Earthquake Resistant Design of Structures (CE-802 (D)) - Unit 1 Short Notes

UNIT 1: FOUNDATION ENGINEERING PRINCIPLES

I. SUBSURFACE INVESTIGATION & SOIL SAMPLING

A. Planning & Execution of Exploration

• Significance: Determines soil stratigraphy, properties, and groundwater to design safe, economical foundations.

• IS Code Criteria (IS 1892:2016):

  • Depth: Boreholes should penetrate at least 3 m into bedrock or a depth where stress increase < 10% of net foundation stress.

  • Spacing: For uniform sites, grid pattern with spacing ≤ 30 m; for variable sites, ≤ 15 m.

• Boring Methods:

  • Auger Boring: Hand/manual, for shallow depths (< 6 m), soft soils. Disturbed samples.

  • Shell & Auger Boring: For medium depths, loose/soft soils. Uses bailer.

  • Rotary Drilling (Detailed): Uses rotary motion with core barrel/drill bit. Advantages: Fast, good for hard strata, allows undisturbed sampling (using Shelby tubes). Types: Mud rotary, dry rotary.

  • Percussion Drilling: Chiseling action. For boulders/hard strata. Slow, highly disturbed samples.

• Geophysical Methods:

  • Seismic Refraction: Measures velocity of seismic waves to infer stratum boundaries & rock depth.

  • Electrical Resistivity: Measures soil resistivity to detect changes in lithology, water content, contamination.

[!TIP] Exam Focus: IS 1892 criteria for borehole depth/spacing are frequently asked. Differentiate rotary (best for undisturbed samples) vs percussion (for hard strata).

B. In-Situ Testing

• Standard Penetration Test (SPT):

  • Procedure: Drive a split spoon sampler (50 mm ID, 60 mm OD) with a 65 kg hammer falling 750 mm. Count blows for 30 cm penetration after seating drive of 15 cm. N-value = blows for last 30 cm.

  • Significance of N-value: Index of soil density/consistency, correlates with relative density (sand), undrained shear strength (clay), and settlement estimates.

• Corrections to SPT N-value:

  1. Overburden Pressure Correction (Normalization): \( N_{corrected} = N_{observed} \times \frac{100}{\sigma'_{v0}} \) (for sand, \(\sigma'_{v0}\) in kPa). Standardizes N to 100 kPa overburden.

  2. Dilatancy Correction (for Dense Sand/Stiff Clay): \( N_{corrected} = N_{measured} \text{ (if } N_{measured} < 15) \); else \( N_{corrected} = 15 + 0.5(N_{measured} - 15) \). Corrects for negative friction/volume increase.

  3. Energy Correction: \( N_{60} = N_{field} \times \frac{ER_{field}}{60\%} \). Converts to standard energy ratio (60%). Common hammers: Donut (30-42%), Safety (50-60%), Automatic (60-80%).

  \boxed{N_{60} = N_{field} \times \frac{ER_{field}}{60%}}

• Cone Penetration Test (CPT):

  • Equipment: Pushed at constant rate (20 mm/s). Measures tip resistance (q_c) and sleeve friction (f_s) continuously.

  • Types: CPT (mechanical), SCPT (with pore pressure transducer - measures u₂).

  • Advantages over SPT: Continuous profile, faster, more repeatable, provides friction ratio (R_f = f_s/q_c × 100%), better for soft soils & stratification.

C. Soil Sampling

• Disturbed vs. Undisturbed:

  • Disturbed: Structure altered. Used for classification, water content, compaction tests.

  • Undisturbed: Structure preserved. Used for consolidation, permeability, triaxial tests.

• Sampling Tube Design Parameters:

  • Inside Clearance (C_i): \( C_i = \frac{D_i - D_s}{D_s} \times 100\% \). Typical: 0.5-1.0%. Allows sample expansion.

  • Outside Clearance (C_o): \( C_o = \frac{D_h - D_o}{D_o} \times 100\% \). Reduces friction during driving. Typical: < 2%.

  • Area Ratio (A_r): \( A_r = \frac{(D_h^2 - D_i^2)}{D_i^2} \times 100\% \). < 10% for good-quality undisturbed sample. Lower = less disturbance.

• Criteria for Good Quality Undisturbed Sample: Minimal distortion, no air voids, ends parallel, length > 2× diameter.

