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Vibrations of building structures caused by human activities: case

study of a gymnasium

(2)

CANADA INSTITUTE

FOR SCIENTIFIC AND TECHNICAL

INFORMATION

セセ

1JJ-14

jスセl

[a».

rr

'}o}1-ISSN 0077-5606

INSTITUT CANADI EN

DE L'INFORMATIOI'J SCIENTIFIQUE

ET TECHNIQUE

NRC/CNR TT-2077

TECHNICAL TRANSLATION TRADUCTION TECHNIQUE

H. BACHMANN

VIBRATIONS OF BUILDING STRUCTURES CAUSED BY

HUMAN ACTIVITIES, CASE STUDY OF A GYMNASIUM

SCHWEIZER INGENIEUR UND ARCHITEKT, 101 (6): 104-110, 1983

TRANSLATED BY/TRADUCTION DE

W.R. SCHRIEVER

THIS IS NUMBER 250 IN THE SERIES OF TRANSLATIONS

PREPARED FOR THE DIVISION OF BUILDING RESEARCH

TRADUCTION NUMERO 250 DE LA SERlE PREPAREE POUR

LA DIVISION DES RECHERCHES EN BATIMENT

OTTAWA

1984

1+

National Research

Council Canada

Conseil national

de recherches Canada

(3)

CONSEIL NATIONAL DE RECHERCHES CANADA Author/Auteur: TECHNICAL TRANSLATION TRADUCTION TECHNIQUE H. Bachmann. 2077 TitlelTi tre: Reference/Reference: Trans1ator/Traducteur:

Vibrations of building structures caused by human

activities, case study of a gymnasium.

Schweizer Ingenieur und Architekt, 101 (6): 104-110,

1983.

W.R. Schriever.

Canada Jnstitute for Scientific and Technical Information

Institut canadien de l'information scientifique et technique

Ottawa, Canada KlA OS2

(4)

Vibration problems have recently occurred in long-span floor structures

supporting rhythmic activities such as dancing, rock concerts and jumping

exercises. The problems have arisen because the natural frequency of the

floor structure has decreased with an increase in span and approached the

rhythmic frequency of the activity creating a resonance condition. For

certain well-coordinated events such as jumping exercises, however, there can also be resonance above the rhythmic frequency resulting in large vibrations. This paper describes such a case that occurred in practice and the remedial measures that were carried out to correct the problem.

For further information on control of vibrations due to rhythmic activities, see the Supplement to the National Building Code of Canada 1985, Chapter 4, Commentary A.

The Division of Building Research is grateful to Mr. W.R. Schriever for

translating the paper.

Ottawa March 1984

C. B. Crawford, Director

(5)

ACTIVITIES, CASE STUDY OF A GYMNASIUM

by Hugo Bachmann*

Summary:

The widespread belief that resonance vibrations in a structure can be avoided if

its natural frequency is higher than the forcing frequency is not always

」ッイイ・セエN The following example of a gymnasium shows that with certain periodic

impact forces (in this セ。ウ・ running, hopping and jumping exercises for physical

conditioning) lightly damped structures may be forced into significant resonant vibrations if their natural frequency falls on a integer multiple of the forcing frequency.

Soon after the completion of a two-storey gymnasium considerable structural

vibrations were noticed in the building. These occurred mainly when the upper

gymnasium was used for conditioning exercises done to music e.g. running,

hopping, and jumping.

Fear reaction of the occupants.

The vibrations were especially noticeable in the lower gymnasium by visible

deflections of the ceiling, Le. the upper floor on which the exerci.ses were

taking place, by horizontal wall movements and by considerable noise produced by

the swinging and rattling of the entrance doors and the doors to equipment

rooms, and also by the noise of certain equipment parts attached to the ceiling

and the walls. Furthermore strong rhythmic air currents were noticed at the

main door due to the compression and decompression of the building volume by the

floor above. These effects frequently caused people to leave the lower hall as

soon as conditioning training in the upper hall started. In April 1977 the

owners asked the author of this paper to investigate the reasons for these

phenomena and to assess the vibrations.

(6)

A modern building.

