Operations, Logistics & Engineering
Fellenius Ordinary Method of Slices (Swedish Circle) Slope Stability Factor of Safety
Calculate the non-iterative circular-slip Slope Stability Factor of Safety FS = SUM(c'*l_i + MAX(0, W_i*cos(alpha_i) - u_i*l_i)*tan(phi')) / SUM(W_i*sin(alpha_i)) using Wolmar Fellenius’s (1927/1936) Ordinary Method of Slices.
Why Fellenius’s non-iterative Swedish Circle formula is the closed-form benchmark (and seed!) for every slope-stability solver!
Developed by Swedish Harbor Commission engineer Wolmar Fellenius (1927/1936), the Ordinary Method of Slices (OMS / Swedish Circle Method) divides a potential circular slip mass into vertical soil slices of weight W_i (kN/m), base inclination angle alpha_i (degrees), base arc length l_i = b_i / cos(alpha_i) (m), and pore-water pressure u_i (kPa). By neglecting interslice normal and shear resultants, moment equilibrium about the slip-circle center yields a direct, NON-ITERATIVE closed-form Factor of Safety—unlike Bishop’s Simplified Method, which has FS on both sides of the equation!
Summing resisting shear forces along the arc divided by driving tangential gravity components gives FS = SUM(c' * l_i + MAX(0, W_i * COS(alpha_i) - u_i * l_i) * TAN(phi')) / SUM(W_i * SIN(alpha_i)), where clamping effective normal force N'_i = MAX(0, W_i*cos(alpha_i) - u_i*l_i) >= 0 prevents unphysical negative friction at high pore pressures near the toe! Because OMS is ~5–15% conservative relative to Bishop, FS >= 1.50 provides a safe static design gate!
| Fellenius Factor of Safety FS | Geotechnical Slope Stability Tier | Engineering Action |
|---|---|---|
| FS < 1.00 | Active Limit-Equilibrium Failure | Immediate toe buttress, horizontal drains, or regrading |
| 1.00 <= FS < 1.50 | Marginal Static Stability (< 1.50 USACE/Eurocode Target) | Verify with Bishop/Spencer & lower phreatic pore pressure u_i |
| FS >= 1.50 | Adequate Long-Term Static Stability | Passes static slope factor of safety benchmark |
Fellenius Ordinary Method of Slices Factor of Safety FS Across 3 Slip Slices
Sums cohesive resistance c'*l_i plus frictional resistance N'_i*tan(phi') across slices 2:4 and divides by total driving tangential force SUM(W_i*sin(alpha_i)).
=IF(OR(G2 < 0, G3 < 0, G3 >= 50, SUM(E2:E4) <= 0), "Invalid Soil Strength or Driving Moment", ROUND((G2 * SUM(C2:C4) + TAN(RADIANS(G3)) * SUM(D2:D4)) / SUM(E2:E4), 3))Static Slope Stability Design Check (FS >= 1.50 Long-Term Benchmark)
Classifies Fellenius FS into Unstable (< 1.0), Marginal (1.0 to < 1.5), or Stable (>= 1.50).
=IF(OR(G2 < 0, G3 < 0, G3 >= 50, SUM(E2:E4) <= 0), "Invalid Soil Strength or Driving Moment", IF((G2 * SUM(C2:C4) + TAN(RADIANS(G3)) * SUM(D2:D4)) / SUM(E2:E4) < 1, "Unstable Slope (FS < 1.00 — Failure Risk!)", IF((G2 * SUM(C2:C4) + TAN(RADIANS(G3)) * SUM(D2:D4)) / SUM(E2:E4) < 1.5, "Marginal Static FS (1.00 to < 1.50 — Add Drainage/Berm)", "Adequate Long-Term Static Stability (FS >= 1.50)")))Try the example
Use the original values or change them to test your understanding.