# Stress Intensity Factors of a Uniformly Line Loaded Rectangular Plate with Mixed Edge Conditions

## Main Article Content

## Abstract

The main purpose of the paper is to investigate the stress intensity factors at the end points of discontinuous support in rectangular thin elastic plate. The plate is simply supported on two opposite edges, clamped on the third edge and partially simply supported on the fourth edge, and also loaded by a uniformly distributed line load. Since the plate has the partial simple support along one edge leading to a pair of dual-series equations that resulted from the mixed boundary conditions, this causes the existence of moment singularities in the order of an inverse-square-root type. In order to analytically determine the stress intensity factors, the finite Hankel integral transform techniques are applied for solving the dual-series equations which can reduce to finding the solution of Fredholm integral equation. Numerical results concerning the solution of integral equation, stress intensity factors, and additionally, the change in strain energy due to the presence of a partial simple support are given for the case of a square plate in the form of graphs and tables for easy reference.

## Article Details

Copyright @2021 Engineering Transactions

Faculty of Engineering and Technology

Mahanakorn University of Technology

## References

D. Broek, Elementary Engineering Fracture Mechanics, 4th ed., Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1986.

M. L. Williams, “The bending stress distribution at the base of a stationary crack”, J. Appl. Mech. (ASME), vol.28, pp. 78-82, 1961.

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.

H. Boduroglu and F. Erdogan, “Internal and edge cracks in a plate of finite width under bending”, J. Appl. Mech. (ASME), vol.50, pp. 621-629, 1983.

L.P. Pook, Crack Paths, WIT Press, Southampton, UK, 2002.

K. Hellan, Introduction to Fracture Mechanics, McGraw-Hill, New York, 1984.

S. P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill, Singapore, 1970.

P. Kongtong and Y. Sompornjaroensuk, “On the Williams’s solution in plates having stress singularities”, Proceedings of 15th National Convention on Civil Engineering, NCCE15, Ubonratchatani, Thailand, May 12-14, 2010, STR025, 2010.

Y. Sompornjaroensuk, “In-plane stress singularities at the vertex of isotropic wedge-shaped flat plates in extension”, Proceedings of 15th National Convention on Civil Engineering, NCCE15, Ubonratchatani, Thailand, May 12-14, 2010, STR026, 2010.

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

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 Inst. Tech., Chicago, June 1952, pp. 325-329, 1952.

Y. Sompornjaroensuk and K. Kiattikomol, “Dual-series equations formulation for static deformation of plates with a partial internal line support”, Theor. Appl. Mech. (TAM), vol. 34, no. 3, pp. 221-248, 2007.

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 (IJCM), vol. 6, no. 1, pp. 1-21, 2009.

A. Muenkling and Y. Sompornjaroensuk, “Mathematical solutions for bending of uniformly loaded rectangular plates with mixed edge conditions”, Engineering Transactions, 2010 (accepted for publication).

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.

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

Y. Sompornjaroensuk, D. Dy and K. Kiattikomol, “Moment stress intensity factor of uniformly loaded rectangular plates having a partial internal line support”, Proceedings of 14th National Convention on Civil Engineering, NCCE14, Nakhon Ratchasima, Thailand, May 13-15, 2009, pp. 2171-2176, 2009.

G. C. Sih, P.C. Paris and F. Erdogan, “Crack-tip, stress-intensity factors for plane extension and plate bending problems”, J. Appl. Mech. (ASME), vol.29, pp. 306-312, 1962.

E. Kreyszig, Advanced Engineering Mathematics, 9th ed., John Wiley & Sons, Inc., Singapore, 2006.

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.

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