Geotechnical engineering
Geotechnical engineering is a discipline of civil engineering that applies principles of soil mechanics, rock mechanics, and hydrology to the design, analysis, and construction of foundations, earthworks, and other structures interacting with the ground.[1] It focuses on evaluating the physical properties and behavior of earth materials, including soil (such as sand, silt, and clay) and rock, to ensure stability and safety in engineered systems.[2] Developed as a formal field in the early 20th century by Karl Terzaghi, often called the father of soil mechanics, it integrates geology, geophysics, and engineering to address challenges like load-bearing capacity, deformation, and groundwater flow.[2] Key areas within geotechnical engineering include site investigation through borings, geophysical testing, and laboratory analysis to characterize subsurface conditions; foundation design for buildings, bridges, and dams; and slope stability assessments to mitigate risks from landslides and erosion.[3] Practitioners also specialize in computational geomechanics for modeling complex behaviors, earthquake engineering to counteract soil liquefaction, and environmental geotechnics for managing contamination and waste containment.[1] Common applications encompass retaining walls, tunnels, embankments, pipelines, and offshore structures, where soil or rock serves as both a construction material and a supporting medium.[3] The field's importance has grown with global infrastructure demands, as aging systems and natural hazards underscore the need for resilient designs; for instance, the American Society of Civil Engineers' 2025 Infrastructure Report Card assigns the U.S. overall infrastructure a grade of C, highlighting geotechnical vulnerabilities in categories like dams and levees.[4] Advances in ground improvement techniques, such as grouting and reinforcement, continue to enhance sustainability and reduce environmental impacts in urban development and renewable energy projects like wind farms.[2]Introduction
Definition and Scope
Geotechnical engineering is a subdiscipline of civil engineering that focuses on the engineering behavior of earth materials, such as soils and rocks, to support the design, analysis, and construction of foundations, earthworks, and retaining structures.[5] It applies principles of soil and rock mechanics to evaluate how these materials interact with applied loads and environmental conditions.[1] This field ensures the safe and efficient transfer of structural loads to the ground while addressing potential issues like deformation or failure.[6] The scope of geotechnical engineering encompasses the interaction between built structures and the ground, including assessments of stability, settlement, and load-bearing capacity.[7] It plays a critical role in infrastructure projects such as dams, tunnels, bridges, and buildings, where understanding subsurface conditions is essential for preventing hazards like landslides or excessive settlement.[8] Geotechnical engineers investigate site-specific ground properties to inform decisions on foundation types, slope stability, and earth retention systems.[9] Geotechnical engineering is inherently interdisciplinary, integrating principles from geology, mechanics, and hydrology to analyze subsurface behavior.[10] Unlike structural engineering, which primarily deals with above-ground elements, geotechnical engineering emphasizes the unpredictable nature of earth materials and their response to forces, often incorporating geophysical data and water flow dynamics.[5] This integration allows for comprehensive solutions in complex environments, such as coastal or seismic zones.[11] The term "geotechnical" originated in the early 20th century, derived from the Greek "geo" meaning earth and "technical" referring to applied scientific knowledge.[12] It gained prominence with advancements in soil mechanics during that era, formalizing the study of earth materials in engineering practice.[13]Importance and Applications
Geotechnical engineering plays a pivotal role in safeguarding infrastructure against failures such as excessive settlements, landslides, and structural collapses by analyzing soil and rock behavior to inform stable design and construction practices.[14] This discipline ensures the longevity and safety of built environments, mitigating risks that could otherwise lead to catastrophic events and substantial human and financial losses.[15] Economically, the sector underpins global infrastructure development, with the geotechnical services market projected to grow from $2.93 billion in 2025 to $6.95 billion by 2032, reflecting its integral contribution to the multi-trillion-dollar construction industry.[16][17] The applications of geotechnical engineering span a wide array of civil projects, including the design of foundations for high-rise buildings and bridges, which distribute loads effectively to prevent subsidence.[18] It also encompasses the construction of embankments for roads and railways, ensuring slope stability and load-bearing capacity in varied terrains.[19] Critical water management structures like dams and levees rely on geotechnical assessments to handle seepage, settlement, and erosion, while tunnels and underground facilities benefit from evaluations of ground conditions to avoid collapses during excavation.