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Probabilistic Evaluation of Tornado Missile Hazard to Containment Isolation Valve Compartment Equipment,South Texas Project
	ML20077R898
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Site:	South Texas
Issue date:	08/31/1983
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ML20077R884	List: ML20077R891; ML20077R898; ST-HL-AE-1003, Forwards Info Re Comparison of Alternatives for Isolation Valve Cubicle Roof Design & Probabilistic Evaluations of hurricane-generated & Tornado Missile Hazard to Containment Isolation Valve Compartment Equipment;
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ATTACHMENT 3

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ST-HL-AE-1003 s

I l

l South Texas Project Probabilistic Evaluation of Tornado Missile Hazard to the Containment Isolation Valve Compartment Equipment 14926-001

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Risk / Reliability Group Los Angeles Power Division Bechtel Power Corporation August 1983 r-m u

w 9309210050 830913 PDR ADOCK 05000498 PDR 4

TABLE OF CONTENTS Page Section I

Introduction 1

9 II Acceptance Criteria 1

III Summary 1

IV Analysis Approach 2

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V Assumption and Conservatisms 3

VI Results and Conclusions 4

w VII References 5

Appendix A Conditional Probability of Hitting a Target A-1 A.1 The Equation of Motion of a Tornado Missile A-1 A.2 Tornado Missile Motion As a Markovian Process A-6 A.3 Green's Function of the Tornado Missile in the Phase Space A-8 r

A.4 Properties of the Averaged Green's Function A-10 A.5 Equations for the Green's Function A-12 A.6 Hitting Function A-14 i

A.7 Height Distribution of Airborne Missiles A-17 A.8 Conditional Probability of Hitting the Hori-zontal Target Given a Tornado Strike to the Nuclear Power Plant A-23 References A-24 Appendix B Adjustment for Reporting Efficiency B-1 P

References B-5 Appendix C The Probability of Injection of Potential Tornado

. Missile C-1

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C-9

References O

i

1 Section Page Appendix D General Methods D-1 D.1 Preamble D-1 D.2 Tornado Characteristics D-3 D.3 Tornado Missile Description D-4 D.4 Probability of Damage Given Tornado Frequency v, Path area a, Fujita Scale F, Density of Poten-tial Missiles n, Injection Probability q(F) and Height DistFibution +(z,F)

D-8 D.5 Distribution of Random Parameters D-10 D.6 Annual Frequency V of Tornado Occurrence D-11 D.7 Distribution f(a) for Tornado Path Area D-13

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D.8 Joint Distribution of Tornado Path Area a and Fujita Scale F D-14 D.9 Surface Density of Potential Missiles n D-15

~*

p D.10 Probability of Injection q(F)

D-16

~

D.11 Height Distribution of Airborne Missiles $(z,F)

D-17

l'

'[

D.12 Conditional Probability of Damage Given a Hit D-17 D.13 Distribution of Damage Probability P D-17 T

D.14 Point Estimate of Median Value of Damage Probability P D-18 T

References D-20 LIST OF TABLES l-l Table Page I

Probability of Tornado Occurrence at Plant Site 6

II Distribution of Potential Missiles 7

I III

, Probability of Damage to IVC from Tornado-Generated 8

- Missiles Per Year B.1 Reported Number of Tornadoes N and Population Density in the U.S.A.

B-6 l

l I

ii l

Table Page C.1 Probability of Injection q(0)

C-10 C.2 Probability of Injection q(1)

C-11 C.3 Probability of Injection q(2)

C-12 4.

C.4 Probability of Injection q(3)

C-13 C.5 Probability of Injection q(4)

C-14 a.

C.6 Probability of Injection q(5)

C-15 0

C.7 Probability of Injection q(6)

C-16 P

C.8 Sensitivity of q to the Number of Trials C-17 C.9 Maximum and Minimum Values for q(F)

C-18 C.10 Means for q(F)

C-19 C.11 Lognormal Distribution for q(F)

C-20 Lower and Upper Limits and Means for q(")(F)

C-21 C.12 Lognormal Distribution for q(*)(F)

C-21 C.13

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C.14 Lower and Upper Limits and Means for q(F) for Restrained Potential Missiles C-22 y

C.15 Lognormal Distribution for q(F) for Restrained Potential Missiles C-22 D.1 Classification of Tornadoes According to Path Area a D-21 D.2 Relationship Between Fujita Scale F and Damaging Wind Speed w (mph)

D-22 D.3 Specification of Tornado-Generated Missiles D-23 D.4 Acceleration Parameters of Potential Missiles D-24 w

D.5 List of Counties Near the Plant Site of STP D-25 D.6 Annual Number v f Tornado Occurrences for Six 6

Counties D-26 D.7 Parameters of Lognormal Distribution for Annual

Frequency of Tornado Occurrences in Six Counties D-27 D.8 Comparison of Empirical and Fitted Lognormal Distributions for v D-38 6

iii s-

Table Page D.9 Parameters of Distribution for the Adjusted Annual Frequency v D-29 D.10 Distribution of Tornado Path Area D-30 D.11 Best Parameters for Lognormal Distribution of Tornado Path Area for Texas D-31 D.12 Comparison of Empirical and Lognormal Distribution of Tornado Path Area for Texas D-32 D.13 Joint Number of Tornadoes (A-Scale - F-Scale)

D-33 D.14 Joint Probability Distribution for A and F Scales D-34 D.15 Density of Potential Missiles D-35 D.16 Probability of Injection q(F) for Surface Potential Missiles D-36 D.17 Probability of Injection q(F) for Elevated Poten-tial Missiles D-37 D.18 Probability of Injection q(F) for Restrained Elevated Potential Missiles D-38 D.19 Median Value for Height Distribution of Airborne Missiles (Z = 55 ft, b = 20 ft)

D-39 0

D.20 Probability of Damage to IVC from Tornado-Generated D-40 Missiles per Year D.21 Point Estimate of Damage Probability D-41 LIST OF FIGURES Figure Page A-1 Aerodynamic Forces Acting Upon a Tornado Missile A-2 A-2 Geometrical Parameters of a Tornado Missile A-3 A-3 Illustration to Formula (A-49)

A-14 B-1

. Annual Reported Number of Tornadoes for the U.S.A.

B-7 B-8 B-2

Population Density in the U.S.A.

B-9 B-3 Reporting Efficiency Curve iv p

_..7

l c.

Figure Page B-4 Annual Adjusted Number of Tornadoes for the U.S.A.

B-10 t

D-1 Typical Distribution Function D-1 D-2 Cylindrical Missile D-5 D-3 Missile Orientation D-6 i

Se a

LL e

M l

ee e4 h

4 L

V

SOUTH TEXAS PROJECT TORNADO MISSILE EVALUATION REPORT I.

Introduction This study uses the Probabilistic Risk Assessment (PRA) methodology to evaluate the probability of damage to equipment located in the containment isolation valve cubicle (IVC) from tornado generated missiles. Tornado-generated missiles include objects, on or near the plant site, that could become airborne during a tornado and be transported to the top of the IVC.

The study includes an evaluation of the likelihood of a tornado occur-rence, as well as the probability of tornado generated missiles entering through the top of the IVC. The extent of damage is not evaluated, but is conservatively assumed to be certain and total for all missile strikes.

II.

Acceptance Criteria The NRC's acceptance criteria are contained in the " General Design Criteria (GDC) for Nuclear Plants" [1]. Specifically, GDC 2 and 4 apply to this evaluation and are summarized below:

GDC 2 requires that " Structures, systems, and components important a.

to safety shall be designed to withstand the effects of natural phenomena such as - tornadoes - without loss of capability to per-form their safety functions..."

b.

GDC 4 requires that "... structures, systems, and components shall be appropriately protected against dynamic effects, includ-y ing the effects of missiles,... from events and conditions out-side the nuclear power unit."

I The Standard Review Plan (SRP) [2] Section 3.5.1.4 and NRC Regulatory Guide 1.76 [3] provide further guidance in meeting GDC 2 and 4 require-a ments. Specifically, SRP Section 3.5.1.4 refers to the acceptance criteria of SRP 2.2.3 which states "... design basis events include

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each postulated type of accident for which the expected rate of occur-isestimatedtoexceedtheNRCst'affobjectiveofapproximately10}ines rence of potential exposures in excess of the 10 CFR Part 100 guide per year... expected rate of occurrence of potentigl exposures in excess of the 10 CFR 100 guidelines of approximately 10 per year is acceptable if, when combined with reasonable qualitative arguments, the realistic probability can be shown to be lower..."

III.

Summary l

The probability of failure of the equipment located in the IVC to perform its safety function in the event of a torcado is evaluated using the PRA meth'dology. The study quantifies the probability of tornado generated o

l

\\

t missiles entering through the top of the IVC. Since there is consider-e able uncertainty in all factors, probability distributions are propagated throughout the analysis. The results are compared to the NRC acceptance criteria. The results indicate that tornado generated missiles are not a significant threat to the IVC equipment. The results further indicate that no physical barriers are required at the top of the IVC.

IV. Analysis Approach The probability of damage to equipment located in the IVC depends on three factors:

A.

The tornado occurrence rate at the plant site, B.

The conditional probability of one or more tornado generated missiles striking the top of any one of four IVCs, given the tornado occurrence, and 1

C.

The conditional probability of IVC equipment being damaged, given that the tornado-generated missile or missiles have entered the IVC.

r The tornado occurrence rate is based on the National Weather Service record of tornado strikes for the site region between 1953 and 1982

[4]. The probability of tornado occurrence at the plant site and its contributors are shown in Table I.

The distributions of annual frequency, 2

v, of tornado occurrences in area, S = 10000 mi, and tornado path area, a, are based on plant specific historical data. These data are in good agreement with nationwide and regional assessments.

The conditional probability of the missile strike, given the tornado g

occurrence, depends on the following subfactors:

A.

The number of potential missiles, B.

The conditional probability of the potential missile becoming airborne (or injected), given the occurrence of a tornado, n

C.

The conditional probability of missiles being transported from their origin to the target, given that they become airborne, and D.

The target area.

u The number of potential missiles is based on data from Electric Power Research Institute (EPRI) surveys at seven nuclear power plant sites [5].

[

The probability of the potential missiles becoming airborne is calculated L

using a missile model developed at Jet Propulsion Laboratory (JPL) [6].

The conditional probability of missiles being transported from their origin to the target is based on a statistical mechanics model [7], [8].

This approach develops a modified Green's function to quantify the probebility of the airborne missile striking a unit target area at some distance and elevation from its origin. This probability is then 2

I I

la*

b multiplied by the area of the target to get the total probability of strike. The area of the top of each of the four IVCs is 745 square feet (total target area is 2980 square feet). The IVC height is 55 feet above the grade and the grade elevation does not vary significantly within 300 feet of the IVC. The number of potential missiles and the missile density incorporated into this study are shown in Table II.

The conditional probability of IVC equipment being damaged, given that the tornado-generated missile or missiles have entered the IVC, is con-servatively taken to be certain and total. That is, the conditional probability is taken as unity.

Each of the above factors has a considerable amount of uncertainty associated with it.

For some factors, the uncertainty is associated with the statistical nature of the data, and in others, it is associated with the modeling techniques. For this reason, a probability distribu-tion is used for each factor. These uncertainties are propagated throughout the analysis. Therefore, the final results are not a single value for probability of damage, but a distribution of values. The median (50th) and 95th percentile values are reported in Table III.

The median is then compared to the NRC acceptance criteria.

The NRC acceptance criteria values require _9areful egnsideration

r.

because they are given as point values (10 and 10 per year).

]!

However, as mentioned above, there are significant uncertainties associated with the probability of damage, ranging over orders of

(,

magnitude. The median is often compared to the acceptance criteria

't.

value because the median is interpreted as the "best estimate" or

" recommended" value [9).

t V.

Assumptions and Conservatisms The following assumptions are used in this study:

A.

The IVC roof area is assumed to be transparent to tornado missiles.

That is, the top of the IVC is assumed to be open and without missile protection of any kind.

B.

A tornado missile strike in the open top of any one IVC compart-ment represents failure (see conservatisms A and D, below).

C.

The distribution of potential tornado missiles by number and length are based on an EPRI survey of seven nuclear plants [5].

Conservatisms incorporated in this study are:

A.

The comparison of the strike probability to the activity release

frequency acceptance criteria, assumes;

1.

Missile inflicted damage is certain sad total and 2.

Damage leads directly to activity releases in excess of 10CFR100.

3

B.

The potential missile model assumes 1.

A missile distribution based on EPRI survey maximum, 2.

A missile density increased by factor of 2.5 over EPRI

survey, l.

3.

One half of missiles are distributed up to 20 feet above grade, remainder at grade, and 4.

The number of unrestrained missiles postulated for this study is equal to the total number of missiles (restrained and unrestrained) in the EPRI survey.

C.

The tornado frequency is based on a 30-year historical record fitted with a more conservative lognormal distribution having a 1arger mean and spread than the empirical distribution.

1 D.

Geometric factors that result in further conservatisms are:

9 1.

Sheltering by other structures is neglected, 2.

A missile strike in any IVC opening results in failure (i.e, no credit is taken for the existence of redundant components or for separation between safety-related trains), and 3.

Safety related target areas are less than IVC open area.

VI.

Results and Conclusions l

The results of the analysis is a probability distribution for tornado missile damage to the IVC equipment. The median (50th percentile) and The upperbound(95thperceygile)valuesarereportedinTabgeIII.

Themedianor"bestestimate"valueof2x10{g6x10per year is very small median value is 2 x 10 and the upper bound, per year.

.I 7

~6 2

compared to the NRC acceptance criteria value of 10 to 10 per year.

The above results indicate that tornado generated missiles are not a significant threat to the IVC equipment. These results further indicate

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that no physical barriers are required at the IVC top opening to protect IVC equipment from potential tornado generated missiles.

m w

e 4

VII.

