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This section is part of: Athanasopoulos, G. A. (2021) "Online Lecture Notes on Soil Mechanics", Geoengineer.org, DOI: https://doi.org/10.48246/GEOENG-EDU-001

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For the retaining wall shown in the figure below, calculate the active earth pressures and their application point and define the direction of the sliding surface. If the wall is characterized by a soil-wall friction angle of * δ=30°*, how much will the active and passive earth pressures be decreased (in %)?

__Solution__

For a smooth wall (** δ=0°**) the factor of the active impulse will be given by the equation:

(Rankine)

For ** φ=30°** we find that

The distribution of the active earth pressures on the wall is illustrated below.

For **z=0**:

For **z=6.0m**:

Therefore, we get:

The distance between the application point of * P_{α}* and the top surface of the wall will be:

The sibling surface will form an angle with respect to the horizontal surface:

If the angle of friction between the wall and the soil was **30°**, the value of * K_{Α}* is calculated using Coulomb’s general relationship:

Because ** i=0°**,

consequently:

with an angle of **30°** with respect to the horizontal.

Therefore, the active earth pressure is decreased by:

The horizontal component of the active earth pressure is:

and we have a decrease of:

Based on the above, it is concluded that that ignoring the friction between the wall - soil, is a conservative option.

For the retaining wall below, calculate the active earth pressures of the soil:

- using Culmann’s graphical method
- using Coulomb’s general relationship

__Solution__

**1)** Culmann’s graphical method is shown in the figure below:

5 trial surfaces were considered BC₁, BC₂,...,BC₅. The surfaces were selected so that they cross the earthfill surface at points C₁, C₂,..., C₅ considering a horizontal distance between them of 2.0 m. In this way, the triangles ABC₁, C₁BC₂,...,C₄BC₅ are characterized by the same area. The weight of the triangle ABC₁ is equal to 2.0 x 6.0 x 0.5 x 1.8 = 10.8T, the weight of ABC₂ is twice the weight of ABC₁, (ABC₂) = 2 (ABC₁), etc.

The angle * ψ* is equal to:

The weights of the trial triangles were estimated assuming a scale on the line BF (points 10.8, 21.6, 32.4, 43.2, 54.0). From each point we draw a parallel line with regard to the BG and we define the crosspoints with the corresponding sides of the triangles. We draw the curve that passes through these points and find the point of the curve farther from line BF in the direction of BG. Thus, we define the direction of the slip surface (BC) and the value of the active earth pressure which is * P_{α} = 8.8 T/m²*.

**2)** Coulomb’s relationship for * φ=38°*,

and the value of the active earth pressure will be:

The values of * P_{α}* that are estimated based on the graphical method, are less than 2% different than those resulting from the analytical solution.

The vertical retaining wall of the figure below has a height of *2.7m*, and it is pulled towards the sandy embankment through an anchor. For the embankment material, we have ** γ_{t}=1.9T/m²** and

_{}

_{}

__Solution__

Since we have a vertical wall and a horizontal embankment the diagrams in Section 7.2 can be used. For ** φ=30°** and

and it will form an angle ** 15°** with respect to the vertical. As a result, the horizontal component of the passive resistance will be:

Also, we could apply Coulomb’s general relationship for the passive earth pressure coefficient by using * i= 0°* and

We observe that Coulomb’s relationship, even though it is using a horizontal sliding surface, gives a value of Kₚ only 0.6% smaller than the value estimated by using fig. 7-2, in which curved sliding surfaces are considered.

For the diaphragm wall of the following figure, calculate the depth * d* so that the factor of safety against overturning is

**Solution**

We assume that the wall will rotate around a point O near its bottom end. From point O and below, passive earth pressures are developed and their resultant is * R*, acting on point O. We consider the moments of forces with regard to point O. The moment of the active earth pressures,

The active earth pressure coefficient is:

(assuming **δ*** =0*).

