Stress = force over area
Stress (sigma) is how heavily a force presses on each unit of area, not just the size of the force: stress = force / cross-sectional area. A given force on a small area gives HIGH stress; a larger area lowers it. Units: N/mm2 (force in N, area in mm2) or pascals (N/m2). Worked example: 4000 N / 20 mm2 = 200 N/mm2. For a round bar the area is pi times radius squared — use the radius, not the diameter.
Strain = extension over length
Strain (epsilon) is how much a member stretches relative to its starting length, not the raw distance moved: strain = change in length / original length. Dividing one length by another, the units cancel, so strain is a pure number with no units — the most-tested fact about it. Both lengths must be in the SAME unit first. Worked example: a rod 2000 mm long that stretches 3 mm has strain 3 / 2000 = 0.0015.
Factor of safety is a margin
The factor of safety is the designer's margin between a structure's real strength and its working load: factor of safety = failure (ultimate) stress / working stress. It is always GREATER than 1 — a value of 1 means loaded right to failure, no margin. Worked example: a cable breaking at 12000 N but carrying 3000 N has a factor of 12000 / 3000 = 4. A larger factor is used where failure is dangerous or loads are uncertain.
Drawn from real examiner reports.
Area from radius, not diameter
For a round bar or wire the cross-sectional area is pi times radius squared. The costliest error is to put the DIAMETER into pi r squared without halving it. Because the radius is squared, using the diameter makes the area four times too big and the stress four times too small. Convert diameter to radius (divide by two) BEFORE squaring.
s23 P42 Q10c(ii)
Factor of safety needs a real reason
Asked why a structure needs a factor of safety, many write "so it is safe" or "to make it strong" and score little. Examiners want the idea of a margin: the structure can carry MORE than its working load, so unexpected loads, hidden flaws, corrosion and variable workmanship cannot push it to failure. Name what the margin protects against.
w23 P42/P43 Q9b
Stress vs strain confusion
The key definition trap. Stress = force / area, units N/mm2. Strain = change in length / original length, a pure ratio with NO units. Candidates swap the two, write "strain = force / area", or give strain a unit. Remember: stress has a force and an area; strain has two lengths, so its units cancel.
Mixing millimetres and metres
A calculation slips when units are mixed mid-way. Keep force in newtons and area in mm2 throughout and stress comes out in N/mm2; switch to metres for the area and the figure is out by a factor of a million. For strain, both lengths must share a unit before dividing. Decide the units at the start and state the unit (or its absence) with the answer.
s23/w23 P42/P43 key message
Factor of safety divided the wrong way
Factor of safety = failure (ultimate) load / working load, so it is always GREATER than 1. Dividing working load by failure load gives a fraction below 1, not a usable factor. The rearrangement traps too: maximum working stress = failure stress / factor of safety, so a bigger factor gives a SMALLER allowed stress. If your factor is below 1, you divided upside down.
Find the working stress first
To find a factor of safety, get the working stress first (load / area), then divide the ultimate stress by it. Candidates divide ultimate stress by the raw load in newtons, mixing a stress with a force. Do the stress step, THEN the ratio: working stress = load / area, factor of safety = ultimate stress / working stress.
Strain has units like stress
The wrong belief: that strain carries a unit such as newtons or N/mm2. In fact strain = change in length / original length divides one length by another of the same kind, so the units cancel and strain is a pure number (for example 0.0015, or 0.15 per cent). It is STRESS, not strain, that has a unit (N/mm2), because it divides a force by an area.
A factor of safety of 1 is safe
Two linked wrong beliefs. First, a factor of safety of 1 is NOT safe: it is loaded right to its failure point, no margin. A usable factor is always above 1. Second, bigger is not always better: a large factor is safer but makes it heavier and costlier, so the designer suits it to the risk — large where failure is dangerous, small where weight and cost matter.
Lay a calculation out in stages
Marked for METHOD, so work in stages: write the formula; work out any area first (convert diameter to radius); substitute; then state the answer with its unit (N/mm2 for stress, none for a ratio). Method is banked at each stage.
Rearrange before you substitute
When the unknown is not the subject, rearrange FIRST, then substitute. For allowed working stress: working stress = failure stress / factor of safety. For the area needed: area = force / allowable stress. Writing it out first earns the method mark.
State the right unit
Units cost marks as often as arithmetic. Quote N/mm2 for a stress (N over mm2), and no unit for a strain or factor of safety, since each is a ratio whose units cancel. A strain written "0.0015 N/mm2" is wrong on units.
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