CELE Geotechnical Engineering — Stresses in Soil MassStudy Notes

The effective-stress principle is the cornerstone of soil mechanics. Terzaghi's principle states that the total vertical stress at any depth in a soil mass is shared between two components: the stress carried by the soil skeleton (effective stress σ') and the stress carried by pore water (pore pressure u). This relationship is expressed mathematically as: σ = σ' + u Rearranging: σ' = σ - u Where: - σ (sigma) = total vertical stress at a given depth (kPa) - σ' (sigma-prime) = effective stress carried by soil particles (kPa) - u = pore water pressure (kPa) PHYSICAL INTERPRETATION: Imagine a column of soil as a network of grains with water-filled spaces between them. The total weight of the soil column (total stress) is distributed across two load paths: the grain-to-grain contact points (effective stress) and the water within the pores (pore pressure). The water, being fluid, cannot sustain shear stress and distributes load equally in all directions (hydrostatic pressure). Only the grain-to-grain contacts can develop friction and resistance to shear. Therefore, effective stress governs all mechanical behavior: strength, settlement, stiffness, and bearing capacity. WHY EFFECTIVE STRESS MATTERS: 1. Soil strength (friction angle φ and cohesion c) depends only on effective stress, not total stress. 2. Compressibility and settlement calculations use effective stress. 3. Bearing capacity formulas are derived in terms of effective stress. 4. Consolidation theory (Terzaghi) is based entirely on effective stress. 5. Two deposits with the same total stress profile may behave very differently if their pore pressures differ. CRITICAL INSIGHT FOR BOARD EXAMS: A common exam trap is to assume that total stress alone determines soil behavior. For example, if you are asked "Will a saturated clay deposit settle more than a dry sand deposit at the same depth?" the answer depends on the pore pressure and effective stress, not just the total stress. A thick layer of water above the soil can significantly reduce effective stress and settlement potential.

Total stress at any depth is simply the cumulative weight of all material above that point, divided by the area. For a vertical column (which is the standard assumption), we calculate by summing the weight contributions layer by layer. FORMULA: σ = Σ(γ_i × z_i) Where: - γ_i = unit weight of the i-th layer (kN/m³) - z_i = thickness of the i-th layer (m) CRITICAL RULE FOR UNIT WEIGHT SELECTION: 1. ABOVE the water table: Use γ (moist unit weight) or γd (dry unit weight). Typical values: γ = 16–18 kN/m³. 2. BELOW the water table: Always use γsat (saturated unit weight). Typical values: γsat = 19–22 kN/m³. 3. For submerged soil (below WT), some engineers use γ' = γsat - γw (buoyant/submerged unit weight) instead of γsat. This is mathematically equivalent: σ = γsat × z = (γ' + γw) × z, and when you subtract pore pressure u = γw × z, you get σ' = γ' × z directly. However, the standard approach (use σ = γsat × z, then subtract u = γw × z) is preferred in exams because it clearly shows the effective-stress calculation. STEP-BY-STEP PROCEDURE: 1. Identify the water table location. 2. Divide the soil profile into layers with different unit weights. 3. For each layer above the WT, use the in-situ (or moist) unit weight. 4. For each layer below the WT, use the saturated unit weight. 5. Sum the stress contributions: σ = γ₁z₁ + γ₂z₂ + ... + γₙzₙ. 6. Subtract pore pressure to get effective stress. COMMON EXAM ERRORS: - Using γsat above the water table (wrong—use γ or γd). - Using dry unit weight γd below the water table (wrong—must use γsat). - Forgetting that pore pressure acts everywhere below the WT, even in fine-grained soils. - Not recognizing that capillary tension above the WT can create negative pore pressure and increase effective stress.

