The shear modulus calculator above returns G, the constant that relates a shear stress to the shear strain it produces. Shear is the deformation you get when a force acts along a surface rather than perpendicular to it: the sideways lean of a stack of paper pushed at the top, the twist in a driveshaft, the deflection of a rubber mount under a lateral load. It is a different quantity from Young's modulus, and materials that are stiff in tension are not automatically stiff in shear.
Arb Digital builds free physics calculators that show the intermediate steps rather than one bare answer. This one gives you three routes to G — from a force-and-deflection test, from a stress and strain pair you already have, or from Young's modulus and Poisson's ratio — and prints the shear stress, the shear strain, the shear angle and the implied Young's modulus alongside, so a bad measurement shows itself.
What This Shear Modulus Calculator Does
It solves G = τ ÷ γ. The shear stress τ is the force divided by the area it acts across, and the shear strain γ is the sideways displacement divided by the distance over which that displacement develops. Because γ is a length divided by a length it has no units, so G comes out in the same units as stress — pascals, or more usefully gigapascals.
The tool also runs the isotropic elastic relation G = E ÷ (2(1 + ν)) in the third method, which lets you get a shear modulus from two constants that are far more commonly published than G itself. In the other two methods that relation runs backwards instead, turning your measured G and the Poisson's ratio you entered into an implied E. That implied figure is the most useful number on the page, because you can check it against a value you already trust.
The shear angle in the grid is the same information as the strain, expressed in degrees. For small strains the two are numerically almost identical in radians, and seeing the angle makes it obvious when a supposed elastic test has produced a deformation far too large to still be elastic.
How to Use It
- Pick a method. Use the force-and-deflection route if you have test data. Use the direct route if someone has already reduced that data to a stress and a strain. Use the E and ν route when you have no shear data at all.
- Get the area right. In shear it is the area the force slides along, not the area a tensile load would pull apart. This is the single most common error on the page.
- Measure L across the force, not along it. The transverse distance is the gap between the loaded face and the fixed one, perpendicular to the direction the force is pushing.
- Enter Poisson's ratio. Most metals sit near 0.3, rubbers approach 0.5, and cork is close to zero. It only affects the implied E in the first two methods, but it drives the answer entirely in the third.
- Sanity-check the implied Young's modulus. If it comes out at a third of the published value for your material, the test was not measuring pure shear.
The Formula: How Shear Modulus Is Calculated
Shear stress is τ = F ÷ A, with the force parallel to the surface of area A. Shear strain is γ = Δx ÷ L, where Δx is the displacement of the loaded face and L is the transverse distance. The shear modulus is their ratio, G = τ ÷ γ = (F ÷ A) ÷ (Δx ÷ L). OpenStax University Physics Volume 1, section 12.3 on stress, strain and elastic modulus, sets out the shear case alongside the tensile and bulk cases and works a bookcase example in exactly this form.
Take the defaults. A shear force of 20 kN acts across an area of 500 mm², so τ = 20,000 ÷ 0.0005 = 40 MPa. The loaded face moves 0.06 mm relative to a fixed face 40 mm away, so γ = 0.06 ÷ 40 = 0.0015. Then G = 40 ÷ 0.0015 = 26,667 MPa, or 26.7 GPa. That is a textbook value for aluminium.
Now the check. With ν = 0.33, the implied Young's modulus is E = 2G(1 + ν) = 2 × 26.7 × 1.33 = 70.9 GPa, which is the accepted figure for aluminium alloys. The two independent routes agree, which is the signal that the test was clean.
Two Routes to G, and Why They Can Disagree
The relation G = E ÷ (2(1 + ν)) is not an approximation. It is exact — but only for a material that is homogeneous, isotropic and linearly elastic. Those three conditions do a lot of work. Steel and aluminium alloys satisfy them well enough that the formula is used without comment. Timber, carbon fibre laminates, rolled sheet with a strong grain direction and most reinforced plastics do not, and for those materials the shear modulus is an independent property that has to be measured, not derived.
Wood is the clearest example. Its stiffness along the grain can be twenty times its stiffness across the grain, and its shear modulus is nowhere near what the isotropic relation would predict from an along-grain E. Applying the formula anyway can be wrong by an order of magnitude, and it will be wrong in the unconservative direction.
The other reason the two routes disagree is that a real shear test rarely produces pure shear. Fixtures flex, bolts slip, and part of the measured Δx is bending rather than shear. Every one of those contaminations makes the deflection larger, which makes the apparent G smaller, which makes the implied E smaller than the published value. That is why the direction of the disagreement is diagnostic and not just noise.
