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8/12/2019 16(2)0098 http://slidepdf.com/reader/full/1620098 1/9 98 THE SEISMIC DESIGN OF AN INDUSTRIAL CHIMNEY WITH ROCKING BASE R.D.Sharpe , R.I. Skinner ABSTRACT: An industrial reinforced concrete chimney, of cruciform cross-section and thirty-five metres tall, has been designed and built with the high degree of seismic protection afforded by allowing the base to rock during large earthquakes. As a precaution a pair of purpose-made yielding steel dampers have been installed to assist in the control of the most violent response that might occur. This paper discusses the advantages in the adoption of this revolutionary design and describes the hand dynamic analysis carried out during the design process. Subsequent time-history computer analysis carried out to confirm the design method is reported on. 1.0 INTRODUCTION: During the design of the new hangar and maintenance facilities at the Air New Zealand Engineering Base at Christchurch there was a requirement for an industrial chimney to be provided. As this chimney would be close to, and associated with, the prestigious hangar and maintenance complex an architectural solution was sought which would be more appropriate than the conventional guyed circular flue. Acceptable architectural and engineering criteria were met with the development of a reinforced concrete structure of cruciform cross-section to which the flues could be attached. The chosen form allows the eventual duplication of the chimney capacity while its freedom to rock gives it the ability to withstand very large earthquakes without requiring the substantial repairs that most reinforced concrete structures would need. its foundations. Over the bottom half the tower was splayed out to provide greater resistance to over-turning and flanges were proposed for the outer edges of the four legs of the cruciform to increase their stability. The final dimensions of the chimney are shown in Figure 1. By being able to rock, the chimney superstructure is protected from attracting larger seismic moments and shears than those required to produce lift-off. The superstructure may then be safely designed to remain elastic under these limited loads. Extra damping is provided by two steel hysteretic dampers, each of 120 kN yield force, placed at opposite corners of the base. Any torsional stiffness of the chimney is greatly enhanced by the addition of horizontal diaphragms or ring beams at three intermediate levels. 2.0 DEVELOPMENT OF THE DESIGN: The preliminary architectural proposals showed a reinforced concrete tower of uniform cross-section with height and integral with its foundations. The cruciform shape gave four quadrants in each of which there could be attached one flue. The application of conventional engineering design practices would mean that this reinforced concrete structure would either have to be detailed to behave as a single ductile cantilever shear wall or be heavily penalised by having to have very large reserves of elastic strength provided. The provision of a ductile region, although it would be indeed difficult to detail, implies that damage might be expected whenever the earthquake induced loads approach those of the design level. In order to make use of the pro tection inherent in a ductile design of such a simple cantilever large deformations are required and the tower may be left with a permanent set. The original shape was therefore modified to allow the chimney to rock on * Chief Engineer, Computer Services, Beca Carter Hollings & Ferner Ltd., Consulting Engineers, Wellington. **Chief Engineer, Physics and Engineering Laboratory, NZ DSIR. BULLETIN OF THE NEW ZEALAND NATIONAL SOCIETY F OR 3 .0 DETAILS OF THE ROCKING MECHANISM: It will be seen from an inspection of a cross section at the base of the chimney (Figure 1) that there are two possible modes of rocking of the superstructure. Rocking about the axis A-A will require a lesser ground acceleration than that about axis B-B as the latter mode involves a greater lever-arm to the centre of gravity of the chimney. The base is, in its 'at rest 1 position, therefore supported at each of four points, one under each of the wall flanges at the corners of a square of side 7050 mm. The supports are lead sheets of 380 x 380 x 16 mm. To produce a positive transfer of horizontal shear between foundation and superstructure a 50 mm dia. stainless steel vertical dowel, wrapped in a soft proprietary tape, protrudes from the foundation through each lead pad into the chimney base. Thus, for the expected rocking displacements, the chimney structure is prevented from walking off the foundation. The size of the lead plates is such that stresses due to the chimney's dead load are not high enough to induce creep strains that would be significant in the life of the structure. EARTHQUAKE ENGINEERING, VOL. 16, NO. 2, JUNE 1983
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T H E S EIS M IC D E S IG N O F A N IN D U S T R IA L C H IM N E Y
WITH ROCKING BASE
R.D.Sharpe , R . I . Sk inner
A B S T R A C T:
A n i n d u s t r i a l r e i n f o r c e d c o n c r e t e c h i m n e y, o f c r u c i f o r m c r o s s - s e c t i o na n d t h i r t y - f i v e me t r e s t a l l , h a s b e e n d e s i g n e d a n d built w i t h t h e h i g h d e g r e eo f s e i s mi c p r o t e c t i o n a f f o r d e d b y a l l o w i n g t h e b a s e t o r o c k d u r i n g l a rg ee a r t h q u a k e s . A s a p r e c a u t i o n a p a i r o f p u r p o s e - ma d e y i e l d i n g s t e e l d a mp e r sh a v e b e e n i n s t a l l e d t o a s s i s t i n t h e c o n t r o l o f t h e mo s t v i o l e n t r e s p o n s et h a t m i g h t o c c u r.
