Research On Structural Geometric Nonlinear Analysis Based On Rigid Body Rule And Orthogonal Iteration Method

In recent years,the structure has developed in a soft and flexible direction,showing obvious nonlinear characteristics in the various stages of design,construction,and service.Nonlinear analysis of the structure is an important means for engineers to accurately grasp the load-bearing performance of the structure.There are two methods for structural nonlinear analysis:one is analytical method;the other is numerical simulation method.Because the analytical method has too strict requirements for structure types and boundary conditions,and has great limitations in its application,Nonlinear finite element method is the most commonly used method.The research of nonlinear finite element mainly exists in two aspects:one is the establishment of nonlinear finite element model,which is on the element level;When establishing nonlinear finite elements based on beam-column theory or virtual work principles,different element models are often obtained due to the different choices of higher-order terms in the element equations.The derivation process is often very tedious,especially the derivation of the geometric stiffness of the structure,and the quality of the model is also difficult to guarantee.The other is the tracking technology for structural equilibrium path,which is on the structure level.At present,there are many tracking technologies for structural equilibrium paths,which can also solve the equilibrium paths of weakly nonlinear structures.However,the technologies require further verification,and the main points of the analysis process are not clear.Aiming at the problem that the derivation process of the nonlinear element model is too complicated,the main points of the nonlinear analysis process are not clear and the stability of most structural equilibrium path tracking technology is insufficient,this study starts from the physical point of view and focuses on the rigid body behavior of the element model and incremental-iterative analysis.The main content and conclusions are:（1）The main process of increment-iteration is discussed in detail,and the accuracy of the entire increment-iteration calculation is mainly controlled by the corrector,and the predictor does not have to be very precise.At the same time,in the process of nonlinear analysis,the rigid body displacement of the element occupies the main part of the element displacement.The element nodal force change caused by the rigid body displacement can be dealt with by the rigid body rule,and the element nodal force change caused by the natural deformation can be determined by the elastic stiffness matrix.Through combing the entire calculation process,it is proposed that use only the elastic stiffness matrix for the geometric nonlinear analysis.The feasibility of the idea is proved through case analysis.It is concluded that the cost is the calculation time and the number of iterations is slightly increased,the calculation accuracy is equivalent to the conventional methods,and it can be applied to most structural nonlinear analyses.If the elastic stiffness matrix is replaced with an elastoplastic stiffness matrix,use only the elastoplastic stiffness matrix can also solve the geometric and material nonlinear problem.（2）Based on the UL formulation,for the three-dimensional beam element,the sectional stress should be strictly defined at the deformation configuration₂.When undergoes three-dimensional rotation,the moment of the node will generate the induced moment matrix[6)6)₄4))],which is the key consideration for the equilibrium of the space frame node,and it is also the key points to simulates the curved beam structure by straight beam element.Based on[6)6)₄4))],the geometric stiffness matrix is constructed.According to the symmetry and the qualification of rigid body,the rigid body qualified geometric stiffness matrix is deduced with 10 parameters.By assigning values to the parameters,they can be well used in nonlinear analysis,which proves that the rigid body qualified geometric stiffness matrix can guide the predictor direction well.（3）The geometric stiffness matrix of three-dimensional beams be simplified to two-dimensional truss,three-dimensional truss,and two-dimensional beam elements and extended to triangular plate elements and I-shaped beam elements.The geometric stiffness matrix for eighteen freedom degree triangular plate element and the geometric stiffness matrix for fourteen freedom degree I-shaped beam element be derived.The parameter assignment proves that the rigid body qualified geometric stiffness matrix can guarantee the accuracy of the predictor direction and has good applicability.（4）The incremental-iterative control method is described in detail,and five expressions for controlling the size of load increments and four expressions for controlling the iterative direction are proposed.The numerical case shows that the index S is a good detector for judging the loading and unloading;the orthogonal iterative scheme has high computational efficiency.The F3T9 combination has the highest efficiency among all the incremental-iteration combinations,and it is recommended to track the nonlinear path of the structure.The innovations of this thesis:（1）The incremental-iterative process is systematically explained,and the method of using only the elastic stiffness matrix for structural geometric nonlinear analysis is proposed.Numerical cases prove that the calculation time and the number of iterations is slightly increased,and the calculation accuracy is almost the same as that of conventional methods.It can be applied to most structural nonlinear analysis,and the method can be extended to geometric and material nonlinear analysis.（2）Based on the induced moment matrix[6)6)₄4))],which is the key factors for the equilibrium of the nodes when the three-dimensional beam element undergoes rotation,the geometric stiffness matrix of the three-dimensional beam element is deduced,and the geometric stiffness matrix is derived to the truss element,two-dimensional beam element,triangular plate element,and obtain the rigid-body qualified geometric stiffness matrix for various elements with including parameters.Regarding the I-shaped section beam as a combination of three rectangular section beam elements,by considering the internal force combination of the section,the rigid body qualified geometric stiffness matrix of the I-shaped beam element containing the warping freedom degree is deduced,which also explains the Saint-Venant torque is the simitangential moment,and the warping torque is regarded as a quasitangential moment.Through parameter assignment and case analysis which further proves that the rigid-body qualified geometric stiffness matrix can guarantee the accuracy of the predictor direction well in the nonlinear analysis.（3）It shows that in an incremental step in the incremental-iterative calculation control method,the first step mainly controls the size and direction of the load increment,and the remaining steps are iterative steps to reduce the error.The two are independent and can be combined arbitrarily.Five expressions for controlling load increment step size and four orthogonal iteration directions,as well as an index S for judging loading and unloading,are proposed.The calculation combination F3T9 has the highest efficiency and is recommended for incremental-iterative nonlinear analysis of structures.