Improved time-dependent seismic fragility estimates for deteriorating RC bridge substructures exposed to chloride attack

Abstract

RC bridge substructures exposed to chloride environments inevitably suffer from corrosion of reinforcement embodied in concrete. This deterioration issue leads to the loss of reinforcement areas and a reduction in seismic capacity of reinforced concrete (RC) bridge substructures. To quantify the effect of steel corrosion on seismic fragility estimates, this paper proposes an improved time-dependent seismic fragility framework by taking into account the increase in the corrosion rate after concrete cracking and the reduction in seismic capacity of RC bridge substructures during the service life. Additionally, an analytical method based on a back propagation artificial neural network (BP-ANN) is proposed to provide probabilistic capacity estimates of deteriorating RC substructures. A three-span T-shaped girder bridge is selected as a case study bridge to provide improved time-dependent seismic fragility estimates that consider uncertainties in the material properties, geometric parameters, deterioration process and ground motions. The obtained fragility curves show that there is a nonlinear increase in the exceedance probability of deteriorating RC bridge substructures for different damage states during the service life. In addition, time-dependent seismic fragility analysis shows that the cases of considering only the effect of an increase in seismic demand or the reduction in seismic capacity as well as neither of them may lead to a significant underestimation of the seismic vulnerability of deteriorating RC bridge substructures.

Keywords

bridge substructures seismic fragility chloride-induced deterioration after-cracking corrosion rate time-dependent capacity probability

Introduction

As transportation hubs, bridge structures play an important role in promoting national economic growth. A reinforced concrete (RC) bridge substructure is a critical bridge component that suffers the lateral force induced by potential seismic hazards (Zhang et al., 2016). However, RC bridge substructures are confronted with deterioration problems caused by the corrosion of the reinforcement, especially for substructures in typical chloride environments, such as exposure to deicing salts and coastal regions (Akiyama et al., 2011). Chloride-induced corrosion gradually weakens the cross-sectional area of the reinforcement, while the accumulation of corrosion products results in the cracking of concrete protective layers. These factors lead to a reduction in the structural stiffness and bearing capacity of RC bridge substructures. As a consequence, a deteriorating RC bridge substructure in a seismic zone will become vulnerable and may not be able to withstand the seismic load due to the effect of chloride contamination.

Recent decades have witnessed tremendous efforts toward investigating the joint effects of structural deterioration and seismic hazards on RC bridge structures and components. Choe et al. (2009) developed a probabilistic deterioration model to quantify the corrosion process of reinforcement embedded in concrete and integrated the model into a seismic demand model to obtain time-dependent seismic fragility curves of RC bridges. In addition to the aging of columns exposed to chloride attack, Ghosh and Padgett (2010) also considered the potential effect of steel bearing corrosion to establish a time-dependent seismic fragility model for aging bridges. Cui et al. (2018) investigated the time-variant seismic fragility of RC bridge substructures by developing a deterioration model that takes into account the increase in steel corrosion after concrete cracking. The experimental results obtained by Cao et al. (2013) and Poursaee and Hansson (2008) suggested an increase in the steel corrosion rate after concrete cracking and further underlined the necessity to consider this factor in the fragility and reliability assessment of deteriorating RC components.

Normally, structural seismic fragility can be defined as a conditional probability that structural demands reach or exceed structural capacities under a given ground motion intensity (Long et al., 2015; Ramanathan et al., 2015). For RC bridge substructures under chloride attack, due to the inevitable deterioration, the seismic demand and capacity vary with time, eventually leading to variation in the seismic damage state or failure probability. However, almost all existing literature (Alipour et al., 2011; Dong et al., 2013; Guo et al., 2014, 2015; Decò and Frangopol, 2013) concentrating on the time-dependent fragility analysis of RC bridge substructures considers only the effect of corrosion-induced deterioration on seismic demand, and little emphasis is placed on the study of seismic capacity reduction in corroding RC substructures. For simplicity, these studies assume that the seismic capacity of RC substructures takes a constant value that is the same as that of the pristine structural state. As a result, this approach may lead to an underestimation of the failure probability of RC substructures under seismic hazards. Currently, some researchers (Aquino and Hawkins, 2007; Enright and Frangopol, 1998; Ghosh and Sood, 2016) have highlighted the significant decrease in seismic capacity of deteriorating RC substructures by experimental approaches. Therefore, to better understand the seismic performance of deteriorating RC bridge substructures, it is necessary to conduct a time-dependent fragility analysis by employing a more precise corrosion rate model while considering the reduction in seismic capacity from a probabilistic perspective.

This paper presents a probabilistic framework to develop the time-dependent seismic fragility curves of deteriorating RC bridge substructures under chloride attack. In this framework, a three-phase corrosion rate model that considers the effect of concrete cracking is integrated into deterioration models to better simulate the influence of steel corrosion on the seismic demand and capacity of RC bridge substructures. In addition, a probabilistic approach that integrates a back propagation (BP) neural network and uniform design (UD) is proposed to obtain the time-varying seismic capacity distributions for the subsequent fragility estimates of RC bridge substructures. As a case study, the proposed framework is applied to the RC substructures of a typical three-span girder bridge to study its time-dependent seismic fragility under chloride-induced corrosion.