T h i s p a p e r d i s c u s s e s t h e a d v a n t a g e s i n t h e a d o p t i o n o f t h i s r e v o l u t i o n a r yd e s i g n an d d e s c r i b e s t h e h a n d d y n a mi c a n a l y s i s c a r r i e d o u t d u r i n g t h e d e s i g np r o c e s s . S u b s e q u e n t t i me - h i s t o r y c o mp u t e r a n a l y s i s c a r r i e d o u t t o c o n f i r mt h e d e s i g n me t h o d i s r e p o r t e d o n .
1 . 0 I N T R O D U C T I O N :
D u r i n g t h e d e s i g n o f t h e n e w h a n g a ra n d ma i n t e n a n c e f a c i l i t i e s a t t h e A i rN e w Z e a l a n d E n g i n e e r i n g B a s e a t C h r i s t c h u r c ht h e r e w a s a r e q u i r e me n t f o r a n i n d u s t r i a lc h i mn e y t o b e p r o v i d e d . A s t h i s c h i mn e yw o u l d b e c l o s e t o , a n d a s s o c i a t e d w i t h ,t h e p r e s t i g i o u s h a n g a r a n d m a i n t e n a n c ec o mp l e x an a r c h i t e c t u r a l s o l u t i o n w a ss o u g h t w h i c h w o u l d b e mo r e a p p r o p r i a t et h a n t h e c o n v e n t i o n a l g u y e d c i r c u l a r f l u e .A c c e p t a b l e a r c h i t e c t u r a l a n d e n g i n e e r i n gc r i t e r i a w e r e me t w i t h t h e d e v e l o p m e n t o fa 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 o fc r u c i f o r m c r o s s - s e c t i o n t o w h i c h t h e f l u e sc o u l d b e a t t a c h e d . T h e c h o s e n f o r m
a l l o w s t h e e v e n t u a l d u p l i c a t i o n o f t h ec h i mn e y c a p a c i t y w h i l e i t s f r e e d o m t or o c k g i v e s i t t h e a b i l i t y t o w i t h s t a n dv e r y l a r g e e a r t h q u a k e s w i t h o u t r e q u i r i n gt h e s u b s t a n t i a l r e p a i r s t h a t mo s t r e i n f o r c e dc o n c r e t e s t r u c t u r e s w o u l d n e e d .
i t s f o u n d a t i o n s . O v e r t h e b o t t o m h a l ft h e t o w e r w a s s p l a y e d o u t t o p r o v i d e g r e a t e rr e s i s t a n c e t o o v e r - t u r n i n g a n d f l a n g e s w e r ep roposed fo r the ou te r edges o f the fou rl e g s o f t h e c r u c i f o r m t o i n c r e a s e t h e i rs t a b i l i t y. T h e f i n a l d i me n s i o n s o f t h ec h i mn e y a r e s h o w n i n F i g u r e 1 .
B y b e i n g a b l e t o r o c k , t h e c h i mn e ys u p e r s t r u c t u r e i s p r o t e c t e d f r o m a t t r a c t i n gl a rg e r s e i s mi c mo me n t s a n d s h e a r s t h a nt h o s e r e q u i r e d t o p r o d u c e l i f t - o f f .T h e s u p e r s t r u c t u r e ma y t h e n b e s a f e l yd e s i g n e d t o r e ma i n e l a s t i c u n d e r t h e s el i m i t e d l o a d s . E x t r a d a mp i n g i s p r o v i d e db y t w o s t e e l h y s t e r e t i c d a mp e r s , e a c h o f1 2 0 k N y i e l d f o r c e , p l a c e d a t o p p o s i t e
c o r n e r s o f t h e b a s e .A n y t o r s i o n a l s t i f f n e s s o f t h e c h i mn e y
i s g r e a t l y e n h a n c e d b y t h e a d d i t i o n o fh o r i z o n t a l d i a p h r a g ms o r r i n g b e a ms a tt h r e e i n t e r m e d i a t e l e v e l s .
