Lecture 5 Notes – Determination of Rate of Settlement
Based on Terzaghi’s one-dimensional consolidation theory
Terzaghi derived a governing differential equation for describing the rate change of
excess pore water pressure along the sample depth
Cv2u/z2 ) =  u/ t , where u = excess pore water pressure, z = depth, t = time
Cv =
,
where k = permeability, av = coefficient of compressibility,
The boundary condition for our test sample is:
unit : L2 T -1 or in2 / sec
[ Solution ] :
∑
f1(Z) : geometry factor, f2(T) : time factor
where Z = z / Hdr, T =
, Hdr2 : longest drained path
Graphic Solution:
∑
Ref. 2
Uz is different for every point at time factor T.
U = Average Consolidation Ratio
(double drained layers)
Figure 2: Average consolidation ratio: linear initial excess pore pressure. (Ref 1)
(a) Graphical interpretation of average consolidation (b) U versus T
We know that 2' & 1' relate directly to u2 & u1 (decreases in u increase in ')
therefore, Eq(2) becomes
∑
where
ui = 2' -1'
Uz = 1
Uz = 0
100% Consolidation u = 0
0% Consolidation u = 
Graphic Solution of Uz (Fig. 27-2)
Uz is a function of sample thickness and time factor.
Define a new variable U = Avg. consolidation ratio
(for entire layer, not for a single point), (see Fig.27.3)
Physically U = S(t) / Sc, S(t) is settlement at time t, Sc is total settlement.
U=f(T), (only T)
Example: Given: Cv = 2 x 10-4 in2/sec, Sc = 20 in. (was found by Eq1 Cc &Cs)
Find:
Time for S(t) = 10 in.
e0 = 0.62
w = 23.2%
γs
4
σ
v
Δσ
[Solution]
U = S(t) / Sc = 10 / 20 = 50%
from Fig. 27.3 
we know T = Cv (t / Hdr2) t = T Hdr2 /Cv 
t = 0.2(30/2210-4) sec = 375 days
In consolidation test, we can obtain Cv value based on one of the following two methods:
(1) Taylor's square root of time fitting method
(2) Casagrande's logarithm of time fitting method
For (1) Cv = T90 Hdr2 / t90 (t90 U = 90%) see Fig 9.9
(2) Cv = T50 Hdr2 / t50 (t50 U = 50%) see Fig 9.7
Ref. 4.
In general, the purposes of consolidation test are:
(1) Find Cc & Cs by plotting e vs. log '
(2) Find c' from e vs. ' plot by using Casagrande's graphical method, and
(3) Find Cv at each stage of ' by plotting square root of time vs. e or log (time)
vs. e and plot Cv vs. ' curve
Reference:
1. Soil Mechanics, T. W. Lambe and R. V. Whitman, Wiley Book Company,1969.
2.
An Introduction to Geotechnical Engineering, Robert D. Holtz and William D.
Kovacs, Prentice-Hall Book Company,1984.
3.
Geotechnical Engineering - Soil Mechanics, John N. Cernica, Wiley Book
Company, 1995.
4.
Engineering Properties of Soils and Their Measurement, Joseph E. Bowles, 4th
Ed., McGraw-Hill Book Company,1992.