Stability Analysis of a Cohesive Soil Slope

7. Slope Stability Analysis 161

7.4 Stability Analysis of a Cohesive Soil Slope

Determining the slope of the failure surface is one of the key requirements in slope stability analysis. It has been proved that the cohesionless soil slope has a plane failure surface and the failure surface of a straight line in the cross-section view. However, the failure surface of the cohesive soil slope tends to be a curved

surface, which is often approximated as a cylindrical surface in theoretical analysis. Thus, the safety factor of slope calculation can be simplified using the assumption of the circular arc failure surface.

This assumption for cohesive soil slope stability analysis proposes a simplified and convenient way for safety factor calculation, i.e. the circular slip surface method.

The circular slip surface method was first proposed by Petterson (1916) and was then extensively studied and improved by Fellenius (1922) and Taylor (1937). The specific methods are the Swedish circular arc method, slice method, friction circle method, total stress method, effective stress method, and stability number method (Taylor, 1937), etc. These methods have different influencing factors and application conditions, but the same assumption of the circular arc slide surface and the limit equilibrium state to calculate the safety factor of the slope. Two types of methods can be summarized for those methods: the first one is the overall stability analysis method with a circular arc failure surface for slope stability analysis, which is mainly applicable to homogeneous and simple soil slope. The second is the slice method, and it can be applied to the soil slope with non- homogeneous soil, which have a complex structure and are submerged in water. This chapter mainly introduces the Swedish slice method with a circular slip surface, the Fellenius method for determining the failure surfaces of the most dangerous surfaces, the Bishop method of the circular slip surface, and the stability number method.

7.4.1 Swedish slice method with circular slip surface

The shear strength of the cohesive soil includes friction resistance and cohesion. The cohesive soil slope does not slide along the failure surface of the cohesionless soil slope. The most dangerous failure surface of the cohesive soil slope always cuts into the interior of the soil, thus yielding a circular arc failure surface.

7.4.1.1 The conception of the slice method

The shear strength of each point on the failure surface is closely related to the normal stress at that point if the value of ϕis larger than zero. Normal stress is caused by the soil weight at each point on the failure surface. Thus, the shear strength at each point on the

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failure surface is also varied. In order to determine the magnitude of the normal stress of the failure surface (or the stress distribution on the failure surface), the commonly used method is to divide the sliding soil into several slices as a result of which the force on each slice can be easily analyzed. The formula of the safety factor can be established using the static force equilibrium conditions of each slice.

This method is called the slicing method, which was proposed and improved by the Swedish railway engineers Petersen and Fellenius.

It can be applied for the slope stability analysis problem with the circular slip surface or the non-circular slip surface and in some complex situations (such as seepage force or seismic force). The slice method is always applied to the plane strain problem (two- dimensional).

7.4.1.2 The basic ideas

Figure 7.5(a) is assumed to be the cross-section of the soil slope drawn in a certain proportion, which represents a slope sliding along the circular arc failure surface. The sliding soil mass is divided into several slices, then the forces acting on the slice include the body force of the sliceW_i, the normal force and tangential resistance of the bottom surface of the slice ¯N_i and ¯T_i, and the horizontal and vertical forces acting on both sides of the sliceE_i,X_i andE_i+1,X_i+1. Slicei was taken out for force equilibrium analysis, as shown in Fig. 7.5(b).

If the number of slices is large enough and the width of each slice is small enough. It can be assumed that ¯N_i acts on the midpoint of

Fig. 7.5. The slice and the force equilibrium conditions.

the bottom surface of the slice, and the resultant forces of the forces acting on the two sides of slice i, i.e. the normal component forces X_i, X_i+1 and the tangential forces E_i, E_i+1, can be approximately considered to be equal in magnitude and opposite in direction.

According to Coulomb’s law, the shear strength of the failure surface should be

τ_fi=c_i+σ_itanϕ_i = c_il_i+ ¯N_itanϕ_i

l_i , (7.7)

where l_i is the arc length (m) of the failure surface of the slice i,c_i is the cohesion (kPa) of the soil on the failure surface of the slice i, and ϕ_i is the friction angle (^◦) of the soil on the failure surface of the slice i. According to the definition, the formula for the safety factor (see Eq. (7.7)) can be written as follows:

K_s = T_f

T¯_i = τ_fil_i

T¯_i = c_il_i+ ¯N_itanϕ_i

T¯_i . (7.8)

Therefore, the relation between ¯T_i and ¯N_i can be figured out as T¯_i= τ_fil_i

K_s = c_il_i+ ¯N_itanϕ_i

K_s . (7.9)

The number of slices is n, and the number of unknown quantities is shown in Table 7.1. There are only two static equilibrium equations for forces, only one equation for moments, and the total number of equations is 3n, so the number of unknowns (n−2) cannot be solved. Therefore, the general soil slope stability analysis is always a statically indeterminate problem. In order to transform it into a statically determinate problem, it is necessary to simplify the assumption of the force acting on the slices.

