missile impact 1
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7/29/2019 Missile Impact 1
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Nuclear Engineering and D esign 64 (1981) 33-38North-Holland Publishing Com pany
D E V E L O P M E N T O F N O N - L I N E A R F L O O R R E S P O N S E S P E C T R A
G . V I T I , M . O L I V I E R I a n d S. T R A V I
A M N S . p . A . 1 -1 6 12 1 G e n o v a , I ta ly
Received 3 April 1980
A com putational scheme for the d evelopm ent of non-linear floor response spectra is presented. Re sults in the c ase ofan assum ed missile impact on a reactor building are shown. Expected redu ctions, with respect to the linear ca se, forseismic or m issile impa ct excitations are discussed.
I . I n t r o d u c t i o n
I n e l a s t i c r e s p o n s e s p e c t r a , o b t a i n e d w i t h a n
i d e a l e l a s t o p l a s t i c s i n g l e d e g r e e o f f r e e d o m
m o d e l , h a v e b e e n d i s c u s s e d i n t h e p a s t b y s o m e
a u t h o r s: N e w m a r k p r e s e n t e d a p p l i c a t io n s to t he
d e v e l o p m e n t o f s e i s m i c i n e l a s t i c d e s i g n r e s p o n s e
s p e c t r a [ 1 ] a n d c o n s i d e r a t i o n s f o r i n e l a s t i c
e q u i p m e n t b e h a v i o u r u n d e r s e is m i c e x c i ta t io n s
[ 2 1 .D a n i s c h a n d G r a u b n e r [3 ] e x a m i n e d t h e a p -
p l i c a t i o n o f i n e l a s t i c s p e c t r a t o t h e d e s i g n o fr e i n f o r c e d c o n c r e t e s t r u c t u r e s s u b m i t t e d t o i n -
d u c e d v i b r a t i o n s o f a i r c r a f t i m p a c t o r g a s c l o u d
e x p l o s i o n a n d K a m i l e t a l . [ 4 ] s t r e s s e d t h e i m -
p o r t a n c e o f i n e l a s t i c s p e c t r a f o r v e r i f i c a t i o n o f
d u c t i l e c o m p o n e n t s u n d e r p u l s i v e l o a d i n g .
B a s i c a l ly N e w m a r k s h o w e d t h a t , i n t h e l o w
f r e q u e n c y r a n g e o f t h e d e s i g n s e i s m i c r e s p o n s e
s p e c t r a ( t y p ic a l ly b e l o w 3 H z ) , t h e a c c e l e r a t i o n s
o f t h e i n e l as t ic c a s e a r e r e d u c e d a l m o s t d i r e c t ly
b y th e a s s u m e d d u c t i li t y f a c t o r / z ; f o r t h e i n-
t e r m e d i a t e r a n g e o f f r e q u e n c i e s ( t y p i c a ll y b e -t w e e n 3 a n d 8 H z ) t h e r e d u c t i o n f a c t o r i s a p -
p r o x i m a t e l y g i v e n b y V ~ - 1 ; i n t h e r e l a t iv e l y
h i g h f r e q u e n c y r a n g e ( h i g h e r t h a n 8 H z ) p r a c -
t ic a ll y n o r e d u c t i o n o c c u r s . O t h e r m e n t i o n e d
a u t h o r s e v i d e n c e t h a t t h e h i g h e r f r e q u e n c y c o n -
t e n t o f e x c i ta t i o n s o t h e r t h a n t h e s e i s m i c o n e ,
c a u s e s s i g n i f i c a n t r e d u c t i o n s o f t h e i n e l a s t i c
s p e c t r a , e v e n f o r f r e q u e n c i e s u p t o 3 0 H z .
F u r t h e r m o r e t h e y p o i n t o u t t h a t b u i l d i n g
r e s p o n s e t o s h o c k t y p e e x c i t a t i o n s , a t s o m e d i s -
t a n c e f r o m t h e i m p a c t r e g i o n , a r e c h a r a c t e r i z e d
b y hi gh f r e q u e n c y c o m p o n e n t s a n d o n l y o n e ( o r
i n a n y c a s e a v e r y l im i t e d n u m b e r ) e x c u r s i o n t o
p e a k v a l u e , s o t h a t t h e y r e s u l t l e s s d a m a g i n g
t h a n c o r r e s p o n d i n g r a n d o m v i b r a t i o n s a s s o c i -
a t e d w i t h a n e a r t h q u a k e . I n s u c h c a s e s a v e r y
s l i g h t a m o u n t o f i n e l a s t i c y i e l d i n g w i l l b e
s u f f i c i e n t t o a c c o m m o d a t e t h e s m a l l d i s p l a c e m e n t s
a s s o c i a t e d w i t h t h e h i g h f r e q u e n c y a c c e l e r a t i o n
r e s p o n s e .
