Shear Strength Mohr Coulomb Interactive Calculator

The Shear Strength Mohr-Coulomb Interactive Calculator determines the shear strength of soils using the Mohr-Coulomb failure criterion, one of the most fundamental relationships in geotechnical engineering. This calculator enables engineers to compute shear strength, normal stress, effective stress parameters, and factor of safety for slope stability, foundation design, and retaining wall analysis. Understanding soil shear behavior is critical for preventing failures in earthworks, embankments, and underground excavations.

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Mohr-Coulomb Diagram

Interactive Shear Strength Calculator

Governing Equations

Theory & Engineering Applications

The Mohr-Coulomb failure criterion represents one of the most enduring and widely applied theories in geotechnical engineering, formulated in the 1770s by Charles-Augustin de Coulomb and later extended by Christian Otto Mohr. This criterion describes the linear relationship between shear strength and normal stress for soils and rocks, providing the foundation for slope stability analysis, bearing capacity calculations, retaining wall design, and underground excavation support.

Fundamental Principles of the Mohr-Coulomb Model

The Mohr-Coulomb criterion posits that shear failure occurs along a plane when the shear stress reaches a critical value determined by two independent material properties: cohesion and friction angle. Cohesion represents the inherent shear strength at zero normal stress, attributed to cementation, electrostatic forces between clay particles, or apparent cohesion from suction in unsaturated soils. The friction angle quantifies the increase in shear strength with increasing normal stress, reflecting particle interlocking, surface roughness, and mineral hardness.

The linear failure envelope intercepts the vertical axis at the cohesion value and slopes upward at an angle equal to the friction angle. This graphical representation on the Mohr diagram—where circles represent stress states and the failure envelope defines the limiting condition—provides intuitive visualization of soil behavior under various loading conditions. Stress states plotting below the envelope represent stable conditions, while those touching the envelope indicate incipient failure.

Effective Stress versus Total Stress Analysis

A critical distinction in geotechnical practice involves effective stress versus total stress analysis. Terzaghi's principle of effective stress states that soil behavior is controlled by the effective stress (σ′ = σ - u), where the total stress is reduced by pore water pressure. For drained conditions—where excess pore pressures dissipate during loading—effective stress parameters (c′ and φ′) govern behavior. In undrained conditions typical of saturated clays loaded rapidly, total stress parameters (c_u and φ_u) apply, with φ_u often assumed to be zero for fully saturated clays.

The distinction becomes particularly important in slope stability analysis following rapid drawdown of reservoirs, earthquake loading, or construction of embankments on soft clay foundations. Engineers must select appropriate parameters based on drainage conditions, loading rate, and soil permeability. Misapplication of total stress parameters in long-term stability analysis or effective stress parameters for rapid loading can lead to unconservative designs and catastrophic failures.

Limitations and Non-Linear Behavior

While the Mohr-Coulomb criterion provides excellent predictions for many practical problems, real soil behavior exhibits important departures from the linear model. At low confining stresses typical of shallow foundations, the failure envelope for many soils curves upward, providing higher shear strength than predicted by the linear model. Conversely, at high confining pressures encountered in deep excavations or beneath tall embankments, particle crushing and pore collapse cause the envelope to curve downward.

Heavily overconsolidated clays demonstrate peak strength significantly exceeding residual strength due to bonding and particle structure developed during previous loading history. Post-peak softening and progressive failure can occur as local stress concentrations cause sequential failure of soil elements, mobilizing lower residual strengths along the failure surface. This behavior necessitates careful consideration in slope stability analysis, where brittle failures may occur with less warning than predicted by peak strength parameters.

Worked Example: Embankment Slope Stability

Consider a highway embankment constructed on a foundation of normally consolidated clay. The embankment is 8.5 meters high with side slopes of 3H:1V (horizontal to vertical). At a critical failure plane located 2.7 meters below the toe of the slope, soil testing indicates effective cohesion c′ = 12.3 kPa and effective friction angle φ′ = 24.7 degrees. The total normal stress at this depth is calculated from overburden as σ = 167.4 kPa. Piezometer measurements indicate pore water pressure u = 38.6 kPa at this location.

Step 1: Calculate Effective Normal Stress

Using Terzaghi's effective stress principle:

σ′ = σ - u = 167.4 - 38.6 = 128.8 kPa

Step 2: Calculate Available Shear Strength

Applying the Mohr-Coulomb criterion with effective stress parameters:

φ′ = 24.7° = 0.4311 radians

tan(φ′) = tan(24.7°) = 0.4592

τ_f = c′ + σ′ tan(φ′) = 12.3 + (128.8)(0.4592) = 12.3 + 59.15 = 71.45 kPa

Step 3: Determine Mobilized Shear Stress

From slope stability analysis using the method of slices, the mobilized shear stress along this critical surface is calculated as τ_mob = 48.7 kPa.

Step 4: Calculate Factor of Safety

FS = τ_f / τ_mob = 71.45 / 48.7 = 1.467

Interpretation: The factor of safety of 1.47 exceeds the minimum acceptable value of 1.3 for temporary highway embankments but falls below the 1.5 typically required for permanent structures. If groundwater levels rise during heavy rainfall, reducing effective stress, the factor of safety could drop below acceptable limits. The design engineer should consider installing horizontal drainage layers or flattening the slope to 4H:1V to achieve FS = 1.5 or greater for long-term stability.

Applications in Foundation Engineering

The Mohr-Coulomb criterion forms the basis for bearing capacity equations developed by Terzaghi, Meyerhof, and others. For shallow foundations, the ultimate bearing capacity depends on cohesion and friction angle through dimensionless bearing capacity factors that are functions of φ. The general bearing capacity equation incorporates both cohesive and frictional resistance, with the relative contribution of each term depending on soil type and foundation depth.

In pile foundation design, the Mohr-Coulomb model governs shaft friction along the pile-soil interface. Effective stress analysis predicts long-term shaft capacity in driven piles, while total stress analysis applies to immediate loading of piles in clay. The β-method and λ-method for estimating pile shaft friction both derive from Mohr-Coulomb principles, relating shaft friction to vertical effective stress through empirically calibrated coefficients related to the friction angle.

Retaining Wall and Earth Pressure Applications

Lateral earth pressure coefficients for retaining wall design derive directly from Mohr-Coulomb theory through consideration of limiting equilibrium states. Rankine's theory provides active and passive earth pressure coefficients as functions solely of the friction angle, applicable to smooth, vertical walls with horizontal backfill. Coulomb's wedge theory extends this to include wall friction, backfill slope, and wall inclination, requiring iterative solution of the Mohr-Coulomb criterion along a trial failure surface.

The transition from at-rest pressure (K₀ condition) to active pressure requires sufficient wall movement to mobilize the full friction angle along the failure wedge. Typical movements range from 0.001H to 0.004H for active conditions in dense sand, where H represents wall height. Passive pressure mobilization requires substantially greater movement—0.01H to 0.05H—making passive resistance unsuitable for serviceability-controlled designs where deformations must be minimized.

For comprehensive engineering resources including additional geotechnical calculators for bearing capacity, consolidation settlement, and earth pressure coefficients, visit the complete engineering calculator library.

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