Geometrically Nonlinear Analysis Of Geometrically Exact Euler-Bernoulli Beams Based On Isogeometric Collocation Method

Isogeometric analysis(IGA)is a recent method of computational analysis with the main objective of integrating Computer Aided Design(CAD)and Finite Element Analysis(FEA)into one model.IGA uses B-Splines or Non-Uniform Rational B-Splines(NURBS)basis functions,which are commonly used in CAD,in order to describe both the geometry and the unknown variables for analysis problems.The isogeometric collocation(IGA-C)method takes advantage of the higher-continuity properties of the basic functions in IGA and the concept of isoparametric to discretize the strong form of the governing partial differential equation,and then an efficient spatial discretization method for nonlinear partial differential equations of continuum mechanics is obtained.In this thesis,the IGA-C approach is extended to solve the geometrically nonlinear problems of the geometrically exact Euler-Bernoulli beam structures with arbitrary initial curvatures and torsion for the first time.Based on Euler-Bernoulli assumptions and the geometrically exact analysis,a total Lagrangian formulation for geometrically exact Euler-Bernoulli beam structures with large displacements and large rotations in three dimensional space is derived from the extended Hamilton principle in this work.And the objective,ge-ometric,work-conjugate and co-rotating Jaumann strains and stresses are adopted.Within the isogeometric framework,a set of nonlinear algebraic equations suitable for the solving geometrically nonlinear statics and dynamics problems is derived.Then the corresponding program of IGA-C is also written.To make the code applicable to beam structures consist-ing of multiple patches,a rigid coupling model is presented by strong enforcement of the continuity of displacements and rotations of the finite rotation matrix as well as the balance of forces and moments at the connection points of the beam reference line.In addition,the Lagrange multiplier method is used to enhance the constraint relations of rigid coupling,so as to improve the accuracy of the solution.In the process of the three-dimensional finite rotation,two Euler angles are used to parameterize the rotation vector,and the higher order continuity of the displacement and the basis functions are fully utilized.Combined with the IGA-C method and the geometrically exact Euler-Bernoulli beam theory,a large number of geometrically nonlinear static and dynamic examples of simple structures are provided in this study.In the examples of 2D statics,the convergence of the proposed method is tested.In the linear case,the order of convergence and the parity of the degree of B-splines or NURBS show correlation,which is also observed in the nonlinear problems.In the case of multiple patches,it not only verifies the correctness of the proposed rigid coupling model,but also shows that the application of present method for nonlinear buckling analysis is feasible.All these numerical simulations are intended to illustrate the good accuracy and effectiveness of the IGA-C program based on the proposed formulation.This study develops the application fields of IGA-C,and it also provides an efficient alter-native to the numerical simulation of geometrically exact beam structures.