UNIT 2: FOUNDATION ENGINEERING & SOIL MECHANICS (Applied to Design)

A. SUBSURFACE INVESTIGATION & SOIL PROPERTIES

A.1 Methods of Boring & Sampling

• Rotary drilling: Uses rotating bit with circulating fluid (mud or water).

  Advantages: Continuous sampling, minimal disturbance, suitable for all soils and rock, can change bits.

• Other boring methods:

  • Auger boring: Hand or power auger, suitable for soft to medium soils.

  • Percussion boring: Chiseling action, for hard soils/rock.

  • Wash boring: Jet of water, for sandy soils.

• Sampling:

  • Disturbed sample: Remolded during sampling, used for classification tests.

  • Undisturbed sample: Preserves in-situ structure, used for strength and consolidation tests.

• Sampling tube design:

  • Inside clearance: $$\displaystyle C_i = \frac{D_i - D_c}{D_i} $$, allows sample expansion.

  • Outside clearance: $$\displaystyle C_o = \frac{D_o - D_c}{D_o} $$, reduces friction between sample and tube.

  • Area ratio: $$\displaystyle A_r = \frac{A_o - A_i}{A_i} \times 100\% $$. Should be $$\displaystyle < 10\% $$ for undisturbed samples.

  [!TIP] Example: Tube ID = 100 mm, thickness = 2 mm → OD = 104 mm. Cutting shoe ID = 100 mm (flush), thickness = 3 mm.

  $$\displaystyle C_i = 0\% $$, $$\displaystyle C_o = 3.85\% $$, $$\displaystyle A_r = 8.16\% $$. Zero inside clearance may cause sample tightness, but area ratio acceptable.

A.2 In-Situ Testing

• Standard Penetration Test (SPT):

  • Procedure: Drive split spoon sampler (50 mm ID) with 63.5 kg hammer falling 760 mm. Record blows for last 300 mm → N-value.

  • Significance: Indicator of soil density/stiffness; correlates with relative density, $\phi$, bearing capacity.

  • Corrections:

    1. Overburden pressure correction:

    \boxed{N_1 = N \left( \frac{P_a}{\sigma'_v} \right)^{0.5}}, \quad $$\displaystyle P_a = 100\, $$kPa.

    2. Dilatancy correction: For dense sands below water table, if $$\displaystyle N_1 > 15 $$ and $$\displaystyle \sigma'_v < P_a $$, then

    \boxed{N_2 = 0.7,N_1}.

    [!TIP] Apply overburden correction first, then dilatancy if conditions met.

• Cone Penetration Test (CPT):

  • Push cone (10 cm² area) at constant rate; measure cone resistance $$\displaystyle q_c $$, sleeve friction $$\displaystyle f_s $$, friction ratio $$\displaystyle R_f = \frac{f_s}{q_c} \times 100\% $$.

  • Advantages over SPT: Continuous profiling, more reliable, faster, provides quantitative soil parameters.

• Plate Load Test:

  • Load rigid plate (typically 0.3 m square), measure settlement.

  • Interpretation:

    • Clay: $$\displaystyle S_2 = S_1 \left( \frac{B_2}{B_1} \right) $$.

    • Sand: $$\displaystyle S_2 = S_1 \left( \frac{B_2}{B_1} \right)^{0.5} $$.

A.3 Bore-log & Reporting

• Bore-log components: Depth, soil description, SPT N-value, water table, sampling details, lab test results.

• IS criteria (IS 1892):

  • Depth: $\ge 1.5$–$2 \times$ footing width or until hard stratum.

  • Spacing: 10–30 m for regular sites; closer for variable soils.

A.4 Geophysical Methods

• Seismic refraction: Measures wave velocity to infer soil/rock layers.

• Electrical resistivity: Detects variations in moisture, density, and layers.

• Others: Ground penetrating radar, magnetic surveys.

B. BEARING CAPACITY OF SHALLOW FOUNDATIONS

B.1 Fundamental Definitions

• Net pressure: $$\displaystyle q_{net} = q - \gamma D_f $$.

• Ultimate bearing capacity ($$\displaystyle q_u $$): Maximum gross pressure before shear failure.

• Net ultimate bearing capacity: $$\displaystyle q_{net,u} = q_u - \gamma D_f $$.

• Net safe bearing capacity: $$\displaystyle q_{net,s} = \dfrac{q_{net,u}}{FOS} $$.

• Allowable bearing pressure: $$\displaystyle q_a = q_{net,s} + \gamma D_f $$.

