CELE Geotechnical Engineering — Shear Strength of SoilsSummary

The Mohr-Coulomb criterion is the foundational equation for predicting soil failure. It states that shear failure occurs when the shear stress (τ) on a plane reaches a value proportional to the normal stress (σ) acting on that plane, plus a material property called cohesion (c). Mathematically: τ_f = c + σ tan(φ), where φ is the angle of internal friction. For saturated soils, we must distinguish between total stress (τ_f = c + σ tan(φ)) and effective stress (τ_f = c' + σ' tan(φ')), where σ' = σ − u (u = pore water pressure). The effective stress form is preferred in modern geotechnical practice because it explicitly accounts for pore pressure effects. The criterion can be visualized graphically as a line on a Mohr circle plot, where the line is tangent to circles representing stress states at failure.

Cohesion is the intercept of the Mohr-Coulomb failure line on the shear stress axis (when normal stress is zero). It represents the shear strength that exists even under zero normal stress, arising from electrostatic attraction between clay particles, cementation, or other bonding mechanisms. Cohesion is typically non-zero in clay soils and essentially zero in clean sands and gravels. In saturated clays under undrained (fast) loading, the effective cohesion c' may differ from the total-stress cohesion c due to pore pressure buildup. Cohesion decreases with increasing pore pressure (for saturated soils) and varies with stress history and soil structure.

The angle of internal friction is the slope of the Mohr-Coulomb failure line (tan(φ) is the slope). It represents the frictional resistance between soil particles due to their interlocking and surface friction. Sands typically have φ = 30° to 45°, depending on grain shape (angular sands are stronger) and density (dense sands stronger than loose). Clays have lower φ values, typically 15° to 35°, and under undrained loading, φ = 0° (no frictional resistance because pore pressures prevent grain-to-grain contact changes). The relationship sin(φ) = (σ₁ − σ₃)/(σ₁ + σ₃) applies for cohesionless soils (c = 0) at failure, allowing direct calculation of φ from principal stresses.

The effective stress is the portion of total stress actually carried by the solid soil particles; the remainder is carried by pore water pressure. Mathematically: σ' = σ − u, where σ is total stress and u is pore water pressure. This principle, established by Terzaghi, is foundational to soil mechanics. In undrained loading of saturated clay, pore pressures increase immediately, reducing effective stress and thus reducing shear strength. In drained loading (allowing time for pore pressure dissipation), effective stresses remain stable and full strength is mobilized. The effective stress principle explains why a saturated clay is weak immediately after fast loading (undrained, φ_u = 0) but regains strength over time as water drains out (drained, φ' > 0).

Drained shear strength (c', φ') applies when pore pressures have time to dissipate during shear, typically in slow loading or long-term conditions. It is found using effective stresses: τ_f = c' + σ' tan(φ'). Undrained shear strength applies when loading is rapid and pore water cannot escape (no time to drain), as in suddenly applied loads or fast construction. For saturated clays in undrained conditions, the pore pressure increases such that the effective stress changes minimally during shearing, resulting in φ_u ≈ 0° (purely cohesive behavior with shear strength = c_u). Sands are always effectively drained because water drains freely from the pores. The undrained shear strength of saturated clay can be measured directly from the unconfined compression test: c_u = q_u/2. For design problems: (1) embankment constructed rapidly on soft clay → use undrained c_u; (2) embankment has been in place for years → use drained c', φ'.

In a direct shear test, a soil sample is placed in a square or circular shear box divided horizontally into upper and lower halves. A normal load (N) is applied vertically, creating normal stress σ = N/A (A = area). The upper half is then sheared horizontally until the sample fails, and the maximum shear force (S) is recorded, giving shear stress τ = S/A. The test is repeated for several different normal loads. Plotting τ_f vs σ yields a straight line (for most soils), and the intercept and slope give c and tan(φ). The direct shear test is simple, economical, and provides direct visualization of the failure plane. However, it cannot control drainage separately from the normal direction, and it cannot measure pore pressures. The test is most reliable for sands and is often used in quality control.

