The Foundation Engineering Handbook Chapter 11

The Foundation Engineering Handbook Chapter 11 Geotechnical earthquake engineering can be defined as that subspecialty within the field of geotechnical engineering that deals with the design and construction of projects in order to resist the effects of earthquakes. Geotechnical earthquake engineering requires an understanding of basic geotechnical principles as well as an understanding of geology, seismology, and earthquake engineering. In a broad sense, seismology can be defined as the study of earthquakes. This would include the internal behavior of the earth and the nature of seismic waves generated by the earthquake.

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11

Stability Analysis and Design of Slopes

Manjriker Gunaratne CONTENTS

11.4 Slope-Stability Analysis Using the Stability Number Method 49611.4.1 Stability Analysis of a Homogeneous Slope Based on an Assumed

Failure Surface

496

11.4.2 Stability Analysis of a Homogeneous Slope Based on the Critical FailureSurface

498

11.6 Reinforcement of Slopes with Geotextiles and Geogrids 506

11.10 Approximate Three-Dimensional Slope-Stability Analysis 514

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11.1 Introduction

Construction of building foundations and highways on sloping ground or embankments canpresent instability problems due to potential shear failure Therefore, geotechnical designersare often required to design stable embankments that would allow additional constructionsuch as highways and buildings on top of them On the other hand, instability could also resultdue to partial excavation of slopes during foundation construction Furthermore, when onedesigns a structure in the vicinity of a slope, then safety considerations would naturally

warrant a stability analysis of that slope Hence, designers are often required to perform aground stability analysis in addition to the foundation design Stability analysis can be

performed more effectively and accurately if the analyst comprehends the specific causes ofpotential slope failure under the given geological conditions

The primary cause of slope instability due to possible shearing is the inadequate

mobilization of shear strength to meet the shear stresses induced on any impending failureplane by the loading on the slope Mathematically, the condition for instability can be

expressed as in the following equation based on the Mohr-Coulomb criterion:

(11.1)

where σis the normal stress on the potential failure plane.

One can identify the following factors that would trigger the above condition (Equation(11.1)):

1 Common factors that cause increased shear stresses in slopes:

a Static loads due to external buildings or highways

b Cyclic loads due to earthquakes

c Steepened slopes due to erosion or excavation

2 Common factors that cause reduction in shear strength of slopes:

a Increased pore pressures due to seepage and artesian conditions

b Loss of cementing materials

c Sudden loss of strength in sensitive clays

The limit equilibrium method is the most popular method adopted in slope-stability analysis

In this approach, it is assumed that the shear strength is mobilized simultaneously along theentire (predetermined) failure plane Then the factor of safety for the predetermined failurewedge can be defined based on either the forces or moments as follows:

(11.2)

It must be noted that the stabilizing force or the stabilizing moment is the maximum force orthe maximum moment that can be generated by the failing soil along the failure plane Hence,these quantities can be determined by assuming that shear strength is mobilized along theentire failure surface As discussed inChapter 1, and employed in Equation (11.1), the shear

foundation engineering.

On the other hand, the destabilizing force or moment active at a given instance can bedetermined in terms of the shear forces required to maintain the current state of equilib-

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TABLE 11.1

Suggested Minimum Factors of Safety from FHWA

Safety (FS)

Slopes affecting significant structures (e.g., bridge

abutments, major retaining walls)

11.1.1 Required Minimum Factors of Safety

The minimum factors of safety as suggested by FHWA and AASHTO are given inTable 11.1and Table 11.2

11.2 Analysis of Finite Slopes with Plane Failures

Finite slopes such as natural embankments that are limited in extent can contain strata ofrelatively weak layers as shown in Figure 11.1(a)and(b)andFigure 11.2 Similar situationscan also occur due to stratified deposits and interfaces between crusts (or shells) of dams thatare typically granular soils and cores of dams made of impervious soils By considering theweak layers to be planar surfaces (Figure 11.1a), one can perform simple stability analysesbased on the limit equilibrium method

TABLE 11.2

Required Minimum Factors of Safety from AASHTO

Required Minimum Factors of Safety (FS)

Highway embankment slopes and retaining walls 1.3 1.5 Slopes supporting abutments or abutments above

retaining walls

Source: From AASHTO, 1996, Standard Specifications for Highway Bridges, American Association for State

Highway and Transportation Officials, Washington, DC With permission.

FIGURE 11.1

(a) Finite slope with a homogeneous failure plane, (b) Finite slope with a homogeneous nonplanar

failure surface.

Case A Homogeneous failure plane

Referring to Figure 11.1a, one can derive the following relations by considering forceequilibrium parallel and perpendicular to the failure plane, respectively,

T=W sin(υ)

(11.3)

N=W cos(υ)

(11.4)The weight of the failure mass can be expressed in terms of the unit weight of the failing soilmass as

(11.5)

where

FIGURE 11.2

Finite slope with a nonhomogeneous failure plane.