Corrosion deterioration modeling

Corrosion initiation

Chloride ions, accumulated in the surface of RC bridge members, gradually permeate to the reinforcement bars through the concrete layers. When the chloride content at the surface of steel reinforcement reaches a critical threshold value, steel corrosion is expected to happen (Kassir and Ghosn, 2002). Fick’s second law is an effective approach to simulate the ingress of chloride ions into RC components. At time t, the chloride ion concentration $C (x, t)$ of concrete at depth x can be expressed by (Choe et al., 2009; Cui et al., 2018; Ghosh and Padgett, 2010):

C (x, t) = C_{0} [1 - e r f (\frac{x}{2 \sqrt{D_{c} t}})]

(1)

where x is the concrete depth from the surface; t is the service life in years; $\erf (\cdot)$ is the Gaussian error equation, which is represented as $\erf (θ) = \frac{2}{\sqrt{π}} \int_{0}^{θ} e^{- p^{2}} dt$ ; $D_{c}$ is the diffusion coefficient; and $C_{0}$ is the chloride ion concentration at the concrete surface and is represented as:

C_{0} = A_{cs} (w / c) + ε_{cs}

(2)

where $w / c$ is the water-cement ratio and $A_{cs}$ and $ε_{cs}$ are model parameters involved in the uncertainty estimation of $C_{0}$ .

When the chloride concentration at the reinforcement surface reaches the critical value $C_{cr}$ , the initial time of corrosion of the reinforcement, denoted as $t_{corr}$ , can be obtained by equation (1); thus, $t_{corr}$ can be written as (Hoffman and Weyers, 1996):

t_{corr} = \frac{x^{2}}{4 D_{c}} {[er f^{- 1} (1 - \frac{C_{cr}}{C_{0}})]}^{- 2}

(3)

It should be noted that the initial time of corrosion is an probabilistic parameter that varies significantly with the structural dimensions and exposure conditions. The critical RC members considered in this study are RC bridge substructures exposed to deicing salt environments since deicing salts are extensively used in bridges for the removal of ice and snow (Vu and Stewart, 2000). The random variables involved in equation (3) and their corresponding statistical parameters, taken from related literature (Ghosh and Padgett, 2010; Li et al., 2016) and field data, are listed in Table 1.

Table 1.

Statistical parameters of the considered random variables.

Parameter	Unit	Distributiontype	Mean	St. dev.
x	mm	Normal	50	6
D₀	mm²/a	Normal	129	12.9
C_cr	kg/m²	Normal	1.4	0.2
A_cs	—	Normal	10.35	0.71
ε_cs	—	Normal	0	0.58
w/b	—	Deterministic	0.4	—

Based on the information given in Table 1, a Monte Carlo simulation (MCS) with a sample size of 20,000 is adopted to analyze the distribution type and statistical parameters of the initial time of corrosion. The simulation results show that a lognormal distribution exhibits a good fit to the random samples generated by MCS. The mean and standard deviation are 11.5 years and 4.1 years, respectively, for the initial time of corrosion of the studied RC substructures.

Corrosion propagation

Once the sustained ingress of chloride ions leads to damage or destruction of the protective passive film formed on the reinforcement surface, reinforcement corrosion initiates, and the effective cross-sectional area diminishes with time. Given that the cross-sectional shape of a corroded rebar is irregular, a uniform corrosion model is an effective approach to account for this for simplicity (Choe et al., 2009; Dong et al., 2013; Ghosh and Padgett, 2010; Stewart and Rosowsky, 1998a; Vu and Stewart, 2000). Then, the time-dependent residual cross-sectional area of steel reinforcement can be represented as follows (Vu and Stewart, 2000):

A (t) = {\begin{matrix} d_{0}^{2} \frac{π}{4} for t \leq t_{corr} \\ {[d (t)]}^{2} \frac{π}{4} for t \geq t_{corr} \end{matrix}

(4)

where $d_{0}$ is the initial diameter of a single steel reinforcement and $d (t)$ is the effective residual diameter of the steel reinforcement at time t and can be expressed as:

d (t) = d_{0} - 2 \int_{t_{corr}}^{t} λ (t) dt

(5)

where $λ (t)$ is the corrosion rate of the steel reinforcement at time t.

After the onset of steel corrosion, the corrosion rate becomes the crucial factor during corrosion propagation and structural deterioration. Some previous researchers (Ghosh and Padgett, 2010; Stewart and Rosowsky, 1998a, 1998b) considered the corrosion rate to be constant during the service life. However, this approach fails to accurately reveal the time-varying law of structural deterioration. Although a widely used time-variant corrosion rate model was proposed by Vu and Stewart (2000), it assumes that the corrosion rate decreases continuously over the full service life of bridges. Experiments conducted by Cao et al. (2013) demonstrated that the reinforcement corrosion rate exhibits a sustained increase after concrete cracking due to easier access for chlorides and other electrochemical reactants. Clearly, the Vu and Stewart model, shown in Figure 1, does not take this increase into consideration and may underestimate the influence of corrosion on long-term structural performance.

Figure 1.

Steel corrosion rate models proposed by Cui et al. (2018) and Vu and Stewart (2000).

To overcome this limitation, Cui et al. (2018) established an improved three-phase corrosion rate model (shown in Figure 1) that fully considers the increase after concrete cracking based on available data and literature. Thus, the improved time-dependent corrosion rate model $λ (t)$ employed in this study can be represented as

\begin{matrix} λ (t) = {\begin{matrix} λ_{1} (t) = 0.0116 \cdot i_{corr, 0} \cdot 0.85 \cdot (t - t_{corr})^{- 0.29} for t_{corr} < t \leq t_{cr} \\ λ_{2} (t) = (t - t_{cr}) \cdot \frac{λ_{3} (t_{Wcr}) - λ_{1} (t_{cr})}{t_{Wcr} - t_{cr}} + λ_{1} (t_{cr}) for t_{cr} < t \leq t_{Wcr} \\ λ_{3} (t) = (4.5 - 26 λ_{1} (t)) \cdot λ_{1} (t) for t > t_{Wcr} \end{matrix} \end{matrix}

(6)

where $i_{corr, 0}$ is the steel corrosion current density at the beginning of corrosion and is expressed by

i_{corr, 0} = 37.8 (1 - w / c)^{- 1.64} / d_{c}

(7)

where $d_{c}$ is the depth of the concrete cover and $w / c$ is the previously defined water-to-cement ratio.

