Performance of corrosion-affected RC structures depends on the spatial variability of steel corrosion of reinforcing bars. The effect of spatial variability of steel corrosion on the remaining service life of concrete structures is significant. In this paper, an experimental study was conducted to investigate the effect of current density on the spatial variability of steel corrosion. This variability is modeled using Gumbel distribution derived from experiments and incorporated with FE method to estimate the yield loading capacity of RC beams. The results show that using Gumbel distribution parameters derived from the specimens with high current density may lead to an overestimation of load capacity of corroded RC beams.This work was supported by Waseda University Grant for Special Research Projects (project number: 2018B-146)

Akiyama, Mitsuyoshi

Frangopol, Dan M.

Lim, Sopokhem

Zhang, Mingyang

SNU Open Repository and Archive

13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  1 Modeling spatial variability of steel corrosion using Gumbel distribution Mingyang Zhang Graduate Student, Dept of Civil and Environmental Engineering, Waseda University, Tokyo, Japan Sopokhem Lim Assistant professor, Dept of Civil and Environmental Engineering, Waseda University, Tokyo, Japan Mitsuyoshi Akiyama Professor, Dept of Civil and Environmental Engineering, Waseda University, Tokyo, Japan Dan M. Frangopol Professor, Lehigh University, 117 ATLSS Drive, Bethlehem, PA, USA ABSTRACT: Performance of corrosion-affected RC structures depends on the spatial variability of steel corrosion of reinforcing bars. The effect of spatial variability of steel corrosion on the remaining service life of concrete structures is significant. In this paper, an experimental study was conducted to investigate the effect of current density on the spatial variability of steel corrosion. This variability is modeled using Gumbel distribution derived from experiments and incorporated with FE method to estimate the yield loading capacity of RC beams. The results show that using Gumbel distribution parameters derived from the specimens with high current density may lead to an overestimation of load capacity of corroded RC beams.  1. INTRODUCTION Steel corrosion is one of the most important factors causing performance degradation of RC structures exposed to marine environment. This has become an important issue over the past decades. There has been substantial evidence that the localized damages of reinforcements have detrimental effects on structural performance of corroded RC structures. Zhu and François (2014) reported that the capacity loss of corroded RC beams was affected by the steel cross-section loss at the failure location. Malumbela et al. (2010) showed that using average cross-section loss in the theoretical model resulted in an overestimation of the loading capacity of RC members. Stewart (2004) found that the failure probability considering the effect of spatial corrosion pits was twice larger than that of uniform corrosion for the reliability analysis for RC members in flexure. Therefore, it is important to model the spatial variability of steel corrosion in reliability analysis of corroded RC structures. Gumbel distribution has been widely used for modeling spatial variability in steel corrosion. Stewart (2009) proposed a pitting factor Rp = Pmax/Pav, where Pmax and Pav are the maximum and average pitting depths, respectively. The pitting factor Rp was assumed as a time-invariant parameter while Rp was observed to decrease with time. Gu et al. (2018) proposed a spatial variability factor R = Aave/Amin, where Aave and Amin are average and minimum cross-section area, respectively. The Gumbel parameters of R shows a linear relationship with the increase of corrosion level. However, these parameters were derived using the data of steel corrosion from different corroded rebars with corrosion level lower than 5%. Lim et al. (2016) applied a non-destructive method using X-ray and imaging processing technique to quantify the spatial variability of steel weight loss. However, the relationship of 13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  2   Figure 1: Front view and cross-section of RC beam  Gumbel distribution parameters for Rswl and mean steel weight loss was proposed with the experimental data without considering the effect of current density. In this paper, Gumbel distribution parameters are derived from two RC beams corroded by two level of current densities (Icorr) 50 μA/cm2 and 500 μA/cm2 to model the steel weight loss distribution of corroded rebar. The effect of current density on the Gumbel distribution parameters for modeling the spatial variability of steel weight loss along the length of rebar is highlighted. 2. EXPERIMENTAL INVESTIGATION 2.1. Experimental program Two RC beams with the dimensions shown in Figure 1 were cast for this study. The mixing proportions of concrete is listed in Table 1. The galvanostatic method using Icorr of 50 and 500 μA/cm2 was applied to simulate corrosion-induced damage for specimens I50 and I500, respectively. During the corrosion process at various corrosion levels, the X-ray apparatus was used to take photograms of the non-corroded and corroded rebars from 8 different viewing angles as shown in Figure 2.  