BC-T0P-9A

Revision 2
SEPTEMBER 1974

TOPICAL REPORT
DESIGN OF STRUCTURES
FOR MISSILE IMPACT

BSIER
BechteE Power Corporation
San Francisco, California
 TOPICAL REPORT

BC-TOP-9-A

Revision 2

DESIGN OF STRUCTURES FOR MISSILE IMPACT

PREPARED BY:

R. B. Linderman
J. V. Rotz
G. C. K. Yeh

APPROVED BY

W. A. Brandes JJJ fj('iO

H. W. Wahl

BECHTEL POWER CORPORATION Issue Date: September 1974

BC-T0P-9-A
Rev. 2

CAVEAT: THIS REPORT HAS BEEN PREPARED BY AND FOR

THE USE OF BECHTEL POWER CORPORATION AND
ITS RELATED ENTITIES. ITS USE BY OTHERS
IS PERMITTED ONLY ON THE UNDERSTANDING
THAT THERE ARE NO REPRESENTATIVES OR
WARRANTIES, EXPRESS OR IMPLIED, AS TO THE
VALIDITY OF THE INFORMATION OR CONCLUSIONS
CONTAINED HEREIN.
 UNITED STATES
ATOMIC ENERGY COMMISSION
WASHINGTON, D.C. 20S49

Mr. John V. Morowski

Vice President-Engineering
Bechtel Power Corporation
Fifty Beale Street
San Francisco, California 94119

Dear Mr. Morowski:

The Regulatory staff has completed its review of Bechtel Power

Corporation's Topical Report, BC-TOP-9, Revision 2, dated September
.1.974 and entitled "Design of Structures for Missile Impact". We
conclude that the design criteria and procedures described by this
report are acceptable to the Regulatory staff and that BC-TOP-9,
Revision 2, i s acceptable by reference in applications for construction
permits and operating licenses. A summary of our evaluation is
enclosed.

BC-TOP-9 does not provide a l l of the pertinent information required

by the Regulatory staff in i t s review of specific applications.
Therefore, the appropriate supplementary information identified in
the Regulatory Position of the enclosed Topical Report Evaluation
will have to be provided in individual Safety Analysis Reports.

The staff does not Intend to repeat i t s review of BC-TOP-9, Revision

2, when I t appears as a reference in a particular license application.
Should Regulatovy criteria or regulations change, such that our
conclusions concerning BC-TOP-9, Revision 2, are invalidated, you
will be notified and given the opportunity to revise and resubmit
your topical report for review, should you so desire.
 ftut 6 J ld74
Mr. John v. Morowski - 2 -

We request that you reissue BC-TOP-9, Revision 2, dated September

1974 in p.ccordance with the provisions of the "Elements of the
Regulatory Staff Topical Report Review Program" which was forwarded
to you on August 26» 1974. If you have any questions in this
regard, plezne let us know.

'R. W. Klecker, Technical Coordinator

for Light Water Reactors Group 1
Directorate of Licensing
Enclosure:
Topical. Report Evaluation
 TO;Jic;a 1 Report Evaluation
Report: BC-TOP-9 Rev.2
Report T i t l e : Design of Structures for Missile Imp-act
Report Date: September 1974
Originating Organization: Bechtel Power Corporation
Reviewed by: Structural Engineering Branch, November 1974

Summary of Report
This report contains the current general procedures and c r i t e r i a
used by Bechtel Power Corporation for design of nuclear power
plant structures and components against the effects of impact of
missiles. The report covers the evaluation of local effects due to
missiles impacting on both concrete and steel structural elements.
I t also covers the procedures used to evaluate the overall structural
response to missile impact loads. Design guidelines related to use
of dynamic capacity increase factors, allowable d u c t i l i t y ratio and
allowable range of steel ratios used in concrete structural elements
are also discussed in the report. Brief discussionsof special
problems related to (a) force-time history for automobile crash and
(b) penetration of a missile through a liquid are included as a
part of the report.

The formulae which can be used to predict the penetration resulting

from missile impact are included in the report. The penetration and
perforation formulae assume that the missile strikes the target normal
to the surface, and the axis of the missile is assumed parallel to
the line of f l i g h t . These assumptions result in a conservative
estimate of local damage to the target. The formula used to predict
the penetration is the Modified Petry equation, while that for per-
foration and spa!ling is the B a l l i s t i c Research Laboratory formula mod-
ified to allow i t s use for concrete strength other than 3000 psi
by replacing the constant coefficient 7.3 by 427/»/f . The wall
thicknesses to prevent perforation and spall ing are that calculated
using the B a l l i s t i c Research Laboratory formula multiplied by factors
of 1.25 and 2.5, respectively. The B a l l i s t i c Research Laboratory
formula for steel is used to predict design thickness requirement
for steal targets. The thicknesses of steel targets to provent
perforation are obtained by multiplying 1.25 by the thicknesses
for threshold perforation as determined by the URL formula.
The report discusses both elastic and plastic modes of overall
structural response of target subjected to a missile impact.
Expressions for (a) velocities of missile and target after impact,
(b) strain energy of a target required to stop a missile after
impact, (c) target effective mass definition and (d) resistance
functions for various target configurations are presented in the
report. The overall structural response of a target is determined
by equating the available target strain energy to the required strain
energy to stop a missile. The resistance function for a structural
element is determined using yield-line theory for concentrated loads
impacting steel and reinforced concrete beam and slab. The allowable
d u c t i l i t y ratios to be used for design are based on the available data
from the literature accepted in the engineering practice. However the
governing requirement for an overall structural response design con-
sideration is that the maximum deflection of the target shall be
limited so as not to impair the function of other safety related
equipment. Due to the complexity of the impact phenomena, the target
affective mass is conservatively derived based on the tests performed
on concrete slabs and beams.

The report covers two types of special problems, i . e . , determination

of an empirical formula for force-time history of automobile crash
and an avaiuation of a missile velocity as i t passes through a l i q u i d .
In deriving the force-time history of an automobile crash under frontal
impact, the automobile is considered as a deformable missile and the
structure a rigid target. The pertinent equations are based on
thaoretical considerations backsd by experimental data.
 -3-
The- derivation of the velocity of a missile after i t has penetrated
through a l i q u i d takes into consideration the buoyant force, which is
variable during the process of immersion of the missile and constant
after the entire missile is immersed in the l i q u i d , and drag force
which may be considered as constant for any particular set of con-
ditions. The non-linear, second order, non homogeneous differential
equation is transformed into a linear differential equation which
is solved by applying pertinent boundary conditions.
For the postulated missiles and their properties as well as for
structures, shields and barriers that, are required to be designed
against effects of missile impacts the report refers to the plant
SAR.

Appendix A provides the cross reference between sections of the AEC's

Standard SAR format and the sections, of BC-TOP-9. Glossary of the
report is given i n Appendix B. A review of existing design formulas
is given in Appendix C whereas Appendix D discusses theoretical der-
ivation fop- force-time history associated with automobile crash
and velocity of a missile penetrating through a l i q u i d . Sample
applications of the procedures presented in the report are shown in
Appendix E with references and bibliography l i s t e d in Appendix F.

Summary of the Regulatory Evaluation

The Structural Engineering Branch of the Directorate of Licensing
has reviewed the subject report and i t s appendices. The procedures
covered by this report with the qualifications stated in the follow-
ing Regulatory Position and augmentation of pertinent information
that is referred to and to be provided in the plant SAR are judged to
represent the present "state of the a r t " in the f i e l d of design of
structures and components against missile impacts. I f properly
utilized in nuclear power plant structural design work, the pro-
cedures and c r i t e r i a contained in the report should provide
 -4-
conservative and acceptable bases for design of structural •:.•! events
against nissile impact effects.
Reg uJatp_ry__Pos_ltion_
The design c r i t e r i a and procedures are acceptable to the Regulatory
s t a f f . The report may be referenced in future casa applications
provided that the following specific information reviewed and
accepted by the Regulatory staff is included in individual SAR:
a) Parameters that define the postulated missiles such as striking
v e l o c i t y , weight, missile configurations and impacting area, etc.
b) Structures) shields and barriers that are required to be designed
f o r missiles with t h e i r pertinent characteristics.
c) I f use o f a d u c t i l i t y ratio greater than 10 ( i . e . , y> 10) is
required to demonstrate design adequacy of structural elements
against missile impact, such a usage should be identified in the
Plant SAR. Information j u s t i f y i n g the use of this relatively high
d u c t i l i t y value may become necessary for inclusion in the plant
SAR. In such a case, the Regulatory staff w i l l request the
applicant t o provide the information on a case by case basis.
d) The evaluation of punching shear effect due to impact of uncon-
ventional missiles, is not included as a part of the overall
structural responsa consideration in the report. The subject
should be adequately addressed in individual plant SAR.
 BC-T0P-9-A
Rev. 2

ABSTRACT

This report contains methods and procedures for evaluating the effects of
missile impact on structures. A means to evaluate the change of velocity
of a missile passing through a liquid i s also included. Missile impact
effects on structures are evaluated in terms of local damage (penetration,
perforation, and spalling) and structural response. Empirical formulae
are used to evaluate local effects. Structural dynamic principles are used
to evaluate structural response.

ACKNOWLEDGMENT

This document is the result of a joint effort on the part of several

contributors.

The following is a chronological account of major participants contrib-

uting to the development of this document:

Revision 0 (issued October, 1972) was prepared by M. Fakhari,

B. Linderman, J. Rotz and M. Suarez and approved by A. J. Bingaman
(Gaithersburg Office Chief Civil Engineer) and D. W. Halligan
(Power and Industrial Division Chief Civil Engineer).

Revision 1 (issued July, 1973) was prepared by R. B. Linderman,

M. Fakhari, J. V. RotE, E. Thomas, G. A. Tuveson, and G. C. K. Yeh;
and approved by W. A. Brandes (Los Angeles Power Division, Chief
Civil Engineer) and L. G. Hinkelman (Thermal Power Organization,
Chief Civil Engineer); Technical Consultant, N. M. Newmark.

Revision 2 (Issued September, 1974) was prepared by R. B. Linderman,

J. V. Rotz and G. C. K. Yeh; and approved by W. A. Brandes (Los
Angeles Power Division, Chief Civil Engineer).and H. W. Wahl (Thermal
Power Organization, Chief Civil Engineer); Technical Consultant,
N. M . Newmark.
 BC-T0P-9-A
Rev. 2

CONTENTS

Section Title

1, INTRODUCTION

1.1 General
1.2 Approach 1-1
.1.3 Missile Characteristics 1-2
1.4 Target Characteristics 1-2

2. LOCAL EFFECTS 2-1

2.1 Reinforced Concrete T a r g e t s 2-1

2.1 . 1 Penetration 2-1
2.1 . 2 Perforation 2-2
2.1 . 3 Spalling 2-3
2.2 Steel Targets 2-3
2.3 Multiple Element Barriers 2-4
2.3 . 1 Reinforced Concrete Barrier 2-5
2.3 . 2 Steel Barrier 2-5

3. STRUCTURAL, RESPONSE TO MISSILE IMPACT LOAD 3-1

3.1 General 3-1

3.2 V e l o c i t y After Impact 3-1
3.3 Required Target S t r a i n Energy Capacity 3-2
3.3.1 E l a s t i c Impact 3-2
3.3.2 Plastic Impact 3-3
3.3.3 Force Time Function Known 3-3
3.4 Target Effective Mass 3-5
3.5 Structural Response by Energy Balance Method 3-7
3.5.1 General Procedures 3-7
3.5.2 Elastic Target Response 3-7
3.5.3 Elasto-Plastic Target Response 3-8
3.5.4 Non-Linear Target Responses 3-9

4. DESIG>: GUIDELINES 4-1

4.1 Allowable S t r e s s e s and Loadings 4-1

4.2 Design Parameters 4-1
4.3 Allowable D u c t i l i t y R a t i o 4-3
 BC-T0P-9-A
Rev. 2

CONTENTS (Cont)

