Stresses in Soil Mass - Revision Notes
Understanding stresses in a soil mass is fundamental to geotechnical engineering and is consistently tested in the PRC Civil Engineer Licensure Examination. Terzaghi's effective-stress principle — the single most important concept in soil mechanics — governs settlement, shear strength, and bearing capacity. This chapter covers three interconnected topics: (1) computing the effective-stress profile in a multi-layer soil deposit, (2) estimating stress increases from surface loads using the Boussinesq point-load equation, and (3) applying the 2:1 approximation for spread footings. Mastery of these tools is a prerequisite for consolidation settlement, slope stability, and foundation design problems that also appear on the board exam.
1. Terzaghi's Effective-Stress Principle
Formulas
σ = Σ(γᵢ × zᵢ)
Example
3 m of soil (γ = 18 kN/m³) over 2 m of saturated soil (γ_sat = 20 kN/m³): σ at 5 m = 18(3) + 20(2) = 54 + 40 = 94 kPa
Variables
γᵢ = unit weight of layer i (kN/m³); zᵢ = thickness of layer i (m)
Application
Compute total vertical stress at any depth by summing contributions of all overlying layers. Use γ_sat for layers below the water table.
u = γ_w × z_w
Example
Point is 2 m below the water table: u = 9.81 × 2 = 19.62 kPa
Variables
γ_w = 9.81 kN/m³; z_w = depth below the water table (m)
Application
Hydrostatic pore-water pressure at a point located z_w metres below the phreatic surface.
σ' = σ − u
Example
σ = 94 kPa, u = 19.62 kPa → σ' = 94 − 19.62 = 74.4 kPa
Variables
σ = total vertical stress (kPa); u = pore-water pressure (kPa)
Application
Effective vertical stress at any point in the soil mass. This is the stress used in all strength and consolidation calculations.
u = γ_w × z_w ± i × γ_w × z (seepage)
Example
Upward seepage with i = 0.5 at a point 3 m below WT: u = 9.81(3) + 0.5(9.81)(3) = 29.43 + 14.72 = 44.15 kPa (higher u → lower σ')
Variables
i = hydraulic gradient; + for upward seepage; − for downward seepage
Application
Modified pore pressure when steady-state seepage is present. Upward seepage (e.g., below a dam) is the critical case for instability.
Exam tips
- Always draw a soil profile sketch showing layer boundaries and the water-table position before computing stresses.
- Build the stress profile at key depths: ground surface (σ = 0), just above and just below the water table, and at the point of interest.
- At the water table: σ = γ × z_WT and u = 0, so σ' = σ. Just below: σ increases with γ_sat, u increases with γ_w; since γ_sat > γ_w, σ' still increases with depth.
- Express answers in kPa (kN/m²); check units at every step.
- If the water table rises to the surface: σ = γ_sat × z, u = γ_w × z, σ' = (γ_sat − γ_w) × z = γ' × z. This is a common board-exam scenario.
Key points
- Effective stress σ' = σ − u is the stress carried by the soil skeleton (grain-to-grain contact) and is the stress that controls strength and volume change.
- Total stress σ is computed by summing the product of unit weight and thickness for each layer above the point of interest.
- Below the water table, use γ_sat (saturated unit weight); above it, use γ (moist/bulk unit weight).
- Pore-water pressure u = γ_w × z_w, where z_w is the depth of the point below the phreatic surface (hydrostatic case).
- γ_w = 9.81 kN/m³ is the standard unit weight of water used in Philippine board exams.
- Upward seepage increases u (reduces σ'); downward seepage decreases u (increases σ'). At the quick condition (piping/heave), σ' = 0.
- Effective stress cannot be negative in granular soils (no tensile strength at grain contacts).
Definitions
Total Stress (σ)
The total vertical pressure at a point in the soil due to the weight of all material (soil + water) above it. Computed as Σ(γᵢzᵢ).
Importance
Starting point for all stress calculations; must be computed correctly before finding effective stress.
Pore-Water Pressure (u)
The pressure of water in the voids of the soil. Under hydrostatic conditions, u = γ_w × z_w. It acts in all directions equally (isotropic).
Importance
Reduces the effective stress that controls strength. Critical in evaluating slope failures, liquefaction, and piping.
Effective Stress (σ')
The portion of total stress transmitted through the soil skeleton at grain contacts. σ' = σ − u. Governs shear strength (Mohr–Coulomb: τ_f = c' + σ'tanφ') and consolidation.
