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Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters
Composite Structures 81 (2007) 283–291 www.elsevier.com/locate/compstruct
Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters Mostafa M. Abdalla, Shahriar Setoodeh *, Zafer Gu¨rdal Aerospace Structures, Delft University of Technology, Delft, The Netherlands Available online 10 October 2006
Abstract This paper considers maximisation of the natural frequency of composite panels. The use of composite materials allows greater freedom in tailoring the response of the panels. The tailoring potential of composite panels is further expanded by introducing the concept of variable stiffness laminates. Variable stiffness implies that fibre paths are not necessarily straight but can be continuously curved. In this fashion, the stiffness properties at every point of the panel can be varied independently leading to optimal tailoring of the composite panel to design requirements. In this paper, the variation of the stiffness properties is parameterised using lamination parameters, and the fundamental frequency is predicted based on classical lamination theory. A new structural approximation scheme is introduced which is referred to as the generalised reciprocal approximation. The optimality conditions for the problem are formulated based on this generalised reciprocal approximation. Results indicate that significant increase can be achieved of the optimal fundamental natural frequency of variable stiffness panels as compared with panels of constant stiffness. 2006 Elsevier Ltd. All rights reserved. Keywords: Frequency optimisation; Variable stiffness panels; Lamination parameters
1. Introduction To avoid vibrational resonance in aerospace and naval structures, laminated plates are usually designed for maximum fundamental frequency constraints. Composite panels are particularly attractive because the vibration response can be optimised by tailoring the fibre angles of different layers without incurring weight penalties. In traditional composite design optimisation, panels are assumed to have straight fibres. For such panels, the stiffness properties are constant in space. The design optimisation problem is usually posed as the problem of finding the optimal lamination sequence (ply angles) to maximise a certain design requirement. The design variables for such problems are one orientation angle per layer, and thus, the number of design variables is small. Traditionally, finite element analysis, or closed form analytic expressions are used to evalu*
Corresponding author. E-mail address: [email protected] (S. Setoodeh).
0263-8223/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.08.018
ate the objective function and constraints. Depending on the objective function, traditional gradient based search techniques or genetic algorithms can be used to solve such problems efficiently. For example, Bert [1] maximised the fundamental frequency of thin symmetric laminates by treating the fibre orientation angle of each layer as a continuous design variable. The frequency analysis was performed using a closed form formula for thin rectangular laminated plates based on the classical lamination theory. Kam and Lai [2] removed the restriction of thin rectangular laminates using a shear deformable finite element frequency analysis. They used a multi-start global optimisation approach to avoid local minima which exist due to the non-convexity of the optimisation problem when formulated in fibre orientation angle space. Although significant increase in natural frequency can be obtained through optimisation of composite panels with straight fibres, the potential of fibrous composites is not fully exploited. A concept for the design of panels with holes
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was introduced in the late eighties by Hyer and Charette [3] in which curvilinear fibres were used to improve structural response instead of straight fibre paths. The concept has been generalised later by Gu¨rdal and Olmedo [4] by designing variable stiffness laminates that use curvilinear fibre paths. For variable stiffness panels, the stiffness properties are continuous functions of position. Ideally, by varying the fibre steering geometry, the stiffness properties at each point in the panel can be independently varied. The additional freedom in locally tailoring the stiffness properties means that the performance of variable stiffness panels can be highly improved over constant stiffness (straight fibres) panels. However, this additional freedom comes at the price of having a significantly enlarged design space. One possible approach to the problem is to define a set of design variables for each element in the mesh. In this case, the total number of design variables is the number of layers multiplied by the number of elements. For variable stiffness panels, finer finite element meshes are needed as compared to constant stiffness panels. Thus, the total number of design variables for this approach can become excessively large when panels with large number of layers are considered. Formulating the design problem using fibre angles requires repeated transformation of the material properties using the classical trigonometric transformations. These transformations are not only expensive in terms of the computational costs, but also can cause convergence problems. Besides, the dimension of the design space grows as the number of layers increase. An increased number of design variables makes the optimisation process computationally expensive and cumbersome. Even for the optimisation of a single layer variable stiffness panel, the number of design variables is so large that traditional gradient based search techniques are abandoned in favour of optimality criteria methods. For a single layer, and for compliance minimisation as the objective, closed form analytic solutions for the local fibre direction in terms of stress resultants can be found, and successful single layer variable stiffness panel optimisation is widely reported in the literature. For example, minimum compliance design of variable-stiffness composite laminae was studied in [5,6]. Another approach to the design of variable stiffness panels is to define parametric fibre paths. In this approach, a linear fibre angle variation along a fixed direction is postulated. For example, Tatting and Gu¨rdal [7] used a three parameter curvilinear fibre path definition to model variable-stiffness laminates along with genetic algorithm to design for buckling load. As such, two design variables per layer are defined for the whole panel, and the number of design variables is only twice as many as the constant stiffness case. An important advantage, in addition to limiting the number of design variables, is that fibre paths can be constructed analytically and are guaranteed to be continuous. On the other hand, one disadvantage of this approach is that it limits the fibre paths to linear fibre angle variations and thus restricts the design space.
In order to circumvent the problem of large numbers of design variables for variable stiffness panels, the design can be parameterised in terms of lamination parameters. For symmetrically laminated panels made of layers of a single orthotropic material, the bending stiffness is a function of only four laminations parameters. In turn, the laminations parameters themselves are functions of the local lamination sequence. It can also be shown that the design problem is convex in the lamination parameters space [8]. These characteristics are highly beneficial in the optimal design of laminated composites in the sense that they substantially improve the computational efficiency as well as the accuracy. Fukunaga et al. [9] used lamination parameters to design constant stiffness symmetric laminated plates for fundamental frequency where the effect of the coupling between bending and twisting was taken into account by including all four bending lamination parameters as design variables. Lamination parameters were later used to design composite laminates for minimum compliance under inplane loadings (see for example [10,11]). In the present work, a number of features of the previous approaches are combined to ensure robustness and numerical efficiency. Laminations parameters are used as design variables to limit the number of variables without sacrificing generality. Moreover, design variables are associated with nodes rather than elements to ensure a continuous distribution of laminations parameters. Presumably, a continuous variation of lamination parameters would lead to continuous fibre paths. Finally, discrete optimality criteria are developed based on a new generalisation of the reciprocal approximation. In this new approximation, the optimality conditions reduce to a local minimisation problem in terms of the laminations parameters. Numerical experiments show that this local minimisation problem is indeed convex and can be solved efficiently. The overall algorithm shows robust monotone convergence behavior. It is shown that significant increase in the fundamental frequency of composite panels can be obtained by varying the fibre paths at no additional weight penalty. The remainder of this paper is organised as follows: first, the optimisation problem is formulated in Section 2. The definition and properties of lamination parameters are explained in Section 3. The proposed generalised reciprocal approximation is presented in Section 4, and the corresponding reciprocal interpolation of the stiffness tensor is presented in Section 5. The optimality criterion based design rule for maximum frequency design is presented in Section 6. Finally, representative results are presented followed by conclusions. 2. Problem formulation Maximisation of the fundamental frequency of a composite panel is considered here. The fundamental frequency is calculated using conforming finite elements based on classical lamination theory.
