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Multiobjective optimization of laminated composite cylindrical shells for maximum frequency and buckling load
Materials and Design 30 (2009) 2584–2594
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Multiobjective optimization of laminated composite cylindrical shells for maximum frequency and buckling load Umut Topal * Gümüsßhane University, Department of Civil Engineering, 29000 Gümüsßhane, Turkey
a r t i c l e
i n f o
Article history: Received 25 July 2008 Accepted 17 September 2008 Available online 24 September 2008 Keywords: B. Laminates E. Mechanical
a b s t r a c t This paper deals with multiobjective optimization of laminated cylindrical shells to maximize a weighted sum of the frequency and buckling load under external load. The layer fibre orientation is used as the design variable and the multiobjective optimization is formulated as the weighted combinations of the frequency and buckling under external load. The first order shear deformation theory is used for the finite element formulation of the laminated shells. Five shell configurations with eight layers are considered as candidate designs. The modified feasible direction method (MFD) is used as optimization routine. For this purpose, a program based on FORTRAN is used for the optimization of the laminated shells. Finally, the effect of different weighting ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal designs is investigated and the results are compared. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Advanced fibre-reinforced composite materials used in modern industries, such as aerospace, automotive, marines, sporting goods and civil engineering applications, have given a promise for better performance in comparison to their isotropic competitors. An increase in the utilization of these materials is due to their physical properties such as higher strength-to-weight ratio, stiffness-toweight ratio and versatility. These physical properties are achieved by combining different materials to meet specific requirement. Enormous benefits that these materials posses attract researchers to explore more into it to script its behavior in a well-defined form to the users. On the other hand, the structures are quite often are subjected to in-plane or external loads which may cause buckling. In addition, the vibration can be problematic when the excitation frequency coincides with the shell’s resonance frequency. Such loadings may occur at different times under in-service conditions, necessitating a design approach which is capable of taking in to account these various loading conditions. In the recent years, optimization approaches are focused on the multiobjective problems of the composite structures. For example, Walker and Smith [1] described a methodology for using genetic algorithms with the finite element method to minimise a weighted sum of the mass and deflection of fibre reinforced structures with several design variables. The design constraint implemented was based on the Tsai–Wu failure criterion. The four fibre orientations and laminae thicknesses were to be determined optimally by * Tel.: +90 462 377 4017. E-mail address: [email protected] 0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2008.09.020
defining a design index comprising a weighted average of the objective functions and determining the minimum. Kere and Koski [2] studied the stacking sequence optimization of laminated plates considering failure margins in each loading case as the objective functions. In some design considerations, different design criteria were to be introduced simultaneously in the optimization problem in a competitive manner in which improvement in one objective function leads to a decrease in another. The same approach was chosen by Rizzo and Spallino [3] in optimization of composite laminates with respect to elastic strength, structural bucking, and thermal buckling conditions. Using evolutionary based optimization algorithms, two objective functions were combined and competitively optimized through the application of game theory in the genetic algorithm. In another research, considering the importance of low weight of composite laminates used in aerospace industry, Wang et al. [4] considered the weight and cost of these structures as the optimization criteria, combination of which is performed using Pareto method and the optimal solution is obtained out of a curve extracted from the method. Walker [5] maximized the weighted sum of the critical buckling load and the resonance frequency for a given laminated plate thickness by optimally determining the fibre orientations. In the optimization stage, golden section method was employed. Walker et al. [6] obtained the multiobjective design of a symmetrically laminated shell with the objectives defined as the maximization of the axial and torsional buckling loads. The ply angle was taken as the optimizing variable and the performance index was formulated as the weighted sum of individual objectives in order to obtain Pareto optimal solutions of the design problem. Single objective design results were obtained and compared with the multiobjective design. The effect
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of weighting factors on the optimal design was investigated. Results were given illustrating the dependence of the optimal fibre angle and performance index on the cylinder length, radius and wall thickness. Adali et al. [7] investigated the optimal design of uniaxially loaded laminated plates subject to elastic in-plane restraints along the unloaded edges for a maximum combination of prebuckling stiffness, postbuckling stiffness and buckling load. The results were also obtained for biaxially loaded plates without elastic restraints. The method of solution involves defining a design index comprising a weighted average of the objective functions and identifying candidate configurations which had to be optimized and compared to determine the best stacking sequence. This paper deals with multiobjective optimization of laminated cylindrical shells to maximize a weighted sum of the frequency and buckling load under external load. The layer fibre orientation is used as the design variable and the multiobjective optimization is formulated as the weighted combinations of the frequency and buckling under external load. The first order shear deformation theory is used for the finite element formulation of the laminated shells. Five shell configurations with eight layers are considered as candidate designs. The modified feasible direction method (MFD) is used as optimization routine. For this purpose, a program based on FORTRAN is used for the optimization of the laminated shells. Finally, the effect of different weighting ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal designs is investigated and the results are compared.
nite number, N, of uniform-thickness orthotropic layers as shown in Fig. 1. The structure is referenced in an orthogonal coordinate system (x, h, z), where x, h and z are the longitudinal, circumferential and radial directions, respectively. Based on the first order shear deformation theory, the in-plane and transverse displacements are assumed in the following form:
uðx; h; z; tÞ ¼ uo ðx; h; tÞ þ z/x ðx; h; tÞ vðx; h; z; tÞ ¼ vo ðx; h; tÞ þ z/h ðx; h; tÞ
ð1Þ
wðx; h; z; tÞ ¼ wo ðx; h; tÞ Here, u0, v0, w0 reply for the displacements of a point in the mid or reference surface in longitudinal, circumferential, and transverse directions, respectively; /x and /h are the rotations of normal to the mid-surface about y- and x-axes, respectively. The in-plane stress resultants Nx, Nh, Nxh, Nhx, the stress-couple resultants Mx, Mh, Mxh, Mhx and the transverse stress resultants Qx, Qh in the laminated shell are
fNx ; Nh ; Nxh ; N hx g ¼
N Z X
N Z X k¼1
fQ x ; Q h g ¼
N Z X k¼1
zkþ1
frx ð1 þ z=RÞ; rh ; rxh ð1 þ z=RÞ; rx gdz
zk
k¼1
fMx ; M h ; Mxh ; M hx g ¼
zkþ1
zkþ1
frx ð1 þ z=RÞ; rh ; rxh ð1 þ z=RÞ; rx gz dz
zk
frxz ð1 þ z=RÞ; rhz gdz
zk
2. Basic equations Consider a fibre-reinforced laminated composite circular cylindrical shell of finite length L, radius R, total wall thickness h P (h ¼ hi ; hi represents thickness of a layer) and composed of fi-
ð2Þ The stress–strain relations are expressed as follows:
8 > > > > > > <
9
38
2
9
rx > Q Q 12 0 0 Q 16 > > > > ex > > 6 11 > 7> > > > > eh > rh > > > 7 6 Q Q 0 0 Q 22 26 > = 6 12 = < 7 7 c rhz ¼ 6 0 0 Q Q 0 44 45 hz 7 6 > 6 > > > > 7> > > > cxz > r > > 4 0 > > > 0 5> 0 Q 45 Q 55 > > > > > > xz > > ; ; : : cxh rxh Q 16 Q 26 0 0 Q 66
ð3Þ
where Q ij is the transformed reduced stiffness component. The strain of the shell is expressed as follows:
8 > > > > > > <
9
9
8
ou ex > > > ox > > > > > > > > ov w> > > > eh > þ > = < Roh R > = ov ow chz ¼ oz þ Roh > > > > > > > ou > > > > cxz > þ ow > > > > > > > oz ox > > > > ; : ov ou > ; : cxh þ ox Roh
ð4Þ
By performing the similar procedure in Eq. (1), the strain components are obtained as the forms:
8 > > > > > > <
9
9
8
8
9
ex > e0x > e1x > > > > > > > > > > > > > > > > > 0 1 > > > > > > eh > > = > = = < eh > < eh > 0 1 chz ¼ chz þ z chz > > > > > > > > > > > > > > > cxz > > > > > > > c0xz > c1xz > > > > > > > > > > > > ; : 0 ; ; : : 1 > cxh cxh cxh
ð5Þ
where
8 > > > > > > <
9
8
9
ou0
8
9
8
2
9
o u > > > > e0x > e1x > ox > > > > > oxoz > > > > > > > > > > > > > > ov w > > > > > > > 0 0 1 > > > > > > o2 v þ e0h > e > > > > > > = = < Roh R = < h = < Rohoz > ov0 ow0 0 1 0 1 fe g ¼ chz ¼ þ Roh ; fe g chz ¼ 0 oz > > > > > > > > > > > > > > > > > > > > > ou0 0 > > > > > > > > > c0xz > c1xz > þ ow 0 > > > > > > > oz ox > > > > > > > > > > > : 0 ; > : 1 ; > 2 2 ; ; : : o v o u ov0 ou0 cxh c þ xh þ oxoz Rohoz
Fig. 1. Geometry of circular cylindrical laminated shell (a) and cross-sectional view of the laminated cylindrical shell (b).
