Dynamic properties of Lions' Gate suspension bridge

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Dynamic properties of Lions' Gate suspension bridge

Rainer, J. H.; Van Selst, A.

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N21d National Research Conseil national

lo. 700

1

Council Canada de recherches Canada

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DYNAMIC PROPERTIES OF

LIONS' GATE SUSPENSION BRIDGE

by J.

H.

Rainer and A. Van Selst

Reprinted, with permission, from

Proceedings, ASCEIEMD Specialty Conference

Dynamic Response of Structures : Instrumentation, Testing Methods. and System Identification

held at the University of California, Los Angeles 30 and 31 March 1976

p. 243 252

s f - -

f s : : , R - . L i - 2 3

DBR Paper No. 700

Division of Building Research

DYNAMIC PROPERTIES OF LIONS1 GATE SUSPENSION BRIDGE by

I

J . H . R a i n e r and A . Van S e l s t I I

INTRODUCTION

The L i o n s ' Gate Suspension Bridge c r o s s e s t h e F i r s t Narrows o f Burrard I n l e t a t Vancouver, B r i t i s h Columbia. The b r i d g e , which opened i n 1938, c a r r i e s t h r e e l a n e s o f v e h i c u l a r t r a f f i c and two s i d e w a l k s . Because o f t h e need f o r e x t e n s i v e r e p a i r s t o t h e b r i d g e deck and r e q u i r e m e n t s f o r w i d e r t r a f f i c l a n e s , a complete r e b u i l d i n g o f t h e deck s t r u c t u r e i s p l a n n e d . This w i l l r e s u l t i n t h e s i d e w a l k s b e i n g moved from i n s i d e t h e s t i f f e n i n g t r u s s e s t o t h e o u t s i d e . A new o r t h o t r o p i c r o a d s u r f a c e i s t o r e p l a c e t h e p r e s e n t c o n c r e t e and s t e e l g r i d s u p p o r t e d on s t e e l beams and g i r d e r s . S i n c e changes i n g e o m e t r i c and s t r u c t u r a l p r o p e r t i e s c o u l d i n f l u e n c e t h e b e h a v i o r o f t h e m o d i f i e d b r i d g e under wind l o a d i n g , an aerodynamic i n v e s t i g a - t i o n i s b e i n g u n d e r t a k e n which i n c l u d e s wind t u n n e l s e c t i o n t e s t s and a f u l l a e r o - e l a s t i c model. To o b t a i n some guidance i n e s t a b l i s h i n g t h e dynamic p a r a m e t e r s f o r t h e model t e s t s and t h e d e s i g n c a l c u l a t i o n s , measurements on t h e e x i s t i n g s t r u c : t u r e were c a r r i e d o u t t o d e t e r m i n e i t s dynamic p r o p e r t i e s . This p a p e r d e s c r i b e s t h e measurement program and t h e r e s u l t s o b t a i n e d , a s well a s comparisons between t h e o b s e r v e d v a l u e s o f n a t u r a l f r e q u e n c i e s and t h o s e from two d i f f e r e n t methods of c a l c u l a t i - o n .

DESCRIPTION OF THE BRIDGE

I

-

I

The b r i d g e c o n s i s t s o f a main span o f 1550 f t (473 m) and two s i d e s p a n s , I _{each o f 614 f t (187 m)}

.

The tower h e i g h t i s 361.5 f t (110 m) from t o p o f c o n c r e t e I

i f o o t i n g t o t h e s a d d l e p o i n t s , w i t h t h e road deck p a s s i n g a t m i d - h e i g h t . An e l e v a - t i o n i s p r e s e n t e d i n F i g . 1, and a t y p i c a l c r o s s - s e c t i o n w i t h some s t r u c t u r a l p r o p e r t i e s i n F i g . 2 .

I

i D i v i s i o n o f B u i l d i n g Research, N a t i o n a l Research Council of Canada, Ottawa, O n t a r i o , . . Canada. - .

I I Buckland 6 T a y l o r Ltd. , C o n s u l t i n g E n g i n e e r s , ~ a n c o u v e r , B r i t i s h Columbia, Canada.

I

A t t h e b r i d g e c e n t e r and a t t h e n o r t h and s o u t h ends o f t h e s u s p e n s i o n b r i d g e , t r a c t i o n rods cbnnect t h e c a b l e t o t h e t o p chord o f t h e s t i f f e n i n g t r u s s . However, t h e s e t r a c t i s n rods were found t o be somewhat l o o s e . The two s t i f f e n i n g t r u s s e s a r e j o i n e d a t t h e lower chords by f l o o r beams t h a t s u p p o r t t h e r o a d deck, and by h o r i z o n t a l c r o s s b r a c i n g t h a t d e c r e a s e s i n s t i f f n e s s toward t h e c e n t e r o f t h e

b r i d g e . The t r u s s e s , t e g e t h e r with t h e c r o s s b r a c i n g s and t h e decking, t h u s e f f e c - t i v e l y form an open channel s e c t i o n . The t r u s s e s r e s t on f i x e d b e a r i n g s a t t h e N and S s u p p o r t s o f t h e b r i d g e and have s l i d i n g b e a r i n g s a t t h e t o w e r s . The g i r d e r has a v e r t i c a l curve w i t h r i s e o f 22.3 f t ( 6 . 8 m) from t h e tower t o t h e b r i d g e c e n t e r l i n e .

