ncci vibrations

NCCI: Vibrations SN036a-EN-EU

NCCI: Vibrations

This NCCI gives rules for the consideration of vibrations by simple (and thus conservative) approaches for verification.

Contents

1. Introduction 2

2. Eigenfrequencies 3

3. Modal Mass 5

4. Damping 7

5. Acceleration from human activities 8

6. Acceptable accelerations 10

7. References 16

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1. Introduction The verification of serviceability limit states should include taking account of criteria concerning vibrations (EN 1990 §3.4(3)). Additional provisions are given in the relevant material ENs. Requirements for modelling to determine dynamic actions are given in EN 1990 § 5.1.3 and reference is made to EN 1990 §A.1.4.4 for guidance on assessing the limits. That clause in turn refers to ISO 10137 for further guidance. The clause is also referred to by EN 1993-1-1 § 7.2.3 and EN 1994-1-1 § 7.3.2 but it is noted that the National Annex may specify limits.

The procedure presented below is based on the recommendations and requirements given by ISO 10137 and ISO 2631. For cases where the guidance in these Standards is unavailable, or incomplete, recommendations that have been developed in the ECSC project entitled “Vibration of Floors” (Project reference: 7210-PR/314) are presented.

An overview of the design procedure is given in Figure 1.1.

Determine Eigenfrequencies By computer calculation By hand (see Section 2)

Determine Modal Mass Mm By computer calculation By hand (see Section 3)

Determine Modal Damping ξ For calculations (see Section 4)

Determine Acceleration By computer calculation By hand (see Section 5)

Compare with Acceptable Accelerations (See Section 6)

Figure 1.1 Design procedure

The most realistic assessment will be obtained by applying the procedure with measured values of the Eigenfrequencies, modal mass and damping.

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http://www.access-steel.com/discovery/linklookup.aspx?id=EC266






For design purposes, the procedure may be applied to computer or simple hand calculations. An approach for calculating the Eigenfrequencies of the floor, which is amenable to hand calculations, is given in Section 2. Values of the modal mass and the damping are given in Section 3 and Section 4 respectively. An approach for predicting the acceleration from walking activities is presented in Section 5, which can be compared with recommended levels of acceptable accelerations in Section 6.

The limits given in Section 6 may be used where the National Annex does not give specific limits or where it gives limits different from those derived from ISO 10137.

2. Eigenfrequencies In conventional composite floor systems, the first Eigenfrequency of a floor may be estimated by using engineering judgement on the likely mode shape and the support, or boundary conditions, this will impose on the individual structural components. For example, on a simple composite floor comprising a slab continuous over a number of secondary beams, which in turn, are supported by stiff primary beams, there are two possible mode shapes that may be sensibly considered:

Secondary (floor) beam mode The primary beams form nodal lines (i.e. they have zero deflection) about which, the secondary beams vibrate as simply-supported members (see Figure 2.1(a)). In this case, the slab flexibility is affected by the approximately equal deflections of the supports. As a result of this, the slab frequency is assessed on the basis that fixed-ended boundary conditions exist.

Primary (main) beam mode The primary beams vibrate about the columns as simply-supported members (see Figure 2.1(b)). Using a similar reasoning as above, due to the approximately equal deflections at their supports, the secondary beams (as well as the slab) are assessed on the basis that fixed-ended boundary conditions exist.

(a) (b)

Figure 2.1 Typical fundamental mode shapes for composite floor systems (a) governed by

secondary beam flexibility (b) governed by primary beam flexibility

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As steel construction is essentially an overlay of one-way spanning elements, the frequency of the whole floor system can be calculated for each mode shape, by summing the deflection calculated from each of the above components, and placing this value within the equation below. The lowest frequency value determined by consideration of these two cases defines the fundamental frequency of the floor f0.

δ18

0 =f

where:

δ is the total deflection (in millimetres) based on the gross second moment of area of the composite beam or slab, with a load corresponding to the self weight, and other permanent loads, plus the variable load multiplied by the frequent variable action factor (it is recommended that, for vibrations, ). 1,01 ≤ψ1ψ

For cases when the floor grid is regular, the value of δ may be determined from Table 2.1.

