Publisher’s version / Version de l'éditeur:
Canadian Acoustics/Acoustique Canadienne, 14, 2, pp. 12-21, 1986-04
READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. https://nrc-publications.canada.ca/eng/copyright
Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.
Questions? Contact the NRC Publications Archive team at
PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.
NRC Publications Archive
Archives des publications du CNRC
This publication could be one of several versions: author’s original, accepted manuscript or the publisher’s version. / La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur.
Access and use of this website and the material on it are subject to the Terms and Conditions set forth at
Vertical dynamic forces from footsteps
Rainer, J. H.; Pernica, G.
https://publications-cnrc.canada.ca/fra/droits
L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.
NRC Publications Record / Notice d'Archives des publications de CNRC:
https://nrc-publications.canada.ca/eng/view/object/?id=306f57be-f464-4040-814c-fb37e92b4dd0 https://publications-cnrc.canada.ca/fra/voir/objet/?id=306f57be-f464-4040-814c-fb37e92b4dd0Ser
T H ~
National Research
Conseil national
H2I-d
I+
Council Canada
de
recherches Canada
no. 1371
I c . 2
Institute for
lnstitut de
B I ~ G
IResearch in
recherche en
-
Construction
construction
Vertical Dynamic Forces from Footsteps
by J.H. Rainer and G. Pernica
Reprinted from
Canadian Acoustics
Vol. 14, No. 2, April 1986
p. 12-21
(IRC Paper No. 1371)
Price $2.00
NRCC 25879
\*
NRC-
C I S T IBLDG. RES.
L I B R A - R Y
86-
41-
3
T h i s p a p e r i s b e i n g d i s t r i b u t e d i n r e p r i n t form by t h e I n s t i t u t e f o r R e s e a r c h i n C o n s t r u c t i o n . A l i s t of b u i l d i n g p r a c t i c e and r e s e a r c h p u b l i c a t i o n s a v a i l a b l e from t h e I n s t i t u t e may be o b t a i n e d by w r i t i n g t o t h e P u b l i c a t i o n s S e c t i o n , I n s t i t u t e f o r R e s e a r c h i n C o n s t r u c t i o n , N a t i o n a l Research C o u n c i l of C a n a d a , O t t a w a , O n t a r i o , K l A
OR6.
Ce document e s t d i s t r i b u g s o u s forme de t i r 6 - a - p a r t p a r 1 ' I n s t i t u t de r e c h e r c h e e n c o n s t r u c t i o n . O n p e u t o b t e n i r une l i s t e d e s p u b l i c a t i o n s de 1 ' I n s t i t u t p o r t a n t s u r l e s t e c h n i q u e s oul e s
r e c h e r c h e s e nmatisre
d e b d t i m e n t e n 6crivant: 3 l a S e c t i o n d e s - - p u b l i c a t i o n s , - 1 _ A c o n s t r u c t i r e c h e r c h e s KIAOR6,
VERTICAL DYNAMIC FORCES FROM FOOTSTEPS J.H. R a i n e r and G . P e r n i c a
Noise and V i b r a t i o n S e c t i o n D i v i s i o n of B u i l d i n g Research N a t i o n a l Research Council Canada
Ottawa, Canada, K1A OR6
ABSTRACT
V e r t i c a l dynamic f o r c e s from w a l k i n g and r u n n i n g were measured by l o a d c e l l s p l a c e d between temporary s u p p o r t s and t h e c e n t r e of a 17-m-long f l o o r s t r i p . The measurements showed t h a t t h e dynamic f o r c e s a r e composed of wave t r a i n s of harmonics of t h e walking o r r u n n i n g r a t e . The
s i g n i f i c a n t low f r e q u e n c y c o n t r i b u t i o n s a r e g e n e r a l l y c o n t a i n e d w i t h i n t h e f i r s t t h r e e t o f o u r harmonics.
