Simply supported beam

Engine vs formula: 5wL⁴/384EI · Relative difference 0 %

Dense solver (6 free DOFs)

Residual 1.14e-15 (6 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 = 5wL⁴/(384EI); δmid = PL³/(48EI) (a = L/2); R₀ = Pb/L; R₁ = Pa/L; Mmax = Pab/L; b = L − a; R₀ = R₁ = wL/2; Mmax = wL²/8; δ(x≤a) = Pbx(L²−b²−x²)/(6LEI); δ(x≥a) = Pa(L−x)[L²−a²−(L−x)²]/(6LEI)

Worked examples

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

Uniform load: w = 10 kN/m → δmax = 0.010546875 m

Point load: P = 10 kN; a = 3 m → δmax = 0.0028125 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.