工程科学与技术   2018, Vol. 50 Issue (5): 122-129

1. 河海大学 岩土力学与堤坝工程教育部重点实验室，江苏 南京 210098;
2. 河海大学 岩土工程科学研究所，江苏 南京 210098;
3. 河海大学 隧道与地下工程研究所，江苏 南京 210098

Non-slice Calculation of Safety Factor and the Determination of Critical Sliding Surface of Homogeneous Ground Embankment
ZHAO Sihan1,2, LIU Xin1,3, SHAN Hao1,2, HONG Baoning1,2
1. Key Lab. of Ministry of Education for Geomechanics and Embankment Eng., Hohai Univ., Nanjing 210098, China;
2. Geotechnical Research Inst., Hohai Univ., Nanjing 210098, China;
3. Inst. of Tunnel and Underground Eng., Hohai Univ., Nanjing 210098, China
Abstract: Around the stability problem of homogeneous ground embankment, a new method for non-slice calculation of safety factor and the determination of the critical sliding surface of embankment slope is proposed. The range of three dimensional independent variables which determine the position of circular sliding surface is given. By taking the landslide body as the research object and using it as the integral domain, the non-slice integral limit equilibrium analysis is carried out, and the function expression of the safety factor is deduced. Based on the extreme condition of binary function, the critical sliding surface position can be directly determined without searching by combining finding the minimum value by lowering the dimension with the iterative calculation. The calculation program is compiled and the proposed method and the slice method are compared by two examples of homogeneous soil slope and embankment slope, and the results show that the safety factor of the same sliding surface is 0.68% smaller than that of M-P method; the deviation of the critical sliding surface position obtained by this method and slice method which strictly satisfying the limit equilibrium condition + traditional search method is small; the minimum safety factor of the embankment slope corresponding to the critical sliding surface is 0.87% higher than that of the M-P method + traditional search method; when the limit equilibrium condition is strictly met, the safety factor is not sensitive to the distribution of normal stress at the bottom of slip surface.
Key words: homogeneous ground embankment    safety factor    non-slice method    limit equilibrium    critical sliding surface

1 滑动面形状和位置的描述 1.1 自变量的选择和滑动面函数的确定

 图1 自变量选取示意图 Fig. 1 Schematic of the selection of independent variables

 ${f_1}:y = kx + k\left( {l - a} \right)$ (1)
 ${f_2}:{x^2} + {\left( {y - a\cot \;\alpha } \right)^2} = \frac{{{a^2}}}{{{{\sin }^2}\alpha }}$ (2)
1.2 自变量各维度取值范围的确定 1.2.1 $l$ 的取值范围

 $0 < l < \frac{L}{2}$ (3)
1.2.2 $a$ 的取值范围

 $\frac{l}{2} < a < l$ (4)
1.2.3 $\alpha$ 的取值范围

1） $\alpha$ 的最小值。

 图2 ${{\alpha}}$ 最小值示意图 Fig. 2 Schematic of the minimum value of ${{\alpha}}$

 ${a^2} +y_{O'\alpha _{\min}}^2 = x_{D\alpha _{\min }}^2 + {\left( {{y_{O'{\alpha _{\min }}}} - H} \right)^2},$

$\alpha$ 的最小值为：

 ${\alpha _{\min }} = \arctan \frac{a}{{{y_{O'{\alpha _{\min }}}}}} = \arctan \frac{{2aH}}{{{{\left( L/2 - l + a \right)}^2} + {H^2} - {a^2}}}$ (5)

2） $\alpha$ 的最大值应根据 $C$ 点的位置分两段考虑。

 ${\alpha _{\max }} = \arctan \;\frac{a}{H},l > \frac{H}{k}$ (6)
 图3 ${{\alpha}}$ 最大值示意图 $({ l} > { H}/{ k})$ Fig. 3 Schematic of the maximum value of ${{\alpha}}({ l} > { H}/{ k})$

 图4 ${\alpha}$ 最大值示意图 $({ l} < { H}/{ k})$ Fig. 4 Schematic diagram of the maximum value of α(l< ${ H}/{ k})$