As shown in equation (6), $t_{cr}$ and $t_{Wcr}$ are two important time parameters that have significant effects on corrosion propagation. $t_{cr}$ is the initial cracking time of concrete and can be determined by:

t_{cr} = {[\frac{p_{cr} \cdot d_{c}}{0.52494 {(1 - w / c)}^{- 1.64}}]}^{1.40845} + t_{corr}

(8)

where $p_{cr}$ is the critical corrosion depth of reinforcement corresponding to the initial cracking state of concrete. Similarly, $t_{Wcr}$ is defined as the severe cracking time of concrete, at which time the corrosion rate reaches its peak value before it starts to decrease to a steady value. This time can be obtained through the following equation by numerical methods (Cui et al., 2018):

p_{Wcr} = p_{cr} + \int_{t_{cr}}^{t_{Wcr}} λ (t) dt

(9)

where $p_{Wcr}$ is the critical corrosion depth of reinforcement that corresponds to the severe cracking state of concrete.

In addition, the decreases in reinforcement strength and ductility caused by corrosion are considered in this study. The model proposed by Du et al. (2005) is used to describe the change trend of the yield strength of corroded rebars and can be written as

f_{y} (t) = (1 - 0.005 Q_{corr}) f_{y 0}

(10)

where $f_{y} (t)$ is the reinforcement yield strength at time t; $f_{y 0}$ is the initial yield strength of reinforcement; and $Q_{corr}$ is the reinforcement corrosion ratio (%) in terms of area loss and can be expressed as

Q_{corr} (t) = \frac{A_{0} - A (t)}{A_{0}} \times 100 %

(11)

where $A_{0}$ is the initial area of reinforcement, calculated from $A_{0} = π d_{0}^{2} / 4$ .

Moreover, the ductility of corroded reinforcement can be considered in terms of a reduction in the ultimate strain of reinforcement. The time-dependent function recommended by Zhang et al. (2007) for the ultimate strain of reinforcement can be written as

ε_{u} (t) = (1 - 0.0137 Q_{corr}) ε_{u 0}

(12)

where $ε_{u} (t)$ is the reinforcement ultimate strain at time t and $ε_{u 0}$ is the initial ultimate strain of the reinforcement.

The reduction in the compressive strength of confined concrete induced by steel corrosion is also considered in this study. For the circular RC columns of the case study bridge, the degradation in compressive strength of confined concrete can be simulated using the following theoretical stress-strain model proposed by Mander et al. (1988):

f_{c, core} (t) = f_{c, cover} [2.254 \sqrt{1 + \frac{7.94 f_{l} (t)}{f_{c, cover}}} - \frac{2 f_{l} (t)}{f_{c, cover}} - 1.254]

(13)

where $f_{c, core} (t)$ is the time-dependent compressive strength of confined concrete, $f_{c, cover}$ is the compressive strength of unconfined concrete, and $f_{l} (t)$ is the time-variant effective lateral confining pressure assumed to be uniformly distributed over the surface of confined concrete.

Improved time-dependent fragility estimates of deteriorating RC substructures

The seismic fragility of an RC bridge substructure located in a seismic zone can be defined as a conditional probability that the seismic demand (D) on the RC substructure exceeds the capacity (C) at a designated seismic intensity (Nielson and DesRoches, 2007). Deteriorating RC substructures are expected to undergo a continuous decrease in seismic performance. thus, a time-dependent fragility expression is a better choice to describe the seismic vulnerability during the service life. After introducing the time variable t and assuming that both the seismic demand and capacity follow a lognormal distribution, the time-dependent fragility estimates of RC substructures can be expressed as (Cui et al., 2018; Ghosh and Sood 2016)

P [D (t) > C (t) | IM] = [\frac{\ln (S_{D} (t) / S_{C} (t))}{\sqrt{β_{D | IM}^{2} (t) + β_{C}^{2} (t)}}]

(14)

where IM is the intensity parameter for the selected earthquakes; $S_{D} (t)$ is the median estimate of seismic demand at time t; $S_{C} (t)$ is the median estimate of seismic capacity at time t; $β_{D | IM} (t)$ is the logarithmic dispersion of seismic demand at time t; and $β_{C} (t)$ is the logarithmic dispersion of seismic capacity at time t.

Probabilistic seismic demand model

Based on the work conducted by Cornell et al. (2002), the time-dependent seismic demand model (PSDM) of a deteriorating RC substructure can be written as

\ln (S_{D} (t)) = a (t) + b (t) \ln (IM)

(15)

where $a (t)$ and $b (t)$ are the regression parameters of the seismic demand at time t.

