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Buckling analysis of symmetrically laminated composite plates by the extended Kantorovich method
Composite Structures 73 (2006) 120–128 www.elsevier.com/locate/compstruct
Buckling analysis of symmetrically laminated composite plates by the extended Kantorovich method Variddhi Ungbhakorn a
a,1
, Pairod Singhatanadgid
b,*
Department of Mechanical Engineering, Mahanakorn University of Technology, Nong chok, Bangkok 10530, Thailand b Department of Mechanical Engineering, Chulalongkorn University, Phatumwan, Bangkok 10330, Thailand Available online 19 March 2005
Abstract An extended Kantorovich method is employed to investigate the buckling problem of rectangular laminated composite plates with various edge supports. The principle of minimum total potential energy along with a separable displacement function is utilized to derive a set of governing ordinary differential equations. The buckling load and mode are determined from iterative calculations of the governing equations using the initial trial function which can be selected arbitrarily. The accuracy of this method is confirmed with the available Le´vy and Rayleigh–Ritz solutions. The results demonstrate that the presented semi-analytical approach can be used to analyze the buckling of laminated unidirectional and cross-ply symmetrical plates with any combinations of simple, clamped, and free supports. Several numerical examples of buckling behavior of composite plates with various boundary conditions are also tabulated for future reference. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Buckling; Laminated plate; Kantorovich; Separation of variables
1. Introduction Due to their high stiffness-to-weight and strength-toweight ratios, laminated composite materials are increasingly considered for engineering applications. Furthermore, composite materials can be designed to have the desired properties in the specified directions without over-designing in other directions. Composite plates are widely used in the mechanical, civil, and aerospace engineering. In many applications the rectangular plates are subjected to compressive forces that may cause instability due to buckling. Therefore, structural instability becomes one of the major failure modes concerned in safe and workable in the design of laminated composite plates. There have been many studies on the *
Corresponding author. Tel.: +662 218 6595; fax: +662 252 2889. E-mail addresses: [email protected] (V. Ungbhakorn), [email protected] (P. Singhatanadgid). 1 Tel.: +662 988 3666; fax: +662 252 2889. 0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.02.007
stability of laminated composite plates but closed-form solutions are possible only for the case which all edges are simply supported [1,2]. For other types of boundary conditions, either approximate methods such as the Galerkin and Rayleigh–Ritz method [2] or numerical methods such as the FEM and BEM are usually employed. In this paper, the Kantorovich method [3] is extended to obtain the critical buckling load of symmetrically laminated composite rectangular plates with various combinations of the boundary conditions. Earlier, Kerr [4] successfully used the extended Kantorovich method for the problem of bending and buckling of an isotropic rectangular plate. The efficiency and accuracy of the method have also been demonstrated in the stress analysis of clamped isotropic plates [5] and clamped orthotropic plates [6], also. Although the extended Kantorovich method is based on the variational principle, Yuan and Jin [7] have shown that initial trial functions are neither required to satisfy the geometric nor
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
the force boundary conditions because the iterative procedure will force the solution to satisfy all boundary conditions eventually. Furthermore, the method reduces the problem of solving the partial differential equations to a set of ordinary differential equations. These two outstanding features make the extended Kantorovich method more attractive than the Galerkin or Rayleigh–Ritz methods for a certain circumstance.
2. Derivation of the iterative differential equations The extended Kantorovich method is a semi-analytical method which requires iterative calculation by reducing the governing partial differential equations to a set of governing ordinary differential equations (ODE). These iterative equations can be derived either by using the Galerkin equation [6] or the principle of minimum total potential energy [7]. In this study, the latter principle is employed. The total potential energy of the symmetrically laminated composite plates subjected to the inplane loads, as shown in Fig. 1, is the summation of the strain energy and the potential energy of external loads, and given by [8]
0
þ N yy X 2 Y 2;y þ 2N xy X ;x Y ;y XY dx dy
ð3Þ
If X(x) is priorly specified, then Eq. (3) can be written as Z 1 b P¼ ½S 1x D11 Y 2 þ 2S 2x D12 YY ;yy þ S 3x D22 Y 2;yy 2 0 þ 4S 4x D66 Y 2;y þ 4ðS 5x D16 YY ;y þ S 6x D26 Y ;y Y ;yy Þ dy
Z
1 2
a
½S 4x N xx Y 2 þ S 3x N yy Y 2;y þ 2S 6x N xy YY ;y dy 0
ð4Þ If Y(y) is priorly specified, then Eq. (3) can be written as P¼
1 2
where S 1x ¼
Substitute Eq. (2) into (1), the total potential energy becomes
S 3x ¼ S 5x ¼
Z
a
½S 3y D11 X 2;xx þ 2S 2y D12 XX ;xx þ S 1y D22 X 2 0
y
Z
X 2;xx dx;
Z0 a
X 2 dx;
S 2x ¼ Z S 4x ¼
Z0 a X ;x X ;xx dx;
S 1y ¼ S 3y ¼
Z Z Z
Y 2;yy dy;
S 2y ¼
0 b
Y 2 dy;
S 4y ¼
0
XX ;xx dx 0 a
Z
0
ð6Þ
b
YY ;yy dy 0 b
Z
Y 2;y dy 0
b
Y ;y Y ;yy dy;
a
X 2;x dx 0 Z a S 6x ¼ XX ;x dx
b
0
Nyy
Z
a
0
S 5y ¼
S 6y ¼
Z
b
YY ;y dy 0
When X(x) is priorly specified, the variational principle [8] requires the stationary condition for the functional Eq. (4), dP = 0. The procedure yields the following governing ODE and the associated boundary conditions (BCs):
Nxy
a
½D11 X 2;xx Y 2 þ 2D12 X ;xx Y ;yy XY
0 2
þ D22 X Y 2;yy þ 4D66 X 2;x Y 2;y þ 4ðD16 X ;xx Y Z Z 1 a b þ D26 XY ;yy ÞX ;x Y ;y dx dy ½N xx X 2;x Y 2 2 0 0
ð1Þ
ð2Þ
b
b
ð5Þ
where comma denotes the differentiation with respect to the subscripted variable and Dij is the bending stiffness of the composite plate. For classical Kantorovich method, the solution w(x, y) is assumed to be separable as wðx; yÞ ¼ X ðxÞY ðyÞ
Z
a
þ 4S 4y D66 X 2;x þ 4ðS 6y D16 X ;x X ;xx þ S 5y D26 XX ;x Þ dx Z 1 a ½S 3y N xx X 2;x þ S 4y N yy X 2 þ 2S 6y N xy XX ;x dx 2 0
P¼U þV Z Z 1 a b P¼ ½D11 w2;xx þ 2D12 w;xx w;yy þ D22 w2;yy 2 0 0 þ 4D66 w2;xy þ 4ðD16 w;xx þ D26 w;yy Þw;xy dx dy Z Z 1 a b ½N xx w2;x þ N yy w2;y þ 2N xy w;x w;y dx dy 2 0 0
Z
1 P¼ 2
121
Nxx x
Fig. 1. Rectangular plate subjected to in-plane load Nxx, Nyy, Nxy.
