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Evaluation of the Dynamic Seismic Analysis Recommended for the
1975 National Building Code
NATIONAL RESEARCH C O U N C I L O F CANADA D I V I S I O N O F B U I L D I N G RESEARCH
EVALUATION O F THE DYNAMIC S E I S M I C A N A L Y S I S RECOMMENDED
FOR THE 1975 NATIONAL BUILDING CODE
by J . H . R a i n e r DBR P a p e r N o . 694 of t h e D i v i s i o n o f B u i l d i n g R e s e a r c h O t t a w a , A u g u s t 1976
EVALUATION OF THE DYNAMIC SEISMIC ANALYSIS RECOMMENDED
FOR THE 1975 NATIONAL BUILDING CODE
by J . H . R a i n e r ABSTRACT
A s a n a l t e r n a t i v e t o t h e s t a t i c s e i s m i c l o a d a n a l y s i s o f b u i l d i n g s , a recommended dynamic s e i s m i c p r o c e d u r e i s p r e s e n t e d i n Commentary K of Supplement No. 4 t o t h e 1975 E d i t i o n of t h e N a t i o n a l B u i l d i n g Code o f
Canada (NBC). I n t h i s p a p e r comparisons a r e p r e s e n t e d between t h e
recommended r e s p o n s e s p e c t r u m and t h e f o l l o w i n g : a ) t h e a v e r a g e r e s p o n s e s p e c t r u m by Housner; b ) t h e r e s p o n s e s p e c t r a by Newmark, Blume and Kapur, and by Newmark and H a l l ; c ) t h e s e i s m i c c o e f f i c i e n t S of t h e 1975 NBC; and d ) t h e p r o d u c t o f t h e c o e f f i c i e n t s SKI a p p l i c a b l e t o c r o s s - b r a c e d w a t e r towers.
D e t a i l e d comparisons of s t o r e y s h e a r s and o v e r t u r n i n g moments are
p r e s e n t e d f o r t h r e e b u i l d i n g s , a s c a l c u l a t e d from t h e 1975 NBC and t h e
recommended dynamic a n a l y s i s . Although t h e r e is o v e r - a l l agreement i n
t h e f o r c e s o b t a i n e d from t h e two methods, f o r r e g u l a r b u i l d i n g s t h e recommended dynamic a n a l y s i s i s more c o n s e r v a t i v e i n t h e s h o r t p e r i o d r a n g e ( a p p r o x i m a t e l y 0.5 s ) , and l e s s c o n s e r v a t i v e i n t h e l o n g p e r i o d range ( a p p r o x i m a t e l y 2 s and g r e a t e r ) t h a n t h e 1975 NBC s e i s m i c
r e q u i r e m e n t s .
UNE
VALUATION
DE L'ANALYSE S ~ I S M I Q U E DYNAMIQUE RECOMD~E EN WE DUCODE NATIONAL DU B ~ T I M E N T DE 1975
p a r J . H . R a i n e r
Devant l ' a n a l y s e s t a t i q u e e x i s t a n t e d e s s u r c h a r g e s dues a u x sGismes, l e commentaire K du suppl6ment no. 4 d e 1' 6 d i t i o n de 19 75 du Code
n a t i o n a l du b 2 t i m e n t du Canada (CNB) recommande une mgthode s g i s m i q u e dynamique. Le p r e s e n t a r t i c l e d g c r i t b r i h e m e n t c e t t e mgthode, fond6e s u r l a s u p e r p o s i t i o n des rgponses modales de l a c o n s t r u c t i o n . L ' a u t e u r
compare l e s p e c t r e d e ri5ponses recommand6 a ) a u s p e c t r e de r6ponses
moyennes de Housner; b) aux s p e c t r e s de rgponses de Newmark, Blume e t
Kapur, e t de Newmark e t H a l l ; c) a u c o e f f i c i e n t s g i s m i q u e S du CNB de 1975; e t d) a u p r o d u i t d e s c o e f f i c i e n t s SKI a p p l i c a b l e a u x r g s e r v o i r s d ' e a u s u r po t e a u x a v e c t r a v e r s e s d e con t r e v e n t e m e n t .
L' a u t e u r p r 6 s e n t e d e s comparaisons d 6 t a i l l 6 e s Y pour t r o i s G d i f i c e s , des c i s a i l l e m e n t s des 6 t a g e s e t d e s moments de renversement, conform6ment
a u CNB de 19 75 e t 2 1
'
a n a l y s e dynamique recommand6e. Bien que l e s f o r c e sc a l c u l 6 e s 2 1' a i d e des deux m6thodes s ' a c c o r d e n t dans l ' e n s e m b l e ,
l ' a n a l y s e dynamique r e c o m n d 6 e e s t p l u s p r u d e n t e , e n c e q u i concerne l e s e d i f i c e s o r d i n a i r e s , pour de brZves p g r i o d e s ( e n v i r o n 0 , 5 s ) , e t moins p r u d e n t e pour l e s p g r i o d e s d ' e n v i r o n 2 s e t p l u s , que ne l e s o n t l e s e x i g e n c e s s 6 i s m i q u e s du CNB de 1975.
