Asymptotic post-buckling analysis of rectangular plates by HC finite elements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 38,2325-2345 (1995)

ASYMPTOTIC POST-BUCKLING ANALYSIS OF RECTANGULAR PLATES BY HC FINITE ELEMENTS

ANTONIO D. LANZO, GIOVANNI GARCEA AND RAFFAELE CASCIARO

Dipartimento di Strutture, Universita della Calabria, 87030 Rende, Cosenza, Italy

SUMMARY The present paper extends the finite element perturbation approach already presented for pin-jointed and framed structuresL5 to rectangular thin plates. Koiter’s asymptotic strategy’ is coupled with a High- Continuity finite element discretization of the plate.” The consistency of the discrete model is discussed from the kinematical and numerical points of view and several numerical tests are reported. It appears that use of the HC elements makes the perturbation algorithm insensitive to the locking phenomenon occurring in the evaluation of the postbuckling behaviour. It also allows the use of very fine discretization meshes at low computational cost.

KEY WORDS: postbuckling; plates; perturbation methods; FEM

1. INTRODUCTION

The analysis of the critical behaviour of thin elastic rectangular plates has been widely discussed in literature’ whereas there have been fewer papers on the postcritical behaviour of these models. There are various reasons for this. First, while the bifurcation analysis of a rectangular plate subjected to simplified kinematical and load conditions can be obtained quite simply in closed form, its post-buckling analysis very often presents great mathematical difficulties. Moreover, rectangular plates loaded in plane exhibit stable postbuckling behavioq2. and so its quantitative description could be considered of less interest, being the carrying capacity of a single plate nominally related to the bifurcation load.

However an accurate evaluation of the postbuckling solution becomes important for slender plates working in the postbuckling range (usually in aeronautical applications) as a base for brittle-collapse criteria. Moreover, the study of simple plate problems can be very valuable, as local step in the study of assemblages of plates (e.g. box-girders and thin section beams) which can exhibit unstable postbuckling behavior and modal interaction phenomena.

Within this context, analysis is usually performed by at-hand approaches, valid only under particular kinematical conditions?, In more general contexts there are no exact analytical and very few numerical solutions are available in literature. This lack is only partially explained by the prevalent analyticalsv6 or semi-analytical’ approaches generally used. A more important reason is the practicability of coupling perturbation methods and finite element techniques, as pointed out by Gallagher.’ So the asymptotic approach is usually seen more as a theoretical framework for qualitative analysis than as a numerical tool to furnish accurate quantitative results.

Extensive research, as outlined in References 9-15, has brought out the essential features that affect the practicability of this coupling. In Reference 11 the accuracy of the perturbation solution has been shown to depend on the kinematical consistency used in defining the deformation

CCC 0029-5981/95/142325-21 0 1995 by John Wiley & Sons, Ltd.

Received 9 February 1994 Revised 29 July 1994

2326 A. D. LANZO, G. GARCEA AND R. CASCIARO

energy. It is also strongly affected by the capability of finite element discretization in avoiding numerical locking phenomena coming from the evaluation of the initial postbuckling curvature. Within this framework, the aim of the paper is to present a finite element procedure for the buckling and postbuckling analysis of generally constrained and loaded rectangular plates as the first step in the analysis of assemblages of plates.16 (The paper follows the first author's doctoral thesis" and the preliminary results of a joint research with the Italian aeronautical industry ALENIA S.p.A.'8-'')

The paper is organized as follows. First, the asymptotic approach used is briefly summarized (Section 2). Then a plate model, based on the classical Karman-Marguerre theory and suitable for postbuckling analysis, is presented (Section 3) and discussed (Section 4). The finite element discretization, based on the high-continuity interpolation of displacement fields proposed in Reference 22, is then described (Section 5). Some comments are given about relevant implementation details (Sections 6 and 7). Finally, a comprehensive numerical test is reported and discussed (Section 8). Further comments and suggestions are given in the conclusions (Section 9).

2. THE ASYMPTOTIC EQUATIONS

The asymptotic approach strictly follows from Koiter's non-linear theory of elastic stability' and may be considered well known in principle (see the excellent review by B~d iansky~~) . In this paper it is implemented, as a finite element tool, in the form described in References 14 and 15. Refer to these papers for further details.

