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A Steel Construction Institute Publication 4—

inassociationwiththe 4

Construction Industry 1 n Association 11 1 _________

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14 ___________________________ 4 1 4

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4 __________________________________________________________________ - .-.:..tz;1 . • ___________________________________________

1.1 1411 4,1 4:

(7is document

containS 50 pae) —

is The Steel Construction Institute. Its aim is to promote the proper and effective use of steel in construction.

Membership is open to all organisations and individuals that are concerned with the use of steel in construction, and members include clients, designers, contractors, suppliers, fabricators, academics and government departments. SC! is financed by subscriptions from its members, revenue from research contracts, consultancy services and by the sales of publications.

SCI's work is initiated and guided through the involvement of its members on advisory groups and technical committees. A comprehensive advisory and consultancy service is available to members on the use of steel in construction.

SCI's research and development activities cover many aspects of steel construction including multi-storey construction, industrial buildings, use of steel in housing, development of design guidance on the use of stainless steel, behaviour of steel in fire, fire engineering, use of steel in barrage schemes, bridge engineering, offshore engineering, development of structural analysis systems and the use of CAD/CAE.

Further information is given in the SCI prospectus available free on request from: The Membership Secretary, The Steel Construction Institute, Silwoód Park, Ascot, Berkshire SL5 7QN. Telephone: (0990) 23345, Fax: (0990) 22944, Telex: 846843.

Although care has been taken to ensure, to the best of our knowledge, that all data and information contained herein are accurate to the extent that they relate to either matters of fact or accepted practice or matters of opinion at the time of publication, the Steel Construction Institute assumes no responsibility for any errors in or misinterpretations of such data and/or information or any loss or damage arising from or related to their use.

© The Steel Construction Institute 1989

Design Guide on the Vibration of Floors

T. A. Wyatt BSc PhD FEng FICE

ISBN: 1 870004 34 5 © The Steel Construction Institute 1989

SCI PUBLICATION 076

The Steel Construction Institute Silwood Park Ascot Berkshire SL5 7QN Telephone 0990 23345 Fax 0990 22944 Telex 846843

Construction Industry Research and Information Association

6 Storey's Gate London SW1 P 3AU Telephone 01-222 8891 Fax 01-222 1708 Telex 24224 Mon Ref G (prefix 2063)

FOREWORD

This publication is intended to provide guidance for designers in an important area of design where information is lacking. It has been prepared by Dr T A Wyatt of Imperial College with assistance from Dr A F Dier of the Steel Construction Institute.

The Guide was drafted in conjunction with the support of a steering committee which commented on and otherwise advised on the draft versions. The members of the steering committee comprised:

Mr B Boys British Steel Structural Advisory Service Mr R Clark Skidmore Owings & Merrill Mr E Dibb-Fuller Building Design Partnersh'ip Mr E Dore CIRIA Mr K Irish Vibronoise Limited Mr R Povey Mitchell McFarlane & Partners Mr M Willford Ove Arup & Partners.

The work leading to this publication has been funded by British Steel General Steels, and the Department of the Environment under a CIRIA research project. Studies are continuing and future editions of the publication will be amended as necessary to account for new results. The Steel Construction Institute will be pleased to receive any comments concerning this publication and subject area.

How to Use this Guide The Guide is divided into seven Sections and two Appendices as shown on the facing page. Section 1 is intended as a broad introduction and has been written in such a way that it is suitable for copying to a Client as an aid to preliminary discussions. The background to the design procedures, which are set out in Section 7, is given in Sections 2 to 6 and a study of these will be an aid, although not normally necessary, in the application of Section 7. The design procedures of Section 7 are self contained as far as is practical, although in some cases reference to Section 5.2 may be required. The examples in

Appendix B will be useful for following the design procedures. Explanation of terms used for describing dynamic behaviour, which may not be familiar to the non-specialist, will be found in Section 4.2 where they are highlighted by italic script. Defmitions essential for the application of the design procedures are given in Section 7.1.

11

CONTENTS

Page

SUMMARY iv

NOTATION iv

1. INTRODUCTION 1

2. SOURCES OF VIBRATION EXCITATION IN BUILDINGS 3

3. HUMAN REACTION TO VIBRATiON 6 3.1 Review of Factors 6 3.2 Specifications 7

4. GENERAL CONSIDERATIONS 10

4.1 Structural and Floor Configurations 10

4.2 Introduction to Dynamics 11

5. EVALUATION OF NATURAL FREQUENCY 16

5.1 Component and System Frequencies 16

5.2 Practical Evaluation 17

6. FLOOR RESPONSE 20 6.1 Low Frequency Floors 20 6.2 High Frequency Floors 21

7. DESIGN PROCEDURES 25

7.1 Definitions 25

7.2 General Considerations 25 7.3 Procedure for Checking Floor Susceptibility 26 7.4 Natural Frequency 26 7.5 Floors of High Natural Frequency 27 7.6 Floors of Low Natural Frequency 28 7.7 Acceptance Criteria 30

REFERENCES 31

APPENDIX A: CALIBRATION STUDY 32

APPENDIX B: DESIGN EXAMPLES 33

111

SUMMARY This publication presents guidance for the design of floors in steel framed structures against unacceptable vibrations caused by pedestrian traffic. It has particular relevance to composite floors comprising permanent metal decking topped with concrete. As well as the design procedures set out in Section 7, the Guide contains background commentary and a general, non-technical, introduction.

Notation a acceleration amplitude

acceleration response (Canadian Code) b floor beam spacing

effective width between floor beams B parameter for effective width (Canadian Code) GB factor for determining natural frequency C Fourier component factor C, effective mass and lateral distribution factor for impulsive loading C effective mass and lateral distribution factor for sustained vibration El flexural rigidity (of composite section where appropriate) f natural frequency f0 fundamental system frequency f1, J, f3 idealised component natural frequencies of f0

g acceleration due to gravity J impulse (= force x time) k stiffness

1, L floor beam span lengths L length of span Leff length for establishing effective mass

Lm main beam span rn distributed mass rn lumped mass at mesh point 'i' M effective modal mass P static load P force amplitude P1 amplitude of fundamental Fourier component of walking force P weight of oscillating mass

distributed loading R multiplication factor applied to human reaction base curve S width for establishing effective mass

S weighting factor t time

smeared concrete thickness

5 deflection amplitude

y deflection at mesh point 'i' y° maximum value of self-weight deflection

y weighted average of self-weight deflection

W floor bay width critical damping ratio

iv

1. INTRODUCTION

The main purpose of this Guide is to provide a practical method for assessing the likely vibrational behaviour of floors in steel framed buildings. The subject of floor vibration is complex and consequently the Guide contains sections dealing with the current 'state of the art', the background to the proposed assessment methods and a commentary so that the designer may develop an appreciation of the phenomenon rather than apply the design method by rote.

Notwithstanding this intention, the design procedure set out in Section 7 and the worked

examples contained in Appendix B have been prepared to permit a conservative design assessment to be executed by those with only a limited knowledge of structural dynamics.

Floor vibration is not a new phenomenon, the 'live' feel of timber floors under pedestrian loading is well established. However, because of the increasing trend towards lighter longer span floors in all forms of construction, but most notably in steelwork, CIRIA and SCI considered it an opportune time to provide interim guidance on this aspect of design pending further research. This Guide has not therefore been prepared in response to any existing problems but rather it is intended that its use will prevent such problems occurring in the future. Vibration in forms of construction other than steelwork may also require consideration.

The use of structural steelwork for multi-storey construction has increased dramatically over the past ten years. Such increase is largely due to the response of the building industry to Clients' demands for buildings that are fast to construct, have large uninterrupted floor areas and are capable of accommodating highly sophisticated air conditioning and other services systems. Modern design and construction techniques enable the industry to satisfy such demands and produce steel framed structures which are competitive in terms of overall cost. This trend towards longer span lightweight floor systems in both steelwork and other forms of construction, with their tendency to lower natural frequencies and less effective natural damping, has created a greater awareness of the dynamic nature of some types of superimposed loadings. Currently the most popular form of floor construction used in conjunction with multi-storey steel frames is the 'composite floor'. This form of floor slab comprises profiled metal decking spanning between beams and topped with insitu concrete. Much of the design guidance given in this publication is directly related to this form of construction.

The vibration of floors can arise from external sources such as road and rail traffic. Where such problems are anticipated, however, it is preferable to isolate the building as a whole. This aspect of vibration control is not taken further in this Guide, which addresses floor vibrations caused by internal sources.

The most usual and important internal source of dynamic excitation is pedestrian traffic. A person walking at a regular pace applies a periodically repeated force to the floor which may cause a build up of response in the structural floor. Other sources of internal excitation such as vigorous rhythmic group activities are not specifically covered in this Guide. However, where such activities are envisaged a robust structure should be provided which has adequate ductility, and special attention should be paid to the beam/column connections. These design features are similar to those considered when preparing good seismic-resistant designs and it is to publications dealing with this subject that the designer's attention is directed.

Human perception of vibration is in one sense very sensitive; the criterion is likely to be set at a low level. In another sense it is very insensitive; a substantial quantitative change in the amplitude of vibration corresponds to a relatively small qualitative change in perception. If a person is asked to express an opinion on his perception of vibration in two different rooms on separate occasions, he will not draw a distinction unless the quantitative difference is at least a factor of 2. There are also substantial differences between persons and there may also be differences between nationalities. Human reaction at these levels is substantially psychological, depending partly on the delicacy of the

1

activity being performed. Response to vibrations is often affected by other stimuli (sight and sound). Although floor vibration may induce a sense of insecurity in some people, it must be stressed that perception of floor vibration does not imply any lack of structural safety. Once constructed, it is very difficult to modify an existing floor to reduce its susceptibility to vibration, since only major changes to the mass, stiffness or damping of the floor system will produce any perceptible reduction in vibration by people regularly trafficking the floor.

It is therefore important that the levels of acceptable vibration be established at the conceptual stage having regard to the anticipated usage of the floors. The Client must be involved in this decision, since the selected design target level for vibrational response will usually have a significant bearing on both the cost and overall floor construction depth for the project. The question is frequently raised of the tolerance of modem computer equipment to ambient vibration. The steering group for this study has been unable to fmd any firm evidence of actual problems resulting from floor vibration. Manufacturers commonly state that their equipment is tolerant of the levels of vibrations acceptable in a good office environment. Consultation with a prominent manufacturer has confirmed that vibrations within the range tolerable for human occupancy would cause no problem to computer equipment. In conclusion, therefore, it is intended that the publication of this Guide will aid both designers and Clients in setting sensible targets for acceptable levels of vibration which can then be incorporated into the design of the floor structure to produce economic, usage-related, buildings.

2

2. SOURCES OF VIBRATION EXCITATION IN BUILDINGS

There are a number of distinct possible causes of dynamic excitation of floors. The important characteristics of these excitations vary to the extent that quite different check procedures may be appropriate depending on which potential cause is most important. The obvious, almost universal, excitation is the effect of walking on the floor. The geometry of the human body walking is (to a first approximation) a straight-leg motion that necessarily causes the main body mass to rise and fall with every pace (see Figure 2.1). This rise and fall is typically about 50 mm, peak to peak, but is sensitive to the angle of the leg at full stretch, and thus to the extent to which the walker is forcing the pace. One is not aware of this movement, because the brain identifies the resulting acceleration signals as correlated with walking and disregards them; it is, however, interesting to note that these accelerations are around 3 m/s2, which is roughly 30 times the value that would be acceptable as the resonant response of a floor, and 100 times the value that would commonly be set as a limit to sustained vibrations. The annoyance caused by floor vibrations is essentially psychological, and is very susceptible to expectation or familiarity; it is none the less a real problem.

