INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 39, 3007-303 1 ( 1996)

KOITER’S ANALYSIS OF THIN-WALLED STRUCTURES BY A FINITE ELEMENT APPROACH

ANTONIO D. LANZO AND GIOVANNI GARCEA

Dipartimento di Strutture, Universita della Calabria, 87035 Arcavacata di Rende, Cosenza, Italy

SUMMARY This paper refers to the analysis of the postbuckling behaviour of thin-walled structures by means of an asymptotic approach based on a finite element implementation of Koiter’s non-linear theory of instability.

The analysis has been accomplished by using the following assumptions: (i) the structure is described as an assemblage of flat slender rectangular panels; (ii) a non-linear Kirchhoff-type plate theory is used to model each panel; (iii) HC finite elements discretization is used; (iv) linear and quadratic extrapolations are assumed for the fundamental and the postbuckling paths, respectively; (v) multimodal buckling is considered; and (vi) imperfection sensitivity analysis is performed in both multimodal and monomodal form based on the steepest-descent path criterion.

Several numerical results are presented and discussed. Comparisons with numerical solution obtained by standard incremental codes are given, which show the accuracy and reliability of the proposed approach.

KEY WORDS: postbuckling; instabilities; multiple modes; perturbation methods; thin-walled structures; FEM

1. INTRODUCTION

Thin-walled structural elements, such as stiffened panels, thin-walled or girder-box structures, are widely used in the fields of civil, mechanical and aeronautical engineering. High-resistant and composite materials, together with an optimization design to high slenderness values, cause a complex unstable postbuckling behaviour of these structures, characterized by strong modal inter- actions and sensitivity imperfection, which can hardly be predicted by standard analyses, based on incremental-iterative numerical approaches.

Because of the technical interest in these types of structures, the study of their behaviour and, in particular, of their coupled modal phenomena, has received in past years considerable attention from many researchers. It is worth citing the pioneering works of van der Neut,’ Graves-Smith? Koiter? and T~ergaard.~ (For a review of recent works, see Reference 5.)

Oriented to analytical6 and, later,’ semi-analytical solutions, the perturbation approach based on Koiter’s non-linear stability theory’ is often followed in these investigations. In fact, while based on Taylor expansions and then limited to the initial postbuckling behaviour, the effectiveness of this approach is connected to the following points:

1. Attention is focused on the main aspects of the mechanical phenomenon. 2. A synthetic insight on the structural behaviour is made possible. 3. A complete and simple prediction of imperfection sensitivity can be obtained.

The transfer of the perturbation approach into a fully automatized context of FEM analysis allows the overcoming of the limits resulting from the use of analytical solutions, based on

CCC 0029-598 119611 73007-25 0 1996 by John Wiley & Sons, Ltd.

Received 3 April 1995 Revised 1 February 1996

3008 A. D. LANZO AND G. GARCEA

'ad hoc' assumptions and valid only under simple boundary conditions. However, the practicability of coupling perturbation methods and finite element techniques needs particular care in setting the perturbation algorithm and developing the finite element used.

Within this framework the present paper describes a powerful and reliable code for the au- tomated analysis of thin-walled structures composed by flat rectangular plates assembled along its longitudinal sides, under different load and kinematical boundary conditions. As such struc- tures are characterized by small prebuckling displacements; the present work is restricted to linear fundamental extrapolated paths. The perturbation strategy followed9* lo refers to simul- taneous or nearly simultaneous buckling mode conditions, with quadratic extrapolation for the bifurcated paths. This allows the reconstruction of multimode buckling interaction phenomena and the execution of imperfection sensitivity analysis whether in multiparametric form within the space of the critical directions, or following the monoparametric criterion of steepest post- critical load descent path? The paper should be considered as the natural continuation of the previous work, where a monomodal postbuckling analysis for single rectangular panel was presented.

Because of the regular geometry of the structures under consideration, rectangular elements are used. The finite element, already basically developed in References 11 and 12, is set on the basis of a Kirchhoff non-linear theory of slender plates and using a high-continuity interp~lation'~ of the three components of displacement field. This discretization scheme is proved in References 11 and 12 to be suitable for perturbation approaches being relatively insensitive to a postcritical numerical locking phenomena that generally disturbs the FEM-compatible implementation of quadratic per- turbation algorithms. As it involves a low number for element of interpolation parameters, it allows also, through very fine discretization grid, an accurate representation of the complex kinematic of the buckling phenomena for structures under consideration.

Accuracy is tested with reference to meaningful examples. The proposed approach proves to be fast and reliable. In the context considered in the present paper, it has to be regarded as an efficient alternative to standard numerical strategies based on incremental-iterative schemes, which are usually characterized by high computational costs. Furthermore, incremental strategies tend to became unreliabile in the case of multimode buckling (simultaneous or nearly simultaneous buck- ling modes) because of its difficulties in recognizing secondary bifurcation phenomena (modal jumping after bifurcation).

2. STRUCTURAL MODEL

We consider thin-walled structures that can be represented as assemblages of rectangular plate interconnected along its longitudinal edges, under general boundary conditions and with load in- creasing linearly with a L parameter. Each plate is assumed to be of constant thickness, which is small compared with the length and the width of the plate. For each plate a local Cartesian co-ordinate system (XI , xz ,x~ ) is introduced, where the XIXZ plane describes the undeformed middle surface. In that system, ( U I , U Z , U J ) will be the components of the displacement field u referred to the middle surfaces and measured from the undeformed state.

