Predictions and Evolution Characteristics of Failure Modes of Degenerate RC Piers

Abstract

During the service process, piers are often in harsh chloride ion erosion environments. The failure mode evolution of reinforced concrete (RC) piers may occur under the action of continuous corrosion. Accurately identifying the failure mode types and evolution characteristics of corroded RC bridge piers is a prerequisite for the lifetime seismic performance evaluations of bridges. First, based on Fisher’s theory and 174 RC pier columns as the analysis samples, a two-stage discrimination formula for the pier failure modes was established and compared with the existing theoretical discrimination methods. Then, based on Fisher’s discriminant grouping, and combined with Bayes’ formula and chloride erosion theory, a failure mode discrimination method for corrosion-damaged bridge piers that considers probability was developed. Finally, taking a medium-span concrete bridge as an example, the failure modes of the corroded pier in different service periods were predicted, and the influences of the various parameters on the failure mode evolution process of the corroded pier were studied. The results show that the accuracy of the proposed discriminant model was significantly improved compared with those of previous theoretical studies. The development of the failure mode features depends on how the distinct RC pier material qualities degrade under the influence of chloride ions. The degradation of the stirrups and concrete accelerates the nonductile failure of RC bridge piers, while the degradation of the longitudinal reinforcements delays it.

1. Introduction

The failure of bridge piers is considered the main cause of the bridge collapses during earthquakes [1]. Investigations and analyses of seismic damage show that RC piers generally fail in three failure modes caused by earthquakes: shear failure, flexure–shear failure, and flexure failure [2]. The nonductile failure of piers, including shear failure and flexure–shear failure, is frequently avoided in practical bridge designs with capacity protection designs [3]. However, actual engineering events have shown that some piers, although originally designed in accordance with the code, have poor anti-seismic performances under ground motion, which result in flexural–shear and shear failures. These piers often serve in chloride ion environments. The stirrup of an RC pier is more susceptible to corrosion than the longitudinal reinforcement because of the chloride-induced corrosion. The shear performances of RC piers degrade more quickly than the flexural performances with the effect of the concrete-spalling protective layer [4]. Therefore, the failure mode of degraded RC piers may change from ductile failure to brittle failure due to chloride-induced corrosion.

Many studies have been carried out on the failure modes of piers. In the early years, the classical fiber beam—column element model was used to simulate the flexure failure of RC columns. Some progress has been made in terms of flexure and shear failures [5,6]. However, due to the complexity of the shear force mechanism, there are no reliable theoretical models to describe these two types of failure characteristics. Recently, the relationship between the shear demand ($V_{\mathrm{P}}$) and shear capacity ($V_{\mathrm{n}}$) of RC pier columns in the plastic hairpin section was found to be different among the different failure modes in previous theoretical studies. Therefore, it is widely accepted that the value of the $V_{\mathrm{P}}/V_{\mathrm{n}}$ coefficient is used to identify the failure modes of piers. ASCE/SEI 41–17 provide a $V_{\mathrm{P}}/V_{\mathrm{n}}$ equal to 0.6 as the criterion for judging flexure and non-flexure (including shear and flexure–shear) failures [7]. Zhu et al. [8] suggested coefficient values of 0.7 and 1.0 for the $V_{\mathrm{P}}/V_{\mathrm{n}}$, respectively, as the critical points for the three failure modes. The approach of combining the shear span ratio, longitudinal reinforcement ratio, and value of the $V_{\mathrm{P}}/V_{\mathrm{n}}$ coefficient was proposed by Liu et al. [9]. Some scholars have advocated for using the lateral deformation as a criterion for identifying the failure mechanisms of RC piers. Based on the modified pressure field theory and fiber beam–column element model, Sun et al. [10] calculated the shear deformation and bending deformation of RC piers, respectively, under lateral loads. The failure mode is determined by the ratio of the shear deformation to the bending deformation. The development of the failure mode discrimination of bridge piers has been greatly enhanced by the abovementioned research. However, the calculation accuracy of the shear-bearing capacity is limited due to the complex shear mechanism of RC piers. In addition, Ma et al. [3] incorporated the distinctive properties of RC piers and put forward the empirical discriminant formula, making use of the benefits of machine learning in the classification recognition and parameter prediction direction. To identify and categorize the failure modes of RC piers, Mangalathu et al. [11] developed machine learning algorithms, such as the quadratic discriminant function, K-nearest neighbor, decision tree, random forest, naive Bayes, and artificial neural network (ANN) algorithms, and they discovered that ANNs are more effective than the other approaches. Feng et al. [12] adopted the ensemble learning algorithm and single learning algorithm to evaluate pier specimens, and they created the ensemble learning algorithm model. Although machine learning in intelligence tasks has shown high efficiency in identifying the failure modes of RC piers, it is difficult to transform the algorithm into a straightforward discriminating pattern, which makes it unsuitable for direct use in actual engineering.

The abovementioned research promotes our understanding of the failure mode identification of bridge piers. However, there is a lack of research on the failure modes of corroded piers. In fact, quite a few bridges are exposed to chloride ion erosion. It is difficult to accurately analyze the seismic performances of bridge piers if the corrosion effect of chloride ions is ignored. To more effectively study the seismic performances of piers in service under adverse conditions, a method for distinguishing the failure modes of corroded piers is necessary.

This paper proposes an analytical method that considers the failure modes of corroded piers in order to study their transformations. Initially, this paper introduces the original RC pier parameters to identify the failure modes of RC piers based on Fisher discrimination analysis (FDA) considering the abovementioned research. Then, in order to identify the failure modes of RC piers quickly and effectively, this paper analyzes the significant influencing factors of the different failure modes of RC piers. A two-stage FDA is proposed to distinguish the three failure modes. Lastly, the proposed Fisher discriminant function is combined with the chloride corrosion theory and Bayes formula to analyze and study the evolution characteristics of the failure modes of degraded bridge piers under the impact of chloride corrosion.

2. Failure Mode Discrimination Method of RC Piers Based on FDA

FDA is a classical linear discriminant method that is often used to solve multifactor classification problems [13]. FDA has a complete mathematical theory, in contrast to traditional “black box” machine learning, and its discriminant results are expressed as straightforward linear functions. As a consequence, if the relationship between the original parameters of the bridge pier and failure mode is analyzed using FDA, then the linear discriminant function of the failure mode of the bridge pier can be found, which can be directly applied in engineering practice.

2.1. Calculation Theory of FDA

FDA is usually used to solve the problem of the binary or multiple classifications of multidimensional data. Its basic idea is dimensionality reduction (or projection), which projects the sample’s multidimensional data onto the low-dimensional space through one or more linear discriminant functions according to the principle of the maximum between-class distance and minimum within-class distance (Figure 1). In this way, samples that belong to different categories are separated as much as possible. A linear discriminant function can be used to classify new samples with unknown categories. The basic calculation model is as follows.

First, some assumptions and definitions in mathematics are provided. We define $n$ as the number of samples, $m$ as the number of sample attributes, and $k$ as the number of classes. All the operation data from the measurement system, including the normal operation and faulty operation, can be classified into different data classes.