ox
Roh
ð6Þ
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The general form of Eq. (2) can be written as
N
¼
M
A 0
0
"
D
# Qx A44 e0 ; ¼ K Qh A45 e1
A45 A55
gives us the element governing equations of motion as follows:
"
c0xz c0xh
#
½M e f€de g þ ½K e fde g ¼ f0g ð7Þ
where Aij and Dij are extensional and bending stiffnesses, respectively, which are defined in terms of the lamina stiffness Q ij as
fAij ; Dij g ¼
N Z X k¼1
zkþ1
Q ij ð1; z2 Þdz
ð8Þ
zk
where zk and zk+1 denote the distances from the shell reference surface to the outer and inner surfaces of the kth layer and K is the shear correction factor. In this study, the shear correction factor is taken 5/6 [12]. In this study, a C0 four-noded quadrilateral shear flexible shell element having five degrees of freedom per node (u, v, w, /x, /y) developed based on the field consistency approach [13] is employed for the laminated cylindrical shells. The interpolation function of the displacement field is defined as
ue0 ¼
Nn X
uei Nei ;
ve0 ¼
Nn X
i¼1
/ex
¼
Nn X
vei Nei ;
i¼1
wexi Nei ;
/ey
¼
i¼1
Nn X
we0 ¼
Nn X
wei Nei
i¼1
ð9Þ
weyi Nei
i¼1
where Ni represents the element interpolation functions and Nn is the number of nodes per element. When the FEA based on the Reissner–Mindlin shell theory is applied to thin shells, the shear locking may occur. Reduced/selective integration technique is employed to evaluate the transverse shear stresses, while full integration is used for bending. The stiffness matrix of the shell is obtained by using the minimum potential energy principle. The kinetic and strain energy of the shell can be found aswarm
T¼
1 2
Z Z "X N Z k¼1
hkþ1 hk
qk fu_ k v_ k w_ k gxfu_ k v_ k w_ k gT 1 þ
# z dz dx dy ð10Þ R
# ) Z Z "(X N Z hkþ1 1 z T U¼ frg feg 1 þ dz dx dy 2 R hk k¼1
ð11Þ
ð12Þ
ð13Þ
e
where [M ] and [K ] are the element mass and stiffness matrices, respectively. After constructing the element matrices, through the finite elements assembly procedure, the global finite element matrices are found to form the finite element equation, the free vibration problem of the shell becomes as follows:
½½K þ x2 ½Mfdg ¼ f0g
ð14Þ
which is an eigenvalue problem of which the solution with an appropriate method will give the natural frequencies (x) of the laminated cylindrical shell. Again following the general assembly procedure in the finite element solution, governing equation for the buckling analysis of the laminated cylindrical shells is obtained as
½½K þ k½K G fdg ¼ f0g
ð15Þ
where [K] and [KG] are the global structural and geometric stiffness matrices, respectively. Through an eigenvalue solution, the eigenvalues k of which the multiplication by the initial in-plane load would give the critical buckling load of the shell. In the all computations, the following non-dimensionalized quantities are used:
N ¼
Ncr ; N0
F ¼
x x0
ð16Þ
where N0 and x0 are the critical buckling load and fundamental frequency, respectively, corresponding to a prescribed lamination angles (0°/0°/ /0°)sym for eight layered laminated cylindrical shell. 3. Optimization problem The optimization problem is formulated in order to find the best orientation angles of fibres in the laminated cylindrical shells so that to simultaneously maximize the fundamental natural frequency and critical buckling load of the shell with the laminate configurations given by a combination of h, 0°, 90° ply angles. The multiobjective design index, MODI, can be describes as follows:
MODI ¼ gF þ nN
_ k g are density and velocity vector of the kth where qk and fu_ k v_ k w layer, respectively. Applying Lagrangian equations of motion:
d=dt½oðT UÞ=od_ i ½oðT UÞ=odi ¼ 0 i ¼ 1 : N
e
ð17Þ
where g and n are the weighting coefficients summing the two objective functions with g, n P 0, g + n = 1. Thus, the optimization problem can be expressed as follows:
max max MODIi i ¼ 1; 2; . . . ; I i
h
MFD 1. q=0, X q = X m 2. q=q+1. Evaluate the objective function F( X q −1 )
3. Calculate the gradient of the objective function ∇ F( X q −1 ) 4. Find the usable-feasible direction Sq 5. Perform a one-dimensional search X q = X q −1 + α Sq 6. Check convergence. If satisfied, go to 7. Otherwise go to 2 7. X m =X q Fig. 2. Flow chart of modified feasible direction method.
ð18Þ
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od. A search direction Sq, is determined, and a one-dimensional search is made to find a. The above procedure is repeated with the new design vectors, until the design satisfies the optimality conditions or some other termination criterion. The objective function F(Xi) is accurately modelled as a quadratic polynomial approximation around the current iterate Xi as in the following equation:
where I denotes the number of candidate laminates. In this study, five laminate configurations are considered as candidate designs. These configurations are specified as {h, h, h, h}sym, {h, h, 90,0}sym, {h, h, 0,0}sym, {h, h, 0, 90}sym and {h, h, 90,90}sym and referred to as laminate type C1, C2, C3, C4 and C5, respectively. 4. Modified feasible direction method
FðX i Þ ¼ a0 þ
The MFD method is one of the most powerful methods for optimization problems. This method takes into account not only the gradients of objective function and constraints, but also the search direction in the former iteration. In this study, there is no any constraint. Fig. 2 shows the iterative process within each optimization process [8–11]. In Fig. 2, Xq and Xq1 are the design variable vectors in two consecutive cycles of iteration. The procedure starts with an initial design vector, X0, i.e. q = 0, q = q + 1, and the objective function F(Xi), is evaluated. rF(Xi), the gradients of objective function with respect to design variable is calculated using finite difference meth-
a
MODI
Nd X
bi X 2i
ð19Þ
i¼1
where Nd and Xi are number of design variables and ith design variable, respectively. ai and bi are the coefficients of polynomial function determined by a least squares regression. After the objective function is approximated, their gradients with respect to the design variables are calculated by finite differences methods. The solving process is iterated until convergence is achieved.Convergence or termination checks are performed at the end of each optimization loop. The optimization process continues until either convergence or termination occurs. The process may be terminated before convergence in two cases:
b
2
η=0
2
η=0.25
1.6
1.2
1.2 C1 C2 C3 C4 C5
0.8
0.4
C1 C2 C3 C4 C5
0.8
0.4 0
2
10
20
30
40
θ
50
60
70
80
0
90
d
η=0.5
10
20
30
40
θ
50
60
70
80
90
60
70
80
90
2
η=0.75 1.6
1.6
1.2
1.2
C1 C2 C3 C4 C5
0.8
C1 C2 C3
0.8
C4 C5
0.4
0.4 0
10
20
30
e
40
θ
50
60
70
80
90
0
10
60
70
20
30
40
θ
50
2
η=1 1.6
MODI
MODI
ai X i þ
i¼1
1.6
c
Nd X
C1 C2 C3 C4 C5
1.2
0.8
0.4 0
10
20
30
40
θ
50
80
90
Fig. 3. The dependence of the MODI on fibre angle for five different laminate configurations.
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jF current F best j sF
The number of design sets so far exceeds the maximum number of optimization loops. If the initial design is infeasible and the allowed number of consecutive infeasible designs has been exceeded. The optimization problem is terminated if all of the following conditions are satisfied: The current design is feasible. Changes in the objective function F: (a) The difference between the current value and the best design so far is less than the tolerance sF:
a
(b) The difference between the current value and the previous design is less than the tolerance:
jF current F current1 j sF Changes in the design variables Xi: (a) The difference between the current value of each design variable and the best design so far is less than the respective tolerance si:
jX icurrent X ibest j si
b
2
2
C2
C1 1.6
MODI
1.6 1.2
η=0 η=0.25 η=0.5 η=0.75 η=1
η=0.25 η=0.5
0.8
0.8
η=0.75 η=1
0.4 0
c
1.2
η=0
10
20
30
40
50
θ
60
70
80
0.4
90
0
d
2
20
30
40
θ
50
60
70
80
90
60
70
80
90
2
C3
C4 1.6
1.6
MODI
10
1.2
1.2 η=0 η=0.25 η=0.5 η=0.75 η=1
η=0 η=0.25 η=0.5
0.8
0.8
η=0.75 η=1
0.4 0
10
20
30
40
0.4 50
60
70
80
90
0
10
20
70
80
90
30
40
θ
e
θ
50
2
C5
MODI
1.6
1.2 η=0 η=0.25 η=0.5 η=0.75 η=1
0.8
0.4 0
10
20
30
40
θ
50
60
Fig. 4. The dependence of the MODI on fibre angle for different weighting ratios.
