A New Framework for Condition and Risk Assessment for Sustainable Management of PT Bridges

Abstract

The state-of-the-art literature in engineering provides information on the current state of technical subjects, shapes the understanding of relevant research, and communicates potential next directions for future exploration in relation with condition and risk assessment and sustainable maintenance of post-tensioned (PT) concrete bridges. However, the formulation of a quantitative methodological framework in selecting suitable safety assessment and management techniques for in-service PT concrete bridges is insufficient in the current literature. The absence of such a framework could render ineffective the sustainability of bridge maintenance strategies. In addition, the failure of structural assets due to deterioration and lack of proper maintenance is regarded as an important risk aspect. This is because structural failure often results in adverse economic, environmental, and safety outcomes. To underscore this, formulating mitigation strategies through sustainable asset maintenance programs is consequential. In recognition of the current limitation, this paper proposes a new framework with a structured methodology for the risk assessment and management of PT concrete bridges. This framework helps the bridge practitioners to navigate through design, inspection results, environmental factors, estimation of probability of failure, and corresponding consequences to calculate the risk and decide on the maintenance. It encompasses the decision on the type and interval of the inspection regime, repair and strengthening, and potentially replacing existing work. The presentation of this framework is original in that it carries the stochastic approach through all steps of risk assessment and maintenance decision making while allowing room for the inclusion of evolving methods. The proposed framework can be used for the development of a complete guide capable of stochastic, reliability-based, data-driven, and quantitative risk assessment and maintenance management for PT concrete bridges.

1. Introduction

A considerable number of post-tensioned (PT) bridges have been constructed to take advantage of their strength, accelerated construction, and the aesthetics afforded by a shallower superstructure. Figure 1 shows the Sunshine Skyway Bridge where post-tensioning was used in the approach and main span. However, certain damages, especially those prompted by corrosion, present a sustained concern for PT tendons, the most critical structural elements in a PT bridge. Such damages are leading causes of tendon weakening, failure, and, ultimately, the decline of the overall structural performance. The strands/wires, the main tension elements of the tendons, are normally encapsulated into protective ducts (fully grouted or not), introducing challenges for inspection. In the last two decades, examples of tendon rupture and significant damages have added to the concern about the safety of PT concrete bridges. In the case of external PT tendons, the lack of bond with the surrounding concrete and inability for the redistribution of loads, and for the internal tendons, the difficulty of access for inspection, have exacerbated this concern. Hence, condition assessment, reliability analysis, and estimation of risk for deteriorating PT bridges are of great importance for their maintenance. Many PT bridges constitute major links in the transportation network and are critical for the economy and environment in ordinary time and essential as evacuation and recovery means when communities are impacted by natural hazards. In that sense, a risk-based maintenance can contribute significantly to sustainability, resiliency, and safety.

Several past investigations have targeted the qualitative risk assessment and reliability-based bridge inspections. These include FHWA HIF-20-041 (Methodology for Risk Assessment of PT Tendons) [1] and NCHRP Report 782 (Proposed Guideline for Reliability-Based Bridge Inspection-RBI-Practices) [2]. Kaewunruen et al. [3] adopted the data obtained from the construction of a bridge in China as a case study to highlight the applications of digital twins to bridge model establishment, information collection and sharing, data processing, and inspection and maintenance planning. Their main discussion mostly pivots around building information modeling (BIM). Although these past studies adopted a general approach to base their risk assessment on the data obtained from bridge inspections, they lack the detailed probability perspective and stochastic approach introduced in this paper. The existing work can be used as a backbone to build upon for the development of a hybrid framework capable of stochastic, data-driven, and quantitative risk assessment and management for PT bridges.

Other investigations have targeted the safety evaluation of different types of existing bridges, considering several corrosion scenarios as a function of the age of the bridge. These studies provided useful information for the risk assessment and maintenance of existing bridges, as the corrosion levels which vary with the bridge’s age also represent their maintenance status. Crespi et al. [4,5] presented a simplified procedure for evaluating the structural behavior of existing bridges subjected to corrosion effects under earthquake loads. The study highlights that deterioration due to corrosion significantly increases seismic risk, especially for older structures. The findings underscore the importance of considering corrosion in risk assessments and suggest that monitoring corrosion levels is crucial for optimizing maintenance strategies and ensuring the long-term safety of bridge networks. By analyzing various risk indices across different corrosion scenarios, Crespi et al. [4,5] provided valuable insights for scheduling maintenance interventions and managing bridge networks effectively.

Although previous works (e.g., FHWA HIF-20-041 [1] and NCHRP Report 782 [2]) have contributed to the general understanding of the risk assessment and maintenance of PT bridges, they lack the formulation of a qualitative methodological framework in selecting suitable safety assessment techniques. Their contribution centers on eliciting experts to identify damage modes, attributes, likelihood of failure and associated risks/costs, as well as devising the inspection scope and intervals and remedial action. The absence of a thorough guide raises the need for a comprehensive formulated approach to create one for the risk assessment and management of PT concrete bridges.

