Mechanics of Materials — Stress & Strain

Stress — force divided by area

ELI5: the same pull is gentle on a thick rod and brutal on a thin wire. Stress σ = F / A measures the intensity: newtons per square metre, which engineers call pascals (and millions of them, MPa). It's the number you compare against what a material can take.

F_{axial} := 50 kN

= 50 kN a pull on a round steel rod

d_{rod} := 20 mm

= 20 mm rod diameter

A_{cross} = \pi \cdot \left(\frac{d_{rod}}{2}\right)^{2}

= 314.16 mm^2 cross-section area ≈ 314 mm²

\sigma _{axial} := F_{axial} / A_{cross} in MPa

= 159.15 MPa stress ≈ 159 MPa

✓ pass

abs(\sigma _{axial} - 159.15 MPa) < 0.5 MPa

force packed into the rod's area

Real-world hook: halve the diameter and the area drops four-fold — so stress quadruples. That square law is why a small nick or thin spot is where parts fail.