Direction of walk

Rise and fall of

-——f main body mass

________ Legs at mid-stride (broken lines)

The vertical accelerations of the body mass are necessarily associated with reactions on the floor, and they will be closely periodic, at the pace frequency. The fluctuation can be resolved as a series of sinusoidal components (i.e. a Fourier series) and it is found that the fundamental term agrees fairly well with the simple visualisation of Figure 2.1, giving a force amplitude between 100 N and 300 N. Walking pace frequency can vary between 1.4 Hz and 2.5 Hz, and the force amplitude tends to increase rather severely with increasing frequency. However, walking pace indoors is most commonly towards the lower end of this range, around 1.6 Hz. The British Standard for bridges" suggests 180 N force amplitude for checking footbridge designs2.

A typical example of the contact force from a single footfall is shown as the light solid curve in Figure 2.2(a). Unless the floor structure is exceptionally sensitive to the precise location of the load (i.e. if one pace-length makes a major difference), the dynamic excitation is given by the sum of the concurrent walker's foot forces, which takes the form shown as the heavy solid curve in Figure 2.2(a). The basic pace frequency is clearly represented but the second Fourier component, representing excitation at twice the pace frequency, is also important. The third component is smaller, and succeeding components can generally be ignored, except that there is a significant impulsive effect of very short duration as the foot contacts the ground. The first three Fourier components are shown in Figure 2.2(b), and the degree of approximation given by the summation of these three components is indicated on Figure 2.2(a). This example is taken from the work of Ohlsson3.

The magnitude of the second Fourier component varies with the walking pace in a similar way to the basic component. Unfortunately, however, these higher frequency effects, especially the contact impulse, vary considerably between persons. The average values of

3

Legs at point of footfall (Solid lines)

'',, ;''-.' "; / Figure 2.1 Simplified geometry of walking

z C.)

0 '300 E C

200

100

0

—100

-200

N

300

200

100

0

—100

-200

(b) Fourier components of reaction on floor

Figure 2.2 Typical walking excitation

the Fourier coefficients reported by Rainer, Pernica and len4 from a Canadian study directed to footbridge loading are shown in Figure 2.3. The contact impulse is typically about 3 Ns (Newton seconds). It is, of course, possible for more than one person to walk in unison, but such augmented excitation is not normally regarded as sufficiently common to be taken as the design check case against comfort criteria.

Much larger impulsive loading can arise in the so-called 'heel drop'. A person standing on tip-toe who returns heavily onto his heels can deliver an impulse of typically 70 Ns, within a duration of some 0.04 s. Although such action can occur in an office or residence, for example when reaching for something on a high shelf, it is probably of greater significance as a standard design-check (or practical measurement) input5, which will give useful guidance on sensitivity to impulsive loadings from any cause, including walking.

4

One pace, period 0.6 s (a) Footfall force and reaction on floor

Amplitude (N) ,0 / \ 10

o

N (presuming body 0.6 400 mass is 67 kg)

/1

0:. / 2

Frequency (Hz)

Figure 2.3 Fourier component amplitudes for regular walking

Running-step frequencies can rise to higher values, but do not commonly exceed 3 Hz. The fundamental Fourier component of the force exerted on the floor is of the order of the body weight (i.e. perhaps three times the corresponding component in walking), with a period of zero force while both feet are off the ground. The 'free flight' phase of body motion becomes even more important when rhythmical activities, such as dancing or aerobic exercises, are considered. The body leaving the ground, with no way of accelerating the return to keep up with the 'beat', imposes a clear upper bound on the combination of impulse and frequency that can be developed6, and for this reason the frequency will not significantly exceed the value quoted for running. Unfortunately, however, such activities clearly offer the likelihood of a large number of persons acting in unison, and the structural effects are potentially severe. Useful quantitative guidance can be found in the National Building Code of Canada. Mechanical excitation is also possible. The classic example is out-of-balance rotating machinery. There is little to be said about such excitation; it is generally strongly preferable to tackle such problems at source rather than in the structure, by reduction of the out-of-balance or by vibration-isolation mountings for the machine.

Impulsive or transient mechanical excitation is more commonly external to the building, possible causes being road or rail traffic, or (in special cases) heavy machinery or use of explosives. Where this effect is likely to be severe, vibration isolation at building foundation level is generally preferable to using control measures at specific floors, especially because user reaction would be dependent on the interaction of vibration (including high frequencies) and acoustic effects. The same comment that the solution does not really lie in the hands of the floor designer applies to the occasional within-building impulsive mechanical loads, such as problems arising from operation of the lifts. In this preliminary survey it is also pertinent to point out that similar problems can arise from vehicle movement in car-parking areas within a building, and again the preferable remedy is to tackle the problem at source by providing a smooth running surface.

5

3. HUMAN REACTION TO VIBRATION

3.1 Review of Factors Given large amplitudes of oscillation at frequencies in the range 2 Hz to 20 Hz there may be significant strains within the human body, possibly including resonance of specific organs, giving rise to acute discomfort, serious impairment of ability to perform mechanical tasks, and even injury. These problems have been studied extensively in relation to tasks involved in national defence, such as piloting high-performance aircraft, and also for the establishment of criteria for working conditions in onerous industrial situations. It is immediately clear that there is a very wide range between the amplitudes of motion associated with such criteria and the threshold of perception; this range is typically one hundred times the threshold. The criteria appropriate to residential or office environments are associated with intermediate levels of vibration at which purely physiological effects take second place to psychological factors.

The importance of psychological factors makes it difficult to quantify human reaction at these levels. Any experiment in which the subjects are aware that their reaction is under test is clearly subject to doubt. There are also wide variations between individuals, a range of amplitude exceeding a factor of 2 exists between the top and bottom 5% of the population for any given reaction.

Reaction at these levels may be influenced by a number of factors. At the lower end of the frequency range, reaction is strongly linked to a feeling of insecurity, based on instinctive association of perceptible motion in a 'solid' building structure with an expectation of structural inadequacy or failure. At the higher end of the frequency range, reaction is strongly linked to associated noise levels. Ohlsson3 has reported a case study in which office workers had mutually agreed that hard shoes would not be worn, and found this highly beneficial. Measurement showed that the difference in vibration was quite insufficient to account for the difference in reaction, which was attributed to the elimination of noise that the occupant would associate with vibration. The floor in question falls seriously short of the acceptance criteria put forward in this Guide.

Because of the wide range to be covered, it is usual to plot contours indicating human reaction on twin logarithmic scales of frequency and amplitude of response; the response can be expressed in terms of either displacement, velocity or acceleration. If amplitude of acceleration is taken as the ordinate, a constant value of displacement plots as a straight line of slope +2. A line of slope —1 corresponds to a constant value of the rate of change of acceleration. It is rational to assume that human reaction would be related to the former at very high frequencies, since the body mass will not follow the floor motion and the perception will be of strain in the legs and spine. At the other extreme of very low frequency, human reaction would be related to the rate of adjustment of the inertia forces on the body, and thus reaction contours should plot to the slope of—i. It is therefore apparent that the contours will have a trough shape.

The most important range of floor frequencies covers the band where the reaction contours are changing from slope zero (acceleration criterion) to slope + 1 (velocity criterion). Typical broad qualitative contours of reaction to sustained uniform vibration are shown in Figure 3.1. A margin of at least a factor of 2 is required before an observer would change his qualitative description of reaction, in addition to the variability between observers.

It is even more difficult to extend the criteria to non-steady vibrations. For continuous random oscillation (i.e. a continuously modulated harmonic motion) it is usual to quote criteria in terms of the root-mean-square value of the motion. It is not clear, however, how far this is a uniform criterion over different rates of modulation, or over oscillations in bursts that are separated by intervals of quiescence. It is certainly not a good criterion for occasional occurrences of oscillation, especially where the oscillation is initiated sharply and damped out rapidly. The rapidity of decay is widely recognised as having a major effect; doubling the effective decay rate may raise the level of a given reaction contour (based on the peak oscillation amplitude) by a factor of three.

6

10 // Quickly tiring /

1.0 strongly perceptible —

tiring over long periods

C Clearly perceptible —

disacting 0.1

Perceptible

0.01 .:

Barely perceptible/

Frequency (Hz) (log scale)

Figure 3.1 Qualitative description of human reaction to sustained steady oscillation

It has been suggested above that noise directly associated with the oscillation is an adverse factor. However, for high-quality environments (residential or office) where an occupant will resent intrusion on his mental concentration, it may be that the appropriate vibration limit would actually be higher where there is substantial ambient noise from other causes.

3.2 Specifications As noted above, studies of human reaction have tended to focus on relatively severe circumstances, and this is reflected in the balance of published specifications. For example, several specifications can be consulted about severe industrial working conditions, but there is very little available with a track record of satisfactory application to assessment of floors in office or residential accommodation.

The Canadian Specification CAN3—S 16.1 Steel Structures for Buildings8 does, however, include a very useful Appendix entitled 'Guide for floor vibrations', although this is not a mandatory part of the Code. The proposed annoyance criteria for floor vibrations are shown in Figure 3.2. The labelling of these curves need interpretation: the curves labelled 'walking vibration' are to be used for assessing the response to heel drop impulse, and the curve labelled 'continuous vibration' is to be used for the assessment of the motion caused by a person walking across the floor. For example, in the latter case, a floor of span 14 m and frequency 6 Hz crossed by a person walking at 2 paces per second (so that there was significant response to the third harmonic in the pace excitation) would show sustained response over about ten paces or 30 cycles. The interpretation of 'average peak' in such a case is left open; the average over the worst 20 cycles might be reasonable.

The three curves in Figure 3.2 labelled 'walking vibration' are specifically linked in the Canadian Code8 with the 'heel drop' impact test. The Canadian Specification suggests 6% of critical damping for typically-furnished floors without partitions. The sensitivity to the level of damping reflects the greatly reduced annoyance caused by an impulsive event when the subsequent decay is very rapid. Unfortunately, it also reflects the application of

7

100 : I 1 I

50 / , - • , -

," Criteria for waiking 20 — Walking vibration , — vibrations:

— — — acceleration determined (12% damping) ,- by heel impact test

10

Walking vibration

c_ • (6% damping)

Co - , -

2 — Walking vibration / —

— Criterionfor

1.0 (3% damping) continuous vibration

0 Continuous vibration

a0

_I_t,,_r • (10 to 30 cycles)

0.1 I 11111 I

1 2 4 6 10 20

Frequency (Hz)

Figure 3.2 Annoyance criteria for floor vibrations (residential, school and office occupancies)

this test to assessing the sensitivity of the floor to walking excitation, where damping has a different action. In this case higher damping primarily causes a reduction of the dynamic magnifier at resonance. The more rapid decay once the source of excitation has moved off the span is only of secondary significance. As noted later in Section 4.2, the effective decay rate from the impulsive event is very commonly enhanced by a lateral dispersion of the energy of oscillation. This may legitimately be included in the effective damping value for identifying the acceptable level of initial response to impulsive excitation, and is presumably so included in the Canadian Specification. The energy dispersion effect is not equally effective under repeated-pace excitation. Care is therefore recommended in the use of these curves. Impulse response criteria which give similar values have also been presented by Murray; some discussion of his proposals is given in Section 6.2

The Supplement to the National Building Code of Canada° postulates limits for human tolerance in cases of group activities, namely an acceleration amplitude of O.02g for dancing and dining, or O.05g for lively concert or sports events. For these activities, the check is applied to the consideration of the fundamental-frequency excitation component only. The response considered is thus at frequencies up to 3 Hz, and floor resonance to high frequency components is not taken into account. A second-component excitation, thus giving an excitation frequency up to 6 Hz, is given for 'jumping exercises'. The most relevant United Kingdom specification is BS 6472 Evaluation of human exposure to vibration in buildings (1 Hz to 80 Hz)9. This is strongly linked to the International Standard ISO 2631 Guide to the evaluation of human exposure to whole body vibration°>, which is in turn to some extent a descendant of German specifications originally drawn up for industrial working conditions. However, it incorporates a substantial recent review in the broader context, including the work of Irwin". BS 6472 defmes a base curve of acceleration as a function of frequency, with multipliers to define the acceptable level as a function of building function and the nature of the excitation. The base curve is identical in shape to the lines of Figure 3.2 (for frequencies exceeding 4 Hz), with numerical values one-tenth of the Canadian curve for sustained oscillation. However, the measure used in BS 6472 is the root-mean-square (r.m.s) value of the

8

acceleration, rather than the peak (or 'average peak'). For a response which is dominated by a single hannonic excitation component the r.m.s. value is l// times the peak, and the Canadian curve is thus equivalent in this case to 7 units (or 'Curve 7' in the notation of BS 6472) according to the British Standard.

