Limit analysis of orthotropic plates

Limit analysis of orthotropic plates§

Leone Corradi*, Pasquale Vena

Department of Structural Engineering, Politecnico diMilano, Piazza Leonardo daVinci 32, 20133Milan, Italy

Received in final revised form 25 April 2002

Abstract

The limit analysis problem for plates in bending is considered. The failure criterion for thematerial is assumed as orthotropic, with possible non-symmetric strength properties.According to Kirchhoff ’s hypothesis, the plate is conceived as a superposition of layers,

individually in plane stress situation, and continuity is enforced by means of a kinematicassumption. By exploiting previous results, recently established by the authors, the expressionof the dissipation power per unit plate area is defined on this basis and the kinematic (upper

bound) theorem of limit analysis is cast in a form suitable for numerical solutions. To thispurpose, efficient algorithms successfully employed in the isotropic case can be used withminor modifications. The effectiveness of the procedure is demonstrated by solving some

homogeneous plate examples. Results permit the assessment of the influence of differentaspects, such as the ratio between strengths along the orthotropy directions, the tensile tocompressive strength differential and the inclination of the orthotropy axes with respect to thesides. The effects of in-plane edge constraints are also discussed and it appears that they are

emphasized considerably by anisotropy. Even if referred to specific cases, some conclusionscan be regarded as fairly general.# 2002 Published by Elsevier Science Ltd.

Keywords: A. Plastic collapse; B. Anisotropic material; B. Ideally plastic material; B. Plates; C. Finite elements

1. Introduction

Limit analysis, dealing with the direct definition of the load bearing capacity ofductile structures, is the basis for methods providing in a comparatively simplemanner a piece of information of prominent engineering significance. As soon as the

International Journal of Plasticity 19 (2003) 1543–1566

www.elsevier.com/locate/ijplas

0749-6419/03/$ - see front matter # 2002 Published by Elsevier Science Ltd.

PI I : S0749-6419(02 )00021 -9

§ Partial results were presented at the Plasticity ’00 Symposium, Whistler, BC, 16–20 July 2000.

* Corresponding author. Tel.: +39-02-2399-4224; fax: +39-02-2399-4220.

E-mail addresses: [email protected] (L. Corradi), [email protected] (P. Vena).

http://www.elsevier.com/locate/ijplas/a4.3d

mailto:[email protected]

mailto:[email protected]

fundamental theorems were established, the procedure was applied to a number ofstructural typologies of interest and plates in bending were among the first to beexamined (see, e.g., Prager, 1959). With few exceptions, closed form solutions onlydefine lower and upper bounds to the limit load, often bracketing it rather loosely.Results can be improved by using numerical procedures based on finite elementdiscretization. In particular, with reference to Kirchhoff’ plates governed by vonMises’ condition, Hodge and Belytschko (1968) obtained comparatively stringentbilateral delimitations. Even if more than 30 years old, their results remained forlong time the best bounds available and were still referred to as such in a recentcollection compiled by Save (1995).Actually, after an initial enthusiasm, the interest on the numerical solution of the

limit analysis problem began to deaden. In the formulation, inequality constraintsare inherent and the first numerical approaches relied on mathematical program-ming algorithms designed in view of different applications. As a result, the compu-tational burden associated to the solution of meaningful structural problems wassignificant and the parallel development of efficient finite element codes, performingelastic-plastic computations up to collapse, diverted the attention of researchersfrom direct methods. Only recently the progress occurred both in finite elementmodeling and in solution algorithms permitted a far better efficiency and a revival ofinterest in limit analysis was observed, as witnessed by several papers on the subject.Most of them (e.g., Liu et al., 1995; Sloan and Kleeman, 1995; Jiang, 1995; Capsoniand Corradi, 1997; Ponter and Carter, 1997; Christiansen and Andersen, 1999)exploit the kinematic (upper bound) theorem, bringing the problem to the search ofthe minimum of a convex, albeit non-smooth, functional. The essential conditionsare compatibility, which is spontaneously accounted for when a displacement finiteelement approach is used, and the requirement that strain rates be plasticallyadmissible, the only possible source of inequalities. In several instances no inequal-ities are present and a free minimum problem is easily obtained. Within this frame-work, numerical upper bounds were computed for Kirchhoff and Mindlin plates(Capsoni and Corradi, 1999), which in a few cases can be regarded as reasonableestimates of the actual limit load.With respect to the pioneering work of Hodge and Belytschko, the formulation

does not entail significant novelties, but the computational efficiency obtainablenowadays stimulates generalizations to other yield conditions. In particular, platesoften exhibit anisotropy in their strength properties and the extension of limit ana-lysis to this context appears of practical relevance: in the different, but related fieldof metal forming the effects of anisotropy are significant (Kim et al., 2000; Knockaertet al., 2002). Actually, the ductility of some anisotropic materials, such as compo-sites, is questionable, but this does not jeopardize the interest of the approach.Independently of ductility, limit analysis provides upper bounds to the load carryingcapacity of structures (Salencon, 1990) and, as experimental evidence shows, oftensuch bounds turn out to be meaningful for engineering purposes.Methods developed within the isotropic framework can be extended without

modifications, except that a single, but crucial, point has to be solved. The effec-tiveness of the solution algorithms demands that the dissipation power be expressed