The triaxial test is the most versatile laboratory test for shear strength. A cylindrical soil sample is enclosed in a rubber membrane and placed inside a pressure chamber. A confining pressure σ₃ is applied to all sides via pressurized fluid. An axial load is then applied to the top of the sample, increasing the axial stress σ₁ until the sample shears. By varying σ₃ and measuring the deviator stress (σ₁ − σ₃) at failure, the test yields shear strength parameters over a wide range of normal stresses. Three drainage conditions are possible: (1) UU (Unconsolidated-Undrained): no drainage before or during shearing, used for quick assessment of saturated clay strength under undrained conditions; (2) CU (Consolidated-Undrained): the sample is first consolidated under σ₃ (allowing pore pressure dissipation), then sheared without drainage while measuring pore pressure; provides both total and effective stress parameters; (3) CD (Consolidated-Drained): the sample is consolidated, then sheared very slowly while allowing continuous pore pressure dissipation, yielding effective stress parameters c' and φ'. Multiple samples at different σ₃ values yield several Mohr circles at failure; the envelope of these circles defines c and φ.

The unconfined compression (UC) test is a special case of the UU triaxial test with σ₃ = 0. A cylindrical clay sample is loaded axially (without any confining pressure) until it fails. The maximum axial stress is the unconfined compressive strength q_u. For saturated clay under undrained loading, the relationship is simple: c_u = q_u/2 and φ_u = 0°. This is because the test is unconfined (σ₃ = 0) and undrained, and the symmetry of the situation yields shear strength = q_u/2. The UC test is rapid, inexpensive, and requires no confining apparatus, making it ideal for field testing and routine quality control. It is particularly useful for estimating quick undrained strength of clay without a full laboratory triaxial setup.

At any point in a stressed soil mass, stresses can be decomposed into principal stresses σ₁ (major, largest), σ₂ (intermediate), and σ₃ (minor, smallest), acting perpendicular to orthogonal planes. In axisymmetric loading (as in triaxial tests), σ₂ = σ₃. The shear stress τ_max on any plane depends on the orientation of that plane relative to the principal stress axes. The maximum shear stress is τ_max = (σ₁ − σ₃)/2, occurring on planes at 45° to the principal axes. For a soil with cohesion c and friction angle φ, the failure plane makes an angle of (45° + φ/2) with the major principal stress direction. The Mohr circle graphically represents all possible stress combinations on different planes: the principal stresses plot as points on the horizontal axis, and the circle's radius equals τ_max. For cohesionless soils (c = 0), the relationship sin(φ) = (σ₁ − σ₃)/(σ₁ + σ₃) provides a direct way to calculate φ from failure stresses.

Clean sands have negligible cohesion (c ≈ 0) under normal conditions. Their shear strength depends entirely on the friction angle φ and the normal stress: τ_f = σ tan(φ). The friction angle of sand depends on particle shape (angular particles lock together better, higher φ), grain size distribution (well-graded soils often stronger), and relative density (dense sand stronger than loose sand). Typical values: loose sand φ = 28° to 32°, medium dense φ = 32° to 36°, dense sand φ = 36° to 45°. Sand is always effectively drained because drainage is rapid; the distinction between undrained and drained behavior is irrelevant for sand. The principal stress relationship sin(φ) = (σ₁ − σ₃)/(σ₁ + σ₃) applies directly to sands. In triaxial tests on sand, the failure angle can be determined from a single test at one confining pressure.

Saturated clays present a complex picture due to the role of pore water. Under undrained loading (short-term, fast), pore pressure increases, reducing effective stress and thus reducing strength. The behavior becomes approximately φ = 0° (purely cohesive), with undrained shear strength c_u constant regardless of confining pressure. Under drained loading (long-term, slow), pore pressures dissipate, effective stresses develop normally, and the full effective strength parameters c' and φ' are mobilized. The relationship between total stress c and effective stress c' is: c = c' + u_r × tan(φ'), where u_r is the initial pore pressure ratio. However, in practice, it is standard to measure c' and φ' directly from CD tests (or infer from CU tests with pore pressure measurement) and to use these for most calculations, switching to c_u only for specific undrained (short-term) scenarios. Many clays have c' ≈ 0 (like sands) and derive their strength primarily from φ'. Normally consolidated clays often have low c' but increasing φ' with depth due to increasing effective stress.

