Vibration Characteristics of a Floor Sample

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Vibration Characteristics of a Floor Sample

Rainer, J. H.; Pernica, G.

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rr

and G

VIBRATION CHARACTERIST

lGS

OF

A FLOOR SAMPLE

by

J.H.

Raine

i.

Pemka

A54 ALY

DBR -Paper

No. 1271

Mvidon

o

f

_{BuUdlng ~eeeardi}

NATIONAL RESEARCH COUNCIL OF CANADA

DIVISION OF BUILDING RESEARCH

ANALYZED

VIBRATION CHARACTERISTICS OF A FLOOR SAMPLE

by

J.H. Rainer and G. Pernica

Ottawa

March

1985

VIBRATION CHARACTERISTICS OF A FLOOR SAMPLE

J . H . Rainer and G. P e r n i c a

ABSTRACT

V i b r a t i o n t e s t s c o n s i s t i n g of h e e l impacts by a person, s h a k e r impacts, and random s h a k e r e x c i t a t i o n s were c a r r i e d o u t on an 8.5 x 8.8

m

s t e e l

j o i s t - c o n c r e t e s l a b f l o o r sample. S t r u c t u r a l v a r i a t i o n s t o t h e f l o o r sample i n c l u d e d r e s t r a i n e d and f r e e edge c o n d i t i o n s , two c o n c r e t e s l a b t h i c k n e s s e s , s e v e r a l j o i s t s t i f f n e s s e s , and t r a n s v e r s e b r a c i n g of t h e s t e e l j o i s t s . The lowest f o u r t o s i x n a t u r a l f r e q u e n c i e s , a s s o c i a t e d mode s h a p e s and modal damping v a l u e s were determined f o r v a r i o u s s t r u c t u r a l c o n f i g u r a t i o n s . Modal damping of t h e f l o o r sample was s i g n i f i c a n t l y i n c r e a s e d by t h e p r e s e n c e of one o r more persons on t h e f l o o r , whereas s t r u c t u r a l m o d i f i c a t i o n s had

r e l a t i v e l y l i t t l e e f f e c t . P r e d i c t e d and measured i n i t i a l peak a c c e l e r a t i o n s due t o h e e l impact and fundamental f l o o r f r e q u e n c i e s were a l s o compared.

INTRODUCTION

A s f l o o r systems i n b u i l d i n g s become s t r u c t u r a l l y more e f f i c i e n t , t h e y a l s o t e n d t o become more r e s p o n s i v e t o common dynamic l o a d i n g s s u c h a s people walking, dancing o r jumping, v e h i c u l a r t r a f f i c , and machinery. The main f a c t o r s c o n t r i b u t i n g t o t h i s i n c r e a s e d v i b r a t i o n s e n s i t i v i t y a r e

r e d u c t i o n s i n mass, s t i f f n e s s , and damping. Since t h e s e parameters dominate t h e dynamic behaviour of s t r u c t u r a l s y s t e m , a b e t t e r u n d e r s t a n d i n g and q u a n t i t a t i v e e v a l u a t i o n of them w i l l a l l o w t h e p r e v e n t i o n o r a l l e v i a t i o n of o b j e c t i o n a b l e v i b r a t i o n s .

An o p p o r t u n i t y t o s t u d y t h e s e parameters a r o s e when a f u l l - s c a l e f l o o r model became a v a i l a b l e f o r dynamic t e s t i n g . The o b j e c t i v e of t h e t e s t s was t o measure t h e dynamic p r o p e r t i e s of t h e composite s t e e l j o i s t - c o n c r e t e s l a b f l o o r sample and t o a s c e r t a i n how t h e s e p r o p e r t i e s were a f f e c t e d by

s t r u c t u r a l m o d i f i c a t i o n s t o t h e f l o o r sample. P a r t i c u l a r emphasis was p l a c e d on t h e e v a l u a t i o n of damping, a s t h i s parameter cannot a s y e t b e c a l c u l a t e d .

Some a s p e c t s of t h i s i n v e s t i g a t i o n have been d e s c r i b e d i n an e a r l i e r paper.l T h i s r e p o r t i s more e x t e n s i v e i n i t s p r e s e n t a t i o n of modal

f r e q u e n c i e s , mode s h a p e s , and modal damping r a t i o s .

EXPERIMENTAL PROCEDURE

The 8.8 x 8.5 m f l o o r sample comprised a nominally 75 mm t h i c k c o n c r e t e deck,* composite w i t h s i x 457

mm

deep s t e e l j o i s t s . The j o i s t s were

*Since t h e measured t h i c k n e s s from c o n c r e t e c o r e samples was 81 mm, t h e s l a b

s u p p o r t e d on s t e e l g i r d e r s , which i n t u r n framed i n t o f o u r c o r n e r s t e e l columns. The edges of t h e sample were a l s o bounded by s t e e l beams ( s e e F i g u r e 1 f o r d e t a i l s ) .

I n s t r u m e n t a t i o n f o r measuring t h e dynamic r e s p o n s e of t h e f l o o r c o n s i s t e d of a c c e l e r o m e t e r s l o c a t e d a t t h e g r i d p o i n t s shown i n F i g u r e 2. The s i g n a l s from t h e a c c e l e r o m e t e r s were f i l t e r e d and a m p l i f i e d b e f o r e b e i n g r e c o r d e d on FM t a p e r e c o r d e r s . A p o r t a b l e o s c i l l o s c o p e and s p e c t r u m

a n a l y z e r were u s e d t o m o n i t o r t h e s i g n a l s d u r i n g t h e e x p e r i m e n t s . A t f i v e p o i n t s on t h e f l o o r ( l a b e l l e d A t o E i n F i g u r e 2 ) , dynamic l o a d s were

a p p l i e d from random s h a k e r e x c i t a t i o n , s h a k e r i m p a c t , and h e e l i m p a c t s by a person.

M o d i f i c a t i o n s t o t h e o r i g i n a l f l o o r sample c o n s i s t e d o f : 1 ) end beams s u p p o r t e d by screw j a c k s of a t y p e commonly used t o s u p p o r t f l o o r s i n t h e basements of h o u s e s , w i t h s u p p o r t e d o r u n s u p p o r t e d edge beams ( p o s i t i o n s shown i n F i g u r e 1 ) ; 2 ) suspended c e i l i n g and gypsum board c e i l i n g on t h e u n d e r s i d e of t h e f l o o r ; 3) d i a g o n a l s t e e l b r a c i n g between t h e j o i s t s ;

4 ) a 152 x 229 mm c o n c r e t e s t i f f e n i n g beam normal t o t h e j o i s t s ;

5 ) s t e e l - n e o p r e n e - s t e e l sandwich s t r i p s ( 6 x 76 mm

-

7 x 75 mm

-

6 x 76 mm) b o l t e d t o t h e lower chord of t h e j o i s t s ; and 6 ) a 25 m c o n c r e t e t o p p i n g added t o t h e s l a b .

