Question

The density index ID in terms of densities is given by ______

(a) \(\frac{(γ_{d,max}-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d}) γ_d}\)

(b) \(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d,min}) γ_d}\)

(c) \(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_d-γ_{d,max}) γ_d}\)

(d) \(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d}) γ_d}\)

The question was asked in an online interview.

My question is taken from Density Index and Relative Compaction topic in section Preliminary Definitions and Relationships of Geotechnical Engineering I

Answer 1

The correct option is (b) \(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d,min}) γ_d}\)

Explanation: The density index is given by,

ID=\(\frac{(e_{max} – e)}{(e_{max}-e_{min})}\)

\(e=(\frac{Gγ_w}{γ_d})-1\)

\(e_{max}=(\frac{Gγ_w}{γ_{d,min}})-1\)

\(e_{min}=(\frac{Gγ_w}{γ_{d,max}})-1\)

substituting the values of voids ratio

ID=\(\frac{((\frac{Gγ_w}{γ_{d,min}}) – (\frac{Gγ_w}{γ_d}))}{((\frac{Gγ_w}{γ_{d,min}})- (\frac{Gγ_w}{γ_{d,max}}))}\)

ID=\(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d,min}) γ_d}.\)