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The eigenvalue problem for vibration is, ðK kMÞ a ¼ 0;
The lamination parameters are defined as ð1Þ
where K is the global stiffness matrix, M is the global mass matrix, k = x2 where x is the natural frequency, and a is the mode shape normalised for unit modal mass such that, a M a ¼ 1:
ð2Þ
The global mass and stiffness matrices are obtained by assembling element matrices. The element stiffness matrix can be written in the standard finite element form Z Ke ¼ B De B dX: ð3Þ X
where B is the strain displacement matrix, and De is the bending stiffness tensor of the element. The bending stiffness is assumed to vary over the domain. Therefore, the bending stiffness at any given point in the domain is a function of four lamination parameters as will be explained in Section 3. Noting that the element stiffness matrix is linear in the stiffness coefficients and that it is zero identically when the stiffness coefficients are zero, the element stiffness matrix as can be expressed as ^e : Ke ¼ Deqr K qr
285
ð4Þ
where Keqr are constant matrices that are independent of the stiffness coefficients and depend only on the finite element shape functions. They can be computed once and for all elements and stored for repeated use as the optimisation proceeds. Here and in the rest of the paper, the usual Einstein summation convention is used where summation is implied on repeated indices. Greek subscripts run from 1 to 3 and denote elements of the bending stiffness or compliance tensors. Latin superscripts and subscripts refer to node or element numbers as implied by the context. The element mass matrix is given by Z e M ¼ qhNT N dX: ð5Þ
ðW 1 ; W 2 ; W 3 ; W 4 Þ Z 1=2 ¼ 12 z2 ðcos 2hðzÞ; sin 2hðzÞ; cos 4hðzÞ; sin 4hðzÞÞ dz; 1=2
ð6Þ in which hðzÞ is the distribution of the fibre orientation angle through the normalized thickness z ¼ z=h (h is the total laminate thickness). The bending stiffness matrix D is a linear function of the lamination parameters, D¼
h3 ðC0 þ W 1 C1 þ W 2 C2 þ W 3 C3 þ W 4 C4 Þ; 12
ð7Þ
where Ci (i = 1, . . . , 4) matrices depend on the properties of the material. For orthotropic materials they can be written in terms of the laminate invariants as 2 3 2 3 0 0 U1 U4 0 U2 6 7 6 7 C0 ¼ 4 U 4 U 1 0 5; C1 ¼ 4 0 U 2 0 5; 0 0 U5 0 0 0 2 3 2 3 0 0 0 U 2 =2 U 3 U 3 6 7 6 7 0 U 2 =2 5; C3 ¼ 4 U 3 U 3 0 5; C2 ¼ 4 0 0 U 2 =2 U 2 =2 0 0 U 3 2 3 0 0 U3 6 7 0 U 3 5; C4 ¼ 4 0 U 3 U 3 0 where the laminate invariants U1 through U5 are defined in terms of lamina reduced stiffnesses [14]. Lamination parameters as defined in (6) cannot be arbitrarily prescribed since the trigonometric functions entering their definition are related. The feasible domain for the bending lamination parameters is known to be convex and defined by [10,15] 2W 21 ð1 W 3 Þ þ 2W 22 ð1 þ W 3 Þ þ W 23 þ W 24 4W 1 W 2 W 4 6 1; W 21 þ W 22 6 1; 1 6 W 3 6 1: ð8Þ
X
where q is the mass density, h is the laminate thickness, and N is the vector of shape functions. Note that rotary inertia has been neglected. 3. Lamination parameters Lamination parameters as initially introduced by Tsai and Hahn [12] represent the laminate lay-up configuration in a compact form. In general, the bending behavior of composite laminates in the classical lamination theory can be fully modeled using only four lamination parameters regardless of the actual number of layers [13]. Moreover, the laminate bending stiffness matrix is linear in terms of the lamination parameters.
4. The generalised reciprocal approximation One of the standard approaches in structural optimisation is the use of successive approximations methods [16]. The idea of the solution algorithm is to construct separable approximations of the objective function and constraints. Separable approximations have the property that each term in the approximation depends on the design variables associated with a single element or node and are suited for dual and optimality criteria solution methods. The optimisation is carried out on the approximation leading to a new design. After the design is updated, the analysis repeated and a new approximation is constructed, and the process is repeated until convergence.
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One popular approximation scheme is the reciprocal approximation [16]. In the standard reciprocal approximation, the objective function is expanded in a Taylor series in terms of the reciprocals of the design variables. The reciprocal approximation is used extensively for truss design in which the design variables are the cross sectional areas of the members. To generalise the reciprocal approximation, it is noted that the cross sectional area for trusses plays the role of the stiffness of the member. In composite panel design for vibration this role is played by the bending stiffness matrix D. The generalised reciprocal approximation is obtained by expanding the objective function in a Taylor series in terms of the inverse tensor of the stiffness tensor, commonly known as the compliance tensor, which is denoted by S = D1. For the problem of maximising the eigenfrequency of a structure, the generalised reciprocal approximation takes the form ok k k þ c ðS cab Scab Þ; ð9Þ oS ab c is known towhere it is assumed that an initial design S gether with the corresponding mode shape ~ a and eigenvalue k. The index c sums over the nodes of the finite element model. 5. Reciprocal interpolation An important requirement in the design optimisation of variable stiffness panels is to ensure the continuity of the distribution of the lamination parameters. Traditionally, design variables are linked to element properties. This is the intuitive approach suggested by the use of the finite element method for analysing the response of the structure. When independent design variables are linked to element properties to simulate the variable stiffness concept, there is no guarantee that the distribution of the lamination parameters is going to be continuous. This issue is wellknown and is especially severe in topology optimisation where it results in checkerboard patterns. To ensure the smoothness of the optimal lamination parameters distribution a methodology was introduced [6,17,18], in which the design variables are associated with nodes rather than elements. For the construction of element matrices, the variation of the stiffness properties over the element is required. While it is possible to use finite element formulations that allow for the variation of stiffness properties over the element, the simplest approach is to use an average value of the stiffness over the element. The use of an average value simplifies the construction of element matrices and the calculation of design sensitivities with respect to nodal variables. Instead of defining the element stiffness tensor as the weighed average of the stiffness tensors at the element nodes, the element compliance tensor is calculated as the average of the compliance tensors at the nodes. This approach, first introduced by Abdalla and Gu¨rdal [17],
was shown to be effective in producing smooth distributions of design variables for topology optimisation problems. Thus, the average element compliance is given by X e ¼ S we;c Sc ; ð10Þ c2Ie
where superscript c denotes node numbers and Ie is the set of nodes connected to element e. The sum is weighed by integration weighing coefficients wc such that for a smooth function f, Z Z X f dX dX we;c f c : ð11Þ Xe
Xe
c2Ie
The effective element stiffness tensor is calculated as the inverse of the average compliance tensor given by (10). 6. Design update rule The reciprocal interpolation scheme introduced in Section 5 couples nicely with the generalised reciprocal approximation introduced in Section 4 to give a simple nodal based design formulation. In this section, an approximation of the eigenvalue of the panel is obtained in terms of the nodal values of the compliance tensor, and derive the corresponding optimality conditions. The derivative of the eigenvalue k with respect to nodal compliance needs to be related to the derivative of k with respect to element compliance tensor. Using the chain rule of differentiation, and by virtue of the interpolation formula (10), it can be deduced that, X ok ok we;c e ; ð12Þ c ¼ oDlr e2Ic oDlr in which Ic is the set of elements connected to node c, and e sums over elements. The calculations of oDoke is detailed in lr Section 7. The following notation is introduced Uclr ¼
ok : oS clr
ð13Þ
Thus, the following compact form of the generalised reciprocal approximation in terms of nodal values is obtained ð14Þ k k þ Ucab Scab Ucab S cab ðW ci Þ: In order to update the design for the next iteration, the reciprocal approximation of the eigenvalue is maximised. Because of separability, the maximisation can be carried out at each node separately. The design update rule for the nodal values of the lamination parameters can thus be expressed as min Ucab S cab ðW ci Þ . . . no sum on c: c Wi
ð15Þ
subject to the feasibility constraints expressed in (8). Note that the design update involves a minimisation problem because of the minus sign in (14).
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When the optimisation is carried out for a constant stiffness panel, the compliance tensors of all nodes are identical, and the update rule is no longer dependent on the node number. The lamination parameters are thus updated as ! X min Ucab S ab ðW i Þ: ð16Þ Wi
c
287
ok ^ qr aÞ: ¼ Dbr Dqa ða K oS eab
ð24Þ
It is important to emphasise that the presented sensitivity analysis is valid regardless of the design parameterisation and element type used. In other words, if fibre orientation angles are used to parameterise the design, the compliance of each node, Se, in Eq. (15) will be computed using the standard transformations in terms of fibre orientation angles rather than lamination parameters.
7. Sensitivity analysis The derivatives of the square of the natural frequency with respect to the components of the compliance tensor of element e can be written using the chain rule as e ok ok oDqr e ¼ e : e oS ab oDqr oS ab
ð17Þ
The derivative of the stiffness tensor with respect to the compliance tensor can be obtained by deriving the identity Dql S lm ¼ dqm ;
ð18Þ
where dqm is the Kronecker delta. Thus, obtaining the expression oDql ¼ Dbr Dqa : oS ab
ð19Þ
Substituting from (19) into (17), the following expression of the derivative of the frequency with respect to the compliance tensor is obtained ok ok ¼ Dbr Dqa e : oS eab oDqr
ð20Þ
The derivative of the eigenvalue with respect to the stiffness coefficients of an element can be obtained from the standard relation [16] ! ok oK oM ¼a k e a: ð21Þ oDeqr oDeqr oDqr For the case of constant thickness variable stiffness panels, the mass matrix is independent of the lamination and the expression reduces to ok oK ¼ a e a: oDeqr oDqr
ð22Þ
It is noteworthy that since the stiffness tensor variation over the element depends only on the stiffness tensor assigned to that particular element, the sensitivity of the global stiffness matrix can be localised over one element. From Eq. (4) it can be concluded that oK ^e : ¼K qr oDeqr
8. Numerical results To demonstrate the performance of the proposed design methodology, design of a rectangular simply supported composite panel is considered first. The following material properties in the numerical simulations E11 ¼ 25; E22
m12 ¼ 0:25:
The frequencies are nondimensionalised using the following relation, 2 rffiffiffiffiffiffiffi b q ¼x x ; ð25Þ E22 h in which b is the shorter side of the panel and q is the mass density. The results are classified in three groups of constant stiffness, balanced variable stiffness, and general variable stiffness panels. For the constant stiffness panels, the bending stiffness is constant throughout the panel and hence there are only four design variables, i.e. W1, W2, W3, and W4, for the entire panel. Balanced variable stiffness panels are designed by treating W1 and W3 as design variables at each finite element node. Finally, all four nodal values of the lamination parameters are treated as design variables for general variable stiffness panels. The optimal fundamental frequencies of constant stiffness plates with aspect ratios of 1 and 2 are listed in Table 1 either simply supported or clamped on all four edges. As this table shows, the present results are in good agreement with results reported in [9,19]. The small differences in frequencies of clamped plates are attributed to different types Table 1 Optimal constant stiffness designs for different aspect ratios and boundary conditions Boundary conditions
Design
a/b
1 x
SSSS
Ref. [19] Present
1
20.571 20.571
SSSS
Ref. [19] Present
2
CCCC
Ref. [19] Present
CCCC
Ref. [19] Present
ð23Þ
Substituting from (4) into (22), the following sensitivity equation for the eigenvalue with respect to the compliance tensor of element e is obtained
G12 ¼ 0:5; E22
W1
W2
W3
W4
0.00 0.00
1.00 1.00
0.00 0.00
0.00 0.00
14.458 14.458
1.00 1.00
0.00 0.00
1.00 1.00
0.00 0.00
1
33.476 33.430
0.00 0.00
0.00 0.00
1.00 1.00
0.00 0.00
2
32.474 32.517
1.00 1.00
0.00 0.00
1.00 1.00
0.00 0.00
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of elements used in the two studies. Diaconu et al. [19] have used a mesh of 4 · 4 quadratic elements for both aspect ratios whereas in the present study grids of 16 · 16 and 32 · 16 of bilinear elements are used for aspect ratios of 1 and 2 respectively. The so called layerwise optimisation approach (LOA) has been used to design constant stiffness panels for maximum fundamental frequency by taking the fibre orientation angle of all layers as design variables [20,21] as opposed to the present study where only four lamination parameters are treated as design variables. Here, present results for constant stiffness panels are compared with published LOA results in [22,20]. Optimal frequencies for different aspect ratios and different boundary conditions are given in Table 2. Symmetric laminates with eight layers were designed for maximum fundamental frequency using the LOA by Narita [20] based on a Ritz eigenfrequency analysis. The normalized frequency parameter X given this table is defined as [20] 1=2 q 2 X ¼ xa ð26Þ D0 in which D0 = E2h3/12(1 m12m21). An examination of the designs given in Table 2 reveals that, firstly, designs reported in [20,22] are not globally optimum designs since lamination parameters designs are known to be the globally optimum designs [15,23]; secondly, the fundamental frequency is not sensitive to changes in W2 near the global optimum (a similar phenomenon was also reported in [24]). For a plate with an aspect ratio of a/b = 2, the optimal lamination parameters obtained are W1 = W3 = 1.0 and ¼ 14:46 (cf. Table 3) W2 = W4 = 0 with a frequency of x which results in a balanced symmetric lay-up configuration. Next, the case of balanced variable stiffness design
Table 3 First three natural frequencies for optimal simply supported designs a/b
1 x
2 x
3 x
% Improvements
1
20.57 20.57 22.22
46.71 47.06 44.50
46.71 47.06 44.50
– 0.0 8.0
1.5
14.83 16.50 17.70
26.63 22.25 22.70
42.22 32.11 32.10
– 11.24 19.33
2
14.46 15.37 16.28
15.23 17.86 17.99
17.04 23.28 22.92
– 6.29 12.60
2.5
14.38 14.89 15.60
14.82 16.20 16.31
15.80 19.40 18.84
– 3.52 8.43
is considered where each node in the grid has two design variables, W1 and W3 (W2 = W4 = 0). The optimal fre ¼ 15:37 for which the correspondquency in this case is x ing distribution of the lamination parameters are depicted in Fig. 1. Note that the figure represent only the lamination parameters distribution, not the fibre orientations. Determination of the fibre orientations for a given lamination parameters distribution is a step that is not addressed in the present paper. There has been, and currently continuing work on that problem [25]. Finally, the general variable stiffness design (four design variable per each node) gives a fundamental frequency of ¼ 16:28 which is improved by more than 12% as comx pared to a constant stiffness design. The optimal distribution of the lamination parameters are given in Fig. 2. The iteration history for this case is depicted in Fig. 3. While this figure shows a typical convergence behavior in structural optimisation, it is important to notice that higher
Table 2 Comparison of the optimal fundamental frequencies for plates with different aspect ratios and boundary conditions (CSP: constant stiffness panels, VSP: variable stiffness panels) Design
Layup
(a) SSSS, a/b = 1 Ref. [20] Ref. [22] Present CSP Present VSP
[45/45/45/45]s [45/45/45/45]s – –
(b) CCCC, a/b = 1 Ref. [20] Ref. [22] Present CSP Present VSP
[0/90/90/90]s [0/90/0/90]s – –
(c) SSSS, a/b = 2 Ref. [20] Present CSP Present VSP (d) CCCC, a/b = 2 Ref. [20] Present CSP Present VSP
W1
W2
W3
W4
X (Ritz)
0.0 0.0 0.0
0.156 0.375 0.000 –
1.0 1.0 1.0 –
0.0 0.0 0.0 –
56.32 55.30 – –
56.32 55.29 55.53 61.26
0.156 0.375 0.000
1.0 1.0 1.0
–
0.0 0.0 0.0 –
0.0 0.0 0.0 –
93.67 93.69 – –
93.62 93.61 93.62 122.84
[90/90/90/90]s – –
1.0 1.0 –
0.0 0.0 –
1.0 1.0
0.0 0.0 –
159.90 – –
159.89 159.89 173.52
[90/90/90/90]s – –
1.0 1.0 –
0.0 0.0 –
1.0 1.0
0.0 0.0 –
354.44 – –
354.44 354.44 362.14
–
–
–
–
Simply supported and clamped boundary conditions on all four edges are denoted by SSSS and CCCC respectively.
