Stress, Strain & Young's Modulus Calculator
Stress, strain, elongation, shaft torsion and elastic constants, with the working shown.
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Results are estimates from the formulas shown, not a professional design or certification. Have a qualified engineer verify anything safety-critical (structures, electrical installations, gas or pressure systems).
About the Stress, Strain & Young's Modulus Calculator
Work out the stresses and deformations of a bar or shaft in seven calculations: axial stress, strain and elongation (σ = P/A, ε = σ/E, δ = PL/AE) with the lateral contraction from Poisson's ratio; Young's modulus from a tensile test; direct shear of pins, bolts and rivets in single or double shear; torsion of solid and hollow shafts (τ = Tr/J and the angle of twist TL/GJ), with the torque typed in or worked out from power and speed; the shaft diameter for an allowable shear stress and twist; the elastic constants E, G, K and ν from any two of them; and the factor of safety against yield or fracture.
Units are yours to choose — newtons or pounds-force, millimetres or inches, MPa, GPa, psi or ksi — and each result shows how it was worked out. Material presets fill in the minimum strengths of common structural steels, stainless steel 304 and aluminium 6061-T6, and every value stays editable. Results are linear-elastic estimates for checking and learning, not a professional design.
Without a pass, the free preview works on your own numbers and shows the label of every figure your calculation gives and the checks that apply, with the stresses, moduli, sizes, factors of safety and the working hidden; the full result, Copy result and the CSV come with a Pro pass.
How to use it
- Choose what to work out, and the units (SI or US customary, then forces, lengths, stresses and moduli one by one if you need to). Changing a unit converts the numbers already typed, so they keep their meaning.
- Enter the load and the dimensions. For an axial bar pick the cross-section (round bar, tube, rectangle or an area you know); positive forces pull, negative forces push. For a shaft, give the torque, or the power and speed.
- Pick a material preset or type E, ν, G and the strengths yourself. Presets use the minimum strengths of the named standard; typing over a value switches the preset to your own values.
- Read the headline result, the other figures and the working, and check the notes: they warn when a stress passes the yield strength, when a strain is beyond the elastic range or when a compression member needs a buckling check.
- Copy the result or download it as a CSV for your notes — with a Pro pass, or after unlocking this result.
Examples
50 kN on a round bar of 20 mm diameter, 2 m long, E = 200 GPa, ν = 0.3, Sy = 250 MPa
A = 314.2 mm², σ = 159.2 MPa, ε = 0.000796 (796 µε), δ = 1.592 mm, the diameter shrinks by 0.0048 mm, factor of safety against yield 1.57
10 kN on a 10 mm bar, 50 mm gauge length stretching 0.032 mm, diameter −0.0019 mm
σ = 127.3 MPa, ε = 0.00064, E = 198.9 GPa, Poisson's ratio ν = 0.297, G = 76.7 GPa
1 kN·m on a 50 mm shaft, 1 m between the torques, G = 80 GPa
J = 613,600 mm⁴, τmax = 16T/πD³ = 40.74 MPa, angle of twist 0.02037 rad = 1.167°
10 kW at 1,450 rpm, allowable shear stress 40 MPa, solid shaft
T = P/ω = 65.86 N·m, D = ∛(16T/πτ) = 20.32 mm — choose the next standard size up
E = 200 GPa and ν = 0.3
G = E/2(1 + ν) = 76.92 GPa, K = E/3(1 − 2ν) = 166.7 GPa, λ = 115.4 GPa
Common uses
- Checking how much a tie rod, bolt or hanger stretches under load, and whether it stays elastic
- Working out E (and ν) from the force and extension readings of a lab tensile test
- Sizing a pin in double shear, or a motor shaft from its power and speed
- Converting between E, G, K and ν for a finite-element or vibration model
- Coursework in mechanics of materials, with every step written out
Formulas used
- Axial: σ = P/A, ε = σ/E, δ = PL/(AE), lateral strain ε_lat = −νε, axial stiffness k = AE/L
- Tensile test: σ = P/A₀, ε = ΔL/L₀, E = σ/ε, ν = −ε_lat/ε
- Direct shear: τ = V/(nA) for n shear planes, γ = τ/G
- Torsion of a circular shaft: J = π(D⁴ − d⁴)/32, τ = Tc/J with c = D/2, φ = TL/(GJ), γ = cφ/L; torque from power T = P/ω with ω = 2πn/60
- Shaft size: D = ∛(16T/(πτ(1 − k⁴))) for the allowable stress and D = ⁴√(32T/(πGθ′(1 − k⁴))) for an allowable twist θ′ per length, with k = d/D
- Elastic constants (isotropic): G = E/2(1 + ν), K = E/3(1 − 2ν), E = 9KG/(3K + G), ν = (3K − 2G)/2(3K + G), λ = K − 2G/3
- Factor of safety: n = S/σ; in shear the yield strength is Sy/√3 (distortion energy, von Mises) or Sy/2 (maximum shear stress, Tresca), and the ultimate shear strength is taken as 0.67 Su
Material presets
- S275 and S355 (EN 10025-2, up to 16 mm thick): minimum yield 275 and 355 MPa, tensile 410 and 470 MPa; E = 210 GPa and ν = 0.3, the values EN 1993-1-1 uses for steel (steel material properties)
- ASTM A36: minimum yield 36 ksi (250 MPa), tensile 58 ksi (400 MPa); E = 29,000 ksi (200 GPa)
- ASTM A992: minimum yield 50 ksi (345 MPa), tensile 65 ksi (450 MPa)
- Stainless steel 304 (ASTM A240, annealed): minimum 0.2 % proof strength 205 MPa, tensile 515 MPa; E = 193 GPa is typical
- Aluminium 6061-T6: minimum yield 240 MPa and tensile 290 MPa for sheet and plate; E = 69 GPa in every temper
Minimum strengths depend on the product form and thickness, and a mill certificate gives the actual values: type those in when you have them.
