Statistical Reliability Analysis for Assessing Bridge Structural Integrity: A Review Paper

Abstract

This article reviews methods for estimating the remaining service life of bridges, focusing on the statistical analysis of reliability indices, which aids in identifying risks and predicting structural failures. Among the methodologies examined, the First-Order Reliability Method (FORM) is highlighted for its effectiveness in calculating failure probabilities based on current deterioration and loading conditions. Sensitivity analysis is also discussed, as it pinpoints the variables that most significantly impact structural stability. Enhanced using the Finite Element Method (FEM), this method allows the simulation of structural behavior across different deterioration scenarios, improving the precision of failure predictions and optimizing maintenance planning. This review provides insight into how the integration of probabilistic methods and sensitivity analysis can enhance failure prediction and support more efficient maintenance planning for bridge structures.

1. Introduction

The highway network of a country plays a vital role in its economy by connecting producers and consumers and facilitating the movement of people. In Brazil, for instance, this network spans over 1.7 million kilometers [1], accounting for approximately 70% of goods transportation and impacting up to 29% of the country’s economy [2]. Due to their extensive lengths worldwide, these routes encounter natural and artificial obstacles, such as topographical features, bodies of water, buildings, and even other routes. To ensure the continuity of these networks, structures such as bridges have been constructed. Bridges are ubiquitous worldwide, with approximately 600,000 in the United States alone [3] and an estimated 120,000 in Brazil [4]. These structures have been built for centuries using a variety of materials, including masonry, concrete, steel, and timber.

Bridges undergo a deterioration process over time due to exposure to various environmental and load conditions [5]. Given their significant socioeconomic importance, ensuring the safety and functionality of bridges is critical and often falls under the responsibility of public jurisdictional bodies. According to [3], approximately 40% of bridges in the United States are over 50 years old (the average service life for such structures), and 9% are classified as structurally deficient.

In Brazil, no precise survey has been conducted to determine the exact quantity, location, and condition of existing bridges [4]. This scenario may be explained by the fact that many bridges are in distant and difficult-to-access locations, not having a safe location for inspection [6]. Ref. [7] mention an inventory of 4725 Brazilian bridges, with an average age of 40 years, in which 4.7% required immediate or mid-term interventions, and 38% exhibited minor structural damage. Until the 1960s, bridge management, maintenance, and repair services were reactive, carried out only as structural issues arose [8]. By the 1970s, with the increasing number of bridges and a growing gap between repair demand and available resources, bridge management programs were developed to assess the current state of bridges and prioritize those in the worst condition [9]. According to [10], these programs typically consist of several stages: bridge inspection, assessment of the current condition, optimized maintenance planning, prioritization of deteriorated structures, and execution of necessary interventions.

Inherent uncertainties arise in the data when analyzing existing structural systems, including geometry, material properties, existing damage, and applied loads. As [11] highlight, since a theoretical model of the structure relies on data with some degree of variability, these inputs can be treated as random variables and analyzed using statistical methods. One statistical approach to evaluating the stability, safety, and functionality of a bridge is applying reliability theory to the structural system. This method enables the estimation of the structure’s probability of failure [12] and provides a quantitative measure of its current state, aiding in comparison and maintenance planning.

This paper aims to provide a review of statistical reliability analysis methodologies applied to bridge structures, focusing on evaluating structural integrity and predicting remaining service life. The main contribution is providing a comprehensive survey and emphasize the discussion in the integration of probabilistic models with advanced simulation techniques and parameter sensitivity assessments. The paper identifies key opportunities to improve failure prediction accuracy and optimize maintenance decision-making.

The paper is organized as follows: Section 2 details the data collection and parameter determination; reliability and sensitivity analysis are approached on Section 3 and Section 4, respectively; Section 5 describes the remaining service life; and finally, the conclusion is in Section 6.

2. Data Collection and Parameter Determination

Since management programs often involve the analysis of existing structures, original design documents, such as blueprints and material specifications, are frequently unavailable, making it challenging to obtain accurate information about the structure’s geometry and materials [13,14]. Therefore, accurate assessment of a bridge’s structural integrity begins with the collection of reliable data and the determination of parameters that influence its performance.

Data collection can be conducted using a variety of methods, ranging from more invasive approaches, such as destructive testing that involves extracting samples for laboratory analysis (e.g., uniaxial compressive tests) [15], to non-destructive testing (NDT) techniques and visual inspections, which aim to obtain critical information without compromising the structure’s integrity [5,15]. Non-destructive methods offer some advantages, including safety, cost-effectiveness, and the ability to provide real-time data while maintaining the structure in service albeit having a higher degree of uncertainty [13]. Given the wide variety of bridges in terms of geometry, materials, loads, and locations, the choice of methods must be tailored on a case-by-case basis, considering the bridge’s specific characteristics and the available resources.