The active earth pressure is:

and the overturning moment:

The passive earth pressure coefficient is:

The passive resistance is:

And the resisting moment:

For ** F’=2** the above equation becomes:

So, since *M _{E}*

Following common practice, the embedment depth is increased by about 20% so that the passive resistance is fully developed. In this way, the final value of * d* is

Check the stability of the cantilever wall of the figure below.

Given:

- unit weight of soil:
,**γ**_{t}=1.70 T/m^{3} - angle of internal friction:
**φ**,**=40°** - angle of friction between wall base and soil:
**φ**_{B}=30° - unit weight of concrete:
**γ=2.35 Τ/m**^{3}

**Solution**

For this type of retaining wall, we assume that the soil between the wall and line AB is part of the retaining wall system. In this way, the active earth pressures are calculated over the imaginary surface ΑΒ by using Rankine’s relationship. The factor * K_{A}* is:

__Active earth pressures__

- From surcharge:

- From the earthfill:

- Thus, in total:

__Vertical forces__

- wall weight,
: 5.0 x 0.3 x 2.35 =*W*_{b1}**3.53 T** - foundation weight,
: 5.0 x 0.4 x 2.35 =**W**_{b2}**2.82 T** - soil weight,
: 5.0 x 1.75 x 1.70 =*W*_{b}**14.88 T** - surcharge,
: 1.75 x 4.0 =*W***7.00 T** - sum,
=*N***28.23 T**

__Check against slip__

- Factor of Safety:

__Check against overturning__

- Driving moment (with respect to point O):

- Resisting moment:

- Factor of Safety:

For the retaining wall of the figure below:

- Calculate the active earth pressure of the soil acting on the wall
- If a uniform surcharge of
is applied on the surface of the backfill, what will be the new active earth pressure?**q**_{s}= 10 T/m² - What is the minimum value of
so that tensile stresses are not developed on the wall.**q**_{s}

The soil and wall characteristics are shown in the figure.

**Solution**

**1)** As the wall is smooth (**δ*** =0*), we can use Rankine’s relationship. The distribution of the active stresses on the wall is given by the equation:

Thus:

the depth in which **σ _{h}**

The area of the triangle of the compressive stresses is:

This is the value of the active earth pressure on the wall. The point at which * P_{A}* is applied, is at:

above the base of the wall.

**2)** When the uniform load * q_{s}* is applied, we will have:

Thus:

The active earth pressure, now, is computed as:

The point of application of * P_{A}* is at distance

**3)** In this case, it is evident that if the value of * σ_{h}* is equal to zero at the top of the wall, then tensile stresses are not developed on the wall. The value of

for ** z = 0**:

For this value of * q_{s}*, the horizontal stress at the base of the wall is:

The value of the active earth pressure in this case is **P _{A}**

For the retaining wall of the following figure: calculate the active earth pressure acting on the wall. The characteristics of the soil and the wall are shown on the figure.

__Solution__

In this case, the active earth pressures acting on the wall are estimated considering first the effective stresses (for depth > 2.5 m) and then adding the pore water pressures. The active earth pressure coefficient, * K_{A}*, will be the same above and below the groundwater table, because

**φ = 30°***δ =30°*-
.*K*_{A}= 0.29

So the active earth pressure distribution on the wall will have the following form:

The active earth pressures are:

and will act at an angle of 30° with respect to the horizontal. The point of application of * P_{A}* is located at distance from the base of the wall:

The horizontal component of * P_{A}* is:

The resultant of the hydrostatic pressures is:

and it is applied at a distance from the base of the wall:

Therefore, we note that the presence of the groundwater table results in large pressures and thus water accumulation behind the wall should be avoided.

Chapter 7: Earth Pressures7.2. Coulomb Theory (1776)7.1. Rankine Theory (1857)7.3. Graphical Methods7.4. Types of Retaining StructuresOnline Lecture Notes on Soil Mechanics

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