(1)$G_i:G_1,G_2,\dots ,G_k(i=1,2,\dots ,k)$

where $G_1$ refers to the normal operation data class, and $G_2$,…, $G_k$ refer to the various faulty data classes. To determine the class of measurement vectors ($x={(x_1,x_2,x_3,\dots ,x_m)}^{\mathrm{T}}$), a discriminant function is given by:

(2)$y(x)=c_1{x}_1+c_2{x}_2+\dots +c_m{x}_m=C^{\mathrm{T}}x$

where $C={(c_1,c_2,\dots ,c_m)}^{\mathrm{T}}$. It then follows that the mean and variance of the $y(x)$ on $G_i$ are ${\overline{y}}^{(i)}=C^{\mathrm{T}}{\overline{x}}^{(i)}$ and ${\sigma }_i^2=C^{\mathrm{T}}{s}^{(i)}C$, respectively. Only when the deviation reaches the maximum vector can $C$ be obtained. The formula for calculating the deviation value is denoted as:

(3)$\lambda =\frac{{\displaystyle \sum _{i=1}^k{({\overline{y}}^{(i)}-\overline{y})}^2}}{{\displaystyle \sum _{i=1}^k(n-1){\sigma }_i{}^2}}$

At the same time, ${\overline{y}}^{(i)}=C^{\mathrm{T}}{\overline{x}}^{(i)}$, ${\sigma }_i^2=C^{\mathrm{T}}{s}^{(i)}C$, and $\overline{y}=C^{\mathrm{T}}\overline{x}$ are substituted into Equation (3):

(4)$\lambda =\frac{C^{\mathrm{T}}AC}{C^{\mathrm{T}}EC}$

where $A$ is the sum of the squared within-group dispersions, and $E$ is the sum of the squared between-group dispersions. If $\partial \lambda /\partial C=0$, then the value of the coefficient of $\lambda$ is the largest:

(5)$\frac{2AC}{C^{\mathrm{T}}EC}-\frac{2\lambda EC}{C^{\mathrm{T}}EC}=0\Rightarrow AC=\lambda EC$

where $\lambda$ is the characteristic eigenvalues and eigenvectors of $A$, and $C$ is the characteristic eigenvalues and eigenvectors of $E$. If we define all the nonzero characteristic eigenvalues as ${\lambda }_1$, ${\lambda }_2$,…, ${\lambda }_m$, then the number of $m$ discriminant functions can be constructed:

(6)$y_l(x)=C^{{(l)}^{\mathrm{T}}}x,\begin{array}{l}\hfill \end{array}l=1,\dots ,m$

2.2. Fisher Discriminant Model for Failure Mode of RC Pier

According to the aforementioned mathematical model, the key to constructing the discriminant function of the bridge pier failure mode is the selection of the appropriate samples and the identification of the primary influencing factors of the bridge pier failure mode. The factors with significant correlations are extracted as discriminant indexes, and the discriminant function of the pier failure mode is acquired by the Fisher discriminant model.

The back-substitution technique is used to assess the discriminant effect of the aforementioned discriminant function, and the discriminant function with the best discriminant effect may be used to determine how a bridge pier will collapse.

2.2.1. Sample Set

The Pacific Seismic Research Center (PEER) database was used to select 174 samples from 154 rectangular RC piers with various failure modes and 20 corroded rectangular RC piers with various failure modes. The samples with various failure modes had 111 flexure failures, 43 combined flexure–shear failures, and 20 shear failures [14,15,16]. PEER and corrosion tests provided the geometric dimensions and material properties of these specimens. The basic parameters of the selected samples are shown in Table 1.

2.2.2. Analysis of Influencing Factors

In order to ensure the accuracy and effectiveness of the FDA, the above sample sets needed to be analyzed by the parameters to select the influential factors with strong correlations.

Numerous parameters, such as the geometric dimensions, material strength, reinforcement features, applied load, and others, influence the failure modes of RC piers during earthquakes [17]. This work considered the geometrically and materially large parameters, including the longitudinal reinforcement configuration, stirrup configuration, shear span ratio, and axial compression ratio, which were separately analyzed for their significances to obtain the significant factors that affect the failure modes of RC pier columns. To improve the efficiency of the Fisher discrimination and retain the characteristics of all the parameters, the abovementioned parameters were normalized so that all their values were between 0 and 1. The normalization formula can be expressed as:

(7)$x_{\mathrm{norm}}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}$

where $x_{\mathrm{norm}}$ is the normalized parameter, $x$ is the initial value of the parameter. $x_{\max }$ and $x_{\min }$ are the maximum and minimum values of the parameters in all the samples, respectively.

Reinforcement Configuration

The mechanical properties of RC pier columns are affected by the longitudinal reinforcement and stirrup configuration during earthquakes. Good longitudinal reinforcement can improve the bending resistances of RC pier columns. The formation of oblique cracks may be prevented by using adequate stirrups, which can also improve the occlusal action of the oblique section aggregates and increase the shear strength. At the same time, it can also constrain the core of the concrete and improve its ultimate strain, as well as increase the ductility of the members [18].

Figure 2 shows the distribution of the longitudinal reinforcement characteristic value (${\rho }_{\mathrm{l}}^{}{f}_{\mathrm{y}}/f_{\mathrm{c}}^{\prime }$) and stirrup characteristic value (${\rho }_{\mathrm{sv}}^{}{f}_{\mathrm{yv}}/f_{\mathrm{c}}^{\prime }$) under the three failure modes: flexure (F), flexure–shear (FS), and shear (S). In the figure, the major distribution of the longitudinal reinforcement characteristic value (${\rho }_{\mathrm{l}}^{}{f}_{\mathrm{y}}/f_{\mathrm{c}}^{\prime }$) exhibits a growing trend from flexure failure to flexure–shear failure to shear failure (F–FS–S), with the average value increasing from 0.22 (F) to 0.32 (FS), and then to 0.36(S). The main distribution (25–75%) of the stirrup characteristic value (${\rho }_{\mathrm{sv}}^{}{f}_{\mathrm{yv}}/f_{\mathrm{c}}^{\prime }$) shows a downward trend, with the average value decreasing from 0.16 (F) to 0.09 (FS), and then to 0.07 (S). In total, there was a linear correlation between the longitudinal reinforcement characteristic value and stirrup characteristic value in the sample set and failure mode of the RC pier column.

Shear Span Ratio

The shear span ratio reflects the relative magnitude of the bending moment and shear force on the section of a member, which is an important parameter that affects the shear-bearing capacity of the member.

The three RC pier failure modes are shown in Figure 3, together with the shear span ratio distribution. With the change in the failure mode (F–FS–S), the main distribution of the shear span ratio (25–75%) showed a decreasing trend. The sample with flexure failure had a greater primary shear span ratio distribution (25–75%) compared with the other two groups of samples, and in particular the (F) group. Its average value decreased from 3.81 (F) to 2.53 (FS), and then to 1.89 (S). Consequently there was an obvious linear correlation between the shear span ratio and failure mode of the RC pier columns in the sample set.

