CELE Geotechnical Engineering — Bearing Capacity of SoilsStudy Notes

A foundation fails when the soil beneath it undergoes shear rupture. The ultimate bearing capacity (q_u) is the maximum pressure at which this failure occurs. In the Philippines, where tropical soils vary widely—from soft clays in lowland areas to dense sands and laterites in upland regions—predicting bearing capacity is essential to prevent catastrophic settlement or collapse. There are two types of bearing-capacity failure: **General Shear Failure**: Occurs in dense soils and stiff clays. The failure surface extends from the footing edges to the ground surface, forming a characteristic wedge-shaped block. This is the classic case assumed in most textbooks and design codes. **Local (Punching) Shear Failure**: Occurs in loose sands and soft clays. The footing punches into the soil with less lateral displacement. The failure surface does not extend fully to the surface. In this case, the bearing capacity is lower than predicted by the general shear equations, and corrections are necessary. **Why It Matters for PRC Exam**: Board questions frequently ask you to: - Identify which failure mode applies given soil properties - Apply the appropriate correction factors (typically 2/3 reduction for local shear) - Compute the safe allowable bearing pressure with proper factors of safety (FS = 2.5 to 3.0) The relationship between applied stress and bearing capacity determines the safety of the foundation. If the applied footing pressure exceeds q_u, the soil yields and uncontrolled settlement occurs, potentially damaging the structure. If it exceeds the allowable bearing capacity q_a (which is q_u divided by a safety factor), the design is unsafe under normal conditions, though the structure may still stand temporarily under reduced loads.

Karl Terzaghi (1943) developed the classic bearing-capacity formula that decomposes the ultimate bearing capacity into three independent components, each contributing to the soil's ability to resist applied pressure: **The Fundamental Equation (Strip Footing)**: q_u = c·N_c + q·N_q + (1/2)·γ·B·N_γ Where: - **c** = cohesive strength (undrained shear strength for clays, kPa) - **q** = overburden pressure at footing depth = γ·D_f (kPa) - **γ** = unit weight of soil above footing base (kN/m³) - **B** = width of footing (m) - **N_c, N_q, N_γ** = bearing-capacity factors (dimensionless, functions of friction angle φ) **Physical Interpretation of the Three Terms**: 1. **c·N_c (Cohesion Term)**: Represents the contribution of soil cohesion to bearing capacity. The factor N_c increases with friction angle; for φ = 0 (pure clay), N_c = 5.7. This term is independent of footing size—a cohesive soil provides the same unit cohesive resistance regardless of footing width. 2. **q·N_q (Surcharge Term)**: Reflects the effect of confining pressure at the footing depth. The deeper the footing (larger D_f), the higher the surcharge q, and the higher the bearing capacity. This accounts for the fact that soil at greater depth is confined by overburden and is stronger. N_q increases exponentially with φ. 3. **(1/2)·γ·B·N_γ (Width/Self-Weight Term)**: Represents the resistance from the weight of soil on either side of the footing (the "passive resistance" of the soil wedge). This term increases with footing width B and with the soil's unit weight γ. The factor N_γ increases rapidly with φ and is zero when φ = 0 (pure clay, no angular resistance). **Bearing-Capacity Factors from Tables** (for various φ values): For φ = 25°: N_c = 25.13, N_q = 12.72, N_γ = 8.34 For φ = 30°: N_c = 37.16, N_q = 22.46, N_γ = 19.13 For φ = 35°: N_c = 57.75, N_q = 41.44, N_γ = 42.16 For φ = 40°: N_c = 95.66, N_q = 81.27, N_γ = 100.41 For φ = 0° (clay): N_c = 5.7, N_q = 1.0, N_γ = 0 Note: These values are taken from standard soil mechanics references (e.g., Das & Sobhan, or the Philippine NSCP) and are widely used in practice. Different formulations (Hansen, Vesic) may give slightly different factors, but Terzaghi's values remain standard for board exams. **Why This Equation Is Important**: - It separates the problem into manageable parts, each reflecting a physical mechanism of load resistance. - It allows engineers to see which factor dominates: in shallow footings on weak clay (low c), the surcharge and width terms matter little; in deep footings, surcharge dominates. - It forms the basis for all subsequent modifications (shape factors, depth factors, inclination factors, etc.).

