A direct shear test is performed on an intact sample of sandstone. The laboratory determines the cohesive strength ($c$) is $5 \text{ MPa}$ and the angle of internal friction ($\phi$) is $30^{\circ}$. Calculate the maximum shear strength ($\tau$) of the rock if it is subjected to a normal stress ($\sigma_n$) of $12 \text{ MPa}$.

During triaxial testing of a limestone core, failure occurs on a predefined joint surface when the normal stress ($\sigma_n$) is $8 \text{ MPa}$ and the corresponding shear stress ($\tau$) is $7 \text{ MPa}$. A second test on a similar joint surface fails at a normal stress of $15 \text{ MPa}$ and a shear stress of $11 \text{ MPa}$. Determine the cohesion ($c$) and friction angle ($\phi$) of the joint.

A massive rock wedge on a slope is resting on a planar fault. The weight of the wedge exerts a driving shear stress ($\tau_d$) of $4 \text{ MPa}$ down the slope, and a normal stress ($\sigma_n$) of $6 \text{ MPa}$ perpendicular to the fault plane. The fault plane has a cohesion ($c$) of $1.5 \text{ MPa}$ and a friction angle ($\phi$) of $25^{\circ}$. Calculate the Factor of Safety (FS) against sliding.

A $10 \text{ m}$ diameter highway tunnel is being excavated through a slightly weathered granite massif. Detailed geological mapping of the tunnel face yields the following parameters for Barton's Q-System:

Calculate the Rock Mass Quality ($Q$).

A deep exploratory tunnel for a proposed high-speed railway is being driven $1,500 \text{ m}$ below the surface through massive, exceptionally hard, unjointed quartzite.

A water diversion tunnel is being excavated at a moderate depth ($300 \text{ m}$) through a fault zone consisting entirely of heavily sheared, highly altered, wet shale. The rock mass is essentially clay-like, with a very low Geological Strength Index (GSI) value under the Hoek-Brown failure criterion.