Axial Deformation (δ = PL/AE)

Also known as elongation under load · PL/AE

\delta = \frac{P L}{A E}

Worked example: 100 kN on 3 m x 1000 mm^2 steel → 1.5 mm — press Try an example to run it live, then adjust anything.

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Axial Deformation (δ = PL/AE) explained

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.

Axial Deformation (δ = PL/AE) formula

\delta = \frac{P L}{A E}

Where

$\delta$ = Elongation (m)
$P$ = Axial force (N)
$L$ = Original length (m)
$A$ = Cross-sectional area (m²)
$E$ = Young's modulus (kPa)

Missing one of these? Work it out first, then come back

Axial force — Normal (Axial) Stress, Combined Axial and Bending Stress
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

Part of this formula set

Stress, strain and the elastic constants

9 formulasNormal and shear stress, strain, axial deformation, and the four elastic constants E, G, K and ν that link them.