...

Povrosnyk-FinalReport.pdf

by user

on
Category: Documents
5

views

Report

Comments

Transcript

Povrosnyk-FinalReport.pdf
TFC: VBA Code for Estimating Target Fragility to Tornado
Generated Missiles
by
Nataliya Povroznyk
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2012
© Copyright 2012
by
Nataliya Povroznyk
All Rights Reserved
ii
ACKNOWLEDGMENT
I’d like to thank my parents Myroslav Povroznyk and Galyna Povroznyk for their
encouragement and support throughout my collegiate career and my RPI professors and
mentors for their teachings and guidance over the years.
iii
CONTENTS
ACKNOWLEDGMENT .................................................................................................. iii LIST OF TABLES ............................................................................................................ vi LIST OF FIGURES ......................................................................................................... vii GLOSSARY ................................................................................................................... viii LIST OF SYMBOLS ........................................................................................................ ix ABSTRACT ..................................................................................................................... xi 1. Introduction.................................................................................................................. 1 1.1 Background ........................................................................................................ 1 2. Methodology ................................................................................................................ 3 2.1 Problem Description .......................................................................................... 3 2.2 Approach Overview ........................................................................................... 3 2.3 Tornado Missile Fragility Analysis.................................................................... 5 2.3.1 Missile Injection ..................................................................................... 5 2.3.2 Site Layout ........................................................................................... 10 2.3.3 Missile Distribution.............................................................................. 13 2.3.4 Missile Simulation ............................................................................... 16 2.3.5 Missile Propagation.............................................................................. 19 2.3.6 Target Interaction ................................................................................. 21 2.3.7 Target Fragility .................................................................................... 22 3. Results........................................................................................................................ 23 3.1 Missile Injection Simulation Results ............................................................... 23 3.2 Missile Strike Simulation Results .................................................................... 24 3.3 Target Damage Probabilities ............................................................................ 30 3.4 Resulting Target Fragilities .............................................................................. 31 4. Conclusion ................................................................................................................. 32 5. References.................................................................................................................. 33 iv
Appendix A: VBA Script................................................................................................. 34 Part 1: Missile Injection ............................................................................................. 34 Part 2: Missile Propagation ........................................................................................ 39 v
LIST OF TABLES
Table 2-1: Results of Missile Injection Simulations.......................................................... 9 Table 2-2: Zone Definition .............................................................................................. 11 Table 2-3: Missile Definition........................................................................................... 14 Table 2-4: Missile Distribution ........................................................................................ 15 Table 2-5: Histogram of Tornado Directions for CPNPP ............................................... 20 Table 2-6: Maximum Travel Distances for Missiles as a Function of Tornado Intensity 21 Table 2-7: Target Definition ............................................................................................ 21 Table 2-8: Sample Results for Missile Propagation Simulation for an F1 Tornado ....... 22 Table 3-1: Missile Simulation ......................................................................................... 23 Table 3-2: Missile Simulation ......................................................................................... 24 Table 3-3: Missile Strikes ................................................................................................ 25 Table 3-4: Calculated Strike Probabilities ....................................................................... 26 Table 3-5: Assumed Target Damage Probabilities .......................................................... 30 Table 3-6: Detailed Fragility Summary, Target 1, F3 Tornado ....................................... 31 Table 3-7: Target Fragility Results .................................................................................. 31 vi
LIST OF FIGURES
Figure 2-1: Missile Injection Script Simplified Flowchart ................................................ 9 Figure 2-2: Comanche Peak Nuclear Power Plant Hypothetical Zone Definition .......... 12 Figure 2-3: Illustration of the Ray Casting Algorithm .................................................... 16 Figure 2-4: Tornado Direction Histogram ....................................................................... 19 Figure 3-1: Missile Strikes for an F0 Simulation ............................................................ 27 Figure 3-2: Missile Strikes for an F1 Simulation ............................................................ 28 Figure 3-3: Missile Strikes for an F2 Simulation ............................................................ 28 Figure 3-4: Missile Strikes for an F3 Simulation ............................................................ 28 Figure 3-5: Missile Strikes for an F4 Simulation ............................................................ 29 Figure 3-6: Missile Strikes for an F5 Simulation ............................................................ 29 vii
GLOSSARY
ANS
American Nuclear Society
ASME
American Society of Mechanical Engineers
CPNPP
Comanche Peak Nuclear Power Plant
EPRI
Electric Power Research Institute
NPP
Nuclear Power Plant
NRC
Nuclear Regulatory Commission
NWS
National Weather Service
PRA
Probabilistic Risk Assessment
RG
Regulatory Guide
SSC
Systems, Structure, or Component
TORMIS
Tornado Missile Risk Evaluation Methodology
VBA
Visual Basic for Applications
WEC
Westinghouse Electric Company
viii
LIST OF SYMBOLS
Symbol
A
Unit
Description
Graph
Units1
Cross sectional area of a missile
CD
None
Drag force coefficient
CL
None
Lift force coefficient
FAx
N
Aerodynamic forces in the x-direction
FAy
N
Aerodynamic forces in the y-direction
FAy
N
Aerodynamic forces in the y-direction
fD
N
Drag force
fL
N
Lift force
FRx
N
Restraining forces in the x-direction
FRy
N
Restraining forces in the y-direction
FRy
N
Restraining forces in the y-direction
g
m/s2
Gravitational constant
KD
None
Drag restraint coefficient
KL
None
Lift restraint coefficient
Kx
N
X-direction restraint coefficient
Ky
N
Y-direction restraint coefficient
Kz
N
Z-direction restraint coefficient
m
kg
Missile mass
Ninj
None
Number of simulated missile injections
NT
None
Total number of missile injection simulations
Pinj
None
u
v
Probability of missile injection into the tornado wind
stream
Graph
X-coordinate of the intersection between a ray and a
Units1
segment
Graph
Y-coordinate of the intersection between a ray and a
1
Units
segment
ix
Symbol
Unit
Description
w
m/s
Tornado wind speed
Graph
x1
Units1
Graph
x2
Units1
Graph
xmax
Units1
Graph
xmin
Units1
Graph
xP
Units1
Graph
y1
Units1
Graph
y2
Units1
Graph
ymax
Units1
Graph
ymin
Units1
Graph
yP
Units1
First x-coordinate of a given segment
Second x-coordinate of a given segment
Maximum of x1 and x2
Minimum of x1 and x2
Simulated missile x-coordinate
First y-coordinate of a given segment
Second y-coordinate of a given segment
Maximum of y1 and y2
Minimum of y1 and y2
Simulated missile y-coordinate
α
Degrees
Attack angle
θ
Degrees
Angle of orientation of the drag force
ρa
kg/m3
Air density
φ
Degrees
Angle of orientation of the drag force
ψ
Degrees
1
Angle between the lift force and plane containing the
drag force
Graph units refer to the arbitrary distance units presented in Figure 2-2.
x
ABSTRACT
This project develops a method that can be used to estimate the impact of tornadogenerated missiles on the safety of pre-defined targets using a Visual Basic for
Applications (VBA) script title Tornado Fragility Calculator (TFC) which simulates the
tornado missile environment at a site as well as the target locations relative to the
defined missiles and outputs the simulated fragility of each target. The target fragility
represents the probability of target damage due to tornado driven missile strikes. Target
fragility varies as a function of tornado wind speed and is a product of several major
factors: probability of missile injection into the tornado wind stream, probability of the
injected missiles striking the target, and probability of target damage that would result in
target inoperability due to the missile strike. The VBA script developed contains two
major parts. Part 1 simulates missile injection for eight different types of missiles. Part 2
simulates missiles contained on the site and computes the probability of the eight missile
types striking each target. Conditional target damage probabilities are assumed. Target
fragility is computed as the product of the number of missiles, the missile injection
probability, missile strike probability, and target damage probability. The Comanche
Peak Nuclear Power Plant site layout is used for this project; however, missile and target
types, counts, and their locations are fabricated.
xi
1. Introduction
External hazards have always been a potential threat to the safety of nuclear power plants and
with the recent events in Fukushima utilities and regulators are examining these hazards closer
than ever. The Nuclear Regulatory Commission (NRC) Regulatory Guide 1.200 Revision 2
(Ref. 1) endorses the ASME/ANS PRA Standard (Ref. 2), Part 7 of which requires the
evaluation of the hazard resulting from wind-generated missiles using a high-wind missile
hazard analysis methodology. With NRC’s endorsement of the ASME/ANS PRA Standard,
Nuclear Power Plants (NPPs) are required to perform these evaluations in order to satisfy the
ASME/ANS PRA Standard. Due to the complex nature of performing a high-wind missile
analysis, there are no specific methodologies currently endorsed by the NRC; rather, each
analysis is peer-reviewed on a case-by-case basis.
This project investigates the elements used to perform a tornado missile analysis and uses that
information to develop Visual Basic for Applications (VBA) code in order to simulate the
injection and propagation of missiles in order to compute target fragilities. The code, contained
in a macro-enable Excel spreadsheet, also contains functions for propagating the results of the
simulations to compute the overall target fragilities.
1.1 Background
Below is a brief summary of two key publications that are relevant to this project. The methods
employed by each publication, as well as the results, are summarized. In addition to these,
Nuclear Regulatory Commission Regulatory Guide 1.76 (Reference 9) defines the design basis
missiles that need to be considered for nuclear power plants.
