Chapter 9
Stability of Slopes and Retaining Structures

9.1 STABILITY OF NATURAL SLOPES, ROCK BUND AND RIP-RAP
9.2 STABILITY OF RETAINING STRUCTURE
9.3 FLEXIBLE RETAINING STRUCTURE
9.4 COMPUTER-AIDED STABILITY ANALYSES

In reclamation projects, most edges of the boundaries are completed with either natural fill slopes or retaining structures. Natural fill slope is only feasible when the reclaimed front is sheltered by a breakwater or headland. Other boundaries which are exposed to wave and current actions are generally protected by a rock bund or seawall. The selection of rock bund or seawall is dependent upon the seabed condition, soil condition, the available foreshore area, future land use, and cost considerations. In any case, stability analyses are deemed necessary.

Most stability analyses involve total stress analysis, assuming undrained conditions soon after construction. Effective stress analyses are only carried out when stage construction technique is applied or the structure is constructed with a marginal safety factor which will be increased during the construction stage or to check long term stability.

9.1 STABILITY OF NATURAL SLOPES, ROCK BUND AND RIP-RAP

9.1.1 Natural slope of sand

The stability of natural sand slopes is conveniently estimated based on the friction angle (ϕ) of sand. If the angle of the slope (i) is greater than ϕ, the natural slope will become unstable. The factor of safety of a natural submerged sand slope (Figure 9.1) of granular fill is given by:

If seepage flow is involved, the factor of safety is reduced to:

where r_b is submerged unit weight and r_t is saturated unit weight.

In this case seepage flow is assumed to be parallel to the slope, as shown in Figure 9.2.

9.1.2 Stability of rock bund, rip-rap, headland and breakwater

The stability of a rock bund or rip-rap can be calculated by applying the limit equilibrium approach. Equilibrium can be either force equilibrium or moment equilibrium.

Slope stability analysis can also be carried out by applying overall moment equilibrium for which the factor of safety is given by:

where M_R is the resisting moment and given as M_R = c_uLr, and M_D is the driving moment and given as M_D = w.d. The dimensions of L, r and w are shown in Figure 9.3.

For the c - ϕ soil resisting moment is given by M_R= [cL + N tanϕ]r where N = w cos α. The determination of α is shown in Figure 9.4.

9.1.3 Method of slices

Slope stability can be calculated by making a soil mass within the failure slip as slices (Figure 9.4). The driving force can be taken as the summation of the weight of the slices where the resisting force can be calculated using the following equation:

where N = w cos α, as shown in Figure 9.4. This method can only be used for circular slip, but also suitable for φ= 0 soil.

9.1.3.1— Ordinary method of slices

The moment equilibrium method satisfies overall equilibrium:

For φ= 0 soils, the factor of safety is given by:

For φ< 0 soils, the factor of safety is given by:

9.1.3.2— Bishop’s modified method

For φ< 0 soils, the factor of safety is given by:

This equation needs to be solved by iteration. The Bishop’s method is excellent for circular slip plane analysis, and good for both φ= 0 and φ< o soils.

9.1.4 Force equilibrium methods

The force equilibrium method satisfies both horizontal and vertical force equilibriums. This method is good for non-circular slip planes. The factor of safety is given by:

This method is called the wedge method. The driving forces can be calculated from the total weight of the sliding mass.

where the resisting force is given by

The characteristics of various limit equilibrium methods for slope stability analysis are shown in Table 9.1. The various methods available for slope stability analysis and equilibrium conditions are also shown in Table 9.2.

The accuracy of various methods for analyzing slope stability was assessed by Wright (1973) for two different types of slopes, as shown in Figure 9.5. Tables 9.3 and 9.4 show comparisons of minimum values of factor of safety for γ_u= 0 conditions and Tables 9.5 and 9.6 show comparisons for γ_u= 0.6 conditions. It can be seen in the tables that the ordinary method of slices gives slightly lower values for both 1.5:1 and 3.5:1 slopes where γ_u= 0, and even lower where γ_u= 0.6 for 3.5:1 slope. The force equilibrium method, using Lowe and Karafiath’s assumptions, provides slightly higher factor of safety for most of the cases.