REFERENCES

[1]

10 CFR Part 50, Appendix A, " Design Basis for Protection Against Natural Phenomena."

[2] Standard Review Plan, U.S. Nuclear Regulatory Commission, NUREG-75087.

[3] Nuclear Regulatory Commission, " Design Basis Tornado for Nuclear Power Plants," Regulatory Guide 1.76, April 1974.

[4]

"U.S. Tornado Breakdown by Counties 1953-1982," U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Weather Service National Severe Storms Forecast Center, Room 1728, Federal Building, 601 E. 12th Street, Kansas City, Missouri 64106.

[5] Twisdale, L. A., et al., " Tornado Missile Risk Analysis," EPRI NP-768, May 1978, EPRI NP-769, May 1978.

[6] Redmann, G. M., et al., " Wind Field and Trajectory Models for Tornado-Propelled Objects," EPRI 308, Technical Report 1, February 1976.

[7] Goodman, J. and Koch., J.

E., " Conditional Probability of the Tornado Missile Impact Given a Tornado Occurrence," Proceedings of w

the International ANS/ ENS Topical Meeting on Probabilistic Risk Assessment, Port Chester, New York, September 20-24, 1981,

~

pp. 419-424.

[8] Goodman, J. and Koch, J. E., "The Probability of a Tornado Missile m'

Hitting a Target," Nuclear Engineering and Design 74, (1983).

[9] Apostolakis, G., et al., " Data Specialization for Plant Specific

.I Risk Studies," Nuclear Engineering and Design 56 (1980), pp. 321-329.

I w

h l.

\\

e 5

l_

Table I Probability of Tornado Occurrence at Plant Site Median Upper Limit (50th Percentile)

(95th Percentile)

Annual Frequency v of Tornado Occurrence 2

in Area S = 10,000 mi 5.25 37.39 Tornado Path Area

-2

-1 (sq. mi.), a 2.22 x 10 1.39 x 10 Probability of Tornado Occurrence at Plant Site,

-5

{L per Year, P, 1.17 x 10 1.71 x 10 '

~

r 4

r9

)'.

a f

[i m

'1 m.

>=

4 W

L b

e 6

Table II Distribution of Potential Missiles Median Upper Limit (50th Percentile)

(95th Percentile)

Number of Potential Missiles on Site and Vicinity 7 2

(2.5 x 10 f t ), N 6,000 6,196 P

Average Surface Density of Potential

-4

-4 Missiles on Site (ft-2) 2.40 x 10 2.48 x 10 Local Surface Density of Potential Missiles NeartheIycCompart-

_4 4

ments (ft

), n 2.40 x 10 6.13 x 10 P

Effective Number of Potential Missiles on Site and Vicinity, N,7 6,000 15,325 he be L

7

l'

l 1.

Table III Probability of Damage to IVC from Tornado-Generated Missiles per Year Median Upper Limit (50th Percentile)

(95th Percentile)

-10

-0 2 x 10 6 x 10 b

e ce 4

e 04 p

6 6

6 I

l l

54 l

j I

i 8

t APPENDIX A CONDITIONAL PROBABILITY OF HITTING A TARGET I

A.1 The Equation of Motion of a Tornado Missile The Newton equation of motion of the center-of-mass of a tornado missile is:

=f*L+ S*

m D

g where:

m = mass of missile, y = velocity vector of missile, v = acceleration vector of missile, 4

3D = aer dynamic drag force, F

n f = aerodynamic lift force, g

f = aerodynamic side force, 3

[ = gravity force.

.[i The expression for the gravity force is:

'L f = - mg h (A.2) where g is the gravitational acceleration r:onstant and k is a unit vector oriented upward. The drag force E acts in the direction of D

4 the relative wind-missile velocity vector u (see Figure A-1).

The a

velocity ($) can be expressed as the vector difference of wind veloc-ity (w) and missile velocity (v) as follows:

$=w-v (A.3)

The absolute value of the vector E is denoted as u.

TheliftforceEg is perpendicular to I and in the plane formed by the vector $ and the l

D t.

4 unit vector p,along the main missile axis. The side force is per-3 pendicslar to the vectors f ""

D L'

A-1

oF t a

$\\

a 7

D u

I a

F r.

g Figure A-1.

Aerodynamic Forces Acting Upon a Tornado Missile f

The empirical expressions for these aerodynamic forces are:

84 PA a

(A.4)

[D*

uu D 2 t.

P,A y

ux xu (A.5) it=C77

-p0 P,A g

(A.6)

I=C

[0 * ".

s S 2 L

where:

p, = air density r

A = missile cross-sectional area (see Figure A-2)

CD = aer dynamic drag coefficient C7 = aerodynamic lift coefficient C3 = aerodynamic side coefficient L

La

~

m A-2

I d:

/m/

L

/

t..

'M Figure A-2.

Geometrical Parameters of a Tornado Missile depend on cer-The empirical aerodynamic coefficients C '

L, and C D

g tain aerodynamic parameters and the angle of attack Ia) (see Fig-ure A-1).

For cylindrical missiles, they are m/A and 1/d, where A, 2, and d are shown in Figure A-2.

Expressions for C C

and C for severalbodyshapesaregivenbyRedmannetal.[k,1]g,Forsomesym-3 metrical bodies (for example, cylinders), the coefficient CS = 0.

The missile trajectory depends on six external parameters:

the yind speed (w), the angle of attack (a), the angle (p) bgtween w and v, two relatige to angles 0 and $ which give the orientation of force Fn

,I laboratory system of coordinates, and the angle ($) Between Ft and the vertical axis (0Z). These parameters are very convenient for a tornado missile problem but the standard mechanical approach uses other param-eters: three quasi-external parameters (Euler angles, which can be determined by using the anEular momentum equations) and tgree truly external parameters (the components of the wind velocity w).

The wind velocity vector field (w(x, y, z, t)) consists of a regular part which depends on tornado parameters (Fujita scale, length and width of tornado, translational velocity vector, etc.) and an irregu-lar or stochastic part which is due to turbulent fluctuations.

m In probabilistic risk analysis, the distribution of the possible tor-ll gado parameters is considered. Therefore, the entire vector field

'L w(x, y, z, t) is a random vector function. Hence, equation (A.1) can be rewritten in the form:

= $ - og k (A.7)

L a

is a random force, equal to I + L +

where D

S*

A-3

For cylindrical missiles, equation (A.7) can be written in the form:

mE = f sin 6 cos $ + ft (cos O cos $ cos $ - sin $ sin $)

(A.8) p my = f sin 6 sin $ + ft (cos 8 sin $ cos $ + cos $ sin $)

(A.9)

D mE=f cos 0 - f sin 8 cos $ - mg (A.10) p g

where:

2 p, Au

(^'

}

fD* D 2

2

~

p* Au (A.12)

L fg=Cg 2

The angles 6 and $ determine the orientation of the drag force f in n

the spherical system of coordinatgs relative to the earth's sur ace.

and the plane containing the ver-The angle between the lift forse Ft tical axis and the drag force F is den ted as $.

D Air density is denoted as p* and missile cross-section area as A.

The e

aerodynamic coefficients C and C for a cylindrical missile are con-g sidered in Redmann [A.1]. pFor the standard missile, the approximate expression is:

3 CD = 0.98 sin a (A.13) 2 (A.14)

Cg = 0.98 sin a cos a I

where the attack angle a has the range:

O _< a _< n (A.15) t The system of equations (A.8) through (A.10) can be rewritten in the form:

E=a (A.16) x (A.17)

=a y

y (A.18)

I=a

-8 w

z or, in the vector notation, as I=$

gI (A.19) where a is a random vector of acceleration with components:

a,=aj[ sin a sin 0 cos $ + sin 2,c,,

y 3

(cos O cos $ cos $ - sin $ sin $)]

(A.20)

A-4 (u

3 2

= a, [ sin a sin 0 sin $ + sin a cos a a

(cos e sin $ cos $ + cos $ sin $)]

(A.21) 3 2

a,= a, [ sin a cos 0 - sin a cos a sin 6 cos $]

(A.22) and a,= 0.003727 (w2,y2 - 2 w v cos p)

(A.23) 4 4

where $ is an angle between wind velocity w and missile velocity v.

Speeds w and y are measured in metric units.

4 The random vector a depends on six random parameters:

w, S, o, 0, $,

and $.

These parameters are functions of otger independent random parameters:

three components of wind field w and three Euler's angles determining the orientation of the missile in the space. Even random-ness of some independeng random parameter (for example, randomness only of the wind field w in the case of nontumbling missiles) can create the randomness of all parr. meters, w, a, p, 0, $ and $.

The number of independent random parameters determines the number of degrees of freedom of randomness. The maximum number of degrees of freedom is six.

For a large number of degrees of freedom of randomness, the distribution of components of acceleration a, a, and a is practically normal

(-

according to the central limit fheofem. Th5 seans of random accelerations are zeros:

L i

a

=0 (A.24)

~

x a

=0 (A.25) y y

L a

=0 (A.26)

~

z r

Le 4

1.

l 1,.

A-5

4 A.2 Tornado Missile Motion As a Markovian Process Every solution to the Newton equation with a random force constituent represents a Markovian process [A.2].

An example is a Brownian movement.

+

To show this, let us discretize the random acceleration vector a.

We assume that a random vector a is constant during the small interval of time T.

At the end of every time period T, this vector is assumed to r

change suddenly and randomly.

Consider gome tornado missile trajectory with an initial point r, a velocity v, a final point r, and a velocity v.

Dividing the miEsile transportafion time T = t - t by n equal intervals of duration i I

each, we can calculate all infermediate points and velocities of the missile trajectory:

w

+ 2 a1

+

3 3 = r, + v, I + 2 r

b-3 = v, + a3 I

v n

2'2

+

2*#1 + "1 I+

r 2

(A.27) 2 * "I * '2 I v

y

+ 2

.I

iT e

+

g = r _3 + v,3 T+

w r

g g

2 F

4

+

4

= v,3 + ag I

g v

g l

+ 2 w

aI l

4 4

n l

r, = rn-1 * "n-1 I*

2 v,= v,_3 + a, t where l

r

=r L.

(A.28) l v, = v A-6

l and a is a random acceleration of the missile during time period t:

g (A.29) t7<t<tg i

where (A.30)

t. = t + i*t 1

o It follows from equation (A.27) that the location r and velocity g

and g

g depend on r 3,

g_3 v of the tornado missile at the moment t g

It is clear from equations (A.8) through (A.10) that a probabil-a.

g ity distribution function for a random vector a could depend only on f

.+

v3 Therefore, the probability density that a missile at moment tg 3

-s

-e

-s will have location r and velocity v depends on r,g_3 and v _3 only:

g g

g r d3 +v 3+

+

g (+r, v

+

dP (+r _3, v _3, r, v ) =

r

_3, v _3) d g

g f

g g

g f

E' (A.31)

I where:

s

-+

+

dr (r 3, v 3, r, v ) = probability that a tornado missile having

~

i g

g g

g at moment t _3 the location r and velocity v _3 will have at moment g

g g

t the location in the range (r, rg+di)andvelocityintherange g

g (v, vg + dv );

g g

+

fg (r, vg g,vg ) = conditional probability density corresponding r

g g, r, v );

to the probability dP (r 3, v g

g g

r d r = dx dy dz g

3 g

g e

= dv (i) dv (g) dv,(g) i x

y 4

4 We see that tornado missile motion is a Markov chain (r, v,; r, v ;

o 3

3

-+n' "n) because the probability (A.31) that a r ' "2I *** #i' "iI *** #

2 l

tornado missile will occur in the i,th state described by r and v g

g given being in the (i-1) state before depends only on the preceded (1-1) state and does not depend on all other previous steps.

If we approach I + 0 and n + ca then our Markov chain will tend to the continuous Markov process.

A-7 l

w A.3 Green's Function of the Tornado Missile in the Phase Space A continuum of tornado missile locations forms the real space (R-space). A continuum of tornado missile velocities forms the veloc-ity space (V-space). A continuum of locations and velocities together forms the phase (or configuration) space (P-space).

A radius-vector o{ missile location r is a point in the R-space. A missile velocity y is a point in,the V-space. A combination of missile location r and velocity v is a point in the P-space.

s-

+

Sequential missile locations r(t) and velocities v(t) putting in the time-ascending order form a phase trajectory or phase path.

In our model with discrete random accelerations, every phase trajec-

+

n tory can be represented by the set of phase points:

r, v,; r, v ;

o 3

3

.L

+

+ +

... r, v,... r,,3, v,_3; r, v.

Trajectories with the same initial E.

g+

+ +

point r,, v, and final point r, y can be distinguished by the sets F

(,

of intermediate points:

+

f f (+r ' "l i#2' "2I **** #i' "iI *** #n-l' "n-1)

(A.32) l

+

+ +

The probability density f(+r,, v,; r, v; f) that a tornado missile follows the phase trajectory T can be estimated as:

. g-

'I*

n

+

+ +

+

p f(+r,,v,;r,v;F)=][f(+r,vg ; r _3, v _3)

(A.33) g g

g g

!L i=1 1

where condition (A.28) is assumed.

Thetotalprobabilitydensity,G({,,v;r,v)thatatornadomissile g

,v wifl reach the final phase point

{rog the initial phase point r,l of the expregsiog (A.33} oyer all r, v is the G(n-1)-fold integra possible phase trajectories T between points r, v, and r, v:

o l

G(?,,0,;i,0)=}ff(?,,0,;?,0;r)dr (A.34) r 1

where 3

3 3

d; (A.35) d 7... d dr = d r dv dr d "2... d rg g

,_3 n-1 3

3 2

+

+ +

The quantity G(+r,, v,; r, v) is the so-called propagation or Green's function.

I l

A-8

m w

w r

w, w

r n

In the previous section, we assumed that a propagation time T = t - t i.

o is fixed.

If we drop this limitation (for a discrete model, it means that the number n of steps can vary) then the Green's function will have

'r a more general form:

G (t, r, v,; t, r, v; F, a, y)

(A'. 36)

+

4 4

+

o o

We explicitly show all parameters that Green's function depends on (F is a Fujita scale, a is a tornado path area, and y is a collective index which stands for all other parameters).

