Stochastic processes for modelling bridge deterioration

Traditionally, bridge management systems were designed using Markov chain models. Recently, researchers applied the gamma process successfully to structural deterioration problems. The stochastic process captures the temporal variability of degradation, and has been applied to a range of problems in structures. We report on a study for the modelling of the condition of bridges in the state of NSW. The study encompasses large amounts of data spanning more than 15 years. We argue for the applicability of the gamma process and other stochastic processes. While the gamma process has been adopted in the past decade on grounds of mathematical tractability and physical motivation, we also observe another distribution for the deterioration at different times. The finding promotes the stochastic process modelling direction taken in the past decade and brings forth new models for the time-dependent reliability analysis of bridges. ters (m 2 ) if it is a surface, in meters (m) for some elements such as railing and joints, or units (ea) for timber elements. All quantities are recorded as integers. It is a practical approximation that facilitates the visual estimation of damage. A unit of quantity of an element is either 1m 2 , 1m or 1. There are 4,945 structures and 66 elements considered in the study, for which over 230,000 inspection records are analyzed. The records go back to 1989, up to the present time. Each record corresponds to an element of a particular structure on a scheduled inspection. Among the recorded entries are q, the total quantity of the element, q1 the quantity in condition state 1, q2 the quantity in state 2, ..., to qn the quantity in condition state n. For example, the concrete posttensioned girder on some bridge, on a particular inspection, was found to have q1=590m 2 , q2=2m 2 , q3=0m 2 and q4=0m 2 for a total q= 592m 2 . Most structures have about ten to twelve elements, with more than 30 in some cases. In the course of the years that the database spans, an element was inspected a number of times, roughly every two years. The elements fall into several categories; concrete elements, steel elements with lead based paint, steel elements with other protective treatment, timber elements, joints, bearings, railings and miscellaneous. Some of the more common types are concrete pre-tensioned girder, concrete reinforced prestressed pile, concrete deck slab, concrete culverts, steel rolled beams/I girders with lead based paint protective coating, timber beam/cross girder, pourable/cork joint seal, elastomeric bearing and metal bridge railing. Each element in each structure is classified as subject to one of three degrees of environmental stress; light, moderate and severe. The same element type can be classified as being in two different environmental stresses in two different structures. Approximately two thirds of the records show a moderate environmental stress, with most of the remaining third being light stress. 3 MODELLING BRIDGE DETERIORATION Effort went into the design and implementation of pavement and bridge management systems (BMS) due to ageing road infrastructure. In the United States, more than 70% of the bridges were built prior to 1935, and a large percentage of the United Kingdom’s current bridge stock was built between the late 1950s and early 1970s. In the state of New South Wales, Australia, around 70% of the operating bridges were built before 1985, with a significant proportion before the 1940’s. The near completion of most of the road networks and the ageing of the bridge stocks shifted the emphasis from developing new networks to the maintenance, repair, rehabilitation and replacement of existing road structures. Bridge management systems were developed as software tools for gathering and analyzing bridge condition data. The goal of a BMS is to predict conditions for bridge stocks and estimate maintenance funding. Pontis (Golabi & Shepard 1997) is one of the most widely used BMS. Pontis was designed and developed at the request of the United States Federal Highway Administration with the collaboration of six states and two national agencies. Bridge inspection data in Pontis are collected in the same fashion as for the data of this study. 