Corrosion current density prediction in reinforced concrete by imperialist competitive algorithm

Reinforced concrete (RC) is one of the most commonly used construction materials in civil engineering industry, and reinforcement corrosion is a major problem. For this reason, inspection techniques are used to evaluate steel corrosion in concrete in order to protect RC structures [1‐9].

Tuutti developed the model for predicting the service life of reinforcing steel [10]. According to this model, the corrosion process has two distinct stages: corrosion initiation and corrosion propagation (Fig. 1a). Once this process has started, the time until damage occurs will mostly depend on relative humidity, the availability of oxygen, and temperature. When corrosion is initiated, active corrosion results in a volumetric expansion of the rust around the reinforcing bars against the surrounding concrete [10]. In the corrosion initiation stage, the required chemical reactions take place in concrete cover to initiate corrosion process. These chemical reactions may be carbonation or chloride ion attack.

Liu and Weyers [11] developed a performance-based service life model presented in Fig. 1b, which uses structural engineering performance criteria: serviceability and strength limit states. In service life model, the corrosion process represents three distinct phenomena: depassivation, propagation, and final state. Depassivation is the loss of oxide layer over the rebar due to the high alkalinity of concrete. Depassivation takes an initiation period T _i (time from completion of the new built structures to the time of corrosion initiation in the structure). The second life cycle is the propagation period T _s from the initiation of corrosion to corrosion-induced unserviceability of the structure. The third life cycle is the time period T _f from loss of serviceability to ultimate failure [12‐15].

It is proper to note that once the corrosion is initiated, concrete resistivity plays an important role in deciding reinforcement corrosion [16‐18]. The concrete resistivity measurement provides additional information to assist in assessing corrosion process. As mentioned in [12], corrosion propagation is the phase in which the accelerated corrosion leads to rust staining, cracking of the concrete cover, and the deterioration mechanism in RC structures generally occurring in the form of longitudinal cracking, spalling, and delamination of concrete cover. Figure 2 shows the deterioration mechanism of RC element as a result of corrosion process.

One of the most useful methods of providing a direct evaluation of the corrosion rate by corrosion current density measurement i _corr is using the linear polarization resistance (LPR). In this method, a specific voltage shift (typically 10 mV) is applied to an electrode in solution. As mentioned in [16], the instantaneous corrosion current density i _corr is obtained by dividing a Stern–Geary [20] constant B by the polarization resistance R _p value:By measuring i _corr, a corrosion rate can be derived. Typical corrosion rates from LPR measurements are presented in Table 1.

LPR has its disadvantages because it can only be effectively performed in relatively clean aqueous electrolytic environments and in terms of destroying concrete lining to measure the amount of steel corrosion [21]. LPR will not work in gases or liquid emulsions where fouling of the electrodes will prevent measurements [22, 23]. It has been presented that corrosion current density is one of the most important input parameter in the corrosion-induced damage models [24]. Researchers developed corrosion current density prediction models based on the electrochemical principles of steel reinforcement corrosion [25], experimental testing [26], and statistical analysis [11].

It seems sensible to employ more parameters in one model to obtain a more accurate answer concerning the corrosion current density. Artificial neural networks (ANN) are massively parallel distributed processors that have a natural propensity for storing experiential knowledge and making it available for use [27, 28]. Nowadays, the ANN are well known and over the last few years have emerged as a powerful device that could be used in many engineering applications such for the prediction of corrosion of steel reinforcement in concrete [29‐35]. ANN has been used, for example, as the prediction of chloride permeability of concretes with obtaining an empirical model having high capability of estimation of permeability for both their experimental data and the ones obtained from the studies of other researchers [36]. It has been also concluded that ANN model has a theoretical value in the prediction of the corrosion current rate of steel in concrete using corrosion current density without the need for a connection to the steel reinforcement [23].

In previous research, backpropagation has been used for optimizing ANN. Although backpropagation has unquestionably been a major factor for the success of past ANN applications, it is plagued with inconsistent and unpredictable performances [37, 38]. Today, the new techniques exploiting ANN model, based on optimization algorithms, are being widely used in engineering fields [39]. The most popular seems to be imperialist competitive algorithms (ICA) and genetic algorithms (GA). The total number of papers related to corrosion of reinforced concrete published in Science Direct database increased from 42 in 2010 to 78 in 2013 (Fig. 3). The number of papers related to corrosion by using neural networks is on the same level since 2010. There were few attempts to use GA for the corrosion modeling. It is hard to find in the existed literature the application of ICA for corrosion current density prediction in steel reinforced concrete. It is proper to note that the applications of ICA for solving various engineering problems increased from 4 applications in 2010 into 39 in 2013.