The Green's function (A.36) has the following properties:

~

+

+ +

1.

G (t,, r,, v,; t, r, v; F, a, y) E O,

or z<0 (A.37) if t<t,

+

+ +

r..

2.

G (t,, r,, v,; t, r, v; F, a, y) + 0 lv-v,l+0 (A.38) l r - r,l + 0 if or Property I reflects the principle of causality (the missile can hit a target only after it is airborne) and boundary condition (no missiles below the ground level z = 0).

a Property 2 means that there is some finite radius of missile transpor-

~

tation and maximum missile speed beyond which the probability of 1

missile occurrence is infinitesimally small.

8

o I

k I'

i L

e e

l P

5 A-9 l

A.4 Properties of the Averaged Green's Function For a given F and a, we have a number of other parameters beyond our e

control. These are the direction of tornado movement, translational velocity, ratio of tornado width to tornado length, number of vor-tices, exact location of trajectories of the centers of every vortex

[

relative to the target, detailed distribution of wind field, etc.

All sets of these parameters are denoted as y.

Because it is impossible to develop the detailed distribution for all sets of parameters y, we average our probability over all y.

Introduce the averaged Green's function G (t,, r, v,; t, r, v; F, a):

o

+

+ +

.i G (t,, r,, v,; t, r, v; F, a) =

P 1

+

+ +

j7 G (t, r,, v,; t, r, v; F, a, y)

(A.39) o e

Y where N is a total number of different sets of parameters y.

The averaged Green's function retains all properties of the ordinary

.l.

Green's function discussed in the previous sections. It also contains L.

some new properties.

These properties are:

(1) Time uniformity (2) Uniformity in the plane x0y (3) Axial symmetry around axis Oz 1

(4) Uniformity in V-space Time uniformity means that the probability distribution for tornado missile propagation does not depend on initial moment of injection.

It depends only on transportation time t - t o

I Uniformity in the plane x0y means that the probability distribution for tornado missile propagation does not depend on the location of the potential missile at the plane x0y. Therefore, the Green's function depends on differences x - x, and y - y,.

Axial symmetry means that there is no preferable direction for tor-

~

nado missile distribution in the plane x0y. Thus the Green's function depends on (x - x,)2, [y, y )2 rather than x - x,and y - y, 9

separately.

Uniformity in V-spa.ce means that the probability distribytion for tornado missile propagation does not depend on initial velocity v, and depends only on the difference, v - v,.

A-10 m

l' Therefore, the averaged Green's function has the structure:

+ +

G (t, r, v,; t, r, v; F, a) s o

o (A.40) 0 (*o ; t - t,,

(x - x,)2 + (y. y )2,;,;

y,

m M

L l

I T

iL E

[

'[

L 1

.. n t

i 1

A.5 Equations for the Green's Function l

The Green's function for a Markovian process satisfies the Chapman-Kolmogorov-Saoluchovski equation [A.2]:

+

.+

G(t, r, v,; t, r, v; F, a) =

o o

I f

+

4 4

+

4 4 +

G(t,, r,, v,;

t',

r', v'; F, a) G(t',

r', v'; t, r, v; F, a)

V' v'

d +r3'dv3+'

(A.41) r-l-

-, -+

where G(t,, r, v,; t, r, v; F, a) is an averaged Green's function o

(A.40).

s The Markovian process is a diffusion Markovian process if

O G(t,, r,, v,;

t,, r, v; F, a) E 6 (r - I,) 6 (v - v,)

(A.42) 4 and there exist continuous and differentiable functions a.

(t, r, v; f.

+1 +

+ +

in F, a), bg (t, r, v; F, a), cik (t, r, v; F, a), dik (t, r, v; F, a) and fik (t, r, v; F, a) satisfying the equations:

L.

+ +

g (t, r, v; F, a) =

a s

3 3+,

(xj-x)G(t,I,v;t',7',v';F,a)dr'd lim g

t' +t t' - t 1

J J

v'4' (A.43) g(t,Y,v;F,a)=

b I

3 3

lim (v! - v.) G(t, r, v; t',

r', v'; F, a) d r' d v' L

t ' -+ t t' - t JJ v' v' (A.44)

.+ -+

ik (t, r, v; F, a) =

c

+ +

+

1 (xj-x)(x{-x)G(t,r,v;t',r',v';F,a) lia j

k t' +t t' - t

['

y, h, 3

3 (A.45) d r' d v'c A-12

. _. ~ _

F 1

1 i

t'

+ +

i.,

dik (t, r, v; F, a) =

r r

1 J (x1 - x ) (vi - v ) G(t, 7, t',

r', v'; F, a) 1p+t t, - t J i

g t

V' v

3 3

d ;,

(A.46) d 7,

., +

fik (t, r, v; F, a) =

I(v{-v)(v{-y)G(t,r,v;t',r',v';F,a)

I 1

+ +

+

lim g

k t' t t' - t V'

v 3

3d+,

(A.47) d r' where i,k = 1, 2, 3 and x, x ' *3 stand for x, y, z and v, v ' '3 I'

~

3 2

v,v,v.

7, The Green's functicn of a diffusion Markovian process satisfies the Fokker-Planck equation:

-,.+

8 G(t,, r, v,; t, r, v; F, a) o 8t

]

3 P

ea

+ -.

a (t, r, v; F, a) G(t,, r, v,; t, r, v; F, a) f gx, g

o j

i=1 3

+ ->

a b (t, r, v; F, a) G(t,, r, v,; t, r, v; F, a)

~

g o

gy,*

i=1 l-3 3

2 1

y a

+ +

+

+ +

2 f,,, 8x. 8x ik( ' "' 'I

' *)

( o' +#o' 'oi

' 'I

~

c i=1 k=1 3

3 2

a ik(t, r, v; F, a) G(t, r, v,, t, r, 9 F,a) d o

o ax. av i=1 k=1 3

3:-

ik(t, 7,

F, a) G(t,, Y,,,; t, 7,

F, a) 11av$av,

=0 f

i=1 k=1 (A.48)

A-13

Conditions (A.43) through (A.47) mean that the radius-vector of the missile r(t) and the velocity v(t) change more or less smoothly without jumps and discontinuities in the P-space. These conditions are satis-fied for tornado :nissile motion because of inertia of movement.

A.6 Hitting Function Let us find the probability that a missile injecting at point r, with

+

velocity v, at moment t, will hit the unit area of a target at moment *

+

t near point r with velocity v.

Consider the small volume 4

dV = dA -

v6 dt (A.49) shown in Figure A-3 Tgis is a skew cylinder with base dA, slant height vdt, and altitude l 4v 0 l dt.

r-All missiles with center-of gass inside volume dV given by equation (A.49) and velocity vector (v) will hit the area dA during time dt.

Thus, the probability of hitting area dA during time dt given velocity

-+

v is:

G(to, r, v ; t, r, v) - v6

dA

dt (A.50) o o

t pi",,,

I

(

I w n

i p

L,,,,,J l

pi'h.,

1 1

(

+

I'h,,,,/

i 4

n.m v,D dt -

Figure A-3.

Illustration to Formula (A.49)

A-14 1

e Dividingexpression(A.50)bydAdt,6perunittimegivenvelocityvthe probability of hitting the.

unit area oriented in the direction is found:

G(t,, r, v,; t, r, v) - v*6 (A.51) g Let q(F) be the probability of injection of a potential missile, and

-s

-+

p (r,, v,, t,) be the density of potential missiles at moment t, near P

point r, with initial velocity at the moment of injection v,.

Then the probability of hitting the unit area near point 5 oriented in the direction 6, or hitting function h (r, d, F, a), is:

h (r, 6; F, a) =

[2 f3+

3+

+

-+

2 q(F) dt dt d r, d v, p (r,, v,, t,) G(t,, r,, v,;

p "o

(vd<0) 1 1

o t, r, v) l v 6 l d y (A.52) 3 4

The integration over velocity v satisfies the condition:

v 6<0 (A.53)

^

i which selects the missiles intersecting the unit area only from one side.

J Assuming that all potential missiles are constrained and uniformly dis-i tributed in the plan x0y (actually, we need the uniformity in the area of radius 300 ft corresponding to the 95th percentile of missile transportation distribution), we obtain the density of potential missiles in the form:

p (,,,, t,) E n p(z,)

(AW p

p where n is the surface density of potential missiles and p(z ) is a dimensi8nless function reflecting the height distribution of p,otential missiles.

If all missiles are on the ground then:

p(z,)f.6(z,)

(A.55) where 6(z,) is the delta-function.

A-15

r If all missiles are uniformly distributed up to elevation h, then I 1 z, < h, (A.56) p(z,) = q z, > h, (0

The expression for the hitting function h(z, 6; F, a) takes the form:

h(z,6;F,a)=

F F

2 r2

,e 3

3 r)(F) n dt, dt d r, d v, p(z,) G(z,;

t-t,,

p t

t V,

< 0) y y

3 (x - x,)2, (y, 7 )2, v - v,, F, a) - v6 dv (A.57) 9

.s e

0 I

~

4 e

km A-16

r A.7 Height Distribution of Airborne Missiles Presentthehittingfunctionh(z,6;F,a)intheform:

h (z, 6; F, a) = n q(F) $(z, 6; F, a)

(A.57) p where $ (z, 6; F, a) is the so-called height distribution of airborne missiles given by the formula:

~

$ (z, 3; F, a) =

4 f2 F2I

{dv

[*

p(z,) G(z,;

t-t,,

(x-x,)2, (y,y )2 3

y,y, y,,)

dt, dt d r,J J(v.6<o) 9 Jt J t Jv v

1 1

o o

v.6 dv (A.58) x To find the equation for the heigh 3 digtribution, we have to multiply

) and l v. 0l and integrate over t ghe equ3 tion (A.48) by p(z, ties of averaged Green's function,o, t, Gauss'o, r

y and v.

Using the proper t0eorem in the V-space and the condition that Green's function is equal to zero before and after the tornado strike, we obtain:

g [a(z, 6; F, a) $(z, 6; F, a)] =

2 2[

(2,

F,a)$(z,6;F,a)]

(A.59)

~'

az l

where:

}

a(z, 6; F, a) = a (z, F, a)

(A4 3

9(z, 6; F, a) = f c33(z, F, a)

(A.61)

Now we can make some conservative assumptions. Consider the direction 6whichgivesthemaximumvaluefor.$givenalletherparameters.

Assuming this value for all other directions, we make the function $

isotropic and conservative.

If the tornado missile lands before it leaves the tornado wind field, the Green's function will not depend on tornado path area, a.

This is true for all tornadoes except some very small ones. Premature leaving of the tornado wind field only reduces the height and distance of missile transportation. Therefore, the assumption that Green's func-tion ddes not depend en tornado path area a will be conservative.

Hence, the equation (A.59) takes the form:

-h[a(z,F)$(z,F)]=

2[

(, F) $(z, F)]

(A.62)

Bz A-17

If all potential missiles have the same original elevat*on z = const (for example, ground distribution of potential missiles with z = 0) the coefficients a(z, F) and 9 (z, F) do not depend on z accor8ing to their definitions (A.43) and (A.45).

Therefore, the solution to equation (A.62) satisfying the boundary condition is:

~

$(z, F) = e (A.63) where (A.64) a,(F) =

)

This solution is good for cases when all potential missiles are at ground elevation.

c If all potential missiles are uniformly distributed from the ground to

~

the elevation h, we assume that coefficients a(z, F) and 9(z, F) are constant but di?ferent in the areas z < h, and z > h,:

ia (F) z<h y

~

(A.65) a(z, F) =

J L a (F) z>h, 2

9 (F) z<h i

3 9(z, F) = L

~

(A.66)

L 9 (F) z>h, 2

~

The solution to the equation (A.62) for this case is:

rB(z)

O<z<h

{

~

lB(0)

(A.67)

(z, F) =

a (F) a (F)[h,-z]-a (F) h, y

2 3

' B(0) 02(F)

o where

~

-a (F) z

'a (F)

-a (F)b 3

y l

(A.68)

B(z) = e

+

-1 e

2(F) a a (F) 3 (A.69) a (F) = 9 (F) y

a (F) 2 (A.70) a (F) = 9 (F) 2 2

A-18

If h + 0, then a (F) = a (F) E a,(F) and the expression (A.67) coin-2 cideE with (A.63)

According to definitions (A.43) and (A.45), parameters a (F) and a (F) y 2

can be found from the formula:

E a = - lia (A.71) 3 2

AI+0 Y (h )

2 where E and (Az ) are average displacement and squared displacement in the vertical direction for time AT.

The differential equation of tornado missile motion in the vertical direction is:

.n E = a, - g (A.72) t where g is the gravitational constant and i;.

R (A.73) a =

z where m is the missile mass and R is the z-component of the random z

aerodynamic force.

For a small increment of time AT:

z = z, + v OI * (*z ~ 8) (AT)

(A.74) nz 2

Because L

v

= 0, and (A.75) oz (A.76) a = 0, z

due to random distribution of velocity and acceleration directions, the average increment in elevation.is:

E = EIOI)

(A.77) 2 Similarly (Az ) 3, y

- (AT)2 +...

(A.78) 2 2

oz where higher degrees of at are omitted.

A-19

m-

.,ge Putting equations (A.77) and (A.78) into equation (A.71), we obtain:

8 a=

(A.79) 2 v,

2 2

where v,,2 is replced by v because the average v does not depend z

z on time.

In the region 01z$h, (A.80) the number of horizontally injected missiles is dominant because the probability of horizontal injection is much higher.

For horizontally injected missiles, it can be assumed, at the moment of injection, the vertical velocity 9f the missile is equal to zero.