• Borelog Report Components: Borehole location/depth, soil description (color, consistency), depth of samples, SPT N-values, water table depth, lab test results, graphical log.

II. BEARING CAPACITY OF SHALLOW FOUNDATIONS

A. Fundamental Concepts & Definitions

• Ultimate Bearing Capacity (q_u): Max pressure soil can withstand before shear failure.

• Net Ultimate Bearing Capacity (q_net,u): \( q_{net,u} = q_u - \gamma D_f \). Excludes overburden pressure.

• Net Safe Bearing Capacity (q_net,s): \( q_{net,s} = \frac{q_{net,u}}{FOS} \). Allowable net pressure.

• Allowable Bearing Pressure (q_all): \( q_{all} = q_{net,s} + \gamma D_f \). Total allowable pressure at foundation base.

• Gross Pressure: Total pressure applied at base (includes surcharge + structural load).

• Factors Influencing Capacity: Soil properties (c, φ, γ), foundation dimensions (B, L), depth (D_f), load inclination, water table.

B. Theories of Failure & Capacity

• Modes of Shear Failure:

  • General Shear: Dense sand/ stiff clay. Sudden failure, well-defined failure surface, large settlements.

  • Local Shear: Medium-dense sand/ medium clay. Progressive failure, limited surface heave.

  • Punching Shear: Very loose sand/ soft clay. Failure surface confined under footing, minimal heave.

  Sketch Required: Show failure surfaces for each mode.

• Terzaghi's Bearing Capacity Theory (1943):

  • Assumptions: Strip footing, φ > 0°, base rough, soil homogeneous, load vertical, failure surface as log spiral + linear rays.

  • Equation (Strip): \( q_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma \)

  • Shape Factors (for Square/Circular):

    • Square: \( q_u = 1.3c N_c + \gamma D_f N_q + 0.4 \gamma B N_\gamma \)

    • Circular: \( q_u = 1.3c N_c + \gamma D_f N_q + 0.3 \gamma B N_\gamma \)

  • IS Code (BIS) Method (IS 6403:1981): Uses shape factors (s_c, s_q, s_γ) and depth factors (d_c, d_q, d_γ). More comprehensive.

    \boxed{q_u = c N_c s_c d_c + \gamma D_f N_q s_q d_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma}

• Water Table Correction:

  • If water table above foundation base, use submerged unit weight (γ' = γ_sat - γ_w) for the term involving γ (N_γ term) and replace γ in N_q term with γ' if D_f is below WT.

  • γ_correction factor (d_γ): Applied to N_γ term based on depth of WT relative to B.

C. Numerical Problems Approach

1. Identify soil type (c-φ, pure c, pure φ).

2. Determine relevant bearing capacity factors (N_c, N_q, N_γ) from tables (Terzaghi or IS).

3. Apply shape, depth, and water table correction factors.

4. Calculate q_u, then q_net,u, q_net,s, q_all.

5. For FOS against shear failure: \( FOS = \frac{q_{net,u}}{q_{net,applied}} \).

[!TIP] Common Pitfall: Forgetting to use γ' (submerged unit weight) when water table is at/below foundation base. Always check water table position relative to D_f and B.

III. SETTLEMENT OF FOUNDATIONS

A. Components of Total Settlement (S_total = S_i + S_c + S_s)

• Immediate Settlement (S_i): Elastic distortion, occurs during/just after construction. Significant in sands & stiff clays.

• Primary Consolidation Settlement (S_c): Pore water expulsion, time-dependent. Major component in saturated clays.

• Secondary Consolidation Settlement (S_s): Creep of soil skeleton after primary consolidation. Important in organic/clayey soils.

B. Immediate Settlement Calculation

• Equation (Cohesive soils, c-φ):

  \boxed{S_i = \frac{q B (1 - \mu^2) I_f}{E_s} \quad \text{(for strip footing)}}

  • q = net applied pressure

  • B = footing width

  • μ = Poisson's ratio

  • I_f = Influence factor (from table/chart based on L/B, φ)

  • E_s = Modulus of elasticity of soil (from lab/empirical correlations)

• For Rectangular Footing: \( S_i = \frac{q B (1 - \mu^2)}{E_s} I_f \). I_f depends on L/B ratio.

• Influence Factor (I_f): Tabulated (e.g., IS 8009-1:2016). Increases with footing size, decreases with depth.

