Meccanica (2012) 47:719–730 DOI 10.1007/s11012-011-9482-5
Stacking sequence optimization of composite plates for maximum fundamental frequency using particle swarm optimization algorithm H. Ghashochi Bargh · M.H. Sadr
Received: 10 June 2010 / Accepted: 19 September 2011 / Published online: 4 November 2011 © Springer Science+Business Media B.V. 2011
Abstract The paper illustrates the application of the particle swarm optimization (PSO) algorithm to the lay-up design of symmetrically laminated composite plates for maximization of fundamental frequency. The design variables are the fiber orientation angles, edge conditions and plate length/width ratios. The formulation is based on the classical laminated plate theory (CLPT), and the method of analysis is the semianalytical finite strip approach which has been developed on the basis of full energy methods. The performance of the PSO is also compared with the simple genetic algorithm and shows the good efficiency of the PSO algorithm. To check the validity, the obtained results are compared with those available in the literature and some other stacking sequences, wherever possible. Keywords Composite plates · PSO algorithm · Finite strip · Optimization 1 Introduction Fiber reinforced composite laminated structures are widely used as structural components in various H. Ghashochi Bargh () · M.H. Sadr Aerospace Engineering Department, Center of Excellence in Computational Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran e-mail:
[email protected] M.H. Sadr e-mail:
[email protected]
branches of engineering, in particular in the aerospace industry due to superior characteristics in design, i.e. variable stacking sequence and ply angles, when compared to conventional materials. However, as a direct consequence of the number of variables involved and of the intrinsic anisotropy of the individual layers, the reliable and economical design of composite laminated structures is usually more complex than the one associated with isotropic material structures. Furthermore, vibration can be a problem when the excitation frequency coincides with the resonance frequency, which is essential to maximize the fundamental frequency in laminated composite structures. Several researchers have reported different studies on vibration of laminated plates. For simply supported plates, Bert proposed an expression for the plate natural frequency which takes account of the effects of bend-twist coupling in an approximate but simple manner. Optimal design to maximize the fundamental frequency were then determined using an invariant formulation [1]. Bert also proposed a suitable equation to determine the fundamental frequencies of composite plates with clamped boundaries [2]. When considering four plies, he [1, 2] found that the optimal fiber orientation varied from 0◦ to 90◦ with increase in the plate aspect ratio. Grenested evaluated the lowest free vibration frequency both numerically, by the finite difference method and analytically, using a perturbation approach [3]. Chow considered symmetrically laminated angle-ply plates with simply supported or clamped boundaries. Two-dimensional orthogonal polynomials
720
are used in this work to derive the governing system of eigenvalue equations [4]. Fan and Cheung used a finite strip method to analyze rectangular plates with complex edge support conditions [5]. Leissa and Martin studied the vibration and buckling response due to variation in the fibre spacing [6]. Qatu presented natural frequencies and mode shapes for plates with various boundary conditions, material parameters and fibre orientations [7]. Using higher-order mixed theory, Rao and Desai presented a semi-analytical method to evaluate the natural frequencies as well as displacement and stress eigenvectors for laminated and sandwich plates [8]. Kam and Chang traced the optimal lamination arrangement of thick composite plates for maximum bukling load and vibration frequency [9]. Chen and Dawe used a semi-analytical finite stripe method to predict the linear transient response of laminated rectangular plates [10]. Chai determined the natural frequencies of vibration pertaining to antisymmetric cross-ply and angle-ply plates [11]. Narita offered a Ritz-based layerwise optimization approach for the symmetrical composite plates [12]. He reduced evidently the searching time and calculation for the optimal solution. However, his approach may yield a local optimum for some edge conditions. Apalak et al. determined the optimal layer sequences of the symmetrical composite plates using Genetic Algorithm, artificial neural networks and finite element method [13]. In the current paper, fundamental frequency optimization of symmetrically laminated plates is studied using the particle swarm optimization (PSO) algorithm. The fitness function is computed with a semianalytical finite stripe model developed originally on the basis of full energy method. The main philosophy of this method is to discretize the structure by finite strips and by assuming shape functions for inplane and out-of-plane displacements, comprising the degrees of freedom, the in-plane strains and mid-plane curvatures are evaluated. Furthermore, the variation of displacements in the longitudinal direction is postulated by harmonic function (i.e., semi-analytical FSM formulation) and in the transverse direction by polynomial function. The geometry of a typical finite strip of layered composite material is shown in Fig. 1 indicating the coordinate axes system pertaining to the strip displacements u, v and w with a length of a and width of bs . It is noted that the finite strip method has been used previously by Khalil et al. [14] and Houlston [15] in analyzing the dynamic response of plate structures.
