Vibrations of Sectorial Plates Having Corner Stress Singularities

A procedure is presented for determining the free vibration frequencies and mode shapes of sectorial plates having re-entrant corners (i.e., vertex angles exceeding 180 degrees). No correct results for such problems have been found in the vast literature of plate vibrations. The procedure is applicable to sectorial plates having arbitrary (but continuous) boundary conditions (e.g., clamped, simply supported, or free) along the two radial edges and the circular edge. It is based upon the Ritz method, but utilizes two sets of admissible functions simultaneously. One set consists of algebraic-trigonometric polynomials. The other is the set of corner functions derived by Williams (1952) to deal with the bending stress singularities which may arise at the corner when the vertex angle becomes large. The method is demonstrated for sectorial plates having all edges simply supported, which yields the strongest singularity in a re-entrant corner. Frequencies are compared with those obtained from an analytical solution involving Bessel functions. It is shown that the latter solution is invalid for re-entrant corners. Analytical solutions are also obtained for annular sectorial plates having very small ratios of inner to outer boundary radii. These solutions are found to be consistent with those using polynomials and corner functions. Accurate fundamental frequency data is presented for simply supported sectorial plates having three values of Poisson’s ratio (0, 0.3, 0.5) and the full range of vertex angles (0 < α ≤ 360 deg).