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, beam bending with shear and moment diagrams, 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.
Yield point or smooth yielding?
Mild and structural steels (A36, 1020) have a yield point. Carbon and nitrogen atoms pin the dislocations, so the stress climbs a little past the yield strength until a band of metal breaks free: that is the upper yield point. The force then drops to the lower yield point and the bar stretches at nearly constant stress while Lüders bands spread along it (the Lüders strain, about 1 to 3% for A36), with small serrations as each band starts. Only then does strain hardening begin, up to the tensile strength (58 to 80 ksi, 400 to 550 MPa for A36, at about 15 to 20% strain) and necking. The upper yield is usually only a few % to about 10% above the lower one, and its height depends on the test: a stiff machine, a fast pull and a smooth, well-aligned specimen give a taller, sharper peak; a soft machine or a slightly bent bar can hide it. The line also bends a little just before the peak (some grains yield early), so the top is rounded rather than a sharp corner. Codes therefore use the lower yield (or 0.2% offset) as the yield strength: 250 MPa (36 ksi) for A36.
Aluminium alloys (2014-T6, 6061-T6), copper, stainless steel and most other metals yield continuously: no peak, no drop, no plateau. The curve just bends away from the straight line, so the yield strength is defined by the 0.2% offset rule: draw a line with slope E starting at 0.2% strain, and where it meets the curve is the yield strength (414 MPa, 60 ksi, for 2014-T6).
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.
Loads drag them along the beam; drag a handle to change the size
The beam is solved exactly for any mix of loads (singularity functions: equilibrium plus the deflection conditions at the supports), so the fixed and propped cases, which statics alone cannot solve, come out right too. It assumes a straight, elastic beam with small deflections (Euler–Bernoulli), bending about the strong axis, no buckling sideways (long, slender I-beams without bracing can twist out of the way: lateral-torsional buckling) and no self-weight unless you add it as a spread load.
Words used here
- Bending moment M
- How much the loads try to bend the beam at a point: force × distance, summed over everything on one side. Sagging (smiling, top squeezed) is drawn upward here; hogging (frowning, top stretched, over supports) downward.
- Shear force V
- The force trying to slide one part of the beam past the next. It jumps at every point load and support, and the slope of the moment diagram equals it, so M is largest where V crosses zero.
- Neutral axis
- The line through the section that is neither stretched nor squeezed. For these symmetric sections it sits at mid-depth; the stress grows in a straight line away from it.
- Second moment of area I
- How the area is spread away from the neutral axis (m⁴, in⁴). Depth counts cubed, which is why I-beams put their metal in the flanges and why a plank is far stiffer on edge.
- c and section modulus S = I/c
- c is the distance from the neutral axis to the outermost fibre, where the stress is largest. S rolls I and c into one number: σmax = M/S.
- Flexure formula σ = Mc/I
- The largest bending stress in the section, valid while the material stays elastic.
- Q and τ = VQ/(It)
- Q is the first moment of the area above the neutral axis and t the width there. The shear stress is largest at the neutral axis; in an I-beam the thin web carries nearly all of it.
- Deflection δ and L/360
- How far the beam sags. Codes limit it as a fraction of the span (L/360 for floors under live load, L/240 or L/180 for roofs) so floors do not feel bouncy and plaster does not crack.
- Simply supported, cantilever, fixed, propped, overhang
- A pin and a roller at the ends; built into a wall at one end only; built in at both ends; built in at one end and resting on a support at the other; resting on two supports with part of the beam sticking out past one of them.
- Point load, spread load (UDL), moment
- A force at one spot (kN, kip); a force spread evenly along a length (kN/m, kip/ft), such as a floor's weight; a twisting couple applied at one spot (kN·m, kip·ft).
- EI (flexural rigidity)
- Young's modulus times I: how hard the beam is to bend. Deflection is inversely proportional to it.
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 318 and Eurocode 2 (EN 1992-1-1, table 3.1) figures for concrete, the Portland Cement Association and the Brick Industry Association for concrete, mortar and brick, rock-mechanics tables (Goodman, Introduction to Rock Mechanics) for stone, the Asphalt Institute for asphalt, ceramic makers' data sheets (CoorsTek, Kyocera) for technical ceramics, Gent's Engineering with Rubber and rubber compounders' data for elastomers (hardness-graded compounds: real compounds vary a lot), 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
Eight 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.
- Bending: beams on any supports (simply supported, cantilever, fixed at both ends, propped, overhang) with point loads, spread loads and moments you drag along the beam: reactions, shear-force and bending-moment diagrams, the deflected shape, the bending stress σ = Mc/I in a rectangle, round bar, tube, box, I-beam or channel, and the L/360 deflection check. Also on its own page: the bending stress and beam deflection calculator.
- 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.