Axial Compression Ratio

The axial compression ratio refers to the ratio of the axial load of the pier to the compressive bearing capacity of the concrete section, which mainly affects the ductility of the pier. The higher the axial compression ratio, the worse the ductility of the pier. The distribution of the axial compression ratio in the different failure modes is shown in Figure 4. In the diagram, the mean and overall distribution range (0–100%) of the axial compression ratio are not significantly different among the three failure modes: flexure failure (F), flexure–shear failure (FS), and shear failure (S). There is no pattern, even if the major distributions of the axial compression ratio (25–75%) that correspond to the three failure types are varied. In a word, the axial compression ratio and RC pier failure mechanisms were not clearly linearly correlated in the sample set.

According to the abovementioned research, a few variables, such as the orientation of the longitudinal reinforcement, design of the stirrups, and shear span ratio, are clearly linearly correlated with the failure modes (F–FS–S). In this paper, the data analysis software SPSS was used to analyze the bivariate correlation between the fundamental parameters, including the reinforcement configuration and strength, concrete strength, and different pair failure modes, in order to further explore the significant factors that affect the failure modes of RC piers. The Pearson correlation coefficient (PCC) between each variable and failure mode is shown in Table 2. The PCC is used to show whether two datasets are on the same line, and to measure the linear relationship between the fixed distance variables. The larger the absolute value of the PCC, the stronger the linear correlation between the two variables.

According to Table 2, the concrete strength ($f_{\mathrm{c}}^{\prime }$), stirrup yield strength ($f_{\mathrm{yv}}$), stirrup ratio (${\rho }_{\mathrm{sv}}$), longitudinal reinforcement characteristic value (${\rho }_{\mathrm{l}}^{}{f}_{\mathrm{y}}/f_{\mathrm{c}}^{\prime }$), stirrup characteristic value (${\rho }_{\mathrm{sv}}^{}{f}_{\mathrm{yv}}/f_{\mathrm{c}}^{\prime }$), and shear span ratio ($\lambda$) were the main influencing variables of the flexure failure (F) and flexure–shear failure (FS). The concrete strength ($f_{\mathrm{c}}^{\prime }$), yield strength of the longitudinal reinforcement ($f_{\mathrm{y}}$), stirrup ratio (${\rho }_{\mathrm{sv}}$), longitudinal reinforcement characteristic value (${\rho }_{\mathrm{l}}^{}{f}_{\mathrm{y}}/f_{\mathrm{c}}^{\prime }$), stirrup characteristic value (${\rho }_{\mathrm{sv}}^{}{f}_{\mathrm{yv}}/f_{\mathrm{c}}^{\prime }$), and shear span ratio ($\lambda$) were the main influencing variables of the flexure failure (F) and shear failure (S) in the sample set. The main influencing variables of the flexure–shear failure (FS) and shear failure (S) in the sample set were the yield strength of the longitudinal reinforcement ($f_{\mathrm{y}}$), stirrup ratio (${\rho }_{\mathrm{sv}}$), shear span ratio ($\lambda$), and stirrup characteristic value (${\rho }_{\mathrm{sv}}^{}{f}_{\mathrm{yv}}/f_{\mathrm{c}}^{\prime }$).

In conclusion, the significant influencing factors between the two failure modes were different. According to Table 2 and Figure 2, Figure 3 and Figure 4, the primary determining factors of the F–FS and F–S failures were the longitudinal reinforcement configuration, stirrup configuration, and shear span ratio. The primary determining factors of the FS–S were the stirrup configuration and shear span ratio. In this research, a two-stage Fisher discriminant was taken into consideration to obtain the best discriminant impact of the Fisher linear discriminant function. In the first stage, the components with flexure failure (F) and those without flexure failure (FS, S) were classified and discriminated. In the second stage, the nonflexure failure (FS, S) components from the first stage were classified and discriminated to achieve the classification of the components of the three failure modes.

2.2.3. Two-Stage Fisher Discriminant Function

First, based on FDA, the first-stage discriminant function that considers the longitudinal reinforcement configuration, stirrup configuration, and shear span ratio is given by:

(8)$\begin{array}{l}{Y}_1=1.773f_{\mathrm{c}}^{\prime }+0.436f_{\mathrm{yv}}+1.577f_{\mathrm{y}}\\ +2.891\lambda -2.899{\rho }_{\mathrm{sv}}+4.148{\rho }_{\mathrm{sv}}{f}_{\mathrm{yv}}/f_{\mathrm{c}}^{\prime }\\ -2.992{\rho }_{\mathrm{l}}{f}_{\mathrm{y}}/f_{\mathrm{c}}^{\prime }-1.12\end{array}$

The nonflexure failure is identified if the result of entering the required specimen variables into the function $Y_1$ is less than 0, and the flexure failure is determined if the result is more than 0. Subsequently, the second-stage discriminant function that considers the stirrup configuration and shear span ratio is expressed as:

(9)$Y_2=18.998{\rho }_{\mathrm{sv}}+5.877\lambda -3.048f_{\mathrm{yv}}{\rho }_{\mathrm{sv}}/f_{\mathrm{c}}^{\prime }-2.724$

The component is classified as flexure–shear failure if the result is more than 0, and it is classified as shear failure if the result is less than 0, after the relevant variables from the indeterminate specimens have been substituted into the function $Y_2$.

2.3. Validity Check

A total of 174 specimens from the sample set were examined using back substitution to determine the discriminant impact of the discriminant approach suggested in this section. The findings are shown in Table 3. This resulted in a total of 111 pier flexure failures, with an accuracy rate of 91.0%, of which 101 were properly diagnosed, 9 were mistaken as bending shear failures, and 1 was misjudged as shear failure. A total of 43 piers experienced flexure–shear failures; the accuracy of 81.4% indicates that 35 of these failures were accurately classified as flexure failures, 3 as shear failures, and 5 as flexure failures. A total of 20 piers in all experienced shear failures, with an accuracy of 80.0%. Of these, 16 were properly assessed, 3 were incorrectly classified as flexure–shear failures, and 1 was incorrectly classified as flexure failure. The overall discrimination accuracy of the 174 specimens was 87.9%, which is relatively high. The identification of 20 corrosion samples is shown in Table 4. Only RC-4 and RC-5 in reference [15] and Z-8 in reference [16] identified errors, with a 15% rate of misjudgment. In addition, the samples with wrong judgments were all from flexure–shear and shear failures, and the flexure failures of the corroded components were all correctly judged.

Therefore, the longitudinal reinforcement configuration, stirrup configuration, and shear span ratio were the discriminant factors in the first stage, whereas the stirrup configuration and shear span ratio were the discriminant factors in the second stage. The RC piers with different failure modes under earthquake conditions were effectively classified by the two-stage Fisher discrimination.

