Design Guide for Floor Vibation-ArcelloMittal

ArcelorMittal Europe - Long Products
Sections and Merchant Bars
Design Guide for
Floor Vibrations

This design guide presents a method for assessing floor
vibrations guaranteeing the comfort of occupants. This
document is based on recent research developments
(RFCS-Project “Vibration of floors”).
Guidance for the structural integrity, given in this
document, is based on common approaches.
©
Claude
Vasconi
Architecte
-
Chambre
de
Commerce
de
Luxembourg

Contents
1. Introduction 3
2. Definitions 7
3. Determination of Floor Characteristics 11
4. Classification of Vibrations 15
5. Design Procedure and Diagrams 19
Annex A Formulas for Manual Calculation 31
Annex B Examples 43
Technical assistance Finishing 52
References 52
Your partners 53
1

Floor structures are designed for ultimate limit
state and serviceability limit state criteria:
l Ultimate limit states are those
related to strength and stability;
l 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.
Guidance is given for:
l Specification of tolerable vibration
by the introduction of acceptance
classes (Chapter 4) and
l Prediction of floor response due to human
induced vibration with respect to the
intended use of the building (Chapter 5).
An overview of the general design procedure
presented in Chapter 5 is given in Figure 1.
For the prediction of 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 design guide.
1. Introduction
The design guide comprehends simple
methods, design tools and recommendations
for the acceptance of vibration of floors
which are caused by people during normal
use. The given design methods focus on
the prediction of vibration. Measurements
performed after erection may lead to
differences to the predicted values so that
one cannot claim on the predicted result.
The design and assessment methods for floor
vibrations are related to human induced resonant
vibrations, mainly caused by walking under
normal conditions. Machine induced vibrations or
vibrations due to traffic etc. are not covered by
this design guide.
The design guide should not be applied to
pedestrian bridges or other structures, which
do not have a structural characteristic or a
characteristic of use comparable to floors in
buildings.

5
1. Introduction
Figure 1 Design procedure (see Chapter 5 )
Determine dynamic floor characteristics:
- Natural Frequency
- Modal Mass
- Damping
(Chapter 3; Annex A)
Read off OS-RMS90 – Value
(Chapter 5)
Determine Acceptance Class
(Chapter 4)

©
ArcelorMittal
Photo
Library

2. Definitions
The definitions given here are
consistent with the application
of this design guide.
Damping D Damping is the energy dissipation of a vibrating system. The total damping consists of
l Material and structural damping,
l Damping by furniture and finishing (e.g. false floor),
l Spread of energy throughout the whole structure.
Modal mass Mmod = Each mode of a system with several degrees of freedom can be represented by a system with
generalised mass a single degree of freedom:
where f is the natural frequency of the considered mode,
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 3.
mod
mod
2
1
M
K
f
π
=

2. Definitions
9
Natural Frequency f = Every structure has its specific dynamic behaviour with regard to shape and duration T[s] of a
Eigenfrequency 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:

where K is the stiffness,
M is the mass.
The determination of frequencies is described in Chapter 3.
OS-RMS90 = RMS- value of the velocity for a significant step covering the intensity of 90% of people’s steps
walking normally
OS: One step
RMS: Root mean square = effective value, here of velocity v:

where T is the investigated period of time.
2
)
(
1
0
2 Peak
T
RMS dt
t
T
=
= ∫
M
K
f
π
2
1
=
v
v
v
v

