A Study On Nonlinear Finite Element Analysis Models For Frame And Thin-walled Bar

Requirements for the accuracy and efficiency of nonlinear structural analysis methods are becoming higher and higher in mechanical engineering, civil engineering and aerospace. Although the research on nonlinear analysis and linear stability analysis for frame structures has been fruitful and much progress has been made, the existing finite element theories and models may fail to achieve satisfactory results for complicated structures that include curved beams and thin-walled members. In view of the current existing problems, the methods for nonlinear analysis and linear stability analysis of frame structures and thin-walled bars are studied in this paper. Taking into account the effects of various factors such as initial shape, transverse shear deformation, warping and distortion of cross section, the nonlinear finite element models which can consider finite rotation and the exact finite element method for buckling analysis are set up. The work completed is as follows:1. The problems of geometrical nonlinear analysis and linear stability analysis for frame structures are studied. Firstly, based on the geometrical exact beam theory, using the rotational vector to parameterize the element displacement fields, a nonlinear spatial beam element model which can consider shear deformation and finite rotation is developed, and a high efficiency method for geometrical nonlinear analysis of spatial frame structures is achieved. As to the problem of the increase in the number of degrees of freedom and sharp increase in calculation workload caused by high order interpolation function, a method for condensing out internal degrees of freedom based on element-level equilibrium iteration procedure is proposed. The numerical examples show that the proposed method can substantially enhance the computational efficiency. In addition, as to the problem of stability for frame structures, an exact beam element considering transverse shear deformation is developed based on the general solutions of the governing equations for buckling of Timoshenko beam, and the corresponding improved solution algorithm is proposed. The numerical examples demonstrate that the proposed element model and algorithm get high efficiency in buckling analysis for planar frames, featuring excellent computational accuracy and reliability.2. The geometrical nonlinear analysis problem of curved beams is studied. Firstly, a curved beam element formulation with planar variable curvature is proposed based on mapping. The description method for initial configuration and the strain expression considering initial curvature are researched and the effect of initial curvature on computational results is discussed. Furthermore, by combining the geometrically exact beam theory and the mixed variational principle, a nonlinear curved beam element which simultaneously takes the effects of finite rotation, shear deformation and spatial shape into consideration is proposed, and meanwhile the element internal degrees of freedom is condensed. The study shows that the proposed curved beam mixed element has excellent computational accuracy and generality.3. The thin-walled beam element models which consider the warping are studied for nonlinear and linear stability analysis. Firstly, based on the theories of finite deformation and finite rotation, the strain expression of the thin-walled beam is derived and a nonlinear thin-walled beam finite element model which considers the complicated warping distribution is proposed, using the Lagrangian interpolation functions to describe the warping distribution on the midline of the cross section. The effects of distribution characterization of warping on linear response, nonlinear response and stability of thin-walled member are investigated. In the study of linear stability, an exact finite element for linear stability analysis of thin-walled member without shear deformation is proposed to obtain reliable reference solutions. Further research shows that the effects of shear deformation and warping characterization on axial compression stability will be more obvious under the flexural-torsional coupling effect. In addition, by using the thin-walled beam element model with specified warping distributions, the effects of nonlinear strain terms on computational results are studied, simplified strain expressions are obtained and the importance of the Wagner effect and the quadratic strain terms of shearing is indicated for stability analysis under axial compression. Finally, a mixed element model based on the simplified strain expression is proposed for geometric and material nonlinear analysis of slender thin-walled members. The results of the examples have proved the effectiveness of the proposed model.4. The nonlinear analysis of thin-walled beams with deformable cross sections is studied. It is pointed out that the distribution characterization of warping in cross section is a critical factor that influences the analysis accuracy of the thin-walled beam elements, with respect to the nonlinear problem which considers the cross section distortion. Based on the theories of finite deformation and finite rotation, two models for thin-walled beams with deformable cross section are proposed which consider the complicated warping distribution. In the first model, the in-plane displacements and rotation angles of the section nodes are taken as distortion parameters and the cross section distortion is described by interpolation function. In the second model, the cross section distortion is described by combining the cross section distortion modes and the amplified functions of each mode are taken as distortion parameters. The numerical examples show that the two proposed models can both reasonably simulate the nonlinear response of the thin-walled beam, attaining high calculation precision, and overcome the defects and deficiencies of the exist cross section deformable thin-walled beam models in the case of small wall thickness.