2 . 0 D E V E L O P M E N T O F T H E D E S I G N :
T h e p r e l i m i n a r y a r c h i t e c t u r a l p r o p o s a l ss h o w e d a r e i n f o r c e d c o n c r e t e t o w e r o fu n i f o r m c r o s s - s e c t i o n w i t h h e i g h t a n di n t e g r a l w i t h i t s f o u n d a t i o n s . T h ec r u c i f o r m s h a p e g a v e f o u r q u a d r a n t s i ne a c h o f w h i c h t h e r e c o u l d b e a t t a c h e d o n ef l u e . T h e a p p l i c a t i o n o f c o n v e n t i o n a le n g i n e e r i n g d e s i g n p r a c t i c e s w o u l d m e a nt h a t t h i s 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 ew o u l d e i t h e r h a v e t o b e d e t a i l e d t o b e h a v ea s a s i n g l e d u c t i l e c a n t i l e v e r s h e a r w a l lo r b e h e a v i l y p e n a l i s e d b y h a v i n g t o h a v ev e r y l a rg e r e s e r v e s o f e l a s t i c s t r e n g t hp r o v i d e d . T h e p r o v i s i o n o f a d u c t i l er e g i o n , a l t h o u g h i t w o u l d b e i n d e e dd i f f i c u l t t o d e t a i l , i mp l i e s t h a t d a ma g em i g h t b e e x p e c t e d w h e n e v e r t h e e a r t h q u a k ei n d u c e d l o a d s a p p r o a c h t h o s e o f t h e d e s i g nl e v e l . I n o r d e r t o ma k e u s e o f t h e p r ot e c t i o n i n h e r e n t i n a d u c t i l e d e s i g n o fs u c h a s i mp l e c a n t i l e v e r l a rg e d e f o r ma t i o n sare requ i red and the tower may be l e f tw i t h a p e r m a n e n t s e t .
T h e o r i g i n a l s h a p e w a s t h e r e f o r emo d i f i e d t o a l l o w t h e c h i mn e y t o r o c k o n
* C h i e f E n g i n e e r, C o mp u t e r S e r v i c e s , B e c aC a r t e r H o l l i n g s & F e r n e r L t d . , C o n s u l t i n gE n g i n e e r s , We l l i n g t o n .
* * C h i e f E n g i n e e r , P h y s i c s a n d E n g i n e e r i n gL a b o r a t o r y, N Z D S I R .
B U L L E T I N O F T H E NE W Z E A L A N D N A T I O N A L S O C I E T Y F OR
3 .0 D E TA I L S O F T H E R O C K I N G M E C H A N I S M :
I t w i l l b e s e e n f r o m a n i n s p e c t i o n o fa c r o s s s e c t i o n a t t h e b a s e o f t h e c h i mn e y( F i g u r e 1) t h a t t h e r e a r e t w o p o s s i b l emo d e s o f r o c k i n g o f t h e s u p e r s t r u c t u r e .R o c k i n g a b o u t t h e a x i s A - A w i l l r e q u i r ea l e s s e r g r o u n d a c c e l e r a t i o n t h a n t h a ta b o u t a x i s B - B a s t h e l a t t e r mo d e i n v o l v e sa g r e a t e r l e v e r - a r m t o t h e c e n t r e o fg r a v i t y o f t h e c h i mn e y. T h e b a s e i s , i ni t s ' at re s t 1 p o s i t i o n , t h e r e f o r e s u p p o r t e da t e a c h o f f o u r p o i n t s , o n e u n d e r e a c ho f the wa l l f l ange s a t the co r ners o f as q u a r e o f s i d e 7 0 5 0 mm. T h e s u p p o r t sare lead sheets of 380 x 380 x 16 mm.To p r o d u c e a p o s i t i v e t r a n s f e r o f h o r i z o n t a ls h e a r b e t w e e n f o u n d a t i o n a n d s u p e r s t r u c t u r ea
5 0 mm d i a . s t a i n l e s s s t e e l v e r t i c a l d o w e l ,w r a p p e d i n a s o f t p r o p r i e t a r y t a p e , p r o t r u d e sf r o m t h e f o u n d a t i o n t h r o u g h e a c h l e a d p a di n t o t h e c h i mn e y b a s e . T h u s , f o r t h ee x p e c t e d r o c k i n g d i s p l a c e m e n t s , t h e c h i m n e ys t r u c t u r e i s p r e v e n t e d f r o m w a l k i n g o f f t h e
f o u n d a t i o n .