©
SINGLE
Speed
IMAGE
-
Peter
Vanderwarker
-
Paul
Pedini

11
11
3. Determination of
Floor Characteristics

3. Determination of
Floor Characteristics
The determination of floor characteristics can
be performed by simple calculation methods,
by Finite Element Analysis (FEA) or by testing.
As the design guide is intended to be used
for the design of new buildings, testing
procedures are excluded from further
explanations and reference is given to [1].
Different finite element programs can perform
dynamic calculations and offer tools for
the determination of natural frequencies.
The model mass is in many programs also
a result of the analysis of the frequency.
As it is specific for each software what elements
can be used, how damping is considered and
how and which results are given by the different
programs, only some general information is
given in this design guide concerning FEA.
If FEA is applied for the design of a floor with
respect to the vibration behaviour, it should
be considered that the FE-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 different consideration
of boundary conditions in vibration analysis.
If compared to ULS design: a connection
which is assumed to be a hinged connection
in ULS may be rather assumed to provide full
moment connection in vibration analysis.
For concrete, the dynamic modulus of
elasticity should be considered to be 10%
higher than the static modulus Ecm.
For manual calculations, Annex A gives
formulas for the determination of
frequency and modal mass for isotropic
plates, orthotropic plates and beams.
Damping has a big influence on the vibration
behaviour of a floor. Independent on the
method chosen to determine the natural
frequency and modal mass the damping values
for a vibrating system can be determined
with the values given in Table 1. These values
are considering the influence of structural
damping for different materials, damping due
to furniture and damping due to finishes. The
system damping D is obtained by summing
up the appropriate values for D1 to D3.
In the determination of the dynamic
floor characteristics, a realistic fraction
of imposed load should be considered in
the mass of the floor (m, M). Experienced
values for residential and office buildings
are 10% to 20% of the imposed load.

13
3. Determination of Floor Characteristics
Table 1 Determination of damping
Type Damping (% of critical damping)
Structural Damping D1
Wood 6%
Concrete 2%
Steel 1%
Composite (steel-concrete) 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%
Gymnastic 0%
Damping due to finishes D3
Ceiling under the floor 1%
Free floating floor 0%
Swimming screed 1%
Total Damping D = D1 + D2 + D3

15
15
4. Classification
of Vibrations

4. Classification
of Vibrations
The perception of vibrations by persons and
the individual feeling of annoyance depends
on several aspects. The most important are:
l The direction of the vibration,
however in this design guide only
vertical vibrations are considered;
l Another aspect is the posture of people
such as standing, lying or sitting;
l The current activity of the person considered
is of relevance for its perception of vibrations.
Persons working in the production of a factory
perceive vibrations differently from those
working concentrated in an office or a surgery;
l Additionally, age and health of affected
people may be of importance for
feeling annoyed by vibrations.
Thus the perception of vibrations is a
very individual problem that can only
be described in a way that fulfils the
acceptance of comfort of the majority.
It should be noted that the vibrations
considered in this design guide are relevant
for the comfort of the occupants only. They
are not relevant for the 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
annoying floor vibrations. The OS-RMS
values correspond to the harmonic vibration
caused by one relevant step onto the floor.
As the dynamic effect of people walking on a
floor depends on several boundary conditions,
such as weight and speed of walking of the
people, their shoes, flooring, etc., the 90%
OS-RMS (OS-RMS90) value is recommended as
assessment value. The index 90 indicates that
90 percent of steps on the floor are covered by
this value.
The following table classifies vibrations into
several classes and gives also recommendations
for the assignment of classes with respect
to the function of the considered floor.

17
Table 2 Classification of floor response and recommendation for the application of classes
Recommended
Critical
Not recommended
Class
Lower
Limit
Upper
Limit
Critical
Workspace
Health
Education
Residential
Office
Meeting
Retail
Hotel
Industrial
Sport
OS-RMS90 Function of Floor
A 0.0 0.1
B 0.1 0.2
C 0.2 0.8
D 0.8 3.2
E 3.2 12.8
F 12.8 51.2
4. Classification of vibrations

19
19
5. Design Procedure
and Diagrams

5. Design Procedure
and Diagrams
An overview of the general design procedure
is given in Figure 2. The design is carried
out in 3 steps where the determination
of the dynamic floor characteristics is the
most complex one. Thus Annex A gives
detailed help by simplified methods; general
explanations are given in Chapter 3.
When modal mass and frequency are
determined, the OS-RMS90-value as well as
the assignment to the perception classes may
be determined with the diagrams given below.
The relevant diagram needs to be selected
according to the damping characteristics of
the floor in the condition of use (considering
finishing and furniture), see Chapter 3.
The diagrams have been elaborated by TNO
Bouw, the Netherlands, in the frame of [1].
Determine dynamic floor characteristics:
- Natural Frequency
- Modal Mass
- Damping
(Chapter 3; Annex A)
Read off OS-RMS90 – Value
(Chapter 5)
Determine Acceptance Class
(Chapter 4)
Figure 2 Design procedure

