Dr. Chandrasekhar PutchaProfessor of Civil and Environmental EngineeringCalifornia State University, FullertonFullerton, CA 92834USA
A method has been developed in this paper for calculation of probability of occurrence of a disaster associated with calculation of reliability levels for various well known disasters. These are based on the limit state chosen, functional relationship between various dependent and independent parameters and the uncertainty in the associated parameters. This approach is based on probabilistic methods. In addition, an idea abut the intensity of disaster can also be had from traditional deterministic methods. Both these methods are discussed and the disaster levels are calculated for various well-known disasters are calculated in this paper.
The most important thing to be noted in the case of any disaster (natural or otherwise), is to find its intensity so that precautions can be taken to mitigate the effect of the disaster. This paper deals with well known natural disasters. Two methods are discussed. One is a deterministic method and the other is a probabilistic method. In the following section, both methods are discussed. First, the basic concepts of both methods are discussed followed by a specific example in each. These examples deal for each state in United States of America. The point to be noted here is that, the methods discussed here are general in nature and hence can be applied to any data from any part of the world.
In a deterministic method, all quantities are considered as fixed, in the sense that the exact values are supposed to be known.
The method is as follows, as applicable to different state in USA
1.Get square mileage of each state that might be affected by the disaster tornado, for example. This can be obtained from the literature(http://www.erh.noaa.gov/cae/svrwx/tornadobystate.htm)2.Obtain the frequency of death, injury, number of disasters and cost of damages for that region.3.Divide the square mileage by any of the four factors mentioned in step 2.4.Get the ranking of each state by these individual categories.5.Add the total of each states individual rankings and divide by the number of factors. In this, the number of factors are four.6.The number obtained is cumulative risk factor for that state.
Based on the above methodology, the equation for calculating the intensity factor can be written as,IF = (AREA / (CD x FNDS x FNF) (1)Where, AREA = area of region under consideration in square milesCD = cost of damage for the region under considerationFNDS = Frequency of number of disasters FNF = frequency of number of fatalitiesUsing the above method the following intensity factors have been calculated for each state (Disaster center report,1975) for tornadoes.
RankStateIntensity Factor11Florida12.7512Georgia13.2513Connecticut13.2514Iowa15.2515Missouri15.2516Tennessee1617Texas1718Michigan17.2519Delaware18.520South Carolina18.75
RankStateIntensity Factor21Kentucky19.2522Nebraska20.2523North Carolina20.2524Pennsylvania20.2525Wisconsin20.7526Minnesota2227Maryland2528South Dakota29.7529Virginia29.7530North Dakota30.25
RankStateIntensity Factor31New Jersey30.7532New York31.2533Rhode Island32.534Colorado3535West Virginia3536New Hampshire3837Wyoming3938Arizona39.2539Washington39.2540New Mexico40
These values of intensity factors for each state give an idea of the intensity of damage that a tornado could do to that state if it were to occur. This is based on the past data. Once this information is available, the authorities can take steps to mitigate the situation.This approach can also be used for any state even if the data is available for only few factors for any disaster.For example, the following data is available for flood for the State of Texas.
AREA = 268,601 square milesTotal payments for the flood damage (CD) = $2,249,450,933.34FNDS (number of disasters tornadoes) = 137/365 = 0..3753FNF (frequency of number of fatalities) = 8/365 = 0.0219Hence, intensity factor for flooding for Texas = [268601/(2,249,450,933.34 x 0.3753 x 0.0219)] x 10,000 = 145 per 10,000 miles
Similarly, for the tornado disaster, this factor for USA = [3537441/[2249450933.34 x 50 x 2.739 x 0.219 ] x 100000 = 5.2 per 100,000 miles
In the above equation, FNDS = 1000/365 = 2.739 and FNF = 80/365 = 0.219However, this does not give an idea about the probability of occurrence of a tornado. To do this, one has to use probabilistic methods discussed in the next section.
