Time-dependent reliability assessment and optimal design of corroded reinforced concrete beams

Abstract

It remains a significant challenge to quantitatively describe the corrosion of reinforced concrete (RC) structures under chloride penetration. Moreover, when considering uncertainties throughout the life cycle of corroded RC structures for assessing their safety, reliability, and optimal design, the complexity of the problem intensifies. To address these issues, this paper studies the time-dependent reliability analysis and optimal design of corroded RC beams. At first, the time-dependent reliability of beams is investigated by considering both the serviceability limit state (SLS), which corresponds to the corrosion initiation of the reinforced steel, and the ultimate limit state (ULS), associated with the bending failure of the beam. This analysis takes into account the time-dependent chloride diffusion coefficient and incorporates a stochastic process. The reliability is evaluated using the Monte Carlo Simulation (MCS) method and the cumulative distribution function (CDF) method. Subsequently, a time-dependent reliability-based design optimization (TRBDO) problem is formulated, and the PSO-MCS, a methodology incorporating a particle swarm optimization (PSO) algorithm and MCS is adopted to solve it. After optimization, the initial cost of the specific RC beam is reduced from 1351.879€ to 1247.075€, while the time-dependent reliability within [0, 100] years is improved from 0.6057 to 0.6508. The effectiveness of the CDF, MCS and PSO-MCS methods are demonstrated through reliability analysis and design examples of corroded RC beams.

Keywords

chloride penetration uncertainties time-dependent reliability analysis time-dependent reliability-based design optimization corroded reinforced concrete beams

Introduction

Reinforced concrete (RC) structures, such as bridges (Li et al., 2022), buildings (Li et al., 2020a), dams (Moran, 1956), and pipelines (Kong and Pesinis, 2017), generally have complicated structural forms, high construction costs, and harsh working environments. Once an accident happens, it will cause adverse social impacts and enormous economic losses. Therefore, it is of vital importance to evaluate the performance of RC structures (Bagheri et al., 2020; Ben Seghier et al., 2021). In this way, reliability and safety of structures can be ensured. There are two ways to improve the performance of the structure. One way is to increase the behaviour of RC structures in the design stage (Beulah et al., 2018), another way is to repair the old structures during the operation stage (Beulah et al., 2021).

However, the corrosion of steel bars in concrete is a significant cause of failure in RC structures. This damage can be categorized into two types: concrete carbonation and chloride penetration (Li et al., 2018a; Liu and Fang, 2012; Nogueira and Leonel, 2013; Paul et al., 2018). In practical engineering, chloride penetration occurs at a much higher rate than carbonation (Xu et al., 2011). Therefore, chloride penetration is more critical in causing reinforcement corrosion and structural damage.

Chloride ions have two primary sources (Hong, 1999): the first is chloride added during the mixing and pouring process, such as calcium chloride and sodium chloride. The second source is the external environment during the service life of the structure, including deicing salts used in winter, seawater, and sea breeze. Chlorides can diffuse into the concrete due to concentration gradients. When the chloride concentration reaches a certain threshold at the depth of a steel bar, the protective film on the surface of the bar is damaged by water, oxygen, and chloride (Otieno et al., 2011; Shao et al., 2020; Stambaugh et al., 2018). Consequently, the steel bar starts to corrode. Corrosion of steel reinforcements can lead to concrete fracture through cracking, delamination, and spalling of the concrete cover, reducing the cross-sectional areas of both the concrete and reinforcements. This significantly diminishes the serviceability, strength, safety, and service life of concrete structures (Saassouh and Lounis, 2012).

The penetration of chloride ions in concrete can be described well by Fick’s second law. In 1972, Collepardi et al. (1972) proposed a method to calculate the chloride diffusion coefficient based on Fick’s second law. This method gained widespread use and subsequent improvements (Bader, 2003; Chatterji, 1995; Chen and Qian, 2017; Ding and Chen, 2019; Mangat and Molloy, 1994; Suryavanshi et al., 2002). The chloride diffusion coefficient plays a crucial role in the transport of chloride in concrete. Many diffusion models based on Fick’s second law assume a constant chloride diffusion coefficient (Amey et al., 1998; Collepardi et al., 1972; Kong and Pesinis, 2017; Nogueira and Leonel, 2013; Saassouh and Lounis, 2012). However, in practice, especially in marine environments, the diffusion of chloride ions into concrete is nonlinear and unsteady (Kim et al., 2016; Wu et al., 2017). Thus, the actual chloride diffusion coefficient is time-dependent rather than constant.

Due to the complex construction process, changeable service environment, and diverse load forms of RC structures, there are various uncertainties such as geometric dimensions, material properties, and environmental loads in the design, construction, and use of the structure (Li et al., 2023b). These uncertainties will seriously affect the performance of the structure, leading to fluctuations in its performance and even failure. Therefore, how to properly describe these uncertainties plays a pivotal role in practical engineering. Ding et al. (2023c, 2023e, 2023f, 2023g) proposed many methods to effectively quantify the uncertainty and get its statistical distribution, such as the multivariant joint probability density distribution (PDF) function based on copula theory (Ding et al., 2023e), the model based on Bayes’ theorem (Ding et al., 2023f), and the Angular-linear model (Ding et al., 2023g). In addation, Li et al. (2023c) developed the phase-type fitting method to approximate the PDF of the positive dataset.

