Design of floor structures for human
induced vibrations
M. Feldmann, Ch. Heinemeyer, Chr. Butz, E. Caetano, A. Cunha, F. Galanti, A. Goldack,
O. Hechler, S. Hicks, A. Keil, M. Lukic, R. Obiala, M. Schlaich, G. Sedlacek, A. Smith, P. Waarts
Background document in support to the implementation, harmonization and
further development of the Eurocodes
Joint Report
Prepared under the JRC – ECCS cooperation agreement for the evolution of Eurocode 3
(programme of CEN / TC 250)
Editors: G. Sedlacek, Ch. Heinemeyer, Chr. Butz
EUR 24084 EN - 2009
Design of floor structures for human
induced vibrations
M. Feldmann, Ch. Heinemeyer, Chr. Butz, E. Caetano, A. Cunha, F. Galanti, A. Goldack,
O. Hechler, S. Hicks, A. Keil, M. Lukic, R. Obiala, M. Schlaich, G. Sedlacek, A. Smith, P. Waarts
Background document in support to the implementation, harmonization and
further development of the Eurocodes
Joint Report
Prepared under the JRC – ECCS cooperation agreement for the evolution of Eurocode 3
(programme of CEN / TC 250)
Editors: G. Sedlacek, Ch. Heinemeyer, Chr. Butz
EUR 24084 EN - 2009
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JRC 55118
EUR 24084 EN
ISBN 978-92-79-14094-5
ISSN 1018-5593
DOI 10.2788/4640
Luxembourg: Office for Official Publications of the European Communities
© European Communities, 2009
Reproduction is authorised provided the source is acknowledged
Printed in Italy
Acknowledgements
This report is based on the results of two European research projects funded by
the Research Fund for Coal and Steel (RFCS), namely:

Generalisation of criteria for floor vibrations for industrial, office and
public buildings and gymnastic halls – VOF prepared by ArcelorMittal,
TNO, SCI, RWTH Aachen University [1]

Human induced vibrations of steel structures – HIVOSS prepared by
ArcelorMittal, TNO, SCI, RWTH Aachen University, CTICM, FEUP and
Schlaich Bergermann und Partner [2]
The project partners gratefully acknowledge the financial contributions of RFCS
as well as their agreement to publish the results in a “JRC-Scientific and
Technical Report” to support the maintenance, further harmonization, further
development and promotion of the Eurocodes.
Foreword
(1)
The EN Eurocodes are a series of European Standards which provide a
common series of methods for calculating the mechanical strength of
elements playing a structural role in construction works, i.e. the structural
construction products.
They enable to design construction works, to check their stability and to
give the necessary dimensions to the structural construction products.
(2)
They are the result of a long procedure of bringing together and
harmonizing the different design traditions in the Member States. In the
same time, the Member States keep exclusive competence and
responsibility for the levels of safety of works.
(3)
Sustainability requirements for buildings often lead to structural concepts,
for which the mechanical resistance and stability of construction works is
not governing the design, but serviceability criteria can control the
dimensions. A typical example are long span lightweight floor structures,
for which the design for vibrations to avoid discomfort provides the main
design parameters.
(4)
So far for floor structures the Eurocodes give only recommendations for
estimated limits for eigenfrequencies, e.g. 3 Hz or 8 Hz depending on the
construction material, or they give reference to ISO-standards as ISO/DIS
10137 and ISO 2631, which give general criteria for the perception of
vibrations and could be the basis to develop more detailed design rules for
vibrations specific to particular structures and types of excitation.
(5)
This report is intended to fill this gap and to provide an easy-to-use design
guide with background information that shall help to specify comfort
requirements for occupants and to perform a design that guarantees the
specified comfort.
(6)
It applies to floors in office and/or residential buildings that might be
excited by walking persons and which can affect the comfort of other
building users.
(7)
This report may be considered as a supplement to EN 1990 and may also
be used as a source of support to:
-
further harmonization of the design rules across
structural materials and construction procedures,
-
further development of the Eurocodes.
different
(8)
The rules for the “Design of floor structures for human induced vibrations”
given in this report are the result of two international projects, the VOFproject and the HIVOSS-project, both funded by the Research Fund for
Coal and Steel (RFCS), initiated and carried out by a group of experts from
RWTH Aachen University, Germany, ArcelorMittal, Luxembourg, TNO, The
Netherlands, SCI, United Kingdom, CTICM, France, FEUP Porto, Portugal
and Schlaich, Bergermann und Partner, Germany [1], [2]
(9)
The agreement of RFCS and the project partners to publish this report in
the series of the “JRC-Scientific and Technical Reports” in support of the
further development of the Eurocodes is highly appreciated.
(7)
The examples given in this guideline mainly covers light-weight steel
structures, where the consideration of human induced vibrations is part of
the optimization strategy for sustainable constructions. Therefore, the
publication has been carried out in the context of the JRC-ECCScooperation agreement in order to support the further harmonization of
National procedures and the further evolution of the Eurocodes.
Aachen, Delft, Paris and Ispra, September 2009
Gerhard Sedlacek
ECCS-Director of Research
Frans Bijlaard
Chairman of CEN/TC 250/SC3
Jean-Armand Calgaro
Chairman of CEN/TC 250
Michel Géradin, Artur Pinto, Humberto Varum
European Laboratory for Structural Assessment, IPSC, JRC
Design of floor structures for
human induced vibrations
List of Contents
1
Objective ........................................................................................... 5
2
General procedure .............................................................................. 6
3
Description of the loading..................................................................... 7
4
Dynamic floor response ......................................................................13
5
Comfort assessment of the floor structures ............................................17
6
Development of design charts ..............................................................26
7
Guidance for the design of floors for human induced vibrations using design
charts ....................................................................................................29
7.1
Scope ..........................................................................................29
7.2
Procedure.....................................................................................29
7.3
Determination of dynamic properties of floor structures ......................30
7.4
Values for eigenfrequency and modal mass .......................................31
7.4.1 Simple calculation formulas for isotropic plates and beams.............31
7.4.2 Simple calculation methods for eigenfrequencies of orthotropic floors34
7.4.3 Natural frequencies from the self-weight approach ........................35
7.4.4 Natural frequency from the Dunkerley approach ...........................36
7.4.5 Modal mass from mode shape ....................................................37
7.4.6 Eigenfrequencies and modal mass from FEM-analysis ....................39
8
7.5
Values for damping ........................................................................39
7.6
Determination of the appropriate OS-RM90-value................................40
7.7
Vibration performance assessment...................................................51
Design examples................................................................................52
8.1
Filigree slab with ACB-composite beams (office building).....................52
8.1.1 Description of the floor .............................................................52
8.1.2 Determination of dynamic floor characteristics .............................56
8.1.3 Assessment .............................................................................58
8.2
Three storey office building .............................................................58
8.2.1 Description of the floor .............................................................58
8.2.2 Determination of dynamic floor characteristics .............................60
8.2.3 Assessment .............................................................................63
9
References ........................................................................................64
1
Table of definitions and frequently used symbols
Definitions
The definitions given here are oriented on the application of this guideline.
Damping D
Damping is the energy dissipation of a vibrating
system. The total damping consists of

material and structural damping

damping by furniture and finishing (e.g. false
floor)

geometrical radiation (propagation of energy
through the structure)
Modal mass Mmod
In many cases, a system with n degrees of freedom
generalised mass
can be reduced to a n SDOF systems with frequency:
fi 
1
2
K mod,i
M mod,i
where:
fi
is the natural frequency of the i-th system
Kmod,i
is the modal stiffness of the i-th system
Mmod,i
is the modal mass of the i-th system
Thus, the modal mass can be interpreted to be the
mass activated in a specific mode.
The determination of the modal mass is described in
section 7.
2
Natural frequency f =
Each mode of a structure has its specific dynamic
Eigenfrequency
behaviour with regard to vibration mode shape and
period T [s] of a single oscillation. The frequency f is
the reciprocal of the oscillation period T (f = 1/T).
The natural frequency is the frequency of a free
decaying oscillation without continuously being driven
by an excitation source.
Each structure has as many natural frequencies and
associated mode shapes as degrees of freedom. They
are commonly sorted by the amount of energy that is
activated by the oscillation. Therefore, the first natural
frequency is that on the lowest energy level and is thus
the most likely to be activated.
The equation for the natural frequency of a single
degree of freedom system is
f 
where:
OS-RMS90
1
2
K
M
K
is the stiffness
M
is the mass
One-Step-RMS-value of the acceleration resp. velocity
for a significant single step, that is larger than the 90%
fractile of peoples’ walking steps.
OS:
One step
RMS:
Root mean square = effective value of the
acceleration a resp. velocity v:
1
T
a RMS 
where:
T
T
 a(t )
0
2
dt 
a Peak
2
is the period.
3
Variables, units and symbols
a
Acceleration
[m/s²]
B
Width
[m]
D
Damping ratio (% of critical damping)
[-]
D1
Structural damping ratio
[-]
D2
Damping ratio from furniture
[-]
D3
Damping ratio from finishings
[-]
(x,y)
Deflection at location x,y
[m]

Deflection
[m]
E
Young’s modulus
[kN/cm2]
f, fi
Natural frequency
[Hz]
fs
Walking frequency
[Hz]
G
Body weight
[kg]
K, k
Stiffness
[N/m]
l
Length
[m]
Mmod
Modal mass
[kg]
Mtotal
Total mass
[kg]

