Vibration Design of Floors

Transcription

RFS2-CT-2007-00033
Human induced Vibrations of Steel Structures
Guideline
Guideline_Floors_EN02.doc – 27.10.2008
Guideline
Table of Contents
Summary ...................................................................................3
1. Introduction .........................................................................4
1.1.
General ........................................................................4
1.2.
Scope ..........................................................................4
1.3.
References ...................................................................5
1.4.
Definitions ....................................................................6
1.5.
Variables, units and symbols ...........................................8
2. Design of floors against vibrations ...........................................9
2.1.
Design procedure ..........................................................9
2.2.
Related design methods .................................................9
2.2.1. Hand calculation method using measurements ...............9
2.2.2. Transfer function method .......................................... 10
2.2.3. Modal superposition ................................................. 10
3. Classification of Vibrations .................................................... 11
3.1.
Quantity to be assessed................................................ 11
3.2.
Floor classes ............................................................... 11
4. Hand calculation method ...................................................... 13
4.1.
Determination of eigenfrequency and modal mass............ 13
4.1.1. Finite Element Analysis ............................................. 13
4.1.2. Analytical formulas .................................................. 13
4.2.
Determination of damping............................................. 14
4.3.
Determination of the floor class ..................................... 14
4.3.1. Systems with more than one eigenfrequency ............... 15
4.4.
OS-RMS90 graphs for single degree of freedom systems .... 16
A. Calculation of natural frequency and modal mass of floors and other
structures .......................................................................... 25
A.1.
Natural Frequency and Modal Mass for Isotropic Plates...... 25
A.2.
Natural Frequency and Modal Mass for Beams.................. 27
A.3.
Natural Frequency and Modal Mass for Orthotropic Plates .. 27
A.4.
Self weight Approach for natural Frequency..................... 28
A.5.
Dunkerley Approach for natural Frequency ...................... 29
A.6.
Approximation of modal mass ....................................... 30
B. Examples ........................................................................... 33
B.1.
Filigree slab with ACB-composite beams (office building) ... 33
B.1.1. Description of the Floor............................................. 33
B.1.2. Determination of dynamic floor characteristics ............. 37
B.1.3. Assessment ............................................................ 38
B.2.
Three storey office building ........................................... 39
B.2.1. Description of the Floor............................................. 39
B.2.2. Determination of dynamic floor characteristics ............. 40
B.2.3. Assessment ............................................................ 42
2
Guideline
Summary
Modern large span floors and light weight floors show a tendency to vibrate
under service conditions. This guideline presents a method with which floors
can be easily designed for these vibrations.
The scope of this guideline are floors in office and/or residential buildings
that might be excited by persons walking normally and which can effect the
comfort of other building users. The aim of the design guide is to help
specify comfort requirements for occupants and to perform a design that
guarantees the specified comfort.
This guideline is accompanied by a background document which also
presents alternative and more general ways for the determination of the
floor response to dynamic human induced forces.
The theoretical methods presented here and in the background document
have been elaborated/investigated in the RFCS-Project “Vibration of Floors”.
The guideline and background document are here disseminated under the
grant of the Research fund for Coal and Steel within the project “HIVOSS”.
Guidance for the determination of the relevant dynamic floor characteristics
and application examples are given in the annexes of this document.
3
1.
Introduction
1.1.
General
Guideline
Floor structures are designed for ultimate limit states and serviceability limit
state criteria:
•
Ultimate limit states are those related to strength and stability;
•
Serviceability limit states are mainly related to vibrations and hence
are governed by stiffness, masses, damping and the excitation
mechanisms.
For slender floor structures, as made in steel or composite construction,
serviceability criteria govern the design.
This guideline gives guidance for:
•
Specification of tolerable vibration by the introduction of acceptance
classes and
•
Prediction of floor response due to human induced vibration with
respect to the intended use of the building.
For the prediction of floor vibration several dynamic floor characteristics
need to be determined. These characteristics and simplified methods for
their determination are briefly described. Design examples are given in
annex B of this guideline.
1.2.
Scope
The procedure provided in this guideline provides a simplified method for
determining and verifying floor designs for vibrations due to walking. The
guideline focuses on simple methods, design tools and recommendations for
the acceptance of vibration of floors which are caused by people during
normal use. The given design and assessment methods for floor vibrations
are related to human induced vibrations, mainly caused by walking under
normal conditions. Machine induced vibrations or vibrations due to traffic etc.
are not covered by this guideline.
The guideline should not be applied to pedestrian bridges or other structures
which do not have a structural characteristic or a the characteristic of use
comparable to floors in buildings.
The guideline focuses on the prediction and evaluation of vibration at the
design level.
4
1.3.
Guideline
References
[1]
European Commission – Technical Steel Research: Generalisation of
criteria for floor vibrations for industrial, office, residential and public building
and gymnastic halls, RFCS Report EUR 21972 EN, ISBN 92-79-01705-5,
2006, http://europa.eu.int
[2]
Hugo Bachmann, Walter Ammann. Vibration of Structures induced by
Man and Machines” IABSE-AIPC-IVBH, Zürich 1987, ISBN 3-85748-052-X
[3]
Waarts, P. Trillingen van vloeren door lopen: Richtlijn voor het
voorspellen, meten en beoordelen. SBR, September 2005.
[4]
Smith, A.L., Hicks, S.J., Devine, P.J. Design of Floors for Vibrations: A
New Approach. SCI Publication P354, Ascot, 2007.
[5]
ISO 2631. Mechanical Vibration and Shock, Evaluation of human
exposure to whole-body vibration. International Organization for
Standardization.
[6]
ISO 10371. Bases for design of structures – Serviceability of buildings
and walkways against vibrations. International Organization for
Standardization.
5
1.4.
Guideline
Definitions
The definitions given here are oriented on the application of this guideline.
Damping D
Modal mass Mmod =
generalised mass
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)
In many cases, a system with several degrees of
freedom can be reduced to a system with a single
degree of freedom with frequency:
f =
where
1
2π
Kmod
M mod
f
is the natural frequency
Kmod
is the modal stiffness
Mmod
is the modal mass
Thus the modal mass can be interpreted to be the
mass activated in a specific mode shape.
The determination of the modal mass is described
in chapter 0.
6
Natural Frequency f
=
Eigen frequency
Guideline
Every structure has its specific dynamic behaviour
with regard to vibration mode shape and duration
T[s] of a single oscillation. The frequency f is the
reciprocal of the oscillation time T (f=1/T).
The natural frequency is the frequency of a free
oscillation without continuously being driven by an
exciter.
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
2π
K
M
K
is the stiffness
M
is the mass
One step RMS- value of the acceleration for a
significant step covering the intensity of 90% of
peoples’ steps walking normally.