The s i ze o f the l ead p la tes i s sucht h a t s t r e s s e s d u e t o th e c h i mn e y ' s d e a dl o a d a r e n o t h i g h e n o u g h t o i n d u c e c r e e ps t r a i n s t h a t w o u l d b e s i g n i f i c a n t i n t h el i f e o f t h e s t r u c t u r e .
E A R T H Q U A K E E N G I N E E R I N G , V O L . 1 6 , N O . 2 , J U N E 1 9 8 3
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TYPICAL ELEVATION SECTIONBetween Ground
and Level 1
Figure 1 •• CHIMNEY SUPERSTRUCTURE LAYOUT
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4 H Y S T E R E T I C D A M P E R S : o f t h e f l e x i b l e c h i mn e y s u p e r s t r u c t u r e .
A p a r t f r o m t he i n h e r e n t ma t e r i a l d a mping there i s an energy lo s s to the sys temeach t ime one o f the rock ing ' fee t ' comesdown on i t s l ead pad and rad ia tes energyi n t o t h e h a l f - s p a c e b e l o w. E x t r a d a mp i n gi s a c h i e v e d b y p l a c i n g t w o h y s t e r e t i c d a mp e r s( F i g u r e 2 ) d i a me t r i c a l l y o p p o s i t e e a c ho t h e r a t t h e e n d s o f o n e w a l l o f t h e c r u c if o r m. Wi t h t h i s c h o i c e o f p o s i t i o n s atl e a s t o n e d a mp e r w i l l b e e f f e c t i v ew h a t e v e r t h e d i r e c t i o n o f r o c k i n g .D e v e l o p e d b y t h e N Z D S I R an d ma n u f a c t u r e df r o m mi l d s t e e l p l a t e t h e h y s t e r e t i cd a mp e r s a r e t a p e r e d c a n t i l e v e r s s o t h a ty i e l d i n g i s p r o mo t e d a l o n g t h e t a p e r w h e r et h e mo me n t c a p a c i t y v a r i e s a t t h e s a mer a t e a s t h e mo me n t . E a c h d a mp e r r e q u i r e sa for ce of 120 kN to y ie ld i t and s oa s s i s t s i n p r e v e n t i n g t h e c h i mn e y r o c k i n gp r e m a t u r e l y d u r i n g c o m p a r a t i v e l y c o m m o n
w i n d s t o r m s .T h e d e t a i l i n g o f t h e d a mp e r s i s s u c h
t h a t t h e h o l d i n g d o w n b o l t s a r e p o s i t i o n e da w a y f r o m th e c r i t i c a l y i e l d i n g a r e a d u r i n gt h e i r o p e r a t i o n .
5 . 0 W I N D L O A D I N G :
A s w e l l a s b e i n g c h e c k e d b yc o n v e n t i o n a l e m p i r i c a l m e t h o d s f o rp o s s i b l e s u s c e p t i b i l i t y t o v o r t e x s h e d d i n ga n d g a l l o p i n g , a s c a l e mo d e l w a s t r e a t e db y t h e N Z D S I R i n t h e i r w i n d t u n n e l w i t h o u ta n y u n u s u a l e f f e c t s b e i n g r e c o r d e d .
6 .0 M E T H O D O F A N A L Y S I S :
6 . 1 I n t e r n a l F o r c e s -
A l t h o u g h t r u e n o r ma l mo d e s d o n o te x i s t i n a n o n - l i n e a r s y s t e m, s u c h a s ar o c k i n g t o w e r, mo d a l c o n c e p t s ma y b e u s e d t oo b t a i n a p p r o x i ma t e s e i s mi c f o r c e s a n d d e f o r ma t i o n s f o r a r e l a t i v e l y r i g i d t o w e r . T h ec h i mn e y i s t h u s t r e a t e d a s r i g i d w h e nc a l c u l a t i n g f o r c e s and t h e d e f o r ma t i o n so f on ly th i s f i r s t ( rock ing ) mode a rei n c l u d e d w h e n c a l c u l a t i n g s i d e s w a y. I t i st h i s a p p r o a c h w h i c h w a s u s e d w h e n d e s i g n i n gt h i s c h i m n e y.