Table 1 The best configurations and (MODI)max of the five different candidates for weighting ratios Laminate configuration
C1 C2 C3 C4 C5
hopt (°)
(MODI)max
g=0
g = 0.25
g = 0.5
g = 0.75
g=1
g=0
g = 0.25
g = 0.5
g = 0.75
g=1
90.0 32.1 37.0 39.8 81.1
90.0 37.5 43.0 50.0 80.2
90.0 42.5 47.0 49.0 78.0
37.2 45.5 49.5 49.0 76.2
37.4 48.2 50.8 49.5 40.4
1.89 1.24 1.22 1.13 1.90
1.76 1.34 1.33 1.34 1.78
1.64 1.45 1.46 1.45 1.66
1.65 1.58 1.59 1.57 1.54
1.82 1.71 1.73 1.68 1.58
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(b) The difference between the current value of each design variable and the previous design is less than the respective tolerance:
almost the same for C2, C3, C4 and C1, C5 laminate configurations for 0.25 6 g 6 0.75 and g > 0:75, respectively. As seen from Fig. 4, the curves for C1 and C5 laminate configurations intersect approximately at 68.3° and 56.1°, respectively. After the intersecting points, the trend in the MODI differs suddenly. As the weighting ratio increases, the (MODI)max increases for C2, C3 and C4 laminate configurations. As seen from Table 1, the optimum fibre angles increase with increase in the weighting ratio for C2 and C3 laminate configurations. The optimum fibre angles decrease suddenly for C1 and C5 laminate configurations for g > 0:5 and g > 0:75, respectively. But, the optimum fibre angle remains constant for C4 laminate configuration for g > 0.
jX icurrent X icurrent1 j sF We resolved the optimization process to obtain global maximum from different initial points to check if other solutions are possible. The converge tolerance ratio is considered 0.01 for objective function. 5. Numerical results and discussion Numerical results are given for T300/5208 graphite/epoxy material for which E1 = 181 GPa, E2 = 10.3 GPa, G12 = 7.17 GPa, m12 = 0.28, q = 1600 kg/m3. The laminated cylindrical shell is constructed of equal thickness layers. H/R = 0.2 and L/R = 1 are considered.
5.2. Effect of shell lengths on the optimal designs The effect of shell lengths on the optimal designs is given in Figs. 5–7 for five laminate configurations. As seen from the figures, as the shell length increases, the MODI decreases. But, this decrease in the MODI diminishes with increase in the shell length. It can be said that, the MODI remains fairly constant for larger shell length. As the fibre angle increases, the MODI generally decreases for L/R = 2, 4 and 6. The optimum fiber angles and (MODI)max are given for all laminate configurations for shell lengths in Tables 2–4. As seen, the (MODI)max remains practically constant at all
5.1. Effect of weighting ratios on the optimal designs The effect of five different weighting ratios on the optimal results is given for different five laminate configurations in Figs. 3 and 4 and Table 1. As seen from Fig. 3, the best laminate configuration is C5 for g = 0, 0.25 and 0.5. On the other hand, the best laminate configuration is C1 for g = 0.75 and 1. The (MODI)max is
b
2
a
η=0.25, C1
1.6
η=0.25, C2 1.2 L/R=1
1.2
MODI
L/R=2
0.4
0.4 0 0
10
20
30
40
θ
50
60
70
80
0
90
d
1.6 η=0.25, C3 1.2
0
10
20
30
40
θ
50
60
70
L/R=1 L/R=2
0.4
0.4 0
L/R=4 L/R=6
0.8
L/R=4 L/R=6
90
η=0.25, C4
L/R=2
0.8
80
1.6 1.2
L/R=1
0 0
10
20
30
40
θ
e
50
60
70
80
90
0
10
20
30
40
2 η=0.25, C5
1.6
MODI
MODI
L/R=1 L/R=2 L/R=4 L/R=6
0.8
L/R=4 L/6=6
0.8
c
1.6
L/R=1
1.2
L/R=2 L/R=4
0.8
L/R=6
0.4 0
0
10
20
30
40
θ
50
60
70
80
90
Fig. 5. The dependence of the MODI on fiber angle for shell lengths for g = 0.25.
θ
50
60
70
80
90
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a
b
2
η=0.75, C1
MODI
1.6
2
η=0.75, C2
1.6 L/R=1
1.2
L/R=1
1.2
L/R=2
L/R=2 L/R=4
L/R=4 L/R=6
0.8
0.8
0.4
L/R=6
0.4 0
0 0
10
20
30
40
50
60
70
80
0
90
10
20
30
40
θ
c
2
MODI
d
η=0.75, C3
1.6
50
60
70
80
90
θ 2
η=0.75, C4
1.6 L/R=1
1.2
L/R=4
0.8
L/R=1
1.2
L/R=2
L/R=2 L/R=4
0.8
L/R=6
0.4
L/R=6
0.4
0 0
10
20
30
40
50
60
70
80
0
90
0
10
20
30
40
θ
50
60
70
80
90
θ
e
2
η=0.75, C5
1.6
MODI
L/R=1
1.2
L/R=2 L/R=4 L/R=6
0.8 0.4 0 0
10
20
30
40
50
60
70
80
90
θ Fig. 6. The dependence of the MODI on fibre angle for shell lengths for g = 0.5.
laminate configurations for all weighting ratios for L/R = 4 and 6. The (MODI)max increases with increase in the weighting ratio for L/R = 2. The optimum fibre angles increase as the weighting ratio increases for L/R = 2 and 4. The optimum fibre angles are zero for L/R = 6 except for C1 and C3 laminate configurations for g = 0.75. The best laminate configuration is mostly C4 for L/R = 2, 4 and 6 regardless of the weighting ratios. The optimum fibre angles and the best laminate configurations which must be selected in order to maximize the MODI and obtain the ultimate structural performance is demonstrated by these figures and tables. 5.3. Effect of shell thickness on the optimal designs The effect of shell thickness on the optimal designs is given in Fig. 8 for five different laminate configurations. As seen from Fig. 8, as the shell thickness increases, the MODI increases. But this increase in the MODI diminishes for larger H/R ratios. The optimum fibre angles and (MODI)max are given for all laminate configurations for shell thickness in Table 5. As seen, the optimum fibre orientation is 90° for C1 and C5 laminate configurations for all H/R ratios except for H/R = 1 for C5 laminate configuration. On the
other hand, the optimum fibre orientation increases with increase in the H/R for C2, C3 and C4 laminate configurations. The best laminate configuration is C5 for H/R = 0.2 and 0.4 for g = 0.5. However, the best laminate configurations are C1 and C5 for H/R = 0.5 and 0.6 for g = 0.5. 5.4. Effect of boundary conditions on the optimal designs The different combinations of free (F), simply supported (S) and clamped (C) boundary conditions are considered, viz. clamped/ clamped (CC), clamped/simply supported (CS), simply supported/ free (SF) and clamped/free (CF). The boundary conditions are defined as below: 1. (SS) boundary condition At x = 0 and x = L, u0 = w0 = /x = 0. 2. (CC) boundary condition At x = 0 and x = L, u0 = v0 = w0 = /x = /y = 0.
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a
η=0.5, C1
1.6
MODI
b 1.6
2
L/R=1
L/R=1
1.2
L/R=2
L/R=2
0.8
L/R=4
0.8
L/R=4 L/R=6
L/R=6
0.4
0.4 0 0
c
η=0.5, C2
1.2
10
20
30
40
θ
50
60
70
80
0
90
0
d 1.6
1.6
20
30
40
θ
50
60
70
80
1.2
L/R=1
L/R=1 L/R=2
MODI
L/R=2 L/R=4
0.8
90
η=0.5, C4
η=0.5, C3 1.2
10
L/R=4
0.8
L/R=6
L/R=6
0.4
0.4
0
0 0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
θ
e
60
70
80
90
2
η=0.5, C5
1.6
MODI
50
θ
L/R=1
1.2
L/R=2 L/R=4
0.8
L/R=6
0.4 0
0
10
20
30
40
50
60
70
80
90
θ Fig. 7. The dependence of the MODI on fibre angle for shell lengths for g = 0.75.