The stabilizing force is determined by the available strength based on the Mohr-Coulombcriterion as

(11.6)

where c and are the shear strength parameters of the weak soil layer (of length L) along the failure plane It is also seen that under the current state of equilibrium, Fdestab is equal and

opposite to T.

Then, it follows from Equations (11.2) and (11.6) that

where u is the average pore pressure along the failure plane.

By using Equations (11.3)−(11.5), the factor of safety can be simplified to

(11.7)

A reasonable value of u representative of the pore pressures in the failure plane can be

estimated by any of the methods outlined inSection 1.3

Under “undrained” conditions Equation (11.7) simplifies to

(11.8)

Case B Nonhomogeneous failure plane

When the relatively weak stratum defines only a part of the potential failure plane, thendestabilization of a slope occurs only if the stronger soil composing the failure mass allowsthe rest of the failure plane to form within itself, as shown inFigure 11.2 However, in suchcases, it is difficult to obtain a closed-form solution for the safety factor without making anumber of assumptions regarding the distribution of shear stresses on the entire failure planecomprising two different materials undergoing shear failure, i.e., weak stratum and the

relatively stronger soil forming the rest of the failure surface

Such assumptions are typically made in the “method of slices” described inSection 11.3.Hence, it is the method of slices that would be most suitable for analyzing the stability ofslopes where conditions are nonuniform throughout the plane of failure or the failing soilmass

11.3 Method of Slices

The method of slices is a numerical procedure that has been developed to handle stabilityanalysis of slopes where conditions are nonhomogeneous within the soil mass making itimpossible to deduce closed-form solutions Some such nonhomogeneous conditions that arecommonly encountered are as follows:

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1 Irregularity of failure planes, i.e., failure planes cannot be defined by simple geometricshapes This situation arises when relatively weak strata are randomly distributed within theslope or when the slope contains different soil types along the potential failure surface

2 Presence of two different soil types within the failure wedge requiring the use of differentsoil properties in analysis

3 Significant variation in the distribution of pore pressure along the failure plane, even understatic groundwater conditions

4 Irregularity of the slope geometry

5 Significant variation in the buoyancy effects due to artesian conditions and seepage ofgroundwater

The analysis requires the selection of a trial failure plane and discretization of the resultingfailure wedge into a convenient number of slices as shown in Figure 11.3 The analyst isrequired to device the slicing in a manner that can incorporate any nonhomogeneity within theslope so that each resulting slice would be a homogeneous entity Then, the stability of eachslice can be analyzed separately using the limit equilibrium method and principles of statics,

as done inSection 11.2

The free-body diagram for each slice is illustrated inFigure 11.4where it is seen that the

side forces on the slices (X i and Y i) introduce additional unknowns into the analysis making it

a statically indeterminate problem Hence, the analyst needs to make simplifying assumptions

to reduce the number of unknowns to facilitate a statically determinate solution This

flexibility has given rise to a variety of different analytical procedures, some of which will beoutlined in this section It is also realized that although the number of slices used in the

analysis determines the accuracy of the solution, today’s availability of superior

computational devices and effective algorithms enable one to achieve solutions with

reasonable accuracy for even the most complicated situations

Without the need for any assumptions a simple expression can be derived for the safetyfactor by considering the equilibrium of the entire failure wedge as follows

Under normal equilibrium conditions (Figure 11.3 andFigure 11.4), the destabilizingmoment about the center of the trial failure surface is given by

where n is the total number of slices.

FIGURE 11.3

Illustration of the method of slices.

FIGURE 11.4

Free-body diagram for any slice i.

On the other hand, the stabilizing moment obtained from Equation (11.6) is based on theavailable strength and can be given as

where l i is the arc length of slice i which can be expressed as

(11.9b)

A reasonable value of u can be estimated by any of the methods outlined in Section 1.3.2 It is the elimination of N ithat requires simplifying assumptions giving rise to several differentapproaches used in the method of slices

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11.3.1 Ordinary Method of Slices

Fellinius (1937) oversimplified the problem by neglecting the side forces completely Then,based on the free-body diagram inFigure 11.4, one notes that

11.3.2 Bishop’s Simplified Method

Bishop (1955) assumed the side forces in all of the slices to be horizontal This provides thefollowing additional equations For the vertical force equilibrium of each slice:

Equations (11.11) and (11.12) can be used to solve for N i in terms of F Subsequent

substitution for N i in Equation (11.9b) yields an implicit expression for the safety factor as

(i) the ordinary method of slices and (ii) Bishop’s method of simple slices for:

Case 1 Dry embankment conditions and using the ordinary method of slices only for

(Table 11.3 andTable 11.4)

FIGURE 11.5

(a) Illustration for Example 11.1 (drawn to scale), (b) Investigation of slope failure under rapid

drawdown.

