Introduction to Geotechnical Engineering - VOCABULARY Flashcards

2.1 Minerals and Rocks

This section introduces minerals and rocks as the building blocks of geotechnical materials. Minerals are naturally occurring inorganic, solid crystalline substances with a fixed structure and defined chemical composition (which may vary within limits). Rocks are naturally occurring aggregates of one or more minerals (or mixtures of rocks) and may include mineraloids such as obsidian or coal when crystalline structure is absent. The rock cycle describes how igneous, sedimentary, and metamorphic rocks transform under varying conditions of temperature, pressure, and fluids, producing a wide range of rock textures and properties that influence engineering behaviour. Igneous rocks form by cooling of magma, with intrusive/plutonic rocks cooling slowly at depth (coarse-grained) and extrusive/volcanic rocks cooling rapidly at the surface (fine-grained or glassy). Metamorphic rocks arise from existing rocks subjected to heat and pressure, leading to new minerals and structures (foliation in many metamorphics). Sedimentary rocks form through lithification of deposited sediments, with clastic, biogenic, chemical, and volcaniclastic categories. Silt, clay, sand, gravel, and boulders are major grain-size classifications; grain size, sorting, and mineralogy strongly influence rock strength and deformation. Weathering alters rocks chemically and physically and drives the rock cycle toward weaker deposits or new rock forms.

Key concepts and details to remember:

• Rocks are classified into igneous, sedimentary, and metamorphic, each with distinct formation processes and textural attributes (aphanitic vs. phaneritic, porphyritic textures, foliations, bedding, etc.).

• The rock cycle links magmatic processes, sedimentation, lithification, and metamorphism, highlighting how deposition environment and post-depositional changes control engineering properties.

• Rock textures reflect cooling history: aphanitic (rapid cooling, fine grains), phaneritic (slow cooling, coarse grains), and porphyritic textures (mixed grain sizes from multi-stage cooling).

• Rocks are described by type, age, place, and structure. The rock mass can differ markedly from the intact rock due to defects, fractures, weathering, and porosity.

• Weathering types (mechanical/physical, chemical, biological) control strength, permeability, and susceptibility to dissolution and weathering-related hazards such as karst.

• Lithification comprises compaction and cementation, reducing pore space and increasing strength.

• Clastic rock classification depends on grain size, sorting, and shape (gravel, sand, silt, clay); examples include greywacke, limestone, mudstone, conglomerate, sandstone, siltstone, shale, and vein-filled rocks.

• Engineering implications: strength and deformation depend on rock type, presence of discontinuities (bedding, cleavage, joints), weathering, and depositional/tectonic history. Weathering can degrade strength and alter permeability; volcanogenic rocks may exhibit strong variability and problematic clay deposits after weathering.

Representative concepts and equations (for quick reference):

• Rock types:

  • Igneous: plutonic (intrusive) vs volcanic (extrusive); textures reflect cooling history.

  • Sedimentary: lithification via compaction and cementation; grain-size controls mechanical properties.

  • Metamorphic: changes in mineralogy and foliation due to heat/pressure; foliated metamorphics are weaker along foliation planes.

• Rock mass vs intact rock: rock mass includes defects and discontinuities; these control engineering behaviour alongside the intrinsic rock properties.

• Phase relationships are central to understanding rock behaviour under stress, fluid flow, and weathering processes.

Key references/links noted in the content (for further reading): OpenGeology material on historical geology and rock types; field/diagrammatic resources for rock textures and formation processes.

2.2 Rock Deformation & Defects

Rocks are rarely homogeneous; rock masses contain defects such as fractures, joints, faults, bedding planes, and foliation. Deformation arises from crustal processes including tectonic movement, faulting, folding, and changes in loading due to gravity and tectonics. Deformation manifests as compressive, tensile, and shear forces, producing folds and faults that structure the rock mass and influence stability, permeability, and strength.

Common deformation structures and defects:

• Folds: produced by ductile deformation at depth; hinge points, axial planes, limbs, and fold axes describe their geometry. Folding is influenced by temperature, pressure, strain rate, and rock type. Fold structures include monoclines, anticlines, and synclines.

• Planar defects: fractures and joints (no displacement), bedding (layering in sedimentary rocks), and cleavage (foliation in metamorphic rocks). Joints are often sets with varying spacing and aperture. Bedding and lithologic boundaries mark interfaces between different rock units.

• Discontinuity orientation and spacing are critical. Dip and strike describe orientation in 3D space; spacing measures the distance between discontinuities; aperture describes the opening size of the discontinuity.

