Stress, strain and the line that connects them
Load a bar along its axis and two things happen at once. Internally, the material carries a stress equal to the force divided by the area it acts on. Externally, the bar stretches by an amount that depends on how long it was to begin with. Dividing that stretch by the original length gives strain, a pure ratio with no units, usually quoted as a percentage or in microstrain.
Below a certain stress, these two quantities are proportional. Double the load and you double the stretch; remove the load and the bar returns exactly to its original length. The constant of proportionality is Young's modulus, and it is a property of the material, not of the part. Steel is about 29,000,000 psi (200 GPa), aluminium about 10,000,000 psi (69 GPa), titanium about 16,500,000 psi.
The single most useful consequence, and the one most often missed, is that modulus barely responds to heat treatment or grade. A mild A36 bar and a hardened alloy steel bar of the same size stretch by the same amount under the same load. Strength decides when the bar stops behaving elastically; stiffness decides how much it moves before then. If your problem is deflection, a stronger steel will not help. If your problem is yielding, it will.
Combining the definitions gives the working formula for elongation: ΔL = F × L₀ / (A × E). Every term is intuitive. More force stretches more. A longer bar stretches more, because the same strain acts over more length. More area or a stiffer material stretches less.
Engineering stress, true stress, and where the line ends
Engineering stress divides force by the original area. True stress divides by the instantaneous area, which shrinks as the specimen stretches. Below yield the difference is negligible — a steel bar at 0.1 percent strain has contracted laterally by about 0.03 percent, so its area is 0.06 percent smaller. After yield, and especially once necking begins, the two diverge sharply: an engineering stress-strain curve turns over and comes down while the true stress curve keeps rising. Every specification value you will meet — yield, tensile strength, elongation at fracture — is an engineering value.
The elastic line ends at the proportional limit, close to but not identical with the yield point. Because most metals curve away gradually rather than breaking sharply, yield is defined by convention as the 0.2 percent offset yield: draw a line parallel to the elastic slope but displaced by 0.002 strain, and take the stress where it crosses the curve. Low-carbon steels are the exception; they show a genuine yield plateau with distinct upper and lower yield points.
Strain at yield is worth committing to memory, because it is small. A36 steel yields at 36 ksi against a modulus of 29,000 ksi, so it yields at 36/29,000 = 0.124 percent strain. Over a 24 in bar that is 0.030 in of total stretch before permanent deformation begins. All elastic design happens inside that tiny window.
Aluminium behaves differently for a reason worth understanding. 6061-T6 yields at 35 ksi against a modulus of 10,000 ksi, so it yields at 0.35 percent strain — nearly three times the strain of steel. That is not because it is stronger; it is because it is more compliant. An aluminium part carries a similar stress but moves three times as far getting there.
Poisson's ratio links axial and lateral strain: stretch a bar axially by strain ε and it contracts laterally by νε, with ν about 0.30 for steel and 0.33 for aluminium. It is why a tensile specimen gets thinner as it stretches, and why the area used in a true-stress calculation shrinks.
Worked example: a tie rod and a tensile test
Sizing a rod. A 0.75 in diameter A36 steel tie rod, 24 in long, carries 10,000 lbf in tension.
- Area. π/4 × 0.75² = 0.441786 in².
- Stress. 10,000 ÷ 0.441786 = 22,635 psi, which is 156.1 MPa.
- Strain. 22,635 ÷ 29,000,000 = 0.00078053, or 0.0781 percent.
- Elongation. 0.00078053 × 24 = 0.018733 in. Less than twenty thousandths over two feet.
- Factor of safety. 36,000 ÷ 22,635 = 1.59 against yield.
Back-solving the modulus. Now run it the other way, as a tensile test does. A specimen with a 1.000 in² cross-section and a 10.000 in gauge length is loaded to 29,000 lbf, and the extensometer reads 0.0100 in.
- Stress. 29,000 ÷ 1.000 = 29,000 psi.
- Strain. 0.0100 ÷ 10.000 = 0.001000, or 0.100 percent.
- Modulus. 29,000 ÷ 0.001000 = 29,000,000 psi, exactly the handbook value for steel.
That second calculation is only as good as the strain measurement, which is why ASTM E111 requires a calibrated extensometer clamped to the specimen rather than crosshead travel. Crosshead motion includes the compliance of the load frame, the grips and any seating of the specimen in the jaws, and on a stiff specimen those can be a larger share of the reading than the specimen's own stretch. A modulus that comes out 30 percent low is almost always a measurement artefact rather than an unusual material.
Modulus, yield and strain at yield for common metals
| Material | E (ksi) | E (GPa) | Yield (ksi) | Strain at yield (%) | Poisson's ratio |
|---|---|---|---|---|---|
| A36 carbon steel | 29,000 | 200 | 36 | 0.124 | 0.30 |
| A572 Gr 50 / A992 | 29,000 | 200 | 50 | 0.172 | 0.30 |
| 304 stainless, annealed | 28,000 | 193 | 30 | 0.107 | 0.29 |
| 6061-T6 aluminium | 10,000 | 68.9 | 35 | 0.350 | 0.33 |
| 7075-T6 aluminium | 10,400 | 71.7 | 63 | 0.606 | 0.33 |
| Ti-6Al-4V | 16,500 | 114 | 120 | 0.727 | 0.34 |
| Copper 110, annealed | 17,000 | 117 | 10 | 0.059 | 0.34 |
| Brass 260, annealed | 16,000 | 110 | 15 | 0.094 | 0.35 |
| Grey cast iron | 14,000 | 96.5 | — | — | 0.26 |
Grey cast iron has no defined yield point; it is brittle and is designed against ultimate strength instead, with different values in tension and compression.
