Keywords

1 Introduction

Bridges are important assets of the transport infrastructure network that play a crucial role in ensuring the well-being of populations and the economy. Concrete is the popular material used in bridges, but after many years of service, concrete bridge structures are aging and deteriorating, which becomes a hazard to traffic users in the event of structural failure or falling apart. The causes of concrete deterioration have been intensively studied, such as the carbonation of concrete and chloride attack [1, 2]. Contributing factors to deterioration include traffic volume, exposure to corrosive soil and airborne chloride near the coastline and acidic gases (including airborne carbon dioxide, nitrous and sulfurous oxides) in urban area. The deterioration process can be single, such as concrete creep, or combined processes, such as stress corrosion cracking in steel, and coupled with random damage events such as flooding and earthquake. The search for better concrete materials against natural and man-made hazards is ongoing. Recent advances in nanotechnology and its application in concrete bridge construction can help not only ensure safety and serviceability throughout service life but also cost-effective maintenance, rehabilitation and replacement (MRR).

Nano-concrete has been developed in the past decade to improve key characteristics of normal concrete such as tensile and compressive strength, anticorrosion and durability [3, 4]. The efficiency of the nano-concrete depends on its density, which can be maximized by minimizing the particle gaps. The optimal particle gap within the concrete mass can be achieved by incorporating a homogeneous gradient of fine and coarse particles in the mixture. In this regard, nanomaterials such as nano-silica, nano-graphite platelets, carbon nanotubes, graphene, nano-titanium dioxide and nano clay have been used to reinforce cementitious composites (cement paste, mortar, and concrete) [5]. They can be highly effective because, with their extremely small size, nanomaterials can fill the voids between cement and silica fume particles, leading to higher level of compaction and generating a denser binding matrix. This high level of compaction of concrete particles can significantly improve both the durability and mechanical properties of the nano-concrete. Several studies have reported significantly improved properties of the nano-concrete. For example, the use of 0.02% graphene oxide in ultra-high performance concrete can increase its strength characteristics, such as compressive, tensile, and flexural strength, up to 197%, 160%, and 184%, respectively [5]. Kancharla et al. reported that replacing 0.5 and 1.0% cement with nanosilica showed good improvement in bending strength of 7.8 and 15.7%, respectively, in the crushing stage and slight improvement in bending strength of 0.42 and 1.26%, respectively, in the failure stage [6]. Mostafa et al. found that nano glass waste can increase the bending strength of ultra-high performance concrete up to 1.5-fold if it is added at 1% [7].

With the increasing use of nano-concrete, it is useful to compare deterioration rates between traditional concrete and nano-concrete over service life. Such a comparison is useful for decision making on the use of nano-concrete in bridge engineering and to provide better knowledge for managing the maintenance and life cycle costs for current and future use of the nano-concrete. One method is to use deterioration models to compare the deterioration rates. Deterioration models can be divided into knowledge-based, data-driven and mechanism-based models. Few bridge management agencies use knowledge and experience to determine future deterioration from inspected defects [8]. The data-driven models use inspection data to predict future deterioration [9,10,11]. The mechanism-based model is based on deterioration mechanisms such as corrosion of reinforcing steels [1, 12]. The physical models are considered more advanced and accurate but they require intensive and detailed data, which can be costly and difficult to obtain. Therefore, they were not selected for this study. The deterministic linear model is well known for its ease of use and implementation, but is criticized for failure to capture the uncertainty of the deterioration process, which was addressed by the stochastic Markov model in this study.

We aimed to investigate the deterioration rate of traditional concrete as the benchmark for future study of deterioration of nano-concrete used for bridge components such as bridge girders and the bridge deck. For traditional concrete, visual inspection data of bridge components using traditional concrete are available and collected from the bridge agency for deterioration modeling. However, such data are not available for the nano-concrete. Therefore, scenario analysis was conducted for the deterioration rate of the nano-concrete based on its reported performance in public literature to assess its potential economic benefit. The outcomes of this study will demonstrate the benefit of condition monitoring and data collection for deterioration modeling traditional concrete for expanding the application of nano-concrete in bridge engineering.

