Explain the methods for the determinations of Coefficient of Consolidation.

The coefficient of consolidation is determined by methods based on the comparison between the characteristics of the theoretical relation between the  time factor, `T_v` and degree of consolidation, `U` to the relation between elapsed time,`t` and degree of consolidation, `U` obtained  for soil specimen in laboratory test. Two commonly used methods are:

1. Logarithm of time fitting method

2. Square root of time fitting method

Logarithm of Time Fitting Method:

The method, given by Casagrande, uses the theoretical curve between `U` and `logT_v` as shown as in the figure. The curve consists of three parts:

1. Initial portion which is parabolic in shape,

2. a middle portion which is almost linear,

3.  the last portion to which the horizontal axis is an asymptote.

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The coefficient of consolidation is determined by methods based on the comparison between the characteristics of the theoretical relation between the  time factor, `T_v` and degree of consolidation, `U` to the relation between elapsed time,`t` and degree of consolidation, `U` obtained  for soil specimen in laboratory test. Two commonly used methods are:

1. Logarithm of time fitting method

2. Square root of time fitting method

Logarithm of Time Fitting Method:

The method, given by Casagrande, uses the theoretical curve between `U` and `logT_v` as shown as in the figure. The curve consists of three parts:

1. Initial portion which is parabolic in shape,

2. a middle portion which is almost linear,

3.  the last portion to which the horizontal axis is an asymptote.

   
   
   

For the given incremental loading of the laboratory test, the specimen deformation against log of time plot is shown in the figure. The following procedures are adopted to determine coefficient of consolidation:

1. Plot the dial readings for a given load increment against time on semi log graph paper.

2.  Select two points  `B` and `C` on the parabolic portion of the consolidation curve which corresponds time `t_1` and `t_2` respectively. Note that `t_2=4t_1` .

3. Measure the vertical distance between B and C. Say, it is equal to `a` which is at a distance `a` from B.

4.  Set off an equal distance, which is at a distance `a` above the point `B`. Draw a horizontal line, which fixes the point at `R_c`. The dial gauge reading corresponding to this line `R_c`  is the corrected zero reading which corresponds to degree of consolidation as zero. The difference between the dial gauge reading at `R_0` and `R_c` is called initial compression.

5.  Project the straight line portion of the two lines `MD` and `DE` to intersect at `F`. The dial gauge reading corresponding to `F` is `R_(100)` i.e, 100% primary consolidation.

6. Determine the point `M` on the consolidation curve which corresponds to the dial reading`R_(50)` or `(R_c+R_(100))/2`. The time corresponding to the point `M` is `t_(50)` i.e, time for 50% consolidation.

7. Determine `C_v` from the equation`T_v=(C_v*t)/d^2`. The value of `T_v` for `barU=50%` is `0.197`.Thus, `C_v=0.197d^2/t_(50)` 

   
   
   

Square root Time Fitting Method:

This method was developed by Taylor based on the observation of dial gauge reading and time as shown in the figure. The figure shows the form of theoretical and practical curve. The theoretical curve is linear upto about 60% consolidation and at 90% consolidation, the abscissa is `ac=1.15ab`. The practical curve usually consists of a short curve comprising initial portion, a linear part and a second curve.

The step by step procedure is given as follows:

1.  plot the dial reading as ordinate with the corresponding is square root of time, `root (2)t` as abscissa.

2. Produce backward the initial linear part of the curve to intersect at`R_c`. The distance between `R_0` and `R_c` is called initial compression.

3.  Draw a tangent to the line `AB`. Draw the line `AC` such that its abscissa is `1.1 5` times that of the linear portion of `AB`.

4.  The intersection of `AC` with the consolidation  curve at `C` will locate `R_(90)` corresponding to  `root (2)(t_(90)) ` i.e,the square root of time for 90% of consolidation.

5.  By knowing the`0` and `90%` consolidation, the `100%` consolidation can be obtained.

6.  Beyond `100%` consolidation, it is assumed that the consolidation settlement is completed and secondary consolidation starts.

7.  Find `C_v` by using the equation:`T_v=(C_v*t)/d^2`

8.  The value of `T_v` for `barU=90%` is `0.848`.

So, `C_v=(0.848*d^2)/t_(90)`