Formula of Stress: σ = F/A, Types, SI Unit and Examples

The formula of stress is $\sigma = \frac{F}{A}$ : the force acting on a solid divided by the area of the cross section that carries it. When you pull a wire, its atoms pull back; stress is the size of that restoring pull on each square metre. Its SI unit is the pascal, the same as pressure.

Stress is the first half of every elasticity calculation. Divide it by strain and you get a modulus such as Young's modulus. Engineers compare it with the breaking stress of a material to decide how thick a beam or cable must be. You will find the elasticity formulas side by side in the solids and fluids formula tables.

Quick facts

Formula
$\sigma = \frac{F}{A}$
SI unit
pascal (Pa) = N/m²; engineers often use MPa
Dimensional formula
[M¹ L⁻¹ T⁻²]
Types
tensile, compressive, shear, and hydraulic (volume) stress
Shear stress
$\tau = \frac{F_{\parallel}}{A}$
Round wire
$A = \pi r^2$

Stress formula in physics, symbol by symbol

What σ = F / A means
Symbol	Meaning	SI unit
$\sigma$	Stress in the material	Pa
$F$	Applied force (equal in size to the internal restoring force at equilibrium)	N
$A$	Cross sectional area perpendicular to a tensile or compressive force, or parallel to a shear force	m²

Strictly speaking, stress measures the restoring force inside the solid, per square metre. In a body at rest that restoring force equals the applied load, so in practice you divide the load by the area. For a wire or rod with a round section, use A = πr², taking care to halve a given diameter first.

Types of stress

The four kinds of stress in Class 11
Type	How the force acts	Effect
Tensile	pulls along the length, perpendicular to the section	stretches the body
Compressive	pushes along the length	shortens the body
Shear (tangential)	acts parallel to a face	changes the shape, like a pushed book
Hydraulic (volume)	equal pressure from all sides	changes the volume

Tensile and compressive stress together are called longitudinal or normal stress. Shear stress pairs with the modulus of rigidity, and hydraulic stress, which is simply the extra pressure on the body, pairs with the bulk modulus.

Solved examples

Example 1: a loaded wire

A wire of radius 1.0 mm supports a load of 100 N. Find the tensile stress.

Area: $A = \pi (1.0\times10^{-3})^2 = 3.14\times10^{-6}\ \text{m}^2$

$\sigma = \frac{100}{3.14\times10^{-6}} = 3.2 \times 10^7\ \text{Pa}$

Example 2: a lamp on a cable, from mass and diameter

A 50 kg lamp hangs from a steel cable 4.0 mm in diameter. Find the stress in the cable. Take g = 9.8 m/s².

Force: $F = mg = 50 \times 9.8 = 490\ \text{N}$ . Radius: 2.0 mm = 2.0 × 10⁻³ m.

Area: $A = \pi (2.0\times10^{-3})^2 = 1.257\times10^{-5}\ \text{m}^2$

$\sigma = \frac{490}{1.257\times10^{-5}} = 3.9 \times 10^7\ \text{Pa} = 39\ \text{MPa}$

Example 3: compressive stress in a pillar

A square concrete pillar 25 cm × 25 cm carries a load of 2.0 × 10⁵ N. Find the compressive stress.

$\sigma = \frac{2.0\times10^5}{0.25 \times 0.25} = \frac{2.0\times10^5}{0.0625} = 3.2 \times 10^6\ \text{Pa}$

Turn the mass into a weight, mg, before dividing by the area.

Stress, pressure and the breaking point

Stress and pressure share a formula and a unit, but they are not the same idea. A fluid's pressure always pushes at right angles to any surface it touches. Stress lives inside a solid and can act along the normal or along the surface. On the stress strain graph of a metal wire, stress rises in a straight line at first (Hooke's law region), then curves past the elastic limit, and the wire finally snaps at its breaking stress. The maximum load a wire can carry is breaking stress × area, so a wire with twice the radius can hold four times the load. The dimensional working is on the dimensional formula of stress page.

Mistakes with the formula of stress

Treating the diameter as if it were the radius. That quadruples the area.
Leaving area in mm². 1 mm² = 10⁻⁶ m².
Thinking a longer wire has more stress. Length does not appear in σ = F/A; it affects the extension, not the stress.
Using the side area for a tensile force. For pulling and pushing, use the cross section the force passes straight through.

Frequently asked questions

What is the formula of stress?

Stress σ = F / A, the force per unit cross sectional area. For shear stress, F is the force parallel to the face and A is the area of that face.

What is the SI unit of stress?

The pascal, which is one newton per square metre. Large stresses are given in megapascals (10⁶ Pa) or gigapascals (10⁹ Pa).

Is stress a scalar or a vector?

Stress is neither a simple scalar nor a vector; at advanced level it is a tensor. For school problems you treat its size, F/A, as a single number in pascals.

What is the difference between stress and strain?

Stress is the cause, force per area, measured in Pa. Strain is the effect, the fractional change in size, with no unit.