1544 L. Corradi, P. Vena / International Journal of Plasticity 19 (2003) 1543–1566

as an explicit function of strain rates and general expressions are not available atpresent and hardly are foreseen to be produced. Specific expressions must be workedout case by case, accounting for the features of the yield condition at hand. Thisoperation was performed in a previous contribution (Capsoni et al., 2001) withreference to the tensor polynomial form for failure criteria proposed by Tsai and Wu(1971) truncated to the quadratic term in stresses. Even if the criterion does notaccount for the most general anisotropy, it can represent with acceptable accuracymost orthotropic behaviors (Barlat et al., 1991). The plane stress version of thisresult represents the starting point for the present study.The outline of the paper is as follows. First some basic limit analysis concepts are

recalled and the failure criterion to be used is presented, with the relevant expression forthe dissipation power. Subsequently, the plate model is introduced and the problemformulated in this context. An approximation, which permits significant computationalsaving, is also proposed at this point (the solution procedure is outlined in Appendix).A number of examples, referring to homogeneous plates, is next examined. Thediscussion of results permits some conclusions, which stem from the examplesconsidered, but enlighten some features of orthotropic behavior and probably possessgeneral validity.

2. Summary of existing results

Consider a structure subjected to body forces kB on its volume � and surfacetractions kt on the free portion @F� of its boundary. The constrained boundary @U�is fixed. Loads are defined as basic values B and t, affected by a load multiplier k.The constitutive law for the material includes the requirement that stresses are con-fined within the domain � �ð Þ4 1, which is supposed to be convex (�=1 is the limitsurface). The value s of k for which failure occurs (collapse multiplier) is sought.The problem above is the object of limit analysis. Within the framework of perfect

plasticity, theoretical results exist permitting the computation of the collapse multi-plier. For perfectly plastic materials � �ð Þ plays the role of a yield function. As longas the stress point is strictly inside the limit surface (� <1), the detailed features ofthe constitutive law are immaterial and it is conveniently assumed that no defor-mation occurs. Strain rates may develop for �=1 and obey the normality law

":¼

@�

@�l:

l:5 0 ð1a; bÞ

In addition, it is assumed that strains can accumulate to any required amountwithout rupture (unlimited ductility).According to the kinematic theorem of limit analysis, the collapse multiplier is the

optimal value of the minimum problem

s ¼ min":;u:

ð�

D ":

ð Þdx ð2aÞ

L. Corradi, P. Vena / International Journal of Plasticity 19 (2003) 1543–1566 1545

subject to

":¼ rSs

:in � s

:¼ 0 on @U� ð2bÞ

":2 D in � ð2cÞ

s:ð Þ ¼

ð�

B�s:dx þ

ð@F�

t�s:dx ¼ 1 ð2dÞ

In Eq. (2a), D ":

ð Þ is the dissipation power per unit volume, a uniquely defined func-tion of strain rates. Eq. (2b) expresses compatibility, associating to velocity fields s

:,

vanishing on the constrained boundary, the consequent strain rate distributions (rSs:

is the symmetric part of the of the velocity gradient). Inclusion (2c) enforces con-sistency with the constitutive law, by confining strain rates within the convexdomain D, the subspace spanned by the outward normals to the limit surface. Thiscondition establishes the plastically admissible nature of strain rates and togetherwith (2b) defines a mechanism. s

:ð Þ denotes the power of basic loads, which (2d)

normalizes to unity.Eqs. (2) are a convenient basis for numerical solutions, provided that the expres-

sion of the dissipation power as function of strain rates is available. Such anexpression was produced for yield conditions of the form

� �ð Þ ¼ F : � þ1

2P : �ð Þ : �4 1 ð3aÞ

or, in matrix notation

� �ð Þ ¼ Ft� þ1

2�tPs4 1 ð3bÞ

where � (as " in the sequel for strains) is a vector collecting stress components.Vector F accounts for possible tensile to compressive strength differential and P is asymmetric and positive definite, or at least semidefinite, matrix. Let

p ¼ �20 P þ1

2FFt

� �f ¼

�02F ð4a; bÞ

(�0 indicates a suitable reference strength). If matrix p is positive definite, one canwrite

� ¼ 2p1 ð5Þ

and the dissipation power is expressed by the relations (Capsoni et al., 2001)


D ¼�0

1 ftH f

ffiffiffiffiffiffiffiffiffiffiffi": tQ"

:q

ftH":

� �

with Q ¼ 1 ftH f�

H þH fftH if detP > 0 ð6aÞ

D ¼ �0": tH"

:

2ftH": subject to "

:2 D ¼ "

:jftH"

:5 0 �

if detP ¼ 0 ð6bÞ

In the first case D is the strain rate space itself, since the domain Eq. (3) forallowable stresses is bounded in every direction and any strain rate is normal to it atsome point. The restriction that "

:be plastically admissible plays an effective role

only when (6b) applies.Orthotropic solids in plane stress are now considered. Let the stress plane be z=0,

where z is an orthotropy axis, and x and y be the two in-plane principal directionsfor the material. In this reference frame odd terms in �xy vanish and Eq. (3) becomes

� ¼ Fx�x þ Fy�y þ1

2Pxx�

2x þ 2Pxy�x�y þ Pyy�

2y þ 6H�2xy

h i4 1 ð7Þ

where Fi;Pij and H are parameters defining the strength properties of the material.This is but a special case and all the results presented previously apply unaltered.If written for (7), Eqs. (4–6) define the dissipation power in the orthotropy frame

(x, y). When this is rotated by some angle # with respect to the reference frame usedin computations, denoted by (X, Y), rules of tensor calculus must be applied tostresses to transform the failure criterion or, equivalently, to strain rates to trans-form the resulting dissipation power. In the present, matrix notation the relationsread

�x ¼ Tt�X ":x ¼ T"

:X ð8Þ

In plane problems, T is a 3�3 matrix depending on #.

