FOUNDATIONS AND EARTH STRUCTURES

A slope intersecting multiple soil layers (properties: $C_i$,${\mathrm{\gamma }}_i$,${\mathrm{\phi }}_i$). The stability analysis assumes plane strain conditions (two-dimensional problem).

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 $W_i$, the acting forces include:

Weight components:

Normal component ($N_i$) perpendicular to the circular arc AB

Tangential component ($T_i$) parallel to AB

Boundary forces:

Subgrade reaction ($R_i$) along arc AB

Neighboring slice reactions:

Horizontal components ($H_i$, $H_{i+1}$)

Vertical components ($V_i$,$V_{i+1}$)

Pore water pressure force ($U_i$) acting along AB

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

$H_i-H_{i+i}=0$

$V_i-V_{i+i}=0$

Consequently, only two forces remain active on arc AB:

The soil weight $W_i$

The resultant reaction Rᵢ (where $W_i=R_i$)

leaving only $W_i=R_i$

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

Resisting moment: Maximum available shear resistance along AB

According to Coulomb's failure criterion:

${(R_i)}_t=C_i\cdot \mathit{AB}+N_i\cdot \tan {\mathrm{\phi }}_i$

Sum of moments for all slices:

$\sum _{i=1}^{I=m}R(C_i\cdot \mathit{AB}+N_i\cdot \tan {\mathrm{\phi }}_i)$

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:

$F_s=\frac{\sum _{\mathit{EF}}\mathit{Maximum}\mathit{Resisting}\mathit{Moments}}{\sum _{\mathit{EF}}\mathit{Driving}\mathit{Moments}}$

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

$F_s=\frac{\sum _{i=1}^{i=m}(C_i\cdot \mathit{AB}+N_i\cdot \tan {\mathrm{\phi }}_i)}{\sum _{i=1}^{i=m}{T}_i}=\frac{\sum _{i=1}^{i=m}(C_i\cdot (\frac{b}{\cos \mathrm{\alpha }})+W_i\cdot \cos \mathrm{\alpha }\tan {\mathrm{\phi }}_i)}{\sum _{i=1}^{i=m}{W}_i\cdot \sin \mathrm{\alpha }}$

Important Notes:

In the safety factor ($F_s$) formula, $\sum _{i=1}^{i=m}{T}_i$ represents an algebraic sum.

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

$\begin{array}{c}{C}_i^{\mathrm{*}}=\frac{C_i}{F_{\mathit{sa}}}\\ \tan {\mathrm{\phi }}_i^{\mathrm{*}}=\tan \frac{{\mathrm{\phi }}_i}{F_{\mathit{sa}}}\end{array}$

Where:

$C_i,{\mathrm{\phi }}_i$ : Design cohesion and friction angle

$F_{\mathit{sa}}$ : Required minimum safety factor

The slope stability condition then simplifies to:

$\frac{\sum _{i=1}^{i=m}(C_i^{\mathrm{*}}\cdot \mathit{AB}+N_i\cdot \tan {\mathrm{\phi }}_i^{\mathrm{*}})}{\sum _{i=1}^{i=m}{T}_i}>1$