E-02 — strain & deformation

stress tells you how hard. strain tells you how far.

δ = PL / AE
strain + deformation lab

pull the bar. watch it stretch.

Drag the section, slide the force, or swap materials — stiffer material means less strain for the same stress.

P 50 kips
b 4.0 in
d 4.0 in
deformation (in)
ε = 0.000108
live equation
σ = P/A = 3.13 ksi · ε = σ/E = 0.000108 δ = εL = 0.0129 in (amplified 1,500× in diagram) Elastic — remove the load and the bar springs back.
ε = Deformation per unit length
σ = Force per unit area
E = Material stiffness
δ = Total change in length: δ = εL
explained
Strain is how much the material stretches per unit length. Stiffer materials (higher E) deform less for the same stress. Multiply strain by the original length to get the actual elongation in inches.
key concepts
overview Every loaded member deforms — the question is how much

Steel deforms under load — always. The question is how much. δ = PL/AE tells you everything: more load means more stretch, a longer member stretches more, a bigger cross-section or stiffer material fights back. That's why swapping from lumber to steel for the same member shrinks the deformation by 17×. The deformation is elastic — release the load and the member springs back. Structural engineers only design in this range.

stress causes strain Deformation per unit length and Hooke's Law

When you apply stress, the material deforms. Strain is the measure of that deformation: ε = δ / L (change in length divided by original length). It's dimensionless — typically a very small number. Steel at yield: ε ≈ 0.0017 (0.17%). In the elastic range, stress and strain are proportional: σ = Eε (Hooke's Law), where E is the elastic modulus (29,000 ksi for steel). Remove the load and the deformation disappears.

the stress-strain curve From elastic to yield to fracture

The stress-strain curve tells the complete story of a material. Steel: linear elastic up to Fy (yield strength), then a flat yield plateau where it deforms without additional stress, then strain hardening until it reaches Fu (ultimate strength), then necking and fracture. This curve explains why steel is ductile — it can absorb enormous deformation after yielding before it breaks. Concrete's curve is very different: it peaks and drops quickly, which is why concrete is brittle without reinforcement.

the formula unpacked Four variables, four design levers in δ = PL/AE

Every term in δ = PL/AE is a design lever. Increase any numerator term, get more deformation. Increase any denominator term, get less. They're all linear — except that L compounds quickly because it governs both force and geometry.

P
Applied Force
The axial load in the numerator. Tension stretches the member; compression shortens it.
↑ P → ↑ δ (linear)
L
Member Length
Longer members deform more under the same load. A 20ft column stretches twice as much as a 10ft one.
↑ L → ↑ δ (linear)
A
Cross-Section Area
More area distributes the same load over more material, reducing stress and deformation proportionally.
↑ A → ↓ δ (linear)
E
Elastic Modulus
The material's resistance to deformation per unit stress. Steel (29,000 ksi) is ~17× stiffer than lumber (1,700 ksi).
↑ E → ↓ δ (linear)

Deformation in a member is one thing. Next: how much a beam bends under load — and why span dominates everything.

E-03: Deflection →