Hello this is Professor Kitch. Welcome to this webcast which is the second
of two on section 10.5. This webcast covers soil compressibility parameters
After this webcast, you should be able to define the compression index, recompression
index, compression ratio and recompression ratio
And you should be able to compute these parameters given lab test data
We discussed in the previous module, that we can plot consolidation test data as either
void ratio versus log of effective stress or strain versus log of effective stress.
We�ll first look at the void ratio versus log effective stress plot.
The compression curve in this plot starts at the initial void ratio, e-naught
and follows the recompression and virgin curves to the final void ratio, e-f. It is important
to note that the void ratio decreases from e-naught to e-sub-f
For computation purposes we have placed two points on the virgin curve a and b,
and two points on the recompression curve c and d
The slope of the compression curve is a measure of compressibility. The steeper the curve
the greater the compressibility. That is why the virgin curve is steeper than the recompression
curve The slope of the virgin curve per log cycle
is called the compression index and denoted C-sub-c
By definition this is the change in void ratio per change in the log of effective stress.
The definition contains a negative because the void ratio decreases as the soil compresses
and we desire a positive value of C-sub-c to represent a positive compression.
Using points a & b to compute C-sub-c as the slope of the virgin curve, C-sub-c is the
void ratio at a minus the void ratio at b divide by log of the effective vertical stress
at b minus the log of the effective vertical stress at a. We use e-sub-a minus e-sub-b
in the numerator, rather than the other way round, to account for the negative sign.
Since the log of b minus the log of is equal to the log of b over a, we can rewrite the
equation for the compression index as e-sub-a minus e-sub-b over the log of the effective
stress at b divide by the effective stress at a.
The recompression index is formulated in the same fashion as the compression index.
Using points c and d, the recompression index is e-sub-c minus e-sub-d divided by the log
of the effective vertical stress at d over by the effective vertical stress at c.
If we use the method of plotting the strain versus the log of effective stress, which
is the more common method, the compression curve starts at a vertical
strain of zero and ends at a final vertical strain of epsilon-z-f.
In this formulation, the slope of the virgin curve is called the compression ratio and
equal to C-sub-c over 1 pluse e-naught And we compute the compression ratio as the
vertical strain at b minus the vertical strain at a divided by the log of the ratio of the
vertical effective stress at b over the vertical effective stress at a.
Since the strain is increasing from epsilon-z equal zero at the start to epsion-z-f at the
end we don�t need a negative sign so the numerator in the compression ratio is epsilon-z
at b minus epsilon-z at a. This is opposite from the way we computed the numerator in
the compression index where we subtracted the void ratio at b from the void ratio at
a. The recompression ratio is then the recompression
index divide by 1 plus e-naught and using points d and c, computed as epsilon-z
at d minus epsilon-z at c divided by the log of the ratio of the vertical effective stress
at d over the vertical effective stress at c.
All we�re doing in the these last two slides is computing the slopes of the recompression
and virgin curves, but we have to do it in semi-log space.
I�ll do a quick example and show you a trick to making the computations very simple
In this case we�re using the more common form with the vertical strain as a function
of the log of effective stress. And we�ll be computing the compression and recompression
ratios. It�s easiest to compute the slopes of the
two lines over a full log cycle, so we start off by drawing to vertical lines one log cycle
apart. We�ll choose to draw the lines at an effective
stress of 100 and 1000. We could have picked any two points
one cycle apart such as 50 and 500, or 200 and 2000.
We then extend the virgin curve so it intersects both vertical lines.
And we�ll compute the slope of the virgin curve where it intersects the two vertical
lines a points a and b. At b the strain is 0.202
and at a it�s 0.030 The compression ratio is then 0.202 minus
0.030 divided by the log of 1000 over 100 which is 0.202 minus 0.030 over the log of
ten. But the log of 10 is simply one.
So the compression ratio is simply 0.202 minus 0.030.
So this is the trick. If you compute the slope using points that are one log cycle apart,
the slope will simply be the difference between the two strain values.
Or in this case, 0.172 The recompression ratio is computed in a similar
fashion. In this case we extend the recompression curve
until it intersects our two vertical lines. The recompression curve intersects these two
lines at strains of 0.097 and 0.042
And the recompression index is simply 0.097 minus 0.042
or 0.053. Notice the both indices are dimensionless
so we don�t need to know the units of effective stress.
That�s all there is to it. It�s really quite simple, especially if you use the trick
of picking points that are a log cycle apart. It�s always good to check our computations
to see if they seem reasonable. We have a few ways to do this.
From Kulhawy and Mayne�s EPRI report on estimating soil properties
we can use these two correlations between the plasticity index and the compression and
recompression indices. Most soils will lie within 50% of the value computed form these
correlations, so if you compute a value outside the range, you should double check your computations.
Table 10.2 of your text provides a qualitative description of the ranges of compression or
recompression ratios. It is also useful for checking computed values.
We�ll that�s all there is for this webcast so let�s summarize
We use the parameters C-sub-c and C-sub-r or C-sub-c over 1 plus e-naught and C-sub-r
over 1 plus e-naught to represent the compressibility of soil in our semi-logarithmic stress-strain
space. C-sub-c and C-sub-r are used if we plot by
void ratio and C-sub-c over 1 plus e-naught and C-sub-r
over 1 plus e-naught are used if we plot by vertical strain.
Don�t mix the two up, or your answers will be wrong.
We can make our job easier by picking points on the curves that are one log cycle apart
since the log of 10 is equal 1 and the number 1 in our friend.