EVALUATION OF THE DYNAMIC SEISMIC ANALYSIS RECOMMENDED
FOR THE 1975 NATIONAL BUILDING CODE*
by J . H . Rainer
The primary purpose o f t h e s e i s m i c requirements o f t h e National Building Code o f Canada i s t o reduce t o an acceptably low l e v e l t h e r i s k o f i n j u r y o r death t o occupants o f b u i l d i n g s s u b j e c t t o earthquakes. This implies t h e p r e v e n t i o n o f p a r t i a l o r t o t a l c o l l a p s e o f a b u i l d i n g under t h e a c t i o n o f earthquakes t h a t a r e l i k e l y t o occur i n a given l o c a t i o n . In most seismic r e g i o n s o f t h e world t h e usual method o f achieving t h i s i s t o endow t h e b u i l d i n g s with a c e r t a i n l e v e l o f r e s i s t a n c e t o s t a t i c l a t e r a l loads o f t h e o r d e r o f those induced by earthquakes. Acceptable design methods a r e a l s o p r e s c r i b e d s o t h a t adequate d u c t i l i t y i s achieved f o r severe overloads. These a r e a l s o t h e p r i n c i p l e s followed i n t h e seismic p r o v i s i o n s o f t h e National Building Code o f Canada. ( There a r e c a s e s , however, f o r which t h e s t a t i c load method i s inadequate o r i n a p p r o p r i a t e , and t h e r e i s p r o v i s i o n f o r t h e use of a dynamic method a s an a1 t e r n a t i v e t o t h e Code procedure. Such a method, based on an average spectrum approach, i s p r e s e n t e d i n Commentary K , Supplement No. 4 t o t h e National Building Code of Canada 1975. ( 2 )
I t i s t h e purpose o f t h i s p a p e r t o b r i e f l y d e s c r i b e t h i s method, t o compare t h e spectrum used i n t h e recommended procedure with o t h e r design s p e c t r a , and t o compare t h e loads derived from t h e recommended dynamic procedure with those o b t a i n e d from t h e s e i s m i c p r o v i s i o n s of t h e 1975 National Building Code ( h e r e i n a f t e r r e f e r r e d t o a s 1975 NBC)
.
PRINCIPAL FEATURES OF
THE
RECOMMENDED DYNAMIC ANALYSISThe recommended dynamic a n a l y s i s p r e s e n t e d i n Commentary K o f Supplement No. 4 t o t h e 1975 NBC ( 2 ) i s one t h a t i s based on t h e
p r i n c i p l e of s u p e r p o s i t i o n o f t h e s p e c t r a l responses o f t h e normal modes o f v i b r a t i o n o f t h e s t r u c t u r e . The spectrum employed i s a smoothed envelope response spectrum, sometimes c a l l e d t h e average response spectrum.
*
Presented a t t h e UPADI-
EIC Congress, Toronto, O n t a r i o . 6- 12 October 1974.The p r i n c i p a l s t e p s i n t h e recommended dynamic procedure a r e , b r i e f l y , a s follows:
1. E s t a b l i s h the seismic zone and corresponding ground a c c e l e r a t i o n t o be used a t t h e given l o c a t i o n . The recommended values, which correspond t o t h e 100-year r e t u r n p e r i o d , a r e given i n 1975 NBC,
Supplement No. 1 on Climatic Infarmation. ( 3 ) The information i s
p r e s e n t e d i n a t a b l e and a l s o i n t h e form o f t h e zoning map reproduced h e r e a s Figure 1. C a l c u l a t i o n s of expected ground a c c e l e r a t i o n s f o r p a r t i c u l a r l o c a t i o n s can a l s o be o b t a i n e d a s o u t l i n e d i n Commentaries J and
K
o f Supplement No. 4 t o t h e 1975NBC. The recommended spectrum reproduced i n Figure 2 i s then
s c a l e d i n p r o p o r t i o n t o t h e peak design ground a c c e l e r a t i o n .
2. Determine t h e dynamic p r o p e r t i e s o f t h e s t r u c t u r e , i n c l u d i n g n a t u r a l f r e q u e n c i e s , mode shapes, and p a r t i c i p a t i o n f a c t o r s . 3 . S e l e c t design values of modal damping and s t r u c t u r a l d u c t i l i t y ,
following the recommendations given i n Commentary K .