7.4.1.3 Basic assumptions

The Swedish slice method of circular slip surface is assumed as follows:

(1) The failure surface is a cylinder and the sliding soil mass is a rigid body (no deformations).

(2) The forces acting on both sides of the slice are not considered.

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Table 7.1. Unknown quantities and the corresponding number of the circular arc slip surfaces.

Unknown quantities Number

Safety factor,Ks 1

Normal force of slice bottom, ¯Ni n Normal force between the slices,Ei n−1 Shear force between the slices,Xi n−1 Location of the force between the slices,zi n−1

Total 4n−2

According to the above assumptions, the unknowns quantities in Table 7.1 are reduced to (3n−3). We thus have only (n+ 1) unknown quantities. The value of K_s and ¯N_i can be obtained by the force equilibrium equation of each slice and the moment equilibrium equation of the whole sliding soil mass.

7.4.1.4 Derivation of a formula for the Swedish slice method with a circular slip surface

A homogeneous soil slope is used (shown in Fig. 7.6(a)) for deriv- ing the formula of the Swedish slice method with a circular slip surface.

A proportional slope cross-sectional is shown in Fig. 7.6(a). AC is a circular slip surface and the center of the circle isO, the radius is R, and sliding soil massABC is a rigid block that slides around the circle centerO along theAC surface. The sliding soil massABC can be divided into several slices with uniform width. Generally speaking, the smaller the width of the slice is, the higher the calculation accuracy will be. However, in order to simplify the calculation and meet the design requirements, the width of the slice is selected to be 2–6 m or 0.1R is usually used.

As shown in Fig. 7.6(b), a sliceiwas selected to analyze the force equilibrium. According to the basic assumption of the Swedish slice method with a circular slip surface, the forces acting on the slice i are as follows: the body force of the sliceW_i, the normal reaction ¯N_i, and the tangential reaction ¯T_i acting on the bottom surface of the slice. These forces are discussed as follows:

Fig. 7.6. Calculation scheme of the Swedish method.

(1) The gravity of the slice W_i is vertically downward, and its value is

W_i =γb_ih_i, (7.10)

whereγ is the unit weight of the soil (kN/m³) andb_i and h_i are the width and average height of the slice (m), respectively.

Let the unit weight beW_i along its centroid action line to the failure surface of the slice, and it can be divided into the normal force N_i passing through the center of the circular slip surface and the tangential force T_i acting on the circular slip surface.

If θ_i represents the intersection angle between the normal and the vertical line at the midpoint of the bottom surface of the slice, the body force of the slice in the normal direction and the tangent direction is as follows:

N_i =W_icosθ_i, (7.11) T_i =W_isinθ_i, (7.12) whereT_i is the force pushing the soil down. However, as shown in Fig. 7.6(b), if the slice iis located to the left of the vertical line through the center of the slip arc, then T_i is the slip resistance.

The value of slip resistance T_i is very small, which is conducive to stability, so it can be neglected.

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(2) The normal reaction force ¯N_i on the bottom of the slice is equal to the normal component of gravityN_i obtained by Eq. (7.12), but the direction is opposite.

(3) The maximum value of the tangential reaction ¯T_i acting on the bottom surface of the slice should be equal to the shear strength τ_fi of the slip surface multiplied by the length of the sliding arc l_i, and the direction is opposite to the sliding direction.

It is assumed that the safety factor of the slip surface at the bottom of each slice is equal to the safety factor of the whole slip surface when the slope is stable. The tangential reaction ¯T_i can be calculated from Eq. (7.9).

According to the moment equilibrium equation of the sliding soil mass, the sum of the moment in the center of the circleO generated by the external forces of sliding mass ABC should be zero, i.e.

M_oi= 0. (7.13)

The sum of the sliding moment generated by the body force W_i of each slice is

M_si =

W_iRsinθ_i =

γb_ih_iRsinθ_i. (7.14) The normal reaction on the circular slip surface passes through the center of the circle O, and no moment will be yielded. According to Eq. (7.9), the slip resistant moment generated by the tangential reaction ¯T_i on the circular slip surface is

M_ri = T¯_iR=

(cl_i+ ¯N_itanϕ)

K_s R

=

(cl_i+W_icosθ_itanϕ)

K_s R

=

(cl_i+γb_ih_icosθ_itanϕ)

K_s R. (7.15)

If the sliding moment and the anti-sliding moment are in equilibrium and Eqs. (7.14) and (7.15) are equivalent, then

γb_ih_iRsinθ_i=

(cl_i+γb_ih_icosθ_itanϕ)

K_s R. (7.16)