d4 Y d2 Y þ ð2S D 4S D þ S N Þ 2x 12 4x 66 3x yy dy 4 dy 2 þ ðS 1x D11 S 4x N xx ÞY ¼ 0
S 3x D22
ð7Þ
122
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
Boundary conditions along y = 0 and y = b are
3. Solutions of the iterative ODE
Either: 4
dY dY þ ðS 2x D12 4S 4x D66 þ S 3x N yy Þ dy 3 dy ð2S 5x D16 S 6x N xy ÞY ¼ 0
S 3x D22
ð8Þ
or dY ¼ 0
ð9Þ
ð10Þ
or dY ;y ¼ 0
ð11Þ
These conditions, (8)–(11), correspond to the following edge supports: Simply supported edge: conditions (9) and (10) Clamped edge: conditions (9) and (11) Free edge: conditions (8) and (10)
ð17Þ
k1 ¼
ðS 2x D12 2S 4x D66 Þ S 3x D22
ð18Þ
k2 ¼
ðS 1x D11 S 4x N xx Þ S 3x D22
ð19Þ
The characteristic equation of Eq. (17) is s4 þ 2k 1 s2 þ k 2 ¼ 0
ð20Þ
whose four roots are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1;2;3;4 ¼
When Y(y) is priorly specified, the variational principle requires the first variation of the functional Eq. (5) equals to zero. The procedure yields the following governing ODE and the associated BCs: d4 X d2 X S 3y D11 4 þ ð2S 2y D12 4S 4y D66 þ S 3y N xx Þ 2 dx dx þ ðS 1y D22 S 4y N yy ÞX ¼ 0
d4 Y d2 Y þ 2k þ k2Y ¼ 0 1 dy 4 dy 2 where
And either: d2 Y dY þ S 2x D12 Y ¼ 0 S 3x D22 2 þ 2S 6x D26 dy dy
Hereafter, only the case of rectangular plates subjected to uniform axial compression is considered. Setting Nyy = Nxy = 0 in Eq. (7), one has the following fourth order ODE:
k 1
k 21 k 2
ð21Þ
Depending on the values of k1 and k2, Eq. (20) has four kinds of roots in Eq. (21). In general, the solution can be written in one of the following four forms: Y ðyÞ ¼ Ay sinðp1 yÞ þ By cosðp1 yÞ þ C y sinðp2 yÞ þ Dy cosðp2 yÞ ð22Þ
ð12Þ
Y ðyÞ ¼ Ay sinðp1 yÞ þ By cosðp1 yÞ þ C y sinhðp2 yÞ ð23Þ
þ Dy coshðp2 yÞ
Boundary conditions along x = 0 and x = a are Either:
Y ðyÞ ¼ ½Ay cosðp2 yÞ þ By sinðp2 yÞ coshðp1 yÞ þ ½C y cosðp2 yÞ þ Dy sinðp2 yÞ sinhðp1 yÞ
3
dX dX þ ðS 2y D12 4S 4y D66 þ S 3y N xx Þ 3 dx dx ð2S 5y D26 S 6y N xy ÞX ¼ 0
S 3y D11
ð13Þ
Y ðyÞ ¼ Ay sinhðp1 yÞ þ By coshðp1 yÞ þ C y sinhðp2 yÞ ð25Þ
þ Dy coshðp2 yÞ or
The conditions for each type of solution are as follows:
dX ¼ 0
ð14Þ
And either: S 3y D11
ð24Þ
d2 X dX þ S 2y D12 X ¼ 0 þ 2S 6y D16 dx2 dx
1. If k1 > 0 and ðk 21 k 2 Þ > 0, all four roots are imaginary, Eq. (22) is the solution with s1;2 ¼ ip1 ;
ð15Þ
or
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 ¼
dX ;x ¼ 0
k1 þ
ð16Þ
Two sets of ODE, one set for X(x) priorly specified and the other set for Y(y) priorly specified, are now complete for iterative calculations. Next, both derived sets of differential equations are solved.
s3;4 ¼ ip2
k 21
k2;
p2 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1
k 21 k 2
ð26Þ 2. If k1 < 0 and k2 < 0, two roots are imaginary and the other two are real, Eq. (23) is the solution with s1;2 ¼ ip1 ;
s3;4 ¼ p2
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
where rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 21 k 2 þ k 1 ;
p1 ¼
p2 ¼
s1;2 ¼ ip3 ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 21 k 2 k 1
ð27Þ 3. If k1 < 0 and ðk 21 k 2 Þ < 0, roots are in complex conjugate pairs, Eq. (24) is the solution with s1;2 ¼ p1 ip2 ;
s3;4 ¼ p1 ip2
where 1 p1 ¼ pffiffiffi 2
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k2 k1;
1 p2 ¼ pffiffiffi 2 k 21 ,
4. If k1 < 0 and 0 < k 2 < (25) is the solution with s1;2 ¼ p1 ;
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k2 þ k1
ð28Þ
all four roots are real, Eq.
p2 ¼
k 21 k 2
k3
k 23 k 4
ð37Þ 2. If k3 < 0 and ðk 23 k 4 Þ < 0, roots are in complex conjugate pairs, Eq. (35) is the solution with s3;4 ¼ p3 ip4
where 1 p3 ¼ pffiffiffi 2
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k4 k3;
s1;2 ¼ p1 ;
1 p4 ¼ pffiffiffi 2
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k4 þ k3
ð38Þ
s3;4 ¼ p2
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 21
k 1
p4 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3. If k3 < 0 and k 23 k 4 > 0, Eq. (36) is the solution with
s3;4 ¼ p2
k 1 þ k2; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 23 k 4 ;
k3 þ
s1;2 ¼ p3 ip4 ;
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 ¼
s3;4 ¼ ip4
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p3 ¼
123
p3 ¼
k 3 þ
k 23
ð29Þ
k4;
p4 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 3
k 23 k 4
ð39Þ
Similarly, if Y(y) is known, Eq. (12) can be arranged as d4 X d2 X þ 2k þ k4X ¼ 0 3 dy 4 dy 2
ð30Þ
To obtain the solution, the following algorithm is devised:
where k3 ¼ k4 ¼
ðS 2y D12 2S 4y D66 þ S 3y N xx =2Þ S 3y D11
ð31Þ
S 1y D22 S 3y D11
ð32Þ
whose four roots are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1;2;3;4 ¼
k 3
k 23 k 4
ð33Þ
Since k4 is always greater than zero, it can be shown that there are only three possible solutions for this case as given by X ðxÞ ¼ Ax sinðp3 xÞ þ Bx cosðp3 xÞ þ C x sinðp4 xÞ þ Dx cosðp4 xÞ
ð34Þ
X ðxÞ ¼ ½Ax cosðp4 xÞ þ Bx sinðp4 xÞ coshðp3 xÞ þ ½C x cosðp4 xÞ þ Dx sinðp4 xÞ sinhðp3 xÞ
ð35Þ
X ðxÞ ¼ Ax sinhðp3 xÞ þ Bx coshðp3 xÞ þ C x sinhðp4 xÞ þ Dx coshðp4 xÞ
4. Iteration procedures
ð36Þ
The conditions for each type of solution are as follows: 1. If k3 > 0 and ðk 23 k 4 Þ > 0, Eq. (34) is the solution with
1. Assume an initial solution X(x) in the x direction which may or may not satisfy any boundary conditions, and then evaluate k1 and k2 according to Eqs. (18) and (19). 2. Calculate the four roots in Eqs. (26)–(29) and select the form of solution from Eqs. (22)–(25) corresponding to the roots. 3. Apply the BCs to the solution Y(y) and determine the eigenvalue Nxx and eigenvector Y(y). This completes the simple Kantorovich method. 4. For the extended Kantorovich method, use the Y(y) obtained in step 3 to evaluate k3 and k4 according to Eqs. (31) and (32). 5. Calculate the four roots in Eqs. (37)–(39) and select the form of solution from Eqs. (34)–(36) corresponding to the roots. 6. Apply the BCs to the solution X(x) and determine the eigenvalue Nxx and eigenvector X(x). 7. Compare the eigenvalue Nxx obtained from step 6 with the previous one. If the difference satisfies the specified tolerance, the last Nxx is taken as the final solution. This completes the extended Kantorovich procedure. Otherwise continue the iterative calculation by repeating steps 1–3 using the eigenvector X(x) obtained from the previous step.
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Observe that if the assumed initial solution in step 1 satisfies all BCs, the solution for simple Kantorovich will give good result, depending on the form of the initial solution as in the Rayleigh–Ritz or Galerkin method.
y
y
F
S
C S
x C Boundary conditions: SCSF
5. Numerical verification and accuracy The iteration procedure outlined in the previous section may apply to rectangular plates with any combinations of simple support (S), clamped support (C), and free edge (F). A simple-clamped–simple-free (SCSF) plate as shown in Fig. 2, is a specimen with simple support on x = 0 and x = a, and clamped and free on y = 0 and y = b, respectively. The other example in Fig. 2 is a CSSC specimen which is simply supported on y = 0 and
S
C
x S Boundary conditions: CSSC
Fig. 2. Boundary conditions of plate denoted by S, C, and F.
x = a, and clamped on the other two edges. The iteration example for an isotropic SCSF plate is illustrated in Table 1. Mechanical properties and plate dimensions are E = 70 GPa, v = 0.25, a = 0.5 m, aspect ratio = 3, and thickness = 0.0023 m. The first iteration begins with assuming function X(x) as x2 and solving for Y(y)
Table 1 Iteration example for an isotropic plate with SCSF boundary condition Iteration no.
Assumed solution X(x) or Y(y) 2
Solution Eigenvalue (kN/m)
Eigenvector X(x) or Y(y)
65.27
A
1
x
2
A
36.84
sinð2px a Þ
3
sinð2px a Þ
36.02
B
4
B
36.02
sinð2px a Þ
Note. E = 70 GPa, v = 0.25, a = 0.5 m. Aspect ratio = 3, and thickness = 2.3 mm. A sinð11:61yÞ 1:103 cosð11:61yÞ 0:8954 sinhð12:97yÞ þ 1:103 coshð12:97yÞ B sinð10:78yÞ 0:5448 cosð10:78yÞ 0:5186 sinhð20:79yÞ þ 0:5448 coshð20:79yÞ.
