing), d

�(sti!ener depth), t

�( sti!ener

thickness), R ( radius of the shell) and h ( thickness of shell). These values are given byb/t"40, d

�/t"10, t

�/t"2, R"400t, h"0.5t, where t is the unit length.

The structure is subjected to uniform axial compressive force. It is simply supported at its ends inthe `classicala sense, i.e. v"w"�"0; and normal de#ections are arrested along the outerlongitudinal edges too. The following two materials are considered:

Case 1. Isotropic material (modulus of elasticity, E and Poisson's ratio, �"0.3).Case 2. Specially orthotropic : 8-layered [(03/903)

�]�composite laminate with

E�"32900ksi, E

�"1800ksi, G

��"G

��"G

��"880 ksi, �

��"0.24.

The symmetry of response exists with respect to either center line of the shell.


Fig. 3. Five-bay sti!ened cylindrical shell.

In each case the length of the shell, L, is so chosen as to make the overall critical stress close to thelocal critical stress. These resulted in the following characteristics in each case:

Case 1: L"450t, m"18, �/E"1.159�10�,

�/E"1.176�10�, �"

�/

�"0.986,

Case 2: L"495t, m"15, �/E

�"0.328�10�,

�/E

�"0.328�10�, �"

�/

�"1.0.

Analysis is carried out using 8 elements, with p"5 in either direction. Only one local mode isconsidered in the analysis.

The responses of the structure in terms of nondimensional load versus maximum downwardcentral de#ection (W

��) for three di!erent levels of imperfections are shown in Fig. 4(a) and (b) for

the two cases considered. The structure exhibits considerable imperfection-sensitivity, losing morethan 20% and 30% under purely local imperfections of 0.05t and 0.1t, (��

�"0.05 and 0.1),

respectively. With a minute overall imperfection of about L/1000, (��"!0.5) superposed on

a local imperfection of 0.1t, it loses nearly 50% of its buckling capacity. In the presence of smalllocal imperfections only, the structure tends to bend outwards initially. The results given by thepresent model agree well with those given by ABAQUS obtained using a mesh of 16�24, 8-nodedshell elements.

3.4. Doubly symmetric columns

Next, we present examples of thin-walled columns with doubly symmetric cross-sections forwhich two companion local modes become active in the interactive buckling response. In all thecalculations presented herein, we include the two relevant local modes having the same wavelength.These results are compared against previously published results obtained using beam elementscarrying local buckling information embedded in them [15] as well as ABAQUS. Finally, weexamine the role of the secondary local mode in the present approach where considerable freedomfor modi"cation of the primary local buckling mode is already available as we let the amplitudemodulating function to vary from element to element.


Fig. 4. A comparison of the models: (a) isotropic sti!ened shell; (b) composite layered shell. Maximum de#ection vs. axialstress.

3.5. Square box Column

3.5.1. Geometry and buckling dataSquare box columns made of isotropic material with simply supported ends under uniform axial

compressive stress are studied. The cross-section is de"ned by b/t"60, b is the width of section andt, the thickness ("1). Two cases are considered:

(i) �" �/

�"1.022, L"2400t, m"40,

(ii) �" �/

�"1.807, L"1800t, m"30.

The primary local mode is symmetric with respect to both the lines of symmetry of the section.


Fig. 5. Response of a square box column as given by di!erent models.

The secondary local mode is anti-symmetric with respect to the axis of overall buckling andsymmetric with respect to the other axis of symmetry (Fig. 1). The ratio of the critical stress of thesecondary local mode to that of the primary one, �"

�/

�"1.432.

3.5.2. Details of the analysisImperfection in the form of the primary local mode is taken as 0.025t in all the calculations

(��"0.025) and that in the form of Euler bucklingmode is taken as a fraction of the r, the radius of

gyration of the cross-section i.e., ��"0.02r/t and 0.08r/t. Note that no initial imperfections are

present in the secondary local mode, but this mode will be triggered automatically in the course ofinteraction.

Because of symmetry, the model consisted of one-half of the section over one-half of the length ofthe column. The model consisted of only four elements one for each of the top and bottom #angesand the other two representing the web. The analysis was performed with p"4 ( cubic polynomialsrepresenting the displacement variables) in both the longitudinal and transverse directions. Thelocal buckling amplitude modulating functions for the four elements were treated as independent.This makes the model somewhat more compliant- a point that will be discussed later on.

The results of the analysis are compared with those obtained using beam elements carrying localbuckling information and ABAQUS. In the former analysis classical plate theory was employed, 24"nite strips were used to represent the cross-section in the determination of local buckling "eldsand 5 beam elements were used in the "nal nonlinear analysis. The amplitude modulating functionswere solely functions of x. This means that the transverse transformation of local bucklingdeformation must come entirely from an appropriate linear combination of the two participatingmodes. Abaqus model consisted of a 8�60 mesh of 8-noded shell elements over the same model.

