Strength of Materials and Theory of Machines
What you must command
Stress and strain (normal, shear, thermal), Hooke's law, elastic constants and their relations, bending and shear force diagrams, bending and torsion formulae, deflection, columns (Euler and Rankine), principal stresses and Mohr's circle, fatigue and stress concentration. In theory of machines: kinematic pairs, linkages, cams, gears, governors, flywheels, and balancing.
Traps that appear in questions
- Section modulus: for a rectangle, Z = bd²/6 about the centroidal axis; using bd²/3 (the plastic modulus) is wrong for elastic design.
- Torsion: τ/r = T/J = Gθ/L; polar moment for a solid shaft is πd⁴/32.
- Thermal stress arises only when expansion is restrained: σ = EαΔT.
- Euler vs. Rankine: Euler applies to long (slender) columns; Rankine–Gordon to intermediate ones.
- Mohr's circle: principal stresses are the extreme normal stresses; maximum shear equals the radius.
- Gear trains: velocity ratio is the product of driven teeth over driving teeth.
Worked rule of thumb
A simply supported beam of span L carries a central point load W: maximum bending moment = WL/4 at midspan, maximum deflection = WL³/(48EI). Example: W = 10 kN, L = 4 m, E = 200 GPa, I = 5×10⁻⁶ m⁴. M = 10 000 × 4/4 = 10 000 N·m; δ = 10 000 × 64/(48 × 200×10⁹ × 5×10⁻⁶) = 640 000/48 000 ≈ 0.0133 m, or 13.3 mm. Check units: N·m³ divided by N·m² gives metres.
For a flywheel, the energy fluctuation ΔE = I ω² C_s, where C_s is the coefficient of fluctuation of speed - a compact formula that answers many numerical questions.