Buckling mode
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
according to Eq. (17). The assumed function X(x) is not required to satisfy the S–S boundary conditions. The eigenvalue and eigenvector Y(y) obtained from the first iteration are 65.27 kN/m and the function denoted as ‘‘A,’’ respectively. The obtained function Y(y) is forced to satisfy the boundary conditions in the y-direction automatically. The out-of-plane displacement w(x, y) is plotted in the last column of Table 1. The second iteration employs function Y(y) obtained from the first iteration as an assumed function. The governing equation, Eq. (12), is reduced to an equation of unknown X(x) in form of Eq. (30). The solutions of the second iteration are eigenvalue of 36.84 kN/m and eigenvector of sinð2px Þ. a The next iteration is performed using the function X ðxÞ ¼ sinð2px Þ from the second iteration. The solution a of Y(y) from the third iteration, represented as ‘‘B,’’ is different from that of the first iteration if comparing term by term. However, the buckling modes obtained in the second and third iteration are almost the same, as shown in the last column. The iteration processes conclude when the eigenvalues converge to a value, i.e. 36.02 kN/m for this case, which is the buckling load. The eigenvectors are also converged to the buckling mode, w(x, y). The Kantorovich method is verified by comparing the solutions from this method with those of the known solutions. In Tables 2–4, nondimensional buckling loads 2 defined in term of pN2xxDb22 of SSSF, SCSF, and SCSC for isotropic and laminated composite plates are compared with the Le´vy solution[9]. The numerical verifications in Tables 2–4 are buckling load of specimens with v = 0.25 for isotropic plates, and v = 0.25, G = 0.5E2 for orthotropic plates. The nondimensional buckling loads and buckling modes obtained from both methods are identical. The buckling configuration for SSSF plates are said to be ‘‘mode 1’’ because there is only one point of local maximum out-of-plane displacement. The buckling modes for SCSF and SCSC plates change from mode 1 for plates with low aspect ratios to higher mode for
Table 2 Nondimensional buckling loads of rectangular plate (SSSF) determined from Levy solution and Kantorovich method a b
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
E1 E2
¼1
[0]8,
E1 E2
¼3
[0/90]2s,
E1 E2
¼ 10
Le´vy solution
Present solution
Le´vy solution
Present solution
Le´vy solution
Present solution
4.4036 1.4342 0.8880 0.6979 0.6104 0.5630 0.5345 0.5161 0.5034 0.4944
4.4036 1.4342 0.8880 0.6979 0.6104 0.5630 0.5345 0.5161 0.5034 0.4944
12.5334 3.5626 1.9075 1.3307 1.0648 0.9208 0.8341 0.7780 0.7395 0.7121
12.5334 3.5626 1.9075 1.3307 1.0648 0.9208 0.8341 0.7780 0.7395 0.7121
7.6905 2.0396 0.9941 0.6284 0.4592 0.3673 0.3119 0.2759 0.2513 0.2336
7.6905 2.0396 0.9941 0.6284 0.4592 0.3673 0.3119 0.2759 0.2513 0.2336
125
Table 3 Nondimensional buckling loads of rectangular plate (SCSF) determined from Levy solution and Kantorovich method a b
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
E1 E2
¼1
[0]8,
E1 E2
[0/90]2s,EE12 ¼ 10
¼3
Le´vy Present Solution Solution
Le´vy Present Solution Solution
Le´vy Present Solution Solution
4.5182(1) 1.6983 1.3392 1.3862 1.4319(2) 1.3392 1.3355 1.3862 1.3392(3) 1.3295
12.6646(1) 3.8609 2.4070 2.0764 2.1100 2.4070(2) 2.1738 2.0764 2.0640 2.1100
7.7670(1) 2.2294 1.3489 1.2079 1.3255 1.3489(2) 1.2318 1.2079 1.2448 1.2581(3)
4.5182(1) 1.6983 1.3392 1.3862 1.4319(2) 1.3392 1.3355 1.3862 1.3392(3) 1.3295
12.6646(1) 3.8609 2.4070 2.0764 2.1100 2.4070(2) 2.1738 2.0764 2.0640 2.1100
7.7670(1) 2.2294 1.3489 1.2079 1.3255 1.3489(2) 1.2318 1.2079 1.2448 1.2581(3)
Note. Buckling modes are shown in the parenthesis only for an aspect ratio that bucking mode changes to the higher one.
Table 4 Nondimensional buckling loads of rectangular plate (SCSC) determined from Levy solution and Kantorovich method a b
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
E1 E2
¼1
[0]8,
E1 E2
[0/90]2s,EE12 ¼ 10
¼3
Present Le´vy solution solution
Le´vy solution
Present solution
Le´vy Present solution solution
7.6913(1) 7.6913(2) 7.1159(2) 6.9716(3) 6.9989(4) 7.0552(5) 7.0008(5) 6.9716(6) 6.9778(7) 6.9989(8)
16.2261(1) 11.1626(1) 11.2219(2) 11.1626(2) 10.8973(3) 11.1626(3) 10.8659(4) 10.9810(5) 10.8817(5) 10.8973(6)
16.2261(1) 11.1626(1) 11.2219(2) 11.1626(2) 10.8973(3) 11.1626(3) 10.8659(4) 10.9810(5) 10.8817(5) 10.8973(6)
9.6276(1) 7.8342(1) 7.0500(2) 7.3323(3) 7.0922(3) 7.0500(4) 7.1726(5) 7.0431(5) 7.0500(6) 7.0922(6)
7.6913(1) 7.6913(2) 7.1159(2) 6.9716(3) 6.9989(4) 7.0552(5) 7.0008(5) 6.9716(6) 6.9778(7) 6.9989(8)
9.6276(1) 7.8342(1) 7.0500(2) 7.3323(3) 7.0922(3) 7.0500(4) 7.1726(5) 7.0431(5) 7.0500(6) 7.0922(6)
plates with higher aspect ratios. The number in the superscripted parenthesis refers to buckling mode. The Kantorovich solution is also verified with other available solutions as shown in Table 5. In [10], composite plates with CCCC and CSCS boundary conditions were solved for buckling loads using the Rayleigh–Ritz method and trigonometric functions. Buckling loads of square cross-ply plates determined from the two methods are compared in Table 5. The buckling loads presented in the table are in term of total load in order to be consistent with those shown in [10]. Solutions from the Kantorovich iteration agree very well with those of the previous study. The solutions from the present study are approximately 0.6% higher than the solution of the Rayleigh–Ritz method in all cases of study. Convergence of the solutions of the Kantorovich method is very fast. In this study, the eigenvalue is assumed to be the buckling load of the problem if it is less than 0.01% different from the solution of the previous iteration. Table 6 shows number of iteration required to obtain the buckling load and a comparison of the
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Table 5 Comparing the buckling loads of square plates with solutions from the Rayleigh–Ritz method Boundary condition
Stacking Thickness sequence (in.)
Rayleigh–Ritz Present solution [10] solution (103 lbs) (103 lbs)
% difference
CCCC
[0/90]5s
17.5509 12.2464 8.6962 17.0906 11.9252 8.4681 11.7625 8.2074 5.8282 10.0538 7.0152 4.9815
0.57 0.57 0.57 0.55 0.55 0.55 0.60 0.60 0.60 0.60 0.60 0.60
[90/0]5s
CSCS
[0/90]5s
[90/0]5s
0.115 0.102 0.091 0.115 0.102 0.091 0.115 0.102 0.091 0.115 0.102 0.091
17.6505 12.3159 8.7456 17.1851 11.9911 8.5150 11.8328 8.2565 5.8630 10.1141 7.0572 5.0114
Note. E11 = 215 GPa (31.18 Msi), E22 = 23.6 GPa (3.42 Msi), G12 = 5.2 GPa (0.754 Msi), v12 = 0.28, a = b = 25.4 cm (10 in.).
