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Vibrations of building structures caused by human activities: case
study of a gymnasium
CANADA INSTITUTE
FOR SCIENTIFIC AND TECHNICAL
INFORMATION
セセ
1JJ-14
jスセl
[a».
rr
セ'}o}1-ISSN 0077-5606
INSTITUT CANADI EN
DE L'INFORMATIOI'J SCIENTIFIQUE
ET TECHNIQUE
NRC/CNR TT-2077
TECHNICAL TRANSLATION TRADUCTION TECHNIQUEH. 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
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
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
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.
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 MP211MP3uSᄋᄋGQヲセ」ᄋmセャ[セセオM[Gセセᄋ
_
mセs イMセSMMM;;;-.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
,__
セ
EINGANG ⦅セ Wセ セLGI|NBN s[MIZセS Bセ アセRCracks 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
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.
, ,
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 frequencyhowever 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
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 theactual 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
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 designcalculations (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 atmid-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.
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
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.
246Hz
The classic analytical approach using a
1.53Hz
エZj
G セ G.,
I' I ' .. , I ' I " _ ---Lo
05 10 155o
05 1.0 15 20 25 305The 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' 1185This 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
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 2J
.
IT T p(t)·COS(I/·_·tj·dt p 0 Tp 2 T" . 2IT b; =_·jp(t),s1I1(n·-·tj·dtr,
0r,
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'/["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 asp
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 . セ .p")2 a". (\ - ーセI - b".2 .セ .p"*"
(\ _
ーセIR +(2 . セ . p")2replaced 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 valueobtained 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.
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 1/3A
1/4 - - ...J/\
L l)
_
RセM[[MMQ
p(1 -p SIn-·--·t ,M
Tp IpITpU-J
I ! Tp 1---1m
IpITpM
M
un
Q_u
.
I,
I ,j\'
1/4 1/3 I • / ,:\ 2/3 i'/
'
I \ I ' aNセ|L 6/5/ 1/ 2セ
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 oL.
o 0+ o 'JePn2+" , "T'n Figure 6. Amplitudes ofvarious 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
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. Thisp
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 theforcing 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 mightexample, to high jumps with relatively long lasting
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
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
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.
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.
T
21-Literature
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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
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[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
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[5J Dutertr. R. 119Rll "Modelie lur dJS
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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.