100

Structure B has the larger factor of safety and is, therefore, safer.

Taking a probabilistic approach for two statistically independent, normally distributed random variables, the safety is expressed in terms of a reliability index (b).

Structure A

y u Capacity ~ u Demand

Structure B

V102 + 302

where /x and ax are the mean and standard deviation of the random variable X. The random variables in this problem are capacity and demand.

Under these conditions, the reliability index is translated to a probability of failure as follows: pf = F(—b), where F is the distribution function of the standard normal variate. From a Table of Standard Normal Probability, one observes that the probability of failure for Structure A is approximately pf = 0.0000038 and for Structure B, pf = 0.00079. By this analysis, Structure A is the safer structure by a considerable margin. Figure 36.3 illustrates the demand and capacities for the two structures. Failure occurs when demand exceeds capacity. The reliability analysis becomes more complex, as illustrated in Section II, Chapter 12 of the Handbook, when the types of probability density distributions vary, the number of random variables increases, and correlation is considered.

While the safety of both structures has been quantified, the question remains as to whether either structure is safe enough to meet societal expectations. Acceptable risk, especially when considering the value of a human life and injuries is difficult to quantify. People die on the highways every day; bridges occasionally collapse; and hurricanes, wildfires, and earthquakes seem to destroy homes and lives every year. For each of these events, there is some level at which the public will demand improvement from their government. The issue of a threshold level of safety is not trivial. Menzies (1996) contends after considerable analysis that society will accept a risk to life of 1 in 1,000,000 per year for the specific example of short-span bridges in the United Kingdom. He further states that acceptable risk levels depend on value judgments and societal norms and cannot be found solely by analysis.

Load, kN

FIGURE 36.3 The capacity and demand for Structures A and B. Failure occurs when demand exceeds capacity.

Load, kN

FIGURE 36.3 The capacity and demand for Structures A and B. Failure occurs when demand exceeds capacity.

Fortunately, every engineer does not have to examine societal values and make this determination; those engineers who write the building codes provide that service. As the United States has transitioned to probability-based codes in steel, concrete, and wood, the code committees have wisely calibrated these new codes to those design requirements that have been successful in the past. The American Association of State Highway and Transportation Officials (AASHTO) load and resistance factor design (LRFD) code (AASHTO 1998) calibrates the design of elements of a new bridge in the reliability index range b = 3.0 to b = 4.0. The exact value is determined by a load factor modifier z, which accounts for ductility, redundancy, and importance of the structure. If the modifiers, for example, dictated a reliability requirement of b = 3.5 for Structures A and B in the previous example, it would appear that Structure B (b = 3.16) was inadequate and Structure A (b = 4.47) was substantially overdesigned.

While the codes provide a great means to compare different designs, they do not necessarily yet measure society's preferences and risk tolerances. Since there will always be a scarcity of resources to fund everything the public desires, every decision to improve the safety of a bridge or building is a tradeoff and a consent to accept additional risk in some other area such as disease prevention or police protection. Risk assessment ultimately needs to be viewed in this broader context. People face risks every day and most of these risks are quantifiable. Technology Review (1979) lists smoking 1.4 cigarettes, drinking 0.51 of wine, traveling 300 miles by car, and flying 1000 miles by airplane as activities that increase the chance of death by one in a million. Even with these statistics, people will not make rational preferences such as the family who drives for three days rather than fly to a location because they perceive it to be safer. Corotis (2003a,b) contends that people's perceptions of risk are affected by whether the risk is objective or subjective, aleatoric or epistemic, familiar or unfamiliar, and voluntary or involuntary. Real world decisions are based on that perception of risk which can include dread, familiarity, number of people exposed, trust, and technological stigma. Corotis further suggests that the public uses several standards to assess risk (cost-benefit analysis, revealed preferences, expressed preferences, and calibrated values) and that future building codes should take all of these into account if society's preferences are to be effectively addressed. This remains a fruitful area for future research.

Even if one concedes that the current building codes provide a reasonable assessment of society's tolerance for risk acceptance, there are additional issues with life cycle cost analysis. Any structure will start deteriorating from the day it is placed in service and, if not maintained, will continually worsen over time. In a design situation, the optimization of a life cycle cost must consider the level to which a structure should be designed and to what level it will be allowed to deteriorate before a repair or replacement is needed. Should the reliability specified in the codes be an average reliability over the lifetime of the structure (blifetime), the reliability to which a structure is designed (bdesign),

^design Plan A

Life cycle

Plan A

Life cycle Plan B

Life cycle

Plan A

Life cycle Plan B

^design Plan

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