Example for Lateral Earth Pressure

Now let’s give a simple example to wrap things up. This example will use many things we have learned so far, not only in soil mechanics, but structural basics section too.

Solving retaining wall forces as a result of active and passive earth pressure

We have a gravity retaining wall here, which is a type of retaining wall that resists overturning effect of retained soil by its weight.

In the figure above, you first see the wall and the soil it retains.

The figure in the middle shows the free body diagram of the wall, drawn with same principles as we learned in structural basics section.

Note here that the Fp, passive force, and Fa, active force is a triangular distribution, which is only logical because pressure increases with depth.

Before we said,  

Lateral pressure of soil at depth h = γsoil x h x K

So Fa , the active earth pressure, the the force of the higher side soil, would be equal to the area of the triangle:

Fa = 1/2 x h x lateral pressure at depth h = 1/2 x h x γsoil x h x K


And as seen in third figure on the right, for simplification we can think of them as a single force acting on the center of gravity of the triangle, that is why they act 1/3 h from the bottom.

When taking h1 to calculate active earth pressure, it is from the soil top surface, not from the top of the wall. But when calculating the weight of the wall, which acts at the wall center of gravity, the whole wall height is taken.

We also see other forces in the free body diagram, such as,

  • The weight of the wall for instance, shown as W, can be thought to act as a single force acting on the center of gravity of the dam,
  • The friction force generated at the bottom of the wall, shown as V, as a reaction force to the soil’s pushing of the wall horizontally from right, the upwards vertical reaction force from the bottom of the wall,
  • The weight of the soil the walls over the left side of the wall, between points A and D, which is usually ignored in such cases like this, as it is a conservative simplification anyway,
  • The normal force (reaction force) that acts from the bottom of the dam, as a soil reaction force. Note that this is not uniformly rectangular, but more on the left side as a trapezoid, as the retained soil is trying to overturn the wall, and so the left toe of the wall presses more on the soil, and hence the trapezoidal reaction, which is more on the left side. This is also simplified as one normal force in the third figure, shown as N. This normal force acts through the center of trapezoid, which is somewhere in between points a and B, closer to A.

Note that all forces shown here, are forces per one unit length of wall. In other words, they are 1 feet or 1 meter towards inside of the page you are looking at, as 3rd dimension. You can see for yourself if you look at units. For example we multiplied lateral pressure of soil at depth h = γsoil x h x K, which has a unit of Force / area, by only 1/2 h, to find force, which makes a unit of Force / length. So, all these forces here are force / length, which is taken towards z axis, if the paper plane you are looking at right now is x-y plane. But that is enough for most hand calculations. as we can design the cross section and just apply over the whole length of the wall.

To calculate the stability of the wall, just like we did in structural section, we must make sure that the sum of all forces acting on the wall in horizontal and vertical directions must be zero, and also the net moment acting on the wall is zero, which can be taken with respect to any point. For simplicity, for these types of problems, we take moment (calculate moments of all forces) with respect to point A in the figure and make sure that the wall will not overturn. So rewriting all these, we must make sure that:

Sum Fx = 0

Sum Fy = 0

Sum M = 0

Although we will not give a numerical example, here we showed how the forces act on a wall, and the free body diagram of the wall, in order to check the overall stability of the wall. Of course a retaining wall design involves much more things than just the overall stability check, which include considerations such as for the adequacy of soil below the wall, the strength of the wall material and cross section itself, earthquake effects… Also often times a surcharge load, such as a load of a street with vehicles or building may act on the higher side of the wall, which we did not show here to keep the forces simple. In some cases water may exist, which is a very unwanted situation and adds greatly to the load that wall must carry, and it be drained by weep holes in the wall, or must be prevented from entering into the system in the first place.

During design, we must apply safety factors, in other words, we can not just be satisfied, if the stability is barely ok. We apply safety factors of at least 1.5 or more, depending on the circumstances and specific problem, whether it is for overturning or sliding.  

Finally, we have just mentioned angle of internal friction in this section, which we will cover more in the next section too. One practical use of angle of internal friction can be seen below. The active and passive soil regions (called Rankine active and passive zones) of soil forms an angle depending on internal friction angle of soil, as seen in the figure below:

Active zone and passive zone of soil in front and behind of retaining wall

As can be seen here, angle of internal friction of soil affects the angle of wedges which they form with horizontal. The wedges shown here are for ideal conditions, for a smooth wall, meaning, ignoring friction between the wall and soil. If the friction is not ignored, the angles will be slightly different. If the soil reaches this state, it means the soil is mobilized.

Also, as noted in the figure, if wall anchors are needed, these anchors must extend beyond the active zone, meaning that steep sloped line that forms the border of active zone of soil. Because all of the soil within that wedge is assumed to be unstable and moving. An anchor must extend beyond that sloped wedge line, to hold to the firm, not moving soil beyond, and when calculating its contribution to overall stability, only the anchor’s length that falls beyond the wedge in stable soil is taken into account. The apostrophe sign near Ø, indicates that the angle comes from not total but effective strength measurements, that is why we write it as: Ø’. In many cases in soil mechanics, we deal with effective stress parameters, and not total, as the effective stress (soil skeleton stress) is what matters for the strength of soil, as we mentioned before.

In the next post of this series, we will discuss “Angle of Internal Friction”

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