Mathematical Solutions for Bending of Uniformly Loaded Rectangular Plates with Mixed Edge Conditions

Main Article Content

Adisak Muenkling
Yos Sompornjaroensuk

Abstract

The present paper deals with the method of finite Hankel integral transforms for solving two specific cases of uniformly loaded rectangular plate simply supported on two opposite edges, and mixed between partially simply supported and free on the third edge where the remaining fourth edge may be specified by: (a) clamped support or (b) free support. The mathematical solution for the problems analyzed can be written in terms of single Fourier series satisfying the fourth-order partial differential equation governing to the plate behaviors. Therefore, the mixed boundary conditions on the third edge are formulated through dual-series equations, which can be reduced to determining the solution of inhomogeneous Fredholm integral equation of the second kind for the unknown auxiliary function. The most important consideration is that the inverse-square-root moment singularities are taken into account in the analysis at the points of transition from a simple support to a free edge and treated analytically. The solutions of integral equation are evaluated numerically for two different cases of the plate. The obtained results are demonstrated graphically and also given numerically in tabular form for assessing other analytical or numerical methods.

Article Details

Section
Research Articles

References

S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed., McGraw-Hill, Singapore, 1959.

A. W. Leissa, Vibration of Plates, Reprinted ed., Acoustical Society of America, Washington, DC, 1993.

D. O. Brush and B.O. Almroth, Buckling of Bars, Plates, and Shells, McGraw-Hill, New York, 1975.

M. L. Williams, “Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending”, Proceedings of the 1st U.S. National Congress of Applied Mechanics, American Society of Mechanical Engineers, ASME, Illinois Institute of Technology, Chicago, June 1952, pp. 325-329, 1952.

L. M. Keer and C. Sve, “On the bending of cracked plates”, Int. J. Solids Structures, vol. 6, pp. 1545-1559, 1970.

B. Stahl and L. M. Keer, “Vibration and stability of cracked rectangular plates”, Int. J. Solids Struct., vol. 8, pp. 69-91, 1972.

W. Nowacki and Z. Olesiak, “Vibration, buckling and bending of a circular plate clamped along part of its periphery and simply supported on the remaining part”, Bull. Acad. Pol. Sci., C1, IV, vol.4, no. 4, pp. 247-258, 1956.

C. C. Bartlett, “The vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder”, Quarl. J. Mech. Appl. Math., vol. 16, pp. 431-440, 1963.

B. Noble, “The vibration and buckling of a circular plate clamped on part of its boundary and simply supported on the remainder”, Proceedings of the 9th Midwestern Mechanics Conference, Univ. of Wisconsin, Madison, Wisconsin, April 16-18, 1965, vol.3, no. 2, pp. 141-146, 1965.

L. M. Keer and B. Stahl, “Eigenvalue problems of rectangular plates with mixed edge conditions”, J. Appl. Mech. (ASME), vol.39, pp. 513-520, 1972.

Y. Sompornjaroensuk and K. Kiattikomol, “Exact analytical solutions for bending of rectangular plates with a partial internal line support”, J. Eng. Math., vol. 62, pp. 261-276, 2008.

Y. Sompornjaroensuk and K. Kiattikomol, “Inverse square root moment singularities in rectangular plates containing an internal line support”, Int. J. Comp. Methods, vol. 6, no. 1, pp. 1-21, 2009.

Y. Sompornjaroensuk and A. Muengkling, “On two-coupled Fredholm integral equations for rectangular plate resting on angle-leg corner supports”, Proceedings of 15th National Convention on Civil Engineering, NCCE15, Ubonratchatani, Thailand, May 12-14, 2010, STR012, 2010.

K. Kiattikomol, L. M. Keer and J. Dundurs, Application of dual series to rectangular plate, J. Eng. Mech. Div. (ASCE), vol.100, pp. 433-444, 1974.

D. Damang, Y. Sompornjaroensuk and K. Kiattikomol, “Bending of rectangular plate having a partial edge support under a uniformly distributed strip load”, Proceedings of 14th National Convention on Civil Engineering, NCCE14, Nakhon Ratchasima, Thailand, May 13-15, 2009, pp. 2141-2146, 2009.

Y. Sompornjaroensuk, K. Kiattikomol and D. Damang, “Numerical results for strip loaded rectangular plate with a partial edge support”, Proceedings of 15th National Convention on Civil Engineering, NCCE15, Ubonratchatani, Thailand, May 12-14, 2010, STR024, 2010.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series 55, Dover Publications, New York, 1964.

C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, 1978.

I. N. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., Academic Press, New York, 1956.

A. Gilat, MATLAB An Introduction with Applications, 3rd ed., John Wiley & Sons, Inc., New Jersey, 2008.