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Shahunyants

The Shahunyants method is a general method of slices developed on the basis of limit equilibrium. It requires a satisfying equilibrium of forces and moments acting on individual blocks. The blocks are created by dividing the soil above the slip surface by dividing planes. Forces acting on individual blocks are displayed in the following figure:

Static scheme - Shahunyants method

Each block assumes action of the following forces:

The following assumptions are adopted in the Shahunyants method to calculate the limit state on a given block:

• dividing planes between blocks are always vertical
• slope of the forces E_i acting between blocks is zero, forces act horizontally

Solution procedure:

Forces P_yi and P_xi are first transformed with the help of expressions (1) and (2) into directions of forces T_i and N_i. For a positive angle α_i (the same way as in the schema) the force P_Ni acts in the direction opposite to N_i, the force P_Qi acts in the direction opposite to T_i.

(1)

(2)

The forces acting along the slip surface segment are related by:

(3)

The force equations of equilibrium are fulfilled on the block:

The equilibrium condition in the direction normal to the slip surface segment:

(4)

The equilibrium condition in the direction parallel to the slip surface segment:

(5)

Introducing Eq. (3) into Eq. (5) gives:

(6)

Next, substituting Eq. (4) into Eq. (6) gives:

(7)

After some algebra:

(8)

Exploiting the following mathematical expression:

(9)

yields Eq. (8) in the form:

(10)

This can be modified as:

(11)

to provide the recurrent expression for E_i forces acting between blocks as:

(12)

At this stage, the analysis enters the factor of safety K_u. The factor of safety is the value which brings the forces acting on individual blocks of soil into the state of limit states. This is achieved by multiplying active forces, i.e. forces contributing to sliding of the soil mass above the slip surface, by the factor of safety. Active forces are in Eq. (12) contained within the term P_Qi. This term contains, on the one hand, active forces that contribute to sliding and on the other hand forces that resist sliding. The contributing forces will be denoted as P_Qi,sd whereas the resisting forces as P_Qi,ud. Eq. (12) then becomes:

(13)

Providing the value of P_Qi is positive then it contributes to sliding and will be assumed as active force P_Qi,sd. Providing the value of P_Qi is negative then it resists to sliding and will be assumed as force P_Qi,ud. Therefore subtracting the value P_Qi,ud, which is negative, in Eq. (13) means essentially adding the positive value, so we can formally write:

(14)

At the slip surface origin the value of E₀ = 0. The value of E₁ is then given by:

(15)

The value of E₂ is then given by:

(16)

Similarly, we may determine the values of all forces acting between blocks. It further holds at the endpoint of the slip surface we have E_n = 0. Exploiting the previous expressions this can be written as:

(17)

This equation directly provides the factor of safety K_u in the form:

(18)