Shallow and Deep Foundations

A foundation must not fail in shear (bearing capacity) and must not settle too much. Shallow footings spread the load near the surface; piles carry it deeper.

Bearing capacity (Terzaghi)

For a strip footing of width B at depth D_f:

q_u = c N_c + γ D_f N_q + 0.5 γ B N_γ

• Square footing: q_u = 1.3cN_c + γD_fN_q + 0.4γBN_γ. Circular: 1.3cN_c + γD_fN_q + 0.3γBN_γ.
• Bearing capacity factors grow quickly with φ. For φ = 0 (saturated clay, undrained): N_c = 5.7 (Terzaghi), N_q = 1, N_γ = 0; so q_u = 5.7 c_u + γD_f for a strip. (Skempton uses N_c = 5.14 for a strip and 6.2 for a square at the surface.)
• Net ultimate q_nu = q_u − γD_f; net safe q_ns = q_nu/FS (FS usually 3); safe = q_ns + γD_f.
• Water table: at or above the base, use γ' in the N_γ term (that term roughly halves). At the surface, use γ' in both terms.
• General shear failure in dense soils; local shear in loose soils (use reduced c and φ: c' = 2c/3, tan φ' = (2/3) tan φ).

Settlement

Total settlement = immediate (elastic) + consolidation + secondary. Allowable values (IS 1904) are about 25 mm (isolated footings on sand) to 65–100 mm (rafts, clay).

Plate load test: settlement of the footing on sand from the plate settlement: S_f = S_p [B_f(B_p + 0.3)/(B_p(B_f + 0.3))]².

Piles

• Static formula: Q_u = Q_b + Q_s = q_b A_b + Σ f_s A_s.
• Clay: Q_b = 9 c_u A_b; skin friction f_s = α c_u (adhesion factor α).
• Sand: Q_b = σ'_v N_q A_b; f_s = K σ'_v tan δ.
• Dynamic formulas (Engineering News): Q_a = W h/(6(s + C)) with s = set per blow.
• Negative skin friction: soft soil settling around a pile drags it down.
• Pile groups: efficiency η = group capacity/(n × single pile capacity). Converse–Labarre: η = 1 − (θ/90)[(n−1)m + (m−1)n]/(mn), θ = tan⁻¹(d/s). In clay, the group capacity is the smaller of n × single and block failure.

Site investigation

• SPT (standard penetration test): N-value = blows for 300 mm penetration; corrected for overburden and dilatancy (for N > 15 in fine sands below water: N' = 15 + (N − 15)/2).
• Cone penetration test, boring, sampling. Area ratio of a sampler A_r = (D_o² − D_i²)/D_i² should be under 10–20% for undisturbed samples.

Worked example

A strip footing on saturated clay (c_u = 50 kPa, φ = 0) at D_f = 1 m, γ = 18 kN/m³.

• q_u = 5.7 × 50 + 18 × 1 × 1 = 285 + 18 = 303 kPa; q_nu = 285 kPa; with FS 3, q_ns = 95 kPa.

Practice