Bearing Capacity of Shallow Foundations

Theory: Bearing Capacity

Where the bearing capacity equation comes from, how Terzaghi, Meyerhof and Vesić differ, and how the footing's shape and depth, the load's inclination and eccentricity and the water table enter it.

Bearing capacity failure

As the load on a footing grows, the soil under it fails in shear along a surface that runs from the footing's edges down and out to the ground. In general shear failure, typical of dense sand and stiff clay, the surface is well defined and the ground heaves beside the footing. In loose or soft soil the failure is local or by punching, with large settlement before any clear peak.

Prandtl (1921) described the mechanism for a strip: an active wedge pushed down under the footing, a radial fan bounded by a log spiral, and a passive Rankine wedge pushed up at each side. The bearing capacity is the pressure that brings this mechanism to failure.

Terzaghi's equation

Terzaghi (1943) split the ultimate bearing capacity into three terms — the soil's cohesion, the overburden beside the footing (treated as a surcharge q = γD_f, with the soil above the base giving no strength) and the weight of the soil in the failure zone — each with a factor that depends only on φ′.

q_u = c' N_c\, s_c + q N_q + \tfrac{1}{2} \gamma B N_\gamma\, s_\gamma

Strip: s_c = s_γ = 1 · square: s_c = 1.3, s_γ = 0.8 · circular: s_c = 1.3, s_γ = 0.6 (B the diameter)

N_q = \frac{e^{2\left(3\pi/4 - \phi'/2\right) \tan\phi'}}{2 \cos^2\left(45^\circ + \phi'/2\right)}

N_γ has no closed form; Das uses Kumbhojkar's (1993) values. For φ′ = 0, N_c = 5.7

For local shear, Terzaghi suggested the same equation with c′ and tan φ′ reduced to two thirds.

Meyerhof and the general equation

Meyerhof (1963) took Prandtl and Reissner's N_c and N_q, with a rough-footing wedge at 45° + φ′/2, and added factors for the footing's shape, its depth (counting the shear strength of the soil above the base) and the load's inclination.

q_u = c' N_c\, s_c d_c i_c + q N_q\, s_q d_q i_q + \tfrac{1}{2} \gamma B' N_\gamma\, s_\gamma d_\gamma i_\gamma

s: shape · d: depth · i: inclination factors

N_q = e^{\pi \tan\phi'} \tan^2\left(45^\circ + \frac{\phi'}{2}\right)

Prandtl (1921) and Reissner (1924). For φ′ = 0, N_c = π + 2 = 5.14

	Meyerhof (1963)	Vesić / general equation
N_γ	(N_q − 1) tan(1.4φ′)	2(N_q + 1) tan φ′
s_c	1 + 0.2K_p B/L	1 + (N_q/N_c)(B/L)
s_q	1 + 0.1K_p B/L	1 + (B/L) tan φ′
s_γ	1 + 0.1K_p B/L	1 − 0.4 B/L
d_c	1 + 0.2√K_p D_f/B	1 + 0.4 D_f/B
d_q	1 + 0.1√K_p D_f/B	1 + 2 tan φ′(1 − sin φ′)² D_f/B
d_γ	1 + 0.1√K_p D_f/B	1
i_c = i_q	(1 − β/90°)²	(1 − β/90°)²
i_γ	(1 − β/φ′)²	(1 − β/φ′)²

K_p = tan²(45° + φ′/2). Meyerhof's s_q, s_γ, d_q and d_γ hold from φ′ = 10°; below it they're taken as 1. In the general equation, Hansen's depth factors replace D_f/B by arctan(D_f/B) (in radians) when D_f > B. The N_c and N_q of the two are the same; they differ in N_γ, Vesić's being the larger, and in the factors.

Water table

Above the base (D_w ≤ D_f): q = γD_w + γ′(D_f − D_w), and the γ term takes the buoyant γ′ = γ_sat − γ_w.
Within B below the base (D_f < D_w ≤ D_f + B): q = γD_f, and the γ term takes γ̄ = γ′ + (d/B)(γ − γ′), d being D_w − D_f.
Deeper than D_f + B: the water table doesn't affect the bearing capacity.

Eccentric loads

A load at an eccentricity e from the center is taken, after Meyerhof (1953), as centered on an effective footing of width B′ = B − 2e. The γ term and the shape factors use B′, and the load the footing can carry is q_u over the effective area A′ = B′L.

Allowable bearing capacity

q_{\mathrm{all}} = \frac{q_u}{\mathrm{FS}}

FS: factor of safety, usually 3 for shallow foundations

The allowable value also has to keep settlement within limits, which often governs on wide footings and in soft clays.

References

Terzaghi, K. Theoretical Soil Mechanics. New York: Wiley, 1943.
Meyerhof, G. G. Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal, v. 1, n. 1, p. 16–26, 1963.
Vesić, A. S. Analysis of ultimate loads of shallow foundations. Journal of the Soil Mechanics and Foundations Division, ASCE, v. 99, n. SM1, p. 45–73, 1973.
Hansen, J. B. A revised and extended formula for bearing capacity. Danish Geotechnical Institute, Bulletin 28, 1970.
Kumbhojkar, A. S. Numerical evaluation of Terzaghi's Nγ. Journal of Geotechnical Engineering, ASCE, v. 119, n. 3, p. 598–607, 1993.
Das, B. M.; Sobhan, K. Principles of Geotechnical Engineering. 9th ed. Boston: Cengage, 2018.

Enter the footing — shape, width, depth, and the load's inclination and eccentricity — and the soil's c′, φ′ and unit weights to get the ultimate and allowable bearing capacity by Terzaghi, Meyerhof and Vesić, side by side, with each method's factors.

q_all Terzaghi: 415 kPa
q_all Meyerhof: 519 kPa
q_all Vesić: 526 kPa

Footing

Shape

Width, BmDepth, D_fmInclination, β0°Eccentricity, em

β is the load's angle from the vertical; e is measured along B. Circular footings take centered loads.

Soil

c′kPaφ′30°γ (above WT)kN/m³γ_sat (below WT)kN/m³Water table, D_wmγ_wkN/m³

Leave the water table blank when it's deeper than D_f + B; it's measured from the ground surface.

Design

Terzaghi's failure

FS is the factor of safety on q_u. Local shear, for loose or soft soils, takes ⅔c′ and tan φ′ reduced to ⅔ in Terzaghi's equation.

Homogeneous soil under the footing; the water table follows Das's three cases.
Terzaghi's N_γ is Kumbhojkar's (1993), as Das tabulates it. His shape factors are extended to rectangles linearly in B/L, and he has no depth or inclination factors.
Vesić's method is the general equation as in Das: Vesić's N_γ, De Beer's shape factors, Hansen's depth factors and Meyerhof's inclination factors.
An eccentric load is carried on Meyerhof's effective width B′ = B − 2e, used in the γ term, the shape factors and the area; depth factors use B.
q_all = q_u/FS on the gross bearing capacity.