Table 2 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 2)
Table 3 Optimum fibre angles and (MODI)max for for five different laminate configurations (L/R = 4)
Laminate configuration
Laminate configuration
hopt (°)
g = 0.25
g = 0.5
g = 0.75
g = 0.25
g = 0.5
g = 0.75
C1 C2 C3 C4 C5
7.0 26.0 1.0 26.4 25.5
9.0 27.4 5.0 27.3 27.0
11.5 29.0 9.0 30.0 29.4
0.43 0.47 0.43 0.49 0.48
0.43 0.47 0.43 0.49 0.48
0.44 0.47 0.44 0.49 0.48
C1 C2 C3 C4 C5
hopt (°)
(MODI)max
g = 0.25
g = 0.5
g = 0.75
g = 0.25
g = 0.5
g = 0.75
15.5 18.5 17.6 20.5 17.5
35.5 29.6 34.4 31.1 27.9
42.9 38.1 44.0 38.5 35.4
0.81 0.79 0.80 0.82 0.78
0.82 0.79 0.79 0.82 0.79
0.91 0.83 0.82 0.85 0.82
3. (CS) boundary condition At x = 0, u0 = v0 = w0 = /x = /y = 0. At x = L, u0 = w0 = /x = 0. 4. (SF) boundary condition At x = 0, u0 = w0 = /x = 0. 5. (CF) boundary condition At x = 0, u0 = v0 = w0 = /x = /y = 0.
(MODI)max
The effect of end conditions on the optimal designs is given for laminated shells in Fig. 9 and Table 6 (L/R = 1, H/R = 0.2, g = 0.5). As seen from Fig. 9, the best laminate configuration is C5 for (CC) and (CS) boundary conditions, on the other hand, the best laminate configuration is generally C4 for (CF) and (SF) boundary conditions. The optimum fibre angles and (MODI)max are given for all laminate configurations for boundary conditions in Table 6. As seen, the (CC) boundary condition has the largest (MODI)max, on the other hand, (SF) boundary condition has the smallest ones. The optimum fibre
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Table 4 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 6) Laminate configuration
hopt (°)
C1 C2 C3 C4 C5
a
14
η=0.5,C1
12
g = 0.25
g = 0.5
g = 0.75
g = 0.25
g = 0.5
g = 0.75
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
19.0 0.0 21.0 0.0 0.0
0.38 0.38 0.38 0.41 0.38
0.38 0.39 0.38 0.41 0.39
0.38 0.40 0.38 0.42 0.40
b
H/R=0.2 H/R=0.4
10
MODI
(MODI)max
5
η=0.5,C2
4
H/R=0.5 H/R=0.6
8
3
6
H/R=0.2 H/R=0.4
2
H/R=0.5
4 2 0
c
0
4
10
20
30
40
50
θ
60
70
MODI
80
0
90
d
η=0.5,C3
3
H/R=0.2 H/R=0.4
2
0
5
10
20
30
η=0.5,C4
40
θ
50
60
70
80
90
H/R=0.2
4
H/R=0.4
3
H/R=0.6
H/R=0.5
H/R=0.5
2
H/R=0.6
1 0
H/R=0.6
1
1
0
10
20
30
40
50
60
70
80
0
90
0
10
20
30
40
θ
e
50
60
70
80
90
14 H/R=0.2
η=0.5,C5
12
H/R=0.4 H/R=0.5
10
MODI
θ
H/R=0.6
8 6 4 2 0
0
10
20
30
40
θ
50
60
70
80
90
Fig. 8. Effect of shell thickness on the optimal results for different five laminate configurations (L/R = 1, g = 0.5).
Table 5 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 1, g = 0.5) Laminate configuration
C1 C2 C3 C4 C5
hopt (°)
(MODI)max
H/R = 0.2
H/R = 0.4
H/R = 0.5
H/R = 0.6
H/R = 0.2
H/R = 0.4
H/R = 0.5
H/R = 0.6
90.0 42.5 47.0 49.0 78.0
90.0 56.1 65.5 71.2 90.0
90.0 64.1 69.7 78.2 90.0
90.0 80.0 72.5 80.5 90.0
1.64 1.45 1.46 1.45 1.66
6.16 3.11 2.88 3.26 6.66
9.57 3.88 3.32 3.95 9.57
12.76 4.68 3.81 4.66 12.76
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a
b
3
(CC)
2.5
(CS) 2
2
MODI
2.5
1.5
1.5
C1
1
C3 C5
0 0
10
20
30
40
50
C2 C3
0.5
C4
0.5
C1
1
C2
C4 C5
0 60
70
80
90
0
10
20
30
40
θ 1
MODI
c
1
d
(CF) 0.8
0.8
0.6
0.6 C1
0.4
C3 C5
0
10
20
30
40
θ
50
70
80
80
90
60
70
80
90
C2 C3 C4 C5
0
90
70
C1
0 60
60
(SF)
0.2
C4
0
50
0.4
C2
0.2
θ
10
20
30
40
θ
50
Fig. 9. Effect of boundary conditions for optimal designs for different five laminate configurations (L/R = 1, H/R = 0.2, g = 0.5).
Table 6 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 1, H/R = 0.2, g = 0.5) Laminate configuration
hopt (°) Boundary condition (CC)
Boundary condition (CS)
Boundary condition (CF)
Boundary condition (SF)
Boundary condition (CC)
(MODI)max Boundary condition (CS)
Boundary condition (CF)
Boundary condition (SF)
C1 C2 C3 C4 C5
90.0 47.5 51.0 55.8 78.2
90.0 45.8 49.8 53.5 90.0
24.1 22.5 26.3 23.7 22.8
28.5 20.8 30.0 21.7 21.5
2.67 1.85 1.78 1.88 2.69
2.14 1.63 1.60 1.64 2.14
0.70 0.77 0.70 0.78 0.78
0.65 0.72 0.66 0.74 0.72
orientation suddenly decreases as the rigidity of the laminated shell decreases. 6. Conclusions In this study, the multiobjective optimization of laminated cylindrical shells is carried out to maximize a weighted sum of the frequency and buckling load under external load. Five shell configurations with eight layers are considered as candidate designs. The best is chosen under the given loading and geometric parameters. The best design has the highest design index, which comprises a weighted sum of the individual objectives. This leads to optimal laminate configuration which can be function in a satisfactory manner under a range of loading conditions. The effect of different weighting ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal designs is investigated. Graphs showing the effect of the fibre angle on the design index illustrate that the maximum design index occurs at a specific value of the fibre angle (referred to as the optimum fibre angle) and this value can be several times higher than the design index at other fibre angles. This fact emphasises the importance of carrying out optimization in design work of this nature to obtain the best performance of laminated composite shells. On the other
hand, generally speaking, as the weighting ratio increases, the maximum design index increases. That is, the fundamental frequency generally has a more significant effect than the buckling load on the maximum design index. The weighting ratio generally has not a marked effect on the optimum fibre angles. As the shell length increases, the maximum design index decreases. The shell length has a marked effect on the optimum fibre angles but hopt = 0° for larger L/R ratios. As the shell thickness increases, the MODI increases. It is hoped that the tabulated results could be useful to engineers and designers and may also serve as benchmark solutions for other academic research workers. References [1] Walker M, Smith RE. A technique for the multiobjective optimisation of laminated composite structures using genetic algorithms and finite element analysis. Comput Struct 2003;62:123–8. [2] Kere P, Koski J. Multi-criterion stacking sequence optimization scheme for composite laminates subjected to multiple loading conditions. Comput Struct 2001;54:225–9. [3] Spallino S, Rizzo S. Multi-objective discrete optimization of laminated structures. Mech Res Commun 2002;57:17–25. [4] Wang K, Kelly D, Dutton S. Multi-objective optimization of composite aerospace structures. Comput Struct 2002;57:141–8. [5] Walker M. Multi-objective design of laminated plates for maximum stability using finite element method. Comput Struct 2001;54:389–93.
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[6] Walker M, Reiss T, Adali S. Multiobjective design of laminated cylindrical shells for maximum torsional and axial buckling loads. Comput Struct 1997;62(2):237–42. [7] Adali S, Walker M, Verijenko VE. Multiobjective optimization of laminated plates for maximum prebuckling, buckling and postbuckling strength using continuous and discrete ply angles. Comput Struct 1996;35(1):117–30. [8] Topal U, Uzman Ü. Optimum design of laminated composite plates to maximize fundamental frequency using MFD method. Struct Eng Mech 2006;24(4):479–91. [9] Topal U, Uzman Ü. Thermal buckling load optimization of laminated composite plates. Thin-Walled Struct 2008;46:667–75.
[10] Topal U, Uzman Ü. Frequency optimization of laminated composite angle-ply plates with circular hole. Mater Des 2008;29(8):1512–7. [11] Topal U, Uzman Ü. Maximization of buckling load of laminated composite plates with central circular holes using MFD method. Struct Multidiscip Optim 2008;35(2):131–9. [12] Qian W, Liu GR, Chun L, Kam KY. Active vibration control of composite laminated cylindrical shells via surface-bonded magnetostrictive layers. Smart Mater Struct 2003;12(6):889–97. [13] Pratap G. A C0 continuous 4-noded cylindrical shell element. Comput Struct 1985;21:995–9.
