GLE / Morgenstern-Price

The General Limit Equilibrium (GLE) method, in its Morgenstern-Price form, is the most general rigorous method in JW Slope. It satisfies both force and moment equilibrium and lets you choose the shape of the interslice force function. The solve runs in a C++ kernel compiled to WebAssembly.

Theory summary

• Equilibrium satisfied: complete static equilibrium — both force and moment equilibrium.

• Interslice forces: the interslice shear X and interslice normal E are related by a selectable function:

  X = lambda * f(x) * E

  where f(x) is the chosen interslice function (see Interslice Functions) and lambda is a single scalar the solver determines. lambda scales the function; f(x) sets its shape along the surface. Spencer is the special case where f(x) is constant.

• Surface type: circular or non-circular.

Solving for lambda

For any fixed lambda, the kernel iterates two parallel equilibrium states from the same starting factor:

• a force-equilibrium factor FS_force, and
• a moment-equilibrium factor FS_moment.

The interslice shear from the previous iteration enters each slice's normal-force balance. The two factors generally disagree at an arbitrary lambda. The GLE solution is the value of lambda at which they coincide:

FS_moment(lambda) - FS_force(lambda) = 0

This is the classic FS-vs-lambda crossing: plotting FS_force and FS_moment against lambda gives two curves whose intersection is the rigorous factor of safety. JW Slope can return a sampled FS-lambda curve (up to 32 points) around the solution for plotting.

Root finding

The solver first brackets the sign change of FS_moment - FS_force over the range 0 <= lambda <= 6 by sampling a grid (with extra resolution near zero), then refines with a bracket-preserving root finder. The default is the Brent-Dekker method, chosen for robustness across the regression suite; TOMS 748 is available as an alternative through result options. Refinement stops when the force/moment factors agree to within 0.005 for standard (Mohr-Coulomb) materials. A candidate is rejected if the residual |FS_moment - FS_force| exceeds the accepted tolerance (0.005 for Mohr-Coulomb, 0.02 when Generalized Hoek-Brown material is present, where the equivalent tangent strength makes the matching less smooth).

Practical guidance

• GLE / Morgenstern-Price is the method to use when you want a rigorous result and control over the interslice-force assumption. If you have no specific reason to prefer a shape, the default half-sine function is the common engineering choice; selecting constant reproduces Spencer.
• For most slopes the factor of safety is only weakly sensitive to the choice of interslice function, but it is good practice to confirm this for your case (see Interslice Functions).
• A converged GLE result reports the factor of safety, the solved lambda, the per-slice forces, and, when requested, the line of thrust and the FS-lambda plot.