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Fractional model of concrete hereditary viscoelastic behaviour

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Abstract

The evaluation of creep effects in concrete structures is addressed in the literature using different predictive models, supplied by specific codes, and applying the concepts of linear viscoelastic theory with ageing. The expressions used in the literature are mainly based on exponential laws, which are introduced in the integral expression of the Boltzmann principle; this approach leads to the need of finding approximated numerical solutions of the viscoelastic response. In this study, the hereditary fractional viscoelastic model is applied to concrete elements, underlining the convenience of using creep or relaxation functions expressed by power laws. The full reciprocal character of creep and relaxation, described through the power laws, is shown in Laplace domain, and the basic theorems of linear viscoelastic theory are re-written. In order to simplify the solution of fractional expressions, the Grünwald–Letnikov approach is followed and a discretization of fractional operators is applied through a simple matrix procedure. Two cases are examined: The first is the application of the fractional model to the creep laws supplied by international codes, finding the power law which approximates in the best way the creep curve given by the B3 predictive model. The second one is the application to the experimental data of an actual concrete mixture, for which the corresponding power law is found and introduced in the fractional model. A best-fitting procedure is applied to obtain the two parameters of power law in order to describe the hereditary character of concrete creep; afterwards, ageing is interpreted through the variation of coefficients with time. A third application on a concrete beam shows the suitability of the proposed model to the estimation of the delayed deformations of actual concrete structures.

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References

  1. Chiorino, M.A., Casalegno, C.: Evaluation of the structural response to the time-dependent behaviour of concrete: part I—an internationally harmonized format. Indian Concr. J. 86(12), 25–36 (2012)

    Google Scholar 

  2. Dilger, W.H.: Creep analysis of prestressed concrete structures using creep-transformed section properties. PCI J. 27(1), 98–117 (1982)

    Article  Google Scholar 

  3. Bažant, Z.P.: Inelastic Analysis of Structures. Wiley, Chichester (2002)

    Google Scholar 

  4. Chiorino, M.A.: A rational approach to the analysis of creep structural effects. In: Gardner & Weiss (ed) Shrinkage and Creep of Concrete. ACI SP-227, pp. 107–141 (2005)

  5. Chiorino, M.A., Creazza, G., Mola, F., Napoli, P.: Analysis of aging viscoelastic structures with n-redundant elastic restraints. In: Bažant, Z.P. (ed) Fourth RILEM International Symposium on Creep and Shrinkage of Concrete: Mathematical Modelling, pp. 623–644. Northwestern University, Evanston (1986)

  6. Mola, F., Giussani, F.: Long-term behaviour of cable stayed bridges. Stud. Res. 24, 153–187 (2003). (Fratelli Pesenti, Politecnico di Milano)

    Google Scholar 

  7. Nutting, P.G.: A new general law deformation. J. Frankl. Inst. 191, 678–685 (1921)

    Article  Google Scholar 

  8. Gemant, A.: A method of analyzing experimental results obtained by elasto-viscous bodies. Physics 7, 311–7 (1936)

    Article  Google Scholar 

  9. Scott-Blair, G.W., Gaffyn, J.E.: An application of the theory of quasi-properties to the treatment of anomalous strain–stress relations. Philos. Mag. 40, 80–94 (1949)

    Article  Google Scholar 

  10. Di Paola, M., Pirrotta, A., Valenza, A.: Viscoelastic behavior through fractional calculus: an easier method for best fitting experimental results. Mech. Mater. 43(12), 799–806 (2011)

    Article  Google Scholar 

  11. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  12. Mainardi, F., Gorenflo, R.: Time fractional derivatives in relaxation processes: a tutorial survey. Fract. Calc. Appl. Anal. 10(3), 269–308 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. Trans. ASME 51(2), 294–298 (1984)

    Article  MATH  Google Scholar 

  14. Celauro, C., Fecarotti, C., Pirrotta, A., Collop, A.C.: Experimental validation of a fractional model for creep/recovery testing of asphalt mixtures. Constr. Build. Mater. 36, 458–466 (2012)

    Article  Google Scholar 

  15. Di Paola, M., Heuer, R., Pirrotta, A.: Fractional visco-elastic Euler–Bernoulli beam. Int. J. Solids Struct. 50, 3505–3510 (2013)

    Article  Google Scholar 

  16. Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3(4), 359–386 (2000)

    MathSciNet  MATH  Google Scholar 

  17. ACI 209.2R08: Guide for Modelling and Calculating Shrinkage and Creep in Hardened Concrete, Report by ACI Committee209, pp. 1–45 (2008)

  18. Dutron, R.: Deformations lentes du beton arme sous l’action des charges permanent. Annales des Travaux Publics de Belgique, Tome 37, 2nd Series (1936)

  19. Granata, M.F., Margiotta, P., Arici, M.: Simplified procedure for evaluating the effects of creep and shrinkage on prestressed concrete girder bridges and the application of European and North American prediction models. J. Bridg. Eng. ASCE 18(12), 1281–1297 (2013). doi:10.1061/(ASCE)BE.1943-5592.0000483

    Article  Google Scholar 

  20. Park, Y.-S., Lee, Y.-H., Lee, Y.: Description of concrete creep under time-varying stress using parallel creep curve. Adv. Mater. Sci. Eng. Article ID 9370514, 1–13. doi:10.1155/2016/9370514 (2016)

  21. Bažant, Z.P., Baweja, S.: Creep and shrinkage prediction model for analysis and design of concrete structures: model B3. In: Al-Manaseer, A. (Ed.) Adam Neville Symposium: Creep and Shrinkage-Structural Design Effects, ACI SP-194, pp. 1–83. Farmington Hills, Michigan (2000)

  22. Di Paola, M., Fiore, V., Pinnola, F.P., Valenza, A.: On the influence of the initial ramp for a correct definition of the parameters of fractional viscoelastic materials. Mech. Mater. 69, 63–70 (2014)

    Article  Google Scholar 

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  1. Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale e dei Materiali (DICAM), Università di Palermo, Viale delle Scienze, 90127, Palermo, Italy

    Mario Di Paola & Michele Fabio Granata

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  1. Mario Di Paola
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  2. Michele Fabio Granata
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Correspondence to Michele Fabio Granata.

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Di Paola, M., Granata, M.F. Fractional model of concrete hereditary viscoelastic behaviour. Arch Appl Mech 87, 335–348 (2017). https://doi.org/10.1007/s00419-016-1196-7

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