Critical α — sweep

The failure plane dip α is the most influential single parameter for FS. The α critical search automatically sweeps α through the kinematically admissible range and finds the value that minimises FS.

Why

In a real rock mass, the discontinuity that triggers planar failure can have a range of orientations. To bracket the worst case, the designer wants to know:

• the α that gives the lowest FS
• the corresponding FS_min

If FS_min ≥ FS_required, the wedge is safe under all admissible α; if not, the design must be revised.

How to use it

In the Block geometry card, next to the α input, click α crit. The software:

1. Determines the admissible range: α ∈ (ψ + 0.5°, β − 0.5°)
2. Runs a coarse sweep at 1° steps
3. Refines around the minimum at 0.05° steps
4. Sets the α field to the critical value
5. Recomputes the full model with α = α_crit

A blue banner shows:

α crit ≈ XX.X°  ·  FS_min = X.XXX  (required ≥ X.XX)  ·  range [a°…b°]

with colour:

• 🟢 green if FS_min ≥ FS_required
• 🟡 yellow if 1.0 ≤ FS_min < FS_required
• 🔴 red if FS_min < 1.0

Physical meaning

The α_crit typically lies between φ and (β − some margin). For pure friction (c = 0), the classical result is α_crit = (β + φ) / 2.

When water, seismic or reinforcement are added, α_crit can shift significantly. The sweep handles all of them automatically.

Inadmissible cases

The sweep skips α values that:

• produce a degenerate wedge (area ≤ 0)
• produce a negative resisting τ (unphysical)
• produce a negative FS (unphysical)

If the project has reinforcement (especially R1 meshes), for α very close to β the wedge becomes a thin sliver, the mesh pressure dominates the small W, and the calculation can become unphysical. The sweep filters those points; a warning is shown if no admissible α exists.

See also the verification page →