The Probability of a Tornado Missile Hitting a Target (Ref. 4)
Published in 1982 by Goodman and Koch, this paper documents an approach for determining
the probability of a missile hitting a target, which in this case is a nuclear power plant
component. The authors demonstrate that missile transport in a tornado is a Markovian process,
which provides a basis for application of statistical methods. The problem is broken down into
two segments: the probability of missile take-off which is calculated using simulated injection
conditions, and the transport of the missile using the Markovian process and an averaged
1
Green’s function. A simulation using Monte Carlo methods is performed to find probabilities
of injection for a “standard” missile type for various tornado intensities as a function of vertical
and horizontal restraint coefficients.
EPRI Tornado Missile Simulation and Design Methodology (Ref. 3)
Published in 1981 by the Electrical Power Research Institute, this project included
experimental work to measure missile aerodynamics, analytic developments of tornado wind
fields and missile trajectory models, and missile impact tests on reinforced concrete barriers.
The end product of this work was simulation-based software titled TORMIS which combined
the National Weather Service tornado data sets and EPRI’s experimental results with a Monte
Carlo simulation method for propagating the transport of tornado missiles. TORMIS and other
variations of the code are currently the software of choice for performing high wind missile
analyses. However, TORMIS is an expensive option for utilities. In addition to budgetary
concerns, it is a proprietary product to EPRI and its use is limited to EPRI members.
2
2. Methodology
2.1 Problem Description
The goal of this project is to outline a methodology that can be used to perform a site-specific
analysis of target fragilities at a Nuclear Power Plant (NPP) site. The methodology must be
sufficiently accurate in determining probability of target damage due to tornado generated
missiles while maintaining its practicality for use in Probabilistic Risk Assessment (PRA)
applications. The final result of the methodology is expressed as the probability of target
damage as a function of tornado intensity. The Comanche Peak Nuclear Power Plant (CPNPP)
site layout is used as a “pilot” for this project to determine the site specific result.
2.2 Approach Overview
The characteristics of tornado generated missiles are highly stochastic in nature, which makes
the problem of solving their impact on nearby targets nearly impossible using deterministic
approaches. Due to this, the impacts of tornado generated missiles on targets have previously
been solved by performing simulations of the various factors involved. The simulations are
performed many times until the result convergence is achieved. Similar to such previous
approaches, the methodology developed during the course of this project uses a Monte Carlo
approach. Consistent with the goal, the methodology developed provides a template that can be
used at any NPP site, which defines the surrounding environment and simulates the impact of
missiles as a function of tornado intensity. The script that performs the simulations is written in
Visual Basic for Applications (VBA), and works in conjunction with Microsoft Excel.
Target fragilities are a product of missile injection, missile strike, and target damage due to the
strike. The scope of this project is limited to computing the missile injection and missile strike
probabilities. Conditional target damage probabilities are assumed by starting with assigning a
damage probability of 1.0 for a F5 tornado, and probabilities that decrease by an order of
magnitude for each decrease in tornado intensity (e.g., probability of .01 is assigned to an F4
tornado, probability of .001 to an F3 tornado, etc.).
3
The missile injection is based on the methodologies provided in Reference 4. Subroutine
“Missile_Injection” calculates the probability of a missile’s injection into the tornado wind
stream as a function of the missile’s cross sectional area, mass, tornado wind speed, and any
restraint coefficients. The developed subroutine is re-run for each missile type, which produces
the injection probabilities that can be used in computing the final target fragilities.
Conservatively, it is assumed that there are no restraint coefficients, other than gravity, that
would inhibit missile injection for all but one missile types. Trees are an exception to this
assumption, where the restraint coefficient in the vertical direction is assumed to be 2.0 and in
the horizontal directions it is assumed to be 1.0.
The missile propagation script computes the probability of each missile type striking each
target, in which The CPNPP site layout is used as a pilot for the software simulation. A
coordinate system is superimposed over the site layout, as shown in Figure 2-2. The site is
divided into zones, which are defined by their vertices on an (x,y) coordinate system, as shown
in Figure 3.2-2. Each zone contains a specified missile distribution, as would be determined
during site walkdowns. For the purposes of code development, missile distributions were
assumed. The setup allows the missile distributions to be easily changed within the excel
spreadsheet when the precise missile distributions are known. For each zone, subroutine
“Missile_Propagation” simulates missile origins by generating a random (x,y) coordinate such
that the x-coordinate is between the minimum of the zone’s x-coordinates and the maximum
of the zone’s x-coordinates, and similarly the y-coordinate is between the minimum of the
zone’s y-coordinates and the maximum of the zone’s y-coordinates. Using a ray-casting
algorithm, it is determined whether the simulated point is actually within the defined zone
boundaries. The coordinate is ‘discarded’ if it is not within the boundaries of the zone. All
simulated missile origins are written to two arrays xP1() and yP1(). After all missile origins are
simulated, the missile type is determined.
The distance and direction of travel of each missile are simulated. The maximum distance a
missile can be travel varies depending on tornado intensity. Table 2-6 lists the maximum travel
distances for missiles as a function of tornado intensity. The missile travel distance is assumed
to be uniformly distributed between 0 and the maximum travel distance. The direction of travel
4
of each missile depends on the direction of tornado travel. Using the script developed in
Reference 5, tornado records pertaining to the Comanche Peak site are selected. Most records
have an associated starting and ending coordinate, which allows for determining the overall
direction of the tornado travel. Out of all the tornado records that have occurred within 200
miles of the Comanche Peak site, approximately 1500 had starting and ending coordinate
information. Section 2.3.5 describes how missile direction is simulated using this information.
Once the distance and angle of missile travel are known, the missile’s end point is calculated.
An algorithm is then used to check whether the missile’s trajectory intersects with any of the
target areas. Similar to zone definitions, target areas are defined by their vertices on the same
coordinate system shown in Figure 3.2-2. An intersection with any of the target boundaries is
counted as a missile strike. The number and type of each missile strike incurred by each target
is tallied into an array. The probability of target strike is provided for each missile and tornado
intensity.
Target fragility for missile type “a”, for example, is calculated as the product of the total
number of missiles “a”, the probability of missile “a” injection, and the probability of missile
“a” strike on the target. The details associated with each aspect of this approach are provided in
the sections below.
2.3 Tornado Missile Fragility Analysis
2.3.1
Missile Injection
Simplified methods for determining missile injection probabilities for various vertical and
horizontal restraint coefficients are presented in Appendix B of Reference 4. These methods
are applied by reproducing the simulations presented in Reference 4 for to each missile type
listed in Table 2-3. The methodology presented in Reference 4 is reproduced in this section for
record. As noted in Reference 4, results were shown to converge when the simulation was run
10,000 times, which is the number of simulations performed to determine the missile typespecific injection conditions.
5
For tornado missile injection the restraining forces (FR) must be overcome by aerodynamic
forces (FA) before motion is possible. The expressions for the aerodynamic forces in each
direction are presented in Equation 2.3‐1 through Equation 2.3‐3, where FAz represents the
vertical direction.
FAx  f D sin  cos   f L (cos  cos  cos  sin  sin ) Equation 2.3‐1 FAy  f D sin  sin   f L (cos  sin  cos  cos  sin ) Equation 2.3‐2 FAz  f D cos   f L sin  cos Equation 2.3‐3 The angles θ and φ give the orientation of the drag force fD in the spherical system of
coordinates relative to the earth surface. The angle between the lift force fL and the plane
containing the vertical axis and the force FD is denoted as ψ. The coefficients fD and fL are
determined by Equation 2.3‐4 and Equation 2.3‐5.
f D  CD
f L  CL
 a Aw 2
2
 a Aw 2
2
Equation 2.3‐4 Equation 2.3‐5 where,
ρa = air density (0.0748 lbm/ft3, or 1.1959 kg/m3),
A = missile cross-section area (length * width),
w = tornado wind speed,
CD = aerodynamic drag coefficient,
CL = aerodynamic lift coefficient.
CD and CL for a cylindrical missile are considered in Reference 8. Tabulating and using the
values presented in Reference 5 is a potential area of improvement in determining missile
injection probabilities. At this stage of the project, an approximate expression from Reference
6
4 is used to determine the drag and lift coefficients, as shown in Equation 2.3‐6 and Equation 2.3‐7, respectively.
C D  0.98 sin 3  Equation 2.3‐6 C L  0.98 sin 2  cos  Equation 2.3‐7 where,
α = the attack angle; 0 ≤ α ≤.
The restraining forces include gravity, frictional, structural, and interlocking forces which tend
to resist motion. The restraining forces in each direction can be summed up by the expressions
in Equation 2.3‐8.
(a) . FRx   K x mg (b) FRy   K y mg Equation 2.3‐8 (c) FRz   K z mg where,
m = missile mass,
g = gravitational constant,
Kx, Ky, Kz = restraint coefficients.
For potential missiles located on the ground the injection condition is expressed by Equation 2.3‐9.
FA z  FR z
or Equation 2.3‐9 f D cos  f L sin  cos  K z mg Since Kz includes gravity, Kz ≥ 1. For missiles located at an elevated height horizontal
displacement is possible, in which case conditions for horizontal injection presented in
Equation 2.3‐10 apply.
7
FA x  FA y  FR x  FR y
2
2
2
2
or Equation 2.3‐10 2
2
2
2
FA x  FA y   K x  K y mg 

where,
K x  K y  K D , the drag restraint coefficient.
2
2
To determine missile injection the following variables are simulated using random number
generation. Because the range of a random number is [0,1] for each random number (x)
generated, the parameters are calculated by Equation 2.3-11. A uniform distribution is assumed
for each parameter.