Owing to the heterogeneity of materials, complexity of environments, and uncertainties associated with them, it is necessary to employ probabilistic methods to study the corrosion damage of RC structures (Xiao, 1995). Furthermore, time-dependent reliability analysis of RC structures becomes necessary when considering the time-dependent chloride diffusion coefficient and stochastic characteristics of loads. With the development of machine learning methods (Ding et al., 2023b), it is possible to effectively construct the mapping relationship between the input parameters and the output responses, making the reliability analysis easier. Machine learning methods mainly include the radial basis function (Ding et al., 2023a), the neural network (Ding et al., 2023d), the Kriging surrogate model (Li et al., 2023a) and so on(Li et al., 2024).

In evaluating the structural reliability, two types of limit states are considered: the ultimate limit state (ULS) and the serviceability limit state (SLS) (Saassouh and Lounis, 2012). ULS focuses on safety, while SLS emphasizes applicability and durability (Li et al., 2020c). Both limit states need to be investigated. In addition to reliability analysis, the optimal design of RC structures is crucial for saving materials and striking a balance between safety and cost. Previous studies have examined the cost minimization problem of RC frame structures (Chutani and Singh, 2017) and considered seismic reliability and life-cycle costs for RC buildings (Chiu et al., 2013). It is essential to conduct an in-depth study on the optimal design of RC structures incorporating uncertain factors.

Considering the limitations of the aforementioned research, this paper aims to study the time-dependent reliability analysis and optimal design of corroded RC beams. The innovative points are threefold: 1) a reliability model of RC structures is formulated by considering the time-dependent chloride diffusion coefficient and incorporating a stochastic process; 2) the time-dependent reliability of RC structures is solved under both SLS (corrosion initiation of reinforced steel) and ULS (bending failure of the beam) scenarios; 3) the CDF method is developed to efficiently and accurately evaluate the time-dependent reliability.

Review of time-dependent reliability analysis

In engineering applications, it is common to encounter various uncertainties in the design, construction, and operation of structures. These uncertainties are often represented by random variables or stochastic processes. In the context of time-dependent reliability analysis, a general limit state function (LSF) that incorporates time factors can be formulated as G(X, Y(t), t), where X represents a vector of random variables, Y(t) represents a vector of stochastic processes, and t represents the time parameter. Within a defined time interval [0, T_L], a failure event is said to occur if there exists a time instant t within the interval [0, T_L] such that the value of the LSF becomes less than zero (Li et al., 2020b):

\exists t \in [0, T_{L}], G (X, Y (t), t) \leq 0

(1)

Therefore, the time-dependent failure probability, denoted as P_f (0, T), within a time period [0, T] (where 0 ≤ T≤T_L), is a monotonic increasing function with respect to T. This failure probability is defined as:

P_{f} (0, T) = \Pr (\exists t \in [0, T], G (X, Y (t), t) \leq 0), 0 \leq T \leq T_{L}

(2)

And time-dependent reliability is given by:

\begin{array}{l} R (0, T) = \Pr (\forall t \in [0, T], G (X, Y (t), t) > 0), 0 \leq T \leq T_{L} \\ = 1 - P_{f} (0, T), 0 \leq T \leq T_{L} \end{array}

(3)

In cases where stochastic processes are involved in the LSF, performing time-dependent reliability analysis becomes a complex task. This is because, initially, discretization is required to convert the stochastic processes into a set of random variables (Li and Kiureghian, 1993; Loève, 1977; Sudret and Kiureghian, 2000; Zhang and Ellingwood, 1994). One such discretization strategy is the Expansion Optimal Linear Estimation method (EOLE) (Li et al., 2019). The time interval [0, T] is discretized into s time nodes, represented as t_i, where i = 1, 2, …, s (with t₁ = 0 and t_s = T). Subsequently, EOLE is employed to reconstruct a Gaussian process Y(t) as follows:

Y (t) \approx μ_{Y} (t) + σ_{Y} (t) \sum_{i = 1}^{p} \frac{Z_{i}}{\sqrt{I_{i}}} \cdot ϕ_{i}^{T} \cdot C_{Y} (t)

(4)

where μ_Y(t), σ_Y(t) and ρ_Y(t_i,t_j) represent the mean function, standard deviation function, and autocorrelation function of Y(t), respectively. Z_i refers to uncorrelated standard normal variables. The eigenvalues and eigenvectors of the correlation matrix C = [ρ_Y(t_i,t_j)]_s×s are denoted as I_i and ϕ_i, respectively. C_Y (t) = [ρ_Y (t, t₁), ρ_Y (t, t₂), …, ρ_Y (t, t_s)]^T represents a time-dependent vector, and p (≤s) represents the number of terms in the summation, corresponding to the p major eigenvalues of C. These terms satisfy the following relation:

\sum_{i = 1}^{p} I_{i} / \sum_{i = 1}^{s} I_{i} > θ

(5)

where the value of θ is set to 0.99 in this paper. The stochastic process Y(t) is reconstructed through the aforementioned process, representing a linear summation of deterministic functions and a set of independent standard normal variables Z. As a result, [X, Y(t)] is transformed into [X, Z]. If there are ny stochastic processes included in the LSF, denoted as Y(t) = [Y₁(t), Y₂(t), …, Y_ny(t)]^T, and p₁, p₂, …, p_ny represent the numbers of standard normal random variables required to reconstruct each stochastic process, the dimension of Z becomes nz = p₁+p₂+…+p_ny. Therefore, Z = [Z₁, Z₂, …, Z_nz]^T. For simplicity, let W represent the random input parameters [X, Z]. If X = [X₁, X₂, …, X_nx]^T, then W = [X₁, …, X_nx, Z₁, …, Z_nz]^T = [W₁, …, W_nx, W_nx+1, …, W_nx+nz]^T, and the dimension becomes n = nx + nz.

