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Stiffened plates and cylindrical shells under interactive buckling
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 analyzed using a novel "nite elements in which local buckling deformation has been embedded. Amplitude modulation, a key feature of the interactive buckling has been incorporated in the element formulation. The model has the following additional features: (i) the inclusion of a key secondary local mode where the cross-section has complete or approximate double symmetry; and (ii) the introduction of a simple approach for capturing localization of local buckling; this involves incorporating a single local buckling mode in the analysis, but letting the amplitude modulation function to be di!erent for di!erent elements. Numerical examples of plate and shell structures are presented to throw light on these aspects of the methodology as well as to demonstrate the accuracy and e$ciency of the model. 2001 Elsevier Science B.V. All rights reserved. Keywords: Finite elements; Buckling; Plate structures; Cylindrical shells; Modal interaction; Local buckling; Amplitude modulation; Localization of buckling; Imperfection sensitivity
1. Introduction Cylindrical shell and panels are often reinforced with stringers to enhance their sti!ness in resisting axial compression. The sti!ening elements not only enhance the buckling resistance but also reduce the imperfection-sensitivity of the shells. Because of the resistance o!ered by the sti!eners to radial movement, &local' buckling modes whose nodal lines do not coincide with the location of the sti!eners are simply eliminated. This has the e!ect of minimizing the nonlinear modal interactions which are the source of imperfection-sensitivity in unsti!ened shells.
* Corresponding author. Former doctoral student. 0168-874X/01/$ - 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
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However there remain principally two distinctive modes of buckling which dominate the behavior 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 with it. The optimum design of the shells often leads to a con"guration for which the critical stresses are close to each other. Thus, a study of nonlinear modal interaction of local and overall instabilities is of considerable signi"cance in the context of optimal design of such shells. The problem of interaction of an Euler buckling with local plate buckling was studied in the 1960s and 1970s by several investigators (see for example, [1}3]). Tvergaard presented a detailed analysis of sti!ened plates under interactive buckling [4]. The problem was studied Byskov and Hutchinson [5] using an asymptotic approach and solutions in the classical mold. Koiter and Pignatrao [6,7] authored two papers of fundamental importance to the interaction of local and overall buckling in sti!ened panels and sti!ened shells. They introduced the technique of amplitude modulation 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 his co-workers. A "nite strip approach was used [8] to extract the buckling modes and the second order "elds to simplify the analysis. Next, a special beam element was developed to deal with the interaction of local and overall buckling . This is an Euler}Bernoulli beam element in which the local buckling deformation was embedded. The local buckling mode(s) and the periodic part of second order "elds are determined a priori by the asymptotic procedure. These e!ects are built into a beam "nite element in terms of two additional degrees of freedom, for each local buckling mode considered, thus allowing automatically for the amplitude modulation [9,10]. The theoretical basis of such an approach has been discussed by Sridharan and Peng [11]. It is shown that the amplitude modulation is the key factor in the interaction; it performs the function of capturing the contributions of several neighboring modes of the same longitudinal description as the fundamental local mode, but with di!ering circumferential wave numbers. For doubly symmetric cross-sections it is vitally important to consider a companion secondary local mode which is triggered by the interaction of the overall buckling mode and the primary local mode. The use of amplitude modulated local modes and the induction of key secondary modes as full participants in the interaction pulls in all the additional patterns of deformation generated in the interaction of an overall mode with the primary local mode. So much so, it is super#uous to consider the mixed second order "eld arising by the interaction of these two modes. Use of a beam element in the formulation 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 in which local buckling deformation was embedded. However, only relatively simple examples were considered for illustration, such as sti!ened panels studied previously by Tvergaard. The method was 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 means of nonlinear "nite element analysis. Such an analysis would obviously entail considerable
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computational resources as it would require the use of a su$ciently "ne mesh to capture the local deformation (such as sinusoidal ripples of small wavelength) in a structure long and broad enough for the overall instability to be of importance. From the point of view of modeling interactive buckling using a modal approach , prismatic structural components can be divided into two broad categories: (i) Structures whose cross-section exhibits strong asymmetry with respect to the axis of bending or overall buckling. Typical examples of these would be slender plates attached to stocky sti!eners, e.g. Tvergaard panels [4]. (ii) Structures which exhibit near or complete symmetry with respect to axis of bending or overall buckling. Typical examples of these are rectangular box- or I-section columns. Consider case (i): In its simplest form, the structure has only two signi"cant modes of buckling, viz., Euler buckling mode and the plate buckling mode with scaling parameters and respec tively. The potential energy function governing the interaction can be written as: # higher order terms, (1) "a ! b #a ! b #c 2 where the key term of interaction is the c term. This term involves the energy of interaction of overall bending associated with Euler buckling with the midsurface stretching associated with local buckling. Such a potential energy function was employed by Tvergaard [4] and subsequently thoroughly investigated by Hunt [14]. Next consider the case (ii). In this case, the key term c vanishes because of symmetry of the cross-section with respect to the axis of bending: the overall buckling stress/strain is antisymmetric with respect to the axis of bending while the mid-surface stretch associated with local buckling is symmetric with respect to the same axis. In order to compute a higher order term involving both and , one has to compute the mixed second order "eld which arises by an interaction of the two modes. Such a procedure will yield a biquadratic term, viz., [10]. The inherent numerical di$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 (associated with 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 deformation in the compression #ange of the square box column, i.e. the local deformation in the #ange which su!ers additional compression due to overall buckling is further accentuated while that in the opposite tension #ange is alleviated (Fig. 1). In order to ensure accuracy, post-local buckling e!ects must be accounted for in the analysis. Thus not only the secondary local mode, but also the second order "eld associated with it and the mixed second order "eld that arises by the interaction of 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 local buckling deformation embedded in them. It would again, seem necessary to incorporate degrees of freedom to represent the secondary local mode, together with the relevant second order "elds. This
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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 of achieving the same result is also presented. This, is considering just a single local mode but letting the amplitude modulation function be di!erent for di!erent plate elements. Thus the crosssectional pattern of local buckling deformation of the structure evolves freely under the in#uence of overall buckling deformation. Another issue that is relevant is the relative importantance of the short wave contribution of the second order "elds for obtaining an accurate estimate of the maximum load. In this study an attempt 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 of localization 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 the computations. (iii) Finally, the accuracy of the model by comparison with detailed FE analysis and earlier simpli"ed formulations. In order to make the paper complete, we present a brief outline of the formulation of &locally buckled' plate/shell elements. This is followed by several worked examples and commentary thereof. 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.
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2.1. Displacement, strain and stress vectors The displacement variables are u2"u, v, w, , ,
(3)
where u, v and w are the displacement components in the axial (x-), transverse (y-) and outward normal (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: 2" , , , , , , , . (4) V W VW V W VW VX WX Of these, " , , are the inplane strain components, " , , are the curvature V W VW V W VW components, and " , are the transverse shearing strain components. VX WX The following strain}displacement relations are assumed for the shell/sti!ener: u 1 " # V x 2
v w , # x x
v w 1 " # # W y R 2
u w # , y y
(5a) (5b)
u v w w " # # , VW y x x y
(5c)
" , V x
(5d)
" , W y
(5e)
" # , VW y x
(5f)
Fig. 2. Coordinate axes of a cylindrical shell.