The use of the perturbation strategy needs particular care in setting the kinematical model of the structure. In fact, a consistent definition of the energetic terms involved in the strategy strictly depends on the coherency of its kinematics. In Reference 11 the classical von-Karman model of slender plates was proved to be sufficiently adequate for the purpose of the analysis. In the present paper we refer basically to this model, enriching its non-linear terms in order to take into account the non-linear effects connected to in-plane rotations, generally not negligible in the buckling of the structures under consideration. For in-plane strains and curvatures, we assume then the following

KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES 3009

kinematical relations:

where the comma notation for partial derivatives is assumed.

expressed by Using an implicit summation convention on repeated indices, the strain energy of the plate is

where the stress states { N i j , M i j } are connected to the deformation parameters (1) through the following linear constitutive relations:

both Ci jhk and D i j h k being totally symmetric and positive definite. It defines the strain energy (2) as a functional of the displacement field u.

Oriented to the implementation of the perturbation strategy described in subsequent sections, we are interested in the evaluation of the variation of the functional @ [ u ] . According to Reference 11, we observe the following.

1. In the perturbation strategy followed, these energetic variations are computed in configura- tion belonging to a fundamental path (u'[A] = At;) linearly extrapolated in the load param- eter 2 on the basis of the initial behaviour of the structure in the undeformed configura- tion.

2. It is meaningful to assume that the displacements from the initial undeformed configuration to the fundamental deformed configuration are negligible, simplifying the concerning terms from the deformation measures that appear in the energy quantities of the problem.

3. Out of the context where the above hypotheses are strictly verified, we have in general an error in the representation of the problem. In any case, however, this error does not condition greatly the solution being a-posteriori recovered in the perturbation strategy as equilibrium residual along the fundamental path through suitable imperfection (implicit) coefficients that will be added to the asymptotic equilibrium equations of the problem.

The computation of all relevant terms required by the analysis (i.e. second, third and fourth variation of the strain energy) is performed in agreement with these hypotheses. We refer to Reference 11 for more details.

3. FEM DISCRETIZATION

The discrete model of the structure is based on the use of compatible High Continuity (HC) finite elements, already proposed by the authors in References 11 and 12 for the non-linear analysis of a single slender panel. The discretization is obtained by splitting each panel of the structure into homogeneous rectangular elements with sides I I and 12 and interpolating each of the displacement components ( U I , U ~ , U J ) by the HC shape hc t ions (quadratic splines) suggested by Aristodemo in

3010 A. D. LANZO AND G. GARCEA

Figure 1 . Boundary nodes and grid for HC discretization

Reference 13:

41[<] = - + it2, 42[tl = - <’, 63[<] = + i t + it2 A forced interelement continuity of the fbnction and its derivative reduces the parameters needed for this local displacement representation. It follows a fbnction approximation of C’ continuity in the domain of the panel using a minimal number of interpolation parameters (approximately one parameter for element for each displacement component interpolated). This feature enables us to handle very fine C’ discretization grid by a relatively small number of parameters, and obtains a great effectiveness in describing the complex geometries of the postbuckling deformed configurations of the structures.

The continuity between panels is obtained by means of the map

uz” + u; u; + u; 2436 - u; , 4 = - A1 , v 2 = - , v 3 = - v1 = - Uf + up

2 2

that transforms the variables placed near the edges (see Figure. 1) in boundary variables ( v l , v2,03,4)

directly meaningful with the scope of forcing the C’ continuity of displacement components be- tween the panels, needed in a compatible formulation of Kirchhoffs plate problems. The natural hierarchy of the variables of the problem (variables internal to each panel and variables meaningful for the global representation of the kinematic compatibility between the panels) is easily handled by means of standard substructuring techniques.

The variations of the strain energy needed for the analysis are computed by means of a direct assemblage of contributions in each finite element, using analytical and numerical integration tech- niques as described in References 11, 12, 14 and 15. In particular, in Reference 11, the reliability of the proposed numerical model in perturbation approaches (mainly referring to a postcritical numerical locking phenomenon) has been positively tested for a large number of examples.

KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES 301 1

4. THE PERTURBATION STRATEGY For slender structures of hyperelastic materials subjected to conservative loads increasing linearly with a I parameter ( p [ I ] = Aj), the equilibrium problem is expressed by the stationary potential energy condition in the space of compatible displacement field u (here and in the following we refer to References 10, 11 and 16 for the notation),

(4)

where @[u] denotes the strain energy and Abu the load potential (linear in u). The condition (4) is expressed in virtual work terms by

( 5 1 The solutions of this problem are curves in the space (u,A), named equilibrium paths of the structure.

The essential feature of the perturbation strategy of analysis followed in References 11 and 16 is the reproduction of a bifurcation phenomenon between a fundamental path u'[A] (already known or extrapolated starting from a known initial point) and a bifurcated path (reconstructed asymptotically). In the cited works there was the implicit hypothesis that the buckling phenom- ena are dominated by the unique non-linearity related to critical configuration at smaller I I value along uf [ A ] . However, simple isolated bifurcation represents a borderline case in the complexity of phenomena observable in the structures. For the case of structures composed of slender pan- els, because of a strong optimization generally carried out in their design, the fundamental path presents clusters of critical points at very close values of the load. In these conditions the different critical modes can interact, greatly affecting the equilibrium paths of the structure. In particular, these interaction phenomena can give to the structure a strong sensitivity, in terms of limit load reduction, to small geometrical or load imperfections. Therefore, these phenomena must be taken into account in the analysis.