Section Title Page

5. SPECIAL PROBLEMS 5-1

5.1 Force-Time H i s t o r y for Automobile Crash 5-1

5.2 P e n e t r a t i o n of a M i s s i l e Through a Liquid 5-1
5.2. 1 Liquid Depth i s Less Than o r Equal to M i s s i l e
Length 5-2
5.2. 2 Liquid Depth i s G r e a t e r Than M i s s i l e Length (H > L) 5-2
5.2. 3 D e f i n i t i o n s of Notations 5-3

x u
 BC-T0P-9-A
Rev. 2

LIST OF APPENDICES

Appendix Title Page

APPENDIX A Cross Reference Listing to AEC Standard SAR Format A-l

APPENDIX B Glossary B-l

APPENDIX C Review of Existing Formulas C-l

APPENDIX D Derivations D-l

APPENDIX E Sample Applications E-l

APPENDIX F References and Bibliography F-l

xiii
 UC-TOI'-y-A
Ruv. 2

LIST OF TABLES

Table Title rny,e

4-1 Dynamic Increase Factor (DIF) 4-4
4-2 Resistance-Yield Displacement Values for
Beams 4-5
4-3 Reaistance-YIeltl Displacement Vuiuea for
Slabs 4-6
4-4 Ductility Ratios (From Reference 28) 4-7
5-1 Drag Coefficient for Variously Shaped Bodies
in Incompressible Flow 5-4
C-.l Concrete Penetration, Perforation, and
Spalling Formulas C-5
C-2 Perforation in Steel Formulas C-3

xv
 UC-TOI'-'J-A
Ih-v. 2

LIST OF

Figure Title
2-1 Vnlucu of I'cnetrntr ion Coof f Irvlont. (K.) for
Reinforced Concrete 'J 2-6
2-2 Penetration of Reinforced Concrete for Various
Mlouilcn (Modified f'etry) 2-7
2-U l'urforation oi' Kulnfotccd CoticrciLt1 for Vnrffiun
(fkillifitl.cn RcHenrch T-nboratory) 2-8

1 2-4 Penetration, Perforation, and Spalling of

Reinforced Concrete T.-iryet by Postulated

Tornndo Missiles 2-9
3-1. RcBiHtance-Disptacemcnt Functions With
Associated Structural Re.qponHe With and Without
The Effects of Other Loads 3-12
3-2 Energy-Displacement Functions - Impact Loads Only 3-13
3-3 Energy-Displacement Functions - Impact Combined
With Other Loads 3-14
2 | 4-1 Coefficients for Moment of Inertia of Cracked
• Sections 4-8
5-1 Penetration of a Missile in a Liquid 5-^
C-l Typical Crater Profiles C-9
I C-2 Deleted
2 | thru
I C-22

xvii
 UC-T0|'-9-A
I

Strctlcn J

INTRODUCTION

1.1 GENERAL

The detilgn of nuclear power f a c i l i t i e s Includes the effects of missile

Impact on s t r u c t u r e ^ systems, and equipment.. Kxturnal building surfaces,
Interior wnltfi nnd floortu nnd special b a r r i e r s (constructed of concrete
and/or s t e e l ) that will reulut or dcllcct mltt»( lct> may l>u uHi'd tu (jrotct t
system?} and t;n'-ipn"*n*' WIHT«

This roport contains mcthodo and preferred procedures to evaluate missile

impact on structures and b a r r i e r s . Missile effects /ire evaluated In ternui
of local damage (penetration, perforation, und upnlllng) and HtructuraJ
response.

Mlasilca may be generated by an event that la not related to plant operation,

or by the failure of plant equipment. The primary sources of mlasllen, not
related to plant operations are debris transported by tornudo winds, and
falling objects generated by a c t i v i t i e s near the plant s i t e (such as com-
mercial, i n d u s t r i a l , o r military a c t i v i t i e s ) . Missiles that may result
from the failure of equipment generally result from the uncontrolled release
of energy and forces from a pressurized system or rotating machinery.
Missiles that nay result from the failure of equipment are f i t t i n g s , valve
p a r t s , various nuts and b o l t s , and parts of rotating machinery, e t c .

1.2 APPROACH

Determining tlte effect of missile Impact i s outlined in the following

general steps. However, there are many interactive effectu in each step
that nhould be considered In the complete a n a l y s i s .

• Determine missile characteristics.

• Define target, considering impact in combination with other loads

and requirements (piollmlnary propertl'-s).

e Determine local effects of missile on t a r g e t .

• Determine target c h a r a c t e r i s t i c s for s t r u c t u r a l response and

stability.

• Determine equivalent: target mass during impact.

• Determine structural, response.

• Evaluate structural integrity.

• Verify that the maximum deflection does not impair the function
of other safety related systems.

1-1
BLANK PAGE
 M>T0r-9-A
Rev. 2

1.3 HI5S1UE SHAKACTKRiy ICfl

parameturu required for mlouile Impact unalyoia Include t r a j e c t o r y ,

maun, velocity, geometry, and deformation c h a r a c t e r i s t i c s . The geometry
Bhoulil incUulo contact aruy, projected frontal area and variation of
area with roupect t o length, Deformation chnrncti'r lnt ICH Include If the
mlauile will deform or In rigid nnd if I t l« ductile or b r i t t l e . Missile
geometry and deformation chai'acferlHtlca Imve a significant effect on pene-
t r a t i o n or perforation of a t a r g e t . A pointed ralBslie will penetrate deeper
Into a t a r g e t than n blunt misuilai I t w i l l also perforate n thicker t a r g e t .
Deformation of a miooilu during Impact consumes energy, which r e s u l t s in
diminished l o c a l

Postulated mlosllua and their properties may vary with each plant and are
defined in the Safety Analysis Report (SAR) for nuclear power p l a n t s .

1.4 TARGET CHARACTERISTICS

Structures or b a r r i e r s ( t a r g e t s ) , providing missile protection, act aa

energy absorbers. The target absorbs the energy by local damage at the
location of Impact ( i . e . penetration of the missile into the b a r r i e r ) and
by the s t r u c t u r a l response of the t a r g e t .

Local damage depends on missile c h a r a c t e r i s t i c s , target material p r o p e r t i e s ,

and s t r u c t u r a l response. Empirical methods are used to estimate local
damage because of the complex phenomena associated with missile impact.
The a b i l i t y of a target t o absorb energy by s t r u c t u r a l response dependo on
the dynamic properties of the t a r g e t , support conditions and other Imposed
loads at the time of impact. Structural dynamic p r i n c i p l e s are used to
estimate the s t r u c t u r a l response and determine if the target w i l l remain
s t a b l e during and a f t e r missile impact.

Structures, shields and bnrriors that are required to be designed for a

missile are given in the Safety Analysis Reports.
 IJC-TOI'-y--A
Rev. 2

Section 2

LOCAL EFFECTS

Predicting local damage in the impact area includes estimating depth of

penetration, minimum thickness required to prevent perforation, and minimum
thickness required to preclude spalling. The penetration and perforation
formulae in this section assume that the missile strikes the target normal
to the surface, and the axis of the missile is assumed parallel to the line
of flight. These assumptions result in a conservative estimate of local
damage to the target. Appendix C has information an the mere cctniuOi* l«e«I
effect formula and a discussion oi the effects on the penetration for a
missile striking a target at oblique angle.

2.1 REINFORCED CONCRETE TARGETS

2.1.1 PENETRATION

The depth to which a rigid missile will penetrate a reinforced concrete

target of infinite thickness is estimated by the following formula

X . 12 Kp A p Log1Q \1 - 21^000/ <2-«

where

X = Depth of missile penetration into concrete element of infinite

thickness (inches)

Note: Usually this equation expresses the depth of pene-

tration in feet; however, for this document it has been
modified to express it in inches.

K = Penetration coefficient for reinforced concrete (see Figure 2-1).

A . W.= Missile weight (psf)

p A Projected frontal area of missile
V = Striking velocity of missile (ft/sec). (Limit V £. 1000 ft/sec)
a S

This formula is known as the Modified Petry formula.

When the element has a finite thickness the depth of penetration is:
X, (t

2-1
KC-T0P-9-A
Kuv. 2

where

X. » Depth of p e n e t r a t i o n of m i s s i l e Into a concrete; clement of finite

thlcknt'HB ( I n c h e s ) .
a
e Bane of Napierian Logarithms

t « Thlckncan of concrete element (Inches)

P e n e t r a t i o n s for various I l l u s t r a t i v e examples of mlHHIlew are uliown in

2-2 and 2 - 4 .

2.1.2 PERFORATION

The thickness of a concrete clement that will Just be perforutod by u

mlsollo is given by:

V c

where

T • Thickness of concrete element to be Just perforated (inches)

W - Weight of missiles (lb)

D » Diameter of missiles (Inches)

Note: For Irregularly shaped missiles, an equivalent

diameter Is used. The equivalent diameter is taken as
the diameter of a circle with an area equal to the cir-
cumscribed contact, or projected frontal area, of the
non-cylindrical missile.

V «• Striking veloc}»-v of missile (ft/sec)

f • Compress Ive strength of concrete (psi)

Th( ! '.mil is kuuwn as; the Ballistic Research Laboratory, BRl,, formula.

The thickness, tp, of a concrete element required to prevent perforation

must be greater than T. It is recommended to increase T by 25 percent, but
not more than 10 Inches, to obtain the t , required to prevent perforation

t « 1.25T S T + 10 (in inches) (2-4)

2-2
 BC-T0P-9-A
Rev. 2

The threshold of perforation, T, for various i l l u s t r a t i v e examples of

misoiles io shown in figures 2-3 and 2-4.

2.1.3 SPALL1NG
Spalling of concrete from the side opposite the contact surface of the e l e -
ment may occur even if the miBsile will not perforate the element. For an
estimate of the thickness that will Just s t a r t s p a l l i n g , i t id recommended
that the following equation be used:
Ts - 2T (2-5)

where
T •> Concrete element thickness that will j u s t s t a r t spalling (Inches)
8

T i Concrete thickness to be just perforated (inches).

See Equation (2-3)
The thickness, t 8 ) of a concrete element required to prevent spalling must
be greater than T8. It is recommended to increase Tg by 25 percent, but
not more than 10 inches, to prevent spalling.
t - 1.25 T D S T + 10 ( i n Inches) (2-6)
s s s

2.2 STEEL TARGETS

Steel t a r g e t s , such as pipes and mechanical equipment vessels, may be per-
forated by a missile. Sometimes, protruding elements of a missile may
puncture a s t e e l target when the e n t i r e missile does not perforate or pass
through the target. The minimum contact area of a missile protrusion is
used to calculate puncture thickness and the projected area o£ the e n t i t e
missile is used to calculate perforation thickness.
The BRL Formula is shown below, modified by setting a material constant
K * 1 and solving directly for steel plats thickness, T, which will j u s t
be perforated by the missile,

2/3

672— <2~7>

2-3
 RC-T0P-9-A
Rev. 2

T a Steal plate thickness t o Just perforate (Inches).

M • Mass of the Missile (lb s e c 2 / f t )
W - Weight of the Missile (lb)
V. • Striking Velocity of the Missile Normal to Target Surface (ft/sec)

D = Diameter of the Missile (In.)