We refer to hyperelastic structures subjected to proportionally increasing conservative loads. With usual symbols, we denote with @[u] and A#u the strain energy and the load potential of the structure, and use primes and hats to express (FrechCt's) differentiations with respect to the displacement field u and to the load multiplier A. The equilibrium problem is then expressed by the virtual work equation

w [u] su - npsu = 0, vsu, (1)

and implicitly defines a curve or equizibriurn path in the (u, A) space that is the goal of the analysis. Following Reference 14 and using dots to express differentiation with respect to <, the

asymptotic solution of equation (1) is obtained in the form

U[A, <] := + <& + w[n, <] (2) In this expression, the buckling mode ilb corresponds to the first point (ub, &,) along the fundamental path

Uf [A] := u[A, 01

@I'[uf] = 0, vsu

where the bifurcation condition

holds. The linear extrapolation li and the quadratic orthogonal corrections w are obtained by the linear equations

mb'tisu = fisl.4, vsu

w i w s w = - f t2ob"il:sw - f A2cpb"li:8w, w E w, v s w E w W being the space defined by the orthogonality condition

w = ( w : m'b"tibljbW = O}

HC FINITE ELEMENTS 2327

The 1-5 relationship which defines the equilibrium path is then obtained by a Galerkin reduction of equation (l), i.e.

@’[u [A, (11 db - p [A] db = 0 (3) where the energy term W[u[A, 5]]& is expressed by its asymptotic expansion from the bifurcation point (Ub, &).

The overall procedure is computationally simple (actually, very simple if using linear extrapola- tions, see Reference 15). However, it can allow high accuracy in both path description and limit (snapping) load evaluation (see Reference 24).

Applications described in the present paper are all characterized by fundamental solutions small enough to allow its linearization

uf[A] = Ail

and such that

(1 - n b ) @ r f i t d t < wCdb6:

(A - A,) @rGbf i : 4 wcfi: By these assumptions, the manifold (2) simplifies as

u[A, 51 = + + f t 2 w b

and the path equatian (3) becomes

$ A2@zc2f ib + t2@,b”fi: + <(A - Ab)@g’dfit + & 53(Wl‘dt - 3@;62) = 0 (4) It is worth noting that all energy terms appearing in equation (4) reduce to scalar factors which

can be simply computed as integrals of known functions. By introducing the imperjiectionfactor ,U[AI and the postcritical coejicients &, and 1,

As discussed in Reference 15, an useful feature of the asymptotic approach is that the presence of small additional imperfections, such as geometric imperfection ti and load imperfection q[A], can be taken into account by simply adding the additional factors

to the (implicit) imperfection term p[A] in equation (6). It makes it possible to carry out efficiently the imperfection sensitiuity analysis of the structure , i.e. to correlate the imperfection amplitude to the limit load reduction. This treatment of additional imperfections, while leading to a very simple formula, somehow tends to decrease the accuracy of the algorithm in comparison with a complete re-analysis. However, the accuracy may be considered generally fairly good for small imperfections and can be considerably improved by a deeper insight into the nature of the imperfections (see Appendix I).


3. THE PLATE MODEL

We consider a rectangular plate of constant thickness, assumed to be small compared with the length and the width of the plate. We introduce a Cartesian co-ordinate system (xl, x2, x3) where the xlxz plane describes the undeformed middle surface of the plate. Let (uI,u2,uj) be the components of the displacement vector of the middle surface, measured from the undeformed state. Then, for the in-plane strains and the flexural curvatures of the plate we assume the following kinematical relations:

where eij and wi are defined as

eij = f (ui, + u ~ , ~ ) , wi = u3,i (9) according to the classical Karman-Marguerre theory of slender plates. In (8) and from now on, the synthetic notation

and implicit summation on repeated indices will be used. The strain energy of the plate is defined by means of

( N i j ~ i j + M i j ~ i j } dA

where the stress states { N i j , M , } are connected to the deformation parameters (8) through the following linear constitutive relations:

both C i j h k and Dijhk being totally symmetric and positive definite.

