Euler-Bernoulli beam

Euler-Bernoulli beam theory relates a beam’s bending to its curvature through M = E I (d2w/dx2), where M is bending moment, E the elastic modulus, I the second moment of area, and w the deflection, assuming plane sections stay plane and perpendicular to the neutral axis. In naval architecture it is the basis of classical hull-girder analysis, treating the ship as a free-free beam loaded by the difference between weight and buoyancy along its length to give the longitudinal bending moment and the resulting deck and keel stresses. It neglects shear deformation, so for short deep girders or fine local checks Timoshenko theory or finite elements are used instead.