Mass distribution per unit of length or per unit of [kg/m] or
area
[kg/m²]
OS-RMS
One step root mean square value of the effective [-]
velocity resp. acceleration
OSRMS90
90 % fractile of OS-RMS values
p
Distributed load (per unit of length or per unit of [kN/m] or
area)
[kN/m²]
T
Period (of oscillation)
[s]
t
Time
[s]
t
Thickness
[m]
v
Velocity
[mm/s]
[-]
4
1 Objective
Sustainability requires multi-storey buildings built for flexible use concerning
space arrangement and usage. In consequence large span floor structures with a
minimum number of intermediate columns or walls are of interest.
Modern materials and construction processes, e.g. composite floor systems or
prestressed flat concrete floors with high strengths, are getting more and more
suitable to fulfil these requirements.
These slender floor structures have in common, that their design is usually not
controlled by ultimate limit states but by serviceability criteria, i.e. deflections or
vibrations.
Whereas for ultimate limit state verifications and for the determination of
deflections design codes provide sufficient rules, the calculation and assessment
of floor vibrations in the design stage has still a number of uncertainties.
These uncertainties are related to:
-
a suitable design model including the effects of frequencies, damping,
displacement amplitudes, velocity and acceleration to predict the dynamic
response of the floor structure with sufficient reliability in the design stage,
-
the characterisation of boundary conditions for the model,
-
the shape and magnitude of the excitation,
-
the judgement of the floor response in light of the type of use of the floor
and acceptance of the user.
This report gives a procedure for the determination and assessment of floor
responses to walking of pedestrians which on one side takes account of the
complexity of the mechanical vibrations problem, but on the other side leads –
by appropriate working up-to easy-to-use design charts.
5
2 General procedure
The procedure for the determination of an acceptable floor response to excitation
induced by walking persons is based on the following:
1.
the characteristics of the loading by identifying the appropriate features
of the walking process by describing the load-time-history as a function of
body weight, step frequency and their statistical demographic distribution,
2.
the identification of the dynamic floor response from representative
“Single degree of freedom”-models for different typologies of floors, to
which actions in the form of parameterized time-histories of step forces
are applied; these responses are given as time-histories or frequency
distributions for further evaluations,
3.
the comfort assessment of the floor responses taking into account
human perception and condensation of data to a single representative
response parameter (OS-RMS-value90) which defines a certain fractile of
the distribution of responses to actions and is suitable for being
compared with response requirements depending on the type of building
and its use.
The procedure has been used to develop design diagrams, the use of which is
demonstrated by worked examples.
6
3 Description of the loading
Walking of a person differs from running, because one foot keeps continuously
contact to the ground while the other foot moves. It can be described by the
time history of walking induced contact forces.
The movement phases of a single leg, as illustrated in Figure 3-1, are the
following:
a)
The right foot touches the ground with the heel. This is the starting point
of the contact forces.
b)
The right leg is stretched; it transmits the full body weight.
c)
Rocking: the right foot rocks while the left leg swings forward.
d)
The left foot touches the ground while the right leg swings forward.
a
b
Right leg:
Ground contact
c
d
Streched,
Rocking
full body weight
Swing
Figure 3-1: Movement phases of legs and feet during walking
v in mm/s
A typical velocity time history measured at a representative point of a floor
structure excited by a walking person is given in Figure 3-2.
2
Original Signal
0
-2
0
1
2
3
4
5
6
Time in s
Figure 3-2: Typical velocity response time history of a floor to walking loads
Due to the periodicity of the contact forces it is possible to consider the time
history of the contact force of a single step according to Figure 3-1 only and to
describe this force-time history in a normalised way.
7
Figure 3-3 gives an example for the time history of the contact forces for two
different step frequencies, where the amplitudes are normalized by relating them
to the body weight G of the person.
Contact force related to body weight (normalized force)
1.5
Step frequency
1.5 Hz
2.2 Hz
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
time in s
Figure 3-3: Example for the time history of the normalised contact forces for
two different step frequencies
The standard walking load of a person can then be described as a series of
consecutive steps, where each step is given by a polynomial function, as given in
Table 3-1.
8
Polynomial function for the contact force due to a single step:
F t 
 K 1t  K 2 t 2  K 3 t 3  K 4 t 4  K 5 t 5  K 6 t 6  K 7 t 7  K 8 t 8
G
Coefficient
step frequency ranges
fs ≤ 1.75 Hz
1.75 < fs < 2 Hz
fs ≥ 2 Hz
K1
-8 × fs + 38
24 × fs – 18
75 × fs - 120
K2
376 × fs – 844
-404 × fs + 521
-1720 × fs + 3153
K3
-2804 × fs +
6025
4224 × fs – 6274
17055 × fs - 31936
K4
6308 × fs –
16573
-29144 × fs +
45468
-94265 × fs +
175710
K5
1732 × fs +
13619
109976 × fs –
175808
298940 × fs –
553736
K6
-24648 × fs +
16045
-217424 × fs +
353403
-529390 × fs +
977335
K7
31836 × fs –
33614
212776 × fs –
350259
481665× fs –
888037
K8
-12948× fs +
15532
-81572× fs +
135624
-174265× fs +
321008
Table 3-1: Determination of the normalized contact forces
The load duration t s of a single footfall is given by
Ts  2.6606  1.757 f s  0.3844 f s2 .
Figure 3-4 gives an example of a standard walking load history which is
1
composed by a repetition of normalized contact forces at intervals of
.
fs
9
Normalized force -
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
time in s
Figure 3-4: Example of a walking load function composed of normalized contact
forces
In order to obtain information on the statistical distributions of walking
frequencies f s and body weights G of persons, measurements of step
frequencies were carried out in the entrance area of the TNO building in Delft (in
total 700 persons) and the distribution of step frequencies were correlated with
the distribution of body mass, as published for Europe, assuming that step
frequencies and body masses would be statistically independent.
Figure 3-5 gives the distribution of step frequencies and body mass and Table
3-2 gives the associated cumulative distributions.
Figure 3-5: Frequency distribution of body mass and step frequency for a
population of data of 700
10
Classes of step frequency fsm
Classes of masses Mn
m = 1  35
n = 1  20
Cumulative
probability
Step frequency fs
(Hz)
Cumulative
probability
Step frequency fs
(Hz)
0,0003
1,64
0,0000
30
0,0035
1,68
0,0002
35
0,0164
1,72
0,0011
40
0,0474
1,76
0,0043
45
0,1016
1,80
00,146
50
0,1776
1,84
0,0407
55
0,2691
1,88
0,0950
60
0,3679
1,92
0,1882
65
0,4663
1,96
0,3210
70
0,5585
2,00
0,4797
75
0,6410
2,04
0,6402
80
0,7122
2,08
0,7786
85
0,7719
2,12
0,8804
90
0,8209
2,16
0,9440
95
0,8604
2,20
0,9776
100
0,8919
2,24
0,9924
105
0,9167
2,28
0,9978
110
0,9360
2,32
0,9995
115
0,9510
2,36
0,9999
120
0,9625
2,40
1,0000
125
0,9714
2,44
0,9782
2,48
0,9834
2,52
0,9873
2,56
0,9903
2,60
0,9926
2,64
0,9944
2,68
0,9957
2,72
0,9967
2,76
0,9975
2,80
0,9981
2,84
0,9985
2,88
0,9988
2,92
0,9991
2,96
0,9993
3,00
Table 3-2: Cumulative probability distribution functions for step frequency fs.m
and body mass Mn
11
The functions for contact forces
frequency and body mass are
responses of floor structures. The
step frequency as given in Table
develop design charts.
in Figure 3-3 and the distributions of step
the input data for calculating the dynamic
20 classes of body mass and the 35 classes of
3-2 were used (in total 700 combinations) to
12
4 Dynamic floor response
The dynamic response of a floor structure to persons walking is controlled by the
loading characteristics, as described in Section 3, and by the structural dynamic
properties of the floor.
The dynamic properties of the floor structure relevant to the floor response are,
for each vibration mode i:
-
the eigenfrequency fi ,
-
the modal mass Mmod,i ,
-
the damping value Di .
The various modes i are normally arrayed according to their energy contents.
The first mode (i = 1) needs the smallest energy content to be excited.
When the eigenfrequency of a mode and the frequency of steps are identical,
resonance can lead to very large response amplitudes. Resonance can also occur
for higher modes, i.e. where the multiple of the step frequency coincides with a
floor frequency.
The response amplitudes of floor structures due to walking of persons are in
general limited by the following effects:
-
the mass of the floor structure. As the number of step impulses is limited
by the dimensions of the floor (walking distances), the ratio of the body
mass to the exited floor mass influences the vibration,
-
the damping D that dissipates excitation energy. The damping Di consists
of the structural damping D1 , e.g. due to inner friction within the floor
structure or in connections of the floor to other structural components
such as supports, of the damping D2 from furniture and equipment and of
the damping D3 from further permanent installations and finishings.
Table 4-1 gives an overview on typical damping values as collected from various
sources of literature [6].
13
Type
Damping
(% of critical damping)
Structural Damping D1
Wood
6%
Concrete
2%
Steel
1%
Composite
1%
Damping due to furniture D2
Traditional office for 1 to 3 persons
with separation walls
2%
Paperless office
0%
Open plan office.
1%
Library
1%
Houses
1%
Schools
0%
Gymnasium
0%
Damping due to finishings D3
Ceiling under the floor
1%
Free floating floor
0%
Swimming screed
1%
Total Damping D = D1 + D2 + D3
Table 4-1: Components of damping
Figure 4-1 demonstrates by means of a flow chart how floor responses in terms
of time histories or frequency spectra of velocity have been calculated for various
floor systems k, which were used for further evaluation.
14
1 Floor system with index k
2 Single mass oscillator
representative for the deck k
with the structural properties
Mk, fk , Dk
3 Body mass Mn
and associated probability
distribution function HM,n
30 ... 125 kg
For each step frequncy fm (m=1...35)
For each body mass Mn (n=1...20)
4 Step frequency fs,m
and associated probability
distribution function Hf,m
1,6 ... 3,0 Hz
5 Generation of load
function
 Fn,m(t)
Mk
Ck
Dk
HM
M
Hf
f
F(t)
700
600
500
400
300
200
100
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
6 Time step analysis
M k x( t )  D k x( t )  C k x( t )  F ( t )
 x( t ); x( t ); x( t )
7 … 11 Determination of
OS-RMS value, associate
with joint probability of
frequency and mass
HOS-RMS = HM,n * Hf,m
t
HOS-RMS
OS-RMS
Next step frequency
Next body mass
12 Determination of the 90%
fractile OS-RMS90
from the cumulated
probability function
Figure 4-1: Flow chart for calculation of dynamic floor responses to walking
excitations by a person with the mass M n and the frequency f m , see also Figure
5-5
15
In these calculations the excitation point is assumed to be stationary , i.e. the
walking path is not taken into consideration. In general, the location of the
stationary excitation and hence the location of the response are selected where
the largest vibration amplitudes are expected (for regular floors it is usually the
middle of the floor span).
Apart from excitation by the regular walking also the excitation from single
impacts, e.g. from heel drop may occur that leads to transient vibrations. This
report only refers to excitation from regular walking because experience shows
that for floor structures with lowest eigenfrequency fs  7 Hz walking is the
relevant excitation type, whereas heel drop is only relevant for fundamental
eigenfrequencies fs > 7 Hz.
In general, the time response of a floor system to regular excitation by walking
take the form of one of the plots given in Figure 4-2.
acceleration
acceleration
time
time
a)
b)
Figure 4-2: Possible envelopes of dynamic responses of a floor to regular
excitation a) resonant response, b) transient response
If the excitation frequency (or higher harmonics of the excitation) is similar to an
eigenfrequency of the floor, the response takes the form as shown in Figure 4-2
a): a gradually increasing of the response envelope until a steady-state level.
This response is known as either resonant response or steady state response.
This kind of response can occur for floors with a fundamental natural frequency
inferior to 9-10 Hz.
If the excitation frequency is significantly lower than the natural frequency of the
floor, the response envelope shown in Figure 4-2 b) is typical, known as
transient response. In this case, the floor structure responds to the excitation as
if it is a series of impulses with the vibration due to one foot step dying away
before the next step impulse.
16
5 Comfort assessment of the floor structures
The purpose of the comfort assessment of the floor structures is a design, by
that vibrations are so small, that adequate comfort of the users is obtained.
This comfort assessment implies the use of a single response parameter that
reflects both, the comfort perception of users and the dynamic response of the
floor structure.
The definition of such a parameter requires various assumptions:
1.
a weighting of the frequencies obtained from the response of the floor
structure to take the frequency dependence of human perception into
account. In a similar way to human hearing, the human perception of
vibration varies with the frequency.
The weighting function used applies to the response in terms of velocity,
see Figure 5-1:
B(f)
B(f) 
1.2
1
v0
1
1( f
f0