A generalization of the pembation strategy to the case of multiple simultaneous or nearly simultaneous critical modes is described in References 10, 17 and 18. Referring to this theoretical framework and with the same simplifying hypothesis used in Reference 11, the equilibrium paths are expressed asymptotically in the form

I I [ U , I ] = @[u] - n&l= Stat(,)

@"U]SU - I j S u = 0, v s u

( 6 ) m 1 m

u = Iu + c t i u i + - c t j t j w i j i = I 2 i , j = l

where

(1) uf[A] = I2 is the fundamental (prebuckling) path, defined (as suggested in References 19 and 20 and recently in Reference 16) by a Taylor extrapolation starting from the natural undeformed configuration of the structure, i.e. setting the tangential term li by means of the incremental equilibrium condition in this configuration

@ p s u - j s u = 0, vsu (8)

( 2 ) u(i=~,...,,,,) are the m buckling modes considered in the analysis, i.e. solutions of the following

(9)

bitkcation problem along the fundamental path

@"[Iiii]diSu = 0 , V 6 u , -@:u^buiuj = Sij (Si, is the Kronecker's symbol)

3012 A. D. LANZO AND G. GARCEA

(3) Go are the secondary buckling modes associated to the primary ones, i.e. belonging to the space W = {G: @ b i t i i G = 0) orthogonal to the m modes tii and solutions of the problems

@ : ~ v d u + @ruiujdu = 0, ( W i j , V a u ) E W ( i , j = {I, . . . , m}) (10)

(4) the coefficients d,$ and B j j h k of the cubic and quartic multilinear form of (7) are completely symmetrical in their indices and defined in terms of variations of the strain energy by means of

(11) (12)

d.. - @“’V.lj.i)

g.. rjhk - - b @““vqj.u I J h v k - @E(wi j%hk + w i h “ i j j k f w j k w j h )

r j k - b r j k

( 5 ) the coefficients p k [ n ] of implicit imperfection

/.&[A] = (@’[nil - n j ) i k M in2@YG2dk (13)

take into account, in the projecting manifold (6), the equilibrium residual connected to the extrapolation made in defining the fundamental path.

A general procedure of automatic reconstruction of the non-linear behaviour of the structure can

1. Linear extrapolation of the fundamental path uf [A] = li solving the linear problem (8). 2. Computation of the bifurcation points and buckling modes ( A i , tji) along the fundamental path

3. Computation of the secondary buckling modes wjj as solutions of the linear equations (10). 4. Computation of the coefficients of the cubic and quartic multilinear forms using, respec-

tively, expressions (1 1 ) and (12). The implicit imperfection coefficients p k [ n ] are computed according to (13).

be drawn according to the following steps:

solving the eigenvalue problem (9).

5. Resolution of the non-linear equation system (7) for (A, < I , . . . ,(,) relationships. 6. Finally, the behaviour of the structure is reconstructed in their equilibrium paths in the

This strategy gives a synthetic description, in the bifurcation manifold. of the complex energy behaviour of the structure and then of the relative mode interaction phenomena. It reminds the mode reduction technique proposed by Noor2’ in the same context of non-linear analysis of slender structures. The present method however attains a notable saving of the computational costs (limited mainly to the bifurcatin problem). Furthermore, the higher articulation of the manifold of projection and the consistency of the development followed, guarantee a better accuracy in the obtained results.

manifold (6).

5. IMPERFECTION SENSITIVITY ANALYSIS

The presence of small additional imperfections in the geometry EU” or in the load ~ j j [ A ] alters inevitably the perfect structure scheme, i.e. structure showing bifurcation phenomena along its natural equilibrium path. This does not modify the dominant nature of the structural behaviour, still strictly correlated to the behaviour of the perfect structure, but conditions some of its rel- evant features and, generally, involves a reduction of the load-carrying capacity of the structure (imperfection sensitivity).

Because of the random distribution of imperfections in a structure, an accurate sensitivity analysis needs to be performed for a large range of possible imperfections, varying both their shapes

KOITER’S ANALYSIS OF THIN-WALLED STRUCTURES 3013

( j [ A ] , ii) and entity E . This discourages the use of incremental-iterative numerical approaches which are strongly penalized by prohibitive computational costs.

On the contrary, a sufficiently approximated evaluation of equilibrium paths of imperfect struc- tures can be easily obtained in the asymptotic Koiter’s approach, looking again for the solution of the equilibrium problem ( 5 ) of the imperfect structure in the same bifurcation manifold (6) of the perfect one. In this approach the presence of imperfections simply redefines the final (A, 5 1 , . . . , 5 , ) relationships by adding the coefficients

dmriilidl, & j [ L ] d i

respectively, for geometric and load imperfections. Setting the shape of imperfections, this performs an efficient parametrization in their entity of the sensitivity analysis, requiring the only resolution of

{ k = I , ..., m} (14) for different values of E .

However, while in the case of unique buckling mode this direction exhausts the space of mean- ingful imperfection shapes, in the case of m multiple buckling modes, a complete analysis needs to be performed in the relative m-dimensional space (a” possible imperfections). We need then a criterion for selecting a narrower range of meaningful directions, leading again the analysis for borderline case to the monoparametric form of the simple bifurcation. This aim can be obtained in the frame of minimum path theory of Ho,~* Koiter’ and S a l e r n ~ . ” ~ ’ ~

5.1. Minimum path theory

Along the various equilibrium paths branching from the fundamental one, the stationariness of total potential energy is satisfied for definition. Among these, paths satisfying the more restrictive condition of minimum of total potential energy assume an important role.