MEASUREMENT PROGRAM

The measurements were performed i n two main s t a g e s : Phase I was c a r r i e d o u t d u r i n g t h e w i n t e r o f 1974-75, and Phase 11, i n September 1975. The s u b s e q u e n t d e s c r i p t i o n and r e s u l t s o b t a i n e d r e f e r t o Phase 11, e x c e p t where o t h e r w i s e n o t e d . I n s t r u m e n t a t i o n

The motions o f t h e b r i d g e were monitored w i t h b a t t e r y - p o w e r e d s e r v o - d r i v e a c c e l e r o m e t e r s o f 5 V/g s e n s i t i v i t y . These were mounted on b r a s s b a s e s and p l a c e d on t h e t o p f l a n g e o f t h e c r o s s beam c l o s e t o t h e i n s i d e o f t h e s t i f f e n i n g t r u s s a t t h e numbered l o c a t i o n s shown i n F i g . 1. Cables c a r r i e d t h e s i g n a l s t o t h e r e c o r d - i n g s t a t i o n i n t h e o b s e r v a t i o n c a b i n a t t h e b r i d g e c e n t e r . D-C compensation was a p p l i e d , > t h e s i g n a l s p a s s e d through 1 Hz low p a s s f i l t e r s and 20 dB a m p l i f i e r s and were r e c o r d e d on a 7-channel FM t a p e r e c o r d e r . The t r a n s d u c e r s , s i g n a l c o n d i t i o n - i n g and r e c o r d i n g channels were c a l i b r a t e d i n t h e l a b o r a t o r y and a s t a t i c i n v e r s i o n check o f t h e t r a n s d u c e r s was performed a t t h e r e c o r d i n g s i t e .

S t a t i o n 1N was used a s a r e f e r e n c e ; t h e remaining s t a t i o n s were covered by s u c c e s s i v e l y moving t h e t r a n s d u c e r s t o t h o s e s t a t i o n s . A s p e c i a l r u n was made with t r a n s d u c e r s p l a c e d a t S t a t i o n s 2 N and 2s.

Ambient V i b r a t i o n s .

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Ambient v i b r a t i o n s d u r i n g Phase I 1 c o n s i s t e d o f v e h i c u l a r t r a f f i c o n l y , mainly p a s s e n g e r c a r s and t r a n s i t b u s e s . Wind s p e e d s ranged from zero t o about 5 mph (8 km/h). During Phase I , measurements were a l s o made under v a r i o u s wind c o n d i t i o n s d u r i n g e a r l y morning h o u r s when t r a f f i c was minimal o r a b s e n t . One l i m i t e d s e t o f measurements was o b t a i n e d a t winds g u s t i n g t o 50 mph

(80 km/h). For each s e t u p t h e motions were monitored f o r a p p r o x i m a t e l y 45 min. Vehicle Impacts.

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A s e r i e s of s i m u l a t e d impacts were a p p l i e d t o t h e b r i d g e from a s i n g l e r e a r a x l e sand t r u c k and a small p a s s e n g e r c a r . _{The t r u c k was d r i v e n i n} t h e o u t s i d e l a n e a t a b o u t 10 mph, swerved normal t o t h e b r i d g e and b r o u g h t t o a sudden b r a k i n g s t o p . T h i s produced t o r s i o n a l and l a t e r a l r e s p o n s e s o f t h e b r i d g e . Another s i m u l a t e d impact t e s t f o r t o r s i o n a l and l a t e r a l v i b r a t i o n s c o n s i s t e d o f a small c a r d r i v e n normal t o t h e roadway and b r o u g h t t o a sudden b r a k i n g s t o p . V e r t i c a l impact r e s p o n s e s were induced by s l o w l y d r i v i n g t h e r e a r wheels o f t h e t r u c k o f f a 6 - i n . (15-cnl) ram?. However, t h i s proved t o b e s u c c e s s f u l i n o n l y one of t h r e e a t t e m p t s . Driving t h e t r u c k from t h e main s p a n o n t o t h e s i d e s p a n a t approximately- 40 mph (64 km/h) produced some s a t i s f a c t o r y v e r t i c a l motions.

