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57 CHAPTER SEVEN PROBABILITY AND RISK DETERMINISTIC, PROBABILISTIC, AND RISK-BASED APPROACHES Engineering design involves calculations and decisions that are associated with uncertainty. Three approaches can be selected: 1. The deterministic approach, where best estimates of parameter values are selected, typically the mean value, and a global factor of safety is applied to the result to minimize the possibility of failure. 2. The probabilistic approach, where in addition to making calculations using the mean values of the parameters, the uncertainty associated with the answer is quantified by calculating the probability of failure, usually through the use of standard devi- ations. Then an acceptable probability of failure is selected and the design requirements are back- calculated to satisfy this very low target probability of failure. 3. The risk approach, where in addition to calculating the probability of failure, the value of the consequence is introduced in calculating the risk expressed as the product of the probability of failure times the value of the consequence. Then an acceptable risk value is selected and the design requirements are back-calcu- lated to satisfy this very low target risk. Fifty years ago, engineering design was dominated by the deterministic approach; today, the probabilistic approach is well on its way to becoming the dominant approach. In the future, the risk-based approach is likely to be chosen by the profession. The advantages and drawbacks of each approach are discussed next. Further reading on the subject topic should include the following: ⢠Baecher and Christian (2003), Reliability and Statistics in Geotechnical Engineering ⢠Fenton and Griffiths (2008), Risk Assessment in Geotechnical Engineering ⢠Phoon and Ching (2015), Risk and Reliability in Geotechnical Engineering. ADVANTAGES AND DRAWBACKS OF THE DETERMINISTIC APPROACH An advantage of the deterministic approach is that the calcula- tions correspond well with what happens in reality. For exam- ple, the best estimate of the water velocity in the river during a big flood is used and the engineer gets a sense of what is hap- pening near the embankment. At the same time, this estimate is associated with uncertainty, which is absorbed in a single factor of safety; this factor of safety is the same regardless of the level of uncertainty. Consider Figure 99, which shows the probability density function for two cases. In the case of curve A, the soil properties of the embankment are relatively well known, the soil is relatively uniform, and the standard deviation is small. In the case of curve B, the soil properties of the embankment are not as well known, the soil is more heterogeneous, and the stan- dard deviation is larger. By definition, the probability of failure is the area under the curve to the left of the axis corresponding to a factor of safety equal to 1. As the figure illustrates, this area is much larger in the case of curve B than in the case of curve A. Yet the mean factor of safety is the same at 1.5. It does not make sense to use the same factor of safety in both cases. This is why the probabilistic approach was developed. FIGURE 99 Difference in probability of failure for the same factor of safety. ADVANTAGES AND DRAWBACKS OF THE PROBABILISTIC APPROACH As pointed out earlier, the probability of failure is the area under the curve to the left of the axis corresponding to a fac-
58 tor of safety equal to 1. In the probabilistic approach, the first step is to determine the probability density function for the soil in the embankment, for example. This is often reduced to evaluating the mean and the standard deviation for the data available. Then calculations include determining the factor of safety, which corresponds to the mean of the distribution plus the probability of failure. The probability of failure is added information that helps the engineer to incorporate the variability of the input data into the engineering decision. Now the engineer can select an allowable target probability of failure and back-calculate the factor of safety that will satisfy not only the mean value but also the allowable prob- ability of failure. This is, for example, what is practiced with the use of the Load and Resistance Factor Design, where an allowable target probability of failure of the order of 0.001 (1 chance of failure in 1,000) is considered acceptable for most structural components. Table 15 shows data on probability of death for humans to be compared with the previously men- tioned target value in civil engineering of 0.001. This target probability of failure is a very useful concept, but it does not make sense to design a warehouse in the middle of nowhere for the same probability of failure as a nuclear power plant next to a megacity. The value of the consequence, should failure occur, is important. This is what prompted engineers to think in terms of risk instead of only probability of failure. ADVANTAGES AND DRAWBACKS OF THE RISK APPROACH The risk R is defined as R PoF C (7.1) Where PoF is the probability of failure and C the value of the consequence. This consequence can be evaluated in terms of the number of fatalities F or in terms of the number of dollars lost D. ( )R fatalities PoF F (7.2) ( )R dollars lost PoF D (7.3) The probability is usually an annual probability of fail- ure; the risk R is therefore in units of fatalities per year or dollars lost per year. Often Equation 7.1 is more properly written as R T V C (7.4) Where T is the threat, V the vulnerability, and C the value of the consequence. As can be seen in this case, the prob- ability of failure is split into two components. The threat is the probability that a certain event will occur (big flood or big earthquake), whereas the vulnerability is the probability that failure will occur if the event occurs. Vulnerability is the part of the system where one has the most control. Fragil- ity curves (Figure 100) link the probability of failure to the severity of the threat; they quantify the vulnerability of the system through the function V. FIGURE 100 Fragility curves. TABLE 15 PROBABILITY OF DEATH FOR HUMANS (BRIAUD 2013) Activity Probability of Death Heart disease 0.25 Cancer 0.23 Stroke 0.036 Car accident 0.012 Suicide 0.009 Fire 0.0009 Airplane 0.0002 Bicycle 0.0002 Lightning 0.00001 Earthquake 0.000009 Flood 0.000007 Note that it is not possible to design a structure (for exam- ple, an embankment) that has zero risk associated with its engineering life. This is because any calculation is associ- ated with some uncertainty; that the engineering professionâs knowledge, though having made great strides, is still incom- plete in many respects; that human beings are not error free; and that the engineer designs the structure for conditions that do not include extremely unlikely events such as a fall- ing satellite hitting the structure at the same time as an earth- quake, a hurricane, and a 500-year flood during rush hour. The choice of an acceptable risk is difficult because so many factors enter into the decision including the replace- ment cost and the number of lives at risk. The choice of an acceptable risk also involves other disciplines such as philos- ophy, politics, and social sciences. One of the very difficult steps required in estimating an acceptable risk is deciding what price to put on human life. It is not uncommon to use a number such as $1 million, because that is an average life insurance value for many people.