To r e p r e s e n t t h e c o n t r i b u t i o n of e a c h harmonic t o t h e t o t a l dynamic f o r c e s , a 'dynamic l o a d f a c t o r , a ' was d e f i n e d a s t h e r a t i o of dynamic f o r c e a m p l i t u d e of t h e harmonic t o t h e w e i g h t of t h e person. F o o t s t e p f o r c e s were produced by t h r e e male s u b j e c t s ; f o r walking, t h e maximum dynamic l o a d f a c t o r s measured and c o r r e s p o n d i n g f r e q u e n c i e s are:
al = 0.56 a t 2.4 Hz f o r t h e f i r s t harmonic, a2 = 0.28 a t
5.4
Hz f o r thesecond, a 3 = 0.12 a t around 7.8
Hz
f o rthe
third, and ab = 0.08 f o r the f o u r t h harmonic. For r u n n i n g , maximum values of a1 were 1.45 a t3.0
to3.5 Hz
f o r t h e f i r s t harmonic,a2
= O,4 f o r t h e second, a3 = 0.2for the
t h i r d , and a,, = 0.1 o r less f o r t h e f o u r t h harmonic.Les f o r c e s dynamiques v e r t i c a l e s du p a s d e marche e t du p a s d e c o u r s e o n t 4 t 6 mesurges a u moyen d e c e l l u l e s e x t e n a o m 6 t r i q u e s p l a c 6 e s e n t r e d e s s u p p o r t s t e m p o r a i r e s e t l e c e n t r e d'une p i s t e d e 17 m. Les mesures o n t r 6 v b l 6 que l e s f o r c e s dynamiques s e composent d e t r a i n s d'harmoniques d e l a f r d q u e n c e du pas. Les c o n t r i b u t i o n s i m p o r t a n t e s 3 b a s s e f r d q u e n c e s o n t ggnbralement a s s o c i g e s aux t r o i s ou q u a t r e p r e m i z r e s harmoniques.
Pour r e p r d s e n t e r l a c o n t r i b u t i o n d e chacune d e s harmoniques aux
f o r c e s dynamiques t o t a l e s , on a d 6 f i n i un " f a c t e u r d e c h a r g e dynamique, a " comme d t a n t l e r a p p o r t d e l ' a m p l i t u d e d e l a f o r c e dynamique d e
l'harmonique e t du p o i d s d e l a personne. Les f o r c e s d e p a s o n t 6 t 6
produites par t r o i s hommes. Au pas d e marche, les f a c t e u r s d e c h a r g e
dynamlque maximaux mesur&s e t les f r g q u e n c e s c o r r e s p o n d a n t e s 6 t a i e n t : a ] =
0,56
3 2,4 H z pour l a premlsse harmonique, a 2 = 0 , 2 8 35,4
Hz pour l adeuxiZme, ag = 0,12
3
7,8 Hz e n v i r o n pour l a t r o i s i s m e , e t a,, = 0 , 0 8 pourl a quarrisme. Au p a s d e c o u r s e , Les f a c t e u r s maximaux 6 t a i e n t : a1 = 1,45
entre 3,O Hz e t
3,5
Hz pour l a p r e m i ~ l r e harmonique, a2 = 0 , 4 pour l a deuxidme, a 3 = 0 , 2 pour l a t r o i s i d m e , e t a 4 = 0 , l ou moins pour l a quatrieme.INTRODUCTION
Walking and r u n n i n g a r e two of t h e most cormon forms of dynamic e x c i t a t i o n produced by occupants i n b u i l d i n g s , y e t r e l a t i v e l y l i t t l e q u a n t i t a t i v e d a t a i s
a v a i l a b l e a b o u t them, To p r o v i d e some d a t a on t h i s t o p i c , a s t u d y of t h e f o r c e s produced by w a l k e r s and r u n n e r s was u n d e r t a k e n a t t h e D i v i s i o n of B u i l d i n g Research of t h e N a t i o n a l Research Council of Canada.
A number of p r e v i o u s s t u d i e s of f o r c e s induced by human a c t i v i t y c a n be found i n Refs. 1 t o 4. These s t u d i e s s i m u l a t e d t h e f o r c e s due t o a c o n t i n u o u s s e r i e s of s t e p s by combining d u p l i c a t e s of t h e f o r c e produced by a s i n g l e s t e p s e p a r a t e d by t i m e d e l a y s c o r r e s p o n d i n g t o t h e walking r a t e ( 3 , 4 ) . The p r e s e n t i n v e s t i g a t i o n measures t h e dynamic f o r c e s produced by a c o n t i n u o u s series of s t e p s ; p o s s i b l e i n a c c u r a c i e s a r i s i n g from t h e c r e a t i o n of sequences c o n s i s t i n g of s i n g l e s t e p p u l s e s a r e t h u s avoided. A knowledge of t h e f o r c e s produced by f o o t s t e p s i s of p r a c t i c a l s i g n i f i c a n c e : a ) i n u n d e r s t a n d i n g t h e n a t u r e of u n d e s i r a b l e f l o o r v i b r a t i o n s produced by o c c u p a n t s ; and b ) i n a f f o r d i n g t h e p o s s i b i l i t y of p r e d i c t i n g d u r i n g t h e d e s i g n s t a g e t h e p r o b a b l e b e h a v i o u r of a f l o o r s u b j e c t e d t o walking o r running. T h i s i s of e v e r - i n c r e a s i n g importance s i n c e l i g h t e r s t r u c t u r a l s y s t e m s and l o n g e r s p a n s t e n d t o make f l o o r s more prone t o occupancy-induced v i b r a t i o n s .