X B (f)  B(f)  X(f)
1
)
2
v 0  1,0 mm/s
f 0  5,6 Hz
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
f in Hz
Figure 5-1: Weighting function for the spectrum of vibration velocities
The weighting
dimensionless.
2.
function
achieves
that
the
weighted
response
is
Use of RMS-values (Root mean square values) as effective response
values by evaluation of a time window Ts :
17
RMS n ,m 
t Ts

t
x B2 t dt
Definition of the time window T = Ts. If Ts is too long, the results are
smeared, if Ts is too short, the results are arbitrarily.
v in mm/s
The well-proven definition of the time window Ts is the time interval of
standard contact force for a single step according to Figure 3-3, see
Figure 5-2.
2
Original Signal
0
-2
0
v in mm/s
3.
1
Ts
1
2
3
4
5
6
2
3
4
5
6
2
RMS = 0.56 mm/s
0
-2
0
1
time in s
Figure 5-2: Selection of the time window
Ts for the RMS-value of the
weighted velocity response
This definition leads to the “one step-root mean square value”, so called
OS-RMS-value, which is independent on the step frequency and duration
of time interval:
OS  RMSn , m 
1 t  Ts 2
 x B t dt
Ts t
Figure 5-3 gives as an example for a floor structure with the dynamic
properties f = 2.8 Hz, Mmod = 20000 kg, D = 3% the OS-RMS-value as a
function of the step frequency and of the body mass.
18
Figure 5-3: Example for OS-RMS-values as a function of step frequency
and body mass
The results in Figure 5-3 do however not yet consider effects of the
frequency distributions of the step frequency fs and of the body mass G.
They may be agglomerated to a cumulative frequency distribution, see
Figure 5-4.
4.
Accounting for the frequency distribution Hfm of the step frequency fs and
the body mass G.
The classes of OS-RMS-values HOS-RMS in Figure 5-3, are multiplied with
the cumulative probability distributions Hfm. In conclusion a cumulative
distribution of OS-RMS-values is obtained according to Figure 5-4, that
also contains the results.
19
90%
1
f = 2,8 Hz / M = 20 000 kg / D = 3 %
accumulated frequency
distribution
0.8
0.6
0.4
With accounting for demographic frequency occurence
Without accounting for demographic frequency occurence
0.2
0
0
OS-RMS90
4
8
12
16
OS-RMS
20
24
28
Figure 5-4: Cumulative frequency distribution of OS-RMS-values with and
without taking the frequency distribution Hfm into account
5. Definition of a representative OS-RMS-value to obtain the desired reliability.
This representative value is defined as the 90 %-fractile of OS-RMS-values
from the cumulative frequency distribution, as indicated in Figure 5-4, which
is denoted as OS-RMS90.
Figure 5-5 gives an overview of the various steps to obtain the OS-RMS90 values
by means of a flow chart.
20
6 Time step analysis
M k x( t )  D k x( t )  C k x( t )  F( t )
 x( t ); x( t ); x( t )
t
7 Transformation from
time to frequency domain
(FFT)
t
↓
x( t )  X ( f )
f
8 Frequency weighting
according to perception
1
1

XB(f ) 
 X ( f )
v 0 1  ( f0 / f )2
f
v 0  1,0 mm / s ; f 0  5,6 Hz
f
9 Transformation into time
domain (iFFT)
↓
X B ( t )  x B ( f )
10 Determination of the
effective value for the
duration Ts of a single step
(OS-RMS-value) and and
allocation to a OS-RMSclass
OS  RMS n ,m 
1
Ts
f
t
OS-RMS
t
t Ts
 x
2
B
( t ) dt
t
11 Improvement of the
distribution function
HOS-RMS = HM,n * Hf,m
HOS-RMS
OS-RMS
Next step frequency
Next body mass
12 Determination of the 90%
fractile OS-RMS90
from the accumulated
frequency distribution
90%
OS-RMS90
Figure 5-5: Flow chart for the evaluation of dynamic floor-response to walking
excitations by a person with the mass Mn and the frequency fm to obtain the OSRMSn,m values and their distribution
21
The limits for the OS-RMS90-values for comfort are based on various standards for
standardizing human perception [6], [7], [8], [9], [10], [11].
In general, the perception and the individual judgement, whether vibrations are
disturbing or not (discomfort), are based on the same criteria but can lead to
different limits, as certain persons can detect vibrations without being
discomforted by them.
The governing parameters are e.g.:

momentary activity of the user (manual work or sleeping),

age and state of health of the user,

posture of the user (sitting, standing, laying down), see Figure 5-6

Relation between the user and the source of excitation (are vibrations
expected or not),

