# The Application of Affine/Interval Algebra to Determine the Time of Concrete Cover Damage in Reinforced Concrete Due to Corrosion

### Abstract

This paper presents the application of affine numbers to determine the time of concrete cover cracking in reinforced concrete elements from the initiation of the reinforcement corrosion. This issue is crucial for evaluating sustainability and durability of reinforced concrete structures. The proposed general approach has been used to forecast displacements and crack propagation versus time in reinforced concrete elements subjected to corrosion tests. The affine approach equations describing changes caused by the formation of corrosion products are defined and the corresponding tensor of the volumetric strain rate is formulated. The time of cover cracking has been analysed using the Finite Element Method (FEM) and the Monte Carlo (MC) method to verify the correctness of calculations.

### Keywords

corrosion, FEM, elastic-plastic, cover cracking, affine numbers,### References

1. U. Angst, B. Elsener, C.K. Larsen, Ø. Vennesland, Critical chloride content in reinforced concrete – A review, Cement and Concrete Research, 39(12): 1122–1138, 2009, doi: 10.1016/j.cemconres.2009.08.006.2. U.M. Angst, Predicting the time to corrosion initiation in reinforced concrete structures exposed to chlorides, Cement and Concrete Research, 115: 559–567, 2019, doi: 10.1016/j.cemconres.2018.08.007.

3. I. Balafas, C.J. Burgoyne, Environmental effects on cover cracking due to corrosion, Cement and Concrete Research, 40(9): 1429–1440, 2010, doi: 10.1016/j.cemconres.2010.05.003.

4. A. Bentur, M.G. Alexander, A review of the work of the RILEM TC 159-ETC: Engineering of the interfacial transition zone in cementitious composites, Materials and Structures, 33: 82–87, 2000, doi: 10.1007/BF02484160.

5. G. Bilotta, Self-verified extension of affine arithmetic to arbitrary order, Le Matematiche, 63: 15–30, 2008.

6. A. Bossio, S. Imperatore, M. Kioumarsi, Ultimate flexural capacity of reinforced concrete elements damaged by corrosion, Buildings, 9(7), 160, 2019, doi: 10.3390/buildings9070160.

7. H.G. Campos Silva, P. Garces Terradillos, E. Zornoza, J.M. Mendoza-Rangel, P. Castro-Borges, C.A. Juarez Alvarado, Improving sustainability through corrosion resistance of reinforced concrete by using a manufactured blended cement and fly ash, Sustainability, 10(6), 2004, 2018, doi: 10.3390/su10062004.

8. S. Chen, H. Lian, X. Yang, Interval static displacement analysis for structures with interval parameters, International Journal for Numerical Methods in Engineering, 53(2): 393–407, 2002, doi: 10.1002/nme.281.

9. L.T.N. Dao, V.T.N. Dao, S.H. Kim, K.Y. Ann, Modeling steel corrosion in concrete structures – part 1: A new inverse relation between current density and potential for the cathodic reaction, International Journal of Electrochemical Science, 5(3): 302–313, 2010.

10. L.T.N. Dao, V.T.N. Dao, S.H. Kim, K.Y. Ann, Modeling steel corrosion in concrete structures – part 2: A unified adaptive finite element model for simulation of steel corrosion, International Journal of Electrochemical Science, 5(3): 314–326, 2010.

11. L.H. De Figueiredo, J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37(1–4): 147–158, 2004, doi: 10.1023/B:NUMA.0000049462.70970.b6.

12. D. Degrauwe, G. Lombaert, G. De Roeck, Improving interval analysis in finite element calculations by means of affine arithmetic, Computers and Structures, 88(3–4): 247–254, 2010, doi: 10.1016/j.compstruc.2009.11.003.

13. H. El-Owny, Hansen’s generalized interval arithmetic realized in C-XSC, Bergische Universitat Wuppertal, Wuppertal, Germany, pp. 1–52, 2006.

14. S. Guzmán, J.C. Gálvez, J.M. Sancho, Cover cracking of reinforced concrete due to rebar corrosion induced by chloride penetration, Cement and Concrete Research, 41(8): 893–902, 2011, doi: 10.1016/j.cemconres.2011.04.008.

15. P.O. Iqbal, T. Ishida, Modeling of chloride transport coupled with enhanced moisture conductivity in concrete exposed to marine environment, Cement and Concrete Research, 39(4): 329–339, 2009, doi: 10.1016/j.cemconres.2009.01.001.

16. Y. Liu, R.E. Weyers, Comparison of guarded and unguarded linear polarization CCD devices with weight loss measurements, Cement and Concrete Research, 33(7): 1093–1101, 2003, doi: 10.1016/S0008-8846(03)00018-8.

17. W. López, J.A. González, Influence of the degree of pore saturation on the resistivity of concrete and the corrosion rate of steel reinforcement, Cement and Concrete Research, 23(2): 368–376, 1993, doi: 10.1016/0008-8846(93)90102-F.

