CANTILEVER RETAINING WALL

What this page computes

The calculator verifies the global stability and ground bearing pressure of a reinforced-concrete cantilever retaining wall, to EN 1997 (geotechnical verifications) and EN 1990 (combinations of actions). It covers the three ultimate-limit-state checks at foundation level: sliding, overturning and contact pressure on the ground. The model follows worked example 2.5 of the Eurocode 2 Worked Examples.

Earth pressure (Rankine)

Active pressure is taken on the virtual vertical plane through the back edge of the heel, over the total height \(H = H_{stem} + t_{footing}\):

\(K_a = \tan^2\!\left(45^\circ - \frac{\varphi}{2}\right), \qquad H_{k,terr} = \tfrac{1}{2}\,K_a\,\gamma\,H^2, \qquad H_{k,q} = K_a\,q\,H\)

The earth pressure is a triangle with its resultant at \(H/3\) above the base; the surcharge pressure is a rectangle with its resultant at \(H/2\).

The three verifications

Sliding. The design horizontal force (destabilizing permanent \(\gamma_G = 1.1\), variable \(\gamma_Q = 1.5\)) is compared with the friction mobilised at the base, from the favourable permanent weights (\(\gamma_G = 0.9\)):

\(H_{Ed} = 1.1\,H_{k,terr} + 1.5\,H_{k,q}, \qquad R_d = \mu\,(0.9)\,(G_{stem} + G_{footing} + G_{terr})\)

Overturning about the toe: the destabilizing moment from the thrusts is compared with the stabilizing moment from the weights, each with its lever arm about the toe.

Bearing pressure (STR/GEO, Set B — Approach 2): moments are taken about the centre of the footing, and the eccentricity \(e = M_{tot}/N_{tot}\) gives the contact pressure

\(\sigma_{max} = \frac{N_{tot}}{B} + \frac{6\,M_{tot}}{B^2} \quad (e \le B/6), \qquad \sigma_{max} = \frac{2\,N_{tot}}{3\,(B/2 - e)} \quad (e > B/6)\)

The four partial-factor combinations (self-weight of concrete and of soil, each with \(\gamma_G \in \{1.0,\,1.35\}\)) are evaluated automatically, and the page reports the combination giving the largest \(\sigma_{max}\).

Design pressure vs. allowable pressure

The \(\sigma_{max}\) from the combinations above is a design value: the actions are already factored (\(1.35\) / \(1.5\)). It is correctly compared with a design bearing resistance \(R_d\) (ultimate resistance divided by \(\gamma_R\)).

The allowable pressure usually given by the geotechnical report, however, is a service value that already contains a safety factor (often ~3) against the ultimate capacity. Comparing it with the design \(\sigma_{max}\) would count the safety twice and produce a needlessly wide footing.

That is why the page also reports the characteristic pressure (unfactored), computed from the same actions without partial factors — that is the one to compare with an allowable pressure. For the default geometry the difference is significant: \(\sigma_{max} = 99.8\) kN/m² design versus \(\sigma_{k,max} = 72.3\) kN/m² characteristic.

In short: design value ↔ design \(R_d\), characteristic value ↔ allowable pressure. Do not cross them.

Conventional pressure method (Romanian practice)

If you work with the conventional pressure \(p_{conv}\) from the geotechnical report (STAS 3300/2-85, carried into NP 112), the check under eccentric loading has two criteria, not one — enable the dedicated switch on the page:

\(p_{med} \le p_{conv} \qquad \text{and} \qquad p_{max} \le 1.2\,p_{conv}\)

The 20% increase is allowed because the edge pressure peak is local, over a narrow strip; the average over the whole base stays limited to \(p_{conv}\). The second criterion alone is therefore not sufficient. Both are checked on characteristic (service) values, not factored ones.

For the special (seismic) combination the limits rise to \(p_{med} \le 1.2\,p_{conv}\) and \(p_{max} \le 1.4\,p_{conv}\) — this page has no seismic combination, so it does not compute them. Check the current edition of NP 112 for the values applicable to you.

Why the threshold is 1.0 and not 1.5

The checks above work in limit states: safety is introduced through partial factors and the criterion becomes \(E_d \le R_d\) — that is, a ratio ≥ 1.0. Requiring 1.5 on top of already factored values would count the safety twice.

Overturning and sliding are not the same limit state

The distinction is easy to lose, and losing it makes the sliding check about 12% wider than it should be.

Overturning is an EQU limit state: loss of equilibrium of a rigid body, in which the strength of the ground is not what provides resistance — the wall rotates about the toe, it does not shear anything. Factors: \(\gamma_{G,dst} = 1.10\), \(\gamma_{G,stb} = 0.90\), \(\gamma_Q = 1.50\).

Sliding is a GEO limit state: everything that holds the wall in place is the friction mobilised under the footing, which is precisely the strength of the ground. The definition of EQU excludes it explicitly. It is verified with Design Approach 2 (A1 + M1 + R2) — the same one the page already uses for contact pressure: actions factored \(1.35\) / \(1.50\), stabilising weights at \(1.00\), and resistances divided by the factors of EN 1997-1 Table A.13 — \(\gamma_{R;h} = 1.10\) for base friction and \(\gamma_{R;e} = 1.40\) for passive earth resistance.

