What is Terzaghi's bearing capacity formula?

For a strip footing qu = c·Nc + q·Nq + 0.5·γ·B·Nγ, where c is cohesion, q = γ·Df is the overburden pressure at footing level, γ is the soil unit weight below the footing, B the footing width and Nc, Nq, Nγ the bearing capacity factors. Square footing: qu = 1.3·c·Nc + q·Nq + 0.4·γ·B·Nγ. Circular footing (B = diameter): qu = 1.3·c·Nc + q·Nq + 0.3·γ·B·Nγ.

What are Terzaghi's bearing capacity factors?

Nq = a² / (2·cos²(45° + φ/2)) with a = e^((3π/4 − φ/2)·tan φ), Nc = (Nq − 1)·cot φ (about 5.7 for φ = 0), and Nγ, which has no closed form, from Kumbhojkar's numerical solution. For φ = 30°: Nc = 37.16, Nq = 22.46, Nγ = 19.13. The full table for φ = 0–50° is below the calculator.

How do Meyerhof, Hansen and Vesić factors differ from Terzaghi's?

They share Prandtl–Reissner factors Nq = e^(π tan φ)·tan²(45° + φ/2) and Nc = (Nq − 1)·cot φ (5.14 for φ = 0), and differ in Nγ: Meyerhof (Nq − 1)·tan(1.4φ), Hansen 1.5·(Nq − 1)·tan φ, Vesić 2·(Nq + 1)·tan φ. At φ = 30° they give Nq = 18.40, Nc = 30.14 and Nγ = 15.67, 15.07 and 22.40. They also add shape and depth factors, so they are not interchangeable with Terzaghi's 1.3/0.8/0.6 shape coefficients.

How does the water table affect bearing capacity?

If the water table is above the footing base, q = γ·Dw + γ′·(Df − Dw) and the Nγ term uses γ′ = γsat − γw. If it lies a depth d below the base with d < B, q is unchanged and the Nγ term uses γ′ + (d/B)(γ − γ′). Deeper than B below the base, there is no correction. A water table at the surface can roughly halve the Nγ term.

What is the difference between gross, net and allowable bearing capacity?

Gross ultimate capacity qu is the total pressure at failure. Net ultimate capacity qu − q removes the overburden that was there before excavation. Allowable capacity divides by the factor of safety: qa = qu / FS (gross) or (qu − q) / FS (net). Use the net value when comparing with the net added load from the structure.

What factor of safety should I use?

For shear failure of shallow foundations a factor of safety of 2.5 to 3.0 on the ultimate capacity is commonly used for permanent structures, with lower values only where soil properties are well defined. Settlement often governs before bearing failure, so check it separately.

When should I use local shear failure?

Terzaghi suggested local shear for loose sands and soft clays that compress a lot before failing. The calculation uses reduced strength c* = ⅔·c and φ* = tan⁻¹(⅔·tan φ) in the same equation.

Bearing Capacity (Terzaghi) Calculator

Enter cohesion, friction angle, unit weight, footing width and depth to get Terzaghi's ultimate bearing capacity qu, net capacity, allowable qa and the allowable load. Includes the water table correction, local shear option, Meyerhof / Hansen / Vesić alternatives and a table of bearing capacity factors for φ = 0–50°.

Soil & Footing Parameters

Method

Footing Shape

Cohesion c (kPa)

Friction angle φ (°)

Unit weight γ (kN/m³)

Width B (m)

Depth Df (m)

Factor of safety FS

Water table depth Dw (m)

γsat (kN/m³)

Depth below ground surface; leave blank if deeper than Df + B. γw = 9.81 kN/m³.

Local shear failure (loose sand / soft clay)

Ultimate q_u

—

kPa (gross)

Allowable q_a

—

kPa = q_u / FS

Bearing Capacity Factors

N_c

—

N_q

—

N_γ

—

Calculation Breakdown

Bearing capacity factors by friction angle φ

Terzaghi: Nq = a²/(2cos²(45°+φ/2)), a = e^((0.75π−φ/2)tanφ); Nc = (Nq−1)cotφ; Nγ from Kumbhojkar (1993). Meyerhof, Hansen and Vesić share Nq = e^(π tanφ)tan²(45°+φ/2) and Nc = (Nq−1)cotφ.

