Buckling of Plates with Intermediate Rigid Supports

Abstract

T WRITING of the present note was suggested to the author by a paper by Saibel which deals with the same subject. Mr. Saibel's aim was to express the buckling modes of the plate with intermediate supports as a combination of the known buckling modes of the plate without the intermediate supports. The coefficients of the series representing the unknown modes are to be determined by the energy method, in which the intermediate rigid supports are introduced by means of the method of the Lagrangian multiplier. Unfortunately, Mr. Saibel's use of the multiplier method is incorrect so far as two-dimensional problems are concerned; as a consequence, his paper contains conclusions that cannot be accepted. As such may be mentioned the last (unnumbered) equation on page 401 of the paper and the last two sets of equations on page 402 of the paper. In what follows it will be shown how results can be obtained for the problems under consideration which are natural extensions of the Weinstein-Trefftz procedure' 3 for the buckling and vibrations of the rectangular plate with clamped edges. These results are valid for rectilinear, as well as for curvilinear, rigid intermediate supports. The supports may be concentrated in a number of points along lines or distributed over portions of the plate area. Only the formal aspects of the problem are discussed here, and no actual applications of the formulas obtained are made. The difference between Saibel's and the present approach may perhaps be characterized by saying that the former assumes the constraints accurately and intends to solve the problem approximately, while in the present note constraints are assumed which approximate the real constraints and the resultant problem is solved rigorously. I t is sufficiently general for the present purpose to consider a plate that is simply supported all along its boundary and which is acted upon by a uniform thrust N in the plane of the plate. Taking coordinates x, y in the plane of the plate and letting the direction of the thrust N coincide with the ^-direction, it is known that buckling modes 4>ij and corresponding buckling loads Ntj are obtained as the characteristic functions and characteristic values of the following variational problem: