BECHTEL Engineering Design Guide Vibration Sensitive Structures

BECHTEL CIVIL/STRUCTURAL ENGINEERING DESIGN GUIDE FOR VIBRATION SENSITIVE STRUCTURES 3DG C01 007, Rev. 02, 06/94 Prepared by: T. Yen Approved by: C. C. Elizondo TABLE TAB LE OF CONTENTS CONTENTS Page No. 1.0

GENERAL GENERAL

5

1.1 1.2 1.2 1.3 1.3 1.4 1.4

Purpos Purp ose e Codes, Standards, and Reference Reference Documents Docum ents Desig Design n Metho Methodol dol ogy Use of Consultant Consu ltant Servi Servi ces

5 5 7 8

2.0

VIBRATION VIBRA TION DESIGN CRITERIA

8

2.1 2.1 2.2 2.2 2.3 2.3

General General Requ Requirement irements s Project-Specifi c Criteria Criteri a Generic Generic Vibration Vibrati on Criteria Criteri a

8 9 10

3.0

SOURCES OF VIBRATION VIBRA TION

11

3.1 3.1 3.2 3.2 3.3 3.4 3.4 3.5 3.6 3.6

The Vibration Vibrati on Environm Envir onment ent Ambient Ambi ent Ground Vibrations Machine Machin e Generated Vibration Vibr ation s Football Excitation Traffic Excit ation Air bor ne Noise Effects

11

O:\WINWORD\3DG\C01\007-02.DOC

12 13 13 14 14

PAGE 1 OF 53

Page No. 4.0

METHODOLOGY FOR VIBRATION VIBRA TION CONTROL

15

4.1 4.1 4.2 4.3 4.3 4.4 4.4 4.5 4.5 4.6 4.6 4.7 4.7

General General Consideratio Consi deratio ns Site Selection Selecti on Physical Physic al Separatio Separatio n of Sources Sourc es and Receivers Receivers Structur al Isolation of Sensiti Sensitive ve Floors Use of Massi Massive ve Constr uction uct ion s Isolatio n of Servi Service ce Equipment Equip ment Isolatio n of Sensi Sensitiv tive e Tools

15 16 17

5.0

VIBRATION ANALYSIS ANA LYSIS

20

6.0

VIBRATION VIBRA TION SURVEY

21

6.1 6.1 6.2 6.2 6.3 6.3

Site Characterization Intermediate Intermedi ate Vibratio n Surveys Final (Certif (Certificati ication) on) Vibratio n Survey

21 22 23

7.0

REFERENCES

23

APPENDIX APPENDIX A

ESSENTI ESSENTIAL AL CONCEPTS CONCEPTS IN VIBRATION THEORY

APPENDIX APPENDIX B

TYPICAL TYPICAL VIBRATION VIBRA TION CRITERI CRITERIA A

APPENDIX APPENDIX C

VIBRATIONAL FORCES FORCES


17 18 19 20

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Page No. 4.0

METHODOLOGY FOR VIBRATION VIBRA TION CONTROL

15

4.1 4.1 4.2 4.3 4.3 4.4 4.4 4.5 4.5 4.6 4.6 4.7 4.7

General General Consideratio Consi deratio ns Site Selection Selecti on Physical Physic al Separatio Separatio n of Sources Sourc es and Receivers Receivers Structur al Isolation of Sensiti Sensitive ve Floors Use of Massi Massive ve Constr uction uct ion s Isolatio n of Servi Service ce Equipment Equip ment Isolatio n of Sensi Sensitiv tive e Tools

15 16 17

5.0

VIBRATION ANALYSIS ANA LYSIS

20

6.0

VIBRATION VIBRA TION SURVEY

21

6.1 6.1 6.2 6.2 6.3 6.3

Site Characterization Intermediate Intermedi ate Vibratio n Surveys Final (Certif (Certificati ication) on) Vibratio n Survey

21 22 23

7.0

REFERENCES

23

APPENDIX APPENDIX A

ESSENTI ESSENTIAL AL CONCEPTS CONCEPTS IN VIBRATION THEORY

APPENDIX APPENDIX B

TYPICAL TYPICAL VIBRATION VIBRA TION CRITERI CRITERIA A

APPENDIX APPENDIX C

VIBRATIONAL FORCES FORCES


17 18 19 20

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LIST OF TAB TABLES LES

Table Table 1

Application Appl ication Guide for the Generic Generic Vibration Criteria


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LIST OF FIGURES

Figure 1

Generic Generic Vibration Criteria

Figur e 2

An Example Veloc Velocity ity Time-Hist Time-History ory

Figure 3

An Example Example Constant-Bandwi Constant-Bandwidth dth Spectral Plot

Figur e 4

An Example One-Third One-Third-O -Octave ctave Band Spectral Plot

Figure A-1 Frequency, Acceleration, Velocity, and Displacement Nomograph (SI Units) Figure A-2 Frequency, Acceleration, Velocity, and Displacement Displacement Nomogr aph (Engli (English sh Units) Figur e B-1 Vend Vendor or Crit Criteria eria for Selected Selected Steppers Steppers Figure B-2 Vendor Vendor Criteria for Selected Selected Scannin Scanning g Electronic Microscopes Figure B-3 Vendor Vendor Criteria for Selected Selected Electro Electron n Beam Machin Machines es Figur e B-4 B-4 BBN Vibration Vibrati on Crit eria (One-T (One-Thir hir d Octave Analy An alysi sis) s) Figur e B-5 FHA FHA Vibration Vibr ation Criteria (FFT (FFT Analysis ) Figur e C-1 C-1 An Idealized Foot Footfall fall Time-Histor y Figure C-2 C-2 Floor L oading Caused Caused by Walking Walking


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1.0

GENERAL

1.1

Purpose

Modern high-technology facilities increasingly require the use of sophisticated tools and instruments that can only be operated in a nearly vibration free environment. Examples of such equipment include photolithographic tools used in producing very large scale integration (VLSI) chip circuits, platforms for testing inertial guidance systems, and systems that employ lasers or electron beams. Low level floor vibrations caused by the ambient ground vibration, routine movement of workers on the floor, and operation of ancillary equipment can render a facility totally unsuitable for its intended functions, unless the facility is carefully designed to mitigate the effects of these sources of disturbances. The subject area is sometimes referred to as "microvibration design" because the vibration of concern is of an extremely low level, usually way below the threshold of human sensation. Successful operation of high-technology equipment depends on achievement of design goals by the equipment developer and on effective control of the vibration environment within the user's facility. This design guide covers the general considerations essential to the planning and design of highly vibration-sensitive vibration-sensitiv e facilities. For the requisite dynamic analyses, the designer should refer to the Bechtel Design Guides listed in Section 1.2.3. When dealing with an extremely low level vibration environment, these guides may not provide adequate solution to the design problem. In most cases, vibration specialists must be consulted during the planning and design of the facility. 1.2 1.2