B y ma k i n g t h e r i g i d b o d y a s s u mp t i o n
the shape o f the f i r s t ( rock ing ) mode mayb e t a k e n a s t r i a n g u l a r w h e n c o mp u t i n g f o r c e s .T h e h i g h e r r o c k i n g m o d e s , w i t h t h e i r i n s i gn i f i c a n t d e f o r m a t i o n s h a v e n o n - r e s o n a n tr e s p o n s e s w h i c h a r e i n p h a s e . 1 1 g 1
h o r i z o n t a l l o a d s w e r e a p p l i e d t o c o mp u t e t h ef o r c e s i n t h e f i r s t t r i a n g u l a r mo d e( F i g u r e 3 ) . H a v i n g a s s u me d t h a t a l l t h er o c k i n g a c t i o n i s i n c l u d e d i n t h e f i r s t mo d ei t i s a s i mp l e ma t t e r t o e q u a t e t h e b a s em o m e n t c o r r e s p o n d i n g t o t he t r i a n g u l a rmo d e s h a p e t o t h e o v e r - t u r n i n g mo me n t o ft h e r i g i d c h i mn e y .
U n d e r t h e c o n d i t i o n w h e r e t h e r e i sb e i n g a c o n s t a n t g r o u n d a c c e l e r a t i o n o f1 g a p p l i e d t o t h e b a s e o f t h e c h i m n e y,i t f o l l o w s t h a t t h e s u m o f t h e c h i mn e y 1 sl a t e r a l f o r c e s a r i s i n g f r o m mo d e s h i g h e rt h a n t h e f i r s t ( t r i a n g u l a r ) mo d e mu s t b et h e d i f f e r e n c e o f t h e 1 g l a t e r a l l o a d s a n dt h e f i r s t mo d e l o a d s c o mp u t e d a s s u mi n g at r i a n g u l a r mo d e s h a p e . T h e r i g i db o d y r o c k i n g mo d e h a s t h e r e f o r e b e e nd i v o r c e d f r o m th e m o r e c o n v e n t i o n a l m o d e s
The fo rces tha t a re ac t ing on thec h i mn e y w h e n i t i s r o c k i n g a r e t h e r e f o r ema d e u p o f a f i x e d c o mp o n e n t d e r i v e df r o m t h e f i r s t mo d e r o c k i n g c r i t e r i o n a n da c o m p o n e n t d e r i v e d b y c o n v e n t i o n a lr e s p o n s e s p e c t r a c o n s i d e r a t i o n s a p p l i e dt o a l l s i g n i f i c a n t mo d e s h i g h e r t h a n t h ef i r s t*
6 .2 D i s p l a c e me n t s -
To f i n d t h e ma x i mu m l i k e l y d i s p l a c eme n t o f t h e c h i mn e y t h e a p p r o x i ma t e r i g i dr o c k i n g mo d e l o f F i g u r e 4 w a s u s e d . T h ef i r s t s t e p w a s t o o b t a i n t h e ma x i mu md i s p l a c e me n t o f t h e e q u i v a l e n t s i n g l emass m. I t i s shown by New mark e t a l(1) tha t an app ro x im ate va l ue fo r the
m a x i m u m d i s p l a c e m e n t o f a s i n g l e - m a s s n o nl inear sy s tem can be g iven by tha t o f an
e q u i v a l e n t l i n e a r r e s o n a t o r f o r w h i c ha p p r o p r i a t e v a l u e s o f e q u i v a l e n t p e r i o da n d e q u i v a l e n t d a mp i n g a r e d e f i n e d . T h i st e c h n i q u e h a s b e e n u s e d b y S k i n n e r e t a l ( 2 )a n d t h e a p p r o x i ma t e c o r r e c t n e s s o f t h eme t h o d h a s b e e n c o n f i r m e d b y d e t a i l e d a n a lys i s .
T h e l a t e r a l r e s i s t a n c e o f t h e mo d e lo f F i g u r e 4 i s o f a c o n s t a n t s t i f f n e s sf o r m u p t o t h e p o i n t a t w h i c h p i v o t i n ga b o u t o n e s u p p o r t c o mme n c e s a n d t h e n i s o fc o n s t a n t f o r c e f o r g r e a t e r l a t e r a l d i sp l a c e m e n t s . T h i s l o a d - d i s p l a c e m e n tr e l a t i o n s h i p i s t h e s a me a s t h a t f o r a ne l a s t o p l a s t i c s i n g l e - m a s s r e s o n a t o r.N e w ma r k (1 ) s h o w s i t s e f f e c t i v e p e r i o d t obe
_ 13 u 1 + ] j? (i)
w h e r e
and
= p e r i o d f o r d e f o r ma t i o n s l e s st h a n t h o s e w h i c h w i l l c a u s er o c k i n g .
