Non-uniform corrosion of steel rebar and its influence on reinforced concrete elements` reliability

Abstract Remarkable place of reinforced concrete structures in construction field has been noted in wide number of recent researches. Subsequently, their degradation due to aggressive environment has become the topical problem nowadays. Therefore, the formulation of reliable technique for corroded element strength decrement is of great importance, and could be achieved only with the use of complex experimental and theoretical analysis. In this article an attempt is made to propose the mathematical approach to corrosive process modelling, taking into consideration the specifics of its development. According to thorough literature review on existing studies, main specifics of the process were indicated for further suppositions and assumptions formulation. Accordingly, the complex theoretical investigation with corresponding mathematical computations was conducted and results of analytical modelling were discussed. As the initial data for analytical modelling results of previously conducted experiments were used. Analysis of the obtained results shows rather high correspondence with the real conditions of structural element exploitation, taking into consideration material anisotropy and complexity of the corroded zone spread along the rebar cross-section. Proposed methodology for limit force decrease evaluation in general demonstrates reliable results and could be used for further evaluation of corrosion impacts on reinforced concrete elements bearing capacity.


Introduction
In the last few decades reinforced concrete structures have occupied remarkable place on the construction market and have become, probably, the most widely used construction material (Bobalo et. al., 2019;Christodoulou and Goodier, 2014). Considering the noteworthy prevalence of reinforced concrete structures in various application fields, issues of their reliability, possible damages and material deterioration have reached high topicality. As it was affirmed by Cherinin L. and Val D., 2012, the reinforcement corrosion is one of the major threads causing reduction of the construction safety. The same statement could be found in the number of other works (Tantele, et. al., 2017;Luo et. al., 2019;Li and Ye, 2018). Thus, the authors assume that the long-term performance of the corroded RC element is affected due to degradation chemical processes and corresponding decreasing of the effective area of the steel rebar. In recent years, a great number of experimental and theoretical investigations was conducted, aiming to thoroughly investigate this complex process, focusing on different micro-and macro-scale aspects (Yogalakshmi et. al., 2020;Matesová et.al. 2007;Sadeghi et. al., 2019). In general, authors agreethat the corrosion process in RC structures could occur in correspondence with various degradation mechanisms. As it was described in articles of Silva M. et. al., 2015 andGoya A. et. al., 2015. this process could be identified as the surface deterioration aused by environmental influences, reduction reaction, which could result into conversion of the metal into another material. Subsequently, the geometrical, chemical and mechanical properties of the structural component are changed. It is also important to note, that the damaged layer spreading is mostly non-uniform, and depends on environmental conditions. The reliable determination of strength decrement could be achieved only with the use of appropriate complex technique of corrosion process simulation, which could take into consideration all the circumstances. In this article an attempt is made to propose the mathematical approach to corrosive process modeling, taking into account the specifics of its development.

Aims
The main purpose of this work is to conduct a thorough analysis of the corrosion process in the steel rebar and evaluate its impact on structural element strength reduction. In order to provide a complex theoretical investigation the corresponding mathematical computations will be conducted and results of analytical modeling will be discussed.

Analytical investigation
The exceptional aspects and specifics of the reinforcement material degradation are presented on the basis of particular samples with predetermined material properties. The samples are thermally 20 mm A500C  steel bars. The attention should be paid to specific non-uniform properties of the modeled samples, described in the previous study (Blikharskyy, 2019). On the basis of experimental investigation (Blikharskyy, 2019) results following assumptions are made in order to predefine the marginal conditions for theoretical modeling.
The rebar cross section could be considered as the composite heterogeneous surface which consists of three zones with different physical, mechanical and chemical properties. The approximate model of the sample cross-section can be seen in Fig.1. 2) The transitional layer with the averaged properties is considered with the use linearization rules. The layer thickness is  -the corresponding layers` thicknesses.
3) The third layer is situated in the middle of the rebar crosssection and, therefore, is not subjected to microstructural transformations during the thermal strengthening. The layer thickness is 6.5 mm III   and its mechanical properties are sufficiently lower than those of the first layer. Thus, the yield strength of the material is During the mathematical computations the elastic state of work of the material, it will be considered where the stress does not exceed the yield strength of the material ( 0.2

