Rankine theory

Rankine theory, also known as Rankine's earth pressure theory, is a foundational analytical method in geotechnical engineering for calculating the lateral pressures exerted by cohesionless soils on retaining structures such as walls and dams. Developed by the Scottish civil engineer William John Macquorn Rankine in 1857, it derives from the principles of plastic equilibrium and Mohr-Coulomb failure criterion to predict active earth pressure (the minimum pressure when soil expands laterally), passive earth pressure (the maximum pressure when soil is compressed laterally), and, by extension, at-rest earth pressure (the pressure under no lateral wall movement).^[1]^[2]^[3] The theory relies on several key assumptions to simplify the complex behavior of soil: the soil is dry, homogeneous, isotropic, and cohesionless (c = 0); the retaining wall is vertical, rigid, and frictionless (no wall-soil adhesion or friction angle δ); the backfill surface is horizontal or at a known slope (β); and failure occurs along planar slip surfaces at an angle of 45° ± φ/2 relative to the horizontal, where φ is the soil's internal friction angle.^[2]^[3] These assumptions enable a stress-field solution where lateral earth pressure varies linearly with depth, acting parallel to the backfill slope with its resultant force located at one-third the wall height from the base.^[2] Central to Rankine theory are the earth pressure coefficients, which quantify the ratio of horizontal to vertical stress. The active earth pressure coefficient is given by
K_a = \tan^2 \left(45^\circ - \frac{\phi}{2}\right) = \frac{1 - \sin \phi}{1 + \sin \phi},
resulting in the active pressure per unit length of wall:
P_a = \frac{1}{2} K_a \gamma H^2,
where γ is the soil unit weight and H is the wall height. Conversely, the passive earth pressure coefficient is
K_p = \tan^2 \left(45^\circ + \frac{\phi}{2}\right) = \frac{1 + \sin \phi}{1 - \sin \phi},
yielding
P_p = \frac{1}{2} K_p \gamma H^2.
For at-rest conditions, the coefficient is typically
K_0 = 1 - \sin \phi,
though this is an empirical extension often attributed to Jaky (1944) rather than Rankine's original formulation.^[2]^[3] These expressions assume level ground; for sloping backfill, modified coefficients account for the angle β.^[2] Rankine theory is widely applied in the design of retaining walls, sheet piles, and excavations to ensure structural stability against soil thrust, particularly for granular soils like sands and gravels. It provides a conservative estimate for active pressures but underpredicts passive pressures in cases with wall friction or sloping ground, where Coulomb's wedge theory is preferred as a complement. Limitations include its inapplicability to cohesive soils (c > 0), submerged conditions without adjustments for pore water pressure, or non-planar failure modes in complex geometries.^[2]^[3] Despite these, it remains a cornerstone of modern geotechnical practice due to its simplicity and alignment with limit equilibrium principles.

Historical Background

Development by William Rankine

William John Macquorn Rankine (1820–1872) was a prominent Scottish civil engineer and physicist, recognized as a key figure in the development of thermodynamics and engineering mechanics. Born in Edinburgh on July 5, 1820, he received his education at the University of Edinburgh before embarking on a career that included practical engineering projects such as railway construction and waterworks improvements from 1839 to 1841 under John Benjamin MacNeill.^[4] In 1855, Rankine was appointed as the first Regius Professor of Civil Engineering and Mechanics at the University of Glasgow, where he pursued extensive research and teaching until his death on December 24, 1872.^[4] His work bridged theoretical science and practical engineering, particularly during the rapid infrastructure expansion of the 19th century.^[5] Rankine's seminal contribution to soil mechanics came in 1857 with the publication of "On the Stability of Loose Earth" in the Philosophical Transactions of the Royal Society of London, following receipt by the society on June 10, 1856, and presentation on June 19, 1856.^[1] In this paper, he formulated a mathematical theory for the stability of granular, cohesionless masses based on frictional resistance, avoiding the assumption-heavy approaches of earlier works.^[1] He later expanded these ideas in his Manual of Applied Mechanics (1858) and Manual of Civil Engineering (1861), applying concepts of conjugate pressures to analyze stresses in soil against vertical retaining structures.^[5] Rankine's motivation stemmed from the need to create a rational foundation for soil theory amid the limitations of prior hypotheses, such as those relying on arbitrary assumptions about soil behavior.^[5] This effort was driven by practical engineering challenges during the Industrial Revolution's railway boom, where retaining walls for embankments and cuttings required reliable methods to assess lateral earth pressures in cohesionless soils.^[5] Building on Charles-Augustin de Coulomb's 1776 wedge theory, Rankine simplified the analysis for non-cohesive materials, eliminating the need to assume a specific failure plane and instead deriving pressures from equilibrium conditions throughout the soil mass.^[5] His approach emphasized graphical methods, drawing analogies to electrical equipotential lines for visualizing pressure distribution.^[5] The theory quickly influenced British engineering practices in the post-1850s era of infrastructure growth, providing a more accessible tool for designing stable retaining structures in railways and canals compared to Coulomb's more complex wedge method.^[6] Rankine's work laid essential groundwork for modern geotechnical engineering, later refined by continental European scholars like Joseph Boussinesq in 1882, and remains a cornerstone in the evolution of soil mechanics.^[5]