A n a l y s i s o f .Data

Ambient U b r a t i o n s . - The ambient v i b r a t i o n d a t a were a n a l y z e d on a 5 0 0 - l i n e r e a l - t i m e s p e c t r u m a n a l y z e r u s i n g t a p e s p e e d s o f 8 , 1 0 , and 16 times t h e r e c o r d i n g speed and w i t h 2 and 4 s p e c t r u m a v e r a g e s . Phase r e l a t i o n s h i p s between two s t a t i o n s were o b t a i n e d by a d d i t i o n and s u b t r a c t i o n o f t h e r e s p e c t i v e s i g n a l s . The a m p l i t u d e and phase r e l a t i o n s h i p s were a l s o confirmed by c r o s s - s p e c t r a l d e n s i t y c a l c u l a t i o n s u s i n g 512 p o i n t s and f o u r s p e c t r u m a v e r a g e s .

Impact Response.

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The i m p a c t r e s p o n s e s i g n a l s were d i g i t i z e d and a n a l y z e d on a d i g i t a l computer u s i n g FFT r o u t i n e s . The t i m e decay c u r v e f o r v a r i o u s modes o f v i b r a t i o n was o b t a i n e d by i n v e r t i n g i n t o t h e time &main t h e f i l t e r e d s p e c t r u m f o r a p a r t i c u l a r modal f r e q u e n c y band. From t h i s decay, modal damping was computed u s i n g t h e l o g decrement r e l a t i o n s h i p .

CALCULATIONS OF MODAL PROPERTIES

Two methods were u s e d t o c a l c u l a t e t h e modal p r o p e r t i e s o f t h e b r i d g e : a continuum model, where t h e s o l u t i o n s t o t h e d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e v i b r a t i o n problem a r e e v a l u a t e d , and a lumped mass, l i n e a r s t i f f n e s s model, f o r which eigenmodes were found.

Continuum model.

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The t h e o r y f o r v i b r a t i o n o f s u s p e n s i o n b r i d g e s a s

p r e s e n t e d by S e l b e r g ( 3 ) was u t i l i z e d i n c a l c u l a t i n g t h e mode s h a p e s and f r e q u e n - c i e s f o r t h e e x i s t i n g b r i d g e . The f o l l o w i n g assumptions a r e made t o s i m p l i f y t h e t r e a t m e n t o f tlie m a t h e m a t i c a l t h e o r y : a ) a x i a l d e f o r m a t i o n s o f towers and

g i r d e r a r e n e g l e c t e d ; b) t h e i n d i v i d u a l h a n g e r s a r e r e p l a c e d by an e q u i v a l e n t membrane h a v i n g no s h e a r r e s i s t a n c e ; c ) dead l o a d i s u n i f o r m l y d i s t r i b u t e d a l o n g t h e s p a n , i m p l y i n g a p a r a b o l i c c a b l e p r o f i l e ; d) g i r d e r h a s c o n s t a n t s t i f f n e s s w i t h i n each s p a n ; e ) c u r v a t u r e o f g i r d e r i s n e g l i g i b l e compared t o c a b l e s a g ; f ) s h e a r d e f o r m a t i o n i n g i r d e r i s n e g l i g i b l e ; g) d e f l e c t i o n s a r e s m a l l , i . e . , l i n e a r d i s p l a c e m e n t t h e o r y

i s

used. The s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n s i s a r a p i d l y c o n v e r g i n g i n f i n i t e s e r i e s . A g e n e r a l computer program was w r i t t e n t o s o l v e t h e s e r i e s by a t r i a l and e r r o r method u t i l i z i n g t h e f i r s t 5 terms f o r t h e s i d e s p a n s and 7 terms f o r t h e main s p a n . The t o r s i o n a l modal p r o p e r t i e s were o b t a i n e d a n a l o g o u s l y t o t h e v e r t i c a l ones by r e p l a c i n g t h e v e r t i c a l s t i f f n e s s and mass w i t h t h e t o r s i o n a l s t i f f n e s s and mass moment o f i n e r t i a o f t h e g i r d e r . The f u l l t o r s i o n a l s t i f f n e s s o f t h e roadway i s assumed t o b e a c t i v a t e d and t h e t r a c - t i o n r o d s a r e assumed t o b e u n s t r e s s e d .

The above method i s s i m i l a r t o t h e t h e o r y p r e s e n t e d e a r l i e r by B l e i c h e t a l . (1) b u t i s more r e f i n e d .

Lumped mass, l i n e a r s t i f f n e s s model.

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The s t i f f n e s s m a t r i x f o r t h e d i s c r e t e model o f some 750 members and 400 j o i n t s h a v i n g o v e r 1,000 d e g r e e s o f freedom was assembled w i t h a space-frame a n a l y s i s program. Symmetry p r o p e r t i e s a l o n g t h e l e n g t h and t h e width o f t h e b r i d g e were u t i l i z e d s o t h a t o n l y one q u a r t e r o f t h e s t r u c t u r e needed t o be encompassed. The e i g e n v a l u e problem w i t h t h e d i a g o n a l mass m a t r i x was s o l v e d by a sweeping t e c h n i q u e . _ - - - -

- . -

The method o f a n a l y s i s o f t h e problem i m p l i e s l i n e a r d i s p l a c e m e n t t h e o r y . 'Ihe t o r s i o n a l r i g i d i t y u s e d f o r t h e b r i d g e g i r d e r i n c l u d e s f u l l p a r t i c i p a t i o n o f t h e roadway and s t r i n g e r s and t h e assumption t h a t t h e e x p a n s i o n j o i n t s have s e i z e d .