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
Multiobjective optimization of laminated composite cylindrical shells for maximum frequency and buckling load Umut Topal * Gümüsßhane University, Department of Civil Engineering, 29000 Gümüsßhane, Turkey
a r t i c l e
i n f o
Article history: Received 25 July 2008 Accepted 17 September 2008 Available online 24 September 2008 Keywords: B. Laminates E. Mechanical
a b s t r a c t This paper deals with multiobjective optimization of laminated cylindrical shells to maximize a weighted sum of the frequency and buckling load under external load. The layer fibre orientation is used as the design variable and the multiobjective optimization is formulated as the weighted combinations of the frequency and buckling under external load. The first order shear deformation theory is used for the finite element formulation of the laminated shells. Five shell configurations with eight layers are considered as candidate designs. The modified feasible direction method (MFD) is used as optimization routine. For this purpose, a program based on FORTRAN is used for the optimization of the laminated shells. Finally, the effect of different weighting ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal designs is investigated and the results are compared. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Advanced fibre-reinforced composite materials used in modern industries, such as aerospace, automotive, marines, sporting goods and civil engineering applications, have given a promise for better performance in comparison to their isotropic competitors. An increase in the utilization of these materials is due to their physical properties such as higher strength-to-weight ratio, stiffness-toweight ratio and versatility. These physical properties are achieved by combining different materials to meet specific requirement. Enormous benefits that these materials posses attract researchers to explore more into it to script its behavior in a well-defined form to the users. On the other hand, the structures are quite often are subjected to in-plane or external loads which may cause buckling. In addition, the vibration can be problematic when the excitation frequency coincides with the shell’s resonance frequency. Such loadings may occur at different times under in-service conditions, necessitating a design approach which is capable of taking in to account these various loading conditions. In the recent years, optimization approaches are focused on the multiobjective problems of the composite structures. For example, Walker and Smith [1] described a methodology for using genetic algorithms with the finite element method to minimise a weighted sum of the mass and deflection of fibre reinforced structures with several design variables. The design constraint implemented was based on the Tsai–Wu failure criterion. The four fibre orientations and laminae thicknesses were to be determined optimally by * Tel.: +90 462 377 4017. E-mail address: [email protected] 0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2008.09.020
defining a design index comprising a weighted average of the objective functions and determining the minimum. Kere and Koski [2] studied the stacking sequence optimization of laminated plates considering failure margins in each loading case as the objective functions. In some design considerations, different design criteria were to be introduced simultaneously in the optimization problem in a competitive manner in which improvement in one objective function leads to a decrease in another. The same approach was chosen by Rizzo and Spallino [3] in optimization of composite laminates with respect to elastic strength, structural bucking, and thermal buckling conditions. Using evolutionary based optimization algorithms, two objective functions were combined and competitively optimized through the application of game theory in the genetic algorithm. In another research, considering the importance of low weight of composite laminates used in aerospace industry, Wang et al. [4] considered the weight and cost of these structures as the optimization criteria, combination of which is performed using Pareto method and the optimal solution is obtained out of a curve extracted from the method. Walker [5] maximized the weighted sum of the critical buckling load and the resonance frequency for a given laminated plate thickness by optimally determining the fibre orientations. In the optimization stage, golden section method was employed. Walker et al. [6] obtained the multiobjective design of a symmetrically laminated shell with the objectives defined as the maximization of the axial and torsional buckling loads. The ply angle was taken as the optimizing variable and the performance index was formulated as the weighted sum of individual objectives in order to obtain Pareto optimal solutions of the design problem. Single objective design results were obtained and compared with the multiobjective design. The effect
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of weighting factors on the optimal design was investigated. Results were given illustrating the dependence of the optimal fibre angle and performance index on the cylinder length, radius and wall thickness. Adali et al. [7] investigated the optimal design of uniaxially loaded laminated plates subject to elastic in-plane restraints along the unloaded edges for a maximum combination of prebuckling stiffness, postbuckling stiffness and buckling load. The results were also obtained for biaxially loaded plates without elastic restraints. The method of solution involves defining a design index comprising a weighted average of the objective functions and identifying candidate configurations which had to be optimized and compared to determine the best stacking sequence. This paper deals with multiobjective optimization of laminated cylindrical shells to maximize a weighted sum of the frequency and buckling load under external load. The layer fibre orientation is used as the design variable and the multiobjective optimization is formulated as the weighted combinations of the frequency and buckling under external load. The first order shear deformation theory is used for the finite element formulation of the laminated shells. Five shell configurations with eight layers are considered as candidate designs. The modified feasible direction method (MFD) is used as optimization routine. For this purpose, a program based on FORTRAN is used for the optimization of the laminated shells. Finally, the effect of different weighting ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal designs is investigated and the results are compared.
nite number, N, of uniform-thickness orthotropic layers as shown in Fig. 1. The structure is referenced in an orthogonal coordinate system (x, h, z), where x, h and z are the longitudinal, circumferential and radial directions, respectively. Based on the first order shear deformation theory, the in-plane and transverse displacements are assumed in the following form:
uðx; h; z; tÞ ¼ uo ðx; h; tÞ þ z/x ðx; h; tÞ vðx; h; z; tÞ ¼ vo ðx; h; tÞ þ z/h ðx; h; tÞ
ð1Þ
wðx; h; z; tÞ ¼ wo ðx; h; tÞ Here, u0, v0, w0 reply for the displacements of a point in the mid or reference surface in longitudinal, circumferential, and transverse directions, respectively; /x and /h are the rotations of normal to the mid-surface about y- and x-axes, respectively. The in-plane stress resultants Nx, Nh, Nxh, Nhx, the stress-couple resultants Mx, Mh, Mxh, Mhx and the transverse stress resultants Qx, Qh in the laminated shell are
fNx ; Nh ; Nxh ; N hx g ¼
N Z X
N Z X k¼1
fQ x ; Q h g ¼
N Z X k¼1
zkþ1
frx ð1 þ z=RÞ; rh ; rxh ð1 þ z=RÞ; rx gdz
zk
k¼1
fMx ; M h ; Mxh ; M hx g ¼
zkþ1
zkþ1
frx ð1 þ z=RÞ; rh ; rxh ð1 þ z=RÞ; rx gz dz
zk
frxz ð1 þ z=RÞ; rhz gdz
zk
2. Basic equations Consider a fibre-reinforced laminated composite circular cylindrical shell of finite length L, radius R, total wall thickness h P (h ¼ hi ; hi represents thickness of a layer) and composed of fi-
ð2Þ The stress–strain relations are expressed as follows:
8 > > > > > > <
9
38
2
9
rx > Q Q 12 0 0 Q 16 > > > > ex > > 6 11 > 7> > > > > eh > rh > > > 7 6 Q Q 0 0 Q 22 26 > = 6 12 = < 7 7 c rhz ¼ 6 0 0 Q Q 0 44 45 hz 7 6 > 6 > > > > 7> > > > cxz > r > > 4 0 > > > 0 5> 0 Q 45 Q 55 > > > > > > xz > > ; ; : : cxh rxh Q 16 Q 26 0 0 Q 66
ð3Þ
where Q ij is the transformed reduced stiffness component. The strain of the shell is expressed as follows:
8 > > > > > > <
9
9
8
ou ex > > > ox > > > > > > > > ov w> > > > eh > þ > = < Roh R > = ov ow chz ¼ oz þ Roh > > > > > > > ou > > > > cxz > þ ow > > > > > > > oz ox > > > > ; : ov ou > ; : cxh þ ox Roh
ð4Þ
By performing the similar procedure in Eq. (1), the strain components are obtained as the forms:
8 > > > > > > <
9
9
8
8
9
ex > e0x > e1x > > > > > > > > > > > > > > > > > 0 1 > > > > > > eh > > = > = = < eh > < eh > 0 1 chz ¼ chz þ z chz > > > > > > > > > > > > > > > cxz > > > > > > > c0xz > c1xz > > > > > > > > > > > > ; : 0 ; ; : : 1 > cxh cxh cxh
ð5Þ
where
8 > > > > > > <
9
8
9
ou0
8
9
8
2
9
o u > > > > e0x > e1x > ox > > > > > oxoz > > > > > > > > > > > > > > ov w > > > > > > > 0 0 1 > > > > > > o2 v þ e0h > e > > > > > > = = < Roh R = < h = < Rohoz > ov0 ow0 0 1 0 1 fe g ¼ chz ¼ þ Roh ; fe g chz ¼ 0 oz > > > > > > > > > > > > > > > > > > > > > ou0 0 > > > > > > > > > c0xz > c1xz > þ ow 0 > > > > > > > oz ox > > > > > > > > > > > : 0 ; > : 1 ; > 2 2 ; ; : : o v o u ov0 ou0 cxh c þ xh þ oxoz Rohoz
Fig. 1. Geometry of circular cylindrical laminated shell (a) and cross-sectional view of the laminated cylindrical shell (b).
