Documentation

Methods & References

Every formula used in SoilLab — with its original author, the published source, and the assumptions implemented in code. Symbols follow standard geotechnical notation.

1. Shallow foundations — ultimate bearing capacity

The general bearing-capacity equation used (Meyerhof / Hansen / Vesic family):

q_ult = c·N_c·s_c·d_c·i_c + q·N_q·s_q·d_q·i_q + ½·γ'·B'·N_γ·s_γ·d_γ·i_γ

where q = γ·D_f is the effective overburden at the founding depth, B' is the effective footing width, and N_c, N_q, N_γ are the bearing-capacity factors.

Bearing-capacity factors

N_q = e^{π·tan φ'} · tan²(45° + φ'/2)

N_c = (N_q − 1) · cot φ' (= 5.14 for φ' = 0)

For N_γ the method selected determines the form:

Terzaghi (1943): classical formulation with Kumbhojkar's N_γ approximation.
Meyerhof (1963): N_γ = (N_q − 1) · tan(1.4 φ').
Hansen (1970): N_γ = 1.5·(N_q − 1)·tan φ'.
Vesic (1973): N_γ = 2·(N_q + 1)·tan φ'.

Net ultimate and allowable bearing pressures are reported as q_net,ult = q_ult − γ·D_f and q_net,allow = q_net,ult / FoS (default FoS = 3.0).

2. Shape, depth & inclination factors

Shape factors (Meyerhof / Hansen / Vesic), with r = B'/L':

s_c = 1 + (N_q/N_c)·r | s_q = 1 + r·tan φ' | s_γ = max(0.6, 1 − 0.4·r)

Depth factors (Hansen), with k = D_f/B' if D_f/B' ≤ 1, else k = atan(D_f/B'):

d_q = 1 + 2·tan φ'·(1 − sin φ')²·k | d_c = d_q − (1 − d_q)/(N_q·tan φ') | d_γ = 1

Inclination factors (Meyerhof) for a load inclined α° from vertical:

i_c = i_q = (1 − α/90°)² | i_γ = (1 − α/φ')²

3. Effective area for eccentric loads (Meyerhof, 1953)

For a footing loaded with eccentricities e_B, e_L:

B' = B − 2·e_B L' = L − 2·e_L A' = B'·L'

All capacity terms use the effective dimensions B' and L'.

4. Groundwater corrections

Three classical cases based on water-table depth d_w relative to D_f and B:

Case I (d_w ≤ D_f): q = γ·d_w + γ'·(D_f − d_w); use γ' in the N_γ term.
Case II (D_f < d_w < D_f + B): blend γ for the N_γ term linearly between γ' and γ.
Case III (d_w ≥ D_f + B): no correction.

5. Settlement

Elastic (immediate) settlement of a flexible rectangular footing (Bowles, Das):

S_i = q·B·(1 − ν²)·I_w / E_s

with influence factor I_w ≈ 0.88 (square), 1.12 (L/B = 2), 1.7 (strip-like). Poisson's ratio ν defaults to 0.3.

Consolidation settlement in clay (Terzaghi, 1925):

S_c = (C_c · H) / (1 + e₀) · log₁₀((σ'₀ + Δσ) / σ'₀)

Stress increase Δσ is computed by the 2:1 Boussinesq approximation under the footing centre.

6. Pile axial capacity

Total ultimate axial capacity:

Q_u = Q_s + Q_p − W_p, Q_all = Q_u / FoS (default FoS = 2.5)

Shaft resistance in sand (effective-stress / β-method)

Q_s = π·D · K_s·tan δ · γ'·L²/2 (per layer, integrated)

K_s ≈ 1.0 (bored) – 1.4 (driven); δ = (0.7–0.8)·φ'. An optional critical-depth cap L_c = 15·D limits the effective stress used in shaft computations (Vesic, Meyerhof).

Shaft resistance in clay (α-method, Tomlinson 1957)

f_s = α · c_u, Q_s = Σ π·D·ΔL · f_s

Default α = 0.55 (Tomlinson). A β-method alternative is available.

End bearing

Q_p = A_p · σ'_L · N_q* (sand, Meyerhof / Berezantsev)

Q_p = A_p · 9·c_u (clay, Skempton)

7. SPT correlations for piles (Meyerhof, 1976)

f_s (kPa) = 2·N̄₆₀ (driven), N̄₆₀ (bored)

q_p (kPa) = min(40·N₆₀·(L/D), 400·N₆₀)

8. Pile group efficiency (Converse–Labarre)

η = 1 − (θ/90°) · [ (n−1)·m + (m−1)·n ] / (n·m), θ = atan(D/s)

n × m piles at centre-to-centre spacing s; D = pile diameter.

9. Field SPT corrections

Energy-corrected blow count and overburden-corrected (N₁)₆₀:

N₆₀ = N · (E_m/0.60) · C_B · C_S · C_R

(N₁)₆₀ = C_N · N₆₀, C_N = √(p_a/σ'_v) (Liao & Whitman, 1986)

10. CPT — Robertson Soil Behaviour Type

Normalised tip resistance Q_tn, friction ratio F_r and the SBT index I_c (Robertson, 1990; 2009):

I_c = √[ (3.47 − log Q_tn)² + (log F_r + 1.22)² ]

Soil Behaviour Type zones are assigned from I_c thresholds (gravelly sand < 1.31 < sand < 2.05 < sand mixture < 2.60 < silt mixture < 2.95 < clay < 3.60 < organic).

References

Terzaghi, K. (1943). Theoretical Soil Mechanics. Wiley.
Meyerhof, G. G. (1963). Some recent research on the bearing capacity of foundations. Canadian Geotech. J., 1(1), 16–26.
Meyerhof, G. G. (1976). Bearing capacity and settlement of pile foundations. JGED, ASCE, 102(GT3), 197–228.
Hansen, J. B. (1970). A revised and extended formula for bearing capacity. Danish Geotechnical Institute Bulletin No. 28.
Vesic, A. S. (1973). Analysis of ultimate loads of shallow foundations. JSMFD, ASCE, 99(SM1), 45–73.
Tomlinson, M. J. (1957). The adhesion of piles driven in clay soils. Proc. 4th ICSMFE, 2, 66–71.
Skempton, A. W. (1951). The bearing capacity of clays. Building Research Congress, London.
Bowles, J. E. (1996). Foundation Analysis and Design, 5th ed. McGraw-Hill.
Das, B. M. (2016). Principles of Foundation Engineering, 8th ed. Cengage.
Coduto, D. P. (2001). Foundation Design: Principles and Practices, 2nd ed. Prentice Hall.
Robertson, P. K. (1990). Soil classification using the CPT. Canadian Geotech. J., 27(1), 151–158.
Robertson, P. K. (2009). Interpretation of cone penetration tests — a unified approach. Canadian Geotech. J., 46, 1337–1355.
Liao, S. S. C. & Whitman, R. V. (1986). Overburden correction factors for SPT in sand. JGE, ASCE, 112(3), 373–377.
Converse, F. & Labarre, E. (1963). Group efficiency of piles. (As cited in Bowles, 1996.)
British Standards Institution. BS 1377: Methods of test for soils for civil engineering purposes.

SoilLab is an engineering aid; results must be reviewed and signed off by a competent geotechnical engineer. Calibrate parameters against site-specific testing.