 ${a^2} + y_{O'\alpha _{\max}}^2 = x_{D\alpha _{\max }}^2 + {\left( {{y_{O'{\alpha _{\max }}}} - H} \right)^2},$

 ${\alpha _{\max }} \!=\! \arctan \frac{a}{{{y_{O'{\alpha _{\max }}}}}} \!=\! \arctan \frac{{2aH}}{{{{\left( {{H/k} \!-\! l \!+\! a} \right)}^2} \!+\! {H^2} \!-\! {a^2}}},l < \frac{H}{k}$ (7)
2 无条分稳定安全系数公式

 图5 潜在滑动体受力分析示意图 Fig. 5 Mechanical analysis of landslide body

2.1 力矩平衡

 $\sum {M = } {M_\sigma } + {M_G}{\rm{ + }}{M_{\tau '}} = 0$ (8)

 ${M_G} = - \iint\limits_{{S_{CDEF}}} {x{\gamma _1}{\rm d}x{\rm d}y} = - {\gamma _1}\int\limits_0^H {{\rm d}y\int\limits_{{f_1}\left( y \right)}^{{f_2}\left( y \right)} {x{\rm d}x} }$ (9)

 ${M_{\tau '}} = \left( {2\alpha + \theta } \right){R^2}\tau$ (10)

$\theta$ 为半径 $O'C$ $O'D$ 的夹角，将 $y = H$ 代入式（2），可得 $D$ 点的横坐标为：

 ${X_D} = \sqrt {\frac{{{a^2}}}{{{{\sin }^2}\alpha }} - {{\left( {H - a\cot \;\alpha } \right)}^2}} {\rm{ = }}\sqrt {{a^2} - {H^2} + 2Ha\cot \;\alpha }$ (11)

 $\theta = \arcsin \frac{{\sqrt {\left( {{a^2} - {H^2}} \right){{\sin }^2}\alpha + Ha\sin\left( 2\alpha \right)} }}{a} - \alpha$ (12)

 $\tau {\rm{ = }}\tau '{\rm{ = }}\frac{{{\gamma _1}{{\sin }^2}\alpha }}{{2{a^2}}}\frac{{ - \displaystyle\frac{{1 + {k^2}}}{{3{k^2}}}{H^3} + \left( {a\cot \;\alpha - \frac{{a - l}}{k}} \right){H^2} + \left( {2a - l} \right)lH}}{{\left( {\alpha + \arcsin \displaystyle\frac{{\sqrt {\left( {{a^2} - {H^2}} \right){{\sin }^2}\alpha + Ha\sin \left(2\alpha \right)} }}{a}} \right)}}$ (13)
2.2 水平方向的静力平衡

 $\int_{ - \alpha }^{\alpha + \theta } {\tau 'R\cos\; \eta {\rm d}\eta } - \int_\alpha ^{\alpha + \theta } {{\sigma _1}R\sin \;\eta {\rm d}\eta } {\rm{ = }}0$ (14)

 ${\sigma _1}{\rm{ = }}\frac{{\sin\; \alpha + \sin \left( {\alpha + \theta } \right)}}{{\cos\; \alpha - \cos \left( {\alpha + \theta } \right)}}\tau '$ (15)

 ${\sigma _1}{\rm{ = }}\frac{{\displaystyle\frac{a}{R} + \displaystyle\frac{{{X_D}}}{R}}}{{\cos \;\alpha - \displaystyle\frac{{R\cos\; \alpha - H}}{R}}}\tau ' = \frac{{a + \sqrt {{a^2} - {H^2} + 2Ha\cot \;\alpha } }}{H}\tau '$ (16)
2.3 竖直方向的静力平衡

 $\sum {{F_Y}} = {F_{\tau 'Y}} + {F_{{\sigma _1}Y}} + {F_{{\sigma _2}Y}} + {F_{GY}} = 0$ (17)

 ${F_{\tau 'Y}} = \int_{ - \alpha }^{\alpha + \theta } {\tau 'R\sin\; \eta {\rm d}\eta } = \tau 'R\left( {\cos \;\alpha - \cos \left( {\alpha + \theta } \right)} \right)$ (18)