Pore water pressure, also called pore pressure or neutral stress, is the pressure of water in the void spaces of soil. Understanding and calculating pore pressure correctly is essential because it directly affects effective stress and therefore all aspects of soil behavior. BASIC HYDROSTATIC PORE PRESSURE: In static conditions (no groundwater flow), pore pressure at any depth is simply the weight of the water column above that point: u = γ_w × z_w Where: - γ_w = unit weight of water = 9.81 kN/m³ (or 10 kN/m³ in quick estimates) - z_w = depth below the water table (m) IMPORTANT POINTS: 1. Pore pressure is zero above the water table (in dry soil). 2. Pore pressure increases linearly with depth below the water table at a rate of ~10 kPa per meter. 3. Pore pressure acts equally in all directions (hydrostatic); it does not depend on soil type or stiffness. 4. Pore pressure cannot sustain shear stress—only hydrostatic (normal) stress. 5. Pore pressure is independent of total stress once you are below the water table; it depends only on the water column height. CAPPLARY RISE AND NEGATIVE PORE PRESSURE: Above the water table, in fine-grained soils (clay, silt), capillary action can draw water upward, creating a capillary zone. In this zone, pore pressure is negative (suction), which increases effective stress. Heights of capillary rise vary widely: - Sand: 0.3–1 m - Silt: 1–5 m - Clay: >5 m (sometimes very high) In the capillary zone: u = -γ_w × h_c (negative, where h_c is height above the water table). Example: If capillary rise in a clay is 4 m and you are 2 m above the water table, then u = -9.81 × 2 = -19.62 kPa (suction). This increases effective stress: σ' = σ - (-19.62) = σ + 19.62 kPa. This is why clay can be very hard and stiff above the water table but becomes soft and weak when saturated. SEEPAGE AND SEEPAGE PRESSURE: When groundwater flows (seepage), additional pressure develops in the direction of flow. The seepage pressure (or seepage force) can increase pore pressure in the direction of flow and decrease it in the opposite direction. This is critical for: - Slope stability analysis (upward seepage beneath a slope) - Piping failure in dams (critical gradient) - Bearing capacity under flowing groundwater For downward seepage with hydraulic gradient i: u_seepage = u_hydrostatic + ρ_w × g × i × distance_in_flow_direction For upward seepage (most dangerous): u = γ_w × (z_w + i × distance) where i is the hydraulic gradient When upward seepage is significant, the "quick condition" (also called "boiling") can occur when the seepage force equals the weight of the soil, making it behave like a fluid. This is critical in excavation and dam analysis.

Now that we understand total stress, pore pressure, and the effective-stress principle, we can systematically calculate effective stress at any depth and construct an effective-stress profile. STEP-BY-STEP PROCEDURE: 1. Determine the soil layers and water table location. 2. Calculate total stress at the depth of interest: σ = Σ(γ_i × z_i). 3. Calculate pore pressure at the same depth: u = γ_w × z_w (or u = 0 if above WT). 4. Compute effective stress: σ' = σ - u. 5. Repeat for multiple depths to construct a profile. PROPERTIES OF EFFECTIVE-STRESS PROFILES: - Above the water table: σ' = σ (since u = 0). The effective-stress line coincides with the total-stress line. - Below the water table: σ' < σ because u > 0. The effective-stress line lags behind the total-stress line. - At the water table: σ' = σ and u = 0 (transition point). - Below the WT, as depth increases, both σ and u increase, but σ increases faster (because of soil weight). The gap between σ and σ' remains constant at γ_w × (z - z_w), which grows linearly with depth below the WT. - In a uniform soil below the WT, effective stress increases at a rate of (γ_sat - γ_w) = γ' per meter of depth. EFFECTIVE STRESS UNDER SLOPING GROUNDWATER TABLE: If the water table is not horizontal (e.g., near a river or coastal area), the vertical distance from any point to the water table varies. The effective-stress calculation remains the same—u = γ_w × (vertical distance to WT)—but the shape of the pore-pressure profile becomes non-linear when plotted against absolute depth. DRAWING EFFECTIVE-STRESS DIAGRAMS: In professional practice and on exams, effective-stress profiles are often plotted with depth on the vertical axis and stress on the horizontal axis. Two diagrams are typically drawn side by side: 1. Total stress (σ) vs. depth: A single line increasing with depth. 2. Pore pressure (u) vs. depth: u = 0 above WT, then increasing linearly below WT. 3. Effective stress (σ') vs. depth: Follows σ above WT, then diverges below WT. Alternatively, all three are plotted on the same axis (with different scales) for comparison.

