The probability of a tornado missile hitting a target

Nuclear Engineering and Design 75 (1982) 125-155 North-Holland Publishing Company

125

T H E PROBABILITY OF A T O R N A D O M I S S I L E H I T r l N G A TARGET J. G O O D M A N and J.E. KOCH Bechtel Power Corporation, Norwalk, California 90650, USA Received 12 November 1982

It is shown that tornado missile transportation is a diffusion Markovian process, Therefore, the Green's function method is applied for the estimation of the probability of hitting a unit target area. This probability is expressed through a joint density of tornado intensity and path area, a probability of tornado missile injection and a tornado missile height distribution.

1. Introduction

Generation and propulsion of missiles by tornadoes is a commonly recognized natural hazard to the safe operation of nuclear power plants. All safety related category I buildings and structures are designed to withstand a tornado missile impact; other structures, not so designed, can be damaged. For an assessment of the risk associated with tornado missiles, we have to know the probability of the damage per year (PT). This is expressed as PT = P0 PH PD, where P0 is the probability of the tornado occurrence at the nuclear plant site, PH is the conditional probability of hitting a target given the tornado occurrence, and PD is the conditional probability of target damage assuming a hit. The probability P0 is addressed by Thom [ 1]. The calculation of probability PD is a problem of structural engineering involving the spectrum of missile kinetic energies. In many cases, for the sake of simplicity and conservatism, we can adopt PD = 1 for unprotected targets. Therefore, the remaining concern is the calculation of the conditional probability of strike, PH, given the tornado occurrence. The probability PH can be readily calculated knowing the probability of hitting any arbitrary oriented unit area. Thus, the purpose of this paper is to develop a method of calculating PH. Twisdale et al. [2] have presented a detailed study on the calculation of the probability PH. However, the results of the Twisdale study are highly specific and cannot readily be applied to other targets. No explicit relationships are provided. All information was generated by a Monte Carlo computer code which prevents further analysis and transfer of results to new problems. The method we have developed for calculating PH is very simple. To hit a target, any potential tornado missile has to be injected from the place of storage; i.e., it must become airborne, and transported to the target. Because the probability of injection is dependent on the tornado intensity, we first consider the conditional probability of hitting a target at given intensities. We then multiply these probabilities by the relative frequencies of a tornado of each intensity and add the products. The sum of the products is the probability pn. The probability of missile transportation is described in terms of Green's (or propagation) function. The expression for Green's function is complicated and depends on the set of tornado parameters. Because we don't know in advance what kind of tornado may strike a plant site, it is reasonable from a probabilistic 0029-5493/83/0000-0000/$03.00 © 1983 North-Holland

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J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

point of view to average all sets of tornado parameters that reflect our expectation. This averaging simplifies significantly the entire analysis and results in formulations that allow simple intuitive explanation. We first demonstrate that tornado missile transportation is a Markovian process. This provides a basis for application of statistical mechanics methods. With these results, we express the probability of missile strike through Green's function. Finally, we conclude the main formulation using the technique of averaged Green's function.

2. T h e e q u a t i o n o f m o t i o n o f a t o r n a d o m i s s i l e

The N e w t o n equation of motion of the center-of-mass of a tornado missile in the metric system is:

(i)

m i ~ = Ft) + FL + Fs + Fg

where: m = v = /~ = Ft) = FL = Fs = Fg = The

mass of missile, velocity vector of missile, acceleration vector of missile, aerodynamic drag force, aerodynamic lift force, aerodynamic side force, gravity force. expression for the gravity force is: (2)

Fg = - m g k ,

where g is the gravitational constant and k is a unit vertical vector oriented upward. The drag force F D acts in the direction of the relative wind-missile velocity vector u (see fig. 1). The velocity (u) can be expressed as the vector difference of wind velocity (w) and missile velocity (v) as follows: u = w - v.

(3)

The absolute value of the vector u is denoted as u. The lift force F L is perpendicular to F D and in the plane formed by the vector u and the unit vector #0 along the missile main axis. The side force F s is

I',

FO

u

d

E////////~ ~/////////I ///////// ~/H//////I

¢/////////,

////, //// /¢///////,zAz///////. ¢///////// r/////////. ~///////// ~///////// ,///////// ,/////////t

~,///////// r$

Fig. 1. Aerodynamic forces acting upon a missile tornado.

Fig. 2. Geometrical parameters of a tornado missile.

J. Goodman, J.E, Koch / The probability of a tornado missile hitting a target

127

perpendicular to the vectors F D and F L. The empirical expressions for these aerodynamic forces are: Pa A FD = CD T uu,

r,. =

[,, x p~ A

(4)

x ,,]],

(5)

.

F s = Cs---~ - [/z 0 x u],

(6)

where: = air density A = missile cross-sectional area (see fig. 2). C D = aerodynamic drag coefficient C L = aerodynamic lift coefficient cs = aerodynamic side coefficient The empirical aerodynamic coefficients CD, C L and C s depend on certain aerodynamic parameters and the angle of attack (a) (see fig. 1). For cylindrical missiles they are m / A and l / d , where A, 1 and d are shown in fig. 2. Expressions for C D, CL and C s for several body shapes are given by Redman et al. [3]. For some symmetrical bodies (for example, cylinders) the coefficient C s = 0. The missile trajectory depends on six external parameters: the wind speed (w), the angle of attack (a), the angle (fl) between w and v, two angles O and ff which give the orientation of force F D relative to laboratory system of coordinates and the angle ( ~ ) between F L and the vertical axis (OZ). These parameters are very convenient for a tornado missle problem but the standard mechanical approach uses other parameters: three quasi-external parameters (Euler angles, which can be determined by using the angular m o m e n t u m equations) and three 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 irregular or stochastic part which is due to turbulent fluctuations. In probabilistic risk analysis, the distribution of the possible tornado parameters is considered. Therefore, the entire vector field w(x, y, z, t) is a random vector function. Hence, eq. (1) can be rewritten in the form: (7)

mi~ : R - m g k ,

Table 1 Missile classification Missile type a c c o r d i n g to [5]

m /A

B C D E F "'Standard Missile"

152 156 186 242 179 170

l/d

( k g / m 2)

40 36 30 15 31 33

CD

CL

(for a - 90 °)

(for a - 55 °)

1.01 0.99 0.96 0.84 0.97 0.98

0,389 0,381 0.370 0.323 0.373 0.377

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J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

where R is a random force, equal to F D + F L + F s. The range of the random vector variable R depends on the Fujita F-scale [4] and the aerodynamic parameters m / A and l i d . For most types of tornado missiles considered in section 3.5.1.4 of the Standard Review Plan [5], the aerodynamic parameters and coefficients are similar. The exceptions are wood planks (type A) and automobiles (type G). Typical values for the aerodynamic parameters and coefficients are shown in table 1. The parameters m / A and l / d are calculated according to data given in the Standard Review Plan [5], the aerodynamic coefficients are calculated according to formulas given by Redman et al. [3]. For C D the maximum value corresponds to an angle of about 90 ° and for C L to an angle of about 55 °. Table 1 shows that the variation of aerodynamic coefficients is within 15%. So, instead of considering five different types of tornado missiles, only one standard type of missile with average characteristics is considered. The problem of tornado missile movement has three time scales:

(s) (9) (10)

1"1 = l / w , T2 = ~ / l / g , T~ = R / w ,

where l is the length of the missile and R is the characteristic size of the tornado "footprint". T] is a characteristic fluctuation time, T2 is a characteristic rotation time and T3 is a characteristic translation time. These characteristic times satisfy the inequality: T, < Tz < T 3.

(11)

During a small time period ( - T]), all the parameters determining the random force R ( w , ~, B, O, q,, q~) are practically constant. During a larger time period ( - Tz), these parameters can change significantly because they depend on missile orientation. For simplicity, all parameters determining R are assumed constant during any time period (~'), satisfying the condition: T, < ' r < T2.

