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Bearing Capacity & Phase Relations

Two exam workhorses in one place: Terzaghi's ultimate bearing capacity for strip, square, and circular footings, and a three-phase soil solver that fills in any phase quantities it can from whatever you know.

Terzaghi ultimate bearing capacity

q_ult = s_c c N_c + γD_fN_q + s_γ ½ γB N_γ q_all = q_ult / FS

N_c, N_q, N_γ	Terzaghi bearing-capacity factors (functions of φ only)
s_c, s_γ	shape factors — strip: 1.0, 1.0; square: 1.3, 0.8; circular: 1.3, 0.6
c	cohesion
φ	friction angle, in degrees
γ	unit weight of the soil above the base (use submerged γ′ if the water table is at the base)
D_f, B	embedment depth and footing width (diameter for circular)

N_c, N_q, N_γ are Terzaghi's rough-base values at 5° intervals (0°: 5.7, 1.0, 0.0 · 25°: 25.1, 12.7, 9.7 · 30°: 37.2, 22.5, 19.7 · 40°: 95.7, 81.3, 100.4); the calculator interpolates linearly between them. General shear failure is assumed — match the failure mode to your reference tables on exam day.

Unit system Footing shape Cohesion c (kPa) Friction angle φ (degrees) Unit weight γ (kN/m³) Embedment D_f (m) Width B (m) Factor of safety FS

Soil phase relations solver

n = e / (1 + e) S = w G_s / e

γ_d = G_sγ_w / (1 + e) γ_sat = (G_s + e)γ_w / (1 + e) γ = γ_d(1 + w)

e	void ratio = V_v/V_s
n	porosity = V_v/V (enter as percent)
w	water content = W_w/W_s (enter as percent)
G_s	specific gravity of solids (typically 2.65–2.75)
S	degree of saturation = V_w/V_v (enter as percent)
γ_w	unit weight of water: 9.81 kN/m³ (SI) or 62.4 pcf (US)

Enter any of the values you know — the solver propagates through every relation it can. Two phase quantities alone do not always fix the soil (e.g. e and w leave S and G_s linked by S = wG_s/e), so the solver reports exactly what is determined and what still needs one more measurement.

Unit system Void ratio e Porosity n (%) Water content w (%) Specific gravity G_s Saturation S (%)

For study and checking only. These calculators run entirely in your browser — nothing is sent anywhere. They are meant to verify your hand calculations while you study; on the exam you will work by hand with the NCEES reference handbook.

Worked example: Terzaghi bearing capacity of a square footing

Square footing, B = 1.5 m, embedment D_f = 0.8 m. Clean sand: c = 0, φ = 30°, γ = 17 kN/m³. Bearing factors for φ = 30°: N_c = 37.2, N_q = 22.5, N_γ = 19.7.

Square shape factors: s_c = 1.3, s_γ = 0.8
Cohesion term: 1.3 × 0 × 37.2 = 0
Surcharge term: γD_fN_q = 17 × 0.8 × 22.5 = 306 kPa
Self-weight term: 0.8 × ½ × 17 × 1.5 × 19.7 = 200.9 kPa
q_ult = 306 + 200.9 ≈ 507 kPa; with FS = 3, q_all ≈ 169 kPa

Exam tip: shape factors are strip 1.0/1.0, square 1.3/0.8, circular 1.3/0.6 — applying square factors to a strip footing is a favourite distractor.

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Preparing for exam day? The 110-question FE Civil practice exam is a full 5-hour-20-minute rehearsal with detailed solutions.