The two-storey gymnasium measures approximately 19 x 31 m in plan (see figure

1). The longitudinal walls consist of low walls of reinforced concrete of

different heights, on which steel columns spaced 6 metres apart carry the upper

floor made of reinforced concrete. The rest of the upper storey construction

consists mainly of steel members. The longitudinal facades consist of

continuous window sections as well as prefabricated facade elements. The end

walls of the hall consist of reinforced concrete walls 25 and 37 em thick which

have an 18 em thick brick lining on the inside. The floor of the upper

gymnasium which is the main subject of this paper consists of reinforced

concrete with main beams 100 em deep and 35 em wide with ordinary reinforcing

(Ul to U9 in the figure). Their span is lS .52 m, The floors lab is 12 em

thick. The spacing of the main members is 3 m, so that every second beam only

is supported directly by columns in the facade walls while every other beam is

indirectly supported by edge beams (RTl and RT2). In addition the floorslab is

supported in the longitudinal direction by three rows of secondary beams So em

deep and 35 em wide (Ll, L2, L3) at a spacing of 5.5 m again reinforced with

ordinary bars.

Zセ

--11

JL

Figure 1. Cross section and plan view (actually

view of the underside) of the upper floor of a two-storey gymnasium ...ᄋtエセ]]ゥL」セセイ . : :! U9,I __-- ..... ' L . . . MMBMMMMBセN⦅ GtM⦅NセNMゥAMNM ⦅NNNNNNNN[セA __ t, uPL セMセMセZZゥMMMMMMMMMMMMイイMMM II mpセ ': U7 I:

__

」ェZセ

__

エイMM]セセ⦅セ」」

- -'1----:1---0 MP211MP3uS

ᄋᄋGQヲセ」ᄋmセャ[セセオM[Gセセᄋ

_

mセs イMセSMM

M;;;-.1 [1 _J .. セI セ. セ . ll ⦅NNcjNR⦅セ⦅N⦅ セ ---r;--- MMセNLNLN⦅⦅⦅M MMLNセM⦅NM N セ ';:J iZiZセ Ul ___M M B セ ⦅ N __ L L __. ._

-::

ttMZセッjM

,__

EINGANG ⦅セ Wセ セLGI|NBN s[MIZセS Bセ アセR

(7)

Cracks and damages.

A first inspection showed that the strongest vibrations had the character of

resonance vibrations which were observed when the exercises upstairs were

carried out at a frequency of 2.3 to 2.8 Hz. A first approximate calculation of

the natural frequency of the main reinforced concrete beams assumed to be simply

supported at the walls showed values of 3.3 to 4.4 Hz depending on the

assumptions made regarding existing cracks.

A crack survey was then carried out on the underside of the floor with the floor

at rest, using a moveable tower scaffold. This survey showed that the

longitudinal beams (Ll, L2 and L3) had cracks up to 0.6 mm in width, some of the

cracks were in locations where a shrinkage joint had been arranged during

construction. The main beams in the central part of the hall (U3 to U7) were

cracked at fairly regular intervals, at spacings of 15 to 40 cm and maximum

crack width of 0.25 mm. Towards the end of the hall (U2, U8), the cracks

diminished and the two end members (Ul, U9) were almost crack free. This

indicated that the cracking was related to the dynamic loading of the floor.

The influence of the exercises is greater in the central portion of the floor

and less in the end zones. In addition the end zones are stiffened somewhat by

the longitudinal beams running from the end walls so that the loading and the

deflections are reduced. Other observations showed that at several of the

attachment points of facade elements spalling of concrete had occurred.

Presumably this was also related to the dynamic loading of the floor.

Dynamic tests with the aid of exercising people.

Based on the above observations the author suggested to carry out dynamic tests

and appropriate measurements. The aim was to determine the vibration properties

of the floor and to measure the effects of conditioning exercises. Very good

cooperation developed between the measuring team and the users and the required

dynamic loading was readily produced by the exercising people. Regarding the

type and frequency of the forcing exercises the following partly speci fically

(8)

a) exercises with prescribed forcing frequency:

"Hopping in place" during approximately 20 seconds whereby the hopping frequency was regulated by means of a metronome whose sound was augmented by a loudspeaker: Increase of the hopping frequency from 2.0 to 3.2 Hz in steps of 0.2 Hz in order to determine approximately the resonance frequency;

Variation of the hopping frequency within the range of the resonance

frequency in very small steps of 0.05 and 0.02 Hz in order to determine the resonance frequency more accurately;

Constant hopping frequency at approximately the resonance frequency with

several repetitions in order to check the reproducibility.

