 Original article
 Open Access
Reliability of Reinforced Concrete Structures Subjected to CorrosionFatigue and Climate Change
 Emilio BastidasArteaga^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s400690180235x
© The Author(s) 2018
 Received: 26 November 2016
 Accepted: 7 January 2018
 Published: 31 January 2018
Abstract
Durability of reinforced concrete (RC) structures is affected by certain environmental conditions and operational actions which can reduce their lifetime significantly. Among these actions, this paper proposes a stochastic model that accounts for the combined effects of chlorideinduced corrosion, climate change and cyclic loading. Separately, corrosion leads to crosssection reduction, climate change produces changes in temperature and humidity and fatigue induces nucleation and propagation of cracks in the rebars. When considered together, pitting corrosion nucleates cracks while environmental factors affect the kinematics of chloride ingress and corrosion propagation. The proposed approach is illustrated with the reliability analysis of a bridge girder subjected to cyclic loading under various environmental conditions. The overall results indicate that climate change effect induces lifetime reductions ranging between 1.4 and 2.3% if fatigue load is neglected. Under cyclic loading, total lifetime reduction increases up to 7%.
Keywords
 reliability
 corrosionfatigue
 reinforced concrete
 climate change
 chloride ingress
1 Introduction
Reinforced concrete (RC) civil infrastructure systems are critical assets for the socioeconomic development of any country. Designing and maintaining these systems for a particular service lifetime have been recognized as critical issues worldwide. RC structures are characterized by high durability; however, during their operational life, they are subjected to internal and external actions that affect performance, serviceability and safety (Imam et al. 2015; Kim et al. 2016; MarquezPeñaranda et al. 2016; Morga and Marano 2015; SánchezSilva and Klutke 2016). Nowadays, many deteriorated structures are evaluated for possible repair and continued service because their replacement would be economically unfeasible. For example, about 173,000 bridges in the United States are structurally deficient or functionally obsolete due in part to corrosion (Bhide 1999; Pritzl et al. 2014; Radlińska et al. 2014). Regarding costs, Koch et al. (2016) reported that the global cost of corrosion is US$2.5 trillion (about 3.4% of the global Gross Domestic Product). Thus, developing robust models for prediction and strategies for periodic inspection and maintenance plays a significant role in enabling target reliabilities to be met over a period of continued service (BastidasArteaga et al. 2009; BastidasArteaga and Schoefs 2015; Clifton 1993; Mori and Ellingwood 1995).
This paper focuses on a combined corrosionfatigue deterioration mechanism. Corrosion is induced by chloride penetration that results in turn from a complex interaction between physical and chemical processes that are driven by environmental surrounding conditions (BastidasArteaga and Stewart 2016; Nguyen et al. 2017; Saetta et al. 1993). Combined corrosionfatigue deterioration results from the action of cycling stresses in corrosive environments. Localized corrosion leading to pitting may provide sites for fatigue crack initiation. For example, several experimental studies have shown that pitting corrosion has been responsible for the nucleation of fatigue cracks in a wide range of steels and aluminum alloys (Ahn et al. 1992; Chen and Duquette 1992; Kondo 1989). Corrosive agents (e.g., seawater) increase the fatigue crack growth rate (Gangloff 2005), whereas the morphology of metals/alloys at microlevel governs the pit nucleation sites (Rajasankar and Iyer 2006).
There exists a limited amount of experimental tests on corrosionfatigue in RC structures. Ahn and Reddy (2001) performed an experimental study to evaluate the durability of RC beams subjected to fatigue loading and chloride ingress. The tests included 16 beams and accounted for the influence of static and cyclic loading for different water/cement ratios. Alternate filling and draining of a tank simulated the marine tidal zone, and a galvanostatic corrosion technique was used to accelerate corrosion of the reinforcement. The ultimate strength of the beams was tested after 78,000 cycles by applying fourpoint flexural loading. The results indicated that beams subjected to cyclic loading during the exposure period showed lower ultimate strength than those subjected to static loading. Taking into account the imminent reduction of ultimate strength, other experimental studies focused on estimating the effectiveness of using composite repair materials (AlHammoud et al. 2011; ElSafty et al. 2014; Masoud et al. 2005; Song and Yu 2015). More recently, Wang et al. (2018) performed a comprehensive experimental study to quantify the effects of cyclic load on the chloride ingress process. They found that considering fatigue loading conditions have important effects in lifetime assessment depending on the aggressiveness of the surrounding environment.
Modeling the combined effect of corrosion and fatigue remains still an open challenge. BastidasArteaga et al. (2009) proposed a first probabilistic corrosionfatigue model for RC structures subjected to chloride ingress; however, this model uses a simplified solution of Fick’s law to simulate the chloride ingress process that does not always represent reality. On the other hand, experimental evidence indicates that the chloride ingress is highly influenced by the weather conditions at the surrounding environment—i.e. temperature and humidity. Since climate change studies predict several changes in the climate (IPCC 2013), the impact of global warming on chloride ingress, and therefore on structural reliability, should be also considered in the assessment of the structural behavior.
Within this context, the main goal of this paper is to improve the model proposed by BastidasArteaga et al. (2009) to be able to consider the influence of realistic exposure conditions (including climate change) on failure probability. To accomplish this objective, the proposed lifetime prediction approach includes a numerical solution for the transport governing equations in the assessment of the corrosion initiation time.
The paper starts describing the deterioration model used to combine the interaction between corrosion and fatigue (Sect. 2). Section 3 presents a stochastic weather model, including global warming. Section 4 describes the stochastic approach to the problem, and Sect. 5 applies the proposed methodology to the reliability assessment of a bridge girder subjected to various environmental and cyclic loading conditions.
2 Deterioration Model
The proposed deterioration model does not account for the loss of adhesion at the steel/concrete interface caused by corrosion propagation as well as the effect of permanent charges on the longterm mechanical behavior. Further model developments will be necessary in the future to integrate these aspects and improve lifetime assessment.
2.1 Corrosion Initiation and Pit Nucleation
Correspondence between Eq. (1) and the governing differential equations.
Physical problem  \( \psi \)  \( \zeta \)  \( J \)  \( J' \)  \( q_{\psi }^{s} \) 