${F_{{\sigma _1}Y}}$ 为滑动面 $\overset\frown{CD}$ 上法向力在 $Y$ 方向的分力，在滑动面 $\overset\frown{CD}$ 上任取一角度为 ${\rm d}\eta$ 的微圆弧，在滑动面 $\overset\frown{CD}$ 范围内进行积分，得：

 ${F_{{\sigma _1}Y}} = \int_\alpha ^{\alpha + \theta } {{\sigma _1}R\cos\; \eta {\rm d}\eta } = {\sigma _1}R\left( {\sin \left( {\alpha + \theta } \right) - \sin \;\alpha } \right)$ (19)

${F_{{\sigma _2}Y}}$ 为滑动面 $\overset\frown{ABC}$ 上法向力在 $Y$ 方向的分力，在滑动面 $\overset\frown{ABC}$ 上任取一角度为 ${\rm d}\eta$ 的微圆弧，在滑动面 $\overset\frown{ABC}$ 范围内进行积分，得：

 ${F_{{\sigma _2}Y}} = \int_{ - \alpha }^\alpha {{\sigma _2}R\cos \;\eta {\rm d}\eta } = 2{\sigma _2}R\sin \;\alpha$ (20)

${F_{GY}}$ 为潜在滑动体重力在 $Y$ 方向的分力，如图5所示，潜在滑动体的重力由 $ABC$ $CDEF$ 两部分土体的重力组成，即

 ${F_{GY}}{\rm{ = }}{\gamma _2}{S_2} + {\gamma _1}{S_1}$ (21)

$ABC$ 部分土体的面积 ${S_2}$ 可由扇形 $O'ABC$ 与三角形 $O'AC$ 的面积之差得到：

 \begin{aligned}[b]& {S_2} = {S_{O'ABC}} - {S_{O'AC}} = \frac{1}{2} \times 2\alpha \times {R^2} - \\& \quad\quad \frac{1}{2} \times 2a \times a\cot \;\alpha = {a^2}\left( {\frac{\alpha }{{{{\sin }^2}\alpha }} - \cot \;\alpha } \right)\end{aligned} (22)

 \begin{aligned} {S\!_1}\!\! =\!\!\!\! & \iint\limits_{{S_{CDEF}}} \!\!\!{{\rm d}x{\rm d}y}\!\! =\!\!\!\!\int\limits_0^H \!\!\!{{\rm d}y\!\!\!\int\limits_{{f_{1\left( y \right)}}}^{{f_{2\left( y \right)}}}\!\!\!{{\rm d}x} } \!\!=\!\!\!\!\int\limits_0^H \!\!{\left(\!\!\! {\sqrt {\frac{{{a^2}}}{{{{\sin }^2}\alpha }}\!\!-\!\! V{{\left( {y\!\! -\!\! a\cot \;\alpha } \right)}^2}}\!\!-\! \frac{y}{k}}\right.} \!- \\ & \Bigg.{a + l} \Bigg)\cdot {\rm d}y=- \frac{1}{{2k}}{H^2}\! +\! \left( {l \!-\! a \!+\! \frac{1}{2}\sqrt {{a^2} \!-\! {H^2}\! +\! 2aH\cot \;\alpha } } \right)H + \\ & \frac{{{a^2}}}{{2{{\sin }^2}\alpha }}\left( {\arctan \left( {\cot \;\alpha } \right) - \arctan \frac{{a\cot \;\alpha - H}}{{\sqrt {{a^2} \!-\! {H^2} \!+\! 2aH\cot \;\alpha } }}} \right)+\\ & \frac{1}{2}a\cot \;\alpha \left( {a - \sqrt {{a^2} - {H^2} + 2aH\cot \;\alpha } } \right)\end{aligned} (23)

 ${\sigma _2}{\rm{ = }}\frac{{{\gamma _1}{S_1} + {\gamma _2}{S_2}}}{{2a}} - \tau '\cot\; \alpha$ (24)
2.4 安全系数公式