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w "# , VX x
(5g)
w "# . WX y
(5h)
These may be viewed as Donnell's strain}displacement relations modi"ed to account for transverse shear deformation and the large in-plane motions of sti!eners such as would occur under overall buckling. Thus, the expression is augmented to include a nonlinear term in v*a key term which V enables the modeling of overall bending/buckling phenomena. The nonlinear terms in u and v can be neglected in dealing with a purely local buckling problem. The Eqs. 5(a)}5(h) can be written in an abbreviated form "¸ GH (u )#¸ GH (u ) H G H
(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 2, moment resultants M"M M M 2, and transverse shear forces V W VW V W VW Q"Q Q 2. V W The stress}strain relations take the form
(7a)
Q"kGM t ,
(7b)
N
A B
B
"
M
D
,
where k is the shear correction factor (taken as 5/6), GM is the averaged transverse shear modulus and t is the thickness of the shell. The stress}strain relations may be written in the abbreviated form
"H . G GH H
(8)
2.2. Determination of the local buckling xelds 2.2.1. Notation The 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 to a 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 critical stresses; double superscripts (11) and (22) indicate the second order "elds associated with the primary local mode and secondary local mode, respectively, and (12) stands for the mixed second order "eld (m.s.o.f.) arising by the interaction of local modes 1 and 2. The local buckling problem is solved following the standard "nite strip procedure [17]. The salient features thereof will now be mentioned.
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2.2.2. The linear stability (eigenvalue) problem The potential energy function corresponding to the nth local mode (u) can be written in the form (uL)], L" [H ¸ G I (uL) ) ¸ H l (uL l )# H ) ¸ I G G I I GH
(9)
where represents the stress in the unbuckled state, and a dot operation indicates multiplication G 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 uL, "uL, L (y) sin G G G
mx , ¸
vL, wL, L"vL, wL, L (y) cos G G G G
mx . ¸
(10)
Here u , v , 2 are the degrees of freedom (d.o.f.) and are appropriately chosen polynomial shape G functions. In the present work, the functions are chosen in a hierarchical form as explained later. G For su$ciently large m, the boundary conditions at the end are deemed not to in#uence the local buckling process. Designating the d.o.f.'s of the local buckling problem as qL, the potential energy function G (Eq. (9)) may be expressed in the form L"aL!bL qL qL GH G H GH
(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 aL! bL qLo "0. GH GH H
(12)
Solution of the linear eigenvalue problem in Eq. (12) gives the critical stress for local buckling and the eigenmode, qL . G 2.2.3. Second order xeld problem The potential energy function governing the second order "eld problem is given by )# H ) ¸ GI (uLL) LL"[H ¸ GI (uLL) ) ¸ Hl (uLL l G I GH I )#2¸ GI (uL) ) ¸ Hl (uL , uLL)], #H ¸ GI (uLL) ) ¸ Hl (uL l I l l GH I
(13)
where uLL 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
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equations consists of three sets of terms in general: r"r(y)#r(y) cos
2mx 2mx #r(y) sin , ¸ ¸
(14)
The last two terms are rapidly varying trigonometric functions (m 1) in x, while the "rst term is independent of x. The second order "eld problem may, then, be viewed as that of a cylindrical shell subjected to two loads which vary sinusoidally in the x-direction, together with a third term which remains constant in the x-direction. The solution takes the form uLL"u#u cos
2mx 2mx #u sin , ¸ ¸
(15)
where u is a function of x and y, whereas u and u are functions of y only and can be viewed as the particular solution of the di!erential equation. u is a slowly varying function with respect to x and contains additional terms needed to enforce the end boundary conditions. Note that the solution cannot contain a component in the form of the buckling mode in the asymptotic procedure [18]. Also because of the slowly varying nature of u, it is decoupled in the solution process from u and u. Unlike in the asymptotic procedure for the initial postbuckling analysis [17], it is not necessary to compute u at this stage. Rather we shall let it arise, by the interaction of L (uL) terms with the degrees of freedom associated with the "nite element mesh to be introduced later. Because of its slowly varying character, a relatively coarse "nite element mesh must be able to pick up the deformation associated with u. So, we need to compute only the solution vectors associated with the trigonometric terms at this stage. For a specially orthotropic material, the displacement "elds to be computed take the simpli"ed form uLL, LL"uLL, LL (y) sin G G G
2mx , ¸
vLL, wLL, LL"vLL, wLL, LL (y) cos G G G G G G
2mx , ¸
(16)
where u,2 are the d.o.f.'s second order "eld. G The potential energy function (Eq. (10)) can now be expressed in terms of the d.o.f.'s qL and P qLL de"ning the "rst and second order "elds, respectively, and takes the form G LL"(aLL!bLL)qLLqLL#c qLLqLqL (r, s"1, 2 , n ); (i, j"1,2, 2 , n ), (17) GH G H GPQ G P Q GH 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 (aLL! bLL) qLL"!c qL qL, GH GH H GPQ P Q is set equal to in the calculations.
(18)
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2.3. Mixed second order xeld The potential energy function governing this problem can be expressed in the form "[H ¸ GI (u) ) ¸ GJ (u)# H ) ¸ GI (u) GH J J G I #H 2¸ GI (u) ) ¸ HJ (u, u)#2¸ GI (u) ) ¸ HJ (u, u) GH I J J I J J #2¸ GI (u) ) ¸ HJ (u, u)]. (19) I J J Here we consider the second order "eld arising out of interaction of two local modes of the same wave length. As discussed in a later section, signi"cant interaction occurs only between the overall mode and two local modes of the same (or nearly the same) wavelength. Thus, we assume the two local modes have the same number of half-waves, m. This would lead to a di!erential equation which has the same structure as Eq. (15) and results in a solution of the type given in Eq. (16). Once again 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 written in the form ul "u #u #u #u #u , (20) where u"u v w 2; 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 the double 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 longitudinal direction as the local buckling deformation comes under the in#uence of overall buckling of the structure. Thus, the local buckling amplitudes are `modulateda to enable the model to depict the accentuation of local buckling in regions where compressive stresses develop due to overall buckling. Apart from this, because of its `slowly varyinga character, the amplitude modulation performs the role of accounting for the neighboring local modes of the same transverse description as the associated local mode but of slightly di!ering half-wavelength, which are liable to be triggered by the interaction [11,12]. Thus we let the amplitude vary with respect to x- according to a `slowly varyinga modulating function. The local buckling deformation is thus represented in the form u"u G #u G (x)#u G H #u G H #u G H (x) (x), G G H &s represent the degrees of freedom associated with amplitude modulating functions &s.
