8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability
PMC2000-113
A FAULT TREE MODEL OF BRIDGE DETERIORATION K. H. LeBeau, P.E., Ph.D. Student and S. J. Wadia-Fascetti, M. ASCE Northeastern University, Boston, MA 02115
[email protected],
[email protected] Abstract Bridge management systems (BMS) have provided State and Metropolitan Planning Organizations with a systematic approach to bridge programming. While BMS are continually improving, those available fail to address the issue of element interaction. The fact that the deterioration of one bridge element may accelerate that of another is not acknowledged. Fault tree analysis, which is a graphical depiction of the various failure paths that lead to an undesirable outcome, offers a systematic method of organizing the element interactions that contribute to bridge deterioration. In addition to visually uncoupling the interaction of the bridge system components, the tree sets forth logical interrelationships that may be applied qualitatively to explain the system deterioration process and quantitatively to arrive at various probabilities of bridge deterioration. The integration of fault tree analysis into current BMS provides the missing link between component condition and structural system performance for structures cataloged in the bridge management systems. The fault tree analysis presented in this paper gives an approximate 95% probability that an individual bridge comprised of elements in “poor” condition will require rehabilitation if no maintenance action is taken. Replacement of the deck, joints, and bearings will reduce the probability of required rehabilitation to around 50%.
Introduction and Element Interaction Modeling in BMS System performance is a term used in rating the overall function of complex systems. In a bridge system, each element contributes to the overall “health” of the system. Bridges are continually subjected to environmental factors that cause deterioration and possible damage. The decline of the system performance of a bridge can be attributed to deterioration mechanisms such as corrosion and fatigue. Due to their interconnectedness, the breakdown of one element affects others and ultimately the system performance of the bridge. State and Metropolitan Planning Organizations have the responsibility of maintaining their population of bridges, which are key elements in the transportation infrastructure. A bridge management system (BMS) serves as a decision-making tool in the prioritizing of maintenance, repair and rehabilitation projects based on the current and future condition of the bridges and budget constraints. A number of BMS have been developed including PONTIS and BRIDGIT. While BMS serve their original goal well, there is a continual call for improvement. One such improvement could be providing the missing link between component deterioration and system performance. The objective of this paper is to present a fault tree analysis to quantify the effect of component deterioration to the overall bridge performance and the need for rehabilitation. The most widely used BMS is PONTIS (from Latin pons: bridge), developed by Optima Incorporated and Cambridge Systematics and originally funded by the Federal Highway Administration (FHWA) (Golabi et al., 1993). In PONTIS, a population of bridges is represented on a network level by the individual bridge elements (deck, girder, bearings, etc.) with field inspection data providing numerical condition states for each element. The prediction model, a probabilistic second order Markovian chain, is applied at the
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network level, estimating the proportion of each bridge element that is expected to deteriorate in the next inspection cycle. A rank order of the bridge element condition states in any inspection cycle leads to an application at the bridge level. PONTIS (Golabi et al., 1993) supports “most interactive issues between elements” through four environmental effects categories: (1) Benign: element condition will not be significantly affected; (2) Low: environmental effects and/or operating practices do not adversely influence element condition; (3) Moderate: expected deterioration due to environmental effects and/or operating practices; and (4) Severe: rapid deterioration due to environmental effects and/or operating practices. The PONTIS user accounts for possible accelerated deterioration by assigning the element into a higher environmental effect category. One shortcoming of this approach is that the accelerated deterioration of one element due to the deterioration of another element is not considered (Sianipar and Adams, 1997). BRIDGIT, another popular BMS developed under the National Cooperative Highway Research Project (NCHRP) 12-28(2) and sponsored by the American Association of State Highway and Transportation Officials (AASHTO), is similar to PONTIS in that a Markovian prediction model is applied at the element level (Hawk and Small, 1998). The primary difference between the two systems lies in the optimization model, which is more bridge specific in BRIDGIT. The same environment states used in PONTIS are defined (Benign, Low, Moderate, Severe). In addition to the environmental effects, they also account for element interactions and loading conditions. However, BRIDGIT addresses the issue of element interaction more extensively than PONTIS. Protective systems are grouped separately from their underlying elements, yet the two are linked in their deterioration models. For example, the condition of the asphaltic overlay slows the rate of deterioration of a concrete deck. While this is an improvement, a comprehensive representation of all the relationships between elements that contribute to bridge deterioration is still lacking. PONTIS and BRIDGIT are proficient in individual element deterioration models. However, deterioration of a bridge system as a whole could be better represented by adequately modeling the interconnectedness between elements. The use of environmental effects categories would fail to account for element interaction if such environmental factors did in fact exist, forcing the user to assign the element into the “severe” category (Sianipar and Adams, 1997). Furthermore, correlation between environmental factors and deterioration rate is considered statistically in the PONTIS prediction model, but does not consider the relationship between environment, its effect on bridge elements, and the overall system. Also, the limited element interaction modeling of these BMS is highly dependent on the unavoidably subjective input data of the user. Fault Tree Modeling of Interaction of Bridge Elements An appropriate representation of the interaction between bridge elements requires a highly complex model to completely capture all the different element interdependencies.