After reconstructing the stochastic processes Y(t), the time-dependent LSF G(X, Y(t), t) is transformed into G(X, Z, t) = G(W, t). The reliability can then be evaluated using conventional reliability analysis methods. The following procedure outlines the time-dependent reliability analysis using Monte Carlo Simulation (MCS) in combination with the EOLE method:

Step 1: Generate N MC samples within the design space W^*:{W^*₁, W^*₂, …, W^*_N}, where W^*_j = [W^*_j1, W^*_j2, …, W^*_jn]^T for j = 1, 2, …, N.

Step 2: Calculate the LSF values at each time node t_i (i = 1, 2, …, s) using W^*_j (j = 1, 2, …, N). The number of function evaluations is given by N_call = s × N.

Step 3: Obtain the time-dependent failure probability P_f (0, T) based on the statistical analysis of G(W^*_j, t_i) for i = 1, 2, …, s and j = 1, 2, …, N:

P_{f} (0, T) = \frac{1}{N} \sum_{j = 1}^{N} I (W_{j}^{*}) |_{[0, T]}

(6)

I (W_{j}^{*}) |_{[0, T]} = {\begin{cases} 1, & \min_{1 \leq i \leq s} G (W_{j}^{*}, t_{i}) \leq 0 \\ 0, & o t h e r w i s e \end{cases}

(7)

Time-dependent reliability analysis of corroded reinforced concrete beams

Different LSFs can yield varying reliability results. In this section, we conduct time-dependent reliability analysis for corroded RC beams considering both the SLS and the ULS.

Time-dependent reliability analysis with serviceability limit state

The primary focus of this paper is on the corrosion of RC beams caused by chloride ion attack, where the penetration of chloride ions in concrete is described using Fick’s second law (Saassouh and Lounis, 2012):

\frac{\partial C_{x, t}}{\partial t} = \frac{\partial}{\partial x} (D \frac{\partial C_{x, t}}{\partial x})

(8)

where x represents the depth from the concrete surface, C_x,t denotes the concentration of chlorides at depth x and time t, and D represents the chloride diffusion coefficient.

In the case of a constant chloride diffusion coefficient D, and with the following initial condition and boundary condition, the Crank’s solution (Crank, 1975) can be obtained:

C_{0, t} = C_{s}, C_{x, 0} = 0

(9)

C_{x, t} = C_{s} [1 - \erf (\frac{x}{2 \sqrt{D t}})]

(10)

\erf (z) = 2 Φ (z \sqrt{2}) - 1

(11)

where C_s is the surface chloride concentration and erf() denotes the error function or Gauss error function.

However, in practical applications, the chloride diffusion coefficient may vary over time. A power law is commonly recommended to describe the time-dependent behavior of the chloride diffusion coefficient (Li and Ye, 2018):

D (t) = D_{r e f} {(\frac{t_{r e f}}{t})}^{m}

(12)

where m is the age factor, and D_ref is the chloride diffusion coefficient at the reference time t_ref = 28 days. Based on the experimental study by Mangat and Molloy (1994), m can be estimated as follows:

m = 2.5 (W / C) - 0.6

(13)

where W/C represents the water cement ratio.

As mentioned in reference (Saassouh and Lounis, 2012), the SLS refers to the normal use of a structure and includes considerations of excessive deformation, vibration, and local damage (e.g., cracking, spalling, corrosion, etc.). Once the chloride concentration at the location of the steel exceeds the chloride threshold level, the reinforcement starts to corrode, leading to cracking and spalling of the concrete cover. Thus, the LSF for corrosion initiation (i.e., SLS) can be formulated as follows (Yu et al., 2017; Zhang, 2018):

G (X, Y (t), t) = C_{c r} - C_{c, t} = C_{c r} - C_{s} [1 - \erf (\frac{c}{2 \sqrt{D (t) t}})]

(14)

where C_cr is the chloride threshold level, c represents the depth of the concrete cover over the steel, and C_c,t is the concentration of chlorides at depth c and time t.

By setting G(X, Y(T_i), T_i) = 0, the corrosion initiation time, T_i, can be solved using the following equation:

T_{i} = {\frac{c^{2}}{4 D_{r e f} t_{r e f}^{m}} {[e r f^{- 1} (1 - \frac{C_{c r}}{C_{s}})]}^{- 2}}^{\frac{1}{1 - m}}

(15)

Since the parameters C_cr, C_s, c, and D_ref are all random variables, T_i is also a random variable. Therefore, the SLS can be rewritten as:

G (X, Y (t), t) = T_{i} - t = {\frac{c^{2}}{4 D_{r e f} t_{r e f}^{m}} {[e r f^{- 1} (1 - \frac{C_{c r}}{C_{s}})]}^{- 2}}^{\frac{1}{1 - m}} - t

(16)

The time-dependent reliability can be computed using the MCS method. Note that in equation (16), the random variable T_i and the time parameter t are completely separated in form, allowing for the evaluation of reliability using the cumulative distribution function (CDF) method as described below.

By substituting equation (16) into equation (2), we obtain:

P_{f} (0, T) = \Pr (\exists t \in [0, T], T_{i} \leq t)

(17)

It can be observed that the essence of the above formula is the CDF of T_i. In other words, the time-dependent failure probability P_f (0, T), can be evaluated through the CDF of T_i, denoted as F_Ti(t). Let W = [X, Y(t)] = [X, Z]. The procedure for the CDF method is as follows:

Step 1: Generate N MC samples in the design space W^*: {W^*₁, W^*₂, …, W^*_N}, where W^*_j = [W^*_j1, W^*_j2, …, W^*_jn]^T for j = 1, 2, …, N.