In this method, all the functional variables connected to the physical phenomenon are treated as Random variables (RV) whose outcome is not certain. The details are given below.This concept can be explained through two parameters Resistance and Strength associated with a system. Any uncertainty in materials and loads can be incorporated into these two variables of Resistance and strength. Probability of failure of a system is defined as the probability of resistance being less than the corresponding strength. Recently, reliability index () is being used by reliability engineers, instead of probability of failure, to express the margin of safety in a structure.
For resistance and strength being normal, the reliability index is given as Ang and Tang,1980), = [(R - S)/(R * R + S * S)]. (2)where, R and S represent the expected value (mean values) in resistance and strength respectively. For resistance and strength being lognormal-lognormal, the reliability index is given as, =[log R log S /(VR *VR+ VS*VS)]. (3)Where, VR and VS represent the coefficient of variation in resistance and strength respectively.
These equations for reliability index are based on the well known FOSM (First Order Second Moment) approach. When the two basic parameters of Resistance and strength follow some other statistical distribution, the reliability index can be calculated from numerical integration. A variation of the above procedure can also be used using the following equation (Ang and Tang,1980),
P (C ) = P (A U B)(4)Where , P(A)= probability of occurrence of event A (fatality due to the disaster) P(B)= probability of occurrence of event B (occurrence of disaster tornado in this case)The above EQ. 4, can be rewritten as,P (C ) = P(A) + P(B) P (A) P(B) (5)This method can be applied to the data shown in Table 2 below.
Avg. # of Avg. # of Avg. # ofStatetornadoes/year deaths/year tornadoes/10,000 miles (A)(B)Alabama2364.53Alaska000Arizona400.35Arkansas2153.95California400.26Colorado2402.32Connecticut102.05Delaware105.18Florida5229.59Table 2 number of tornadoes and associated fatalities(http://www.erh.noaa.gov/cae/svrwx/tornadobystate.htm)
Avg. # of Avg. # of Avg. # ofStatetornadoes/year deaths/year tornadoes/10,000 miles (A)(B)Georgia2113.61 Hawaii101.56Idaho200.24Illinois2754.86Indiana2376.41Iowa3506.25Kansas3624.65Kentucky1032.52Louisiana2726.07Maine202.22
Avg. # of Avg. # of Avg. # ofStatetornadoes/year deaths/year tornadoes/10,000 miles (A)(B)Maryland30.073.05 Massachusetts303.83Michigan1833.16Minnesota1922.39Mississippi26105.51Missouri2723.92Montana500.34Nebraska360.74.70Nevada100.09New Hampshire202.22
Avg. # of Avg. # of Avg. # ofStatetornadoes/year deaths/year tornadoes/10,000 miles (A)(B)New Jersey304.02 New Mexico900.74New York501.06North Carolina1422.87North Dakota2002.39Ohio1653.90Oregon100.10Pennsylvania1022.23Rhode Island0.2302.22South Carolina1013.31
Avg. # of Avg. # of Avg. # ofStatetornadoes/year deaths/year tornadoes/10,000 miles (A)(B)South Dakota2803.69 Tennessee1232.91Texas13785.23Utah200.24Vermont101.08Virginia601.51Washington100.15West Virginia1013.31Wisconsin2113.86Wyoming1101.13
Assuming normal distribution, the probabilities P (A) and P (B) can be expressed as (Ang, 2007),P (A)= [ (Aactual A-)/A) (4)P (B)= [ (Bactual B-)/B) (5)
For the data in Table 2, this data is given as,A- = 73.77, A = 7,377, Aactual = 82B- = 141.67,B = 21.25, Bactual = 180
Using the above information, P (A) = P (Aactual>= 82) = 1.0 - [ (82.0 73.77/7.377)] = 1.0 -0.8643= 0.134
Similarly, for the event B,P (B) = P (Bactual>= 180) = 1.0 - [ (180.0 141.67/21.25] = 1.0 - (1.80) = 1.0 0.9640= 0.036Hence, the probability of the combined event, C can be easily calculated from Eq.4 as,P (C ) = 0.134 + 0.036 (0.134 x 0.036) = 0.165
The corresponding reliability index () can be obtained from the following equation (Ellingwood et al., 1980),
Pf = (- ) (6)Hence, = - (1-0.165)Hence, = - (0.835) From the standard normal distribution tablesHence, = - (-0.95) = 0.95
Another popular method, as mentioned earlier is the use of FOSM (First Order Second Moment Method). Before this method is applied to the present data of tornado in Table 2, a brief example is discussed below. This example is taken from (Ellingwood et al.,1980).