BS 6472 gives (inter alia) values for the multiplying factor to apply to the base curve for the assessment of continuous vibration, as shown in Table 3.1.

Table 3.1 Multiplying factors to apply to the base curve

Environment Reaction level A* Reaction level B

Offices 4 8 Residential — day 2 to 4 4 to 8 Residential — night 1.4 3

* See text for explanation of 'reaction level'

The values in column A are postulated as 'magnitudes below which the probability of adverse comment is low', and it is postulated that the values in column B 'may result in adverse comment'. A note is added to the effect that tolerance in residential accommodation is strongly influenced by 'social and cultural factors, psychological attitudes and the expected degree of intrusion'.

It will be seen that the levels B and A for offices correspond roughly to the Canadian recommendation (Figure 3.2), and to one-half that level, respectively. However, there is a strong implication that the term 'continuous vibration' is to be interpreted rigorously in BS 6472. These values are thus reasonably applicable only to very heavily trafficked floors with walkers continually present. In such cases occasional peaks due to concurrent excitation by more than one person can probably be traded off against the number of people not moving regularly or at resonant-pace frequency. BS 6472 offers the suggestion that intermittent vibration can be equated to an equivalent continuous level by the root-mean-quad, i.e.:

T 114

aeq = (J a4(t)&)

where a(t) is the value of acceleration at time t.

The root-mean-quad of a sinusoidal vibration modulating as a person walks across a floor taking six seconds, repeated once per minute, is about one-third of the peak amplitude. As this root-mean-quad is used in substitution for the root-mean-square value of continuous oscillation, which would be 1 times the peak amplitude, a floor subject to a person walking at the resonant frequency once per minute could reasonably be permitted to show peak response of twice the peak value acceptable for continuous oscillation.

BS 6472 notes that there may be locations where it is necessary to restrict vibrations to the level of the base curve (factor 1). 'Some hospital operating theatres' and 'some precision laboratories' are put forward as examples.

9

4. GENERAL CONSIDERATIONS

4.1 Structural and Floor Configurations The following discussion of steel flooring configurations is presented to indicate the terminology used in discussion of floor vibrations and the approximate parameter ranges; it is not intended to constitute guidance on the selection of the parameters. The essential objective of flooring is to provide a flat load-carrying surface. The floor slab construction is generally either steel-concrete composite, timber or concrete, and usually carries some form of fmishing or furnishing (carpeting and underlays, hardwood surfacing or similar, and, in the case of concrete slabs, a screed). There is little evidence that finishes have much effect on vibration problems, except through the resulting increase of mass. There is possibly a marginal increase in damping and a marginal cushioning of impulsive loads by appropriate fmishes, but a finish soft enough to have a marked cushioning action will be too soft to have much structural damping action. However, the acoustic and walking comfort factors of various fmishes are likely to interact in the expressed opinion of users relating to the vibration environment as discussed in Section 3.1.

Timber floors are certainly susceptible to vibration problems, which have been studied in both Canada8 and Swederi3. It will be shown that higher mass is generally favourable, and in this respect timber floors are inherently more at risk than concrete floors. Nevertheless, in view of the current balance of the market in the U.K., attention will be focused in this Guide on concrete floors, but with emphasis on recent design trends leading to a reduction of the mass per unit area. In particular, there is increasing use of permanent steel formwork (profiled decking of various configurations) and of lightweight concrete, often in conjunction with each other.

The density of lightweight concrete commonly adopted in the U.K. is around 1800 kg/m3; lower values are not uncommon in North America. A composite slab comprising a 70 mm continuous thickness of lightweight concrete on 60 mm steel decking may thus have a mass of about 220 kg/m2, excluding finishes. It may be noted here that references to floor thicknesses in the U.K. generally refer to the total slab depth; a 'smeared' thickness equal to (mass of concrete per unit area)/(concrete density) is often used in North American literature, including design guides. Such a slab is typically supported on floor beams (commonly called 'joists' in North America) at about 3 m spacing. The short-term modulus of elasticity should be used for all dynamic calculations, and current specifications and design guides tend to present rather conservative (low) values, bearing in mind the influence of the age of the concrete and the area participating in the critical circumstances. For normal density concrete the dynamic modulus of elasticity can be taken as 38 kN/mm2, and for lightweight concrete at around 1800 kg/rn3 the dynamic modulus can be taken as 22 kN/mm. A stiffness parameter of the form El1/L4 can be considered as an aid to the appreciation of the importance of slab stiffness, in which El1 is the flexural rigidity per unit width. For the application of the design guidance in Section 7, the rigidity may be computed from a smeared thickness of concrete with decking as appropriate (see design example No 1 in

Appendix B). The actual stiffness under distributed load would be obtained by multiplying the stiffness parameter by a coefficient depending on support conditions and load distribution. Considering the span between adjacent floor beams, so that the effective span L,, is set equal to the beam spacing b, this parameter is commonly in the range 30—100 kN/m3. On the other hand, considering the ability of the slab to support load over the full bay width, Le = W, this parameter very rarely exceeds 1 kN/m3 and for wide bays continuous over (say) 8 floor beams it will be less than 0.01 kN/m3. The corresponding stiffness parameter EI/bL4 for the floor beams is typically in the range 1—10 kN/m3.

The relative stiffness of slab and floor beams indicated by these parameters has the effect that under a global distributed loading the slab deflection between beams is relatively small. The slab is also sufficient to give significant resistance to differential deflection of the floor beams, although clearly not thereby causing a major departure from the basic

10

concept that the dominant load path is via the floor beams as a 'one-way' span. The net result in terms of dynamic action is that the floor behaves broadly as a strongly orthotropic plate (see Section 4.2) and a strip containing one or two floor beams can be considered as the dominant structural unit when considering walking excitation.

Precast 'Omnia type' planks, 50—65 mm in thickness with an insitu concrete topping and supplementaiy continuity reinforcement, will behave in a similar manner to a metal decking composite floor system considered above. However, greater caution must be exercised when assessing the continuity and stiffening effects of other forms of precast floor construction. Where hollow-cored precast units are required to mobilise the composite action of the supporting beams, then the ends of the units should be 'notched' and supplementary tying reinforcement used in conjunction with an insitu concrete topping should be provided. The implementation of these measures will, in addition, have a stiffening effect on the floor slab such that the floor system will tend to act as an orthotropic plate. Conversely, if 'dry construction' precast flooring is used, without such measures being implemented, then the supporting beams should not be considered to act compositely with the slab nor should the slabs be assumed to assist in reducing any differential deflection between beams or in distributing any local effects. This form of construction therefore, through Ick of stiffness, contributes only by virtue of its mass to the vibration characteristics of the floor as a whole.

For very long spans, or where very high standards are sought, the floor system may comprise beams of comparable stiffness in the two orthogonal directions, constituting an effective 'two-way' span, and thus a nearly isotropic dynamic system. Subject to the above limitation on deflection of the slab between beams, this mobilises the whole floor in resisting dynamic excitation, and is thus a very favourable configuration. The floor beams themselves will very often be supported by main beams, which form part of the principal structural framing of the building. The resulting additional deflection under a global distributed loading may be comparable to the floor beam deflection between main beams.

It should be noted that the deflection and stress levels tolerable in dynamic response are low, typical stress amplitudes being less than 1% of the static design stress, so that conventional design provisions for simple supports will not generally in practice act as such in dynamic situations. Large floor areas may thus act as if structurally continuous.

The greater effective structural continuity, under dynamic loading, has the effect that column stiffness commonly contributes significant end restraint, even where the beam connections are of a form that would normally be regarded as permitting rotation. Column stiffness is particularly likely to be significant in high-rise buildings. An adequate analysis can commonly be achieved by the 'substitute-frame' procedure. Cantilever forms of construction are relatively uncommon. Although the methods presented in Section 5 for evaluating natural frequencies are broadly applicable to cantilever construction, this form gives a rather ineffective mobilisation of mass if dynamic excitation is applied near the free end, and the evaluation of response presented in Section 6 may be non-conservative. Specialist advice should be taken if a reliable estimate is required.

4.2 Introduction to Dynamics The classic text-book model of a dynamic system, shown in Figure 4.1, is characterised by a mass, a spring stiffness, and a damper. For mathematical convenience, the damper is usually imagined to develop a force opposing the direction of movement in proportion to the velocity. Except in very rare cases where some identifiable damper has been fitted to tackle a specific oscillation problem, real floors do not incorporate such elements, but nevertheless there will be some ways in which energy is dissipated in the event of oscillation. This is usually by friction which commonly depends heavily on non-structural components such as partitions. It also depends on structural behaviour differing from the designer's model, such as nominally non-moment-resisting connections that actually develop considerable frictional resistance. Human occupants also add damping, although a high density of occupation would be necessary to have any substantial effect on a floor

11

/

///// Stiffness I Viscous k

[...]

damper

Mass m

Figure 4.1 Simple dynamic model

with a concrete or composite slab; this effect is most noticeable where high occupation density is combined with low mass, as in a schoolroom with a timber floor.

Damping is thus generally recognisable only as a global property, most directly measured and expressed by the 'logarithmic decrement' of the decay of the free oscillation after excitation has ceased. For the moderate or small levels of damping inherent in engineering structures, a logarithmic decrement of (say) 0.2 means that the amplitude falls by 20% in each successive cycle. An alternative measure, especially popular in North America, is the 'fraction of critical damping' or 'critical damping ratio', which is 1/2 times the logarithmic decrement. These quantities are non-dimensional, and care is necessary to avoid confusion when reading design guides or test reports. The frequency of free oscillation of the system shown in Figure 4.1 depends on the stiffness in comparison with the mass according to the following equation: iF f=;; wheref is the natural frequency (in Hz, i.e. Hertz = cycles/second).

It is usually convenient to work in kN and t (tonne) units; in this case the stiffness k would be expressed in kN/m and the mass in tonnes. Damping has very little effect on the natural frequency, or vice-versa.

It can be seen that the static deflection caused by the weight of the mass m (presumed to act in the appropriate direction, in line with the spring) would be y, = mg/k, and thus the frequency equation can also be expressed in the form:

-1[-- 21ry

The self-weight deflection is a quantity which the engineer can generally characterise quite closely without the need for detailed calculation, and which will follow a consistent pattern as a function of span for any given structural form. This equation thus offers a useful general approach to evaluating frequencies, and shows that conventional static design procedures, which include a limit on y,, actually constrain very strongly the value that will result for natural frequency.

For 'multi-degree-of-freedom' systems with several masses elastically interconnected, and especially the continuously distributed mass system such as the beam shown in Figure 4.2, there will be a series of natural frequencies, each associated with its own mode shape. The various modes are dynamically independent (orthogonal, or 'normal' modes) so that response can be synthesised by adding modal solutions computed independently.

The lowest frequency mode is the fundamental. This mode has the simplest shape, and its frequency will still be strongly constrained as above, and there is a procedure (Rayleigh's energy method) for estimating an appropriate weighted average of the self-weight

12

3 2 1

Figure 4.2 Beam mode shapes

deflection. For many beam and plate problems, y, in the above equation should be taken as about 3/4 of the maximum value of the self-weight deflection. For continuous beams greater care is required in this approach, which is discussed further below.