ox
Roh
ð6Þ
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The general form of Eq. (2) can be written as
N
¼
M
A 0
0
"
D
# Qx A44 e0 ; ¼ K Qh A45 e1
A45 A55
gives us the element governing equations of motion as follows:
"
c0xz c0xh
#
½M e f€de g þ ½K e fde g ¼ f0g ð7Þ
where Aij and Dij are extensional and bending stiffnesses, respectively, which are defined in terms of the lamina stiffness Q ij as
fAij ; Dij g ¼
N Z X k¼1
zkþ1
Q ij ð1; z2 Þdz
ð8Þ
zk
where zk and zk+1 denote the distances from the shell reference surface to the outer and inner surfaces of the kth layer and K is the shear correction factor. In this study, the shear correction factor is taken 5/6 [12]. In this study, a C0 four-noded quadrilateral shear flexible shell element having five degrees of freedom per node (u, v, w, /x, /y) developed based on the field consistency approach [13] is employed for the laminated cylindrical shells. The interpolation function of the displacement field is defined as
ue0 ¼
Nn X
uei Nei ;
ve0 ¼
Nn X
i¼1
/ex
¼
Nn X
vei Nei ;
i¼1
wexi Nei ;
/ey
¼
i¼1
Nn X
we0 ¼
Nn X
wei Nei
i¼1
ð9Þ
weyi Nei
i¼1
where Ni represents the element interpolation functions and Nn is the number of nodes per element. When the FEA based on the Reissner–Mindlin shell theory is applied to thin shells, the shear locking may occur. Reduced/selective integration technique is employed to evaluate the transverse shear stresses, while full integration is used for bending. The stiffness matrix of the shell is obtained by using the minimum potential energy principle. The kinetic and strain energy of the shell can be found aswarm
T¼
1 2
Z Z "X N Z k¼1
hkþ1 hk
qk fu_ k v_ k w_ k gxfu_ k v_ k w_ k gT 1 þ
# z dz dx dy ð10Þ R
# ) Z Z "(X N Z hkþ1 1 z T U¼ frg feg 1 þ dz dx dy 2 R hk k¼1
ð11Þ
ð12Þ
ð13Þ
e
where [M ] and [K ] are the element mass and stiffness matrices, respectively. After constructing the element matrices, through the finite elements assembly procedure, the global finite element matrices are found to form the finite element equation, the free vibration problem of the shell becomes as follows:
½½K þ x2 ½Mfdg ¼ f0g
ð14Þ
which is an eigenvalue problem of which the solution with an appropriate method will give the natural frequencies (x) of the laminated cylindrical shell. Again following the general assembly procedure in the finite element solution, governing equation for the buckling analysis of the laminated cylindrical shells is obtained as
½½K þ k½K G fdg ¼ f0g
ð15Þ
where [K] and [KG] are the global structural and geometric stiffness matrices, respectively. Through an eigenvalue solution, the eigenvalues k of which the multiplication by the initial in-plane load would give the critical buckling load of the shell. In the all computations, the following non-dimensionalized quantities are used:
N ¼
Ncr ; N0
F ¼
x x0
ð16Þ
where N0 and x0 are the critical buckling load and fundamental frequency, respectively, corresponding to a prescribed lamination angles (0°/0°/ /0°)sym for eight layered laminated cylindrical shell. 3. Optimization problem The optimization problem is formulated in order to find the best orientation angles of fibres in the laminated cylindrical shells so that to simultaneously maximize the fundamental natural frequency and critical buckling load of the shell with the laminate configurations given by a combination of h, 0°, 90° ply angles. The multiobjective design index, MODI, can be describes as follows:
MODI ¼ gF þ nN
_ k g are density and velocity vector of the kth where qk and fu_ k v_ k w layer, respectively. Applying Lagrangian equations of motion:
d=dt½oðT UÞ=od_ i ½oðT UÞ=odi ¼ 0 i ¼ 1 : N
e
ð17Þ
where g and n are the weighting coefficients summing the two objective functions with g, n P 0, g + n = 1. Thus, the optimization problem can be expressed as follows:
max max MODIi i ¼ 1; 2; . . . ; I i
h
MFD 1. q=0, X q = X m 2. q=q+1. Evaluate the objective function F( X q −1 )
3. Calculate the gradient of the objective function ∇ F( X q −1 ) 4. Find the usable-feasible direction Sq 5. Perform a one-dimensional search X q = X q −1 + α Sq 6. Check convergence. If satisfied, go to 7. Otherwise go to 2 7. X m =X q Fig. 2. Flow chart of modified feasible direction method.
ð18Þ
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U. Topal / Materials and Design 30 (2009) 2584–2594
od. A search direction Sq, is determined, and a one-dimensional search is made to find a. The above procedure is repeated with the new design vectors, until the design satisfies the optimality conditions or some other termination criterion. The objective function F(Xi) is accurately modelled as a quadratic polynomial approximation around the current iterate Xi as in the following equation:
where I denotes the number of candidate laminates. In this study, five laminate configurations are considered as candidate designs. These configurations are specified as {h, h, h, h}sym, {h, h, 90,0}sym, {h, h, 0,0}sym, {h, h, 0, 90}sym and {h, h, 90,90}sym and referred to as laminate type C1, C2, C3, C4 and C5, respectively. 4. Modified feasible direction method
FðX i Þ ¼ a0 þ
The MFD method is one of the most powerful methods for optimization problems. This method takes into account not only the gradients of objective function and constraints, but also the search direction in the former iteration. In this study, there is no any constraint. Fig. 2 shows the iterative process within each optimization process [8–11]. In Fig. 2, Xq and Xq1 are the design variable vectors in two consecutive cycles of iteration. The procedure starts with an initial design vector, X0, i.e. q = 0, q = q + 1, and the objective function F(Xi), is evaluated. rF(Xi), the gradients of objective function with respect to design variable is calculated using finite difference meth-
a
MODI
Nd X
bi X 2i
ð19Þ
i¼1
where Nd and Xi are number of design variables and ith design variable, respectively. ai and bi are the coefficients of polynomial function determined by a least squares regression. After the objective function is approximated, their gradients with respect to the design variables are calculated by finite differences methods. The solving process is iterated until convergence is achieved.Convergence or termination checks are performed at the end of each optimization loop. The optimization process continues until either convergence or termination occurs. The process may be terminated before convergence in two cases:
b
2
η=0
2
η=0.25
1.6
1.2
1.2 C1 C2 C3 C4 C5
0.8
0.4
C1 C2 C3 C4 C5
0.8
0.4 0
2
10
20
30
40
θ
50
60
70
80
0
90
d
η=0.5
10
20
30
40
θ
50
60
70
80
90
60
70
80
90
2
η=0.75 1.6
1.6
1.2
1.2
C1 C2 C3 C4 C5
0.8
C1 C2 C3
0.8
C4 C5
0.4
0.4 0
10
20
30
e
40
θ
50
60
70
80
90
0
10
60
70
20
30
40
θ
50
2
η=1 1.6
MODI
MODI
ai X i þ
i¼1
1.6
c
Nd X
C1 C2 C3 C4 C5
1.2
0.8
0.4 0
10
20
30
40
θ
50
80
90
Fig. 3. The dependence of the MODI on fibre angle for five different laminate configurations.
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jF current F best j sF
The number of design sets so far exceeds the maximum number of optimization loops. If the initial design is infeasible and the allowed number of consecutive infeasible designs has been exceeded. The optimization problem is terminated if all of the following conditions are satisfied: The current design is feasible. Changes in the objective function F: (a) The difference between the current value and the best design so far is less than the tolerance sF:
a
(b) The difference between the current value and the previous design is less than the tolerance:
jF current F current1 j sF Changes in the design variables Xi: (a) The difference between the current value of each design variable and the best design so far is less than the respective tolerance si:
jX icurrent X ibest j si
b
2
2
C2
C1 1.6
MODI
1.6 1.2
η=0 η=0.25 η=0.5 η=0.75 η=1
η=0.25 η=0.5
0.8
0.8
η=0.75 η=1
0.4 0
c
1.2
η=0
10
20
30
40
50
θ
60
70
80
0.4
90
0
d
2
20
30
40
θ
50
60
70
80
90
60
70
80
90
2
C3
C4 1.6
1.6
MODI
10
1.2
1.2 η=0 η=0.25 η=0.5 η=0.75 η=1
η=0 η=0.25 η=0.5
0.8
0.8
η=0.75 η=1
0.4 0
10
20
30
40
0.4 50
60
70
80
90
0
10
20
70
80
90
30
40
θ
e
θ
50
2
C5
MODI
1.6
1.2 η=0 η=0.25 η=0.5 η=0.75 η=1
0.8
0.4 0
10
20
30
40
θ
50
60
Fig. 4. The dependence of the MODI on fibre angle for different weighting ratios.
