Mechanics of Materials and Engineering Materials
This area tests the stress–strain relationships and the geometric properties behind them; the materials portion tests structure–property relationships and failure modes.
Core rules
- Normal stress σ = P/A, strain ε = δ/L, and axial deformation δ = PL/(AE) for a uniform member.
- Torsion: τ = Tc/J and φ = TL/(JG), with J = πd⁴/32 for a solid round shaft.
- Bending: σ = Mc/I, with I = bd³/12 for a rectangle; maximum bending stress occurs at the extreme fiber.
- Principal stresses: σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²].
- Euler buckling: P_cr = π²EI/(KL)², valid for slender columns; K depends on end conditions.
- Fatigue: stress concentrations raise local stress; the endurance limit and S–N curves govern life.
- Materials: higher carbon content in steel raises hardness and lowers ductility; quenching and tempering trade strength against toughness; aluminum alloys strengthen by precipitation hardening; polymers creep viscoelastically.
Worked example
A 50 kN tensile load on a 500 mm² bar that is 2 m long, with E = 200 GPa. δ = (50,000 N)(2 m)/[(500×10⁻⁶ m²)(200×10⁹ Pa)] = 100,000/1.0×10⁸ = 0.001 m = 1.0 mm. Stress is 50,000/500×10⁻⁶ = 100 MPa, comfortably below typical steel yield.
Traps
- Using diameter where radius belongs in J or I.
- Forgetting that MPa = N/mm², not N/m².
- Applying Euler buckling to a stocky column that yields first.
- Confusing engineering stress, based on original area, with true stress.