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Bending analysis of thick laminated plates using extended Kantorovich method
Composite Structures 62 (2003) 279–283 www.elsevier.com/locate/compstruct
Bending analysis of thick laminated plates using extended Kantorovich method M.M. Aghdam *, S.R. Falahatgar Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., Tehran 15914, Iran
Abstract This study is concerned with bending of moderately thick rectangular laminated plates with clamped edges. The governing equations, based on Reissner first-order shear deformation plate theory; in terms of deflection and rotations of the plate include a system of three second-order, partial differential equations (PDEs). Application of extended Kantorovich method (EKM) to the system of partial differential equations reduces the governing equations to a double set of three second-order ordinary differential equations in the variables x and y. These sets of equations were then solved in an iterative manner until convergence was achieved. Normally three to four iterations are enough to get the final results with desired accuracy. It is demonstrated that, unlike other weighted residual methods, in the extended Kantorovich method initial guesses to start iterations are arbitrary and not even necessary to satisfy the boundary conditions. Results of this study also reveal that the convergence of the EKM is rapid and the method is an efficient way to solve system of PDEs of the same type. To compare the results of this study, the problem was also analyzed using commercial finite element software, ANSYS. Results show reasonably good agreement with the finite element analysis. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Extended Kantorovich method; Laminated composite plate; Bending analysis
1. Introduction The classical plate theory (CLT) is the simplest plate theory and has been used extensively for bending analysis of plates. However, this theory is valid for thin plates where the ratio of the thickness to length of the plate is small. This is due to the fact that the theory is based on various simplifying assumptions [1], such as neglecting effects of transverse shear deformations and normal stress. In particular, the theory shows inaccurate results for laminated plates due to the low shear modulus of the composite materials. Therefore, it has been accepted that the CLT should be modified in order to include shear deformations and normal stress effects. Amongst the first and well-known attempts are firstorder theories presented by Reissner [2,3] and Mindlin [4]. Since then, there have been considerable interests to present higher-order theories in order to reduce inaccuracies [5–7]. These theories have also been extended to use for bending analysis of laminated plates [8–11].
*
Corresponding author. Tel.: +98-21-64543429; fax: +98-21-6419736. E-mail address: [email protected] (M.M. Aghdam). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2003.09.026
Solution to the governing equations in various theories normally involves complicated numerical procedures. Only in certain cases, such as simply supported special orthotropic plates, closed form solutions exist. One of the approximate methods to solve the governing equations of plates is the extended Kantorovich method (EKM). It was first introduced by Kerr [12] to obtain approximate solution for torsion of an isotropic beam with a rectangular cross section. Since then, EKM, the same as other weighted residual methods, was used to solve various boundary-value [13–15] and eigenvalue [16,17] problems. In these applications, the EKM has been used to solve a single partial differential equation (PDE). The EKM has also been employed to a system of PDE’s [18] where bending of Reissner plate with clamped edges was analyzed. In this study, the attention is focused on the bending of thick laminated plates subjected to uniform distributed load. The solution method includes converting the system of PDE’s as the governing equations to a dual system of ordinary differential equations using the EKM. These systems of equations were then solved in an iterative approach. The same problem has also been studied using the commercial finite element code ANSYS. Comparison of the
280
M.M. Aghdam, S.R. Falahatgar / Composite Structures 62 (2003) 279–283
results with the finite element solutions shows good agreement.
