Symmetric Catenary of a Uniform Elastic Cable of Neo-Hookean Material

The Cartesian equations of the symmetric catenary of a uniform nonlinear elastic cable of neo-Hookean material are determined in parametric form. The catenary is plotted here in nondimensional terms for a number of values of the two constants on which it depends: One being that of an inextensible catenary and the other specifying the thermoelastic properties of the neo-Hookean material. The resolution of the equations to find these values from the geometrical and material data requires much more sophisticated numerical algorithms than do the inextensible catenary and the elastic one based on Hookeʹs law, but this drawback is circumvented by adjusting an accurate analytical approximation that evens the difficulty of the two problems and allows the symmetric catenary of a neo-Hookean cable to be determined and drawn by using any graphing program endowed with numerical algorithms for basic calculus. A closed form of the Cartesian equations of the catenary is also provided on the basis of this analytical approximation.