Some Limitations of Modal Analysis in Seismic Design

8/10/2019 Some Limitations of Modal Analysis in Seismic Design

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Research news and notes

SOME LIM IT TION S OF MOD L N LYS IS

IN SEISM IC D ESIGN

R. Shepherd*

1 Synopsis

I n t h e n o r m a l - m o d e , r e s p o n s e - s p e c t r u m a p p r o a c h t o e a r t h

q u a k e r e s i s t a n t d e s i g n o f m u l t i s t o r e y b u i l d i n g s t h e e x t e n d e d

ela sti c seis mic design loads are fre que ntl y cal cul ated as the

squ are roo t of the sum of the squ are s of the mod al r es po ns es .

T h e i n d i v i d u a l m e m b e r f o r c e s a r e t h e n d e t e r m i n e d u s i n g t h e s e

s e i s m i c d e s i g n l o a d s . P r e v i o u s r e s e a r c h w o r k e r s h a v e e x a m i n e d

the li mit ati ons of this techn ique and it is accep ted as be in g

g e n e r a l l y a p p l i c a b l e i n p r a c t i c a l d e s i g n p r o c e d u r e s .

R e c e n t c o m p u t e r a n a l y s e s of p r o j e c t e d N e w Z e a l a n d h i g h -

r i s e b u i l d i n g s h a v e i l l u s t r a t e d t w o c o n d i t i o n s i n w h i c h t h e

8

s q ua re root of the sum of the mo da l res pon ses squared * rul e

i s i n a p p l i c a b l e .

I n t h i s n o t e t h e s e s i t u a t i o n s a r e d e s c r i b e d a n d s u g g e s t i o n s

a r e m a d e o f a n a l t e r n a t i v e a p p r o a c h w h i c h m a y b e a d o p t e d w h e n

d e r i v i n g d e s i g n l o a d s in s u c h c a s e s .

2

I n t r o d u c t i o n

1 2

T h e n o r m a l - m o d e

#

r e s p o n s e - s p e c t r u m t e c h n i q u e ' h a s b e e n

ap pl ied suc ces sfu lly , so far as can ye t b e asce rta ined , to the

s e i s m i c d e s i g n o f m a n y h i g h - r i s e b u i l d i n g s i n c l u d i n g s e v e r a l

N e w Z e a l an d o n e s

3

*

4

*

5

*

I n t h e i r r e p o r t o n m e t h o d s o f m o d e c o m b i n a t i o n . M e r c h a n t

and Hud son co ncl ude d that a sui tab le we ig ht ed aver age of the

sum of the abs olu te val ues of the in div idu al mode s and the s qua re

ro ot of the sum of the squa res of the mo de s wi ll give a pra ct ic al

d e s i g n c r i t e r i o n f o r t h e b a s e s h e a r f o r c e s i n m u l t i s t o r e y

b u i l d i n g s . T h e y p o i n t e d o u t t h a t f o r c r i t i c a l c a s e s , t h e

w e i g h t e d a v e r a g e r e d u c e s t o t h e a b s o l u t e s u m o f t h e m o d e s .

* U n i v e r s i t y o f C a n t e r b u r y , C h r i s t c h u r c h .


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S k i n n e r

1

h a s p r o p o s e d a s i m p l e a p p r o x i m a t e r u l e f o r c o m

b i n i n g t h e m a x i m u m e a r t h q u a k e r e s p o n s e s of t h e n o r m a l m o d e s ,

n a m e l y t o c o m p u t e t h e s q u a r e r o o t o f t h e s u m o f t h e s q u a r e s o f

t h e v a l u e s f o r t h e i n d i v i d u a l n o r m a l m o d e s . C o m p a r i s o n o f t h e

r e s u l t s o b t a i n e d u s i n g t h i s a p p r o a c h w i t h t h o s e c a l c u l a t e d b y

d i r e c t i n t e g r a t i o n o f th e e q u a t i o n o f m o t i o n h a v e s h o w n t h a t

f o r m a n y p r a c t i c a l d e s i g n c a s e s S k i n n e r

1

s s i m p l e a p p r o x i m a t i o n

i s e n t i r e l y a d e q u a t e .