F l o o r r e s p o n s e s produced by random and impact e x c i t a t i o n s were a n a l y z e d on a Hewlett-Packard 54238 s p e c t r u m a n a l y z e r t o o b t a i n n a t u r a l f r e q u e n c i e s and phase between t r a n s d u c e r l o c a t i o n s . Mode s h a p e s were drawn from

s p e c t r a l a m p l i t u d e s and phase. Modal damping r a t i o s were found by two methods: 1 ) a c u r v e - f i t t i n g p r o c e d u r e of t h e s p e c t r u m a n a l y z e r a p p l i e d t o t h e s p e c t r a l peaks of t h e F o u r i e r s p e c t r u m of t h e f l o o r r e s p o n s e ; and 2 ) f o r s h a k e r and h e e l impact s i g n a l s , t h e f r e e v i b r a t i o n decay c u r v e s were

hand-pass f i l t e r e d around t h e n a t u r a l f r e q u e n c y and l o g a r i t h m i c decrements computed. Examples of F o u r i e r s p e c t r a f o r s h a k e r random and impact

e x c i t a t i o n a r e shown i n F i g u r e s 3 and

4;

time r e c o r d s of f r e e v i b r a t i o n decay c u r v e s on l i n e a r s c a l e s a r e shown i n F i g u r e s 5 and 6 , and on l i n e a r and l o g a r i t h m i c s c a l e s i n F i g u r e 7 a and b. The v i b r a t i o n decay c u r v e s were o b t a i n e d by a d d i n g t h e s i g n a l s from s y m m e t r i c a l l y - p l a c e d t r a n s d u c e r s f o r mode 1 , a s y m m e t r i c a l mode, and s u b t r a c t i n g them f o r mode 2 , a n

a n t i - s y m m e t r i c mode. T h i s e l i m i n a t e d a n t i - s y m m e t r i c modes a l m o s t c o i n c i d e n t i n f r e q u e n c y from t h e f o r m e r , and symmetric modes from t h e l a t t e r . O t h e r u n d e s i r a b l e modal components were removed by s u i t a b l e band-pass f i l t e r i n g .

RESULTS

N a t u r a l F r e q u e n c i e s and Mode Shapes

T a b l e s 1 and 2 g i v e n a t u r a l f r e q u e n c i e s and a s s o c i a t e d mode s h a p e s a l o n g t h e j o i s t s and a c r o s s t h e j o i s t s , a s d e p i c t e d i n t h e t a b l e h e a d e r . N a t u r a l f r e q u e n c i e s w i t h a s t e r i s k s i d e n t i f y t h o s e modes whose s h a p e s c o u l d n o t be p o s i t i v e l y i d e n t i f i e d from t h e d a t a . They have been f i t t e d i n t o t h e t a b l e s by c o n s i d e r i n g t h e i r p o s i t i o n i n t h e s e q u e n c e of modal f r e q u e n c i e s .

For some of t h e h i g h e r modes i n T a b l e 1, r e l a t i v e l y l a r g e modal

a m p l i t u d e s were measured on t h e end o r edge beams, c o n t r a r y t o what might be e x p e c t e d f o r s u p p o r t e d end and edge c o n d i t i o n s . Although some s m a l l

amplitudes w i l l always be p r e s e n t due t o e l a s t i c s h o r t e n i n g of t h e j a c k s , t h e r e l a t i v e l y l a r g e amplitudes a r e thought t o be due t o l o c a l s e p a r a t i o n of t h e s l a b from t h e t o p f l a n g e of t h e end and edge beams. It i s t h e r e f o r e q u e s t i o n a b l e whether f u l l y - s u p p o r t e d end and edge c o n d i t i o n s apply t o t h e h i g h e r modes.

Table 2 p r e s e n t s n a t u r a l f r e q u e n c i e s and mode s h a p e s f o r t h e f l o o r sample w i t h s u p p o r t e d o r unsupported end and edge beams. A s a r e s u l t of reduced p e r i m e t e r r e s t r a i n t , t h e r e were more modes w i t h n a t u r a l f r e q u e n c i e s below 22 Hz f o r t h e unsupported t h a n f o r t h e f u l l y - s u p p o r t e d c o n d i t i o n s . It was d i f f i c u l t t o o b t a i n t h e mode shape f o r mode 3 because t h e e x c i t a t i o n

p o i n t was n e a r a node, e x c e p t f o r Run No. 111, w i t h e x c i t a t i o n a t l o c a t i o n E.

Measured mode shapes of t h e o r i g i n a l f l o o r sample a r e p r e s e n t e d i n F i g u r e s 8 t o 10 f o r v a r i o u s end and edge s u p p o r t c o n d i t i o n s . The p l o t t e d p o i n t s a r e t r a n s d u c e r l o c a t i o n s f o r which modal a m p l i t u d e s were determined. E x t r a p o l a t e d p o r t i o n s of t h e mode shapes a r e shown by dashed l i n e s . For s u p p o r t e d end and edge c o n d i t i o n s , modes w i t h s u c c e s s i v e l y h i g h e r n a t u r a l f r e q u e n c i e s c o n t a i n e d an i n c r e a s i n g number of n e a r l y s i n u s o i d a l l o b e s , o c c u r r i n g f i r s t a c r o s s t h e j o i s t s , and t h e n a l o n g t h e j o i s t s , a s shown i n F i g u r e 9.

The r e s u l t s of Tables 1 and 2 and F i g u r e s 8 t o 10 showed t h e f o l l o w i n g t r e n d s :

1)

Supporting t h e end beams p r i m a r i l y i n c r e a s e d t h e n a t u r a l f r e q u e n c i e s of modes w i t h nodal l i n e s p e r p e n d i c u l a r t o t h e j o i s t s . The l a r g e s t i n c r e a s e i n frequency o c c u r r e d i n t h e fundamental and t h e f i f t h mode.

2 ) Supporting t h e edge beams had l i t t l e e f f e c t on t h e frequency of t h e fundamental mode, b u t s u b s t a n t i a l l y i n c r e a s e d t h e f r e q u e n c i e s of t h e second mode and h i g h e r modes w i t h nodal l i n e s p a t a l l e l t o t h e

j o i s t s . 3 ) A s i n 2 ) , t h e a d d i t i o n of s t i f f e n i n g elements p e r p e n d i c u l a r t o t h e j o i s t s p r i m a r i l y r a i s e d t h e f r e q u e n c i e s of t h o s e modes w i t h n o d a l l i n e s p a r a l l e l t o t h e j o i s t s . 4 ) The a d d i t i o n of 25 mm of c o n c r e t e t o t h e s l a b reduced t h e n a t u r a l frequency of a l l modes by a t l e a s t 5%.