Multiplying equation (A.72) by v = z and integrating from the moment of injection to the moment of stfiking the ground yields the following:

2

~

v

-l- = a v, t + gz, (A.81) z s

where z,= initial elevation of missile, t = flight time of missile, v = vertical velocity at the ground Because o

av

=0 (A.82) zz due to equation (A.75) and equation (A.76) and independence of a and z

vz 2 = 2 gz (A.83)

^

vg o

It is clear that 2=1y 2 (A.84) vz 2 g

' 2 where v is an average squared velocity in a vertical direction for a missile

falling from elevation z

0 A-20 L

Therefore, (A.85)

8*o vz For uniform distribution of initial elevation z, from z = 0 to z, = h, the average for all missiles is:

=fgh, (A.86)

Putting (A.86) into (A.79) yields:

2 (A.87) a3=p o

In this approximation, the parameter a (F) does not depend on Fujita 3

scale F.

~

Now, consider the region z>h, (A.88)

In this region there are only vertically infected missiles. Let w be the damaging wind speed. According to [A.3. and [A.4], the range for the maximum missile velocity is:

fWivaax 5 (A.89) w Because l

U=yv (A.90) 1 2

V aax and h

(A.91) v

=

z The following expression is obtained:

1 2 v'*Cw (A.92) z

Hence, a (F) :: h (A.93) 2 w

G A-21

.t 1

j-where coefficient C is in the range:

24 $ C $ 96 (A.94)

Coefficient C is assumed t'o have lognormal distribution with a median i

value of 48 and an error factor of 2.

If we consider any Fujita scale, F, a corresponding middle value from the intervals given in Table 1, then this velocity w can be found as:

w = 6.30 (F + 2.5) *

(A.95)

_p-putting equation (A.95) into equation (A.93) we obtain:

1 :

C a (F) =

(A.96) 2 (F + 2.5)3 r

where

= 9. 1 C

(A.97)

C 2

6.30 se e9

?

I D

e e

6 A-22 g

i

A.8 Conditional Probability of Hitting the Horizontal Target Given a Tornado Strike to the Nuclear Power Plant Multiplying the expression (A.57) for the conditional probability of hitting the unit area of a target given a tornado strike to the nuclear power plant by the horizontal area of target at the elevation z, we obtain the conditional probability of hitting the horizontally oriented target:

H * "p A q(F) $ (z, F)

(A.98)

~

P In the formula (A.98), we took into account the above-mentioned conservative assumptions yhich eliminated the dependence of height

~

distribution $ (z, F) on 0 and a.

The formula (A.98) can be applied to the IVC roof.

,1 9

9 a

a be e

ig

=

l s

A-23

REFERENCES

[A.1] Redmann, G. H., et al., " Wind Field and Trajectory Model for Tornado-Propelled Objects," EPRI 308, Technical Report 1, February 1976.

[A.2] Papoulis, A., " Probability, Randon Variables, and Stochastic Processes," McGraw-Hill Book Company, New York, 1965.

~

[A.3] Standard Review Plan, U.S. Nuclear Regulatory Commission, NUREG-75/087.

[A.4] Twisdale, L. A., et al., " Tornado Missile Risk Analysis,"

EPRI NP-768, May 1978, EPRI NP-769, May 1978.

e e

W e

w 9

I F

hee e

4 4

A-24

~

APPENDIX B ADJUSTMENT FOR REPORTING EFFICIENCY To determine the reporting efficiency (

), defined as the ratio of that have occurred, the followiEg) to the real umber of tornadoes (N,)

reported number of tornadoes (N expression is used:

e N

C

N

^

RE n

C is meaningful only when the number N is quite large. Thezefore, RE C

can be determined for small area and a lo,ng period of time or a short RE period of time and a large area.

4 d

Because a population bias that depends on time is being analyzed, the number of tornadoes per year must be considered. Therefore, the largest area In this is takEn(or observed number N ) has to be considered.

for estimation of N to be the total number of tornadoes per year in the analysis, N U.S.A.

The" numbers of reported tornadoes per year N and the population density,I?, are shown in Table B-1.

The numbers N are taken from [B.1) and IP from [B.2]. Corresponding graphs are shown [n Figures B-1 and B-2.

1 Actually, the reporting efficiency C depends on many factors but all RE of them are assumed to be statistically correlated with population density.

The time trend of N is assumed to depend on reporting efficiency CRE '

r which depends on population density,IP. Fluctuations of N depend on climatology.

t and Following the work of Abbey and Fujita [B.3), the relationship of CRE population density (f?) takes the form:

}

E

-c (f?+ f?,)

(*)

CRE

I ~

f, 1-whereeandfp;areconstants.

Abbey and Fujita applied the equation (B.2) to the relatively small area (10,000 square miles ) and found a low saturation densityJG',, which is determined from the condition:

C II2) (:)1 (B.3)

RE s

According to [B.3], GP = 1 person per square mile.

So, the formula (B.2) provides no correctio$ to the STP data with original coefficients assumed by Abbey and Fujita.

B-1 i

In this study the nationwide trend is checked. Therefore, the coeffi-cients e and9 have to be found from national data, shown in Table B-1

[B.1,B.2].

Using the least square method, the curve shown in Figure B-1 is fit to the data and is given by the following equation:

c9 N = a - be (B.4) r where:

}I-smoothed number of reported tornadoes

=

a,b, and c = constants population density 9

=

~

Constant (c) from formulas (B.2) and (B.4) are identical and constant, is:

9,=-fin h (B.5) 4 Let index i numerate the year. Index i=1 corresponds to 1950 and index and 9 be the reported number of tornadoes i=30 corresponds to 1979. Lg Ng 3

and population density for i year, from Table B-1.

The coefficients a, b, and c have to be chosen to minimize the expression S:

30

-c 9 g

g

)2 S (a,b.c) =

i a - be

- N.

1 (B.6)

\\

/

i=1

-}

4 The minimum conditions are satisfied when:

8 S (a,b,c) = 0 (B.7)

Da 8 S (a,b,c) = 0 (B.8)

Ob B S (a,b,c) = 0 (B.9) ac e

4 B-2 m

-,-.7 h, _.

Using (B.7), (B.8) and (B.9), the constants are found as follows:

a = N + b E (c)

(B.10) y N E (c) - N (c) y y

b=

(B.ll) 2 2

1 (c)

E (c) - E N D (c) - F(c) y

(.12) b=D2 (c) - Dy (c) E (c) y where:

W = h {30 N

(B.13) g i=1 1

-c 9 E (c) = h {30 1

e (B.14)

A y

i=1 i

-2c 9

.I E (c) = h {30 i

e (B.15) 2 i=1 y

-c 9

I 30 1

e (B.16)

Dy (c) = 30 9i i=1 O

-2c 9

~

30 1

j y

D (C)

M Ee (B.17) i 2

i i=

[.

e l.

B-3

-c9 30 i

N(c)=h Ne (B.18) y i

i

-c9 F(c) = -h {30 i

(B.19)

N e

g i=1 Equating (B.ll) and (B.12), an equation for constant (c) is obtained.

Once constant (c) is evaluated, constant (a) is found using (B.10) and constant (b) found using (B.11) or (B.12).

Using a computer code for least square fit, the coefficients are determined to be:

a = 917.584091 b = 104264.8753 (B.20) c = 0.118188864 i

Hence, (B.21) 9,= -40.04561369 NOTE:

High precision is required to obtain the absolute minimum for expression S(a,b,c), EQN (B-6). Once the constants

[

are determined, fewer significant figures will be used.

The formula (B.4) is applicable if 7

9>

9 (B.22) e o

The approximate saturated density ( @ is:

(B.23) 9, ~ 100 persons /per square mile le

,4 u

I, B-4 l

I

4 The smoothed function NE (SP) is shown in Figure B-1 and the reporting The corrected number of tornadoes (N ) is efficiency (

) in Figure B 3.

c determined by he following equation:

N (B.24)

N *

-c (9+ 9,)

c i

1-e The graph of N is shown in Figure B-4.

c REFERENCES

[B.1)

U.S. Tornado Breakdown by Counties 1950-1980, U.S. Department of l.

Commerce, National Oceanic and Atmospheric Administration, National Weather Service, National Severe Storms Forecast Center.

[B.2)

"The World Almanac and The Book of Facts 1981," Newspaper Enterprise Association, Inc., New York,1981

[B.3)

Abbey, R. F., Jr. and Fujita T. T., "1he Dapple Method for Computing Tornado Hazard Probabilities: Refinements and Theoretical Considerations,"

lith Conference on Severe local Storms, Oct. 1979, Kansas City.

o-9 l

vd e

4 1

b B-5

TABLE B-1 Reported Number of Tornadoes N and Population r

Density EP in the U.S. A.

Year N,

IP 1950 202 42.6 1951 260 43.5 1952 265 44.3 1953 427 45.1 1954 557 45.9

~

1955 626 46.8 1956 509 47.4 v

1957 861 48.2 1958 568 49.0 1959 607 49.8 1960 617 50.6 1961 695 51.5 1962 661 52.3 1963 458 53.1 1964 759 53.9 1965 909 54.7 1966 590 55.3 1967 931 55.9 1968 666 56.5 1969 609 56.9 1970 660 57.4 f

1971 901 57.9 1972 743 58.4 1973 1105 59.0 1974 950 59.6

}

1975 920 60.2 1976 829 60.6 1977 856 61.1 1978 780 61.7 1979 855 62.2 m

I M

4 e

'e B-6

l 1

1200 l

1100 1000 i

\\

R, m

aa 4

800 g

4 2 700 O

~

A

/

~'

a I

}

a 1

^

l 400 300 i I I I I I I I I I I I I I I I I I I i g

1950 53' 56 59

.52 GE 85 71 74 77 80 t (yest)

Figure B-1.

Annual Reported Number of Tornadoes for the U.S.A.

B-7

64

~

/

~,

/

2 56 b 52 l.

48 g

/

o.

I I I I I I I I I I I I I I I I I I I I 40 1950 53 56 59 62 65 68

'71 74 77 80 t (year)

Figure B-2.

Population Density in the U.S.A.

M e

m 4

B-8

1

[

/

0.9 0.8

(

~

)

7 0.7 E

~g 0.6 u

l.

r; 0.5 7

I I O.4 O.3 I. '

I I

I J

0.2

~

88' 100 -

- 112

~

'T24 40 52 64 76 l

o l'

I Figure B-3.

Reporting Efficiency Curve

.f

't.

l 1

B-9 I

1400 g

1300 1200 1100 I

\\

.a-7 A

i A

g*

l' 900 1

800

~

700

\\

20 I I I I I I I I I I I

I I I I I I I I I 500 1950 53'

,5)

.59

,j2 65 68,

.71 74 77 80

,. lLY**V Figure B-4.

Annual Adjusted Number of Tornadoes for the U.S. A.

l B-10

1 1

APPENDIX C THE PROBABILITY OF INJECTION OF POTENTIAL TORNADO MISSILES Introduction In this appendix, the probability of injection of potential missiles or the probability that a potential missile will become airborne is considered.

The model of missile injection is similar to that developed in Twisdale [C.1].

Because Twisdale did not calculate the injection probability in en explicit form, this study develops an explicit expression and presents the numerical results for the injection probability.

Fortornadomissileinjection,therestrainingforces(f)mustbeovercome by aerodynamic forces before motion is possible. TheseIerodynamicforces are the lift and drag forces discussed in Appendix A.

The expressions for the aerodynamic forces are:

[I

'F

I sin 0 cos $ + fg (cos 8 cos $ cos $ - sin $ sin $)

(C.1)

Ax D

1 F

I sin 8 sin $ + ft (cos 6 sin $ cos $ + cos ( sin $)

(C.2)

Ay D

F

I cos 0 - f sin 0 cos Q (C.3)

Az D

g The angles 0 and $ give the orientation of the drag force f in the spherical and plane con-system of coordinates and the angle between the lift force g l

taining the axis OZ and the force I is denoted as d.

D The angles 0, $ and $ are the Euler's angles (sometimes other definitions 1

are used, for example, $ + n - $, $ + n - $). The ranges for these angles are:

0$0in (C.4) 0 $ $ < 2n (C.5) 0 $ $ < 2n (C.6) h4 W

e f

7 C-1

The coefficients f and f are determined by the formulas:

D g

2 p Aw (C.7) fp=CD 2

. p, A w (C.8) fg=CL 2

where:

p, = air density, A = missile cross-section area, w = tornado wind speed, Cy = aerodpanic drag coefficient,

~

Cg = aerodynamic lift coefficient.

The aerodynamic coefficients C and C, for a cylindrical missile are con-D sidered in Redmann [C.2]. For the " standard missile" (see Appendix B), the approximate expression is:

3 a (C.9)

CD = 0.98 sin 2 a cos a (C,10)

Og = 0.98 sin where the attack angle a has range:

O$a$n (C.ll)

The restraining forces f include gravity force and frictional, structural, andinterlockingforceshhichtendtoresistnotion. The expression for restraining forces can be written in the following form:

Fh * ~K mg (C.12) x F

~K og (C.13)

Ry y

T

"K mg (C.14) h z

where a is the missile mass, g is the gravitational constant and K,, K and K are restraint coefficients which show how many times greater (or lels) t$e restraining forces are than pure gravity (ag).

m E

C-2

Because F includes the gravity force, the coefficient K* satisfies the

~

inequalith!

Kz11 (C.15) i The coefficients K,and K satisfy the following inequalities:

y K, > 0 (C.16)

K

>0 (C.17) y For potential missiles lying on the ground, the injection condition is F,1 F

(C.18) g Rz or mg (C.19) f cos 0 - f sin 6 cos $ 1 K p

g g

where:

Kg=K is the lift restrain coefficient satisfying the inequality:

z Kg11 (C.20)

.;a The expression (C.19) is the condition of vertical injection.

- [.

For the case where the potential missiles are located at some elevated height, the potential missile can become airborne due to horizontal dis-placement.