Case 2 Immediately after compaction (assuming a pore pressure coefficient, ru=0.4)

Case 3 Under completely submerged conditions (groundwater table at the level CD)

Case 4 On sudden drawdown of the groundwater table to level AB for the trial failure

surface drawn inFigure 11.5(b)

TABLE 11.3

Data for the Ordinary Slices Method ( Example 11.1 , Case 1)

Slice (i) b i (m) h i (m) W i =γ b i h i α i W i sin α i l i =b i sec α i C i l i W i cos α i tan Φi

Undrained cohesion=20 kPa

Dry unit weight=16.0 kN/m3

clay

Cohesion=20kPa

Undrained cohesion=30 kPa

Dry unit weight=17.0 kN/m3

First, it is noted that slicing is done to separate the soil layers

Solution

FromSection 1.6, γdry=Gsyw/(1+e), and assuming Gs to be 2.65 and knowing that γw=9.8kN/m3,

For silty clay, e=0.623

Also γsat=γw(Gs+e)/(1+e)

Case 2 Immediately after compaction

The Bishop and Morgenstern pore pressure coefficient, ru, is defined as (Table 11.5)

TABLE 11.5

Pore Pressure Estimations for Example 11.1 , Case 2

Slice (i) h i(m) W i =γ h i u i(kPa) l i =b i sec α i U i l i

Case 3 Under completely submerged conditions

In this case, one can assume undrained conditions to be the most critical and both porepressure and the surcharge water can be incorporated together in the analysis by consideringthe submerged weights of the slices (Table 11.6)

Using Equation (11.10), F=(652)/(301.57)=2.16.

Case 4 Rapid drawdown

In cohesive soils rapid drawdown conditions promote slope failures as the one shown inFigure 11.5(b) due to the transient seepage condition developing at the face of the slope asshown inFigure 11.5(b) Then, the safety factor for the trial base failure surface shown inFigure 11.5(b) can be computed as illustrated inTable 11.7 It must be noted that the porepressure values have been computed using the flownet principles discussed in Section 13.2Since pore pressures are separately computed, the weights of slices are evaluated based on thesaturated unit weight of 19.8 kN/m3 It is also reasonable to assume that drained conditionsoccur in a transient flow regime Hence, an effective stress analysis is performed using the

evaluated pore pressures From Equation (11.10), F=1.1.

The method of slices can be employed to perform stability analysis of slopes under state seepage conditions as well In such cases, one can conveniently predict the pore

steady-pressures using Bernoulli’s equation (Section 13.2, Equation 13.1)

TABLE 11.6

Pore Pressure Estimations for Example 11.1 , Case 3

Slice (i) b i (m) h i (m) W i =γsubb i h i α i W i sin α i l i =b i sec α i (Cu) i l i W i cos α itan Φi

4 5.06 5.82 429.7 60° 372.13 10.12 202.4 0

Σ=301.57 Σ=652

TABLE 11.7

Pore Pressure Estimations for Example 11.1 , Case 4

Slice (i) α i b i l i u i(kPa) u i b i u i l i W i sin α i

11.4 Slope-Stability Analysis Using the Stability Number Method

In the stability number method (Taylor, 1948), the limit equilibrium computations are based

on an assumed linear or circular rupture surface Then the safety factor, F, is defined as the

ratio of maximum shear strength that can be mobilized to the shear strength required on theassumed failure surface to maintain the slope in equilibrium Based on the Mohr-Coulomb

criterion, F can be written by

(11.15)

where c and are the strength parameters and the subscript “d” indicates the strength

parameters required for equilibrium or the developed strength σnis the average normal stress

on the failure surface

The individual safety factors with respect to cohesion and friction can be also defined,respectively, as follows:

(11.16)

(11.17)Observation of Equations (11.15)–(11.17) shows that the actual (true) safety factor is obtainedunder the following condition:

Step 1 Assume a reasonable for the assumed failure surface.

Step 2 Use Equation (11.17) to estimate knowing the available friction

Step 3 Knowing the developed frictional strength perform a static equilibrium

analysis of the failure mass to determine the developed cohesive strength cd This is

conventionally designated as a nondimensional cohesive strength or the stability number for

the given failure mass using γ(unit weight) and H (height of slope).

11.4.1.1 Plane Failure Surface

In terms of the notation inFigure 11.6, the stability number can be expressed as follows:

(11.20)

11.4.1.2 Circular Failure Surface

For assumed circular trial failure surfaces, the stability numbers can be expressed in

Equations (11.21a) and (11.21b), in terms of the notation inFigure 11.7 for toe failure (Taylor,1937)

FIGURE 11.7

Equilibrium of failure wedge on a circular failure surface.

height H (for base failure).