TEST PROCEDURE AND ANALYSIS
The f o r c e s f r o m w a l k i n g and r u n n i n g were measured a t mid-span of a s i m p l y
s u p p o r t e d f l o o r s t r i p c o n s i s t i n g of two open-web j o i s t s 914 mm deep, spaced a t 1.76 m and s p a n n i n g 17.04 m. The j o i s t s a r e c o v e r e d by p r e - c a s t c o n c r e t e p a n e l s of 100 mm t h i c k n e s s , e a c h measuring 2.13 m a c r o s s and 1.19 m a l o n g t h e f l o o r span. A p l a n view and e l e v a t i o n a r e shown i n F i g u r e 1.
A t c e n t r e s p a n under e a c h j o i s t a p i e z o e l e c t r i c f o r c e t r a n s d u c e r was i n s e r t e d between a temporary s u p p o r t and t h e bottom c h o r d , and pre-loaded t o a p p r o x i m a t e l y
I
PLAN PRECAST CONCRETE , ,STEEL TRUSSES I E L E V A T I O N1
'FORCE T R I \ N ~ C ~ ON TEMPORARY SUPPORT D I M E N S I O N S I N m m F i g u r e 1. T e s t f l o o r s t r i p .4000
N.
The s i g n a l s from t h e two t r a n s d u c e r s were added, a m p l i f i e d and r e c o r d e d . A n a l y s i s commenced w i t h h i g h - p a s s f i l t e r i n g t h e s i g n a l s a t 0.2 Hz t o e l i m i n a t e t h e low frequency d r i f t a s t h e s u b j e c t moved a c r o s s t h e f l o o r s t r i p . The s i g n a l s were t h e n d i g i t i z e d and a n a l y z e d on a F o u r i e r a n a l y z e r and by F a s t F o u r i e r Transform (FFT) r o u t i n e s on a minicomputer. D i g i t a l f i l t e r i n g was c a r r i e d o u t by 'windowing' i n t h e f r e q u e n c y domain. The peak a m p l i t u d e s n e a r t h e middle of t h e s i g n a l t r a i n were u s e d a s a measure of s i g n a l magnitudes ( s e e F i g s . 2 and 3 ) .- 6 0 0 1 3 5 7 9 360 TIME. S 180 0 0 5 F R E Q U E N C Y . H z b ) SECOND H A R M O N I C , 4 . 8 Hz
H
36,-,t
c l THIRD H A R M O N I C . 7 . 2 Hz1
T I M E , sF i g u r e 2. Walking f o r c e s a t F i g u r e 3, Harmonic components of f o r c e 2.4 s t e p s / s e c o n d , s u b j e c t A. from walking a t
2.4 s t e p s / s e c o n d , s u b j e c t A.
The t e s t s were conducted by p l a y i n g p r e - r e c o r d e d p u l s e s a t t h e d e s i r e d walking o r running r a t e through l o u d s p e a k e r s , and r e q u e s t i n g t h e s u b j e c t t o walk o r r u n a l o n g t h e f l o o r s t r i p a t t h e s p e c i f i e d r a t e u s i n g a s t r i d e of h i s own choosing. For walk t e s t s i n which s t r i d e was v a r i e d , t h e s u b j e c t walked f i r s t a t h i s n a t u r a l s t r i d e , and t h e n a t a s h o r t e r and a l o n g e r one.
To compensate f o r t h e dynamic a m p l i f i c a t i o n e f f e c t s n e a r t h e lowest r e s o n a n c e f r e q u e n c y of t h e s u p p o r t e d f l o o r s t r i p , c o r r e c t i o n f a c t o r s were determined from 2 t o
10 Hz by comparing t h e l o a d c e l l o u t p u t w i t h t h e known s h a k e r f o r c e t h a t was a p p l i e d a t c e n t r e span.