The higher modes, which may be referred to as harmonics (although their frequencies are not in general exact integer multiples of the fundamental frequency), have shapes of increasing complexity. For beams, the second mode frequency is commonly at least three times the fundamental, depending on the support conditions, mass and stiffness distribution and (where applicable) span ratios. For the simply supported uniform beam (Figure 4.2) the second frequency is four times the fundamental.

A useful insight into the behaviour of some floors is given by the behaviour of an orthotropic plate, shown in Figure 4.3. The fundamental mode shape resembles the corresponding beam mode shape in both directions. This principle applies also to the higher modes, but if the stiffness is highly orthotropic, the weak direction deformation has relatively little effect on the frequency, and a basic family of modes retaining the fundamental shape in the strong direction can occur at rather close frequencies.

Low bending stiffness

For continuous beams the fundamental frequency is clearly associated with a shape of the form shown in Figure 4.4. The inertial loads act in the sense shown and enhance the deflections, whereas in the static design process the self-weight effects on adjacent spans combine to reduce the corresponding stresses and deflections. Thus, if designed to the same static criteria, continuous construction with fairly closely uniform spans may have a significantly lower fundamental frequency than a simple structure. For the self-weight deflection approach, effective self-weight loads should be applied in an upwards direction in alternate spans.

For each mode, it is possible to establish an effective mass and stiffness, which can be used in broadly the same way as the mass and stiffness of the simple system of Figure 4.1. These are referred to as the modal generalised values. The modal generalised mass for each mode of a simple beam is one-half of the actual mass. For plates and for continuous beams the fraction is smaller, but the effective mass of a continuous beam system may still be larger than the corresponding simple structure, because the factor is applicable to

13

I 1

A

—

— ..—-— / -.-..' "

/ / / Mode shapes

High bending stiffness

Figure 4.3 Orthotropic plate mode shapes

Orthotropic plate simply supported on edges

Inertial load

—

—

Fundamental mode shape

Figure 4.4 Continuous beam fundamental mode shape

the total mass in motion, i.e. all spans. A word of caution is necessary: the values referred to here presume that the mode shape functions are each defmed to give a maximum value of unity, but some authors and some computer programs adopt other criteria by which to scale the shape functions.

The response to various sources of vibration is discussed in detail later (Section 6). In this introduction to dynamics it is sufficient to note the analytical solutions to two classes of excitation: impulsive loads, and harmonic (sinusoidal variation with time) loads. In both cases the motion is approximately a sinusoidal function of time, at the free vibration frequency and the loading frequency respectively. The amplitude is then the peak value in the current cycle, and the response envelope is the smooth curve indicated by the peaks, as shown in Figure 4.5. Note that the amplitude is 'mean-to-peak', i.e. closely equal to one-half of the peak-to-peak value. Amplitudes may be quoted for displacement, velocity or acceleration; displacement may be implied if no other indication is given. Change with time of the response envelope ordinate is referred to as modulation. Progressive reduction of the envelope ordinate (usually by damping) is referred to as attenuation.

a) C 0 0. Ca a)

Time

impulse is defmed as a change of momentum; it is usually implied that this is produced by a large force of short duration. For a single-degree-of-freedom system (Figure 4.1) the solution is very simple for an impulse (value J, say, with units Ns consistent with expressing mass in kg, or kNs for use with tonne) of short duration by comparison with the natural period of vibration. Within the duration of the impulse the mass acquires velocity Jim. Subsequently it is in free vibration and, ignoring the attenuation due to damping, the displacement y can be written in terms of the initial amplitude 9:

y = ysin(2irft)

By differentiating the above displacement equation the maximum velocity is obtained as 2rf 9 and equating this to Jim yields:

14

J = 2irfm

.— Response envelope

Response

Figure 4.5 Definition of response envelope

Precise calculations are neither justified nor required for floor vibrations, and this impulse solution is adequate for force pulses of duration up to at least one-third of the natural period of vibration, and thus for the impulsive components of walking forces and for heel-drop excitation for natural frequencies up to about 10 Hz.

Unfortunately the response of a practical distributed-mass floor system is not so simple. Potentially, all modes are excited and, as the effective masses (modal generalised masses) of the various modes are of similar order, the solution by summation of the modal responses will converge rather slowly. Response in higher modes will make a major contribution to the accelerations of the floor but the adequacy of the impulsive model of the excitation becomes questionable for such frequencies. Modal analysis is not often recommended for impulsive actions, for which a stress-wave solution would in principle be preferable, but there is no simple answer for this structural form.

As noted earlier, the simple floor which approximates dynamically to an orthotropic plate will have a family of modes as illustrated in Figure 4.3. The response in each mode may be predicted from the simple impulse response solution given above, and the total obtained by summation. Several members of this family may start with similar amplitudes, and they will start in phase. However, they will rapidly get out of phase according to the frequency differences, and the result may be a rapid attenuation of the response envelope, perceptually equivalent to high damping. The solutions for harmonic excitation are perhaps more familiar. The most important case is resonance, when the frequency of the load (or of a periodic component in the load) coincides with a natural frequency of the structure. In this case the response builds up over successive cycles. The steady-state displacement amplitude for long-continued excitation is given by:

P 5 = x (magnification factor)

where .P = amplitude of resonant Fourier component of force k = stiffness.

The magnification factor is:

amplitude — _______________ — 0.5

static deflection —

damping logarithmic —

critical damping by same force decrement ratio

This magnification may typically be fifteen-fold. A magnification of ten-fold (or two- thirds of the steady-state value, if smaller) would be reached within five cycles of excitation.

It would normally be quite unacceptable for a floor to have a fundamental frequency within the range of walking- or running-pace frequency. However, as noted in Section 2, walking contains significant second and third Fourier components which may coincide with the natural frequency and cause resonant response. Response as the sum of several modes creates less of a problem with continued excitation, because exact resonance will clearly only occur in one mode. Repeated impulsive effects at the natural frequency or at integer sub-multiples of that frequency (one-half, one-third, etc.) can also cause resonant build-up of response.

15

5. EVALUATION OF NATURAL FREQUENCY

5.1 Component and System Frequencies It has been noted above that a floor usually comprises three identifiable elastic components: a concrete or composite slab, floor beams, and main beams. These components are basically connected in series, and for the evaluation of static deflections (for example) it is appropriate to consider each component separately and estimate the total deflection as the sum of the component deflections. A similar division may well be useful for dynamic analyses, but with greater caution because the interactions between component deformations are commonly more subtle.

Once the components are connected to form a floor structure it will generally no longer be possible to identify specific component frequencies. Floor frequencies are a property of the assembled structure, and in principle each mode involves motion of all parts of the system. However, idealised component natural frequencies can be defined:

• the frequency of oscillation of the slab, presuming no deflection of the floor beams; • the frequency of the floor beams, presuming no deflection of the main beams and

that a mass associated with a strip of slab of width equal to the floor beam spacing moves with each beam;

• the frequency of the main beams, presuming that the motion of the floor corresponds to the deflection of the main beams only.

These component frequencies can be used for an approximate evaluation of the fundamental frequency of the total floor system by Dunkerly's method. Denoting the component frequencies byf, f2, andf3 (Hz) respectively, the fundamental system frequencyf0 is obtained from:

1 1 1 1 = — +— +— f f12 f22 f2 With the exercise of some engineering judgement concerning support conditions, this procedure can give good estimates. The support conditions assumed for each component must be compatible with the concept of 'inertia loading' which acts in the direction of the total system deflection. For example, for a simple floor comprising a slab continuous over a number of floor beams supported by stiff main beams, there are perhaps two possibilities that may sensibly be considered for the fundamental mode shape. The interaction of the floor beams and the slab would normally give a fundamental system mode as shown in Figure 5.1(a) and the slope of the slab is only small where it is supported on the floor beams. The slab component frequency should thus be based on fixed-end conditions.

L

(a)

(b) k. Deck alone

Figure 5.1 Possible deck and floor beam interactions

16

Floor beams

Main / beams

I Deck and floor beam interaction

Mode shapes on Section AA: stiff main beams

The lowest possible frequency for the slab alone would arise with the shape shown in Figure 5.1(b), corresponding to no rotational restraint. This would not combine with the floor beam deflection in this example; it might nevertheless correspond to the lowest natural frequency of the floor, but this is very rare in practice. Similar considerations apply to the interaction between floor beams and main beams, and in this case it may be necessary to sketch different possibilities. The valid solution is the combination giving the lowest value off0, given compatible assumptions for support conditions for each combination.

It can be seen from the form of Dunkerly's equation that the result is not especially sensitive to the interactions. Any component frequency that is more than twice the lowest component has little effect. Bearing in mind the comments on relative stiffness in Section 4.1, together with the relation between stiffness and frequency, it can be seen that the composite slab component frequency generally has little influence on the floor fundamental frequency. The component frequency for the floor beam is closely constrained (as a function of span) in conventional designs by the application of conventional limits on deflection or on span/depth ratios in combination with normal design stresses. However, these limits are based principally on live loading, whereas the governing factor for natural frequency is the stiffness in relation to mass or self-weight. It is generally appropriate to assess floor dynamics on the assumption that only a small fraction, say 10%, of the specified (characteristic) live load (including 10% for partitions where allowed) will be operative in addition to the mass of the slab, ceiling, services and any raised floor. There is also a consistent trend for a high specified live loading to result in relatively high natural frequency because beam stiffness increases by a larger amount than the corresponding increase in oscillating mass.

Since the main beam layout and structural form varies greatly from building to building, the importance of the main beam component frequency is variable, ranging from being negligible to being similar to that of the floor beams.

5.2 Practical Evaluation Four levels of approach for evaluating natural frequencies can usefully be discussed. In increasing order of refmement:

a) from a global estimate of the self-weight deflection; b) from a combination of component frequencies estimated from self-weight deflection

or tabulated frequency formulae; c) by iterative application of static analysis, using common static analysis software at

the desk-top; d) by use of dynamic analysis software packages, possibly including finite element

modelling of the structure.

The first three levels are generally limited to an evaluation of the fundamental frequency but an extended sequence of modes will be output by the fourth approach. Where response amplitudes are to be studied, most practical floors show the mode sequence effect discussed in Section 4.2 and illustrated by Figure 4.3. To a relatively poor approximation this problem can be circumvented by using an empirical estimate of an effective strip width of floor (see Section 6). if a better estimate of response is required, dynamic analysis must be more detailed, generally calling for an appropriate established software package. Details of the four approaches are given below:

a) The self-weight deflection approach was introduced in Section 4.2. Taking the suggested weighted average value of the deflection y = -y0, where y0 is the maximum value, the equation can be rewritten for convenience:

1 flj 18

17

Cu

3.0

2.0

1.0

It should be noted that this is a dimensional form, in which y0 must be expressed in mm. The foregoing comments concerning the appropriate load pattern for continuous spans and concerning the assumptions to be made on support conditions, Young's modulus of concrete and the contribution of superimposed ('live') loads should be borne in mind. Long-term effects such as shrinkage and creep deflections are excluded. This method is likely to be quite sufficient for the estimation of the fundamental frequency of a slab and floor beam system on stiff main beams.

b) The component frequency approach is likely to be helpful where there is a significant interaction with main beam deflections, especially where this results in a fundamental mode shape with significant deflections in further bays. In such cases a careful sketch of the mode shape is recommended. if the component frequencies are estimated by the self-weight deflection method, this becomes effectively the same as the global self-weight deflection approach but aids a clearer judgement of the critical mode shape. In some cases the analytical solution for the natural frequency of uniform beams can be used; this will generally be preferable for regular continuous beams. The analytical solution may be written as:

El 1/2

f= CB() where m is the mass per unit length (units in t/m if El is expressed in kNm2, or kg/m

if El is expressed in Nm2) L is the span in m (for continuous beams take the longest span).

