On the analytical approach to the linear analysis of an arbitrarily curved spatial Bernoulli-Euler beam

G. Radenkovic, A. Borkovic*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The equilibrium and kinematic equations of an arbitrarily curved spatial Bernoulli–Euler beam are derived with respect to a parametric coordinate and compared with those of the Timoshenko beam. It is shown that the beam analogy follows from the fact that the left-hand side in all the four sets of beam equations are the covariant derivatives of unknown vector. Furthermore, an elegant primal form of the equilibrium equations is composed. No additional assumptions, besides those of the linear Bernoulli–Euler theory, are introduced, which makes the theory ideally suited for the analytical assessment of big-curvature beams. The curvature change is derived with respect to both convective and material/spatial coordinates, and some aspects of its definition are discussed. Additionally, the stiffness matrix of an arbitrarily curved spatial beam is calculated with the flexibility approach utilizing the relative coordinate system. The numerical analysis of the carefully selected set of examples proved that the present analytical formulation can deliver valid benchmark results for testing of the purely numeric methods.
Original languageEnglish
Pages (from-to)1603-1624
Number of pages22
JournalApplied Mathematical Modelling
Issue number2
Publication statusPublished - Jan 2020
Externally publishedYes


  • Curved spatial beam
  • Beam equations
  • Analytical solution
  • Parametric coordinate
  • Curvature change
  • Big-curvature beam

ASJC Scopus subject areas

  • Applied Mathematics
  • Modelling and Simulation

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