CELE Geotechnical Engineering — Bearing Capacity of SoilsRevision Notes

Exam Tips

• If the problem says 'dense sand' or 'stiff clay' → general shear, use full c and φ.
• If the problem says 'loose sand' or 'soft to medium clay' → local shear, replace c with (2/3)c and tan φ with (2/3) tan φ.
• Punching shear is rarely tested numerically; focus on identifying the mode conceptually.

Key Points

• Bearing capacity failure is a shear failure in the soil mass beneath the footing — the soil cannot resist the applied contact pressure.
• Three classical failure modes: (1) General Shear Failure — dense/stiff soils, well-defined slip surface, sudden collapse; (2) Local Shear Failure — medium-density soils, significant compression before collapse, less defined slip surface; (3) Punching Shear Failure — loose/soft soils, vertical shearing around the perimeter, no heave observed.
• Shallow footing condition: embedment depth D_f ≤ B (width) — Terzaghi's classical assumption.
• Deep footings (D_f > B) require Meyerhof or Hansen equations — outside standard Terzaghi scope.
• The contact pressure q at the base of the footing equals Q / A (applied column load divided by footing area) for a centrically loaded footing.

Definitions

Section Title

1. Fundamental Concepts and Failure Modes

Common Mistakes

• Forgetting to classify the failure mode before applying formulas — dense soils use full c and φ; loose soils use reduced values (2/3 reduction).
• Confusing shallow and deep footing criteria — Terzaghi is valid only for D_f ≤ B.
• Using the wrong failure mode chart when the problem does not explicitly state 'general shear' or 'local shear.'

Formulas

Exam Tips

• Memorize the shape coefficient table: strip=(1.0, 1.0, 0.5); square=(1.3, 1.0, 0.4); circular=(1.3, 1.0, 0.3).
• For φ=0 problems: write q_u=5.7c+q (strip) or 7.41c+q (square) directly — saves time.
• Always write q=γD_f as your first step; it prevents the most common error of forgetting the surcharge term.
• When the problem asks for 'allowable column load', use Q_a = q_a,net × B² (square) or q_a,net × A (rectangular).
• Bearing-capacity factors are given in all board exam problem stems or supplementary tables — you do not need to memorize exact values for every angle, but memorize φ=0 values (5.7, 1, 0) and know the trend.

Key Points

• Terzaghi (1943) derived the first rational bearing-capacity equation for shallow continuous (strip) footings, later extended to square and circular shapes.
• The equation has three terms: cohesion term (cN_c), surcharge/overburden term (qN_q), and self-weight/width term (½γBN_γ).
• Strip footing: q_u = cN_c + qN_q + ½γBN_γ
• Square footing: q_u = 1.3cN_c + qN_q + 0.4γBN_γ
• Circular footing (diameter B): q_u = 1.3cN_c + qN_q + 0.3γBN_γ
• For purely cohesive soil (φ = 0): N_c = 5.7, N_q = 1, N_γ = 0; so q_u = 5.7c + q (strip) or q_u = 1.3(5.7)c + q = 7.41c + q (square).
• For sand (c = 0): q_u = qN_q + ½γBN_γ (strip), with the cohesion term vanishing completely.

Definitions

Section Title

2. Terzaghi's Bearing Capacity Equation

Common Mistakes

• Using 0.5 (strip coefficient) instead of 0.4 (square) or 0.3 (circular) on the γBN_γ term.
• Forgetting to multiply by 1.3 on the cN_c term for square and circular footings.
• For φ=0 clay, leaving N_γ=0 but still multiplying by 0.4 or 0.3 — result is still zero, but students sometimes apply 1.3 to the wrong term.
• Computing q_u then reporting it as the allowable load without dividing by FS.
• Omitting the surcharge term qN_q entirely when D_f is given.

Exam Tips

• φ=0 values (5.7, 1, 0) must be memorized — they appear in virtually every clay problem.
• The ratio N_q/N_c ≈ 0.5 at φ=0 and increases with φ — use as a quick sanity check.
• If factors are not given and φ is given, the problem will provide them in the given data section.

Key Points

• N_c, N_q, and N_γ are tabulated functions of the internal friction angle φ; they increase as φ increases.
• At φ=0°: N_c=5.7, N_q=1.0, N_γ=0.0 (Terzaghi values for saturated clay).
• At φ=10°: N_c≈9.61, N_q≈2.69, N_γ≈1.22.
• At φ=20°: N_c≈17.69, N_q≈7.44, N_γ≈3.64.
• At φ=25°: N_c≈25.13, N_q≈12.72, N_γ≈8.34.
• At φ=30°: N_c≈37.16, N_q≈22.46, N_γ≈19.13.
• At φ=35°: N_c≈57.75, N_q≈41.39, N_γ≈42.92.
• The exponential growth of N_q and N_γ with φ explains why dense, highly frictional soils (gravels, dense sands) have dramatically higher bearing capacity.
• Board exam problems always provide the required N values in the problem statement — focus on correctly applying them, not deriving them.

