Total Stress, Effective Stress, Pore Water Pressure

Before we move on here with soils, let’s talk about fluid pressure first.

The pressure of any fluid (liquid or gas), such as water, increases constantly with depth such as:

Pressure = Unit Weight of the Material x Depth

Written in correct terms:

Pressure = γfluid x d,

where,

γ: the unit weight of the material,

and,

d: depth in which we are interested. We can call it h for height too.

So, it means, the pressure of a fluid, increases directly proportional to depth, and if we want to show it as a graph, it will look like this:

Fluid pressure with depth

And, again for a fluid, the pressure in all directions is the same. So, at a certain depth, the pressure would be equally acting downwards, upwards, or towards side, all being equal to γ x h, as seen in figure.

Now we can proceed with soils…

Total stress in soil is made of the stress between solid particles, plus the pore water pressure.

Diagram showing Total Stress, Effective Stress and Pore Water Pressure in soils

As shown in figure above,

Total stress in saturated soil = Stress on soil skeleton + Pore water pressure in the soil

This is one the few most important relationships in soil mechanics. Soil skeleton’s stress is called effective stress (σ’). That is the stress carried by soil grains contact, which is what matters for strength of soil.  

Writing this again in proper symbols,

σT = σ’ + u 

As you have seen before, the Greek symbol sigma σ is used for stress. So here

σis total stress,

σ’ is stress on soil skeleton, which is effective stress. (We put an apostrophe sign for effective stress)

u is pore water pressure

Important note here, is that if the soil was not saturated, the pore pressure can not develop and therefore will be zero, which means, all total stress will be equal to soil skeleton (effective) stress.

So in a non saturated soil, where water pressure is zero, total stress would be equal to effective stress as:

σT = σ’  

This would mean, all of the load imposed on that soil is entirely carried by soil skeleton (which is what is supposed to happen in the long run anyway, as water will sooner or later flow away). So the real strength of a soil comes from soil skeleton, and not the pore water. The more water pressure in a saturated soil, the less stress will be carried by the soil skeleton and the weaker soil, from the equation above.

In the figure above you also see how the stress is calculated.

The stress at a certain depth is found by:

σ =γ.h

Here γ is the unit weight of soil. It has units of pounds per cubic feet (lbs/ft3) or Newtons per cubic meter (N/m3) or their conversions.

And, h is the height.

So as you see, when you multiply unit weight and height, you again reach stress unit, which is lbs/ft2 or N/m2 

Unit weight can be just like above for the soil’s wet or dry but not saturated state, which is for the portion above the water table.

γsat is the unit weight of saturated soil, which is below the water table as you see in the figure. So for below the water table, for total stress we must use that.

γ’ is the effective unit weight, which is also called the buoyant or submerged unit weight, which considers the effect of pore water pressure. It is as if you are trying to submerge an object into the water and water exerts uplift force on it, (that is how water carries ships).

It is found as

 γ’ = γsat – γwater.

That is the one we use for calculating effective stress as you see in the figure.

And  for calculating u, pore water pressure, we simply multiply unit weight of water by water depth at the desired level.

So what do all these mean in practice?

One implication is settlement and compaction of soil. As we mentioned above, load on soil is only carried by soil skeleton. Water pressure contributes nothing to load carrying capacity. If the load on soil skeleton (effective stress) is too much, the soil will fail and settle, settle further, until the soil skeleton is dense enough,  which means now more particles contact each other, so that it can carry the new load. Now we have just obtained a stronger soil, but only after settling. If this was under a foundation it was bad. But sometimes we do it on purpose to make soil denser and stronger, such as soil compaction, before laying out footings or laying layers of a road.

Another implication is a phenomenon called liquefaction, which is a term we often hear in soil mechanics and foundation engineering.