X (FE)
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W 1
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
W 3
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
Fig. 1. Optimal distribution of the lamination parameters W1 and W3 for a/b = 2; simply supported, balanced variable stiffness design (16 · 32 elements).
order frequencies may decrease while the fundamental frequency is increasing. Moreover, notice that no modal switching is happening. Mode shapes for both simply supported and clamped designs are depicted in Fig. 4. The first three frequencies for simply supported and clamped plates with different aspect ratios are given in
Tables 3 and 4. For each aspect ratio, there are three rows corresponding to constant stiffness, balanced variable stiffness, and general variable stiffness cases. As the aspect ratio increases, the first and second modes become closer to one another and hence there is possibility of repeated modes or mode switching for higher aspect ratios. This phenomena is specially more evident for clamped plates. For such cases, necessary modifications have to be made in the problem formulation to accommodate for the mode switching [26]. Now to investigate the effect of the mesh density, design of a panel with aspect ratio of 2 is considered with simply supported and clamped boundary conditions. The nondimensionalised frequencies for different mesh sizes for simply supported and clamped panels are given in Tables 5 and 6 respectively. The rows corresponding to each mesh size give the first three frequencies of constant, balanced variable stiffness, and general variable-stiffness designs respectively. As expected, mesh refinement has nearly no effect on the frequencies of constant stiffness designs. However, mesh refinement results in a larger design space and thereby increases the improvement in fundamental frequency. This effect is more pronounced specially for the simply supported plate. Finally, the effect of the degree of orthotropy on the optimal frequencies is studied. Consider an orthotropic layer for which E11/E22 varies from 2 to 40 and the rest of the material properties fixed as G12 ¼ 0:5; E22
W2
0.6 0.4 0.2 0 -0.2 -0.4 -0.6
W4 W3
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5
1st Mode 2nd mode 3rd mode 25
20
15
0
Fig. 2. Optimal distribution of the lamination parameters W1, W2, W3, and W4 for a/b = 2; simply supported, general variable stiffness design (16 · 32 elements).
m12 ¼ 0:25:
The nondimensionalised frequencies are listed in Table 7 for a square simply supported plate. Rows corresponding to each ratio represent constant, balanced variable stiffness, and general variable stiffness designs respectively. As this table shows, the improvements in the fundamental frequency increases significantly as the degree of the orthotropy increases.
Nondimensional Frequency
W1
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
289
10
20
30
40
50
Iteration
Fig. 3. Iteration history for the first three frequencies of a simply supported plate with aspect ratio of 2 (16 · 32 elements).
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(a) First mode
(b) Second mode
(c) Third mode
(d) First mode
(e) Second mode
(f) Third mode
Fig. 4. The first three mode shapes of the optimal variable stiffness plate with aspect ratio of 2; (a)–(c) simply supported, (d)–(f) clamped.
Table 4 First three natural frequencies for optimal clamped plate designs
Table 6 Mesh convergence of optimal designs for a clamped plate with aspect ratio of 2
a/b
1 x
2 x
3 x
% Improvements
1
33.43 41.93 44.26
69.18 75.01 73.54
69.18 75.01 73.54
– 25.44 32.39
Mesh
1 x
2 x
3 x
% Improvements
10 · 20
32.64 34.21 35.48
33.96 39.36 40.70
37.20 50.55 53.00
– 4.80 8.69
32.43 32.75 33.33
32.85 33.97 34.47
33.96 36.96 37.62
– 0.98 2.78
16 · 32
32.47 32.98 33.84
33.02 34.64 35.10
34.30 38.56 38.99
– 1.55 4.19
32.47 32.98 33.84
33.02 34.63 35.21
34.30 38.54 39.17
– 1.55 4.21
20 · 40
32.41 32.62 33.16
32.70 33.35 33.69
33.34 35.00 35.28
– 0.64 2.30
32.48 33.05 34.04
33.05 34.82 35.43
34.35 39.03 39.61
– 1.76 4.81
30 · 60
32.49 33.15 34.33
33.07 35.10 35.68
34.40 39.87 40.21
– 2.04 5.66
1.5
2
2.5
Table 5 Mesh convergence of optimal designs for a simply supported plate with aspect ratio of 2 Mesh
1 x
2 x
3 x
% Improvements
10 · 20
14.46 15.29 15.96
15.23 17.74 17.93
17.04 22.92 22.57
– 5.73 10.39
16 · 32
14.46 15.37 16.28
15.23 17.86 18.02
17.04 23.29 22.93
– 6.29 12.61
20 · 40
14.46 15.39 16.38
15.23 17.90 17.97
17.04 23.42 22.86
– 6.42 13.29
14.46 15.41 16.50
15.23 17.95 17.86
17.04 23.60 22.66
– 6.55 14.09
30 · 60
The present variable stiffness designs provide a theoretical upper bound on the maximum value of the fundamental frequency. The manufacturability of such designs, however, is not readily known. The authors believe that it is possible to retrieve a continuous fibre path from the lamination parameters in a post processing step. Current work by the authors [25] concentrates on using curve fitting
Table 7 Effect of the degree of orthotropy on optimal frequencies of a simply supported square plate (16 · 16) 2 x
3 x
% Improvements
2
6.75 7.09 7.45
17.22 17.42 17.19
17.22 17.42 17.19
– 5.05 10.36
5
8.35 9.93 10.70
15.94 23.48 22.51
27.34 23.48 22.51
– 18.99 28.25
10
10.50 13.41 14.52
17.14 31.09 29.39
30.44 31.09 29.40
– 27.72 28.29
20
13.83 18.49 20.02
19.35 42.41 40.12
31.71 42.41 40.14
– 33.70 44.74
30
16.51 22.46 24.28
21.34 51.28 48.55
32.95 51.28 48.57
– 36.05 47.09
40
18.81 25.82 27.92
23.16 58.83 55.81
34.15 58.87 55.82
– 37.30 48.49
E1/E2
1 x
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techniques to find fibre paths for any arbitrary number of plies that approximate the optimised lamination parameters distribution. In such a scheme, manufacturing constraints such as minimum allowed radius of curvature, can be included as well. 9. Concluding remarks The maximisation of the fundamental frequency of variable stiffness composite panels is considered in this paper. The design is parameterised using lamination parameters. An optimality criteria algorithm is developed for the solution of the problem. The main ingredient of the method is a new structural approximation scheme, the so called generalised reciprocal approximation. This new approximation scheme is demonstrated to lead to robust stable convergence of the design iteration. The generalised reciprocal approximation has several advantages. The approximation is constructed based on element by element calculations which makes it highly suitable for parallel processing. The approximation is physically meaningful since it is built around the use of the compliance tensor as an intermediate variable. Last, the approximation captures the interaction between the lamination parameters at the local level. The numerical results for rectangular panels with various aspect ratios show that a significant increase in the optimal fundamental frequency can be achieved using variable stiffness panels compared to constant stiffness panels. References [1] Bert CW. Optimal design of composite material plates to maximize its fundamental frequency. J Sound Vibrat 1977;50:229–37. [2] Kam TY, Lai FM. Design of laminated composite plates for optimal dynamic characteristics using a constrained global optimization technique. Comput Methods Appl Mech Eng 1995;120:389–402. [3] Hyer MW, Charette RF. Use of curvilinear fiber format in composite structure design. In: 30th structures, structural dynamics, and material conference, AIAA, Mobile, AL, 1989, p. 1011–15. [4] Gu¨rdal Z, Olmedo R. In-plane response of laminates with spatially varying fibre orientations: variable stiffness concept. AIAA J 1993;31(4):751–8. [5] Pedersen P. On thickness and orientational design with orthotropic materials. Struct Optim 1991;3:69–78. [6] Setoodeh S, Abdalla MM, Gu¨rdal Z. Combined topology and fiber path design of composite layers using cellular automata. Struct Multidisciplinary Optim 2005;30(6):413–21. [7] Tatting BF, Gu¨rdal Z. Analysis and design of tow-steered variable stiffness composite laminates. In: AHS Meeting, Williamsburg, VA, 2001.