Examples of NDT techniques include ultrasonic testing for detecting internal flaws, ground-penetrating radar (GPR) [16,17] and Terrestrial Laser Scanner (TLS) [17,18] for assessing subsurface conditions. These methods, along with technologies such as drones and laser scanning, provide comprehensive information on geometric properties, material conditions, existing damage, all of which are essential for creating an accurate and reliable structural model. However, the data collection methodologies must consider the type of bridge and construction material, as these factors directly affect the quality and interpretation of data.

In addition, the implementation of Digital Twin frameworks [19,20,21] has enabled continuous data acquisition, since they are a virtual representation of a physical structure that mirrors its current state, behavior and performance. In the context of bridge health monitoring, Digital Twins integrate sensor data, inspection records and inputs to create an accurate and evolving digital model. Ref. [19] applied a Digital Twin approach to study a railway bridge by embedding sensors that collected strain, temperature and acceleration data that were fed into a digital replica of the structure, showing how this technique can facilitate data acquisition and support decision making and predictive maintenance strategies.

One of the primary challenges in this process is ensuring the precision and consistency of the collected data. Traditional visual inspections, while widely used, often suffer from subjectivity and may miss critical details in hard-to-reach areas [22]. Recent advancements, such as aerophotogrammetry using drones, have mitigated some of these challenges by enabling high-resolution imaging and the creation of detailed 3D models which allow for pathological location and geometry acquisition [23,24]. These technologies allow for comprehensive documentation of structural conditions, reducing the risks associated with manual inspections and improving data accuracy.

Once the data is collected, key parameters must be determined to facilitate statistical reliability analysis. These parameters include the geometric properties of the bridge, such as span length, cross-sectional dimensions, and material thicknesses, as well as material properties, including strength, elasticity, and degradation rates. These parameters are often estimated based on existing initial data, tests performed and comparisons with literature values and similar structures. Additionally, loading conditions must be quantified, encompassing live loads (e.g., traffic) and dead loads (e.g., structural weight). These loads are typically based on data collected by relevant regional authorities or calculated according to national design standards. Environmental factors, such as temperature variations, humidity, and exposure to corrosive agents, may be also considered as they impact the structure’s deterioration over time.

According to [11], the theoretical models used to describe the behavior of a structure are based on information such as geometry and material properties, which exhibit a certain degree of natural variation and can be treated as random variables. To properly assess these variations, it is crucial to identify appropriate probability density functions (PDFs) for each variable, considering uncertainties in material properties, geometry, and applied loads. Table 1 provides a bibliographic survey of the probability density functions commonly considered for these variables.

Due to the vast diversity of structural types and materials, it is not feasible to establish global statistical parameters for each variable. Instead, these parameters must be determined experimentally or sourced from the literature on a case-by-case basis, depending on the specific structure analyzed. This approach ensures that the reliability analysis reflects realistic conditions, providing a robust foundation for structural assessment and decision-making.

In this context, these variables are managed through statistical analyses, which enable the computation of structural resistance, the probability of failure of the structure under study and thus remaining service life. This type of analysis is typically conducted based on an estimation of the decrease in structural strength reliability over time [29]. To achieve this, modeling is usually performed using software that employs the Finite Element Method (FEM), such as DIANA FEA [26,29] and ATENA [14], where all geometric and material parameters can be applied, allowing the observation of the structure’s behavior as loading is progressively applied. According to [95], probabilistic analysis in finite elements often encounters challenges related to model convergence, making it necessary to perform case-specific adjustments, such as reducing mesh size or modifying time increments.

Additionally, the data collection methodology must be selected based on the type of bridge being analyzed, since it directly influences the feasibility of certain types of testing and the quality and interpretation of collected data. This is due to the inherent differences in materials and construction techniques used, i.e., concrete bridges tend to exhibit more uniform and continuous behavior, while masonry bridges are more heterogeneous, with greater variability in geometry and material composition, leading to greater data uncertainty. Consequently, the setup and calibration of the FEM model must account for these distinctions, reinforcing the importance of a case-by-case approach to ensure accurate structural assessment.

3. Reliability Analysis

Ref. [96] describes the concept of reliability in statistics as the ability of a given system to perform its intended function adequately over a specified period without the occurrence of undesirable events (failures) during operation. Structural reliability analysis seeks to compute the probability of failure of an entire structure or one of its elements, considering uncertainties in certain parameters, i.e., loads, materials and geometrical dimensions [12,96].

Since reliability analyses can be applied to various structures subjected to diverse types of analyses and considerations, different formulations may be employed. However, these analyses are often based on performance/failure criterion equations $G\left(R,S\right)$, which represent either the difference (1) or the ratio (2) between resistance $R$ and load $S$. The outcome of the failure criterion equations is that it equals zero at the limit state (imminent failure), yields a negative value for failure conditions, and a positive value otherwise.