Table 2.1 Total deflection of a floor panel for a variety of framing arrangements

Framing arrangement

Secondary beam mode of vibration

Condition when mode shape is governed by the motion of the primary beams

Primary beam mode of vibration

b L⎟⎟⎠

⎞⎜⎜⎝

⎛+=

slab

3

b

45384 I

bIL

Ebωδ - -

Lb

As above b3

3

p16 I

LbI ≤

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

slab

3

b

4

p

364384 I

bIL

ILb

Ebωδ

Lb

As above b3

3

p92 I

LbI ≤ ⎟

⎟⎠

⎞⎜⎜⎝

⎛++=

slab

3

b

4

p

3368384 I

bIL

ILb

Ebωδ

Notes: ω is the load per unit area, E is the Young’s modulus for steel, b is the spacing of the secondary beams, L is the span of the secondary beams, Ib and Ip is the dynamic second gross second moment of area of the composite secondary beam and primary beam respectively (which may conservatively taken as that used in the static design and increased by 10%) and Islab is the second moment of area of the composite slab in steel units (which may be determined from Table 2.2).

Table 2.2 Dynamic second moment of area for composite slabs with different deck types

Profile type Deck height, hp Dynamic second moment of area per metre width, Islab

NWC LWC

0.37 h3.7 0.65 h3.5 Re-entrant deck hp = 51 mm 0.23 h3.7 0.40 h3.5 hp = 60 mm Trapezoidal deck 0.19 h3.7 0.37 h3.5 hp = 80 mm Trapezoidal deck 0.05 h3.7 0.12 h3.5 hp = 225 mm Trapezoidal deck

Notes: NWC normal weight concrete, LWC lightweight concrete and h is the overall slab depth.

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3. Modal Mass The effective vibrating mass M may be taken as equal to mSLeff / 4, where m is mass per unit area (kg/m²) of the floor plus any loading that is considered to be permanent. The values of S and Leff should be taken from Table 3.1, where:

RFp is the relative flexibility of the primary beam *S is the effective width of the floor participating in the vibration, calculated from the

effective slab stiffness, given by: 4/1

20

slab* 5,4 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

mfEIS (m)

where is the dynamic flexural rigidity of the slab in Nm² per metre width (e.g. from

slabEITable 2.2, for a 140 mm NWC slab with a 60 mm trapezoidal deck

= 210 × 0,23 × 1403.7 × 10-3 = 4 213 000 Nm²) bEIsla

*L is the effective span of the secondary beam participating in the vibration, calculated from effective composite beam stiffness, given by:

4/1

20

* 8,3 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

mbfEIL b (m)

where is the dynamic flexural rigidity of the composite secondary beam (Nm²) and b is the secondary beam spacing (m)

bEI

W is the width of the floor plate under consideration (m)

is the span of the primary beam (m) mL

is the total length of the secondary beam when considered to act continuously (m) maxL

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Table 3.1 Values for dimensions Leff and S used in determining the effective mass of the floor

Indicative floor layout Qualifying conditions

Leff (m) S (m)

RFp < 0,2 L S* but ≤ W

Case 1

Lm

S*w

L

RFp > 0,2 L Greater of S* or

Lm but ≤ W

L = L 2 L

0,8 L < l < L 1,7 L

Mod

e sh

ape

gove

rned

by

mot

ion

of

seco

ndar

y be

ams

Case 2

wLS*

As for Case (1) above

l < 0,8 L L

RFp < 0,6 2L

Case 3

wL

L

RFp > 0,6 L* but ≤ Lmax

W

W2 = W1 2 W1

W2 > 0,8 W1 1,7 W1

Mod

e sh

ape

gove

rned

by

mot

ion

of

prim

ary

beam

s

Case 4

L

Lw

w

w

1

2

As for Case (3) above

W2 < 0,8 W1 W1

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4. Damping For calculations, assumptions on the magnitude of the damping values have to be made. As damping depends on different boundary conditions, which cannot be defined in general for all buildings, Table 4.1 gives components for damping values for different types of structures, furniture and finishes.

The resulting damping value is the sum of these three components.