A risk and reliability tool is needed to process the information on the condition of an existing bridge structure which was not necessarily known in the design phase for life-cycle risk and reliability assessment, and to assist the bridge owners in their maintenance decision making. The risk and reliability assessment of the population of bridges in a network plays a key role in identifying the maintenance priorities. The material properties, measurements of actual bridge geometry, load data, and the results of the periodic inspections performed within the lifetime of the bridge are the major sources of such information. The maintenance of deteriorating bridges is indisputably imperative in improving their serviceability and safety. Bridge maintenance prevents and/or delays the bridge performance deterioration. Consequently, a proper maintenance regime can improve the bridge condition, safety, and service life. The life-cycle bridge management can be optimized by exploring the effects of bridge maintenance. Given the high costs associated with bridge maintenance and limited financial resources, exploring a cost-effective maintenance management approach is essential. Because of the uncertainties involved in this type of investigation, this requires establishing the probability that an event might occur and the potential outcome of each event. Risk evaluation compares the magnitude of each risk and ranks them according to prominence and consequence. Some past research work has introduced a general understanding of this risk assessment approach [1,2].

The research work included in this paper aims at introducing a framework/guide for an integrated and stochastic, data-informed risk and reliability assessment and maintenance decision making for in-service PT concrete bridges, utilizing probability tools. The proposed framework aims to guide bridge engineering practitioners and owners through processes for risk review and assessment, as well as management of the risk and safety assurance. There is an emphasis on the most critical element of PT bridges, i.e., the tendons, with respect to damage modes, especially corrosion, and the means and methods for damage detection and condition assessment. The framework will provide user-friendly yet advanced tools and processes that ensure accuracy and reliability, consistently enabling a stochastic, data-driven approach throughout the entire process from the selection of inspection methods and intervals to decision making. Such data-driven guides can rely on the results of inspection, analysis for demand and strength, formulations and software for risk and reliability estimation, life-cycle cost analysis for various maintenance scenarios, and decision making. The process outlined in this framework would enable calibration through expert elicitation, owner-defined criteria, and experimental data. This framework differs from the previous works in concept and scope, in that it is geared toward stochastic and risk-based inspection and condition assessment, uses inspection data and deterioration models for model updating and reliability estimation and projection, adds processes for maintenance decision making, carries the stochastic approach throughout, and considers the risk for the entire bridge rather than individual tendons.

To this end, the proposed framework is populated by detailed modules to be used for facilitating the risk-based, data-driven process ending with a decision on the optimal maintenance strategy. The steps taken in this study to achieve this goal include a literature review, identifying damage modes, expanding the probability approach using proper deterioration models, a risk-based selection of inspection methods, condition updating and estimating the reliability index using advanced analytical tools, and life-cycle analysis and determining the optimal maintenance strategy. The process also aims at integrating expert elicitation using reliable experimental data relating to the risk assessment and management of bridges.

2. Materials and Methods

2.1. Past Experiences

The previous work as described above has relied heavily or solely on expert opinions [1,2]. The first step in the process adopted by the past investigations is the determination of the failure/damage modes in the structure and structural elements. NCHRP Report 782 [2] suggests a process for determining the failure mode through consensus among RAP based on experiences and past research. The expert elicitation process has also been used to estimate the likelihood of modes being the cause of failure. From past experiences and the available data on damage models, RAP has then suggested occurrence factors (OFs). Table 1 shows the rating scale for the OFs [2] that have been used in this process. The rating scales can be translated to quantitative probability of occurrence through expert elicitation, past experiences, and available data. A similar process is carried out to determine the consequence factors (CFs) for the bridge and bridge elements. Table 2 shows a rating scale for the CFs [2] used in this process. With the OFs and CFs determined, one can use the reliability matrix for inspection shown in Figure 2 to define the intervals for inspection consistent with this qualitative reliability-based bridge inspection (RBI). Intervals need to be such that the likelihood of failure is kept low and reduced as the consequence of failure increases. NCHRP Report 782 [2] suggested the maximum inspection time intervals shown in Table 3. The process is simple, effective, and offers a great starting point to determine inspection intervals for newly constructed PT bridges. However, inherently, it lacks the framework for the application of a holistic, data-driven stochastic approach for determining the inspection methods, current and future condition based on deterioration models, and risk and reliability built on probabilistic variables and incorporated into risk-based decision making.

Washer et al. [6] have used case studies to verify the RBI process for the bridges described above. The process adopted by their suggested RBI method is illustrated in the flowchart shown in Figure 3.