9.2 STABILITY OF RETAINING STRUCTURE

For reclamation projects, four major types of retaining walls are used. They are:

1. Gravity and reinforced concrete cantilever wall
2. Counterfort wall
3. Flexible sheet pile wall
4. Caisson

To ensure the stability of the walls, the following factors need to be checked:

• the moment equilibrium must be satisfied so that there is no overturning of the structure.
• the horizontal force equilibrium must be satisfied so that no sliding of the structure will occur.
• the vertical force equilibrium must be satisfied so that no bearing failure will occur.
• the earth pressure will not overstress the wall so that no shear or bending failure will occur.

A schematic diagram of the types of retaining structure failure is shown in Figure 9.6.

In order to calculate the earth pressure, it is necessary to understand earth pressure theory. Two well-known earth pressure theories—Rankine and Coulomb earth pressure theories—will be explained here briefly.

9.2.1 Rankine earth pressure theory

Rankine earth pressure theory assumes that soil is isotropic and possesses only internal friction and no cohesion. The theory considers the state of the

plastic equilibrium of soil under active and passive earth pressures. The backfill surface is horizontal, and the back of the wall is considered vertical and smooth.

For cohesionless soil, active pressure (P_a) is given by:

where γis the unit weight of soil, H is the height of fill, K_a is active earth pressure coefficient, given by:

Passive earth pressure (P_P) is given by:

For backfill with cohesive soil, the lateral pressure on the wall is given by:

9.2.2 Coulomb earth pressure theory

In Coulomb earth pressure theory, the soil is isotropic and homogeneous and posseses both internal friction and cohesion. The static equilibrium of an assumed wedge failure is used to determine active and passive earth pressures. The backfill and back of the wall can be inclined. Wall friction is also taken into consideration in this theory.

For cohesionless soil, the lateral active earth pressure is again given by:

where α, δ and βare shown in Figure 9.7 and δis the wall friction angle.

The passive earth pressure is given as:

Example 9.1

A gravity wall of 5m height was backfilled with sand. The unit weight of sand is 17kN/m³ and the internal angle of friction is 30°. (i) Calculate the active pressure on the wall if the back of the wall is vertical and the backfill is horizontal. (ii) Calculate the active and passive pressure on the wall if the back of the wall is 15° from the vertical and the backfill is 25° from the horizontal.

1. Since the back of the wall is vertical and backfill is horizontal, Rankine's solution is applied, using Equations 9.15 and 9.16:

2. Since the back of the wall and backfill are inclined, Coulomb's solution is applied using Equations 9.20 and 9.22 for active pressure, and 9.22 and 9.23 for passive pressure.

After learning about earth pressure theories and limit equilibrium, the stability of a retaining structure can be designed. In the following section, an example of a calculation for the stability of a cantilever retaining wall and sheet pile will be explained. Details of the design of the retaining structures are referred to in Bowles (1988), Kaniraj (1988), and Conduto (1994).

9.2.3 Bearing capacity of retaining structure

1. The bearing capacity of the foundation can be calculated by applying the shallow foundation method. The bearing capacity of a shallow foundation is given by Terzerghi as follows:

where, q'_u is net ultimate bearing capacity

C is soil cohesion

δ' _D is effective stress at depth D below ground surface

γ is unit weight of soil

D is depth of footing below the ground surface

B is width of footing

N_c,N_q and Nγ are bearing capacity factors. These bearing capacity factors may vary depending upon the friction angle, as shown in Table 9.7.

The resultant force below the wall generally falls within the middle third of the base and is usually compressing. Therefore, eccentricity is limited to:

In such a case, the maximum and minimum soil reaction is given by:

where V is the total vertical reaction, and + sign is used for maximum and - sign is used for minimum reaction. The total vertical reaction can be obtained by summing up all vertical forces (Figure 9.10). If the maximum is less than the allowable bearing pressure, the bearing capacity is acceptable.

9.2.4 Stability against overturning

The stability of the wall is important to prevent overturning and is given as:

Generally, a factor of safety of about 1.5 – 2 is adequate for stability against overturning.

