The Theory Of Closed Thin-Walled Beams Considering The Distortion Effect And Numerical Analysis

The traditional Euler beam or Timoshenko beam did not take into account both warping and distortion, therefore that can not describe the deformation of closed thin-walled beams accurately. The torsion theory for the thin-walled open cross-sections was developed by Vlasov where the warping deformation effect was considered but the transverse shear strain was neglected. Umansckii theory considered the transverse shear strain, but ignored the distortion deformation. For thin-walled beams’node connectivity problems, the traditional methods were based on one-dimensional beam theory without considering the effect of the distortion on the nodal displacements delivery. In view of this, on the basis of classical beam theory, we carried out a more detailed analysis of thin-walled beam distortion effect on both members and structures, established a more comprehensive and accurate method to describe the thin-walled beam structures’ deformation. The main contents of this thesis are as follows:(1) Based on the Umansckii theory, used the homogeneous solution of torsion angle differential equation as the shape function, derived a precise torsional element stiffness matrix, used ANSYS’ shell63element in numerical analysis, compared the effects of different loading methods on the calculation results. Analyzed the distribution of warping stress on the cross-section, verified that the rigid cross-section assumption is the main cause of the calculation error.(2) Compared the distortion theory under loads decomposition and Vlasov’ generalized coordinate method with each other, and derived the finite element stiffness matrixes of this two methods. Verified that the correctness of this two matrixes by the numerical examples, and compared with calculation results of ANSYS. Analyzed warping deformation of cross-section, in which the distortion warping is the dominant part, while the influence of torsional warping is quite small.(3) Based on Timoshenko beam and generalized coordinate method, created a finite beam element matrix which considered the deflection/bending, torsion deformation, warping and distortion. After that, a transformation matrix at a joint of thin-walled box beams has been derived, so simplify a three-dimensional problem by a one-dimensional approach which considering the distortion effects at node junction. With the plate frame examples, the validity of the derived transformation matrix is demonstrated by showing good agreement between the Shell63element results. Finally, using the mixed finite element in numerical simulation based on MPC method, under the ensuring of the accuracy, that greatly reduced the number of discrete elements, and also consider the distortion effect at nodes and the impaction on the other nodes displacement.