Direct stress and strain, Young's modulus relating them, the factor of safety, and selecting materials by their mechanical properties.

What this key area is asking

The SQA wants you to calculate stress and strain in a loaded member, use Young's modulus to relate them, apply a factor of safety, and select materials by their mechanical properties. Once you know the force in a member (from the structures analysis), this is how you check whether the material can carry it safely.

Stress and strain

Stress tells you how hard the material inside a member is being worked: the same force in a thinner member (smaller area) gives a higher stress, which is why thin members fail first. The almost universal slip is units: an area in square millimetres must be converted to square metres ($1\ \text{mm}^2 = 10^{-6}\ \text{m}^2$) before dividing, or the stress comes out a million times wrong. Strain is the relative stretch and has no units because it is one length divided by another.

Young's modulus

Young's modulus is a property of the material, not the shape, so all steel has roughly the same $E$ (about 200 GPa) regardless of the size of the bar. It lets you predict the strain (and hence the extension) a member will suffer under a known stress, $\varepsilon = \sigma / E$, which matters when a structure must not deflect too much.

The factor of safety

Engineers never load a material right up to its failure stress. The factor of safety builds in a margin.

A factor of safety of 4 means the component is designed to work at only a quarter of the stress that would break it. The margin covers the real-world uncertainties: loads larger than expected, flaws or variation in the material, wear, corrosion and fatigue (weakening under repeated loading). Critical or life-bearing structures use larger factors of safety.

Selecting materials by their properties

Choosing a material weighs several mechanical properties against the job and the cost: strength (stress it withstands before failing), stiffness (Young's modulus), ductility (how much it deforms plastically before breaking, so it can be drawn into wire), toughness (resistance to sudden fracture or impact), and hardness (resistance to scratching and wear), alongside density, corrosion resistance and cost. A tie can be a slender high-strength rod; a part that must not deflect needs a high Young's modulus; a part subject to impact needs toughness. This links back to the contexts area, where material choice also weighs environmental impact and life-cycle cost.

Examples in context

A suspension bridge cable is steel chosen for high tensile strength, carrying enormous stress at a safe working level set by the factor of safety. An aircraft component uses aluminium alloy or composite for a high strength-to-weight ratio, with a carefully calculated factor of safety because the loads and fatigue are critical. A drinks can uses thin aluminium: enough strength for the job, ductile so it can be formed, and cheap and recyclable. In each case the member force and area set the stress, Young's modulus sets the deflection, and the factor of safety keeps the working stress safely below failure.

Try this

Q1. A force of 5000 N acts on an area of $25\ \text{mm}^2$. Find the stress in MPa. [2 marks]

• Cue. $\sigma = \frac{5000}{25 \times 10^{-6}} = 2.0 \times 10^{8}\ \text{Pa} = 200\ \text{MPa}$.

Q2. A 2 m rod stretches by 1 mm under load. Find the strain. [2 marks]

• Cue. $\varepsilon = \frac{\Delta L}{L} = \frac{0.001}{2} = 5 \times 10^{-4}$.

Q3. A material fails at 360 MPa and a factor of safety of 6 is used. Find the safe working stress. [1 mark]

• Cue. $\frac{360}{6} = 60\ \text{MPa}$.