Terzaghi's equation was originally derived for strip (infinitely long, rectangular in cross-section) footings. Real footings are square, rectangular, or circular. Empirical research and theory show that the bearing capacity of a non-strip footing differs from that of a strip footing of the same width. Shape factors account for this difference. **Shape Factor Definition**: The bearing capacity of a non-strip footing is computed by multiplying certain terms in Terzaghi's equation by shape factors s_c, s_q, s_γ: q_u = s_c·c·N_c + s_q·q·N_q + s_γ·(1/2)·γ·B·N_γ Where the shape factors apply only to their respective terms. **Standard Shape Factors** (widely adopted in practice): **Square Footing** (B = L): - s_c = 1.3 - s_q = 1.0 - s_γ = 0.8 (some sources use 0.4 for conservative design; 0.8 is from Vesic; use 0.4 if specified) **Rectangular Footing** (B < L): - s_c = 1 + (B/L)·(N_q/N_c) - s_q = 1 + (B/L)·tan(φ) - s_γ = 1 - 0.4·(B/L) **Circular Footing** (diameter D): - s_c = 1.3 - s_q = 1.0 - s_γ = 0.6 (or 0.3 from some sources; use 0.3 for conservative design) **Physical Reason for Shape Factors**: 1. **Cohesion Term** (s_c = 1.3 for square/circular): Non-strip footings provide additional confining pressure at the corners, increasing the effective cohesive resistance. This is why s_c > 1.0 for square and circular footings. 2. **Surcharge Term** (s_q = 1.0 for square/circular): The surcharge effect is not significantly influenced by footing shape, so s_q ≈ 1.0 for most shapes. 3. **Width Term** (s_γ < 1.0): The width term is most sensitive to shape. For a square footing, s_γ = 0.8 (or 0.4), meaning the width contribution is 80% (or 40%) of that for a strip footing. For rectangular footings, s_γ decreases as the footing becomes more elongated (larger B/L ratio), meaning a long, narrow footing has less benefit from width resistance than a square one. **PRC Exam Context**: Board questions often specify "use shape factors" or give the footing dimensions and ask you to select the appropriate factors. Common mistakes include: - Using strip footing factors (1.0, 1.0, 0.5) for a square footing—this overestimates bearing capacity. - Confusing s_γ = 0.4 (conservative Terzaghi) with s_γ = 0.8 (Vesic); always use the factors specified in your problem or code. - Forgetting that shape factors apply only to their respective terms; some students multiply the entire equation by a single factor, which is incorrect.