small enough to have We assume the fundamental path to be characterized by very small out-of-plane rotations

ofofi x 0

By this assumption, related to an essentially plane loading hypothesis, all stresses came to be linear in A along the fundamental path:

NC[A] = A f i i j , M;j[A] = A M , (1 2) where

N.. = C.. 6 -1, aJhk hk, M i j = D i j h k f h k

By using equations (1 1) and (12) and differentiating equation (10) with respect to 5 we derive the expression for the second variation of the energy:


Likewise, by further differentiations we obtain the explicit expressions for the higher-order energy terms used by the analysis:

where

4. KINEMATICAL CONSISTENCY

As already stated in Reference 11, the use of the perturbation strategy needs particular care in structure modelling. In fact, a consistent definition of the energy terms (13-17) strictly depends on the coherency of its kinematics.

The proposed plate model (8)-( lo), derived by a technical approach to structural mechanics, does not represent a coherent kinematical definition. This can be shown by the non-objectivity of the deformation measures (8) with regard to rigid-body motions. For example, reference to a rigid-rotation around the axis x2 = 0 belonging to the plate plane, results in

u': [xl, xz] = x1 (cos CL - 1) uXx1, xzl = 0 =. = - 1 +cosct +$sin2cr#0 u ; [ x l , x 2 ] = x1 sincl I

However, by referring to the discussion in Reference 1 1 where the one-dimensional analogous beam model

@[u] = j;AE" + EJXZdx, E = E~ + u , ~ + f o2 , = - (18)

is described, this kinematical incoherency has negligible effects on the first three terms of the strain energy (errors of order c0 with respect to the unity) and practically affects only the fourth-order term

- 3m;w; (19)

which defines the post-buckling curvature by equation (5). In the one-dimensional case (see Reference 11) the error is essentially expressed by a term

,l

3 N b k t dx J Nb being the buckling axial force and k b the initial post-buckling rotation (i.e. rotations associated to the buckling mode 4). While small compared with Wrtjt (of order ~ ~ / 1 ) this term


may not be negligible in the full expression (19) where the two energy contributions tend to compensate. In particular, when using (18), they compensate exactly for some isostatic schemes such as in Eulero’s classical rod problem. In this case, from equation (18) we obtain f&tz = 0 in spite of the exact expression

x,t2 = nbX2(6/L)2/8

6 being the midspan postbuckling displacement and L the rod length. This corresponds to a 100 per cent relative error. We have to consider, however, that:

1. This effect is small globally (for Euler’s rod it results in an increment of 1 per cent of load for a midspan displacement of 10 per cent of the length).

2. Furthermore, it proves to be of minor importance for hyperstatic structures where buckling is followed by a redistribution of the in-plane stresses and consequently the two terms in equation (19) do not compensate.

3. Typically plates present high hyperstatic behaviour.

For these reasons, the technical plate model described in the previous section may be considered sufficiently adequate for the purposes of this paper.

It is worth mentioning that are not available plate models both computationally simple and completely consistent. For instance, the (simple) use of the Green-like measure

l 3 2,=1

~ i j = e i j + - 1 u k , i u k , j , (i,j={1,2})

for the in-plane deformations, while consistent with respect to rigid rotations, expresses incorrect- ly the flexural state as shown in Reference 11 (for the Eulero’s beam, it furnishes the worst value )I,<’ = - 52,7~~(6/L)~/8) .

5. PLATE DISCRETIZATION

The asymptotic analysis can be performed numerically through a finite element discretization of the plate. The High-Continuity (HC) interpolation technique suggested by Aristodemo in Refer- ence 22 (also see References 25 and 26) is used in this paper.

The main advantages of HC interpolation are: (i) we get C’ displacement continuity using a minimal number of interpolation parameters; (ii) the use of rectangular elements leads to very simple algebra. Both of these are very useful in allowing a simple management of very fine C1 discretization grids.

According to the Aristodemo’s paper,22 the rectangular domain of the plate is split in rectangular elements with sides I l and l2 and nodes are located in the centre of each element. Then the displacement components (u l , u2, u3) are interpolated by the following rules:

where the nodal parameters uv are associated with the elementary grid 3 x 3 of Figure 1, the interpolation functions 4j[ti] (see Figure 2) are defined by

42[51 = 9 - t2, 41 [<I = 6 - 3 t -k 4 t2, $‘s[cl = 6 + f 5 + f c2 and < t2 are two adimensional abscissae referred to the nodal position.