Interesting considerations can be attained in the simple context of cubic or quartic structural sys- tems, i.e. perfect structures in multiple simultaneous buckling modes, characterized by the preva- lence in the (A, 4 1 , . . . ,t,) relationships, respectively, of the cubic or quartic multilinear form

For these systems, bifurcated equilibrium paths are connected to stationary directions in the unitary sphere cy=, eie; = 1 of the cubic or quartic form:

3014 A. D. LANZO AND G. GARCEA

Then paths of minimum total potential energy correspond simply to minimum directions e; of the relative multilinear form in the unitary sphere, resulting in steepest descend load postcritical paths:

For these systems, Ho22 and Koiter’ proved that limit loads associated with geometric imper- fections in the minimum directions

m

i ii = ECefdi

are lower bound for the stability limit of imperfect structures with arbitrary directions of the imper- fections under the same norm lliill = -@r4ii2. Further conclusions can be found in S a l e r n ~ , ” * ~ ~ where minimum equilibrium paths are proved to be attractors with reference to imperfect paths belonging to their neighbourhoods. It means that paths connected to generic directions of imper- fections, after a complex articolation of bifurcation phenomena (‘jumping mode after bifurcation’), converge to the nearest minimum equilibrium path, in accordance with a more general trend of physical phenomena to achieve a minimum of the energy of the system.

These conclusions suggest that structures are particularly sensitive to dominant imperfection distribution. The restriction to these minimum directions of the imperfection sensitivity analysis leads again to the monoparametric forms, respectively, for cubic and quartic systems,

These conclusions are strictly valid only for cubic or quartic structural systems. For more general system, such as the structures under consideration, the minimum directions represent in all cases useful information to orient the analysis in the broader space of possible imperfections. To that order, it is worth considering that, near the biiimation point, the contribution of cubic form {the tangents to postbuckling paths) is generally prevalent, while as the distance fiom the bifurcation point increases, the contribution of the quartic form (the curvatures of postbuckling paths) becames prevalent in structural behaviour.

6. SOME COMPUTATIONAL REMARKS

The more peculiar computational aspects of the perturbation strategy sketched in the previous sections concern the resolution of eigenvalue problem (9) and to obtain the minimum directions of the cubic and quartic multilinear form by (15) and (16).

Because of the hypotheses made, the first problem is linear in the I parameter. The solution is obtained by using the following residual iteration algorithm already suggested in Reference 16 for the more general case of non-linear eigenvalue problem

KOITER’S ANALYSIS OF THIN-WALLED STRUCTURES 3015

To the aim of stabilizing and accelerating the sequence of iteration, the algorithm (17) is coupled with a subspace iteration technique, referring to the hyperplane of critical modes di where locally a Jacobi method is used.

For the evaluation of the minimum directions of the cubic and quartic form, we refer to the problem written for an arbitrary order of the multilinear symmetric form

m ...

(c;:,X,x,=l) f [ x l , . . . ,x,] = C 9jj...kx;xj . . .xk = min . . I , J. .... k = 1

From a computational point of view, its solution may be obtained by using the iterative algorithm described in Reference 23 and summarized here. It is based on the following iteration:

The iteration is performed a first time for p = 0, converging to the solution (t;,.. .,t:) of the problem

For odd (cubic) systems the solution of problem (‘18) is directly obtained by setting

+xi* X;min) = if f [ x ; ,..., x;] < 0 { -xI* if f [ x ; , ..., x i ] > 0

For even (quartic) systems, ([T,. . . , [k) correspond to the solution searched if / [ x i , . . . , x i ] < 0. Otherwise, the iteration (19) has to be repeated with p = ( 2 / p ) f [ x ; , . . . ,x:J, p being the order of the multilinear form.

A last observation concerns the resolution of vectorial equation of (A, (1 , . . . ,5,) relationships. It is worth noting that the systems (7), (14), condensing the greatest non-linearity of the problem, are highly non-linear. However, because of its reduced number of variables, it can be easily solved by standard step-by-step numerical techniques.

7. A DISCUSSION ON THE NON-LINEAR PLATE MODEL

The plate model (1) which we refer to in the present paper is used in most applications in this field of non-linear structural mechanics. It takes into account rotations in the plane of the plate by means of a Green-Lagrange strain measure, simplified in some of its non-linear terms. We named it ‘Simplified Lagrangian’ (SL). As an exception can be cited the paper of Goltermann and M~llmann,~ where the in-plane non-linear behaviour of the model is expressed by Green-Lagrange strain measures that we named ‘Complete Lagrangian’ (CL),

In order to study the influence of a different choice of the continuum model, two different isostatic cases for a simple rectangular plate uniformly compressed in its longitudinal direction are considered in this section. (see Figure 2).

By forcing the out-of-plane displacement to be zero (243 = 0), the reduced in-plane behaviour is tested using the different plate models. As limit case (I >> a ) for the geometry considered for our plate, comparison of the result can be obtained with the known postcritical behaviour of an equivalent one-dimensional Eulero’s beam24 in the ( X I ,x2)-plane. Setting the maximum transverse

3016 A. D. LANZO AND G. GAUCEA

Figure 2. Rectangular plate uniformly compressed

Table I. In-plane plate behaviour

Grid SL CL (0.25)

33 x 3 2.00560 0.26097 51 x 5 1.99804 0.25200 8 9 x 11 1.99711 0,25087

*In parentheses are given the value of the relative equivalent one-dimensional beam model

Table 11. Out-of-plane plate behaviour.

z Jib*

Grid SL (0.0) CL (-0.75)

33 x 3 0.03874 -0.70989 51 x 3 0.00523 -0.74336 89 x 3 0.00072 -0.74785

* In parentheses are given the value of the relative equivalent onedimensional beam model

displacement component as perturbation parameter (t = max { UZ}), the relative postcritical per- turbation coefficients @& t are reported in Table I for different discretization grids. In this case the ‘Complete Lagragian’ is a geometrically exact strain measure and its results perfectly agree with that of exact analytical beam model (i/& = 0.25). The simplified model hrnishes a value (j/& = 2), about eight times greater. However, we can note that the effect of this error remains globally small: it results in an increment of about 1 per cent of load for transverse displacements of 3 per cent of the wavelength which, in case of in-plane buckling, is usually related to the global size of the structure.

