Coupled Bending And Torsional Analysis Of Thin-walled Members Based On First-order Thin-walled Beam Theory

Thin-walled components are widely used in the fields of construction,bridges,aerospace and machinery.The theory and method of bending and torsion analysis of thin-walled components have become the focus of current research.The traditional beam theory mainly includes Euler-Bernoulli beam theory,Timoshenko beam theory,Vlasov thin-wall beam theory and Benscoter theory.There are currently two methods for establishing beam elements,one is based on the displacement interpolation function,the element stiffness matrix is derived using the energy principle,and the other is based on the transfer matrix.In this paper,the bending and torsion problems of thin-walled components are studied on the basis of first-order thin-walled beam theory,and the element stiffness matrix of thin-walled beams with bending-torsion coupling considering bending,torsional deformation,warpage and shear deformation is derived.First consider the case where the cross-sectional shear center and the centroid coincide.According to the characteristics of the shear wall of the thin-walled member,bending and torsion can be considered separately,that is,bending and torsion are not coupled.The first-order thin-walled beam theory bending problem is written as a system of mixed differential equations in the form of state vectors.The exponential expansion expression is used to obtain the expression of the transfer matrix.The transfer matrix during axial tension and compression can be obtained in a similar way.For the transfer matrix of the thin-walled beam torsion,using the relationship between the initial parameter vector and the unknown parameter vector given by the initial parameter method of the first-order thin-walled beam theory,the state vector and the transition matrix of the thin-walled beam torsion can be obtained by matrix transformation.The state vector and transfer matrix of bending,tension,compression and torsion are combined to obtain the total state vector and transfer matrix of the first-order thin-walled beam theory.Furthermore,the transfer matrix transformation method is used to obtain the uncoupled element stiffness matrix.Second,consider the case where the shear center and the centroid of the thin-walled section do not coincide.When deriving the element stiffness matrix,the coordinate conversion matrix between the shear center and the centroid is used to obtain the relationship between the displacement vector based on the shear center and the displacement vector based on the centroid.Using the principle of virtual work,the coordinate transformation matrix between the nodal force vector based on shear center and the nodal force vector based on centroid is obtained.The coupled element stiffness matrix can be obtained based on the coordinate conversion matrix of displacement and nodal force and the previous uncoupled element stiffness matrix.In order to convert the element stiffness matrix in the local coordinate system to the global coordinate system,a coordinate conversion matrix based on azimuth angle is given.By using Python as the programming language of object-oriented programming,the analysis program was compiled based on the deduced beam unit.Finally,through the analysis of the calculation example and comparison with the results of finite element software and classical beam theory,the reliability of the current element is proved.