The variations of the strain energy used by the asymptotic method are computed by standard assemblage of contributions in each finite element. For the second variation in the unloaded state

HC FINITE ELEMENTS

Figure 1. Local grid

233 1

du, this contribution can be directly computed by analytical integration on the element domain. The result is

2332 A. D. LANZO. G. GARCEA AND R. CASCIARO

Figure 2. HC interpolation

using the following (3 x 3) matrices:

4i[(]$j[c]d< => [doo] = 120

+ 112 d y =

L J

A similar analytical integration is possible for higher order variations (14)-(17). However from a computational point of view it is more efficient to use numerical integration which leads to equivalent results. A scheme involving centre and corner points (five points for element) is used,


assuming weights 8/12 and 1/12, respectively. Being each corner-point related to four elements, such a choice provides considerable algebraic simplifications.

It is worth mentioning that, in comparison with standard FE discretization, the HC technique needs a topological regularity of the discretization grid. Furthermore, a certain management of the boundary condition is needed because the function values are not directly represented by the nodal parameters (see Reference 22 for more details). Both aspects prove to be irrelevant in dealing with rectangular plates, as in the present case.

6. SOME IMPLEMENTATION DETAILS

With reference to the HC finite element discretization described in Section 5, let u be the nodal vector defining the displacement field u. Furthermore let @ be the load vector and K[u] the tangent stiffness matrix, respectively defined by the following energy equivalences:

pu:= f i UTK[u] 6 ~ : = W'[U] USU

The numerical evaluation of the structural equilibrium paths by the perturbation strategy of Section 2 develops as follows:

1. The initial stiffness matrix of the structure KO is assembled (by analytical integration). The fundamental path

u"L] = AQ is then obtained by solving in Q the linear system

KoQ = fl (20)

2. The bifurcation load Ab and the buckling mode i b are obtained as solutions of the eigenvalue problem

where K[A] is the tangent stiffness matrix in correspondence with the configuration uf [A].

K[A]+b = 0 (21)

3. The vector f is computed on the grounds of the definition

fTGu := - w;'d:Gu

Then, denoting the matrix representation of the orthogonality operator W;'Q( * ) (. ) with M, the secondary mode f i b is obtained by minimum condition

3 WEKbWb - fTWb = min, i i M W b = 0

i.e. by solving the linear system

a being defined by: KbWb - f = aMib

+Ef - a + i M + b = i t K W b E 0

4. The following energy terms are computed art i+2 ~ r d f i ; , %fig, Wrr&

b b ,

The coefficients ( i b , &,,pi) which are present in equation (6) are then obtained by their expressions (5).


5. The A-t relationship expressed by (6) is solved by points. The desired equilibrium path is finally obtained making use of the expression

u = 10 + ( i b + f ('fib

6. The imperfection sensitivity of the structure can be investigated by computing, for each load p or geometrical ii additional imperfection, the corresponding coefficients p" [A] and pq [A] defined by (7) and solving the A-( relationship again. When a geometrical imperfection ii is present, the equilibrium path is obtained in the form

To comment further, the solution of the linear system (20) is simply obtained by means of Cholesky's factorization. The same factorization is utilized for the eigenvalue problem (21) which is solved using the residual iteration algorithm suggested in Reference 15 for the general case of a non-linear bifurcation equation:

Note that due to expression (13), K[A] is actually linear in 1. However the scheme (23), based on the direct assemblage of vector K [Aj] i,, proves to be computationally more efficient than a standard implementation of the inverse power method.

Problem (22) is solved by computing the reduced vector

and then using the following iteration scheme:

rj = I - K[&]Wj

iij+ = wj + K, 'rj which is convergent (usually rapidly convergent) because the matrices K[A] and KO are quite similar in all deformation modes except for buckling mode ir, which is filtered by the process.

Note that both iteration schemes, (23) and (24), use as a basic iteration matrix the initial stiffness matrix KO already available in factorized form from the solution of the linear system (20).

