Settlement¶

A pile of length \(L\), with its base at a distance \(C\) from the surface of the incompressible stratum — the bedrock or a layer so stiff that the deformations below it can be neglected —, undergoes two types of deformation under the vertical load \(P\):

\[ \rho = \rho_e + \rho_s \]

Term	What it is
\(\rho_e\)	Elastic shortening of the pile itself, as a compressed structural member, with the base held fixed
\(\rho_s\)	Soil settlement — the compression of the strata between the pile base and the incompressible stratum

As a result, the length becomes \(L - \rho_e\) and the distance to the incompressible stratum becomes \(C - \rho_s\).

Elastic shortening¶

The axial force diagram¶

The axial force is not constant along the shaft: it drops from \(P\) at the head to \(P_p\) at the base, as the load is transferred to the soil by friction.

The methodology is that of Aoki (1979), and it starts from three assumptions:

The applied load is greater than the shaft resistance and less than the bearing capacity: \(R_L < P < R\).
All shaft friction is mobilised.
The toe reaction balances what is left, and is lower than the toe resistance at failure: \(P_p = P - R_L < R_p\).

Assuming a linear variation of \(P(z)\) in each segment corresponding to a layer, the average axial force in each segment is:

[ P_1 = P - \frac{R_{L1}}{2} ] [ P_2 = P - R_{L1} - \frac{R_{L2}}{2} ] [ P_3 = P - R_{L1} - R_{L2} - \frac{R_{L3}}{2} ]

— and so on: the load already transferred above the segment, plus half of the load transferred within it.

Hooke's law¶

\[ \rho_e = \frac{1}{A \, E_c} \sum \left(P_i \, L_i\right) \]

with \(A\) the cross-sectional area of the shaft and \(E_c\) the modulus of elasticity of the concrete, assumed constant.

Modulus of elasticity¶

In the absence of a specific value:

Pile type	\(E_c\)
Precast	28 to 30 GPa
CFA, Franki and large bored pile	21 GPa
Strauss and dry bored	18 GPa

For reference: steel 210 GPa, timber around 10 GPa.

Comparison with a column

In a column, the axial force diagram is constant and equal to \(P\), and the shortening is simply \(P L / A E_c\). The pile is different because the soil progressively unloads the shaft — which is why the calculation needs the diagram, not just the head load.

Soil settlement¶

By the principle of action and reaction, the pile applies the loads \(R_{Li}\) to the soil along the shaft and the load \(P_p\) at the base. The layers between the base and the incompressible stratum deform under this loading.

The methodology is that of Aoki (1984).

Stress increase¶

Assuming a 1:2 stress spread, the increase at the mid-height of a layer of thickness \(H\), located at a vertical distance \(h\) from the point of application, is:

For the toe reaction:

\[ \Delta\sigma_p = \frac{4\,P_p}{\pi\left(D + h + \dfrac{H}{2}\right)^{2}} \]

where \(D\) is the diameter of the pile base.

For each shaft resistance term, applied at the centroid of its segment:

\[ \Delta\sigma_i = \frac{4\,R_{Li}}{\pi\left(D + h + \dfrac{H}{2}\right)^{2}} \]

where \(D\) is the diameter of the shaft. The total increase in the layer is

\[ \Delta\sigma = \Delta\sigma_p + \sum \Delta\sigma_i \]

All terms, layer by layer

The procedure is repeated for each layer between the pile base and the incompressible stratum. It is not a single stress spread over a fixed area: each layer receives the contribution of the toe and of every shaft segment, each attenuated by its own distance.

Settlement by the Theory of Elasticity¶

\[ \rho_s = \sum \left(\frac{\Delta\sigma}{E_s}\,H\right) \]

Soil deformation modulus¶

By the adapted expression of Janbu (1963):

\[ E_s = E_0 \left(\frac{\sigma_0 + \Delta\sigma}{\sigma_0}\right)^{n} \]

Symbol	Meaning
\(E_0\)	Soil modulus before the pile is installed
\(\sigma_0\)	Geostatic stress at the centre of the layer
\(n\)	Exponent that depends on the nature of the soil

\[ n = \begin{cases} 0.5 & \text{granular materials} \\ 0 & \text{hard and stiff clays} \end{cases} \]

In sand, the modulus increases with the stress increase; in clay, it does not — and that is what the exponent expresses.

For \(E_0\), Aoki (1984) considers:

Pile type	\(E_0\)
Driven	\(6 \, K \, N_{SPT}\)
CFA	\(4 \, K \, N_{SPT}\)
Bored	\(3 \, K \, N_{SPT}\)

with \(K\) the empirical coefficient of the Aoki-Velloso method, a function of the soil type.

Load × settlement curve¶

Aoki (1979) proposes predicting the curve from one point on it, using the expression of Van der Veen (1953):

\[ P = R\left(1 - e^{-a\rho}\right) \]

Once the bearing capacity \(R\) has been calculated and the settlement \(\rho\) estimated for a load \(P\), the parameter that defines the shape of the curve is obtained from:

\[ a = \frac{-\ln\left(1 - P/R\right)}{\rho} \]

The range in which the anchoring point is valid

The load used to anchor the curve must lie between the shaft resistance and half the capacity:

\[ R_L < P \le \frac{R}{2} \]

It is the same condition as the assumptions of the elastic shortening — all friction mobilised, and the toe still far from failure. Outside it, the curve no longer represents the behaviour.

The curve is not an independent prediction: it is the Van der Veen interpolation anchored at a single point. It serves to visualise the margin to failure and the expected non-linearity, not as a substitute for a load test.

Group effect¶

Pile groups always settle more than a single pile under the same load:

\[ \rho_g = \alpha \, \rho_i \]

Experimental values indicate \(\alpha\) between 1.6 and 4.0, depending on the size and shape of the group, for models of driven piles in medium-dense sand (Cintra, 1987).

Geometric formulas are not reliable

Formulas from the literature that estimate \(\alpha\) only from the group's geometric parameters are not reliable: the most important variables are the deformability of the stratum between the pile bases and the incompressible stratum and the thickness of that stratum — neither of which appears in the geometry.

There are field cases in which large groups settled the same as a single pile, because the piles were close to the incompressible stratum.

SPX applies the simplified estimate \(\rho_g = \rho_{max}\sqrt{n}\), which is one of those geometric formulas. Treat the result as an order of magnitude. For large or critical groups, the most comprehensive method is that of Aoki & Lopes (1975), which considers the interaction between all the elements.

Allowable settlement¶

For usual pile foundations, the values of Meyerhof (1976):

Soil	Allowable settlement
Sand	25 mm
Clay	50 mm

Limits of validity¶

Range of application

The method estimates immediate settlement. In saturated clays consolidation continues for years and is not covered.
The elastic shortening assumptions require \(R_L < P < R\) with all friction mobilised. For loads below \(R_L\), the axial force diagram is different and the calculation overestimates the shortening.
\(E_0\) comes from a correlation with \(N_{SPT}\), with the scatter inherent to an empirical method. Treat the result as an order of magnitude.
The position of the incompressible stratum governs \(\rho_s\): without knowing where it is, there is no way to delimit the layers that compress.
The allowable settlement is a property of the structure, not of the foundation.
Differential settlements between supports — which are what actually damage structures — require comparing the foundations with each other, and are outside the scope of the single pile analysis.
Collapsible or expansive soils, or soils subject to water table lowering, are outside the model.