Complex Eigenfunction Method for Bending Analysis of Rectangular Plates

The present article will give a Fadle eigenfunction analysis of rectangular plates in flexure by means of complex matrix algebra. Rectangular plates have many structural applications, and their flexural analysis is thus of considerable importance. The solution of the flexural problems of classical elasticity generally involves the satisfaction of the homogeneous biharmonic equation and the imposed boundary conditions. Although these boundary value problems have been the subject of many investigators and the literature is replete with numerous solu-tions, many problems of practical interest have not been solved with respect to the actual imposed boundary conditions. Fadle and Papkovitch were the first to present a method for solving rectangular plate problems by the use of complex biharmonic eigenfunction. The utility of a representation in terms of a Fadle eigenfunction series is contingent on the ability to express arbitrary functions in terms of the series. Each term of a series of these functions satisfies the governing differential equation (nabla)4w = 0 and certain homogeneous boundary conditions on two parallel edges identically. In addition, each term of the general eigenfunction series, when written for finite rectangular plates, contains two arbitrary complex constants which can be used to satisfy arbitrary boundary conditions on the remaining two edges. Thus, the use of these eigenfunction permits the simultaneous satisfaction of the boundary conditions on all four sides of the rectangular plate. An approximate expansion formula is developed and applied to the flexural rectangular plate problem. The analysis can be made for complex quantitiesas as these appear, and needs not to separate real parts from imaginary ones, because these can be evaluated numerically with digital computers.Article信州大学工学部紀要 32: 23-42 (1972)departmental bulletin pape