Frequency and amplitude of vibration (as taken into account by the
weighting function).
z
z
y
y
x
x
Supporting
surface
Supporting
surface
x
z
Supporting
surface
y
Figure 5-6: Directions for vibrations defined in ISO 10137 [6]
22
1
0.1
Weighting factor
Weighting factor
Figure 5-7 gives examples for curves of same perception for z-axis vibration
( Wb curve) and x-and y-axis vibrations ( Wd curve); e.g. according to the Wb
curve a sine wave of 8 Hz is equivalent to a sine wave with 2.5 Hz or 32 Hz with
double amplitude.
1
10
Frequency (Hz)
Wb Weighting
100
1
0.1
1
10
Frequency (Hz)
Wd Weighting
100
Figure 5-7: Wb and Wd -weighting curves
These parameters can be allocated to various classes of perception defined by
lower and upper threshold values for the OS-RMS90-values, that are suitable for
being associated to certain typical usages of floors, see
Table 5-1.
23
D
0.8
3.2
E
3.2
12.8
F
12.8
51.2
Sports facilities
0.8
Industrial Workshops
0.2
Hotels
C
Senior citizens’
Residential building
0.2
Meeting rooms
0.1
Office buildings
B
Schools, training centers
Upper limit
0.1
Hospitals, surgeries
Lower limit
0.0
Critical areas
Class
A
Residential buildings
Usage of the floor structure
OS-RMS90
Recommended
Critical
Not recommended
Table 5-1: Allocation of classes of perception A to F to threshold values of OSRMS90-values and relation of occupancies of floors to comfort limits
Table 5-2 gives the background to Table 5-2 from limits specified in ISO 10137
[6].
24
Usage
Time
Multiplying
Factor
OS-RMS90
equivalent
Critical working areas (e.g. hospitals
operating-theatres, precision
laboratories, etc.)
Day
1
0.1
Night
1
0.1
Day
2 to 4
0.2 to 0.4
Night
1.4
0.14
Day
2
0.2
Night
2
0.2
Day
4
0.4
Night
4
0.4
Day
8
0.8
Night
8
0.8
Residential (e.g. flats, homes, hospitals)
Quiet office, open plan
General office (e.g. schools, offices)
Workshops
Table 5-2: Vibration limits specified in ISO 10137 [6] for continuous vibration
As it depends on the agreement between designer and client to define the
serviceability limits of comfort for floor structures, the allocation of perception
classes to comfort classes for various occupancies (Table 5-1) has the character
of recommendations.
25
6 Development of design charts
The procedure described in sections 2 to 5 may be used as assumed in this
report to calculate for other excitation mechanisms, e.g. for heel drop, the
structural response and the associated OS-RMS90-values. But it has been used for
the particular excitation by walking persons to develop design charts, which give
a relationship between
-
the modal mass Mmod of the floor structure [kg],
-
the eigenfrequency fi of the floor structure [Hz],
-
the OS-RMS90-values and their association to perception classes A to F
all for a given damping ratio D.
Figure 6-1 gives an example for such a design diagram for a damping ratio of
3 %.
Each point in this design chart is based on the statistical evaluation of 700
combination functions of step frequency and body mass.
26
Classification
based
damping
ratio of 3%
Klassifizierung
bei on
eineraDämpfung
von 3%
20 10
9
19
7
11
8
18
5
17
6
12
13
16
17
14
13 25
3
10
9
11
15
1.61.4
3.2 2.6
2.8
4
2 1.8
2.2
2.4
3.2 2.6
2.8
5 4
8
7
21
6
12
13
12
5
9
45
49 41
0.7 0.6
0.5
0.2
0.8
0.3
3.2 2.6
4
2.8
7
11
1.6
0 .7 0.6
1.4
B
0.4
1.2 1
2 1.8
2.2
2.4
8
33
0.5
0.2
0.8
0.3
3
21
25
9
6
8
13
10
Eigen frequency
of floor
in inHz
Eigenfrequenz
der Decke
Hz
7
37
29
156136
6
56
76
1.6
3.2 2.6
2.8
11
17
4
5
9
116
8
3
21
0.2
0.8
1.8
D
2.2
2.4
33
5
0.4
0.2
0.70.6 0.5
1 0 .8
1.61.4 1.2
216196
4
0.3
6
25
276
0.1
0.2
0.3
0 .4
0.5
0.70.6
1
1.2
1.4
2
7
45
41
49
96
0.1
C
12
7,1 Hz
A
0.1
0.4
0.1
17
10
0.3
1.2
1
2 1.8
2.2
2 .4
10
1137
29
0.2
0. 8
1.6 1.4
3
0.1
0 .7
0. 6
0.4
0. 5
1.2 1
236
256 176
F
0.6
2.6
3.2
2 .8
13
4
10
0.3
0.5
E
12
0.4 0.3
2.2
5
0.4
0.7
2
1.8
0.3
0.50. 4
0 .3
0.8 0.6
0.2
2.4
1
3
9
37
196
216
27 6
316
356
456
2
7
29
0.7
0.5 0.4
0.3
0.6
0. 8 0.7 0. 0.4
5
1
0.6
0.8
2 .6 21.8
1.2
.2
1.4
1
2.8 22.4
1.6
3.2
1.2
1.8
2
1.4
3
4
2.6 2.2 1.6
1 .41.2
1.6
11 8
17
116
79
6
696
856
5 36 476 336
836
616
676
576
876
776
756
416 296
816
736
59 6496
636
716
396
656
436
556
516 376
3
6
5
2.8
2.4
136
156
12
33
256236
7
21
56
76
96
1
100
200
500
4
10
98
13
45
41
6
25
2.2 1.8
1.6
1.4
1.2
2
1
2 000
5000
1.8
1.6
1.4
1.2
1
0. 8
0.6
0.5
0.4
3
2.2
2.6
1.8
2.4 2 1.6
1.4
1 .2
1
0.80.7
0.6
0.5
2.4
1000
2.8
3.2
5
3 2.6
49
2
3.2
10000
0.8
20000
0.7
0.4
50000
0.3
100000
17220 kg
Modale Masse der Decke in kg
Modal mass of floor in kg
Figure 6-1: Example of a design chart for the vibration assessment of floor
structures for a damping ratio D = 3 %
27
The design procedure based on these design charts provides the following steps,
see Figure 6-2:
1.
Determination of the basic floor characteristics (natural frequency, modal
mass, damping) for input,
2.
Determination of the OS-RMS90-value (90 % one-step RMS-value) from the
design chart, which characterizes the floor response to walking,
3.
Compare the OS-RMS90-value with the recommended or required limits for
the floor occupancy.
Determine dynamic floor
characteristics:
Natural Frequency
Modal Mass
Damping
Read off OS-RMS90 value
Determine and verify
floor class
Figure 6-2: Design procedure using the proposed design charts
If the floor response is characterized by more than one natural frequency, the
OS-RMS90-value should be determined as a combination of OS-RMS90-values
obtained for each mode of vibration i:
OS  RMS 90 
 OS  RMS 
2
90 i
i
28
7 Guidance for the design of floors for human
induced vibrations using design charts
7.1 Scope
This guidance provides a simplified method for determining and verifying floor
design for vibrations due to walking developed with the procedure given in
Section 2 to 6.
The guidance focuses on recommendations for the acceptance of vibration of
floors which are caused by people during normal use. Human induced vibrations
from rhythmic movements as dancing, gymnastic activities, jumping, machine
induced vibrations or vibrations due to traffic etc. are not covered by this
guidance.
The use of the guidance should be restricted to floors in buildings; it is not
applicable to pedestrian bridges or other structures not comparable with floors.
The guidance focuses on the prediction and evaluation of vibration at the design
level.
7.2 Procedure
The procedure used in this guidance needs the determination of the following
values:
1.
Dynamic properties of the floor structure:
-
eigenfrequency,
-
modal mass,
-
damping.
The dynamic properties should include a realistic assumption of the
mechanical behaviour at the level of the vibration amplitudes expected
(elastic behaviour), of the permanent mass and of the quasi permanent
part of the mass of variable loads.
In case of very light floor structures also the mass from persons should
be included in the floor mass.
2.
The appropriate OS-RMS90-value.
29
3.
The relevant occupancy class or classes of the floor.
4.
The requirement for comfort assessment.
7.3 Determination
structures
of
dynamic
properties
of
floor
In general, the method for the determination of dynamic properties of floor
structures should not be disproportionately more refined than the method for the
vibration limit state assessment, which is basically a hand calculation method.
Hence, this method is part of the package agreed between the designer and the
client in the design stage.
The hand calculation method for the determination of dynamic properties of
floor-structures assumes that the dynamic response of the floor can be
represented by a single degree of freedom system based on the fundamental
eigenfrequency.
The eigenfrequency, modal mass and damping of this system can be obtained by
-
calculation on the basis of the project documents or by
-
measurements carried out at floor-structures which have been built and
are used in a similar way as those projected and are suitable to be used
as prototypes.
For the calculation of the stiffness of the structure and of the connections the
initial elastic stiffness should be used, e.g. for concrete the dynamic modulus of
elasticity should be considered to be 10 % larger than the static tangent modules
Ecm.
For calculation of the masses on the basis of project documents experienced
values for the quasi permanent part of imposed loads for residential and office
buildings are 10 % to 20 % of the mass of the characteristic values. For lightweight floors the mass of one person with a minimum mass of 30 kg is
recommended to be added to the mass of the structure.
30
7.4 Values for eigenfrequency and modal mass
7.4.1 Simple calculation formulas for isotropic plates and
beams
Table 7-1 gives hand formulas for the determination of the first natural
frequency and the modal mass of isotropic plates for different supporting
conditions. For the application of this table it is assumed that all four edges of
the plate are linearly supported (no deflection of edges).
Table 7-2 gives hand formulas for beams for various support conditions.
Supporting Conditions:
clamped
Frequency ; Modal Mass
f 
hinged

L2
E t3
; M mod    M tot
12   (1   2 )

9.00
8.00
  1.57  ( 1  2 )
7.00
6.00
5.00
B
4.00
3.00
2.00
1.00
L
0.00
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Ratio  = L/B
 ≈ 0.,25 for all 

16.00
14.00
  1 . 57
1  2 . 5  2  5 . 14  4
12.00
10.00
8.00
B
6.00
4.00
2.00
L
0.00
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Ratio  = L/B
 ≈ 0.20 for all 
Table 7-1: Natural frequencies and modal mass for isotropic plates
31
Frequency ; Modal Mass
Supporting Conditions:
clamped
f 
hinged

L2
E t3
; M mod    M tot
12   (1   2 )

14.00
12.00
  1 . 57
5 . 14  2 . 92  2  2 . 44  4
10.00
8.00
B
6.00
4.00
2.00
L
0.00
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Ratio  = L/B
 ≈ 0.18 for all 

12.00
10.00
  1 . 57
1  2 . 33  2  2 . 44  4
8.00
6.00
B
4.00
2.00
L
0.00
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Ratio  = L/B
 ≈ 0.22 for all 

12.00
  1 . 57
2 . 44  2 . 72  2  2 . 44  4
10.00
8.00
6.00
B
4.00
2.00
L
0.00
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Ratio  = L/B
 ≈ 0.21 for all 
Table 7-1 (continued): Natural frequencies and modal mass for isotropic
plates
32
Frequency ; Modal Mass
Supporting Conditions:
clamped
f 
hinged

L2
E t3
; M mod    M tot
12   (1   2 )

18.00
16.00
  1 . 57
5 . 14  3 . 13  2  5 . 14  4
14.00
12.00
10.00
B
8.00
6.00
4.00
2.00
0.00
L
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Ratio  = L/B
 ≈ 0.17 for all 
E
Young’s Modulus in N/m²
t
Thickness of Plate in m

mass of floor including
furniture in kg/m²

Poisson ratio
Mtot
Total mass of floor including finishings and
representative variable loading in kg
finishing
and
Table 7-1 (continued): Natural frequencies and modal mass for isotropic plates
33
Supporting Conditions
Natural
Frequency
f 
l
f 
l
4

2

f 
l
f 
l
2

1
2
Modal Mass
3EI
0.37 l 4
M mod  0,41  l
3EI
0.2 l 4
M mod  0,45  l
3EI
0.49 l 4
M mod  0,5  l
3EI
0.24l 4
M mod  0,64  l
Table 7-2: Natural frequencies and modal mass for beams
7.4.2 Simple calculation methods for eigenfrequencies of
orthotropic floors
Orthotropic floors as e.g. composite floors with beams in the longitudinal
direction and a concrete plate in the transverse direction, see Figure 7-1, have
different stiffness in length and width EI y  EI x


b
y
l
x
z
Figure 7-1: Dimensions and axis of an orthotropic plate
The first natural frequency of the orthotropic plate being simply supported at all
four edges can be determined from
34
f1 

EI y
2
 l4
  b  2  b  4  EI x
1  2     
  l   l   EI y
where:

is the mass per m² in kg/m²,

is the length of the floor in m (in x-direction),
b
is the width of the floor in m (in y-direction),
E
is the Young’s Modulus in N/m²,
Ix
is the moment of inertia for bending about the x-axis in m4,
Iy
is the moment of inertia for bending about the y-axis in m4.
7.4.3 Natural frequencies from the self-weight approach
The self-weight approach is a very practical approximation in cases where the
maximum deflection  max due to self-weight loads has been determined, e.g. by
finite element calculation.
The natural frequency may be obtained from
f 
1
2
K
1

