CELE Geotechnical Engineering — Foundations (Shallow and Deep)Study Notes

A foundation is the lowest part of a structure that transmits loads from the superstructure, through the foundation element, into the supporting soil or rock. The primary role is to: • **Safely distribute loads** to the ground such that soil bearing capacity is not exceeded. • **Control settlement** so that differential movement does not damage the structure or its contents. • **Prevent erosion and scour** in water-bearing or flood-prone areas. • **Accommodate ground conditions** — weak or compressible soils require different approaches than firm soils. **Two Major Categories:** 1. **Shallow Foundations** — where the depth of embedment is less than or equal to the width of the footing (typically $D_f \leq B$). Load is transferred by direct bearing on the soil immediately beneath the footing. 2. **Deep Foundations** — where the depth of embedment is significantly greater than the footing width, and load is carried by a combination of end bearing (at the pile tip or base) and skin friction (along the pile shaft). Used when surface soils are weak or compressible. The choice between shallow and deep depends on: • Allowable bearing capacity of surface soils. • Expected settlement magnitude and tolerance. • Depth to firm bearing stratum. • Cost and feasibility of construction. • Building code requirements (NSCP 2015, Section 1802).

**2.1 Types of Shallow Foundations** **Isolated (Spread) Footings** — individual footings under single columns, spaced apart so their stress bulbs do not overlap. Most common and economical. **Combined Footings** — two or more columns share one footing, used when: • Columns are close together. • An isolated footing for one column would be unusually large or eccentric (e.g., edge or corner column). **Strap (Cantilever) Footings** — two footings connected by a stiff beam (strap), allowing load redistribution. Used for edge or corner columns. **Raft or Mat Foundation** — a single large footing supporting all or most columns, used when: • Allowable bearing is very low ($q_a < 100$ kPa). • Building footprint is compact and total footing area would exceed ~50% of plan area. • Soil is compressible and differential settlement is a concern. **2.2 Footing Sizing and Bearing Capacity Check** The required area to keep bearing stress within allowable limits is: $$A_{\text{req}} = \frac{P_{\text{service}}}{q_a}$$ where: • $P_{\text{service}}$ = service (working) load (dead + live, in kN). • $q_a$ = allowable bearing pressure (kPa), typically from a geotechnical investigation or NSCP 2015 Table 1804.2. For a square footing, $B = \sqrt{A_{\text{req}}}$. For a rectangular footing, choose $L$ and compute $B = A_{\text{req}}/L$. **2.3 Allowable Bearing Pressure** NSCP 2015 Section 1804 provides tabulated allowable bearing pressures for common soils: • **Crystalline bedrock** — 4000–8000 kPa (firm rock). • **Gravel, sand & gravel, and sand** — 150–300 kPa depending on density. • **Clay** — 50–150 kPa depending on consistency. • **Silt** — 25–100 kPa. When direct test data (SPT, CPT, lab tests) are available, compute $q_a$ from bearing capacity theory: $$q_u = c N_c F_{cs} F_{cd} F_{ci} + q N_q F_{qs} F_{qd} F_{qi} + \frac{1}{2}\gamma B N_\gamma F_{\gamma s} F_{\gamma d}$$ then $q_a = q_u / FS$ with $FS = 2.5$–$3$ for static loads (ACI 336, NSCP 2015). **2.4 Additional Checks** **Settlement Check:** Even if bearing is safe, settlement must be within tolerance (typically <25 mm for rigid structures, <50 mm for flexible). Use immediate (elastic) and consolidation settlement formulas. **Structural Design:** Once $B$ and $L$ are determined, design the footing as a rigid or flexible slab using shear and bending moment under the net bearing pressure: $$q_{\text{net}} = q_a - \gamma_c h$$ where $h$ is footing depth. Check one-way shear (critical section at distance $d$ from column face) and two-way (punching) shear per ACI 318. **Eccentricity Control:** For edge or corner columns, eccentric loads must be checked to ensure the resultant falls within the footing's middle third (or kern): $$e = \frac{M}{P}$$ If $e > B/6$, the footing must be enlarged or combined/strapped.