Meccanica (2012) 47:719–730
Fig. 1 Strip geometry and coordinate axes system
However, these studies were limited to homogeneous, isotropic materials. 2 Formulation In this section, the fundamental elements of the theory of the semi-analytical finite strip method (FSM) are briefly presented. The finite strips are assumed to be simply supported or clamped out-of-plane at two ends, and be thin so that the classical laminated plate theory (CLPT) assumptions are applied in this paper. In the CLPT, the Kirchhoff normalcy conditions are assumed and, thus, the displacement parameters of an arbitrary point anywhere on the plate strip can be introduced as follows: ∂w(x, y, t) u(x, ¯ y, z, t) = u(x, y, t) − z ∂x ∂w(x, y, t) (1) v(x, ¯ y, z, t) = v(x, y, t) − z ∂y w(x, ¯ y, z, t) = w(x, y, t) where u, ¯ v¯ and w¯ are components of displacement at the arbitrary point, whilst u, v and w are corresponding ones on the middle surface (z = 0). On the assumption that the plate is in a state of plane-stress, the stress-strain relationship at a general point for the plate becomes: ⎧ ⎫ ⎡ ⎤ ¯ 12 Q ¯ 16 Q¯ 11 Q ⎨ σ¯ x ⎬ ¯ 22 Q ¯ 26 ⎦ .¯ε ; σ¯ = σ¯ y = ⎣ Q¯ 12 Q ⎩ ⎭ ¯ ¯ ¯ 66 τ¯xy Q16 Q26 Q ⎧ ⎫ ⎨ ε¯ x ⎬ ε¯ = ε¯ y (2) ⎩ ⎭ γ¯xy
Meccanica (2012) 47:719–730
721
where Q¯ ij (i, j = 1, 2, 6) are plane-stress stiffness coefficients. The linear strain-displacement relationship can be expressed as follows: ε¯ = ε + zψ
the following form:
1 {ε T .[A].ε + ψ T .[D].ψ}dxdy Us = 2
(3a)
1 = {d}T [k]{d} 2
(7)
where The general expression for the strip kinetic energy ε = {u,x , v,y , u,y + v,x }T ψ = {−w,xx , −w,yy , −2w,xy }T
(3b)
in the above equation ψ is the curvature vector and ε is the mid-plane strains. The constitutive equations for a plate can be obtained through the use of (2) and (3) and by performing appropriate analytical integration through the uniform thickness. The outcome, which is given by the equations: ⎧ ⎫ ⎧ ⎫ Nx ⎪ σ¯ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ny ⎪ ⎪ ⎪ ⎪ σ¯ y ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ h/2 ⎪ ⎨ ⎬ Nxy τ¯xy = dz Mx ⎪ zσ¯ x ⎪ ⎪ −h/2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M ⎪ zσ¯ ⎪ ⎪ ⎪ ⎩ y ⎪ ⎭ ⎩ y⎪ ⎭ Mxy zτ¯xy [A] [B] {ε} = × (4) [B] [D] {ψ} where Nx , Ny and Nxy are the membrane direct and shearing stress resultants per unit length and Mx , My and Mxy are the bending and twisting stress couples per unit length. Moreover, the plate stiffness coefficients are defined as: (Aij , Bij , Dij ) =
n
zk
(Q¯ ij )k (1, z, z2 )dz,
k=1 zk−1
i, j = 1, 2, 6
(5)
Because of the symmetry condition about the mid surface of the subject laminates, the in plane and out of plane coupling stiffness coefficients (Bij ) are zero. The strain energy per unit volume is: 1 Us = σ¯ T ε¯ 2
(6)
using (2)–(5) and integrating through the thickness of the structure with respect to z gives an expression for the strain energy of a finite strip, which can be put into
is: 1 Ts = ρh 2
1 ˙ T ˙ {u˙ 2 + v˙ 2 + w˙ 2 }dxdy = {d} [m]{d} 2 (8)
where [k] is the strip stiffness matrix and [m] is the strip mass matrix and {d} is a column matrix which contains the strip’s degrees of freedom and ρ is a mean mass per unit area of the plate. For the whole structure, the total strain energy, kinetic energy are obtained by summations of the corresponding energy components of all strips. In this way, the whole structural matrices are generated by following the standard FEM assembly procedure. Thus, the equivalents of (7) and (8) for the whole structure can be expressed as: 1 ¯ T ¯ U = {d} [K]{d}, 2 1 ˙¯ T ˙¯ T = {d} [M]{d} 2
(9) (10)
where [K] and [M] are the square symmetric, positivedefinite structure stiffness and consistent mass matri¯ is a vector, which includes the degrees of ces and {d} freedom for the whole structure. The structural equation of motions can be obtained by applying the Lagrange equations as: ¨¯ + [K]{d} ¯ = {0} [M]{d}
