Shear Force Formula: F = τA, Beam Shear and Examples

The shear force formula is $F = \tau A$ : a shear force equals the shear stress τ multiplied by the area over which it acts, with the force lying parallel to that area. A steel pin of cross section 50 mm² that carries a shear stress of 80 MPa is resisting a shear force of 4000 N.

Shear forces try to slide one part of a body past the neighbouring part, like scissors cutting paper or a deck of cards pushed sideways. In physics they lead to shear stress and the modulus of rigidity. In engineering they appear as the shear force at a section of a beam. Both views use the newton and both start from force balance, the core of the laws of motion.

Quick facts

Stress form
$F = \tau A$
Beam section
V = sum of transverse forces on one side of the cut
Cantilever, end load P
V = P along the whole length
Simply supported, central load P
V = P/2 on each half
SI unit
newton (N); kN in engineering
Link to bending
$V = \frac{dM}{dx}$

Formula for shear force from shear stress

Symbols in F = τA
Symbol	Meaning	SI unit
$F$	Force parallel to the surface	N
$\tau$	Shear stress, force per unit area along the surface	Pa
$A$	Area that the force acts along	m²

Normal stress acts at right angles to a surface; shear stress acts along it. The area in this formula is the one the force slides across. For a bolt joining two plates, that is the bolt's cross section. Shear stress then causes a shear strain φ, and the ratio $G = \frac{\tau}{\phi}$ is the modulus of rigidity. The general stress formula page compares the two kinds of stress.

Worked examples on the shear force formula

Example 1: a bolt in shear, with mm²

A bolt of cross section 50 mm² carries a shear stress of 80 MPa. Find the shear force on it.

Convert: 50 mm² = 50 × 10⁻⁶ m², and 80 MPa = 80 × 10⁶ Pa.

$F = \tau A = 80 \times 10^6 \times 50 \times 10^{-6} = 4000\ \text{N} = 4.0\ \text{kN}$

Example 2: shear stress in a rubber pad

A 200 N sideways force acts on the top face of a rubber pad of area 0.010 m². Find the shear stress.

$\tau = \frac{F}{A} = \frac{200}{0.010} = 2.0 \times 10^4\ \text{Pa}$

Example 3: a simply supported beam

A 6.0 m beam rests on supports at both ends and carries a uniform load of 2.0 kN/m. Find the shear force at each support, at 1.0 m from the left end, and at midspan.

Total load: 2.0 × 6.0 = 12 kN, shared equally, so each support reaction is 6.0 kN.

At the left support, V = 6.0 kN. At x = 1.0 m: V = 6.0 − 2.0 × 1.0 = 4.0 kN. At midspan, x = 3.0 m: V = 6.0 − 6.0 = 0.

Shear changes sign at midspan, which is exactly where the bending moment is greatest.

Shear force in beams

Imagine cutting a beam at some point. The shear force V at that section is the sum of all the forces perpendicular to the beam on one side of the cut. A common convention takes upward forces on the left part as positive; check the convention your course uses, because books differ. Adding up from the other side must give the same size of V.

Shear force for standard beams
Beam and load	Shear force
Cantilever, point load P at the free end	V = P, constant along the length
Cantilever, uniform load w per metre, length L	V = wx at distance x from the free end, maximum wL at the wall
Simply supported, central point load P	V = P/2, changing sign at the load
Simply supported, uniform load w over length L	V = wL/2 − wx, zero at midspan

The slope of the bending moment diagram equals the shear force, $V = \frac{dM}{dx}$ . Where V passes through zero, the bending moment reaches its maximum, which is the point a designer checks first.

Shear force compared with other forces

A shear force is still just a force: dimensionally [M¹ L¹ T⁻²], measured in newtons. What makes it special is its direction relative to the surface. Friction between two blocks is a shear force on each contact face. The viscous force between fluid layers is also a shear force, with the stress given by $\eta\frac{dv}{dx}$ .

Mistakes with shear

Using an area perpendicular to the force instead of parallel to it.
Leaving mm² or MPa unconverted, or converting only one of them.
Mixing shear force and bending moment units: kN for V, kN m for M.
Forgetting the support reactions before summing forces along a beam.

A bolt that joins three plates is cut at two planes. It is in double shear, so each plane carries only half the load.

Frequently asked questions

What is the formula for shear force?

In terms of stress, shear force F = τA, shear stress times the area parallel to the force. In a beam, the shear force at a section is the sum of transverse forces on one side of it.

What is the unit of shear force?

The newton, N. Engineers often quote beam shear in kilonewtons, kN.

What is the difference between shear force and normal force?

A normal force acts perpendicular to a surface and squeezes or pulls it. A shear force acts parallel to the surface and tries to slide layers past each other.

Where is shear force zero in a beam?

For a simply supported beam with a uniform load, the shear force is zero at midspan. In general V is zero where the bending moment is at a maximum or minimum.