Computational Mechanics

, Volume 53, Issue 4, pp 611–623 | Cite as

Variational formulation of curved beams in global coordinates

  • Peter Hansbo
  • Mats G Larson
  • Karl LarssonEmail author
Original Paper


In this paper we derive a variational formulation for the static analysis of a linear curved beam natively expressed in global Cartesian coordinates. Using an implicit description of the beam midline during derivation we eliminate the need for local coordinates. The only geometrical information appearing in the final expressions for the governing equations is the tangential direction. As a consequence, zero or discontinuous curvature, for example at inflection points, pose no difficulty in this formulation. Kinematic assumptions encompassing both Timoshenko and Euler–Bernoulli beam theories are considered. With the exception of truly three-dimensional formulations, models for curved beams found in the literature are typically derived in the local Frenet frame. We implement finite element methods with global degrees of freedom and discuss curvature coupling effects and locking. Numerical comparisons with classical solutions for straight and curved cantilever beams under tip load are given, as well as numerical examples illustrating curvature coupling effects.


Curved beams Global coordinates Finite elements  Linear elasticity Vector distance function 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringJönköping UniversityJönköpingSweden
  2. 2.Department of Mathematics and Mathematical StatisticsUmeå UniversityUmeåSweden

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