Context in Early Soil Mechanics

Early efforts to understand earth pressure originated in the context of military engineering during the 17th century, particularly with Sébastien Le Prestre de Vauban's empirical rules for designing fortification walls up to 25 meters in height in France.^[5] These guidelines relied on practical observations rather than analysis of soil properties, providing a foundational but rudimentary approach to retaining earth in defensive structures. By the late 18th century, Charles-Augustin de Coulomb advanced the field with his 1776 wedge theory, which modeled the active and passive earth pressures on retaining walls by considering frictional resistance and cohesion along a planar failure surface.^[2] However, Coulomb's method was limited to vertical walls with horizontal backfill, assuming a straight slip line and neglecting complexities like inclined surfaces or sloping backfills.^[5] The mid-19th century marked the emergence of soil mechanics as a distinct engineering discipline in Europe, spurred by the rapid expansion of infrastructure projects such as railways and bridges that demanded reliable predictions of soil behavior under load.^[5] This period saw increased experimental investigations, including full-scale tests by engineers like John Burgoyne in 1834, which highlighted the need for systematic theories to address earth pressure stability in large-scale constructions.^[5] The growth of civil engineering societies across Europe facilitated the exchange of ideas, transitioning from empirical rules to more analytical methods amid the industrial boom. In the 1850s, debates within engineering societies focused on earth pressure stability, contrasting traditional empirical approaches with emerging analytical models, where Rankine's 1857 publication in the Philosophical Transactions of the Royal Society represented a pivotal advancement by introducing a continuum-based theory for cohesionless soils.^[1]^[5] Following its introduction, Rankine's theory gained traction and was adopted into French and British engineering practices by the 1860s, influencing design codes for retaining structures and infrastructure.^[5] This adoption contrasted with later extensions, such as A.L. Bell's 1915 modifications, which adapted the framework to account for cohesive soils by incorporating tension cracks and adjusted pressure coefficients.^[7]

Fundamental Assumptions

Soil and Geometry Assumptions

Rankine theory relies on several key assumptions regarding the soil properties to simplify the analysis of earth pressures on retaining structures. The soil is modeled as dry and cohesionless, consisting solely of granular materials such as sand or gravel, where resistance to shear arises entirely from interparticle friction without any adhesive forces or cohesion (c = 0).^[1] This assumption excludes the effects of moisture, ensuring no pore water pressures or water table influences that could alter effective stresses.^[8] Additionally, the soil is treated as homogeneous and isotropic, meaning its frictional properties and unit weight γ are uniform throughout the mass, allowing for consistent behavior under stress.^[8] The geometric configuration of the system is idealized to facilitate analytical solutions under plane strain conditions. The backfill surface is assumed to be horizontal, with no sloping ground or additional surcharge loads applied, which maintains a uniform vertical stress distribution.^[1] The retaining wall is vertical, rigid, and smooth, implying no frictional resistance between the wall and the soil (δ = 0), and it extends infinitely in the direction perpendicular to the plane of analysis to eliminate three-dimensional effects.^[8] The soil mass itself is considered semi-infinite, extending indefinitely behind the wall without boundaries that could influence pressure development.^[9] These assumptions align the theory's failure behavior with the Mohr-Coulomb criterion for cohesionless soils, emphasizing frictional resistance as the primary mechanism.^[8]

