Efficient Analysis Method For Skeletal Structures With Elasto-Plastic Large Displacement Based On The Inelasticity-separated Theory

Skeletal structures,which are widely applied to the civil and industrial buildings,have been getting more and more attention on its mechanical characteristic and failure mode.The nonlinear analysis method which is a powerful tool for investigating the disaster mechanism and performance evaluation of skeletal structures under environmental loads has received popularly.Therefore,various methods have been developed for analyzing accurately and efficiently nonlinear behaviors,and can be classified into two categories:(1)Macroscopic analysis method.This kind of methods has some characteristics,such as modeling simply and calculating efficiently.However,the refined simulation for the local domains of skeletal structures cannot be realized by employing these methods.(2)Nonlinear finite element method(FEM).Although the FEM can accurately analyze the nonlinear behaviors of skeletal structures,it is time-consuming and its modeling is complicated.Thus,seeking the balance scheme between the calculation precision and efficiency all along is the purpose of researchers.Material nonlinearity and geometric nonlinearity(hybrid nonlinearities)for skeletal structures subjected to environmental loads occur frequently and the numerical simulation of complex nonlinear force-deformation relationship becomes increasingly important and significant.For investigating the disaster mechanism of skeletal structures,the elasto-plastic large displacement analysis is implemented,in which the elasto-plastic constitutive model is used to simulate the material nonlinear behaviors and geometric nonlinearity only involves the large displacement and small strain.In recent years,the inelasticity-separated theory,as a new nonlinear analysis theory,is proposed,which is used to solve the nonlinear problems of engineering structures.Its advantage is that the calculation efficiency can be improved greatly,while the precision can satisfy the engineering requirement.The core of the inelasticity-separated theory is that the global stiffness matrix can keep unchanged throughout the whole computational process by decomposing the deformation of members or elements into the linear and nonlinear parts.This method has obvious efficiency advantage over the variable stiffness method.In this paper,an efficient analysis method for skeletal structures with elasto-plastic large displacement is proposed by resorting the inelasticity-separated theory.At the macroscopic level,an analysis model of nonlinear buckling behavior for axial compression members is established by decomposing the axial displacement into its linear-elastic and nonlinear components,and is applied to the static and dynamic response analyses of skeletal structures.To achieve refined finite element simulation,an inelasticity-separated fiber beam element model considering the hybrid nonlinear behaviors is established by decomposing the section deformation into its linear-elastic and plastic components.Meanwhile,an efficient solution method is proposed to analyze the elasto-plastic large displacement problems of skeletal structures.The main research contents are shown as follows:(1)An analysis model of nonlinear buckling behavior for axial compression members is presented based on the displacement decomposition concept,which used a plastic hinge(RH)and a slide hinge(SH)to simulate the behavior of plastic bending deformation produced by buckling and the axial plastic elongation caused by yield and growth effect,respectively.In addition,the elastic bending deformation representing geometric nonlinearity is considered.In the proposed model,RH and SH have clear physical meaning,and their nonlinear behaviors only depend on the material properties.Particularly,the rotation of RH can be acquired and provides a direct evaluation on the buckling behavior state.Subsequently,the proposed model is applied to simulate the hybrid nonlinearities of skeletal structures.It can be found from the governing equations that the global stiffness matrix is maintained constant and needs not be factorized in real time during the analysis process.Therefore,the present research can be used to analyze the elasto-plastic large displacement behaviors of skeletal structures,and is an accurate,efficient,and stable algorithm.(2)A refined analysis model for skeletal structures with elasto-plastic large displacement behaviors is established within the framework of the inelasticity-separated theory.The section deformation of fiber beam elements can be decomposed into the linear-elastic and plastic components based on the material strain decomposition concept,and the elemental governing equation considering the hybrid nonlinear behaviors is derived by using the principle of virtual work with the total Lagrangian formulation.The global stiffness matrix,which is equal to the sum of the initial stiffness matrix and geometric stiffness matrix,changes in real time.To efficiently solve the governing equation,the efficient Woodbury-CA hybrid(WCH)method is proposed based on the Woodbury formula and the combined approximations(CA)method.Its advantage is that the factorization of the large-scale global matrix is avoided and the factorization of the Schur complement matrix with a small rank is implemented during the nonlinear analysis.Further,the time complexity theory is used to evaluate the calculation efficiency of the present method,and the results show that it is obviously superior to the conventional finite element method for analyzing local material nonlinearity and geometric large displacement behaviors of skeletal structures.(3)A developed Woodbury approximation(DWA)method is presented to analyze the nonlocal material nonlinearity and geometric large displacement behaviors of skeletal structures.The numerical characteristic of some matrices in the inelasticity-separated governing equation and the solution process of the WCH method are investigated.It is indicated that the factorization of the Schur complement matrix in Woodbury formula costs substantial computational time,and the WCH method is only suitable for the local material nonlinearity and geometric nonlinearity problems.For this reason,an efficient analysis method is proposed for more general material and geometric nonlinear situations,where the governing equation is optimized by incorporating the updated Lagrangian formulation into the framework of inelasticity-separated theory.To solve the governing equation,the DWA method is presented as an efficient solver,in which a linear equation related to the Schur complement matrix is solved by the CA method and the changing global stiffness matrix is approximated as constant matrix for small periods of time.To eliminate the additional error stemming from the approximation,an adaptive iteration strategy based on the energy norm is adopted,in which the difference between the approximate solution of the DWA method and the exact solution of the FEM is evaluated.Furthermore,time complexity analysis is employed to verify the high efficiency of the DWA method,which demonstrates that the proposed method can be implemented for nonlocal material nonlinearity and geometric large displacement analyses of skeletal structures.(4)An efficient stability analysis method for skeletal structures is proposed by using the displacement control algorithm with multiple point constraints.The inelasticity-separated governing equation including the multiple point displacement constraints is derived based on the Woodbury formula.It is solved by the DWA method in the proposed scheme,resulting in an obvious advantage in efficiency for analyzing the stability of skeletal structures.The feasibility and validity of the proposed method is verified by the experimental data and numerical examples.Finally,the proposed scheme is applied to solve the nonlinear stability problems of some skeletal structures,such as steel frame,steel braced frame and single layer reticulated shell.Numerical results show that the proposed method can accurately acquire the critical load value in this case where the nonlinear equilibrium path has snap-through buckling situations.Thus,the proposed method has greater potential for obtaining the complete loaddisplacement curves and solving the negative stiffness problems of skeletal structures.