A closed-form correction for three-point bend testing of functionally graded beams, with application to endovascular devices

Abstract

Three-point bend testing offers simple, non-destructive quantification of the flexural rigidity of

a slender structure, but the single apparent stiffness that it returns misrepresents the true local properties if

flexural rigidity varies along the span. The error can be severe in structures with deliberately engineered

stiffness gradients, such as endovascular catheters and guidewires whose graded transition zones govern

clinical performance, and which are thus the regions most in need of accurate characterization. We solve the

Euler-Bernoulli bending of a linearly graded beam in closed form and show that the true midspan rigidity

exceeds the apparent value by a correction factor with the compact expansion

W = 1 + m²/10 + O(m⁴), where m is the dimensionless change in stiffness across the support span. The

leading measurement error term therefore grows quadratically with the local gradient. To recover m from

discrete, noisy data we estimate the local gradient by windowed linear regression with optional Gaussian

pre-smoothing, and identify the sampling and smoothing needed for stable inversion. On synthetic profiles

spanning the gradient range of real devices, the correction reduces transition-zone error from as much as 10%

to a fraction of a percent; applied to a commercial graded long sheath it removes a systematic transition-zone

bias of order 1%. We further distinguish where the correction helps and where standard three-point testing

already suffices, yielding a simple decision rule for the recovery of local properties from bending of any

functionally graded beam.