ICA is a randomized population algorithm inspired from of the human political–social evolution [40‐45]. A number of colonial countries along with their colonies try to find a general optimal point in solving optimization problem. Different methods have been introduced to solve optimization problems. Some find the cost function optimum point iteratively, based on the gradient. In spite of the high rate of these methods, there is still the problem of falling into the local optimum trap [46]. The main objective of this research is using ICA to optimize the weights of ANN as a new optimization algorithm in determining steel corrosion in concrete. The advantages of this method [46] have been listed below:

In the last few years, there were few attempts to apply ICA for the engineering problems modeling like for the prediction of soil compaction in soil bin facility [47], for the prediction oil flow rate of the reservoir [48], or for optimum cost design of cantilever retaining walls [49].

GA is inspired by nature. Nature evolution or Darwin’s theory is the basis of its formation in which the bests have the right to survival. In this method, chromosomes with high competence have a higher chance to repeat in the selected population in the replication process. This takes place by the selection process. Various methods have been proposed, and the wheel method is the most famous one. Also, the elitist selection is used to determine how many of the most graceful persons were transferred to the next generation, unchangeably [50, 51]. In the last few years, there were few successful attempts to apply GA for modeling the concrete structures such as finding optimum reliability-based inspection plans for the service life of the hypothetical bridge deck [52], finding optimal placements of control devices and sensors in seismically excited civil structures [53] or active control of high-rise buildings [54].

Considering the above in this paper, steel corrosion is determined and predicted in concrete using ICA–ANN model. Thus, the different aspects of the network will be checked with 2, 3, and 4 inputs; temperature, AC resistivity over the steel bar, AC resistivity remote from the steel bar, and the DC resistivity over the steel bar are as input parameters in the ANN, and corrosion current density is considered as an output parameter. The model ICA–ANN accuracy is evaluated compared with a GA, in three phases of training, testing, and prediction.

ICA is a method in the field of evolutionary computing that seeks to find the optimal solutions in various optimization issues, and it offers an algorithm for solving mathematical optimization problems. ICA forms an initial set of possible responses with a particular process improving the initial responses (countries) gradually and providing the appropriate response of optimization problem (the ideal country). The foundations of this algorithm are consisted of assimilation policies, imperialistic competition (IC), and revolution [46, 55]. ICA begins with random initial population, and some of the best elements of the population are selected as imperialists. The remaining population is considered as a colony. Depending on the colonial power, the colonies are absorbed by imperialists with a special process. The total power of the empire depends on both constituent parts, i.e., the imperialist country (as the central core) and its colonies [46]. This dependence has been modeled as the sum of the power of the imperialist country, plus a percentage of its average colonial power. With the formation of the early empires, IC between them will begin. Each empire unable to compete, succeed, and increase its power (or at least prevents reduction of its influence) will be removed in the IC. So the survival of an empire depends on its ability to absorb the colonial empires of the rivals and bringing them under control. As a result, during the IC, gradually, the power of larger empires will increase, and weaker empires will be removed. In Fig. 4a, the flowchart of ICA process has been presented [46].

Policy of assimilation was performed with the aim of analyzing the culture and social structure of colonies in culture of central government. Colonial countries, to increase their influence, began to build developmental infrastructures. In line with this policy, the colony will move as x units to the line connecting the colony to imperialist and will be drawn to a new position. X is a random number with uniform distribution. The distance between the imperialist and the colonized is shown by “d,” and the value of parameter d is shown by [46] in Eq. (2):where β is a number greater than one and close to two. A good choice can be β = 2. β ≥ 1 coefficient causes the colony country to near the imperialist country from different directions during moving to it. In addition to this move, a small angular deviation is added to the path with a uniform distribution [46]. Every empire unable to increase its strength and lose its competitiveness power will be removed in the IC. This process will be done gradually. This means that over time, weakened empires will lose their colonies and more powerful empire, will conquer them and increase their power—the empire being removed is the weakest one. Thus, when repeating the algorithm, one or more of the weakest colonies of the weakest empire will be taken, and to take possession of the considered colonies, a competition among all empires will be created. The colonies, not necessarily will be seized by the most powerful empire, but the stronger empires, are more likely to acquire them.