Codes, Standards, And Reference Reference Document s

1.2.1 Codes and Standards At the the prese present nt time time,, there there exist exists s no specific specific code or stand standard ard that governs governs the design of vibration-sensitive vibration-sensitiv e facilities. The following standards serve as general guides in the evaluation of vibrations in buildings: 

International Standards Organization (ISO) ISO 4866 Mechanical Vibration and Shock - Vibration of Buildings Guidelines for the Measurement of Vibrations and Evaluation of Their Effect on Buildings


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



American National Standards Institute (ANSI) ANSI S3.29-1983, Guide to Evaluation of Human Exposure to Vibration in Buildings National Environmental Balancing Bureau (NEBB) Procedural Standards for Measuring Sound and Vibration, 1977.

1.2.2 General Reference Documents General references are provided in Section 7. 1.2.3 Bechtel Design Guides The technical backgrounds which are relevant to the performance of dynamic analyses requisite for meeting the intent of this document are contained in the following Bechtel Design Guides: 







Bechtel Design Guide C-101, Wind and Earthquake Design Bechtel Design Guide C-104, Seismic Analyses Structures and Equipment for Nuclear Plants

of

Bechtel Design Guide C-301, Concrete Framed Structures Supporting Vibrating Equipment Bechtel Design Guide C-303, Rigid Block Foundation for Rotating Equipment


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1.3

Design Methodology

1.3.1 Technical Requisites To adequately address a micro-vibration design problem, the engineer should have a working knowledge in the fields of structural dynamics, vibration analysis, vibration measurement, and principles of vibration control. Some important concepts are explained in Appendix A. For general background information, the reader is referred to the references listed in Section 7. 1.3.2 Overview of Methodology Successful control of vibration in facilities requires close attention be paid to the vibration issues during site selection, facility layout, structural design, and mechanical system design and installation. The recommended design methodology for the facility structure involves the following steps: 



1.4

Establishment of maximum acceptable levels of facility floor vibration Identification of vibration sources and strategies to reduce their impact on operation



Design optimization through dynamic modelling and analysis



On-site vibration measurements



Design modification, if required



Certification survey of floor vibration.

Use Of Consultant Services

Because of the specialized nature of the subject area, the services of a vibration specialist should be engaged at the inception of the project and continuing through its successful completion. The scope of services to be provided by the vibration specialist may encompass all or part of the following:


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















Perform vibration measurements to characterize the ambient ground vibration at the site; Develop project-specific vibration criteria based on the operational requirements; Assist in facility planning to minimize vibration impact on the sensitive structures; Assist in configuring the foundations and structures supporting the sensitive equipment; Develop vibration control measures for the facility equipment and the process equipment; Assist in performing the structural dynamic analyses; Perform vibration measurements to verify that the design vibration criteria have been met; Recommend a proper course of action to correct any unacceptable conditions.

2.0

VIBRATION DESIGN CRITERIA

2.1

Appr oach To Criteria Specification

Vibration criteria can be stated in terms of a variety of different kinematic variables and parameters. For example, some criteria may be stated in terms of the maximum allowable peak-to-peak displacement, or peak velocity, or peak acceleration, or a combination thereof, while others may be stated in terms of their root-mean-squared (or rms) values. A more complete specification would contain a statement as to the frequency range of importance and how the allowable limit varies with frequency. In that case, the criteria may be expressed in terms of the maximum allowable spectral density values or the maximum allowable narrow-band spectral amplitudes over the range of frequencies of interest. The latter specification may be based either on proportional bandwidth analyses or on constant bandwidth analyses. In terms of proportional bandwidth analyses, typical


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industry practice uses one-third octave spectra to describe the frequency dependence of the criteria. Some criteria may impose requirements on the vertical floor vibrations that are different from those imposed on the horizontal floor vibrations. Still others might place a limit not only on the linear motion but also on the angular motion of the structure as well. The nature of the operation for which the facility is designed will dictate how the criteria can best be stated in a given case. 2.2

Project-Specifi c Criteria

A rational basis needs to be established in developing vibration criteria for a given facility. Typical criteria developed from vendor specifications may be a combination of the example criterion curves shown in Appendix B. If the vibration criteria are overly relaxed, the usefulness of the facility could be impaired. On the other hand, unreasonably stringent criteria can result in excessive cost. When the vibration criteria are furnished by the client, the criteria should be examined in relationship to the known conditions of the site. The client should be informed if implementation of the imposed criteria can be expected to lead to excessive cost. To avoid possible misunderstanding, the environment conditions and facility operations or activities that are covered by or excluded from the application of a criterion should be unequivocally stated in the criterion document. For example, a given vibration criterion may apply to floor vibration caused by the general ambient conditions at the site but not that caused by an exempt event such as the passing of a train on a nearby track or the operation of emergency diesel generators within the facility. In most cases, acceptance of the structural design should be based on the measurement of vibrations related to all identified sources but excluding those vibrations caused by machines or tools installed by the client or by others. 2.3

Generic Vibration Criteria

Two different generic vibration criteria have been proposed. One was developed by Acentech/Bolt Beranek and Newman (BBN) Laboratories and the other, by Frank Hubach Associates (FHA).