Note: For irregularly shaped missiles the equivalent
diameter is used. The equivalent diameter i s taken as the
diameter of a c i r c l e with an area equal to the circum-
scribed contact, or projected frontal area of the
non-cylindrical missile .
'file thickness, t , of a steel b a r r i e r required t o prevent perforation should
exceed the thickness for threshold of perforations. It is recommended to
increase the thickneso, T, by 25 percent to prevent perforation.
t p * 1.25T (2-8)

2.3 MULTIPLE ELEMENT BARRIERS

It may be desirable to construct a missile barrier of several thinner e l e -
montB, instead of one thick element. Analysis of missile b a r r i e r s composed
of neveral elements involves determining the residual velocity (V r ) after
perforation of one element and using t h i s value for the striking velocity
(Vq) on the next element. The following formula is used to determine the
residual velocity, Vr (see Appendix C)
1/2
/2 2\
V
r " (Vs ~ V p ) For
<Vp * V
(2-9)
For (Vp 2 Vs)

where
Vr « residual velocity of missile after perforation of an element of
21 thickness t . (fps)
V • striking velocity of the missile normal to target surface (fps)

V • velocity required to just perforate an element (fps)

2-4
 BC-T0P-9-A
Rev. 2

2.3.1 REINFORCED CONCRETE BARRIER

Combining equations (2-3) and (2-9), the residual velocity of a missile
perforating a concrete target is
-ll/2

V = (2-10)
t

where ts thickness of concrete element (inches)

2.3.2 STEEL BARRIER

Combining equations (2«-7) and (2-9), the residual velocity of a missile
perforating a steel target is

F,,2 u
1.12 x 10 6 (Dt) 1>3 1
L''
where t = thickness of steel element (inches)
J (2-11)

2-5
BC-T0P-9-A
Rev. 2

0.U05O

2,000 3,000 4,000 5,000 6,000 7,000

28 - DAY COMPRESSIVE STRENGTH OF CONCRETE

Figure 2-1

VALUES OF PENETRATION COEFFICIENT (K p ) FOR REINFORCED CONCRETE

( R e f e r e n c e 14)

2-6
 COMPRESSIVE.STRENGTH OF
V) CONCRETE = 3000 PSI
W
U
z
NOTE: 1. W i n pounds
2. A in inches ^
UJ
3. Equation 2-1 uses
LU Ap in Ib/ft 2
cc
O
O
O
o
s
>•

to
•vl

X
Q.
Ul
Q
2
O
<
m
z So
ID

I
200 300 400 500
MISSSLE VELOCITY (FT/SEC)
Figure 2-2
PENETRATION OF REI?iFORCED CONCRETE FOR
VARIOUS MISSILES (MODIFIED PETRY)
 n ca

en
UJ
COMPRESSIVE STRENGTH OF
I >
CONCRETE = 3000 PS!
o

Q NOTE: W in pounds
LU
0 in inches

O
IX.
cc
UJ
o.

I
UJ
CQ

oo
O

8
OJ
2
O

5E
UJ
UJ
cc
o
I
200 300 400 500
MISSILE VELOCITY (FT/SEC)
Figure 2-3

PERFORATION OF REINFORCED CONCRETE FOR

VARIOUS MISSILES (BALLISTIC RESEARCH LAB)
 6-Z

PENETRATION DEPTH
IN INFINITE THICKNESS CONCRETE, EQ. 2-1 (INCHES)
ro
o

no)
CO j^ a
- X m
•©• •©•
-^ z
CO CO ro Q
—1 H *2
m rn o
m m o
r- r- a
33 -o
•o O r-
rn a |>
00•z.
00 7s
00 ro
00 CO o
C/5 o
CO
CO

(s © C5
CO ro
•©-
; •o-
O)
co
H m

UTI
m
ni
rn r~
i~ r~
H
^^ m
-o m
OLE
00
en
CO
p.. CO
ro CO
CO CO
o
LBS.

CONCRETE THICKNESS FOR THRESHOLD OF PERFORATION, EQ. 2-3

ro en
o o

CONCRETE THICKNESS FOR THRESHOLD OF SPALLING, EQ, 2-5 (INCHES)

V-6--cI01-0?<
 HC-TOl'-O-A
Ht!V. 2

STRUCTURAL RESPONSE TO MISSILE IMPACT LOAD

3.1 GENERAL

When a missile strikes a target, large forces develop at the missile -

target Interface, which decelerate the missile and accelerate the target.
If the interface forcing function is known, (experimentally determined),
the target structure can be modeled mathematically and conventional numeri-
cal techniques can be used to predict structural respond?: F<?? v\pf*t cases,
the forcing function is not known, and a rational method involving energy
balance techniques is used to estimate structural respond. This involves
using the strain energy of the target at maximum response to balance the
residual kinetic energy of the target (or target-misHile combination)
resulting from missile impact.

For investigation purposes, it is convenient to model the event as a missile

of mass, M,,,, and striking velocity, V 8 , impacting a spring-backed target
mass, M e . The spring may be linear, bilinear, or non-linear, depending on
the target structure resistance-displacement function. Since the actual
coupled mass varies during impact, an estimated average effective target
mass, M e is used to evaluate inertia effects during Impact.

The impact may be either elastic or plastic, depending on whether or not

significant energy losses are sustained during impact. These losses are
associated with inelastic deformations, local damage in the impact zone,
etc.

Plastic impact is characterized by the missile remaining in contact with

the target, subsequent to impact. In an elastic impact, the missile and
target remain in contact for a very short period of time, and then disen-
gage due to elastic interface restoring forces.

An clastic missile impact case is rarely encountered in nuclear plant

design. For example, baaed on information available, a plastic collision
can be considered for all postulated tornado-gentrated missiles.

3.2 VELOCITY AFTER IMPACT

Since the duration of impact is very short, (usually less than a few milli-
seconds), the target mass displacement and the corresponding spring force
are also very small. Neglecting the spring force effect during impact,
(a slight conservatism), the velocities of the missile and target after
impact are calculated from the following relationships:

V
s <Mm - eMe)
V - (3-1)
m M + M
e
m

V
' Vm,(1+e)
T~ M •
m
3-1
AN
BC-T0P-9-A
Kav. 2

V - Missile velocity after impact

V,- « Target velocity after impact

V Q - Missile striking velocity

M - Mass of missile
m

M.^ • Effective mass of target during impact

e » Coefficient of restitution

3
-3 REQUIRED TARGET STRAIN ENERGY CAPACITY

3.3.1 ELASTIC IMPACT

Equations (3-1) and (3-2)^ ' show that the velocity of the missile after
impact is opposite to that of the target if M m is less than eMe. For this
case, the strain energy, E s , of the responding target spring required to
diminish the target ma9s velocity to zero (maximum target response) is
numerically equal to the kinetic energy of the target mass at the end of
the impact duration.

(3-3)

If the Impact is determined to be elastic and the coefficient of restitution

is not known, a conservative value of e equal to unity can be assumed.
Making this substitution in equation (3-2), and substituting this value for
Vq> into equation 3-3, the required strain energy of the responding target
is;

2 2
2MZM V
p _ •" e s
s ~7
7 +TTM )?
(M (3-4)
3 \ m e/ 2
( )
Referring again to equations (3-1) and (3-2), the velocity of the missile after
impact is in the same direction as that of the target if M,,, is greater than
eM e . In this case, the target spring decelerates the target mass, allowing
the missile to overtake the target, which results in multiple impact.

If the impact is purely elastic (e = 1 ) , the target will eventually stop the
missile through a series of impacts and absorb all the initial kinetic

*References are in appendix F.

3-2
 BC-T0P-9-A
Rev. 2

energy oi the m i s s i l e . The required a t t a i n energy of the responding t a r g e t

is then equal to the i n i t i a l kinetic energy of the missile.

M V2
E
s

3.3.2 PLASTIC IMPACT | 2

For a p l a s t i c collision, the coefficient of r e s t i t u t i o n reduces to zero
(e = 0) and Lhe (ulrislle and target masses attain the same velocity at the
end of impact duration. If the impact is of short duration, the target
displacement and corresponding spring force effect during impact are small,
and can be conservatively neglected. The strain energy required to stop
the target-mi3sile combination is then the sum of the kinetic energy of the
missile and the target masses at the end of the duration of impact.

From equations 3-1 and 3-2

Vm - V, - jffi- (3-7,
Substituting the value for V and VT from equation (3-7) into equation ( 3 - 6 ) ,
the required target s t r a i n energy i s

M2V2
E
s * 2 (M V M ) <3"8>
m e

3.3.3 FORCE TIME FUNCTION KNOWN

In some isolated cases, (such as for frontal impact of an automobile, see
section 5 . 1 ) , sufficient experimental data are available to enable d e f i n i -
tion of a force-time function, F ( t ) , a t the interface between the missile
and t a r g e t . This enables d i r e c t solution of the equation of motion:
F(t) - R(x) = Mex (3-9)
F(t) = Force-time function
R(x) = Resisting spring force as a function of
displacement, x
x = Acceleration of target mass
Me = Effective target mass

3-3
UC-T0P-9-A
Hev. 2

Numerical methods arc usually used for solution of equation 3-9 which is
solved for the maximum value of displacement xm. The target strain energy
la then;

R(x)dx
s Jo

An abbreviated conservative solution for required target strain energy can

be obtained ii! R(x) during impact is small compared to F(t) and plastic or
permanent deformation is dominant at the missile-target interface

The velocity of the target mass at time, t, is;

*<t> - C k'dt - C M
dt
Jo Jo
Jo e

The kinetic energy of the target mass at time t is then

M [i (t)] 2
E(t)

or (3-10)
E(t) - [F(t) - R(x)] dt
2M

Equation (3-10) shows that deletion of the R(x) term will result in a
conservative overestimate of E(t). If R(x)<< F(t) during impact, t, the
inaccuracy Is usually negligible. For this condition, the kinetic energy
of the target mass at time tj is conservatively estimated as;

n2
F(t) dt

t. ZCl
(3-11)
1 e
The applied impulse, I, is by definition, the area under the force-time
curve.

J F(t) dt

3-4
 BC-T0P-9-A
Rev. 2

Making this substitution into equation (3-11);

I2 2
Et = 2M = d/2)M V* (3-12)
i e

If the elastic restoring forces at the missile-target interface are small,

the velocity of the missile approaches that of the L^rget at the end of
time, t^, equal to the duration of impact. The strain energy of the target
required to stop the missile-target combination is then;

For a plastic collision,

From equation (3-12):

v2 - * 2
T ~—2"

and,

Making this substitution into equation (3-13):

(Ms + M ) I.2
E -— § (3-14)
S
2M2
e

3.4 TARGET EFFECTIVE MASS

The effective target mass during impact varies from a low value at initial
contact and generally increases to an upper limit during or at the end of
the impact duration. Due to the complex phenomenology associated with
missile impact, no general analytical solution is available to evaluate the
effective coupled mass on a continuous time basis. The average effective
mass can, however, be estimated, utilizing the results of impact tests on
reinforced concrete beamsO) wherein the measured maximum structural
response was used to back-calculate the average mass during impact.

3-5
I1C-T0P-9-A
Ilev. 2

Based on these data, the following formulae provide a lower limit estimate
of M c (which results in on upper limit estimate of kinetic energy after
impact).

Fur concrete beams:

2T) -|-. [ « B <(Dy + 2T)J

y T r , (3-15)
l f lfUU
Me - (D x + 2T)(Dy + 2T) - ||- . [[ 2«>
«> y +
+ 2T)]
2T)]
For concrete s l a b s :
YCT
Me - (D x + T) <Dy + T) - £ - (3-16)

For steel beams:

M e - (Dx + 2d) M x (3-17)

For steel places

M " Average effective mass of target during impact

Mv " Mass per unit length of steel beam

D - Maximum missile contact dimension in the x direction (longitudinal

axis for beams or slabs)

D " Maximum missile contact dimension In the y direction (transverse to

' longitudinal axis for beams or slabs)

T - -Thickness or depth of concrete element

t - Thickness of steel plate

d = Depth of steel beam

B » Width of concrete beam (not to exceed D + 2T)

Y * Weight per unit volume of concrete

Y o * Weight per unit volume of steel

g a Acceleration of gravity

3-6
 BC-T0P-9-A
Rev. 2

3.5 STRUCTURAL RESPONSE BY ENERGY BALANCE METHOD

3.5.1 GENERAL PROCEDURES

The strain energy, E 8 , required to stop the target (or missile-target

combination), Is determined from the relationships in sections 3.2 and 3.3.

The resistance-displacement function, R(x), for a concentrated load at the

area of impact is determined from the target structure physical configura-
tion and material properties.