OS:
One step
RMS: Root mean square = effective value, of the
acceleration a:
T
aRMS =
a
1
a (t ) 2 dt ≈ Peak
∫
T 0
2
where: T is the investigated period of time.
7
1.5.
Guideline
Variables, units and symbols
a
acceleration
[m/s²]
B
width
[m]
f, fi
natural frequency under consideration
[Hz]
δ(x,y)
Deflection at location x,y
[m]
D
Damping (% of critical damping)
[-]
D1
Structural damping
[-]
D2
Damping from furniture
[-]
D3
Damping from finishes
[-]
l
length
[m]
K, k
stiffness
[N/m]
Mmod
Modal mass
[kg]
Mtotal
Total mass
[kg]
OS-RMS
One step root mean
effective velocity
OSRMS90
90 percentile of OS-RMS values
[-]
T
period (of oscillation)
[s]
t
time
[s]
p
distributed load (per unit length or per unit [kN/m]
area)
[kN/m²]
δ
deflection
μ
mass distribution per unit length or per unit [kg/m]
area
[kg/m²]
square
value
of
the [-]
or
[m]
or
8
Guideline
2.
Design of floors against vibrations
2.1.
Design procedure
The design procedure described in this guideline corresponds to a simplified
procedure with which a floor design can be verified for vibrations due to
human walking loads. The first step in the procedure is to determine the
basic floor characteristics or parameters. Using these parameters and a set
of graphs, a quantity called the 90% one-step RMS value, which
characterizes the floor response due to walking, is obtained. This value is
then compared to recommended values for different floor destinations. These
three steps are visualized in Figure 1. This method is referred to the
simplified procedure or hand calculation method and is treated in chapter 4.
Determine dynamic floor
characteristics:
Natural Frequency
Modal Mass
Damping
Read off OS-RMS90 value
Determine and
verify floor class
Figure 1:
Design procedure.
2.2.
Related design methods
2.2.1.
Hand calculation method using measurements
The hand calculation method can also be used in those cases where the floor
characteristics have been determined by experiment. For a discussion on
how to extract dynamic floor characteristics from tests refer to [1] and [3].
9
2.2.2.
Guideline
Transfer function method
In the transfer function method, instead of using modal parameters such as
modal mass stiffness and damping, the floor's characteristics are described
in terms of a frequency response function, FRF, or transfer function. Starting
from a statistical description of walking loads, a probabilistic analysis is
carried out to determine the OS-RMS90 value of the floor. The method
described in more detail in [1] and [3].
In the hand calculation method, being a simplified version of the transfer
function method, the classical transfer function for a single degree of
freedom mass-spring-dashpot system is used and the probabilistic analysis
to obtain the OS-RMS90 has been carried out in advance.
The transfer function method can be applied where the floor response is
obtained by finite element calculations or by measurements. The same set of
acceptance criteria as described in this guideline are used.
2.2.3.
Modal superposition
Two methods of analysis based on modal superposition are presented in the
guideline by the Steel Construction Institute, [4]. The general method uses
finite element software to determine the modal properties of the floor for a
number of modes, and then applies design walking forces to the floor to
determine a response, in terms of acceleration. The simplified method is
based on a parametric study using the general approach, and is presented as
analytical formulae. Unlike the hand calculation method, the approach used
for the modal superposition method is deterministic and the results are
comparable directly to limits supplied by international standards in [5] and
[6]. This method also allows the influence of separation between the walking
activity and the receiver to be considered, which enables the isolation of
critical areas such as hospital operating theatres to be determined. Further
information on this approach is given in the background document.
10
3.
Classification of Vibrations
3.1.
Quantity to be assessed
Guideline
The perception of vibrations by persons and the associated annoyance
depends on several aspects. The most important are:
•
The direction of the vibration. In this guideline only vertical vibrations
are considered.
•
The posture of people such as standing, laying or sitting.
•
The current activity of an occupant is of relevance for his or her
perception of vibrations, for example, persons working in the
production of a factory will perceive vibrations differently to those
working in an office.
•
Additionally, age and health of affected people may play a role in
determining the level of annoyance perceived.
Thus the perception of vibrations varies between individuals and can only be
judged in a way that fulfils the expectations of comfort for the majority of
people.
It should be considered that the vibrations levels considered in this guideline
are relevant for the comfort of the occupants only. They are not relevant for
structural integrity.
Aiming at an universal assessment procedure for human induced vibration it
is recommended to adopt the so-called one step RMS value (OS-RMS) as a
measure for assessing floor vibrations. The OS-RMS values corresponds to
the vibration caused by one relevant step onto the floor.
As the dynamic effect of people walking on a floor depends on several
different factors, such as weight and speed of walking people, their shoes,
flooring, etc., the 90% OS-RMS (OS-RMS90) value is recommended as
assessment value. This value is defined as the 90 percentile of all the OSRMS values obtained for a set of loads representing all possible combinations
of persons' weights and walking speeds.
3.2.
Floor classes
The following table classifies vibrations into six floor classes (A to F) and
gives also recommendations for the assignment of classes with respect to
the function of the considered floor.
11
Guideline
Table 1: Classification of floor response and recommendation for the application of
classes
A
B
C
D
E
F
Sport
Industrial
Prison
Hotel
Retail
Meeting
Office
Residential
Education
Health
Function of Floor
Critical Workspace
Upper Limit
Lower Limit
Class
OS-RMS90
0.0
0.1
0.1
0.2
0.2
0.8
0.8
3.2
3.2 12.8
12.8 51.2
Recommended
Critical
Not recommended
Limits on vibration are also given in International Standard ISO 10137[6],
which is referenced in the Eurocodes. These limits are reproduced in Table 2,
with the equivalent OS-RMS90 limit.
Table 2: Vibration limits specified by ISO 10137 for continuous vibration
Place
Time
Critical working areas (e.g. hospital
operating-theatres, precision
laboratories, etc.)
Residential (e.g. flats, homes,
hospitals)
Quiet office, open plan
General office (e.g. schools, offices)
Workshops
Day
Multiplying
Factor
1
OS-RMS90
equivalent
0.1
Night
Day
Night
Day
Night
Day
Night
Day
Night
1
2 to 4
1.4
2
2
4
4
8
8
0.1
0.2 to 0.4
0.14
0.2
0.2
0.4
0.4
0.8
0.8
It is considered that these limits are unnecessarily harsh, and testing on a
number of subjects found the limits in Table 1 to be more appropriate (see
[1]).
12
4.
Guideline
Hand calculation method
The hand calculation method assumes that the dynamic response of a floor
can represented by a single degree of freedom system. The eigenfrequency,
modal mass and damping can be obtained by calculation as set out in this
chapter. As discussed previously in section 2.2.1, the modal parameters can
be also obtained from a measurement. As this guideline is intended to be
used for the design of new buildings, testing procedures are excluded from
it.
4.1.