Stress and Failure Assumptions

In Rankine theory, the soil mass behind a retaining wall is assumed to reach a state of plastic equilibrium at failure, where the soil elements are on the verge of shear failure along predefined slip surfaces.^[10] This condition implies that the entire soil mass behaves as if it is fully mobilized to its ultimate shear resistance without further increase in stress, forming a coherent failure zone.^[11] The slip planes develop at an angle of $45^\circ + \phi/2 to the horizontal, where \phi is the soil's angle of internal friction, representing the orientation of maximum shear strain in the active case for cohesionless soils.^[10] These planes emerge due to the extension of the soil mass, assuming a horizontal backfill geometry that simplifies the stress field. The theory posits that there are no shear stresses acting on vertical planes within the soil mass behind the wall, which allows the vertical and horizontal stresses to act as the principal stresses.^[11] The vertical principal stress \sigma_v is taken as the overburden pressure \gamma z, where \gamma is the unit weight of the soil and z is the depth, arising from the self-weight of the soil above.^[10] The horizontal principal stress \sigma_h then becomes the minor principal stress in the active state, determined by the soil's frictional properties without wall friction influence.^[2] Failure in Rankine theory is triggered when the retaining wall yields sufficiently to induce horizontal strain in the adjacent soil, thereby mobilizing the full shear strength along the potential slip planes. This mobilization occurs as the wall movement—typically a small rotation or translation—allows the soil to expand laterally, reducing the horizontal stress until the Mohr-Coulomb failure criterion is met.^[2] The theory assumes an elastic-perfectly plastic soil behavior, where the soil deforms elastically up to the yield point and then plastically without hardening, ignoring intermediate stress states or partial mobilization.^[10] This idealization simplifies analysis for granular soils but presumes uniform properties and neglects effects like dilation or intermediate principal stresses.^[11]

Theoretical Framework

Mohr-Coulomb Failure Criterion

The Mohr-Coulomb failure criterion defines the conditions under which a soil reaches shear failure, serving as the foundational envelope for stress states in Rankine theory. It posits that failure occurs when the shear stress \tau along a plane exceeds the sum of the cohesive strength c and the frictional component dependent on the normal stress \sigma, given by the equation:

\tau = c + \sigma \tan \phi

where \phi is the internal friction angle of the soil. In the context of Rankine theory, which primarily addresses cohesionless granular soils, cohesion is assumed to be zero (c = 0), simplifying the criterion to \tau = \sigma \tan \phi and emphasizing frictional resistance as the dominant mechanism.^[1] This criterion was originally developed by Charles-Augustin de Coulomb in 1776, who applied it to the stability of retaining structures by considering the maximum shear resistance along potential slip planes in granular materials. Otto Mohr later graphicalized and generalized it in 1900, introducing the concept of an envelope in the \sigma-\tau plane to represent failure conditions across various stress orientations. In Mohr's representation, failure is visualized using Mohr's circles, where each circle corresponds to a stress state defined by the principal stresses \sigma_1 (major) and \sigma_3 (minor). Failure initiates when the Mohr's circle becomes tangent to the linear Coulomb envelope, with the point of tangency indicating the critical plane of shear. The radius of this tangent circle relates to the maximum shear stress, and the center to the mean normal stress, providing a geometric interpretation of the \tau = c + \sigma \tan \phi relation. Within Rankine theory, the Mohr-Coulomb criterion is adapted to relate the principal stresses at failure in a soil element under plastic equilibrium, particularly for the active earth pressure case where the soil tends to expand laterally.^[1] Rankine derived that the ratio of the major to minor principal stress satisfies:

\frac{\sigma_1 - \sigma_3}{\sigma_1 + \sigma_3} = \sin \phi

which rearranges to \sigma_1 = \sigma_3 \frac{1 + \sin \phi}{1 - \sin \phi} for cohesionless soils, establishing the proportional limit for stress redistribution behind a retaining structure.^[1] This adaptation by Rankine in 1857 extended Coulomb's and Mohr's work to continuous soil masses, assuming uniform frictional properties without cohesion.^[1] In Rankine theory, the criterion applies to zones of plastic equilibrium, where soil elements reach this failure limit to determine overall earth pressure distribution.

