Slope stability analysis: Bishop, Spencer, Morgenstern-Price.

Why global slope stability matters for retaining walls

Most retaining-wall failures in the historical record are not pure overturning, sliding, or bearing failures. They are compound failures in which a slip surface initiates at depth, propagates around the wall foundation, and emerges at the retained soil surface behind the wall. The wall rides the failure as a rigid body, often coming away intact while the slope around it collapses. Examples: the 1993 Highland Towers tower-1 collapse (Hulu Klang) was a global slope failure; the 2008 Bukit Antarabangsa landslide involved global failure of the slope on which retaining walls had been built.

The mathematical reason: external stability checks examine the wall as a free body, with the soil treated as a load. Global stability examines the wall and the soil as a single body, with internal slip surfaces examined. The two analyses look at different failure mechanisms.

The limit-equilibrium method (the foundation of all three approaches)

All slope-stability methods start with the same idea: assume a slip surface, compute the ratio of resisting forces to driving forces along that surface, and the result is the FoS for that surface. Repeat for many trial surfaces; the lowest FoS is the design value, and the corresponding surface is the "critical" slip surface.

Mathematically, on a circular slip surface of radius R passing through the slope:

FoS = (Σ resisting moments) / (Σ driving moments)

The slip surface is divided into vertical slices. Each slice has weight W, base length l, base inclination α from horizontal, and shear resistance τ available along its base. The driving moment is W · sin α · R; the resisting moment is τ · l · R, where τ = c' + (σ_n − u) · tan φ'.

The difficulty: the normal stress σ_n on each slice base depends on the inter-slice forces between slices, which the slice-by-slice formulation does not directly solve. The three methods differ in how they handle this indeterminacy.

The three methods, ranked

Bishop's simplified method (1955)

Bishop's simplified method assumes the inter-slice forces are purely horizontal and equal in magnitude on both sides of each slice. This gives a closed-form expression for normal stress on each slice base, and FoS becomes the solution to a one-variable iterative equation. Easy to compute by hand or spreadsheet.

Restricted to circular slip surfaces. Accurate for natural slopes where the critical slip surface is approximately circular, which describes most homogeneous slope failures.

Where it falls down: layered soils with strong contrast in shear strength, slopes with weak horizontal layers, slopes with internal reinforcement (MSE walls). For these cases the critical slip surface is not circular, and Bishop will miss or under-estimate it.

Spencer's method (1967)

Spencer's method assumes the inter-slice forces have a single constant inclination across all slices (the inclination is determined by the analysis, not assumed in advance). This produces both force and moment equilibrium and gives a unique FoS for an arbitrary slip surface shape.

Spencer is the rigorous benchmark for slope stability and is the default in modern commercial software. Slightly more computationally intensive than Bishop but feasible on any computer.

Morgenstern-Price method (1965)

Morgenstern-Price is similar to Spencer but assumes a more general functional form for the inter-slice force inclination (the inclination can vary along the slip surface following an analyst-chosen function f(x), where x is the distance along the slip surface). The analyst typically uses a half-sine function f(x), which is the practical default in commercial software.

Equivalent in accuracy to Spencer for most geometries. Slightly more flexible for unusual geometry. Same computational class.

Other methods

• Janbu's simplified method, for non-circular slip surfaces, force-equilibrium only (no moment), conservative bias. Use as a check.
• Fellenius / Swedish slice method, ignores inter-slice forces entirely. Conservative bias, mostly of historical interest now.
• Finite-element strength reduction analysis, the modern alternative to limit-equilibrium. PLAXIS, ABAQUS, FLAC. Captures stress-deformation effects but more demanding on input parameters.

What conditions to analyse

Effective vs total stress

For long-term permanent stability of granular fills and weathered residual soils, use effective stress analysis with drained strength parameters c', φ', and an assumed groundwater table position.

For short-term temporary stability of cohesive soils (during construction, before consolidation), use total stress analysis with undrained shear strength c_u and φ_u = 0. Total stress is the conservative-bound check during cut excavation.

For Malaysian residual soils, both checks are required: effective for permanent works, total for temporary cut faces.

Groundwater conditions

Three groundwater scenarios are typically analysed:

1. Dry / dry-season: groundwater table well below the slip-surface depth, no pore-pressure contribution
2. Long-term steady-state: typical wet-season groundwater table, with flow lines computed by a seepage analysis
3. Saturated / 1-in-25-year rainstorm: groundwater raised to near the surface, maximum pore pressure on the slip surface

The saturated case is design-critical for Malaysian slopes and is the case where FoS often drops below 1.0 if drainage and reinforcement are inadequate.

Seismic loading

For Malaysian sites, pseudo-static seismic analysis with horizontal coefficient k_h = 0.05 to 0.10 (peninsular) or 0.10 to 0.15 (Sabah). The seismic coefficient adds a horizontal body force k_h · W to each slice, increasing the driving moment and reducing FoS.