(12)

At the end of every time period (~-), these parameters are assumed to change suddenly and randomly. The random step changes in force parameters are represented in fig. 3. This model was used by Twisdale et al [2] for simulation of a tornado missile trajectory (without the above mentioned justification of the time ~-). The movement of a tornado missile is a typical Markov chain because the probability of transition from one point to another depends only on coordinates of these points and the velocity of the tornado missile at the initial point and does not depend on previous history. It follows directly from eq. (7). Because the tornado missile motion is a Markovian process, the methods of statistical mechanics can be employed for its description. Due to the condition (11) this is a diffusion Markovian process. RAROOM PARAMETER i I t I

I I

I

I I

I i i I i

1 Fig. 3. Random step changes in a force parameter.

J. Goodman,J.E. Koch / The probability of a tornado missile hitting a target

129

3. Propagation function for a tornado missile

The probability that an airborne missile located at point (to) at time (to) will hit the unit area of some target at point (r) oriented in the direction (J2) during unit time at moment (t), given tornado occurrence with characteristics F and 3' is denoted as: G ( r 0' to; r - ro, t - to; 1"2;F, 3').

(13)

In expression (13) ro and r are the radius-vectors of the missile center-of-mass, I2 is the unit vector showing the orientation of unit area of the target, F is the Fujita scale of tornado intensity [4] and 3' is a collective index describing all other tornado characteristics. The function G(r o, to; r - r 0' t - t o ; ~2; F, 3') is the propagation or Green's function for a tornado missile. It has the following properties:

1. 2.

G(ro, t o ; r - r o ; t - t o ; ~ ; F , G(ro, t o ; r - r o ; t - t o ; ~ ; F ,

3")=-O, i f t < t o, 3")-*O, if [ r - ro[ --, ~ ,

3.

f£2f~,fG[ro, t o ; r ( X , t ~ ) - r o , t - t o ; I 2 ( X , l ~ ) ' F ,

3"]~ao(X,t~)dXd~tdt=m.

(14) (15) (16)

Property 1 reflects the principle of causality: the missile can hit a target only after it is airborne. Property 2 means that there is some finite radius of missile transportation beyond which the probability of missile propagation is infinitesimally small. In expression (16) the integral is taken over a closed surface ( ~ ) not intersecting the earth surface (see fig. 4). The tornado duration starts at time t] and ends at time t 2. The expression dA = a0(X,/~)dXd~ is the infinitesimal element of the surface • in the curvelinear system of coordinates X and/L. Expressions ao(X, /x) and ~2(X,/L) are given by: [ a(z, x) a( 2' a0(~ ' /~)= [ 0(Y'Z) ]2+ [~7/~)]2+[0(X,/~)x'Y)]

1 ) {a(y,z)

a(z,x), a(x,y) }

(17)

(18)

where i, j, and k are unit vectors of Cartesian coordinates and, for example, ax 0(x,y)_ a(x, ~)

O)_ 2,

Oh

0#

ax Oh

ay 0F

(19)

TRAJECTORY

GROUND

Fig. 4. Illustrationto property 3.

J. Goodman, J.E. Koch / The probability o f a tornado missile hitting a target

130

Property 3 means that every missile eventually intersects the enveloping surface ( ~ ) because it has to fall down to the ground. Consider a small volume dVo = dxodyodz o within a plant site. Let pp(X o, to) be the density of potential missiles, i.e., the number of potential missiles per unit volume near a point ( ~ ) and per unit time at time (to) where: t I <

to <

t2.

The number of potential missiles in the volume dVo during time increment (dto) is: pp(r 0 ,

to)dVodt o.

(20)

The probability that a potential missile becomes airborne (probability of missile injection) given a tornado strike of a given Fujita scale is denoted as:

vl( F, ro).

(21)

The rate (per year) at which tornados strike the point (r0) is:

Po(F, y, ro).

(22)

Thus, the probability per year of missile injection is:

Po( F, y, ro)~( r, ro).

(23)

The probability of a tornado missile injected at time t o at point r0 hitting the arbitrary unit area at point r during time interval (dt) at time (t) is given by the expression:

G( ro, t o , r - ro, t - to; 12; F, 7)dt.

(24)

Therefore, the probability, d P ( r , t, 12; ro, to, F, y), per year of hitting the unit area at point (r) oriented in the direction (12) during time interval (dt) at time (t) by tornado missiles generated during time (dt 0) in the volume (dVo) is the product of the number of potential missiles in the volume (dVo), eq. (20), times the probability that the potential missile becomes airborne, eq. (23), and the probability, eq. (24), of hitting the unit area of some target, given missile injection. This is given by:

d P ( r , t, 12; r0, t 0, F, "y) = P0(F, 2/, ro) ~/(F,

ro)Pp(ro, to)G(ro,

to; r -

r0, t -- t0; 12; F, y)dVodtodt. (25)

The probability P(r, 12; F, y), per year of hitting the unit area at point r oriented in the direction 12 by all potential missiles during the tornado with given characteristics F and ~, is:

P(r, 12; F, V) = f

? f'2eo(g, v, ro)*l( g, ro)Pp(ro, to)G(ro,

to; r - ro, t - to; 12; F, y)dVodtodt.

Vo~,tl ." t I

(26) The inner integral (26) has to be taken over variable t from t o to t2, but due to Green's function Property 1, eq. (14), the lower limit of t I, can be used because the interval from t I to t o gives a zero contribution. Formally, the volume integral fvo". dV0 can be considered as an integral over infinite space. But practically, this integral has a finite volume. The density of potential missiles is equal to zero below and above some elevations. This limits the volume of integration in the vertical direction. Due to Property 2, eq. (15), the volume of integration at the plane X O Y is finite. According to Twisdale [2], the main contribution to the probability of hitting the unit area at point r are missiles located within an area around the point of radius about 300 ft (or 100 m). The maximum credible distance of missile transportation is about 1000 ft

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

131

(or 300 m). The estimates, in this study, of these distances are in good agreement with the results of Monte Carlo simulation [2]. In the area which contributes in the integral, eq. (26), the probability P0(F, ~,, r0) does not depend on the point r0 (the area of constant tornado occurrence rate is about 4000-10000 mi2). The probability ~(F, r0) depends on the orientation distribution of potential missiles. If we assume that this distribution is the same at all points r0 then the function ~ ( F ) will not depend on the radius-vector r0. Eq. (26) takes the form:

P(r'12; F' Y)=P°(F' Y)~l(F) fv, ftt2i ftl zpp(r°'t°)G(r°'t°; r-r°'t-t°;12; F'

(27)

The total probability, P(r, I2), per year of hitting a unit area at the point r oriented in the direction J2 can be found by the summation of eq. (27) over all the possible values of the parameters F and y:

o, = z

O;, ,, dVod,od,

F

y

)

I

The only information about the dependence of Po(F, 3') on F and 3' that is available now is the joint distribution Po(F, a) where " a " is the tornado path area. Excluding parameter " a " from parameter set 3' and denoting the new set as 3":

~Po(F, 3")G(ro, to;r-ro, t-to;12;F, 3")=Po(F,a)G(ro, to;r-ro, t-to;12;F,a ),

(29)

3'

where G is an avearge Green's function determined by the formula: G(r0,/0;

r - to, t - to; 12; F, a) = ]~_,G(ro, to;

r - r0' t - to;/2; F, 3").

(30)

y'

The average Green's function possesses a higher degree of symmetry that simplifies the calculations. Using eq. (29), eq. (28) is rewritten in the following form:

P(r,~2)=~_~Po(F,a)~(F) f f"-f"-Pp(ro, to)G(ro, to;r-ro, , Vo tl

F

'0; 12;

F,a)dVodtodt.