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The BBN criteria (Reference 9) were modeled after the ISO guidelines for evaluating building vibrations. The ISO standard establishes acceptable vibration levels in terms of a constant vibrational acceleration between 4 and 8 Hz and a constant vibrational velocity between 8 and 100 Hz. The BBN criteria are expressed by constant one-third octave band velocity levels over the frequency range of 4 to 100 Hz with an option of specifying a less restrictive constant acceleration level between 4 and 8 Hz. See Appendix A.1 for a definition of the standard one-third octave bands. Five levels of allowable vibrations (ranging from A to E) are proposed for different types of equipment operation and technology, as described in Appendix B. The FHA criteria (Reference 11) specify that the vibration be described by FFT (Fast Fourier Transform) or narrow-band spectra with an effective bandwidth of 0.1875 Hz. The criteria define five levels of allowable vibration (ranging from D to AA), each corresponding to different types of vibrationsensitive equipment and technology, over a frequency range of 5 to 50 Hz. The criteria are described in Appendix B. These two criterion specifications are basically different. For example, the BBN criterion D level most closely corresponds to the FHA criterion A level. However, if narrow-band vibrations are present at lower frequencies, the BBN criterion would be the less restrictive. But, if broad-band vibrations predominate at the higher frequencies, the BBN criterion would be the more restrictive of the two. In the absence of specific client-imposed requirements, a set of vibration criteria similar to those proposed by BBN is recommended herein for the design of vibration-sensitive facilities. An appropriate criterion level may be selected from among the standard vibration classes defined in Figure 1. The criteria are formulated in terms of rms velocity spectral amplitudes in each of the 21 one-third octave bands from 1 Hz to 100 Hz. The spectrum of a standard vibration class has a constant velocity spectral value from 8 Hz up. At lower frequencies, it assumes a constant acceleration spectral amplitude. Each vibration class is identified by the constant rms velocity spectral amplitude above 8 Hz. For example VC 25 corresponds to the criterion class having an allowable velocity spectral amplitude of 25 microns/sec. This method of designation is consistent with the practice of ANSI and ISO in similar circumstances. The suggested application for each vibration class is indicated in Table 1.


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For sensitive equipment which are supported by pneumatic vibration isolators, relaxed vibrational velocity requirement at frequencies below 8 Hz could lead to excessive motion of the mounted equipment. In that case, a constant allowable velocity spectral value from 1 Hz to 100 Hz has been proven to be more appropriate. A vibration criterion class so extended is given the designation of VCX. 3.0

SOURCES OF VIBRATION

3.1

The Vibration Environment

Vibration-sensitive systems in a high-technology facility are affected by vibrations that originate both outside of and within the facility. Internal sources generate vibrations near these systems and can have a direct impact on their operation. These are also classified as direct sources. In many cases, these vibrations are generated by the vibration-sensitive systems themselves as well as by the mechanical systems directly serving them. External sources consist of activities that occur on site or off the site; such as, road and rail traffic, heavy manufacturing, construction, wind and ultra-low frequency sound, and other uses of machinery. All these external activities pump vibrational energy into the soil which transmits the vibrations to the foundations of the facility. The ground vibration measured on the facility foundation represents the combined effect of disturbances from various sources. Internal sources that create vibrations within a facility include personnel activities and operation of service and production-related equipment. Personnel activities include walking, closing doors, pushing handcarts, and operating forklifts, etc. Experiences (Reference 10) have indicated that on elevated floors the most significant source is the footfalls associated with personnel walking on the floor on which the sensitive equipment is located. Repair and construction activities will contribute to the vibration on an irregular basis and will be hard to quantify. Large equipment items that service the building and the clean rooms are generally located some distance away from the vibration-sensitive areas. As a result, their effect may be less than that produced by the production-related equipment. Direct sources that have a substantial effect on the vibration-sensitive systems include infrasound generated by the cleanroom air-conditioning


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system, airflow, and the operation of the systems themselves. For some ultra-sensitive systems, the internal vibrating elements within the systems, usually considered as minor sources, might constitute a major internal factor in determining the upper limit of system performance. The relative significance of the various sources depends on many factors, including the site condition, the arrangement of the facility, the design of the structure, and the governing vibration criteria. The commonly occurring sources of vibration are discussed below. 3.2

Ambient Ground Vibrations

The ambient ground vibration at a given site is the aggregate result of vibrations caused by a myriad of sources near and far. It can be expected to vary significantly from site to site. Therefore, the ambient ground vibration of a given site can only be characterized through measurements conducted at the site. Since the vibration typically will be different in different directions, the data normally should be comprised of vibrations measured in three mutually perpendicular directions at each selected sampling location. The vibration measurement results can be expressed in the form of time histories of the vibrational acceleration, velocity, or displacement at the measurement locations. An example velocity time history is shown in Figure 2. The vibration data may also be represented in the frequency domain in the form of spectral functions of any one of the three kinematic variables. The most commonly used spectral formats are the spectral density functions and the one-third-octave-band spectra. Example spectral functions are shown in Figures 3 and 4. It should be noted that time history data can always be converted into spectra data if need be, but the reverse is not true. 3.3

Machine Generated Vibrations

Machine generated vibrations can come either from the service machinery or the production-related machines. Service machinery includes all mechanical and electrical equipment that either is part of the building system or is installed by the user. It includes air-conditioning and distribution fans, chillers, cooling towers, furnaces, liquid pumps, compressors and vacuum pumps, etc. as well as elevators and mechanically actuated doors and loading platforms. The production-related machinery and sources of


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disturbances include the cleanroom air-conditioning system, airflow, X-Y stages, cooling fans, vacuum pumps, coolant piping, etc. The effects of vibration resulting from machinery operation can be reduced by keeping the machines as far away as possible from the sensitive areas. With proper structural isolation, the vibration from the service-related machines usually can be effectively dissipated by the soil. Grounding of the vibrational energy transforms it into a part of the ambient ground motion, which is addressed in Section 3.2. The forces generated by various machines or tools can be found in the Bechtel Design Guides listed in Section 1.2.3. A few commonly encountered forcing functions are provided in Appendix C. 3.4

Footfall Excitation

Footfall induced vibration is a major design factor for elevated floors, but is usually less significant for slabs on grade. On elevated floors, the footfall impacts on the floor set the floor structure into motion, subjecting any equipment resting on it to corresponding vibrations. Such vibrations tend to be most severe at mid-span and less so near supports of the floor. The forcing function generated by footfalls is given in Appendix C. 3.5

Traffic Excitation

The most severe vibration associated with road traffic results from heavy vehicles moving along roads with surface irregularities; such as speed bumps, potholes, and expansion joints. These effects are particularly significant for on-site traffic in the vicinity of vibration-sensitive areas of the facility. Similarly, the joints in rails are the major cause of vibrations induced by rail traffic. 3.6

Airbor ne Noise Effects

Some types of high-technology equipment exhibit considerable sensitivity to airborne noise, air pressure fluctuations associated with ventilation air flow, and pressure pulsations produced by the sudden opening or closing of doors in the pressurized areas. In facilities that involve "clean room" or "climatic test chamber" installations which require large amounts of air flow, audible


