1.5. FLOOR VIBRATION (2240 Words)

Movement of floors caused by occupant activities can present a serious serviceability problem if not properly considered and prevented by the design of the structural system. Humans are very sensitive vibration sensors – vertical floor movement of as little as forty thousandths of an inch can be very annoying. Post-construction repairs of floors that vibrate are always very expensive, and sometimes cannot be done because of occupancy limitations. This reinforces the necessity of addressing potential vibration problems in the original design.

The response of individuals to floor motion depends on the environment, occupant age, and location. People are more sensitive in quiet environments, such as a residence or quiet office, as compared to a busy shopping mall. The elderly are more sensitive than young adults, and sensitivity appears to increase when sitting as compared to standing or reclining.

Stiffness and resonance are dominant considerations in the vibration serviceability design of steel floor structures and footbridges. The first known stiffness criterion appeared nearly 170 years ago. In 1828, an English carpenter named Tregold published a book on carpentry writing that girders over long spans should be "made deep to avoid the inconvenience of not being able to move on the floor without shaking everything in the room." The traditional stiffness criterion for steel floors limits the live load deflection of beams or girders supporting plastered ceilings to span/360. This limitation, along with restricting span-to-depth ratios of members to 24 or less, have been widely applied to steel-framed floor systems in an attempt to control vibrations, but with limited success.

Traditionally, soldiers "break step" when marching across bridges to avoid large, potentially dangerous, resonant vibrations. Until recently, resonance had been ignored in the design of floors and footbridges. Approximately 30 years ago problems arose with the vibrations induced by walking on steel-joist supported floors that had satisfied traditional stiffness criteria. Since that time much has been learned about the loading function due to walking and the potential for resonance. More recently, new rhythmic activities, such as aerobics and high impact dancing, have caused serious floor vibrations due to resonance.

A number of analytical procedures have been developed which allow a structural designer to assess the floor structure for occupant comfort for a specific activity and for suitability for sensitive equipment. Generally, the analytical tools require the calculation of the first natural frequency of the floor system and the maximum amplitude of acceleration, velocity, or displacement for a reference activity or excitation. An estimate of the damping in the floor is also generally required. A human comfort scale or sensitive equipment criterion is then used to determine whether the floor system meets serviceability requirements. Some of the analytical tools incorporate limits on acceleration into a single design formula whose parameters are estimated by the designer.

Before presenting a technical explanation of floor design principles, basic terminology is listed and explained. A review of this terminology will greatly assist in the understanding of the structural design principles that follow.

Basic Vibration Terminology

Dynamic Loadings. Dynamic loadings can be classified as harmonic, periodic, transient and impulsive as shown in Figure 18. Harmonic or sinusoidal loads are usually associated with rotating machinery. Periodic loads are caused by rhythmic human activities such as dancing and aerobics, and by impactive equipment. Transient loads occur from movement of people and include walking and running. Single jumps and heel-drop impacts are examples of impulsive loads.

Period and Frequency. Period is the time, usually in seconds, between successive peak excursions in repeating events. Period is associated with harmonic (or sinusoidal) and repetitive time functions as shown in Figures 18a and 18b. Frequency is the reciprocal of period and is usually expressed in Hz (Hertz or cycles per seadcond).

Steady State and Transient Motion. If a structural system is subjected to a continuous harmonic driving

force (see Figure 18a), the resulting motion will have a constant frequency and constant maximum amplitude and is referred to as steady state motion. If a real structural system is subjected to a single impulse, damping in the system will cause the motion to subside as illustrated in Figure 19. This is one type of transient motion.

Natural Frequency and Free Vibration. Natural frequency is the frequency at which a body or structure

will vibrate when displaced and then quickly released. This state of vibration is referred to as free vibration. All structures have a large number of natural frequencies; the lowest or "fundamental" natural frequency is of most concern.

Damping and Critical Damping. Damping refers to the loss of mechanical energy in a vibrating system. Damping is usually expressed as the percent of critical damping or as the ratio of actual damping to critical damping. Critical damping is the smallest amount of viscous damping for which a free vibrating system that is displaced from equilibrium and released comes to rest without oscillation.

Resonance. If a frequency component of an exciting force is equal to a natural frequency of the structure, resonance will occur. At resonance, the amplitude of the motion can become very large as shown in

Figure 20.

Step Frequency. Step frequency is the frequency of application of a foot or feet to the floor, e.g., walking, dancing or aerobics.

Harmonic. A harmonic multiple is an integer multiple of the frequency of application of a repetitive force (e.g.,multiple of step frequency for human activities or multiple of rotational frequency of reciprocating machinery). Harmonics can also refer to natural frequencies, e.g., of strings or pipes.

Mode Shape. When a floor structure vibrates freely in a particular mode, it moves up and down with a certain configuration or mode shape. Each natural frequency has a mode shape associated with it. Figure 21 shows typical mode shapes for a simple beam and for a slab/beam/girder floor system.

Modal Analysis. Modal analysis refers to a computational analytical or experimental method for determining the natural frequencies and mode shapes of structures, as well as the responses of individual modes to a given excitation.

Spectrum. A spectrum shows the variation of relative amplitude with frequency of the vibration components that contribute to the load or motion. Figure 22 is an example of a frequency spectrum.

Acceleration Ratio. The acceleration of a system divided by the acceleration of gravity is referred to as the acceleration ratio. Usually the peak acceleration of the system is used.

Floor Panel. A rectangular plan portion of a floor encompassed by the span and an effective width is

defined as the floor panel.

Bay. A rectangular plan portion of a floor defined by four column locations.

