Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-04T20:25:43.059Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of composite plates with variable stiffness using Galerkin Method

Published online by Cambridge University Press:  03 February 2016

E. Senocak  and
H. Tanriover
E. Senocak
Affiliation:
Department of Mechanical Engineering, Istanbul Technical University, Istanbul, Turkey
H. Tanriover
Affiliation:
Department of Mechanical Engineering, Istanbul Technical University, Istanbul, Turkey
Get access Rights & Permissions [Opens in a new window]

Abstract

A solution methodology is developed to solve plane stress problem of composite plates with variable stiffness by using Galerkin technique and polynomials as trial functions. In the solution process, analytical computation has been done wherever it is possible, and analytical-numerical type approach has been made for all problems. The methodology is applied to two known case problems, composite plate with variable fibre content and laminated plate with spatially varying fibre orientations. The formulation of these problems results into coupled partial differential equations (with variable coefficients). The solutions of these equations are obtained using the polynomials as trial functions in the Galerkin method. The results are compared to that of Ritz and collocation technique published elsewhere. The method is found to determine closely both the displacements and the stresses with a few number of terms and in good agreement with other approximating methods. Computations on some examples show that, the method with the help of a symbolic math package is simple and efficient for solving these types of problems in engineering applications.

Type
Research Article
Information
The Aeronautical Journal , Volume 111 , Issue 1118 , April 2007 , pp. 247 - 255
Copyright
Copyright © Royal Aeronautical Society 2007 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Setoodeh, S., Abdalla, M.M. and Gürdal, Z., Design of variable stiffness laminates using lamination parameters, Composites Part: B Engineering, 2006, 37, pp 301–309.Google Scholar
2. Reddy, J.N., Applied Functional Analysis and Variational Methods in Engineering, 1987, Second edition, McGraw-Hill, Singapore.Google Scholar
3. Sokolnikoff, I.S., Mathematical Theory of Elasticity, 1983, Reprint edition, Krieger, Florida.Google Scholar
4. Zienkiewicz, O.C. and Morgan, K., Finite Elements and Approximation, 1983, John Wiley & Sons, Singapore.Google Scholar
5. Becker, E.B., Carey, G.F. and Oden, J.T., Finite Elements, An Introduction, Volume I, 1981, Prentice-Hall, New Jersey.Google Scholar
6. Bhat, R.B., Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method, J Sound Vibr, 1985, 102, (4), pp 493–499.Google Scholar
7. Dickinson, S.M. and Di Blasio, A., On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates, J Sound Vibr, 1986, 108, (1), pp 51–62.Google Scholar
8. De Sampaio, P.A.B., A Petrov-Galerkin/modified operator formulation for convection-diffusion problems, Int J Numer Methods Eng, 1990, 30, pp 331–347.Google Scholar
9. Westerink, J.J. and Shea, D., Consistent higher degree Petrov-Galerkin methods for the solution of the transient convection-diffusion equation, Int J Numer Methods Eng, 1989, 28, pp 1077–1101.Google Scholar
10. Dennis, S.T., A Galerkin solution to geometrically nonlinear laminated shallow shell equations, Computers and Structures, 1997, 63, pp 859–874.Google Scholar
11. Saadatpour, M.M. and Azhari, M., The Galerkin method for static analysis of simply supported plates of general shape, Computers and Structures, 1998, 69, pp 1–9.Google Scholar
12. Tanriover, H. and Senocak, E., Large deflection analysis of unsymmetrically laminated composite plates: analytical-numerical type approach, Int J of Non-linear Mechanics, 2004, 39, pp 1385–1392.Google Scholar
13. Storch, J. and Strang, G., Paradox lost: natural boundary conditions in the Ritz-Galerkin method, Int J Numer Methods Eng, 1988, 26, pp 2255–2266.Google Scholar
14. Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates, 1987, Technomic, Dayton, OH.Google Scholar
15. Martin, A.F. and Leissa, A.W., Application of the Ritz method to plane elasticity problems for composite plates with variable fibre spacing, Int J Numer Methods Eng, 1989, 28, pp 1813–1825.Google Scholar
16. Wolfram, S., MathematicaTM: A System for Doing Mathematics by Computer, 1991, Addison-Wesley, Redwood City, CA.Google Scholar
17. Gürdal., Z. and Olmedo, R., In-plane response of laminates with spatially varying fibre orientations: variable stiffness concept, AIAA J, 1993, 31, (4), pp 751–758.Google Scholar
18. Finlayson, B.A., The Method of Weighted Residuals and Variational Principles, 1972, Academic Press, New York.Google Scholar