Chloride ingress  C _{ fc }  1  \( D_{c}^{*} \vec{\nabla }C_{fc} \)  \( C_{fc} D_{h}^{*} \vec{\nabla }h \)  \( q_{h}^{s} \) 
Moisture diffusion  h  ∂w_{ e }/∂h  \( D_{h}^{*} \vec{\nabla }h \)  0  0 
Heat transfer  T  ρ _{ c } c _{ q }  \( \lambda \vec{\nabla }T \)  0  0 
For moisture diffusion, the humidity diffusion coefficient D_{ h } is estimated by accounting for the influence of the parameters presented in Eq. (3). The term ∂w_{ e }/∂h (Table 1) represents the moisture capacity which relates h and w_{ e }. For a given temperature this relationship has been determined experimentally by adsorption isotherms. According to the Brunauer–Skalny–Bodor (BSB) model (Brunauer et al. 1969) and the empirical expressions of Xi et al. (1994), the adsorption isotherm depends on temperature, water/cement ratio, w/c, and the equivalent hydration (curing) period, t_{ e }. This work adopts the BSB model to represent the moisture capacity.
Finally, for heat transfer (Table 1), ρ_{ c } is the density of the concrete, c_{ q } is the concrete specific heat capacity, λ is the thermal conductivity of concrete and T is the temperature inside the concrete matrix after time t.
The flow of chlorides into concrete is estimated by solving simultaneously the system of equations described by Eq. (1) and Table 1. The numerical approach used to solve the coupled system of PDEs combines a finite element formulation with finite difference to estimate the spatial and temporal variation of C_{ fc }, h and T. Then, the time to corrosion initiation, t_{ cini }, is estimated by comparing the chloride concentration at the cover depth, c_{ t }, with a threshold concentration for corrosion initiation C_{ th }.
From a comparison between the timesspans of the time to corrosion initiation and the time to pit nucleation, BastidasArteaga et al. (2009) found that the length of t_{ pn } can be neglected.
2.2 PittoCrack Transition
It is important to mention that the crack growth process does not account for effects of temperature and relative humidity. Further research is required to improve the modeling of crack propagation in rebars under realistic environmental conditions.
2.3 Crack Growth
3 Modeling Weather Including Climate Change Effects
3.1 Basic Considerations and Model Description
Until recently all corrosionrelated research assumed constant average climatic conditions for the development of deterioration models. However, it is expected a temperature rise up to 2 °C by 2100 even under an optimistic scenario where CO_{2} emissions are abated (IPCC 2013). Rises in temperature increase the rate of infiltration of deleterious substances (increased material diffusivity) and increase the corrosion rate of steel. Optimum relative humidity levels may also increase the rate of infiltration of deleterious substances (Stewart et al. 2011).

