Mechanics of Materials Calculator: Stress, Strain, Young's Modulus, Shear, Torsion
Free mechanics of materials calculators: stress and strain with Young's modulus, a tensile test simulator with the stress-strain curve, shear stress in bolts and pins, torsion of shafts, Poisson's ratio and 3D Hooke's law, thermal stress, and a table of 30 materials. SI and US units.
Your own material
Start from a material in the list and change what you know. Yield is not used for brittle materials or rubber.
Type what you know
Any four values give the rest. The value you typed last always stays; an older one is worked out again when the newer ones fix it. Clear a box (×) to leave it unknown.
Side by side
The curves are a model fitted to each material's typical Young's modulus, yield strength, tensile strength and elongation (Hollomon / Ramberg–Osgood hardening, a Lüders plateau for mild steel). Specimen: a standard round bar, 12.5 mm across with a 50 mm gauge length (0.500 in and 2.000 in in US units). Real curves also depend on the test speed, temperature and the exact alloy.
For an isotropic material (the same in every direction) any two elastic constants fix all the others. Wood, carbon fibre laminates and bone are not isotropic, so these formulas do not link their E, G and ν.
Click a row to use that material in every tool. Click a column heading to sort.
Stiffness against weight
Both scales are logarithmic (each step is ten times the last). Click a dot to use that material. Dashed lines join materials with the same stiffness per kilogram (E/ρ): steel, aluminium and titanium sit on the same line.
Where the numbers come from
These are typical room-temperature values for learning and first estimates. They change with the exact alloy, heat treatment, temper, bar size, moisture (wood, nylon), direction (wood, composites, bone) and supplier. Yield and tensile strengths of the A36 steel are the specification minimums. Elongation is in a 50 mm (2 in) gauge length. Reduction of area is a rough figure that only shapes the neck in the tensile test.
Sources: published supplier data sheets (Special Metals for nickel alloys, mill and distributor sheets for steels, aluminium and copper alloys), the ASTM / UNS property collections summarised by MatWeb and MakeItFrom, ASM handbooks, the USDA Forest Products Laboratory Wood Handbook (clear wood at 12% moisture), ACI and Eurocode figures for concrete, textbook tables (Hibbeler, Gere & Goodno, Callister) and biomechanics reviews for bone. For design, use the certified values for your material and the rules of your design code.
For learning and first estimates. Real parts need certified material values, the design code that applies (AISC, Eurocode, ASME…) and a check by a qualified engineer.
Mechanics of materials calculators
Seven tools for strength of materials, in one place and with one material list, so the material you pick is used in every tab. Each tool shows the formula, the numbers put into it, a live picture drawn from the same numbers and a warning when the material would yield or break. SI and US units, a shareable link for every tab, and nothing is uploaded: your numbers stay in your browser.
What is here
- Stress & strain (axial loading): type any four of F, A, L, ΔL, σ, ε and E and get the rest, with a draggable picture of a stretched bar, the safety factor and the buckling load. Also on its own page: the Young's modulus calculator.
- Tensile test: the stress-strain curve drawn as a bar is pulled to destruction, with unloading, true stress, resilience and toughness, and three materials side by side. Bigger on the tensile test simulator page.
- Shear of bolts and pins (single and double shear), glued joints and punching, and Torsion of solid and hollow shafts with the angle of twist and motor power. Together on the shear stress calculator page.
- Poisson & 3D: how much a bar gets thinner when it is pulled, the change in volume, Hooke's law for stresses in all three directions (and back from strains to stresses), and a converter between E, G, ν and the bulk modulus K.
- Thermal: how much a bar grows when it is heated, and the stress when walls stop it, with or without a gap.
- Materials table: about 30 materials (steels, stainless, aluminium, titanium, copper, brass, bronze, cast iron, nickel alloys, tungsten, lead, gold, silver, wood, concrete, glass, plastics, rubber, carbon fibre and bone) with E, G, ν, yield and tensile strength, elongation, density and thermal expansion, a search, sorting and a chart of stiffness against weight.
The key formulas
- Stress and strain: \(\sigma = F/A\), \(\varepsilon = \Delta L / L\), \(E = \sigma/\varepsilon\), so \(\Delta L = FL/(AE)\).
- Shear: \(\tau = V/A\), \(\gamma = \tau/G\), \(G = E/(2(1+\nu))\).
- Torsion: \(\tau_{max} = Tc/J\), \(\theta = TL/(GJ)\), \(P = T\omega\).
- Poisson's ratio and volume: \(\varepsilon_{lat} = -\nu\,\varepsilon\), \(\Delta V / V \approx \varepsilon\,(1 - 2\nu)\), \(K = E/(3(1-2\nu))\).
- Thermal: \(\delta_T = \alpha\,\Delta T\,L\), and between rigid walls \(\sigma = -E\,\alpha\,\Delta T\).
Worked examples
A steel rail held at both ends, warmed by 40 °C. With α = 11.7 × 10⁻⁶ per °C and E = 200 GPa the stress is 200,000 × 11.7 × 10⁻⁶ × 40 = 93.6 MPa in compression. Free, a 10 m rail would have grown 4.68 mm. That is why rails and bridges have expansion joints, or are laid with the stress planned in.
A 20 mm steel bar pulled to 100 MPa. The strain is 0.0005 and, with ν = 0.26, the bar gets 0.00013 thinner: its diameter shrinks by 2.6 µm. Its volume grows by about 0.024%, because ν is below 0.5.
Questions
What is mechanics of materials?
The part of engineering (also called strength of materials) that works out how solid parts stretch, bend, twist and break under loads: stress, strain, the material properties that link them, and how to size a part so it stays safe.
How accurate are the material values?
They are typical room-temperature values from data sheets, handbooks and textbooks, good for learning and first estimates. Real values change with the alloy, heat treatment, size, moisture and supplier; use certified values and your design code for real parts.
Can I use my own material?
Yes: pick "Your own material" at the end of the list and type its Young's modulus, Poisson's ratio, strengths, elongation, density and thermal expansion. Every tool, including the tensile test curve, uses it.