Transcript Slide 1

Dr. Chandrasekhar Putcha
Professor of Civil and Environmental Engineering
California State University, Fullerton
Fullerton, CA 92834
USA
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
http://www.disastercenter.com/to
rnado/rank.htm
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 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.
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.
Rank
State
Intensity Factor
1
Indiana
4.25
2
Massachusetts
4.25
3
Mississippi
6.75
4
Oklahoma
8.25
5
Ohio
8.25
6
Illinois
8.75
7
Alabama
8.75
8
Louisiana
9.5
9
Arkansas
11
10
Kansas
11.75
Rank
State
Intensity Factor
11
Florida
12.75
12
Georgia
13.25
13
Connecticut
13.25
14
Iowa
15.25
15
Missouri
15.25
16
Tennessee
16
17
Texas
17
18
Michigan
17.25
19
Delaware
18.5
20
South Carolina
18.75
Rank
State
Intensity Factor
21
Kentucky
19.25
22
Nebraska
20.25
23
North Carolina
20.25
24
Pennsylvania
20.25
25
Wisconsin
20.75
26
Minnesota
22
27
Maryland
25
28
South Dakota
29.75
29
Virginia
29.75
30
North Dakota
30.25
Rank
State
Intensity Factor
31
New Jersey
30.75
32
New York
31.25
33
Rhode Island
32.5
34
Colorado
35
35
West Virginia
35
36
New Hampshire
38
37
Wyoming
39
38
Arizona
39.25
39
Washington
39.25
40
New Mexico
40
Rank
State
Intensity Factor
41
Maine
40.25
42
Hawaii
41.25
43
Vermont
42
44
Montana
42.75
45
California
44.5
46
Idaho
46.25
47
Oregon
46.5
48
Utah
48
49
Nevada
48.75
50
Alaska
49.75
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 miles
Total payments for the flood damage (CD) =
$2,249,450,933.34
FNDS (number of disasters – tornadoes) =
137/365 = 0..3753
FNF (frequency of number of fatalities) =
8/365 = 0.0219
Hence, 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 = 0.219
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.
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.
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.
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),
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)
This method can be applied to the data shown in
Table 2 below.
(5)
Table 2 number of tornadoes and associated fatalities
(http://www.erh.noaa.gov/cae/svrwx/tornadobystate.htm)
State
Avg. # of
Avg. # of
Avg. # of
tornadoes/year deaths/year
tornadoes/10,000 miles
(A)
(B)
Alabama
23
6
4.53
Alaska 0
0
0
Arizona 4
0
0.35
Arkansas
21
5
3.95
California
4
0
0.26
Colorado
24
0
2.32
Connecticut
1
0
2.05
Delaware
1
0
5.18
Florida 52
2
9.59
State
Avg. # of
Avg. # of
Avg. # of
tornadoes/year deaths/year
tornadoes/10,000 miles
(A)
(B)
Georgia 21
1
3.61
Hawaii 1
0
1.56
Idaho
0
0.24
Illinois 27
5
4.86
Indiana 23
7
6.41
Iowa
35
0
6.25
Kansas 36
2
4.65
Kentucky
10
3
2.52
Louisiana
27
2
6.07
Maine
0
2
2
2.22
State
Avg. # of
Avg. # of
Avg. # of
tornadoes/year deaths/year
tornadoes/10,000 miles
(A)
(B)
Maryland
3
0.07
3.05
Massachusetts
3
0
3.83
Michigan
18
3
3.16
Minnesota
19
2
2.39
Mississippi
26
10
5.51
Missouri27
2
Montana
5
0
0.34
Nebraska
36
0.7
4.70
Nevada 1
0
New Hampshire 2
3.92
0.09
0
2.22
State
Avg. # of
Avg. # of
Avg. # of
tornadoes/year deaths/year
tornadoes/10,000 miles
(A)
(B)
New Jersey
3
0
4.02
New Mexico
9
0
0.74
New York
5
0
1.06
North Carolina
14
2
2.87
North Dakota
20
0
2.39
Ohio
5
3.90
Oregon 1
0
0.10
Pennsylvania
10
2
2.23
Rhode Island
0.23
0
2.22
South Carolina
10
1
3.31
16
State
Avg. # of
Avg. # of
Avg. # of
tornadoes/year deaths/year
tornadoes/10,000 miles
(A)
(B)
South Dakota
28
0
3.69
Tennessee
12
3
2.91
Texas
137
8
5.23
Utah
2
0
0.24
Vermont 1
0
1.08
Virginia 6
0
1.51
Washington
1
0
0.15
West Virginia
10
1
3.31
Wisconsin
21
1
3.86
Wyoming
11
0
1.13
Assuming normal distribution, the probabilities P (A)
and P (B) can be expressed as (Ang, 2007),
P (A) = Ф [ (Aactual– A-)/ σA)
P (B) = Ф [ (Bactual – B-)/ σB)
(4)
(5)
For the data in Table 2, this data is given as,
A- = 73.77, σA = 7,377, Aactual = 82
B- = 141.67,σB = 21.25, Bactual = 180
Using the above information, P (A) = P (Aactual>= 82)
= 1.0 - Ф [ (82.0 – 73.77/7.377)]
= 1.0 -0.8643
= 0.134
Similarly, for the event B,
P (B) = P (Bactual>= 180) = 1.0 - Ф [ (180.0 – 141.67/21.25]
= 1.0 - Ф (1.80) = 1.0 – 0.9640
= 0.036
Hence, 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 (β) can be
obtained from the following equation
(Ellingwood et al., 1980),
Pf = Ф (- β)
(6)
Hence, β = - Ф (1-0.165)
Hence, β = - Ф (0.835)
From the standard normal distribution tables
Hence, β = - (-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 (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.
The statistics is given as follows:
Fy = 38 ksi, VFy = 0.10
Z- = 54 ksi, VZ = 0.05.
V represents the coefficient of the random variable.
Eq.2 is used to calculate the reliability index β.
This works out to be 5.14.
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= 141.67, σS = 21.25.
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).
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*)])
Xi-N
=X*
i
- φ-1 [ Fi (xi*)] σiN(9)
(8)
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)
A variation of this procedure has been suggested
by Grimaldi et al. (2010). In this method, Risk,
R, is expressed as,
R= HVD
(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
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, 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.
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.