3. Comparison of Common Failure Mode Discrimination Methods

3.1. Based on Shear Demand and Shear Strength

It is a highly accepted method to identify the failure modes of RC piers based on the shear demand and shear strength, which should determine the shear demand ($V_{\mathrm{P}}$) and shear strength ($V_{\mathrm{n}}$) of the section in the plastic hinge area. The demand for the shear force can be determined by $V_{\mathrm{P}}=M_{\max }/a$, where $M_{\max }$ is the maximum bending moment of the flexure capacity of the plastic hinge section, which can be obtained from the moment–curvature curve of the pier column, and $a$ is the shear span. However, there is currently no unified calculation method for the $V_{\mathrm{n}}$ due to the different theoretical models proposed by various scholars. In order to determine the shear strength of stirrups, Aschheim et al. [19] employed a truss model with a constant angle of 30°, and they published a formula that takes the displacement ductility and axial pressure into account. Priestly et al. [20] also addressed the impact of the shear span ratio, as well as the contribution of the axial forces on the shear strength, based on the truss mode with a fixed angle of 30°. In this paper, the $V_{\mathrm{n}}$ was calculated based on the 40° fixed angle truss model proposed by Sezen et al. [21] and the Mohr Coulomb theory to consider the ductility growth of RC piers. Thus, the formula for the shear strength ($V_{\mathrm{n}}$) can be calculated by:

(10)$V_{\mathrm{n}}=k\frac{A_{\mathrm{sv}}{f}_{\mathrm{yv}}{h}_0}{s}+k\left(\frac{0.5\sqrt{f_{\mathrm{c}}^{\prime }}}{\lambda }\sqrt{1+\frac{P}{0.5\sqrt{f_{\mathrm{c}}^{\prime }}{A}_{\mathrm{g}}}}\right)0.8A_{\mathrm{g}}$

where $A_{\mathrm{sv}}$ is the cross-sectional area of the stirrup, $h_0$ is the effective height of the section, and $A_{\mathrm{g}}$ is the cross-sectional area of the RC pier. $k$ is the influence coefficient of the displacement ductility, which is determined by:

(11)$k=\{\begin{array}{ll}0.7\hfill & \mu \ge 6.0\hfill \\ 1.15-0.075\mu \hfill & 2.0\le \mu <6.0\hfill \\ 1.0\hfill & \mu \le 2.0\hfill \end{array}$

where $\mu$ is displacement ductility factor.

Using the method in this section, the values of the $V_{\mathrm{P}}/V_{\mathrm{n}}$ for the RC pier column samples in the database were calculated. According to Zhu et al. [8], the pier column is classified as shear failure when $V_{\mathrm{P}}/V_{\mathrm{n}}\ge 1.0$, as flexure failure when $V_{\mathrm{P}}/V_{\mathrm{n}}<0.7$, and as flexure–shear failure when $0.7\le V_{\mathrm{P}}/V_{\mathrm{n}}<1.0$.

Using correlation analyses, some scholars have concluded that the factors that affect the failure modes of RC piers are not only the $V_{\mathrm{P}}/V_{\mathrm{n}}$, but also the shear span ratio ($\lambda$), reinforcement ratio (${\rho }_{\mathrm{l}}$), ratio of the stirrup spacing to the section height ($s/h$), and other factors. Liu et al. [9] performed a bivariate correlation analysis and partial correlation analysis on the influencing variables, taking the $V_{\mathrm{P}}/V_{\mathrm{n}}$, ${\rho }_{\mathrm{l}}$, and $\lambda$ into account, in order to distinguish between the three failure types. Qi et al. [17] established a linear discriminant function using a step-by-step method, taking the $V_{\mathrm{P}}/V_{\mathrm{n}}$, $s/h$, and $\lambda$ as the discriminant factors. Figure 5 displays the influences of the three abovementioned discriminating approaches on the RC pier failure modes in the sample set used for this work.

Figure 5 displays that the foregoing three discrimination approaches, which are mostly based on the $V_{\mathrm{P}}/V_{\mathrm{n}}$, had discrimination accuracies of more than 70% for the RC pier failure scenarios in the database. The three techniques efficiently identified the RC piers with flexure failures and had strong overall identification impacts. However, they could not efficiently discriminate the RC piers with flexure–shear or shear failures in the sample set in this paper because of the complex mechanism of the shear action on the RC pier columns. At the moment, the research on the two flexure–shear and shear failure modes is not perfect. Although the failure mode may potentially be determined based on the link between the shear demand ($V_{\mathrm{P}}$) and shear strength ($V_{\mathrm{n}}$) of the plastic hinge region of RC piers, no consensus has been reached on the proper method for calculating the $V_{\mathrm{n}}$. Moreover, in order to consider the influence of the displacement ductility of the pier on the shear strength, the displacement ductility coefficient ($\mu$) should be determined in the calculation of the $V_{\mathrm{n}}$. Numerical simulation methods determine the $\mu$ differently, adding another degree of uncertainty to the calculation of the $V_{\mathrm{n}}$. As a result, determining the discriminant impact of piers and columns with visible shear action is difficult in reality, even if the discriminant technique with the $V_{\mathrm{P}}/V_{\mathrm{n}}$ as the primary factor is correct in principle.

The two-stage Fisher discriminant approach in this study is an empirical formula that is based on the fundamental characteristics of RC piers, and on the linear correlation between the distinct failure modes. The two-stage Fisher discriminant approach is simpler and avoids the analysis of the bending moment curvature and the calculation of the displacement ductility coefficient of the pier when compared with the aforementioned shear-demand- and shear-strength-based discriminating techniques. Moreover, the speed of the failure mode identification of the RC pier is improved, as well as the recognition accuracy.

3.2. Based on Displacement

Because the proportion of the bending and shear deformation of bridge piers is obviously different under different failure modes, the shear deformation of bridge piers with flexure failures is significantly smaller than that of those with shear failures. Thus, Sun et al. [10] suggested categorizing the various failure modes according to the proportion of the shear deformation to the bending deformation. Based on the modified compression field theory (MCFT) [22], the shear and displacement relationships of the bridge pier are calculated to determine its shear deformation. Pushover analysis is used to determine the bridge pier’s bending deformation after developing the fiber beam–column element model. The discrimination effects of this method on the RC pier columns in the sample set are shown in Figure 6.

As shown in Figure 6, the overall accuracy of the displacement-based failure mode discrimination approach was 78.74%, and it was 87.39% and 70.00% for the RC piers with flexure failures and shear failures, respectively, which indicates a good discriminating effect. The two-stage FDA improves the accuracy of the detection of the three failure modes over the displacement-based technique, and it avoids the more efficient MCFT computation.

The two-stage FDA is simpler to use and has greater discriminant accuracy than the conventional approach when compared with the conventional theoretical calculation model.

4. Analysis of Failure Mode Evolution of Corroded Bridge Pier

4.1. Corrosion Theory

The degradation process of RC structures that are subjected to chloride-induced corrosion can be divided into the diffusion stage, corrosion stage, and degradation stage [4]. In the analysis process of the chloride ion erosion of RC, it is an important to determine the initial corrosion time. In this paper, Fick’s second one-dimensional law was used to describe the chloride concentration ($C\left(x,t\right)$) at depth ($x$) at time ($t$). The formula [23] is given by:

(12)$C(x,t)=C_{\mathrm{s}}\left[1-\mathrm{erf}\left(\frac{x}{2\sqrt{Dt}}\right)\right]$

The initial corrosion time of the reinforcement is expressed as:

(13)$T_{\mathrm{corr}}=\frac{x^2}{4D}{\left[{\mathrm{erf}}^{-1}\left(\frac{C_{\mathrm{s}}-C_{\mathrm{cr}}}{C_{\mathrm{s}}}\right)\right]}^{-2}$

where $x$ is the depth from the steel bar surface; $\mathrm{erf}$ is the error function; $D$ is the reference chloride diffusion coefficient of the concrete age; $C_{\mathrm{s}}$ is the chloride ion concentration on the outer surface of the concrete protective layer; $C_{\mathrm{cr}}$ is the chloride ion concentration when the steel bar begins to rust.

