In-plane vibration of plates with curvilinear plan-forms by a trigonometrically enriched curved triangular p-element

A trigonometrically enriched curved triangular p-element is developed and applied to free in-plane vibration analysis of plates with curvilinear plan-forms. New sets of trigonometric shape functions are presented. The main feature of these functions is their ability to overcome ill-conditioning which may occur using orthogonal polynomials of high order. In this method, meshes are constructed from a few large elements and curved segments are represented accurately by using the blending function method. The accuracy of the solution is improved simply by increasing the trigonometric order while keeping the mesh constant. Frequency values are first found for a closed sectorial plate and comparisons are made with exact solutions and with solutions from conventional linear and isoparametric quadratic triangular finite elements. Frequency values are also found for an open sectorial plate and comparisons are made with exact solutions. Furthermore, highly accurate frequency values are given for a square plate with a central circular opening. It is found that the method converges very fast downwards to the exact solution as the trigonometric order is increased and produces a much higher accuracy than linear and isoparametric quadratic triangular finite element solutions with far fewer d.o.f.s.