Fractional and fractal derivative-based creep models for concrete under constant and time-varying loading

Abstract

In this paper, fractional and fractal derivative-based viscoelastic laws are adopted to develop linear creep models for reinforced and prestressed concrete under constant and time-varying loading. The model parameters are determined by curve-fitting the creep compliances of the spingpot and Kelvin-Voigt viscoelastic models to experimental data of basic creep in plain concrete specimens. Equilibrium and strain compatibility equations are formulated for reinforced and prestressed concrete under concentric loading which are solved simultaneously with the constitutive stress–strain equation of the selected viscoelastic model. For the case of a constant load, adoption of the fractal viscoelastic models lead to first-order differential equations that, when solved, yield analytical expressions for basic creep in reinforced and prestressed concrete. A semi-analytical solution is derived when adopting the fractional viscoelastic laws, obtained using the Laplace transform technique. Both models ensure strain compatibility and consider stress redistribution from concrete to the steel reinforcement bars. A high-level of agreement is made between the derived fractional and fractal-derivative based models and existing experimental data of creep in reinforced and pre-stressed concrete members under constant load. When the applied loading varies with time, numerical procedures are employed to approximate the fractional and fractal derivatives. It is found that creep in plain concrete members subjected to time-varying loads is accurately predicted when using the fractional derivative models. Comparisons are then made to other methods of modelling creep under time-varying stress. The primary advantage of the derived models is that only up to three parameters require calibration using basic creep tests for an accurate representation of creep and that closed-form expressions for creep can be obtained for the case of constant, sustained loading.