Determination of eigenfrequency and modal mass
In practice, the determination of floor characteristics can be performed by
simple calculation methods (analytical formulas) or by Finite Element
Analysis (FEA).
In the determination of the dynamic floor characteristics also a realistic
fraction of imposed load should be considered in the mass of the floor.
Experienced values for residential and office building are 10% to 20% of the
imposed load. For very light floors it is recommended to include also the
mass of one person. A minimum representative person's mass of 30 kg is
recommended.
4.1.1.
Finite Element Analysis
Different finite element programs can perform dynamic calculations and offer
tools for the determination of natural frequencies. Many programs also
calculate the modal mass automatically in a frequency analysis.
As the element types, the modelling of damping and the output is program
specific, only some general information can be given in this guideline
concerning FEA.
If FEA is applied for the design of a floor with respect to the vibration
behaviour, it should be considered that the FEA-model for this purpose may
differ significantly to that used for ultimate limit state (ULS) design as only
small deflections are expected due to vibration.
A typical example is the selection of boundary conditions in vibration analysis
compared to ULS design. A connection which is assumed to be a hinged
connection in ULS may be assumed to provide full moment connection in a
vibration analysis.
For concrete the dynamic modulus of elasticity should be considered to be
10% higher than the static tangent modulus Ecm.
4.1.2.
Analytical formulas
For calculations by hand, Annex A gives formulas for the determination of
frequency and modal mass for isotropic plates, orthotropic plates and
beams.
13
4.2.
Guideline
Determination of damping
Damping has a great influence on the vibration behaviour of a floor.
Independently of the way of determining natural frequency and modal mass,
damping values for vibrating systems can be determined using Table 3 for
different structural materials, furniture and finishing. The system damping D
is obtained by summing up the appropriate values for D1 to D3.
Table 3: Determination of damping
Type
Damping (% of critical
damping)
Structural Damping D1
Wood
Concrete
Steel
Steel-concrete
6%
2%
1%
1%
Damping due to furniture D2
Traditional office for 1 to 3 persons with
separation walls
Paperless office
Open plan office.
Library
Houses
Schools
Gymnastic
Damping due to finishes D3
Ceiling under the floor
Free floating floor
Swimming screed
2%
0%
1%
1%
1%
0%
0%
1%
0%
1%
Total Damping D = D1 + D2 + D3
4.3.
Determination of the floor class
When modal mass and frequency are determined, the OS-RMS90 value as
well as the assignment of the floor classes can be obtained with the
diagrams given in section 4.4. The relevant diagram needs to be selected
according to the damping characteristics of the floor in the condition of use
(considering finishing and furniture).
14
0. 1
3
0.
0.4
0.
0.3
4
7
0.
1.4 1.2
1
0. 7
0. 8
5
1. 6
12
9
11 13
10
6
1.
4
3
D
2 1.8
2.2
2.4
2.
2. 86
3.2
4
0.6 0.5
0. 7
1
1.2
2
3.
2 1
2.24.2 .8
22..6
8
0.3
0
0.6 0.5 .4
0. 8
C
7
33
5
2
0.
5
0.
6
0.
1.6
8
6
21
25
Frequency [Hz]
0.2
A
0. 2
B
1
1.2
1.4
3
4
29
7
17
8
6
9 12
11
10 13
9
1.8
5
7
10
22. 2
2.4
2.6
2.8
3.2
8
11
4
12
1. 6
3
13
0. 7 0. 8
14
0.5
0.6
0.8 1
1.2
5
1. 4
4
3.2
1.8
1.6 22.2
2.4
2.6
2.8
0.3
0.4
0.5
0.6 0.7
1
1.2
1.4
3
15
1.8
2
2.2
2.4
2.6
20
19
18
17
16
Guideline
1
1.2
1.4
1.6
2.2
8
6
109
21
25
5
8 7
11
12
13
3
6
9
10
11
17
33
136
216
176196
15 6
96
11
6
F
1. 6
2. 4
2.6
3.2 2.8
4
7
29
2
2
41
1.8
37
45
E
3
3
5
17
56
49
4
2
2.4
2.6
3.22.8
4
5
6
87
12
41
49
29
10
11
9
21
12
17
8
45
37
56
7
13
76
25
13
10
9
4
7
3.2
1000
2000
3000
3 2.8
2.62.42.2
6
1
250
6
5
4
8
7
6
5
4000 5000 6000
8000 10000
Modal Mass [kg]
Figure 2: Application of diagrams
The diagram is applied by entering the x-axis with the modal mass and the
y-axis with the corresponding eigenfrequency. The OS-RMS90 value and the
acceptance class can be read-off at the intersection of the lines extending
from both entry points (see Figure 2).
4.3.1.
Systems with more than one eigenfrequency
In some cases, the floor response may be characterized by more than one
natural frequency. In these cases, the OS-RMS90 must be determined as a
combination of OS-RMS90 values obtained for each mode of vibration. The
procedure is as follows:
a) Determine the eigenfrequencies.
b) Determine the modal mass and damping corresponding to each
eigenfrequency.
c) Determine the corresponding OS-RMS90 value for each eigenfrequency.
d) Approximate the total (or combined) OS-RMS90 value using:
OS − RMS90 =
∑ OS − RMS
2
90; i
i
e) Read off the corresponding floor class from Table 1.
15
4.4.
Guideline
OS-RMS90 graphs for single degree of freedom systems
Classification based on a damping ratio of 1%
20
19
18 21
12
13
11
17
8
7
6
16 29
15
11
8
7 6
41 33
37
13
45
49
56
25
76
7
6
1312
17
8
7
6
45
49
A
1
0.5
0.6
0.4
0.8
0.2
0.3
0.1
0.1
0.7
1.41.2
2.6
2.4
1.8
2.2
2
5
10
37
1
0.3
0.8
0.1
2.8
9
1.6
C
0.5
0.4
0.6
1 0.8
29
96
12
13
17
25
176
216
236
276
256
196156
136
7
6
8
2.62.4
4
5
11
76
1.41.2
1.8
2.2 2
33
5
1.4
10
9
37
0.8
49
376 316
396
416
236
336
356276 256
296
E
4
6
7
29
13
5
3 2.42.2
2.6
8
12
1.6
3.2
1.2
F
0.3
0.5
0.8
1
1.2
1.4
96
25
56
256
21
296
476 376
496
336
196
756
596 456396 316 216
876
736
636
956
796
776
936
616 576
176
856676
916896
536
436 356
836
816
556
716
656 516 416
696
2
76
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
0.3
33
41
37
276
45
156
17
29
116
236
25
8
136
49
10
21
5
12
11
9
6
0.8 0.6
0.7
2.8
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
4
3.2 3
1.81.6
1
0.6
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
7
9
5 4
10 8
6
3.2
3
2.4
4
2.8
2.6 2.2
5
7
3.23
2
1.8
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
0.8 0.60.7
0.7
0.5 0.4
1.4
1.2
2.82.6
0.2
0.4
0.3
0.6
0.7 0.5 0.4
17
3
0.3
0.4
0.6
0.7
2.41.82.22
2.6
0.3
0.4
0.5
1.2 1
33.2
45
196
216
0.2
0.3
0.4
0.5
0.7 0.6
D
2.8
116
0.1
0.2
1.6
21
0.2
0.3
0.7
56
B
0.2
0.5
0.6 0.4
3.2
3
41
4
0.1
1.6
4
11
33
8
Eigenfrequency of the floor (Hz)
2.8
0.7
21
96
0.2
0.3
3.2
3
41
9
5
29
11
10
2.6
2.4 1.8 1.41.2
2.2
2
4
109
12
0.4
1.6
21
14
0.5
0.6
0.8
3.2
3
1312
17
25
0.7
1
2.8
10 9
17
1.41.2
2.6
2.4
2.2 1.8
2
4
5
0.3
0.5
1
100
200
500
1000
2000
5000
10000
20000
50000
100000
Modal mass of the floor (kg)
Figure 3: OS-RMS90 for 1% Damping
16
Guideline
20
19
18
13
11 10 8
9
12
4
5
7
3.2
2.8
3
6
1.81.6
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2
17 17
1110
15
9
8
13 33
29 25
12
4
5
7
13
12
14
10
0.5
0.6
2.6
1621
11
0.7
1
1.2
0.3
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1.6
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1
1.8
3
1.4
2.42.2
1.2
2
6
0.2
0.4
0.7
0.5
0.6
0.2
0.4
0.3
17
2.6
21
1110 8
9
4137
13
12
45
5649
4
1.6
1.8
3.2
2.8
3
5
7
0.1
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2.4
2.2
2
6
0.7
1
1.4
0.2
0.5
0.6
1.2
29
17
25
2.6
C
0.8
8
21
9
4137
136
5
96
56
6
0.2
0.5
0.6
1.4
0.4
0.1
0.3
0.1
0.2
1.2
2.4
2.2 2
13
12
45
49
0.7
1.6 1
1.8
3.2 2.8
3
7
116
156
176
4
11 10 8
76
6
0.1
0.3
33
196
B
0.4
9
7
A
0.1
0.2
0.3
D
0.8
0.5 0.4
0.7
0.2
0.6
5
29
4
276216
336296
196
356316
236
256
176
17
F
76
1.2
25
E
21
3
8
11
10
0.7
1.8
6
13
116
45
276
156
49
56
1
96
7
25
12
21
13
8
200
500
1
0.6
0.50.4
0.8 0.7
0.3
0.60.5 0.4
0.3
0.2
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
2.4
1.6
0.7
3
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
1.2 1
0.6
1.8
0.8
0.7 0.5
0.4
2.62.2 1.6 1.4
0.6
3
1
12
33 29
1
100
0.3
0.4
2.6
376
356
236
396
516 416
676596
216
456 336
556
616
576
896
916
536
256
856
836736
696636 476
296
876
756656 496436
776
816
796
716
196
1.6 1.4 1.2
7
0.3
0.50.4
0.8 0.70.6
0.5
2
2.42.2
9
136
316
2.8
3.23
4
5
4137
2
1
1.6
1.8 1.4
2.6
33
1000
2000
2
5
1.2
4
5000
10000
20000
0.8
0.7
0.5
0.4
50000
0.3
100000
17
Guideline
20 10
9
19
7
11
8
18
5
17
6
12
13
16
13 25
8
12
13
10
45
49 41
2 1.8
2.2
2.4
0.1
0.4
0.5
0.2
0.3
0.7 0.6
1.2
1
0.2
0.8
0.3
5
9
11
17
8
3.2 2.6
2.8
4
7
33
21
25
1.6
B
0.4
1.2 1
2
1.8
2.2
2.4
3
6
0.7 0.6
1.4
0.5
0.2
0.8
0.3
8
13
10
7
37
116
156
96
56
76
45
41
49
11
17
4
5
9
29
136
8
2.8
2
7
3
D
0.4
236
256 176
0.2
0.70.6 0.5
1 0.8
1.61.4 1.2
E
12
F
13
4
2.2
5
37
3
7
29
0.7
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
6
5
2.8
2.4
136
96
1
100
200
12
33
21
56
49
500
45
41
10
98
13
6
7
4
3 2.6
25
2.2 1.8
1.6
1.4
1.2
2
1
5000
10000
3
2.2
2.6
1.8
2.4 2 1.6
1.4
1.2
1
0.80.7
0.6
1.8
1.6
1.4
1.2
1
0.8
0.6
0.5
0.4
0.5
2.4
2000
2.8
3.2
5
1000
2
3.2
156
76
0.2
1.41.2
1.6
11 8
17
116
256236
0.3
0.50.4
0.3
0.8 0.6
1
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
0.4
2.4
9
2
0.3
0.5
0.7
2
1.8
3
196
216
0.4 0.3
0.6
2.6
3.2
2.8
10
276
316
356
456
0.3
6
196
216
4
0.2
0.8
1.8
25
276
0.1
0.2
0.3
0.4
0.5
2.2
2.4
21
0.7 0.6
1
1.2
1.6 1.4
3.2 2.6
33
5
0.1
C
12
6
A
0.1
0.4
0.5
0.1
10
11 37
29
0.8
1.6 1.4
3
6
0.70.6
1.2 1
3.2 2.6
2.8
7
21
12
9
2 1.8
2.2
2.4
5 4
9
11
17
14
3
10
15
1.61.4
3.2 2.6
2.8
4
0.8
20000
0.7
0.4
50000
0.3
100000
18
Guideline
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
98
7
6
5
2.6 2
1.6
2.4
2.2
4
1.8
12
13
11
0.4
29
37
25
41 33
0.1
B
10
98
17
7
2.6 2 1.6
2.4
2.2
4
5
6
1.2 1
0.5
0.80.7
0.4
0.2
0.3
1.4
0.1
3
21
13
C
2.8
10
87
9
7
17
56
96
0.6
3.2
1.8
8 49
45
2
2.6
4
5
6
1.21
1.6
3.2
0.6 0.5
0.7
0.8
2.4
0.4
0.3
0.2
1.8 1.4
29
37
25
41 33