3. Orthotropic plates in bending

The limit analysis problem is now formulated for plates in bending. The materialyield function is expressed by Eq. (3) and its dissipation power is provided by therelevant of Eq. (6). Unless differently stated, it is supposed that (6a) applies.Consistently with Kirchhoff’s hypothesis, the plate is regarded as a superposition

of layers, each in plane stress situation, connected by a kinematic model expressingvelocities on the thickness as linear functions of z. The relations read

s:x ¼ u

:x z

@w:

@xs:y ¼ u

:y z

@w:

@yð9a; bÞ


where u:i and w

:are the membrane and transverse velocities of the middle plane z=0

(note that velocities do not vanish on the middle plane). It follows, for strain rates

":¼ �

:þ z�

:ð10Þ

where the vectors

�:¼

@u:x

@x

@u:y

@y

@u:x

@yþ@u:y

@x

� �t�:¼

@2w:

@x2@2w

:

@y22@2w

:

@x@y

� �tð11a; bÞ

collect the rates of membrane stretching and of (bending and twisting) curvatures,respectively. Account taken of (6a) and (10), the dissipation power per unit platearea is

D �:;�:

ð Þ ¼

ðh=2h=2

D ":

ð Þdz ¼

ðh=2h=2

�01 ftH f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaz2 þ bzþ c

p Azþ Bð Þ

� �dz ð12Þ

with

a ¼ �: tQ�

:b ¼ 2�

: tQ�:

c ¼ �: tQ�

:ð13aÞ

A ¼ ftH�:

B ¼ ftH�:

ð13bÞ

Attention is focused on homogeneous plates, with material properties independentof z. In this case (12) can be integrated in closed form (the expression is given inAppendix), permitting the formulation of the limit analysis problem for the plate.One obtains

s ¼ minu:;w:;�:;�:

ð�

D �:;�:

ð Þdx ð14aÞ

subject to:

�:¼ rsu

:�:¼ �2w

:; ð14bÞ

w:

ð Þ ¼

ð�

pw:dx ¼ 1 ð14cÞ

along with the boundary conditions on @U�. Eq. (14b) stand for (11) and condition(2c) was eliminated since no constraints on �

:, �:are present as long as (6a) applies.

When writing (14c) only transverse distributed loads, with basic value p x; yð Þ, areconsidered, but point or edge loads can be introduced with obvious modifications.


A discrete form of the problem is obtained by standard finite element techniques.In each element e e ¼ 1; � � � ;Nð Þ velocities and associated compatible stretching andcurvature rates are expressed as

u:e xð Þ ¼ Nem xð ÞU

:w:e xð Þ ¼ Neb xð ÞW

:ð15a; bÞ

�:e xð Þ ¼ Bem xð ÞU

:�:e xð Þ ¼ Beb xð ÞW

:ð15c; dÞ

where Ne xð Þ are shape function matrices, from which Be xð Þ follow. U:, W

:are the

vectors of free parameters, i.e. the assemblage operation is understood and is meantto enforce both interelement continuity and displacement boundary conditions.After substitution, problem (14) assumes the form

s ¼ minU:;W:

XNe¼1

De U:;W:� �

subject to RtwW:¼ 1 ð16a; bÞ

where De is the integral over the element area of the dissipation power (12),expressed as function of nodal velocities by means of (15c, d). Rw is the vector ofnodal forces equivalent to basic loads. The essentially unconstrained minimumproblem (16) can be solved by means of the procedure indicated in Capsoni andCorradi (1999) for isotropic plates, slightly modified to account for membranedegrees of freedom. The main steps of the solution method are summarized inAppendix.In-plane velocities, governed by vector U

:, are required to account for tensile to

compressive strength differential. However, the plate being subjected to transverseloads only, it is reasonable to expect that the collapse mechanism is essentially dic-tated by the parameters governing the transverse velocity field, collected in vectorW:, and that U

:plays a marginal role. This suggests the search for a formulation

involving W:only, with significant computational saving.

To this purpose, the strain model (10) is replaced by the expression

":zð Þ ¼ z

h

2

� ��:

j j4 1 ð17a; bÞ

Now strain rates vanish on a neutral surface, identified by the local values of ,which Eq. (17b) requires to be inside the plate. By integrating over the thickness, oneobtains

D ¼M0

1 ftHf1þ 2� ffiffiffiffiffiffiffiffiffiffiffiffi

�: tQ�

:pþ 2 ftH�

:h i

ð18Þ

where M0 ¼ �0h2=4 is a reference fully plastic moment.