The estimated maximum target response is determined by equating the avail-

able target strain energy to the required strain energy and solving for the
maximum displacement x . (See Figure 3-1.)

3.5.2 ELASTIC TARGET RESPONSE

For elastic response,

R(x) - kx

k • Elastic spring constant

If no other loads are acting concurrently with the missile impact loading,
the maximum response is

(3-19)

If other loads are present on the target structure which will act concurrently
with missile impact loads, the maximum combined displacement is determined
as follows:

Then

X * X + X1
m o
Since

•2 v w

Let

x 1 - Displacement due to missile impact (See Figure 3-1)

x » Displacement due to other loads

x • Maximum combined displacement

m
 BC-T0P-9-A
Rev. 2

2
, i t follows that
I To „ n l / 2
x - x - < 3 ~ 20 >
m o

3.5.3 ELASTO-PLASTIC TARGET RESPONSE

For elasto-plastic target response with no other concurrent loads acting:
R(x) » kx, C0<x<x e )

R(x) = k x e - R m , ( x e < x < xm)

where
x = Yield displacement

R = Plastic resistance,
m
Then / j
E =• R ( x - -
s m\ m <
or Ex
x . _ £ + _J
m

The required ductility ratio, p r , i s obtained from equation(3-21) by dividing

both sides of the equation by x .
X
m

E
+
s 1
TIT
em
Y

If other loads are present on the target structure which will act concurrent
with missile impact loads, the maximum combined displacement Is determined
as follows:
Let
x 1 = xx - xx (see figure 3-1)

x = displacement due to other loads

 BC-T0P-9-A
Rev. 2

x • yield displacement

x «• maximum combined displacement

R * plastic resisting force

k
• elastic spring constant
Then

^2
F + kx
'c * ^ 9 " ' <xm ' O < s e e ^ g u r e 3-1)

in kx1 2 e
Substituting x* = x - x in the above equation feives
e o

\ - k <»e -s,o)+ H ^ (3
-">
The required ductility ratio, p r , is obtained by dividing both sides of
equation (3-23) by x_.

(3
~24)

The values of u r should be less than the allowable d u c t i l i t y ratios p given

in section 4.

3.5.4 NON-LINEAR TARGET RESPONSES

If the resistance-displacement function i s nonlinear (figure 3-1) the
determination of s t r u c t u r a l response i s f a c i l i t a t e d by f i r s t defining the
strain energy-displacement function, (see figure 3-2).

E - I R(x) dx (3-25)
•'o
E * strain energy at displacement x

E = strain energy at displacement x

e

3-9
BC-T0P-9-A.
Rev. 2
V.hen no other concurrent loads are acting* the maximum displacement occurs
at the value of x where E e I s equal to E s . The correct value of XQ I S there-
fore the value of x, which will satisfy the following relationship:

E « f ° R(x) dx (3-26)

A typical graphical solution is shown In figure 3-2.

When other loads are acting concurrent with missile impact loading, the
correct value of xm will satisfy the following relationship:

E = / R(x)dx - R (x - x ) (3-27)
s J o m o

R = equivalent s t a t i c resistrance required for other loads

0
(see figure 3-1)
xo =* displacement associated with Ro .

A typical graphical solution for x is shown schematically in figure 3-3.

To provide an adequate margin of safety the values of E should satisfy the
condition
E
s 1 FsEf (3-28)

E, = impact strain energy capacity

F «• safety factor
8
F a 0.5 if R(x) is well defined from tests
a
F = 0.25 if R(x) i s approximately determined (such as by failure
analysis)
Fof impact only:
x
• f
Ef = / R(x) dx (3-29)
o

Xc = displacement at failure

3-10
 BG-TOP-9-A
Rev. 2

For impact combined V7ith other loads:

dx - R (* - x ) (3-30)
•/ O E O
X
and o
R < R,
o— f

R. --= resistance at failure

3-11
 70 a
RESISTANCE - AVAILABLE STRAIN AVAILABLE STRAIN
RESPONSE DISPLACEMENT ENERGY WITHOUT ENERGY WITH
FUNCTION OTHER LOADING OTHER LOADING

ELASTIC fi - kx

x'

ELASTO-PLASTIC
A

NON-LINEAR

SHADED AREA (STRAIN ENERGY) MUST EQUAL E, (FROM SECTIONS 3.2 AND 3.3)

Figure 3-1
RESISTANCE-DISPLACEMENT FUNCTIONS WITH ASSOCIATED STRUCTURAL
RESPONSE WITH AND WITHOUT THE EFFECT OF OTHER LOADS
 BC-T0P-9-A
Rev. 2

<r
111

DISPLACEMENT X

Figure 3-2
ENERGY-DISPLACEMENT FUNCTIONS-
IMPACT LOADS ONLY

3-13
BC-T0P-9-A
Rev. 2

UJ

(S
K

"ffl "f

DISPLACEMENT X

Figure 3-3
ENERGY=DISPLACEMENT FUNCTIONS -
IMPACT COMBINED WITH OTHER LOADS

3-14
 BC-T0P-9-A
Rev. 2

Section 4

DESIGN GUIDELINES

4.1 ALLOWABLE STRESSES AND LOADINGS

The combination of loadings, allowable stress and strain limits, and

applicable codes used with the missile impact loading are given in the
Safety Analysis Report. The resistance of a structural component must be
baaed on its minimum strength, I.e., the minimum of its flexural or shear-
ing capacity. The dynamic capacity of the structural elements must be
based on mafcurrial dynamic sLtfength properties which arc obtained by applying
a dynamic Increase factor (DIF) to the static strength value:

f. - (DIF) f , (4-1)

where

f = allowable dynamic strength value

f = specified static strength value

DIF » dynamic increase factor

The dynamic increase factor for various materials are given in table 4-1.

4.2 DESIGN PARAMETERS

The resistance of typical structural elements, whose flexural strength

defines the minimum capacity, and their yield displacement approximations
are presented in tables 4-2 and 4-3. Similar equations can be developed
for the load at other location on the structural element. It is prefer-
able that the limiting capacity of an element be in the flexural mode not
in shear. In evaluating the yield displacement with the usual elastic
analysis, the foment of inertia must account for cracking of concrete
sections. The empirical relation for this type of loading is an average
moment of inertia I a calculated as follows is:

h-iK + ^'V1^*™! (A 2)
-
where

J * moment of inertia of gross concrete cross section of thickness t

about its centroid (neglecting steel areas)

I c - moment of inertia of the cracked concrete section

4-1
BC-T0P-9-A
Rev. 2

b =» width of concrete section

F = coef Ccient for moment of inertia of cracked section with

ion reinforcing only. (See figure 4-1.)

t = concrete thickness

d • distance from extreme compression fiber to centroid of tension

reinforcing

The moment of inertia I , as calculated by equation (4-2), must be used in

the displacement equation in tables 4=2 and 4-3 for all reinforced concrete
members. The ultimate moment capacity of a concrete section shall be con-
sidered as the moment strength

M
u = °' 9A s fdy(d " a / 2 ) (4
" 3)
where

A = area of tensile reinforcing steel

f. = allowable dynamic yield stress for reinforcing steel

d = distance from extreme compression fiber to centroid of tension

reinforcing

a = depth of equivalent rectangular stress block

If the element has compression steel, it should be considered and the

appropriate equation used.

The amount of reinforcing steel in a concrete members must satisfy the

following criteria:

For members with tension steel only:

V* °'25 *c (4-4)-
f \d' - bd - f
y y
For members with tension and compression steel:

f \d/ - bd i, , \
y (4-4a)
A - A1 / f 2 0.25 V
bd 'd' - f

4-2
 BC-T0P-9-A
Rev. 2

where

f' = compression strength of concrete

A' = area of compressive reinforcement of concrete

4.3 ALLOWABLE DUCTILITY RATIO

The maximum allowable ductility ratios for concrete and steel members are
presented in Table 4-4. However, the maximum deflection shall be limited
so as not to impair the function of other safety related equipment.

4-3
BC-T0P-9-A
Rev. 2
Table 4-1
DYNAMIC INCREASE FACTOR
(DIF)
(From Ref. 19)

I, Reinforced or Prestresaed Concrete

Concrete DIF
Compression 1.25
Diagonal Tension & Dlrecc Shear (Punch Out) 1.0
Bond 1.0
Reinforcing Steel
Tension & Compression For AOksi yield strength steel 1.2
60ksl yield strength steel 1.0
Diagonal Tension & Direct Shear (Stirrups) 1.0

II. Structural Steel

Flexure, Tension, & Compression for AOksi yield strength steel 1.2
60ksi yield strength steel 1.0
Shear 1.0

4-4
 BC-TOP-9-A
Rev. 2

Table 4 - 2

RESISTANCE-YIELD DISPLACEMENT
VALUES FOR BEAMS

YIELD
DESCRIPTION RESISTANCE DISPLACEMENT

(1) CANTILEVER

HI

(21 SIMPLY SUPPORTED

- RL 3
T5

ffnt
k-

(3) FIXED SUPPORTS

iff
4<M++M 5 ) X - BL 3
192EI

LI2 ' L/2

(4) MULTI-SPAN

R- X. - 0.011RLJ
e E,

t t .|. ff. Where M+ ULTIMATE POSITIVE MOMENT CAPACITY

ULTIMATE NEGATIVE MOMENT CAPACITY
MOMENT OF INERTIA (in*)
FOR REINFORCED CONCRETE I - la,
SEE EQUATION 4-2.

4-5
 BC-T0P-9-A
Rev. 2

Table 4 - 3

RESISTANCE YIELD DISPLACEMENT

VALUES FOR SUBS
AND PLATES

YIELD
DESCRIPTION RESISTANCE (29.30,31. 32) DISPLACEMENT (33)

ID SIMPLY SUPPORTED ON ALL

4SIDES WITH LOAD AT
CENTER

R - • ana 2 i\-v2)
12EI

b/a 1.0 1.1 1.2 1.4 1.6 1.8 2.0 3.0 00

a .1390 .1518 .1624 .1781 .1884 .1944 .1981 .2029 .2031

(2) FIXED SUPPORTS ON ALL V POISSON'S RATIO

A SIDES WITH LOAD AT
t THICKNESS (in)
CENTER
E MODULUS OF ELASTICITY (Ib/in2)
I MOMENT OF INERTIA PER UNIT WIDTH (in4/in)
FOR REINFORCED CONCRETE SECTION I = la,
SEE EOUATION 4-2
ULTIMATE POSITIVE MOMENT CAPACITY (in Ib/in)
"S
ULTIMATE NEGATIVE MOMENT CAPACITY (in Ib/in)

m
X-
27T
12EI

b/a 1.0 1.2 1.4 1.6 1.8 2.0 oo

a .0671 .0776 .0830 .0854 .0864 .0866 O.0671

4-6
 BC-T0P-9-A
Rev. 2

Table 4-4

DUCTILITY RATIOS
(From Reference 28)

Max. Allowable Value of u

Reinforced Concrete

Flexure

Beams

Slabs 2ii2. < 30

P-P "

Compression

Malls & Columns 1*

where
A
s
p is the ratio of tensile reinforcement • -rr

A'
p 1 is the ratio of compressive reinforcement • -TTT
bd

Steel Elements

Members proportioned to preclude lateral

and local buckling

Flexure, compression and shear 20

Steel Columns

Proportioned to preclude elastic buckling 1.3

e
j.
Members stressed in tension only 0.5 —
"y

e = ultimate strain

e = yield strain

4-7
BC-T0P-9-A
Rev. 2

1.0

cr = F bd' >
ri. 0 > A
0.
(JL), 0. 50.X X y
0. 25 N
.0. 00 N
1 N
F ^****~+
10-
a*
j *

y
/
10-2
10-2 101 1.0
RATIO pn

A's Es
n =
bd' bd E£

- j p - a 1.9. -j-as 0.10, K = -m + (m2

P' P'

Figure 4-1
COEFFICIENTS FOR MOMENT OF INERTl/i
OF CRACKED SECTIONS
 BC-T0P-9-A
Rev. 2

SECTION 5

SPECIAL PROBLEMS

Two special problems are the determination of an empt:.' ^.-al formula for
force-time history of automobile crash, and the evaluation of a missile's
velocity as i t passes through a liquid.