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[8] Foldager J, Hansen JS, Olhoff N. A general approach forcing convexity of ply angle optimization in composite laminates. Struct Optim 1998;16:201–11. [9] Fukunaga H, Sekine H, Sato M. Optimal design of symmetric laminated plates for fundamental frequency. J Sound Vibrat 1994;171(2):219–29. [10] Hammer VB, Bendsøe MP, Lipton R, Pedersen P. Parameterization in laminate design for optimal compliance. Int J Solids Struct 1997;34(4):415–34. [11] Setoodeh S, Abdalla MM, Gu¨rdal Z. Design of variable-stiffness laminates using lamination parameters. Composites Part B: Eng 2006;195(9–12):836–51. [12] Tsai SW, Hahn HT. Introduction of composite materials. Lancaster: Technomic; 1980. [13] Gu¨rdal Z, Haftka RT, Hajela P. Design and Optimization of Laminated Composite Materials. John Wiley & Sons Inc.; 1999. [14] Jones RM. Mechanics of Composite Materials. second ed. Taylor & Francis Inc.; 1998. [15] Grenestedt JL, Gudmundson P. Layup optimization of composite material structures. In: Pedersen P, editor. Proceedings of IUTAM symposium on optimal design with advanced materials. Amsterdam: Elsevier Science; 1993. p. 311–36. [16] Haftka RT, Gu¨rdal Z. Elements of Structural Optimization (Solid Mechanics and Its Applications). Kluwer Academic Publishers; 1992. [17] Abdalla MM, Gu¨rdal Z. Structural design using optimality based cellular automata. In: Proceedings of the 43rd AIAA/ASME/ASCE/ AHS structures, structural mechanics and materials conference, AIAA, Denver, CO, 2002, p. 1676–2002. [18] Rahmatalla S, Swan CC. Continuum topology optimization of buckling-sensitive structures. AIAA J 2003;41:1180–9. [19] Diaconu CG, Sato M, Sekine H. Layup optimization of symmetrically laminated thick plates for fundamental frequencies using lamination parameters. Struct Multidisciplinary Optim 2002;24:302–11. [20] Narita Y. Layerwise optimization for the maximum fundamental frequency of laminated composite plates. J Sound Vibrat 2003;263(5):1005–16. [21] Narita Y, Hodgkinson JM. Layerwise optimisation for maximising the fundamental frequencies of point-supported rectangular laminated composite plates. Compos Struct 2005;69(2):127–35. [22] Zhao X, Narita Y. Maximization of fundamental frequency for generally laminated rectangular plates by the complex method. Trans JSME 1997;63C:364–70 (in Japanese). [23] Svanberg K. On local and global minima in structural optimization. In: Atrek E, Gallagher RH, Ragsdell KM, Zienkiewicz OG, editors. New directions in optimum structural design. New York: Wiley; 1984. p. 327–41. [24] Grenestedt JL. Layup optimization and sensitivity analysis of the fundamental frequency of composite plates. Compos Struct 1989;12:193–209. [25] Setoodeh S, Blom AW, Abdalla MM, Gu¨rdal Z. Generating curvilinear fiber paths from lamination parameters distribution. In: 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Newport, Rhode Island, 2006. [26] Abdalla MM, Setoodeh S, Gu¨rdal Z. Design of variable stiffness composite panels for maximum buckling load, in preparation.
Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters Mostafa M. Abdalla, Shahriar Setoodeh *, Zafer Gu¨rdal Aerospace Structures, Delft University of Technology, Delft, The Netherlands Available online 10 October 2006
Abstract This paper considers maximisation of the natural frequency of composite panels. The use of composite materials allows greater freedom in tailoring the response of the panels. The tailoring potential of composite panels is further expanded by introducing the concept of variable stiffness laminates. Variable stiffness implies that fibre paths are not necessarily straight but can be continuously curved. In this fashion, the stiffness properties at every point of the panel can be varied independently leading to optimal tailoring of the composite panel to design requirements. In this paper, the variation of the stiffness properties is parameterised using lamination parameters, and the fundamental frequency is predicted based on classical lamination theory. A new structural approximation scheme is introduced which is referred to as the generalised reciprocal approximation. The optimality conditions for the problem are formulated based on this generalised reciprocal approximation. Results indicate that significant increase can be achieved of the optimal fundamental natural frequency of variable stiffness panels as compared with panels of constant stiffness. 2006 Elsevier Ltd. All rights reserved. Keywords: Frequency optimisation; Variable stiffness panels; Lamination parameters
1. Introduction To avoid vibrational resonance in aerospace and naval structures, laminated plates are usually designed for maximum fundamental frequency constraints. Composite panels are particularly attractive because the vibration response can be optimised by tailoring the fibre angles of different layers without incurring weight penalties. In traditional composite design optimisation, panels are assumed to have straight fibres. For such panels, the stiffness properties are constant in space. The design optimisation problem is usually posed as the problem of finding the optimal lamination sequence (ply angles) to maximise a certain design requirement. The design variables for such problems are one orientation angle per layer, and thus, the number of design variables is small. Traditionally, finite element analysis, or closed form analytic expressions are used to evalu*
Corresponding author. E-mail address: [email protected] (S. Setoodeh).
0263-8223/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.08.018
ate the objective function and constraints. Depending on the objective function, traditional gradient based search techniques or genetic algorithms can be used to solve such problems efficiently. For example, Bert [1] maximised the fundamental frequency of thin symmetric laminates by treating the fibre orientation angle of each layer as a continuous design variable. The frequency analysis was performed using a closed form formula for thin rectangular laminated plates based on the classical lamination theory. Kam and Lai [2] removed the restriction of thin rectangular laminates using a shear deformable finite element frequency analysis. They used a multi-start global optimisation approach to avoid local minima which exist due to the non-convexity of the optimisation problem when formulated in fibre orientation angle space. Although significant increase in natural frequency can be obtained through optimisation of composite panels with straight fibres, the potential of fibrous composites is not fully exploited. A concept for the design of panels with holes
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was introduced in the late eighties by Hyer and Charette [3] in which curvilinear fibres were used to improve structural response instead of straight fibre paths. The concept has been generalised later by Gu¨rdal and Olmedo [4] by designing variable stiffness laminates that use curvilinear fibre paths. For variable stiffness panels, the stiffness properties are continuous functions of position. Ideally, by varying the fibre steering geometry, the stiffness properties at each point in the panel can be independently varied. The additional freedom in locally tailoring the stiffness properties means that the performance of variable stiffness panels can be highly improved over constant stiffness (straight fibres) panels. However, this additional freedom comes at the price of having a significantly enlarged design space. One possible approach to the problem is to define a set of design variables for each element in the mesh. In this case, the total number of design variables is the number of layers multiplied by the number of elements. For variable stiffness panels, finer finite element meshes are needed as compared to constant stiffness panels. Thus, the total number of design variables for this approach can become excessively large when panels with large number of layers are considered. Formulating the design problem using fibre angles requires repeated transformation of the material properties using the classical trigonometric transformations. These transformations are not only expensive in terms of the computational costs, but also can cause convergence problems. Besides, the dimension of the design space grows as the number of layers increase. An increased number of design variables makes the optimisation process computationally expensive and cumbersome. Even for the optimisation of a single layer variable stiffness panel, the number of design variables is so large that traditional gradient based search techniques are abandoned in favour of optimality criteria methods. For a single layer, and for compliance minimisation as the objective, closed form analytic solutions for the local fibre direction in terms of stress resultants can be found, and successful single layer variable stiffness panel optimisation is widely reported in the literature. For example, minimum compliance design of variable-stiffness composite laminae was studied in [5,6]. Another approach to the design of variable stiffness panels is to define parametric fibre paths. In this approach, a linear fibre angle variation along a fixed direction is postulated. For example, Tatting and Gu¨rdal [7] used a three parameter curvilinear fibre path definition to model variable-stiffness laminates along with genetic algorithm to design for buckling load. As such, two design variables per layer are defined for the whole panel, and the number of design variables is only twice as many as the constant stiffness case. An important advantage, in addition to limiting the number of design variables, is that fibre paths can be constructed analytically and are guaranteed to be continuous. On the other hand, one disadvantage of this approach is that it limits the fibre paths to linear fibre angle variations and thus restricts the design space.