18. G. Manson, Calculating frequency response functions for uncertain systems using complex affine analysis, Journal of Sound and Vibration, 288(3): 487–521, 2005, doi: 10.1016/j.jsv.2005.07.004.

19. N. Metropolis, S. Ulam, The Monte Carlo method, Journal of the American Statistical Association, 44(247): 335–341, 1949, doi: 10.2307/2280232.

20. A. Michel, B.J. Pease, A. Peterová, M.R. Geiker, H. Stang, A.E.A. Thybo, Penetration of corrosion products and corrosion-induced cracking in reinforced cementitious materials: Experimental investigations and numerical simulations, Cement and Concrete Composites, 47: 75–86, 2014, doi: 10.1016/j.cemconcomp.2013.04.011.

21. R.E. Moore, Interval analysis, Mathematics of Computation, 22(101): 219–222, 1968, doi: 10.2307/2004792.

22. O. Na, Y. Xi, Parallel finite element model for multispecies transport in nonsaturated concrete structures, Materials, 12(17), 2764, 2019, doi: 10.3390/ma12172764.

23. J. Ožbolt, F. Oršanic, G. Balabanic, Modeling influence of hysteretic moisture behavior on distribution of chlorides in concrete, Cement and Concrete Composites, 67: 73–84, 2016, doi: 10.1016/j.cemconcomp.2016.01.004.

24. J. Ožbolt, F. Oršanic, G. Balabanic, Modelling processes related to corrosion of reinforcement in concrete: coupled 3D finite element model, Structure and Infrastructure Engineering, 13(1): 135–146, 2017, doi: 10.1080/15732479.2016.1198400.

25. S.J. Pantazopoulou, K.D. Papoulia, Modeling cover-cracking due to reinforcement corrosion in RC structures, Journal of Engineering Mechanics, 127(4): 342–351, 2001.

26. K.G. Papakonstantinou, M. Shinozuka, Probabilistic model for steel corrosion in reinforced concrete structures of large dimensions considering crack effects, Engineering Structures, 57: 306–326, 2013, doi: 10.1016/j.engstruct.2013.06.038.

27. S.M. Rump, M. Kashiwagi, Implementation and improvements of affine arithmetic, Nonlinear Theory and Its Applications, IEICE, 2(3): 1101–1119, 2015, doi: 10.1588/nolta.2.1101.

28. A. Saetta, R. Scotta, R. Vitaliani, Mechanical behavior of concrete under physicalchemical attacks, Journal of Engineering Mechanics, 124(10): 1100–1109, 1998, doi: 10.1061/(ASCE)0733-9399(1998)124:10(1100).

29. J. Stolfi, L.H. de Figueiredo, Self-validated numerical methods and applications, Cite-SeerX, pp. 1–122, 1997.

30. C. Suwito, Y. Xi, The effect of chloride-induced steel corrosion on service life of reinforced concrete structures, Structure and Infrastructure Engineering, 4(3): 177–192, 2008, doi: 10.1080/15732470600688699.

31. J. Szymenderski, W. Machczynski, K. Budnik, Modeling effects of stochastic stray currents from D.C. traction on corrosion hazard of buried pipelines, Energies, 12(23), 4570, 2019, doi: 10.3390/en12234570.

32. K. Tuutti, Corrosion of Steel in Concrete, Swedish Cement and Concrete Research Institute, Stockholm, 1982.

33. Z. Wang, W. Jin, Y. Dong, D.M. Frangopol, Hierarchical life-cycle design of reinforced concrete structures incorporating durability, economic efficiency and green objectives, Engineering Structures, 157: 119–131, 2018, doi: 10.1016/j.engstruct.2017.11.022.

34. B. Wieczorek, T. Krykowski, Application of damage mechanics rules to evaluate the growth of corrosive deformations in transition layer [in Polish: Zastosowanie reguł mechaniki uszkodzen do oceny wzrostu odkształcen korozyjnych w warstwie przejsciowej], Ochrona przez korozja, 60(1): 5–8, 2017, doi: 10.15199/40.2017.1.1.

35. W. Zhu, K. Yu, Y. Xu, K. Zhang, X. Cai, A prediction model of the concrete cracking induced by the non-uniform corrosion of the steel reinforcement, Materials, 13(4): 830, 2020, doi: 10.3390/ma13040830.

36. T. Krykowski, T. Jasniok, F. Recha, M. Karolak, A cracking model for reinforced concrete cover, taking account of the accumulation of corrosion products in the ITZ layer, and including computational and experimental verification, Materials, 13, 2020 [in print].

**Computer Assisted Methods in Engineering and Science**, [S.l.], v. 27, n. 4, p. 265–284, dec. 2020. ISSN 2956-5839. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/297>. Date accessed: 14 sep. 2024. doi: http://dx.doi.org/10.24423/cames.297.