When a load is applied at the ground surface (e.g., a building footing, a wheel load, an embankment), it creates stresses that propagate downward and outward through the soil. These stress increases, superimposed on the in-situ stress from the soil's own weight, determine settlement and influence stability. The most rigorous method for calculating these stress increases is the Boussinesq solution, derived for an elastic half-space under a point load. BOUSSINESQ FORMULA FOR POINT LOAD: For a concentrated vertical load Q (in Newtons or kN) applied at the ground surface, the vertical stress increase Δσ at a depth z and horizontal distance r from the load is: Δσ = (3Q / 2πz²) × [1 / (1 + (r/z)²)]^(5/2) Special case — directly below the load (r = 0): Δσ_max = 3Q / (2πz²) Where: - Q = concentrated load (kN) - z = depth below surface (m) - r = horizontal distance from the vertical line through the load (m) - Δσ = vertical stress increase (kPa) CHARACTERISTICS OF THE BOUSSINESQ SOLUTION: 1. At r = 0 (directly below the load), Δσ is maximum: Δσ_max = 1.5Q / (πz²). 2. As r increases (moving horizontally away from the load), Δσ decreases rapidly. 3. As z increases (going deeper), Δσ decreases, proportional to 1/z². 4. The stress spreads with depth—at twice the depth (2z), the stress is one-quarter as large (for r = 0). 5. The Boussinesq solution assumes the soil is elastic, homogeneous, isotropic, and semi-infinite (no bottom boundary). It is an upper bound for most natural soils. LIMITATIONS OF BOUSSINESQ: - Assumes elastic material (not plastic deformation). - Assumes homogeneous soil (same properties everywhere). - Assumes no bottom boundary (very deep deposits). - Does not account for soil layering or variations. - Does not account for time-dependent consolidation. DESPITE LIMITATIONS, Boussinesq is widely used because: - It provides a closed-form solution (no numerical integration needed). - Results are reasonably accurate for design purposes. - It is conservative (often overestimates stress increases). - It is the standard in most geotechnical textbooks and codes. FOR PRACTICE AND EXAMS: The Boussinesq formula is usually applied in two contexts: 1. Directly below a point load: Δσ = 3Q / (2πz²). 2. At an offset from the load, using the full formula with the ratio (r/z). Many reference tables and charts (influence factor charts) simplify the calculation by expressing Δσ in terms of an influence factor I_z: Δσ = I_z × (Q / z²) Where I_z is tabulated as a function of r/z ratio.

While Boussinesq is rigorous and widely used, many practical problems involve distributed loads over finite areas (footings, slabs, embankments) rather than point loads. For such cases, rigorous solutions require integration over the loaded area (very tedious) or use of influence charts. A simpler and widely accepted approximation is the 2:1 (or 45°) method, which assumes the load spreads at a slope of 2 vertical to 1 horizontal (or equivalently, at a 45° angle to the vertical). 2:1 APPROXIMATION FORMULA: For a uniformly loaded rectangular footing of dimensions B × L carrying a total load Q: Δσ = Q / [(B + z) × (L + z)] Where: - Q = total load on footing (kN) - B = width of footing (m) - L = length of footing (m) - z = depth below the footing base (m) - Δσ = average vertical stress increase at depth z (kPa) PHYSICAL INTERPRETATION: The method assumes that the loaded area, when projected to depth z, has expanded by z in all directions (spreading at 1:1 slope in plan), so the width increases from B to (B + z) and the length from L to (L + z). The stress "spreads out" over this larger area, so the intensity decreases. WHEN TO USE 2:1 VS. BOUSSINESQ: 1. Point loads (concentrated, non-distributed): Use Boussinesq. 2. Small footings (B, L are small compared to z): Use Boussinesq or influence charts. 3. Large footings (B, L are significant compared to z): Use 2:1 method or influence charts. 4. For exams, if the problem doesn't specify which method, use 2:1 for footings and Boussinesq for concentrated loads. ACCURACY OF 2:1 METHOD: - Generally gives reasonable estimates of average stress at depth z directly below the footing. - Most conservative (slightly overestimates) in the near field (z < B). - Less accurate for points far from the footing or at shallow depths. - Underestimates stresses at points far laterally from the footing. VARIATION: STRESS AT POINTS AWAY FROM CENTER The 2:1 method as written gives the average stress directly below the footing. For points away from the footing centerline, more sophisticated methods (influence charts, numerical integration) are needed. Some approximate extensions exist but are beyond the scope of licensure exams. COMPARISON WITH BOUSSINESQ: For a distributed load, the 2:1 method and Boussinesq give similar results at moderate depths (z > B/2 or so) but diverge at shallow depths. For practical foundation design, both are acceptable. The 2:1 method is preferred because it is simpler and requires no complex calculations or tables.

