Derivation of the Euler-Bernoulli Beam Equation EI·y″ = M(x)
From plane sections to the moment–curvature relation EI·y″ = M(x), step by step.
Derivation
Derivation of the Euler Critical Buckling Load Pcr = π²EI/L²
From equilibrium on the deflected geometry to the Euler critical load Pcr = π²EI/L², step by step.
Derivation of the Curvature Formula κ = y″/(1+y′²)^(3/2)
From the geometric definition of curvature to κ = y″/(1+y′²)^(3/2), and when it reduces to y″.
Derivation of Cantilever Tip Deflection δ = PL³/(3EI)
From the moment–curvature relation to the cantilever tip deflection δ = PL³/(3EI), step by step.