Structural rigidity optimization of thin laminated shells

Composite Structures 95 (2013) 35–43

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Structural rigidity optimization of thin laminated shells A. Jibawy a,c, B. Desmorat a,b,⇑, A. Vincenti a a

IJLRA, UPMC Univ. Paris 06/CNRS UMR 7190, France Univ. Paris-Sud 11, France c Segula Technologies, France b

a r t i c l e

i n f o

Article history: Available online 31 July 2012 Keywords: Laminate Shell Compliance Angle ply

a b s t r a c t In this work, we present an optimization methodology used in order to optimize laminated composite shell structures with variable stiffness. Considering the maximization of the structural global rigidity measured by the compliance, a topology optimization problem of the anisotropy fields for thin laminated plates as well as the associated optimization algorithm are extended to thin laminated shells. Numerical examples with quasi-homogeneous angle-ply stacking sequences showing the optimization methodology feasibility are presented. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Laminated composite shell optimization is a research domain that can be split by considering diverse design objectives and parametrizations related to constant or variable stiffness designs [1,2]. Considering the parametrization, two opposite approaches use optimization parameters related to the homogenized shell behaviour (lamination parameters [3,4], polar parameters [5,6]) or related directly to the stacking sequence (e.g. by considering the angles of the plies in the stacking sequence). The weight can be considered as an optimization criterion for shape optimization (shape of the midsurface) or topology optimization (distribution of the thickness) of the shell. A design objective for shell structures may be the fundamental eigenfrequency [7,8]. A third important shell design problem is the maximization of the buckling load [9–11]. Usually, this problem is related to thin shells, as when the thickness increases, the design objective may become the ply failure or delamination, or more generally damage in the laminated composite [7]. A fifth design objective is the shell structural rigidity optimization [12–14]. This objective is related to the wish of the smallest variation of the shell shape with respect to one or multiple loadings, e.g. for the use of the shell in an air or liquid flow. This five types of optimization objectives can be considered at the same time if needed by choosing an optimization objective and multiple constraints or by doing multi-criteria optimization [15]. The present paper interest is in variable stiffness design (anisotropy distribution) of laminated composite shell for ⇑ Corresponding author at: IJLRA, UPMC Univ. Paris 06/CNRS UMR 7190, France. E-mail addresses: [email protected] (B. Desmorat), angela.vincenti@ upmc.fr (A. Vincenti). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.07.014

structural rigidity optimization, assuming that this objective is of greater interest than the others in terms of structure functionality, and that the obtained optimal layup will be verified in order to satisfy buckling and/or damage and/or frequency constraints in a second stage. The proposed optimization methodology is then related to a preliminary design phase (it has to be a very efficient numerical tool) and must be able to treat any 3D surface geometry (and then unstructured meshes). Considering the compliance as a measure of the global rigidity of the shell structure, a new result demonstrated in this paper is the extension of the compliance optimization problem formulated as a minimum–minimum problem with respect to optimization parameters and statically admissible fields [16,17] to the case of laminated shells, and the definition of the associated minimal set of assumptions. In this framework, the numerical tool used for the examples considers quasi-homogeneous angle-ply stacking sequences: it leads to a very high numerical efficiency while the use of a more general set of stacking sequences is still under study. In a first section, we introduce the elasticity problem and the variational formulation of a shell structure under the small perturbation assumption, considering symmetric generalized forces and moments and assuming uncoupled behaviour law of the layup. In a second section, we write the optimization problem and algorithm for this shell structure as an extension of a minimum–minimum formulation of compliance optimization. In a third section, we show which assumptions are necessary to use this formulation: the membrane and bending behaviours are uncoupled and the homogenized behaviour does not depend on the shell curvature. In order to derive a semi-analytical solution for the sensitivity analysis, we consider specific stacking sequences that are quasi-homogeneous and have isotropic transverse shear behaviour. Finally, two numerical examples are presented: a shell assembly (cylinder with an

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inner floor) and a 3D elliptic shell geometry with an unstructured mesh. 2. Thin laminated shell model In this section, we introduce necessary notations of the description of the shell geometry. Then we present the elasticity problem in terms of symmetric generalized forces and moments to finally derive the variational formulations used in the optimization problem and algorithm [18,19]. 2.1. Description of the shell geometry A point P inside the shell C, close to the midsurface S, is defined using the point M of the midsurface and the curvilinear coordinates (n1, n2, n3):

OP ¼ Uðn1 ; n2 ; n3 Þ ¼ OMðn1 ; n2 Þ þ n3 a3 ðn1 ; n2 Þ

ð1Þ

At each point M of the midsurface S, the covariant base of the tangent plane to S is defined:

