1.1.2. Square root of time method

The coefficient of three graphical procedure are used:

1. Logarithm of time method
2. Square root of time method
3. Hyperbola method

The square root of time method is one of three techniques used to determine the coefficient of consolidation (Cv) from laboratory data. This method, along with the logarithm of time method and the hyperbola method, provides different frameworks for analyzing soil consolidation behavior. The square root of time method focuses on the relationship between consolidation time and the amount of consolidation that occurs, leveraging the mathematical property that consolidation rate is proportional to the square root of time.

Imagine you are planting a small tree in your backyard and want to understand how quickly the soil settles around it. You can observe that as time passes, the soil compresses around the roots. Just like with soil consolidation, the more time passes, the more tightly the soil will settle. In the square root of time method, we essentially quantify this 'settling process' so that we can predict how long it will take for the soil to fully consolidate around the tree.

Log – time curve fitting method:

1. Plot the dial reading of compression for a given pressure increment versus time to log scale.
2. Plot two points P and Q on the upper portion of the consolidation curve (say compression line) corresponding to time t1 and t2 such that t2=4t1.
3. Let x be the difference in dial reading between P and Q. locate R at a vertical distance x above point P.
4. Draw a horizontal line RS the dial reading corresponding to this line is d0 which corresponds with 0% consolidation.

The steps of the square root of time method involve plotting data from a soil compression test. First, you collect data on how much the soil compresses under a specific amount of pressure and plot this data against time on a logarithmic scale. Then, select two specific timestamps (t1 and t2) related by the formula t2 = 4t1. Next, calculate the change in dial readings (x) between these time points and mark it on the graph to visualize the current state of consolidation. The horizontal line drawn at this step represents the degree of consolidation up to that point, labeled as d0.

Think of setting up dominoes in a line. When you push the first domino, they start to fall one by one. If you time how long it takes for the first half of the dominoes to fall (t1) and then observe the second half (t2 = 4t1), you can see how the total effect increases over time. In the experiment, you plot these times and the changes in height like you would plot which dominoes have fallen and their corresponding times.

5. Project the straight line portion of primary and secondary consolidation to intersect at point T. The dial reading corresponding to T is d100 and this corresponds to 100% consolidation.

In the final step, you will extrapolate the data plotted to visualize the overall trend of consolidation. This involves projecting the plotted line segments from the initial points towards where they would theoretically intersect if continued, ending at point T. The reading at this new point (d100) reflects that the soil has fully consolidated and can be interpreted as reaching 100% consolidation.

Imagine you’re tracking how far a car travels over time. You start measuring distance at various points and notice that at some point, you slow down just before hitting the maximum speed. By extending your observations on a graph, you can predict when you would reach your top speed, similar to predicting the total consolidation of soil using extrapolated points in the graph.

--