The resisting moment at the toe at point A can be calculated if all the moment arms are known. Therefore, the sum of the resisting moment is given by:

The sum of the resisting moment at the toe = ΣM_R

The overturning moment about A is given as:

where moment arm about A is given as:

9.2.5 Stability against sliding

To calculate stability against sliding all the horizontal forces need to be calculated.

A safety factor above 1.5 – 2 is considered safe. Generally, passive resistance is ignored in the calculation.

The resisting horizontal forces against sliding is given by:

where, tan φ* = tan φ to 0.67 tan φ, c* = 0.5c to 0.75c.

The sum of the driving horizontal forces is equivalent to horizontal forces P_a which can be obtained from Equation 9.20.

Therefore, the factor of safety is given as:

This type of calculation for stability can also be used for gravity wall and counterfort wall. An appropriate weight calculation needs to be made according to the geometry of the wall. If the backfill is partially submerged under water, appropriate density values and water pressure need to be taken into consideration. Details on the stability of the wall, taking into consideration the groundwater level, the wall geometry and the backfill geometry are found in Kaniraj (1988).

Example 9.2

The retaining structure, as shown in Figure 9.11, is constructed on soft cohesive soil with c’ of 5 kN/m², and φ’ is 27º. Calculate the allowable bearing capacity.

From the table, N_c= 29.2

N_q= 15.9

Nγ= 12.5

Groundwater table is greater than D.

Therefore,

Using Equation 9.24,

Example 9.3

A cantilever retaining wall with dimensions as shown in Figure 9.12,

is constructed on soft clay and backfilled with sand with density of 18 kN/m3 and φ is 35º. (i) Calculate the maximum and minimum soil reaction. (ii) Calculate the stability against overturning, and (iii) the stability against sliding.

In order to calculate the stability of the wall, all the forces on the wall must first be determined. The weight of the wall itself and the weight of the soil on the heel are the major resisting forces.

Therefore, W₁ = 9 × 5 × 18 = 810 kN/m

W₂ = 1 × 9 × 24 = 216 kN/m

W₃ = 1/2 × 1 × 9 × 24 = 108 kN/m

W₄ = 1 × 8 × 24 = 192 kN/m

The major driving force is the earth pressure on the wall.

φ = 35º

therefore,

If the soil above the toe is also sand,

Pp = ½ × 3.690 × 18 × 1² = 33.21 kN/m

1. By summing up the weight of the retaining structure, the total vertical reaction V is:
   V = W₁+W₂+W₃+W₄
   = 810 + 216 + 108 + 192 = 1326 kN/m
   The total moment is on the active side:
   ΣMA = 810 x 5.5 +216 x 2.5 + 108 x 1.67 +192 x 4 - 243.9 x 3.33
   
   e = 4 − 3.876 = 0.124m
   Therefore, the maximum and minimum soil reactions are:
   
2. Stability against overturning is calculated using Equation 9.27:
   
3. Stability against sliding is calculated using Equation 9.31:
   
   

9.3 FLEXIBLE RETAINING STRUCTURE

An alternative type of retaining structure is the flexible retaining structure, which is usually used as a temporary structure but sometimes as a permanent one. In addition to the structural stiffness, the embedded depth is an important factor to be considered for the stability analysis of a sheet pile wall. A sheet pile wall in cohesionless soil is shown in Figure 9.13. Assuming that the rotation of the sheet pile is close to the bottom, the embedded depth is given as:

where H is height of wall above dredged line

D is embedded depth

The point of rotation below the dredged line is given as:

where d is the depth of rotation below the dredged line.

Therefore, the maximum moment is given as:

When water level is involved, the embedded depth is given as:

where, γ_b is submerged density, h₁ and h₂ are the heights of fill above the water level and dredged line respectively. K_a1 and K_a2 are coefficients of active earth pressure for soil above water and below water respectively. However, since the water level does not affect φvalue.

A simple calculation for sheet pile with the same water level on both sides of the wall is given below.

Example 9.4

A sheet pile wall of 8 meters height is backfilled with granular material of density 18 kN/m³, and friction angle of 35°. The water level is 3 meters below the top of the backfill and the density of the water is 10 kN/m³. Calculate the depth of embedment required.