Values of GB for a single span with various end conditions are: pinned/pinned ('simply supported') 1.57

fixed/pinned 2.45 fixed both ends 3.56 fixed/free (cantilever) 0.56

Values for continuous beams are given in Figure 5.2. The component frequencies are combined by the formula given in Section 5.1, namely:

Jo = 1

1

1)1/2

0.6 0.8 1.0

Span ratio, ilL Figure 5.2 Frequency factor CB for continuous beams

c) Where the layout is insufficiently regular to permit idealisation as uniform beam components acting in series, and/or a convincing picture of the fundamental mode shape cannot be obtained by simple judgement, the fundamental mode shape can be found by successive approximation using desk-top static analysis procedures. The fundamental frequency can then be obtained with excellent accuracy by a subsequent

18

(1 I —+— +— 1,2 f2 2 Vi J2 J3

0 0.2 0.4

summation or numerical integration stage that is amenable to either 'spread sheet' computation or hand calculation. The objective is to discover a distribution of loading, q (say), which produces deflections, y (say), such that the productym is in the same proportion to the load q at all points. The loads q and the masses m can be regarded as continuous variables or functions of co-ordinates defming position on the floor, or they can be discretised as a set of point loads and corresponding 'lumped' masses. If attention is focused on one bay of the floor, comprising a number of floor beams, a lumped model with three points (i.e. at the quarter and mid-span positions) on each floor beam in this bay and an equal number on deck points midway between the floor beams would usually be appropriate. In the adjoining bays a coarser mesh should be acceptable. One point at mid-span on each floor beam is often sufficient, but more points are desirable if it is anticipated that the mode shape will show nearly equal deflections in the various bays. The mode shape is sketched by judgement and values, y1 (say), assigned at each point. Loads in proportion to m, y, (where m, is the lumped mass attributed to load-point i) can then be estimated. The constant of proportionality is arbitrary at this stage, so the loads can be written as Pq1 where P is a convenient value (say 1 kN) and q1 are non-dimensional coefficients expressing the variation according to the variation of the target n1y1. The static deflections resulting from this loading are now computed; these will be a better approximation to the mode shape than the initial set. In principle, this process can be continued to convergence at the true shape. In practice, it will be sufficient when values of y11y0 at all points (wherey0 is the biggest value) are changing by less than 0.1 in one cycle of this process to proceed to estimate the frequency from:

1/2 1 (Pq1y, —

27rlEmy This will often only require two or three cycles of iteration. If P is expressed in kN, then m1 must be expressed in tonne and y in metre; alternatively, N, kg, m can be used respectively.

d) All the established commercial structural analysis packages (e.g. ASAS, NASTRAN, PAFEC, STRUDL, etc.) include appropriate dynamic capability, generally with provision for fmite element modelling, and these are readily available through computer bureaux. The degree of refinement in modelling should generally be somewhat superior to that indicated for the iterative approach, above. The mathematical solution can be obtained to any desired accuracy. It should be borne in mind, however, that this will outstrip the quality of the input data, including joint and support continuity, stiffness prediction for concrete elements and modelling of the excitation processes.

19

6. FLOOR RESPONSE

6.1 Low Frequency Floors When any person walks onto any floor, a full description of the response of the structure inevitably involves dynamic (that is to say, inertial) effects, because the basic process of walking inescapably produces forces which vary through each pace, as described in Section 2. The likelihood that the floor is strongly orthotropic, and has a basic family of modes sharing the property of a similar mode shape along the direction parallel to the floor beams, identifies a potentially critical event when the person walks at a steady pace parallel to a floor beam. This is especially so when a small integer multiple (i.e. less than 4, say) of the pace frequency comes within the close band of frequencies of the basic family of modes. The corresponding Fourier component of the pace force will then give rise to a resonant, or nearly resonant, response. The effective modal input varies with the location of the walker in proportion to the mode shape function at that location, and thus typically increases as the walker moves onto the span, reaches a maximum when he is near mid-span and will then fall off. With practical structural damping values and the walker advancing (say) 0.8 m per pace, the maximum response will occur a few paces after passing mid-span, and of the order of 10 paces will have been applied. If the floor has a reasonant frequency between about 4.8 Hz and 7 Hz, i.e. it is susceptible to the third Fourier component of the pace, it will be subjected to some 30 cycles of that component. This would be sufficient to give a response very nearly equal to the steady-state response to a sinusoidal force of the given amplitude applied continuously at mid-span.

It was noted in Section 4.2 that the displacement amplitude, 5 (say), was then given by:

P1

where is the damping, expressed as the critical damping ratio.

Now the response will be nearly sinusoidal, so the acceleration amplitude, a (say), is 4ir2f025; but the stiffness k is 4ir2f02M where M is the effective modal mass (compare this with the first equation in Section 4.2). Thus:

- PCs a= — mWL 2

where P = amplitude of near-resonant Fourier component of force m = mass per unit area of floor W= bay width L = floor beam span

and C, is a factor which takes account of the ratio of the effective modal mass, M, to the value mWL and also of the interaction of the modes making up the basic family.

The bay width, W, is difficult to define both concisely and rigorously, but the objective is very simple: to define the mass which must be significantly set in motion. The bay thus defined is very commonly larger than the rectangle marked off by the grid of adjacent columns. The coupling of the motion of the floor beams clearly depends on the relative stiffness of the slab. For floors of conventional proportions, coupling will be effectively interrupted by a floor beam giving a stiffness exceeding 2.5 times that of its neighbour; it should be noted that effective stiffnesses commonly vary between adjacent beams due to elastic support on main beams by comparison with direct framing into columns.

For a basic floor bay which approximates a simply supported orthotropic plate, the generalised mass is approximately mWL. If the bay is roughly square, and given conventional slab and beam stiffnesses, the interaction between modes is weak. Thus C. 4. For wider bays, the frequency differential between the modes shown in Figure 4.3 becomes very narrow and significant dynamic magnification may occur in more than one mode. C, is thus increased. A restricted parametric study of the steady-state response of

20

floors of conventional proportions has suggested that the effective mass of wide simply supported orthotropic plates is approximately --mSL, in which S can be computed as a function of the relative orthogonal flexural rigidities. This has been generalised for cases where the actual floor beam stiffnesses need to be modified according to their support conditions by substituting their stiffness as 4ir2f02M (see above), thus making use of the allowance made for support conditions in computing the natural frequency. The result is set out in Section 7.6.

The floor beam support conditions and continuity may also directly affect the effective mass by bringing into play more than one floor beam span. This may occur in two ways. Firstly, the floor beam may be continuous such that the fundamental mode takes the form shown in Figure 4.4. With two equal spans the deflections of the two spans are equal in magnitude and the effective mass is doubled. The response is thus halved. This effect, however, falls off rapidly if the spans are of dissimilar length; the design procedure given in Section 7.6 postulates a reduction of 0.6 (= 1/1.7) provided the adjoining span is not less than 0.8L Alternatively, the main beam may be sufficiently flexible that the fundamental mode has similar deflections in both floor beam spans. A similar increase in the effective mass is applicable. This condition generally only applies where the column layout has been selected to give long clear spans in both directions. In either case, floor beam continuity is accounted for in Section 7.6 through an effective length parameter, LCff.

It will be recalled from earlier sections: • that the Fourier components of pace forces are identifiable up to the third (frequency

up to about 7 Hz) but with diminishing amplitude, roughly in inverse proportion to their respective frequencies, and

• that the reaction (or criterion) curve for acceleration is flat up to about 8 Hz.

It follows that the criterion of reaction to sustained oscillation set up by regular walking becomes increasingly onerous in the frequency bands 4.8—7 Hz (third-component frequency), and 3—4.8 Hz (second-component frequency). A natural frequency in the range 4—4.8 Hz is particularly likely to result in perceptible response to walking. It can be seen thatfor a given natural frequency this criterion in effect leads to a minimum acceptable participating mass of floor. However, it should be noted that an increase of mass must be accompanied by a pro-rata increase of stiffness if the same natural frequency is to be maintained.

Simplified rules based on this analysis are given in Section 7.6. The dimensions S and Lth, dependent on the relative orthogonal stiffness, are given explicitly by incorporating C, =4 into the overall numerical factor. The base values of the exciting force and the acceleration perception criterion have also been taken into the numerical factor.

It may be noted that a low frequency floor will also respond to the transient forces due to heel strikes. Indeed, this may be more often perceived for actual floors than the resonant effect discussed above. However, from a design point of view, resonance defmes a more onerous event.

6.2 High Frequency Floors The behaviour described above is not seen as an appropriate model for floors where the natural frequency exceeds that of the third Fourier harmonic of the walking pace. For higher frequencies, an impulsive excitation can be considered.

The simple expression for the response to an impulsive excitation given in Section 4.2 can likewise readily be expressed in terms of acceleration. The effect of lateral continuity on this response is relatively weak, because the criterion is based on the first response peak, before lateral dispersion takes substantial effect. It is therefore sensible to base the equation on the mass of one panel (i.e. width b, the floor beam spacing) together with a coefficient, C (say), which is to be determined empirically:

J a= —2irfC, mbL

21

For the simple panel as above, roughly square in plan, C, 1.7. Longitudinal (floor beam) continuity should be beneficial but this has not been explored to the stage at which positive recommendations can be made. It is therefore suggested that C1 = 1.7 is an appropriate default value for all cases. For the design rules set out in Section 7.5, it has, however, been felt prudent to set an upper limit on the effective width to ensure coverage of slabs which are more slender than current normal practice.

Impulsive excitation has direct significance for relatively high frequency floors. A regular pace impulse would lead to a pace-by-pace response which each time would be substantially damped (including any lateral dispersion effect) during the ensuing pace interval, only to be renewed to a similar level. This effect is not dependent on the regularity of pacing or the exact synchronism, but the repeated decay and renewal would moderate the subjective reaction compared with sustained oscillation at the predicted peak acceleration amplitude. Applying the root-mean-quad procedure (Section 3.2) would give the effective acceleration amplitude as between 0.6 and 0.75 times the maximum.

With a pace impulse of 3—4 Ns and C = 1.7, the effective acceleration amplitude would thus be:

4 200f 2 a= —2irf= — (±25%; m/s given mbl inkg); mbL mbL 8

This effect is principally significant at frequencies above 8 Hz where the acceptable level of acceleration increases in proportion to frequency. This implies that the mass mbL should be not less than some limiting value which is in turn proportional to the multiplying factor, R (say), (or the selected 'curve R') as used in BS 6472. The permitted acceleration is 0.005R rn/s2 r.m.s. or 0.007R rn/s2 amplitude atf= 8 Hz, and thus the requirement is:

200 30000 mbL< = (kg) 0.007R R

WhereR=7 is acceptable: mbL <4300 (kg)

This can be compared with the North American specifications based on the predicted response to a heel-drop impulse of 70 Ns of duration 0.05 s. The Canadian Specification8 gives a response, a (say):

600f 2 = (rn/s given mBL in kg) mBL where B is the equivalent fully-participating width of slab.

When the main beam deflection is only a small part of the total, the effective width is specified as B = 40t where t, is the 'smeared' concrete thickness. Where main beam deflection is significant, there is a further enhancement of the effective value of BL. B is thus generally larger than the floor beam spacing b. The value a,, is to be assessed against the criterion for isolated, rapidly damped events, for example 0.062f(m/s2) for a fully furnished floor (with ceiling, ducts, flooring and furniture; specified curve for effective damping 6%). This leads to:

mBL <9600 (kg)

The Canadian Specification is thus similar in form to this Guide for high frequency floors, although suggesting a greater sensitivity to floor thickness; the effect is often more onerous than this Guide. For low frequency floors, however, there is a significant difference of principle and this Guide is generally more onerous.

