Mohr Master
Professional Mohr’s circle stress transformation across the full mechanics-of-materials curriculum — plane stress, plane strain, 3D tensors, strain rosettes, Hooke’s law and failure theories — with step-by-step solutions in every mode.
Six modes, one circle
Every mode shares the same geometry engine and produces Formula → Substitution → Result steps with a live interactive diagram.
Stress 2D
Plane stress transformation
σ₁, σ₂, τ_max and θp from any σx, σy, τxy — plus the stress state at any rotated angle and the absolute max shear via the 3-circle construction with σ₃ = 0.
Strain 2D
Plane strain transformation
ε₁, ε₂ and γ_max = 2R with the circle plotted as γ/2 — following the Philpot Ch. 13 convention.
Failure & safety factor
4 failure theories
Rankine (max normal), Tresca, von Mises and Mohr–Coulomb with material presets and interactive failure envelopes.
Hooke's law
Stress ↔ strain, both directions
Modulus conversion with G = E/[2(1+ν)], Poisson effects and the out-of-plane strain εz for biaxial states.
Strain rosettes
Gauge analysis
45° rectangular (0/45/90), 60° delta (0/60/120) and tee arrangements — plus custom gauge sets solved by matrix inversion.
Stress 3D
Full tensor analysis
All six stress components, the cubic characteristic equation, three Mohr’s circles, octahedral stresses and von Mises / Tresca evaluation.
The transformation set
σ_avg = (σx + σy) / 2
R = √[((σx − σy)/2)² + τxy²]
σ₁,₂ = σ_avg ± R
tan(2θp) = 2τxy / (σx − σy)
τ_max = R
I₁ = σx + σy = σ₁ + σ₂ · I₂ = σxσy − τxy² = σ₁σ₂
Built to teach, engineered to verify
Methodology follows Beer & Johnston, Hibbeler and Philpot — every solution closes with an invariant check.
7-step solutions
Given → Center → Radius → Principal → Angles → Max Shear → Verification, with an invariant check closing every cycle.
Formula → Substitution → Result
Every step shows the symbolic expression, the substituted numbers, and the reduced result.
14 curated examples
5 basic, 4 intermediate and 5 advanced problems spanning the mechanics-of-materials curriculum.
Practice mode
Three difficulty levels with generated problems and guided feedback.
8-question quiz
Timed concept and computation checks with scored results.
Tutorials
Methodology walkthroughs following Beer & Johnston, Hibbeler and Philpot.
20-item history
Every solved state is retained for review and re-export.
Export everywhere
PDF, PNG, JPEG, CSV and JSON at 150 / 300 / 600 DPI.
Verified presets, one tap away
Hooke’s law presets
| Material | E | ν |
|---|---|---|
| Steel | 200 GPa | 0.30 |
| Aluminum | 70 GPa | 0.33 |
| Copper | 120 GPa | 0.34 |
| Titanium | 116 GPa | 0.32 |
| Concrete | 30 GPa | 0.20 |
| Brass | 100 GPa | 0.34 |
Failure-theory presets (MPa)
| Material | σy | σut | σuc |
|---|---|---|---|
| Structural Steel | 250 | 400 | 400 |
| Aluminum 6061-T6 | 276 | 310 | 310 |
| Gray Cast Iron | 150 | 150 | 600 |
| Concrete C30 | 3 | 3 | 30 |
Reference outputs, regression-locked
These five states reproduce hand-calculated and textbook reference solutions exactly.
Standard biaxial tension
σx = 80 MPa · σy = 40 MPa · τxy = 20 MPa
Pure shear
σx = σy = 0 · τxy = 50 MPa
Uniaxial tension
σx = 100 MPa · σy = 0 · τxy = 0
adv-04 — shaft with torque & bending
σx = 120 MPa · σy = 0 · τxy = 85 MPa
adv-03 — steel beam (ksi)
σx = 30 ksi · σy = 0 · τxy = 21 ksi
Offline-first, responsive, private
Mohr Master is a fully client-side application: no account, no ads, no data collection — your inputs never leave the device.
Fully offline
Runs entirely in the browser
Desktop
3-column workspace
Tablet
2-column layout
Mobile
Tabbed layouts
Draw your first Mohr’s circle in seconds
6 modes, 14 curated examples, practice mode and exportable reports — all offline, all free.