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A fault tree is a viable approach at modeling this complexity (Sianipar and Adams, 1997; Johnson, 1999). A fault tree is a graphical depiction of the various failure paths that lead to an undesirable outcome (Ang and Tang, 1984). In addition to visually uncoupling possible events within the system, the tree sets forth logical interrelationships that may be applied qualitatively to explain its process and quantitatively to arrive at various probabilities of bridge deterioration, synonymous with need for rehabilitation. Fault trees use the symbolic notation shown in Table 1. Symbol
Name
Usage
Event
Top and intermediate positions of tree
Basic Event
Bottom positions of the tree
OR Gate
Representing the union of two or more events
AND Gate
Representing the intersection of two or more events
Table 1. Symbolic notation used in fault trees ( Ang and Tang, 1984 ).
The bridge considered is a single span, steel girder, reinforced concrete deck bridge supported by concrete cantilever abutments on spread footings carrying an interstate highway over a local street. The bridge is comprised of three basic components: (1) deck, (2) superstructure and (3) substructure as shown in Figure 1. The overall performance of the bridge decreases if any one of its main components (deck, superstructure, or substructure) deteriorates. Poor condition of the deck may be attributed to faulty joints or decay of the deck material. The condition of the superstructure is determined by the amount of wear and corrosion found in its girders or bearings. Considering the specific bridge type modeled in this paper, the deterioration of the substructure is directly related to the condition of the abutments. These described relations are modeled in the fault tree shown in Figure 2. Deck Condition Superstructure Condition
DETERIORATION OF BRIDGE PERFORMANCE (failure = need for rehabilitation)
Deck Condition
Superstructure Condition
Substructure Condition
Substructure condition Joints Condition
Figure 1. Three basic bridge components.
Deck Material Condition
Girders Condition
Bearings Condition
Abutments Condition
Figure 2. Top-level fault tree events.
The undesirable top event or failure of this fault tree is the deterioration of bridge performance, specifically requiring rehabilitation. The probability of deterioration of bridge performance (failure = need for rehabilitation) (T) is evaluated as a function of deck condition (A), superstructure condition (B), and substructure condition (C) through the Boolean equation:
p(T ) = p( A U B U C ) LeBeau and Wadia-Fascetti
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As shown in Figure 2, the fault tree can be further decomposed to show the sub-events that lead to the events. For example, deterioration of the deck is due to decay of the deck material, which is concrete in this case, or the condition of the joints. Then the probability of deck condition (A), which is a function of joint condition (A1) and deck material condition (A2), is obtained through the Boolean equation:
p( A) = p ( A1 U A2 )
(2)
Superstructure condition (B) is attributed to the condition of the girders (B1) and the condition of the bearings (B2) yielding:
p( B ) = p ( B1 U B2 )
(3)
Abutment condition has a direct influence on the amount of deterioration experienced by the substructure. Further decomposition of the sub-events can consider the joints’ ability to function, the condition of the seal, the ride quality of the transition from the roadway to the bridge and the condition of the joint anchorage devices as shown in Figures 3 and 4. For example, the basic events causing an expansion joint to malfunction are: (1) the joint could be partially or completely paved over inhibiting its movement, (2) the joint could be improperly aligned, (3) abutment settlement could affect the joints ability to function, (4) excessive dirt and debris could be packed into the joint “freezing” its movement. (5) the expansion joints could be impaired by the wear and tear (ripped seal, anchorage assemblies torn out, portions of the joint missing) caused by traffic impacts. Function of Expansion Joint
Joints Condition
Function of Expansion Joint
Transition from Roadway to Bridge Condition of Seal
Condition of Joint Anchorage Devices
Figure 3. Intermediate-level fault tree event.
J1
J2
J3
J1 = Indiscriminate Overlay J2 = Improper Alignment J3 = Abutment Settlement
J4
J5
J4 = Dirt and Debris J5 = Impact Damage
Figure 4. Basic events.