(a) 0     ;
  Rnd *  (b) 0     ;
  Rnd *  (c) 0    2 ;
(d) 0   2 ;
  Rnd * 2   Rnd * 2 Equation 2.3‐11 Missile injection conditions are simulated for each type of missile presented in Table 2-3. The
simulation is repeated 10,000 times for each tornado intensity, for each missile type to
determine the average missile injection probability, as determined by Equation 2.3-12.
PInj 
N Inj
NT
Equation 2.3‐12 8
Figure 2-1: Missile Injection Script Simplified Flowchart
All missile types, except trees, are assumed to have no restraints. The results of the missile
injection simulations are provided in Table 2-1.
Table 2-1: Results of Missile Injection Simulations
Missile
Type
a
b
c
d
e
f
g
h
F0
Description Material Injection
Prob
Rebar
Steel
1.68E-01
Utility Pole
Wood
1.88E-01
3” Pipe
Steel
2.01E-01
6” Pipe
Steel
1.51E-01
12” Pipe
Steel
1.18E-01
Wood Beam Wood
4.80E-01
Vehicle
Steel
1.18E-01
Tree
Wood
0.00E+00
F1
Injection
Prob
4.21E-01
4.40E-01
4.76E-01
3.93E-01
3.11E-01
7.26E-01
3.06E-01
0.00E+00
9
F2
Injection
Prob
6.19E-01
6.51E-01
6.75E-01
6.12E-01
5.24E-01
8.23E-01
5.37E-01
2.60E-03
F3
Injection
Prob
7.41E-01
7.55E-01
7.67E-01
7.32E-01
6.74E-01
8.73E-01
6.84E-01
1.04E-01
F4
Injection
Prob
8.06E-01
8.24E-01
8.33E-01
8.09E-01
7.70E-01
8.99E-01
7.61E-01
3.08E-01
F5
Injection
Prob
8.52E-01
8.56E-01
8.75E-01
8.48E-01
8.14E-01
9.21E-01
8.20E-01
4.71E-01
2.3.2
Site Layout
The tornado missile fragility analysis methodology presented herein uses a pre-defined site
layout as a basis for the analysis. As such, the starting point of the analysis is to determine the
appropriate site area that needs to be accounted for. For the purposes of this project the
Comanche Peak Nuclear Power Plant Site was used, as shown in Figure 2-2. This layout was
pre-determined at the beginning of methodology development. The area that needs to be
included in the site layout is land within the maximum determined distance of the specified
targets. Reference 4 postulates that the maximum distance a tornado-driven missile can travel
is 1000 ft. For the purpose of this project the maximum missile travel distances used are listed
in Table 2-6. The distance traveled by each missile is assumed to have a uniform distribution
from 0 to the maximum listed in Table 2-6.
In order to determine the appropriate area that needs to be considered in a realistic scenario,
targets susceptible to tornado missiles need to be identified by performing walkdowns, as
required per Supporting Requirement WFR-A2 of the ASME/ANS PRA Standard. Then, the
general area in the vicinity of the identified targets can be outlined for use in the missile
analysis. This area can then be divided into zones. A single zone needs to have an
approximately uniform missile distribution, which is categorized during site missile
walkdowns for each site (see Section 2.3.3). It is acceptable to have areas which don’t contain
any missiles not defined as part of a zone (e.g., if there is a parking lot that is typically empty,
this area can be excluded from consideration). It is also acceptable to have zones which
overlap.
There were 28 total zones identified for the pilot, as documented in Table 2-2. Each zone is
defined using the coordinates of their vertices, estimated from Figure 2-2. The number of
vertices allowed for each zone is limited to 5.
10
Table 2-2: Zone Definition
Zone 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 # Vertices 4 4 4 4 4 3 4 4 4 5 4 4 4 4 4 3 3 4 4 4 5 4 4 4 4 4 5 3 Vertex Coordinates X1 17.9 15.8 15.5 15.5 19 20.5 18.7 14.8 14.2 15.4 16.1 20.3 20.8 24.4 26.4 22.8 28.5 13 10.6 10.6 13.8 7.4 8.7 5.0 3.7 3 3 13.7 Y1 8.2 10.3 8.4 7.7 6.5 9.0 11.6 11.6 6.5 6.4 4.9 6.1 11.7 5.8 5.6 5.9 0.7 2.6 4.5 4.5 11.7 12.7 16.6 16.2 12.4 10 10 3.4 X2 19.3
19.5
17.9
19.1
20.3
20.8
20.8
18.7
15.4
17.5
18
24.4
29.3
26.4
30.7
28.5
30.7
14.9
13
13.8
14.8
13.8
14.3
8.2
7.4
4.8
10.6
17.5
Y2 8.1
10.1
8.1
7.2
6.1
11.7
11.7
11.6
6.4
6
4.1
5.8
10.6
5.6
5.4
0.7
5.4
1.5
2.6
4.4
11.6
11.7
15.6
13.8
12.7
10.0
4.5
6
11
X3 19.5
18.7
18.2
19.3
20.5
18.7
21.2
18.7
16.0
19.0
20.3
24.2
27.4
26.4
29.3
28.5
28.5
18
13.7
14.8
15.5
14.3
10.4
8.7
8.2
7.4
11.5
14.2
Y3 10.1
11.6
10.2
8.1
9.0
11.6
14.4
12.7
11.6
6.5
6.1
11.2
13.3
10.9
10.6
5.5
5.5
4.1
3.4
11.6
12.6
15.6
19.4
16.6
13.8
12.7
11.9
6.5
X4 18.2
16.0
15.8
15.5
19.5
18.7
15.5
14.8
19.1
19.0
20.8
21.2
24.2
26.4
16.1
13.8
11.5
16.4
8.7
8.6
8.6
5.0
3.7
7.4
Y4 10.2 11.7 10.3 8.4 10.1 X5 15.0 12.6 11.6 7.1 15.5
6.5 11.7 14.4 11.2 10.9 4.9 4.4 11.9 12.6 14.3
16.6 19.4 18.7 16.2 12.4 12.7 4.8
Y5 7.7
15.6
10.0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
0
1
2
3
4
5
6
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
12
Figure 2-2: Comanche Peak Nuclear Power Plant Hypothetical Zone Definition
7
2.3.3
Missile Distribution
Objects with the potential to become wind-generated missiles can be loose materials that are
stored near the plant, contents from structures destroyed by the high winds (e.g., trailers), and
nearby trees. To identify and catalogue the missile population it is necessary to perform a site
walkdown. Appendix G of NUREG/CR-4458 or NUREG/CR-4710 contains a list of
vulnerable SSCs for a Westinghouse 2-loop reactor or a Combustion Engineering 2-loop
reactor, respectively, which may be used as a starting point for the identification of potential
targets. The objectives of the site missile survey walkdowns include the following:
1. Identify all potential missiles within 1000 ft from SSCs
2. (Optional) Estimate missiles from other time periods of plant conditions. This
applies to cases such as refueling outages, when there is additional equipment on
site. However, this increase in missile density does not apply to an at-power PRA
analysis and as such may be an over-conservative estimate of the site missile
population for the purposes of this analysis, which intends to estimate the target
fragility for an at-power PRA.
Following the completion of the site walkdowns the missiles should be catalogued and binned
into zones. As a result, each zone should contain the types of missiles and the missile density in
the zone for each type such that it can be appropriately assumed that the zone containing these
trailers has a uniform density of the missiles identified. Table 2-3 shows the missiles and their
associated properties used in this project.
To eliminate the uncertainty associated with potential missile types at a site this project
assumes that plant walkdowns will be conducted to catalogue all missile types within 1000 ft
of the identified targets, approximate counts of each type, and their general locations. This is
required per Supporting Requirement WHA-A5 of Part 7 of the ASME/ANS PRA Standard
(Ref. 2). Since a zone is defined as an area where a uniform missile distribution assumption is
applicable, the missile distribution is defined for each zone. A zone can contain multiple
missile types, which will be the case in many instances. For this project, 8 specific missiles
types were identified, as listed in Table 2-3. The missile distribution in each zone is defined by
13
specifying the missile type and the number of missiles located in that zone, as shown in Table
2-4. To decrease computer run times the missile counts were kept very low during testing. In a
typical missile analysis the number of missiles can be anywhere from 10,000 to 20,000.
Table 2-3: Missile Definition
Missile Type a b c d e f g h Missile Description (Typical) Rebar Utility Pole 3” Pipe 6” Pipe 12” Pipe Wood Beam Vehicle Tree Length Width Depth Material L (ft) (in) (in) Mass (lbs) Steel 3.00
1.00 8.01
Wood 35.00 13.50 1122.10
Steel 10.00
3.50 75.80
Steel 15.03
6.63 284.07
Steel 15.00 12.75 744.00
Wood 12.00 12.00 114.00
Steel 15.95 66.00 66.00 3987.50
Wood 20.00
8.00 700.00
14
Table 2-4: Missile Distribution
Total Missiles by Zone Missile Counts by Type Zone b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 a 25 25 50 50 50 50 50 50 50 50 50 50 50 25 25 50 50 c 25
25
50
50
50
50
50
50
50
50
50
Total Missiles by Type 600 150 500
d 25
50
50
50
50
e 25
225
25
15
f g h 25
50
50
50
50
50
25 25 50 50 50 50 25 50 50 50 50 50 50 50 50 50 50 50 50 100
100
100
100
150
150
150
50
150
150
150
150
150
50
100
50
100
50
50
150
50
50
50
50
50
100
50
50
275
250 625 2650
2.3.4
Missile Simulation
As shown in Figure 2-2 the site is fit onto an (x,y) coordinate system. For each zone,
subroutine “Missile_Propagation” simulates missile origins by generating a random (x,y)
coordinate such that the x-coordinate is between the minimum of the zone’s x-coordinates and
the maximum of the zone’s x-coordinates, and similarly the y-coordinate is between the
minimum of the zone’s y-coordinates and the maximum of the zone’s y-coordinates. Using a
ray-casting algorithm, it is determined whether the simulated point is actually within the
defined zone boundaries.
