Elasticity in engineering

Engineers use these ideas to keep buildings, bridges and machines safe. The basic rule: the stress must stay below the yield strength, so that nothing is permanently bent. In practice they leave a big safety factor, often about ten, for wear, shocks and faults in the metal.

Example: a crane must lift 2 tonnes with a steel rope of yield strength 3 × 10⁸ Pa and a safety factor of 10. The rope needs A ≥ 10 × Mg/σ_y = 10 × 2000 × 9.8/(3 × 10⁸) ≈ 6.5 × 10⁻⁴ m², a radius of about 1.4 cm.

Such a rope is not one solid rod but many thin wires braided together, like a plait of hair. That makes it flexible and easy to wind, and one broken wire does not bring the load down.

A beam of length l, breadth b and depth d, loaded by W at its middle, sags by δ = Wl³/(4bd³Y):

• Doubling the depth d cuts the sag to 1/8, because of the d³.
• Doubling the breadth b only halves it.
• So beams are made deep, and from a material with a large Y.

But a very deep, thin beam tends to buckle sideways. The answer is the I-shaped girder: broad top and bottom flanges carry the load, and a thin upright web gives the depth, with much less metal and weight. You see them in railway bridges and building frames.

Pillars work under compression. A pillar with spread-out ends carries more load than one with rounded ends.

Even mountains obey these limits. The rock at the base of a mountain of height h is under a stress of about hρg. Rock begins to flow at about 3 × 10⁸ Pa, so a mountain can be at most about 10 km high. Everest, at 8.8 km, is close to that limit.