GATE Civil Engineering Geotechnical Engineering Syllabus

How does temperature affect the coefficient of permeability $k$? Give the correction relation.

$$k=\frac{K\,\gamma_w}{\mu}$$ where $K$ is absolute permeability (property of soil only) and $\mu$ is the fluid viscosity. As temperature rises, $\mu$ falls so $k$ increases; values are normalized to $27^{\circ}$C in India.

Give the equivalent permeability for stratified soils for flow parallel to and perpendicular to the bedding planes.

Parallel (horizontal): $$k_H=\frac{\sum k_i H_i}{\sum H_i}$$ Perpendicular (vertical): $$k_V=\frac{\sum H_i}{\sum \frac{H_i}{k_i}}$$ Always $k_H\ge k_V$.

Which lab test suits coarse-grained soils for $k$ and which suits fine-grained soils? Give the falling-head formula.

Constant-head test → coarse soils; falling-head test → fine soils. Falling head: $$k=\frac{aL}{At}\ln\frac{h_1}{h_2}=2.303\frac{aL}{At}\log_{10}\frac{h_1}{h_2}$$

State the Laplace equation governing steady two-dimensional seepage in isotropic soil.

$$k_x\frac{\partial^2 h}{\partial x^2}+k_z\frac{\partial^2 h}{\partial z^2}=0$$ For isotropic soil ($k_x=k_z$): $\dfrac{\partial^2 h}{\partial x^2}+\dfrac{\partial^2 h}{\partial z^2}=0$.

What two families of curves make up a flow net, and how do they intersect?

Flow lines (paths of water particles) and equipotential lines (lines of equal total head). In an isotropic medium they intersect orthogonally (at $90^{\circ}$) forming approximate curvilinear squares.

Give the formula for seepage discharge through a flow net, defining each term.

$$q=k\,H\frac{N_f}{N_d}$$ where $H$ = total head loss, $N_f$ = number of flow channels, $N_d$ = number of equipotential drops, $k$ = permeability. ($q$ is per unit length normal to the section.)

How is the head loss per equipotential drop and the shape factor of a flow net defined?

Head loss per drop $\Delta h=\dfrac{H}{N_d}$. The shape factor of the flow net is $\dfrac{N_f}{N_d}$, a property of the geometry independent of $k$ and $H$.

For anisotropic soil, what coordinate transformation makes the flow net valid, and what effective permeability is used?

Transform the horizontal scale: $x'=x\sqrt{\dfrac{k_z}{k_x}}$ (or scale by $\sqrt{k_z/k_x}$). The equivalent isotropic permeability used in $q$ is $$k'=\sqrt{k_x\,k_z}$$

How is uplift (pore) pressure at a point under a hydraulic structure found from a flow net?

Determine the number of drops $n_d$ from upstream up to the point. Pressure head $=$ total head $-$ elevation head, where total head at the point $=H-n_d\,\Delta h$. Uplift pressure $u=\gamma_w\times$ pressure head.

What is the exit gradient $i_e$ at the downstream end of a flow net and why does it matter?

$$i_e=\frac{\Delta h}{l}$$ where $l$ is the length of the last flow field at exit. Piping occurs when $i_e$ approaches the critical gradient $i_c$; structures are designed so $i_e \ll i_c$.

Define the critical hydraulic gradient $i_c$ and give its typical value.

$$i_c=\frac{\gamma'}{\gamma_w}=\frac{G-1}{1+e}$$ It is the gradient at which effective stress becomes zero; typically $i_c\approx 1$ for most soils.

What is the quicksand (boiling) condition and when does it occur?

Quicksand is a hydraulic condition (not a soil type) where upward seepage makes effective stress zero, so the soil loses all shear strength. It occurs when the upward hydraulic gradient $i$ reaches the critical gradient $i_c=\dfrac{G-1}{1+e}$.

Define the factor of safety against piping/heave using the critical gradient.

$$F=\frac{i_c}{i_e}$$ where $i_e$ is the actual exit gradient. A value $F\ge 4$–$5$ is commonly required for safety against piping at the downstream toe.