For more accurate calculations of stress increase under distributed loads—especially at points away from the footing center—engineers use influence factors (also called influence coefficients) or Newmark's chart. These are graphical or tabular tools that account for the geometry of the loaded area and the position of the point of interest. INFLUENCE FACTOR METHOD: The stress increase under a loaded area is expressed as: Δσ = I_z × q Where: - I_z = influence factor (dimensionless), typically in the range 0 to ~0.5 - q = uniform pressure on the loaded area (σ_footing = Q/[B × L], in kPa) The influence factor depends on: - The depth z relative to footing dimensions B and L - The horizontal position relative to the footing corners - The shape of the loaded area (rectangle, circle, etc.) INFLUENCE FACTORS FOR RECTANGULAR AREAS: For a rectangular loaded area B × L with uniform pressure q, the maximum influence factor (directly below the center) is given by Boussinesq-based solutions or tables. Common values: - At z = 0 (footing base): I_z ≈ 0.5 to 1.0 (depending on geometry) - At z = B (depth = footing width): I_z ≈ 0.1 to 0.3 - At z = 2B: I_z ≈ 0.05 to 0.1 - At z → ∞: I_z → 0 NEWMARK'S CHART: Newmark's chart is a graphical tool (influence chart) that simplifies the calculation of stress increase at any point below a uniformly loaded rectangular area. The chart is typically divided into small squares, each representing an influence factor of 0.001 to 0.01 (depending on the scale). To use the chart: 1. The loaded area is drawn to scale. 2. The point of interest (depth z) is marked. 3. A transparent overlay (with a grid scaled to depth z) is placed on the chart. 4. The number of grid squares within the loaded area is counted. 5. The stress increase is calculated as: Δσ = (number of squares / total squares) × q × influence factor per square. IN PRACTICE: Newmark's chart is less commonly used now because: - Digital calculators and spreadsheet programs have made tabular integration easy. - Boussinesq-based formulas are available in most software. - The chart requires careful drafting and is prone to reading errors. For PRC licensure exams, you should be familiar with the concept but may not need to use Newmark's chart in detail. However, understanding that stress is non-uniform below a footing and varies with lateral distance is important. FOR BOARD EXAMS: Questions on influence factors typically ask you to: 1. Look up or calculate I_z for given B, L, z dimensions. 2. Apply Δσ = I_z × q to find stress increase. 3. Understand that I_z decreases with depth and with distance from the footing center. 4. Compare results with the 2:1 method (which gives average stress).

