Cantilever beam

Engine vs formula: wL⁴/8EI · Relative difference 0 %

Dense solver (3 free DOFs)

Residual 8.37e-16 (3 free DOFs)

Assumptions

Linear elastic, small deflection, prismatic Euler–Bernoulli member. SI units: m, MPa, m⁴, kN, kN/m; E is converted to kN/m². Self-weight excluded. Elastic buckling only; not a code design check. Maxima are absolute values; x is measured from the left.

Closed form

δmax = PL³/(3EI) (a = L); δmax = Pa²(3L − a)/(6EI); δmax = wL⁴/(8EI); R = P / wL; |M₀| = Pa / (wL²/2)

Worked examples

L = 6 m; E = 200000 MPa; I = 0.00008 m⁴; EI = 16000 kN·m²

Uniform load: w = 10 kN/m → δmax = 0.10125 m

Point load: P = 10 kN; a = 6 m → δmax = 0.045 m

References

Timoshenko & Gere, Theory of Elastic Stability, 2 ed. (1961); Young & Budynas, Roark’s Formulas for Stress and Strain, 7 ed. (2002), Chapter 8.