5.1 FORCE-TIME HISTORY FOR AUTOMOBILE CRASH

In d e r i v i n g t h e f o r c e - t i m e h i s t o r y of an automobile crash under f r o n t a l

impact, t h e automobile i s considered as a deformable m i s s i l e and t h e
s t r u c t u r e as a r i g i d t a r g e t . According t o Appendix D, Paragraph D . I ,
which i s based on a t h e o r e t i c a l c o n s i d e r a t i o n and c o n s i d e r a b l e e x p e r i -
mental d a t a , the f o r c e - t i m e h i s t o r y under such a c o n d i t i o n i s a p p r o x i -
mately as f o l l o w s :

F ( t ) = 0.625 V W 3 l n 2 0 t , ( 0 < t < 0.0785 s e c )

s ra
(5-1)

F(t) = 0 (t > 0.0785 sec)

where

t = time from the instant of initial contact (sec)

F(t) = time-dependent force on target (lb)

V = striking velocity of the automobile (ft/sec)
s

W =• weight of automobile (lb)

m
References on derivations of more elaborate force-time histories for auto-
mobile crashes are given in reference 11.

5.2 PENETRATION OF A MISSILE THROUGH A LIQUID

To evaluate the effect of a missile on a target that Is submerged in &

liquid, determine the striking velocity of the missile, V, after it has
penetrated through a depth, H, of liquid covering the target (figure 5-1).
This involves evaluating the velocity change due to missile weight, the
buoyant force, and the drag force.

The penetration of a missile as It enters a liquid depends on the geometric

shape of the missile. For the vertical entry of a missile with uniform
horizontal cross-sectional area A-, and length L, the depth of penetration
and the velocity at a depth, x, are in terms of two functions of x. (The
functions are evaluated at x - H or L.)

5-1
BC-TOP-9-A
Rev. 2

Z ^ x ) - g/a + bA 0 (l-2ax)/2a Z + e (VQ -g/a-bA 0 /2a^), (0 < x < L) (5-1)

2aL
( bAQ [e 2 a L (l-2aL)-l] /2a2 2 +
+ V
VQ22
+ (e2aL
+ gg(e
2V
2 2
2

<x > W (5-2)

Notations used above are defined at the end of t h i s section. Missile pene-
t r a t i o n in a liquid can be catagorized by the following c a s e s :

5.2.1 LIQUID DEPTH IS LESS THAN OR EQUAL TO MISSILE LENGTH (H< L)

5.2.1.1 I f Zj^Cx) is Negative or Zero a t Depth x • H (Z.(H) < 0)

The missile w i l l not s t r i k e the t a r g e t . I t w i l l penetrate a depth H < H

such that Z (H^ = 0, and then float to the liquid surface.

5.2.1.2 If Z^x) is Positive a t Depth x = H ^ (H) > 0)

The s t r i k i n g velocity at depth H is

V- [Zx (H)] 11/2

/2
(5-3)

5.2.2 LIQUID DEPTH IS GREATER THAN MISSILE LENGTH (H > L)

5.2.2.1 If Z»(x) is Negative or Zero at Depth x = L (Z2(L) < 0)

The missile will not s t r i k e the target. It will penetrate a depth HL < L
such that Z1 (H- ) = 0, and then float to the liquid surface.

5.2.2.2 If Z2(x) is Positive at Depth x = L (Z2(L) > 0)

The missile will penetrate the liquid deeper than L. There are two
possibilities:

A. If 2 (x) i s Negative or Zero at Depth x = H (Z (H) < 0)

The missile will not strike the target. I t will penetrate a

depth H (L< H < H) such that Z (H ) = 0, and then float to the
l l
liquid s u r f a c e /

B. If Z2(x) i s Positive at Depth x => H (Z2(H) > 0)

The striking velocity at depth H is

V = [Z2(H)I 1 / 2 (5-4)

5-2
 BC-T0P-9-A
Rev. 2

In case the missile shape does not have a uniform cross-sectional area,
refer to equations (D-23) and (D-36) in Appendix D.2 for more general
solutions.

5.2,3 DEFINITIONS OF NOTATIONS

a « Y A 0 CD/2W (5-5)

b - Yft/W (5-6)

g = gravitational acceleration

(g - 32.17 ft/ 8 ec 2 at sea level)

W » weight of missile

Y =» weight density of liquid

( Y = 62.4 lb/ft3 for water at 80°F)

Y
m =• weight density of the missile

x = depth of missile e.g. below the initial eg,, as shown in figure 5-1.
A n • horizontal cross-sectional area of the missile (constant over
Length L)
C^ = drag coefficient (given in table 5-1 or other references on fluid
mechanics) which is a function of L/d,, R and shape of the missile.

L « vertical length of the missile

d = characteristic dimension of the missile as shown in table 5-1.

V
R • Reynolds number » ^ (5-7)

v e kinematic viscosity-of the liquid

( - 0.95 x. 10-5 ftVsac for water at 80°F)

V « initial velocity of the missile at x • 0 (See figure 5-1)

V » striking velocity of the missile at x - H (See figure 5-1)

V, - terminal velocity - !g(l - Y / Y )/a|1^2 (5-8)

z i m j

5-3
 UC-T0P-9-A
Rev. 2

Table 5-1
(25)
DRAC COEFFICIENT FOR VARIOUSLY SHAPED BODIES IN INCOMPRESSIBLE FLOW

Form of Body L/d R

S
Circular disk >103 1.12

Tandem disks, 0 >io3 1.12

L " spacing 1 0.93
d » diameter 2 1.04
3 1.54

Rectangular p l a t e , 1 >io 3 1.16

L * length 5 1.20
d = width 20 1.50
00
1.95

Circular cylinder (axis II to flow) 0 >io 3 1.12

L = length 1 0.91
d • diameter 2 0.85
4 0.87
7 0.99

Circular cylinder (axis 1 to flow) 1 io5 0.63

L = length 5 0.74
d •- diameter 20 0.90
1.20

5 >5 x 10 5 0.35
CD
0.33

Streamlined foil (1 : 3 airplane strut) 00 >4 x 10 4 0.07

L » span
d = chord

Hemisphere: Hollow upstream >10 3 1.33

Hollow downstream 0.34

Sphere io 5 0.5
>3 x 10 5 0.20

'I Ellipsoid ( 1 : 2 , major axis || to flow) >2 x 10 5 0.07

Airship hull (model) >2 x 1 0 3 0.05

5-4
 BC-T0P-9-A
Rev. 2

MISSILE

"VQ *"0 CO
LIQUID SURFACE
V
a-L t-t,

MISSILE
.e.g.
H
t't

We v2/g

NOTE: SEE APPENDIX D. PARAGRAPH

D.? FOR AN ANALYSIS OF THIS
CASE.

Figure 5-1

PENETRATION OF A MISSILE IN A LIQUID

5-5
 BC-T0P-9-A
Kuv. 2

APPENDIX A

CROSS REFERENCE LISTING TO AEC STANDARD SAR FORMAT

This appendix shows the cross reference between s e c t i o n s of AEC's Standard

SAR format and the s e c t i o n s o f thin t o p i c a l report.

AEC SAR Format BC-TOP-9

3.5.4 2.0, 3.0, 4.0

A-l
 BC-T0P-9-A
Rev. 2

APPENDIX B

GLOSSARY

B.I PENETRATION

Penetration is the displacement of the missile into the target. If. is a

measure of the depth of the cratur formed at the zone of impact.

11.2 PERFORATION

Perlorution ia "full Penetration" or where the missile passes through the

target with or without exit velocity (of missile).

B.3 SPALLING OF CONCRETE

Spalling is the peeling off of the back face of the target opposite to the
face of impac<:.

B.4 DUCTILITY RATIO

The ductility ratio is the ratio of the maximum deflection to the

deflection at the "effective yield point."

B.5 EFFECTIVE YIELD POINT

That point on an idealized bilinear resistance function separating the

elastic and perfectly plastic portion of the function. The effective yield
point is based on the strength of the structure by ultimate (or plastic)
design methods.

B.6 ELASTIC IMPACT

An elastic collision is characterized by elastic deformations at the

missile-target interface.

B.7 PLASTIC IMPACT

A plastic collision is characterized by inelastic deformation and local

damage of the missile and/or target in the impact zone. For a purely
plastic collision, elastic restoring forces at the missile-target inter-
face and associated elastic rebound energy release converge to zero.

B-l
 BC-T0P-9-A
Rev. 2

APPENDIX C

REVIEW OF EXISTING FORMULAS

C.I PENETRATION AND PERFORATION

The most common formulas used in determining the local effects of a missile
<jn a target, such as penetration, perforation, and spalling for missiles
striking either a concrete or steel target, are given in tables C-l and C-2.
These tables include equations C-l through C-ll. These are the current
state-of-the-art formulas on impact analysis, which consists primarily of
empirical methods based on experiments conducted for specific and limited
applications. Generally, the experiments were conducted for the Government
using missiles, 9uch as bombs and bullets, and having velocities above
1000 ft/sec. Current impact analysis assumes that the missile impinges
the target normal to the surface. The effects of the oblique angle of
striking at various velocities are illustrated in figure C-l. It can be
seen that assuming normal striking of the target is conservative, since a
small deviation from a normal impact decreases the depth of penetration
considerably.

The Army Corps of Engineers and National Defense Research Committee

equations (table C-l) for penetration, perforation, and spalling have a
term, which depends only on the diameter of the missile. However, this
term provides overly conservative results when a low velocity and large
diameter missile is considered. For example: as V s -> 0 the penetration
approaches 0.5D; perforation approaches (1.8)D; and spalling approaches
(2.8)D, which is not realistic.

Experimental data with velocities below 500 ft/sec are just beginning to
develop, with the emphasis on the effect of impact on the target. Some
experiments have been completed with missile velocities in the range of
interest. However, the tests were not necessarily conducted for target
information.(21) Therefore, available pertinent data are limited.

The modified Petry formula has had the widest application for determining
the penetration of a missile into concrete targets and is adopted for use
at the present time. It was developed by the Poncelet theory, provides
estimate of penetration, and has functioned best in the velocity range of
interest. Also, conservatism is built into this approach because of the
following:

A. The angle of striking the target has a large effect if the angle
is greater than 20°. A normal angle of strike :ls assumed.

C-l
LANK PAGE
BC-T0P-9-A
Rev. 2

B. The probability of a missile being oriented in a manner chat, would

produce the greatest penetration Is remote. In addition, any
rotational effect tends to Increase the area of Impact.

C. Conservative estimates for weight, velocity, area of Impact, and

target strength provide conservatism.

Even though tne modified Petry formula was developed In 1910, the material
coefficient for penetration, Kp, has been revised by experiments and is
reported by Amirikian(l^) and shown In figure 2-1.

The BRL formula for perforation of concrete targets is used. It is

selected instead of the modified Petry formula of T = 2X because the BRL
formula was developed for perforation and not as an approximation from a
penetration.

The BRL equation, given in equation (2-3), has been modified to account for
concrete strength other than 3000 ipsi by replacing the constant coefficient
7.8 by 427/N/fT in equation (C-7).1

Two steel perforation formulas are available, the Ballistic Research

Laboratories (BRL) formula^ 2 )O) a n d the Stanford Research Institute
formula, known as the Stanford Equation.(2") The Stanford Equation is
based on experimental data, using missile velocities within the range of
interest. However, its limits of applicability are very restrictive
because most missiles encountered fall outside the range of the Stanford
Equation.

The Ballistic Research formula, table C-2, is used with an assigned value of
K equal to unity. Rearranging terms and solving directly for T leads to the
formula for calculating the threshold of perforation.

