Dual variables system analysis for Euler beam

Regarding displacement and internal force as dual variables, Euler beam problems can be steered to dual variables system. Based on eigenvalue problems of Hamiltonian dual equation of Euler beam, eigenvector expansion method and modal expansion method of Euler beam are deduced, and the unification of elastic wave problems and vibration problems is revealed. According to transfer form solution of Hamiltonian dual equation of Euler beam, equivalent stiffness and equivalent flexibility of beam end are proposed, elemental stiffness equation and the shape functions are inferred, and boundary integral equation and fundamental solutions are derived. In dual variables system, dynamic characteristics of elastic beam are more obvious, the analysis and calculation of beam system are intuitional and simple, and the intrinsic relationships among the analysis and solving methods of elastic beam are embodied profoundly.