**3.1 When Deep Foundations Are Required** Deep foundations are used when: • Surface soils have very low bearing capacity (clay with cu < 30 kPa, or silt). • Significant depth of soft or compressible layers exists above firm strata. • Differential settlement would be excessive with shallow footings. • Scour or erosion risk in water areas requires embedment below expected scour depth. • Seismic or high-wind uplift forces demand deep anchorage. **3.2 Pile Types and Installation** **Driven Piles:** • Steel H-piles — open cross-section, displace soil, suitable for dense sand and clay. • Precast concrete piles — durable, high capacity, cause ground heave during driving. • Timber piles — economical but limited load capacity, subject to rot if exposed. • Closed-ended steel pipe or shell — displace soil, good for scour zones. **Drilled Shafts (Caissons):** • Excavated holes (0.6–3.5 m diameter, often larger) filled with reinforced concrete. • No vibration or heave, suitable in urban areas and near sensitive structures. • Require stable hole (slurry support in soft soils, casing in sandy layers). • Higher capacity than piles of same length due to larger diameter. **Installation Methods:** • Drive (impact hammer, vibrator, or jacking). • Drill/auger with continuous or segmental auger. • Wash boring or jet drilling for soft soils. **3.3 Pile Capacity Components** The ultimate bearing capacity of a pile is the sum of: $$Q_u = Q_p + Q_s$$ where: • $Q_p$ = **end bearing** (resistance at the pile tip). • $Q_s$ = **skin friction** (resistance along the shaft). **End Bearing ($Q_p$):** At the pile tip, bearing capacity is similar to shallow foundations but with different factors due to the deep, confined nature: $$Q_p = A_p \cdot q_p$$ where $A_p$ = cross-sectional area of pile tip. **In Clay (Undrained, φ = 0 condition):** $$q_p = c_u N_c^*$$ For deep piles, bearing capacity factor $N_c^* \approx 9$ (not the 5.7 or 5.14 of shallow footings). Thus: $$Q_p = 9 c_u A_p$$ Typically, $c_u$ is determined from unconfined compression tests or from SPT blow counts: $c_u \approx 12.5 N$ (kPa), where $N$ is the SPT N-value. **In Sand (Drained, φ > 0 condition):** $$q_p = q' N_q^* + \frac{1}{2}\gamma' B N_{\gamma}^*$$ For piles, $N_q^*$ and $N_{\gamma}^*$ are taken from tables. A simplified formula often used is: $$q_p = \sigma'_v \tan(\phi') N_q^*$$ where $\sigma'_v$ = effective vertical stress at pile tip, and $N_q^*$ ≈ 40–50 for φ' = 35°. **Skin Friction ($Q_s$):** Friction acts along the pile perimeter and is the product of average friction stress and surface area: $$Q_s = \sum f_s \cdot A_s$$ where $A_s$ = lateral surface area of pile shaft in a given soil layer. **In Clay (α-Method, Undrained):** $$f_s = \alpha c_u$$ where α = adhesion factor (0.3 to 1.0, typically 0.5–0.9 depending on cu and pile roughness). Common guideline: • cu < 50 kPa: α ≈ 0.8–1.0. • 50–100 kPa: α ≈ 0.6–0.8. • > 100 kPa: α ≈ 0.3–0.6. For a pile of diameter D and length L in a single clay layer: $$Q_s = \alpha c_u \cdot (\pi D) \cdot L$$ **In Sand (β-Method, Drained):** $$f_s = K \sigma'_v \tan \delta$$ where: • K = coefficient of lateral earth pressure (typically 0.5–1.5, depends on relative density and installation). • $\sigma'_v$ = effective vertical stress at the depth of interest. • δ = friction angle between pile and sand (typically 0.7φ to 0.9φ). For varying $\sigma'_v$ with depth, integrate over pile length: $$Q_s = \int_0^L K \sigma'_v(z) \tan \delta \cdot (\pi D) \, dz$$ Often simplified as: $$Q_s = K (\overline{\sigma'_v}) \tan \delta \cdot (\pi D) \cdot L$$ where $\overline{\sigma'_v}$ = average effective stress along the pile. **3.4 Allowable Pile Capacity** The allowable capacity (safe working load) applies a factor of safety: $$Q_a = \frac{Q_u}{FS}$$ Typical factors of safety per NSCP 2015 and common practice: • **Static loads** — FS = 2.5 to 3.0 (more conservative for piles than footings due to uncertainty in installation). • **Dynamic/Impact loads** — FS = 3.5 to 4.0. For design, use $Q_a$ to determine the number and arrangement of piles.