In this case, the condition for vertical injection is:

1 F

+F 1

F

+F (C.21) h Ay Rx Ry a

or 2

2 (C.22)

F

+F 1X mg h

Ay D

~

where K

K

+K (C.23)

D x

y The drag restrain coefficient K satisfies the inequality:

g (C.24)

KD>0 i

O C-3

Using results of Appendix A, the variables w, K, K, a, 6, $, $ are n

g treated as random functions with uniform distri5ution in the ranges:

w (F) $ w I w (F)

(C.25) g 2

K iKgiKy + AK (C.26) 3 K iKgiK2 + AK (C.27) 2 01ain (C.28) 0161n (C.29) 01$<2n (C.30) 01$<2n (C.31)

(F) are lower and upper levels of wind speed corresponding where w (f) and w,le of tornado intensity.

y to some Fujita sci For angles a, 0, $, and $, the uniform distribution is a good approximation.

For wind speed (w), the uniform distribution is conservative because it over-

~

estimates the contribution of higher wind speeds. The real distributions of restrain coefficients K and K are not known, but, if the interval AK is n

g quite narrow, then the tiniform distribution is appropriate.

r In the seven-dimension space of variables w, K, K, a, e, t), and $, the p

7 expressions (C.25) through (C.31) specify a right parallelepiped of volume V :

y V7=4n (AK)2 w (F) - w (F)

(C.32) y

]

The equation (C.33)

F

=K og g

g I t*

is determined by a seven-dimension surface which divides the volume VF into two parts V (V) and V f)

In the volume V (Y}, the condition of vertical y

y y

injections is satisfied.

It is obvious that b

V f)+V I)=V (C.34) y y

7 C-4

.,n

The equation F

+F 2=

(C.35) ag h

gy is determined on another seven-dimension surface which divides the volume V into F

another two. parts V ( } and V (H)

The condition of horizontal injection y

y (C.22)issatisfiedinthevolumeVf).

Inthesamemanner,thevolumeV[a(T)canbeintroducedwhereeitherofthe ccaditions (C.19) or (C.22) are tisfied.

It is clear that V()+V(}*V (C.36) y F

F and V (T), y (i) = V (C.37) y y

7

~

but, generally 4

V ("}

V ( }

Y (C.38)

I)

F F

F because in some subvoltme both conditions (C.19) and (C.22) can be satisfied

~

simultaneously.

Because of the uniform distribution of all parameters, the probability of horizontal injection q(H)(F), vertical injection q(V)(F), and total injec-

)

tion q( )(F) are calculated according to the formulas:

o y (H) rr q( )(F) =

F (C.39)

V7 V (V) q(V)(F) =

F (C.40)

F q(T)(F) = y((T)

(C.41)

F i

L C-5 n--

---m-r -

Forcalculationoftheprobabilitiesq(H)(p),q(V)(F),andh)(F),the Monte Carlo method is used. For this punose, it is convenient to use the scaled variables:

y=g-K 1

(C.42) x g

K

-K g

2 (C.43) 2*

M x

w-w y

(C.44) 3

w ~ "1 x

2 x4 = f (C.45) e (C.46) 5=g x

(C.47) 6=

x 7=h (C.48) x All variables x, (i=1, 2,.. 7) are random variables with uniform distri-bution in the range from 0 to 1.

Computerprogram[3]generatedtherandomvectorx(xddc$n,ditiond)(,C.19)

,a

.. 2 cal-culated the variables w, K, E, a, e, $, $ and check g

and (C.22).

.I trials for F-scale tornado, the condition (C.22) is met N (H)

After N p

F times, the condition (C.19) is met N (Y) times, and either of conditions p

(C.19) and (C.22) is met N (T) times.

p I ) is proportional Because for a large number of trials N, the number N F

F to V (H), the number N (Y) is proportional to V I), the number N (T) y F

p is proportional to V I ) and the number N is proportional to V.

p g

7 1

I 41 3 k

C-6

Hence, the probability in question can be calculated by formulas:

y (H) q( (F) =

(C.49)

F yF

(V) q(V)(F) =

(C.50)

F y

F y (T) 1 q( }(F) =

(C.51)

In further considerations, index T for total injection is omitted, but 9

indices H and V (for horizontal and vertical) are retained.

1, The results of the calculation for AK = 0.5, Kn = 0, 1, 2, 3, 4, 5, Kg =In 2, 3, 4, 5 and F = 0, 1, 2, 3, 4, 5, 6 are shoDn in Tables C-1

.o C-7.

these tables, the letter H stands for horizontal injection, V for vertical L

injection, and T for total injection. The nunoer of trials for every case is taken as 10,000. This corresponds to an accuracy of about 1%.

In Table C-8, the sensitivity of the results as a function of the number of g

simulation trials is shown. The result, rounded to two digits in paren-thesis, stabilizes between 10,000 and 100,000 trials. This suggests that 10,000 trials is a good approximation.

The upper limit for the probability of injection corresponds to the case when the restraining force for horizontal injection is friction and for 4

vertical injection is gravity. The lower limit for q(F) is quite uncertain because it depends on maximum values of K and K which we assign to the g

7

]

potential missiles. The number of potential misEiles derived from [4] was based on maximum values for K = 5 and K = 5.

D g

The maximum and minimum values of q(F) extracted from Tables C-1 to C-7 are shown in Table C-9.

Assuming a uniforro distribution for K 'and K in the range D

g O<K I5 (C.52)

D 1<K p. 5 (C.53) g we will find men values for q(F) given in Table C-10.

~

Now the distribution for q(F) is fit with a lognormal distribution, which has the sami uppe r limit as that given in Table C-9 and the mean and median are as close as possible to the data in Tables C-1 through C-7.

The results are shown on Table C-11.

C-7 j

k

n For q(3), q(4), q(5), and q(6), the distribution generated by uniform distri-butions of ( and K, is exactly lognormal. For q(0), q(1), and q(2), the lognormal diItributTon is a conservative approximation.

The same methodology is applied to the vertical injection probability, q( )(F). The lower and upper limits and means are extracted from Tables C-1 through C-7,and,are shown in Table C-12.

The data for the lognormal fit is shown in Tab,le C-13.

The results, shown in Tables C-11 and C-13, indicate that the probability of potential missiles being injected (i.e., becoming airborne) is rather small, except for the larger F-scale tornadoes.

During the operation period, there are no unrestrained potential missiles at i

any elevation above the ground. Although the equation (C.52) is valid for the ground potential missiles, for the elevated potential missiles we should use the range for KD 15K 55 (C.54)

D i

The lower and upper limits and means for this case are shown in Table C-14.

i L

The lognormal fit is shown in Table C-15.

i i

i C-8 l

n--

References

[C.1)

Twisdale, et. al., " Tornado Missile Risk Analysis", EPRI NP-768, NP-769, May (1978).

[C.2]

Redmann, G. M. et. al., " Wind Field and Trajectory Models for Tornado - Propelled Objects", EPRI 308, Technical Report 1, Feb. (1976).

[C.3]

IMSL Library Reference Manual, Edition 8, IMSL LIB-0008, International Mathematical and Statistical Libraries, Inc.,

June, 1980, 1

I l

u i

i I

l i

l 1

i C-9 i.

.,-e

-, e. - - - - - -

n --

a t

Table C-1 Probability of Injection n (0)

L Type 1

2 3

4 5

K D

H

.1756

.1756 1756

.1756 0

V

.0000

.0000 0000

.0000 T

.1756

.1756 1756

.1756 l

H

.0000

.0000 0000

.0000 1

V

.0000

.0000 0000

.0000 T

.0000

.0000 0000

.0000 f

l H

.0000

.0000 0000

.0000 2

V

.0000

.0000 0000

.0000 T

.0000

.0000 0000

.0000 i

H

.0000

.0000 0000

.0000 3

V

.0000

.0000 0000

.0000 T

.0000

.0000 0000

.0000 H

.0000

.0000 0000

.0000 4

V

.0000

.0000 0000

.0000 T

.0000

.0000 0000

.0000 H

.0000

.0000 0000

.0000 l

5 V

.0000

.0000 0000

.0000 l

T

.0000

.0000 0000

.0000 t

0 5

I C-10 l.

i

/

Table C-2 Probability of Injection n (1) l I

EL

}-

K Type 1

2 3

4 5

g

)

H

.4274

.4274 0

V

.0000

.0000 T

.4274

.4274 l

H

.0000

.0000 1

V

.0000

.0000 T

.0000

.0000 H

.0000

.0000 2

V

.0000

.0000 T

.0000

.0000 l

H

.0000

.0000 3

V

.0000

.0000 T

.0000

.0000 H

.0000

.0000 4

V

.0000

.0000 T

.0000

~ I i

H

.0000

.0000 f

5 V

.0000

.0000 T

.0000

.0000 0

4

{

i l

C-11 l

I Table C-3 l

I Probability of Injection n (2) l 5

Type 1

2 3

4 5

E p

H

.6474

.6474 0

V

.0305

.0000

.0000 T

.6528

.6474

.6474 H

.0647

.0647 1

V

.0305

.0000

.0000 T

.0941

.0647

.0647 H

.0000

.0000 2

V

.0305

.0000

.0000 T

.0305

.0000

.0000 l

H

.0000

.0000 3

V

.0305

.0000

.0000 g

T

.0305

.0000

.0000 N

.0000

.0000 4

V

.0305

.0000

.0000 T

.0305

.0000

.0000 i

H

.0000

.0000 5

V

.0305

.0000

.0000 T

.0305

.0000

.0000 i

i I

i f

i.

C-12

- ~ _.... - _, _ _

m

+

Table C-4 l

Probability of Injection n (3)

I k

I l

L Type 1

2 3

4 5

(

E D

H

.7591

.7591 0

V

.1230

.0281

.0001

.0000

.0000 T

.7657

.7610

.7592

.7591

.7591 H

.2895

.2895 l

1 V

.1230

.0281

.0001

.0000

.0000 T

.3654

.3104

.2896

.2895

.2895 H

.0592

.0592 2

V

.1230

.0281

.0001

.0000

.0000 T

.1771

.0873

.0593

.0592

.0592 i

H

.0004

}

3 V

.1230

.0281

.0001

.0000

.0000 T

.1234

.0285

.0005

.0004

.0004 i

i H

.0000

.0000 4

V

.1230

.0281

.0001

.0000

.0000 T

.1230

.0281

.0001

.0000

.0000 H

.0000

.0000 5

V

.1230

.0281

.0001

.0000

.0000 l

T

.1230

.0281

.0001

.0000

(

l i

i C-13 8

Table C-5 Probability of Injection r, (4) h I

h I

Type 1

2 3

4 5

g H

.8184

.8184 0

V

.1943

.1100

.0466

.0102

.0000 T

.8218

.8206

.8198

.8188

.8184 H

.4871

.4871 1

V

.1943

.1100

.0466

.0102

.0000 T

.5428

.5230

.5065

.4934

.4871 1

I H

.2554

.2554 2

V

.1943

.1100

.0466

.0102

.0000 T

.3798

.3321

.2935

.2651

.2554 l

I H

.1046

)

3 V

.1943

.1100

.0466

.0102

.0000 T

.2751

.2063

.1504

.1148

.1046 i

H

.0188

.0188 4

V

.1943

.1100

.0466

.0102

.0000 T

.2102

.1285

.0654

.0290

.0188 H

.0000

.0000 l

5 V

.1943

.1100

.0466

.0102

.0000 T

.1943

.1100

.0466

.0102

.0000 I

l i

t I

I t

C-14 l

t

l Table C-6 Probability of Injection n (5) l L

g Me 1

2 3

4 5

H

.8584

.8584 0

V

.2449

.1728

.1192

.0751

.0377 T

.8602

.9598

.9596

.8594

.8593 H

.6208

.6208 l

1 V

.2449

.1728

.1192

.0751

.0377 i

T

.6545

.6442

.6396

.6359

.6313 H

.4300

.4300 2

V

.2449

.1728

.1192

.0751

.0377 T

.5212

.4926

.4773

.4652

.4514 i

H

.2784

.2784 3

V

.2449

.1728

.1192

.0751

.0377 l

T

.4260

.3840

.3566

.3339

.3088 H

.1715

.1715 f

4 V

.2449

.1728

.1192

.0751

.0377 T

.3584

.3074

.2709

.2399

.2074 H

.0827

.0827 i

5 V

.2449

.7728

.1192

.0751

.0377 T

.3036

.2424

.1964

.1564

.1204 l

7 t

!i 1

C-15 j

Table C-7 Probability of Injection r1 (6)

J K

(

Type 1

2 3

4 5

H

.8857

.8857 0

V

.2847

.2214

.1737

.1360

.1034 T

.8865 8864

.8863

.8862

.8862 H

.6999

)

1 V

.2847

.2214

.1737

.1360

.1034 l

T

.7206

.7148

.7122

.7107

.7093 H

.5623

.5623 2

V

.2847

.2214

.1737

.1360

.1034 T

.6210

.6041

.5954

.5903

.5862 i

H

.4346

.4346 3

V

.2847

.2214

.1737

.1360

.1034 T

.5423

.5135

.4964

.4860

.4773 I

H

.3208

.320S 1

4 V

.2847

.2214

.1737

.1360

.1034 T

.4774

.4404

.4153

.3979

.3841 H

.2384

.2384 1

5 V

.2847

.2214

.1737

.1360

.1034 T

.4285

.3863

.3568

.3352

.3167 1

q f

C-16

. ~.

Table C-8 i

Sensitivity of n to the Number of Trials G

Q 5

I 1

N-10 q(H)(1) for K w 0, Kg=1

/8 6=

l N

10 l

100

.47000 (.47)

.1000

.0830 1000

.44600 (.45)

.0316

.0277 10,000

.42740 (.43)

.0100

.0151 I

100,000

.43397 (.43)

.0032 t

I I

i 1

b I

I I

C-17

- - ~. _. _, _ _ _ _ _ _ _ - - - - _, - _ -, - - - _ _.. _ _ _ _

Table C-9 Maximum and Minimum Values for rl(F)

Maximum Minimum I

F.

q(F) q(F) k 0

0.1756 0.0000 1

0.4274 0.0000 2

0.6528 0.0000 3

0.7657 0.0000 4

0.8218 0.0000 5

0.8602 0.1204 6

0.8865 0.3167 i

l

\\

i l

(

i

)

i k

C-18

\\

=

1 Table C-10 Means for Q(F)

J F

Nean l

i l.