Groundwater has a profound effect on bearing capacity. In the Philippines, where tropical monsoons and high water tables are common, accounting for water-table position is crucial to design accuracy. Water affects bearing capacity through two mechanisms: 1. **Reduction in Effective Unit Weight**: Above the water table, soil has full unit weight γ. Below the water table, the effective unit weight is γ' = γ_sat - γ_w, where γ_w = 9.81 kN/m³ (unit weight of water). Saturated soil weighs less effectively because water provides buoyancy. 2. **Pore Pressure Effects**: Water in pores reduces effective stress and thus reduces friction and cohesion. However, Terzaghi's formula uses total unit weight and total (or undrained) strength, so these effects are implicitly accounted for if c and φ are measured under the relevant conditions (undrained for clay, drained for sand). **Correction Rules** (Standard Practice): The adjustment concerns the unit weight in the width/self-weight term (1/2)·γ·B·N_γ. The cohesion and surcharge terms remain unchanged. **Case 1: Water Table at or Above the Footing Base** If the water table is at depth D_f (at the footing base) or shallower, the entire soil beneath the footing is saturated. Replace γ with γ' = γ_sat - γ_w in the width term: q_u = c·N_c + q·N_q + (1/2)·γ'·B·N_γ where q = γ·D_f (the overburden to depth D_f may be partially dry and partially wet; use the weighted average). **Case 2: Water Table at Depth (D_f + B) Below the Footing Base** If the water table is deeper than D_f + B, it has minimal effect on bearing capacity. Use the original equation with γ (dry or moist unit weight): q_u = c·N_c + q·N_q + (1/2)·γ·B·N_γ **Case 3: Water Table Between D_f and (D_f + B)** The failure surface extends below the footing base into a zone that transitions from dry (or moist) to saturated. Use an interpolated unit weight: γ_avg = γ_above + [(depth_to_WT - D_f) / B]·(γ' - γ_above) Then: q_u = c·N_c + q·N_q + (1/2)·γ_avg·B·N_γ Or, more conservatively, use the weighted average of the dry portion and the saturated portion of the failure zone. **Practical Importance**: In the Philippines, the dry season may see water tables drop several meters below the surface, while the wet season (June–October) brings water tables close to or above the ground surface in many areas. A footing designed for dry-season water table may fail during the monsoon if the design does not account for saturation. Always check the critical (worst-case) water-table position for your design. **Example Calculation**: If a soil has: - Dry unit weight: γ = 17 kN/m³ - Saturated unit weight: γ_sat = 20 kN/m³ - γ' = 20 - 9.81 = 10.19 kN/m³ A 2 m footing at 1 m depth with water table at the base: q_u (dry) = c·N_c + q·N_q + (1/2)(17)(2)N_γ = ... + 17·N_γ (for the width term) q_u (saturated) = c·N_c + q·N_q + (1/2)(10.19)(2)N_γ = ... + 10.19·N_γ The saturated case gives about 40% less bearing capacity from the width term (10.19 vs 17), which can be significant in granular soils where N_γ is large.

The ultimate bearing capacity q_u is a theoretical limit. In engineering practice, we apply a **factor of safety (FS)** to reduce q_u to an **allowable (safe, admissible) bearing capacity** q_a that can be used for design: q_a = q_u / FS The allowable bearing capacity is the maximum pressure that the footing can exert on the soil under normal service conditions. Structures are designed so that the actual applied pressure does not exceed q_a. **Standard Factors of Safety**: For shallow foundations on soils where bearing-capacity failure is the governing limit: - **FS = 2.5** to **3.0** for common designs (most typical) - **FS = 2.0** may be used for relatively predictable soil conditions and high-quality site investigations - **FS = 3.5** to **4.0** for highly uncertain soils or important structures The choice of FS depends on: 1. **Soil Variability**: If soil properties vary widely across the site, use higher FS. 2. **Site Investigation Quality**: Better data → lower FS is justified. 3. **Consequence of Failure**: Critical structures → higher FS. 4. **Uncertainty in Load Estimation**: If future loads are uncertain, use higher FS. **Gross vs Net Allowable Bearing Capacity**: **Gross Allowable Bearing Capacity** q_a,gross = q_u / FS This is the total pressure (including the weight of the footing itself) that the soil can support. **Net Allowable Bearing Capacity** q_a,net = (q_u - γ·D_f) / FS This subtracts the overburden pressure that would have existed if the footing were not there. The net bearing capacity is the additional pressure that the footing introduces. In design, the net bearing pressure from the applied loads must not exceed q_a,net. **Why the Distinction Matters**: When a footing is excavated to depth D_f, the soil is removed and replaced by the footing structure. The soil beneath the footing "feels" a relief of pressure equal to γ·D_f from the removed soil, but a new pressure from the footing. The net effect is the difference. For deep footings or when allowable settlement is a constraint, using net bearing capacity is more accurate and often more economical. **Relationship to NSCP 2015 and Philippine Practice**: The NSCP does not prescribe a specific FS for bearing capacity but references the ACI 318 and other international codes, which recommend FS = 2.5–3.0. Philippine foundation design practice typically uses FS = 3.0 as a starting point, with adjustments based on site-specific conditions. This is aligned with the earlier reference documents which state FS = 2.5–3.0. **Settlement as an Alternative Criterion**: Bearing capacity is not the only criterion. In many Philippine soils—particularly soft clay in lowland areas—**settlement** governs the design more restrictively than bearing capacity. The allowable bearing capacity based on settlement is often less than that based on shear failure. Engineers must compute both and use the lower (more restrictive) value. **Typical Settlement Limits** (from building codes): - Total settlement: 25–50 mm - Differential settlement: 15–25 mm - Tilting: less than 1/500 to 1/1000 of building dimension For a 3-story residential building on soft clay, the settlement-based allowable bearing capacity might be 100–150 kPa, while the shear-failure-based capacity could be 300+ kPa. In such cases, settlement governs, and the 300+ kPa capacity is not usable.

