Non-overlapping friction element for soil-structure interaction and its application
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摘要:
为了解决Goodman单元法向嵌入的缺陷，提出了一种无嵌入摩擦单元，将该单元加入到由土和结构共同作用的有限元模型中，实现了编程计算。以岩土体作为实体弹塑性单元，以斜坡或基坑边坡的支护桩作为梁单元，将土和梁接触处的Goodman 4节点8自由度的无厚度单元概化为4节点6自由度的无嵌入摩擦单元，即将Goodman单元在受压时，两个相邻的法相自由度合二为一，切向自由度仍为两个，该单元相当于切向可以滑动，法向不嵌入也不分离，由此定义了一种无嵌入摩擦单元。通过构造无嵌入摩擦单元刚度矩阵和组配总刚度矩阵建立了相应的有限元算法，编制了计算程序。算例分析表明，当Goodman单元法向刚度足够大时，其计算结果与无嵌入摩擦单元计算结果基本一致，无嵌入摩擦单元计算结果是Goodman单元法向刚度不断增大的极限解。无嵌入摩擦单元在算法上避免了法向刚度取值不确定引起的嵌入问题，为土与结构相互作用提供了一种严谨的解决方案。
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关键词:
- Goodman接触单元 /
- 无嵌入接触 /
- 有限元 /
- 土-结构相互作用
Abstract:To address the issue of overlap in the Goodman element, this study introduces a non-overlapping friction element, which is integrated into a finite element model of soil-structure interaction and programmed for calculation. The model represents the soil mass as a solid elasto-plastic element and the retaining pile of slopes or foundation pits as beam elements. The Goodman four-node, eight-degree-of-freedom element at the interface between soil and beam is simplified to a four-node, six-degree-of-freedom non-overlapping friction element by merging adjacent normal degrees of freedom under compression condition. The non-overlapping friction element allows for tangential sliding without normal overlapping or separation. The corresponding finite element algorithm is established by constructing the element stiffness matrix of the non-overlapping friction element and assembling it into the global stiffness matrix. A computational program is subsequently implemented. Case studies indicate that when the normal stiffness of the Goodman element is sufficiently large, its computational results are essentially consistent with those of the embedded frictionless element, with the latter being the limiting solution as the normal stiffness of the Goodman element continuously increases. The non-overlapping friction element algorithmically avoids the overlapping caused by the normal stiffness, and provides a rigorous solution for soil-structure interaction.
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Key words:
- Goodman element /
- non-overlapping /
- FEM /
- soil-structure interaction
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表 1 单元刚度矩阵与总刚度矩阵对应关系
Table 1. Correspondence between element stiffness matrix and total global stiffness matrix
总节点号 单元节点 总刚度 单元刚度 (74,74) 58(3,3)，68(2,2) K74,74 $k^{(58)}_{3,3}+k^{(68)}_{2,2} $ (74,75) 58(3,4)，68(2,1) K74,85 $k^{(58)}_{3,4}+k^{(68)}_{2,1} $ (74,85) 68(2,3) K74,85 $k^{(68)}_{2,3} $ (74,86) 68(2,4) K74,86 $k^{(68)}_{2,4} $ (74,90) 无对应关系 0 无对应关系 (74,91) 无对应关系 0 无对应关系 (75,75) 58(4,4),68(1,1) K75,75 $k^{(58)}_{4,4}+k^{(68)}_{1,1} $ (75,85) 68(1,3) K75,85 $k^{(68)}_{1,3} $ (75,86) 68(1,4) K75,86 $k^{(68)}_{1,4} $ (75,90) 无对应关系 0 无对应关系 (75,91) 无对应关系 0 无对应关系 (85,85) 68(3,3),78(2,2),82(2,2) K85,85 $k^{(68)}_{3,3}+k^{(78)}_{2,2}+k^{(82)}_{2,2} $ (85,86) 68(3,4),78(2,1),82(2,1) K85,86 $k^{(68)}_{3,4}+k^{(78)}_{2,1}+k^{(82)}_{2,1} $ (85,90) 82(2,3) K85,90 $k^{(82)}_{2,3} $ (85,91) 82(2,4) K85,91 $k^{(82)}_{2,4} $ (86,86) 68(4,4),78(1,1),82(1,1) K86,86 $k^{(68)}_{4,4}+k^{(78)}_{1,1}+k^{(82)}_{1,1} $ (86,90) 82(1,3) K86,90 $k^{(82)}_{1,3} $ (86,91) 82(1,4) K86,91 $k^{(82)}_{1,4} $ (90,90) 82(3,3),86(2,2) K90,90 $k^{(82)}_{3,3} +k^{(86)}_{2,2}$ (90,91) 82(3,4),86(2,1) K90,91 $k^{(82)}_{3,4} +k^{(86)}_{2,1} $ (91,91) 82(4,4),86(1,1) K91,91 $k^{(82)}_{4,4} +k^{(86)}_{1,1} $ 注：单元节点表示方法为单元号(节点号，节点号)；K为总刚度，下标为总节点号；k为单元刚度，上标为单元号，下标为单元节点号。 
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表 2 土体和支护桩的参数
Table 2. Parameters of soil and piles
土体参数 取值 支护桩参数 取值 重度/（kN·m−3） 16.0 抗弯刚度/（GPa·m4） 0.3 弹性模量/MPa 100.0 泊松比 0.3 内摩擦角/（°） 20.0 拉压刚度/（GPa·m2） 10.0 黏聚力/kPa 10.0
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