After the chloride ion concentration on the steel bar surface reaches a critical value, the corrosion of the steel bar may be viewed as the loss of the steel bar section caused by the decrease in the steel bar diameter in the corrosion stage. The uniform corrosion can be calculated by:

(14)$d_{\mathrm{s}}\left(t\right)=\{\begin{array}{l}{d}_{\mathrm{s}0}\hfill \\ d_{\mathrm{s}0}-2\lambda \left(t-T_{\mathrm{corr}}\right)\hfill \\ 0\hfill \end{array}\begin{array}{l}\hfill \end{array}\begin{array}{l}t\le T_{\mathrm{corr}}\hfill \\ T_{\mathrm{corr}}<t\le d_{\mathrm{s}0}/2\lambda +T_{\mathrm{corr}}\hfill \\ t>d_{\mathrm{s}0}/2\lambda +T_{\mathrm{corr}}\hfill \end{array}$

where $d_{\mathrm{s}0}$ is the diameter of the steel bar without corrosion, $d_{\mathrm{s}}(t)$ is the residual diameter of the reinforcement after time ($t$), and $\lambda$ is the corrosion rate of the reinforcement, which is calculated by:

(15)$\lambda =\{\begin{array}{l}0.0117i_{\mathrm{corr}}\hfill \\ 2.5\times 0.0117i_{\mathrm{corr}}\hfill \end{array}\begin{array}{cc}& \begin{array}{c}t<T_{\mathrm{corr}}\\ t\ge T_{\mathrm{corr}}\end{array}\end{array}$

where $i_{\mathrm{corr}}$ is the current density at the beginning of corrosion, which is defined by:

(16)$i_{\mathrm{corr}}(t)=0.85\times \frac{37.8{(1-w/c)}^{-1.64}}{d_{\mathrm{c}}}\times {(t-T_{\mathrm{corr}})}^{-0.29}$

The steel bar not only undergoes uniform corrosion, but also pitting corrosion in the process of chloride ion erosion. Val et al. [24] proposed the simplification of the shape of the pit corrosion into a quadrilateral to consider the pit erosion area of reinforcement (Figure 7). Through this simplified model, the pit corrosion area ($A_{\mathrm{corr},\mathrm{p}}$) can be calculated by:

(17)$A_{\mathrm{corr},\mathrm{p}}(t)=\{\begin{array}{ll}{A}_1+A_2\hfill & P(t)\le \frac{d_{\mathrm{s}0}}{\sqrt{2}}\hfill \\ \frac{\pi d_{\mathrm{s}0}^2}{4}-A_1+A_2\hfill & \frac{d_{\mathrm{s}0}}{\sqrt{2}}<P(t)\le d_{\mathrm{s}0}\hfill \\ \frac{\pi d_{\mathrm{s}0}^2}{4}\hfill & P(t)>d_{\mathrm{s}0}\hfill \end{array}$

(18)$A_1=\frac{1}{2}\left[{\theta }_1{\left(\frac{d_{\mathrm{s}0}}{2}\right)}^2-\frac{b}{2}\sqrt{d_{\mathrm{s}0}^2-b^2}\right]$

(19)$A_2=\frac{1}{2}\left[{\theta }_{2}P{\left(t\right)}^2-\frac{bP{\left(t\right)}^2}{d_{\mathrm{s}0}}\right]$

(20)${\theta }_1=2\arcsin \left(\frac{b}{d_{\mathrm{s}0}}\right),{\theta }_2=2\arcsin \left(\frac{b}{2P(t)}\right)$

(21)$b=2R\lambda \sqrt{1-{\left(\frac{P(t)}{d_{\mathrm{s}0}}\right)}^2}$

where $R$ is the pitting corrosion coefficient, which, according to the statistical consideration of Stewart et al. [25], conforms to the extreme value type I through experimental observation.

The irregular cross section of the steel bar after corrosion causes stress concentration, which reduces its mechanical qualities. The yield strength of the corroded steel bar ($f_{\mathrm{y}}$) can be expressed as:

(22)$f_{\mathrm{y}}=(1-0.0049Q_{\mathrm{corr}})f_{\mathrm{y}0}$

where $f_{\mathrm{y}0}$ is the yield strength of the steel bar without corrosion, and $Q_{\mathrm{corr}}$ is the corrosion rate of the reinforcement, which can be obtained by:

(23)$Q_{\mathrm{corr}}=\frac{d_{\mathrm{s}0}{}^2-d_{\mathrm{s}}{(t)}^2}{d_{\mathrm{s}0}{}^2}\times 100\%$

The steel bar corrodes and expands as a result of chloride ion erosion. After a certain amount of rust swelling has occurred, the concrete protective layer breaks, which results in a decrease in the concrete compression strength. Coronelli et al. [26] proposed a method for calculating the compressive strength of the concrete protective layer after the steel bar rusts and expands, which may be computed by:

(24)$f_{\mathrm{c}}=\frac{f_{\mathrm{c}0}}{1+K\frac{n_{\mathrm{bars}}2\pi (v_{\mathrm{rs}}-1)x}{b_0{\epsilon }_{\mathrm{c}0}}}$

where $f_{\mathrm{c}0}$ is the compressive strength of the concrete before the rust expansion of the steel bar; $K$ is the correlation coefficient between the reinforcement size and roughness, which is generally 0.1; $n_{\mathrm{bars}}$ is the number of longitudinal reinforcement bars; $v_{\mathrm{rs}}$ is the corrosion volume expansion coefficient of the reinforcement, which is generally 2.0; $x$ is the depth of the chloride corrosion; $b_0$ is the width of the concrete section; ${\epsilon }_{\mathrm{c}0}$ is the peak strain of the concrete before corrosion.

The restraint effect of the core concrete decreases and the peak stress ($f_{\mathrm{cc}}$) of the core concrete degrades due to the corrosion and degradation of the stirrup. When the condition of the stirrup rust is known, the degradation of the constrained concrete can be calculated by the Mander constitutive model.

4.2. Analysis Process

According to the abovementioned research, the Fisher discrimination function based on the basic parameters of RC piers can effectively classify and identify piers with different failure modes. Furthermore, the corrosion degradation of a pier is reflected as the strength deterioration of the reinforcement and the decrease in the cross-sectional area, as well as the strength degradation of the concrete produced by stirrup corrosion. Pier rust is thus considered to be the deterioration of the fundamental characteristics of RC piers when paired with the process of chloride erosion, rust theory, and the Fisher discriminant function. The discriminant models of the failure mechanisms of corroded piers are generated by putting these parameters into the discriminant function.