= m a x i m u m l a t e r a l d i s p l a c e m e n to f s i n g l e ma s sl a t e r a l d i s p l a c e me n t a tc o mme n c e me n t o f r o c k i n g
Equ a t i on ( i) may be rea r ra nge d to g ive
4
3/g 1 2 T Ffo_
D (ii)
w h e r e
and
m a x i m u m l a t e r a l d i s p l a c e m e n t o fma s s
l a t e r a l f o r c e a t c o mme n c e me n to f r o c k i n gw e i g h t o f s i n g l e ma s s
T h e h y s t e r e t i c d a mp e r s f o r t h e s t e p p i n gc h i mn e y w e r e c h o s e n t o gi v e an e f f e c t i v ed a mp i n g o f f i v e p e r c e n t o f c r i t i c a l f o rt h e r o c k i n g ' mo d e ' o f t h e c h i mn e y a n dt h i s , t h e r e f o r e w a s t h e v a l u e a s s u me d f o rt h e e q u i v a l e n t l i n e a r r e s o n a t o r t or e p r e s e n t t h e mo d e l o f F i g u r e 4 .
6 .3 H y s t e r e t i c D a mp e r S i z e -T h e v e r t i c a l c o u l o mb d a mp e r f o r c e
r e q u i r e d t o g i v e a n e f f e c t i v e d a mp i n gf a c t o r o f f i v e p e r c e n t o f c r i t i c a l w a so b t a i n e d b y n o t i n g t h a t a r e c t a n g u l a rh y s t e r e s i s l o o p g i v e s a d a mp i n g f a c t o ro f 6 3 % (N e w ma r k ( 1 ) ) . T h i s ma y b ee x p r e s s e d a s
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101
PLAN VIEW OF DAMPER
Figure 2 : DETAILS OF HYSTERETIC DAMPERS
i \\ /
/ /
/ /
/
/
RIGID
^
IN
1st Mode Higher Modes
Figure IDEALISATION OF ROCKING MODES
Figure MODEL OF SINGLE MASS ROCKING CHIMNEY
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102AXIS LATERAL FORCE
FOR UPLIFT,f W0
MAX. MASSDISPLACEMENT (mm)
DAMPERUPLIFT (mm)
A-A(minor)
0.165 W ± 104 25
B-B(major)
0 233 W± 69 23
TABLE : MAXIMUM DISPLACEMEN TS AND UPLIFTS FOREL CENTRO SIZED EARTHQUAKES
a? 200
Q
150
100
50
0-0
T0 - 1.
Dampng Factor of 0 05
•10 20Resonator Perio d, ^ (s)
Figure 5 : AVERAGE DISPLACEMENT RESPONSE SPECTRUM
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103
Q 400
300
200
100
0
ResondDr Spine Curve
Lhecr for Lateral Force < f 0 W/P a
Zero Slope for Lateral Force > foW/P a
Z I Mass Displacement
— Mass Displacementdue to uplift
0 0 1 0 2
Normalised Uplift Factor
Figure 6 : MAXIMUM DISPLACEMENT vs UPLIFT FACTOR
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104
F +F
w h e r e
__A63 (i i i )
and
A
v e r t i c a l f o r c e a t d a mp e rt o l i f t w e i g h t o f c h i mn e y
h y s t e r e t i c d a mp e r f o r c e
e f f e c t i v e d a m p i n g p e r c e n t a g e
A p p r o x i m a t e l y t h e n
F_ (1 + —± 63 63 r V (iv)
A s s u mi n g t h a t t h e c h i mn e y a n d f o u n da t i o n h y s t e r e s i s p r o v i d e a n e f f e c t i v edamping o f 5% fo r mo t ion up to the onse to f s t e p p i n g t h e n e q u a t i o n ( iv ) r e d u c e st o
0. 0 86 F.
d i r e c t i o n s o f r o c k i n g i n Ta b l e 1 .
T h e l i k e l y d i s p l a c e me n t f o r a n yr e c o r d e d e a r t h q u a k e f o r w h i c h t h e r e i sa d i s p l a c e me n t s p e c t r u m a v a i l a b l e ma y b e
s i mi l a r l y c a l c u l a t e d f o r t h e r o c k i n gs y s t e m.
I t s h o u l d b e n o t e d t h a t l o a d si n c r e a s e o n l y s l o w l y w i t h i n c r e a s e dma x i mu m a c c e l e r a t i o n ( i . e . w i t h s c a l i n gf a c t o r P ^ ) w h i l e d i s p l a c e me n t a n d r o c k i n gi n c r e a s e s u b s t a n t i a l l y w i t h i n c r e a s e ds p e c t r a l v e l o c i t y.