T h e a l l o w a b l e a m o u n t o f i n e l a s t i c y i e l d i n gf o r a s t r u c t u r a l e l e m e n t , a c o m p o n e n t o r i t s
s u p p o r t i n g s t r u c t u r e i s s p e c i f i e d b y p r o p e r d u c -
t i l i t y f a c t o r s .
I n t h i s p a p e r t h e m a i n c h a r a c t e r i s t i c s o f a
n u m e r i c a l p r o c e d u r e f o r th e d e v e l o p m e n t o f
n o n - l i n e a r f l o o r r e s p o n s e s p e c t r a a r e d e s c r i b e d
a n d f i n a l l y t h e a p p l i c a t i o n t o t h e c a s e o f a n
a s s u m e d m i s s i l e i m p a c t o n t h e r e a c t o r b u i l d i n g
o f a n u c l e a r p o w e r p l a n t , n o w u n d e r c o n s t r u c -
t i o n i n I t a l y , i s s h o w n .
2 . N u m e r i c a l p r o c e d u r e
I n e l a s t i c r e s p o n s e s p e c t r a a r e o b t a i n e d c o n -
s i d e r i n g th e s i n gl e d e g r e e o f f r e e d o m s y s t e m o f
f i g . 1 . T h e e l a s t o - p l a s t i c s t i f f n e s s o f t h e s p r i n g i s
r e p r e s e n t e d i n f ig . 2 , w h e r e Yem i s t h e m a x i m u m
e l a s t i c d i s p l a c e m e n t , y p t h e i n e l a s t i c d i s p l a c e -
m e n t , F 'e m = K ye m t h e m a x i m u m s p r i n g f o r c e .
33
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34 G. Vi t i e t a l . / No n - l in ear f loor re sponse spec t ra
. . . . . Y b
F i g . I .
F ~
% ' - - ? " 7,i /
i / / /
F i g . 2 .
T h e e q u a t i o n s o f m o t i o n o f t h e s y s t e m , d e s -
c r i b in g t h e e l a st i c b e h a v i o u r :
y + 2 1 to o y ~ + t o g ( y - yp) = --j;b,
Yp - - Yem <-- Y ~ Yp + Yem
a n d t h e p l a s t i c o n e :
+ 2 1 t o o ~ + Fern = - - Y b ,
~ . ~ , _ > o ,
( 1 )( l a )
( 2 )( 2 a )
w he re t o, , = ( k / M ) ' n , l : ½ ( c 2 / k M ) 1 /2 , IFem[ = F'em
a n d )~, i s t h e v e l o c i t y a t t h e t r a n s i t i o n f r o m t h e
e l a s t i c b e h a v i o u r t o t h e p l a s t i c o n e , a r e i n -
t e g r a t e d a s s u m i n g t h e b a s e a c c e l e r a t i o n j~b t o b e
a p i e c e - w i s e l i n e a r f u n c t i o n o f t i m e t a b u l a t e d a t
e q u a l i n t e r v a l s A . W i t h t h e a b o v e a s s u m p t i o n
t h e d i s p l a c e m e n t i s e x p r e s s e d b y a f u n c t i o n o f
t h e t y p e
y = e - t ~ ( K i s in t o t + K 2 c o s t o t ) + K 3 t + K 4 , (3 )
( e l a s t i c r e s p o n s e ) o r
y = h t e ~' + h 2 t 2 + h 3 t + h 4 , (4 )
( p l as t ic r e s p o n s e ) w h e r e s = l w o , w = t o0 ( l - / 2 ) i n
an d ~r = 21too . E x p r e s s i o n s f o r v e l o c i t y a n d a c -
c e l e r a t i o n a r e d e r i v e d f r o m t h e s e . T h e d i s -
c r e t i s a ti o n i n t e r v a l A is c h o s e n t o b e o n e t e n t h