Mistakes that make a stress calculation wrong
- Using the gross area where a net area applies. A bar with a bolt hole through it carries its load on the remaining metal. Deduct the hole, and add a stress concentration factor for the local peak at the hole edge.
- Ignoring the thread on a threaded rod. A tensile check on a bolt uses the tensile stress area, which is smaller than the shank area — for a 1/2-13 bolt it is 0.1419 in² against 0.1963 in² for the plain shank.
- Applying this to a slender member in compression. A long column buckles at a load far below its yield stress. Above roughly twenty diameters of length, Euler buckling governs and a stress calculation is irrelevant.
- Confusing modulus with strength. Heat treating a steel part raises its yield strength substantially and its stiffness essentially not at all. A deflection problem is not solved by a stronger grade.
- Measuring strain from crosshead travel. Frame compliance and grip seating inflate the apparent elongation and depress the calculated modulus. Use a calibrated extensometer on the gauge length.
- Reading elongation predictions beyond yield. The linear relationship stops at the proportional limit. Past it, the real deflection is larger than the elastic formula predicts and part of it is permanent.
- Forgetting thermal strain. Steel expands about 6.5 microstrain per degree Fahrenheit. A restrained member that warms by 100 °F sees roughly 0.065 percent strain — half its yield strain — with no mechanical load at all.
Key terms
- Proportional limit
- The highest stress at which stress and strain remain proportional. Slightly below the yield strength for most metals.
- 0.2% offset yield
- The conventional definition of yield: the stress where a line parallel to the elastic slope, offset by 0.002 strain, crosses the curve.
- Microstrain
- Strain × 10⁻⁶. Strain gauges read in microstrain, so 1,000 µε is 0.1 percent strain.
- Resilience
- Elastic energy stored per unit volume, equal to the area under the elastic line, σ²/2E at the proportional limit.
- Poisson's ratio
- The ratio of lateral contraction to axial extension, about 0.30 for steel. It is why a tensile specimen thins as it stretches.
Where axial stress fits among the other checks
Pure axial stress is the simplest loading case, and real parts rarely see only that. Bending distributes stress linearly across the section, so the extreme fibre carries far more than the average and the governing quantity becomes section modulus rather than area. Torsion produces shear rather than direct stress and is governed by the shear modulus, roughly 0.385 times Young's modulus for steel. Combined loading requires a yield criterion such as von Mises rather than a simple comparison against yield.
The stiffness insight carries into every one of those cases. Because modulus is nearly constant across steel grades, the deflection of a beam, the twist of a shaft and the stretch of a rod are all set by geometry and by material family rather than by strength grade. That is why a section is chosen for stiffness and a material for strength, and why the metal weight calculator is often the better first stop when comparing candidate sections — weight and area come first, capacity second.
On the materials side, if the alloy is unlabelled you can get an approximate tensile strength from a hardness reading using the hardness conversion calculator, and it is worth repeating that hardness tracks strength but says nothing about stiffness. In fabrication, a welded joint introduces a region whose properties differ from the parent metal; the fillet weld strength calculator handles the joint capacity and the welding heat input calculator covers the thermal cycle that decides what those local properties become.
Forming operations exploit the other side of the curve deliberately. A press brake bend takes the outer fibre well past yield on purpose, and springback — the elastic recovery when the ram lifts — is governed by the ratio of yield strength to modulus. That ratio is why 6061-T6 springs back roughly three times as far as mild steel for the same bend, and it is the same 0.35 percent versus 0.124 percent strain-at-yield comparison in the table above. The press brake tonnage calculator covers the force side of that operation.
This factor of safety is against yield, not against fatigue
The factor of safety this calculator reports compares the applied stress to yield strength under one static application of load. That is the right comparison for a load applied once, or a handful of times over a part's life. It is the wrong comparison for anything that cycles — a bracket flexing with every pass of a machine, a fastener under vibration, a rod loaded and unloaded thousands of times a day — because those parts fail by fatigue at stresses well below yield, and a safety factor comfortably above 1 against yield says nothing at all about how many cycles the part will survive.
Fatigue starts as a crack at a stress concentration this net-section calculation cannot see — a fillet radius, a hole edge, a weld toe — and grows a little further with every cycle until the remaining section can no longer carry the static load, at which point it fails suddenly, with none of the visible yielding this page's elastic model assumes. Steel has a genuine endurance limit: a stress below which fatigue life becomes effectively unlimited. Most non-ferrous metals, aluminium among them, do not, so a stress this calculator shows as comfortably safe against yield can still fail after enough repetitions if the load cycles.
If the load repeats, use the stress computed here as the starting point for a stress-life (S-N) fatigue check rather than as the final answer, and apply the stress concentration factor at any hole or fillet to the nominal stress this page reports — not to the yield strength it is being compared against.