Surcharges on a retaining wall

What this page computes

The lateral pressure on a retaining wall from loads applied at the ground surface behind it: a uniform surcharge, a strip (a road, a stockpile), a line load (a wall, a crane rail) or a point load (a column base, a scaffold leg). It gives the pressure diagram, the resultant, its point of application and the moment at the base.

The two models are not the same model

This is the page's fundamental division, and the reason it exists separately from the earth pressure calculation.

A uniform surcharge over the whole surface is treated by limit equilibrium: the load above is equivalent to an extra layer of soil, and

\(\sigma_h = K \cdot q\)

constant over the full depth. It depends on \(\varphi\), through \(K\).

Local loads — strip, line, point — cannot be treated that way. They do not produce a constant pressure but one that rises from zero, peaks at an intermediate depth and dies away below. They are computed with elastic solutions (Boussinesq for the point load, Flamant for the line load, their integral for the strip).

The consequence that surprises people: the elastic solutions do not contain \(\varphi\) at all. A weak soil and a stiff one receive exactly the same pressure from this source. That is not a simplification made here, it is a property of the model — elasticity theory knows nothing about shear strength.

Why the diagram is not triangular

The habit from earth pressure is a triangular diagram with the maximum at the base. From local surcharges the maximum is at mid-height. The figure on the page shows this directly.

It matters in practice: if the wall is designed assuming the governing section is at the base, a nearby surface load can make a mid-height section governing instead — where the reinforcement is usually lighter. The page states the depth of the maximum explicitly.

The factor of 2

The Boussinesq and Flamant solutions are for the free elastic half-space — with no wall. A rigid, unyielding wall reflects the deformation, and current practice doubles the resulting pressure.

Here the factor is a visible switch, not a constant buried in coefficients. The reason: for a flexible wall that can yield (a thin sheet pile), the doubling stays conservative but may be excessive — and dropping it should be a deliberate decision, not an oversight.

For a uniform surcharge the factor does not apply: that is not an elastic solution but limit equilibrium with \(K\), which already presupposes the wall.

The near-wall correction

For loads very close to the wall the elastic solution overestimates. On the basis of Spangler's tests, Terzaghi proposed that below \(m = a/H = 0.4\) the ratio \(m\) be frozen at 0.4.

The published form for the line load is

\(\sigma_h = \frac{q}{H} \cdot \frac{0.203\,n}{(0.16 + n^2)^2}\)

and it can be shown to be exactly the theoretical form with \(m = 0.4\): the constant 0.203 is the rounding of \(4 \cdot 0.4^2/\pi = 0.20372\), and \(0.16 = 0.4^2\). The ratio between the two expressions is 0.99647 for any \(n\) — a constant, not an approximation that degrades. That check is in a test.

This is why the correction is implemented for the line load.

What it does not cover

• The empirical correction for the point load. A published form exists for this case too, but its structure differs from the frozen theoretical one (\(n^2\) and power 3, against \(n\) and power 5/2), so it is not the theoretical form with \(m\) locked and could not be verified internally the way the line load was. It is not implemented. In the zone \(m \le 0.4\) the page shows the elastic value — on the conservative side — and says so explicitly.
• Superposition with earth pressure. The page gives only the surcharge part; it adds on top of the earth pressure diagram computed separately.
• Arching and the real stiffness of the wall. The model is either a rigid wall or a free half-space, with no interaction.
• Dynamic loads and moving-vehicle loads, which require impact factors.

The strip solution is verified through its limits: beneath the centre of a very wide strip it gives \(\sigma_x = q\), and a narrow strip converges to the equivalent line load, with the error falling from \(10^{-3}\) to \(10^{-9}\) as the width decreases.

Related calculations

• Earth pressure — \(K_a\), \(K_0\), \(K_p\), to which the result here is added.
• Retaining wall — the stability checks, which the base moment feeds.
• Stress bulb — the same Boussinesq theory, but for the vertical stress under a footing.