2. Governing equations The mixed variational formulation of Reissner for isotropic plates has been extended to obtain the governing equations for a rectangular plate composed of special orthotropic layers. Details of derivation of the governing equations can be found elsewhere [19]. The governing equations in terms of unknown displacement and rotations are A55 ðux;x þ w;xx Þ þ A44 ðuy;y þ w;yy Þ ¼ q;
where ðxb ; yb Þ is an arbitrary point on the boundary of the plate. Once the solutions of the governing equations (1) are obtained, one can determine bending and twisting moments together with transverse shear forces within the plate using 6 m13 þ m12 m23 Mxx ¼ D11 ux;x þ m12 uy;y þ q ; 5h E1 6 m23 þ m21 m13 q ; Myy ¼ D22 uy;y þ m21 ux;x þ 5h E2 ð4Þ Mxy ¼ D66 ðux;y þ uy;x Þ; Qx ¼ A55 ðux þ w;x Þ; Qy ¼ A44 ðuy þ w;y Þ:
D11 ux;xx þ D66 ux;yy þ ðD12 þ D66 Þuy;xy A55 ðux þ w;x Þ ¼ Cx q;x ;
ð1Þ
D22 uy;yy þ D66 uy;xx þ ðD12 þ D66 Þux;xy
3. Application of the EKM
A44 ðuy þ w;y Þ ¼ Cy q;y ; where comma denotes differentiation with respect to x or y, w is deflection and ux and uy are rotations of the plate and q denotes distributed lateral force. Aij and Dij are extensional and flexural stiffness components, respectively, and related to ply stiffness using Z h=2 ðkÞ Qij ð1; z2 Þ dz; ðAij ; Dij Þ ¼ h=2 ðkÞ
where Qij are reduced stiffness components of the kth layer. Effects of normal stress were also included in Eq. (1) through Cx and Cy , which can be defined as 6 m13 þ m12 m23 Cx ¼ D11 5h E1 6 m23 þ m21 m13 Cy ¼ D22 5h E2
w ¼ kn1 ðxÞ:w1 ðyÞ; ux ¼ n2 ðxÞ:w2 ðyÞ;
ð2Þ
ð3Þ
ð5Þ
uy ¼ n3 ðxÞ:w3 ðyÞ; in which, ni ðxÞ and wi ðyÞ are unknown dimensionless 1=2 functions of x and y to be determined and k ¼ ðLx Ly Þ with Lx and Ly as dimensions of the plate in the x and y directions. The boundary conditions (3) in terms of separable forms are ni ð0Þ ¼ ni ðLx Þ ¼ 0; wi ð0Þ ¼ wi ðLy Þ ¼ 0;
in which Ei and mij are Young’s modulus and Poisson’s ratios of the layers and subscript 1 denotes properties parallel to the fibre direction and 2 and 3 are normal to the fibre. It should be noted that ignoring normal stress effects in the governing equations (1), assuming Cx ¼ Cy ¼ 0, leads to the equations derived by Whitney and Pagano [10] for bending analysis of anisotropic plates. This is due to the point that effects of normal stress were ignored in [10]. Furthermore, assuming similar properties for plate material in the x and y directions reduces the governing equations (1) to the equations obtained by Reissner [6] for bending of isotropic plates. The boundary conditions for a clamped rectangular plate are wðxb ; yb Þ ¼ ux ðxb ; yb Þ ¼ uy ðxb ; yb Þ ¼ 0;
In order to apply the EKM to the governing equations (1), the deflection and rotations should be considered as product of separate functions of x and y as
i ¼ 1; 2; 3:
ð6Þ
Introducing (5) into (1) yields to the new form of governing equations in terms of ni and wi A55 ðn2;x w2 þ kn1;xx w1 Þ þ A44 ðn3 w3;y þ kn1 w1;yy Þ ¼ q; D11 n2;xx w2 þ D66 n2 w2;yy þ ðD12 þ D66 Þn3;x w3;y A55 ðn2 w2 þ kn1;x w1 Þ ¼ Cx q;x ; D22 n3 w3;yy þ D66 n3;xx w3 þ ðD12 þ D66 Þn2;x w2;y A44 ðn3 w3 þ kn1 w1;y Þ ¼ Cy q;y : ð7Þ According to the general procedure of the weighted residual method, it is necessary to multiply each of Eq. (7) by an appropriate weighting function. These functions would be w1 ðyÞ, w2 ðyÞ, and w3 ðyÞ for the first, second and third of Eq. (7), respectively. Now starting with arbitrary functions for wi ðyÞ, as initial guess, and integrating over the entire length of the plate in the y direction, Ly , in conjunction with the boundary conditions (6) yields to
M.M. Aghdam, S.R. Falahatgar / Composite Structures 62 (2003) 279–283
kðF1 d2 þ c1 F7 Þn1 þ F4 dn2 þ c1 F5 n3 ¼ B1 ; kc3 F4 dn1 þ ðF2 d2 þ c6 Þn2 þ c4 F6 dn3 ¼ B2 ;
ð8Þ
2
kc1 c3 F5 n1 c4 F6 dn2 þ ðc5 F3 d þ c7 Þn3 ¼ B3 ; where d d=dx and F ’s, B’s and c’s are defined in Appendix A. Solving the system of ordinary differential equations ODE (8) together with boundary data (6) results in the first approximation for ni ðxÞ. Now, using the same procedure in the x direction in (7) leads to another system of ODE in wi ðyÞ: kðc1 G1 d2 þ G7 Þw1 þ G5 w2 þ c1 G4 dw3 ¼ E1 ; kc3 G5 w1 þ ðc5 G2 d2 þ J1 Þw2 þ c4 G6 dw3 ¼ E2 ;
281
achieved. Normally, three to four iterations are enough to provide a good convergence. Figs. 1 and 2 depict convergence of n1 ðxÞ=ðmax jn1 ðxÞjÞ and w1 ðyÞ=ðmax jw1 ðyÞjÞ versus x=Lx and y=Ly respectively, for a square plate with width to thickness ratio of 5. As can be seen in the figures, the convergence of the EKM is rapid. The convergence of deflection of the plate at ðx; Ly =2Þ is shown in Fig. 3 which was determined using n1 ðxÞ and w1 ðyÞ in the first of Eq. (5) and normalized with respect to the maximum value of the deflection. For validation purpose, results of the presented study were compared with finite element analysis. The finite
ð9Þ
2
kc1 c3 G4 dw1 c4 G6 dw2 þ ðc2 G3 d þ J2 Þw3 ¼ E3 ; where d d=dy and G’s, E’s and J ’s are defined in Appendix A. This procedure shows how the governing system of PDE (1) reduces to two separated systems of ODEs, (8) and (9), by the EKM.
4. Results and discussions The procedure explained in previous section was applied to a four layer glass/epoxy laminate [0°, 90°]s with ply properties [20] E1 ¼ 43:5 GPa; E2 ¼ E3 ¼ 11:5 GPa; m12 ¼ m13 ¼ 0:27; m23 ¼ 0:4; G12 ¼ G13 ¼ 3:45 GPa;
Fig. 1. Convergence of n1 ðxÞ.
ð10Þ
G23 ¼ 4:12 GPa:
The plate was square with 0.5 m length and 0.1 m thickness. The following functions were selected as initial guesses to start iterations: w1 ðyÞ ¼ sinðpy=Ly Þ; w2 ðyÞ ¼ ð1 y=Ly Þ; w3 ðyÞ ¼ sinðpy=Ly Þ:
ð11Þ
Unlike other weighted residual methods, in EKM initial guesses are arbitrary and are not necessary to satisfy the boundary data (6). This is due to the iterative nature of the EKM, which means the boundary conditions will be satisfied in the subsequent iterations. As can be seen in Eq. (11), the applied initial guess function for w2 ðyÞ does not satisfy the boundary condition (6) at y ¼ 0. Now, it is possible to determine constants B’s and F ’s in Eq. (8), using initial guesses (11). Solving the system of ODE (8) in conjunction with boundary conditions (6) yields to the first approximations for ni ðxÞ. The obtained ni ðxÞ can then be used to determine new expressions for wi ðyÞ using (9). This completes the first iteration. The iterations should be continued until convergence is
Fig. 2. Convergence of w1 ðyÞ.
Fig. 3. Convergence of deflection for square plate at wðx; Ly =2Þ.
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Acknowledgements The first author wishes to acknowledge fruitful discussions with Dr. S.J. Fariborz of the Mechanical Engineering Department of Amirkabir University.
Appendix A
Fig. 4. Comparison of the plate deflection at wðx; Ly =2Þ between EKM and FEM.