5. Design Procedure and Diagrams
21
Figure 3 Application of diagrams The diagram is used by entering the modal
mass on the x-axis and the corresponding
frequency on the y-axis. The OS-RMS value
and the acceptance class can be read-off at the
intersection of extensions at both entry points.
Frequency
of
the
floor
[Hz]
Modal Mass of the floor [kg]
B
F
E
D
C
A
: means out of the range of a tolerable assessment

Modal mass of the floor [kg]
Eigenfrequency
of
the
floor
[Hz]
Classification based on a damping ratio of 1%
B
F
E
D
C
A
Figure 4 OS-RMS90 for 1% Damping

23
Eigenfrequency
of
the
floor
[Hz]
B
F
E
D
C
A

Eigenfrequency
of
the
floor
[Hz]
B
F
E
D
C
A

25
Eigenfrequency
of
the
floor
[Hz]
B
F
E
D
C
A

Eigenfrequency
of
the
floor
[Hz]
B
F
E
D
C
A

27
Eigenfrequency
of
the
floor
[Hz]
B
F
E
D
C
A

Eigenfrequency
of
the
floor
[Hz]
B
F
E
D
C
A

29
Eigenfrequency
of
the
floor
[Hz]
B
F
E
D
C
A

B
F
E
D
C
A
Eigenfrequency
of
the
floor
(Hz)
Modal mass of the floor (kg)

Annex a formulas FOR
MANUAL CALCULATION
A.1 Natural Frequency and Modal Mass for Isotropic Plates 32
A.2 Natural Frequency and Modal Mass for Beams 34
A.3 Natural Frequency and Modal Mass for Orthotropic Plates 35
A.4 Self Weight Approach for Natural Frequency 36
A.5 Dunkerley Approach for Natural Frequency 37
A.6 Approximation of Modal Mass 38
31

M
; M = β
mod
Frequency; Modal Mass
α
β λ Ratio = L/B
λ
)
(
. 2
1
57
1 λ
α +
⋅
=
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
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
α
β λ Ratio = L/B
λ
14.00
12.00
10.00
8.00
6.00
4.00
2.00
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
α
Ratio = L/B
λ
4
2
14
5
5
2
1
57
1 λ
λ
α .
.
. +
+
=
4
2
44
2
92
2
14
5
57
1 λ
λ
α .
.
.
. +
+
=
β λ
9.00
8.00
7.00
6.00
5.00
4.00
3.00
2.00
1.00
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
α
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
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
α
14.00
12.00
10.00
8.00
6.00
4.00
2.00
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
α
4
2
14
5
5
2
1
57
1 λ
λ
α .
.
. +
+
=
4
2
44
2
92
2
14
5
57
1 λ
λ
α .
.
.
. +
+
=
)
(
. 2
1
57
1 λ
α +
=
)
1
(
12 2
3
2
υ
α
−
=
m
t
E
f
β λ
β λ
β λ
b
b
b
l
l
l
l
Ratio = l/b
λ
Ratio = l/b
λ
Ratio = l/b
λ
A.1 Natural Frequency
and Modal Mass for
Isotropic Plates
The following table gives formulas for the
determination of the first natural frequency
(acc. to [2]) and the modal mass of plates
for different support conditions.
For the application of the given equations,
it is assumed that no lateral deflection
at any edges of the plate occurs.
Support Conditions:
clamped hinged