(C-12)

The Stanford Equation (table C-2) has the following defined limits of
applicability:

0.1 < T/D < 0.8,

0.002 < T/L < 0.05,

10 < L/D < 50,

5 < W/D < 8,

8 < W/T < 100,

C-2
 BC-T0P-9-A
Rev. 2

70 < V < 400,

3
L • length of cylindrical missile

V • striking missile velocity normal to the target surface for the

s
threshold of perforation (ft/sec)

Solving equation (C-ll) directly for plate thickness gives,

/ W V2 /u v2 u I
T « J 0.045 -JL5-+ 0.0022 / •£• \ - 0.047 J- (C-13) I 2

where,

WmVB22
2g

W - weight of missiles (pounds)

ra

A parametric study comparing the BRL formula and the Stanford Equation,
within the limits of applicability of the Stanford Equation, showed the
BRL and SRI formula are generally in good agreement for the shorter spans.
But, for longer spans the SRI formula is less conservative. Considering
this and the narrow band of limits for the SRI equation the BRL equation
is used for design.

C.2 MULTIPLE ELEMENT BARRIER EQUATION

Equation (2-9) assumed the residual kinetic energy of the missile after
perforation (Er) is the difference between the kinetic energy of the missile
before impact (E^) and the energy required to perforate the steel (Ep)

2 2 2
MV WJ MV

where

M =* mass of the missile ( lb-sec j

^ /

C-3
BC-T0P-9-A
Rev. 2

Solving for Vr

V -

(27)
This equation neglects the mass of the plug which may be punched out of
the target, which would be very small for a steel target; for a concrete
target, the concrete would fracture and not act in conjunction with the
missile mass.

Vp can be obtained from equations (2-3) and (2-7) by solving for Va, which
will be the velocity to just perforate, V_, when a given thickness of
target, t, is used.

C-4
 Table C-1

CONCRETE PENETRATION, PERFORATION, AND SPALLING FORMULAS (Sheet 1 of 3)

Equation
Identification Formula Remarks No.

A. Penetration into Reinforced Concrete

v 2
X = 12K A p 1 O g 10
f s
For infinitely thick slab C-1
P i 1 + 215 ,0OC .)
Modified Petry
(Refs. 13, 14, 15)

Depth of penetration for

? h •[• X slabs vith Finite thick-
ness. X.-> X when t -* 3X

Army Corps of Engineers 5

282 s
and National Defense WD * I
Research Conmi ttee C—L
(Refs. 13, 16, 17) V'c D2 \ 1000 ) ' •

Amnann & Whitney 282 N W D 0 ' 2 / Vs 8

Y m /•• o
t<— J
(Refs. 18, 19). •\/f' Q2 I 1000

B. Concrete Thickness to be Just Perforated

73 CD
Modified Petry X is obtained from
(Refs. 139 14, 15)
t = 2X
Equation (C-1)
C-4
i

>
 73 EO
Table C-l (0 n
o
CONCRETE PENETRATION, PERFORATION, AND SPALLING FORMULAS (Sheet 2 of 3) I
VO
I

Identification Formula Remarks

Equation
No.
Is
B. Concrete Thickness to be Just Perforated (Con't)

Army Corps of Engineers X is obtainted from

T =» 1.35D + 1.24X C-5
(Refs. 13, 16) Equation (C-2)

National Defense Research

X is obtained from
Committee T « 1.23D + 1.07X Equation (C-2)
C-6
(Refs. 13, 17)

? Ballistic Research
T= 7 8
' ^ O Vioooj
For f' = 3000 psi
c
Laboratories C-7
(Modified)
(Ref. 13)
T m 427 W ( _%_ ) For any value f' Ref. 3
1
/I' J..8 \1000/
V cD
C. Concrete Thickness to be Just Spalled

Army Corps of Engineers X is obtained from

T = 2.2D + 1.35X C-8
(Refs. 13, 16) s Equation (C-2)

National Defense Research

X is obtained from
Committee T = 2.28D + 1.13X C-9
S Equation (C-2)
(Refs. 13, 17)

I
 Table C-l

COXCEETE PENETRATION, PERFORATION, AND SPALLING FORMULAS (Sheet 3 of 3)

w = Weigifc of Missile (lb.)

V Striking Velocity of Missile (ft/sec.)

s

D = Diameter of Missile (in.)

Missile Weight
A = Projected Frontal Area of Missile
P (psf)
X = Depth of Penetration into Slab of Infinite Thick Concrete (in.)
Depth of Penetration into a Finite Thickness Slab of Concrete (in.)
o
Thickness of the Slab (in.)
Compressive Strength of Concrete (psi)

K Experimentally Obtained Material Coefficient for Penetration (See Figure 2-l>

P
N Nose Factor = 0.72 + 0.25 (n - 0.25)1/'2
radius of nose section
diameter of missile

T = Thickness To Be Just Perforated (in.)

50
Thickness To Be Just Spalled (in.) 5
NOTE: Some of the equations have been rewritten to reflect consistent units and terminology. 3
I
 Table C-2
PERFORATION IN STEEL FORMULAS

Equation
Identification Formula Remarks No.

Ballistic Research Lab T3/2 °'5 ml C-10

2 3/2
(Refs. 2, 3, 13) 17,400 K V
Stanford Research
Institute See Limits page C-3 C-11
D ~ 46^00 (16'00° T ' + 1 » 5 0 C % T )
(Ref. 20)

T = steel thickness to be just perforated (in.)

? M =
2
mass of the missile (lb-sec /ft)*
03

V = striking velocity of the missile normal to target surface (ft/sec),

s

K = constant depending on the grade of the steel, (K is usually * 1,)

D = diameter of the missile (in.)

E = critical kinetic energy required for perforation (ft-lb),

S = ultimate tensile strength of the target minus the tensile stress in the steel (psi)

W = length of a square side between rigid supports (in.),

W = length of a standard width (4 in.). (See Ref. 20)

s
 BC-T0P-9-A
Rev. 2

Effect of Oblique Strike

Spading

L 7 - W/Perforation
WO/Porforation
». * / 20* 28*

'2M0 'Sill

r
V$= '2535 for -

V$= f,2OI«
T
*2057 •^002 RIC

A..
'I4$B 1311 UtO
/ \ 7 x
I4«2 RIC. 1474 me

/
v,= '960 '949 J72 me. ^ 9 0 TflC »37>lC.

?
V,= 724 729 RIC. «B» NIC

37 MM. M80 Projectile 2

Concrete ThickneaB = 22". Compresaive strength = 5700 lbs/in.
Striking velocity (Va) and angle of obliquity ( 0 ) shown.
Stuck projectiles and path of ricochet projectilea shown.

Figure C~l
TYPICAL CRATER PROFILES

C-9
 Rev. 2

APPENDIX D

DERIVATIONS

D.I DERIVATION OF FORCE-TIME HISTORY FOR AUTOMOBILE CRASH, EQUATION (5-1)

An approximate r e l a t i o n s h i p has been observed In experiments on automobile

c r a s h e s . ( 2 2 ) xhe deceleration per unit deformation associated with the
c r v h i n g force was observed to be approximately the same for a wide variety
of standard-size U.S. automobile makes and models. The d e c e l e r a -
tion during a frontal impact i s as follows:

- x = 12.5g x (D-l)

where

-x - deceleration (ft/sec )

x • distance automobile crushes into target (ft)

g • gravitational acceleration (ft/sec )

Newton's law of motion and equation (D-l) give the relation

W
_ m .. to e it (D-2)
p s x = 12.5 W x
g m
where
W « weight of automobile (lb)
m
Equation (D-l) is the motion for an undamped linear oscillator with a unit
mass and a spring constant equal to 12.5g. Its solution with initial zero
deformation is
1/2
x - C sin (12.5g)L/* t (D_3)

To determine the constant, C, consider the balance of the input kinetic

energy, E m , by the striking automobile with work done by the impact force
plus energy lost, E^t by other phenomena such as target response

E » I -SUL „ I F x +E (D-4)
m 2 g 2 max max L
where

V - striking velocity of the automobile (ft/sec).

s

D-l
 BC-T0P-9-A
Rev. 2

In the conservative case of EL " 0 the constant C can be determined by

substituting equations (D-2) and (D-3) into equation (D-4)
/2
/ 1
\12.5g
Finally substituting equations (D-3) and (D-5) into equation (D-2) gives
the force-time history

• 1 \^^ 1/2
(- - i - . ) VB sin (lif.Sgr'* t
k12.5g/ a
(D-6)
- 0.625 V W ain 20 t
s m
This is a sine wave of frequency u) • 20 rad/sec and period T - 2TT/U)
= 0.314 sec. The maximum force occurs at t • T/4 a 0.0785 sec when the
velocity of the striking automobile is zero relative to the rigid surface
21 and then rapidly reducing to zero. Thus under the condition of plastic col-
lision ( i . e . , missile and target acquire same velocity after impact) the
duration of the Impact force i s from t = 0 to t = T/4 = 0.0785 sec. At
22 t a 0.0785 s e c , t h e force diminishes from a maximum value to zero.

As an example of using the r e s u l t i n g expressions, consider the experimental

data in reference 23. Test No. 505-IW for a 1963 Plymouth automobile
s t r i k i n g a r i g i d wall yielded the following d a t a .

W - 3270 lb
m
V - 53.3 raph - 78.17 f t / s e c
s
x - 3.82 ft
max
gF
° ave/Wm » 25g°
(average over distance)

From equations (D-3) and (D-5) arid the above data the stopping distance i s

\1/2
max(lOi) <78'17> s3
- 9 1 ft
2|
According to the forcing function equation (D-6) the average deceleration
(average over distance, not over time) for Test No. 505-IW is-
8F
ave/Wm " *Fmax/2Wm " (0.625)<78.17)g/2 - 24.42g
which agrees with the test result (25g) quite closely.

D-2
 BC-T0P-9-A
Rev. 2

D.2 DERIVATION OF THE VELOCITY OF A MISSILE AFTER IT HAS PENETRATED

THROUGH A LIQUID

Consider the motion of a m i s s i l e , length L, entering a l i q u i d pedium and

s t r i k i n g a target at depth H from the l i q u i d s u r f a c e , as shown in f i g -
ure 5 - 1 . When the i r i s a i l e f i r s t h i t s the l i q u i d , a compressive shock wave
may be generated i n the l i q u i d with a r e s u l t i n g l o s s of m i s s i l e v e l o c i t y .
This i s c a l l e d the "compression phase" of l i q u i d entry i n reference 24,
(page 1 8 ) . As the m i s s i l e d i s p l a c e s the l i q u i d i t experience? a hydrody-
namlc f o r c e with v a r i a b l e impact drag c o e f f i c i e n t C_. This " l i q u i d -
displacement phase" further reduces the v e l o c i t y . After the maximum m i s s i l e
c r o s s - s e c t i o n a l area i s Immersed, the "cavity drag phase" 1B i n i t i a t e d in
which the drag c o e f f i c i e n t Cn may be considered c o n s t a n t . In t h i s appendix
the v e l o c i t y o f the m i s s i l e during l i q u i d entry i s analyzed on the assump-
t i o n s that the v e l o c i t y iooo in the "compression phase" i s n e g l i g i b l e and
that the impact drag c o e f f i c i e n t Cp in the "liquid-displacement phase" i s
equal t o the drag c o e f f i c i e n t Cn in the "cavity drag phase." Since CQ i s
always smaller than Cp ( s e e reference 24, page 30 and figure 2-7) t h e s e
assumptions give more conservative (high) r e s u l t s for the m i s s i l e v e l o c i t y .
Only the case of v e r t i c a l entry (normal to the h o r i z o n t a l l i q u i d surface)
i s considered.

Under t h e s e assumptions, the equation of m i s s i l e motion i s

— « - W - P. - P. (D-7)

g o d

where

W - Weight of missile

g • gravitational acceleration

x - depth of missile e.g.' below the Initial e.g. as shown in figure 5-1

t - time after initial contact of missile with liquid

F, • buoyant force

F, •" drag force

and a dot denotes differentiation with respect to t.

