Bearing Capacity of Soils - Cheat Sheet
Your final 30-minute revision companion for PRC CE exam questions on footing design, ultimate bearing capacity, allowable bearing pressure, and Terzaghi's equation. Master the formulas, shape factors, water-table corrections, and common pitfalls.
Terzaghi's Bearing Capacity Equation — Core Formula
Formulas
q_u = cN_c + qN_q + (1/2)γBN_γ
Meaning
q_u = ultimate bearing capacity (kPa); c = cohesion (kPa); N_c, N_q, N_γ = bearing-capacity factors (function of φ); q = γD_f = overburden pressure at footing depth (kPa); γ = unit weight (kN/m³); B = footing width or diameter (m)
Watch out
DO NOT forget the surcharge term qN_q — it is critical and often omitted. Always include q = γD_f. The (1/2)γBN_γ term applies only to strip; shape factors modify it for square/circular.
When to use
Strip footing (infinite length) on c-φ soil; this is the base equation for all bearing-capacity problems.
Common values
Value
N_c = 5.7, N_q = 1, N_γ = 0
Symbol
Terzaghi (undrained condition)
Quantity
φ = 0 (undrained clay)
Value
N_c ≈ 17.7, N_q ≈ 7.4, N_γ ≈ 5.1
Symbol
Loose sand/soft soil range
Quantity
φ = 20°
Value
N_c ≈ 25.1, N_q ≈ 12.7, N_γ ≈ 8.3
Symbol
Medium sand range (Terzaghi tables)
Quantity
φ = 25°
Value
N_c ≈ 37.2, N_q ≈ 22.5, N_γ ≈ 19.1
Symbol
Dense sand range
Quantity
φ = 30°
Value
9.81 kN/m³
Symbol
Use for γ' = γ_sat − γ_w
Quantity
γ_w (water)
Value
2.5–3.0
Symbol
Depends on uncertainty and code; 3.0 common in Philippines
Quantity
Typical FS (safety factor)
Value
1.3
Symbol
Increases cohesion contribution vs strip
Quantity
Shape factor square (c-term)
Value
0.4
Symbol
Reduces footing-width contribution vs strip (0.5)
Quantity
Shape factor square (γ-term)
Important facts
- Strip footing: q_u = cN_c + qN_q + (1/2)γBN_γ (base equation, L → ∞)
- Square footing: q_u = 1.3cN_c + qN_q + 0.4γBN_γ (shape factor 0.4 on width term)
- Circular footing: q_u = 1.3cN_c + qN_q + 0.3γBN_γ (shape factor 0.3 on width term)
- Rectangular footing (L × B): use shape factors s_c, s_q, s_γ; factors approach 1.0 as L → ∞
- q = γD_f is ALWAYS the overburden at footing base depth; never omit this in N_q term
- φ = 0 clay (undrained): N_c = 5.7, N_q = 1, N_γ = 0 — only cohesion and surcharge apply
- Allowable bearing (net) = (q_u − γD_f) / FS — subtracts the already-applied overburden
- Allowable bearing (gross) = q_u / FS — includes footing self-weight effect
- Water table correction: replace γ with γ' (submerged unit weight) in the (1/2)γBN_γ term if water table is at/above footing base or within depth B below
- Settlement often governs design over shear bearing capacity — use serviceability limits
Key definitions
Ultimate Bearing Capacity (q_u)
Example
If q_u = 750 kPa, footing will fail (shear) at applied pressure > 750 kPa.
Maximum pressure the soil can sustain before shear failure occurs; foundation design keeps applied pressure below this.
Allowable Bearing Capacity (q_a)
Example
If q_u = 750 kPa and FS = 3, then q_a = 250 kPa (safe for design).
Safe design pressure = q_u / FS, where FS is factor of safety (typically 2.5–3.0).
Bearing-Capacity Factors (N_c, N_q, N_γ)
Example
For φ = 25°: N_c ≈ 25.1, N_q ≈ 12.7, N_γ ≈ 8.3 (from Terzaghi tables).
Dimensionless coefficients derived from soil friction angle φ; account for cohesion, surcharge, and footing width contributions.
