Question

To obtain an undisturbed clay soil sample, an Area Ratio of 10 % needs to be achieved for a thin walled sampling tube. If the outer diameter of the tube is 50.8 mm, the inner diameter (in mm) is (rounded off to one decimal place).

Hint:

Area Ratio = (Do^2 - Di^2)/Di^2 x 100; solve for Di from the given Do and Ar.

Answer 1

Correct Answer: 48.4

Step 1: Recall the Area Ratio formula for a sampling tube.
The Area Ratio (\(A_r\)) of a soil sampling tube measures how much the tube wall displaces the soil relative to the sample it collects, and it controls how undisturbed the recovered sample is. It is defined as
\[ A_r (\%) = \frac{D_o^2 - D_i^2}{D_i^2} \times 100 \]
where \(D_o\) is the outer diameter and \(D_i\) is the inner diameter of the cutting edge of the tube. A lower area ratio (usually under 10-15%) means less soil disturbance, which is why thin walled tubes are used for undisturbed clay sampling.

Step 2: Substitute the given values.
Here \(A_r = 10\%\) and \(D_o = 50.8\) mm.
\[ 0.10 = \frac{(50.8)^2 - D_i^2}{D_i^2} \]
\[ 0.10 \, D_i^2 = (50.8)^2 - D_i^2 \]

Step 3: Collect terms and solve for \(D_i\).
\[ D_i^2 + 0.10\, D_i^2 = (50.8)^2 \]
\[ 1.10 \, D_i^2 = 2580.64 \]
\[ D_i^2 = \frac{2580.64}{1.10} = 2346.04 \]
\[ D_i = \sqrt{2346.04} = 48.44 \ \text{mm} \]

Final Answer:
Rounded to one decimal place, the required inner diameter is 48.4 mm.
\[ \boxed{D_i = 48.4 \ \text{mm}} \]