(11)
a general solution of (11) may be written as: ¯ = { }eiωt {d}
(12)
where { } are the amplitudes of displacement and ω is the natural frequency in rad/sec. the real port of (12) simply represents a harmonic response of the structure. Substitution of (12) into (11) gives: [[K] − ω2 [M]]{ } = {0}
(13)
722
Meccanica (2012) 47:719–730
This is a general linear eigenvalue problem, which involves determining the natural frequencies ω and mode shape { }. Natural frequency is normalized as a frequency parameter: 1/2 ρ 2 (14) = ωa D0 where the reference bending rigidity is: D0 = E2 h3 /12(1 − ν12 ν21 )
(15)
types and orders, involving undetermined displacement coefficients in the side bounds. In representing the in-plane displacement variations (u, v) across a strip, the linear Lagrange polynomials are used, whilst in representing w the cubic Hermitian polynomial is utilized as in most previous finite strip studies [16–19]. The assumed in-plane displacement and out-ofplane displacement in the full-energy semi-analytical method are: u=
ru ((1 − η)u1i + ηu2i ) cos(iζ x),
(18a)
i=1
The optimum problem consists in finding the stacking sequence which maximizes the normalized frequency for the first natural mode of the symmetric laminated plate. It should be observed that the symmetry requirement it easily enforced by optimizing only one half of the laminate and deriving from symmetry conditions the other half. The optimal design problem can be stated as follows: Find
θ = (θ1 , θ2 , . . . , θk )
Maximize
= (θ1 , θ2 , . . . , θk )
Subject to
−90◦
≤ θk
(16)
≤ 90◦
where k is half of the layer number. The optimal stacking sequences and ply angles are searched with the Elitist-Genetic algorithm. The displacement fields adopted for the vibration analysis are expressed as: u=
rv
Ui (x)giu (y)diu (t),
(17a)
Vi (x)giv (y)div (t),
(17b)
i=1
w=
(18b)
i=1
w=
rw ((1 − 3η2 + 2η3 )w1i + bs (η − 2η2 + η3 )θ1i i=1
+ (3η2 − 2η3 )w2i + bs (η3 − η2 )θ2i )Wi (x) (18c) sin(iζ x) end simply supported Wi (x) = sin(ζ x). sin(iζ x) end clamped where u1i , u2i , v1i and v2i are the undetermined inplane nodal displacement parameters and w1i , w2i , θ1i and θ2i are the undetermined out-of-plane nodal displacement parameters along the edges of the strip and η = y/bs and ζ = π/a.
3 PSO Algorithm
ru i=1
v=
rv v= ((1 − η)v1i + ηv2i ) sin(iζ x),
rw
Wi (x)giw (y)diw (t)
(17c)
i=1
where the Ui (x), Vi (x) and Wi (x) are longitudinal functions that satisfy the kinematic conditions prescribed at the two ends of the strip. In the present paper, the trigonometric functions that present any order of continuity are assumed for Ui (x), Vi (x) and Wi (x). It is noted that ru, rv and rw represent the number of longitudinal terms assumed for the corresponding displacement functions. The gi (y) functions are transverse polynomial interpolation functions of various
Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Eberhart and Kennedy in 1995 [20, 21], inspired by social behavior of bird flocking or fish schooling. PSO shares many similarities with evolutionary computation techniques such as Genetic Algorithms (GA). However, unlike GA, PSO has no evolution operators such as crossover and mutation. Compared to GA, PSO is based on social biology which requires cooperation while GA is based on competition. The advantages of PSO are that PSO is easy to implement and there are few parameters to adjust. In PSO, each single solution is a “bird” in the search space. We call it “particle”. All of particles have fitness values which are evaluated by the fitness function to be optimized, and have velocities which direct the flying of the particles.
