Shear Stress Calculator
This shear stress calculator computes the shear stress due to transverse loads and the shear stress due to torsion (torque) applied on a circular shaft. Switch between the two modes, pick your unit system (Metric SI or American/Imperial), enter the section dimensions, and get the maximum shear stress instantly.
The shear stress from transverse forces is critical in the design of thin-walled members. For non-thin-walled members subjected to transverse loads, the most important quantity is usually the axial stress caused by the bending moment. If that's your case, use a bending stress calculator instead; otherwise, this calculator is exactly what you need.
The shear stress equation to use depends on whether we apply a transverse load to a beam or a torsion couple to a circular shaft. The sections below show the formulas for both conditions, how to calculate the maximum shear stress for the most common beam shapes, and how to find the stress at any point.
Transverse Shear Stress Formula
By definition, shear stress is force acting tangent to an area. In a beam of cross-sectional area A subjected to a shear force V, the average shear stress is:
τavg = V / A
This is just an average. The shear stress across a beam subjected to transverse shear varies widely, and we usually need to know its maximum value rather than the average. You can find the exact shear stress at any vertical distance from the neutral axis with:
τ = V·Q / (I·t)
- τ — Shear stress at a specific distance from the neutral axis
- V — Internal shear force at the section of interest in the beam
- Q — First moment, about the neutral axis, of the cross-sectional area that lies above the point where we calculate the stress
- t — Section width at the point where we calculate the shear stress
- I — Moment of inertia of the entire cross-sectional area about the neutral axis
Q is a tricky quantity. Mathematically, it is the area above the point of interest multiplied by the distance from the centroid of that area to the neutral axis (Q = ȳ′·Ā). Because computing Q correctly and locating the point of maximum shear stress takes care, this calculator gives you τmax directly for the most common shapes by only asking for their dimensions and the shear force.
Maximum Transverse Shear Stress by Cross-Section
| Cross-Section | Maximum shear stress (τmax) |
|---|---|
| Solid Rectangle (b × h) | 1.5 · V / A = 3V / (2bh) |
| Solid Circle (diameter d) | (4/3) · V / A = 16V / (3πd²) |
| I-Beam / Wide Flange | V·Q / (I·tweb) at the neutral axis |
For the rectangle and circle, τmax occurs at the neutral axis. For an I-beam, the maximum shear stress also occurs at the neutral axis and is carried almost entirely by the web, so the calculator evaluates V·Q/(I·t) using the web thickness as t.
Torsional Shear Stress Equation — Shear Stress Due to Torsion
When a torque (twisting couple) is applied to a circular shaft, the shear stress varies linearly from zero at the center to a maximum at the outer surface. The torsional shear stress at radius r is:
τ = T·r / J
- T — Applied torque (N·m, kN·m, lb·ft, lb·in, kip·in)
- r — Radial distance from the shaft axis (maximum at the outer radius)
- J — Polar moment of inertia of the cross-section
| Shaft | Polar moment of inertia (J) | τmax |
|---|---|---|
| Solid Circular (diameter d) | π·d⁴ / 32 | 16T / (π·d³) |
| Hollow Circular (D outer, d inner) | π·(D⁴ − d⁴) / 32 | T·(D/2) / J |
Important Considerations When Using This Calculator
- The transverse-shear formula τ = V·Q/(I·t) assumes the shear stress is uniform across the width t and that the material is linear-elastic and homogeneous.
- The torsion formula τ = T·r/J applies to circular cross-sections (solid or hollow). Non-circular shafts warp and require different theory.
- Make sure your shear force V or torque T is the internal value at the section of interest, not simply the external applied load.
- Keep units consistent. In Metric mode, lengths are in mm, force in N/kN, and stress comes out in MPa (N/mm²). In Imperial mode, lengths are in inches, force in lbf/kip, and stress comes out in psi.
- For short, deep beams shear stress can govern the design; for long, slender beams the bending stress usually governs.
Frequently Asked Questions
What is the difference between average and maximum shear stress?
The average shear stress τavg = V/A simply divides the shear force by the whole cross-sectional area. The actual shear stress is not uniform — it is zero at the top and bottom faces and peaks at the neutral axis. The maximum value τmax = V·Q/(I·t) is what you use for design, and it is always larger than the average (1.5× for a rectangle, 1.33× for a circle).
Where does the maximum shear stress occur?
For transverse loading of symmetric beams, the maximum shear stress occurs at the neutral axis (the centroidal axis), where Q is largest. For a circular shaft under torsion, the maximum shear stress occurs at the outer surface, where the radius r is largest.
What is the first moment of area Q?
Q is the first moment, about the neutral axis, of the portion of the cross-section that lies beyond (above or below) the point where you want the shear stress. It equals the area of that portion multiplied by the distance from its centroid to the neutral axis: Q = ȳ′·Ā. Q is largest at the neutral axis, which is why the shear stress peaks there.
Does this calculator support American (Imperial) and Metric units?
Yes. Use the Unit System selector to switch between the SI metric system (mm, N, MPa, N·m) and the US customary / Imperial system (inches, lbf, psi, lb·ft). All inputs and outputs adjust automatically.
Can I use the torsion mode for a hollow shaft (tube)?
Yes. Choose the torsion mode and select "Hollow Circular Shaft", then enter the outer and inner diameters. The calculator uses J = π·(D⁴ − d⁴)/32 and reports the maximum shear stress at the outer surface.