Different results are obtained for a plate simply supported along its transverse side and free along the longitudinal ones, such as to be equivalent to a one-dimensional beam in the (x,,x3)-plane. In this case the ‘Complete Lagrangian’ plate gives an unstable (descending) postcritical path, in spite of the stable behaviour of the exact one. This can be observed in Table I1 where, setting the maximum transverse displacement component as perturbation parameter (5 = max { u3}) , the relative post-critical coefficients i/& for different discretization grids are reported. The results of

KOITER’S ANALYSIS OF THIN-WALLED STRUCTURES 3017

the ‘Complete Lagrangian’ model reply the wrong value ;i/& = -0.75 of the equivalent one- dimensional technical model, while better results are given by the ‘Simplified Lagragian’ plate model that reply the value @& = 0.0 of the equivalent Biot beam model.

As a conclusion, the ‘Complete Lagrangian’ plate model performs better for in-plane modes, while the ‘Simplified Lagrangian’ model is well suited for describing the flexural ones. The latter being usually more relevant in the overall buckling phenomenon, the ‘Simplified Lagrangian’ model could be preferred. However, as shown in the following, the two models give quite equivalent results in real cases of thin-walled structures where, due to their internal hyperstaticity, buckling is followed by a strong redistribution of the stresses.

8. NUMERICAL RESULTS

This section reports the results obtained by a computer code, developed on the basis of the described perturbation algorithm, in the multi-mode interation analyses of three examples of com- monly used thin-walled structural elements. The first example concerns with a beam of symmetric ‘C’ cross-section uniformly compressed at its ends. In the second example the buckling behaviour of a beam of ‘T’ section loaded by a concentrate force at mid-length is evaluated. The last ex- ample concerns with a uniformly compressed column of square box-section. In all the cases, the beams are simply supported at the ends and an isotropic material is assumed, with non-zero elastic coefficients C$,k and D,jhk of the panels expressed by

Eh vEh 2Eh c1212 = - l + v c1122 = - 1 -v2’ Cllll = c 2 2 2 2 = - 1 - 9 ’

Eh3 vEh3 Eh3 Dl212 = ~ 6(1 + v) 12(1 - 9)’ 0 1 1 2 2 = 12(1 - VZ)’ Dilii =D2222 =

with (E, v, h ) denoting Young’s modulus, Poisson’s coefficient and the thickness of the plate, respectively.

The analyses have been accomplished using both the simplified plate model (denoted by ’SL’ ) and complete plate model (denoted by ‘CL’). The results are compared with solutions available in the literature and with numerical values obtained by means of step-by-step strategies implemented in the commercial code MSC-NASTRAN (Lev. 68).t This allows an extensive discussion on the behaviour of these structures and on the performances of the different solutions in term of accuracy, computational costs and reliability.

8.1. Example I : Uniformly compressed beam of ‘C’ symmetric cross-section

The beam, whose geometric details are described in Figure 3, is subjected to a I-increasing uniformly compressed axial load such that $ = 125. The results of the perturbation analysis refer to a discretization of 75 x (3 + 9 + 3) = 1124 HC finite elements. Solutions based on Koiter’s approach and finite strip discretization are available in the literature due to A1i-Sridharan2’ and van E1-p.2~ Comparison of the results are also made with numerical solutions obtained with the codes NASTRAN using a discretization of 144 x (4 + 12 + 4) = 2880 CQUAD4 finite elements with linear interpolation.

Eleven buckling modes ranging from I I = 1423.92 to 1 1 1 = 1603.04 have been taken into account in the analysis. The modes, described in Figure 4, embrace three classes of behaviour:

t NASTRAN’s analyses were carried out by Ing. G. Attanasio of ALENIA S.p.A

3018 A. D. LAN20 AND G. GARCEA

E=2 100000 u 0.51.0,0)=0 “I0,O.O) =v( I ,o ,O)=O

X,U b=75 v.v e . B

cross section I = 900

Figure 3. Simply supported beam of ‘C’ cross section (Example 1 )

Figure 4. Buckling modes of Example I

flexural, torsional and local modes with different number of half-waves (we have to outline that these distinctions as ‘global’ and ‘local’ modes, or ‘symmetric’ or ‘double symmetric’ cross-section, are however inessential and meaningless to the general computational framework followed in our perturbation approach). The critical values are in good agreement with the NASTRA”s values as reported in Table 111. They agree also with References 25 and 26, where only the flexural mode tj,, the torsional mode tj,, and the local mode tj, at lowest A-value are taken into account.

It is worth noting that the coefficients d i , k of the cubic multilinear form do not vanish: this reveals that asymmetric postcritical behaviour of the considered beam must be expected. In Figure 5

KOITERS ANALYSIS OF THIN-WALLED STRUCTURES

Table 111. Critical values of Example 1

3019

Ant.

Mode Pert. Nastran Ali-Sr. van Erp Mode type

dl 1423.9 1410.2 1419.6 1419 Local ( I 1 half-waves) liz 1429-4 1415.0 - - Local (12 half-waves) d3 1437.1 1423.7 - - Local ( 1 0 half-waves) d4 1456.7 1437.1 - - Local (1 3 half-waves) us 1462-2 1465.6 1493-1 1463 Flexural d6 1466.5 1456.9 - - Local (9 half-waves) d7 1492.9 1468.5 - - Local (14 half-waves) Us 1530.5 1522.3 - - Local (8 half-waves) d9 1538.8 1509.1 - - Local (15 half-waves) dl0 1594.1 1582.2 - - Local (16 half-waves) irl I Torsional 1603.0 1602.8 1614.9 -

Figure 5. Some secondary buckling modes of Example 1

some of the secondary modes calculated at &, = A1 are shown. As can be expected, the secondary modes connected to primary local modes (e.g. G1.1) present a lower wavelength and, then, force to set a close discretization grid in order to obtain a reliable postcritical description. From executed tests, no important changes are observed in the coefficients Bij*n of the quartic multilinear form for a different choice of the reference value Ab in the range of the A-value concerned in the analysis. This agrees with the observation of Reference 5, all the modes being considered in the present analysis.