3
2
α
=
t
E
f ; M = β
mod
α
β λ Ratio = L/B
λ
)
(
. 2
1
57
1 λ
α +
⋅
=
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
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
α
β λ Ratio = L/B
λ
14.00
12.00
10.00
8.00
6.00
4.00
2.00
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
α
Ratio = L/B
λ
4
2
14
5
5
2
1
57
1 λ
λ
α .
.
. +
+
=
4
2
44
2
92
2
14
5
57
1 λ
λ
α .
.
.
. +
+
=
12.00
10.00
8.00
6.00
4.00
2.00
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
α
12.00
10.00
8.00
6.00
4.00
2.00
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
α
18.00
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
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
α
clamped hinged
3
2
α
=
t
E
f M
; M = β
mod
Frequency; Modal Mass
Support Conditions:
clamped hinged
E Young's modulus [N/m²]
t Thickness of plate [m]
m Mass of floor including finishing
and a representative amount of
imposed load (see chapter 3)
[kg/m²]
υ Poisson ratio
M Total mass of floor including
finishes and representative
amount of imposed load
(see chapter 3) [kg]
4
2
44
2
33
2
1
57
1 λ
λ
α .
.
. +
+
=
4
2
44
2
72
2
44
2
57
1 λ
λ
α .
.
.
. +
+
=
4
2
14
5
13
3
14
5
57
1 λ
λ
α .
.
.
. +
+
=
β λ
β λ
β λ
β λ
0.17
)
1
(
12 2
υ
−
m
l
b
b
b
l
l
l
Ratio = l/b
λ
Ratio = l/b
λ
Ratio = l/b
λ
Annex A Formulas for Manual Calculation
33

and Modal Mass
for Beams
Support Conditions Natural Frequency Modal Mass
l
M µ
41
0
mod =
4
37
0
3
4
l
EI
f
µ
π .
=
l
M µ
45
0
mod =
4
2
0
3
2
l
EI
f
µ
π .
=
l
M µ
5
0
mod =
4
49
0
3
2
l
EI
f
µ
π .
=
l
M µ
64
0
mod =
4
24
0
3
2
1
l
EI
f
µ
π .
=
Table 3 Determination of the first Eigenfrequency of Beams
.
.
.
.
l
l
l
l
The first Eigenfrequency of a beam can be
determined with the formula according to the
supporting conditions from Table 3 with:
E Young's modulus [N/m²]
I Moment of inertia [m4]
µ Distributed mass m of the floor
(see page 33) multiplied by the floor
width [kg/m]
Length of beam [m]
l

y
x
y
EI
EI
l
b
l
b
l
m
EI
f














+






+
=
4
2
4
1 2
1
2
π
35
and Modal Mass for
Orthotropic Plates
Orthotropic floors as e.g. composite floors
with beams in longitudinal direction and a
concrete plate in transverse direction have
different stiffnesses in length and width
(EIy EIx). An example is given in Figure 13.
The first natural frequency of the orthotropic
plate being simply supported at all four
edges can be determined with:
Figure 13 Dimensions and axis of an orthotropic plate
Where:
m is the mass of floor including finishes and a representative amount
of imposed load (see chapter 3) [kg/m²],
l is the length of the floor (in x-direction) [m],
b is the width of the floor (in y-direction) [m],
E is the Young's modulus [N/m²],
Ix is the moment of inertia for bending about the x-axis [m4],
Iy is the moment of inertia for bending about the y-axis [m4].
Formulas for the approximation of the modal mass for orthotropic plates
are give in Annex A.6.
x
x
x
x
x
x
x
x
z
y
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
l
b

A.4 Self Weight
Approach for
Natural Frequency
The self weight approach is a very practical
approximation in cases where the maximum
deflection δmax due to the mass m is already
determined, e.g. by finite element calculation.
This method has its origin in the
general frequency equation:
The stiffness K can be approximated
by the assumption:
where:
M is the total mass of
the vibrating system [kg],
is the gravity [m/s2] and
is the average deflection [mm].
The approximated natural frequency is
where:
δmax is the maximum deflection due to
loading in reference to the mass m.
M
K
f
π
2
1
=
4
3
δ
g
M
K =
81
.
9
g =
4
3
δ
]
[
18
3
4
2
1
2
1
max
max mm
g
M
K
f
δ
δ
π
π
=
=
=

