## Fellenius' Slice Method

### Method : a. Stability for a Given Circle

A slope intersecting multiple soil layers (properties: C i C_i , γ i %gamma _i , φ i %varphi _i ). The stability analysis assumes plane strain conditions (two-dimensional problem).

Division of a slope into elementary slices.

For an arbitrary slip circle with center O and radius R, we evaluate the safety factor against sliding. The method involves dividing the potentially unstable soil mass above the arc EMF into vertical slices.

Key Observations:

- The slicing should ensure that any intersection between the slip circle and layer boundaries coincides with slice boundaries

- Field experience demonstrates that satisfactory accuracy can be achieved with a limited number of slices

For any given slice ABCD with total weight W i W _i , the acting forces include:

Weight components:

Normal component ( N i N_i ) perpendicular to the circular arc AB

Tangential component ( T i T _i ) parallel to AB

Boundary forces:

Subgrade reaction ( R i R _i ) along arc AB

Neighboring slice reactions:

Horizontal components ( H i H_i , H i + 1 H _{i+1} )

Vertical components ( V i V _i , V i + 1 V _{i+1} )

Pore water pressure force ( U i U_i ) acting along AB

Forces acting on a typical slice ABCD.

Fellenius[1] introduced a key hypothesis that significantly simplifies calculations by considering that the internal forces H i H _i , H i + 1 H _{i+1} , V i V _i , and V i + 1 V _{i+1} become self-balancing when analyzing the complete slice system. Their moment contributions are neglected, yielding:

H i − H i + i = 0 H _i-H _{i+i}=0

V i − V i + i = 0 V _i-V _{i+i}=0

Consequently, only two forces remain active on arc AB:

The soil weight W i W _i

The resultant reaction Rᵢ (where W i = R i W _i= R_i )

leaving only W i = R i W _i= R_i

Driving moment, Generated solely by tangential component T i T _i is equal to T i . R T_i.R (Normal component N i N_i passes through center O, producing zero moment).

Resisting moment: Maximum available shear resistance along AB

According to Coulomb's failure criterion:

( R i ) t = C i ⋅ AB + N i ⋅ tan φ i (R_i)_t =C_i cdot AB+ N_i cdot tan %varphi_i

Sum of moments for all slices:

∑ i = 1 I = m R ( C i ⋅ AB + N i ⋅ tan φ i ) sum from{i=1} to{I=m} R(C_i cdot AB+ N_i cdot tan %varphi_i )

Where:

m: Total number of slices, cᵢ' and ϕᵢ': Effective shear strength parameters (cohesion and friction angle) of the layer containing arc AB.

The failure surface being bounded by points E and F, the global safety factor Fₛ is defined by the ratio:

F s = ∑ EF Maximum Resisting Moments ∑ EF Driving Moments F_s= { sum from{EF} Maximum Resisting Moments } over { sum from{EF} Driving Moments }

The factor of safety (Fₛ) is then defined as:

F s = ∑ i = 1 i = m ( C i ⋅ AB + N i ⋅ tan φ i ) ∑ i = 1 i = m T i = ∑ i = 1 i = m ( C i ⋅ ( b cos α ) + W i ⋅ cos α tan φ i ) ∑ i = 1 i = m W i ⋅ sin α F_s= { sum from{i=1} to{i=m} ( C_i cdot AB+N_i cdot tan %varphi_i ) } over { sum from{i=1} to{i=m} T_i } ={ sum from{i=1} to{i=m} ( C_i cdot ({b} over {cos %alpha}) +W_i cdot cos %alpha tan %varphi_i ) } over { sum from{i=1} to{i=m} W_i cdot sin %alpha}

Important Notes:

In the safety factor ( F s F_s ) formula, ∑ i = 1 i = m T i sum from{i=1} to{i=m} T_i represents an algebraic sum.

F s F_s can be directly applied to mechanical properties through strength reduction:

C i * = C i F sa tan φ i * = tan φ i F sa C_{i}^{ %etoile } = C_i over F_sa ~ newline tan %varphi_i^{ %etoile } = tan %varphi_i over F_sa

Where:

C i , φ i C_i, %varphi_i : Design cohesion and friction angle

F sa F_sa : Required minimum safety factor

The slope stability condition then simplifies to:

∑ i = 1 i = m ( C i * ⋅ AB + N i ⋅ tan φ i * ) ∑ i = 1 i = m T i > 1 {sum from{i=1} to{i=m} (C_i^%etoile cdot AB+ N_i cdot tan %varphi_i^%etoile)}over{sum from{i=1} to{i=m} T_i } ` ` ` > 1

### Note :

Suitable for rapid assessments; used to find critical slip circle with lowest safety factor ( F s F_s ).

### b. Determining the Minimum Safety Factor

To identify the true safety factor ( F s F_s ) of a slope, the critical slip circle yielding the lowest F s F_s must be found, as failure is most likely to occur along this surface. There is no exact analytical method to predict the position of this critical circle a prior analysis.

### Simulation :

Industry-Standard Approach:

Most commercial software (e.g., Geo-Slope[2], shown in the figure below) uses an iterative trial-and-error process:

1. Grid Search: Computes F s F_s for numerous slip circles

2. Geometric Constraints: Tests only topographically plausible failure surfaces

3. Optimization: Identifies the circle with the minimal F s F_s as critical

Sample slope stability analysis output from Geo-Slope software.

## Notes

1. 3

   Fellenius, W. (1936). Calculation of the Stability of Earth Dams. Transactions of the 2nd Congress on Large Dams, Washington, Vol. 4, pp. 445–462.

2. 2

   www.geo-slope.com — Information about slope stability software and analysis methods.