Signs and units
Tension and elongation are positive, compression and shortening negative. Shear stresses and torques are worked with their size; the direction does not change the stress. All arithmetic is done in SI base units, so you can mix units freely — kN with mm, or lbf with in — and the results use the units chosen above.
Sources
- Beer, Johnston, DeWolf and Mazurek, Mechanics of Materials, chapters 2 (axial loading), 3 (torsion) and 7 (stress transformation)
- Budynas and Nisbett, Shigley's Mechanical Engineering Design, chapter 3 (load and stress analysis) and chapter 5 (failure theories)
- ISO 6892-1 and ASTM E111: tensile testing and the measurement of Young's modulus
- EN 10025-2, ASTM A36, ASTM A992 and ASTM A240: minimum strengths of the preset materials
Limitations
- Linear-elastic, isotropic and small deformations: beyond the yield strength the stress–strain relation is no longer linear, and the results stop applying.
- No buckling, fatigue, creep or stress-concentration checks. A compression member, a shaft with a keyway or a part under repeated load needs those as well.
- Torsion formulas are for solid and hollow circular sections only; rectangular and open sections (channels, angles) twist differently.
- The shaft size covers torque alone. Most shafts also carry bending, which a combined-stress check (for example the von Mises stress in the Mohr’s circle calculator) has to include.
- An estimate for learning and checking — not a professional design. Have a qualified engineer verify anything that carries people or property.
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Frequently asked questions
What do I get without a pass?
Without a pass, Stress, Strain & Young's Modulus Calculator shows the labels of every figure your calculation gives, with the stress, modulus, shaft size or safety factor, the working and every other figure hidden. Until you unlock it, the result can’t be downloaded or copied. A Pro, Premium or Ultimate pass, a one-time payment that never renews, unlocks the full result. The pricing page lists the passes and their prices.
What is the difference between stress and strain?
Stress is the internal force per unit area, σ = P/A, in pascals (N/m²), MPa or psi. Strain is the resulting stretch per unit length, ε = ΔL/L, and has no unit (often written in microstrain, µε = 10⁻⁶). In the elastic range they are proportional: σ = Eε, where E is Young’s modulus.
How do I calculate Young's modulus from a tensile test?
Divide the stress by the strain at a reading in the straight, elastic part of the curve: E = (P/A₀) ÷ (ΔL/L₀). For a better value, use the change in force and extension between two readings in that range, as ISO 6892-1 and ASTM E111 describe. Steel gives about 200 GPa, aluminium about 69 GPa.
How are E, G and Poisson’s ratio related?
For an isotropic material G = E/(2(1 + ν)) and K = E/(3(1 − 2ν)). Steel with E = 200 GPa and ν = 0.3 has G = 76.9 GPa. Any two of E, G, K and ν fix the other two; the calculator rejects pairs that would need ν outside −1 to 0.5.
How do I find the angle of twist of a shaft?
φ = TL/(GJ) in radians, with J = πD⁴/32 for a solid shaft or π(D⁴ − d⁴)/32 for a hollow one. A 50 mm steel shaft 1 m long under 1 kN·m twists 0.0204 rad, or 1.17°. The maximum shear stress at the surface is τ = T(D/2)/J.
Why is the shear yield strength Sy/√3?
The distortion-energy (von Mises) criterion predicts yielding in pure shear at τ = Sy/√3 ≈ 0.577 Sy; the more conservative maximum-shear-stress (Tresca) criterion gives Sy/2. Choose the criterion in the factor-of-safety calculation.
What factor of safety should I use?
It depends on how well the loads and the material are known and on what failure would cost; building codes, pressure-vessel and lifting standards set the factors their designs must meet. This calculator reports the factor your numbers give — it does not choose one for you.