Step 2: Calculate the corresponding T_i values using W^*_j for j = 1, 2, …, N. This allows for the estimation of the probability distribution based on the data of T_i.

Step 3: Obtain the time-dependent failure probability P_f (0, T), based on the CDF of T_i (i.e., F_Ti(t)).

In this regard, the number of function calls for the CDF method is N_call = N. Compared with the MCS method (N_call = s × N) described earlier, the computational costs can be significantly reduced.

Time-dependent reliability analysis with ultimate limit state

As is well known, the corrosion of reinforcement is followed by cracking and spalling of the concrete cover, leading to a reduction in the bearing capacity of the structure and eventual collapse.

Chloride-induced corrosion is typically characterized by highly localized corrosion, such as pitting corrosion. A corrosion rate of 1 μA/cm² results in a section loss of 11.6 μm/year (Jones, 1992). The reduction in diameter of a steel bar can be expressed as follows:

p (t) = 0.0116 λ i_{c o r r} (t) (t - T_{i})

(18)

where the corrosion current intensity or corrosion rate i_corr(t) is given in μA/cm², the unit of p(t) is mm, and λ is the ratio coefficient that typically varies from 4 to 8 (Stewart, 2004). Stewart and Mullard (2007) suggest expressing the corrosion rate as follows:

i_{c o r r} (t) = i_{c o r r} (1) α {(t - T_{i})}^{β}

(19)

where i_corr(1) represents the corrosion rate at the start of corrosion propagation. The constants α and β have values of α = 0.85 and β = −0.29 (Li et al., 2018b; Mohammadi Farsani and Keshtegar, 2015; Tarighat and Jalalifar, 2014). Substituting these values, the corrosion rate can be written as:

i_{c o r r} (t) = 0.85 i_{c o r r} (1) {(t - T_{i})}^{- 0.29}

(20)

Considering the influence of the water cement ratio W/C and the concrete cover depth c, the expression for i_corr(1) can be empirically defined as follows:

i_{c o r r} (1) = \frac{γ {(1 - W / C)}^{- 1.64}}{c}

(21)

where γ is a humidity-related parameter. It has been suggested that γ takes the value of 27 for an ambient relative humidity of 80% (Suo and Stewart, 2009) and 37.8 for 75% humidity (Vu and Stewart, 2000). By linear interpolation, a value of γ = 48.6 is adopted for a humidity ratio of 70%, which is a typical value for cities in Asia, Europe, America, and Australia (Vu et al., 2005).

The pit configuration of a steel bar is illustrated in Figure 1, and the cross-sectional area loss can be calculated using the following equations (Mahmoodian and Alani, 2015; Mohammadi Farsani and Keshtegar, 2015; Stewart, 2004; Val and Melchers, 1997):

A_{p} (t) = {\begin{cases} A_{1} + A_{2}, & p (t) \leq \frac{D_{0}}{\sqrt{2}} \\ \frac{π D_{0}^{2}}{4} - A_{1} + A_{2}, & \frac{D_{0}}{\sqrt{2}} < p (t) \leq D_{0} \\ \frac{π D_{0}^{2}}{4}, & p (t) > D_{0} \end{cases}

(22)

A_{1} = \frac{1}{2} [θ_{1} {(\frac{D_{0}}{2})}^{2} - a | \frac{D_{0}}{2} - \frac{p {(t)}^{2}}{D_{0}} |]

(23)

A_{2} = \frac{1}{2} [θ_{2} p {(t)}^{2} - a \frac{p {(t)}^{2}}{D_{0}}]

(24)

a = 2 p (t) \sqrt{1 - {[\frac{p (t)}{D_{0}}]}^{2}}

(25)

θ_{1} = 2 \arcsin (\frac{a}{D_{0}}), θ_{2} = 2 \arcsin (\frac{a}{2 p (t)})

(26)

Figure 1.

Pit configuration.

Here, D₀ represents the initial diameter of the steel bar. The net cross-sectional area of a corroded rebar at time t can be calculated using the following equations:

A (t) = A (0) - A_{p} (t) = \frac{π D_{0}^{2}}{4} - A_{p} (t) = {\begin{cases} \frac{π D_{0}^{2}}{4} - A_{1} - A_{2}, & p (t) \leq \frac{D_{0}}{\sqrt{2}} \\ A_{1} - A_{2}, & \frac{D_{0}}{\sqrt{2}} < p (t) \leq D_{0} \\ 0, & p (t) > D_{0} \end{cases}

(27)

If the reinforcement layout comprises n_s reinforcing bars with the same diameter D₀, the remaining cross-sectional area of reinforcing steel at time t due to pitting corrosion of n_s bars is:

A_{s} (t) = n_{s} A (t)

(28)

The loss of cross-sectional area of reinforcing bars in a steel beam can lead to safety problems, and the LSF for the ULS can be written as follows:

G (X, Y (t), t) = M_{u} (t) - M_{a} (t)

(29)

where M_u(t) and M_a(t) represent the resistance and applied load effect at time t, respectively. M_u(t) depends on the value of A_s(t), while M_a(t) may include a stochastic process modeling. Unlike the SLS of equation (16), where the CDF method can be applied, the time parameter t implicitly exists in M_u(t) and M_a(t). It is necessary to transform a stochastic process into a set of random variables using discretization strategies (e.g., EOLE) for the reliability analysis of the structure.