Table 1 The best configurations and (MODI)max of the five different candidates for weighting ratios Laminate configuration
C1 C2 C3 C4 C5
hopt (°)
(MODI)max
g=0
g = 0.25
g = 0.5
g = 0.75
g=1
g=0
g = 0.25
g = 0.5
g = 0.75
g=1
90.0 32.1 37.0 39.8 81.1
90.0 37.5 43.0 50.0 80.2
90.0 42.5 47.0 49.0 78.0
37.2 45.5 49.5 49.0 76.2
37.4 48.2 50.8 49.5 40.4
1.89 1.24 1.22 1.13 1.90
1.76 1.34 1.33 1.34 1.78
1.64 1.45 1.46 1.45 1.66
1.65 1.58 1.59 1.57 1.54
1.82 1.71 1.73 1.68 1.58
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(b) The difference between the current value of each design variable and the previous design is less than the respective tolerance:
almost the same for C2, C3, C4 and C1, C5 laminate configurations for 0.25 6 g 6 0.75 and g > 0:75, respectively. As seen from Fig. 4, the curves for C1 and C5 laminate configurations intersect approximately at 68.3° and 56.1°, respectively. After the intersecting points, the trend in the MODI differs suddenly. As the weighting ratio increases, the (MODI)max increases for C2, C3 and C4 laminate configurations. As seen from Table 1, the optimum fibre angles increase with increase in the weighting ratio for C2 and C3 laminate configurations. The optimum fibre angles decrease suddenly for C1 and C5 laminate configurations for g > 0:5 and g > 0:75, respectively. But, the optimum fibre angle remains constant for C4 laminate configuration for g > 0.
jX icurrent X icurrent1 j sF We resolved the optimization process to obtain global maximum from different initial points to check if other solutions are possible. The converge tolerance ratio is considered 0.01 for objective function. 5. Numerical results and discussion Numerical results are given for T300/5208 graphite/epoxy material for which E1 = 181 GPa, E2 = 10.3 GPa, G12 = 7.17 GPa, m12 = 0.28, q = 1600 kg/m3. The laminated cylindrical shell is constructed of equal thickness layers. H/R = 0.2 and L/R = 1 are considered.
5.2. Effect of shell lengths on the optimal designs The effect of shell lengths on the optimal designs is given in Figs. 5–7 for five laminate configurations. As seen from the figures, as the shell length increases, the MODI decreases. But, this decrease in the MODI diminishes with increase in the shell length. It can be said that, the MODI remains fairly constant for larger shell length. As the fibre angle increases, the MODI generally decreases for L/R = 2, 4 and 6. The optimum fiber angles and (MODI)max are given for all laminate configurations for shell lengths in Tables 2–4. As seen, the (MODI)max remains practically constant at all
5.1. Effect of weighting ratios on the optimal designs The effect of five different weighting ratios on the optimal results is given for different five laminate configurations in Figs. 3 and 4 and Table 1. As seen from Fig. 3, the best laminate configuration is C5 for g = 0, 0.25 and 0.5. On the other hand, the best laminate configuration is C1 for g = 0.75 and 1. The (MODI)max is
b
2
a
η=0.25, C1
1.6
η=0.25, C2 1.2 L/R=1
1.2
MODI
L/R=2
0.4
0.4 0 0
10
20
30
40
θ
50
60
70
80
0
90
d
1.6 η=0.25, C3 1.2
0
10
20
30
40
θ
50
60
70
L/R=1 L/R=2
0.4
0.4 0
L/R=4 L/R=6
0.8
L/R=4 L/R=6
90
η=0.25, C4
L/R=2
0.8
80
1.6 1.2
L/R=1
0 0
10
20
30
40
θ
e
50
60
70
80
90
0
10
20
30
40
2 η=0.25, C5
1.6
MODI
MODI
L/R=1 L/R=2 L/R=4 L/R=6
0.8
L/R=4 L/6=6
0.8
c
1.6
L/R=1
1.2
L/R=2 L/R=4
0.8
L/R=6
0.4 0
0
10
20
30
40
θ
50
60
70
80
90
Fig. 5. The dependence of the MODI on fiber angle for shell lengths for g = 0.25.
θ
50
60
70
80
90
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a
b
2
η=0.75, C1
MODI
1.6
2
η=0.75, C2
1.6 L/R=1
1.2
L/R=1
1.2
L/R=2
L/R=2 L/R=4
L/R=4 L/R=6
0.8
0.8
0.4
L/R=6
0.4 0
0 0
10
20
30
40
50
60
70
80
0
90
10
20
30
40
θ
c
2
MODI
d
η=0.75, C3
1.6
50
60
70
80
90
θ 2
η=0.75, C4
1.6 L/R=1
1.2
L/R=4
0.8
L/R=1
1.2
L/R=2
L/R=2 L/R=4
0.8
L/R=6
0.4
L/R=6
0.4
0 0
10
20
30
40
50
60
70
80
0
90
0
10
20
30
40
θ
50
60
70
80
90
θ
e
2
η=0.75, C5
1.6
MODI
L/R=1
1.2
L/R=2 L/R=4 L/R=6
0.8 0.4 0 0
10
20
30
40
50
60
70
80
90
θ Fig. 6. The dependence of the MODI on fibre angle for shell lengths for g = 0.5.
laminate configurations for all weighting ratios for L/R = 4 and 6. The (MODI)max increases with increase in the weighting ratio for L/R = 2. The optimum fibre angles increase as the weighting ratio increases for L/R = 2 and 4. The optimum fibre angles are zero for L/R = 6 except for C1 and C3 laminate configurations for g = 0.75. The best laminate configuration is mostly C4 for L/R = 2, 4 and 6 regardless of the weighting ratios. The optimum fibre angles and the best laminate configurations which must be selected in order to maximize the MODI and obtain the ultimate structural performance is demonstrated by these figures and tables. 5.3. Effect of shell thickness on the optimal designs The effect of shell thickness on the optimal designs is given in Fig. 8 for five different laminate configurations. As seen from Fig. 8, as the shell thickness increases, the MODI increases. But this increase in the MODI diminishes for larger H/R ratios. The optimum fibre angles and (MODI)max are given for all laminate configurations for shell thickness in Table 5. As seen, the optimum fibre orientation is 90° for C1 and C5 laminate configurations for all H/R ratios except for H/R = 1 for C5 laminate configuration. On the
other hand, the optimum fibre orientation increases with increase in the H/R for C2, C3 and C4 laminate configurations. The best laminate configuration is C5 for H/R = 0.2 and 0.4 for g = 0.5. However, the best laminate configurations are C1 and C5 for H/R = 0.5 and 0.6 for g = 0.5. 5.4. Effect of boundary conditions on the optimal designs The different combinations of free (F), simply supported (S) and clamped (C) boundary conditions are considered, viz. clamped/ clamped (CC), clamped/simply supported (CS), simply supported/ free (SF) and clamped/free (CF). The boundary conditions are defined as below: 1. (SS) boundary condition At x = 0 and x = L, u0 = w0 = /x = 0. 2. (CC) boundary condition At x = 0 and x = L, u0 = v0 = w0 = /x = /y = 0.
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a
η=0.5, C1
1.6
MODI
b 1.6
2
L/R=1
L/R=1
1.2
L/R=2
L/R=2
0.8
L/R=4
0.8
L/R=4 L/R=6
L/R=6
0.4
0.4 0 0
c
η=0.5, C2
1.2
10
20
30
40
θ
50
60
70
80
0
90
0
d 1.6
1.6
20
30
40
θ
50
60
70
80
1.2
L/R=1
L/R=1 L/R=2
MODI
L/R=2 L/R=4
0.8
90
η=0.5, C4
η=0.5, C3 1.2
10
L/R=4
0.8
L/R=6
L/R=6
0.4
0.4
0
0 0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
θ
e
60
70
80
90
2
η=0.5, C5
1.6
MODI
50
θ
L/R=1
1.2
L/R=2 L/R=4
0.8
L/R=6
0.4 0
0
10
20
30
40
50
60
70
80
90
θ Fig. 7. The dependence of the MODI on fibre angle for shell lengths for g = 0.75.