Meccanica (2012) 47:719–730
The particles fly through the problem space by following the current optimum particles. Figure 2 shows a basic flow chart of PSO algorithm. PSO is initialized with a group of random particles and then searches
723
for optima by updating generations. In every iteration, each particle is updated by following two “best” values. The first one (p i ) is the best position attained by the particle i in the swarm so far. Another “best” g value (pk−1 ) is the global best position attained by the swarm at iteration k − 1. After finding the two best values, the particle updates its velocity and positions with following (19) and (20) [22–24]. g
i i i + c1 r1 (p i − xk−1 ) + c2 r2 (pk−1 − xk−1 ), vki = wvk−1 (19) i + vki xki = xk−1
Fig. 2 Flow chart of PSO algorithm
(20)
where the superscript i denotes the particle and the subscript k denotes the iteration number; v denotes the velocity and x denotes the position and it is the real number; r1 and r2 are uniformly distributed random numbers in the interval [−1 1]; c1 and c2 are the acceleration constants; w is the inertia weight. In the different reference, it is mentioned that the choice of these constants is problem dependent. In this work, c1 = c2 = w = 1 are chosen which give better optimal results in lesser iterations, and the results also are rounded to nearest integer values after optimization. The performance of the PSO is shown in Fig. 3a and b in comparison with the GA and it shows the good efficiency of the PSO algorithm. In the 8 and 10 layered cases, the optimal values for GA converge in the generation of 13 and 15 with around the initial population of 35 and 43, respectively. The GA parameters such as crossover rate and mutation rate also are
Fig. 3 Comparison of the PSO algorithm and GA results for CCCF and SFCF edge conditions for symmetrically laminated 8-layered rectangular (a/b = 2) plates
724
Meccanica (2012) 47:719–730
selected to be 0.6, and 0.02, respectively. In addition, it is concluded that using of PSO provides a much higher convergence and reduced the CPU time in comparison with the GA.
4 Numerical results and discussions The optimization results of the laminated composite plates (h/a = 0.01) are given for AS/3501 graphite/
Table 1 Comparison of optimal stacking sequences and natural frequency parameter of symmetric 8-layerd composite plates (a/b = 1, increment 1◦ ) Case
Edges BCs
Optimal stacking
opt Narita [12]
Present study
Narita [12]
Present study
1
SFSF
38.69
38.692
[0/0/0/0]S
[0/0/0/0]S
2
SFCF
60.47
60.977
[0/0/0/0]S
[0/0/0/0]S
3
CFCF
87.77
87.919
[0/0/0/0]S
[0/0/0/0]S
4
SSSF
39.84
39.835
[0/0/0/0]S
[0/0/0/0]S
5
SCSF
40.28
40.280
[0/0/0/0]S
[0/0/0/0]S
6
SSCF
61.49
61.595
[−5/0/−5/−5]S
[3/−3/0/0]S
7
SCCF
61.88
62.415
[−5/−5/0/0]S
[−4/−4/0/−2]S
8
CSCF
88.41
88.557
[0/0/0/0]S
[0/0/0/0]S
9
CCCF
88.63
88.777
[0/0/0/0]S
[0/0/0/0]S
10
SSSS
56.32
56.372
[45/−45/−45/−45]S
[−45/45/46/46]S
11
SSSC
65.27
66.885
[90/75/−60/−60]S
[60/60/−58/61]S
12
SSCC
68.72
71.505
[0/45/−45/−45]S
[−42/42/−42/42]S
13
SCSC
90.89
90.894
[90/90/90/90]S
[90/−90/90/−89]S
14
CCCS
91.99
92.152
[0/0/0/0]S
[0/0/0/0]S
15
CCCC
93.67
93.769
[0/90/0/90]S
[0/0/0/1]S
Table 2 Comparison of optimal stacking sequences and natural frequency parameter of symmetric 8-layerd composite plates (a/b = 2, increment 1◦ ) Case
Edges BCs
Optimal stacking
opt Narita [12]
Present study