C. Plate Load Test

• Procedure: Load a rigid plate (usually 0.3 m²) at ground level, measure settlement. Load incrementally until failure/specified settlement.

• Ultimate Bearing Capacity (q_u,plate): From load-settlement curve (e.g., where tangent slope = 0.5× initial slope, or sudden failure).

• Settlement Extrapolation (Terzaghi & Peck):

  • Sand: \( S_{footing} \approx S_{plate} \times \frac{B_{footing}}{B_{plate}} \)

  • Clay: \( S_{footing} \approx S_{plate} \times \frac{B_{footing}}{B_{plate}} \) (for flexible footing on saturated clay).

D. Numerical Problems Approach

1. For S_i: Identify E_s, μ, q, B, L/B → get I_f from table → compute S_i.

2. For plate load: Use extrapolation formula based on soil type and footing flexibility.

IV. PILE FOUNDATIONS

A. Classification & Functions

• By Material: Timber, Concrete, Steel, Composite.

• By Function:

  • End-bearing Pile: Rest on hard stratum.

  • Friction Pile: Derive capacity from shaft friction.

  • Combined Pile: Both end-bearing & friction.

• By Installation:

  • Driven: Precast, displacement (hammers).

  • Bored: Cast-in-situ, non-displacement (drilling).

  • Screw: Helical piles.

• Functions: Transfer load, resist uplift, control settlement, stabilize slopes.

B. Load Carrying Capacity of Single Pile

• Static Formulae:

  • Cohesive Soils (α-method): \( Q_{up} = \alpha c_u A_s + c_b A_b \)

    \boxed{Q_{up} = \alpha c_u (\pi d L) + c_b \left(\frac{\pi d^2}{4}\right)}

    • α = adhesion factor (0.5-1.0, decreases with depth/sensitivity)

    • c_u = undrained shear strength

    • A_s = shaft area, A_b = base area

    • c_b = base adhesion (≈ 9c_u for bored, 6c_u for driven in soft clay)

  • Cohesionless Soils (β-method): \( Q_{up} = \beta \sigma'_{v0} A_s + q_b A_b \)

    \boxed{Q_{up} = \beta K \sigma'_{v0} (\pi d L) + \gamma D_f N_q \left(\frac{\pi d^2}{4}\right)}

    • β = skin friction factor (≈ K tanδ, K = lateral earth pressure coeff, δ = friction angle)

    • σ'_{v0} = avg vertical effective stress along shaft

    • q_b = bearing pressure at pile tip (≈ σ'_{v0} N_q)

• Dynamic Formulae (Drop Hammer):

  • Energy Principle: \( \text{Energy input} = \text{Work done} + \text{Losses} \)

  • Drop Hammer Formula (Engineering News Record):

    \boxed{Q_{safe} = \frac{W h}{S + C} \times \frac{W + n P}{W} \times \frac{1}{FOS}}

    • W = hammer weight, h = fall, S = final set (penetration per blow), C = constant (25 mm for drop hammer, 2.5 mm for steam hammer), n = coefficient (1 for single-acting, 2 for double-acting), P = ram weight.

  • Pile Set: Total elastic + plastic compression.

• Capacity from SPT/CPT:

  • Sand: \( q_b = N_q \sigma'_{v0} \), \( f_s = K \sigma'_{v0} \tan\delta \) (K from SPT N-value correlations).

  • Clay: \( q_b = 9 c_u \), \( f_s = \alpha c_u \) (α from SPT N-value or CPT sleeve friction).

C. Pile Group Capacity & Settlement

• Group Efficiency (η): \( \eta = \frac{Q_{ug}}{n Q_{us}} \). η < 1 for soft clays (due to overlapping stress zones), η ≈ 1 for dense sand/rock.

• Group Settlement Ratio (GSR): \( GSR = \frac{S_g}{n S_s} \). GSR > 1 for clays (group settlement > sum of single pile settlements).

• Ultimate Capacity of Pile Group:

  • Block Failure (cohesive soils, close spacing): Treat group as a single large foundation.

    \( Q_{ug} = c N_c (B_g L_g) + \gamma D_f N_q (B_g L_g) + 0.5 \gamma B_g N_\gamma (B_g L_g) \)

  • Individual Failure (spacing > 3-4d): \( Q_{ug} = \eta \times n \times Q_{us} \)

• Pile Group Settlement: Usually governs design. Can be estimated from settlement of equivalent raft at depth of pile cap.