The ray-casting algorithm is a simple way of finding whether a point is located inside of a
polygon by testing how many times a ray originating from this point intersects with the edges
of the polygon. If the ray intersects with the polygon an even number of times, it follows that
its originating point is located outside of the polygon; if the ray interests with the polygon an
odd number of times the originating point is located inside the polygon. This principle is
illustrated in Figure 2-3, where a ray originating at point B intersects with an arbitrarily shaped
polygon an even amount of times and a ray originating at point A intersects with the polygon
an odd number of times. Note that in this project, the ray is assumed to travel toward the yaxis, which is to the left by convention, and not as displayed in Figure 2-3. Figure 2-3 is used
simply for demonstrative purposes.
Figure 2-3: Illustration of the Ray Casting Algorithm
16
To simplify the problem, zone definitions are limited to five vertices (and therefore five sides).
For each simulated point the script tests the number of intersections that the simulated missile’s
ray has with the zone. If the number of intersections is odd, the point is kept and is assigned to
the zone. If the number of intersection is even, the point is ‘discarded.’
Equations associated with testing intersections are provided below. Each side of a particular
zone is defined by two sets of vertices (x1, y1) and (x2, y2). Equation 2.3-13 through Equation
2.3-14 determine the maximum x- and y-values for each side.
x max  max( x1 , x 2 ) Equation 2.3‐13 x min  min( x1 , x 2 )
Equation 2.3‐14 y max  max( y1 , y 2 )
y min  min( y1 , y 2 )
Equation 2.3‐15 Equation 2.3‐16 The ray cast for each simulated missile origin is a ray that is parallel to the x-axis and is
directed toward the y-axis. The slope of any line segment (i.e., side of a zone) can be either
vertical, horizontal, or of a non-zero slope. Intersections are counted by testing the following
conditions, in the order presented:
1. If the line segment is vertical (x1=x2) then an intersection is counted if xP ≥ xmin and
ymin ≥ yP ≥ ymax.
2. If the line segment is horizontal (y1=y2) then no intersections are counted. A note here
is that zone vertices (x1, x2, y1, and y2) are defined to the tenth decimal point, whereas
the simulated missile origin is rounded to the thousandths decimal point. Due to this it
is very improbable that a missile origin that falls directly onto a zone boundary will be
simulated; in most cases the missile origin will fall either above or below the zone
boundary.
3. If the line segment has a nonzero slope ((y1/y2) / (x1/x2) ≠ 0) then define point (u, v) as
the point of intersection for the ray and boundary segment. Since the ray is defined by
17
Equation 2.3-17, v = yP. Equation 2.3-18 solves for u. An intersection is counted if xP
≥ xmin and ymin ≥ yP ≥ ymax, and if u ≤ xP.
x  yP ,
u
x  xP
( y p  y1 )( x 2  x1 )
( y 2  y1 )
Equation 2.3‐17 x1
Equation 2.3‐18 If it is determined that the simulated point indeed falls into a zone, the next step is to assign a
missile type to this point, which will in effect simulate a specific object (e.g., car) with a
specific starting coordinate. Since missile densities are expected to vary per zone, a counter,
ZMC(zone), of the number of missiles simulated in each zone is kept. If ZMC(zone) reaches
the number of missiles specified for that zone, the simulated coordinate is not used further.
For simulating the missile type several arrays are used. Array MCounts(zone, missile type)
counts the number of each missile type in each zone, as shown in Table 2-4. Array ZMT(zone)
counts the total number of missiles in each zone. Array FracZM(zone, missile type), calculated
by Equation 2.3-19, is the percentage of each missile type in each zone.
FracZM ( zone, missile type) 
MCounts( zone, missile type)
ZMT ( zone)
Equation 2.3‐19 Array CFracZM(zone, missile type) represents the cumulative percentage of missiles types in
the zone and is calculated by Equation 2.3-20.
CFracZM ( zone, missile type)  FracZM (i, j ),
FracZM (i, j )  CFracZM (i, j  1) ,
i 1
i  1 Equation 2.3‐20 A missile is simulated by sampling a random number between 0 and 1. The program then
checks which of the cumulative percentage ranges this random number is in, and assigns the
corresponding missile type to the simulated missile origin point.
18
2.3.5
Missile Propagation
There are two main factors simulated that determine the missile end point: direction and
distance traveled.
Direction Determination
The National Oceanic and Atmospheric Administration (NOAA) maintains a national tornado
database, which contains tornado records for the contiguous United States starting with the
year 1950. The most recent data is available through 2011. The database is comprehensive,
with most tornado records containing starting and ending coordinates of each record (path
length information is not available). Using software developed for tornado hazard analyses in
Reference 5 all tornado records that have occurred within 200 miles of the Comanche Peak site
were extracted from the database. If both starting and ending coordinates of the tornado were
available, the direction of each tornado record was calculated. The histogram shown in Figure
2-4 displays the distribution of the directions in which tornado segments have been shown to
travel. The data suggests that most tornadoes in this area tend to travel in the northeastern
direction.
Figure 2-4: Tornado Direction Histogram
19
This insight is incorporated into the simulation of the missile travel direction by skewing the
simulated direction of the missile path to reflect the data presented in Figure 2-4. This skewing
is achieved by simulating a number between 0 and 1435, which is used to determine which bin
will be used for the angle simulation. For example, if the simulated number is 1155, a random
number between 60 and 90 will be simulated to determine the angle at which the missile
travels.
Table 2-5: Histogram of Tornado Directions for CPNPP
Bin 0
30
60
90
120
150
180
210
240
270
300
330
360
Frequency 269
358
411
108
4
10
12
18
18
21
20
80
106
Cumulative Frequency 269
627
1038
1146
1150
1160
1172
1190
1208
1229
1249
1329
1435
Distance Determination
While there is a lot of uncertainty associated with determining the distance that certain objects
can travel via tornado winds, there is some guidance available from studies and simulations
performed in the past. Conservatively, simulations have used 2500 ft as the zone of interest to
consider. According to Reference 4 the maximum credible distance of missile transportation is
about 1000 ft. The distance is also dependent on missile type, however for the simplified
purposes of this project missile type is not taken into account when simulating the distance
traveled by each missile. The maximum travel distance is varied as a function of tornado
intensity, as listed in Table 2-6.
20
Table 2-6: Maximum Travel Distances for Missiles as a Function of Tornado Intensity
Tornado Intensity Maximum Travel Distance
F0 1 F1 2 F2 3 F3 5 F4 8 F5 13 2.3.6
Target Interaction
The objective of target identification is to identify all systems, structures, and components
(SSCs) important for to decay heat removal that may be vulnerable to tornado generated
missiles. As a starting point, all SSCs related to safe shutdown located outside or within nonClass I structures should be examined for their susceptibility to wind-generated missiles. Note
that other failure modes may exist for site SSCs, such as high wind load failures. These failure
modes are not in the scope of this project and are not analyzed. Since there were no sitespecific walkdowns performed, sizable targets were assumed. The target areas identified are
listed in Table 2-7.
Table 2-7: Target Definition
Vertex Coordinates Target # Description X1 Y1 X2 Y2 Target Area
X3 Y3 X4 Y4 1 Target 1 15.5 8.4 16.0 8.4 16.0
9.0 15.5
9 0.60 2 Target 2 16.9 8.1 17.3 8.1 17.3
8.5 16.9
8.5 0.32 3 Target 3 17.2 9.2 17.5 9.2 17.5
9.6 17.2
9.6 0.24 4 Target 4 18.1 9.7 18.3 9.7 18.3 10.1 18.1 10.1 0.16 With missile direction of travel and distance traveled known, it can be calculated whether the
missile path interacts with any of the targets by checking for intersection with each of the
target’s boundaries. Each missile strike is recorded to an array. For a given tornado intensity,
21
the probability that missile type “a” will strike Target 2 is simply the number of type “a”
missiles that struck Target 2 divided by the total number of type “a” missiles. Table 2-8 lists
shows sample results for a single simulation. The results are color-coded to show that missile
strike probabilities are generally higher for larger targets (refer to Table 2-7 for target areas)
and that results are typically less stable for missile types with fewer missiles.
Table 2-8: Sample Results for Missile Propagation Simulation for an F1 Tornado
Missile Type Total # Missiles Target 1 Target 2 Target 3 Target 4 a 600.00 1.00E‐02 5.00E‐03 5.00E‐03 1.67E‐03 b 150.00 6.67E‐03 2.67E‐02 0.00E+00 0.00E+00 c 500.00 1.20E‐02 1.80E‐02 2.00E‐03 1.00E‐02 d 225.00 3.11E‐02 8.89E‐03 8.89E‐03 4.44E‐03 e 25.00 4.00E‐02 0.00E+00 0.00E+00 0.00E+00 f 275.00 1.45E‐02 3.64E‐03 0.00E+00 3.64E‐03 g 250.00 2.80E‐02 8.00E‐03 4.00E‐03 0.00E+00 h 625.00 9.60E‐03 3.20E‐03 6.40E‐03 1.60E‐03 2.3.7
Target Fragility
With missile injection and strike probabilities known, it is possible to calculate the target
fragility, assuming that P(damage|strike) is 1. The target fragility for each missile type is the
product of the total number of missiles, and the injection, strike, and damage probabilities,
since these events are assumed to be independent from each other. The total fragility is the sum
of the fragilities of each missile type.