Based on this model, the Bayes formula [27] is proposed to compute the likelihood of the various pier failure modes to further research the evolution features of the corroded RC pier failure modes. Let us assume that there are $k$ groups with the algebraic representations ${\pi }_1,{\pi }_2,\dots ,{\pi }_{\mathrm{k}}$; then, the posterior probability that sample $x$ belongs to group ${\pi }_i$ can be calculated by:

(25)$P\left({\pi }_i|x\right)=\frac{p_i{f}_i(x)}{{\displaystyle \sum _{i=1}^k{p}_i{f}_i(x)}}$

where $p_i$ is the prior probability that $x$ belongs to group ${\pi }_i$, and $f_i(x)$ is the probability density function of group ${\pi }_i$.

Figure 8 depicts the failure mode evolution analysis process of the corroded bridge pier, including the probability, based on the findings of the FDA performed on the sample set and integrated with the Bayesian formula.

5. Analysis of Failure Mode Evolution Characteristics of Corroded Bridge Pier

5.1. Example Model

The example is a typical medium-span concrete continuous beam bridge on a highway. The piers are made of C30 concrete with double piers. The section sizes of the piers are 1.2 m $\times$ 1.2 m, and the heights of the piers are 10 m. The pier section is reinforced with 36 HRB335 longitudinal reinforcement bars, each having a diameter of 32 mm and a reinforcement ratio of 2.01%. The bridge pier is supported by an HRB335 rectangular stirrup with a diameter of 10 mm, 8 cm spacing, and a stirrup ratio of 1%. The schematic diagram of the pier and its reinforcement is shown in Figure 9.

5.2. Failure Mode Evolution Characteristics

The service environment of the example bridge was assumed to be a chloride-salt environment based on the mechanisms of chloride ion erosion and steel corrosion; thus, only the effect of the chloride corrosion on the bridge pier was studied. Every 20 of 100 years of service time was taken as the time node, and a total of six time nodes were calculated for the example bridge. The material parameters of the bridge piers at each time node are shown in Table 5. It should be noted that the ratio of the area occupied by the concrete protective layer and the core concrete in the pier section determines the concrete strength because the rust expansion of the steel bar causes the concrete protective layer to fracture and lose strength.

The mechanical properties of the concrete, longitudinal reinforcement, and stirrup were degraded to varying degrees over the 100 years of bridge pier service. By the 100th year, the stirrup yield strength had decreased by 47.26%, the longitudinal reinforcement yield strength had decreased by 5.24%, and the concrete strength had decreased by 20.4%. Among them, the concrete strength and yield strength of the stirrups were obviously degraded. Moreover, the reinforcement ratio and stirrup ratio also had different degrees of loss with the corrosion of the longitudinal reinforcement and stirrup. In the 100th year, the corrosion rate of the longitudinal reinforcement was 10.22%, and the corrosion rate of the stirrups was 96.19%. As the material mechanical qualities and reinforcement configuration are the major indications that influence the failure modes of RC pier columns, the failure modes of bridge piers may vary when the aforementioned material factors deteriorate.

The failure mode evolution features of the pier over 100 years of operation in a chlorine environment were determined using two-stage Fisher discrimination after normalizing the pier parameters at each time node and substituting them into the aforementioned discriminant equation, as shown in Table 6.

Table 6 shows that the failure mode of the pier of the example bridge from 0 to 40 years of service was flexure failure (F). The failure mode changed from flexure failure (F) to flexure–shear failure (FS), and the failure mode of the flexure–shear failure did not change up until 100 years of service, between 40 and 60 years. As seen in Figure 10, the posterior probabilities of the three failure types during pier service were determined using the Bayesian formula.

As can be seen in Figure 10, as the amount of chloride ion erosion of the bridge pier increased with the service time, the corrosion condition became progressively worse. The chance of the flexure failure (F) of the bridge pier in the example decreased from 80.67% at the start of the operation to 37.01% after 100 years. In the 100th year, the chance of the flexure–shear failure (FS) increased from 19.33% to 62.68%, and the variation amplitude reached above 43%. Although the probability of shear failure (S) increased year by year, it always remained at a low level due to the large shear span ratio of the bridge pier in the example. The probability of flexure failure (F) was much higher than those of the other two failure modes at the beginning of service. However, the probability of brittle bridge pier failure increased with the increase in the service time. The chance of flexure–shear failure (FS) exceeded that of flexure failure (F) after 54 years of service, and the probability of nonductile failure reached 62.97% when the service term approached 100 years. Consequently, the bridge pier had the substantial potential for brittle failure over its service life when subjected to seismic activity.

5.3. Parameter Analysis

The pier material characteristics deteriorate under the influence of chloride ion erosion, which results in the development of the pier column failure mechanism, as can be seen from the study above. First, the parameters were divided into five categories, ranging from small to large, and the values of each parameter after 100 years of chloride corrosion were determined in order to further research the impacts of the various material qualities on the evolution of the failure modes. (The corrosion rate of the reinforcement was the same as the example bridge. The corrosion rate of the longitudinal reinforcement was 10.22%, and the corrosion rate of the stirrups was 96.19%.) Next, all the parameters were analyzed by control variables to study the influence of the single-parameter numerical variation on the pier failure modes. The parameters and their variation ranges are shown in Table 7. In the end, the variables in Table 7 were gradually plugged into the two-stage Fisher discriminant function and Bayes formula, and the discriminant findings indicated that the bridge pier was deemed to suffer flexure failure if just one parameter deterioration was evaluated. The relationship between the parameter changes and the probability of bending failure is represented in Figure 11.

According to Table 6 and Figure 11, as the corrosion depths of piers increase in chlorine environments, the compressive strength of the concrete ($f_{\mathrm{c}}^{\prime }$), yield strength of the longitudinal reinforcement ($f_{\mathrm{y}}^{}$), yield strength of the stirrup ($f_{\mathrm{yv}}^{}$), reinforcement ratio (${\rho }_{\mathrm{l}}$), and stirrup ratio (${\rho }_{\mathrm{sv}}$) all decrease, and the influence trends of the different parameter degradations on the pier failure modes vary. There was a significant effect of the deterioration of $f_{\mathrm{c}}^{\prime }$, $f_{\mathrm{yv}}^{}$, ${\rho }_{\mathrm{l}}$, and ${\rho }_{\mathrm{sv}}$ on the pier, while the degradation of $f_{\mathrm{y}}^{}$ was negligible. Reductions in the parameters $f_{\mathrm{c}}^{\prime }$, $f_{\mathrm{yv}}^{}$, and ${\rho }_{\mathrm{sv}}$ reduce the probability of ductile failure and increase the probability of nonductile failure. The reason is that the shear strength of a bridge pier is mainly borne by the concrete and stirrups. Their degradation reduces the shear strength of the bridge pier and makes it harder to meet the shear demand, and thus nonflexure failure is more likely to occur. The influence of the ${\rho }_{\mathrm{l}}$ and other parameters on the pier failure modes showed different trends. The probability of ductile failure increased and the probability of nonductile failure decreased as the value of the ${\rho }_{\mathrm{l}}$ decreased. This is because the reduction in the ${\rho }_{\mathrm{l}}$ reduces the shear demand that corresponds to its bending capacity, while the shear strength remains unchanged. Therefore, the shear strength more easily meets the shear demand, and the pier is more prone to flexure failure.