3.1 The Markov Model The decision model of Pontis, and similar BMS such as the BRIDGIT bridge management system (Hawk & Small 1998), include a Markov chain as a statebased deterioration model that drives the prediction and maintenance optimization. The Markov chain model can be found in the maintenance and repair problems since the early 60’s. It is introduced to the maintenance of road infrastructure by Golabi et al. (1982). The model is appealing for its handling of uncertainty and its formulation of solvable maintenance optimization problems. A number of criticism points have been made against the usefulness of the model (Frangopol et al. 2001); bridge element performance is not addressed from a reliability viewpoint, maintenance and repair are driven by economic reasons, and the Markovian assumption does not take into account the history of the bridge deterioration. Time can be used in the assessment of the transition probabilities of the Markov chain, but the model does not explicitly capture the deterioration process. Madanat et al. (1995) make a number of observations. They point to the fact that the methods used in estimating these probabilities are ad-hoc and suffer from important methodological limitations; (i) the change in condition from one inspection to the next is not modeled explicitly, failing to capture the structure of the deterioration process, (ii) consequently, the model fails to capture the inherent nonstationarity of the deterioration process, and (iii) the approach does not recognize the latent nature of deterioration. Since deterioration is an unobservable process, it is not the state of the observable condition that should be modeled, but rather the process that generates these conditions. A practical difficulty with the estimation of the transition probabilities is the lack of data for some condition states. The severe conditions of elements of a bridge are rarely observed, due to the maintenance effort. Only the first states transition probabilities are estimated properly. Change in condition from one point in time to another is not explicitly modeled, and the Markov approach does not capture the inherent nonstationarity nature of the deterioration process. 3.2 Time-Based Models Time based models consider the time between condition changes. If the condition states are discrete, the transition times can vary from one structure to another. The Markov chain, a state-based deterioration model, assumes that the time of transition from one state to another is exponentially distributed. This assumption permits the application of the memorylessness property by which an old unit of element and a newer one are equally likely to move to the next condition state (Golabi & Shepard 1997). A departure from the Markov assumption is the reliability based approach of Estes & Frangopol (1999). They proposed a system reliability based approach for optimising repair strategies for bridges, consisting of identifying relevant failure modes of the bridge, developing a system model of the overall bridge as a series-parallel combination of individual failure modes and computing the system reliability of the bridge. The idea is carried over by Yang et al. (2004). This approach applies mainly to bridges where the times to reach severe condition states can be estimated. In addition, the assumption of independence of the lifetimes of elements of bridges is a hard one to make, while modelling correlation is a harder task. It was shown that the optimum maintenance cost is strongly dependent on the system model used to represent the structure. 3.3 Stochastic Deterioration Process Many factors contribute to the deterioration of bridge components; the quality and time of construction, traffic load, environmental stress. There are inherent deterioration processes in the materials of bridges that can be caused by chloride ingress, carbonation, a large sulphate attack, freeze-thaw action or alkali silica reaction (Gall 2004). Gaal et al. (2002) relate the deterioration of concrete bridges in the Netherlands to the use of de-icing salt resulting in chloride induced corrosion of the reinforcement. The work is cited by van Beek et al. (2003) in their study of the deterioration of concrete structures. A stochastic process that considers non-linear rates of deteriorations is more appropriate to model the deterioration of structures (Pandey et al. 2005). The stochastic process can capture the temporal variability of degradation. The argument is made in a series of papers by Pandey & van Noortwijk (2004), van Noortwijk et al. (2005) and Pandey at al. (2007). A particularly convenient stochastic process is the gamma process. Empirical studies showed that the expected deterioration in some cases followed the power law at b , t being the time. This function of time is incorporated into the gamma process, along with a time-dependent variance σ 2 t, and used to model structural deterioration (van Noortwijk 2007). The gamma process was first applied by the Australian scientist Moran in the 1950’s to model water flow into a dam (Moran 1954). In recent times, AbdelHameed (1975) was the first to propose the gamma process as a deterioration model. The gamma process is suitable for modelling gradual wear and degradation in cumulative amounts. This is recognized and applied in many structural studies, for example Nicolai et al. (2007) and van Noortwijk et al. (2007). 4 THE DETERIORATION DATA Deterioration is most likely to suit the measuring model of a continuous variable, except possibly in the case of some timber elements. In this study, the condition data (q1, ..., qn) of an element is converted to a univariate measure C, using the notion of a ‘Condition Index’. The perc