(1)$G\left(R,S\right)=R-S$

(2)$G\left(R,S\right)={\displaystyle \frac{R}{S}}-1$

This equation serves as the basis for calculating the failure probability $P_f$ (3), which represents the probability that $R$ is greater than $S$. This can be computed as the integral of the system’s joint Probability Density Function (PDF) $f_x\left(x\right)$, which accounts for all variables involved, whether random or deterministic. Given the high complexity of real structures, an analytical solution to this integral is unfeasible, thus requiring alternative methods. One of the main challenges lies in determining the joint PDF of the system, for which three main groups of methodologies can be highlighted: sampling-based methods, metamodeling-based methods and approximate methods.

(3)$P_f=P\left(R-S\le 0\right)={\int }_{R-S\le 0}{f}_x\left(x\right)·dx$

3.1. Sampling-Based Methods

Sampling-based methods are computational techniques designed to estimate the failure probability by exploring the performance function across the probabilistic space, simulating random samplings of the variables involved. These methods are primarily based on Monte Carlo simulations (MCSs). MCS is a statistical method used to estimate the outcomes of complex and uncertain processes by generating a large number of random samples to approximate the solution of a given problem. In this approach, the failure probability is obtained from the fraction of simulations in which failure occurs [97,98,99,100]. However, the traditional MCS approach requires a substantial number of simulations to achieve accurate convergence, resulting in high computational costs. To address this limitation, various sampling strategies have been developed to reduce the number of required samples without introducing spurious correlations among the variables.

Among these, Latin Hypercube Sampling (LHS) [101,102,103] enhances sample representativeness by stratifying the input variable’s distribution and ensuring that each stratum is adequately sampled, thereby reducing the statistical variability and enhancing convergence efficiency. Ref. [101] utilized LHS to obtain the training points for a Kriging model, combining both techniques to study the structural health of a truss bridge and an arch bridge, obtaining results that were considered accurate and with great reduction of computational cost when compared to a generic algorithm.

Subset Simulation [104,105] decreases computational complexity in estimating small failure probabilities by decomposing the event into a sequence of more probable intermediate events and then generating samples within these subset events. Ref. [104] employed this approach in the reliability analysis of dynamic responses of train–bridge systems subjected to track irregularities. The study significantly reduced the number of simulations needed while maintaining accurate estimates for low-probability events, demonstrating computational advantage over traditional MCSs.

Importance Sampling [106,107,108] enhances computational efficiency by concentrating sample generation from a biased distribution focused on the failure region. Each sample is weighted according to the ratio between the original and biased densities, improving the accuracy of statistical estimations. Ref. [106] demonstrated that, when the sampling density is well chosen, Importance Sampling can reduce the number of simulations needed by several orders of magnitude when compared to standard MCSs.

3.2. Metamodeling-Based Methods

Metamodeling-based methods, also known as surrogate modeling, are approaches used to estimate the failure probability by replacing the system’s joint PDF with a simpler approximated function that is easier to solve [109]. In this context, metamodels (or surrogate models) are introduced to approximate the system’s behavior through a simplified representation of the actual response, thereby significantly reducing the associated computational cost [110]. This is shown by [111], who concluded that, for 10⁵ 3D train–bridge interaction simulations, the computational time required without surrogate models was 20,000 times greater than when surrogate models were employed. Surrogates can be built using various models, such as Kriging, Polynomial Chaos Expansion (PCE), Artificial Neural Networks (ANNs), Bayesian Networks and Support Vector Machines (SVMs).

Among the available metamodeling techniques, the Polynomial Chaos Expansion method [12,46,112] represents the structural response as a sum of orthogonal polynomials of the different random variables, enabling efficient uncertainty propagation and analytical derivation of statistical moments and sensitivity indices. Ref. [112] applied PCE to quantify the uncertainty of an existing prestressed bridge with a nonlinear FEM model. The surrogate model was used to compute the statistical parameters (such as mean and standard deviation) of the model outputs.

Kriging [101,113,114,115,116] models the response of a system based on limited data points, assuming that the output varies smoothly and using both the distance between points and trends in the data. This technique provides an estimate value alongside a measure of prediction uncertainty. Ref. [113] developed a Kriging-based surrogate model calibrated on nonlinear FEM simulations to assess the scour-induced fragility of a masonry arch bridge, significantly reducing the computational costs while preserving accuracy.

ANNs are data-driven models inspired by the human brain’s structure. These models consist of interconnected layers of nodes which transform input data through a series of weighted combinations, making them well suited for modeling highly nonlinear structural responses. Ref. [117] utilized ANNs, combined with Kriging, to predict the embodied energy of prestressed slab bridges.

