Geotechnical Engg (CE-701) - Unit 4 Short Notes

Unit 4: Geotechnical Engineering - Comprehensive Short Notes

1. Fundamental Soil Properties and Phase Relationships

Definition: Soil is a three-phase system consisting of solids, water, and air. Phase relationships quantitatively describe the arrangement and proportions of these phases.

Key Parameters & Definitions:

Fundamental Relationships:

1. Relationship between $e$, $w$, $G$, $$\displaystyle S_r $$:

$$e = S_r \cdot w \cdot G$$

> [!TIP] This is a **fundamental identity**. Use it to find any unknown parameter when three are known. Ensure $w$ is in decimal form (not %).

2. Unit Weight Relationships:

$$\gamma = \frac{(G+e)}{(1+e)} \gamma_w \quad \text{(for saturated soil, } S_r=1)$$

$$\gamma_d = \frac{G \gamma_w}{1+e}$$

$$\gamma = \gamma_d (1+w)$$

Problem-Solving Approach (Interconversion):

Given: $\gamma$, $w$, $$\displaystyle G_s $$. Find: $$\displaystyle \gamma_d $$, $e$, $n$, $$\displaystyle S_r $$.

1. Find $$\displaystyle \gamma_d = \frac{\gamma}{1+w} $$.

2. Find $$\displaystyle e = \frac{G \gamma_w}{\gamma_d} - 1 $$.

3. Find $$\displaystyle n = \frac{e}{1+e} $$.

4. Find $$\displaystyle S_r = \frac{w G}{e} $$ (from $$\displaystyle e = S_r w G $$).

2. Soil Classification Systems

A. Grain Size Analysis

• Sieve Analysis: For coarse-grained soils ($$\displaystyle d > 0.075 $$ mm). Results plotted as % finer vs. particle size (log scale) on a semi-log graph.

• Hydrometer Analysis: For fine-grained soils ($$\displaystyle d < 0.075 $$ mm). Based on Stokes' Law ($$\displaystyle v = \frac{\gamma_w d^2 (G_s-1)}{18 \mu} $$).

• Effective Size: $$\displaystyle D_{10} $$ = diameter at 10% finer. Indicates permeability.

• Representative Sizes: $$\displaystyle D_{30} $$, $$\displaystyle D_{60} $$.

B. Coefficient of Uniformity & Curvature (Coarse Soils)

• Uniformity Coefficient: $$\displaystyle C_u = \frac{D_{60}}{D_{10}} $$

• Coefficient of Curvature: $$\displaystyle C_c = \frac{(D_{30})^2}{D_{10} \cdot D_{60}} $$

• Gradation Types:

  • Well-graded (WG): $$\displaystyle C_u > 4 $$ (gravel) or $$\displaystyle >6 $$ (sand) AND $$\displaystyle C_c $$ between 1 and 3.

  • Poorly-graded (SP/GP): Either $$\displaystyle C_u $$ is low OR $$\displaystyle C_c $$ is outside 1-3.

  • Gap-graded: Missing intermediate sizes.

C. Textural Classification (USDA)

Based on % sand, silt, clay. Uses a triangular chart.

D. IS Classification (Two-Group System)

• Division: Coarse-grained (>50% retained on 75µm sieve) vs. Fine-grained (>50% passing 75µm).

• Coarse-Grained Subdivision:

  • Gravel (G): >50% retained on 4.75mm sieve.

  • Sand (S): >50% passing 4.75mm sieve.

  • Further subdivided by gradation (W/P) and cleanliness (Fines content).

• Fine-Grained Subdivision:

  • Silts (M) & Clays (C) based on Plasticity Chart.

  • Liquid Limit ($$\displaystyle w_L $$) & Plastic Limit ($$\displaystyle w_P $$).

  • Plasticity Index (PI): $$\displaystyle I_p = w_L - w_P $$.

  • A-line Equation: $$\displaystyle I_p = 0.73 (w_L - 20) $$.

  • Regions:

    • CL/CH: Above A-line, $$\displaystyle w_L > 50\% $$ is CH, else CL.

    • ML/MH: Below A-line, $$\displaystyle w_L > 50\% $$ is MH, else ML.

    • OL/OH: Organic soils (low-medium-high plasticity).

• Consistency Limits: Liquid, Plastic, Shrinkage limits define soil state.

Problem-Solving (Classification):

1. Compute $$\displaystyle C_u $$, $$\displaystyle C_c $$ → determine gradation (WG/PG).

2. Plot on Plasticity Chart using $$\displaystyle w_L $$ and $$\displaystyle I_p $$ → find region (CL, CH, ML, etc.).

3. Combine coarse/fine fraction info → final IS symbol (e.g., SW-SC, CI).

3. Effective Stress Principle

Total Stress ($\sigma$): Force per unit area including both solids and fluid.