Table 6 Number of iteration required for each boundary condition Boundary condition
SSSF SCSF SCSC CCCC CCCF CSSC CSCS
No. of iteration
1 1 1 3 3 3 3
Nondimensional buckling load Present study
Past study
2.0396 2.2294 7.8342 12.1023 7.8494 6.5576 8.9399
2.0396 2.2294 7.8342 – – – –
nondimensional buckling loads determined from the present and past studies. The specimen used in Table 6 is a [0/90]2s square plate with E1 = 10E2. The assumed function for the first iteration is X ðxÞ ¼ sinðpxaÞ for all cases of boundary conditions. From Table 6, the first three cases require only one iteration to obtain the buckling load because the initial assumed function happens to satisfy the simply supported boundary condition. If other functions were chosen for the first iteration, it would need additional iterations to reach the buckling load. Similarly, the other four boundary conditions, which are C–C and C–S support in the x-direction, required three iterations to converge to the buckling load. Therefore, the extended Kantorovich method predicts buckling load of isotropic and orthotropic plates with high accuracy compared with several available solutions. The convergence of the solution is very fast, although only one term is used and the initial assumed function is not required to satisfy the boundary conditions. If the correct function is chosen for the first iteration, the extended Kantorovich method returns the buckling load and mode in the first iteration.
6. Other numerical examples Additional nondimensional buckling loads of plates with CCCF, CSSC, and CSCS boundary conditions are tabulated in Tables 7–9, respectively. Nondimensional buckling loads of all cases decrease when plate aspect ratio is increased. The buckled configuration of plates with aspect ratio of 0.5 is mode 1, and buckling
Table 7 Nondimensional buckling load of rectangular plate (CCCF) determined from Kantorovich method a b
Isotropic
[0]8, E1 E2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
16.4899 4.6504 2.5872 1.9916 1.8332 1.7392(2) 1.5897 1.5289 1.5150 1.4711(3)
¼3
48.6355 12.8083 6.3169 4.1821 3.3301 2.9976 2.9067 2.8999 2.5447(2) 2.4358
[0/90]2s, E1 E2
¼ 10
160.6444 40.8181 18.7725 11.1958 7.8297 6.1443 5.2708 4.8404 4.6660 4.6278
E1 E2
¼3
23.8068 6.4015 3.3033 2.3459 2.0277 1.9590 1.7922(2) 1.6611 1.6064 1.5937
E1 E2
¼ 10
30.3764 7.8494 3.7880 2.4895 2.0158 1.8747 1.8610 1.5981(2) 1.4980 1.4592
Table 8 Nondimensional buckling load of rectangular plate (CSSC) determined from Kantorovich method a b
Isotropic
[0]8, E1 E2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
10.9106 6.2304 5.8750(2) 5.6726 5.5561(3) 5.5301(4) 5.4888 5.4709(5) 5.4617 5.4479(6)
¼3
27.7747 10.9092 9.5951 9.0057(2) 8.6893 8.5895(3) 8.4635 8.4356(4) 8.3732 8.3627(5)
[0/90]2s, E1 E2
¼ 10
85.0875 25.2817 16.3439 15.2296 14.6191(2) 13.7375 13.5798 13.4360(3) 13.2045 13.1670
E1 E2
¼3
14.1555 6.6595 6.2952(2) 5.8419 5.7674(3) 5.6755 5.6353(4) 5.6119 5.5858(5) 5.5790
E1 E2
¼ 10
16.6687 6.5576 6.0568(2) 5.4602 5.3863(3) 5.2368 5.2142(4) 5.1563 5.1452(5) 5.1185
Table 9 Nondimensional buckling load of rectangular plate (CSCS) determined from Kantorovich method a b
Isotropic
[0]8, E1 E2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
18.1874 6.7432 5.3747 4.8471(2) 4.5224 4.4065(3) 4.2786 4.2367(4) 4.1737 4.1542(5)
¼3
50.6458 15.2061 9.4526 8.2868 7.6426(2) 6.9645 6.7939 6.5965(3) 6.4210 6.3845
[0/90]2s, E1 E2
¼ 10
162.6750 43.2368 21.9452 15.4429 13.3894 13.0294 11.6246(2) 10.7697 10.4462 10.3932
E1 E2
¼3
25.0849 8.1043 5.7404 5.3087(2) 4.6619 4.5360 4.3391(3) 4.2568 4.1950(4) 4.1380
E1 E2
¼ 10
31.0038 8.9399 5.6477 5.2293 4.3659(2) 4.1019 3.9650(3) 3.7948 3.7647 3.6634(4)
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
127
Fig. 3. Buckling modes of plates with various edge supports.
mode increases with plate aspect ratios. The superscripted buckling mode is shown only for plates with aspect ratios that have a change of buckling mode to a higher one. Buckling modes of several plate configurations are represented in Fig. 3. The plots are buckling modes of a composite plate with stacking sequence of [0/90]2s, E1 = 10E2, and an aspect ratio of 3. It is observed that buckling modes of SCSF and CCCF are mode 2 and mode 1, respectively. The only difference of these two specimens is the boundary conditions on the loading edges which are simple and clamped supports respectively. The buckling mode is reduced from 2 to 1 when the loading edges are constrained, i.e. the specimen edge is not free to rotate. The CCCF specimen may buckle into mode 2 if the plate aspect ratio is increased to 4. For CSSC specimen, boundary conditions on the loading edges are different, i.e. clamped and simple supports. As a result, buckling configuration on both sides of the support is not identical, i.e. displacement on the side of clamped support is smaller than that of the other.
7. Discussion and conclusion Buckling loads and modes of laminated composite plates are solved using the extended Kantorovich method. The boundary conditions of the rectangular plate can be any combinations of simple support, clamped support, or free edge. The present study is limited to symmetrical composite plates with unidirectional or
cross-ply fibers. Although the solution of the Kantorovich method is obtained from solving a set of ODE, this method requires a series of iterative calculation, so it is considered as a semi-analytical method. By assuming that the out-of-plane displacement of the buckled plates is separable and either one of X(x) or Y(y) is known beforehand, applying the principle of minimum total potential energy along with the variational principle to the total potential energy yields a set of ODE. These ODE and the prescribed boundary conditions are used in iterative calculations for eigenvalues and eigenvectors. The procedure is repeated until the eigenvalue converges to a specific value of buckling load. The final product of X(x) and Y(y) indicates the buckling mode. The extended Kantorovich method is verified numerically by comparing the buckling load and mode with known solutions. Buckling behaviors of SSSF, SCSF, and SCSC specimens obtained from the present study are identical to the Le´vyÕs solutions in [9]. Differences in buckling load in case of CCCC and CSCS specimen when compared to solutions from the Rayleigh–Ritz method [10] are less than one percent. Therefore, the extended Kantorovich method is validated for isotropic rectangular plates or composite plates with unidirectional or cross-ply symmetric stacking sequence, i.e. specimen with D16 = D26 = 0. For specimens with a presence of D16 and D26 i.e. angle-ply panels, the approximation of the displacement functions is required to include some additional terms in order to simulate the actual buckled mode. The approximate function could be in
128
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
form of w(x, y) = X1(x)Y1(y) + X2(x)Y2(y) which will lead to a set of simultaneous ODE. Solving these ODE are tedious and not in the scope of this study. New numerical examples of specimen with combinations of simple, clamped, and free boundary conditions are also included in this work. An advantage of this method is that there is no need to solve the governing partial differential equations. They are transformed to a set of ODE with an assumed displacement function. Another advantage is that the initial assumed displacement function in the first iteration can be arbitrarily selected regardless of the type of boundary conditions. The displacement functions are automatically forced to satisfy the boundary conditions in the next iterations.
References [1] Jones RM. Mechanics of composite materials. Washington, DC: Scripta Book Co.; 1975.
[2] Iyengar NGR. Structural stability of columns and plates. Chichester, England: Ellis Horwood Limited Publishers; 1988. [3] Kantorovich LV, Krylov IV. Approximate method of higher analysis. New York: Interscience Publishers Inc.; 1964. [4] Kerr AD. An extended Kantorovich method for the solution of eigenvalues problems. Int J Solids Struct 1969;5: 559–72. [5] Kerr AD, Alexander H. An application of the extended Kantorovich method to the stress analysis of a clamped rectangular plate. Acta Mech 1968;6:180–96. [6] Dalaei M, Kerr AD. Analysis of clamped rectangular orthotropic plates subjected to a uniform lateral load. Int J Mech Sci 1995;37(5):527–35. [7] Yuan S, Jin Y. Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method. Comput Struct 1998;66(6):861–7. [8] Reddy JN. Mechanics of laminated composite plates and shells. Boca Raton: CRC Press; 2004. [9] Reddy JN. Theory and analysis of elastic plates. Philadelphia, PA: Taylor & Francis; 1999. [10] Chai GB. Buckling of generally laminated composite plates with various edge support conditions. Comput Struct 1994;29(3): 299–310.
