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SOME LIMITATIONS OF MODAL ANALYSIS
IN SEISMIC DESIGN
R. Shepherd*
1. Synopsis
In the 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 to 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 of m u l t i s t o r e y b u i l d i n g s the e x t e n d e d
e l a s t i c s e i s m i c d e s i g n loads are f r e q u e n t l y c a l c u l a t e d as the
s q u a r e r o o t of the sum of the s q u a r e s of the m o d a l r e s p o n s e s .
T h e i n d i v i d u a l m e m b e r forces are t h e n d e t e r m i n e d u s i n g these
seismic design loads.
Previous 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
t h e l i m i t a t i o n s of this t e c h n i q u e and it is accepted as b e i n g
g e n e r a l l y a p p l i c a b l e in 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 analyses 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 two c o n d i t i o n s in w h i c h the
s q u a r e r o o t of the sum of the m o d a l r e s p o n s e s squared* r u l e
is i n a p p l i c a b l e .
8
In this n o t e t h e s e s i t u a t i o n s are d e s c r i b e d and s u g g e s t i o n s
a r e m a d e of an 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 loads in such c a s e s .
2. Introduction
1 2
The n o r m a l - m o d e response-spectrum technique ' has been
a p p l i e d s u c c e s s f u l l y , so far as c a n y e t b e a s c e r t a i n e d , to the
s e i s m i c d e s i g n of m a n y h i g h - r i s e b u i l d i n g s including s e v e r a l
New Zealand o n e s * * *
#
3
4
5
In their r e p o r t
on m e t h o d s of m o d e c o m b i n a t i o n . M e r c h a n t
and H u d s o n c o n c l u d e d that a s u i t a b l e w e i g h t e d a v e r a g e of the
s u m of the a b s o l u t e v a l u e s of the i n d i v i d u a l m o d e s and the s q u a r e
r o o t of the sum of the squares of the m o d e s w i l l g i v e a p r a c t i c a l
d e s i g n c r i t e r i o n for the b a s e shear forces in m u l t i s t o r e y
buildings.
T h e y p o i n t e d out that for c r i t i c a l c a s e s , the
w e i g h t e d a v e r a g e r e d u c e s to the a b s o l u t e sum of the m o d e s .
* U n i v e r s i t y of C a n t e r b u r y ,
Christchurch.
S k i n n e r 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 for c o m b i n i n g the 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 the n o r m a l m o d e s ,
n a m e l y to c o m p u t e the s q u a r e r o o t of the s u m of the s q u a r e s of
t h e v a l u e s for 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 of the
r e s u l t s o b t a i n e d u s i n g this 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 of the e q u a t i o n of m o t i o n h a v e s h o w n t h a t
for 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 s s i m p l e a p p r o x i m a t i o n
is e n t i r e l y a d e q u a t e .
1
1
H o w e v e r two 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 b u i l d i n g s
h a v e i l l u s t r a t e d that a p p l i c a t i o n of the 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 in c e r t a i n c i r c u m s t a n c e s .
These are described
below.
3. The Buildings and the Problems
The first high-rise building analysed had a flexible r e i n forced concrete frame, fourteen storeys h i g h and three bays b y
f o u r b a y s on p l a n .
Its 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
unnecessary.
ing
The 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
table.
Direction
are l i s t e d in the
Mode
Period
(Seconds)
Longitudinal
1
1.-28
0.28
Transverse
1
1.38
0.26
Longitudinal
2
0.45
0.70
Transverse
2
0.49
0.67
accompany-
Earthquake Amplification
Factor
(10% C r i t i c a l D a m p i n g )
T h e c o m p u t e d lg storey forces in the first 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 in t h e second m o d e e x c e p t just a b o v e
b e l o w the u p p e r n o d e p o i n t in the h i g h e r m o d e .
and
O n m u l t i p l y i n g b y the 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
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 to b e s o m e 2.5 times g r e a t e r
t h a n the f i r s t m o d e f o r c e s .
W h e n the s q u a r e r o o t of the sum of the 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 , the 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 the b a s e of the b u i l d i n g and h e n c e the
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 the s e i s m i c load
determination process.