In order to circumvent the problem of large numbers of design variables for variable stiffness panels, the design can be parameterised in terms of lamination parameters. For symmetrically laminated panels made of layers of a single orthotropic material, the bending stiffness is a function of only four laminations parameters. In turn, the laminations parameters themselves are functions of the local lamination sequence. It can also be shown that the design problem is convex in the lamination parameters space [8]. These characteristics are highly beneficial in the optimal design of laminated composites in the sense that they substantially improve the computational efficiency as well as the accuracy. Fukunaga et al. [9] used lamination parameters to design constant stiffness symmetric laminated plates for fundamental frequency where the effect of the coupling between bending and twisting was taken into account by including all four bending lamination parameters as design variables. Lamination parameters were later used to design composite laminates for minimum compliance under inplane loadings (see for example [10,11]). In the present work, a number of features of the previous approaches are combined to ensure robustness and numerical efficiency. Laminations parameters are used as design variables to limit the number of variables without sacrificing generality. Moreover, design variables are associated with nodes rather than elements to ensure a continuous distribution of laminations parameters. Presumably, a continuous variation of lamination parameters would lead to continuous fibre paths. Finally, discrete optimality criteria are developed based on a new generalisation of the reciprocal approximation. In this new approximation, the optimality conditions reduce to a local minimisation problem in terms of the laminations parameters. Numerical experiments show that this local minimisation problem is indeed convex and can be solved efficiently. The overall algorithm shows robust monotone convergence behavior. It is shown that significant increase in the fundamental frequency of composite panels can be obtained by varying the fibre paths at no additional weight penalty. The remainder of this paper is organised as follows: first, the optimisation problem is formulated in Section 2. The definition and properties of lamination parameters are explained in Section 3. The proposed generalised reciprocal approximation is presented in Section 4, and the corresponding reciprocal interpolation of the stiffness tensor is presented in Section 5. The optimality criterion based design rule for maximum frequency design is presented in Section 6. Finally, representative results are presented followed by conclusions. 2. Problem formulation Maximisation of the fundamental frequency of a composite panel is considered here. The fundamental frequency is calculated using conforming finite elements based on classical lamination theory.
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The eigenvalue problem for vibration is, ðK kMÞ a ¼ 0;
The lamination parameters are defined as ð1Þ
where K is the global stiffness matrix, M is the global mass matrix, k = x2 where x is the natural frequency, and a is the mode shape normalised for unit modal mass such that, a M a ¼ 1:
ð2Þ
The global mass and stiffness matrices are obtained by assembling element matrices. The element stiffness matrix can be written in the standard finite element form Z Ke ¼ B De B dX: ð3Þ X
where B is the strain displacement matrix, and De is the bending stiffness tensor of the element. The bending stiffness is assumed to vary over the domain. Therefore, the bending stiffness at any given point in the domain is a function of four lamination parameters as will be explained in Section 3. Noting that the element stiffness matrix is linear in the stiffness coefficients and that it is zero identically when the stiffness coefficients are zero, the element stiffness matrix as can be expressed as ^e : Ke ¼ Deqr K qr
285
ð4Þ
where Keqr are constant matrices that are independent of the stiffness coefficients and depend only on the finite element shape functions. They can be computed once and for all elements and stored for repeated use as the optimisation proceeds. Here and in the rest of the paper, the usual Einstein summation convention is used where summation is implied on repeated indices. Greek subscripts run from 1 to 3 and denote elements of the bending stiffness or compliance tensors. Latin superscripts and subscripts refer to node or element numbers as implied by the context. The element mass matrix is given by Z e M ¼ qhNT N dX: ð5Þ
ðW 1 ; W 2 ; W 3 ; W 4 Þ Z 1=2 ¼ 12 z2 ðcos 2hðzÞ; sin 2hðzÞ; cos 4hðzÞ; sin 4hðzÞÞ dz; 1=2
ð6Þ in which hðzÞ is the distribution of the fibre orientation angle through the normalized thickness z ¼ z=h (h is the total laminate thickness). The bending stiffness matrix D is a linear function of the lamination parameters, D¼
h3 ðC0 þ W 1 C1 þ W 2 C2 þ W 3 C3 þ W 4 C4 Þ; 12
ð7Þ
where Ci (i = 1, . . . , 4) matrices depend on the properties of the material. For orthotropic materials they can be written in terms of the laminate invariants as 2 3 2 3 0 0 U1 U4 0 U2 6 7 6 7 C0 ¼ 4 U 4 U 1 0 5; C1 ¼ 4 0 U 2 0 5; 0 0 U5 0 0 0 2 3 2 3 0 0 0 U 2 =2 U 3 U 3 6 7 6 7 0 U 2 =2 5; C3 ¼ 4 U 3 U 3 0 5; C2 ¼ 4 0 0 U 2 =2 U 2 =2 0 0 U 3 2 3 0 0 U3 6 7 0 U 3 5; C4 ¼ 4 0 U 3 U 3 0 where the laminate invariants U1 through U5 are defined in terms of lamina reduced stiffnesses [14]. Lamination parameters as defined in (6) cannot be arbitrarily prescribed since the trigonometric functions entering their definition are related. The feasible domain for the bending lamination parameters is known to be convex and defined by [10,15] 2W 21 ð1 W 3 Þ þ 2W 22 ð1 þ W 3 Þ þ W 23 þ W 24 4W 1 W 2 W 4 6 1; W 21 þ W 22 6 1; 1 6 W 3 6 1: ð8Þ
X
where q is the mass density, h is the laminate thickness, and N is the vector of shape functions. Note that rotary inertia has been neglected. 3. Lamination parameters Lamination parameters as initially introduced by Tsai and Hahn [12] represent the laminate lay-up configuration in a compact form. In general, the bending behavior of composite laminates in the classical lamination theory can be fully modeled using only four lamination parameters regardless of the actual number of layers [13]. Moreover, the laminate bending stiffness matrix is linear in terms of the lamination parameters.
4. The generalised reciprocal approximation One of the standard approaches in structural optimisation is the use of successive approximations methods [16]. The idea of the solution algorithm is to construct separable approximations of the objective function and constraints. Separable approximations have the property that each term in the approximation depends on the design variables associated with a single element or node and are suited for dual and optimality criteria solution methods. The optimisation is carried out on the approximation leading to a new design. After the design is updated, the analysis repeated and a new approximation is constructed, and the process is repeated until convergence.
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One popular approximation scheme is the reciprocal approximation [16]. In the standard reciprocal approximation, the objective function is expanded in a Taylor series in terms of the reciprocals of the design variables. The reciprocal approximation is used extensively for truss design in which the design variables are the cross sectional areas of the members. To generalise the reciprocal approximation, it is noted that the cross sectional area for trusses plays the role of the stiffness of the member. In composite panel design for vibration this role is played by the bending stiffness matrix D. The generalised reciprocal approximation is obtained by expanding the objective function in a Taylor series in terms of the inverse tensor of the stiffness tensor, commonly known as the compliance tensor, which is denoted by S = D1. For the problem of maximising the eigenfrequency of a structure, the generalised reciprocal approximation takes the form ok k k þ c ðS cab Scab Þ; ð9Þ oS ab c is known towhere it is assumed that an initial design S gether with the corresponding mode shape ~ a and eigenvalue k. The index c sums over the nodes of the finite element model. 5. Reciprocal interpolation An important requirement in the design optimisation of variable stiffness panels is to ensure the continuity of the distribution of the lamination parameters. Traditionally, design variables are linked to element properties. This is the intuitive approach suggested by the use of the finite element method for analysing the response of the structure. When independent design variables are linked to element properties to simulate the variable stiffness concept, there is no guarantee that the distribution of the lamination parameters is going to be continuous. This issue is wellknown and is especially severe in topology optimisation where it results in checkerboard patterns. To ensure the smoothness of the optimal lamination parameters distribution a methodology was introduced [6,17,18], in which the design variables are associated with nodes rather than elements. For the construction of element matrices, the variation of the stiffness properties over the element is required. While it is possible to use finite element formulations that allow for the variation of stiffness properties over the element, the simplest approach is to use an average value of the stiffness over the element. The use of an average value simplifies the construction of element matrices and the calculation of design sensitivities with respect to nodal variables. Instead of defining the element stiffness tensor as the weighed average of the stiffness tensors at the element nodes, the element compliance tensor is calculated as the average of the compliance tensors at the nodes. This approach, first introduced by Abdalla and Gu¨rdal [17],
was shown to be effective in producing smooth distributions of design variables for topology optimisation problems. Thus, the average element compliance is given by X e ¼ S we;c Sc ; ð10Þ c2Ie
where superscript c denotes node numbers and Ie is the set of nodes connected to element e. The sum is weighed by integration weighing coefficients wc such that for a smooth function f, Z Z X f dX dX we;c f c : ð11Þ Xe
Xe
c2Ie
The effective element stiffness tensor is calculated as the inverse of the average compliance tensor given by (10). 6. Design update rule The reciprocal interpolation scheme introduced in Section 5 couples nicely with the generalised reciprocal approximation introduced in Section 4 to give a simple nodal based design formulation. In this section, an approximation of the eigenvalue of the panel is obtained in terms of the nodal values of the compliance tensor, and derive the corresponding optimality conditions. The derivative of the eigenvalue k with respect to nodal compliance needs to be related to the derivative of k with respect to element compliance tensor. Using the chain rule of differentiation, and by virtue of the interpolation formula (10), it can be deduced that, X ok ok we;c e ; ð12Þ c ¼ oDlr e2Ic oDlr in which Ic is the set of elements connected to node c, and e sums over elements. The calculations of oDoke is detailed in lr Section 7. The following notation is introduced Uclr ¼
ok : oS clr
ð13Þ
Thus, the following compact form of the generalised reciprocal approximation in terms of nodal values is obtained ð14Þ k k þ Ucab Scab Ucab S cab ðW ci Þ: In order to update the design for the next iteration, the reciprocal approximation of the eigenvalue is maximised. Because of separability, the maximisation can be carried out at each node separately. The design update rule for the nodal values of the lamination parameters can thus be expressed as min Ucab S cab ðW ci Þ . . . no sum on c: c Wi
ð15Þ
subject to the feasibility constraints expressed in (8). Note that the design update involves a minimisation problem because of the minus sign in (14).