Floor Vibration Principles

Although human annoyance criteria for vibration have been known for many years, it has only recently become practical to apply such criteria to the design of floor structures. The reason for this is that the problem is complex, the loading complex, and the response complicated - involving a large number of modes of vibration. Experience and research have shown, however, that the problem can be simplified sufficiently to provide practical design criteria.

Most floor vibration problems involve repeated forces caused by machinery or by human activities such as dancing, aerobics or walking, although walking is a little more complicated than the others because the forces change location with each step. In some cases, the applied force is sinusoidal or nearly so. AISC's Steel Design Guide No. 11: Floor Vibrations Due to Human Activities explains in detail the required engineering calculations and assessment techniques. These techniques use acceleration, as a percent of acceleration due to gravity, to measure human perception of floor movement. For example, the tolerance level for quiet environments, residences, offices, churches, etc. is 0.5 percent of gravity (0.005g).

Figure 23 shows tolerance levels for a number of situations. Note that the scale is a function of frequency and acceleration. Also, note that the tolerance acceleration level increases as the environment becomes less quiet. For instance, the tolerance level for people participating in aerobics (rhythmic activities) is ten times greater than if they are in a quiet office. To use the scale, the natural floor frequency and the estimated acceleration for an activity must be calculated.

The acceleration of a floor system depends on the activity, the natural frequency for the floor, the amount of mass that moves when the floor vibrates, and the damping in the floor. Floor acceleration increases as energy in the activity increases; thus, floor acceleration is greater for aerobics than for walking. Acceleration decreases with increasing weight; the acceleration for a lightweight concrete floor will be greater than that for the same normal weight concrete floor for the same activities. Acceleration decreases with increasing damping.

Evaluation of a floor system for potential annoying vibration requires careful estimation of the weight supported by the floor on a typical day. A fully loaded floor will never be a problem; most occupant complaints are received when the problem floor is slightly loaded. The design dead load for mechanical equipment and ceiling should never be used, nor should the design live load. An estimate of the real mechanical loading (for instance, 2 psf not 5 psf as may be used for strength design) and ceiling is required. Recommended live loads in the Floor Vibrations design guide are 11 psf for office live loading (not 50 psf as used for strength design), 6 psf for residences, and 0 psf for shopping malls.

Frequency is the rate at which a floor vibrates and is expressed in cycles per second (Hz). Floor systems generally have a frequency between 3 and 20 Hz. For a typical steel framed 30 ft by 30 ft office building bay, the frequency will be in the 5-8 Hz range. Frequency is a function of span (the longer the span, the lower the frequency) and weight supported (the heavier the floor and the supported contents, the lower the frequency). Thus, a floor constructed using normal weight concrete will vibrate at a lower frequency than the same floor constructed with lightweight concrete. When the frequency is above 15 Hz, as occurs in very short spans (say less than 15 ft), floor vibrations are generally not felt.

Damping is energy loss due to relative movement of floor components or fixtures on the floor. Damping causes a freely vibrating floor system to come to rest and is usually expresses as a percent of critical damping. Critical damping is the amount of damping required to bring a vibrating system to rest in one-half cycle. Damping for floors is usually between 2 percent and 5 percent. The lower value is for floors supporting few non-structural components, like for open work areas and churches. The larger value is for floors supporting full-height partitions. A typical office floor with movable, half-height partitions has about 3 percent damping.

Particular attention should be given to office floors with open spaces, no fixed partitions, and light loads. This situation is what results in problem floors if the design is not done correctly. Also, floors with high design loads (say 125 psf) and light actual loads (say less than 15 psf) do not have the same amount of damping as floors designed for normal office loading (say 50 psf). In this case, a lower estimate of damping should be used (e.g., 1-2 percent).

The design of floors supporting rhythmic activities, dancing, aerobics, etc. require consideration of the entire structure, not just the supporting floors. These activities introduce very high energy levels into the structure and can cause annoying floor motion quite some distance from the activity area. Aerobics on the 60th floor of a building have caused excessive floor motion twenty floors below. When a rhythmic activity floor is located above approximately six stories, column deflections must be considered.

To avoid annoying vibrations in floors supporting rhythmic activities, the fundamental natural frequency must be above frequencies associated with harmonics of the activity and the tolerance acceleration ratio. The tolerance acceleration ratio is a function of both the rhythmic activity and the affected occupancy. For instance, when dancing and dining are considered, the tolerance acceleration ratio is 0.02g. The tolerance level is increased to 0.05g for participants in lively concerts or sports events.

To satisfy the criterion, a relatively large fundamental natural frequency is required. For example, if jumping exercises are shared with weightlifting with an acceleration tolerance level of 0.02g and floor weight of 50 psf, the required frequency is 10.6 Hz. The economical solution for this example is lightweight concrete and deep, lightweight supporting members.

Floors supporting sensitive equipment, such as operating room equipment, electron microscopes, and microelectronics manufacturing equipment must be very stiff and heavy. Tolerance levels for this type of equipment are usually expressed in velocity with numbers like 100 to 8,000 micro-in./second. The means of accommodating sensitive equipment are readily available, but usually require specialists in this area to produce a satisfactory design.

Summary

The determination of potentially annoying floor motion for a proposed design requires careful consideration of the structural system, the anticipated activities, and the finished space. Art, as well as science, is required on the part of the designer. The most important parameter to be determined is the fundamental natural frequency of the floor structure. This calculation requires a careful estimate of the supported weight on an average day. Floor system damping, which depends on the components of the building systems, as well as occupancy furnishings and partitions, also must be estimated. Finally, an acceleration tolerance criterion must be selected and compared to the predicted acceleration of the floor structure.

(Design with Structural Steel - A Guide for Architects)