3.5.3. DiscussionFig. 5 shows a plot of the applied stress versus the end-shortening as measured at the middle

point of the web, �, as given by the three models for case (i). The responses are close to each otheruntil the limit point, but the present model predicts the smallest maximum load carrying capa-city*a value which is about 3��% less than that predicted by the Abaqus model*which isdeemed to be more accurate. The reason for this lower prediction is the simpli"ed treatment of theamplitude modulation of the local buckling modes which is treated as independent for each plate


Fig. 6. Square box columns; beam model vs present model.

element. This allows for considerable freedom for the local mode to change at the expense of someadditional compliance because of lack of compatibility of the local buckling rotations at thejunctions which is implied in such an assumption. We thus have a slightly conservative predictionof the maximum load carrying capacity at considerable savings of computational e!ort.

Fig. 6 shows the results for both the cases ((i) and (ii) ) for two levels of overall imperfections, viz.,2% and 8% of the radius of gyration, as given by the beam model and the present one. As beforethe present model exhibits a more compliant behavior. It is seen that the responses given by thepresent model near the maximum load are less peaky. It is also seen that for case (i) (coincidentbuckling) the smaller the imperfections the more abrupt the unloading past the peak.

Fig. 7 shows the deformed con"guration of the column just past the peak load. Clearly, there issevere localization of deformation over a relatively small length near the center of the top #angewhich comes under combined compression due to axial load and overall buckling. Local bucklingdeformations are accentuated here and elsewhere they are not noticeable.

3.6. I-section column

3.6.1. Geometry and buckling dataCross-section (Fig. 8) is de"ned by the following parameters: t, thickness of #ange, b, width of

#ange, t thickness of web, and depth of web, h. In the present study, we take t

"2t, b"h"80t.

For near coincident buckling, L"2000t with �"1.06. The primary and secondary local modes aredisplayed in Fig. 9. Note that overall buckling takes place about the web, which is the weaker axis.These are anti-symmetric and symmetric with respect to the web. The ratio of the secondary to theprimary local critical stress, �"1.22.

3.6.2. Details of the AnalysisImperfections in the form of the primary local mode and Euler buckling mode are taken as 0.1t

and L/4000, respectively (��"0.1, ��

�"¸/(4000t)). Because of symmetry, the model consisted of

one-half of the section over one-half of the length of the column. Only six elements were employed


Fig. 7. Localization of deformation in a box column.

Fig. 8. Geometry of the I-section column.

with p"4 in both the directions. The local buckling amplitude modulating functions for the threeplate elements involved were treated as independent.

The results of the analysis are compared, as before with those obtained using beam elementscarrying local buckling information [15] and Abaqus. In the former analysis 24 "nite strips wereused to represent the cross-section and 5 beam elements were used over one-half of the column.ABAQUS model consisted of a 12�18 mesh of 8-noded shell elements over one quarter of thecolumn.

3.6.3. DiscussionFig. 10 shows the variation of applied compressive stress with end shortening �, measured at the

center of the web. As before, the beammodel overestimates the maximum load carrying capacity by


Fig. 9. Local buckling modes of the I-section column.

12%, but the two responses tend to come together in the advanced post-peak response. This isbelieved due to the de"ciency of the beam element in which the freedom for the cross-sectional localbuckling deformation to modify itself was restricted. The results of ABAQUS and the presentmodel agree more closely in this example with the latter once again giving a more compliantresponse. Note for this case of near-coincident buckling, there is an erosion of load carryingcapacity of 40% under imperfections often unavoidable in practice.


Fig. 10. Response of the I-section column as given by the di!erent models.

3.7. Ewect of secondary local mode

In earlier models (see for example, [9,15]), the secondary local mode plays a pivotal role inmodeling the interaction. In these models, each of the local modes was associated with anamplitude modulating function and this was a function of x only. Thus, the secondary local modewas essential to model the localization the local buckling deformation in the cross-section . Asmentioned earlier, in the analysis this gives rise to a nonvanishing trilinear term of interaction. Thepresent model with its `locally buckleda elements has built into it ample freedom for the localbuckling mode to undergo modi"cation in the transverse direction because each element isassociated with an independent amplitude modulating function. As a result it may be inferred thatthe role of the secondarymode is seriously diminished and the primary local mode, because it is freeto modify itself under the in#uence of compression which varies from element to element , usurpsthe role of the secondary buckling mode.

The problem of the square box column under near-coincident buckling previously studied isreexamined. Details are the same as given earlier. The results for case (i) are reworked with thesecondary local mode suppressed in the formulation. Fig. 11(a) and (b) show the results of the loadplotted against maximum overall de#ection W

��and maximum amplitude of local buckling

de#ection, ��

, respectively. The di!erences in the results are too small to be noticeable in the"gures. Thus it would seem that all that is necessary is to identify the principal local mode andbuild into the model the buckling mode and its second order e!ects.


Fig. 11. (a) E!ect of secondary mode in the present model. Axial stress vs. maximum `overalla de#ection; (b) E!ect ofsecondary mode in the present model. Axial stress vs. maximum local buckling amplitude.