${c_1}$ ${\varphi _1}$ ${c_2}$ ${\varphi _2}$ 分别为路堤填土和地基土的黏聚力和内摩擦角，由式（16）和（24），得滑动面 $\overset\frown{CD}$ 和滑动面 $\overset\frown{ABC}$ 上的平均抗剪强度 ${\tau _{\rm f1}}$ ${\tau _{\rm f2}}$ 为：

 \left\{ \begin{aligned}& {\tau _{\rm f1}} = {c_1} + \frac{{a + \sqrt {{a^2} - {H^2} + 2Ha\cot \;\alpha } }}{H}\tau '\tan \;{\varphi _1},\\& {\tau _{\rm f2}}{\rm{ = }}{c_2} + \left( {\frac{{{\gamma _1}{S_1} + {\gamma _2}{S_2}}}{{2a}} - \tau '\cot \;\alpha } \right)\tan\; {\varphi _2}\end{aligned} \right. (25)

 ${\tau _{\rm f}} = \frac{{\theta R{\tau _{\rm f}}_1 + 2\alpha R{\tau _{\rm f}}_2}}{{\left( {2\alpha + \theta } \right)R}} = \frac{{\theta {\tau _{\rm f}}_1 + 2\alpha {\tau _{\rm f}}_2}}{{2\alpha + \theta }}$ (26)

 \begin{aligned} F\!\! =\!\! & \frac{{{\tau _{\rm f}}}}{\tau }\!\! =\!\! \frac{\theta }{{\left( {2\alpha \!\!+\!\! \theta } \right)\tau }}\!\left(\!{{c_1} \!\!+\!\! \frac{{\tau '\tan\; {\varphi _1}}}{H} \!\left(\! {a \!\!+\!\! \sqrt {{a^2} - {H^2} + 2Ha\cot \;\alpha } } \right)}\!\!\right)\!+ \\ & \quad\quad \frac{{2\alpha }}{{\left( {2\alpha + \theta } \right)\tau }}\left( {{c_2} + \left( {\frac{{{\gamma _1}{S_1} + {\gamma _2}{S_2}}}{{2a}} - \tau '\cot \;\alpha } \right)\tan \;{\varphi _2}} \right)\end{aligned} (27)

3 临界滑动面的确定 3.1 基本思路

3.2 降维度求极值

$l$ $0 < l < L/2$ 中某一值时，式（27）中仅有 $a$ $\alpha$ 两个未知量，因式（27）为比值的形式，对其求偏导较复杂，在此引入目标函数 $G$

 $G\left( {a,\alpha } \right) = {\tau _{\rm f}} - \tau = \frac{{\theta {\tau _{\rm f}}_1 + 2\alpha {\tau _{\rm f}}_2}}{{2\alpha + \theta }} - \tau$ (28)

$F$ 的极小值转化为求 $G$ 的极小值，由二元函数的极值条件， $G$ 取得极小值时， $a$ $\alpha$ 必须满足式方程组（29）。

$\left( {a,\alpha } \right)$ 的各组数值解中，令函数 $G$ 取得最小值的解，也即令函数 $F$ 取得最小值的解，该解所确定的滑动面即为该 $l$ 值下的临界滑动面（过点 $C$ 的临界滑动面），对应安全系数即为该 $l$ 值下的安全系数 ${F_{SC}}$

