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TECHNICAL PAPERS

# A Novel Finite-Element– Numerical-Integration Model for Composite Laminates Supported on Opposite Edges

[+] Author and Article Information
Tarun Kant

Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, Indiatkant@civil.iitb.ac.in

Sandeep S. Pendhari

Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, Indiaspendhari@iitb.ac.in

Yogesh M. Desai

Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, Indiadesai@civil.iitb.ac.in

J. Appl. Mech 74(6), 1114-1124 (Jan 03, 2007) (11 pages) doi:10.1115/1.2722770 History: Received May 03, 2006; Revised January 03, 2007

## Abstract

An attempt is made here to devise a new methodology for an integrated stress analysis of laminated composite plates wherein both in-plane and transverse stresses are evaluated simultaneously. The method is based on the governing three-dimensional (3D) partial differential equations (PDEs) of elasticity. A systematic procedure is developed for a case when one of the two in-plane dimensions of the laminate is considered infinitely long ($y$ direction) with no changes in loading and boundary conditions in that direction. The laminate could then be considered in a two-dimensional (2D) state of plane strain in $x-z$ plane. It is here that the governing 2D PDEs are transformed into a coupled system of first-order ordinary differential equations (ODEs) in transverse $z$ direction by introducing partial discretization in the finite inplane direction $x$. The mathematical model thus reduces to solution of a boundary value problem (BVP) in the transverse $z$ direction in ODEs. This BVP is then transformed into a set of initial value problems (IVPs) so as to use the available efficient and effective numerical integrators for them. Through thickness displacement and stress fields at the finite element discrete nodes are observed to be in excellent agreement with the elasticity solution. A few new results for cross-ply laminates under clamped support conditions are also presented for future reference and also to show the generality of the formulation.

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Copyright © 2007 by American Society of Mechanical Engineers
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## Figures

Figure 7

Variation of normalized (a) inplane normal stress σx¯, (b) inplane displacement u¯, (c)transverse shear stress τzx¯, and (d) transverse normal stress σz¯ through thickness of 0deg∕90deg∕0deg symmetric laminate under cylindrical bending

Figure 10

Variation of normalized (a) in-plane normal stress σx¯ and (b) transverse shear stress τzx¯ through thickness of 0deg∕90deg∕0deg symmetric laminate for clamped supported boundary conditions

Figure 5

Variation of normalized transverse displacement (w¯) with respect to a∕h ratios of 0deg∕90deg unsymmetric laminates under cylindrical bending

Figure 6

Variation of normalized (a) in-plane normal stress σx¯, (b) in-plane displacement u¯, (c) transverse shear stress τzx¯, and (d) transverse normal stress σz¯ through thickness of 0deg∕90deg unsymmetric laminate under cylindrical bending

Figure 8

Variation of normalized transverse displacement w¯ through thickness of (a) 0deg∕90deg unsymmetric and (b) 0deg∕90deg∕0deg symmetric laminates under cylindrical bending

Figure 9

Variation of normalized (a) in-plane normal stress σx¯ and (b) transverse shear stress τzx¯ through thickness of 0deg∕90deg unsymmetric laminate for clamped supported boundary conditions

Figure 1

Figure 2

Linear finite element with dependent variables

Figure 3

Linear elements (concept of partial discritization)

Figure 4

Convergence of (a) maximum transverse shear stress (τzx¯) and (b) midplane transverse displacement (w¯) with number of elements for a 0deg∕90deg∕0deg laminate under cylindrical bending

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