(31)

tl

4. Hitting function for a tornado missile

The joint distribution for the probability of a tornado strike P0(F, a) can be represented in the form: P0(F, a) =

Po(a)q,(Fla),

where ~ ( F l a ) is the relative frequency of occurrence of F-scale tornado given path area (a) and the probability of tornado with area ( a ) striking per year, given by T h o m [1]. The expression (31) can be rewritten as:

P(r, 12) = Po(a)h(r, where

h(r,

121a),

(32)

Po(a)

is

(33)

121a) is a hitting function determined by the formula:

F t~ t~ h(r, 12ia)=~-,+(Fla)il( )f f-f-Pp(ro, to)G(ro, to;r-ro, t to;12;F,a)dVodtodt. F Vo tl tl

(34)

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J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

The hitting function h(r, l2[a) is the probability of hitting the unit area at point r oriented in the direction ~2 given a tornado strike with path area a. Due to randomization over thousands of tornado histories the average Green's function will satisfy the condition of plane and time uniformity and axial symmetry. Because the distribution of tornado parameters ~,' are the same for all points of the plant site and all moment of time, the average Green's function (G) does not depend on x 0, Y0 and t 0. There is no uniformity in the vertical direction due to the gravitational field. Therefore, the average Green's function can be written as:

G(ro, to;r-ro, t-to;g~;F)=g(Zo;Z-Zo,¢(X-Xo)2+(y-yo)2,t-to,

COsX;F),

(35)

where X is the angle between the vertical direction given by unit vector k and the vector I'/: cos X = k . J~.

(36)

Because the Green's function (g) is a scalar and there is only one preferred direction k, the only scalar which can be constructed from the vectors J2 and k is the expression (36). Define R = ~-xT=

Xo) 2 + ( y - y o ) 2 ,

(37)

t - to,

(38)

Then x f r o f h o ft2 I'tz-to

h(r'121a)=Ee~(Fla)n(F)Jo .Io J,, Jo

/

PP[Z°'R't°)g(z°;z-z°'R'T'c°sx;F)

F

× 2~rRdRdzodtodt,

(39)

where R 0 is the radius inside which the Green's function is non-zero and h 0 is the maximum height of the distribution of potential missiles. Expression (39) is not sensitive to the detailed distribution of the density pp(Z 0, R, to). It can be described by integral parameters: the local surface density of potential missiles (r/p) and the maximum height of distribution of potential missiles (h 0). The simplest expressions for pp(Z 0' g,, R, t 0) are: np

PP-ho(t2-t,)'

ifz°
= 0,

(40)

if z o > h o,

for potential missiles distributed over a range of elevations up to maximum height (ho) and Up

PP

(t 2 -

t,) 3(z°)

(41)

for surface distribution on the ground. Putting (40) or (41) into (39) gives:

h(r, k. ~[a) = npY',o( Fla)~l( F)q/(z, F, cos X),

(42)

F

where, for vertically distributed missiles:

't'(z,F, cosx)=t2 lftt2fo'2-'¢, t--~l d t 0

dT~oJodz0J ° lrho

rRo

g(z, zo, R,Y, cosx;F)21rRdR,

(43)

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

133

or, for surface distribution:

1 qt(z,F,c°sx)-t2_t~Jt,

ft2__

ft2--to__

CltoJo

fro aTJo gtZ, Zo,R,T,

cosx;F)2~rRdR.

(44)

Consider the case cos X = 1.

(45)

It corresponds to the horizontal unit area hit by falling missiles. It is easy to see that

fo'~-t" I Zo,¢(X-Xo)2+(y-yo)2,r, - d T Lc+~ ~ L c+~ g[Z, fol2--lo drJ 1* +~~ Ji = * + ~ g[Z, [

Zo, ¢ ( x - Xo)2 + ( y - y 0 )

cosx;F ) dxodyo -= 2 , T, c o s

x;

F)dxdy.

(46)

The identity (46) reflects the so-called principle of reciprocity. The left side of (46), given (45), is the probability that the unit horizontal area at point x, y, z will be hit by a tornado missile during a tornado of intensity ( F ) and path area (a). The right side of (46), given (45), is the probability that a missile injected at point (x0, Y0, z0) at moment (to) intersects the horizontal plane at the elevation (z) from the upper surface. The expression (43) for q'(z, F, 1) is the same probability of intersecting the upper surface of a horizontal plane at elevation (z) averaging over initial elevations of injection point (z0) betwen 0 and h 0 and initial injection times between t~ and t 2. It is clear that g'(0, F, 1 ) = 1,

(47)

simply because every injected missile has to fall down (neglecting hits of other targets resulting in capture). Mathematically, it can be derived from formula (16) if the closed surface ( ~ ) is chosen as a circle of radius greater than R 0 covered by a hemisphere of the same radius. So the quantity '/'(z, F, 1) represents the fraction of airborne missiles exceeding elevation z.

5. Conclusion

Formula (42) gives the probability of a tornado missile hitting a unit area target.

h(r, k.

12]a) =

np~ ep( F]a)~( F)g,'(z, F, cos X). F

The probability PH of hitting any target of surface area q) is given by:

p. : £ f h [ r( ?t, ~t), k . 12( )t, ~t)l a ] a0( ~,/.t )d ~d/.t.

(48)

The expression (42) depends on two plant specific parameters reflecting the distribution of potential missiles: the surface density of potential missiles np in the target vicinity and the maximum height h o of the potential missile distribution. All other parameters can be considered as generic. The first attempt to calculate the function 4,(Fla) was done by Wen and Chu [6]. The continuation of this study [7] extends their work and increases the accuracy. The probability r/(F) depend on the distributions of potential missile orientations and restraint forces. Because these distributions can vary from time even for the same plant, the generic totally randomized distributions are more appropriate for this case. The height distribution of airborne missiles '/'(z, F, cos X) is a generic function depending on one plant

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

134

specific parameter h o. There are three methods of calculation of this function: (a) Model approach based on some simplifying assumptions about tornado missile trajectories; (b) Statistical mechanics approach based on integral and differential equations for Green's functions; (c) Simulation by Monte Carlo method. The calculations of functions ~ ( F l a ) , ~/(F) and ~/'(z, F, cos X) will be considered in Appendix A and Appendix B. Appendix A. A statistical mechanics approach for calculating the height distribution of tornado missiles AI. Introduction The probability (h) of an airborne tornado missile hitting a unit area target is addressed by us before (see section 4) using the Green's function method expressed through a joint density of tornado intensity and path area, a probability of tornado missile injection, and a tornado heigh distribution. The purpose of this appendix is to develop a method of calculating the height distribution of tornado missiles using a statistical mechanics approach. In the process of solving this problem, we also show how to generalize our approach to calculate the probability of target damage. The probability hitting any target of (PH) can be readily expressed through the probability (h). For the unit area located at point r(x, y, z) and oriented in the direction given by unit vector 12(~2,, f2,, ~2=) the probability of a hit given a tornado occurrence with path area (a) is:

h(z, k. Ilia ) = np~.,dp( Fla)~l( F ) ~ ( z , F, cos X),

(1)

F

where: k k.Q F

= unit vector in the vertical direction, =

cosx,

= Fujita scale intensity, F/p surface density of potential missile; ~(F[a) = relative frequency of occurrence of a F-scale tornado given that the path area is a; probability that a potential missile becomes airborne; n(F) ~ ( z , F , c o s x ) = height distribution of airborne missiles. The quantities rip, q~(Fla ) and 7/(F) are addressed by Goodman and Koch [7] (see Appendix B). Here the attention is focused on the calculation of height distribution q'(z, F, cos X)- Because ~/'(z, F, cos X) is the significant part of hitting function h(z, k.f2la ) which can be expressed through Green's (or propagation) function G, alternate methods of calculating Green's function are considered. There are three alternatives for calculting Green's functions (see section 5). A statistical mechanics approach is chosen because it results in an explicit analytical expression that provides some insight into the physical aspects of the problem. The height distribution q'(z, F, cos X) is expressed through the so-called averaged modified Green's function in the coordinate space. The Fokker-Planck equation is developed for the usual Green's function. That is why we consider the relationships between different types of Green's functions and, using these relationships and the Fokker-Planck equation, obtain the equation for height distribution 9 ( z , F, cos X)A2. The relationships between different types of Green's functions and the hitting function According to section 4 the hitting function h(r, ~[a) can be expressed through an averaged Green's function G ( ro, to; r - r0, t - to; 12; F, a) by the formula:

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

135

F

The average Green's function is determined by the expression:

G(ro, to;r-ro, t-to;12;V,a)= ~ G(ro, to,r-ro, t-to;12;r,y ),

(3)

y (y:':a)

where: y is the set of parameters which the Green's function depends on. The function G ( v0' to; r - r o, t - to; 12; F, ~,) is the so-called modified reduced Green's function ( M R G F ) . The function G(r0, to; r - r 0' t - to; 12; F, a) is an averaged modified reduced Green's function (AMRGF). The original Green's function (GF) gives the probability of hitting per unit volume in real space and unit volume in velocity space. The dimensions of GF are:

[GF] =

L - 6 T 3.