@OMðn1 ; n2 Þ aa ¼ @na

ða ¼ 1; 2Þ

a1 ^ a2 a3 ¼ ka1 ^ a2 k

ð2Þ

i 1hb b ab ¼ uajb  bab u3 b ba with U U ab þ U 2 1 b ma qab ðu; hÞ ¼ ½bh ab þ bh ba  with bh ab ¼ hajb  bma U 2 k fa ðu; hÞ ¼ ha þ u3;a þ ba uk

cab ðuÞ ¼

ua;b Ccab

(Ccab

ð10Þ ð11Þ ð12Þ

 ac

in which uajb ¼ ¼ aa;b are the surface Christoffel symbols). Let’s note rab the components of the stress tensor in the 3D covariant base (at the point P). The application of the virtual work principle leads to the definition of the symmetric internal generalized stress tensors in membrane N, in bending M and in transverse shear Q. Their components in the covariant base of the midsurface S are

Nab ¼

Z

rab l dn3 Mab ¼

H

Z

n3 rab l dn3

Qa ¼

Z

H

ra3 l dn3

ð13Þ

H

in which H denotes the thickness of the shell. The application of the virtual work principle also leads to the definition of the local equilibrium equations

h i a a ab Njb  bk M kb  bb Q b þ pa ¼ 0 jb

ab  Q a þ Ma ¼ 0 M jb

ð14Þ

This base (a1, a2, a3) is in general not orthogonal nor normed. The contravariant base (a1, a2, a3) of the midsurface is such that

Q aja þ bab Nab  cab M ab þ p3 ¼ 0

aa  ab ¼ dba ;

and of the boundary conditions in terms of forces on the part @ S1 of the external shape of S:

a; b ¼ 1; 2

ð3Þ

The second fundamental form (symmetric) of the midsurface S, called curvature tensor of S, is defined by

h i a N ab  bk Mkb mb ¼ T a

b ¼ bab aa  ab

aa þ p3~ a3 and Ma~ aa are the surface forces and moments in which pa~ aa is the normal vector to the applied on the midsurface, ~ m ¼ ma~ external shape of the midsurface, T a~ aa þ T 3~ a3 and C a~ aa are the line forces and moments applied on the part @S1 of the external shape of the midsurface. Clamped boundary conditions in terms of displacements on the part @S0 of the external shape of S are also considered:

a

b

¼ bb aa  a

with bab ¼ aa;b  a3 a

ð4Þ

a

with bb ¼ a;b  a3

ð5Þ

At each point P of the shell, the 3D covariant base (g1, g2, g3) is defined as

ga ¼

lka

 @OP  k k ¼ da  n3 ba ak @na

g3 ¼

@OP @n3

¼ a3

ð6Þ

Let’s note l the determinant of the surface mixed tensor   k ¼ dka  n3 ba . The elementary volume dV around the point P

Q a ma ¼ T 3

M ab mb ¼ C a

ui ¼ 0 ha ¼ 0

ð15Þ

ð16Þ

2.3. Laminated shell behaviour law

of the shell is defined by

dV ¼ l dn3 dS

ð7Þ

pffiffiffi in which dS ¼ a dn1 dn2 is the elementary surface around the point M of the midsurface.

ð17Þ

Hijkl ¼ Hklij ¼ Hjikl ¼ Hijlk

The Reissner–Mindlin kinematic assumption reads

ð18Þ

and the monoclinic symmetry imply the supplementary conditions

with uðn1 ; n2 Þ

¼ ui ðn1 ; n2 Þai

ð8Þ

The shell model is obtained by a linearisation with respect to n3 of the strain tensor obtained using the kinematic assumption (8). The linearised strain tensors reads then

e ¼ eab ga  gb

r¼H:e

The components Hijkl of H in the 3D covariant basis (at the point P) satisfy the symmetry conditions

2.2. Shell local constitutive equations

Uðn1 ; n2 ; n3 Þ ¼ uðn1 ; n2 Þ þ n3 ha ðn1 ; n2 Þaa

Each ply of the laminated shell is made of a linear elastic material with a monoclinic symmetry. The associated behaviour law is

8 3 > < eab ¼ cab ðuÞ þ n qab ðu; hÞ ða; b ¼ 1; 2Þ with 2ea3 ¼ fa ðu; hÞ > : e33 ¼ 0

H3bcd ¼ H333d ¼ 0

ð19Þ

Assuming that the transverse stress r33 is equal to zero, the behaviour law (17) reads

(

rab ¼ C abcd ecd ra3 ¼ Cb ab 2eb3

with

C abcd ¼ Habcd  H b ab ¼ Ha3b3 C

33cd

Hab33 H3333

ð20Þ

Integrating over the thickness Eq. (13), the shell behaviour law reads

ð9Þ in which the components of the membrane strain tensor cab, the bending strain tensor qab and the shear strain tensor fa are defined by

N ¼A:cþB:q M ¼B:cþD:q

ð21Þ

in which the membrane stiffness A, the coupling stiffness B and the bending stiffness D reads

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A. Jibawy et al. / Composite Structures 95 (2013) 35–43

8 abcd R abcd A ¼ HC l dn3 > > < R Babcd ¼ H C abcd n3 l dn3 > > : abcd R abcd 3 2 D ¼ HC ðn Þ l dn3

ð22Þ

The transverse shear behaviour law is considered in the following form:

Q ¼ kF  f

ð23Þ

Relation between compliance L(u, h) and complementary energy J(N, M, Q) If (u, h) and (N, M, Q) are solutions to the elasticity problem ðPÞ, then