The Terzaghi bearing-capacity equation assumes **general shear failure**, which occurs in dense, stiff soils. In loose soils and soft clays, the failure mode is **local (or punching) shear**, and a correction to the bearing-capacity factors is necessary. **General Shear Failure** (Dense Soil, Stiff Clay): - The failure surface extends from the footing edges to the ground surface as a continuous shear band. - Lateral displacement of the footing is significant; the soil is "squeezed out" from beneath the footing. - Bearing capacity is maximum; use Terzaghi factors directly. - Typical soil condition: φ ≥ 30° for sand, undrained N > 15 for clay (where N = SPT blows per 300 mm), or c_u ≥ 100 kPa. **Local Shear Failure** (Loose Soil, Soft Clay): - The footing punches into the soil with vertical settlement dominating horizontal displacement. - The failure surface does not fully extend to the ground surface; lateral restraint is limited. - Bearing capacity is lower than predicted by the general shear equation. - Typical soil condition: φ < 30° for sand, undrained N < 15 for clay, or c_u < 100 kPa. **Correction for Local Shear Failure**: When local shear failure is expected, reduce the soil parameters before computing bearing capacity: **Modified friction angle** φ' = arctan((2/3)·tan(φ)) **Modified cohesion** c' = (2/3)·c Then compute the bearing-capacity factors N'_c, N'_q, N'_γ using φ' instead of φ, and use these in the bearing-capacity equation with the modified cohesion c': q_u = c'·N'_c + q·N'_q + (1/2)·γ·B·N'_γ **Why This Correction?** The 2/3 reduction factor is empirical and reflects the observation that in loose soils, the shear stress distribution is more uniform (less stress concentration at the footing edge), and the full cohesion and friction angle are not mobilized. **Alternative Approach** (Some Codes): Instead of modifying soil parameters, some designers reduce the bearing-capacity factors directly by a factor that varies with the relative density or consistency of the soil. However, the 2/3 parameter reduction is more common in textbooks and is what you'll encounter on the PRC exam. **Criteria for Identifying Local Shear** (Decision Rules): 1. **SPT Blow Count** (most common in Philippine practice): - N < 10: Local shear is likely; use correction. - 10 ≤ N ≤ 20: Intermediate; check other indicators. - N > 20: General shear; no correction needed. 2. **Friction Angle**: - φ < 28°–30°: Local shear likely. - φ > 30°–32°: General shear likely. 3. **Undrained Shear Strength** (Clay): - c_u < 50 kPa: Local shear likely; consider correction. - 50 < c_u < 150 kPa: Check other data. - c_u > 150 kPa: General shear likely. 4. **Relative Density** (Sand): - Dr < 50%: Local shear likely. - Dr > 75%: General shear likely. **Important Note for PRC Exam**: The problem statement often specifies the failure mode or gives enough information (e.g., φ value, SPT N, c_u) for you to identify it. If local shear is indicated, you MUST apply the 2/3 reduction; omitting it will give an unsafe (overestimated) bearing capacity. **Relationship to Settlement**: Local shear failure is associated with significantly larger settlements than general shear failure because the footing displacement is more vertical and less restrained. Thus, a soil that exhibits local shear failure often has settlement governs the allowable bearing pressure, not shear failure.