It is important to note that the RBI method as adopted by NCHRP Report 782 [2] prioritizes the inspection activities by comparison of risk among a population of bridges that compete for inspection resources. As noted by the authors of the prior work [1], decisions based on such an approach would be subjective and rely heavily on engineering judgement. It also limits itself to the decision on individual tendons and does not discuss the cumulative effect of potential tendon issues on the performance of the entire bridge. The risk determination of an individual tendon does not represent the risk level of a bridge. This is because the majority of bridge designs include reserve capacity, and tendon failure does not equate to a high risk for the safety of the bridge as a whole. Therefore, an approach to account for the individual tendon damage for the total performance of the bridge is necessary.

Concurrently, extensive work has been performed on stochastic approaches for estimating the risk and reliability of bridges in general and PT bridges specifically. These include several recent works with a strong focus on risk–reliability-based maintenance decision-making systems [7,8,9]. The latter is based on the application of well-established probability concepts on all elements of the system, including the selection of the inspection method, treatment of inspection data, digital modeling, condition updating, failure probability, cost/risk estimation, and life-cycle analysis for obtaining the optimal remedial solution.

2.2. Proposed Approach for Development of the New Framework

It is quite reasonable to assume that neither an expert-elicited qualitative ranking method nor stochastically formulated risk/reliability analysis alone can offer the best solution. However, a coalition of the two methods can prove to be a more effective alternative. Advances in statistical approaches to risk and reliability along with enhanced numerical computational methods offer a helpful means to quantify the effect of attributes and translate them to probability distributions. New technologies such as Artificial Intelligence (AI), Machine Learning (ML) algorithms, and utilizing the digital twin concept can significantly benefit this process. Such approaches also allow formulation and automation and present themselves very attractively for use in decision-making processes. On the other hand, calibrating these methods to ensure their rationality is only possible with reliable experimental data and expert elicitation. This paper will use a combination of the two approaches to develop the framework/guide for risk assessment and maintenance decision making for PT bridges. A snapshot of the proposed hybrid concept and the details of how it can be achieved are shown in Figure 4 below. Such a combined approach has attributes that fit well within the development of a user-friendly computer software for this purpose and will be a potential next direction for future exploration.

The overall assessment that follows this framework has a practical goal of determining scheduled preventive maintenance, timely retrofits, and efficient resource allocation, all aimed at improving operational efficiency. In the meantime, special attention is paid to bridge safety during service life through bridge inspection and structural condition monitoring, bridge maintenance, and bridge life-cycle analysis. This is expected to minimize the probability of failure and optimize maintenance schedules effectively to reduce the overall costs. Reliability-based methods are particularly important variables in achieving this goal. In other words, this proactive strategy ensures targeted upkeep to avert failures, while extending the life of the structure and ensuring the effectiveness of maintenance strategies.

This study results in a framework for assessing risk and reliability and supporting maintenance decision making for in-service post-tensioned concrete bridges. This framework can be used by practitioners and owners for an effective, accurate, and data-informed maintenance decision making. This will help alleviate the concerns of bridge owners regarding uncertainties associated with the performance and maintenance of in-service PT bridges, and as such, will positively impact their safety, integrity, and operation. The framework will also impact the design and construction by highlighting best and worst practices. Using the flowchart developed in this study for PT bridge risk assessment, the engineer in charge can use the processes and tasks described in each subtask to enhance and refine the assessment process. In the absence of data for any of the tasks, the engineer can use the expert-elicited process to bypass that specific task to complete the risk assessment procedure.

3. Results: Modules for Quantitative Stochastic Procedure

3.1. Literature Review

To accomplish the objectives of this project, a literature review has been performed to determine the state-of-the-art in the general area of risk assessment and maintenance decision making, with qualitative and quantitative results including expert elicited and informed stochastic approaches. The results of this review are described below within each module, covering the available tools and methodologies that would allow performing the tasks within each module. As more information and literature becomes available, the number of options for the users will increase and more relevance can be established. Regardless, the framework provided in this paper serves as a live document for guiding engineers in their assessment and management activities.

3.2. Overall Approach

It would be a prudent choice to build on the previously developed framework (NCHRP Report 782 [2] and FHWA HIF-20-041 [1]) and to complement/amend/refine individual modules with the relevant new methods and approaches. The goal is to transform the existing framework into a smart learning system. This system exploits the field data, expert input, and owner constraints, and develops an array of quantitative indices via statistical analysis to achieve a viable maintenance strategy. Figure 4 shows the activities envisioned to achieve this objective. The existing backbone/framework for reliability-based risk assessment and decision making is at the center of the graph, surrounded by the complementary envisioned/proposed tasks/modules for accomplishing the risk assessment and arriving at the optimal bridge maintenance strategy. Each satellite module/task uses the available information and advances a stochastic quantitative approach for supplementing the otherwise qualitative risk assessment. The user will have the choice to utilize the satellite tasks in the guide to improve, update, or quantify each of the backbone modules with their specific information as they become available. A research advisory panel (RAP) consisting of experts in the main areas can also be elicited during the assessment to assure the practicality of the steps.