S i n c e t h e lowest r e s o n a n c e frequency of t h e i n s t r u m e n t e d f l o o r s t r i p was 12 Hz, t h e u s e f u l r e s u l t s were t h u s l i m i t e d t o f r e q u e n c i e s below a b o u t 10 Hz. Above t h a t I f r e q u e n c y , t h e dynamic a m p l i f i c a t i o n becomes l a r g e and r a t h e r s e n s i t i v e t o damping. A f r e q u e n c y r a n g e from 0.2 t o 10 Hz was c o n s i d e r e d a d e q u a t e f o r p r e s e n t p u r p o s e s s i n c e t h e m a j o r i t y of problems a s s o c i a t e d w i t h f o o t s t e p s have o c c u r r e d i n f l o o r s w i t h fundamental f r e q u e n c i e s below a b o u t 8 Hz (5).
CHARACTERISTICS OF FOOTSTEP FORCES
The d i s t i n c t i o n between w a l k i n g and r u n n i n g i s t h a t d u r i n g w a l k i n g t h e p e r s o n always h a s one f o o t i n c o n t a c t w i t h t h e f l o o r o r ground, whereas d u r i n g r u n n i n g , c o n t a c t i s l o s t . The w a l k i n g o r r u n n i n g f r e q u e n c y i n Hz i s d e f i n e d a s t h e r a t e a t which t h e f e e t make c o n t a c t w i t h t h e f l o o r . Walking and r u n n i n g were performed by t h r e e male subjects, whose p h y s i c a l c h a r a c t e r i s t i c s a r e g i v e n i n Table 1.
TABLE 1
P h y s i c a l C h a r a c t e r i s t i c s of Male T e s t S u b j e c t s
Weight Height Age
S u b j e c t
(N
(m) ( Y )-
A 7 3 4 1.82 48
walking
A t y p i c a l t i m e r e c o r d of w a l k i n g f o r c e s , l o w - p a s s f i l t e r e d a t 10 Hz, i s shown i n F i g u r e 2a. The r e c o r d c o n t a i n s o n l y t h e dynamic p o r t i o n of t h e induced f o r c e s , a s t h e s t a t i c components have been e l i m i n a t e d by t h e 0.2 Hz h i g h p a s s f i l t e r . These f o r c e s a r e bounded by a p a r a b o l i c a l l y shaped e n v e l o p e which i s t h e s t a t i c i n f l u e n c e l i n e f o r t h e mid-span s u p p o r t of t h e f l o o r s t r i p ( 6 ) . The F o u r i e r a m p l i t u d e s p e c t r u m of t h i s f o r c e r e c o r d i s shown i n F i g u r e 2b. The spectrum shows t h a t t h e f o r c e s
produced by walking c o n s i s t of d i s t i n c t f r e q u e n c y components a t i n t e g e r m u l t i p l e s of t h e walking r a t e , w i t h s p e c t r a l a m p l i t u d e s t h a t d e c r e a s e w i t h i n c r e a s i n g frequency. The f i r s t t h r e e o r f o u r harmonics comprise t h e main dynamic components of walking f o r c e s i n t h e frequency r a n g e of i n t e r e s t . A s t h e t i m e r e c o r d i n F i g u r e 2a shows, t h e f o r c e s a r e p e r i o d i c , w i t h a r e p e t i t i o n r a t e e q u a l t o t h e w a l k i n g r a t e . Runnine The t i m e s i g n a t u r e s of f o r c e s from r u n n i n g a r e s i m i l a r t o t h o s e f o r w a l k i n g , e x c e p t t h a t t h e p u l s e s a s s o c i a t e d w i t h e a c h s t e p a r e z e r o d u r i n g t h e t i m e t h e r u n n e r i s a i r b o r n e . N e v e r t h e l e s s , t h e F o u r i e r s p e c t r u m of t h e f o r c e r e c o r d a g a i n c o n t a i n s d i s c r e t e f r e q u e n c y components a t i n t e g e r m u l t i p l e s of t h e running r a t e , a s shown i n F i g u r e 4.
3
6
9
F R E Q U E N C Y ,
H z
Figure
4.
Fourier amplitude spectrum
for running at
3.0 stepslsecond.
MATHEMATICAL REPRESENTATION OF FOOTSTEP FORCES
Walking
The analysis of force records indicates that the walking forces (F(t))
can be
represented by the following expression:
F(t)
=P(1
+
an sin (n2xft
+
.
n=
1
Here,
P
=static weight of subject;
a
=Fourier amplitude or Fourier coefficient;
n
=order of harmonic of the walking rate (n
=1, 2, 3..
