# The Effect of Shape on Chloride Penetration of Circular Reinforcement Concrete Columns and Its Durability Design

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Circular Diffusion Model for a Cross-Section of an RC Circular Column

#### 2.1. Theoretical Derivation

_{ref}is the diffusion coefficient at the referenced time t

_{ref}and m represents the exponent coefficient.

_{0}represents the Bessel’s first function with order zero and Y

_{0}denotes the Bessel’s second function with order zero. The parameters A, B, and C are undetermined coefficients, and on the basis of Equations (7) and (8), the specific formulation of $C(\rho ,t)=F(\rho )T(t)$ is:

_{1}is Bessel’s first function with order one, and ${\alpha}_{m}$ depends on the solution of ${J}_{0}\left(R{\alpha}_{m}\right)=0$.

#### 2.2. Statistical Properties of Model Parameters

_{0}and age factor m as presented in Table 1.

_{28}represents the diffusion coefficient for 28 days of t

_{ref}and w/c is the water-to-cement ratio.

#### 2.3. Validation Using a Numerical Model

_{28}is 4.35 cm

^{2}/a. The mean values of C

_{0}and m selected from Table 1 are 2% of the weight of the binder and 0.53, respectively. The diffusion depth is set at 5 cm. The final analysis result is presented in Figure 1. Figure 1 clearly indicates that the estimation results of the circular diffusion model keep stable when zero numbers exceed 20.

#### 2.4. The Effect of a Time-Variant Diffusion Coefficient on Chloride Diffusion

_{28}and C

_{0}, stay the same as in Section 2.3 with a series of assumed m of 0, 0.3, 0.6, and 0.9. For a given diffusion time, the chloride distribution and the diffusion depth under different m values are exhibited in Figure 3. In particular, the diffusion coefficient attains a constant value over time for m = 0. Clearly, in Figure 3 the distribution of chloride concentration is significantly overestimated at m of 0. The larger diffusion time and smaller age factor the more evident that is. Consequently, it will lead to an underestimation of the service life of the RC column. The higher the m, the faster the diffusion coefficient degrades simultaneously. This explains the lower concentrations at the same diffusion depth for higher m.

## 3. Comparison between Circular and Slab Diffusion Models

#### 3.1. Chloride Concentration Estimation

_{28}is set at 4.35 cm

^{2}/a, with an age factor m of 0.53, and the surface chloride concentration C

_{0}is assumed at 2% of the binder weight. The effect of the shape on the chloride concentration on the surface of rebar during diffusion process is analyzed, as shown in Figure 4 and Figure 5. Figure 4 and Figure 5 highlight that for an RC circular section, the slab diffusion model underestimates the chloride concentration. As the radius increasing, the difference between the circular diffusion and slab diffusion models deceases gradually, implying that the effect of the circular cross-section diminishes slowly.

_{cir}/C

_{slab}is introduced to measure the error on the work of using the slab diffusion model. The impact of the model parameters on C

_{cir}/C

_{slab}is disscused according to the sensitivity analysis methodology. Diffusion time, radius of the RC circualr column, and concrete cover thickness are labeled as t, R, and x, respectively. Obvioulsy, according to Equations (11) and (12), it can be found that C

_{cir}/C

_{slab}has no relation with C

_{0}. The change interval of other model parameters are listed in Table 2. During this analysis, one model parameter varies, with the others kept fixed. The final result is presented in Figure 6. It can be seen in Figure 6 that both higher x and smaller R will lead to the increase in C

_{cir}/C

_{slab}. Nevertheless, m, D

_{28}, and t have a very slight influence on this value. Distinctly, both R and x are more sensitivitive to C

_{cir}/C

_{slab}than the other model parameters.

_{cir}/C

_{slab}is rewritten as a function using the radius of column and the concrete cover thickness. Assuming that D

_{28}, m, t, and R are fixed at the mean value, the C

_{cir}/C

_{slab}variation against x is plotted in Figure 7. The regression fitting result indicates that C

_{cir}/C

_{slab}can also be modeled using a linear function. Hence, C

_{cir}/C

_{slab}is expressed as follows:

_{cir}/C

_{slab}is expressed as follows:

_{s}represents the shape influence coefficient of circular section, K

_{s}= 1.8 R

^{−1.3}x + 1.

