The deformation of reinforced concrete columns and beams is controlled by the variation of the flexural rigidity (EI) both along the member and with applied loads and time. Currently, the moment-curvature (M/χ) approach is used to quantify EI. Prior to cracking, the M/χ approach provides a pure mechanics based solution for EI; that is, the only components of the model that have to be determined empirically are the material stress-strain relationships. However after cracking, the M/χ approach has to be semi-empirical, that is EI has to be determined empirically because the M/χ approach cannot simulate the mechanics of tension-stiffening. An alternative approach for quantifying EI using a moment-rotation (M/θ) approach is described in this paper. It is shown that the M/θ approach gives exactly the same results as the M/χ approach prior to cracking but after cracking has an advantage over the M/χ approach in that it can quantify the mechanics of tension-stiffening, that is allow for bond slip and its effect on crack spacing and crack widths. This paper deals with the mechanics of incorporating creep, shrinkage, prestress, relaxation and thermal gradients (broadly referred to as residual strains) on the flexural rigidity of RC beams and columns at all levels of loading prior to concrete softening.
properties are varied over time. Being strain based and applied at a cross-section, the M/Θ approach cannot directly simulate slip between the reinforcement and the adjacent concrete. While this approach can provide reasonable results for steel-reinforced beams that lie within the experimental tests from which EI eff was calibrated, when extended beyond this range, a poor correlation between predicted and observed deflections often occurs. This poses a significant problem for beams reinforced with FRP where, due to the high strength of the tendons, reinforcement ratios are low, resulting in tension stiffening being commonly overestimated. Despite extensive research on the tension stiffening behaviour of FRP reinforced concrete [9,10,17], empirically derived equations are still only applicable and restricted to the bounds of the test data from which they are derived.To overcome this empiricism associated with the strain-based M/χ approach, a displacement-based segmental M/Θ approach has been developed. This displacementbased approach was first developed for beams reinforced with either steel or FRP [19][20][21][22][23][24][25] and then extended for both the short-term loading of PC beams [26] and the longterm loading of ungrouted PC beams [27]. The benefit of the M/Θ approach is that it can directly simulate the mechanisms associated with tension stiffening and also concrete softening [28][29][30][31], which is in contrast to the M/χ approach, which requires empiricisms to allow for these mechanisms. Furthermore, the purely mechanics-based M/Θ results can be easily converted to moment/equivalent curvatures and, consequently, equivalent flexural rigidities EI equ that can be used in standard strain-based design practices. This method thereofre obviates the need for empirically derived values for curvatures or flexural rigidities. The M/Θ approach is now further extended here to accommodate the time-dependent behaviour of preand post-tensioned PC beams with grouted FRP or steel tendons.This paper first shows how, through the analysis of a segment of the beam, the M/Θ approach can simulate the effects of pre-and post-tensioning as opposed to the current M/χ strain-based approach, which deals with a section of the member. This is then followed by the analysis of the member subjected to applied loads and prior to cracking. Up to this point the M/Θ approach gives exactly the same results as the M/χ approach. The only benefit of the M/Θ approach at this stage is that the behaviour of the
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