influence of global warming,

seasonal variation of weather parameters, and

random nature of weather.
Climate change effect is modeled by assuming a linear variation of the weather parameters (humidity or temperature); while seasonal variations of humidity or temperature follow a sinusoidal shape. The uncertainties related to weather are treated in the following section.
3.2 Selected Scenarios
The IPCC Fifth Assessment Report (AR5) (IPCC 2013) uses Representative Concentration Pathways (RCPs) where RCP 8.5, RCP 6.0 and RCP 4.5 are roughly equivalent to A1FI or A2, A1B, and A1B to B1 emission scenarios, respectively (Inman 2011). These RCPs were considered to be representative of the literature, and included a mitigation scenario leading to a low forcing level (RCP 2.6), two medium stabilization scenarios (RCP 4.5/RCP 6) and one high baseline emission scenarios (RCP 8.5) (Moss et al. 2010).

carbon dioxide, methane and nitrous oxide emissions;

global population growth;

introduction of new and clean technologies leading to the reduction of the impact of global change; and

use of fossil sources of energy.

the difference between the annual means of temperature for the initial year T(t_{ 0 }) and the year of the end of the forecast T(t_{ a }), ΔT,

the difference between the annual mean of relative humidity for h(t_{ 0 }) and h(t_{ a }), Δh, and

the difference between the normalized durations of cold seasons for R(t_{ 0 }) and R(t_{ a }), ΔR.
Climate change scenarios.
Scenario  ΔT (°C)  Δh  ΔR 

Without climate change is neglected  0  0  0 
Expected Use of alternative and fossil sources of energy, birthrates follow the current patterns and there is no extensive employ of clean technologies (equivalent to scenarios RCP2.6–RCP4.5 depending on the location)  2.5  0.05  − 0.1 
Pessimistic Vast utilization of fossil sources of energy, appreciable growth of population and there are no policies to develop and extend the use of clean technologies (equivalent to scenarios RCP6–RCP8.5 depending on the location)  6.5  0.1  0.2 
Although in general terms the presented model simulates the effect of seasonal variations and climate on temperature and humidity, it is important to stress that predicted values only represent an overall behavior which does not include the randomness of the phenomena. The following section addresses this and other random aspects.
4 Probabilistic Approach
4.1 Modeling Weather Uncertainties
4.2 Probability of Failure
Closedform solutions for both the CDF of the total corrosionfatigue lifetime and the failure probability are very difficult to obtain. Therefore, Monte Carlo simulations and Latin hypercube sampling are used herein to deal with this problem.
5 Numerical Example
5.1 Problem Description
Design load and material constants.
Variable  Value 

Characteristic punctual design load, P_{ k }  150 kN 
Elastic modulus of steel, E_{ st }  200 GPa 
Characteristic concrete compression strength, f’_{ ck }  30 MPa 
Characteristic steel strength, f_{ yk }  500 MPa 
Concrete Poisson ratio, ν_{ c }  0.2 
Water to cement ratio, w/c  0.5 
Curing period, t_{ e }  28 days 

two environments: oceanic and tropical (Table 4);Table 4
Description of the studied environments.
Climate
Latitude
T _{ min }
T _{ max }
h _{ min }
h _{ max }
b ^{a}
Oceanic
Middle
5 °C
25 °C
0.6
0.8
0.1 year
Tropical
Equatorial
20 °C
30 °C
0.7
0.9
0.1 year

two scenarios of global warming: without and pessimistic (Table 2); and
The time to corrosion initiation considers flow of chlorides in one dimension where the Langmuir isotherm is used to account for chloride binding. The constants of the isotherm are α_{ L } = 0.1185 and β_{ L } = 0.09. The effect of temperature on the binding process is not considered in the study. For a concrete elaborated with ordinary Portland cement the adsorption isotherm (BSB model) only depends on the values of w/c and t_{ e } reported in Table 3 (BastidasArteaga et al. 2011). Since Eq. (6) does not account for the influence of environmental variations on the corrosion rate, this paper only considers the interaction between surrounding weather and the chloride ingress process.
Probabilistic models of the random variables (BastidasArteaga et al. 2011).
Variable  Units  Distribution  Mean  COV 