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noise can be a significant problem from both the equipment performance and personnel comfort standpoints. Noise sensitivity data are available for only a very limited number of equipment items. Experience has shown that the noise sensitivity of microelectronic production equipment typically is the greatest at frequencies in the 20 to 250 Hz range. Typical noise levels in operating facilities have been found to be acceptable for most equipment as well as the personnel. Optical equipment, which can be affected by density fluctuations of the air in the optical path, may require a considerably more stringent limitation on the airborne noise. In some situations, pressure fluctuations associated with low-frequency sound, with air flow, or with flow pulsations also may constitute a significant source of vibration of the floors that support sensitive equipment. This is the case particularly for a facility that requires a large amount of air flow and is located on an upper floor with large spans and relatively flexible construction. Thus, it would be appropriate to address the noise control requirements simultaneously with the vibration control requirements. 4.0

METHODOLOGY FOR VIBRATION CONTROL

4.1

General Considerations

Successful control of vibration in a facility requires paying close attention to potential vibration problems during site selection, facility layout, structural design, and mechanical system design and installation. As a prerequisite, the ground vibration level of the selected site should be less than the vibration level to be achieved within the building. Subsurface soil conditions generally dictate the propagation of vibrational energy through the soil. The amount of vibration attenuation depends on soil type, compaction, moisture content and height of water table. A detailed vibration study of a site should be conducted to define the vibrational characteristics of the site, and to locate a region of minimum vibration on the basis of survey data. Through careful planning, the vibration-sensitive facility can be located in a low-vibration region and would provide the best assurance for meeting the restrictive vibration criteria imposed on the facility.


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The layout of the facility can have a significant impact on vibrations originating from internal sources. The basic concepts for minimizing vibrations are fairly simple. Sources of high vibrations; such as compressors and fans for air distribution, and generators for emergency power; should be located on grade and as far away from the sensitive areas as possible. Routing of pipes and ducts should be accomplished without hanging them from the bottom of the sensitive floor slabs. Isolation hangers are only partially effective for reducing transmission of vibration into the structure. Silencers should be used to control airborne noises. Adequate space for ducts and silencers should be provided during the planning stage. Corridors that will be the major thoroughfares in the facility should be located away from the vibration-sensitive areas. Routes for handcarts and electric vehicles must be carefully chosen to avoid traversing over critical areas. Passageway surfaces must be smooth and free from bumps at the isolation joints. The design of the structural system is an important component in the overall design process for an ultra-low vibration facility. The basic design concept is to increase the structural stiffness as much as practical. In a conventional building, the fundamental structural frequencies usually fall in the range of 5 to 7 Hz. To obtain a sufficiently stiff floor system for supporting the clean rooms, deep waffle slabs with closely spaced columns are normally used. This type of construction can result in vertical fundamental frequencies greater than 35 Hz. Shear walls may be required to achieve an adequate horizontal structural stiffness. Vibrations due to service equipment can be reduced through proper location, selection and maintenance. Reciprocating compressors typically generate more vibration than screw type compressors and should be avoided if possible. Specifications for rotating equipment should include more restrictive balancing criteria than normal. Air distribution systems should be designed for low air flow velocity in the ducts which would allow selection of low-speed fans. Service machinery should be mounted on vibration isolators to reduce the amount of vibration transmitted to the structure. 4.2

Site Selection

The vibration environment of a site reflects the geological features of the site and its proximity to cultural sources of disturbances such as highway and railway traffic and other industrial operations. In extreme cases, natural sources of disturbance such as wind, waterfalls and tidal waves may also


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play a role. Because the allowable vibration of a facility can approach the ambient vibration of a bare site, the existing vibration environment of a site usually is an important consideration in site selection. Attempts at attenuating ground vibration through the use of barriers or trenches have been found to be by and large futile for the range of frequency that is of concern to typical vibration-sensitive operations. The vibration level of the selected site must be less than the vibration level desired within the building, since the building will tend to amplify ground vibration and will also introduce new vibration sources. It is suggested that the ambient ground vibration level for the site be 10 dB (i.e., a factor of 3) below the allowable level within the building. A site having a high water-table should be avoided because the saturated soil will reduce the ability of the soil to attenuate vibrations. In general, these facilities require the underlying soils to be stiff and free of features that might expand or settle over time. 4.3

Site Planning

At the site, roadways, both on and off site, should be paved and free of potholes and undulations. Routes of delivery trucks and location of delivery docks need careful consideration to avoid a close approach to the sensitive areas. On-site roads should be kept free of speed bumps. The highly vibration-sensitive operations of a facility should be grouped together and be located as far away as possible from the vibration generating activities. A method for calculating vibration attenuation through the soil is given in Appendix C. 4.4

Structural Isolation Of Sensitive Floors

Floors and foundations supporting the sensitive equipment should be structurally isolated by adequate air gaps from other structural elements of the building, including the perimeter walls and the columns that extend to the roof. The mechanical equipment, to the extent feasible, should be located off the foundation pads for the sensitive equipment or tools. Incidental bridging of the gaps by partitions or equipment straddling the gaps should be guarded against so as not to detract from the effectiveness of the isolation joints. Fluctuating wind load acting on the building frame can shake a building and cause it to vibrate. This source of excitation is more important for tall and slender structures than for structures that have a squat profile. To mitigate


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the effects of wind, it may be necessary to structurally isolate the vibration sensitive floors from the building framework which is designed to resist the wind load. This means that the vibration sensitive floors normally should be placed near the center of the building and away from the perimeter walls and the columns that support the roof structure. It also means that a building with a large clear span would tend to be better than one that contains many interior columns. 4.5

Use Of Massive Construc tions

4.5.1 Slab-on-Grade Construction In most cases, placing the vibration sensitive floor directly on the ground is likely to achieve the lowest vibration for a given site. The underlying soil should be competent and be properly compacted. A relatively thick slab is usually required to provide the needed stability. The controlling factor in designing a soil supported slab is the lumped-mass natural frequencies of the soil-foundation system. These frequencies should be designed to be substantially different from the primary excitation frequencies. A frequency separation of "50% is normally desirable. An isolation joint should be created around the entire perimeter of the sensitive slab. Interior columns should be placed on isolated footings rather than resting on the slab. For a well constructed slab and a quiet site, it should be feasible to meet the VC 3.15 criterion class. Use of vibration isolators underneath the concrete mat is usually undesirable. 4.5.2 Table-Top Construction A table-top type structure consisting of an elevated floor supported by columns is the most common type of construction used for manufacturing purposes. The columns may be supported by individual footings or by a monolithic concrete mat foundation. Both the floor and the foundation should be completely isolated from the rest of the building. This type of structure is more susceptible than slab-on-grade construction to ground vibration or other types of disturbances such as workers walking on the floor. The controlling factor is the natural frequencies of the structure including the effect of the soil springs. It is usually necessary to proportion the structure so that the minimum structural frequencies would be above the region where the ground vibration tends to reach a peak. Since ambient ground vibrations