The above o b s e r v a t i o n s g e n e r a l l y a g r e e w i t h what might be e x p e c t e d from p r i n c i p l e s of v i b r a t i o n t h e o r y p e r t a i n i n g t o p l a t e s .

A s m a l l d e c r e a s e i n t h e f i r s t and second modal f r e q u e n c i e s i s

d i s c e r n i b l e i n Table 2, between what were o r i g i n a l l y thought t o be i d e n t i c a l r u n s , Run 5 and Runs 8, 9 and 10. These d i f f e r e n c e s c a n be a t t r i b u t e d t o s l i g h t changes i n t h e p e r i m e t e r c o n d i t i o n s of t h e f l o o r sample. It i s thought t h a t j a c k i n g o p e r a t i o n s on t h e end and edge beams and t h e j o i s t s caused l o c a l c o n c r e t e c r a c k i n g and l o s s of bond between t h e s t e e l beams and t h e c o n c r e t e s l a b . F u r t h e r s e p a r a t i o n between t h e c o n c r e t e s l a b and t h e s t e e l beams may a l s o have o c c u r r e d a s t h e c o n c r e t e d r i e d and shrank. The r e s u l t i n g l o s s of bond would s l i g h t l y d e c r e a s e t h e end and edge s t i f f n e s s of t h e f l o o r sample and r e s u l t i n lower modal f r e q u e n c i e s .

Damping of t h e F l o o r Sample

Modal damping r a t i o s f o r s h a k e r impact and h e e l impact response were o b t a i n e d from t h e f r e e v i b r a t i o n decay u s i n g t h e r e l a t i o n s h i p :

where :

5

= modal damping r a t i o , f r a c t i o n of c r i t i c a l ;

Xm = amplitude of t h e mth peak;

Xmh = a m p l i t u d e of t h e peak n c y c l e s l a t e r ;

6 = l o g a r i t h m i c decrement.

V i b r a t i o n decays f o r t h e f i r s t two modes a r e shown i n F i g u r e s 5 , 6 and

7.

A convenient d i s p l a y of t h e decay i n l o g a r i t h m i c a m p l i t u d e , shown i n

F i g u r e 7b, provided a simple method f o r c a l c u l a t i n g t h e l o g a r i t h m i c decrement ( 6 ) by f i t t i n g a s t r a i g h t l i n e through t h e peaks. It a l s o

a f f o r d e d t h e o p p o r t u n i t y t o observe any changes i n t h e l o g a r i t h m i c decrement w i t h p r o g r e s s i v e l y s m a l l e r amplitudes; i . e . , t o d e t e c t amplitude-dependent damping. The l o g a r i t h m i c decrement f o r each mode and t h e modal damping r a t i o were computed f o r t h e f i r s t 20 c y c l e s e x c e p t where o t h e r w i s e noted.

Modal damping r a t i o s f o r s h a k e r impact and s h a k e r random e x c i t a t i o n were a l s o o b t a i n e d u s i n g t h e c u r v e - f i t t i n g a l g o r i t h m f o r F o u r i e r spectrum peaks provided by t h e Hewlett-Packard 53238 S t r u c t u r a l Dynamics Analyzer. Some r e s u l t s u s i n g b o t h p r o c e d u r e s a r e p r e s e n t e d i n T a b l e 3 f o r d i f f e r e n t c o n f i g u r a t i o n s of t h e f l o o r sample. I n g e n e r a l , comparable modal damping r a t i o s were o b t a i n e d u s i n g b o t h methods. On _{o c c a s i o n , however, t h e}

agreement was less t h a n s a t i s f a c t o r y (Run 183, Mode 2 ) . T h i s may be due t o i n a d e q u a t e s p e c t r a l r e s o l u t i o n i n t h e F o u r i e r spectrum o r i n a p p r o p r i a t e placement of t h e c u r s o r s f o r bounding t h e s p e c t r a l peak of t h e c u r v e - f i t t i n g a l g o r i t h m on t h e spectrum a n a l y z e r .

Damping c o n t r i b u t e d by people: The h e e l impact has l o n g been a f a v o u r i t e s o u r c e of f l o o r e x c i t a t i o n because of i t s s i m p l i c i t y and i t s

resemblance t o o t h e r dynamic l o a d i n g s on f l o o r s such a s walking and j ~ m p i n g . ~ , ~ However, t h i s t e s t r e q u i r e s t h e p r e s e n c e of a t l e a s t one person on t h e f l o o r . A comparison of modal damping r a t i o s d e r i v e d from t e s t s demonstrated t h a t a s i g n i f i c a n t i n c r e a s e i n modal damping c a n r e s u l t when people a r e p l a c e d on a f l o o r t h a t h a s i n h e r e n t l y l i t t l e damping.

T a b l e

4

i l l u s t r a t e s t h i s e f f e c t by comparing modal damping r a t i o s d e r i v e d from h e e l impact w i t h t h o s e from s h a k e r impact t e s t s w i t h and without people on t h e f l o o r . The e f f e c t of one, two o r t h r e e p e r s o n s on t h e measured

damping r a t i o s i s demonstrated t h e r e .

The l o c a t i o n of a person on t h e f l o o r a l s o a f f e c t e d t h e amount of i n c r e a s e i n t h e t o t a l measured damping r a t i o . The r e s u l t s f o r t h e h e e l impact t e s t , Run No. 155, i n T a b l e s

4

and 5 show t h a t t h e modal damping r a t i o f o r t h e fundamental mode d e c r e a s e d a s t h e p e r s o n moved from l o c a t i o n A, t h e c e n t r e of t h e f l o o r , t o l o c a t i o n D. T h i s d e c r e a s e i n damping is

a t t r i b u t e d t o d e c r e a s e s i n modal amplitude w i t h l o c a t i o n . I n T a b l e 5 t h e modal damping r a t i o s measured a t l o c a t i o n s B , C and D were a d j u s t e d by t h e s q u a r e of t h e r a t i o of modal a m p l i t u d e s r e l a t i v e t o l o c a t i o n A , r e s u l t i n g i n damping r a t i o s f o r p e r s o n s a s i f they were s t a n d i n g a t l o c a t i o n A. The

mathematical basis for this adjustment is presented in Figure

11

_{and is}

discussed in a later section. For the floor sample with supported end and

edge beams, each person standing at position A (the centre of the floor

sample), contributes between

0 . 8 0

_and

1 . 0 0 %

_{of critical to the damping ratio}

for the fundamental mode of the floor sample. Aspects of the mathematical

modelling of damping due to people are also treated in a later section.