• Faults: rock rupture with relative movement. Major types include strike-slip (horizontal movement), normal (extensional, hanging wall moves down), reverse and thrust (compressional, hanging wall moves up). The presence and movement of faults alter slope stability and groundwater flow.

• Engineering implications: defects control strength and stability of rock masses, water movement along discontinuities, and potential for abrupt or gradual movement, with earthquake-induced shaking a major consideration in fault zones.

Engineering considerations for projects:

• The orientation and density of discontinuities along with weathering state heavily influence stability and design approaches for dams, tunnels, and slopes.

• In constructing, one aims to select material with favorable continuity and to exploit best available layers while avoiding damaged or weathered zones.

Key connections to prior material:

• The interaction between rock type, weathering, and defects defines the structural integrity of rock masses, informing slope design, dam foundations, and underground excavations.

Key examples and visual references described in the slides:

• Dams and tunnels examples show how abrupt changes in deposits (planar and zonal defects) can affect stability and permeability.

2.3 Geomorphic Processes

Geomorphic processes describe how landscapes evolve under weathering, erosion, transportation, and deposition, driven by climate, vegetation, gravity, tectonics, and fluid systems. These processes continually modify landforms and deposit characteristics, which in turn influence geotechnical properties.

Key processes:

• Weathering: mechanical (physical) and chemical, sometimes biological. Mechanical weathering reduces rock to smaller fragments and increases surface area for chemical processes. Chemical weathering alters rock composition via reactions with water and atmospheric constituents. Biological weathering involves organisms (lichens, plant roots) that open joints and joints widen over time.

• Erosion and degradation: weathered material is removed by water, wind, ice, and waves. Degradation lowers ground surfaces, causes undercutting, and can produce features like sinkholes in soluble rocks.

• Transportation: movement of eroded material via water, wind, ice, and waves. Materials may travel as bedload, suspended load, or dissolved load.

• Deposition and aggradation: constructive processes that accumulate sediments, forming new soils and rocks through lithification over time.

• Mass movement: downslope movement of soil and rock due to gravity and moisture (rockfalls, landslides, debris flows, creep).

• Endogenetic processes: diastrophism/tectonism (earthquakes, faulting, folding) and volcanism; metamorphism is linked to internal processes and earlier sections.

Important conceptual outcomes:

• Geomorphic processes occur at different scales and durations, continuously reshaping the landscape and creating potential hazards (landslides, erosion rates, sinkholes).

• Weathering rates depend on rock type, climate, presence of soil, bedding and defects, and time.

• The type of depositional environment (coastal, fluvial, glacial, desert, etc.) strongly influences the resulting soil and rock characteristics.

Practical implications for geotechnical engineering:

• Understanding past and present geomorphic processes helps anticipate soil/rock properties at a site, including potential weathering profiles, soil formation histories, and risk of geohazards.

2.4 Rock Classification

Rock classification targets describing rock mass properties relevant to engineering, recognizing that rock masses differ from intact rocks due to defects, weathering, and discontinuities.

Key principles:

• Intact rock vs rock mass: intact rock is a continuous solid; rock mass includes defects and discontinuities that dominate engineering behaviour.

• Weathering indicators: colour, fabric, weathering grade, and presence of defects influence strength and stiffness.

• Weathering grades: a system ranges from Unweathered (UW) to Completely Weathered (CW) with intermediate grades (Slightly Weathered, Moderately Weathered, Highly Weathered) to describe how weathered the rock mass is. These grades reflect the decrease in strength and alteration in fabric.

• Discontinuities: orientation (strike and dip), spacing, persistence, roughness, aperture, and infill influence shear strength and hydraulic behaviour. Planar defects include fractures, joints, bedding planes, and cleavage; zonal defects include faults and zones of closely spaced joints; voids/cavities may form within rock masses.

• Bedding and lithologic boundaries indicate interfaces between deposited units and can guide excavation strategies.

• Engineering implications: the rock type, weathering state, and discontinuity network determine strength, stability, and permeability, influencing excavation quality, stability of slopes, and seepage characteristics.

• Example classification examples demonstrate how weathering state, rock name, discontinuitites, and geological information are combined to produce a main description with qualifying paragraphs.

Takeaways for field descriptions:

• Use a systematic approach to describe rock mass, not just the intact rock. Include weathering state, discontinuities, bedding, and fabric to characterize behaviour.

• Recognize that a rock mass with strong intact rock properties may still behave poorly if many discontinuities or weathered zones are present.