The dissipation power (18) depends on the value of . If the neutral surface isassumed to coincide with the middle plane, one has


1 ¼ 0 : D1 ¼M0

ffiffiffiffiffiffiffiffiffiffiffiffi�: tQ�

:p1 ftH f

ð19aÞ

In general, the lowest collapse multiplier corresponds to the values of that mini-mize (18) locally. One obtains

2 ¼ ftH�

:


:p : D2 ¼M0�: tH�

:


:p ð19bÞ

The expression of D2 was simplified by exploiting the definition of matrix Q givenin (6a). On the same basis, it is readily verified that the inequality 2j j4 1 holdseverywhere, consistently with (17b). For symmetric materials (f ¼ 0;Q ¼ H) the twoexpressions coincide. Eqs. (19) express the dissipation power per unit plate area asfunction of the curvature rates and the limit analysis problem can be cast in terms ofthe variables involved in classical Kirchhoff ’s plate formulation.It should be noted that the strain model (17) is unable to enforce the compatibility

requirements for in-plane velocities unless it is =0. Therefore (19b) must be regardedmerely as an approximation and results do not necessarily represent upper boundsto the collapse multiplier. On the contrary, (19a) always defines a legitimatemechanism, providing strict, maybe coarse, upper bounds.Nevertheless, (19b) reflects the natural tendency of the collapse mechanism at

minimizing bending dissipation and the approximation seems reasonable if is notkinematically restrained, i.e. when boundary conditions permit in-plane sliding. Onthe other hand, when in-plane velocities are constrained to be zero on the boundary,it is likely that the collapse mechanism would reduce the more demanding mem-brane dissipation as much as possible with respect to the bending contribution andstrain rates are expected to vanish close to the middle plane, so that the upper boundobtained with (19a) should be reasonably close to the correct value. Obviously, theconjectures above must be validated by means of comparisons with the solutions ofthe fully kinematic problem (16).

4. Illustration and discussion of numerical examples

A few numerical solutions are presented now. To describe material failure, threedifferent criteria are employed, which are listed below, with the expressions ofmatrices defining the relevant dissipation power Eq. (6). In all cases, these appear inthe form

H ¼

�xx �xy 0Hxy H yy 0

0 0 H

24

35 f ¼ f

110

8<:

9=; ð20a; bÞ

Matrix Q can be computed on this basis, as indicated in (6a).


a: � ¼1

�20�2x

1

�2�x�y þ

1

�2�2y þ 3��2xy

� �0:5 < � < 1ð Þ ð21aÞ

�xx ¼4�2

4�2 1�xy ¼

2�2

4�2 1�yy ¼

4�4

4�2 1

� ¼1

3�f ¼ 0 ð21bÞ

The criterion refers to a symmetric material, with strengths of opposite sign equalin absolute values in any direction. In particular, for the two in-plane orthotropyaxes, it is

�!T;Cx ¼ �0 �!T;Cy ¼ ��0 ð22aÞ

and along the biaxial stress path �x ¼ �y ¼ S (�xy ¼ 0) failure is attained for

S! ¼ �0 ð22bÞ

b: � ¼1

1 �2�

�0�x þ �y�

þ1

�20�2x �x�y þ �2y þ 3��2xy

� ��

ð1 < � < 1Þð23aÞ

�xx ¼ �yy ¼1

31 �2�

4 3�2�

�xy ¼1

31 �2�

2 3�2�

� ¼ 1 �2� 1

3�f ¼

�

2 1 �2ð Þð23bÞ

For �xy ¼ 0, the limit surfaces are von Mises’ ellipses shifted so that their centerslocate at the point �x ¼ �y ¼ �. Isotropy is recovered for �=1, symmetry for �=0.

c: � ¼1

�0�x þ �y�

þ1

�202�2x þ 4

1

�2

� ��x�y þ 2�2y þ 3��2xy

� �

ð0:354 < � < 1Þ

ð24aÞ

�xx ¼ �yy ¼9�4

9�2 1�xy ¼ �2

9�2 2

9�2 1� ¼

1

3�f ¼

1

2ð24bÞ

As for the case before, (24a) refers to a material with equal properties along the twoorthotropy directions (tetratropic material, Zheng and Betten, 1995). The tensile tocompressive strength ratio is equal to one half. Namely


�!Tx ¼ �!Ty ¼1

2�0 �!Cx ¼ �!Cy ¼ �0 ð25aÞ

and along the stress path �x ¼ s, �y ¼ s the limit value is

s! ¼ ��0 ð25bÞ

All criteria depend on two material parameters only, denoted as � and �. Thelatter governs the shear strength in the orthotropy frame, proportional to �0=

ffiffiffiffiffi3�

p,

while � has different meaning in each criterion. Eq. (6a) applies for � > 0 and � inthe intervals indicated in each case. Fig. 1 depicts two of the limit curves in the plane�xy ¼ 0. The criteria are not meant to represent real materials. Rather, they arechosen because the influence of individual parameters is singled out easily.Square (side a), simply supported plates under uniform pressure (basic value

p ¼ 4M0=a2) are examined. To model bending behavior, the 16 d.o.f., C1-con-

tinuous BFS element (Zienkiewicz and Taylor, 1989) is used. Its well known sensi-tivity to shape distortion has no influence since only regular subdivisions intosquare, equal elements are employed, with no attempt at adapting meshes to pre-dictable collapse mechanisms. The membrane behavior, when considered, is modeled bymeans of the four node, 8 d.o.f. bilinear plane element. Whenever possible, symmetry isexploited and only one quarter of the plate analyzed. As for the isotropic case (Capsoniand Corradi, 1999) convergence with mesh refinement is very rapid: the results obtainedwith 5�5 and 10�10 elements for a quarter of plate differ by less than 0.2%.

