框架几何非线性分析的若干问题
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摘要
本文主要研究了框架几何非线性分析的若干问题及一些问题解决办法。
平面梁单元修正拉格朗日列式分析中,以截面形心位移函数为参数,推导得出了截面转角与形心线横向位移函数和轴向位移函数的准确关系。考虑二阶影响后,得出截面转角不仅与截面形心横线位移导数有关,还与纵向位移导数有关。推导得出了修正拉格朗日列式几何非线性分析的切线刚度矩阵。分析表明,梁单元几何刚度矩阵能否通过刚体检验与截面转角与横向和纵向位移导数之间的关系近似程度有关,通常分析中,仅考虑截面转角与横向位移导数的线性关系,因而得出的几何刚度矩阵不能通过刚体检验。分析了采用同一切线刚度矩阵计算增量节点位移和增量节点力问题,研究表明,采用同一切线刚度矩阵计算增量节点力,与已有节点力的和是欲求构形节点力,这个力是用已知构形坐标描述的,必须采用由增量节点位移确定的近似坐标变换,把这个节点力变换到欲求构形,才得欲求构形单元真实节点力;采用同一刚度矩阵计算单元增量节点力,数值计算结果表明几何刚度矩阵能否通过刚体检验对分析结果并无明显影响。平面梁单元分析中,推导了考虑加载变形弯曲影响的修正拉格朗日列式,计算增量节点力采用了协同转动列式,分析表明对于某些情形考虑加载弯曲变形,会有效减少单元划分数量;但对于某些情形,考虑加载弯曲会导致膜锁。以悬臂曲梁为例,分析了产生膜锁的原因,得出忽略初始弯曲变形二阶项以及高阶项是产生膜锁的主要原因,这些项虽然很小,但系数很大,其乘积可能比有关的线性项还大,从而影响了分析结果,导致误差甚至错误。
推导分析了绕定轴旋转、绕从轴旋转、半切向旋转的数学表达式,得出绕定轴转角、绕从轴转角与转角顺序有关;由空间梁单元变形后单元两端的旋转矩阵,精确得出了单元定位矩阵。分析了由于截面转动导致的节点弯矩增量和节点弯矩虚功,得出保守力矩的虚功与转角类型和力矩类型有关。以半切向转角为变量,采用虚功原理推导得出了空间梁单元修正拉格朗日列式的切线刚度矩阵;计算增量节点力采用了协同转动列式,精确考虑增量节点位移确定的刚体旋转,导致计算增量节点力的刚度矩阵是不对称矩阵;若仅考虑增量节点位移二阶项确定的刚体旋转,得到的刚度矩阵为对称矩阵。对于保守弯矩,节点外荷载弯矩的虚功产生外荷载刚度矩阵,这个矩阵会改变几何刚度矩阵一些项。全面搜集整理了空间梁单元几何非线性分析算例,众多的数值计算结果表明,对于空间梁单元采用修正拉格朗日列式计算单元增量节点位移,用协同转动列式计算单元的增量节点力是可行、有效的。
以杆件非线性理论为基础,推导得出了杆件稳定分析的平衡方程。以悬臂梁在悬臂端作用弯矩的临界弯矩分析为例,推导得出了半切向弯矩和两类准切向弯矩的临界弯矩;运用有限元软件采用实体单元,分析了悬臂梁在悬臂端作用弯矩的临界荷载,理论分析结果和有限元数值计算结果吻合很好。
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.
平面梁单元修正拉格朗日列式分析中,以截面形心位移函数为参数,推导得出了截面转角与形心线横向位移函数和轴向位移函数的准确关系。考虑二阶影响后,得出截面转角不仅与截面形心横线位移导数有关,还与纵向位移导数有关。推导得出了修正拉格朗日列式几何非线性分析的切线刚度矩阵。分析表明,梁单元几何刚度矩阵能否通过刚体检验与截面转角与横向和纵向位移导数之间的关系近似程度有关,通常分析中,仅考虑截面转角与横向位移导数的线性关系,因而得出的几何刚度矩阵不能通过刚体检验。分析了采用同一切线刚度矩阵计算增量节点位移和增量节点力问题,研究表明,采用同一切线刚度矩阵计算增量节点力,与已有节点力的和是欲求构形节点力,这个力是用已知构形坐标描述的,必须采用由增量节点位移确定的近似坐标变换,把这个节点力变换到欲求构形,才得欲求构形单元真实节点力;采用同一刚度矩阵计算单元增量节点力,数值计算结果表明几何刚度矩阵能否通过刚体检验对分析结果并无明显影响。平面梁单元分析中,推导了考虑加载变形弯曲影响的修正拉格朗日列式,计算增量节点力采用了协同转动列式,分析表明对于某些情形考虑加载弯曲变形,会有效减少单元划分数量;但对于某些情形,考虑加载弯曲会导致膜锁。以悬臂曲梁为例,分析了产生膜锁的原因,得出忽略初始弯曲变形二阶项以及高阶项是产生膜锁的主要原因,这些项虽然很小,但系数很大,其乘积可能比有关的线性项还大,从而影响了分析结果,导致误差甚至错误。
推导分析了绕定轴旋转、绕从轴旋转、半切向旋转的数学表达式,得出绕定轴转角、绕从轴转角与转角顺序有关;由空间梁单元变形后单元两端的旋转矩阵,精确得出了单元定位矩阵。分析了由于截面转动导致的节点弯矩增量和节点弯矩虚功,得出保守力矩的虚功与转角类型和力矩类型有关。以半切向转角为变量,采用虚功原理推导得出了空间梁单元修正拉格朗日列式的切线刚度矩阵;计算增量节点力采用了协同转动列式,精确考虑增量节点位移确定的刚体旋转,导致计算增量节点力的刚度矩阵是不对称矩阵;若仅考虑增量节点位移二阶项确定的刚体旋转,得到的刚度矩阵为对称矩阵。对于保守弯矩,节点外荷载弯矩的虚功产生外荷载刚度矩阵,这个矩阵会改变几何刚度矩阵一些项。全面搜集整理了空间梁单元几何非线性分析算例,众多的数值计算结果表明,对于空间梁单元采用修正拉格朗日列式计算单元增量节点位移,用协同转动列式计算单元的增量节点力是可行、有效的。
以杆件非线性理论为基础,推导得出了杆件稳定分析的平衡方程。以悬臂梁在悬臂端作用弯矩的临界弯矩分析为例,推导得出了半切向弯矩和两类准切向弯矩的临界弯矩;运用有限元软件采用实体单元,分析了悬臂梁在悬臂端作用弯矩的临界荷载,理论分析结果和有限元数值计算结果吻合很好。
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.
引文
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