(4)

The probability d P ( r o, Vo, to; r, v, t) that a missile that becomes airborne at m o m e n t t o at point r o with velocity vo hits the volume dV near point r at m o m e n t t with velocity between values v and v + d v expressed through GF is: d P ( r 0 , v 0' to; r, v, t) =

G(ro, Vo, to;

r - to, v - v o, t - to; F, ),)dVd3v.

(5)

If an unprotected target with conditional probability of damage given a hit close to unity is considered, the information about the missile velocity distribution is excessive. After eliminating this excessive information the reduced Green's function is:

GR (to, Vo, to; r - to, t - to; F, 7) = f

v)

G(ro, Vo, to; r - ro, v - Vo, t - to; F, y)d3v.

(6)

If the probability of damage assuming a hit is considered, the only information of interest is the velocity distribution of tornado missiles. This information is provided by another reduced Green's function:

GR(r°'v°'t°'v--v°'t •

- t°;F'Y)=

f( v) G(ro,vo,to;r-ro,v-vo,t-to;F,v)dV.

(7)

To distinguish them, the first one is denoted as RLGF and the second one as RvGF (where suscripts L and V denote distributions in real space and velocity space, respectively)• The dimensions of these Green's functions are:

Fig. A I.

136

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

[ReGF ] = L

3,

(8)

[ R v G F ] = L - 3 T 3.

(9)

Consider the small volume d V = d A I v . ~21dt

(10)

shown in fig. A1. This is a skew cylinder with base dA, slant height v d t and altitude Iv. ~2[dt. All missiles with center-of-mass inside volume dV, given by eq. (10), and velocity vector (v) will hit the area dA during time dt. Hence, the probability of hitting area dA during time dt given velocity v is: G(r0, v0, to; r - r0, v - v0, t - to; F, "/)Iv" J21dAdt.

(1 1)

Dividing expression (1 1) by d A d t , the probability of hitting the unit area oriented in the direction 12 per unit time given velocity v is found. Integration of overall velocities v satisfy the condition: v. J2< 0.

(12)

It is found that M R LGF:

G ( r°' v°' t° ; r - r° ' t - t° ; J2; F' "/ ) = f(v.~
(13)

The function G ( r o, vo, to; r - t o ; t - to; 1'2; F, "/) still depends on the initial velocity vo. Averaging expression (13) according to the spectrum of initial velocities, it is found that the final expression for function M R LGF is:

G(ro,to;r-ro,t-to;I2;F,"/)=

f ~(vo)

d%o(vo)f

G(ro,Vo, t o ; r - r o , v - v o , t - t o ; F , " / )

(v.12<0)

× Iv~21d3v.

(14)

where p ( v o ) d 3 v o is the probability that the initial velocity is between vo and vo + d v o. The dimension of M R LGF is:

[MRLGF ] = L-2T -'.

(15)

The dimension of A M R L G F is the same, and the dimension of the hitting function h(x, $21a ) is L -2. The expression for the hitting function through GF is:

h(r,f'/la)=,~__., ~ F

O(Fla).'F)f'2dtof'2dtfv,,dVofv,,d3vof., .,

.,

q

q

"Z (y~a)

Pp ( r ° ' v ° ' , ° ) ~-J2<0)

x G(ro, Vo, to; r - ro, v - vo, t - to; F, y ) l v . I21d3v.

(16)

C o m p a r i n g expressions (1) and (16), the formula for 9 ( z . F. cos X) is derived as follows: t~ ut,

t~

3

f~dtof-dtf to) ut, -'(vo) dVof "(v,,) d-vof U(v.#
±

• ( z , F , c o s X ) = r/p y=~a

× G(ro, Vo, to; r - to, v - v0, t - t o, F,

v)lv. J2ld3v.

(17)

Let the probability of damage given a hit be PD(V, 8) where ~ is the set of parameters other than missile

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

137

velocity (mass, shape, etc.). The probability of hitting the unit area of target h D resulting in d a m a g e is:

hD(r, 12la)=Y'. ~, F

"f -t..a

~,ep(Fla)n(F)f dtof-dt[ dWof dvof t2

8

t~

"t I

~t I

× G(ro, %, to; r - ro, V - V o , t -

3

" (Vo)

~'(Vo)

pp(ro,Vo, to)PD(V, 8 )

"(v-Q
(18)

to" F, 7)

The ratio (18) to (16) gives the averaged probability of damage given a hit, P o .

A3. The equation for the Green's function Green's function GF satisfies the Smoluchowski (or C h a p m a n - K o l m o g o r o v ) equation [8]:

G( ro, Vo, to; r - ro, V - Vo, t - to, F, y ) = ftv,)f~v),G( ro, Vo, to, r' - ro, v' - Vo, t' - to; F, 7) × G(r', v', t'; r - r', v - v', t - t'; F, y ) d V ' d 3 v '.

(19)

This equation is applicable to all M a r k o v i a n processes despite the nature of the objects. Eq. (19) has a simple interpretation. Consider the successive transition from the point r0 at m o m e n t t o with initial velocith v0 to the point r at m o m e n t t with final velocity v through point r ' at m o m e n t t' with intermediate velocith v'. The time of transition from ro to r ' is t' - t o and from r ' to r is t - t'. The total time of transition from r0 to r is t - t 0. Because the processes of transition from ro to r ' and from r ' to r are independent and the probability of transitions depends only on the initial and final points, the expression for the probability of transition from point r0 to r through r ' can be written as:

G(ro, Vo, to; r ' - r0, v' - v0, t' - to; r , ),)G(r', v', t'; r - r', v - v', t - t'; r , ),)

(20)

The transition from ro to r through different intermediate points r ' are mutually exclusive events. Thus, the total probability of transition from ro to r, according to the addition rule of probability, is an integral of expression (20) over all possible points r', i.e., the formula (19). In the case of tumbling tornado missiles, for which the r a n d o m force R applied upon them is changed m a n y times during the flight time, the F o k k e r - P l a n c k a p p r o x i m a t i o n [see A p p e n d i x B] to eq. (19) is used:

3 G ( ro , Vo , to; r - ro , V - Vo , t - t o ; F , y ) 3t 3

+ ~, ~---~i[ai(r,v,t;F,y)G(ro,Vo,

to;r-ro,V-Vo,

t-to;F,'~)]

to;r-ro,V-Vo,

t-to;F,Y)]

i=1 3

+ ~_, O - ~ i [ b i ( r , v , t ; F , y ) G ( r o , V o , i=l 3

3

32

- ½ ~., ~_, 3 x , 3 v ~ [ C i k ( r , v , t ; F , ~ ' ) G ( r o , V o , i~l

to;r-ro,V-Vo,

--½i~=l k=l ~'3 3Xi3Vk32 [ d i k ( r , v , t ; F, y)G(ro,Vo, to; r - r o , V - V o , 3

-½ E i=l

t-to;F,Y)]

k=l

3

32

~-, 3vi3v~ . [ ~ k ( r , v , t ; g , y ) G ( r o , v k=l

t - t o ; F, Y)]

o,to;r-ro,v-v

o,t-to;g,y)]

=0,

(21)

138

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

where x~ and v~ are components of vectors r and v respectively and coefficients a~, b~, c,k, d~k and ~k are determined by formulas: 1 t {f~ v')f v')( x , _ x , ) G ( r , v , t ; r , _ r , v , _ v , t , _ t ; F , ai( r' v' t; F' Y ) = }im, t-7-~-

7)dV,d3v,),

bi(r,v,t;F,y)=lim[.1-~f~ f~ ( v : - v i ) G ( r , v , t ; r ' - r , v ' - v , t ' - t , F , y ) d V ' d 3 v ' ) , t" ~ t ~ t -- t V') v') c,k(r,v,t;F,~,)=liml~f

(22) (23)

((x;-xi)(x'~-xk)G(r,v,t;r'-r,v'-v,t'-t;F,y)

~ t - t J~v'/~')

,,-~,

)< dV'd3v'},

(24)

dik(r'v't;F'y)=lim(1--~--f
v')f~

× dV'd3v'}, f,.k(r,v,t;F,y)=lim

(25)

%)(v'k--vk)G(r,v,t;r'--r,

~

t" ~ t

V')

--v,

-t;F,'y)

× dV'd3v'}.