-1
c4 ¼ ðD12 þ D66 Þ=D11 ; Z Z F6þj ¼ wj wj;yy dy ¼ w2j;y dy;
-2 Lx /Ly =3 Lx /Ly =2.5 Lx /Ly =2 Lx /Ly =1.5 Lx /Ly =1
3
4
wh E2 × 100 / (qLy)
0
-3
-4 0.5
0.6
0.7
0.8
0.9
Fi , Bi and ci , in Eq. (8) are Z Fi ¼ w2i dy ði ¼ 1; 2; 3Þ; c1 ¼ A44 =A55 ; Z F4 ¼ w1 w2 dy; c2 ¼ D22 =D11 ; Z Z F5 ¼ w1 w3;y dy ¼ w3 w1;y dy; c3 ¼ A55 =D11 ; Z Z F6 ¼ w2 w3;y dy ¼ w3 w2;y dy;
1.0
y / Ly
Fig. 5. Deflection of the plate with various aspect ratios.
element results were obtained using commercial finite element software ANSYS. Comparison of the nondimensional deflection of the mid-plane of the square plate at wðx; Ly =2Þ is presented in Fig. 4. As can be seen in the figure, the maximum deviation is about 5%, which was occurred at the center of the plate. Fig. 5 depicts deflection of the non-dimensional deflection of the plate for various aspect ratios from Lx =Ly ¼ 1–3. It seems that there is not major difference for deflection of the plate with aspect ratios greater than 3.
ðj ¼ 1; 2; 3Þ;
c5 ¼ D66 =D11 ; Z 1 qw1 dy; c6 ¼ c5 F8 c3 F2 ; B1 ¼ A55 Z Z Cx Cy q;x w2 dy; B3 ¼ q;y w3 dy; B2 ¼ D11 D11 c7 ¼ c2 F9 c1 c3 F3 : Except for G4 and G5 , all other Gi and Ei in Eq. (9) can also be found using the same expressions for Fi and Bi by interchanging w, y, B and F with n, x, E, G, respectively, and G4 and G5 are as follows: Z Z G4 ¼ n1 n3 dx; G5 ¼ n1 n2;x dx: All integrals should be determined from y ¼ 0 to y ¼ Ly for Fi and Bi and from x ¼ 0 to x ¼ Lx for Gi and Ei . In Eq. (9) J1 and J2 are defined as J1 ¼ G8 c3 G2 ;
J2 ¼ c5 G9 c1 c3 G3 :
5. Conclusion References Bending of moderately thick laminated plates was analyzed using the EKM. It was shown that convergence of the method is fast. The method is equally efficient for both single partial differential equation and system of equations. The method can easily be employed for non-linear loading of rectangular plates as long as integrals for Bi and Ei in Eqs. (8) and (9) can be determined. It is also possible to solve the resulting dual system of ordinary differential equations, (8) and (9), in a closed form manner, as it was done in [18] for the system of governing equations of thick isotropic plates.
[1] Whitney JM. Structural analysis of laminated anisotropic plates. Lancaster, Pennsylvania: Technomic; 1987. [2] Reissner E. On the theory of bending of elastic plates. J Math Phys 1944;23:184–91. [3] Reissner E. The effects of transverse shear deformation on bending of elastic plates. J Appl Mech 1945;12:69–77. [4] Mindlin RD. Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J Appl Mech 1951;18:31–8. [5] Reissner E. On transverse bending of plates including the effects of transverse shear deformation. Int J Solids Struct 1975;11:569–73. [6] Reissner E. Reflections on the theory of elastic plates. Appl Mech Rev 1985;38:1453–64.
M.M. Aghdam, S.R. Falahatgar / Composite Structures 62 (2003) 279–283 [7] Lo KH, Christensen RM, Wu EM. A higher order theory of plate deformation. Part 1: homogeneous plates. J Appl Mech 1977;44: 663–8. [8] Lo KH, Christensen RM, Wu EM. A higher order theory of plate deformation. Part 2: laminated plates. J Appl Mech 1977;44:669– 76. [9] Nelson RB, Lorch DR. A refined theory of laminated orthotropic plates. J Appl Mech 1974;41:177–83. [10] Whitney JM, Pagano NJ. Shear deformation in heterogeneous anisotropic plates. J Appl Mech 1970;37:1031–6. [11] Reddy JN. A simple higher order theory for laminated composite plates. J Appl Mech 1984;51:745–52. [12] Kerr AD. An extension of the Kantorovich method. Quart Appl Math 1968;26:219–29. [13] 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.
