【作者】 刘永方；

【导师】 李忠学；

【作者基本信息】 浙江大学 ， 结构工程， 2008， 硕士

【摘要】 本文发展了一种新型9节点四边形曲壳单元。单元的局部坐标系采用了先进的协同转动框架,它能随单元的刚体转动和平动而运动,但不受单元变形的影响,从而在计算单元节点位移时可以排除其中的刚体位移成分,降低了建立单元切线刚度矩阵的复杂性。单元每个节点包含3个平动自由度和2个转动自由度,其中转动自由度为单元节点处中性面法向矢量的两个较小分量,在增量求解过程中它们是可以直接累加的。采用矢量型转动变量,通过简单的矢量变换即可建立起局部变量与整体变量之间的关系,从而非常方便地将单元局部切线刚度矩阵转换到整体坐标系下。此外,引进适用于浅壳的Green-Lagrange应变来计算单元应变能,然后对单元应变能进行一阶和二阶微分,即可得到单元内力矢量和对称的切线刚度矩阵。采用降阶积分法可以克服闭锁现象,但有时也会使单元因缺秩而导致伪零能模式出现,从而使单元切线刚度矩阵产生病态现象。为了消除伪零能模式,引进Hellinger-Reissner函数,用假设应变部分代替协调应变。假设应变由低阶应变和高阶应变组成,其中低阶应变由协调应变采用2×2积分得到;而作为稳定应变的高阶应变由假设应变法构造而得。由Hellinger-Reissner变分原理得到的切线刚度矩阵仍是对称的。最后,本文采用8个经典算例对单元的可靠性、计算效率和精度进行测试,它们由3个小变形算例和5个大变形算例(包括3个屈曲问题)组成。非线性增量方程组的求解分别采用了广义位移法和位移控制法,以越过结构屈曲后平衡路径中的极值点和反跳点,从而有效跟踪结构的屈曲后响应。分析结果表明:本文所发展的9节点四边形曲壳单元已有效地消除了闭锁现象和伪零能模式,单元的可靠性、收敛性和计算效率也是非常满意的。更多还原

【Abstract】 An advanced 9-node curved quadrilateral shell element for large displacement and large rotation analysis is developed, where the local coordinate system is a co-rotational framework, it translates and rotates rigidly with the element, but does not deform with the element. Thus, the contribution of rigid-body motion to nodal displacements and rotations can be excluded in advance, making it simpler to obtain the element tangent stiffness matrix. There are 3 translational degrees of freedom and 2 rotational degrees of freedom at each node, and the latters are the two smallest components of the mid-surface normal vector at each node, and additive in an incremental solution procedure, taking advantage in updating the tangent stiffness matrix. Furthermore, the Green-Lagrange strains specialized for the shallow curved shell are employed, and the internal force vector and the element tangent stiffness matrix are calculated respectively as the first derivative and the second derivative of the strain energy with respect to local nodal variables. Considering the commutativity of all nodal variables, the achieved element tangent stiffness matrix is symmetric.The uniformly reduced integration method can eliminate/alleviate locking problems, but sometimes it may lead to the occurrence of element rank deficiency and spurious singular modes. To overcome these problems, the Hellinger-Reissner functional is adopted in which part of conforming strains are replaced by assumed strains. The assumed strains consist of the lower-order strain and the higher-order strain, and the former is interpolated linearly by using the corresponding displacement-based strain, while the latter playing the role of stabilizing strain is formulated based on an assumed strain procedure. The element tangent stiffness matrix derived from Hellinger-Reissner variational principle is still symmetric.Finally, 8 elastic flat plate and curved shell problems are solved to evaluate the performance of the proposed element. The generalized displacement control method is employed in solving these problems. It is shown that locking phenomena has been eliminated/alleviated in the present element, and the reliability, computational efficiency and convergence of the element are satisfying.更多还原