Static And Dynamic Analysis Of Sandwich Beams By The Weak Form Quadrature Element Method

Sandwich structure is prevalent in engineering as its high stiffness and strength to weight ratio, where its mechanical performance varies in terms of core compliance. It is necessary to establish an efficient and precise numerical method for analysis of sandwich structures since existing numerical methods for analysis of sandwich structures with soft cores are either lack of precision or time consuming. The newly developed weak form quadrature element method(WQEM or QEM) is an efficient and precise numerical method thus is to be employed in present investigation.Based on the Extended High-order Sandwich Panel Theory and differential quadrature rule, a one dimensional N-node weak form quadrature sandwich beam element is established. Gauss–Lobatto–Legendre(GLL) points are used as the element nodes as well as integration points. The weighting coefficients of derivatives at integration points are calculated by using the explicit formulas in the differential quadrature method. Based on the potential and kinetic energy of the sandwich beam, element stiffness and mass matrix are derived. The convergence performance and numerical accuracy are verified by comparing the results with analytical solutions of sandwich beam with soft core. Compared with the 2-node finite element method which is derived based on the same Extended High-order Sandwich Panel Theory, it is found that the weak form quadrature element method needs fewer degrees of freedom and has higher solution accuracy. The WQEM is then applied to static analysis of sandwich beams with different LTR(length to thickness ratio) and FCSR(face to core stiffness ratio) as well as different boundary conditions. The results are verified by comparing them to two dimensional finite element data obtained with very fine meshes. It is demonstrated that the proposed one-dimensional element can yield accurate results. After that, the WQEM is further extended to static analysis of sandwich beams with functionally graded material(FGM) or non-homogeneous cores. Based on the exponential law, constitutive equation is modified and the elemental stiffness matrix is re-derived for analysis of FGM sandwich structure. A convergence study is performed. Its application on different functional graded material and boundary conditions is demonstrated. Finally, the WQEM is extended to dynamic analysis of sandwich beams with different core materials. The convergence and solution accuracy are proved by comparing numerical data with analytical results of 2D elasticity. The mass matrix is optimized for efficiency in dynamic analysis. Comparisons are made with results obtained by using commercial FEM software. It is shown that the WQEM is of high accuracy and efficiency in analyzing the static and dynamic behavior of sandwich structures with either orthotropic homogeneous or non-homogeneous cores.