The degrees of the effects of the aforementioned factors adjusted to the various grades on the failure modes were also different for the corroded pier in comparison with the noncorroded pier. If the longitudinal reinforcement corrosion rate is 10.22% and the stirrup corrosion rate is 96.19%, then increasing the parameters ${\rho }_{\mathrm{sv}}$ and ${\rho }_{\mathrm{l}}$ results in an increase in the flexure failure risk of the pier before and after corrosion. (For instance, if the configuration level of the parameter ${\rho }_{\mathrm{sv}}$ is level 1, or when ${\rho }_{\mathrm{sv}}=0.30\%$, then ${\mathrm{P}}_{\mathrm{F}}^{1-0}-{\mathrm{P}}_{\mathrm{F}}^{1-100}=1.5\%$; if the configuration level is level 5, or when ${\rho }_{\mathrm{sv}}=3.00\%$, then ${\mathrm{P}}_{\mathrm{F}}^{1-0}-{\mathrm{P}}_{\mathrm{F}}^{1-100}=7.52\%$.) The probabilities of flexure failure before and after corrosion were not significantly different under the various grades of the parameters $f_{\mathrm{y}}^{}$ and $f_{\mathrm{yv}}^{}$. The change law of the failure modes of RC piers after corrosion states that the failure modes of RC piers with greater stirrup or reinforcement ratios are more significantly impacted by corrosion, assuming that the diameter of the steel reinforcement remains constant (the corrosion rate is the same). As the value of parameter $f_{\mathrm{c}}^{\prime }$ increases, the chance of bridge pier flexure failure decreases for the concrete strength ($f_{\mathrm{c}}^{\prime }$) both before and after corrosion. If the configuration level of parameter $f_{\mathrm{c}}^{\prime }$ is level 1, or when $f_{\mathrm{c}}^{\prime }$ = 30 MPa, then ${\mathrm{P}}_{\mathrm{F}}^{1-0}-{\mathrm{P}}_{\mathrm{F}}^{1-100}=11.77\%$; if the configuration level is level 5, or when $f_{\mathrm{c}}^{\prime }$ = 70 MPa, then ${\mathrm{P}}_{\mathrm{F}}^{1-0}-{\mathrm{P}}_{\mathrm{F}}^{1-100}=1.06\%$. The change law of the concrete strength impacted by corrosion to the failure mode states that the greater the concrete strength, the smaller the change in the bridge pier failure mode caused by corrosion.

6. Conclusions

In this study, we analyzed the correlation between the original parameters of a rectangular RC pier and its failure mode using the Fisher discriminant method. The significant factors that affected the failure mode of the RC pier were identified, and the two-stage Fisher discriminant model was developed. Additionally, the development features of the RC pier failure mode and the effect of the factors in the chloride environment were explored in combination with the chloride erosion mechanism and corrosion theory. The main conclusions are as follows:

• (1). The two-stage Fisher discriminant formula described in this study has a superior accuracy of 87.4% when compared with the classic discriminant approach based on the shear strength and displacement. In addition, the failure mode of the corroded pier was also effectively distinguished. The proposed discriminant formula not only ensures efficient discrimination, but it also has the advantage of convenient application in engineering practice;

• (2). The results show that the analysis method presented in this paper is suitable for analyzing the evolution characteristics of the lifespan failure modes of bridge piers. After 100 years of operation in a chloride environment, the likelihood of a bridge pier failing due to flexure failure is 37.03%, while the likelihood of nonflexure failure increases to 62.97%. The evolution of the failure mode from flexure failure to flexure–shear failure presents a great possibility;

• (3). The degradation of the different material parameters has different effects on the evolution characteristics of the pier failure modes. The deterioration of the concrete and stirrups accelerates the brittle failure of the pier, whereas the degradation of the longitudinal reinforcement delays it.

Author Contributions

Data analysis, S.H.; Funding acquisition, S.H.; Formal analysis, S.H. and B.C.; Resources, S.H.; Writing Articles, K.S. and X.L.; Writing—review & editing, K.S. and X.L.; Experiment, Z.M.; Data curation, Z.M.; Revise the article, B.C. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data is contained within the article.

Acknowledgments

The authors would like to thank all the partners of the School of Engineering and Construction of Nanchang University. The authors wish to thank S.H. for his help in data analysis and formal analysis of this paper. In addition, the authors would like to thank X.L. and K.S. for their contribution to the writing of this paper. The authors wish to thank Z.M and B.C. for their contribution to the revision of this article.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

Variable	Meaning
$V_{\mathrm{P}}$	Shear demand
$V_{\mathrm{n}}$	Shear capacity
$n$	Number of samples
$m$	Number of sample attributes
$k$	Number of classes
$G_1$	Normal operation data class
$G_2$, …, $G_k$	Faulty data classes
$A$	Sum of squared within-group dispersions
$E$	Sum of squared between-groups dispersions
$\lambda$	Eigenvalues and eigenvectors of $A$
$C$	Eigenvalues and eigenvectors of $E$
$L$	Column height
$f_{\mathrm{c}}^{\text{'}}$	Concrete crushing strength
$f_{\mathrm{y}}$	Yield strength of longitudinal reinforcement
${\rho }_{\mathrm{l}}$	Longitudinal reinforcement ratio
$f_{\mathrm{yv}}$	Stirrup yield strength
$s$	Stirrup spacing
${\rho }_{\mathrm{sv}}$	Stirrup ratio
$\lambda$	Shear span ratio
$u$	Axial compression ratio
$x_{\mathrm{norm}}$	Normalized parameter
$x$	Initial value of parameter
$x_{\max }$	Maximum values of parameters in all samples
$x_{\min }$	Minimum values of parameters in all samples
$M_{\max }$	Maximum bending moment of flexure capacity of plastic hinge section
$A_{\mathrm{sv}}$	Cross-sectional area of stirrup
$h_0$	Effective height of section
$A_{\mathrm{g}}$	Cross-sectional area of RC pier
$k$	Influence coefficient of displacement ductility
$\mu$	Displacement ductility factor
$s/h$	Ratio of stirrup spacing to section height
$D$	Reference chloride diffusion coefficient of concrete age
$C_{\mathrm{s}}$	Chloride ion concentration on outer surface of concrete protective layer
$C_{\mathrm{cr}}$	Chloride ion concentration when steel bar begins to rust
$d_{\mathrm{s}0}$	Diameter of steel bar without corrosion
$d_{\mathrm{s}}(t)$	Residual diameter of reinforcement after time ($t$)
$i_{\mathrm{corr}}$	Current density at beginning of corrosion
$R$	Pitting corrosion coefficient
$f_{\mathrm{y}0}$	Yield strength of steel bar without corrosion
$Q_{\mathrm{corr}}$	Corrosion rate of reinforcement
$f_{\mathrm{c}0}$	Compressive strength of concrete before rust expansion of steel bar
$K$	Correlation coefficient between reinforcement size and roughness
$n_{\mathrm{bars}}$	Number of longitudinal reinforcement bars
$v_{\mathrm{rs}}$	Corrosion volume expansion coefficient of reinforcement
$x$	Depth of chloride corrosion
$b_0$	Width of concrete section
${\epsilon }_{\mathrm{c}0}$	Peak strain of concrete before corrosion
$p_i$	Prior probability that $x$ belongs to group ${\pi }_i$
$f_i(x)$	Probability density function of group ${\pi }_i$

Footnotes

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Figures and Tables

Figure 1. Schematic of Fisher discriminant analysis.