 
). Thus, the limit strength of the material will be equal to the yield strength of the particular layer, which at this moment is subjected to aggressive impact, namely . Such assumption is taken into consideration because further increment of the stress in the steel bar could result in interruption of the joint work, as strain in the reinforcement reaches the creep stage.
The corrosion process could be associated with complicated mechanism, which depends on various environmental and timescale factors and is identified by different development scenarios. Namely, the damaged layer could spread uniformly, when the effect of aggressive action is applied equally, along the perimeter of the rebar. However, such type of the steel deterioration is unlikely to occur in real-life conditions, as the corrosion impact usually has more localized unsymmetrical form. Therefore, the study will be focused on the situation, when the effective cross-section of the rebar is reduced by dismemberment of separate segments, as illustrated by Figures 2 and 3. As could be observed when such type of the corrosion occurs, the separate zones are accessed by aggressive environment, when the other part of the cross-section could still effectively bear the load.  Such type of the impairments` distribution is associated with complicated geometrical changes and offset of the gravity center from O to O` (Fig. 4). Therefore, a certain eccentricity of the load takes place. A particular stress state could be identified as non-central tension; thus, certain complication of the sample`s performance should be taken into account. Generally, the limit state of the steel rebar will occur when the following condition is fulfilled: Main geometrical parameters of the rebar cross-section (r, c, h, d), identified in Fig. 4, 5 will be used in following computations (eqs.4-19).
After corrosion occurs, the cross-section area of the rebar will be reduced according to following formula: -the initial value of the area and the corroded field area respectively, hx  the thickness of the corroded layer n each particular stage, r -the initial radius of the steel bar ( fig.4). Parameter c could be defined as: The load of the limit value of max i N is initially applied to the cross-section gravity center ( fig. 4); however, due to corrosion the bending moment appears: where the offset of the force application point: i gc y y r    The coordinate of the cross-section gravity could be calculated as: where the gravity center of the corroded segment is: where the maximum distance from the neutral line is: The value of inertia moment on each particular corrosion stage will be equal to: where xinit I , segm x I are the corresponding inertia moments of the initial cross-section and the corroded part respectively.
After the corresponding calculations being conducted the critical load value, which could perceive the steel bar is: Minor mathematical computations allow to transverse the formula (14) as the following: where the coefficients, mentioned in the equation (15) could be calculated as: 2 1 sin cos After that the actual values of the corroded layer thickness are substituted into the eqs. (14-19) and corresponding results of the limit force reduction are received (Table 1). The data analysis includes also indication the following functional dependencies: The above cited dependencies (20-22) could be approximated in the form of 6-order polynom (eq.23): The polynomial coefficients 17 .... aa were obtained through approximation with high accuracy level (determination coefficient 2 1 R  ). Thus, the functional dependencies (20-22) will be the following:

Results and discussion
Results of mathematical modeling were presented in the graphical form (Fig. 6-7), demonstrating the kinetics of steel bar bearing capacity changes at each corrosion rate. The graphs outline the major parameters of strength changes, taking into account the anisotropic character of material properties along the rebar cross-section. Therefore, the destruction process could be analyzed and thorough discussion of obtained data could be conducted. Graphs 5-6 indicate relative decrease of the rebar bearing capacity, depending on corrosion depth and rebar cross-section reduction.
It could be seen that graph on fig. 7 shows relatively linearized pattern of rebar strength decrease, depending on cross-section decrease. However, according to graph in Figure 6 the limit force decrease on the first corrosion stages is smoother with more sharp degradation after the corrosion depth reaches the value of 2 mm (10% sample diameter). Such divergences could be explained by functional argument choice. Therefore, it could be assumed the relative cross-section area reduction, used in Figure 7 does not fully reveal the complicity of sample geometrical properties changes.
Thereby, in general, the strength changes are non-linear, due to geometrical complicity of the process and material anisotropy.
It is important to notice that the proposed method does not fully represent the real-life corrosion process, as the number of external factors is not taken into account. Thus, the material structure on the micro-scale should be taken into consideration, as different layers of the rebar have different chemical and mechanical properties and will have different respond to aggressive impact. Another indicator, which could contribute to the research, is the time-scale factor which could demonstrate the kinetics of strength changes more completely.

Summary and conclusion
The analytical research on the basis of experimental data was conducted regarding the issue of steel bar non-uniform corrosion; received results were analysed, divergences and possible ways for research improvement were indicated. In the article the major emphasize was made towards the non-uniform corrosion of the steel bar, which generally demonstrates the real exploitation conditions of the reinforcement. In addition the thorough literature review and theoretical research on the issue of corrosion development was conducted and main specifics of the process were indicated for further suppositions and assumptions formulation. Corresponding mathematical computation were conducted and functional dependencies were found, which reveal non-linear character of the sample bearing capacity reduction.
It could be concluded that the process of thermally strengthened steel bar degradation due to aggressive impacts is rather complex and depends on various factors: environmental conditions, type of the reinforcement, initial stress-strain state, etc. In order to provide more veracious modelling of the problem could be recommended to include into research exact data on material response to aggressive environment, as well as take into account external factors.
The proposed methodology for limit force decrease evaluation and analysis of its influence on structural element bearing capacity in general demonstrates reliable results and wide perspectives for its further improvement and implementation. More thorough analysis of the problem could provide more complex investigation and complete technique for corrosion impacts assessment, which will be of great significance in both theoretical fields as well as in practical application during existing structures reconstruction.