Using Equations 9.36 to 9.40

a = 3×0.0162×0.271((18×3) + (18×5)) = 1.897

b = 3×0.0162×((0.271×18×(3)²) + (2×0.271×18×3×5) + (0.271×18×(5)

b = 15.173

D³−aD²−bD−C = 0

Therefore, D = 5.45m

Example 9.5

Calculate the embedded depth if there is no groundwater level behind the wall.

9.3.1 Sheet pile wall in purely cohesive soil

For a sheet pile wall in cohesive soil, the embedded depth is given as:

where, is the effective surcharge at the level of the dredged line, c is the cohesion of soil.

Example 9.6

A sheet pile wall is driven in cohesive soil, with cohesion at 10 kN/m².

The height of the sheet pile wall above the dredged line is 8 meters and the wall friction δ is 18. The backfill material is sand and its bulk density is 18 kN/m³ and f is 35º. The groundwater level is at 3 meters below the backfill level. Calculate the depth of embedment required if the factor of safety required is 1.5.

q = 18 x 3 + 8 × 5 = 94 kN/m²

R_a= ½ × 18 × 3² × K_a + 18 × 3 × K_a × 5 + ½ × 8 × 5² × K_a

R_a= K_a (451) = 0.271 × 451 = 122.221 kN/m

y = [(½ × l8 × 3² × K_a) x 6] + [(l8 × 3 × K_a × 5) × 2.5] + [(½ × 8 × 5² × K_a) × l.67]

y = 359.888

The depth of embedment required = 1.5 x 37 = 55.5m

It seems therefore that great embedded depth is required for the construction of a flexible retaining wall in purely cohesive soil. As such, either a tie-back or anchor is introduced in the flexible wall in order to reduce the embedded depth (Figure 9.18). The anchor can also be inclined,

as shown in Figure 9.19. Two common methods of analysis are available: (i) free earth support method, and (ii) fixed earth support method. The first method is generally applied for short penetration, and the second is applied for deep penetration. Details of these methods will not be discussed again here and can be found in Kaniraj (1988).

9.4 COMPUTER-AIDED STABILITY ANALYSES

Nowadays with the help of the computer, more complicated stability cases can be analyzed within a short time. There are several powerful finite different and finite element programs available in the market. The approach of the finite different programs is usually based on limit equilibrium and that of the finite element programs is either based on limit equilibrium or stress and deformation analysis. Some examples of finite different programs are (i) Stabr, developed by Duncan and Wong (ii) Stabl, developed by GEO-SLOPE International Ltd., Canada, and (iii) Geosolve (slope and wall), developed by Geosolve, UK.

Some examples of finite element programs are (i) SAGE CRISP, developed by the CRISP Consortium, Ltd., UK (ii) Plaxis, developed by Plaxis BV, the Netherlands, and (iii) FREW, developed by Oasys Geosolve, etc. Most finite element programs require sophisticated and advanced geotechnical parameters.

The conventional methods are always confined to limit equilibrium analysis, based on active and passive states, and no information about displacement is suggested.

Numerical methods can handle complex boundary conditions. In addition, information about displacements, stress and failure zones are

available. They can also handle the initial non-zero stress. Undrained, drained, seepage as well as consolidation cases also can be handled. In addition, non-linear stress-strain soil behaviors can be modeled, and construction stages can be introduced in the analysis. Details of finite element analyses using commercial softwares can be found in the relevant manuals for softwares.

Table 9.8 shows a comparison of a slope stability analysis using various software for geometry of slopes and soil parameters, as shown in Figure 9.20. It can be seen that the various methods provide slightly different values of safety factors. Some examples of program output are shown in Figures 9.21 to 9.28. Finite element programs can provide various types of deformation and stress values, as well as the safety factor using φ-c reduction method. The Plaxis programme provides values for mesh deformation, horizontal, vertical and total displacement, stresses, and bending moment. It also calculates the axial force on the anchors for retaining structures. Safety factors can also be obtained for both undrained and drained conditions by applying phi-c reduction methods.