7. A LOCKING PHENOMENON

FEM analysis introduces unavoidable (discretization) errors in the solution. Usually the errors rapidly decrease as the mesh size becomes smaller. However, in some cases, when the aspect ratio (e.g. thickness/length) degenerates and consequently the structure tends to behave as if internally constrained (e.g. axially undeformable), the discretization error can become pathological (locking) and the accuracy of the solution is completely destroyed. In the linear context this phenomenon arises, for example, in thin shell or Mindlin's plate analysis.

In the quadratic perturbation algorithm described in Section 2, a quite similar locking phenomenon can arise from the evaluation of the fourth-order term

- 3@i;w;


that defines the initial curvature of the postbuckling path. In fact it is the result of a small difference between two large terms. These being computed separately through different interpolations, the discretization errors produced in the evaluation of the single terms by separate approximations cannot exactly compensate. The total error may be of the same order, or greater, than the small value of the difference. Obviously, the numerical response given by the perturbation algorithm is completely unreliable in this case.

The size of the error produced by this locking pathology depends naturally on the finite elements used and, in particular, it decreases for an appropriate balancing of the in-plane and out-of-plane interpolations as produced by HC discretization. This can be shown in Figure 3 which refers to Eulero's rod case and reports numerical results for the postbuckling factor &,/A,, obtained, for different values of the ratio EAL*/EJ between the axial and the flexural stiffness, by using HC and standard beam elements (linear and cubic interpolation for the axial and transversal displacements, respectively).

Note that, for EAL2/EJ = 1.2 x lo5, 20 HC elements (22 total DOF) are sufficient to contain under 1 per cent the error in 2, while standard discretizations do not yield reliable results even using a large number of elements. In both cases a fourth-order exact expansion of the strain energyI3 and exact analytical integration have been used.

This test may be considered meaningful for the general behaviour of the HC interpolation technique. In particular, we conclude that the use of finite elements based on different polynomial interpolations for flexural and axial displacements, can appreciably reduce the accuracy of the perturbation algorithm in reproducing the postbuckling behaviour (see also Reference 27 where a complete revision of the approach was necessary for avoiding such a problem). From this point of view, the simple description (quadratic only) and the uniform treatment of the different displacement components obtained from HC interpolation gives remarkable advantages.

Finally it can be observed that using a numerical interpolation technique, as suggested in Reference 11, gives a further filter tool for the residual locking effects.

.. * HC (EAI2/EJ=1.2~1OJ) + HC (EAI2/EJ=1.2x1O5) * standard + standard ""Ab

1000, 8 4

2

0 10 20 30 40 number of elements

Figure 3. Locking in Eulero's rod case


8. NUMERICAL RESULTS

This section reports the results obtained from the proposed algorithm in buckling and postbuckling analysis of rectangular plates for a series of geometric, load and constraint conditions. In all cases, a hysotropic material is assumed, with the non-zero elastic constant Cilhk and Dijhk expressed by

Eh vEh 2Eh c1212 = - 1 + v c1122 = - 1 - v2’ Cll l l = c2222 = - 1 - v2’

Eh3 vEh3 Eh3 Dl212 = ~ 6(1 + v) 12(1 - v2)’ Dl122 = 12(1 - v2)’ D l l l l = D2222 =

E, v and h being Young’s modulus, Poisson’s coefficient and the thickness of the plate, respectively. When possible, results are expressed in adimensional form. In particular, the following

expression is generally assumed for the modal amplitude parameter

max out-of-plane displacement plate thickness 5 =

Tables are given which collect, for each test case, the bifurcation factor A, and the quadratic postbuckling coefficient (due to symmetry, all the analysed problems present &, = 0). Tables also report the values of the ratio K/Ko between the postbuckling and prebuckling global stiffness as defined by Budiansky in Reference 23:

When possible, the numerical values are compared with exact and numerical solutions available in literature.

Additional geometric imperfections are considered in the form of the buckling mode with different amplitudes expressed by the ratio

max out-of-plane imperfection plate thickness Y =

and the refined formula

(see Appendix I) is implemented to express the global A-5 relationship. In order to facilitate and promote comparison with results developed by different analysis

strategies (experimental, numerical incremental-iterative, use of different finite elements, etc.), we also report the equilibrium paths obtained for four different values of geometric imperfection ( y = 0.0,O.Ol h, O.lh, lh) for a certain number of sample cases. The results are given in the form of load vs. out-of-plane and load vs. in-plane displacement, with reference to representative points of the plate. For each case the buckling mode i)b and the quadratic orthogonal correction w b are also displayed. More extensive and complete results can be found in Reference 21.

