A Numerical Approach for Limit Analysis of Orthotropic Composite Laminates

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 70:71–93Published online 5 September 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1872

A numerical approach for limit analysis of orthotropiccomposite laminates

A. A. Pisano∗,† and P. Fuschi

Dipartimento Arte Scienza e Tecnica del Costruire, University Mediterranea of Reggio Calabria,via Melissari, 89124 Reggio Calabria, Italy

SUMMARY

A numerical approach for limit analysis of structures whose constituent material exhibits orthotropicbehaviour is presented and discussed. Attention is focused on orthotropic composite laminates underplane stress conditions. The proposed approach is an extension, in the context of orthotropic materials, ofthe linear matching method (LMM). The latter is based on a sequence of linear analyses performed onthe analysed structure made of a fictitious linear viscous material with spatially varying moduli. Here theLMM is applied to structures made of materials obeying the Tsai–Wu criterion. An appropriate choice ofthe fictitious material, which in this case is assumed linear, viscous, orthotropic and suffering a distributionof assigned initial stresses, reduces the number of parameters to be spatially varied thus rendering thewhole procedure applicable and reliable. The results obtained are highly promising as witnessed by anumber of numerical examples which are carried out to verify the effectiveness of the proposed approach.Copyright q 2006 John Wiley & Sons, Ltd.

Received 9 February 2006; Revised 20 June 2006; Accepted 18 July 2006

KEY WORDS: limit analysis; orthotropic materials; Tsai–Wu criterion; linear matching procedure; FEiterative procedure

1. INTRODUCTION

The definition of the load bearing capacity of a structure has always been considered an essentialdata from an engineering point of view. Limit analysis, based on two fundamental theorems dueto Drucker et al. [1], is an effective tool for the direct definition of the load bearing capacity of astructure. Although the validity of such a direct method was confined to ductile structures madeof standard perfectly plastic materials, limit analysis was applied from the beginning to a number

∗Correspondence to: A. A. Pisano, DASTEC Dipartimento Arte Scienza e Tecnica del Costruire, UniversityMediterranea of Reggio Calabria, Via Melissari, 89124 Reggio Calabria, Italy.

†E-mail: [email protected]

Contract/grant sponsor: Italian Ministero dell’Istruzione, dell’Universita e della Ricerca (MIUR)

Copyright q 2006 John Wiley & Sons, Ltd.

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72 A. A. PISANO AND P. FUSCHI

of engineering problems (see e.g. [2, 3]). As noted in Del Piero [4], the tendency of using limitanalysis approaches and Drucker’s theorems mentioned above to evaluate the bearing capacity ofsoils and structures made of stone or concrete is witnessed by a number of papers, coeval of thefounding one, e.g. [5–8]. More recently, in [4], the applicability of the two fundamental theoremsof limit analysis was proved outside the domain of perfect plasticity for a class of materials called‘normal linear elastic’ including no-tension materials as a particular case. In the context of non-standard materials, namely for soils, a theory of limit analysis was proposed by Radenkovic inthe early 1960s (see e.g. [9]) and, on these bases, by Josselin de Jong [10] and Palmer [11]. Inthis context, with different approaches, limit analysis theory has been proposed by Atkinson andPotts [12] and recently by a number of researchers [13–16].

On the other hand, the development of numerical procedures, based on finite element formu-lations, was a further essential contribution to the success of such a direct method (see e.g. theearly contributions of Hodge and Belytschko [17, 18] up to the comprehensive work of Save [19]).Combining the mathematical programming algorithms and the finite element technique, a numberof simplified analytical methods have also been used to compute the lower and upper bound limitloads according to static and kinematic theorems. However, the non-linearity and non-smoothnessof the objective function in the upper bound procedure and the strong physical non-linearity andunidirectionality of the constraints in lower bound analysis render most of these methods time-consuming and costly for meaningful structural problems. To this concern, alternative approaches,based on the finite element method (FEM) and mathematical programming technique combined inan iterative fashion, have recently been proposed to avoid some of the above drawbacks [20, 21].

Indeed, the development in the last few decades of elastic–plastic step-by-step numerical anal-yses, able to follow the structural response up to collapse, attracted the researcher’s interest in theevolutive analyses against the direct methods and this at least for structures whose constitutivematerial behaviour is isotropic and governed by well-established yield conditions. If it is possibleto share the opinion that limit analysis on structures made of elastic perfectly plastic materials is,or may be, obsolete—it is very often more cumbersome than an evolutive step-by-step analysis—such a belief cannot be accepted when dealing with materials whose available constitutive lawsare unable to catch the complexity of the phenomena characterizing the actual post-elastic materialbehaviour. Limit analysis on structures made of such materials is, in the authors’ opinion, an effec-tive tool for defining, although approximately, the actual bearing capacity of the structures and istherefore useful for design purposes. This is even more valid for composite material structures forwhich the evolutive numerical analyses, although grounded on subtle constitutive material modelsthat take into account phenomena like interlaminar behaviour, delamination, damage evolution,etc., turn out to be effective only for solving specific case-studies. Moreover, in the context ofcomposites, almost all the available constitutive models are based on material constants that aredifficult to identify via laboratory tests thus resulting useless for practical engineering applications.The current interest in research on limit analysis approaches in the field of composite materialstructures is witnessed by several contributions, see e.g. [15, 22–25].

The present paper belongs to this research line whose main goal is the extension to orthotropiccomposite laminates of a procedure known in literature as linear matching method, whose generalcharacteristics are the same of the elastic compensation method [26–28]. The LMM has beensuccessfully applied for isotropic yield conditions [29, 30] also in the case of pressure dependencyand non-associativity [31–33] and is here applied to one of the most popular criterion for compositelaminates, namely the Tsai–Wu failure criterion [34, 35] and, in particular, to a second-order tensorpolynomial form of it, the latter assumed as yield condition. The use of LMM in conjunction with

Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 70:71–93DOI: 10.1002/nme

LIMIT ANALYSIS OF ORTHOTROPIC COMPOSITE LAMINATES 73

such yield criterion guarantees the convergence of the whole procedure on the base of the sufficientcondition given in [31] for a general class of yield conditions including the one here assumed.In such a context, the numerical strategies involved in the whole procedure have been suitablymodified and, as shown by a few simple examples, they seem to be effective, flexible and easilyapplicable to other criteria for anisotropic materials.

