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Determination of foundation flexibilities of structures
Rainer, J. H.
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DETERMINATION O F FOUNDATION FLEXIBILITIES O F STRUCTURES by T
J . H . R a i n e r SYNOPSIS
A r e l a t i v e l y s i m p l e and d i r e c t method is p r e s e n t e d f o r d e t e r m i n i n g quantitatively t h e foundation f l e x i b i l i t i e s u n d e r s t r u c t u r e s . Data r e - q u i r e d a r e t h e m o d a l p r o p e r t i e s and m a s s e s . T h e p r o c e d u r e i s i l l u s
-
t r a t e d by findingthe r o t a t i o n a l and h o r i z o n t a l foundation s t i f f n e s s e s f o r a1 5 - s t o r e y c o n c r e t e building on clay, a 9 - s t o r e y c o n c r e t e building on d e n s e sand, and a b r i d g e p i e r founded on clay. On t h e b a s i s of t h e f i x e d - b a s e f r e q u e n c i e s obtained, a n a s s e s s m e n t is m a d e of t h e i m p o r t a n c e of the f0undatio.n flexibilities on t h e m o d a l c h a r act e r i s t i c s of t h e s e s t r u c t u r e s .
INTRODUCTION
A knowledge of t h e foundation flexibility u n d e r s t r u c t u r e s is i m p o r - tant f o r t h e followings r e a s o n s : 1. Flexibility i n a foundation changes t h e dynamic c h a r a c t e r i s t i c s of t h e s t r u c t u r e , and consequently the r e - s p o n s e t o dynamic d i s t u r b a n c e s s u c h a s wind, t r a f f i c and e a r t h q u a k e s . 2.' F l e x i b l e foundations m u s t be c o n s i d e r e d i n t h e i n t e r p r e t a t i o n of e x - p e r i m e n t a l m e a s u r e m e n t s of s t r u c t u r a l p r o p e r t i e s . 3 . Q u a n t i t a t i v e values of foundation s t i f f n e s s e s a r e r e q u i r e d in dynamic s t r u c t u r a l d e s i g n and a n a l y s i s .
Although m a n y t h e o r e t i c a l s t u d i e s on t h e e f f e c t s of flexible foundations under dynamic l o a d s h a v e been r e p o r t e d , t h e r e is a s c a r c i t y of q u a n t i t a - t i v e values of foundation s t i f f n e s s e s a s s o c i a t e d with e x i s t i n g s t r u c t u r e s .
In a previous study, a method w a s p r e s e n t e d f o r d e t e r m i n i n g q u a n t i t a - t i v e l y t h e foundation s t i f f n e s s of s t r u c t u r e s , which was b a s e d on m a t c h -
ing m e a s u r e d and c a l c u l a t e d s t r u c t u r a l modal p r o p e r t i e s . T h i s p a p e r p r e s e n t s a s i m p l e r and m o r e d i r e c t method that r e q u i r e s only m e a s u r e d modal p r o p e r t i e s and t h e m a s s e s of t h e s t r u c t u r e . THEORY T h e d i f f e r e n t i a l equation f o r l a t e r a l modal v i b r a t i o n x of a s t r u c t u r e having r o t a t i o n a l foundation s t i f f n e s s i s
+J
w2( ~ m . Y .h.+
I o 8 ) = K 8 1 1 1 cp (1)w h e r e wis the a n g u l a r f r e q u e n c y of t h e mode,
rn
a r e t h e m a s s e sa t s t o r e y i, h . is t h e height f r o m t h e b a s e , and 8'is t h e b a s e rotation.
1
T h e t e r m
I
i s t h e r o t a t i o n a l i n e r t i a of the m a s s e s and c a n u s u a l l y be 0neglected. F o r t h e l a t e r a l motion of a building foundation a s well a s v e r t i c a l v i b r a t i o n of a b r i d g e p i e r t h e f o r c e s a c t i n g on t h e soil-footing i n t e r f a c e a r e t h e r e a c t i o n f o r c e s R a s s o c i a t e d with a p a r t i c u l a r m o d a l vibration of t h e s t r u c t u r e . If
16
is t h e modal' a m p l i t u d e of t h e footing. IR e s e a r c h Officer, Division of Building R e s e a r c h , National R e s e a r c h Council of Canada, Ottawa, Canada, KIA OR6
t h e n the footing stiffness K c a n be found f r o m h
w h e r e R = Cm. x. f o r a building. F o r v i b r a t i o n s of buildings a s well a s 1 1
b r i d g e s , a l l t e r m s can be d e t e r m i n e d f r o m f u l l - s c a l e m e a s u r e m e n t s and t h e m a s s e s c a n be calculated easily.