Figure 11.7 also shows the forces that ensure the equilibrium of the assumed failure wedge

These are the weight, W, the design cohesion, Cd, and the resultant of the normal and

frictional forces, P The force P must be tangent to a circle of radius and centered at o(Figure 11.14) This circle is known as the friction circle

Step 4 Knowing the maximum available cohesion, c, estimate Fcfrom Equations (11.21)

Step 5 If F c and are not equal, repeat the procedure from Steps 1 to 4 for different

values until the condition of is satisfied Once it is satisfied, this F c (or ) is the truesafety factor FS (Equation (11.18))

11.4.2 Stability Analysis of a Homogeneous Slope Based on the Critical Failure

Surface

If the ultimate goal is to find the failure surface with the minimum safety factor, i.e., thecritical failure surface, then the above procedure has to be repeated for a number of differenttrial failure surfaces

The above failure plane would provide the highest potential for sliding

11.4.2.2 Use of Taylor’s Stability Charts

surfaces with respect to a given slope, can be determined usingFigure 11.8andFigure 11.9.Accordingly, if the analyst is in search of the minimum possible safety factor for the givenslope and not a safety factor with respect to a specific failure surface, Figure 11.8andFigure11.9will immensely reduce the volume of computations in Step 3 of the above iterativeprocedure.

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FIGURE 11.8

Stability chart for soils with friction angle (From Taylor, D.W., 1948, Fundamentals of Soil

Mechanics, John Wiley, New York With permission.)

Taylor’s simplified method of stability analysis will be illustrated in Examples 11.2–11.4

FIGURE 11.9

Stability chart for soils with zero-friction angle (From Taylor, D.W., 1948, Fundamentals of Soil

Mechanics, John Wiley, New York With permission.)

Trial 2

For an assumed of 5, a similar procedure produces F c=7.77

Trial 3

Finally, for an assumed of 6.2, one obtains F c=1.97

Then the results of the above iterative procedure can be plotted inFigure 11.11, from which

it is seen that the true factor of safety

FIGURE 11.10

Illustration for Example 11.2

FIGURE 11.11

Plot of Fc vs for Example 11.2

Example 11.3

With respect to the slope shown inFigure 11.12, estimate the minimum safety factor

corresponding to a critical failure plane passing through the toe Assume the following soilproperties:

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Solution

Trial 1

Assume an of 1.5 Then, from Equation (11.17)

FromFigure 11.8, D=0 (toe failure), i=31°, and yield

Assume an of 1.6 Then, from Equation (11.17),

FromFigure 11.8, D=0 (toe failure) and yield

Thus, F c=1.608

Hence, considering a toe failure, the true minimum safety factor for the slope inFigure11.12is about 1.6

Example 11.4

With respect to the slope shown inFigure 11.12, estimate the minimum safety factor

corresponding to a critical failure plane that touches the bedrock Assume the following soilproperties:

Undrained cohesion=25 kPa

Dry unit weight=17.5 kN/m3

11.5 Stabilization of Slopes with Piles

The use of piles as a restraining element has been applied successfully in the past and proven

to be an effective solution, since piles can often be installed without disturbing the

equilibrium of the slope Piles used to stabilize slopes are in a passive state and the lateralforces acting on the piles are dependent on the soil movements that are in turn affected by thepresence of piles Due to their relatively low cost and the insignificant axial strength andlength demand in this particular application, timber piles are ideal for stabilization of slopes

11.5.1 Lateral Earth Pressure on Piles

Poulos (1973) first suggested a method to determine the lateral forces on piles Ito and Matsui(1975) proposed a different theoretical approach to analyze the growth mechanism of lateralforces acting on stabilizing piles assuming that soil is forced to squeeze between the piles.This condition is applicable to relatively small gaps between piles Then, the passive force onthe pile per unit length (2) (Figure 11.13) can be computed by the following equation based

on Ito et al (1975):

(11.23)

where

FIGURE 11.13

Plastically deforming ground around stabilizing piles (From Hassiotis, S., Chameau, J.L., and

Gunaratne, M., 1997, Journal of Geotechnical and Geoenvironmental Engineering, ASCE,

123(4) With permission.)

D1is the pile spacing, D2is the opening between the piles, c is cohesion and is the friction

angle

Under undrained conditions Equation (11.23) reduces to

(11.24)

where γ z is the overburden stress.

11.5.2 Analysis Using the Friction-Circle Method

Modified stability number For a slope of inclination i and height H (Figure 11.14), the

stability number can be expressed as in the following equations (Hassiotis et al., 1997)

FIGURE 11.14

Forces on a slope reinforced with piles (From Hassiotis, S., Chameau, J.L., and Gunaratne, M., 1997,

Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123(4) With

permission.)