Narita [12]
Present study
1
SFSF
38.66
38.661
[0/0/0/0]S
[0/0/0/0]S
2
SFCF
60.44
60.952
[0/0/0/0]S
[0/0/0/0]S
3
CFCF
87.74
87.895
[0/0/0/0]S
[0/0/0/−1]S
4
SSSF
45.26
48.575
[0/−30/40/35]S
[−38/38/37/37]S
5
SCSF
61.94
64.203
[90/70/−55/−55]S
[60/59/−61/60]S
6
SSCF
64.84
65.413
[−10/0/−5/25]S
[−10/−13/25/17]S
7
SCCF
69.88
71.835
[−10/65/−35/−35]S
[−33/59/−31/57]S
8
CSCF
90.28
90.447
[0/0/0/0]S
[0/0/0/0]S
9
CCCF
92.28
92.446
[0/0/0/0]S
[0/0/0/0]S
10
SSSS
159.9
159.886
[90/90/90/90]S
[90/90/−89/90]S
11
SSSC
245.7
245.736
[90/90/90/90]S
[90/−90/−90/90]S
12
SSCC
246.4
246.376
[90/90/90/90]S
[90/89/90/90]S
13
SCSC
353.9
353.943
[90/90/90/90]S
[90/−90/90/90]S
14
CCCS
247.1
247.233
[90/90/90/90]S
[90/90/89/89]S
15
CCCC
354.9
355.082
[90/90/90/90]S
[90/90/−90/−90]S
Meccanica (2012) 47:719–730
725
Fig. 4 Comparison of the optimum frequency opt and frequencies of symmetric ten-layered rectangular plates for various stacking sequences, a/b ratios and edge conditions: (a) SFSF, (b) SFCF, (c) CFCF, (d) SSSF, (e) SCSF, (f) SSCF, (g) SCCF, (h) CSCF
726
Meccanica (2012) 47:719–730
Fig. 5 Comparison of the optimum frequency opt and frequencies of symmetric ten-layered rectangular plates for various stacking sequences, a/b ratios and edge conditions: (a) CCCF, (b) SSSS, (c) SSSC, (d) SSCC, (e) SCSC, (f) CCCS, (g) CCCC
Meccanica (2012) 47:719–730
727
Table 3 Optimum designs of symmetric 10-layered plates obtained by PSO algorithm BCs
a/b
Optimal stacking
opt
BCs
Optimal stacking
opt
SFSF
0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4
[0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [−43/−43/43/−43/43]S [58/58/58/−58/−57]S [90/−90/−89/90/89]S [−90/−90/89/90/88]S [−1/−1/−1/−1/−1]S [−4/−4/−4/−3/−3]S [−10/−11/−9/−11/−9]S [−30/58/−29/−30/60]S [−88/−90/90/−88/−87]S [−90/−90/−90/87/89]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/2]S [90/−90/−89/90/88]S [−90/−90/90/90/90]S [0/0/0/2/0]S [−60/−60/−60/60/ − 61]S [90/90/90/−88/−88]S [90/−90/−90/−90/−87]S [−90/−90/−90/90/−90]S [−90/−90/−87/−90/−87]S [0/0/0/2/−1]S [90/−90/−90/90/−87]S [−90/−90/−90/−88/−89]S [−90/−90/90/−90/90]S [−90/−90/−90/89/−90]S [90/−90/−90/90/88]S [0/0/0/0/0]S [0/0/0/0/0]S [90/90/90/87/89]S [90/−90/90/89/90]S [−90/−90/90/90/−87]S [90/90/90/88/90]S
38.712 38.692 38.674 38.661 38.665 38.656 87.934 87.919 87.906 87.895 87.865 87.832 39.051 40.280 45.883 64.232 129.737 225.992 61.301 62.426 64.264 71.897 130.542 226.691 88.118 88.777 90.139 92.446 132.060 227.956 40.199 66.888 139.958 245.722 548.163 971.373 40.489 90.894 200.375 353.943 792.866 1407.484 88.885 93.769 201.962 355.077 793.673 1408.029
SFCF
[0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [−18/−18/17/18/−19]S [−37/37/37/−37/38]S [−42/42/42/−42/−41]S [−43/43/43/−44/43]S [−1/−1/1/−1/−2]S [−3/−3/−3/−4/−3]S [−7/−8/−7/−8/−10]S [−11/−11/16/−13/15]S [−31/−31/40/42/−25]S [−38/−38/46/47/45]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/−1]S [0/0/0/0/0]S [−22/22/23/−21/−21]S [37/−37/−37/37/−40]S [0/0/−1/0/−1]S [−45/45/45/46/−46]S [−64/64/−64/63/−62]S [90/−90/−90/−87/−89]S [−90/−90/90/−88/90]S [−90/90/87/90/−88]S [0/0/0/0/0]S [44/−43/−43/44/44]S [−90/90/90/89/88]S [−90/90/−90/−89/−89]S [90/−90/90/−90/−90]S [90/89/90/88/90]S [0/0/0/0/0]S [0/0/0/0/0]S [90/90/89/−90/88]S [90/−90/90/−87/88]S [90/−90/−90/90/90]S [−90/−90/90/89/89]S
