Stability of Earth Slopes | Soil Engineering

In this article we will discuss about:- 1. Introduction to Earth Slope 2. Definition of Factor of Safety 3. Types 4. Friction Circle Method 5. Taylor’s Stability Number and Stability Charts 6. Improving the Stability.

Introduction to Earth Slope:

An earth slope is an unsupported inclined surface of a mass. Earth slopes may be formed naturally, as for a hill slope; or artificially, as for a railway formation, a highway embankment, an earth dam, a canal bank, etc.

Thus, the classification of earth slopes is as follows (see Fig. 17.1):

1. Natural slopes:

i. Natural Hill Slopes:

Its stability is of interest to a civil engineer when it exists near a highway/railway line.

ii. Side-Hill Slopes:

When a natural hill slope is used as the formation for a highway or railway line, it may be referred as a side-hill slope. In this case, the slope will be subjected to moving/impact loads due to vehicular traffic.

iii. Cut Slopes:

When the formation level for a highway/railway line is below the natural ground surface, the soil is excavated with stable-side slopes up to the required depth.

iv. Open Excavations:

Open pits may be excavated for different purposes, such as for the construction of a foun­dation/ structure below the ground level or for soil exploration. The stability of the sides and bottom of the excavation should be checked and ensured in this case.

2. Man-Made Slopes:

i. Embankments:

Embankments are constructed for various purposes and in situations when the formation level of a highway/railway line, or the floor level of a building or other structure is above the natural ground sur­face. The side slopes of an embankment should be designed in such a way that they are safe and stable, and a minimum quantity of borrowed soil should be used for the construction of the embankment for economy.

ii. Earth Dams:

An earth dam is similar to an embankment, but larger in size and trapezoidal in cross section, it is constructed for the purpose of water storage and withdrawal. The upstream (u/s) side is subjected to the hydrostatic pressure that varies with the depth of water in the reservoir at different times. The body of the dam is also subjected to seepage forces. The stability of u/s and downstream (d/s) slopes of an earth dam should be properly analyzed to ensure safe and economical slopes.

iii. Canal Banks:

The side banks of a canal, which carries water from a reservoir for irrigation/water supply, should be designed to ensure safety and economy.

iv. Waterfront Structures:

There may be other types of structures such as sheet piles or retaining walls, which need to take into consideration the stability of the soil, which is similar to a slope stability problem.

The cost of earth-work will be the least, if the slopes are made steep. However, very steep slopes may not be stable.

To strike a balance between safety and economy, the steepest slopes, which are stable, must be designed. Hence, earth slopes are designed with a predefined factor of safety.

Definition of Factor of Safety:

There are the following three types of factor of safety with respect to shear strength, cohesion and friction:

1. Factor of Safety With Respect to Shear Strength:

Factor of safety with respect to shear strength is defined in –

or

where c_m is the mobilized cohesion, ɸ_m is mobilized angle of friction, c is available cohesion, ɸ is available friction angle, and σ’ is the normal stress on a critical surface.

2. Factor of Safety With Respect to Cohesion:

Factor of safety with respect to cohesion is defined by –

F_c = c/c_m …(17.3)

3. Factor of Safety With Respect to Friction:

Factor of safety with respect to friction is defined by –

F_ɸ = tanɸ/ tanɸ_m …(17.4)

where ɸ is the angle of shearing resistance of soil and ɸ_m is the angle of mobilized shearing resistance.

Types of Slope Failures:

An earth slope may have any one of the following modes of failure:

1. Rotational Failure:

This type of failure occurs by rotation of unstable mass of soil along a slip surface by the downward and outward movement of soil, above a slip surface from the rest of the soil mass. The slip surface is generally circular for homogeneous soils and non-circular for non-homogeneous conditions, as shown in Fig. 17.2(a).

Rotational failures are further divided into the following three types:

i. Toe Failure – Failure occurs along a slip surface that passes through the toe.

ii. Slope Failure – Failure occurs along a slip surface that intersects the slope above the toe.

iii. Base Failure – Failure surface passes below the toe.

A slope failure occurs when the soil at the base of the slope is stronger than the soil within the slope. A base failure occurs if a weak stratum lies beneath the toe. In all other cases, the failures are generally toe failures. Toe failure occurs when the soil in the slope and the base are of approximately the same strength. Toe failures are the most common among all modes of failure.

2. Translational Failure:

This type of failure generally occurs by sliding along a plane surface. In translational failure, the failure surface is not curved but a plane surface, as shown in Fig. 17.2(b). Translational failure occurs in the case of infinite slopes. Infinite slopes are those which have a constant slope for a considerable extent and in which the soil characteristics on all verticals are adequately represented by the average soil conditions. Translational failures may also occur along slopes of layered materials.

3. Compound Failure:

A compound failure is a combination of rotational and translational failures. A compound failure surface is curved at the two ends with a plane in the middle portion, as shown in Fig. 17.2(c). A compound failure generally occurs when a hard stratum exists at considerable depth below the toe.

4. Wedge Failure:

A failure along an inclined plane is known as a wedge failure, as shown in Fig. 17.2(d). A wedge failure is also known as a plane failure or a block failure. It occurs when distinct blocks or wedges of soil mass become separated from the rest of the earth slope.