Between x » 0 and x - L the buoyant force varies with x

Fb - Y J A(xx) dx x - yf(x). (0 < x i L) (D-8)

D-3
BC-T0P-9-A
Rev. 2

where

y - weight density of the liquid

A(x.\ - horizontal croas-sectional area of the missile at vertical

* f distance x. from the tip

When x > L the buoyant force is a constant

F
b "W Y / V (X > L) CD-9)
where

Y « weight density of missile

m

The drag force is given by the expression

F
d " y AmCDv2/28 (D-10)

where

A - maximum horizontal cross-sectional area of missile

m
v =• x • velocity of missile at depth x

If the liquid is assumed to be incompressible, the drag coefficient, C Q , in

equation (D-10) is a function of the missile shape and the Reynolds number
R, defined as

V
R - -~- (D-ll)

where

d » characteristic dimension of missile as shown in cable 5-1

V Q - initial velocity (at t-0 and x-0) of missile

V - kinematic viscosity of liquid

Table 5-1 from reference 25 lists some typical values of C Q for variously
shaped bodies in incompressible fluid flow. Reference 2A (page 35) presents
some C D values for a family of nose shapes. Other references on fluid
mechanics can also be consulted.

D-4
 BC-T0P-9-A
Rev. 2

Substituting equations (D-8), (D-9) and (D-10) into equation (D-7) r e s u l t s

In the following two forms of the equation of motion and solutions:
A.. For 0 < x < L

x + ax 2 + bf(x) - g = 0 , (0 £ x _< L) (D-12)

where
a = yA C_/2W (D-13)
m D
b
= Yg/W (D-14)
and f (x) i s given in equation (D-8) .
This i s a nonlinear, second order, nonhomogeneous, ordinary d i f f e r -
ential equation for x(t) .
According to reference 26 (page 551) i t can be solved as follows:
Let
y(x) = i 2 = v 2 (D-15)
Then i f a prime denotes differentiation with respect to x,
y ' ( x ) - 2x(x)' = 2x x/x = 2 X (D-16)
Equation (D-12) becomes
y'(x) + 2ay(x) = 2g - 2bf(x) (D-17)
which is a l i n e a r , f i r s t order, nonhomogeneous, ordinary differen-
t i a l equation, for y(x), and has the solution

y(x) - J2 yii(x) [g-bf(x)] dx + c}/u(x) (I

where c is the Integration constant and

/2adx _ ,.
u(x) - J »e2ax °

D-5
BC-T0P-9-A
Rev. 2

Substituting equation (0-19) Into equation (D-18) gives

y(x) - v 2 -e" Zax [2g Je2ax dx - 2b J e 2 a x f(x) dx + c ]

(D-20)
2ax 2a
- g/a - 2be" G(x)+ c e ~ * , ( 0 < x < L )

where

G(x) - fe 2 a X f(x) dx - /"e 2 a X / * A/x.\ dx. dx (D-21)

J J |_0 IV lj
in which equation (D-8) has been used.
At the initial position (See figure 5-1) x - 0, v - Vfl, and
equation (D-20) gives

- g/a + 2bG(0) (D-22)

Then equation (D-20) becomes

y(x) - v 2 - g/a + e" 2 a x [VQ2 - g/a

(D-23)
+ 2b [G(0) - G(x)]| , (0 < x < L)

At x » L equation (D-21) gives

G(L) -
[r * dx
x-L
(D-2A)

and equation (D-23) gives

y(L) gY/Yma + e" 2 a L {v Q 2 - g/a

(D-25)
2b [G(0) -

D-6
 BC-T0P-9-A
Rev. 2

where V.. Is the missile velocity at x » L (See figure 5-»l) and

(D-26)

Consider the special case of a missile with uniform horizontal

cross-sectional area A .• Then A(x-) - AQ. Equation (D-21) gives

x
G(x) - f^aA2ax I/ /T A. ..,„
^ j\ dx
. . .-. .A Jf .._2ax
Q xe"~dx
(D-27)
2
(2ax-l)Ma , (0 < x < L)

from which

G(0) - -A Q /4a 2 (D-28)

and

G(L) - Aoe*<*1J (2aL-l)/4a2 (D-29)

Equation (D-23) becomes

v 2 - g/a + bA_ (1 - 2ax)/2a 2 + e" 2 a x (v 2

- a/a '
(D-30)
, (0 £ x < L)

Formulas for other missile shapes can be derived similarly.

For x > L

x + ax 2 + gY/Y o - 8 " 0 » (s >. D (D-31)

This is a special case of equation (D-12) with

f (x) - gY/Ymb , (x > L) (D-32)

which, when substituted into equation (D-20)» gtvee

v 2 - V22 + ke~ 2 a x , (x > L) (D-33)

D-7
BC-T0P-9-A
Rev. 2

The Integration constant k. can be determined by the condition that

at x • L, v - V- obtained in equation (D-25)

(D-34)

Hence the missile velocity at x >. L is given by

V 2 +
2 ( V 1 2 -V ? 2 ) e" 2 a ( x ~ L ) f , (x > L) (D-35)

Substituting V, from equation (D^25) into equation (D-35) gives

2
v - IV-.- + e — [2b (G(0) - G(L» + V(0

11/2 <D"36>

In the special case of a missile with uniform horizontal cross-

sectional area AQ equations (D-28) and (D-29) are substituted into
equation (D-36) to give

v V 2 2 + e " 2 a x [bAQ ( e 2 a L (1 - 2aL) - l ) /2a 2

(D-37)
2 2aL
0 + 8 (e Y /Y m -l)/a] , (x > L)

At x - H, when the missile strikes the target (See figure 5-1) the
velocity V is given by equation (D-36) or equation (D-37) with
x replaced by H.

D-8
 BC-T0P-9-A
Rev. 2

APPENDIX E

SAMPLE APPLICATIONS

E.I CONCRETE (PENETRATION, PERFORATION AND SPALLING)

A 4 - i n c h x 12-lnch wooden p l a n k , weighing 108 pounds, s t r i k e s a t 300 mph

(440 fpa) in a normal head-on c o l l i s i o n with a r e i n f o r c e d concrete
(f* c - 3000 p s i ) w a l l . The plank ha9 a 48 square inch c r o s s - s e c t i o n a l
area with the equivalent diameter of 7.8 I n c h e s .

E.1.1 PENETRATION

P e n e t r a t i o n i s given by e q u a t i o n ( 2 - 1 ) :

V2 \
12 It A Log i n 1 +
p p *10 \ * " 215000

For 3O00 p s i c o n c r e t e K » 0.00348 ( f i g u r e 2-1)

and

457144
Then

X - 12 x 0.00348 x 324 x Log1Q ((l + - 3.77 in.

When the thickness of a wall i s l e s s than 3 x 3.77 = 11.3 i n . , the depth of

penetration I s given by equation ( 2 - 2 ) :

1 + e

For example, for a wall with thickness t ° 8 in., we get:

x 3.77 - 6.08 in.

E-l
BC-T0P-9-A
Rev. 2

E.1.2 PERFORATION

The thickness of a wall to be just perforated is given by formula 2-3:

A27 W a.33
/Va V
T-

For f - 3000 psi,

T - ~ — ~-s- Ir^) - 7.01 in.

•V/SOOO" 7.8 1 * 1

Therefore, the concrete thickness required to prevent perforation according

to equation 2-4 is:

t - 1.25 x 7.01 - 8.76 in.

P

E.I.3 SPALL

The thickness of a wail to be just spalled is given by equation (2-5).

T » 2 T • 2 x 7.01 " 14.02 in.

s

Therefore, the concrete thickness required to prevent spelling according to

equation (2-6) is:

t - 1.25 x 14.02 - 17.53 in.

E.2 STEEL TARGETS

Given: A ten pound missile one inch in diameter impacts a target at

200 ft/sec.

Question: Find the thickness of steel plate, T, to just perforate and the
thickness t required to prevent perforation.

Solution: Use equation (2-7) and (2-8)

Then

5/3

" 0 . 5 inches

and t - 1.25 x 0.5 - 0.625 inches.

P

E-2
 BC-T0P-9-A
Rev. 2

E.3 STRUCTURAL RESPONSE

Consider a 10 l b s o l i d metal m i s s i l e of 1-inch diameter s t r i k i n g with

200 f t / s e c v e l o c i t y at the mid-span of a simply-supported s t e e l I-beam of
10 f t span and AISC designation W6xl2( 8 ) with s t a t i c y i e l d strength
fy • 50,000 p a l . I t i s required to evaluate the structural response of the
beam according t o Section 3 under the condition of p l a s t i c impact.

According t o equation (3-17) the e f f e c t i v e mass of the s t e e l beam may be

conservatively estimated a s the mass of a 13-inch length of the beam ( s i n c e
the depth of beam d - 6" and D x « 1", the m i s s i l e diameter) which i s for
W6xl2 beam(8),

Me . O2H13J . 13/g

According t o equation (3-3) for p l a s t i c impact, the required target s t r a i n

energy t o absorb the Impact energy i s

M2VZ m 2 (200 x 12)2

E
s " ,Z/Mni ^ B"< m- \™
g /' / , n TTX -m" 32,440 in.-lb

The resistance-displacement function of a simply-supported beam under

central loading can be idealized as a bilinear function (figure 3-1 and
table 4-2) with

R . IS* . !iffi . 8(21.7) (50y000)(1.2) „ , ^w ±

m L Ld (10 x 12) (6) " ' '

and

.. _ RmL . (14 r A67) (10 x 1 2 ) 3

~ /OPT ****"* 4."" "" "*°** 0.HO i n .
e 48EI
48 (30 x 1O6)(21.7)
where the value of the moment of inertia, I , for the beam cross-section i s
taken from reference 8, and modulus of elasticity E - 30 x 10& and dynamic
Increase factor D1F » 1.2 (table 4-1) have been used.

According to figure 3-1 the maximum strain energy for purely elastic
structural response i s

E - ~ R x - -7 (1A, 467) (0.80) - 5,787 in.-lb

e i m e Z

E-3
BC-T0P-9-A
Rev. 2
which Is l e s s than E8 •- 32,440 ln.-lb,eo the structural response i s
elasto-plastic. Then according to equation (3-22) the required d u c t i l i t y
ratio is

+
"r-TT i - (t.d)\i"ui) + °- 5 " 3 - 3 0
em
S i n c e , according to table 4-4 the allowable d u c t i l i t y r a t i o for a s t e e l
beam under l a t e r a l loads i s 20, t h i s beam can withstand the postulated
m i s s i l e impact i f no other loads are acting simultaneously. In case other
loads are present a s m i s s i l e impacts and remain in e f f e c t throughout the
s t r u c t u r a l reuponse the required d u c t i l i t y r a t i o should be evaluated by
equation (3-24) instead of equation ( 3 - 2 2 ) .

E.4 MISSILE PENETRATION THROUGH WATER

Consider the postulated accident condition of a fuel shipping cask (the

m i s s i l e ) f a l l i n g from an overhead crane and possibly damaging the spent fuel
pool floor slab (the target) underneath. The cask i s a cylinder with length
L - 17 f t , diameter d - 7 f t , end weight W • 2 x 10^ i b . [The spent fuel
pool contains water of depth H - 37 f t . If the cask i s to drop h » 11 f t
t o j u s t h i t the water surface the i n i t i a l v e l o c i t y i s

VQ = (2 g h ) 1 / 2 - [2(32.17) ( l l ) ] 1 ' ' 2 - 26,6 f t / s e c .

The Reynolds number i s , according to equation (5-7),

R -2.0X107.
V 3
0.93 x 10"
Sines L/d - 17/7 • 2.43 the drag coefficient i s , according to table 5-1 for
the case of circular cylinder with axis parallel to flow and with R >. 10-*,

GD - 0.854

the horizontal cross-sectional area i s

AQ - ird2/r - 7r(7)2/4 - 38.5 f t 2 .

