利用变分法探索地震作用隧道掌子面破坏形式

Exploration of face failure curve under earthquakes using calculus of variations

介绍基于变分法给出地震作用下隧道掌子面坍塌破坏极值曲线解析表达式的方法。首先建立隧道开挖纵向参考平面,通过引入地震系数k给出地震作用下掌子面坍塌破坏块体拟静力计算模型,然后由Hoek–Brown破坏准则及关联流动给出破坏块体的内力耗散功率,从而根据上限定理建立关于掌子面破坏曲线的泛函,再由Euler–Lagrange方程得到相应破坏极值曲线的非线性偏微分方程,最后给出B=0.5时掌子面破坏极值曲线特解。经与数值分析对比表明解析结果在预测掌子面前方最远距离zmax与数值结果较为一致,但破坏区域偏大。亦探讨了滑动方向角?与掌子面压力T?等参数变化对破坏极值曲线形状的影响。分析表明:该方法预测地震作用下掌子面破坏曲线的具体表达式可以较为简单且有效准确的反映掌子面前方区域受扰动特征,亦可以作为地震多发区隧道施工风险评价定量依据之一。

A solution of face collapse failure under earthquakes is obtained in the realm of plasticity theory with the help of the calculus of variations. A simplicity face collapse failure model is introduced. Based on the Hoek–Brown failure criterion and the upper bound theorem, an exact function of face failure curves under earthquakes is presented. The extreme collapse curves derived by the Euler–Lagrange equation are solved in a nonlinear PDF. And then the exact extreme curves of face collapse failure are got when B=0.5. It is shown that the most remote distances（zmax） are consistent with the numerical results, but the area of collapse zone is too large. Also, a sensitivity analysis to some direction angles（ ？） of the face detachment and several face pressures（T？） is presented. The resulting expressions are so simple and rational to show the characteristics of face collapse failure, and can be used to make comparisons of the empirical and numerical analyses.