-1;
f =
rate of walking in
Hz;
t
=time variable;
$ =
relative phase angle;
N =
total number of harmonics.
The dynamic component of the walking force in Eq. (1) is represented by the
summation term, which is a Fourier series with Fourier coefficients (an) at the
discrete frequencies (nf).
From the analysis of time records, the relative phase
angle
(9,)
was determined to be about 0°, 90°, and 0
'
for n
=1, 2, and 3,
respectively, for walking rates between 2.0 and
2.4
Hz.The time signatures of the
three lowest and most significant harmonics that comprise the walking forces in
Figure 2 are presented in Figures 3a, b and
c.These were obtained from the time
record shown in Figure 2a by digital bandpass filtering in the frequency domain using
Fourier transform techniques
(7).
The centre amplitudes of the harmonics, normalized
by the weight
(P), then represent the Fourier coefficients (
a
,
)
.
Running
Periodic dynamic forces produced by running can also be represented by Eq.
( 1 ) .However, the relative phase angles of the harmonics are not as clearly defined as
those for walking.
VARIATION OF DYNAMIC FORCES WITH STEP FREQUENCY Walking
The key p a r a m e t e r s i n Eq. (1) t h a t d e s c r i b e t h e dynamic f o r c e from walking a r e t h e F o u r i e r c o e f f i c i e n t s ( a n ) and t h e walking r a t e o r s t e p frequency ( f )
.
I n amanner s i m i l a r t o t h a t used i n t h e d e s c r i p t i o n of r h y t h m i c f o r c e s ( 8 ) , t h e 0. h F o u r i e r c o e f f i c i e n t s ( a ) a r e c a l l e d H A R M O N l C 'dynamic l o a d f a c t o r s , '"defined a s t h e r a t i o 1 of t h e dynamic f o r c e a m p l i t u d e of e a c h 2 0 3 0 harmonic t o t h e weight of t h e s u b j e c t . I n 4 v t h i s p a p e r , t h e v a r i a t i o n of a w i t h s t e p
0.2 ,a-d 0-9
-
f r e q u e n c y was s t u d i e d f o r walking r a t e s f r o m1.0 t o 3.0 Hz u s i n g t h r e e d i f f e r e n t male
0 s u b j e c t s (A, B and C, Table 1 ) . The r e s u l t s
0 1 2 3 4 5 6 7 8 9 1 0 a r e shown i n F i g u r e s 5a, b, and c.
a
F R E Q U E N C Y , H z
F i g u r e 5. Dynamic l o a d f a c t o r f o r walking.
A t v e r y low w a l k i n g r a t e s ( n e a r 1.0 Hz) t h e f i r s t f o u r harmonics have comparable dynamic l o a d f a c t o r s . A d d i t i o n a l h i g h e r harmonics a r e a l s o e v i d e n t , a s shown i n F i g u r e 6 by t h e F o u r i e r a m p l i t u d e s p e c t r u m of a walking r e c o r d a t 1.0 Hz. A t h i g h e r f o o t s t e p f r e q u e n c i e s t h e dynamic l o a d f a c t o r of t h e f i r s t harmonic ( a l ) i s t h e l a r g e s t and r e a c h e s a maximum n e a r 2.4 Hz f o r a l l t h r e e s u b j e c t s . For s u b j e c t s A and B t h e f o l l o w i n g c a n be observed from F i g u r e s 5a and
b.