D _{ c,ref }  m^{2}/s  Lognormal  3 × 10^{−11}  0.20 
C _{ th }  wt % cem.  Normal^{a}  0.5  0.20 
c _{ t }  mm  Normal^{b}  50  0.25 
D _{ h,ref }  m^{2}/s  Lognormal  3 × 10^{−10}  0.20 
λ  W/(m °C)  Beta on (1.4; 3.6)  2.5  0.20 
c _{ q }  J/(kg °C)  Beta on (840; 1170)  1000  0.10 
ρ _{ c }  kg/m^{3}  Normal^{a}  2400  0.05 
P  kN  Lognormal  115  0.20 
f’ _{ c }  MPa  Normal^{a}  40  0.15 
f _{ y }  MPa  Normal^{a}  600  0.10 
5.2 Results
5.2.1 Influence of Weather Model
Three types of environmental inputs are included in the analysis. The stochastic one includes seasonal variations and randomness for weather inputs (temperature and humidity). For the timevariant one, a sinusoidal function is used to consider the seasonal variations of temperature and humidity. Finally, the constant case supposes that temperature and humidity are constant in time—i.e., T = 15 °C, and h = 0.7. C_{ env } and i_{ th } are considered as constant values for all the cases.
In general, failure probability depends on the type of input entries and load frequency (f). Constant and timevariant models underestimate failure probabilities. For instance, for a traffic frequency of 50 cycles per day, a constant level of failure probability (i.e., p_{ f } = 0.5) is reached at 58, 69 and 113 years for stochastic, timevariant and constant entries, respectively. Taking as a reference the stochastic case, this implies that the timevariant and constant cases overestimate the time to reach p_{ f } = 0.5 respectively of 11 and 55 years. A similar behavior is observed when traffic frequency is increased. Since the total fatigue lifetime is highly dependent on time to corrosion initiation, it is expected that the type of input entries becomes a paramount parameter in the assessment of failure probability (BastidasArteaga et al. 2011). These results justify the use of a more representative chloride ingress model including realistic environmental inputs for improving lifetime assessment with respect to the findings presented in (BastidasArteaga et al. 2009).
5.2.2 Failure Probabilities for Different Levels of Aggressiveness
It is observed that the failure probability increases, in all cases, when the level of aggressiveness and the frequency tend to higher values. This increment is caused by: (i) higher concentrations of chlorides and corrosion rates of aggressive environments and (ii) quicker crack propagation under greater traffic frequencies. If the failure probabilities are evaluated for a lifecycle length of 50 years, it is noted that the structural reliabilities are very low for all levels of aggressiveness. By considering that the failure probability should be lower than a critical value to ensure a given safety level—i.e., p_{ f } < p_{ ft } = 10^{−4}, the studied structural configuration only guarantees this condition for the moderate level when there is no cyclic load (p_{ f } = 10^{−4}) and for the low level when there is no fatigue loading (p_{ f } = 7 × 10^{−5}) and traffic frequency is 50 cycles per day (p_{ f } = 4 × 10^{−4}). These results are not surprising because in real structures close to the sea, such as ports or quays, appreciable levels of deterioration have been reported after 15 or 20 years of exposure. The behavior for the oceanic environment is similar, but the failure probabilities are lower for all exposures and frequencies. This behavior indicates that a sophisticated model of chloride ingress calibrated with experimental observations and monitored with inspections should be included in the management of RC bridges to assure appropriate levels of safety during their lifecycle.
5.2.3 Influence of Climate Change
Mean (in years) and coefficient of variation (between brackets) of the total lifetime without fatigue effects.
Environmental aggressiveness  Tropical  Oceanic  

Without (%)  Pessimist (%)  Without (%)  Pessimist (%)  
High  58.2 (15)  57.9 (15)  62.2 (15)  60.8 (15) 
Moderate  120.5 (16)  119.6 (16)  124.8 (16)  123 (16) 
Low  409.4 (18)  407.7 (18)  411.2 (18)  405.2 (19) 
Mean (in years) and coefficient of variation (between brackets) of the total lifetime for the tropical environment.
Frequency (cycles/day)  High aggressiveness  Moderate aggressiveness  Low aggressiveness  

Without (%)  Pessimist (%)  Without (%)  Pessimist (%)  Without (%)  Pessimist (%)  
50  50.2 (15)  49.8 (15)  82.9 (21)  81.8 (21)  157.7 (36)  153.8 (35) 
500  32.6 (22)  32.2 (22)  49.5 (27)  48.4 (28)  103.9 (43)  101.3 (42) 
1000  30 (23)  29.5 (22)  45.8 (28)  45.1 (28)  97.1 (43)  94.7 (42) 
2000  28.3 (23)  27.7 (23)  43.5 (28)  42.5 (27)  91.7 (42)  88.8 (42) 
Mean (in years) and coefficient of variation (between brackets) of the total lifetime for the oceanic environment.
Frequency (cycles/day)  High aggressiveness  Moderate aggressiveness  Low aggressiveness  

