Determination of coefficient of consolidation

and

du z

0 i

2 d =area under initial isochrone

ò

(For a half-closed layer, the limits of integration are 0 and d in the above equations.)

The initial variation of excess pore water pressure in a clay layer can usually be approximated in practice to a linear distribution. Curves 1, 2 and 3 in Figure 4.15 represent the solution of the consolidation equation for the cases shown in Figure 4.16, where

1 Represents the initial conditions u_i in the oedometer test, and also the field case where the height of the water table has been changed (water table raised = positive u_i; water table low- ered = negative u_i);

2 Represents virgin (normal) consolidation;

3 Represents approximately the field condition when a surface loading is applied (e.g. placement of surcharge or construction of foundation).

Consolidation

The log time method (due to Casagrande)

The forms of the experimental and theoretical curves are shown in Figure 4.17. The experimental curve is obtained by plotting the dial gauge readings in the oedometer test against the time in min- utes, plotted on a logarithmic axis. The theoretical curve (inset) is given as the plot of the average degree of consolidation against the logarithm of the time factor. The theoretical curve consists of three parts: an initial curve which approximates closely to a parabolic relationship, a part which is linear and a final curve to which the horizontal axis is an asymptote at U_v = 1.0 (or 100%). In the experimental curve, the point corresponding to U_v = 0 can be determined by using the fact that the initial part of the curve represents an approximately parabolic relationship between compression and time. Two points on the curve are selected (A and B in Figure 4.17) for which the values of t are in the ratio of 1:4, and the vertical distance between them, ζ, is measured. In Figure 4.17, point A is shown at one minute and point B at four minutes. An equal distance of ζ above point A fixes the dial gauge reading (a_s) corresponding to U_v = 0. As a check, the procedure should be repeated using dif- ferent pairs of points. The point corresponding to U_v = 0 will not generally correspond to the point (a₀) representing the initial dial gauge reading, the difference being due mainly to the compression Figure 4.16 Initial variations of excess pore water pressure.

of small quantities of air in the soil, the degree of saturation being marginally below 100%: this compression is called initial compression. The final part of the experimental curve is linear but not horizontal, and the point (a₁₀₀) corresponding to U_v = 100% is taken as the intersection of the two linear parts of the curve. The compression between the a_s and a₁₀₀ points is called primary consoli‑

dation, and represents that part of the process accounted for by Terzaghi’s theory. Beyond the point of intersection, compression of the soil continues at a very slow rate for an indefinite period of time and is called secondary compression (see Section 4.8). The point a_f in Figure 4.17 represents the final dial gauge reading before a subsequent total stress increment is applied.

The point corresponding to U_v = 50% can be located midway between the a_s and a₁₀₀ points, and the corresponding time t₅₀ obtained. The value of T_v corresponding to U_v = 50% is 0.196 (Equation 4.24 or Figure 4.15, curve 1), and the coefficient of consolidation is given by

c d

v= 0 196t ²

50

. (4.27)

the value of d being taken as half the average thickness of the specimen for the particular pres- sure increment, due to the two-way drainage in the oedometer cell (to the top and bottom). If the average temperature of the soil in-situ is known and differs from the average test temperature, a correction should be applied to the value of c_v, correction factors being given in test standards (see Section 4.2 for references).

The root time method (due to Taylor)

Figure 4.18 shows the forms of the experimental and theoretical curves, the dial gauge readings being plotted against the square root of time in minutes and the average degree of consolidation Figure 4.17 The log time method.

Consolidation

against the square root of time factor respectively. The theoretical curve is linear up to approxi- mately 60% consolidation, and at 90% consolidation the abscissa (AC) is 1.15 times the abscissa (AB) of the extrapolation of the linear part of the curve. This characteristic is used to determine the point on the experimental curve corresponding to U_v = 90%.

The experimental curve usually consists of a short curved section representing initial com- pression, a linear part and a second curve. The point (D) corresponding to Uv = 0 is obtained by extrapolating the linear part of the curve to the ordinate at zero time. A straight line (DE) is then drawn having abscissae 1.15 times the corresponding abscissae on the linear part of the experi- mental curve. The intersection of the line DE with the experimental curve locates the point (a₉₀) corresponding to U_v = 90% and the corresponding value t₉₀ can be obtained. The value of T_v corresponding to U_v = 90% is 0.848 (Equation 4.24 or Figure 4.15, curve 1), and the coefficient of consolidation is given by

c d

v = 0 848t ²

90

. (4.28)

If required, the point (a₁₀₀) on the experimental curve corresponding to U_v = 100%, the limit of primary consolidation, can be obtained by proportion. As in the log time plot, the curve extends beyond the 100% point into the secondary compression range.

Figure 4.18 The root time method.

The root time method requires compression readings covering a much shorter period of time compared with the log time method, which requires the accurate definition of the second linear part of the curve well into the secondary compression range. Robinson and Allam (1996) showed for a range of natural soils that c_v from the log time method is consistently lower than that from the root time method (by approximately 20%) across a wide range of applied stresses. In most cases, the root time method is preferred for practical reasons; however, a straight-line portion is not always obtained on the root time plot, and in such cases the log time method should be used instead. Other methods of determining c_v have been proposed by Naylor and Doran (1948), Scott (1961) and Cour (1971).

Compression ratios

The relative magnitudes of the initial compression, the compression due to primary consolidation and the secondary compression can be expressed by the following ratios (refer to Figures 4.17 and 4.18).