Numerical analysis of time-dependent reliability with different limit state functions

Numerical simulations are conducted through the commercial software MATLAB (Ferreira, 2009), and all analyses are carried out using the same computer with a Intel(R) Core(TM) i7-4790K CPU processor at 4.0 GHz with 16 GB RAM. Different LSFs may lead to different reliability results. In this section, the numerical analyses for corroded RC beams with SLS and ULS are all considered.

Reliability analysis with serviceability limit state

Numerical analysis of reliability with SLS is first presented, focusing on the corrosion initiation of reinforced steel. The parameter values used in this analysis, as reported in the literature (El Hassan et al., 2010; Shao et al., 2018; Weyers et al., 2003; Zhang et al., 2015), are summarized in Table 1.

Table 1.

Distribution of Random Parameters for the Analysis With SLS.

Parameter	Distribution	Mean	Coefficient of variation	Source
C _cr	Uniform	0.9 kg/m³	0.2	El Hassan et al. (2010)
C _s	Lognormal	3.5 kg/m³	0.2	Zhang et al. (2015)
c	Normal	30 mm	0.2	Weyers et al. (2003)
D _ref	Lognormal	12 × 10⁻¹² m²/s	0.2	Shao et al. (2018)

The age factor m in equation (13) is influenced by the water cement ratio W/C. For this example, we set W/C values of 0.3, 0.4, and 0.5, resulting in m values of 0.15, 0.4, and 0.65, respectively, as calculated using equation (13). The chloride diffusion coefficient D(t) is plotted in Figure 2 for D_ref = 12 × 10⁻¹² m²/s.

Figure 2.

The variation of D(t).

From Figure 2, it can be observed that D(t) decreases rapidly at the early stages and then decreases slowly over time. To ensure that D(t) decreases with time (Wu et al., 2017), the value of m should be limited to the range of 0-1. Gjorv (2011) suggested setting m to 0.4, which is consistent with empirical formulas (Lu et al., 2015). Therefore, we adopt m = 0.4 (i.e., W/C = 0.4) for this work. With this value, the equation (16) becomes:

G (X, Y (t), t) = T_{i} - t = {\frac{c^{2}}{4 D_{r e f} {(\frac{28}{365})}^{0.4}} {[e r f^{- 1} (1 - \frac{C_{c r}}{C_{s}})]}^{- 2}}^{\frac{1}{1 - 0.4}} - t

(30)

where t represents the time parameter ranging from 0 to 100 years. X = [C_cr, C_s, c, D_ref]^T are random variables, and Y(t) = [].

Reliability analysis with ultimate limit state

Second, numerical analysis of reliability with ULS is presented, focusing on the bending failure of a beam. As shown in Figure 3, we consider a rectangular cross-section beam with width b, height h, and n_s = 4 steel bars embedded in the concrete. A moment M_a(t) is applied to the beam.

Figure 3.

Layout of a RC beam cross-section.

The distribution parameters for this analysis, obtained from various sources (Aoues et al., 2009; Bastidas-Arteaga et al., 2008; El Hassan et al., 2010; Mohammadi Farsani and Keshtegar, 2015; Shao et al., 2018; Weyers et al., 2003; Zhang et al., 2015), are summarized in Table 2.

Table 2.

Distribution of Random Parameters for the Analysis With ULS.

Parameter	Distribution	Mean	Coefficient of variation	Autocorrelation function	Source
C _cr	Uniform	0.9 kg/m³	0.2		El Hassan et al. (2010)
C _s	Lognormal	3.5 kg/m³	0.2		Zhang et al. (2015)
c	Normal	30 mm	0.2		Weyers et al. (2003)
D _ref	Lognormal	12 × 10⁻¹² m²/s	0.2		Shao et al. (2018)
λ	Gumbel	6	0.2		Mohammadi Farsani and Keshtegar (2015)
D ₀	Normal	Φ18 mm	0.05		Mohammadi Farsani and Keshtegar (2015)
b	Normal	600 mm	0.05		Bastidas-Arteaga et al. (2008)
h	Normal	800 mm	0.05		Bastidas-Arteaga et al. (2008)
f _y	Lognormal	400 MPa	0.1		El Hassan et al. (2010)
f _c	Lognormal	30 MPa	0.15		El Hassan et al. (2010)
M_a(t)	Gaussian process	200 kN·m	0.25	exp(-((t₂- t₁)/τ)²)	Aoues et al. (2009)

Note: τ = 1 year in the autocorrelation function.

The bending moment capacity of the RC beam is expressed by:

M_{u} (t) = f_{y} A_{s} (t) [h_{0} - \frac{f_{y} A_{s} (t)}{2 f_{c} b}]

(31)

where f_y and f_c are the steel yield stress and concrete compressive strength, respectively. The effective height h₀ = h-c-D₀/2 is a normally distributed random variable. In this case, the equation (29) can be represented as:

G (X, Y (t), t) = M_{u} (t) - M_{a} (t) = f_{y} A_{s} (t) [h_{0} - \frac{f_{y} A_{s} (t)}{2 f_{c} b}] - M_{a} (t)

(32)

where t is the time parameter ranging within [0, 100] years. X = [C_cr, C_s, c, D_ref, λ, D₀, b, h, f_y, f_c]^T are random variables, and Y(t) = [M_a(t)] represents a Gaussian process. Substituted with γ = 48.6 and W/C = 0.4 into equations (18)–(21), one can get:

\begin{array}{l} p (t) = 0.0116 λ \cdot 0.85 \cdot \frac{48.6 \cdot {(1 - 0.4)}^{- 1.64}}{c} \cdot {(t - T_{i})}^{- 0.29} \cdot (t - T_{i}) \\ = 1.1075 \frac{λ}{c} {(t - T_{i})}^{0.71} \end{array}

(33)

Results and discussion

Numerical analysis of reliability with serviceability limit state

Since Y(t) is not involved in this case, we have W = [X, Y(t)] = [X, Z] = [C_cr, C_s, c, D_ref]^T. The MCS and CDF methods are all used to compute the time-dependent reliability. In the MCS method, N = 10⁵ MC samples of W are directly generated and the time-dependent failure probability can be evaluated using equations (6) and (7).

