ASTM D3080 / Eurocode 7
Intermediate
Geotechnical Engineering
Mohr’s Circle of Stress & Soil Shear Slip Failure
Transform 2D normal and shear stresses to principal axes (σ₁, σ₂) and evaluate Coulomb failure envelopes.
Visualizes 2D plane stress transformation on representative material and soil elements. Plot Mohr’s stress circle, compute principal stresses, and check against the Mohr-Coulomb shear slip rupture envelope.
Governing Physical Law & Equations
\tau_f = c + \sigma_n \tan \phi, \quad R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}
Mohr-Coulomb shear failure criterion and Mohr circle radius.
Law: Stress Tensor Coordinate Transformation & Coulomb Friction | Standard Reference: ASTM D3080 / BS 1377-7 / Eurocode 7 (EN 1997-1) (Direct Shear & Soil Triaxial Tests)
Adjustable System Parameters
| Parameter |
Nominal Value |
Dynamic Range |
Physical Role |
| Normal Stress σx (σ_x) |
140 kPa |
20 to 250 kPa |
Normal stress applied on vertical face X |
| Normal Stress σy (σ_y) |
50 kPa |
0 to 180 kPa |
Confining normal stress applied on horizontal face Y |
| Shear Stress τxy (τ_{xy}) |
35 kPa |
0 to 80 kPa |
In-plane shear stress applied along element faces |
| Plane Rotation θ (θ) |
35 ° |
0 to 180 ° |
Angle of inclined inspection failure plane |
Analytical Proof & Derivation
From 2D stress transformation: σ_θ = (σ_x + σ_y)/2 + ((σ_x - σ_y)/2) cos(2θ) + τ_xy sin(2θ) and τ_θ = -((σ_x - σ_y)/2) sin(2θ) + τ_xy cos(2θ). These form a circle (σ - σ_avg)² + τ² = R² with center σ_avg = (σ_x + σ_y)/2 and radius R = √[((σ_x - σ_y)/2)² + τ_xy²]. Failure occurs if the Mohr circle intercepts or touches the Coulomb envelope τ_f = c + σ_n tan(φ).
Verification Benchmark
ASTM D3080 triaxial benchmark: σ_x = 140 kPa, σ_y = 50 kPa, τ_xy = 35 kPa. Center σ_avg = 95.0 kPa, radius R = √[45² + 35²] = 57.01 kPa. Major principal stress σ₁ = 152.01 kPa, minor principal stress σ₂ = 37.99 kPa, max shear τ_max = 57.01 kPa. Principal angle θ_p = 18.94°.
Field Engineering Insights
Eurocode 7 mandates partial safety factors on soil parameters: γ_φ = 1.25 on tan(φ) and γ_c = 1.25 on cohesion c. Pore water pressure u reduces effective stress σ' = σ - u, shifting the Mohr circle leftward toward premature shear slip liquefaction.