Bayesian Networks [118,119,120] are probabilistic graphical models that encode conditional dependencies among random variables through a directed acyclic graph, facilitating efficient uncertainty quantification and decision-making under incomplete data. In the context of structural engineering, they are used for modeling interdependent parameters where computational or experimental data are sparse, enabling risk assessment and reliability analysis without relying solely on costly simulations. Ref. [119] utilized Bayesian networks to evaluate the scour risk in submerged foundations of bridges, incorporating both structural and hydraulic variables, updating failure probabilities as new monitoring data became available.

Support Vector Machines [121,122,123] apply supervised machine learning techniques that separate data into categories by finding the most effective boundary between different classes. SVMs are usually used to identify whether a certain input leads to a safe or failed outcome, making it useful to approximate the limit-state surface. Ref. [121] integrated SVMs with Digital Twin models to classify bridge conditions based on real-time sensor data, enabling predictive maintenance strategies.

3.3. Approximate Methods

Approximate methods, also referred to as gradient-based methods, aim to compute the system’s failure probability by transforming the problem from the original space (failure surface) into a standardized normal space, allowing the random variables involved to be appropriately mapped. Among the various approaches, structural reliability analysis stands out as a technique intended to compute the failure probability of a structure or structural element considering the uncertainty associated with certain parameters [12]. Within the different reliability analysis techniques, the First-Order Second-Moment (FOSM) method—also known as the First-Order Reliability Method (FORM), as proposed by [124]—is particularly noteworthy.

This method can be used to quantitatively estimate the safety level of a structure. One example of this is [98], who utilized FORM (combined with MCS) to evaluate the structural reliability of a prestressed bridge in Brazil, showcasing how variations in resistance and traffic loads impact the safety index.

Both the resistance and load values of the structure can be analyzed as random variables resulting from the interactions between identified parameters. Consequently, various simulations can be conducted for different combinations of random and deterministic variables. These simulations can be used to construct a histogram [5,15,29,32,43,60] of the results, facilitating the identification of the most suitable probability distribution and relevant statistical parameters, i.e., fitting a PDF.

It is important to note, however, the need for careful consideration in how resistance and load simulations will be conducted. This is due to the high computational cost and time required for executing a large number of analyses, necessitating alternative approaches, as stated before. Furthermore, it is imperative that the permutation of random variables in the simulations be performed in a way that effectively represents the system and avoids the emergence of spurious correlations between variables [103].

On the other hand, while it is possible to assume independence among the considered random variables to simplify the analysis, this assumption may fail to accurately represent the real-world behavior of the structure and lead to misleading results [125]. This is due to the fact that variables may exhibit significant correlation due to shared factors, such as environmental exposure and construction process. The correlation between variables might amplify or mitigate risk depending on the nature of dependency. To address this, Ref. [126] employed copula-based methods to model non-linear dependencies among variables, demonstrating improved accuracy in reliability estimation.

From the simulations conducted, a reliability index is calculated to quantitatively represent the safety level of the analyzed structure. One of the most common methods [127] for this calculation involves FORM. This approach allows the determination of the probability of failure of the structure (5) through the Cumulative Density Function (CDF) of a standard normal distribution. The reliability index at the time of analysis $\beta$ (4) is calculated based on the mean μ and standard deviation σ of the system’s resistance and imposed load.

(4)$\beta ={\displaystyle \frac{{\mu }_R-{\mu }_S}{\sqrt{{\left({\sigma }_R\right)}^2+{\left({\sigma }_S\right)}^2}}}$

(5)$P_f=\mathsf{\Phi }(-\beta )$

Examples of studies employing this method in the context of bridge analysis include [15,26,57]. Similarly, studies conducted by [14,17,61,74] calculated the probability of failure using the standard normal CDF of the difference between resistance and load. The reliability index was obtained by applying the inverse CDF function to the probability that the structure will not fail.

Ref. [25], on the other hand, conducted Monte Carlo simulations using combinations of random variables to determine the percentage of failure scenarios. The reliability index was obtained by applying this probability to the inverse function of the standard normal PDF. Refs. [34,95] used a similar formulation (6), where the reliability index is calculated based on the mean and coefficient of variation $V$ of resistance and load. According to these authors, despite similarities, this formulation is more suitable for cases dominated by a lognormal PDF.

(6)$\beta ={\displaystyle \frac{\mathit{log}\left(\frac{{\mu }_R\sqrt{1+V_S^2}}{{\mu }_S\sqrt{1+V_R^2}}\right)}{\sqrt{log\left[\right(1+V_R^2\left)\right(1+V_S^2\left)\right]}}}$

However, as noted by [127], if the identified PDFs do not conform to normal or lognormal distributions, the first-order reliability index becomes inapplicable, necessitating the use of second- or third-order reliability indices. The third-order method is the only one suitable when the PDFs are unknown. The authors present three third-order method calculations based on kurtosis (7) and the second-order method (8). These include a lognormal-based distribution (9), a normal-based distribution (10) and an intermediate form between the two (11).