However, the CDF method utilizes N = 10⁵ MC samples to calculate the corresponding corrosion initiation time T_i, and the PDF can then be estimated using kernel density estimation, Gaussian fitting and lognormal fitting. These results are compared with the empirical PDF (i.e., the frequency histogram contour) in Figure 4.

Figure 4.

Comparison of different approximation methods for PDFs.

Figure 4 shows that the corrosion initiation time T_i can be well approximated as a lognormal random variable, consistent with the situation described in reference (Mohammadi Farsani and Keshtegar, 2015). Therefore, T_i is assumed to be a random variable following a lognormal distribution, where ln(T_i)∼N(μ, σ²). The maximum likelihood estimate is used to estimate the parameters as follows:

\hat{μ} = \frac{1}{N} \sum_{j = 1}^{N} \ln T_{i}^{(j)} = 2.6014

(34)

{\hat{σ}}^{2} = \frac{1}{N} \sum_{j = 1}^{N} {(\ln T_{i}^{(j)} - \frac{1}{N} \sum_{j = 1}^{N} \ln T_{i}^{(j)})}^{2} = 0.4756

(35)

where T_i^(j) (j = 1, 2, …, N = 10⁵) represents the j-th sample of T_i. The results of time-dependent failure probability using different methods are given in Figure 5 and Table 3.

Figure 5.

Comparison of the MCS and CDF methods for the analysis with SLS.

Table 3.

Failure Probability Over different Time Intervals for the Analysis With SLS.

Time interval [0, t]	MCS		CDF
Time interval [0, t]	P_f (0, t)	N _call	P_f (0, t)	N _call	Error (%)
[0, 10]	0.3471	11 × 10⁵	0.3324	10⁵	4.2351
[0, 20]	0.7020	21 × 10⁵	0.7163	10⁵	2.0370
[0, 30]	0.8665	31 × 10⁵	0.8769	10⁵	1.2002
[0, 40]	0.9396	41 × 10⁵	0.9426	10⁵	0.3193
[0, 50]	0.9709	51 × 10⁵	0.9713	10⁵	0.0412
[0, 60]	0.9859	61 × 10⁵	0.9848	10⁵	0.1116
[0, 70]	0.9921	71 × 10⁵	0.9915	10⁵	0.0605
[0, 80]	0.9960	81 × 10⁵	0.9951	10⁵	0.0904
[0, 90]	0.9977	91 × 10⁵	0.9970	10⁵	0.0702
[0, 100]	0.9988	101 × 10⁵	0.9982	10⁵	0.0601

The results obtained from the MCS method are so accurate that can be considered as the reference. Compared with the MCS method, it is shown from Figure 5 and Table 3 that the CDF method can evaluate the time-dependent reliability in an accurate and efficient manner. The time-dependent failure probability within [0, 70] years is greater than 0.99, indicating that the initial corrosion of reinforced steel is highly likely to occur.

Numerical analysis of reliability with ultimate limit state

It can be seen from equation (32) that Y(t) = [M_a(t)] is involved in the ULS, and the EOLE is first employed to reconstruct the Gaussian process M_a(t). Δt = 1 year is used to discretize the time interval [0, 100] years, resulting in a total of s = 101 time nodes. The first 98 dominant eigenvalues are considered to satisfy equation (5), which gives p = 98. Thus, 98 uncorrelated standard normal variables [Z₁, Z₂, …, Z₉₈]^T are used to reconstruct M_a(t). Three typical trajectories of the stochastic process are shown in Figure 6.

Figure 6.

Trajectories of the stochastic process.

Let W = [X, Y(t)] = [X, Z] = [C_cr, C_s, c, D_ref, λ, D₀, b, h, f_y, f_c, Z₁, Z₂,…, Z₉₈]^T, and N = 10⁵ MC samples of W are generated with the LSF values being computed accordingly at each time node. After that, the time-dependent failure probability can be calculated using equations (6) and (7). The results are presented in Figure 7.

Figure 7.

The results using the MCS method for the analysis with ULS.

It is clear that the variation trends of time-dependent failure probability are consistent with reference (Mohammadi Farsani and Keshtegar, 2015). In addition, the results of ULS are compared with that of SLS, as depicted in Figure 8.

Figure 8.

Comparison of the results with SLS and ULS.

It is evident from Figure 8 that different LSFs (SLS or ULS) yield varying results in reliability analysis. Comparing the results, it can be observed that the SLS tends to be more conservative than the ULS. This means that when using the SLS criterion, the lifetime of RC structures may be underestimated. Although the reinforcement in the RC beam undergoes corrosion, it still retains its bearing capacity. Consequently, the time-dependent failure probability over the [0, 100] years is close to 1 with SLS, whereas it is approximately 0.85 with ULS. In the early stages, the time-dependent failure probability with ULS exceeds that with SLS. This difference arises from the fact that ULS takes into account the load effect on the structure, which introduces additional uncertainties into the analysis compared to SLS.

Time-dependent reliability-based design optimization of corroded reinforced concrete beams

The previous research discussed the time-dependent reliability analysis of corroded RC beams. However, in practical engineering, it is important to consider not only the safety but also the economic aspects of the structure. Therefore, the optimization design of RC structures plays a significant role in material savings and cost reduction while ensuring safety. In this section, the TRBDO of corroded RC beams is discussed.