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tend to peak in the frequency range of 5 to 15 Hz, it would thus usually be desirable to keep the structural frequencies above 15 Hz. Shear walls below the floor usually will be required to achieve such a high frequency for lateral vibrations. For vertical floor vibrations, still higher frequencies are usually required to control the vibration caused by footsteps or carts rolling over the floor. A natural frequency of above 25 Hz is usually required. The vertical floor vibration can be more effectively controlled with closely spaced columns than with a thick floor slab. In principle, high natural frequencies can be accomplished with high stiffness or low mass. For typical structures, these two parameters are interrelated and can not be varied independently. To achieve a low level of vibration, the floor structure should be both stiff and massive. This means that heavy concrete structures should in general perform better than heavy steel structures. A concrete structure normally will also exhibit less response at resonant frequencies due to its inherently higher damping capacity. In practice, a well designed table-top structure should be able to achieve the vibration class of VC 12.5. Lower vibration levels can be achieved by supporting the floor on piers that extend down to a stable rock formation, provided that the piers are isolated from the surrounding soil. 4.6

Isolatio n Of Service Equipment

A typical high-technology facility will require a large array of high-powered mechanical equipment to support the operation of the facility. In general, such equipment should be placed on structures isolated from the vibration sensitive floors. Nonetheless, such equipment normally would represent a significant source of disturbance and must be properly dealt with no matter where they are located. The conventional approach to isolating a vibrating machine is to place it on resilient mounts. If the forcing frequency is sufficiently higher than the natural frequency of the mounted machine, the force transmitted to the foundation would be significantly reduced. The amount of force reduction can be augmented with the use of an inertial base for the machine. The inertial base is usually made of concrete and should weigh much more than the weight of the machine itself. Lateral stops should be used to ensure adequate lateral stability of the resiliently mounted equipment.


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Ducts or pipes carrying air or water can be a significant source of vibration, unless the flow speed is kept low. In general, isolation (spring) hangers can be used to reduce the transmission of vibrations into the structure. However, they are usually not effective for isolating low frequency vibrations. In most situations it would be best to avoid supporting major pipes or ducts directly off the sensitive floors. 4.7

Isolatio n Of Sensitive Tools

A facility invariably will contain tools or machines with varying degrees of sensitivity to vibration. In many cases, it may be more practical to vibration isolate the few most sensitive tools from the floor structure rather than to impose a very stringent vibration criteria on the entire floor. Vibration isolation of tools is usually done with pneumatic springs in order to attain an ultra low resonant frequency and yet maintain good stability vibration. However, isolation of tools that contain internal sources of vibration is generally undesirable. The vendor of the tools should be consulted if vibration isolation of their tools is deemed necessary. 5.0

VIBRATION ANALYSIS

When the input disturbances are defined, whether in terms of ground vibration or applied forces, the response of the structure can be computed using standard structural dynamic analysis procedures described in the Bechtel Design Guides referenced in Section 1.2.3. The analysis may involve a step-wise numerical integration of the equations of motion. In that case, the excitation function would be given in the form of time-histories. The integration time-step, ∆ t in seconds, should satisfy the condition: ∆t ≤

1 5 f max

1

where f max is the maximum frequency of interest, in Hz. The analysis may also be carried out in the frequency domain. The excitation functions would then need to be expressed in terms of the spectral functions or the spectral density functions. This type of analysis is more expedient when the vibration criteria are expressed in terms of spectral values.


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However, due to the usually complex nature of the structure, the wide frequency band of interest, and the many different types of sources of vibration involved, detailed finite-element modelling of the structure is usually impractical, except in extreme cases. In most cases, a lumped-parameter type of analysis may be the only prudent approach. Considerable experience is required in determining what type of analysis is required for a particular facility and how it should be performed. 6.0

VIBRATION SURVEY

Because of the high degree of uncertainty that is inherent in designing a facility to achieve a very low level of vibration, vibration survey should be considered as an integral part of the design and construction process. Vibration surveys are needed to provide essential input data and to verify satisfactory performance of individual components or the facility as a whole. 6.1

Site Characterization

The existing ambient ground vibration of a site depends on the geology of the area and its proximity to natural and cultural sources of excitation. The ground vibration at a bare site must be ascertained through a vibration survey. Generally, the survey needs to establish the prevailing and the worst-case vibrations at the site. Vibration measurement may be obtained either with velocity sensing transducers or with accelerometers. When a velocity transducer is used, the output would be a time history of the vibrational velocity. When an accelerometer is used, the output would be a time history of vibrational acceleration. The acceleration time history can be converted into an equivalent velocity time history through an integration over time which can be accomplished either digitally or in an analog fashion. The equivalent displacement time history can be obtained from the velocity time history in a similar fashion. However, if displacement data are desired, they should be obtained from the velocity transducer data rather than from the accelerometer data, because double integration of the accelerometer data can lead to excessive error at low frequencies. At each survey point, the data should be obtained in 3 mutually perpendicular directions (e.g., vertical, east-west, and north-south). The sensors should be


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installed at a depth that corresponds to the anticipated elevation of the bottom of the foundations because sensors placed on the ground surface tend to be unduly influenced by nearby activities. The site vibration data may be presented in the form of time histories or spectral functions. Time histories are more appropriate for documenting transient events such as the passing of a heavy truck on a nearby roadway. For steady-state vibrations, spectral data are preferred because the spectral amplitudes can be averaged over many samples to obtain a more representative picture of the condition at the site. Enveloping of the sample spectra can be done to generate the worst-case design condition. 6.2

Intermediate Vibration Surveys

Intermediate vibration measurements should be made to confirm the design calculations and to identify potential deficiencies at an early stage of construction. Because the interaction of the foundations and the underlying soil is the least understood element of the design, it is prudent to measure the vibration of the foundations as soon as they are complete. The vibration data obtained on the foundations may be used to validate the adequacy of the design of the upper structures. Should the outcome of the measurements indicate that the allowable vibration might be exceeded, appropriate mitigation measures must then be devised and implemented. Examples of the mitigation measures include the addition of shear walls or intermediate columns for the floor supporting the sensitive equipment. Vibration measurements should also be made as soon as the major service equipment become operable. This would provide an early indication of potential problem areas and an opportunity to resolve these problems with minimal cost and schedule implications. 6.3

Final (Certificatio n) Vibration Survey

Upon completion of a vibration sensitive facility, a complete vibration survey of the facility should be conducted. The survey would serve to document the performance of the facility against the design criteria and would provide a valuable benchmark for future reference. 7.0

REFERENCES7.0


REFERENCES7.0

REFERENCES

PAGE 21 OF 53

1.