B.2 Theories & Failure Modes

• Modes of shear failure:

  1. General shear: Continuous failure surface, sudden failure (dense soils).

  2. Local shear: Failure limited to under footing (medium dense).

  3. Punching shear: Soil punches into footing (very loose/stiff stratum).

  [!TIP] Sketches essential: show failure surfaces for each mode.

• Terzaghi's Theory:

  \begin{align}

  \text{Strip:} \quad & \boxed{q_u = c N_c + \gamma D_f N_q + \frac{1}{2} \gamma B N_\gamma} \

  \text{Square:} \quad & \boxed{q_u = 1.3,c N_c + \gamma D_f N_q + 0.4,\gamma B N_\gamma} \

  \text{Circular:} \quad & \boxed{q_u = 1.3,c N_c + \gamma D_f N_q + 0.3,\gamma B N_\gamma}

  \end{align}

  $$\displaystyle N_c, N_q, N_\gamma $$ depend on $\phi$ (from tables).

• IS Method (IS 6403): Uses shape/depth factors; similar form. Terzaghi's equations often accepted in exams.

• Water Table Correction:

  • If water table at/above foundation base → use submerged unit weight $\gamma'$ for $\gamma$ in third term and effective overburden for $q$.

  • If water table below foundation base → use total $\gamma$.

  [!TIP] For water table at ground level: $$\displaystyle q_u = c N_c + \gamma' D_f N_q + \frac{1}{2} \gamma' B N_\gamma $$ (effective stress).

B.3 Application Problems

• Compute $$\displaystyle q_u $$ for given shape, $c$, $\phi$, $\gamma$, $$\displaystyle D_f $$, $B$.

• Adjust for water table location.

• Factor of safety: $$\displaystyle FOS = \dfrac{q_{net,u}}{q_{net}} $$.

C. SETTLEMENT OF FOUNDATIONS

C.1 Components of Settlement

• Immediate (elastic): Rapid, due to elastic deformation.

• Primary consolidation: Due to water expulsion from saturated clays, time-dependent.

• Secondary compression: Creep of soil skeleton after primary consolidation.

C.2 Calculation of Immediate Settlement

• Cohesive soils:

  \boxed{S_i = q \cdot B \cdot \frac{1 - \nu^2}{E} \cdot I_z}

  $$\displaystyle I_z $$ = influence factor (given or from $L/B$ table).

  [!TIP] For square footing, $$\displaystyle I_z $$ often provided (e.g., 1.06).

• Granular soils: Use elastic theory with modulus $E$ (pressure-dependent; estimated from SPT/relative density).

C.3 Settlement Prediction from Plate Load Test

• Clay: \boxed{S_{footing} = S_{plate} \left( \frac{B_{footing}}{B_{plate}} \right)}.

• Sand: $$\displaystyle S_{footing} = S_{plate} \left( \frac{B_{footing}}{B_{plate}} \right)^{0.5} $$.

D. PILE FOUNDATIONS

D.1 Classification & Functions

• By material: Concrete, steel, timber.

• By function: End-bearing, friction, combined.

• By installation: Driven, bored, drilled (CFA).

D.2 Estimation of Pile Capacity

• Static Formulae:

  • Clay:

    \boxed{Q_u = c_u N_c A_b + \alpha c_u A_s}

    $$\displaystyle N_c = 9 $$ (deep), $\alpha$ = adhesion factor (0.7–1.0).

    [!TIP] Bored piles may have lower $$\displaystyle N_c $$ (6–9).

  • Sand:

    $$\displaystyle Q_b = \sigma'_v N_q A_b $$, \quad $$\displaystyle Q_s = \sum (f_s A_s) $$,

    $$\displaystyle f_s = K \sigma'_v \tan \delta $$, $K \approx 1 - \sin\phi$ (NC), $\delta \approx \phi$ (rough piles).

• Dynamic Methods:

  • Drop hammer (Hiley's formula):

    \boxed{Q_{net} = \frac{W H e}{s + 0.5 \Delta}}

    $$\displaystyle e = \dfrac{1 - \alpha}{1 + \alpha} $$, $\alpha$ = restitution (0.25–0.5).

    Safe load: $$\displaystyle Q_{safe} = \dfrac{Q_{net}}{FOS} $$.

• Negative Skin Friction (NSF):

  • Causes: Clay consolidation, fill, water table drop.

  • Single pile: $$\displaystyle Q_{ns} = f_{ns} A_s $$.

    • Cohesive: $$\displaystyle f_{ns} = c_u $$ (undrained) or $c' + \sigma' \tan\delta'$ (drained).

    • Granular: $$\displaystyle f_{ns} = \gamma D \tan\delta $$ (simplified).

  [!TIP] Consider depth where soil settles relative to pile.

