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**Reliability Levels for Various Well-known Disasters**

Dr. Chandrasekhar Putcha Professor of Civil and Environmental Engineering California State University, Fullerton Fullerton, CA 92834 USA

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Abstract 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.

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Introduction 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.

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**Deterministic Methods:**

In a deterministic method, all quantities are considered as fixed, in the sense that the exact values are supposed to be known.

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**Disaster Center Method**

The method is as follows, as applicable to different state in USA

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**Disaster Center Method**

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 Get the ranking of each state by these individual categories. 5. Add the total of each state’s 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.

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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 miles CD = cost of damage for the region under consideration FNDS = Frequency of number of disasters FNF = frequency of number of fatalities Using the above method the following intensity factors have been calculated for each state (Disaster center report,1975) for tornadoes.

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**Rank State Intensity Factor 1 Indiana 4. 25 2 Massachusetts 4**

Rank State Intensity Factor 1 Indiana Massachusetts Mississippi Oklahoma Ohio Illinois Alabama Louisiana Arkansas Kansas 11.75

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**Rank State Intensity Factor 11 Florida 12. 75 12 Georgia 13**

Rank State Intensity Factor 11 Florida Georgia Connecticut Iowa Missouri Tennessee Texas Michigan Delaware South Carolina 18.75

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**Rank State Intensity Factor 21 Kentucky 19. 25 22 Nebraska 20**

Rank State Intensity Factor 21 Kentucky Nebraska North Carolina Pennsylvania Wisconsin Minnesota Maryland South Dakota Virginia North Dakota 30.25

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**Rank State Intensity Factor 31 New Jersey 30. 75 32 New York 31**

Rank State Intensity Factor 31 New Jersey New York Rhode Island Colorado West Virginia New Hampshire Wyoming Arizona Washington New Mexico 40

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**Rank State Intensity Factor 41 Maine 40. 25 42 Hawaii 41**

Rank State Intensity Factor 41 Maine Hawaii Vermont Montana California Idaho Oregon Utah Nevada Alaska 49.75

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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.

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AREA = 268,601 square miles Total payments for the flood damage (CD) = $2,249,450, FNDS (number of disasters – tornadoes) = 137/365 = FNF (frequency of number of fatalities) = 8/365 = Hence, intensity factor for flooding for Texas = [268601/(2,249,450, x x )] x 10,000 = 145 per 10,000 miles

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Similarly, for the tornado disaster, this factor for USA = [ /[ x 50 x x ] x = 5.2 per 100,000 miles In the above equation, FNDS = 1000/365 = and FNF = 80/365 = However, 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.

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Probabilistic Method 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 (β) is being used by reliability engineers, instead of probability of failure, to express the margin of safety in a structure.

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For resistance and strength being normal, the reliability index is given as Ang and Tang,1980), β = [(µR - µS)/(σR * σR + σS * σS)]. (2) where, µR and µS represent the expected value (mean values) in resistance and strength respectively. For resistance and strength being lognormal-lognormal, the reliability index is given as, β =[log µR – log µS /(VR *VR+ VS*VS)]. (3) Where, VR and VS represent the coefficient of variation in resistance and strength respectively.

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These equations for reliability index β 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 β can be calculated from numerical integration. A variation of the above procedure can also be used using the following equation (Ang and Tang,1980),

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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.

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**Table 2 number of tornadoes and associated fatalities**

(http://www.erh.noaa.gov/cae/svrwx/tornadobystate.htm) Avg. # of Avg. # of Avg. # of State tornadoes/year deaths/year tornadoes/10,000 miles (A) (B) Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida

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Avg. # of Avg. # of Avg. # of State tornadoes/year deaths/year tornadoes/10,000 miles (A) (B) Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine

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Avg. # of Avg. # of Avg. # of State tornadoes/year deaths/year tornadoes/10,000 miles (A) (B) Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire

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Avg. # of Avg. # of Avg. # of State tornadoes/year deaths/year tornadoes/10,000 miles (A) (B) New Jersey New Mexico New York North Carolina North Dakota Ohio Oregon Pennsylvania Rhode Island South Carolina

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Avg. # of Avg. # of Avg. # of State tornadoes/year deaths/year tornadoes/10,000 miles (A) (B) South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming

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**P (A) = Ф [ (Aactual– A-)/ σA) (4) P (B) = Ф [ (Bactual – B-)/ σB) (5) **

Assuming normal distribution, the probabilities P (A) and P (B) can be expressed as (Ang, 2007), P (A) = Ф [ (Aactual– A-)/ σA) (4) P (B) = Ф [ (Bactual – B-)/ σB) (5) For the data in Table 2, this data is given as, A- = 73.77, σA = 7,377, Aactual = 82 B- = ,σB = , Bactual = 180 Using the above information, P (A) = P (Aactual>= 82) = Ф [ (82.0 – 73.77/7.377)] = = 0.134

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**Similarly, for the event B, P (B) = P (Bactual>= 180) = 1**

Similarly, for the event B, P (B) = P (Bactual>= 180) = Ф [ (180.0 – /21.25] = Ф (1.80) = 1.0 – = Hence, the probability of the combined event, C can be easily calculated from Eq.4 as, P (C ) = – (0.134 x 0.036) = 0.165

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The corresponding reliability index (β) can be obtained from the following equation (Ellingwood et al., 1980), Pf = Ф (- β) (6) Hence, β = - Ф ( ) Hence, β = - Ф (0.835) From the standard normal distribution tables Hence, β = - (-0.95) = 0.95

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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).

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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 (β). 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.

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**The statistics is given as follows: Fy = 38 ksi, VFy = 0**

The statistics is given as follows: Fy = 38 ksi, VFy = 0.10 Z- = 54 ksi, VZ = V represents the coefficient of the random variable. Eq.2 is used to calculate the reliability index β. This works out to be Eq.2 can be used for the data in Table 2 to calculate the reliability index β. The necessary data is given as: µR = 73.77, σS = 7.377, µs= , σS =

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**The value of reliability index ( β) works out to be 3**

The value of reliability index ( β) 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).

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1. Define the appropriate limit state function: g (x1,x2---,xn) =0 (7) 2. Make an initial guess at the reliability index (β). 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. αiN = φ (φ-1 [ Fi (xi*)]) / Fi(x*)]) (8) Xi-N = Xi* - φ-1 [ Fi (xi*)] σiN(9)

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**5. Compute partial derivatives ðg/ðxi evaluated at the point Xi. 6**

5. Compute partial derivatives ðg/ðxi evaluated at the point Xi* . 6. Compute the direction cosines αi as, αi = (ðg/ðxiσiN) /[ ∑ (ðg/ðxiσiN)2 ]0.5 (10) 7. Compute new values of Xi* from Xi* = Xi-N - αiβσiN (11) And repeat steps 4 through 7 until the estimates of αistabilize. 8. Compute the value of β necessary for g (x1* , x2*, ……, xn* ) = 0 (12)

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**A variation of this procedure has been suggested by Grimaldi et al**

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 of potentially damaging phenomenon V = degree of loss resulting from the occurrence of the phenomenon D = total amount of both the direct and indirect losses due to the occurrence of the phenomenon

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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.

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**Figure 1. Illustration of the Reliability Index Concept**

Figure 1. Illustration of the Reliability Index Concept. (Ellingwood et al, 1980)

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Figure 2. Formulation of Safety Analysis in Original and Reduced Variable Coordinates. (Ellingwood et al, 1980)

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**Discussion of Results and Conclusions**

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, β, 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.

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References 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

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