(21)
2.5. The wavelength of the secondary local mode It has been assumed in the foregoing treatment that the two local modes which together describe the local buckling deformation have the same number of half-wavelengths, m. This requires some
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elucidation. The most signi"cant term in the interaction of two local modes with the overall mode is 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 nearly constant. Thus, the said integral tends to vanish unless the longitudinal wavelengths (or number of half-waves, m) are the same. Thus only the interaction of local modes of the same wavelength with the 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 of orthogonal transverse description but of the same wave length and all the associated second order e!ects. To this we add yet another feature in the present study : The amplitude modulating functions are given freedom to di!er from element to element, thus giving the the local buckling deformation freedom to vary across the section. This tends to make the secondary local mode somewhat redundant*an issue which will be considered later. 2.7. Neglection of mixes second order xelds (m.s.o.f.), uL The m.s.o.f uL arising of out the interaction of a typical local mode (n"1,2) with the overall mode, 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 the authors have [12] have discussed the pitfalls in the evaluation of the m.s.o.f. arising from the interaction of local and overall modes. In brief, accuracy is lost due to the fact interaction in a long prismatic structure gives rise to a number of local modes with eigenvalues close to that of the principal local mode at which the m.s.o.f. is evaluated. On the other hand, the m.s.o.f. can help identify the local modes that are triggered in the interaction [11,15]. Once these modes are identi"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 amplitude modulation of the local modes and the freedom given for the local buckling deformation to undergo modi"cation freely in the transverse direction*all these fully account for any change of the local buckling deformation. Further, we have a "nite element mesh that would automatically account for any changes in the overall deformation under the in#uence of the local modes. Also note 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 evaluated requiring it to be orthogonal with respect to the totality of the "rst order "eld (which includes the fundamental 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 can therefore be conveniently neglected. Note that a typical m.s.o.f. strain takes the form "¸ (uL)#¸ (u,u), (22) L where the "rst term represents the contribution of mixed second order displacement "eld and the second term arises by the mixing of the two fundamental modes. If only the "rst term on R.H.S. is
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neglected, Eq. (22) is clearly inconsistent and therefore the second term which acts as the driver of the m.s.o.f. must also be neglected. Thus, we set all the m.s.o.f stresses, strains and displacements are set to zero. 2.8. Finite element formulation 2.8.1. Choice of shape functions A 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), the problems of shear and membrane locking associated with lower order elements become inconsequential. A relatively small number of elements (in comparison to the h-version approach) would be su$cient to accurately model the structure and this results in considerable savings of e!ort in data input. The type of polynomials chosen in the present work are integrals of the Legendre G functions, advocated by Szabo and Babuska [19]. These have been successfully used in previous work by the authors (see for example [17]). 2.8.2. Strain-displacement matrix As stated earlier, a cylindrical shell element based on Donnell's theory admitting sheardeformation via Reissner-Mindlin theory (Eq. 5(a)}5(h)) is employed. The displacement functions take the form u2"u , v , w , , 2 (x) (y). GH GH GH GH GH G H
(23)
As a "rst step in the formulation we set up the B-matrix which relates the incremental strains to the incremental degrees of freedom. To this end, each strain component is expressed in terms of displacement variables of the local and overall "elds. Thus the mid-plane strain take the form V "u #(w w l #v v l ) l GI H G G I V GI G I GI H # H # G V G V G # G H # H H # G H , V G H V G H V G H
(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 "uL (y) cos ( x), V G G K K
(25a)
LL" K wLwL (y) (y)# 2 uLL (y)# K wLwL (y) (y) cos (2 x), H K G G H K V 4 G H G 4 G H G
(25b)
where "m/¸, and a prime denotes di!erentiation w.r.t. x. K Expressions similar to Eq. (25) are written down for all the strain components. The incremental strain vector " can be related to the incremental V W VW V W VW VX WX
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degrees of freedom. The full set of d.o.f's q can be divided into two categories: those that depict overall action q and others that control the amplitude modulating functions associated with local buckling ql, i.e. q2"q2ql2. The increments of strains are expressed in terms of the q's with the aid of B-matrices as follows l"B q #B q q , G GH H GHI H I
(26a)
l" [B ] cos(n x)#[B ] sin(n x) ql G LGH K LGH K G L # [B ] cos (n x)#[B ] sin(n x) ql ql . LGHI K LGHI K H I L
(26b)
The elements of the matrix [B] can be grouped under two distinct categories (i) those that are which describe the variations of the overall strain quantities and (ii) those which describe the variations of the strain associated with local buckling via amplitude modulating function. (The latter comes from the quadratic L (wL) and L (w, w) terms.) Both these variations are treated as `slowa compared to cos ( x) or sin ( x) associated with local buckling. K K Current stresses too are arranged in a similar form and these must be available for every integration point on the surface of the structure. " [ cos (i x)# sin (i x)] G K G K G
(27)
The tangential sti!ness matrix [K] is a sum of two matrices [K ] and [K ] whose elements are derived from:
[KM ]" GH
1 [Kl ]" GH 2
B H B dx dy# NG NO OH
B dx dy NGH N
(28a)
B H B #B H B dx dy KNG NO KOH KNG NO KOH K
1 # 2
B #B dx dy. KNGH KN KNGH KN L
(28b)
Note that [K ] houses overall, local and interactive terms whereas [K ] contains only local J buckling terms. The simpli"ed form of Eq. (28) is due to the `slowly varyinga nature of the overall displacements and the amplitude modulating functions. The vector of unbalanced forces at any stage in the analysis are given by
f " f !
[B]2 dx dy,
(29)
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where f is the externally applied force. From this point on, standard procedures of nonlinear analysis are used to trace the load de#ection relationship. Appropriate imperfections are used to de"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 or divided longitudinally into two elements. A su$ciently high polynomial ( up to p"5) is used in the x-direction. Quadratic functions are employed for the representation of the amplitude modulating function. Thus, the number of elements are of the same order as the longitudinal plate strips constituting the structure. 3.2. Objectives In this section the results of a numerical study are presented with the following broad objectives in view: 1. Compare the results obtained by the present simpli"ed model with that produced by a general purpose commercial program to illustrate the accuracy and e$ciency of the former. The example 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 be considered in a modal interactive buckling analysis. Examples of box and I-section columns are considered for this purpose. 3. To examine the roles respectively of the secondary local mode and the periodic part of the second order "elds in the determination of the maximum load carrying capacity under interactive 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 the cross-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 by b/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 in the `classicala sense, i.e. v"w""0; and normal de#ections are arrested along the outer longitudinal 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 "32900 ksi, E "1800 ksi, G "G "G "880 ksi, "0.24. The symmetry of response exists with respect to either center line of the shell.