[20][21] In disaster-prone regions, geotechnical engineering is essential for risk mitigation, particularly against earthquakes and floods, by incorporating seismic-resistant foundations and flood-resilient soil stabilization techniques to protect communities and infrastructure.[22] A classic example is the stabilization of the Leaning Tower of Pisa, where geotechnical interventions, including soil extraction beneath the foundation, reduced the tilt by approximately 44 cm between 1999 and 2001, averting collapse through precise management of compressible soils.[23] These efforts demonstrate how geotechnical solutions can preserve historical and modern assets in challenging conditions.[24] Geotechnical engineering increasingly integrates with environmental and transportation fields to promote sustainable urban development, such as by optimizing ground improvement methods to minimize resource use and environmental impact while enhancing transportation networks' resilience.[25] This collaboration supports eco-friendly practices, like using recycled materials in earthworks and designing low-carbon foundations, aligning infrastructure with broader sustainability goals.[26]Historical Development
Early Foundations
The origins of geotechnical engineering trace back to ancient civilizations, where empirical knowledge of soil behavior was essential for constructing enduring structures. In ancient Egypt around 2600 BCE, builders utilized rammed earth techniques, compacting layers of Nile silt and clay to form stable foundations and internal fills for the pyramids, demonstrating an intuitive understanding of soil compaction to support massive stone superstructures.[27] Similarly, the Romans engineered extensive infrastructure with gravel-based foundations; their roads featured layered sub-bases of compacted gravel and sand over stabilized earth, while aqueducts employed gravel-filled trenches to distribute loads and prevent settlement on varied terrains.[28] During the medieval and Renaissance periods, geotechnical practices evolved through trial-and-error adaptations to challenging site conditions. In China, the Dujiangyan irrigation system, constructed in 256 BCE but maintained and expanded through subsequent eras, incorporated earthen dams and diversion weirs made from compacted local soils to manage river flow without large reservoirs, showcasing advanced empirical control of water-soil interactions.[29] In Europe, castle foundations often addressed soft, compressible soils by employing timber piles or raft-like bases of driven wood and stone, as seen in structures like those in marshy regions, where builders drove stakes into underlying firm layers to mitigate differential settlement.[30] The 18th and 19th centuries marked a transition toward scientific approaches, laying groundwork for modern geotechnical principles. William Smith's 1815 geological map of England and Wales provided the first systematic depiction of subsurface strata, enabling engineers to anticipate soil variability for foundation design and excavation.[31] In the 1840s, French engineer Jean-Victor Poncelet advanced early theories on soil mechanics by extending Coulomb's earth pressure concepts with graphical methods for analyzing lateral forces in retaining structures and compacted embankments.[32] Culminating these developments, Henry Darcy's 1856 experiments on water flow through sand filters established the principle of laminar seepage in porous media, which became foundational for analyzing groundwater effects on soil stability and foundation performance.[33]Modern Evolution
The formalization of geotechnical engineering as a distinct scientific discipline gained momentum in the early 20th century, transitioning from empirical practices to rigorous theoretical frameworks. Karl Terzaghi, often hailed as the father of soil mechanics, published his groundbreaking consolidation theory in 1925, which explained the time-dependent settlement of saturated soils under load through the principles of effective stress and pore water pressure dissipation.[34] This work, detailed in his book Erdbaumechanik, established the foundational principles for analyzing soil behavior and earned Terzaghi enduring recognition for elevating geotechnics to a systematic engineering science.[35] Concurrently, academic institutions began institutionalizing the field, with Terzaghi founding the world's first dedicated soil mechanics institute at TU Wien in 1929, reflecting the growing need for specialized education amid rapid industrialization.[36] The mid-20th century witnessed a post-World War II surge in global infrastructure projects, propelling geotechnical engineering into widespread application and institutional support. The founding of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE) in 1948, stemming from the second International Conference on Soil Mechanics and Foundation Engineering in Rotterdam, fostered international collaboration, knowledge exchange, and standardization of practices.[37] This era's infrastructure boom—encompassing highways, bridges, dams, and urban expansions—drove key advancements in pile foundations, where driven and bored piles became essential for transferring loads in variable soils, and in slope stability analysis, employing methods like the Bishop's simplified procedure to mitigate failures in embankments and excavations.