Applied Mechanics and Strength of Materials
Applied Mechanics and Strength of Materials
The exam tests equilibrium of forces (Lami's theorem, moments), simple stress and strain, and shear force and bending moment values.
Rules
- Lami's theorem applies to three coplanar concurrent forces in equilibrium: each force divided by the sine of the angle between the other two is constant.
- Equilibrium: ΣF = 0 and ΣM = 0 about any point.
- Stress σ = P/A (N/mm² = MPa); strain ε = δ/L (dimensionless); E = σ/ε.
- Hooke's law holds up to the proportional limit; Poisson's ratio μ = lateral strain / longitudinal strain, about 0.3 for steel.
- Simply supported beam, central point load W over span L: each reaction = W/2, maximum bending moment = WL/4 at mid-span. For a uniformly distributed load w per metre, maximum BM = wL²/8.
Worked example
A 20 mm diameter steel rod, 2 m long, carries 50 kN. Take E = 2 × 10⁵ N/mm². A = π/4 × 20² = 314.2 mm²; σ = 50 000 / 314.2 = 159 N/mm². ε = 159 / (2 × 10⁵) = 7.95 × 10⁻⁴; extension δ = εL = 7.95 × 10⁻⁴ × 2000 = 1.59 mm.
Traps
- Using diameter where radius belongs: A = πd²/4.
- In Lami's theorem taking the angle between two forces instead of the angle opposite the third.
- Giving strain a unit - it has none.
- Assuming equal reactions when the point load is off-centre; take moments about one support instead.
- Ignoring the sagging/hogging sign convention in bending moment diagrams.