**4.1 Why Pile Groups Are Not Simply n × Single-Pile Capacity** When multiple piles are driven close together, their stress zones overlap in the surrounding soil. This interaction causes: • Reduced skin friction on adjacent piles (soil in between is already mobilized). • Possible reduction in end bearing if piles are spaced very closely. • A group efficiency factor **η < 1** such that: $$Q_{u,\text{group}} = \eta \cdot n \cdot Q_{u,\text{single}}$$ where: • η = group efficiency (typically 0.5–0.95). • n = number of piles in the group. **4.2 Estimating Group Efficiency** **Empirical Formula (Converse–Labarre, for clay):** $$\eta = 1 - \left( \frac{\theta(m-1)n + (n-1)m}{90mn} \right)$$ where: • m, n = number of rows and columns. • θ = angle of stress distribution cone (typically 30–45°, use 45° for clay). **Simplified Thumb Rules:** • **Close spacing** (center-to-center distance $s \leq 3D$): η ≈ 0.5–0.7. • **Moderate spacing** ($3D < s < 5D$): η ≈ 0.7–0.85. • **Wide spacing** ($s > 5D$): η ≈ 0.85–1.0 (approaching individual behavior). **4.3 Block Failure Check (Clay)** For pile groups in clay, also check if the entire group fails as a single large pier (block failure): $$Q_{u,\text{block}} = (B' + 2L)h \cdot c_u + q' B' + (\text{perimeter} \times h) \alpha c_u$$ where B', L are the equivalent dimensions of the group perimeter, h = embedded depth. The governing capacity is the **lesser of**: • Individual pile group capacity: $Q_{u,\text{group}} = \eta \cdot n \cdot Q_{u,\text{single}}$. • Block capacity: $Q_{u,\text{block}}$. **4.4 Allowable Group Capacity** $$Q_{a,\text{group}} = \frac{Q_{u,\text{group}}}{FS}$$ If block failure governs, use the block capacity instead. **4.5 Pile Group Settlement** Group settlement is generally larger than single-pile settlement because: • The stress bulb from the group penetrates deeper and wider. • Soil consolidation or elastic compression affects a larger zone. Settlement is estimated by integrating compression of soil layers beneath the group, treating the group as a raft of equivalent size. **4.6 Pile Arrangement and Spacing Guidelines** • **Minimum spacing** — at least 2.5D to 3D (center-to-center) to ensure adequate load transfer and avoid excessive congestion during driving or construction. • **Typical spacing** — 3D to 5D for reasonable efficiency and constructability. • **Thick piles (D > 0.6 m)** — may require larger spacing (5D+) to avoid interference.

