Mohr Circle: The Complete Civil Engineering Encyclopedia
Derivation · 3D Representation · Failure Envelopes · Strain Circle · Pole Method · Interactive Mastery
📐 Mathematical Derivation of Mohr Circle (Full Proof)
Consider a 2D stress element with stresses σₓ, σᵧ, τₓᵧ. On a plane inclined at angle θ (counterclockwise from X-axis), the transformed stresses are:
σ_θ = (σₓ+σᵧ)/2 + (σₓ-σᵧ)/2 · cos2θ + τₓᵧ·sin2θ
τ_θ = – (σₓ-σᵧ)/2 · sin2θ + τₓᵧ·cos2θ
Let σ_avg = (σₓ+σᵧ)/2, and let R = √[((σₓ-σᵧ)/2)² + τₓᵧ²]. Then the equations can be rewritten as:
(σ_θ – σ_avg) = R·cos(2θ – 2β) and τ_θ = R·sin(2θ – 2β) where tan(2β) = 2τₓᵧ/(σₓ-σᵧ).
Squaring and adding: (σ_θ – σ_avg)² + τ_θ² = R² → equation of a circle with center (σ_avg,0) and radius R. This is the Mohr Circle.
⚙️ Advanced Interactive: Mohr Circle + Mohr-Coulomb Failure Envelope
Adjust stresses and also the soil/rock parameters (cohesion c and friction angle φ). The envelope line is drawn; if the circle touches or crosses it, failure is indicated. Real-time safety assessment.
The failure envelope is τ = c + σ·tanφ. If the Mohr circle lies entirely below the line → safe. Touching → critical failure.
🧊 3D Mohr Circles & The Pole Method (Advanced)
Three Mohr Circles for 3D Stress
Given principal stresses σ₁ ≥ σ₂ ≥ σ₃, the three circles represent the possible shear-normal stress combinations. The largest circle (σ₁-σ₃) gives the absolute maximum shear stress τ_abs_max = (σ₁-σ₃)/2. This is fundamental for triaxial testing and failure criteria in rock mechanics.
Pole (Origin of Planes) Method
The pole is a unique point on the Mohr circle. To find stresses on any plane with orientation α (measured from X-axis): draw a line from the pole at angle α; its intersection with the circle gives the stress state. No double-angle transformation – faster and less error-prone.
📏 Mohr Circle for Strain (Analogous Method)
Exactly the same geometry, but with normal strain ε on horizontal axis and γ/2 (half the engineering shear strain) on vertical axis. Principal strains ε₁, ε₂ and maximum shear strain γ_max = ε₁ – ε₂. Essential for strain gauge rosette analysis in structural monitoring and experimental stress analysis.
ε_θ = (εₓ+εᵧ)/2 + (εₓ-εᵧ)/2·cos2θ + (γₓᵧ/2)·sin2θ
γ_θ/2 = – (εₓ-εᵧ)/2·sin2θ + (γₓᵧ/2)·cos2θ → yields same circle radius R = √[((εₓ-εᵧ)/2)² + (γₓᵧ/2)²].
📌 Detailed Construction & Sign Conventions (Civil vs. Mechanical)
- Standard mechanics (tension positive): τₓᵧ positive when acting on positive face in positive direction. Plot point A(σₓ, τₓᵧ) and B(σᵧ, -τₓᵧ).
- Geotechnical (compression positive): Often plot σ (compression) positive to right, shear positive as defined in soil mechanics. The circle geometry remains consistent but orientation flips.
- Effective stress application: For saturated soils, use σ’ = σ – u (pore pressure). Mohr circle in terms of effective stresses directly used in shear strength analysis.
Step-by-step (with numbers): (1) Compute center and radius, (2) Draw axes to scale, (3) Plot points A and B, (4) Draw circle, (5) Read principal stresses at intersections with σ-axis, (6) Maximum shear = radius, (7) Angle 2θₚ from center to point A.
✅ Advantages (Deep List)
- Visualizes invariant quantities – σ_avg, τ_max, principal directions.
- No trigonometry needed – after construction, read values directly.
- Handles combined loading easily – just compute resultant σₓ,σᵧ,τₓᵧ.
- Direct link to failure theories – Mohr-Coulomb, Tresca, von Mises (via principal stresses).
- Useful for both stress and strain – same graphical technique.
⚠️ Limitations (Expanded)
- Only 2D full representation – 3D requires three circles (still informative but not a single circle).
- Assumes linear elasticity / homogeneity – not for complex non-linear materials directly.
- Shear sign convention must be consistent – common source of mistakes.
- Pole method requires careful identification – but once mastered, it’s powerful.
