Retaining structure design: Cantilever embedded wall

Code	EN 1997
Category	Geotechnics
File name	RET_1.CLC
Publishing date	7. 4. 2014
Check description	Calculation of the embedment depth of the retaining cantilever wall.

1. Objectives

The purpose of the application is to determine the embedment depth of the Cantilever embedded wall based on the moment and horizontal equilibrium conditions. Cantilever walls are often used for stabilizing of shallow excavations or as retaining walls along cuts. The important feature of the application is that it is actually a program - the required result (the depth of embedment) is not an input which is further checked, but it is an computation output based on chosen geometry, geology and partial safety factors.

2. Applied theoretical assumptions

2.1 Calculation of earth pressure coefficients and earth pressures

Coefficients of active and passive earth pressures are calculated according to the formulas (1) and (2) . These formulas are based on Rankine theory of earth pressures (Rankine, 1857).

(1)

(2)

Active and passive earth pressures are calculated according to the formulas (3) and (4).

(3)

(4)

Active, passive earth forces and relevant moment arms are determined consequently (fig.1).

Fig. 1 Acting earth pressures and forces

2.2 Determination of embedment depth

Initial embedment depth d0 is computed based on the assumption that a construction rotates around its toe. Therefore, sum of moments crated by active and passive earth forces must be zero at this point (5).

(5)

In order to fulfill also horizontal equilibrium conditions, it is necessary to increase the depth of embedment by 15% - 20% (Bond and Harris, 2008). The actual value of embedment depth increase is optional (the name of the variable is corr) according to the formula (6).

(6)

3. Implementation of design approaches according to EN 1997

It is possible to use 6 different partial factors:

Partial factor of safety for the surface load (Action),

(7)

Partial factor of safety for the active earth pressure (Action),

(8)

Partial factor of safety for the unit weight (Material),

(9)

Partial factor of safety for the cohesion (Material),

(10)

Partial factor of safety for the angle of internal friction (Material),

(11)

Partial factor of safety for the passive earth pressure (Resistance).

(12)

Appropriate combination of partial factors can be chosen for required design approach.

4. Program features and its limitations

No friction between the wall and soil is considered (basic assumption of Rankine, 1857 theory). Resultant active and passive forces are perpendicular to the construction geometry. This is an reasonable assumption for smooth walls (f.e. sheet piles in clay immediately after driving).
The retaining wall is considered as vertical and the soil surface as horizontal.
Applied distribution of active and passive earth pressure is valid for the situation when the rotation center is in the retaining wall toe. Additionally, wall is considered as ideally rigid with no deflection due to the bending.
Be aware, that effective strength parameters are utilized throughout the determination of earth pressures (3,4). Chosen calculation method is therefore only appropriate for long-term conditions (Briaud and Kim, 1998).
In order to determine the embedment depth, iteration procedure is utilized. The embedment depth is gradually increased (in 0.01m step). Relevant earth pressures, forces and moments arms are calculated per each step. Moment equilibrium equation is formulated according to (5) for the each step based on the assumption, that the center of the rotation is in the retaining wall toe.
Maximum number of steps in iteration process is 2000. This value is fixed and can not be changed. For 0.01m step in each iteration, upper theoretical limit of embedment depth is 20m. If the limit is reached, moment equilibrium condition is not met. This situation might occur if the free height of the construction is unrealistically high. Free height of the construction should not exceed approximately 4m and 6.5m in the case of soldier pile walls and sheet pile walls, respectively, based on recommendations stated in Masopust (2004).
Small numerical tolerances (f.e. 0,5kN in forces from earth pressures) might occur - this is due to a rounding in outputs.
Because of the assumptions applied during the calculation, design embedment depth is preliminary and must be further checked. For this purpose, methods based on a beam column approach or numerical methods (finite element, finite difference method) are usually adopted.

5. Comments to the usage of the program

It is possible to input either one or two soil layers. The governing parameter is the width of the layer in the Subsoil parameters table. If the width is set to 0, layer is not further taken into account.
If only one layer is required, 1st row in the Subsoil parameters table must be inputed and its width should be larger than the assumed overall height of the construction (H+D).
If two layers are required, 1st and 2nd row should be fulfilled in the Subsoil parameters table. DO NOT input width of the 1st layer larger than the assumed overall construction height (H+D). This would result in an error in runtime.
Poisson´s ratio in the Subsoil parameters table is used to determine the coefficient of earth pressure at rest K₀ (for informational purposes only).

6. Future developments

Adding third layer,
Calculation taken into account groundwater table,
Adding different theories for earth pressure calculation (f.e. theory based on Coulomb, 1777), involvement of interface friction and it´s influence on magnitude of earth pressures.