The structural ultimate load-bearing capacity plays an influential role in engineering applications. Melan’s static shakedown theorem offers a valuable approach for predicting the lower bound of shakedown loading factors and providing a safer shakedown domain when the structures are subjected to cyclic variable loads. However, the associated nonlinear mathematical programming is plagued by substantial computational expenses due to excessive design variables and constraints. Inspired by the data-driven FEM Cluster-based Analysis (FCA) [44–47] for predicting nonlinear effective properties of RVE of heterogeneous materials very efficiently, this paper introduces a novel FEM cluster-based basis reduction method to predict the shakedown domain of structures. Firstly, the FEM elements of discretized structures are grouped into several clusters using the elastic strain tensor under different load vertex cases, which differs from the orthogonal loading conditions for clustering RVE. In this way, similar mechanical behavior in each cluster of the structures is expected in future loading. Then, the cluster eigenstrain-driven algorithm is employed to construct the self-equilibrium stress (SES) basis vectors, which should satisfy the equilibrium equation and statical boundary condition. Furthermore, the essential time-independent beneficial residual stress in the static shakedown analysis is represented as a linear combination of the constructed SES basis vectors based on the basis reduction method, which can reduce a large number of time-independent residual stress to several linear combination coefficients. In addition, the reduced-order model (ROM) is constructed by the cluster SES, which are volume averaged stresses of the element SES basis vectors within each cluster. Based on the ROM, a constraint reduction strategy (CRS) is introduced to selectively remove stress constraints significantly below yield stress from the enormous element-wise yield constraint set. These innovations decrease the number of design variables and nonlinear constraints in the shakedown optimization, thus significantly enhancing computational efficiency. Several numerical examples illustrate the effectiveness and efficiency of the proposed shakedown analysis method of FCA.

中文翻译：

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基于集群的有限元元基折减方法，用于结构在可变载荷下的极限承载力预测


结构极限承载能力在工程应用中起着影响作用。Melan 静力稳定定理为预测稳定载荷因子的下限提供了一种有价值的方法，并在结构承受循环可变载荷时提供更安全的稳定域。然而，由于过多的设计变量和约束，相关的非线性数学规划受到大量计算费用的困扰。受数据驱动的FEM Cluster-based Analysis （FCA） [44\u201247]用于非常有效地预测异质材料RVE的非线性有效特性的启发，本文引入了一种新颖的基于FEM簇的基约简方法来预测结构的稳定域。首先，利用弹性应变张量将离散化结构的有限元元在不同载荷顶点情况下分为几个簇，这与聚类 RVE 的正交载荷条件不同。这样，在未来的载荷中，预计每个结构集群的机械行为相似。然后，采用聚类特征应变驱动算法构建自平衡应力 （SES） 基向量，该基向量应满足平衡方程和静态边界条件。此外，静态稳定分析中必不可少的时间无关有益残余应力表示为基于基约简法构建的 SES 基向量的线性组合，可以将大量与时间无关的残余应力简化为多个线性组合系数。此外，降阶模型 （ROM） 由簇 SES 构建，它是每个簇内元素 SES 基向量的体积平均应力。 基于 ROM，引入了一种约束缩减策略 （CRS），以选择性地从巨大的单元级屈服约束集中明显低于屈服应力的应力约束。这些创新减少了 Shakedown 优化中的设计变量和非线性约束的数量，从而显著提高了计算效率。几个数值示例说明了所提出的 FCA 安定分析方法的有效性和效率。