M 2
4g
3 max

18
 max mm
where the following assumptions have been made:
K
M g
3
4
 max
where:
M is the total mass of the vibrating system,
g  9.81 sm2 is gravity and
3
4
 max is the average deflection.
35
7.4.4 Natural frequency from the Dunkerley approach
The Dunkerley approach is an approximation for the case that the relevant mode
shape is complex and can be considered as a superposition of simple modes, for
which the natural frequencies can be determined, e.g. according to section 7.4.1
and 7.4.2.
Figure 7-2 gives an example for a composite floor with two simply supported
beams and a concrete plate without stiff supports.
Initial System:
Mode of concrete slab:
Mode of composite beam:
Figure 7-2: Example for mode shape decomposition
The expected mode shape may be divided into a beam mode with the frequency
f1 for the composite beam and a plate mode with the frequency f2 for the concrete
slab.
The natural frequency accounting for the interaction of the beam mode and the
plate mode would be
1
1
1
 2  2
2
f
f1
f2
36
7.4.5 Modal mass from mode shape
Where an approximation of the mode shape by a normalized function   x, y  with
  x, y  max  1,0 is available, e.g. from calculation of deflection due to a
distribution of mass forces, see Figure 7-3, the modal mass may be obtained
from:
M mod  μ  δ 2(x,y) dF
F
where:

is the distribution of mass
(x,y)
is the vertical deflection at location x, y
Expected mode shape:
Application of loads:
Figure 7-3: Example for the application of mass load distributions to obtain an
approximation of mode shape
In case of FEM calculations the modal mass results from:
M mod 
δ
2
i
 dM i
Nodes i
where:
i
is the vertical deflection at node i (normalised to the maximum deflection)
dMi
is the mass attributed to the node i of the floor.
Examples for the use of these approximations, that in the case of exact solution
for the mode shape give the exact modal mass, are given in Table 7-3.
37
Example
1
Approximation of mode shape
  x, y   sin
Ly
x
x
sin
y
y
;   x, y  max  1,0
Mass
distribution

M total
x  y
Modal mass
M mod     2  x, y dF 

Lx
M total
 x y
 
M
2  x 
2 y


sin
sin
ly lx   x    y dx dy  4total
 
Ly  Lx
2
Lx
Ly >> Lx
3
lx
l
and l y  x  y  l y
2
2
x
y
 x, y   sin  sin ;   x, y  max  1,0
x
y
1. 0  y 
Ly

M total
x y

lx
l
 y  ly  x
2
2
x
  x, y   sin 1,0   x, y  max  1,0
x
2.
  x, y  
Ly
Lx
x
x  y
y
sin sin sin ;   x, y  max  1,0
x
y




M total
x y
where:
x
= deflection of the beam
Plate and beams simply
supported
y
M mod     2  x, y dF 
= deflection of the slab
assuming stiff supports by
the beams  x  0
 , x / 2
M total x
2
x y 
 x , y   x / 2
  sin
2
 sin
2
 y 
 x 
   sin 2  dx dy 
 