The probability of each basic event is needed in order to calculate the probability of the top event. There are many methods of obtaining these probabilities. An analysis of inspection data of a population of bridges would be ideal. One could also create analytical reliability models of each failure event. Still another approach is knowledge acquisition from experts (Dym and Levitt, 1991; Ang and Tang, 1984). The latter method is the one used in this study. This approach has its merits, in that the knowledge of actual bridge engineers and inspectors is elicited, and large amounts of historical data are not required. Seven bridge engineers and inspectors participated in this study, each having varied background and years experience. Multiple experts were used in order to eliminate the subjectivity associated with only one expert. It must be noted, though, that the
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probabilities arrived at in this study are biased owing to the fact that the inspection and design experience of the experts is localized to the states of Massachusetts and Rhode Island. A series of questions were asked in order to obtain the basic event probabilities used in this fault tree. The responses were then averaged using weight factors considering the approximate number of bridges each participant had inspected and his number of years of experience. The quality of these heuristic estimates was reviewed by one of the experts. The resulting probabilities for 32 basic events that contribute to the "poor" condition of elements are listed in Table 2. In this study, all of the basic events are assumed to be independent, thus simplifying the calculations associated with the fault tree. Using substitution and Boolean algebra rules, Equation 1 is converted into minimal cut set (MCS) form (Ang and Tang, 1984). A minimal cut set represents a combination of basic probabilities that leads to the occurrence of the top event. In other words, Equation 1 is expressed in terms of basic events only. Basic Event 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Paving over Expansion Joint Improper Alignment of Expansion Joint Abutment Settlement Excessive Dirt and Debris Traffic Impact Damage of Joints Clogged Deck Drains Leakage Corrosion of Joints Improper Installation of Joint Deck Cracking Deck Spalls Corroding Reinforcement in Deck Delamination (Deck) Poor Condition of Wearing Surface Efflorescence (Deck) Damaged Drainage Outlet Pipes
Probability 0.06 0.13 0.07 0.21 0.12 0.44 0.18 0.14 0.18 0.14 0.15 0.16 0.10 0.25 0.12 0.43
Basic Event 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Corrosion of Girders Fatigue Cracking Poor Alignment of Girders Collision Damage of Girders Worn Bearing Elements Incomplete Bearing Assemblies Corroded Bearings Deteriorated Concrete Pedestals Differential Vertical Movement (Abutment) Rotational Movement (Abutment) Cracks in Abutment Spalls in Abutment Corroded Reinforcement (Abutment) Delamination (Abutment) Efflorescence (Abutment) Severe Environmental Exposure
Probability 0.16 0.05 0.14 0.07 0.36 0.07 0.15 0.14 0.03 0.03 0.05 0.13 0.11 0.09 0.06 0.57
Table 2. Basic event probabilities obtained from interviews.
The probability of bridge performance deterioration (failure = need for rehabilitation) is evaluated using the basic probabilities obtained from the experts and Equation 1. The fault tree analysis yields a 94% chance that the bridge needs some sort of rehabilitative work. This high probability of deterioration is owing to the 58% probability of joint deterioration and the 59% probability of bearing deterioration, while the deck, girders and abutments are more on order of a 30% probability of poor condition. The fault tree points to the joints and bearings as the main culprits of bridge performance deterioration. Using the fault tree to investigate the effects of rehabilitation, hypothetical repairs can be applied to this example bridge. Table 3 lists different hypothetical repairs and the new probability of bridge performance deterioration. This exercise demonstrates a logical manner in which the fault tree can be used to evaluate rehabilitation alternatives. The top event, “Deterioration of Bridge Performance” is a broad, albeit vague event, and the sources of the deterioration of bridge performance are not evident unless the tree is traversed downwards. The fault tree is effective, however, in comparison of different maintenance scenarios in order to see which would be most beneficial.
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Maintenance Action No action Replace joints Replace joints and bearings Replace joints, bearings, and deck
Probability of Event T 94% 87% 67% 51%
Table 3. Probability of “Bridge Performance Deterioration” due to maintenance actions.
Conclusions Current BMS do not properly model element interaction in their deterioration models. One possible approach to modeling element interaction is fault tree analysis. The fault tree model of the bridge considered in this study, results in an approximate 95% probability of the bridge experiencing performance deterioration. Replacement of the deck, joints and bearings will reduce the probability of deterioration of bridge performance to around 50%. The fault tree may be used as a tool to compare rehabilitation scenarios. Different alternatives can be evaluated based on effectiveness of reducing the failure probability, which is the need for rehabilitation. The integration of fault tree analysis into current BMS provides the missing link between component condition and system performance. Acknowledgements Bridge engineers/inspectors are acknowledged for their participation in this study: John Doucette, PE, Witold Kloczkowski, PE, Michael E. LeBeau, PE, SE, James Onysko, Vartan Sahakian, PE, Benjamin Soares, and David Titus. Support from NSF Career Development Award # CMS-9702656 is appreciated. References Ang, A.H.-S., and W.H. Tang (1984), Probability concepts in engineering planning and design: Vol II, Wiley, New York. Bridge inspector’s training manual/90 (1991). Rep. No. FHWA-PD-91-015. Federal Highway Administration, Washington, D.C. Dym, C. L. and R. E. Levitt (1991), Knowledge-Based Systems in Engineering, McGraw-Hill, Inc., New York. Golabi, K., P. D. Thompson, and, W. A. Hyman (1993). “Pontis technical manual.” Tech. Rep. No. FHWASA-94-031. Optima Inc. and Cambridge Systematics, Inc., Cambridge, Mass. Hawk, H., and E. Small (1998), “The BRIDGIT Bridge Management System,” Structural Engineering International: Journal of the International Association for Bridge and Structural Engineering, 8, 309314. Johnson, P. (1999), “Fault Tree Analysis of Bridge Failure due to Scour and Channel Instability,” ASCE Journal of Infrastructure Systems, 5(1), 35-41. Sianipar, P., and T. Adams (1997), “Fault-Tree Model of Bridge Element Deterioration Due to Interaction,” ASCE Journal of Infrastructure Systems, 3(3), 103-110.
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