Reliability Analysis (FEM)

Reliability analysis quantifies the probability that a slope will not fail by propagating the uncertainty in the soil parameters through to the factor of safety. This page describes the finite-element reliability analysis, which is identical in method to the Taylor Series Probability Method used for limit equilibrium, but computes each factor of safety with the finite-element Shear Strength Reduction Method (SSRM) instead of a limit-equilibrium search.

Because the method, the parameter-uncertainty guidance (standard deviation and COV estimation, typical COV values), and the reliability equations are the same, they are not repeated here — see the Reliability Analysis overview and the Taylor Series Probability Method page for the full theory. This page covers only what is specific to the FEM version.

Method

The analysis uses the same Taylor Series Probability Method (TSPM):

1. \(F_{MLV}\) — the factor of safety at the most-likely parameter values, computed with solve_ssrm.
2. For each uncertain strength parameter, compute \(F_i^+\) and \(F_i^-\) with that parameter set to \(MLV \pm \sigma_i\) (all others held at their most-likely value), then \(\Delta F_i = |F_i^+ - F_i^-|\).
3. Combine: \(\sigma_F = \sqrt{\sum_i (\Delta F_i / 2)^2}\), \(COV_F = \sigma_F / F_{MLV}\), and finally the lognormal reliability index \(\beta_{LN}\) and the reliability \(R = \Phi(\beta_{LN})\).

The analysis runs \(1 + 2N\) SSRM solves (one at the most-likely values plus a \(F^+\)/\(F^-\) pair for each of the \(N\) uncertain parameters), all on a single shared mesh — only the material-to-element mapping is rebuilt per perturbation. The auto-expanding SSRM bracket is what makes this practical: each perturbation shifts the factor of safety, and the bracket adjusts itself so a fixed F_min/F_max does not have to bracket every perturbed case in advance.

Numerical precision and reproducibility

A plain SSRM factor of safety is the midpoint of the final bisection band, so it carries a \(\pm\text{tolerance}/2\) imprecision whose exact value depends on the starting F_min/F_max (different brackets subdivide the axis on different grids). The reliability index amplifies this: \(\dfrac{d\beta}{dF} = \dfrac{1}{F\sqrt{\ln(1+COV_F^2)}}\), which is \(\approx 9\) at \(COV_F = 0.1\) — so a small bracket-dependent change in \(F_{MLV}\) produces a proportionally larger change in \(\beta\), enough to flip the last shown digit of the reliability between two bracket choices.

To remove that entirely, reliability_fem runs each SSRM on a fixed global grid (grid = tolerance, default 0.001; see solve_ssrm). Instead of halving your bracket, it locates the single global grid cell that straddles the failure threshold — a fact of the slope and mesh, not of the bracket — so every starting bracket lands in the same cell. The result is identical to every decimal regardless of F_min/F_max, at the same cost (~log₂ solves) and precision (the grid step) as ordinary bisection.

Two things still legitimately change the result and are not numerical noise: the mesh (a finer or different-element mesh gives a slightly different factor of safety, as with any FE analysis — use a converged mesh and report which one), and genuinely larger parameter uncertainty (a higher \(COV_F\) both lowers the reliability and reduces the \(\beta\) sensitivity above).

Which parameters are perturbed

The FEM reliability perturbs the same strength parameters as the LEM reliability — the cohesion \(c\) and friction angle \(\phi\) for a Mohr-Coulomb (mc) material, or \(c\) and the rate \(c_p\) for the depth-varying undrained (cp) model, plus the unit weight \(\gamma\) for both — using the standard deviations sigma_c, sigma_phi / sigma_cp, and sigma_gamma from the materials table. Using the same standard deviations keeps the FEM and LEM reliability results directly comparable.

Why the elastic parameters E and ν are excluded

Unlike a limit-equilibrium analysis, a finite-element analysis also takes the elastic modulus \(E\) and Poisson's ratio \(\nu\) as inputs. It is conventional wisdom that these affect the computed displacements but have little effect on the factor of safety, which is governed by the strength. We confirmed this directly on Griffiths & Lane (1999) Example 1 (\(c/\gamma H = 0.05\), \(\phi = 20°\)), holding the strength fixed and varying \(E\) and \(\nu\) across their full plausible ranges:

Variation	FS	ΔFS vs. base
Base (\(E\) = 700,000, \(\nu\) = 0.30)	1.394	—
\(E \times 0.5\) (350,000)	1.394	0.000
\(E \times 2.0\) (1,400,000)	1.394	0.000
\(\nu\) = 0.20	1.406	+0.013
\(\nu\) = 0.40	1.381	−0.013

\(E\) has no effect on the factor of safety — halving and doubling it give an identical FS, because SSRM converges to the same critical strength-reduction factor regardless of the elastic stiffness. \(\nu\) changes the FS by only ~1% across its entire plausible range (0.2–0.4), which is at the resolution of the bisection itself; at a realistic \(\sigma_\nu\) (Poisson's ratio rarely varies by more than about ±0.05) its contribution to \(COV_F\) is negligible.

For these reasons \(E\) and \(\nu\) are treated as deterministic and are not perturbed, and the materials table has no sigma_E / sigma_nu columns. The same strength-parameter standard deviations drive both the LEM and the FEM reliability analyses.

Data Input

Provide the standard deviations for the uncertain strength parameters in the Materials table exactly as for Taylor series reliability — the main parameter values are the most-likely values, and sigma_c, sigma_phi / sigma_cp, and sigma_gamma give their standard deviations. At least one non-zero standard deviation is required. As in the LEM case, a standard deviation may not exceed its mean (a negative \(MLV - \sigma\) is non-physical and the analysis stops with an error).

Usage

In XSLOPE Studio, build a mesh, then choose Run FEM → Analysis: Reliability (SSRM). It uses the F min/F max bracket from the dialog (auto-expanding) and the material standard deviations from the mat sheet; the bisection tolerance, however, is set internally to the tight reliability default (not the dialog's single-run Tolerance field — see Numerical precision). The FEM Results view shows the deformation at the most-likely values, the reliability summary appears in a dialog and the status bar, and the full per-parameter ΔF table is written to the Log pane. See Reliability analysis for the dialog and the engine comparison shared with the Taylor series and Monte Carlo pages.

From Python:

from xslope.fileio import load_slope_data
from xslope.advanced import reliability_fem

slope_data = load_slope_data("my_slope.xlsx")
success, result = reliability_fem(
    slope_data,
    element_type="quad8",   # quadratic elements (tri6/quad8/quad9); see Overview
    target_size=3.5,        # omit to auto-size from the domain width
    F_min=1.0, F_max=2.0,   # starting bracket; auto-expands if the guess is off
    # tolerance defaults to a tight 0.001 here (see Numerical precision above)
)
# reliability_fem already prints the per-parameter ΔF table and the summary
# (F_MLV, COV_F, beta, reliability, Pf); all of these are also in `result`.

reliability_fem returns the same reliability keys as the LEM reliability function (F_MLV, sigma_F, COV_F, beta_ln, reliability, prob_failure, param_info), plus mlv_solution (the SSRM result at the most-likely values) and the mesh used for every trial. Only the most-likely-values solve — the one result that is plotted — captures the at-failure mechanism for the deformation/vector/strain figures; the perturbation solves report only their factor of safety. Building the mesh is the same as for a single SSRM run: it is built here from the geometry (or an attached slope_data['mesh'] / a mesh= argument is reused), and all \(1 + 2N\) trials share it so the only thing that changes between trials is the perturbed material property. Because the factor of safety — and hence the reliability — depends on the mesh, report the element type and size used for a given result.