Upper-bound Limit Analysis of Seismic Quasi-static Stability of Multi-stage Slopes
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摘要: 基于极限分析上限定理，结合非线性Mohr-Coulomb破坏准则，运用地震拟静力分析方法，构建三级边坡在地震效应影响下对数螺旋线破坏极限分析模型，并推导出边坡安全系数计算方程。通过MATLAB编程计算，采用序列二次规划方法优化求解，分析了地震效应下多种因素对多级边坡稳定性的影响。研究结果表明，边坡坡角和台阶宽度是影响地震作用下多级边坡稳定性的重要因素，为提高地震区域的边坡稳定性，应合理设置边坡台阶宽度并放缓坡角；地震效应对边坡稳定性的影响在非线性条件下更符合工程实际，非线性系数m的增加将降低边坡稳定性；水平地震效应明显影响边坡稳定性，随着水平地震系数kh的增大，边坡安全系数FS显著减小，且减小速率逐渐加快；同时当地震效应比例系数λ越大，即竖向地震作用越大时，边坡在水平和竖向地震效应作用下更易失稳。因此，进行多级边坡工程设计时有必要同时考虑水平地震效应和竖向地震效应。Abstract: Based on the upper limit theorem of limit analysis and the nonlinear Mohr-Coulomb damage criterion, a limit analysis model for the logarithmic helix damage of three-stage slopes under seismic effects is developed. This model applies the seismic static analysis method and derives the equation for calculating the slope safety factor. The calculations are performed using MATLAB programming, with sequential quadratic programming employed to optimize the solution. The study analyzes the influence of various factors on the stability of multi-stage slopes under seismic effects. The results show that slope angle and step width are key factors influencing stability; thus, these parameters should be carefully selected to enhance slope stability in seismic regions. The effect of seismic forces on slope stability is better represented under nonlinear conditions, with an increase in the nonlinear coefficient (m) leading to a reduction in slope stability. Horizontal seismic effects significantly affect stability; as the horizontal seismic coefficient (kh) increases, the slope safety factor (FS) decreases notably, with the rate of decrease accelerating over time. Additionally, when the seismic ratio coefficient (λ) is large—indicating higher vertical seismic forces—slope stability becomes more prone to destabilization under both horizontal and vertical seismic effects. Therefore, both horizontal and vertical seismic forces should be considered in the design of multi-stage slopes in seismic regions.
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图 1 非线性Mohr-Coulomb破坏强度准则示意
Figure 1. Diagram of a nonlinear Mohr-Coulomb failure criterion
图 2 地震拟静力分析法在边坡中的作用力示意
Figure 2. Schematic diagram of the force at slope by seismic quasi-static analysis
图 3 三级台阶边坡对数螺旋线破坏计算简图
Figure 3. Simplified diagram of log spiral failure of three-step slope
图 6 江西省遂大高速公路拟建多级边坡示意图
Figure 6. Fig. 6 Schematic photo of the proposed multi-step slope for the Suiding expressway in Jiangxi province
图 8 不同λ取值下FS变化曲线
Figure 8. The variation of safety factor FS with different values of λ
图 9 不同β1取值下FS随kh变化曲线
Figure 9. The variation of FS with horizontal seismic coefficient kh for different values of β1
图 10 不同β2取值下FS随kh变化曲线
Figure 10. The variation of FS with horizontal seismic coefficient kh for different values of β2
图 11 不同β3取值下FS随kh变化曲线
Figure 11. The variation of FS with horizontal seismic coefficient kh for different values of β3
图 12 不同β1和β3取值下FS变化曲线
Figure 12. The variation of safety factor FS with seismic effect for different values of β1 and β3
图 13 不同台阶宽度下FS变化曲线
Figure 13. The variation of safety factor FS under different step widths
表 1 边坡安全系数计算结果与典型算例结果对比
Table 1. Comparison of safety factors between this study and other typical computation example solution
算例 黏聚力c
/kPa内摩
擦角φ
/（°）重度γ
/（kN·m−3）坡高比
α1、α2、α3坡角
β1、β2、β3
/（°）边坡
高度H
/m边坡安全系数FS 邓东平等（2010） 高连生等（2014） 本文研究 1 28 25 18.5 1/3、1/3、1/3 45、45、45 12 2.212（Bishop法）
2.197（单一任意曲线法）2.071 9 2.039 2.210（折线法）
2.220（Janbu法）2 36 25 18 2/7、3/7、2/7 30、45、60 14 2.236（Bishop法）
2.221（单一任意曲线法）2.165 8 2.133 2.248（Janbu法）
2.223（折线法）
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表 2 非线性破坏准则下边坡稳定性系数NS计算结果与典型算例对比
Table 2. Comparison of slope stability coefficient NS from computation example under nonlinear failure criterion in this study with other typical computation example solution
坡角β/(°) 非线性系数m Zhang等（1987） 李得建等（2015） 本文研究 减小百分比/% 75 1.2 6.77 6.79 6.82 0.74 1.4 6.33 6.36 6.39 0.95 1.6 6.04 6.07 6.11 1.16 1.8 5.82 5.86 5.91 1.55 2.0 5.60 5.70 5.71 1.96 60 1.2 8.95 8.98 9.01 0.67 1.4 8.13 8.18 8.19 0.74 1.6 7.61 7.65 7.68 0.92 1.8 7.24 7.29 7.32 1.10 2.0 6.97 7.02 7.07 1.43 45 1.2 12.55 12.61 12.67 0.96 1.4 10.82 10.87 10.93 1.02 1.6 9.70 9.84 9.82 1.24 1.8 9.10 9.17 9.24 1.54 2.0 8.78 8.69 8.95 1.94
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