To obtain these regression parameters, various levels of ground motions have to be considered. These ground motions must be representative of the seismic characteristics of the site. For earthquake-prone areas, the existing earthquake data are the preferred choice. For regions where actual earthquake data are scarce, ground motion records from other places and artificial ground motions are alternatives. Similar with the previous researches (Cui et al., 2018; Li et al., 2020; Wu et al., 2016), the selection principle used in this study is to guarantee that the mean acceleration spectra of the selected ground motions matches well the design response spectrum given by the seismic design codes (JTG/T B02-01, 2008). Additionally, brand ranges of magnitude and hypocentral distance are required for the selected ground motions to account for the uncertainty of earthquakes. Then, the selected ground motions are paired with finite element models of deteriorating RC substructures that consider the uncertainties inherent in the deterioration process, material properties and structural geometric characteristics. Following a nonlinear time-history analysis, the maximum seismic demands placed on RC substructures are recorded. Regression analyses are used to obtain PSDMs of RC substructures with different service lives.

Probabilistic seismic capacity model

In fragility theory, capacity limit states, also known as damage limit states, are usually defined as some specific levels beyond which a bridge structure or component will lose part or full functionality (Long et al., 2015; Wu et al., 2016). The seismic capacity limit states of deteriorating RC substructures under chloride attack inevitably change with time. However, current studies (Alipour et al., 2011; Decò and Frangopol, 2013; Dong et al., 2013; Guo et al., 2014, 2015) tend to assume that the seismic capacity is invariant over time in fragility analysis for simplicity. To address this issue, this paper presents a probabilistic method to provide seismic capacity estimates of deteriorating RC substructures by considering different sources of uncertainties.

Definition of seismic capacity levels

Curvature at the base section of an RC column is a widely recognized damage measure that describes the different damage limit states of deteriorating RC substructures (Alipour et al., 2011; Cui et al., 2018; Ghosh and Padgett, 2010). Following the research work done by Pan et al. (2007), four critical points, shown in a typical moment-curvature diagram in Figure 2, correspond to slight, moderate, extensive, and complete damage states of RC substructures, respectively.

Figure 2.

Definitions of different damage states (capacity levels).

As illustrated in Figure 2, the first critical curvature point $ϕ_{S}$ , associated with the slight damage state of RC substructures, corresponds to the first yielding of steel reinforcement in columns. The second critical curvature point $ϕ_{M}$ , associated with the moderate damage state of RC substructures, can be obtained by an elastic-perfectly plastic idealization of the moment-curvature diagram. This point is determined by (Pan et al., 2007):

ϕ_{M} = \frac{M_{n}}{M_{y}} ϕ_{S}

(16)

where $ϕ_{S}$ and $M_{y}$ are the curvature and corresponding moment at the first yielding of steel reinforcement and $M_{n}$ and $ϕ_{M}$ are the equivalent moment and curvature, respectively.

The third critical curvature point $ϕ_{E}$ , associated with the extensive damage state, corresponds to the moment reaching its peak value. The fourth critical curvature point $ϕ_{C}$ , associated with the complete damage state, corresponds to the maximum flexural deformation that RC substructures can resist.

For RC substructures at different deterioration levels, these four critical curvature points can be not only determined by an experimental approach but also calculated by pushover analysis. However, an experimental approach requires considerable manpower and material resources. Thus, pushover analysis, which is used in this paper, is a more economical and efficient way to obtain the damage limit states or capacity levels of deteriorating RC substructures (Ghosh and Sood, 2016).

BP artificial neural network

Different sources of uncertainty exist in the structural dimensions and material properties of deteriorating RC substructures and the external load applied to structures (Cui and Ghosn, 2019). These factors necessarily affect the probabilistic characteristics of the seismic capacity of RC substructures. MCS can be directly used with pushover analysis to provide a probabilistic seismic capacity estimate. However, each sample generated by MCS needs a time-consuming deterministic finite element analysis, which significantly reduces the efficiency of probabilistic capacity estimates (Bichon et al., 2008).

To increase the efficiency of seismic capacity estimates of deteriorating RC substructures, a BP-ANN is used herein as an alternative scheme to time-consuming pushover analysis by establishing a nonlinear function model between the seismic capacity measure and the associated random variables.

A BP-ANN is one kind of feed-forward network structure that includes an input layer, hidden layer, and output layer (Yi et al., 2007). Nodes or neurons in different layers are connected through weight parameters. In the generation process of a BP-ANN, all the weight parameters are continuously updated by an error BP mechanism to minimize the gap between the actual outputs and the expected outputs (Cheng et al., 2008; Sharifi et al., 2019). A typical architecture of a 3-layer BP-ANN is shown in Figure 3. Each node or neuron in different layers performs by summing up its weighted inputs and then passing this result to the next layer through a predefined transfer function.

Figure 3.

An architecture of a 3-layer BP-ANN.

For an RC substructure at a specific degree of deterioration, the random variables affecting the seismic capacity are chosen as input samples, whereas the critical curvature points of the damage limit states are taken as output samples. Subsequently, both sets of samples are used as training samples to generate the BP-ANN model for seismic capacity estimates of deteriorating RC substructures. Given that each component of the training samples may be on a different scale, all the training data need to be scaled or normalized before importing them into the neural network (Cheng et al., 2008).

Uniform design

As discussed above, each training sample used in the BP-ANN model needs a time-consuming pushover analysis; therefore, the size of the training samples is a major factor that influences the generation efficiency of the BP-ANN. The traditional method of generating the samples randomly has a very high requirement for sample size, and the sample quality cannot be guaranteed (Cheng, 2013). To overcome this problem, UD is adopted herein to generate the training samples used for the BP-ANN model for the seismic capacity estimates of deteriorating RC substructures.

