How to use this calculator

Choose the cross-section and enter its dimensions and the internal shear force V at the beam location you are checking (read it from a shear diagram). Enter y, the distance from the neutral axis to the level where you want the stress; y = 0 gives the maximum. The graph shows how the shear stress varies from the bottom fiber to the top.

Formulas

Shear formulaτ = V QI t Q = ∫_A′ y dA = A′ ȳ′

Rectangle b × hQ = b2(h²4 − y²) τ_max = 32 VA

Solid circle, radius rQ = ⅔(r² − y²)^3/2 t = 2√(r² − y²) τ_max = 43 VA

I-beam, in the web (h_w = d − 2t_f)Q = b_ft_f d − t_f2 + t_w2(h_w²4 − y²)

Q is the first moment, about the neutral axis, of the part of the cross-section beyond the level y. t is the width of the section cut at y. Q is largest at the neutral axis and zero at the outer fibers, so shear stress follows the same pattern.

Worked example

A 50 × 150 mm rectangle carrying V = 10 kN (the default)

I = 50 × 150³/12 = 14.06 × 10⁶ mm⁴.
At the neutral axis: Q = (50/2)(150²/4) = 140,625 mm³ and t = 50 mm.
τ = 10,000 × 140,625 / (14.06 × 10⁶ × 50) = 2.00 MPa, exactly 1.5 times the average V/A = 1.33 MPa.

An I-beam: d = 300, b_f = 150, t_f = 12, t_w = 8 mm, V = 10 kN

I = 88.71 × 10⁶ mm⁴ and τ_max = 4.73 MPa at the neutral axis.
The quick web estimate V/(d t_w) = 4.17 MPa is within 12%, which is why steel codes use it.
Where the web meets the flange, τ jumps from 3.65 MPa in the web to 0.19 MPa in the flange because t widens from 8 to 150 mm.

When shear stress governs

For long, slender beams bending stress almost always controls the design. Shear matters for short, deep beams with heavy loads near a support, for thin webs, for wooden beams (wood is weak in horizontal shear along the grain) and for the connectors that join built-up sections, where the shear flow q = VQ/I sets the nail, bolt or weld spacing.

Assumptions and limits

Linear elastic, prismatic beam bending about a principal axis, with shear stress uniform across the width t.
Accurate for rectangles that are deeper than they are wide. For wide, shallow sections the stress at the edges of the width is higher than the average the formula gives.
For a solid circle the formula gives the vertical component averaged across each chord; the 4V/(3A) maximum is the standard textbook value.
The I-beam model uses sharp corners with no root fillets, so it slightly underestimates I for rolled sections.

Frequently asked questions

Why is the shear stress zero at the top and bottom?

The top and bottom surfaces are free, so there is no shear on them. Complementary shear means the vertical shear stress at those fibers must also be zero.

What is the difference between shear stress and shear flow?

Shear flow q = VQ/I is force per unit length along the beam. Dividing by the width t gives the stress τ. Shear flow is what fasteners and welds must carry.

Where do I get V?

From a shear diagram of the beam. The shear and moment diagram calculator finds the largest V for common loadings.