H o w e v e r t w o r e c e n t c o m p u t e r a n a l y s e s o f p r o j e c t e d b u i l d i n g s

h a v e i l l u s t r a t e d t h a t a p p l i c a t i o n of t h e s i m p l e c o m b i n a t i o n r u l e

c a n p r o v e m i s l e a d i n g i n c e r t a i n c i r c u m s t a n c e s . T h e s e a r e d e s c r i b e d

b e l o w .

3. The B u i l d i n g s and the

roblems

T h e f i r s t h i g h - r i s e b u i l d i n g a n a l y s e d h a d a f l e x i b l e r e i n

f o r c e d c o n c r e t e f r a m e , f o u r t e e n s t o r e y s h i g h a n d t h r e e b a y s b y

f o u r b a y s o n p l a n . I t s s y m m e t r y m a d e t o r s i o n a l c o n s i d e r a t i o n s

u n n e c e s s a r y .

T h e c o m p u t e d d y n a m i c p r o p e r t i e s a r e l i s t e d i n t h e a c c o m p a n y

i n g t a b l e .

D i r e c t i o n M o d e P e r i o d E a r t h q u a k e A m p l i f i c a t i o n

S e c o n d s ) F a c t o r

(10

C r i t i c a l D a m p i n g )

L o n g i t u d i n a l 1

T r a n s v e r s e 1

L o n g i t u d i n a l 2

T r a n s v e r s e 2

1.-28 0.28

1.38 0.26

0.45 0.70

0.49 0.67

T h e c o m p u t e d lg s t o r e y f o r c e s i n t h e f i r s t m o d e w e r e

c o m p a r a b l e w i t h t h o s e i n t h e s e c o n d m o d e e x c e p t j u s t a b o v e a n d

b e l o w t h e u p p e r n o d e p o i n t i n t h e h i g h e r m o d e .

O n m u l t i p l y i n g b y t h e e a r t h q u a k e a m p l i f i c a t i o n f a c t o r t h e

s e c o n d m o d e f o r c e s w e r e p r e d i c t e d t o b e s o m e 2.5 t i m e s g r e a t e r

t h a n t h e f i r s t m o d e f o r c e s .

W h e n t h e s q u a r e r o o t o f t h e s u m o f t h e m o d a l r e s p o n s e s

s q u a r e d w a s c o m p u t e d , t h e s e c o n d m o d e p o s i t i v e a n d n e g a t i v e s t o r e y

f o r c e s a c c u m u l a t e d t o w a r d s t h e b a s e o f t h e b u i l d i n g a n d h e n c e t h e

s e c o n d m o d e r e s p o n s e c o m p l e t e l y d o m i n a t e d t h e s e i s m i c l o a d

d e t e r m i n a t i o n p r o c e s s .


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In the cas e of one elem en t the

8

sq ua r e root of the sum of

t h e s q u a r e s

1

b as e shear pro ved to be five tim es the fir st m od e

b a s e s h e a r s .

T h e d e s i r a b i l i t y o f a v o i d i n g t h e s i t u a t i o n i n w h i c h c o m p o n e n t

e l e m e n t s c a n f i g h t b e t w e e n t h e m s e l v e s w h e n r e s p o n d i n g t o l a t e r a l

l o a d s w a s v e r y c l e a r l y i l l u s t r a t e d i n t h i s d e s i g n . N e v e r t h e l e s s

in this par tic ul ar case a sol uti on to the anal ysis pr ob le m ha d to

b e f o u n d . T h e n e c e s s i t y t o c o n s i d e r t o r s i o n a l e f f e c t s p r e c l u d e d

a d i r e c t i n t e g r a t i o n s o l u t i o n f o r t h i s b u i l d i n g .

Ins tea d the follo wing app roa ch wa s ado pte d. A ser ies of

s t a t i c a n a l y s e s w e r e u n d e r t a k e n t o d e t e r m i n e t h e e l e m e n t m e m b e r

a c t i o n s c o r r e s p o n d i n g , f or e a c h m o d e , t o t h e l g m o d a l s h e a r s

m u l t i p l i e d b y t h e a p p r o p r i a t e e a r t h q u a k e a m p l i f i c a t i o n f a c t o r .

The ove ral l mem ber a ction s we re then com put ed as the squ are ro ot

of the sum of the square s of the mo da l mem ber ac ti on s.