Let’s exaggerate things a little here, to visualize this…

Assume we have a volume of saturated soil, but the soil is so loose that the soil particles barely touch  or do not even contact each other… so in the formula above, stress on soil skeleton (effective stress σ’) will be zero and total stress will be equal to pore water pressure:

σT = σ’ + u = 0+u = u

This basically means as if, you just have a container of water now. This is basically liquid, and liquids can not carry anything on them (as we will see why in following sections). This is the logic behind liquefaction of soil.

Liquefaction is especially dangerous for sands because unlike clays, sands are sensitive to shaking, such as during an earthquake. Especially if that sand is loose, and it contains a lot of water, such as saturated loose sand, when the earthquake hits, there is not enough time for water to escape from pores, the stress is at first entirely taken over by water, which creates the situation we just saw above as:

σT = σ’ + u = 0+u = u

which turns the whole sand mass into practically a liquid. But the building load from above (total stress) doesn’t change, it is always there. So suddenly pore water pressure takes on everything and climbs to a very high value, and per the formula above, all stress is now carried by water. It means, the soil has now practically turned into a liquid. So in this instant no the load from structure above can not be carried at all, it is basically as if the structure is now standing on water.

As can be seen, liquefaction is a dangerous phenomenon and may cause small or even catastrophic failures to even structurally very well designed structures and foundations, which would not fail otherwise during an earthquake. Entire structures can sink as a whole, as if placed on swamp, if liquefaction occurs.

Loose sands are vulnerable to liquefaction. And since clay soils are not permeable, it also does not apply to clay soils except under certain specific circumstances such as low plasticity clay under certain conditions, or “quick clay” which is found in Scandinavian region and Russia.

Liquefaction potential should be checked for any type of structure and the codes dictate it. For example for piers or wharfs that have considerable area and heavy loading over water, or bridge footings in rivers, liquefaction is one of the first things to look at for foundations.

If you paid attention, we said “loose” sandy soils are most vulnerable. If the sand is dense, some settlement can still occur during earthquakes but this is small in proportion to the settlement that occurs after structure is fully erected with all its dead load and live loads on it. In situations where groundwater table is close to footings these effects can be more pronounced, but in general footings on dense sands are designed for permanent dead load settlements. Here the effect of earthquake can be covered by allowing for a higher bearing pressure on soil for example (after all engineering is all about making practical assumptions as long as they can be supported by evidence).

As can be concluded from paragraphs above, water should be drained and kept away from any foundation. Otherwise, the soil will get weaker, as it will support less of the overall (total) stress, and will start to settle under same loads, that it would have resisted otherwise, if there was no water. Excessive settlement may cause significant damage to structures, especially if it is uneven and differ from footing to footing (which is called differential settlement). Keeping the water away is important not only for foundations but for retaining walls too, as water exerts unnecessary lateral burden on walls. That is also why you see drain holes on faces of retaining walls for example, where water comes out, in order to keep it from accumulating behind the wall. We will talk about retaining structures later in this series.

One final note in this section is that sometimes pore water pressure can also be negative, which means suction on that soil. This can happen when water climbs above water table underground, (although this can be a few feet at the most) through capillary action (by sticking to particles of soil). So in that parts where water climbs above prevailing underground water table, water basically only “hangs” itself to soil particles. In a way, they pull the sand particles down. Look at the formula in the previous page again, if water pressure (u) is negative, for the same fixed amount of total stress, this means more effective stress.

Another case where negative pressure can exist, is when we excavate a clay soil. After rapid unloading of material such as excavation, the water in the soil around the foundation pit will have negative pressure. Why is that? Let’s look at our famous formula again:

 σT = σ’ + u

Now, when you excavated, suddenly the burden of the soil which existed there for ages, is suddenly removed. It means, the total stress, σT is suddenly less. But the soil skeleton cannot immediately respond to this change, so at first effective stress σ’ remains same. But since the left side of the equation has decreased, so must right side. This reduction is possible by water pressure (u) being negative, which means suction. By time, the soil will heave a little, as it has less stress on it now than in the past. We will also cover this phenomenon under consolidation settlement section in the following posts of this series.  

In the next post of this series, we discuss “Shear Strength of Soil”

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