D.3 Pile Groups

• Geometrical properties: Spacing $s$, diameter $d$, group dimensions.

• Group efficiency: $$\displaystyle \eta = \dfrac{Q_{ug}}{n Q_{u,individual}} $$.

• Cohesive soils:

  • $$\displaystyle s > 3d $$ → $\eta \approx 1$.

  • Closely spaced → block failure possible.

  • Ultimate group capacity: $$\displaystyle Q_{ug} = \min(n Q_{u,individual}, Q_{block}) $$.

  • Block failure (neglect end bearing):

    \boxed{Q_{block} = \alpha c_u \cdot (\text{perimeter of block}) \cdot L}

    Perimeter $$\displaystyle = 2(B_g + L_g) $$ for rectangular group; $$\displaystyle B_g = (n-1)s + d $$ for $n \times n$ group.

  [!TIP] For $3\times3$ group, $$\displaystyle s=0.9 $$ m, $$\displaystyle d=0.3 $$ m → $$\displaystyle B_g = 2.1 $$ m.

• Sand: $\eta$ often $$\displaystyle < 1 $$; rarely $$\displaystyle > 1 $$ due to compaction.

D.4 Special Pile Types

• Under-reamed Piles:

  • Enlarged bases (under-reams) for expansive soils; provide uplift resistance.

  • Ultimate tensile capacity:

    \boxed{Q_t = \alpha c_u A_s + c_u N_c A_b}

    $$\displaystyle N_c \approx 9 $$ for uplift.

    [!TIP] Note if suction or under-ream contribution to shaft adhesion is neglected.

D.5 Design Problems

• Friction pile group in clay: Determine $n$, $s$ from load, $$\displaystyle c_u $$, $\alpha$, FOS.

• Layered clay: Find pile length to penetrate firm layer or based on capacity.

• Dynamic data: Use Hiley's formula for safe load.

E. RETAINING WALLS & LATERAL EARTH PRESSURE

E.1 Types of Lateral Earth Pressure

• At-rest ($$\displaystyle K_0 $$): No wall movement.

• Active ($$\displaystyle K_a $$): Wall moves away → minimum pressure.

• Passive ($$\displaystyle K_p $$): Wall moves into soil → maximum pressure.

E.2 Earth Pressure Theories

• Rankine's Theory:

  • Assumptions: $$\displaystyle \delta = 0 $$, vertical wall, horizontal backfill, isotropic soil.

  • Cohesionless:

    \boxed{K_a = \tan^2\left(45^\circ - \frac{\phi}{2}\right)}, \quad \boxed{K_p = \tan^2\left(45^\circ + \frac{\phi}{2}\right)}.

  • Cohesive:

    Active: \boxed{\sigma_a = \gamma z K_a - 2c \sqrt{K_a}}.

    Passive: $$\displaystyle \sigma_p = \gamma z K_p + 2c \sqrt{K_p} $$.

    Tension crack depth: $$\displaystyle z_c = \dfrac{2c}{\gamma \sqrt{K_a}} $$.

• Coulomb's Theory:

  • Includes wall friction $\delta$, backfill slope $\beta$, planar failure.

  • For level backfill ($$\displaystyle \beta=0 $$) and $$\displaystyle \delta>0 $$:

$$K_a = \frac{\cos^2(\phi - \delta)}{\cos^2\delta \left(1 + \sqrt{\frac{\sin(\phi+\delta)\sin(\phi-\delta)}{\cos^2\delta}}\right)^2}$$

Reduces to Rankine when $$\displaystyle \delta=0 $$.

• Merits: Accounts for $\delta$, $\beta$; more realistic for passive pressure.

• Culmann's Graphical Method: For non-horizontal backfill; graphical construction for active pressure.

E.3 Calculation of Earth Pressures & Thrust

• Pressure at depth $z$:

  • Water table: Use $\gamma'$ below WT, add pore pressure $u$.

  • Surcharge $q$: Add $$\displaystyle q K_a $$ (active) or $$\displaystyle q K_p $$ (passive) uniformly.

• Total thrust:

  • Uniform $\gamma$, $$\displaystyle c=0 $$: $$\displaystyle P = \frac{1}{2} \gamma H^2 K_a $$ (active), acts at $H/3$ from base.

  • Cohesive $$\displaystyle c>0 $$: $$\displaystyle P = \frac{1}{2} \gamma H^2 K_a - 2c H \sqrt{K_a} $$ (active); find centroid by area/moment.

  • Stratified backfill: Compute pressure at each interface, draw diagram, find area and centroid.

• Effect of water table: Use effective stress or separate soil/water pressures.