The assumption of a perfectly plastic material behaviour obeying a yield condition given bythe Tsai–Wu criterion can appear too coarse for a certain class of composites, but it is definitivelyacceptable for composites such as metal matrix reinforced by metal fibres, plastics reinforced bykevlar or glass fibres that, as is known, undergo plastic flow and considerable ductility. The lackof associativity, on the other hand, is overcome by the ‘non-standard limit analysis’ approachproposed by Radenkovic, see Lubliner [36]. In truth, the gap between the plastic collapse load andthe failure load for non-associative materials exhibiting limited ductility, may be large. However,the proposed method also gives a lower bound to the collapse load and the adopted constitutivecriterion seems to be very effective for the evaluation of the load bearing capacity of a class ofstructural problems for which the above gap is reasonably small. Framing the present study in awider context, the proposed approach is concerned with limit analysis of a class of anisotropicstructures characterized by a yield function in the form of a general quadratic stress function.

The outline of the paper is as follows. After this introductory Section, in Section 2 the adoptedconstitutive material model and all the hypotheses concerning the material behaviour are given.Some basic concepts of non-standard limit analysis are also briefly summarized and the basicformulae for the upper and lower bound collapse multipliers evaluation are set up. Section 3,which is the core of the paper, is devoted to the generalization of the LMM to orthotropicmaterials. The fundamental assumptions are reported and the iterative procedure is explained indetail using flow-chart style. Two numerical examples, with the main purpose of validating thewhole procedure, are given in Section 4. Section 5 closes the paper drawing some conclusions andforecasting some possible developments of the present study.

Notation: The subscripts denote Cartesian components and the repeated index summation ruleis applied. Bold face symbols denote vectors or tensors. Cartesian orthogonal co-ordinates areemployed. The symbol := means equality by definition. Other symbols will be defined in the textwhere they appear for the first time.

2. CONSTITUTIVE ASSUMPTIONS AND PROBLEM POSITION

For anisotropic materials Tsai and Wu [34], see also [35], proposed a (failure) criterion in a tensorpolynomial form. They postulated that a failure surface in six-dimensional stress space exists inthe form:

Fi�i + Fi j�i� j = 1 (i, j = 1, . . . , 6) (1)

where: Fi and Fi j are strength tensors of the second and fourth rank, respectively, and the con-tracted notation, usually adopted in this context [37] is used (i.e. �4 := �23, �5 := �31, �6 := �12).In this form it is difficult to untangle the Tsai–Wu criterion and also for that which concerns theexperimental identification of the material parameters entering the tensors Fi , Fi j pertaining toa given material. Nevertheless, as is known, composite laminates consist of many layers stackedup with different fibre reinforcement orientation. Each layer can be viewed as a unidirectionalorthotropic lamina and the orthotropic properties of individual layers result in a material that



is anisotropic. Subsequently attention is then focused on a unidirectional orthotropic laminateadding the further simplifying hypothesis of plane stress conditions. This restriction, in theauthors’ opinion, does not affect the general applicability of the whole procedure to be developedhereafter and referred to orthotropic laminates. Possible improvements are certainly conceivableif each layer of a multilayered laminate is analysed in conjunction with concepts like ‘first plyfailure’ [38] or ‘progressive failure scenario’ [39]. This point is not addressed here because it isoutside the scope of the present paper.

For a undirectional reinforced lamina in plane stress state (i, j = 1, 2, 6) the Tsai–Wu criterionsimplifies as follows (1 and 2 denoting the principal directions of orthotropy):

F11�21 + F22�

22 + F66�

26 + 2F12�1�2 + 2F16�1�6 + 2F26�2�6 + F1�1 + F2�2 + F6�6 = 1 (2)

where: Fi and Fi j (i, j = 1, 2, 6) have to be determined by tensile, compressive and shear tests.Taking into account that the unidirectional laminate is referred to its orthotropic axes and that thestrength should be unaffected by the direction or sign of the shear stress component, all the termsin Equation (2) containing first-degree shear stresses have to be neglected. Equation (2) simplifiessubsequently, i.e.:

F11�21 + F22�

22 + F66�

26 + 2F12�1�2 + F1�1 + F2�2 = 1 (3)

where, see e.g. [37]:F1 := 1

X t+ 1

Xc(4a)

F2 := 1

Yt+ 1

Yc(4b)

F11 := − 1

X tXc(4c)

F22 := − 1

YtYc(4d)

F66 := 1

S2(4e)

F12 := − 12

√F11F22 (4f)

with X t, Xc the longitudinal tensile and compressive strengths, respectively; Yt, Yc the transversetensile and compressive strengths, respectively, and S the longitudinal shear strength. As deducibleby inspection of Equations (4a)–(4e), five of the six coefficients required for the definition of thecriterion are given by performing simple tests. The sixth, namely F12, related to the interactionbetween the two normal stress components �1 and �2 requires a biaxial test. This experimentaltask is not easy to perform as the simple uniaxial or shear tests and simplified assumptions, as theone adopted herein with position (4f), are usually made [40, 41].

It is worth noting that the estimation of the Tsai–Wu strength parameters is not straightforward(see e.g. [42, 43]); moreover, considering that a failure process in a laminate involves a combinationof failure mechanisms due to matrix crushing, fibre breaks, fibre buckling, delaminations [44]



modified versions of the Tsai–Wu criterion have been presented to take into account some itsinternal incoerencies [40, 45].

Despite the above remarks the Tsai–Wu criterion, in the quadratic form adopted herein, issimple; it allows one to apply the standard rules of transformation, invariance and symmetry; italso contemplates interactions among the stress or strain components analogously to the von Misescriterion for isotropic materials. As noted by who conceived this criterion [35], failure criteria forcomposite laminates should provide a convenient framework or model for mathematical operations.The framework should remain the same for different definitions of failures, such as ultimate strength,yielding, endurance limit, etc.; the criteria are not intended to explain the complex mechanisms offailure that in composite laminates are characterized by the concurrent and sequential occurrence ofmany failure modes. With this conjecture the Tsai–Wu criterion is used for defining an admissiblestress states domain. Points within the domain locate stress state pertaining to an anisotropic linearelastic behaviour of the material. Points lying on the domain boundary locate stress states at whichthe material has exhausted its strength capabilities.