5
3
21
12
196
2.8
11
156
176
216
13
4945
F
3
1.2
2
10
0.5
0.4
2.2
1.8
1.4
3
3
2.8 2.2
2.4
2.6
116
76
5
1211
29
25
10
7
8
37
21
41
13
6
3.2
32.8 2.4
2.6
33
1.2
21.8 1.4
4
9
Figure 6: OS-RMS90
1000
2000
5000
0.6
0.7
10000
20000
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
2.2
1 0.8
500
0.7 0.5 0.4
0.8 0.6
1
0.7
1.2
0.8
1
1.6
2 1.4 1.2
1.8
3.2
136
200
0.6
4
216
1
100
0.4 0.3
2 1.6
1.4
2.4
2.21.8
2.6
2.8
0.3
0.2
0.5
1
1.2
6
56
96
0.7
3.2
17
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
0.6
0.8
5
0.2
0.3
0.4
7
9
176
0.3
1.6
2.6
2.4
4
8
0.4
0.60.5
0.7
0.8
1
E
236
2
0.1
D
76
4
0.1
0.2
2.2
116
6
0.2
0.3
2.8
11
1211
0.6 0.5
0.80.7
1.4
3
21
12
9
1.21
3.2
14
10
A
10
15 17
13
0.2
0.3
0.5
0.4
50000
0.3
100000
for 4% Damping
19
Guideline
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
0.5
0.1
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
29
4
25
6
1.6
1.4
3.2
3 2.8
2.2
5
12
B
0.1
0.80.7 0.6
1
1.2
0.5
1.8
2.42
2.6
21 17
33
0.4
0.2
0.3
9
C
10
11
8
9
13
7
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
49
56
96
37
45
25
29
6
6
1
1.4
1.6
3.2
3 2.8
4
0.80.7 0.6
0.1
0.2
0.5
1.2
0.4
0.3
2.2 1.8
5
2.4 2
21 17
41
0.1
12
33
0.2
D
2.6
5
10
176
4 196
1
11
F
156
3
136
1.6
1.4
1.2
E
9
13
4
7
8
0.5
2
416
276
556
596
516
496 436356
536
336
476 396
616 456
636
576
376 296
316 256
236
96
2.6
29
216
2.6
4
41
2.2
10
9
116
5
7
2.8
3
2.6
33
11
8
2
2.4 1.8 1.4
1
1.8
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
6
1.2
1.4
1.6
17
12
176
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
21
0.2
0.4
0.8
1
2.4
25
56
0.3
0.6
0.7
6 5
37
45
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
0.8
2
0.7
0.4
0.3
0.6
0.5
1
100
200
500
1000
2000
5000
10000
20000
50000
100000
20
Guideline
20
7
6
19
18
4
8
17 10
7
13
12
6
17
1110
8
21
10
25
29
1.6
1.8
1.2
0.5
7 6
0.6
4
2.2
2.4
3.22.8
0.7
1.2
49
1.4
21
7
2.6
2.4
4
25
2
3329
8
0.8
D
0.2
F
0.2
1.2
2.6
6
2.2
13
2
0.5
1.6
0.4
1.8
2.4
5
2.8
3.2
4
12
1
0.7
2
1.4
196
216
236
3
316
456 356
436
596
636
576
496
336
556
476
536
616
516
416
396
376
0.3
0.8
7
21
0.2
0.4
0.6
E
176
256
0.5
3
9
4
3
0.3
1.4
10
37
49
0.1
1
0.7
11
41
156
0.2
0.4
0.6
17
5
0.1
0.3
1.6
1.8
3.22.8
45
C
1.2
2.2
6
56
0.7
5
12
76
0.1
0.2
0.5
6
13
96
0.4
3
7
B
0.3
1
2
9
8
0.5
0.6
1110
41
0.1
0.2
0.8
1.6
1.8
8
17
0.4
1
2.6
5
A
0.3
0.8
3
37
0.7
0.1
0.2
1.4
13
12
11
0.4
2
9
12
9
4
0.6
1
1.4
2.2
2.4
3.22.8
5
0.3
0.8
2.6
14
13
1.2
3
9
16
15
2.6
2.2
2.4 1.6
1.8
3.2 2.8
2
5
76
0.2
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
0.3
136
1
296
276
2.8
25
56
1.8
3.2
176
45
2.2
11 10
96
33
29
17
9
6
3
4
7
2
1.2
1.6
1.4
5
1.6
1.41.2
1
0.8
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
Figure 8:OS-RMS90 for 6% Damping
21
Guideline
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
16
15
14
17
13
6
11
13
12 10
9
12
11
10
29
21
25
5
2.2
2.4
2.8
4
2
3
3.2 2.6
1.4
1.8
0.3
0.4
0.5
0.7
1
0.1
0.2
0.8 0.6
0.3
1.6
0.6
0.8
0.4
8
6 5
2.2
2.4
2.8
4
10
3.2
17
1.4
2
3
1.8
0.5
0.7
1
1.6
0.8
0.6
0.4
0.1
4133
5649
37
6
11
5
2.2
2.4
2.8
4
13
45
6
12
76
10
21
25
29
96
B
0.3
8
7
0.1
0.2
1.2
2.6
7
9
8
A
0.1
0.2
1.2
7
11
13
12
9
0.5
0.7
1.4
C
1
0.2
0.5
0.7
0.1
0.3
1.8
2
D
1.2
3
3.2
2.6
17
0.1
0.4
0.6
0.2
0.8
5
1.6
7
E
8
4
33
49
1.4
F
3
136
0.2
5
4
37
2.4
11
2.8
1.8
2
2.6
3.2
13
2
416
376
336
516
616
496 356
576
476
536
396
556
456
436
176
236196
0.3
0.6
0.8
1.2
3
256
296
0.4
1
2.2
6
0.5
2.2
1 0.8
1.8
2
1.6
2.4
156
316
0.3
0.6 0.5 0.4
1
0.8
0.7 0.6
1.4 1.2
45
216
0.4
0.7
1.6
12 10
276
0.2
0.7
41
56
0.3
0.5
9
116
21
76
7
1.4 1.2
1
2.8
2.6
4
25
29
17
9
8
3
5
1.8
1.6
1.2
1.4
0.7
0.8 0.6
0.7 0.5
0.4
1
0.6
3.2
96
0.8
0.3
0.8
2
6
0.5 0.4
1
100
200
500
1000
2000
5000
10000
20000
50000
100000
22
Guideline
20
197
5
6
3.2
3
4
2 1.8
1.6
2.4
2.6
2.8
18
1.21
0.5 0.4
0.7
0.8
1.4
17 9
16
8
2.2
15 13
14
5
7 6
10
4
3.2
3
2
1.81.6
2.4
2.6
2.8
11
12
13
0.5 0.4
1.21
17
5
13
4
7 6
10
3.2
3
2
1.8 1.6
2.4
2.6
2.8
10
0.6
98
5
21
7
13
0.1
4
3
6
2.4
2.6
1.8
1.6
1
0.7
1.2
0.2
0.3
0.1
0.6
0.8
33
C
0.5 0.4
2
3.2
10
76
6
B
2.2
17
45
0.3
0.6
1.4
8 4137
29
0.1
0.2
0.7
0.8
12
49
56
0.5 0.4
1
1.2
11
25
9
A
0.1
0.2
2.2
12
7
0.3
0.7
0.8
1.4
9 8
11 21
0.1
0.2
0.3
0.6
2.8
D
11
25
1.4
5
12
0.4 0.3
0.5
9 8
E
4137
96
1
2
0.7
3.2
3
29
F
276
236
196
4
0.6
0.8
1.81.6 1.2
3
2.4
2.6
7 6
21
1.4
2.8
13
45
216
356
316
496
416
396 296
596 436
476
376
576
556
516
536
456
336 256
5
17
49
56
10
2.2
0.70.6
1 0.8
1.2
1.6
21.8
1.4
76
0.3
0.4
0.5
0.6
0.7
0.8
1
33
136
11
1.2
2.4
3.2
3
25
12
0.7
8
2000
0.5
0.4
0.3
1.2
5
0.8
0.5
0.4
2
9
1000
1.8
0.7
0.6
0.6
2.8
2.2
0.8
1
1.6
1.4
116
500
0.2
0.8
4
200
0.3
0.4
0.5
176
2.6
156
1
100
0.2
2.2
4
2
0.1
0.2
5000
10000
20000
50000
100000
23
