1. 第七章 边坡稳定 Slop Stability
7.1 Introduction (前言)
Slopes are classified into natural slopes (e.g. hill slopes and river banks) as well as man-made slopes (e.g. earth dams and embankments). (边坡分为天然边坡和人工边坡)
Ensuring stability of slope is very important because slope failures usually result in much death and destruction. (边坡的稳定性十分重要因为滑坡导致人命与经济的损失)

2. Las Colinas landslide, El Salvador
(2001)
Alpine debris flow (2000)

3. Major slope-instability related catastrophes of the 20th century
Kansu (China) Loess flow 200,000 killed
Gros Ventre (Wyoming) 1925 Rockslide ~40 killed
Madison (Montana) Rockslide >100 killed
Vaiont (Italy) Rockslide 2,600 killed
Aberfan (Wales) Debris-slide 144 killed
Huascaran (Peru) Complex 25,000 killed
Nevado del Ruiz (Colombia) 1985 Debris flow 23,000 killed
Casitas (Nicaragua) Debris flow +2,000 killed
Venezuela Complex +20,000 killed
Swiss/Italian Alps Debris flow 38 killed

4. 7.2 Types of slope failures (滑坡的类型)
(i) Translational slides
failure of a slope along a weak zone of soil (沿软弱层滑动)
failure surface tends to be plane and roughly parallel to the slope (滑动面为平面与坡面平行 )
Sliding mass (滑体)
Sliding surface or weak zone (滑动面或软弱层)
Detached landslide deposit

5. (ii) Rotational slides
Sliding surface is an arc which is common in cohesive soils
(粘性土的滑动面为一曲面)
Sliding surface (滑动面)

6. 7.3 Methods of slope stability analysis (边坡稳定分析方法)
The stability of a slope can be analysed using one or more of the following methods:
Limit equilibrium method - equilibrium of forces (极限平衡法)
Limit analysis based on plasticity - equilibrium of stresses (极限应力分析法)
Finite difference method (有限差分法)
Finite element method (有限元法)

7. 1. Limit equilibrium method (极限平衡法)
In the limit equilibrium method (LEM), the soil is considered to be on the verge of failure along an assumed or a known sliding surface. (在某一假设滑动面上,整个块体处于极限平衡状态)
In general, the sliding surface is assumed to be a circular arc for clays or a logarithmic spiral for sands and gravels.
Factor of safety is defined as the ratio of shear strength over shear stress acting on the soil mass at the assumed slip plane in order to maintain a condition of limiting equilibrium. (安全糸数定义为在极限平衡条件下抗剪强度与下滑力之比)

8. For the slope stability analysis using the LEM, two analyses are considered for each slope:
Effective Stress Analysis (c´ and ´) that represents drained or long-term behaviour of the slope (有效应力分析对应边坡长期的稳定)
Total Stress Analysis (cu) that represents undrained or short-term behaviour of the slope (总应力分析对应边坡短期的稳定)

9. 2. Slop in cohesionless soil 无粘土边坡
1）. General case 常规情况
For an infinite slope, the depth of the failure surface is small compared with the length of the slope (滑动面深度比边坡长度小)
Assumptions (假定):
no ground water table (没有地下水)
homogeneous cohesionless soil (均质无粘性土)
failure surface is parallel to the surface of the slope (滑动面与坡面平行)
The slope is inclined at angle  to the horizontal and the depth of failure plane is z (坡角为, 滑动面深度为z) z b

10. Consider a slice of soil element of width b (土单元体的宽度为b)
Assume the side forces on the soil element can be neglected in the stability analysis (假设不考虑该单元体两侧应力对稳定性的影响)
The force diagram acting on the element is shown as follows: z b W N T
W is weight of soil element (土单元体的重量), T is shear force on the slip plane (滑动面上的剪切力) and N is normal force on the slip plane (滑动面上的法向支撑力)

11. Force equilibrium parallel to the slope surface (顺坡方向静力平衡)z b W N T
Force equilibrium normal to the slope surface (坡面法线方向静力平衡)
Shear strength along the slip plane (滑动面上的抗剪力)
where  is angle of internal friction (内摩擦角)

12. Factor of safety is defined as the ratio of the shear strength over the shear stress: (安全糸数定义为抗剪力与下滑剪切力之比)
For homogeneous cohesionless soil slope, the slope is stable if the slope angle is smaller than the angle of internal friction of the soil (对于均质无粘性土土坡,只要坡角小于土的内摩擦角, 土体就是稳定)

