  Membrane locking in finite element approximations of thin beams and shells
has remained an unresolved topic despite four decades of research. In this
article, we utilize Fourier analysis of the complete spectrum of natural
vibrations and propose a criterion to identify and evaluate the severity of
membrane locking. To demonstrate our approach, we utilize standard and mixed
Galerkin formulations applied to a circular Euler-Bernoulli ring discretized
using uniform, periodic B-splines. By analytically computing the discrete
Fourier operators, we obtain an exact representation of the normalized error
across the entire spectrum of eigenvalues. Our investigation addresses key
questions related to membrane locking, including mode susceptibility, the
influence of polynomial order, and the impact of shell/beam thickness and
radius of curvature. Furthermore, we compare the effectiveness of mixed and
standard Galerkin methods in mitigating locking. By providing insights into the
parameters affecting locking and introducing a criterion to evaluate its
severity, this research contributes to the development of improved numerical
methods for thin beams and shells.