A plane failure is similar to a translational failure in many respects. However, unlike translational failure, which occurs in an infinite slope, a plane failure may occur even in a finite slope consisting of two different materials or in a homogeneous slope having cracks, fissures, joints, or any other specific planes of weakness.

5. Miscellaneous Failures:

Complex failures in the form of spreads and flows fall under this category.

Friction Circle Method:

Friction Circle:

The friction circle method considers the overall moment equilibrium of the wedge of soil using the arc of a circle as the failure surface. Figure 17.17 shows the failure surface of radius r and center O. As the wedge of soil slides along the failure surface, a frictional force is generated along the failure surface.

Let F be the frictional force opposing the movement of the sliding mass acting tangential to the failure surface, N be the normal reaction acting perpendicular to the tangent to the failure surface, and R be the resultant of the frictional force, F and the normal reaction, N.

It is known that the resultant, R, which makes an angle ɸ_m with the normal to the failure surface. Now, if a small circle is drawn with O as center and r sinɸ_m as radius, the resultant R will be tangential to this circle. This small circle of radius r sinɸ_m is known as the friction circle.

Forces Acting on the Sliding Mass:

The following are the forces acting on the wedge of the sliding mass:

1. W is the weight of sliding mass. It acts vertically downwards through the centroid of the wedge of sliding mass.

2. The resultant (of F and N) R acting tangential to the friction circle.

3. The cohesive force C_m acting along the curved failure surface at a radial distance r.

Thus, the sliding mass is in equilibrium under the action of only three forces W, C_m, and R. It is known from the principles of mechanics that when a body is in equilibrium under the action of three forces, the three forces are concurrent.

Thus, in the friction circle method, the factor of safety is determined by considering the equilibrium of the slid­ing mass, subjected to the three concurrent forces W, C_m, and R.

Equivalent Cohesive Force:

The cohesive force, C_m, is replaced by an equivalent cohesive force, C_m‘, acting parallel to the slope, such that both of them cause the same moment at the center of the failure surface. Let c_m be the mobilized cohesion of the soil, I be the length of the failure surface, l_c be the length AB of the slope, and a is the distance from the center perpendicular to the slope at which the equivalent force C_m‘ acts. Equating the moment of C_m and C_m‘ about the center O –

Determination of Factor of Safety:

Following is the procedure to determine the factor of safety using the friction circle method:

1. A trial value of factor of safety with respect to friction, Fɸ, is assumed and the mobilized friction angle (ɸ_m) is determined from –

tanɸ_m= tanɸ/Fɸ …(17.36)

2. The slope AB of height H and the trial slip surface of radius r with O as center are constructed to some scale. The friction circle is constructed with O as center and r sinɸ_m as radius.

3. A vertical line is drawn through the centroid of the sliding wedge ABDA, representing the line of action of W, to intersect the trial slip surface at point E.

4. A tangent ET is drawn to the friction circle from point E. The line ET gives the direction of R

5. The weight of the sliding wedge of soil mass is determined from –

W = γ.A.1 = γ. A …(17.37)

where A is the area of the wedge of soil ABDA and γ the bulk density of the soil.

6. The force diagram is now constructed by drawing vertical line, ab, representing W to some scale. A line is drawn, representing the direction of C’_m, from point a parallel to the slope AB. Another line is drawn, repre­senting the direction of R, from point b parallel to the line ET. These two lines intersect at point c, completing the force diagram abc.

7. The length of the line ac is measured, which gives the force C’_m, to the scale of the force diagram. The mobilized cohesion is computed from –

c_m = C’_m,/l_c …(17.38)

8. The factor of safety with respect to cohesion (F_c) is computed from –

F_c = c/c_m …(17.39)

9. The value of F_c computed from Eq. (17.39) is compared with the assumed value of Fɸ. If they do not agree, the procedure is repeated from steps 1 to 8, by assuming another value of Fɸ. The solution is found to be conver­gent, so that it is useful to use the value of F_c computed in a given trial as F_ɸ in the next trial. It was found that three to four trials are sufficient to obtain a reasonably accurate value of F_c = F_ɸ.

10. The final value of the factor of safety obtained in step 9 is applicable for the assumed trial slip surface. The proce­dure is repeated from steps 1 to 9, by using a number of possible trial slip surfaces with different radius and center.

11. The minimum value of factor safety obtained in step 10 is taken as the factor of safety for the slope and the corresponding slip surface is taken as the critical slip surface.

Special Cases – Case 1 – Slope with Pure Cohesive Soil:

For a pure cohesive soil, ɸ = 0 and the friction circle reduces to a point, the factor of safety is computed from

Special Cases – Case 2 – Submerged Soil Slope under Steady Seepage:

If the soil slope is submerged, the submerged density is used to calculate the weight of the wedge, W. The three neutral forces namely (a) the neutral part of the weight, (b) the water pressure acting along the slope AB, and (c) the resultant water pressure acting along the curved surface, ABDA, balance themselves and do not affect the stabil­ity.

