1.1.3. Hyperbola method

The coefficient of three graphical procedure are used: Logarithm of time method, Square root of time method, Hyperbola method.

The Hyperbola method is one of the three graphical procedures designed to determine the coefficient of consolidation (Cv) from laboratory data. The three methods are: 1) Logarithm of time method, 2) Square root of time method, and 3) Hyperbola method. Each of these methods utilizes different mathematical approaches to analyze the consolidation data.

Think of it like trying to find out how fast a sponge absorbs water. Each method—like different techniques for measuring the absorption—gives you a way to see just how quickly the sponge consolidates the water. The Hyperbola method represents one of those techniques.

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.

To utilize the Hyperbola method, you start by plotting dial readings that record how much compression occurs in response to a pressure increase over a logarithmic scale of time. Then, you mark two distinct points, P and Q, on this graph. These points correspond to specific times: t1 and t2, with t2 being four times longer than t1. This plotting lays the groundwork for analyzing how quickly consolidation occurs at different stages.

Imagine recording how much a sponge compresses as it absorbs water. The first reading (t1) is noted right after a few seconds, while the second reading (t2) is taken later, four times the time of the first. This helps you see how the sponge's compression rate changes over time.

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.

After plotting points P and Q, you calculate the difference in their dial readings, which we denote as x. Now, you place a new point, R, above point P at the height of this difference x. You then draw a horizontal line from R, called line RS. The level this line reaches on the dial reading scale corresponds to the position of d0, signifying the state of 0% consolidation. This serves as a reference point in your analysis.

Think of measuring how much a sponge has squished after absorbing water. The difference in height between two measurements indicates how much more water it can absorb. The line RS is like drawing a baseline, showing how much compression there is at the very start, which we label as 0%.

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.

Finally, you extend the straight-line portions of both primary and secondary consolidation lines on your plot until they cross at point T. This crossing point is critical because it represents the full consolidation of the material. The dial reading at T, referred to as d100, indicates that the material has achieved 100% consolidation—an essential metric in understanding the material's behavior under pressure.

If you think of the sponge absorbing water until it's completely full, point T is the moment where it can’t hold any more water—the maximum level. The measurement at T (d100) indicates the point where the sponge is fully consolidated, meaning it's absorbed all it can.

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