Module 9 · Lesson 9.1

The elastic calculation

Rigid, flexible, and which number you are comparing.

Why this matters

Module 1 showed that settlement usually governs. This is the calculation that produces the number, and it is disarmingly simple — which is exactly the danger, because its simplicity has nothing to do with its reliability.

By the end of this lesson you should be able to

Compute elastic settlement for a footing on a deep uniform layer
Apply the rigid and flexible cases correctly
Identify which input carries the uncertainty

Immediate settlement on an elastic half-space

s_{i} = \frac{q _{n e t} B ( 1 - ν ^{2} ) I _{f}}{E}

What it calculates: Settlement occurring as the load is applied, before any drainage

$q_{n e t}$: Net bearing pressure (kPa)
B: Foundation width (m)
$ν$: Poisson's ratio — 0.5 undrained, 0.2–0.3 drained (—)
I_f: Influence factor for shape and rigidity (—)
E: Young's modulus of the soil (kPa)

This assumes

Homogeneous, isotropic, linearly elastic soil — none of which soil is
A layer deep relative to the foundation, so the half-space idealisation holds

In plain terms: Settlement rises in proportion to pressure and to width, and falls in proportion to stiffness. Because E appears alone in the denominator, the answer is only ever as good as the modulus.

Rigid or flexible changes both the shape of the settlement and the number you should quote.

A flexible foundation — a thin raft, an embankment, a tank base — applies uniform pressure and settles into a dish, deepest at the centre. A rigid foundation settles uniformly and applies a non-uniform pressure instead, concentrating it at the edges.

For a 3 m square footing carrying 120 kPa on soil with E = 20 MPa, the flexible centre settles 18.3 mm and the rigid case 17.1 mm — about 7% less. Small, but the comparison matters: quoting a flexible-centre settlement against a limit written for uniform movement compares two different quantities.

Now the part that dominates everything. Halve the modulus in that example, from 20 MPa to 10 MPa, and settlement goes from 18.3 mm to 36.7 mm — an exact doubling, since E appears linearly in the denominator.

Soil stiffness is rarely known to better than a factor of two. So the honest position is that the settlement calculation has a factor-of-two uncertainty attached to it before any question of shape, rigidity or layering arises. Refining I_f while guessing E is polishing the wrong term.

Practice

A flexible 3.0 m square foundation carries a net pressure of 120 kPa on soil with E = 20 MPa and ν = 0.3. The settlement is 18.3 mm. If the modulus turns out to be 10 MPa, what is the settlement, in mm?

Check yourself

A rigid pad and a flexible raft apply the same net pressure over the same plan area on the same ground. How do they differ?

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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