Importance
The single most important concept in soil mechanics — every strength and settlement problem depends on it.
Phreatic Surface (Water Table)
The surface at which pore-water pressure equals atmospheric (u = 0). Depth to water table determines where to switch from γ to γ_sat in stress computations.
Importance
Its position determines the effective-stress profile; seasonal fluctuations affect foundation design.
Quick Condition
The state when upward seepage pressure equals the submerged weight of the soil, making σ' = 0. The soil loses all strength and behaves like a liquid (boiling sand/piping).
Importance
Critical for excavation dewatering, dam safety, and cofferdam design — all board-exam topics.
Common mistakes
- Using γ (moist unit weight) instead of γ_sat below the water table — always switch to γ_sat once the water table is crossed.
- Forgetting to subtract pore pressure: students sometimes report total stress as effective stress for points below the water table.
- Using γ_w = 10 kN/m³ when the problem expects 9.81 kN/m³ — read the problem statement carefully; many board problems specify γ_w.
- Above the water table, assuming u = 0 always — capillary rise creates negative pore pressure (suction), which increases σ'. However, most board problems ignore capillarity unless specifically stated.
- Confusing depth below ground surface (for σ) with depth below water table (for u) — keep track of both reference datums.
2. Stress Increase from Surface Loads — Boussinesq Theory
Formulas
Δσ = (3Q / 2πz²) × [1 / (1 + (r/z)²)]^(5/2)
Example
Q = 100 kN, z = 2 m, r = 0: Δσ = 3(100)/(2π × 4) × [1]^(5/2) = 300/25.13 = 11.94 kPa
Variables
Q = point load (kN); z = depth below surface (m); r = horizontal distance from the load axis (m)
Application
Vertical stress increase at a point (z, r) in the soil mass due to a concentrated surface load Q. Used when exact geometry is needed.
Δσ = 3Q / (2πz²) [when r = 0]
Example
Q = 200 kN, z = 3 m: Δσ = 3(200)/(2π × 9) = 600/56.55 = 10.61 kPa
Variables
Q = point load (kN); z = depth (m). This is the maximum stress increase on the load axis.
Application
Quick calculation for the stress directly below a column or point load — the most common board-exam case.
Δσ = q × I (rectangular load, Newmark/influence factor)
Example
For m = 1, n = 1 (square footing, point at corner directly below): I ≈ 0.175; if q = 100 kPa, Δσ = 100 × 0.175 = 17.5 kPa
Variables
q = contact pressure (kPa); I = influence factor (function of m = B/z, n = L/z); tabulated in Bowles or Das
Application
Stress increase under the corner of a uniformly loaded rectangular area. For a point not at the corner, use superposition by subdividing the rectangle.
Exam tips
- For point-load problems directly below the load (r = 0), the formula reduces cleanly: Δσ = 3Q/(2πz²). Memorize this form.
- The factor 3/(2π) ≈ 0.4775 — a useful shortcut: Δσ = 0.4775 Q/z² for r = 0.
- Boussinesq problems on board exams almost always set r = 0 (directly below the load); the full formula with r is less common but possible.
- When given a uniformly loaded circular footing, use the circular Boussinesq formula (influence charts); for rectangular footings, use the 2:1 method or Newmark chart.
- Sketch the geometry (top view and cross-section) before substituting into formulas — it prevents confusion between r, z, and dimensions B and L.
Key points
- Boussinesq (1885) derived stress distribution in a homogeneous, isotropic, linearly elastic semi-infinite half-space — the theoretical foundation for stress increase from surface loads.
- The stress increase Δσ (vertical) diminishes with depth z and horizontal offset r from the load.
- Directly below the point load (r = 0), the formula simplifies to Δσ = 3Q/(2πz²).
- For distributed loads over rectangular or circular areas, Boussinesq is integrated; influence factor charts (Newmark's chart) are used in practice.
- The Boussinesq approach is more accurate but requires knowledge of r and z; the 2:1 method is a practical approximation for footings.
- Δσ from multiple loads/footings can be superimposed (elastic principle of superposition).
Definitions
Boussinesq Stress Distribution
Theoretical vertical stress increase in an elastic half-space due to a concentrated surface load, varying with both depth z and radial offset r. The stress bulb shape is a fundamental visualization tool.
Importance
Provides the theoretical basis for all stress-distribution charts used in settlement analysis.
