Axial Deformation (δ = PL/AE)
Also known as elongation under load · PL/AE
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Chain σ = P/A, ε = δ/L and E = σ/ε together and everything collapses into one line: δ = PL/AE. The product AE is the axial stiffness of the member, and dividing it by L gives the spring constant of a rod — a structural member really is just a very stiff spring. A 3 m steel hanger rod of 1000 mm² carrying 100 kN stretches δ = (100 000 × 3) ÷ (0.001 × 200 × 10⁹) = 0.0015 m, one and a half millimetres.
Two practical consequences. First, length is linear, so the elevator ropes at the bottom of a 400 m shaft stretch dramatically more than the same ropes near the top — the reason high-rise lifts need compensating ropes and re-levelling. Second, in a run of pipe supported by rods of different lengths, the long rods stretch more under the same load and quietly shed their share onto the short ones; in a redundant hanger array the load distributes by stiffness, not by fairness. If several segments are in series with different A, E or P, compute δ for each and add them.
- = Elongation
- = Axial force
- = Original length
- = Cross-sectional area
- = Young's modulus
- Elongation — Speed, Distance & Time, Displacement (Uniform Acceleration)
- Axial force — Normal (Axial) Stress, Newton's Second Law
- Original length — Normal Strain (ε = δ/L), Thermal Linear Expansion
- Cross-sectional area — Normal (Axial) Stress, Radius of Gyration (r = √(I/A))
- Young's modulus — Young's Modulus (E = σ/ε), Relation Between E, G and ν