286
T h e d e s i g n shears so e s t a b l i s h e d w e r e c o n s i d e r e d to b e of
v e r y little v a l u e . A factor of about eight w a s n e e d e d to r e d u c e
t h e b a s e shears to the code r e q u i r e m e n t s *
7
In this case the p r o b l e m w a s o v e r c o m e b y r e v e r t i n g to a
direct integration procedure.
The v a r i a t i o n of the seismic
f o r c e s w i t h r e s p e c t to time at each floor w a s e s t a b l i s h e d for a
c h o s e n (El C e n t r o ) d i g i t i s e d e a r t h q u a k e r e c o r d .
It w a s found
t h a t the w o r s t cases of l o a d i n g o c c u r r e d over a r e l a t i v e l y s h o r t
t i m e i n t e r v a l and so a static load a n a l y s i s w a s u n d e r t a k e n
a s s u m i n g the w o r s t case loading o c c u r r e d at e a c h floor s i m u l t aneously.
T h e r e s u l t i n g m e m b e r f o r c e s w e r e only a b o u t h a l f those
d e r i v e d from the m o d a l a n a l y s i s a p p r o a c h and a factor of a b o u t
f o u r w a s found n e c e s s a r y to r e d u c e the b a s e s h e a r s to the c o d e
requirements.
The second h i g h r i s e b u i l d i n g a n a l y s e d c o m p r i s e d a r e i n f o r c e d c o n c r e t e c o m p o s i t e tower and frame s t r u c t u r e of n i n e
storeys.
T h e two towers and t h e four p e r i m e t e r f r a m e s w e r e n o t
s y m m e t r i c a l l y p l a c e d and so t o r s i o n h a d to b e c o n s i d e r e d .
This
in i t s e l f did n o t p o s e any s p e c i a l p r o b l e m s s i n c e the a n a l y s i s
technique has been well developed^' •
T h e f u n d a m e n t a l p e r i o d of the b u i l d i n g w a s c o m p u t e d to b e
a b o u t 0.4 s e c o n d s and so the h i g h e r m o d e s c o u l d n o t h a v e the p r e d o m i n a t i n g e f f e c t w h i c h t h e y h a d in t h e case of the first b u i l d i n g
r e f e r r e d to a b o v e .
In this s e c o n d s t r u c t u r e it w a s the i n t e r a c t i o n b e t w e e n the e l e m e n t s w h i c h c a u s e d c o n s i d e r a b l e d i f f i c u l t y
in s u b s t a n t i a t i n g the a n a l y s i s p r o c e d u r e i n i t i a l l y u s e d .
A s the frames and t o w e r s s u f f e r e d o v e r a l l l a t e r a l d i s p l a c e m e n t s , in r e s i s t i n g the l a t e r a l loading, it w a s e v i d e n t from t h e
m o d a l a n a l y s e s • r e s u l t s t h a t in a d d i t i o n to the forces n e c e s s a r y
i n the m a i n s t r u c t u r a l e l e m e n t s to r e s i s t the e x t e r n a l l o a d i n g ,
e x t r a loads e x i s t e d b e c a u s e of the i n t e r a c t i o n of the e l e m e n t s
on each other.
T h i s a d d i t i o n a l e f f e c t w a s e m p h a s i s e d b y the e x i s t e n c e of
b o t h p o s i t i v e and n e g a t i v e first m o d e forces in the r e s i s t i n g
elements.
On t a k i n g the s q u a r e r o o t of the sum of the m o d a l r e s p o n s e s
s q u a r e d , the sense of the c o m p o n e n t forces w a s lost and so all
t h e s t o r e y f o r c e s a c c u m u l a t e d d o w n the b u i l d i n g to p r o d u c e
r e l a t i v e l y large b a s e s h e a r s .
In the c a s e of one e l e m e n t the s q u a r e root of the sum of
the s q u a r e s b a s e shear p r o v e d to b e f i v e t i m e s the f i r s t m o d e
base shears.
8
1
T h e d e s i r a b i l i t y of a v o i d i n g the s i t u a t i o n in w h i c h c o m p o n e n t
e l e m e n t s can fight 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 to l a t e r a l
loads 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 in this d e s i g n .