  [!TIP] Always check for tension cracks in active state with cohesive soil.

E.4 Retaining Wall Design & Analysis

• Total lateral pressure: For vertical back (masonry), use Rankine active.

• Pressure distribution: Linear with depth; intercept at $$\displaystyle z_c $$ if $$\displaystyle c>0 $$.

• Modes of failure:

  1. Overturning: Moment from earth pressure.

  2. Sliding: Horizontal thrust exceeds friction.

  3. Bearing capacity failure: Excessive foundation pressure.

  [!TIP] Sketches required for each mode.

• Tension cracks: Depth $$\displaystyle z_c = \frac{2c}{\gamma \sqrt{K_a}} $$ in active state.

F. SPECIAL SOILS & SOIL IMPROVEMENT

F.1 Expansive & Collapsible Soils

• Expansive soils:

  • Characteristics: High montmorillonite clay, large swell-shrink with moisture.

  • Problems: Heave, cracking, foundation damage.

  • Preventive measures: Moisture control (impermeable barriers, drainage), deep foundations (piles), under-reamed piles, lime treatment, sand cushions.

• Collapsible soils:

  • Characteristics: Loess, low density, cemented particles, collapse on wetting.

  • Problems: Sudden settlement.

  • Preventive measures: Pre-wetting, compaction, replacement, chemical stabilization.

F.2 Geosynthetics

• Types & functions:

  • Geotextiles (woven, non-woven): Separation, reinforcement, filtration.

  • Geomembranes: Impermeable liners.

  • Geogrids: Reinforcement (high tensile strength).

  • Geocells: Confinement, erosion control.

  • Geocomposites: Combinations (e.g., geotextile + geomembrane).

• Uses in foundation engineering: Soil reinforcement, layer separation, drainage, liner protection, erosion control.

F.3 Soil Stabilization

• Need: Improve strength, reduce compressibility, control swell.

• Methods:

  • Mechanical: Compaction, preloading, vertical drains.

  • Chemical: Lime, cement, fly ash, bitumen.

  • Electrical: Electro-osmosis for fine-grained soils (injects water, consolidates).

G. MISCELLANEOUS & DEFINITIONS

G.1 Floating Foundation & Raft Foundation

• Floating foundation: Structure weight = excavated soil weight → net pressure increase zero. Used for heavy structures on soft clay.

• Raft foundation: Combined footing covering entire area.

  Proportioning: Soft soil → increase size to reduce pressure; rock → reduce size to minimize differential settlement.

G.2 Types of Footings & Basic Performance Criteria

• Types: Isolated, combined, strip, raft, mat.

• Performance criteria: Adequate bearing capacity, acceptable total/differential settlement, structural integrity.

G.3 Well Foundations

• Components (

  DiagramCANVAS: Sketch circular well with labeled parts
  ):

  1. Well curb: Bottom cutting edge.

  2. Well steining: Tapered portion above curb.

  3. Well lining: Shaft walls (brick/concrete).

  4. Bottom plug: Seals bottom.

  5. Top plug: Supports pier.

  6. Sand filling: Between lining and pier.

G.4 Comparison: Sheet Pile vs. Retaining Wall; Uses of Sheet Piles

• Sheet piles: Flexible, thin sections; temporary earth retention (cofferdams, bulkheads, excavation support). Driven or pushed.

• Retaining walls: Rigid, substantial; permanent (gravity, cantilever, anchored).

• Uses of sheet piles: Cofferdams, waterfront structures, temporary shoring, slope protection.

G.5 Boussinesq vs. Westergaard Theories

• Boussinesq: Isotropic elastic half-space; stress spreads laterally.

  $$\displaystyle \sigma_z = \dfrac{3P}{2\pi} \dfrac{z^3}{(r^2+z^2)^{5/2}} $$.

• Westergaard: Stratified medium with vertical cracks (no lateral strain); stress more concentrated vertically.

  $$\displaystyle \sigma_z = \dfrac{P}{\pi z^2} \dfrac{1}{(1 + 2(r/z)^2)^{3/2}} $$.

• Difference: Westergaard gives higher stresses near load, lower at distance.

G.6 Well Foundations vs. Pile Foundations

• Well foundations: Open caissons, sunk by excavation; suitable for hard strata at moderate depth; large bearing area; used for bridges/heavy structures.

• Pile foundations: Transfer load to deeper strata; suitable for soft soils over hard stratum; faster installation.

NOTE: All formulas and concepts are aligned with past RGVP exam questions. Focus on problem-solving for bearing capacity, pile capacity (static/dynamic), earth pressure (Rankine/Coulomb), and settlement calculations.