"Mixed exercises" such as running hops, high knee lifts, hopping etc. for

approximately 15 to 20 seconds, to music over loudspeakers as closely as

possible at the determined resonance frequency.

b) exercises with random forcing frequency:

"Normal exercises" with warm-up etc. to music over the loudspeakers, chosen

freely by the various gym teachers in their training program.

These tests using the exercises a) and b) mentioned above took place in February

1978. 40 to 130 people took part in the exercises. In order to keep expenses

low only a few measurements were made at a few selected points. The arrangement

of instruments and measuring points (MP) is shown in figure 1:

MP1: Inductive crack-width meter on U5

MP2: Strain measuring bridge of 24 cm length on U5

MP3: Inductive meter for vertical deflection on U5

MP4: Vertical acceleration meter on U5

MP5: Inductive crack width meter on L2 between U3 and U4

MP6: Inductive meter for vertical deflection on U7

MP7: Horizontal acceleration meter on steel column under U4.

The time function of all measured values was recorded either on magnetic tape with the possibility of replaying this on a chart recorder or recorded directly on the latter.

(9)

, ,

Unusual measurement results.

This initially

The fundamental frequency of the structural vibration

times of the maximum of the impact forces (see arrows) and the forcing period

. . . , .., . ., . • t· ... . 1+1:

セaZ[

::: ::: : " : :;

lセ GZZZゥNZNZセ

,:::L

エcゥAZ[AZセ ..

+

1::::I::L::\:::Y:

t '

I

;'::1:

B

I ' o j . ' t

...

, .

セB

1

1

ゥセセセィ

,..j ,.", .•

ZNiNZセヲゥエN

ZZセZセKAセZZK

11"

""1.

1, 'j • • • • j • • " " , ' " ' ' . , , , , t" . . .· · 1 • t o , . , . , . " " • ' ' ' ' • [セZ I ;

ii

i ,: セ

:'::!t

Natural frequency of structure

=

double the forcing frequency

however was 4.9 Hz, that is double the forcing frequency.

T and the natural period T

=

T /2.

p p

approximately 2.45 Hz.

The tests and measurements yielded results which were not entirely as expected.

Figure 2. Section of a deflection record at mセS with indication of the assumed

The first important conclusion from the recordings of all measuring points was

that the strongest resonance vibrations occurred at a forcing frequency of

surprising fact indicates clearly that the exercising people produce an impact at every second wave trough of the structural vibration (with a certain phase

sh.ift). Figure 2 shows a portion of the deflection record at measuring point

(10)

Surprising bending stiffness.

Prior to the tests and measurements i t had been estimated that the fundamental

natural frequency of the floor supported by a system of simple beams was

approxima tely 3.3 to 4.4 Hz. The difference, compared to the actual value of

approximately 4.9 Hz can be attributed to the following stiffening influences: Contribution of the un cracked concrete between the cracks in the main beams, relatively high dynamic E-modulus of the concrete,

contribution of the concrete topping (4 cm),

contribution of the floor (so-called resilient floor, see section on

loading function),

effect of the longitudinal beams (slab effect, orthotropic plate), frame action (partial fixity of the slab at the steel columns).

The remarkably high contribution of the concrete between the cracks is of

special signi ficance, especially its contribution to the bending stiffness of

the beam. By recalculation from the measured deflection it was possible to

determine a "composite action coefficient" [6] of x

=

0.5. This means that the

actual bending stiffness in the cracked state - assumed for the area between the

longitudinal members Ll and L3 - was approximately 200% of the pure cracked

stiffness or approximately 80% of the stiffness of the uncracked section.

Running exercises the most critical.

Furthermore it was found that the largest effects on the structure took place,

not as initially expected, during hopping exercises, but during running

exercises, both carried out at the resonance frequency, that is half of the

natural frequency of the structure. This observation was actually made

accidentally, in that for exercise type b) with people running at 2.4 Hz, that

is a little below the critical resonance frequency, almost the same defl.ections

and accelerations were recorded as for exercise type a) with a resonance

frequency of 2.45 Hz. It was, however, not possible to verify this observation

(11)

J

This value is

Figure 3. Example of a resonance curve:

Floor deflection at the midpoint of concrete beam US (MF3) as a function of the forcing frequency l/T .

p

As can be seen from figure 3, no completely

The damping could be determined, however, from the decay curves

uncertain.

with an average of 0.024, that is 2.4% of critical damping.

within the range of values measured in other reinforced concrete constructions

[4]. It is however, considered quantitatively a rather low value which also

contributes to a rather high potential for vibration problems.