**5.1 What Is Negative Skin Friction?** Negative skin friction occurs when surrounding soil moves **downward relative to the pile**, creating a **downward drag force** on the pile shaft rather than upward resistance. This is counterintuitive but critical in certain soil conditions. **Common Causes:** • **Consolidation of soft layers** — newly filled embankments or preloading on clay settles, dragging down any piles that pass through it. • **Creep or secondary compression** — long-term settlement in soft soils continues after pile installation. • **Groundwater lowering** — dewatering increases effective stress and causes consolidation. • **Adjacent excavation** — lateral stress relief causes ground movement toward the excavation. **5.2 Neutral Plane Concept** In a pile passing through consolidating soil, there is a **neutral plane** at depth $z_n$ where pile and soil move together (no relative displacement, so no friction). Above the neutral plane, the pile is dragged downward; below, the pile supports the overlying soil against downward movement. **5.3 Estimating Negative Skin Friction** For a pile in consolidating clay: $$f_s^{\text{neg}} = \alpha c_u$$ (same formula as positive friction, but applied from surface to neutral plane depth $z_n$) Total negative force on pile: $$Q_s^{\text{neg}} = \alpha c_u \cdot (\pi D) \cdot z_n$$ The ultimate pile capacity becomes: $$Q_u = Q_p + Q_s^{\text{pos}} - Q_s^{\text{neg}}$$ where $Q_s^{\text{pos}}$ = positive friction in stable layers below the neutral plane. In severe cases, negative friction can reduce the pile capacity by 20–50%, or even make the pile a liability (net capacity < 0) if the consolidation is very large. **5.4 Prevention and Mitigation** • **Preload the site** before pile driving so consolidation occurs before pile construction. • **Use low-displacement piles** (drilled shafts, H-piles) instead of high-displacement closed-end pipe piles to minimize soil disturbance. • **Install a slipping layer** — coat the upper part of the pile shaft with bitumen or synthetic sleeves to reduce friction. • **Extend piles deeper** into stable strata to increase positive friction and bearing capacity. • **Account for negative friction in design** — reduce the allowable capacity or increase the number of piles. **5.5 NSCP 2015 Guidance** NSCP 2015 Section 1805.7 requires consideration of negative skin friction in designs where: • Piles are driven through consolidating soils or fills placed above the original ground. • Groundwater table is expected to change during the service life. • Nearby construction or dewatering will induce settlement. For Philippine projects (especially near Manila's clay deposits or in new reclamation areas), negative friction is a common and often underestimated issue.

**6.1 Overall Foundation Design Process** Step 1: **Geotechnical Investigation** • Boreholes, SPT, CPT, lab tests (cu, φ, density, settlement parameters). • Identify soil profile, groundwater, and bearing strata. • Estimate or determine allowable bearing pressure per NSCP 2015. Step 2: **Select Foundation Type** • If qa is high and settlement is acceptable → shallow footing. • If qa is low or deep bearing strata exist → deep foundation (piles/shafts). • Compare lifecycle cost (material, installation, risk). Step 3: **Size Foundation** • Shallow: compute $A = P / q_a$, then B and L. • Deep: compute $Q_u$ per soil method (clay α, sand β), apply FS, then $Q_a = Q_u / FS$. Number of piles = Total Load / $Q_a$. Step 4: **Check Settlement** • Shallow: use elastic + consolidation settlement formulas (typically < 25–50 mm). • Deep: typically less settlement concern (piles reach firm strata), but verify by calculation. Step 5: **Structural Design** • Shallow footings: treat as RC slab, check shear