Axial Stress & Strain
σ = F/A and ε = σ/E, with an optional yield-strength safety check.
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The engineering
Force per area is the one-line summary of strength of materials, but the units are where careers stumble: newtons over square millimeters is conveniently MPa exactly, and mild steel yields around 250 MPa while its modulus is 200,000 MPa — that 800:1 gap is why elastic strains are always fractions of a percent and why 'it looks fine' means nothing.
The safety-factor row is the honest output. A computed SF of 1.1 on nominal loads is not a margin, it's a coin flip against real-world scatter in loads, material, and geometry; codes want 1.5–4 depending on how well you know your loads and how loudly the failure would be reported.
Where this math comes from
Hooke's 1678 law said force and extension are proportional, but it took 130 years to divide both sides by the right geometry: Thomas Young's 1807 lectures defined the modulus that bears his name, making stiffness a *material* property instead of a property of one particular bar. Augustin-Louis Cauchy then generalized everything in 1822 with the stress tensor — the moment 'stress' became a defined mathematical object rather than a metaphor.
The subject matured into engineering practice through the great strength-of-materials texts, above all Stepan Timoshenko's, which trained the generation that built the twentieth century. Every FEA package today is printing Cauchy's tensor onto Timoshenko's homework problems.
- 1678Robert HookeLinear elasticity — 'ut tensio, sic vis'.
- 1807Thomas YoungElastic modulus defined as a material property.
- 1822Augustin-Louis CauchyThe stress tensor — stress made rigorous.
- 1930Stepan Timoshenko'Strength of Materials' canonizes modern practice.
See the full timeline of the math behind every calculator →
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