o f t h e p e r i o d T = 2 7 r/ to , o f t h e s y s t e m a n d t h e
r e s p o n s e i s c o m p u t e d a t t h e e n d o f e a c h i n t e rv a la s s u m i n g a s i n i t i a l c o n d i t i o n s t h o s e c o m p u t e d a t
t h e e n d o f t h e p r e c e d i n g o n e . W h e n t r a n s it i o n
f r o m e l a s t i c ( p l as t i c ) t o p l a s t i c ( e l a s ti c )
b e h a v i o u r i s d e t e c t e d , t h e i n s t a n t o f t r a n s i t i o n i s
e v a l u a t e d b y i te r a t i o n a n d t h e n t h e r e s p o n s e i s
c o m p u t e d a t t h e e n d o f e a c h s u b - i n te r v a l . T h i s
t e c h n i q u e p r o v e d t o b e v e r y a c cu r a t e a n d
s t a b le : n o p r a c t i ca l d i f f e r e n c e s w e r e o b s e r v e d
p e r f o r m i n g c o m p u t a t i o n s i n s i n g l e a n d d o u b l e
p r e c i s i o n a n d r e d u c i n g t h e d i s c r e t i s a t i o n i n t e r v a l
A.
R e s p o n s e s p e c t r a S ( l , w o , I x ) a r e c o m p u t e d a t
d i s c r e t e v a l u e s o f d a m p i n g l, f r e q u e n c y f =
~ o0 /27r a nd duc t i l i t y f a c t o r :
Ix = ym/yem (5 )
w h e r e Y m is t h e m a x i m u m d i s p l a c e m e n t , u t i l iz -
i n g a s b a s i c a l g o r i t h m t h e a b o v e i n t e g r a t i o n
t e c h n i q u e . S i n c e t h e d u c t i l i t y f a c t o r c a n n o t b e
i n s e r t e d d i r e c t l y i n t o t h e e q u a t i o n s o f m o t i o n
(Y m i s k n o w n o n l y a t t h e e n d o f t h e i n t e g r a t i o n )
t h e s e a r e i n t e g r a t e d , f o r e a c h d a m p i n g a n d e a c h
f r e q u e n c y , a t s u i t a b l e v a l u e s o f t h e m a x i m u m
e l a s t i c d i s p l a c e m e n t Y o r e . Y e m 2 . . . . o b t a i n i n g t h e
c o r r e s p o n d i n g v a l u es o f r e s p o n s e S ~ , $ 2 . . . . a n d
d u c t i l i t y f a c t o r I x ~, Ix2 . . . , t h e r e s p o n s e c o r -r e s p o n d i n g t o t h e r e q u e s t e d v a l u e s o f d u c t i l i t y
f a c t o r Ix~ . . . . # , is t h e n c o m p u t e d b y i n t e r -
p o l a ti o n . T h e m a x i m u m e la s ti c d i s p l a c e m e n t
v a l u e s a r e c h o s e n i n s u c h a w a y t h a t t h e
r e q u e s t e d d u c t i li t y f a c t o r s b e s e p a r a t e d b y
c o m p u t e d d u c t i l i t y f a c t o r s a c c o r d i n g t o t h e f o l -
l o w in g p r o c e d u r e .
S t e p 1 . I n t r o d u c e t w o e x t r a - v a l u e s o f d u c ti l it y
fa c t o r : Ix ( )= (1 + I x l ) / 2 , / z n + l = 1 .5 Ix , , fo r bound-
i n g p u r p o s e s .
S t e p 2 . C o m p u t e Y e m ] , S 1 c o r r e s p o n d i n g t o ac o m p l e t e l y e l a s t i c b e h a v i o u r o f t h e s y s t e m a n d
se t ~1 = 1 , Y e m 2 : O , t ~ 2 = + o o .