and flexure per ACI 318. • Piles: check axial capacity, design pile cap as stiff slab, connect to superstructure. • Include load factors per NSCP 2015 Section 1605. Step 6: **Address Special Conditions** • Negative friction, scour, slope stability, liquefaction risk (seismic). • Water table, erosion, durability (corrosion, sulfate attack). Step 7: **Construct and Monitor** • Inspect footings (excavation, soil removal, concrete placement). • Perform pile tests (static load test, dynamic monitoring during driving). • Monitor settlement and performance during construction and early operation. **6.2 Material Durability and Protection** **Concrete Footings and Piles:** • In marine or aggressive environments, use high-performance concrete (fc' ≥ 40 MPa, low water–cement ratio). • Minimum concrete cover: 75 mm for footings, 100 mm for piles in water. • Curing time: 28 days before loading (full strength). **Steel Piles:** • Coat with 3–5 mm epoxy or polyurethane in corrosive environments. • For permanent water immersion, use cathodic protection or premium coatings. • Above-ground portions should be painted every 5–10 years. **Timber Piles:** • Treat with creosote or salt-based preservative. • Submerged portions in fresh water are durable; brackish/saline waters cause rot without treatment. **6.3 Quality Control During Construction** **Pile Driving:** • Monitor blow count (N_SPT equivalent) to detect change in soil or obstructions. • Use dynamic formulas (e.g., Hiley, Gates) to estimate capacity if static load tests are not done. • Photograph each pile, record drive log (depth, blow count, set). **Static Load Test:** • For critical piles or large projects, perform one test per 10–20 piles or per NSCP 2015. • Load to 1.5× design load or 2× if failure proof required. • Measure settlement; acceptable: < 25 mm at design load, < 50 mm at 1.5× load (no excessive creep). **Pile Caps:** • Ensure contact with all piles; no gaps or voids (concrete honeycombing). • Minimum depth to satisfy shear and bending (typically 0.8–1.0 m for 3–4 piles). **6.4 Common Failures and Lessons Learned** **Bearing Failure:** • Soil bearing capacity underestimated due to poor investigation. • Remedy: deeper boreholes, lab testing, conservative assumptions. **Excessive Settlement:** • Compressible layer not detected; piles too short. • Remedy: SPT/CPT refusal, engineer review, design for actual subsurface. **Negative Friction Not Addressed:** • Piles fail or settle unexpectedly in consolidating fills. • Remedy: preload fill, account for downdrag in calculations, consider slipping layer. **Pile Group Interaction Ignored:** • Group capacity assumed as n × single, causing overload and settlement. • Remedy: apply group efficiency η, consider block failure in clay. **Corrosion/Durability Issues:** • Premature failure of steel or concrete in aggressive environments. • Remedy: proper concrete cover, coatings, monitoring, replacement schedule. **6.5 Summary Table: Shallow vs. Deep Foundations** | Aspect | Shallow (Footings) | Deep (Piles) | |--------|-------------------|---------------| | **Typical depth** | Df ≤ B (0.5–2 m) | Df >> B (5–50+ m) | | **Suitable soils** | Medium-to-stiff clay, dense sand | Soft clay, silt, layered soils | | **Cost** | Lower material; simpler construction | Higher (drilling/driving); longer installation) | | **Settlement** | Can be significant (20–100 mm); design tolerance needed | Minimal if reaches firm strata; larger group settlement | | **Capacity check** | Bearing capacity formula (Terzaghi, Meyerhof) | End bearing + skin friction (α or β method) | | **Interaction** | Not applicable (isolated or combined) | Group efficiency η < 1; block failure in clay | | **Negative friction** | Not a concern | Critical in consolidating fills (downdrag) | | **Typical load range** | 300–2000 kN per footing | 300–5000 kN per pile (single) | Choose shallow if feasible (cost); use deep for poor soils or large loads.