Section Title

3. Bearing-Capacity Factors — Reference Table

Common Mistakes

• Interpolating incorrectly between table values — always use the values given in the problem.
• Using Meyerhof N factors (different values) when the problem specifies Terzaghi — always check which formulation is specified.

Formulas

Exam Tips

• Draw a quick sketch of the footing and water table position before selecting the correction case.
• Always compare d_w (depth to WT below footing base) with B to identify Case 2 or Case 3.
• If WT is exactly at the footing base, use γ' in the width term and γ_moist·D_f for the surcharge.
• A quick memory hook: 'Below the base, within B — interpolate; beyond B — ignore.'

Key Points

• Groundwater reduces the effective unit weight of soil, thereby reducing the bearing capacity.
• Case 1 — Water table at or above the footing base (d_w ≤ D_f): The surcharge q in the N_q term uses the moist/saturated unit weight above the water table; the width term ½γBN_γ uses the effective (submerged) unit weight γ' = γ_sat − γ_w.
• Case 2 — Water table within depth B below the footing base (0 < d_w ≤ B): Linear interpolation applies; the effective unit weight in the width term is adjusted proportionally: γ_eff = γ' + (d_w/B)(γ − γ'). For d_w=0 use γ'; for d_w=B use γ (moist).
• Case 3 — Water table deeper than B below the footing base: No correction required; use the moist unit weight γ throughout.
• Typical values: γ_w = 9.81 kN/m³; γ' = γ_sat − γ_w (commonly 8–11 kN/m³ for typical saturated soils).
• The N_q surcharge term is also affected when the water table is above the footing base — use effective stress in q = γD_f when fully submerged above the base.

Definitions

Section Title

4. Water-Table Correction

Common Mistakes

• Applying γ' to the surcharge term q=γD_f when the water table is below the footing base — the surcharge uses moist unit weight above the table.
• Using γ_sat instead of γ' in the width term when water table is at the footing base.
• Forgetting that Case 3 (water table > B below base) requires no correction at all.
• Using γ_w = 10 kN/m³ instead of 9.81 kN/m³ — use 9.81 unless the problem specifies otherwise.

Formulas

Exam Tips

• When a problem asks for 'allowable column load', always use the NET allowable approach: Q_a = [(q_u − γD_f)/FS] × A.
• If FS is not stated, use FS=3 as the default Philippine practice value.
• Remember: q_a,net × A gives the load the SOIL can support from the COLUMN; footing self-weight is additional.

Key Points

• The factor of safety for bearing capacity is typically FS = 3 for permanent structures; FS = 2.5 may be used with careful site investigation.
• Gross allowable: q_a = q_u / FS — compared against the total contact pressure including the weight of footing and overburden.
• Net allowable: q_a,net = (q_u − γD_f) / FS — represents the net additional pressure the footing imposes on the soil beyond the original overburden.
• Allowable column load: Q_a = q_a,net × A_footing.
• Settlement governs design in many cases — particularly for normally consolidated clays and loose sands — even when the shear bearing capacity is sufficient.
• The PRC exam may ask you to check both bearing capacity (shear failure) and settlement separately; settlement calculations use consolidation theory (covered in a separate chapter).
• In Philippine practice, the NSCP 2015 Section 304 provides prescriptive allowable bearing pressures for common soil types as a starting point for design, but computed q_a governs for actual design.

Definitions

Section Title

5. Allowable Bearing Capacity and Settlement Consideration

Common Mistakes

• Applying FS directly to q_u to get allowable load without subtracting overburden first (net vs. gross distinction).
• Forgetting that Q_a is the column load only — the weight of the footing itself must be added when checking soil pressure if required.
• Not considering settlement when the problem states soft clay or loose sand conditions.

Formulas

Exam Tips

• On the board exam, write the formula first, substitute values, then compute — partial credit is awarded for correct setup even if arithmetic errors occur.
• Box your final answer with units (kPa for pressures, kN for loads).
• Always state which footing shape (strip/square/circular) before writing the formula — it signals to the examiner you know the correct form.

Key Points

• Example A: Strip footing, c-φ soil, general shear.
• Example B: Square footing, purely cohesive clay (φ=0).
• Example C: Square footing, net allowable load.
• Example D: Strip footing with water table at base.

Section Title

6. Comprehensive Worked Examples (Board-Exam Style)

Common Mistakes

• In Example D, applying γ' also to the surcharge term q=γD_f when the water table is exactly at the base (not above it) — the moist unit weight governs the surcharge in this case.
• In Example C, multiplying q_a (gross) by area instead of q_a,net — overestimates the net column load capacity.