37
A.5 Dunkerley
Approach for
Natural Frequency
The Dunkerley approach is an approximation
for manual calculations. It is applied when
the expected mode shape is complex but
can be subdivided into different single
modes for which the natural frequency can
be determined, see A.1, A.3 and A.2.
Figure 14 shows an example of a composite
floor with two simply supported beams and no
support at the edges of the concrete plate.
The expected mode shape is divided into two
independent single mode shapes; one of the
concrete slab and one of the composite beam.
Both mode shapes have their own natural
frequency (f1 for the mode of the concrete
slab and f2 for the composite beam).
According to Dunkerley, the resulting
natural frequency f of the total system is:
...
1
1
1
1
2
3
2
2
2
1
2
+
+
+
=
f
f
f
f
Figure 14 Example for mode shape decomposition
Initial System
Mode of concrete slab
Mode of composite beam

A.6 Approximation
of Modal Mass
Figure 15 Application of load to obtain
approximated load shape (example)
Expected mode shape:
Application of loads:
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 normalised to the maximum deflection.
As the mode shape cannot be determined
by manual calculations, approximations
for the first mode are commonly used.
As an alternative to manual calculations,
Finite Element Analysis is commonly used. If
the Finite Element software 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 15.
If the mode shape of a floor can be
approximated by a normalised function
f (x,y) (i.e. |f (x,y)|max = 1.0) the
corresponding modal mass of the floor can
be calculated by the following equation:
Where:
µ is the distribution of mass
f(x,y) is the vertical deflection at location x,y
When mode shape deflections
are determined by FEA:
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
If the function f (x,y) represents the exact
solution for the mode shape the above described
equation also yields to the exact modal mass.
dF
y
x
f
M
F
∫
= )
,
(
2
mod µ ∑
=
i
Nodes
i
i dM
f
M 2
mod

39
The following gives
examples for the
determination of
modal mass by
manual calculation:
Example 1
Plate simply supported along
all four edges, ~
l Approximation of the first mode shape:
l Mass distribution
l Modal mass
















=
y
x l
y
l
x
y
x
f
π
π
sin
sin
)
,
(
y
x l
l
M
=
µ
4
sin
sin
)
,
( 2
2
2
mod
l l
y
x
y
x
F
M
dy
dx
l
y
l
x
l
l
M
dF
y
x
f
M
y x
=
















=
= ∫∫
∫
π
π
µ
x
l
y
l
x
l
y
l
0
1
,
( max
=
y
x
f .
(
0 0

Example 2
Plate simply supported along all
four edges, lxly
1. and
2.
l Mass distribution
l Modal mass
2
0 x
l
y ≤
≤ :
y
x
y l
y
l
l ≤
≤
−
2 















=
y
x l
y
l
x
y
x
f
π
π
sin
sin
)
,
(
2
2
x
y
x l
l
y
l
−
≤
≤ : 0
.
1
sin
)
,
( 







=
x
l
x
y
x
f
π
y
x l
l
M
=
µ
0
1
)
,
( max
=
y
x
f .
0
1
)
,
( max
=
y
x
f .
= ∫
F
dF
y
x
f
M )
,
(
2
mod µ








−
=
y
x
l
l
M
2
4
















+
















= ∫ ∫
∫ ∫
−
= l
l
l
x
y
y
x
y
x
dy
dx
l
x
dy
dx
l
y
l
x
l
l
M x
x
x
sin
sin
sin
2
0
2
2
0
2
0 0
2
2 y π
π
π =
y
x
l
y
l
x
l x
l
2 2
lx
2
l

41
Example 3
Plate spans uniaxial between beams,
plate and beams simply supported
With
δx = Deflection of the beam
δy = Deflection of the slab assuming the
deflection of the supports (i.e. the
deflection of the beam) is zero
δ = δx + δy
l Mass distribution
l Modal mass








+








=
y
y
x
x
l
y
l
x
y
x
f
π
δ
δ
π
δ
δ
sin
sin
)
,
(
y
x l
l
M
=
µ
x
l
y
l
0
1
)
,
( max
=
y
x
f .