S t e p 3. C on s id e r ea ch in te rv a l Yemi, Yem,+~ (s ta r t -
i n g w i t h t h e o n e d e f i n e d a b o v e ) i n tu r n a n d i f
m o r e t h a n o n e r e q u e s t e d I x k i s c o m p r i s e d b e -
t w e e n / L a n d / 2 i + ] b r e a k t h e i n t e r v a l c o m p u t i n g
r e s p o n s e a n d d u c t i l i t y f a c t o r c o r r e s p o n d i n g t o
0 . 5 ( Y e m i -4 - Y e m i * l ) "
S t e p 3 i s r e p e a t e d u n t i l s e p a r a t i o n o f d u c t i l i t y
f a c t o r s # , . . . . . Ix n.t h as b e e n o b t a i n e d o r a
m a x i m u m n u m b e r o f i te r a ti o n s h a s b e e n p e r -
f o r m e d .
3 . A p p l i c a t i o n
T h e n u m e r i c a l p r o c e d u r e d e s c r i b e d a b o v e
h a s b e e n a p p l i e d t o t h e c a s e o f a n a s s u m e d
i m p a c t o f a m i s s i l e o f u n s p e c i f i e d o r i g i n o n t h e
i n t eg r a t e d R e a c t o r - A u x i l i a r y - F u e l- B u i l d i n g o f a
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G. V i t i e t a l . / N on - l i nea r loo r response spec t ra 35
i F o n s )
/ / /~ t ( m se c )
0 1 5 6 0
Fig . 3 . P r e s c r i b e d d y n a m i c f o r c i n g f u n c t i o n .
BWR M ark I I I nuc lear power p lant , ac tua l ly
under construct ion in I taly .
The dynamic exc i ta t ion , s hown in f ig . 3 , has
been app l ied vert ica l ly on a s mal l cen tra l por-
t ion o f the Sh ie ld Bu i ld ing dome or hor izonta l ly
on a small region at the top of i t s cyl indrical
port ion . A typ ica l ax i s ymmetr ic s he l l model o f
the bu i ld ing , w i th the as s umed impact locat ionsis shown in f ig . 4 .
L inear and non- l inear f loor res pons e s pectra
have been eva luated a t s e lec ted locat ions for
various damping rat ios and duct i l i ty factors .
F igs . 5 -8 s how, for example , ver t ica l s pectra for
2% damping and hor izonta l s pectra for 7%
damping a t the top o f the reactor pres s ure ves -
o.
u~N
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Z OCD
QC
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IL lU'3Z
O -
LIJ
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2 0 .
C O N T .T O P
~ 1 i m p a c t
t / . - ,
t 1 _Fig . 4 . B u i l d i n g a x i s y m m e t r i c s h e l l m o d e l .
R L T O L R Z I O N P S - R R F B U I L D I N GM I S S I L E IM P R C T - V E R T I C R L D I R E C T I O NF LO O R RE S P O NS E S P E CT RRR P V T O P ( O . D E G . )
E Q U IP M E N T O R M P I N G 0 . 0 2 0
f / . 1 . 0 0
.0 0
4 0 . 6 0 .
E Q U I P M E N T F R E Q U E N C Y - C P 5
8 0 . t O 0 .
Fig . 5 .
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3 6 G . V i t i e t a l . / N o n - l i n e a r f l o o r r e s p o n s e s p e c t ra
I
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MISSILE IMPRCT - VERTICRL DIRECTION
FLOOR RESPONSE SPECTRR
ST EEL C ON T . T OP { O . D E G. )
EQU IPMEN T D R MPIN G 0 .020
1 . 0 0
2 0 , 4 0 , 6 0 . O 0 ,
EQUIPMENT FREQUENCY - CPS
Fig 6.
I 0 0
R L T O L R Z I O N P S - R R F B U I L D I N g
M I S S I L E I M P R C T - H O R I Z O N T R L D I R E C T I O N
F L O O R R E S P O N S E S P E C T R R
R P V T O P ( O , D E G . )
E Q U I P M E N T D R M P I N O 0 ,0 9 0
j ¢ / ! . 0 0
)O ° 2 0 , 4 0 . 6 0 •
EQUIPMENT FREQUENCY - CPS
F i g . 7 .
8 0 , t O 0 .
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G . V i t i e t a l . / N o n - l i n e a r f l o o r r e s p o n se s pe ct r a 37
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M I S S I L E IM P RC T - H O R I Z O N T R L D I R E C T I O N
F L O O R R ES P O N S E S P E C T R R
S T E E L C O N T . T O P ( O . D E G . )
E O U I P M E N T O R M P IN G 0 . 0 7 0
t • 5 0
P . 0 0
• 0 0
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EQUIPMENT FREQUENCY - CP$
Fig. 8.