Plastic Equilibrium Concept

In Rankine theory, the concept of plastic equilibrium describes the idealized condition at failure where the entire soil mass behind a retaining wall reaches a state of impending collapse, with the soil deforming along a network of conjugate slip planes. These slip planes are oriented such that shear stress is zero on the principal planes, allowing the soil to mobilize its full shear strength without actual rupture. This equilibrium state assumes a semi-infinite, cohesionless soil mass under the influence of the wall's movement, as originally formulated by William Rankine in his analysis of loose earth stability.^[1] In the active case, the retaining wall yields away from the backfill, inducing horizontal extension throughout the soil mass and reducing the lateral (horizontal) stress to its minimum limiting value. Here, the vertical stress \sigma_v acts as the major principal stress, while the horizontal stress \sigma_h becomes the minor principal stress, related by \sigma_h = \sigma_v K_a, where K_a is the coefficient of active earth pressure derived from the soil's friction angle. The conjugate slip planes in this state are inclined at $45^\circ + \phi/2 to the horizontal.^[12] Conversely, in the passive case, the wall is pushed toward the soil, compressing the mass horizontally and elevating the lateral stress to its maximum. In this configuration, the horizontal stress \sigma_h serves as the major principal stress, with the vertical stress \sigma_v as the minor, given by \sigma_h = \sigma_v K_p, where K_p is the coefficient of passive earth pressure. The slip planes orient at $45^\circ - \phi/2 to the horizontal, enabling the soil to resist further intrusion through plastic flow. This state is bounded by the Mohr-Coulomb failure criterion, which defines the shear strength envelope.^[12] In Rankine theory, the soil mass is assumed to be in plastic equilibrium throughout, with slip lines at $45^\circ \pm \phi/2 to the horizontal forming the characteristic pattern, unlike Coulomb's approach which assumes discrete triangular failure wedges. In the basic theory, these slip lines are idealized as planar surfaces, neglecting more refined approximations such as logarithmic spirals that account for curved failure paths in advanced analyses.^[13]

Active Earth Pressure

Derivation Process

In Rankine's theory for active earth pressure, the derivation begins with the assumption of a cohesionless soil mass behind a smooth, vertical retaining wall, where the wall movement is directed away from the soil, inducing extension and mobilizing the minimum lateral pressure. The vertical effective stress at depth z is given by \sigma_v = \gamma z, where \gamma is the unit weight of the soil; in the active state, this vertical stress acts as the major principal stress (\sigma_1), while the horizontal stress (\sigma_h) becomes the minor principal stress (\sigma_3) at failure.^[1]^[2] To determine the stress state at failure, Rankine's method employs the concept of plastic equilibrium, interpreted through the Mohr-Coulomb failure criterion, where the soil reaches its shear strength along potential slip surfaces. The Mohr circle of stress is constructed with \sigma_v as the center coordinate for the major stress, and the circle contracts as \sigma_h decreases until it is tangent to the Mohr-Coulomb failure envelope, defined by the soil's internal friction angle \phi. This tangency condition yields the ratio of the minor to major principal stresses as \frac{\sigma_h}{\sigma_v} = \frac{1 - \sin \phi}{1 + \sin \phi}, defining the active earth pressure coefficient K_a.^[1]^[3] The equivalent geometric representation confirms this ratio through the orientation of the failure planes. In the active state, the slip planes form at an angle of $45^\circ + \phi/2 to the horizontal (major principal stress direction), ensuring the equilibrium configuration that minimizes resistance to wall movement. This orientation arises from the double-angle relationship in the Mohr circle, where the pole of planes indicates the critical shear direction for plastic flow.^[2]^[14] Combining these elements, the active earth pressure coefficient is expressed as:

K_a = \tan^2 \left(45^\circ - \frac{\phi}{2}\right) = \frac{1 - \sin \phi}{1 + \sin \phi}.

The resulting active earth pressure at depth z is then p_a = K_a \gamma z, representing the horizontal stress required to maintain plastic equilibrium in the extended soil mass.^[1]^[3]

Pressure Formulas and Distribution

In Rankine theory, the active earth pressure represents the minimum horizontal thrust that a cohesionless soil mass exerts on a retaining structure when the soil expands laterally. The horizontal active pressure p_a at any depth z below the surface increases linearly with depth, given by p_a = K_a \gamma z, where K_a is the active earth pressure coefficient, \gamma is the unit weight of the soil, and z ranges from 0 at the surface to H at the base of the wall of height H.^[2]^[15] This distribution forms a triangular pressure diagram, with zero pressure at the ground surface and a maximum value of K_a \gamma H at the base, illustrating the soil's reduction in horizontal stress to accommodate wall movement away from the soil mass.^[1] The total active force P_a acting on the wall is the integral of this pressure distribution over the wall height, resulting in P_a = \frac{1}{2} K_a \gamma H^2.^[2]^[15] This force acts horizontally through the centroid of the triangular distribution, located at a distance of H/3 above the base of the wall.^[2] In graphical representations, the active pressure profile depicts a standard triangular loading on the wall, emphasizing the soil's thrusting capacity, which is crucial for calculating stability in free-standing or gravity retaining structures.^[15] For a typical granular soil with an internal friction angle \phi = 30^\circ, K_a \approx 0.33, yielding an active pressure p_a = 0.33 \gamma z that demonstrates the reduction of vertical stress into horizontal thrust compared to at-rest conditions.^[2]^[1]

