13.7 Floor Vibration

Back

13.7.1 Introduction

Annoying floor vibrations may be caused by occupant activities. Walking, dancing, jumping, aerobics,

and audience participation at music concerts and sporting events are some prime examples of occupant

activities that create floor vibrations. The operation of mechanical equipment is another cause for

concern. Heating, ventilation, and air-conditioning systems, if not properly isolated, can cause serious

vibration problems. The current trend towards longer spans and lighter floor systems has resulted in a

significant increase in the number of floor vibration complaints by building owners and occupants.

Most of the sources contributing to reported human discomfort rest on the floor system itself.

However, human activities or machinery off a floor can cause significant floor vibrations. On more than

one occasion, aerobics on one floor of a high-rise building has been reported to cause vibration

discomfort on another level in the building. The vibrations caused by automobiles on parking levels

below have been reported to disrupt sensitive laboratory work on upper floors. Other equipment and

activities off the floor that can contribute to a floor vibration problem are ground or air traffic, drilling,

the impact of falling objects, and other construction-related events.

When the natural frequency of a floor system is close to a forcing frequency and the deflection of the

system is significant, motion will be perceptible, and perhaps even annoying. Perception is related to the

activity of the occupants: a person at rest or engaged in quiet work will tolerate less vibration than a

person performing an active function, such as dancing or aerobics. If a floor system dissipates the

imparted energy in a very short period of time, the motion is likely to be perceived as less annoying.

Thus, the damping characteristics of the system affect acceptability.

In design guidelines for floor vibration analysis, limits are stated as a minimum natural frequency of a

structural system. These limits depend on the permissible peak accelerations (as a fraction of

gravitational acceleration) on the mass affected by an activity, the environment in which the vibration

occurs, the effectiveness of interaction between connected structural components, and the degree of

damping, among other factors.

Recently, excessive floor vibrations have become a common problem due to a decrease in the natural

frequency at which buildings vibrate due, in turn, to increased floor spans and a decrease in the amount

of damping and mass used in standard construction practice, because of the availability of stronger and

lighter materials. Some methods have been developed in the recent past to check the floor vibrations of

structures. These methods are summarized in this section. More details can be found in the texts given in

the list of references.

13.7.2 Types of Vibration

13.7.2.1 Walking

A walking person’s foot touching the floor causes a vibration of the floor system. This vibration may be

annoying to other persons sitting or lying in the same area, such as an office, a church, or a residence.

Although more than one person may be walking in the same area at the same time, their footsteps are

normally not synchronized. Therefore, the analysis is based on the effect of the impact of the individual

walking.

13.7.2.2 Rhythmic Activities

In some cases, more than a few people may engage in a coordinated activity that is at least partially

synchronized. Spectators at sporting events, rock concerts, and other entertainment events often move in

unison in response to music, a cheer, or other stimuli. In these cases, the vibration is caused by many

people moving together, usually at a more or less constant tempo.

The people disturbed by the vibration may be those participating in the rhythmic activity, or those in

nearby part of the structure engaged in a more quiet activity. The people engaged in the rhythmic activity

Vibration and Shock Problems of Civil Engineering Structures 13-51

© 2005 by Taylor & Francis Group, LLC

have higher level of tolerance for the induced vibrations, while those nearby will have a lower level of

tolerance.

13.7.2.3 Mechanical Equipment

Mechanical equipment may produce a constant impulse at a fixed frequency, causing the structure to

vibrate.

13.7.2.4 Analysis Methods

Because the nature of the input varies for these three types of loads, each of the three requires a somewhat

different solution. However, all cases require knowledge of an important response parameter of the floor

system, its natural frequency of vibration, and all three analysis methods are based on finding a required

minimum frequency.

13.7.3 Natural Frequency of Vibration

The natural frequency of a floor system is important for two reasons. It determines how the floor system

will respond to forces causing vibrations. It is also important in determining how human occupants will

perceive the vibrations. It has been found that certain frequencies set up resonance with internal organs

of the human body, making these frequencies more annoying to people.

Figure 13.35 shows the human sensitivity over a range of frequencies during various activities. The

human body is most sensitive to frequencies in the range of 4 to 8 Hz. This range of natural frequencies is

commonly found in typical floor systems. Recommended acceleration limits for vibrations due to

rhythmic activities are given in Table 13.10.