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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 the local 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 is considered in the analysis. The responses of the structure in terms of nondimensional load versus maximum downward central 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 more than 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 small local imperfections only, the structure tends to bend outwards initially. The results given by the present model agree well with those given by ABAQUS obtained using a mesh of 16;24, 8-noded shell elements. 3.4. Doubly symmetric columns Next, we present examples of thin-walled columns with doubly symmetric cross-sections for which two companion local modes become active in the interactive buckling response. In all the calculations presented herein, we include the two relevant local modes having the same wavelength. These results are compared against previously published results obtained using beam elements carrying local buckling information embedded in them [15] as well as ABAQUS. Finally, we examine the role of the secondary local mode in the present approach where considerable freedom for modi"cation of the primary local buckling mode is already available as we let the amplitude modulating function to vary from element to element.
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Fig. 4. A comparison of the models: (a) isotropic sti!ened shell; (b) composite layered shell. Maximum de#ection vs. axial stress.
3.5. Square box Column 3.5.1. Geometry and buckling data Square 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 and t, 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.
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The secondary local mode is anti-symmetric with respect to the axis of overall buckling and symmetric with respect to the other axis of symmetry (Fig. 1). The ratio of the critical stress of the secondary local mode to that of the primary one, " / "1.432. 3.5.2. Details of the analysis Imperfection 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 buckling mode 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 of interaction. Because of symmetry, the model consisted of one-half of the section over one-half of the length of the column. The model consisted of only four elements one for each of the top and bottom #anges and the other two representing the web. The analysis was performed with p"4 ( cubic polynomials representing the displacement variables) in both the longitudinal and transverse directions. The local 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 local buckling 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 "elds and 5 beam elements were used in the "nal nonlinear analysis. The amplitude modulating functions were solely functions of x. This means that the transverse transformation of local buckling deformation must come entirely from an appropriate linear combination of the two participating modes. Abaqus model consisted of a 8;60 mesh of 8-noded shell elements over the same model. 3.5.3. Discussion Fig. 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 other until the limit point, but the present model predicts the smallest maximum load carrying capacity*a value which is about 3 % less than that predicted by the Abaqus model*which is deemed to be more accurate. The reason for this lower prediction is the simpli"ed treatment of the amplitude modulation of the local buckling modes which is treated as independent for each plate
Fig. 5. Response of a square box column as given by di!erent models.
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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 some additional compliance because of lack of compatibility of the local buckling rotations at the junctions which is implied in such an assumption. We thus have a slightly conservative prediction of 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 before the present model exhibits a more compliant behavior. It is seen that the responses given by the present model near the maximum load are less peaky. It is also seen that for case (i) (coincident buckling) 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 is severe localization of deformation over a relatively small length near the center of the top #ange which comes under combined compression due to axial load and overall buckling. Local buckling deformations are accentuated here and elsewhere they are not noticeable. 3.6. I-section column 3.6.1. Geometry and buckling data Cross-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 are displayed 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 the primary local critical stress, "1.22. 3.6.2. Details of the Analysis Imperfections 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
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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 three plate elements involved were treated as independent. The results of the analysis are compared, as before with those obtained using beam elements carrying local buckling information [15] and Abaqus. In the former analysis 24 "nite strips were used 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 the column. 3.6.3. Discussion Fig. 10 shows the variation of applied compressive stress with end shortening , measured at the center of the web. As before, the beam model overestimates the maximum load carrying capacity by
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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 is believed due to the de"ciency of the beam element in which the freedom for the cross-sectional local buckling deformation to modify itself was restricted. The results of ABAQUS and the present model agree more closely in this example with the latter once again giving a more compliant response. Note for this case of near-coincident buckling, there is an erosion of load carrying capacity of 40% under imperfections often unavoidable in practice.
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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 in modeling the interaction. In these models, each of the local modes was associated with an amplitude modulating function and this was a function of x only. Thus, the secondary local mode was essential to model the localization the local buckling deformation in the cross-section . As mentioned earlier, in the analysis this gives rise to a nonvanishing trilinear term of interaction. The present model with its `locally buckleda elements has built into it ample freedom for the local buckling mode to undergo modi"cation in the transverse direction because each element is associated with an independent amplitude modulating function. As a result it may be inferred that the role of the secondary mode is seriously diminished and the primary local mode, because it is free to modify itself under the in#uence of compression which varies from element to element , usurps the role of the secondary buckling mode. The problem of the square box column under near-coincident buckling previously studied is reexamined. Details are the same as given earlier. The results for case (i) are reworked with the secondary local mode suppressed in the formulation. Fig. 11(a) and (b) show the results of the load plotted 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 and build into the model the buckling mode and its second order e!ects.
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Fig. 11. (a) E!ect of secondary mode in the present model. Axial stress vs. maximum `overalla de#ection; (b) E!ect of secondary 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 computationally and 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 data The 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:
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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 apply as the overall buckling mode is now antisymmetric with respect to x}x (consisting of two lobes across the section). The primary and secondary local modes are, respectively, symmetric and antisymmetric with respect to the center line. 3.8.2. Details of modeling Each 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 the shape functions, p"5. Independent amplitude modulating functions are associated with each panel consisting of a sti!ener and the shell elements on either side. 3.8.3. Discussion The 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 bends
downward due to overall buckling and therefore experiences higher compressive stress. Fig. 12 plots the load versus maximum de#ection for the two cases obtained, respectively, including and neglecting (the periodic part of) the second order "elds. While this makes only a slight di!erence
Fig. 12. E!ect of the periodic component of the second order "eld.
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(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 case with "1.829. Thus, one may conclude that the neglection of these "elds results in a nontrivial overestimate of the load carrying capacity. (Note however, the `slowly varyinga part of the second order "elds is automatically generated by the interaction of the buckling modes with the degrees of freedom 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 studied using a specially formulated shell element . The element has additional degrees of freedom which can trigger and modulate the relevant local buckling modes together with the associated (periodic components of ) second order "elds. In numerical computations, the element is seen to be extremely e$cient and produces satisfactory 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 amplitude modulating function and this greatly facilitates the capturing of localization of deformation both longitudinally and transversely. Because of this feature the incorporation of a higher local mode*which plays a pivotal role in modal interaction analysis*is found to be redundant. The primary 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 of the second order "eld is necessary to obtain accurate estimates of the maximum load carrying capacity. In shell structures, neglecting this "eld could result in an unconservative prediction, i.e. an overestimate of the maximum load.
References [1] T.R. Graves Smith, The ultimate strength of locally buckled columns of arbitrary length, Ph.D. Thesis, Cambridge University, England, 1966. [2] J.M.T. Thompson, G.M. Lewis, On the optimum design of thin-walled compression members, J. Mech. Phys. Solids 22 (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 WTHD-19, Delft University of Technology, Delft, The Netherlands, 1976. [8] Benito, Sridharan Interactive buckling analysis with "nite strips (with R. Benito), Int. J. Numer. Methods Eng. 21 (1985) 145}161. [9] S. Sridharan, M.A. Ali, An improved interactive buckling analysis of thin-walled columns having doubly symmetric sections, Int. J. Solids Struct. 22 (4) (1986) 429}443.
