Mohr’s Circle Calculator
Principal stresses are the centre of Mohr’s circle plus and minus its radius: for σx = 80, σy = −40 and τxy = 45 MPa the major principal stress is 95 MPa and τmax = 75 MPa.
Tension positive.
95 MPa
- Minor principal stress σ₂
- -55 MPa
- Maximum in-plane shear stress τmax = R
- 75 MPa
- Absolute maximum shear (σ₃ = 0)
- 75 MPa
- Centre of the circle C
- 20 MPa
- Principal plane angle θp
- 18.4349 °
- Maximum shear plane θs = θp − 45°
- -26.5651 °
- Normal stress on plane at 30°
- 88.9711 MPa
- Shear stress on plane at 30°
- -29.4615 MPa
Step-by-step solution
- C = (σx + σy)/2 = (80 + -40) ÷ 2 = 20 MPa
- R = √[((σx − σy)/2)² + τxy²] = √[60² + 45²] = 75 MPa
- σ₁,₂ = C ± R = 95 MPa, -55 MPa
- tan 2θp = 2τxy/(σx − σy) → θp = 18.4349°
Mohr’s Circle Formula
σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]
centre C = (σx + σy)/2, radius R = τmax
Worked Example
σx = 80 MPa, σy = −40 MPa and τxy = 45 MPa:
- C = (σx + σy)/2 = (80 + -40) ÷ 2 = 20 MPa
- R = √[((σx − σy)/2)² + τxy²] = √[60² + 45²] = 75 MPa
- σ₁,₂ = C ± R = 95 MPa, -55 MPa
- tan 2θp = 2τxy/(σx − σy) → θp = 18.4349°
Maximum in-plane shear stress τmax = R: 75 MPa · Absolute maximum shear (σ₃ = 0): 75 MPa · Centre of the circle C: 20 MPa · Principal plane angle θp: 18.4349 ° · Maximum shear plane θs = θp − 45°: -26.5651 ° · Normal stress on plane at 30°: 88.9711 MPa · Shear stress on plane at 30°: -29.4615 MPa
More Mechanics of Materials Calculators
-
Bending Stress Calculator
σ = M y / I = M / Z
M bending moment, y distance from the neutral axis, I second moment of area, Z = I/y
Example: A 100 × 200 mm rectangular beam carrying a 20 kN·m bending moment: Maximum bending stress σ = 30 MPa.
-
Shaft Torsion Calculator
T/J = τ/r = Gθ/L
J = π(D⁴ − d⁴)/32, r = D/2, G modulus of rigidity
Example: A 50 mm solid steel shaft (G = 80 GPa), 1 m long, carrying 1 kN·m: Maximum shear stress τmax = 40.7437 MPa.
-
Beam Deflection Calculator
δ = PL³/3EI, wL⁴/8EI, PL³/48EI, 5wL⁴/384EI
P point load, w load per length, L span, E Young’s modulus, I second moment of area
Example: A 2 m steel cantilever (E = 200 GPa, I = 8 × 10⁶ mm⁴) with 10 kN at the free end: Maximum deflection δ (free end) = 16.6667 mm.
-
Euler Buckling Calculator
Pcr = π²EI / Le²
Le effective length: L pinned, 2L fixed–free, L/2 fixed–fixed, L/√2 fixed–pinned
Example: A 3 m pinned steel column with I = 8 × 10⁶ mm⁴ and A = 5000 mm²: Critical buckling load Pcr = 1754.6 kN.
Frequently asked questions
What is the shear stress on a principal plane?
Zero. Principal planes carry only normal stress; maximum shear acts on planes at 45° to them.
How is absolute maximum shear different from in-plane maximum shear?
In plane stress σ₃ = 0, so the absolute maximum shear is the largest of |σ₁|/2, |σ₂|/2 and |σ₁ − σ₂|/2. It exceeds R when σ₁ and σ₂ have the same sign.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Mohr’s Circle Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/mohrs-circle-calculator/
IEEE
Scientific Converters, "Mohr’s Circle Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/mohrs-circle-calculator/