4. S c a l e recommended spectrum values f o r t h e a p p r o p r i a t e ground a c c e l e r a t i o n and p l a s t i c behaviour, and compute probable response by combining t h e modal c o n t r i b u t i o n s .
5 . Design t h e s t r u c t u r a l members and connections f o r expected performance. This g e n e r a l l y means p r o v i s i o n s f o r t h e d u c t i l e behaviour assumed i n t h e a n a l y s i s .
I t i s expected t h a t t h e recommended dynamic procedure w i l l be used f o r b u i l d i n g s having l a r g e t o r s i o n a l e c c e n t r i c i t i e s , major s e t b a c k s i n p l a n dimensions, unusual mass o r s t i f f n e s s d i s t r i b u t i o n s , unusual
foundation c o n d i t i o n s , and g e n e r a l l y f o r c a s e s where t h e Code p r o v i s i o n s can b e expected t o be i n a p p l i c a b l e o r too crude. The dynamic procedure i s n o t intended t o r e p l a c e t h e s t a t i c Code p r o v i s i o n s , b u t r a t h e r t o supplement them.
RECOMMENDED RESPONSE SPECTRUM
The b a s i s o f t h e recommended procedure l i e s i n t h e p r e s c r i b e d response spectrum. A s e i s m i c response spectrum i s t h e envelope o f maximum responses o f single-degree-of-freedom o s c i l l a t o r s o f d i f f e r e n t n a t u r a l f r e q u e n c i e s o r p e r i o d s when s u b j e c t e d t o a given ground motion. A s every earthquake w i l l have d i f f e r e n t s p e c t r a , and because t h e
c h a r a c t e r i s t i c s of a p o t e n t i a l earthquake cannot be p r e d i c t e d , a
s t a t i s t i c a l e v a l u a t i o n o f t h e s p e c t r a o f a number o f p r e v i o u s l y recorded earthquakes h a s r e s u l t e d i n t h e p r e s e n t a t i o n o f various average s p e c t r a . These f a l l i n t o two main c a t e g o r i e s :
1) an a l g e b r a i c average, i . e . , t h e mean o f various s p e c t r a of earthquakes,
2) a s c a l e d average, o r t h e mean p l u s some f r a c t i o n o f t h e s t a n d a r d
d e v i a t i o n from t h e mean o f t h e c o n t r i b u t i n g s p e c t r a .
I t i s important t o understand t h e b a s i s f o r t h e d e r i v a t i o n s o f t h e v a r i o u s s p e c t r a and t h e subsequent d i f f e r e n c e s i n t h e r e s u l t s .
Included i n t h e f i r s t c a t e g o r y i s t h e well-known average spectrum
by Housner, ( 4 ) reproduced i n Figure 3 with o t h e r information. This
spectrum was o b t a i n e d by averaging t h e computed s p e c t r a l responses corresponding t o 7 C a l i f o r n i a earthquake r e c o r d s . The r e s u l t i s a mean response spectrum, i . e . , t h e r e i s a 50 p e r c e n t chance of t h e response being g r e a t e r o r s m a l l e r than t h e average curves p r e s e n t e d .
Among t h e s p e c t r a i n t h e second category i s t h e spectrum advanced by Newmark, Blume and Kapur, (5) i l l u s t r a t e d i n Figures 4 and 5 f o r
h o r i z o n t a l and v e r t i c a l d i r e c t i o n s , r e s p e c t i v e l y . These s p e c t r a r e s u l t e d from t h e s t a t i s t i c a l e v a l u a t i o n o f about 30 earthquake r e c o r d s
-
mostly ( b u t n o t e x c l u s i v e l y ) from C a l i f o r n i a.
They r e p r e s e n t t h e maximumresponse a t one s t a n d a r d d e v i a t i o n from t h e mean o f a log-normal d i s t r i - b u t i o n . This means t h a t t h e r e i s a p r o b a b i l i t y o f 84 p e r c e n t t h a t t h e response o f a given single-degree-of-freedom system t o a p a r t i c u l a r earthquake used i n t h a t s t u d y would be l e s s than t h e proposed spectrum. Conversely, t h e r e i s a 16 p e r c e n t p r o b a b i l i t y t h a t i t would be exceeded.
I t w i l l be e v i d e n t t h a t t h e average s p e c t r a o f t h e second category, e . g . , one s t a n d a r d d e v i a t i o n above t h e mean, a r e a t a . h i g h e r l e v e l and t h u s more c o n s e r v a t i v e than t h e s p e c t r a d e r i v e d from t h e a l g e b r a i c mean. COMPARISON OF SPECTRA
The spectrum i n t h e recommended procedure i s c l o s e l y r e l a t e d t o t h e spectrum o f t h e second category, and s p e c i f i c a l l y t o t h e one from Ref. ( 5 ) . Comparisons o f t h e 5 p e r c e n t damping curve with v a r i o u s types of s p e c t r a a r e p r e s e n t e d i n Figures 4 , s and 6. Except where otherwise
s t a t e d , a l l s p e c t r a a r e s c a l e d t o 1.0 g peak ground a c c e l e r a t i o n i n o r d e r t o f a c i l i t a t e comparisons.
