GATE Agricultural Engineering Soil and Water Conservation Engineering Flashcards

24 sample cards from the Soil and Water Conservation Engineering deck

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

1. State Manning's equation for uniform open channel flow.

   $$V = \frac{1}{n} R^{2/3} S^{1/2}$$ where $n$ is Manning's roughness coefficient, $R = A/P$ the hydraulic radius, and $S$ the bed slope. Discharge is $Q = AV = \dfrac{1}{n} A R^{2/3} S^{1/2}$.

2. What defines critical depth in an open channel and the critical condition?

   Critical depth is the flow depth at which specific energy is minimum for a given discharge, where $Fr = 1$. The general condition is $$\frac{Q^{2} T}{g A^{3}} = 1$$ where $T$ is the top width. For a rectangular channel, $y_{c} = \left(\dfrac{q^{2}}{g}\right)^{1/3}$.

3. What is a hydraulic jump and what does it dissipate?

   A hydraulic jump is a sudden transition from supercritical ($Fr>1$) to subcritical ($Fr<1$) flow, marked by an abrupt rise in depth and turbulence. It dissipates excess kinetic energy as heat, and is used downstream of spillways/structures for energy dissipation.

4. State Buckingham's $\pi$-theorem.

   If a physical problem involves $n$ variables expressed in $m$ fundamental dimensions, the relationship can be reduced to $(n - m)$ independent dimensionless $\pi$ terms. Each $\pi$ group is formed by combining repeating variables with one of the remaining variables.

5. Give the definition and physical meaning of the Weber number.

   $$We = \frac{\rho V^{2} L}{\sigma}$$ It is the ratio of inertial force to surface-tension force. It is significant where surface tension dominates, such as in capillary flows, droplet formation, and thin sheets of liquid.

6. Define the Euler number and the Mach number.

   Euler number: $Eu = \dfrac{V}{\sqrt{p/\rho}}$ (or $\dfrac{\Delta p}{\rho V^{2}}$), the ratio of inertial to pressure force. Mach number: $M = \dfrac{V}{C}$, the ratio of fluid velocity to the speed of sound $C$, indicating compressibility effects ($M>1$ = supersonic).

7. State the three-phase relationship void ratio (e) versus porosity (n).

   Void ratio $e = \dfrac{V_{v}}{V_{s}}$ (voids to solids); porosity $n = \dfrac{V_{v}}{V}$ (voids to total). They relate as $$n = \frac{e}{1+e}, \qquad e = \frac{n}{1-n}.$$

8. Define degree of saturation and the basic phase relationship $S e = w G_s$.

   Degree of saturation $S = \dfrac{V_{w}}{V_{v}}$ (fraction of voids filled with water). The fundamental identity linking the phases is $$S\, e = w\, G_{s}$$ where $w$ is water content, $G_{s}$ specific gravity of solids, and $e$ void ratio.

9. Give the formula relating bulk (moist) unit weight to specific gravity, void ratio, and saturation.

   $$\gamma = \frac{(G_{s} + S e)\,\gamma_{w}}{1 + e}$$ For saturated soil ($S=1$): $\gamma_{sat} = \dfrac{(G_{s}+e)\gamma_{w}}{1+e}$; for dry soil ($S=0$): $\gamma_{d} = \dfrac{G_{s}\gamma_{w}}{1+e}$.

10. What are the Atterberg limits and their associated indices?

    The Atterberg limits are the liquid limit ($w_L$), plastic limit ($w_P$), and shrinkage limit ($w_S$), marking water-content boundaries between liquid, plastic, semi-solid, and solid states. Plasticity index $I_{P} = w_{L} - w_{P}$, the range of water content over which soil is plastic.

11. Define consistency index and liquidity index.

    Liquidity index $I_{L} = \dfrac{w - w_{P}}{I_{P}}$ and consistency index $I_{C} = \dfrac{w_{L} - w}{I_{P}}$, where $w$ is natural water content. Note $I_{L} + I_{C} = 1$. High $I_{L}$ (near 1) means soft soil near liquid limit; high $I_{C}$ means stiff soil.

12. State Darcy's law for flow through soils.

    $$q = k\, i\, A, \qquad v = k\, i$$ where $q$ is discharge, $k$ the coefficient of permeability, $i = h/L$ the hydraulic gradient, and $A$ the cross-sectional area. $v$ is the discharge (superficial) velocity. Valid for laminar flow.