Passive Earth Pressure

Derivation Process

In Rankine's theory for passive earth pressure, the derivation begins with the assumption of a cohesionless soil mass behind a smooth, vertical retaining wall, where the wall movement is directed toward the soil, inducing compression and mobilizing the maximum lateral resistance. The vertical effective stress at depth z is given by \sigma_v = \gamma z, where \gamma is the unit weight of the soil; in the passive state, this vertical stress acts as the minor principal stress (\sigma_3), while the horizontal stress (\sigma_h) becomes the major principal stress (\sigma_1) at failure.^[1]^[2] To determine the stress state at failure, Rankine's method employs the concept of plastic equilibrium, interpreted through the Mohr-Coulomb failure criterion, where the soil reaches its shear strength along potential slip surfaces. The Mohr circle of stress is constructed with \sigma_v as the center coordinate for the minor stress, and the circle expands as \sigma_h increases until it is tangent to the Mohr-Coulomb failure envelope, defined by the soil's internal friction angle \phi. This tangency condition yields the ratio of the major to minor principal stresses as \frac{\sigma_h}{\sigma_v} = \frac{1 + \sin \phi}{1 - \sin \phi}, defining the passive earth pressure coefficient K_p.^[1]^[3] The equivalent geometric representation confirms this ratio through the orientation of the failure planes. In the passive state, the slip planes are inclined at an angle of $45^\circ - \phi/2 to the horizontal, ensuring the equilibrium configuration that maximizes resistance to wall thrust. This orientation arises from the double-angle relationship in the Mohr circle, where the pole of planes indicates the critical shear direction for plastic flow.^[2]^[16]^[17] Combining these elements, the passive earth pressure coefficient is expressed as:

K_p = \tan^2 \left(45^\circ + \frac{\phi}{2}\right) = \frac{1 + \sin \phi}{1 - \sin \phi}.

The resulting passive earth pressure at depth z is then p_p = K_p \gamma z, representing the horizontal stress required to maintain plastic equilibrium in the compressed soil mass.^[1]^[3]

Pressure Formulas and Distribution

In Rankine theory, the passive earth pressure represents the maximum horizontal resistance that a cohesionless soil mass can mobilize against a retaining structure when the soil is compressed laterally. The horizontal passive pressure p_p at any depth z below the surface increases linearly with depth, given by p_p = K_p \gamma z, where K_p is the passive earth pressure coefficient, \gamma is the unit weight of the soil, and z ranges from 0 at the surface to H at the base of the wall of height H.^[2]^[15] This distribution forms a triangular pressure diagram, with zero pressure at the ground surface and a maximum value of K_p \gamma H at the base, illustrating the soil's buildup of high horizontal stress to resist wall movement into the soil mass.^[1] The total passive force P_p acting on the wall is the integral of this pressure distribution over the wall height, resulting in P_p = \frac{1}{2} K_p \gamma H^2.^[2]^[15] This force acts horizontally through the centroid of the triangular distribution, located at a distance of H/3 above the base of the wall.^[2] In graphical representations, the passive pressure profile depicts a steep, inverted triangular loading on the wall, emphasizing the soil's resisting capacity, which is crucial for calculating embedment depths in anchored or embedded retaining structures.^[15] For a typical granular soil with an internal friction angle \phi = 30^\circ, K_p \approx 3.0, yielding a passive pressure p_p = 3.0 \gamma z that demonstrates the threefold amplification of vertical stress into horizontal resistance compared to at-rest conditions.^[2]^[1]