Equation (3) can be rearranged in terms of the following dimensionless parameters:

X := √F11�1 (5a)

Y := √F22�2 (5b)

Z := √F66�6 (5c)

f12 := F12√F11F22

(6a)

f1 := F1√F11

(6b)

f2 := F2√F22

(6c)

With the above positions Equation (3) reads:

X2 + Y 2 + Z2 + 2 f12XY + f1X + f2Y = 1 (7)

which, in the dimensionless space (X, Y, Z), individuates an ellipsoid whose major axis lies onZ = 0 plane and it is rotated by a counterclockwise angle of 45◦ with respect to the X -axis.Assuming that (·)TW stands for a quantity (·) pertaining to the Tsai–Wu criterion and/or surface;denoting with �TW, �TW and �TW the X , Y , Z co-ordinates of the ellipsoid centre, respectively, itis easy to verify that the following expressions hold true:

�TW = − f1 − f2 f122(1 − f 212)

(8a)

�TW = − f2 − f1 f122(1 − f 212)

(8b)

�TW = 0 (8c)



Moreover, denoting with aTW, bTW and cTW the semi-axes ellipsoid dimensions (aTW referring tothe major axis, cTW to the axis parallel to Z ) it is

aTW = [(1 + f12)/(1 + �2TW + �2TW + 2 f12�TW�TW)]−1/2 (9a)

bTW = [(1 − f12)/(1 + �2TW + �2TW + 2 f12�TW�TW)]−1/2 (9b)

cTW = [1 + �2TW + �2TW + 2 f12�TW�TW]1/2 (9c)

The Tsai–Wu surface in the shape of Equation (3), or in the equivalent dimensionlessEquation (7), is assumed as yield surface for the orthotropic material here considered. Thisassumption, unfortunately, is not sufficient to proceed further. Even though it postulates the exis-tence of a yield surface for the composite laminate, which will be treated as an elastic perfectlyplastic material, it does not imply the associativity of the yield criterion. In spite of that, in thecontext of non-standard materials, namely for soils, a theory of limit analysis was actually pro-posed by Radenkovic [9] with several modifications (see e.g. [36] and references therein). The limitanalysis fundamental theorems have in practice been restated in the shape of ‘upper’ and ‘lowerbound theorems’. Precisely, after [36], Radenkovic’s first theorem—or upper bound theorem—states: the limit loading for a body made of a non-standard material is bounded from above bythe limit loading for the standard material obeying the same yield criterion. Radenkovic’s secondtheorem—or lower bound theorem—states: the limit loading for a body made of a non-standardmaterial is bounded from below by the limit loading for the standard material obeying the yieldcriterion g(r) = 0. g(r) = const. being a convex function lying entirely within the yield surface ofthe non-standard material, say f (r) = 0, and complying with the condition that to any r at whichf (r) = 0 there corresponds on g(r) = 0 a r′ such that the plastic strain rate ep is normal at r′ tothe surface g(r) = 0 and the inequality (�i j − �′

i j )�pi j�0 holds true. Proof of the above theorems

is reported in [9] (see also [36]). After all, every value of the limit load for a non-standard bodyis located between two fixed boundaries defined by the values of the limit loads for two corre-sponding standard materials. Obviously, Radenkovic’s two theorems locate a range of collapseload multiplier values, because for non-standard materials even the uniqueness of the limit load isuncertain due to the absence of an uniqueness for the stress field.

Following the directions of the above theorems and taking into account the strict convexityof the Tsai–Wu surface, which for Radenkovic’s lower bound theorem can itself play the role ofg(r) = 0—thus satisfying condition (�i j −�′

i j )�pi j�0 always as an equality—it is therefore possible

to search for an upper and a lower bound on the collapse load multiplier with reference to theTsai–Wu surface.

2.1. Problem position: upper and lower bound multipliers evaluation

Consider a body of volume V , external surface �V , referred to a Cartesian co-ordinate system(xi , i = 1, 2, 3) and subjected to loads Pp(x), where: P is the scalar load multiplier; p(x) thereference load vector collecting all the surface force components, pi , acting on points of a portionof the body surface, namely �Vt ; for simplicity, only surface forces are considered. The remainingpart of �V , namely �Vu = �V − �Vt , is assumed to suffer displacements u= 0; plane stressconditions are also assumed. The material is, by hypothesis, orthotropic, homogeneous and witha constitutive behaviour obeying the Tsai–Wu criterion in the form given by Equation (3). In



the assumed hypothesis of associated flow rule and rewriting Equation (3) in the abridged formf (� j ) = 0 ( j = 1, 2, 6), the strain rate components at collapse can be expressed in the form:

� j = �� f

�� j(10)

where �>0 is a scalar multiplier and � j are the components of the outward normal to the yieldsurface f (� j ) = 0.

For a given distribution of compatible strain rates � j , say �cj , i.e. such that the related displacement

rates uci satisfy the condition uci = 0 on �Vu , an upper bound to the collapse limit load multiplieris given by

PUB

∫�Vt

pi uci d�V =

∫V

�yj �cj dV (11)

where: PUB denotes the upper bound multiplier; �yj the stresses at yield associated to givencompatible strain rates �cj ; u

ci the related displacement rates. The set (�cj , u

ci ) defines a collapse

mechanism.If at every point within V a stress field � j exists satisfying the condition f (� j )�0 and in

equilibrium with Pp(x) on �Vt for a value of P , say PLB, then PLB is a lower bound on thecollapse limit load multiplier.

