Weak-Form Quadrature Element Analysis Of Geometrically Nonlinear Framed Structures

Engineering structures exhibit large displacement and rotation inevitably,which is closely related to geometrically nonlinear analysis.Studies of geometrically nonlinearity analysis have a long history and have been a classical topic in the field of computation and analysis.With the development of nonlinear theories and computational methods,high precision and efficiency of computing are attached importance by researchers.In order to gain the precise equilibrium equations,the present thesis combined with a weak form quadrature element method for geometrically nonlinear analysis based on incremental virtual work increment equation.According to elements types,the structures composed of truss elements,plane beam elements and space beam elements were analyzed respectively.Geometrically nonlinear analysis is based on incremental iteration process usually which can be described as follows.Firstly,choosing an incremental step as analysis procedure,the incremental virtual work equation was derived during the incremental step based on principle of virtual displacement.Secondly,combing the weak form quadrature element method with the incremental virtual work equation,equilibrium equations were derived by numerical integration and differential quadrature for the incremental virtual work equation.Finally,a generalized displacement control method was chosen as a nonlinear solution scheme for solving the equilibrium equations derived by the second step.Internal virtual work of the incremental virtual work equation can be divided into stain energy and potential energy on account of virtual work physical significance.Therefore,elastic stiffness matrix and geometric stiffness matrix can been gained by two parts of the internal virtual work,which are important components of stiffness matrix in equilibrium equations.For the purpose of insuring accuracy of the stiffness matrix,rigid body test was taken for testing the stiffness matrix derived.Some representative examples were analyzed by the equilibrium equations derived to verify the accuracy and efficiency of the suggested method.Furthermore,geometrically nonlinearity of cured beams structures were analyzed by vast straight beam elements.This paper thesis studied the relationship between high-order interpolations using variational numbers of joints and computing convergence,and discussed the reasons causing by the phenomenon and law.Overserving results analyzed by the suggested method,conclusions can be drawn that equilibrium equations derived can solve geometrically nonlinearity accurately,and the generalized displacement control method chosen can track post-buckling equilibrium path of structures automatically.Meanwhile,the weak form quadrature element method having wide application in the field of geometric nonlinearity can be verified.