Mechanical Systems and Structural Design: Statics, Strength of Materials, Mechanisms and Machine Elements
This is the most heavily weighted calculation block of the PAU paper: free-body analysis of beams, stress checks on sections, and the kinematics of gears, belts and linkages. Marks are awarded for the diagram and the units as much as for the final number.
Statics and internal forces
- Equilibrium: ΣFx = 0, ΣFy = 0, ΣM = 0. A pin gives two reaction components, a roller one, an embedment three (plus a moment).
- Work along the beam using the sign conventions for axial force N, shear V and bending moment M; M is maximum where V = 0.
- Standard results: simply supported beam with central load P and span L gives Mmax = P·L/4; with uniformly distributed load q it gives Mmax = q·L²/8.
Strength of materials
- Bending: σ = M·y/I, with I = b·h³/12 for a rectangle and I = π·d⁴/64 for a circle.
- Torsion: τ = T·r/J, with J = π·d⁴/32 for a solid shaft.
- Buckling: Euler load Pcr = π²·E·I/Le², where Le = L for pinned–pinned and 0.5L for built-in ends.
- The safety factor is n = σy/σworking or σr/σworking; it must exceed 1 with a margin set by the brief.
Mechanisms and elements
- Gear train ratio i = z₂/z₁ = n₁/n₂ = M₂/M₁; power is conserved apart from losses.
- Four-bar linkage: Grashof condition s + l ≤ p + q guarantees at least one full rotation.
- Belt drives slip and are used for long centre distances; chains and gears transmit more torque without slip.
Worked example
A 4 m simply supported beam carries a 10 kN central point load. Reactions are 5 kN each and Mmax = P·L/4 = 10·4/4 = 10 kN·m. For a 100 mm × 200 mm rectangular section, I = 0.1·0.2³/12 = 6.67×10⁻⁵ m⁴ and y = 0.1 m. Then σ = M·y/I = 10 000 × 0.1 / 6.67×10⁻⁵ = 15 MPa. With σy = 120 MPa the safety factor is 8.
Traps that cost marks
- Omitting self-weight or a reaction when the beam is built in at one end.
- Using half the load as Mmax when the load is uniformly distributed.
- Taking y from the wrong fibre in an asymmetric section.
- Mixing metres and millimetres inside I, which throws the result off by orders of magnitude.