Shallow Footing
Example
A 2 m wide footing at 1 m depth (D_f/B = 0.5) is shallow.
Depth D_f ≤ 3–4 × width B; Terzaghi's equation applies directly.
General Shear Failure
Example
Dense sand or stiff clay under normal loading.
Clear shear surfaces develop; occurs in dense/stiff soil; use standard N-factors.
Local (Punching) Shear Failure
Example
Loose sand or soft clay; use reduced φ' = arctan(2/3 tan(φ)).
Footing punches into soil without clear shear surfaces; occurs in loose/soft soil; reduce c and tan(φ) to 2/3 of values.
Diagrams to know
- Stress distribution beneath footing (pressure bulb)
- Shear failure surface (Terzaghi's log-spiral wedge)
- Effect of water table on bearing capacity (three zones: above base, within B below, deep)
- Bearing-capacity factor variation with φ (N_c, N_q, N_γ curves)
Reactions or equations
q = γD_f
Note
This is the overburden pressure; it is the 'q' in the Terzaghi equation's qN_q term, NOT the applied footing pressure.
Conditions
D_f = depth from ground surface to footing base; γ = unit weight of soil above footing
q_a = q_u / FS
Note
Gross allowable; does not subtract overburden. Use net allowable (q_a,net) if subtracting self-weight effects.
Conditions
FS = 2.5–3.0 typical; design applied pressure must be ≤ q_a
q_a,net = (q_u − γD_f) / FS
Note
Often used for embankment loading; compares net applied pressure to net ultimate.
Conditions
FS on net basis; accounts for surcharge already in place
γ' = γ_sat − γ_w
Note
Submerged unit weight; used when water table is at or above footing base.
Conditions
γ_sat = saturated unit weight; γ_w = 9.81 kN/m³ (water)
Shape Factors & Depth Corrections
Formulas
q_u = s_c × c × N_c + s_q × q × N_q + s_γ × (1/2) × γ × B × N_γ
Meaning
s_c, s_q, s_γ = shape factors (1.0 for strip, >1.0 for square/circular); all other terms as defined
Watch out
Strip footing is reference (s = 1.0); do NOT apply shape factors twice. Circular uses 1.3 and 0.3 (not 1.0 and 0.5).
When to use
Any footing shape (square, circular, rectangular); accounts for geometry.
s_c = 1 + (B/L) × (N_q / N_c), s_q = 1 + (B/L) × tan(φ), s_γ = 1 − 0.4(B/L)
Meaning
B = width (shorter dimension), L = length (longer dimension); correction for finite footing length.
Watch out
Use L ≥ B; if B > L swap them. For strip L >> B so s → 1.0.
When to use
Rectangular footing; for square B = L so s_c ≈ 1.3, s_q ≈ 1.2, s_γ ≈ 0.6 (approximate).
d_c = 1 + (D_f / B) × tan(φ), d_q = 1 + (D_f / B) × tan(φ), d_γ = 1.0
Meaning
d_c, d_q, d_γ = depth factors; D_f = footing depth, B = width.
Watch out
d_γ = 1 for most cases (depth does not affect width term); d_c and d_q increase with D_f. Limit D_f ≤ 3B for shallow assumption.
When to use
Shallow foundations; D_f / B typically 0.25–2.0; deeper footings get higher bearing capacity.
Common values
Value
1.3, ~1.2, 0.6 (approximate; varies with φ)
Symbol
Standard for exam; exact values from NSCP or tables
Quantity
Square footing (s_c, s_q, s_γ)
Value
1.3, ~1.2, 0.3 (approximate)
Symbol
Use D = B (diameter as width)
Quantity
Circular footing (s_c, s_q, s_γ)
Value
1.0, 1.0, 1.0 (reference)
Symbol
L → ∞; base case
Quantity
Strip footing (s_c, s_q, s_γ)
Important facts
- Standard shape factors: square s_c = 1.3 (or 1 + tan(φ) × (B/L)); circular s_c = 1.3
- Square footing width term: use 0.4 (not 0.5) in Terzaghi base equation for N_γ
- Rectangular footing: B/L ratio matters; shape factors approach 1.0 as L → ∞ (becomes strip)
- Depth factors increase bearing capacity 5–50% depending on D_f/B ratio and φ
- d_c and d_q are typically equal for most φ values
- d_γ = 1.0 in standard analysis (depth does not significantly reduce the weight term)
- For very deep footings (D_f >> B), bearing capacity becomes insensitive to width (gravity term fades)
Key definitions
Shape Factor (s_c, s_q, s_γ)
Example
Square footing on c-φ soil: s_c = 1.3, so the cohesion term is 1.3 × c × N_c instead of c × N_c.
