What this calculator does
The shear stress formula is the direct-shear equivalent of the ordinary stress equation, except the force acts parallel to the surface it is loaded across rather than straight into it. A rivet resisting two plates trying to slide past each other, a pin in a clevis joint, and a bolt in a lap joint are all loaded this way. Average shear stress is the shear force divided by the area it acts over: tau = V / A.
Average shear stress spreads the load evenly across the whole area, which is a reasonable approximation for bolts, pins and rivets. Inside a solid beam or shaft, the true distribution is not even: it peaks at the centroid and falls to zero at the outer edge, so the maximum shear stress is higher than the average by a shape-dependent factor. This calculator gives both figures, using the standard factors for solid rectangular and solid circular cross-sections.
The formula
Divide the shear force V by the cross-sectional area A it acts across to get the average shear stress. For a solid rectangular section the true maximum, at the centroid, runs 1.5 times the average; for a solid circular section it runs 4/3 (about 1.33) times the average. These factors come from the parabolic shear stress distribution across the section and are a standard result in mechanics of materials, distinct from the full transverse shear stress formula (tau = VQ / (I b)), which needs the first moment of area Q and second moment of area I for the exact shape and location.
| Term | Meaning |
|---|---|
| tau (τ) | Shear stress: force per unit area acting parallel to the surface, in MPa (N/mm²), Pa or psi. |
| V | The shear force, acting parallel to the cross-section rather than perpendicular to it. |
| A | The cross-sectional area the shear force acts across. |
| Shape factor | The ratio of maximum to average shear stress for a given solid cross-section: 1.5 for rectangular, 1.33 for circular. |
The inputs explained
| Field | What to enter |
|---|---|
| Shear force (N) | The shear force acting on the section, parallel to it. |
| Cross-sectional area (mm²) | The cross-sectional area the force acts across. |
| Cross-section shape | Choose rectangular or circular if you need the estimated maximum shear stress at the centroid. Choose "other" if the shape is unknown or irregular; average shear stress alone still applies. |
When to use it
Checking a bolted or riveted joint
A lap joint transfers load by shearing the fastener rather than stretching it. Dividing the load by the fastener's cross-sectional area, using the shear stress equation directly, gives the average shear stress to compare against the fastener material's allowable shear strength.
Sizing a pin in a clevis or bracket
A pin loaded in double shear carries the load across two cross-sections rather than one, so the effective area used in the calculation is doubled before dividing. The result is still the same average shear stress formula applied to the correct total area.
Sense-checking a beam or shaft design
Solid round shafts and rectangular beams see a higher shear stress at the centroid than the section average suggests. Applying the 1.33 or 1.5 shape factor here gives a quick estimate of that peak, useful for an early check before running the full transverse shear stress formula on a specific section.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What shear stress results from a range of forces on a 200 mm² rectangular section?
The same 200 mm² cross-sectional area under a range of shear forces.
| Shear force | Average shear stress | Maximum shear stress (rectangular) |
|---|---|---|
| 2,000 N | 10.00 MPa | 15.00 MPa |
| 4,000 N | 20.00 MPa | 30.00 MPa |
| 6,000 N | 30.00 MPa | 45.00 MPa |
| 8,000 N | 40.00 MPa | 60.00 MPa |
| 10,000 N | 50.00 MPa | 75.00 MPa |
| 12,000 N | 60.00 MPa | 90.00 MPa |
How does cross-section shape change the maximum shear stress?
The same force and area, compared across the two shapes this calculator supports.
| Cross-section shape | Average shear stress | Maximum shear stress | Shape factor |
|---|---|---|---|
| Rectangular | 25.00 MPa | 37.50 MPa | 1.50 |
| Circular | 25.00 MPa | 33.33 MPa | 1.33 |
Questions
What is the shear stress formula?
Average shear stress equals the shear force divided by the area it acts across: tau = V / A. It is the same shear stress equation used for both a single fastener and a rough estimate on a beam or shaft cross-section.
What is the maximum shear stress formula for a beam or shaft?
For a solid rectangular cross-section the true maximum, at the centroid, is 1.5 times the average shear stress. For a solid circular cross-section it is 4/3 (about 1.33) times the average. Both come from the parabolic shear stress distribution across a solid section under transverse loading.
How is this different from the transverse shear stress formula, tau = VQ / (I b)?
That full formula gives the shear stress at any point through the depth of a beam, using the first moment of area Q, the second moment of area I and the width b at that point. The shape factors used here (1.5 and 1.33) are the specific centroid-line answer that formula gives for solid rectangular and circular sections, so they agree at the point of maximum stress without needing Q and I entered separately.
How is shear stress different from the stress calculated elsewhere on this site?
The mechanical stress calculator covers normal stress, where the force acts straight into the cross-section (tension or compression). Shear stress is the same force-over-area arithmetic, but the force instead acts parallel to the surface, which is a physically different mode of loading even though both use MPa or psi.
For force acting perpendicular to a section, see the mechanical stress calculator. For bending stress in a loaded beam, see the bending stress calculator.