The amount of modal damping also varied with the individual. Although

no systematic study was done in this series of tests, there are indications

that the damping contributed by people varied from person to person,

depending on weight and physique.

Changes in damping due to floor modifications: Major floor

modifications can be expected to result in different modal damping ratios as

new sources of damping ire introduced and others diminish. The following

modifications to the floor sample were investigated: end beams and edge

beams fully supported; suspended ceiling on underside of floor; gypsum board

ceiling on underside of floor; steel-neoprene damping strips bolted to

bottom chord of the joists. The results are summarized in Table

6 .

For assessing the influence of various floor modifications, modal

damping ratios were determined from shaker impact tests. Heel impact tests

could not be utilized since they contained the additional damping of the

person doing the test. Modal damping was calculated from the logarithmic

decrement of the free vibration decay curve, as it was considered to give a

more reliable estimate than applying the curve-fitting algorithm of the

spectrum analyzer to the Fourier spectrum peak.

The results presented in Table

6

_{show that for the floor without}

modifications, and for all combinations of supported and unsupported end and

edge beams, the damping ratio for the fundamental mode

(CS)

was

0.37

_or

0 . 4 %

of critical. The addition of a gypsum board ceiling raised this value to

0 . 4 4 % .

_{A suspended ceiling with simulated lights increased the damping}

ratio to a greater degree, to

0 . 6 2 %

_{of critical for unsupported end and edge}

beams and

0 . 7 1 %

_{for supported end beams. The largest change in damping}

ratio occurred with the addition of the steel-neoprene sandwich strips,

which for supported end beams resulted in

1 . 0 %

_{modal damping ratio, and for}

the unsupported beams gave

0 . 7 6 %

_{of critical.}

_A

_{larger modal mass as a}

result of end and edge beam movement without a commensurate increase in

damping mechanisms may have been responsible for the smaller damping ratios

for the unsupported end and edge beams. It must also be noted that a

substantial increase in stiffness occurred with the application of the

sandwich damping strips, as can be seen by the increase in the fundamental

frequency from

6 . 4

_{to 7.8 Hz for the supported end beams.}

Response to Heel Impact

A

number of vibration criteria for performance of floors incorporate

the initial peak amplitude of floor acceleration to an actual or

mathematical representation of a heel impact. This applies to the criteria

presented by ~ e n z e n ~

and the criteria in the appendix of CSA Steel

standard.

For floors with fundamental frequency less than about

10

_{Hz, the}

initial peak acceleration amplitude (ao) in units of gravity (g) can be

calculated from:

where f = fundamental frequency i n Hz, I = e q u i v a l e n t impulse of h e e l impact = 67 N s , and ME = e q u i v a l e n t mass i n kg. An approximate e q u i v a l e n t mass MJ f o r a simply-supported j o i s t f l o o r can be o b t a i n e d by assuming a t r i b u t a r y s t r i p 60 t wide, , h i v i n g :

An examination of t h e fundamental mode shapes of t h e f l o o r sample i n d i c a t e s , however, t h a t t h e e n t i r e w i d t h of t h e sample s h o u l d be used f o r t h e

c a l c u l a t i o n of t h e e q u i v a l e n t mass r a t h e r t h a n 60 tc, s i n c e t h e e n t i r e sample width c o n t r i b u t e s t o a h a l f - s i n e wave. I f one sums t h e e l e m e n t a l i n e r t i a s i n p r o p o r t i o n t o t h e f i r s t mode shape, t h e n t h e e q u i v a l e n t i n e r t i a l mass (MI) is o b t a i n e d :

M~ = p

7

)'

s i n x s i n y dx dy 0 0

= 0.405 pab

= 0.405 times t o t a l mass of t h e f l o o r sample where p = mass d e n s i t y p e r u n i t a r e a , and a , b = p l a n dimensions of t h e

f l o o r .

C a l c u l a t e d and measured peak i n i t i a l a c c e l e r a t i o n a m p l i t u d e s ( a o ) a r e g i v e n i n Table 7. The measured v a l u e s shown were d e r i v e d from a v e r a g i n g t h e r e s u l t s of f o u r s u c c e s s i v e h e e l impacts. C a l c u l a t e d v a l u e s of a. f o r

l o c a t i o n s B , C and D were o b t a i n e d by a d j u s t i n g t h e r e s u l t s of Equation ( 2 ) i n p r o p o r t i o n t o t h e r a t i o of modal amplitudes a t t h e s e l o c a t i o n s , and t h o s e a t l o c a t i o n A , a s i s c o n s i s t e n t w i t h t h e p r o p e r t i e s of modal f o r c e p r e s e n t e d i n F i g u r e 11.

The r e s u l t s i n Table 7 show t h a t t h e u s e of e q u i v a l e n t i n e r t i a l mass (MI) from Equation ( 4 ) f o r Run No. 33 g i v e s e x c e l l e n t agreement between measurements and c a l c u l a t i o n s . For t h e o t h e r two f l o o r c o n f i g u r a t i o n s (Run Nos. 155 and 190), t h e measured v a l u e s a r e somewhat l a r g e r t h a n t h e c a l c u l a t e d ones. The l a t t e r two r u n s were performed on t h e f l o o r modified by t h e t r a n s v e r s e c o n c r e t e beam, and t h e response t o impulsive l o a d s may be somewhat d i f f e r e n t from t h a t assumed i n t h e s i m p l e s p r i n r m a s s model i m p l i e d by Equation ( 2 ) . On t h e o t h e r hand, u s e of MJ of Equation ( 3 ) , t h e

e q u i v a l e n t mass w i t h a s t r i p width of 60 tc, g i v e s peak a c c e l e r a t i o n s t h a t a r e c o n s i d e r a b l y l a r g e r t h a n t h e measured ones. T h i s i s perhaps n o t

s u r p r i s i n g , s i n c e t h e behaviour of t h e f l o o r sample i s obviously c l o s e r t o t h e assumptions i n h e r e n t i n t h e d e r i v a t i o n of t h e e q u i v a l e n t i n e r t i a l mass

(MI)

t h a n t h o s e of t h e approximate e q u i v a l e n t mass (MJ).

The measured i n i t i a l peak a c c e l e r a t i o n amplitudes f o r t h e unsupported end and edge beams (Run Nos. 8 t o 11) i n T a b l e 7 a r e c o n s i d e r a b l y l a r g e r t h a n f o r t h e comparable c a s e s w i t h supported beams (Run No. 33). There i s

a l s o a l a r g e d i s c r e p a n c y i n a. d e r i v e d from u s i n g t h e e q u i v a l e n t masses MJ

t h a t t h e method of c a l c u l a t i n g impulse r e s p o n s e f o r f l o o r s w i t h f l e x i b l e s u p p o r t beams needs t o be i n v e s t i g a t e d f u r t h e r .