Real soil deposits often consist of multiple layers with different properties (sand, clay, silt) and different unit weights. When calculating effective stress in a layered profile subject to surface loads, we must: 1. Calculate the total stress: σ = Σ(γ_i × z_i) + Δσ_load, where Δσ_load is the stress increase from surface loads. 2. Calculate pore pressure: u = γ_w × z_w (depends only on depth below WT). 3. Calculate effective stress: σ' = σ - u. MULTI-LAYER STRESS CALCULATION: Procedure: - At each layer boundary and within each layer, calculate σ by summing weights down to that depth, then add the Δσ from the applied load (using Boussinesq or 2:1 method). - Calculate u based on the depth below the water table (same calculation regardless of layers). - Compute σ' = σ - u. IMPORTANCE FOR FOUNDATION DESIGN: The stress distribution through layers determines: - Settlement potential: If a compressible layer (clay) has high σ', it will settle; if another clay at the same depth has low σ' (high water table), it may not. - Bearing capacity: Different layers have different strengths; the critical layer (weakest or most stressed) governs. - Lateral stress: Horizontal stress in each layer depends on K_0 (coefficient of lateral stress at rest), which varies by layer type. DRAWING PROFILES FOR LAYERED SOILS: For a complete analysis, plot three diagrams for each layer: 1. Total stress σ (increases in steps at layer boundaries) 2. Pore pressure u (increases linearly with depth below WT) 3. Effective stress σ' = σ - u (step changes at layer boundaries, adjusted by u) 4. Stress increase Δσ from applied load (decreases with depth) 5. Total stress including load: σ_total = σ_initial + Δσ 6. Effective stress with load: σ'_total = σ_total - u

PITFALL 1: FORGETTING PORE PRESSURE BELOW THE WATER TABLE One of the most common errors is calculating total stress correctly but then either: - Not subtracting pore pressure (incorrectly stating that σ' = σ below the WT). - Using pore pressure as if it acts like soil stress (it doesn't—water has no shear strength). Reminder: σ' = σ - u always applies. Below the WT, u is always positive and reduces σ'. PITFALL 2: USING THE WRONG UNIT WEIGHT - Using γ_moist above the WT is correct. - Using γ_sat above the WT is wrong—leads to overstated total stress. - Using γ_d (dry) below the WT is wrong—gives zero stress. - Always use γ_sat below the WT. Example error: "A layer is 4 m thick and has γ = 20 kN/m³. Below the WT, I'll use γ = 20..." Wrong! If 20 is the in-situ (moist) value, and the soil is saturated below the WT, the actual γ_sat is likely 19–21 kN/m³, and this should be used. PITFALL 3: MISAPPLYING THE 2:1 METHOD - The formula is Δσ = Q / [(B + z)(L + z)], not Q / (B × L) (which gives the footing pressure). - Some students write Δσ = Q / [(B + z)(L - z)] or other variants—these are wrong. - The 2:1 method is an approximation; it doesn't work well at very shallow depths (z < B/2) or for highly non-uniform loads. PITFALL 4: CONFUSING BOUSSINESQ FORMULAS - For a point load directly below: Δσ = 3Q / (2πz²). This is correct. - At an offset: Use the full formula with (r/z) term. Don't forget the fifth power in the denominator. - Don't apply the direct formula to distributed loads without integration. PITFALL 5: NOT ACCOUNTING FOR WATER TABLE LOCATION CORRECTLY - Pore pressure depends on depth below the WT, not absolute elevation or depth below the surface. - If the WT is at 3 m and you are at 7 m depth, then z_w = 7 - 3 = 4 m, and u = 9.81 × 4. - If the WT is above the surface (e.g., in flooded conditions), treat the water layer as having γ = γ_w. PITFALL 6: IGNORING CAPILLARY EFFECTS - Above the WT, in fine-grained soils, capillary rise creates negative pore pressure (suction). - This suction increases effective stress and can make clay very strong and stiff. - When the capillary water is removed (e.g., by construction drainage), the soil loses this suction and settles or becomes weaker. - For exams, if the problem doesn't explicitly mention capillary rise, assume u = 0 above the WT. But be aware of this concept. PITFALL 7: SEEPAGE AND THE QUICK CONDITION - Upward seepage increases pore pressure, reducing effective stress and creating instability. - The critical hydraulic gradient i_c ≈ 1.0 for most sands (varies from 0.8 to 1.2). - If the applied hydraulic gradient exceeds i_c, the quick condition develops, and the soil behaves like a fluid. - This is critical for excavations, dams, and any situation with upward groundwater flow. BOARD EXAM STRATEGY: 1. Always draw a clear sketch of the problem showing depths, WT location, layers, and loads. 2. List known values: γ values, depths, WT location, applied loads. 3. Calculate in order: total stress → pore pressure → effective stress (→ stress increases if loads are present). 4. Check units (kPa, kN, m) consistently. 5. For multiple-choice, check if the answer makes physical sense: effective stress should decrease with rising WT, increase with depth, and be reduced by pore pressure. 6. For free-response, show all steps and unit conversions. 7. If using Boussinesq or 2:1, state the method clearly and justify (e.g., "For this concentrated load, Boussinesq applies...").