The function of UD is to produce a group of experimental points that are uniformly distributed in the design space. This is realized through the successful application of a number-theoretic method (Fang, 1981). The experimental points generated by UD have great representativeness in the given design domain. In addition, this method can accommodate the largest possible number of levels for each factor (random variable) in the experimental design (Zhang et al., 1998).

When the UD method is used in the generation of training samples, the entire design space is uniformly divided by the number of levels of random variables that affect the seismic capacity of deteriorating RC substructures. Then, a particular UD table is adopted to produce the desired training samples for the BP-ANN for seismic capacity estimates.

Procedure of the proposed method

The procedure of the proposed method for obtaining probabilistic seismic capacity estimates of deteriorating RC bridge substructures is summarized as follows:

Select the RC substructure deteriorating states of interest, and obtain the material and structural parameters of the corresponding states based on the improved deterioration model of RC substructures;

Determine the statistical characteristics of the random variables that affect the seismic capacity of RC substructures; then, the UD method is used to generate the input samples for the BP-ANN;

Substitute the input samples into finite element software for pushover analysis, and then obtain the corresponding output samples ( $ϕ_{S}$ , $ϕ_{M}$ , $ϕ_{E}$ , $ϕ_{C}$ );

Scale the input and output samples, and import the scaled samples for training to establish a BP-ANN model that represents the relationship between the selected random variables and capacity measures of deteriorating RC substructures;

A large number of input samples are generated randomly by MCS, and then the generated BP-ANN serves as an alternative to finite element software to predict the damage states of RC substructures. Finally, the results are used to obtain the statistical parameters and distribution types by statistical analysis.

Case study

Bridge description and modeling

The bridge selected for the case study is a representative three-span prestressed concrete bridge under chloride attack. This kind of bridge is chosen due to its prevalence and accounts for over one-third of all bridges in China. A cross section of the superstructure of the selected bridge consists of five T-shaped girders with a height of 1.5 m, and the substructure is a double-column bridge pier with a bent cap. The longitudinal reinforcement ratio of each RC column is approximately 0.7%. The substructure is supported by a cast-in-situ pile foundation with a height of 15.6 m. The superstructure and substructure are connected by plate-type elastomeric bearings. Figure 4 presents the configuration and simulation details of the case study bridge.

Figure 4.

Layout and simulation details of the case study bridge.

OpenSees software is employed herein to establish a finite element model of the case study bridge and to conduct a subsequent nonlinear time-history analysis. An elastic beam-column element is used to simulate the superstructure since it is expected to behave linearly under ground motions. A nonlinear beam-column element with fiber-defined cross sections is adopted to model the seismic behavior of RC bridge substructures (Ghosh and Padgett, 2010). For the fiber cross sections of RC columns, the stress-strain relationship of the unconfined and confined concrete is simulated using Concrete 04 material, whereas the nonlinearity of longitudinal steel rebars is modeled using Steel 02 material. Both of these material models are readily available in the OpenSees database (OpenSees Manual, 2009). Equivalent soil springs with linear behavior are utilized to model the soil-structure interaction. The stiffness of these springs is determined through the “m” method recommended by the guidelines for seismic design of Chinese highway bridges (JTG/T B02–01, 2008). The plate-type elastomeric bearings are simulated by a zero-length element with perfectly elastic-plastic material, where the initial stiffness is determined by the plate geometry (Nielson and DesRoches, 2007). The dead load caused by the structural weight is directly applied to the elements of the superstructure and substructure.

Although the superstructure and substructure of RC bridges suffer reinforcement corrosion simultaneously, substructure deterioration has the most significant effect on the seismic performance of RC bridges. Thus, the abovementioned deterioration model is used only in RC substructures for simplicity. The finite element model of RC bridges is continuously updated with the deterioration process.

Deterioration modeling

The initial time of corrosion $t_{corr}$ of the studied RC substructures was obtained in section 1.1. After reinforcement corrosion, the initial cracking time $t_{cr}$ and severe cracking time $t_{Wcr}$ of the concrete can be determined to be 15.47 and 19.31 through equations (8) and (9), respectively. By substituting these values into equation (6), the improved time-dependent corrosion rate model that considers concrete cracking can be determined as

λ (t) = {\begin{matrix} λ_{1} (t) = 0.01723 (t - 11.5)^{- 0.29} for 11.50 a < t \leq 15.47 a \\ λ_{2} (t) = 0.0075 t - 0.1044 for 15.47 a < t \leq 19.31 a \\ λ_{3} (t) = 0.0775 (t - 11.5)^{- 0.29} - 0.00772 (t - 11.5)^{- 0.58} for t > 19.31 a \end{matrix}

(17)

To investigate the effect of concrete cracking simulation on the steel corrosion rate, Figure 5 shows the complete corrosion rate curves for cases with considered and unconsidered concrete cracking.

Figure 5.

Comparison of steel corrosion rate curves.

As shown in Figure 5, the two kinds of steel corrosion rate models exhibit a significant difference when concrete cracking is considered. For the RC columns of the case study bridge at the severe cracking time, the corrosion rate accounting for cracking is approximately four times larger than that neglecting concrete cracking.

By incorporating these two corrosion rate models into equation (4), the residual cross-sectional areas of steel reinforcement in RC columns can be calculated. The results are plotted in Figure 6.

Figure 6.

Comparison of residual cross-sectional areas of steel rebar with time.

As can be observed in Figure 6, the cross-sectional areas of steel reinforcement in both cases decrease with time after the initial corrosion. The rate of decrease in the reinforcement area with consideration of concrete cracking is much faster than that without considering cracking. After 20 years, the difference in the steel residual areas is only 1.2%, while this difference reaches 30.3% after 100 years. This highlights the importance of accounting for the effect of concrete cracking on the deterioration estimates of RC substructures under chloride attack.