The Mohr-Coulomb failure line is derived from the principal stresses at failure. For a cohesionless soil (c = 0), the relationship sin(φ) = (σ₁ − σ₃)/(σ₁ + σ₃) directly connects the friction angle (from the Mohr-Coulomb line slope) to the principal stresses. The failure plane orientation (45° + φ/2 from σ₁) is also derived from principal stress geometry. Thus, the Mohr-Coulomb criterion and principal stress analysis are two equivalent views of the same failure mechanism.

The effective stress principle (σ' = σ − u) underpins the drained-vs-undrained distinction. In undrained loading, pore pressure (u) increases, so effective stress (σ') decreases, and strength drops (because strength depends on σ' not σ). In drained loading, u remains low (or zero), so σ' ≈ σ, and full strength is mobilized. This relationship explains why the same clay exhibits weak behavior in short-term (undrained) conditions but stronger behavior in long-term (drained) conditions.

Different laboratory tests extract different parameters: (1) Direct shear directly yields c and φ from a plot of τ_f vs σ; (2) Triaxial UU gives total stress cu and φu ≈ 0 for clay; (3) Triaxial CU with pore pressure gives both total and effective parameters; (4) Triaxial CD yields effective c' and φ' over a range of stresses; (5) Unconfined compression gives c_u = q_u/2 for quick undrained assessment. Choosing the correct test depends on whether you need total or effective parameters and whether you are analyzing short-term or long-term conditions.

Sands derive strength from friction (φ, no cohesion) between hard particles interlocking under normal stress. Clays derive strength from both friction (φ') between particles and cohesion (c') from electrostatic attraction and cementation. Under undrained loading, clay pore pressures prevent friction from developing (φ_u ≈ 0), leaving only cohesion (c_u). Under drained loading, both mechanisms work, and full strength (c' + σ' tan(φ')) is mobilized. Sand is always effectively drained, so only drained friction governs.

For saturated soils, total cohesion c and effective cohesion c' are related by pore pressure conditions. In undrained loading, pore pressure increases, which can increase total stress cohesion measurement (apparent strength) but actually represents reduced effective stress (reduced true strength). In drained loading, pore pressure is minimal, and measured c' represents true bonding. This relationship shows why undrained c_u is not the same as drained c' and why we must be careful about distinguishing total stress from effective stress in clay analysis.

A Mohr circle represents all possible stress combinations (normal and shear) on all possible planes within a soil element at a given state of principal stress (σ₁, σ₂, σ₃). The failure envelope (Mohr-Coulomb line) separates safe stress states (inside the envelope) from unsafe states (outside). Soil fails when a Mohr circle touches or crosses the failure envelope. By conducting multiple tests at different confining pressures and plotting the corresponding Mohr circles and their common tangent envelope, we graphically determine c and φ.

The choice of laboratory test (UU, CU, or CD) and interpretation (total or effective parameters) must match the design scenario. For an embankment immediately after construction on soft clay (fast loading, no drainage time): use UU test or UC test to get c_u, and analyze with φ = 0 (undrained analysis). For the same embankment years later (pore pressures dissipated, drained conditions): use CD test to get c' and φ', and analyze with drained parameters. This mapping between field conditions and laboratory selection is critical for safe design.

The bearing capacity of a shallow foundation (footings, rafts) in clay depends on whether failure is undrained (short-term, immediate after construction) or drained (long-term, after consolidation). For short-term capacity in soft clay (typical in Philippine alluvial soils), use undrained shear strength c_u from UC tests: q_ult ≈ c_u × N_c + γD_f × N_q + 0.5 × γB × N_γ (where N_c, N_q, N_γ are bearing capacity factors modified for φ_u = 0). For long-term capacity after pore pressures dissipate, use effective stress parameters c' and φ'. In sands, use drained analysis only (always drained). The NSCP 2015 requires both short-term and long-term checks for clayey soils.