Buckling analysis of symmetrically laminated composite plates by the extended Kantorovich method Variddhi Ungbhakorn a
a,1
, Pairod Singhatanadgid
b,*
Department of Mechanical Engineering, Mahanakorn University of Technology, Nong chok, Bangkok 10530, Thailand b Department of Mechanical Engineering, Chulalongkorn University, Phatumwan, Bangkok 10330, Thailand Available online 19 March 2005
Abstract An extended Kantorovich method is employed to investigate the buckling problem of rectangular laminated composite plates with various edge supports. The principle of minimum total potential energy along with a separable displacement function is utilized to derive a set of governing ordinary differential equations. The buckling load and mode are determined from iterative calculations of the governing equations using the initial trial function which can be selected arbitrarily. The accuracy of this method is confirmed with the available Le´vy and Rayleigh–Ritz solutions. The results demonstrate that the presented semi-analytical approach can be used to analyze the buckling of laminated unidirectional and cross-ply symmetrical plates with any combinations of simple, clamped, and free supports. Several numerical examples of buckling behavior of composite plates with various boundary conditions are also tabulated for future reference. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Buckling; Laminated plate; Kantorovich; Separation of variables
1. Introduction Due to their high stiffness-to-weight and strength-toweight ratios, laminated composite materials are increasingly considered for engineering applications. Furthermore, composite materials can be designed to have the desired properties in the specified directions without over-designing in other directions. Composite plates are widely used in the mechanical, civil, and aerospace engineering. In many applications the rectangular plates are subjected to compressive forces that may cause instability due to buckling. Therefore, structural instability becomes one of the major failure modes concerned in safe and workable in the design of laminated composite plates. There have been many studies on the *
Corresponding author. Tel.: +662 218 6595; fax: +662 252 2889. E-mail addresses: [email protected] (V. Ungbhakorn), [email protected] (P. Singhatanadgid). 1 Tel.: +662 988 3666; fax: +662 252 2889. 0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.02.007
stability of laminated composite plates but closed-form solutions are possible only for the case which all edges are simply supported [1,2]. For other types of boundary conditions, either approximate methods such as the Galerkin and Rayleigh–Ritz method [2] or numerical methods such as the FEM and BEM are usually employed. In this paper, the Kantorovich method [3] is extended to obtain the critical buckling load of symmetrically laminated composite rectangular plates with various combinations of the boundary conditions. Earlier, Kerr [4] successfully used the extended Kantorovich method for the problem of bending and buckling of an isotropic rectangular plate. The efficiency and accuracy of the method have also been demonstrated in the stress analysis of clamped isotropic plates [5] and clamped orthotropic plates [6], also. Although the extended Kantorovich method is based on the variational principle, Yuan and Jin [7] have shown that initial trial functions are neither required to satisfy the geometric nor
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
the force boundary conditions because the iterative procedure will force the solution to satisfy all boundary conditions eventually. Furthermore, the method reduces the problem of solving the partial differential equations to a set of ordinary differential equations. These two outstanding features make the extended Kantorovich method more attractive than the Galerkin or Rayleigh–Ritz methods for a certain circumstance.
2. Derivation of the iterative differential equations The extended Kantorovich method is a semi-analytical method which requires iterative calculation by reducing the governing partial differential equations to a set of governing ordinary differential equations (ODE). These iterative equations can be derived either by using the Galerkin equation [6] or the principle of minimum total potential energy [7]. In this study, the latter principle is employed. The total potential energy of the symmetrically laminated composite plates subjected to the inplane loads, as shown in Fig. 1, is the summation of the strain energy and the potential energy of external loads, and given by [8]
0
þ N yy X 2 Y 2;y þ 2N xy X ;x Y ;y XY dx dy
ð3Þ
If X(x) is priorly specified, then Eq. (3) can be written as Z 1 b P¼ ½S 1x D11 Y 2 þ 2S 2x D12 YY ;yy þ S 3x D22 Y 2;yy 2 0 þ 4S 4x D66 Y 2;y þ 4ðS 5x D16 YY ;y þ S 6x D26 Y ;y Y ;yy Þ dy
Z
1 2
a
½S 4x N xx Y 2 þ S 3x N yy Y 2;y þ 2S 6x N xy YY ;y dy 0
ð4Þ If Y(y) is priorly specified, then Eq. (3) can be written as P¼
1 2
where S 1x ¼
Substitute Eq. (2) into (1), the total potential energy becomes
S 3x ¼ S 5x ¼
Z
a
½S 3y D11 X 2;xx þ 2S 2y D12 XX ;xx þ S 1y D22 X 2 0
y
Z
X 2;xx dx;
Z0 a
X 2 dx;
S 2x ¼ Z S 4x ¼
Z0 a X ;x X ;xx dx;
S 1y ¼ S 3y ¼
Z Z Z
Y 2;yy dy;
S 2y ¼
0 b
Y 2 dy;
S 4y ¼
0
XX ;xx dx 0 a
Z
0
ð6Þ
b
YY ;yy dy 0 b
Z
Y 2;y dy 0
b
Y ;y Y ;yy dy;
a
X 2;x dx 0 Z a S 6x ¼ XX ;x dx
b
0
Nyy
Z
a
0
S 5y ¼
S 6y ¼
Z
b
YY ;y dy 0
When X(x) is priorly specified, the variational principle [8] requires the stationary condition for the functional Eq. (4), dP = 0. The procedure yields the following governing ODE and the associated boundary conditions (BCs):
Nxy
a
½D11 X 2;xx Y 2 þ 2D12 X ;xx Y ;yy XY
0 2
þ D22 X Y 2;yy þ 4D66 X 2;x Y 2;y þ 4ðD16 X ;xx Y Z Z 1 a b þ D26 XY ;yy ÞX ;x Y ;y dx dy ½N xx X 2;x Y 2 2 0 0
ð1Þ
ð2Þ
b
b
ð5Þ
where comma denotes the differentiation with respect to the subscripted variable and Dij is the bending stiffness of the composite plate. For classical Kantorovich method, the solution w(x, y) is assumed to be separable as wðx; yÞ ¼ X ðxÞY ðyÞ
Z
a
þ 4S 4y D66 X 2;x þ 4ðS 6y D16 X ;x X ;xx þ S 5y D26 XX ;x Þ dx Z 1 a ½S 3y N xx X 2;x þ S 4y N yy X 2 þ 2S 6y N xy XX ;x dx 2 0
P¼U þV Z Z 1 a b P¼ ½D11 w2;xx þ 2D12 w;xx w;yy þ D22 w2;yy 2 0 0 þ 4D66 w2;xy þ 4ðD16 w;xx þ D26 w;yy Þw;xy dx dy Z Z 1 a b ½N xx w2;x þ N yy w2;y þ 2N xy w;x w;y dx dy 2 0 0
Z
1 P¼ 2
121
Nxx x
Fig. 1. Rectangular plate subjected to in-plane load Nxx, Nyy, Nxy.