If for a given vibrating system one assumes linear behaviour and viscous

(proportional to velocity) damping, one can calculate the damping either from

the shape of the resonance curve for forced vibration or from the decay of the

amplitude for free vibration.

Significant dynamic loading.

which were determined from the sudden stopping of the exercise "hopping in

place" at the resonance frequency. The values obtained were t;, = 0.022 to 0.026

satisfactory resonance curve was available, since especially the peak value was

The floor had been designed by the structural engineer according Swiss Standard

SIA 160 for the permanent load plus a static occupancy load of 5.5 kN/m;:> (550

kg/m2) . According to the s t.r-uct.ur-aI

ウケセNエ\[ュ

assumed for the design

calculations (simply supported beams, because only very small negative moments

resul t from the steel columns) the permanent load produces stresses in the ten

34 mm reinforcing bars of each of the beams Ul to U9 a stress of (J '" 135

2 e,g

N/mm at mid-span. The measurements made on beam U5 showed that for the

exercise type a), "hopping in place" with a constant hopping ヲイ・アオ・セ」ケ

approxima tely equal to the resonance frequency a maximum de flection 0f' 'J :::: +

5.5 rom and a maximum acceleration of a

o

=

+ 5.15 m/s2 were measured at

(12)

mid-span. The variation in the crack width at mid-span was w

d :.: .:!:. 0.04 mm ,

If one assumes a parabolic shape of the acceleration along the main beams one

obtains an additional dynamic stress of a + 58 N/mm2•

e,g

For the same exercise type the longitudinal beam L2 showed a maximum variation

in the crack width (at MP5) of w

d = + 0.15 mm , This rather large value is

probably related to several factors: to shrinkage effects, dynamic loading

producing a weakening of composite action, indirect support. and thus a slightly

different deflection of every second beam leading to a par·ticipation in the

vibration of the edge beams RTl and RT2. The stress values were estimated at

(J '" 145 N/mm

2

and a d + 45 N/mm2•

e,g e,

-Discussion.

Based on the results of the measurements obtained it was possible to evaluate

the condition and the usefulness of the structure and to make recommendations

for remedial action. These recommendations are further supported by some

theoretical considerations mentioned further below.

Fatigue risks.

To assess the stresses in the reinforcing with regard to fatigue a maximum

estimated acce Ler-ati on from the running exercises of a

o = 7.0

2

m/s was

The same conclusion was reached for the

the requirements of SIA standard 162

is 0; + 78 2 N/mm • SIA 162 for fatigue diameter = mm Standard 78 34 + of Swiss '" 135 bars reinforcing the

an additional dynamic stress of

to to

Thus

to a total stress of

ae ,max

the allowable stress according to

corresponds this thus This to 3.07, extrapolated 2 '" 177 N/mm • ae, all

in the reinforcing are not satisfied. Compared

Article assumed.

N/mm2 and

maximum acceleration measured during the

2

mis, although with a somewhat smaller

"hopping exercises" of a = + 5.15

o

excess over the allowable stress.

(13)

MMMMMMMMMMMMMMMMMセMMMMM

10

-The effects of the vibration on the facade elements, especially on their

supports where some damage had already been observed, could not be determined. However it was felt that here, too, there was some danger of fatigue failures.

Similar conclusions seemed to be justified for the windows and for the wall

cladding, as well as for certain elements and gym equipment attached to the

walls, such as basket ball hoops for which the bolts had frequently needed

retightening.

Reduced usefulness.

People react to vibrations and also to any resulting noise very critically and soon reach a point where they feel that there must be some overloading of the

building or even a danger of collapse. In the present case it was especially

the swinging of the entrance door and of the door to the equipment room that

produced noises which were felt to be very disturbing and alarming. As already

stated there were many instances when people left the lower gym as soon as the

upper gym was used for conditioning training that produced to noticeable

vibrations of the ceiling. For this reason and because the upper gym could no

longer be used for the same number of people as before one had to admit that the usefulness of the gym bUilding had been reduced.

Recommendations for remedial action.

After it had been observed that even with small groups of approximately 30

participants the building could shake rather badly and that the reduction in the

number of people did not provide the solution the owners decided in 1979 to

proceed with basic remedial action.