.0 0.0293 1

0.0712 i

2 0.1239 1

3 0.2082

, l 4

0.3281 i

5 0.4708 6

0.5817 l

t 1

i 1

i s

.I I

e I

C-19 i

l

i i

Table C-11 Lognormal Distribution for r1(F)

F Lower Limit.

Median Mean Upper Limit i

O d.0008 0.0119 0.0454 0.1756 l

1 0.0020 0.0292 0.1105 0.4274 2

0.0029 0.0435 0.1687 0.6528 1

3 0.0098 0.0866 0.2083 0.7657 4

0.0789 0.2546 0.3282 0.8218 l

5 0.2160 0.4310 0.4708 0.8602 6

0.3529 0.5593 0.5817 0.8865 L

I 1

I k

9

?

C-20 l

Table C-12 Lower and Upper Limits and Means for q(Y)(F) l l

l F

Lower Limit Mean Upper Limit 0

0 0

0 1

0 0

0 g

2 0

0.0061 0.0305 3

0 0.0302 0.1230 4

0 0.0722 0.1943 5

0.0377 0.1199 0.2449 l

6 0.1034 0.1838 0.2847 f

i Table C-13 Lognormal Distribution for q(Y)(F) i l

I F

Lower Limit Median Mean Upper Limit I

2 0.0001 0.0017 0.0079 0.0305 1

3 0.0006 0.0086 0.0318 0.1230 l

4 0.0143 0.0527 0.0722 0.1943 5

0.0450 0.1050 0.1199 0.2449 et 6

0.1089 0.1761 0.1838 0.2847 6

f I

e I

C-21 f

4 1

i Table C-14 l

Lower and Upper Limits and Means for q(F) for Restrained Potential Missiles e

I F

Lower Limit Mean Upper Limit 0

0 0

0 j

1 0

0 0

2 0

0.0141 0.0941 3

0 0.0977 0.3654 l

4 0

0.2297 0.5428 i

8 5

0.1204 0.3930 0.6545 I

6 0.3167 0.5207 0.7206 i

Table C-15 l

Lognormal Distribution for q(F) for Restrained Potential Missiles F

Lower Limit Median Mean Upper Limit 2

0.0002 0.0043 0.0249 0.0941 9

3 0.0038 0.0375 0.0976 0.3654 4

0.0635 0.1857 0.2297 0.5428 5

0.2093 0.3701 0.3930 0.6545 l

6 0.3598 0.5092 0.5207 0.7206 I

3 I

I h

C-22

Appendix D.

General Methods D.1 Preamble The purpose of the PRA analysis of a tornado missile's hazard is to evaluate the probability, P, of damaging some target per year. However, due to randomness of natural factors and uncertainty of our knowledge about many parameters, we have an estimate for the probability, P '

T which has uncertainty associated with it.

The best approach is to develop a distribution function, f(P ), for 7

the damage probability, P. The typical curve of the distribution T

of the damage probability (in logarithmic scale) is shown in Figure D-1.

The distribution function, f(P,), allows us to evaluate the "best" estimate for P, which is a me8ian value, and a confidence interval T

that shows the spread of the most likely values for the probability, i

P, ar und the median.

T J

L funP I T

s I

i In P g

l 4

-7 +, -6

-5

-4

-3

-2,

-1 0

i Figure D-1.

Typical Distribution Function f

f D-1

~-

Therefore, we can specify three major steps of PRA analysis:

A.

Development of the model for calculating the damage probability,

'P.7, as a function of some random parameters.

B.

Development of the distributions for all random parameters that the damage probability, P.7, depends on.

C.

Calculation of the final distribution of the damage probability, j

f(P,), using the distribution for all random parameters (propagation l

of Gncertainties).

There are three sources of randomness and uncertainty:

A.

Natural phenomena are essentially random.

B.

Data are incomplete.

C.

Mathematical models or solutions for these models are approximate.

Classical statisticians believe that a complete set of data is always available and the uncertainty could be reduced to zero. Classical statistics has developed very well-established methods of dealing with randomness.

Recently, some analysts have started to recognize (see [D.1)-[D.2]) that

{

in many areas of research the uncertainty cannot be completely eliminated.

The reason is not based only on difficulty of collecting data.

In early PRA analyses, classical statistics was applied with poor data bases. Naturally, all these results were questionable and only compro-mised the credibility of PRA methods.

The Bayesian approach improved the practice of dealing with uncertainties but the major flaw contained in the arbitrary a priori distribution is still present.

We have to recognize that classical statistics was developed for problems dealing with randomness but not with uncertainty.

In cases when uncer-tainty is not removable, a new approach has to be developed. This i

approach has some features in common with the Bayesian approach, but it is much broader.

The major features of the new approach are:

A.

Every random parameter can be described in a way that reflects both randomness and uncertainty.

B.

Distribution of the random parameter has to satisfy the principle iof maximum entropy given the knowledge.

I.

D-2

~-_ _ - -

The entropy is a measure of the expected uncertainty.

If the entropy of the probability distribution given some knowledge is maximal, it means that the shape of the probability curve exactly reflects what j

we know.

For example, the cogtinuous distribution having the largest entropy for a given variance, o, is the normal distribution [D.3]. However, if we know.that.a random parameter is positive, then a maximum entropy distribution is presumably lognormal. The entropy of the lognormal distribution is less than for the normal one because we have some additional information (a parameter could be only positive). However, the entropy of the legnormal distribution is believed to be the highest in the class of distribution functions with a positive domain of i

definition of a random parameter and a given variance.

c If the distribution satisfies the principle of maximum entropy, we are guaranteed that a confidence interval for a damage probability can only be reduced when additional information becomes available.

The principles discussed are incorporated in this study.

I

~

D.2 Tornado Characteristics l

The main characteristics of tornadoes t* at determine the probability u

of damage are the tornado path area, a, and the Fujita scale, F.

The tornado path area, a, is a product of the length, L, and width, I

W, of the tornado's destructive track on the earth's surface:

a=LxW (D.1)

The classification of tornadoes according to their damage areas was proposed by Fujita [D.4]. He used a decimal logarithm of the tornado path area, a, measured in square miles. However, a computerized file at the National Severe Storms Forecast center keeps information about tornado length, L, in tenths of miles and tornado width, W, in tens of feet. Because all data are rounded, it will be more accurate to classify all tornadoes by areas estimated in miles-feet.

The computerized record naturally groups all tornadoes at the decimally j

logarithmic scale of tornado area, a, measured in miles-feet.

If we translate miles-feet to square miles and calculate the number of tornadoes into new intervals, we have to somehow divide the number of tornadoes in old intervals between the new ones. This procedure requires us to use some hypothesis of tornado distribution by areas and introduces additional sources of error.

l Therefore, in our classification of tornadoes by areas, we will use tornado path area, a, measured in miles-feet. We will call this clastification an A-scale (see Table D.1).

I D-3 4

- ~-

- -, -,. -. - - -..... -...,, -.. - - -, ~..-,-.- - -,,

--..n-

The upper and lower limits of tornado area belonging to some scale, A, can be determined by formulae:

A (D.2)

= 10 a,

A-1 (D.3) a

= 10 The median value assuming a lognormal distribution is:

l (D.4) med = 10 a

f where A = 1, 2,..

7.

Fujita also proposed classifying tornadoes by intensity [D.5].

The Fujita scale, F, is a characteristic of to nado intensity that depends on the damaging wind speed according to Table D.2.

The relationships between upper, lower, and median wind speeds with corresponding Fujita scale, F, are given by the following approximate formulae:

14.1 x (F + 3) /

(mph)

(D.5) w

=

,p 14.1 x (F + 2)3/2 (mph)

(D.6) w

=

y,y eed = 14.1 x (F + 2.5)

(aph)

(D.7)

I w

For wind speed, w, given in meters per second (m/s), the coefficient in formulae (D.5) through (D.7) should be 6.30.

i D.3 Tornado Missile Description The spectrum of potential tornado missiles at the plant site and in the nearby area is described in the Standard Review Plan, section 3.5.1.4

[2]. A more detailed spectrum, based on seven-plant survey data, is given in an EPRI study [5].

From the point of view of missile injection and transportation, the classification of potential missiles should be based on parameters determining the missile acceleration. The equations of missile motion can be written in the form:

I p* "2 (D.8) v = C l

1 D-4

p* "2 4=C (D.9) v 2 (;)

v p* "2 (D.10) v

=C

-g

'

2 ("-)

l Where a is the missile mass; v,, v,, and v are components of the j

missile velocity; C, C, and C,are enpirical aerodynamic coefficients; y

is the air density; A is the snissile cross-section area; u is the p,lative wind-missile speed; and g is the gravitational acceleration re I

near the earth's surface.

I Missile acceleration components depend on aerodynamic coefficients C,

g C, and C,, and parameter m/A. Aerodynamic coefficients depend on missile l

oYientation,windvelocity,andmissileshape.

For a cylindrical missile with length,1, and diameter, d, the shape i

parameter is A/d (see Figure D-2).

0 l

I g

/_ N 4

k i

f Figure D-2.

Cylindrical Missile i

9 D-5

Consider a simplified case when the wind velocity, w, is directed along axis Ox and a cylindrical missile is located in the plane x0z with l

the angle of attack, a, (see Figure D-3).

For this case, we have:

(

.(D.ll)

C, = CD C = 0 (D.12)

(p.13)

C, = Cg, i

1 Where C is the drag coefficient and C is the lift coefficient (the D

g side coefficient for a cylinder is equal to zero).

In general, these i

equations are considered in Appendix A.

Z g

l

}

F l

t o

I FD a

=

r l

~

i i

4

X 0

Figure D-3.

Missile Orientation I

l i

D-6 9

and C Y

According to [6), tfae_ coefficients Cg D * *"

formulae:

t 2

C esa cos a sin a (D.14) sin,c,,

Cg=Cy 2

3 3

(D.15)

C

=.C sin a + C. cos a D

y 2

l where (D.16)

Cy = 1.2 k (D.17)

~

C =

C 2

and K=0.59+0.41exp'(-20f)

(D.18) k

~

1.16, f 51 C =( 0.84 + 0.32 exp [-2

-1

],

15 f < 4 (D.19) l I

4 i

(0.79 + 0.0125 f,

f>4 I

For long missiles (D.20) f<1 i

we have approximately 9

2 (D.21)

C

~C sin a cos a g

y (D.22)

C sin a D

1 The aerodynamic coefficient, C, is maximal for the angle of attack g

a ~ 55' and coefficient C f r the angle a ~ 90*.

D The specification of tornado-generated missiles is given in Table D.3.

We adopted the effective diameter for a wooden plank, d = 0.65 ft, and for in automobile, d = 6 ft.

9 D-7

According to equations (D.8) through (D.10), the missile acceleration depends on parameters C /(m/A) and 4/(m/A). These' parameters for y

different missiles are nown in Tab 1E D.4.

Averaging these parameters over the spectrum of potential missiles given in the EPRI study [5], we find so-called " standard missile" parameters. The major contributors to the spectrum of potential missiles are: type C (46%) and types B, D, and E (21% altogether).

1 Our analysis assumes, for simplicity of explanation, that we have only one type of potential missiles: so-called " standard" missiles.

However, the final results are not sensitive to the particular values of standard missile parameters. Actually, the numerical values of the parameters under consideration will affect the parameters of height distribution, $(z, F), (see section D.11) and the probability of injection, q(F), (see section D.10).

The uncertainties of $(z, F) and q(F) adopted in our study are much broader than those created by dispersion of aerodynamic characteristics.

l D.4 Probability of Damage Given Tornado Frequency v, Fath Area a, Fujita Scale F, Density of Potential Missiles n, Injection Probability P

q(F) and Height Distribution $(z, F)

Let us assume that a tornado striking the plant site has a given path area, a, and Fujita scale, F.

We assume also that the missile's J

characteristics are known. Therefore, the. density of potential missiles near a target, n, injection probability, q(F), and height

}

p distribution, $(z, F) are certain values.

I Under these conditions, the probability of damage, P, can be written T

in the form:

(D.23)

P P

}

PT=P0 H

D i

Where:

O = Probability per year that a tornado with given characteristics I

P a and F strikes the plant site.

H = Conditional probability of hitting a target given that the P

i tornado strikes the plant site.

D = Conditional probability of a target damage given a hit.

P j

i I

i D-8

is:

According to Thom [D.6], the probability PO (D.24)

PO=

where v is the annual frequency that a tornado having the same occurrence characteristics as a tornado at the plant site will strike the area S.

For the probability of damage, 'r, we conservatively assume:

g (D.25)

PD=1 This assumption will provide a conservatism of several orders of magnitude.

For the calculation of the conditional probability of hitting, P '

H a special statistical mechanics approach was developed. This approach is based on the following assumptions:

1.

Tornado missile motion is a diffusion Markovian process.

l I

2.

Propagation or Green's function for a tornado missile is uniform in space and time and has axial symmetry.

and justification fo-the The derivation of the expression for Pu assumptions are given in Appendix A.

The final expression fsr the I

hitting probability, P, is:

H (D.26)

H * "p A q(F) $(z, F)

P Where A is a target area (in our case, it is the area of the IVC roof) and 2 is a target elevation above the ground.

Putting (D.24) through (D.26) into (D.23), we obtain:

1 (D.27)

PT*

"p A q(F) $(z, F)

The probability of damage, P, depends on tornad2 Path area, a, T

tornado Fujita scale, F, and set of parameters, {, that determine e

the distributions of v, a, q(F).and $(z, F).

i p

The distributions of these parameters will be discussed in the g

next sections.

g i

l i

D-9

D.5 Distribution of Randon Parameters The probability of damage, P, depends on two groups of random T

parameters:

1.

v, a, F 2.