Lðu; hÞ ¼ 2JðN; M; Q Þ

ð30Þ

3. Optimization problem and algorithm

with

F ab ¼

Z

b ab l dn3 C

ð24Þ

H

k is the shear coefficient factor, that is assumed to be independent of the shell curvature and of the stacking sequence (this coefficient is equal to 5/6 if we consider a transverse shear behaviour that is homogeneous along the thickness of the shell). In the behaviour law (21), we will assume through the rest of the paper that the membrane and bending behaviour are uncoupled:

B¼0

ð25Þ

(necessary assumptions related to this choice will be presented in Section 4.1). The inverse behaviour law reads then

1 k

c ¼ a : N q ¼ d : M f ¼ fQ

ð26Þ

d ¼ D1

f ¼ F 1

ð27Þ

The elasticity problem ðPÞ is defined by Eqs. (14)–(16), (21), and (23). Let’s define the kinematically admissible displacements field Uad, i.e. satisfying the boundary conditions in terms of displacements (16), and the statically admissible stress field Rad, i.e. satisfying the equilibrium Eq. (14) and the boundary conditions in terms of forces (15). Variational formulation in terms of displacements If (u, h) is a displacement field solution to the elasticity problem ðPÞ then

ðu; hÞ 2 U ad aððu; hÞ; ðu0 ; h0 ÞÞ ¼ Lðu0 ; h0 Þ 8ðu0 ; h0 Þ 2 U ad

ð28Þ

with 0

0

aððu; hÞ; ðu ; h ÞÞ ¼

Z Z

ðA

S

abcd

cab ðuÞccd ðu Þ

ab

þ kF fa ðu; hÞfb ðu0 ; h0 ÞÞdS Z Z S

 i 0  p ui þ Ma h0a dS þ

Z @S1

  T i u0i þ C a h0a ds

Complementary energy theorem If (N, M, Q) is a stress field solution to the elasticity problem ðPÞ then



Z Z

ðN; M; Q Þ 2 Rad ; JðN; M; Q Þ 6 JðN0 ; M0 ; Q 0 Þ 8ðN0 ; M0 ; Q 0 Þ 2 Rad

ð29Þ

with

JðN; M; Q Þ ¼

1 2

ðpi ui þ Ma ha ÞdS þ

S

Z

 ðT i ui þ C a ha Þds

ð31Þ

@S1

Because the compliance is equal to twice the complementary energy (Eq. (30)), the use of the complementary energy theorem (29) leads to an optimization problem formulated as a double minimization with respect to the optimization parameters and to statically admissible generalized membrane, bending and transverse shear stresses:

min

min

0 bi 2½bmin ;bmax ðN 0 ;M 0 ;Q Þ2Rad i i



Z Z   1 aabcd N0ab N0cd þ dabcd M 0ab M 0cd þ fab Q 0a Q 0b dS k S

ð32Þ

3.2. Optimization algorithm The optimization problem (32) is numerically solved by using the following optimization algorithm [16]:  Initialization: The mesh, the boundary conditions in terms of displacements and forces applied to the shell, and the optimizað0Þ tion parameters initial field bi are defined in order to calculate using a finite element method the generalized stresses (N(0), M(0), Q(0))  Iteration: Each iteration is composed of two parts: 1. Local minimization with fixed stresses. The local criterion is minimized considering fixed stress fields (N(n), M(n), Q(n)) by considering the following optimization problem:

0

þ    Dabcd qab ðu; hÞqcd ðu0 ; h0 Þ

Lðu0 ; h0 Þ ¼

Structural rigidity optimization of a thin laminated shell subjected to membrane and bending loadings is formulated by the minimization of the compliance with respect to distributed (i.e. continuously variableh over the imidsurface S of the shell) optimization parameters bi 2 bmin : ; bmax i i

bi 2½bmin ;bmax  i i

2.4. Variational formulations



3.1. Optimization problem

min

with

a ¼ A1

Using the variational formulations of the shell elasticity problem introduced in the previous section in terms of symmetric generalized forces and moments, we introduce the optimization problem and algorithm for the maximization of the shell structural rigidity. This development follows the classical 3D elasticity topology optimization problem [16] and is possible thanks to the assumption of uncoupled behaviour law (26).

Z Z  1 aabcd Nab Ncd þ dabcd M ab Mcd þ fab Q a Q b dS k S

min

bi 2½bmin ;bmax  i i

h

NðnÞ : aðbi Þ : NðnÞ þ M ðnÞ : dðbi Þ : M ðnÞ

1 þ Q ðnÞ :f ðbi Þ:Q ðnÞ k



2. Global minimization with fixed optimization parameters. The ðnþ1Þ elasticity problem ðPÞ associated to the parameters bi that were determined at the local minimizations step is solved in order to define the new generalized stress fields N(n+1), M(n+1) and Q(n+1).