Target seismic FoS ≥ 1.1 to 1.2 with the design seismic event. For high-consequence sites, dynamic time-history analysis with finite-element software may be required.

Including reinforcement in the analysis

For an MSE wall or a soil-nail wall, the reinforcement elements are tensile members that cross potential slip surfaces. The slope stability analysis includes their resistance as an additional resisting force.

MSE reinforcement

Each reinforcement layer is modelled as a horizontal tensile force at its depth. The tensile capacity is the smaller of:

• The long-term tensile design strength of the reinforcement (T_d = T_ult / γ partial factors)
• The pullout resistance behind the slip surface (P_r = friction or anchorage contribution behind the slip)

For AnchorSOL®'s anchored MSE, the pullout resistance depends on whether the slip surface intersects ahead of or behind the deadman block. If ahead, the full passive resistance of the deadman is available; if behind, only the friction along the tendon up to the slip surface contributes.

Soil nails

Each nail is modelled as an inclined tensile force at its installation angle (typically 10 to 30° from horizontal). Tensile capacity is the smaller of bar tensile strength, grout-bar bond, and grout-soil bond.

Tied-back anchors (post-tensioned)

Each anchor contributes its applied prestress force as a resisting force along its angle of installation. The anchor's design load capacity (typically 60 to 75% of ultimate tendon strength) is the resisting force.

Typical FoS targets in Malaysian practice

Software in Malaysian practice

The standard tools for slope stability analysis on Malaysian engineering practice:

• Slide2 / Slide3 (Rocscience), the most-common limit-equilibrium tool. Bishop, Spencer, Morgenstern-Price, Janbu all available. Supports reinforcement, distributed loads, seismic, water table, multiple soil layers.
• SLOPE/W (GeoStudio), the long-time alternative. Same method coverage, broadly comparable workflow.
• PLAXIS 2D / 3D, finite-element analysis. Strength-reduction-FoS option for slope stability, plus deformation analysis. Used for high-consequence projects.
• FLAC / FLAC3D, finite-difference analysis. Specialist use, large-deformation problems.
• Geo5, integrated geotechnical suite popular in some Malaysian practices.

For routine retaining-wall design, Slide2 or SLOPE/W is sufficient. PLAXIS or FLAC is justified where deformation prediction matters (sensitive existing structures), where complex soil constitutive models are needed (creep, anisotropy), or where the project is high-consequence.

The check sequence for an AnchorSOL® hillside wall

1. Characterise the soil profile from the site investigation: layered strength, unit weights, groundwater table, residual vs intact strength parameters
2. Define the wall geometry: facing height, embedment depth, reinforcement length and spacing, deadman position
3. Run drained-condition slope stability with Bishop's simplified for the initial check, then Spencer or Morgenstern-Price for the critical surface refinement. Include reinforcement.
4. Run saturated-condition slope stability with raised groundwater table. Re-check FoS targets.
5. Run seismic pseudo-static analysis for sites with design seismic event > M5.5
6. Iterate the wall design: adjust reinforcement length, deadman position, or wall geometry until FoS targets are met in all conditions
7. For high-consequence sites: run finite-element verification in PLAXIS with strength-reduction option, plus deformation prediction

The output: a design report with critical slip surface, FoS in each condition, reinforcement layout, and the input parameters justified from the site investigation. This is the engineering certification deliverable for the project.

Frequently asked questions

What is the difference between FoS and partial factor design?

FoS (factor of safety) is a single overall ratio of resisting to driving forces. Partial factor design applies separate factors to each parameter (γ_φ on friction angle, γ_c on cohesion, γ_R on resistance, γ_f on actions). Modern codes (Eurocode 7, BS 8006) use partial factors. For routine Malaysian slope stability work, the legacy FoS framework is still common because of its conceptual simplicity, but the partial-factor framework is more rigorous.

How deep is the critical slip surface usually?

Depends on the slope geometry, soil layering, and reinforcement. For homogeneous slopes, the critical surface depth scales with slope height. For layered slopes, the critical surface often follows a weak layer interface. For reinforced slopes, the critical surface is often forced deeper than the reinforcement zone or above it.

Do I need to do slope stability for a flat-site retaining wall?

Yes, always. Even on a flat site, the foundation soil may have a weak layer below that initiates a deep-seated failure surface. The minimum routine check is a Bishop's simplified analysis for surfaces passing through the wall base and back up to the slope surface (or to the original ground level for flat sites). FoS > 1.5 should hold easily on competent foundations; if it doesn't, the foundation is the problem.

Can I use Bishop's method alone for an MSE wall design?

For routine projects, yes, as long as the critical surface is approximately circular. For atypical geometry (tall walls, sloping backfill, weak foundation layer below), use Spencer or Morgenstern-Price. Modern software runs both with the same input data so there's little reason not to.