On the Love strain form of naturally curved and twisted rods

Abstract

Love proposed in 1944 [A.E.H. Love, <i>A Treatise on the Mathematical Theory of Elasticity</i>. Dover Publications, New York, 1944] that the nonvanishing (linear) strain components of a naturally curved and twist spatial rod, whose centroidal axis is along x and cross-section is in yz plane, can be represented nicely in the form ϵxx = e₁ + zk₂ − yk₃ϵ_xy = e₂ − zk₁ϵ = e₃ + yk₁ where e₁, e₂, e₃ are the strain components at y = z = 0 and k₁, k₂, k₃ are the curvatures. Functions e₁, e₂, e₃, k₁, k₂, k₃ depend on x alone. Mottershead [J. E. Mottershead, “Finite elements for dynamical analysis for helical rods”, International Journal of Mechanical Sciences, 22, (1980), pp 252–283], Pearson and Wittrick [D. Pearson and W.H. Witrick “An exact solution for the vibration of helical springs using a Bernoulli-Euler Model”, International Journal for Mechanical Sciences, 28, (1986), pp 83–96], Leung [A.Y.T. Leung “Exact shape functions for helix- elements”, Finite Elements in Analysis and Design, 9, (1991), pp 23–32], and Tabarrok and Xiong [B. Tabarrok and Y. Xiong, “On the buckling equations for spatial rods”, International Journal for Mechanical Sciences, 31, (1980), pp 179–192] have made use of the Love form. We shall show that the Love form is not even valid for two-dimensionally curved beams when shear deformation is considered. The fact that the differential length ds at point P, on the cross-section with distance y, z away from the centroidal axis is different from the differential length dx at point S on the centroidal axis has been neglected. In fact ds = (1 − k₃y + k₂z)dx, where k; are initial curvatures, which contribute to the strain components of the first order of curvatures.