The stress and strain calculator above takes an axial load and a cross-section and returns the normal stress in pascals, megapascals and pounds per square inch. Give it an original length and a measured elongation as well and it returns engineering strain and the Young's modulus those two figures imply. That last number is the useful one, because it lets you check a measurement against a known material property and see immediately whether the specimen is behaving elastically or whether something else is going on.
Arb Digital builds free tools that handle the geometry rather than making you do it first. Round bars, hollow tubes and rectangles are all built in, so you enter the dimensions you actually have rather than pre-computing an area, and every unit is switchable. The section that trips people up most is the hollow tube, where the load-bearing area is the annulus of metal and not the circle described by the outer diameter, so the tool computes that difference explicitly.
What This Stress and Strain Calculator Does
Stress is force per unit area, and strain is the fractional change in length that force produces. OpenStax University Physics Volume 1, section 12.3 on stress, strain and elastic modulus, sets out both definitions and the relationship between them, and lists Young's modulus for steel as 20.0 × 1010 pascals, which is 200 gigapascals. That value is the reference the implied-modulus figure in the grid is most usefully compared against.
The tool handles tension and compression identically in magnitude. A negative force produces a negative stress and, with a negative elongation, a negative strain; the modulus that comes out is still positive, because both quantities change sign together. Shear stress and bending stress are deliberately out of scope, since those involve a different distribution of force across the section and a different modulus entirely.
The unit for stress is the pascal, one newton per square metre, which is a coherent derived unit in the system defined by the SI Brochure published by the BIPM. A pascal is tiny in engineering terms — atmospheric pressure is about 101,325 of them — so structural work is done almost entirely in megapascals, and one megapascal is exactly one newton per square millimetre. That equivalence is worth memorising, because it makes most section calculations mental arithmetic.
How to Use It
- Enter the axial force and its unit. Kilonewtons, newtons, pounds-force and kilograms-force are all accepted. Use a negative value for compression.
- Pick the cross-section type. The two dimension fields relabel themselves in meaning: diameter alone for a round bar, outer diameter and wall thickness for a tube, width and height for a rectangle.
- Set the section unit. Millimetres is the default because that is what drawings use. A directly entered area is converted using the square of the same factor.
- Add the original length and elongation for strain. Both use the length unit selector, so a length in metres and an elongation in millimetres needs one of them converting first.
- Compare the implied modulus with the published value. Around 200 GPa suggests steel behaving elastically; a much lower figure usually means the material has yielded or the measurement includes slack somewhere in the rig.
The Formula: How Stress and Strain Are Calculated
Normal stress is σ = F ÷ A, with force in newtons and area in square metres giving a stress in pascals. Engineering strain is ε = ΔL ÷ L0, a pure ratio with no units. Young's modulus follows as E = σ ÷ ε, which is simply the slope of the straight part of a stress-strain curve.
The areas are standard geometry. A solid round bar of diameter d has an area of πd² ÷ 4. A tube of outer diameter d and wall thickness t has an inner diameter of d − 2t, and the load-bearing area is the difference between the two circles. A rectangle is width times height. Every division is guarded, so a zero area or a zero original length returns a message rather than an infinity.
Work the defaults. A 20 mm round bar has an area of π × 0.02² ÷ 4 = 3.1416 × 10−4 m², which is 314.16 mm². Under 50 kN the stress is 50,000 ÷ 0.00031416 = 1.5915 × 108 Pa, or 159.15 MPa, which is 23,083 psi. If a 2,000 mm length of that bar stretches by 1.6 mm, the strain is 1.6 ÷ 2,000 = 0.0008, or 0.08 per cent, and the implied Young's modulus is 159.15 × 106 ÷ 0.0008 = 1.989 × 1011 Pa, or 198.9 GPa. That is within one per cent of the published figure for steel, which is exactly the sanity check the grid is there to provide.
Engineering Stress Versus True Stress
Everything on this page is engineering stress, which divides the force by the original cross-sectional area. As a specimen stretches it also gets thinner, so the area actually carrying the load shrinks. True stress divides by the instantaneous area and is therefore always higher than engineering stress once deformation becomes significant.
Below the yield point the difference is negligible. Steel strains by around 0.1 per cent before yielding, and a 0.1 per cent extension narrows the section by a few hundredths of a per cent, which is far below measurement noise. Past yield the gap opens rapidly, and this is why textbook stress-strain curves appear to turn downwards after the ultimate tensile stress: the engineering curve falls because the load falls, while the true stress in the necking region is still climbing.
The practical consequence is that this calculator is trustworthy in the elastic range and increasingly optimistic outside it. If the implied modulus comes out far below the published value for your material, the specimen has almost certainly left the elastic region and the engineering-stress framework no longer describes what is happening inside it.
Why the Load-Bearing Area Is Not Always Obvious
Force divided by area is trivial arithmetic; deciding which area is not. A hollow tube is the first trap. A 50 mm tube with a 3 mm wall has an outer circle of 1,963 mm² but a load-bearing annulus of only 443 mm², so using the outer figure understates the stress by a factor of more than four. The tube mode exists precisely to prevent that.