The maximum dynamic l o a d f a c t o r s f o r t h e second harmonic ( a 2 ) a r e somewhat l e s s t h a n h a l f t h e maxima of a l . T h i s peak o c c u r s a t a b o u t 5.4 Hz, s l i g h t l y more t h a n t w i c e t h e f r e q u e n c y of t h e maximum f o r a l . The dynamic l o a d f a c t o r s of t h e t h i r d harmonic ( a g ) a r e a l s o somewhat l e s s t h a n h a l f of a*, w i t h t h e peak o c c u r r i n g n e a r 7.8 Hz. T h i s i s s l i g h t l y more t h a n t h r e e t i m e s t h e f r e q u e n c y of t h e maximum dynamic l o a d f a c t o r f o r t h e f i r s t harmonic. T h i s s h i f t i n f r e q u e n c i e s of t h e maximum dynamic l o a d f a c t o r s f o r t h e second and t h i r d harmonics from i n t e g e r m u l t i p l e s of t h e O51
1
f r e q u e n c y a t which t h e maximum ( a , ) o c c u r s Y c a n be a t t r i b u t e d t o t h e f a c t t h a f h i g h e r n 3 walking r a t e s produce l a r g e r f o r c e s d u r i n g +-
t h e h e e l l a n d i n g and t o e push-off p h a s e s o f Z ~3 impact. The r e s u l t i s a r e l a t i v e l y l a r g e r 4 I p r o p o r t i o n of t h e h i g h e r harmonics t h a n a t lower walking r a t e s . n 0 10 F R E Q U E N C Y , H z The r e s u l t s f o r s u b j e c t C ( F i g . 5c)d i f f e r somewhat from t h o s e of s u b j e c t s A and B ( F i g s . 5a and b ) i n t h a t t h e second and F i g u r e 6. F o u r i e r a m p l i t u d e s p e c t r u m t h i r d harmonics do n o t e x h i b i t t h e same
f o r walking a t 1.0 s t e p s / shape a s t h o s e f o r t h e o t h e r two s u b j e c t s . second, s u b j e c t B. S u b j e c t C shows a r e l a t i v e l a c k of second
i
and t h i r d harmonic components i n h d s walking f o r c e s a t walking 2.6 Hz, and t h e dynamic l o a d f a c t o r s r e a c h a maximum a t f r e q u e n c i e s of 6 Hz f o r t h e rates between 2 and 4!
second harmonic and 9 Hz f o r t h e t h i r d , c o r r e s p o n d i n g t o a 3 Hz walking r a t e . On t h e Io t h e r hand, t h e r e i s good agreement i n t h e v a r i a t i o n of a l among t h e t h r e e s u b j e c t s ,
I
b o t h i n t h e g e n e r a l s h a p e and t h e frequency a t which t h e maxima o c c u r . Maximum-
v a l u e s of t h e dynamic l o a d f a c t o r s among t h e t h r e e t e s t s u b j e c t s were: a l = 0.56,I
a2 0.28, a3 = 0.12, and a4 = 0.08. This e x c l u d e s t h e extreme v a l u e s n e a r 9 Hz f o r
I s u b j e c t C. a ) 2 . 0 H z 0 . 7 0 . 8 0.9 1 . 0 S T R I D E , rn The d a t a f o r t h e t h r e e s u b j e c t s a t e a c h S T R I D E , rn c 0.6 u oc- E 0 . 4 - i5 n a 9 0 . 2
-
2z I=;
F i g u r e 8. V a r i a t i o n of dynamic l o a d f a c t o r w i t h s t e p l e n g t h and w a l k i n g r a t e . The dynamic l o a d f a c t o r s ( a ) f o r r u n n i n g a r e p l o t t e d a g a i n s t r u n n i n g r a t e f o r t h e t h r e e male s u b j e c t s i n F i g u r e s 9a, b and c. The r e s u l t s show t h e f i r s t harmonic t o be t h e most s i g n i f i c a n t , i n c r e a s i n g m o n o t o n i c a l l y from a p p r o x i m a t e l y 0.5 a t 1.5 Hz t o a maximum of 1.5 a t a f r e q u e n c y of 3.6 Hz. T h e r e a f t e r a g r a d u a l d e c r e a s e i n a1 i si n d i c a t e d . The second harmonic i n i t i a l l y d e c r e a s e s from a h i g h of a b o u t 0.4 a r o u n d 3.0 Hz, t o a low of 0.1 around 4.0 Hz, and t h e n g r a d u a l l y i n c r e a s e s t o between 0.35 and 0.47 around 7.0 Hz. The t h i r d harmonic remains r e l a t i v e l y c o n s t a n t a t l e s s t h a n
0 1 2 3 4 5 6 7 8 9 1 0 t h e above v a l u e s . T h i s a g a i n i g n o r e s t h e FREQUENCY. Hz 9 Hz d a t a p o i n t , which is s t r o n g l y i n f l u e n c e d by t h e c o n t r i b u t i o n from s u b j e c t C. F i g u r e 7. Averaged dynamic l o a d f a c t o r s f o r walking, V a r i a t i o n of a w i t h l e n g t h of s t e p The s u b j e c t s A, B and C. v a r i a t i o n of a w i t h s t e p l e n g t h w a s i n v e s t i g a t e d a t t h e 2 and 2.4 Hz walking r a t e s u s i n g s u b j e c t A. The r e s u l t s a r e shown i n F i g u r e 8. The 0.9 m s t e p l e n g t h was t h e n a t u r a l s t r i d e of t h e walker. For t h e t h r e e s t e p l e n g t h s i n v e s t i g a t e d h e r e , t h e dynamic l o a d f a c t o r s i n c r e a s e w i t h s t e p l e n g t h f o r b o t h t h e 2 and t h e 2.4 Hz w a l k i n g r a t e s .