Multiplier on each Terzaghi term; accounts for footing geometry (strip, square, circular, rectangular).
Depth Factor (d_c, d_q, d_γ)
Example
If D_f / B = 1 and φ = 25°, d_q ≈ 1 + tan(25°) ≈ 1.47 (47% increase).
Multiplier that increases bearing capacity with footing depth; deeper footings confined by surrounding soil.
Inclination Factor (i_c, i_q, i_γ)
Example
Inclined load (e.g., sloping footing) requires i-factors; vertical load: i = 1.0.
Reduces bearing capacity if load is inclined (not vertical); rarely tested in standard foundations.
Diagrams to know
- Shape factor variation with B/L ratio
- Depth factor vs D_f/B for different φ angles
- Comparison of strip, square, and circular footing bearing capacities
Water-Table Corrections & Saturated Soil
Formulas
Use γ' in (1/2)γBN_γ term if water table at footing base or within B below
Meaning
γ' = γ_sat − γ_w = submerged unit weight; replace γ with γ' only in the width term.
Watch out
DO NOT replace γ in the q = γD_f term (surcharge always includes water weight above footing). Only modify the (1/2)γBN_γ term.
When to use
Water table at or above footing depth; use γ (or γ_sat) above, γ' below.
If water table between footing base and depth B below: interpolate γ in (1/2)γBN_γ
Meaning
Partially submerged; use weighted average or linear interpolation for unit weight.
Watch out
Interpolation adds complexity; check NSCP 2015 or textbook for exact method. Many exams simplify to 'at base' or 'well below'.
When to use
Water table depth d below footing (0 < d < B); partial saturation zone.
γ' = γ_sat − 9.81 kPa (or 9.81 kN/m³ in density)
Meaning
γ' = effective (submerged) unit weight; γ_sat = total saturated weight; γ_w = 9.81 kN/m³.
Watch out
DO NOT use effective stress σ' in bearing-capacity equation; use unit weights γ, γ_sat, γ'.
When to use
Any soil below water table.
Common values
Value
9.81 kN/m³
Symbol
Constant; used in γ' = γ_sat − γ_w
Quantity
γ_w (water unit weight)
Value
18–22 kN/m³
Symbol
Varies; check soil properties
Quantity
Typical γ_sat (sand/clay)
Value
8–12 kN/m³
Symbol
Much lower than dry; reduces bearing capacity
Quantity
Typical γ' (submerged sand/clay)
Important facts
- Water table above footing base: bearing capacity DECREASES significantly (buoyancy reduces effective unit weight)
- Water table well below (>B from base): negligible effect on bearing capacity
- Use γ_sat or γ above water table; γ' below water table
- The q = γD_f term (surcharge) does NOT change with water table — it always includes water above footing
- Only the (1/2)γBN_γ term is corrected for water table (use γ' if submerged)
- Cohesion term cN_c is unaffected by water table position (cohesion is internal soil strength)
- For φ = 0 undrained clay, water-table position is irrelevant (N_γ = 0 anyway)
- Saturated fine sand & silt: bearing capacity significantly reduced vs dry case
Key definitions
Saturated Unit Weight (γ_sat)
Example
Typical clay: γ_sat ≈ 19 kN/m³.
Total weight of soil + water per unit volume, typically 18–22 kN/m³.
Submerged (Effective) Unit Weight (γ')
Example
If γ_sat = 19 kN/m³, then γ' = 19 − 9.81 ≈ 9.2 kN/m³.