F o r t h i s c h i mn e y , d i s p l a c e me n t s o ft w i c e t h o s e g i v e n b y Ta b l e 1 w e r e a d o p t e db e c a u s e o f t h e p o s s i b i l i t y t h a t t h eC h r i s t c h u r c h a r e a m a y e x p e r i e n c e r a t i o so f s p e c t r a l v e l o c i t y t o m a x i m u m a c c e l e ra t i o n s o me w h a t g r e a t e r t h a n t h o s e f o r t h eE l C e n t r o M a y 1 8 , 1 9 4 0 t y p e e a r t h q u a k e s .
V 7.0 T I M E - H I S T O RY A N A LY S I S :
The va lue o f F Q can then be found
s i mp l y b y c o n s i d e r a t i o n o f t h e o v e rt u r n i n g mo me n t t o c a l c u l a t e F ^ .
6 .4 S p e c t r a l M o d a l R e s p o n s e
S p e c t r a l c u r v e s w h i c h r e l a t e , f o ra s p e c i f i e d r e c o r d e d e a r t h q u a k e , t h em a x i m u m d i s p l a c e m e n t s t h a t w o u l d h a v eb e e n e x p e r i e n c e d b y s i n g l e ma s s r e s o n a t o r sw i t h v a r i o u s p e r i o d s a n d d a mp i n g f a c t o r sa r e a v a i l a b l e ( 3 ) . F o r t h e p u r p o s e s o f
t h i s d e s i g n t h e w e i g h t e d a n d a v e r a g e dres pon se spe c t r a (Tab le 1 (3 ) ) o f an u mb e r o f r e c o r d e d e a r t h q u a k e s w e r e u s e d .T h e w e i g h t i n g w a s i n t e n d e d t o g i v e e a c ho f t h e c o n s t i t u e n t e a r t h q u a k e s a p p r o xi ma t e l y t h e s a me 1 s i ze ' a s tha t wh i cho c c u r r e d a t E l C e n t r o , C a l i f o r n i a o nMa y 18 , 1940 (N-S c o m p o n e n t ) . F i g u r e 5s h o w s t h e r e s u l t i n g d i s p l a c e m e n t s p e c t r u mf o r 5 % o f c r i t i c a l d a mp i n g . A l s o p l o t t e do n t h i s f i g u r e a r e t h e r e l a t i o n s h i p so f equ a t i on ( i i) fo r To = 0 and T 0 = 1s e c o n d . T h e s e r e l a t i o n s h i p s a r e g i v e nf o r v a r i o u s v a l u e s o f f 0 whe re P i s
p -a
a l i n e a r f a c t o r b y w h i c h t h e a v e r a g e de a r t h q u a k e a n d i t s r e s u l t i n g d i s p l a c e m e n ts p e c t r u m mi g h t b e s c a l e d .
I n t e r c e p t s w i t h t h e s p e c t r a l c u r v eg i v e t h e r e l a t i o n s h i p s p l o t t e d i n F i g u r e 6 .
T h e f i r s t n a t u r a l p e r i o d o f t h e c h i mn e y(befo re rock ing ) was found to be T 0 = 0 434
s e c o n d s i n b o t h p r i n c i p a l d i r e c t i o n s .T h e g e o me t r y o f t h e c h i mn e y d i c t a t e s t h ed i m e n s i o n l e s s f o r c e s f 0 t o j u s t p r o d u c eroc k i ng to be 0 .165 and 0 .2 33 fo r them i n o r a n d m a j o r a x es r e s p e c t i v e l y.
A n e s t i m a t e o f th e l i k e l y d i s p l a c eme n t r e s p o n s e t o t h e a v e r a g e d e a r t h q u a k e s
may then be found f rom F ig u re 6 . Ther e s u l t s f o u n d a p p l y t o t h e e q u i v a l e n ts i n g l e - ma s s r o c k i n g mo d e l o f F i g u r e 4 .A s s u mi n g a g a i n t h e a p p r o x i ma t i o n o f t h ec h i m n e y 1 s r o c k i n g mo d e s h a p e b e i n gt r i a n g u l a r, i t i s t h en a s i mp l e m a t t e rt o p r o d u c e e q u i v a l e n t m a x i m u m l a t e r a ld i s p l a c e me n t s f o r t h e c h i mn e y t o p .T h e s e a r e p l o t t e d f o r b o t h p r i n c i p a l
A l t h o u g h i t w a s t h o u g h t t h a t t h ep r e v i o u s l y d e s c r i b e d h a n d a n a l y s i s w a ss u f f i c i e n t l y a c c u r a t e f o r t h e d e s i g n o ft h e r o c k i n g c h i mn e y, t h e r e l a t e r a r o s e a no p p o r t u n i t y t o p e r f o r m a c o mp u t e r a n a l y s i st o g i v e t h e t i me - h i s t o r y r e s p o n s e o f t h es t r u c t u r e s u b j e c t e d t o t h e N - S c o mp o n e n to f t i me - h i s t o r y r e s p o n s e o f t h e s t r u c t u r es u b j e c t e d t o t h e N - S c o mp o n e n t o f t h eE l C e n t r o , M a y 1 8 , 1 9 4 0 e a r t h q u a k e . F o rs i mp l i c i t y t h e c h i mn e y w a s mo d e l l e d a sa t w o - d i m e n s i o n a l s t r u c t u r e . T h e p a r ti c u l a r e l e v a t i o n c h o s e n w a s t h a t c o r r e s
p o n d i n g t o th e s u p e r s t r u c t u r e r o c k i n ga b o u t i t s ma j o r a x i s .
7 . 1 T h e C o mp u t e r M o d e l -
T h e i n e l a s t i c d y n a mi c a n a l y s i sp r o g r a m D R A I N - 2 D d e v e l o p e d b y K a n a a nand Powel l (4) was u se d to mod e l a s imp ler e p r e s e n t a t i o n o f t h e p r o p o s e d c h i m n e y.T h i s mo d e l , d e p i c t e d i n F i g u r e 7 , i s ma d eu p o f a n a s s e mb l a g e o f s i mp l e b e a m t y p em e m b e r s w h o s e i n d i v i d u a l s t i f f n e s s e s a r ec o mb i n e d t o g i v e a n o v e r a l l s t i f f n e s sf o r t h e s t r u c t u r e . M a s s i s a p p o r t i o n e dto the model in a lumped fo rm a t the jo in t sb e t w e e n m e m b e r s w h e r e a p p r o p r i a t e . T h i sma s s w i l l a t t r a c t b o t h h o r i z o n t a l a n dv e r t i c a l i n e r t i a l o a d i n g s . T h e a b i l i t yt o r o c k i s g i v e n t o t h e mo d e l b y mo d e l l i n gt h e s u p p o r t s w i t h a d e v i c e w h i c h w i l l r e s i s tc o mp r e s s i o n l o a d s o n l y. T h i s i s a c h i e v e db y i n c l u d i n g o n e o f t h e c o mp u t e r p r o g r a m ' st r u s s e l e me n t s a t e a c h s u p p o r t . T h e s ee l e me n t s c a n b e s p e c i f i e d t o a s t o r e s i s tt e n s i l e f o r c e s o n l y . A n i n s p e c t i o n o fF i g u r e 7 w i l l s h o w h o w t h i s f e a t u r e c a nb e e f f e c t i v e l y r e v e r s e d .
N o a t t e mp t w a s ma d e t o i n c l u d e t h er e s t o r i n g e f f e c t o f t h e p o t e n t i a l e n e rg ya s s o c i a t e d w i t h t h e s ma l l l i f t i n g o f t h ec e n t r e o f t o t a l ma s s o f t h e c h i mn e yd u r i n g r o c k i n g . L i k e w i s e , t h e d i s s i p a t i o no f e n e rg y o n t h e i mp a c t o f t h e f o u n d a t i o nb e a m o n t h e l e a d p a d s w a s i g n o r e d . B e c a u s et h e e x p e c t e d ma g n i t u d e o f t h e p e r i o d o fr o c k i n g w o u l d i mp l y r e a s o n a b l e s l o w i mp a c tv e l o c i t i e s , an e c o n o m i c a l i n t e g r a t i o n t i m e -s t e p o f 0 . 01 s w a s s e l e c t e d .
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105
Chimney
LunrpedMass
Support Pad
Truss member whichbuckles undercompression
(See below for hysteress)
ChimneySuperstructure
725.
DETAIL ADamper modelled ascantilever with cbi-linear hysteretic myielding n bending ^
K Stiffnessof lead pad
\f Displacement
Damper r CHARACTERISTICS OF TRUSS) SUPPORT
Figure 7 : ROCKING CHIMNEY MODEL FOR TIME- HISTORY ANALYSIS
Top of Chimney horizontal dispL)
North Foot'(vertical displ.
South'Foot' vertical displ.)
Earth qua keE l Centro, May 18.1940N-S Component
Figure 8 -DISPLACEMENT RESPONSE OF CHIMNEY SUPERSTRUCTURE
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