Bearing Capacity Factors — N_c, N_q, N_γ

By Michael Flynn, PE · water resources & dam safety engineer · Last updated Sep 9, 2026

Ultimate bearing capacity of a shallow foundation resolves into three contributions — cohesion, surcharge, and the self-weight of the failure wedge — each scaled by a dimensionless factor that depends only on the friction angle φ. Two families are in common use and they are not interchangeable: Terzaghi's original 1943 factors, and the Meyerhof/Vesić "general" factors that most modern codes build on.

Terzaghi's Equations

Strip q_u = c·N_c + q·N_q + 0.5·γ·B·N_γ Square q_u = 1.3·c·N_c + q·N_q + 0.4·γ·B·N_γ Circular q_u = 1.3·c·N_c + q·N_q + 0.3·γ·B·N_γ

where q = γ·D_f is the effective surcharge at founding level, B is the footing width (or diameter), and c is cohesion. The 1.3 / 0.4 / 0.3 multipliers are Terzaghi's empirical shape allowances, already built into the equations above — do not apply separate shape factors on top of them.

Terzaghi Bearing Capacity Factors

φ (deg)	N_c	N_q	N_γ	Typical soil
0	5.70	1.00	0.00	saturated clay, undrained
5	7.34	1.64	0.14	soft silty clay
10	9.61	2.69	0.56	silt, clayey silt
15	12.86	4.45	1.52	loose silty sand
20	17.69	7.44	3.64	loose sand
25	25.13	12.72	8.34	medium sand
30	37.16	22.46	19.13	medium–dense sand
35	57.75	41.44	45.41	dense sand, gravel
40	95.66	81.27	115.31	very dense sand / gravel
45	172.28	173.28	325.34	dense angular gravel
50	347.50	415.14	1072.80	rarely justified from field data

N_c and N_q above are Terzaghi’s closed forms. The N_γ column is the Kumbhojkar (1993) numerical evaluation of Terzaghi’s wedge solution — the set used in current editions of Das, Principles of Foundation Engineering (Table 3.1). Interpolate linearly between rows only where the interval is small; N_γ grows faster than linearly above φ = 35°, so prefer the closed forms or the calculator there.

Two different tables are both published as “Terzaghi N_γ”, and they do not agree. Terzaghi’s original 1943 values were read off a graph; Kumbhojkar later evaluated the same wedge solution numerically. Bowles, Foundation Analysis and Design (Table 4-1) carries a reconstruction closer to the original graph. The two diverge sharply in low-friction soils:

φ (deg)	Kumbhojkar / Das (used here)	Bowles Table 4-1	Difference
5	0.14	0.5	+257%
10	0.56	1.2	+114%
15	1.52	2.5	+64%
20	3.64	5.0	+37%
30	19.13	19.7	+3%
40	115.31	100.4	−13%

They agree near φ = 30° and part company on either side. If a reviewer checking your work against Bowles gets a different number, this is why — say which table you used. On cohesive soils the N_γ term is usually small enough that the divergence does not govern; on loose granular soils at low φ it can.

Sizing a structure, not just a footing? HydroComplete handles the hydraulic side of the same site — outlet structures, headwalls and basin embankments — with the loads and tailwater carried through.

Meyerhof / Vesić General Factors

φ (deg)	N_c	N_q	N_γ (Vesić)
0	5.14	1.00	0.00
5	6.49	1.57	0.45
10	8.35	2.47	1.22
15	10.98	3.94	2.65
20	14.83	6.40	5.39
25	20.72	10.66	10.88
30	30.14	18.40	22.40
35	46.12	33.30	48.03
40	75.31	64.20	109.41
45	133.88	134.88	271.76
50	266.89	319.07	762.89

N_c at φ = 0 is the tell. Terzaghi gives 5.70; the general solution gives 5.14 (= 2 + π). If you are checking someone's undrained clay calculation and see q_u = 5.14·c_u + q, they used Meyerhof/Vesić. Terzaghi's factors are roughly 10–15 percent higher across the board and are the less conservative choice — but Terzaghi's method carries no depth, inclination or base-tilt factors, so a full Meyerhof/Vesić check with those applied usually lands lower.

Closed-Form Expressions

Factor	Terzaghi	Meyerhof / Vesić
N_q	e^{2(3π/4 − φ/2)tanφ} / (2cos²(45° + φ/2))	e^πtanφ · tan²(45° + φ/2)
N_c	(N_q − 1)·cotφ	(N_q − 1)·cotφ
N_γ	no closed form — tabulated from the wedge solution (Kumbhojkar 1993)	2(N_q + 1)·tanφ (Vesić)

N_c is indeterminate at φ = 0 by the cotφ form; the limiting values are 5.70 (Terzaghi) and 5.14 (general). Meyerhof's own N_γ = (N_q − 1)tan(1.4φ) differs from Vesić's and runs lower at high φ — state which one you used.

Allowable Capacity and Factor of Safety

Quantity	Expression	Note
Gross allowable	q_all = q_u / FS	FS = 3 typical for shallow foundations
Net ultimate	q_u,net = q_u − q	subtract the surcharge already there
Net allowable	q_all,net = (q_u − q) / FS	the value to compare against net applied pressure

Bearing capacity is rarely what governs. For footings on sand and on stiff clay, settlement almost always controls the design before shear failure does. Size for capacity, then check settlement — and expect the settlement check to drive the footing dimension.

Sources: Terzaghi, K. (1943). Theoretical Soil Mechanics. Meyerhof, G.G. (1963). "Some recent research on the bearing capacity of foundations," Canadian Geotechnical Journal. Vesić, A.S. (1973). "Analysis of ultimate loads of shallow foundations," JSMFD ASCE. Kumbhojkar, A.S. (1993). “Numerical evaluation of Terzaghi’s N_γ,” Journal of Geotechnical Engineering ASCE 119(3) — the source of the N_γ column here. Das, B.M., Principles of Foundation Engineering, Table 3.1. Bowles, J.E., Foundation Analysis and Design, Table 4-1 — a different N_γ reconstruction. Verify φ against actual site investigation data rather than the "typical soil" column, which is orientation only.

Need to run the numbers? Open the Terzaghi bearing capacity calculator → · Then check consolidation settlement.

Related cheat sheets and tools

Getting φ and c right matters more than the factor table — start from USCS classification and unit weights, and see presumptive bearing values for the code-table shortcut where a full analysis isn't warranted. For retaining and embankment work see lateral earth pressure and slope stability. For dam and basin embankment design that ties the geotechnical section to the hydraulics, see HydroComplete, the SaaS sister product to PE-Calc.

Bearing Capacity Factors — Nc, Nq, Nγ

Terzaghi's Equations

Terzaghi Bearing Capacity Factors

Meyerhof / Vesić General Factors

Closed-Form Expressions

Allowable Capacity and Factor of Safety

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Bearing Capacity Factors — N_c, N_q, N_γ