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When the optimisation is carried out for a constant stiffness panel, the compliance tensors of all nodes are identical, and the update rule is no longer dependent on the node number. The lamination parameters are thus updated as ! X min Ucab S ab ðW i Þ: ð16Þ Wi
c
287
ok ^ qr aÞ: ¼ Dbr Dqa ða K oS eab
ð24Þ
It is important to emphasise that the presented sensitivity analysis is valid regardless of the design parameterisation and element type used. In other words, if fibre orientation angles are used to parameterise the design, the compliance of each node, Se, in Eq. (15) will be computed using the standard transformations in terms of fibre orientation angles rather than lamination parameters.
7. Sensitivity analysis The derivatives of the square of the natural frequency with respect to the components of the compliance tensor of element e can be written using the chain rule as e ok ok oDqr e ¼ e : e oS ab oDqr oS ab
ð17Þ
The derivative of the stiffness tensor with respect to the compliance tensor can be obtained by deriving the identity Dql S lm ¼ dqm ;
ð18Þ
where dqm is the Kronecker delta. Thus, obtaining the expression oDql ¼ Dbr Dqa : oS ab
ð19Þ
Substituting from (19) into (17), the following expression of the derivative of the frequency with respect to the compliance tensor is obtained ok ok ¼ Dbr Dqa e : oS eab oDqr
ð20Þ
The derivative of the eigenvalue with respect to the stiffness coefficients of an element can be obtained from the standard relation [16] ! ok oK oM ¼a k e a: ð21Þ oDeqr oDeqr oDqr For the case of constant thickness variable stiffness panels, the mass matrix is independent of the lamination and the expression reduces to ok oK ¼ a e a: oDeqr oDqr
ð22Þ
It is noteworthy that since the stiffness tensor variation over the element depends only on the stiffness tensor assigned to that particular element, the sensitivity of the global stiffness matrix can be localised over one element. From Eq. (4) it can be concluded that oK ^e : ¼K qr oDeqr
8. Numerical results To demonstrate the performance of the proposed design methodology, design of a rectangular simply supported composite panel is considered first. The following material properties in the numerical simulations E11 ¼ 25; E22
m12 ¼ 0:25:
The frequencies are nondimensionalised using the following relation, 2 rffiffiffiffiffiffiffi b q ¼x x ; ð25Þ E22 h in which b is the shorter side of the panel and q is the mass density. The results are classified in three groups of constant stiffness, balanced variable stiffness, and general variable stiffness panels. For the constant stiffness panels, the bending stiffness is constant throughout the panel and hence there are only four design variables, i.e. W1, W2, W3, and W4, for the entire panel. Balanced variable stiffness panels are designed by treating W1 and W3 as design variables at each finite element node. Finally, all four nodal values of the lamination parameters are treated as design variables for general variable stiffness panels. The optimal fundamental frequencies of constant stiffness plates with aspect ratios of 1 and 2 are listed in Table 1 either simply supported or clamped on all four edges. As this table shows, the present results are in good agreement with results reported in [9,19]. The small differences in frequencies of clamped plates are attributed to different types Table 1 Optimal constant stiffness designs for different aspect ratios and boundary conditions Boundary conditions
Design
a/b
1 x
SSSS
Ref. [19] Present
1
20.571 20.571
SSSS
Ref. [19] Present
2
CCCC
Ref. [19] Present
CCCC
Ref. [19] Present
ð23Þ
Substituting from (4) into (22), the following sensitivity equation for the eigenvalue with respect to the compliance tensor of element e is obtained
G12 ¼ 0:5; E22
W1
W2
W3
W4
0.00 0.00
1.00 1.00
0.00 0.00
0.00 0.00
14.458 14.458
1.00 1.00
0.00 0.00
1.00 1.00
0.00 0.00
1
33.476 33.430
0.00 0.00
0.00 0.00
1.00 1.00
0.00 0.00
2
32.474 32.517
1.00 1.00
0.00 0.00
1.00 1.00
0.00 0.00
288
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of elements used in the two studies. Diaconu et al. [19] have used a mesh of 4 · 4 quadratic elements for both aspect ratios whereas in the present study grids of 16 · 16 and 32 · 16 of bilinear elements are used for aspect ratios of 1 and 2 respectively. The so called layerwise optimisation approach (LOA) has been used to design constant stiffness panels for maximum fundamental frequency by taking the fibre orientation angle of all layers as design variables [20,21] as opposed to the present study where only four lamination parameters are treated as design variables. Here, present results for constant stiffness panels are compared with published LOA results in [22,20]. Optimal frequencies for different aspect ratios and different boundary conditions are given in Table 2. Symmetric laminates with eight layers were designed for maximum fundamental frequency using the LOA by Narita [20] based on a Ritz eigenfrequency analysis. The normalized frequency parameter X given this table is defined as [20] 1=2 q 2 X ¼ xa ð26Þ D0 in which D0 = E2h3/12(1 m12m21). An examination of the designs given in Table 2 reveals that, firstly, designs reported in [20,22] are not globally optimum designs since lamination parameters designs are known to be the globally optimum designs [15,23]; secondly, the fundamental frequency is not sensitive to changes in W2 near the global optimum (a similar phenomenon was also reported in [24]). For a plate with an aspect ratio of a/b = 2, the optimal lamination parameters obtained are W1 = W3 = 1.0 and ¼ 14:46 (cf. Table 3) W2 = W4 = 0 with a frequency of x which results in a balanced symmetric lay-up configuration. Next, the case of balanced variable stiffness design
Table 3 First three natural frequencies for optimal simply supported designs a/b
1 x
2 x
3 x
% Improvements
1
20.57 20.57 22.22
46.71 47.06 44.50
46.71 47.06 44.50
– 0.0 8.0
1.5
14.83 16.50 17.70
26.63 22.25 22.70
42.22 32.11 32.10
– 11.24 19.33
2
14.46 15.37 16.28
15.23 17.86 17.99
17.04 23.28 22.92
– 6.29 12.60
2.5
14.38 14.89 15.60
14.82 16.20 16.31
15.80 19.40 18.84
– 3.52 8.43
is considered where each node in the grid has two design variables, W1 and W3 (W2 = W4 = 0). The optimal fre ¼ 15:37 for which the correspondquency in this case is x ing distribution of the lamination parameters are depicted in Fig. 1. Note that the figure represent only the lamination parameters distribution, not the fibre orientations. Determination of the fibre orientations for a given lamination parameters distribution is a step that is not addressed in the present paper. There has been, and currently continuing work on that problem [25]. Finally, the general variable stiffness design (four design variable per each node) gives a fundamental frequency of ¼ 16:28 which is improved by more than 12% as comx pared to a constant stiffness design. The optimal distribution of the lamination parameters are given in Fig. 2. The iteration history for this case is depicted in Fig. 3. While this figure shows a typical convergence behavior in structural optimisation, it is important to notice that higher
Table 2 Comparison of the optimal fundamental frequencies for plates with different aspect ratios and boundary conditions (CSP: constant stiffness panels, VSP: variable stiffness panels) Design
Layup
(a) SSSS, a/b = 1 Ref. [20] Ref. [22] Present CSP Present VSP
[45/45/45/45]s [45/45/45/45]s – –
(b) CCCC, a/b = 1 Ref. [20] Ref. [22] Present CSP Present VSP
[0/90/90/90]s [0/90/0/90]s – –
(c) SSSS, a/b = 2 Ref. [20] Present CSP Present VSP (d) CCCC, a/b = 2 Ref. [20] Present CSP Present VSP
W1
W2
W3
W4
X (Ritz)
0.0 0.0 0.0
0.156 0.375 0.000 –
1.0 1.0 1.0 –
0.0 0.0 0.0 –
56.32 55.30 – –
56.32 55.29 55.53 61.26
0.156 0.375 0.000
1.0 1.0 1.0
–
0.0 0.0 0.0 –
0.0 0.0 0.0 –
93.67 93.69 – –
93.62 93.61 93.62 122.84
[90/90/90/90]s – –
1.0 1.0 –
0.0 0.0 –
1.0 1.0
0.0 0.0 –
159.90 – –
159.89 159.89 173.52
[90/90/90/90]s – –
1.0 1.0 –
0.0 0.0 –
1.0 1.0
0.0 0.0 –
354.44 – –
354.44 354.44 362.14
–
–
–
–
Simply supported and clamped boundary conditions on all four edges are denoted by SSSS and CCCC respectively.