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[14] Dalaei M, Kerr AD. Analysis of clamped rectangular orthotropic plates subjected to a uniform lateral load. Int J Mech Sci 1995;37:527–35. [15] Yuan S, Yan J, Williams FW. Bending analysis of Mindlin plates by extended Kantorovich method. ASCE J Eng Mech 1998;124: 1339–45. [16] Kerr AD. An extended Kantorovich method for the solution of eigenvalue problems. Int J Solids Struct 1969;5:559–72. [17] Dalaei M, Kerr AD. Natural vibration analysis of clamped rectangular orthotropic plates. J Sounds Vib 1996;189:399–406. [18] Aghdam MM, Shakeri M, Fariborz SJ. Solution to the Reissner plate with clamped edges. ASCE J Eng Mech 1996;122:679–82. [19] Aghdam MM, Fariborz SJ, Shakeri M. Bending of orthotropic plates with shear deformation. In: Proceedings of the First Annual Conference on Boilers and Pressure Vessels, Iran, 1992, p. 56–67. [20] Herakovich CT. Mechanics of fibrous composites. NY: John Wiley & Sons; 1988.
Bending analysis of thick laminated plates using extended Kantorovich method M.M. Aghdam *, S.R. Falahatgar Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., Tehran 15914, Iran
Abstract This study is concerned with bending of moderately thick rectangular laminated plates with clamped edges. The governing equations, based on Reissner first-order shear deformation plate theory; in terms of deflection and rotations of the plate include a system of three second-order, partial differential equations (PDEs). Application of extended Kantorovich method (EKM) to the system of partial differential equations reduces the governing equations to a double set of three second-order ordinary differential equations in the variables x and y. These sets of equations were then solved in an iterative manner until convergence was achieved. Normally three to four iterations are enough to get the final results with desired accuracy. It is demonstrated that, unlike other weighted residual methods, in the extended Kantorovich method initial guesses to start iterations are arbitrary and not even necessary to satisfy the boundary conditions. Results of this study also reveal that the convergence of the EKM is rapid and the method is an efficient way to solve system of PDEs of the same type. To compare the results of this study, the problem was also analyzed using commercial finite element software, ANSYS. Results show reasonably good agreement with the finite element analysis. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Extended Kantorovich method; Laminated composite plate; Bending analysis
1. Introduction The classical plate theory (CLT) is the simplest plate theory and has been used extensively for bending analysis of plates. However, this theory is valid for thin plates where the ratio of the thickness to length of the plate is small. This is due to the fact that the theory is based on various simplifying assumptions [1], such as neglecting effects of transverse shear deformations and normal stress. In particular, the theory shows inaccurate results for laminated plates due to the low shear modulus of the composite materials. Therefore, it has been accepted that the CLT should be modified in order to include shear deformations and normal stress effects. Amongst the first and well-known attempts are firstorder theories presented by Reissner [2,3] and Mindlin [4]. Since then, there have been considerable interests to present higher-order theories in order to reduce inaccuracies [5–7]. These theories have also been extended to use for bending analysis of laminated plates [8–11].
*
Corresponding author. Tel.: +98-21-64543429; fax: +98-21-6419736. E-mail address: [email protected] (M.M. Aghdam). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2003.09.026
Solution to the governing equations in various theories normally involves complicated numerical procedures. Only in certain cases, such as simply supported special orthotropic plates, closed form solutions exist. One of the approximate methods to solve the governing equations of plates is the extended Kantorovich method (EKM). It was first introduced by Kerr [12] to obtain approximate solution for torsion of an isotropic beam with a rectangular cross section. Since then, EKM, the same as other weighted residual methods, was used to solve various boundary-value [13–15] and eigenvalue [16,17] problems. In these applications, the EKM has been used to solve a single partial differential equation (PDE). The EKM has also been employed to a system of PDE’s [18] where bending of Reissner plate with clamped edges was analyzed. In this study, the attention is focused on the bending of thick laminated plates subjected to uniform distributed load. The solution method includes converting the system of PDE’s as the governing equations to a dual system of ordinary differential equations using the EKM. These systems of equations were then solved in an iterative approach. The same problem has also been studied using the commercial finite element code ANSYS. Comparison of the
280
M.M. Aghdam, S.R. Falahatgar / Composite Structures 62 (2003) 279–283
results with the finite element solutions shows good agreement.