Figure 4 shows t h e spectrum curve f o r 5 p e r c e n t damping drawn on t h e h o r i z o n t a l spectrum from Ref. ( 5 ) . I t can be s e e n t h a t t h e
recommended spectrum a g r e e s reasonably w e l l with t h e corresponding one from Ref. (5)
.
The recommended spectrum can t h e r e f o r e be s a i d t o have s i m i l a r p r o b a b i l i t i e s of exceedance of maximum response. A s i m i l a r comparison f o r t h e v e r t i c a l component i s p r e s e n t e d i n Figure 5.Figure 6 shows a comparison with t h e spectrum advanced by Newmark
and H a l l . ( 6 ) Again, t h e recommended spectrum curve f o r 5 p e r c e n t
damping agrees reasonably we1 1.
From t h e previous d i s c u s s i o n on t h e v a r i o u s average s p e c t r a i t
might be a n t i c i p a t e d t h a t t h e recommended spectrum would be s u b s t a n t i a l l y
i n Figure 3. The o r d i n a t e s f o r t h e 5 p e r c e n t damped recommended spectrum a r e about twice a s l a r g e a s t h e corresponding ones on t h e
Housner spectrum. The major p a r t o f t h e d i f f e r e n c e can be a t t r i b u t e d t o t h e d i f f e r e n t c r i t e r i a f o r exceedance o f response t h a t were employed i n d e r i v i n g t h e s e two s p e c t r a .
I n Figure 7 t h e recommended average spectrum f o r 5 p e r c e n t damping i s compared with t h e response spectrum f o r t h e E l Centro earthquake, 1940, N-S component. The s p e c t r a correspond t o a peak ground a c c e l e r a - t i o n o f 0.33 g. I t may be observed t h a t t h e recommended design spectrum provides almost a n envelope t o t h e peak values f o r t h e E l Centro e a r t h - quake spectrum. In t h e r e g i o n o f long p e r i o d s o r low frequency, however, i . e . , t h e displacement bounds, t h e recommended v a l u e s a r e s u b s t a n t i a l l y l a r g e r . This i n d i c a t e s t h a t i f a time h i s t o r y a n a l y s i s is c a r r i e d o u t using t h i s p a r t i c u l a r earthquake record, s c a l i n g f a c t o r s s l i g h t l y l a r g e r than those based merely on peak ground a c c e l e r a t i o n s should b e used i n o r d e r t o a r r i v e a t answers t h a t a r e reasonably compatible w i t h t h e r e s u l t s o b t a i n e d from t h e recommended dynamic procedure i n Commentary K . COMPARISON BETWEEN S AND RECOMMENDED SPECTRUM
The s e i s m i c c o e f f i c i e n t S = 0.5/T i n S e c t i o n 4.1.9 o f t h e 1975 NBC i s p l o t t e d i n Figure 8 a l o n g with t h e recommended e l a s t i c spectrum. Also shown i s t h e corresponding e l a s t i c - p l as t i c response spectrum f o r a c c e l e r a t i o n ( o r f o r c e ) f o r damping r a t i o A = 5 p e r c e n t and d u c t i l i t y f a c t o r p = 3 . A d i r e c t comparison can be made between t h e e l a s t i c - p l a s t i c a c c e l e r a t i o n design spectrum and S only f o r systems t h a t a r e p h y s i c a l l y w e l l modelled by a single-degree-of-freedom o s c i l l a t o r . Under t h e assumption t h a t f o r such a s t r u c t u r e , p = 3 and A = 5 p e r c e n t corresponds roughly t o s t r u c t u r a l response f a c t o r K = 1.0 i n t h e Code, t h e Code i s seen t o give design f o r c e s roughly one-half those obtained from t h e dynamic procedure a t p e r i o d T = 0 . 3 s . On t h e o t h e r hand, f o r p e r i o d s above about 1 s t h e Code f o r c e s a r e s u b s t a n t i a l l y l a r g e r than those from t h e dynamic procedure.
For water towers on cross-braced s t e e l columns, t h e 1975 NBC
r e q u i r e s K = 3.0, with SKI between 1.2 and 2.5 ( I = importance f a c t o r . ) The corresponding curve o f SK i s p l o t t e d i n Figure 8 . The r e s u l t i n g
design f o r c e s a r e seen t o demand l i t t l e p l a s t i c a c t i o n i n t h e s h o r t - p e r i o d range; they a r e g r e a t e r than t h e e l a s t i c dynamic requirements i n t h e p e r i o d range above 0 . 8 s .