This framework of tasks and modules serves as a guide for risk and reliability assessment and maintenance decision making for in-service post-tensioned concrete bridges. It provides the original expert-elicited, qualitative RBI framework as a default and starting point to be augmented with modules that can be used collectively or individually for a more accurate and quantitative approach, leading to a better-informed and accurate decision.

3.3. Expected and Observed Damages

Using the literature review and recent experiences, damage modes and types have been identified. It is understood that the initial risk assessment and determination of inspection intervals rely heavily on the identification of sources and location of damages. Additionally, the selection of the inspection methods, processes, and remedial options depends on the damage sources. Therefore, the identification of damage sources and locations (etiology) is an essential part of this task.

Like in any other structure types, a variety of damages can be expected to occur in all bridge elements of PT concrete bridges. However, in this case, tendons seem to present the highest challenge not only regarding their unique characteristics and persistent damages but also the accessibility for inspection. Premature failures of PT tendons, along with persistent factors contributing to these damages (see Figure 5), have highlighted the need for focused bridge risk assessment and maintenance decision-making processes. A variety of damage types and mechanisms have been associated with PT tendons [10]. Among these, the most prevalent are reported to be the severe corrosion in external tendons resulting in the rupture of strands (Figure 5a), corrosion and other damages at the anchorages (Figure 5b), corrosion and rupture of internal tendons visible only after major damage to concrete (Figure 5c), wire and strand corrosion visible only by coring and the use of borescopes (Figure 5d), and evidence of grout mix issues and workmanship problems causing separation and grout bleeding (Figure 5e). The existing PT systems changing over time and by each supplier without a uniform standardization have aggravated the situation. The performance of grouted tendons has also been inconsistent, with some showing drastic flaws due to material, design, and workmanship. The condition assessment of PT tendons has also presented major challenges because of inaccessibility for inspection. The consequences of various damages for the safety and strength of the bridge have therefore been difficult to quantify. The owners ask the following basic questions: “What can go wrong and how likely is it?”, “What are the consequences?”, and “What should be the remedial actions and future inspection scope and time interval?” [2]. To guide through these uncertain venues, a holistic risk-based decision support system is needed that is simple to use but quantitative as well as statistically accurate.

PT systems are arguably complex, and a variety of factors may be involved in their deterioration including environmental, materials, construction and workmanship, and structure-related factors. Among the damages expected for PT tendons, corrosion has been the most outstanding. Research indicates that the primary factors leading to corrosion include the presence of chloride ions, moisture, and oxygen; all are essential for the onset and continuation of the corrosion process [11]. The ions that disrupt the protective passive layer on the steel surfaces trigger active corrosion. The ions most identified in studies as culprits are those resulting from chloride [12]. Figure 6 illustrates how various factors contributing to the corrosion of PT components correlate with the fundamental causes of corrosion [7]. A knowledge of the potential causes present for each bridge can help in determining which deterioration model is more applicable to the case and how the bridge will age over time.

As described earlier, previous work [1,2,6] has determined the occurrence factor corresponding to the probability of failure as well as the risk factor that corresponds to risks and consequences based on the input from experts. In those investigations, six damage mechanisms were focused on, including breached duct or anchorage, construction and workmanship quality, the environment, inadequate specifications and detailing, poor or improper materials, and grout voids. The attributes identified by the prior investigation that corresponded to these mechanisms included the PT tendon and profile, joints and closures, PT system materials and components, installation quality, and environmental factors that included loading. The key attributes of bridge and tendons potentially influencing the risk of damage, mainly corrosion, were then ranked in a scoring scheme to obtain the occurrence factor. A cross-check of the expert-elicited findings and the wider set of parameters illustrated in Figure 6 can be performed to narrow down the affecting parameters to facilitate the adoption/development of deterioration models and probability of failure [7].

3.4. Probability of Failure

To provide a stochastic approach for describing the uncertainties involved, the existing deterioration models for the damages and failure modes identified by the previous task can be reviewed, and the model applicable to the case in hand can be selected. Using the most applicable model and the existing condition of the tendons expressed in probability terms through risk-based inspection, the probability of failure can be estimated. The results can serve as the stochastic conjugate of the expert-elicited occurrence factor. Since the occurrence factor is directly related to the likelihood of failure, as indicated in Table 1, one can relate the probability of failure to the occurrence factor, as per the user’s established criteria.

Asset management and structural health monitoring greatly benefit from deterioration models, especially for PT systems where the need is more pronounced. However, addressing this need is challenging due to the diverse factors influencing the degradation of PT components. Simplifying the issue through logical assumptions is essential. One challenge is determining the corrosion’s location and rate within post-tensioning elements for both local and uniform corrosion.