3. THE LINEAR MATCHING METHOD FOR ORTHOTROPIC MATERIALS

The LMM, [29–33], is a programming technique involving an iterative FE-based numerical pro-cedure which performs a sequence of linear analyses on the structure made, by hypothesis, of alinear viscous fictitious material with spatially varying moduli. At each iteration an adjustment ofthe fictitious moduli is carried out so that the computed fictitious stresses are brought on the yieldsurface at a fixed strain rate distribution. This allows one to define a collapse mechanism, the relatedstresses at yield and, consequently, an upper and a lower bound to the collapse load multiplier.

In the present context the LMM utilizes a fictitious linear viscous material which is orthotropicand subjected to a distribution of imposed initial stresses. The key ideas of the proposed generaliza-tion are summarized as follows with reference to orthotropic laminates under plane stress conditions.

Let us consider the body, of volume V , made of a fictitious, linear, viscous, orthotropic material,with spatially varying moduli and suffering a distribution of imposed initial stresses. This fictitiousmaterial has a complementary dissipation rate given by

W (� j ) = 1

2

[�21E1

+ �22E2

+ �26E6

− 2�12�1�2E2

− 2

(�1E1

− �12�2E2

)�1 − 2

(�2E2

− �12�1E2

)�2 − 2

�6E6

�6

+ �21E1

+ �22E2

+ �26E6

− 2�12�1�2E2

](12)



where � j are imposed initial stresses, E j are spatially varying moduli and �12 is the Poisson ratio.For this fictitious material and at a fixed value of the load multiplier a linear analysis is performedto compute: the distribution of strain rates, �ej = �W (�ej/��ej ), related stresses, �ej , and compatibledisplacement rates, uei , of the points at which surface loads are applied. This fictitious solution iscomputed, in principle, at each point of the body volume V and, in practice, at each Gauss point(GP) of each finite element (FE) of the discretized domain V . At each GP, the fictitious kinematicsolution (�ej , u

ei ) is forced to represent a collapse mechanism, namely it is forced to identify with

(�cj , uci ) of Equation (11). To this aim, if �ej is kept fixed and assumed as �cj , it is sufficient to

compute the stress at yield associated to �cj ≡ �ej , namely �yj , by varying the fictitious moduli and

initial stresses so that �ej coincides with �yj , uei being the compatible displacements associated to

�cj . The linear material is so matched to the yield surface and this, performed to within a discretizedFE approach, is carried out in an iterative fashion.

At operative level, grounding on the formal analogy between the linear viscous problem and thelinear elastic problem, the fictitious linear solution can be computed as a fictitious elastic solution,W (� j ) of Equation (12) playing the role of complementary energy potential of a fictitious elasticmaterial. The fictitious elastic analyses, performed at each iteration, can be carried on by anycommercial FE-code rendering the whole procedure easy to be performed.

From a geometrical point of view, conceivable in plane stress hypothesis, the matching proceduremerely states that the complementary energy equipotential surface of the fictitious elastic material,W (� j ) = const., by appropriate updating of the spatially varying elastic parameters and initialstress values, is brought to be tangential to the Tsai–Wu surface at the stress point ry whoseexternal normal is ec. In Figure 1 the matching is schematically depicted with reference to theTsai–Wu surface and the W (� j ) = const. surface. The dependence of the latter surface on the setof elastic parameters and initial stress fictitious values is highlighted: (·)(0) denoting an arbitraryinitial set of such values; (·)(∗) denoting the modified values which achieve the matching.

Figure 1. Matching procedure at the generic Gauss point: geometrical sketch in the �6 = 0 plane.



In the present case, such matching is easily obtainable by taking advantage of the ellip-soidal shapes of the Tsai–Wu surface and of the equipotential surface W (� j ) = const.; thesesurfaces can, in fact, be made coincident if the fictitious material is, from the beginning,endowed with a complementary energy equipotential surface homothetic to the Tsai–Wu sur-face. To this aim, rewriting Equation (12) in the dimensionless space (X, Y, Z), i.e. usingEquations (5a)–(5c), it is

W (X, Y, Z) = 1

2

[X2

E1F11+ Y 2

E2F22+ Z2

E6F66− 2�12

E2√F11

√F22

XY

− 2√F11

(X

E1√F11

− �12Y

E2√F22

)X− 2√

F22

(Y

E2√F22

− �12 X

E2√F11

)Y− 2Z

E6F66Z

+ X2

E1F11+ Y 2

E2F22+ Z2

E6F66− 2�12

E2√F11

√F22

X Y

](13)

For a given load multiplier initial value, say P(0)UB, and any fixed set of elastic parameters and initial

stresses, namely (E (0)1 , E (0)

2 , E (0)6 , �(0)

12 , X (0), Y (0), Z (0)), the above expression individuates in the

(X, Y, Z) space an ellipsoid of the form W [E (0)j , �(0)

12 , (0)j ] = W (0) ≡ const. The latter abridged

form points out the dependence of the ellipsoid location and amplitude on the elastic parametersand initial stress values, W (0) being the pertinent complementary energy equipotential value corre-sponding to the given loads. For brevity, j for j = 1, 2, 6 identifies with X, Y, Z , respectively. If

the initial choice is made by imposing that W [E (0)j , �(0)

12 , (0)j ] = W (0) is homothetic to the Tsai–Wu

surface given by Equation (7), i.e.: the semi-axes ratios are equal (three conditions); the two ellip-soids have the same centre (three conditions) and the main axis is rotated by a counterclockwiseangle of 45◦ with respect to the X axis (one condition); it easy to verify that the following positionshold true:

E (0)1 = 1

2F11(14a)

E (0)2 = 1

2F22(14b)

E (0)6 = 1

2F66(14c)

�(0)12 = − f12

√F11√F22

(14d)

X (0) = �TW (15a)

Y (0) = �TW (15b)

Z (0) = 0 (15c)



Figure 2. Matching procedure at the generic Gauss point for W [E (0)j , �(0)

12 , (0)j ] = W (0) homothetic to T–W

surface: geometrical sketch in the Z = 0 plane.