For multi-degree-of-freedom s t r u c t u r e s t h e comparison i s n o t a s s t r a i g h t f o r w a r d . Both modes h i g h e r than t h e fundamental and modal masses have t o be taken i n t o c o n s i d e r a t i o n . A d e t a i l e d t r e a t m e n t o f t h i s a s p e c t
w i l l not be p r e s e n t e d h e r e , although numerical comparisons f o r t h r e e s p e c i f i c b u i l d i n g s a r e given i n t h e next s e c t i o n .
COMPARISON OF SHEAR AND MOMENTS WITH COIIE VALUES
The s t a t i c e q u i v a l e n t s e i s m i c load requirements i n most b u i l d i n g codes have been derived from r u l e - o f - thumb p r i n c i p l e s , experience gained
from p r e v i o u s e a r t h q u a k e s , and t h e o r e t i c a l and economic c o n s i d e r a t i o n s . With t h e i n t r o d u c t i o n o f e x p e c t e d maximum ground a c c e l e r a t i o n a s a b a s i s f o r s e i s m i c zoning and i t s use i n t h e l a t e r a l s e i s m i c l o a d c a l c u l a t i o n s o f t h e 1975 NBC, t h e p o s s i b i l i t y p r e s e n t s i t s e l f o f a c h i e v i n g a r a t i o n a l c o n s i s t e n c y between t h e s t a t i c and dynamic p r o c e d u r e . I n o r d e r t o
a s s e s s such a p o s s i b l e agreement, s e i s m i c l o a d c a l c u l a t i o n s were
performed on t h r e e d i f f e r e n t b u i l d i n g s , u s i n g t h e p r o v i s i o n s o f t h e 1975
NBC 'and t h e recommended dynamic procedure i n Commentary K
.
The b u i l d i n g s were :- A 1 4 - s t o r e y r e i n f o r c e d c o n c r e t e s h e a r - w a l l b u i l d i n g w i t h a e l l i p t i c a l p l a n s e c t i o n . The s h e a r w a l l s r e s i s t
t h e e n t i r e l a t e r a l l o a d and a r e assumed t o b e d u c t i l e f l e x u r a l w a l l s . F u r t h e r s t r u c t u r a l d e t a i l s a r e g i v e n i n Ref. ( 9 ) . The
foundation was assumed r i g i d .
B u i l d i n g No. 2 . - A 1 5 - s t o r e y r 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 of combined s h e a r - w a l l and frame s y s tem, w i t h s t r u c t u r a l d e t a i l s d e s c r i b e d i n Ref. (8)
.
B u i l d i n g No. 3 . - A h y p o t h e t i c a l d e s i g n example o f a 2 5 - s t o r e y r e i n f o r c e d c o n c r e t e frame b u i l d i n g , a s given i n Ref. ( 7 ) .
A summary o f t h e p r o p e r t i e s o f t h e t h r e e b u i l d i n g s i s p r e s e n t e d i n Table 1.
A l l c a l c u l a t i o n s were c a r r i e d o u t f o r a peak d e s i g n ground a c c e l e r a - t i o n o f 1 . 0 g. The Code f o r c e s were determined a c c o r d i n g t o t h e f o l l o w - i n g formula from t h e 1975 NBC, S e c t i o n 4 . 1 . 9 . : V = ASKIFW ( 1 where V = b a s e s h e a r A = d e s i g n ground a c c e l e r a t i o n , a s a f r a c t i o n o f g r a v i t a t i o n a l a c c e l e r a t i o n K = s t r u c t u r a l r e s p o n s e f a c t o r S = s e i s m i c c o e f f i c i e n t = 0 . 5 / T I = importance f a c t o r F = f o u n d a t i o n f a c t o r W = weight o f s t r u c t u r e
The b a s e s h e a r V was t h e n d i s t r i b u t e d throughout t h e b u i l d i n g h e i g h t a s p r e s c r i b e d by t h e Code. A summary o f numerical v a l u e s i s p r e s e n t e d i n Table 2 .
The o v e r t u r n i n g moments were determined s i m i l a r l y u s i n g t h e a p p l i c - abl c v a l u e o f J , t h e o v e r t u r n i n g moment r e d u c t i o n c o e f f i c i e n t ; t h e s e and t h e r e s u l t i n g b a s e o v e r t u r n i n g moments a r e a l s o p r e s e n t e d i n Table 2 .