Stress Bulb
The zone of significant stress increase (typically Δσ > 0.1q or 10% of the applied pressure) beneath a loaded area. For a square footing, significant stresses extend to a depth of about 2B.
Importance
Defines the zone of soil that must be investigated for settlement — critical for site investigation planning.
Influence Factor (I)
A dimensionless coefficient that relates the applied surface pressure q to the stress increase Δσ at a point in the soil mass. I = f(m, n) for rectangular loads; tabulated in standard references.
Importance
Allows rapid computation of Δσ under any point relative to a loaded rectangle using charts or tables.
Common mistakes
- Applying the point-load formula to a distributed footing load — use the 2:1 method or influence-factor approach for spread footings, not the point-load Boussinesq formula.
- Forgetting to convert the total column load Q (kN) to contact pressure q (kPa) when using influence factor charts: q = Q/(B×L).
- Confusing r (horizontal offset from load) with z (depth below surface) in the Boussinesq formula — keep the geometry clear.
- Not using superposition when a point is not at the corner of the loaded area — subdivide the rectangle into four sub-rectangles sharing the point of interest as a common corner.
- Neglecting that Boussinesq assumes linear elastic, homogeneous, isotropic soil — real soils are layered, which is why the 2:1 method is often preferred for quick estimates.
3. The 2:1 Approximation Method for Spread Footings
Formulas
Δσ = Q / [(B + z)(L + z)]
Example
2 m × 2 m footing, Q = 800 kN, z = 3 m: Δσ = 800/[(2+3)(2+3)] = 800/25 = 32 kPa
Variables
Q = total column/footing load (kN); B = footing width (m); L = footing length (m); z = depth below footing base (m); Δσ in kPa
Application
Quick estimate of vertical stress increase at depth z below a rectangular footing. Used in consolidation settlement calculations for clay layers beneath a footing.
Δσ = Q / (B + z)² [square footing]
Example
3 m × 3 m footing, Q = 1 800 kN, z = 3 m: Δσ = 1800/(3+3)² = 1800/36 = 50 kPa
Variables
Q = total load (kN); B = side length (m); z = depth below footing (m)
Application
Simplified form for square footings — the most common case in board exam problems.
Δσ = q_s / (B + z) [strip footing per unit length]
Example
Strip footing B = 1.5 m, q_s = 200 kN/m, z = 2 m: Δσ = 200/(1.5+2) = 200/3.5 = 57.1 kPa
Variables
q_s = load per unit length of strip (kN/m); B = strip width (m); z = depth (m)
Application
Stress increase below a strip footing (wall footing). Common in retaining wall and continuous footing problems.
Exam tips
- Board exam problems on consolidation settlement almost always require computing Δσ using the 2:1 method at the mid-depth of the compressible layer — be ready to identify this depth.
- When multiple footings are present, check if their 2:1 stress zones overlap at depth — if they do, add the Δσ contributions (superposition).
- Always verify: at z = 0, Δσ = Q/(B×L) = q₀ (contact pressure). Use this as a sanity check before computing Δσ at greater depths.
- For a strip footing, the per-unit-length load q_s (kN/m) is used — note this is different from contact pressure q (kPa). Make sure units are consistent.
- The 2:1 method is acceptable for board-exam settlement problems. Boussinesq is used when the problem specifically asks for it or when computing stress at an offset point.
Key points
- The 2:1 method assumes the applied load spreads at a 2 (vertical) : 1 (horizontal) slope from each edge of the footing.
- For a rectangular footing B × L carrying total load Q, the stress increase at depth z below the footing base is Δσ = Q / [(B+z)(L+z)].
- For a square footing (B = L): Δσ = Q / (B+z)².
- For a strip footing (L → ∞, load per unit length q per meter): Δσ = q / (B+z).
- The 2:1 method overestimates Δσ at shallow depths and slightly underestimates at large depths compared to Boussinesq, but is widely accepted for settlement calculations.
- The depth z is measured from the footing base, not the ground surface — critical distinction when footings are embedded.
- Δσ diminishes rapidly with depth: at z = B, the stress is already only Q/[(2B)(2B)] = Q/(4B²) — one-quarter of the contact pressure for a square footing.
Definitions
2:1 Stress Distribution
An empirical approximation in which the area over which load is distributed increases by z/2 on each side of the footing for every metre of depth z. The loaded area at depth z becomes (B+z) × (L+z).
Importance
The most widely used practical method for computing stress increases in settlement calculations — appears on almost every consolidation problem in board exams.
Contact Pressure (q₀)
The average pressure at the footing-soil interface: q₀ = Q/(B×L) for a rectangular footing. This is the Δσ at z = 0 (at the base of the footing).
Importance
The starting value of the Δσ profile; it decreases with depth following the 2:1 spread.
Common mistakes
- Measuring z from the ground surface instead of from the footing base — the 2:1 spread starts at the footing level, not at the ground surface.
- Using (B×z)(L×z) or (B+2z)(L+2z) instead of (B+z)(L+z) — the correct spread is z/2 from each edge, adding z total to each dimension.
- Dividing the contact pressure q₀ = Q/(BL) by (1+z/B)² — wrong rearrangement. Always substitute directly: Q/[(B+z)(L+z)].
- Forgetting to account for footing embedment depth D_f: the footing base is at depth D_f, so z is measured from D_f downward.
- Applying the 2:1 method to layered soils without recognizing that the load must still pass through each layer — the formula gives Δσ at any depth z regardless of soil type.
4. Building the Effective-Stress Profile (Worked Procedure)
Formulas
γ' = γ_sat − γ_w
Example
γ_sat = 20 kN/m³ → γ' = 20 − 9.81 = 10.19 kN/m³. Effective stress 2 m below WT: σ' = σ'_WT + γ'×2 = σ'_WT + 20.38 kPa
Variables
γ_sat = saturated unit weight (kN/m³); γ_w = 9.81 kN/m³
Application
Buoyant (submerged) unit weight — the rate at which effective stress increases per metre of depth below the water table. Shortcut for computing σ' without computing σ and u separately.
Exam tips
- A tabular format (depth | σ | u | σ') is the fastest and most error-free approach for computing the effective-stress profile on board exams.
- If the water table is at the surface (fully saturated deposit), the formula simplifies to σ' = γ' × z throughout. This is a very common board-exam scenario.
- For a partially saturated deposit above the WT, use the moist/bulk unit weight γ and set u = 0 to get σ' = σ.
Key points
- An effective-stress profile is a plot of σ, u, and σ' versus depth — a standard deliverable in geotechnical reports and a frequent board-exam computation.
- Key depths to compute: ground surface, each layer boundary, and the water table.
- At the water table: u = 0. Just above: u could be negative (capillary) — usually ignored in board problems.
- In saturated soil below the WT: σ increases at γ_sat per metre; u increases at γ_w per metre; σ' increases at (γ_sat − γ_w) = γ' per metre.
- The submerged (buoyant) unit weight γ' = γ_sat − γ_w is the rate of effective-stress increase below the WT.
- Typical values: γ_sat = 18–22 kN/m³; γ' = 8–12 kN/m³; γ_w = 9.81 kN/m³.
Definitions
Buoyant (Submerged) Unit Weight (γ')
γ' = γ_sat − γ_w. The effective unit weight of soil submerged below the water table, accounting for the upward buoyant force of water. Numerically equal to the rate of increase of effective stress with depth below the water table.
Importance
Streamlines effective-stress computation in saturated layers; avoids computing σ and u separately.
Effective-Stress Profile
A diagram or table of σ, u, and σ' at key depths in a soil profile. Essential for settlement, bearing capacity, and lateral earth-pressure calculations.
Importance
Required output for virtually all geotechnical design calculations — the foundation of all subsequent analysis.
Common mistakes
- Confusing γ' with γ_d (dry unit weight) — they are different quantities with different uses.
- Not resetting the pore pressure to zero at the water table when the problem involves a perched water table or multiple aquifer levels.
- Assuming σ' is constant below the water table — it increases at γ' per metre, not γ_sat per metre.
Connections
- Consolidation Settlement: The stress increase Δσ from the 2:1 method is directly used in Terzaghi's consolidation equation ΔH = [Cc/(1+e₀)] × H × log[(σ'₀ + Δσ)/σ'₀] — you cannot compute settlement without first finding Δσ and σ'₀ (initial effective stress).
- Shear Strength — Mohr–Coulomb Criterion: Peak shear strength τ_f = c' + σ'tanφ'. Effective stress σ' must be correctly computed before any slope stability or bearing capacity analysis. A rise in u (pore pressure) reduces σ' and therefore reduces τ_f — this is why saturated slopes fail during heavy rainfall.
- Bearing Capacity (Terzaghi, Meyerhof): The ultimate bearing capacity q_ult = cN_c + q̄N_q + 0.5γBN_γ where q̄ = effective overburden pressure at the footing level = σ' at depth D_f. Effective stress at the footing base is a direct input.
