If D1 & D2 are inside and outside diameters of the cutting edge then the Area ratio of the sampler is determined using the relation:
Calculating Soil Sampler Area Ratio
The area ratio of a soil sampler is an important parameter in geotechnical engineering. It provides an indication of the disturbance caused to the soil sample during the sampling process. A lower area ratio generally means less disturbance, leading to a more representative sample.
The area ratio is defined with respect to the cutting edge of the sampler. The cutting edge has an inside diameter (D1) and an outside diameter (D2). The area ratio is calculated as the ratio of the area displaced by the sampler tube wall to the area of the collected soil sample, expressed as a percentage.
- D1: Inside diameter of the cutting edge. This defines the area of the collected soil sample.
- D2: Outside diameter of the cutting edge. This defines the total area occupied by the sampler tube at the cutting edge.
Understanding the Area Ratio Formula
The area of the soil sample collected is based on the inside diameter D1. This area is given by:
Area of sample \( = \frac{\pi}{4} D_1^2 \)
The area displaced by the sampler tube wall is the difference between the area enclosed by the outside diameter D2 and the area enclosed by the inside diameter D1. This area is given by:
Area displaced \( = \frac{\pi}{4} D_2^2 - \frac{\pi}{4} D_1^2 = \frac{\pi}{4} (D_2^2 - D_1^2) \)
The area ratio (Ar) is the ratio of the displaced area to the sample area, multiplied by 100%:
Area Ratio \( = \frac{\text{Area displaced}}{\text{Area of sample}} \times 100 \) %
Substituting the expressions for the areas:
\( \text{Area Ratio} = \frac{\frac{\pi}{4} (D_2^2 - D_1^2)}{\frac{\pi}{4} D_1^2} \times 100 \)
The \( \frac{\pi}{4} \) terms cancel out, leaving the formula for the area ratio:
\( \text{Area Ratio} = \frac{D_2^2 - D_1^2}{D_1^2} \times 100 \)
Evaluating the Given Options
Let's compare this derived formula with the provided options:
| Option | Formula |
|---|---|
| 1 | \( \frac{D_2^2 - D_1^2}{2D_1^2} \times 100 \) |
| 2 | \( \frac{D_2^3 - D_1^3}{D_1^2} \times 100 \) |
| 3 | \( \frac{D_1^2 - D_2^2}{D_1^2} \times 100 \) |
| 4 | \( \frac{D_2^2 - D_1^2}{D_1^2} \times 100 \) |
Comparing the derived formula \( \frac{D_2^2 - D_1^2}{D_1^2} \times 100 \) with the options, we see that Option 4 matches the correct relation for the area ratio of the sampler using the inside diameter D1 and outside diameter D2 of the cutting edge.
Option 1 has an extra factor of 2 in the denominator. Option 2 uses cubic terms of diameters instead of square terms. Option 3 incorrectly subtracts \( D_2^2 \) from \( D_1^2 \), which would result in a negative area ratio since \( D_2 \) is always greater than \( D_1 \) for a standard sampler cutting edge.
Therefore, the correct relation is based on the difference in the squares of the outside and inside diameters, divided by the square of the inside diameter, multiplied by 100.
Important Questions from Site Investigation and Sub-Soil Exploration
In order to minimize sampling disturbance the area ratio A r,\(\left( {{{\rm{A}}_{\rm{r}}} = \frac{{{\rm{D}}_{\rm{e}}^2 - {\rm{D}}_{\rm{i}}^2}}{{{\rm{D}}_{\rm{i}}^2}}{\rm{\;}}} \right)\) should be
Shelby tube is one of the most widely used devices for:
Which one of the following is NOT a soil exploration technique?
If the sampler tube is pushed at the bottom of the bore hole to a distance of 585 mm with length of the sample recovered being 535 mm. Then the value of recovery ratio is ______.
If the inside diameters of the cutting edge and sample tube are 68 mm & 70 mm respectively and 76 mm & 74 mm are the outside diameters of the cutting edge and sample tube respectively, then outside clearance of the sampler is _______.