RESULTS
The r e s u l t s f o r t h e s h e a r d i s t r i b u t i o n s of t h e t h r e e b u i l d i n g s a r e p r e s e n t e d i n F i g u r e s 9 , 10 and 11. I n a d d i t i o n t o t h e Code v a l u e s and t h o s e from t h e recommended dynamic procedure, r e s u l t s from e l a s t i c a n a l y s i s and t h o s e from o t h e r methods of modal s u p e r p o s i t i o n s a r e given. The l a t t e r a r e p r e s e n t e d f o r p u r p o s e s o f comparison o n l y . S i m i l a r r e s u l t s a r e p r e s e n t e d f o r o v e r t u r n i n g moments i n F i g u r e s 12, 1 3 and 1 4 .
I t should b e noted t h a t t h e p e r i o d T r e q u i r e d f o r t h e c a l c u l a t i o n o f S and c o n s e q u e n t l y o f V was t h a t o f t h e fundamental mode a n d n o t t h a t c a l c u l a t e d by t h e Code formula.
DISCUSSION OF RESULTS
a) Design Forces
The r e s u l t s o f d e s i g n s t o r e y s h e a r s (Figure 9) f o r b u i l d i n g No. 1
i n d i c a t e t h a t t h e Code b a s e s h e a r s f o r
K
= 1.00 a r e a p p r o x i m a t e l y 40 p e r c e n t s m a l l e r t h a n t h o s e from t h e dynamic method. A comparison of t h e s h e a r d i s t r i b u t i o n o v e r t h e b u i l d i n g h e i g h t shows a s i m i l a r v a r i a t i o n , w i t h t h e p e r c e n t a g e d i f f e r e n c e si n
t h e upper 213 of t h e b u i l d i n g b e i n g s l i g h t l y l a r g e r t h a n a t t h e b a s e .A comparison o f t h e b a s e o v e r t u r n i n g moments i n Figure 12 shows t h a t t h e Code y i e l d s v a l u e s 17 p e r c e n t h i g h e r t h a n t h e dynamic a n a l y s i s . The d i f f e r e n c e becomes 1 a r g e r , however, above t h e midheigh t o f t h e b u i 1 ding, r e a c h i n g between 30 t o 40 p e r c e n t .
The s t o r e y s h e a r s f o r b u i l d i n g No. 2 ( F i g u r e 10) show s u b s t a n t i a l agreement between t h e 1975 NBC and t h e dynamic procedure f o r t h e b a s e o f t h e s t r u c t u r e . Near midheight, t h e s h e a r s c a l c u l a t i o n from t h e Code r e q u i r e m e n t s a r e l a r g e r , whereas i n t h e upper 2/3 o f t h e b u i l d i n g they a r e s m a l l e r by about 40 p e r c e n t than t h o s e from t h e dynamic method. The o v e r t u r n i n g moments b a s e d on t h e Code f o r b u i l d i n g No. 2 and
p r e s e n t e d i n Figure 1 3 a r e s e e n t o b e about 20 p e r c e n t l a r g e r t h a n t h e dynamic procedure a t t h e b a s e , b u t t h e t r e n d r e v e r s e s i n t h e upper h a l f o f t h e s t r u c t u r e .
For b u i l d i n g No. 3 t h e s h e a r s based on t h e Code ( F i g u r e 11) a r e seen t o b e 1 . 6 times a s l a r g e a s t h o s e from t h e dynamic p r o c e d u r e . T h i s d i f f e r e n c e p e r s i s t s i n t h e lower h a l f o f t h e b u i l d i n g b u t g r a d u a l l y t h e y approach each o t h e r a t t h e t o p o f t h e s t r u c t u r e . For t h e o v e r t u r n i n g moments i n Figure 14 a s i m i l a r d i f f e r e n c e e x i s t s between t h e Code v a l u e s and t h e dynamic p r o c e d u r e .
b) E l a s t i c R e s u l t s
The e l a s t i c * r e s u l t s from t h e dynamic a n a l y s i s shown i n F i g u r e s 9
through 14 a r e s u b s t a n t i a l l y l a r g e r t h a n t h e d e s i g n v a l u e s o b t a i n e d from t h e recommended dynamic procedure o r from t h e i n f o r m a t i o n i n t h e Code.
T h i s i s due, o f c o u r s e , t o t h e f a c t t h a t t h e design f o r c e s a r e o b t a i n e d from t h e e l a s t i c modal s h e a r s and o v e r t u r n i n g moments by d i v i d i n g by a d u c t i l i t y c o e f f i c i e n t t o account f o r p l a s t i c a c t i o n i n t h e s t r u c t u r e .