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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, Int. J. Solids Struct. 31 (7) (1994) 2347}2366. [13] ABAQUS, Version 5.8, Hibbit, Karlsson and Sorenson Inc., Newark, California, USA 1998. [14] G.W. Hunt, Imperfection-sensitivity of semi-symmetric branching, Proceedings of the Royal Society, London, A357, 1977 193}211. [15] S. Sridharan, M.A. Ali, Interactive buckling in thin-walled beam columns, J. Eng. Mech. Div. ASCE 111 (12) (1985) 1470}1486. [16] M.A. Ali, S. Sridharan, A versatile model for interactive buckling of columns and beam}columns, Int. J. Sol. Struct. 24 (5) (1988) 481}496. [17] S. Sridharan, Z. Madjid, J.H. Starnes, Postbuckling response of sti!ened composite cylindrical shells, AIAA J. 30 (12) (1992) 2897}2905. [18] B. Budiansky, Dynamic buckling of elastic structures: criteria and estimates, in: G. Herrman (Ed.), Dynamic Stability of Structures, Pergamon, Oxford, UK, 1965, pp. 83}106. [19] B.A. SzaboH , I. Babus\ ka, Finite Element Analysis, John Wiley, New York, 1991.
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 analyzed using a novel "nite elements in which local buckling deformation has been embedded. Amplitude modulation, a key feature of the interactive buckling has been incorporated in the element formulation. The model has the following additional features: (i) the inclusion of a key secondary local mode where the cross-section has complete or approximate double symmetry; and (ii) the introduction of a simple approach for capturing localization of local buckling; this involves incorporating a single local buckling mode in the analysis, but letting the amplitude modulation function to be di!erent for di!erent elements. Numerical examples of plate and shell structures are presented to throw light on these aspects of the methodology as well as to demonstrate the accuracy and e$ciency of the model. 2001 Elsevier Science B.V. All rights reserved. Keywords: Finite elements; Buckling; Plate structures; Cylindrical shells; Modal interaction; Local buckling; Amplitude modulation; Localization of buckling; Imperfection sensitivity
1. Introduction Cylindrical shell and panels are often reinforced with stringers to enhance their sti!ness in resisting axial compression. The sti!ening elements not only enhance the buckling resistance but also reduce the imperfection-sensitivity of the shells. Because of the resistance o!ered by the sti!eners to radial movement, &local' buckling modes whose nodal lines do not coincide with the location of the sti!eners are simply eliminated. This has the e!ect of minimizing the nonlinear modal interactions which are the source of imperfection-sensitivity in unsti!ened shells.
* Corresponding author. Former doctoral student. 0168-874X/01/$ - 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
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However there remain principally two distinctive modes of buckling which dominate the behavior 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 with it. The optimum design of the shells often leads to a con"guration for which the critical stresses are close to each other. Thus, a study of nonlinear modal interaction of local and overall instabilities is of considerable signi"cance in the context of optimal design of such shells. The problem of interaction of an Euler buckling with local plate buckling was studied in the 1960s and 1970s by several investigators (see for example, [1}3]). Tvergaard presented a detailed analysis of sti!ened plates under interactive buckling [4]. The problem was studied Byskov and Hutchinson [5] using an asymptotic approach and solutions in the classical mold. Koiter and Pignatrao [6,7] authored two papers of fundamental importance to the interaction of local and overall buckling in sti!ened panels and sti!ened shells. They introduced the technique of amplitude modulation 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 his co-workers. A "nite strip approach was used [8] to extract the buckling modes and the second order "elds to simplify the analysis. Next, a special beam element was developed to deal with the interaction of local and overall buckling . This is an Euler}Bernoulli beam element in which the local buckling deformation was embedded. The local buckling mode(s) and the periodic part of second order "elds are determined a priori by the asymptotic procedure. These e!ects are built into a beam "nite element in terms of two additional degrees of freedom, for each local buckling mode considered, thus allowing automatically for the amplitude modulation [9,10]. The theoretical basis of such an approach has been discussed by Sridharan and Peng [11]. It is shown that the amplitude modulation is the key factor in the interaction; it performs the function of capturing the contributions of several neighboring modes of the same longitudinal description as the fundamental local mode, but with di!ering circumferential wave numbers. For doubly symmetric cross-sections it is vitally important to consider a companion secondary local mode which is triggered by the interaction of the overall buckling mode and the primary local mode. The use of amplitude modulated local modes and the induction of key secondary modes as full participants in the interaction pulls in all the additional patterns of deformation generated in the interaction of an overall mode with the primary local mode. So much so, it is super#uous to consider the mixed second order "eld arising by the interaction of these two modes. Use of a beam element in the formulation 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 in which local buckling deformation was embedded. However, only relatively simple examples were considered for illustration, such as sti!ened panels studied previously by Tvergaard. The method was 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 means of nonlinear "nite element analysis. Such an analysis would obviously entail considerable
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computational resources as it would require the use of a su$ciently "ne mesh to capture the local deformation (such as sinusoidal ripples of small wavelength) in a structure long and broad enough for the overall instability to be of importance. From the point of view of modeling interactive buckling using a modal approach , prismatic structural components can be divided into two broad categories: (i) Structures whose cross-section exhibits strong asymmetry with respect to the axis of bending or overall buckling. Typical examples of these would be slender plates attached to stocky sti!eners, e.g. Tvergaard panels [4]. (ii) Structures which exhibit near or complete symmetry with respect to axis of bending or overall buckling. Typical examples of these are rectangular box- or I-section columns. Consider case (i): In its simplest form, the structure has only two signi"cant modes of buckling, viz., Euler buckling mode and the plate buckling mode with scaling parameters and respec tively. The potential energy function governing the interaction can be written as: # higher order terms, (1) "a ! b #a ! b #c 2 where the key term of interaction is the c term. This term involves the energy of interaction of overall bending associated with Euler buckling with the midsurface stretching associated with local buckling. Such a potential energy function was employed by Tvergaard [4] and subsequently thoroughly investigated by Hunt [14]. Next consider the case (ii). In this case, the key term c vanishes because of symmetry of the cross-section with respect to the axis of bending: the overall buckling stress/strain is antisymmetric with respect to the axis of bending while the mid-surface stretch associated with local buckling is symmetric with respect to the same axis. In order to compute a higher order term involving both and , one has to compute the mixed second order "eld which arises by an interaction of the two modes. Such a procedure will yield a biquadratic term, viz., [10]. The inherent numerical di$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 (associated with 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 deformation in the compression #ange of the square box column, i.e. the local deformation in the #ange which su!ers additional compression due to overall buckling is further accentuated while that in the opposite tension #ange is alleviated (Fig. 1). In order to ensure accuracy, post-local buckling e!ects must be accounted for in the analysis. Thus not only the secondary local mode, but also the second order "eld associated with it and the mixed second order "eld that arises by the interaction of 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 local buckling deformation embedded in them. It would again, seem necessary to incorporate degrees of freedom to represent the secondary local mode, together with the relevant second order "elds. This
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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 of achieving the same result is also presented. This, is considering just a single local mode but letting the amplitude modulation function be di!erent for di!erent plate elements. Thus the crosssectional pattern of local buckling deformation of the structure evolves freely under the in#uence of overall buckling deformation. Another issue that is relevant is the relative importantance of the short wave contribution of the second order "elds for obtaining an accurate estimate of the maximum load. In this study an attempt 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 of localization 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 the computations. (iii) Finally, the accuracy of the model by comparison with detailed FE analysis and earlier simpli"ed formulations. In order to make the paper complete, we present a brief outline of the formulation of &locally buckled' plate/shell elements. This is followed by several worked examples and commentary thereof. 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.