2 Case Study

VicRoads is the registered business name of the Roads Corporation in the State of Victoria, Australia (www.vicroads.com.au). It is a Victorian statutory authority established under the Transport Act 1983 and continued in the Transport Integration Act 2010. One of VicRoads’ core services is to plan, develop and manage the arterial road network, including roads, bridges, culverts and traffic signs. The bridge management of VicRoads is supported through its computerized database, which contains basic data of 6207 bridge structures including bridges, culverts and tunnel with their attributes such as number of spans, length and width. The oldest structure was built in 1899. The database also stores 26,1324 records of Level 2 inspections as of 2017. The first recorded inspections in the computerized database began in 1995.

The VicRoads’s inspection practice is published in an open-access inspection manual, which basically has three levels of inspection [13]. Level 1 is considered a screening inspection with a maximum interval of 6 months. Level 2 is a routine inspection with a typical interval of 2 years and Level 3 is an in-depth inspection for special cases. The VicRoads’ inspection manual describes the breakdown of bridge structures into superstructures and substructures and into structural elements such as piles, decks and bearings. Each bridge element is coded by its function and one of five types of material (i.e., timber, steel, in-situ concrete, precast concrete and others); for example, the 2C mean open girder/stringer with the concrete material. The Level 2 inspection provides q condition rating for individual elements based on their inspected defects as per inspection guidelines. The condition rating is a 1–4 scale, with 1 being good and 4 being worst. The condition state percentage distribution is used to provide inspection reports. For example, an inspection report of [90% 10% 0% 0%] of 2C means 90% of the open girders in condition 1, 10% in condition 2 and 0% in conditions 3 and 4. The case study dataset of Level 2 inspection data from 1997 to 2017 was used for deterioration modeling of bridge components in this study.

A collaborative research project between VicRoads and RMIT University with support from the ARC Nanocom Hub was established to develop deterioration models using Level 2 inspection data for bridge structures. The outcomes of the collaborative project could provide justification for future state funding of bridge management and also provide support for more effective and efficient asset management programs.

2.1 Deterioration Models

2.1.1 Linear Regression Model

Equation (1) shows the linear relation between bridge condition (output) with age (input) [9]. More input factors can be added into Eq. (1) in a similar manner.

$$ Y = a_{0} + a_{1} *{\text{age}} + \varepsilon $$
(1)

where a0 is a constant coefficient, a1 is slope coefficient of age (year) and ɛ is the measurement error assumed to have zero mean and independent constant variance.

The interpretation of Eq. (1) is that (a) for the case of 1 input and 1 output, the relation is a straight line; (b) for the case of ≥2 inputs, the relation is a plane and hyperplane; (c) a linear relation means a constant deterioration rate over time; (d) measurement error means that for the same values of the inputs, every time the output is measured or assessed, the output value might not be the same due to equipment sensitivity or human subjectivity; and (e) deterioration increases with age, meaning the unchanged condition is not valid.

For calibrating or estimating the model coefficients, the least square method is often used to minimize the error term between the observed output and model output. The predictive performance of the linear model is often assessed using the coefficient of determination R2 between observed values and model outputs.

2.1.2 Markov Model

The Markov model is based on the stochastic theory of Markov chain [14], which describes a system that can be in one of several defined condition states at any time, and it can stay still or move to another state at each time step with some transition probabilities over time. This theory is well suited for observation of Level 2 condition data of bridge elements because the snapshot inspection reveals the condition state at the time of inspection and condition state movement over time can be captured with the transition probabilities.

The most important property of the Markov chain is that the probability of movement depends only on the current condition regardless of the history of movement, called memoryless property. This means that (a) the currently known condition can represent the accumulated deterioration up to the current time and (b) the future condition does not depend on how long it stays in previous conditions. This property is well suited for long-life assets such as concrete. The opposite is usage-based assets such as light bulbs and machinery in which their future condition depends on how long they have been used in the past.