Table 4.1 Components of damping

Type Damping (% of critical damping)

Structural damping of bare floors

Wood joist floors 2%

Reinforced concrete, monolithic 1,5%

Prestressed concrete, precast 1,3%

Steel 1,3%

Steel-concrete composite beams with shear connectors 1,8%

Damping due to furniture

Traditional office for 1 to 3 persons with separation walls 2%

Paperless office 0%

Open plan office. 0,5%

Library 1%

Houses 1%

Schools 0%

Gymnastic 0%

Damping due to finishes

Ceiling under the floor 0,5%

Free floating floor 0%

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5. Acceleration from human activities In comparing the predictions calculated in Section 5.2 and Section 5.3 with the acceptable accelerations in Section 6, the predicted accelerations should be converted from peak values to root-mean-square (rms) values. The rms acceleration is defined as:

2/1

0

2rms )(1

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫

T

dttaT

a

where

a(t) is the acceleration time-history

T is the integration time in seconds (taken as 1-second in accordance with ISO 2631-1: 1997)

For continuous steady-state sinusoidal motion, the magnitude of the rms acceleration may simply be taken as the peak acceleration amplitude, apeak / √2 ≈ 0,707 apeak.

Once the predicted rms acceleration has been calculated, the acceptability of a floor may assessed by:

1. comparing the calculated rms acceleration with the appropriate frequency-weighted base curve given in Figure 6.3, Figure 6.4 and Figure 6.5; or

2. filtering, or frequency-weighting, the calculated rms acceleration and comparing this with the base values in Table 6.1.

The appropriate frequency-weighting factors for different vibration directions are as follows (N.B. the base curves given in Figure 6.3 and Figure 6.4 have been derived from dividing the base values in Table 6.1 by the frequency-weighting factors given below):

z-axis vibrations

ofaa 5.0rmsrms ×= for 3 Hz < f0 < 4 Hz Weighted

or Weighted for 4 Hz ≤ f0 ≤ 8 Hz rmsrms aa =

or

0rmsrms

8f

aa ×= for f0 > 8 Hz Weighted

x- and y-axis vibrations

20

rmsrmsfaa ×= for f0 > 3 Hz Weighted

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5.2 Resonant response (low frequency floors) For floor structures where the first Eigenfrequency is between approximately 3 to 10 Hz, the floor is described as ‘low frequency’. In this case, harmonic components of the walking force can coincide with the floor frequency causing resonant excitation. By assuming that the response is sinusoidal, the peak acceleration is given by:

ζα

210

peak MPa n=

where

αn is the Fourier coefficient of the nth harmonic. Owing to the fact that the largest accelerations are generated when the first Eigenfrequency of a floor is an integer multiple (harmonic) of the pacing frequency, the appropriate Fourier coefficient may be taken from in Table 5.1 by equating the first Eigenfrequency of the floor to the common range of the forcing frequency (for walking activities, it may be assumed that the floor is effectively tuned out of the range of the first harmonic component provided that the first Eigenfrequency of the floor, f0 > 3,55 Hz).

P0 is the static force exerted by an ‘average person’ (normally taken as 76 kg × 9,81 = 746N)

M is the modal mass (kg) taken from Section 3.

ζ is the damping ratio, which may be taken from Table 4.1.

The forces produced by a person walking or running depend mainly on the physique of the person and on the rate of walking or running, and to a lesser degree on the type of shoes and floor construction. Examples of the Fourier coefficients, αn for a continuous series of steps for various ranges of walking or running rates are given in Table 5.1.

Table 5.1 Examples of design parameters for moving forces due to one person

Activity Harmonic number,

n

Common range of forcing frequency,

nfp (Hz)

Fourier coefficient for vertical direction

αn

Walking 1 1,5 to 2,5 0,37 (f – 1,0) 2 3,0 to 5,0 0,1 3 4,5 to 7,5 0,06 4 6,0 to 10,0 0,06

Running 1 2,0 to 4,0 1,4 2 4,0 to 8,0 0,4 3 6,0 to 12,0 0,1

5.3 Impulsive response (high frequency floors) For floor structures where the first Eigenfrequency is greater than approximately 10 Hz, the floor is described as ‘high frequency’. In this case, the response is dominated by a train of impulses, which correspond to the heel impacts made by the walker. The peak acceleration on floors of this type may be calculated from:

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MIfa 0peak 2π=

where:

I is the impulsive force from the person in Newton-seconds (Ns).

From Young[8], it is recommended that the following equation may be used to establish the effective impulsive force from walking activities:

3.1m

43.1p

006.0ff

PI =

where:

fp is the pace frequency (taken as 1,5 to 2,5 Hz)

fm is the Eigenfrequency of the first mode of vibration

P0 is the static force exerted by an ‘average person’ (normally taken as 76 kg × 9,81 m/s² = 746 N).