(7)$a_{3g}=E\left[{\left({\displaystyle \frac{G-{\mu }_G}{{\sigma }_G}}\right)}^3\right]$

(8)${\beta }_{2M}={\displaystyle \frac{{\mu }_G}{{\sigma }_G}}$

(9)${\beta }_{3M-1}=-{\displaystyle \frac{a_{3g}}{6}}-{\displaystyle \frac{3}{a_{3g}}}·\mathit{ln}\left(1+{\displaystyle \frac{1}{3}}{a}_{3g}{\beta }_{2M}\right)$

(10)${\beta }_{3M-2}={\displaystyle \frac{3-\sqrt{9+a_{3g}^2-6a_{3g}{\beta }_{2M}}}{a_{3g}}}$

(11)${\beta }_{3M-3}={\displaystyle \frac{1}{3}}{\beta }_{2M}\left\{2+\mathit{exp}\left[{\displaystyle \frac{1}{2}}{a}_{3g}\left({\beta }_{2M}-{\displaystyle \frac{1}{{\beta }_{2M}}}\right)\right]\right\}$

Subsequently, the calculated reliability index is used to assess the safety of structures. Due to the fact that the reliability index is calculated to reflect the structure’s failure limit state, it is undesirable for it to approach zero. To ensure safety, the reliability index should exceed a target value (usually represented as ${\beta }_{\mathrm{t}}$ or ${\beta }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}$). Ref. [61] suggests that one method to determine ${\beta }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}$ involves minimizing the lifecycle costs of the structure, accounting for construction costs, direct maintenance costs, and indirect costs associated with the loss of roadway functionality. It was observed that higher initial construction costs correspond to higher initial reliability indices, which in turn reduce maintenance costs and failure-related expenses. On the other hand, lower construction costs are correlated with lower reliability indices, higher maintenance costs and failure-related expenses.

Standards such as Refs. [128,129,130,131], alongside Ref. [132], provide minimum reliability index values to ensure structural safety and performance. However, each standard can assign different values of ${\beta }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}$ to the same structure and thus affect the results of a reliability analysis. In addition, these values depend on numerous factors, such as the bridge type and traffic flow, necessitating a case-specific study for each structure or element under consideration. An example of this difference is presented by [61], in which ${\beta }_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}$ varies depending on the road classification (traffic flow), corrosion rate scenario and type of element.

Ref. [42] also highlights that analyses often depend on the structural design standards in effect in the country where the structures are located. This factor influences the results due to differences in calculation methodologies, load considerations, and levels of conservatism across standards. The authors observed that, when analyzing the historical progression of standards that were in effect in the UK [133,134,135,136,137], there was an increase in the expected design loads and a consequent reduction in the estimated reliability indices.

4. Sensibility Analysis

Due to the considerable number of variables that can simultaneously influence the structure, an analysis considering all of them would entail high computational cost, making it time-consuming and offering low cost-effectiveness. Furthermore, a portion of the random variables, although present, do not necessarily have a significant impact on structural behavior and can be omitted, or their accuracy relaxed [57] in the analysis, without considerable loss of output accuracy.

Sensitivity analysis is often used to identify which parameters have the most significant impact on structural reliability, enabling engineers to focus their efforts on refining the most critical inputs. This systematic approach ensures that reliability analysis is both robust and reflective of real-world conditions, laying the groundwork for accurate service life prediction and maintenance planning.

This sensitivity analysis can be performed in diverse ways, depending on the formulation adopted. Based on the research of [97], Refs. [14,15,26,29,32] utilized the formulation presented in (12) to calculate the importance ($b_k$) of each random variable ($k$) analyzed. This importance is determined for each variable based on its coefficient of variation ($COV$), mean ($x_{m,k}$), and standard deviation ($∆x_{i,k}$), as well as the mean ($y_{m,k}$) and standard deviation ($∆y_{i,k}$) of the structural response. These authors established a minimum threshold value of 10% [14,29,32], 20% [26] or 35% [15] for their importance to be considered relevant in the reliability analysis.

(12)$b_k=COV·\sum _{i=0}^n\left({\displaystyle \frac{∆y_{i,k}}{y_{m,k}}}\right)/\left({\displaystyle \frac{∆x_{i,k}}{x_{m,k}}}\right)$

Ref. [42] used a parametric sensitivity factor (${\alpha }_i$)—based on [108,138,139]—to measure the sensitivity of the reliability index ($\beta$) in relation to small variations in the mean value (${\mu }_i$) of the variable ($i$) (13). Furthermore, the variable’s importance (${\alpha }_i^2$) (14) was used to quantitatively compare and indentify which parameter has a greater impact on the structure, aiming for a minimum threshold of 10% for parameter selection.