Guideline
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
3.2
1.6 1.2
1.4
2.2
2.6
2.8
2.4
3
13 10
12
9
87
0.1
A
0.3
0.2
0.1
0.5
0.6
B
0.7
2
6
9
8
45
7
0.1
0.2
29
17
5
0.4 0.3
4
11
3.2
2.62.2
2.8
2.4
3
49
76
0.2
0.3
0.4
0.8
1
1.8
25 21
0.1
0.7
2
5 4
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
4
1.2 1 0.8
1.6 1.4
3.2
2.2
2.6
2.8 2.4
3
11
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
376 296
436
316
13
76
216
6
37
1
56
2.21.8
176
21
7
3.2 2.8
2.62.4
2
25
Figure 11: OS-RMS90
500
1000
2000
1.2
0.8
0.7
0.6
0.5
5000
0.6
0.5
0.4
0.3
4
1.6
200
1.4
1
3
8
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
10000
20000
0.8
0.7
0.4
50000
100000
for 9% Damping
24
Guideline
A.
Calculation of natural frequency and modal mass of floors
and other structures
A.1.
Natural Frequency and Modal Mass for Isotropic Plates
The following table gives hand formulas for the determination of the first
natural frequency (according to [2]) and the modal mass of plates for
different supporting conditions.
For the application of the given equations it is assumed that no lateral
deflection at any edges of the plate occurs.
Frequency ; Modal Mass
Supporting Conditions:
clamped
f =
hinged
α
L2
E t3
12 ⋅ μ (1 − υ 2 )
; M mod = β ⋅ M tot
α
9.00
α = 1.57 ⋅ ( 1 + λ2 )
8.00
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 λ = L/B
β ≈ 0,25 for all λ
α
16.00
14.00
α = 1 . 57
1 + 2 . 5 λ 2 + 5 . 14 λ 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 λ = L/B
α
14.00
12.00
α = 1 . 57
5 . 14 + 2 . 92 λ 2 + 2 . 44 λ 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 λ = L/B
25
Guideline
Supporting Conditions:
clamped
Frequency ; Modal Mass
f =
hinged
α
L
2
E t3
12 ⋅ μ (1 − υ 2 )
; M mod = β ⋅ M tot
α
12.00
α = 1 . 57
1 + 2 . 33 λ 2 + 2 . 44 λ 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 λ = L/B
α
12.00
α = 1 . 57
2 . 44 + 2 . 72 λ 2 + 2 . 44 λ 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 λ = L/B
α
18.00
16.00
α = 1 . 57
5 . 14 + 3 . 13 λ 2 + 5 . 14 λ 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 λ = L/B
E
t
μ
ν
Mtot
Youngs Modulus in N/m²
Thickness of Plate in m
mass of floor including finishing and
furniture in kg/m²
Poisson ratio
Total mass of floor including finishes and
representative variable loading in kg
26
A.2.
Guideline
Natural Frequency and Modal Mass for Beams
The first Eigenfrequency of a beam can be determined with the formula
according to the supporting conditions from Table 4 with:
E
I
μ
ℓ
Youngs-Modulus [N/m²]
Moment of inertia [m4]
distributed mass [kg/m]
length of beam
Table 4: Determination of the first Eigenfrequency of Beams
Supporting Conditions
Natural
l
Frequency
4
3EI
f =
π 0.37 μl 4
f =
l
2
π
f =
l
f =
l
A.3.
3EI
0. 2 μ l 4
2
π
1
2π
Modal Mass
M mod = 0,41 μ l
M mod = 0,45 μ l
3EI
0.49 μl 4
M mod = 0,5 μ l
3EI
0.24μl 4
M mod = 0,64 μ l
Natural Frequency and Modal Mass for Orthotropic Plates
Orthotropic floors as e.g. composite floors with beam in longitudinal direction
and a concrete plate in transversal direction have a different stiffness in
length and width (EIy>EIx). An example is given in Figure A.1.
27
Guideline
b
y
l
x
z
Figure A.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 with:
f1 =
π
EI y
2
μ l4
⎡ ⎛ b ⎞ 2 ⎛ b ⎞ 4 ⎤ EI
1 + ⎢2⎜ ⎟ + ⎜ ⎟ ⎥ x
⎝ l ⎠ ⎦⎥ EI y
⎣⎢ ⎝ l ⎠
Where:
μ
ℓ
b
E
Ix
Iy
A.4.
is the mass per m² in kg/m²,
is the length of the floor in m (in x-direction),
is the width of the floor in m (in y-direction),
is the Youngs-Modulus in N/m²,
is the moment of inertia for bending about the x-axis in m4,
is the moment of inertia for bending about the y-axis in m4.
Self weight Approach for natural Frequency
The self weight approach is a very practical approximation in cases where
the maximum deflection δmax due to self weight loading is already
determined, e.g. by finite element calculation.
This method has it origin in the general frequency equation: f =
1
2π
K
M
The stiffness K can be approximated by the assumption:
K=
M ⋅g
3
4
δ max
,
where
M is the total mass of the vibrating system,
g = 9.81 sm2 is gravity and
3
4
δ max is the average deflection.
The approximated natural frequency is
28
f =
A.5.
1
2π
Guideline
K
1
=
M 2π
4⋅ g
18
=
3 ⋅ δ max
δ max [ mm]
Dunkerley Approach for natural Frequency
The Dunkerley approach is an approximation for hand calculations. It is
applied when the expected mode shape is complex but can be subdivided
into different single modes of which the natural frequency can be
determined, see A.1, A.3 and A.2.
Figure 12 shows an example of a composite floor with two simple supported
beams and no support at the edges of the concrete plate.