6. Acceptable accelerations NOTE: The limits given below may be used where the National Annex does not give specific limits or where it gives limits different from those derived from ISO 10137.

When vibration serviceability is assessed the usual variable to measure is acceleration. However, acceptable vibration levels vary with the frequency of the motion; therefore, it is necessary to filter the acceleration. The appropriate filters or frequency weighting parameters are given in ISO 2631-1 for situations where the critical vibration direction is specified or ISO 2631-2: 2003 if the critical direction is unknown. ISO 2631-1 also describes how suitable values of rms. acceleration can be determined from the filtered acceleration. If the ratio of the peak value to the rms value of the filtered acceleration (taken for the full period of vibration exposure which it is desired to assess) is greater than 6, the rms values may not be appropriate; in this case, Vibration Dose Values (VDVs), which are based upon a root-mean-quad (rmq) evaluation, can be used. The appropriate VDV values should then be compared with the acceptance criteria. See ISO 2631-1:1997 for definitions of rms and VDV, ISO 2631-2:1989 (Annex B) for definition of rm. q

The recommended acceptable filtered accelerations in buildings are determined from a multiple of the base rms acceleration levels. The base levels for rms acceleration are given in Table 6.1.

Table 6.1 Base rms acceleration for various directions

Direction (Figure 6.2) Base rms acceleration (m/s²)

5 × 10-3 z-axis (foot-to-head direction)

x-axis (side-to-side direction) 3,6 × 10-3

y-axis (back-to-chest direction)

Notes: These accelerations correspond to the lowest values in Figure 6.3, Figure 6.4 and Figure 6.5.

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y

z

x

y

x

x

z

y

z

1 1

1

Key: 1. Supporting surface

Figure 6.2 Directions of basicentric coordinate systems for vibrations influencing humans

6.2 Continuous vibrations The multiplication factors for rms acceleration are given in Table 6.2. For VDVs the multiplication factors for continuous vibration in Table 6.2 are also appropriate.

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Table 6.2 Multiplying factors used in several countries to specify satisfactory magnitudes of building vibration with respect to human response

Place Time Multiplying factors to base curve (Figure 6.3, Figure 6.4 and Figure 6.5)1)

Continuous vibration and intermittent

vibration2)

Impulsive vibration excitation with several occurrences per day

Critical working areas (eg hospital operating theatres, precision laboratories, etc.)

Day 1 1 13) Night 1

2 to 44) 30 to 904) Residential (eg flats, homes, ward areas in hospitals)

Day Night 1.4 1,4 to 20

Quiet office, open plan Day 2 60 to 128 Night 2 60 to 128

General offices (eg offices, schools, laboratories)

Day 4 60 to 128 Night 4 60 to 128

Busy offices and workshops Day 8 90 to 128 Night 8 90 to 128

Notes: 1) These factors lead to magnitudes of vibration below which the probability of adverse comment is low (any acoustic noise caused by structural vibration is not considered). 2) Doubling of the suggested vibration magnitudes may result in adverse comment and this may increase significantly if the magnitudes are quadrupled (where available, dose/response curves may be consulted). "Continuous vibrations" are those with duration of more than 30 min per 24 h; "intermittent vibrations" are those of more than 10 events per 24 h. 3) Magnitudes of impulsive shock in hospital operating-theatres and critical working places pertain to periods of time when operations are in progress or critical work is being performed. At other times, magnitudes as high as those for residences are satisfactory provided there is due agreement and warning. 4) Within residential areas, people exhibit wide variations of vibration tolerance. Specific values are dependent upon social and cultural factors, psychological attitudes and expected degree of intrusion.

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0.001

0.010

0.100

1.000

1.0 10.0 100.0

Frequency (Hz)

rms

acce

lera

tion

(m/s

²)

Figure 6.3 Building vibration z-axis base curve for acceleration (foot-to-head vibration direction) [Taken from ISO 10137]

0.001

0.010

0.100

1.000

1 10 100

Frequency (Hz)

rms

acce

lera

tion

(m/s

²)

Figure 6.4 Building vibration x- and y-axis base curve for acceleration (side-to-side and back-to-chest vibration direction) [Taken from ISO 10137]

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0.001

0.010

0.100

1.000

1 10 100

Frequency (Hz)

rms

acce

lera

tion

(m/s

²)