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A. Jibawy et al. / Composite Structures 95 (2013) 35–43

It is proved that this algorithm is convergent [17]. Its numerical performance is linked to the local sensitivity analyses (no global sensitivity analysis has to be performed). 4. Laminated thin shell optimization

For consistency, we will also neglect the influence of curvature in the transverse shear behaviour (23), that is

b ab ’ C b ab j 3 ¼ C b ab C M n ¼0 and then

In the structural rigidity optimization framework introduced in the previous section, the necessary assumptions in terms of behaviour law and stacking sequences in order to perform shell structural rigidity optimization are defined. Then, the solution of the local minimization problem that leads to the implementation of a numerically efficient optimization tool is derived.

F ab ¼

The components C and in Eqs. (22) and (24) are the comb expressed in the 3D covariant base ponents of the tensors C and C (i.e. at the point P). They are function of the coordinate n3 and of curvature radii. Noting with an index M the tensor components expressed at the point M in the covariant base of the midsurface, those components at the point P are linked to the components b ab of the tensor C and C b (at the point M in the covariant C aMbcd and C M base of the midsurface S) by the following relations 0 0 0 0



with

lac mcb ¼ dab



a0 b0

b b ab ¼ m0a m0b C C a b M

ð34Þ

Thus, considering a shell made of an homogeneous isotropic linear elastic material, the behaviour law (21) is coupled (B – 0). Considering a laminated shell, it is then vain to search in the general case for some particular stacking sequences leading to an uncoupled shell behaviour law. In order to obtain a uncoupled laminate behaviour law, it is then necessary to neglect the influences of the curvature on A, B and D. This leads to assume that abcd C abcd ’ C abcd jn3 ¼0 ¼ C M

and

l ¼ detðlba Þ ’ 1

ð35Þ

The components of A, B and D in the covariant base of the midsurface S reads then

8 abcd R abcd 3 A ¼ H C M dn > > < R abcd B ¼ H C aMbcd n3 dn3 > > : abcd R abcd 3 2 3 D ¼ H C M ðn Þ dn

ð36Þ

Let’s note (i1, i2, a3) the orthonormal base in which is defined the behaviourlaw (17) in terms of physical components. If one notes wba and uba wca ubc ¼ dba the matrices relating this base to the covariant base of the midsurface S:

aa ¼ uba ib

ia ¼ wba ab

ð37Þ

The components in the orthonormal base (i1, i2, a3) will be noted with a tilde (). It is proved that (see below):

8 e abcd ¼ R C e abcd dn3 > A > H M > < R e abcd 3 3 e abcd ¼ C B n dn H M > > > : e abcd R e abcd 3 2 3 D ¼ H C M ðn Þ dn

b ab dn3 C M

e F ab ¼

Z H

~ ab 3 b C M dn

ð40Þ

Proof of Eq. (38)

Z 0c 0d e a0 b0 c0 d0 C aMbcd dn3 ¼ wa0a w0b dn3 b wc wd C M H H Z 0c 0d e a0 b0 c0 d0 dn3 ¼ wa0a w0b C b wc wd M

Aabcd ¼

Z

H

b ab C

0c 0d a b c d C abcd ¼ ma0a m0b b mc md C M

Z H

4.1. About uncoupled behaviour and optimization abcd

ð39Þ

ð38Þ

and the obtained (approximate) shell behaviour laws are identical to those of plates. The uncoupling of membrane/bending behaviour is then obtained with the same stacking sequences than those of laminated plates.

by definition 0c 0d e a0 b0 c0 d0 Aabcd ¼ wa0a w0b ; b wc wd A

then

e abcd ¼ A

Z H

e abcd dn3 C M

Remark 1. If the previous assumptions (35) and (39) are not considered in the behaviour law, it is not possible to get an equivalent equation for the relation (38). 4.2. Choice of specific orthotropic stacking sequences Let’s consider stacking sequences made of identical plies (identical material in each ply, identical ply thicknesses). The elementary layer is supposed to have an orthotropic linear elastic behaviour. Let’s assume that the stacking sequence implies an uncoupled behaviour (B = 0) and orthotropy in membrane and in bending. The local minimization problem (33) is analytically solved considering plates in the case of a membrane local energy or of a bending local energy [20]. It is demonstrated that  three cases of optimal orthotropic behaviour are defined with rII increasing values of jrrII  (in which rI and rII are the two prinþrII j cipal stresses, with rI > rII),  in case 1: there exist an infinite number of optimal stacking sequences,  in cases 1 and 2: the angle-ply is optimal,  in cases 1, 2 and 3: the cross-ply is optimal (and is the only optimal staking sequence in case 3). Considering laminates with identical ply thicknesses, the cross ply is not feasible because a change in relative ply thicknesses will imply a 90° fibre orientation change inside the plies. Considering smooth variations of stress field inside the plate domain, the angle-ply is feasible in the first two cases presented above, but is sub-optimal in case 3 and provide discontinuous orientation variations inside plies in the third stress case (the stacking sequence jumps from unidirectional to equilibrated cross-ply at a given varII lue of jrrII  that is a function of the elementary layer material þrII j parameters). For shells (or uncoupled plates with simultaneous membrane and bending loadings), the membrane and bending energies coexist and no analytical solution to the local minimization of the sum of membrane and bending energies has been found yet. We will then consider in this work specific stacking sequences in order to solve the local minimization problems.

