Weak Form Quadrature Element Method And Its Applications In Analysis Of FGM Structures

Weak form quadrature element method(abbreviated as QEM)is regarded as a numerical method based on the principle of minimum potential energy and can be used in the weak form description of a problem.With the help of the differential quadrature(DQ)law to compute the strain at integration points,the equations of quadrature element with arbitary number of nodes can be explicitly given.QEM has the capability of obtaining highly accurate solutions with a minimal computational effort.It has successfully solved numerous problems in structural mechanics,including problems of static,buckling,free vibration and dynamic response of structural members as well as structures.In this dissertation,some existing shortages of the QEM and its applications in functionally graded material structures are studied.Innovative research results are obtained.Firstly,the Lagrange type and Hermite type quadrature element with variable number of nodes are explicitly obtained for cases when the nodal points are different from the integration points.The number of integral points can be changed according to the accuracy requirement.The limitation of that the nodal points must be the same as the integration points is completely removed.This lays the foundation to establish quadrature elements with various types of nodes.Secondly,various types of FGM(functional graded material)quadrature elements are established,including the bar element with the transverse effect,plane Euler beam element,plane Timoshenko beam element,3D(three-dimensional)parallelepiped element,rectangular thin plate bending element and skewed plate bending element.Several FORTRAN programs are developed.The dynamics responses of FGM bars,beams,plates and 3D parallelepipeds are analyse by using the proposed quadrature elements,including the dynamic responses of FGM beams under a moving point load,free vibtations and wave propagation in a bar.Accurate results are obtained.The new data may serve as references for researchers to develop new numerical methods.Numerical examples show that accurate numerical results can be obtained for structures with isotropic materials and functionally graded materials by the QEM with relatively smaller number of nodes or elements as compared to conventional finite element method.This shows the strong computing capability of the QEM.Present research improves the weak form quadrature element method and extends its application range to functionally graded material structures.