🛡️ Safety Assessment Using Mohr Circle: Practical Cases
In geotechnical engineering, the Mohr-Coulomb failure envelope is drawn from triaxial test results (c, φ). For any in-situ stress state (e.g., beneath a foundation), the Mohr circle is constructed using total or effective stresses. Safety factor = (distance from circle top to envelope) / (radius). In structural engineering, the Tresca criterion (max shear stress) uses τ_max from Mohr circle; if τ_max < τ_yield/2 → safe. The interactive tool above demonstrates this in real time.
🏗️ Advanced Applications in Civil Engineering Sub-disciplines
🏞️ Geotechnical (Earth dams, slopes)
Stress paths, critical state soil mechanics, liquefaction analysis – all rely on Mohr circles of effective stress.
🏗️ Structural (RC beams, prestressed)
Principal tensile stress direction for crack control; Mohr circle determines angle of diagonal tension cracks.
⛏️ Rock mechanics (underground excavations)
Kirsch equations for stress around tunnels → plotted as Mohr circles to assess borehole breakouts.
🛣️ Pavement engineering
Critical shear stress from wheel loading at different depths; Mohr circle aids in fatigue design.
📝 Worked Example – Full Hand Calculation
Solution: σ_avg = (120+50)/2 = 85 kPa. R = √[((120-50)/2)² + 40²] = √[35² + 1600] = √(1225+1600)=√2825=53.15 kPa.
σ₁ = 85+53.15 = 138.15 kPa, σ₂ = 85-53.15 = 31.85 kPa. τ_max = 53.15 kPa. 2θₚ = atan2(2*40, 120-50)=atan2(80,70)=48.81° → θₚ=24.4°.
Safety: Draw envelope τ = 15 + σ·tan20°. At σ=85 kPa, τ_envelope = 15+85·0.364 = 45.94 kPa. Circle top τ_max =53.15 > 45.94 → unsafe (circle crosses envelope). Need improvement.
❓ Extended FAQ (50+ critical insights condensed)
1. What is the physical meaning of the Mohr circle radius?
Radius = maximum in-plane shear stress. It also equals the distance from the center to any point on the circle, representing the maximum deviatoric stress (σ₁-σ₂)/2.
2. How does the Mohr circle change under hydrostatic stress?
Hydrostatic stress (σₓ=σᵧ, τₓᵧ=0) collapses the circle to a point on the σ-axis (radius 0) → no shear stress on any plane.
3. What is the “pole” and how to locate it?
The pole is found by drawing a line from a known stress point (e.g., point A) parallel to the plane on which those stresses act. Intersection with circle is pole.
4. Can Mohr circle be used for anisotropic materials?
Directly only for isotropic. For anisotropic, you need a “modified Mohr circle” or use of transformation tensors; however, failure criteria like the Hoek-Brown still use principal stresses from circles.
5. How do I incorporate pore water pressure in Mohr circle?
Compute effective stresses σ’ₓ = σₓ – u, σ’ᵧ = σᵧ – u, τ’ₓᵧ = τₓᵧ (shear unaffected). Draw circle with effective stresses; failure envelope uses effective stress parameters c’, φ’.
6. What is the difference between Mohr circle and stress path?
A Mohr circle represents the state at a point; a stress path is a series of Mohr circles (usually tracking the center and radius) during loading – often plotted as p’-q diagrams.
7. How to get the absolute maximum shear in 3D?
Absolute max shear = (σ₁-σ₃)/2 from the largest of the three Mohr circles.
8. Why is the angle 2θ on the circle?
Because the transformation equations involve 2θ; a physical plane rotation θ corresponds to a 2θ rotation on the Mohr circle (double angle).
9. Can I use Mohr circle for dynamic loading (cyclic)?
Yes, for each time step you compute instantaneous stresses and plot circle; often used in liquefaction assessment (cyclic stress ratio).
10. What software uses Mohr circle internally?
Plaxis, Abaqus, RS2, and many FEA packages show Mohr circles in post-processing for stress invariants.
11. How to determine if a point is in tension or compression?
If the circle crosses the vertical axis (σ=0) into negative σ region, tensile stresses exist. Principal stresses sign indicates tension/compression.
12. What is the “stress path” in terms of Mohr circle?
The trajectory of the center (σ_avg) and radius (q) as loading proceeds – common in critical state soil mechanics.
13. How accurate is graphical Mohr circle compared to analytical?
If drawn precisely, accuracy equals calculations. Modern tools use analytic formulas but the graphical concept remains key for intuition.
14. Does Mohr circle work for strain hardening materials?
Yes, it describes the stress state at any instant, independent of material nonlinearity; however, the failure envelope might evolve.
15. What is the relation between Mohr circle and von Mises criterion?
von Mises uses the deviatoric stress √(J₂) which relates to the radius of the Mohr circle in 3D (octahedral shear).