13. 2）. With Seepage 有渗流情况
If ground water table exists and coincides with the surface of the slope (地下水位与坡面相同)
Force equilibrium parallel to the slope surface (顺坡方向静力平衡) z b W N’ T U
Force equilibrium normal to the slope surface (坡面法线方向静力平衡)
Shear strength along the slip plane (滑动面上的抗剪力)

14. Factor of safety (FS) is expressed as:z b W N’ T U
Weight W is expressed as:
Hydrostatic force U is expressed as:

15. Factor of safety (FS) is expressed as:
The factor of safety is reduced by a factor ’/sat if the water table rises to the slope surface.

17. 7.4. Slope in cohesive soil -Total stress analysis, (粘土边坡的总应力分析)
This total stress analysis covers the case of a fully saturated clay under undrained condition, or for the condition immediately after construction. (总应力分析合适用于饱和粘土在不排水条件下或短期的稳定分析)
Only moment equilibrium is considered in the analysis. (满足力矩平衡条件)
In section, the potential failure surface is assumed to be a circular arc. (滑动面为弧形)

18. A trial failure surface: center O, radius r and length L is shown below. (下图为一假设滑动面:圆心为O,半径为r,弧长为L)

19. Consider the moment at point O, the distributing moment of W is expressed as
where d is the moment arm of W from point O (d是W对滑弧圆心的力臂)
The resisting moment of Tf at point O is expressed as:
where cu is undrained shear strength of the soil, L is length of the circular arc and r is radius of the circular arc

20. Factor of safety (Fs) is given by:
If tension crack exists at the crest of the slope, how to evaluate the factor of safety for the slope? (当裂缝在坡顶出现,怎样计算边坡的安全糸数?)

21. Depth of tension crack is 2cu/
If tension crack exists at the crest of the slope, the arc length L will be shortened. [当裂缝在坡顶出现,滑弧长度便会减小]
Tension crack
Depth of tension crack is 2cu/

22. If the crack is filled with water, a hydrostatic force will act normal to the crack. The additional moment of this hydrostatic force must be added to equation (7.11) for calculating the factor of safety of the slope. (当裂缝积水,计算安全糸数时公式(7.11)中必需考虑静水压力对滑弧圆心O的力矩) h
Hydrostatic force U W d L’

23. 7.5 Swedish Method of slices (瑞典条分法)
For   0, method of slices can be used where the soil mass above the assumed slip surface is divided into a series of vertical slices (当  0时,滑动块体分成多个垂直的土条)
The factor of safety (Fs) is defined as N’ W T

24. The forces acting on a vertical slice of soil mass are shown as follows: (某一个土条上力的分布如下)

25. Total weight of the slice (土条重量) W = bh
Effective normal force on the base (有效法向力) N’ = ’l
Hydrostatic force on the base (静水力) U = ul where u is pore-water pressure at the center of the base
Shear force on the base (剪切力) T
Total normal forces on the sides (法向条间力) E1 and E2
Shear forces on the sides (切向条间力) X1 and X2

26. For n slices, the number of unknowns are
N’: n
T: n
E: n-1
X: n-1
Point of application of E: n-1
Point of application of N: n
Factor of safety: 1
Total no. of unknown 6n-2
For n slices, number of equations are 4n
If N’ is assumed acting in the middle of the base of the slice, we need a further n–2 assumptions for the problem to be statically determinate

28. It is assumed that the resultant of the interslice forces is zero and the slip surface is a circular arc. (滑动面为圆弧和不考虑间条力 )
Consider equilibrium of forces normal to the base of the slice (土条底面法向力的平衡 ) b W h T a N’ l
The factor of safety (Fs) is expressed as ul

29. 7.6 Bishop solution (毕肖普条分法)
It is assumed that the resultant forces on the sides of the slices are horizontal and the slip surface is a circular arc. (滑动面为圆弧和不考虑切向间条力 )
Fs is defined as follows: (安全糸数定义抗切强度与剪切力的比) h W b l a N T ER EL ul
Consider force equilibrium in the vertical direction (垂直方向力的平衡)

30. Substitute l = b sec  and put N’ to the left hand side of the expression
The factor of safety (Fs) is expressed as