8.1, Plates compressed in the longitudinal direction

The first set of tests in Figure 4 refers to rectangular plates uniformly compressed in the longitudinal direction by a (1 x A) load distribution along the transversal edges. Different boundary conditions are considered for the out-of-plane displacements, while a free edges condition is


I Test A ’ Y 0

L

I Test A 3

h

rest A2

Test A4

Figure 4. Tests Al-A4. Plates uniformly compressed in the longitudinal direction

Table 1. Test results Al-A4 (E = 2.1 x lo6, v = 025)

Es. Case Mesh b2 12(1 - v’) Illb K / K ,

Test A1 a/b = 1 (25 x 25) a/b = 2 (49 x 21) a/b = 3 (21 ~ 7 )

(33 x 11) (49 x 15)

Test A2 a/b = 1 (33 x 33) a/b = 2 (49 x21)

Test A3 a/b = 2 (33 x 17) Test A4 a/b = 2 (49 x 21)

4.00263 (4) 4.00323 (4) 4,03339 (4) 4.01 359 4,00674 7.71346 ( ~ 7 . 6 9 ) 7.01 142 ( x 7.02) 1.38808 ( x 1.38) 4.85495 1

036488 0.42354 0.44067 044259 0.44333 039151 0.53081 0.01 76 1 0.52165

0-34 179 037587 0.38722 0.38712 038710 0.24165 0.4 1 607 0-06103

045738

always assumed for the in-plane-displacements. The main results obtained are reported in adimensional form in Table I.

For the bifurcation load, analytical results available in literature (see Reference 1) are given in parentheses in Table I. In particular, for the simply supported case (test Al), the exact value of the bifurcation load (A, = 4(n2/b’) Eh3/12(l - v’)), allows the evaluation at less than 0.1 per cent the error obtained with a 40 x 21 ( x loo0 out-of-plane DOF) discretization mesh. For tests A2 and A3 the analytical (approximate) bifurcation value verify however the reliability of our numerical value.

There are few references in literature relative to the analysis of the postcritical behaviour. A comparison value K/Ko = 0.408 is reported from Reference 6 for the exact solution of simply

2338 A. D. LANZO, G. GARCEA A N D R. CASCIARO

supported infinitely long flat plate with longitudinal edges free for in-plane displacements. It is worth noting (see Figure 5 ) that this value is the limit of the values numerically obtained in the test A1 for increasing values of the length/width ratio u/b.

Plates with straight-line constraint on loaded edges (which avoids differential displacements of the edge points) are considered in Figure 9. These tests may be useful for a direct comparison with experimental results where, due to load technology, the simple boundary conditions can be more easily reproduced. The critical and postcritical coefficients obtained are reported in Table 11. It is worth noting that, because of the additional constraints, precritical KO and post-critical stiffness K are slightly greater than the values of the tests A1 and A4 which present similar geometrical and load conditions. However, due to the Poisson's effect on the loaded edges, the critical values are slightly less (the same effect can be observed in Reference 28).

For sample tests, Figures 6 8 and Figure 10 represent the equilibrium path obtained in terms of load (A parameter) vs. in-plane and out-of-plane displacements of representative points indicated for each diagram. With reference to the test A3, Figure 6 shows the equilibrium paths obtained by means of numerical step-by-step analysis obtained from NASTRAN code. It is worth noting that,

a/b

Figure 5. Test A l . Postcritical behaviour for long plates

240

P'-

0 - 120

0 5 , , , . . . . , ~ . , , , , . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . .

max shortening (u1[0,0.5b]) 0.68 0.01 0% 0.03 O . k 0.05

Figure 6. Load vs. displacement of test A1 (u/b = 3, h = 01, E = 2.1 x lo6, v = 0.25)

1 I I I 1 I I

400

528

a z 240 - 160

0 0.00 0.06 0.10 0.16 0.20 0.25 0.30 0.35

max deflection (u,[O.l7a,0.5b])

