Stability of Slopes(边坡稳定)

CHAPTER 9

Stability of Slopes

9.1 Introduction

Gravitational and seepage forces tend to cause instability in natural slopes, in slopes formed by excavation and in the slopes of embankments and earth dams. The most important types of slope failure are illustrated in Fig.9.1.In rotational slips the shape of the failure surface in section may be a circular arc or a non-circular curve．In general，circular slips are associated with homogeneous soil conditions and non-circular slips with non-homogeneous conditions．Translational and compound slips occur where the form of the failure

surface

is

influenced

by

the presence of an adjacent stratum of significantlydifferent strength．

Translational slips tend to occur where the adjacent stratum is at a relatively shallow depth below the surface of the slope:the failure surface tends to be plane and roughly parallel to the slope.Compound slips usually occur where the adjacent stratum is at greater depth，the failure surface consisting of curved and plane sections．

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In practice, limiting equilibrium methods are used in the analysis of slope stability. It is considered that failure is on the point of occurring along an assumed or a known failure surface．The shear strength required to maintain a condition of limiting equilibrium is compared with the available shear strength of the soil，giving the average factor of safety along the failure surface．The problem is considered in two dimensions，conditions of plane strain being assumed．It has been shown that a two-dimensional analysis gives a conservative result for a failure on a three-dimensional(dish-shaped) surface．

9.2 Analysis for the Case of φu =0

This analysis, in terms of total stress，covers the case of a fully saturated clay under undrained conditions, i.e. For the condition immediately after construction．Only moment equilibrium is considered in the analysis．In section, the potential failure surface is assumed to be a circular arc. A trial failure surface(centre O，radius r and length La)is shown in Fig.9.2. Potential instability is due to the total weight of the soil mass(W per unit Length) above the failure surface．For equilibrium the shear strength which must be mobilized along the

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failure surface is expressed as

where F is the factor of safety with respect to shear strength．Equating moments about O：

Therefore

(9.1)

The moments of any additional forces must be taken into account．In the event of a tension crack developing ，as shown in Fig.9.2，the arc length La is shortened and a hydrostatic force will act normal to the crack if the crack fills with water．It is necessary to analyze the slope for a number of trial failure surfaces in order that the minimum factor of safety can be determined．

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Based on the principle of geometric similarity，Taylor[9.9]published stability coefficients for the analysis of homogeneous slopes in terms of total stress．For a slope of height H the stability coefficient (Ns) for the failure surface along which the factor of safety is a minimum is

(9.2)

For the case ofφu =0，values of Ns can be obtained from Fig.9.3.The coefficient Ns depends on the slope angleβand the depth factor D，where DH is the depth to a firm stratum．

Gibson and Morgenstern [9.3] published stability coefficients for slopes in normally consolidated clays in which the undrained strength cu(φu =0) varies linearly with depth．

Example 9.1

A 45°slope is excavated to a depth of 8 m in a deep layer of saturated clay of unit weight 19 kN／m3:the relevant shear strength parameters are cu =65 kN／m2 andφu =0．Determine the factor of safety for the trial failure surface specified in Fig.9.4.

In Fig.9.4, the cross-sectional area ABCD is 70 m2.

Weight of soil mass=70×19=1330kN／m

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The centroid of ABCD is 4.5 m from O．The angle AOC is 89.5°and radius OC is 12.1 m．The arc length ABC is calculated as 18.9m．The factor of safety is given by：

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This is the factor of safety for the trial failure surface selected and is not necessarily the minimum factor of safety．

The minimum factor of safety can be estimated by using Equation 9.2.

From Fig.9.3，β=45°and assuming that D is large，the value of Ns is 0.18.Then

9.3 The Method of Slices

In this method the potential failure surface，in section，is again assumed to be a circular arc with centre O and radius r．The soil mass (ABCD) above a trial failure surface (AC) is divided by vertical planes into a series of slices of width b, as shown in Fig.9.5.The base of each slice is assumed to be a straight line．For any slice the inclination of the base to the horizontal isαand the height, measured on the centre-1ine,is h. The factor of safety is defined as the ratio of the available shear strength(τf)to the shear strength(τm) which must be mobilized to maintain a condition of limiting equilibrium, i.e.

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The factor of safety is taken to be the same for each slice，implying that there must be mutual support between slices，i.e. forces must act between the slices．

The forces (per unit dimension normal to the section) acting on a slice are：

1.The total weight of the slice，W=γb h (γsat where appropriate)．

2.The total normal force on the base，N (equal to σl)．In general this

force has two components，the effective normal force N'(equal toσ'l ) and the boundary water force U(equal to ul )，where u is the pore water pressure at the centre of the base and l is the length of the base．

3.The shear force on the base，T=τml.

4.The total normal forces on the sides, E1 and E2.

5.The shear forces on the sides，X1 and X2.