Problem:A 16WF31 steel section with yield stress Fy = 36 ksi is used for construction of a bridge. The section modulus of the section of the bridge is S = 54 in. Assuming an applied moment of 1140 in-kip, calculate the reliability index ().
Solution:The limit state function g for this structure is given as, g = Fy Z 1140 = 0. Both F and Z are probabilistic random variables in this problem.
The statistics is given as follows:Fy = 38 ksi, VFy = 0.10Z- = 54 ksi, VZ = 0.05.V represents the coefficient of the random variable.Eq.2 is used to calculate the reliability index . This works out to be 5.14.Eq.2 can be used for the data in Table 2 to calculate the reliability index . The necessary data is given as:R = 73.77, S = 7.377, s= 141.67, S = 21.25.
The value of reliability index ( ) works out to be 3.01 ,which is much higher than the previous value obtained.This is because, an inherent assumption of normality has been made in using Eq. 2.For getting more accurate results, one can use AFOSM (Advanced First Order method) also. This is well documented in literature (Ellingwood et al., 1980). The procedure is as follows (Ellingwood et al.,1980).
1.Define the appropriate limit state function:g (x1,x2---,xn) =0 (7)
2.Make an initial guess at the reliability index ().
3.Set the initial checking point values Xi* = X- for all i.
4.Compute the mean and standard deviation of the equivalent normal distribution for those variables that are non-normal as per equation given below.
iN = (-1 [ Fi (xi*)]) / Fi(x*)])(8)
Xi-N = Xi* - -1 [ Fi (xi*)] iN(9)
5.Compute partial derivatives g/xi evaluated at the point Xi* .
6.Compute the direction cosines i as,
i = (g/xiiN) /[ (g/xiiN)2 ]0.5 (10)
7.Compute new values of Xi* from Xi* = Xi-N - iiN (11)
And repeat steps 4 through 7 until the estimates of istabilize.
8.Compute the value of necessary forg (x1* , x2*, , xn*) = 0(12)
A variation of this procedure has been suggested by Grimaldi et al. (2010). In this method, Risk, R, is expressed as,R = H V D(13)Where,H =probability of occurrence ofpotentially damaging phenomenonV =degree of loss resulting from theoccurrence of the phenomenon D =total amount of both the direct and indirect losses due to the occurrence of the phenomenon
The phenomenon that is being referred in the above equations can be any of the disasters tornado, flood or landslide. The above equation is based on the following definition of Risk (Ayyub,1990):Risk = P(occurrence of the event) x Consequences of occurrence (14)In the above Eq. 2, H is the only random variable. The other two variables - H and D are considered are deterministic.
Figure 1. Illustration of the Reliability Index Concept. (Ellingwood et al, 1980)
Figure 2. Formulation of Safety Analysis in Original and Reduced Variable Coordinates. (Ellingwood et al, 1980)
The probability of failure of death due to tornado, the probability of occurrence of tornado and the combined probability of the two events has been calculated in this paper. The safety index, , is also calculated. The results obtained using the FOSM method and the traditional probability of failure approach checked well.
This shows the versatility of the probabilistic methods as compared to the traditional deterministic methods.
Ayyub, B.A. and McCuen,R. (1997). Probability, Statistics, & Reliability for Engineers. CRC Press.
Ang, A. H-S. and Tang, W.H. (2007). Probability Concepts in Engineering. John Wiley &Sons,Inc.
Ellingwood, B, Galambos, T.V.,MacGregor, J.G. and Cornell, C.A. (1980). Development f a Probability Based Load Criterion for American National Standard A 58, NBS Special Publication 577,Washington, D.C.
Grimaldi, S., Vesco, R.D. , Patrocco,D. and Poggi, D. (2010). A Synthetic Method for Assessing The Risk of Small Dam Flooding, 30th Annual USSD conference on Collaborative Management of Integrated Watersheds, Sacramento, April 12-16.