D. Special Topics

• Negative Skin Friction (NSF):

  • Concept: Downward drag on pile due to settlement of compressible soil around it (e.g., fill, soft clay, lowering WT).

  • Calculation for Single Pile:

    \boxed{Q_{nsf} = \gamma_{fill} \cdot K \cdot \tan\delta \cdot (A_s){drag} \quad \text{or} \quad Q{nsf} = \bar{c}u \cdot (A_s){drag}}

    • (A_s)_{drag} = area of pile in settling layer.

    • δ = interface friction angle (≈ φ for sand, 3/4 φ for clay-fill).

  • For Pile Group: Consider group as a block; NSF acts on block perimeter.

• Under-reamed Piles:

  • Concept: Pile with bulbs (under-reams) at intervals. Provides tensile capacity & swell pressure resistance in expansive soils.

  • Components: Shaft, under-ream bulb (diameter 2-3× shaft), collar (to prevent soil intrusion).

  • Suitability: Expansive soils (black cotton soil), collapsible soils, loose sands.

  • Ultimate Tensile Capacity:

    \boxed{T_u = \sum (A_b \cdot q_b) + \sum (A_s \cdot f_s)}

    • q_b = bearing capacity of bulb base (≈ 9c_u for clay).

    • f_s = shaft friction (αc_u).

E. Numerical Problems Approach

1. Single Pile (Cohesive): Compute Q_s (αc_uA_s) and Q_b (c_bA_b or 9c_uA_b). Sum = Q_ult. Safe load = Q_ult / FOS.

2. Single Pile (Cohesionless): Compute Q_s (βKσ'v0A_s) and Q_b (q_bA_b). Use β from table or β = K tanδ.

3. Pile Group (Neglect End Bearing): \( Q_{ug} = \eta \times n \times (\alpha c_u A_s) \) or use block failure if spacing very close (<3d).

4. Negative Skin Friction: Identify drag layer, compute average effective stress/c_u, calculate f_drag, multiply by affected surface area.

5. Under-reamed Pile Tensile Capacity: Calculate capacity of each bulb (base + shaft above/below bulb) and sum.

V. LATERAL EARTH PRESSURE & RETAINING STRUCTURES

A. Types & Fundamentals

• At Rest (K₀): Wall rigid, no movement. \( K_0 = 1 - \sin\phi' \) (Jaky's formula for normally consolidated cohesionless soil).

• Active (K_a): Wall moves away from soil. Minimum lateral pressure.

• Passive (K_p): Wall pushed into soil. Maximum lateral pressure.

• Relationship: \( K_a = \frac{1}{K_p} \) for φ > 0°.

B. Rankine's Earth Pressure Theory (1875)

• Assumptions: Wall smooth & vertical, backfill horizontal, cohesionless (c=0) or homogeneous c-φ soil, failure surface planar.

• Coefficients:

  • Cohesionless: \( K_a = \tan^2(45° - \phi/2) \), \( K_p = \tan^2(45° + \phi/2) \)

  • Cohesive (c>0): \( \sigma_a = K_a \gamma z - 2c \sqrt{K_a} \) (tension crack depth \( z_c = \frac{2c}{\gamma \sqrt{K_a}} \))

• Pressure Distribution:

  • Horizontal Backfill: Linear for c-φ, triangular for φ>0, trapezoidal for c>0 (tension zone at top).

  • Sloping Backfill (β): \( K_a(\beta) = \frac{\cos\beta \sqrt{1+\sin\phi'\cos^2\beta}}{1-\sqrt{\sin\phi'\sin(\phi'-\beta)}} \) (complex). Pressure not linear.

  • Sloping Wall: Use β = wall inclination.

C. Coulomb's Earth Pressure Theory (1776)

• Assumptions: Wall rough (δ = wall friction), planar failure surface through toe, backfill dry/cohesionless, wedge in equilibrium.

• Active Thrust (P_a):

  \boxed{P_a = \frac{1}{2} \gamma H^2 K_a \quad \text{(for horizontal backfill, vertical wall)}}

  • \( K_a = \frac{\cos^2(\phi' - \delta)}{2\cos^2\delta \cos(\delta + \beta) \left[1 + \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' - \beta)}{\cos\delta\cos(\beta - \delta)}}\right]^2} \)

  • Point of Application: From base, \( \bar{z} = \frac{H}{3} \) (for horizontal backfill, vertical wall, δ=0).

