Transverse Shear Stress in Beams

τ = \frac{V Q}{I t}

Horizontal (and complementary vertical) shear stress in a beam. Q is the first moment about the NA of the area beyond the fiber; t is the width at that fiber. For a rectangle, τ_max = 1.5 V/A at the NA and τ = 0 at the top and bottom fibers.

$τ$ (Pa): Shear stress
$V$ (N): Transverse shear force
$Q$ (m³): First moment about the NA
$I$ (m⁴): Moment of inertia about NA
$t$ (m): Width at the fiber

Step-by-step solved example

A 100×200 mm rectangular beam, V = 30 kN. Maximum shear stress?

Horizontal shear τ = VQ/(Ib) — first moment Q of the area beyond the cut.

1. Area and average τ
A = 0.10×0.20 = 0.020 m². τ_avg = V/A = 30e3/0.020 = 1.50 MPa.
2. Rectangle peak
τ_max = 1.5 V/A = 1.5 × 1.50 = 2.25 MPa at the NA.
$τ_{m a x} = 2.25 MPa$

Answer: 2.25 MPa

10 practice questions

0/10 correct

1.For a rectangle, τ_max / τ_avg =

2.Q is

3.τ from VQ/(It) is typically maximum at the

4.t in τ = VQ/(It) is

5.V = 40 kN, rectangle A = 20000 mm². τ_max =

6.At the top fiber of a rectangle, Q =

7.I in the shear formula is

8.The web of a thin I-beam carries

9.Doubling V (same section) doubles

10.SI units of Q are