FOUNDATION DESIGN (NSCP 2015 AND LOCAL PRACTICE): The National Structural Code of the Philippines (NSCP 2015) emphasizes effective-stress calculations for foundation design. When calculating bearing capacity, the designer must: 1. Determine the effective stress profile at the foundation level and at critical depths below. 2. Use effective stress in bearing-capacity formulas (e.g., Terzaghi's formula adjusted for shape and depth). 3. Account for groundwater conditions; if the water table is high, the bearing capacity is reduced. For example, in the Manila area where water tables are often high (1–3 m), effective stress is significantly reduced, limiting bearing capacity. This is why deep foundations (piles, caissons) are common in Manila and Cebu. SETTLEMENT ANALYSIS (TERZAGHI'S CONSOLIDATION THEORY): When a load is applied to clay, the soil compresses. Settlement occurs in two phases: 1. Immediate settlement (elastic, occurs instantly). 2. Consolidation settlement (time-dependent, as pore water is squeezed out and effective stress increases). The rate of consolidation depends on: - The thickness of the clay layer - The drainage conditions (is water free to escape?) - The permeability (how easily water flows through the clay) - The increase in effective stress (Δσ') from the applied load Without correct calculation of Δσ' from the footing load, consolidation estimates are useless. SLOPE STABILITY AND WATER TABLE MANAGEMENT: In the Philippines, during the monsoon season, water tables rise significantly. This is a major cause of slope failures on roads and embankments. The mechanism: 1. Rising WT increases pore pressure u. 2. Higher u reduces effective stress σ'. 3. Lower σ' reduces friction strength (tan φ decreases in terms of effective stress). 4. Slope fails. Solutions include: - Controlling drainage (interceptor drains, subsurface drainage) - Reducing slopes (flatter inclination) - Removing weight from slope top - Using pile reinforcement (soil nails, geo-grids) All these require understanding how u and σ' vary with depth and groundwater conditions. EXCAVATION AND EARTH PRESSURE: When an excavation is made, the stress conditions change: - The in-situ stress above the excavation is removed. - Lateral earth pressure on retaining walls decreases if the water table is above the wall. - But if water seeps into the excavation, upward seepage pressure acts on the base, reducing effective stress and potentially causing base heave or piping failure. Correct calculation of pore pressure is essential to: 1. Design safe retaining walls. 2. Prevent base heave (instability of the excavation bottom). 3. Prevent piping (where water seeps along a path, removing soil). DAM AND LEVEE DESIGN: For dams and levees, especially in flood-prone areas of the Philippines, effective stress in the dam material and at the foundation is critical: - Rapid drawdown (quick removal of water from the reservoir) can cause slope failure because the pore pressure inside the dam doesn't drop as fast as the outside water level, creating a hydraulic gradient that points outward. Effective stress decreases, and the slope fails. - During seepage through the dam, high pore pressures in zones where seepage is concentrated can cause piping and failure. For PRC ENGINEERS IN PRACTICE: Before submitting a design, a PE (Professional Engineer) in civil engineering must: 1. Verify that all stress calculations are correct (total, pore, effective). 2. Check that bearing capacities, settlements, and stability factors are based on correct effective stress. 3. Ensure that water-table assumptions are reasonable and conservative (usually assume worst-case, highest WT during design life). 4. Document all assumptions clearly for future reviewers and inspectors. FAILURE TO DO SO can result in: - Unsafe structures (settlements, failures) - Legal liability (RA 544 accountability) - Damage to public safety - Professional sanctions from the Board of Examiners