+
+
= 2
2
2
2
8
2 δ
δ
δ
π
δ
δ
δ y
x
y
x
M 2
















+








=
= ∫ ∫
∫
2
2
mod sin
sin
)
,
(
π
δ
δ
π
δ
δ
µ
l l
y
y
x
x
y
x
F
dy
dx
l
y
l
x
l
l
M
dF
y
x
f
M
x y
0 0

43
43
Annex B Examples
B.1 Filigree slab with ACB-composite beams (office building) 44
B.1.1 Description of the Floor 44
B.1.2 Determination of dynamic floor characteristics 47
B.1.3 Assessment 47
B.2 Three storey office building 48
B.2.1 Description of the Floor 48
B.2.2 Determination of dynamic floor characteristics 50
B.2.3 Assessment 51

B.1 Filigree slab with ACB-composite
beams (office building)
In the first worked example, a filigree slab
with false-floor in an open plan office is
checked for footfall induced vibrations.
The slab spans uniaxially by over 4.2 m between
main beams. Its overall thickness is 160 mm.
The main beams are ArcelorMittal 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 18. For a vibration analysis it
is sufficient to check only a part of the floor
(representative floor bay). The representative
part of the floor to be considered in this example
is indicated by the hatched area in Figure 18.
B.1.1 Description
of the Floor
Figure 16 Building structure
Figure 18 Floor plan
Figure 17 Beam to column
connection
y
x
1.0
1.0
4.2
16.8
23.0
3.2 4.2 1.0
4.2 4.2 4.2 4.2 4.2 4.2 4.2 1.0 8.4
1.0
1.0
21.0
1.0
16.8
39.8
15.8
1.0
4.2 4.2 4.2 4.2 4.2 4.2 4.2 1.0
1.0
31.4
[m]

.
45
Annex B Examples
For the main beams (span of 16.8 m)
ACB/HEM400/HEB400 profiles in steel
S460 have been used, see Figure 19.
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.
Figure 19 Cross section
The nominal material properties are
As stated in Chapter 3, the nominal
elastic modulus of the concrete will be
increased for the dynamic calculations:
The expected mode shape of the considered
part of the floor which corresponds to the
first Eigenfrequency is shown in Figure 20.
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
in Figure 17) it is assumed that for small
amplitudes as they occur in vibration analysis,
the beam-column connection provides
sufficient rotational restrain, i.e. the main
beams are considered to be fully fixed.
l Steel S460: Es = 210000 N/mm², fy = 460 N/mm²
l Concrete C25/30: Ecm = 31000 N/mm², fck = 25 N/mm²
²
/
100
34
1
.
1
, mm
N
E
E cm
dyn
c =
=
Figure 20 Expected mode
shape of the considered
part of the floor
corresponding to the
first Eigenfrequency
160
HE400B
HE400M
287.95
304.05
592
76.95
422
93.05
[mm]

Section properties Loads
Slab
l Self weight (includes 1.0 kN/m²
for false floor):
l 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
Main beam
l Self weight (includes 2.00 kN/m for ACB):
l Live load:
3
5
0
.
1
25
10
160
gslab =
+
×
×
= −
3
.
0
0
.
3
1
.
0
qslab =
×
=
00
.
23
0
.
2
2
2
2
.
4
0
.
5
gbeam =
+
×
×
=
.
2
2
2
.
4
3
.
0
qslab =
×
×
=
kN 2
m
kN m
kN m
Slab
The relevant section properties of
the slab in global x-direction are:
Main beam
Assuming the previously described first
vibration mode, the effective width of
the composite beam may be obtained
from the following equation:
The relevant section properties of
the main beam for serviceability
limit state (no cracking) are:
Aa,net = 21936 mm2
Aa,total = 29214 mm2
Ai = 98320 mm2
Ii = 5.149 x 109 mm4
mm
mm
A x
c
2
, 160
=
mm
mm
I x
c
4
5
, 10
41
.
3 ×
=
l
l
b
b
b eff
eff
eff
8
8
0
0
2
,
1
,
=
+
=
+
=
m
94
.
2
8
8
.
16
7
.
0
2 =
×
×
kN 2
m
1 26
2

