Dynamic Stiffness Formulation for Plates Using First Order Shear Deformation Theory

Abstract

The dynamic stiffness method for plates is developed to carry out an exact free vibration analysis by using both classical theory and first order shear deformation theory. Hamiltonian mechanics is used to provide a systematic general procedure for the development of the method. Explicit expressions for the elements of the dynamic stiffness matrices have been derived with the help of symbolic computation. Details of the assembly procedure and application of boundary conditions using the dynamic stiffness elements have been explained when investigating the free vibration characteristics of complex structures modelled by plate assemblies. The usually adopted Wittrick-Williams algorithm has been modified to avoid the requirement of computing the clamped-clamped natural frequencies of individual plates and yet converging upon any number of natural frequencies of the overall structure within any desired accuracy. The results using both classical and first order shear deformation theories are rigorously validated by published results for both uniform and stepped plates with various boundary conditions. Representative mode shapes are presented and the numerical accuracy and computational efficiency of the method are demonstrated. Significant plate parameters are varied and their subsequent effects on the accuracy of classical plate theory when compared to the first order shear deformation theory are investigated. For both uniform and stepped plates, the circumstances when the classical theory leads to inaccurate results are identified and discussed. The investigation offers the prospects for dynamic stiffness development of anisotropic plates using Hamiltonian mechanics and symbolic algebra.