Apparent weight in water; γ' = γ_sat − γ_w ≈ γ_sat − 9.81 kN/m³, typically 8–12 kN/m³.
Water Table at Footing Base
Example
Footing at 1.5 m depth, water level at 1.5 m: use γ' in width term.
Water surface is at depth D_f; full saturation below footing.
Water Table Above Footing Base
Example
Footing at 1.5 m, water at 1.0 m: use γ (or γ_sat) for 0–1.5 m zone, γ' for 1.5+ m.
Water surface is shallower than footing depth; buoyancy effects.
Diagrams to know
- Water table position vs footing depth (three zones: above base, within B, deep)
- Unit weight profile with depth (dry, saturated, submerged regions)
Reactions or equations
γ' = γ_sat − γ_w = γ_sat − 9.81
Note
Effective (submerged) unit weight; always positive but much lower than dry γ.
Conditions
All soils below water table; γ_w = unit weight of water (constant, 9.81 kN/m³)
Failure Modes: General vs Local Shear
Formulas
For local shear: use c' = (2/3)c and φ' = arctan((2/3)tan(φ)) in Terzaghi equation
Meaning
Reduce cohesion and friction angle by 2/3 factor; recalculate N-factors with φ'.
Watch out
DO NOT use local-shear reduction unless problem explicitly states it or soil is very loose (relative density <50%, SPT N < 5). Most standard problems assume general shear.
When to use
Loose sand, soft clay, or footing punching into soil; absence of clear shear surface.
Common values
Value
Dr > 70% → general; Dr < 50% → local
Symbol
Indicates failure mode from boring data
Quantity
Relative Density (Dr) threshold
Value
N > 30 → general; N < 10 → local
Symbol
Rough guideline; context-dependent
Quantity
SPT N-value threshold
Important facts
- General shear is standard assumption in Terzaghi's equation and exams unless stated otherwise
- Local shear occurs in loose/soft soils; bearing capacity is 5–30% lower than general shear estimate
- Relative density Dr or SPT N-value indicates mode: Dr > 70% or N > 30 → general; Dr < 50% or N < 10 → local
- For local shear, reduce tan(φ) to (2/3)tan(φ), then recalculate N-factors (do not just scale final q_u)
- Undrained clay (φ = 0): failure mode distinction is less critical (N_γ = 0 anyway)
- Drained sand: choice of general vs local can change q_u by 20–40%
Key definitions
General Shear Failure
Example
Dense sand (Dr > 70%), stiff clay (cu > 100 kPa).
Clear shear surfaces (log-spiral wedge) develop; footing moves vertically with well-defined collapse load; dense/stiff soil.
Local Shear Failure
Example
Loose sand (Dr < 50%), soft clay (cu < 50 kPa).
Footing settles into soil without clear slip surfaces; soil punching or punching shear; loose/soft soil.
Punching Shear
Example
Weak foundation soil; footing sinks rather than tilts.
Footing penetrates soil vertically; lateral soil displacement is minimal; associated with local shear.
Diagrams to know
- Shear failure surface in dense soil (Terzaghi log-spiral wedge, general shear)
- Punching pattern in loose soil (local shear, minimal lateral displacement)
Reactions or equations
φ' = arctan((2/3) × tan(φ))
Note
Use φ' to recalculate N-factors from tables, then apply to Terzaghi equation.
Conditions
Local shear reduction; φ' is reduced friction angle for loose/soft soil
Net Bearing Capacity & Allowable Load Calculations
Formulas
q_u,net = q_u − q = q_u − γD_f
Meaning
q_u,net = net ultimate bearing capacity (subtracts overburden already in place); q_u = gross ultimate.
Watch out
DO NOT confuse: q_u is gross (includes all contributions from footing level down); q_u,net is the increment above existing q.
When to use
Design accounting for surcharge effect; net applies only applied load increment, not total pressure.
Q_a = q_a × A
Meaning
Q_a = allowable column load (kN); q_a = allowable bearing capacity (kPa); A = footing area (m²).
Watch out
Use q_a (or q_a,net) in consistent units; area must be in m² if q_a is in kPa.