~.n, q(F), $(z, F)

The first group reflects the randomness of natural phenomena and depends on tornado characteristics.

The second group depends on tornado missile characteristics and reflects the uncertainty of data.

The distributions for the first group of parameters are based on historical data, and those for the second group are partially appealing to the state-of-knowledge, engineering judgment, and physical limitations.

The two groups of parameters are completely independent. Among the first group of parameters, we did not find any credible correlation between frequency, v, and the other two parameters, a and F.

Therefore, the distribution of these parameters can be written in the form:

f(V, a, F) = f (v) f (a, F)

(D.28) y 2

It is convenient to write the joint distribution, f (a, F) in the 2

following way:

f(a,F)=f(a)$(Fla)

(D.29) 2 When f(a) is a marginal distribution of the tornado path area, and

$(F a) is a conditional distribution of Fujita scale given the path area.

i The distribution, f(a), is different for different geographical areas.

The typical area containing enough data to develop a reliable distribution is about the area of one or several states.

In our case, the data for Texas are sufficient.

The distribution, f(a), depends on meteorological conditions which set the scale of tornado size. Therefore, every large meteorological i

zone requires its own distribution.

Ontheotherhand,theconditionaldistribution,$(Fla), depends on the physical model of the tornado. Essentially, the physical model of the tornado as a natural phenomenon is the same for all geogtaphical areas. All varieties of tornado characteristics I

depend on ratios of fundamental parameters: total energy, I

momentum, and angular.aomentum of the tornado. Therefore, the nationwide data were used for the calculation of the conditional I

distribution,$(Fla).

I D-10 i

For the second group of parameters, we remove all possible correlations j

by using the principle of the superior estimate.

Let { ((bibu$ ions of,n, r)(F) and $(z, F)., &,... ( ) be the set of random the dist The joint distributions of all parameters are:p R(&., & ' *** I )

8 (b ) 8 (b b ) *** 8 fb b b *** b -1)

(D.30) i 2

n 1 l 2 2 l

a n l 2 n

Define the set of superior functions:

5(b)18(blb)I#*11b 2 2 2 2 l l

(D.31)

E (b ) 1 8 (b lb b ) f r all (1, (2 3 3 2 3 l 2 k (b ) 18 (b $ b '

b -1) I # *11 b ' b ' * *

b -1 n n n n 1 2 n

l 2

n Then the conservative estimate of the joint probability g(&y, (2' * *

b ) (in the meaning of higher probability of damage) n is:

g(&y, & ' *** b )

8 (b ) 2(b)***k(b)

(D.32) 2 n

1 1 2

n n The specific details of all distributions will be discussed in the corresponding sections of this report. The total joint distribution of all parameters takes the form:

f(v, a, F, l) = f (v) f(a)

$(F a) g ((y) i (b ) *** b (b ) (D.33) y y

2 2 n n The distribution (D.33) generates the distribution for damage probability l

which will be considered in section D.13).

P T

1 D.6 Annual Frequency v of tornado occurrence The nuclear power plant at the South Texas Project site is located in Matagorda county with coordinates of units:

Unit I Unit II i

Latitude 28* 47' 42" 28* 47' 42"

)

i 96* 02' 53" 96* 02' 59" Longitude Because both units are located near the coast, a 10,000-square-mile area with its center within the plant site will contain about 40%

sea with low efficiency of tornado counting.

Therbfore, we determine the annual frequency, v, using the historical reccLrd for six counties located near the plant site.

I The list of selected counties is given in Table D.S.

g l

D-11

Our computer code estimates the annual number of tornado occurrences, finds the 25th percentile, median, 75th percentile, mean, and standard deviation for the empirical distribution. It then fits this distribution by a lognormal distribution having the same median and ratio of 25th and 75th percentiles as the empirical distribution.

The snaval number of tornado occurrences for si'x counties listed in Table D.5 are given in Table D.6.

Tornado segments are conservatively treated as separate occurrences for purposes of deteruining frequency.

Parameters of the fitted legnormal distribution are given in Table D.7.

The comparison of the empirical and fitted lognormal distribution is shown in Table D.8.

We see that the lognormal fit is quite conservative.

The annual frequency, v, of tornado occurrences for six counties g

has to be adjusted to t5e area S = 10,000 square miles according to the formula:

S' (D.34) v' = v6 *T6 Where v' and v are medians for the areas S and S, corre8Pondly, 6

g and S6 = 5880 Iquare miles.

The annual frequency v' has to be adjusted for reporting efficiency (see Appendix B) according to the formula:

i v=f (D.35) k is the reporting Where v is an adjusted annual frequency and CE efficiency:

i f

I (D.36)

C

=1-e j

E i

Where:

= Population density for six counties (69.97) i D

D,= -40.04561369 C = 0.118188864 The calculation according to the formulae (D.35) and (D.36) for the f

median value gives:

(D.37) v = 5.25 D-12

\\

The parameters of distribution for the adjusted annual frequency V are given in Table D.9.

The local data for Matagorda county are not sufficient to develop a reliable distribution. However, we can estimate the mean for Matagorda County and adjust it to the area S = 10,000 square miles.

j The adjusted mean annual frequency v for Matagorda County:

v=.h.394 (D.38) i This number is lower than the number 10.704 incorporated in the fitted lognormal distribution.

D.7 Distribution f(a) for Tornado Path Area To collect sufficient data to develop the distribution, f(a), of the tornado path area, we use data for Texas for 30 years (1953-1982) containing 2730 records [D.9].

The distribution of the path area according to the A-scale is given in Table D.10.

This distribution is compared with the nationwide distribution.

Mean tornado path area a for Texas is 3.5 times less than a for the U.S.

However, this difference is significantly less chan the difference between Thom data for Kansas and Iowa [D.6] and nationwide data. The mean path area a for these states is 20 times greater than for the U.S.

A thorough analysis of data presented in Table D.10 shows that a real number of tornado occurrences in each A-scale for Texas could vary in the range up to 15%. Therefore, these data should be corrected by using the appropriate analytical fit.

We assume that the distribution, f(a), of the tornado path area is exactly lognormal. The best parameters of lognormal distributions p and o are shown in Table D.ll.

These parameters were found j

with the least-squared method ID.3].

The empirical and lognormal distributions for A-scale for Texas are I

compared in Table D.12.

We see that the lognormal distribution I

has a tendency to overestimate the probability of large tornadoes.

Therefore, the use of the lognormal distribution will be conservative I

because it will increase the estimated probability of a tornado j

strike to the nuclear power plant site.

l I

D-13

._.- ~.. _.,. _,.. _. _ _ _ -

D.8 Joint Distribution of Tornado Path Area a and Fujita Scale F The correlation coefficient between the tornado path area and Fujita scale, F, is very sensitive to data and reflects some generic tornado characteristics rather than local peculiarities. Therefore, the nationwide statistics are used for its determination.

Data.from.the computerized file of National Severe Storms Forecast Center [D.9) contain 14,563 complete records for a 30-year period (1953-1982). The joint distribution of A-and F-scales is shown in Tables D.13 and D.14.

Following Wen and Chu [D.7], we will fit this discrete distribution by a continuous one:

1 1

-(An a - p,) 2 I

f(a, w) =

exp i

2

( 2(1 - p )

j o,

j 21r a, o aw fEnw-p) fina-p,\\[fnw-p)"

y y

l (D.39) l 1

- 2p

\\

"w

}

N "a

) 'Q "w

).

I Where the tornado path area, a, is related to the A-scale according to Table D.1, and tornado wind speed, w, is related to the Fujita scale according to Table D.2.

The best fit for parameters of the distribution is:

p, = -2.8175 o, = 2.8759 i

(D.40) 1 p,= 4.6390 1

0,= 0.4040 l

p = 0.59396 s

Now, present the distribution, f(a, w), in the form:

I (D.41) l f(a, w) = f(a)

$(wja)

I 1

e i

i

?

D-14 i

i

where:

in a - p*i 2"

-f (D.42) l f(a) =

exp h, y g

a a

and i

~

1 11finw-pla 2 w

(D.43)

$(wla) = N Uwla" exp 2(

o, l, p,, = p, + p o" (in a - p,)

(D.44) a I

2 (D.45) o,

= Y1 - p o

We assume that parameters p, o, and p are generic. Parameters p, y

and o, are region-specific.

If we compare parameters p, o, and p for Texas, we find that g

nationwide parameters are more conservative because the median wind y

speed distribution for the U.S. is higher than for Texas.

D.9 Surf ce Density of Potentia'l Missiles n P

In the EPRI study [D.10], the number of potential missiles in the 7

I missile origin zone (2.5 x 10 square feet including the plant i

site) is provided. For a two-unit plant with both units under operation, the possible range for the number, N, of potential P

snissiles is:

i 5836 < N < 6196 (D.46) p-Dividing relationship (D.46) by the missile origin area, we find the range for the average density of potential missiler:

~0 (D.47) 2.33 x 10 ' i Ep 5 2.48 x 10

~

ih Assuming the lognormal distribution for average density na90percentconfidenceint we can readily find the parameters of this distribution (see Table D.15).

An analyrj s of zone distribution of potential missiles in the area of i

l l

2.5 x 10 square feet based on data given in [D.10] shows that the j

maxisium deviation from the average density for a plant under operation is 2)S5.

The local density of potential missiles a can be presented in the form:

i P

n

=K 6

(D.48) i P

n p I

where K is the noanniformity coefficient.

i D-15

Assuming for coefficient K, a lognormal distribution with upper limit 2.55 and median equal to 1, we find the distribution for n as a product of two lognormal distributions for n and K. The parameters of the distribution for n are given in TabIe D.15" P

D.10 Probability of Injection q(F)

The probability that a potential tornado missile could become airborne or the probability of injection was considered in [D.'ll).

This proba-bility is different for potential missiles located on the surface and at some elevation. For the surface potential missiles, the probability of injection is lower because the minimum restraining force is gravity. For elevated potential missiles, the minimum restraining force is friction, which is less than gravity. For both cases, the maximum restraining force is assumed five times greater than gravity. This restraint can be overcome only by tornadoes F5 and F6.

We assume that 50% of all potential missiles are lying on the ground and 50% are distributed uniformly up to elevation 20 ft.

Twenty i

percent of the elevated missiles are restrained, (i.e., the minimum restraining coefficient is 1). Ten percent of the elevated missiles are unrestrained (i.e., the minimum restraining coefficient is equal to zero).

For the surface potential missiles (so-called vertically injected missiles) the distribution of q(F) is shown in Table D.16.

This distribution is created by random orientation of potential missiles and random distribution of restraining coefficients.

(See Appendix C.) Tornadoes of Fujita scales F0 and F1 cannot lift 7

the potential missiles specified in [D.12] from the surface.

For the elevated unrestrained potential missiles (so-called horizon-j tally injected missiles), the distribution of injection probability q(F) is shown in Table D.17.

For the elevated restrained potential missiles, the distribution of injection probability q(F) is shown in Table D.18.

These missiles also cannot be lifted by tornadoes of Fujita scales F0 and F1.

Actually, the assumption that 10% of elevated missiles are not restrained introduces some additional conservatisms.

I

\\

l

~

D-16 t

-.n.,

i l.

D.11 Height Distribution of Airborne Missiles $ (z, F)

The height distribution of airborne missiles $ (z, F) for a uniformly spread source of potential s;issiles is addressed by [D.ll]:

, O f z $ h, i

(D.49)

$(z, F) =

2 (F) + (h, - z) - a (F) h, "l(F) a y

1 B(0) a (F) e

, z > h, y

where B(z), a (F) and a,(F) are defined in section A.7.

The median j

value of $(z,F) for z ='55 ft and h,= 20 ft is shown in Table D.19.

D.12 Conditional Probability of Damage Given a Hit For the sake of simplicity and conservatism, we assumed that the con-ditional probability of damage PD given a hit is equal to unity:

(D.50)

P

=1 D

Because not every missile entering the IVC will hit the sensitive part of the equipment and because damage of at least two redundant elements is required to incapacitate the system, this assumption results in a safety margin of several orders-of-magnitude.

D.13 Distribution of Damage Probability PT g

I The Monte Carlo simulation method was used for the propagation of uncertainty. Using distribution functions for all random parameters described in Sections D.6 through D.12, we developed the distribution l

for the damage probability P. Based on this distribution, the median T

(the best estimate) and 95% upper licit are reported. The best i

estimate should be compared with the acceptance criteria.

g Our computer code uses the standard procedure generally accepted for this sort of problems with the best available "on-the-market" generator of random numbers and effective sorting procedure.

We used 10,000 simulations per run and reran the code 10 times with 1

different seed numbers.

It assures us that relative error for the l

best estimate is within the range 2-3%, which is very good for such spread distributions. For comparison, the best result that the SAMPLE code used in WASH-1400 can give is 12% accuracy [D.8].

l The result of the calculation is shown in Table D.20.

I i

D-17 l.

l l

D.14 Point Estimate of Median Value of Damage Probability P T To give some idea about the numerical value of every contributor to damage probability P and independently check our computer code, we provide an easy-to-follow manual point estimate of the median of f

T damage probability P

T The median,of frequency, v, of tornado occurrence in area S

)

= 10000 square miles is calculated according to the formula:

1 E

i v

(D.51) v=e where parameter p = 1.659 is taken from Table D.9.

I y

The median of tornado path area a is estimated according to a similar formula:

a (D.52) a=e where parameter p, = -3.8076 is taken from Table D.11.

I Tornado freo ncy, P, at the plant site is calculated using formula D.24.

The median wind velocity can be calculated as w a (D.53) w=e I

where the parameter p,, is determined by formula D.44.

However, for madian value, a i

(D.54) l p,, = p, For the U.S., p = 4.6390 which yields w = 103.4 mph (for Texas,

= 4.505 and w = 90.5 mph).