The corrosion in PT bridges affects structural performance deterioration by reducing the cross-sectional area and bond strength of the concrete reinforcement, as shown in Figure 7 below [8,13]. Corrosion susceptibility increases in aggressive surroundings when exposed, for example, to deicing salts or a marine environment [14]. The progression of corrosion in PT tendons can be attributed to various factors, including the presence of sulfates and the soft grout phenomenon. However, concrete carbonation and chloride penetration into concrete play a major role in the corrosion of PT tendons.

Taeby and Mehrabi in 2022 included both localized and uniform corrosion in their study [7]. Typically, localized corrosion, a severe consequence of excessive bleed water, manifests within a decade of construction [11,15]. The layout and elevation of the tendons in the PT system affect the likelihood of grout segregation and, consequently, localized corrosion. Uniform corrosion, in contrast, involves a consistent corrosive effect across the entire surface, which is mitigated by corrosion protection barriers on the strands. Taeby and Mehrabi [7] proposed a deterioration model that accounts for factors related to both localized and uniform corrosion, and developed equations to estimate the failure risk from both types of corrosion, individually and collectively, drawing on a range of small-scale experiments and literature reviews. An example of how the failure probability can be calculated based on predicted deterioration can be found in [7,8,9]. A corrosion deterioration function was defined for tendons as the ratio of the cross-sectional area at any time to the initial cross-sectional area. This function describes the reduction in the cross-section based on the corrosion rate in different conditions of sealed and exposed steel elements, and the probability of failure of corrosion barriers. This deterioration function was used to calculate the ratio of capacity to demand for each tendon. With such information, the probability of failure over time, i.e., the probability of the capacity to demand a ratio falling below 1 at time t, was calculated using the deterioration function tuned by the established corrosion rates in a specific environment, the percentage of tendons affected by local versus uniform corrosion, and an estimate of the time of failure for corrosion barriers. Figure 8 shows an example of the probability of failure from combined local and uniform corrosion for a bridge with a t₀ (expected remaining service life of corrosion protection system) of 10 years and t₁ (the time corrosion protection system fails).

A similar approach, let it be with some modification to fit the case in hand, can be taken to determine the probability of failure in tendons for other bridge circumstances. The probability of failure estimated in this manner can be used for determining the probabilistic load-carrying capacity of the bridge during its life cycle. The probability of failure estimated in this manner can be compared with the occurrence factor determined from expert-elicited input, as explained above, and modifications can be applied, if justified.

3.5. Consequences and Costs

Risk involves assessing the probability of an undesirable event and the severity of its potential adverse effects. For bridges, especially those using post-tensioning (PT), designing and operating under a ‘no-risk’ assumption is not feasible. It is accepted that a certain level of risk is inevitable. Acceptable risk levels are defined with the understanding that not all failures result in severe consequences, and incidents with serious consequences are likely to have a very low probability of occurrence [7].

Damage consequences and their importance need to be identified through a stochastic process and cost to be associated to each occurrence for further analysis. For a stochastic representation of consequence and cost, a twin numerical model for the bridge needs to be constructed. Such a model can be updated with inspection results. The model can be used to estimate the probability distribution of the bridge strength and compared to the loading in each instance to estimate the reliability index. This index, with the input from experts, can be translated to a risk matrix similar to that offered by the current RBI methods, but with an advantage of being data-driven and informed rather than opinion-based.

It is well understood that a risk-based life-cycle analysis must account for “consequences”. As noted previously, the probability of failure is determined using the inspection data on bridge components, in this case, with a focus on PT tendons. Combining the results for individual components to extrapolate for the system condition assessment requires a complex approach. Taeby et al. proposed a procedure for carrying the stochastic condition assessment from the population of individual tendons to the entire bridge [8,9]. It is also understood that the inspection results, i.e., from non-destructive testing (NDT), shall be expressed in probability terms to facilitate stochastic condition assessment. A sample of a PT bridge span with PT tendons is shown in Figure 9 [14]. By applying two sets of elimination, Taeby et al. [8,9] were able to obtain probabilistic aggregation to determine the updated nominal resistance of the bridge span. To obtain the nominal (flexural) strength of the bridge span, a twin-digital model of the span was constructed, and finite element analysis was conducted. Tabiatnejad et al. also performed an extensive analytical investigation on damage detection in PT tendons utilizing advanced finite element analysis tools and damage detection methods [14]. Figure 10 shows the finite element model used for this study. They concluded that changes in the force of a tendon can be associated to damage in that tendon alone if the original compression zone in the section remains in compression [8].

By the probabilistic expression of the nominal strength and demand, one can determine the reliability index and translate to risk and, therefore, consequence at each instance based on the condition of the bridge at that time. Consequence can be monetized according to a predefined criterion. Projection to later times within the context of life-cycle analysis will require the activation of probability of failure and deterioration models, and model updating utilizing the digital twin for future predictions.

