The problem of the minimum reinforcement condition in fibre-reinforced and hybrid-reinforced concrete flexural elements is addressed in the framework of fracture mechanics by means of the Updated Bridged Crack Model (UBCM). The model describes the crack propagation process occurring in the critical cross-section of the reinforced member, by assuming the composite as a multiphase material, whereby the toughening contribution of the cementitious matrix and of the reinforcements are independently evaluated. The key-point of the discussion is that, when the influence of the matrix nonlinearities on the response is neglected, the minimum reinforcement condition is defined by a linear relationship between the critical values of two dimensionless numbers: (i) the bar- reinforcement brittleness number, NP , proportional to the steel-bar area percentage, ρ; (ii) the fibre- reinforcement brittleness number, NP,f, proportional to the fibre volume fraction, Vf. The model is applied to several experimental campaigns of the literature, in order to assess its suitability in the minimum reinforcement design of reinforced members in a unified fracture mechanics-based framework.

中文翻译：

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纤维增强和混合钢筋混凝土梁最小配筋设计的断裂力学方法

纤维增强和混合增强混凝土弯曲元件的最小配筋条件问题通过更新桥接裂纹模型（UBCM）在断裂力学框架中得到解决。该模型通过假设复合材料为多相材料，描述了在增强构件的临界截面中发生的裂纹扩展过程，从而独立评估了水泥基体和增强材料的增韧贡献。讨论的要点是，当忽略矩阵非线性对响应的影响时，最小配筋条件由两个无量纲数的临界值之间的线性关系定义：（i）钢筋脆性数量，N磷，与钢筋面积百分比ρ成正比； (ii) 纤维增强脆性值，NP,f，与纤维体积分数 V 成正比F。该模型应用于文献中的多个实验活动，以评估其在基于统一断裂力学的框架中加固构件最小加固设计的适用性。