  In this paper we derive a variational formulation for a linear curved beam
which is natively expressed in global Cartesian coordinates. During derivation
the beam midline is assumed to be implicitly described by a vector distance
function which eliminates the need for local coordinates. The only geometrical
information appearing in the final expressions for the governing equations is
the tangential direction, and thus there is no need to introduce normal
directions along the curve. 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 literature are typically derived in the Frenet frame defined by
the geometry of the beam midline. While it is intuitive to formulate curved
beam models in these local coordinates, the Frenet frame suffers from ambiguity
and sudden changes of orientation in straight sections of the beam.
  Based on the variational formulation we implement finite element models using
global Cartesian degrees of freedom and discuss curvature coupling effects and
locking. Numerical comparisons with classical solutions for both straight and
curved cantilever beams under a tip load are given, as well as numerical
examples illustrating curvature coupling effects.