Also shown i n F i g u r e s 10 and 1 3 f o r b u i l d i n g No. 2 a r e comparisons o f d i f f e r e n t ways o f combining t h e modal c o n t r i b u t i o n s . For t h e e l a s t i c r e s u l t s t h e root-sum-square (R .S . S .) method g i v e s v a l u e s f o r t h e s h e a r s s u b s t a n t i a l l y s m a l l e r t h a n t h e maximum o f any two modes; t h e
a b s o l u t e maximum o f a l l c o n t r i b u t i n g modes i s i n t u r n s u b s t a n t i a l l y
l a r g e r than t h e maximum o f any two modes. The maximum o f any two modes
emphasizes t h e c o n t r i b u t i o n s o f t h e second mode, whereas i n t h e R.S .S. r e s u l t s t h e c o n t r i b u t i o n s o f t h e modes h i g h e r t h a n t h e fundamental a r e reduced and smoothed.
The e l a s t i c r e s u l t s f o r t h e o v e r t u r n i n g moments f o r b u i l d i n g No. 2 shown i n Figure 1 3 a r e s e e n t o vary l e s s among t h e methods o f modal combinations t h a n t h o s e f o r t h e s h e a r s i n F i g u r e 10. This c a n b e e x p l a i n e d by t h e f a c t t h a t t h e fundamental mode c o n t r i b u t e s a g r e a t e r p r o p o r t i o n t o t h e t o t a l o v e r t u r n i n g moments than t o t h e s t o r e y s h e a r s . SUWRY AND CONCLUSIONS
The spectrum i n t h e recommended dynamic procedure given i n
Commentary K o f Supplement No. 4 t o t h e 1975 NBC a g r e e s s u b s t a n t i a l l y
w i t h t h e spectrum recommended by Newmark, Blume and Kapur. (5) The
given spectrum d i f f e r s ~ ' u b s t a n t i a l l ~ i n amplitude b u t n o t i n shape, from t h e a v e r a g e spectrum p r e s e n t e d by Housner. ( 4 j This c a n b e
a t t r i b u t e d t o t h e d i f f e r e n t methods employed i n d e r i v i n g t h e s p e c t r a . The comparison o f s h e a r s and o v e r t u r n i n g moments between t h e p r o v i s i o n s o f t h e 1975 NBC and t h e recommended dynamic procedure i n d i c a t e s t h e - f o l l o w i n g t r e n d s :
1. For t h e b u i l d i n g w i t h a n a t u r a l p e r i o d i n t h e o r d e r o f 1 s, t h e two methods a r e i n s u b s t a n t i a l agreement, w i t h s m a l l d i f f e r e n c e s
o c c u r r i n g through t h e h e i g h t o f t h e b u i l d i n g .
2 . For t h e s h o r t - p e r i o d s t r u c t u r e , i n t h e o r d e r o f 0.5 s , t h e recom- mended dynamic procedure 'gives f o r c e s t h a t a r e approximately 30 p e r c e n t l a r g e r t h a n t h a t o f t h e Code.
3 . For t h e l o n g p e r i o d s t r u c t u r e , o f about 2 s , t h e Code g i v e s l a r g e r f o r c e s a t t h e b a s e t h a n t h e recommended dynamic procedure, b u t comparable f o r c e s n e a r t h e t o p .
I t i s b e l i e v e d t h a t t h e t r e n d s e s t a b l i s h e d i n t h e s e r e s u l t s a r e g e n e r a l l y v a l i d f o r s i m i l a r t y p e s o f b u i l d i n g s . A d d i t i o n a l examples would b e needed, however, t o c o r r o b o r a t e t h i s .
SUGGESTED CHANGES OF ?HE DYNAMIC PROCEDURE
1. The spectrum, a s p r e s e n t e d i n Supplement No. 4 t o t h e 1975 E d i t i o n o f t h e National Building Code, ranges from a p e r i o d o f 0.1 s t o 15 s . Although t h i s should be adequate f o r most s t r u c t u r a l a p p l i c a t i o n s , t h e need may a r i s e f o r an e x t e n s i o n i n t h e s h o r t - p e r i o d range. I n t h a t c a s e i t i s suggested t h a t a procedure s i m i l a r t o t h a t given i n Ref. (5) be followed, namely, t h a t a s t r a i g h t l i n e be drawn j o i n i n g the spectrum l i n e a t 0 . 1 s t o t h e peak ground a c c e l e r a t i o n bound a t 0.03
s
.
T h e r e a f t e r t h e spectrum follows t h e peak ground a c c e l e r a t i o n bound. This i s i l l u s t r a t e d i n Figure 8 .2. I t i s suggested t h a t a t l e a s t t h r e e modes be used i n t h e recom- mended dynamic a n a l y s i s .
