地下工程开挖计算的广义有限元法

GENERALIZED FINITE ELEMENT OF EXCAVATION COMPUTATION FOR UNDERGROUND ENGINEERING

基于传统有限元理论,将每个结点位移的Lagrange型插值空间推广为具有任意多个广义位移的函数展开式,在不增加结点个数的前提下,仅通过提高结点插值函数的阶数,达到提高有限元精度的目的,建立了三维广义8结点等参单元和一维广义杆单元。首先推导单元的广义形函数,并给出单元的位移模式;然后进一步推导单元应变矩阵和劲度矩阵以及单元等效荷载列阵等有限元列式。针对地下工程开挖问题,提出联合运用广义有限元和传统有限元,在开挖边界附近和加锚区围岩采用高精度的广义有限元,在远离开挖区域岩体采用高效率的传统有限元,既改善有限元的计算精度,又提高有限元的计算效率,并探讨了广义有限元的程序实施细则。通过对若干算例及工程实例的计算,其结果表明广义有限元法的优越性,为地下工程开挖计算提供了一种合理的数值方法。

Based on the conventional finite element, Lagrange type interpolation space of displacements on each node is extended to an arbitrary function expansion with any variable number of generalized displacements. The accuracy of the numerical computations is improved only through increasing the order of interpolation functions without increasing the number of nodes. Three-dimensional generalized isoparametric element of eight nodes and one-dimensional generalized bar element are established. The formulae of generalized shape function and the element displacement formulation are given. The strain matrix, stiffness matrix and load vector of element are formulated. In response to the excavation problem of underground engineering, the combined usage of conventional finite element with generalized finite element is suggested. The generalized finite element is applied to the surrounding rock mass of excavation boundary and the anchored rock mass. The conventional finite element is applied to the rock mass far from excavation boundary. Not only the computation precision, but also the computation efficiency is improved. The numerical implement programs of generalized finite element are discussed. From the computational examination questions as well as the engineering example, the numerical results indicate the advantage of generalized finite element. The proposed method is rational for analysis and computations of underground engineering.