BENDING OF A SQUARE PLATE WITH TWO ADJACENT EDGES FREE AND THE OTHERS CLAMPED OR SIMPLY SUPPORTED

Abstract

T LITERATURE abounds with applications of classical thin elastic plate theory to rectangular plates having a multitude of combinations of boundary conditions and transverse loading. By far the greatest number of these papers deal with plates having only simply-supported and/or clamped boundary conditions, which requires restrictions upon the deflection and its first or second derivatives along the edges. Considerably fewer solutions have been obtained when a free edge is involved because of the relatively difficult combination of third derivatives of deflection which is encountered in the boundary conditions. When two adjacent free edges are involved, so that a free corner is created, the problem becomes exceedingly difficult. No closed form of solution is known to exist for any of this latter class of problems. Some solutions have been obtained for cantilevered rectangular plates—i.e., three edges free and the other clamped. Finite difference solutions for various aspect ratios and loadings were presented by Holl, Barton, MacNeal, Nash, and Livesly and Birchall. Nash also solved the problem of the uniformly loaded contilevered plate having a span-to-chord ratio of 1/2 by using a form of collocation known as point matching. This method depends upon choosing deflection functions which satisfy the partial differential equation of the continuum exactly, while matching the boundary conditions at only a finite number of discrete points. Nash solved the problem twice, once using an algebraic polynomial and, again, using a hyperbolic-trigonometric series. A recent paper by Leissa and Niedenfuhr presents the solution for the uniformly loaded cantilevered square plate using two other approaches: point matching, using an algebraic-trigonometric polynomial, and a Rayleigh-Ritz minimal-energy formulation. Doubtlessly, much of the foregoing work with cantilevered plates has been the result of the need for accurate information for the design of aircraft wings. However, approximate deflections and stresses may be obtained for these problems by using elementary beam theory. In marked contrast, the rectangular plate having two adjacent edges free and the others clamped or simply supported is in no way representable by beam theory. These problems are no more difficult than the cantilevered plate, but very little has been done with them up to the present. The problem of the uniformly loaded square plate with two adjacent edges free and the others clamped was solved by Huang and Conway. This involved a skillful superposition of five problems and the partial solution of an infinite set of simultaneous equations. Yeh generalized this same problem by including the reaction due to an elastic foundation. The Rayleigh-Ritz method was employed. In the present work the writers exhibit solutions for the four problems of a square plate with two adjacent edges free and the others clamped or simply supported subjected to either a uniform transverse loading or a concentrated force at the free corner. The method of point matching is used to solve the problems and the results are compared with other known solutions for two of the cases.