φ	Terzaghi			Meyerhof / Hansen / Vesić
φ	Nc	Nq	Nγ	Nc	Nq	Nγ (M)	Nγ (H)	Nγ (V)
0°	5.71	1.00	0.00	5.14	1.00	0.00	0.00	0.00
1°	6.00	1.10	0.01	5.38	1.09	0.00	0.00	0.07
2°	6.30	1.22	0.04	5.63	1.20	0.01	0.01	0.15
3°	6.62	1.35	0.06	5.90	1.31	0.02	0.02	0.24
4°	6.97	1.49	0.10	6.19	1.43	0.04	0.05	0.34
5°	7.34	1.64	0.14	6.49	1.57	0.07	0.07	0.45
6°	7.73	1.81	0.20	6.81	1.72	0.11	0.11	0.57
7°	8.15	2.00	0.27	7.16	1.88	0.15	0.16	0.71
8°	8.60	2.21	0.35	7.53	2.06	0.21	0.22	0.86
9°	9.09	2.44	0.44	7.92	2.25	0.28	0.30	1.03
10°	9.60	2.69	0.56	8.34	2.47	0.37	0.39	1.22
11°	10.16	2.98	0.69	8.80	2.71	0.47	0.50	1.44
12°	10.76	3.29	0.85	9.28	2.97	0.60	0.63	1.69
13°	11.41	3.63	1.04	9.81	3.26	0.74	0.78	1.97
14°	12.11	4.02	1.26	10.37	3.59	0.92	0.97	2.29
15°	12.86	4.45	1.52	10.98	3.94	1.13	1.18	2.65
16°	13.68	4.92	1.82	11.63	4.34	1.37	1.43	3.06
17°	14.56	5.45	2.18	12.34	4.77	1.66	1.73	3.53
18°	15.52	6.04	2.59	13.10	5.26	2.00	2.08	4.07
19°	16.56	6.70	3.07	13.93	5.80	2.40	2.48	4.68
20°	17.69	7.44	3.64	14.83	6.40	2.87	2.95	5.39
21°	18.92	8.26	4.31	15.81	7.07	3.42	3.50	6.20
22°	20.27	9.19	5.09	16.88	7.82	4.07	4.13	7.13
23°	21.75	10.23	6.00	18.05	8.66	4.82	4.88	8.20
24°	23.36	11.40	7.08	19.32	9.60	5.72	5.75	9.44
25°	25.13	12.72	8.34	20.72	10.66	6.77	6.76	10.88
26°	27.09	14.21	9.84	22.25	11.85	8.00	7.94	12.54
27°	29.24	15.90	11.60	23.94	13.20	9.46	9.32	14.47
28°	31.61	17.81	13.70	25.80	14.72	11.19	10.94	16.72
29°	34.24	19.98	16.18	27.86	16.44	13.24	12.84	19.34
30°	37.16	22.46	19.13	30.14	18.40	15.67	15.07	22.40
31°	40.41	25.28	22.65	32.67	20.63	18.56	17.69	25.99
32°	44.04	28.52	26.87	35.49	23.18	22.02	20.79	30.21
33°	48.09	32.23	31.94	38.64	26.09	26.17	24.44	35.19
34°	52.64	36.50	38.04	42.16	29.44	31.15	28.77	41.06
35°	57.75	41.44	45.41	46.12	33.30	37.15	33.92	48.03
36°	63.53	47.16	54.36	50.59	37.75	44.43	40.05	56.31
37°	70.07	53.80	65.27	55.63	42.92	53.27	47.38	66.19
38°	77.50	61.55	78.61	61.35	48.93	64.07	56.17	78.02
39°	85.97	70.61	95.03	67.87	55.96	77.33	66.76	92.25
40°	95.66	81.27	115.31	75.31	64.20	93.69	79.54	109.41
41°	106.81	93.85	140.51	83.86	73.90	113.99	95.05	130.21
42°	119.67	108.75	171.99	93.71	85.37	139.32	113.96	155.54
43°	134.58	126.50	211.56	105.11	99.01	171.14	137.10	186.53
44°	151.95	147.74	261.60	118.37	115.31	211.41	165.58	224.63
45°	172.29	173.29	325.34	133.87	134.87	262.74	200.81	271.75
46°	196.22	204.19	407.11	152.10	158.50	328.73	244.65	330.34
47°	224.55	241.80	512.84	173.64	187.21	414.33	299.52	403.65
48°	258.29	287.85	650.67	199.26	222.30	526.45	368.67	496.00
49°	298.72	344.64	831.99	229.92	265.50	674.92	456.40	613.14
50°	347.51	415.15	1,072.80	266.88	319.06	873.86	568.57	762.86