The excepted mode shape is divided into two independent single mode
shapes. Both mode shapes have their own natural frequency (f1 for the
vibration of the concrete slab and f2 for the composite beam).
Initial System:
Mode of concrete slab:
Mode of composite beam:
Figure 12: Example for mode shape decomposition
According to Dunkerley the resulting natural frequency f of the total system
is:
1
1
1
1
= 2 + 2 + 2 + ...
2
f
f1
f2
f3
29
A.6.
Guideline
Approximation of modal mass
The modal mass may be interpreted as the fraction of the total mass of a
floor that is activated when the floor oscillates in a specific mode shape.
Each mode shape has its specific natural frequency and modal mass.
For the determination of the modal mass the mode shape has to be
determined and to be normalized to the maximum deflection. As the mode
shape cannot be determined by hand calculations approximations for the
first mode are commonly used.
As an alternative to hand calculations Finite Element Analysis is commonly
used. If the Finite Element Program does not give modal mass as result of
modal analysis the mode shape may be approximated by the application of
loads driving the plate into the expected mode shape, see Figure 13.
Expected mode shape:
Application of loads:
Figure 13: Application of load to obtain approximated load shape (example)
If the mode shape of a floor can be approximated by a normalised function
δ(x,y) (i.e. |δ(x,y)|max. =1,0) the corresponding modal mass of the floor can
be calculated by the following equation:
∫
Mmod = μ δ2 (x, y) dF
F
Where
μ
δ(x,y)
is the Distribution of mass
is the vertical deflection at location x,y
When mode shape deflections are determined by FEA:
M mod =
∑δ
2
i
× dMi
Nodes i
Where
fi
is the vertical deflection at node i (normalised to the maximum
deflection)
dMi
is the mass of the floor represented at node i
30
Guideline
If the function δ(x,y) represents the exact solution for the mode shape the
above described equation also yields to the exact modal mass.
The following gives examples for the determination of modal mass by hand
calculation:
Example 1: Plate simply
supported at all four
edges, Lx~Ly
-
t
Lx
Ly
Approximation of the first mode shape:
⎛ π× y ⎞
⎛ π× x ⎞
⎟ , δ(x, y)
⎟⎟ × sin⎜
δ(x, y) = sin⎜⎜
= 1,0
max .
⎜ l ⎟
⎝ lx ⎠
⎝ y ⎠
-
-
Mass distribution
M
μ = total
lx × l y
Modal mass
∫
Mmod = μ× δ 2 (x, y) × dF =
F
⎛ π× y ⎞
⎛ π× x ⎞
M total
M
⎟ × dx× dy = total
⎟⎟ × sin 2 ⎜
sin 2 ⎜⎜
×
⎜ l ⎟
l x × l y ly lx
l
4
⎝ x ⎠
⎝ y ⎠
∫∫
Example 2: Plate simply
supported at all four
edges, Lx<<Ly
-
t
Ly
Lx
l
l
1. 0 ≤ y ≤ x and l y − x ≤ y ≤ l y :
2
2
⎛π ×y ⎞
⎛π ×x ⎞
⎟
⎟⎟ × sin⎜
f ( x, y ) = sin⎜⎜
f ( x, y ) max . = 1,0
⎜ l ⎟,
⎝ lx ⎠
⎝ y ⎠
2.
lx
l
≤ y ≤ ly − x :
2
2
⎛ π× x ⎞
⎟⎟ × 1.0 , δ(x, y) max . = 1,0
δ(x, y) = sin⎜⎜
⎝ lx ⎠
-
-
Mass distribution
M
μ = total
lx × l y
Modal mass
31
Guideline
∫
Mmod = μ× δ 2 (x, y) × dF
F
lx
y =l2
⎛ π× y ⎞
⎛ π× x ⎞
M total ⎡
⎟ × dx× dy +
⎟⎟ × sin 2 ⎜
× ⎢2 ×
sin 2 ⎜⎜
⎜ l ⎟
0
0
l x × l y ⎢⎣
l
x
y
⎠
⎝
⎝
⎠
⎛
M
l ⎞
= total × ⎜ 2 − x ⎟
⎜
l y ⎟⎠
4
⎝
∫∫
=
lx
∫∫
0
Example 3: Plate spanning in
one direction between
beams, plate and
beams simply
supported
-
l
l2 − 2 x
2
0
⎤
⎛ π× x ⎞
⎟⎟ × dx× dy ⎥
sin 2 ⎜⎜
⎥⎦
⎝ lx ⎠
Ly
Lx
δ(x, y) =
⎛ π× y ⎞
⎛ π× x ⎞ δ y
δx
⎟ , δ(x, y)
⎟⎟ +
× sin⎜⎜
× sin⎜
= 1,0
max .
⎜ l ⎟
δ
l
δ
x
y
⎝
⎠
⎝
⎠
With
δ x = Deflection of the beam
δ y = Deflection of the slab assuming the deflection of the supports (i.e.
the deflection of the beam) is zero
δ = δx +δy
-
-
Mass distribution
M
μ = total
lx × l y
Modal mass
Mmod
M
= μ× δ (x, y) × dF = total ×
× l y lx ly
l
x
F
∫
2
∫∫
2
⎡δx
⎛ π× y ⎞⎤
⎛ π× x ⎞ δ y
⎟ × dx× dy
⎟⎟ +
× sin⎜
⎢ × sin⎜⎜
⎜ l ⎟⎥⎥
⎢⎣ δ
⎝ lx ⎠ δ
⎝ y ⎠⎦
⎡δx 2 + δx 2
8 δx× δy ⎤
= M total × ⎢
+ 2 ×
⎥
π
δ 2 ⎦⎥
⎣⎢ 2 δ
32
Guideline
B.
Examples
B.1.
Filigree slab with ACB-composite beams (office building)
B.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 14: Structure building
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 shown in Figure 15. In Figure 15
the part of the floor which will be considered for the vibration analysis is
indicated by the hatched area.
33
Guideline
y
x
Figure 15: Floor plan
For the main beams with a span of 16.8 m ACB/HEM400 profiles in material
S460 have been used, the main beams with the shorter span of 4.2 m are
ACB/HEM360 in 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
As stated in chapter 0 the nominal Elastic modulus of the concrete will be
increased for the dynamic calculations:
34
Guideline
E c , dyn = 1.1 × E cm = 34100 N / mm ²
The expected mode shape of the considered part of the floor which
corresponds to the first eigenfrequency is shown in Figure 17. 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 16) it is assumed that for small amplitudes as they occur
in vibration analysis the beam-column connection provides sufficient
rotational restrained, i.e. the main beams are considered to be fully fixed.