T h e p a t t e r n o f m e m b e r a c t i o n s s o c o m p u t e d w a s s i g n i f i c a n t l y

dif fer ent fro m that obt aine d from the square root of the sum of

t h e s q u a r e s o f t h e m o d a l s t o r e y f o r c e s

1

a p p r o a c h . T h o s e t o t a l

r e s p o n s e s w h i c h h a d p r e v i o u s l y a p p e a r e d u n j u s t i f i a b l y l a r g e w e r e

red uce d by a facto r of be tw ee n 2.5 and 3.0 on rec al cu la ti on and

this mu ch mo re reas ona ble set of mem ber a ctio ns wa s ju dged to be

o f v a l u e i n t h e s u b s e q u e n t s e i s m i c d e s i g n p r o c e s s .

4 C o n c l u s i o n

The two inst ance s des cri bed above illust rat e that the

ge ner al ly sat isf act ory 'square roo t of the sum of the mo da l

r e s p o n s e s s q u a r e d

1

rul e is not alwa ys app li ca bl e. As long as

des ign er s appr eci ate that limita tio ns on its use do exi st it

wi ll stil l be used in the ma jo ri ty of exte nded elas tic sei smi c

d e s i g n c a l c u l a t i o n s .

Th e objec t of thi s not e is to dra w attent ion to t wo

cir cum st anc es in wh ic h its use on the mo da l forces lead s to

e r r o n e o u s c o n c l u s i o n s .

5 R e f e r e n c e s

1 . S k i n n e r , R . I . E a r t h q u a k e - G e n e r a t e d F o r c e s a n d M o v e m e n t s

i n T a l l B u i l d i n g s , N e w Z e a l a n d D . S . 1 . R .

B u l l e t i n 1 6 6 , 1 9 6 4 .

2 . S h e p h e r d , R . T h e D e t e r m i n a t i o n o f S e i s m i c D e s i g n L o a d s

i n a F r a m e d S t r u c t u r e . N . Z . E n g i n e e r i n g ,

1 9 6 7 , 2 2 ( 2 ) , 5 6 - 6 1 .


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288

3. She ph erd , R.

T h e D y n a m i c A n a l y s i s o f a n A p a r t m e n t

B u i l d i n g , B u l l e t i n of t h e S e i s m o l o g i c a l

S o c i e t y o f A m e r i c a , 1 9 6 6 , 56

(1),

1 3 - 3 6 .

4 . S h e p h e r d , R

L a t e r a l L o a d A n a l y s e s o f t h e A u c k l a n d

C u s t o m s H o u s e . N . Z . E n g i n e e r i n g 1 9 6 7 ,

22(7), 213-211.

5 She phe rd, R.

S e i s m i c L a t e r a l L o a d A n a l y s i s o f a S t e e l

F r a m e d B u i l d i n g . N . Z . E n g i n e e r i n g 1 9 6 7 ,

22(10), 4 0 7 - 4 1 3 .

7 .

M e r c h a n t , H . C

(

& Hudson, D.E,

M o d e S u p e r p o s i t i o n i n M u l t i - d e g r e e o f

F r e e d o m S y s t e m s U s i n g E a r t h q u a k e R e s p o n s e

S p e c t r u m D a t a . B u l l e t i n o f t h e S e i s m o l o g i c a l

S o c i e t y o f A m e r i c a 1 9 6 2 , 52(2), 4 0 5 - 4 1 6 .

B a s i c D e s i g n L o a d s . N . Z . S . S . 1 9 0 0 , C h a p t e r

8 , 1 9 6 4 . N e w Z e a l a n d S t a n d a r d s I n s t i t u t e .

8. She phe rd, R. &

D o n a l d , R . A . H .

S e i s m i c R e s p o n s e o f T o r s i o n a l l y U n b a l a n c e d

B u i l d i n g s . J o u r n a l o f S o u n d a n d V i b r a t i o n

1 9 6 7 , 6 ( 1 ) , 2 0 - 3 7 .

9. She phe rd, R. P r e d i c t i o n o f t h e R e s p o n s e o f a T o r s i o n a l l y

U n b a l a n c e d H i g h - R i s e B u i l d i n g t o E a r t h q u a k e

L o a d i n g . P r o c e e d i n g s , F i r s t A u s t r a l a s i a n

C o n f e r e n c e o n t h e M e c h a n i c s of S t r u c t u r e s

a n d M a t e r i a l s . S y d n e y 1 9 6 7 , 1 6 - 3 1 .

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Some Limitations of Modal Analysis in Seismic Design