G i r d e r b e a r i n g s a t t h e towers a r e t a k e n a s f i x e d and t h e t r a c t i o n r o d s a r e assumed t o b e i n e f f e c t i v e . These assumptions were found t o b e s t f i t t h e e x p e r i m e n t a l d a t a . RESULTS

*

The r e s u l t s o f t h e measurements and c a l c u l a t i o n s a r e p r e s e n t e d i n Tables 1, 2 and 3 f o r each i d e n t i f i e d mode. The measuring p o i n t s on t h e i n t e r p o l a t e d modes s h a p e s a r e shown by c i r c l e s f o r t h e v e r t i c a l and t o r s i o n a l d i s p l a c e m e n t s , and

t r i -

a n g l e s f o r h o r i z o n t a l g i r d e r d i s p l a c e m e n t . Damping v a l u e s a r e given where p o s s i b l e , as w e l l a s t h e computed f r e q u e n c i e s from t h e two mathematical models a l r e a d y des- c r i b e d . The damping v a l u e s from ambient v i b r a t i o n s were c a l c u l a t e d u s i n g t h e

method o f half-power hand w i d t h . Dampine v a l u e s c a l c u l a t e d from t h e f i l t e r e d decay curves vary s l i g h t l y o v e r t h e l e n g t h o f t h e decaying v i b r a t i o n r e c o r d and conse- q u e n t l y t h e v a l u e s n e a r t h e b e g i n n i n g o f t h e impact and t h o s e a f t e r a b o u t 15 c y c l e s a r e given. A c o r r e c t i o n was a p p l i e d t o t h e a m p l i t u d e s . o f t h e decaying s i g n a l o f

t h e Phase I measurements by s u b t r a c t i n g t h e a m p l i t u d e s o f t h e ambient s i g n a l due t o wind e x c i t a t i o n .

C a t e e o r i z a t i o n o f Modes

The normal modes i d e n t i f i e d i n t h e measurement program have been c a t e g o r i z e d i n t o v e r t i c a l , h o r i z o n t a l and t o r s i o n a l modes. However, t h e u s u a l c l e a r d i s t i n c - t i o n between h o r i z o n t a l and t o r s i o n a l modes becomes o b s c u r e d because both t o r s i o n and h o r i z o n t a l motion i s p r e s e n t i n most o f t h e s e modes. Consequently, o n l y t h e lowest mode, which i s c l e a r l y i d e n t i f i a b l e a s a h o r i z o n t a l mode, was d e s i g n a t e d a s a h o r i z o n t a l mode, and a l l o t h e r modes w i t h t o r s i o n a l components were d e s i g n a t e d a s t o r s i o n a l modes.

V e r t i c a l Modes.

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A sample s e t o f F o u r i e r s p e c t r a o f v e r t i c a l s i g n a l s monitored a t S t a t i o n 1N a r e shown i n Fig. 3 . The modes a s s o c i a t e d w i t h t h e v a r i o u s s p e c t r u m peaks a r e i n d i c a t e d n e a r t h e t o p o f t h e f i g u r e .

The v e r t i c a l mode shapes, t o g e t h e r w i t h r e l e v a n t damping v a l u e s ' a n d t h e c a l - c u l a t e d f r e q u e n c i e s , a r e p r e s e n t e d i n Table l . The modes a r e p r e s e n t e d sequen- t i a l l y a c c o r d i n g t o i n c r e a s i n g f r e q u e n c y . I t i s b e l i e v e d t h a t i n t h e v e r t i c a l d i r e c t i o n t h e modes i d e n t i f i e d r e p r e s e n t a complete s e t o f t h e major n a t u r a l modes o f v i b r a t i o n f o r t h e s u s p e n s i o n b r i d g e .

H o r i z o n t a l Modes.

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Because t h e h o r i z o n t a l motions a r e c o u p l e d w i t h t h e t o r s i o n o f t h e deck and towers and v i c e v e r s a , a p u r e h o r i z o n t a l mode i s u n l i k e l y t o o c c u r . I n f a c t , o n l y t h e lowest h o r i z o n t a l mode a t 0.12 Hz p r e s e n t e d i n T a b l e 3 was w i t h - o u t measurable r o t a t i o n s under ambient v i b r a t i o n s . Only d u r i n g t h e s e v e r e s t impact t e s t s could r o t a t i o n a l components b e d e t e c t e d f o r t h i s mode.