\(R_d = \frac{\mu \sum G_k}{\gamma_{R;h}} + \frac{k_{mob} E_{p,k}}{\gamma_{R;e}} \qquad H_{Ed} = 1.35\,H_{k,terr} + 1.50\,H_{k,q}\)

Equivalence with the classical FS

The 1.5 many engineers know comes from the global safety factor method that predates the Eurocodes and works on characteristic (unfactored) values. The page also reports this classical FS, as an informative row below the normative one — but the verdict (the badge) stays with the Eurocode check.

The equivalence is not fixed; it depends both on the check and on the permanent/variable mix:

Sliding lands essentially on top of the classical 1.5 rule — the two methods agree, which is reassuring. Overturning stays more permissive, but it rarely governs: sliding and eccentricity usually decide.

Soil in front and passive resistance

If there is fill in front of the wall, it can optionally be taken into account. You enter the height \(h_f\) of the soil above the top of the footing, and the embedment depth over which passive pressure acts is \(D = h_f + t_{footing}\), measured from the front ground surface down to the base of the footing:

\(K_p = \tan^2\!\left(45^\circ + \frac{\varphi}{2}\right), \qquad E_{p,k} = \tfrac{1}{2}\,K_p\,\gamma\,D^2\)

The contribution comes through three routes:

• Sliding — the passive resistance adds directly to the base friction; this is the largest gain, because \(K_p = 1/K_a\) and it grows with the square of the depth.
• Overturning — the passive force acts at \(D/3\) above the base and opposes the rotation, while the soil resting on the short toe projection adds a stabilizing moment with lever arm \(b_{toe}/2\).
• Bearing pressure — the weight of the front soil enters as vertical load; the passive resistance is ignored here, as a conservative assumption.

Mind the mobilisation. Full passive pressure requires large movements, and the fill in front may later be excavated (services, drains) or eroded. The page therefore applies a mobilisation factor (0.5 by default) on top of the favourable permanent factor of 0.9. If the soil in front is not guaranteed for the design life, set the factor to 0.

Reinforcement design (EN 1992-1-1)

The page also sizes the main bending reinforcement per metre width, at three critical sections:

• Stem — vertical cantilever fixed at the base of the stem, loaded by the active thrust over its own height \(H_{stem}\); tension on the earth face.
• Toe (front of footing) — cantilever loaded upward by the (factored) ground pressure, relieved by self-weight; tension at the bottom.
• Heel (back of footing) — cantilever loaded downward by the soil column, surcharge and self-weight, relieved by the ground pressure; tension at the top.

The design moment comes from the factored actions (permanent \(1.35\), variable \(1.5\)), and the steel area is found in the singly-reinforced regime:

\(K = \frac{M_{Ed}}{b\,d^2 f_{ck}}, \qquad z = d\left(0.5 + \sqrt{0.25 - \tfrac{K}{1.134}}\right) \le 0.95\,d, \qquad A_s = \frac{M_{Ed}}{f_{yd}\,z}\)

Minimum reinforcement \(A_{s,min} = \max\!\left(0.26\,\tfrac{f_{ctm}}{f_{yk}};\,0.0013\right) b\,d\) is checked, and the page proposes a practical bar Ø / spacing arrangement covering \(\max(A_{s,req};\,A_{s,min})\). If \(K > K'\) the section is flagged as too small (increase the thickness).

Why eccentricity matters

As long as the resultant stays within the middle third (\(e \le B/6\)), the whole footing is in compression and the pressure diagram is trapezoidal. Once eccentricity exceeds \(B/6\), a corner lifts off (⚠), the pressure redistributes triangularly over the remaining contact area and \(\sigma_{max}\) rises sharply — a situation to avoid by widening the footing or enlarging the heel.

Input data

• Geometry: stem height and thickness, footing thickness, toe and heel projections. Footing width \(B\) and total height \(H\) are computed automatically.
• Soil: unit weight \(\gamma\), angle of internal friction \(\varphi\) (giving \(K_a\)), footing–soil friction coefficient \(\mu = \tan\delta\).
• Surcharge \(q\), optionally over the whole embankment (which also loads the heel vertically).
• Soil in front (optional): height \(h_f\) above the footing and the passive-resistance mobilisation factor.
• Design bearing resistance of the ground, for the pressure check.

Assumptions and limitations

• Homogeneous, dry soil (dry effective thrust). For a water table see earth pressure on a wall.
• Rankine active pressure, level backfill, no friction on the back of the stem (\(\delta = 0\) on the virtual plane).
• Designs the main bending reinforcement (stem, toe, heel); it does not cover footing shear, secondary / distribution steel or crack-width — those are treated separately.
• Partial factors follow Design Approach 2; check the National Annex for the applicable values.

Related calculations

• Earth pressure on a wall — lateral pressure diagram, with water and surcharge
• Ground pressures (Winkler) — contact distribution under a rigid footing, biaxial
• Isolated footing (full design) — pressures + reinforcement + punching + report + DXF drawing