13. How is the coefficient of permeability of layered soil computed for flow parallel and perpendicular to bedding?

    Parallel to layers (equivalent horizontal): $$k_{H} = \frac{\sum k_{i} H_{i}}{\sum H_{i}}$$ Perpendicular to layers (equivalent vertical): $$k_{V} = \frac{\sum H_{i}}{\sum (H_{i}/k_{i})}$$ Always $k_{H} \geq k_{V}$.

14. What is the seepage discharge from a flow net, and what is its formula?

    From a flow net, $$q = k\, H\, \frac{N_{f}}{N_{d}}$$ where $H$ is the total head loss, $N_{f}$ the number of flow channels, and $N_{d}$ the number of equipotential drops. $N_f/N_d$ is the shape factor of the net.

15. Define the critical hydraulic gradient and the quicksand condition.

    The critical hydraulic gradient is $$i_{c} = \frac{G_{s} - 1}{1 + e} = \frac{\gamma'}{\gamma_{w}}$$ Quicksand (boiling) occurs when the upward seepage gradient reaches $i_c$, making effective stress zero so the soil loses shear strength.

16. State Terzaghi's principle of effective stress.

    Total stress equals effective stress plus pore water pressure: $$\sigma = \sigma' + u$$ Effective stress $\sigma'$ (carried by the soil skeleton) governs shear strength and volume change; $u$ is the pore water pressure.

17. State the Mohr-Coulomb shear strength equation.

    $$\tau_{f} = c + \sigma' \tan\phi$$ where $\tau_f$ is shear strength on the failure plane, $c$ the cohesion, $\sigma'$ the effective normal stress, and $\phi$ the angle of internal friction. For sands $c \approx 0$; for saturated clays (undrained) $\phi_u \approx 0$.

18. On a Mohr's circle of stress, what do the centre and radius represent?

    For principal stresses $\sigma_{1}$ and $\sigma_{3}$, the centre lies at $\left(\dfrac{\sigma_{1}+\sigma_{3}}{2}, 0\right)$ and the radius is $\dfrac{\sigma_{1}-\sigma_{3}}{2}$, which equals the maximum shear stress $\tau_{max}$.

19. Give the normal and shear stress on a plane inclined at angle $\theta$ from Mohr's circle.

    $$\sigma_{\theta} = \frac{\sigma_{1}+\sigma_{3}}{2} + \frac{\sigma_{1}-\sigma_{3}}{2}\cos 2\theta$$ $$\tau_{\theta} = \frac{\sigma_{1}-\sigma_{3}}{2}\sin 2\theta$$ where $\theta$ is measured from the major principal plane.

20. What is the orientation of the failure plane in terms of the friction angle $\phi$?

    The failure plane makes an angle $$\theta_{f} = 45^{\circ} + \frac{\phi}{2}$$ with the major principal plane (the plane on which $\sigma_1$ acts), where $\phi$ is the angle of internal friction.

21. State the relationship between major and minor principal stresses at failure (Mohr-Coulomb).

    $$\sigma_{1} = \sigma_{3} \tan^{2}\!\left(45^{\circ}+\frac{\phi}{2}\right) + 2c\,\tan\!\left(45^{\circ}+\frac{\phi}{2}\right)$$ Using $N_{\phi} = \tan^{2}(45^{\circ}+\phi/2)$: $\sigma_{1} = \sigma_{3} N_{\phi} + 2c\sqrt{N_{\phi}}$.

22. What is the uniformity coefficient and coefficient of curvature in soil grading?

    $$C_{u} = \frac{D_{60}}{D_{10}}, \qquad C_{c} = \frac{(D_{30})^{2}}{D_{60}\, D_{10}}$$ where $D_{10}, D_{30}, D_{60}$ are particle sizes at 10%, 30%, 60% passing. A soil is well graded if $C_{u}$ is high and $1 < C_{c} < 3$.

23. Differentiate the discharge velocity and seepage velocity in soil.

    Discharge (superficial) velocity $v = ki$ is based on total cross-sectional area. Seepage velocity is the actual velocity through the voids: $$v_{s} = \frac{v}{n} = \frac{ki}{n}$$ where $n$ is porosity. Since $n<1$, $v_{s} > v$.

24. What are the three common laboratory tests for shear strength of soil and the drainage conditions they represent?

    (1) Unconsolidated-Undrained (UU) test — no drainage during consolidation or shear. (2) Consolidated-Undrained (CU) test — drainage during consolidation only. (3) Consolidated-Drained (CD) test — full drainage throughout. They are run in a triaxial cell (or direct shear/vane for field).

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