3.1 Soil Origins

Soils originate from rock weathering and subsequent processes that produce distinct soil deposits. Soils differ from rocks in composition, structure, and multi-phase nature and are typically formed via transported processes or weathering in place (residual soils).

Transported soils include:

• Alluvium: deposited by rivers and streams; grain size ranges from clay to gravel, with distinct depositional zones along rivers (headwaters, transfer zone, depositional zone) leading to stratified soils.

• Colluvium: transported downslope by gravity and surface water, from clay to boulder-sized fragments; commonly mantling slopes and forming colluvial aprons.

• Glacial tills: deposited by glacier ice, ranging from silt to boulder-sized particles; heterogeneous and often highly variable.

• Aeolian (wind-blown) deposits: loess and fine sands; typically fine-grained and well-sorted.

Residual soils form in place by weathering of the underlying parent rock, with degradation and disintegration producing soils with properties strongly tied to the parent material and climate.

Soil formation involves parent material, climate, topography, time, and biological/human activity. The transport history and depositional environment shape the resulting soil's grain-size distribution, stratification, and degree of consolidation.

Key soil deposit examples include:

• Alluvium, Colluvium, Glacial, Aeolian, Volcanic soils for transported soils.

• Residual soils arising from in-situ weathering of parent rock.

• Organic soils: formed from accumulating organic material, peat, and other organic matter.

Key conceptual points:

• Soils are three-phase materials (solids, water, air) and exhibit time-dependent behaviour. The solid phase properties depend on grain contacts and interactions with fluids.

• Soil classification and understanding origin help predict engineering behaviour, such as shear strength, compressibility, and permeability.

3.2 Basic Soil Types

Soils are broadly categorized by grain size, fabric, and mineralogy. The main soil types are coarse-grained soils (gravel, sand) and fine-grained soils (silts, clays), with organics forming a special class. Soil phase is three-dimensional, consisting of solids, liquids (water), and gases (air), each occupying voids within the soil.

Grain-size criteria and gradation:

• Boulders, Cobbles (coarse); Gravel (coarse to medium); Sand (coarse to fine); Silt (fine); Clay (very fine); Organics.

• Gradation descriptors: well graded, poorly graded, gap graded.

• Particle sizes (approximate ranges):

  • Gravel: > 4.75 mm (coarse) down to 2 mm, etc.

  • Sand: 0.06 to 2 mm (various fractions: coarse, medium, fine)

  • Silt: 0.002 to 0.06 mm

  • Clay: < 0.002 mm

Clay minerals and fabric:

• Clays are built from layers (tetrahedral and octahedral sheets). 1:1 clays have one tetrahedral and one octahedral sheet per layer; 2:1 clays have an octahedral sheet between two tetrahedral sheets. The interlayer bonding differs between 1:1 (strong hydrogen bonds) and 2:1 (water can enter between layers). This structure gives rise to properties such as plasticity, shrink-swell behaviour, and high specific surface area.

• Common clay minerals include kaolinite (1:1) and montmorillonite (2:1). Volcanic clays (e.g., halloysite, allophane) exhibit unique behaviour.

Soil behaviour concepts:

• Soil is a porous three-phase material; capillary rise depends on pore size and surface tension, with sands and gravels showing only a few cm rise while clays can exhibit tens of metres.

• Water content, porosity, void ratio, and density relationships govern the phase composition and the soil’s mechanical response.

• Organic soils often have distinctive odours and lower stiffness/strength due to organic content.

3.3 Soil Description – Laboratory Tests

Laboratory tests extend field (visual) classification to quantify soil properties and fractions:

• Particle Size Distribution (PSD): Determines the relative amounts of gravel, sand, and fines (silt+clay). PSD can be assessed using sieve analysis for coarse fractions (75 mm to 0.06 mm) and hydrometer analysis for fines (0.06 mm to 0.0002 mm).

• Sieve analysis computes Mtotal and % retained on each sieve, enabling the calculation of percent passing.

• Gradation indices: well-graded vs poorly graded; Cc and Cu are used for coarse-grained soils to assess gradation quality. Cc and Cu formulas require D-values (D10, D30, D60).

• Hydrometer test: describes particle settling with time according to Stoke’s law; larger particles settle faster and smaller particles remain in suspension longer.

• Atterberg limits (for fine-grained soils): LL (Liquid Limit), PL (Plastic Limit), PI (Plasticity Index) with PI = LL - PL. Liquid Limit is determined by Casagrande apparatus or a cone penetrometer; Plastic Limit is the moisture content at which a thread of soil crumbles when rolled to 3 mm diameter; PI measures the range of plastic behaviour.