C. M. Harris and C. E. Crede, Eds., Shock and Vibration Handbook, 2nd Ed. McGraw-Hill, 1976.

2.

F. E. Richart, Jr., J. R. Hall, Jr., and R. D. Woods, Vibrations of Soils and Foundations, Prentice-Hall, 1970.

3.

D. D. Barkan, Dynamics of Bases and Foundations, McGraw-Hill, 1962.

4.

E. E. Ungar, D. H. Sturz, and C. H. Amick, "Vibration Control Design of High Technology Facilities", Sound and Vibration, pp. 20-27, July, 1990.

5.

S. Wu and E. V. French, Concrete Framed Structures Supporting Vibrating Equipment, Bechtel Civil Design Guide C-301.

6.

S. Wu and J. L. Suderman, Rotating Equipment Foundations, Bechtel Civil Design Guide C-303.

7.

A. H. Hadjian, Ed., Seismic Analyses of Structures and Equipment for Nuclear Power Plants, Bechtel Civil Design Guide C-104.

8.

N. Owen and R. Hale, "Factors in the Design and Selection of Vibration-Sensitive Equipment," SPIE Vol. 1619 Vibration Control in Microelectronics, Optics, and Metrology, 1991.

9.

C. G. Gordon and H. Amick, "Vibration and Noise Control in State-ofthe-Art Clean Rooms," Proceedings of the Microcontamination Conference, 1989.

10.

C. G. Gordon, "A Study of Low-Frequency Ground Vibration in Widely Differing Geographic Areas," Proceedings of Noise-Con 87, 1987.

11.

Frank Hubach Associates, Building Vibration Criteria, FHA Brochure.


PAGE 22 OF 53

Table 1 Application Guide for the Generic Vibration Criteria (After Ref. 4) Vibration Class(6)

Equipment or Use Operating rooms; Surgery, Bench microscopes at up to 100x magnification; Laboratory robots

VC 100

Bench microscopes at up to 400x magnification; Optical and other precision balances; Coordinate measuring machines; Metrology laboratories; Optical comparators; Microelectronics manufacturing equipment (1)

VC 50

Micro-surgery, eye-surgery, neuro-surgery; Bench microscopes at up to 10,000x magnification; Optical equipment on isolation tables; Microelectronics manufacturing equipment (2)

VC 25

Electron microscopes at up to 25,000x magnification; Microtomes; Magnetic resonance imagers; Microelectronics manufacturing equipment (3)

VC 12.5

Electron microscopes at up to 50,000x magnification; Mass spectrometers; Cell implant equipment; Microelectronics manufacturing equipment (4)

VC 6.3

Laser and optical research systems; Microelectronics manufacturing equipment

VC 3.15

(5)

(1) Inspection, probe test, and other manufacturing support equipment. (2) Aligners, steppers and other critical equipment for photolithography with line widths of 3 microns or more. (3) Aligners, steppers and other critical equipment for photolithography with line widths of 1 micron.


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(4) Aligners, steppers and other critical equipment for photolithography with line widths of 1/2 micron; including electron-beam systems. (5) Aligners, steppers and other critical equipment for photolithography with line widths of 1/4 micron; including electron-beam systems. (6) For equipment supported by pneumatic isolators the VC class should be replaced by the corresponding VCX class.


PAGE 24 OF 53

Figure 1 Generic Vibration Criteria (After Ref. 4)


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Figure 2 An Example Velocity Time-History


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Figure 3 An Example Constant-Bandwith Spectral Plot


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Figure 4 An example One-Third-Octave Band Spectral Plot


PAGE 28 OF 53

APPENDIX A ESSENTIAL CONCEPTS IN VIBRATION THEORY


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A.1

DESCRIPTION OF VIBRATION

Vibration is a small-amplitude motion of an object about its mean position, i.e.: u = u(t) 2

and

1 Lim

T

T

_ o u(t)dt = 0 3

T→∞

where u is the instantaneous displacement from the mean position. A similar relationship holds true for the corresponding instantaneous velocity or acceleration of the moving object which are alternative descriptors of the same motion. The most direct way to represent the vibration of a structure is simply to provide the function that describes the variation of the chosen variable or descriptor over time. This time-varying function is often referred to as a time history. For vibration control considerations, the attributes of a given time history that is of the greatest interest usually include the peak amplitude of the descriptor, Xpk, and the root-mean-squared (rms) value, X rms , of the same function. The value of Xrms is defined as: ⎡ 1 T 2 ⎤ _ o x (t)dt ⎥1/ 2 4 Xrms = ⎢ ⎣ T ⎦

where T is the duration of integration. A vibration that is simple harmonic in nature is most naturally described in the time domain. However, for vibrations that are random in nature it is generally more expedient to describe the motion in terms of its spectral attributes. The spectral representations commonly used to describe a random motion include the constant bandwidth spectra, the constant percentage bandwidth spectra, and the (power) spectral density functions. In a constant bandwidth spectral plot, the rms magnitude of the descriptor is plotted against the center frequencies of a series of contiguous frequency bands which span the frequency range of interest. In principal, any bandwidth can be used. However, the preferred bandwidth is 1 Hz in order to


PAGE 30 OF 53

standardize the presentation. A linear frequency scale is normally used in such a plot. The amplitude scale may be either linear or logarithmic. In a constant percentage bandwidth spectral plot, the frequency band of interest is broken into a number of contiguous bands whose bandwidth is a constant percentage of the individual band center frequencies. The most commonly used constant percentage bandwidths are the octave bands and the one-third octave bands. The one-third octave band presentation is preferred over the octave band presentation because the former provides greater frequency resolution. The relationship of the lower and upper limiting frequencies, f l and f u , of a one-third octave band to the center frequency, f o , is given by the following equation: f o