7
7
47
Annex B Examples
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.
With
the maximum deflection is
Thus the first Eigenfrequency may be
obtained (according to Annex A.4) from
Modal Mass
The total mass of the considered floor bay is
According to Chapter A.6, Example 3, the modal
mass of the considered slab may be calculated as
B.1.2 Determination of
dynamic floor characteristics
Damping
The damping ratio of the steel-concrete slab
with false floor is determined according to
table 1:
With
D1 = 1.0 (steel-concrete slab)
D2 = 1.0 (open plan office)
D3 = 1.0 (false floor)
Based on the above calculated modal properties
the floor is classified as class C (Figure 6). The
expected OS-RMS value is approx. 0.5 mm/s.
According to Table 2, class C is classified
as being suitable for office buildings,
i.e. the requirements are fulfilled.
B.1.3 Assessment
beam
slab δ
δ
δ +
=
mm
slab 9
.
1
10
41
.
3
34100
384
4200
10
)
3
.
0
0
.
5
(
5
5
4
3
=
×
×
×
×
×
+
×
=
−
δ
mm
beam .
4
10
149
.
5
210000
384
16800
)
0
.
23
(
1
9
4
=
×
×
×
×
+
×
=
δ
mm
.
6
.
4
9
.
1 =
+
=
δ
Hz
f .
7
.
6
18
1 =
=
kg
M 37397
2
.
4
8
.
16
10
)
3
.
0
5
( 2
=
×
×
×
+
=
kg
M 17220
4
.
6
5
.
4
9
.
1
8
4
.
6
2
5
.
4
9
.
1
37397 2
2
2
2
2
mod =





 ×
×
+
×
+
×
=
π
max
max
.
1 26
0
6
6

B.2 Three storey
office building
The method leads in general to conservative
results when applied as single bay method using
the mode related to the fundamental frequency.
However, in special cases in which the modal
mass for a higher mode is significantly
low, also higher modes need to be
considered, see the following example.
B.2.1 Introduction Figure 21 Building overview
Figure 22 Plan view of floor with choice of sections
The floor of this office building, Figure 21,
spans 15 m from edge beam to edge beam.
In the regular area these secondary floor
beams are IPE600 sections, spaced in
2.5 m. Primary edge beams, which span
7.5 m from column to column, consist
also of IPE600 sections, see Figure 22.
B.2.2 Description
of the Floor
IPE600 IPE600 IPE600
IPE600 IPE600 IPE600
IPE600
IPE600
IPE600
IPE600
IPE600
IPE600
IPE600
IPE600
IPE600
IPE600
9 x 2.5
7.5
A B C D
1
2
[m]

49
Annex B Examples
Figure 23 Floor set up
with COFRASTRA 70
Section properties
Slab (transversal to beam, E = 210000 N/mm2)
A = 1170 cm²/m
I = 20355 cm4/m
Composite beam
(beff = 2,5 m; E = 210000 N/mm²)
A = 468 cm²
I = 270089 cm4
Loads
Slab
Self weight:
g = 3.5 kN/m²
g = 0.5 kN/m²
g + g = 4.0 kN/m² (permanent load)
Life load
q = 3.0 x 0.1 = 0.3 kN/m²
(10% of full live load)
Composite beam
Self weight:
g = (3.5+0.5) x 2.5 + 1.22 = 11.22 kN/m
Life load:
q = 0.3 x 2.5 = 0.75 kN/m
l Steel S235: Es = 210000 N/mm², fy = 235 N/mm²
l Concrete C25/30: Ecm = 31000 N/mm², fck = 25 N/mm²
The slab is a composite plate of 15 cm total
thickness with steel sheets COFRASTRA 70,
Figure 23.
More information concerning the COFRASTRA
70 are available on www.arval-construction.fr.
The nominal material properties are:
For dynamic calculations (vibration analysis) the
elastic modulus will be increased according to
Chapter 3:
Ec, dyn = 1.1 Ecm = 34100 N/mm²