[38] By the late 20th century, computational innovations and environmental imperatives further shaped the discipline. The 1970s introduction of finite element methods in geotechnical analysis enabled numerical modeling of nonlinear soil responses, complex geometries, and stress distributions, marking a shift from limit equilibrium approaches to more precise simulations of phenomena like earth pressures and settlements.[39] The 1980s emergence of environmental geotechnics addressed escalating concerns over pollution, focusing on engineered waste containment systems such as landfills with impermeable liners and leachate collection to prevent groundwater contamination from hazardous materials.[40] Entering the 21st century, geotechnical engineering has integrated advanced geospatial technologies, particularly since the 2010s, to improve data acquisition and risk management. The incorporation of Geographic Information Systems (GIS) and remote sensing techniques, including satellite imagery and LiDAR, has revolutionized site characterization by enabling large-scale mapping of soil variability, landslide susceptibility, and subsidence patterns, thus supporting proactive infrastructure planning.[41] In response to intensifying climate-driven events and seismic hazards in the 2020s, the second-generation revisions to Eurocode 7— with the third and final part published in April 2025 and full CEN publication expected by 2027—have enhanced provisions for seismic resilience and climate adaptation, including updated partial factors and design scenarios for extreme weather impacts on foundations and retaining structures.[42][43][44]Fundamental Concepts
Soil and Rock Mechanics
Soil classification systems provide a standardized framework for identifying and categorizing soils based on their physical characteristics, which is essential for geotechnical analysis. The Unified Soil Classification System (USCS), developed by Arthur Casagrande during World War II as the Airfield Classification System and formally adopted in 1952 by the U.S. Bureau of Reclamation and the U.S. Army Corps of Engineers, groups soils into coarse-grained (gravels and sands) and fine-grained (silts and clays) categories using criteria such as particle size distribution and plasticity.[45] Grain size distribution is determined through sieve analysis for coarse particles and hydrometer analysis for fines, revealing percentages of gravel (>4.75 mm), sand (0.075–4.75 mm), silt (0.002–0.075 mm), and clay (<0.002 mm) that influence engineering behavior.[45] For fine-grained soils, Atterberg limits, introduced by Swedish chemist Albert Atterberg in the early 1900s, define boundaries of consistency: the liquid limit (water content at which soil transitions from plastic to liquid state), plastic limit (minimum water content for plastic behavior), and shrinkage limit (water content below which further drying causes no volume change).[46] These limits, standardized in ASTM D4318, help distinguish clays (high plasticity) from silts (low plasticity) and predict volume change potential. Phase relationships describe the volumetric and mass interrelations among solid particles, water, and air in a soil mass, forming the basis for quantifying soil composition. The void ratio e is defined as the ratio of the volume of voids V_v to the volume of solids V_s, typically ranging from 0.3 for dense sands to over 1.0 for loose clays, indicating packing density.[47] Water content w is the ratio of the mass of water M_w to the mass of solids M_s, expressed as a percentage, and reflects moisture state critical for compaction and strength.[47] The degree of saturation S represents the fraction of void volume occupied by water, S = V_w / V_v, where V_w is water volume; S=0 for dry soil and S=1 for fully saturated conditions.[47] Porosity n is the ratio of void volume to total volume, related to void ratio by n = e / (1 + e), and typically 30–50% for most soils.[47] The specific gravity of soil solids G_s, the ratio of solid density to water density, averages approximately 2.65 for quartz-based sands and gravels, though it ranges from 2.60 to 2.80 for common inorganic soils.[48] Key soil properties govern hydraulic, mechanical, and deformation responses in geotechnical contexts. Porosity quantifies void space availability for fluid flow and storage, directly linked to phase relations and influencing compressibility. Permeability k, a measure of the soil's capacity to transmit water under hydraulic gradient (as in Darcy's law), varies widely: high (k > 10^{-3} cm/s) for clean gravels, moderate ($10^{-3} to $10^{-5} cm/s) for sands, and low (< 10^{-7} cm/s) for clays, determined via constant-head or falling-head tests per ASTM D2434 and D5084. Shear strength parameters include cohesion c, the intercept of the Mohr-Coulomb failure envelope representing inherent particle bonding (often 0 for granular soils but up to 50 kPa for clays), and the friction angle \phi, the slope indicating frictional resistance (typically 25–35° for sands, 0–30° for clays), derived from triaxial or direct shear tests.[49] Rock mechanics fundamentals address the behavior of intact rock and rock masses, where discontinuities significantly alter overall properties. The Rock Mass Rating (RMR) system, developed by Z.T. Bieniawski in 1973, provides a quantitative index (0–100 scale) for engineering assessment by evaluating six parameters: uniaxial compressive strength of intact rock, rock quality designation (RQD), spacing and condition of discontinuities, groundwater influence, and orientation effects relative to engineering structures.