60.863 60.978 60.962 60.948 60.930 61.000 39.000 39.835 40.508 48.742 68.103 88.771 61.242 62.054 63.235 65.405 78.321 96.086 88.094 88.557 89.343 90.447 94.122 109.383 39.971 56.528 93.383 159.890 353.384 624.325 62.086 72.407 140.902 246.403 548.572 971.957 88.745 92.152 142.109 247.201 549.154 972.465
CFCF
SCSF
SCCF
CCCF
SSSC
SCSC
CCCC
SSSF
SSCF
CSCF
SSSS
SSCC
CCCS
728
Meccanica (2012) 47:719–730 Table 4 Optimum designs of symmetric 10-layered plates obtained by PSO algorithm for cases which can be manufactured
BCs
a/b
Optimal stacking
SFSF
0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4 0.5 1 1.5 2 3 4
[0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [−45/−45/45/−45/45]S [60/60/60/−60/−60]S [90/−90/−90/90/90]S [−90/−90/90/90/90]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [−30/60/−30/−30/60]S [−90/−90/90/−90/−90]S [−90/−90/−90/90/90]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [90/−90/−90/90/90]S [−90/−90/90/90/90]S [0/0/0/0/0]S [−60/−60/−60/60/−60]S [90/90/90/−90/−90]S [90/−90/−90/−90/−90]S [−90/−90/−90/90/−90]S [−90/−90/−90/−90/−90]S [0/0/0/0/0]S [90/−90/−90/90/−90]S [−90/−90/−90/−90/−90]S [−90/−90/90/−90/90]S [−90/−90/−90/90/−90]S [90/−90/−90/90/90]S [0/0/0/0/0]S [0/0/0/0/0]S [90/90/90/90/90]S [90/−90/90/90/90]S [−90/−90/90/90/−90]S [90/90/90/90/90]S
CFCF
SCSF
SCCF
CCCF
SSSC
SCSC
CCCC
opt 38.712 38.692 38.674 38.661 38.665 38.656 87.934 87.919 87.906 87.895 87.865 87.832 39.051 40.280 45.863 64.217 129.614 225.920 61.244 62.136 63.971 71.827 130.531 226.684 88.118 88.777 90.139 92.444 132.062 227.956 40.202 66.887 139.951 245.725 548.163 971.377 40.480 90.897 200.364 353.943 792.868 1407.486 88.885 93.769 201.949 355.071 793.675 1408.021
BCs
Optimal stacking
opt
SFCF
[0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [−30/−30/30/30/−30]S [−45/45/45/−45/45]S [−45/45/45/−45/−45]S [−45/45/45/−45/45]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/30/0/30]S [−30/−30/45/45/−30]S [−45/−45/45/45/45]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [0/0/0/0/0]S [−30/30/30/−30/−30]S [30/−30/−30/30/−45]S [0/0/0/0/0]S [−45/45/45/45/−45]S [−60/60/−60/60/−60]S [90/−90/−90/−90/−90]S [−90/−90/90/−90/90]S [−90/90/90/90/−90]S [0/0/0/0/0]S [45/−45/−45/45/45]S [−90/90/90/90/90]S [−90/90/−90/−90/−90]S [90/−90/90/−90/−90]S [90/90/90/90/90]S [0/0/0/0/0]S [0/0/0/0/0]S [90/90/90/−90/90]S [90/−90/90/−90/90]S [90/−90/−90/90/90]S [−90/−90/90/90/90]S
60.863 60.978 60.962 60.948 60.930 61.000 39.000 39.835 39.186 47.834 67.913 88.365 61.209 61.831 62.875 63.213 78.312 93.060 88.094 88.557 89.345 90.447 93.167 108.812 39.963 56.527 93.099 159.886 353.381 624.315 62.086 72.381 140.890 246.393 548.572 971.938 88.745 92.152 142.097 247.187 549.154 972.458
SSSF
SSCF
CSCF
SSSS
SSCC
CCCS
Meccanica (2012) 47:719–730
epoxy material [25]. The material properties are given as below: E1 = 138 GPa,
E2 = 8.96 GPa
G12 = 7.1 GPa,
ν12 = 0.3
Each of the lamina is assumed to be same thickness. The performance of the PSO is shown in Tables 1 and 2 in comparison with the results of numerical values by Narita [12] with the use of the Ritz-based layerwise carried out optimization method for symmetrically laminated 8-layered square (a/b = 1) and rectangular (a/b = 2) plates with various plate edge conditions. It is evident that the PSO is successful in the determination of the optimum, in contrast to layerwise optimization method. The fiber angle of each ply in the composite plates is allowed