8 0 • t O O •
s e l a n d t h e s t e e l c o n t a i n m e n t , f o r d u c t i l i ty f a c -
t o r s o f 1 - 1 , 5 - 2 a n d 3 . T a b l e 1 s u m m a r i z e s
c o r r e s p o n d i n g p e a k r e d u c t i o n c o e f f i c i e n t s w i t hi n c r e a s i n g d u c t i l i t y f a c t o r s .
I n g e n e r a l , f u n d a m e n t a l p e a k s a r e s t ro n g l y
r e d u c e d ; w h e r e a s f o r a n y d u c t i l i t y f a c t o r , v a l u e s
a t v e r y h i g h f r e q u e n c i e s t e n d t o t h e l in e a r
s t r u c t u r a l r e s p o n s e . W i t h d u c t i l i t y f a c t o r s l a r g e r
t h a n t h r e e a n y d y n a m i c a m p l i f i c a t i o n , w i t h r e s -
p e c t t o t h e l i n e a r s tr u c t u r a l r e s p o n s e , i s p r a c -
t i c a l l y e l i m i n a t e d • F u n d a m e n t a l p e a k s o f
h o r i z o n t a l s p e c t r a o c c u r a t r e l a t i v e l y l o w
f r e q u e n c i e s ( l es s t h a n l 0 H z ) a n d a r e a s s o c ia t e d
w i t h b e a m t y p e f le x u r a l m o d e s o f v ib r a t i o n ,
e x c i t e d b y h o r i z o n t a l i m p a c t ; a s a c o n s e q u e n c e ,
p e a k r e d u c t i o n c o e f f i c i e n t s l i s t e d in ta b l e 1 , f o r
a h o r i z o n t a l i m p a c t , a r e r a t h e r c l o s e t o t h o s e
p r e d i c t e d b y N e w m a r k a s a l i n e a r r e d u c t i o n
w i t h t h e d u c t i l i t y f a c t o r • F o r v e r t i c a l s p e c t r a ,
f u n d a m e n t a l p e a k s o c c u r a t h i g h e r f r e q u e n c i e s
( in t h e r a n g e b e t w e e n 2 5 a n d 3 0 H z ) a n d a r e
a s s o c i a t e d w i t h a x i a l b e a m t y p e a n d s h e l l t y p e
v i b r a t i o n m o d e s p e c i f i c a l l y e x c i t e d w i t h a v e r t i -
T a b l e I
Peak reduct ion coef f i c i ent s
Vert i cal impact
D a m p i n g R P V t o p C o n t a i n m e n t t o p
(% of crit.) Du cti l i ty factor
1.5 2 3 1.5 2 3
2% 0.38 0.28 0.18 0.5 0.38 0.23
4% 0.44 0.31 0.22 0.59 0.41 0.26
7% 0.52 0.39 0.29 0.64 0.52 0.33
H o r i z o n t a l i m p a c t
D a m p i n g R P V t o p C o n t a i n m e n t t o p(% of crit.) Du cti l i ty factor
1.5 2 3 1.5 2 3
2% 0.65 0.48 0.40 0.64 0.44 0.35
4% 0.65 0.52 0.43 0.61 0.46 0.38
7% 0.68 0.58 0.47 0.68 0.51 0.42
1//z 0.67 0.50 0.33 0.67 0.50 (I.33
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38 G . V i t i e t a l . / N o n - l i n e a r f l o o r r e s p o n s e sp e c t r a
cal central impact; in th is case peak reduction
fac tor s ar e ge n e r a l l y h i gh e r th an th ose p r e d i c te d
for s e i sm i c c on d i t i on s .
[ 2] N . M . N e w m a r k , S M i R T - 4 , p a p e r K 4 / 1.
[ 3] R . D a n i s c h a n d N . G r a u b n e r , S M i R T - 4 , p a p e r K 9 /2 .
[ 4 ] H . K a m i l , N . K r u t z i k , G . K o s t a n d R . S h a r p e , N u c l .
E n g n g . D e s . 4 6 0 9 7 8 ) 1 1 ) 9 - 1 2 1 .
Re fe r e n c e s
[ I ] N . M . N e w m a r k , S M i R T - 3 , p a p e r K 5/ 1 .