MATHEMATICAL MODELLING OF VIBRATING FLOORS

S i n c e t h e t h e o r e t i c a l p r e d i c t i o n s of i n i t i a l peak a c c e l e r a t i o n a. h a v e d i s p l a y e d some a p p a r e n t i n c o n s i s t e n c i e s , a c l o s e r e x a m i n a t i o n of t h e b a s i c a s s u m p t i o n s i n t h e m a t h e m a t i c a l m o d e l l i n g p r o c e s s a p p e a r s w a r r a n t e d .

The main p a r a m e t e r s i n t h e v i b r a t i o n s y s t e m ( i . e . , mass, damping, and n a t u r a l f r e q u e n c y ) w i l l s u b s e q u e n t l y be d i s c u s s e d . I n p r i n c i p l e , t h e dynamic r e s p o n s e of a f l o o r s y s t e m t o a n impact c a n be o b t a i n e d by t h e method of modal s u p e r p o s i t i o n , a s is p o s s i b l e f o r any v i b r a t i n g l i n e a r

e l a s t i c system. D e t a i l s of t h e method c a n be found i n s t a n d a r d t e x t b o o k s on

vibration^.^

A summary of modal p r o p e r t i e s f o r a s i m p l e beam s t r u c t u r e i s p r e s e n t e d i n F i g u r e 11.

Mass

Modal mass and modal damping a r e f u n c t i o n s of t h e s q u a r e of t h e mode s h a p e ( Y , F i g u r e 11). T h i s i s i n c o n t r a s t t o e q u i v a l e n t i n e r t i a l mass i n E q u a t i o n ( 4 ) , which i s a f u n c t i o n of t h e mode s h a p e o n l y t o t h e power of 1. F o r

a

simply-supported p l a t e , t h e modal mass of t h e fundamental mode e q u a l s

0.250 M , whereas t h e e q u i v a l e n t i n e r t i a l mass ( M ~ ) a c c o r d i n g t o E q u a t i o n ( 4 ) i s 0.405 M (M i s t h e t o t a l mass of t h e p l a t e ) . Modal mass i s t h e

m a t h e m a t i c a l l y c o r r e c t form f o r s o l v i n g v i b r a t i o n problems a s d e s c r i b e d by t h e e q u a t i o n of motion i n F i g u r e 11. E q u i v a l e n t i n e r t i a l mass a r i s e s f r o m t h e e m p i r i c a l f o r m u l a t i o n of t h e i m p u l s e r e s p o n s e of a s y s t e m h a v i n g

d i s t r i b u t e d mass, and

i s

o n l y a p p l i c a b l e t o t h e c a l c u l a t i o n of peak i m p u l s e r e s p o n s e . It c a n n o t be u s e d a s t h e mass f o r t h e d e t e r m i n a t i o n of n a t u r a l frequency.

The a c c u r a c y i n p r e d i c t i n g a. f o r t h e f l o o r sample u s i n g e q u i v a l e n t i n e r t i a l mass (MI) seems t o depend on t h e end and edge c o n d i t i o n s of t h e f l o o r ( T a b l e 7 ) . C o n s i d e r i n g t h e i n h e r e n t v a r i a b i l i t y of t h e t e c h n i q u e , agreement between t h e c a l c u l a t e d and measured v a l u e s of a. f o r t h e s u p p o r t e d end and edge c o n d i t i o n s was e x c e l l e n t . For u n s u p p o r t e d end and edge beams, however, t h e e q u i v a l e n t i n e r t i a l mass model seems t o b e u n s u i t a b l e , a s i s e v i d e n t from t h e l a r g e d i f f e r e n c e between p r e d i c t e d and measured r e s u l t s . A more c o n s i s t e n t f o r m u l a t i o n i s needed, b u t t h i s i s beyond t h e s c o p e of t h i s r e p o r t .

Damping

A s was shown b e f o r e , t h e v a r i a t i o n s i n damping r a t i o s f o r t h e

f u n d a m e n t a l mode w i t h changing p o s i t i o n of t h e p e r s o n c a n be a s s e s s e d u s i n g t h e m a t h e m a t i c a l model shown i n F i g u r e 11. However, t h i s m a t h e m a t i c a l model h a s t o be viewed somewhat a s a n a r t i f a c t , s i n c e t h e p e r s o n s t a n d i n g on t h e f l o o r i s s u r e l y n o t w e l l r e p r e s e n t e d by a s i n g l e d a s h p o t c o n n e c t e d t o t h e ground. A r e a l i s t i c r e p r e s e n t a t i o n of a p e r s o n would r e q u i r e t h e u s e of a multi-degree-of-freedom s y s t e m w i t h many masses and s t i f f n e s s e s , and damping d a s h p o t s n o t connected t o t h e ground. The c o n s i d e r a t i o n of s u c h a model i s , however, beyond t h e s c o p e of t h i s r e p o r t . D e s p i t e t h e a p p a r e n t s h o r t c o m i n g s

on a physical basis, the simple dashpot model gives answers that are in

general agreement with the observed values (Table

5).

Fundamental Natural Frequency

The fundamental natural frequency of the floor is another important

parameter in assessing the vibrational behaviour of a floor system. For a

simply-supported joist floor, the frequency in

Hz can be calculated from:

where

E

=

Young's modulus (MPa),

I

=

moment of inertia

(mm4),

w

=

loading

per unit length (N/mm) and

L

=

span length

(mm).

The fundamental frequency

of the floor sample was calculated assuming fully composite action between

the joist and the concrete slab and a tributary floor area equal to the

joist spacing. Comparisons of calculated and measured frequencies given in

Table 8 show excellent agreement for the cases where end and edge beams were

supported.

For unsupported end and edge beams, the fundamental frequency (f) of

the floor sample can be estimated by Dunkerly's approximation:

L L

where fl

=

fundamental frequency of floor with edge and end beams

supported, and f2

=

fundamental frequency of unsupported end beams. The

calculation of f2 requires some assumptions regarding the composite nature

of the steel beanrconcrete slab system, and the size of its tributary area.

It was assumed here that one quarter of the floor slab contributes to both

the mass and the stiffness of the end beam element. For the two cases,

fully composite and noncomposite, the following frequencies were obtained

for the

81

mm

slab:

f2 (noncornposite) =

6.2

HZ

f2 (composite)

=

10.6

HZ

.