In order to study the imperfection sensitivity of this beam, different shape of the initial geometric imperfections are considered in the following. However, we have to note that, because of non-zero coefficients of both the cubic and quartic multilinear forms, no minimum directions of this forms will be attractive for the postbuckling path of the imperfect structure.

3020 A. D. LANZO AND G . GARCEA

8.1.1. Case A : Local-Jlexural imperfection. In this case an initial geometric imperfection is set in accordance with that used in References 25 and 26, i.e.

u" = l l U l + 5"5U5, = 0.38095, p5 = 0.325423

with amplitudes ( f l , & ) such that the maximum cross-sectional displacements are uc = -0.05 and % = -0-4, respectively, in the local mode f l d , and in the global-flexural mode &ti,.

The results of the analyses are compared in Figure 6 with those reported in References 25 and 26, plotting the non-dimensional load A / ~ I ( X I = 1419.6 is the same value used in the cited papers) against the local 51 and flexural 5 5 buckling mode amplitudes. A direct comparison with NASTRAN's results and with that reported in Reference 25 is carried out in Figure 7, plotting the load versus meaningful displacement component of fixed points of the beam. No noticeable different behaviour is observed in using the Complete Lagrangian and the simplified plate models (the first gives only a reduction of about 0.8 per cent of the limit load). The results perfectly agree with that of NASTRAN, proving the reliability of our approach. They are in enough agreement with that obtained in References 25 and 26, particularly in terms of limit load as reported in Table IV. However, locally a different kinematic evolution of the postbuckling configuration can be observed in Figure 6 with reference to the results of References 25 and 26, due to the narrower buckling space used in References 25 and 26 (the modes effectively excited in our analysis are those with component V A # 0, i.e. {U1,U4,u5,u6,U9}). As a proof of that, a test was carried out forcing the perturbation analysis only to the modes (Ul, Us) considered in the previous citations (see curves '2 modes' in Figure 6), obtaining results now also locally in agreement with each other.

30

Figure 6. Example I . Load vs. ( ( 1 , r ~ ) buckling modes amplitudes for case A

Table IV. Limit load values of Example 1 (case A)

Pert. Nastran Ah-Sr. van Erp

0.8403 0.8421 0.846 0.880

KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES 302 1

Figure 7. Example 1. Load vs. displacements for case A

8.1.2. Case B : Flexural-torsional imperfection. To the aim to investigate the participation of the torsional buckling mode (mode ull in our analysis) in the phenomena of interaction, an analysis is carried out using the following initial geometric imperfection:

u'= t 1 d l + f s & + t l l U l l , = 0-76191, & = 0.32542, E l l = -0.48842

with amplitudes ( P I , ts, t , I ) such as to have the cross-sectional displacements

vc = -0.1 , W B = -0 .4 , wc = -0-6546

respectively, along the modes ( i r l , ds, dl 1 ). Also in this case the equilibrium path, plotted in Figure 8 in the load vs. displacement components, is in good accordance with that obtained from the code NASTRAN. With reference to the previous case, a slightly reduction (= 7 per cent) of the maximum load of the structure is obtained, confirming the conclusions in Reference 25: although the phenomena of interaction between local and flexural buckling modes, the torsional mode can play a significant role in the load-carrying capacity of the structure.

8.1.3. Case C : Flexural imperfection. For the beam with a 'C' section, a last case, a geometric imperfection, along the flexural mode

t 5 V 5 , E s = -0.081356

is considered, such that WB = 0. I , The results are compared with those obtained from the code NASTRAN, plotting in Figure 9 the load versus meaningful displacement components of fixed points of the beam.

Our postbuckling analyses follow initially the direction of the flexural mode, directly excited by the imperfection. At the point of maximum II value, a secondary bifurcation takes place, address- ing the equilibrium path also to different local directions (the modes { U I , u4, &, d6, u9) are excited). This can be observed in the sharp fold of the curves of Figure 10 where the equilibrium path is represented with reference to its (51,54,45, 5 6 , 5 9 ) buckling mode amplitudes. The secondary bifur- cation greatly conditions the 1 value of the equilibrium path, rapidly decreasing in the postbuckling range. This proves the influence of local-flexural modal interaction in the postcritical behaviour

3022 A. D. LANZO AND G. GARCEA

-e W

- PERT. B - NAST. B - PERT. C - NAST. C - PERT. F - NAST. F I I I

'0 0.40 0.60 0.80 U

Figure 8. Example 1. Load vs. displacements for case B

++-+ PERT.E +--+ PERT. E ( 1 mo - NAST.E - PERT.D

0 3.

W U

Figure 9. Example I . Load vs. displacements for case C

of this structures, and confirms already well-known conclusions. No meaningful differences are pointed out using the simplified or complete plate models.

These results are in good agreement with those of NASTRAN up to the critical zone, where for the load parameter a maximum value Amax = 1389.5 is obtained against the value Amax = 1416.8 of NASTRAN. Slightly different situation is depicted by the step-by-step strategy in the postbuckling range, with the equilibrium path characterized by less rapidly &descending values, while no secondary bifurcations are recordered, continuing the path along the flexural direction excited initially from the imperfection. In fact, on the contrary of the deformed configuration relative to the perturbation analysis, no local distorsions can be observed in the configuration relative to the NASTRAN's analysis. Furthermore, it is worth noting that these results essentially

KOITERS ANALYSIS OF THIN-WALLED STRUCTURES 3023

D

Figure 10. Example 1. Path description in (51.54,55,56,r9) mode amplitudes for case C

agree with those obtained by forcing the perturbation analysis only to the flexural mode, reported in Figure 9 as ‘1 mode’.