The various components of a bridge each play distinct roles in the overall performance of the structural system [7,15]. Effective and efficient maintenance management can be achieved by evaluating the contributions of these components to the overall bridge system. The significance of each structural component can be assessed both independently and in the context of the overall system’s reliability. For an example of the application of this approach to determine the consequences through stochastic evaluation, reference can be made to the work by Taeby et al. [8,9].

3.6. Inspection

The promising and available methods for the inspection of the bridge under consideration should be identified and ranked based on their accuracy and associated to a probability distribution. Accordingly, a risk-based method for the selection of the optimum method can be introduced for a trade-off between accuracy and cost. With such a method, the inspection results can be expressed stochastically. The criteria for risk reassessment can therefore be quantified as per the significance of the statistical difference between the current and previous condition.

Although past investigations have explored means for the selection of the inspection methods and intervals to address the level of risk in structural components [1,2,6], there has not been much focus on a risk-based selection of the inspection methods, mainly the NDT techniques. In other words, the notion of risk has not taken back to the selection of NDT methods. Recent investigations however, including refs [7,8,9], have introduced a procedure for the risk-based selection of NDT methods in general and in PT tendons specifically. It was shown that NDT methods that could be applicable to PT tendons can be categorized into visual, mechanical waves and vibration, infrared thermography, electrochemical, electromagnetic, ground penetration radar, radiography, other unclassified methods, and sensors [10]. Figure 11 shows the methods applicable to PT tendons.

Currently, the selection of inspection methods mostly relies on previous experience and anecdotal evidence. As such, the results of the inspection do not really reflect the level of the accuracy for the method employed in each case. This is an apparent deviation from a risk-based procedure. NDT methods, like any other, carry some level of error and uncertainty in their results. Normally, more accurate methods would be more expensive. There are no generally accepted criteria to rank the NDT methods as per accuracy and cost. Taeby and Mehrabi [7] rated the NDT methods applicable to PT tendons for their accuracy as per the literature and past experiences into four categories: poor, fair, good, and very good. They then associated the probability of normal distribution with each category, ranging from, for example, 2σ to 5σ for poor to very good, respectively, as shown in Figure 12 [7], where σ is the standard deviation and the vertical axis is the density of probability. The estimation of the probability of error for NDT methods is performed based on the accuracy of the NDT method, the condition of the bridge, and the accuracy of condition. The condition of the bridge (or component) is normally determined as per the rating adopted by the Specifications for the National Bridge Inventory (SNBI) [17] or AASHTO Manual for Bridge Evaluation [18]. By combining the accuracy of the NDT method, the condition of the bridge (or component), and the accuracy with which the condition is determined, Taeby and Mehrabi [7] formulated the probability of error. One can therefore calculate the risk associated with the use of the NDT methods that are available for each bridge by multiplying the probability of error by the cost, adding the initial cost of the method, and choosing the method that offers the least risk. For a detailed process and implementation of a risk-based selection of inspection methods for PT tendons, reference can be made to the work by Taeby and Mehrabi [7]. There have been other investigations on reliability-based inspection and related topics that can be referenced as well [19].

3.7. Maintenance Decision Making

The last step in the risk assessment and management process is the decision making for maintenance. Following the previous proposed tasks for risk-based inspection, a probability distribution for the current bridge condition can be determined, and projection of the condition for the bridge service life can be performed using the applicable deterioration models. These are the basic information pieces needed for a life-cycle analysis and maintenance decision making.

Maintenance is essential for mitigating structural performance deterioration over time and for extending the service life of the structure. However, appropriate maintenance depends on the outcome of inspections. Structural performance improvement as a result of inspections and maintenance is illustrated in Figure 13 below. It can be seen from the figure that if a damage is not discovered, or an inspected damage does not result in maintenance, the structural performance will not improve, and the anticipated service life may not be achieved. In contrast, the figure shows that structural performance can be updated through on-time inspection or deterioration can be delayed through appropriate maintenance. Two different maintenance scenarios of A and B and their effects on the performance are shown in this figure.

In recent decades, bridge maintenance techniques and management approaches have evolved significantly in response to the growing number of structurally deficient bridges [20,21]. Effective bridge maintenance management should be grounded in a thorough understanding of how maintenance impacts bridge performance, service life, and life-cycle costs [22]. Ultimately, achieving well-balanced bridge maintenance management is possible through an optimization process [23,24].

