  In this paper, we take a fresh look at using spectral analysis for assessing
locking phenomena in finite element formulations. We propose to "measure"
locking by comparing the difference between eigenvalue and mode error curves
computed on coarse meshes with "asymptotic" error curves computed on "overkill"
meshes, both plotted with respect to the normalized mode number. To demonstrate
the intimate relation between membrane locking and spectral accuracy, we focus
on the example of a circular ring discretized with isogeometric curved
Euler-Bernoulli beam elements. We show that the
transverse-displacement-dominating modes are locking-prone, while the
circumferential-displacement-dominating modes are naturally locking-free. We
use eigenvalue and mode errors to assess five isogeometric finite element
formulations in terms of their locking-related efficiency: the
displacement-based formulation with full and reduced integration and three
locking-free formulations based on the B-bar, discrete strain gap and
Hellinger-Reissner methods. Our study shows that spectral analysis uncovers
locking-related effects across the spectrum of eigenvalues and eigenmodes,
rigorously characterizing membrane locking in the displacement-based
formulation and unlocking in the locking-free formulations.