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Any external forces must also be included in the analysis．

The problem is statically indeterminate and in order to obtain a solution assumptions must be made regarding the interslice forces E and X：the resulting solution for factor of safety is not exact．

Considering moments about O，the sum of the moments of the shear forces T on the failure arc AC must equal the moment of the weight of the soil mass ABCD．For any slice the lever arm of W is rsinα,

therefore

∑Tr=∑Wr sinα

Now,

For an analysis in terms of effective stress，

Or

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(9.3)

where La is the arc length AC．Equation 9.3 is exact but approximations are introduced in determining the forces N'．For a given failure arc the value of F will depend on the way in which the forces N' are estimated．

The Fellenius Solution

In this solution it is assumed that for each slice the resultant of the interslice forces is zero．The solution involves resolving the forces on each slice normal to the base，i.e.

N'=WCOSα-ul

Hence the factor of safety in terms of effective stress (Equation 9.3) is given by

(9.4)

The components WCOSαand Wsinαcan be determined graphically for each slice．Alternatively，the value of αcan be measured or calculated．Again，a series of trial failure surfaces must be chosen in order to obtain the minimum factor of safety．This solution underestimates the factor of safety：the error，compared with more accurate methods of analysis，is usually within the range 5-2%.

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For an analysis in terms of total stress the parameters Cu andφu are used and the value of u in Equation 9.4 is zero．If φu=0 ,the factor of safety is given by

(9.5)

As N’ does not appear in Equation 9.5 an exact value of F is obtained．

The Bishop Simplified Solution

In this solution it is assumed that the resultant forces on the sides of the

slices are horizontal，i.e.

Xl-X2=0

For equilibrium the shear force on the base of any slice is

Resolving forces in the vertical direction:

(9.6)

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It is convenient to substitute

l=b secα

From Equation 9.3，after some rearrangement，

(9.7)

The pore water pressure can be related to the total ‘fill pressure’ at any

point by means of the dimensionless pore pressure ratio，defined as

(9.8)

(γsat where appropriate)．For any slice,

Hence Equation 9.7 can be written:

(9.9)

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As the factor of safety occurs on both sides of Equation 9.9,a process of successive approximation must be used to obtain a solution but convergence is rapid．

Due to the repetitive nature of the calculations and the need to select an adequate number of trial failure surfaces，the method of slices is particularly suitable for solution by computer．More complex slope geometry and different soil strata can be introduced．

In most problems the value of the pore pressure ratio ru is not constant over the whole failure surface but，unless there are isolated regions of high pore pressure，an average value(weighted on an area basis) is normally used in design．Again，the factor of safety determined by this method is an underestimate but the error is unlikely to exceed 7％and in most cases is less than 2％．

Spencer [9.8] proposed a method of analysis in which the resultant Interslice forces are parallel and in which both force and moment equilibrium are satisfied．Spencer showed that the accuracy of the Bishop simplified method，in which only moment equilibrium is satisfied, is due to the insensitivity of the moment equation to the slope of the interslice forces．

Dimensionless stability coefficients for homogeneous slopes，based on Equation 9.9，have been published by Bishop and Morgenstern [9.2].It can be shown that for a given slope angle and given soil properties the factor of safety

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varies linearly with γu and can thus be expressed as

F=m-nγu (9.10)

where，m and n are the stability coefficients．The coefficients，m and n are

functions ofβ,φ’，the dimensionless number c'/γand the depth factor D.

Example 9.2

Using the Fellenius method of slices，determine the factor of safety，in terms of effective stress，of the slope shown in Fig.9.6 for the given failure surface．The unit weight of the soil，both above and below the water table，is 20 kN／m 3 and the relevant shear strength parameters are c’=10 kN/m2 andφ’=29°.

The factor of safety is given by Equation 9.4.The soil mass is divided into slices l.5 m wide. The weight (W) of each slice is given by

W=γbh=20×1.5×h=30h kN／m

The height h for each slice is set off below the centre of the base and the

normal and tangential components hcosαand hsinαrespectively are determined graphically，as shown in Fig.9.6.Then

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Wcosα=30h cosα

W sinα=30h sinα

The pore water pressure at the centre of the base of each slice is taken to beγwzw，where zw is the vertical distance of the centre point below the water table (as shown in figure)．This procedure slightly overestimates the pore water pressure which strictly should be) γwze，where ze is the vertical distance below the point of intersection of the water table and the equipotential through the centre of the slice base．The error involved is on the safe side．

The arc length (La) is calculated as 14.35 mm．The results are given in

Table 9.1

∑Wcosα=30×17.50=525kN／m

∑W sinα=30×8.45=254kN／m

∑(wcos α-ul)=525—132=393kN／m

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9.4 Analysis of a Plane Translational Slip

