Quadrature Element Analysis Of Geometrically Nonlinear Curved Beam Structures

In recent years,human beings have made great breakthroughs in the height and span of civil engineering structures,which has led increasing geometric nonlinearity to the structure.Solving geometric the nonlinearity accurately and efficiently is increasingly demanded in engineering practice.At the same time,curved beam structures have been used more and more in the design because of their aesthetic appearance and good mechanical performance.Consequently,the geometrically non-linear problem of the curved beam structure has attracted more and more attention.The purpose of this thesis is to establish the stiffness matrix of the curved beam element from its incremental virtual work equation by using quadrature element method(QEM)that has features of high accuracy and high efficiency,the geometric nonlinearity of curved beam structure is studied using the stiffness matrix.In this thesis,the incremental iteration method is used to solve the geometrically nonlinear problem.In each incremental step,the incremental virtual work equation is numerically integrated and differentiated by the QEM that uses the same set of discrete nodes for the integration and differentiation.Then elastic stiffness matrix and geometric stiffness matrix of the curved beam element are derived,and their correctness is preliminarily validated by the rigid body test.Last,the generalized displacement control method is used to solve the equilibrium equation iteratively,so as to obtain the entire equilibrium path of the loaded structure.Aiming at the derived stiffness matrices,several typical examples exhibiting large deflection and rotation are selected to verify the solution.In addition,the influence of the number of interpolation nodes on the computational results is also studied.It is found that the results converge rapidly with the increase of the number of interpolation nodes,which proves the high-order advantage of the QEM in solving the curved beam structures.Finally,by making the radius of the curved beam element infinite and the central angle infinitesimal,the curved beam element is reduced to a straight beam element.Their stiffness matrices are thus unified.And the correctness of the beam element is verified by a few typical geometrically nonlinear examples of straight beam structures.In summary,the stiffness matrices of the curved beam element deduced in this thesis are highly accurate and the derivation is simple.The process combines the characteristics of high-order approximation of the QEM with the complex deflection of the curved beam element,and makes up for the shortcomings of the conventional finite element method in solving the curved beam structures.The results of the typical geometrically non-linear examples prove that the QEM has wide applicability in treating this kind of problems.