The Markov model is mathematically expressed as a matrix M of transition probabilities:

$$ M = \left[ {\begin{array}{*{20}c} {P_{11} } & {P_{12} } & {P_{13} } & {P_{14} } \\ {P_{21} } & {P_{22} } & {P_{23} } & {P_{24} } \\ {P_{31} } & {P_{32} } & {P_{33} } & {P_{34} } \\ {P_{41} } & {P_{42} } & {P_{43} } & {P_{44} } \\ \end{array} } \right] $$
(2)

where Pij is the probability of moving from condition state i to condition state j over a unit time step. In this study, Pij was assumed to be 0 if i > j, meaning improved condition was not modeled due to the lack of maintenance data.

The memoryless property and the theory of total probability can be used to predict the probability vector [P1 P2 P3 P4] at any future time T, given the known current condition with certainty or probabilities [C1 C2 C3 C4] as shown in the Chapman–Kolmogorov equation [15]:

$$ \left[ {P1\;P2\;P3\;P4} \right] = \left[ {C1\;C2\;C3\;C4} \right]*M^{T} $$
(3)

Equation (3) is a matrix multiplication of the current condition with the transition matrix powered to future time T. To utilize Eq. (3), the transition matrix M of Eq. (2) needs to be estimated. Among several calibration/estimation techniques, we used the proven Bayesian Markov chain Monte Carlo simulation technique [16] to estimate the transition matrix M of the Markov model. In brief, the Bayesian technique transforms the unknown elements Pij of the transition matrix M into a multivariate distribution by using observed data, Eq. (3) and the Bayesian theory. The Markov chain Monte Carlo simulation is then used to generate sampling data of the elements, which become the estimator of the unknown elements (see [16]).

The predictive performance of the Markov model is commonly validated using the Chi-square test on a separate dataset that is not used in the calibration of the Markov model [15, 16]. The validation dataset is often randomly selected from 15 to 20% of the entire dataset.

3 Results

The deterioration models were applied to the case study and the results are demonstrated for a concrete open girder.

3.1 Model Fitness and Effect of Long-Term Data Collection

The effect of long-term and short-term data collection was investigated using the linear and Markov models. Table 1 shows that the Markov model passed the fitness test for both datasets. The negative values for R2 of the linear model on both datasets indicate the poor fitness of the linear model and it should not be used. The poor fitness of the linear model can be seen in Fig. 1 where the observed data have high uncertainty as shown by the unpatterned scattering. Instead of trying to fit an impossible line through the observed percentage in condition 1 with age, the Markov model takes a different approach by capturing the percentage changes between condition states. This shows the good generality of the Markov model in such cases.

Table 1 Results of fitness test for Markov and linear models for percentage prediction on condition 1
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Fig. 1
Two scatter plots along with a linear model fitted labeled a and b depict percentages in condition 1 versus age. R square equals negative 0.544, as labeled in graph a. R squared equals negative 0.247, as labeled in graph b. Both graphs tend toward negative correlations.

Linear model fitted with long-term (a) and short-term (b) data for percentage in condition 1

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Despite the unacceptable fitness of the linear models, Fig. 1 can still be used to illustrate the significant effect of long-term data collection. This effect showed the predicted deterioration rate is sensitive to the range of calibration data. The predicted deterioration rate for long-term data collection (Fig. 1a) has a mild slope as compared with the steep slope of the predicted deterioration rate for short-term data collection (Fig. 1b).

The calibrated transition matrix for an old bridge is shown as a demonstration in Eqs. (4) and (5) for long-term and short-term data collection. Figure 2a, b shows the effect of long-term data collection on the Markov model. It appears that removal of the first inspection accelerated the deterioration. For example, at the age of 100 years, the percentages in condition 1 were 45 and 18% between the long-term and short-term data. Similarly, the percentage of 20 and 30% in condition 4 can be observed between the long-term and short-term data, implying a 15% difference.