--
H A R M O N I C/"4,
1 A 2-
P
hi"
3 o 4 0-
P
-
o ,pdOp0 a' o 0.l2-'p'
m"~~+."r-rP=O-wvrv
- f r e q u e n c y were a v e r a g e d and a r e p l o t t e d i n F i g u r e 7; a f o u r t h d e g r e e polynomial c u r v e was f i t t e d t o t h e r e s u l t i n g p o i n t s . The maximum v a l u e s f o r t h i s polynomial f i t a r e : a1 = 0.52 a t 2.4 Hz, a = 0.24 a t 5.6 Hz, aa = 0.06 a t around 682,
and a,, = 0.05 a t around 8 Hz. ag and a,, c o u l d w e l l be0
0 2 3 - 4 5 6 7 8 9 1 0
F R E Q U E N C Y , H z
F i g u r e 9. Dynamic l o a d f a c t o r f o r running.
0.2 between 4 and 8 Hz. For t h e f o u r t h
harmonic, t h e dynamic l o a d f a c t o r a i s less
t h a n 0.09 f o r a l l f r e q u e n c i e s i n v e s t i g a t e d . The p l o t of t h e a v e r a g e s of t h e dynamic l o a d f a c t o r s among t h e t h r e e t e s t s u b j e c t s i s shown i n F i g u r e 10. A f o u r t h d e g r e e polynomial f i t g i v e s a maximum v a l u e of
al
= 1.40 a t 3.6 Hz, a2 = 0.40 a t 3Hz
and 0.35 a t 6.5 Hz; a3 and a are r e l a t i v e l y c o n s t a n t a t 0.12 and 0.08, r e s p e c t i v e l y . A summary of t h e dynamic l o a d f a c t o r s f o r w a l k i n g and r u n n i n g i s p r e s e n t e d i n T a b l e 2.DISCUSSION AND SUMMARY
The t e c h n i q u e u s e d t o measure s t e p f o r c e s r e s t r i c t s t h e u s e f u l d a t a t o below 12 H z , t h e l o w e s t r e s o n a n c e f r e q u e n c y of t h e t e s t s t r u c t u r e . While t h e r e a r e h i g h
f r e q u e n c y components p r e s e n t i n f o o t s t e p s , i t i s t h e low f r e q u e n c y ones t h a t c a u s e most v i b r a t i o n problems due t o w a l k i n g o r r u n n i n g i n b u i l d i n g s . The F o u r i e r s e r i e s r e p r e s e n t a t i o n of t h e dynamic components of f o o t s t e p f o r c e s s i m p l i f i e s a n a l y t i c a l c a l c u l a t i o n s of dynamic r e s p o n s e s of f l o o r s and o t h e r b u i l d i n g components. T h i s p e r m i t s t h e t r e a t m e n t of a dynamic p r o c e s s t h a t h a s s o f a r e l u d e d s i m p l e a n a l y s i s . Not o n l y i s t h e
1. The maximum dynamic l o a d f a c t o r of t h e f i r s t harmonic f o r r u n n i n g i s
a p p r o x i m a t e l y t h r e e times a s l a r g e a s t h e c o r r e s p o n d i n g a l f o r walking. Theie maxima o c c u r around 2.4
Hz
f o r w a l k i n g and between 3.0 and 3.5Hz
f o r r u n n i n g .n u m e r i c a l e v a l u a t i o n of t h e r e s p o n s e c 1.6 u 2 1 . 4 1 . 2 - 2 l . o L 0 . 8 - 3 0 . 6 - 0 . 4 -
42
0 . 2 - > n 0 1-
H A R M O N l C p o s s i b l e , b u t a r e a l i s t i c i n t u i t i v e d a s s e s s m e n t of t h e c a u s e s of v i b r a t i o n/ problems from f o o t s t e p s becomes f e a s i b l e .