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2.1. Displacement, strain and stress vectors The displacement variables are u2"u, v, w, , ,
(3)
where u, v and w are the displacement components in the axial (x-), transverse (y-) and outward normal (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: 2" , , , , , , , . (4) V W VW V W VW VX WX Of these, " , , are the inplane strain components, " , , are the curvature V W VW V W VW components, and " , are the transverse shearing strain components. VX WX The following strain}displacement relations are assumed for the shell/sti!ener: u 1 " # V x 2
v w , # x x
v w 1 " # # W y R 2
u w # , y y
(5a) (5b)
u v w w " # # , VW y x x y
(5c)
" , V x
(5d)
" , W y
(5e)
" # , VW y x
(5f)
Fig. 2. Coordinate axes of a cylindrical shell.
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w "# , VX x
(5g)
w "# . WX y
(5h)
These may be viewed as Donnell's strain}displacement relations modi"ed to account for transverse shear deformation and the large in-plane motions of sti!eners such as would occur under overall buckling. Thus, the expression is augmented to include a nonlinear term in v*a key term which V enables the modeling of overall bending/buckling phenomena. The nonlinear terms in u and v can be neglected in dealing with a purely local buckling problem. The Eqs. 5(a)}5(h) can be written in an abbreviated form "¸ GH (u )#¸ GH (u ) H G H
(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 2, moment resultants M"M M M 2, and transverse shear forces V W VW V W VW Q"Q Q 2. V W The stress}strain relations take the form
(7a)
Q"kGM t ,
(7b)
N
A B
B
"
M
D
,
where k is the shear correction factor (taken as 5/6), GM is the averaged transverse shear modulus and t is the thickness of the shell. The stress}strain relations may be written in the abbreviated form
"H . G GH H
(8)
2.2. Determination of the local buckling xelds 2.2.1. Notation The 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 to a 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 critical stresses; double superscripts (11) and (22) indicate the second order "elds associated with the primary local mode and secondary local mode, respectively, and (12) stands for the mixed second order "eld (m.s.o.f.) arising by the interaction of local modes 1 and 2. The local buckling problem is solved following the standard "nite strip procedure [17]. The salient features thereof will now be mentioned.
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2.2.2. The linear stability (eigenvalue) problem The potential energy function corresponding to the nth local mode (u) can be written in the form (uL)], L" [H ¸ G I (uL) ) ¸ H l (uL l )# H ) ¸ I G G I I GH
(9)
where represents the stress in the unbuckled state, and a dot operation indicates multiplication G 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 uL, "uL, L (y) sin G G G
mx , ¸
vL, wL, L"vL, wL, L (y) cos G G G G
mx . ¸
(10)
Here u , v , 2 are the degrees of freedom (d.o.f.) and are appropriately chosen polynomial shape G functions. In the present work, the functions are chosen in a hierarchical form as explained later. G For su$ciently large m, the boundary conditions at the end are deemed not to in#uence the local buckling process. Designating the d.o.f.'s of the local buckling problem as qL, the potential energy function G (Eq. (9)) may be expressed in the form L"aL!bL qL qL GH G H GH
(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 aL! bL qLo "0. GH GH H
(12)
Solution of the linear eigenvalue problem in Eq. (12) gives the critical stress for local buckling and the eigenmode, qL . G 2.2.3. Second order xeld problem The potential energy function governing the second order "eld problem is given by )# H ) ¸ GI (uLL) LL"[H ¸ GI (uLL) ) ¸ Hl (uLL l G I GH I )#2¸ GI (uL) ) ¸ Hl (uL , uLL)], #H ¸ GI (uLL) ) ¸ Hl (uL l I l l GH I
(13)
where uLL 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
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equations consists of three sets of terms in general: r"r(y)#r(y) cos
2mx 2mx #r(y) sin , ¸ ¸
(14)
The last two terms are rapidly varying trigonometric functions (m 1) in x, while the "rst term is independent of x. The second order "eld problem may, then, be viewed as that of a cylindrical shell subjected to two loads which vary sinusoidally in the x-direction, together with a third term which remains constant in the x-direction. The solution takes the form uLL"u#u cos
2mx 2mx #u sin , ¸ ¸
(15)
where u is a function of x and y, whereas u and u are functions of y only and can be viewed as the particular solution of the di!erential equation. u is a slowly varying function with respect to x and contains additional terms needed to enforce the end boundary conditions. Note that the solution cannot contain a component in the form of the buckling mode in the asymptotic procedure [18]. Also because of the slowly varying nature of u, it is decoupled in the solution process from u and u. Unlike in the asymptotic procedure for the initial postbuckling analysis [17], it is not necessary to compute u at this stage. Rather we shall let it arise, by the interaction of L (uL) terms with the degrees of freedom associated with the "nite element mesh to be introduced later. Because of its slowly varying character, a relatively coarse "nite element mesh must be able to pick up the deformation associated with u. So, we need to compute only the solution vectors associated with the trigonometric terms at this stage. For a specially orthotropic material, the displacement "elds to be computed take the simpli"ed form uLL, LL"uLL, LL (y) sin G G G
2mx , ¸
vLL, wLL, LL"vLL, wLL, LL (y) cos G G G G G G
2mx , ¸
(16)
where u,2 are the d.o.f.'s second order "eld. G The potential energy function (Eq. (10)) can now be expressed in terms of the d.o.f.'s qL and P qLL de"ning the "rst and second order "elds, respectively, and takes the form G LL"(aLL!bLL)qLLqLL#c qLLqLqL (r, s"1, 2 , n ); (i, j"1,2, 2 , n ), (17) GH G H GPQ G P Q GH 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 (aLL! bLL) qLL"!c qL qL, GH GH H GPQ P Q is set equal to in the calculations.