State Terzaghi's principle of effective stress equation.

$$\sigma=\sigma'+u$$ where $\sigma$ = total normal stress, $\sigma'$ = effective stress (carried by the soil skeleton), and $u$ = pore water pressure. Effective stress controls strength and volume change.

How does the seepage force per unit volume act in a soil, and what is its magnitude?

$$j=i\,\gamma_w$$ The seepage force per unit volume acts in the direction of flow. Downward seepage increases effective stress; upward seepage reduces it.

How do upward and downward seepage change effective stress at depth $z$ compared to no flow?

No flow: $\sigma'=\gamma' z$. Upward seepage: $\sigma'=\gamma' z - i\gamma_w z$ (reduced). Downward seepage: $\sigma'=\gamma' z + i\gamma_w z$ (increased).

Explain capillary rise in soils and give the maximum height formula.

Surface tension draws water up the soil pores. Maximum capillary rise: $$h_c=\frac{4T\cos\alpha}{\gamma_w\,d}$$ where $T$ = surface tension, $d$ = pore (tube) diameter. Finer soils → smaller $d$ → greater $h_c$.

What is the nature of pore pressure in the capillary zone, and how does it affect effective stress?

Pore water pressure is negative (suction) in the capillary zone: $u=-\gamma_w h_c$. This negative pressure increases effective stress and produces apparent cohesion in moist sands.

State the relationship between dry unit weight $\gamma_d$, bulk unit weight $\gamma$ and water content $w$ used in compaction.

$$\gamma_d=\frac{\gamma}{1+w}$$ Compaction is the densification of soil by mechanical energy expelling air, increasing $\gamma_d$ at constant solids.

Describe the compaction curve and define optimum moisture content (OMC) and maximum dry density (MDD).

Plot of $\gamma_d$ vs $w$ is bell-shaped. The peak gives the maximum dry density (MDD) at the corresponding optimum moisture content (OMC). Beyond OMC, added water occupies volume and reduces $\gamma_d$.

Write the equation of the zero-air-voids line on a compaction plot.

$$\gamma_{d}=\frac{G\,\gamma_w}{1+wG}\quad(\text{for }S=100\%)$$ The compaction curve always lies below this theoretical line because some air remains.

Compare Standard Proctor and Modified Proctor compaction tests in terms of energy and result.

Modified Proctor imparts much higher compactive energy (≈ 4.5 times Standard). Higher energy gives a higher MDD and a lower OMC, shifting the compaction curve up and to the left.

Differentiate consolidation from compaction.

Compaction: rapid densification by mechanical energy expelling AIR from partially saturated soil. Consolidation: gradual volume reduction of a saturated soil under sustained load by expelling WATER (and dissipating excess pore pressure) over time.

Define the coefficient of compressibility $a_v$ and coefficient of volume change $m_v$.

$$a_v=-\frac{\Delta e}{\Delta \sigma'},\qquad m_v=\frac{a_v}{1+e_0}$$ $a_v$ is the slope of the $e$–$\sigma'$ curve; $m_v$ is the volumetric strain per unit stress increase.

Give the compression index $C_c$ and the consolidation settlement formula for a normally consolidated clay.

$$C_c=\frac{\Delta e}{\log_{10}(\sigma'_2/\sigma'_1)},\qquad S_c=\frac{C_c\,H}{1+e_0}\log_{10}\frac{\sigma'_0+\Delta\sigma'}{\sigma'_0}$$ Skempton's empirical relation: $C_c=0.009\,(w_L-10)$ for undisturbed clay.

Distinguish normally consolidated (NC) and overconsolidated (OC) clay using the overconsolidation ratio (OCR).

$$OCR=\frac{\sigma'_p}{\sigma'_0}$$ where $\sigma'_p$ = preconsolidation pressure, $\sigma'_0$ = present effective overburden. NC: $OCR=1$; OC: $OCR>1$ (soil was once loaded to a higher stress).