Pore Water Pressure ($u$): Pressure of water in voids (positive for seepage, negative for suction).

Effective Stress ($\sigma'$): Stress carried by soil skeleton.

$$\boxed{\sigma' = \sigma - u}$$

• Significance: Governs shear strength ($$\displaystyle \tau = c' + \sigma' \tan \phi' $$), compressibility, and volume change.

• Calculation with Water Table:

  • Below WT: $$\displaystyle u = \gamma_w \cdot h $$ (where $h$ = depth below WT).

  • Above WT (unsaturated): $u$ is negative (suction) or taken as zero if $$\displaystyle S_r $$ unknown. Often $\sigma' \approx \sigma$ if $$\displaystyle S_r $$ is low.

  • Partially Saturated: $u$ is complex; for simple analysis, often use $$\displaystyle \sigma' = \sigma - \gamma_w \cdot h_{sat} $$ only for saturated zones.

• Stress Distribution: Always consider total stress first, then subtract pore pressure to get effective stress diagram.

4. Stress Distribution in Soils from Surface Loads

A. Boussinesq's Theory (Point Load)

• Assumptions: Homogeneous, isotropic, elastic half-space; vertical point load $Q$ on surface.

• Vertical Stress Increase at depth $z$, radial distance $r$:

$$\Delta \sigma_z = \frac{3Q}{2\pi z^2} \cdot \frac{1}{\left[1 + \left(\frac{r}{z}\right)^2\right]^{5/2}}$$

*   **Below point load ($$\displaystyle r=0 $$):** $$\displaystyle \Delta \sigma_z = \frac{3Q}{2\pi z^2} $$

*   **At depth $z$, horizontal distance $r$:** Use full equation.

B. Westergaard's Theory (Vertical Cracks)

• Assumptions: Soil has vertical, incompressible elements (like in stratified deposits); no lateral strain.

• Vertical Stress Increase:

$$\Delta \sigma_z = \frac{Q}{\pi z^2} \cdot \frac{1}{\left[1 + 2\left(\frac{r}{z}\right)^2\right]^{3/2}}$$

• Comparison: Westergaard gives higher stress near the load ($r/z$ small) but lower at large $r/z$ compared to Boussinesq.

C. Stress under Uniformly Loaded Areas - Equivalent Point Load Method

• Concept: Replace loaded area with an equivalent point load $$\displaystyle Q_{eq} = q \times A $$ at the center of pressure.

• Application: For a rectangular footing $B \times L$ with load $q$:

  1. Find $$\displaystyle \Delta \sigma_z $$ at point of interest using Boussinesq for point load $$\displaystyle Q_{eq} = qBL $$.

  2. Correction Factor ($F$): Often a factor $$\displaystyle F < 1 $$ is applied to account for load distribution. For center of rectangular area:

$$F = \frac{1}{[1 + (2a/B)^2]^{3/2}} \cdot \frac{1}{[1 + (2b/L)^2]^{3/2}}$$

    where $a, b$ are distances from center to edges in $B$ and $L$ directions.

3.  $$\displaystyle \Delta \sigma_z = F \times \Delta \sigma_z (\text{point load at center}) $$.

• Common Exam Problem: Calculate $$\displaystyle \Delta \sigma_z $$ at a point (e.g., below corner, below center) under a footing.

[!TIP] For points below the center of a rectangular footing, use the 2:1 distribution method as a simpler approximation: $$\displaystyle \Delta \sigma_z = \frac{q \cdot B \cdot L}{(B+z)(L+z)} $$.

5. Permeability and Seepage

A. Permeability Coefficient ($k$)

• Darcy's Law: $$\displaystyle q = k \cdot i \cdot A $$

  • $q$ = discharge, $i$ = hydraulic gradient, $A$ = cross-sectional area.

• Factors Affecting $k$:

  • Viscosity & Temperature: $$\displaystyle k \propto \frac{1}{\mu} $$; increases with temperature.

  • Void Ratio: $$\displaystyle k \propto \frac{e^3}{1+e} $$ (Kozeny-Carman for sands).

  • Soil Structure: Fabric, particle shape.

  • Degree of Saturation: Must be 100% saturated for accurate measurement.

B. Laboratory Determination

• Constant Head Method (Sands/Gravels):

  • Setup: Soil specimen of length $L$, area $A$. Apply constant head $h$.

  • Equation: $$\displaystyle k = \frac{QL}{A h t} $$

    • $Q$ = volume of water collected in time $t$.