**Problem 1: Spread Footing Sizing and Bearing Check** A reinforced concrete column (P = 800 kN service load) is founded on a square footing in sandy soil with: • Allowable bearing pressure qa = 180 kPa (per NSCP 2015). • Column load eccentricity e = 0 (centered). **Find:** (a) Required footing area and dimension B. (b) Actual bearing pressure for a 2.2 m × 2.2 m footing. (c) Check eccentricity limit (e ≤ B/6). **Solution:** (a) Required area: $$A_{\text{req}} = \frac{P}{q_a} = \frac{800}{180} = 4.44 \ \text{m}^2$$ For square footing: $B = \sqrt{4.44} = 2.11 \ \text{m}$; use **B = 2.2 m**. (b) Actual bearing pressure: $$q_{\text{actual}} = \frac{P}{A} = \frac{800}{2.2 \times 2.2} = \frac{800}{4.84} = 165.3 \ \text{kPa} < 180 \ \text{kPa} \ ✓$$ (c) Eccentricity: Given e = 0 (centered load) → e = 0 < B/6 = 2.2/6 = 0.37 m → **OK** ✓ **Answer:** Use 2.2 m × 2.2 m footing; bearing and eccentricity checks pass. --- **Problem 2: Pile Capacity in Clay** A cylindrical pile: • Diameter D = 0.5 m • Length L = 14 m • Soil: uniform clay with cu = 70 kPa, adhesion factor α = 0.75 • Bearing capacity factor for deep pile: Nc* = 9 • Factor of safety: FS = 2.5 **Find:** (a) End bearing capacity Qp. (b) Skin friction capacity Qs. (c) Ultimate capacity Qu. (d) Allowable capacity Qa. **Solution:** (a) End bearing: $$A_p = \frac{\pi}{4} D^2 = \frac{\pi}{4}(0.5)^2 = 0.1963 \ \text{m}^2$$ $$Q_p = c_u N_c^* A_p = 70 \times 9 \times 0.1963 = 123.7 \ \text{kN}$$ (b) Skin friction: $$\text{Perimeter} = \pi D = \pi(0.5) = 1.571 \ \text{m}$$ $$Q_s = \alpha c_u (\pi D) L = 0.75 \times 70 \times 1.571 \times 14 = 1160.9 \ \text{kN}$$ (c) Ultimate capacity: $$Q_u = Q_p + Q_s = 123.7 + 1160.9 = 1284.6 \ \text{kN}$$ (d) Allowable capacity: $$Q_a = \frac{Q_u}{FS} = \frac{1284.6}{2.5} = 513.8 \ \text{kN}$$ **Answer:** Qa ≈ **514 kN** (can support service loads up to this value). --- **Problem 3: Pile Group Efficiency** A 3×2 group of piles (3 rows, 2 columns) is arranged with: • Center-to-center spacing s = 1.5 m • Pile diameter D = 0.4 m • Each pile single capacity: Qu,single = 900 kN • Clay soil (use θ = 45° in Converse–Labarre formula) **Find:** (a) Spacing ratio s/D. (b) Group efficiency η using Converse–Labarre. (c) Ultimate group capacity Qu,group. (d) Allowable group capacity with FS = 2.5. **Solution:** (a) Spacing ratio: $$\frac{s}{D} = \frac{1.5}{0.4} = 3.75$$ (b) Converse–Labarre formula (m = 2 columns, n = 3 rows, θ = 45°): $$\eta = 1 - \left( \frac{\theta(m-1)n + (n-1)m}{90 m n} \right) = 1 - \left( \frac{45(2-1) \times 3 + (3-1) \times 2}{90 \times 2 \times 3} \right)$$ $$= 1 - \left( \frac{45 \times 3 + 2 \times 2}{540} \right) = 1 - \left( \frac{135 + 4}{540} \right) = 1 - \frac{139}{540} = 1 - 0.257 = 0.743$$ Use **η ≈ 0.74** (or round to **0.75**). (c) Group ultimate capacity: $$Q_{u,\text{group}} = \eta \times n \times Q_{u,\text{single}} = 0.74 \times 6 \times 900 = 3996 \ \text{kN}$$ (d) Allowable group capacity: $$Q_{a,\text{group}} = \frac{3996}{2.5} = 1598.4 \ \text{kN}$$ **Answer:** Qa,group ≈ **1598 kN** (or **1600 kN** rounded). --- **Problem 4: Negative Skin Friction** A pile (D = 0.4 m, L = 16 m) passes through: • 0–8 m: newly placed fill with cu = 45 kPa, α = 0.8 • 8–16 m: original soft clay with cu = 65 kPa, α = 0.9 Estimate neutral plane at zn = 6 m (consolidation in upper fill). **Find:** (a) Negative friction force Qs,neg. (b) Positive friction below neutral plane Qs,pos. (c) End bearing Qp (at 16 m in cu = 65 kPa). (d) Net ultimate capacity accounting for negative friction. **Solution:** (a) Negative friction (0–6 m in fill, cu = 45 kPa, α = 