 x 
 y
 x 
M
 dx dy   total
4
 x 

2   x

y





M mod     2  x, y dF 
M
 total
 x y
 M total
2
 x
x  y
y 

sin
sin

    x   y  dx dy
lx l y 


2
2
 x   y
8 x  y 

 2

2

2 
 2
  x  y
Table 7-3: Examples for the determination of modal mass by hand calculation
38
7.4.6 Eigenfrequencies and modal mass from FEM-analysis
Various FEM-programs can perform dynamic calculations and offer tools for the
determination of natural frequencies. Many programs also calculate the modal
mass automatically in the frequency analysis.
If FEM is applied for determining the dynamic properties for vibration, it should
be considered that the FEM-model for this purpose may differ significantly from
that used for ultimate limit state verification as only small deflections in the
elastic range are expected.
A typical example is the selection of boundary conditions in vibration analysis
compared with that for ULS design. A connection which is assumed to be hinged
in ULS may be assumed to provide a full moment connection in the vibration
analysis (due to initial stiffness).
7.5 Values for damping
Independently of the way of determining the natural frequency and modal mass,
damping values for vibration systems can be determined using Table 7-4 for
different construction materials, furniture and finishing in the condition of use.
The system damping is obtained by summing up the appropriate values for D1 to
D3.
39
Type
Damping
(% of critical damping)
Structural Damping D1
Wood
6%
Concrete
2%
Steel
1%
Composite
1%
Damping due to furniture D2
Traditional office
separation walls
for
1
to
3
persons
with
2%
Paperless office
0%
Open plan office
1%
Library
1%
Houses
1%
Schools
0%
Gymnastic
0%
Damping due to finishings D3
Ceiling under the floor
1%
Free floating floor
0%
Swimming screed
1%
Total Damping D = D1 + D2 + D3
Table 7-4: Determination of damping
7.6 Determination of the appropriate OS-RM90-value
When frequency and modal mass are determined, the OS-RMS90-value can be
obtained with the design charts given in Figure 7-4 to Figure 7-12. The
relevant diagram needs to be selected according to the damping characteristics
of the floor.
40
The diagrams also contain an allocation of OS-RMS90 values to the floor classes.
In case various natural frequencies are relevant, the total (combined) OS-RMS90value may be determined from
OS  RMS 90 
 OS  RMS 
2
90 i
i
41
Classification based on a damping ratio of 1%
20
19
18 21
12
13
11
17
8
7
6
5
0.5
0.6
0.4
0.8
15
1312
17
25
11
7 6
8
2.6
2.4 1.8 1.41.2
2.2
2
4
5
0.7
1
21
14
41 33
37
13
45
49
2.8
109
12
11
56
25
1312
17
7
6
A
0.5
0.6
0.4
0.8
41
96
45
49
8
0.1
0.1
5
33
10
37
1
0.3
0.8
0.1
2.8
9
1.6
96
56
176
216
236
276
256
196156
136
6
8
1.41.2
1.8
2.2 2
2.62.4
4
5
11
76
33
1.4
10
9
0.8
49
1.6
E
6
13
5
3 2.42.2
2.6
8
12
3.2
1.2
F
1.2
1.4
96
0.3
256
21
296
476 376
496
336
196
756
596 456396 316 216
876
736636
956
796
776
936
616 576
176
856676
916
896
536
436 356
836
816
556
716
656 516 416
696
2
0.7
0.8
1.8
1
1.6 1.4
2
4 2.8
2.2
2.4
32.6
3.2
9
6 5
10
7
8
11
25
56
76
33
41
37
17
276
29
116
236
45
156
12
11
25
9
6
8
136
49
4
3.2 3
1.81.6
21
1
2.82.6
0.6
0.3
0.4
0.8
2
0.2
0.4
0.3
0.5
0.6
0.4
0.7
0.5
0.8
0.6
1.2 1
0.7
1.8 1.6
1.4 1.2 0.8
1
2
1.4
2.8 2.2
1.6
2.4
1.8
2.6
2
3
4
3.2
2.2
2.82.4
2.6
5
3
6
4 3.2
12
13 11
9 7
5 4
10 8
6
3.2
3
2.4
4
2.8
2.6 2.2
5
7
3.23
21.8
2.4
1.6
2.2
1.4
2.8
2.6
1.2
1
2 1.8
1.6
1.4
0.8
2.4
1.21
2.2
0.7
0.8
0.6
0.7 0.5 0.4
1.4
1.2
5
10
0.8 0.6
0.7
1
1.8 1.2
0.8
0.7
0.6
2 1.6
1.4 1 0.80.70.6 0.5 0.4
0.3
0.4
0.5
0.6
17
3
0.2
0.4
0.3
0.6
0.7 0.5 0.4
2.8
7
29
0.3
0.5
0.8
1
4
0.3
0.4
0.6
0.7
45
0.5
1.2 1
2.4
2.61.82.22
376 316
396
416
236
336
356276 256
296
0.3
0.4
33.2
37
196
216
0.2
0.3
0.4
0.5
0.7 0.6
D
2.8
116
5
0.2
1.6
21
41
0.1
0.3
0.7
7
0.2
C
0.5
0.4
0.6
1 0.8
29
12
13
17
25
B
0.2
0.5
0.6 0.4
3.2
3
8
4
0.3
0.7
1.41.2
2.6
2.4
1.8
2.2
2
21
10
9
0.2
1.6
4
11
76
6
0.1
3.2
3
29
7
0.2
0.3
1.6
3.2
3
16 29
Eigenfrequency of the floor (Hz)
0.7
1
2.8
10 9
17
1.41.2
2.6
2.4
2.2 1.8
2
4
0.5
1
100
200
500
1000
2000
5000
10000
20000
50000
100000
Modal mass of the floor (kg)
Figure 7-4: OS-RMS90 for 1 % damping
42
Classification based on a damping ratio of 2%
20
19
18
11 10 8
9
13
12
4
5
7
3.2
2.8
3
6
1.81.6
1.4
2.42.2
2
17 17
1110
9
15
13 33
29 25
12
4
8
0.2
0.4
0.1
1.6
1
1.8
1.4
2.42.2
1.2
2
6
0.7
0.5
0.6
0.2
0.4
0.3
17
2.6
21
1110 8
9
4137
4
0.1
1.6
1.8
1.4
2.4
2.2
2
6
0.7
1
0.2
0.5
0.6
1.2
0.4
0.1
0.3
9
33
29
17
25
2.6
C
0.8
8
21
Eigenfrequency of the floor (Hz)
3.2 2.8
3
5
96
56
0.2
0.5
0.6
1.4
0.4
0.1
0.1
0.3
0.2
1.2
2.4
2.2 2
13
12
45
49
0.7
1.6 1
1.8
7
116
156
176
9
4137
136
6
4
11 10 8
76
196
B
0.8
3.2
2.8
3
5
7
13
12
45
5649
7
A
0.3
0.8
3.22.8
3
5
7
13
12
14
10
0.5
0.6
2.6
1621
11
0.7
1
1.2
0.2
0.3
6
0.8
D
0.5 0.4
0.7
0.2
0.6
5
17
29
4
1.2
25
276216
336296
196
356316
236
256
176
E
21
F
76
3
8
11
10
1.8
7
6
13
116
276
156
49
56
96
7
25
1
12
21
13
12
1000
2000
2
5
8
500
0.8
0.4
0.3
0.60.5
0.4
0.7
2
0.5
1.2 1 0.8 0.6
2.2
2.8
1.4
1.6
0.7
3 2.4
3.2
1.8
4
1.2 10.8
1.4
2
2.6 2.2
1.6
2.4 1.8
5
2.8
3
3.2
2
2.62.2
2.4
4
8 7 6
2.8
3
17
10 9
3.2 2.62.4
11
2.2
5
2.8
2 1.8
4
1.6
32.6 2.4
1.4
3.2
2.2 1.8
1.2 1
2 1.6
1.4
0.8
6
2.8
2.4
1
1.2
0.6
1.8
0.8
0.7 0.5
0.4
2.62.2 1.6 1.4
0.6
3
1
33 29
200
1
0.6
0.50.4
0.8 0.7
0.3
0.60.5 0.4
0.3
0.2
2.6
45
1
100
0.3
0.4
0.7
9
376
356
236
396
516 416
676596
216
456 336
556
616
576
896
916
536
256
856
836
736
696656
476
636
296
876
756
496436
776
816
796
716
196
0.3
0.50.4
0.8 0.70.6
0.5
2
2.42.2
1.6 1.4 1.2
136
316
2.8
3.23
4
5
4137
2
1
1.6
1.8 1.4
2.6
33
1.2
4
5000
10000
20000
0.8
0.7
0.5
0.4
50000
0.3
100000
Modal mass of the floor (kg)
Figure 7-5: OS-RMS90 for 2 % damping
43
Classification based on a damping ratio of 3%
20 10
9
19
7
11
8
18
5
17
6
12
13
16
13 25
2 1.8
2.2
2.4
8
7
21
29
45
49 41
0.3
0.7 0.6
A
0.1
0.4
0.5
1.2
1
0.2
0.8
0.3
0.1
5
9
11
17
8
3.2 2.6
2.8
4
7
1.6
0.7 0.6
1.4
1.8
2.2
2.4
B
0.4
1.2 1
2
33
0.5
0.2
0.8
0.3
3
21
25
9
0.2
1.6 1.4
10
11 37
10
0.1
0.4
0.5
0.8
2 1.8
2.2
2.4
3
6
12
13
12
0.70.6
1.2 1
3.2 2.6
2.8
5 4
9
11
17
14
3
10
15
1.61.4
3.2 2.6
2.8
4
6
0.1
C
12
8
13
10
Eigenfrequency of the floor (Hz)
7
37
116
6
156
136
56
76
5
9
29
11
17
4
8
2.8
2
7
0.2
0.8
1.8
D
2.2
3
5
21
0.4
0.3
6
0.2
0.70.6 0.5
1 0.8
1.61.4 1.2
25
196
216
276
4
0.1
0.2
0.3
0.4
0.5
2.4
33
45
41
49
96
0.7 0.6
1
1.2
1.6 1.4
3.2 2.6
236
256 176
F
0.6
2.6
3.2
2.8
13
4
10
0.4
0.7
2
1.8
2.2
5
0.8 0.6
0.50.4
0.3
2.4
3
1
9
37
196
216
276
316
356
456
2
3
7
29
0.7
6
136
156
12
33
256236
76
7
21
56
13
45
41
10
98
6
3.2
5
3 2.6
96
1
100
200
49
500
25
2000
5000
0.5 0.4
0.3
0.6
0.8 0.7 0.50.4
1
0.6
0.8
2.6 21.8
1.2
1.4
1
2.8 2.2
1.6
3.2
2.4
1.2
1.8
1.4
2
3
4
2.6 2.2 1.6
5
2.8
2.4
1.8
2
3.2
1.6
1.4
3
2.2
2.6
1.8
1.2
2.4 2 1.6
4
1
1.4
2.8
0.8
1.2
1
2.2 1.8
0.80.7 0.6
0.5
1.6
1.4
0.4
0.6
1.2
0.5
2
1
2.4
1000
0.3
0.2
1.41.2
1.6
11 8
17
116
796
696 576
856
536 476
836
616
676
876
776
756
416336296
816
736
596 496
636
716
396
656
436
556
516 376
0.3
0.5
E
12
0.4 0.3
10000
0.8
20000
0.7
0.4
50000
0.3
100000
Modal mass of the floor (kg)
Figure 7-6: OS-RMS90 for 3 % damping
44
Classification based on a damping ratio of 4%
20
8 7
19 9
6 5
2.6 2
1.6 1.2
1
2.4
3.2
2.2
1.8 1.4
3
2.8
4
18
17 12
11
13
16
0.60.5
0.4
0.80.7
0.1
7
98
6
5
2.6 2
2.4
4
3.2
1.6
1.21
0.6 0.5
0.80.7
2.2
14
1.8
0.4
1.4
0.2
0.3
3
21
12
11
0.1
2.8
11
13
12
10
98
17
7
2.6 2 1.6
2.4
3.2
2.2
4
6 5
10
29
37
25
41 33
9
1.2 1
1.8
21
B
0.5
0.4
0.2
0.3
1.4
0.1
1211
13
C
2.8
10
87
9
7
17
56
96
0.6
0.80.7
3
8 49
45
Eigenfrequency of the floor (Hz)
A
10
15 17
13
0.2
0.3
2
2.6
4
0.6 0.5
0.7
0.8
2.4
5
6
1.21
1.6
3.2
0.4
0.3
0.1
0.2
0.1
0.2
2.2
116
6
76
1.8 1.4
29
37
25
41 33
D
3
5
21
12
196
11
156
176
216
13
4945
F
3
2
10
8
0.4
0.5
0.4
2.2
1.8
1.4
0.7
5
1
1.2
6
3
0.7 0.5 0.4
0.8 0.6
1
0.7
1.2
0.8
1
1.6 1.4 1.2
2
1.8
3.2
3
2.8 2.2
2.4
2.6
116
76
5
1211
29
25
10
7
8
37
21
3.2
32.8 2.4
13
2.2
1.2
21.8 1.4
4
9
2000
5000
0.6
0.7
1 0.8
1000
1.61.4
1.2
2
1.8
1
1.6 1.41.2
0.8
1
0.80.7 0.6
0.50.4
6
2.6
33
500
0.6
4
136
200
0.2
0.4 0.3
2 1.6
1.4
2.4
2.21.8
2.6
2.8
216
1
100
0.3
0.5
0.8
3.2
56
41
0.3
0.6
17
96
0.2
0.3
7
9
196
356
476416
396
436
376 296236
556
816
616
576 496
796
736
636 456
836
776
716
696
676
656 536
336
596516
276
316
256
0.3
1.6
2.6
2.4
4
176
2
1.2
E
236
4
0.4
0.60.5
0.7
1
0.8
2.8
10000
20000
0.5
0.4
50000
0.3
100000
Modal mass of the floor (kg)
Figure 7-7: OS-RMS90 for 4 % damping
45
Classification based on a damping ratio of 5%
20
19
7
8
1.6 1.4 1
3.2 2.8
3
1.2
2.21.8
2.4
2.6 2
4
6 5
18
0.8 0.7 0.6
0.1
0.5
0.4
0.2
0.3
17
10
11
9
16 13
6
14 17
5
12
0.4
0.2
10
11
9
13
12
8 7
11
4
25
6
29
1.6
1.4
3.2
3 2.8
0.1
0.5
1.8
2.42
2.6
12
B
0.80.7 0.6
1
1.2
2.2
5
21 17
33
0.4
0.2
0.3
9
C
10
11
8
9
13
7
Eigenfrequency of the floor (Hz)
A
0.1
0.5
0.3
13
10
1.6
1 0.80.7 0.6
1.4 1.2
2.2 1.8
2.4
2.6 2
4 3.2
3 2.8
87
15
87
76
56
96
37
45
25
29
6
6
1
1.4
1.6
3.2
3 2.8
4
49
0.1
0.80.7 0.6
0.4
0.3
2.2 1.8
5
0.1
0.2
0.5
1.2
2.4 2
0.2
21 17
41
D
2.6
12
33
5
10
4
176
196
1
11
9
F
156
3
136
1.6
1.4
1.2
E
13
4
7
8
0.5
2
2.6
316 256
236
96
56
29
21
216
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
0.8
1
1.61.41.2
1.8
2.22
2.8
3 2.4
3.2
25
0.2
0.4
0.8
1
2.4
276
556
596
516
496
356
536
336
476436396
616 456
636
576
376 296
0.3
0.6
0.7
6 5
37
45
416
0.3
2
49
0.2
0.4
1.8
3.2
3 2.8 2.2
76
196
0.3
0.4
0.70.6 0.5
0.8
1.2
1.4
1.6
1
17
2.6
12
176
1.8
4
41
2.2
10
9
116
5
7
2.8
3
2.6
33
11
8
6
2.4 1.8 1.4
1.2
1
1.6
0.8
1.2
1
0.80.7
0.6
0.5 0.4
1.4
1.2 1
1.6
3.2
2
0.8
2
0.7
0.4
0.3
0.6
0.5
1
100
200
500
1000
2000
5000
10000
20000
50000
100000
Modal mass of the floor (kg)
Figure 7-8: OS-RMS90 for 5 % damping
46
Classification based on a damping ratio of 6%
20
7
19
5
18
4
8
17 10
7
13
12
4
1.6
1.8
1.2
7
4
1
2.2
2.4
3.22.8
0.8
0.6
49
1.4
7
2.6
2.4
56
4
25
45
2
3329
8
0.8
0.2
D
1
0.2
11
0.3
0.5
0.7
41
1.4
10
0.6
9
4
0.2
1.2
7
21
3
2.6
6
F
49
2
0.5
2.2
13
1.6
0.4
0.3
1.8
2.4
5
2.8
3.2
4
12
1
2
1.4
3
76
0.6
0.8
0.5 0.4 0.3
0.7 0.6 0.5
1.2 1
0.8
0.7
1.6
1.8 1.4
2.2
1.2 1
2.6
2.4
2
8
116
0.2
0.7
196
216
236
316
456 356
436
596
636
576
496
336
556
476
536
616
516
416
396
376
0.3
0.8
E
176
256
0.2
0.4
3
37
156
0.1
0.4
0.6
17
5
0.1
0.3
1.6
1.8
3.22.8
6
C
1.2
2.2
5
12
76
0.5
6
13
0.1
0.2
0.7
3
21
7
96
0.4
1
2
9
8
B
0.3
1.2
1110
37
Eigenfrequency of the floor (Hz)
0.7
1.6
1.8
8
17
0.1
0.2
0.5
5
41
0.4
2.6
6
13
12
A
0.3
0.6
3
21
25
29
0.5
1.4
9
0.1
0.2
0.8
2
8
11
0.4
0.7
1110
12
9
0.6
1
1.4
2.2
2.4
3.22.8
5
17
10
0.3
0.8
2.6
6
14
13
1.2
3
9
16
15
2.6
2.2
2.4 1.6
1.8
3.2 2.8
2
6
136
1
296
276
2.8
25
56
1.8
3.2
176
45
1
0.8
2.2
11 10
96
6
3
4
17
33
29
9
7
2
1.2
1.6
1.4
5
1.6
1.41.2
0.7
0.6
0.5
0.4
2.6
2.4
0.8
0.3
1
1
100
200
500
1000
2000
5000
10000
0.8
0.7
0.6
0.5
0.4
20000
50000
100000
Modal mass of the floor (kg)
Figure 7-9: OS-RMS90 for 6 % damping
47
Classification based on a damping ratio of 7%
20
6
19
2.2
1.4
2.4
1
2.8
1.8
2
1.2
3
2.6
3.2
1.6
5 4
18
7
8
17 9
0.5
0.7
0.1
0.2
0.3
0.4
0.8 0.6
16
15
14
17
13
6
11
13
12 10
9
5
2.2
2.4
2.8
4
2
3
3.2 2.6
1.4
1.8
0.5
0.7
1
1.2
A
0.3
1.6
7
0.1
0.2
0.6
0.8
0.4
8
12
11
10
29
21
25
11
13
12
6 5
2.2
2.4
2.8
4
10
1.4
3.2
1.2
2.6
17
9
B
0.3
0.6
1.6
7
9
0.2
1.8
2
3
0.1
0.5
0.7
1
0.4
0.8
0.1
8
C
8
4133
5649
7
6
37
11
5
2.2
2.4
2.8
4
Eigenfrequency of the floor (Hz)
13
45
6
1.4
1
0.2
0.5
0.7
0.1
0.3
1.8
2
12
76
10
21
96
D
3
2.6
25
29
3.2
1.2
17
0.1
0.4
0.6
0.2
0.8
5
1.6
7
0.3
0.5
9
116
E
8
4
0.2
0.7
0.4
33
1.4
1
41
49
F
56
3
136
0.2
2.2
6
0.3
5
4
37
2.4
11
1.8
2
2.6
3.2
13
2
416
376
336
516
616
496 356
576
476
536
396
556
456
436
0.5
2.8
3
256
296
0.6
0.8
1.2
176
236196
2.2
1.6
2.4
156
316
1 0.8
1.8
2
45
216
0.3
0.6 0.5 0.4
1
0.8
0.7 0.6
1.4 1.2
12 10
276
0.4
0.7
1.6
21
76
7
1.4 1.2
1
2.8
2.6
4
25
17
29
9
8
3
5
1.8
1.6
1.2
0.8 0.6
0.7 0.5
0.4
1
1.4
0.6
3.2
96
0.8
0.7
0.3
0.8
2
6
0.5 0.4
1
100
200
500
1000
2000
5000
10000
20000
50000
100000
Modal mass of the floor (kg)
Figure 7-10: OS-RMS90 for 7 % damping
48
Classification based on a damping ratio of 8%
20
197
5
6
3.2
3
4
2 1.8
1.6
2.4
2.6
2.8
18
17 9
16
1.21
0.1
0.2
0.3
0.6
1.4
8
2.2
15 13
7 6
10
14
5
4
3.2
3
2
1.81.6
2.4
2.6
2.8
11
12
13
0.5 0.4
1.21
A
0.1
0.2
0.3
0.7
0.8
0.6
1.4
98
2.2
12
11 21
0.5 0.4
0.7
0.8
17
5
4
7 6
13
3.2
3
2
1.8 1.6
2.4
2.6
2.8
10
10
0.5 0.4
1
1.2
0.8
11
25
0.3
B
0.6
1.4
12
9
0.1
0.2
0.7
98
49
56
7
17
29
5
21
45
Eigenfrequency of the floor (Hz)
0.1
2.2
8 4137
7
13
3
1.8
1.6
6
2.4
2.6
1
0.7
1.2
0.3
0.1
0.6
0.8
33
C
0.5 0.4
2
3.2
10
76
6
4
0.2
2.8
0.1
D
11
25
0.2
1.4
5
12
0.4 0.3
0.5
9 8
0.2
2.2
4
4137
96
E
1
2
0.7
3.2
3
5
17
49
29
56
F
4
2.4
2.6
7 6
21
45
216
2
276
236
196
1.4
2.8
13
356
316
496
416
396 296
596 436
476
376
576
556
516
536
456
336 256
0.6
0.8
1.81.6 1.2
3
10
2.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1
1.4
76
0.8
33
136
1.2
2.4
3.2
3
0.7
1.6
1.4
116
4
12
25
8
1000
2000
0.7
0.6
0.5
0.4
0.6
2.8
0.3
2.2
1.8
1.2
0.8
0.5
0.4
5
2
9
500
0.8
1
2.6
156
200
0.70.6
1 0.8
1.2
1.6
21.8
176
11
1
100
0.3
0.4
0.5
5000
10000
20000
50000
100000
Modal mass of the floor (kg)
Figure 7-11: OS-RMS90 for 8 % damping
49
Classification based on a damping ratio of 9%
20
19
18
5
4 3.2 2.82.6 2.2
2.4
3
6
17 8 7
9
16
11
15
13 12 10
14
5
13
17
10
0.4
0.5
0.6
0.8
0.1
0.2
0.3
0.1
A
0.7
2
5 4
3.2
1.6 1.2
1.4
2.2
2.6
2.8
2.4
3
13 10
12
0.4
0.8
1
0.3
0.2
0.1
0.5
0.6
1.8
25 21
B
0.7
2
6
9
87
9
8
0.1
0.2
29
17
45
5
0.4 0.3
4
11
7
3.2
2.62.2
2.8
2.4
3
49
76
Eigenfrequency of the floor (Hz)
0.2
0.4 0.3
0.6 0.5
0.7
1.61.4 1.2
1
1.8
6
11
11
0.8
3.2
2.2
2.62.4
2.8
3
4
8 7
9
12
1.6
1.4 1.2 1
1.8
2
6
13
12 10
41
56
3733
1.6
1.41.2 1
0.5
0.6
0.8
1.8
21
25
C
0.7
0.2
2
6
5
0.1
87
0.1
0.3
0.4
D
9
4
E
29
F
45
3
136
17
0.2
0.5
0.6
5
1.2 1 0.8
1.6 1.4
4
3.2
0.5
0.6
0.80.7
1
1.2
1.4
1.6
1.8
49
2
236
2
356
336
416
256
396
476
456
296
376
436
316
13
76
6
37
1
56
2.21.8
176
216
21
7
3.2 2.8
2.62.4
2
500
1000
2000
5000
0.6
0.5
0.4
0.3
4
1.6
200
0.8
0.7
1
3
8
1.2
0.6
0.5
1.4
25
1
100
0.7
0.8
33
116 96
276
10
0.3
0.4
0.5
0.6
12
41
196156
0.3
0.4
0.7
2.2
2.6
2.8 2.4
3
11
10000
20000
0.8
0.7
0.4
50000
100000
Modal mass of the floor (kg)
Figure 7-12: OS-RMS90 for 9 % damping
50
7.7 Vibration performance assessment
In the serviceability assessment for the vibration performance, the performance
requirement expressed in terms of floor-class according to Table 7-5 should be
compared with the performance capacity resulting from the OS-RMS90-value in
Figure 7-4 to Figure 7-12.
The performance requirement as well as the use of this guidance should be
agreed with the designer and the client.
0.8
3.2
E
3.2
12.8
F
12.8
51.2
Sport
D
Industrial
0.8
Prison
0.2
Hotel
C
Retail
0.2
Meeting
0.1
Office
B
Residential
0.1
Education
Upper Limit
0.0
Health
Lower Limit
A
Function of floor
Critical Workspace
Class
OS-RMS90
Recommended
Critical
Not recommended
Table 7-5: Recommendations for performance requirements
51
8 Design examples
8.1 Filigree slab
building)
with
ACB-composite
beams
(office
8.1.1 Description of the floor
In the first worked example a filigree slab with false-floor in an open plan office
is checked for footfall induced vibrations.
Figure 8-1: Building structure
It is spanning one way over 4.2 m between main beams. Its overall thickness is
160 mm. The main beams are Arcelor Cellular Beams (ACB) which act as
composite beams. They are attached to the vertical columns by a full moment
connection. The floor plan is shown in Figure 8-2. In Figure 8-2 the part of the
52
floor which will be considered for the vibration analysis is indicated by the
hatched area.
Figure 8-2: Floor plan (dimensions in [m])
For the main beams with a span of 16.8 m ACB/HEM400 profiles made of
material S460 have been used. The main beams with the shorter span of 4.2 m
are ACB/HEM360 made of S460.
The cross beams which are spanning in global x-direction may be neglected for
the further calculations, as they do not contribute to the load transfer of the
structure.
The nominal material properties are
-
Steel S460:
Concrete C25/30:
Es = 210 000 N/mm²,
Ecm = 31 000 N/mm²,
fy = 460 N/mm²
fck = 25 N/mm²
53
As required in section 7 of these guidance the nominal Elastic modulus of the
concrete will be increased for the dynamic calculations:
E c ,dyn  1.1  E cm  34100 N / mm ²
The expected mode shape of the part of the floor considered which corresponds
to the first eigenfrequency is shown in Figure 8-4. From the mode shape it can
be concluded that each field of the concrete slab may be assumed to be simply
supported for the further dynamic calculations. Regarding the boundary
conditions of the main beams (see beam to column connection, Figure 8-3), it is
assumed that for small amplitudes as they occur in vibration analysis the beamcolumn connection provides sufficient rotational restraint, so that the main
beams may be considered to be fully clamped.
Figure 8-3: Beam to column connection
54
Figure 8-4: Mode shape expected for the part of the floor considered
corresponding to the first eigenfrequency
Section properties
-
Slab:
The relevant section properties of the slab in global x-direction are:
Ac , x  160
mm 2
mm
I c , x  3.41  10 5
-
mm 4
mm
Main beam:
Assuming the first vibration mode described above the effective width of
the composite beam may be obtained from the following equation:
beff  beff ,1  beff , 2 
l0 l0
0.7  16.8
  2
 2.94 m
8 8
8
The relevant section properties of the main beam for serviceability limit
state (no cracking) are:
Aa ,netto  21936mm 2
Aa brutto  29214 mm 2
Ai  98320 mm 2
55
I i  5.149  10 9 mm 4
Loads
-
Slab:

Self-weight (includes 1.0 kN/m² for false floor):
g slab  160  10 3  25  1.0  5

Live load: Usually a characteristic live load of 3 kN/m² is
recommended for floors in office buildings. The fraction of the
live load considered for the dynamic calculation is assumed to
be approx. 10% of the full live load, i.e. for the vibration
check it is assumed that
q slab  0.1  3.0  0.3
-
kN
m2
kN
m2
Main beam:
 Self-weight (includes 2.00 kN/m for ACB):
g beam  5.0 

kN
4.2
 2  2.0  23.00
m
2
Live load:
q slab  0.3 
4.2
kN
 2  1.26
2
m
8.1.2 Determination of dynamic floor characteristics
Eigenfrequency
The first eigenfrequency is calculated on the basis of the self-weight approach.
The maximum total deflection may be obtained by superposition of the deflection
of the slab and the deflection of the main beam, i.e.
 total   slab   beam
with
 slab 
5  (5.0  0.3)  10 3  4200 4
 1.9 mm
384  34100  3.41  10 5
56
 beam 
1  (23.0  1,26)  16800 4
 4.5mm
384  210000  5.149  10 9
The total deflection is
 total  1.9  4.5  6.4 mm
Thus, the first eigenfrequency may be obtained from the self-weight approach
(section 7.4.3):
f1 
18
6.4
 7.1 Hz
Modal mass
The total mass of the slab is
M total  (5  0.3)  10 2  16.8  4.2  37397 kg
According to Table 7-3, example 3, the modal mass of the slab considered may
be calculated as
1.9 2  4.5 2
8 1.9  4.5 
M mod  37397  
 2
  17220kg
2
6.4 2 

