Approximate Fundamental Frequency Solutions Under Initial Load Effect for 6 Typical Plates

Based on two sets of dynamic equilibrium differential equations for plates under initial load effect, which were respectively expressed as general and polar coordinate forms to fit different boundary conditions. The approximate solutions of fundamental frequencies under initial load effect for the simply supported rectangular plate, the clamped rectangular plate, the simply supported equilateral triangular plate, the clamped elliptic plate, the clamped circular plate and the simply supported circular plate, were derived with the Galerkin method. These approximate solutions were verified with the finite element method under initial load effect, which clearly illustrated the initial load effect and corresponding factors that influence the plates’ fundamental frequencies. Initial load effect on fundamental frequencies of the above 6 typical plates was analyzed with these solutions. Due to initial load effect, bending stiffnesses of the plates increased, and their fundamental frequencies rose. The key physical factors governing the initial load effect on the plates are the initial load magnitude，the ratio of span to thickness and the boundary conditions, etc. The bigger the initial loads and the smaller the bending stiffnesses of the plates are, the higher the initial load effect on the fundamental frequencies is. This initial load effect is obvious and should not be neglected in the design and analysis of plates.