When to use
Convert bearing pressure to total allowable load on footing.
q_a = q_u / FS or q_a,net = (q_u − γD_f) / FS
Meaning
q_a = allowable (gross); q_a,net = allowable (net); FS = safety factor (2.5–3.0).
Watch out
MOST COMMON EXAM ERROR: using FS = 2 or forgetting FS entirely. Typical FS = 3 in Philippines; check code.
When to use
All footing design; choose gross or net depending on problem context.
FS ≥ (q_u / q_applied) or q_applied ≤ q_u / FS
Meaning
Check if applied bearing pressure is safe; equivalently, applied must be ≤ allowable.
Watch out
FS on bearing capacity is safety margin; separate from settlement criteria (which also limit design).
When to use
Verification problem: given applied pressure, check safety margin.
Common values
Value
2.5–3.0
Symbol
Philippine standard; FS = 3 most common in exams
Quantity
Typical FS (bearing capacity)
Value
1.5–2.0
Symbol
More stringent; limits differential settlement
Quantity
Typical FS (settlement)
Value
2.5–3.0 on bearing capacity
Symbol
Philippine requirement; may vary by occupation
Quantity
RA 544 (Building Code) FS
Important facts
- Net bearing capacity is used when foundation is embedded (depth D_f > 0); represents the 'new' load being added
- Gross bearing capacity includes the surcharge q = γD_f; total pressure at failure
- For shallow foundations in Philippines (RA 544, building code), typical FS = 2.5–3.0 for shear
- Settlement check is SEPARATE from bearing-capacity check; often settlement is more restrictive
- q_a,net is commonly used for allowable bearing in embankment analysis or deep placement
- Q_a = q_a × A: ensure units are consistent (kPa × m² = kN)
- Factor of safety on net basis typically 1.5–2.0 for settlement (much stricter than shear FS = 3)
Key definitions
Gross Allowable Bearing Capacity (q_a)
Example
If q_u = 750 kPa and FS = 3, then q_a = 250 kPa (includes footing self-weight in the calculation).
Safe design pressure including self-weight effect; q_a = q_u / FS.
Net Allowable Bearing Capacity (q_a,net)
Example
If q_u = 750 kPa, γD_f = 20 kPa, FS = 3, then q_a,net = 730 / 3 ≈ 243 kPa.
Safe design pressure for applied load only, excluding overburden; q_a,net = (q_u − γD_f) / FS.
Net Ultimate Bearing Capacity (q_u,net)
Example
q_u = 750 kPa, q = 20 kPa → q_u,net = 730 kPa.
Shear failure pressure minus overburden; q_u,net = q_u − γD_f.
Safety Factor (FS)
Example
FS = 3 means footing can carry 3× the design load before failure.
Ratio of failure load to safe load; FS = 2.5–3.0 typical; higher for uncertain soils.
Diagrams to know
- Stress distribution: applied footing pressure + overburden = total pressure
- Allowable bearing vs ultimate bearing: q_a = q_u / FS concept
Reactions or equations
q_u = q_u,net + q = q_u,net + γD_f
Note
q_u is the total stress at footing level when failure occurs; q_u,net is the increment.
Conditions
Relationship between gross and net ultimate bearing capacities
Special Cases: Cohesionless & Cohesive Soils
Formulas
For φ = 0 (undrained clay): N_c = 5.7, N_q = 1, N_γ = 0
Meaning
Undrained condition; cohesion dominates; surcharge has minimal effect; footing width term vanishes.
Watch out
N_γ = 0 means (1/2)γBN_γ = 0 — footing width does NOT increase bearing capacity. Only depth (surcharge) helps.
When to use
Saturated clay in undrained (short-term) condition; total stress analysis.
For cohesionless soil (φ > 0, c = 0): q_u = qN_q + (1/2)γBN_γ
Meaning
Cohesion term drops; only surcharge and footing width contribute.
Watch out
Footing width effect is large in sand; shallow wide footings are preferred. Depth effect is also critical.
When to use
Sand, gravel, or drained soil; no cohesion.