A f t e r t h e foundation s t i f f n e s s e s a r e found, t h e uncoupled rocking p e r i o d T and t r a n s l a t i o n a l p e r i o d T c a n be computed f r o m
cp h
T h e fixed-based p e r i o d T i s then d e t e r m i n e d f r o m t h e well-known a p p r o x i m a t i o n f
w h e r e T i s t h e m e a s u r e d p e r i o d of t h e s t r u c t u r e with t h e f l e x i b l e base. RESULTS
T h e p r o c e d u r e i s used t o d e t e r m i n e the foundation s t i f f n e s s e s f o r t h r e e s t r u c t u r e s .
( 1 ) 15 - 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 wall building.
-
T h e building c o n s i s t s of 7-in. (17. 8 c m ) s l a b s , a n 8-in. (20. 3 c m ) roof s l a b , and one b a s e m e n t , with a 3 -ft (0. 92 c m ) t h i c k foundation m a t and 10-in. (25. 4 c m )p e r i m e t e r b a s e m e n t w a l l s . A typical h o r i z o n t a l s e c t i o n i s shown in Fig.
1. T h e s o i l p r o f i l e beneath t h e r a f t c o n s i s t s of 40 ft (12 m ) of clay, 10 f t ( 2 . 5 m ) of t i l l , then shale. F u r t h e r s o i l d e t a i l s a r e given i n Ref. ( 2 ) .
T h e ambient m o v e m e n t s of the s t r u c t u r e w e r e monitored and mode s h a p e s and f r e q u e n c i e s d e t e r m i n e d . Rocking motion was m e a s u r e d at opposite ends of t h e building, with a r e f e r e n c e i n s t r u m e n t at t h e 15th f l o o r level. T h e fundamental mode s h a p e in the E-W d i r e c t i o n i s shown in Fig. 2, and in t h e N-S direction in Fig. 3. Using the m e a s u r e d mode s h a p e s , with interpolated values f o r locations without e x p e r i m e n t a l a m p l i - t u d e s , t h e foundation s t i f f n e s s e s w e r e d e t e r m i n e d f r o m Eqs. 1 and 2, and a r e p r e s e n t e d in T a b l e I. A l s o shown a r e t h e uncoupled rocking and t r a n s - l a t i o n a l p e r i o d s and t h e fixed-based p e r i o d T computed f r o m Eqs. 3, 4
f
and 5. A c o m p a r i s o n between t h e m e a s u r e d and t h e fixed-base periods shows that t h e foundation flexibility a f f e c t s t h e fundamental p e r i o d of t h e s t r u c t u r e significantly, a s t h e rocking period i s a p p r e c i a b l y l a r g e r than t h e t r a n s l a t i o n a l p e r i o d .
( 2 ) 9 - s t o r e y reinforced c o n c r e t e building,
-
By way of c o n t r a s t , a build- ing founded on f i r m s o i l was investigated. F r o m t h e r e s u l t s of a f o r c e dv i b r a t i o n t e s t on t h e Millikan L i b r a r y , California Institute of T e c h n o 1 0 ~ ~ ( ~ ) foundation s t i f f n e s s e s and uncoupled p e r i o d s a r e obtained f o r t h e funda-
m e n t a l modes f r o m E q s . 1 t o 5, and a r e p r e s e n t e d i n T a b l e I. The m o d a l amplitudes of f l o o r s not m e a s u r e d w e r e e s t i m a t e d by interpolation. A
c o m p a r i s o n b e t w e e n t h e m e a s u r e d a n d f i x e d b a s e p e r i o d s c o n f i r m s q u a n t i t a t i v e l y t h a t t h e i n f l u e n c e of t h e b a s e f l e x i b i l i t i e s on t h e s t r u c t u r a l p e r i o d s i s n e g l i g i b l e . ( 3 ) S t e e l t r u s s r a i l w a y b r i d g e . - T h e b r i d g e c o n s i s t s of f o u r s i m p l y s u p p o r t e d , p a r a l l e l - c h o r d , t h r o u g h - t r u s s s p a n s c a r r y i n g a s i n g l e r a i l w a y t r a c k . T h e s o l i d m a s o n r y p i e r s w e r e c o n s t r u c t e d i n t h e 1 8 8 0 ' s on a c l a y r i v e r b e d ; t h e o r i g i n a l s p a n s w e r e r e p l a c e d i n t h e 1 9 2 0 ' s by t h e p r e s e n t
9
- p a n e l s t e e l t r u s s e s , 204 f t9
