Hello this is Professor Kitch.
Welcome to this webcast on section 11.2.
In this webcast we'll discuss how to compute the time rate of settlement
When you're finished with this webcast, you should be able to define the degree of consolidation
for a soil layer and contrast the degree of consolidation with the degree of pore pressure
dissipation--they're not exactly the same thing.
We'll discuss to methods for determining the degree of consolidation in this webcast and
you should be able to use the simplified solution to either determine how much settlement has
occurred at a given time, or determine how long it will take to reach a certain level
of settlement.
The solution to the simplified method is presented in Figure 11.6, Table 11.1 and Equations 11.24-11.25
In our previous modules on the rate of consolidation, we solved Terzaghi's 1-D consolidation theory
and used this solution to determine how excess pore pressures change at points within the
consolidating layer.
This is valuable information, but it doesn't answer some important questions our clients
need to know, questions related to the rate of settlement.
Questions like "How much settlement will occurs a year after construction," or "When will
the consolidation be done?"
These are clearly key questions to be answered.
So let's answer them.
Computing settlement versus time is simply a process of combining the magnitude of settlement
computations we learned in Chapter 10 with the pore pressure dissipation computations
we just learned in module 11.1 An example will illustrate the process best.
Let's consider the doubly drained soil layer shown on here
with the strain versus log effective stress curve shown below it.
We'll assume a fill has just been placed on the ground surface and we have this initial
excess pore pressure distribution.
Our objective is to compute the consolidation settlement as a function of time.
The process starts by computing the ultimate consolidation settlement using the techniques
we learned in Chapter 10.
We then need to divide the compressible layer up into sub-layers for computations.
For accurate computations, we'll need as many as 50 sub-layers, but for illustration, we'll
show just a few.
We then start a cyclical process were we increment time by a small amount delta-t.
The excess pore pressure will start to dissipate as show by the new isochrone.
Now we perform a series of computations for each sub-layer.
First we compute the pore pressure for each layer using the static pore pressure plus
the current excess pore pressure determined from our isochrone.
We then compute the vertical effective stress for each layer using our effective stress
equation.
And with the effective stress known, we can compute the strain in each sub-layer using
our strain versus effective stress relationship.
With the strain we can compute the settlement for each sub-layer.
Once we've done these computations for all the sub-layers, we can add up the settlement
of each sublayer to determine the total consolidation settlement at the current time.
And we can plot that point on our graph of settlement versus time.
We then go back to step three, increment time and repeat the whole process.
At each time step we'll be able to compute a new consolidation settlement.
And we continue the process.
Until consolidation is done and we have a complete consolidation versus time curve.
We'll define the degree of consolidation, U, as the consolidation settlement at any
given time, divided by the ultimate consolidation settlement.
While this is conceptually a simple process, it's computationally very tedious and can't
be reasonably done by hand.
It requires a computer program to accurately perform the calculations.
So we need a simpler solution which we can do with hand calculations.
To get to a simpler solution, let's look at how pore pressure dissipates in our consolidating
layer.
We know that as the excess pore pressure dissipates the soil settles.
We've already learned how to compute the excess pore pressure at any point in the layer at
any time using the isochrones in Figure 11.4.
With this information we can define the degree of pore pressure dissipation at a point as
the initial excess pore pressure minus the current excess pore pressure divided by the
initial excess pore pressure.
At the start of consolidation u-i is equal to u-e and the degree of excess pore pressure
dissipation is zero.
At the end of consolidation u-e is zero and the degree of excess pore pressure dissipation
is 100 % This is helpful, but what we really want to
know is the average degree of excess pore pressure dissipation for the entire consolidating
layer, U-sub-e To calculate U-e we simply integrate the excess
pore pressure across the entire height of the soil layer or U-e is the integral of the
initial excess pore pressure from zero to 2H minus the integral of the current excess
pore pressure from zero to 2 H divided by the integral of the initial excess pore pressure
We can illustrate this graphically.
The integral of the initial excess pore pressure is simply the yellow rectangle.