  • Suitable: Coarse soils with high $k$.

• Falling Head Method (Clays/Silts):

  • Setup: Standpipe of area $a$ connected to specimen. Head falls from $$\displaystyle h_1 $$ to $$\displaystyle h_2 $$ in time $t$.

  • Equation: $$\displaystyle k = \frac{aL}{A t} \ln \left( \frac{h_1}{h_2} \right) $$.

C. Stratified Soil Deposits

• Flow Parallel to Layers (Horizontal): Arithmetic mean weighted by thickness.

$$k_H = \frac{\sum (k_i \cdot H_i)}{\sum H_i}$$

*   **Derivation:** Total discharge $$\displaystyle q = \sum q_i = \sum (k_i \cdot i \cdot A) $$. Since $i$ and $A$ are same for all layers, $$\displaystyle k_H $$ is weighted average.

• Flow Normal to Layers (Vertical): Harmonic mean weighted by thickness.

$$\frac{1}{k_V} = \frac{\sum (H_i / k_i)}{\sum H_i}$$

*   **Derivation:** Total head loss $$\displaystyle \Delta H = \sum \Delta h_i = \sum (q \cdot \frac{H_i}{k_i \cdot A}) $$. Since $q$ and $A$ constant, $$\displaystyle \frac{1}{k_V} $$ is weighted average of $$\displaystyle \frac{1}{k_i} $$.

D. Flow Nets

• Definition: Graphical representation of 2D steady seepage. Network of flow lines (path of water) and equipotentials (lines of equal head).

• Construction Rules:

  1. Flow lines & equipotentials are perpendicular.

  2. Form curvilinear squares (each element has same shape/size ratio).

  3. Boundary conditions: Flow lines along impermeable boundaries & phreatic line; Equipotentials along constant head boundaries.

• Application - Discharge Calculation:

$$q = k \cdot H \cdot \frac{N_f}{N_d} \quad \text{(per unit width)}$$

*   $$\displaystyle N_f $$ = number of flow channels.

*   $$\displaystyle N_d $$ = number of equipotential drops.

*   $H$ = total head loss.

• Seepage Pressure & Quick Condition:

  • Seepage Pressure: $$\displaystyle j = i \cdot \gamma_w $$ (force per unit volume, acts in direction of flow).

  • Quick Condition (Boiling): Occurs when upward seepage pressure equals submerged unit weight, causing effective stress to become zero.

  • Critical Hydraulic Gradient:

$$i_c = \frac{\gamma'}{\gamma_w} = \frac{G_s - 1}{1 + e}$$

    > [!TIP] **Derivation:** At failure, $$\displaystyle \sigma' = \sigma - u = 0 \Rightarrow u = \sigma $$. For a soil element of height $z$, $$\displaystyle \sigma = \gamma_{sat} \cdot z $$, $$\displaystyle u = i_c \gamma_w z $$. Set equal: $$\displaystyle i_c \gamma_w z = \gamma_{sat} z \Rightarrow i_c = \frac{\gamma_{sat}}{\gamma_w} = \frac{G_s+e}{1+e} \cdot \frac{1}{?} $$ Wait, correct derivation: $$\displaystyle \gamma_{sat} = \frac{(G_s+e)\gamma_w}{1+e} $$. So $$\displaystyle i_c = \frac{\gamma_{sat}}{\gamma_w} = \frac{G_s+e}{1+e} $$. But standard formula is $$\displaystyle i_c = \frac{G_s-1}{1+e} $$. **Correction:** At boiling, effective stress at a point becomes zero: $$\displaystyle \sigma' = \gamma_{sat} z - i \gamma_w z = 0 \Rightarrow i \gamma_w z = \gamma_{sat} z $$. But $$\displaystyle \gamma_{sat} = \frac{(G_s+e)\gamma_w}{1+e} $$. So $$\displaystyle i_c = \frac{G_s+e}{1+e} $$. However, **submerged unit weight** $$\displaystyle \gamma' = \gamma_{sat} - \gamma_w = \frac{(G_s-1)\gamma_w}{1+e} $$. The condition for **heaving** is when upward seepage force $$\displaystyle i \gamma_w $$ equals $\gamma'$. So $$\displaystyle i_c \gamma_w = \gamma' \Rightarrow i_c = \frac{\gamma'}{\gamma_w} = \frac{G_s-1}{1+e} $$. **This is correct.** The confusion arises because $\sigma'$ at the base of a soil column of height $z$ is $$\displaystyle (\gamma_{sat} - i\gamma_w)z $$. Setting to zero gives $$\displaystyle i = \gamma_{sat}/\gamma_w $$, but that's for the entire column. For **boiling at the surface**, the critical condition is when the seepage force just overcomes the submerged weight of the soil particle, i.e., $$\displaystyle i_c \gamma_w = \gamma' $$. So use $$\displaystyle i_c = \frac{G_s-1}{1+e} $$.

