CELE Geotechnical Engineering — Slope Stability and Soil ImprovementRevision Notes

Formulas

Exam Tips

• If a problem states 'undrained' or 'short-term' condition for clay, use c_u (undrained cohesion) and φ = 0.
• If a problem states 'long-term' or 'drained' condition, use c' and φ' with effective stresses.
• Always report FS to two decimal places in board-exam solutions.

Key Points

• FS is defined as the ratio of the maximum shear strength available to the shear stress mobilized along the failure surface.
• FS > 1.0 means the slope is stable; FS = 1.0 means the slope is at the verge of failure (limit equilibrium); FS < 1.0 means failure has already occurred.
• Design targets in Philippine practice: FS ≥ 1.3 for temporary cuts, FS ≥ 1.5 for permanent slopes and earth dams.
• A higher FS is required when soil parameters are uncertain, consequences are severe, or seismic loading must be considered.
• FS is always computed with respect to a specific failure surface — different assumed surfaces give different FS values; the critical surface gives the minimum FS.

Definitions

Section Title

1. Factor of Safety (FS) — The Core Concept

Common Mistakes

• Using FS = 1.3 for all cases — the required FS depends on the consequence of failure and quality of soil data.
• Forgetting that FS must be computed for multiple trial surfaces; one trial is never sufficient.
• Confusing total stress analysis (undrained, φ = 0) with effective stress analysis (drained, c', φ') — these apply to different drainage conditions.

Formulas

Exam Tips

• Memorize the two key formulas: FS_dry = tanφ'/tanβ and FS_cohesive = [c' + γz cos²β tanφ']/(γz sinβ cosβ).
• For seepage problems, replace γ with γ' in the resistance term and keep γ_sat in the driving term.
• Note: sin β cos β = (sin 2β)/2 — can simplify denominator calculations.

Key Points

• Applicable to long, uniform slopes where the failure plane is parallel to the ground surface — common in residual soil slopes and road cuts in the Philippines.
• The failure plane depth z is measured perpendicular to the slope surface.
• For a dry cohesionless slope (c' = 0), FS depends only on φ' and β — it is independent of depth z. The slope is stable as long as β < φ' (angle of repose).
• Cohesion adds resistance; the presence of seepage parallel to the slope dramatically reduces FS because pore pressure acts along the entire failure plane.
• With full seepage (water table at the surface, flow parallel to slope): FS_cohesionless = (γ'/γ_sat) × (tan φ'/tan β) ≈ 0.5 × FS_dry — seepage approximately halves FS for cohesionless soils.
• The cos²β term in the cohesive formula is frequently mis-remembered in board exams — it is cos²β, not cosβ.

Definitions

Section Title

2. Infinite-Slope Analysis

Common Mistakes

• Using cosβ instead of cos²β in the cohesive infinite-slope formula — the correct term is γz cos²β tanφ' in the numerator.
• Applying the dry formula when a seepage condition is stated — always check whether seepage is present.
• Forgetting that for a dry cohesionless slope, FS is depth-independent — adding more soil above the failure plane doesn't help.
• Using γ (total unit weight) instead of γ' (buoyant unit weight) in the friction term when seepage is present.

Formulas

Exam Tips

• In Taylor problems, if FS ≠ 1.0 is required: H_design = c/(γ N_s FS) — the FS simply appears in the denominator.
• For φ = 0 (undrained clay) quick check: N_s ≈ 0.261 for a vertical cut (β = 90°); H_cr ≈ 3.83c/γ.
• When a board problem says 'use Taylor's chart with N_s = ___', treat it as given data and apply the formula directly.
• Slice method: if the problem gives net driving moment ΣWsinα and net resisting moment, FS = Resisting/Driving — no need to track individual slices.

Key Points

• Used when the failure surface is a circular arc — applicable to most embankments, natural slopes, and earth dams.
• The failure mass above the trial circular arc is divided into vertical slices; forces on each slice are analyzed.
• Fellenius (Swedish) Method: Neglects inter-slice forces — simple but slightly conservative (underestimates FS by 5-15%).
• Bishop's Simplified Method: Accounts for horizontal inter-slice forces — more accurate and widely used in practice.
• Multiple trial circles must be analyzed; the one giving minimum FS is the critical circle.
• Taylor's Stability Number (Ns) provides a rapid solution for homogeneous slopes — ideal for board exam problems.
• Critical height H_cr is the maximum height of an unsupported slope at FS = 1.0.
• For a vertical cut in undrained clay (φ = 0): H_cr = 4c_u/γ (Terzaghi's approximation).

