FOUNDATIONS AND EARTH STRUCTURES

The components $V_n$, $V_{n+1}$, $H_n$,$H_{n+i}$ contribute to the forces acting on slice (i) .

• Vertical Force Equilibrium:

$W_i+\mathrm{\Delta }{V}_i-(N_i^{\text{'}}+u_i\cdot l_i)\cos {\mathrm{\alpha }}_i-(\frac{C_i^{\text{'}}}{F_s}{l}_i+N_i^{\text{'}}\frac{\tan {\mathrm{\phi }}_i^{\text{'}}}{F_s})\sin {\mathrm{\alpha }}_i=0$

$N_i^{\text{'}}(\cos {\mathrm{\alpha }}_i+\frac{\tan {\mathrm{\phi }}_i^{\text{'}}\sin {\mathrm{\alpha }}_i}{F_s})=W_i+\mathrm{\Delta }{V}_i-u_i{l}_i\cos {\mathrm{\alpha }}_i-\frac{C_i^{\text{'}}{l}_i}{F_s}\sin {\mathrm{\alpha }}_i$

$N_i^{\text{'}}=\frac{W_i+\mathrm{\Delta }{V}_i-u_i{l}_i\cos {\mathrm{\alpha }}_i-\frac{C_i^{\text{'}}{l}_i}{F_s}\sin {\mathrm{\alpha }}_i}{\cos {\mathrm{\alpha }}_i+\frac{\tan {\mathrm{\phi }}_i^{\text{'}}\cdot \sin {\mathrm{\alpha }}_i}{F_s}}$

• Moment Equilibrium:

Driving moment:

$\sum _1^n{W}_{i}R\sin {\mathrm{\alpha }}_i$

Resisting moment (opposing movement):

$\sum _1^{n}R(\frac{C_i^{\text{'}}{l}_i}{F_s}+N_i^{\text{'}}\frac{\tan {\mathrm{\phi }}_i^{\text{'}}}{F_s})$

By equating moments, substituting (Nᵢ'), and simplifying by R:

$F_s(\sum _1^n{W}_i\sin {\mathrm{\alpha }}_i)=\sum _1^n{C}_i^{\text{'}}{l}_i+\frac{W_i+\mathrm{\Delta }{V}_i-u_i{l}_i\cos {\mathrm{\alpha }}_i-\frac{C_i^{\text{'}}{l}_i}{F_s}\sin {\mathrm{\alpha }}_i}{\cos {\mathrm{\alpha }}_i+\frac{\tan {\mathrm{\alpha }}_i^{\text{'}}}{F_s}}$

The numerator terms can be rewritten with common denominator:

$C_{\mathit{iˇ}}\text{'}{l}_i\cos \mathrm{\alpha }+\frac{1}{F_s}{C}_i^{\text{'}}{l}_i\tan {\mathrm{\phi }}_i^{\text{'}}\sin {\mathrm{\alpha }}_i+(W_i+\mathrm{\Delta }{V}_i-u_i{l}_i\cos {\mathrm{\alpha }}_i-\frac{1}{F_s}{C}_i^{\text{'}}{l}_i\sin {\mathrm{\alpha }}_i)\tan {\mathrm{\phi }}_i^{\text{'}}$

Safety factor formula is given by:

$F_s=\frac{1}{\sum _1^n\sin {\mathrm{\alpha }}_i}\sum \frac{C_i^{\text{'}}{l}_i\cos {\mathrm{\alpha }}_i+(W_i+\mathrm{\Delta }{V}_i-u_i{l}_i\cos {\mathrm{\alpha }}_i)\tan {\mathrm{\phi }}_i^{\text{'}}}{\cos {\mathrm{\alpha }}_i+\frac{1}{F_s}\tan {\mathrm{\phi }}_i^{\text{'}}\sin {\mathrm{\alpha }}_i}$

This is called the exact Bishop formula.

Solution Procedure:

• Requires iterative calculations since Fs^[1] appears on both sides

• Requires additional assumptions to define Vᵢ

Simplified Bishop Method

Assuming$\mathrm{\Delta }{V}_i=V_i-V_{i+1}=0$, the equation  becomes:

$F_s=\frac{1}{\sum _1^n\sin {\mathrm{\alpha }}_i}\sum \frac{C_i^{\text{'}}{l}_i\cos {\mathrm{\alpha }}_i+(W_i-u_i{l}_i\cos {\mathrm{\alpha }}_i)\tan {\mathrm{\phi }}_i^{\text{'}}}{\cos {\mathrm{\alpha }}_i+\frac{1}{F_s}\tan {\mathrm{\phi }}_i^{\text{'}}\sin {\mathrm{\alpha }}_i}$

Or $F_s=\frac{1}{\sum _1^n{W}_i\sin {\mathrm{\alpha }}_i}\sum _1^n\frac{C_i^{\text{'}}{b}_i+(W_i-u_i{b}_i)\tan {\mathrm{\phi }}_i^{\text{'}}}{\cos {\mathrm{\alpha }}_i+\frac{1}{F_s}\tan {\mathrm{\phi }}_i^{\text{'}}\sin {\mathrm{\alpha }}_i}$

The expression for $F_s$is not explicit. Therefore, $F_s$ cannot be calculated directly. An implicit method will be used in the form$F_{m+1}=f(F_m)$.

The initial value of$F_s$can be taken as $F_{s0}$, the value obtained using the Fellenius method. Convergence is generally quite fast. The process is stopped when$(F_{m+1}-F_m)$ is less than a pre-defined threshold.