• Culmann's Graphical Method:

  • Used for non-horizontal backfill or sloping wall with Coulomb.

  • Draw failure planes at various angles, compute weight of wedge & corresponding thrust. Envelope gives maximum P_a and its angle.

D. Comparison & Application

• Numerical Problems Steps:

  1. Draw cross-section, identify soil layers, water table, surcharge.

  2. Compute effective unit weights below WT.

  3. Determine K (Rankine/Coulomb) for each layer.

  4. Calculate pressure at layer interfaces.

  5. Draw pressure diagram (triangular/trapezoidal/compounded).

  6. Integrate to get total thrust (P) and point of application (from base).

     • For triangular: \( P = \frac{1}{2} \sigma_{base} H \), \( \bar{z} = H/3 \).

     • For trapezoidal: \( P = \frac{1}{2}(\sigma_{top} + \sigma_{base})H \), \( \bar{z} = \frac{H(2\sigma_{base} + \sigma_{top})}{3(\sigma_{top} + \sigma_{base})} \).

E. Retaining Walls

• Sheet Pile vs. Retaining Wall:

  | Sheet Pile | Retaining Wall | | :--- | :--- | | Thin, embedded section | Massive, gravity/ cantilever | | Used for temporary cofferdams, shoring | Permanent structures | | Resists lateral pressure by soil-structure interaction (flexure) | Resists by self-weight & base friction | | Uses: Excavation support, waterfront structures | Uses: Highway cuts, bridge abutments, basement walls |

• Modes of Failure:

  1. Overturning: Moment about toe > resisting moment. Check FOS > 1.5.

  2. Sliding: Horizontal thrust > friction + cohesion at base. \( FOS = \frac{\mu W + c_b B}{P_a} > 1.5 \).

  3. Bearing Capacity Failure: Excessive pressure under toe. Check q_max < q_all.

  4. Deep-seated failure: Global slope failure.

• Earth Pressure with Tension Cracks: In cohesive backfills (c>0), active pressure at top may be tensile → crack develops until pressure = 0. Depth of crack \( z_c = \frac{2c}{\gamma \sqrt{K_a}} \). Pressure diagram starts from z_c.

VI. SPECIAL SOILS & GROUND IMPROVEMENT TECHNIQUES

A. Problematic Soils

• Expansive Soils (e.g., Black Cotton Soil):

  • Characteristics: High montmorillonite clay, high shrink-swell potential, low bearing capacity when wet, high when dry.

  • Problems: Differential heave/shrinkage, cracking in foundations/floors, seasonal damage.

  • Preventive Measures: Moisture control (watering, barriers), under-reamed piles, chemical stabilization (lime), replacement, raft foundations.

• Collapsible Soils (e.g., Loess):

  • Characteristics: Metastable, loose, cemented (carbonate/salt), low moisture content. Sudden collapse upon wetting.

  • Problems: Sudden settlement, differential collapse.

  • Preventive Measures: Pre-wetting, deep foundations (piles), compaction, chemical stabilization.

B. Soil Stabilization

• Need: Improve strength, reduce swell/shrink, increase durability.

• Mechanical: Compaction (increases density), Reinforcement (geosynthetics).

• Chemical:

  • Lime: Best for high-plasticity clays. Reduces plasticity, increases strength via pozzolanic reactions.

  • Cement: For sandy/silty soils. Increases strength, reduces permeability.

  • Bitumen: For waterproofing, base courses.

• Electrical (Electro-osmosis): Apply DC current to move water from anode to cathode in fine-grained soils. Used for dewatering/slope stability.

C. Geosynthetics

• Functions in Detail:

  • Separation: Prevent mixing of dissimilar soils.

  • Reinforcement: Tensile strength to resist loads.

  • Filtration: Allow water flow but retain soil particles.

  • Drainage: In-plane flow of water/leachate.

  • Erosion Control: Protect slopes from runoff.

  • Containment: Barrier to fluids/gases.

D. Compaction

• Field Equipment:

  • Smooth Wheel: Granular soils, finishing.

  • Sheepsfoot: Cohesive soils, deep compaction.

  • Pneumatic: Granular & cohesive, uniform pressure.