[50] Discontinuity analysis involves characterizing joints, faults, and bedding planes through orientation, spacing, persistence, roughness, infilling, and weathering, as these planes of weakness control deformability, strength, and permeability in rock masses far more than intact material properties.[51] Intact rock strength is primarily assessed via uniaxial compressive strength (UCS), the maximum axial stress a cylindrical core can withstand before failure, tested per ASTM D7012 or ISRM suggested methods, with values ranging from 50 MPa for weak rocks to over 200 MPa for strong igneous types, serving as a baseline for mass behavior predictions.[52]Stress-Strain Behavior
In geotechnical engineering, the response of soils and rocks to applied stresses is fundamentally governed by the effective stress principle, which distinguishes between total stress and the stress borne by the soil skeleton. Introduced by Karl Terzaghi in 1925, this principle states that the effective stress \sigma' is equal to the total stress \sigma minus the pore water pressure u, expressed as \sigma' = \sigma - u.[53] This concept is crucial because it determines both the strength and deformation behavior of saturated soils; changes in pore pressure due to loading or drainage directly influence the interparticle forces, thereby controlling shear resistance and compressibility.[54] Without accounting for effective stress, predictions of soil stability and settlement would be inaccurate, particularly in water-saturated conditions common in foundations and embankments. The stress-strain behavior of soils exhibits distinct phases, transitioning from elastic recovery to plastic deformation under increasing load. In the elastic range, soils deform proportionally to applied stress with minimal permanent strain, but beyond the yield point, plastic flow occurs, leading to irreversible deformation and potential failure. This behavior is often characterized by stress-strain curves obtained from triaxial or direct shear tests, which reveal the soil's stiffness and ductility depending on its composition—granular soils like sands show more frictional response, while cohesive clays exhibit initial cohesion-dominated yielding. Failure in soils is commonly modeled using the Mohr-Coulomb criterion, which posits that shear failure occurs when the shear stress \tau on a plane reaches \tau = c + \sigma' \tan \phi, where c is the cohesion (typically 0–100 kPa for most soils, zero for clean sands) and \phi is the effective friction angle (ranging from 20° to 45°, higher for dense granular materials).[55] This linear envelope provides a practical envelope for predicting limiting equilibrium in design, though it assumes isotropic conditions and does not capture post-peak softening in some soils.[56] A key aspect of soil stress-strain response under sustained loading is consolidation, where saturated fine-grained soils undergo time-dependent settlement as excess pore pressures dissipate and effective stresses increase. Primary consolidation involves the expulsion of water from soil voids, driven by Terzaghi's one-dimensional theory, which models the process as diffusion of pore pressure governed by the coefficient of consolidation c_v, relating the rate of settlement to soil permeability, void ratio, and compressibility.[57] Secondary consolidation follows, characterized by creep under constant effective stress, contributing to long-term settlements in organic or highly plastic clays. This time-rate behavior is critical for predicting total settlement in structures like dams or buildings, ensuring that designs accommodate both immediate and delayed deformations without excessive risk.[58] For rocks, stress-strain behavior contrasts with soils due to their higher stiffness and tendency toward brittle failure under compression, though ductile responses can occur in weaker or confined rock masses. Intact rock typically fails abruptly with low strain (0.1–1%), exhibiting linear elastic behavior up to peak stress followed by sudden drop-off, while rock masses with discontinuities may show more progressive, ductile yielding influenced by jointing and weathering. The Hoek-Brown failure criterion, developed empirically for rock masses, captures this non-linear strength by relating the major principal stress at failure \sigma_1 to the minor principal stress \sigma_3 as \sigma_1 = \sigma_3 + \sigma_{ci} (m_b \sigma_3 / \sigma_{ci} + s)^{0.5}, where \sigma_{ci} is the uniaxial compressive strength of intact rock, m_b is the reduced value of the material constant m_i (accounting for fracturing), and s is a cohesion-like parameter (ranging from 1 for intact rock to near 0 for highly fractured masses).[59] This criterion has become a standard for tunneling and slope stability analyses, emphasizing the role of geological quality in scaling intact rock properties to mass behavior.[60]Geotechnical Investigation
Site Exploration Methods