(26)

A4. The equation for '~'(z, F, cos X)

Eq. (17) expresses the function '/'(z, F, cos X) through Green's function GR. Multiplying eq. (21) by (l/np)pp(ro, Vo, to)V. 12 integrating over variables t o, t, ro, r, v0, v (v-12 < 0) and averaging over all parameters y(y ~ a), an equation is derived for q'(z, F, cos X). The main steps in the derivation of the equation for g'(z, F, cos X) are as follows: multiply eq. (21) by p(Vo)V. 12, integrating over vo and v' (v. 12 < 0) and averaging over parameters 7(7 ~ a), then eq. (21) takes the form: O~(Vo, to; 3

r-

Vo, t Ot

3

to; 12; F,

a)

+

0 [ag(r, t; F, a, 12)(ff(ro,

to; r - ro, t -

to; 12; F, a)]

i=1

~2

~i2=l k=, ~-~ 3x,3xk[Cik(r't; F'a'12)G(r 0 , t o ; r - r

o,t-to;12;F,a)]=0,

(27)

where:

ai(v, t; F, a, 12) Y~ f~)P(vo)f~ 3' (y~a)

ai(r,v,t;F, 7)G(ro,Vo, to;r-ro,v-Vo, t-to;F,y)lv'I2l d3v v-,fi'< O)

¢ff(ro, to; r - ro, t - to; 1'2; F, a)

(28)

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

139

cik(r, t; F, a,

~-" f

P(v°)f,

~,(-y~ a) (vo)

(v.~<0)

cik(r,v,t;F,y)G(ro,Vo, to;r-ro,V-Vo, t-to;F,'~)lv'I'ltd3vd3vo t~(ro, to; r - r0, t - to; 1'2; F, a ) (29)

Gauss' theorem in the velocity space and the condition

G(ro,Vo, t;r-ro, V-Vo, t-to;F,y)~O were used. According to section 4, the uniformity:

AMRLGF has

if Ivl ---' ~

axial symmetry and satisfies the condition of space and time

G(ro, to;r-ro, t-to;~2;F,a)-g(zo; ~/(X-Xo)2+(y-yo)2,Z-Zo, t-to;COsx;F,a) It is shown that, when

a3(z , F, cos X),

~z

AMR LGL satisfies condition (1/np)pp (to, to) and

[a(z, F, cos X)'P(z, F, cos X)]

(31)

(31), the only nonzero coefficients are:

Cll(Z, F, cos X), c22(z, F, cos X), c33( z, F, cos X).

Multiplying eq. (27) by F, cos X) is obtained:

(30)

(32)

integrating over r0, t o and t, the equation for function ~ ( z ,

02S [ ~ ( z , F, cos X ) ' / ' ( z , F, cos X)] = Oz----

(33)

where:

a(z, F, cos X) = a3(z, F, cos X),

(34)

°~(z, F, cos X) = ½c33( z, F, cos X).

(35)

The condition that Green's function is equal to zero at times t I and t 2 is used because the probability of transportation of missiles before and after tornado passage is equal to zero. If the near ground layer is excluded, Green's function depends on displacement in the vertical direction but not on the initial height z o. The individual displacement of any missile relative to the initial point for given values of random parameters does not depend on initial elevation z o because the gravitational force, Fg, does not depend on z 0 because it is significantly less then Earth radius. For near ground distribution of a potential missile, the number of potential missiles located below elevation z 0 is the same for all z0's. Therefore, Green's function (31) depends on the difference z - z 0. Under this condition a and 6~ do not depend on z o, and eq. (33) is simplified.

a(F, cos X)

a ~ ( z , F, cos X) 3z

6~(F, cos X)

a2fft(z, F, cos X) ~Z 2

(36)

denoting

a(F, cosx)

a(F, cos X) 6 ) ( F , cos X) •

(37)

The solution of eq. (36) is found as follows: • (z, F, cos X) =

C(F, cos X)

e - " ' r ....

x,:,

(38)

140

J. Goodman. J.E. Koch / 7"heprobability of a tornado missile hitting a target

where, according to section 4, eq. (47): C ( F , 1)= 1.

(39)

For near ground distribution of potential missiles the number of missiles intersecting the horizontal plane in the upward and downward direction is the same. Therefore C(F,

- t ) - - 1.

(40)

In this case, the isotropic distribution of airborne missiles is a good approximation: • (z, F ) = e ,o(r)z

(41)

Expression (41) is a conservative approximation for the more general formula (38) if it is assumed that:

ao(F ) = a ( r , 1)

(42)

because the maximum number of intersecting missiles corresponds to the horizontal orientation of unit area such that the number of falling missiles cannot be exceeded. To take into account the near ground effect, the following simple model is considered. Let h 0 be the maximum height of potential missiles. Above height h 0 there are only vertically injected missiles (see fig. A2). Below height h 0 there are either vertically or horizontally injected missiles. If a and @ are assumed to be constant, but different, in the areas z < h 0 and z > h 0, the analytical expressions for a and @ are: a(z,F)=a](r),

= a 2 ( F ), @(z,F)=@,(F),

=@2(F),

ifz h 0.

(43)

ifz h o.

(44)

As previously explained, the assumption that cos X = 1 is conservative, denoting:

a,( F) = a,( F )/@,(F), a2(F ) = a2(F)/@2(F).

(45) (46)

The solution of eq. (33) for this case is: e

1 e ,,(F)<,

a,(F)z +

az(F)

• (z, F ) =

, 1+

2(F)

1 e

VERTICALLY INJECTEDMISSILE

HORIZONTALLY INJECTEDMISSILE

GROUN0

Fig, A2.

if0
J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

a l ( F ) e[a2tF)-a.(F)lho a2(F)

e a2(F)z,

if z > h o.

141

(47)

1 + [ ~xl(F)ot2(F) 1] e-a'(•)h° If h o ~ 0, formula (47) reduces to formula (41).

A5. Conclusion

The height distribution of potential missiles is given by formulas (41) or (47). For targets that are tall compared to the maximum height of potential missiles, formula (41) is adequate. Expression (41) depends on one generic height distribution parameter, a 0. For targets that are low compared to the maximum height of potential missiles, formula (47) is more appropriate. Expression (47) depends on two generic height distribution parameters, a I and a 2, and one plant specific parameter, h 0, to determine the maximum height of potential missile distribution. Appendix B. The probability of injection of a potential tornado missile

BI. Introduction

The scenario for a tornado missile hazard consists of three parts: the probability that a tornado of given intensity will strike the plant site, the probability that a potential missile becomes airborne, and the propagation of airborne missiles to the target. We addressed the general methodology of tornado missile propagation in the head article and the height distribution in Appendix A. The purpose of this appendix to develop an explicit expression for the probability of injection of potential missiles (or the probability that potential missiles become airborne).