• Liquidity Index (LI) is a field- or lab-derived indicator of soil state, typically LI = (W - LL)/(LL - PL) (note: the slides show a similar concept; standard definition is LI = (W - LL)/(LL - PL)). LI helps classify soils as soft/softening (LI > 1 indicates sensitive soil) or stiff (LI near or below 1).

• Phase relationships and phase diagrams are used to relate elementary soil properties such as water content, void ratio, and degree of saturation using densities and unit weights.

Practical content connections:

• PSD and Atterberg limits underpin the classification and prediction of soil behaviour in engineering contexts, such as liquefaction potential, shear strength, and compressibility.

• Laboratory testing informs field design by providing quantitative inputs for phase diagrams and subsequent stress analyses.

4.1 Elementary Definitions and Phase Relationships

Soil is treated as a multi-phase material (solids, water, air) with a phase diagram that helps relate key properties and guide the development of relationships among them. Core definitions include:

• Voids Vv and solids Vs, total volume V, and the three-phase volumes sum to V = Vs + Vv (and for partially saturated soils, V = Vs + Va + Vw).

• Volume fractions: void ratio e = Vv/Vs and porosity n = Vv/V. The total volume V comprises the solid, water, and air phases in various proportions depending on saturation.

• Mass considerations: Ms is the mass of solids, Mw is mass of water, Ma is mass of air (often neglected). Densities and unit weights connect mass and volume through γ = ρg (unit weight) for each phase: γw is the unit weight of water, γs for solids, and γ for the total soil.

• Phase diagrams: show how V and M of each phase relate, and how changes in one phase (e.g., capillary rise, saturation) alter others. In dry soils, Va and Vw = 0; in unsaturated soils Va > 0 and Vw > 0; in saturated soils Va = 0 and Vw > 0.

• Specific gravities: Gs = ρs/ρw = γs/γw. This links the solid density to water density and to unit weights.

• Modified phase diagrams: alternative representations that use e, w_c (water content), and Gs or other combinations to develop relationships independent of a fixed reference frame.

Key practical exercises:

• Build a phase diagram by listing knowns (e.g., total volume, constituent masses) and solving for unknowns using relations among Vs, Vv, Va, Vw, Ms, Mw, and densities.

• Use the phase diagram to derive relationships between properties such as e, n, w_c, Gs, and γ values.

Testing and measurement methods discussed:

• Bulk density and unit weight via laboratory tests; field methods include sand cone test and nuclear densitometry to estimate density and water content.

5.1 Stresses in Soils

Soil stress analysis relies on three interrelated stresses: total vertical stress σv, pore (pore water) pressure u, and effective vertical stress σ'v. The effective stress principle states that the stress carried by the soil skeleton (the particle framework) is σ'v = σv − u. This principle governs compression, shear strength, and stability in saturated soils and is central to soil mechanics.

Key concepts and equations:

• Total vertical stress at depth z in a stratified layer system:
  ext{Total stress (layer 1)}: \ σv = γ1 z1 \text{and for layered materials} \ σv(z) = γ1 z1 + γ2 (z - z1) \text{when depth z exceeds the first layer thickness } z_1.

• Pore pressure in saturated zones below the water table:
  u(z) = γw (z - z1) \,\text{for } z>z_1 \,\text{(below the water table)}.
  In simpler terms, u increases linearly with depth below the water table at a rate equal to the unit weight of water γw.

• Effective stress: the stress borne by the solid skeleton is
  σ'v = σv - u.

• Lateral earth pressure at rest (K0): the ratio of horizontal to vertical effective stress under undisturbed conditions:
  K0 = rac{σ'h}{σ'_v} = 1 - \, ext{sin} φ'
  where φ' is the effective angle of internal friction.

• Horizontal stress relation:
  σ'h = K0 \, σ'_v.

• Capillary and capillary rise effects: above and near the water table, capillary suction or rise can alter the effective stress state by changing the pore water pressure, with higher capillary rise in finer materials and smaller pores (clays) and much less in coarse materials (sands and gravels).

Two-layer example considerations:

• In a two-layer soil profile, total stress at depth in each layer is a sum of layer-wise contributions, and pore pressure depends on the vertical location relative to the water table. For layered conditions, the total stress can be represented as
  σv = γ1 z1 + γ2 (z - z1) \text{and if the second layer is saturated, } u = γw (z - z_1).

