Stress, Strain & Young’s Modulus Calculator
Work out tensile stress, strain, Young’s modulus and how far a bar stretches. Add the load for a given stretch, Poisson’s contraction and a safety factor against yield, for steel, aluminium, copper, titanium, nylon or your own material.
How to Use
- Choose what to work out: how far the bar stretches (Elongation), the Stress from a load, the Strain from a measured stretch, Young’s Modulus from a test, or the Load that gives a stretch.
- Pick a material to fill in its typical Young’s modulus, yield strength and Poisson’s ratio, or type your own values (the material then switches to Custom).
- Enter the load, the cross-section (a round bar’s diameter, a rectangle’s width and thickness, or the area) and the original length, each with its unit.
- The answer appears in the highlighted field and the first readout, with the stress, the strain and the safety factor against yield beside it.
- Read the stress–strain line under the drawing: the dot is your working point, and it turns red if the stress goes past yield.
- Press a preset to load a steel tie rod, an aluminium flat bar, a tensile test, a copper wire or a titanium rod.
Worked Example
A steel tie rod. A 12 mm A36 rod, 2 m long, carries 20 kN. Its area is π × 12² ÷ 4 = 113.1 mm², so the stress is 20,000 ÷ 113.1 = 176.8 MPa. With E = 200 GPa the strain is 176.8 ÷ 200,000 = 0.000884 (884.2 µε), and the rod stretches 0.000884 × 2,000 mm = 1.768 mm. Against the 250 MPa yield strength that is a safety factor of 250 ÷ 176.8 = 1.41.
An aluminium flat bar, in inches. A 1 × ¼ in 6061-T6 bar has 0.25 in², so 5,000 lbf puts it at 20,000 psi (20 ksi). With E = 10,000 ksi the strain is 0.002, and over 5 ft (60 in) it stretches 0.120 in, with a safety factor of 2 against the 40 ksi yield.
The common mistake: putting the diameter into πr². Using 12 mm as if it were the radius gives an area of π × 12² = 452.4 mm², four times too big. The stress comes out as 44.21 MPa instead of 176.8 MPa and the stretch as 0.442 mm instead of 1.768 mm, which would make a rod that is close to its limit look four times safer than it is. Use A = πd² ÷ 4 with the diameter, or πr² with half of it.
Show Work
Formulas
From Hooke’s Anagram to Young’s Modulus
Robert Hooke published his law of springs in 1676 as an anagram, ceiiinosssttuv, so he could claim it without giving it away. In 1678 he revealed the answer, ut tensio, sic vis: “as the extension, so the force”. Hooke wrote about springs, wires and whole objects; he had no notion yet of stress or strain inside a material.
Thomas Young described the stiffness of a material in his Course of Lectures on Natural Philosophy and the Mechanical Arts in 1807, and the modulus is named after him, although Leonhard Euler had used the same idea decades earlier. Augustin-Louis Cauchy gave the modern definitions of stress and strain in the 1820s, and Siméon Denis Poisson worked out at the end of that decade that a stretched bar must also get thinner, the effect now measured by Poisson’s ratio.
Today the modulus and the yield strength come from tensile tests on standard specimens, for example to ASTM E8 or ISO 6892-1, in which a bar is pulled while its load and extension are recorded. The straight part of that record gives E; where it bends over is the yield point.
About This Tool
This calculator solves Hooke’s law in tension for the elongation, the stress, the strain, Young’s modulus or the load, with the cross-section given as a round bar, a rectangle or an area. It adds the safety factor against yield, the load at which the bar would yield, the narrowing from Poisson’s ratio, the change in volume and the strain energy stored. The drawing shows the bar stretched (exaggerated) and an idealised stress–strain line with your working point on it.
The material values are typical, not guaranteed: E and Poisson’s ratio are handbook figures (ASM International, the Copper Development Association, DuPont for nylon), the A36 yield of 250 MPa and tensile strength of 400 MPa are the ASTM A36 specified minimums, and nylon’s stiffness falls a long way as it absorbs moisture. The line is elastic–perfectly plastic: real metals keep strengthening after yield. Use a supplier’s certified figures for design. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Column Buckling Calculator, Beam Calculator, and Moment of Inertia & Centroid Calculator.
Frequently Asked Questions
How do you calculate stress?
Divide the load by the cross-sectional area: σ = F ÷ A. A 12 mm round rod has A = π × 12² ÷ 4 = 113.1 mm², so a 20 kN pull gives 20,000 ÷ 113.1 = 176.8 N/mm², which is 176.8 MPa or 25,648 psi.
How do you calculate strain?
Strain is the stretch divided by the original length, ε = ΔL ÷ L₀, so it has no units. A bar that stretches 0.5 mm over 1 m has ε = 0.0005, often written 500 µε or 0.05%. For steel with E = 200 GPa that strain means a stress of 200,000 × 0.0005 = 100 MPa.
How do I find Young’s modulus from a tensile test?
Take a point on the straight part of the curve and divide stress by strain, E = σ ÷ ε. A 10 mm bar (78.54 mm²) carrying 10 kN is at 127.3 MPa; if it stretches 0.0318 mm over a 50 mm gauge length the strain is 0.000636, so E = 127.3 ÷ 0.000636 = 200.2 GPa, which is steel.
How much will a steel rod stretch under load?
Use ΔL = F L₀ ÷ (A E). A 12 mm A36 rod 2 m long carrying 20 kN stretches 20,000 × 2,000 ÷ (113.1 × 200,000) = 1.768 mm. The stress is 176.8 MPa against a 250 MPa yield, a safety factor of 1.41, so it springs back when the load comes off.
What is Poisson’s ratio?
It is how much a bar narrows for each unit it stretches: lateral strain = −ν × axial strain. Steel has ν of about 0.30, so the 12 mm rod stretched to 884.2 µε narrows by 0.30 × 884.2 = 265.3 µε, which is just 0.00318 mm off its diameter. Rubber is close to 0.5, the value at which volume does not change.
How do I use the Stress, Strain & Young’s Modulus Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Tie rods and hangers
A 12 mm A36 steel rod 2 m long holding 20 kN works at 176.8 MPa and stretches 1.768 mm; it would start to yield at 28.27 kN.
Tensile testing
From a 10 kN load and a 0.0318 mm extension over a 50 mm gauge on a 10 mm bar, the modulus comes out at 200.2 GPa, the value for steel.
Aluminium structures
A 1 × ¼ in 6061-T6 flat bar 5 ft long carrying 5,000 lbf is at 20 ksi and stretches 0.120 in, half its 40 ksi yield.
Wires and cables
A 2 mm annealed copper wire carrying 200 N is at 63.66 MPa, with a safety factor of only 1.08 against its typical 69 MPa yield.
Preloading and fitting
Stretching an 8 mm Ti-6Al-4V rod 0.5 mm over 300 mm takes 9.55 kN and 190 MPa, a safety factor of 4.63 against its 880 MPa yield.
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