B2. Model of injection

The model of missile injection used for this paper is similar to that developed in Twisdale [2]. Because Twisdale did not calculate the injection probability in an explicit form, this study develops an explicit expression and presents the numerical results for the injection probability. For tornado missile injection, the restraining forces (FR) must be overcome by aerodynamic forces before motion is possible. These aerodynamic forces are the lift and drag forces discussed in Twisdale [2]. The expression for the aerodynamic forces are: FA~,- fD sin 0 cos q~+ fL (cos 0 cos 4~ cos ~k-- sin ,~ sin +),

(1)

FA,. = f D sin 0 sin q~+fL(COS 0 sin ,p cos ~p+ cos ~ sin +),

(2)

FAz = f D COS 0 --fL sin 0 COS 4'.

(3)

=

The angles 0 and q, give the orientation of the drag force F D in the spherical system of coordinates relative to the earth surface. The angle between the lift force F L and the plane containing the vertical axis and the force F D is denoted as ~b. The angles 0, q~ and + are the Euler's angles. Sometimes other definitions are used, for example,

J. Goodman, ,I.E. Koch / The probability of a tornado missile hitting a target

142

q~ --* ~r - +, + ~ ~r - 0. The ranges for these angles are:

(4) (5) (6)

0<0<~', 0 <4~ < 27r, 0 <~p < 2~', The coefficients f o and fL are determined by the formulas: PaAW 2

fD = CD

~

,

,OaA W 2

fL = CL

2

'

(7)

(8)

where: Oa = air density, A = missile cross-section area, W tornado wind speed, aerodynamic drag coefficient, C D = aerodynamic lift coefficient. C L = The aerodynamic coefficients C D and C L for a cylindrical missile are considered in Redmann [3]. For the "standard missile", the approximate expression is: C D = 0.98 sin3a, CL = 0.98 sin2a cos a,

(9) (10)

where the attack angle a has range: 0 _~< O¢ __< 'ft.

(11)

The restraining forces F R include gravity, frictional, structural, and interlocking forces which tend to resist motion. The expression for restraining forces can be written in the following form: FRx = - K ~ r n g ,

(12)

FRy = - K ~ m g ,

(13)

F~ = -K.mg,

(14)

where m is the missile mass, g is the gravitational constant and K , , K~., and K. are restraint coefficients which show how many times greater (or less) the restraint forces are than pure gravity force (rng). The minus sign indicates that the restraint force is in the opposite direction from the aerodynamic force. Because FR~ includes the gravity force, the coefficient K_ satisfies the inequality: K= > 1.

15)

The coefficients K x and Kv satisfy the following inequalities: Kx>0,

16)

K~. > 0.

17)

For potential missiles lying on the ground, the injection condition is FA. > IF~I,

18)

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

143

or

fD COS O--fL sin 0 cos kO> KLmg ,

(19)

where:

K c - K. is the lift restraint coefficient satisfying the inequality: K L _>

1.

(20)

Expression (19) is the condition for vertical injection. For the case where potential missiles are located at some elevated height, a potential missile can become airborne due to horizontal displacement. In this case, the condition for horizontal injection is:

~FA2~+ FA2,,> F ~ + F~v

(21)

CFAZx+ FA2v> KDmg,

(22)

Ko =

(23)

or

where

The drag restraint coefficient KI) satisfies the inequality: K D

> 0.

(24)

Using previous results (see section 2), the variables w, K D, K L, a, 0, q~, ~b are treated as random functions with uniform distribution in the ranges:

W,(F)
Wz(F ),

(25)

K t < K D_
(26)

K 2 < K L < K 2 + AK,

(27)

0 _< a _< ~r,

(28)

0 < 0 < ~r,

(29)

0 _< 4, < 2~r,

(30)

0<~<2~r,

(31)

where W t ( F ) and W2(F ) are lower and upper levels of wind speed corresponding to some tornado intensity on the Fujita scale [4]. For angles a 0, q~ and ~b, the uniform distribution is a good approximation. For wind speed (w), the uniform distribution is conservative because it overestimates the contribution of higher wind speeds. The real distribution of restrain coefficients K D and K L are not known but, if the interval AK is quite narrow, the uniform distribution is appropriate. In the seven-dimension space of variables w, K D, K L, a, 0, q~ and q~, expressions (25) through (31) specify a right parallelepiped of volume Vv:

Vv = 4¢r4( AK )2[Wz( F) - W t ( F ) ] .

(32)

144

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

The equation (33)

FAz = K L m g

is determined by a seven-dimension surface which divides volume VF into two parts, V~v) and Vv~). In volume V~v), the condition of vertical injections is satisfied, therefore:

Vv~V)+V e

- V v.

(34)

The equation ~FA2x + F~y = K D m g

(35)

is deter_mined on another seven-dimension surface which divides volume V v into another two parts, Vv~H) and Vv~H). The condition of horizontal injection (22) is satisfied in volume V~H~. In the same manner, volume Vv~T) can be introduced where either of the conditions (19) or (22) are satisfied, and V~vH) + V(v~ ) = Vv,

(36)

vv~T'+ Vv~¥'= V v,

(37)

but, generally vv(H)+ vv(V)~: V~T'

(38)

because in some subvolume both conditions (19) and (22) can be satisfied simultaneously. Because of the uniform distribution of all parameters, the horizontal injection ~/(H)(F), vertical injection ~/~V)(F), and total injection ~/(X~(F) probabilities are calculated according to the formulas: rI(H)(F) = V~vH)/Vv,

(39)

T/tV)(F) = v~vV'/vF,

(40)

r/(T)(F) = v(vT)/vv.

(41)

For calculation of the probabilities */(H)(F), ~/~V)(F), and ~/(T)(F), the Monte Carlo method is used. For this purpose, it is covenient to use the scaled variables: x~ = ( K D - K , ) / A K ,

(42)

x2 = ( K L - K 2 ) / A K ,

(43)

x 3 = (w-

(44)

W ~ ) / ( W 2 - W~),

X 4 = a/qT",

(45)

x s = 0/~r,

(46)

x 6 = 4~/27r,

(47)

x 7 = +/2~r.

(48)

All variables x, (i = 1, 2 . . . . . 7) are random variables with uniform distribution in the range from 0 to 1. A computer program [9] generated the random vector x (x~, x 2 . . . . . x 7), calculated the variables W, K D, KL, Or, 0, ~b, ~, and checked conditions (19) and (22).

,I. Goodman, ,I.E. Koch / The probability of a tornado missile hitting a target

145

Table B1 Propability of injection "O(0)for Fujita scale F = 0

Type

.1756 .0000 .1756

.1756 .0000 .1756

.1756 .0000 .1756

.1756 .0000 .1756

.1756 .0000 .1756

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

After N v trials for F-scale tornado, condition (22) is met NF(HI times, condition (19) is met N~vvl times, and either condition (19) or (22) are met N(Fx~ times. Because, for a large n u m b e r of trials N v, the n u m b e r Nv~H) is proportional to Vv(H~, the n u m b e r N(vv) is proportional to V~ v). N u m b e r Nv(T) is proportional to Vv~T), and n u m b e r N v is proportional to VF. Hence, the probability in question can be calculated by the formulas: 7/(H)( r ) = Uv~H) I N v,

(49)

~/(v)(F) = N ( v v ) / u v ,

(50)

r / ( T ' ( F ) = N~vT'/NF.

(51)

The results of calculations for A K = 0.5; K D = 0, 1, 2, 3, 4, 5; K L = 1, 2, 3, 4, 5 and F = 0, 1, 2, 3, 4, 5, 6 are shown in tables B1 to B7. In these tables, the letter H stands for horizontal injection, V for vertical injection, and T for total injection. The n u m b e r of Monte Carlo trials for every case is 10000. This corresponds to an accuracy of about 1%. Table B8 shows the sensitivity of the results as a function of the n u m b e r of simulation trials. The results, rounded to two digits in parenthesis, stabilizes between 10 000 and 100000 trials, and suggests that 10000 trials is a good approximation.

