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.

翻译：尽管经过四十年的研究，薄梁与薄壳有限元近似中的膜锁定问题仍未完全解决。本文利用自然振动全谱的傅里叶分析，提出了一种识别和评估膜锁定严重程度的判据。为验证该方法，我们采用标准伽辽金公式和混合伽辽金公式，对采用均匀周期B样条离散的圆形欧拉-伯努利环进行计算。通过解析计算离散傅里叶算子，我们获得了特征值全谱归一化误差的精确表示。本研究探讨了膜锁定的关键问题，包括模态敏感性、多项式阶数的影响、壳/梁厚度与曲率半径的作用。此外，我们比较了混合伽辽金法和标准伽辽金法在缓解锁定方面的有效性。通过揭示影响锁定的参数并引入评估其严重程度的判据，本研究为发展薄梁与薄壳的改进数值方法做出了贡献。