Table 2 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 2)
Table 3 Optimum fibre angles and (MODI)max for for five different laminate configurations (L/R = 4)
Laminate configuration
Laminate configuration
hopt (°)
g = 0.25
g = 0.5
g = 0.75
g = 0.25
g = 0.5
g = 0.75
C1 C2 C3 C4 C5
7.0 26.0 1.0 26.4 25.5
9.0 27.4 5.0 27.3 27.0
11.5 29.0 9.0 30.0 29.4
0.43 0.47 0.43 0.49 0.48
0.43 0.47 0.43 0.49 0.48
0.44 0.47 0.44 0.49 0.48
C1 C2 C3 C4 C5
hopt (°)
(MODI)max
g = 0.25
g = 0.5
g = 0.75
g = 0.25
g = 0.5
g = 0.75
15.5 18.5 17.6 20.5 17.5
35.5 29.6 34.4 31.1 27.9
42.9 38.1 44.0 38.5 35.4
0.81 0.79 0.80 0.82 0.78
0.82 0.79 0.79 0.82 0.79
0.91 0.83 0.82 0.85 0.82
3. (CS) boundary condition At x = 0, u0 = v0 = w0 = /x = /y = 0. At x = L, u0 = w0 = /x = 0. 4. (SF) boundary condition At x = 0, u0 = w0 = /x = 0. 5. (CF) boundary condition At x = 0, u0 = v0 = w0 = /x = /y = 0.
(MODI)max
The effect of end conditions on the optimal designs is given for laminated shells in Fig. 9 and Table 6 (L/R = 1, H/R = 0.2, g = 0.5). As seen from Fig. 9, the best laminate configuration is C5 for (CC) and (CS) boundary conditions, on the other hand, the best laminate configuration is generally C4 for (CF) and (SF) boundary conditions. The optimum fibre angles and (MODI)max are given for all laminate configurations for boundary conditions in Table 6. As seen, the (CC) boundary condition has the largest (MODI)max, on the other hand, (SF) boundary condition has the smallest ones. The optimum fibre
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Table 4 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 6) Laminate configuration
hopt (°)
C1 C2 C3 C4 C5
a
14
η=0.5,C1
12
g = 0.25
g = 0.5
g = 0.75
g = 0.25
g = 0.5
g = 0.75
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
19.0 0.0 21.0 0.0 0.0
0.38 0.38 0.38 0.41 0.38
0.38 0.39 0.38 0.41 0.39
0.38 0.40 0.38 0.42 0.40
b
H/R=0.2 H/R=0.4
10
MODI
(MODI)max
5
η=0.5,C2
4
H/R=0.5 H/R=0.6
8
3
6
H/R=0.2 H/R=0.4
2
H/R=0.5
4 2 0
c
0
4
10
20
30
40
50
θ
60
70
MODI
80
0
90
d
η=0.5,C3
3
H/R=0.2 H/R=0.4
2
0
5
10
20
30
η=0.5,C4
40
θ
50
60
70
80
90
H/R=0.2
4
H/R=0.4
3
H/R=0.6
H/R=0.5
H/R=0.5
2
H/R=0.6
1 0
H/R=0.6
1
1
0
10
20
30
40
50
60
70
80
0
90
0
10
20
30
40
θ
e
50
60
70
80
90
14 H/R=0.2
η=0.5,C5
12
H/R=0.4 H/R=0.5
10
MODI
θ
H/R=0.6
8 6 4 2 0
0
10
20
30
40
θ
50
60
70
80
90
Fig. 8. Effect of shell thickness on the optimal results for different five laminate configurations (L/R = 1, g = 0.5).
Table 5 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 1, g = 0.5) Laminate configuration
C1 C2 C3 C4 C5
hopt (°)
(MODI)max
H/R = 0.2
H/R = 0.4
H/R = 0.5
H/R = 0.6
H/R = 0.2
H/R = 0.4
H/R = 0.5
H/R = 0.6
90.0 42.5 47.0 49.0 78.0
90.0 56.1 65.5 71.2 90.0
90.0 64.1 69.7 78.2 90.0
90.0 80.0 72.5 80.5 90.0
1.64 1.45 1.46 1.45 1.66
6.16 3.11 2.88 3.26 6.66
9.57 3.88 3.32 3.95 9.57
12.76 4.68 3.81 4.66 12.76
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a
b
3
(CC)
2.5
(CS) 2
2
MODI
2.5
1.5
1.5
C1
1
C3 C5
0 0
10
20
30
40
50
C2 C3
0.5
C4
0.5
C1
1
C2
C4 C5
0 60
70
80
90
0
10
20
30
40
θ 1
MODI
c
1
d
(CF) 0.8
0.8
0.6
0.6 C1
0.4
C3 C5
0
10
20
30
40
θ
50
70
80
80
90
60
70
80
90
C2 C3 C4 C5
0
90
70
C1
0 60
60
(SF)
0.2
C4
0
50
0.4
C2
0.2
θ
10
20
30
40
θ
50
Fig. 9. Effect of boundary conditions for optimal designs for different five laminate configurations (L/R = 1, H/R = 0.2, g = 0.5).
Table 6 Optimum fibre angles and (MODI)max for five different laminate configurations (L/R = 1, H/R = 0.2, g = 0.5) Laminate configuration
hopt (°) Boundary condition (CC)
Boundary condition (CS)
Boundary condition (CF)
Boundary condition (SF)
Boundary condition (CC)
(MODI)max Boundary condition (CS)
Boundary condition (CF)
Boundary condition (SF)
C1 C2 C3 C4 C5
90.0 47.5 51.0 55.8 78.2
90.0 45.8 49.8 53.5 90.0
24.1 22.5 26.3 23.7 22.8
28.5 20.8 30.0 21.7 21.5
2.67 1.85 1.78 1.88 2.69
2.14 1.63 1.60 1.64 2.14
0.70 0.77 0.70 0.78 0.78
0.65 0.72 0.66 0.74 0.72
orientation suddenly decreases as the rigidity of the laminated shell decreases. 6. Conclusions In this study, the multiobjective optimization of laminated cylindrical shells is carried out to maximize a weighted sum of the frequency and buckling load under external load. Five shell configurations with eight layers are considered as candidate designs. The best is chosen under the given loading and geometric parameters. The best design has the highest design index, which comprises a weighted sum of the individual objectives. This leads to optimal laminate configuration which can be function in a satisfactory manner under a range of loading conditions. The effect of different weighting ratios, shell aspect ratio, shell thickness-to-radius ratios and boundary conditions on the optimal designs is investigated. Graphs showing the effect of the fibre angle on the design index illustrate that the maximum design index occurs at a specific value of the fibre angle (referred to as the optimum fibre angle) and this value can be several times higher than the design index at other fibre angles. This fact emphasises the importance of carrying out optimization in design work of this nature to obtain the best performance of laminated composite shells. On the other
hand, generally speaking, as the weighting ratio increases, the maximum design index increases. That is, the fundamental frequency generally has a more significant effect than the buckling load on the maximum design index. The weighting ratio generally has not a marked effect on the optimum fibre angles. As the shell length increases, the maximum design index decreases. The shell length has a marked effect on the optimum fibre angles but hopt = 0° for larger L/R ratios. As the shell thickness increases, the MODI increases. It is hoped that the tabulated results could be useful to engineers and designers and may also serve as benchmark solutions for other academic research workers. References [1] Walker M, Smith RE. A technique for the multiobjective optimisation of laminated composite structures using genetic algorithms and finite element analysis. Comput Struct 2003;62:123–8. [2] Kere P, Koski J. Multi-criterion stacking sequence optimization scheme for composite laminates subjected to multiple loading conditions. Comput Struct 2001;54:225–9. [3] Spallino S, Rizzo S. Multi-objective discrete optimization of laminated structures. Mech Res Commun 2002;57:17–25. [4] Wang K, Kelly D, Dutton S. Multi-objective optimization of composite aerospace structures. Comput Struct 2002;57:141–8. [5] Walker M. Multi-objective design of laminated plates for maximum stability using finite element method. Comput Struct 2001;54:389–93.
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[6] Walker M, Reiss T, Adali S. Multiobjective design of laminated cylindrical shells for maximum torsional and axial buckling loads. Comput Struct 1997;62(2):237–42. [7] Adali S, Walker M, Verijenko VE. Multiobjective optimization of laminated plates for maximum prebuckling, buckling and postbuckling strength using continuous and discrete ply angles. Comput Struct 1996;35(1):117–30. [8] Topal U, Uzman Ü. Optimum design of laminated composite plates to maximize fundamental frequency using MFD method. Struct Eng Mech 2006;24(4):479–91. [9] Topal U, Uzman Ü. Thermal buckling load optimization of laminated composite plates. Thin-Walled Struct 2008;46:667–75.
[10] Topal U, Uzman Ü. Frequency optimization of laminated composite angle-ply plates with circular hole. Mater Des 2008;29(8):1512–7. [11] Topal U, Uzman Ü. Maximization of buckling load of laminated composite plates with central circular holes using MFD method. Struct Multidiscip Optim 2008;35(2):131–9. [12] Qian W, Liu GR, Chun L, Kam KY. Active vibration control of composite laminated cylindrical shells via surface-bonded magnetostrictive layers. Smart Mater Struct 2003;12(6):889–97. [13] Pratap G. A C0 continuous 4-noded cylindrical shell element. Comput Struct 1985;21:995–9.