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FIGURE 36.28 Bridge network consisting of 14 highway bridges in the Denver-Boulder corridor of Colorado. (Agkul and Frangopol 2003. Reprinted with permission from the American Society of Civil Engineers.)

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FIGURE 36.28 Bridge network consisting of 14 highway bridges in the Denver-Boulder corridor of Colorado. (Agkul and Frangopol 2003. Reprinted with permission from the American Society of Civil Engineers.)

this study is described in Akgül and Frangopol (2004). A study of networks of bridges (Frangopol et al. 2000a) offers new opportunities for life cycle cost. Akgül and Frangopol (2003) relate the system reliability of the bridges to the load rating for which there are existing data on all bridges. Bridges can be prioritized based on their importance to the overall network and historical inspection data become more useful for maintenance planning of a large category of bridges.

While all three examples in this section involve highway bridges, the concepts are applicable for any structures. Highway bridges are common examples because bridges are critical structures; there exist over 600,000 in the national database, and the Federal Highway Administration and the State DOTs have invested greatly in bridge management systems that provide much of the supporting data needed to support life cycle analyses.

### 36.12 Conclusions

A life cycle analysis of a structure introduces a number of issues that are not considered in deterministic or probabilistic time invariant solutions. Factors such as deterioration rates, inspection capabilities, repair effects, failure costs, discount rates, and threshold reliabilities present a number of challenges — many of which have not yet been solved. Life cycle analyses are necessarily more complex and require additional input data than those used in conventional time-invariant analyses. It is important to dismiss any misconception that very extensive data are required to perform probabilistic life cycle structural analysis. In fact, probabilistic methods allow the analysis of incomplete data in a much more reliable and consistent manner than conventional deterministic methods. Such analyses also reveal that the costs of maintenance, repair, and inspection may greatly exceed the original cost of construction. In times of scarce and competing resources for an aging infrastructure, these costs cannot be ignored. This chapter has used examples to illustrate the issues involved in condition assessment and life cycle cost analysis and design of deteriorating structures and highlighted some studies that have attempted to address these challenges. There is an urgent need for civil engineers to (a) acquire a good understanding of financial management under uncertainty, (b) have a stronger grasp of reliability-based optimal decision making, and (c) increase their role in the ultimate decisions on life cycle management of our decaying civil infrastructure systems.

### Acknowledgments

The partial financial support of the U.S. National Science Foundation through grants CMS-9506435, CMS-9912525, and CMS-0217290, of the U.S. Army Corps of Engineers Construction, Engineering Research Laboratory, of the U.K. Highways Agency and of the Civil Engineering Division of the Netherlands Ministry of Transport, Public Works, and Water Management is gratefully acknowledged. The support of the Colorado Department of Transportation is also gratefully acknowledged. The second author thanks his former and current students Dr. Michael Enright, Dr. Emhaidy Gharaibeh, Dr. Jung Kong, Dr. Seung Yang, Mr. Masaru Miyake, Dr. Ferhat Akgül, Mr. Luis Neves, and Dr. Aruz Petcherdchoo for their cooperation. The opinions and conclusions presented in this chapter are those of the authors and do not necessarily reflect the views of the sponsoring organizations.

### Glossary

Bias factor — A factor that relates the nominal deterministic value of a quantity to the mean value of that quantity when it is considered a random variable. Coefficient of variation — Ratio of the mean value to the standard deviation of a random variable; it is a convenient nondimensional measure of variability (Ang and Tang 1975). Component (with one failure mode) — A portion of a system represented by a single failure mode and thus a single limit state equation.

Condition rating — A rating assigned by an inspector to identify the structural condition, usually by matching the visual condition of a structure to the most appropriate written description of distress.

Correlation (linear) — A measure of the (linear) relationship between two random variables, usually indicated by a correlation coefficient that ranges from p = 1 (perfect positive correlation) to p = —1 (perfect negative correlation), where p = 0 indicates no correlation.

Deterioration model — A mathematical relationship that describes how a structure is expected to deteriorate over time, usually obtained from laboratory data, theory, or other studies.

Deterministic — No (aleatory and epistemic) uncertainties are present; everything is certain.

Discount rate — The interest rate to be used to convert future monetary sums to present value to allow alternatives to be compared over time.

Ductility — A measure of the ability of a structural member to elongate prior to rupture; a ductile member is still able to carry a load after yielding and allows warning prior to failure, which normally produces a safer structure.

Expected cost — A mean value of the cost; expected cost is used when there is uncertainty associated with the cost and is often a weighted average of the costs associated with various alternatives and their probability of occurrence.

Factor of safety — The ratio of the capacity of a structure (resistance) to the demand placed upon it (load or load effect).

Failure — When a structure no longer performs as intended; using the performance function, failure is defined as g(X) < 0.

Failure cost — Costs incurred by a structural failure multiplied by the probability of failure.

Hazard function — Provides the instantaneous rate of failure; expresses the likelihood of failure in a specific time interval given that the structure has not already failed.

Importance — A measure of the criticality of a structure, usually determined by the degree of consequence if the structure fails.

Independent (statistically) — Two events are independent if the occurrence or (nonoccurrence) of one event does not affect the probability of the other event (Ang and Tang 1975); if two variables are independent, their correlation coefficient is zero.

Initial cost — Those costs associated with the design and construction of a structure (e.g., permits, architect fees, materials, equipment, labor); typically, those costs associated with placing a structure into service.

Life cycle cost — Total costs associated with a structure throughout its entire life; typically includes initial, maintenance, inspection, repair, and failure costs.

Limit state — The boundary between the desired and undesired performance of a structure; the boundary is typically represented by a limit state equation g(X) = 0.

Mean value — The central tendency of a random variable; in the discrete case, the mean value is the average value of the data, in the continuous case where fX(X) is the density function of a random variable, the mean value is m = J—1 xfx(x) dx (Rao 1992).

Nondestructive evaluation — The assessment or inspection of a structure conducted without damaging the structure; the evaluation can be visual or involve testing equipment.

Parallel system — A system where all members must fail for the system to fail.

Preventive maintenance — Precautionary maintenance actions, design to forestall damage and deterioration of a structure.

Probability density function — For a continuous random variable, fX(X) is a function that defines the probability of detecting X in the infinitesimal interval (x, x + dx); therefore, fX(X) dx =

Random variable — A function that maps events from the sample space into the axis of real numbers; it can be continuous, discrete, or mixed.

Redundancy — Measure of reserve capacity; failure of a single member will not cause failure of a redundant structure.

Reliability — Probability of safe performance of a structure.

Reliability index — A measure of reliability, denoted as b; the shortest distance from the origin to the failure surface in standard normal space; when both load and resistance are normally distributed, pf = F(—b), where F is the distribution function of the standard normal variate.

Risk — A function of the probability of occurrence of an adverse event and the consequence of the event; often risk is defined by the probability of failure of the adverse event and its consequence (Ang and De Leon 2005).

Safe — A structure performing as intended.

Series system — A system that fails if any member of the system fails.

Survivor function — Defines the probability that a structure is safe at any particular time.

Visual inspection — A periodic inspection where an inspector observes a structure and classifies its condition in terms of predefined condition ratings.

Notation

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