And the integral of the current excess pore pressure at a certain point in time (in this
case when T-v is 0.4) is the brown area.
There is a series solution for U-e shown here.
It's simply an integration of the series solution to Terzaghi's consolidation theory we covered
in module 11.1 If we assume the soil is linearly elastic,
then we can show that the degree of consolidation is equal to the degree of pore pressure dissipation
or U is equal to U-e.
We call this the simplified solution in your text.
Using this solution we can compute the consolidation settlement at any point in time as the degree
of pore pressure dissipation time the ultimate consolidation settlement.
The assumption that the soil is linear elastic is a bit confusing.
We're still using our non-linear strain versus log effective stress curve to determine the
ultimate consolidation settlement.
We just using this simplifying assumption to compute the time rate of settlement.
As with our solutions for excess pore pressure dissipation, we can compute the degree of
consolidation as a function of our dimensionless time factor, T-v.
This graph shows this relationship along with two equations that have been fit to the solution.
So we can use either this graph or these equations to compute the degree of consolidation versus
the time factor.
This graph is presented in your text as Figure 11.6 and the equations are 11.24 and 11.25
This same information is presented in tabular form as Table 11.1.
So we can use either Figure 11.6, Equations 11.24 and 11.25, or Table 11.1 to determine
the simplified version of the average degree of consolidation.
And then compute the consolidation settlement, delta-c as U times delta-c-ultimate
Remember this is a simplified solution and there will be some difference between it and
the more accurate computer based solution.
This figure will give you an idea of how much these two solutions can differ.
The actual difference depends upon the specific problem being solved, but the gray area shown
gives you a general idea of the range of answers possible.
Note that as time increases the simplified solution becomes more accurate
Let's illustrate how to do these computations with an example.
We'll solve a portion of Example 11.3 by hand.
Here were given a 10 meter thick compressible clay layer with the properties shown and a
3.2 meter thick fill is to be placed on the ground surface.
We'll skip the computations for the ultimate consolidation settlement, they're shown in
the text.
The value of delta-c-ultimate is 471 millimeters.
The first question we'll answer is how much settlement occurs 5 years after the fill is
placed First we must find the time factor representing
5 years in our problem.
This will be c-v (0.0021 meters squared per day) times time (5 years times 365 days per
year), divide by 5 meters squared.
Which will be 0.153.
We'll use Equation 11.24 to compute U from the time factor and find that the simplified
degree of consolidation is 44.2 percent.
We'll then compute the degree of consolidation at 5 years to be 44.2 percent of 471 or 208
millimeters.
Another question we might ask is how long will it take to reach 95% consolidation?
Using Table 11.1 we'll find that the time factor, T-v, for 90% consolidation is 1.13
We then use our equation for the time factor to calculate the actual time.
In this case, 1.13 time 5 meters squared, divide by c-v (0.0021 meters squared per day).
Which will be 13,450 days or 36.9 years An important point to make here is that the
theoretical time to 100 percent consolidation is infinity.
It's not very helpful to tell our clients that it will take in infinite amount of time
for the consolidation to be complete on their project.
We need to remember that this is just a theoretical solution and, from a practical standpoint
consolidation will be complete sooner.
So we generally assume that the theoretical time to 95 percent consolidation represent
complete consolidation.
Just to give you an idea of the difference between our simplified solution and the more
accurate computer solution, we compare the two in this figure, taken from Example 11.3
in your text.
You'll see here that the two solutions are very similar.
There's more uncertainty in our soil properties than in the difference between these two solutions.
So let's summarize The proper definition of the degree of consolidation
is the current consolidation settlement divided by the ultimate consolidation settlement.
Computing the degree of consolidation is tedious and requires using a computer program
The simplified solution assumes that the degree of excess pore pressure dissipation is equal
to the degree of consolidation This solution allows for easy hand computations
using charts, tables, or simple equations.
It implicitly assumes that the soil is linearly elastic.
And it's generally an adequate solution for simple problems.
Theoretically consolidation goes on forever For practical purposes we usually assume 95%
consolidation represents full consolidation.