Shape and depth factors used

Method	Shape (B/L = 1 for square and circle)	Depth (k = Df/B, or tan⁻¹(Df/B) if Df > B)
Terzaghi	sc = 1.3; sγ = 0.8 square, 0.6 circle; sq = 1	none
Meyerhof	sc = 1 + 0.2Kp·B/L; sq = sγ = 1 + 0.1Kp·B/L (φ > 10°)	dc = 1 + 0.2√Kp·Df/B; dq = dγ = 1 + 0.1√Kp·Df/B (φ > 10°)
Hansen	sc = 1 + (Nq/Nc)B/L; sq = 1 + (B/L)sinφ; sγ = 1 − 0.4B/L	dc = 1 + 0.4k; dq = 1 + 2tanφ(1 − sinφ)²k; dγ = 1
Vesić	sc = 1 + (Nq/Nc)B/L; sq = 1 + (B/L)tanφ; sγ = 1 − 0.4B/L	same as Hansen

Kp = tan²(45° + φ/2). Worked example (Terzaghi, strip): c = 20 kPa, φ = 25°, γ = 18 kN/m³, B = 1.5 m, Df = 1 m → Nc = 25.13, Nq = 12.72, Nγ = 8.34; qu = 20×25.13 + 18×12.72 + 0.5×18×1.5×8.34 = 502.69 + 228.97 + 112.59 = 844.25 kPa; qa = 844.25 / 3 = 281.4 kPa.

Summary

Terzaghi's bearing capacity equation for a strip footing is qu = c·Nc + q·Nq + 0.5·γ·B·Nγ, with q = γ·Df the effective overburden at footing level. Square footings use 1.3·c·Nc + q·Nq + 0.4·γ·B·Nγ and circular footings 1.3·c·Nc + q·Nq + 0.3·γ·B·Nγ. Terzaghi's own factors are Nq = a²/(2cos²(45° + φ/2)) with a = e^((0.75π − φ/2)tanφ), Nc = (Nq − 1)cotφ (5.7 at φ = 0) and Nγ from Kumbhojkar's (1993) numerical solution — they are not the Meyerhof/Vesić factors many calculators use under Terzaghi's name. This tool keeps the methods separate, applies the groundwater correction to q and γ, and reports gross, net and allowable capacity. It is a preliminary design aid; final foundation design needs site investigation, settlement checks and the applicable code.

How it works

Choose the method (Terzaghi by default, or Meyerhof, Hansen, Vesić) and the footing shape.
Enter cohesion c, friction angle φ, unit weight γ, footing width or diameter B and embedment depth Df.
If groundwater is within Df + B of the surface, enter its depth and the saturated unit weight; q and the γ in the Nγ term are corrected.
For loose or soft soil, tick local shear: Terzaghi uses c* = ⅔c and φ* = tan⁻¹(⅔ tan φ).
Read qu, net qu − q, allowable qa = qu / FS (gross and net) and the allowable load on the footing.

Use cases

Preliminary sizing of shallow strip and pad footings in geotechnical design.
Cross-checking hand calculations during foundation engineering coursework.
Comparing Terzaghi with Meyerhof, Hansen and Vesić results for the same soil.
Checking how a rising water table reduces bearing capacity.
Looking up bearing capacity factors Nc, Nq and Nγ for a given friction angle.

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu