In-structure Damping and Energy Dissapation

IN-STRUCTURE DAMPINGAND

ENERGY DISSIPATION

Revision 0: July, 2001

DESIGN GUIDELINESDESIGN GUIDELINESDESIGN GUIDELINESDESIGN GUIDELINESTrevor E Kelly, S.E.Trevor E Kelly, S.E.Trevor E Kelly, S.E.Trevor E Kelly, S.E.

Holmes Consulting GroupHolmes Consulting GroupHolmes Consulting GroupHolmes Consulting Group

© Holmes Consulting Group LtdLevel 1

11 Aurora TerraceP O Box 942Wellington

New Zealand

Telephone 64 4 471 2292Facsimile 64 4 471 2336www.holmesgroup.com

The Holmes Group of Companies

Company Offices In ServicesHolmes Culley San Francisco, CA Structural EngineeringHolmes Consulting Group New Zealand (Auckland, Wellington,

Christchurch, Queenstown)Structural Engineering

Holmes Fire & Safety New Zealand (Auckland, Wellington,Christchurch)Australia (Sydney)

Fire EngineeringSafety Engineering

Optimx New Zealand (Wellington) Risk AssessmentHolmes Composites San Diego, CA Structural Composites

Copyright © 2001. This material must not be copied, reproduced or otherwise used withoutthe express, written permission of Holmes Consulting Group.

2001

http://www.holmesgroup.com/

DISCLAIMERDISCLAIMERDISCLAIMERDISCLAIMER

The information contained in these Design Guidelines has been prepared by Holmes ConsultingGroup Limited (Holmes) as standard Design Guidelines and all due care and attention has been takenin the preparation of the information therein. The particular requirements of a project may requireamendments or modifications to the Design Guidelines.

Neither Holmes nor any of its agents, employees or directors are responsible in contract or tort or inany other way for any inaccuracy in, omission from or defect contained in the Design Guidelines andany person using the Design Guidelines waives any right that may arise now or in the future againstHolmes or any of its agents, employees or directors.

Copyright © 2001. This material must not be copied, ireproduced or otherwise used without the express, writtenpermission of Holmes Consulting Group.

CONTENTSCONTENTSCONTENTSCONTENTS

1111 INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION 1111

1.1 OUR COMPANY INVOLVEMENT 11.2 CURRENT STATUS OF THESE GUIDELINES 21.3 BACKGROUND 21.4 MARKET PARTICIPANTS 31.5 HOW GOOD IS THE TECHNOLOGY? 41.6 IMPEDIMENTS TO USE OF THE TECHNOLOGY 41.7 AVAILABLE DESIGN TOOLS 51.8 SCOPE OF THESE GUIDELINES 5

2222 PRINCIPLES OF IPRINCIPLES OF IPRINCIPLES OF IPRINCIPLES OF IN-STRUCTURE DAMPINGN-STRUCTURE DAMPINGN-STRUCTURE DAMPINGN-STRUCTURE DAMPING 7777

2.1 DAMPING OF STRUCTURES 72.2 EQUIVALENT VISCOUS DAMPING 82.3 EFFECT OF DAMPING ON RESPONSE 9

3333 DAMPER PROPERTIDAMPER PROPERTIDAMPER PROPERTIDAMPER PROPERTIESESESES 13131313

3.1 HYSTERETIC METAL YIELDING 133.1.1 DESCRIPTION OF DAMPER 133.1.2 DAMPER PROPERTIES 15

3.1.2.1 GENERIC HYSTERETIC PROPERTIES 163.1.2.2 SPECIFIC BRACE PROPERTIES 19

3.1.3 SUMMARY OF HYSTERETIC DAMPERS 233.2 HYSTERETIC FRICTION 23

3.2.1 DESCRIPTION OF DAMPER 233.2.2 DAMPER PROPERTIES 243.2.3 SUMMARY OF FRICTION DAMPER 24

3.3 VISCOUS 263.3.1 DESCRIPTION OF DAMPER 263.3.2 DAMPER PROPERTIES 273.3.3 INTERACTION OF STRUCTURE WITH VISCOUS DAMPER 303.3.4 SUMMARY OF VISCOUS DAMPER 32

3.4 VISCO-ELASTIC 323.4.1 DESCRIPTION OF DAMPER 32

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3.4.2 DAMPER PROPERTIES 343.4.3 SUMMARY OF VISCO-ELASTIC DAMPER 35

3.5 OTHER TYPES OF DAMPER 373.6 DAMPING WIND LOADS 37

4444 ANALYSIS OF DAMPINANALYSIS OF DAMPINANALYSIS OF DAMPINANALYSIS OF DAMPING DECAYG DECAYG DECAYG DECAY 39393939

4.1 PROCEDURE FOR EVALUATING DAMPING DECAY 394.2 VISCOUS DAMPING IN THE STRUCTURE 404.3 10 STORY MODEL 424.4 DAMPING VARIATIONS 431.5 DAMPING DECAY CURVES 441.6 EVALUATION OF DAMPING 45

1.6.1 STRUCTURE WITHOUT DAMPING DEVICES 461.6.2 HYSTERETIC DAMPERS 471.6.3 FRICTION DAMPERS 491.6.4 VISCOUS DAMPERS 501.6.5 VISCO-ELASTIC DAMPERS 51

1.7 SUMMARY OF DAMPING DECAY 52

5555 TIME HISTORY ANALYTIME HISTORY ANALYTIME HISTORY ANALYTIME HISTORY ANALYSISSISSISSIS 54545454

5.1 OBJECTIVE 545.2 PROTOTYPE BUILDINGS 545.3 SEISMIC INPUT 56

5.3.1 BASIS FOR SELECTING RECORDS 565.4 DAMPER VARIATIONS 605.5 TIME HISTORY EVALUATION PROCEDURE 625.6 RESPONSE OF BUILDING WITHOUT DAMPERS 63

5.6.1 EFFECT OF VISCOUS DAMPING 645.7 DAMPER EFFECTIVENESS 66

5.7.1 EFFECT ON DRIFTS 665.7.2 EFFECT ON BASE SHEAR 735.7.3 EFFECT ON FLOOR ACCELERATIONS 77

5.8 EQUIVALENT VISCOUS DAMPING 785.9 OPTIMUM DEVICES 805.10 SUMMARY OF PERFORMANCE 82

6666 PRACTICAL DEVICE PPRACTICAL DEVICE PPRACTICAL DEVICE PPRACTICAL DEVICE PROPERTIESROPERTIESROPERTIESROPERTIES 86868686

6.1 HYSTERETIC DEVICES 866.2 FRICTION DEVICES 886.3 VISCOUS DAMPERS 896.4 VISCO-ELASTIC DEVICES 92

Copyright © 2001. This material must not be copied, iiireproduced or otherwise used without the express, writtenpermission of Holmes Consulting Group.

7777 DAMPING DESIGN PRODAMPING DESIGN PRODAMPING DESIGN PRODAMPING DESIGN PROCEDURESCEDURESCEDURESCEDURES 94949494

7.1 APPLICABLE CODES 947.2 SECTION OF DEVICE TYPE AND PROPERTIES 947.3 DEVICE DESIGN 997.4 EVALUATION OF PERFORMANCE 100

7.4.1 NSP FOR DISPLACEMENT DEPENDENT DEVICES 1017.4.2 NSP FOR VELOCITY DEPENDENT DEVICES 1017.4.3 NDP FOR ALL DEVICES 102

7.5 EXAMPLE 10 STORY BUILDING 1027.5.1 NDP RESPONSE 1037.5.2 NSP RESPONSE 105

7.6 DESIGN RECOMMENDATIONS 108

8888 SUMMARYSUMMARYSUMMARYSUMMARY 110110110110

8.1 IN-STRUCTURE DAMPING AND ENERGY DISSIPATION 1108.2 DAMPER TYPES AND PROPERTIES 1118.3 DAMPING DECAY 1118.4 TIME HISTORY ANALYSIS 1128.5 DESIGN PROCEDURES 1138.6 RECOMMENDATIONS 115

9999 BIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHY 116116116116

AAAA TIME HISTORY RTIME HISTORY RTIME HISTORY RTIME HISTORY RESULTSESULTSESULTSESULTS A-1A-1A-1A-1

Copyright © 2001. This material must not be copied, ivreproduced or otherwise used without the express, writtenpermission of Holmes Consulting Group.

LIST OF FIGURESLIST OF FIGURESLIST OF FIGURESLIST OF FIGURES

FIGURE 2-1 EFFECT OF DAMPING ON DECAY ....................................................................................... 7FIGURE 2-2 EQUIVALENT VISCOUS DAMPING ....................................................................................... 9FIGURE 2-3 EFFECT OF DAMPING ON RESPONSE SPECTRUM ................................................................ 11FIGURE 2-4 FEMA SPECTRUM DEFINITION.................................................................................. 12

FIGURE 3-1 CONFIGURATIONS OF HYSTERETIC DAMPERS ...................................................................... 14FIGURE 3-2 YIELDING DAMPER HYSTERESIS ......................................................................................... 15FIGURE 3-3 DAMPING AS A FUNCTION OF BRACE PROPERTIES................................................................ 18FIGURE 3-4 HIGH STIFFNESS AND STRENGTH HYSTERETIC DAMPERS.......................................................... 19FIGURE 3-5 DAMPING IN YIELDING BRACE AT 0.5% DRIFT ..................................................................... 21FIGURE 3-6 DAMPING IN YIELDING BRACE AT 2.5% DRIFT ..................................................................... 22FIGURE 3-7 FRICTION DAMPER HYSTERESIS ......................................................................................... 24FIGURE 3-8 FRICTION DAMPER FORCE < STRUCTURE FORCE .................................................................. 25FIGURE 3-9 FRICTION DAMPER FORCE > STRUCTURE FORCE ................................................................. 25FIGURE 3-10 PHASING OF DISPLACEMENT AND VELOCITY ...................................................................... 26FIGURE 3-11 DAMPER COEFFICIENT, C............................................................................................ 28FIGURE 3-12 DAMPER EXPONENT, α, FOR CONSTANT C ..................................................................... 28FIGURE 3-13 DAMPER EXPONENT, α, FOR CONSTANT DAMPER FORCE.................................................... 29FIGURE 3-14 VELOCITY CUT-OFF ON VISCOUS DAMPER....................................................................... 29FIGURE 3-15 LOADING FREQUENCY................................................................................................ 30FIGURE 3-16 DISPLACEMENT AMPLITUDE............................................................................................ 30FIGURE 3-17 HYSTERETIC DAMPER IN PARALLEL WITH VISCOUS DAMPER.................................................... 31FIGURE 3-18 COUPLING OF VISCOUS DAMPER AND STRUCTURE α = 1.0................................................ 32FIGURE 3-19 VISCO-ELASTIC DAMPER .............................................................................................. 33FIGURE 3-20 FORCE-DISPLACEMENT RELATIONSHIP FOR VISCO-ELASTIC DEVICE ......................................... 33FIGURE 3-21 STORAGE MODULUS OF VISCO-ELASTIC DAMPER AT 21-26°C.............................................. 35FIGURE 3-22 LOSS MODULUS OF VISCO-ELASTIC DAMPER AT 21-26°C .................................................. 35FIGURE 3-23 TEMPERATURE DEPENDENCE OF VISCO-ELASTIC DAMPER..................................................... 36FIGURE 3-24 STRAIN DEPENDENCE OF VISCO-ELASTIC DAMPER .............................................................. 36FIGURE 3-25 FREQUENCY DEPENDENCE OF VISCO-ELASTIC DAMPER ...................................................... 37

FIGURE 4-1 ANALYTICAL DECAY CURVE ............................................................................................. 39FIGURE 4-2 RAYLEIGH DAMPING ..................................................................................................... 41FIGURE 4-3 MODEL USED FOR DECAY STUDIES ................................................................................... 42FIGURE 4-4 PUSHOVER CURVE FOR EXAMPLE 10 STORY BUILDING .......................................................... 43FIGURE 4-5 DAMPING DECAY CURVES.............................................................................................. 45FIGURE 4-6 NO DEVICES FITTED DECAY CURVE................................................................................... 47FIGURE 4-7 BEAM MOMENTS .......................................................................................................... 47FIGURE 4-8 HYSTERETIC DAMPERS FITTED DECAY CURVE ....................................................................... 48FIGURE 4-9 FRICTION DAMPERS FITTED DECAY CURVE .......................................................................... 49FIGURE 4-10 VISCOUS DAMPERS FITTED DECAY CURVE......................................................................... 51

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FIGURE 4-11 VISCO-ELASTIC DAMPERS FITTED DECAY CURVE................................................................. 52

FIGURE 5-1 PROTOTYPE BUILDINGS .................................................................................................. 55FIGURE 5-2 5% DAMPED SPECTRUM OF EL CENTRO TIME HISTORY........................................................ 59FIGURE 5-3 5% DAMPED SPECTRUM OF NORTHRIDGE TIME HISTORY ....................................................... 59FIGURE 5-4 5% DAMPED SPECTRUM OF FREQUENCY SCALED EL CENTRO TIME HISTORY .............................. 60FIGURE 5-5 DAMPER DISTRIBUTION WITH HEIGHT ............................................................................... 62FIGURE 5-6 TIME HISTORY OF ROOF DISPLACEMENT (EL CENTRO RECORD)............................................... 64FIGURE 5-7 EFFECT OF VISCOUS DAMPING ON DRIFTS ....................................................................... 65FIGURE 5-8 VISCOUS DAMPING EFFECT ON BASE SHEAR (EL CENTRO)..................................................... 66FIGURE 5-9 SUMMARY OF ALL DAMPERS : UNIFORM DISTRIBUTION ......................................................... 70FIGURE 5-10 SUMMARY OF ALL DAMPERS : TRIANGULAR DISTRIBUTION..................................................... 71FIGURE 5-11 SUMMARY OF ALL DAMPERS : REVERSE TRIANGULAR DISTRIBUTION ......................................... 72FIGURE 5-12 EFFECT OF DAMPERS ON TOTAL BASE SHEAR..................................................................... 74FIGURE 5-13 EFFECT OF DAMPERS ON FRAME SHEAR............................................................................ 74FIGURE 5-14 HYSTERETIC DAMPER SHEAR (EL CENTRO) ......................................................................... 75FIGURE 5-15 FRICTION DAMPER SHEAR (EL CENTRO) ........................................................................... 76FIGURE 5-16 VISCOUS DAMPER SHEAR (EL CENTRO) ........................................................................... 76FIGURE 5-17 VISCO-ELASTIC DAMPER SHEAR (EL CENTRO) .................................................................... 77FIGURE 5-18 FLOOR ACCELERATIONS .............................................................................................. 78FIGURE 5-19 EQUIVALENT DAMPING (EL CENTRO)............................................................................... 80

FIGURE 6-1 HYSTERETIC DAMPER DISPLACEMENT ................................................................................. 87FIGURE 6-2 FRICTION DAMPER DISPLACEMENT .................................................................................... 88FIGURE 6-3 VISCOUS DAMPER VELOCITY........................................................................................... 89FIGURE 6-4 VISCOUS DAMPER FORCE ............................................................................................... 90FIGURE 6-5 TAYLOR DEVICES 225 KN VISCOUS DAMPERS..................................................................... 91FIGURE 6-6 TAYLOR DEVICES 5850 KN AND 9000 KN VISCOUS DAMPERS.............................................. 91FIGURE 6-7 VISCO-ELASTIC DAMPER FORCE........................................................................................ 93FIGURE 6-8 VISCO-ELASTIC DAMPER DISPLACEMENT ............................................................................. 93

FIGURE 7-1 5% DAMPED SPECTRUM FOR EVALUATION........................................................................ 103FIGURE 7-2 NDP RESPONSE DETAILS .............................................................................................. 104FIGURE 7-3 PUSHOVER CURVES ..................................................................................................... 105FIGURE 7-4 NSP TARGET DISPLACEMENTS METHOD 1 ...................................................................... 106FIGURE 7-5 NSP TARGET DISPLACEMENTS METHOD 2 ...................................................................... 106

FIGURE A-1 EFFECTIVENESS OF HYSTERETIC DAMPERS EQ1................................................................... A-8FIGURE A-2 EFFECTIVENESS OF HYSTERETIC DAMPERS EQ2................................................................... A-9FIGURE A-3 EFFECTIVENESS OF HYSTERETIC DAMPERS EQ3................................................................. A-10FIGURE A-4 EFFECTIVENESS OF FRICTION DAMPERS EQ1................................................................... A-11FIGURE A-5 EFFECTIVENESS OF FRICTION DAMPERS EQ2................................................................... A-12FIGURE A-6 EFFECTIVENESS OF FRICTION DAMPERS EQ3................................................................... A-13FIGURE A-7 EFFECTIVENESS OF VISCOUS DAMPERS EQ1 .................................................................... A-14FIGURE A-8 EFFECTIVENESS OF VISCOUS DAMPERS EQ2 .................................................................... A-15FIGURE A-9 EFFECTIVENESS OF VISCOUS DAMPERS EQ3 .................................................................... A-16FIGURE A-10 EFFECTIVENESS OF VISCO-ELASTIC DAMPERS EQ1 ......................................................... A-17FIGURE A-11 EFFECTIVENESS OF VISCO-ELASTIC DAMPERS EQ2 ......................................................... A-18FIGURE A-12 EFFECTIVENESS OF VISCO-ELASTIC DAMPERS EQ3 ......................................................... A-19

Copyright © 2001. This material must not be copied, vireproduced or otherwise used without the express, writtenpermission of Holmes Consulting Group.

LIST OF TABLESLIST OF TABLESLIST OF TABLESLIST OF TABLES

TABLE 2-1 DAMPING REDUCTION FACTORS ........................................................................................ 10

TABLE 3-1 DAMPING PROVIDED BY BRACE OPTIONS ............................................................................ 20TABLE 3-2 EFFECT OF NECKED BRACE ............................................................................................... 20

TABLE 4-1 DAMPER VARIATIONS ...................................................................................................... 43TABLE 4-2 DAMPING IN STRUCTURE WITHOUT DAMPING ...................................................................... 46TABLE 4-3 HYSTERETIC DAMPERS ...................................................................................................... 48TABLE 4-4 FRICTION DAMPERS........................................................................................................ 49TABLE 4-5 VISCOUS DAMPERS......................................................................................................... 50TABLE 4-6 VISCO-ELASTIC DAMPERS................................................................................................. 51TABLE 4-7 SUMMARY OF DAMPING DECAY ........................................................................................ 52

TABLE 5-1 DESIGN PARAMETERS FOR PROTOTYPE BUILDINGS .................................................................. 55TABLE 5-2 SCALE FACTORS FOR VARIOUS EARTHQUAKES........................................................................ 57TABLE 5-3 VARIATIONS IN DAMPER PROPERTIES ................................................................................... 61TABLE 5-4 MAXIMUM RESPONSE QUANTITIES - NO DAMPERS.................................................................. 63TABLE 5-5 OPTIMUM DEVICES FOR 3 STORY BUILDING......................................................................... 83TABLE 5-6 OPTIMUM DEVICES FOR 5 STORY BUILDING......................................................................... 84TABLE 5-7 OPTIMUM DEVICES FOR 10 STORY BUILDING....................................................................... 85

TABLE 6-1 STEEL AREA FOR HYSTERETIC DAMPERS ACTING AS BRACES ....................................................... 86

TABLE 7-1 DAMPER PROPERTIES TO REDUCE DRIFT > 15%.................................................................... 96TABLE 7-2 DAMPER PROPERTIES TO REDUCE DRIFT > 30%.................................................................... 97TABLE 7-3 EXAMPLE DEVICES IN 10 STORY BUILDING .......................................................................... 104TABLE 7-4 COMPARISON OF NDP AND NSP RESULTS........................................................................ 107

TABLE A-1 RESPONSE RATIOS FOR TIME HISTORY ANALYSES ................................................................... A-1

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1111 INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION

1.1 OUR COMPANY INVOLVEMENT

Holmes Group has been involved in the design and supply of base isolation systems for almost 20 years. Thisis one form of passive protection for earthquake loads. The other form of passive protection, in-structuredamping and energy dissipation, has not been developed or implemented to the same extent as base isolationbut has applications where isolation is not suitable. This is a potential growth area for the provision ofstructural engineering services for earthquake damage mitigation.

To date, we have implemented energy dissipation on one structure, a frame building at the University ofCanterbury was strengthened using yielding brace dampers by the Christchurch office. We have investigatedsupplemental damping for a number of other structures but have been hampered by a lack of designinformation.

We have a program underway to extend our capabilities in base isolation and performance based design toincorporate this technology with two aims:

1. To provide damping hardware, through our associated companies, Holmes Devices (which has developedan improved yielding brace damper) and Holmes Composites (which provides Fyfe Co. visco-elasticdampers).

2. To establish Holmes Consulting Group and Holmes Culley as recognized experts in providing designservices, analysis services and documentation for projects incorporating in-structure damping and energydissipation.

The methodology we are following to achieve these aims requires that we:

1. Become familiar with the latest developments in the technology.

2. Refine our analysis procedures as required to be able to implement promising types of device.

3. Develop design procedures so that we can implement the devices in projects.

4. Provide marketing support to the devices we intend to supply.

The intention is to develop these Design Guidelines into a comprehensive resource containing the fullmethodology described in the task list above. This will be a continuing process and so revisions to thisdocument will be issued as progress is made.


1.2 CURRENT STATUS OF THESE GUIDELINES

To date, we have completed a series of analysis studies. A non-comprehensive literature survey has beenperformed; some trial analyses have checked the capability of our in-house software and theoreticalderivations of damping have attempted to define desirable device properties; the effect of different types ofdampers on a limited range of structures has been assessed.

The guidelines at this stage are not conclusive. They contain background material and my interpretation ofthe properties and use of devices in the current market. They are being circulated for comment beyond ourcompany so that we can participate in developments that may be underway in other parts of the world.

1.3 BACKGROUND

In-structure damping, or energy dissipation, encompasses any component used to reduce the movement ofstructures under lateral loads such as wind and earthquakes. This strategy attempts to reduce the demand on astructure, rather than the more usual approach of adding capacity. The basic aim of structural engineeringmay be expressed as:

CAPACITY > DEMAND

Usual structural engineering processes attempt to achieve this requirement by increasing the capacity of thestructure. Passive protection takes the opposite approach and attempts to reduce the demand on thestructure.

Energy dissipation is not a new technology in that some devices have been promoted since the 1970’s.However, as with so much in structural engineering it has been very slow to progress. As far as I can tell, theState-of-the-Art paper from the 1993 ATC-17-1 seminar could almost be re-published today and be called thesame.

In terms of earthquake mitigation hardware, the three general classifications are Seismic Isolation, PassiveEnergy Dissipation and Active Control. We have fully developed capabilities in isolation and active control isprobably too experimental for us to get involved in yet. These guidelines are restricted to the range of deviceswithin the Passive Energy Dissipation classification. Within this, there are four main categories of device:

1. Yielding metal devices, such as steel cantilevers, yielding braces and lead extrusion dampers. The force isdisplacement dependent and energy dissipation is through hysteretic yielding.

2. Friction devices, such as brake pads clamped with bolts at brace intersections. As for the yielding metal,the force is displacement dependent and energy dissipation is through a frictional hysteresis.

3. Viscous dampers, usually fluid forced through an orifice. The force is velocity dependent and energydissipation is by the fluid viscosity.

4. Visco-elastic dampers, usually a solid copolymer such as the product developed by 3M which was basicallythick Scotch tape bonded between steel plates. These materials have an elastic stiffness, with a


displacement dependent force, as well as a viscous component which produces a velocity dependent force.Some visco-elastic devices are liquid. Damping is through the material viscosity.

There are other more exotic passive devices such as shape memory alloys but these guidelines are restricted tothe four types above.

All energy dissipation devices basically perform the same function, they convert kinetic energy from externalloads into heat energy.

Some devices seem promising but are not yet widely available in our target markets. These include a rubber-based visco-elastic device from the UK and several from Japan – a wall damper, where a visco-elastic materialis placed between multiple plates within a wall, a low yield point steel damper acting in shear and a rubbermodified asphalt visco-elastic damper. These dampers all fall within one of the categories listed above and soconclusions reached would also apply to these devices.

1.4 MARKET PARTICIPANTS

The field of passive energy dissipation seems to have reached a peak in terms of research interest in the late1980’s and early 1990’s and since then has retrenched. Early participants were:

1. 3M. Sponsored much research on visco-elastic devices and were active at conferences for about 5 years.Abandoned the market in the mid 1990’s.

2. Roger Scholl, formed CounterQuake which worked with Bechtel on the ADAS yielding steel damper. Heis deceased.

3. Pall Dynamics, a Canadian company with friction based products. Have probably the longest project listin North America. Still active and publish many case studies.

4. Taylor Devices, a U.S. manufacturer of fluid viscous dampers. These devices are declassified militaryhardware used for missile silo protection and aeroplane arresters. They tend to be a high cost item. Acompetitive manufacturer of similar devices, Enidine, appears to have less market penetration.

5. Several Japanese companies, such as Oiles, Sumitomo, Bridgestone and others. They have developed awide variety of devices of all types. Most appear to have had one or two installations within Japan ininstrumented buildings. These companies have not had much presence in the U.S other than presentingconference papers.

6. S.E. companies with a reputation for being early adopters of new technology, for example, Gary HartConsultants and Nabih Yousef & Associates.

In the 1980’s academics moved their interest from seismic isolation to passive energy dissipation, and similarlyin the 1990’s they assessed the passive technology as mature and have moved on to active control. Also as forisolation, they did not leave behind a technology developed to the point where practising engineers can use itand so there are opportunities for device suppliers or specialist designers to fill this gap.


1.5 HOW GOOD IS THE TECHNOLOGY?

Although seismic isolation is a subset of the general field of passive energy dissipation, in-structure dampingvaries from isolation in two major respects:

1. In-structure damping is distributed up the height of the building rather than concentrated at one plane.

2. Most of the effectiveness of isolation is the period shift effect, lengthening the period of response, with alesser effect from damping. In-structure damping has a minor effect on period and in fact shortens theperiod if anything. Response reductions rely entirely on energy dissipation.

From an engineering mechanics viewpoint, a fundamental difference is that an isolation system acts in serieswith the structure whereas in-structure damping acts in parallel with the structure. An isolation systemabsorbs energy and filters the motion before it passes into the structural system. For a structure with in-structure damping, all energy passes into the combined system which then dissipates this energy depending onthe characteristics of each of the components (structural system and devices). This requires that the dampingbe tuned to the structure for optimum performance, a more complex design problem than isolation.

The response reductions from in-structure damping are much less dramatic than from isolation. Isolation canreduce structural forces and deformation by a factor of from 4 to 6. In-structure damping provides reductionsby factors of 1.5 to 2 at best. However, it is less intrusive than isolation and cheaper to install.

Almost by definition, buildings not suitable for base isolation are the best candidates for in-structure damping.It is most effective on flexible buildings with slender lateral load systems and is also suitable for soft soil sites.

The suitability of flexible buildings arises from the fact that in-structure damping is activated by inter-storymovement, either velocity or displacement. The greater the movement the greater the damping which givesrise to a paradox in that the aim of the damping is to reduce the movements which give rise to the damping.

For near fault type earthquakes, buildings with in-structure damping are probably no better or worse thanconventional or isolated buildings. However, this will need to be one topic for our development efforts as thedampers are unlikely to be effective for a single pulse.

The design of in-structure damping is difficult and it is only suitable for a restricted range of buildings.Unfortunately, this range is not well defined and so a lot of effort may be expended simply to prove that abuilding is not suited to added damping. These guidelines are intended to eventually ensure that we filter outunsuitable projects before we expend all this effort.

The more efficient types of damper, at least in theory, are the most expensive – fluid viscous dampers.Hysteretic dampers tend to merge with structural elements and for some types if is difficult to differentiatebetween a structural brace and a damper.

1.6 IMPEDIMENTS TO USE OF THE TECHNOLOGY

Passive earthquake protection functions by changing the dynamic characteristics of the structure. Mostengineers prefer not to deal with the dynamics of response and use equivalent static loads or, at most, a


response spectrum analysis. These methods are not really suitable for assessing most types of device althoughsources such as FEMA-356 and the SEAOC Blue Book attempt to provide means to use them.

Time history analysis with explicit modelling of the devices is the only procedure to accurately assessperformance and the structural engineering profession resists this type of analysis. Impediments to timehistory analysis (onerous code requirements, lack of suitable software, requirements for peer review) becomeimpediments to the use of in-structure damping.

Most damper manufacturers have tested their devices and published these test results. The researchinstitutions do similar tests, often sponsored by manufacturers. These test programs generally involve either asingle device or devices installed in a one story, one bay frame. They do not deal with the distribution ofdevices within the structure or the selection of device characteristics relative to structural properties such asmass, stiffness and period.

Attempts at developing design procedures to bridge this gap between device test results and the design ofdevices for a real structure do not seem to be successful. An engineer does not have a realistic starting pointas to type of device and device properties, and in fact no reliable way exists to even assess whether to considerin-structure damping. This is probably the major impediment to adoption of the technology. Theseguidelines, once complete, are intended to remove this impediment for our company.

1.7 AVAILABLE DESIGN TOOLS

Design of most devices either follows the usual design rules for specific materials (e.g. steel dampers) or usesinformation provided by suppliers of proprietary devices (e.g. viscous dampers). We will probably developour normal design aids such as spreadsheets but have no major design tool developments planned.

For evaluation we will generally follow the FEMA 273 and SEAOC guidelines as they are the mostcomprehensive sources of code type rules. These documents generally allow for static analysis in very limitedapplications and non-linear procedures for all other applications (NSP or pushover analysis and NDP or timehistory analysis).

Our Performance Based Design tools (ModelA, ANSR-L and ProcessA) provide a means to implement boththe NSP and NDP. The input spreadsheet has been updated to include sections to define different dampertypes and their connectivity. The ANSR-L analysis program has element types suited for dampers (yieldingbraces and viscous dashpots). The Users Guide is being updated to provide details of this.

There are some technical issues which need to be resolved for the analysis of structures with viscous dampers.A complete model of some damper types requires a dashpot with a spring in parallel connected to thestructure through a further spring in series. The series spring seems to be causing numerical problems. Wecan continue to do studies without this complete model for now but the problem will have to be solvedeventually.

1.8 SCOPE OF THESE GUIDELINES

Chapter 2 of these Guidelines summarises the principles of in-structure damping, the concept of equivalentviscous damping and the effect of damping on response. There is a brief description of the effect of damping


on wind loads, but note that these guidelines are almost entirely devoted to the damping of earthquake loads.Wind load damping is an important topic and we may later develop this in more detail.

Chapter 3 provides the properties of the different types of dampers and in Chapter 4 the damping decayprovided by each type is quantified on an example 10 story structure.

Three example structures are used for a series of parametric time history analysis in Chapter 5. These are usedto evaluate the effect on seismic response of each damper type on the types of building for which we wouldmost likely consider in-structure damping. Chapter 6 examines practical damper device properties in relationto the optimum properties determined from the time history analyses.

Chapter 7 provides an initial effort to develop damping design procedures. At this stage these are moreguidance for designers rather than explicit procedures. It was clear from the limited evaluations performedthat the complexity of in-structure damping, and the number of options available to the designers, makedevelopment of comprehensive design procedures a very difficult exercise. This area will be the focus forfuture development.

A summary of these guidelines in provided in Chapter 8, followed by a bibliography that provides referencesources for further information. An appendix provides details of the time history analysis results.


2222 PRINCIPLES OF IN-STRUCTURE DAMPINGPRINCIPLES OF IN-STRUCTURE DAMPINGPRINCIPLES OF IN-STRUCTURE DAMPINGPRINCIPLES OF IN-STRUCTURE DAMPING

2.1 DAMPING OF STRUCTURES

To damp is defined as to reduce or stop the vibration of. In structural engineering, damping can be defined as theinherent property of materials which tends to oppose movement. The higher the damping of a system thequicker it will return to rest from a displaced position, as shown in Figure 2-1. Viscous damping, β, alsochanges the period of response for the undamped system, T, to the damped period, TD, as

2β1

TTD−

= ......................................................................................................................................(2-1)

For typical levels of structural damping the change in period is negligible. For 5% damping the change is only0.1% and even at 20% damping the period increases by only 2%.

FIGURE 2-1 EFFECT OF DAMPING ON DECAY

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 1 2 3 4 5 6

TIME

DIS

PLA

CEM

EN

T

Damping =2%Damping =5%Damping =10%Damping =20%


For mechanical systems damping is expressed as a fraction, usually a percentage, of critical damping, Cc. Asystem is critically damped if, when released, it returns to rest without vibration. Critical damping is a functionof the stiffness, K, and the mass, M, of a system:

KMCc = ..............................................................................................................................................(2-2)

For dynamic motions the damping forces are proportional to the velocity of the mass, hence the name viscousdamping.

2.2 EQUIVALENT VISCOUS DAMPING

Although it is convenient to use viscous damping for dynamic analysis, much of the energy dissipation instructural systems is not truly viscous in nature. The concept of equivalent viscous damping is used to convertdamping arising from sources such as material yielding to an equivalent viscous damping ratio.

Equivalent viscous damping, β, is defined as:

S

D

WWπ

β4

= .............................................................................................................................................(2-3)

where WD is the cyclic energy dissipated (the shaded area in Figure 2-2) and WS is the strain energy (thehatched area in Figure 2-2).

This formulation appears to be a simple method of calculating equivalent viscous damping where the area ofthe hysteresis loop of a device is known. And, in fact, implementation is straightforward for base isolationsystems where both the strain energy and energy dissipated are functions of the isolator properties and allisolators can be assumed to have the same displacement.

For distributed damping the strain energy is much more difficult to calculate as it is the summation of strainenergy throughout the structure plus the strain energy in all devices, all of which may have differentdisplacements. Because of this, equivalent viscous damping can provide only a very approximate indication ofthe effectiveness of the devices.


FIGURE 2-2 EQUIVALENT VISCOUS DAMPING

DISPLACEMENT

FOR

CE

Shaded area = energy dissipated

Hatched area = strain energy

2.3 EFFECT OF DAMPING ON RESPONSE

In general, increased damping reduces response, as shown in the acceleration and displacement responsespectra in Figure 2-3. However, the reduction is not constant over the full period range of response and italso varies with earthquake (see Base Isolation Design Guidelines for further discussion). At zero periods,damping has no effect as the spectrum value is equal to the maximum ground acceleration. At very longperiods damping also tends to have little effect on accelerations but has more effect on displacements.

Codes such as UBC and FEMA approximate the effect of damping by defining a damping coefficient, B,which is a function of the equivalent viscous damping, β. Table 2-1 lists the values from FEMA, whichdefines BS as the coefficient to adjust short period spectral response and B1 to adjust the one-second periodresponse for the effect of viscous damping.

The factor BS applies to periods up to T0, which is the characteristic period of the response spectrum, definedas the period associated with the transition from the constant acceleration segment of the spectrum to theconstant velocity segment of the spectrum (see Figure 2-4). For periods longer than T0, B1 applies.


TABLE 2-1 DAMPING REDUCTION FACTORS

Effective Dampingββββ

% of critical

BS B1

< 2510203040

> 50

0.81.01.31.82.32.73.0

0.81.01.21.51.71.92.0


FIGURE 2-3 EFFECT OF DAMPING ON RESPONSE SPECTRUM

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000

PERIOD (Seconds)

ACC

ELE

RATI

ON

(g)

Damping = 5.0%Damping = 10.0%Damping = 30.0%

0

50

100

150

200

250

300

350

400

450

500

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000

PERIOD (Seconds)

DIS

PLA

CEM

EN

T (m

m)

Damping = 5.0%Damping = 10.0%Damping = 30.0%


FIGURE 2-4 FEMA SPECTRUM DEFINITION

PERIOD

SPE

CTRA

L RE

SPO

NSE

ACC

ELE

RATI

ON

S

XSa BSS =

TBSS

1

X1a =

T00.2 T0


3333 DAMPER PROPERTIESDAMPER PROPERTIESDAMPER PROPERTIESDAMPER PROPERTIES

3.1 HYSTERETIC METAL YIELDING

3.1.1 DESCRIPTION OF DAMPER

Hysteretic yielding dampers are generally steel, which may be configured to yield in bending, shear or axially,or lead which yields in shear. The dampers are configured such that the metal is deformed by the seismicstory drifts, as shown by the layouts in Figure 3-1.

Axially yielding dampers are generally configured as diagonal braces although they may also be placedhorizontally from the top of a partial height wall to an adjacent column. Shear or flexural yielding damperscan be configured to connect the top of a wall panel to the soffit of the girder of the floor above. The wallpanel is a cantilever from the wall below, with a gap between the top of the wall and the floor above. As analternative to a wall panel, the shear and flexural dampers can be mounted on a steel frame.

Proposals have been made to use the cladding panels of a building to mount the shear or flexural dampers butthere is no record of this being implemented.

Lead is generally elasto-plastic with no strain hardening. The mild steel used for dampers has a yield plateaufollowed by strain hardening to ultimate strength. Depending on the strain levels at which a steel damperoperates there may be an increase in damper force with displacement. The development of properties in thissection assumes zero strain hardening. The effects of strain hardening are considered later in these guidelines.

Some hysteretic damper configurations, such as the yielding brace, are indistinguishable from a structuralmember and in fact, as will be seen later, they may act much as a structural member. The intent of in-structure damping is to install devices in which the energy dissipation function is more dominant than theadded stiffness and/or strength. Unless carefully designed, hysteretic dampers may not meet this intent. Aswill be seen later, the prime determinant of whether they provide meaningful damping is the initial elasticstiffness. The higher the stiffness, the higher the energy dissipation.

Even if the hysteretic damper acts as a structural member the design may need to be based on damper designprocedures rather than usual methods for the design of strengthening elements. This is because the hystereticdamper will usually be designed to yield before the existing structure. There will be non-linearity at the designload level, whereas linear elastic behaviour is assumed for conventional design.


FIGURE 3-1 CONFIGURATIONS OF HYSTERETIC DAMPERS

Yielding Brace

Shear Yielding Damper

Yielding SteelCantilever


3.1.2 DAMPER PROPERTIES

The yielding damper is defined by a yield force, Fy, and an elastic stiffness, KD, as shown in Figure 3-2. Theperformance of the damper is a function of these damper properties and the elastic stiffness of the structure,KE.

FIGURE 3-2 YIELDING DAMPER HYSTERESIS

Damper

Structure

Forc

e

Displacement

Fy

KD

KE

FE

Define the damper properties in terms of the structure properties as follows:

E

D

KKf = = the ratio of damper stiffness to total structure stiffness............................................(3-1)

E

y

FF

g = = the ratio of damper yield force to total structure force ..............................................(3-2)

These definitions can be used to calculate equivalent viscous damping using the formula

S

D

WWπ

β4

= ................................................................................................................................................(3-3)


where WD is the hysteretic energy dissipation, equal to the area under the hysteresis loop which at adisplacement ∆ is calculated as:

)(4 yyD FW ∆−∆= ................................................................................................................................(3-4)

where ∆y is the yield deformation of the damper = Fy/KD.

The strain energy, WS, is calculated as

)(21

yDES KKW ∆+∆= .......................................................................................................................(3-5)

NOTE : There are some differences in the literature as to how the portion of strain energy due to the damperis included in equation (3-5). However, this makes a relatively minor difference to the trends developed withthe procedure.

From equations (3-4) and (3-5) the damping is defined as

)()(2

yDE

yy

KKF

∆+∆∆−∆

=π

β ..........................................................................................................................(3-6)

substitute Fy = KD∆y and KD = fKE

)()(2

yEE

yyE

fKKfK

∆+∆∆−∆∆

=π

β ........................................................................................................................(3-7)

Cancelling out provides an equation for damping as a function of the maximum displacement and the damperproperties relative to the structure:

)()(2

y

yy

ff

∆+∆∆∆−∆∆

=π

β ............................................................................................................................(3-8)

3.1.2.1 Generic Hysteretic Properties

The ultimate displacement, ∆, may be expressed in terms of the elastic structure properties as

E

E

KF

=∆ ..................................................................................................................................................(3-9)

and the brace yield displacement, ∆y, may also be expressed in terms of the elastic structure properties as:


fg

fKgF

KF

E

E

D

yy ∆===∆ ........................................................................................................... (3-10)

Substitute equations (3-9) and (3-10) in equation (3-8):

)(

)(2

fgf

fg

fgf

∆+∆∆

∆−∆∆=

πβ .................................................................................................................... (3-11)

Cancelling out provides an equation for damping which is a function solely of the ratio of damper yield forceto elastic force, g, and the ratio of damper elastic stiffness to the structure elastic stiffness, f, as shown inequation (3-12).

)1(

)1(2

gfgg

+

−=

πβ .................................................................................................................................. (3-12)

Equation (3-12) can be used to generate a family of curves as a function of f and g as shown in Figure 3-3.This figure shows some general trends:

• The higher the stiffness of the damper relative to the structure, f, the higher the damping. Practically, it isdifficult to achieve values of f much greater than 1 and so damping of the order of 10% to 15% is arealistic target.

• For a realistic value of the stiffness ratio, f, there is an optimum value of the brace strength to the elasticstructure force. This increases with increasing stiffness, from a value of 0.12 at f = 0.25 to 0.72 at f =2.0.


FIGURE 3-3 DAMPING AS A FUNCTION OF BRACE PROPERTIES

0%

5%

10%

15%

20%

25%

30%

35%

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.36

0.40

0.44

0.48

0.52

0.56

0.60

0.64

0.68

0.72

0.76

0.80

0.84

0.88

0.92

0.96

1.00

g = RATIO OF DAMPER YIELD / ELASTIC FORCE

EQ

UIV

ALE

NT

VIS

COU

S D

AM

PIN

G

f = Kd/Ks = 100

f = Kd/Ks = 10

f = Kd/Ks = 5

f = Kd/Ks = 2

f = Kd/Ks = 1

f = Kd/Ks = 0.5

f = Kd/Ks = 0.25

f = Kd/Ks = 0.1

Note that the value of g in Figure 3-3 has a maximum value of 1.0, that is, the damper yield is equal to theelastic force in the structure. That implies that the damper resists a force equal to that resisted by thestructure, or one-half of the total force in the system, not the entire force.

If the value of g exceeds unity, it is implied that the damper resists more force than the structure. In the limit,the entire force would be resisted by the damper. Figure 2-8 plots the increased damping as the damper takessuccessively more of the total load. In the limit, the damping shown in Figure 3-4 is the damping whichwould be provided by any structural system with a stable hysteresis function.


FIGURE 3-4 HIGH STIFFNESS AND STRENGTH HYSTERETIC DAMPERS

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

0.00 3.00 6.00 9.00 12.00 15.00

RATIO OF DAMPER YIELD / ELASTIC FORCE

EQ

UIV

ALE

NT

VIS

COU

S D

AM

PIN

G

Kd/Ks = 10

Kd/Ks = 100

3.1.2.2 Specific Brace Properties

Consider a yielding brace with a yield displacement of 11 mm but a variable ratio of damper stiffness tostructure stiffness. For a prismatic yielding brace (that is, a constant section along the brace length) the yielddisplacement is a function of the steel yield stress and brace length but not of the section size. This is becausethe yield strain is a constant and yield displacement is this strain times length. A yield displacement of 11 mmcorresponds to the yield of a chevron brace with a steel strength of 260 MPa in a 7.500 m bay with a 4.150 mstory height.

Figure 3-5 plots the hysteresis curve and the equivalent viscous damping of a series of braces in a structurewith a total elastic force of 4000 KN at a story drift of 0.5%. The braces plotted in Figure 3-5 have increasingyield strength and so an increasing stiffness relative to the structure stiffness (increasing f where f = KD/KE).In Figure 3-6 the same braces are plotted at a drift level of 2.5%.

Table 3-1 summarises the damping from each option plotted in Figures 3-5 and 3-6. As expected from thegeneric plots above, the damping is quite small for braces with low stiffness (f = 0.25) and a brace stiffnessapproximately equal to the structure stiffness is required to get reasonable levels of damping. However, thiscan require substantial braces, in this case plate sections approximately 100 mm x 80 mm thick.


Another feature of hysteretic damping shows up in Table 3-1, that is, damping becomes less effective withincreasing displacements. The aim of hysteretic dampers is generally to reduce drifts though and so they willusually be designed to act at low levels of displacement.

TABLE 3-1 DAMPING PROVIDED BY BRACE OPTIONS

Drift f = KDAMPER/KELASTIC

Brace YieldForce (KN)

BraceArea (mm2)

EquivalentViscous

Damping, ββββ0.50% 0.25

1.002.00

51620654131

1,9867,94515,891

3.52%10.48%15.64%

2.50% 0.251.002.00

51620654131

1,9867,94515,891

1.45%5.38%9.84%

The brace dampers in Table 3-1 and Figures 3-5 and 3-6 are for prismatic dampers where the full length of thebrace yields. A design option for this type of damper is to reduce the yielding length to only a portion of thelength by defining a region of reduced section. For example, if the central 20% of the brace is permitted toyield and the remainder of the brace has an area of two times the central portion then the yield displacementwill be 60% of the yield displacement of a prismatic brace and damping will increase as shown in the secondcolumn of Table 3-2. Damping is increased by about 40% at small displacements but only by 5% at largerdisplacements.

TABLE 3-2 EFFECT OF NECKED BRACE

Drift Brace YieldForce (KN)

PrismaticEquivalent

ViscousDamping, ββββ

NeckedEquivalent

ViscousDamping, ββββ

0.50% 51620654131

3.52%10.48%15.64%

5.02%14.96%22.32%

2.50% 51620654131

1.45%5.38%9.84%

1.52%5.64%10.30%


FIGURE 3-5 DAMPING IN YIELDING BRACE AT 0.5% DRIFT

Damping 3.52% f = 0.25

-5000-4000-3000

-2000-1000

010002000

300040005000

-30 -20 -10 0 10 20 30

Damping 10.48% f = 1

-5000-4000-3000

-2000-1000

010002000

300040005000

-30 -20 -10 0 10 20 30

Damping 15.64% f = 2

-5000-4000-3000

-2000-1000

010002000

300040005000

-30 -20 -10 0 10 20 30


FIGURE 3-6 DAMPING IN YIELDING BRACE AT 2.5% DRIFT

Damping 1.45% f = 0.25

-25000-20000

-15000

-10000-5000

05000

10000

1500020000

25000

-150 -100 -50 0 50 100 150

Damping 5.38% f = 1

-25000-20000

-15000

-10000-5000

05000

10000

1500020000

25000

-150 -100 -50 0 50 100 150

Damping 9.84% f = 2

-25000-20000

-15000

-10000-5000

05000

10000

1500020000

25000

-150 -100 -50 0 50 100 150


3.1.3 SUMMARY OF HYSTERETIC DAMPERS

The formulas developed in this section for the damping provided by hysteretic dampers appear simple but arevery difficult to implement in practice, for a number of reasons:

• The two damper parameters, the stiffness and yield strength, are normalised to the structure stiffness andelastic force level, which are difficult to define for any except the simplest single story structure. For anymulti-story structure the stiffness and elastic forces need to be integrated over the height of the building.

• The elastic force is a function of the earthquake loading.

• Most structures requiring dampers will not respond within their elastic limit and so some hysteretic energywill be dissipated by the structural system.

• Most importantly, the studies above suggest that the yielding dampers require a high stiffness and highyield strength for maximum effectiveness. With high stiffness and strength, the yielding dampers actuallyform an alternate structural system and modify the dynamic characteristics of the structure beyond simplyadding damping. Typically, they will reduce the period which in most buildings will increase the baseshear.

In practice, for actual applications of yielding dampers it is difficult to separate the effects of added stiffnessfrom the effects of added damping on response as both tend to reduce the displacement response.

These factors make it more difficult to develop a design procedure than would originally appear. In laterchapters of these guidelines the response of actual buildings with yielding dampers installed is assessed todevelop empirical rules for design.

3.2 HYSTERETIC FRICTION


A variety of proprietary friction dampers are available with various materials used for the sliding surface.Examples include brake pad material on steel, steel on steel or steel on brass in slip bolted connections andother metal alloys.

Friction dampers have most commonly been placed within diagonal braces, as for yielding metal dampers, butcan also be placed horizontally between the top of a wall and the beam above, again as for yielding metaldampers.

Most friction devices produce a stable rectangular hysteresis although some are configured to produce a self-centring force and provide non-rectangular hysteresis shapes with slip load proportional to displacement.These guidelines include only the most common types which provide a rectangular hysteresis as shown inFigure 3-7.


FIGURE 3-7 FRICTION DAMPER HYSTERESIS

Damper

Structure

Forc

eDisplacement

Fy

KE


Considering the damper alone, the equivalent viscous damping can be calculated by modifying equation (2-15)by setting the ratio of the damper stiffness to structure stiffness, f, to ∞, giving the formula in equation 2-16:

)1(2gg+

=π

β ..................................................................................................................................... (3-13)

In Figure 3-8 this function is plotted for values of g ≤ 1.0, where the damper resists up to one-half the totalforce, as applies in most applications. Figure 3-9 extends this case for the damper providing more resistancethan the structure, in the limit becoming a purely friction damped frictional structural system. The equivalentviscous damping converges to a limiting value of 2/π = 63.66%.

3.2.3 SUMMARY OF FRICTION DAMPER

The damping plotted in Figures 3-8 and 3-9 are for the device itself. For in-structure damping thedisplacements are due to story drifts applied to the friction damper. This requires that the damper extendsfrom floor to floor, connected by a structural element such as a brace or wall panel. This element will have afinite stiffness and will act in series with the friction damper. This has the effect of providing a finite initial


loading stiffness to the overall friction damping component, rather that the rigid assumption used to developFigures 3-8 and 3-9.

Because all practical applications will have non-rigid elements to mount the damper, the actual hysteresis willresemble that of the yielding damper hysteresis in Figure 3-2 rather than the rectangular hysteresis in Figure 3-7. Therefore, the derivations and comments in the previous section, applying to the yielding damper, are alsoapplicable to the friction damper.

FIGURE 3-8 FRICTION DAMPER FORCE < STRUCTURE FORCE

0%

5%

10%

15%

20%

25%

30%

35%

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00


EQ

UIV

ALE

NT

VIS

COU

S D

AM

PIN

G

FIGURE 3-9 FRICTION DAMPER FORCE > STRUCTURE FORCE

0%

10%

20%

30%

40%

50%

60%

70%

0 10 20 30 40 50 60 70 80 90 100


EQ

UIV

ALE

NT

VIS

COU

S D

AM

PIN

G


3.3 VISCOUS


Viscous dampers are devices that provide a resisting force that is proportional to the applied velocity ratherthan applied displacement. Most viscous dampers are fluid dampers, similar to the shock absorbers inautomobiles. These devices have low resistance to deformation when loads are applied very slowly but theresistance increases as the speed at which the deformation is applied increases.

The dampers are described by the general formula:

)sgn(|| uuCFDα= ...................................................................................................................... (3-14)

where FD is the damper force, C is the damper coefficient, u is the applied velocity, α is the damper exponentand sgn is the signum function which defines the sign of the relative velocity term. The value of α generallyranges between 0.3 and 1.0. Some dampers have a relief valve which provides a velocity limit. For allvelocities beyond the limit the damping force is constant.

Viscous dampers are attractive from a theoretical viewpoint because the velocity is out of phase with thedisplacement. Figure 3-10 show the velocity corresponding to an applied sine wave of displacement with aperiod of 1 seconds. At peak displacement the velocity is zero and, conversely, the peak velocity occurs whenthe displacement is zero.

In theory, forces from a viscous damper will not add to the total elastic forces in a structure because themaximum damping forces occur when the elastic forces due to building deformation are small. In practice,the two forces do couple to some extent and so the total force often does increase.

FIGURE 3-10 PHASING OF DISPLACEMENT AND VELOCITY

-8.000

-6.000

-4.000

-2.000

0.000

2.000

4.000

6.000

8.000

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000

TIME (Seconds)

DIS

PLA

CEM

EN

T

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

2.000

VE

LOCI

TY

Velocity

Displacement



As shown by equation (3-14), the damper parameters that define the damping force are the coefficient C, theexponent α and a velocity limit, if any. The impact of these parameters, and the effect of the characteristics ofthe loading regime, is illustrated in the following Figures 3-11 to 3-16, each generated for a sinusoidaldisplacement trace. The formulas for the displacement, ∆, and velocity, u, are:

∆=∆ tTπ2sin0 ............................................................................................................................................ (3-15)

∆=∆= tTTdt

du ππ 2cos20 ......................................................................................................................... (3-16)

In each figure the legend identified the values of the parameters; T is the period of the applied sine wave, C isthe damping coefficient and a is the damping exponent, α.

• Figure 3-11 shows the effect of varying C. As expected from the form of equation (3-14), the dampingforce is linear with C. If C is doubled, the damping force is doubled for the same velocity. The shape ofthe damping versus displacement curve is elliptical, following the shape of the velocity trace.

• Figure 3-12 keeps the coefficient C constant and varies the exponent α from 0.3 to 1.0, the normal rangeof practical devices. As α reduces from 1.0 to 0.3 the damping force reduces and the damping forcefunction tends from an elliptical toward a more rectangular form.

• Figure 3-13 also varies α but the coefficient C is adjusted such that the total damping force is the same.To maintain the damping force provided by α = 1.0 when α is reduced to 0.3 the damping coefficientmust increase from 5.5 to 20. This plot clearly shows the changing of the ellipse to a rectangle as theexponent is reduced.

• Figure 3-14 shows the effect of a velocity limit which limits the damper force to 50 units as C increasesfrom 5 to 20 (see also Figure 3-11, the same curves without the velocity limit). As the value of C increasesthe limit truncates the ellipse. This has a similar effect to reducing the exponent in that the elliptical shapebecomes more rectangular.

• Figure 3-15 shows the effect on damping force of varying the period of the sine curve displacement butretaining the same amplitude. From equation (3-16), the velocity is inversely proportional to the period ofresponse, T. For the same displacement, a shorter period provides a larger damping force.

• Figure 3-16 plots the damping forces for varying displacements. The velocity is directly proportional tothe displacement for a constant period. The damper force is proportional to the displacement to thepower of the exponent. In this case, the exponent is 0.5 and so if the displacement is increased by a factorof 4 the damping force will increase by a factor of √4 = 2.

In terms of available damper properties, the coefficient C can be selected to be almost any value as it can bevaried by simply installing more or less dampers in the structure. The exponent α can vary between limits of


0.3 and 1.0. In general, the higher value, which provides a linear relationship between damping force andvelocity, will provide the best results and a value of 1.0 is most commonly used.

Although the velocity limit may be useful in limiting forces, this feature will remove part of the desirablecharacteristics of viscous dampers, forces that are out of phase with displacements.

FIGURE 3-11 DAMPER COEFFICIENT, C

-150

-100

-50

0

50

100

150

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

DISPLACEMENT

DA

MPI

NG

FO

RCE

T = 1 C = 5 a = 1T = 1 C = 10 a = 1T = 1 C = 20 a = 1

FIGURE 3-12 DAMPER EXPONENT, α, FOR CONSTANT C

-80

-60

-40

-20

0

20

40

60

80

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

DISPLACEMENT

DA

MPI

NG

FO

RCE

T = 1 C = 10 a = 0.3T = 1 C = 10 a = 0.65T = 1 C = 10 a = 1


FIGURE 3-13 DAMPER EXPONENT, α, FOR CONSTANT DAMPER FORCE

-40

-30

-20

-10

0

10

20

30

40

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

DISPLACEMENT

DA

MPI

NG

FO

RCE

T = 1 C = 20 a = 0.3T = 1 C = 10.5 a = 0.65T = 1 C = 5.5 a = 1

FIGURE 3-14 VELOCITY CUT-OFF ON VISCOUS DAMPER

-60

-40

-20

0

20

40

60

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

DISPLACEMENT

DA

MPI

NG

FO

RCE

T = 1 C = 5 a = 1T = 1 C = 10 a = 1T = 1 C = 20 a = 1


FIGURE 3-15 LOADING FREQUENCY

-150

-100

-50

0

50

100

150

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

DISPLACEMENT

DA

MPI

NG

FO

RCE

T = 0.2 C = 20 a = 0.5T = 0.5 C = 20 a = 0.5T = 1 C = 20 a = 0.5

FIGURE 3-16 DISPLACEMENT AMPLITUDE

-60

-40

-20

0

20

40

60

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

DISPLACEMENT

DA

MPI

NG

FO

RCE

T = 1 C = 20 a = 0.5T = 1 C = 20 a = 0.5T = 1 C = 20 a = 0.5

3.3.3 INTERACTION OF STRUCTURE WITH VISCOUS DAMPER

As for the friction damper, the damping function of a viscous damper may be modified by the flexibility ofthe connection between the damper and the structure. For example, a viscous damper in a brace will have theproperties of the damper plus a spring in series. Part of the story drift will cause deformation in the springwhich will reduce the relative movement of the damper. This will reduce the damping force by a constantfactor, the magnitude of which will be a function of the stiffness of the connection.


The dampers will also act in parallel with the structure which they are damping. If the structure is elastic thenthe effect will be to “tilt” the ellipse, as discussed in the next section for visco-elastic dampers. If the structureyields, the usual case, then a combined force-displacement trace of the form shown in Figure 3-17 will beexhibited.

In the example plotted in Figure 3-17, the maximum force from the structure is 60 and the peak damper forceis 31.4. The maximum force in the combined system is 83.2 and so the “coupling” is 83.2 – 60 = 23.2, whichmeans that the maximum force in the structure in increased by 74% of the damper force. If the exponent α isless than one then the coupling is increased. For this example, if α = 0.3 and the peak damping force is thesame then the coupling increases to 81%.

FIGURE 3-17 HYSTERETIC DAMPER IN PARALLEL WITH VISCOUS DAMPER

-150

-100

-50

0

50

100

150

-1.500 -1.000 -0.500 0.000 0.500 1.000 1.500

DISPLACEMENT

DA

MPI

NG

FO

RCE

TotalViscousHysteretic

For a constant damper of exponent of 1.0, the degree of coupling is a function of the damper coefficient, C.Figure 3-18 plots the amount of damping provided (defined as the ratio of peak force in the damper to thepeak force in the structure) versus the extent of coupling (defined as the ratio of the peak total force minus thepeak structure force divided by the peak damper force).

For relatively small amounts of damping (damper force 10% of the structure force) there is not muchcoupling, less than 30%. However, the coupling increases rapidly with the damping force and when thedamper force equals the structure force there is 85% coupling, that is, the structure force is increased by 85%of the damper force.


FIGURE 3-18 COUPLING OF VISCOUS DAMPER AND STRUCTURE α = 1.0

0%

20%

40%

60%

80%

100%

120%

20% 30% 40% 50% 60% 70% 80% 90%

EXTENT OF COUPLING

DA

MPE

R FO

RCE

/ S

TRU

CTU

RE F

ORC

E

3.3.4 SUMMARY OF VISCOUS DAMPER

The viscous damper provides damping forces that are out of phase with the displacements and so these forcesare not directly additive to the structure forces. This makes the velocity dependent damper more efficient, intheory.

In practice, although the velocity and displacements are out of phase, there is some degree of couplingbetween the two sets of forces, especially if the exponent of the damper is near the lower limit of 0.3. Theextent of coupling increases with the amount of damping. In effect, the more the damping provided, thesmaller the benefit of having the damper force out of phase with the structure force.

3.4 VISCO-ELASTIC


Visco-elastic dampers provide a velocity dependent damping force but have an elastic stiffness in addition tothis damping. The most common type is formed of two layers of polymer bonded between a central drivingplate and two outer plates (Figure 3-19). The force is this type of device may be expressed as:


CukF effD +∆= ............................................................................................................................... (3-17)

where keff is the effectivestiffness of the damper, ∆ is thedisplacement, C is the dampingcoefficient and u is the velocity.Unlike the viscous damper thevelocity dependent damping isa linear function of velocity,that is, the exponent α =1 .0for all devices.

This equation provides a force-displacement function of theform shown in Figure 3-20.

FIGURE 3-20 FORCE-DISPLACEMENT RELATIONSHIP FOR VISCO-ELASTIC DEVICE

Displacement

Forc

e

Keff

FIGURE 3-19 VISCO-ELASTIC DAMPER

Steel PlatesVisco-ElasticMaterial


The terminology used in describing visco-elastic dampers is different from that used for other devices. Theshear stiffness is defined in terms of G’, the shear storage modulus, and the effective stiffness is defined fromthis as:

tAGK b

eff'

= .............................................................................................................................................. (3-18)

where Ab is the bonded area of the device and t is the total thickness of visco-elastic material in the device(sum of all layers).

The damping coefficient, C, is defined in terms of G”, the shear loss modulus:

tAG

C b

ω"

= .................................................................................................................................................. (3-19)

where ω is the frequency. The loss modulus is generally normalised by the frequency, as G”/ω so that it canbe factored directly by damper dimensions Ab/t, as for the storage modulus.


The damper properties G’ and G” are dependent on the frequency, temperature and strain. The amount ofdependence is a function of the specific material used for the damping. The results summarised in this sectionare taken from system qualification tests for Tyfo® Visco-elastic Dampers. Material supplied by othermanufacturers may differ.

Figures 3-21 and 3-22 plot the variations in these parameters with frequency and shear strain. Figure 3-23plots the variations with temperature. Figure 3-24 illustrates the effect of strain on the hysteresis curve andFigure 3-25 the effect of frequency on this curve.

The first point to note is that design of this type of device will be complex and most probably iterative.However, there is some simplification in that for most projects the frequency can be determined early in thedevelopment stage, on the assumption that damping makes only a small change to the frequency. An estimatecan also generally be made of the story drifts, using the elastic drifts and reducing them by an estimate of theeffect of damping. This allows preliminary properties to be selected.

The temperature effects will be project specific. For typical earthquake duration the heat change in thedampers is relatively small but the possible change in ambient temperatures must be considered becauseproperties are sensitive to temperature change (Figure 3-23).

Design limits are generally based on a strain of about 150% under the DBE and up to 250% for the MCEalthough this may be restricted by the range of test properties available. Tyfo® dampers have a failure strainin excess of 500% at failure but the response is highly non-linear for strains exceeding 250%.

The strain limit defines the required damper thickness, for example, the story drift at DBE divided by 1.5 tokeep strains to 150%. The dampers are quite small in plan dimension, up to about 250 mm square maximum,and so typically a large number of dampers will be used.


3.4.3 SUMMARY OF VISCO-ELASTIC DAMPER

The visco-elastic damper combines the properties of an elastic spring and a viscous damper. The damper isthe most complex of the type considered in these guidelines as the properties are a function of strain levels,frequency and temperature. In most cases this will involve an iterative design procedure and multiple analysesto bound the likely range of properties.

FIGURE 3-21 STORAGE MODULUS OF VISCO-ELASTIC DAMPER AT 21-26°C

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

FREQUENCY (hz)

SHE

AR

STO

RAG

E M

OD

ULU

S (M

Pa)

G' Strain 25%G' Strain 50%G' Strain 75%G' Strain 100%G' Strain 150%

FIGURE 3-22 LOSS MODULUS OF VISCO-ELASTIC DAMPER AT 21-26°C

0.000

0.050

0.100

0.150

0.200

0.250

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

FREQUENCY (hz)

SHE

AR

LOSS

MO

DU

LUS/

w (

MPa

-sec

)

G"/w Strain 25%G"/w Strain 50%G"/w Strain 75%G"/w Strain 100%G"/w Strain 150%


FIGURE 3-23 TEMPERATURE DEPENDENCE OF VISCO-ELASTIC DAMPER

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

15 20 25 30 35 40

TEMPERATURE (DEGREES CELSIUS)

STO

RAG

E /

LO

SS M

OD

ULU

S G' Strain 50% @ 0.50 hzG"/w Strain 50% @ 0.50 hz

FIGURE 3-24 STRAIN DEPENDENCE OF VISCO-ELASTIC DAMPER

-3

-2

-1

0

1

2

3

-25 -20 -15 -10 -5 0 5 10 15 20 25

DISPLACEMENT (mm)

FORC

E (K

N)

f = 0.5hz Strain = 150%

f = 0.5hz Strain = 75%


FIGURE 3-25 FREQUENCY DEPENDENCE OF VISCO-ELASTIC DAMPER

-3

-2

-1

0

1

2

3

-25 -20 -15 -10 -5 0 5 10 15 20 25

DISPLACEMENT (mm)

FORC

E (K

N)

f = 2hz Strain = 150%

f = 0.5hz Strain = 150%

3.5 OTHER TYPES OF DAMPER

A number of other damping devices have been proposed but the four categories listed above have accountedfor the majority of damping projects to date. As we become aware of alternative types of dampers which havesystem characterisation tests available we will expand these guidelines to include them.

3.6 DAMPING WIND LOADS

In principle, dampers operate independent of the source of the loads and so in theory will provide amplitudereductions for wind loads as well as earthquake loads. However, there are two main differences betweenserviceability loads such as wind and earthquake loads that make this difficult to achieve in practice:

1. Deformations under wind loads are much less than earthquake movements. Because practical dampingdevices provide damping forces that are a function of either displacements or velocities, the dampingeffectiveness is lower for small movements.

2. Wind loads provide many more cycles of movement than earthquake loads. Over the design life of abuilding there probably be 106 or more wind load cycles compared to less than 102 earthquake load cycles.


For many materials, particularly yielding metals, the number of wind cycles would be sufficient to causefatigue failure.

One other difference is that the response of the structure under wind load is linear elastic whereas mostbuildings are designed to yield under earthquake loads.

Of the damper types considered here, hysteretic dampers are usually yielding metals and could not be designedto operate beyond yield under wind loads because of the potential for fatigue failure. This type of damperwould provide added stiffness under wind loads but no energy dissipation. Some forms of hysteretic damperare promoted as suitable for wind loads, for example, the lead shear damper. Lead strained into the plasticrange re-crystallises at room temperature and retains its original properties and so in theory this type ofdamper may be able to be used to damp wind loads. However, design procedures for this type of damper arenot readily available and are not covered in these guidelines.

It is unlikely that friction dampers could be designed so as to slip under wind loads. Most friction materialsare subject to wear and would lose efficiency under the number of cycles typical for wind. If slip did notoccur, the friction dampers would add stiffness but not energy dissipation, as for the hysteretic dampers.

Viscous dampers provide a damping force equal to the product of the damping coefficient and the velocity.Regardless of velocity, a specific damping force can be obtained by installing dampers with sufficient dampingcoefficient. However, for low velocities this would require such a large number of dampers that the costwould likely be prohibitive.

Visco-elastic dampers are probably the most practical types for reducing wind response. For this type ofdamper the layer thickness is a function of maximum displacement. For small displacements thin layers ofpolymer can be used. As the damper stiffness and damping coefficient are both inversely proportional to thelayer thickness, this implies that these dampers could be effective if designed for the displacement likely tooccur.

If a wind damper is practical then it is unlikely to be also suitable to operate as an earthquake damper.Dampers designed for small displacements would generally fail if subjected to large seismic displacements.Therefore, the design would have to be force limited or designed to fail at a specified displacement level. Aviable strategy might be to use visco-elastic and friction dampers in series, with the slip force set to limit theload in the visco-elastic damper to a safe limit.

The tools used in later chapters of these guidelines to evaluate performance could be adapted for wind loads.Damping decay using the elastic structure could measure damping provided by devices. Although feasible, thetime history analysis method would probably not be best for assessing wind load response. Some form offrequency response method would be better. This would be possible because the structure remains elastic.

There is demand for a wider range reliable wind dampers. The most common method currently used is thetuned mass damper which is not usually suitable for seismic damping as it requires a linear elastic structure.For future developments, it is intended to assess potential wind dampers in more detail.


4444 ANALYSIS OF DAMPING DECAYANALYSIS OF DAMPING DECAYANALYSIS OF DAMPING DECAYANALYSIS OF DAMPING DECAY

The theoretical equations for the dampers described in the preceding section provide a means of calculatingthe properties of the devices and estimating the damping they provide. However, damping calculated this wayis at best a very approximate estimate due to the difficulties in defining the strain energy of most realstructures.

This section evaluates the damping provided by a variety of devices by duplicating analytically a physicalmethod of measuring damping, the snap-back test, in which is a structure is released from a deformed positionand the decay in displacements over successive cycles is measured.

4.1 PROCEDURE FOR EVALUATING DAMPING DECAY

The displacement pushover option of ModelA has an added option for load type, termed Decay. When thisoption is selected a pushover displacement is applied for ¼ cycle, that is, the first loading sequence, to thedisplacement amplitude selected. The load is then released and the structure allowed to vibrate freely.

The procedure is the analytical equivalent of the experimental ‘snap back’ method used to measure damping insome types of structure. This option provides a displacement trace of the form shown in Figure 4-1. Thefinal portion of the plot, after release, demonstrates the damping decay.

FIGURE 4-1 ANALYTICAL DECAY CURVE

-250

-200

-150

-100

-50

0

50

100

150

200

250

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00

TIME (Seconds)

DIS

PLA

CEM

EN

T (m

m)

Measured Displacement TraceFitted Damping CurveFitted Damping Curve


The effective damping ratio, ξ as a fraction of critical damping can be determined from the logarithmic decayusing peaks m cycles apart. The peak displacements are extracted for cycle n and cycle n + m and the dampingcalculated using the following formula:

ξπ mmn

n 2ln =∆∆

+

...........................................................................................................................(4-1)

The damping can be displayed on the plot by over-laying an exponential curve representing the dampingcalculated above. This curve, an example of which is overlain on Figure 4-1, has an equation of

te ωξ−∆=∆ 0 ..........................................................................................................................................(4-2)

4.2 VISCOUS DAMPING IN THE STRUCTURE

In-structure damping or energy dissipation adds extra damping to the damping inherent in the structuralsystem. The analysis procedure applies 5% viscous damping to the structure in addition to the dampers, basedon the assumption underlying most seismic codes.

The viscous damping specified in ANSR-L is Rayleigh damping where the damping matrix, [C], is constructedfrom the mass matrix [M] and stiffness matrix [K] as

[C] = α[M] + β [K].................................................................................................................................(4-3)

where α and β are user-specified coefficients. These two constants may be calculated for two periods ofresponse, T1 and T2, which have associated viscous damping ratios, λ1 and λ2 as:

)T(T)T(T4

21

22

2211

−−

=λλα ...................................................................................................................(4-4)

)T(T)T(TTT

11

22

211221

−−

=λλβ ...............................................................................................................(4-5)

The use of two constants, α and β, allows damping to be specified exactly at two periods. At all periodsbetween these two periods the damping will be less than specified and for periods outside the range of thesetwo periods the damping will be greater than specified. At any period, T, the viscous damping can becalculated as:

TT βππ

λ +=4a

.......................................................................................................................................(4-6)


Figure 4-2 shows the total damping, plus the components from mass and stiffness damping respectively,where α and β have been set to provide damping λ of 5% at periods of 0.10 seconds and 3.0 seconds. Themass damping increases with increasing periods whereas the stiffness damping decreases with increasingperiod. The correct damping is applied at the two periods selected. The minimum damping applied isapproximately 1.8% at a period of 0.5 seconds.

As shown in Figure 4-2, the damping increases rapidly for periods less than or greater than the specifiedvalues. The longer period is specified so as to allow for period lengthening due to yielding. The shorterperiod is set to the shortest period likely to be important in response, with a lower limit approximately equal tothe shortest period for which the response can be captured by the time step, say 5 to 10 times the time step, or0.05 to 0.10 seconds for the usual time step of 0.01 seconds.

The coefficients are generally applied as scalar quantities, that is, the same coefficient is applied to allcomponents of the mass and stiffness matrices. ANSR-L does have the capability of varying the twocoefficients so that different damping is generated by different mass points and elements. It is quite complexto solve for the vector values of {α} and {β} for these situations although ModelA procedures do providesome facilities for this – see Users Manual.

FIGURE 4-2 RAYLEIGH DAMPING

0%

1%

2%

3%

4%

5%

6%

7%

8%

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

PERIOD (Seconds)

DA

MPI

NG

(% o

f Crit

ical)

Mass DampingStiffness DampingTotal DampingTarget Damping


4.3 10 STORY MODEL

The decay studies used a single model, a 10 story, 3 bay concrete plane frame, as shown in Figure 4-3. Theframe was designed for a low seismic zone.

Design was based on a 3.16 second period with a ductility factor µ = 3 which provided for a base shearcoefficient of 0.024. The building was modelled in ANSR-L using concrete elements with a stable elasto-plastic yield function (see ModelA user manual). Figure 4-4 shows the pushover curve developed for thisbuilding. The elastic limit is approximately 0.04. The building has a peak strength of approximately 0.075times the weight, which occurs at an average story drift of 1.75%. At this stage a mechanism has formed witha negative stiffness due to P-∆ effects. At the limit, although the average drift is 1.75% the displacements areconcentrated at lower levels with a maximum drift of 3.09% at the 3rd story.

FIGURE 4-3 MODEL USED FOR DECAY STUDIES


FIGURE 4-4 PUSHOVER CURVE FOR EXAMPLE 10 STORY BUILDING

0.000.010.020.030.040.050.060.070.08

0.00% 0.50% 1.00% 1.50% 2.00%

AVERAGE STORY DRIFT

FORC

E /

WE

IGH

T

4.4 DAMPING VARIATIONS

Damping decay curves were generated for two benchmark analyses plus 11 damping variations. Thebenchmark analyses, for the structure without dampers, were (1) the building with element strengths set sothat the building did not yield and (2) the building with yielding at the calculated element strengths. Alldamper variations were added to the second model, with element yielding.

Variations are as listed in Table 4-1. For the hysteretic dampers, it was assumed that damping was providedby a diagonal brace element with a constant stiffness throughout its length. The hysteretic and frictiondamper types (HD and FD) used similar properties except that the friction damper had the elastic stiffnessincreased by a factor of 10.

TABLE 4-1 DAMPER VARIATIONS

Name DamperType

Property1

Value Property2

Value

ELASTIC NoneYIELDING NoneHD 200HD 400

HystereticHysteretic

Yield LevelYield Level

200 KN400 KN

FD 200FD 400

FrictionFriction

Yield LevelYield Level

200 KN400 KN


Name DamperType

Property1

Value Property2

Value

VD 1000VD 2000VD 3000VD 5000

ViscousViscousViscousViscous

Coefficient CCoefficient CCoefficient CCoefficient C

1000200030005000

Exponent αExponent αExponent αExponent α

1.01.01.01.0

VE 400 200VE 1000 500VE 5000 2500

Visco-elasticVisco-elasticVisco-elastic

Coefficient CCoefficient CCoefficient C

2005002500

Stiffness KEFF

Stiffness KEFF

Stiffness KEFF

40010005000

4.5 DAMPING DECAY CURVES

Figure 4-5 plots the damping decay curves for the benchmark structure and the four damping device types.Each plot has curves for two variations of a particular damper type; where more than two variations for adevice (VD and VE) were used, only the lowest and highest values are plotted. The vertical axis of all plots isthe displacement at the top of the building (mm) and the horizontal axis time, in seconds. The initialdisplacement is applied over a 20.48 second time duration and the decay measured for a further 20 secondduration of free vibration.

These plots illustrate some characteristics of the structure and the damping devices which affect furtherprocessing of results:

• The elastic structure produces a “classical” decay curve with displacements reducing in successive cyclestoward a zero displacement value.

• When yielding is permitted, the decay is no longer centred about a zero displacement position as there is apermanent set of about 50 mm in the building caused by element yielding.

• None of the damper devices have sufficient restoring force to overcome this permanent set. The devicesall modify the rate of reduction in displacements but tend toward a non-zero value.

• The two HD devices plotted provide a similar rate of decay but there is a significant period difference.This is caused by the elastic stiffness, which is proportional to the yield force of the dampers. The devicewith the higher yield level has the higher stiffness and shorter period.

• The FD devices provide a very rapid initial reduction in displacements but the vibrations then continue ata reduced period. As for the HD devices, the period is shorter for the device with the higher yield level,for the same reasons.

• The VD devices have a similar period regardless of the level of damping. The difference between the twodevices plotted is solely in the rate of reduction of amplitude.

• The VE decices have a slight period dependence because of the difference in stiffness but this is lessmarked that for the HD and FD devices.


FIGURE 4-5 DAMPING DECAY CURVES

4.6 EVALUATION OF DAMPING

A two step procedure was followed to evaluate the damping from the decay traces listed in the precedingsection:

1. The displacements at successive positive and negative peaks we used to calculate the damping from thelogarithmic decay using equation (4-1).

2. A curve was fitted to the decay curve using the form of equation (4-2). As discussed later, most devicesdo not produce constant damping and so judgment was used to develop this curve. The fitted curve wasadjusted for the permanent set by adding the average displacement to the positive and negative curves.

-200-150

-100-50

050

100150

200250

0 10 20 30 40

ELASTICYIELDING

-100

-50

0

50

100

150

200

250

0 10 20 30 40

HD 400HD 200

0

50

100

150

200

250

0 10 20 30 40

FD 400FD 200

-100

-50

0

50

100

150

200

250

0 10 20 30 40

VD 1000VD 5000

-100

-50

0

50

100

150

200

250

0 10 20 30 40

VE 400 200VE 5000 2500


4.6.1 STRUCTURE WITHOUT DAMPING DEVICES

Table 4-2 summarizes the calculation of damping for the structure without devices for both elastic andyielding response. There is little difference between the two cases, with damping of 6.7% at the first cyclereducing to 4.4% at the 4th cycle. The target damping was 5%. Figure 4-6 shows the elastic trace with a 6%damping decay curve fitted, which provides a reasonable match.

The actual damping of 6% exceeds the target value of 5%. This is because analysis using step-by-stepintegration is not a closed form solution but rather a numerical solution. As discussed in Section 3.2, thespecification of Rayleigh damping is imprecise. There are other influences such as the discrete time step,significant digits in output and the change in period caused by P-∆ effects such that an exact correlation oftarget and actual damping cannot be expected.

TABLE 4-2 DAMPING IN STRUCTURE WITHOUT DAMPING

Peak ELASTIC YIELDINGPeriod Damping Period Damping

Positive1234

3.023.033.243.05

6.68%6.43%5.06%4.38%

3.023.023.233.07

6.77%6.30%5.24%4.41%

Negative1234

3.043.183.133.03

6.08%6.06%4.35%4.61%

3.043.163.143.04

6.02%6.12%4.48%4.59%

The similarity of the elastic and yielding cases is unexpected as yielding is usually associated with energydissipation, which should correspond to higher apparent damping. Referring to Figure 4-5, comparin theyielded to the elastic structure, after the load is released the yielded structure does not rebound as far on thefirst ½ cycle as the elastic structure but the next positive peak is higher.

Figure 4-7 shows the time history of moments in the beam with maximum plastic rotation and the moment-rotation plot of this same beam. Under the initial loading phase, the beam reaches its yield moment anddeforms to a plastic rotation of 0.008 radians. When the load is released the beam moment reduces but doesnot reach its negative moment capacity. Therefore, the hysteresis loop is not closed and the free vibrationcauses the beam to vibrate along its elastic stiffness curve. This adds no damping to the system.

This type of response, where the yielding element has insufficient strain energy to reverse the sign of yielding,influences attempts to develop damping from the decay curve for hysteretic and friction type dampers.


FIGURE 4-6 NO DEVICES FITTED DECAY CURVE

ELASTIC Damping 6.00%

-250-200-150-100-50

050

100150200250

20 25 30 35 40

FIGURE 4-7 BEAM MOMENTS

4.6.2 HYSTERETIC DAMPERS

The hysteretic dampers provided approximately 8% damping in the first cycle, reducing to between 4% and5% in subsequent cycles, as listed in Table 4-3. Figure 4-8 shows that an 8% decay curve fits the first cyclesbut overestimates the damping as free vibrations continue. The period of response is reduced because of theelastic stiffness of the dampers, from 3 seconds for the elastic structure to 2.6 seconds for the HD 200 and 2.3seconds for the HD 400. The damper configuration is assumed to be prismatic brace elements and sostiffness is proportional to yield strength.

The high initial cycle decay is caused by hysteretic cycling but for subsequent cycles the damper is linear elasticand so the damping reverts to that for the elastic structure. The Rayleigh damping coefficients were kept thesame for all models. As the period reduced (from 3.02 seconds to 2.3 seconds for the HD 400) the dampingsupplied would be less than the target value of 5% (see Figure 4-2).

-600

-400

-200

0

200

400

600

0 10 20 30 40

MOMENT

TIM

E

Positive Yield MomentNegative Yield MomentMoment

-100

0

100

200

300

400

500

600

0 0.002 0.004 0.006 0.008 0.01 0.012

ROTATION

MO

ME

NT


TABLE 4-3 HYSTERETIC DAMPERS

Peak HD 200 HD 400Period Damping Period Damping

Positive1234

2.532.592.582.68

8.50%4.75%5.33%4.35%

2.252.312.252.34

8.25%4.17%4.19%4.71%

Negative1234

2.602.542.672.65

5.57%4.81%4.94%3.90%

2.282.292.262.39

4.66%3.82%4.55%4.01%

FIGURE 4-8 HYSTERETIC DAMPERS FITTED DECAY CURVE

HD 400 Damping 8.00%

-100

-50

0

50

100

150

200

250

20 25 30 35 40


4.6.3 FRICTION DAMPERS

The hysteretic dampers provided very high damping in the first cycle, over 30%, but this reduces to between2½% and 4% in subsequent cycles, as listed in Table 4-4. Figure 4-9 shows that a 22% decay curveapproximately fits the first cycle but overestimates the damping as free vibrations continue by a large amount.The period of response is reduced because of the high elastic stiffness of the friction dampers. The initial 3second period for the elastic structure reduces to 1.8 seconds for the FD 200 and 1.45 seconds for the HD400. This is an unavoidable consequence of providing a high initial stiffness to ensure a high degree ofhysteretic energy dissipation.

As for the hysteretic dampers, the high initial cycle decay is caused by hysteretic cycling but for subsequentcycles the damper is linear elastic and so the damping reverts to that for the elastic structure. The periodreduction is such that the damping supplied by the Rayleigh coefficients for the elastic model is much lessthan 5%.

TABLE 4-4 FRICTION DAMPERS

Peak FD 200 FD 400Period Damping Period Damping

Positive1234

2.301.791.791.76

39.89%4.18%4.16%4.58%

1.801.451.461.46

30.36%3.36%3.37%3.28%

Negative1234

1.881.801.771.76

16.72%2.79%2.58%2.59%

1.451.461.461.46

2.66%2.66%2.42%2.24%

FIGURE 4-9 FRICTION DAMPERS FITTED DECAY CURVE

FD 400 Damping 22.00%

-50

0

50

100

150

200

250

20 25 30 35 40


4.6.4 VISCOUS DAMPERS

The results from the four configurations of viscous dampers, listed in Table 4-5, show some markeddifferences from the hysteretic and viscous dampers:

1. The period remains essentially independent of cycle and increases slightly as the damping constantincreases. This is expected, refer to equation (2-1).

2. The damping also remains reasonably constant over each cycle. Figure 4-10 shows that a decay curve canbe fitted to match the decay across the full range of cycles. There are some anomalies, such as cycle 4 forVD 5000, but these are caused by significant digit issues as the displacement is damped to close to zero.

The average damping increases successively from 8% to 10%, 12% and 17% as the damping coefficient isincreased from 1000 to 2000, 3000 and 5000, which suggests that the damping increases with coefficient butmore slowly than the coefficient. This is at least partly because the velocity is proportional to displacementand so reduces as the increased value of C reduces displacements.

TABLE 4-5 VISCOUS DAMPERS

Peak VD 1000 VD 2000 VD 3000 VD 5000Period Damping Period Damping Period Damping Period Damping

Positive1234

3.053.103.093.08

9.41%7.80%7.53%7.63%

3.083.093.083.09

11.91%10.01%10.07%10.33%

3.093.093.093.09

14.35%12.45%12.77%13.58%

3.113.103.103.10

19.36%17.81%19.81%30.54%

Negative1234

3.103.103.083.08

8.11%7.39%7.29%7.24%

3.113.083.083.09

10.07%9.59%9.44%9.16%

3.103.093.083.09

12.17%11.75%11.33%10.53%

3.113.103.103.10

16.50%15.61%13.65%10.07%


FIGURE 4-10 VISCOUS DAMPERS FITTED DECAY CURVE

VD 5000 Damping 17.00%

-150

-100

-50

0

50

100

150

200

250

20 25 30 35 40

4.6.5 VISCO-ELASTIC DAMPERS

The visco-elastic damper results, summarised in Table 4-6, exhibit some of the characteristics of the viscousdampers but are modified by the effect of the spring in parallel with the dashpot:

1. The period remains effectively constant for a given damper but decreases as the damper coefficientsincrease. The effect of an increased spring stiffness in reducing the period more than counteracts theeffect of the dashpot in increasing the period.

2. The damping remains more constant than for the hysteretic and friction dampers but there is somedecrease in damping with increasing number of cycles. This is slight as the plotted decay curve (Figure 4-11) shows a reasonable match for all cycles.

TABLE 4-6 VISCO-ELASTIC DAMPERS

Peak VE 400 200 VE 1000 500 VE 5000 2500Period Damping Period Damping Period Damping

Positive1234

3.013.063.143.05

7.26%6.50%5.55%5.27%

3.003.063.073.04

8.01%6.84%6.18%6.16%

2.922.922.912.92

12.37%10.41%10.49%10.77%


Peak VE 400 200 VE 1000 500 VE 5000 2500Period Damping Period Damping Period Damping

Negative1234

3.053.133.093.04

6.50%6.08%5.18%5.29%

3.043.093.053.04

7.08%6.37%6.00%6.07%

2.932.922.912.92

10.43%10.00%9.83%9.52%

FIGURE 4-11 VISCO-ELASTIC DAMPERS FITTED DECAY CURVE

VE 5000 2500 Damping 11.00%

-150

-100

-50

0

50

100

150

200

250

20 25 30 35 40

4.7 SUMMARY OF DAMPING DECAY

Table 4-7 summarises the results of the damping decay analyses. These studies have provided some data onthe effect of various damping devices but has also identified some difficulties in quantifying damping for sometypes of device using this type of analysis.

TABLE 4-7 SUMMARY OF DAMPING DECAY

Cycle 1 Average of 4Name Damper

TypePeriod Damping Period Damping

ELASTIC None 3.02 6.7% 3.09 5.5%YIELDING None 3.02 6.8% 3.09 5.5%HD 200HD 400

HystereticHysteretic

2.532.25

8.5%8.2%

2.612.30

5.3%4.8%

FD 200FD 400

FrictionFriction

2.301.80

39.9%30.4%

1.861.50

9.7%6.3%


Cycle 1 Average of 4Name Damper

TypePeriod Damping Period Damping

VD 1000VD 2000VD 3000VD 5000

ViscousViscousViscousViscous

3.053.083.093.11

9.4%11.9%14.3%19.4%

3.093.093.093.10

7.8%10.1%12.4%17.9%

VE 400 200VE 1000 500VE 5000 2500

Visco-elasticVisco-elasticVisco-elastic

3.013.002.92

7.3%8.0%12.4%

3.073.052.92

5.9%6.6%10.5%

A summary of the response of the different types of damper identified the following characteristics:

• The ANSRL program modelled the target damping of 5% damping reasonably well, with an averagedamping of 5.5%.

• Structural yielding (beams and columns) had only a very slight effect on damping as measured by decay.This is because the structure immediately unloads to its elastic state over one-half cycle and then vibratesas for the non-yielding model.

• The hysteretic dampers provided increased damping, about 8%, for the first cycle but in subsequent cyclesthe damping reduced to that for the base structure with the elastic stiffness of the dampers. This isbecause the dampers did not cycle plastically after the initial release.

• The friction dampers produced a similar response to the hysteretic dampers but with much higherdamping in the initial cycle, over 30%.

• The viscous dampers produced relatively constant damping, from 8% to 18% for the properties includedin this study. The damping did not increase linearly with the damping coefficient; increasing the dampingcoefficient by a factor of 5 increased damping by a factor of 2.2.

• The visco-elastic dampers provided almost constant damping but with some decrease with decreasingamplitude because of the stiffening effect of the elastic component. These devices seemed to provideapproximately as much damping as a viscous damper with the same coefficient. For example, C = 2500produced 10½% damping for the visco-elastic device, compared to 10% for the C = 2000 viscous deviceand 12% for the C = 3000 viscous device.

At first examination, these results appear to indicate much better performance from viscous devices (VD andVE) than from hysteretic devices (HD and FD) in that the damping for the latter only applies for the firstcycle. However, this more likely identifies problems with quantifying damping using this procedure ratherthan necessarily ineffectiveness of the devices. The intention of using supplemental dampers for seismicprotection is generally to reduce the peak amplitude of response and the HD and FD dampers may beeffective in this. The time history analyses described in the following section were intended to better definethis effect.


5555 TIME HISTORY ANALYSISTIME HISTORY ANALYSISTIME HISTORY ANALYSISTIME HISTORY ANALYSIS

5.1 OBJECTIVE

Previous sections have examined the damping provided by different devices by considering their properties(Section 3) and evaluating the decay curve from a snapback test (Section 4). Each of these procedures haveidentified difficulties with quantifying the response reductions achieved with the devices, particularly forhysteretic and friction dampers.

The aim of these guidelines is to develop design procedures for the use of dampers to reduce seismicresponse. The definitive method of determining whether this has been achieved is to calculate the response ofa structure with the damper installed. As the damper properties are non-linear and modify the dynamicproperties of the structure the most suitable method to quantify response is to use a time history analysis withthe dampers explicitly modelled.

Three prototype buildings were used for this study, each concrete frame structures with heights of 3, 5 and 10stories respectively. The buildings were designed for a low seismic zone and the performance evaluated withvarying devices for earthquake records corresponding to a high seismic zone. The aim of this study was todetermine which devices and configurations could improve the performance so as to be satisfactory for thehigher load.

5.2 PROTOTYPE BUILDINGS

Three buildings were selected and plane frames from these buildings used for the evaluation. Each buildingwas three bays with a constant 7.500 m bay length. Bottom story heights were 4.570 m and all upper stories3.650 m high. The dampers were assumed to be in a diagonal configuration in the central bay, as shown inFigure 5-1.

The buildings were designed to be flexible and with a strength required for a low seismic zone as this is thecondition of most buildings for which supplemental damping will be considered. Table 5-1 summarises thedesign parameters to NZS4203 for Z = 0.6, intermediate soil type and a ductility of 3. The design base shearcoefficients were 0.024, 0.033 and 0.047 for the 10 story, 5 story and 3 story buildings respectively.


FIGURE 5-1 PROTOTYPE BUILDINGS

DamperLocations

TABLE 5-1 DESIGN PARAMETERS FOR PROTOTYPE BUILDINGS

No. Storeys 10 5 3Columns: Breadth (mm) Depth (mm)

700700

600600

500500

Beams: Breadth (mm) Depth (mm)

500600

400600

400600

qWt base (kN) 18943 9084 5337µx

Sp

RZSoilKm

Lu

30.67

10.6I

0.81

30.67

10.6I

0.81

30.67

10.6I

0.81

Period (seconds) 3.16 1.86 1.36


No. Storeys 10 5 3Ch(T1,µ)SpRZLu

C(T)USEC(T)Wt

0.0520.4020.0210.030568

0.0890.4020.0360.036324

0.1210.4020.0490.049260

Vbase(1) 1127 901 753Km

Sm1

Sm2

Sm

0.8000.3300.4030.403

0.8000.3300.2870.330

0.8000.3300.2760.330

Vbase (kN) 455 297 248Vbase/Wt 0.024 0.033 0.047SmSpRZLu 0.162 0.133 0.133

5.3 SEISMIC INPUT

The objective of the study was to determine which devices could improve the performance of this building ina high seismic zone and so time histories were selected for the highest NZ zone, Z = 1.2, corresponding totwo times the design level. The spectrum as defined by NZS4203 for this zone and soil type is equivalent to aUBC spectrum for Z = 0.4 soil type SC and near fault factors of unity (CA = 0.40, CV = 0.56).

The New Zealand code, NZS4203, provides only very general requirements for time history scaling and so themore explicit requirements of the UBC were used. The UBC requires that the time history be scaled such thatthe average value of the SRSS spectrum of the two components does not fall below 1.4 times the 5% dampedspectrum over a period range of 0.2T to 1.5T where T is the fundamental period of the structure.

5.3.1 BASIS FOR SELECTING RECORDS

The procedure used to select time histories is based on a spreadsheet database of the 5% damped responsespectra from 36 pairs of components. The process is:

1. Define the fundamental structural period, T, in this case 3 seconds.

2. Set the UBC limits of 0.2T to 1.5T, in this case 0.6 seconds to 4.5 seconds.

3. For each pair of spectra, calculated the scaling factor such that the average ratio of the SRSS values at eachperiod within this range is 1.4.

4. Calculate the standard deviation of the ratio of SRSS to design value at each period within the range.

5. Select time histories, generally those that provide the smallest standard deviation although also usingjudgement based on a visual examination of the match.


Table 5-2 lists the factors for the 36 earthquake considered. The 1st 10 records are those recommended byATC-40 for soil sites greater than 10 km from sources (ATC-40 Table 4-9). The remaining 26 records arefrom the SMARTS earthquake database of earthquakes up to the 1971 San Fernando quake.

The three earthquakes with the lowest standard deviations are No. 11, 1940 El Centro, No. 28, the 1933Vernon Command Building from the Long Beach Earthquake and No. 9, from the 1954 Eureka earthquake.

The 1940 El Centro N-S component was one of the earliest recorded strong motion accelerograms andformed the basis for much seismic design code development. Since 1940 thousands of strong motion recordshave been processed and are available for use and so it is perhaps surprising that this record still provided thebest match. However, it is not a coincidence that El Centro provides the best match in that the shape of theresponse spectra in codes such as NZS4203, UBC and FEMA-273 all indirectly have their origin in thisearthquake.

The characteristic of these codes which makes El Centro the best match for medium to long periods (greaterthan about 1 second) is that the spectral acceleration is inversely proportional to the period. This implies aconstant spectral velocity. It may be time to re-visit this constant velocity assumption by evaluating the widedatabase of records now available. Some codes have an exponent on the reciprocal to the period, T (e.g.AASHTO has acceleration proportional to 3/2/1 T and the Turkish code to 8.0/1 T ). That is outside ourcurrent scope.

The records selected for this evaluation were the El Centro 1940 N-S record and the Century City record fromthe 1994 Northridge earthquake. This latter record has a larger standard deviation than other records in thelist but it was considered prudent to use a more modern record in addition to the 1940 record. A third recordused was the El Centro record frequency scaled to match the target spectrum. For the 1940 El Centroearthquake, the scaling procedure requires a scaling factor of 1.59 to match NZS4203 intermediate soil forZ=1.2 and Sp = 1. For the Northridge record the scaled factor was 2.20. Figures 5-2, 5-3 and 5-4 showcomparisons between the scaled earthquake response spectra and the design spectrum.

TABLE 5-2 SCALE FACTORS FOR VARIOUS EARTHQUAKES

Record ScaleFactor

StandardDeviation

1 1949 Western Washington Station 325 2.49 0.362 1971 San Fernando, California Station 241 1.58 0.303 1989 Loma Prieta, California Gilroy #2 1.57 0.644 1992 Landers, California Yermo 1.98 0.445 1989 Loma Prieta, California Hollister, South & Pine 1.24 0.416 1992 Landers, California Joshua Tree 1.95 0.587 1994 Northridge, California Century City LACC North 2.20 0.448 1994 Northridge, California Moorpark 2.80 0.679 1954 Eureka, California Station 022 2.11 0.2810 1971 San Fernando, California Station 458 1.78 0.5711 1940 El Centro Site Imperial Valley Irrigation District 1.59 0.23


Record ScaleFactor

StandardDeviation

12 1952 Pasadena – Caltech Athenaeum 7.54 0.5213 1952 Taft Lincoln School Tunnel 3.45 0.4914 1952 Santa Barbara Courthouse 3.60 0.5715 1952 Hollywood Storage Basement 6.56 0.3916 1952 Hollywood Storage p.e. lot 6.63 0.3717 1957 San Francisco Golden Gate Park 12.80 1.4618 1933 Vernon Cmd Bldg 3.02 0.2619 1934 El Centro Imperial Valley 3.87 0.6120 1935 Helena S00w Helena Montana 6.06 0.7921 1949 Seattle S02w Western Washington 8.77 1.0022 1965 Olympia S04e Puget Sound Washington 4.68 0.8023 1966 Cholame N05w Parkfield California 2.50 0.9424 1966 Cholame N50e Parkfield California 3.97 0.9725 1966 Temblor N50e Parkfield California 7.71 0.4026 1966 Temblor N65w Parkfield California 3.52 1.1627 1971 Pacoima Dam S16e San Fernando 0.74 0.5928 1971 250 E First Street Basement N36e San Fernando 3.56 0.3129 1971 445 Figueroa Street N52w San Fernando 3.10 0.2830 1971 Hollywood Storage Bsmt. s00w San Fernando 2.74 0.3331 1971 Caltech Seismological lab. s00w San fernando 5.40 0.8732 1971 Caltech Athenaeum N00e San Fernando 4.70 0.3733 1971 Caltech Millikan Library N00e San Fernando 3.63 0.5234 1971 Jet Propulsion Lab. s82e San Fernando 4.59 0.7035 1971 Palmdale Fire Station S60e San Fernando 4.21 0.6436 1971 15250 Ventura Blvd. n11e San Fernando 2.00 0.57


FIGURE 5-2 5% DAMPED SPECTRUM OF EL CENTRO TIME HISTORY

1940 EL CENTRO SITE IMPERIAL VALLEY IRRIGATION DISTRICT A001 x 1.59

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

PERIOD (Seconds)

ACC

ELE

RATI

ON

(g)

Design Spectrum

Component 1

Lower Period Limit

Upper Period Limit

FIGURE 5-3 5% DAMPED SPECTRUM OF NORTHRIDGE TIME HISTORY

1994 Northridge, California Century City LACC North x 2.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

PERIOD (Seconds)

ACC

ELE

RATI

ON

(g)

Design Spectrum

Component 1

Lower Period Limit

Upper Period Limit


FIGURE 5-4 5% DAMPED SPECTRUM OF FREQUENCY SCALED EL CENTRO TIME HISTORY

Frequency Scaled El Centro 1940 N-S Component

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

PERIOD (Seconds)

ACC

ELE

RATI

ON

(g)

Design Spectrum

Lower Period Limit

Upper Period Limit

El Centro N-S Seed

5.4 DAMPER VARIATIONS

Five damper variations were considered for this part of the study. Table 5-3 lists the damping parametersused for each variation. Definitions of the damper types and properties in Table 5-3 are:

1. Type H is a hysteretic steel damper, modelled as a yielding diagonal brace with an elastic – perfectly plasticyield function. The value listed in Table 5-3 is the yield force, Py, in KN. The brace was assumed to havea yield strength of 250 MPa and the brace was assigned an area of Py/250. The maximum force, 1000KN, corresponds to a brace area of 4000 mm2.

2. Type F is a friction damper, also modelled as a yielding diagonal brace. The value listed is the frictionforce, Ff, in KN. An area was defined as for type F but the elastic stiffness was increased by a factor of 10compared to Type H (yield displacement approximately 1.3 mm compared to 13 mm for H).

3. Type S is an hysteretic steel damper as for Type H except that the brace has a strain hardening ratio of 1%of the initial elastic stiffness. Values are the yield force, as for Type H.

4. Type V is a viscous damper, linking adjacent floors and oriented on the diagonal. Values listed in Table 5-3 are the damping coefficient, C, in units of KN-sec/m. The exponent, α, was assumed to be 1.0 for allanalyses.

5. Type VE is a visco-elastic damper, also linking adjacent floors and oriented on the diagonal. The valueslisted in Table 5-3 are the damping coefficient, C, in units of KN-sec/m, the same as for Type V. These


dampers also had a corresponding effective stiffness, KEFF, with a value numerically equal to 2 times C, inunits of KN/m. This is a reasonable ratio of the loss modulus to the storage modulus for low frequencyresponse (see Figures 3-20 and 3-21).

TABLE 5-3 VARIATIONS IN DAMPER PROPERTIES

Variation TypesH,F and S

TypesH,F and S

TypesH,F and S

TypesV and VE

10 Story 5 Story 3 Story All1 0 0 0 02 50 25 15 5003 100 50 30 10004 150 75 45 15005 200 100 60 20006 250 125 75 25007 300 150 90 30008 350 175 105 35009 400 200 120 400010 450 225 135 450011 500 250 150 500012 550 275 165 550013 600 300 180 600014 650 325 195 650015 700 350 210 700016 750 375 225 750017 800 400 240 800018 850 425 255 850019 900 450 270 900020 950 475 285 950021 1000 500 300 10000

Each damper type and property variation was modelled with the three different distributions shown in Figure5-5:

1. Distribution U = Uniform Distribution. The damper property listed in Table 5-3 was used in the deviceat each story level.

2. Distribution T = Triangular Distribution. The damper property listed in Table 5-3 was used to define thedevice at the uppermost story. The device at the base was defined by using a value ¼ of the value at thetop. Linear interpolation was used at intermediate stories.


3. Distribution R = Reversed Triangular Distribution. The damper property listed in Table 5-3 was used todefine the device at the bottom story. The device at the top used a value ¼ of the base value. Linearinterpolation was used at intermediate stories.

FIGURE 5-5 DAMPER DISTRIBUTION WITH HEIGHT

1.0 0.25

1.0 1.0

1.0

0.25

UniformU

TriangularT

ReverseTriangular

R

5.5 TIME HISTORY EVALUATION PROCEDURE

The ModelA spreadsheet was used to develop models of each of the three buildings with the nominalstrengths of the beam and column elements. At each level two elements were added in the centre bay inparallel, one a truss element to model added stiffness and the second a damper element to model addeddamping. A template ANSR-L file was produced for each of the three prototype buildings.

A QuickBasic “driver” program was set up to evaluate response for each earthquake record, building, dampertype, damper distribution and damper parameters:

1. Read in the template file and modify the properties of the spring and/or damper elements depending onthe type of damper.

2. Shell the ANSR-L program to run the time history.

3. Read the ANSR-L output files and summarise maximum displacements, drifts and element actions to adisk file.

The end product was a disk file with one line per variation for each building. The procedure was also used toproduce a series of benchmark results, based on the response of the buildings without added damping but


incrementally increased viscous damping. This was implemented by modifying the α and β factors definingthe Rayleigh damping.

The drifts were calculated using an approximate procedure from the envelope displacement profile rather thanthe instantaneous displacement profiles at every time step. The buildings responded primarily in first modeand so this has little effect for the 3 and 5 story building. This method tended to underestimate the drifts forthe 10 story building by up to about 5%. This would not affect conclusions from this phase of the study.

5.6 RESPONSE OF BUILDING WITHOUT DAMPERS

Table 5-4 summarises the response of the as-designed buildings under each of the three earthquake records.The maximum drifts increase with height of the building, from 1.5% for the 3 story building, 2.0% for the 5story building and 2.4% for the 10 story building. Maximum beam plastic rotations follow a similar pattern.

The 5% damped traces of roof displacement for each building in Figure 5-6 for the El Centro record show adifference in behaviour of the 3 and 10 story buildings compared to the 5 story building. The former twodemonstrate a permanent set, of magnitude approximately 50 mm for the 3 story and 400 mm for the 10 storybuilding. This occurs when a side-sway mechanism forms, comprising plastic hinges at all beam ends plus thecolumn base hinges. Under this earthquake record, column base hinges did not occur in the 5 story buildingand so a mechanism did not form. This avoided the permanent set.

TABLE 5-4 MAXIMUM RESPONSE QUANTITIES - NO DAMPERS

EQ 1 EQ 2 EQ 33 Story Displacement (mm) Drift (m/m) Column Plastic Rotation (rads) Beam Plastic Rotation (rads) Base Shear (KN)

1060.0130.0090.0161136

(0.212W)

1130.0110.0060.0151189

(0.222W)

1270.0150.0110.0191138

(0.213W)

5 Story Displacement (mm) Drift (m/m) Column Plastic Rotation (rads) Beam Plastic Rotation (rads) Base Shear (KN)

1720.0150.0000.0221448

(0.159W)

2760.0200.0100.0311494

(0.164W)

2310.0170.0050.0271330

(0.146W)

10 Story


EQ 1 EQ 2 EQ 3 Displacement (mm) Drift (m/m) Column Plastic Rotation (rads) Beam Plastic Rotation (rads) Base Shear (KN)

5860.0240.0070.0301811

(0.096W)

4950.0240.0020.0331796

(0.095W)

3770.0200.0010.0291659

(0.088W)

5.6.1 EFFECT OF VISCOUS DAMPING

In order to provide benchmarks for the damping devices, the building without dampers was analysed forincreasing levels of viscous damping, from 0% to 40%.

Figure 5-6 illustrates the effect of increasing damping from 5% to 25% on the roof displacements for the ElCentro record. The higher damping reduces displacements throughout the record but, more importantly,reduces the permanent set occurring in the 3 and 10 story buildings.

FIGURE 5-6 TIME HISTORY OF ROOF DISPLACEMENT (EL CENTRO RECORD)

-150

-100

-50

0

50

100

150

0 5 10 15 20

TIME (Seconds)

DIS

PLA

CEM

EN

T (m

m) 3 Story 5% Damping

3 Story 25% Damping

-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20

TIME (Seconds)

DIS

PLA

CEM

EN

T (m

m)

5 Story 5% Damping5 Story 25% Damping

-700

-600

-500

-400

-300

-200

-100

0

100

200

0 5 10 15 20

TIME (Seconds)

DIS

PLA

CEM

EN

T (m

m)

10 Story 5% Damping10 Story 25% Damping


Figure 5-7 plots the maximum drifts in each building for each of the three buildings as viscous damping isincreased from 0% to 60%. Although drifts tend to decrease with increasing damping, there are largedifferences between buildings and between earthquake records:

• For the El Centro record increased damping reduces drift for all buildings but the maximum effect is forthe 10 story building. For this building the drift at 60% damping is about one-fifth the zero damped valuewhereas for the 3 and 5 story buildings the 60% damped drift is about one-half the zero damped value.

• For the Northridge record the damping has most effect on the 5 story building. The drifts in the 10 storybuilding are largely unchanged for damping from 0% to 10% but then reduce for higher values ofdamping.

• The frequency scaled record produces a more consistent effect over all three buildings with maximumdrifts reducing by approximately the same factor in all buildings as damping increases from 0% to 60%.

The variation in the effect of the viscous damping is a feature of the non-linearity of these structures underthis level of earthquake loading. Figure 5-8 shows the effect of viscous damping on base shear. There ismuch less variation in base shear than there is in drifts. This is because the base shear is limited by thestrength of the beam hinging mechanism which forms in each building.

FIGURE 5-7 EFFECT OF VISCOUS DAMPING ON DRIFTS

Scaled El Centro

0.000

0.010

0.020

0.030

0.040

0.050

0% 10% 20% 30% 40% 50% 60%

DAMPING (% of Critical)

DRI

FT (m

/m) 3 Story Drift

5 Story Drift10 Story Drift

Scaled Northridge

0.0000.0050.0100.0150.0200.0250.0300.035

0% 10% 20% 30% 40% 50% 60%


DRI

FT (m

/m)

3 Story Drift5 Story Drift10 Story Drift

Frequency Scaled El Centro

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0% 10% 20% 30% 40% 50% 60%


DRI

FT (m

/m)

3 Story Drift5 Story Drift10 Story Drift


FIGURE 5-8 VISCOUS DAMPING EFFECT ON BASE SHEAR (EL CENTRO)

0

500

1000

1500

2000

2500

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0%

VISCOUS DAMPING (% of Critical)

BASE

SH

EA

R (K

N)

3 Story Base Shear Force5 Story Base Shear Force10 Story Base Shear Force

5.7 DAMPER EFFECTIVENESS

Type S, which is a hysteretic damper with a strain hardening ratio of 1% of the elastic stiffness, producedresults almost identical to the hysteretic damper with zero strain hardening. Differences were in almost everyinstance less than 1% and so the results are not reported further here.

The objective of installing supplemental dampers into a building is generally to reduce displacements underearthquake loads and so the effectiveness of dampers is primarily measured by the degree to whichdeformations are reduced. Secondary issues that may be important include base shear and floor accelerations.

5.7.1 EFFECT ON DRIFTS

The effectiveness of each type of damper in modifying performance was assessed by calculating the maximumdrifts from each time history analysis of damper configurations and computing the ratio of this value to theequivalent value for the building with no dampers. The plots in appendix A provide this ratio for eachearthquake and each damper distribution.

