Hello this is Professor Kitch. Welcome to this webcast on section 11.1. This
is the first of two webcasts on Terzaghi's consolidation theory. In this webcast we'll
derive the differential equation governing one dimensional consolidation. In the second,
we'll solve the equation Up to now, we've discussed only the magnitude
of consolidation settlement, that is how much will occur.
In this section, we'll start our discussion about how long it takes. As geotechnical engineers,
we need to be able to answer two questions about consolidation for our clients, how much
and how long. This will be one of the few times we derive
differential equations in this course. It's not an objective for you to be able to derive
the equation yourself. I think that's a tedious memorization exercise and not particularly
valuable. I do think it's worth going through the derivation because it gives you a chance
to understand the theories and rules applied and the assumptions made. Ultimately, I think
it makes you better equipped to use the solution if you understand where it came from.
With that in mind, the learning objectives are geared to this level of understanding.
Specifically, you should be able to list the key principles and rules used in deriving
Terzaghi's consolidation theory and most of the assumptions used.
Also, it's key that you understand what the coefficient of consolidation, c-sub-v, is
and what soil properties it includes Before we start the derivation, let's briefly
review the consolidation process. Consolidation starts with a change in total
stress that initially generates excess pore pressure
As time passes water flows out of the soil As water flows out, three things happen simultaneously:
excess pore pressure decreases, effective stress increases, and the soil volume decreases
The rate at which consolidation occurs will depend up, soil properties, specifically the
soil's compressibility and hydraulic conductivity. What the initial distribution of excess pore
pressure is in the soil. And what the drainage conditions are at the
soil boundaries. So during the consolidation process, excess
pore pressure is decreasing, effective stress in increasing and settlement is increasing.
Here's a graphical picture of the consolidation process. We'll consider a saturated compressible
soil layer with thickness 2 H, and we'll track the changes in applied vertical stress, total
vertical stress, pore pressure and effective vertical stress all as a function of time.
Before any construction, the total stress, pore pressure and effective stress are all
at their initial values and there is no applied stress, delta-sigma-z.
We then construct a wide fill on top of our compressible soil.
During fill placement, the applied vertical stress increases, and both the total stress
and pore pressures increase by the same amount. This is because the soil has a finite hydraulic
conductivity and the fill placement is fast compared to the soil's ability to drain. Therefore
loading is essentially undrained, and there's no change in effective stress.
As time passes after completion of the fill, settlement starts.
There is no increase in delta-sigma-z, so the total stress doesn't change.
The pore pressure decreases as the effective stress increases.
Eventually the pore pressure returns to its initial condition and the effective stress
increases an amount equal to the increase in total stress, delta-sigma-z.
Not let set the stage for deriving Terzaghi's consolidation theory by presenting his assumptions.
First, as we often do in engineering, we'll assume our material is homogeneous and isotropic
The entire theory is based on flow of water out of the soil, so we assume that it's saturated
We assume all the volume change in the soil comes from decrease in the void space, therefore
both the water and soil particles are incompressible. We'll be using Darcy's law to define the flow
of water. We assume we've got a really great construction
crew who can instantaneously apply our surcharge load to the soil.
The change in vertical stress is immediately transferred to the pore water generating a
uniform initial excess pore pressure throughout the soil.
We also assume our soil properties don't change as the soil consolidates.
And finally, this is a one dimensional process. Both the flow of water and compression occur
only in a vertical direction. We'll revisit these assumptions at the end
of the next webcast and discuss how valid they are.
OK, buckle you seatbelts, sharpen your pencils, here we go.
We'll start with a soil element dx by dy by dz, and the total volume of the element will
be dx dy dz. There will be some flow into the volume, Q-in
And some flow out, Q-out, and both with be vertical.
Now we'll apply our first principle, write it down, continuity, or conservation of mass.
We'll assume that the change in volume of the soil element is a function of only water
flow out of the soil. Remember we've already assumed the soil is saturated and the water
and solids are incompressible. In this case, the volume change over time, dv-dt will be
negative and equal to Q-out minus Q-in. In partial differential form, Q-out minus
Q-in, will the partial of Q with respect to z times the height of the element dz.
For clarity I'll be numbering the some of the equations in this derivation. There will
be seven of them. If God can create the world in seven days we should be able to derive
consolidation theory in seven equations. So there was evening and there was morning and
this is the first equation. Now our second principle, good old Darcy's
law, Q equal k i A. Minus because the flow is opposite the direction of the gradient.
Remember that the gradient, i, is the change in head per length, or the partial of h with
respect to z. And remember that h is equal to the pore pressure
divide by the unit weight of water, gamma-w. Here we use excess pore pressure, so the gradient
is one over gamma-w times the partial of u-excess with respect to z.