Bridge asset management generally involves actions and strategies aimed at preventing unexpected structural failures, extending service life, delaying and reducing deterioration, and improving structural performance and condition [24,25]. This management is divided into three categories: preventive maintenance (PM), essential maintenance (EM, also known as rehabilitation), and replacement (RP) [26]. Figure 14 illustrates the time-dependent conditions of a bridge along with the corresponding maintenance types. The four conditions—good, fair, poor, and severe—correlate with the Specifications for the National Bridge Inventory (SNBI) [17] condition rating values, ranging from 0 to 9. A good condition corresponds to rating values of 7, 8, and 9; a fair condition corresponds to values of 5 and 6; a poor condition is indicated by a value of 4; and a severe condition is represented by values between 0 and 3 [25]. For bridges in a good or fair condition, PM can be scheduled or applied based on predefined criteria to slow the deterioration of performance. EM is conducted when a bridge’s performance declines to a fair or poor condition. Replacement, undertaken when a bridge reaches a poor or severe condition, involves fully replacing a structurally deficient bridge with a new one, thereby restoring it to its original performance level [25,27].

Given that maintenance incurs costs and financial resources are limited, it is crucial to explore cost-effective strategies for bridge maintenance management [28]. It is also important to recognize the effect of each maintenance strategy in a probabilistic manner, with proper mean values and distributions.

As noted above, the condition of the bridge (or tendons) can be predicted with the current condition and application of the deterioration models and described in stochastic form. It has been shown that the reliability of bridges correlates with the corresponding rating of the bridge (e.g., as per SNBI) [29]. Hence, the condition obtained from the inspection and reliability of the bridge can be interchangeably estimated during bridge service life, as per the procedure discussed above. By applying various maintenance strategies to the bridge at intervals, the owners can maintain the condition/reliability of the bridge above an “acceptable” level. Each strategy carries a cost/risk and improvement to the condition. By comparing various strategies as per their risk, one can choose the strategy which provides the minimum risk. Various maintenance strategies can be described and their effect on improving the risk can be discussed. The strategies can range from “doing nothing” with a specific inspection interval/method to the “replacement” of the tendons. Life-cycle cost analysis can be introduced, which utilizes the deterioration models and the effect of various maintenance scenarios to estimate the cost and, therefore, assist the bridge owner in the selection of the optimum or desirable maintenance strategy [8].

Other work on risk- and reliability-based inspection and maintenance can also be referred to for the implementation of the risk- and reliability-based inspection and maintenance strategies [30,31,32,33].

4. A New Framework to Build on

Previous sections of this paper described the new framework for the risk assessment and management of PT bridges, with sufficient details for accomplishing each module in the framework. However, a robust, clear, and step-by-step recipe is essential to facilitate qualitative and stochastic risk assessment and maintenance decision making by bridge engineering practitioners and owners. Figure 15 shows this recipe in the form of a flowchart. Although the steps to follow will remain unchanged, the content and references to methods for achieving each goal can evolve over time, making this framework a live document. The ideal case is to have all the necessary data for conducting each step in a data-driven quantitative and stochastic manner; however, if any step lacks adequate data, the qualitative alternative can be used as default.

5. Summary and Conclusions

Post-tensioned (PT) bridges constitute a large inventory within the surface transportation and often form the critical links within the network. The condition and risk assessment of these bridges are critical for decision making and prioritizing crucial maintenance work to address their sustainability, resilience, and safety. This paper addresses the existing gap in PT bridge maintenance and proposes a new structured quantitative methodology framework for suitable risk assessment and management of PT concrete bridges. This framework assists bridge practitioners in navigating design considerations, inspection results, environmental factors, and the estimation of failure probabilities and their consequences. By integrating these elements, it enables the estimation of risk and informed maintenance decisions with a stochastic approach throughout the entire process. It also guides the selection of inspection types and intervals, repair and strengthening strategies, and potential replacements for existing structures. These strategies encompass various aspects, such as the type and frequency of inspection regimes, repair and strengthening actions, and, if necessary, replacement work. To achieve this, this paper reviews the concepts and methods related to life-cycle bridge safety, risk assessment, and maintenance management of in-service PT concrete bridges. The topics include bridge safety and service life prediction, bridge inspection and maintenance, and life-cycle bridge management. It describes the probabilistic time-dependent structural performance based on suitable deterioration methods and the service life prediction of deteriorating bridges. Representative inspection methods and their application to bridge management are noted. The paper also explains the effects of bridge maintenance on performance, service life, and cost within a stochastic framework, and shows the optimization of bridge management based on these effects. Finally, the study presents applications of risk- and reliability-based optimum life-cycle bridge management planning along with examples of the implementation of the proposed approach. The relevant reference sources for further information are also highlighted throughout the paper.

The following inference can be drawn from this study:

• There are uncertainties associated with the maintenance and management of bridge structures. Projecting bridge performance and cost assessment involves considerable unpredictability. To pinpoint and address these uncertainties, utilizing probabilistic concepts and methods is inevitable. Condition updating based on stochastic inspection and monitoring information can help to describe these uncertainties for bridge management.