Two possible different answers can be given to this problem:

(a) The incremental code was not able to recognize the secondary bifurcation phenomenon, con- tinuing along the natural equilibrium path, also if this doesn’t represent the path of minimum potential energy of the system. As a proof that these secondary path bifurcations are very severe conditions for step-by-step strategies, we have to note that the code STAGS (whose results are not reported for this test) was not able to by-pass the secondary bihrcation point.

(b) The flexural equilibrium path followed by NASTRAN represents a real minimum for the complete potential energy of the system, whereas the results of perturbation analysis, refer- ring to the first fourth-order terms of the potential energy, are attracted by a minimum path meaningfully different from the real one. In other words, higher energy terms, neglected in the perturbation approach, can greatly condition some aspects of the postbuckling behaviour for this kind of thin-walled structures.

3024 A. D. LANZO AND G. GARCEA

We have not been able to give a one-for-all solution to this problem and fiuther investigations are needed for a deeper insight into the question.

8.2. Example 2: Trasversally loaded beam of ‘T’ symmetric cross-section

In this example a simply supported T-beam of aluminium material loaded at midspan by a concentrate force is considered. The example, whose geometric details are furnished in Figure 11, has been widely studied by van Erp in Reference 26 and 27. The discretization used in our perturbation analyses is based on 75 x (6 + 6 + 10) = 1650 HC finite elements. The results of NASTRAN’s analyses refer to grid of 120 x (5 + 5 + 16) = 3120 elements CQUAD4.

The bifurcation analysis gives two buckling modes at close 1 values (see Figure 12 and Table V), in good agreement with that of NASTRAN and only slightly differing from the critical value of van Erp. The secondary modes given in Figure 12 agree with that obtained in References 26 and 27. It is worth noting that, being equal to zero all the coefficients d i j k of the cubic multilinear form, the structure will present symmetric postbuckling behaviour.

=

0.0011 concentrated in the point ‘C’ normally to the plane (x,z) of the problem. The results of the perturbation analysis are in good agreement with that of the step-by-step analysis and reveal a slight imperfection sensitivity of the structure.

All the equilibrium paths of the imperfect structure are attracted along one of the two directions of minimum of quartic multilinear form B ; j h k . To highlight the attractivity phenomena connected to these minimum directions, analyses have been carried out using different shapes of geometric imperfections. A first imperfection f21j2 is set along the buckling direction lj2 such that vc = 0.1. A second imperfection is obtained summing to the first a small component l l l j l along the buckling direction l j 1 such that uc = 0.001. As can be observed in Figure 14, where the paths are plotted in their components (51 , tz ) in the buckling plane, the relative equilibrium paths converge asymptotically to different directions of minimum. This reveals that, because of the presence of two directions of minimum for the problem, also a slightly variation of the imperfection greatly conditions the postbuckling behaviour of the structure.

The analyses reported in Figure 13 refer to the beam with an initial imperfection of load

Constraints u(0.0.5b.0)=0 v(0.0.5b,z)=v(1,0.5b,z)=O cross section

b=38 E=70960 w(O,y,z) =w(I,y,z)=O - r x,u v=0.321

\I

2.w

2.W

1 = 450

Figure 1 1 . Simply supported beam of ‘T’ cross section (Example 2)

Table V. Critical values of Example 2

Amit. Mode Pert. Nastran van Erp

dl 2843.72 2821.1 I 2730 d2 3 102.47 3098.74 2999

KOITER’S ANALYSIS OF THIN-WALLED STRUCTURES 3025

Figure 12. Primary and secondary buckling modes of Example 2

Figure 13. Example 2. Load vs. displacements

8.3. Example 3: Uniformly compressed square box-column

As last example we consider the square box-column of Figure 15 already studied in Reference 5. The column is simply supported and subjected to a &increasing uniformly compressed axial load such that ) = 700. The perturbation analysis has been executed with a discretization of 65 x (5 + 5 + 5 + 5) = 1300 HC finite elements, while NASTRAN’s analyses refer to a grid of 134 x (6 + 6 + 6 + 6) = 3216 elements CQUAD4.

3026 A. D. LANZO AND G. GARCEA

Figure

1 .OO

0 .80

0.60

t 2

0.40

0.26

0.00 -0- 0.30 8.50

( 1

I

14. Example 2. Path description in ( ( l , c 2 ) mode amplitudes

c cross section I = 4000

Figure 15. Simply supported square box-column (Example 3)

Only the first 12 buckling modes ranging from 1, = 3182.782 to 1 1 2 = 334852 have been taken into account in the analyses, also if other critical modes are present to slightly higher 1 values. The modes are described in Figure 16 (see also Table VI). Except the flexural modes d, and d,, they are typically local modes with different half-waves. NASTRAN’s bifurcation analysis proves the accuracy of the critical values. A symmetric postbuckling behaviour has to be expected because of the zero values of the coefficients d ; , k of the cubic multilinear form. The secondary modes and the coefficients a i j h k of the quartic multilinear form are computed at 11, = 1,. It is worth noting that ‘ad hoc’ approximations are not needed for these calculations (on the contrary of that made in Reference 5 ) , but they are developed automatically by a simple computer code procedure.

In the following, analyses have been carried out for different initial load and geometric imper- fections with the aim of highlighting the unstable behaviour connected to local-global buckling mode interaction in comparison to the stable one when only local or global modes are excited.