Site exploration methods in geotechnical engineering involve a range of field techniques to characterize subsurface conditions, including soil and rock profiles, groundwater levels, and potential hazards, prior to design and construction. These methods provide essential data for mapping stratigraphy and mechanical properties without extensive excavation, enabling engineers to assess site suitability and mitigate risks. Geophysical surveys offer non-invasive reconnaissance, while direct drilling and sampling yield precise material recovery for further analysis. Effective exploration integrates multiple approaches to balance cost, accuracy, and coverage, often following standardized guidelines to ensure comprehensive subsurface profiling.[61] Geophysical methods are widely used for initial, non-destructive subsurface mapping in geotechnical site investigations. Seismic refraction and reflection techniques determine layer depths and velocities by analyzing the propagation of compressional and shear waves through the ground; refraction identifies refractor depths like bedrock interfaces, while reflection provides detailed stratigraphic imaging for deeper profiles. Electrical resistivity surveys detect groundwater tables and subsurface voids by measuring soil resistance to electrical current, with applications in identifying contaminated zones or clay layers that alter conductivity. Ground-penetrating radar (GPR) excels in shallow feature detection, such as utilities, voids, or thin soil layers, by transmitting electromagnetic pulses and interpreting reflections, typically effective to depths of 5-10 meters in low-conductivity soils. These methods are often combined for enhanced resolution, as in integrated surveys that correlate seismic and resistivity data to refine soil profiles.[62][63][64][65][66] Drilling and sampling methods provide direct access to subsurface materials through boreholes, allowing recovery of samples for property assessment. Standard penetration tests (SPT) conducted during borehole drilling involve driving a split-spoon sampler into the soil using a 63.5 kg hammer dropped from 760 mm, with the number of blows required for 300 mm penetration indicating soil density and strength; this in-situ test is standardized under ASTM D1586 and commonly used for granular soils. Rotary drilling employs a rotating bit with circulating drilling fluid to advance boreholes to depths exceeding 100 meters, suitable for rock coring and deep profiling in varied lithologies. Undisturbed sampling via Shelby tubes—thin-walled steel cylinders pushed hydraulically into cohesive soils—preserves sample integrity for shear strength evaluation, minimizing disturbance in soft clays as per ASTM D1587. These techniques ensure representative sampling, with borehole logs documenting stratigraphy, water levels, and recovery ratios.[67][68][69] Exploration planning requires strategic borehole spacing and integration of remote sensing to optimize coverage and efficiency. According to Eurocode 7 (EN 1997-2, Annex B), borehole spacing for buildings typically ranges from 25-75 meters, depending on structure size and soil variability, with closer intervals (e.g., 15-40 meters) for high-rise or sensitive sites to capture lateral heterogeneity. Remote sensing technologies, including drones for aerial photogrammetry and LiDAR for high-resolution topographic mapping, have been integrated since the mid-2010s to supplement traditional methods; LiDAR generates digital elevation models to identify surface anomalies like sinkholes, while drone surveys enable rapid, cost-effective site reconnaissance over large areas. These tools enhance planning by providing preliminary data that guides borehole locations, reducing unnecessary drilling.[70][71][72][73] Hazard identification during site exploration targets geological features that could compromise stability, such as karst formations and fault zones, through targeted geophysical and drilling surveys. Karst features, including cavities and pinnacles in soluble rocks like limestone, are detected via electrical resistivity or GPR to map low-resistivity voids, with borehole confirmation via coring to assess collapse risks. Fault zones are identified by seismic reflection profiling to delineate shear planes and offset layers, informing seismic hazard assessments. Phased investigations—starting with geophysical reconnaissance followed by targeted drilling—optimize cost-benefit by refining exploration based on initial findings, minimizing over-investigation while ensuring comprehensive hazard mitigation.[74][75][61][76][77]Testing and Analysis Techniques
Geotechnical testing and analysis techniques are essential for quantifying the mechanical properties of soil and rock, enabling engineers to assess site suitability and inform design decisions. These methods encompass both laboratory-based procedures on retrieved samples and in-situ tests conducted directly in the ground, often obtained through site exploration methods such as drilling. Laboratory tests provide controlled measurements of parameters like shear strength and compressibility, while in-situ tests capture field-scale behavior, accounting for natural variability. Analysis involves interpreting raw data to derive design parameters, incorporating statistical considerations and computational tools for reliable profiling.