A family of spatial beam finite elements based on the fixed pole approach is developed and presented in this work . The family consists of three different interpolation options which arise as a consequence of the fact that the virtual position vector has been interpolated in a non linear manner and depends on the actual position vector not only at the nodal points but also at the integration points . All three formulations turn out to be non invariant and path dependent . In two of them however this problem may be easily solved by interpolating the total local rotations which also makes the procedure more robust in handling large load increments . The topic of this paper is the development of a family of spatial beam finite elements based on the geometrically exact beam theory given by Reissner for 2D beams and extended to 3D beams by Simo . Implementation of this theory in a finite element framework becomes non trivial because of the properties of rotations in 3D . These are defined by orientation matrices elements of a special orthogonal group SO which is also a Lie group . By definition a Lie group is also a differentiable manifold and in order for the solution to stay on the manifold we must acknowledge existence of the so called exponential map a specific operation between a Lie group and a related Lie algebra . This theory was first implemented by Simo and Vu Quoc interpolating the infinitesimal rotations the so called spin vectors . Similar results can be obtained interpolating the incremental rotational changes as proposed by Cardona and G radin or the total rotations as given in . The additive interpolation of any of these rotational fields was shown to be the source of non invariance of strain measures with respect to a rigid motion while Romero thoroughly examined and discussed how different interpolations affect the invariance properties . An invariant formulation was proposed in which not only solved the problem of non invariance of the strain measures but also the problem of path dependence which also existed in some of those formulations by interpolating only the relative rotations between specific nodes i.e . the rotations from which the rigid rotation is removed . In a different vein Betsch and Steinmann have proposed a formulation where interpolation of the rotational degrees of freedom is circumvented by reparametrising the orientation matrix using and interpolating its base triad while Zupan and Saje developed a finite element where the strain vectors are interpolated . In this work we want to shed more light on the fascinating and highly promising fixed pole concept and investigate alternative options of its implementation into the geometrically exact 3D beam theory to those currently available . In the context of this theory the fixed pole concept was first introduced by Borri and Bottasso in 1994 and thoroughly researched in a series of subsequent papers . In the context of simple material and polar material elastomechanics the concept. 


Highlights:
We analyse fixed pole approach in geometrically exact beam theory. We propose a new formulation using standard degrees of freedom. We assess the new formulation with respect to invariance and path dependence.