There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.

中文翻译：

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下限分析中的一类基本应力元素

使用有限元方法进行严格的下限分析的公式化存在主要限制。一旦应力场被离散化，屈服准则和平衡条件必须应用在有限数量的点上，以便在整个离散化结构的任何地方都满足它们。到目前为止，只有线性应力元素才能满足多种类型的载荷和结构条件的要求。然而，也有一些标准类型的问题，比如在均匀分布载荷下的板之一，下界定理的实现仍然是不可能的。本文首次证明了一类应力插值单元满足下界定理的所有要求。而且，多项式插值阶数没有上限。通过平面应变、平面应力和基尔霍夫板示例检查效率。对三维和其他结构条件的推广也很简单。因此，这种基于伯恩斯坦多项式的插值方案有望在未来的发展和应用中发挥重要作用。