(13)${\alpha }_i={\displaystyle \frac{\partial \beta }{\partial {\mu }_i}}$

(14)$\sum _{i=1}{\alpha }_i^2=1$

Ref. [59], based on the [128], in turn, used a sensitivity coefficient (${\alpha }_i$) (15) of a variable ($X_i$) based on the structure’s performance function ($g\left(x^{*}\right)$) and the variable’s standard variation (${\sigma }_{X_i}$). Thus, per this formulation, the sum of the squares of the sensitivity coefficients of each variable is equal to 1, similarly to what is shown in (14). Additionally, the authors indicated that there are various ways to represent the sensitivity of a variable, which can be measured with respect to different parameters of that variable, such as the mean and/or standard deviation. These indicators differ not only in their conceptual formulation but also in their mathematical structure and interpretation. This variety of indicators may lead to confusion or incompatibility in comparisons, particularly due to potential differences in units of measurement. Therefore, a unified formulation should be employed.

(15)${\alpha }_i=-{\displaystyle \frac{{\sum }_{i=1}^n\frac{\partial g\left(x^{*}\right)}{\partial X_i}·{\sigma }_{X_i}}{\sqrt{{\sum }_{i=1}^n{\left[\frac{\partial g\left(x^{*}\right)}{\partial X_i}\right]}^2·{\sigma }_{X_i}^2}}}$

5. Remaining Service Life

Once the current safety level of a structure is assessed, this allows for the prioritization of the structures in the worst condition so that maintenance and repair services can be carried out in a way to optimize the available resources. However, considering that the reliability index for measuring safety is time-dependent—since it tends to decrease over time due to the progression of damage [13,140]—and aiming to avoid constantly repeating this safety analysis for all structures, it becomes important to estimate the remaining service life based on the current conditions.

The remaining service life, according to [13], can be defined as the estimated period remaining until the structure (or component) reaches a safety level considered insufficient, representing a risk of failure and/or the need for intervention. This estimation is based on a projection of the expected progression of existing pathological manifestations and may consider possible intervention scenarios for the structure. According to [140], this progression is responsible for the decline in the structure’s performance level. However, maintenance actions can either increase this level (essential maintenance) or temporarily reduce the deterioration rate (preventive maintenance).

The application of reliability analysis to bridge structures was first proposed by [141], and the formulation was later developed by [142,143,144]. Building on this, Ref. [140] presented a formulation to estimate the remaining service life based on the initial reliability index.

This analysis involves estimating a new reliability index over time, $\beta \left(t\right)$ (16), based on the initial reliability, ${\beta }_0$; the age relative to the initial reliability, $t$; the age since the initiation of pathologies, $t_I$; and the reliability deterioration rate, $\alpha$. Additionally, (16) and (17), as shown in Figure 1, consider variables such as ${\beta }_1$: the reliability index after the first maintenance; ${\gamma }_1$: the increase in reliability index after the first maintenance; ${{\beta }^{\prime }}_1$: the reliability index after the period $t_{PD}$ since the first maintenance; $\theta$: the reliability deterioration rate during $t_{PD}$; $t_{PI}$: the age at the first maintenance; $t_{PD}$: the duration of maintenance; and $t_P$: the interval between maintenance actions.

These equations demonstrate that performing maintenance and repairs improves reliability and delays its deterioration for a certain period, requiring a new reliability index to be calculated/estimated after maintenance.

(16)$\beta \left(t\right)=\left\{\begin{array}{c}{\beta }_0, 0\le t\le t_I\\ {\beta }_0-\left(t-t_I\right)·\alpha , t_I<t\le t_{PI}\\ {\beta }_1-\left(t-t_{PI}\right)·\theta , t_{PI}<t\le t_{PI}+t_{PD}\\ {{\beta }^{\prime }}_1-\left[t-\left(t_{PI}+t_{PD}\right)\right]·\alpha , t_{PI}+t_{PD}<t\le t_P\\ ⋮\end{array}\right.$

(17)${\beta }_1=\beta \left(t_I\right)+{\gamma }_1$

The authors point out that these variables can also be treated as random variables, given that uncertainties exist throughout the structure’s lifecycle. For instance, according to [140,145,146], the initial reliability index ${\beta }_0$ and the time of pathology initiation $t_I$ are typically modeled as lognormal distributions, while the deterioration rate $\alpha$ follows a uniform distribution, reflecting linear degradation under constant environmental and loading conditions, which justifies the linear graphs shown in Figure 1. Maintenance-related parameters, such as the application time of interventions $t_{PI}$ and their effective duration $t_{PD}$, are often described by triangular and lognormal distributions, respectively. Notably, the increase in reliability after maintenance ($\gamma )$ and the reduction in the deterioration rate ($\alpha -\theta )$ during maintenance are also treated as lognormal and uniform variables. In addition, Refs. [145,146] indicate that these variables present different statistical parameters depending on dominant failure mode—whether shear or moment governs the structural behavior. This variability underscores the importance of context-dependent parameter selection to ensure accurate remaining service life predictions.