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A. Jibawy et al. / Composite Structures 95 (2013) 35–43

Introducing the homogenized membrane and bending tensors A ¼ H1 A and D ¼ H123 D, we choose to work with angle-ply stacking sequences that satisfy the quasi-homogeneous property:

A ¼ D

ð41Þ

This quasi-homogeneity property adds an optimization constraint but this difficulty can be easily circumvented if we limit the stacking sequences to those which assure this property [21]. As an example, the following eight plies stacking sequence is uncoupled, quasihomogeneous and orthotropic:

½þa=  a=  a= þ a=  a= þ a= þ a=  a

ð42Þ

For this type of stacking sequences, the optimization parameters are the lamination angle a 2 [0°, 45°] and the stiffness principal direction of orthotropy of the homogenized laminate U1 2 [0°, 90°]. The transverse shear energy being very small in comparison to those of membrane and bending, we assume that the transverse shear behaviour of the elementary layers is isotropic and has no effect on the local minimization solution. In terms of engineering moduli, this leads to equal shear moduli GTT and GLT, and the transverse shear behaviour law (23) reads

Q¼

5 b b H C  f ðF ¼ H CÞ 6

ð43Þ

Considering the assumptions of isotropic transverse shear behaviour (43) and of uncoupled behaviour (26), the local minimization reduces to the minimization of the sum of the membrane and bending energies because the transverse shear energy remains constant during this local minimization with fixed generalized stresses. Considering the assumptions (35) and (39) that the homogenized behaviour does not depend on the shell curvature, the part of the optimization criterion that depends on the optimization parameters is written in the orthonormal base (i1, i2, a3) defined in Eq. (37) and the local minimization problem reads

ðU1 ;aÞ



1 e e 12 e e ~ ~abcd M ab M cd a N ab N cd þ 3 a H abcd H

c ðU1 Þ @W ¼ 2k1 cos 2U1  2k2 sin 2U1 þ 4k3 cos 4U1  4k4 @ U1  sin 4U1

 ð44Þ

in which the homogenized membrane compliance tensor e a is defined in terms of material parameters only (i.e. does not depend on the shell curvature). Because of the uncoupled behaviour  1 e a ¼ A , and the membrane and bending assumption (26), e homogenized compliance tensors are identical because of the quasi-homogeneity assumption (41). Missing an analytical solution, we solve numerically this optimization problem using a fixed point algorithm. The choice of multiple initializations allows to find the global optimum in the case of multiple local minima. The numerical cost of the resolution is very low in comparison to a finite element calculation thanks to the existence of semi-analytical solutions to the local minimization problems presented in the next subSections 4.3.1 and 4.3.2.

ð46Þ

Considering the first derivative equal to zero and taking the square of the previous equation, we get

l0 þ l1 Y þ l2 Y 2 þ l3 Y 3 þ l4 Y 4 ¼ 0 with Y ¼ cos 2U1

ð47Þ

c ðU1 Þ is continuous and periodic, the global Considering that W minimum is found numerically in three steps: 1. The Y solutions of (47) with Y 2 [0, 1] are found. b 2. Corresponding angles U1 solutions not satisfying @ W@UðU1 1 Þ ¼ 0 are eliminated. 3. The criterion values for the remaining angles U1 are compared in order to find the global minimum. 4.3.2. Local minimization with respect to the lamination angle with a fixed principal orthotropy direction The local criterion is put in the form (the expressions of the coefficients jij and ki function of the stiffness principal orthotropy direction U1 are given in annex):

P4 i i¼0 j1i ðYÞ c ðaÞ ¼ 1 W H3 k0 þ k2 Y 2 þ k4 Y 4

4.3. Solution to the local minimization problem

min

The first derivative reads

with Y ¼ cosð2aÞ

ð48Þ

The first derivative reads

c ðaÞ c ðYÞ @W @W with ¼ 2 sinð2aÞ @a @Y P6 i 1 i¼0 j2i ðYÞ ¼ 3 2 H ðk0 þ k2 Y þ k4 Y 4 Þ2

c ðYÞ @W @Y ð49Þ

c ðaÞ is continuous and periodic, the global Considering that W minimum is found numerically in three steps: b 1. a = 0 is one solution of @ W@ aðaÞ ¼ 0. b ðYÞ @W 2. The Y solutions of @Y ¼ 0 with Y 2 [0, 1] are found. 3. The criterion values for the remaining angles a are compared in order to find the global minimum. 5. Numerical examples 5.1. Finite elements calculation In order to simulate numerically thin shell behaviour, we use a mitc4 shell element [18]. The calculation of the through the

4.3.1. Local minimization with respect the principal orthotropy direction with a fixed lamination angle The local criterion is written in the following form (expressions of the coefficients ki and li function of the lamination angle a are given in annex):

c ðU1 Þ ¼ k0 þ k1 sin 2U1 þ k2 cos 2U1 þ k3 sin 4U1 þ k4 W  cos 4U1

ð45Þ

Fig. 1. Boundary conditions in terms of forces (cylinder with an inner floor).