Substitution of f2 into Equation (4) gives:

f =

4.7 Hz (noncomposite)

and

f

=

5.5

Hz (composite)

.

The comparison with measured values for other floor configurations is

presented in Table

8.

The composite assumption for the end beam gives

better agreement with measurements than the noncomposite one. The effect of

the transverse beam on the frequencies of the fundamental mode in Table 8

was ignored.

DISCUSSION

The s t u d y of t h e dynamic p r o p e r t i e s of t h e f l o o r sample provided a n o p p o r t u n i t y t o a s s e s s t h e i n f l u e n c e of v a r i o u s p a r a m e t e r s t h a t cannot

u s u a l l y be i n v e s t i g a t e d on a c o n s t r u c t i o n s i t e o r i n completed and occupied s t r u c t u r e s . However, t h e f l o o r sample s u f f e r s from one major d i s a d v a n t a g e i n t h a t i t does n o t r e p r e s e n t an a c t u a l f l o o r system. Real f l o o r s e x t e n d o v e r s e v e r a l bays i n both d i r e c t i o n s and c o n t a i n many more s o u r c e s of

s t i f f n e s s and damping such a s w a l l s , p a r t i t i o n s and o t h e r s u p p o r t i n g beams. These a d d i t i o n a l s o u r c e s of s t i f f n e s s can produce s m a l l t o major changes i n mode shapes and n a t u r a l f r e q u e n c i e s and t h u s r e s u l t i n changes t o t h e

v i b r a t i o n r e s p o n s e of t h e f l o o r t o dynamic loading. Modal damping r a t i o s o b t a i n e d f o r people and f o r v a r i o u s s t r u c t u r a l m o d i f i c a t i o n s of t h e f l o o r should t h e r e f o r e n o t be d i r e c t l y employed i n o t h e r s i t u a t i o n s w i t h o u t a p p r o p r i a t e v e r i f i c a t i o n of t h e i r a p p l i c a b i l i t y .

SUMMARY

AND _CONCLUSIONS

The dynamic t e s t s on a f u l l s c a l e l a b o r a t o r y f l o o r sample of composite s t e e l j o i s t - c o n c r e t e s l a b c o n s t r u c t i o n i n d i c a t e d t h e following:

1. Mode shapes and n a t u r a l f r e q u e n c i e s can be e v a l u a t e d s a t i s f a c t o r i l y by e x c i t i n g t h e f l o o r system w i t h e i t h e r h e e l impact, s h a k e r impact, o r s h a k e r random e x c i t a t i o n . S e v e r a l p o i n t s of e x c i t a t i o n a r e needed i f a complete s e t of modal p r o p e r t i e s i s d e s i r e d .

2.

Modal damping r a t i o s can be e v a l u a t e d from e i t h e r t h e l o g a r i t h m i c decrement method of f r e e v i b r a t i o n decay from impact, o r from t h e half-power bandwidth o r c u r v e - f i t t i n g methods a p p l i e d t o F o u r i e r

spectrum peaks d e r i v e d from t h e f l o o r r e s p o n s e t o random e x c i t a t i o n s o r impacts. The p r e s e n c e of people on t h e f l o o r a f f e c t s t h e measured modal damping r a t i o , e s p e c i a l l y f o r f l o o r s w i t h i n h e r e n t l y low damping.

3 . The c o n t r i b u t i o n of people t o modal damping and t h e amplitude of t h e i n i t i a l peak a c c e l e r a t i o n t o h e e l impact a r e b o t h dependent on t h e l o c a t i o n of t h e p e r s o n s on t h e f l o o r and t h e p o i n t of e x c i t a t i o n . C o r r e c t i o n s f o r t h e s e two e f f e c t s c a n be made from a knowledge of t h e mode shape of t h e f l o o r s l a b .

4. S i g n i f i c a n t d i f f e r e n c e s were found between t h e c a l c u l a t e d e q u i v a l e n t i n e r t i a l mass (MI) and t h e c a l c u l a t e d e q u i v a l e n t mass (MJ) of t h e f l o o r sample. Good agreement was o b t a i n e d between c a l c u l a t e d and measured v a l u e s of t h e i n i t i a l peak a c c e l e r a t i o n a. t o a h e e l impact, f o r t h e c a s e of supported end and edge beams, when t h e e q u i v a l e n t i n e r t i a l

mass

was used. T h i s agreement

i s

a s c r i b e d t o t h e s i m i l a r i t y between t h e a c t u a l mode shape of t h e f l o o r sample and t h e shape implied by t h e i n e r t i a l m a s s model. On t h e o t h e r hand, t h e agreement was poor f o r unsupported edge and end beams. T h i s would i n d i c a t e t h a t t h e i n e r t i a l mass model

i s

i n a p p r o p r i a t e f o r t h e c a l c u l a t i o n of i n i t i a l peak

a c c e l e r a t i o n f o r t h e f l o o r sample w i t h unsupported end and edge beams. 5. A d d i t i o n of a suspended c e i l i n g o r a gypsum board c e i l i n g r a i s e d t h e

damping r a t i o of t h e fundamental mode from 0.4% t o a maximum of 0.71% of c r i t i c a l . Although t h i s r e p r e s e n t s a s u b s t a n t i a l p e r c e n t a g e i n c r e a s e ,

i t i s o n l y ' a r e l a t i v e l y s m a l l i n c r e a s e i n a b s o l u t e terms. The most s i g n i f i c a n t i n c r e a s e i n t h e damping r a t i o of t h e fundamental mode occurred when t h e s t e e l - n e o p r e n e sandwich s t r i p s were b o l t e d t o t h e

bottom chord of t h e j o i s t s . For t h e c a s e w i t h supported end and edge beams, a modal damping r a t i o of 1% of c r i t i c a l was obtained.

6 . The agreement between measured and c a l c u l a t e d fundamental frequency of t h e f l o o r sample was good f o r c a s e s where b o t h end and edge beams were e i t h e r supported o r unsupported. For t h e l a t t e r c a s e , t h e assumption of f u l l composite a c t i o n was r e q u i r e d i n c a l c u l a t i n g t h e frequency of t h e end beams. Assuming non-composite a c t i o n f o r t h e end beams gave

(1) Rainer, J.H. and Pernica, G., "Damping of a Floor Sample,"

Proceedings, ASCE-EMD Specialty Conference on Dynamic Response of

Structures, Atlanta, GA, January, 1981, 859-873, NRCC 19457.

(2) Lenzen, K.H., "Vibration of Steel Joist-Concrete Slab Floors," AISC

Engineering Journal,

-

3, 3, 1966, 133-136.

(3) Canadian Standards Association, "Steel Structures for Buildings

-

Limit States Design," CAN3-S16.1-M84, Appendix G, Canadian Standards

Association, Rexdale, Ontario, 1984.