 2  6.4
Damping
The damping ratio of the steel-concrete slab with false floor is determined
according to Table 7-4:
D  D1  D2  D3  1%  1%  1%  3 %
with
57
D1 = 1.0 % (composite slab)
D2 = 1.0 % (open plan office)
D3 = 1.0 % (false floor)
8.1.3
Assessment
Based on the modal properties calculated above, the floor is classified as class C
(Figure 7-6). The expected OS-RMS90 value is approx. 0.5 mm/s.
According to Table 7-5 class C is classified as being suitable for office buildings,
i.e. the requirements are fulfilled.
8.2 Three storey office building
8.2.1
Description of the floor
The floor of this office building, Figure 8-5, has a span of 15 m from edge beam
to edge beam. In the regular area these secondary floor beams have IPE600
sections and are laying in a distance of 2.5 m. Primary edge beam which span
7.5 m from column to column have also IPE600 sections, see Figure 8-6.
Figure 8-5: Building overview
58
Figure 8-6: Steel section of the floor
The floor is a composite plate with steel sheets COFRASTRA 70 with a total
thickness of 15 cm, as represented in Figure 8-7.
Figure 8-7: Floor set up
The nominal material properties are:
-
Steel S235:
Concrete C25/30:
Es = 210 000 N/mm²,
Ecm = 31 000 N/mm²,
fy = 235 N/mm²
fck = 25 N/mm²
E c ,dyn  1.1  E cm  34100 N / mm ²
Section properties
59
-
-
Slab (transversal to beam):
A
= 1170 cm²/m
I
= 20 355 cm4/m
g
= 3.5 kN/m²
g
= 0.5 kN/m²
Composite beam (beff = 2,5m; E=210000 N/mm²):
A
= 468 cm²
I
= 270 089 cm4
g
= (3.5+0.5) x 2.5 + 1.22 = 11.22 kN/m
Loads
-
Slab (transversal to beam):
g + g = 4.0 kN/m² (permanent load)
-
q
= 3.0 x 0.1 = 0.3 kN/m² (10% of full live load)
ptotal
= 4.3 kN/m²
Composite beam (beff = 2.5m; E=210000 N/mm²):
g
= 11.22 kN/m
q
= 0.3 x 2.5 = 0.75 kN/m
ptotal
= 11.97 kN/m
8.2.2 Determination of dynamic floor characteristics
Supporting conditions
The secondary beams are connected to the primary beams which have open
sections with low torsional stiffness. Thus these beams may be assumed to be
simple supported.
Eigenfrequency
For this example the supporting conditions are determined in two ways.
60
The first method is the application of the beam formula neglecting the
transversal stiffness of the floor.
The second method is the self-weight method considering the transversal
stiffness.

Application of the beam equation (Table 7.2):
p  11.97 [kN / m]    11.97  1000 [kg m / s ² / m] / 9.81[m / s ²]  1220 [kg / m]
f 


3  210000  10 6 [ N / m²]  270089  10 8 [m 4 ]
 4.77 Hz
0.49  1220 [kg / m]  15 4 [m 4 ]
Application of the equation for orthotropic plates (section 7.4.2):

f1
3EI
2

4

0.49 l
2

EI y
2
 l4
  b  2  b  4  EI
1  2      x
  l   l   EI y
  2.5  2  2.5  4  3410  20355
210000  10 6  270089  10 8
1  2

 
 
2
1220  15 4
  15   15   21000  270089
 4.76  1.00  4.76 Hz


Application of the self-weight approach (section 7.4.3):
 total   slab   beam
 slab 
5  4.3  103  25004
 0.3 mm
384  34100  2.0355  105
 beam 
5  11.97  150004
 13.9mm
384  210000  270089  104
 total  0.3  13.9  14.2 mm

f1 
18
 4.78 Hz
14.2
Modal mass
The determination of the eigenfrequency, as presented above, shows that the
load bearing behaviour of the floor can be approximated by a simple beam
61
model. Thus, this model is taken for the determination of the modal mass; see
Figure 7-2:
M mod  0,5  l  0,5  1220  15  9150 kg
Damping
The damping ratio of the steel-concrete slab with false floor is determined
according to Table 7-4:
D  D1  D2  D3  1%  1%  1%  3 %
with
D1 = 1.0 % (composite slab)
D2 = 1.0 % (open plan office)
D3 = 1.0 % (ceiling under floor)
62
8.2.3 Assessment
Based on the modal properties calculated above, the floor is classified as class D
(Figure 7-6). The expected OS-RMS90 value is approx. 3.2 mm/s.
According to Table 7-5 class D is classified as being suitable for office buildings,
i.e. the requirements are fulfilled.
63
9 References
[1]
G. Sedlacek, Chr. Heinemeyer, Chr. Butz, B. Völling, S. Hicks, P. Waarts, F. van
Duin, T. Demarco: “Generalisation of criteria for floor vibrations for industrial, office
and public buildings and gymnastic halls - VOF”, RFCS Report EUR 21972 EN, 2006,
ISBN 92-76-01705-5, http://europa.eu.int.
[2]
M. Feldmann, Chr. Heinemeyer, E. Caetano, A. Cunha, F. Galant, A. Goldack, O.
Hechler, S. Hicks, A. Keil, M. Lukic, R. Obiala, M. Schlaich, A. Smith, P. Waarts:
RFCS-Project: ”Human induced Vibration of Steel Structures – HIVOSS“: Design
Guideline
and
Background
Report“
Internet:
http://ww.stb.rwth-
aachen.de/projekte/2007/HIVOSS/HIVOSS.html.
[3]
H. Bachmann, W. Amman: “Vibration of Structures induced by Man and Machines”
IABSE-AIPC-IVBH, Zürich, ISBN 3-85748-052-X, 1987.
[4]
Waarts, P. “Trillingen van vloeren door lopen: Richtlijn voor het voorspellen, meten en
beoordeelen”, SBR, September 2005.
[5]
A. Smith, S. Hicks, P. Devine: “Design of Floors for Vibration: A new Approach”, SCIPublication P354, ISBN 10:1-85942-176-8, ISBN 13:978-1-85942-176-5, 2007.
[6]
ISO/DIS 10137: “Basis for design of structures – Serviceability of buildings and
walkways against vibrations”, International Organisation for Standardisation, 2007.
[7]
ISO 2631-1: Mechanical vibration and shock - Evaluation of human exposure to
whole body vibration; Part 1: General requirements, 1997 International Organisation
for Standardisation.
[8]
ISO 2631-2: Mechanical vibration and shock - Evaluation of human exposure to
whole body vibration: Part 2. Vibration in buildings (1 Hz-80 Hz), April 2003.
[9]
Verein Deutscher Ingenieure: “VDI-Richtlinie 2057: Einwirkungen mechanischer
Schwingungen auf den Menschen”, VDI-Verlag GmbH, Düsseldorf, 2002.
[10]
NS 8176 E ”Vibration and shock – Measurement of vibration in buildings from land
based transports and guidance to evaluation of its effect on human beings“,
Standards Norway (1999, 2005).
[11]
DIN 4150 “Erschütterungen im Bauwesen – Teil 2: Einwirkungen auf Menschen in
Gebäuden”, June 1999.
64
European Commission
EUR 24084 EN – Joint Research Centre – Institute for the Protection and Security of the Citizen
Title: Design of floor structures for human induced vibrations
Author(s):
M. Feldmann, Ch. Heinemeyer, Chr. Butz, E. Caetano, A. Cunha, F. Galanti, A.
Goldack, O. Hechler, S. Hicks, A. Keil, M. Lukic, R. Obiala, M. Schlaich, G.
Sedlacek, A. Smith, P. Waarts
Luxembourg: Office for Official Publications of the European Communities
2009 – 64 pp. – 21 x 29.7 cm
EUR – Scientific and Technical Research series – ISSN 1018-5593
ISBN 978-92-79-14094-5
DOI 10.2788/4640
Abstract
In recent years, the introduction of new structural materials and innovative
construction processes, associated to architectural and space arrangement
requirements, in multi-storey buildings construction have produced significantly
more flexible floor structural systems. The design of these floor systems is
usually controlled by serviceability criteria, deflections or vibrations. Recognizing
a gap in the design codes, this report gives a procedure for the determination
and assessment of floor response for human induced vibrations.
First, the proposed procedure is presented, giving particular attention to the
human induced loading characterization, dynamic properties and the comfort
criteria for the verification of floor structural systems. Design charts are derived.
Finally, it is presented a guidance manual to use the simplified procedure
proposed for the design of building floors for human induced vibrations. Two
worked examples of the proposed design procedure are given, namely a filigree
slab with ACB-composite beams and a composite slab with steel beams.
Keywords: Floor structures; Design procedures; Human induced vibrations;
Structural safety; Human comfort; Dynamic properties; Vibration control
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