Where This Sits Against Normal Stress and Young's Modulus
Arb Digital already publishes a stress and strain calculator, and the boundary between the two pages is precise. That tool works in normal stress and engineering strain: an axial force pulling or pushing along the length of a bar, an elongation measured in the same direction, and Young's modulus as the constant that relates them. This page works in shear: force along the surface, displacement across it, and G as the constant. Same structure of equation, different physics, different number.
The Young's modulus calculator covers the tensile modulus on its own, including running the relation backwards to predict an extension. If you need the shear stress itself rather than the modulus, the shear stress calculator handles the average and transverse cases without asking for any deflection at all. For a plane stress state where normal and shear components act together, the von Mises stress calculator combines them into a single equivalent stress.
The Elastic Limit Is the Boundary of This Calculation
G is only a constant while the material is behaving elastically. Past the proportional limit, stress and strain stop being proportional, and the ratio τ ÷ γ keeps producing a number but that number is no longer the shear modulus — it is a secant slope that falls as the load rises. OpenStax University Physics Volume 1, section 12.4 on elasticity and plasticity, shows how the curve departs from the straight line and why the modulus is only defined on the straight part.
Practically, this means the answer is trustworthy only when the shear strain is small. A shear angle of a fraction of a degree in a metal is fine. A shear angle of several degrees in a metal means yielding has already happened, and the number this tool returns is a description of a damaged sample rather than a material property. Elastomers are a different case entirely, because they stay elastic to very large strains but stop being linear long before they stop being elastic.
Why We Publish No Table of Material Values
You will notice this page has no table of shear moduli to pick from. That is deliberate, and it follows the same rule as the breaker size calculator, which publishes no ampacity table. A material property that is wrong by an alloy designation, a temper, a grain direction or a temperature is worse than no value at all, because it looks authoritative. Every property here is a field you fill from your own material data sheet or standard, with the reason stated on the page.
Where a shear modulus feeds a design decision rather than a homework answer, the working figure normally comes from a governing standard rather than a generic handbook, and the margin against it is set by a safety factor. The factor of safety calculator covers that side; this page stops at the material property.
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 cross-sectional area instead of the sheared area — in shear the force acts along the surface, so the area is the one sliding, not the one being cut across.
- Applying G = E ÷ (2(1 + ν)) to wood or composites — the relation holds only for isotropic materials, and for anisotropic ones it can be wrong by a factor of ten.
- Measuring L along the force rather than across it — the transverse distance is perpendicular to the push, and swapping the two directions inverts the strain.
- Reporting a modulus from a test that yielded — once past the proportional limit the ratio is a secant slope, not a material constant.
- Confusing shear modulus with shear strength — the modulus describes stiffness, how much it deflects; the strength describes when it breaks. They are unrelated numbers.
Related Free Tools From Arb Digital
For the tensile side of the same problem use the Young's modulus calculator and the stress and strain calculator. If you want the stress on its own, the shear stress calculator is the direct route, and the spring rate calculator shows what G does in a real component, since a coil spring's stiffness is governed almost entirely by it. The elastic potential energy calculator covers the energy stored in an elastic deformation, and the pressure converter moves modulus figures between pascals, psi and bar. The full list is on the free online tools hub.
Frequently Asked Questions
Young's modulus relates a normal stress to the strain along the same direction, so it describes stretching and compressing. Shear modulus relates a stress acting along a surface to the sideways distortion it causes, so it describes twisting and sliding. For most metals the shear modulus is roughly 38 per cent of Young's modulus.
Yes. Shear modulus, modulus of rigidity, and the symbols G and S all refer to the same quantity. Older textbooks and some engineering standards prefer modulus of rigidity, and OpenStax uses S, but the definition and the units are identical.
Because it is a displacement divided by a length. That also makes it equal, for small deformations, to the angle in radians through which the material has been distorted, which is why the tool reports a shear angle alongside it.
Not without Poisson's ratio. The relation needs both. If you have no measured ratio, using a typical value for the material class gets you close for metals, since a change from 0.29 to 0.33 moves G by only about three per cent. For polymers and composites that assumption is much less safe.
For structural metals, well under a degree. Shear strains at yield are typically in the region of one or two thousandths, which is roughly a tenth of a degree. A test showing several degrees of distortion in a metal has already passed the elastic range.
Almost always because the measured deflection contains something other than shear. Fixture flex, bolt slip and bending all add displacement, which inflates the strain, which deflates G and therefore the implied E. A result that is too high is far rarer and usually means the transverse distance was mismeasured.
Yes. For metals G falls gradually as temperature rises, typically a few per cent per hundred degrees Celsius, and much faster near the melting point. For polymers the change is dramatic around the glass transition, where the modulus can drop by orders of magnitude over a narrow temperature range.
This tool is provided for educational and preliminary study use. It models linear elastic behaviour only, publishes no material property values of its own, and is not a structural design. Any load-bearing application must be checked by a qualified engineer against the governing standard.