Without (%)  Pessimist (%)  Without (%)  Pessimist (%)  Without (%)  Pessimist (%)  
50  54.1 (16)  52.7 (15)  86.9 (21)  85 (21)  159.3 (35)  153.1 (36) 
500  36.4 (22)  35.1 (22)  53.7 (27)  51.3 (27)  105.8 (41)  99.4 (42) 
1000  33.8 (23)  32.6 (22)  50.3 (27)  47.9 (27)  99.3 (41)  92.5 (42) 
2000  32.3 (24)  30.8 (23)  48.1 (27)  45.8 (27)  93.2 (40)  86.7 (40) 
When the girder is not subjected to cyclic loading or corrosionfatigue damage is neglected, Table 7 shows that the mean and the COV of the total lifetime, t_{ T }, decrease for more aggressive environments. It is observed in general that while the mean of t_{ T } is lower for tropical than for oceanic environments, the COV remains almost constant for both environments. This behavior seems logical because tropical environments are characterized by higher temperature and humidity that accelerate chloride ingress reducing corrosion initiation time. It is also noted that the effect of global warming is more important for structures located in oceanic environments. This analysis includes only the pessimistic scenario of global warming because there is no major difference between the expected and pessimistic scenarios. While for the tropical environment global warming only induced reductions from 0.4% to 0.7% in the mean of t_{ T }, for the oceanic environment this reduction ranges between 1.4% and 2.3%. It is concluded that climate change has more influence in environments where humidity and temperature are characterized by important seasonal variations.
The results of Tables 8 and 9 include the effect of corrosionfatigue damage and climate change for tropical and oceanic environments, respectively. This analysis also includes the effect of traffic frequencies. Similar to the previous case, the mean and COV of t_{ T } decrease when the level of aggressiveness increases. By comparing both cases (with and without fatigue damage) it is observed that the mean of the PDF is reduced when the assessment considers the effect of the cyclic load. The reduction of the lifetime induced by cyclic load can be estimated by setting the case without fatigue damage as the reference one. For example, for a case without climate change, oceanic environment, low aggressiveness, and f = 50 cycles/day, the mean of t_{ T } is reduced by 61%. For the same conditions but f = 2000 cycles/day this value corresponds to 77%. These results justify the consideration of combined damage mechanisms in lifetime assessment.
Similar to the case without fatigue damage, the structures located in oceanic environments are more susceptible to the effects of global warming. The reduction of the mean of t_{ T } varies between 0.8% and 2.5% for the tropical environment and ranges between 2.6% and 3.9% for the oceanic environment when traffic frequency is f = 50 cycles/day. The impact of climate change increases for higher traffic frequency (f = 2000 cycles/day) where reductions from 2.1% to 3.2% were observed for the tropical environment and from 4.4% to 7% for the oceanic environment. High traffic frequencies reduce the length of the stages of pittocrack transition and crack propagation (Eqs. 13 and 14), and consequently, the participation of the stage of time to corrosion initiation and pit nucleation becomes more important.
For all cases and environments, it is observed that the reduction induced by global warming is maximum 7% of total lifetime; however, it is important to clarify that these results are conservative because they do not include the effect of global warming after corrosion initiation. Nowadays, it is known that corrosion rate can be influenced by temperature and humidity, and therefore should be affected by climate change. Nevertheless, there is no a consensus about a comprehensive corrosion model that considers this interaction in a comprehensive manner. Further research in this area is required to improve the prediction after corrosion initiation.
6 Conclusions
This paper presented a deterioration model that integrates the effects of chlorideinduced corrosion, climate change and cyclic loading for RC structures. The total corrosionfatigue life was divided into three stages: (i) corrosion initiation and pit nucleation, (ii) corrosion initiation and pit nucleation, and (iii) crack growth. Since some of these stages are sensitive to climatic conditions, a simplified model of weather that includes global warming was also included. The whole deterioration model was introduced into a stochastic framework to take the inherent uncertainties into account. Finally, a numerical example illustrated the consequences of the deterioration process in the reliability of a RC bridge girder. It was found that for traffic frequencies between 500 and 2000 cycles/day, the combined effect of corrosion and fatigue leads to appreciable lifetime reductions. When no fatigue damage is considered, the climate change effect only induces lifetime reductions ranging between 1.4 and 2.3%. Under cyclic loading, total lifetime could be reduced up to 7% by global warming action. These results highlight the importance of including the combined effect of corrosion and fatigue for comprehensive lifetime assessment.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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