Formulation Of Quadrilateral Element Based On The New First-order Shear Deformation Theory

Plates and shells have been widely used as basic structural elements in various engineering fields such as aerospace,marine industry,engineering machinery and nuclear energy.However,the analytical solutions are usually difficult to obtain and have a small range of applicability.Therefore,it is important to find a high-precision and efficient numerical method for engineering problems with more complex geometries and boundary conditions,as well as higher-order governing equations.The finite element method is one of the most effective numerical methods for solving structural static and dynamics problems;however,the traditional low-order finite element method suffers from slow convergence,excessive dependence on the mesh quality and self-locking.In order to ensure the effectiveness and reliability of the finite element method for solving practical problems,the development of elements with high computational accuracy,and superior convergence is still an important research topic.In the present work,two elements for both thick and thin plates with partial C~1 continuity have been constructed based on a new first-order shear deformation theory,under the framework of the traditional low-order finite element method and the Hierarchical quadrature element method,respectively.And the performances of these two elements have been verified by free vibration and buckling analysis.The main work can be summarized as follows:1.In the new first-order shear deformation theory,the total deflection of a plate can be decomposed into bending deflection and shear deflection,and the two rotations can be expressed as the differential of bending deflection.Under the assumption of this theory,a four-node rectangular element with 16 degrees of freedom is constructed for both thick and thin plates.The element is formulated by simply adding one degree of freedom for each node on the basis of the classical thin plate element.And this element can not only overcome shear locking but also degenerate to the thin plate element when the plate thickness is very small.To evaluate the performance of the element,the good convergence of the element was first verified by free vibration analysis.Then the natural frequencies of rectangular plates at various thickness ratios and aspect ratios were further calculated,and the high accuracy of the element was verified by comparison with data in literatures.2.By combining the new first-order shear deformation theory with the Hierarchical quadrature element method,a fully conforming differential quadrature hierarchical element is constructed.This method allows the number of nodes on the edges of the element varies freely,which makes this element suitable for adaptive analysis.The shear deflection of the element only needs to satisfy the C~0 continuity condition,and the Serendipity functions based on the Gauss-Lobato nodes are used;while the bending deflection needs to satisfy the C~1 continuity,and its shape function is constructed by the 5th-order Hermite interpolation function;the shape functions corresponding to the normal rotation are constructed by the 3rd-order Hermite interpolation;for the interior of the element,the hierarchical shape functions in the form of orthogonal polynomial tensor product is constructed by the hybrid function interpolation method.Then the transformation of degrees of freedom,the node configuration,the conformity and the method of imposing boundary conditions are discussed.Finally,the fast convergence characteristics and superior computational accuracy of the element are verified by the static analysis of a circular plate and the free vibration analysis of a square plate.