Abstract This paper proposes an extension of the Piecewise Rigid Displacement (PRD) method based on a new dual linear programming problem that minimises the complementary energy. Before, the PRD method had been applied to solve the kinematical problem for masonry-like structures composed of normal, rigid, no-tension (NRNT) material minimising the total potential energy. Specifically, the PRD method frames this minimum-energy search as a linear programming problem whose solutions are displacements and singular strain fields (cracks). Here, we show that the corresponding dual linear programming problem discretises the minimum of the complementary energy and returns, as solutions, admissible internal stress states compatible with the crack pattern obtained by solving the primal problem. Thus, these two minimum-energy criteria are dually connected, and their combined use allows coupling mechanisms and internal forces with settlements or homogeneous boundary displacements. This allows addressing different mechanical problems: equilibrium and stability of the reference configuration, effects of settlements, and mechanisms due to overloading (e.g. horizontal forces). Since the NRNT material represents the extension to continuum media of Heyman's material model, the PRD method offers an extremely fast, limit analysis-based, displacement approach that allows simultaneously finding mechanisms and compatible internal forces for any boundary condition, loads and geometry.

中文翻译：

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分段刚性位移 (PRD) 方法：一种基于极限分析的方法，通过两个双能量标准检测机构和内力

摘要 本文基于最小化互补能量的新对偶线性规划问题，提出了分段刚性位移 (PRD) 方法的扩展。之前，PRD 方法已被应用于解决由正常、刚性、无张力 (NRNT) 材料组成的砌体类结构的运动学问题，以最小化总势能。具体而言，PRD 方法将这种最小能量搜索框架化为线性规划问题，其解是位移和奇异应变场（裂缝）。在这里，我们展示了相应的对偶线性规划问题将互补能量的最小值离散化，并返回与通过求解原始问题获得的裂纹模式兼容的可容许内应力状态作为解。因此，这两个最小能量标准是双重连接的，它们的组合使用允许耦合机制和内力与沉降或均匀边界位移。这允许解决不同的机械问题：参考配置的平衡和稳定性、沉降的影响以及由于过载（例如水平力）引起的机制。由于 NRNT 材料代表 Heyman 材料模型对连续介质的扩展，PRD 方法提供了一种极其快速、基于极限分析的位移方法，允许同时找到任何边界条件、载荷和几何形状的机制和兼容的内力。参考配置的平衡和稳定性、沉降的影响以及由于过载（例如水平力）引起的机制。由于 NRNT 材料代表 Heyman 材料模型对连续介质的扩展，PRD 方法提供了一种极其快速、基于极限分析的位移方法，允许同时找到任何边界条件、载荷和几何形状的机制和兼容的内力。参考配置的平衡和稳定性、沉降的影响以及由于过载（例如水平力）引起的机制。由于 NRNT 材料代表 Heyman 材料模型对连续介质的扩展，PRD 方法提供了一种极其快速、基于极限分析的位移方法，允许同时找到任何边界条件、载荷和几何形状的机制和兼容的内力。