State Terzaghi's one-dimensional consolidation governing equation and define the coefficient of consolidation.

$$\frac{\partial u}{\partial t}=C_v\frac{\partial^2 u}{\partial z^2},\qquad C_v=\frac{k}{m_v\,\gamma_w}$$ where $u$ is excess pore pressure and $C_v$ governs the rate of consolidation.

Define the time factor $T_v$ and the degree of consolidation $U$, and give the $T_v$–$U$ relations.

$$T_v=\frac{C_v\,t}{d^2}$$ ($d$ = max drainage path). For $U<60\%$: $T_v=\dfrac{\pi}{4}U^2$; for $U>60\%$: $T_v=1.781-0.933\log_{10}(100-U\%)$. Key values: $U=50\%\Rightarrow T_v=0.197$; $U=90\%\Rightarrow T_v=0.848$.

For a clay layer of thickness $H$, what is the drainage path $d$ for single and double drainage?

Double drainage (permeable on both faces): $d=H/2$. Single drainage (permeable on one face only): $d=H$. Since $t\propto d^2$, double drainage consolidates four times faster.

State the Mohr–Coulomb failure criterion in terms of effective stress.

$$\tau_f=c'+\sigma'\tan\phi'$$ where $\tau_f$ = shear strength on the failure plane, $c'$ = effective cohesion, $\phi'$ = effective angle of internal friction, $\sigma'$ = effective normal stress.

On a Mohr's circle, give the formulas for the center, radius, and the orientation of the failure plane.

Center $=\dfrac{\sigma_1+\sigma_3}{2}$, radius $=\dfrac{\sigma_1-\sigma_3}{2}$. The failure plane makes angle $\theta=45^{\circ}+\dfrac{\phi}{2}$ with the major principal plane.

Write the relation between major and minor principal effective stresses at failure (Rankine) for a $c'$–$\phi'$ soil.

$$\sigma'_1=\sigma'_3\tan^2\!\left(45^{\circ}+\tfrac{\phi'}{2}\right)+2c'\tan\!\left(45^{\circ}+\tfrac{\phi'}{2}\right)$$ Often written $\sigma'_1=\sigma'_3 N_\phi+2c'\sqrt{N_\phi}$ with $N_\phi=\tan^2(45+\phi'/2)$.

Compare the three triaxial test drainage conditions: UU, CU, and CD.

UU (unconsolidated-undrained): no drainage in either stage, gives total-stress $\phi_u\approx 0$ for saturated clay. CU (consolidated-undrained): consolidate then shear undrained, pore pressures measured give $c',\phi'$. CD (consolidated-drained): full drainage, directly gives effective parameters $c',\phi'$ (slow test).

For a saturated clay under undrained loading, what is $\phi_u$ and how is undrained shear strength related to unconfined compressive strength $q_u$?

$\phi_u=0$ (total-stress). Undrained shear strength $$c_u=\frac{q_u}{2}$$ since in the unconfined compression test $\sigma_3=0$ and failure occurs at $q_u=\sigma_1$.

Define Skempton's pore pressure parameters $A$ and $B$.

$$\Delta u=B\left[\Delta\sigma_3+A(\Delta\sigma_1-\Delta\sigma_3)\right]$$ $B$ relates pore pressure to all-round stress ($B=1$ for fully saturated soil); $A$ relates it to deviator stress and depends on soil type and strain.

Contrast the stress–strain and volume-change behavior of dense vs loose sand during shear.

Dense sand: high peak strength then strain-softens to a lower critical (ultimate) value; it dilates (volume increase). Loose sand: no peak, strain-hardens to the same critical state; it contracts (volume decrease). Both converge at the critical void ratio.

What is the critical void ratio in sands, and what does it imply for liquefaction?

The critical void ratio is the void ratio at which a sand shears at constant volume (zero net dilation/contraction). Sands looser than critical (contractive) can generate large positive pore pressures under undrained cyclic loading and are prone to liquefaction.

Contrast peak and residual (ultimate) shear strength in overconsolidated clays.

Heavily overconsolidated clay shows a high peak strength then drops to a much lower residual strength at large displacement, as clay particles reorient parallel to the shear plane. Residual $\phi'_r$ is used for long-term stability of pre-sheared slopes.