# Stress and strain

In engineering, the terms stress and strain mean quite different things to what we mean when we say we are stressed or under a lot of strain.

## Stress (engineering stress)

When a member has a force applied to it — say, a compressive force acting at each end of a bar — there is an internal reaction to that force. This follows Newton’s Third Law: to every action there is an equal and opposite reaction. This internal reaction has a magnitude equal to the applied force.

It is more useful for engineers to consider the reactive force equally distributed over the cross-sectional area of the bar. This is what is termed stress, given by:

Stress (σ) = force (*F*) ÷ cross-sectional area (*A*)

**Example:** What is the stress in a 20 mm diameter round bar loaded with an axial compression of 10 kN?

## Strain (engineering strain)

When a force is applied to a member, that member will be deformed either elastically or permanently (plastically).

In elastic structures such as buildings and bridges, the change in shape will disappear if the force is removed. The degree of deformation (extension, compression, shear or torsion) as a proportion of the original size of a member is termed **strain**:

strain (ε) = change in length (δ*L*) ÷ orignial length (*L*)

Strain is a proportion, or ratio, so it is dimensionless — *ie* it does not have any units. Depending on the amount of deformation, strain may be expressed as a percentage, or as parts per thousand (written °%), or as parts per million (called microstrain, usually written µs).

**Example:** The bar in the example above is reduced in length from *L* = 1 m to 0.995 m. The change of length, δ*L*, is therefore 5 mm. The strain in the bar is given by:

Calculations of stress and strain enable engineers to make direct comparisons between the behaviour of materials of different size and shape when they are subjected to different loading conditions. The use of stress and strain in place of load and deformation also makes calculations easier.

## Young’s modulus

Young’s modulus, *E*, is a measure of the stiffness of a material when it deforms elastically — hence it is also sometimes called the **elastic modulus** or **modulus of elasticity**. *E* is the ratio of stress to strain, with units that are the same as stress (usually GPa), since strain itself has no units, as noted above.

From **Hooke’s law** we know that when materials deform elastically, stress is proportional to strain:

σ = constant × ε

Young’s modulus is the constant linking stress and strain — the stiffer the material, the higher the value of Young's modulus.

It is a useful property for engineers because each material has its own value of *E*. We can therefore calculate the stress if we know the Young’s modulus and we measure the deformation — conversely, we can calculate the strain if we know the Young’s modulus and the applied stress.

The value of *E* is determined from the slope of the stress-strain curve plotted during tensile or compressive tests on a sample of the material. (But engineers don’t have to test a piece of steel, say, every time they need to do a calculation! Tables of *E* values for different materials are available in text books or manuals, or at websites like engineered materials.

**Note:**

No real material is perfectly elastic (or homogeneous or isotropic), but for practical purposes metals commonly used in engineering structures are very nearly elastic up to their yield point or yield strength. Surprisingly, even most massive rocks behave more or less elastically, and the above rules apply. This is very useful for solving geotechnical engineering problems involving the stability of rock — in railway and road tunnels or mines, for example, and the ‘skewbacks’ in the sandstone forming the Sydney Harbour Bridge abutments (see section on Bridge supports).