Supporting conditions
The secondary beams are ending in the
primary beams which are open sections with
low torsional stiffness. Thus these beams
may be assumed to be simply supported.
Eigenfrequency
In this example, the Eigenfrequency is
determined according to three methods: the
beam formula, neglecting the transversal
stiffness of the floor, the formula for
orthotropic plates and the self-weight method
considering the transversal stiffness.
l Application of the beam
equation (Chapter A.2):
l Application of the equation for
orthotropic plates (Chapter A.3):
l Application of the self weight
B.2.3 Determination of
dynamic floor characteristics
Hz
l
EI
f
m
kg
m
kN
p
4
15
1220
49
.
0
10
270089
10
210000
3
2
49
.
0
3
2
/
1220
81
.
9
/
1000
97
.
11
/
97
.
11
4
8
6
4
=
×
×
×
×
×
×
=
=
=
×
=
⇒
=
−
π
µ
π
µ
4
00
.
1
.
4
270089
21000
20355
3410
15
5
.
2
15
5
.
2
2
1
15
1220
10
270089
10
210000
2
2
1
2
4
2
4
8
6
4
2
4
1
=
×
=
×
×














+






+
×
×
×
×
=














+






+
=
−
π
π
y
x
y
EI
EI
l
b
l
b
l
m
EI
f
Hz
.
.
8
8 8
×

51
Annex B Examples
approach (Chapter A.4):
Modal Mass
The determination of the Eigenfrequency
shows 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:
Damping
The damping ratio of the steel-
concrete slab with false floor is
determined according to Table 1:
With
D1 = 1.0 (steel-concrete slab)
D2 = 1.0 (open plan office)
D3 = 1.0 (ceiling under floor)
Based on the above calculated modal
properties the floor is classified as class
D (Figure 6). The expected OS-RMS90
value is approximately 3.2 mm/s.
According to Table 2, class D is suitable for office
buildings, i.e. the requirements are fulfilled.
B.2.4 Assessment
beam
slab δ
δ
δ +
=
mm
slab 3
.
0
10
0355
.
2
34100
384
2500
10
3
.
4
5
5
4
3
=
×
×
×
×
×
×
=
−
δ
mm
beam 9
.
13
10
270089
210000
384
15000
97
.
11
5
4
4
=
×
×
×
×
×
=
δ
mm
2
.
14
9
.
13
3
.
0 =
+
=
δ
Hz
f .
4
2
.
14
18
1 =
=
⇒
kg
l
M 9150
15
1220
5
0
5
0
mod =
×
×
=
= µ
%
3
1
1
1
3
2
1 =
+
+
=
+
+
= D
D
D
D
.
.
8
max
max

Technical
assistance
Finishing
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
Finishing
As a complement to the technical capacities
of our partners, we are equipped with
high-performance finishing tools and
offer a wide range of services, such as:
l drilling
l flame cutting
l T cut-outs
l notching
l cambering
l curving
l straightening
l cold sawing to exact length
l welding and fitting of studs
l shot and sand blasting
l surface treatment
Technical assistance
We are happy to provide you with free technical
advice to optimise the use of our products
and solutions in your projects and to answer
your questions about the use of sections
and merchant bars. This technical advice
covers the design of structural elements,
construction details, surface protection,
fire safety, metallurgy and welding.
Our specialists are ready to support your
initiatives anywhere in the world.
To facilitate the design of your projects, we also
offer software and technical documentation that
you can consult or download from our website:
sections.arcelormittal.com
Construction
At ArcelorMittal we also have a team of
multi-product professionals specialised in
the construction market.
A complete range of products and
solutions dedicated to construction in all
its forms: structures, façades, roofing,
etc. is available from the website
www.constructalia.com

Your partners
Prof. Dr.-Ing. Markus Feldmann
Dr.-Ing. Ch. Heinemeyer
Dr.-Ing. B. Völling
RWTH Aachen University
Institut und Lehrstuhl für Stahlbau
und Leichtmetallbau
Although every care has been taken during the production of this brochure, we regret
that we cannot accept any liability in respect of any incorrect information it may contain
or any damages which may arise through the misinterpretation of its contents.
Authors
ArcelorMittal
Commercial Sections
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L-4221 Esch-sur-Alzette
Luxembourg
Tel: +352 5313 3010
Fax: +352 5313 2799
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ArcelorMittal
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L-4221 Esch-sur-Alzette
LUXEMBOURG
Tel. + 352 5313 3010
Fax + 352 5313 2799
Version
2014-1

Design Guide for Floor Vibation-ArcelloMittal

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