GATE Mining Engineering Geomechanics and Ground Control Flashcards

24 sample cards from the Geomechanics and Ground Control deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is the hydraulic fracturing method of in-situ stress measurement, and what does it determine?

   A sealed borehole interval is pressurized until the rock fractures; the breakdown, shut-in, and reopening pressures are recorded. The shut-in pressure equals the minimum horizontal stress, and the maximum horizontal stress is computed from the breakdown pressure and tensile strength.

2. State the Mohr-Coulomb failure criterion for rock in terms of shear strength.

   $$\tau = c + \sigma_n \tan \phi$$ where $\tau$ is shear strength on the failure plane, $c$ is cohesion, $\sigma_n$ is the normal stress, and $\phi$ is the angle of internal friction.

3. Express the Mohr-Coulomb criterion in terms of principal stresses at failure.

   $$\sigma_1 = \sigma_c + \sigma_3 \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)$$ where $\sigma_c = \dfrac{2c\cos\phi}{1-\sin\phi}$ is the uniaxial compressive strength, and $\sigma_1$, $\sigma_3$ are major and minor principal stresses.

4. Write the empirical Hoek-Brown failure criterion for intact rock.

   $$\sigma_1 = \sigma_3 + \sigma_c \sqrt{ m_i \frac{\sigma_3}{\sigma_c} + 1 }$$ where $\sigma_c$ is intact UCS and $m_i$ is a material constant characterising the rock type.

5. On what failure plane angle (relative to $\sigma_1$) does Mohr-Coulomb failure occur?

   Failure occurs on a plane inclined at $$\theta = 45^\circ + \frac{\phi}{2}$$ to the direction of the major principal stress $\sigma_1$ (i.e., at angle $45^\circ - \phi/2$ to the failure plane normal).

6. What is the peak particle velocity (PPV) and why is it the key parameter for blast-induced ground vibration?

   PPV is the maximum velocity of a ground particle as a vibration wave passes, usually in mm/s. It correlates best with structural damage potential, so blasting damage criteria and statutory limits are specified in terms of PPV.

7. Write the standard scaled-distance equation for predicting peak particle velocity from blasting.

   $$v = K \left( \frac{R}{\sqrt{Q}} \right)^{-\beta}$$ where $v$ is PPV, $R$ is distance, $Q$ is maximum charge per delay, $R/\sqrt{Q}$ is the scaled distance, and $K$, $\beta$ are site constants.

8. Define maximum charge per delay and explain its role in controlling ground vibration.

   It is the largest explosive weight detonating within any single delay interval (typically within 8 ms). Because vibration amplitude depends on charge per delay rather than total charge, using delay detonators to limit it reduces PPV and protects nearby structures.

9. According to the Kirsch solution, what is the tangential stress at the boundary of a circular opening in a hydrostatic stress field $p$?

   For a hydrostatic field ($\sigma_h = \sigma_v = p$), the boundary tangential stress is uniform: $$\sigma_\theta = 2p$$ This is the classic stress concentration factor of 2 around a circular hole.

10. State the Kirsch boundary tangential stresses around a circular opening under a uniaxial far-field stress $\sigma$ at the sidewall and at the roof.

    At the boundary ($r=a$): at the sidewall (perpendicular to load), $\sigma_\theta = 3\sigma$; at the roof/floor (parallel to load), $\sigma_\theta = -\sigma$ (tensile). General boundary value: $\sigma_\theta = \sigma(1 - 2\cos 2\theta)$.

11. How does the ratio of horizontal to vertical stress $k$ affect tangential boundary stresses around a circular opening?

    At the boundary, the sidewall tangential stress is $\sigma_\theta = (3k - 1)\sigma_v$ and the roof tangential stress is $\sigma_\theta = (3 - k)\sigma_v$. Tensile roof stress (instability) appears when $k > 3$, and tensile sidewall stress when $k < 1/3$.

12. Define subsidence and distinguish trough (continuous) subsidence from discontinuous subsidence.

    Subsidence is the lowering of the ground surface due to underground extraction. Trough subsidence is a smooth, continuous dish-shaped settlement typical of longwall mining; discontinuous subsidence involves abrupt features like sinkholes, pits, or steps, common over shallow workings and pillar collapse.