 \left\{ \begin{aligned} & \frac{{\partial G}}{{\partial a}} = \frac{{\left( {{\tau _{\rm f}}_1\displaystyle\frac{{\partial \theta }}{{\partial a}} + \theta \displaystyle\frac{{\partial {\tau _{\rm f}}_1}}{{\partial a}} + 2\alpha \displaystyle\frac{{\partial {\tau _{\rm f}}_2}}{{\partial a}}} \right)\left( {2\alpha + \theta } \right) - \left( {\theta {\tau _{\rm f}}_1 + 2\alpha {\tau _{\rm f}}_2} \right)\displaystyle\frac{{\partial \theta }}{{\partial a}}}}{{{{\left( {2\alpha + \theta } \right)}^2}}} - \displaystyle\frac{{\partial \tau }}{{\partial a}} = 0 , \\ & \frac{{\partial G}}{{\partial \alpha }} = \displaystyle\frac{{\left( {{\tau _{\rm f}}_1\displaystyle\frac{{\partial \theta }}{{\partial \alpha }} + \theta \frac{{\partial {\tau _{\rm f}}_1}}{{\partial \alpha }} + 2{\tau _{\rm f}}_2 + 2\alpha \displaystyle\frac{{\partial {\tau _{\rm f}}_2}}{{\partial \alpha }}} \right)\left( {2\alpha + \theta } \right) - \left( {\theta {\tau _{\rm f}}_1 + 2\alpha {\tau _{\rm f}}_2} \right)\left( {2 + \displaystyle\frac{{\partial \theta }}{{\partial \alpha }}} \right)}}{{{{\left( {2\alpha + \theta } \right)}^2}}} - \displaystyle\frac{{\partial \tau }}{{\partial \alpha }} = 0 \end{aligned} \right. (29)
 \left\{ \begin{aligned}& \frac{{\partial \tau }}{{\partial a}} = \frac{{\gamma H\sin \;\alpha }}{{6{a^3}{k^2}{{(2\alpha + \theta )}^2}}}\left( \Bigg({\left( {3Hk(a - l) + 3{k^2}l(l - 2a) + {H^2}\left( {{k^2} + 1} \right)} \right)\sin\; \alpha - 3aH{k^2}\cos \;\alpha } \right)a\frac{{\partial \theta }}{{\partial a}}+\Bigg. \\& \quad\quad \Bigg. (2\alpha + \theta )\left( {\left( {3Hk(a - 2l) + 6{k^2}l(l - a) + 2{H^2}\left( {{k^2} + 1} \right)} \right)\sin \;\alpha - 3aH{k^2}\cos \;\alpha } \right)\Bigg) , \\& \frac{{\partial \tau }}{{\partial \alpha }} = \frac{{\gamma H}}{{6{a^2}{k^2}{{(2\alpha + \theta )}^2}}}\Bigg( \left( {\left( {3Hk(a - l) + 3{k^2}l(l - 2a) + {H^2}\left( {{k^2} + 1} \right)} \right)\sin \;\alpha - 3aH{k^2}\cos \;\alpha } \right)\sin \;\alpha \left(\frac{{\partial \theta }}{{\partial \alpha }} + 2\right)-\Bigg.\\& \quad\quad \Bigg. (2\alpha + \theta )\left( {\left( {3Hk(a - l) + 3{k^2}l(l - 2a) + {H^2}\left( {{k^2} + 1} \right)} \right)\sin \left(2\alpha\right) - 3aH{k^2}\cos \left(2\alpha \right)} \right)\Bigg) \end{aligned} \right. (30)
 \left\{ \begin{aligned}& \frac{{\partial {\tau _{\rm f1}}}}{{\partial a}} = \frac{{\tan \;{\phi _1}}}{H}\left( {\left( {\sqrt {{a^2} - {H^2} + 2aH\cot \;\alpha } + a} \right)\frac{{\partial \tau }}{{\partial a}} + \tau \left( {\frac{{a + H\cot \;\alpha }}{{\sqrt {{a^2} - {H^2} + 2aH\cot \;\alpha } }} + 1} \right)} \right) , \\& \frac{{\partial {\tau _{\rm f1}}}}{{\partial \alpha }} = \tan\; {\phi _1}\left( {\frac{{\left( {\sqrt {{a^2} - {H^2} + 2aH\cot\; \alpha } + a} \right)}}{H}\frac{{\partial \tau }}{{\partial \alpha }} - \frac{{\tau a{{\csc }^2}\alpha }}{{\sqrt {{a^2} - {H^2} + 2aH\cot\; \alpha } }}} \right) \end{aligned} \right. (31)
 \left\{ \begin{aligned}& \frac{{\partial {\tau _{\rm f2}}}}{{\partial a}} = \frac{{\tan\; {\phi _2}}}{{2{a^2}}}\left( {a\gamma \frac{{\partial {S_1}}}{{\partial a}} - 2{a^2}\cot \;\alpha \frac{{\partial \tau }}{{\partial a}} - \gamma \left( {{a^2}\left( {\cot\; \alpha - \alpha {{\csc }^2}\alpha } \right) + {S_1}} \right)} \right) , \\& \frac{{\partial {\tau _{\rm f2}}}}{{\partial \alpha }} = \frac{{\tan \;{\phi _2}}}{{2a}}\left( {\gamma \frac{{\partial {S_1}}}{{\partial \alpha }} - 2a\cot\; \alpha \frac{{\partial \tau }}{{\partial \alpha }} + 2a{{\csc }^2}\alpha (a\gamma - a\gamma \alpha \cot\; \alpha + \tau )} \right) \end{aligned} \right. (32)
 \left\{ \begin{aligned} & \frac{{\partial {S_1}}}{{\partial a}} = \left( {a - \sqrt {{a^2} - {H^2} + 2aH\cot \;\alpha } } \right)\cot \;\alpha + \left( {\arctan \frac{{H - a\cot\; \alpha }}{{\sqrt {{a^2} - {H^2} + 2aH\cot\; \alpha } }} + \arctan \left( {\cot \;\alpha } \right)} \right)a{\csc ^2}\alpha - H , \\& \frac{{\partial {S_1}}}{{\partial \alpha }} = - a{\csc ^2}\alpha \left( {\left( {\arctan \frac{{H - a\cot\; \alpha }}{{\sqrt {{a^2} - {H^2} + 2aH\cot\; \alpha } }} + \arctan \left( {\cot \;\alpha } \right)} \right)a\cot \;\alpha + a - \sqrt {{a^2} - {H^2} + 2aH\cot \;\alpha } } \right) \end{aligned} \right. (33)
 \left\{ \begin{aligned}& \frac{{\partial \theta }}{{\partial a}} = \frac{{H\sin \;\alpha \left( {H\sin \;\alpha - a\cos\; \alpha } \right)}}{{a\sqrt {{{\left( {a\cos \alpha - H\sin \alpha } \right)}^2}} \sqrt {\left( {{a^2} - {H^2}} \right){{\sin }^2}\alpha + aH\sin \left(2\alpha \right)} }} , \\& \frac{{\partial \theta }}{{\partial \alpha }}\!\! =\!\! - 1 \!\!+\!\! \frac{{aH\cos \left(2\alpha \right)\!\!+\!\! \left( {{a^2}\!\! - \!\!{H^2}} \right)\cos\; \alpha \sin\; \alpha }}{{\sqrt {{{\left( {a\cos \;\alpha \!\!-\!\! H\sin\; \alpha } \right)}^2}} \sqrt {\left( {{a^2} \!\!-\!\! {H^2}} \right){{\sin }^2}\alpha \!\!+\!\! aH\sin \left(2\alpha \right)} }} \end{aligned} \right. (34)
3.3 迭代计算