in. (63 m ) long. T h e v e r t i c a l m o d e s of v i b r a t i o n of o n e end s p a n a n d p i e r a n d t h e a d j a c e n t s p a n w e r e o b t a i n e d f r o m w i n d - i n d u c e d v i b r a t i o n s . T h e m a s s e s of t h e t r u s s a n d p i e r w e r e c o m p u t e d f r o m c o n s t r u c t i o n d r a w i n g s a n d t h e t r i b u t a r y w e i g h t s l u m p e d a t e a c h p a n e l p o i n t . With t h e m e a s u r e d m o d a l a m p l i t u d e s f o r t h e f i r s t v e r t i c a l m o d e , t h e f o u n d a t i o n f l e x i b i l i t y u n d e r t h e p i e r w a s d e t e r m i n e d t o b e 1. C, x 1 C,'7 i n . / l b f r o m E q . 2. F r o m a n e i g e n v a l u e f o r m u l a t i o n , t h e t h e o r e t i c a l m o d e s h a p e s a n d n a t u r a l f r e q u e n c i e s w e r e c o m p u t e d u s i n g t h e a b o v e f o u n d a t i o n f l e x i b i l i t y . A s shown in T a b l e 11, a n o t h e r m o d e h a s b e e n i n s e r t e d b e t w e e n t h e f i r s t a n d s e c o n d f i x e d - b a s e d m o d e s . DISCUSSION In t h i s m e t h o d , t h e m a s s e s c a n u s u a l l y b e c o m p u t e d w i t h i n c l o s e t o l e r a n c e s ; t h u s t h e a c c u r a c y of t h e a n s w e r s d e p e n d s p r i m a r i l y on t h e a c c u r a c y of t h e m e a s u r e m e n t s of m o d a l a m p l i t u d e a n d t h e s q u a r e of t h e f r e q u e n c y . T h e s e c a n b e o b t a i n e d t o within l e s s t h a n 10% v a r i a t i o n , a n d c o n s e q u e n t l y t h e v a l u e s of f o u n d a t i o n s t i f f n e s s e s a r e of t h e s a m e o r d e r of a c c u r a c y . E q u a l l y i m p o r t a n t a s t h e m e a s u r e m e n t a c c u r a c y is t h e p l a c e - m e n t of t h e t r a n s d u c e r s t o m e a s u r e f o u n d a t i o n m o v e m e n t . The p o s s i b i l i t y of m e a s u r i n g p u r e l y l o c a l d e f o r m a t i o n s h a s t o b e g u a r d e d a g a i n s t . T h e a n s w e r s o b t a i n e d a r e a p p l i c a b l e to t h e l e v e l of d e f o r m a t i o n s u n d e r g o n e by t h e s t r u c t u r e a n d f o u n d a t i o n d u r i n g t h e t e s t . F o r a m b i e n t v i b r a t i o n s t h e s t r a i n s a r e v e r y s m a l l . T h e m o v e m e n t s a t s t r a i n l e v e l s a s s o c i a t e d with e a r t h q u a k e s could b e o b t a i n e d by p l a c i n g t w o s t r o n g - m o t i o n s e i s m o g r a p h s a t o p p o s i t e s i d e s in t h e b a s e m e n t t o m e a s u r e r o c k - ing, a n d a t l e a s t o n e i n s t ~ u m e n t n e a r t h e top s t o r e y f o r t r a n s l a t i o n . T h e r e l e v a n t d e f o r m a t i o n s at i n t e r m e d i a t e l e v e l s c o u l d b e o b t a i n e d f r o m e x t r a - p o l a t i o n of low l e v e l v i b r a t i o n t e s t s o r t h e o r e t i c a l c a l c u l a t i o n s . R E F E R E N C E S 1. Ward, H. S. a n d R a i n e r , J. H . , E x p e r i m e n t a l D e t e r m i n a t i o n of S t r u c-
t u r e a n d F o u n d a t i o n P a r a m e t e r s U s i n g Wind-Induced V i b r a t i o n s . P r o c.
I n s t . C i v i l E n g r s . , -- 5 3 , S e p t . 1972, p. 3 0 5 - 3 2 2 . 2 . E d e n , W . J . , M c R o s t i e , G . C . a n d Hall, J . S . , M e a s u r e d C o n t a c t P r e s - s u r e s below R a f t S u p p o r t i n g a Stiff B u i l d i n g . C a n . G e o t e c h . J . , 10,-
2, M a y 1973, p. 1 8 0 - 1 9 2 . '3. J e n n i n g s , P. C. a n d Kuroiwa, J . H . , V i b r a t i o n a n d S o i l S t r u c t u r e I n t e r a c t i o n T e s t s of a N i n e - S t o r y R e i n f o r c e d C o n c r e t e B u i l d i n g . Bull. S e i s n l . Soc. Am.,-
58, 3 , J u n e 1968, p. 891-916.ACKNOWLEDGEMENTS
The 15-storey building is owned by t h e Ontario Housing Corporation and operated by' t h e Ottawa Housing Authority. T h e i r cooperation, a s well a s that of J, S. Hall and Associates, S t r u c t u r a l Consultants, Ottawa, is gratefully acknowledged. T h e bridge vibrations w e r e m a d e possible through t h e a s s i s t a n c e of t h e Engineering Department, Canadian P a c i f i c
Limited. This p a p e r is a contribution f r o m the Division of Building R e s e a r c h , National R e s e a r c h Council of Canada, and is published with t h e approval of t h e Director of the Division.
N O l T H F A C E
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10 5 0 10
b V e r l l c a l M e a r u r e m e n l
P o l n t s l n Basemenl Scale In Feel
1- Hprizonfal M e a r u r e m e n l 3 2 1 0 3 P o l n l r o n F l m r s