*   **Factor of Safety (FoS) against boiling:** $$\displaystyle FoS = \frac{i_c}{i_{actual}} $$.

6. Consolidation and Settlement

A. Primary Consolidation

• Process: Expulsion of water from saturated, low-permeability soils under load, causing volume decrease.

• Terzaghi's 1D Consolidation Theory Assumptions:

  1. Soil is homogeneous, fully saturated.

  2. Small strains, constant $$\displaystyle c_v $$.

  3. Darcy's law valid.

  4. Water is incompressible.

  5. Load is applied instantaneously and remains constant.

  6. One-dimensional flow (vertical only).

B. Settlement Calculation

• Ultimate Primary Consolidation Settlement ($$\displaystyle S_c $$):

$$\boxed{S_c = \frac{C_c}{1+e_0} H \log_{10} \left( \frac{\sigma'_f}{\sigma'_0} \right)}$$

*   $$\displaystyle C_c $$ = Compression Index (from $e$-$\log \sigma'$ curve).

*   $$\displaystyle e_0 $$ = Initial void ratio at $$\displaystyle \sigma'_0 $$.

*   $H$ = Initial thickness of compressible layer.

*   $$\displaystyle \sigma'_0 $$ = Initial effective overburden pressure.

*   $$\displaystyle \sigma'_f $$ = Final effective pressure after construction.

> [!TIP] For **recompression** (if $$\displaystyle \sigma'_f < \sigma'_{preconsolidation} $$), use **Recompression Index ($$\displaystyle C_r $$)** instead of $$\displaystyle C_c $$.

C. Coefficient of Consolidation ($$\displaystyle c_v $$)

• Definition: $$\displaystyle c_v = \frac{k}{m_v \gamma_w} $$

  • $$\displaystyle m_v $$ = Coefficient of volume compressibility.

• Determination from Oedometer Test (√t & log t methods):

  • √t Method (for $$\displaystyle T_v $$ at 50%):

    • Plot deformation vs. $\sqrt{t}$.

    • Draw line through initial linear portion and final asymptote.

    • Time for 50% consolidation $$\displaystyle t_{50} $$ corresponds to point where vertical separation between lines is $$\displaystyle \frac{\Delta H}{2} $$.

    • $$\displaystyle T_v(50\%) = 0.197 $$ (for double drainage), $0.287$ (for single drainage).

    • $$\displaystyle c_v = \frac{T_v H_{dr}^2}{t_{50}} $$.

  • Log t Method:

    • Plot deformation vs. $\log t$.

    • Find primary consolidation endpoint ($$\displaystyle U=100\% $$) from intersection of recompression and virgin compression lines.

    • Find time for 50% consolidation ($$\displaystyle t_{50} $$) from midpoint of vertical line between $$\displaystyle U=0 $$ and $$\displaystyle U=100\% $$.

    • Use same $$\displaystyle T_v $$ values as above.

D. Drainage Conditions & Time for Consolidation

• Drainage Path ($$\displaystyle H_{dr} $$):

  • Double Drainage: $$\displaystyle H_{dr} = \frac{H}{2} $$ (water can escape from both top and bottom).

  • Single Drainage: $$\displaystyle H_{dr} = H $$ (impermeable layer at one boundary).

• Time for a Degree of Consolidation ($U$):

$$\boxed{t = \frac{T_v H_{dr}^2}{c_v}}$$

*   $$\displaystyle T_v $$ depends on $U$. For $$\displaystyle U=50\% $$, $$\displaystyle T_v \approx 0.197 $$ (double drainage).

*   For $$\displaystyle U=90\% $$, $$\displaystyle T_v \approx 0.848 $$.

• Significance: Doubling $$\displaystyle H_{dr} $$ quadruples the time for same $U$.

7. Shear Strength of Soils

A. Mohr-Coulomb Failure Criterion

• Effective Stress Form:

$$\boxed{\tau = c' + \sigma' \tan \phi'}$$

*   $c'$ = Effective cohesion.

*   $\phi'$ = Effective angle of internal friction.

• Total Stress Form (for undrained clays, $$\displaystyle \phi=0 $$):

$$\tau = c_u \quad (\text{where } c_u \text{ is undrained cohesion})$$

B. Triaxial Shear Tests (Drainage Conditions)

[!TIP] UU test is quick, gives total stress parameters. CU with pore pressure measurement gives effective stress parameters. CD is slow but gives best $c'$, $\phi'$.

