Question $1[12$ marks$]$

A sample of dry sand was tested in a direct shear box test under a constant vertical effective stress of $80$ kPa $.$ The

initial dimensions of the sample were $60mm60mm20mm($length x width x height$).$ The test data is presented

in the table below. In this question, ignore any shear box area correction.

a$)$ Plot a graph of shear stress $()$ versus shear strain $().$ You can use Microsoft Excel. Include appropriate

axis labels.

b$)$ Calculate the peak friction angle ${}_{\text{p}\text{e}\text{a}\text{k}}^{\text{'}}$ and critical state friction angle ${}_{\text{c}\text{r}\text{i}\text{t}}^{\text{'}}$ of the sand in this test.

c$)$ Estimate:

i$)$ The shear stress at a shear strain of $3\mathrm{.}0\%.($You can read this off the graph.$)$

ii$)$ The horizontal force applied to the shear box at a shear strain of $3\mathrm{.}0\%.$

iii$)$ The mobilized friction angle ${}_{\text{m}\text{o}\text{b}}^{\text{'}}$ at $3\mathrm{.}0\%$ shear strain.

d$)$ Predict the expected shear stress at critical state ${}_{\text{c}\text{r}\text{i}\text{t}}$ if the test had been carried out at a constant vertical

effective stress of $250$ kPa $.$

e$)$ The exact same sample of sand $($that was previously $60mm60mm20mm)$ is now compacted into

the shear box so that its initial dimensions are $60mm60mm18mm.$ Explain $($qualitatively$)$ what

would happen to the measured ${}_{\text{p}\text{e}\text{a}\text{k}}$ and ${}_{\text{c}\text{r}\text{i}\text{t}}$ when tested at the same vertical effective stress? Provide

reasoning.