Geometrically exact isogeometric Bernoulli–Euler beam based on the Frenet–Serret frame

Aleksandar Borković*, Michael Helmut Gfrerer, Benjamin Marussig

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A novel geometrically exact model of the spatially curved Bernoulli–Euler beam is developed. The formulation utilizes the Frenet–Serret frame as the reference for updating the orientation of a cross section. The weak form is consistently derived and linearized, including the contributions from kinematic constraints and configuration-dependent load. The nonlinear terms with respect to the cross-sectional coordinates are strictly considered, and the obtained constitutive model is scrutinized. The main features of the formulation are invariance with respect to the rigid-body motion, path-independence, and improved accuracy for strongly curved beams. A new reduced beam model is conceived as a special case, by omitting the rotational DOF. Although rotation-free, the reduced model includes the torsion of the beam axis, which allows simulations of spatial beams that are predominantly bent with respect to the binormal. The applicability of the obtained isogeometric finite element is verified via a set of standard academic benchmark examples. The formulation is able to accurately model strongly curved Bernoulli–Euler beams that have well-defined Frenet–Serret frames.

Original languageEnglish
Article number115848
JournalComputer Methods in Applied Mechanics and Engineering
Publication statusPublished - 15 Feb 2023


  • Frenet–Serret frame
  • Geometrically exact analysis
  • Rotation-free beam
  • Spatial Bernoulli–Euler beam
  • Strongly curved beam

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mechanics


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