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A. Jibawy et al. / Composite Structures 95 (2013) 35–43

membrane/bending uncoupling is formulated as for plates. This finite element allows to calculate the generalized stress tensors in membrane N, in bending M and in transverse shear Q in an orthonormal base per element fixed by the user and used to define the material behaviour. 5.2. Example of a cylinder with an inner floor

Fig. 2. Initial fibre orientation (cylinder with an inner floor).

thickness homogenized behaviour law is performed by taking into account the assumptions (35) and (39): the homogenized behaviour does not depend on the shell curvature and the

The structure is made of a cylinder with an inside floor diametrally placed in the plane x = 0. The structure is clamped for z = 0. An inner pressure is applied on the lower half of the cylinder (x < 0, Fig. 1). The floor and the cylinder are quasihomogeneous uncoupled angle-ply laminates with an elementary layer made of Carbon/Epoxy. The optimization procedure is initialized with an unidirectional oriented circumferentially for the cylinder and oriented along the diameter of the cylinder for the floor (Fig. 2). Convergence is obtained after 11 iterations with a relative error of 0.01%. For each iteration, the local minimization step (with four initial starting points) and the finite element resolution step (16,000 DOFs) take respectively 18 and 5 s.

Fig. 3. Optimal plies orientation at the angles U1 + a and U1  a (cylinder with an inner floor).

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A. Jibawy et al. / Composite Structures 95 (2013) 35–43 Table 1 Maximal values of normalized criterion and displacements (cylinder with an inner floor). (Normalized values)

Initial state

Optimal state

Decrease (%)

Compliance Membrane energy Bending energy Transverse shear energy Max. displ.

100 95.3 4.6 0.1 100

15.0 10.8 4.1 0.1 25

85 89 11 0 75

Fig. 3 shows the optimal orientation of the plies oriented at

U1 + a and U1  a. The unidirectional laminate (a = 0) is optimal close to the clamped and free edges of the cylinder and of the floor. It gradually changes to an angle-ply with an increasing lamination angle a that increases from 0 to p/4. This increase occurs in certain regions in a discontinuous manner, which leads to a straight change from a unidirectional to an equilibrated cross-ply with a = p/4 (as it was already mentioned in Section 4.2, this result is not satisfactory from a feasibility point of view). Table 1 presents the optimization results of the cylinder with a floor. The decrease in terms of compliance and maximal displacements are respectively 85% and 75%. 5.3. Example of an elliptic geometry shell The structure is made of a 3D surface with two straight edges and two edges made with a quarter of an ellipse. The structure is clamped on the two straight edges. Considering an unstructured mesh, a constant vertical surface load is applied on the entire surface (Fig. 4). The shell is made of quasi-homogeneous uncoupled angle-ply laminates with an elementary layer made of Carbon/ Epoxy. The optimization procedure is initialized with an unidirectional oriented from one clamped edge to the second clamped edge (Fig. 5). Convergence is obtained after 34 iterations with a relative error of 0.01%. For each iteration, the local minimization step (with four initial starting points) and the finite element resolution step (9000 DOFs) take repectively 14 and 3 s.

Fig. 6. Optimal fibre orientation at the angles U1 + a and U1  a (elliptic geometry shell).

Fig. 6 shows the optimal orientation of the plies oriented at

U1 + a and U1  a. The unidirectional laminate (a = 0) is optimal in some parts of the shell. It gradually changes to an angle-ply with an increasing lamination angle a that increases from 0 to p/4. As already noticed in the previous cylinder with an inner floor example, this increase occurs in certain regions in a discontinuous manner, which leads to a straight change from a unidirectional to an equilibrated cross-ply with a = p/4. Table 2 presents the optimization results of the elliptic geometry shell. The decrease in terms of compliance and maximal displacements are respectively 42% and 50%. 6. Concluding remarks

Fig. 4. Boundary conditions in terms of forces (elliptic geometry shell).

We present in this paper an optimization methodology for laminated shell structures with any arbitrary geometry (the finite element analysis being performed using an unstructured mesh). The optimization algorithm has a great numerical efficiency thanks to the semi-analytical resolution of local sensitivity analysis that has been obtained considering quasi-homogeneous uncoupled angle ply laminates (known to be quasi-optimal for laminated uncoupled plates with membrane only or bending only loadings). Two main assumptions were needed: independence of the homogenized

Table 2 Maximal values of normalized criterion and displacements (elliptic geometry shell).

Fig. 5. Initial fibre orientation (elliptic geometry shell).

(Normalized values)

Initial state

Optimal state

Decrease (%)

Compliance Membrane energy Bending energy Transverse shear energy Max. displ.

