Materials, stress & strain

Choosing materials

Select from a table of properties and justify with properties linked to the job: "Material C, because it is corrosion resistant, so it won't degrade in sea water, and has high strength, so it resists the forces on the mast." "Strong" alone doesn't score.

Stress and strain

• Direct stress σ = F ÷ A, in N/mm² (force in N, area in mm²). E.g. 26 000 N on a 710 mm² cable: σ = 37 N/mm².
• Strain ε = Δl ÷ l: change in length ÷ original length. It has no units. E.g. 0.00036 = Δl ÷ 120 gives Δl = 0.043 mm.
• A bigger force on the same area gives a bigger stress; a bigger area reduces stress.

Stress and strain worked examples

• Rectangular bar 180 × 120 mm with 500 N: A = 21 600 mm², σ = 500 ÷ 21 600 = 0.023 N/mm².
• Wire diameter to carry 8 kN at no more than 200 N/mm²: A = 8000 ÷ 200 = 40 mm²; d = √(4 × 40 ÷ π) = 7.1 mm (round up for safety).
• Strain: a 4.0 m wire stretches 2.5 mm: ε = 2.5 ÷ 4000 = 0.000 63 (no units). Put both lengths in the same units first.
• Change in length: ΔL = ε × L = 0.0075 × 2500 mm = 19 mm.

Design challenge: climbing wall anchor

The brief. An outdoor centre in Aviemore needs a steel tie rod that will carry 12 kN without the stress going above 150 N/mm².

1. Calculate the minimum cross-sectional area.
2. Calculate the minimum diameter of the rod.
3. The rod is 400 mm long and stretches 0.3 mm. Calculate the strain.
4. Justify the choice of material.

Stress–strain graphs and properties HIGHER

• Limit of proportionality: up to here stress is proportional to strain (straight line).
• Elastic limit: below this the material returns to its original length when the load is removed (elastic).
• Yield point: the material starts to stretch permanently (plastic deformation).
• Ultimate tensile stress (UTS): the maximum stress before it necks and breaks.
• Properties: brittle (breaks with little deformation, e.g. cast iron, glass); ductile (can be drawn out into wire); malleable (can be hammered or pressed into shape); elastic; plastic; strength.

Young's modulus, factor of safety and strain energy HIGHER

• Young's modulus E = σ ÷ ε (N/mm²): the stiffness of a material, from the straight part of the graph.
• Factor of safety = UTS ÷ working stress. Working stress = UTS ÷ FoS. Engineers use a FoS to allow for flaws in materials, unexpected loads, corrosion and wear, and inaccuracy in calculations.
• Strain energy E_s = ½ F x (J), with x the extension in m.
• Finding a size: area = force ÷ working stress, then d = √(4A ÷ π). For a tube, outer area = load-bearing area + inner area, then outer diameter, then thickness = (D − d) ÷ 2.

Why structures fail HIGHER

• Overloading: a load bigger than the design load raises the stress above the material's strength.
• Material or joint failure: flaws in the material, poor welds or failed fixings.
• Fatigue: repeated loading and unloading (vibration, traffic, wind) makes cracks grow, so a part can fail at a stress well below its UTS.
• A factor of safety allows for all of these. Typical values are higher where failure would put lives at risk (lifts, cranes) or loads are uncertain.
• Material data such as UTS and Young's modulus is read from tables in the data booklet: use the right row and check the units.

Strain gauges HIGHER

A strain gauge's resistance increases when stretched. It is bonded to a structure and wired into a bridge with a second, passive (dummy) gauge that isn't strained. The passive gauge cancels resistance changes caused by temperature, so only strain is measured. The bridge output feeds a difference amplifier.

Key terms

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