In this method, chromosomes with high competence have a higher chance to repeat in the selected population in the replication process. This takes place by the selection process [50]. After completion of the selection process, the operator is applied on the selected direction for reproduction. In the transplant process, with a constant transplant rate, a random number is generated for each chromosome. If the generated random number is less than the transplant rate, this chromosome is selected to intercourse the next chromosome with the above conditions. In this method, uniform transplantation has been used among different types of transplantation. Then, the mutation operator is applied [50]. The aim of this work is to create more dispersion in the range of design space. In the mutation process, a random number is generated, with a constant mutation rate, for all bits of the chromosomes. If the generated random number is smaller than the mutation rate, the value of that bit changes, i.e., the value of zero becomes one and vice versa. The basic operators of natural genetics are reproduction, crossover, and mutation. The basic operators of natural genetics are reproduction, crossover, and mutation [56, 57]. The GA can be expressed as in Fig. 4b. GA ends when certain criteria such as certain number of generation or the average standard deviation performance of individuals are satisfied [50, 58, 59].

The data used for development of the models were obtained from the past experimental researchers [23, 61]. Exemplary steel corrosion data have been presented in Table 2. Of 68 data patterns, 80 % of samples (54 patterns) were used for training and 20 % of the selected samples (14 patterns) were used to test the network.

Specimens sized 400 mm × 300 mm × 100 mm were available, with each specimen containing a single, short steel bar 30 mm in diameter, made from steel class A-III grade 34GS. The slabs were made from concrete class C 20/25 and from Portland cement CEM I 42.5R and aggregate of maximum size—5 mm. Since the relative humidity of concrete has a significant influence on the concrete resistivity, the specimens were stored in a laboratory under constant relative concrete humidity conditions of 65 ± 1 % up to the time of the tests. A view of the concrete resistivity measurement system, laboratory stand, and resistivity measurement locations on concrete specimen is presented in Fig. 5. The study was carried out based on a database of concrete slabs of two different conditions with high and moderate corrosion rates, respectively, with laboratory-induced corrosion as presented previously [61].

To include all the input parameters in a numerical range as the inputs of ANN to provide more accurate and suitable results, all data, according to formula (3), will be normalized:where xN are the normalized input data, xs are the input data, MinX is the minimum of all data, and MaxX is the maximum of all the data.

Moreover, in the output parameter, the formula (4) will be used in normalization:where, yN are the normalized output data, ys are output data; MinY are minimum of all data, MaxY are maximum of all data. Therefore, all normalized data should be located within numerical distance [−1, +1].

ANN used in this research is called “Feed Forward” with the input parameters of temperature, AC resistivity over the steel bar, AC resistivity remote from the steel bar and the DC resistivity over the steel bar, and the output parameter corrosion current density; the network is shown in Fig. 6.

The hidden layer nodes numbers were determined through using [62] the empirical formula (5).where N _H is the maximum number of nodes in the hidden layers, and N _I is the number of inputs. Considering that the number of effective inputs obtained is equal to 4, the maximum number of nodes in the hidden layer will be equal to 9. The ICA is used to determine the ANN model weights.

A new proposition of corrosion current density prediction of reinforced concrete specimens subjected to corrosion test is presented in this numerical study. The proposed formulations are derived from the most popular algorithms used in ANN, namely ICA and GA. For this, available experimental data presented in the existing literature were gathered and used for prediction. Based on the analysis of the results, the following conclusions can be drawn:

The purpose of this paper is to predict corrosion current density in concrete using ANN combined with ICA used to optimize weights of ANN. The ICA–ANN model has a theoretical value and potential practical significance in the prediction of the corrosion current rate of steel in concrete using corrosion current density without the need for a connection to the steel reinforcement, and it may help the engineers in practice to make a comparison for the corrosion performance of steel in concrete in RC elements. However, to increase the generalization capability, the database should be further enhanced by increasing the number of training samples, a wider range of concrete humidity, corrosion current density, or the concrete’s resistivity.