For partially saturated or drained clay: use c' and φ' from triaxial tests; full Terzaghi applies
Meaning
Effective stress; cohesion is inter-particle bonding; friction is primary resistance.
Watch out
DO NOT mix drained (φ') and undrained (φ = 0) — they are different failure conditions.
When to use
Long-term behavior; drained design.
Common values
Value
N_c = 5.7, N_q = 1, N_γ = 0 (strip); multiply by 1.3 for square
Symbol
Terzaghi classic values
Quantity
Undrained clay (φ = 0)
Value
25–50 kPa
Symbol
Undrained shear strength; low for soft clay
Quantity
Typical c_u (soft clay)
Value
75–150 kPa
Symbol
Higher for stiff/hard clay
Quantity
Typical c_u (stiff clay)
Value
c = 0, φ = 25–35°
Symbol
No cohesion; friction angle is sole parameter
Quantity
Cohesionless soil (sand)
Important facts
- Undrained clay (φ = 0): q_u is independent of footing width B — design width for settlement, not bearing
- Cohesionless soil (c = 0, sand): bearing capacity highly sensitive to width and depth; shallow wide footings preferred
- Drained clay (c', φ'): both terms contribute; typical φ' = 20–30°, c' = 5–20 kPa
- For φ = 0: N_γ = 0 → no benefit from increasing B; must rely on c and q = γD_f
- NSCP 2015 allows net bearing capacity method for long-term conditions; use c', φ'
- Undrained analysis is short-term; drained analysis is long-term (post-consolidation)
- Mixed soil (c ≠ 0, φ ≠ 0): use full Terzaghi with both c and φ terms
Key definitions
Undrained Soil (φ = 0)
Example
Saturated clay under quick loading; c_u = 50 kPa, φ = 0.
No drainage during loading; pore pressure increases; total stress analysis; cohesion c_u (undrained shear strength) governs.
Drained Soil
Example
Sand or gravel; long-term clay settlement; c' = 0 kPa, φ' = 30°.
Pore water can escape during loading; effective stress controls; use c' and φ' (effective parameters).
Cohesionless Soil
Example
Clean sand: c = 0, φ = 30°.
No inter-particle bonding (c = 0); friction angle φ is sole strength parameter; sand, gravel.
Cohesive Soil
Example
Clay: c_u = 75 kPa, φ = 0 (undrained); or c' = 10 kPa, φ' = 25° (drained).
Inter-particle bonding (c > 0); clay, silt; may have both c and φ (drained) or only c_u (undrained).
Diagrams to know
- Stress-strain curve: undrained (constant volume) vs drained (increasing volume)
- Bearing-capacity factor comparison: c-φ soil vs φ = 0 vs c = 0
Reactions or equations
q_u = c_u × N_c for φ = 0
Note
Simplified when N_q = 1 and N_γ = 0; qN_q term is small relative to cN_c in practice.
Conditions
Undrained clay; N_c = 5.7 (Terzaghi) for strip; 1.3 × 5.7 ≈ 7.4 for square
Exam Problem-Solving Workflow
Important facts
- Step 1: Identify footing type (strip, square, circular, rectangular) and soil type (c-φ, φ = 0, cohesionless)
- Step 2: Determine failure mode (general shear standard; local shear if loose/soft)
- Step 3: Check water-table position (above, at, or below footing base)
- Step 4: Gather soil parameters: c, φ, γ, D_f, B (or L for rectangular); look up or calculate N_c, N_q, N_γ
- Step 5: Calculate q = γD_f (surcharge); apply water-table correction to unit weight if needed
- Step 6: Calculate q_u using Terzaghi equation with shape factors and depth factors if applicable
- Step 7: Calculate q_a = q_u / FS (or q_a,net if using net basis) with FS = 2.5–3.0
- Step 8: Convert q_a to allowable load Q_a = q_a × A if needed
- Step 9: Check settlement separately (SBC, differential settlement limits) — often more restrictive than bearing
- Step 10: Report design allowable bearing capacity or safe column load
Must remember
- Terzaghi's equation: q_u = cN_c + qN_q + (1/2)γBN_γ is the CORE formula; memorize exact form (surcharge q = γD_f, shape factors modify each term).
