CELE Geotechnical Engineering — Bearing Capacity of SoilsCheat Sheet
A printable cheat sheet for Bearing Capacity of Soils, built for CELE reviewers who want one go-to reference in the final stretch. Covers formulas, key definitions, common question types, and the Professional Regulation Commission (PRC) — Board of Civil Engineering-specific twists you will see on CELE day.
Exam context
On the CELE 2026, the Geotechnical Engineering subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Bearing Capacity of Soils lands at position 9th out of 11 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Geotechnical Engineering on a typical CELE paper.
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.
Sections
Formulas
Formula
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)
Section Title
Terzaghi's Bearing Capacity Equation — Core Formula
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
Term
Ultimate Bearing Capacity (q_u)
Example
If q_u = 750 kPa, footing will fail (shear) at applied pressure > 750 kPa.
Definition
Maximum pressure the soil can sustain before shear failure occurs; foundation design keeps applied pressure below this.
Term
Allowable Bearing Capacity (q_a)
Example
If q_u = 750 kPa and FS = 3, then q_a = 250 kPa (safe for design).
Definition
Safe design pressure = q_u / FS, where FS is factor of safety (typically 2.5–3.0).
Term
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).
Definition
Dimensionless coefficients derived from soil friction angle φ; account for cohesion, surcharge, and footing width contributions.
Term
Shallow Footing
Example
A 2 m wide footing at 1 m depth (D_f/B = 0.5) is shallow.
Definition
Depth D_f ≤ 3–4 × width B; Terzaghi's equation applies directly.
Term
General Shear Failure
Example
Dense sand or stiff clay under normal loading.
Definition
Clear shear surfaces develop; occurs in dense/stiff soil; use standard N-factors.
Term
Local (Punching) Shear Failure
Example
Loose sand or soft clay; use reduced φ' = arctan(2/3 tan(φ)).
Definition
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
Note
This is the overburden pressure; it is the 'q' in the Terzaghi equation's qN_q term, NOT the applied footing pressure.
Equation
q = γD_f
Conditions
D_f = depth from ground surface to footing base; γ = unit weight of soil above footing
Note
Gross allowable; does not subtract overburden. Use net allowable (q_a,net) if subtracting self-weight effects.
Equation
q_a = q_u / FS
Conditions
FS = 2.5–3.0 typical; design applied pressure must be ≤ q_a
Note
Often used for embankment loading; compares net applied pressure to net ultimate.
Equation
q_a,net = (q_u − γD_f) / FS
Conditions
FS on net basis; accounts for surcharge already in place
Note
Submerged unit weight; used when water table is at or above footing base.
Equation
γ' = γ_sat − γ_w
Conditions
γ_sat = saturated unit weight; γ_w = 9.81 kN/m³ (water)
Formulas
Formula
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.
Formula
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).
Formula
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)
Section Title
Shape Factors & Depth Corrections
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
Term
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.
Definition
Multiplier on each Terzaghi term; accounts for footing geometry (strip, square, circular, rectangular).
Term
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).
Definition
Multiplier that increases bearing capacity with footing depth; deeper footings confined by surrounding soil.
Term
Inclination Factor (i_c, i_q, i_γ)
Example
Inclined load (e.g., sloping footing) requires i-factors; vertical load: i = 1.0.
Definition
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
Formulas
Formula
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.
Formula
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.
Formula
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)
Section Title
Water-Table Corrections & Saturated Soil
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
Term
Saturated Unit Weight (γ_sat)
Example
Typical clay: γ_sat ≈ 19 kN/m³.
Definition
Total weight of soil + water per unit volume, typically 18–22 kN/m³.
Term
Submerged (Effective) Unit Weight (γ')
Example
If γ_sat = 19 kN/m³, then γ' = 19 − 9.81 ≈ 9.2 kN/m³.
Definition
Apparent weight in water; γ' = γ_sat − γ_w ≈ γ_sat − 9.81 kN/m³, typically 8–12 kN/m³.
Term
Water Table at Footing Base
Example
Footing at 1.5 m depth, water level at 1.5 m: use γ' in width term.
Definition
Water surface is at depth D_f; full saturation below footing.
Term
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.
Definition
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
Note
Effective (submerged) unit weight; always positive but much lower than dry γ.