Fig. 2
Two graphs labeled a and b depict probability value versus age in years for conditions 1 to 4. In both graphs, condition 1 trends in a decreasing pattern, and conditions 2, 3, and 4 trends in an increasing pattern.

Markov model using long-term data (a) and short-term data (b)

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$$ M\left( {{\text{all}}} \right) = \left[ {\begin{array}{*{20}c} {0.9934} & {0.0052} & {0.0008} & {0.0006} \\ 0 & {0.9984} & {0.0009} & {0.0007} \\ 0 & 0 & {0.9972} & {0.0028} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] $$
(4)
$$ M\left( {{\text{second}}} \right) = \left[ {\begin{array}{*{20}c} {0.9894} & {0.0082} & {0.0018} & {0.0006} \\ 0 & {0.9889} & {0.0104} & {0.0007} \\ 0 & 0 & {0.9991} & {0.0009} \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] $$
(5)

3.2 Potential Benefit of Nano-concrete

The deterioration rate of traditional concrete was estimated using visual inspection data. There were 3000 bridge structures in the case study. If it is assumed that at current year 2022, all bridge structures in failure condition 5 have already been repaired or replaced, then the budget for proactive asset management over the next 10 years (selected as an example) can be estimated as: (a) unit cost of major repair or replacement AUD 100,000 per bridge girder for its average length of 30 m. The penalty cost is assumed being equal to unit cost of inspection; (b) Markov deterioration model predicts the increase by 4.0% of 3000 bridge structures (i.e. ≈120 structures) that will have its girder in failure condition 5 and require major repair or replacement. The replacement budget is therefore AUD 12 million (which is 120 × $100,000); and (c) if nano-concrete is used instead of traditional concrete, the rate of deterioration could be 20% lower because of its better strength and durability, resulting in 4%*0.8 = 3.2*3000 = 96 girders that require major repair or replacement. The budget cost is AUD 9.6 million dollars and the saving is AUD 2.4 million over 10 years.

It should be noted that the cost figures are hypothetical values that were used in this study only for demonstration of methodology.

4 Discussion

This study was limited to a case of concrete material with a demonstration of an open girder 2C. However, the methodology can be extended to all bridge elements with different materials. The effect of contributing factors such as traffic volume and exposure condition will be investigated in further studies.

This study revealed the weakness of the linear model and thereby supports the use of the stochastic Markov model in modelling deterioration of bridge elements. Despite this advantage, the Markov model still shows deterioration in the first few years, which is considered unrealistic as found in a previous study [17]. Sobanjo and Thompson claimed that new structures are expected to stay in condition 1 for the first 10–30 years before transitioning into a deteriorated condition. If all structures follow this pattern, then the Markov model should be applied to the data after transition start. If some new structures start to move to condition 2 after a few years due to unknown causes such as uncertainty of material batch, inconsistent construction practice or damage events, then the Markov model can reflect that. Therefore, it is not the shortfall of the Markov model but how valid data are compiled for the deterioration models. The comparison between the Markov model and other stochastic models such as the Gamma process model [11] will be conducted in further works.

5 Conclusions

Nanomaterials including nano-silica, nano-alumina and nano-titanium oxide are used in current research into developing nano-concrete. The addition of nanomaterials to concrete can be viewed as having a similar effect to that with micro-based materials such as metakolin and silica fume. The effect comes from pore refinement and thereby increases the strength and durability of concrete. The long-term performance of nano-concrete used in bridge structures is still unknown. We used deterioration models to estimate the deterioration rate of traditional concrete and conducted a scenario analysis to understand the potential benefit of the nano-concrete. This can significantly improve the efficiency of MRR management of bridge networks. We also compared the deterministic linear model against the stochastic Markov model. The demonstration on a concrete open girder showed that the Markov model is more suitable than the linear model. Further investigations will be conducted to study the effect of contributing factors, and to compare with other stochastic models.