The elastic curve of a beam is governed by the differential equation relating curvature to bending moment, EI d²y/dx² = M (equivalently EI d⁴y/dx⁴ = -q), whose successive integration with appropriate boundary and continuity conditions yields the deflection y(x) and slope dy/dx at any point along the beam. Because this governing relation is linear in the applied loading for a beam in the linear-elastic range, the deflection produced by several simultaneous loads can be obtained by superposing the deflections produced by each load acting separately — the superposition method. This belongs to the theory of beam deflection within mechanics of materials/structural mechanics, extending the earlier derivation of the elastic-curve differential equation to different beam and loading configurations.
The elastic curve of a beam is governed by the differential equation relating curvature to bending moment, EI d²y/dx² = M (equivalently EI d⁴y/dx⁴ = -q), whose successive integration with appropriate…