A KINEMATICALLY EXACT SPACE FINITE STRAIN BEAM MODEL - FINITE-ELEMENT FORMULATION BY GENERALIZED VIRTUAL WORK PRINCIPLE
A KINEMATICALLY EXACT SPACE FINITE STRAIN BEAM MODEL - FINITE-ELEMENT FORMULATION BY GENERALIZED VIRTUAL WORK PRINCIPLE
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DOI：
10.1016/0045-7825(94)00056-s
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发表时间：
1995-01-01
影响因子：
7.2
通讯作者：
SAJE, M
中科院分区：
文献类型：
--
作者：
JELENIC, G;SAJE, M
本文提出了一种用于线性弹性空间框架结构静态分析的新型有限元公式，扩展了 Simo 和 Vu-Quoc 给出的公式[结合剪切和扭转翘曲变形的几何精确杆模型，Int. J. Solids Structures 27 (3) (1991) 371-393]，沿着 Saje 提出的平面梁理论工作的思路[直细长弹性梁的有限平面变形的变分原理，Internat。 J.固体与结构26（1990）887-900]。我们应用空间有限应变梁理论的精确非线性运动学关系，假设伯努利假设并忽略横截面的翘曲变形。公式中考虑了有限位移和旋转以及有限拉伸、剪切、扭转和弯曲应变。梁的变形构型由变形质心轴的位移矢量和刚性连接到梁横截面的正交移动框架来描述。移动框架相对于固定参考系的位置由正交矩阵指定，并由旋转矢量参数化，该旋转矢量一步将移动框架从任意位置旋转到变形配置。此外，还引入了增量旋转矢量，它将移动框架从先前迭代步骤中获得的配置旋转到梁的当前配置。其相对于固定全局坐标系的分量被视为节点处的旋转自由度。由于在3维空间中，轴向力矩和从动力矩都是非保守的，因此引入的不是变分原理而是虚功原理作为有限元离散化的基础。在这里，我们通过程序包含精确的运动学方程，提出了虚功原理的广义形式。类似于拉格朗日乘子。这使得从原理中消除位移矢量场成为可能，使得增量旋转矢量场的三个分量仍然是在原理的有限元实现中需要近似的唯一函数。另一方面，其他研究人员采用增量旋转矢量场的三个分量和增量位移矢量场的三个分量。结果，获得了用于空间框架非线性分析的更准确和更有效的梁有限元族。单域公式导致这样一个事实：在当前的有限元中，锁定永远不会发生。变形状态的任何组合都同样精确地描述。这与文献中开发的元素形成对比，在文献中，为了避免锁定，必须应用减少的数值积分，不幸的是，这降低了解决方案的准确性。选择多项式来近似旋转矢量的分量。在这种情况下，可以合理地估计数值积分的阶数，并且可以以多项式的次数不需要限于特定值的方式来编码计算机程序。牛顿法用于非线性平衡方程的迭代求解。在非平衡配置中，通过使用方向导数对控制方程进行线性化而获得的切线刚度矩阵即使对于保守载荷也是非对称的。只有达到平衡状态，切向刚度矩阵才变得对称。这样，得到的切向刚度矩阵可以对称，而不影响牛顿法的收敛速度。然而，对于非保守载荷，切线刚度矩阵始终是非对称的。数值例子证明了本公式准确确定空间框架非线性行为的能力。在数值示例中，确定了面外屈曲载荷，并追踪了悬臂和直角框架的整个临界前和临界后载荷位移路径。在空间梁标准验证实例问题的分析中，即使仅采用一个单元来描述结构本身尺寸的位移、2 pi 的旋转以及远远超出线弹性材料实际值的拉伸应变，也显示了解决方案的出色精度。
The present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by Simo and Vu-Quoc [A geometrically-exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids Structures 27 (3) (1991) 371-393], along the lines of the work on the planar beam theory presented by Saje [A variational principle for finite planar deformation of straight slender elastic beams, Internat. J. Solids and Structures 26 (1990) 887-900]. We apply exact non-linear kinematic relationships of the space finite-strain beam theory, assuming the Bernoulli hypothesis and neglecting the warping deformations of the cross-section. Finite displacements and rotations as well as finite extensional, shear, torsional and bending strains are accounted for in the formulation. A deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame, rigidly attached to the cross-section of the beam. The position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix, parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step. Also, the incremental rotational vector is introduced, which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam. Its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points. Because in 3-D space both the axial and the follower moments are non-conservative, not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization. Here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure. similar to that of Lagrangian multipliers. This makes possible the elimination of the displacement vector field from the principle, so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle. Other researchers, on the other hand, employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field. As a result, more accurate and efficient family of beam finite elements for the non-linear analysis of space frames has been obtained. A one-field formulation results in the fact that in the present finite elements the locking never occurs. Any combination of deformation states is described equally precisely. This is in contrast with the elements developed in literature, where, in order to avoid the locking, a reduced numerical integration has to be applied, which unfortunately, diminishes the accuracy of the solution. Polynomials have been chosen for the approximation of the components of the rotational vector. In this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value. The Newton method is used for the iterative solution of the non-linear equilibrium equations.In an non-equilibrium configuration, the tangent stiffness matrix, obtained by the linearization of governing equations using the directional derivative, is non-symmetric even for conservative loadings. Only upon achieving an equilibrium state, the tangent stiffness matrix becomes symmetric.Thus, obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the Newton method. For non-conservative loadings, however, the tangent stiffness matrix is always non-symmetric. The numerical examples demonstrate capability of the present formulation to determine accurately the non-linear behaviour of space frames. In numerical examples the out-of-plane buckling loads are determined and the whole pre-and post-critical load-displacement paths of a cantilever and a right-angle frame are traced. These, in the analysis of space beams standard verification example problems, show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself, the rotations of 2 pi, and extensional strains much beyond the realistic values of linear elastic material.