Equation
γ' = γ_sat − γ_w = γ_sat − 9.81
Conditions
All soils below water table; γ_w = unit weight of water (constant, 9.81 kN/m³)
Formulas
Formula
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
Section Title
Failure Modes: General vs Local Shear
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
Term
General Shear Failure
Example
Dense sand (Dr > 70%), stiff clay (cu > 100 kPa).
Definition
Clear shear surfaces (log-spiral wedge) develop; footing moves vertically with well-defined collapse load; dense/stiff soil.
Term
Local Shear Failure
Example
Loose sand (Dr < 50%), soft clay (cu < 50 kPa).
Definition
Footing settles into soil without clear slip surfaces; soil punching or punching shear; loose/soft soil.
Term
Punching Shear
Example
Weak foundation soil; footing sinks rather than tilts.
Definition
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
Note
Use φ' to recalculate N-factors from tables, then apply to Terzaghi equation.
Equation
φ' = arctan((2/3) × tan(φ))
Conditions
Local shear reduction; φ' is reduced friction angle for loose/soft soil
Formulas
Formula
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.
Formula
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.
Formula
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.
Formula
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
Section Title
Net Bearing Capacity & Allowable Load Calculations
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
Term
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).
Definition
Safe design pressure including self-weight effect; q_a = q_u / FS.
Term
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.
Definition
Safe design pressure for applied load only, excluding overburden; q_a,net = (q_u − γD_f) / FS.
Term
Net Ultimate Bearing Capacity (q_u,net)
Example
q_u = 750 kPa, q = 20 kPa → q_u,net = 730 kPa.
Definition
Shear failure pressure minus overburden; q_u,net = q_u − γD_f.
Term
Safety Factor (FS)
Example
FS = 3 means footing can carry 3× the design load before failure.
Definition
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
Note
q_u is the total stress at footing level when failure occurs; q_u,net is the increment.
Equation
q_u = q_u,net + q = q_u,net + γD_f
Conditions
Relationship between gross and net ultimate bearing capacities
Formulas
Formula
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.
Formula
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.
Formula
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)
Section Title
Special Cases: Cohesionless & Cohesive Soils
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
Term
Undrained Soil (φ = 0)
Example
Saturated clay under quick loading; c_u = 50 kPa, φ = 0.
Definition
No drainage during loading; pore pressure increases; total stress analysis; cohesion c_u (undrained shear strength) governs.
Term
Drained Soil
Example
Sand or gravel; long-term clay settlement; c' = 0 kPa, φ' = 30°.
Definition
Pore water can escape during loading; effective stress controls; use c' and φ' (effective parameters).
Term
Cohesionless Soil
Example
Clean sand: c = 0, φ = 30°.
Definition
No inter-particle bonding (c = 0); friction angle φ is sole strength parameter; sand, gravel.
Term
Cohesive Soil
Example
Clay: c_u = 75 kPa, φ = 0 (undrained); or c' = 10 kPa, φ' = 25° (drained).
Definition
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
Note
Simplified when N_q = 1 and N_γ = 0; qN_q term is small relative to cN_c in practice.
Equation
q_u = c_u × N_c for φ = 0
Conditions
Undrained clay; N_c = 5.7 (Terzaghi) for strip; 1.3 × 5.7 ≈ 7.4 for square
Section Title
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
Rows
Values
- 1.0, 1.0, 1.0
- 0.5
- Wall foundations, long structures
- Simple; reference case
Property
Strip
Values
- 1.3, ~1.2, 0.6 (approx.)
- 0.4
- Building columns, symmetric loads
- Compact; more capacity than strip
Property
Square
Values
- 1.3, ~1.2, 0.3 (approx.)