Designing a foundation using bearing capacity involves a systematic workflow that integrates all the concepts covered above. Here is a step-by-step guide aligned with Philippine practice and the PRC exam expectation: **Foundation Design Workflow** **Step 1: Site Investigation and Soil Characterization** - Perform borings at depth ≥ D_f + 2B (or as specified by code). - Collect soil samples and conduct laboratory tests: - Undrained shear strength tests (UU, CU, or vane tests for clay). - Drained friction angle (direct shear or triaxial) for sand. - Unit weights (γ, γ_sat, γ'). - Natural water-table depth and seasonal variations. - Conduct SPT to identify relative density and consistency profiles. **Step 2: Estimate Footing Dimensions (Preliminary)** - Based on the expected column loads and a reasonable q_a estimate (e.g., 100–200 kPa for Philippine soils unless proven otherwise), estimate the footing area A = Q_applied / q_a,assumed. - Determine if a square, rectangular, or circular footing is appropriate based on architectural constraints. **Step 3: Check Failure Mode** - Based on SPT N, friction angle, or undrained strength, determine if general or local shear failure governs. - If local shear, prepare to apply the 2/3 reduction. **Step 4: Determine Overburden and Unit Weights** - Calculate overburden q = γ·D_f at the footing depth. - Determine whether water-table correction is needed and at what depth. - If water table is above footing base, use γ' = γ_sat - 9.81 in the width term; otherwise, use γ. **Step 5: Select Bearing-Capacity Factors** - For the friction angle (or modified φ' if local shear), determine N_c, N_q, N_γ from tables or charts. - For the footing shape (strip, square, rectangular, circular), select appropriate shape factors s_c, s_q, s_γ. **Step 6: Calculate Ultimate Bearing Capacity** q_u = s_c·c·N_c + q·N_q + s_γ·(1/2)·γ_eff·B·N_γ Where: - c = cohesion (or c' = 2/3·c if local shear) - γ_eff = γ or γ' depending on water-table position - q = γ·D_f - N_c, N_q, N_γ are from tables for the friction angle (or φ') **Step 7: Apply Factor of Safety** - Select FS = 2.5–3.0 (or higher if justified). - Compute gross allowable: q_a,gross = q_u / FS. - Compute net allowable: q_a,net = (q_u - q) / FS. **Step 8: Check Settlement** - Estimate settlement using Terzaghi-Peck, Janbu, Vesic, or other methods. - If settlement exceeds code limits (typically 25–50 mm for total, 15–25 mm for differential), reduce q_a to the settlement-based allowable. - Use the lower of bearing-capacity and settlement allowables. **Step 9: Verify Footing Size** - Calculate required footing area: A_required = Q_total / q_a,net (if using net) or Q_applied / (q_a,gross - γ_footing·D_f) (if using gross). - Adjust footing dimensions and recalculate if necessary. - Ensure footing depth D_f is adequate (typically ≥ 1.0 m below grade, or deeper if specified). **Step 10: Design the Footing Structure** - Design the reinforced concrete footing using ACI 318 bending and shear provisions. - Ensure the footing can safely distribute the column load to the soil without internal failure. **Step 11: Document and Draw** - Prepare foundation plan, section, and details showing: - Footing dimensions, depth, and reinforcement. - Allowable bearing pressure used. - Soil profile and water-table location. - Note any special provisions (e.g., drainage, underpinning, soil improvement). **Common Pitfalls in Filipino Practice** 1. **Underestimating Water-Table Effect**: Many sites in low-lying areas have shallow water tables, especially during monsoon. Ignoring saturation can lead to overestimated bearing capacity. 