Figure 6.5 Building vibration combined direction (x-, y-, z- axis) acceleration base curve

[Taken from ISO 10137]

6.3 Vibration Dose Values The multiplying factors presented in Table 6.2 are based on continuous vibrations, and are therefore appropriate for floors that are very heavily trafficked with walkers continually present. For less heavily trafficked floors, walking activities will produce intermittent vibrations, and a cumulative measure of the floor response may be made through the use of vibration dose values (VDVs). In these circumstances it can sometimes be shown that the floor would be acceptable, even when the calculated multiplying factor is greater than the values given in Table 6.2. (N.B. it is not considered appropriate to use a dose value assessment on sensitive floors, such as within an operating theatre).

From a recent study of steel-framed floors, Ellis showed that VDVs can be estimated from the following equation:

25.0rms68,0 taVDV ××=

where:

arms is the weighted root-mean-square (rms) acceleration

t is the total duration of the vibration exposure (in seconds).

Should the designer wish to undertake a VDV assessment, the calculated dose values should be less than or equal to the values presented in Table 6.3, which correspond to a ‘low probability of adverse comment’.

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Table 6.3 Vibration dose values below which there is a low probability of adverse comment

Vibration dose value (m/s1.75) Place Time

z-axis x- & y-axis

Residential Day 0,2 to 0.4 0,14 to 0,28 Night 0,13 0,09

Quiet office, open plan 0,2 0,14

General offices 0,4 0,28

Busy offices and workshops 0,8 0,56

For cases when the designer wishes to take advantage of the fact that floor vibrations occasioned by walking activities are intermittent, the graphs shown in Figure 6.6 may be used. In Figure 6.6 the time taken for a walker to walk from one end of a corridor to another is maximized by considering the slowest pace frequency that may be practically achieved (1,5 Hz). Based on multiplying factors that exceed the requirements for continuous vibrations in Table 6.2, the number of crossings per hour have been calculated which, according to Table 6.3, would provide VDVs consistent with a low probability of adverse comment.

1

2

3

4

5

6

7

8

1 100 10000 1000000Crossings per hour

Mul

tiply

ing

fact

or

Corridor length= 5 m




1

2

3

4

5

6

7

8

10 100 1000 10000Crossings per hour

Mul

tiply

ing

fact

or





(a) (b)

Figure 6.6 Maximum number of walking crossings per hour for various multiplying factors and

corridor lengths for: (a) z-axis vibrations in office, residential and general laboratory environments during a 16-hour day (VDV = 0,4 m/s1.75); and (b) x- and y-axis vibrations in residential and hospital ward environments during an 8-hour night (VDV = 0,09 m/s1.75)

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7. References 1 Ellis, B.R. Serviceability evaluation of floor vibration induced by walking loads. The

Structural Engineer, 79(21), 2001, pp 30-36

2 EN 1990, Basis of structural design, §5.1.3, Structural analysis and design assisted by testing, Dynamic actions

3 EN 1990, Basis of structural design, Annex 1.4.4, Application for Buildings, Vibrations

4 EN 1993-1-1, Design of steel structures – General rules and rules for buildings, §7.2.3, Serviceability limit states for buildings, Dynamic effects

5 ISO 2631-1: Mechanical vibration and shock – Evaluation of human exposure to whole-body vibration: Part 1: General requirements, 1997 International Organisation for Standardization, Geneva.

6 ISO 2631-2: Mechanical vibration and shock – Evaluation of human exposure to whole-body vibration: Part 2: Vibration in buildings (1 Hz to 80 Hz), 2003 International Organisation for Standardization, Geneva.

7 ISO/CD 10137, Bases for design of structures - Serviceability of buildings and pedestrian walkways against vibration

8 Young, P.: ‘Improved floor vibration prediction methodologies’ Engineering for Structural Vibration – Current developments in research and practice, Arup Vibration Seminar, Institution of Mechanical Engineers, 2001

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Quality Record RESOURCE TITLE NCCI: Vibrations

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ORIGINAL DOCUMENT

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Created by Stephen Hicks SCI 28/04/2006

Technical content checked by P Devine SCI

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Technical content endorsed by the following STEEL Partners:

1. UK G W Owens SCI 10/7/06

2. France A Bureau CTICM 10/7/06

3. Sweden B Uppfeldt SBI 10/7/06

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5. Spain J Chica Labein 10/7/06

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