4.1. Symmetric material

The failure criterion (21) is first examined. In this situation (f=0,Q=H) it is �:¼ 0

at solution, so that problem (16) and that obtained from (19a) provide the sameresults. For #=0, when the orthotropy directions are parallel to the plate edges,results for different values of � and � are compared to lower bound (staticallyadmissible) estimates, obtained from elastic, orthotropic solutions with elasticparameters adjusted so as to reach simultaneous yielding at the center of the plateand at its vertices. The collapse multipliers computed for � ¼ 1 and different � in theallowable range are plotted in Fig. 2a. The minimum value occurs for a situationclose to isotropy (� ¼ 1, s=6.256). In fact, as � departs from one the yield domainwidens either along the �y axis (�> 1) or in the direction �y ¼ 2�x (�<1, Fig. 1a).The influence of � is depicted in Fig. 2b: as expected, the collapse multiplier dimin-ishes with increasing �, i.e. with decreasing shear strength.The level lines of the collapse mechanisms and the relevant dissipation maps are

illustrated in Fig. 3. The picture suggests a qualitative analogy between the behaviorof a square orthotropic plate and that of an isotropic, but rectangular one: the col-lapse mechanism, symmetric also about the diagonal when �=1, evolves withincreasing � as if the ratio between the edge lengths was augmented.The case in which the orthotropy axes are rotated by an angle # with respect to the

edges is also considered. It is assumed �=1, so that deviations from isotropy follow


Fig. 1. Limit curves in the plane �xy ¼ 0. (a) Symmetric material Eq. (21a). (b) Tetratropic material Eq.

(24a).


Fig. 2. Collapse multipliers for the failure criterion (21).


from different properties in the two orthotropy directions only and are subsumed bythe value of �. The result being unaffected if x and y are interchanged, only therange 0� < # < 45� is explored. The collapse multiplier is plotted as function of # inFig. 4, referring to �=2.0: the load carrying capacity of the plate increases up to12% for # ¼ 450, when the uniaxial strengths parallel to the edges become equal andhigher than the mere average of the two principal values. The dissipation map in

Fig. 3. Level lines (left) and dissipation maps (right) of the collapse mechanisms for the failure criterion

(21). (a) �=1 (isotropic), (b) �=3, (c) �=6.


Fig. 5 (now referred to the entire plate) shows how the 450 inclination of the ortho-tropy axes affects the collapse modality.

4.2. Von Mises’ criterion shifted

The failure criterion (23) is used to assess the influence of the tensile to compres-sive strength differential, governed by the material parameter �. The solutions ofproblem (16) are indicated in Fig. 6 for supports permitting in-plane sliding (dia-

Fig. 4. Collapse multiplier as function of the orthotropy axes rotation.

Fig. 5. Collapse mechanism for # ¼ 45� (�=2.0, �=1.0).


monds) and for in-plane velocities constrained to be zero at the boundary (dots). Thecollapse multipliers computed by means of approximations (19) are also shown. Eq.(19a) implies that D1 is independent of � and the collapse multiplier is unaffected(dashed lines). In fact, since the neutral surface is forced to coincide everywhere withthe middle plane, the decrease in tensile dissipation is exactly compensated by thecorresponding increase in compression. Solid lines depict the results stemming from(19b). For �=0.25 (shear yield strength twice the isotropic value) approximatedresults compare reasonably well with those provided by Eq. (16) for both boundaryconditions. Assumption (19a) provides reasonable upper bound estimates for�<0.5, i.e. for tensile to compressive strength differentials up to 0.57, even if differ-ences increase with increasing �, indicating that membrane dissipation becomes lessdemanding as the shear yield strength diminishes. The solid lines, referring toapproximation (19b), show qualitative agreement with the behavior of platesunconstrained in the plane (diamonds), but values get considerably lower as � and �



increase. It must be remembered that Eq. (19b) does not enforce in-plane compat-ibility and results are not upper bounds in general.Fig. 7 depicts the collapse mechanism obtained on the basis of (19b) for �=0.25

and �=0.4, corresponding to a tensile to compressive ratio in the orthotropy direc-tions of about 0.65. Comparison with the isotropic case (Fig. 3a) shows that themechanism gets closer to the yield line model proposed by Johansen (1943) forreinforced concrete plates, which also exhibit a markedly dissymmetric behavior.

Fig. 7. Collapse mechanism for the failure criterion (23). �=0.4, �=0.25.



4.3. Tetratropic material (24)

In this failure criterion, the parameter � dictates the shape of the limit curves inthe plane �xy ¼ 0 (Fig. 1b). As � increases from its minimum allowable value of0.354, strengths along the biaxial path �x ¼ �y ¼ S diminish, but for �=1.0 they arealready nearly equal to the limiting values S!C ¼ 0:5�0, S!

T ¼ 0:25�0 and no sig-nificant decay occurs as � grows further. On the other hand, in the direction�x ¼ s, �y ¼ s the domain widens considerably with increasing �. This loading pathcorresponds to simple shear in a frame rotated by 450 with respect to orthotropy axes.These features are reflected by a marked sensitivity of the collapse multiplier to the

rotation of the orthotropy axes. The critical points in the plate are its center where,because of symmetry and of the tetratropic nature of the failure criterion, no twist ispresent and bending moments attain their maximum values (with MX ¼MY), andcorners, subject to significant twisting moments. For #=0 bending moments at thecenter, after an initial decrease with �, attain a very nearly constant value, whiletwisting moments depend on � only: in fact, as the relevant curve in Fig. 8 shows,the collapse multiplier decreases little if at all for � > 1. On the contrary, if theorthotropy axes are rotated by 450 the shear strength and, hence, the twistingmoment capacity at corners, grow linearly with �, with a corresponding, remarkableincrease in the limit load. The collapse mechanisms in the two cases are illustrated in

Fig. 9. Collapse mechanisms for the failure criterion (24). �=2.0. (a) # ¼ 0�, (b) # ¼ 45�.