4 算例验证 4.1 安全系数公式 $F$ 的正确性验证

4.2 临界滑动面确定的准确性验证

 图6 Geo-Slope计算模型 Fig. 6 Calculation model of Geo-Slope

 图7 各方法的临界滑动面位置示意图 Fig. 7 Critical sliding surface position of each method

5 结　论

1）以滑弧与坡脚内地平线的交点与坡脚的距离、该交点与圆心在地平线投影点的距离、该交点和圆心投影点所在半径的夹角，共3个维度的自变量可唯一确定圆弧形滑动面的位置，其取值范围不需经验假定，可实现临界滑动面确定范围的全局性。

2）严格满足极限平衡条件的情况下，安全系数对滑面底部正应力的分布并不敏感，对正应力进行合理的处理或假设，以潜在滑动体为研究对象进行整体极限平衡分析，可得出合理的安全系数值。

3）基于二元函数的极值条件，对积分形式的平衡方程解出的安全系数函数表达式，通过降维度求极小值与迭代计算结合，可直接确定临界滑动面位置，克服了传统搜索方法易陷入局部极小值的缺点。

4）算例结果表明：同一滑动面的安全系数，本文方法比M-P法小0.68%；本文方法确定的临界滑动面位置与严格满足极限平衡条件的条分法+传统搜索方法所确定的临界滑动面位置偏差较小；临界滑动面对应的路堤边坡最小安全系数比M-P条分法+传统搜索方法所得结果大0.87%。

