| Symbol | Variable | SI Unit | Imperial Unit |
|---|---|---|---|
| σ | Normal Stress | MPa (N/mm²) | psi |
| τ | Shear Stress | MPa | psi |
| F | Axial Force | N or kN | lbf or kip |
| V | Shear Force | N or kN | lbf or kip |
| A | Cross-Sectional Area | mm² | in² |
| E | Young’s Modulus (Elastic Modulus) | GPa | ksi |
| ε | Axial Strain | mm/mm (dimensionless) | in/in |
| σ_y | Yield Strength | MPa | psi |
| FOS | Factor of Safety | Dimensionless | Dimensionless |
- Assumes uniform stress distribution across the cross-section — not valid near stress concentrations (holes, notches, fillets)
- Hooke’s Law applies only in the elastic region — below the yield strength of the material
- Does not account for buckling in slender compression members — check Euler buckling separately
- Shear stress calculated as average shear — actual maximum shear stress at the neutral axis is 1.5× average for rectangular sections
- Factor of safety values: FOS ≥ 1.5 typical for static loads, FOS ≥ 2.0–4.0 for dynamic/fatigue loads per ASME design codes
- All structural designs must be verified by a licensed engineer per applicable codes
What Is Stress and Strain?
Stress and strain are the two most fundamental concepts in mechanics of materials. Stress (σ) is the internal force per unit area within a material resulting from an applied external load. Strain (ε) is the resulting deformation — the change in dimension divided by the original dimension. Together, they describe how materials respond to loading and form the basis of all structural and mechanical design.
The Fundamental Formulas
Worked Examples
Example 1 — Steel Tension Rod
A 25mm diameter steel rod (A36, σ_y = 250 MPa) carries a tensile load of 80 kN. Calculate stress, strain, elongation over 2m, and factor of safety.
Example 2 — Bolt Shear in a Lap Joint
Two steel plates are connected with a single 16mm bolt in single shear. The joint carries 35 kN. Is the bolt adequate (Grade 8.8, τ_y = 480 MPa)?
Example 3 — Aluminum Column Compression
A 6061-T6 aluminum column (50mm × 50mm square, L = 0.5m) carries 200 kN compressive load. Check stress and elongation.
Real World Applications
Common Mistakes Engineers Make
Frequently Asked Questions
Yield strength (σ_y) is the stress at which a material begins to deform plastically — permanent deformation occurs beyond this point. Ultimate tensile strength (σ_u or UTS) is the maximum stress the material can withstand before necking begins, leading to fracture. Design calculations use yield strength for ductile materials (preventing permanent deformation) and UTS with a larger factor of safety for fracture prevention.
Factor of safety selection depends on: load certainty (known vs estimated), material variability, consequence of failure, and applicable design code. Typical values: static known loads on ductile materials = 1.5–2.0, static loads with uncertainty = 2.0–3.0, dynamic or fatigue loads = 3.0–4.0+, life-safety applications = per applicable code (AISC, ASME, etc.). Always follow the governing code for your application.
Poisson’s ratio (ν) describes lateral contraction when a material is axially loaded. A bar under tension not only elongates axially but contracts laterally by ε_lateral = −ν × ε_axial. For steel ν ≈ 0.30, aluminum ν ≈ 0.33. In multi-axial stress states, Poisson effects must be included in strain calculations using the generalized Hooke’s Law equations.