A Nonlinear Integral Equation for Finite-Deflection Analysis of an Euler–Bernoulli Beam with Variable Bending Stiffness

Document Type : Research Paper

Author
Independent Researcher, Zgorzelec, Poland
Abstract
This paper derives an equivalent nonlinear Volterra integral equation with Hammerstein-type nonlinearity for the analysis of a planar, inextensible Euler–Bernoulli beam with variable bending stiffness subjected to a conservative compressive dead load. The proposed integral equation is obtained by transforming the governing nonlinear differential equation into an equivalent integral form. Selected properties of the resulting nonlinear operator are analysed, and the formulation is shown to be consistent with the classical Euler buckling theory in the limit of small rotations. The proposed equation is solved numerically using the Nyström method and illustrated by a beam with locally degraded bending stiffness. The numerical example determines the compressive load corresponding to a prescribed finite-deflection equilibrium configuration. The results demonstrate that the proposed integral formulation provides a convenient framework for the analytical and numerical investigation of geometrically nonlinear equilibrium problems involving Euler–Bernoulli beams with variable bending stiffness.
Keywords
Subjects

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Available Online from 02 September 2026