13. Define the angle of draw in subsidence engineering.

    The angle of draw is the angle between the vertical drawn through the edge of the underground extraction and the line to the point of zero (negligible) surface subsidence at the limit of the subsidence trough. It defines the lateral extent of surface influence.

14. What is the subsidence factor, and where do tensile and compressive strains occur in a subsidence trough?

    The subsidence factor is the ratio of maximum surface subsidence to extracted seam thickness ($S_{\max}/m$). The trough develops tensile strain over the edges/ribside (convex curvature) and compressive strain over the centre (concave curvature), causing structural damage.

15. State the factor of safety definition used in slope stability analysis.

    $$FoS = \frac{\text{resisting (shear strength) forces}}{\text{driving (shear stress) forces}}$$ A slope is considered stable when $FoS > 1$, typically designed to 1.3-1.5; the resisting term uses $c + \sigma_n \tan\phi$.

16. For plane (planar) failure of a rock slope, what kinematic conditions must the failure plane satisfy?

    The failure plane must strike nearly parallel to the slope face (within about $\pm 20^\circ$), dip less steeply than the slope face (daylight on the face), and dip more steeply than its own friction angle ($\psi_p > \phi$), with release surfaces at the lateral limits.

17. Give the limit-equilibrium factor of safety for a dry planar slope failure surface.

    $$FoS = \frac{c\,A + W\cos\psi_p \tan\phi}{W\sin\psi_p}$$ where $\psi_p$ is the failure-plane dip, $W$ the weight of the sliding block, $A$ the plane area, $c$ cohesion, and $\phi$ friction angle. Water pressure reduces the normal (frictional) term.

18. Name the four common modes of rock slope failure controlled by discontinuities.

    Planar failure, wedge failure (intersection of two discontinuities), toppling failure (steeply dipping blocks rotating), and circular/rotational failure (in heavily fractured or soil-like masses).

19. Write the tributary area method formula for the average vertical stress on a square mine pillar.

    $$\sigma_p = \sigma_v \frac{(W_p + W_o)^{2}}{W_p^{2}}$$ where $\sigma_v = \gamma H$ is the vertical (overburden) stress, $W_p$ is pillar width, and $W_o$ is opening (roadway) width. In terms of extraction ratio $e$: $\sigma_p = \dfrac{\sigma_v}{1-e}$.

20. How is the extraction ratio related to pillar stress, and what is the pillar factor of safety?

    Pillar stress $\sigma_p = \dfrac{\sigma_v}{1 - e}$, where $e$ is the area extraction ratio. The factor of safety is $$FoS = \frac{S_p}{\sigma_p}$$ the ratio of pillar strength $S_p$ to the average pillar stress.

21. State the general form of an empirical pillar strength formula and explain the width-to-height effect.

    A common form is $$S_p = S_1 \left( a + b\frac{W}{H} \right)$$ (e.g., Obert-Duvall, Bieniawski), where $S_1$ is unit pillar strength and $W/H$ is the width-to-height ratio. Strength increases with $W/H$ because confinement of the pillar core grows as pillars become squatter.

22. Compare active and passive roof support systems with examples.

    Active supports apply a pre-load to the rock immediately (e.g., tensioned/pre-tensioned rock bolts, hydraulic powered supports/chocks), reinforcing before deformation. Passive supports develop load only as the rock deforms onto them (e.g., steel arches, timber/wooden props, grouted untensioned bolts, cribs).

23. Describe the three principal mechanisms by which rock bolts support a mine roof.

    (1) Suspension - anchoring weak immediate roof to a stronger stratum above; (2) beam building - clamping thin laminations into a thicker composite beam to increase bending stiffness; (3) keying/friction - increasing interlock and shear resistance across discontinuities in fractured ground.

24. What is mine filling (backfilling), and what are its main functions and types?

    Mine filling is placing waste material into mined-out voids. Functions: regional ground/pillar support, surface subsidence control, waste disposal, and reduced fire/spontaneous combustion risk. Main types: hydraulic fill, rock/dry fill, paste fill, and cemented (cemented hydraulic or paste) fill for added strength.

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