Then equation (5-5) gives

- (62.4)(0.854)(38.5) m -1
2 (2 x 10 T )

E-4
 BC-T0P-9-A
Rev. 2

and equation (5-6) gives

b = J i = (62.4) (32.17)
W
2 x 10 5

The weight density of the cask is

W 2 x 10 5
= =
?m°A^L (38^17)

According to equation (5-8) the terminal velocity is

-a/2
[g (1 - Y/YB)/aJ

r f/2
= (32.17) (1 - 62.4/305.6)/0.0051 = 70.9 ft/sec,

Since H > L, and according to equation (5-2)

2 (L) - V2 + e" 2aL ~ 4 fa2aL (1 " 2aL) -ll

(70.9) 2 + e-2(0.0051)(17) (0.01)(33.5) ' "734 ( 1 _ Q

2
[ 2(0.005t)

5027 + (0.8A08) [- 4193J - 1502 > 0,

E-5
BC-T0P-9-A
Rev. 2

the value of Z.(H) should be calculated:

Z 2 ( H) . v / + e" 2aH {bAQ [e2aL(l - 2aL) - l]/2a2 + V Q 2

+ B (e2aLY/Ym-l)/a - (70.9)2 + e ^ 0 ' 0 0 5 1 * < 3 7 ) (-4193) -

5027 + (0.6856)(-4193) « 2152 > 0

Finally the striking velocity of the cask on the spent fuel pool floor slab
is, according to equation (5-4).

[Z2(H)T/2 - 1/2
(2152) x / z - 46.4 ft/sec

It is interesting to note that if the spent fuel pool is dry the striking
velocity would be

r i i/2 r f/2
V - 2g (h + H) » |2 (32.17) (11 + 3 7 ) - 55.6 ft/sec

For missiles of lighter weights, the reduction of striking velocity due to

the presence of a liquid would be more pronounced.

E-6
 BC-T0P-9-A
Rev. 2

APPENDIX F

REFERENCES AND BIBLIOGRAPHY

F.I REFERENCES

1. Rotz, J. V., Yeh, G. C. K., Bertwell, W., Tornado and Extreme Wind
Design Criteria for Nuclear Power Plants, Topical Report, BC-TOP-3,
Revision 3, Bechtel Power Corporation, August 1974.

2. Russell, C. R., Reactor Safeguards, MacMillan, New York, 1962.

3. Fundamentals of Protective Design, TM 5-855-1, Headquarters,

Department of the Army, Washington, D.C., July 1965.

4. Norris, C. H., et a.l., Structural Design for Dynamic Loads, McGraw-Hill

Book Company, Inc., New York, New York, 1959.

5. Goldsmith, W., Impact, Edward Arnold, Ltd., London, 1960.

6. John M. Biggs, Introduction to Structural Dynamics, McGraw-Hill 1964,

pp. 202-244.

7. Newmark, N. M., and Richart, F. E., Impact Tests of Reinforced

Concrete Beame, NDRC Report No. A-125, A-213, and A-304, 1941-1946.

8. A.I.S.C. Steel Conntruetion Manual, 7th Edition, American Institute of

Steel Construction, N, Y., N. Y., 1970.

9. Building Code Requirements for Reinforced Concrete, ACI Standard 318-71, |2

American Concrete Institute, Detroit, Michigan, 1971.

10. N. H. Burns, C. P. Siess, Plastic Hinging in Reinforced Concrete,

ASCE Proc. V92 (J. Struct. Div.) n ST 5 Ccf.. 5^

11. Vail, C. F., "Dynamic Modeling of Automobile Structures from Test

Data," System Identification of Vibrating Structures - Mathematical
Models from Test Data, The American Society of Mechanical Engineers,
N. Y. (1972) pp. 149-177.

12. Harris, C. ,M., and Crede, C. E. , Shock and Vibration Handbook,

McGraw-Hill Book Company, N. Y., 1961.

13. Gwaltney, R. C , Missile Generation and Protection in Light-Water-

Cooled Power Reactor Plants, ORNL NSIC-22, Oak Ridge National
' Laboratory, Oak Ridge, Tennessee, for the U. S. Atomic Energy
Commission, September 1968.

14. A. Amirikian, Design of Protective Structures, Report NP-3726, \2

Bureau of Yards and Docks, Department of the Navy, August 1950.

F-l
BC-T0P-9-A
Rev. 2

15. Samuely, F. J., and Hamann, C. W., Civil Protection, The Architectural
Press, London, 1939.

16. Fundamentals of Protective Design, Report ATI207821, Array Corps of

Engineers, Office of the Chief of Engineers, 1946.

17. Nationul Defense Research Committee, EffectB of Impact and Explosion,

Summary Technical Report of Division 2, Volume 1, Washington, D.C.

18. Industrial Engineering Study to Establish Safety Design Criteria for

Use in Engineering of Explosive Facilities and Operations Wall
Response, a Report Submitted "to T^ocfiss Engineering Branch,
A.P.M.E.D. Picatinny Arsenal, ' .iver, N. J. , Voril 1963.

19. Structures to Resist the Eff"-- - f Acci<*-.-ry ri E*p3k>9JQng? TM 5-1300,

Department of the Army, Washin,,. i..,i, D.C,, .Jt:Xy 1965*

20. White, R. W., and Botsford, K, »., C-. u'calamtnxt of Fragments from a
Runaway Reactor, Report SKIA-113, $z&n¥avS~R&search Institute,
September 15, 1963.

21. Hardrock Silo Debris Impact Program, Phase I Testing, Test Report,
Physics International Co./Bechtel Corp., (unpublished). ™

22. Emori, R. I., Analytical Approach to Automobile Collisions. Paper

No. 680016, Automotive Engineering Congress, Detroit, Michigan (1968).

23. Ivey, D. L«, Ruth, E., and Hirsch, T. J., Feasibility of Lightweight
Cellular Concrete for Vehicle Crash Cushions, Paper presented at the
Annual Meeting of the Highway Research Board, Washington, D.C. (1970).

24. Kornhauser, M., Structural Effects of Impact, Spartan, Baltimore,

Maryland, 1960.

25. Rouse, H., and Howe, J. W., Basic Mechanics of Fluids, John Wiley and
Son Inc., New York (1953).

26. Kamke, E., Differentialgleichungen Losungsmethoden und L'dsungen,

Vol. I. Gewohnliche Differentialgleichungen (1942), Akademische
Verlagsgeselishaft, Becker & Erler Kotn.-Gea., Leipzig.

27. Recht, R. F., and Ipson, T. W., Ballistic Perforation Dynamics,

J. of Appl. Mechanics, ASME, Sept. 1963. ""

28. Newmark, N. M., and Haltiwanger, J. D., Air Force Design Manual,
AFSWC-TDR-62-138, prepared by the University of Illinois for
Air Force Special Weapons Center, Kirtland Air Force Base, N. M.,
1962.

F-2
 Rev. 2

29. Johansen, K. W., Yield-Line Formulae For Slabs, Cement & Concrete
Association, London (Translation by Paulin M. Katborg).

30. Ferguson, P. M., Reinforced Concrete Fundamentals, 3rd Edition,

John Witey 1973.

31. Hognestad, E., Yield-Line Theory For the Ultimate Flexural Strength
of Reinforced Concrete Slabs, ACF Journal 24 No. 7 March 1953.

32. Wood, R. H. Plastic and Elastic Design of Slabs and Plates, Ronald
Press Co., 1961.

33. Timoshenko, S. and Woinowsky-Kriegers, S., Theory of Plates and Shells,

Mc-Graw-Hill, 1959.

34. Cowell, W. L., Dynamic Tests of Concrete Reinforcing Steels, Technical

Report R394, U.S. Naval Civil Engineering Laboratory, 1965.

35. Watstein, D., Effect of Straining Rate on the Compressive Strength and
Elastic Properties of Concrete, Journal of the American Concrete
Institute, Vol. 24, No. 8, 1953.

36. McHenry, D., Shideler, J. J., Review of Data on Effect of Speed in

Mechanical Testing of Concrete, Bulletin D9, Portland Cement Associ-
ation Research and Development Laboratories, (also reprint, Special
Technical Publication No. 185, by ASTM, 1956).

37. Rao, N. R. N., Lohrmann, M., Tall, L., Effect of Strain Rate On The
Yield Stress Of Structural Steels, Fritz Laboratory Reprint No. 293,
Lehlgh University Institute of Research, (also Journal of Materials,
Vol. 1, No. 1, American Society for Testing and Materials,
March 1966).

38. Gaston, J. R., Siess, C. P., and Newmark, N. M., An Investigation of

the Load-Deformation Characteristics of Reinforced Concrete Beams
Up to the Point of Failure, University of Illinois, December 1952,
Reprint July 1959.

39. Denton, D. R., A Dynamic Ultimate Strength Study of Simply Supported

Two-Way Reinforced Concrete Slabs, TR 1-789, U.S. Army Engineers
Waterways Experiment Station, Corps of Engineers, Vicksburg,
Mississippi, July 1967.

40. Albritton, G. E., Response of Deep Reinforced and Unreinforced Con-

crete Slabs to Static and Dynamic Loading, ASCE National Meeting on
Structural Engineering, September 30 - October 04, 1968.

41. Untrauer, R. E., Behavior and Design of Deep Structural Members,

Part 4, Dynamic Tests of Reinforced Concrete Deep Beams, University
of Illinois, May 1960.

42. Corley, W. G., Rotational Capacity of R/C Beams, ASCE Proceedings,

Journal of Structural Division, October 1966.

F-3
BC-T0P-9-A
Rev. 2

43. Feldman, A., Siess, C. P., Investigation of Resistance and Behavior

of Reinforced Concrete Beams Subjected to Dynamic Loading, University
of Illinois, September 1956.

44. Chelapati, Kennedy E. Wall Probabilistic Assessment of Aircraft

Hazard for Nuclear Power Plants, First International Conference on
Structural Mechanics in Reactor Technology Berlin 20-24 September
1971.

F-4
 BC-T0P-9-A
Rev. 2

F. 2 BIBLIOGRAPHY

1. A.I.S.C. P l a s t i c Design i n S t e e l , American I n s t i t u t e of Steel

Construction, N.Y. , N.Y., 1959.

2. Beedle, Lynn S . , P l a s t i c Design of Steel Frames, John Wiley & Sons,

I n c . , 1958.

3. Commentary of P l a s t i c Design in Steel, A.S.C.E. Manual of Engineering

P r a c t i c e , No. 41, 1961.

4. Design for Pipe Break Effects, BN-TOP-2, Bechtel Corp., August 1972.

5. Design of Structures to ReBist the Effects of Atomic Weapons, The,

EM 1110-345-414 t o 421, Massachusetts I n s t i t u t e of Technology for the
Office of Chief of Engineers, U.S. Army, Washington, D.C.. 1957.

6. Horger, 0. J . , Metala Engineering Design. American Society of Mechan-

i c a l Engineers Handbook, McGraw-Hill, New York, 1953.

7. Johansen, K. W., "Pladeformler Fortnelsamling", Polyteknish Forening,

Copenhagen, 2nd Edition, 1954.

8. Johansen, K. W., "Pladeformler", Polyteknish Forening, Copenhagen,

2nd Edition, 1949.

9. Lorenz, Hanz, Gilbert Associates, I n c . , Aircraft Impact Design,

Power Engineering, Nov. 1970.

10. Newmark, N. M., e t a l . Notes on Blast Resistant Design, Bechtel

Associates Symposium, New York, 1968 (unpublished).

11. Rinder, R., Saffian, L. W., Wachtell, S., Cohen, E., Dobbs, N.,
Manual for Design of Protective Structures Used in Exposive Proc-
essing and Storage F a c i l i t i e s TR38O8, Picatinny Arsenal, Dover,
New Jersey/Amman and Whitney, New York, New York, November 1968.
J2
12. Wilkins, M. L . , Calculation of E l a s t i c - P l a s t i c Flow, UCRL-7322,
Lawrence Radiation Laboratory, Livermore, California, January 24,
1969, Rev 1.

F-5