(4) Allen, D.E. and Rainer, J.H., "Vibration Criteria for Long-Span

Floors," Canadian Journal of Civil Engineering,

3,

2, 1976, 165-173,

NRCC 15207.

(5) Clough, R.W. and Penzien, J., Dynamics of Structures, McGraw-Hill

Book Company, New York, 1975, 634 p.

TABLE

1

Modal Frequencies for Floor Sample With Supported End and Edge Beams

Mode No.

1

2

3

4

5

Across joist

Mode

,--&A

s

Shape

Along joist

- - -

_{w -}

_u

Floor Conditions Run No. Modal Frequencies,

Hz

81

mm slab

2

9

6.50

9.60

15.75*

17.5

22.0

3

0

81

mm slab with

159

6.75

13.25

20.62

17.12

15

x

23

mm

161

13.50

transverse concrete beam

106

mm

slab with

15

x

23

mm transverse concrete beam

TABLE 2

Modal Frequencies of Floor Sample With Supported or Unsupported End and Edge Beams

Mode No. 1

2

3

4

5

₆

- -

Across joist

Mode

-

+-

7 4 - = a -

Shape

Along joist

- - -

-

f

JC

End, Edge Beam

Support

Floor Condition

Conditions**

Run No.

Modal Frequencies,

Hz

81

mm

slab

u,

u

5

5.47

8.98

12.8*

16.4

11.7

22.0

8, 9, 10

5.43

8.79

81

mm

slab

s,

u

16-19

6.4

8.9

12.8*

16.4

17.6

21.9*

81

mm

slab

s,

u

11

1

6.35

8.79

12.70

16.1

17.2

22.0

81

mm

slab

u,

u

146

5.4

8.8

12.6*

16.1

11.4

81

mm

slab

u,

u

147

5.6

9.8

with transverse

concrete beam

81

mm

slab

u,

s

with transverse

concrete beam

106

mm

slab

u,

u

166

5.3

9.2

with transverse

169

concrete beam

*Mode shape could not be identified with certainty from available data.

**U

-

unsupported;

TABLE 3

Modal Damping Ratios of Floor Sample

Modal Damping Ratio,

6 ,

%

Critical

From

Modal

Logarithmic

End, Edge Beam

Frequency,

Decrement,

Floor Condition Support Conditions* Run No. Mode No.

Hz

Shaker Impact

81

mm slab

S V

s

159

1

6.72

0.48

with transverse

161

1

6.72

0

-46

concrete beam

2

13.50

0.81

81

mm

slab

Curve-Fitting

Algorithm of

Spectrum

Analyzer

Shaker

Shaker

Impact

Random

106

mm

slab

S V

s

182

1

6.25

0.55

0.56

with transverse

183

1

6.25

0.50

0.56

concrete beam

2

11.62

1.06

0.82

- ---

*U

-

unsupported;

S

-

supported.

TABLE 4

I n f l u e n c e of P e r s o n s o n Modal Damping R a t i o s of F l o o r Sample, End and Edge Beams S u p p o r t e d

T o t a l Damping

Modal Frequency, R a t i o , % of D i f f e r e n c e From Hz C r i t i c a l Bare F l o o r

Type o f E x c i t a t i o n

Mode 1 Mode 2 Run No. F l o o r C o n d i t i o n and L o c a t i o n Mode 1 Mode 2 Mode 1 Mode 2

6.72 13.5 160 81 mm s l a b w i t h Shaker impact a t A , 0.46 0.81 t r a n s v e r s e c o n c r e t e b a r e f l o o r beam 3 p e r s o n s n e a r A 3.39 13.5 155 8 1 mm s l a b w i t h Heel impact a t A 1.36 t r a n s v e r s e c o n c r e t e B 1.02 1.33 beam C 1.03 D 0.94 1.21 11.62 182 106 mm s l a b w i t h Shaker impact a t A, 0.55 t r a n s v e r s e c o n c r e t e b a r e f l o o r be am 1 p e r s o n n e a r A 1.40 0.85 2 p e r s o n s n e a r A 2.25 1.70

TABLE 5

Damping R a t i o s f o r Fundamental Mode With One P e r s o n on F l o o r , End and Edge Beams Supported

Damping R a t i o ,

6 ,

% of C r i t i c a l A d j u s t e d f o r

Heel Comparable Mode Shape Average f o r F l o o r Impact Bare F l o o r T o t a l Minus R e l a t i v e t o 1 P e r s o n C o n d i t i o n Run No. a t T o t a l (Run No.) Bare F l o o r L o c a t i o n A a t L o c a t i o n A

8 1

mm

s l a b 7 7 A B C D 8 1 mm s l a b 93 A w i t h B suspended C c e i l i n g p l u s D " l i g h t s " 81 mm s l a b 155 A B C D

TABLE 6A

Damping Ratios ( 6 , % of Critical) of Fundamental Mode for Various Floor Modifications, 81 mm Slab Unsupported End and Edge Beams

Shaker Impact Heel Impact Shaker Random

From Curve-Fitting From Curve-Fitting Logarithmic Algorithm of Logarithmic Half-Power Algorithm of

Decrement Specturm Analyzer Decrement Bandwidth* Spectrum Analyzer

Natural Record No., Record No., Record No., Record No., Record No., Frequency, Location of Location of Location of Location of Location of Floor Condition Hz Excitation

tS

Excitation SSP Excitation SH Excitation SW Excitation

~ W P a) no modifi- 5.41 (0.4)** 135, A 0.91 146, E 0.37 cat ions C Q b) suspended 5.30 99, E 0.62 99, E 0.99 98, A 1.10 97, E 0.47 97, E 0.68 ceiling with 0.91 B 0.90 "lights" C 0.88 D 0.85 c) gypsum board 5.38 104, E 0.44 104, E ceiling d) steel-neoprene 5.8 double 121, E 0.76 121, E 1.07 119, A 0.97 118, E 0.65 1 1 8 , E 0.39

damping strips 6.2 peak 0.77

*Analysis bandwidth = 0.060 Hz.

TABLE 6B

Damping Ratios (5, % of Critical) of Fundamental Mode for Various Floor Modifications, 81 mm Slab Supported End Beams, Unsupported Edge Beams

Shaker Impact Heel Impact Shaker Random

From ~ u r G e Fitting From Curve Fitting Logarithmic Algorithm of Logarithmic Half-Power Algorithm of

Decrement Spectrum Analyzer Decrement Bandwidth* Spectrum Analyzer Natural Record No., Record No.

