Civil, Structural & Geotechnical Engineering
Bishop Simplified Method of Slices Formula: Circular Slope Factor of Safety & m_alpha
Compute Alan W. Bishop’s (1955) Simplified Method of Slices circular slope stability Factor of Safety FS = SUM(((c'*b + (W - u*b)*TAN(phi')) / m_alpha)) / SUM(W * SIN(alpha)) using vertical force equilibrium.
Why Bishop’s Simplified Method eliminates Fellenius’s 10–15% underprediction in just 2–3 iterations!
Whereas Fellenius’s Ordinary Method resolves forces perpendicular to each slice base (ignoring interslice normal forces and underpredicting FS on deep circular arcs with pore pressure), Alan W. Bishop (1955, Géotechnique) enforced exact vertical force equilibrium on every slice (zero vertical interslice shear), introducing the base-obliquity factor m_alpha,i = COS(alpha_i) + SIN(alpha_i) * TAN(phi') / FS_trial!
Because FS appears inside m_alpha,i, engineers evaluate FS_updated = SUM((c' * b_i + (W_i - u_i * b_i) * TAN(phi')) / m_alpha,i) / SUM(W_i * SIN(alpha_i)) starting from an initial Fellenius seed FS_trial: fixed-point substitution converges to 3–4 decimal places in 2 to 3 worksheet columns and matches rigorous Spencer/Morgenstern-Price circular solutions within ~1–2%!
| Slice i (c' = 15 kPa, phi' = 28 deg, FS_trial = 1.40) | Width b_i (m) | Weight W_i (kN/m) | Angle alpha_i (deg) | Pore u_i (kPa) | Bishop m_alpha,i |
|---|---|---|---|---|---|
| Slice 1 (Crest) | 4.0 | 280 | 36 | 12 | 1.0323 |
| Slice 2 (Mid-Slope) | 4.0 | 390 | 22 | 24 | 1.0693 |
| Slice 3 (Toe) | 4.0 | 210 | 8 | 15 | 1.0430 |
Bishop Base-Obliquity Factor m_alpha = COS(alpha) + SIN(alpha) * TAN(phi') / FS_trial
Evaluates m_alpha for Slice 1 (angle C2 deg, friction angle G3 deg, trial FS G4; values below 0.2 near steep passive toes flag deep-circle numerical pullout).
=IF(G4 <= 0, "Requires Positive Trial FS", ROUND(COS(RADIANS(C2)) + SIN(RADIANS(C2)) * TAN(RADIANS(G3)) / G4, 4))Updated Bishop Simplified Factor of Safety FS_updated Across Slices 1–3
Uses live per-slice m_alpha helper cells E2:E4 to compute total mobilized shear resistance divided by total driving moment weight SUM(W_i * SIN(alpha_i)).
=IF(OR(G2 < 0, G3 < 0, G4 <= 0, MIN(E2:E4) <= 0.2), "Invalid Parameters or Low m_alpha Toe", ROUND(((G2 * A2 + MAX(0, B2 - D2 * A2) * TAN(RADIANS(G3))) / E2 + (G2 * A3 + MAX(0, B3 - D3 * A3) * TAN(RADIANS(G3))) / E3 + (G2 * A4 + MAX(0, B4 - D4 * A4) * TAN(RADIANS(G3))) / E4) / (B2 * SIN(RADIANS(C2)) + B3 * SIN(RADIANS(C3)) + B4 * SIN(RADIANS(C4))), 3))Try the example
Use the original values or change them to test your understanding.