Bishop Simplified

The Bishop Simplified method is the default analysis method in JW Slope. It is a moment-equilibrium method of slices for circular slip surfaces and is one of the most widely used and trusted methods in routine slope-stability practice.

Theory summary

The factor of safety is obtained from moment equilibrium of the entire sliding mass about the center of the trial circle.

• Equilibrium satisfied: overall moment equilibrium (about the circle center) and vertical force equilibrium of each slice. Horizontal force equilibrium of the whole mass is not explicitly satisfied.
• Interslice forces: interslice forces are assumed horizontal — that is, the interslice shear force is taken as zero. Only the interslice normal force is implied. This single assumption closes the problem.
• Surface type: circular.

Because the available base shear depends on the base normal force, and the normal force in turn depends on the mobilized shear (which depends on FOS), the method is iterative. Each slice normal force is found from a vertical force balance in which the term

m_alpha = cos(alpha) + sin(alpha) * tan(phi) / F

appears in the denominator (alpha is the base inclination, phi the friction angle, F the current factor of safety). A new factor of safety is then computed from moment equilibrium, and the cycle repeats until F converges. In JW Slope the iteration converges when successive factors of safety differ by 1e-4, with a hard limit of 100 iterations.

The initial factor of safety is 1 when Steffensen acceleration is enabled (default) and 1.5 otherwise. Steffensen's method only accelerates the same fixed-point iteration; it does not change the converged result.

Applicability

• Use for circular surfaces. Bishop Simplified is formulated for circular slip surfaces and is the natural companion to the circular grid-and-radius search.
• For non-circular or block surfaces, prefer a rigorous method (Spencer or GLE / Morgenstern-Price), which satisfies full equilibrium.

Limitations

• Only one equilibrium condition (moment) and vertical slice equilibrium are satisfied; horizontal force equilibrium of the mass is not.
• Interslice shear is neglected, so the method does not resolve the internal stress state the way the rigorous methods do.
• The m_alpha term can become small or negative for steep base inclinations (typically near the toe), which makes the slice normal force unreliable.