It corresponds to Fujita Scale Fl.

y p

i y

Tornadoes of Fujita Scale F1 cannot lift surface and restrained ele-l vated missiles. Only unrestrained elevated missiles could be lifted.

The fraction, f, of unrestrained missiles is 0.1.

Therefore, the density of available missiles, n, can be estimated according to the formula:

(D.55) n

=f n

a p

[

The median of density, n (D.55), of potential missiles is taken from Table D.15.

The probabiSity of injection, q(F), and the height dis-tribution, 4(z,F), for F1 tornadoes are taken from Tables C.11 and

[

D.19, respectively.

g l

D-18

t The total probability of damage is estimated according to formula D.27.

We have to note that the multiplication of medians is also the median only in the case when all multipliers are distributed log-normally.

In our case, the height distribution $(z,F) is not dis-tributed lognormally. Therefore, we get some approximation of the median value of P

T The', result of this calculation is shown in Table D.21.

This estimate is close to the exact value for the median reported in Table D.20.

I l

I i

4 e

i j

l 9

l f

D-19 i

I i

References

[D.1] Kaplan, S. and Garrick, B.

J., "On the Quantitative Definition of Risk", Risk Analysis 1, 11-27 (1981).

Kaplan, S., Apostolakis, G., Garrick, B. J., Bley, D. C., and

[D.2] ' Woodward, K., " Methodology for Probabilistic Risk Assessment of Nuclear Power Plants," Draft, PLG-0209 (June 1981).

[D.3] Korn, G. A. and Korn, J. M., Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York, 1968.

[D.4] Fujita, T. T., " Estimate of Areal Probability of Tornadoes from Inflationary Reporting of their Frequencies," SMRP Research Paper No. 89, the University of Chicago,1970.

I

[D.5] Fujita, T. T., " Proposed Characterization of Tornadoes and I

Hurricanes by Area and Intensity," SMRP Research Paper No. 91, the University of Chicago, 1970.

I l

[D.6] Thom, M. C.

S., " Tornado Probabilities," Monthly Weather Review, Oct.-Dec., 730, (1963).

[D.7) Wen, Y. K. and Chu, S. L., " Tornado Risks and Design Wind Speed,"

Journal of the Structural Division, Proceedings, ASCE, Vol. 99, No. C. T. 12, Dec. 1973.

l

[D.8) Goodman, J., " Accuracy and Efficiency of Monte Carlo Method,"

Proceedings of International Conference on Numerical Methods in Nuclear Engineering, September 6-9, 1983, Montreal, Quebec, Canada.

"U.S. Tornado Breakdown by Counties 1953-1982," U.S. Department of

[D.9)

Commerce, National Oceanic and Atmospheric Administration, National g

g Weather Service National Severe Storms Forecast Center, Room 1728, Federal Building, 601 E. 12th Street, Kansas City, Missouri 64106.

t l

[D.10] Twisdale, L.

A., et al., " Tornado Missile Risk Analysis," EPRI

~

NP-768, May 1978, EPRI NP-769, May 1978.

i Goodman, J. and Koch, J.

E., "The Probability of a Tornado Missile l

[D.11]

Hitting a Target," Nuclear Engineering and Design 74, (1983).

[D.12] Standard Review Plan, U.S. Nuclear Regulatory Commission, F

7 NUREG-75087.

I t

I, D-20 y

I Table D.1 i

Classification of Tornadoes According to Path Area a Classification Range of Tornado Path Area a (mi-ft)

Al l < a 5 10 2

A2 10 < a $ 10 3

2 < a 5 10 A3 10 0

A4 103<a 10 10' < a 5 10 5

A5 6

5 < a 5 10 A6 10 0 < a 5 10 7

A7 10 i

I 1

I i

I i.

D-21

_. - ~ -.. _,. _ -

c I

Table D.2 Relationship Between Fujita Scale F and f

Damaging Wind Speed w (mph)

J l

Fujita Scale F Range for the Damaging Wind Speed (mph) i FO 40 < w < 72 F1 72 < w $ 112 F2 112 < w $ 157 F3 157 < w $ 206 l

F4 206 < w $ 260 i

F5 260 < w 5 318 F6 318 < w i 380

-I i

i I

i i

f t

D-22

l

~

Table D.3 Specification of Tornado-Generated Missiles UL D

2 Type

~ Description (kg/m )

(a=55')

(a=90')

i A.

Wood plank, 4 in. x 12 in. x 12 ft, weight 200 lb.

125 18.5

.335 0.87 B.

Steel pipe, 3 in. diameter,

)

schedule 40, 10 ft long, veight 78 lb.

152 40

.389 1.01 l

C.

Steel rod, 1 in. diameter x 3 ft long, weight 8 lb.

156 36

.381 0.99 D.

Steel pipe, 6 in. diameter,

}

schedule 40, 15 ft long, g

weight 285 lb.

186 30

.370 0.96 E.

Steel pipe, 12 in. diameter, schedule 40, 15 ft long, weight 743 lb.

242 15

.323 0.84 i

F.

Utility pole, 13-1/2 in, diameter, 35 ft long, weight 1490 lb.

179 31

.373 0.97 2

G.

Automobile, frontal area 20 ft weight 4000 lb.

199 2.75

.273 0.71 l

i.

l i

l I

i i

l

{

I D-23

Table D.4 i

Acceleration Parameters of Potential Missiles C

'C g

D

)

()

Type of Missile

{

C C

f g

D

-3

-3 A

125 0.335 0.87 2.68 x 10 6.96 x 10

-3

-3 B

152 0.389 1.01 2.56 x.10 6.64 x 10

-3

-3 C

156 0.381 0.99 2.44 x 10 6.35 x 10

-3

-3 D

186 0.370 0.96 1.99 x 10 5.16 x 10

-3 l

E 242 0.323 0.84 1.33 x 10'3 3.47 x 10

-3

-3 F

179 0.373 0.97 2.08 x 10 5.42 x 10

-3

-3 f

G 199 0.273 0.71 1.37 x 10 3.57 x 10

-3

-5

" Standard" 170 0.369 0.98 2.17 x 10 5.76 x 10 t

i i

i I

I i

l l

i 1

j D-24 9

. -.. v

- - - -. - ~ - -

\\

Table D.5 List of Counties Near the Plant Site of STP 2

County Population Area (mi )

l Natagorda 37,828 1,127 i

Brazoria 169,587 1,407 Fort Bend 130,846 876 Wharton 40,242 1,086 Jackson 13,352 844 Calhoun 19,574 540 f

Total 411,429 5,880 1

i l

i i

l l

b f

t

~

l 1

I D-25

Table D.6 Annual Number v, of Tornado Occurrences for Six Counties Year v

Year v

6 6

i

.1953 1

1968 5

1954 1

1969 4

1955 4

.1970 8

1956 2

1971 0

1 1957 3

1972 15 1958 1

1973 5

1959 4

1974 4

1960 1

1975 5

1961 5

1976 8

I 1962 1

1977 4

I 1963 1

1978 3

1964 2

1979 2

)

1965 1

1980 3

1 1966 3

1981 7

1967 27 1982 3

1 I

t I

i i

l i

\\

.1 I

i i

l D-26

l Table D.7 Parameters of Loanormal Distribution for Annual Frequency of Tornado Occurrences In Six Counties Lower Limit (5%)

Median (50%)

Mean Upper Limit (95%)

0.42 3.00 7.39 21.35 i-p o

1.099 1.193 l

l 1

1 i

I

(

i r

9 h

l l

l D-27

Table D.8 Comparison of Empirical and Fitted Loanormal Distributions for v, w

Percentile Empirical Lognormal l

25 1

1.34 50 3

3.00 75 5

6.71 90 8

13.87 95 15 21.35 98.33 27 38.10 i

't i

}

I 1

l-t I

i I

.I i

e I

D-28

s,

l Table D.9 Parameters of Distribution for the Adjusted Annual Frequency v Lower Limit (5%)

Median (50%)

Mean Upper Limit (95%)

0.736 5.254 10.704 37.393 O

N 1.659 1.193 i

I e

d l

t l

1 i

D-29

l Table D.10 Distribution of Tornado Path Area Number of Tornadoes for 30 years Probability Probability A-Scale in Texas for Texas for U.S.A.

Al 354

.1297

.1100 A2 906

.3319

.2371 g

A3 844

.3092

.3028 A4 478

.1751

.2567

)

A5 144

.0527

.0907 A6 4

.0015

.0027 A7 0

.0000

.0000 i

f I

l.

9 5

D-30

Table D.ll Best Parameters for Loanormal Distribution of Tornado Path Area for Texas l

P

-3.8076 i

o 2.5697 I

i I

f 4

i I.

i 9

e i

n i

i i

i t

I D-31

4 Table D.12 J

6 Comparison of Empirical and Lornormal Distributions of Tornado Path Area for Texas Empirical Distribution Lognormal Distribution Cumulative Cumulative A-Scale Probability Probability Probability Probability Al

.1297

.1297 1371

.1371 A2

.3319

.4616 3060

.4431 A3

.3092

.7708 3224

.7655 t

A4

.1751

.9459 1605

.9260 A5

.0527

.9986 0377

.9637 l'

A6

.0015 1.0000 0041

.9678 l

A7

.0000 1.0000 0002

.9680 5

f I

I I

[

D-32

4 4

i Table D.13 Joint Number of Tornadoes (A-Scale - F-Scale)

Total Number F0-F1 F2 F3 F4 F5 F6 ny Path Area

'571 94 7

0 0

0 1602 Al 936 i

A2 1081 1736 586 48 2

0 0

3453 A3 531 2162 1429 259 28 0

0 4409 A4 131 1018 1645 732 202 11 0

3739 A5 24 171 413 419 242 52 0

1321 A6 1

1 10 16 10 1

0 39 t'

A7 0

0 0

0 j

8 3

l Total Number by Fujita Scale Total Number I

of Cases 2698 5659 4177 1481 484 64 0

14563 i

t I

i D-33

T

^ ~

Table D.14 Joint Probability Distribution for A and F Scales Marginal Probability FO.

Il F2 F3 F4 F5 F6 by Fath Area Al

.0639

.0392

.0065

.0005

.0000

.1100 A2

.0742

.1192

.0402

.0033

.0001

.0000

.2371 A3

.0365

.1485

.0981

.0178

.0019

.0000

.3028 A4

.0090

.0699

.1130

.0503

.0139

.0008

.0000

.2567 A5

.0016

.0117

.0284

.0288

.0166

.0036

.0000

.0907 A6

.0001

.0007

.0011

.0007

.0001

.0000

.0027 i

i A7

.0000

.0000 Marginal Probability by Fujita Scale I

.1853

.3886

.2868

.1017

.0332

.0044

.0000 1

1 e

i

'Ag s

d e

i I

I i

f D-34

Table D.15 Density of Potential Missiles Lower Limit hedian Upper Limit (5th Percentile)

(50th Percentile)

(95th Percentile)

~4

~4 Averagedensity(n) 2.33 x 10 2.40 x 10

2.48 x 10 P

Nonuniformity (K )

0.39 1.00 2.55 Coefficient Local density (n )

9.42 x 10 2.40 x 10 6.13 x 10 '

-5

~4

~

p i

r

(

l l

It l l 1

l l

D-35 l

e 1

Table D.16 Probability of Injection O(F) For Surface Potential Missiles Fujita Scale Lower Limit Median Upper Limit F2 0.0001 0.0017 0.0305 F3 0.0006 0.0086 0.1230 F4 0.0143 0.0527 0.1943 FS 0.0450 0.1050 0.2449 F6 0.1089 0.1761 0.2847 I

C i

i I

i i

e I

l l

s f

9 1

D-36 i

Table D.17 Probability of Injection n(F) For Elevated Potential Missiles i

Fujita Scale Lower Limit Median Upper Limit F0 O.0008 0.0119 0.1756 F1 0.0020 0.0292 0.4274 I.

F2 0.0029 0.0435 0.6528 F3 0.0098 0.0866 0.7657 F4 0.0789 0.2546 0.8218 F5 0.2160 0.4310 0.8602 F6 0.3529 0.5593 0.8865 t

l e

1 1

i i

D-37

Table D.18 Probability of Injection n(F) for Restrained Elevated Potential Missiir" Fujita Scale Lower Limit Median Upper Limit I

F2 0.0002 0.0043 0.0941 F3 0.0038 0.0375 0.3654 F4 0.0635 0.1857 0.5428 F5 0.2093 0.3701 0.6543 F6 0.3598 0.5092 0.7206

.i l

k

?

I i

l D-38

l l

Table D.19 Median Value for Heiaht Distribution of Airborne Missiles (z = 55 ft, b = 20 ft)

$(z,F)

Fujita Scale (Formula D-49, z > h,)

FO 0.00002 F1 0.00818 F2 0.07053 F3 0.19531 L

l F4 0.34255 FS 0.47867 F6 0.59131 1

t l l e

t i

I i

I D-39

I, j

l Table D.2G i

Probability of Damage to IVC from Tornado-Generated Missiles per Year Median Upper Limit f.

(50th Percentile)

(95th Percentile)

-10 6xId-6 2 x 10 i

i l

t i

i r

i D-40

Table D.21 Point Estimate of Damage Probability Description Notation Value Frequency of t rnado striking area v

5.2541 l

S = 10000 mi2 (per year)

Tornado path area (mi )

a 0.0222

-5 Frequency of tornado striking the P

1.1664 x 10 plant site (per year) 2.40 x 10 '

~

Loca})densityofpotentialmissiles n

(ft P

Fraction of available missiles f

0.1 Density of available missiles (ft-2)

-5 n,

2.40 x 10 2

IVC target area (ft )

A 2980 Probability of injection (F = 1) q(F) 0.0292 Height distribution (z = 55 ft,

$(z, F) 0.00818 F = 1)

-10 Probability of damage P

. 993 x 10 T

I i

' I I

I I

1 1

1 i

D-41 i

.

ML20077R898

Text

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