The test results had shown that per-Lodi.c .i.mpa c' f'or-ces produced by running and

hopping exercises could cause a gym floor to vibrate whenever its natural

frequency is an integer multiple of the forcing frequency (that is twice, three

times etc., but with increasing effective damping). By a frequency analysis of

different exercise programs it was determined that many of the common running

(14)

recommended by the author that it would be sufficient to raise the natural

frequency of the floor to approximately two and a half times the maximum

possible forcing frequency of 3.2 Hz (actually in [2J values of up to 3.6 Hz are

mentioned), Le. to raise the frequency to approximately 8 Hz. This might not

completely eliminate the possibility of some vibrations of the floor after the

remedial action, especially for forcing frequencies of one third, one quarter

etc. of the natural frequency (see following chapter), but it was considered

sufficient, taking into account the cost of the remedial action. The resulting

loading effects on the structure (risk of fatigue) would thus be reduced to

acceptable limits, even if the structure did vibrate somewhat. It was realized

that with this somewhat limited remedial action there would remain a risk of

some vibration of facade elements, equipment parts etc. but this remaining risk was considered acceptable.

Theoretical considerations.

Loading function

The force versus time function of the load produced by the human foot on the

floor during walking and running, or also during hoppi.ng and other combined

running and jumping exercises, is of very complex nature. This function depends

on various parameters such as the type of exercise, the frequency of walki.ng or

hopping, the shoes, the properties of the flooring etc. Figure 4 shows as an

example the loading functions that were measured according to [lJ (as given in

[2] during walking and running with ordinary shoes on a surface called "sand

over rubber". For relatively low walking frequencies of 0.8 to 1.3 Hz and also

for high frequencies of 3.0 to 3.6 Hz a single force maximum was observed. In

contrast to this the loading function for medium frequencies of approximately

1.5 to 2.7 Hz has two more or less dis tinct force maxima which presumably are caused by the "rolling off" (or rather the setting down and stepping off) of the

foot. The maximum value of the load, called p from now on, increases from

o

the lower to the higher frequencies to approximately 1.5 or 2 times the original load.

(15)

246Hz

The classic analytical approach using a

1.53Hz

エZj

G セ G

.,

I' I ' .. , I ' I " _ ---L

o

05 10 155

o

05 1.0 15 20 25 305

The loading functions which are produced during conditioning

1----

MMセMMMセM

---I N 0.85 Hz 1800

f

YPセエ セOMMMMMMI\MM 10-- Mセ Tp' 1185

This is a relatively soft type of construction which consists in

linoleum.

Figure 4. Loading functions of the force of a person on the floor during walking

and running according to Ref. 1 (as quoted in REf. 2) Case "sand

over rubber". ordinary walking shoes. 12

-viscous damping will be considered.

Damped single mass oscillator.

training, all sorts of running and hopping exercises, on this resilient floor

and then passed on to the structural members, are not known. thick.

principle of several layers of slats which are arranged at right angles which are covered with tight floorboards, large particleboard sheets and finally cork

In the present case the floor consisted of a so-called resilient floor 11 cm

Fourier analysis of the periodic loading according to [3] will be used since

this approach leads to especially graphic results and thus to an improved

understanding of the important relationships. The Fourier representation of an

arbitrary periodic loading pet) is:

In the following, for reasons of si.mp.lici ty, " single mass oscillator with

Figure 4 shows the development with time of the individual loading functions due

to periodic walking or stepping (dashed lines). The corresponding period T

p

of the loading is also shown. Whereas for walking there is always contact with

the ground -that is the loading functions are overlapping a bit- this contact

disappears almost completely for fast walking or completely for hopping or

(16)

00 セG 21t p(t) = ao + L. a ;: cos(n . - ' t) n= I Tp 00

+ \'

L. bn ' S1I1 (n . -

.

2IT .t) n = I T,

Where 2 1T ITP = w1 is the angular frequency of the periodic loading. The

Fourier coefficients are

I

r,

Go = - . j p(t) . dt セL 0 Gn R セ G 2

J

.