• Effective stress in each layer combines these terms via σ'v = σv − u. In particular conditions the relative magnitudes of γ1, γ2, γw, and z determine whether σ'v increases with depth or shows a change at the interface.

Practical implications:

• The concept of effective stress explains why saturated soils respond differently to loading than dry soils: the same total stress can produce different deformations depending on the pore water pressure.

• The K0 parameter helps estimate the initial in-situ horizontal stresses, which are critical for tunnel stability, slope stability, and foundation design.

• Capillary effects matter near ground surfaces, where unsaturated conditions occur, and can temporarily modify effective stresses during loading and drainage events.

Key points to remember:

• Effective stress is the controlling mechanism for deformation and strength in saturated soils.

• Changes in groundwater conditions (water table position) directly affect σ'v via u, while changes in overburden weight affect both σv and u, but the net change in σ'v depends on how u changes with depth.

• The relationship between vertical and horizontal effective stresses is often encapsulated by K0, which depends on soil friction properties (φ') and is used in estimating in-situ stress states for earthworks and underground structures.

4.1 Phase Relationships and Phase Diagrams (Expanded)

A concise synthesis of the phase diagrams used to relate soil constituents:

• For a dry soil, only solids are present in the phase diagram, with V = Vs and no contributions from Vv or Va.

• For unsaturated soils, air and water exist in the voids; water content Wc and degree of saturation determine Vv and Vw, with the phase diagram guiding how these volumes relate to Vs.

• For saturated soils, the voids are filled with water: Va = 0, Vw > 0, and the phase diagram is dominated by the water-solid system with u > 0.

• Phase diagrams relate weight and volume of each phase, enabling the derivation of relationships such as e, n, γ, γw, γs, and unit weights. Specific gravity Gs ties the solid density to water density and helps relate mass-based properties to volume-based properties.

• Modified phase diagrams reframe the relationships using variables such as e (void ratio), w_c (water content), and Gs to derive new expressions for properties like γ, γsat, and other engineering parameters.

Phase diagram steps (practical approach):
1) List knowns (masses and volumes of phases, densities).
2) Draw a phase diagram showing the presence/absence of air/water/solids in the deposit.
3) Fill in known properties (e, n, w_c, γs, γ, γw, etc.).
4) Move from one side of the diagram to the other using the relationships between densities and volumes.
5) Check units and consistency.

Key relationships and definitions to reproduce in notes:

• Phase volumes and total volume: $V = Vs + Vv + Va$ (with Va often zero in saturated conditions) and $V = Vs + V_v$ in dry/unsaturated scenarios when Va = 0.

• Void ratio and porosity definitions:
  e = \frac{Vv}{Vs}, \quad n = \frac{V_v}{V}

• Mass/weight relationships:
  Ms = γs Vs, \quad Mw = γw Vw, \quad M = Ms + Mw + M_a

• Densities and unit weights: γi = ρi g for i ∈ {solid, water, air} with the solid and water phases being the primary contributors to mass in geotechnical problems.

Laboratory and field links:

• Lab methods (bulk density, Sand Cone Density, Nuclear Densitometer) help estimate γ and density-related parameters in the field.

• PSD and Atterberg limits provide empirical inputs to the phase diagrams via fractions of solids and water contents.

5.1 Stresses in Soils (Summary and Key Equations)

The core objective is to understand how stresses distribute with depth in soils, how pore pressure develops, and how the effective stress drives deformation and strength.

Important definitions:

• Total vertical stress at depth z: $\,σ_v(z)$

• Pore pressure: $u(z)$, the pressure carried by water in the voids

• Effective vertical stress: $σ'v(z) = σv(z) - u(z)$

Typical expressions for layered soils:

• If layering exists with layer 1 of unit weight γ1 extending to depth z1 and layer 2 with unit weight γ2 thereafter, the total vertical stress is
  σv(z) = γ1 z1 + γ2 (z - z1) \, \text{for} \, z > z1.

• The pore pressure below the water table is linear with depth in the saturated portion,
  u(z) = γw (z - z1) \, \text{for} \, z > z_1.

• Therefore the effective stress in the saturated portion of the second layer is
  σ'v(z) = σv(z) - u(z) = [γ1 z1 + γ2 (z - z1)] - γw (z - z1).

K0 (lateral earth pressure at rest):

• The horizontal effective stress is related to the vertical by
  σ'h = K0 σ'_v

• For normally consolidated/at-rest conditions, a common expression is
  K_0 = 1 - \, \sin φ'
  where φ' is the effective angle of internal friction.