• The existing process of RBI methods that are qualitative and based on expert elicitation can be augmented by the proposed modules to transition to a quantitative and data-informed, risk- and reliability-based condition assessment and maintenance decision making.

• The risk-based selection of inspection methods for the condition assessment of PT bridges provides a means for determining the probability distribution for the existing condition as well as prediction of the future condition of the bridge.

• Determining the probabilistic models for bridge deterioration and condition rating is essential for determining the reliability index during the life cycle of the bridge. For this reason, digital twins and efficient computational methods offer the necessary tools for condition updating.

• Using the probability distribution for the bridge condition over time and established distribution for the loading demand, the reliability index can be estimated throughout the life cycle. The reliability index can play the determining factor for maintenance decision making.

• Because financial resources to administer bridge maintenance are limited, it becomes necessary to first recognize the maintenance strategies and their impact on the bridge condition, and then derive an optimized solution. This becomes possible with risk- and reliability-based decision making for deteriorating bridges that follows the proposed framework of activities.

• An inclusive and comprehensive framework was developed in a user-friendly format in this study for the first time. The newly proposed framework offers a structured and data-driven process with consistent steps, while also allowing for the integration of evolving methodologies.

• The proposed framework is a live document that can be augmented with new information and processes as well as the integration of evolving methodologies. This can be considered as a starting point to develop a guide for risk assessment and maintenance management for PT concrete bridges.

Author Contributions

Conceptualization, M.F., S.S.K.D. and A.M.; methodology, M.F., S.S.K.D. and A.M.; software, M.F., S.S.K.D., A.M., P.M. and D.T.; validation, M.F., S.S.K.D. and A.M.; formal analysis, M.F., S.S.K.D. and A.M.; investigation, M.F., S.S.K.D. and A.M.; resources, M.F., S.S.K.D., A.M., P.M. and D.T.; data curation, M.F., S.S.K.D., A.M., P.M. and D.T.; writing—original draft preparation, M.F., S.S.K.D. and A.M.; writing—review and editing, M.F., S.S.K.D. and A.M.; visualization, M.F., S.S.K.D., A.M., P.M. and D.T.; supervision, S.S.K.D. and A.M. 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

Dataset available on request from the authors.

Acknowledgments

The authors greatly acknowledge the internal support by the Department of Civil and Environmental Engineering at Florida International University. The contents of this paper reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. The Sunshine Skyway Bridge.

Figure 2. Flowchart, bridge RBI process (refer Table 3 for inspection intervals of categories I-V) [2].

Figure 3. Simple risk matrix for inspection intervals [2].

Figure 4. The proposed set of comprehensive activities for turning the qualitative risk assessment and management of PT bridges into a quantitative procedure.

Figure 5. Types of damage to PT tendons: (a) corrosion and rupture of strands; (b) corrosion in anchorage; (c) corrosion and rupture of internal tendons and concrete spalling; (d) corrosion of wires visible by borescope and (e) free water inside tendons [photo archives of Armin Mehrabi].

Figure 6. Factors involved in corrosion of external post-tensioning system [7].

Figure 7. Uniform and localized corrosion models (Reprinted with permission from [13]; published by Taylor and Francis Group, 2022).

Figure 8. Probability of failure of a tendon in poor condition t0 = 25, t1 = 10 [16].

Figure 9. Details of a post-tensioned concrete span [14].

Figure 10. FEM (3D) of segmental bridge [14].

Figure 11. Non-destructive evaluation methods appropriate to external post-tensioning system [7].

Figure 12. Accuracy levels of NDE methods [7].

Figure 13. Influence of inspection and maintenance on structural performance during service life (Reprinted with permission from [13]; published by Taylor and Francis Group, 2022).

Figure 14. Bridge conditions and maintenance types (Reprinted with permission from [13]; published by Taylor and Francis Group, 2022).

Figure 15. The framework and procedure for quantitative and stochastic risk assessment and management of PT bridges [2,6,7,8,9,10,13,14,20,21,22,23,24].

Table 1

Occurrence factors rating scale [taken from [2]].

Level	Category	Description
1	Remote	Remote likelihood of occurrence, unreasonable to expect failure to occur
2	Low	Low likelihood of occurrence
3	Moderate	Moderate likelihood of occurrence
4	High	High likelihood of occurrence

Table 2

General description of consequence factors [taken from [2]].

Level	Category	Consequence on Safety	Consequence on Serviceability	Summary Description
1	Low	None	Minor	Minor effect on serviceability, no effect on safety
2	Moderate	Minor	Moderate	Moderate effect on serviceability, minor effect on safety
3	High	Moderate	Major	Major effect on serviceability, moderate effect on safety
4	Severe	Major	Major	Structural collapse / loss of life

Table 3

Maximum inspection intervals [taken from [2]].

Category	Maximum Interval
I	12 months or less
II	24 months
III	48 months
IV	72 months
V	96 months

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