Natural slopes tend to slide or fall to their lower side, given long enough time. Depending on the saturation degree of soil, the slope angle, overall 3D geometry, and the soil material itself, this sliding or falling can happen in very different ways and speeds, some can be very slow and virtually harmless, and some can be very sudden and catastrophic.
Slope failure can be in different forms. It can be in the form of:
- an avalanche or continuous flow of soil,
- just the creep of soil with much slower speeds when the soil just spreads laterally,
- sudden rockfalls and toppling off of both soil and rock,
- a landslide which looks like it is rotating around a point,
- sliding along a straight line.
Significant loss of life and property can occur, if these landslides are not predicted and precautions are not taken. Although it is difficult to predict landslides, still the slope can show some hints, even before any slope engineering analysis is done. For example, evidence of displacements, such as tilted or warped soil strata, bowed trees can be a hint. Slopes with sensitive clays, or the ones formed with the cutting action of eroding rivers carry high landslide risks. The projects should be designed to minimize slopes and excavations (not just for soil stability but economical reasons too) and drainage should be provided at the top and toe of slopes.
Natural slopes are vulnerable to failure after heavy rains, which makes the soil saturated, and increase pore water pressure, and decrease effective stress, as per the formula we saw before,
σT = σ’ + u ,
which means, reduction of resistance of the soil to failure. Because when effective strength decreases, shear strength also decrease, as we saw before. And when shear strength is less, which as we mentioned before is the key to soil strength, failures happen.
That is why after heavy rains, sometimes slopes that are on hills or mountains fail. Same condition can occur, after a rapid drawdown of water near an earth embankment dam, which leaves the slope saturated and vulnerable to failure with same logic.
For the slopes that are close enough to human life and property, depending on soil type, depth of water table, and slope angle and geometry, different precautions must be taken to keep the slopes stable.
Slope stability analyses are made to determine the nature of the slope, to assess short and long term slope stability including under seismic effects.
For a slope to be stable, the driving forces must be less than the strength of slope, and not only that it must be smaller by a certain factor, the factor of safety. Here by strength we basically mean the shear strength of that soil along the failure plane.
So for example, if we call the driving shear force as τd and the shear strength as τf, upon performing our slope stability analysis, we can find a factor of safety as:
This FS is what we aim to find, in a slope stability analysis.
Driving force is a result of the weight of the slope itself, that tends to slide or fall down plus any additions such as rainwater or structures built on it or even water seepage. Driving force, creates shear stress along a failure surface, which is resisted by the shear strength of the soil. Note that for slope analysis we ignore any plants and trees that are often present in slopes, their roots provide natural anchorage in some cases which affects the depth of actual slide surface, but they can also just move as a whole with a sliding slope.
We can classify slopes broadly into two, finite and infinite slopes.
Now let’s consider the infinite slope first.
Infinite slopes has length much greater than their height, so that their height is not even comparable to their length, and therefore they can be assumed for our slope analysis purposes that they continue indefinitely. The figure below shows the main components of an infinite slope and how we approach to its analysis.
Soil strength 24
Although we will not present the formula for analysis of this type of slope, we will list the items that goes into the equation. In other words, what factors affect the stability of such as slope as seen above, and how. These factors are:
- Cohesion of the soil (c’): As we have
seen, cohesion is one of the main ingredients of the general formula for shear
strength, which was:
Since we said that analysing slopes is all about comparing shear strength of ailure surface versus the destailizing forces, we must know the shear strength, and hence, cohesion of the soil. Again as usual we put apostroph sign near c, as only the effective cohesion is what matters, a concept we explained before. As explained before, cohesion is only present in clays. Not sands. So for sandy soils, this term doesn’t apply, unless there is considerable amount of clay in sand.
- Angle of internal friction (Ø’): For the same reasons above, since angle of internal frictoin is a main ingredient in shear strength formula, the slope stability equation depends on this angle. And for the same reasons as above, we use effective angle.
- Slope angle with the horizontal (β): As also can be seen intuitively from figure above, the steeper the slope, the less stable it would be. So the stability equation is dependent on this angle beta as well.
- Thickness of the soil layer (H): This effects the weight of the block as seen in the figure and therefore goes into stability equation. How this H is determined is beyond the level of this text.
- Unit weight of soil. So this time let’s ask before explaining… What can be the reason to include unit weight of the soil? …. We include it because of two reasons… To calculate the weight of the soil block and also the vertical stress in the soil at the failure plane. And do you think we must take the total (saturated) unit weight or the effective? We must take the saturated unit weight, because the driving force of the slope is the whole weight of the soil block, including whatever water it has in it. That is why, the more water, the heavier the soil and the more driving force. That is why slopes fail more often after heavy rains. (Rain also reduces effective stress in the soil and thus shear strength, so it has a negative effect to stability in two ways both in terms of increasing detsbilizing force and also reducing stabilizing force)
- Any water seepage. Seeping water exerts pressure in soil. Therefore its effects must be considered, if it is fast enough as it can exert reasonable force on soil particles.
Now let’s mention another important point, before we move on to finite slopes…
We already discussed shear strength of soil before, which was given in general form as,
After we divide driving stress, by the shear strength, for sands we reach an important conclusion. Taking cohesion as zero for sands, and after some simple mathematics, we see that for sands, an infinite slope is stable, if, the slope angle β (as also seen in figure above) is less than the angle of internal friction Ø of sand.
In other words, if,
β < Ø
the slope is stable, when it is made of sand.
As can be seen, this is an important conclusion and it is totally independent of the height of the slope or other dimensions or the soil unit weight.
This is another significant use of the internal friction angle. (and this is the situation where internal friction angle means angle of repose, as we talked about before, which is the angle a sand soil can keep its natural posture)
Now let’s take a look at finite slopes…
A finite slope is where, the height of the failing soil amounts to a considerable portion of the whole slope length. The general shape of a finite slope is as below and it can fail in different modes as shown:
Soil strength 25a
here for different cases, an arc can be drawn and its center can be established. After that, the analysis is done as if the arch is rotating around the center point. Cohesion, height of the slope, angle of internal friction, distance to base rock are determining factors for stability. Again we compare the driving forces such as the weight of the slope that tries to rotate the arc around its center point, versus stabilizing force, such as the shear strength of the soil along the failure surface (one of the arcs as shown above).
Below, we see the situation where bedrock is at reasonably close distance to the failing slope. Then the arc can only form upto the bed rock, as can be seen in the figure.
Soil Strength 25b
An alternate method when analysing finite slopes is, dividing the failure arc into vertical pieces, and analyse the stability of pieces, and then adding their effect overall and come up with a result of driving and stabilizing forces. It is shown in the figure below. The width of slices can be taken as convenient and need not be the same. The important thing is that we reasonably divide the arc into slices and analyse and add the driving forces and stabilizing forces of each piece together, to come up with an overall result. The forces here are not shown for clarity of the figure but are drawn very similar to what we drew in figure 24, meaning, each slice has the weight and the resisting shear force at the bottom and the reaction force from below and its sides. which are used in the analysis. As always, angle of internal friction and cohesion plays important role in determining stability, as they make up the shear strength.
Soil strength 26
Finally, when we also consider earthquake effects, that force must also be taken into account as a destabilizing force and will make slope analysis considerably more complex.
In the next post of this series, we will discuss “Critical State of Soil”