What is the shear formula τ = VQ/(Ib)?

The shear formula gives the average shear stress τ on a horizontal plane at a distance y from the neutral axis. V is the shear force (N), Q is the first moment of area of the portion of the cross-section above the plane of interest (mm³), I is the second moment of area of the entire cross-section (mm⁴), and b is the width of the section at the plane of interest (mm). The result is in N/mm² = MPa.

Where does maximum shear stress occur?

For most solid cross-sections (rectangular, circular), maximum shear stress occurs at the neutral axis (y = 0) where Q/b is largest. For I-beams, the neutral axis is in the thin web, so shear stress there is much higher than in the wide flanges — the web carries nearly all the shear. For hollow sections, the maximum also occurs at the mid-height of each wall.

What is the first moment of area Q?

Q is the first moment of area of the cross-sectional area above (or below) the point of interest, taken about the neutral axis: Q = ∫y dA. For a rectangular section above a horizontal cut at distance y from the neutral axis: Q = b(h/2 − y)(y + (h/2 − y)/2) = b(h²/4 − y²)/2. Q is always zero at the extreme fibers and maximum at the neutral axis.

What is shear flow, and when is it used?

Shear flow q = VQ/I (N/mm) is the shear force per unit length along a longitudinal seam. It is used to design fasteners (welds, bolts, nails) that connect flanges to webs in built-up sections. The fastener spacing is found from s = F_fastener / q, where F_fastener is the allowable shear load per fastener.

Why is maximum shear stress in a rectangular beam 1.5× the average?

For a rectangular cross-section, the maximum shear stress at the neutral axis is τ_max = (3/2)(V/A) = 1.5 V/A, while the average shear stress is V/A. This factor of 1.5 arises from the parabolic distribution of shear stress over the height, which reaches its peak at mid-height and drops to zero at the top and bottom surfaces.

Does this calculator handle hollow circular (tube) sections?

This calculator covers hollow rectangular (box) sections directly. For hollow circular tubes, the maximum shear stress is approximately τ_max = 2V/A for thin-walled tubes. You can use the solid circular mode as an approximation for thick-walled tubes, but results will be conservative. A dedicated hollow-tube mode is not included here.

Beam Shear Stress Calculator

Name: Beam Shear Stress Calculator
Availability: InStock
Author: Nham Vu

Compute shear stress at any location in a beam cross-section using τ = VQ/(Ib) for rectangular, circular, hollow box, and I-beam sections.

Cross-Section Type

Cross-Section Dimensions

Width, b (mm)

Horizontal dimension of the rectangular section.

Height, h (mm)

Vertical dimension (in the plane of loading).

Loading

Shear Force, V (N)

Transverse shear force at the cross-section of interest.

Location y from Neutral Axis (mm) (optional)

Leave blank or enter 0 for maximum shear stress at neutral axis.

Maximum Shear Stress

—

τ_max = VQ/(Ib) at neutral axis (MPa)

Shear Stress at y

—

τ(y) = VQ_y/(Ib_y) (MPa)

Shear Flow at Neutral Axis

—

q = VQ/I (N/mm)

Select a cross-section, enter dimensions and shear force, then click Calculate.

Shear Formula

τ = VQ / (I × b)

q = VQ / I (shear flow)

V = shear force (N)

Q = first moment of area above cut (mm³)

I = second moment of area (mm⁴)

b = section width at the cut (mm)

Summary

Compute shear stress at any location in a beam cross-section using τ = VQ/(Ib) for rectangular, circular, hollow box, and I-beam sections.

How it works

Select the cross-section shape: rectangular, solid circular, hollow box, or I-beam.
Enter the cross-section dimensions in millimeters.
Enter the applied transverse shear force V in Newtons (N).
Optionally enter the distance y from the neutral axis (0 = neutral axis) to compute local shear stress.
Click Calculate to see maximum shear stress, shear flow, and the full stress breakdown.
Use Reset to clear all inputs and start a new calculation.

Use cases

Checking shear stress in timber or steel beams against allowable design limits.
Verifying that transverse shear stress is negligible compared to bending stress.
Determining the location and magnitude of peak shear stress in an I-beam web.
Computing shear flow for the design of welded or bolted flange connections.
Teaching mechanics of materials — illustrating the parabolic shear distribution.
Preliminary cross-section sizing before full structural analysis.
Evaluating shear stress in machine shafts and frames under transverse loads.
Comparing shear stress intensity across different cross-section shapes.

Frequently Asked Questions

Related tools

Beam Bending Stress Torque Calculator Gear Tooth Calculator

Last updated: 2026-05-23 · Reviewed by Nham Vu