2. Governing equations The mixed variational formulation of Reissner for isotropic plates has been extended to obtain the governing equations for a rectangular plate composed of special orthotropic layers. Details of derivation of the governing equations can be found elsewhere [19]. The governing equations in terms of unknown displacement and rotations are A55 ðux;x þ w;xx Þ þ A44 ðuy;y þ w;yy Þ ¼ q;
where ðxb ; yb Þ is an arbitrary point on the boundary of the plate. Once the solutions of the governing equations (1) are obtained, one can determine bending and twisting moments together with transverse shear forces within the plate using 6 m13 þ m12 m23 Mxx ¼ D11 ux;x þ m12 uy;y þ q ; 5h E1 6 m23 þ m21 m13 q ; Myy ¼ D22 uy;y þ m21 ux;x þ 5h E2 ð4Þ Mxy ¼ D66 ðux;y þ uy;x Þ; Qx ¼ A55 ðux þ w;x Þ; Qy ¼ A44 ðuy þ w;y Þ:
D11 ux;xx þ D66 ux;yy þ ðD12 þ D66 Þuy;xy A55 ðux þ w;x Þ ¼ Cx q;x ;
ð1Þ
D22 uy;yy þ D66 uy;xx þ ðD12 þ D66 Þux;xy
3. Application of the EKM
A44 ðuy þ w;y Þ ¼ Cy q;y ; where comma denotes differentiation with respect to x or y, w is deflection and ux and uy are rotations of the plate and q denotes distributed lateral force. Aij and Dij are extensional and flexural stiffness components, respectively, and related to ply stiffness using Z h=2 ðkÞ Qij ð1; z2 Þ dz; ðAij ; Dij Þ ¼ h=2 ðkÞ
where Qij are reduced stiffness components of the kth layer. Effects of normal stress were also included in Eq. (1) through Cx and Cy , which can be defined as 6 m13 þ m12 m23 Cx ¼ D11 5h E1 6 m23 þ m21 m13 Cy ¼ D22 5h E2
w ¼ kn1 ðxÞ:w1 ðyÞ; ux ¼ n2 ðxÞ:w2 ðyÞ;
ð2Þ
ð3Þ
ð5Þ
uy ¼ n3 ðxÞ:w3 ðyÞ; in which, ni ðxÞ and wi ðyÞ are unknown dimensionless 1=2 functions of x and y to be determined and k ¼ ðLx Ly Þ with Lx and Ly as dimensions of the plate in the x and y directions. The boundary conditions (3) in terms of separable forms are ni ð0Þ ¼ ni ðLx Þ ¼ 0; wi ð0Þ ¼ wi ðLy Þ ¼ 0;
in which Ei and mij are Young’s modulus and Poisson’s ratios of the layers and subscript 1 denotes properties parallel to the fibre direction and 2 and 3 are normal to the fibre. It should be noted that ignoring normal stress effects in the governing equations (1), assuming Cx ¼ Cy ¼ 0, leads to the equations derived by Whitney and Pagano [10] for bending analysis of anisotropic plates. This is due to the point that effects of normal stress were ignored in [10]. Furthermore, assuming similar properties for plate material in the x and y directions reduces the governing equations (1) to the equations obtained by Reissner [6] for bending of isotropic plates. The boundary conditions for a clamped rectangular plate are wðxb ; yb Þ ¼ ux ðxb ; yb Þ ¼ uy ðxb ; yb Þ ¼ 0;