d4 Y d2 Y þ ð2S D 4S D þ S N Þ 2x 12 4x 66 3x yy dy 4 dy 2 þ ðS 1x D11 S 4x N xx ÞY ¼ 0
S 3x D22
ð7Þ
122
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
Boundary conditions along y = 0 and y = b are
3. Solutions of the iterative ODE
Either: 4
dY dY þ ðS 2x D12 4S 4x D66 þ S 3x N yy Þ dy 3 dy ð2S 5x D16 S 6x N xy ÞY ¼ 0
S 3x D22
ð8Þ
or dY ¼ 0
ð9Þ
ð10Þ
or dY ;y ¼ 0
ð11Þ
These conditions, (8)–(11), correspond to the following edge supports: Simply supported edge: conditions (9) and (10) Clamped edge: conditions (9) and (11) Free edge: conditions (8) and (10)
ð17Þ
k1 ¼
ðS 2x D12 2S 4x D66 Þ S 3x D22
ð18Þ
k2 ¼
ðS 1x D11 S 4x N xx Þ S 3x D22
ð19Þ
The characteristic equation of Eq. (17) is s4 þ 2k 1 s2 þ k 2 ¼ 0
ð20Þ
whose four roots are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1;2;3;4 ¼
When Y(y) is priorly specified, the variational principle requires the first variation of the functional Eq. (5) equals to zero. The procedure yields the following governing ODE and the associated BCs: d4 X d2 X S 3y D11 4 þ ð2S 2y D12 4S 4y D66 þ S 3y N xx Þ 2 dx dx þ ðS 1y D22 S 4y N yy ÞX ¼ 0
d4 Y d2 Y þ 2k þ k2Y ¼ 0 1 dy 4 dy 2 where
And either: d2 Y dY þ S 2x D12 Y ¼ 0 S 3x D22 2 þ 2S 6x D26 dy dy
Hereafter, only the case of rectangular plates subjected to uniform axial compression is considered. Setting Nyy = Nxy = 0 in Eq. (7), one has the following fourth order ODE:
k 1
k 21 k 2
ð21Þ
Depending on the values of k1 and k2, Eq. (20) has four kinds of roots in Eq. (21). In general, the solution can be written in one of the following four forms: Y ðyÞ ¼ Ay sinðp1 yÞ þ By cosðp1 yÞ þ C y sinðp2 yÞ þ Dy cosðp2 yÞ ð22Þ
ð12Þ
Y ðyÞ ¼ Ay sinðp1 yÞ þ By cosðp1 yÞ þ C y sinhðp2 yÞ ð23Þ
þ Dy coshðp2 yÞ
Boundary conditions along x = 0 and x = a are Either:
Y ðyÞ ¼ ½Ay cosðp2 yÞ þ By sinðp2 yÞ coshðp1 yÞ þ ½C y cosðp2 yÞ þ Dy sinðp2 yÞ sinhðp1 yÞ
3
dX dX þ ðS 2y D12 4S 4y D66 þ S 3y N xx Þ 3 dx dx ð2S 5y D26 S 6y N xy ÞX ¼ 0
S 3y D11
ð13Þ
Y ðyÞ ¼ Ay sinhðp1 yÞ þ By coshðp1 yÞ þ C y sinhðp2 yÞ ð25Þ
þ Dy coshðp2 yÞ or
The conditions for each type of solution are as follows:
dX ¼ 0
ð14Þ
And either: S 3y D11
ð24Þ
d2 X dX þ S 2y D12 X ¼ 0 þ 2S 6y D16 dx2 dx
1. If k1 > 0 and ðk 21 k 2 Þ > 0, all four roots are imaginary, Eq. (22) is the solution with s1;2 ¼ ip1 ;
ð15Þ
or
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 ¼
dX ;x ¼ 0
k1 þ
ð16Þ
Two sets of ODE, one set for X(x) priorly specified and the other set for Y(y) priorly specified, are now complete for iterative calculations. Next, both derived sets of differential equations are solved.
s3;4 ¼ ip2
k 21
k2;
p2 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1
k 21 k 2
ð26Þ 2. If k1 < 0 and k2 < 0, two roots are imaginary and the other two are real, Eq. (23) is the solution with s1;2 ¼ ip1 ;
s3;4 ¼ p2
V. Ungbhakorn, P. Singhatanadgid / Composite Structures 73 (2006) 120–128
where rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 21 k 2 þ k 1 ;
p1 ¼
p2 ¼
s1;2 ¼ ip3 ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 21 k 2 k 1
ð27Þ 3. If k1 < 0 and ðk 21 k 2 Þ < 0, roots are in complex conjugate pairs, Eq. (24) is the solution with s1;2 ¼ p1 ip2 ;
s3;4 ¼ p1 ip2
where 1 p1 ¼ pffiffiffi 2
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k2 k1;
1 p2 ¼ pffiffiffi 2 k 21 ,
4. If k1 < 0 and 0 < k 2 < (25) is the solution with s1;2 ¼ p1 ;
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k2 þ k1
ð28Þ
all four roots are real, Eq.
p2 ¼
k 21 k 2
k3
k 23 k 4
ð37Þ 2. If k3 < 0 and ðk 23 k 4 Þ < 0, roots are in complex conjugate pairs, Eq. (35) is the solution with s3;4 ¼ p3 ip4
where 1 p3 ¼ pffiffiffi 2
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k4 k3;
s1;2 ¼ p1 ;
1 p4 ¼ pffiffiffi 2
qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi k4 þ k3
ð38Þ
s3;4 ¼ p2
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 21
k 1
p4 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3. If k3 < 0 and k 23 k 4 > 0, Eq. (36) is the solution with
s3;4 ¼ p2
k 1 þ k2; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 23 k 4 ;
k3 þ
s1;2 ¼ p3 ip4 ;
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 ¼
s3;4 ¼ ip4
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p3 ¼
123
p3 ¼
k 3 þ
k 23
ð29Þ
k4;
p4 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 3
k 23 k 4
ð39Þ
Similarly, if Y(y) is known, Eq. (12) can be arranged as d4 X d2 X þ 2k þ k4X ¼ 0 3 dy 4 dy 2
ð30Þ
To obtain the solution, the following algorithm is devised:
where k3 ¼ k4 ¼
ðS 2y D12 2S 4y D66 þ S 3y N xx =2Þ S 3y D11
ð31Þ
S 1y D22 S 3y D11
ð32Þ
whose four roots are rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1;2;3;4 ¼
k 3
k 23 k 4
ð33Þ
Since k4 is always greater than zero, it can be shown that there are only three possible solutions for this case as given by X ðxÞ ¼ Ax sinðp3 xÞ þ Bx cosðp3 xÞ þ C x sinðp4 xÞ þ Dx cosðp4 xÞ
ð34Þ
X ðxÞ ¼ ½Ax cosðp4 xÞ þ Bx sinðp4 xÞ coshðp3 xÞ þ ½C x cosðp4 xÞ þ Dx sinðp4 xÞ sinhðp3 xÞ
ð35Þ
X ðxÞ ¼ Ax sinhðp3 xÞ þ Bx coshðp3 xÞ þ C x sinhðp4 xÞ þ Dx coshðp4 xÞ
4. Iteration procedures
ð36Þ
The conditions for each type of solution are as follows: 1. If k3 > 0 and ðk 23 k 4 Þ > 0, Eq. (34) is the solution with
1. Assume an initial solution X(x) in the x direction which may or may not satisfy any boundary conditions, and then evaluate k1 and k2 according to Eqs. (18) and (19). 2. Calculate the four roots in Eqs. (26)–(29) and select the form of solution from Eqs. (22)–(25) corresponding to the roots. 3. Apply the BCs to the solution Y(y) and determine the eigenvalue Nxx and eigenvector Y(y). This completes the simple Kantorovich method. 4. For the extended Kantorovich method, use the Y(y) obtained in step 3 to evaluate k3 and k4 according to Eqs. (31) and (32). 5. Calculate the four roots in Eqs. (37)–(39) and select the form of solution from Eqs. (34)–(36) corresponding to the roots. 6. Apply the BCs to the solution X(x) and determine the eigenvalue Nxx and eigenvector X(x). 7. Compare the eigenvalue Nxx obtained from step 6 with the previous one. If the difference satisfies the specified tolerance, the last Nxx is taken as the final solution. This completes the extended Kantorovich procedure. Otherwise continue the iterative calculation by repeating steps 1–3 using the eigenvector X(x) obtained from the previous step.
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Observe that if the assumed initial solution in step 1 satisfies all BCs, the solution for simple Kantorovich will give good result, depending on the form of the initial solution as in the Rayleigh–Ritz or Galerkin method.
y
y
F
S
C S
x C Boundary conditions: SCSF
5. Numerical verification and accuracy The iteration procedure outlined in the previous section may apply to rectangular plates with any combinations of simple support (S), clamped support (C), and free edge (F). A simple-clamped–simple-free (SCSF) plate as shown in Fig. 2, is a specimen with simple support on x = 0 and x = a, and clamped and free on y = 0 and y = b, respectively. The other example in Fig. 2 is a CSSC specimen which is simply supported on y = 0 and
S
C
x S Boundary conditions: CSSC
Fig. 2. Boundary conditions of plate denoted by S, C, and F.
x = a, and clamped on the other two edges. The iteration example for an isotropic SCSF plate is illustrated in Table 1. Mechanical properties and plate dimensions are E = 70 GPa, v = 0.25, a = 0.5 m, aspect ratio = 3, and thickness = 0.0023 m. The first iteration begins with assuming function X(x) as x2 and solving for Y(y)
Table 1 Iteration example for an isotropic plate with SCSF boundary condition Iteration no.
Assumed solution X(x) or Y(y) 2
Solution Eigenvalue (kN/m)
Eigenvector X(x) or Y(y)
65.27
A
1
x
2
A
36.84
sinð2px a Þ
3
sinð2px a Þ
36.02
B
4
B
36.02
sinð2px a Þ
Note. E = 70 GPa, v = 0.25, a = 0.5 m. Aspect ratio = 3, and thickness = 2.3 mm. A sinð11:61yÞ 1:103 cosð11:61yÞ 0:8954 sinhð12:97yÞ þ 1:103 coshð12:97yÞ B sinð10:78yÞ 0:5448 cosð10:78yÞ 0:5186 sinhð20:79yÞ þ 0:5448 coshð20:79yÞ.