- Shape factors: square uses 1.3 on c-term and 0.4 on γ-term; circular uses 1.3 and 0.3; strip uses 1.0 and 0.5 (reference). DO NOT apply shape factors twice.
- Surcharge q = γD_f is ALWAYS the pressure at footing base depth from soil above; it multiplies N_q and is critical for deep footings.
- Water table: replace γ with γ' = γ_sat − 9.81 ONLY in the (1/2)γBN_γ term if water is at or above base. The q = γD_f term is unaffected.
- φ = 0 undrained clay: N_c = 5.7, N_q = 1, N_γ = 0 — footing width does NOT increase capacity; only c and q matter. Design for settlement, not bearing.
- Allowable bearing: q_a = q_u / FS with FS = 2.5–3.0 (use 3.0 for typical exam). Net basis: q_a,net = (q_u − γD_f) / FS.
- Local vs general shear: general is standard in exams. Use local (reduce φ by 2/3) ONLY if problem states loose/soft soil or Dr < 50%.
- Factor of safety is NOT the same as utilization ratio. FS = 3 means design pressure is 1/3 of ultimate. Check: FS = q_u / q_applied ≥ 2.5–3.0.
- Settlement check is SEPARATE from bearing-capacity check. Settlement often is MORE restrictive (FS = 1.5–2.0 on SBC). Always address both.
- Allowable load on footing: Q_a = q_a × Area (m²). Ensure units: kPa × m² = kN. Most common exam mistake: unit inconsistency or omitting FS entirely.
Last minute tips
- READ THE FOOTING TYPE FIRST: strip, square, circular, or rectangular. If not stated, problem usually specifies or asks you to compare. Shape factors change everything.
- WATER TABLE: Look for depth statement. If water table is not mentioned, assume it is well below (no correction). If mentioned, apply γ' in width term ONLY, never in surcharge term.
- UNDRAINED vs DRAINED: If φ = 0, use N_c = 5.7, N_q = 1, N_γ = 0 (do not look up); width does not help. If φ > 0, use tables to find N-factors. This distinction kills ~30% of student answers.
- FS IS MANDATORY: If problem does not specify FS, use 3.0 (Philippine standard). Always report q_a = q_u / 3, not just q_u. Many exams fail students who give ultimate instead of allowable.
- SURCHARGE = γ × D_f: Depth of footing matters hugely for sand (N_q term is large). Deeper footings are more stable. Do not forget q in the qN_q term — this is a high-frequency error that loses easy points.
Comparison tables
Strip vs Square vs Circular Footings — Bearing Capacity
| Footing Type | Shape Factors (s_c, s_q, s_γ) | Width-Term Multiplier | Typical Use | Advantage |
|---|---|---|---|---|
| Strip | 1.0, 1.0, 1.0 | 0.5 | Wall foundations, long structures | Simple; reference case |
| Square | 1.3, ~1.2, 0.6 (approx.) | 0.4 | Building columns, symmetric loads | Compact; more capacity than strip |
| Circular | 1.3, ~1.2, 0.3 (approx.) | 0.3 | Tank foundations, pole supports | All-directional symmetry |
| Rectangular (B < L) | 1 + (B/L)(N_q/N_c), varies, 1 − 0.4(B/L) | 0.5 | General buildings | Between strip and square |
φ = 0 (Undrained Clay) vs c = 0 (Cohesionless Sand) — Key Differences
| Property | Undrained Clay (φ = 0) | Cohesionless Sand (c = 0) |
|---|---|---|
| N-Factors | N_c = 5.7, N_q = 1, N_γ = 0 | Varies with φ; N_c large, N_q moderate, N_γ significant |
| Footing Width Effect | NONE (N_γ = 0); width does not increase q_u | VERY IMPORTANT (N_γ ≠ 0); larger B → higher q_u |
| Depth Effect (q = γD_f) | Moderate (qN_q term, but N_q ≈ 1 is small) | Large (qN_q term is significant for φ ≥ 25°) |
| Design Approach | Short-term (undrained); cohesion governs | Long-term (drained); friction governs |