Initial compression ratio ^s

0 f

:r a a a a

0 = 0-

- (4.29)

Primary compression ratio (log time) p s f

:r a a a a

= -

-

100 0

(4.30)

Primary compression ratio (root time) p s f

:r a a

a a

=

(

-

)

(

-

)

10 9

90 0

(4.31) Secondary compression ratio:rs= -¹

(

r⁰+rp

)

^(4.32)

In-situ value of c

_v

Settlement observations have indicated that the rates of settlement of full-scale structures are generally much greater than those predicted using values of c_v obtained from oedometer tests on small specimens (e.g. 75 mm diameter × 20 mm thick). Rowe (1968) has shown that such discrep- ancies are due to the influence of the soil macro-fabric on drainage behaviour. Features such as laminations, layers of silt and fine sand, silt-filled fissures, organic inclusions and root-holes, if they reach a major permeable stratum, have the effect of increasing the overall permeability of the soil mass. In general, the macro-fabric of a field soil is not represented accurately in a small oedometer specimen, and the permeability of such a specimen will be lower than the mass perme- ability in the field.

In cases where fabric effects are significant, more realistic values of c_v can be obtained by means of the hydraulic oedometer developed by Rowe and Barden (1966) and manufactured for a range of specimen sizes. Specimens 250 mm in diameter by 100 mm in thick are considered large enough to represent the natural macro-fabric of most clays: values of cv obtained from tests on specimens of this size have been shown to be consistent with observed settlement rates.

Details of a hydraulic oedometer are shown in Figure 4.19. Vertical pressure is applied to the specimen by means of water pressure acting across a convoluted rubber jack. The system used to apply the pressure must be capable of compensating for pressure changes due to leakage and specimen volume change. Compression of the specimen is measured by means of a central spindle passing through a sealed housing in the top plate of the oedometer. Drainage from the speci- men can be either vertical or radial, and pore water pressure can be measured during the test.

Consolidation

The apparatus can also be used for flow tests (i.e. as a permeameter), from which the coefficient of permeability can be determined directly (see Section 2.2).

Piezometers installed into the ground (see Chapter 6) can be used for the in-situ determina- tion of cv, but the method requires the use of three-dimensional consolidation theory. The most satisfactory procedure is to maintain a constant head at the piezometer tip (above or below the ambient pore water pressure in the soil) and measure the rate of flow into or out of the system. If the rate of flow is measured at various times, the value of c_v (and of the coefficient of permeability k) can be deduced. Details have been given by Gibson (1966, 1970) and Wilkinson (1968).

Another method of determining cv is to combine laboratory values of mv (which from experi- ence are known to be more reliable than laboratory values of cv) with in-situ measurements of k, using Equation 4.18.

Example 4.3

The following compression readings were taken during an oedometer test on a saturated clay specimen (Gs = 2.73) when the applied pressure was increased from 214 to 429 kPa:

TABLE G

Time (min) 0 0.25 0.5 1 2.25 4 9 16 25

Gauge (mm) 5.00 4.67 4.62 4.53 4.41 4.28 4.01 3.75 3.49

Time (min) 36 49 64 81 100 200 400 1440

Gauge (mm) 3.28 3.15 3.06 3.00 2.96 2.84 2.76 2.61

After 1440 min, the thickness of the specimen was 13.60 mm and the water content was 35.9%. Determine the coefficient of consolidation from both the log time and the root time Figure 4.19 Hydraulic oedometer.

methods and the values of the three compression ratios. Determine also the value of the coefficient of permeability.

Solution

Total change in thickness during increment = 5.00 − 2.61 = 2.39 mm Average thickness during increment = 13.60 + (2.39/2) = 14.80 mm Length of drainage path; d = (14.80/2) = 7.40 mm

From the log time plot (data shown in Figure 4.17), t

c d

t

50

2 50

2

6

12 5

0 196 0 196 7 40 12 5

1440 365

10 0 45

=

= = ´ ´ ´ =

. min

. . .

. .

v m²²

0

p

/year r

r

= -

- =

= -

- =

5 00 4 79

5 00 2 61 0 088 4 79 2 98

5 00 2 61 0

. .

. . .

. .

. . .7757

1 0 088 0 757 0 155 r_s= -

(

^. + ^.

)

^{= .}

From the root time plot (data shown in Figure 4.18) t90 =7 30. and therefore t

c d

t

90

2 90

2

6

53 3

0 848 0 848 7 40 53 3

1440 365

10 0 46

=

= = ´

´ ´

= . min

. . .

. .

v m²²

0

p

/year r

r

= -

- =

=

(

-

)

- 5 00 4 81

5 00 2 61 0 080 10 4 81 3 12

9 5 00 2

. .

. . .

. .

. .661 0 785 1 0 080 0 785 0 135

( )

⁼

= -

(

+

)

⁼

.

. . .

rs

In order to determine the permeability, the value of m_v must be calculated.

Final void ratio, 0.359 2.73 0.98 Initial void ratio,

1 1 s

0

e w G e

= = ´ =

=ee1+ De Now

D D

e H

e

= 1H+ 0 0

Consolidation

i.e.

De De

2 39 1 98

15 99 .

.

= .+ Therefore

De=0 35. ande0=1 33. Now

m e

e e

v

2

m /kN 0.70 m /M

= + × -

¢ - ¢

= ´ = ´

=

-

1 1

1 2 33

0 35

215 7 0 10

0

0 1

1 0

4 2

s s .

. .

N N Coefficient of permeability:

k c m=

= ´ ´

´ ´ ´

= ´ ^-

v v w

m/s g

0 45 0 70 9 8 60 1440 365 10 1 0 10

3 10

. . .

.