With the above positions Equation (13) reduces to

X2 + Y 2 + Z2 + 2 f12XY − 2(�TW + f12�TW)X − 2(�TW + f12�TW)Y

= W (0) − �2TW − �2TW − 2 f12�TW�TW (16)

which is the searched complementary energy equipotential surface homothetic to the Tsai–Wusurface. The matching and therefore the whole procedure is now more easily realizable actingonly on the elastic moduli values of the fictitious material that control the axes amplitude of theellipsoid given by Equation (16). In Figure 2 a geometrical sketch of such matching is given inthe Z = 0 plane and at a generic GP.

Looking at the sketch of Figure 2, the following can be stated: an elastic analysis on the structureloaded by P(0)

UB pi and made of a fictitious material whose complementary energy is given by (16)

produces, at each GP, an elastic solution of the form (e(0)j , �e(0)j ), (point A in Figure 2), lying

on the surface W [E (0)j , �(0)

12 , (0)j ] = W (0). Assuming �e(0)j as �cj the (adimensionalized) stress at

yield, y(∗)

j , associated to the (normal) �cj is computed (point B in Figure 2). The fictitious elastic

solution, e(0)j , is then forced to identify with the one at yield, namely e(∗)j ≡ y(∗)

j , by rescalingthe fictitious elastic moduli.



In detail, based on the homothetic condition, a rescaling of the fictitious elastic moduli can becarried out on setting:

E (∗)j = E (0)

j �

W (0), j = 1, 2, 6 (17)

while keeping the load fixed, namely P(0)UB, the initial stresses (0)

j , the Poisson coefficient �(0)12 and

W (0). In Equation (17), � := 1+�2TW +2 f12�TW�TW +�2TW>0 is the known term of the Tsai–Wuellipsoid equation rewritten in a cartesian reference system with the origin at the ellipsoid centreand equipollent to the (X, Y, Z) system. It is easy to verify that on substituting (17) in (13) theT–W ellipsoid, given by Equation (7), is obtained.

On the other hand, the stresses at yield, y(∗)

j , can be computed (again referring to Figure 2)with the following equations:

Xy(∗) = [1 − �(0)]�TW + �(0)X e(0) (18a)

Y y(∗) = [1 − �(0)]�TW + �(0)Y e(0) (18b)

Zy(∗) = �(0)Z e(0) (18c)

where, for clarity, all the stress components, y(∗)

j for j = 1, 2, 6, have been explicitly reported;

�TW and �TW are given by Equations (8a), (8b) and �(0) denotes the homothety ratio between thetwo ellipsoids, namely:

�(0) := aTW

a(0)W

= bTW

b(0)W

= cTW

c(0)W

(19)

In Equation (19): aTW, bTW, cTW are the lengths of the T–W ellipsoid semi-axes given byEquations (9a)–(9c) and a(0)

W , b(0)W and c(0)

W are the analogous quantities for the ellipsoid

W [E (0)j , �(0)

12 , (0)j ] = W (0). It easy to verify that

�(0) =√

�

W (0)(20)

However, the stresses at yield �y(∗)

j , given by Equations (18a)–(18c) by application of

Equations (5a)–(5c), will not satisfy the equilibrium conditions pertaining to the loads P(0)UB pi

but, remembering Equation (11), they will satisfy the equilibrium requirements for loads piamplified by

P(∗)UB =

∫V �y(∗)

j �cj dV∫�Vt pi u

ci d�V

(21)

that is the load multiplier value pertinent to the E (∗)j distribution actuating the matching at each GP.

A new elastic analysis, performed with loads P(∗)UB pi and E (∗)

j distribution of Equation (17), will

give at each GP a fictitious elastic solution lying on the surface W [E (∗)j , �(0)

12 , (0)j ] = W (∗), the latter



obviously will not coincide with W [E (∗)j , �(0)

12 , (0)j ] = W (0). The rationale can then be repeated in

an iterative fashionmaking use, at the kth elastic analysis, of an elastic moduli distribution given by

E (k)j = E (k−1)

j �

W (k−1), j = 1, 2, 6 (22)

and for loads amplified by

P(k)UB =

∫V �y(k−1)

j �c(k−1)j dV∫

�Vt pi uc(k−1)i d�V

(23)

The recursive formulae for stresses at yield can easily be derived by looking at Equations(18a)–(18c) and, remembering Equations (5a)–(5c), they read

�y(k)1 = Xy(k)

√F11

(24a)

�y(k)2 = Y y(k)

√F22

(24b)

�y(k)6 = Zy(k)

√F66

(24c)

where

Xy(k) = [1 − �(k)]�TW + �(k)X e(k) (25a)

Y y(k) = [1 − �(k)]�TW + �(k)Y e(k) (25b)

Zy(k) = �(k)Z e(k) (25c)

Finally, the homothety ratio can be given the recursive expression:

�(k) := aTW

a(k)W

=√W (k−1)

W (k)(26)

By substituting (22) in (13) it is easy to verify that the following expression is obtained (at thegeneric GP):

X2 + Y 2 + Z2 + 2 f12XY + f1X + f2Y = 1 + �

[W (k)

W (k−1)− 1

](27)

the latter, in few iterations, identifies with the T–W Equation (7) and this when, at a certain k,W (k) identifies with W (k−1) and the matching is accomplished. At this k it is also �(k) = 1 andy(k)j ≡ e(k)j as it has to be and as deducible by Equations (25a)–(25c) and (26). The iterative

procedure is monitored by means of the computed P(k)UB value, i.e. it stops when the difference

|P(k)UB − P(k−1)

UB | is less than a fixed tolerance.



Concerning the lower bound collapse multiplier, PLB, the LMM, even in its original formulation(see e.g. [29]), provides a method for an approximate evaluation of such a bound. This issue, withdifferent approaches, has been addressed in [46] (see also references therein) and in [20] until,although the list is not exhaustive, the recent contribution given in Hamilton and Boyle [47].Hereafter the PLB evaluation is supplied within the above discussed iterative procedure and, inthis sense, it supplements the extension to structures made of orthotropic materials of the originalversion of the LMM.