T o r s i o n a l Modes.

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The h o r i z o n t a l components o f t h e modes shown by d a s h - d o t t e d l i n e s i n T a b l e s 2 and 3 z i v e t h e modal amplitude on t h e t o p f l a n g e o f t h e f l o o r beam, i . e . , j u s t below t h e roadway l e v e l . The p l o t t e d t o r s i o n a l component shown by s o l i d l i n e s r e p r e s e n t s t h e sum o f t h e v e r t i c a l motions a t t h e i n s i d e o f t h e s t i f f e n i n g t r u s s .

S i n c e t h e a c c e l e r o m e t e r s i n t h e h o r i z o n t a l d i r e c t i o n a r e s e n s i t i v e t o a n g u l a r

movements, a c o r r e c t i o n has t o b e a p p l i e d t o t h e measured h o r i z o n t a l component o f _f

t.

Table I : Dynamic P r o p e r t i e s o f V e r t i c a l Modes L Item No. 1 2 3 4 5 6 7 8 Measured Mode Shape Measured Frequency, Hz 0.200 0.225 0.30 0.358 0.420 0.590 0.760 0.845

-

-0 2

$

Lumped 0.199 0.214 0.298 0.343 0.422 2 % ~Parameter 2 u = Continuum Mode 1 0.185 0.214 0.273 0.345 0.408 F o u r i e r # 0-4 1 . 7 1 . 6 1 . 7 1 . 0 0.7 0 . 6 0.6 . ~ m p u l s e 0.9+,0.6' 0 . 9 * , 0 . S f n e : u u Decay 1.1'+,0.9**1.05**,0.95** + Uncorrected * * C o r r e c t e d f o r ambient v i b r a t i o n s

Table 2: Dynamic P r o p e r t i e s o f a n t i - s y m m e t r i c T o r s i o n a l Modes

-

Item No. 1 2 3 4 5 6 Measured Mode Shape

-

D i f f e r e n c e o f V e r t i c a l Motion ( E a s t + up) -.+- H o r i z o n t a l (+ t o West) _i I Measured Frequency, Hz 0.310 0.32 0.495 0.515 0.600 0.745 0.765 (0.33) 0.507 '0

-

4) _X .y 0 Lumped _{0.317 0.463 0.568} m F -4 4) Parameter 3 1 N ' w f Continuum 3 4) m &4 0 ., 31 H* C' u. _{Model 0.411} Damping R a t i o from F o u r i e r S p e c t r a 1.25 0.79 % c r i t i c a l C a l c u l a t e d h o r i z o n t a l mode

t h e mode s h a p e . T h i s c o r r e c t i o n depcnds on t h e modal frequency and t h e a m p l i t u d e o f t h e a n g u l a r motion a t t h e measuring p o i n t . No c o r r e c t i o n c o u l d b e a p p l i e d t o t h e modal a m p l i t u d e s a t t h e tower t o p s f o r t h e v e r t i c a l and t h e t o r s i o n a l modes. Consequently, t h e p l o t t e d tower a m p l i t u d e s a r e t o o l a r g e .

A sample spectrum o f t h e r o t a t i o n a l component f o r S t a t i o n 1N i s shown i n

F i g . 4 . The a n t i - s y m e t r i c a n d symmetric t o r s i o n a l modes t h a t c o u l d b e i d e n t i f i e d a r e p r e s e n t e d i n Tables 2 and 3 , r e s p e c t i v e l y , t o g e t h e r w i t h damping v a l u e s a n d t h e o r e t i c a l f r e q u e n c i e s where a v a i l a b l e . A s can b e s e e n , e v e r y t o r s i o n a l mode

= Item No. 1 2 3 8 Measured Mode Shape

-

D i f f e r e n c e o f V e r t i c a l Motion ( E a s t + up) -.- H o r i z o n t a l (+ t o West) Measured 0.38 Frequency, Hz 0.12 0.36 (0.37, 0.39) 0.44 0.465 0.475 0.525 0.535 -0

-

a A _Lumped w u Parameter 0.125 0.364 0.441f 0.477 0.533 m c 4 al

3

s2"

4 al m k Continuum u ~ r . Model 0.12 [I** 0.347 0 . 5 9 3 0.646 w e

.;;;

F o u r i e r c

-

u S p e c t r a 3 . 3 1 . 7 0.9 0.34 0 . 6 3 0.57 .PI 0 .d

B 3

.':

a I-, _cr

El

Impulse 3.0 1 . 3 , 1 . 6 Decay 3.9 I tem No. 9 10 11 12 1 3 14 15 Measured Mode , Shape Measured Frequency, H z 0.555 0.615 0.635 0.65 0.715 0.94 1 .OO C a l c u l a t e d Frequency, lumped 0.575 0.618 0.654 Parameter Model, Hz

is a l s o accompanied by a h o r i z o n t a l component.

