Mechanics of Materials and Structural Analysis
Core content
Normal stress σ = P/A; shear stress τ = V/A; strain ε = δ/L; Hooke's law σ = Eε with Poisson's ratio ν = −εlateral/εaxial. Torsion: τ = Tr/J and θ = TL/GJ. Bending: σ = My/I, maximum at the extreme fibre, with section modulus S = I/c giving σ = M/S. Memorise a few deflections, such as δ = 5wL⁴/384EI for a uniformly loaded simply supported beam and δ = PL³/48EI at midspan of a simply supported beam with a central point load. Euler buckling: Pcr = π²EI/(KL)², with K depending on end conditions (fixed–fixed 0.5, pinned–pinned 1.0, fixed–pinned 0.7, cantilever 2.0).
Traps: using diameter instead of radius in I or J; confusing the second moment of area I (bending) with the polar moment J (torsion); applying a static stress safety factor to a buckling or fatigue problem; forgetting stress concentrations Kt at holes, fillets and keyways; and using the wrong effective length factor in buckling.
Worked example. A 20 mm diameter steel rod carries 50 kN in tension. A = π(0.02)²/4 = 3.14 × 10⁻⁴ m², so σ = 50 000/3.14 × 10⁻⁴ ≈ 159 MPa. With a yield strength of 250 MPa the factor of safety is 250/159 ≈ 1.57, below the usual 2 for static design, so the rod would be upsized.
Rule of thumb: design stress = strength/factor of safety; use yield strength for ductile static loading and ultimate strength for brittle materials.