C. Unconfined Compression Test (UCT)

• Procedure: Axial load applied on cylindrical specimen without any confining pressure ($$\displaystyle \sigma_3 = 0 $$), no drainage.

• Suitable for: Saturated clays (friction angle ≈ 0°).

• Failure Stress: $$\displaystyle \sigma_f $$ (axial stress at failure).

• Undrained Cohesion: $$\displaystyle c_u = \frac{\sigma_f}{2} $$ (for $$\displaystyle \phi=0 $$).

• Significance: Quick index test for clay strength.

D. Liquefaction

• Definition: Loss of shear strength in saturated, loose, fine-grained soils (typically sands/silts) due to cyclic loading (e.g., earthquake), causing pore water pressure buildup and $$\displaystyle \sigma' \rightarrow 0 $$.

• Mechanism: Reversal of effective stress → soil behaves like a liquid.

• Factors Influencing:

  • Relative Density: Loose sands more susceptible.

  • Confining Pressure: Higher pressure increases resistance.

  • Stress History: Previously loaded (overconsolidated) soils less susceptible.

  • Cyclic Stress Ratio: Magnitude & number of cycles.

• Consequences: Loss of bearing capacity, large settlements, lateral spreading, flow failures.

E. Problem-Solving from Triaxial Data

Given: $$\displaystyle \sigma_3 $$ (confining pressure), $$\displaystyle \sigma_1 $$ (failure axial stress).

1. Total Stress Analysis (UU): Plot Mohr circle at failure ($$\displaystyle \sigma_1 $$, $$\displaystyle \sigma_3 $$). For $$\displaystyle \phi_u=0 $$, $$\displaystyle c_u = \frac{\sigma_1 - \sigma_3}{2} $$.

2. Effective Stress Analysis (CU with $u$ or CD): Compute $$\displaystyle \sigma'_1 = \sigma_1 - u_f $$, $$\displaystyle \sigma'_3 = \sigma_3 - u_0 $$ (usually $$\displaystyle u_0=0 $$). Plot Mohr circle in $\tau$-$\sigma'$ plane. Draw best-fit envelope → $c'$, $\phi'$.

8. Compaction of Soils

A. Purpose vs. Consolidation

B. Standard Proctor vs. Modified Proctor

C. Compaction Curve & Zero Air Void Line

• Compaction Curve: Plot $$\displaystyle \gamma_d $$ vs. $w$. Peak = Maximum Dry Density (MDD) at Optimum Moisture Content (OMC).

• Zero Air Void Line: Theoretical curve for 100% saturation ($$\displaystyle S_r=1 $$).

$$\boxed{\gamma_d = \frac{G_s \gamma_w}{1 + w}}$$

*   **Significance:** Shows theoretical maximum dry density for a given $w$. Actual MDD lies below this line.

9. Ground Improvement and Geosynthetics

A. Soil Stabilization

• Objectives: Increase strength, reduce compressibility/permeability, improve durability.

• Methods:

  • Mechanical: Compaction, reinforcement (geosynthetics, fibers).

  • Chemical: Lime, Cement, Fly ash, Bitumen.

  • Physical/Other: Geosynthetics, thermal (freezing), vibro-compaction, preloading.

B. Cement Stabilization

• Process: Mixing cement with soil + water.

  • Hydration: $$\displaystyle C_3S + H_2O \rightarrow C-S-H + CH $$ (strength).

  • Pozzolanic: $$\displaystyle CH + SiO_2/Al_2O_3 \rightarrow C-S-H $$ (long-term strength).

• Improvement: Increases strength, durability, reduces permeability & plasticity.

• Applications: Road subgrades, foundations, soil-cement liners, slope protection.

C. Geosynthetics

• Types: Geotextiles (woven/non-woven), Geomembranes, Geogrids, Geocells, Geocomposites.

• Functions & Applications:

  | Function | Description | Typical Application | | :--- | :--- | :--- | | Separation | Prevents mixing of dissimilar materials | Road subgrades, railway ballast | | Reinforcement | Provides tensile strength | Retaining walls, slopes, pavements | | Filtration | Allows flow, retains particles | Drainage systems, behind retaining walls | | Drainage | Conveys fluids (plane or transmissivity) | Edge drains, leachate collection | | Protection | Cushioning against puncture | Geomembrane liners, rock layers |

[!TIP] Remember: Geotextiles often serve separation/filtration, Geogrids for reinforcement, Geomembranes for containment (liners).