The recommendations of Murray12' 13) are similarly based on evaluation of response to a heel drop impulse of 70 Ns, but include specific allowance for the effect of the duration of the impulse in comparison with the fundamental natural period. The acceptance criterion is based on lines which have constant slope on the double logarithmic plot over the whole range of practical significance, corresponding to a specific peak velocity. Murray refers to these lines as a 'modified Reiher-Meister scale', with qualitative descriptions similar to

22

Figure 3.1 (which included, inter alia, the results published by Reiher and Meister for continuous oscillation) but set at levels 5 to 10 times higher to reflect the difference between rapidly-damped impulsive events and constant-amplitude motion. The absence of the change to a specific acceleration criterion for frequency below 8 Hz has the effect that the Murray/Reiher-Meister criteria become relatively more onerous for frequencies below 8 Hz. Murray does not suggest that this is viewed as an allowance for the possible cumulative effects of successive paces including the potential resonance with a Fourier component of the pace force as discussed above, but it would seem to provide a degree of cover for this effect.

Murray's recommendations are particularly influential in the U.S.A. The most widely used acceptance level (for the peak predicted velocity response to the 70 Ns impulse) is 14 mm/s. in the middle of the qualitatively 'distinctly perceptible' range. For frequencies exceeding 8 Hz this is marginally above the level set for 6% damping in the Canadian Specification but as noted above it becomes more onerous at lower frequencies (Figure 6.1). Murray has also discussed the importance of damping'31.

C 0

5) C.) C., (3

Co 1) 0.

Velocity 1 mm/s / (R=7) /

1 2 5 10 20 50

Frequency (Hz) Figure 6.1 Impulse reaction criteria

A different approach to assessment on the basis of response to a single force pulse has been presented by Ungar and White4. In this case the force resulting from a single foot in contact with the floor, as if in regular walking motion, is considered as the excitation. The pulse duration is thus relatively long, being more than the pace period, and the duration of the initial part of the pulse in which the foot force is rising rapidly proves to have a dominant influence on the predicted response. Ungar and White use an idealisation of this phase which is still of relatively long duration (about 0.2 s) by comparison with the contact-impulse effects considered earlier (0.03—0.05 s), and do not consider the concurrent lifting of the other foot which is clearly inherent in walking. The effect of this difference in the force model is that Ungar and White conclude that stiffness rather than mass is the governing parameter for satisfactory behaviour of high frequency floors. The cited referenc&t4 includes valuable discussion of vibration acceptance criteria for the operation of precision scientific equipment.

23

(m/s') 5

Suggested 2 criterion

/

(% g)

50

20

10 -

5.

2-

1—

0.5—

/ / / / / Specification (6% damping) / / /

0.1

/ / / / Acceleration 50 mm/s2 (R = 7)

It should be noted that the various procedures all involve assumptions and generalisations both on prediction and assessment of reaction, and are intended by their authors to be used as individual packages, validated by experience. Parts of the various procedures should not be taken in isolation, or recombined.

24

7. DESIGN PROCEDURES

7.1 Definitions Bay: An area of floor substantially participating in the motion. The width

of the bay should be delimited by stiff lines of support or by a free edge where appropriate. External cladding and internal shear walls should normally be regarded as providing stiff support. Any floor beam of stiffness such that its deflection under a uniformly distributed load applied to the whole floor is less than 0.4 times the deflection of the adjacent parallel floor beam should likewise be regarded as delimiting the bay.

Bay width: The bay dimension perpendicular to the floor beams.

Floor beam: The beam directly supporting the floor slab and which, for composite slabs, is orthogonal to the deck span. This may be the secondary beam when part of a grillage of beams.

Floor slab: The two-dimensional element of the floor structure: for example, a concrete slab composite with steel decking, precast pianks with topping, precast prestressed floor units.

Floor structure: An assembly generally comprising a floor slab, floor beams and main beams.

Main beam Any beam perpendicular to and principally loaded by the floor beams. This may be the primary beam when part of a grillage of beams.

Relative flexibility: The relative contribution by deflection of a given structural element to the maximum deflection in the governing natural mode of oscillation. This may be expressed in terms of the respective deflections in the self-weight approach or the respective contributions (1/f 2) in Dunkerly' s method (discussed in Section 5) for the evaluation of the natural frequency. For example, if the self-weight deflections of slab, floor beam and main beam are 1, 6 and 3 mm respectively, the relative flexibility of the floor beam is 6/(l ÷6+3)=0.6.

7.2 General Considerations Floors should not normally be designed to have a natural frequency less than 3 Hz. Floors that may be subject to rhythmic group activities (dancing, 'pop' concerts, exercises) should not normally be designed to have a natural frequency less than 5 Hz. All such floors must be designed to reduce to a negligible level the possibility of any risk of personal injury resulting from excessive vibration, including the possibility of wilful resonant excitation. This condition can be met by ensuring robust connection of the floor members to their supports and attention to the general requirements for ductile behaviour at high deformations. Attention should also be paid to robust attachment of any items that could fall to the storey below and to the protection of services from the effects of sustained vibration. Some guidance can be found in the National Building Code of Canada. Where possible, floor layouts for rhythmic group activities should be arranged to minimise the transmission of vibration to parts of the floor occupied by people not directly participating in the activity; for example, diners in a region adjoining a dance floor. Where a quantitative assessment of vibration is required in such cases, a detailed procedure is required that will take account of mode shape ordinates at the point of reception as well as at the point of excitation.

25

The possible perception of vibration by the occupants in normal office and residential buildings should be checked by evaluating the response that may be set up by a single person walking on the floor, as set out below. The given procedure for the evaluation of response may also be used for buildings for specialised occupancy (laboratories, hospitals, etc.) but the acceptance criteria may be considerably more onerous and detailed liaison with the Client may be necessary to establish such criteria.

The response predicted by the equations in Sections 7.5 and 7.6 may be exceeded locally near unsupported edges, whether these be associated with cantilevered floor beams or with a floor beam at the edge of a floor area that is not dynamically restrained, by cladding or otherwise. The floor edge beside an atrium may be particularly at risk in this respect. Reservations as to the applicability of this Guide to certain forms of precast-unit floor slabs are discussed in Section 4.1.

7.3 Procedure for Checking Floor Susceptibility The first step is to identify the critical locations to be checked. For this purpose the floor is divided into 'bays' (see definition above), each representing an area of which the mass will be substantially mobilised by resonant dynamic excitation within that area. Where a bay is dynamically continuous (see last paragraph of Section 7.4 below) with a bay having longer span floor beams (span ratio exceeding 1.2), the check may be based on the bay with the longer spans. The next step is to estimate the fundamental natural frequency of vibration, as set out in Section 7.4. Where this natural frequency is greater than 7 Hz, the response parameter R should be evaluated according to Section 7.5. Where this natural frequency is less than 7 Hz, R should be evaluated according to Section 7.6.

The acceptable values of the parameter R are indicated in Section 7.7, according to the function of the floor. A flexible approach is desirable to ensure that the correct balance is maintained between structure cost and the refinement of environmental quality in the building. A clear appreciation of the needs and wishes of the Client may be assisted by specific discussion on this point. If the predicted value of R exceeds the acceptable value, it will commonly be preferable to increase the mass of the floor.

7.4 Natural Frequency Before proceeding to a numerical evaluation, a sketch should be made of the likely relevant mode shape or shapes. The deflected shape produced by a uniformly distributed load on the single floor beam having the longest span in the floor under consideration will generally give a useful guide to this shape. In most cases the floor natural frequency, f0, can be evaluated from the modified sell-weight deflection using the formula:

18 f0=— (Hz)

where y0 is the maximum short-term deflection in mm resulting from loads S P unilormly distributed over each area bounded by adjacent column centrelines (note that the centrelines may run parallel with or normal to the floor beams, depending on the assumed mode shape under consideration — see Example 2 in Appendix B) in which: P is self-weight in the given area, comprising the structure dead load, the

superimposed dead load, and 10% of the characteristic imposed load S. is a weighting factor, equal to unity in the area that includes the point of

maximum deflection and selected in other areas according to the magnitude of the maximum deflection in those areas.

The optimum solution is given if factors S are directly proportional to the total respective deflections in the areas (including negative values as appropriate); this condition can be approached by successive modification. The full development of this principle as an

26

iterative procedure is presented as Method (c) in Section 5.2. For initial assessment S can be taken as unity (positive or negative according to the direction of deflection) in areas where the central deflection exceeds one-third of the maximum deflection y0, and zero in all other areas.

The assumed relative values of S should be re-examined if the frequency thus estimated is less than 3.6 Hz (6.0 Hz for floors which may be subject to rhythmic group activities). They should also be re-examined if the frequency thus estimated is either in the range 4.5—5.5 Hz or in the range 7.0—8.0 Hz and additionally the value of R thus predicted exceeds one-half of the permitted value. If the evaluation of y0 is sensitive to variations of the relative values of S, the evaluation should be repeated with values more closely approaching the optimum. Where the fundamental mode is governed by floor beam flexibility, and the bay width extends over a number of floor beams having a repeated sequence of stiffnesses (for example, beams alternately framing directly to columns and framing to main beams), the average of the computed floor beam deflections may be used, subject to the addition of the slab deflection as appropriate. For this purpose the deflection of the main beams can be computed as if fixed ended.

Frequencies may alternatively be estimated by summation of analytical solutions for the component elements according to Dunkerly's method (this procedure is described, with pertinent analytical results, as Method (b) in Section 5.2) or by any of the recognised computer packages offering dynamic eigenvalue analysis for which quality assurance can be obtained.

Calculation of the natural frequency should be based on the dynamic Young's modulus, typically 38 kN/mm2 for normal weight concrete or 22 kN/mm for lightweight concrete (density around 1800 kg/rn3). Support conditions should be assessed bearing in mind the small magnitude of dynamic deflections. Composite floor slabs with steel decking can normally be regarded as dynamically continuous over the floor beams irrespective of the provision of top reinforcement. Conventional end-plate or cleated beam connections at points where the slab is continuous can normally be regarded as giving continuity. Edges of bays co-linear with walls or cladding can normally be regarded as stiffly supported.

7.5 Floors of High Natural Frequency If the fundamental natural frequency exceeds 7 Hz the floor should be assessed on the response to the heel-contact impulse component of the walking pace. The response factor R is then given by:

30000 R = mbL

where m = floor mass (kg/rn2) including allowance for occupancy loading as above (Section 7.4)

b = the lesser of either the floor beam spacing b (m) or 40 times the average slab thickness (m)

L = floor beam span (m). For continuous construction L can be taken as the length of the span adjoining the span under consideration, if this is larger.

This formulation may be rather conservative for concrete slab floors where bL < 20 m2

provided that the deflection of the slab component does not make a major contribution to the determination of the natural frequency. It may be non-conservative for any floor area where the deflection of the slab component constitutes the largest contribution in the determination of the system natural frequency.

27

7.6 Floors of Low Natural Frequency If the fundamental natural frequency is less than 7 Hz, the floor should be assessed on the basis of response to the appropriate near-resonant component of regular walking forces. Writing the component force amplitude, P, in terms of the typical basic fundamental Fourier component amplitude P1 = 240 N:

P=C1P1 =240 C1

The value of the Fourier component factor Cf should be taken as a function of the floor frequencyf0 (Hz), i.e:

• ifjisbetween3Hzand4Hz Cf=0.4 • ifj is between 4 Hz and 4.8 Hz C1= 1.4—0.25f0 • ifj is over 4.8 Hz C1 = 0.2

For normal office or residential environments the occurrence of critical excitation on floors of frequencyf0 between 4.0 Hz and 4.8 Hz will be sufficiently uncommon that assessment can be based on the reduced value of Cf shown; this reduction (from Cf = 0.4) should not be allowed if the circumstances of use are such that very vigorous walking (pace frequency exceeding 2.0 Hz) is likely to be a significantly frequent occurrence, and it should also be noted that running can lead to excitation significantly in excess of Cf = 0.4 in this frequency range.

The natural damping of the floor, expressed as the critical damping ratio, should in general be taken as = 0.03 for normal, open-plan, well-furnished floors. A lower value may be appropriate for exceptionally bare floors, noting that the damping of an unfumished floor of composite-deck construction may be as low as = 0.015. A higher value, such as = 0.045 for a floor with partitions, should only be assumed if the designer is confident that partitions will be appropriately located to interrupt all relevant modes of vibration.