Capillary effects and water table influences:

• Capillary suction above the water table can reduce effective stress (apparent cohesion effects in unsaturated soils).

• Capillary rise height depends on pore size; finer soils (clays) exhibit larger rises (potentially meters) while coarse soils (sand) show only a few centimeters.

Key takeaways for practice:

• The three stresses—total stress, pore pressure, and effective stress—balance to determine deformation and strength.

• Groundwater conditions and layering produce depth-dependent stress states that significantly influence design of foundations, slopes, cuttings, tunnels, and embankments.

5.1 Key Relations and Practical Notes

• In saturated soils, pore pressure contributes to the total stress and reduces effective stress. The crucial balance is σ'v = σv − u.

• The vertical stress in layered soils accumulates with depth based on the overburden unit weights and thicknesses of layers; the presence of a water table modifies effective stress via pore pressure.

• The lateral earth pressure ratio K0 is rooted in soil friction properties and is used to estimate horizontal stresses in earthworks, supports, and retaining structures.

• Capillarity and capillary rise are particularly important near the ground surface, where unsaturated conditions prevail and moisture states can change with rainfall, drying, and drainage.

• The overall framework emphasizes correlation between measurable soil properties (unit weights, densities, moisture contents, grain sizes) and the resulting stress states that govern settlement, shear strength, and stability.

4.1 Connections to Engineering Practice and Assessment

Across these topics, the course links rock and soil properties to practical engineering outcomes:

• Rock types and weathering states guide excavation planning, slope stability, groundwater control, and rock mass classification for tunnels and foundations.

• Discontinuities and defects strongly influence rock stability, seepage paths, and potential failure modes, including landslides and collapses.

• Soil origins (transported vs residual) and grain-size distributions determine shear strength, compressibility, and drainage characteristics; Atterberg limits and PSD are essential for predicting behavior under loading and environmental changes.

• Phase relationships provide a conceptual tool to relate density, porosity, water content, and saturation to effective stress and to understand how changes in moisture and density affect strength and compressibility.

• Practical measurement methods (PSD analysis, Atterberg limits, bulk densities, cone tests, nuclear densitometry) connect laboratory values to field design and performance.

Key Takeaways (Condensed)

• Rocks vary widely by origin (igneous, sedimentary, metamorphic) and by texture, weathering, and defects; rock mass properties are dominated by discontinuities and weathering.

• Soils are three-phase media whose properties depend on grain size, fabric, moisture, and voids. They exhibit time-dependent and non-linear behaviour, especially clays and organics, and require both field and laboratory testing to characterize.

• Geomorphic processes continually shape soils and rocks, creating landscapes and hazards that engineers must anticipate in design and risk assessment.

• Phase diagrams are essential tools to link elementary soil properties (e.g., e, n, w, LL, PL, PI) and to develop relationships used in design calculations.

• The principle of effective stress is central to soil mechanics and governs strength, settlement, and response to loading; groundwater (pore pressure) is a key variable in determining σ' and the stability of soils and structures.

Symbols and Equations (Quick Reference)

• Void ratio and porosity:
  e = \frac{Vv}{Vs}, \quad n = \frac{V_v}{V}

• Phase volumes (general):
  V = Vs + Vv + V_a \,\text{(total)}

• Unit weights and densities:
  γi = ρi g, \, i ∈ {s, w, a}
  Gs = \frac{ρs}{ρw} = \frac{γs}{γ_w}

• Three-phase relationships (mass/volume balance):
  Ms = γs Vs, \quad Mw = γw Vw

• Effective stress (vertical):
  σ'v = σv - u

• Total vertical stress in layered soils:
  σv(z) = γ1 z1 + γ2 (z - z1) \, (z > z1)

• Pore pressure in saturated zone:
  u(z) = γw (z - z1) \, (z > z_1)

• Effective vertical stress in layered soils:
  σ'v(z) = σv(z) - u(z)

• Lateral earth pressure at rest (K0):
  σ'h = K0 σ'v, \ K0 = 1 - \sin φ'

• Capillary rise (qualitative): finer soils exhibit higher capillary rise; Sands/gravels have minimal rise.

• Plasticity and consistency (Atterberg limits):
  PI = LL - PL, \, LI = \frac{W - LL}{LL - PL} \quad(\text{typical form})

If you want, I can tailor these notes to a specific exam section or expand any of the sections with more examples and worked problems.

Title: Comprehensive Study Notes — Introduction to Geotechnical Engineering (CIVIL 200)