In designing retaining structures, the most critical thing to know is how much horizontal (lateral) pressure the retained soil will exert on the retaining structure, such as a retaining wall. In previous post, we learned how to calculate the vertical stresses soils. Here, we must somehow derive horizontal pressure from it.

To do this, we can use the parameter called “Coefficient of Lateral Earth Pressure”. This is the key component when we want to know lateral earth pressure, and in turn designing our lateral support systems, such as retaining walls.

Lateral Earth Pressure Coefficient (denoted with “K”) which is the ration of horizontal to vertical stress can be calculated as:

K = Horizontal Earth Pressure / Vertical Earth Pressure

In other words,

To visualize this K value better, let’s give an example.
Fill a bucket with sand. And step on it. It will carry your weight. Now, turn
the bucket upside down, and remove the bucket from top *(let’s assume you made the sand wet a little, so the sand can stand by
itself just by the surface attraction forces water caused)*. Now try
stepping on it again. It will fail immediately. It is the same sand. Same
shape, same volume. But what happened here is, for the same K value, when you
removed the bucket which was restraining it horizontally, the horizontal stress
resisting capacity decreased dramatically, as there is nothing retaining it now,
so the corresponding vertical stress carrying capacity also reduced greatly, as
K remained the same for that soil. This is the way K works. It is the ratio of
horizontal to vertical stress. This example also shows another point that restraining
earth horizontally, not only prevents it from falling towards the side, but it
also greatly increases a soil’s vertical load carrying capacity.

As we have seen in previous post, the horizontal (lateral) pressure of a fluid at a certain depth, is equal to its vertical pressure at the same depth. *(From now, for simplicity, we will say liquids, specifically water, instead of fluids, although same is true for all fluids -liquids and gasses)*. So the water pressure at a certain depth, is the same in all directions, whether it is horizontal or vertical or diagonal.

Now let’s see the case for soils. Unlike water, for soil, the pressure does not act equally in all directions at a certain depth. In other words, the lateral pressure of a soil at a certain depth, is NOT equal to its vertical pressure at that depth.

As we have seen,

Vertical pressure of soil = γ_{soil}
x h

This is the vertical pressure, calculated exactly in same way as fluid pressure, which you saw when we talked about total and effective stresses at a certain depth.

So, now since we know the vertical pressure, we can derive the lateral pressure from it, using lateral earth pressure coefficient K.

Lateral pressure of soil at depth h = Vertical pressure depth h x Lateral earth pressure coefficient.

So,

Lateral pressure of soil at depth h = γ_{soil} x
h x K

So it means, that: K value determines how much of the vertical pressure is converted into lateral pressure.

As you may guess, this is a very important information to know for a soil.

So for example,

if K is smaller than 1, then it means, only some of the vertical pressure is converted to lateral pressure,

if K is equal to 1, the lateral pressure and vertical pressure is the same (such as in fluids, that we talked about)

if K value is greater than 1, then it means, the lateral pressure will be greater than vertical pressure.

All these different cases have different effects when designing retaining structures.

We defined the K value as:

K = Lateral Earth Pressure / Vertical Earth Pressure

But we also said, in order to know lateral earth pressure, we must know the value of K first, which seems like a paradox…

Here is how it works…

We gave this

K = Lateral Earth Pressure / Vertical Earth Pressure

formula in the beginning, to tell you about the logic of K, which is correct, but actually the K value is calculated from something called “angle of internal friction”, which we will cover in the coming subsection.

This angle, denoted as Ø, is one of the most important properties of a soil, when talking about soil strength. It is basically a measure of how much internal friction a soil can generate, when load is applied on it. The more internal friction the soil can generate inside, in other words, the larger this angle Ø, the stronger the soil will be to support loads.

For example, angle of internal friction of water is zero.
That is why, you can not walk on water. Because its particles slide on one
another without any internal frictional resistance. As we said, we will cover
angle of internal friction in more detail in next subsection, but gave enough
definition for our purpose in this section. *(Our
not being able to walk on water is also because water has no cohesion between
particles but lets disregard it for our purpose for now, which you will
understand more when we cover the other subsection called soil cohesion)*

We have three types of K values, and they are:

- Active earth pressure coefficient, (K
_{a}) - At rest earth pressure coefficient, (K
_{o}) - Passive earth pressure coefficient (K
_{p})

Do not memorize it, as you will understand it below, but the
values of K from largest to smallest go as: K_{p > }K_{o > }K_{a. }In other words, passive coefficient is the
larger than at rest coefficient, and then the smallest one is active
coefficient…

Now let’s try to understand what each K value means…

K_{a}:
When the retaining structure tends to move away from the retained earth,
*(as the earth pushes the wall, so earth
is active here)* this is called active condition, and we must use the active
earth pressure coefficient, K_{a.}

K_{o}: Here the wall is too heavy to move at all, or it is restrained by a slab, or braced by a suitable means. In this case, we use the at rest pressure, K_{o. }The wall and soil do not move here._{ }

K_{p}:
When the retaining structure tends to move towards the earth, by pushing
the soil, this is called the passive condition of soil, *(as in being pushed, so, passive)* and we must use the passive earth
pressure coefficient K_{p. }For
a soil to develop full passive resistance, some deformation must have taken
place already, in other words, the retaining wall must already have pushed the
soil, which is a step that is easy to miss.

At-rest coefficient is also used to know the original stress state in the soil, prior to any excavation or modification whatsoever. By initial stress state, we mean the ratio of horizontal to vertical stress, in the original, untouched soil. Or for example, after compaction, if a soil is compacted too much, it may have large horizontal stress locked inside of it, even larger than vertical stress. Designing a retaining structure near such soil, needs to consider that locked in horizontal stress.

The formulas for calculating these K values are below. How these formulas are derived is beyond this book’s scope, but interested readers can refer to any basic soil mechanics text for it. The value, Ø is the angle of internal friction here, as we mentioned above, which we will cover in more detail in next section.

In actual design, it must be kept in mind that choosing the proper K value, for a particular retaining structure design is not a complete exact science, as K is based on angle of internal friction, which is actually found by interpretation of test results, and it is reliable to the degree that the results are interpreted correctly. In addition, earth is not a homogeneous factory made material in controlled environment, therefore this is a place where engineer judgment comes into play. There are however widely accepted, established range of values for different kinds of situations and a huge amount of past data available, which makes the job of the designing engineers easier.

In the next post of this series, we see an example of solving of forces of a simple retaining wall.

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