,

Record No., Record No., Record No.,

Frequency, Location of Location of Location of Location of Location of Floor Condition Hz Force 6~ Force 6~~ Force CH Force 5~ Force 5~~ a) no modifi- 5.41 (0.4)** 135, A 0.91 146, E 0.37

cations, unsupported end and edge

beams

i) end and 76, E 77, A 1.19

edge beams 0.91 B 0.85

supported C 0.82

D 0.72

ii) only end 6.38 (0.4)** 112, A 1.25 111, E 0.20 111, E 0.35

beams B 0.95 supported D 0.65 b) suspended 6.20 92, E 0.71 92, E 0.72 93, A 1.67 94, E 0.58 94, E 0.58 ceiling with B 1.35 "lights" C 1.17 D 1.06 c) gypsum board 6.36 109, E 0.44 109, E ceiling d) steelaeoprene 7.8 115, E 1 .O 115, E 114, A 1.74 113, E 0.95 113, E 1.01 damping strips C 1.65 *Analysis bandwidth = 0.060 Hz.

1

TABLE 7

Comparison of Measured and C a l c u l a t e d I n i t i a l Peak A c c e l e r a t i o n Amplitudes from H e e l Impacts

I n i t i a l Peak A c c e l e r a t i o n , a.

,

E q u i v a l e n t % g r a v i t y Measured I n e r t i a l E q u i v a l e n t

Run No. and Fundamental Mass, Mass, C a l c u l a t e d F l o o r L o c a t i o n of Frequency, MI, Eq. ( 4 ) ,

MJ,

Eq. ( 3 ) , from Eq. ( 2 )

C o n d i t i o n Heel Impact Hz kg kg Using MI Using MJ Measured

End and Edge Beams Supported

8 1 mm s l a b 33 A 6.57 6350 3540 4.69 7.18 4.72 B 3.66 3.65 C 3.50 3.65 D 3.15 3.15 81 mm s l a b , 155 A t r a n s v e r s e B c o n c r e t e beam C

D

106 mm s l a b , 190 A t r a n s v e r s e B c o n c r e t e beam C

D

End and Edge Beams Unsupported

81 mm s l a b 8 A 5 -43 9 B

10 C

11 D

TABLE

8

Comparison of Measured and Calculated Fundamental

Frequencies for Floor Sample

Fundament

a1

Natural Frequency, Hz

-- -

Floor Condition

Run No.

Measured

Calculated

End and Edge Beams Supported

81

mm slab

33 81

mm slab

transverse

concrete beam

106 mn

slab,

transverse

I

concrete beam

183 6 . 2 5

End and Edge Beams Unsupported

Composite Noncomposit

e

81

mm slab

81

mm slab,

transverse

concrete beam

106

mm slab,

transverse

concrete beam

166-180 5 . 3

4.3

5 . 2

3 CONCRETE BEAM ADDED UNDER S U B W 8 x 24 COLUMNS 1 829 rnm HIGH 76 mrn CONCRETE SLAB, INCREASED TO 102 rnm AFTER RECORD N O . 166

X TEMPORARY JACK SUPPORTS

Figure 1 . Layout of f l o o r sample

0 ACCELEROMETER STATIONS ( I TO 28) LOCATION OF DYNAMIC FORCE (A TO E)

Figure 2 . Locations of accelerometer s t a t i o n s (

0

_{) and dynamic f o r c e}

F R E Q U E N C Y , H z

Figure

3.

Fourier amplitude spectrum at location 21, run 118, shaker random excitation at location

E,

81 mm concrete slab,

steel-neoprene-steel sandwich strips, unsupported end and edge beams

F R E Q U E N C Y , H z

Figure

4.

Fourier amplitude spectrum at location 21, run 115, shaker impact at location

E,

81 mm concrete slab, steel-neoprene- steel sandwich strips, supported end and edge beams

0 1 2 3 4 5 6 7

T I M E , s

F i g u r e 5. F r e e v i b r a t i o n decay a t l o c a t i o n s 4 , 5, r u n 182, s h a k e r impact a t l o c a t i o n A, 2 p e o p l e s t a n d i n g a t l o c a t i o n A, 106 mm

c o n c r e t e s l a b w i t h 152 x 229 mm t r a n s v e r s e c o n c r e t e beam, mode 1 , u n s u p p o r t e d end and edge beams

0 . 2 5

0 1 2 3 4 5 _{6 7}

TIME, s

F i g u r e 6. F r e e v i b r a t i o n decay a t l o c a t i o n s 4 , 5, r u n 182, s h a k e r impact a t l o c a t i o n A , mode 1 , 106 mm c o n c r e t e s l a b w i t h 152 x 229 mm t r a n s v e r s e c o n c r e t e beam, u n s u p p o r t e d end and edge beams

a ) L I N E A R A M P L I T U D E 0 . 3 > I-

-

> a @L C3 Z 0

-

0 C a @L W -I W 0 0 a -0.3 0 1 2 3 4 5 T I M E , s T I M E , s b ) L O G A R I T H M I C A M P L I T U D E S C A L E ( d B )

Figure 7 . Free v i b r a t i o n decay a t l o c a t i o n s 3 , 6 , run 1 6 1 , shaker impact a t l o c a t i o n B , mode 2 , 81 mm concrete s l a b with 152 x 229 mm transverse concrete beam, unsupported end and edge beams

M O D E S H A P E S A C R O S S J O I S T S

Figure 8. Mode shapes and natural frequencies of f l o o r sample, run 5 ,

shaker random e x c i t a t i o n a t l o c a t i o n D , 81 mm concrete s l a b , unsupported end and edge beams

M O D E S H A P E S

A L O N G J O l S T S M O D E S H A P E S A C R O S S J O I S T S

Figure 9. Mode shapes and natural frequencies of f l o o r sample, run 2 9 , shaker impact a t l o c a t i o n D , 81 mm concrete s l a b , unsupported end and edge beams

M O D E S H A P E S A C R O S S J O I S T S

Figure 10. Mode shapes and natural frequencies of f l o o r sample, run 111, shaker random e x c i t a t i o n a t l o c a t i o n E , 81 mm concrete s l a b , supported end beams, unsupported edge beams

US l NG GENERAL1 ZED COORD

I NATES, THE D l SPLACEMENT I S

u(x, y, t)

=$!Ax,

y) z(t)

THE EQUATIONS OF M O T I O N ARE

/ljz

t

ci

+

Kz

=

WHERE

q l x , y)

=

MODE SHAPE

z ( t )

=

GENERALIZED COORD INATE

M

=

MODAL MASS

=bq2

d~

+C

m.9.'

I I

P

=

MODAL FORCE =lpt,!fd~

+I

pi

qi

AND

A

=

AREA OF PLATE

E l

=

FLEXURAL STIFFNESS