Nevertheless
in this p a r t i c u l a r case a s o l u t i o n to the analysis p r o b l e m h a d to
be found.
T h e n e c e s s i t y to 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 for this b u i l d i n g .
I n s t e a d the following a p p r o a c h w a s a d o p t e d .
A s e r i e s 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 to d e t e r m i n e the 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 , for e a c h m o d e , t o the lg m o d a l s h e a r s
m u l t i p l i e d b y the 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 o v e r a l l m e m b e r actions w e r e then c o m p u t e d as the s q u a r e r o o t
of the sum of the squares of the m o d a l m e m b e r a c t i o n s .
T h e p a t t e r n of member a c t i o n s so c o m p u t e d w a s s i g n i f i c a n t l y
d i f f e r e n t from that o b t a i n e d from the "square root of the sum of
the s q u a r e s of the m o d a l s t o r e y f o r c e s a p p r o a c h .
Those total
responses w h i c h had previously appeared unjustifiably large w e r e
r e d u c e d b y a factor of b e t w e e n 2.5 and 3.0 on r e c a l c u l a t i o n and
this m u c h m o r e r e a s o n a b l e set of m e m b e r a c t i o n s w a s judged to b e
of v a l u e in the 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 .
1
4. Conclusion
The t w o instances d e s c r i b e d above i l l u s t r a t e that the
g e n e r a l l y s a t i s f a c t o r y 'square r o o t of the sum of the m o d a l
r e s p o n s e s s q u a r e d r u l e is n o t a l w a y s a p p l i c a b l e .
A s long as
d e s i g n e r s a p p r e c i a t e that l i m i t a t i o n s on its u s e do e x i s t it
w i l l s t i l l b e u s e d in the m a j o r i t y of e x t e n d e d e l a s t i c s e i s m i c
design calculations.
1
T h e o b j e c t of t h i s n o t e is to d r a w a t t e n t i o n
c i r c u m s t a n c e s in w h i c h its u s e on the m o d a l forces
erroneous conclusions.
to t w o
leads to
5. References
1.
Skinner, R . I .
E a r t h q u a k e - G e n e r a t e d Forces and M o v e m e n t s
in 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 .
Bulletin 166, 1964.
2.
Shepherd,
The D e t e r m i n a t i o n of S e i s m i c D e s i g n L o a d s
in a F r a m e d S t r u c t u r e .
N.Z. Engineering,
1967, 2 2 ( 2 ) , 56-61.
R.
288
3.
Shepherd,
R.
The D y n a m i c A n a l y s i s of an 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 the S e i s m o l o g i c a l
S o c i e t y of A m e r i c a , 1 9 6 6 , 56 ( 1 ) , 1 3 - 3 6 .
4.
Shepherd,
R,
L a t e r a l Load A n a l y s e s of the A u c k l a n d
Customs House.
N . Z . Engineering 1967,
22 ( 7 ) ,
213-211.
5•
Shepherd,
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 of a S t e e l
Framed Building.
N . Z . E n g i n e e r i n g 1967 ,
22 ( 1 0 ) , 4 0 7 - 4 1 3 .
Merchant, H.C
& H u d s o n , D.E,
(
7.
M o d e S u p e r p o s i t i o n in M u l t i - d e g r e e of
Freedom Systems Using Earthquake Response
Spectrum Data.
B u l l e t i n of the S e i s m o l o g i c a l
S o c i e t y of A m e r i c a 1 9 6 2 , 52 ( 2 ) , 4 0 5 - 4 1 6 .
Basic D e s i g n L o a d s .
N.Z.S.S. 1900, Chapter
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.
Shepherd, R. &
Donald, R.A.H.
S e i s m i c R e s p o n s e of T o r s i o n a l l y U n b a l a n c e d
Buildings.
J o u r n a l of S o u n d and V i b r a t i o n
1967, 6 (1), 20-37.
9.
Shepherd,
P r e d i c t i o n of the R e s p o n s e of 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 to E a r t h q u a k e
Loading.
Proceedings, First Australasian
C o n f e r e n c e on the M e c h a n i c s of S t r u c t u r e s
and M a t e r i a l s .
Sydney 1967, 16-31.
R.