59 Figure 101 shows the annual risk associated with vari- ous engineering activities and in everyday life. The annual probability of failure (PoF) is on the vertical axis, and there are two scales on the horizontal axis: lives lost or fatalities per year (F) and dollars lost per year (D). Because the two do not necessarily correspond, the activities are shown as bubbles rather than precise points on the graphs. Because the risk is the product of the probability times the value of the consequence, two risk values can be defined: R (fatalities) = PoF Ã F and R (dollars lost) = PoF Ã D. There- fore, the annual risk is constant on diagonals in Figure 101. The dotted, dashed, and solid lines correspond to a high, medium, and low annual risk. The numbers are shown in Table 16. These data indicate that, in the United States, 0.001 fatalities per year and $1,000 per year may be accept- able target risk values. FIGURE 101 Risk associated with various engineering and human activities (Briaud 2013). TABLE 16 ANNUAL RISKS FOR THE UNITED STATES (RISK = POF X VALUE OF THE CONSEQUENCE) Annual Risk Level Fatalities/Year in the United States Dollars Lost/Year in the United States Low 0.001 1,000 Medium 0.01 10,000 High 0.1 100,000 Risk = PoFx value of the consequence. The advantage of the risk approach is that it is the most effective decision tool available today. The drawback is that it requires complex calculations that require a fair amount of information not always available to the engineer. DESIGN FLOOD AND ASSOCIATED PROBABILITY OF EXCEEDANCE One important engineering decision that affects risk is the choice of the design flood. This design flood can be the 10-year flood, 50-year flood, 100-year flood, 500-year flood, or other timespan. The probability of exceedance associated with this choice can be calculated as follows. As explained previously, the probability of exceeding the 100-year flood is 0.01 or 1 chance in 100 in any one year. This means that each year the probability is the same, regardless of whether a 100- year flood has occurred in the previous year. However, as can be imagined, if an embankment is going to be designed for, say, 75 years, as is the case for the design life of a bridge, then the probability that the 100-year flood will occur or be exceeded during those 75 years is higher than during any one year. The probability of exceedance in 75 years PoE75 is given by 11 1 Y YPoE PoE (7.5) Where PoEY is the probability of exceedance in Y years, and PoE1 is the probability of exceedance in any one year. Let us consider the 100-year flood for example together with the 75-year typical design life. Equation 7.6 gives 751 1 0.01 0.53YPoE (7.6) Therefore, the probability that the 100-year flood will occur or be exceeded during a 75-year design life is over 50%. For the 500-year flood, this probability becomes 0.14 or about 1 chance in 7 that the 500-year flood will occur or be exceeded during the design life of the embankment. If we go back to the acceptable probability of exceedance of 0.001 typical in civil engineering, a back calculation using Equation 7.7 gives 75 1 1 10.001 1 1 75151Y PoE PoE or PoE (7.7) In other words, the 75,000-year flood corresponds to a PoE of 0.001 for a design life of 75 years. This appears extreme, but remember that this only requires to consider an 83% increase in velocity compared with the velocity for the 100-year flood (see the chapter four section âImpact of Recurrence Interval on Velocity and Water Depthâ). SUMMARY This chapter explained the deterministic, probabilistic, and risk approaches that are adopted to deal with the fac- tor of uncertainty presented in the design. The advantages and drawbacks of each of the approaches were discussed. Because the selection of a design flood is a very important decision in the design process, the design flood and the asso- ciated probability of exceedance are also explained. The next chapter presents a summary of the survey results obtained from the DOT engineers.