Steel weight loss (Rw) was estimated per 5 mm along the length of rebar using digital image processing technique of X-ray images which is defined as   1 n nnW WRwk W    (1) where Wθn and W’θn are the weights of the non-corroded and corroded bars per 5 mm, respectively; k = 8 is the number of viewing angles; and the subscript θn represent the viewing angle. More details are given in Lim et al. (2016).  Table 1: Mixing proportions (kg/m3) Water Cement Sand Stone AE* 181 362 754 961 2715 *Air entrainment (mL/m3)   Figure 2: X-ray photography   At the end of the experimental procedure, a four-point bending test was conducted by applying monotonic loads at two points symmetrically as indicated in Figure 1. The applied load P and the mid-span deflection δ were recorded during the bending test. 2.2. Results and discussion Figure 3 illustrates the Rw distributions for specimens I50 and I500 at various corrosion levels. In this study, comparison of Rw was conducted based on same mean steel weight loss (MRw). It can be seen that the I50 specimen shows a more non-uniform Rw distribution than the I500 specimen at three corrosion levels. A few large corrosion pits concentrating at three different regions (i.e. from 310 to 430 mm, 610 to 790 mm, and 1030 to 1090 mm) are found in specimen I50 at MRw = 10.96%. However, the Rw of specimen I500 distributes more evenly over the length of rebar. As a result, the standard deviation (SD) of Rw for specimen I50 is higher than that for specimen I500 at different corrosion levels. These results suggest that the stochastic field of modeled spatial steel corrosion which is 1401006-DB6@100490 1004906-DB6@100280P/2 P/21408020Longitudinal rebar (1-DB13)Wood stickUnit: mm250 mm1090 mmCaptured by X-ray Image intensifier and acquisition0%        2.61%     7.77%   13.53%    X-ray sourceRC specimen8 viewing angles of X-ray Continuous growth of steel corrosionManipulator for horizontaland vertical translation and rotation of specimen θ3 = 90 θ2 = 30 θ1 = 0 θ8 = 330 θ7 = 270 θ6 = 210 θ5 = 180 θ4 = 150 RC specimen13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  3   Figure 3: Spatial variability of the steel weight loss for specimens I50 and I500  generated using the statistical data of accelerated steel corrosion in the laboratory is significantly affected by the impressed current density. 3. PROBABILITY MODEL OF STEEL WEIGHT LOSS FOR CORRODED RC BEAM 3.1. Probability model A probability model is established to estimate the spatial variability of the steel weight loss using the Gumbel distribution of the maximum steel weight loss ratio Rswl = Wmax/MRw, where Wmax is the local maximum steel weight loss at an interval of 50 mm. To represent the spatial variability of steel weight loss at different corrosion levels using Gumbel distribution, the location parameter µ and scale parameter σ can be determined via the expected value and variance of Rswl (i.e. E (Rswl) and D (Rswl)).  ( )swlE R     (2)  2 2( ) 6swlD R    (3) where γ = 0.5772 is Euler’s constant. The probability density function is expressed as   1 1 1( ) exp[ ( )]exp[exp( ( ))]f x x x           (4) The relationships between Gumbel distribution parameters (i.e. µ and σ) and MRw for   Figure 4: Relationship between Gumbel distribution parameter µ, σ and mean steel weight loss   Figure 5: Example of generated steel weight loss  two specimens are illustrated in Figure 4. It can be seen that the scale parameter decreases with the increase of MRw. It is apparent that the Gumbel distribution scale parameter σ derived from the specimen using lower current density is higher than that derived from the specimen with higher current density. 3.2. Representation of Gumbel distributed variables Normal distributed variables are commonly used as standard random variables because of mathematical tractability. When the random variables are independent, the uncertainty in the i-th variable xi (i.e. Rswl) can be expressed as a function of the i-th standard normal distribution variable Ui.  To represent Gumbel distributed variables (i.e. Rswl), the transformation function 051015202530250 370 490 610 730 850 970 1090Rw(%)Distance from left edge of specimen I50 (mm)MRw = 2.54%; SD =1.58%MRw = 5.27%;SD = 3.01%MRw = 10.96%; SD = 4.53%051015202530250 370 490 610 730 850 970 1090Rw(%)Distance from left edge of specimen I500 (mm)MRw = 3.50%; SD = 1.53%MRw = 7.02%; SD = 1.97%MRw = 12.81%; SD = 3.81%00.511.520 0.05 0.1 0.15Location parameter, µMean steel weight loss, MRwI50I500Regression liney = 1.1178x-0.012y = 0.1726x-0.349y = 0.0588x-0.54100.20.40.60.810 0.05 0.1 0.15Scale parameter, σMean steel weight loss, MRwI50I5000102030401 11 21RswlI500I50Location along the length of RC beamSteel weight loss (%)13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  4  Figure 6: Constitutive model and mesh discretization of RC beam in FE analysis   ln[ln( ( ))]x U     (5) is used where Ф(·) is the standard normal cumulative distribution function. Examples of generated steel weight loss using two scale parameters are shown in Figure 5. 4. FINITE ELEMENT MODELING OF CORRODED RC BEAM 4.1. FE model Figure 6 illustrates a two-dimensional finite element model used to simulate the structural performance of a corroded RC beam. The concrete was modeled using a four-node plane stress element with 50 mm length. A two-node truss element was used to model reinforcement. The bond between reinforcement and concrete was modeled as a line-plane interface element. The concrete compression behavior of concrete was modeled based on Japan Society of Civil Engineers (JSCE) (2007) and softening curve of Hordijk, as described in DIANA10.2 User’s manual (2017), was used for modeling tensile behavior. The bond-slip model used in Lim et al. (2016) was also applied herein to model the deteriorated bond between reinforcement and concrete. The data of Rswl was used as the input for modeling the reduced cross-section area of reinforcement (from 250 mm to 1090 mm from the left end of beam).  (a)  (b) Figure 7: Experimental versus FE results for the flexural response of (a) beam I50 and (b)I500  4.2. Validation of FE model The experimental and simulated load-deflection curves for specimens I50 and I500 are shown in Figure 7. Numerical results show a good agreement with the experiments. Therefore, the load capacity can be predicted with reasonable accuracy.  The FE method incorporated with the maximum steel weight loss at the interval of 50 mm can give the flexure responses which are compared to those of the test beams. This comparison provides a good accuracy. Therefore, FE method is applied herein to reflect the effects of current density on the Gumbel distribution parameters. 4.3. Modeling Rswl using Gumbel distribution Ten samples of spatial distribution of steel weight loss with an assumption of MRw = 15% were generated based on Gumbel distribution parameters µ and σ calculated by two regression  Stress-strain relationshipof ConcreteStress-strain relationshipof reinforcement Bond stress-slip relationshipσɛɛcoɛcuf'cGF/hɛuftEs/100Esfufyɛsu ɛσɛy SlipσUmax,DSα Smax0.7Umax,DConcrete 4-node plane stress element Bond-slip interface elementReinforcement truss element1460 mm50 mm0510152025300 5 10 15 20 25 30Load, P (kN)Mid-span deflection, δ (mm)FEExperiment0510152025300 5 10 15 20 25 30Load, P (kN)Mid-span deflection, δ (mm)FEExperiment13th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP13 Seoul, South Korea, May 26-30, 2019  5   Figure 8: Yield load capacity distribution for two cases  equations shown in Figure 4 for two cases. The generated Rswl were used to calculate cross-section area of reinforcement as the input for the FE model to obtain the yield load capacity. Figure 8 shows that the yield load capacity results obtained for two cases with different scale parameters. It can be seen that the yield load capacity in case 1 is lower and more scatter than that in case 2. This comparison suggests that the Gumbel distribution parameter is significantly affected by the impressed current density. Also, using Gumbel distribution parameter derived from the specimen with higher current density may underestimate the non-uniformity of steel weight loss distribution. This can lead to an overestimation of load capacity of a corroded RC structure. 5. CONCLUSIONS 1. Low current density can induce a more non-uniform steel weight loss distribution than using higher current density. This result indicates that the stochastic field of modeled spatial steel corrosion which is generated using the statistical data of specimens using galvanostatic method in the laboratory is significantly affected by the impressed current density. 2. Based on the FE modeling results, the corroded rebar samples generated by the Gumbel distribution parameters derived from the specimen with lower current density present a more scatter and lower yield load capacity, compared with that from a specimen with higher current density. Hence, caution should be taken when applying the galvanostatic method to simulate the corrosion of rebar to obtain the Gumbel distribution parameters. If these parameters are derived from specimens using higher current density, the non-uniformity of steel corrosion may be underestimated. ACKNOWLEDGEMENT This work was supported by Waseda University Grant for Special Research Projects (project number: 2018B-146). REFERENCES Gu, X., Guo, H., Zhou, B., Zhang, W., and Jiang, C. (2018). “Corrosion non-uniformity of steel bars and reliability of corroded RC beams.” Engineering Structures, 167, 188–202. Japan Society of Civil Engineers (JSCE). (2007). “Standard specification for concrete structures.” Tokyo: Maruzen. Lim, S., Akiyama, M., and Frangopol, D. M. (2016). “Assessment of the structural performance of corrosion-affected RC members based on experimental study and probabilistic modeling.” Engineering Structures, 127, 189–205. Malumbela, G., Alexander, M., and Moyo, P. (2010). “Variation of steel loss and its effect on the ultimate flexural capacity of RC beams corroded and repaired under load.” Construction and Building Materials, 24(6), 1051–1059. Stewart, M. G. (2004). “Spatial variability of pitting corrosion and its influence on structural fragility and reliability of RC beams in flexure.” Structural Safety, 26(4), 453–470. Stewart, M. G. (2009). “Mechanical behaviour of pitting corrosion of flexural and shear reinforcement and its effect on structural reliability of corroding RC beams.” Structural Safety, 31(1), 19–30. TNO. (2017). DIANA10.2 User’s manual. “TNO Building and Construction Research.” Delft. Zhu, W., and François, R. (2014). “Corrosion of the reinforcement and its influence on the residual structural performance of a 26-year-old corroded RC beam.” Construction and Building Materials, 51, 461–472. 051015200 1 2 3Yield load capacity (kN)σ = 0.33 σ = 0.16 Case 1     Case 2                     µ = 1.14 

Modeling spatial variability of steel corrosion using Gumbel distribution

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Modeling spatial variability of steel corrosion using Gumbel distribution

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