[78] Laboratory tests form the cornerstone of property determination, typically performed on undisturbed or remolded samples to evaluate stress-strain responses under simulated field conditions. The triaxial shear test is widely used to measure shear strength parameters, including the effective friction angle (φ) and cohesion (c), through variants such as consolidated undrained (CU), consolidated drained (CD), and unconsolidated undrained (UU) procedures. In CU and CD tests, samples are consolidated under confining pressure before shearing, allowing pore pressure measurements to distinguish effective stresses, whereas UU tests simulate rapid loading for undrained conditions. These tests adhere to standardized protocols to ensure reproducibility, with φ typically ranging from 20° to 45° for soils depending on composition.[79][80][81] The oedometer test assesses one-dimensional consolidation behavior, determining the compression index (C_c), which quantifies void ratio reduction with increasing effective stress. The formula for C_c is given by: C_c = \frac{\Delta e}{\Delta \log \sigma'} where Δe is the change in void ratio and Δlogσ' is the change in logarithm of effective vertical stress; typical C_c values for clays range from 0.1 to 1.0. This test simulates overburden loading to predict settlement in fine-grained soils. Complementing these, the direct shear test evaluates interface friction between soil and structural elements, such as geosynthetics or piles, by shearing along a predefined plane under normal load, yielding friction angles that inform design for stability.[81][82][83] In-situ tests provide direct measurements of soil response without sample disturbance, capturing stratigraphy and variability at depth. The cone penetration test (CPT) involves pushing a conical probe into the ground at a constant rate, recording tip resistance (q_c), which can reach up to 100 MPa in dense sands, alongside sleeve friction for soil classification and strength estimation. The piezocone (CPTu) extends CPT by incorporating pore pressure sensors (u), enabling detection of drainage conditions and undrained shear strength in clays via normalized parameters like B_q = (u - u_0)/(q_c - σ_v0), where u_0 is hydrostatic pressure and σ_v0 is total vertical stress. The plate load test applies incremental loads to a steel plate on the ground surface to determine bearing capacity and modulus, simulating foundation behavior; ultimate bearing pressures from these tests guide shallow foundation design, with settlements monitored to 10-25 mm typically.[84][85][86] Data analysis integrates test results for geotechnical profiling, starting with borelog interpretation, where drilling records detail stratigraphy, water levels, and recovery to construct subsurface models. Statistical variability is critical, as standard penetration test (SPT) N-values exhibit coefficients of variation (COV) of 20-50%, reflecting spatial heterogeneity; this informs probabilistic assessments to select conservative design parameters. Software like PLAXIS facilitates preliminary profiling through finite element modeling of test data, simulating stress distributions for initial parameter calibration.[87][88][89][90] Standards ensure consistency and quality in testing. ASTM D698 outlines laboratory compaction methods using standard effort (Proctor test) to relate water content to dry unit weight, achieving maximum densities around 95-100% for field control. In the 2020s, updates emphasize sustainability, such as reduced sample volumes in triaxial testing to minimize material use and waste, aligning with environmental guidelines while maintaining accuracy.[91][91]Design Practices
Foundation Design
Foundation design in geotechnical engineering ensures that structures transfer loads to the ground without exceeding the soil's bearing capacity or causing excessive settlements that could lead to structural distress. Shallow foundations, such as strip footings under load-bearing walls and isolated footings under columns, are typically used when surface soils have sufficient strength, with embedment depths generally less than the foundation width. Deep foundations, like piles, extend into deeper, more competent strata to bypass weak surface layers. Designs incorporate site-specific soil properties obtained from geotechnical investigations to calculate ultimate capacities and apply factors of safety, often ranging from 2.5 to 3.0 for bearing capacity.[92] For shallow foundations, the ultimate bearing capacity q_u is determined using Terzaghi's theory, which assumes general shear failure and a rough base on level ground. The equation is: q_u = c N_c + \gamma D N_q + 0.5 \gamma B N_\gamma where c is soil cohesion, \gamma is the soil unit weight, D is the foundation depth, B is the foundation width, and N_c, N_q, N_\gamma are dimensionless bearing capacity factors dependent on the soil's friction angle \phi. These factors are derived from plasticity theory and tabulated for various \phi values; for example, at \phi = 30^\circ, N_c = 37.2, N_q = 22.5, and N_\gamma = 19.7. Shape and depth modifications may be applied for non-strip footings, such as increasing N_c by 1.3 for square footings. The net allowable bearing pressure is then q_{net,all} = (q_u - \gamma D)/FS, ensuring the factored load does not exceed this limit.[92]| Friction Angle \phi (degrees) | N_c | N_q | N_\gamma |
|---|---|---|---|
| 0 | 5.7 | 1.0 | 0.0 |
| 10 | 8.4 | 1.2 | 0.4 |
| 20 | 14.8 | 2.5 | 3.6 |
| 30 | 37.2 | 22.5 | 19.7 |
| 35 | 46.1 | 33.3 | 42.2 |
| 40 | 75.3 | 64.2 | 109.4 |