Equivalent property between the one-half order and first-order shear deformation theories under the simply supported boundary conditions

It was verified that if the boundary conditions for all edges of a plate or both ends of a beam are assumed to be simply supported the theoretical framework of the one-half order shear deformation plate or beam theory with total deflection w being assumed as the sum of the bending and shearing deflections wb and ws results in being equivalent to that of the Mindlin plate theory or the Timoshenko beam theory. Based on its equivalent property, exact frequency relationships between the classical Kirchhoff plate and the above two equivalent shear deformable plate theories were deduced for general polygonal plates with all edges simply supported, and in addition an approximate frequency relationship which has a very simple form and almost enough accuracy for practical use was obtained based on the series-type synthetic-frequency method.