Layer-wise dynamic stiffness solution for free vibration analysis of laminated composite plates

The dynamic stiffness method has been developed by using a sophisticated layer-wise theory which complies with the Cz0 requirements and delivers high accuracy for the analysis of laminated composite plates. The method is versatile as it derives the dynamic stiffness matrix for plates with any number of layers in a novel way without the need to re-derive and re-solve the equations of motion when the number of layers has changed. This novel procedure to manipulate and solve the equations of motion has been referred to as the L matrix method in this paper. The Carrera unified formulation (CUF) is employed to derive the equations of motion through the use of a first-order layer-wise assumption for a plate with a single layer first. The method is then generalised and extended to multiple layers. Essentially by writing the equations of motion of one single layer in the L matrix form, the system of equations of motion of a laminated plate with any number of layers is generated in an efficient and automatic way. A significant feature of the subsequent work is to devise a method to solve the system of differential equations automatically in closed analytical form and then obtain the ensuing dynamic stiffness matrix of the laminated plate. The developed dynamic stiffness element has been validated wherever possible by analytical solutions (based on Navier's solution for plates simply supported at all edges) for the same displacement formulation. Furthermore, the dynamic stiffness theory is assessed by 3D analytical solutions (scantly available in the literature) and also by the finite element method using NASTRAN. The results have been obtained in an exact sense for the first time and hence they can be used as benchmark solutions for assessing approximate methods. This new development of the dynamic stiffness method will allow free vibration and response analysis of geometrically complex structures with such a level of computational efficiency and accuracy that could not be possibly achieved using other methods.