IT T p(t)·COS(I/·_·tj·dt p 0 Tp 2 T" . 2IT b; =_·jp(t),s1I1(n·-·tj·dt

r,

0

r,

Since the periodic loading - apart from the constant part a - is represented

o

exclusively by sine and cosine functions, the response of the linear oscillator

is determined by superposition of the response of the harmonic loads:

00 I

L

I l'(t) = - (a + k 0 n= I (I - pJ)l+(2 .セ .p.)l . IIan. 2 .セ . Pn +b; .(I - P;,l] .sin (n .

w, .

tj -+ Ian.(I - pセI ..b, .2 .セ . Pnl . cos (n .W, . Of) where v = deflection k = spring constant

!; = ratio of damping to critical damping

Sn

= n wi/w = n . 1'/["

(17)

This equation can also be written as follows:

In the following the shape of the actual, but unknown, loading function will be

Here the observed resonant vibrations corresponded to the value T/T P

=

1/2

("impact at every second wave trough"). In principle it was to be expected that

To deal with the resulting vibrations it is furthermore important to consider the ratio of the period of the oscillator T to a period of the loading T :

p

wave This had to be considered for the

To represent different exercises various

t and of the period of loading Twill

p P

in principle be expected for T/T

=

1 as

p

however, not relevant to the floor considered.

1/3, 1/4 etc. ("impact at every third, fourth etc.

also values of T/T =

P

trough") would lead to resonant vibrations.

v(t)=&la +

L

{セBGウゥョHョGキャGHI k 0 n= 1 + "'" .cos(n .WI .

on

an.2 .セB p" + b" .(1 - /1;) where: セB] (1 - ーセIR +(2 . .p")2 a". (\ - ーセI - b".2 .セ .p"

*"

(\ _

ーセIR +(2 . セ . p")2

replaced by half of a sine wave.

conditions of the loading duration

be considered (figure 5):

The strongest vibrations can

resonance vibrations. This is,

floor after remedial action because .its nacura ' frequency was to be raised to

approximately two and a half times the maximum possible forcing frequency. In

the numerical evaluation of the equation for 'J (t) the damping was assumed to

have a value of セ

=

0.020. This value is somewhat lower than the average value

obtained of during the test of セ = 0.024. Based on the results presented in a

dissertation [5] it was to be expected that the damping would be slightly

smaller after the remedial action, becaus'3 the contribution of frictional

damping due to presence of the crack would be largely eliminated and be replaced by a smaller value due to viscous damping only.

(18)

n . _ - - ----MMMMMMMMMセ 5 2 4 6

Representation of the loading function by sine half waves with different ratios of the loading

duration t to the period of loading T .

P P

I

I

サ[セK エセ

I Figure 5. 5/6 1/2 2/3 1/3

A

1/4 - - ...J

/\

L l

)

_

RセM[[MMQ

p(1 -p SIn-·--·t ,

M

Tp IpITp

U-J

I ! Tp 1---1

m

IpITp

M

M

un

Q_u

.

I,

I ,

j\'

1/4 1/3 I / ,:\ 2/3 i

'/

'

I \ I ' aNセ|L 6/5/ 1/ 2

O Q O '-.\\ セ NOセ

,.

M⦅NセM]MセセセN LNセ

...セM

1 2 3 4 5 n 5 1/';; 1/2 1/3 1 2/3 6/5

"

2

Rセ I I Rセ i i o

L.

o 0+ o 'JePn2+" , "T'n Figure 6. Amplitudes of

various vibration contri-butions of a single mass oscillator for different

ratios T/T and t /T ,

P p P

calculated for damping

セ :::: 0.020

(19)

Figure 6 shows the amplitudes of various vibration contributions of iセ[L +Qェjセ for

TIT = 1/2, 1/3 and 1/4 of tiT = 6/5, 1, 2/3, 1/2, 1/3 and 1/4,

p P P

respectively. As expected it was found that the vibration contributions that

correspond to the natural frequency of the oscillator at n

=

2 for TIT

=

1/2,

P

n

=

3 for TIT

=

1/3 etc. dominate which indicates a resonant vibration. This

p

confirmed the statement formulated earlier that the floor vibrates especially

when the natural frequency (fundamental frequency) is a integer multiple of the

forcing frequency. The influence of t ITp p must, however, still be

discussed. The resonant vibration at n times the forcing frequency contains

some additional frequencies which are not very significant in the present case

because of low damping. Actually in figur e 2 a superposed vibration can be

detected since the amplitude between the impacts do not decrease monotonically

as would be the case for a pure resonance vibration. This superposed vibration

corresponds in figure 6 for TITp

=

1/2 to the value of n

=

1 (vibration at the

forcing frequency).