Figure 16: Beam to column connection
35
Guideline
Figure 17: Expected mode shape of the considered part of the floor corresponding
to the first eigenfrequency
Section properties
-
-
Slab:
The relevant section properties of the slab in global x-direction are:
mm 2
Ac , x = 160
mm
mm 4
I c , x = 3.41 × 10 5
mm
Main beam:
Assuming the above described first vibration mode the effective width
of the composite beam may be obtained from the following equation:
l
l
0.7 × 16.8
beff = beff ,1 + beff , 2 = 0 + 0 = 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 = 29214mm 2
Ai = 98320mm 2
I i = 5.149 × 10 9 mm 4
36
Guideline
Loads
-
Slab:

-
Self weight (includes 1.0 kN/m² for false floor):
kN
g slab = 160 × 10 −3 × 25 + 1.0 = 5 2
m
Live load: Usually a characteristic live load of 3 kN/m² is
recommended for floors in office buildings. The considered
fraction of the live load 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
kN
q slab = 0.1 × 3.0 = 0.3 2
m
Main beam:
Self weight (includes 2.00 kN/m for ACB):
kN
4.2
g beam = 5.0 ×
× 2 + 2.0 = 23.00 2
2
m
Live load:
4.2
kN
q slab = 0.3 ×
× 2 = 0.63 2
2
m
B.1.2.
Determination of dynamic floor characteristics
Eigenfrequency
The first eigenfrequency is calculated based on 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.
δ total = δ slab + δ beam
With
δ slab =
5 × (5.0 + 0.3) × 10 −3 × 4200 4
= 1.9 mm
384 × 34100 × 3.41 × 10 5
δ beam =
1 × ( 23.0 + 0.63) × 16800 4
= 4.5mm
384 × 210000 × 5.149 × 10 9
the total deflection is
δ total = 1.9 + 4.5 = 6.4 mm
Thus the first eigenfrequency may be obtained from
37
f1 =
Guideline
18
= 7.1 Hz
6.4
Modal Mass
The total mass of the slab is
M total = (5 + 0.3) × 10 2 × 16.8 × 4.2 = 37397 kg
According to chapter A.6, example 3 the modal mass of the considered slab
may be calculated as
M mod
⎡1.92 + 4.52
8 1.9 × 4.5 ⎤
= 37397 × ⎢
+ 2×
⎥ = 17220kg
2
6.42 ⎦
π
⎣ 2 × 6.4
Damping
The damping ratio of the steel-concrete slab with false floor is determined
according to table 1:
D = D1 + D2 + D3 = +1 + 1 + 1 = 3%
With
D1 = 1,0 (steel-concrete slab)
D2 = 1,0 (open plan office)
D3 = 1,0 (false floor)
B.1.3.
Assessment
Based on the above calculated modal properties the floor is classified as
class C (Figure 5). The expected OS-RMS value is approx. 0.5 mm/s.
According to Table 1 class C is classified as being suitable for office buildings,
i.e. the requirements are fulfilled.
38
B.2.
Three storey office building
B.2.1.
Description of the Floor
Guideline
The floor of this office building, Figure 18, span 15m from edge beam to
edge beam. In the regular area these secondary floor beams have IPE600
sections are laying in a distance of 2.5m. Primary edge beam which span
7.5m from column to column have also IPE600 section, see Figure 19.
Figure 18: Building overview
Figure 19: Steel section of the floor
The plate of the floor is a composite plate of 15cm total thickness with steel
sheets COFRASTRA 70, Figure 20.
39
Guideline
Figure 20: Floor set up
The nominal material properties are
-
Steel S235:
Concrete C25/30:
Es = 210 000 N/mm²,
Ecm = 31 000 N/mm²,
E c , dyn = 1.1 × E cm = 34 100 N / mm ²
fy = 235 N/mm²
fck = 25 N/mm²
Section properties
-
-
Slab (transversal to beam):
A
= 1170 cm²/m
I
= 20 355 cm4/m
g
= 3.5 kN/m²
Δg
= 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/m²
Loads
-
-
Slab (transversal to beam):
g + Δg = 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/m²
q
= 0.3 x 2.5 = 0.75 kN/m
ptotal = 11.97 kN/m
B.2.2.
Determination of dynamic floor characteristics
40
Guideline
Supporting conditions
The secondary beam are ending in the primary beams which are 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 on two ways. 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 (Chapter A.2):
p = 11.97 [ kN / m] ⇒ μ = 11.97 × 1000 [ kg m / s ² / m] / 9.81[ m / s ²] = 1220 [ kg / m]
f =
•
π
3EI
2
=
4
0.49 μl
π
3 × 210000 × 10 6 [ N / m²] × 270089 × 10 − 8 [ m 4 ]
= 4,77 Hz
0.49 × 1220 [ kg / m] × 154 [ m 4 ]
Application of the equation for orthotropic plates (Chapter A.3):
=
f1
2
π
EI y
2
m l4
⎡ ⎛ b ⎞ 2 ⎛ b ⎞ 4 ⎤ EI x
1 + ⎢2⎜ ⎟ + ⎜ ⎟ ⎥
⎣⎢ ⎝ l ⎠ ⎝ l ⎠ ⎦⎥ EI y
⎡ ⎛ 2.5 ⎞ 2 ⎛ 2.5 ⎞ 4 ⎤ 3410 × 20355
210000 × 106 × 270089 × 10 − 8
=
1 + ⎢2⎜
⎟ +⎜
⎟ ⎥
2
1220 × 154
⎢⎣ ⎝ 15 ⎠ ⎝ 15 ⎠ ⎥⎦ 21000 × 270089
= 4.76 × 1.00 = 4,76
π
•
Application of the self weight approach (Chapter A.4):
δ total = δ slab + δ beam
5 × 4.3 × 10−3 × 25004
δ slab =
= 0.3 mm
384 × 34100 × 2.0355 × 105
5 × 11.97 × 150004
δ beam =
= 13.9mm
384 × 210000 × 270089 × 104
δ total = 0.3 + 13.9 = 14.2 mm
⇒
f1 =
18
= 4.78 Hz
14.2
41
Guideline
Modal Mass
The determination of the eigenfrequency above showed that the load bearing
behaviour of the floor can be approximated by a simple beam model. Thus
this model is taken for the determination of the modal mass:
M mod = 0,5 μ l = 0,5 × 1220 × 15 = 9150 kg
Damping
The damping ratio of the steel-concrete slab with false floor is determined
according to table 1:
D = D1 + D2 + D3 = +1 + 1 + 1 = 3%
With
D1 = 1,0 (steel-concrete slab)
D2 = 1,0 (open plan office)
D3 = 1,0 (ceiling under floor)
B.2.3.
Assessment
Based on the above calculated modal properties the floor is classified as
class D (Figure 5). The expected OS-RMS value is approx. 3.2 mm/s.
According to Table 1 class D is classified as being suitable for office
buildings, i.e. the requirements are fulfilled.
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