Generalized Euler – Bernoulli Beam Theory with Return Potential

In 1749, L. Euler, building on the ideas of Jakob and Daniel Bernoulli, formulated beam theory in an exact formulation with the hypothesis of plane sections. Later, P.-S. Girard linearized the curvature, simplifying the derivation of analytical solutions, and B.P.E. Clapeyron expressed it in terms of derivatives of the deflection function. As a result, the Euler ± Bernoulli model split into two classes: the linear (classical) formulation with Girard’s curvature and the so-called "exact" geometrically nonlinear formulation with Euler ± Clapeyron curvature. This work demonstrates that the class of geometrically nonlinear problems is a methodological fallacy. The function y(x), traditionally interpreted as the deflection function, is in fact a mapping of a topological space onto a plane in the Cartesian system and describes the distance from the topological abscissa to the neutral axis of the deformed beam. The initial segment of the topological abscissa is nearly rectilinear, which justifies the use of the classical model for small deformations. However, for large deformations, even the "exact" curvature formula proves to be incorrect. A new force component ³ the restoring potential P ³ is introduced, which closes the system of equations and links the rotation angles to the external transverse load. The generalization of beam theory in rectilinear and curvilinear (topological) coordinate systems using a generalized variable i has revealed a deep connection between these computational spaces and enables the reconstruction of the exact geometry of the deformed beam based on classical Euler ± Bernoulli solutions. Thus, this work resolves the fundamental problem posed by Jakob Bernoulli (1694), establishing a generalized beam theory in which linearity and the hypothesis of plane sections are preserved throughout the entire range of elastic behavior.