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.
Structural-Mechanics
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 Cantilever Tip Deflection δ = PL³/(3EI)
From the moment–curvature relation to the cantilever tip deflection δ = PL³/(3EI), step by step.
Geometric Instability — Snap-Through of the von Mises Truss
A von Mises truss snaps through under load even though its bars stay straight and elastic. A reduced 0D model shows the instability is purely geometric: the location of the limit point depends on geometry alone, not on the material.
Hybrid Joint Optimization — Analytical Pipeline (Croccolo 2012)
From torque requirement and materials to optimal hub ratio, coupling geometry, and design merit — six-step closed-form pipeline with three canonical scenarios.
Thick-Walled Cylinder Stress Analysis — Analytical Pipeline (Croccolo 2009)
From internal pressure and geometry to radial/hoop stress distribution — Lamé equations, pipeline, notebook, and numerical verification.