Buckling mode
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according to Eq. (17). The assumed function X(x) is not required to satisfy the S–S boundary conditions. The eigenvalue and eigenvector Y(y) obtained from the first iteration are 65.27 kN/m and the function denoted as ‘‘A,’’ respectively. The obtained function Y(y) is forced to satisfy the boundary conditions in the y-direction automatically. The out-of-plane displacement w(x, y) is plotted in the last column of Table 1. The second iteration employs function Y(y) obtained from the first iteration as an assumed function. The governing equation, Eq. (12), is reduced to an equation of unknown X(x) in form of Eq. (30). The solutions of the second iteration are eigenvalue of 36.84 kN/m and eigenvector of sinð2px Þ. a The next iteration is performed using the function X ðxÞ ¼ sinð2px Þ from the second iteration. The solution a of Y(y) from the third iteration, represented as ‘‘B,’’ is different from that of the first iteration if comparing term by term. However, the buckling modes obtained in the second and third iteration are almost the same, as shown in the last column. The iteration processes conclude when the eigenvalues converge to a value, i.e. 36.02 kN/m for this case, which is the buckling load. The eigenvectors are also converged to the buckling mode, w(x, y). The Kantorovich method is verified by comparing the solutions from this method with those of the known solutions. In Tables 2–4, nondimensional buckling loads 2 defined in term of pN2xxDb22 of SSSF, SCSF, and SCSC for isotropic and laminated composite plates are compared with the Le´vy solution[9]. The numerical verifications in Tables 2–4 are buckling load of specimens with v = 0.25 for isotropic plates, and v = 0.25, G = 0.5E2 for orthotropic plates. The nondimensional buckling loads and buckling modes obtained from both methods are identical. The buckling configuration for SSSF plates are said to be ‘‘mode 1’’ because there is only one point of local maximum out-of-plane displacement. The buckling modes for SCSF and SCSC plates change from mode 1 for plates with low aspect ratios to higher mode for
Table 2 Nondimensional buckling loads of rectangular plate (SSSF) determined from Levy solution and Kantorovich method a b
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
E1 E2
¼1
[0]8,
E1 E2
¼3
[0/90]2s,
E1 E2
¼ 10
Le´vy solution
Present solution
Le´vy solution
Present solution
Le´vy solution
Present solution
4.4036 1.4342 0.8880 0.6979 0.6104 0.5630 0.5345 0.5161 0.5034 0.4944
4.4036 1.4342 0.8880 0.6979 0.6104 0.5630 0.5345 0.5161 0.5034 0.4944
12.5334 3.5626 1.9075 1.3307 1.0648 0.9208 0.8341 0.7780 0.7395 0.7121
12.5334 3.5626 1.9075 1.3307 1.0648 0.9208 0.8341 0.7780 0.7395 0.7121
7.6905 2.0396 0.9941 0.6284 0.4592 0.3673 0.3119 0.2759 0.2513 0.2336
7.6905 2.0396 0.9941 0.6284 0.4592 0.3673 0.3119 0.2759 0.2513 0.2336
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Table 3 Nondimensional buckling loads of rectangular plate (SCSF) determined from Levy solution and Kantorovich method a b
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
E1 E2
¼1
[0]8,
E1 E2
[0/90]2s,EE12 ¼ 10
¼3
Le´vy Present Solution Solution
Le´vy Present Solution Solution
Le´vy Present Solution Solution
4.5182(1) 1.6983 1.3392 1.3862 1.4319(2) 1.3392 1.3355 1.3862 1.3392(3) 1.3295
12.6646(1) 3.8609 2.4070 2.0764 2.1100 2.4070(2) 2.1738 2.0764 2.0640 2.1100
7.7670(1) 2.2294 1.3489 1.2079 1.3255 1.3489(2) 1.2318 1.2079 1.2448 1.2581(3)
4.5182(1) 1.6983 1.3392 1.3862 1.4319(2) 1.3392 1.3355 1.3862 1.3392(3) 1.3295
12.6646(1) 3.8609 2.4070 2.0764 2.1100 2.4070(2) 2.1738 2.0764 2.0640 2.1100
7.7670(1) 2.2294 1.3489 1.2079 1.3255 1.3489(2) 1.2318 1.2079 1.2448 1.2581(3)
Note. Buckling modes are shown in the parenthesis only for an aspect ratio that bucking mode changes to the higher one.
Table 4 Nondimensional buckling loads of rectangular plate (SCSC) determined from Levy solution and Kantorovich method a b
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
E1 E2
¼1
[0]8,
E1 E2
[0/90]2s,EE12 ¼ 10
¼3
Present Le´vy solution solution
Le´vy solution
Present solution
Le´vy Present solution solution
7.6913(1) 7.6913(2) 7.1159(2) 6.9716(3) 6.9989(4) 7.0552(5) 7.0008(5) 6.9716(6) 6.9778(7) 6.9989(8)
16.2261(1) 11.1626(1) 11.2219(2) 11.1626(2) 10.8973(3) 11.1626(3) 10.8659(4) 10.9810(5) 10.8817(5) 10.8973(6)
16.2261(1) 11.1626(1) 11.2219(2) 11.1626(2) 10.8973(3) 11.1626(3) 10.8659(4) 10.9810(5) 10.8817(5) 10.8973(6)
9.6276(1) 7.8342(1) 7.0500(2) 7.3323(3) 7.0922(3) 7.0500(4) 7.1726(5) 7.0431(5) 7.0500(6) 7.0922(6)
7.6913(1) 7.6913(2) 7.1159(2) 6.9716(3) 6.9989(4) 7.0552(5) 7.0008(5) 6.9716(6) 6.9778(7) 6.9989(8)
9.6276(1) 7.8342(1) 7.0500(2) 7.3323(3) 7.0922(3) 7.0500(4) 7.1726(5) 7.0431(5) 7.0500(6) 7.0922(6)
plates with higher aspect ratios. The number in the superscripted parenthesis refers to buckling mode. The Kantorovich solution is also verified with other available solutions as shown in Table 5. In [10], composite plates with CCCC and CSCS boundary conditions were solved for buckling loads using the Rayleigh–Ritz method and trigonometric functions. Buckling loads of square cross-ply plates determined from the two methods are compared in Table 5. The buckling loads presented in the table are in term of total load in order to be consistent with those shown in [10]. Solutions from the Kantorovich iteration agree very well with those of the previous study. The solutions from the present study are approximately 0.6% higher than the solution of the Rayleigh–Ritz method in all cases of study. Convergence of the solutions of the Kantorovich method is very fast. In this study, the eigenvalue is assumed to be the buckling load of the problem if it is less than 0.01% different from the solution of the previous iteration. Table 6 shows number of iteration required to obtain the buckling load and a comparison of the
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Table 5 Comparing the buckling loads of square plates with solutions from the Rayleigh–Ritz method Boundary condition
Stacking Thickness sequence (in.)
Rayleigh–Ritz Present solution [10] solution (103 lbs) (103 lbs)
% difference
CCCC
[0/90]5s
17.5509 12.2464 8.6962 17.0906 11.9252 8.4681 11.7625 8.2074 5.8282 10.0538 7.0152 4.9815
0.57 0.57 0.57 0.55 0.55 0.55 0.60 0.60 0.60 0.60 0.60 0.60
[90/0]5s
CSCS
[0/90]5s
[90/0]5s
0.115 0.102 0.091 0.115 0.102 0.091 0.115 0.102 0.091 0.115 0.102 0.091
17.6505 12.3159 8.7456 17.1851 11.9911 8.5150 11.8328 8.2565 5.8630 10.1141 7.0572 5.0114
Note. E11 = 215 GPa (31.18 Msi), E22 = 23.6 GPa (3.42 Msi), G12 = 5.2 GPa (0.754 Msi), v12 = 0.28, a = b = 25.4 cm (10 in.).