Frequency (Hz)

1 2 4 8 12 20

0.05

0.10

0.25

0.50

1.00

1.50

3.00

5.00

10.00

ISO Baseline curve of

RMS acceleration for

human reaction

Operating rooms

Offices, residences,

churches

Indoor footbridges,

shopping malls, dining

and dancing

Rhythmic activities,

outdoor footbridges

Peak Acceleration (% gravity)

FIGURE 13.35 Recommended permissible peak vibration acceleration levels acceptable for human comfort while

in different environments. (Source: Data from Mast, R.F., Vibration of precast prestressed concrete floors, PCI J.,

Nov – Dec, 2001. With permission.)

13-52 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

13.7.3.1 Computing the Natural Frequency

The natural frequency of a vibrating beam is determined by the ratio of its stiffness to its mass (or

weight). The deflection of simple-span beam is also dependent on its weight and stiffness. A simple

relationship exists between the self-weight deflection and the natural frequency of a uniformly loaded

simple-span beam on rigid supports:

fn ¼ 0:18

ffiffiffiffiffi

g=Dj

q

ð13:47Þ

where

fn ¼ natural frequency in the fundamental mode of vibration

g ¼ acceleration due to gravity

Dj ¼ instantaneous simple-span deflection of floor panel due to dead load plus actual live load

13.7.3.2 Computing Deflection

The equation for the deflection Dj for a uniformly loaded simple-span beam is

Dj ¼

5wl 4

384EI ð13:48Þ

where

l ¼ span length of member

I ¼ gross moment of inertia, for prestressed concrete members

Many vibration problems are more critical when the mass (or weight) is low.

For continuous spans of equal length, the natural frequency is the same as for simple spans. This may

be understood by examining Figure 13.36. For static loads, all spans deflect downward simultaneously,

and continuity significantly reduces the deflection. But for vibration, one span deflects downward while

the adjacent spans deflect upward. An inflection point exists at the supports, and the deflection and

natural frequency are the same as for a simple span.

For unequal continuous spans, and for partially continuous spans with supports, the natural frequency

may be increased by a small amount.

Simple Span

Continuous Spans

Inflection point at

support

FIGURE 13.36 Natural frequency of simple and continuous spans.

TABLE 13.10 Recommended Acceleration Limits for Vibrations Due to

Rhythmic Activities

Occupancies Affected by the Vibration Acceleration Limit (%g)

Office and residential 0.4 – 0.7

Dining and weightlifting 1.5 – 2.5

Rhythmic activity only 4 – 7

Source: Data from Alen, D.E., Building vibration from human activities,

Concr. Int., 66 – 73, 1990.

Vibration and Shock Problems of Civil Engineering Structures 13-53

© 2005 by Taylor & Francis Group, LLC

13.7.3.3 Damping

Damping determines how quickly a vibration will decay and die out. This is important because human

perception and tolerance of vibration or motion is dependent on how long it lasts. Damping of a floor

system is highly dependent on the nonstructural items (partitions, ceilings, furniture, and other items)

present. The modal damping ratio of a bare structure undergoing low-amplitude vibrations can be

very low, on the order of 1%. Nonstructural elements may increase this damping ratio up to 5%

(see Table 13.11). It must be appreciated that the results of a vibration analysis are highly influenced by

the choice of the assumed damping, which can vary widely.

13.7.3.4 Resonance

Resonance occurs when the frequency of a forcing input nearly matches the natural frequency of a

system. In order to avoid excessive amplification of vibration, the natural frequency must be higher than

the frequency of the input forces by an amount related to the damping of the floor system.

13.7.4 Vibration Caused by Walking

Vibrations caused by walking can often be objectionable in lighter constructions of wood or steel.

Because of the greater mass and stiffness of concrete floor systems, vibrations caused by walking are

seldom a problem in these systems. However, when designing concrete floor systems of long span, the

serviceability requirement on vibrations may become critical.

13.7.4.1 Minimum Natural Frequency

People are most sensitive to vibrations when engaged in sedentary activities while seated or lying. Much

more vibration is tolerated by people who are standing, walking, or active in other ways. Thus, different

criteria are given for offices, residences, and churches than for shopping malls and footbridges.

