A drained shearbox test was carried out on a sample of saturated sand. The normal effective stress of $41\mathrm{.}67$ kPa was

constant throughout the test, and the initial sample dimensions were $60$ mm x $60$ mm on plan x $30$ mm thick. In the

vicinity of the peak shear stress, the data recorded were:

Shear stress, $($kPa$)$

$42\mathrm{.}5$

$43\mathrm{.}1$

$42\mathrm{.}8$

Relative horizontal displacement, x $($mm$)$

$0\mathrm{.}30$

$0\mathrm{.}40$

$0\mathrm{.}80$

Upward movement of the shearbox lid, y $($mm$)0\mathrm{.}05$

$0\mathrm{.}0750\mathrm{.}105$

$($a$)$ Assuming the horizontal plane is the plane of maximum stress obliquity, draw the Mohr circle of stress

for the soil specimen when the shear stress is at a maximum. Determine the orientations of the major

and minor principal effective stresses from vertical. Determine the peak effective friction angle $($peak$),$

the maximum dilation angle $($max$),$ and then estimate the critical state friction angle $($cs$)$ based on an

empirical relationships between the $$peak and max $($i$.$e$.,$ using Bolton $1986).$

$($b$)$ Assuming that the principal stresses are coincident with the directions of the principal plastic strain

increments $($Rowe $1969),$ draw the Mohr circles of strain increments and stress when the shear stress is

at a maximum. Determine the orientations of the major and minor principal effectives stresses from

vertical. Determine the $$peak and the orientation of the plane of maximum stress obliquity.

$($c$)$ Compare the $$peak from the above two assumptions and state how they are algebraically related.

$($d$)$ Three further drained tests on similar samples of the same soil were carried out, at different normal

effective stresses. The peak and critical state shear stresses were:

Normal effective stress, $$v $($kPa$)$

$20$

$100$

$200$

Peak shear stress, peak $($kPa$)$

$23\mathrm{.}8$

$83\mathrm{.}9$

$132\mathrm{.}0$

Critical state shear stress, crit $($kPa$)$

$12\mathrm{.}6$

$63\mathrm{.}2$

$126\mathrm{.}4$

For all four tests, plot the peak and critical state shear stresses, peak and crit$,$ as functions of $$v$.$ Sketch

failure envelopes for both peak and critical states, and comment briefly on their shapes. Which would

you use for design, and why?