In order to apply the EKM to the governing equations (1), the deflection and rotations should be considered as product of separate functions of x and y as
i ¼ 1; 2; 3:
ð6Þ
Introducing (5) into (1) yields to the new form of governing equations in terms of ni and wi A55 ðn2;x w2 þ kn1;xx w1 Þ þ A44 ðn3 w3;y þ kn1 w1;yy Þ ¼ q; D11 n2;xx w2 þ D66 n2 w2;yy þ ðD12 þ D66 Þn3;x w3;y A55 ðn2 w2 þ kn1;x w1 Þ ¼ Cx q;x ; D22 n3 w3;yy þ D66 n3;xx w3 þ ðD12 þ D66 Þn2;x w2;y A44 ðn3 w3 þ kn1 w1;y Þ ¼ Cy q;y : ð7Þ According to the general procedure of the weighted residual method, it is necessary to multiply each of Eq. (7) by an appropriate weighting function. These functions would be w1 ðyÞ, w2 ðyÞ, and w3 ðyÞ for the first, second and third of Eq. (7), respectively. Now starting with arbitrary functions for wi ðyÞ, as initial guess, and integrating over the entire length of the plate in the y direction, Ly , in conjunction with the boundary conditions (6) yields to
M.M. Aghdam, S.R. Falahatgar / Composite Structures 62 (2003) 279–283
kðF1 d2 þ c1 F7 Þn1 þ F4 dn2 þ c1 F5 n3 ¼ B1 ; kc3 F4 dn1 þ ðF2 d2 þ c6 Þn2 þ c4 F6 dn3 ¼ B2 ;
ð8Þ
2
kc1 c3 F5 n1 c4 F6 dn2 þ ðc5 F3 d þ c7 Þn3 ¼ B3 ; where d d=dx and F ’s, B’s and c’s are defined in Appendix A. Solving the system of ordinary differential equations ODE (8) together with boundary data (6) results in the first approximation for ni ðxÞ. Now, using the same procedure in the x direction in (7) leads to another system of ODE in wi ðyÞ: kðc1 G1 d2 þ G7 Þw1 þ G5 w2 þ c1 G4 dw3 ¼ E1 ; kc3 G5 w1 þ ðc5 G2 d2 þ J1 Þw2 þ c4 G6 dw3 ¼ E2 ;
281
achieved. Normally, three to four iterations are enough to provide a good convergence. Figs. 1 and 2 depict convergence of n1 ðxÞ=ðmax jn1 ðxÞjÞ and w1 ðyÞ=ðmax jw1 ðyÞjÞ versus x=Lx and y=Ly respectively, for a square plate with width to thickness ratio of 5. As can be seen in the figures, the convergence of the EKM is rapid. The convergence of deflection of the plate at ðx; Ly =2Þ is shown in Fig. 3 which was determined using n1 ðxÞ and w1 ðyÞ in the first of Eq. (5) and normalized with respect to the maximum value of the deflection. For validation purpose, results of the presented study were compared with finite element analysis. The finite
ð9Þ
2
kc1 c3 G4 dw1 c4 G6 dw2 þ ðc2 G3 d þ J2 Þw3 ¼ E3 ; where d d=dy and G’s, E’s and J ’s are defined in Appendix A. This procedure shows how the governing system of PDE (1) reduces to two separated systems of ODEs, (8) and (9), by the EKM.
4. Results and discussions The procedure explained in previous section was applied to a four layer glass/epoxy laminate [0°, 90°]s with ply properties [20] E1 ¼ 43:5 GPa; E2 ¼ E3 ¼ 11:5 GPa; m12 ¼ m13 ¼ 0:27; m23 ¼ 0:4; G12 ¼ G13 ¼ 3:45 GPa;
Fig. 1. Convergence of n1 ðxÞ.