100 62.5 37.2 0.3 100

58.2 35.5 22.4 0.3 50.3

42 43 40 0 50

42

A. Jibawy et al. / Composite Structures 95 (2013) 35–43

behaviour with the shell curvature and uncoupled membrane and bending behaviours. The quasi-homogeneous uncoupled angle ply stacking sequences considered are not quite completely adequate for feasibility reasons because there exist some unidirectional to equilibrated cross-ply transitions that cannot be manufactured with laminates made of identical layers. For this reason, a natural perspective to this work would be to consider in a first step an arbitrary uncoupled quasi-homogeneous (or not) homogenized behaviour using for example the polar representation to define the optimization parameters, and to retrieve in a second step feasible stacking sequences made of identical layers which match the optimal distribution of the parameters obtained previously. Appendix A In the local orthonormal base (i1, i2, a3) (defined in Eq. (37)), the components of the homogenized stiffness tensor A⁄, the homogenized compliance tensor a⁄, the membrane forces N and bending e ; a e ab and M e ab . Let U1 be ~ abcd ; N moments M are respectively A abcd the rotation angle of the orthotropy orthonormal base with respect to the local orthonormal base (i1, i2, a3). In this orthotropy base, the components of the homogenized stiffness tensor, homogenized compliance tensor, membrane forces and bending moments are ? ? ? respectively A ? abcd ; aabcd ; N ab and M ab . Using the polar representation [6], the components of the homogenized stiffness tensor A⁄ are defined in the local orthonormal base (i1, i2, a3) by K e A 1111 ¼ T 0 þ 2T 1 þ ð1Þ R0 cos 4U1 þ 4R1 cos 2U1 K e A 1122 ¼ T 0 þ 2T 1  ð1Þ R0 cos 4U1 e ¼ ð1ÞK R0 sin 4U1 þ 2R1 sin 2U1 A 1112

K e A 2222 ¼ T 0 þ 2T 1 þ ð1Þ R0 cos 4U1  4R1 cos 2U1 K e A 2212 ¼ ð1Þ R0 sin 4U1 þ 2R1 sin 2U1 e ¼ T 0  ð1ÞK R0 cos 4U1 A

ð50Þ

1212

In the polar representation of 2D elastic behaviour, five polar components are defined: T0, T1, (1)KR0, R1 and U1. Four polar components are invariants: T0 and T1 (linked to the isotropic part of the behaviour) and (1)KR0 and R1 (linked to the anisotropic part of the behaviour). The angle U1 is the stiffness principal direction of orthotropy. Using the polar representation, the components of the homogenized compliance tensor a⁄ are defined in the local orthonormal base (i1, i2, a3) by k

~ 1111 ¼ t 0 þ 2t1 þ ð1Þ r 0 cos 4u1 þ 4r 1 cos 2u1 a k

¼ ð1Þ r 0 sin 4u1 þ 2r1 sin 2u1

~ 2222 ¼ t 0 þ 2t1 þ ð1Þk r 0 cos 4u1  4r 1 cos 2u1 a

ð51Þ

~ 2212 ¼ ð1Þk r 0 sin 4u1 þ 2r 1 sin 2u1 a ~ 1212 ¼ t 0  ð1Þk r 0 cos 4u1 a The polar components of orthotropic stiffness and compliance tensors are linked by the relation

8 u1 ¼ U1 þ p2 > > > > > T T R2 > > t 0 ¼ 4 0 1D 1 > > < T 2 R2 t1 ¼ 0 D 0 > > > R2 T ð1ÞK R0 > > ð1Þk r 0 ¼ 4 1 1 D > > > > K : R0 r 1 ¼ 2R1 T 0 ð1Þ D

    D ¼ 16T 1 T 20  R20  32R21 T 0  ð1ÞK R0

ð52Þ

ð53Þ

The elementary layer is supposed to be orthotropic. Using the polar representation defined above, the invariant polar compoK EL EL nents of the elementary layer will be noted T EL 0 ; T 1 ; ð1Þ EL EL R0 ; R1 . For classical fibre/matrix elementary layers (carbon/ epoxy, glass/epoxy, aramid/epoxy), it is experimentally observed that KEL = 0. For an angle-ply laminate with identical elementary layers (with KEL = 0), it is proved that

8 > T 0 ¼ T EL > 0 > > < T ¼ T EL 1 1 K EL > > ð1Þ R0 ¼ R0 cos 4a > > : EL R1 ¼ R1 cos 2a

ð54Þ

Considering Eqs. (51)–(54), the components of homogenized compliance tensor of the angle-ply laminate are

~ 1111 a ~ 1122 a ~ 1112 a ~ 2222 a ~ 2212 a ~ 1212 a

¼ V 1 þ V 2 cos 2U1 þ V 3 cos 4U1 ¼ V 4  V 3 cos 4U1 ¼ V 2 sin 2U1 þ V 3 sin 4U1 ¼ V 1  V 2 cos 2U1 þ V 3 cos 4U1 ¼ V 2 sin 2U1  V 3 sin 4U1 ¼ V 5  V 3 cos 4U1

8 V 1 ¼ t 0 þ 2t 1 > > > > > V > < 2 ¼ 4r 1 with V 3 ¼ ð1Þk r0 > > > > V 4 ¼ t 0 þ 2t1 > > : V 5 ¼ 4t0  8  2 EL EL EL > > =D ¼ 4 T T  R cos 2 a t > 0 0 1 1 > > > >  >  2 > > 2 EL > < t1 ¼ ðT EL =D 0 Þ  R0 cos 4a and   2 > > EL EL > > ð1Þk r 0 ¼ 4 REL =D cos 2 a  T R cos 4 a > 1 1 0 > > > >   > > : r1 ¼ 2REL cos 2a T EL  REL cos 4a =D 1 0 0 in which

