Shear Stress Calculator

What the shear stress calculator does

Shear stress is what a material feels when a force tries to slide one part of it past another, like the force on a bolt holding two plates that are being pulled in opposite directions. This calculator finds the shear stress from the force acting along the surface and the area that resists it.

Below is what shear stress is, how it differs from ordinary stress, the equation behind it, how it relates to the way a material twists, and a worked example.

How to use it

1. Enter the force acting along the surface, the shearing force.
2. Enter the area over which it acts, the cross-section being sheared.
3. Press Calculate for the shear stress, or Reset to clear it.

What shear stress is

Stress comes in two basic kinds, set by the direction of the force relative to the surface. When the force acts straight onto a surface, stretching or squashing the material, it is normal stress. When the force acts along the surface, trying to slide one layer over the next, it is shear stress. The difference is the difference between pulling a rope apart and cutting it with scissors.

Shear stress is what scissors, punches, and cutting tools rely on, and what a bolt or rivet resists when the plates it joins are pulled sideways. It is also what acts inside a shaft being twisted. Like all stress it is a force spread over an area, measured in pascals, but the force here runs parallel to the area rather than pressing onto it.

The equation it uses

Shear stress, written with the Greek letter tau, is the shearing force F divided by the area A it acts over:

τ = F ÷ A

The form is the same as for normal stress, force over area, but the meaning of the force differs: here it is the part of the force that runs along the surface, the part trying to cut or slide rather than stretch. When a load arrives at an angle, only its component along the surface contributes to the shear stress, while the component pressing onto the surface produces normal stress instead.

Shear strain and the shear modulus

Just as a pulling stress produces a stretching strain, a shear stress produces a shear strain, a change in shape rather than length. Picture a rubber block glued down and pushed sideways along the top: it skews into a slanted shape, and the angle of that skew is the shear strain. The material distorts without necessarily changing its volume.

The stiffness that connects the two is the shear modulus, the ratio of shear stress to shear strain. A large shear modulus means a material strongly resists being skewed, while a small one skews easily. This is the shear counterpart of the Young's modulus that governs stretching, and it is the property that decides how much a shaft twists under a given torque, or how a rubber mount gives under a sideways load.

Units and precision

Shear stress is measured in pascals, the same unit as normal stress, since both are a force divided by an area. Engineering shear stresses are usually large, running to megapascals. The shear modulus is also in pascals, typically gigapascals for metals. The essential inputs for the shear stress itself are the shearing force and the area it acts across; the result follows directly from their ratio.

A worked example: a bolt in shear

Suppose a bolt with a cross-sectional area of 0.0002 m², that is two square centimetres, carries a shearing force of 5,000 N as the plates it joins are pulled sideways.

The shear stress on the bolt is τ = 5,000 ÷ 0.0002 = 25,000,000 Pa, or 25 MPa. Comparing this against the bolt material's shear strength tells you whether it will hold or be sheared through, which is the everyday job of this calculation in design.

Questions people ask

What is the formula for shear stress?

Shear stress is the shearing force divided by the area it acts over, τ = F/A, where the force runs along the surface rather than onto it.

What is the difference between shear and normal stress?

Normal stress comes from a force acting straight onto a surface, stretching or compressing it. Shear stress comes from a force acting along the surface, sliding one part past another, like a cut.

What is the shear modulus?

It is the ratio of shear stress to shear strain, a measure of how strongly a material resists being skewed in shape. It is the shear counterpart of the Young's modulus for stretching.

Where does shear stress matter?

In bolts, rivets, and pins joining parts pulled sideways, in cutting and punching operations, and in shafts under torsion, where the twisting load is carried entirely as shear.

References

A quick note on where the physics comes from. Shear stress as a force acting parallel to a surface divided by the area, and the shear modulus relating it to shear strain, are standard mechanics of materials, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The pascal and the other SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 12.3, Stress, Strain, and Elastic Modulus (shear stress and shear modulus). https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus
2. HyperPhysics, Georgia State University, Elasticity, Shear. http://hyperphysics.phy-astr.gsu.edu/hbase/permot3.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811