22
3. Results
3.1 Missile Injection Simulation Results
The missile injection simulation was performed separately for each missile type, resulting in a
total of eight simulations. Convergence for a typical simulation was determined to occur
around 10,000 runs per simulation. The complete results of the missile injection simulations,
which include the injection probabilities for vertical restraint coefficients 1 through 5 and
horizontal restraint coefficients 0 through 5, are contained in sheet “Injection.” Conservatively,
it is assumed that all vertical restraint coefficients are 1 (due to gravity) and all horizontal
restraint coefficients are 0. Trees are an exception, which are assumed to have a restraint
coefficient of 2 vertically and 1 horizontally. Table 3-1 summarizes the results of the missile
injection simulations.
Table 3-1: Missile Simulation
Intensity: Missile Type a b c d e f g h F0 F1 Summary of Injection Probabilities 4.21E‐01
6.19E‐01
7.41E‐01
8.06E‐01 8.52E‐01
4.40E‐01
6.51E‐01
7.55E‐01
8.24E‐01 8.56E‐01
4.76E‐01
6.75E‐01
7.67E‐01
8.33E‐01 8.75E‐01
3.93E‐01
6.12E‐01
7.32E‐01
8.09E‐01 8.48E‐01
3.11E‐01
5.24E‐01
6.74E‐01
7.70E‐01 8.14E‐01
7.26E‐01
8.23E‐01
8.73E‐01
8.99E‐01 9.21E‐01
3.06E‐01
5.37E‐01
6.84E‐01
7.61E‐01 8.20E‐01
0.00E+00
2.60E‐03
1.04E‐01
3.08E‐01 4.71E‐01
1.68E‐01 1.88E‐01 2.01E‐01 1.51E‐01 1.18E‐01 4.80E‐01 1.18E‐01 0.00E+00 F2 F3 23
F4 F5 3.2 Missile Strike Simulation Results
The missile strike probabilities are calculated by sub “Missile_Propagation.” For demonstrative
purposes Table 3-2 lists the first 25 missiles simulated. Because the script performs the
simulation by zone, all of these points belong to zone 1. The simulated missile type,
determined using the defined missile distribution, is printed as well.
Table 3-2: Missile Simulation
Simulation # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 X1 19.26806
19.30797
18.87562
18.65258
18.05343
18.13492
19.33281
19.09388
18.78386
19.23729
19.2108
18.45723
19.41172
18.40723
18.21252
18.47145
18.90376
19.31209
18.04529
18.96792
18.98605
18.18902
18.11305
19.15625
19.2877
Zone # Missile Type Y1 10.04414
1 b 10.10665
1 h 10.08513
1 f 9.649048
1 e 9.104545
1 e 8.375818
1 g 8.91349
1 d 9.368485
1 h 8.34412
1 c 8.900963
1 a 9.610518
1 c 9.824008
1 f 9.958334
1 h 9.572267
1 f 9.592714
1 d 8.682132
1 g 8.485457
1 b 8.352414
1 c 8.23835
1 g 9.611706
1 c 8.921249
1 f 8.317387
1 d 8.282505
1 f 10.09369
1 d 8.770439
1 h After all needed missile origins are simulated (1 origin for each defined missile), the distance
and angle of travel are also simulated. Using this distance and angle of travel, the ending
24
coordinates of the missile trajectory (X2, Y2) are calculated. The script tests whether each
missile intersects any of the target boundaries. Intersections are counted as a target strikes and
printed out in the “Output” sheet. Table 3-3 lists the first 25 intersections for demonstrative
purposes. The missile type, struck target, and distance that the missile traveled are all displayed
in Table 3-3. The simulation # refers to the numbered missile origin simulation.
Table 3-3: Missile Strikes
Simulation # 62 203 205 208 208 209 215 219 219 219 225 226 230 233 234 236 236 238 244 245 246 247 251 254 256 X1 18.122 17.111 17.280 15.965 15.965 16.674 16.495 15.919 15.919 15.919 17.461 18.101 17.869 17.043 15.566 16.230 16.230 17.058 16.579 17.923 17.697 16.779 16.971 17.730 16.784 Y1 8.738
8.869
8.478
8.693
8.693
8.918
8.472
8.757
8.757
8.757
9.155
9.682
9.890
8.682
8.460
8.957
8.957
9.339
8.700
9.700
9.780
8.324
9.362
9.641
8.248
X2 18.412
22.637
24.210
25.471
25.471
19.629
24.510
24.463
24.463
24.463
19.037
20.519
26.004
26.366
28.049
19.410
19.410
23.540
26.817
24.639
19.258
17.064
23.611
23.393
23.288
Y2 Missile Type
16.095
e
18.370
a
14.077
a
12.543
h
12.543
h
11.431
b
8.818
g
12.980
b
12.980
b
12.980
b
15.779
a
12.540
a
9.913
h
17.668
b
9.071
b
10.750
a
10.750
a
14.519
a
14.367
a
17.568
h
10.915
b
8.578
a
9.848
b
14.914
b
9.931
h
Target 4 3 2 1 3 3 2 1 3 4 3 4 4 4 1 3 4 3 3 4 4 2 3 4 2 Distance 7.36
10.99
8.91
10.26
10.26
3.88
8.02
9.53
9.53
9.53
6.81
3.74
8.14
12.95
12.50
3.65
3.65
8.30
11.70
10.34
1.93
0.38
6.66
7.74
6.72
A tally is kept for each missile-target interaction, as a function of missile type and target
number. The strike probability is calculated is the total number of interactions divided by the
total number of missiles simulated. Table 3-4 provides the results of the strike probability
25
simulations performed. The simulations for various tornado intensities vary only by the
maximum missile travel distance, which is included in Table 3-4 for reference.
Table 3-4: Calculated Strike Probabilities
Intensity F0 Missile Type a b c d e f g h Intensity F1 Missile Type a b c d e f g h Intensity F2 Missile Type a b c d e f g h Maximum Missile Propagation Distance 1 Target 1 Target 2 Target 3 Target 4 4.17E‐03 8.33E‐04 8.33E‐04 0.00E+00
3.33E‐03 2.67E‐02 6.67E‐03 3.33E‐03
4.00E‐03 4.00E‐03 0.00E+00 2.00E‐03
0.00E+00 0.00E+00 0.00E+00 0.00E+00
0.00E+00 0.00E+00 0.00E+00 0.00E+00
3.64E‐03 0.00E+00 0.00E+00 0.00E+00
8.00E‐03 2.00E‐03 4.00E‐03 2.00E‐03
8.00E‐04 8.00E‐04 3.20E‐03 0.00E+00
Maximum Missile Propagation Distance 2 Target 1 Target 2 Target 3 Target 4 1.08E‐02 4.17E‐03 4.17E‐03 2.50E‐03
3.33E‐03 4.33E‐02 1.00E‐02 6.67E‐03
7.00E‐03 9.00E‐03 0.00E+00 0.00E+00
0.00E+00 0.00E+00 0.00E+00 0.00E+00
0.00E+00 0.00E+00 0.00E+00 0.00E+00
1.09E‐02 0.00E+00 0.00E+00 0.00E+00
0.00E+00 4.00E‐03 6.00E‐03 2.00E‐03
8.00E‐04 8.00E‐04 1.60E‐03 1.60E‐03
Maximum Missile Propagation Distance 3 Target 1 Target 2 Target 3 Target 4 7.50E‐03 5.00E‐03 2.50E‐03 8.33E‐04
3.33E‐03 2.67E‐02 3.33E‐02 1.00E‐02
7.00E‐03 1.30E‐02 4.00E‐03 5.00E‐03
4.44E‐03 0.00E+00 0.00E+00 0.00E+00
0.00E+00 0.00E+00 0.00E+00 0.00E+00
5.45E‐03 3.64E‐03 1.82E‐03 0.00E+00
2.00E‐03 1.00E‐02 6.00E‐03 4.00E‐03
8.00E‐04 8.00E‐04 2.40E‐03 1.60E‐03
26
Intensity F3 Missile Type
a b c d e f g h Intensity F4 Missile Type
a b c d e f g h Intensity F5 Missile Type
a b c d e f g h Maximum Missile Propagation Distance 5 Target 1 Target 2 Target 3 Target 4 8.33E‐03 3.33E‐03 4.17E‐03 4.17E‐03
3.33E‐03 3.67E‐02 2.67E‐02 1.33E‐02
1.20E‐02 9.00E‐03 1.00E‐03 2.00E‐03
4.44E‐03 2.22E‐03 2.22E‐03 0.00E+00
0.00E+00 0.00E+00 0.00E+00 2.00E‐02
2.00E‐02 3.64E‐03 7.27E‐03 3.64E‐03
1.40E‐02 1.40E‐02 2.00E‐02 1.60E‐02
0.00E+00 0.00E+00 3.20E‐03 2.40E‐03
Maximum Missile Propagation Distance 8 Target 1 Target 2 Target 3 Target 4 5.00E‐03 5.00E‐03 7.50E‐03 7.50E‐03
3.33E‐03 5.00E‐02 1.67E‐02 2.00E‐02
1.60E‐02 1.30E‐02 8.00E‐03 6.00E‐03
4.44E‐03 4.44E‐03 0.00E+00 0.00E+00
0.00E+00 0.00E+00 0.00E+00 4.00E‐02
2.18E‐02 1.82E‐03 1.64E‐02 3.64E‐03
1.40E‐02 1.60E‐02 1.00E‐02 1.40E‐02
3.20E‐03 0.00E+00 8.00E‐04 3.20E‐03
Maximum Missile Propagation Distance 13 Target 1 Target 2 Target 3 Target 4 6.67E‐03 8.33E‐03 7.50E‐03 8.33E‐03
2.33E‐02 2.00E‐02 2.33E‐02 2.33E‐02
1.00E‐02 2.00E‐02 8.00E‐03 6.00E‐03
4.44E‐03 2.22E‐03 4.44E‐03 2.22E‐03
0.00E+00 0.00E+00 0.00E+00 2.00E‐02
2.73E‐02 1.82E‐03 1.27E‐02 9.09E‐03
1.40E‐02 2.40E‐02 1.40E‐02 1.20E‐02
1.60E‐03 4.00E‐03 3.20E‐03 5.60E‐03
Missile origins that resulted in target strikes are plotted in Figure 3-1 through Figure 3-6. The
main difference between the simulations for tornado intensities is the maximum distance of
travel that’s assumed. As a result, the number of missiles that interact with a target increases
with tornado intensity. Also note that most of the missiles travel in the northeast direction, as
specified by the missile direction histogram in Figure 2-4.