_{c}corresponding to the concrete cover thickness of 4, 5, and 6 cm estimated by using Equation (16), is displayed in Figure 10. Figure 10 shows that the maximum error is over 20%. And for errors within 5%, the modified slab diffusion model (Equation (15)) is preferable for a circular column radius below 60 cm.

#### 3.2. The Pre-Corrosion Initiation Time for the RC Circular Column

_{cr}is closely related to the time required for corrosion initiation of a concrete structure. The JSCE (Japan Society of Civil Engineering) proposed a value of 1.2 kg/m

^{3}for the C

_{cr}[27], but Stewart et al. [24] emphasized that, according to numerous studies, the value ranges from 0.6 to 1.2 kg/m

^{3}. Since this value depends on the steel material, concrete material composition, and external environment, the statistical variation in the property is unsurprising, but highlights the need for further research on the C

_{cr}. The C

_{cr}applied in the durability design of the HZM project adopted here is summarized in Table 3.

_{cr}. The time to corrosion initiation of RC circular column is calculated by modified slab diffusion model (Equation (15)), which, calculated by original slab diffusion model (Equation (11)), are expressed as:

_{i}represents the difference value between T

_{i-cir}and T

_{i-slab}. A sensitivity analysis of model parameters for ΔT

_{i}is also performed. The ranges of the intervals for C

_{0}and C

_{cr}are presented in Table 4, and the ranges of the interval of the other parameters in Table 2 are employed. In Table 4, the mean values of the model parameters are selected from the atmospheric zone, kept the same background with Table 2. The final analysis result is presented in Figure 11. Figure 11 reveals that higher C

_{cr}, c, x, and m increase ΔT

_{i}, whereas higher C

_{0}, D

_{28}, and R decrease ΔT

_{i}. Compared with other parameters, R and D

_{28}only mildly affect ΔT

_{i}.

#### 3.3. Durability Design of the RC Circular Column against Chloride Degradation

#### 3.3.1. Basic Model

_{d}are rewritten as follows:

_{f}is the failure probability corresponding to the DLS.

_{f}is calculated using the following expression:

_{i}) is:

#### 3.3.2. The Effect of RC Circular Section Shape on the Estimation of Reliability Index

^{−12}m

^{2}/s and the mean concrete cover thickness was 36 mm whereas the mean chloride diffusion coefficients and concrete cover thickness were 4.38 × 10

^{−12}m

^{2}/s and 52 mm for atmospheric zone. And water/cement ratio used in marine environment is 0.33. The statistical properties of model parameters in Table 2 and Table 4 used in the HZM project are suitable for structural concrete having w/c = 0.35. Thus, these model parameters are available for service life prediction of RC circular column. The design reliability index β

_{d}of the RC facilities in the HZM Project was set to 1.3 [29]. The same target reliability index is adopted in this paper. Sampling numbers of 100,000 is generated for Monte Carlo simulation. Under a series of radius of circular section, the reliability index for different service time is calculated, as shown in Figure 12. It can be seen from Figure 12 that the reliability index is overestimated when circular column is regarded as slab element. The data collected in Table 5 indicates that the service life of circular column is also overestimated by slab diffusion model. For the radius of column less 50 cm, the effect of RC circular column shape should be considered.

#### 3.3.3. Chloride Diffusion Coefficient D_{28} and Concrete Cover Thickness x_{d}

_{28}, the β

_{d}is satisfied by adjusting the minimum concrete cover thickness x

_{d}. The preliminary durability design result of x

_{d}corresponding to a series of D

_{28}is shown in Figure 13. Clearly, for the RC circular column, the value of the minimum cover thickness designed by the original slab diffusion model is lower than that designed by the modified slab one. The Δx

_{d}is labeled as the difference between the design values of the minimum cover concrete thickness. Based on Figure 13, the Δx

_{d}as a function of D

_{28}is displayed in Figure 14, with an evident linear relationship. Meanwhile, the reduction of the radius also increases the Δx

_{d}. For an expected lifetime of 50 years, the maximum value of Δx

_{d}is 2 mm when the radius is 30 cm. This value increases to 4 mm when the lifetime is 100 years. These data illustrate that the RC circular column shape minimally affects the design results. As a conservative choice, the minimum concrete cover thickness increases by 5 mm based on the design result of the original slab diffusion model.

_{d}of 100 years, with a fixed 28-day diffusion coefficient of 6 × 10

^{−12}m

^{2}/a, and the relationship between Δx

_{d}and β

_{d}is shown in Figure 15. Obviously, the effect of β

_{d}on Δx

_{d}can be overlooked.

