---
title: "Square root of time method for cv"
subject: "Soil Mechanics"
chapter: 9
key: "semester_4/soil_mechanics:sqrt-time-method-cv"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/sqrt-time-method-cv
site: BCT Engineering
---

# Square root of time method for cv

Soil Mechanics · Chapter 9: Compressibility of Soil

- Asked in: 2078 Chaitra, 4 marks
- Video: [Chapter 11 Consolidation - The square-root-of-time method](https://www.youtube.com/watch?v=ck2bgPilMMg) — uSeeGeo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/sqrt-time-method-cv
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-9

## Question

With the help of a neat sketch, describe the method of determination of coefficient of consolidation by the square root of time method.

## Short answer

Taylor's square root of time method plots dial reading against √t, extends the initial straight line to get corrected d0, draws a second line from d0 with slope 1/1.15, and takes its intersection with the curve as √t90. Then cv = 0.848 Hdr^2 / t90.

## Answer

**Taylor's square root of time method** finds $c_v$ from a consolidation test for a load increment, using the observed dial reading against $\sqrt t$.

```text
 dial                  
 reading |*.
         |  *.          straight line
    d0   |-----*.______  (extended to √t axis → d0 corrected)
         |       \ *  .
         |  line OB  \   *  *  *
         |  at 1.15x  C
         +-----------|--------------> √t
                    √t90
```

#### Procedure

1. Plot the dial gauge reading (settlement) $d$ on the y-axis against the square root of time $\sqrt t$ on the x-axis.
2. The initial part of the curve is a straight line (theoretical curve up to $U = 60\%$). Extend it back to cut the y-axis at the corrected zero reading $d_0$ (this corrects for initial compression).
3. Draw a second line from $d_0$ with abscissa 1.15 times the first (the slope of the line is 1/1.15 of that of the first line).
4. The intersection of this second line with the experimental curve gives $\sqrt{t_{90}}$, for 90% consolidation.
5. Compute the coefficient of consolidation, using $T_v = 0.848$ at $U = 90\%$:

$$
c_v = \frac{0.848\,H_{dr}^2}{t_{90}}
$$

where $H_{dr}$ is the average drainage path during the load increment (half the average specimen thickness for double drainage).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In Taylor's method, the dial reading is plotted against
   A. log t
   B. t
   C. sqrt(t)
   D. 1/t
   Answer: C. Reading is plotted against the square root of time. (Key point: Plot)

2. The initial straight portion is extended back to the y-axis to get
   A. the d100 reading
   B. the corrected zero reading d0 (initial compression correction)
   C. t90
   D. cv
   Answer: B. The intercept gives the corrected zero reading. (Key point: Corrected zero)

3. The line from d0 with abscissa 1.15 times the first line's cuts the curve at
   A. sqrt(t50)
   B. sqrt(t90)
   C. sqrt(t100)
   D. sqrt(t0)
   Answer: B. Its intersection with the experimental curve gives sqrt(t90) for 90% consolidation. (Key point: 1.15 line)

4. cv is calculated as
   A. 0.197 Hdr^2/t90
   B. 0.848 Hdr^2/t90
   C. 0.848 t90/Hdr^2
   D. 0.848 Hdr/t90
   Answer: B. Tv = 0.848 at U = 90%, so cv = 0.848 Hdr^2 / t90. (Key point: cv formula)

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