- 0.3
- Tank foundations, pole supports
- All-directional symmetry
Property
Circular
Values
- 1 + (B/L)(N_q/N_c), varies, 1 − 0.4(B/L)
- 0.5
- General buildings
- Between strip and square
Property
Rectangular (B < L)
Columns
- Footing Type
- Shape Factors (s_c, s_q, s_γ)
- Width-Term Multiplier
- Typical Use
- Advantage
Table Title
Strip vs Square vs Circular Footings — Bearing Capacity
Rows
Values
- N_c = 5.7, N_q = 1, N_γ = 0
- Varies with φ; N_c large, N_q moderate, N_γ significant
Property
N-Factors
Values
- NONE (N_γ = 0); width does not increase q_u
- VERY IMPORTANT (N_γ ≠ 0); larger B → higher q_u
Property
Footing Width Effect
Values
- Moderate (qN_q term, but N_q ≈ 1 is small)
- Large (qN_q term is significant for φ ≥ 25°)
Property
Depth Effect (q = γD_f)
Values
- Short-term (undrained); cohesion governs
- Long-term (drained); friction governs
Property
Design Approach
Values
- Less sensitive to geometry; mainly c_u
- Highly sensitive to B, D_f, and φ
Property
Bearing Capacity Sensitivity
Values
- Primary concern (consolidation settlement)
- Secondary (elastic settlement usually small)
Property
Settlement Check
Columns
- Property
- Undrained Clay (φ = 0)
- Cohesionless Sand (c = 0)
Table Title
φ = 0 (Undrained Clay) vs c = 0 (Cohesionless Sand) — Key Differences
Rows
Values
- 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
Property
Above footing base (z < D_f)
Values
- γ' in (1/2)γBN_γ term
- 20–40% reduction in q_u
- Replace γ with γ' in width term only
Property
At footing base (z = D_f)
Values
- Interpolate or use weighted γ
- Moderate reduction; ~10–20%
- Linear interpolation; often simplified in exams
Property
Within depth B below base (D_f < z < D_f + B)
Values
- γ (dry weight above, no correction below)
- Negligible
- No correction; use γ as normal
Property
Well below footing (z >> D_f + B)
Columns
- Water-Table Position
- Unit Weight Used in q_u Term
- Effect on Bearing Capacity
- Correction Method
Table Title
Water-Table Position — Effect on Bearing Capacity
Rows
Values
- 5.7
- 1.0
- 0
Property
0° (undrained clay)
Values
- 8.3
- 2.5
- 0.8
Property
10°
Values
- 11.0
- 3.9
- 1.7
Property
15°
Values
- 14.8
- 6.4
- 3.5
Property
20°
Values
- 20.7
- 10.7
- 6.8
Property
25°
Values
- 30.1
- 18.4
- 15.1
Property
30°
Values
- 46.1
- 33.3
- 48.0
Property
35°
Columns
- Friction Angle φ
- N_c (Terzaghi)
- N_q (Terzaghi)
- N_γ (Terzaghi)
Table Title
Bearing-Capacity Factors — Selected Values for Common φ Angles
Rows
Values
- 2.5–3.0
- 1.5–2.0
- Philippine building code
Property
Standard building (RA 544, NSCP 2015)
Values
- 2.5
- 1.5
- Low uncertainty
Property
Stable soil, well-defined properties
Values
- 3.0
- 2.0
- High uncertainty; conservative
Property
Uncertain soil, variable deposits
Values
- 3.5–4.0
- 2.0–2.5
- Very high risk
Property
Waterfront, expansive, or weak soil
Values
- 3.0
- 2.0 (often not directly tested)
- Most common assumption
Property
Exam default (when FS not specified)
Columns
- Condition / Code
- FS (Bearing Capacity Shear)
- FS (Settlement)
- Context
Table Title
Safety Factors (FS) — Common Values & Applications
Rows
Values
- Clear tip-over; abrupt failure
- Gradual settlement; punching
Property
Footing Behavior
Values
- Well-defined log-spiral wedge
- Diffuse; minimal lateral displacement
Property
Shear Surface
Values
- Dr > 70% or N > 30
- Dr < 50% or N < 10
Property
Relative Density / SPT N
Values
- Dense sand, stiff clay
- Loose sand, soft clay
Property
Soil Type Example
Values
- Use φ as-is; N-factors standard
- Use φ' = arctan(2/3 × tan(φ))
Property
Reduction Factor for φ
Values
- Baseline (reference)
- 5–30% lower than general shear
Property
Typical q_u Reduction
Columns
- Characteristic
- General Shear (Dense/Stiff Soil)
- Local Shear (Loose/Soft Soil)
Table Title
General Shear vs Local Shear Failure
Previous chapter
Lateral Earth Pressure and Retaining Structures
Next chapter
Foundations (Shallow and Deep)
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