- Lateral Earth Pressure (Rankine/Coulomb): Active and passive pressures are computed using effective stresses: σ'_h = K × σ'_v. Water pressure is added separately as u. Wrong effective stress → wrong earth pressure → wrong retaining wall design.
- Piping and Heave (Upward Seepage): The quick condition (σ' → 0) occurs when upward seepage gradient i = i_cr = γ'/γ_w. This connects effective-stress principles to dam safety and cofferdam design.
- Foundation Engineering (Embedded Footings): Net stress increase from a footing includes the effect of excavation (which removes overburden) — the net Δσ = q_applied − γD_f. This requires the effective-stress framework to correctly account for the pre-existing stress state.
- Liquefaction: During earthquake loading, rapid pore-pressure buildup (Δu) causes σ' → 0 in loose saturated sands — effective stress principles explain why saturated loose sand can suddenly liquefy and lose all strength, a critical consideration in Philippine seismic design (per NSCP 2015 Section 208).
Quick review questions
A saturated clay layer is 6 m thick with γ_sat = 19 kN/m³. The water table is at the surface. What is the effective stress at 4 m depth?
Answer
σ' = (γ_sat − γ_w) × z = (19 − 9.81) × 4 = 9.19 × 4 = 36.76 kPa
When the water table is at the surface, σ = γ_sat × z = 19 × 4 = 76 kPa and u = γ_w × z = 9.81 × 4 = 39.24 kPa. Therefore σ' = 76 − 39.24 = 36.76 kPa. This equals γ' × z = (19 − 9.81) × 4 = 36.76 kPa — the buoyant unit weight shortcut confirms the answer.
A 2 m × 3 m footing carries a total load of 900 kN. Using the 2:1 method, what is the vertical stress increase at 2 m below the footing base?
Answer
Δσ = Q / [(B+z)(L+z)] = 900 / [(2+2)(3+2)] = 900 / (4 × 5) = 900/20 = 45 kPa
The load spreads from 2 m × 3 m at the footing base to (2+2) m × (3+2) m = 4 m × 5 m at 2 m depth. The stress is uniformly distributed over this larger area: 900/20 = 45 kPa. Note that the contact pressure at the footing base is 900/(2×3) = 150 kPa — the stress has decreased significantly with depth.
A point load Q = 150 kN acts at the ground surface. Using Boussinesq theory, find the vertical stress increase directly below the load at a depth of 3 m.
Answer
Δσ = 3Q/(2πz²) = 3(150)/(2π × 9) = 450/56.55 = 7.96 kPa
For r = 0 (directly below the load), the Boussinesq formula simplifies to Δσ = 3Q/(2πz²). The factor 3/(2π) ≈ 0.4775, so Δσ = 0.4775 × 150/9 = 7.96 kPa. This can be used as a quick check: Δσ ≈ 0.4775Q/z².
The water table rises from 3 m depth to the ground surface in a deposit where γ_sat = 20 kN/m³. What is the change in effective stress at 5 m depth?
Answer
Original σ' = 18(3) + (20−9.81)(2) = 54 + 20.38 = 74.38 kPa (assuming γ_moist = 18 kN/m³ above WT). After WT rises to surface: σ' = (20−9.81)(5) = 9.19 × 5 = 45.95 kPa. Change = 45.95 − 74.38 = −28.43 kPa (decrease).
When the WT rises, pore pressure increases throughout the saturated zone. Above the original WT, the previously dry/moist soil now contributes γ_sat instead of γ_moist, but the gain in σ is offset by the increase in u. The net effect is a reduction in effective stress, which reduces shear strength — this is why rising water tables cause slope failures and foundation problems.
For a point load Q = 100 kN, at depth z = 2 m and horizontal offset r = 2 m, compute the vertical stress increase using Boussinesq theory.
Answer
Δσ = (3Q/2πz²) × [1/(1+(r/z)²)]^(5/2) = (3×100)/(2π×4) × [1/(1+1)]^(5/2) = 11.94 × (0.5)^(5/2) = 11.94 × 0.1768 = 2.11 kPa
Here r/z = 2/2 = 1. The term [1/(1+1²)]^(5/2) = [1/2]^(5/2) = (0.5)^(2.5) = 0.5² × 0.5^(0.5) = 0.25 × 0.7071 = 0.1768. Compare to the on-axis value of 11.94 kPa — the stress at the same depth but 2 m offset is only 2.11 kPa, illustrating how rapidly the Boussinesq stress decreases with horizontal distance.