| Bearing Capacity Sensitivity | Less sensitive to geometry; mainly c_u | Highly sensitive to B, D_f, and φ |
| Settlement Check | Primary concern (consolidation settlement) | Secondary (elastic settlement usually small) |
Water-Table Position — Effect on Bearing Capacity
| Water-Table Position | Unit Weight Used in q_u Term | Effect on Bearing Capacity | Correction Method |
|---|---|---|---|
| Above footing base (z < D_f) | γ (or γ_sat) for depth zone above water | Significant reduction; buoyancy | Use γ for zone above water; γ' for zone below in width term |
| At footing base (z = D_f) | γ' in (1/2)γBN_γ term | 20–40% reduction in q_u | Replace γ with γ' in width term only |
| Within depth B below base (D_f < z < D_f + B) | Interpolate or use weighted γ | Moderate reduction; ~10–20% | Linear interpolation; often simplified in exams |
| Well below footing (z >> D_f + B) | γ (dry weight above, no correction below) | Negligible | No correction; use γ as normal |
Bearing-Capacity Factors — Selected Values for Common φ Angles
| Friction Angle φ | N_c (Terzaghi) | N_q (Terzaghi) | N_γ (Terzaghi) |
|---|---|---|---|
| 0° (undrained clay) | 5.7 | 1.0 | 0 |
| 10° | 8.3 | 2.5 | 0.8 |
| 15° | 11.0 | 3.9 | 1.7 |
| 20° | 14.8 | 6.4 | 3.5 |
| 25° | 20.7 | 10.7 | 6.8 |
| 30° | 30.1 | 18.4 | 15.1 |
| 35° | 46.1 | 33.3 | 48.0 |
Safety Factors (FS) — Common Values & Applications
| Condition / Code | FS (Bearing Capacity Shear) | FS (Settlement) | Context |
|---|---|---|---|
| Standard building (RA 544, NSCP 2015) | 2.5–3.0 | 1.5–2.0 | Philippine building code |
| Stable soil, well-defined properties | 2.5 | 1.5 | Low uncertainty |
| Uncertain soil, variable deposits | 3.0 | 2.0 | High uncertainty; conservative |
| Waterfront, expansive, or weak soil | 3.5–4.0 | 2.0–2.5 | Very high risk |
| Exam default (when FS not specified) | 3.0 | 2.0 (often not directly tested) | Most common assumption |
General Shear vs Local Shear Failure
| Characteristic | General Shear (Dense/Stiff Soil) | Local Shear (Loose/Soft Soil) |
|---|---|---|
| Footing Behavior | Clear tip-over; abrupt failure | Gradual settlement; punching |
| Shear Surface | Well-defined log-spiral wedge | Diffuse; minimal lateral displacement |
| Relative Density / SPT N | Dr > 70% or N > 30 | Dr < 50% or N < 10 |
| Soil Type Example | Dense sand, stiff clay | Loose sand, soft clay |
| Reduction Factor for φ | Use φ as-is; N-factors standard | Use φ' = arctan(2/3 × tan(φ)) |
| Typical q_u Reduction | Baseline (reference) | 5–30% lower than general shear |
Diagrams
Bearing Capacity Equation — Complete Structure
Mind map showing the three components of Terzaghi's bearing-capacity equation and their modifiers. Helps students recall which term applies when and how shape/depth factors adjust each component.
Bearing Capacity Problem-Solving Decision Tree
Step-by-step decision tree for solving bearing-capacity problems. Guides students through footing identification, water-table correction, soil classification, N-factor selection, and final allowable bearing calculation.
Foundation Failure Sequence — General vs Local Shear
State diagram showing the progression from initial footing contact through to failure, contrasting general shear (dense soil) and local shear (loose soil) modes. Illustrates why failure mode selection is critical for design.
Water-Table Correction — Unit Weight Selection
Flowchart for determining correct unit weight to use based on water-table position. Shows when γ, γ', or interpolation applies to each term, and qualitatively how bearing capacity changes.