Cantilever sheet pile — embedment depth

The depth required below the excavation level, from moment equilibrium about the toe, in cohesionless soil.

The excavation

H — excavation depth [m] q — surcharge on the retained soil [kPa]

The soil

γ' — effective unit weight [kN/m³]

φ' — angle of internal friction [°]

Safety

FS on K_p Usually 1.5–2.0. It is applied to the passive coefficient, not to the resulting depth: since the relationship between D and moment is non-linear, the two are not equivalent. Maximum D searched [m] The upper bound of the search range. If equilibrium cannot be reached within it, an error is reported rather than silently capping the result.

▸ Fill in the data on the left and press Calculate

Tool information

What this page computes

The page determines the embedment depth required below the excavation level for a cantilever sheet pile wall in cohesionless soil, together with the maximum bending moment in the wall and the total sheet pile length.

"Cantilever" means the wall has no anchor and no prop. The only thing holding it is the soil in front of it, below the excavation level.

Why the depth comes out so large

For a cantilever sheet pile, $D$ is on the order of $1.5$–$2 \times H$. This is not a calculation error but a direct consequence of the structural scheme: the moment from the active pressure, which grows with the square of the retained height, has to be balanced solely by the rotational restraint of the soil in front, and the lever arm of that resistance is short, because it develops right next to the point of rotation.

Cantilevers are therefore used for shallow excavations. Beyond about 5 m they become uneconomical: the embedment depth and the moment grow faster than linearly with $H$, and at some point an anchored or propped wall is cheaper even with the cost of the anchor included. The page flags the threshold.

Moment equilibrium

The depth follows from moment equilibrium about the toe of the wall. On the retained side, active pressure acts over the full length $L = H + D$: a triangular diagram from the self-weight of the soil plus a rectangular one from the surcharge. On the excavated side, passive pressure acts over the depth $D$ only.

$$M_p(D) = \frac{1}{2} K_{p,d}, \gamma' D^{2 \cdot \frac{3}
\qquad
M_a(D) = \frac{1}{2} K_a, \gamma' L}2 \cdot \frac{3} + K_a, q, L \cdot \frac{2}$$

The equation $M_p(D) = M_a(D)$ is of high order in $D$ and has no useful closed form, so it is solved numerically by bisection. That is not a matter of convenience: over the physical range the net moment is monotonically increasing in $D$ — the passive contribution grows as $D^3$ about the toe, the active one only as $L^2$ — so the solution is unique and bisection is guaranteed to converge.

If equilibrium cannot be reached within the maximum depth searched, an error is reported rather than silently capping the result at the limit of the range.

The factor of safety goes on $K_p$, not at the end

This is where two seemingly equivalent approaches are not:

reduce the passive coefficient, $K_{p,d} = K_p / FS$, and only then write equilibrium;
write equilibrium with the full $K_p$, then increase the resulting depth by the factor of safety.

Because the relationship between $D$ and moment is non-linear, the two give different answers. The page applies the first, which is standard practice, and says so explicitly in the results. Usual values for $FS$ are 1.5–2.0.

The physical reason safety is placed on the passive side: passive resistance is the uncertain part of the scheme. It only mobilises at large displacements, it can be compromised by accidental over-excavation in front of the wall, and it depends on a $\varphi$ that is known less precisely than the loading.

The maximum moment

The maximum bending moment in the wall occurs where the shear force vanishes — below the excavation level, at the depth where the accumulated passive pressure equals the accumulated active pressure above it. The page locates this level numerically and reports both the moment and its position, measured from the top of the wall. The sheet pile section follows from $M_{max}$, by checking $M_{max} \le W_{pl} f_y / \gamma_{M0}$.

What the calculation does not cover

The assumptions are those of the simple scheme: cohesionless, homogeneous soil, no groundwater. Outside the scope:

verification of the sheet pile section — the page gives $M_{max}$, choosing the profile is left to the user;
wall displacements, which for a cantilever are often the governing criterion rather than strength;
base heave and hydraulic failure, where water is present;
layered soil and cohesion, which change the shape of the diagrams;
anchored or propped walls, which have a different structural scheme.

The method is standard (Das, chapter 14), but it has not been checked against a published worked example, unlike earth pressure and bearing capacity, which have external benchmarks on this platform. The internal tests cover equilibrium, monotonicity of the solution and orders of magnitude. Read the result as preliminary sizing.