I I I I


15 max shortening (ul[0,0.5b])

Figure 7. Load vs. displacement of test A2 (a/b = 2, h = 01, E = 2.1 x lo6, v = 025)

96 %

80 80

64 64

a a g 48 g 48 3 3

32 32

16 16

0 0 0 .m 0.40 O . m 1 . 2 0 1.60 2.m 0.00 0.02 0.04 0.06 0 .m 0.10 0.12 0.14 0

max deflection (us[0.5a,b]) max shortening (u1[0,0.5b])

Figure 8. Load vs. displacement of test A3 (a/b = 2, h = 0.1, E = 2.1 x lo6, v = 0.25)

Test A l -b is Test A4-bis Y Y

Figure 9. Test Al/A4-bis. Plates compressed in the longitudinal direction with constrained transversal edges

6

while the quadratic perturbation algorithm gives a highly accurate reproduction of the out-of- plane postcritical behaviour, it is unable to reproduce the long distance plate shortening behaviour. Actually, the in-plane displacements being related to the quadratic correction 4 wt2, only the initial postbuckling tangent of the load vs. in-plane displacements curve is reproduced exactly.


Table IT. Test results Al/A4-bis ( E = 2.1 x lo6, v = 0-25)

Es. Case Mesh b2 12(1 - v') Xp, KIKO

Test Al-bis u/b = 2 (49 x 21) 3.93439 0.51762 041387 Test A4-bis u/b = 2 (49 x 21) 4.80090 0.53531 0.45695

280 I I I I

max shortening (u1[0,0.5b])

Figure 10. Load vs. displacement of test Al-bis (a/b = 2, h = 0-1, E = 2.1 x lo6, v = 0.25)

a I Test E2 Test E l

Figure 11. Tests B1 and B2. Plates loaded (pure bending) in the longitudinal direction

8.2. Other cases

Tests B1 and B2 of Figure 11 refer to plate loaded in the longitudinal direction by a load distribution of pure in-plane bending on the transversal edges. The results obtained for different boundary constraints and length/width ratio are given in Table 111, where exact values for the buckling load available are reported for comparison (in parentheses). Equilibrium paths and related perturbation modes are also shown for sample tests in Figure 12.

Finally tests C 1 4 4 of Figure 13 refer to plates uniformly loaded by a shear distribution (1 x A) along the boundary. Results are collected in Table IV and Figure 14. Table IV also reports (in parentheses) upper bound values of the critical load as defined in Reference 1. For the test C2 in


Table 111. Test results B1 and B2 (E = 2.1 x lo6, v = 0-25)

Eh3 ) - l

Es. Case Mesh 6 12(1 - v') x/Ab K / K o

Test B1 a/b = 1 (9x9) 2617768 (25.6) 044720 0.56839 (15 x 15) 25.76175 0.45037 0.56444 (25 x 25) 25.61239 0-43884 056364 (33 x 33) 2557659 0.43869 056349

a/b = 2 (49 x 23) 23.94518 (23.9) 0.43919 0.70223 a/b = 3 (49 x 15) 24-25424 0.42296 074856

Test B2 a/b = 1 (9 x9) 53.48732 0.67992 082125 (15 x 15) 4973677 0-59567 0.79042 (25 x 25) 48-45892 0.59598 078200 (33 x 33) 48.15739 0.5848 1 0.78020

a/b = 2 (49 x 23) 42.2 1 176 057165 0.83938 a/b = 3 (49 x 15) 41.72252 0.55307 0.85268

4 m 0 I I I I I I 1

+ 32000

40000

a $240&3 3

max deflecti

Figure 12. Load vs. displacement of test B2 (a/b = 2, h = 01, E = 2.1 x lo6, v = 0.25)

Test C l l k I . LLLLLLLL

I -

I Test c2

I I

Figure 13. Tests C1 and C2. Plates under the action of shearing stresses


Table IV. Test results C1 and C2 (E = 2-1 x lo6, v = 03)

Es. Case

Test C1 aJb = 1 aJb = 2

a/b = 3 Test C2 a/b = 1

alb = 2

aJb = 3

Mesh

(25 x 25) (15 x 7)