to change with a step of θ = 1◦ between (−90◦ ≤ θk ≤ 90◦ ) to demonstrate the effectiveness and ability of the PSO to provide high-quality solutions. The higher natural frequencies than those predicted by Narita are also achieved for some composite plates. In addition, it is observed in the tables that the agreement between the results of the present method and those of Ritz-based method are very good. The effects of stacking sequences on the optimum design are shown for symmetric 10-layerd composite plates in Figs. 4 and 5 with various aspect ratios. As inferred from the results, the good efficiency of the PSO algorithm and its ability to provide high-quality solutions is confirmed for various aspect ratios. It is clear from the results that the structural design engineer who is unaware of the effects of stacking sequences on frequency response could choose a stacking sequence which is far from the optimum. Table 3 represents the optimal layer sequences and maximum fundamental frequency of the symmetrical composite plates for 90 cases. As seen, with increase in the a/b ratios, the fundamental frequency increases for a symmetric composite plate. The optimal fiber orientations vary from 0◦ to 90◦ or 0◦ to −90◦ with increase in the a/b ratios and are associated with a smooth transitional region for the case of simply supported plates, such as SSSF. For the case of clamped plates, the optimal fiber orientations change from 0◦ to 90◦ or 0◦ to −90◦ in a sudden manner, such as CCCS and CCCC. It can be said from the results that, the effects of a/b ratios on the stacking sequences decrease for larger a/b ratios too.
729
Table 4 shows the optimal layer sequences which are obtained after rounded to the nearest integer discrete design values as 0, ±30, ±45, ±60, and 90. The vibration analysis is then performed with these integer ply angles. Laminates with such stacking sequences can be fabricated using automated composite manufacturing techniques. During composite manufacturing, small errors may be incurred in the ply layup. For the optimum design to be practical, small variations of the ply angles from the optimum design should not cause a large change in the objective function. It is found that round of the ply angles causes very small change in the objective function [26–28]. In this study, the different combinations of free (F), simply supported (S) and clamped (C) edge conditions are also considered. As seen, the effects of edge conditions on the optimum design are given for symmetric laminated plates in Table 3. It can be noticed from the results that, the maximal natural frequency occurs at the CCCC edge condition, whereas the minimal natural frequency occurs at SFSF edge condition. This can be explained that the clamped edges provide less degrees of freedom and it is effective to stiffen the plates.
5 Conclusions In this work, fundamental frequency optimization of symmetrically laminated plates was studied using the PSO algorithm for various plate edge conditions, a/b ratios. As seen from the results, the FSM and PSO was successful in the determination of the fundamental frequency and optimal layered sequences in comparison with the Ritz-based layerwise optimization method [12]. In addition, the maximum fundamental frequency and the optimum stacking sequences are substantially influenced for edge conditions and a/b ratios.