 [1] Chen Zuyu,Shao Changming. The use of the method of optimization for minimizing safety factors in slope stability analysis[J]. Chinese Journal of Geotechnical Engineering, 1988, 10(4): 1-13. [陈祖煜,邵长明. 最优化方法在确定边坡最小安全系数方面的应用[J]. 岩土工程学报, 1988, 10(4): 1-13. DOI:10.3321/j.issn:1000-4548.1988.04.001] [2] Wang Chenghua,Xia Xuyong. State-of-the-art:Methods for searching critical slip surface in slope stability analysis[J]. Building Science Research of Sichuan, 2002, 28(3): 34-39. [王成华,夏绪勇. 边坡稳定分析中的临界滑动面搜索方法述评[J]. 四川建筑科学研究, 2002, 28(3): 34-39. DOI:10.3969/j.issn.1008-1933.2002.03.014] [3] Baker R,Garber M.Variational approach to slope stability[C]//Proceedings of the 9th International Conference on Soil Mechanics and Foundation Engineering.Tokyo:The Japanese Society of Soil Mechanics and Foundation Engineering,1977,2:9–12. [4] Revilla J,Castillo E. The calculus of variations applied to stability of slopes[J]. Geotechnique, 1977, 27(1): 1-11. DOI:10.1680/geot.1977.27.1.1 [5] Ramamurthy T,Narayan C G P,Bhatkar V P.Variational method for slope stability analysis[C]//Proceedings of the 9th International Conference on Soil Mechanics and Foundation Engineering.Tokyo:The Japanese Society of Soil Mechanics and Foundation Engineering,1977:139–142. [6] De Josselin,De Jong G. Application of the calculus of variations to the vertical cut off cohesive frictionless soil[J]. Geotechnique, 1980, 30(1): 1-16. DOI:10.1680/geot.1980.30.1.1 [7] Luceno A,Castillo E.Evaluation of variational methods in slope analysis[C]//International Symposium on Landslides.New-Delhi:Sarita Prakashan,1980:255–258. [8] Luo Wenqiang,Zhang Zhuoyuan,Huang Runqiu,et al. Model of calculus of variation used for determination of sliding surface[J]. Journal of Yangtze River Scientific Research Institute, 2000, 17(3): 35-37. [罗文强,张倬元,黄润秋,等. 滑动面确定的变分法模型[J]. 长江科学院院报, 2000, 17(3): 35-37. DOI:10.3969/j.issn.1001-5485.2000.03.009] [9] 杨庚宇. 土坡稳定分析中条分法的解析计算[J]. 力学与实践, 1995, 17(2): 59-61. [10] 杨庚宇,赵少飞.土坡稳定分析圆弧滑动法的解析解[J].工程力学,1998(增刊):440–444. [11] Jiang Binsong,Lyu Aizhong,Cai Meifeng. Analysis of stability for cohesive soil slopes[J]. Engineering Mechanics, 2003, 20(5): 204-208. [蒋斌松,吕爱钟,蔡美峰. 纯粘土边坡稳定性的解析计算[J]. 工程力学, 2003, 20(5): 204-208. DOI:10.3969/j.issn.1000-4750.2003.05.039] [12] 卢廷浩.土力学[M].南京:河海大学出版社,2002. [13] 殷宗泽.土工原理[M].北京:中国水利水电出版社,2007. [14] Cao Ping,Zhang Ke,Wang Yixian,et al. Mixed search algorithm of critical slip surface of complex slope[J]. Chinese Journal of Rock Mechanics and Engineering, 2010, 29(4): 814-821. [曹平,张科,汪亦显,等. 复杂边坡滑动面确定的联合搜索法[J]. 岩石力学与工程学报, 2010, 29(4): 814-821.] [15] Zou Guangdian. A global optimization method of the slice method for slope stability analysis[J]. Chinese Journal of Geotechnical Engineering, 2002, 24(3): 309-312. [邹广电. 边坡稳定分析条分法的一个全局优化算法[J]. 岩土工程学报, 2002, 24(3): 309-312. DOI:10.3321/j.issn:1000-4548.2002.03.009]