Topological Beam Theory (TBT)

The idea of relating deformation to internal force dates back to Galileo (1638), who first examined the elongation of a bar under load, and to Hooke (1678), who formulated the fundamental law of linear elasticity. Building on this foundation, Jacob Bernoulli (1694) — one of the founders of strength-of-materials theory — was the first to pose the problem of the elastic curve, attempting to extend the laws of axial deformation to bending. However, the absence of a closed geometric relation between curvature and internal forces prevented him from completing a general theory. Euler (1744), developing Bernoulli’s ideas, introduced a variational principle based on the minimization of curvature, but employed two critical assumptions — constant horizontal projection and small rotations — which led to the classical linear Euler – Bernoulli beam theory. These ap-proximations removed axial deformation from the energy balance and introduced a hidden geometrically induced axial force not represented in the strain-energy functional. In this work, we propose the Topological Beam Theory (TBT) — the first geometrically exact model of bending formulated in the natural arc-length coordinate and employing the exact curvature definition of Huygens. The model incorporates axial deformation directly into the variational principle, introduces a topological curvature modifier 1/(1 + N/EA), and yields a closed system of equations for the rotation angle, axial force, and bending moment. Thus, the present study completes the conceptual line initiated by Galileo, Hooke, Bernoulli, and Euler: for the first time in over 330 years since Jacob Bernoulli posed the problem, we obtain a fully exact solution for the elastic curve that consistently accounts for both bending and axial deformation within a unified, energetically coherent topological model.

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