Figure 3-1: Missile Strikes for an F0 Simulation
27
Figure 3-2: Missile Strikes for an F1 Simulation
Figure 3-3: Missile Strikes for an F2 Simulation
Figure 3-4: Missile Strikes for an F3 Simulation
28
Figure 3-5: Missile Strikes for an F4 Simulation
Figure 3-6: Missile Strikes for an F5 Simulation
29
3.3 Target Damage Probabilities
Calculation of target damage probabilities is outside the scope of this project, and as such,
target damage probabilities are assumed. For an F5 tornado the damage probability of a target,
given that it is struck by a missile, is assumed to be 1.0. The damage probabilities are assumed
to decrease by an order of magnitude for each decrease in tornado intensity.
Table 3-5 lists the assumed conditional target damage probabilities.
Table 3-5: Assumed Target Damage Probabilities
Missile Type a b c d e f g h F0 1.00E‐05 1.00E‐05 1.00E‐05 1.00E‐05 1.00E‐05 1.00E‐05 1.00E‐05 1.00E‐05 F1 1.00E‐04
1.00E‐04
1.00E‐04
1.00E‐04
1.00E‐04
1.00E‐04
1.00E‐04
1.00E‐04
F2 1.00E‐03
1.00E‐03
1.00E‐03
1.00E‐03
1.00E‐03
1.00E‐03
1.00E‐03
1.00E‐03
30
F3 1.00E‐02
1.00E‐02
1.00E‐02
1.00E‐02
1.00E‐02
1.00E‐02
1.00E‐02
1.00E‐02
F4 1.00E‐01 1.00E‐01 1.00E‐01 1.00E‐01 1.00E‐01 1.00E‐01 1.00E‐01 1.00E‐01 F5 1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
1.00E+00
3.4 Resulting Target Fragilities
The resulting target fragility for a specific missile type is calculated as the product of: the total
number of missiles at the site, the missile injection probability, missile strike probability, and
target damage probability. The target fragility is capped at 1.0 (i.e., the target fragility is the
minimum of 1 and the target fragility, since calculated target fragilities may exceed 1 in some
cases). The overall target fragility is the sum of the fragilities due to each missile type. Table 36 shows an example calculation for Target 1 in an F3 tornado.
Table 3-6: Detailed Fragility Summary, Target 1, F3 Tornado
Target 1 Missile Type a b c d e f g h F3 Total # Missiles Injection Prob Strike Prob Damage Prob Fragility 600.00 7.41E‐01
8.33E‐03
1.00E‐02 3.71E‐02
150.00 7.55E‐01
3.33E‐03
1.00E‐02 3.78E‐03
500.00 7.67E‐01
1.20E‐02
1.00E‐02 4.60E‐02
225.00 7.32E‐01
4.44E‐03
1.00E‐02 7.32E‐03
25.00 6.74E‐01
0.00E+00
1.00E‐02 0.00E+00
275.00 8.73E‐01
2.00E‐02
1.00E‐02 4.80E‐02
250.00 6.84E‐01
1.40E‐02
1.00E‐02 2.39E‐02
625.00 1.04E‐01
0.00E+00
1.00E‐02 0.00E+00
1.66E‐01
Table 3-7 lists the overall target fragility results. Note that while target fragilities are
proportional to the target area, the trends are slightly erratic due to the fact that missile strike
simulations were not repeated to achieve results convergence.
Table 3-7: Target Fragility Results
Target 1 Target 2 Target 3 Target 4 F0 F1 1.63E‐05 1.30E‐05 3.90E‐06 3.54E‐06 F2 6.80E‐04
6.36E‐04
2.17E‐04
1.22E‐04
F3 7.59E‐03
1.10E‐02
6.76E‐03
3.51E‐03
31
F4 1.66E‐01
1.27E‐01
1.10E‐01
8.23E‐02
F5 1.00E+00 1.00E+00 1.00E+00 1.00E+00 1.00E+00
1.00E+00
1.00E+00
1.00E+00
4. Conclusion
In summary, this project successfully developed a method to simulate the propagation of
tornado-generated missiles and their interactions with defined targets. The major capabilities of
the program developed include:
1. Ability to calculate missile injection probabilities as a function of vertical and
horizontal restraint coefficients.
2. Ability to simulate missile origin points in a 2D environment, given site definitions.
3. Ability to define missile distributions at a site.
4. Ability to define target locations relative to the defined geometry.
5. Ability to simulate missile propagation using historical tornado data.
6. Ability to determine whether the trajectory of each missile intersects with the boundary
of the specified targets, which is used to calculate the strike probabilities.
In addition to the developing the code to perform each of the tasks listed above, this projects
provides a platform for further developing the methods of tornado missile analysis.
Enhancements to consider in the future include:
1. Use more realistic lift and drag coefficients for various object types, which are expected
to have an impact on the injection probability results. Reference 8 provides
experimental data for wind tunnel tests.
2. Take into account the site topography, which impacts how missiles are injected into the
wind stream.
3. Prior to performing walkdowns, determine the largest credible distance missiles are
postulated to travel and limit the missile surveys to this radius.
4. Take into account any barriers that missiles may encounter prior to reach a target
(including other targets).
5. Perform the simulations using 3D geometries.
32
5. References
1. Regulatory Guide 1.200, Rev. 2, “An Approach for Determining the Technical
Adequacy of Probabilistic Risk Assessment Results for Risk-Informed Activities,”
Nuclear Regulatory Commission, March 2009.
2. ASME/ANS RA-Sa-2009, “Standard for Level 1/Large Early Release Frequency
Probabilistic Risk Assessment for Nuclear Power Plant Applications,” American
Society of Mechanical Engineers, 2009.
3. EPRI NP-2005, “Tornado Missile Simulation and Design Methodology,” Electric
Power Research Institute, August 1981.
4. J. Goodman and J.E. Koch, “The Probability of a Tornado Missile Hitting a Target,”
Bechtel Power Corporation, November 1982.
5. LTR-RAM-II-12-011, “Tornado Hazard Analysis Software,” Westinghouse Electric
Company, October 2012.
6. Appendix G of NUREG/CR-4458, “Shutdown Decay Heat Removal Analysis of a
Westinghouse 2-Loop Pressurized Water Reactor,” Reed and Ferrell, 1987.
7. Appendix G of NUREG/CR-4710, “Shutdown Decay Heat Removal Analysis of a
Combustion Engineering 2-Loop Pressurized Water Reactor,” W. R. Cramond, D.
M. Ericson, Jr., G. A. Sanders 1987.
8. EPRI NP-748, “Wind Field and Trajectory Models for Tornado-Propelled Objects,”
Electric Power Research Institute, May 1978.
9. NRC RG 1.76, Rev. 1, “Design-Basis Tornado and Tornado Missiles For Nuclear
Power Plants,” Nuclear Regulatory Commission, March 2007.
