Bearing Capacity of Shallow Foundations (Terzaghi Equation)

NCEES FE Civil spec: Geotechnical Engineering, G. Bearing capacity. FE Reference Handbook: Civil Engineering chapter, Geotechnical section (Ultimate Bearing Capacity).

What the FE Civil Spec Covers

Section G covers the ultimate and allowable bearing capacity of shallow foundations. Typical questions plug values into the general bearing capacity equation, apply a factor of safety, correct for a water table, or find the footing width or load that a soil can support.

Key Concepts

Ultimate bearing capacity $q_{u l t}$ is the pressure that causes a shear failure of the soil under the footing.
Three terms. Cohesion ( $N_{c}$ ), overburden or embedment ( $N_{q}$ ) and soil weight below the footing ( $N_{γ}$ ). All three factors grow quickly with friction angle.
Allowable pressure is $q_{a l l} = q_{u l t} / F S$ , usually with FS = 3. Settlement may govern instead of bearing capacity, especially on clay.
Effective unit weight. Below the water table use $γ^{'} = γ_{s a t} - γ_{w}$ in the terms that involve soil weight.

Core Formulas

Strip (continuous) footing, Terzaghi general shear:

q_{u l t} = c N_{c} + q N_{q} + \frac{1}{2} γ B N_{γ}, q = γ D_{f}

Terzaghi shape modifications:

Square: q_{u l t} = 1.3 c N_{c} + q N_{q} + 0.4 γ B N_{γ}

Circular (diameter B): q_{u l t} = 1.3 c N_{c} + q N_{q} + 0.3 γ B N_{γ}

Saturated clay under undrained loading ( $ϕ = 0$ , so $N_{q} = 1$ and $N_{γ} = 0$ ):

q_{u l t} = 5.7 c + q (Terzaghi strip), q_{u l t} = 5.14 c + q (Prandtl/Meyerhof/Vesic)

Allowable pressure and capacity per unit length of a strip footing:

q_{a l l} = \frac{q _{u l t}}{F S}, Q_{a l l} = q_{a l l} B

Water Table Corrections

Water table deeper than about B below the base: no correction; use moist unit weight in both terms.
Water table at the footing base: the overburden term uses moist unit weight above, but the width term uses $γ^{'}$ .
Water table at the ground surface: use $γ^{'}$ in both terms, so $q = γ^{'} D_{f}$ .
Between those levels: compute q with the actual layers above the base, and interpolate the unit weight in the width term.

Gross vs net. Gross pressure includes the overburden already present at the base level. Net ultimate capacity is $q_{u l t} - γ D_{f}$ . Read whether a problem gives gross or net allowable pressure before sizing a footing. Loose sands and soft clays may fail in local or punching shear rather than general shear, which gives lower capacity; Terzaghi handled this with reduced strength parameters.

Where It Is in the FE Reference Handbook

The Civil Engineering chapter, Geotechnical section, gives the ultimate bearing capacity equation for a strip footing in terms of effective unit weight, with the factor definitions. Retaining wall checks in the same section compare toe pressure with ultimate bearing capacity.

Worked Example: Strip Footing on Sand

A strip footing 1.5 m wide is founded 1.2 m below grade in sand with $γ = 18.5 kN/m^{3}$ , $c = 0$ and $ϕ = 3 0^{\circ}$ . The water table is deep. Use Terzaghi factors $N_{q} = 22.46$ and $N_{γ} = 19.13$ and FS = 3. Find the allowable bearing pressure and the allowable load per meter of wall.

Step 1: overburden pressure at the footing base.

q = γ D_{f} = 18.5 (1.2) = 22.2 kPa

Step 2: each term. The cohesion term is zero.

q N_{q} = 22.2 (22.46) = 498.6 kPa

\frac{1}{2} γ B N_{γ} = 0.5 (18.5) (1.5) (19.13) = 265.4 kPa

Step 3: ultimate and allowable pressure.

q_{u l t} = 498.6 + 265.4 = 764.0 kPa, q_{a l l} = \frac{764.0}{3} = 254.7 kPa

Step 4: allowable load per meter.

Q_{a l l} = 254.7 (1.5) = 382 kN/m

If the water table rose to the footing base, the width term would use the submerged unit weight (roughly half of 18.5), cutting that term about in half.

Common Mistakes

Using the strip footing coefficients (1.0 and 0.5) for a square or circular footing.
Using $D_{f}$ or the footing thickness in place of $B$ in the width term.
Forgetting the overburden term entirely, or using total unit weight below the water table.
Dividing a strip footing pressure by area instead of multiplying by width to get load per unit length.
Mixing factors from different tables (for example, Terzaghi Nq with Vesic N gamma).

Exam Tips

Write the three-term equation first, then cross out terms that vanish: the cohesion term for clean sand, and the width term for undrained clay. Check the footing shape before choosing coefficients, and check the water table before choosing unit weights. If the answer choices differ by about a factor of two, a missed submerged unit weight or a strip-versus-square mix-up is the likely trap. Finally, confirm whether the question wants ultimate pressure, allowable pressure or allowable load, since all three often appear among the choices.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

For a strip footing, q_ult = c·Nc + γ·Df·Nq + 0.5·γ·B·Nγ. Given c = 25 kPa, Nc = 5.7, γ = 18 kN/m³, Df = 1 m, Nq = 1 and Nγ = 0, q_ult is most nearly: See the worked solution or practice the Soils & Foundations deck
A strip footing on sand has c = 0, γ = 18 kN/m³, Df = 1 m, B = 2 m, Nq = 18.4 and Nγ = 15.7. The ultimate bearing capacity is most nearly: See the worked solution or practice the Soils & Foundations deck
Use q_ult = cN_c + qN_q + 0.5γBN_γ with c = 20 kPa, N_c = 30, q = 40 kPa, N_q = 18, γ = 18 kN/m³, B = 2 m, and N_γ = 15. What is the ultimate bearing capacity? See the worked solution or practice the Soils & Foundations deck
A strip footing has an ultimate bearing capacity of 160.5 kPa. With a factor of safety of 3, the allowable bearing pressure is most nearly: See the worked solution or practice the Soils & Foundations deck
For a footing on granular soil, a rise in the groundwater table to near the footing generally causes the bearing capacity to: See the worked solution or practice the Soils & Foundations deck

Related Calculators and Examples

Frequently Asked Questions

Which bearing capacity factors should I use on the FE exam?

Use the factors the problem gives you. Published tables differ, especially for N gamma, so exam problems normally supply Nc, Nq and N gamma or tell you which table to read. For phi = 0, Terzaghi gives Nc = 5.7 while Prandtl, Meyerhof and Vesic give 5.14.

What factor of safety is used for bearing capacity?

A factor of safety of about 3 on ultimate bearing capacity is the common default for shallow foundations, so allowable pressure is roughly one third of the ultimate value.

How does a high water table affect bearing capacity?

Water at or above the footing base reduces the effective unit weight of the soil to about half, which lowers both the overburden term and the width term. Bearing capacity on granular soil drops noticeably.

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