Module 4 · Lesson 4.1

Three terms, three sources of strength

Where the resistance actually comes from.

Why this matters

The bearing capacity equation is usually met as a formula to be memorised. It is much better met as a statement about where resistance comes from — because then you can tell, before calculating anything, which term is going to matter and which is a rounding error.

By the end of this lesson you should be able to

State the three terms and what each represents
Identify which dominates in clay and which in sand
Explain why the width term disappears in undrained clay

The bearing capacity equation

q_{u l t} = c N_{c} + q N_{q} + \frac{1}{2} γ^{'} B N_{γ}

What it calculates: Gross ultimate bearing pressure of a shallow foundation

c: Cohesion — c′ drained, or s_u undrained (kPa)
q: Effective surcharge at founding level, beside the footing (kPa)
$γ^{'}$: Effective unit weight of soil below founding level (kN/m³)
B: Foundation width — the shorter plan dimension (m)
$N_{c}, N_{q}, N_{γ}$: Bearing capacity factors, functions of φ′ alone (—)

This assumes

General shear failure — the mechanism of Module 3 actually forms
A strip footing, vertically and centrally loaded, on level ground
Homogeneous soil to a depth of at least the failure mechanism

In plain terms: Three independent sources of resistance, added. The addition is itself an approximation — the three mechanisms are not truly independent — but a conservative and long-established one.

Taking them in turn.

The cohesion term, c·N_c. Resistance from the soil's shear strength at zero normal stress. In undrained clay this is the whole story.

The surcharge term, q·N_q. Resistance from the weight of soil beside the footing, which must be lifted for the passive zone to move. Note what this means: burying a footing deeper increases capacity, and it does so through the ground alongside rather than beneath.

The self-weight term, ½γ′B·N_γ. Resistance from the weight of the soil within the failure mechanism itself. It is the only term containing B, which is why width helps in sand — and it is the term nobody agrees on.

Worked example

A strip footing in sand, term by term

Given

Strip footing, B = 2.0 m, founded at D = 1.0 m
Dense sand, φ′ = 35°, c′ = 0
Unit weight 19 kN/m³ above and below founding level
Water table well below the failure zone
N_γ taken from the Vesic form

Find

The gross ultimate bearing pressure, and which term dominates

Assumptions

General shear failure — reasonable for dense sand
No shape or depth factors, so the answer is conservative

Practice

A strip footing 1.5 m wide is founded 1.2 m deep in a clay with s_u = 60 kPa and unit weight 20 kN/m³. Using q_ult = 5.14·s_u + q for the undrained case, what is the GROSS ultimate bearing pressure, in kPa?

Check yourself

For a wide raft founded near the surface on dense sand, which term of the bearing capacity equation dominates?

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint