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：摘要：尽管历经四十年研究，薄梁与壳有限元近似中的膜锁紧问题仍未得到解决。本文利用自然振动完整谱的傅里叶分析，提出了一种识别和评估膜锁紧严重程度的准则。为展示该方法，我们采用标准伽辽金格式和混合伽辽金格式，应用于以均匀周期B样条离散的圆形欧拉-伯努利环。通过解析计算离散傅里叶算子，我们获得了整个特征值谱上归一化误差的精确表示。本研究探讨了与膜锁紧相关的关键问题，包括模态敏感性、多项式阶数的影响，以及壳/梁厚度与曲率半径的作用。此外，我们比较了混合伽辽金法与标准伽辽金法在缓解锁紧方面的有效性。通过揭示影响锁紧的参数并引入评估其严重程度的准则，本研究为开发薄梁与壳的改进数值方法做出了贡献。