References 1. Bert CW (1977) Optimal design of a composite material plate to maximise its fundamental frequency. J Sound Vib 50:229–237 2. Bert CW (1978) Design of clamped composite plates to maximise fundamental frequency. J Mech Des 100:274–278 3. Grenested JL (1989) Lay-up optimization and sensitivity analysis of fundamental eigenfrequency of composite plates. Compos Struct 12(3):193–209
730 4. Chow ST, Liew KM, Lam KY (1992) Transverse vibration of symmetrically laminated rectangular composite plates. Compos Struct 20:213–226 5. Fan SC, Cheung YK (1984) Flexural free vibrations of rectangular plates with complex support conditions. J Sound Vib 93:81–94 6. Leissa AW, Martin AF (1990) Vibration buckling of rectangular composite plates with variable fibre spacing. Compos Struct 14:339–357 7. Qatu MS (1991) Free vibration of laminated composite rectangular plates. Int J Solids Struct 28(8):941–954 8. Rao MK, Desai YM (2003) Analytical solutions for vibrations of laminated and sandwich plates using mixed theory. Compos Struct 63:361–373 9. TY Kam, Chang RR (1993) Design of laminated composite plate for maximum buckling load and vibration frequency. Comput Methods Appl Mech Eng 106:65–81 10. Chen J, Dawe DJ (1996) Linear transient analysis of rectangular laminated plates by a finite strip-mode superposition method. Compos Struct 35:213–228 11. Chai GB (1994) Free vibration of generally laminated composite plates with various edge support conditions. Compos Struct 29:249–258 12. Narita Y (2003) Layerwise optimization for the maximum fundamental frequency of laminated composite plates. J Sound Vib 263:1005–1016 13. Apalak MK, Yildirim M, Ekici R (2008) Layer optimisation for maximum fundamental frequency of laminated composite plates for different edge conditions. Compos Sci Technol 68:537–550 14. Khalil MR, Olson MD, Anderson DL (1988) Nonlinear dynamic analysis of stiffened plates. Comput Struct 29:929–941 15. Houlston R (1989) Finite strip analysis of plates and stiffened panels subjected to air blast loads. Comput Struct 32:647–659 16. Loughlan J (2001) The shear buckling behaviour of thin composite plates with particular reference to the effects of bend-twist coupling. Int J Mech Sci 43:771–792
Meccanica (2012) 47:719–730 17. Ovesy HR, Ghannadpour SAM, Morada G (2005) Geometric non-linear analysis of composite laminated plates with initial imperfection under end shortening, using two versions of finite strip method. Compos Struct 71:307–314 18. Assaee H, Ovesy HR (2007) A multi-term semi-energy finite strip method for post-buckling analysis of composite plates. Int J Numer Methods Eng 70:1303–1323 19. Assaee H (2008) Post-local buckling analysis of composite thin-walled sections using semi-finite strip method. PhD Thesis, Amirkabir University of Technology, Iran 20. Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: Proc. IEEE int’l conf on neural networks, vol IV. IEEE Service Center, Piscataway, pp 1942–1948 21. Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, Nagoya, Japan. IEEE Service Center, Piscataway, pp 39–43 22. Kathiravan R, Ganguli R (2007) Strength design of composite beam using gradient and particle swarm optimization. Compos Struct 81:471–479 23. Nadjah N (2006) Swarm intelligent systems. Springer, Berlin 24. Perera R, Fang SE, Ruiz A (2010) Application of particle swarm optimization and genetic algorithms to multiobjective damage identification inverse problems with modeling errors. Meccanica 45:723–734 25. Vinson JR, Sierakowski RL (1986) The behavior of structures composed of composite materials. Martinus Nijhoff, Dordrecht 26. Ganguli R, Chopra I (1995) Aeroelastic optimization of a helicopter rotor with composite coupling. J Aircr 32:1326–1334 27. Ganguli R, Chopra I (1996) Aeroelastic optimization of a helicopter rotor with two-cell composite blades. AIAA J 34:835–841 28. Murugan MS, Ganguli R (2005) Aeroelastic stability enhancement and vibration suppression in a composite helicopter rotor. J Aircr 42:1013–1024
