Consider the following statements, Out of these statements, 1.Coefficient of consolidation cv is dimensionless parameter 2.Time factor Tv is having units of time such as second or years 3.Time factor Tv is dimensionless parameter
- AOnly 1 is correct
- BOnly 2 is correct
- COnly 3 is correctCorrect
- DBoth 1 and 3 are correct
Solution
Concept
Properties of consolidation parameters including time factor and coefficient of consolidation.
Formula
Tv = (cv × t) / (d²)
Step-by-step solution
1. Analyze statement 1: The coefficient of consolidation cv is given by cv = k / (mv × γw), which has dimensions of L²/T (e.g., m²/s). Therefore, it is not dimensionless. 2. Analyze statement 2 & 3: The time factor Tv is a dimensionless number defined as Tv = (cv × t) / d². Thus, Tv is dimensionless, making statement 2 incorrect and statement 3 correct. 3. Conclusion: Only statement 3 is correct.
Explanation
The coefficient of consolidation (cv) has dimensions of area per unit time (e.g., cm²/s or m²/year), so it is not dimensionless. The time factor (Tv) is a dimensionless parameter used in Terzaghi's consolidation theory to relate time, drainage path, and the coefficient of consolidation.
Why the other options are wrong
A: Statement 1 is incorrect because cv has units of L²/T. B: Statement 2 is incorrect because Tv is dimensionless. D: Statement 1 is incorrect.
Shortcut
Remember that time factors in soil mechanics (like Tv) and safety factors are generally dimensionless ratios.
Common mistake
Confusing coefficient of consolidation (cv) with a dimensionless ratio, or assuming time factor has units of time because 'time' is in its name.
Correct answer: C) Only 3 is correct