0.8): $$Q_s^{\text{neg}} = \alpha c_u (\pi D) z_n = 0.8 \times 45 \times \pi(0.4) \times 6 = 0.8 \times 45 \times 1.257 \times 6 = 270.3 \ \text{kN}$$ (b) Positive friction (6–16 m in original clay, cu = 65 kPa, α = 0.9, length = 10 m): $$Q_s^{\text{pos}} = 0.9 \times 65 \times \pi(0.4) \times 10 = 0.9 \times 65 \times 1.257 \times 10 = 736.8 \ \text{kN}$$ (c) End bearing (at 16 m, cu = 65 kPa): $$A_p = \frac{\pi}{4}(0.4)^2 = 0.1257 \ \text{m}^2$$ $$Q_p = 9 \times 65 \times 0.1257 = 73.6 \ \text{kN}$$ (d) Net ultimate capacity: $$Q_u = Q_p + Q_s^{\text{pos}} - Q_s^{\text{neg}} = 73.6 + 736.8 - 270.3 = 540.1 \ \text{kN}$$ Allowable: $Q_a = 540.1 / 2.5 = 216 \ \text{kN}$ **Comparison (if negative friction ignored):** $Q_u = 73.6 + 736.8 = 810.4 \ \text{kN}$ → $Q_a = 324 \ \text{kN}$ (overstated by 50%). **Answer:** Accounting for negative friction, Qa ≈ **216 kN** (vs. 324 kN if neglected—critical difference). --- **Problem 5: Combined Footing** Two columns, 5 m apart: • Interior column: P1 = 700 kN • Edge column: P2 = 400 kN • Total load: 1100 kN • Soil: qa = 160 kPa **Find:** (a) Required footing area and length L. (b) Width B for a rectangular combined footing. (c) Check if resultant (center of gravity) coincides with footing centroid (moment balance). **Solution:** (a) Required area: $$A_{\text{req}} = \frac{P_{\text{total}}}{q_a} = \frac{1100}{160} = 6.875 \ \text{m}^2$$ Assuming column spacing of 5 m plus margins (0.5 m each side for cantilever), choose L ≈ 6 m. (b) Width: $$B = \frac{A_{\text{req}}}{L} = \frac{6.875}{6} = 1.146 \ \text{m} \approx 1.2 \ \text{m}$$ Actual area: $A = 6.0 \times 1.2 = 7.2 \ \text{m}^2$ (slightly > 6.875 m²; OK). (c) Moment balance check: Take moments about the edge (left end) of footing (x = 0): $$M_{\text{applied}} = P_1 \times x_1 + P_2 \times x_2$$ where x1, x2 are distances of columns from left edge. Assuming interior column at x = 1 m and edge column at x = 6 m (spacing = 5 m): $$M_{\text{applied}} = 700 \times 1 + 400 \times 6 = 700 + 2400 = 3100 \ \text{kN·m}$$ For uniform bearing distribution, the resultant must fall at the footing centroid: $$x_{\text{resultant}} = \frac{M_{\text{applied}}}{P_{\text{total}}} = \frac{3100}{1100} = 2.818 \ \text{m}$$ Footing centroid: $x_c = L / 2 = 6 / 2 = 3.0 \ \text{m}$. Small offset (3.0 − 2.818 = 0.182 m) acceptable; if larger, adjust column positions or footing length to balance. In practice, refine the design or use eccentric loading factors. **Answer:** Use a **6.0 m × 1.2 m** combined footing (area = 7.2 m²). Moment balance nearly achieved; acceptable for preliminary design. --- **Problem 6: Sand Pile Skin Friction** A pile in sand: • Diameter D = 0.4 m • Length L = 10 m • K = 0.9 (lateral earth pressure coefficient) • φ = 32°, δ = 0.8φ = 25.6° (pile friction angle) • Effective stress increases linearly from σ'v(0) = 0 to σ'v(10) = 120 kPa **Find:** (a) Average effective stress along pile. (b) Skin friction stress fs. (c) Total skin friction Qs. **Solution:** (a) Average effective stress (linear increase): $$\overline{\sigma'_v} = \frac{0 + 120}{2} = 60 \ \text{kPa}$$ (b) Friction stress: $$f_s = K \overline{\sigma'_v} \tan \delta = 0.9 \times 60 \times \tan(25.6°) = 0.9 \times 60 \times 0.479 = 25.9 \ \text{kPa}$$ (c) Total skin friction (lateral surface area = π D L): $$Q_s = f_s \times A_s = f_s \times (\pi D L) = 25.9 \times (\pi \times 0.4 \times 10)$$ $$= 25.9 \times 12.566 = 325.5 \ \text{kN}$$ End bearing (tip at σ'v = 120 kPa, typically adds ~400–600 kN for this diameter): Qp ≈ 500 kN (simplified; full calculation would use bearing capacity factors). Total ultimate: Qu ≈ 325.5 + 500 = 825.5 kN; Qa ≈ 825.5 / 2.5 ≈ **330 kN**. **Answer:** Qs ≈ **326 kN** from skin friction alone; total capacity with end bearing ≈ **330 kN** (allowable).