I d e n t i f i c a t i o n o f Modes

I

Although t h e i d e n t i f i c a t i o n o f t h e v e r t i c a l mode shapes was f a i r l y s t r a i g h t -

I f o l w a r d , t h e i d e n t i f i c a t i o n o f t h e t o r s i o n a l modes proved t o b e a d i f f i c u l t t a s k .

Some of t h e r e a s o n s a r e : f i r s t , t h e c o u p l i n g o f t o r s i o n and h o r i z o r l t a l motion

I o b s c u r e d t h e u s u a l c l e a r d i s t i n c t i o n between h o r i z o n t a l and t o r s i o n a l modes;

s e c o n d l y , a i a r g e number o f c l o s e l y s p a c e d modes appeared i n t h e s p e c t r a , which t o some e x t e n t i n f l u e n c e d each o t h e r ' s s p e c t r a l a m p l i t u d e s ; and t h i r d l y , some o f t h e modes had such s m a l l a m p l i t u d e s t h a t t h e l i m i t s o f r e s o l u t i o n o f measurement and a n a l y s i s equipment b:cre approached.

The l a r g e number o f t o r s i o n a l modes can b e t r a c e d t o t h e asymmetric s t i f f n e s s

p r o p e r t i e s o f t h e U-section formed by t h e s t i f f e n i n g t r u s s e s and t h e r o a d deck

w i t h h o r i z o n t a l c r o s s b r a c i n g . S i n c e , c o n c e p t u a l l y a t l e a s t , v a r i o u s combinations

o f i n d e p e n d e n t r o t a t i o n a l and h o r i z o n t a l motions o f s i d e spans and main span a r e

p o s s i b l e , t h e appearance o f a l a r g e number o f independent modes i s p e r h a p s n o t

s u r p r i s i n g .

The

c l o s e s p a c i n g o f t h e modes r e q u i r e s h i g h r e s o l u t i o n spectrum

a n a l y s i s and c o n s e q u e n t l y l o n g recording s e s s i o n s . No n o t i c e a b l e d i f f e r e n c e s were

found i n t h e s p e c t r a o f t h e s i g n a l s r e c o r d e d under t r a f f i c e x c i t a t i o n o r predomi- n a n t low s p e e d wind e x c i t a t i o n s o r t h e s t r o n g wind e x c i t a t i o n , a l t h o u g h a more d e t a i l e d examination o f t h e l a t t e r h a s y e t t o b e u n d e r t a k e n .

Some o f t h e u n r e s o l v e d a s p e c t s on t h e i d e n t i f i c a t i o n o f t o r s i o n a l modes a r e t h e f o l l o w i n g .

1. blul t i p l e f r e q u e n c i e s w i t h s u b s t a n t i a l l y t h e same mode shapes were found n e a r

0 . 3 2 , 0 . 3 8 , 0.51 and 0.94 Hz. Whereas some o f t h e s e modes had d i f f e r e n t phases o f

s i d e s p a n and tower motions, o t h e r s e x h i b i t e d no n o t i c e a b l e d i f f e r e n c e s .

2 . From a n examination o f v a r i o u s F o u r i e r and c r o s s - s p e c t r a l c a l c u l a t i o n s , i t i s

s u s p e c t e d t h a t a n o t h e r symmetrical t o r s i o n a l mode e x i s t s very c l o s e t o t h e symme-

t r i c mode a t 0.465 Hz and a n o t h e r one n e a r tl7.e asymmetric mode a t 0.507 Hz. HOW-

e v e r , t h e y c o u l d n o t b e i d e n t i f i e d . I t i s t h o u g h t t h a t t h e p l o t t e d mode a t 0.465

Hz i s a c t u a l l y a composite o f two modes, one a dominant s i d e span mode, and t h e o t h e r a dominant main s p a n mode.

A p o s s i b l e r e a s o n f o r t h e p r e s e n c e o f m u l t i p l e f r e q u e n c i e s n e a r 0.32, 0.38,

0 . 5 1 and 0 . 9 4 Hz might b e t h e changing r e s t r a i n t s a s t h e t i e rods b e g i n t o t i g h t z n and t h e b e a r i n g s s l i p owing t o changing l o a d i n g c o n d i t i o n s on t h e b r i d g e .

Comparison o f Measured and C a l c u l a t e d

A s t h e r e s u l t s i n T a b l e 1 i n d i c a t e , t h e r e i s g e n e r a l l y e x c e l l e n t agreement

between t h e measured and c a l c u l a t e d f r e q u e n c i e s f o r t h e v e r t i c a l modes f o r b o t h t h e

continuum model and t h e lumped p a r a m e t e r model. The s l i g h t l y lower c a l c u l a t e d

f r e q u e n c i e s f o r t h e continuum model may i n p a r t b e due t o t h e n e c e s s a r y s i m p l i f i c a - t i o n s i n t h e t h e o r y , such a s c o n s t a n t mass and s t i f f n e s s p r o p e r t i e s o f t h e g i r d e r

1

and n e g l e c t i n g t h e camber.