Slope stability depends on the balance between shear stress (tending to cause sliding) and shear strength (resisting sliding) along potential failure surfaces. For natural slopes in clay (e.g., cut slopes in residual soils common in the Philippines), immediately after excavation (undrained conditions), use c_u from UC tests; after years of drainage and consolidation (drained conditions), use c' and φ' from CD tests or field measurements. Sands, being drained, are analyzed using φ' only. A simple infinite slope analysis: factor of safety = (c + γh cos²β tan(φ)) / (γh cos β sin β), where h is slope height, β is slope angle. Typical exam problems ask: given c, φ, slope angle, find the factor of safety or the critical height.

Retaining walls must resist lateral earth pressure from soil behind them. The lateral pressure depends on the shear strength of the soil. For actively failing soil (wall moving away from soil), the active earth pressure coefficient K_a depends on φ' (or φ_u if undrained): K_a = (1 − sin(φ))/(1 + sin(φ)) for drained, or K_a ≈ 1 − 2c_u/(γh) for undrained clay. For walls with cohesive backfill (clays), both drained (c', φ') and undrained (c_u) conditions must be checked, depending on construction speed and drainage. The NSCP 2015 and RA 544 (Structural Code of the Philippines) mandate stability analysis of retaining walls; shear strength is the key parameter in these calculations.

Earthen embankments (common for dams, levees, and highway fills in the Philippines) are analyzed for short-term stability (during construction) and long-term stability (after consolidation). Short-term analysis uses undrained shear strength c_u of clay fill. Long-term analysis uses drained parameters c' and φ'. The factor of safety for circular-slip failure is calculated using limit equilibrium methods (e.g., Bishop's method) with shear strength computed as τ = c + σ' tan(φ) along the failure surface. Design practice: construct embankments slowly to allow drainage and consolidation if necessary, or use fill with high φ' (sand and gravel). Soft clay foundations under embankments may fail by bearing capacity or lateral squeeze; shear strength of foundation soil determines capacity.

During soil compaction for fill or subgrades, shear strength increases as the soil is densified. Higher density increases friction angle φ (more interlocking) and may increase cohesion c (better particle bonding). Quality control tests (often UC or direct shear) are used to verify that compacted soil meets minimum strength requirements. For embankments, a target undrained shear strength c_u (from UC tests) or drained friction angle φ' is specified. Wet density, water content (compaction curve), and the number of passes of rolling equipment are adjusted to achieve the target shear strength. Understanding the relationship between compaction, density, and shear strength is essential for controlling quality on construction sites.

Soft clays are prevalent in Philippine alluvial plains, deltas, and coastal areas (e.g., Manila Bay, Laguna de Bay regions). These soils have low undrained shear strength c_u (often 20–50 kPa) and are vulnerable to immediate failure if loaded too rapidly. Design strategies include: (1) light structures (reduces bearing stress), (2) piled foundations (transfers load below soft layer), (3) staged construction with drainage periods to allow consolidation and gain drained strength, (4) ground improvement (preloading, surcharging, or soil replacement). Laboratory testing of soft clay (UC and triaxial CU tests) is critical to determine c_u and c', φ' for design. The distinction between short-term undrained weakness and long-term drained strength recovery is crucial in soft clay design.

In seismic areas, loose saturated sands can lose shear strength under cyclic (earthquake) shaking, a phenomenon called liquefaction. The cyclic triaxial test (CU, with pore pressure measurement during cyclic loading) is used to assess liquefaction potential. The residual shear strength at large strains after liquefaction is much lower than the static undrained shear strength. Design against liquefaction involves increasing friction angle φ' (densification) or reducing excess pore pressures (drainage). Understanding the difference between static shear strength (used in bearing capacity) and residual strength after liquefaction is essential in seismic design per NSCP 2015 provisions.

Professional geotechnical site investigation reports (required by RA 544 for significant projects) typically include boring logs, soil classification, and laboratory test results (UC, triaxial, direct shear). Engineers reviewing these reports must understand which tests were performed and what parameters are available. A typical report might show: c_u from UC tests (for undrained analysis), c' and φ' from CD tests (for drained analysis), and standard penetration test (SPT) N-values (correlable to φ' in sands). Design engineers then select appropriate parameters based on the failure scenario (short-term or long-term) and perform stability calculations. Misinterpreting or misapplying parameters from a site report can lead to unsafe designs or overdesign (wasted cost).