The second trap is the net section. Any hole, notch, keyway or thread reduces the material available to carry load, and the stress at the reduced section is what matters. A 20 mm bar with a 6 mm cross-hole has lost a substantial slice of its area at that point, and the correct area to use is the reduced one. A threaded rod is worse still, since the load passes through the tensile stress area, which is smaller than the area implied by the nominal diameter.
The third and least intuitive is stress concentration. Even after using the correct net area, the local stress at the edge of a hole or the root of a notch is several times the average across that section. Sharp internal corners are the classic failure origin for exactly this reason, and it is why fillet radii appear on well-designed parts. Nothing on this page models concentration factors; it computes the average stress across whatever area you give it, which is the starting point for a proper analysis rather than the end of one.
Reading the Implied Young's Modulus as a Diagnostic
The fourth grid item is the most useful figure here, because it is the only one that can be checked against something external. Published moduli are stable material properties: around 200 GPa for steel, roughly 70 GPa for aluminium, about 110 GPa for copper alloys. If your measured force, area, length and elongation produce something close to the published value, all four measurements are probably sound.
When it comes out too low, the usual cause is not the material but the measurement. Elongation measured across a whole test rig includes take-up in the grips, seating of pins and flexure of the frame, all of which look like extra extension and therefore like a softer material. Measuring across a gauge length on the specimen itself, rather than across the machine, typically doubles the apparent modulus of a poorly instrumented test.
When it comes out too high, suspect the length figures. Mixing a length in metres with an elongation in millimetres inflates the modulus by a thousand, and a modulus in the tens of terapascals is a unit error every time rather than a discovery. The tool applies one unit to both fields deliberately, to make that mismatch harder to commit.
How This Differs From the Adjacent Tools
The boundary in one sentence: this page starts from a load and a physical cross-section and derives stress and strain, while the Young's modulus calculator works the elastic relationship between stress, strain and modulus once you already have those quantities. Use this one when you have a force and a bar; use that one when you have a modulus and want a deformation.
The Hooke's law calculator is the spring version of the same physics, using a stiffness in newtons per metre that already bundles up the material and the geometry, whereas stress and strain separate them. The pressure converter rescales a pascal figure into bar, psi or atmospheres but computes nothing, and the force converter does the same for the load. The force calculator is where to start if the load itself has to be derived from a mass and an acceleration.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the outer diameter of a hollow section — only the metal annulus carries load, and the outer circle can overstate the area several times over.
- Ignoring holes, notches and threads — the net area at the reduced section governs, and local stress there is higher again than the average.
- Mixing length units between original length and elongation — a metre against a millimetre shifts the implied modulus by a factor of a thousand.
- Measuring elongation across the whole test rig — grip take-up and frame flexure both masquerade as extra strain and make the material look far softer than it is.
- Applying engineering stress past yield — the real area has shrunk by then, so the figure shown understates what the material is actually experiencing.
Related Free Tools From Arb Digital
For the elastic relationship from the other direction, use the Young's modulus calculator or the Hooke's law calculator. Rescale results with the pressure converter, the force converter or the area converter. To derive the load in the first place, try the force calculator, and for the energy stored in a stretched member the work calculator covers the force-times-distance side. The full free online tools hub lists everything.
Frequently Asked Questions
Both are force per unit area and both are measured in pascals. Pressure normally describes a fluid pushing equally in all directions, while stress describes internal forces within a solid and can be tensile, compressive or shear.
No. It is a length divided by a length, so it is a pure ratio. It is often quoted as a percentage or in microstrain, where one microstrain is a strain of one millionth, but the underlying quantity is dimensionless.
The annulus of metal between the outer and inner diameters, not the full outer circle. A 50 mm tube with a 3 mm wall has a load-bearing area of about 443 square millimetres against an outer circle of 1,963, so the difference is enormous.
It is a check on your measurements. Steel sits near 200 gigapascals and aluminium near 70, so a result close to the published value suggests the force, area, length and elongation are all sound, while a much lower figure usually points to yielding or slack in the test rig.
Engineering stress, which divides force by the original cross-sectional area. True stress uses the instantaneous area and is higher once the specimen has thinned appreciably, though below the yield point the two are indistinguishable.
No. This page handles axial tension and compression only, where stress is uniform across the section. Bending varies the stress from one face to the other, and shear uses a different modulus entirely.
Because a square millimetre is a millionth of a square metre, so a newton spread over it gives a million pascals. That identity makes stress checks on drawing dimensions quick, since kilonewtons divided by square millimetres land directly in megapascals.
No. It returns the average stress across the area you enter. Local stress at the edge of a hole or the root of a sharp notch can be several times that average, which is why fillet radii and concentration factors exist.
This tool is provided for educational and estimating use. It computes average engineering stress and strain for uniform axial loading, and does not model stress concentrations, buckling, fatigue, temperature effects or any safety factor, so treat its output as a physics result rather than a structural design calculation. Load-bearing structures must be designed and verified by a qualified engineer.