1
It i s a p p a r e n t from t h e c h a r a c t e r i z a t i o n of t h e s p e c t r u m of t h e f o r c e s t h a t a r e s o n a n t o c o n d i t i o n c a n o c c u r when t h e w a l k i n g o r 0 . 0- I I I g-o-ota-y r u n n i n g r a t e , o r a n i n t e g e r m u l t i p l e of i t , 0 1 2 3 4 5 6 7 8 9 1 0 c o i n c i d e s w i t h a n a t u r a l f r e q u e n c y of t h e f l o o r . T h i s c o i n c i d e n c e can c a u s e a l a r g e FREQUENCY. Hz dynamic r e s p o n s e i n f l o o r s h a v i n g low damping. F i g u r e 10. Averaged dynamic l o a d f a c t o r s f o r r u n n i n g , A comparison of dynamic l o a d f a c t o r ss u b j e c t s A, B and C. f o r w a l k i n g and r u n n i n g shows t h e f o l l o w i n g .
TABLE 2
Maximum Dynamic h a d Factors (a) for Walking and Running
-
Maximum from polynomial
Maximum from
fit to averages from
subject
A, B or C*
subjects
A,
B, C*
Harmonic
Frequency
Frequency
Activity
(n)(Hz
a
(Hz
)a
Walking
1
2.4
0.56
2.4
0.52
2
5.2
0.28
5.6
0
-24
3
4.2
-
7.8
0.12
5.8
0.06
4
8
.O
0.08
7.6
0.05
Running
1
3.6
1.50
3.6
1.40
2
6.8
0.47
3.2
0.40
6.6
0.35
3
4.2
0.20
6
-
8
0.12
4
7
-
9
0.09
7 - 9
0.08
*Except for values corresponding to 9.0 Hz and above.
2.
The ratio of the dynamic load factors of the first to the second harmonic (a1/a2)
is considerably larger for running than for walking. In absolute terms, however,
a2 for running is slightly higher than for walking but is shifted to a higher
frequency.
The
low point in a2 occurs around 3.0 Hz for walking and around
4.0
Hz
for running.
3.
The dynamic load factors for the third and fourth harmonics for running are only
slightly larger than those for walking.
The above comparison of dynamic load factors shows that running represents
amore severe dynamic loading than walking for all four harmonics. While this is not
surprising in a qualitative sense, the data provide the quantitative information to
substantiate this widely-held view.
Although the measurements have been taken over a wide frequency range, regular
walking occurs generally at or near 2.0 Hz, and recreational running from about 2.4
to 3.2 Hz. Thus the measured maximum force values of the lowest harmonic occur near
the most common walking or running rates.
ACKNOWLEDGEMENT
This paper is a contribution of the Division of Building Research (DBR),
National Research Council Canada. The measurements were conducted on a test floor
designed by D.E.
Allen of DBR. The assistance of
C.
Pilette, H.H. Ireland,
E.C. Luctkar, and R. Glazer in performing the tests and analyzing the results is
gratefully acknowledged.
REFERENCES
1. Tuan, C.Y. and Saul, W.E. "Loads Due t o S p e c t a t o r Movements." J. S t r u c t . Eng-, ASCE, Vol. 111, No.-2, p. 418-434, 1985.
2. Harper, F.C., Warlow, W.J., a n d - C l a r k , B.L. "The Forces Applied t o t h e Floor by t h e Foot i n Walking." Building Research S t a t i o n , National Building S t u d i e s Research Paper No. 32, HMSO, London,
U.K.,
196 1.3. Ohlsson, S. "Floor V i b r a t i o n s and Human Discomfort." Department of S t r u c t u r a l Engineering, Chalmers U n i v e r s i t y of Technology, Ggteborg, Sweden, 1982.
4.
Nilsson, L. "Impact Loads Produced by Human Motion." Swedish Council f o r Building Research, Stockholm. P a r t 1: Document D13:1976, P a r t 2: Document D20: 1980.5. Allen, D.E. and Rainer, J.H. "Vibration C r i t e r i a f o r Long-Span ~ l o o r s . " Can.
J. Civ. Eng., Vol. 3, No. 2, p. 165-173, 1976.
6. Timoshenko, S. and Young, D.H.
Theory.
McGraw-Hill Book Company,Inc., New York, Chapter 111, 1945.
7. Rainer, J.H. "Applications of t h e F o u r i e r Transform t o t h e Processing of V i b r a t i o n Signals." Building Research Note, I n p r e p a r a t i o n .
8. Allen, D.E., Rainer, J.H., and Pernica, G. "Vibration C r i t e r i a f o r Assembly Occupancies." Can. J. Civ. Eng., Vol. 12, No. 3, p. 617-623, 1985, NRCC 24813.
