Question $1(10$ marks$)$

A rock specimen fails under a shear stress of $15$ MPa $.$ If the angle of internal friction

is $35$ degrees, calculate the normal stress acting on the failure plane.

Question $2(10$ marks$)$

A rock sample is subjected to a total stress of $100$ MPa $.$ The pore pressure within the

rock is measured to be $20$ MPa $.$ Calculate the effective stress.

Question $3(10$ marks$)$

A rock specimen undergoes a uniaxial compressive test. The maximum load applied

before failure is $150$ kN $.$ The diameter of the cylindrical sample is $50$ mm $.$ Calculate the

uniaxial compressive strength $()$ of the rock.

Question $4(25$ marks$)$

A firm is tasked to measure the mechanical properties of a rock. The equipment failed

to measure the peak axial load during their first uniaxial compression test. Still, the

technician did record that the specimen failed by forming a single fracture inclined at

$20$ deg to the loading axis. In a subsequent UCS test, the axial stress, the axial stress,

was recorded as $85$ MPa $.$ Based on these results, propose a failure criterion of the

form ${}_1=a{}_3+b$ for the rock.

Question $5(45$ marks$)$

Direct shear test results on a rock are represented by a peak shear strength envelope

of $=0\mathrm{.}12+$ and a residual strength envelope of $=0\mathrm{.}08+\frac{1}{\sqrt[\phantom{2}]{3}}$ in MPa. For

testing under normal stress of $1$ MPa $.$ It is required to complete the following tasks:

a$.$ Determine peak friction angle $(10$ marks$)$

b$.$ Calculate residual friction angle $(10$ marks$)$

c$.$ Estimate the initial normal closure of the joint surfaces for a normal

stiffness of $500kP\frac{a}{m}m(10$ marks$)$

d$.$ Estimate shear stiffness for a relative shear displacement of $1\mathrm{.}8$ mm $(15$

marks$)$