Table 6 Number of iteration required for each boundary condition Boundary condition
SSSF SCSF SCSC CCCC CCCF CSSC CSCS
No. of iteration
1 1 1 3 3 3 3
Nondimensional buckling load Present study
Past study
2.0396 2.2294 7.8342 12.1023 7.8494 6.5576 8.9399
2.0396 2.2294 7.8342 – – – –
nondimensional buckling loads determined from the present and past studies. The specimen used in Table 6 is a [0/90]2s square plate with E1 = 10E2. The assumed function for the first iteration is X ðxÞ ¼ sinðpxaÞ for all cases of boundary conditions. From Table 6, the first three cases require only one iteration to obtain the buckling load because the initial assumed function happens to satisfy the simply supported boundary condition. If other functions were chosen for the first iteration, it would need additional iterations to reach the buckling load. Similarly, the other four boundary conditions, which are C–C and C–S support in the x-direction, required three iterations to converge to the buckling load. Therefore, the extended Kantorovich method predicts buckling load of isotropic and orthotropic plates with high accuracy compared with several available solutions. The convergence of the solution is very fast, although only one term is used and the initial assumed function is not required to satisfy the boundary conditions. If the correct function is chosen for the first iteration, the extended Kantorovich method returns the buckling load and mode in the first iteration.
6. Other numerical examples Additional nondimensional buckling loads of plates with CCCF, CSSC, and CSCS boundary conditions are tabulated in Tables 7–9, respectively. Nondimensional buckling loads of all cases decrease when plate aspect ratio is increased. The buckled configuration of plates with aspect ratio of 0.5 is mode 1, and buckling
Table 7 Nondimensional buckling load of rectangular plate (CCCF) determined from Kantorovich method a b
Isotropic
[0]8, E1 E2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
16.4899 4.6504 2.5872 1.9916 1.8332 1.7392(2) 1.5897 1.5289 1.5150 1.4711(3)
¼3
48.6355 12.8083 6.3169 4.1821 3.3301 2.9976 2.9067 2.8999 2.5447(2) 2.4358
[0/90]2s, E1 E2
¼ 10
160.6444 40.8181 18.7725 11.1958 7.8297 6.1443 5.2708 4.8404 4.6660 4.6278
E1 E2
¼3
23.8068 6.4015 3.3033 2.3459 2.0277 1.9590 1.7922(2) 1.6611 1.6064 1.5937
E1 E2
¼ 10
30.3764 7.8494 3.7880 2.4895 2.0158 1.8747 1.8610 1.5981(2) 1.4980 1.4592
Table 8 Nondimensional buckling load of rectangular plate (CSSC) determined from Kantorovich method a b
Isotropic
[0]8, E1 E2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
10.9106 6.2304 5.8750(2) 5.6726 5.5561(3) 5.5301(4) 5.4888 5.4709(5) 5.4617 5.4479(6)
¼3
27.7747 10.9092 9.5951 9.0057(2) 8.6893 8.5895(3) 8.4635 8.4356(4) 8.3732 8.3627(5)
[0/90]2s, E1 E2
¼ 10
85.0875 25.2817 16.3439 15.2296 14.6191(2) 13.7375 13.5798 13.4360(3) 13.2045 13.1670
E1 E2
¼3
14.1555 6.6595 6.2952(2) 5.8419 5.7674(3) 5.6755 5.6353(4) 5.6119 5.5858(5) 5.5790
E1 E2
¼ 10
16.6687 6.5576 6.0568(2) 5.4602 5.3863(3) 5.2368 5.2142(4) 5.1563 5.1452(5) 5.1185
Table 9 Nondimensional buckling load of rectangular plate (CSCS) determined from Kantorovich method a b
Isotropic
[0]8, E1 E2
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
18.1874 6.7432 5.3747 4.8471(2) 4.5224 4.4065(3) 4.2786 4.2367(4) 4.1737 4.1542(5)
¼3
50.6458 15.2061 9.4526 8.2868 7.6426(2) 6.9645 6.7939 6.5965(3) 6.4210 6.3845
[0/90]2s, E1 E2
¼ 10
162.6750 43.2368 21.9452 15.4429 13.3894 13.0294 11.6246(2) 10.7697 10.4462 10.3932
E1 E2
¼3
25.0849 8.1043 5.7404 5.3087(2) 4.6619 4.5360 4.3391(3) 4.2568 4.1950(4) 4.1380
E1 E2
¼ 10
31.0038 8.9399 5.6477 5.2293 4.3659(2) 4.1019 3.9650(3) 3.7948 3.7647 3.6634(4)
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127
Fig. 3. Buckling modes of plates with various edge supports.
mode increases with plate aspect ratios. The superscripted buckling mode is shown only for plates with aspect ratios that have a change of buckling mode to a higher one. Buckling modes of several plate configurations are represented in Fig. 3. The plots are buckling modes of a composite plate with stacking sequence of [0/90]2s, E1 = 10E2, and an aspect ratio of 3. It is observed that buckling modes of SCSF and CCCF are mode 2 and mode 1, respectively. The only difference of these two specimens is the boundary conditions on the loading edges which are simple and clamped supports respectively. The buckling mode is reduced from 2 to 1 when the loading edges are constrained, i.e. the specimen edge is not free to rotate. The CCCF specimen may buckle into mode 2 if the plate aspect ratio is increased to 4. For CSSC specimen, boundary conditions on the loading edges are different, i.e. clamped and simple supports. As a result, buckling configuration on both sides of the support is not identical, i.e. displacement on the side of clamped support is smaller than that of the other.
7. Discussion and conclusion Buckling loads and modes of laminated composite plates are solved using the extended Kantorovich method. The boundary conditions of the rectangular plate can be any combinations of simple support, clamped support, or free edge. The present study is limited to symmetrical composite plates with unidirectional or
cross-ply fibers. Although the solution of the Kantorovich method is obtained from solving a set of ODE, this method requires a series of iterative calculation, so it is considered as a semi-analytical method. By assuming that the out-of-plane displacement of the buckled plates is separable and either one of X(x) or Y(y) is known beforehand, applying the principle of minimum total potential energy along with the variational principle to the total potential energy yields a set of ODE. These ODE and the prescribed boundary conditions are used in iterative calculations for eigenvalues and eigenvectors. The procedure is repeated until the eigenvalue converges to a specific value of buckling load. The final product of X(x) and Y(y) indicates the buckling mode. The extended Kantorovich method is verified numerically by comparing the buckling load and mode with known solutions. Buckling behaviors of SSSF, SCSF, and SCSC specimens obtained from the present study are identical to the Le´vyÕs solutions in [9]. Differences in buckling load in case of CCCC and CSCS specimen when compared to solutions from the Rayleigh–Ritz method [10] are less than one percent. Therefore, the extended Kantorovich method is validated for isotropic rectangular plates or composite plates with unidirectional or cross-ply symmetric stacking sequence, i.e. specimen with D16 = D26 = 0. For specimens with a presence of D16 and D26 i.e. angle-ply panels, the approximation of the displacement functions is required to include some additional terms in order to simulate the actual buckled mode. The approximate function could be in
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form of w(x, y) = X1(x)Y1(y) + X2(x)Y2(y) which will lead to a set of simultaneous ODE. Solving these ODE are tedious and not in the scope of this study. New numerical examples of specimen with combinations of simple, clamped, and free boundary conditions are also included in this work. An advantage of this method is that there is no need to solve the governing partial differential equations. They are transformed to a set of ODE with an assumed displacement function. Another advantage is that the initial assumed displacement function in the first iteration can be arbitrarily selected regardless of the type of boundary conditions. The displacement functions are automatically forced to satisfy the boundary conditions in the next iterations.
References [1] Jones RM. Mechanics of composite materials. Washington, DC: Scripta Book Co.; 1975.
[2] Iyengar NGR. Structural stability of columns and plates. Chichester, England: Ellis Horwood Limited Publishers; 1988. [3] Kantorovich LV, Krylov IV. Approximate method of higher analysis. New York: Interscience Publishers Inc.; 1964. [4] Kerr AD. An extended Kantorovich method for the solution of eigenvalues problems. Int J Solids Struct 1969;5: 559–72. [5] Kerr AD, Alexander H. An application of the extended Kantorovich method to the stress analysis of a clamped rectangular plate. Acta Mech 1968;6:180–96. [6] Dalaei M, Kerr AD. Analysis of clamped rectangular orthotropic plates subjected to a uniform lateral load. Int J Mech Sci 1995;37(5):527–35. [7] Yuan S, Jin Y. Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method. Comput Struct 1998;66(6):861–7. [8] Reddy JN. Mechanics of laminated composite plates and shells. Boca Raton: CRC Press; 2004. [9] Reddy JN. Theory and analysis of elastic plates. Philadelphia, PA: Taylor & Francis; 1999. [10] Chai GB. Buckling of generally laminated composite plates with various edge support conditions. Comput Struct 1994;29(3): 299–310.