The response factor R is then given by:

68000Cf R= mSLCffc

where m = floor mass (kg/m2) including allowance for occupancy loading as above (Section 7.4)

S = floor effective width (m) LCff = floor beam effective span (m)

= structural damping (critical damping ratio).

The dimensions S and LCff account for the effective distribution of the effect of the excitation. Their values should be taken from Table 7.1, in which:

RF = relative flexibility (see Section 7.1)

(El1 \h/4 S*=4.5__j (m) mf0)

( EIb L* = 3.81 I (m) m bf02 j where El1 = dynamic flexural rigidity of slab (Nm2 per m width)

Elb = dynamic flexural rigidity of composite floor beam (Nm2) b = floor beam spacing (m)

W = bay width (m) Lm = length of main beam (m)

= total length of continuous floor beam (m).

28

Table 71 Values for dimensions L and S

Indicative floor layout Qualifying conditions L (m) S (m)

Case (1)

w_______ S. •—•

RFnt,am<O.2 L S*bUt�W

RFmainam>O.2 L Greater of S*orL but�W

Case (2)

—

—t

4 L

l=L 2L

Asfor Case(1) above

O.8L<l<L 1.7L

l<O.8L L I.. w

Case (3)

: JI L

RFmnbeam<O.6 2L

RFmnam>O.6 L*but�L

Case (4)

I— w W_H F--

W2=W1

As for Case(3) above

2W1

1.7W1

W1

W2>O.8W1

W2<O.8W1

Cases (1) and (2) of the indicative floor layouts are intended to be applied where the fundamental mode shape is governed by floor beam deflection. Cases (3) and (4) are for mode shapes governed by main beam deflection.

29

For all cases in Table 7.1 it is assumed that the relative flexibility of the slab is less than 0.2, such as would be applicable for metal deck composite slabs. Where this condition is not met, it is suggested that S should be taken as the beam spacing, b, for relative flexibilities equal to or greater than 0.3 and a linearly interpolated value for relative flexibilities betwen 0.2 and 0.3. For 'dry construction' precast flooring systems (as discussed in Section 4.1) S should also be taken as the beam spacing, b.

Floors in which the flexibility of the slab is larger than the flexibility of the floor beams (as expressed by the respective self-weight deflection in the frequency evaluation), and which have a frequency less than 7 Hz, should be assessed by a detailed analytical method that takes proper account of the variations of deflection of the slab from point to point in both directions. This will generally require a fmite element computer model.

7.7 Acceptance Criteria In office accommodation the response factor R should not exceed the values given in Table 7.2.

Table 7.2 R factors for offices

Type of office Response factor, R

General office 8 Special office 4

Busy office 12

The 'general' office classification provides a suitable environment for normal office activities, including the use of computers and normal text operations on VDUs. The 'special' office is suitable for technical tasks requiring prolonged special concentration, including precision operations on VDUs. The 'busy' office is one accessible to a large number of persons, with visual and audible distractions concurrent with any vibration. Occasional adverse comment may arise at levels attributed to the 'general' and 'busy' office above.

For areas subject to much walking traffic with people walking briskly and purposefully, such as may be associated with large public circulation areas (i.e. pedestrian malls, extensive lobbies as at floor level in atria, banking halls, etc.) the response factor (computed for single-person excitation as set out in this Section 7) should not exceed R=4. The daytime response factor for residential accommodation should not exceed the value indicated for the 'special' office. If walking excitation can occur at night on a section of floor supporting bedrooms, it may be necessary to restrict the value of R to half this value. Where the highest environmental quality is required (precision manual operations) lower values may be specified. Much lower values of R may be appropriate for certain critical processes, notably in the semi-conductor manufacturing industry. In these cases guidance on acceptable criteria must be sought from the Client.

It should be noted that although the factor R has been presented in conjunction with BS 6472, the latter is perhaps more focused on reaction to vibration from external sources. In all cases an open-minded approach should be made to selecting the value of R, and a realistic balance struck between the risk and potential consequences of reaction at the 'some adverse comment' level, and the cost of the floor. In many cases adverse comment would not be associated with tangible loss. Changing R by a factor of 2 is equivalent only to the most marginal change of human reaction.

30

REFERENCES

1. BRITISH STANDARDS INSTITUTION BS 5400: 1978 Steel Concrete and Composite Bridges. Part 2— Specification for Loads BSI, 1978

2. BLANCHARD, J., DAVIES, B.L. and SMITH, J.W. Design Criteria and Analysis Behaviour of Bridges Transport and Road Research Laboratory, Supplementary Report SR 275, 1977

3. OHLSSON, S.V. Floor Vibrations and Human Discomfort Department of Structural Engineering, Chalmers University of Technology, Goteborg, Sweden, 1982

4. RAINER, J.H., PERNICA, G. and ALLEN, D.E. Dynamic Loading and Response of Footbridges Canadian Journal of Civil Engineering, Vol. 15, No. 1, February 1988, pp 66—71

5. ALLEN, D.E. and RA1NER, J.H. Vibration Criteria for Long-span Floors Canadian Journal of Civil Engineering, Vol. 3, No. 2, June 1976, pp 165—173

6. WYATF, TA. Floor Excitation by Rhythmic Vertical Jumping Engineering Structures, Vol. 7, July 1985, pp 208—210

7. The Supplement to the National Building Code of Canada National Research Council of Canada, Ottawa, 1985

8. CANADIAN STANDARDS ASSOCIATION CAN3—S 16.1 — M84: Steel Structures for Buildings — Limit States Design Canadian Standards Association, December 1984

9. BRITISH STANDARDS INSTITUTION BS 6472: 1984 Evaluation of Human Exposure to Vibration in Buildings (1 Hz to 80 Hz) BSI, 1984

10. INTERNATIONAL STANDARDS ORGANISATION ISO 2631: Guide to the Evaluation of Human Exposure to Whole Body Vibration ISO, 1985

11. IRWIN, A.W. Human Response to Dynamic Motion of Structures The Structural Engineer, Vol 56A, September 1978, pp 237—244

12. MURRAY, T.M. Design to Prevent Floor Vibrations Engineering Journal, A.I.S.C., Vol. 12, No. 3, 1975

13. MURRAY, T.M. Acceptability Criterion for Occupant-Induced Floor Vibrations Engineering Journal, A.I.S.C., Vol. 18, No.2, 1981

14. UNGAR, E.E. and WHITE, R.W. Footfall-Induced Vibrations of Floors Supporting Sensitive Equipment Sound and Vibration, October 1979

31

Appendix A - CALIBRATION STUDY

The design procedures set out in Section 7 represent a significant departure from previous practice and as such it was considered essential to undertake a calibration study to compare the results of the procedures with the performance of existing buildings. Although the procedures are based on rational principles, it was felt that they would only be readily adopted in practice if it could be shown that they lead to economic floors with satisfactory vibration performance.

For this purpose, 14 existing buildings known to have a satisfactory vibration performance, were back-analysed using the design procedures. Some of the buildings had more than one framing pattern and hence the total number of floor areas analysed (22) exceeded the number of buildings. The floor beam spans ranged from 6—15 m. A plot of calculated response factor against span shows a considerable scatter of points. The results are therefore summarised as in Figure Al below, where the calculated response factor is plotted against the estimated floor frequency. All floor areas were found to be suitable for their intended purpose, as judged by the acceptance criteria of Section 7.7.

8.0 0 E

0. 7.0 ci,

6.0

5.0

000 4.0 0 8

0 So 3.0

2.0 0

1.0

I I I I I I I I I I

0 5 10 15

Floor frequency, f0 (Hz)

Figure Al Summary of results

32

Appendix B — DESIGN EXAMPLES

The Steel Construction Institute

—

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. Sheet of

1Rev. Job Title

A c i Subject EAMP. No. I

Client IMade by se.- Ite

SCL JChecked

b JDate

Exampl4z. I impI Supporfd Lone span Composi1z *ZaTi 5 U iri sIzaJ ctecjç.

Floor oF total arv.. 24n,. 15m, 43s, Pram.r, paffrri as hown blow, or ua s e.n.raJ o9ria.. I I I

PLAFSJ

Conj.* (3rd. o $lal, 0/A Dep44, DcJ Supi 4oo,'ib /j. +hiI.

I.2 Load5 and Masses fo laka- irilo aounf.

kW/m

.0 6. 0 k.J/rn

2.27

o,o 0.b

&Q : 4.22 k.J/n-

1 4.z2'o/'.eg = 430

33

1.1 floor Deciption

I

15.0

(Sc defrsns Ma1n Baani5 1). I

H.o k,.0

Be3fl5 Floor

3.0 Z4o

ImposQ4 Occupanc Pari fions

SIabjCon. ck +

CaiUr9 5vcas R.aj4 Floor

we,ghf, sa io% Inpcs4

______________ and Slab ZnJ+a5. e'prr, Tb nrJ.i or fJ,e beam5 arg cuJc4 norrnalJj, but uIn a snorc appopria.- valu ,4? Øio concrete inodur rolia Par djnan.c bha..'iour con 7.4 0P3 ft recorn,rw.nds a value oi £ oP 2ZkN/mr Par lh' concr, adInq to a moiuIai tajio , OP Zo5/2 z The JP4Tw, w(dfl-, op slat, For compufin9 ec.Jin propzrfIe9 ,houLJ be faken as he Smaller oP jcz one oP fh span or J.hc- bear spac.in.

a Pirsf approimaJion, mnsin 4ie Ir1Q.r4as cakulafz.4 durin sta4ic. de-si9ri io% wi 3)

riormall suPPc&, Par nS-I,l -sfimai'ri +w.- Pjooj r,a4uraJ Peia4c4Zj, g,55 ,TlrIt P1rP05.

Th c npasi4c. brn tr,rhas arc ue4 hQ.se:

Zr,J'ernaj £dQ,

Ploor bzam rairi bean,s

: :

4.4%lÔn-, 1.2 idr (44os (io . IO"cn,.).

IOcni.4).

Arza oI &'v4il sIat (12.s)s/z zeerr Par slab 0/4 dep4J, oP n,rn awiag. 4,;kncs

5in bj 32.5 - ie/i )36,,irr.

The Steel Construction Institute SilwoodPark Ascot BerksSLS 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. Sheet 2. 'Rev. of I Job TitIeAppE,,D B Subject DesPJ E'CAMPL.E. ..io. I Client IMade by

IChec by

I'Jate

Datee4 -, Composife 8eam

U) ______

(11) )ab. The slab inQ4ia (pz.b metre w;dth) maj be calculated

at' uncrack4 sc34or as P0310w5, buf See af fhg. nd oP +I'Is 5+c)ion. Tile PIr5f *P s +0 cstin,afe ++'e 3J3 ncra.tC fhc.k:nes..

l. 5

34

The Steel Construction Institute Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. Sheet 3 'Rev. of I I

Job TitleA PPP%JO .

