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Research Papers

Extended Kantorovich Method for Three-Dimensional Elasticity Solution of Laminated Composite Structures in Cylindrical Bending

[+] Author and Article Information
Santosh Kapuria

Poonam Kumari

Department of Applied Mechanics,  Indian Institute of Technology Delhi, New Delhi 110016, Indiakpmech.iitd@gmail.com

J. Appl. Mech 78(6), 061004 (Aug 22, 2011) (8 pages) doi:10.1115/1.4003779 History: Received September 20, 2010; Revised March 07, 2011; Published August 22, 2011; Online August 22, 2011

The extended Kantorovich method originally proposed by Kerr in the year 1968 for two-dimensional (2D) elasticity problems is further extended to the three-dimensional (3D) elasticity problem of a transversely loaded laminated angle-ply flat panel in cylindrical bending. The significant extensions made to the method in this study are (1) the application to the 3D elasticity problem involving an in-plane direction and a thickness direction instead of both in-plane directions in 2D elasticity problems, (2) the treatment of the nonhomogeneous boundary conditions encountered in the thickness direction, and (3) the use of a mixed variational principle to obtain the governing differential equations in both directions in terms of displacements as well as stresses. This approach not only ensures exact satisfaction of all boundary conditions and continuity conditions at the layer interfaces, but also guarantees the same order of accuracy for all displacement and stress components. The method eventually leads to a set of eight algebraic-ordinary differential equations in the in-plane direction and a similar set of equations in the thickness direction for each layer of the laminate. Exact closed form solutions are obtained for each system of equations. It is demonstrated that the iterative procedure converges very fast irrespective of whether or not the initial guess functions satisfy the boundary conditions. Comparisons of the present predictions with the available 3D exact solutions and 3D finite element solutions for laminated cross-ply and angle-ply composite panels under different boundary conditions show a close agreement between them.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Geometry of a composite laminated infinite panel

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Figure 2

Lay-ups of composite panels (a)–(c)

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Figure 6

Through-thickness distributions of u¯, σ¯x, and τ¯zx for simply supported panels (b) and (c)

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Figure 7

Convergence of deflection and stresses for panel (a) with CF and CS boundary conditions S =10

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Figure 8

Through-thickness distributions of stresses for panel (a) with CS and CC boundary conditions

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Figure 9

Effect of S on the through-thickness distributions of displacements and stresses for CC panel (a)

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Figure 3

Convergence of deflection and stresses for simply supported panel (a)

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Figure 4

Convergence of deflection and stresses for simply supported panels (b) and (c) for S=5

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Figure 5

Through-thickness distributions of u¯, w¯, σ¯x, and τ¯zx for simply supported panel (a)

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