Folding the Carpenter’s Tape: Boundary Layer Effects

Abstract

Abstract The “carpenter’s measuring tape” is a thin spring-steel strip, preformed to a curved cross section of radius R, which is straight when being used for measuring. Under bending moments, it forms a localized hinge, in which the transverse curvature is suppressed, and the longitudinal radius r is approximately equal to R. Rimrott made a simple strain energy analysis of the hinge region for isotropic material, which predicted that r = R. Both experimental observations and finite element computations show that ξ = r/R > 1, where the value of ξ exceeds unity by up to 15%, depending on whether the tape is bent in “equal-sense” or “opposite-sense” curvature; ξ varies linearly with Poisson’s ratio in both cases. We make a minor change to Rimrott’s analysis by introducing a boundary layer, in order better to satisfy the physical conditions at the free edges; this successfully accounts for the observed behavior of the tape.