= 21/6 f l = 2-1/6 f u 5

where f 1 is the lower limiting frequency and f u is the upper limiting frequency of any given band defined by the center frequency f o . The sequence of the standard one-third octave band center frequencies from 1 Hz to 100 Hz is as follows: 1.0, 1.25, 1.6, 2.0, 2.5, 3.15, 4.0, 5.0, 6.3, 8.0, 10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80, and 100 Hz Extension of this sequence to lower and higher frequencies is self evident. In a constant-percentage bandwidth spectral plot the logarithmic scale is usually used for plotting the center frequencies because this would result in regular spacing of the band center frequencies. The vertical axis of a spectrum would represent the rms spectral values, X i , in each frequency band. Depending on the descriptor used, the unit is either (in), or (in/sec) or (in/sec 2). The overall rms value of the variable, X2rms 6 is given by: N

X

2 rms

=

∑

2 Xi 7

i

where N is the total number of bands O:\WINWORD\3DG\C01\007-02.DOC

PAGE 31 OF 53

When the data are presented in the spectral density form, the vertical axis normally would have the unit of either (in) 2/Hz, (in/sec)2/Hz or (in/sec2)2/Hz. If the spectral density function S 2 is known, the corresponding band level values can be computed from S 2 through integration over the appropriate bandwidth, namely: 2

Xi

= _ f l f u S2 df 8

where f l and f u are respectively the lower and upper limiting frequencies of the band in question. Conversely, if the band level values are given, the average spectral density in each band, S i, would be given by: 2

2 i

S

Xi = B

9

where B is the bandwidth. A.2

THE DECIBEL SCAL E

The decibel scale is often used for convenience in expressing the magnitude of spectral values. The decibel value of a quantity, usually referred to as its level, expresses the ratio of the quantity to a standard reference quantity of the same type. Specifically, the level of a "power-like" variable Q, L Q in decibel, would be given by the formula: LQ = 10 log10

Q Q ref

, dB re Q ref 10

where Qref is the reference quantity. To avoid misunderstanding, whenever a decibel value is quoted, the reference quantity must be given, unless the reference quantity is clearly understood by convention. For calculating the level of a "linear" quantity such as the magnitude of displacement, velocity or acceleration, Q is always taken as the quantity in question squared. For example, the level L V of velocity V is given by: 2

LV = 10 log10


V

2

Vref

, dB re Vref 11

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No standard reference values have been established for displacement, velocity, or acceleration. A.3

CONVERSION OF FREQUENCY SPECTRA

In Section 2.3 the generic vibration criteria are expressed in terms of rms velocity in one-third octave bands. However, client or equipment vendor's specifications may be expressed in terms of allowable displacement or acceleration rather than velocity. In such a case, it would be necessary to convert a given velocity spectrum into the corresponding displacement or acceleration spectrum. Since within the bandwidth of a one-third octave band the velocity waveform is nearly sinusoidal, such a conversion can be made using the following formulae: u =

v 2π f

12

a = 2π f • v 13

where f is the frequency, in Hz, of the center frequency of the one-third octave band in question. If v is in microns per second then u would be in microns and a, in microns per second per second. Similarly, if v is in (in/sec) then u would be in (in) and a, in (in/sec 2). This relationship is expressed graphically in the "tripartite nomograph" as shown in Figures A-1 and A-2. This graph contains 3 ordinates versus a single abscissa. The abscissa denotes frequency and the 3 ordinates denote respectively, displacement, velocity, and acceleration. At any point on this graph the values of u, v, a and f satisfy the above two equation. This graph paper is useful for comparing vibration data or criteria which are expressed in terms of different variables. It should be noted that this conversion procedure does not apply to gross attributes such as peak or overall rms values. A.4

RELATING PEAK VALUES TO RMS VALUES

For a simple harmonic motion of the form: u = U pk cos wt


PAGE 33 OF 53

14 where w is the circular frequency in radians per second and t, in seconds. the peak value Upk is related to the corresponding rms value by the relationship: U pk

2 Urms 15

=

For a motion that contains many frequencies there exists no definite relationship between the peak value and the rms value. In the case of an idealized Gausian process, the peak values follow the Raleigh distribution: P ( X pk > Y• Xrms)

⎛ Y2 ⎞ = Exp⎜ - ⎟ 16 ⎝ 2 ⎠

where P is the probability that X pk > Y @ Xrms , Y being an arbitrary positive number. Some example P and Y values are given below: Probability of Exceedance P, %

20

10

5

2.5

1

Y

1.79

2.15

2.45

2.72

3.03

For random vibrations which have a finite bandwidth, the probability of exceeding a given value of K @ X rms would be less than what is calculated from the above formula. For typical ground vibrations and in the absence of information to the contrary, the peak value may be assumed to be 3 times the rms value. The above relationships do not apply to impulsive type phenomena which generally have much higher peak to rms ratios.


PAGE 34 OF 53

A.5

RELATIONSHIP BETWEEN RESPONSE SPECTRA AND THE POWER SPECTRAL DENSITY FUNCTIONS

For evaluating structural response to an imposed ground motion, certain analysis procedures require that the ground motion be expressed in terms of its response spectra. If the ground vibration data are given in the time-history form, this presents no problem because the needed response spectra can be computed from the given time histories. If the ground vibration data is only available in the form of power spectral density functions, the following approximate formula may be used to make the necessary conversion: f(w)

where

=

⎡ -π ⎤ 2 Ln(1 - r) ⎥} 17 R (w)SLASH{ - 2 Ln ⎢ π w ⎣ wT ⎦ 2η

f(w) is the power spectral density function of the given ground acceleration R(w), the response spectrum of the same ground acceleration 0, the damping ratio associated with the response spectrum, R(w) w, the circular frequency T, the duration of the time history r, the probability that the value R(w) will be exceeded. In practice, r may be taken as 0.15 and Ln denotes the natural logarithmic function.

This formula may also be used to find the corresponding power spectral density functions when the floor response spectra are given. Reference: M. K. Kaul, "Stochastic Characterization of Earthquakes Through Their Response Spectra," Earthquake Engineering and Structural Dynamics, Vol. 6, 497-509 (1978).


PAGE 35 OF 53

Figure A-1 Frequency, Acceleration, Velocity, and Displacement Nomograph (SI Units)


PAGE 36 OF 53

Figure A-2 Frequency, Acceleration, Velocity, and Displacement Nomograph (English Units)


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APPENDIX B TYPICAL VIBRATION CRITERIA


PAGE 38 OF 53

Figure B-1 Vendor Criteria for Selected Steppers


PAGE 39 OF 53

Figure B-2 Vendor Criteria for Selected Scanning Electron Microscopes


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Figure B-3 Vendor Criteria for Selected E-Beam Messages


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Figure B-4 BBN Vibration Criteria (One-Third Octave Analysis)


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Figure B-5 FHA Vibration Criteria (FFT Analysis)


PAGE 43 OF 53

APPENDIX C VIBRATIONAL FORCES


PAGE 44 OF 53

C.1

MACHINE GENERATED FORCES

C.1.1 Rotary Machines

The force generated by a rotary machine is primarily related to unbalance of the rotating mass and can be computed from the formula: F(t)

where

=

2 Mew cos (wt) 18

M is the mass of the rotor e, the mass eccentricity of the rotor w, the circular frequency of the rotor.