A typical model of time-dependent reliability-based design optimization

Similar to a reliability-based design optimization (RBDO) problem, a typical TRBDO model incorporates the time factor and can be expressed as follows:

\begin{array}{l} \min f (d) \\ s . t . R (0, T) = \Pr (\forall t \in [0, T], G (d, X, Y (t), t) > 0) \geq R \\ d^{L} \leq d \leq d^{U} \end{array}

(36)

where f(d) is the objective function, R(0,T)≥R is the time-dependent reliability constraint, G(d,X,Y(t),t) represents the LSF that includes the design variables d, the random parameter vector X, a vector of stochastic processes Y(t), and the time parameter t, d^U and d^L are the upper and lower bounds of the design variables d, respectively.

Time-dependent reliability analysis in time-dependent reliability-based design optimization

It is important to note that the LSF in equation (36) includes the design variables d, which differentiates it from the LSF in equation (3). Nonetheless, the reliability analysis procedure is similar. Using the MCS method, for a given design point d₀, the time-dependent reliability calculation formula in TRBDO is as follows:

R (0, T) = 1 - P_{f} (0, T) = 1 - \frac{1}{N} \sum_{j = 1}^{N} I_{j} |_{[0, T]}

(37)

where

I_{j} |_{[0, T]}

is an indicator function defined as:

I_{j} |_{[0, T]} = {\begin{cases} 1, & \min_{1 \leq i \leq s} G (d_{0}, W_{j}^{*}, t_{i}) \leq 0 \\ 0, & o t h e r w i s e \end{cases}

(38)

Equations (37) and (38) are essentially the same as equations (6) and (7) mentioned earlier.

Description of PSO-MCS algorithms

In this paper, the particle swarm optimization (PSO) algorithm (Kennedy and Eberhart, 2001) is utilized as the optimization algorithm due to its simplicity in program implementation and fewer adjustable parameters. In PSO, iterative optimization is carried out using a group of random particles that represent potential solutions. Each particle possesses two vectors: position and velocity. It adjusts its flying pattern based on individual experience and information sharing. Each particle updates its own velocity and position using the following equations:

v_{i j}^{k + 1} = ω v_{i j}^{k} + c_{1} r_{1} (p_{i j} - x_{i j}^{k}) + c_{2} r_{2} (g_{j} - x_{i j}^{k})

(39)

x_{i j}^{k + 1} = x_{i j}^{k} + v_{i j}^{k + 1}

(40)

where i = 1, 2, …, np represents the i-th particle, np denotes the total number of particles, j = 1, 2, …, nd represents the j-th dimension of the particle, nd signifies the dimensionality of the design space, v_ij^k and x_ij^k are the velocity and position of the j-th dimension of the i-th particle in the k-th iteration, ω is an inertia weight, c₁ and c₂ are the learning factors, r₁ and r₂ are random numbers between 0 and 1, pBest_i=(p_i1, p_i2, …, p_ind) represents the individual’s best position found thus far, and gBest=(g₁, g₂, …, g_nd) represents the global best position of the entire particle swarm found thus far.

For TRBDO problems, the PSO-MCS method (Li and Chen, 2019) consists of the following steps:

Step 1: Initialize a particle swarm of size np and set the initial positions and velocities. Each particle represents a feasible design point within the design space.

Step 2: Calculate the time-dependent reliability of each particle using the MCS method. Determine whether or not the reliability constraint is satisfied. If the reliability constraint is not met (δ = 1), apply a penalty function method by adding a large penalty value M to the objective function, i.e., f(d)+δ·M. Otherwise, if the reliability constraint is satisfied (δ = 0), proceed to the next step.

Step 3: Evaluate the fitness value of each particle to determine the historical best position of each particle (pBest) and the global best position of the entire particle population (gBest).

Step 4: Update the velocity and position of particles using equations (39) and (40).

Step 5: If a stopping criterion is met (usually a preset precision or a specified number of iterations), terminate the algorithm and output the optimal solution. Otherwise, return to Step 2.

The flowchart of the PSO-MCS algorithm is shown in Figure 9. It is worth noting that after a certain number of iterations, particles will automatically move away from regions that do not satisfy the constraint, leading to the achievement of an optimal solution.

Figure 9.

Flowchart of the PSO-MCS algorithm.

A time-dependent reliability-based design optimization example

A TRBDO problem for the RC beam is shown in Figure 10. The length of the beam is L = 6 m, and the cross section is rectangular with width b and height h. The number of steel bar inside the concrete is n_s = 4. And the beam is subjected to a moment M_a(t).

Figure 10.

A RC beam.

The initial cost of the RC beam, denoted as C_I, includes the costs of concrete, steel, and the framework. It can be calculated using the following equation:

C_{I} = C_{c} b h L + C_{s} ρ_{s} A_{s} L + C_{f} (2 h + b) L

(41)

where C_c, C_s and C_f represent the unit costs of concrete, steel, and the framework, respectively. The unit weight of steel is denoted as ρ_s, and A_s represents the initial cross-sectional area of reinforcing steel. The values of these parameters are provided in Table 4 and equation (42).

Table 4.

Deterministic Parameters.

