Resistance of Reinforced Concrete Columns Subjected to Axial Force and Bending

Abstract

The paper presents a method for determining the resistance of cross-sections of reinforced concrete (RC) columns subjected to the axial force and bending. It takes account of the effect of concrete softening in plastic range and the mean compressive strength of concrete fcm. Such members are frequently encountered in engineering practice (pillars, bridges, viaducts). The stress-strain relationship for concrete in compression for short term uniaxial loading is assumed according to Eurocode 2 for nonlinear analysis. This stress-strain relation adequately represents the behaviour of the concrete by introducing four parameters. For reinforcing steel characterized by yield stress fyk, linear-elastic model with hardening in plastic range is applied. In the derivation of the resistance of the cross-sections of columns under consideration the following assumptions are introduced:•plane cross-sections remain plane•elasto-plastic stress/strain relationships for concrete and reinforcing steel are used•the tensile strength of concrete is ignored•the ultimate strains for concrete and reinforcing steel are determined a priori.The resistance of the RC cross-section is reached when either ultimate compressive strain in concrete or ultimate tensile strain in steel is reached anywhere in that section. The analytical formulae for the resistance NRm relating to the axial force and MRm relating to the bending moment are derived by integrating the equilibrium equations of the cross-section, taking account of physical and geometrical relationships as well as the condition of the ultimate limit state. On the basis of a combinatorial approach, twelve possible forms of the stress distribution in the section are considered. Using the derived formulae the interaction curves with the values of the normalized, cross-sectional forces nRm = NRm /(b t fcm) and mRm = MRm /(b t2fcm) for the rectangular cross-section have been obtained (b, t – dimensions of the rectangle). The obtained formulae describe the cross-section under consideration in the phase of failure. Replacing the mean values fcm and fyk by the corresponding design values fcd and fyd one obtains formulae determining the design values of the normalized cross-sectional forces nR = NR /(b t fcd) and mR = NR /(b t2fcd). For presentation of the proposed deformation model numerical calculations have been performed. They are presented in the form of interaction diagrams for rectangular cross-sections. Each curve refers to the corresponding value of the reinforcement ratio. The maximum compressive strain in concrete is calculated at the extreme fibre in the compression zone of the section. The points located on the nRm axis are related to pure compression, while on the mRm axis – to pure bending. The occurrence of the tensile strains in the cross-section leads to the crack formation in the concrete. Moreover, these solutions have been compared with those based on the parabolic-rectangular diagram for concrete under compression and with those obtained experimentally by other authors. In a similar way one may obtain interaction diagrams for ring cross-sections. Based on this analysis conclusions are drawn concerning application possibilities of the proposed approach.