This force acts in the plane that is perpendicular to the axis of rotation. The magnitude of rotor eccentricity can usually be obtained from the vendor of the machine. When vendor data are not available, the rotor eccentricity can be estimated based on Civil Design Guide C-301 or C-303. C.1.2 Reciprocating Machines

The forces generated by a single cylinder of a reciprocating machine can be computed from the following formulae: 2

where

Fx (t)

r = ( M rec + M rot ) rw cos wt + M rec w 2 cos 2 wt 19 L

Fy (t)

=

2

2 M rot rw sin wt 20

Mrec is the mass of the reciprocating part Mrot , the mass of the rotating part r, the radius of the crank L, the length of the connecting rod w, the circular frequency of the crank shaft.

The force Fx acts in the direction of the cylinder axis and the force F y acts in the plane perpendicular to the axis. For a multiple-cylinder machine the resultant forces from the combined action of all the cylinders would depend on the design of the machine and must be obtained from the vendor of the machine. O:\WINWORD\3DG\C01\007-02.DOC

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Ref. Don Hartog, Mechanical Vibrations, Chapter 5, p. 177, McGraw-Hill, 1956. C.1.3 Impact-Producing Machines

For a impact type machine, the forcing function can usually be represented by impulses of a triangular shape. Each impulse can be characterized by a duration )t and a peak force F m which must be obtained from the vendor of the machine. In the frequency domain such a force can be described by a spectral density function that is nearly constant from zero to a frequency f max, given by the formula: f max

=

1

∆t

21

The spectral density (i.e. force per unit frequency) is given by: F(w)

=

2 Fm ∆ t 22

Reference: R.B. Randall, Frequency Analysis, pp. 307-308, Bruel and Kjaer, 1987. C.2

FOOTFALL GENERATED FORCES

C.2.1 The Forcing Function

The loading function created by footfalls depends on the walking speed of the walker. An idealized load time-history generated by a single footfall is shown in Figure C-1. Figure C-1(a) depicts the general shape of the forcing function and Figure C-1(b) defines the parameters of the forcing function. The moving loading pattern caused by a walker is as shown in Figure C-2. At a normal gait there is an overlap in dwell time between the left foot and the right foot. The duration of overlap also depends on the walking speed. To calculate the dynamic response of a structure to footfalls, the moving load may be replaced by a load that is fixed in space. In this case, the applied dynamic load should be taken as the difference between the load curve shown in Figure C-1 and the weight of the walker, w. C.2.2 Estimation of Floor Response to Footfalls O:\WINWORD\3DG\C01\007-02.DOC

PAGE 46 OF 53

The response of a floor structure to footfall induced forces can be modelled using the standard computer program for dynamic response calculations. A simplified procedure for estimating the peak response is presented in this section. The dynamic response of a floor slab to an idealized footstep force impulse consists of a series of vibrations superimposed on a static deflection of the slab under the weight of the walker. The vibration is a damped oscillation at the frequency of the first vertical mode of the slab that occurs near the beginning and end of each footfall impulse. In between, the static deflection Ustat has the value of F m k

U stat =

where k is the stiffness of the slab. The dynamic displacement, Umax, can be estimated by the equation U max = Am U stat

where Am =

2 2

_1 - 4 f ot 2o _

in which f o is the fundamental vertical resonant frequency and t o is as shown in Figure C.1(a). In the above expression the damping effect has been ignored in order to be on the conservative side. The corresponding maximum vibrational velocity and acceleration can be calculated using the equation V max = 2π f o U max 2

Amax = (2π f o ) U max

Reference: E. E. Ungar and R. W. White, "Footfall-Induced Vibrations of Floors Supporting Sensitive Equipment," Sound and Vibration, pp. 10-13, Oct. 1979. C.3

DYNAMIC COMPONENT OF WIND LOAD


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The dynamic wind load acting on a building frame can be estimated from the following formulae for the spectral density function, S D (in units of lb2/Hz): SD (f) = 4 D

where Sv (f)

and

in which

D

=

V30

2

Sv (f) 2

L

= 4 K V30

1 2

27

V

_ CD A V

2 ⎡ ⎛ fL ⎞ ⎤ ⎢2 + ⎜ ⎟ ⎥ 5/ 6 ⎝ ⎠ ⎢⎣ V30 ⎥⎦

2

28

29

30is the mean wind velocity at 30 feet above the ground

L, the scale of turbulence of wind which is around 4,000 feet f, the frequency in Hz V ,

31the mean wind velocity acting on the building

k, density of air CD, drag coefficient of the building A, profile area of the building K is the surface drag coefficient which has the values of 0.005 for open country, 0.015 for small towns or suburbs, and 0.05 for large cities. Reference: C. M. Harris and C. E. Crede, Eds., Shock and Vibration Handbook, Chap. 29, McGraw-Hill, 1976.


PAGE 48 OF 53

C.4

GROUND ATTENUATION EFFECT

Attenuation of vibration through the ground can be assumed to follow the equation: A(r) =

A( r o ) r/ r o

e - m(r - r o )

;

r ≥ r o 32

where A(r) is the amplitude of vibration at r and A(r o) is the amplitude at r o, measured from the center of the machine. The parameter m is the attenuation coefficient. The radius r o can be taken to be equal to 33 where A is the area of the foundation block. The attenuation coefficient m for most soil types is frequency dependent. For design purposes, a linear frequency dependency may be assumed, that is: m =

where

πα f_

Cs 34

f is the frequency of the disturbance, Cs , the speed of shear wave in soil, and " may be taken to be a constant.

For common types of soil, " falls in the range of 0.2 to 0.4. Reference: F.E. Richart, Jr., J.R. Hall, Jr., and R.D. Woods, Vibrations of Soils and Foundations, Chapter 8, p. 246, Prentice-Hall, 1970.


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Figure C-1 An Idealized Football Force Time-History


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BECHTEL Engineering Design Guide Vibration Sensitive Structures

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