X (FE)
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W 1
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
W 3
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
Fig. 1. Optimal distribution of the lamination parameters W1 and W3 for a/b = 2; simply supported, balanced variable stiffness design (16 · 32 elements).
order frequencies may decrease while the fundamental frequency is increasing. Moreover, notice that no modal switching is happening. Mode shapes for both simply supported and clamped designs are depicted in Fig. 4. The first three frequencies for simply supported and clamped plates with different aspect ratios are given in
Tables 3 and 4. For each aspect ratio, there are three rows corresponding to constant stiffness, balanced variable stiffness, and general variable stiffness cases. As the aspect ratio increases, the first and second modes become closer to one another and hence there is possibility of repeated modes or mode switching for higher aspect ratios. This phenomena is specially more evident for clamped plates. For such cases, necessary modifications have to be made in the problem formulation to accommodate for the mode switching [26]. Now to investigate the effect of the mesh density, design of a panel with aspect ratio of 2 is considered with simply supported and clamped boundary conditions. The nondimensionalised frequencies for different mesh sizes for simply supported and clamped panels are given in Tables 5 and 6 respectively. The rows corresponding to each mesh size give the first three frequencies of constant, balanced variable stiffness, and general variable-stiffness designs respectively. As expected, mesh refinement has nearly no effect on the frequencies of constant stiffness designs. However, mesh refinement results in a larger design space and thereby increases the improvement in fundamental frequency. This effect is more pronounced specially for the simply supported plate. Finally, the effect of the degree of orthotropy on the optimal frequencies is studied. Consider an orthotropic layer for which E11/E22 varies from 2 to 40 and the rest of the material properties fixed as G12 ¼ 0:5; E22
W2
0.6 0.4 0.2 0 -0.2 -0.4 -0.6
W4 W3
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5
1st Mode 2nd mode 3rd mode 25
20
15
0
Fig. 2. Optimal distribution of the lamination parameters W1, W2, W3, and W4 for a/b = 2; simply supported, general variable stiffness design (16 · 32 elements).
m12 ¼ 0:25:
The nondimensionalised frequencies are listed in Table 7 for a square simply supported plate. Rows corresponding to each ratio represent constant, balanced variable stiffness, and general variable stiffness designs respectively. As this table shows, the improvements in the fundamental frequency increases significantly as the degree of the orthotropy increases.
Nondimensional Frequency
W1
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
289
10
20
30
40
50
Iteration
Fig. 3. Iteration history for the first three frequencies of a simply supported plate with aspect ratio of 2 (16 · 32 elements).
290
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(a) First mode
(b) Second mode
(c) Third mode
(d) First mode
(e) Second mode
(f) Third mode
Fig. 4. The first three mode shapes of the optimal variable stiffness plate with aspect ratio of 2; (a)–(c) simply supported, (d)–(f) clamped.
Table 4 First three natural frequencies for optimal clamped plate designs
Table 6 Mesh convergence of optimal designs for a clamped plate with aspect ratio of 2
a/b
1 x
2 x
3 x
% Improvements
1
33.43 41.93 44.26
69.18 75.01 73.54
69.18 75.01 73.54
– 25.44 32.39
Mesh
1 x
2 x
3 x
% Improvements
10 · 20
32.64 34.21 35.48
33.96 39.36 40.70
37.20 50.55 53.00
– 4.80 8.69
32.43 32.75 33.33
32.85 33.97 34.47
33.96 36.96 37.62
– 0.98 2.78
16 · 32
32.47 32.98 33.84
33.02 34.64 35.10
34.30 38.56 38.99
– 1.55 4.19
32.47 32.98 33.84
33.02 34.63 35.21
34.30 38.54 39.17
– 1.55 4.21
20 · 40
32.41 32.62 33.16
32.70 33.35 33.69
33.34 35.00 35.28
– 0.64 2.30
32.48 33.05 34.04
33.05 34.82 35.43
34.35 39.03 39.61
– 1.76 4.81
30 · 60
32.49 33.15 34.33
33.07 35.10 35.68
34.40 39.87 40.21
– 2.04 5.66
1.5
2
2.5
Table 5 Mesh convergence of optimal designs for a simply supported plate with aspect ratio of 2 Mesh
1 x
2 x
3 x
% Improvements
10 · 20
14.46 15.29 15.96
15.23 17.74 17.93
17.04 22.92 22.57
– 5.73 10.39
16 · 32
14.46 15.37 16.28
15.23 17.86 18.02
17.04 23.29 22.93
– 6.29 12.61
20 · 40
14.46 15.39 16.38
15.23 17.90 17.97
17.04 23.42 22.86
– 6.42 13.29
14.46 15.41 16.50
15.23 17.95 17.86
17.04 23.60 22.66
– 6.55 14.09
30 · 60
The present variable stiffness designs provide a theoretical upper bound on the maximum value of the fundamental frequency. The manufacturability of such designs, however, is not readily known. The authors believe that it is possible to retrieve a continuous fibre path from the lamination parameters in a post processing step. Current work by the authors [25] concentrates on using curve fitting
Table 7 Effect of the degree of orthotropy on optimal frequencies of a simply supported square plate (16 · 16) 2 x
3 x
% Improvements
2
6.75 7.09 7.45
17.22 17.42 17.19
17.22 17.42 17.19
– 5.05 10.36
5
8.35 9.93 10.70
15.94 23.48 22.51
27.34 23.48 22.51
– 18.99 28.25
10
10.50 13.41 14.52
17.14 31.09 29.39
30.44 31.09 29.40
– 27.72 28.29
20
13.83 18.49 20.02
19.35 42.41 40.12
31.71 42.41 40.14
– 33.70 44.74
30
16.51 22.46 24.28
21.34 51.28 48.55
32.95 51.28 48.57
– 36.05 47.09
40
18.81 25.82 27.92
23.16 58.83 55.81
34.15 58.87 55.82
– 37.30 48.49
E1/E2
1 x
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