It is assumed that the potential failure surface is parallel to the surface of the slope and is at a depth that is small compared with the length of the slope. The slope can then be considered as being of infinite length，with end effects being ignored．The slope is inclined at angle βto the horizontal and the depth of the failure plane is z．as shown in section in Fig.9.7.The water table is taken to be parallel to the slope at a height of mz (0- 15 -

and the stress conditions are the same at every point on the failure plane．

In terms of effective stress，the shear strength of the soil along the failure plane is

and the factor of safety is

The expressions forσ,τandμare:

The following special cases are of interest．If c’=0 and m=0 (i.e. the soil

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between the surface and the failure plane is not fully saturated),then

(9.11)

If c’=0 and m=1(i.e. the water table coincides with the surface of the slope)，then：

(9.12)

It should be noted that when c’=0 the factor of safety is independent of

the depth z．If c’ is greater than zero，the factor of safety is a function of z, and βmay exceedφ’ provided z is less than a critical value．

For a total stress analysis the shear strength parameters cu andφu are used with a zero value of u.

Example 9.3

A long natural slope in a fissured overconsolidated clay is inclined at 12°to the horizontal．The water table is at the surface and seepage is roughly parallel to the slope．A slip has developed on a plane parallel to the surface at a depth of 5 m．The saturated unit weight of the clay is 20 kN／m3．The peak strength parameters are c’=10 kN/m2 andφ’=26°；the residual strength parameters are cr’=0 andφr’=18°.Determine the factor of safety along the slip plane(a)in

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terms of the peak strength parameters (b)in terms of the residual strength parameters．

With the water table at the surface(m=1)，at any point on the slip plane，

Using the peak strength parameters，

Then the factor of safety is given by

Using the residual strength parameters，the factor of safety can be

obtained from Equation 9.12:

9.5 General Methods of Analysis

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Morgenstern and Price[9.4]developed a general analysis in which all boundary and equilibrium conditions are satisfied and in which the failure surface may be any shape，circular，non-circular or compound．The soil mass above the failure plane is divided into sections by a number of vertical planes and the problem is rendered statically determinate by assuming a relationship between the forces E and X on the vertical boundaries between each section．This assumption is of the form

X=λf(x)E (9.13) where f(x)is an arbitrary function describing the pattern in which the ratio X/E varies across the soil mass andλis a scale factor．The value ofλis obtained as part of the solution along with the factor of safety F．The values of the forces E and X and the point of application of E can be determined at each vertical boundary．For any assumed function f(x) it is necessary to examine the solution in detail to ensure that it is physically reasonable (i.e. no shear failure or tension must be implied within the soil mass above the failure surface). The choice of the function f(x) does not appear to influence the computed value of F by more than about 5% and f(x)=l is a common assumption．

The analysis involves a complex process of iteration for the values ofλ and F，described by Morgenstern and Price[9.5]，and the use of a computer is essential.

Bell [9.1] proposed a method of analysis in which all the conditions of

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equilibrium are satisfied and the assumed failure surface may be of any shape．The soil mass is divided into a number of vertical slices and statical determinacy is obtained by means of an assumed distribution of normal stress along the failure surface．

Sarma [9.6] developed a method，based on the method of slices，in which the critical earthquake acceleration required to produce a condition of limiting equilibrium is determined．An assumed distribution of vertical interslice forces is used in the analysis．Again，all the conditions of equilibrium are satisfied and the assumed failure surface may be of any shape．The static factor of safety is the factor by which the shear strength of the soil must be reduced such that the critical acceleration is zero．

The use of a computer is also essential for the Bell and Sarma methods and all solutions must be checked to ensure that they are physically acceptable．

References

[9.1]Bell,J,M.(1968):’General Slope Stability Analysis’, Journal ASCE,V01.94,No.SM6.

[9.2]Bishop,A.W.and Morgenstern,N.R.(1960)：‘Stability Coefficients for Earth Slopes Geotechnique，Vo1.10.No.4．

[9.3]Gibson,R.E.and Morgenstern,N.R.(1962):’A Note on the Stability

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of Cuttings in Normally Consolidated Clays’.Geotechnique,Vo1.12.No.3

[9.4]Morgenstern,N.R.and Price,V.E.(1965):’The Analysis of the Stability of General Slip Surfaces’,Geotechnique，Vo1.1 5,No.1.

[9.5]Morgenstern,N.R.and Price,V.E.(1967): ‘A Numerical Method for Solving the Equations of Stability of General Slip Surfaces’Computer Journal，Voi.9,P.388．

[9.6]Sarma,S.K. (1973):’Stability Analysis of Embankments and Slopes’,Geotechnique，Vo1.23，No.2.

[9.7]Skempton,A.W.(1970):’First-Time Slides in Overconsolidated Clays’(Technical Note)，Geotechnique,Vo1.20.No.3

[9.8]Spencer，E．(1 967)：‘A Method of Analysis of the Stability of Embankments Assuming Parallel Inter-Slice Forces’，Geotechnique，Vo1.17.No.1．

[9.9]Taylor,D.W.(1937):’Stability of Earth Slopes’,Journal of the Boston Society of Civil Engineers，Vo1.24,No.3

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