At each iteration and at each GP, the fictitious stresses ve(k), pertinent to loads P(k)UB pi and

Young moduli distribution E (k)j , are located in the (X, Y, Z) space; see also the schematic sketch

given in Figure 3 for three generic GPs in the Z = 0 plane. Among all the stress points ve(k) thusobtained the one further away from the T–W surface is detected, say ve(k)F (point A in Figure 3),and this merely by computing the Euclidean distances from the T–W ellipsoid center. The ratio(k) between the yield stress value measured on the direction ve(k)F /|ve(k)F |, say vy(k)F (point B in

Figure 3), over the stress value ve(k)F allows one to define a lower bound given by

P(k)LB = (k)P(k)

UB (28a)

Figure 3. Fictitious elastic stresses ve(k) at iteration kth and at three generic Gauss points: evaluation ofthe factor (k) for the P(k)

LB computation.



with

(k) := |vy(k)F ||ve(k)F |

(28b)

a rescaling of the applied loads P(k)UB pi by <1 implies that all the fictitious elastic stresses (k)ve(k)

satisfy the admissibility conditions of the static approach for limit analysis.

3.1. The iterative procedure

The whole procedure is summarized hereafter in flow-chart style recalling also the equations tobe utilized in an FE implementation and pointing out a number of operating choices.

• Initialization:

Step #i: Knowing the strength values of the (real) constituent material (Xc, X t, Yc, Yt, S),assign to all FEs in the mesh the initial set of fictitious elastic parameters and initialstresses, namely:

E (0)1 = 1/(2F11), E (0)

2 = 1/(2F22), E (0)6 = 1/(2F66), �(0)

12 = − f12√F11/

√F22

�(0)1 = �TW/

√F11, �(0)

2 = �TW/√F22, �(0)

6 = 0

Step #ii: Set k = 1, P(k−1)UB = P(0)

UB = 1 (for k = 1, P(0)UB can be any arbitrary value) and compute

�= 1 + �2TW + 2 f12�TW�TW + �2TW for later use

• Iteration loop:

Step #1: Perform a fictitious elastic analysis with elastic parameters E (k−1)j , �12 = �(0)

12 , initial

stresses � j = �(0)j and with loads P(k−1)

UB pi , computing a fictitious elastic solution,namely:

�e(k−1)j , u(k−1)

i , �e(k−1)j

· · · at Gauss point levelStep #2: Compute the adimensionalized stresses e(k−1)

j and evaluate (k−1)

Step #3: Compute the value of the complementary potential energy

W (k−1) = 12�

e(k−1)j �e(k−1)

j

Step #4: Compute the homothety ratio, namely

�(k−1) =

⎧⎪⎨⎪⎩

√�/W (0) for k = 1√W (k−2)/W (k−1) for k>1



Step #5: Evaluate stresses at yield:

�y(k−1)1 = [1 − �(k−1)] �TW√

F11+ �(k−1)�e(k−1)

1

�y(k−1)2 = [1 − �(k−1)] �TW√

F22+ �(k−1)�e(k−1)

2

�y(k−1)6 = �(k−1)�e(k−1)

6

Step #6: Compute the E (k)j distribution to be utilized, if necessary, at next iteration, namely:

E (k)j = E (k−1)

j �

W (k−1)

· · · end Gauss point levelStep #7: Set �c(k−1)

j = �e(k−1)j , uc(k−1)

i = ue(k−1)i and evaluate the upper bound multiplier

P(k)UB =

∫V �y(k−1)

j �c(k−1)j dV∫

�Vt pi uc(k−1)i d�V

Step #8: Evaluate lower bound multiplier

P(k−1)LB = (k−1)P(k−1)

UB

Step #9: Plot P(k)UB and P(k−1)

LB versus iterations numberStep #10: Check for convergence

|P(k)UB − P(k−1)

UB |�TOL

⎧⎪⎪⎨⎪⎪⎩YES ⇒ EXIT

NOT ⇒ set(·)(k−1) = (·)(k)

and GOTO step #1

By inspection of the above ten-step procedure the whole analysis, also in this extended versionfor orthotropic materials, is easy to implement and, like the original LMM, can be carried out byany commercial FE code suitably fed, at each iteration, by the fictitious elastic moduli distributionaccomplishing the matching at each GP. Nevertheless some remarks have to be made.

As noted in [29], the correctness of the PUB depends on the kinematic description of thediscretized problem and it is then related to the adopted FE mesh. In this sense the PUB convergesto the minimum upper bound allowed by the class of displacement fields given by the mesh itself.This drawback is easily overcome by using fine meshes in the analysis.

The rationale followed for the PLB evaluation, on the other hand, gives a lower bound to theabove minimum upper bound; Equation (28a) yields, in fact, a pseudo-lower bound. Moreover, thePLB, as evaluated above, appears to be too conservative because it depends on only one stress valueattained at one GP in the whole mesh (the one further away from the T–W surface). Nevertheless,the static approach of limit analysis essentially states that the structure rearranges the internal



stresses to its best possible advantage to withstand the applied loads. Based on this concept, theGP stress values measured to within the single element where ve(k)F has been attained at one GPcan be averaged. It would obviously be incorrect to average across elements since the elasticmoduli of adjacent elements are different. On this averaged stress value Equation (28b) can thenbe applied (at step #2) to evaluate an averaged , say . A weighted lower bound, say PWLB, isthen computed (at step #8), namely:

P(k)WLB = (k)P(k)

UB (29)

A third remark regards the evaluation of the spatially varying elastic moduli E (k)j at step #6. As

stated, Equation (22) yields the updated E (k)j values at each GP in the FE mesh. However, to avoid

accuracy problems, in a FE procedure a unique set of E j is assigned to each single element and

so the E (k)j evaluated at matching on the GPs of each element have been averaged to within the

element itself.A final remark concerns the convergence of the iterative procedure. A theoretic proof of the

convergence for the upper bounds sequence, in the case of Von Mises materials, was givenin [29]; a sufficient condition was then given in [31] for a general class of yield conditionspertinent to pressure dependent materials. To the authors’ knowledge, no such proof exists forthe lower bounds sequence. The convergence of the upper bounds sequence is here assuredby the sufficient condition given in [31] for a general class of yield conditions including theone here adopted. The upper bound FE solution converges, in fact, to the least upper boundcontained within the class of mechanisms described by the FE mesh as it is witnessed by thesatisfactory numerical results obtained. These are presented in the next Section for two nu-merical examples, the first one admitting an explicit analytical expression for the upper boundmultiplier.

