New Elements For Geometrical Nonlinear Analysis Of Flexible Elastic Membrane Structure And Shell Structure

Membrane and thin shells are very important structural types,and they have been widely used in the wings of space shuttles,wind turbines,and large-span building structures.A basic characteristic of the thin shell structure is that the dimensions in the thickness direction are much smaller than those in the other two directions,and have good in-plane film tension effects and out-of-plane bending stiffness.The out-of-plane stiffness of the membrane structure is very small,even without the out-of-plane stiffness.Therefore,under the action of load,the thin shell and membrane structure tend to produce very large deformation,which has obvious geometric nonlinear characteristics.Based on a large-deflection theory of continuum mechanics,such as non-linear analysis,using the FEM on shell and membrane structures usually begins by deriving linear virtual strain energy and non-linear virtual strain energy.Then,shape functions of displacements are introduced and the corresponding elastic stiffness matrix and geometric stiffness matrix can be obtained by using numerical integration.Finally,uses of the incremental-iteration procedure can determine the path of load-deflection equilibrium of structures.However,the derivation of nonlinear virtual strain energy and load potential energy in the conventional finite element method is very tedious,and the resulting geometric stiffness matrix is not accurate due to the approximation of the shape function.In view of the limitations of the element displacement and deformation description in the geometric nonlinear analysis method mentioned above,the rigid body rule is adopted to analyze the large displacement and large rotation of the structure in this paper.According to the rigid body rule,when an initial stressed element undergoes rigid body displacement,the directions of the nodal forces at the element will change with the rigid rotation of the element while the magnitude of the nodal forces will remain unchanged in order to keep the element in balance.In common geometric nonlinear problems,the rigid body displacement accounts for the main part of the element displacement.The rigid body rule can be used to clearly deal with the effect of the element force undergoing the rigid body displacement,which greatly simplifies the calculation process of the element node force.The physical concept of nonlinear analysis process based on rigid body rule is clear and the calculation process is simple,which overcome the disadvantages of traditional methods.Based on the basic idea of rigid body rule,this paper constructs triangular shell elements and triangular membrane elements.The geometric stiffness matrix of the triangular space shell element is derived from the geometric stiffness matrix of the space beam element.the elastic stiffness matrix adopted for the triangular space shell element is constructed as the composition of Cook’s plane hybrid element for membrane actions and the hybrid stress model(HSM)of Batoz et al.for bending actions.Space shell elements can easily degenerate into triangular planar shell elements.This membrane element is composed of three bars to form a pin-joint triangle with a triangle film stretched inside.The material of the bars is assumed the same as that of the film.The geometric stiffness matrix of the proposed membrane element is derived by the geometric stiffness matrix of spatial bar element which is rigid body rule qualified and the elastic stiffness matrix is composed of plane stress elements.By rooting the rigid body rule into the UL incremental-iteration method,the effect of rigid body rotation is fully considered in each stage of analysis and the residual effects of natural deformation is treated by the linearization.The derivation of the geometric stiffness matrix of the proposed membrane element is quite clear without introducing any assumption on the large deformation of the membrane element and can be easily degenerated into a plane membrane element.The accuracy and efficiency of the present shell element and membrane element as well as the rigid-body-rule-rooted nonlinear analysis method is proved by the analyzing of several typical space membrane structures and by the comparison with the results of other methods.