ð10Þ
G23 ¼ 4:12 GPa:
The plate was square with 0.5 m length and 0.1 m thickness. The following functions were selected as initial guesses to start iterations: w1 ðyÞ ¼ sinðpy=Ly Þ; w2 ðyÞ ¼ ð1 y=Ly Þ; w3 ðyÞ ¼ sinðpy=Ly Þ:
ð11Þ
Unlike other weighted residual methods, in EKM initial guesses are arbitrary and are not necessary to satisfy the boundary data (6). This is due to the iterative nature of the EKM, which means the boundary conditions will be satisfied in the subsequent iterations. As can be seen in Eq. (11), the applied initial guess function for w2 ðyÞ does not satisfy the boundary condition (6) at y ¼ 0. Now, it is possible to determine constants B’s and F ’s in Eq. (8), using initial guesses (11). Solving the system of ODE (8) in conjunction with boundary conditions (6) yields to the first approximations for ni ðxÞ. The obtained ni ðxÞ can then be used to determine new expressions for wi ðyÞ using (9). This completes the first iteration. The iterations should be continued until convergence is
Fig. 2. Convergence of w1 ðyÞ.
Fig. 3. Convergence of deflection for square plate at wðx; Ly =2Þ.
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Acknowledgements The first author wishes to acknowledge fruitful discussions with Dr. S.J. Fariborz of the Mechanical Engineering Department of Amirkabir University.
Appendix A
Fig. 4. Comparison of the plate deflection at wðx; Ly =2Þ between EKM and FEM.
-1
c4 ¼ ðD12 þ D66 Þ=D11 ; Z Z F6þj ¼ wj wj;yy dy ¼ w2j;y dy;
-2 Lx /Ly =3 Lx /Ly =2.5 Lx /Ly =2 Lx /Ly =1.5 Lx /Ly =1
3
4
wh E2 × 100 / (qLy)
0
-3
-4 0.5
0.6
0.7
0.8
0.9
Fi , Bi and ci , in Eq. (8) are Z Fi ¼ w2i dy ði ¼ 1; 2; 3Þ; c1 ¼ A44 =A55 ; Z F4 ¼ w1 w2 dy; c2 ¼ D22 =D11 ; Z Z F5 ¼ w1 w3;y dy ¼ w3 w1;y dy; c3 ¼ A55 =D11 ; Z Z F6 ¼ w2 w3;y dy ¼ w3 w2;y dy;
1.0
y / Ly
Fig. 5. Deflection of the plate with various aspect ratios.
element results were obtained using commercial finite element software ANSYS. Comparison of the nondimensional deflection of the mid-plane of the square plate at wðx; Ly =2Þ is presented in Fig. 4. As can be seen in the figure, the maximum deviation is about 5%, which was occurred at the center of the plate. Fig. 5 depicts deflection of the non-dimensional deflection of the plate for various aspect ratios from Lx =Ly ¼ 1–3. It seems that there is not major difference for deflection of the plate with aspect ratios greater than 3.
ðj ¼ 1; 2; 3Þ;
c5 ¼ D66 =D11 ; Z 1 qw1 dy; c6 ¼ c5 F8 c3 F2 ; B1 ¼ A55 Z Z Cx Cy q;x w2 dy; B3 ¼ q;y w3 dy; B2 ¼ D11 D11 c7 ¼ c2 F9 c1 c3 F3 : Except for G4 and G5 , all other Gi and Ei in Eq. (9) can also be found using the same expressions for Fi and Bi by interchanging w, y, B and F with n, x, E, G, respectively, and G4 and G5 are as follows: Z Z G4 ¼ n1 n3 dx; G5 ¼ n1 n2;x dx: All integrals should be determined from y ¼ 0 to y ¼ Ly for Fi and Bi and from x ¼ 0 to x ¼ Lx for Gi and Ei . In Eq. (9) J1 and J2 are defined as J1 ¼ G8 c3 G2 ;
J2 ¼ c5 G9 c1 c3 G3 :
5. Conclusion References Bending of moderately thick laminated plates was analyzed using the EKM. It was shown that convergence of the method is fast. The method is equally efficient for both single partial differential equation and system of equations. The method can easily be employed for non-linear loading of rectangular plates as long as integrals for Bi and Ei in Eqs. (8) and (9) can be determined. It is also possible to solve the resulting dual system of ordinary differential equations, (8) and (9), in a closed form manner, as it was done in [18] for the system of governing equations of thick isotropic plates.
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