Definitions

Section Title

3. Finite Slope Analysis — Method of Slices and Taylor's Chart

Common Mistakes

• Using only one trial circle and reporting that FS — always the minimum governs.
• Forgetting to include pore pressure (u) when computing effective normal force N' in the slices method.
• Treating N_s as having units — it is dimensionless; H_cr = c/(γ N_s) gives meters only when c is in kPa and γ in kN/m³.
• Applying Taylor's chart results for φ = 0 (pure clay) to a c-φ soil without checking the correct φ curve.

Exam Tips

• Board exam questions often ask: 'Which improvement method is best for soft clay?' — Answer: Preloading with PVDs (wick drains) or stone columns.
• For liquefaction mitigation in saturated loose sand: vibroflotation, stone columns, or dynamic compaction.
• For existing slope stabilization: soil nailing, drainage (interceptor drains), or reinforced retaining wall.
• Know the drainage path shortening effect of PVDs: t_drain ∝ H² (Terzaghi), so halving the drainage path reduces time to 90% consolidation to one-quarter.

Key Points

• Soil improvement (or ground improvement) is used when the in-situ soil cannot meet strength, stiffness, or compressibility requirements for the intended structure.
• Four broad categories: Densification, Consolidation Acceleration, Reinforcement, and Stabilization.
• Selection depends on soil type: densification works well for sands; preloading and vertical drains work for soft clays; grouting for fissured rock or loose sand.
• In the Philippines, soft marine clay deposits underlie much of Metro Manila, Pampanga, and coastal cities — preloading with vertical drains is a common solution.
• Geosynthetics (geogrids, geotextiles) are widely used in road embankments over soft ground.
• Soil nailing is common for highway cut slopes in rock and stiff soil (e.g., NLEX, SCTEX slopes).
• Dewatering (wellpoints, deep wells) temporarily improves slope stability by lowering the phreatic surface.
• Lime and cement stabilization are used for subgrade improvement in road construction — relevant to Philippine road projects.

Definitions

Section Title

4. Soil Improvement Techniques

Common Mistakes

• Recommending vibroflotation for clay — it is effective only in clean sands and gravels; clay's low permeability prevents drainage during vibration.
• Confusing geotextile (fabric — for filtration/separation) with geogrid (grid — for reinforcement).
• Thinking preloading alone (without PVDs) is always practical — for thick clay layers, consolidation without drains may take decades.
• Forgetting that lime stabilization works by reducing plasticity (cation exchange) — it does not simply act as a cement binder.

Formulas

Exam Tips

• In the PRC board exam, multiple-choice answers are spaced close together — rounding errors can lead to a wrong answer. Maintain precision throughout.
• If two answers look very close, re-examine whether you used cos²β or cosβ — this is the most common single-point error in this topic.
• Time management: infinite slope problems should take 3-5 minutes; Taylor problems 2-4 minutes; slice problems may take 8-12 minutes — budget accordingly.

Key Points

• Always identify the slope type first (infinite or finite), soil type (c=0, φ=0, or c-φ), and drainage condition (drained or undrained).
• Extract given values carefully: note units (kPa vs kN/m², degrees vs radians).
• Write the governing formula, substitute, and compute — show all intermediate steps for partial credit.
• Verify that the computed FS is reasonable: FS between 1.0 and 3.0 is typical; outside this range, recheck.

Section Title

5. Worked Board-Exam Problems

Common Mistakes

• In Exercise 2, using cos30° instead of cos²30° = (cos30°)² = (0.866)² = 0.750 — a single-step error that changes the entire answer.
• In Exercise 3, applying the formula as H_cr = c/(γ N_s) and forgetting that H_cr is the height at FS=1; for FS=1.5, multiply N_s by FS in the denominator or divide H_cr by FS.
• Rounding intermediate values too aggressively — carry at least 4 significant figures through the calculation.