Although linear deterioration models can be used for simplicity, they might fail to represent the real degradation mechanisms, which are driven by imposed loads and environmental exposure and usually do not progress at a constant rate. Ref. [147] highlighted that the corrosion process can accelerate once cracking initiates, resulting in faster decline in structural performance. Moreover, Ref. [148] demonstrated that coupling mechanical and environmental effects may result in highly nonlinear deterioration patterns, directly affecting the reliability indices over time.

Therefore, another approach to estimating the remaining service life of a structure involves considering material properties, loads and/or damage as time-dependent functions. By adopting these considerations, it becomes possible to calculate the evolution of structural resistances and demands over the years. This enables a safety analysis through the nonlinear progression of the reliability index and/or failure probability.

An example of this is provided by [57], who studied the reliability of a chloride diffusion model that incorporates time into the calculation of ion concentration in concrete. Such considerations were employed in Monte Carlo simulations to calculate the progression of reliability indices and fail probabilities for different damage progression scenarios. It was noted that each progression consideration indicated different reliability indices and failure probabilities, with increasing discrepancy over time, and thus in the remaining service lives. Similarly, Ref. [127] calculated the evolution of structural resistance under various degradation rates using time-dependent action functions. The authors determined the evolution of failure probability, defining the end of the service life as the moment the structure reaches a predetermined maximum allowable failure probability.

6. Conclusions

This review presented a comprehensive synthesis of statistical and probabilistic methods used for assessing the reliability and residual service life of bridge structures. As bridges are essential components to national infrastructure and economic stability, ensuring their safety and functionality is paramount. Through the classification of sampling-based, approximate and surrogate-based models, this paper highlights the diversity of approaches available to help quantify failure probability in the presence of uncertainties in material, load or geometric properties. Among these, the integration of FEM with reliability analysis, especially using FORM, is emphasized as a robust framework for capturing nonlinear structural behavior.

It was noted that data collection is an integral part of the analysis of an existing structure. Additionally, since there are uncertainties associated with geometry, material and load parameters, and this information is not always readily available, the use of destructive and non-destructive testing techniques is an effective tool to obtain reliable data and thus reducing the subjectivity inherent in traditional visual inspections. Moreover, Digital Twin frameworks can offer advantages by continuously acquiring data and updating the model, thus improving accuracy and supporting data-informed maintenance planning.

While a variety of methods are available for structural analysis, their practical application can be restricted by high computational costs or lack of data. Therefore, the trade-off between accuracy and feasibility becomes a critical consideration. Approximate and surrogate models, uncertainty quantification and sensitivity analysis offer a practical alternative, enabling scalable reliability assessments across varying resource contexts even when operating under resource constraints.

In summary, this paper consolidates and structures a wide range of probabilistic methods for assessing the structural reliability and service life of bridge structures. Special attention is given to the time-dependent evolution of reliability indices and the selection of appropriate PDFs for the considered variables. Such an approach is essential for optimizing maintenance strategies and use of resources, ensuring safety and extending the service life of existing bridges.

Author Contributions

Conceptualization, G.S.P.P., R.S., M.H.O. and F.E.J.; validation, G.S.P.P., R.S., M.H.O. and F.E.J.; formal analysis, G.S.P.P., R.S., M.H.O. and F.E.J.; investigation, G.S.P.P., R.S., M.H.O. and F.E.J.; resources, G.S.P.P., R.S., M.H.O. and F.E.J.; data curation, G.S.P.P., R.S., M.H.O. and F.E.J.; writing—original draft preparation, G.S.P.P., R.S., M.H.O. and F.E.J.; writing—review and editing, G.S.P.P., R.S., M.H.O. and F.E.J.; visualization, G.S.P.P., R.S., M.H.O. and F.E.J.; supervision, G.S.P.P., R.S., M.H.O. and F.E.J.; project administration, G.S.P.P., R.S., M.H.O. and F.E.J. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The authors are very thankful to the University of Brasília (UnB); the Computational Methods and Artificial Intelligence Laboratory (LAMCIA); the Civil Engineering Technology Study Group (GETEC-UnB).

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

ANN	Artificial Neural Network
CDF	Cumulative Distribution Function
FEM	Finite Element Method
FORM	First-Order Reliability Method
FOSM	First-Order Second Moment
GPR	Ground-Penetrating Radar
LHS	Latin Hypercube Sampling
MCS	Monte Carlo simulations
NDT	Non-destructive testing
PCE	Polynomial Chaos Expansion
PDF	Probability Distribution Function
SVM	Support Vector Machines
TLS	Terrestrial Laser Scanner

Footnotes

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Figure and Table

Figure 1 Representation of deterioration of the reliability index over the years [140]. Used with permission of ASCE, from Reliability-Based Life-Cycle Management of Highway Bridges, Frangopol, DM; Kong, JS; Gharaibeh, ES, volume 15, issue 1, 2001; permission conveyed through Copyright Clearance Center, Inc., Danvers, Massachusetts, USA.