REFERENCES
A s s o c i a t e Committee on t h e N a t i o n a l B u i l d i n g Code, N a t i o n a l B u i l d i n g Code o f Canada 1975, p u b l i s h e d by t h e N a t i o n a l Research Council o f Canada, Ottawa. (NRC 13982)
A s s o c i a t e Committee on t h e N a t i o n a l B u i l d i n g Code, Commentaries on P a r t 4 , 1975, Supplement No. 4 t o t h e N a t i o n a l B u i l d i n g Code o f Canada 1975, pub1 i s h e d by t h e N a t i o n a l Research Council o f Canada, Ottawa. (NRC 13989)
A s s o c i a t e Committee on t h e N a t i o n a l B u i l d i n g Code, C l i m a t i c I n f o r m a t i o n f o r B u i l d i n g Design i n Canada 1975, Supplement No. 1 t o t h e N a t i o n a l B u i l d i n g Code o f Canada 1975, p u b l i s h e d by t h e N a t i o n a l Research Council o f Canada, Ottawa. (NRC 13986)
Housner, G.W. Design Spectrum. C h a p t e r 5, i n Earthquake
E n g i n e e r i n g , Wiegel, Ed., P r e n t i c e - H a l l , I n c . , Englewood C I i f f s ,
N . J . , 1970.
Newmark,
N.M.,
Blume, J . A . and Kapur, K . K . S e i s m i c Design S p e c t r a f o r N u c l e a r Power P l a n t s . J o u r n a l o f Power D i v i s i o n , ASCE, Vo1.99, No. P02, November 1973, p . 287-303.Newmark, N . M . and H a l l , W . J . P r o c e d u r e s a n d C r i t e r i a f o r Earthquake R e s i s t a n t Design. B u i l d i n g S c i e n c e s S e r i e s 46, N a t i o n a l Bureau o f S t a n d a r d s , Washington, D.C., February 1973, p
.
209-236.Newmark, N . M . , Blume, J . A . and Corning, L . H . Design o f M u l t i s t o r y R e i n f o r c e d Concrete B u i l d i n g s f o r Earthquake Motions. P o r t l a n d Cement A s s o c i a t i o n , Chicago I l l . , 1961.
Ward, H.S. Dynamic C h a r a c t e r i s t i c s o f a Mu1 t i - S t o r e y C o n c r e t e B u i l d i n g . P r o c e e d i n g s , I n s t i t u t i o n o f C i v i l E n g i n e e r s , Vol
.
43, August 1969, p . 553-572. ( P r e p r i n t e d a s NRC 11461)Ward, H .S
.
and R a i n e r , J .H. Experimental D e t e r m i n a t i o n o f S t r u c t u r e and Foundation P a r a m e t e r s Using Wind-Induced V i b r a t i o n s .P r o c e e d i n g s , I n s t i t u t i o n o f C i v i l E n g i n e e r s , Vol. 53, September 1972, p . 305-322. ( P r e p r i n t e d a s NRC 13026)
TABLE I
PROPERTIES OF EXAMPLE BUILDINGS
Assumed Duct- Damp- P a r t i c i - i l i t y i n g Bldg H e i g h t , No. o f P e r i o d s , p a t i o n v F a c t o r R a t i o No. f t S t o r e y s Modes s F a c t o r s * i n . / s K u A, % * f o r mode shapes n o r m a l i z e d t o 1 . 0 a t t h e t o p s t o r e y TABLE 2 SUMMARY OF RESULTS
Recommended Dynamic Procedure 1975 N a t i o n a l B u i l d i n g Code Commentary K (1975 NBC)
Bldg Base Base Base Base
No. S h e a r Moment S h e a r Moment
V M V M
F t / V J x l o 6 l b x 10' i n . / l b x l o 6 l b x l o 9 i n . / l b
3 Assumed 0.80 24.2 4 0 . 8 1 5 . 3 2 6 . 2
F I G U R E 1
P E R I O D T, S E C
F I G U R E
2
A V E R A G E G R O U N D M O T I O N A N D E L A S T I C A V E R A G E R E S P O N S E S P E C T R U M F O R 1 . 0 9 M A X I M U M G R O U N D A C C E L E R A T I O N ( C O M M E N T A R Y
K , 1 9 7 5
N B C )P E R I O D , S E C
F I G U R E
3
FREQUENCY, CPS
F I G U R E 4
H O R I Z O N T A L D E S I G N R E S P O N S E S P E C T R A
-
S C A L E D
T O
l g
H O R I Z O N T A L G R O U N D A C C E L E R A T I O N ( N E W M A R K ,
0.1 0.2 0.5 1 2 5 10 20 50 100 FREQUENCY, cp
F I G U R E
5
V E R T I C A L D E S I G N R E S P O N S E S P E C T R A
-S C A L E D T O
l g H O R I Z O N T A L G R O U N D A C C E L E R A T I O N ( N E W M A R K ,
B L U M E A N D K A P U R , 1 9 7 3 )
Velocity , in./sec.
z m