4. NUMERICAL EXAMPLES

Two numerical examples have been solved to verify the effectiveness of the proposed approach. Thefirst one, taken from [24], has been considered for the sake of comparison. In the quoted paper,following a different approach, an explicit analytical expression of the upper bound multiplieris provided for a simple problem, namely a square plate under plane stress conditions and, byhypothesis, made of a tetratropic material. Despite its simplicity, the example carried out in theabove mentioned paper presents an interesting sensitivity analysis on the influence of the ratiobetween tensile and compressive strength as well as the ‘degree of orthotropy’ of the material. Thesecond example, which analyses the same simple structural plane problem, envisages the case ofa fully orthotropic material and shows all the potentialities of the proposed approach that, even ifup until now has been confined to the simpler context of plane stress conditions, seems to be ofquite general applicability in the realm of orthotropic material structures.

All FE elastic analyses were performed with the ADINA code [48], suitably interfaced witha FORTRAN main programme which controls the iterative procedure described in Section 3.1thus performing the matching at GP level and, in practice, feeding the FE code with theappropriate input parameters at each step. The value of TOL= 10−4 was utilized. In both ex-amples a mesh of 50 isoparametric 16 nodes quadrilateral elements with 16GPs per element wereutilized.



Pσ0

a

ax

y1

2

θo

Pσ0

Figure 4. Example #1—square plate under plane stress conditions made of a tetratropic material.

4.1. Example #1

The square plate of side a and unit thickness shown in Figure 4 is uniformly loaded along twoopposite edges while preventing transverse displacements on the remaining edges (i.e. along yreferring to Figure 4). The load per unit length is specified as P�0, where �0 is a given referencestress value and P is a scalar load multiplier.

The plate, under plane stress conditions, is referred to a Cartesian co-ordinate system (x, y)while the principal directions of orthotropy are individuated by the Cartesian axes (1, 2). �defines the counterclockwise angle between axis 1 and axis x along with the applied loadsact. As stated, the material is tetratropic, i.e. Xc ≡ Yc and X t ≡ Yt, and the material consid-ered in [24] obeys the Tsai–Wu criterion given here by Equation (3) with the followingpositions:

F11 = F22 = 1

1 − �21

�20(30a)

F66 = 3�1

1 − �21

�20(30b)

F12 = − 1

2(1 − �2)

1

�20(30c)

F1 = F2 = �

1 − �21

�0(30d)

In Equations (30a)–(30d): � is the ratio between tensile and compressive material strength, so itrepresents the ‘degree of symmetry’ of the behaviour of the material (� = 0 pertains to a symmetricbehaviour); � defines the ‘degree of orthotropy’ of the material (� = 1 means isotropy). (Refer tothe above mentioned paper for further details on these two parameters.)



0 5 10 15 20 25 30 35 40 450.0

0.4

0.8

1.2

1.6

2.0

Capsoni et al. [24]Present approach

β = 0

β = 0.5

η = 0.25

β = 0.25

P UB

θ [deg](a) (b)

0 5 10 15 20 25 30 35 40 450.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Capsoni et al. [24]Proposed approach

β = 0

β = 0.5

η = 4

β = 0.25

P UB

θ [deg]

Figure 5. Example #1—upper bound multiplier, PUB, versus angle � (in degree) between the orthotropyaxis 1 and the loading direction x : results given by Capsoni et al. [24] (solid lines) and results obtained bythe present approach (lines with diamonds). Plots for �=0; 0.25; 0.5 and: (a) at �=0.25; and (b) at �=4.

0 5 10 15 20 25 30 35 40 450.0

0.3

0.6

0.9

1.2

1.5

1.8

β = 0η = 0.25

PUB

PWLB

β = 0.25

β = 0.5

load

mul

tipl

ier

0 5 10 15 20 25 30 35 40 450.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

β =

4

0

η =

β = 0.25

β = 0.5

load

mul

tipl

ier

(b)(a)

PUB

PWLB

θ [deg] θ [deg]

Figure 6. Example #1—upper bound (solid lines) and weighted lower bound (dashed lines) multipli-ers versus angle � (in degree) between the orthotropy axis 1 and the loading direction x . Plots for

�= 0; 0.25; 0.5 and: (a) at �= 0.25; and (b) at �= 4.

In Figures 5(a) and (b) the results obtained with the proposed approach are compared with thoseof [24] in terms of upper bound multiplier, PUB, versus the angle � and for different values of �and �; precisely for � = 0.25 and 4 the values of � = 0, 0.25, 0.5 were considered. For the same setof � and � values, Figures 6(a) and (b) show the PUB and the PWLB versus angle �. In Figures 7(a)and (b) the PUB, PLB and PWLB sequences are plotted versus iteration number again for � = 0.25and 4 and for �= 0.25 and � = 15◦.

By inspection of Figures 5(a) and (b) the results obtained clearly show a highly satisfactoryagreement with those of [24]. The weighted lower bound values, as shown by Figures 6(a) and (b),



54321 43216 70.0

0.2

0.4

0.6

0.8

1.0

1.2

η = 0.25PUBPWLB β = 0.25

θ = 15

iteration number

load

mul

tipl

ier

PLB

5 60.0

0.2

0.4

0.6

0.8

1.0

1.2

η = 4β = 0.25θ = 15

iteration number

load

mul

tipl

ier

(a) (b)

PUBPWLBPLB

Figure 7. Example #1—upper bound (solid lines), weighted lower bound (dashed lines) and lower bound(dotted lines) multipliers versus iteration number for �=0.25, �=15◦ and: (a) at �=0.25; and (b) at �=4.

are always quite close to the correspondent upper bounds. The gap between the two multipliersdetermined this way is always very narrow and this, which is obviously a consequence of the use ofmean stress values proposed, appears to be a good tool for characterizing the load bearing capacityof the analysed structure. Finally, Figures 7(a) and (b), highlighting the differences between themultiplier values determined, show that the procedure converges rapidly. These positive remarksare mitigated both by the simplifying hypotheses on which the whole approach is based and bythe simplicity of the case-study carried out; nevertheless, as shown by the next example, they alsohold good for a more realistic situation.

