GATE 2024 CE (CE2) – Question 54
The in-situ percentage of voids of a sand deposit is 50%. The maximum and minimum densities of sand determined from the laboratory tests are 1.8 g/cm$^3$ and 1.3 g/cm$^3$, respectively. Assume the specific gravity of sand as 2.7.
The relative density index of the in-situ sand is ______________ (rounded off to 2 decimal places).
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Correct answer: 0.12 to 0.14
Explanation
A porosity of 50% means a void ratio $e = \frac{n}{1 - n} = 1$. The in-situ dry density is $\rho_d = \frac{G_s\rho_w}{1 + e} = \frac{2.7}{2} = 1.35$ g/cm³. The relative density in terms of the densities is $I_D = \frac{\rho_{max}(\rho_d - \rho_{min})}{\rho_d(\rho_{max} - \rho_{min})} = \frac{1.8 \times (1.35 - 1.3)}{1.35 \times (1.8 - 1.3)} = \frac{0.09}{0.675} = 0.133$.