Parameter	C _c	C _s	C _f	ρ_s (_kg/m³)	Source
Value	150.5€/m³	1.46€/kg	47€/m²	7850	Aoues et al. (2009)

A_{s} = n_{s} A (0) = 4 \cdot \frac{π D_{0}^{2}}{4} = π D_{0}^{2}

(42)

The TRBDO problem is formulated as follows:

\begin{array}{l} \min f (d) = 903 b h + 68766 π D_{0}^{2} + 282 (2 h + b) \\ s . t . \Pr (\forall t \in [0, 100], G (d, X, Y (t), t) > 0) \geq R \\ 0 . 65 m \leq b \leq 0.75 m \\ 0.85 m \leq h \leq 0.95 m \\ 0.0 17 m \leq D_{0} \leq 0.021 m \end{array}

(43)

where the time-dependent reliability constraint is set to R = 0.65. The LSF is defined as:

G (d, X, Y (t), t) = f_{y} A_{s} (t) [h_{0} - \frac{f_{y} A_{s} (t)}{2 f_{c} b}] - M_{a} (t)

(44)

In this equation, h₀ = h-c-D₀/2 represents the effective height of the beam, t is the time parameter varying within [0, 100] years, d = [b, h, D₀]^T represents the deterministic design variables, X = [C_cr, C_s, c, D_ref, λ, f_y, f_c]^T represents the random variables, and Y(t) = [M_a(t)] is a Gaussian process. The specific distribution parameters for the example are provided in Table 5.

Table 5.

Distribution of Random Parameters for the TRBDO Example.

Parameter	Distribution	Mean	Coefficient of variation	Autocorrelation function	Source
C _cr	Uniform	0.9 kg/m³	0.2		El Hassan et al. (2010)
C _s	Lognormal	3.5 kg/m³	0.2		Zhang et al. (2015)
c	Normal	30 mm	0.2		Weyers et al. (2003)
D _ref	Lognormal	12 × 10⁻¹² m²/s	0.2		Shao et al. (2018)
λ	Gumbel	6	0.2		Mohammadi Farsani and Keshtegar (2015)
f _y	Lognormal	400 MPa	0.1		El Hassan et al. (2010)
f _c	Lognormal	30 MPa	0.15		El Hassan et al. (2010)
M_a(t)	Gaussian process	200 kN·m	0.25	exp(-((t₂- t₁)/τ)²)	Aoues et al. (2009)

Note: τ = 1 year in the autocorrelation function.

To discretize the time interval, Δt = 1 year is used, resulting in a total of s = 101 time nodes. The EOLE discretization strategy is applied to reconstruct the stochastic process. The first 98 dominant eigenvalues are considered to satisfy equation (5), which gives p = 98. Thus, 98 uncorrelated standard normal variables [Z₁, Z₂, …, Z₉₈]^T are used to reconstruct M_a(t). And then, W = [X, Y(t)] = [X, Z] = [C_cr, C_s, c, D_ref, λ, f_y, f_c, Z₁, Z₂, …, Z₉₈]^T.

After discretizing the stochastic processes, the PSO-MCS method is applied to solve the TRBDO problem. The PSO population size is set to 20, and the maximum number of iterations is 100. The time-dependent reliability of each particle is computed using the MCS method with N = 10⁵ sample points for W = [X, Z]. The response values of the 10⁵ MCS sample points at the 101 time nodes are calculated, and the TRBDO problem defined in equation (43) is solved using the PSO algorithm. The results are presented in Table 6.

Table 6.

Comparison of Results Before and After Optimization.

	Before optimization	After optimization
f (€)	1351.879	1247.075
b (m)	0.7	0.6502
h (m)	0.9	0.8516
D₀ (m)	0.019	0.01965
R(0, 100)	0.6057	0.6508

Table 6 provides a comparison of the results before and after optimization. It can be observed that the reliability of the optimized RC beam is improved, satisfying the reliability constraint. The initial cost is reduced by 7.696%, while the bending moment capacity is improved. This improvement is attributed to the increase in the initial diameter of the steel bar D₀, which enhances the bending performance of the beam. The iterative process is illustrated in Figure 11.

Figure 11.

Iterative process of the PSO-MCS algorithm.

Conclusions

In this study, the time-dependent reliability analysis and optimal design of corroded RC beams are investigated. The time-dependent reliability is analyzed considering both the corrosion initiation of reinforced steel (SLS) and the bending failure of the beam (ULS). The time-dependent chloride diffusion coefficient and a stochastic process are all incorporated into the analysis. Furthermore, a TRBDO model for RC beams is developed and a PSO-MCS algorithm is then employed to solve the TRBDO problem. The MCS method is used to compute the time-dependent reliability, while the PSO algorithm searches for the optimal design point.

The following conclusions can be drawn:

(1) The CDF method is proved to be effective in calculating the time-dependent reliability.

(2) The reliability results obtained using SLS are more conservative compared to those obtained using ULS. This is because the reinforcement in an RC beam still retains bearing capacity even it is corroded.

(3) After optimization, the initial cost of the specific RC beam is reduced from 1351.879€ to 1247.075€, while the time-dependent reliability within [0, 100] years is improved from 0.6057 to 0.6508.

(4) In the TRBDO example, the optimized RC beam exhibits reduced cross-sectional area but increased initial diameter of the steel bar, resulting in improved bending performance.

Due to the fact that this work focuses on the establishment and solution of the reliability model for RC structures, only the results using numerical simulation are obtained. In the next study, the results are supposed to be gained through experiments or real examples. And the numerical results need to be compared with experimental or practical results.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (nos. 52275266, 11832013 and 11572134), Hubei Provincial Department of Education Science and Technology Research Project (no. Q20221714), and the Opening Foundation of Hubei Key Laboratory of Digital Textile Equipment (nos. DTL2023019 and DTL2022012).

ORCID iD

Junxiang Li

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