(18)
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2.3. Mixed second order xeld The potential energy function governing this problem can be expressed in the form "[H ¸ GI (u) ) ¸ GJ (u)# H ) ¸ GI (u) GH J J G I #H 2¸ GI (u) ) ¸ HJ (u, u)#2¸ GI (u) ) ¸ HJ (u, u) GH I J J I J J #2¸ GI (u) ) ¸ HJ (u, u)]. (19) I J J Here we consider the second order "eld arising out of interaction of two local modes of the same wave length. As discussed in a later section, signi"cant interaction occurs only between the overall mode and two local modes of the same (or nearly the same) wavelength. Thus, we assume the two local modes have the same number of half-waves, m. This would lead to a di!erential equation which has the same structure as Eq. (15) and results in a solution of the type given in Eq. (16). Once again 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 written in the form ul "u #u #u #u #u , (20) where u"u v w 2; 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 the double 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 longitudinal direction as the local buckling deformation comes under the in#uence of overall buckling of the structure. Thus, the local buckling amplitudes are `modulateda to enable the model to depict the accentuation of local buckling in regions where compressive stresses develop due to overall buckling. Apart from this, because of its `slowly varyinga character, the amplitude modulation performs the role of accounting for the neighboring local modes of the same transverse description as the associated local mode but of slightly di!ering half-wavelength, which are liable to be triggered by the interaction [11,12]. Thus we let the amplitude vary with respect to x- according to a `slowly varyinga modulating function. The local buckling deformation is thus represented in the form u"u G #u G (x)#u G H #u G H #u G H (x) (x), G G H &s represent the degrees of freedom associated with amplitude modulating functions &s.
(21)
2.5. The wavelength of the secondary local mode It has been assumed in the foregoing treatment that the two local modes which together describe the local buckling deformation have the same number of half-wavelengths, m. This requires some
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elucidation. The most signi"cant term in the interaction of two local modes with the overall mode is 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 nearly constant. Thus, the said integral tends to vanish unless the longitudinal wavelengths (or number of half-waves, m) are the same. Thus only the interaction of local modes of the same wavelength with the 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 of orthogonal transverse description but of the same wave length and all the associated second order e!ects. To this we add yet another feature in the present study : The amplitude modulating functions are given freedom to di!er from element to element, thus giving the the local buckling deformation freedom to vary across the section. This tends to make the secondary local mode somewhat redundant*an issue which will be considered later. 2.7. Neglection of mixes second order xelds (m.s.o.f.), uL The m.s.o.f uL arising of out the interaction of a typical local mode (n"1,2) with the overall mode, 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 the authors have [12] have discussed the pitfalls in the evaluation of the m.s.o.f. arising from the interaction of local and overall modes. In brief, accuracy is lost due to the fact interaction in a long prismatic structure gives rise to a number of local modes with eigenvalues close to that of the principal local mode at which the m.s.o.f. is evaluated. On the other hand, the m.s.o.f. can help identify the local modes that are triggered in the interaction [11,15]. Once these modes are identi"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 amplitude modulation of the local modes and the freedom given for the local buckling deformation to undergo modi"cation freely in the transverse direction*all these fully account for any change of the local buckling deformation. Further, we have a "nite element mesh that would automatically account for any changes in the overall deformation under the in#uence of the local modes. Also note 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 evaluated requiring it to be orthogonal with respect to the totality of the "rst order "eld (which includes the fundamental 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 can therefore be conveniently neglected. Note that a typical m.s.o.f. strain takes the form "¸ (uL)#¸ (u,u), (22) L where the "rst term represents the contribution of mixed second order displacement "eld and the second term arises by the mixing of the two fundamental modes. If only the "rst term on R.H.S. is
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neglected, Eq. (22) is clearly inconsistent and therefore the second term which acts as the driver of the m.s.o.f. must also be neglected. Thus, we set all the m.s.o.f stresses, strains and displacements are set to zero. 2.8. Finite element formulation 2.8.1. Choice of shape functions A 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), the problems of shear and membrane locking associated with lower order elements become inconsequential. A relatively small number of elements (in comparison to the h-version approach) would be su$cient to accurately model the structure and this results in considerable savings of e!ort in data input. The type of polynomials chosen in the present work are integrals of the Legendre G functions, advocated by Szabo and Babuska [19]. These have been successfully used in previous work by the authors (see for example [17]). 2.8.2. Strain-displacement matrix As stated earlier, a cylindrical shell element based on Donnell's theory admitting sheardeformation via Reissner-Mindlin theory (Eq. 5(a)}5(h)) is employed. The displacement functions take the form u2"u , v , w , , 2 (x) (y). GH GH GH GH GH G H
(23)
As a "rst step in the formulation we set up the B-matrix which relates the incremental strains to the incremental degrees of freedom. To this end, each strain component is expressed in terms of displacement variables of the local and overall "elds. Thus the mid-plane strain take the form V "u #(w w l #v v l ) l GI H G G I V GI G I GI H # H # G V G V G # G H # H H # G H , V G H V G H V G H
(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 "uL (y) cos ( x), V G G K K
(25a)
LL" K wLwL (y) (y)# 2 uLL (y)# K wLwL (y) (y) cos (2 x), H K G G H K V 4 G H G 4 G H G
(25b)
where "m/¸, and a prime denotes di!erentiation w.r.t. x. K Expressions similar to Eq. (25) are written down for all the strain components. The incremental strain vector " can be related to the incremental V W VW V W VW VX WX
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degrees of freedom. The full set of d.o.f's q can be divided into two categories: those that depict overall action q and others that control the amplitude modulating functions associated with local buckling ql, i.e. q2"q2ql2. The increments of strains are expressed in terms of the q's with the aid of B-matrices as follows l"B q #B q q , G GH H GHI H I
(26a)
l" [B ] cos(n x)#[B ] sin(n x) ql G LGH K LGH K G L # [B ] cos (n x)#[B ] sin(n x) ql ql . LGHI K LGHI K H I L
(26b)
The elements of the matrix [B] can be grouped under two distinct categories (i) those that are which describe the variations of the overall strain quantities and (ii) those which describe the variations of the strain associated with local buckling via amplitude modulating function. (The latter comes from the quadratic L (wL) and L (w, w) terms.) Both these variations are treated as `slowa compared to cos ( x) or sin ( x) associated with local buckling. K K Current stresses too are arranged in a similar form and these must be available for every integration point on the surface of the structure. " [ cos (i x)# sin (i x)] G K G K G
(27)
The tangential sti!ness matrix [K] is a sum of two matrices [K ] and [K ] whose elements are derived from:
[KM ]" GH
1 [Kl ]" GH 2
B H B dx dy# NG NO OH
B dx dy NGH N
(28a)
B H B #B H B dx dy KNG NO KOH KNG NO KOH K
1 # 2
B #B dx dy. KNGH KN KNGH KN L
(28b)
Note that [K ] houses overall, local and interactive terms whereas [K ] contains only local J buckling terms. The simpli"ed form of Eq. (28) is due to the `slowly varyinga nature of the overall displacements and the amplitude modulating functions. The vector of unbalanced forces at any stage in the analysis are given by
f " f !
[B]2 dx dy,
(29)
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where f is the externally applied force. From this point on, standard procedures of nonlinear analysis are used to trace the load de#ection relationship. Appropriate imperfections are used to de"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 or divided longitudinally into two elements. A su$ciently high polynomial ( up to p"5) is used in the x-direction. Quadratic functions are employed for the representation of the amplitude modulating function. Thus, the number of elements are of the same order as the longitudinal plate strips constituting the structure. 3.2. Objectives In this section the results of a numerical study are presented with the following broad objectives in view: 1. Compare the results obtained by the present simpli"ed model with that produced by a general purpose commercial program to illustrate the accuracy and e$ciency of the former. The example 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 be considered in a modal interactive buckling analysis. Examples of box and I-section columns are considered for this purpose. 3. To examine the roles respectively of the secondary local mode and the periodic part of the second order "elds in the determination of the maximum load carrying capacity under interactive 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 the cross-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 by b/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 in the `classicala sense, i.e. v"w""0; and normal de#ections are arrested along the outer longitudinal 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 "32900 ksi, E "1800 ksi, G "G "G "880 ksi, "0.24. The symmetry of response exists with respect to either center line of the shell.
