基于六节点三角形单元和线性规划模型的上限有限元研究

目前三节点三角形单元常用于极限分析上限有限元，该低阶单元在模拟岩土体失稳破坏时形成的剪切带和塑性区时精度较低。为此，在已有研究成果的基础上，将二阶的六节点三角形单元引入上限有限元，且在单元公共边设置速度间断线并建立线性规划模型，进一步将该单元的应用扩展到服从莫尔–库仑屈服准则的土体。通过对上限有限元进行理论推导并编制程序，利用典型算例研究屈服准则线性化和速度间断线辅助变量线性化对计算结果的影响，并与三节点三角形单元所得结果进行对比分析。研究表明，同等单元数目条件下六节点三角形单元应用于上限有限元能提高计算精度，不过对于某些破坏区域约束不太强烈模型而言，当速度间断线作用显著发挥时，采用三节点三角形单元也可获得良好的解答。

INVESTIGATION OF FINITE ELEMENT UPPER BOUND SOLUTION BASED ON SIX NODAL TRIANGULAR ELEMENTS AND LINEAR PROGRAMMING MODEL

Three nodal triangular elements，of which velocity variables vary linearly，are often used in limit analysis of finite element upper bound solution. However，this low-order element has deficiency in simulating the shear and plastic zones occurred in failure mechanism of rock and soil masses. Based on predecessors? research，the six nodal triangular elements which have quadric changes of variables were introduced into finite element upper bound solution；and velocity discontinuities were also set between adjacent elements. The formulated model with this quadric element was a linear programming and can be used to analyze stability problems of the soil which obeys Mohr-Coulomb yield criterion. The formula of finite element upper bound theory was deduced and its calculating program was compiled. Using two typical calculating examples，the influences of linearization of yield criterion and velocity discontinuities additional variables on calculation results were investigated. The results obtained by six nodal triangular elements were compared to those of three nodal triangular elements. It was indicated that the calculation precision of results were improved by using six nodal triangular elements with equivalent number of elements. However，for the problems that failure zones were not remarkably constrained and velocity discontinuities line playing an key role，the favorable results can also obtained by using three nodal triangular elements.