Figure 6 indicates that the excitation of the oscillator occurs especially for

the smaller values of t IT. This makes sense because the shorter the

p p

periodic impact the more likely it is to produce a vibration at the natural

frequency. If we limit ourselves to the consideration of the higher frequency

components of the loading function and accept the approximation by sine half

waves which makes the loading function at the base somewhat narrower than the

actual one, we conclude from figure 4 for the case of walking values of

tpITp of 1 to approximately 2/3. The relatively strong resonance vibrations

of the floor indicate that for specific running exercises t IT = 1/2 also

p p

enters the picture. For hopping (and running) we must assume, according to

in forma t ion correspond,

given for

in [2] a lower limiting value of t ITp p

=

1/3 which might

example, to high jumps with relatively long lasting

(20)

17

-Maximum expected deflection after remedial action.

On the basis of the above considerations and assumptions as well as of further thoughts it was possible to estimate the vibration amplitudes which would occur

after remedial action, that is after the increase in the stiffness. For the

least favourable case of resonance vibration with T/Tp

=

1/3 and t /Tp p

=

1/3 it was estimated that the deflection would be at the most 10 to 15% of the deflection before remedial action.

Remedial action.

Various possible solutions

A consulting office was asked to prepare a detailed proposal of remedial

action. This proposal consisted of attaching, by means of an epoxy mortar,

large steel angles to the main beams. For fire safety reasons and also in order

to press the steel angles against the concrete beams during the glueing process

vertical anchors were proposed. It was also proposed to lift the whole floor by

means of hydraulic jacks reacting against wooden columns which would make it necessary to remove and reinstall the resilient floor in the lower gymnasium.

This proposal appeared to be very expensive. Also the vertical anchors would

have been very difficult to install because of the dense arrangement of the

reinforcing in the existing concrete beams. In addition there would have been

tolerance problems in connection with the glueing (mean thickness of glue layer

of 5 mm) inspite of the proposed accurate survey of the dimensions of the

reinforced concrete beams. All of this would have been difficult to solve.

80

(21)

18

-Another proposal made by a different firm consisted of building a 12 em thick continuous reinforced concrete slab as a lower flange of the main beams and thus

to increase the plate effect of the floor. This solution would have presented

an esthetically pleasing view from below. It would have involved a considerable

risk however that the existing cracks in the main concrete beams would advance

into the new slab, especially due to tensile and notch effects in connection

with the differential shrinkage and the dynamic influences on the slab. There

was a possibility therefore that the natural frequency of the slab would

gradually drop again to almost the value which existed before the remedial

acti::>n.

In the end the owners agreed with the proposal of the author which is shown in

figure 7 with some minor modifications in the details. The bonding to be

achieved between the existing beams whose concrete surfaces would be roughened and the new steel additions would be achieved conventionally by fresh concrete

and high tensile post-tensioned bolts, the upper set of which would go through

holes drilled through the concrete above the existing longitudinal reinforcing. This method had already been used successfully in other remedial projects which

involved even higher shear strength. This solution would avoid the lifting of

the floor, the removal and reinstallation of the resilient floor of the lower

gym, as well as the rather complicated glueing with its tolerance problems, fire

protection complications etc. It would also mean a considerably reduced cost.

Rapid execution.

After approval of the necessary funds the remedial action project was carried

out during the summer of 1982. Large steel troughs prefabricated by a steel

fabricator, weighing approximately 90 kN each, were hoisted as a single piece by a mobile crane to the under side of the existing beams and supported by a 5 m

wide steel pipe scaffold. This scaffold could be moved in the longitudinal

direction of the hall in a simple way by means of air cushions. After one of

these steel troughs had been located under the appropriate concrete beam it was lifted to its exact position by means of another air cushion and then supported

(22)

finally by the post-tensioning of all high strength ateel bolts. The operation

was completed for all nine members in the per'iod. of four weeks only. Then parts

removed from of the ヲ。N」。、・セ lighting, gym equipment etc. were reinstalled. The

gym was ready again for use after only three months. Figure 8 shows the

completed ceiling with the newly strengthened main beams.

Good results of the rewedi.!!

action.

