Slope Factor of Safety Bishop Simplified Calculator

The Bishop Simplified Method belongs to the family of limit equilibrium methods of slices. A trial circular slip surface is drawn through the slope, the sliding mass above it is cut into vertical slices, and the factor of safety is defined as the ratio of the shear strength available along the arc to the shear stress required for equilibrium. For each slice of weight W, base width b and base inclination alpha, the driving contribution is W sin(alpha) and the resisting contribution is (c b + W (1 - ru) tan(phi)) divided by m-alpha, where m-alpha equals cos(alpha) + sin(alpha) tan(phi) / FS. The pore pressure ratio ru is the base pore pressure divided by the total vertical overburden, running from 0 in a dry slope to roughly 0.5 in a saturated one. Because FS appears inside m-alpha, the equation is implicit and must be solved by iteration, usually converging in five to ten passes. Bishop derived the expression by satisfying vertical force equilibrium on each slice and overall moment equilibrium about the circle centre, while assuming that the shear forces between slices cancel. That assumption is why the method is called simplified, and it is the main limitation: results usually fall within about 5 percent of rigorous methods such as Spencer or Morgenstern-Price, but the method can misbehave when m-alpha approaches zero at steeply inclined slice bases, or where a non-circular weak layer controls failure. Practice requires FS of at least 1.5 for permanent slopes, about 1.3 for temporary cuts, and around 1.1 to 1.2 for pseudo-static seismic checks.

Analysis of slopes by slip circles grew out of Swedish practice after a series of quay and railway embankment failures in the early twentieth century. Following the 1913 collapse of the quay at Gothenburg, the Swedish State Railways set up a Geotechnical Commission whose 1922 report, associated with Wolmar Fellenius and Knut Petterson, formalised the circular arc method of slices. Fellenius published the ordinary or Swedish method in the 1920s and 1930s; it required no iteration because it neglected all inter-slice forces, but it can underestimate the factor of safety by 10 to 60 percent for circles with high pore pressures. Alan Wilfred Bishop, then at Imperial College London, addressed that error in his paper The Use of the Slip Circle in the Stability Analysis of Slopes, published in the first volume of Geotechnique in 1955. By enforcing vertical equilibrium on each slice he obtained a much more accurate normal force at the slice base, at the cost of an implicit equation. Bishop originally described the rigorous version including inter-slice shear, but the simplified form proved almost as accurate and far easier to compute, and it became the profession standard. The method spread further once Bishop and Norbert Morgenstern published stability coefficients in 1960 and once computers made repeated circle searches routine. It remains a required or recommended method in guidance such as the US Army Corps of Engineers manual EM 1110-2-1902 on slope stability.