An empirical formula, based on the resonant effects of walking, has been developed to determine the

minimum natural frequency of a floor system needed to prevent disturbing vibrations caused by walking:

fn $ 2:861 ln

K

bW

􀀏 􀀐

ð13:49Þ

where

K ¼ a constant, given in Table 13.11

b ¼ modal damping ratio

W ¼ weight of area of floor panel affected by a point load

13.7.5 Design for Rhythmic Excitation

Rhythmic excitation may occur when a group of people exercise or respond to a musical beat. Because a

group is acting in unison at a constant frequency, the input forces are much more powerful than those

produced by random walking. Resonance can occur when the input frequency is at or near the

TABLE 13.11 Values of K and b

Occupancies Affected by the Vibration K (kN) b

Offices, residences, churches 58 0.02a; 0.03b; 0.05c

Shopping malls 20 0.02

Outdoor footbridges 8 0.01

a For floors with few nonstructural components and furnishings, open work areas,

and churches.

b For floors with nonstructural components and furnishings, and cubicles.

c For floors with full-height partitions.

Source: Data from Alen, D.E. and Murray, T.M., Design criterion for vibrations due

to walking, Am. Inst. Steel Const. Eng. J., Fourth Quarter, 117 – 129, 1993

13-54 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

fundamental frequency of vibration, and so the fundamental frequency of the floor must be sufficiently

higher than the input frequency as to prevent resonance.

13.7.5.1 Harmonics

A harmonic of frequency is any higher frequency that is equal to the fundamental frequency multiplied by

an integer. For instance, if the frequency of an input excitation is 2.5 Hz, the harmonics are

2.5 £ 2 ¼ 5 Hz, 2.5 £ 3 ¼ 7.5 Hz, and so on. If the fundamental frequency of a floor system is equal to a

harmonic of the exciting frequency, resonance may occur.

This process is less efficient than one which is in resonance striking at each cycle of vibration.

Nevertheless, the 2.5 Hz forcing frequency can cause resonance in the 5 Hz fundamental frequency due to

the input force striking every second cycle in the fundamental frequency.

Higher harmonics should not be confused with higher modes of vibration. The second mode of

vibration of a simple span has a frequency four times the fundamental frequency. This high a frequency is

almost never excited. Harmonics refers to the forcing frequency, compared with the fundamental mode

of vibration.

13.7.5.2 Minimum Natural Frequency

The following design criterion for minimum natural frequency for a floor subjected to rhythmic

excitation is based on the dynamic response of the floor system to dynamic loading. The objective is to

avoid the possibility of being close to a resonant condition:

fn $ f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ

k

a0=g

aiwp

wt

s

ð13:50Þ

where

f ¼ forcing frequency ¼ (i) ( fstep)

fstep ¼ step frequency

i ¼ number of harmonic ¼ 1, 2, 3

k ¼ a dimensionless constant (1.3 for dancing, 1.7 for a lively concert or sport events, 2.0 for aerobics)

ai ¼ dynamic coefficient

a0=g ¼ ratio of peak acceleration limited to the acceleration due to gravity

Wp ¼ effective distributed weight per unit area of participants

Wt ¼ effective total distributed weight per unit

area of participants (weight of participants

plus weight of floor system).

The natural frequency of the floor system, fn, can

be found as discussed previously.

13.7.6 Example — Vibration

Analysis of a Reinforced

Concrete Floor

A concrete floor of a tall building is analyzed in this

example. The plan view and structural configuration

of the building are shown in Figure

13.37. Perimeter columns are spaced at 12 m

centers and are connected by spandrel beams to

support the facade. The example floor will be used

for aerobic exercises and needs to be checked for

vibration. Aerobic exercises are usually undertaken

in the range of 2 to 2.75 Hz, with a maximum

value in the order of 3.0 Hz. Ideally aerobic

FLOOR PLAN

12 m

1 2 3 4 5

FIGURE 13.37 Structural configuration.

Vibration and Shock Problems of Civil Engineering Structures 13-55

© 2005 by Taylor & Francis Group, LLC

exercise floors should be designed so that the floor’s natural frequency exceeds the third harmonic by a

factor of 1.2, resulting in fn . 1:2 £ 3 £ 2:75 ¼ 9:9 Hz. This is not always achievable in practice,

especially for long span floors that have the natural frequency in the range from 4 to 8 Hz. Hence, a floor

with natural frequency greater than 7.5 Hz is considered a minimum standard, although in some cases

floor vibrations may be quite noticeable.

The modal analysis of the floor system was carried out with the assumed damping factor b of

2% (see Table 13.11). It was found that the floor natural frequency is 6.75 Hz, which may result in some

problems in floor vibration. To reduce the vibration problem the following approaches can be used:

* Reduce mass (normally not very effective)

* Increase damping (e.g., using dampers)

* Reduce vibration transmission (stiffening joists at columns may reduce transmission)

Навигация