Topics and Insights on 1-D Tension Stiffening of an RC Member

One-dimensional (1-D) tension stiffening is a fundamental behavior of structural concrete. It refers to the composite uniaxial behavior of a slender, symmetric concrete member of constant cross-section, bonded to a single reinforcing bar (rebar) along its axis. The rebar is subjected to tension by a pair of axial tensile forces applied at its ends. Despite the apparent simplicity of this configuration, the problem represents a cornerstone in RC mechanics. During the loading process, cracks are formed at different cross-sections along the structural member at stages where the tensile stress in the concrete at these cross-sections reaches the concrete tensile strength level. Each crack formation reduces the overall axial stiffness of the RC member, while inducing stress and strain redistributions in both the concrete and the rebar. The interaction between the concrete and the rebar is governed by the bond-slip relationship along their interface, which plays a critical role in controlling the transfer of stresses, the development of strains and the evolution of cracking. Most existing analytical and numerical models addressing this problem are based on simplifying assumptions assuming constant (deterministic) material properties and are denoted herein as "deterministic models". Comparisons between analysis results of such models and experimental observations reveal substantial discrepancies in terms of the number of cracks, their spatial distribution, crack spacing, and the order of crack formation. Considering these inconsistencies, the present study postulates that the inherent variability of concrete properties, particularly its tensile strength, has a decisive influence on the structural response. To address this issue, the tensile strength of concrete is treated as a random variable characterized by the prescribed mean tensile strength and the coefficient of variation (CoV). The "stochastic analyses" with the variable tensile strength are conducted using an exact one-dimensional finite element formulation that explicitly accounts for discrete crack formation within the structural domain. These analyses yield results that differ markedly from those predicted by the deterministic approaches and exhibit characteristics that are in closer agreement with experimental evidence. These analyses indicate a more complex behavior of real structural members. It demonstrates that the CoV significantly influences the magnitude of cracking loads, crack locations, crack spacing, and the order of crack formation. The findings highlight the critical role of even slight material variability in tension stiffening behavior and justify the incorporation of concrete strength variability in tension stiffening modeling.