2. **Assuming General Shear Without Verification**: Soft clays and loose soils are common in Philippine lowlands (e.g., Manila Bay area). Applying Terzaghi's factors without the local shear correction is unsafe. 3. **Neglecting Settlement**: In soft clay regions (Bangkok, Manila, Jakarta), settlement often governs before shear failure. Designing only to bearing capacity and ignoring settlement can result in unacceptable differential settlement and structural damage. 4. **Over-Relying on Old Borings**: Soil conditions can change with time, especially in areas with ongoing subsidence or changes in water regime. Re-boring is recommended every 5–10 years or when site conditions change. 5. **Not Accounting for Seismic Loading**: The Philippines is in a seismic zone (Rank 4–5 per NSCP 2015). Bearing capacity under static loading may be adequate, but under seismic shaking, the effective stress and bearing capacity can be severely reduced (liquefaction risk in loose sand). This is a topic beyond this chapter but crucial for foundation design in the Philippines. **Regional Variations in the Philippines** - **NCR/Metro Manila**: Soft clay, high water table, thick alluvial deposits. FS = 3.0–3.5 typical; settlement governs. - **Coastal Lowlands**: Mixed sand and clay, salt spray concerns, variable water table. Site-specific investigation essential. - **Upland/Volcanic Regions**: Dense laterite, better bearing capacity, but variable weathering. N_c and N_γ factors often higher. - **Limestone/Karst**: Cavities and subsidence risk; bearing capacity highly variable. Specialized investigation required. **Using the Philippine NSCP 2015** The NSCP Chapter 2 (Building Planning) Section 2.3 (Soils and Foundations) references ACI 318 and other standards for bearing-capacity design. The code does not prescribe a specific bearing-capacity formula but permits use of the Terzaghi or other recognized methods. Key provisions: - Minimum depth of footing: 0.9 m below lowest finished grade (or as required for frost/liquefaction protection). - Minimum factor of safety: 2.5 on bearing capacity (but higher FS recommended for uncertain soils). - Settlement limits: Not explicitly specified in NSCP; refer to ACI 318 or client/engineer specifications. - Liquefaction and slope stability: Must be evaluated in seismic zones and sloped terrain. The NSCP 2015 also requires submission of a Geotechnical Engineering Report prepared by a licensed Professional Engineer in the Philippines (RA 544), which must document all design assumptions, soil parameters, and the bearing-capacity calculation.

**Essential Formulas for PRC Exam** 1. **Terzaghi's Bearing-Capacity Equation (Strip Footing)**: q_u = c·N_c + q·N_q + (1/2)·γ·B·N_γ 2. **With Shape Factors**: q_u = s_c·c·N_c + s_q·q·N_q + s_γ·(1/2)·γ·B·N_γ 3. **Overburden**: q = γ·D_f 4. **Allowable Bearing Capacity**: q_a = q_u / FS (gross) q_a,net = (q_u - q) / FS (net) 5. **Effective Unit Weight** (under water table): γ' = γ_sat - γ_w = γ_sat - 9.81 kPa 6. **Modified Friction Angle** (local shear correction): φ' = arctan((2/3)·tan(φ)) 7. **Modified Cohesion** (local shear correction): c' = (2/3)·c 8. **Shape Factors**: Square: s_c = 1.3, s_q = 1.0, s_γ = 0.4–0.8 Circular: s_c = 1.3, s_q = 1.0, s_γ = 0.3–0.6 Rectangular: s_c = 1 + (B/L)(N_q/N_c), s_q = 1 + (B/L)tan(φ), s_γ = 1 - 0.4(B/L) Strip: s_c = 1.0, s_q = 1.0, s_γ = 1.0 (no correction) 9. **Pure Clay (φ = 0)**: N_c = 5.7, N_q = 1.0, N_γ = 0 q_u = c·N_c + q = c(5.7) + γ·D_f **Common PRC Exam Question Types** **Type 1: Calculate q_u for Given Soil and Footing** Given: B, D_f, c, φ (or c_u for clay), γ, and footing shape. Find: q_u Solution: Look up or calculate N_c, N_q, N_γ for φ; select shape factors; apply formula. Common mistake: Forgetting shape factors; using surcharge q incorrectly. **Type 2: Calculate q_a Given q_u and FS** Given: q_u, FS Find: q_a (or q_a,net) Solution: q_a = q_u / FS; if net is asked, subtract q first. Common mistake: Confusing gross and net; using wrong FS. **Type 3: Determine Allowable Load on Footing** Given: Footing dimensions, soil properties, FS Find: Maximum column load Q_a Solution: Calculate q_a, then Q_a = q_a · A (for gross) or Q_a = q_a,net · A (for net). Common mistake: Forgetting to account for footing weight (use net for clarity). **Type 4: Water-Table Effect** Given: Soil properties with and without water table Find: Change in q_u or q_a due to water-table position Solution: Compare γ vs γ' in width term; recalculate q_u; find difference. Common mistake: Applying water correction to all three terms (should be width term only for Terzaghi). **Type 5: Local Shear Correction** Given: SPT N or φ < 30° indicating local shear; soil properties Find: q_u with local shear correction Solution: Reduce c and φ by 2/3; recalculate factors; use in bearing-capacity equation. Common mistake: Forgetting the correction; using general shear factors for loose soil. **Type 6: Shape Factor Selection** Given: Footing dimensions (square, rectangular, circular, strip) Find: Correct shape factors to use Solution: Identify footing type; look up or recall s_c, s_q, s_γ. Common mistake: Using wrong factors (e.g., circular factors for square footing). **Type 7: Comparison Problems** Given: Two different foundation scenarios (different D_f, B, soil, water table) Find: Which has higher bearing capacity? Why? Solution: Calculate q_u for each; explain physical reason (depth effect, width effect, water effect, etc.). Common mistake: Not considering all factors; saying "deeper is always better" without checking water-table effect. **Type 8: Settlement vs Bearing Capacity** Given: Bearing-capacity-based q_a and settlement-based q_a Find: Which governs design? What is design allowable? Solution: Use the lower value. Explain why settlement might govern in soft clay (high compressibility). Common mistake: Assuming bearing capacity always governs; not calculating settlement. **Sample Exam-Style Questions** **Q1**: A square footing 2.5 m × 2.5 m at depth D_f = 1.5 m is placed on soil with c = 20 kPa, φ = 28°, γ = 17.5 kN/m³. Using N_c = 32.67, N_q = 18.92, N_γ = 17.69 for φ = 28°, and shape factors s_c = 1.3, s_q = 1.0, s_γ = 0.8, calculate the ultimate bearing capacity. **Q2**: For the footing in Q1, if the factor of safety is 3.0, what is the net allowable bearing capacity? **Q3**: A circular footing (diameter D = 2.0 m) at D_f = 1.0 m on clay with c_u = 60 kPa, γ = 18 kN/m³. Calculate q_u (use φ = 0 factors and circular shape factors). **Q4**: The water table in Q3 rises from 3 m depth to the footing base due to monsoon. Assuming γ_sat = 20 kN/m³, recalculate q_u. What is the percentage reduction? **Q5**: SPT data shows N = 6 at the footing depth, suggesting local shear failure. If the original φ = 32°, calculate the modified φ' and determine whether the bearing capacity increases or decreases compared to general shear. **Q6**: A strip footing on sand (φ = 35°, c = 0, γ = 18 kN/m³) at D_f = 1.2 m and B = 1.8 m. Using N_c = 57.75, N_q = 41.44, N_γ = 42.16, find q_u and q_a (FS = 2.5). Assume no water-table effect. **Answer Keys** (Sketched; verify with full calculations): Q1: q_u ≈ 850 kPa Q2: q_a,net ≈ 230 kPa Q3: q_u ≈ 350 kPa Q4: q_u ≈ 290 kPa; reduction ≈ 17% Q5: φ' ≈ 23.4°; bearing capacity decreases (local shear governs) Q6: q_u ≈ 1600 kPa; q_a ≈ 535 kPa