4.2. Example #2

The square plate of side a and unit thickness of the previous example is, by hypothesis, madeof a unidirectional laminate with fibres directed along x (i.e. axis 1≡ axis x and axis 2≡axis y). As sketched in Figure 8, the plate is loaded along two opposite edges by a triangularshaped distributed load, while the transverse displacements (along y) are prevented on the otheredges. Two different composite materials were considered, namely: graphite/epoxy (T300/5208)and boron/epoxy (B(4)/5505). Figure 8 shows the reference system, geometry, boundary andloading conditions while the material data, in terms of strength values, are given in Table I. Asbefore, plane stress conditions apply but, in this case, (full) orthotropy is exhibited.

In Figures 9(a) and (b) the upper and weighted lower bound multiplier sequences are givenversus the number of iterations for the two composite materials assumed in the analyses. All theremarks made about the results of example #1 are obviously applicable here showing a satisfactoryperformance of the proposed approach also in this more general case of materials with an orthotropicbehaviour. A further comment concerns the vicinity of the PUB and PWLB values at convergence,exhibited here by the curves of Figures 9(a) and (b). Such closeness, as also indicated in the plotsof Figures 7(a) and (b), against the gap between the upper and the lower (without averaging) boundmultipliers confirms the validity of computing a weighted lower bound whose value does not depend



pk

a

ax

y

o

DATA

a = 10 cm

0p = 20 MN/cm

k = scalar multiplier

Figure 8. Example #2—square plate under plane stress conditions made of a unidirectional laminatewith fibres parallel to axis x .

Table I. Strength values (in GPa) of the composite materials utilized.

Material Xc Yc X t Yt S

Graphite/epoxy (T300/5208) 1.500 0.246 1.500 0.040 0.068Boron/epoxy (B(4)/5505) 2.500 0.202 1.260 0.061 0.067

5 6 7 81.0

1.1

1.2

1.3

1.4

1.5

1.6

PUBPWLB

iteration number(a) (b)

load

mul

tipl

ier

543214321 6 7 8 9 100.5

0.6

0.7

0.8

0.9

1.0

1.1

PUBPWLB

iteration number

load

mul

tipl

ier

Figure 9. Example #2—upper bound (solid lines) and weighted lower bound (dashed lines) multipliersversus iteration number for: (a) graphite/epoxy T300/5208; and (b) boron/epoxy B(4)/5505.

from only one stress value in the whole mesh. Experimental investigations, i.e. laboratory tests onreal prototypes, would be necessary to verify the reliability of the obtained bounds and they arethe object of an ongoing research.



5. CONCLUDING REMARKS

The paper presented a numerical approach for limit analysis of structures made of orthotropicmaterials. In particular, orthotropic composite laminates under plane stress conditions were anal-ysed. The proposed approach can be depicted as an extension, in the wider context of orthotropicmaterials, of the LMM, originally applied for limit analysis of structures made of von Mises typematerials [29] and, successively, applied to geotechnical problems or with reference to materialswhose behaviour is pressure dependent [16, 31].

The LMM basically solves a sequence of linear analyses defined by spatially varying (oradjusting) the moduli of a fictitious linear viscous material the structure is assumed to be madewith. The adjustment is made in an iterative fashion and to within a FE scheme so that thecomputed fictitious stresses at each Gauss point are brought on the yield surface at a fixed strainrate distribution. In this way, at each step, a collapse mechanism can be defined and consequentlyan upper bound collapse multiplier can be evaluated. A definition of a pseudo-lower bound isstraightforward. Moreover, using the formal analogy between the linear viscous problem and thelinear elastic one, at each step and at each point in the domain (GP in a FE context), the methodimplies that the complementary energy equi-potential surface W = W of a fictitious linear elasticmaterial matches the yield surface at a fixed point, the latter representing the fictitious linearsolution.

The extension proposed here is based on three main assumptions, namely: (i) at a constitutivelevel, the material obeys a second-order tensor polynomial form of the Tsai–Wu criterion forcomposite laminates [34] which is assumed to be the yield condition; (ii) at a global analysislevel, the upper and lower bound collapse multipliers are evaluated in the sense of limit analysisfor non-standard materials [9, 36]; (iii) at an operating level, the fictitious linear elastic materialis orthotropic and is suitably defined so that its complementary energy equipotential surface ishomothetic to the Tsai–Wu surface.

The first assumption, criticizable for composite exhibiting limited ductility, is instead adoptablefor a wide class of composite laminates, like plastics reinforced by kevlar for example, undergoingplastic flow and considerable ductility; it is certainly valid in a wider context of orthotropic ductilematerials. The quadratic form of the Tsai–Wu criterion is easily handled allowing the applicationof standard rules of transformation, invariance and symmetry. The second assumption has thepeculiarity that, within the Radenkovic approach for non-standard limit analysis, the adopted T–Wsurface plays the double role of inner and outer surface to the yield criterion itself. The upper andlower bound multipliers can then be evaluated by referring to the same surface, namely the T–Wsurface. The third assumption, finally, has the benefit of reducing the number of elastic parametersto be updated for effecting the matching so improving the computational efficiency of the wholeprocedure.

The convergence of the upper bounds sequence is assured by the sufficient condition givenin [31] for a general class of yield conditions as confirmed by the numerical results obtainedalso for a realistic case-study carried out. Although plane stress conditions were analysed forsimple unidirectional composite laminates, possible extensions to multidirectional laminates, inconjunction with approaches like first ply failure, seem to be applicable; this is certainly asuggestion for future research. A validation of the presented approach by means of labora-tory tests is also desirable. To conclude, the proposed procedure was found to be a computa-tionally efficient tool for locating the load bearing capacity of the typology of structures hereanalysed.



ACKNOWLEDGEMENTS

The financial support of the Italian Ministero dell’Istruzione, dell’Universita e della Ricerca (MIUR) isgratefully acknowledged.

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A Numerical Approach for Limit Analysis of Orthotropic Composite Laminates

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