  • Vibratory: Granular soils, high density.

• Lab Compaction Tests:

  | Standard Proctor (Light) | Modified Proctor (Heavy) | | :--- | :--- | | Hammer: 2.5 kg, Fall: 300 mm | Hammer: 4.5 kg, Fall: 450 mm | | Layers: 3, Blows: 25 | Layers: 5, Blows: 25 | | Compaction Effort: ~600 kN-m/m³ | Compaction Effort: ~2700 kN-m/m³ | | OMC: Higher | OMC: Lower | | MDD: Lower | MDD: Higher |

VII. MISCELLANEOUS FOUNDATION TYPES & CONCEPTS

A. Shallow Foundations

• Types:

  • Spread/Isolated Footing: Under single column.

  • Combined Footing: Under 2+ columns.

  • Strap Footing: Connects isolated footing to column with strap beam.

  • Raft/Mat Foundation: Single thick slab under entire structure. Used when q_all is low or loads heavy.

  • Floating Foundation: Raft designed so net increase in vertical stress = 0 (excavated soil weight = structure weight). For very soft clays.

• Proportioning of Raft Foundations: Thickness based on shear & punching shear; reinforcement for flexure. Check bearing pressure uniformity.

• Basic Criteria for Satisfactory Performance: Adequate bearing capacity, total & differential settlement within limits, structural integrity.

B. Well Foundations

• Components (with sketch):

  1. Well Curb: Bottom cutting edge, concrete/steel.

  2. Well Steining: Vertical masonry/concrete above curb, tapers outward.

  3. Cutting Edge: Steel angle at bottom for sinking.

  4. Apron: Platform around top to prevent soil erosion.

  5. Well Sinking: Process of excavating inside to sink well.

• Uses: Bridges, piers, abutments in sandy soils/rivers.

• Limitations: Slow, difficult in rocky soils, requires skilled labor.

C. Modes of Shear Failure in Soils

• General Shear Failure: Dense sand/stiff clay. Failure surface extends to surface. Sudden, large settlements, well-defined failure wedge. Influenced by: Foundation depth/width ratio (D_f/B), soil compressibility (low).

• Local Shear Failure: Medium-dense sand/medium clay. Failure surface does not reach surface. Progressive, moderate settlements. Influenced by: Moderate D_f/B, moderate compressibility.

• Punching Shear Failure: Loose sand/soft clay. Failure surface confined under footing. Very small settlements, no heave. Influenced by: Low D_f/B, high compressibility.

  Sketch Required: Show failure surfaces for each mode relative to footing.

KEY FORMULAS & DEFINITIONS BOXED FOR EXAM

• SPT N-value Corrections:

  \( N_{60} = N_{field} \times \frac{ER_{field}}{60\%} \)

• Terzaghi's Bearing Capacity (Strip):

  \( q_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma \)

• IS Code Bearing Capacity:

  \( q_u = c N_c s_c d_c + \gamma D_f N_q s_q d_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma \)

• Immediate Settlement (c-φ):

  \( S_i = \frac{q B (1 - \mu^2) I_f}{E_s} \)

• Single Pile Capacity (Cohesive - α-method):

  \( Q_{up} = \alpha c_u (\pi d L) + c_b \left(\frac{\pi d^2}{4}\right) \)

• Drop Hammer Safe Load:

  \( Q_{safe} = \frac{W h}{S + C} \times \frac{W + n P}{W} \times \frac{1}{FOS} \)

• Negative Skin Friction (Single Pile):

  \( Q_{nsf} = \gamma_{fill} \cdot K \cdot \tan\delta \cdot (A_s)_{drag} \)

• Rankine Active Pressure (Cohesive):

  \( \sigma_a = K_a \gamma z - 2c \sqrt{K_a} \)

• Coulomb Active Thrust (Horizontal Backfill):

  \( P_a = \frac{1}{2} \gamma H^2 K_a \)

Critical Definitions to Remember:

• Net Ultimate Bearing Capacity (q_net,u): \( q_u - \gamma D_f \)

• Allowable Bearing Pressure (q_all): \( q_{net,s} + \gamma D_f \)

• Group Efficiency (η): \( \frac{Q_{ug}}{n Q_{us}} \)

• At Rest Earth Pressure Coefficient (K₀): \( 1 - \sin\phi' \) (Jaky's)