33
rhoAir = 0.0748 / 2.2 * (3.28084) ^ 3
pi = 3.1415926
Dim Vprob, Hprob, Tprob As Double
Dim numRuns, v, H, T, tor As Integer
Dim Fax, Fay, Faz As Double
Dim Cd, Cl, fd, fl, rhoAir As Double
'kg/m^3
Dim length, depth, width, area, mass As Double
Dim pi As Double
Dim Kd, Kl, w, alpha, theta, phi, psi As Double
Dim wLow(0 To 5), wHigh(0 To 5) As Double
Dim Mtype As String
Application.ScreenUpdating = False
Public Sub Missile_Injection()
Part 1: Missile Injection
34
Appendix A: VBA Script
'm/s
'm/s
'm/s
'm/s
'm/s
'm/s
'm/s
'm/s
'm/s
'm/s
'm/s
wLow(1) = 73 * 0.44704
wLow(2) = 113 * 0.44704
wLow(3) = 158 * 0.44704
wLow(4) = 207 * 0.44704
wLow(5) = 261 * 0.44704
wHigh(0) = 72 * 0.44704
wHigh(1) = 112 * 0.44704
wHigh(2) = 157 * 0.44704
wHigh(3) = 206 * 0.44704
wHigh(4) = 260 * 0.44704
wHigh(5) = 318 * 0.44704
For tor = 0 To 5
mass = Range("G2").Value / 2.2
'm^2
'kg
'meters
width = Range("E2").Value / 12 / 3.28084
area = length * width
'meters
length = Range("D2").Value / 3.28084
Sheets("Injection").Select
'm/s
'm/s^2
wLow(0) = 40 * 0.44704
g = 9.8
35
fd = Cd * rhoAir * area * w ^ 2 / 2
Cl = 0.98 * (Sin(alpha)) ^ 2 * Cos(alpha)
Cd = 0.98 * (Sin(alpha)) ^ 3
psi = Rnd * 2 * pi
phi = Rnd * 2 * pi
theta = Rnd * pi
alpha = Rnd * pi
w = Rnd * (wHigh(tor) - wLow(tor)) + wLow(tor)
Kl = Rnd * 0.5 + K2
Kd = Rnd * 0.5 + K1
For i = 1 To numRuns
numRuns = 100000
T=0
H=0
v=0
K2 = Range("B5").Offset(0, j).Value
K1 = Range("A7").Offset(k * 3, 0).Value
For k = 0 To 5
For j = 1 To 5
36
Vprob = v / numRuns
Next i
End If
T=T+1
If vertical = True Or horizontal = True Then
End If
horizontal = False
Else
horizontal = True
H=H+1
If Sqr(Fax ^ 2 + Fay ^ 2) >= Kd * mass * g Then
End If
vertical = False
Else
vertical = True
v=v+1
If Faz > Kl * mass * g Then
Faz = fd * Cos(theta) - fl * Sin(theta) * Cos(psi)
37
Fay = fd * Sin(theta) * Sin(phi) + fl * (Cos(theta) * Sin(phi) * Cos(psi) - Cos(phi) * Sin(psi))
Fax = fd * Sin(theta) * Cos(phi) + fl * (Cos(theta) * Cos(phi) * Cos(psi) - Sin(phi) * Sin(psi))
fl = Cl * rhoAir * area * w ^ 2 / 2
Range("I3").Value = Kd
Range("I4").Value = Kl
Range("I5").Value = w / 0.447
Range("I6").Value = alpha / pi * 180
Range("I7").Value = theta / pi * 180
Range("I8").Value = phi / pi * 180
Range("I9").Value = psi / pi * 180
Range("I10").Value = Cd
Range("I11").Value = Cl
Range("I12").Value = fd
Range("I13").Value = fl
Range("I14").Value = Fax
Range("I15").Value = Fay
Range("I16").Value = Faz
Range("I17").Value = Kl * mass * g
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
Range("C8").Offset(k * 3 + 21 * tor, j - 1).Value = Tprob
Range("C7").Offset(k * 3 + 21 * tor, j - 1).Value = Vprob
Range("C6").Offset(k * 3 + 21 * tor, j - 1).Value = Hprob
Tprob = T / numRuns
Hprob = H / numRuns
38
Next j
Next k
Range("I18").Value = Kd * mass * g
Dim TotalMissiles As Integer
Sheets("Output").Range("H2:N10000").Clear
Sheets("Output").Range("A2:E10000").Clear
start = Timer
Dim start As Double
'Application.Calculation = xlCalculationAutomatic
Application.Calculation = xlCalculationManual
Application.ScreenUpdating = False
Public Sub Missile_Propagation()
Part 2: Missile Propagation
End Sub
Next tor
'
39
40
'zone missile type 1+2 count
'zone missile type 1+2+3 count
Dim MCounts(1 To 28, 1 To 8) As Integer
Dim ZMT(1 To 28) As Integer
Dim ZMC(1 To 28) As Integer
Next i
Next j
ZoneDef(i, j) = Range("C2").Offset(i, j).Value
For j = 0 To 9
numVertices(i) = Range("B2").Offset(i, 0).Value
For i = 1 To 28
Sheets("Zones").Select
'read in the zone vertex coordinates
Dim ZoneDef(1 To 28, 0 To 9) As Double
Dim numVertices(1 To 28) As Integer
Dim i, j, k As Integer
numZones = 28
Dim numZones As Integer
TotalMissiles = Sheets("Zones").Range("W31").Value
'read number of each missile type in zone
'read how many missiles in zone
Range("n95").Offset(i, j) = CFracZM(i, j)
'
Dim MTypeTotal(1 To 8) As Integer
Next i
41
'total # missiles by type, a - h
Range("n65").Offset(i, j) = FracZM(i, j)
'
Next j
Range("n35").Offset(i, j) = MCounts(i, j)
'
End If
CFracZM(i, j) = FracZM(i, j) + CFracZM(i, j - 1)
ElseIf j > 1 Then
CFracZM(i, j) = FracZM(i, j)
If j = 1 Then
FracZM(i, j) = MCounts(i, j) / ZMT(i)
MCounts(i, j) = Range("N2").Offset(i, j).Value
For j = 1 To 8
'cumulative fraction of missile types
'fraction of missile types
'set the zone missile counter to 0
ZMT(i) = Range("W2").Offset(i, 0).Value
ZMC(i) = 0
For i = 1 To numZones
Sheets("Zones").Select
Dim CFracZM(1 To 28, 1 To 8) As Double
Dim FracZM(1 To 28, 1 To 8) As Double
Sheets("Output").Select
Dim numInters, sims, CountsNeeded As Integer
Dim zone(1 To 5600) As Integer
Dim xP1(1 To 5600), yP1(1 To 5600) As Double
Dim xP, yP, x1, x2, y1, y2 As Double
Dim u, v As Double
Dim xMax, xMin, yMax, yMin As Double
Dim zoneXmax, zoneXmin, zoneYmax, zoneYmin As Double
For numSims = 1 To numSimsNeeded
Dim numSims As Integer
numSimsNeeded = Sheets("Strike").Range("B1").Value
Dim numSimsNeeded As Integer
Next i
MTypeTotal(i) = Range("N31").Offset(0, i).Value
For i = 1 To 8
Sheets("Zones").Select
42
xP = Rnd * (zoneXmax - zoneXmin) + zoneXmin
End If
43
'random number between zoneXmin and zoneXmax
zoneYmin = Application.Min(ZoneDef(i, 1), ZoneDef(i, 3), ZoneDef(i, 5))
If zoneYmin = 0 Then
End If
zoneYmin = Application.Min(ZoneDef(i, 1), ZoneDef(i, 3), ZoneDef(i, 5), ZoneDef(i, 7))
If zoneYmin = 0 Then
zoneYmin = Application.Min(ZoneDef(i, 1), ZoneDef(i, 3), ZoneDef(i, 5), ZoneDef(i, 7), ZoneDef(i, 9))
zoneYmax = Application.Max(ZoneDef(i, 1), ZoneDef(i, 3), ZoneDef(i, 5), ZoneDef(i, 7), ZoneDef(i, 9))
End If
zoneXmin = Application.Min(ZoneDef(i, 0), ZoneDef(i, 2), ZoneDef(i, 4))
If zoneXmin = 0 Then
End If
zoneXmin = Application.Min(ZoneDef(i, 0), ZoneDef(i, 2), ZoneDef(i, 4), ZoneDef(i, 6))
If zoneXmin = 0 Then
zoneXmin = Application.Min(ZoneDef(i, 0), ZoneDef(i, 2), ZoneDef(i, 4), ZoneDef(i, 6), ZoneDef(i, 8))
zoneXmax = Application.Max(ZoneDef(i, 0), ZoneDef(i, 2), ZoneDef(i, 4), ZoneDef(i, 6), ZoneDef(i, 8))
'find the max and min x and y coordinates that bound the zone
Do While ZMC(i) < ZMT(i)
For i = 1 To numZones
'simulate the missile origins in each zone
'random number between zoneXmin and zoneXmax
'set number of intersections to 0
numInters = numInters + 1
If x2 - x1 = 0 Then
'count 1 intersection
'this is a vertical segment
44
If xP >= xMin And yP <= yMax And yP >= yMin Then 'count intersections with each line segment bounding the zone
'ray intersection algorithm counts intersections
yMin = Application.Min(y1, y2)
yMax = Application.Max(y1, y2)
xMin = Application.Min(x1, x2)
xMax = Application.Max(x1, x2)
End If
y2 = ZoneDef(i, 1)
x2 = ZoneDef(i, 0)
If j = numVertices(i) - 1 Then
End If
y2 = ZoneDef(i, 3 + j * 2)
x2 = ZoneDef(i, 2 + j * 2)
If j < numVertices(i) - 1 Then
y1 = ZoneDef(i, 1 + j * 2)
x1 = ZoneDef(i, j * 2)
For j = 0 To numVertices(i) - 1
numInters = 0
'determine whether simulated point is in zone i
yP = Rnd * (zoneYmax - zoneYmin) + zoneYmin
'this is a horizontal segment. dont count as intersection
'add missile origin to the missile array
'if missile origin falls into the zone
Loop
End If
ZMC(i) = ZMC(i) + 1
45
'add another missile count to the counter array
Range("D1").Offset(sims, 0).Value = i
Range("C1").Offset(sims, 0).Value = yP
Range("B1").Offset(sims, 0).Value = xP
Range("A1").Offset(sims, 0).Value = sims
zone(sims) = i
yP1(sims) = yP
xP1(sims) = xP
sims = sims + 1
If numInters Mod 2 = 1 Then
Next j
End If
End If
End If
numInters = numInters + 1
If u <= xP Then
u = (yP - y1) * (x2 - x1) / (y2 - y1) + x1
ElseIf (y2 - y1) / (x2 - x1) <> 0 Then 'this segment has a slope
numInters = numInters
ElseIf y2 - y1 = 0 Then
MisID(i) = 3
Mtype(i) = "c"
46
ElseIf MTypeRnd > CFracZM(Znum, 2) And MTypeRnd <= CFracZM(Znum, 3) Then
MisID(i) = 2
Mtype(i) = "b"
ElseIf MTypeRnd > CFracZM(Znum, 1) And MTypeRnd <= CFracZM(Znum, 2) Then
MisID(i) = 1
Mtype(i) = "a"
If MTypeRnd <= CFracZM(Znum, 1) Then
Znum = zone(i)
MTypeRnd = Rnd
For i = 1 To sims
Sheets("Output").Select
'assign missile types to the simulated points
Dim MisID(1 To 5600) As Integer
Dim Mtype(1 To 5600) As String
Dim Znum As Integer
Dim MTypeRnd As Double
Next i
'simulate missile propagation
Next i
Range("E1").Offset(i, 0).Value = Mtype(i)
End If
Mtype(i) = "none assigned"
Else
MisID(i) = 8
Mtype(i) = "h"
47
ElseIf MTypeRnd > CFracZM(Znum, 7) And MTypeRnd <= CFracZM(Znum, 8) Then
MisID(i) = 7
Mtype(i) = "g"
ElseIf MTypeRnd > CFracZM(Znum, 6) And MTypeRnd <= CFracZM(Znum, 7) Then
MisID(i) = 6
Mtype(i) = "f"
ElseIf MTypeRnd > CFracZM(Znum, 5) And MTypeRnd <= CFracZM(Znum, 6) Then
MisID(i) = 5
Mtype(i) = "e"
ElseIf MTypeRnd > CFracZM(Znum, 4) And MTypeRnd <= CFracZM(Znum, 5) Then
MisID(i) = 4
Mtype(i) = "d"
ElseIf MTypeRnd > CFracZM(Znum, 3) And MTypeRnd <= CFracZM(Znum, 4) Then
'
'
'direction is 0
'cumulative frequency of tornadoes going in the 0 - 30 degree direction
'1435 tornadoes within a 200 mi radius of CPNPP historically. Use hazard calculator for this value.