Applications and Limitations

Design of Retaining Structures

In the design of gravity retaining walls using Rankine theory, the active earth pressure P_a is calculated to determine the total lateral force acting on the wall, which is essential for stability checks against overturning and sliding. The force P_a is applied at a height of H/3 from the base, where H is the wall height, and is used to compute the overturning moment. Stability against overturning requires a factor of safety (FS) of at least 2.0, calculated as the ratio of resisting moments (from the wall's self-weight and backfill) to the overturning moment from P_a. For sliding resistance, the horizontal component of P_a is compared to the frictional resistance at the base, typically requiring an FS of at least 1.5, often enhanced by keying or battering the base.^[18] For sheet pile walls, Rankine theory facilitates the design by combining active pressure on the retained soil side with passive pressure on the excavated side to find the net lateral force, which informs embedment depth and wall section selection. The active pressure develops behind the wall, while passive resistance mobilizes in front during excavation; a reduced passive coefficient (e.g., K_p' = K_p / 1.5) is often applied for permanent conditions to account for conservatism. The net force is the difference between these pressures integrated over their respective depths, ensuring equilibrium for cantilevered or anchored configurations.^[19] A representative example illustrates the application for a gravity wall: for a 5 m high wall retaining cohesionless soil with unit weight \gamma = 18 kN/m³ and friction angle \phi = 25^\circ, the active force P_a \approx 91.4 kN/m. This force is then used to design the base width, typically around 0.4H to 0.7H (e.g., 2-3.5 m here), ensuring the resultant lies within the middle third of the base to prevent tension and achieve the required FS values. When integrating Rankine pressures with other loads, such as minor uniform surcharges q, the additional lateral force is superimposed as K_a q H, added to P_a for total force determination in stability analyses. This superposition is valid for small surcharges but should be limited to cases without significant wall friction or sloping backfill.^[18]

Extensions and Modern Critiques

One key extension to Rankine theory addresses the assumption of a frictionless interface between the soil and retaining wall by incorporating wall friction through Coulomb's wedge theory, which assumes a planar failure surface but accounts for the friction angle δ between the wall and soil, resulting in a reduced active earth pressure coefficient K_a' compared to Rankine's K_a.^[2] This adjustment is particularly relevant for rough walls, where δ can approach two-thirds of the soil friction angle φ, leading to lower predicted active pressures.^[2] For cohesive-frictional (c-φ) soils, Rankine theory was extended by Bell in 1915 to include the effects of cohesion c, yielding a simplified active earth pressure expression of p_a = K_a γ H - 2c √K_a, where K_a is the Rankine active coefficient and the cohesion term reduces the net pressure, especially near the surface. This formulation maintains the core Rankine assumptions of a smooth vertical wall and horizontal backfill but adjusts for tensile strength in cohesive materials.^[2] Rankine theory has notable limitations, as it neglects wall friction, soil cohesion, and the influence of groundwater, assuming dry, cohesionless, homogeneous soil with a smooth vertical wall and planar failure surfaces, which can lead to inaccuracies in complex field conditions.^[2] For instance, without accounting for wall friction, it overestimates active pressures and underestimates passive pressures; experimental studies in sands indicate that theoretical solutions like Rankine can overestimate passive earth pressures by up to 20% in dense conditions due to unmodeled dilation and non-planar effects.^[20] Modern critiques, emerging from finite element analyses since the 1970s, highlight that Rankine assumes planar failure surfaces, whereas numerical simulations reveal curved, non-planar failure zones, particularly for passive pressures, leading to more accurate but non-linear pressure distributions that deviate from Rankine's uniform triangular profile.^[21] Despite this, Rankine remains highly compatible with finite element results for active pressure magnitudes in simple cases, though it underpredicts pressure centers in walls with movement.^[21] Terzaghi's 1943 refinements in Theoretical Soil Mechanics emphasized arching effects—the transfer of stress from yielding soil zones to adjacent rigid parts via shear— which Rankine does not capture, as it treats the soil mass as failing without such redistribution; Terzaghi's trap-door experiments demonstrated pressure reductions to less than 10% of overburden through arching, improving predictions for confined failures like tunnels.^[22] In contemporary practice, Rankine theory is integrated into design codes like Eurocode 7 (EN 1997-1, Annex C), where active and passive pressures are calculated using Rankine coefficients but verified at ultimate limit states via partial factors on actions (e.g., γ_G = 1.35 for permanent loads) and material properties (e.g., γ_φ = 1.25 on friction angle), ensuring safety margins without altering the core theory.^[23]

Rankine theory

Historical Background

Development by William Rankine

Context in Early Soil Mechanics

Fundamental Assumptions

Soil and Geometry Assumptions

Stress and Failure Assumptions

Theoretical Framework

Mohr-Coulomb Failure Criterion

Plastic Equilibrium Concept

Active Earth Pressure

Derivation Process

Pressure Formulas and Distribution

Passive Earth Pressure

Derivation Process

Pressure Formulas and Distribution

Applications and Limitations

Design of Retaining Structures

Extensions and Modern Critiques

References