Fig. 9, referring to �=2: for #=0 the picture does not differ in essential mannersfrom that relevant to a symmetric, isotropic material (Fig. 3a), but for # ¼ 45� dis-sipation concentrates at corners and the central portion of the plate only undergoesa nearly rigid motion. In computations, (19b) was used, but conclusions are notsignificantly affected by this choice.

5. Concluding remarks and comments

The limit analysis problem for orthotropic plates in bending was considered in thisstudy. Plates were modeled according to Kirchhoff’s hypothesis, slightly modified soas to allow for non-vanishing strain rates on the middle plane. The expression of thedissipation power per unit area was derived on this basis. This permits the use ofsolution algorithms that proved efficient in the isotropic context, exploiting thekinematic theorem of limit analysis in conjunction with finite element discretization.A fully compatible formulation in terms of both transverse and in-plane nodalvelocities was derived. In addition, an approximation involving transverse velocitiesonly was suggested, which provides some computational advantages even if theupper bound nature of results is lost in some instances.A few limitations of the procedure must be pointed out. Results are confined to failure

criteria of the form (3), which can represent with acceptable accuracy certain types oforthotropy, but not general anisotropy (Betten, 1988; Zheng and Betten, 1995). Someanisotropic materials cannot be considered as perfectly plastic and, in this situation, limitanalysis only provides upper bounds to the collapse load. Studies developed in differentcontexts (e.g., Abdi et al., 1995) justify a certain confidence on results, but to assess theirsignificance they should be contrasted to evolutive solutions or to experimental data.The adequacy of Kirchhoff ’s model itself is partly questionable, since transverse sheardeformations might have non-negligible effects also for comparatively slender plates.Nevertheless, results raise some points of interest, which are worth summarizing.

Namely

a. The effects of anisotropy both on the limit load and on the collapse modality
are significant. In the failure criteria considered, deviations from isotropywere controlled by two parameters, one governing the shape of the yielddomain in the �x �y plane, the second the shear strength in the orthotropyframe. The influence of the latter was essentially as predictable. On the con-trary, the yield domain shape turns out to play a, perhaps unexpectedly, sig-nificant role, suggesting that, at least for single-layer plates, considerablebenefits could be gained by a judicious design of the material properties.
b. The tensile to compressive strength ratio also had partly unpredictable con-
sequences. The bending capacity of the plate is essentially dictated by thesmaller of the two values, but shear strength, symmetric in the orthotropyframe, enters the picture when this frame is rotated, with definitely significantconsequences. The remarkable difference between the two curves in Fig. 8 isperfectly understandable on this ground.

c. When the ratio above is different from one, the role of edge constraints
becomes important. Depending on whether in-plane velocities are preventedor free to develop, marked differences show up in computed results.
The comments above are based on a limited number of cases and more extensivenumerical experience is required to assess their general validity. However, it does notseem too daring say that the results obtained enlighten some aspects connected withorthotropic behavior as such in respect of limit analysis.As a final comment, it mentioned that the validity of the proposed model is not

limited to homogeneous plates, which were only considered so far. The same for-mulation can be employed for plates which are actually layered, i.e. whendifferent layers have different properties or their orthotropy axes are differ-ently oriented. In these situations the validity of Kirchhoff’s hypothesis oftenis questioned on the ground that it entails inherent equilibrium violations, butthis plays no role as long as kinematic admissible solutions are sought. Themodel defines valid mechanisms, producing upper bounds to the collapse multi-plier. Such bounds could be too coarse for practical purposes, but are worth con-sidering, since kinematic solutions often produce good estimates even from verycrude mechanisms. In any case, results can be improved by using more refinedmodels, provided that they remain fully compatible. Unfortunately, most of theavailable formulations for layered plates are of mixed nature, as such unsuited forkinematic limit analysis.

Acknowledgements

This study is a part of the project Molecular level instruments for biomaterialinterface design, within the framework of the Large-Scale Computing program ofthe Politecnico di Milano. The financial support of the Institution is gratefullyacknowledged.

Appendix

Some details on problem (16) and on the strategy employed for its solution arenow provided. For homogeneous plates closed form integration of (12) is possible.Matrix Q being positive definite, the equation az2 þ bzþ c ¼ "

:zð ÞtQ"

:zð Þ ¼ 0 either

has no real roots or two coincident roots, the latter case occurring if the three com-ponents of "

:simultaneously vanish for the same value of z (not necessarily within

the plate). This implies

4ac b25 0 ðA1Þ

In this situation one has (see, e.g., Abramowitz and Stegun, 1972)


D0 ¼

ðh=2h=2


pdz

¼2azþ b

4a


pþ4ac b2

8affiffiffia

p arcsinh2azþ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4ac b2

p

� �h=2h=2

ðA2Þ

(the last term vanishes when (A1) holds as an equality). On the other hand, it is,trivially