The frequency f o r t h e l o w e s t h o r i z o n t a l mode, Item I i n Table 3, a l s o shows good a g r e e z e n t between c a l c u l a t i o n s and measurements. A s t h e continuum t h e o r y f o r t h e h o r i z o n t a l modes does n o t t a k e a c c o u n t o f coupled h o r i z o n t a l - t o r s i o n a l motion, t h e agreement f o r t h e h i g h e r modes can b e e x p e c t e d t o become l e s s s a t i s f a c t o r y .

For t h e f r e q u e n c i e s o f t h e t o r s i o n a l modes p r e s e n t e d i n T a b l e s 2 and 3, t h e c a l c u l a t e d v a l u e s f o r t h e continuum model show s u b s t a n t i a l d i f f e r e n c e s from

measured o n e s . Reasons f o r t h i s may b e t h e d i f f i c u l t y i n a c h i e v i n g r e a l i s t i c e s t i m a t e s f o r t o r s i o n a l s t i f f n e s s o f t h e g i r d e r and t h e assumptions o f c o n s t a n t p r o p e r t i e s i n each s p a n . Furthermore, a s mentioned p r e v i o u s l y , s i n c e t h e continuum t h e o r y does n o t account f u l l y f o r coupled t o r s i o n a l - h o r i z o n t a l motion, a c o n s i s t e n t agreement between measurements and c a l c u l a t i o n s cannot be e x p e c t e d .

Damping

The s i m p l e s t method o f o b t a i n i n g modal damping v a l u e s i s t o c a l c u l a t e them from t h e half-power bandwidth o f power ( o r F o u r i e r ) spectrum p e a k s . U n f o r t u n a t e l y t h i s method h a s some p o t e n t i a l shortcomings when a p p l i e d t o t h e s t r u c t u r e u n d e r c o n s i d e r a t i o n . First, changing l o a d i n g c o n d i t i o n s on t h e b r i d g e c a u s e s m a l l v a r i - a t i o n s i n t h e f r e q u e n c i e s . These r e s u l t i n a widening of t h e resonance peak and t h u s y i e l d damping v a l u e s t h a t may b e t o o h i g h . Averaging a number o f s e q u e n t i a l s p e c t r a i s h i g h l y d e s i r a b l e f o r a p r o p e r d e f i n i t i o n of t h e spectrum p e a k s , b u t t h i s a c c e n t u a t e s t h e p o t e n t i a l problem o f g e t t i n g w i d e r peaks due t o s l i g h t f r e q u e n c y s h i f t s . Secondly, f o r some o f t h e modes i t i s d i f f i c u l t o r i m p o s s i b l e t o f i n d a spectrum peak whose width can b e p r o p e r l y measured. Where two modes n e a r l y c o i n - c i d e an e n t i r e l y erroneous p i c t u r e may emerge i f t h i s i s n o t r e c o g n i z e d . F i n a l l y , amplitude dependence o f damping cannot b e d e l i n e a t e d from t h e half-power bandwidth.

Although a f o r c e d resonance t e s t would p o s s i b l y y i e l d t h e most r e l i a b l e v a l u e s o f damping, i t i s i m p r a c t i c a l f o r t h i s type o f s t r u c t u r e . However, t h e v i b r a t i o n decay from a s i m u l a t e d impulse can b e used t o o b t a i n damping v a l u e s . A t t e n t i o n s h o u l d b e p a i d t o t h e f o l l o w i n g a s p e c t s : a ) t h e p r e s e n c e o f ambient v i b r a t i o n s s h o u l d b e r e c o g n i z e d and p o s s i b l e c o r r e c t i o n s made; b) a m p l i t u d e depen- dence o f damping can b e t a k e n i n t o a c c o u n t by c a l c u l a t i n g damping v a l u e s a t v a r i o u s p o i n t s a l o n g t h e decay curve; and c ) when two o r more modes f a l l v e r y c l o s e t o each o t h e r , t h e v i b r a t i o n decay c u r v e may b e a f f e c t e d by modal i n t e r f e r e n c e a n d t h e r e f o r e t h e damping v a l u e s o b t a i n e d have t o b e i n t e r p r e t e d w i t h c a u t i o n . _{T h i s} a p p l i e s t o t h e t o r s i o n a l mode a t 0.38 Hz, where t h e damping v a l u e s d e r i v e d from t h e impulse curve a r e thought t o b e somewhat h i g h e r t h a n i f t h e i n d i v i d u a l f r e q u e n c y components c o u l d have been s e p a r a t e d .

The damping v a l u e s determined f o r t h i s b r i d g e a r e g e n e r a l l y lower t h a n t h o s e r e p o r t e d f o r two o t h e r s u s p e n s i o n b r i d g e s ₍₂₎

.