A soil profile has 2 m of sand (γ = 17 kN/m³) above the water table and 4 m of saturated clay (γ_sat = 18 kN/m³) below it. Compute σ, u, and σ' at the bottom of the clay layer (6 m depth).
Answer
σ = 17(2) + 18(4) = 34 + 72 = 106 kPa; u = 9.81(4) = 39.24 kPa; σ' = 106 − 39.24 = 66.76 kPa
Total stress uses γ (moist) for the sand above WT and γ_sat for the clay below WT. Pore pressure is hydrostatic from the water table: the bottom of the clay is 4 m below the WT so u = 9.81 × 4 = 39.24 kPa. Effective stress = 106 − 39.24 = 66.76 kPa. Alternatively: σ' = σ at WT + γ'×4 = 34 + (18−9.81)×4 = 34 + 32.76 = 66.76 kPa ✓
A 3 m × 3 m square footing carries Q = 1 350 kN. Using the 2:1 method, at what depth z below the footing base does Δσ = 30 kPa?
Answer
30 = 1350/(3+z)² → (3+z)² = 45 → 3+z = 6.71 m → z = 3.71 m
Rearranging Δσ = Q/(B+z)² for z: (B+z)² = Q/Δσ = 1350/30 = 45; B+z = √45 = 6.708; z = 6.708 − 3 = 3.71 m. This type of problem — finding the depth at which stress reaches a specified value — is common in problems that ask for the depth of influence on a clay layer.
Diagrams
Stresses in Soil Mass — Chapter Concept Overview
A comprehensive mind map showing the four main branches of the chapter: the effective-stress principle (with σ, u, σ' relationships), stress increase from loads (Boussinesq and 2:1 methods), key applications (settlement, bearing capacity, shear strength, piping), and common pitfalls. Use this to navigate the chapter and see how all topics interconnect.
Step-by-Step Procedure: Computing Effective Stress at Any Depth
A decision-tree flowchart guiding the user through the exact steps to compute effective stress at any depth in a multi-layer soil profile. The key decision point is whether the point of interest is above or below the water table, which determines the unit weight to use and whether pore pressure is zero or positive.
Choosing the Right Stress-Increase Method
A decision flowchart that guides the user in selecting the correct stress-increase formula based on the type of load (point load, rectangular footing, or strip footing), the location of the point of interest (on-axis or offset), and the accuracy required. This directly answers the common board-exam question: 'Which formula do I use?'
Effective Stress States Under Different Groundwater Conditions
A state diagram illustrating how effective stress changes as groundwater conditions shift from dry (no water table) to hydrostatic, upward seepage, downward seepage, and finally the critical quick condition (piping/heave failure). This is particularly relevant to Philippine engineering practice where high groundwater tables and seepage from typhoon rainfall events are common.
Worked Example: Complete Effective-Stress Profile Calculation
A step-by-step worked flowchart for the classic two-layer effective-stress problem (3 m above WT + 2 m below WT). Shows the parallel computation paths using both the σ − u method and the γ' shortcut method, confirming the same answer of 74.4 kPa. This mirrors the format of board-exam solutions.
Exam strategy
For PRC board exam problems on stresses in soil mass, follow this disciplined four-step approach: (1) DRAW — sketch the soil profile with layer thicknesses, unit weights, and water table position clearly labelled; (2) BUILD THE STRESS TABLE — compute σ, u, and σ' at every key depth (layer boundaries and the point of interest) in a tabular format; this prevents arithmetic errors and makes the work auditable; (3) IDENTIFY THE METHOD — for point loads, use Boussinesq (Δσ = 0.4775Q/z² for r=0); for footings in settlement problems, use the 2:1 method (Δσ = Q/[(B+z)(L+z)]); for distributed areas, use influence-factor tables if provided; (4) CHECK UNITS AND REASONABLENESS — effective stress must increase with depth (below WT it increases at γ' ≈ 8–12 kN/m per metre), and Δσ must decrease with depth. Memorize the key shortcuts: γ' = γ_sat − γ_w, the Boussinesq coefficient 3/(2π) ≈ 0.4775, and the 2:1 formula structure (B+z)(L+z). In the board exam, effective-stress problems almost always connect to a follow-up question on settlement or bearing capacity — be ready to use your computed σ' and Δσ immediately in the next part of the problem.