6.2 Soil Mechanics

Soil mechanics is the backbone of geotechnical engineering. It's all about understanding how soil behaves under different loads and conditions. This knowledge is crucial for designing stable foundations, retaining walls, and slopes.

In this section, we dive into stress distribution, settlement calculations, and stability analysis. These concepts are key to predicting how soil will react to structures built on or in it. Understanding them helps engineers create safe, long-lasting designs.

Stress Distribution in Soils

Effective Stress Principle and Stress Calculation Methods

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• Stress distribution in soils governed by principle of effective stress relates total stress, pore water pressure, and effective stress
• Boussinesq theory calculates stress distribution beneath point loads, line loads, and distributed loads on soil surfaces
• Newmark's influence chart provides graphical method for determining vertical stress increases in soils due to surface loads
• 2:1 method approximates stress distribution beneath foundations assumes stress spreads at a 2 vertical to 1 horizontal ratio
• Westergaard's solution calculates stress distribution in layered soil systems accounts for differences in soil stiffness
• Stress distribution equation using Boussinesq theory for point load: $\Delta\sigma_z = \frac{3Q}{2\pi z^2} \cdot \frac{1}{(1 + (r/z)^2)^{5/2}}$ where Q is the point load, z is the depth, and r is the horizontal distance from the load

Visualization and Analysis of Stress Distribution

• Isobars represent lines of equal vertical stress used to visualize stress distribution patterns in soil masses
• Stress bulbs represent zones of significant stress increase in soils with size and shape dependent on load configuration and soil properties
• Stress bulb for a strip footing extends to a depth of approximately 1.5 times the footing width
• Pressure bulb for a square footing extends to a depth of about 2 times the footing width
• Overlapping stress bulbs from adjacent footings can lead to increased settlement and require special consideration in foundation design

Soil Settlement Calculation

Components and Calculations of Soil Settlement

• Soil settlement consists of three components immediate settlement, primary consolidation, and secondary compression (creep)
• Elastic settlement calculations use theory of elasticity considering soil modulus and Poisson's ratio
• Elastic settlement equation: $S_e = \frac{qB(1-\nu^2)}{E_s} I_s$ where q is the applied pressure, B is the foundation width, ν is Poisson's ratio, Es is the soil elastic modulus, and Is is the influence factor
• Primary consolidation settlement calculated using consolidation theory incorporates void ratio, compression index, and overconsolidation ratio
• Primary consolidation settlement equation: $S_c = \frac{C_c H}{1+e_0} \log_{10}\frac{\sigma'_0 + \Delta\sigma'}{\sigma'_0}$ where Cc is the compression index, H is the layer thickness, e0 is the initial void ratio, σ'0 is the initial effective stress, and Δσ' is the stress increase
• Coefficient of consolidation (cv) determines rate of consolidation obtained from laboratory consolidation tests
• Secondary compression settlement estimated using secondary compression index (Cα) occurs after primary consolidation completes

Time-Dependent Settlement Analysis and Mitigation Techniques

• Time-settlement curves constructed using Terzaghi's one-dimensional consolidation theory and coefficient of consolidation
• Time factor (T) relates to degree of consolidation (U) through equations such as $U = \sqrt{\frac{4T}{\pi}}$ for small T values
• Preloading accelerates consolidation settlement in soft soils by applying temporary surcharge load
• Vertical drains (sand drains or prefabricated vertical drains) reduce drainage path length and accelerate consolidation
• Combination of preloading and vertical drains can significantly reduce time required for consolidation settlement
• Stone columns improve soil bearing capacity and reduce settlement in soft soils

Slope and Retaining Structure Stability

Slope Stability Analysis Methods

• Slope stability analysis determines factor of safety against sliding or rotational failure
• Infinite slope method used for long, shallow slopes with parallel slip surfaces
• Infinite slope stability equation: $F_S = \frac{c'}{\gamma H \sin\beta \cos\beta} + \frac{\tan\phi'}{\tan\beta}$ where c' is effective cohesion, γ is soil unit weight, H is slope height, β is slope angle, and φ' is effective friction angle
• Method of slices (Bishop's simplified method, Fellenius method) divides potential failure surfaces into vertical slices for analysis
• Limit equilibrium methods balance forces and moments acting on potential failure surfaces
• Factor of safety in slope stability typically ranges from 1.3 to 1.5 for long-term stability depending on consequences of failure

Retaining Wall Stability and Earth Pressure Theories

• Retaining wall stability analysis considers overturning, sliding, and bearing capacity failures
• Earth pressure theories (Rankine and Coulomb) calculate lateral earth pressures on retaining structures
• Rankine active earth pressure coefficient: $K_a = \tan^2(45^\circ - \frac{\phi'}{2})$
• Coulomb active earth pressure coefficient: $K_a = \frac{\cos^2(\phi' - \alpha)}{\cos^2\alpha \cos(\delta + \alpha)[1 + \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' - \beta)}{\cos(\delta + \alpha)\cos(\beta - \alpha)}}]^2}$
• Soil reinforcement techniques (geosynthetics, soil nailing) improve slope and retaining wall stability
• Geogrid reinforcement increases soil shear strength and allows construction of steeper slopes
• Soil nails provide additional resisting forces to improve slope stability in both natural and cut slopes

Effective Stress and Pore Water Pressure

Effective Stress Principle and Applications

• Effective stress (σ') stress carried by soil skeleton calculated as difference between total stress (σ) and pore water pressure (u)
• Effective stress equation: $\sigma' = \sigma - u$
• Principle of effective stress governs soil strength, compressibility, and volume change behavior
• Mohr-Coulomb failure criterion in terms of effective stress: $\tau_f = c' + \sigma'_n \tan\phi'$ where τf is shear strength, c' is effective cohesion, σ'n is effective normal stress, and φ' is effective friction angle
• Effective stress analysis crucial in evaluating soil liquefaction potential in saturated, loose sandy soils during earthquakes
• Liquefaction occurs when excess pore water pressure equals total stress, resulting in zero effective stress

Pore Water Pressure and Seepage Analysis

• Pore water pressure can be hydrostatic (in static conditions) or excess (due to loading or unloading)
• Hydrostatic pore water pressure: $u = \gamma_w h$ where γw is unit weight of water and h is depth below water table
• Seepage analysis using flow nets helps determine pore water pressure distribution in soil masses subject to hydraulic gradients
• Flow net consists of flow lines and equipotential lines used to calculate seepage quantity and pore pressures
• Critical hydraulic gradient defines condition at which effective stress becomes zero leading to quicksand conditions
• Critical hydraulic gradient equation: $i_c = \frac{\gamma_s - \gamma_w}{\gamma_w}$ where γs is saturated unit weight of soil
• Capillary rise in soils affects distribution of pore water pressure above water table with height of capillary rise inversely proportional to soil particle size