So for our problem, Darcy's law is Q equal minus k dh by dz times the area of flow dx
dy. And there was evening and there was morning, the second equation.
Now we'll combine equations 1 and 2 by jamming Darcy's equation for Q into the differential
in equation 1 Which will give us this differential equation.
Another of our assumptions was that the soil was homogeneous and isotropic and its properties
didn't change during consolidation. So we can pull both k and gamma-water out of the
differential. And we have the partial of V with respect
to time is k over gamma-water times the second partial of excess pore pressure with respect
to z all times dx dy dz But dx dy dz is just the volume of the soil
element, V, so we have dV by dt is k over gamma-water times the second partial of u-e
with respect to z times V. And there was evening and there was morning, the third equation.
Thus far we've considered continuity and Darcy's law.
Now let's add in our consolidation compressibility relationships. Recall that C-sub-c is the
change in void ratio versus change in log of effective stress or de by d-log-sigma-prime-z
If you remember your differential calculus, you'll recall that derivative of log-a is
a over the derivative of a. Or, if you're like me and don't remember that, you'll look
it up. Either way we'll use this property of logarithms to rewrite the equation as de
= negative C-sub-c over 2.3 times sigma-z-prime times d-sigma-z-prime
Now remember that after the initial stress is applied there's no change in total stress
so all of the change in effective stress is due to a change in the excess pore pressure
or d-sigma-prime-z is equal to minus d-u-sub-e. We make this substitute for d-sigma-prime-z,
and there was evening and there was morning the fourth equation.
Now remember that earlier in chapter 10 we proved that vertical strain was equal to delta-e
over 1 plus e. We can extend that concept to get d-epsilon-z
= de over 1 plus e. Next we'll use our assumption that all the
consolidation is vertical and the change in volume d-V will then be equal to d-epsilon-z
dz which is the change in height of the element time dx dy
We substitute de over 1 plus e for d-epsilon-z And then recognize that dx dy dz is the volume
of the element An there was evening and there was morning,
the fifth equation. Now we substitute equation 4 into equation
five and we'll have dV equal C-sub-c over 2.3 times sigma-prime-z times one over 1 plus
e d-u-e time V We take the partial of this with respect to
time and there was evening and there was morning, the sixth equation.
Now let's go back to the continuity and Darcy part of our derivation and recall equation
3 If we then set equation 3 equal to equation
6 we'll have the following The volumes on each side of the equation cancel
out And then we can rearrange the equation to
get that the first partial of u-e with respect time is equal to this mess of stuff in the
red box times the second partial of u-e with respect to z
Just to make stuff look nicer, we're going to replace the mess in red with a variable
we'll call the coefficient of consolidation, or c-sub-v.
Notice that c-sub-v is composed mostly of soil properties. e, C-sub-c, and k are all
soil properties, gamma-w is constant. The only thing that's not a soil property is sigma-prime-z.
But if we think of this as the initial vertical effective stress, it is sort of like a soil
property. In any case the best way to think of the coefficient of consolidation c-sub-v
is as the soil property that describes how quickly a soil consolidates. There's a bit
of a problem here, because one of our assumptions is that c-sub-v is constant throughout the
consolidation process, but if it's a function of the vertical effective stress that won't
be true. For now we'll ignore this little issue and deal with it later.
With this last simplification there was evening and there was morning, tada the seventh equation!
Terzaghi's differential equation for one dimensional consolidation.
The first partial of u-e with respect time is equal to c-sub-v times the second partial
of u-e with respect to z. If you've been paying attention you'll realize
that God created the earth in 6 days and rested on the seventh, but it took us all seven equations
to derive this theory. That just goes to show that God is better at creating the earth than
we are at figuring out how it works. Still we look around at all that we created and
see that it is very good. Well I don't know if Terzaghi rested after
deriving his differential equation for consolidation. But since God rested on the seventh day of
creation, maybe we should to. Let's summarize or work to this point and we'll pick up the
solution to this differential equation in the next webcast.
First we'll notice that the solution to this differential equation must be a function of
both z and time, since it contains partials of the excess pore pressure with both z and
t. We used the following key principles to derive
Terzaghi's consolidation equation Of the eight original assumptions, the key
ones are that the soil is saturated, Darcy's law governs flow, both the soil and the water
are incompressible and the volume change comes from water flow out of the voids, and both
the flow and consolidation are one dimensional in the vertical direction.
To these we added continuity, we're not creating of destroying mass
And we used our compressibility behavior we learned from the previous chapter. That's
it. Just these few simple principles. Finally we use the parameter c-sub-v as the
material property that tells us how fast a soil will consolidate