FE Mechanical: Mechanics of Materials

Mechanics of Materials

The FE Mechanical mechanics of materials items test stress and strain, axial deformation, torsion, bending, beam deflection, combined loading, failure theories, and buckling. You must track geometry properties such as area, centroid, and moment of inertia, and apply the correct formula for the member and loading.

Core rules and concepts

• Normal stress σ = F/A; normal strain ε = δ/L; Hooke's law σ = Eε. Axial deformation δ = FL/(AE).
• Poisson's ratio ν = -ε_lateral/ε_longitudinal.
• Torsion: τ = Tρ/J, maximum at the outer surface; angle of twist φ = TL/(JG). For a solid circular shaft J = πd⁴/32; for a hollow shaft J = π(D⁴ - d⁴)/32.
• Bending: σ = My/I, with y measured from the neutral axis. For a rectangle I = bh³/12. Maximum bending stress occurs at the farthest fiber.
• Beam shear stress: τ = VQ/(It). Average shear stress is V/A.
• Principal stresses in plane stress: σ1,2 = (σx + σy)/2 ± sqrt(((σx - σy)/2)² + τxy²). Mohr's circle gives the same result graphically.
• Euler buckling: P_cr = π²EI/(KL)². K depends on end conditions; the formula applies to long, slender columns.

Worked example

A solid circular shaft has diameter 40 mm and transmits a torque of 500 N·m. Find the maximum shear stress. Radius c = 0.02 m. J = π(0.04)⁴/32 = 2.513 × 10⁻⁷ m⁴. Then τ = Tc/J = (500)(0.02)/(2.513 × 10⁻⁷) = 39.8 × 10⁶ Pa = 39.8 MPa.

Traps that cost marks

• Using diameter instead of radius in torsion or bending formulas.
• Using the solid-shaft J for a hollow shaft.
• Measuring y from the wrong axis in bending stress.
• Mixing MPa, Pa, and N/mm² without converting.
• Applying Euler buckling to a short column.
• Superimposing stresses without checking linear elasticity and the same load direction.