'between 60 and 90
Direction = Rnd() * 30 + 120
ElseIf DirectionProb > 1150 And DirectionProb <= 1160 Then
Direction = Rnd() * 30 + 90
ElseIf DirectionProb > 1146 And DirectionProb <= 1150 Then
Direction = Rnd() * 30 + 60
ElseIf DirectionProb > 1038 And DirectionProb <= 1146 Then
Direction = Rnd() * 30 + 30
48
'direction is between 0 to 30 degrees
ElseIf DirectionProb > 627 And DirectionProb <= 1038 Then
Direction = Rnd() * 30 + 0.000001
If DirectionProb > 0 And DirectionProb <= 627 Then
Direction = 0
If DirectionProb <= 269 Then
DirectionProb = Rnd() * 1435
'determine missile direction; direction is the same for all wind speeds and missile types
For i = 1 To sims
'simulate direction and distance traveled
MaxDistance = Sheets("Strike").Range("C2").Value
Dim xP2(1 To 5600), yP2(1 To 5600) As Double
Dim DirectionProb, Direction, Angle(1 To 5600), Distance(1 To 5600) As Double
Dim MaxDistance As Double
ElseIf Direction > 90 And Direction < 270 Then
xP2(i) = xP1(i) + Distance(i) * Cos(Angle(i))
If Direction < 90 Or Direction > 270 Then
Distance(i) = Rnd() * MaxDistance
'MaxDistance is a function of tornado intensity
49
'distance is assumed to be a uniform distribution from 0- MaxDistance
Angle(i) = Direction * 3.1415926 / 180
End If
Direction = Rnd() * 30 + 330
ElseIf DirectionProb > 1329 And DirectionProb <= 1435 Then
Direction = Rnd() * 30 + 300
ElseIf DirectionProb > 1249 And DirectionProb <= 1329 Then
Direction = Rnd() * 30 + 270
ElseIf DirectionProb > 1229 And DirectionProb <= 1249 Then
Direction = Rnd() * 30 + 240
ElseIf DirectionProb > 1208 And DirectionProb <= 1229 Then
Direction = Rnd() * 30 + 210
ElseIf DirectionProb > 1190 And DirectionProb <= 1208 Then
Direction = Rnd() * 30 + 180
ElseIf DirectionProb > 1172 And DirectionProb <= 1190 Then
Direction = Rnd() * 30 + 150
ElseIf DirectionProb > 1160 And DirectionProb <= 1172 Then
Dim xTMax, xTMin, yTMax, yTMin As Double
Dim xT1, yT1, xT2, yT2 As Double
Next i
End If
MsgBox "Check yP2 in sub MissilePropagation"
Else
yP2(i) = yP1(i)
50
ElseIf Direction = 0 Or Direction = 180 Or Direction = 360 Then
yP2(i) = yP1(i) - Distance(i) * Sin(Angle(i))
ElseIf Direction > 180 And Direction < 360 Then
yP2(i) = yP1(i) + Distance(i) * Sin(Angle(i))
If Direction > 0 And Direction < 180 Then
'define the missile ending y-coordinate
'determine missile end points.
End If
MsgBox "Check xP2 in sub MissilePropagation"
Else
xP2(i) = xP1(i)
ElseIf Direction = 90 Or Direction = 270 Then
xP2(i) = xP1(i) - Distance(i) * Cos(Angle(i))
If j < numTVertices - 1 Then
yT1 = Range("D2").Offset(target, j * 2).Value
51
'all target zones approximately defined using 4 vertices
'in case the slope is vertical(undef.)
xT1 = Range("C2").Offset(target, j * 2).Value
Sheets("Targets").Select
For j = 0 To numTVertices - 1
numTVertices = 4
For target = 1 To 4
yPMin = Application.Min(yP1(i), yP2(i))
yPMax = Application.Max(yP1(i), yP2(i))
xPMin = Application.Min(xP1(i), xP2(i))
xPMax = Application.Max(xP1(i), xP2(i))
On Error Resume Next
TrajectorySlope = (yP2(i) - yP1(i)) / (xP2(i) - xP1(i))
For i = 1 To sims
'using missile starting and ending coordinates determine intersection with targets
Dim StrikeOffset As Integer
Dim TargetStrikes(1 To 4, 1 To 8) As Integer
Dim numTVertices As Integer
Dim TrajectorySlope, SegmentSlope As Double
Dim target As Integer
Dim xPMax, xPMin, yPMax, yPMin As Double
'yP2(i)
v = TrajectorySlope * (u - xP1(i)) + yP1(i)
u = xT2
Else
'Case 1.2: missile trajectory is not vertical
v=0
u = xT2
If xP2(i) - xP1(i) = 0 Then
52
'Case 1.1: target segment slope is vertical and missile trajectory is vertical
If xT2 - xT1 = 0 Then
'calculate target segment slope
'in case the slope is vertical
'Case 1: target segment slope is vertical
On Error Resume Next
SegmentSlope = (yT2 - yT1) / (xT2 - xT1)
yTMin = Application.Min(yT1, yT2)
yTMax = Application.Max(yT1, yT2)
xTMin = Application.Min(xT1, xT2)
xTMax = Application.Max(xT1, xT2)
End If
yT2 = Range("D2").Offset(target, 0).Value
xT2 = Range("C2").Offset(target, 0).Value
ElseIf j = numTVertices - 1 Then
yT2 = Range("F2").Offset(target, j * 2).Value
xT2 = Range("E2").Offset(target, j * 2).Value
Range("I1").Offset(StrikeOffset, 0).Value = xP1(i)
Range("H1").Offset(StrikeOffset, 0).Value = i
Sheets("Output").Select
StrikeOffset = StrikeOffset + 1
53
TargetStrikes(target, MisID(i)) = TargetStrikes(target, MisID(i)) + 1
yPMin <= v And v <= yPMax Then
xPMin <= u And u <= xPMax And _
yTMin <= v And v <= yTMax And _
If xTMin <= u And u <= xTMax And _
End If
End If
v = SegmentSlope * (u - xT1) + yT1
u = (TrajectorySlope * xP1(i) - SegmentSlope * xT1 + yT1 - yP1(i)) / (TrajectorySlope - SegmentSlope)
Else
'Case 2.2: target segment slope is not vertical and missile trajectory is not vertical
v = SegmentSlope * (u - xT1) + yT1
u = xP2(i)
If xP2(i) - xP1(i) = 0 Then
'Case 2.1: target segment slope is not vertical and missile trajectory is vertical
Else
'Case 2: target segment slope is not vertical
End If
Range("Q1").Offset(StrikeOffset, 0).Value = SegmentSlope
'
Sheets("Strike").Select
Dim TargetStrikeProb(1 To 4, 1 To 8) As Double
Next numSims
'Erase xP1(), xP2(), zone(), Mtype(), MisID(), xP2(), yP2()
Next i
Next target
Sheets("Output").Select
Next j
End If
Sheets("Output").Select
54
Range("P1").Offset(StrikeOffset, 0).Value = TrajectorySlope
'
Exit For
Range("O1").Offset(StrikeOffset, 0).Value = Distance(i)
'
Range("N1").Offset(StrikeOffset, 0).Value = target
Range("M1").Offset(StrikeOffset, 0).Value = Mtype(i)
Range("L1").Offset(StrikeOffset, 0).Value = yP2(i)
Range("K1").Offset(StrikeOffset, 0).Value = xP2(i)
Range("J1").Offset(StrikeOffset, 0).Value = yP1(i)
End Sub
Application.Calculation = xlCalculationAutomatic
'Sheets("Strike").Range("A1").Select
Sheets("Output").Range("Q2").Value = (Timer - start)
Next i
Next j
Range("A3").Offset(j, i).Value = TargetStrikeProb(i, j)
55
TargetStrikeProb(i, j) = TargetStrikes(i, j) / MTypeTotal(j) / numSims
For j = 1 To 8
For i = 1 To 4
Fly UP