Several Problems In Geometrically Nonlinear Analysis For Frames

This thesis studied some problems related the geometrically nonlinear analysis for frames.For the 2D case, it was derived for the exact relation between the cross-sectional rotation and the centroidal displacements; if the effects of the second order terms were considered, the cross-sectional rotations is dependent upon the axial centroidal displacements as well as the transverse centroidal displacements. A geometrical stiffness matrix for a 2D beam was formulated, and the analysis showed that a stiffness matrix qualified by a rigid-body test or not was concerned with the relation between the cross sectional rotations and the centroidal displacements; in most of the literature, it just considered the linear relation between the cross-sectional rotation and the centroidal displacements, which leaded to the geometrical stiffness matrix not qualified by the rigid-body test. In the analysis, the same stiffness matrix was used to calculate the incremental displacements and the incremental nodal forces; after yielded the vectors of the nodal forces in the desired configuration and measured in the known configuration, it is essential for the vectors to perform a transformation defined by the incremental nodal displacements; after the transformation, the vectors of the nodal forces measured in the desired configuration were the real nodal forces. Applied the same stiffness matrix to calculate the incremental nodal displacements and incremental nodal forces, the numerical examples showed that the geometrical stiffness matrix qualified by a rigid-body test had no more significant influence on the numerical results than the one not qualified by a rigid-body test had. For a 2D case, the formulation considered the loaded curvature was derived; for the Williams' toggle, it can efficiently diminish numbers of the meshes, however for the Lee's frame, and it can lead to some errors which term membrane locking. Illustrated by the case of a curved cantilever beam, the cause for membrane locking was analyzed; neglect of the higher order terms of the initial curvature was the primary cause for membrane locking; although the higher order terms of the initial curvature was very small, however their coefficients may be large, and their products may be large than the related linear terms; so the higher order terms can affect significantly the results, even make some error or mistakes.In the thesis, the mathematical descriptions were obtained for the rotational matrices of serial fixed rotations, serial follower rotations and the semitangential rotations. The analysis showed that the last positions for the serial fixed rotations and the serial follower rotations are dependent upon the order of the rotations. By virtue of both rotational matrices for both ends for a beam element, a rotational matrix for a beam element was exactly formulated. After a cross-sectional rotation, it was analyzed the incremental nodal moments and the virtual work for nodal moments, and the analysis showed that the virtual work for conservative moments was dependent upon the styles of the applied moments and of the rotations. Using the semitangential rotations as variables, by means of the principle of virtual work, a tangential stiffness matrix was derived for the updated Lagrangian formulation, and a corotational formulation was used to calculate the incremental nodal forces. If it was exactly analyzed for the rigid-body rotations defined by the incremental nodal displacement, this can make the stiffness matrix for the incremental nodal forces to be asymmetric; however, if the rigid-body rotations just considered the second order terms, the stiffness matrix for nodal forces is symmetric. For conservative external nodal moments, the virtual work for the external moment can result in an external load stiffness matrix, which can significantly modify the geometrical stiffness matrix. The numerical examples for geometrically nonlinear analysis of frames were collected widely and completely, and lots of numerical examples verified that the present solution strategy is correct and efficient.Based on nonlinear theory for a bar, the equations of equilibrium were derived with consideration of the nonlinear effects. Illustrated by a critical moment analysis of a cantilever beam acted upon by a bending moment at the free end, the critical moments were obtained for a semitangential moment and the first and the second quasitangential moments. The cantilever beam was also analyzed by finite element software with solid elements, the theoretical critical moments agreed well with the numerical results by the finite element method.