146

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

Table B2 Probability of injection 7(1) for Fujita scale F = 1

Type

.4274 .0000 .4274

.4274 .0000 .4274

.4274 .0000 .4274

.4274 .0000 .4274

.4274 .0000 .4274

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

The upper limit for the probability of injection corresponds to the case in which the restraining force for horizontal injection is friction and for vertical injection is gravity. The lower imit for ~/(F) is quite uncertain because it depends on the m a x i m u m values K D and K L we assign to the potential missiles. The n u m b e r of potential missiles derived from [2] are based on m a x i m u m values for K D = 5 and K e = 5 which we adopt as m a x i m u m possible values of restraint coefficients for potential missiles. The m a x i m u m and m i n i m u m values of 7/(F) extracted from tables B1 through B7 are shown in table B9. Assuming uniform distribution for K D and K L in the range 0
1
J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

147

Table B3 Probability of injection 7/(2) for Fujita scale F = 2

KD••,

Type

0

.6474 .0305 .6528

.6474 .0000 .6474

.6474 .0000 .6474

.6474 .0000 .6474

.6474 .0000 .6474

.0647 .0305 .0941

.0647 .0000 .0647

.0647 .0000 .0647

.0647 .0000 .0647

.0647 .0000 .0647

.0000 .0305 .0305

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0305 .0305

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0305 .0305

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0305 .0305

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .0000 .0000

exactly lognormal; for */(0), 7/(1), and ~(2) the lognormal distribution is a conservative approximation. The same methodology is applied to the vertical injection probability, ~/(Vl(F). The lower and upper limits and the means are extracted from tables B1 through B7 and shown in table B12. The data, for the lognormal fit, are shown in table B13. The results, shown in tables B l l and B13, indicate that the probability of potential missiles being injected (i.e., becoming airborne) is rather small except for the larger F-scale tornadoes.

Appendix C. Numerical example

CI. Problem description Consider how to apply the method developed in the other parts of this paper to an engineering problem. For example, consider a nuclear plant with cooling towers for the ulitimate heat sink. Fans located inside of a tower could be damaged by tornado missiles if no protection is provided. The question is whether this riks is credible or not?

148

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

Table B4 Probability of iniection 7/(3) for Fujita scale F = 3

Type

.7591 .1230 .7657

.759l .0281 .7610

.7591 .O001 .7592

.7591 .0000 .7591

.759l .0000 .7591

.2895 .1230 .3654

.2895 .028l .3104

.2895 .0001 .2896

.2895 .2895

.2895 .0000 .2895

.0592 .1230 .1771

.0592 .0281 .0873

.0592 .O001 .0593

.0592 .0000 .0592

.0592 .0000 .0592

.0004 .1230 .1234

.0004 .0281 .0285

.0004 .O001 .0005

.0004 .0000 .0004

.0004 .0000 .0004

.0000 .1230 .1230

.0000 .0281 .0281

.0000 .0001 .0001

.0000 .0000 .0000

.0000 .0000 .0000

.0000 .1230 .1230

.0000 .0281 .0281

.0000 .0001 .0001

.0000 .0000 .0000

.0000 .0000 .0000

.0000

To answer this question we have to calculate, besides other things, the probability of hiting the top of a tower by a tornado missile. This probability can be readily found by using the approach considered in this paper.

C2. General formula The conditional probability of hitting the top of a tower given a tornado with path area a striking the plant site can be calculated by formula (48) of section 5:. PH = A n p ~ , @( f l a )~T( F )'t'( z, F ) , F

(1)

where A is the horizontal area of the top of a tower and all other notations are as given before. Some simplifications discussed in Appendix A are implemented. The probability of tornado with path area A striking the plant site is determined by Thorn [1]: Po( a ) = v a / S ,

(2)

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

149

Table B5 Probability of injection 71(4)for Fujita scale F = 4

Type

.8184 • 1943 .8218

.8184 .II00 .8206

.8184

•

.8198

.8184 .0102 .8188

.4871 .1943 .5428

.4871 .1100 .5230

.4871 .0466 .5065

.4871 .0102 .4934

.4871 .0000 .4871

.2554 .1943 .3798

•2554 .II00 .3321

•2554 .0466 .2935

.2554 .0102 .2651

.2554 .0000 .2554

.1046 .1943 .2751

.I046 .II00 .2063

.I046 .1504

.1046 .0102 .1148

.1046 .0000 •1046

•0188 .1943 •2102

.0188 •1100 .1285

.0188 .0466 .0654

.0188 .0102 •0290

.0188 .0000 .0188

.0000 .1943 .1943

.0000 .II00 .II00

.0000 •0466 .0466

.0000 .0102 .0102

.0000 .0000 .0000

.0466

.0466

8184 .0000 • 8184

t

where u is a frequency per year of striking the area S having the same tornado occurrence characteristics as the plant site. We assume S = 4000 mi 2. The distribution f ( a ) for tornado path area was developed by T h o m [1]. All other parameters have some uncertainty too. Therefore, the total probability of hitting the top of a tower PT(a, ~) is a random function of parameters a and ~:

PT( a, ~) = Po( a ) P n ( a, ~),

(3)

where ~ is a set of parameters determining the distribution for ~,, np, ~ ( F ) and ff'(z, F). We developed the special computer code which calculates mean, variance, median, upper and lower limits of any confidence interval for PT" However, for the purpose of illustration we will show a simple point estimate based on manual calculation. Let the distribution function for random parameters be f ( a , ~). Then the expectation fix for probability PT(a, ~) is:

PT = f P. ( a, ~ ) f ( a, f ) d a d f

(4)

150

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

Table B6 Probability of injection r/(5) for Fujita scale F = 5

Type

.8584 .2449 .8602

.8584 .1728 .9598

.8584 .I192 .9596

.8584 .0751 .8594

.8584 .0377 .8593

.6208 .2449 .6545

.6208 .1728 .6442

.6208 .I192 .6396

.6208 .0751 .6359

.6208 .0377 .6313

.4300 .2449 .5212

.4300 .1728 .4926

.4300 .I192 .4773

.4300 .0751 .4652

.4300

.2784 .2449 .4260

.2784 .1728 .3840

.2784 .I192 .3566

.2784 .0751 .3339

.2784 .0377

.1715 .2449 .3584

.1715 .1728 .3074

.1715 .1192 .2709

.1715 .0751 .2399

.1715 .0377 .2074

.0827 .2449 .3036

.0827 .7728 .2424

.0827 .I192 .1964

.0827 .0751 .1564

.0827 .0377 .1204

.0377 .4514

.3088

or pa _ fix = - - ~ - A n p Y ~ F

( F)~I( F)~(

z, F ) ,

(5)

where bars over all quantities are m e a n values and 4(F)=af

1

° a'~(Fla)f(a)da.

(6)

N o w we show step-by-step calculation of/~T a c c o r d i n g to f o r m u l a (5).

C3. Point estimate calculation The first factor in f o r m u l a (5) is ~. A c c o r d i n g to the n a t i o n a l severe storms forecast center file [10] for this p l a n t site ~ = 1.50. The m e a n p a t h area of t o r n a d o d according to T h o m [1] is equal to 2.82 mi 2. A r e a

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

151

Table B7 Probability of injection "0(6) for Fujita scale F = 6

Type

.8857 .2847 .8865

.8857 .2214 .8864

.8857 .1737 .8863

.8857 .1360 .8862

.8857 .1034 .8862

.6999 .2847 .7206

.6999 .2214 .7148

.6999 .1737 .7122

.6999 .1360 .7107

.6999 .]034 .7093

.5623 .2847 .6210

.5623 .2214 .6041

.5623 .1737 .5954

.5623 .1360 .5903

.5623 .]034 .5862

.4346 .2847 .5423

.4346 .2214 .5135

.4346 .1737 .4964

.4346 .1360 .4860

.4346 .1034 .4773

.3208 .2847 .4774

.3208 .2214 .4404

.3208 .1737 .4153

.3208 .1360 .3979

.3208 .1034 .3841

.2384 .2847 .4285

.2384 .2214 .3863

.2384 .1737 .3568

.2384 .]360 .3352

.2384 .1034 .3167

S = 4000 mi 2. Therefore the frequency of t o r n a d o occurrence/~0 at plant site is: fi0 -

~d S

1.50 x 2.82 4000 - 1.06 X 10-3 per year.