As for a bridge column exposed to the deicing salt environments, the chlorides from the bridge leaking joints and the adjacent road traffic could reach any position and surface of the column (Gode and Paeglitis, 2014). An unfavorable situation is considered in this study, where the corrosion rate model is implemented on the reinforcing bars in entire columns. The uncertainty in steel corrosion along the axis of bridge columns will be studied in future work.

Time-dependent seismic demand estimates

To conduct an accurate seismic demand estimate of RC substructures, this paper considers the uncertainties in bridge modeling, including the structural dimensions, material properties, structural damping, deterioration process, etc. The considered random variables used to generate the time-dependent seismic demand are listed in Tables 1 and 2.

Table 2.

Random variables considered in the finite element bridge model.

Parameter (Units)	Distribution	Mean	St. dev.	Reference
Compressive strength of unconfinedconcrete (MPa)	Lognormal	30.8	6.16	Pan (2007); Thanapol et al. (2016)
Strain at peak strength of unconfinedconcrete	Lognormal	0.002	0.0004	Pan (2007); Thanapol et al. (2016)
Ultimate compressive strain of unconfinedconcrete	Lognormal	0.005	0.001	Pan (2007); Thanapol et al. (2016)
Compressive strength of confinedconcrete (MPa)	Lognormal	Equation (13)	0.2·mean	Pan (2007); Thanapol et al. (2016)
Strain at peak strength of confinedconcrete	Lognormal	0.004	0.00088	Pan (2007); Thanapol et al. (2016)
Ultimate compressive strain ofconfined concrete	Lognormal	0.0085	0.0017	Pan (2007); Thanapol et al. (2016)
Elastic modulus of concrete (MPa)	Normal	3.2 × 10⁴	3840	Vu and Stewart, (2000)
Concrete cover depth (mm)	Normal	50	6	Cui et al. (2018)
Yield strength of steel (MPa)	Lognormal	Equation (10)	0.07·mean	Pan (2007); Thanapol et al. (2016)
Ultimate strain of steel (MPa)	Lognormal	Equation (12)	0.2·mean	Pan (2007); Thanapol et al. (2016)
Elastic modulus of steel (MPa)	Normal	2 × 10⁵	4000	Cui et al. (2018)
Bearing shear modulus (MPa)	Uniform	1.365	0.407	Choi et al. (2004)
Translation stiffness of foundation(kN/m)	Uniform	6.2 × 10⁵	1.8 × 10⁵	Barbato et al. (2010)
Rotational stiffness of foundation(kN m/rad)	Uniform	1.8 × 10⁷	5.2× 10⁶	Barbato et al. (2010)
Damping ratio	Normal	0.045	0.0125	Choi et al. (2004)
Deck mass ratios	Uniform	1.0	0.058	Choi et al. (2004)

Since there are not enough ground motions recorded for the location of the case study bridge, a set of 50 ground motions are chosen from the ground motion database of PEER (https://ngawest2.berkeley.edu/). The magnitudes of the selected ground motions are in the range from 5.4 to 6.9, and the source-to-site distances vary from 15 to 90 km. In addition, the mean acceleration spectra (red lines) of these selected ground motions, shown in Figure 7, matches well with the design response spectrum (blue lines) of the case study bridge recommended by the Chinese design code (JTG/T B02–01, 2008). Peak ground acceleration (PGA) is a widely recognized parameter used to reflect the intensity characteristic of ground motions; thus, PGA is selected as the intensity measure (IM) for the seismic demand estimates.

Figure 7.

Response spectrum and PGA distribution of the selected ground motions.

To ease the calculation cost of seismic fragility analysis, the seismic demand model of the studied RC columns is established for 0, 25, 50, 75, and 100 years. At each point in time of interest, the UD method is employed to generate a set of 50 bridge finite element models that consider the different sources of uncertainty. Then, the generated bridge finite element models are randomly paired with the selected ground motions. Nonlinear time-history analysis is subsequently conducted to obtain the maximum RC column curvature demand of all the bridge-ground motion samples. Figure 8 shows the maximum seismic demands placed on the studied RC columns for only the pristine and 100-year corrosion cases, and the red lines present the linear regression relationship between seismic demand and PGA in logarithmic space. A complete list of the PSDMs for the different years is shown in Table 3.

Figure 8.

Results of nonlinear time-history analysis and fitted PSDMs for aging RC columns.

Table 3.

PSDMs of RC columns at different times.

Time (years)	PSDM	R ²	$β_{D \| IM} (t)$
0	$\ln (ϕ) = - 4.014 + 1.038 \ln (PGA)$	0.761	0.386
25	$\ln (ϕ) = - 3.981 + 1.015 \ln (PGA)$	0.758	0.389
50	$\ln (ϕ) = - 3.856 + 1.087 \ln (PGA)$	0.753	0.451
75	$\ln (ϕ) = - 3.727 + 1.094 \ln (PGA)$	0.746	0.456
100	$\ln (ϕ) = - 3.497 + 1.108 \ln (PGA)$	0.739	0.470

Time-dependent seismic capacity estimates

The probabilistic method proposed in section 2.2 is used to provide the time-dependent seismic capacity estimates for the RC substructures of the case study bridge. In accordance with the time-dependent seismic demand estimates, the seismic capacity models of deteriorating RC substructures are also established for time points ranging from 0 years to 100 years with an interval of 25 years.