CONCLUSIONS

Both t h e ambient v i b r a t i o n s u r v e y and t h e impact t e s t s proved u s e f u l and complementary i n o b t a i n i n g dynamic p r o p e r t i e s o f t h e b r i d g e . _{For t h i s p a r t i c u l a r} s t r u c t u r e , however, n e i t h e r method by i t s e l f a p p e a r s t o b e a b l e t o p r o v i d e e n t i r e l y c o n s i s t e n t v a l u e s o f damping. F u r t h e r e f f o r t s toward o b t a i n i n g b e t t e r damping v a l u e s f o r s u s p e n s i o n b r i d g e s a r e i n d i c a t e d .

The modal f r e q u e n c i e s and mode s h a p e s computed from t h e continuum model and

' t h e lumped p a r a m e t e r model showed e x c e l l e n t agreement w i t h t h e measured v a l u e s f o r

t h e v e r t i c a l -and t h e l o w e s t h o r i z o n t a l modes. L a r g e r d i f f e r e n c e s were e n c o u n t e r e d f o r t h e t o r s i o n a l modes, p a r t i c u l a r l y f o r t h e continuum model. T h i s i s a s c r i b e d i n p a r t of, t h e u n c e r t a i n c o n s t r a i n t s t h a t e x i s t i n t h e s t r u c t u r e . The a s s u m p t i o n s i n h e r e n t i n t h e c o n t i n u r n model do n o t a p p e a r t o r e p r e s e n t a d e q u a t e l y t h e t o r s i o n a l b e h a v i o r o f t h i s p a r t i c u l a r suspension b r i d g e .

Acknowledgement

I

1

The L i o n s ' Gate B r i d g e i s owned by t h e Department o f Highways o f B r i t i s h

I

Columbia. The c o o p e r a t i o n o f t h e Department i n t h e measurement program

i s

g r a t e -

I

f u l l y acknowledged, a s i s t h e i r p e n i s s i o n t o p u b l i s h t h e p a p e r . S t r u c t u r a l con- I s u l t a n t s f o r t h e b r i d g e m o d i f i c a t i o n s a r e Buckland and T a y l o r L t d . o f Vancouver,

B . C . The a s s i s t a n c e o f t h e i r s t a f f and o f R . G . Diment, M. Jacob and E . C . L u c t k a r o f t h e D i v i s i o n o f B u i l d i n g Research o f t h e N a t i o n a l Research Council o f Canada

i s

'

a l s o g r a t e f u l l y acknowledged. The r e s u l t s u s i n g t h e lumped mass, l i n e a r s t i f f n e s s model were o b t a i n e d by Hooley E n g i n e e r i n g , Vancouver, B . C .

I

P o r t i o n s o f t h e measurements were t a k e n w h i l e t h e f i r s t a u t h o r ( H J R ) was

on

'

s t a f f p o s t i n g a t t h e F a c u l t y o f Applied S c i e n c e o f t h e U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, B . C .

Re f e r e n c c s :

1 . B l e i c h , F . , e t a l , "The Mathematical Theory o f V i b r a t i o n o f S u s p e n s i o n

B r i d g e s " , Bureau o f Pub1 i c Roads, Department o f Commerce, Washington D . C . , 1950.

2 . McLamore, V . R . , t i a r t , C . C .

,

and S t u b b s ,

I

. R . , "Ambient V i b r a t i o n o f ';bo S u s p e n s i o n P r i d g e s " , J o u r n a l o f t h e S t r u c t u r a l D i v i s i o n , ASCE, Vol. 9 7 ,

S o . ST 10, P r o c . P a p e r 8454, O c t . 1971,

p p .

2567-2582.

3 . S e l b e r g , A., " O s c i l l a t i o n and Aerodynamic S t a b i l i t y o f S u s p e n s i o n Bridges",

Acta P o l y t e c h n i c a S c a n d i n a v i a , C i v i l E n g i n e e r i n g and B u i l d i n g C o n s t r u c t i o n S e r i e s No. 13, Trondheim, Norway, 1961.

S - T O W E R S t - - N N - T O W E R

S - A B U T M E N I

I

FIGURE 1

ELEVATION O F BRIDGE S H O W I N G MEASUREMENT STATIONS

Data for Bridge:

Total Cable area = 144.9 i n 2

Bridge weight

-

4.336 kiplR for side span

= 4.732 kiplR for main .span

Inertia about

Center of Gravity = 31.0 kip sec2 for side span

= 37.0 kip sec2 for main span

FIGURE 2

CROSS SECTION A N D STRUCTURAL CONSTANTS

FIGURE 3 FIGURE 4

FOURIER AMPLITUDE SPECTRUM FOR _{FOURIER AMPLITUDE SPECTRUM FOR} VERTICAL M O T I O N AT S T A T I O N 1 N TORSION AT STATION 1N

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