D ¼ 16T EL 1

   2 2 EL T EL  R cos 4 a 0 0

 2 h i EL  32 REL T EL 1 cos 2a 0  R0 cos 4a The coefficients in Eqs. (45)–(47) are derived as following:

~ 1122 ¼ t0 þ 2t 1  ð1Þk r 0 cos 4u1 a ~ 1112 a

in which

ki ¼ H1 p1i þ H123 p2i ði ¼ f0; 1; 2; 3; 4gÞ   8 e2 þ N e 11 N e 22 þ V 5 N e2 e 2 þ 2V 4 N > p10 ¼ V 1 N > 11 22 12 > > > > > e e e > p ¼ 2V ð N þ N Þ N > 11 2 12 11 22 <   e2  N e2 N ¼ V p 2 12 11 22 > > > > > e e e > p ¼ 4V ð N N > 3 12 11  N 22 Þ 13 > > : 2 e2 e e p14 ¼ V 3 ð N 11  N 22 Þ  4V 3 N 12   8 e2 þM e 11 N22 þ V 5 M e2 e 2 þ 2V 4 M > M ¼ V p > 1 20 11 22 12 > > > > > e e e > > < p21 ¼ 2V2 M 12 ð M 11 þM 22 Þ e2 e2 M p22 ¼ V 2 M 11 22 > > > > > e 12 ð M e 11  M e 22 Þ > p ¼ 4V 3 M > > > 23 : 2 e2 e e p24 ¼ V 3 ð M 11  M 22 Þ  4V 3 M 12

ð55Þ

A. Jibawy et al. / Composite Structures 95 (2013) 35–43

8 > l > > 0 > > > > l 1 > > < l2 > > > > > l3 > > > > :l 4

  2 2 ¼ 4 k2 þ 4k3

References

¼ ð16k1 k3 þ 32k2 k4 Þ   2 2 2 2 ¼ 4 k1  k2  16k3 þ 16k4 ¼ 32ðk1 k3  k2 k4 ÞY 3   2 2 ¼ 64 k3  k4

and the coefficients in Eqs. (48) and (49) are derived using the orthotropy base as following:

   8 2 > > k0 ¼ 16  REL þ ðT EL Þ2 T EL > 0 0 1 > > > >      2 > 2 > EL > > þ 64 REL k2 ¼ 32 REL REL T EL > 1 0 þ T0 1 1 > > >   > 2 > > EL > > k4 ¼ 64REL REL  REL 0 1 0 T1 > > > > > > > < j1i ¼ 12p1i þ p2i H2 ði ¼ f0; 1; 2; 3; 4gÞ j20 ¼ þk0 j11 > > > > > j21 ¼ 2k2 j10 þ 2k0 j12 > > > > > j22 ¼ k2 j11 þ 3k0 j13 > > > > > j23 ¼ 4k4 j10 þ 4k0 j14 > > > > > > j24 ¼ 3k4 j11 þ k2 j13 > > > > > j25 ¼ 2k4 j12 þ 2k2 j14 > : j26 ¼ k4 j13 with

8    ?   EL ? 2 EL > R0  T EL REL > 0 0 þ T0 > p10 ¼ 2 N11 þ N22 > > h     i >   EL > ?2 EL EL ? ? 2 EL > > þ4 4N R þ T  N R þ T þ N T EL > 12 0 0 11 22 0 0 1 > > >   >  ?  ?  EL EL > ? ? EL > p11 ¼ 8 N11  N22 N11 þ N22 R1 R0 þ T 0 > > > >

<  2   ?2 EL REL p þ REL N?11 þ N?2 12 ¼ 8 4N 12 R1 0 22 0 > > >   i >   > 2 > ? ? > T EL  4N?2 > 12 þ N 11  N 22 1 > > >     > ? ? ? ? EL EL > > > > p13 ¼ 16 N 11 þ N 22 N 11  N 22 R0 R1 > >   2 >   > : p14 ¼ 8 N? þ N? 2 REL 11

43

22

0

8    ?   EL ? 2 EL > p R0  T EL REL > 20 ¼ 2 M 11 þ M 22 0 0 þ T0 > > > h     i >   EL > ?2 EL EL ? ? 2 EL > > þ4 4M R þ T  M R þ T þ M T EL > 12 0 0 11 22 0 0 1 > > >   >  ?  ?  EL EL > ? ? EL > p21 ¼ 8 M11  M22 M11 þ M22 R1 R0 þ T 0 > > > >

<  2  2 ?2 EL p þ REL M ?11 þ M?22 REL 22 ¼ 8 4M 12 R1 0 0 > > >   i >   > 2 > ? > T EL  4M ?2 > 12 þ M 11  N 22 1 > > >     > ? ? ? ? EL > > p23 ¼ 16 M 11 þ M 22 M11  M 22 R0 REL > 1 > > > >    2 > : p24 ¼ 8 M ? þ M? 2 REL 11 22 0

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