A total of 14 random variables, including the material constitutive parameters of steel and concrete, the geometric dimensions of the studied RC column, and the axial force placed on it, are considered in this study to estimate the time-dependent seismic capacity of aging substructures. The statistical information of these random variables is presented in Tables 2 and 4. Note that the statistical parameters of the time-dependent random variables, such as the residual area of reinforcement, are assumed to be invariant with time for simplicity.

Table 4.

Statistical parameters of the random variables.

Parameter (Units)	Distribution	Mean	St. dev.	Reference
Cross-sectional area of reinforcement (mm²)	Normal	Equation (4)	0.07·mean	Pan (2007); Thanapol et al. (2016)
Diameter of RC columns (mm)	Normal	1200	60	Pan (2007); Thanapol et al. (2016)
Height of RC columns (mm)	Normal	10,000	500	Pan (2007); Thanapol et al. (2016)
Axial force applied on RC columns (kN)	Normal	3800	950	Choi et al. (2004)

Establishment of the BP-ANN model

For the deterioration state of RC columns at each time point of interest, the UD method is used to generate the experimental points (input samples of the BP-ANN) to be uniformly scattered in a design domain. To achieve this, first, the range of each factor (random variable) is defined by restricting its value to be within three standard deviations of the mean. Then, the range of each factor is uniformly divided by the number of levels of each factor, and the factor levels are coded into values, such as 1, 2, and 3. Furthermore, Data Processing System (DPS) software is employed to generate the required UD table $U_{n} (n^{m})$ , where n represents the number of experimental points and m represents the number of factors (Cheng, 2010). In this study, a $U_{50} (50^{14})$ table is selected. Due to space limitations, a smaller table $U_{5} (5^{3})$ is presented in Table 5 for illustration purposes.

Table 5.

UD table: $U_{5} (5^{3})$ table.

No. of experimental points	Factors (random variables)
No. of experimental points	1	2	3
1	1	2	4
2	2	4	3
3	3	1	2
4	4	3	1
5	5	5	5

Once the UD table is determined, the predefined codes of each factor can be used to transfer the UD table to the input samples for the subsequent pushover analysis of RC columns (Cheng et al., 2008; Pang et al., 2014). Note that each row of the UD table corresponds to a deterministic nonlinear pushover analysis. Then, the four critical curvature values ( $ϕ_{S}$ , $ϕ_{M}$ , $ϕ_{E}$ , and $ϕ_{C}$ ) at different damage states are obtained as the output samples.

The aforementioned BP-ANN model with three layers is employed in this study. A linear transfer function is used to transfer the values from the hidden layer to the output layer, whereas a logistic transfer function is used to transfer the values from the input layer to the hidden layer. The training data that consist of the input and output samples are scaled before they are presented to the model. Other parameters used to generate the BP-ANN model are presented in Table 6. Finally, five BP-ANN models at different deterioration states are established to investigate the probabilistic characteristics of the seismic capacity of RC columns.

Table 6.

Parameters for establishing the BP-ANN.

Parameter	Value	Description
Number of inputlayers	14	Determined by the number ofrandom variables
Number of hiddenlayers	8	Determined by trial and error
Number of outputlayers	4	Determined by the number ofdamage states
Number of trainingsamples	50	Consistent with the number ofground motions
Number of testsamples	20	Generated randomly
Training accuracy ofBP-ANN	10⁻⁸	Desired parameter for trainingof BP-ANN
Allowable error ofBP-ANN	1%	Desired parameter for trainingof BP-ANN

Accuracy validation of the BP-ANN model

To test the effectiveness of the BP-ANN for the use of RC substructure seismic capacity estimates, 20 random test samples are generated to check the accuracy of the predicted values of the BP-ANN model. The results obtained by pushover analysis are regarded as exact values. Then, the relative errors between the predicted values and exact values are shown in Figure 9.

Figure 9.

Relative errors of the predicted values of the BP-ANN model.

As shown in Figure 9, the absolute values of the relative errors of the predicted critical curvature values at different damage states are within the limit of 4%, which satisfies the accuracy requirement of seismic engineering practice. This indicates that the proposed BP-ANN enhanced by the UD method is feasible to predict the seismic capacity of deteriorating RC substructures with a small number of samples.

Results of time-dependent seismic capacity estimates

For the deterioration state of RC substructures at each time point of interest, the MCS method is used to generate 20,000 random input samples. The established BP-ANN models subsequently serve as an alternative to time-consuming pushover analysis to predict the critical curvature values at different damage states. Origin software is used to conduct a statistical analysis of the samples generated by the MCS method. For brevity, Figure 10 displays only the frequency distributions of the curvature values for the slight damage state after 0 and 50 years.

Figure 10.

Lognormal distribution fittings to the curvature results for the slight damage state obtained by pushover analysis.

Figure 10 shows that the critical curvature values of the slight damage state after 0 and 50 years fit a lognormal distribution well. In fact, the same results can also be observed for other critical curvature values at different damage states. Therefore, it is reasonable to assume that the seismic capacity of deteriorating RC columns follows a lognormal distribution, which is consistent with the assumption suggested by Cornell et al. (2002). Thus, Figure 11 gives only the probabilistic density distributions of the curvature values at the slight and extensive damage states.

Figure 11.

Time-dependent probability distributions of the slight and extensive damage states after 0, 50, 100 years.

As shown in Figure 11, the mean values of critical curvature points at different damage states decrease with time. This means that the seismic capacity of the RC columns is expected to degrade as the case study bridge deteriorates. Detailed results of the statistical parameters of the RC columns are summarized in Table 7.