We assume that the horizontal area of the top of the tower A = 6.75 × 103 ft 2 and tower height z = 45 ft. According to survey data given in [2] the average n u m b e r of s t a n d a r d potential missile at area S m = 2.5 x 107 ft 2 a r o u n d p l a n t site is N = 2650. Therefore, the average density np of potential missiles is: N no-

am

2650 - 1.06 x 10 4 per sq. ft. 2.5 X 107

We have to w a r n that local density near a target can vary a r o u n d this n u m b e r by three times. All of the above n u m b e r s are site specific. Now we consider the expression 6 ( ( z ) = Y'~ ~ ( F ) ~ l ( r ) ~ ( z , F=0

r),

(7)

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

152

Table B8 Sensitivity of ~ to the number of trials

q(H)(1) for KD= O, KL = 1

I/v-~

6L °N -

qlO5

qlO5 lO0

.47000 ( . 4 7 )

.lO00

.0830

I000

.44600 ( . 4 5 )

.0316

.0277

lO,O00

.42740 ( . 4 3 )

.OlO0

.O151

lO0,O00

.43397 ( . 4 3 )

.0032

Table B9 Maximum and minimum values for rl(F)

F

Maximum n(F)

Minimum n(F)

0.1756

0.0000

0.4274

0.0000

0.6528

0.0000

0.7657

0.0000

0.8218

0.0000

0.8602

0.1204

0.8865

0.3167

Table BI0 Means for rl(F)

F

Mean 0.0293 0.0712 0.1239 0.2082 0.3281 0.4708 0.5817

I

J. Goodman, J.E. Koch / The probabifity of a tornado missile hitting a target Table B 1 1 Lognormal distribution for 77( F ) F

Lower Limit

Median

Mean

Upper Limit

0.0008

O. Oll9

0.0454

0.1756

0.0020

0.0292

O. ll05

0.4274

0.0029

0.0435

0.1687

0.6528

0.0098

0.0866

0.2083

0.7657

0.0789

0.2546

0.3282

0.8218

0.2160

0.4310

0.4708

0.8602

0.3529

0.5593

0.5817

0.8865

Table B12 Lower and upper limits and means for *I(V)(F) F

Lower Limit

Mean

Upper Limit

0

0

0

0

0

0

0

0.0061

0.0305

0

0.0302

0.1230

0

0.0722

0.1943

0.0377

0. I199

0.2449

0.1034

0.1838

0.2847

Table B13 Lognormal distribution for r/( v ) ( F ) F

Lower Limit

Median

Mean

Upper Limit

2

O.O001

0.0017

0.0079

0.0305

3

0.0006

0.0086

0.0318

0.1230

4

0.0143

0.0527

0.0722

0.1943

5

0.0450

0. I050

0.1199

0.2449

6

0. I089

0.1761

0.1838

0.2847

154

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

Table C1 Calculation c (z = 45 ft) Fujita scale:

FO

F1

F2

F3

F4

F5

F6

~(F) ~(F) 5(F) ~(z = 45 ft, F) ~ ( F ) ~(F) ~(z, F)

0.2013 0.0000 0.0915 0.0163 0.0000

0.4336 0.0000 0.0339 0.2175 0.0000

0.2592 0.0061 0.0159 0.4889 7.73 × 10-4

0.0818 0.0302 0.0087 0.6760 1.67 × 10 3

0.0194 0.0722 0.0053 0.7878 1.10× 10 3

0.0038 0.1199 0.0034 0.8581 3.91 x 10 4

0.0009 0.1838 0.0024 0.8976 1.48 × 10 4

which shows the fraction of total n u m b e r of p o t e n t i a l missiles b e c o m i n g a i r b o r n e a n d elevating a b o v e the target. R e c o m m e n d e d values for ~ ( F ) a n d ~ ( F ) are shown in table C1. W e a s s u m e d that all p o t e n t i a l missiles are at zero elevation a n d considered only vertical injection. U n d e r this a s s u m p t i o n the f o r m u l a for height d i s t r i b u t i o n q'(z, F ) takes the form: ~ - ( z , F ) = e -"(F)z,

(8)

where i f ( F ) and ~ ( z , F ) for z = 45 ft are shown in table C1. A d d i n g the n u m b e r s in the last row of table C1 we get the value: c=4.08×

10 3.

Therefore, the c o n d i t i o n a l p r o b a b i l i t y of hitting the top of a t o w e r / s H is: ffH =AffpC = 6.75 × 10 3. 1.06 × 10 - 4 . 4.08 × 10 - 3 = 2.92 × 10 -3 . T h e total p r o b a b i l i t y of hitting the top of a tower is: f i t = PoPH = 1.06 × 10 - 3 . 2.92 × 10 -3 = 3.10 × 10 -6 per year.

Each of the two r e d u n d a n t cooling towers loses its c a p a b i l i t y to reject heat if two or m o r e of the four fans in each tower is d a m a g e d . Since the fans are s e p a r a t e d physically, one missile is unlikely to d a m a g e two fans. Thus, to i n c a p a c i t a t e one tower two missiles are required. It is obvious that the second missile strike is not an i n d e p e n d e n t event. A l t h o u g h the details are b e y o n d the scope of this a p p e n d i x , the c o n d i t i o n a l p r o b a b i l i t y of hitting two fans PCD, given that the first has hit the top of the tower is: P c D = 4.9 × 10 -3. Hence, the total p r o b a b i l i t y of i n c a p a c i t a t i n g of one cooling tower PD is equal: PD=PcoP-v=4"93×lO

3.3.10×

10 - 6 = 1 . 5 × 10 - 8 p e r y e a r .

C4. Conclusion The p r o b a b i l i t y of d i s a b l i n g one of two r e d u n d a n t cooling towers calculated a b o v e is a b o u t 2 × 10 -8. This n u m b e r is too low to consider the t o r n a d o missile hazard to cooling tower as credible.

J. Goodman, J.E. Koch / The probability of a tornado missile hitting a target

155

Acknowledgement A u t h o r s t h a n k D r . J.L. S h a p i r o , C h i e f N u c l e a r / E n v i r o n m e n t a l E n g i n e e r , B e c h t e l P o w e r C o r p . , for e n c o u r a g e m e n t , s u p p o r t a n d r e v i e w of this w o r k a n d Dr. G. A p o s t o l a k i s for useful discussions.

References [1] H.C.S. Thom, Tornado probabilities, Monthly Weather Review (Oct.-Dec., 1963) 730. [2] L.A. Twisdale et al., Tornado Missile Risk Analysis, EPRI-768, 769 (May, 1978). [3] G.H. Redmann et al., Wind field and trajectory models for tornado-propelled objects, EPRI 308, Technical Report 1 (February, 1976). [4] J.J. Fujita, Proposed characterization of tornadoes and hurricanes by area and intensity, The University of Chicago, SMRP Research Paper No. 91 (February, 1971). [5] Standard Review Plan, U.S. Nuclear Regulatory Commission, NUREG-75/087, section 3.5.1.4. [6] Y.K. Wen and S.L. Chu, Tornado risks and design wind speed, Journal of the Structural Division, Proceeding ASCE, 99 No. S.T. 12 (Dec., 1973) 2409. [7] J. Goodman and J.E. Koch, A generic joint distribution for tornado intensity and path area, to be published. [8] Ta-You Wu, "Kinetic equations of gases and plasmas," Addison-Wesley Publishing Company, Palo Alto-London-Don Mille, Ontario. [9] IMSL Library Reference Manual, Edition 8, IMSL LIB-0008, International Mathematical and Statistical Libraries, Inc. (June, 1980). [10] U.S. Tornado Breakdown by Counties 1950-1980, 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.