8.3.1. Case A : Local imperfection. In this case an initial load imperfection is set such as to make only local modes active. The imperfection consists of a system of two forces acting along the y direction and concentrated at the points A and D, with values FA = 0.71 and Po = 0.7A respectively. In Figure 18 some of the results of the analyses are plotted. No interaction phenomena are excited and stable postbuckling behaviour is recorded. This is in good accordance with the analyses of NASTRAN, a part from a slight difference of the local geometric evolution in the v displacement component due to narrower range of the buckling modes used in the perturbation analysis, all presents on the contrary in an evolutional analysis.

KOITERS ANALYSIS OF THIN-WALLED STRUCTURES 3027

Figure 16. Buckling modes of Example 3

Table VI. Critical values of Example 3

Mode ~~ ~~

G i t . 3182.78 (3112.51) 3185.58 (31 10.95) 3199.49 (3136.79) 321 1.85 (3127.37) 3224.13 (3180.08) 3235.50 (3237.48) 3235.50 (3237.48) 3246.49 (3146.35) 3269.22 (3234.05) 3292.18 (3167-20) 3338.61 (331 1.70) 3348.52 (3205.88)

- ~~ ~~ ~

Mode type

Local (22 half-waves) Local (23 half-waves) Local (21 half-waves) Local (24 half-waves) Local (20 half-waves)

Flexural Flexural

Local (25 half-waves) Local ( 1 9 half-waves) Local (26 half-waves) Local (1 8 half-waves) Local (27 half-waves)

* In parentheses are given the NASTRAN's values.

8.3.2. Case B : Flexural imperfection. A second imperfection is set along the only flexural modes v4 and v7,

u' = + [7d7 , &j = 0.892197, ts = 0-966502

3028 A. D. L A N 2 0 AND G. GARCEA

Figure 17. Some secondary buckling modes of example 3

Figure 18. Example 3. Load vs. displacements for case A

such that UB = 1 . O and WB = 1 -0. As can be seen in Figure 19, only very slightly I-descending postbuckling path is obtained in this case. The results perfectly agree with that of NASTRAN.

8.3.3. Case C : Local-flexural imperfection. Finally, initial imperfections are set such that both flexural and local modes are excited, combining the geometric imperfection of the preceding case B (scaled of a factor 10) and the load imperfection of the case A.

Because of the mode interaction phenomena, an unstable postcritical behaviour is obtained, as reported in Figure 20. It is worth noting that the equilibrium paths are now conditioned by the attractivity of the unique direction of minimum of the quartic multilinear form (see Figure 21, where the paths are plotted in the mode components (&, 1;8,59)). The results are in good agreement

KOITER'S ANALYSIS OF THIN-WALLED STRUCTURES 3029

Figure 19. Example 3. Load vs. displacements for case B

Figure 20. Example 3. Load vs. displacements for case

with that of NASTRAN in term of limit load and initial postcritical path, but differences can be noted for finite displacements in the postbuckling range, giving the step-by-step strategy a steepless &descending path. As already observed, this phenomenon can be explained as either a poor representation of the perturbation strategy or the natural problem of incremental-iterative strategies in the presence of secondary path bifurcations.

9. CONCLUSIONS

A new versatile general numerical approach for the analyses of the postbuckling behaviour of thin-walled elastic structures has been presented. It combines the perturbative Koiter's general theory of stability with a FEM technique based on HC finite elements. This strategy, developed in a computer code, allows a synthetic view over the complex energy phenomena connected to buckling mode interaction that greatly affects the postcritical behaviour of the structure, particularly in terms of limit load reduction. Furthermore, powerful and efficient geometrical representation of the structure model are connected to very fine discretization grid of HC elements, because of the low ratios (number of parameters)/(number of elements) required by HC finite discretization. The overall analysis proves to be fast and reliable, while a simple and complete prediction of imperfection sensitivity can be obtained.

3030 A. D. LANZO AND G. GARCEA T;:#:m

__?i 8 -4.0 -5.0

t 9

Figure 21. Path description of example 3 in (56,58,&3) buckling plane

Numerical results have been developed for a series of examples with various geometrical and load imperfections. Comparisons have been made with solutions already available in the literature or obtained through numerical step-by-step strategies implemented in NASTRAN commercial code. These examples outline the following:

(a) All the critical modes must be taken into account in the analysis: a restriction to a narrower critical space can greatly condition the description of the postbuckling phenomena. Because of this, no noticeable computational problems are involved in our completely automatized approach.

(b) No noticeable differences are observed for real case of thin-walled structures using different non-linear plate models.

(c) In spite of the the simple technical plate model considered, the results obtained using the perturbation approach are very accurate, with computational costs some order lower than the relative costs involved in step-by-step strategies.

(d) The analysis appears simple and effective also in the presence of complex strong modal interaction phenomena, which can hardly be predicted by standard analyses, based on incremental-iterative approaches. To the costs of an accurate implementation, the pertur- bation strategy appears as an efficient alternative, in the contexts considered in this paper, to usual computationally more expensive step-by-step strategies.

Some care is needed in evaluating the results in the finite postbuckling range for some particular cases considered in this paper, where the perturbation approach furnishes a more emphasized unstable (load descending) postbucking behaviour in comparison with step-by-step analysis. A deeper insight into the problem requires hrther investigation.

ACKNOWLEDGEMENT

This work has been supported, in part, by a M.U.R.S.T. 40 per cent grant. The authors are very grateful for the help received from Ing. G. Attanasio and Ing. F. Finizio who followed patiently the development of the work for ALENIA Aeronautica S.p.A. (MASE-DTTE). Finally, a particular

KOITER’S ANALYSIS OF THIN-WALLED STRUCTURES 303 1

acknowledgement is due to Prof. R. Casciaro, from University of Calabria (Italy), for his precious observations and suggestions.

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