Geometric Nonlinear Analysis Based On First-order Thin-walled Beam Theory

In the fields of architecture,bridges,aerospace,and marine vehicles,thin-walled structures have been widely used,so accurate and fast analysis theories and methods are called important research contents in this direction.For ease of application,the widely used analysis method is to build thin-walled beam elements based on existing theories.The main purpose of this paper is to obtain thin-walled beam elements of arbitrary cross-sections based on the first-order thin-walled beam theory for linear problems and geometrically nonlinear problems.First,based on the first-order thin-walled beam theory,the section displacement and rotation angle are divided into two different parts,and the relationships between the deflection and bending angle,the total torsion angle,and the free warping angle are obtained.And give the state vector of the element node,which includes the parameters of the stress field and displacement field,the state vector is 14 th order.Analytical solutions based on the first-order thin-walled theory.Except for the hyperbolic function of the total torsion angle and the free warping angle,the other parameters are polynomial interpolation functions.The interpolation functions of deflection and bending angles,and total torsion angles and free warping angles satisfy the relationship obtained by the first-order theory,avoiding shear locking,and reducing the number of element nodes.According to the theoretical expressions of the displacement field and the stress field,the relationship between the two nodes of the element is established using the migration matrix method,and a new type of thin-walled beam element stiffness matrix is obtained through mathematical transformation.In order to obtain the conversion between the coordinate systems involved in the deformation of the rod,a coordinate conversion matrix based on the azimuth angle is given,a coordinate transformation matrix based on the azimuth angle is given.Based on the relationship between the deflection and bending angle,the total torsion angle and the free warping angle,and the nonlinear displacement-strain relationship,based on von-Karman type nonlinear theory for a rod with large deformations,appropriate local coordinate systems are established.The updated Lagrange method is used to derive the tangent stiffness through the principle of virtual work.In the process of deriving,using the internal force as the generalized stress and the section displacement parameter as the generalized strain,the equations of the internal force and the section displacement parameter can be obtained.The results show that the axial force acts the main part of non-linearity.In addition,since the element configuration changes continuously during the deformation process,the transformation matrix needs to be updated in real time.This paper presents a new method for updating the coordinate transformation matrix.Finally,according to the established beam element,according to the Python language,the corresponding finite element analysis program is compiled.Through the analysis of different types of examples,the comparison and verification with the classical theoretical analytical solution and the finite element software numerical solution,the results show that the accuracy of this element.