Figure 2. Distribution of reinforcement configuration in different failure modes.

Figure 3. Distribution of shear span ratio in different failure modes.

Figure 4. Distribution of axial compression ratio in different failure modes.

Figure 5. Discriminant results based on shear force.

Figure 6. Discrimination results based on deformation.

Figure 7. Schematic diagram of pitting corrosion area.

Figure 8. Flow chart of failure mode probability analysis of corroded bridge pier.

Figure 9. Schematic diagram of pier and its section.

Figure 10. Variation in failure modes of bridge piers during service.

Figure 11. Influence of material parameter variation on failure mode.

Table 1

Ranges of parameters in sample set.

Parameter	Unit	Minimum	Maximum
Column height $(L)$	mm	249	1126
Concrete crushing strength $(f_{\mathrm{c}}^{\prime })$	MPa	16	115.8
Yield strength of longitudinal reinforcement $(f_{\mathrm{y}})$	MPa	318	586.1
Longitudinal reinforcement ratio $({\rho }_{\mathrm{l}})$	%	0.68	6.03
Stirrup yield strength $(f_{\mathrm{yv}})$	MPa	249	1126
Stirrup spacing $(s)$	mm	20	457.2
Stirrup ratio $({\rho }_{\mathrm{sv}})$	%	0.068	6.03
Shear span ratio $(\lambda )$	/	1	6.04
Axial compression ratio $(u)$	/	0	0.9

Table 2

Pearson correlation coefficients between factors and failure modes.

Influence Factor		Failure Mode
	F–FS	F–S	FS–S
$f_{\mathrm{c}}^{\prime }$	−0.353 **	−0.327 **	−0.209
$f_{\mathrm{y}}$	−0.087	−0.335**	−0.073
${\rho }_{\mathrm{l}}^{}$	−0.037	0.081	0.214
$f_{\mathrm{yv}}^{}$	−0.228 **	−0.118	0.091
${\rho }_{\mathrm{sv}}^{}$	−0.486 **	−0.385 **	−0.322 *
$\lambda$	−0.408 **	−0.488 **	−0.418 **
$u$	0.003	−0.040	−0.042
${\rho }_{\mathrm{l}}^{}{f}_{\mathrm{y}}/f_{\mathrm{c}}^{\prime }$	0.409 **	0.475 **	0.239
${\rho }_{\mathrm{sv}}^{}{f}_{\mathrm{yv}}/f_{\mathrm{c}}^{\prime }$	−0.250 **	−0.272 **	−0.255 *

Comments: ** denotes two-sided significance test factor at 0.01 level, * denotes two-sided significance test factor at 0.05 level.

Table 3

Results of back-substitution test.

Failure Mode (Reality)		Failure Mode (Prediction)			Summation
		F	FS	S
I	F	101 (91.0%)	10 (9.0%)		111 (100%)
	FS/S	4 (6.3%)	59 (93.7)		63 (100%)
II	F	101 (91.0%)	9 (8.1%)	1 (0.9%)	111 (100%)
	FS	3 (7.0%)	35 (81.4%)	5 (11.6%)	43 (100%)
	S	1 (5.0%)	3 (15.0%)	16 (80.0%)	20 (100%)

Comments: F: flexure failure; FS: flexure–shear failure; S: shear failure.

Table 4

Identification results of corroded components.

Corrosion Test	Specimen Name	${\mathit{Y}}_1$	${\mathit{Y}}_2$	Prediction	Reality
Reference [14]	U-C-0.1	1.71	/	F	F
	C-A-0.1	1.70	/	F	F
	C-E-0.1	−0.072	2.86	FS	FS
	U-C-0.45	−0.29	1.01	FS	FS
Reference [15]	RC-1	2.35	/	F	F
	RC-2	1.84	/	F	F
	RC-3	−0.89	0.40	FS	FS
	RC-4	−2.50	−1.96	S	FS
	RC-5	−1.46	0.19	FS	S
	RC-6	−2.50	−1.96	S	S
	RC-7	−1.24	−1.98	S	S
	RC-8	−2.63	−2.34	S	S
Reference [16]	Z-1	2.83	/	F	F
	Z-2	2.61	/	F	F
	Z-3	2.47	/	F	F
	Z-4	2.28	/	F	F
	Z-5	1.49	/	F	F
	Z-6	0.55	/	F	F
	Z-7	−0.65	0.12	FS	FS
	Z-8	−1.46	−1.54	S	FS

Table 5

Material parameters of bridge pier over 100 years of service.

$\mathit{T}$ (a)	${\mathit{f}}_{\mathbf{c}}^{\prime }$ (MPa)	${\mathit{f}}_{\mathbf{y}}$ (MPa)	${\mathit{d}}_{\mathbf{s}}$ (mm)	${\mathit{\rho }}_{\mathbf{l}}$ (%)	${\mathit{f}}_{\mathbf{yv}}$ (MPa)	${\mathit{d}}_{\mathbf{v}}$ (mm)	${\mathit{\rho }}_{\mathbf{sv}}$ (%)
0	30	335.00	32.00	2.01	335.00	10.00	1.00
20	27.59	334.07	31.99	2.01	330.51	9.89	0.98
40	24.64	331.48	31.74	1.98	290.95	8.58	0.74
60	24.16	326.86	31.28	1.92	241.55	6.59	0.43
80	23.99	321.40	30.73	1.85	197.27	4.05	0.16
100	23.87	317.45	30.32	1.81	176.67	1.95	0.04

Table 6

Failure modes of bridge piers over 100 years of service.

T (a)	Y1	Y2	Failure Mode
0	0.60	3.13	F
20	0.48	3.04	F
40	0.09	2.53	F
60	−0.15	1.94	FS
80	−0.24	2.35	FS
100	−0.25	1.06	FS

Table 7

Ranges of parameters.

Level	1	2	3	4	5
${\rho }_{\mathrm{sv}}^{}$ (%)	0.30	1.08	1.65	2.22	3.00
	0.01	0.04	0.06	0.08	0.11
${\rho }_{\mathrm{l}}^{}$ (%)	0.70	1.66	2.35	3.04	4.00
	0.63	1.49	2.11	2.73	3.59
$f_{\mathrm{y}}^{}$ (MPa)	300	350	400	450	500
	285.0	332.5	380.0	427.5	475.0
$f_{\mathrm{yv}}^{}$ (MPa)	300	350	400	450	500
	158.6	185.0	211.4	237.9	2643
$f_{\mathrm{c}}^{\prime }$ (MPa)	30	40	50	60	70
	24.01	31.87	39.84	47.73	55.70

Comments: Each parameter corresponds to two rows of variables: the first row is the parameter value without corrosion, and the second row is the parameter value after 100 years of chloride erosion.

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