Subject DE.1GJu E,Ar%4p.E. PJo. 1

iDate Client IMade by l-t- i SC'I jChecked bvj Date4

I2 1Y4Az . __j•

I73

r' P,on ( Deck; rieufr ais posi4ion : manu3cfuNr De are.a p3-f- rne4sc- 2524

¼. Lne-ria per- rne-ftc s96!,.r,' 10 pr wdft,:

Area Avea .j _______ - C.oflctge, JI.6%1OO/, :124.7 11.6/2 59 72,3

21/24 I2.-I.73 10.77 _____ 46.o ____

.. lab riai.stral ax, Yw.o 52/I 46: (o. ,2 Disfancc, ltor,,

_______ MA_-_cm. _____________________________ 124.7 6$2-5.9 072 64. 6 lOOIt.6/I2.(

Dcck 25.24 Io.17-6.52 4.2' ______ ___ = 146c,.o

.. Ir,a4a o 1ab (g tnef ws'dffi) : 44 + 1404 -6 4 & s3mn (s.so n.). t%Jota: If IS rc-omsncnd4 tha+ lha. sial, ;fia I nof calcuJafe'

Por IrifIaI a ei..rne.r'I S cner'afti aisumpJioi, oP 44 Iab dcPlec4Ion can be. tak42, Pot Q.sfiinafiri5 Pleo

4,UMC1. IP 4hc c.c3mafe4 Ppcncq 'S *zates than 7'4, fhe.r, fIw slab na4ia is nof ?e4iied' Por co1cuiJin Pjoor (5pona.. IP +he es4irnalcd Ploor Pue.nC.4j i 1g4$ jJ,an 7,4, fhc. Ploot 1espone- cna 5+,5l b Shown fD be' a3cj btj faking a consa.rs,afve value. Por j+ie. iab iner4 a, Sat (s

so per rneft'c. w1df1

35

The Steel Construction Institute Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

1.4 De.Pini4ioi,

Appl,nq a global dIsfribute4 load Jv fha ssvhole Ploor leà$ J'o 4+e bai i(or) as b9-ing +be.. who$e. rloor ($e.t, c4ion 7.1). Fotb +1-us SrI43k, Are-a S =1 SzcIon 7.4). ThQ mode siape s oP fha Pony,;

1sJauMl ____

Using pproc, (s. Sec4kn 7.4):

____

J'Jot: LP slab incrlIa Is no cakulaj4 (5Q no a ,e. end pat J.i) a consvafi valuq, ma b- assumed,

5aj 0.4 rnni. a bean, pairI oP .orvi.

(ii) Floor ba.an, (snriplj suppoic4 condftIons).

(iii) Main b.aun (Px.4 end ondi3,ons- $ea Sq.-c.-(on 7.4

____ (;/ "4.z2) OOO' • 43rnw J2 Zo5 'cI2o so

Sinc 3ha. J?loor I'eams fenaf.1j Pam n-o columns and main bzarns, +Fie a.'ac- Floor Iec4iorl rriatj be used (ea .+ion 7.4 in t-f). Thus +b- a$c-c4ve-

'b5rn dIc.c4ian 15:

1/2, % o.4,

Job No.

JobTitleA

Sheet 4. of 1$ 1Rev.

Subject DE. IG?J E-'AMPLE- 14o. I

(1) Slab (Pi%eô end C.ondsfion5). W 4.22" = 12. 6kt.1.

= 0.22 vnni.

SF Z 5 'e4F..Z

5,' 12.66 '16% 42o5" 44o-.lo'.' '0

I I 0)Z LI

36

The Job No.

Sheet 5 of I $ 1Rev.

Steel Construction Job Title APPE$.J DIX. B

Institute ___________________________________ Silwood Park Ascot Berks SL5 7QN Subject0 P

Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

The Tofa D41c4on, 0.22w 4. .3 + O.fl = 74mm.

and tk'z- P uenj is jz 5.5N

tJofe: Had +he SLab daPPcc4ion ban a5sumei be 0.4,,,,,, f he riauraJ Prutnc.j would have bea4l Cak.u$af4 to 5.7Hz.

1.6 Floor Respons II

Since l, ',.SHZ < 7 I4, fke. Ploor 1 oP Low J'Jajural arid fI,u j+e, rc.sporic1 *sould b assz4 accbrdIri +o Seion 7.6 0P

The dR.e4pone Faor R- is iwri b Gooo.Cp rn.$.Lpp ç

$ 0.2 since 4.eI4

4ok/rn2 (l'ron part' az). : 0.03 (aumiri9 an open — plan, well- 9urnisI4 Phoor).

S and L9P a. dfrmna4 Prom (,), TjbL, 7. I 0P f4 LdP :1..: Ir,.om.

Sinc.e fRF] main bean, x 022 < o. , $ (lw). 4' r i4 :

4•'f 20 I'.3$031 = (1w:Z4irv.) oi. L43o'S.& J

... ___

37

'' The Steel Construction Institute Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. 1Sheet 6 of Rev.

Job Title DIX.

4

Sub ject DE.1C3%J E%A?'IPLE- No. I

Client IMade by

CL by

IDate

1Date4

•' , 68000 0.2 4o I9. i.o o.o3

TsLofe: Had +bQ- Iab Inz$Ia bn a$umQ4 4o b pZ.rnw4rQ...wkJff as su asfed af

e'ö '9 pt 1.3, S would ha''e bn Ul3fQ4 as I7 rn. and fhe rspon Jah'r R as 4.5

L7 F)oor AccapfabIIij-j The- a(xpfalJQ.- value 0P R- Ior a normal oPPice i as 8 in Thble 7.'Z.

Thus fhe loo s acceptable-. Liofe- fhaf ffi. apprO%?nafc ca$culafk'ns, basa4 on asurne4 Slab in.rfa arid slab deP)ecfior, alSo Show4 fI,e Ptooi jo be acceptable. Thu5, I1or 4i Ploor, fhese ¶uanffies did nof haive fD be atcuIate4.

38

The Steel Cstrction Institute

—

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. 7 of si 1Rev. Job Title A 6 Subject

MPLE. t-4o. 2- 'Date Client IMade by A . SC1 by 1Date.

ampI 2 Drnon,frafioc, OP rnod hap4 and Fraf,nwt oP b,an, boundartj condifIon5.

Z.1 Foor _r-ipfior,

r z -—

x L- 6.o 6.0 J

PLAN 1

Thabo.e shows ilwo i%Jpical 'ba' c'P a PIoor Par 5enraI oPic. T4a1 icr,fh oI? Ploor Is Z4.OVTh

2.2—Loads and Mascs Tlw sartw vaus will b faLc.n c ni E- n-ipk.- 1, I.e.:

= 422 n,: 4,o k43/rii

2.-'Cornpo5ifr 2z-am and slab Inctia5.

FJoor

-fr1a3, —' ems

C

3.0

f"5

2.5

2.5

7.5

fb.rtincnf dcfauls arfz ren in Ean-pJc.. Inz4a. l4rz-, +lie.- In.r4as ar-c mpI, •1. nor- caPajft, btat&

All PJoor barn; Main bgari, ® —®: Main beams ()-©: slab (per in. w;d+b)

* 0.20 (2oJO'cm). I' J.3 "JO irpl (IPoni). I: O.755 r= 2o.o

m4 (7 rr' (z% iOs.sn).

39

The Steel Construction Institute

—

Silwood Park Ascot BerksSL5 7QN Telephone: (0990) 23345 Fax: (0990)22944 Telex: 846843

CALCULATION SHEET

Job No. Iset B of

1Rev. Job TitleAppWO B Subject GJ E-XAMPLE.- LJo.'2 Client IMa by M.- IDate

CL Iecked

by Date

A .

40

2.4 -Mod Sbapas and &an unda,q Condifions.

For an Ploor consis3iri9 oP af Jc,asf a +hrce.- bij fh oIun,ri en1d such as bemn9 considc.t'4 hrc-, 4wo od (A and neQ4 +o 1a vx.ammnQ4.

The- valu 0P (sc SgcIon 7.4) appnopsiafc.. or mcdi. i kowr, in +hc 5keJC, bo4'

S S S.-—

I_' is2sJ

SW,'

sW-l

MODE- e

— Th ur,damn)al modg. I 44iif whlci, fhe Iowii4f r4).w.4ci1, ii.. fh ha.'mn fI,. lara.n 4lc$i.

Mare bQani5 Thg.. pgIgNarif oP vibrafion beasn5 ae fri

Mc'DL A

Mc'o A -I

Porrn nodal lin arid +ht,rc hav zar-o dcl'k4ior,. boundarj ondif ion 'or 4,g. c!loor bearri5 Par fhis mod, s$ Sirnptj upporfed '3n dc.lciori oP a4jacant 44,g. oppasifa nu öd o1'ea.i.iaI manu4.u4z. ot nari ba'n.

The Steel Construction Institute

—

Silwood Park Ascot BerksSLS 7QN Telephone: (0990) 23345 Fax: (0990)22944 Telex: 846843

CALCULATION SHEET

Job No. Sheet of JRev.

Job TitleA

Subject DE.siGM EA1PLE. pjo.'z

Client IMade by )4r bate S CL by

f Date

parfkIpata in In 4S mod fI,a maln Lvacn5 44ia rno4ior, Th boun4arj condifIon or fh P3oør ban-i in fh. mode Wi 33 bc Pd- n4a4 duz fu fh.r QaaJ d Jirr4ion over malr' bizm. For fhc main beam () — a oP 4ie- maximum dPkc1ion can be rrwde bj a5surriiri Ssmple- supports and 14-us will be- dor,a huc-. (ZP this is COnSidQJ04 fo be. oo conse-r.iafive. eount can be. jaken oP confinu.ftj wfh nain bean-i ()—©).

41

Moos B

S4,X —

2.— Uaural Freua.ncj Mode, A

() slab (cIr.4a.rid). W WL I2.6&,ooo O.22mrv. '34E.I 384"2o5 Zooo

V) Ploor bean, (4imupport4

394 EL 394 "2o, ø2oIO'.' lo (i(I) main bgarrt 4PIz.c4ior, i zc.-ro

To3I DeIQ.4IOn, 0.72 + .2 i .

NauaJ Freuen9j or Modz. A

(I)

c? 1)

(lii)

z 19 = ____

Z3" 75. b& 7.21 mm. 64ezos '33

Tofaj Dg,Pkon, 9 = 7.21 .l. 04 + 0.22 z ___

NauaI F ue.nci Pot Mode. 8

: ___ ___ : Mode.- B and henc.

The Steel Construction Institute Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. 10 of II Rev.

Job Title APp.JDlx.

Subject DEs GP.J E.,AMPLE t-Jô. 2.

IDate Client IMade by LA4* CI JChecked

by Date

J2.o6%e, -7c,$j, 5.21 nrni.

0 ____

Mode B.

slab (P%a.d 2nd5): 8 a. 'zz mrr. a abo%'e.

PJoor j,am (8.nd,): SF: WL" I , .21

3ME.I I.04mn,.

main beam (a½un,ina Implj uppotfe4) (beam ®—®)• . 23 P1. p p

'4e 6I #t*./1 L jL/3 jL/3,. L9.e

42

The SteelConstruction Institute

—

Silwood Park Ascot Berks SL5 7QN Telephone: (0990) 23345 Fax: (0990) 22944 Telex: 846843

CALCULATION SHEET

Job No. Iset I I of i

1Rev. Job Title APP 4-i DIX E Subject

D E..S j E.-x. AM PLE. Mo. 2 Client IMade by

5C-1. JChecked

by

ILte

Date1 Floor R.s.spona..

A5 P0 7 +'1' 1e$pOn5Q. iP lo b aS45SQ.4 aordin9 fo 4cc..Jion 7.6 oP +he- J4 R,: 6eooo.Cp

rn. £ LeP9.

Sinc.a. fh PuriainLnfa snod (Modc e) i ovenie4 b3 main b2-arn fh 2fiuJtimate paraç'apl, in cjion 7.6 rP.ss to Cas 4 (and nof C4 2-) dTabIe.- 7.1 as bne appIic.abh- Por fe..rpn,n and Lpj Cf: 0.2 51nC4 P, 4.8 I4. fl: 430 k9/rn

Z: 0.03

For \'/IZ.Orn,W2:7..r,. o.ew1 S = 1.7W1

..Lcc. =j' =

3.e[ E-rb.]4 buf * Lrna,u.

mbPg,

3.Sf2o2ooIO,'14 LoE,.z' .1

(124.ov) o,'.

Rz 6€'oooo.'z 430 ,Ir,.3 Zo.4% 0.03

Z.7" Floor AcptabiJ'J

=3.4

inc4' R = 3.4 < 4, h' Ploor is otl!ke acc.ordin 4o Jable 7.2.

43

t ),7.o {RFJ m3;, 7.21/B. 47 ) 0.6

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