ðh=2h=2

Azþ Bð Þdz ¼ Bh ¼ hftH�:

ðA3Þ

and the integral over the element area of the plate dissipation reads (constantthickness is assumed, besides homogeneity)

De �:e;�

:eð Þ ¼

�01 ftH f

ð�e

D0dx hftH

ð�e

�:edx

� �ðA4Þ

This permits the definition of the objective function of problem (16), expressed interms of nodal velocities by means of Eqs. (15c) and (15d).By writing the Lagrangean function of the problem, namely

L U:;W:;

� �¼

XNe¼1

De U:;W:� �

RtwW: 1

� �ðA5aÞ

the optimality condition reads

�L5 0 8 �U:; �W

:; � ðA5bÞ

The inequality reflects the non-stationary nature of the optimal value, due to the factthat, as (12) shows, the dissipation power per unit area (and, hence, L) is not dif-ferentiable in the regions that do not undergo plastic flow.To illustrate the solution procedure, assume first that the entire structure is affec-

ted by plastic flow at collapse, so that differentiability is ensured everywhere. In thiscase L is stationary at solution and (A5b) produces the nonlinear system

@L

@U: ¼

XNe¼1

@�:e

@U:

� �t@De

@�:e

¼�0

1 ftH f

XNe¼1

ð�e

Btem@D0

@�:e

hHf

� �dx ¼ 0 ðA6aÞ

@L

@W: ¼

XNe¼1

@�:e

@W:

� �t@De

@�:e Rw ¼

�01 ftHf

XNe¼1

ð�e

Bteb@D0

@�:edx Rw ¼ 0 ðA6bÞ

On the other hand, Eqs. (13a) and (15c) and (15d) establish


@D0

@�:e

¼@D0

@c

@c

@�:e

þ@D0

@b

@b

@�:e

¼ 2@D0

@cQ�

:e þ

@D0

@bQ�

:e

� �

¼ 2@D0

@cQBemU

:þ@D0

@bQBebW

:� �

ðA7aÞ

@D0

@�:e¼

@D0

@b

@b

@�:eþ@D0

@a

@a

@�:e¼ 2

@D0

@bQ�

:e þ

@D0

@aQ�

:e

� �

¼ 2@D0

@bQBemU

:þ@D0

@aQBebW

:� �

ðA7bÞ

This permits the explicit expression of Eqs. (A6). By introducing the matrices

Kuu ¼�0

1 ftH f

XNe¼1

ð�e

@D0

@cBtemQBemdx ðA8aÞ

Kuw ¼ Ktwu ¼

�01 ftHf

XNe¼1

ð�e

@D0

@bBtemQBebdx ðA8bÞ

Kww ¼�0

1 ftH f

XNe¼1

ð�e

@D0

@aBtebQBebdx ðA8cÞ

(which depend on the current configuration through the derivatives of D0 with
respect to a, b and c) and the configuration independent vector
Ru ¼�0h

1 ftHf

XNe¼1

ð�e

Btemdx�H f ðA8dÞ

the system assumes the form

Kuu Kuw

Kwu Kww

� �U:

W:

� �¼

Ru

0

� �þ

0

Rw

� �ðA9Þ

or, compactly

kv:¼ ru þ rw ðA10Þ

The system is solved by direct iteration. By denoting with index j quantities eval-uated on the basis of the configuration attained at the j-th iteration, one can write

v:jþ1 ¼ v

:u þ jv

:w ðA11aÞ

with


v:u ¼ k1

j ru v:w ¼ k1

j rw ðA11bÞ

By imposing the condition RtwW:¼ rtwv

:jþ1 ¼ 1 one obtains

j ¼1 rtwv

:u

rtwv:w

ðA11cÞ

Eqs. (A11) define the configuration to be used for the subsequent step of the itera-tion process, which is continued up to convergence. At each step, integration overthe element area is performed numerically, by means of Gauss-Legendre quadratureon a 3�3 grid.As described, the procedure does not consider that plastic flow at collapse often

affects part of the structure only. The remaining portion keeps rigid in the motioncorresponding to the mechanism and L is not differentiable in it. This region isprogressively detected and eliminated from the problem. To this purpose note thatDe, Eq. (A4), is a global value referring to the finite element as a whole. In eachelement, De is either zero and not differentiable (when �

:e ¼ 0 and �

:e ¼ 0 through-

out) or positive and differentiable (in other instances), so that the rigid region of thebody effectively is replaced by the set of rigid elements.In a rigid element, straining modes vanish and rigid body motions only survive.

The two contributions are separated by exploiting the natural approach proposed byArgyris (1966); in particular, straining element modes are singled out by splittingEqs. (15c) and (15d) as follows

�:e xð Þ ¼ bem xð Þq

:em �

:e xð Þ ¼ beb xð Þq

:eb ðA12aÞ

q:em ¼ CemU

:q:eb ¼ CebW

:ðA12bÞ

where q:em, q

:eb are the vectors collecting the natural degrees of freedom (straining

modes) of element e and Cem, Ceb are constant matrices, playing the role of com-patibility operators for the finite element.The computation is started with an arbitrary vector v

:¼ U

:t W

:t

�t, selected so as

to induce plastic flow in all elements. At each iteration, the dissipation power iscomputed separately for each element, and when it gets smaller than a prescribedtolerance the element is predicted to be rigid in the final mechanism. Rigidity isenforced by imposing q

:em ¼ 0 and q

:eb ¼ 0 for the element detected. Through Eqs.