(25 x 11) (33 x 17) (45 x 23) (45 x 15) (25 x 25) (15 x 7)

(25 x 11) (33 x 17) (45 x 23) (45 x 15)

b2 12(1 - v’) XI&

9.35185 (9.34) 6,77725 (6.6) 6.63944 6.58668 6.56822 5.88460 (5.9)

14.78220 (1471) 11.40677 10.66984 10.42467 10,34334 974613

0.22905 0.14593 0.14348 0.14328 0.14341 0.1 5984 0.23371 0.31322 0.27653 0.26808 0.26564 0.1 730 1

KIKo

0.70017 0.66 182 064959 0.64603 0.64505 054548 0.72673 069894 0.65502 064360 0,64037 0,62247

max deflection (u3[0.5a,0.5 -m acement (uz[a,b])

Figure 14. Load vs. displacement of test C1 (a/b = 2, h = 01, E = 2.1 x lo6, v = 0.3)

particular it can be useful to bear in mind that the exact limit value for infinitely long plates

n2 Eh3 Ab = 8.99 -

b2 12(1 - v’)

For all the sample tests it is worth noting the general ability of the proposed numerical approach in approximating the complex geometry of critical mode i)b and of the quadratic orthogonal correction w b .

9. CONCLUSIONS

The paper has described an implementation of the numerical perturbation approach developed in References 14 and 15 to the postbuckling analysis of slender plates. The finite element modelling is derived from von-Karman’s non-linear plate theory using Aristodemo’s high-continuity interpolation of displacement fields. HC discretization technique, as it is allows the use of dense


discretization grid at a very low computational cost and as it is quite insensitive to non-linear- locking phenomena, appears particularly convenient in this context.

The approach appears both simple and computationally efficient in comparison with tradi- tional step-by-step analyses. Numerical results are given for different geometries and load conditions showing the approach's accuracy, reliability and global effectiveness. It also makes available simple benchmarks which could be easily reproduced and tested by different approaches (finite element and analysis strategies).

The paper deals with simple plate problems in order to give prominance to and set out the implementation details of the numerical method. The extension to orthotropic (fibre-reinforced laminated) plates is straightforward.

Extension to the postcritical analysis of box-girders and thin-walled structures requires some care due to the presence of complex mode-interaction phen~mena.'~ A more general perturbation strategy for multiple bifurcation is req~ired. '~ Research on this subject is in progress'9i20 and will be referred to in a forthcoming paper.16

ACKNOWLEDGEMENTS

This work was supported, in part, by an M.U.R.S.T. 40 per cent grant and scholarships from ALENIA Aeronautica (DTTE) and C.N.R. This support is gratefully acknowledged.

APPENDIX I: AN ACCURATE FORMULA FOR GEOMETRICAL IMPERFECTIONS

By reference to the perturbation theory described in Section 2, we now derive an accurate A-c formula for the case of geometrical imperfections given in the form of buckling displacement, i.e.

u' = fdb + f PWb

According to the model in hand (see Section 3) we assume the perfect and imperfect structures are characterized by the energy expressions

@ [ u ] := f E&[U]Z @ * [ u ] := f E ( & [ U ] - E ) Z

&[U] := &6U + + &"U2

E' := &bii + 9 &I'll2

respectively, the deformation fields ~ [ u ] and E" being expressed as:

The displacement field u is defined according to equation (4), i.e.

1( = Ad + (( + f ) d b + f(( + f)2wi;b

Starting from equation (26) and performing the same manipulations used for equation (6), the virtual work equation

@; [u] d, - Afidb = 0

can be rewritten as

f A2@;;'f12db + f ( t + f ) ' @ g ' i ) : + (5 + F ) ( n - Ab)@g'f idi

+ &(t + f ) 3 ( @ : r ' d ; - 30gw3 - E E & ' [ U ] d b = 0


Note that, using equations (26) and (27), the extra term Ek’irb can be expanded as

EE&‘[U]fib = EE&‘[u”]fib + EE&”(U - u“)db

= @’[u“]db + Eb”(u - u”)db

where

and

We have therefore the following expression for the virtual work equation:

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Asymptotic post-buckling analysis of rectangular plates by HC finite elements

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