In November 1982 !'mother set of testa were carried out which were similar to

those which had been carried out in February 1918. Again. approximately 100

people participated in the exercisea. These were carried out with frequencies

varying from 2.4 to 3.5 Hz and consisted mainly of セィッーーゥョァ in ーャ。」・セ according

to exercise type a)0 No obvious resonance vibration could be found, thus no

well defined natural frequencies. The measured lJk1.ximum deflections were barely

0.3 rom. This is approximately 5$ of the value measured before the remedial

action (5.5 mm) and considerably less than had been calculated based on

theoretical considerationsI using a number of unfavourable parameter assumptions

(10 to 15%). Noticeable vibration of facade elements !'ma equipment parts could

no longer be detected. Neither were there any noise ef'fects. The result of the

remedial action can therefore be conSidered satisfactory.

(23)

20

-After these tests the vibration records were analyzed by means of Fourier

amplitude spectra which, however, yielded relatively uncertain values for the

natural frequency of the beams in the range of 7.2 and 7.6 Hz. Therefore

additional tests using a sandbag of 100 kg mass dropped from a height of 1.8 m

were carried out. This resulted in a maximum deflection of 0.1 rom. The

damping was determined at a level of approximately 2% and is thus in the same

range as the one before the remedial action. The natural frequency analysis

for this load yielded a value of 7.3 ::!:. 0.1 Hz. The earlier estimate had

resulted in values between 7.0 and 8.8 Hz depending on the assumptions,

especially those regarding the contribution of the concrete in the tensile

zone. The comparison indicaエセウ that the measured value tends to be in the

lower part of the calculated range.

Conclusions.

Lightly damped structures can readily be forced into resonance vibrations by periodic impact effects (e.g. during conditioning training to rhythmic music)

if the natural frequency (fundamental frequency) of the structure is an

integer multiple of the forcing frequency. This phenomenon which is easily

explained has so far generally gone unnoticed by should be taken into account in the future in the design of structures subject to such dynamic effects.

(24)

T

21

-Literature

Liter.tur

III Goitbraut:and Harron (1970, "Uruund

loading fromfッッャGャ・ーウBLjG|sBTセNQRss

121 Kromer. H.. Kcbe. H.-W 119NI' .Durch

Me nschc n cr zwunge ne B:1UWL'rks·

«hwmgunge n' Dcr Bauingcnicur 54. S.19'-199

III Clough. R. H:. Prnzicn.1.t]l)75I'-n..

narnic« or Structure -,·· \,1( tir.rw-Hu!

Kcg akushu Ltd.. lntcrn.uion.r: Student Edition, Tokyo

[4J DINer/c.R.. Bu,.hmann, H(!l)7R): ..Vc

r--u.he ubc r de n l.intlus-, dcr Kl ...

sbil-dung .ruf die dyn.rrnisc hc n

Eig.C!1-schaftcn von LCIi..h tb e ton- UIHJ

Hetun-bulkc n" Invtnut tur Bnu -,t.ruk und

K o n v t r u k t i o n . F l l I OャjョセィN V c r s u c h v

-hcricht Nr MセエャャMQN Hu kh.ruvc rVert.rg Bascl/Stuug.rrt

[5J Dutertr. R. 119Rll "Modelie lur dJS

Dampfunpvvc rhauen von St;lhlhctnn· trjgi:rn .rn unge risvcne n und gc ncvc-ncn Zu-tund" lnxuuu tur BilU"ltatik

uno Konstrukuon. Llil/ul"il..h.Bc ncht

Nr. Ill,BirLlll'iCr \l..'rL'g

Hd'icl/Slutl-gurt.

Bachmann. If. (14i-lJ .,St.llllhl'tUIl I, II" |ッイォBャlャョセN|[ャャャエH|セゥM[ャーィャエGョ Lrdgc-nuxvisvh c l cchnivch e l Iochschu!e Zu nch

161

L

___

[2] Vibrations of Buildings Produced by People.

[4] Tests on the Influence of Crack Formation on the Dynamic Properties of

Concrete and Lightweight Concrete Beams.

[5] Models for the Damping Behaviour of Reinforced concrete Beams in the

Uncracked and Cracked State.

Figure

Figure 1. Cross section and plan view (actually view of the underside) of the upper floor of a two-storey gymnasium ...ᄋtエセ]]ゥL」セセイ .::!U9,I__-- ..⦅NNNNNNNN[セA..
Figure 2. Section of a deflection record at mセS with indication of the assumed The first important conclusion from the recordings of all measuring points wasthatthestrongestresonancevibrationsoccurredataforcingfrequency of
Figure 3. Example of a resonance curve:
Figure 4. Loading functions of the force of a person on the floor during walking and running according to Ref
+2

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