3.8. Ewect of second order xelds

The question of relative importance of the periodic part of second order "elds is of interest,because if it could be neglected the analysis becomes considerably simpli"ed both computationallyand conceptually. In order to examine the accuracy that would be lost by such a simpli"cation,a numerical example is studied.

3.8.1. Geometry and buckling dataThe example selected is the isotropic sti!ened shell shown in Fig. 5, but with a smaller length.

Two di!erent lengths are considered with their corresponding critical stress ratios:


Fig. 12. E!ect of the periodic component of the second order "eld.

Case (i): L"350t, m"14, �/E"1.176�10�, �"

�/

�"1.006, �"

�/

�"1.01,

Case (ii): L"225t, m"11, �/E"1.176�10�, �"

�/

�"1.829, �"

�/

�"1.01.

For these cases, symmetry condition with respect to longitudinal center line (x}x) does not applyas the overall buckling mode is now antisymmetric with respect to x}x (consisting of two lobesacross the section). The primary and secondary local modes are, respectively, symmetric andantisymmetric with respect to the center line.

3.8.2. Details of modelingEach shell segment between the sti!eners is represented by two elements and each sti!ener by

a single element. Thus the model consists of 16 elements in all. Selected polynomial level for theshape functions, p"5. Independent amplitude modulating functions are associated with eachpanel consisting of a sti!ener and the shell elements on either side.

3.8.3. DiscussionThe interactive buckling responses for the two cases were obtained for imperfections given by:

��"¸/1000t; ��

�"0.05. Note that the sign of overall imperfection is immaterial because of its

antisymmetry. The maximum de#ection, W��

occurs in that portion of the shell which bendsdownward due to overall buckling and therefore experiences higher compressive stress. Fig. 12plots the load versus maximum de#ection for the two cases obtained, respectively, including andneglecting (the periodic part of) the second order "elds. While this makes only a slight di!erence


(about 3%) in the prediction of the maximum load for the case near-coincident buckling(�"1.006), di!erence of 15% is noticed for the case in which local buckling occurs "rst, i.e. the casewith �"1.829. Thus, one may conclude that the neglection of these "elds results in a nontrivialoverestimate of the load carrying capacity. (Note however, the `slowly varyinga part of the secondorder "elds is automatically generated by the interaction of the buckling modes with the degrees offreedom associated with the FE mesh describing the overall response.)

4. Conclusions

The interaction of local and overall buckling in plate structures and sti!ened shells is studiedusing a specially formulated shell element . The element has additional degrees of freedom whichcan trigger and modulate the relevant local buckling modes together with the associated (periodiccomponents of ) second order "elds.

In numerical computations, the element is seen to be extremely e$cient and produces satisfac-tory results with only as many elements as there are shell/plate segments in the structure.

In the present approach each shell/plate segment is associated with an independent amplitudemodulating function and this greatly facilitates the capturing of localization of deformation bothlongitudinally and transversely. Because of this feature the incorporation of a higher localmode*which plays a pivotal role in modal interaction analysis*is found to be redundant. Theprimary local mode essentially usurps the role of the secondary local mode.

Except for the case of near coincident buckling, the incorporation of the periodic component ofthe second order "eld is necessary to obtain accurate estimates of the maximum load carryingcapacity. In shell structures, neglecting this "eld could result in an unconservative prediction, i.e. anoverestimate of the maximum load.

References

[1] T.R. Graves Smith, The ultimate strength of locally buckled columns of arbitrary length, Ph.D. Thesis, CambridgeUniversity, England, 1966.

[2] J.M.T. Thompson,G.M. Lewis, On the optimum design of thin-walled compressionmembers, J. Mech. Phys. Solids22 (1972) 101}109.

[3] A. Van der Neut, The sensitivity of thin-walled compression members to column axis imperfection, Int. J. Sol.Struct. 9 (1973) 999}1011.

[4] V. Tvergaard, In#uence of postbuckling behavior in optimum design of sti!ened panels, Int. J. Sol. Struct. 9 (1973)1519}1534.

[5] E. Byskov, J.W. Hutchinson, Mode Interaction in axially sti!ened cylindrical shells, AIAA J. 15 (7) (1977) 941}948.[6] W.T. Koiter, M. Pignataro, An alternative approach to the interaction of local and overall buckling in sti!ened

panels, in: B. Budiansky (Ed.), Buckling of structures, Springer, New York, 1976, pp. 133}148.[7] W.T. Koiter, M. Pignataro, General theory of mode interaction in sti!ened plate and shell structures, Report

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[10] S. Sridharan, Doubly symmetric interactive buckling of plate structures, Int. J. Solids Struct. 19 (7) (1983) 625}641.[11] S. Sridharan, M.-H. Peng, Performance of axially compressed sti!ened panels, Int. J. Solids Struct. 25 (8) (1989)

879}899.[12] S. Sridharan, Z. Madjid, J.H. Starnes, Mode interaction analysis of sti!ened shells using locally buckled elements,

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Stiffened plates and cylindrical shells under interactive buckling

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