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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 the local 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 is considered in the analysis. The responses of the structure in terms of nondimensional load versus maximum downward central 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 more than 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 small local imperfections only, the structure tends to bend outwards initially. The results given by the present model agree well with those given by ABAQUS obtained using a mesh of 16;24, 8-noded shell elements. 3.4. Doubly symmetric columns Next, we present examples of thin-walled columns with doubly symmetric cross-sections for which two companion local modes become active in the interactive buckling response. In all the calculations presented herein, we include the two relevant local modes having the same wavelength. These results are compared against previously published results obtained using beam elements carrying local buckling information embedded in them [15] as well as ABAQUS. Finally, we examine the role of the secondary local mode in the present approach where considerable freedom for modi"cation of the primary local buckling mode is already available as we let the amplitude modulating function to vary from element to element.
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Fig. 4. A comparison of the models: (a) isotropic sti!ened shell; (b) composite layered shell. Maximum de#ection vs. axial stress.
3.5. Square box Column 3.5.1. Geometry and buckling data Square 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 and t, 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.
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The secondary local mode is anti-symmetric with respect to the axis of overall buckling and symmetric with respect to the other axis of symmetry (Fig. 1). The ratio of the critical stress of the secondary local mode to that of the primary one, " / "1.432. 3.5.2. Details of the analysis Imperfection 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 buckling mode 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 of interaction. Because of symmetry, the model consisted of one-half of the section over one-half of the length of the column. The model consisted of only four elements one for each of the top and bottom #anges and the other two representing the web. The analysis was performed with p"4 ( cubic polynomials representing the displacement variables) in both the longitudinal and transverse directions. The local 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 local buckling 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 "elds and 5 beam elements were used in the "nal nonlinear analysis. The amplitude modulating functions were solely functions of x. This means that the transverse transformation of local buckling deformation must come entirely from an appropriate linear combination of the two participating modes. Abaqus model consisted of a 8;60 mesh of 8-noded shell elements over the same model. 3.5.3. Discussion Fig. 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 other until the limit point, but the present model predicts the smallest maximum load carrying capacity*a value which is about 3 % less than that predicted by the Abaqus model*which is deemed to be more accurate. The reason for this lower prediction is the simpli"ed treatment of the amplitude modulation of the local buckling modes which is treated as independent for each plate
Fig. 5. Response of a square box column as given by di!erent models.
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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 some additional compliance because of lack of compatibility of the local buckling rotations at the junctions which is implied in such an assumption. We thus have a slightly conservative prediction of 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 before the present model exhibits a more compliant behavior. It is seen that the responses given by the present model near the maximum load are less peaky. It is also seen that for case (i) (coincident buckling) 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 is severe localization of deformation over a relatively small length near the center of the top #ange which comes under combined compression due to axial load and overall buckling. Local buckling deformations are accentuated here and elsewhere they are not noticeable. 3.6. I-section column 3.6.1. Geometry and buckling data Cross-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 are displayed 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 the primary local critical stress, "1.22. 3.6.2. Details of the Analysis Imperfections 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
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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 three plate elements involved were treated as independent. The results of the analysis are compared, as before with those obtained using beam elements carrying local buckling information [15] and Abaqus. In the former analysis 24 "nite strips were used 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 the column. 3.6.3. Discussion Fig. 10 shows the variation of applied compressive stress with end shortening , measured at the center of the web. As before, the beam model overestimates the maximum load carrying capacity by
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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 is believed due to the de"ciency of the beam element in which the freedom for the cross-sectional local buckling deformation to modify itself was restricted. The results of ABAQUS and the present model agree more closely in this example with the latter once again giving a more compliant response. Note for this case of near-coincident buckling, there is an erosion of load carrying capacity of 40% under imperfections often unavoidable in practice.
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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 in modeling the interaction. In these models, each of the local modes was associated with an amplitude modulating function and this was a function of x only. Thus, the secondary local mode was essential to model the localization the local buckling deformation in the cross-section . As mentioned earlier, in the analysis this gives rise to a nonvanishing trilinear term of interaction. The present model with its `locally buckleda elements has built into it ample freedom for the local buckling mode to undergo modi"cation in the transverse direction because each element is associated with an independent amplitude modulating function. As a result it may be inferred that the role of the secondary mode is seriously diminished and the primary local mode, because it is free to modify itself under the in#uence of compression which varies from element to element , usurps the role of the secondary buckling mode. The problem of the square box column under near-coincident buckling previously studied is reexamined. Details are the same as given earlier. The results for case (i) are reworked with the secondary local mode suppressed in the formulation. Fig. 11(a) and (b) show the results of the load plotted 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 and build into the model the buckling mode and its second order e!ects.
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Fig. 11. (a) E!ect of secondary mode in the present model. Axial stress vs. maximum `overalla de#ection; (b) E!ect of secondary 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 computationally and 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 data The 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:
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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 apply as the overall buckling mode is now antisymmetric with respect to x}x (consisting of two lobes across the section). The primary and secondary local modes are, respectively, symmetric and antisymmetric with respect to the center line. 3.8.2. Details of modeling Each 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 the shape functions, p"5. Independent amplitude modulating functions are associated with each panel consisting of a sti!ener and the shell elements on either side. 3.8.3. Discussion The 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 bends
downward due to overall buckling and therefore experiences higher compressive stress. Fig. 12 plots the load versus maximum de#ection for the two cases obtained, respectively, including and neglecting (the periodic part of) the second order "elds. While this makes only a slight di!erence
Fig. 12. E!ect of the periodic component of the second order "eld.
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(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 case with "1.829. Thus, one may conclude that the neglection of these "elds results in a nontrivial overestimate of the load carrying capacity. (Note however, the `slowly varyinga part of the second order "elds is automatically generated by the interaction of the buckling modes with the degrees of freedom 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 studied using a specially formulated shell element . The element has additional degrees of freedom which can trigger and modulate the relevant local buckling modes together with the associated (periodic components of ) second order "elds. In numerical computations, the element is seen to be extremely e$cient and produces satisfactory 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 amplitude modulating function and this greatly facilitates the capturing of localization of deformation both longitudinally and transversely. Because of this feature the incorporation of a higher local mode*which plays a pivotal role in modal interaction analysis*is found to be redundant. The primary 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 of the second order "eld is necessary to obtain accurate estimates of the maximum load carrying capacity. In shell structures, neglecting this "eld could result in an unconservative prediction, i.e. an overestimate of the maximum load.
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