## 4. Conclusions

- (a)
- The use of a constant diffusion coefficient causes the overestimation of the chloride concentration distribution, shortening the service life of the structure.
- (b)
- The shape of the circular section element accelerates chloride diffusion compared with the slab element. The error caused by adopting the slab diffusion model shows close relationships with the radius and the diffusion depth. The decrease of the radius of column and the increase of the diffusion depth enlarge this error. In general, the modified slab diffusion model is preferable for a radius below 60 cm.
- (c)
- The pre-corrosion initiation time of the RC circular column is underestimated with the slab diffusion model. Each model parameter shows sensitivity for the difference value between the time estimated with slab diffusion model and that estimated for the modified slab one. The shape of the circular section affects minimally the estimation of the time to corrosion initiation compared with other model parameters.
- (d)
- The service life of RC circular column is overestimated when the circular section is viewed as a slab element. The modified slab diffusion model should be used when the radius of the column below 50 cm.
- (e)
- The RC circular section simplified as a slab element slightly affects the durability design against chloride corrosion. The minimum concrete cover thickness increases by 5 mm for the RC circular column based on the design result of the slab element.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Effects of zero numbers of zero-order Bessel functions J

_{0}on the accuracy of circular diffusion model.

**Figure 3.**The influence of time-variant diffusion coefficients on chloride concentration distribution.

**Figure 4.**The chloride concentration versus concrete cover thickness at a diffusion time of 100 years.

**Figure 13.**Minimum concrete cover thickness as a function of the 28-day chloride diffusion coefficient in different exposure zones.

**Figure 14.**The difference between the minimum concrete cover thickness designed by the modified slab diffusion model and the original slab one.

Parameters | Exposure Condition | Distribution Type | Mean Value | Standard Deviation |
---|---|---|---|---|

C_{0} (%binder) | Atmospheric zone | Lognormal distribution | 2 | 0.31 |

Splashing and tidal zone | 5.4 | 0.82 | ||

Submerged zone | 4.5 | 0.68 | ||

m | Atmospheric zone | Normal distribution | 0.53 | 0.08 |

Splashing and tidal zone | 0.47 | 0.028 | ||

Submerged zone | 0.44 | 0.028 |

Parameters | Mean Value | Lower Limitation | Upper Limitation |
---|---|---|---|

D_{28} | 3.435 cm^{2}/a | 2.51 cm^{2}/a | 4.36 cm^{2}/a |

m | 0.53 | 0.37 | 0.69 |

R | 50 cm | 20 cm | 80 cm |

x | 4 cm | 0 cm | 8 cm |

t | 50a | 0a | 100a |

Exposure Class | Distribution Type | Mean Value (%Binder) | STANDARD Deviation (%Binder) |
---|---|---|---|

Atmospheric zone | Lognormal distribution | 0.85 | 0.13 |

Splashing and tidal zone | Beta distribution (L = 0.45%, U = 1.25%) | 0.75 | 0.23 |

Submerged zone | Beta distribution (L = 1%, U = 3.5%) | 2 | 0.72 |

Parameters | Mean Value | Lower Limitation | Upper Limitation |
---|---|---|---|

C_{0} (%binder) | 2 | 1.38 | 2.62 |

C_{cr} (%binder) | 0.85 | 0.59 | 1.11 |

Time (Years) | R = 30 cm | R = 50 cm | R = 70 cm | Slab Diffusion Model |
---|---|---|---|---|

Atmospheric zone | 39 | 42 | 45 | 46 |

Splashing and tidal zone | 57 | 60 | 61 | 64 |

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**MDPI and ACS Style**

Yin, G.; Pan, L.
The Effect of Shape on Chloride Penetration of Circular Reinforcement Concrete Columns and Its Durability Design. *Appl. Sci.* **2020**, *10*, 459.
https://doi.org/10.3390/app10020459

**AMA Style**

Yin G, Pan L.
The Effect of Shape on Chloride Penetration of Circular Reinforcement Concrete Columns and Its Durability Design. *Applied Sciences*. 2020; 10(2):459.
https://doi.org/10.3390/app10020459

**Chicago/Turabian Style**

Yin, Gu, and Li Pan.
2020. "The Effect of Shape on Chloride Penetration of Circular Reinforcement Concrete Columns and Its Durability Design" *Applied Sciences* 10, no. 2: 459.
https://doi.org/10.3390/app10020459