Analytical modelling of Kirschner wires in Ilizarov circular external fixator as pretensioned slender beams

Transfixing thin Kirschner wires (K-wires) are the key components of the Ilizarov fixator regarding its axial stiffness, which affects the mechanobiological environment in which bone is healed. Mechanically speaking, K-wires are slender beams that are axially tensioned, then fixed and transversely loaded. The existing solutions to such a problem either do not accommodate any axial loading prior to transverse loading, or do not account for the change in the axial load (reaction) due to transverse loading. Their applicability is also limited vis-à-vis applied loads and beam dimensions. This work seeks to address those problems by providing a mathematical formulation for a pretensioned slender beam that accounts for the change in the beam tension due to lateral loading. Central loading of a pretensioned beam was studied and new polynomial equations have been derived, the roots of which yield the final tension for a (i) long, slender and heavily loaded beam and (ii) relatively thicker beam subjected to a lower load. Results were produced and discussed for the specific application of pretensioned K-wires in circular (ring) external fixators in orthopaedics (such as Ilizarov's), which were checked (validated) via two- and three-dimensional finite-element analyses.