Table 1

Bibliographic survey of bridge variables and their PDFs.

Variable	PDF	References
Concrete
Concrete compressive strength	Normal	[5,11,14,25,26,27,28,29,30,31,32,33,34,35,36,37]
	Lognormal	[11,33,37,38,39,40,41,42,43]
Concrete tensile strength	Normal	[11,14,29,30,32,33,37,38,44]
	Lognormal	[11,37,43,44,45]
Concrete modulus of elasticity	Normal	[5,11,12,14,26,29,30,32,33,37,38,46,47,48]
	Lognormal	[43,45]
Concrete density	Normal	[5,14,26,29,34,37,40,43,47,48,49,50]
Pavement density	Normal	[14,34]
Poisson’s ratio	Normal	[5,26]
Concrete tensile strain	Normal	[30]
Concrete compressive strain	Normal	[30]
	Lognormal	[43]
Creep coefficient	Normal	[30]
Water-cement ratio	Triangular	[51]
Curing time	Normal	[51]
Compaction of concrete	Normal	[52]
	Lognormal	[52]
Reinforcing Steel
Cross-sectional area of reinforcement steel	Normal	[5,11,14,26,27,28,29,32,34,35,36,37,41,43,53]
	Lognormal	[42]
Steel yield strength	Normal	[5,11,14,25,26,27,28,29,30,31,32,35,36,37,38,40,43,53,54]
	Lognormal	[11,34,37,41,42,55]
	Beta	[53]
Prestressing steel yield strength	Normal	[11,29,30,35,36,38,56]
	Lognormal	[11]
Steel ultimate tensile strength	Normal	[5,11,14,26,30,37,38,43]
	Lognormal	[11,37]
Prestressing steel ultimate tensile strength	Normal	[11,29,30,36,38,56]
	Lognormal	[11]
Steel modulus of elasticity	Normal	[11,32,34,37,38,43,53,56]
	Lognormal	[45]
Steel ultimate strain	Normal	[11,38,43,56]
	Lognormal	[11]
Geometry
Concrete cover	Normal	[14,25,31,32,36,38,43,55,57,58]
	Lognormal	[41,59]
Pavement thickness	Triangular	[17,58]
	Normal	[34]
Width	Normal	[14,15,17,28,32,34,36,37,43,45,49,55,58,60]
Length	Normal	[43,49]
Height/Thickness	Normal	[14,15,17,27,28,29,30,32,34,36,37,43,55,58,60]
Moment of inertia	Normal	[12,45,46]
Cross-sectional area	Normal	[12,46]
Loading
Dead loads	Normal	[25,27,28,30,31,32,35,36,38,45,61,62,63,64,65,66,67,68,69]
Live loads	Gumbel	[13,14,15,27,30,32,34,36,37,38,43,45,64,65,70,71,72]
	Normal	[5,26,27,28,29]
	Exponential	[40]
Wind load	Normal	[45]
Snow load	Gumbel	[63,68]
Impact load	Lognormal	[73,74,75,76,77]
Explosion load	Lognormal	[74,78,79]
Temperature	Gumbel	[30]
Masonry
Masonry modulus of elasticity	Normal	[26]
Masonry cohesion	Normal	[26]
	Lognormal	[17]
Masonry angle of friction	Normal	[26]
Masonry angle of dilatancy	Normal	[17,26]
Masonry tensile strength	Normal	[26]
Masonry compressive strength	Lognormal	[15,26,60]
	Normal	[72]
Masonry density	Normal	[15,26,49,60,80]
Soil/Backfill
Soil modulus of elasticity	Normal	[26]
Soil density	Normal	[15,26,60]
	Lognormal	[49,72,81]
Soil cohesion	Normal	[15,26,60,72]
	Lognormal	[17,49,81,82,83]
Soil angle of friction	Normal	[15,17,60,72,82,84]
	Lognormal	[49,72]
Slope	Lognormal	[49,85]
Pathologies
Steel corrosion rate	Lognormal	[25]
Surface chloride concentration	Lognormal	[57,59,86]
Critical chloride concentration	Lognormal	[57,59,87]
Chloride transport coefficient	Lognormal	[57,59,86]
Crack initial length	Lognormal	[88,89,90,91,92]
Crack width	Normal	[88,89,90,93]
Crack stress range	Normal	[90,91,94]
Crack smallest detectable length	Normal	[90,91,94]

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