(A12b), this provides some constraints among the components of U:, W

:, which

can be replaced with smaller size vectors U:1, W

:1 by writing U

:¼ Gu1U

:1,

W:¼ Gw1W

:1. The iteration process is continued with the rigid element ignored

and the operation is repeated whenever the dissipation power of a new elementbecomes sufficiently small. Once the r-th rigid element is identified and elimi-nated, one has


U:¼ GurU

:r W

:¼ GwrW

:r ðA13Þ

and matrices and vectors in (A10) are updated by writing

Krhk ¼ Gt

hrKhkGkr Rrh ¼ Gt

hrRh h; k ¼ u;wð Þ ðA14Þ

A sequence of systems with a progressively decreasing number of elements and offree nodal parameters is thus considered. Each system consists of plasticallydeforming elements only and its dissipation power is stationary at solution.It is convenient to consider separately membrane and bending dissipation and to

constrain the relevant degrees of freedom individually, whenever the correspondingcontribution becomes negligibly small. In fact, in a number of situations severalelements turn out to experience bending deformations only. The limit case is that ofa symmetric material, when membrane deformations vanish throughout at collapse.Once the corresponding degrees of freedom are detected and eliminated by the pro-cedure, the same formulation as the one stemming from (19a) is obtained.The presence of membrane degrees of freedom introduces the only significant dif-

ference with respect to the procedure used in Capsoni and Corradi (1999) for iso-tropic plates. When either of Eqs. (19) is employed the two solution methodsidentify, except that for the expression of the dissipation power per unit plate area.

References

Abdi, R., de Buhan, P., Pastor, J., 1995. Calculation of the critical height of a homogenized

reinforced soil wall: a numerical approach. Int. J. Num. Anal. Meths in Geomechanics 18, 485–

505.

Abramowitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions. Dover, New York.

Argyris, J.H., 1966. Continua and discontinua. In: Proceedings of the First Conference onMatrix Methods

in Structural Mechanics. AFFDL, TR 66–88, Dayton, OH, pp. 11–92.

Barlat, F., Lege, D.J., Brem, J.C., 1991. A six-component yield function for anisotropic materials. Int.

J. Plasticity 7, 693–712.

Betten, J., 1988. Applications of tensor functions to the formulation of yield criteria for anisotropic

materials. Int. J. Plasticity 4, 29–46.

Capsoni, A., Corradi, L., 1997. A finite element formulation of the rigid-plastic limit analysis problem.

Int. J. Num. Methods Engng 40, 2063–2086.

Capsoni, A., Corradi, L., 1999. Limit analysis of plates — a finite element formulation. Structural Engi-

neering and Mechanics 8, 325–341.

Capsoni, A., Corradi, L., Vena, P., 2001. Limit analysis of anisotropic structures based on the kinematic

theorem. Int. J. Plasticity 17, 1531–1549.

Christiansen, E., Andersen, K.D., 1999. Computation of collapse states with von Mises type yield condi-

tion. Int. J. Num. Methods Engng 46, 1185–1202.

Hodge Jr., Ph.G., Belytschko, T., 1968. Numerical methods for the limit analysis of plates. Trans. ASME,

J. Appl. Mech. 35, 796–802.

Jiang, G.L., 1995. Nonlinear finite element formulation of kinematic limit analysis. Int. J. Num. Methods

Engng 38, 2775–2807.

Johansen K.W., 1943. Brudlinieteorier, Copenhagen.

Kim, J.-B., Yang, D.-Y., Yoon, J.-W., Barlat, F., 2000. The effects of plastic anisotropy on compressive

instability in sheet metal forming. Int. J. Plasticity 16, 649–676.


Knockaert, R., Chastel, Y., Massoni, E., 2002. Forming limit prediction using rate-independent poly-

crystalline plasticity. Int. J. Plasticity 18, 231–247.

Liu, Y.H., Cen, Z.Z., Xu, B.Y., 1995. A numerical method for plastic limit analysis of 3-D structures. Int.

J. Solids Structures 32, 1645–1658.

Ponter, A.R.S., Carter, K.F., 1997. Limit state solutions based upon linear elastic solutions with a

spatially varying elastic modulus. Comp. Meth. Appl. Mechns. Engng. 140, 237–258.

Prager, W., 1959. An Introduction to Plasticity. Addison-Wesley, Reading, MA.

Salencon, J., 1990. An introduction to yield design theory and its application to soil mechanics. Eur.

J. Mech. A/Solids 9, 477–500.

Save, M., 1995. Atlas of Limit Loads of Metal Plates, Shells and Disks. Elsevier, Amsterdam.

Sloan, S.W., Kleeman, P.W., 1995. Upper bound limit analysis using discontinuous velocity fields. Comp.

Meth. Appl. Mechns. Engng. 127, 293–314.

Tsai, S.W., Wu, E.M., 1971. A general theory of strength for anisotropic materials. J. Compos. Materials

8, 58–80.

Zheng, Q.S., Betten, J., 1995. The formulation of constitutive equations for fiber-reinforced composites in

plane problems. Part II, Archives of Applied Mechanics 65, 161–177.

Zienkiewicz, O.C., Taylor, R.L., 1989. The Finite Element Method, Vol. 2. McGraw-Hill, London, UK.


Documents

Limit analysis of orthotropic plates