The analysis is similar to dry conditions, except that the submerged density, is used to calculate the weight, W instead of bulk density. Alternately, the stability analysis under submerged conditions can also be carried out using Taylor’s stability charts is considered below.

Taylor’s Stability Number and Stability Charts:

Taylor carried out stability analysis of a large number of soil slopes with a wide range of properties such as slope height (H), cohesion (c), friction angle (ɸ), and density (γ) using the friction circle method. Taylor presented the results of the stability analysis in the form of curves which give the relationship between stability number and slope angle (β) for various values of friction angle (ɸ), as shown in Fig. 17.18. Table 17.5 shows the values of Taylor’s stability numbers for toe and slope failures. Taylor’s stability number is defined by –

S_n = c/FγH …(17.41)

where S_n is Taylor’s stability number, c is the unit cohesion of the soil, F is the factor of safety, y is the density of soil, and H is the height of soil slope.

Taylor’s stability charts shown in Fig. 17.18 are applicable to toe failure, where the failure surface passes through the toe of the slope. Toe failure is likely to occur if β > 53°.

For slope angle, β < 53°, and for low values of ɸ, base failure may occur. In case of base failure, the stability number is a function of another parameter known as depth factor D_f, defined by –

D_f = D/H …(17.42)

where D is the depth to the bottom of the slip surface from the top of the slope, as shown in Fig. 17.19 and H is the height of the soil slope.

For soils of ɸ_u = 0 and for β < 53°, the chart shown in Fig. 17.19 is proposed by Taylor, which gives stability number for various values of β and n. Table 17.6 shows the values of Taylor’s stability numbers for these soils.

The following are the uses of Taylor’s stability charts:

1. Determination of Maximum Slope Angle:

The maximum slope angle that can be used for an embankment can be determined from the given values of F, γ, c, and ɸ using the following steps:

i. Determine the mobilized friction angle ɸ_m from Eq. (17.36) –

tanɸ_m = tanɸ_m/Fɸ

ii. Determine the stability number from Eq. (17.41) –

S_n = c/FγH

iii. The slope angle, β can be obtained from the stability chart from the computed values of S_n and ɸ_m.

2. Determination of Factor of Safety:

The factor of safety for a given slope can be determined for the given values of β, γ, c, and ɸ using the following steps:

i. Assume a trial value of factor of safety F_ɸ and determine the mobilized friction angle ɸ_m from Eq. (17.36) –

tanɸ_m = tanɸ_m/Fɸ

ii. Obtain the stability number from the stability chart using the values of β and ɸ_m.

iii. Determine the factor of safety with respect to cohesion from Eq. (17.41) –

F_n = c/S_nγH

If F_c ≠ F_ɸ, the procedure is repeated from the first to the third step by assuming a different value of F_ɸ. A graph in the form of a curve is plotted between F_c and F_ɸ as shown in Fig. 17.20. A 45° line is drawn from the origin to intersect the curve at point P. The abscissa or ordinate of this point P, gives the factor of safety, where F_c = F_ɸ.

3. Special Cases:

Following guidelines may be followed in special cases such as with respect to submerged slopes, sudden drawdown condition, and for cohesionless soil slopes:

i. For a submerged soil slope, the submerged density should be used in Eq. (17.41) for the stability number. The friction angle used should also correspond to submerged conditions

ii. For a sudden drawdown case, the saturated density should be used in computing the stability number. The value of mobilized friction angle (ɸ_m) should be computed from –

tanɸ_m= γ’/γ_sat . tanɸ/F_ɸ …(17.43)

iii. For pure cohesionless soils, c = 0 and stability number also becomes zero. Hence, stability charts cannot be used for pure cohesionless soils.

Improving the Stability of Slopes:

The following are some of the methods that can be adopted for improving the stability of an earth slope:

1. Flattening the Slope:

It is the simplest obvious method for improving the factor of safety of a slope. The slope can be reduced to such a level that the desired factor of safety is obtained in the appropriate method of analysis adopted.

2. Providing a Berm:

A berm can be provided at the mid-height of the slope, thereby breaking the failure surface and improving the slope stability. It is an indirect way of reducing the height without actually doing it.

3. Drainage:

Slope stability can be significantly improved by reducing the seepage forces through effective drain­age measures. The drainage provided should divert the water away from the slope, ensuring that it is not blocked at any point of the earth slope.

4. Construction of a Retaining Wall:

It is one of the usual methods adopted to avoid slope failure, when the factor of safety is very less. This method is costlier than other methods of improving the slope stability, but it is a per­manent solution to frequent slope stability problems.

5. Compaction:

Proper compaction of the earth slope will improve the shear strength of the soil, thereby increasing the factor of safety.

6. Ground Improvement Methods:

Suitable ground improvement methods may be adopted to improve the shear strength of the soil, such as grouting, cement injection, blasting, or electro-osmosis.

7. Soil Nailing or Ground Anchors:

Soil nailing is a common and effective method for improving the stability of earth or rock slopes.

8. Vegetation:

When the earth slope is provided with vegetation with small grassy plants, the roots of the plants penetrate into the earth slope and give the effect of reinforced earth. It also adds to the aesthetic view of the earth structure.