Table 7.

The critical curvature values of different damage states with time.

Time (years)	Slight damage state ( $ϕ_{S}$ )		Moderate damage state ( $ϕ_{M}$ )		Extensive damage state ( $ϕ_{E}$ )		Complete damagestate ( $ϕ_{C}$ )
Time (years)	Mean	$β_{C}$	Mean	$β_{C}$	Mean	$β_{C}$	Mean	$β_{C}$
0	0.00247	0.337	0.00314	0.314	0.0163	0.455	0.0378	0.461
25	0.00238	0.342	0.00305	0.318	0.0158	0.464	0.0342	0.473
50	0.00223	0.364	0.00285	0.327	0.0149	0.481	0.0299	0.486
75	0.00213	0.369	0.00265	0.334	0.0135	0.488	0.0258	0.495
100	0.00204	0.375	0.00252	0.342	0.0127	0.501	0.0230	0.513

Time-dependent seismic fragility curves

Based on the obtained time-dependent seismic demand models and capacity models, the time-dependent seismic fragility curves of the studied RC substructures at different service times are presented in Figure 12.

Figure 12.

Time-dependent seismic fragility curves of the studied RC substructures.

As can be observed in Figure 12, there is a growing trend in the probability that the RC substructures exceed any damage states during the service life. This highlights the significance of considering the effect of steel corrosion on time-dependent seismic fragility assessment. For the slight damage state, the exceedance probability at a PGA of 0.10 g exhibits a maximum increase of 174.6% after 100 years, whereas the exceedance probability of the moderate damage state exhibits a maximum increase that is equal to 126.4% at a PGA of 0.15 g. With regard to the extensive damage state, the probability of exceedance at a PGA of 1.0 g increases by 23.7% after 50 years and reaches an increase of 57.9% after 100 years. In contrast, under ground motions with the same intensity, the probability of reaching the complete damage state rises by 172.7% after 50 years and reaches an increase of 494.5% after 100 years of service. Furthermore, nonlinear changes in the exceedance probability at different damage states are observed. In particular, after 25 years, the probability of exceedance exhibits a significant increase.

Two main factors that have significant effects on the seismic fragility of RC substructures are investigated herein. The first is whether to consider the effect of an increase in seismic demand. The other factor is whether to account for the effect of a reduction in seismic capacity. It should be noted that the seismic fragility analysis conducted in this paper considers these two factors simultaneously. A comparison of seismic fragility results considering different factors is shown in Figure 13.

Figure 13.

Effects of different factors on the seismic fragility of RC substructures.

As shown in Figure 13, after 100 years, the seismic fragility analyses that only account for the increase in seismic demand or the reduction in capacity appear to underestimate the exceedance probability of RC columns at different damage states. Considering only the increase in seismic demand exhibits a smaller underestimation of the seismic fragility than considering only the reduction in seismic capacity. For the moderate damage state, the maximum gap between the fragility curves that consider one or two factors is approximately 24.8% after 100 years, whereas this gap is approximately 30.1% for the complete damage state. These results indicate the importance of simultaneously considering the effects of an increase in seismic demand and reduction in seismic capacity in the seismic fragility assessment of RC bridge substructures.

Conclusion

This paper presents an improved time-dependent seismic fragility framework to study the joint effects of chloride attack and seismic hazards on RC bridge substructures. A typical three-span T-shaped girder bridge with RC columns is selected to demonstrate the application of this time-variant fragility framework.

One the one hand, a three-phase steel corrosion rate model that takes into account the effect of concrete cracking is integrated into the proposed framework to give a more realistic prediction of the residual area of reinforcement with time. On the other hand, an efficient probabilistic approach is proposed to provide time-dependent seismic capacity estimates by incorporating a UD-enhanced BP-ANN, pushover analysis and MCS. The results reveal that the time-dependent capacity limit states of RC columns follow a lognormal distribution, and a nonlinear shift in the mean and standard variation is observed for the column curvature at different damage states during the service life.

Based on the probabilistic seismic demand and seismic capacity estimates of RC columns, the seismic fragility of the RC columns of a case study bridge is investigated by considering the uncertainties in the deterioration process, material properties, structural dimensions, and ground motions. The research results indicate that the effect of steel corrosion cannot be ignored in the seismic fragility assessment of RC bridge substructures. Furthermore, considering only the effect of an increase in seismic demand, a reduction in seismic capacity or neither of them causes a very large underestimation of the seismic vulnerability of deteriorating RC bridge substructures. Considering only the increase in seismic demand exhibits a smaller underestimation of the seismic fragility than considering only the reduction in seismic capacity. In addition, a nonlinear increase is observed in the probability of exceedance of different damage states of the studied RC columns over the entire life. This work gives new insights into the combined effects of chloride attack and seismic hazards on the seismic performance and seismic fragility of RC bridge substructures over time. This work could also provide guidance for decision makers and engineers in making prompt decisions on the maintenance and repair of deteriorating RC bridge substructures. Future work will focus on extending the proposed time-dependent fragility framework to life-cycle loss analysis of aging highway bridges. Additionally, improved deterioration models for other bridge components, such as bearings, need to be studied to comprehensively consider the effect of component aging on the seismic fragility of highway bridges.

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: Thanks for the supports by doctoral startup funding from Shandong Jiaotong University (Grant No. 50004918) and Shandong Jiaotong University “Climbing” Research Innovation Team Program.

ORCID iD

Fengkun Cui

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