In accordance with usual procedures for time history analysis, the maximum values from the three timehistories were used to determine the effect the dampers would have in an actual evaluation:

• The maximum drifts from the three time histories with no devices were set as the benchmark value.

• For each damper configuration, the equivalent maximum drift from the three time histories was extracted.

• The effectiveness was defined as the ratio of the maximum value obtained from the analysis with eachdevice to the benchmark value.

Because the dampers modified the characteristics of the structure, peak values did not necessarily occur forthe same earthquake as for the benchmark structure. Figure 5-9 plots the ratio derived from the three


earthquakes for each damper type and structure for the uniform distribution. Figures 5-10 and 5-11 providethe equivalent results for the triangular and reverse triangular distribution. The detailed plots in Appendix Aprovide full results for each earthquake.

In all the plots, a ratio of 1.0 indicates drifts equivalent to the drifts for the structure with no dampers. A ratiogreater than 1.0 indicates that the devices have increased the drifts, a negative effect, and a ratio less than 1.0indicates reduced drifts, a positive effect.

There were wide variations in effectiveness, both between types and within types.

Hysteretic Dampers

The effectiveness of the hysteretic dampers, (Figures 5-9 to 5-11 and also Figures A-1 to A-3 in appendix A),is extremely building specific and earthquake specific to a lesser degree.

• For the 3 story building the dampers are ineffective regardless of yield level and regardless of damperdistribution. The dampers increase drifts by up to 30%. The details plots in Appendix A show that thereis a slight beneficial effect for EQ1 (El Centro) for low yield levels but a negative effect for the other twoearthquakes.

• The dampers are effective for the 5 story building for low to moderate yield levels but increase drifts forhigher yield levels. This holds for all three distributions of dampers, although the reverse triangular isslightly better then the other distributions. Examining the detailed plots in Appendix A, the dampers areineffective for EQ 1 and EQ 3 but the low yield levels are effective for EQ 2, the Northridge record. Asthis record produces the highest response for the 5 story building (Table 5-4) the effect of the dampers ispositive on maximum response.

• The hysteretic dampers have a beneficial effect on the 10 story building, with the effectiveness generallyincreasing with increasing yield level. The uniform distribution of yield force with height is generallysimilar to the triangular distribution. The reverse triangular distribution has an advantage over the othertwo for high yield levels. With the uniform distribution there is an optimum yield force, beyond whichthe effectiveness remains static. The plots for the individual earthquakes in Appendix A show that lowyield levels have a negative effect for EQ 3 and a neutral effect for EQ 2.

These results suggest that design of this type of damper is likely to be complex and for some buildings thedamper will be of no use at all and may impair earthquake performance if wrongly sized.

Friction Dampers

As applied in this study, the friction dampers are similar to the hysteretic damper except that the initialstiffness is 10 times as high, leading to yield at much lower displacements and a more rectangular hysteresisloop. Figures 5-9 to 5-11 show that this modification to the device characteristics has improved effectivenessin most cases:

• For the 3 story building, the friction dampers are ineffective unless a medium to high friction force isused, exceeding 250 KN. For lower friction forces the dampers are ineffective regardless of distribution.


For high friction force levels, above 250 KN, the uniform distribution produces more benefits than thetwo non-uniform distributions. The plots for each earthquake in Appendix A show that for earthquakes 1and 2 the friction dampers have a negative effect for some yield levels.

• The friction dampers have a positive effect on the 5 story buildings for all friction forces. The uniformand reverse triangular distributions are more effective then the triangular distribution. The plots for eachearthquake in Appendix A show that the friction dampers are most effective for EQ 2, which producesthe greatest response of the three earthquake for the 5 story building.

• The friction dampers are most effective for the 10 story building, provided that a medium to high yieldforce is used. The triangular distribution is less effective than the other two types considered. The plotsfor each earthquake in Appendix A show that the friction dampers are most effective for EQ 1 for the 10story building.

Comparing the friction and hysteretic dampers, the friction dampers are much more efficient than a hystereticdamper with the same yield force. As the only difference between the two is the initial stiffness, this impliesthat the efficiency of the hysteretic damper will be improved if the elastic stiffness is increased.

Viscous Dampers

The performance modification provided by the viscous dampers, Figures 5-9 to Figure 5-11, shows a moreregular trend than the displacement dependent dampers. Effectiveness generally increases with increasingdamping coefficient, although there are some exceptions.

• In the 3 story building, viscous dampers provide decreasing drifts as the coefficient is increased with thehighest coefficient reducing drifts less than 40% of the value with no devices. All distributions ofdampers provide generally similar reductions in response. The plots for each earthquake in Appendix Ashow that this trend occurs for all earthquakes except for EQ 2, where drifts increase slightly for lowvalues of the damping coefficient.

• In the 5 story building, viscous dampers show a similar reduction in drift as the damping coefficientincreases. The increases are not quite as large for the 3 story building. The uniform distribution is moreefficient that the other two distributions. Appendix A shows that for the 5 story building the viscousdampers consistently reduce drifts for all earthquakes although they are more effective for EQ 2 than theother two earthquakes.

• In the 10 story building, viscous dampers are ineffective for small damping coefficients but theeffectiveness increases for higher coefficients. As for the other buildings, the uniform distribution is themost efficient. The plots in Appendix A show that the viscous dampers are more effective forearthquakes 1 and 3 than they are for earthquake 2.

Unexpectedly, the velocity and so the force per damper reduces as the building height increases although withhindsight this is not surprising. All buildings have an equal story weight and so the tributary mass per damperis the same for all buildings. The velocity is related to the period of the building; for equal displacements, thesystem with the shortest period would have the larger velocity.


Visco-Elastic Dampers

The results for the visco-elastic dampers, Figures 5-9 to 5-11, generally show a close correlation to the resultsfor the viscous dampers, which have the same damping coefficients.

• For the 3 story building, the visco-elastic dampers produce the same effective damping as the viscousdampers but need a slightly lower coefficient to achieve the same drift reductions. As for the viscousdampers, all values of the damping coefficient and all distributions produce a positive effect on drifts.

• The 5 story building shows a similar trend, approximately the same effective damping but at a slightlyhigher coefficient compared to the viscous dampers. The differences in the two types are more markedfor the uniform distribution than the non-uniform distributions.

• For the 10 story building the visco-elastic dampers are more efficient than the viscous dampers for alldistributions and provide benefits in terms of drift reductions for low coefficients where the viscousdampers are ineffective.

As for the viscous dampers, the uniform damper distribution and reverse triangular distribution producebetter results than the triangular distributions.


FIGURE 5-9 SUMMARY OF ALL DAMPERS : UNIFORM DISTRIBUTION

0.00

0.20

0.40

0.600.80

1.00

1.20

1.40

0 5 10 15 20

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift

H 3 Story F 3 StoryV 3 Story VE 3 Story

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 5 10 15 20

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift



FIGURE 5-10 SUMMARY OF ALL DAMPERS : TRIANGULAR DISTRIBUTION

0.00

0.20

0.40

0.600.80

1.00

1.20

1.40

0 5 10 15 20

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.000.200.400.600.801.001.201.401.60

0 5 10 15 20Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift



FIGURE 5-11 SUMMARY OF ALL DAMPERS : REVERSE TRIANGULAR DISTRIBUTION

0.00

0.20

0.40

0.600.80

1.00

1.20

1.40

0 5 10 15 20

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift



5.7.2 EFFECT ON BASE SHEAR

The objective of adding dampers is generally to reduce deformations in structures, measured by drifts asdescribed in the preceding section. Depending on the deficiencies in the structure, either or both the totalshear and the proportion of the shear resisted by the structural system may also be important:

• Some damper types, such as those installed in diagonal braces, will add forces to the existing structuralsystem, which must resist the vertical component of the force in the columns. Increased shears mayoverload the structural system. For these systems, the total force is important.

• Many older buildings have shear deficiencies. Shear reaches a peak value when an element yields and sothese deficiencies are not necessarily resolved by reducing drifts, unless they are reduced to below theelastic limit. The dampers studies did not achieve this for the example structures. For these deficiencies,the proportion of the shear force resisted by the structural system is important.

The total base shear force for the different damper types are plotted in Figure 5-12 for the envelope of thethree earthquakes. The plots are for the uniform damper distribution. The shear forces are the maximumvalue from the three earthquakes for a specific damper normalised by dividing by the maximum shear force inthe structure without dampers. Numerical values can be obtained by multiplying the normalised values inFigure 5-12 by 1189, 1494 and 1811 KN for the 3, 5 and 10 story buildings respectively. As for drifts, theeffects are building specific:

• For the three story building all dampers increase base shear, approximately proportionally to the dampingparameter. The viscous and visco-elastic dampers increase base shears by a much greater factor than thehysteretic and friction dampers.

• The dampers have a lesser effect on base shears in the five story building although they follow the samegeneral trend. For this building, the visco-elastic damper increases shear forces by a higher proportionthan the other types.

• The dampers increase shear forces in the 10 story building by a proportion which falls between those forthe 3 and 5 story building. The hysteretic dampers provided high increases for high yield forces.

Figure 5-13 plots the shear resisted by the structural system for each damper type. As for total forces, theplots are for the envelope of the three earthquake normalised by the frame force for the configuration with nodampers. Numerical values can be obtained using the factors listed above for total shear. These show quitedifferent distributions to the total force:

• All damper variations reduced the frame shear for the three story building. The greatest reductions werefor the viscous and visco-elastic dampers which reduced the frame shear by a maximum of 18%.

• For the 5 story building the frame shear was essentially unchanged for the hysteretic dampers but theother types reduced base shear, by a maximum of 27%, again with the best performance being obtainedfrom the visco-elastic damper.


• The 10 story frame shears were increased by up to 6% for hysteretic dampers with a high yield force butwere reduced for all other types. The friction dampers with a high slip force produced the greatestreduction, 25%.

Conclusions from this are that the damper forces increase the total force in the structure but the additionalshears are resisted by the devices themselves and do not usually result in an increase in shear forces in thestructural system. Some devices can reduce the frame shear by up to 25%.

FIGURE 5-12 EFFECT OF DAMPERS ON TOTAL BASE SHEAR

0.00

0.50

1.00

1.50

2.00

2.50

0 5 10 15 20

Damping Parameter

Base

She

ar (K

N)

H-U 3 Story F-U 3 StoryV-U 3 Story VE-U 3 Story

0.000.200.400.600.801.001.201.401.60

0 5 10 15 20

Damping Parameter

Base

She

ar (K

N)


0.00

0.50

1.00

1.50

2.00

0 5 10 15 20

Damping Parameter

Base

She

ar (K

N)


FIGURE 5-13 EFFECT OF DAMPERS ON FRAME SHEAR

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20

Damping Parameter

Base

She

ar (K

N)


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20

Damping Parameter

Base

She

ar (K

N)



0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20

Damping ParameterBa

se S

hear

(KN

)


The differences in the effects on base shear of the viscous dampers between the three story building and thefive and ten story building are very marked. Figures 5-14 to 5-17 plot the shear in the frame and the damperfor each damper type under the El Centro earthquake for the three buildings. The damper coefficients are themid-range values, a damping parameter of 10 in the plots in Figure 5-12 and 5-13.

• The hysteretic dampers, Figure 5-14, provide a parallelogram shaped hysteresis with a relatively large yielddisplacement which reduces the area under the curve, a low efficiency compared to the ideal rectangularshape. The 3 story frame exhibits a more pronounced bi-linear hysteresis than the higher frames,presumambly because a mechanism is present for a longer portion of the response.

• The friction dampers, Figure 5-15, produce a hysteresis shape which is closer to a rectangle than thehysteretic dampers. This provides larger damping and so the maximum displacements are smaller for allbuildings.

• The viscous damper hysteresis, Figure 5-16, provides a generally elliptical shape with maximum forces atthe point of zero displacement. The area under this hysteresis is larger for the 3 story building, indicatinga higher velocity in this building than the 5 and 10 story buildings.

• The visco-elastic damper, Figure 5-17, is generally similar to the viscous dampers although there is aninclination to the hysteresis due to the elastic stiffness of the damper. For the ratio of shear modulus toloss modulus set for these analyses the inclination is relatively small.

FIGURE 5-14 HYSTERETIC DAMPER SHEAR (EL CENTRO)

3 Story H150-U EQ 1

-1500

-1000

-500

0

500

1000

1500

-40 -20 0 20 40 60 80

Displacement (mm)

Shea

r (K

N)

FrameDamper

5 Story H250-U EQ 1

-1500

-1000

-500

0

500

1000

1500

2000

-40 -20 0 20 40 60 80

Displacement (mm)

Shea

r (K

N)

FrameDamper


10 Story H500-U EQ 1

-2000-1500-1000-500

0500

100015002000

-60 -40 -20 0 20

Displacement (mm)Sh

ear (

KN

)

FrameDamper

FIGURE 5-15 FRICTION DAMPER SHEAR (EL CENTRO)

3 Story F150-U EQ 1

-1500

-1000

-500

0

500

1000

1500

-40 -20 0 20 40 60 80

Displacement (mm)

Shea

r (K

N)

FrameDamper

5 Story F250-U EQ 1

-1500

-1000

-500

0

500

1000

1500

-40 -20 0 20 40 60 80

Displacement (mm)

Shea

r (K

N)

FrameDamper

10 Story F500-U EQ 1

-2000

-1500

-1000

-500

0

500

1000

1500

-40 -30 -20 -10 0 10 20

Displacement (mm)

Shea

r (K

N)

FrameDamper

FIGURE 5-16 VISCOUS DAMPER SHEAR (EL CENTRO)

3 Story V5000-U EQ 1

-1500

-1000

-500

0

500

1000

1500

-30 -20 -10 0 10 20 30

Displacement (mm)

Shea

r (K

N)

FrameDamper


-1500

-1000

-500

0

500

1000

1500

-40 -20 0 20 40 60

Displacement (mm)

Shea

r (K

N)

FrameDamper



-2000

-1500

-1000

-500

0

500

1000

1500

-50 -40 -30 -20 -10 0 10 20

Displacement (mm)Sh

ear (

KN

)

FrameDamper

FIGURE 5-17 VISCO-ELASTIC DAMPER SHEAR (EL CENTRO)

3 Story VE5000-U EQ 1

-1500

-1000

-500

0

500

1000

1500

-40 -30 -20 -10 0 10 20 30

Displacement (mm)

Shea

r (K

N)

FrameDamper


-1500

-1000

-500

0

500

1000

1500

-40 -20 0 20 40 60

Displacement (mm)

Shea

r (K

N)

FrameDamper


-2000-1500-1000

-5000

500100015002000

-60 -40 -20 0 20 40

Displacement (mm)

Shea

r (K

N)

FrameDamper

5.7.3 EFFECT ON FLOOR ACCELERATIONS

Floor accelerations are important in the evaluation of existing buildings as they define the forces on buildingcomponents, equipment and contents. Figure 5-15 plots the affect on floor accelerations of each dampertype. As for shears, the accelerations are the maximum value from the three earthquakes normalised by themaximum acceleration in the structure without dampers. All accelerations are the maxima from all floors ofthe building. Numerical values can be extracted by multiplying plotted values by 0.60, 0.56 and 0.52g for the3, 5 and 10 story buildings respectively.

As for the other response quantities, no clear trends are common to all buildings:


• For the 3 story building, floor accelerations generally reduce with increased damping parameter up to themid-point of the damping parameter but then increase. All dampers have a similar effect except thehysteretic dampers, which tend to slightly increase accelerations.

• For the 5 story building, all damper types reduce the floor accelerations except the hysteretic damper.Reductions are greatest for damping parameters in the mid-range.

• Dampers reduce floor accelerations for the 10 story building for low values of the damping parameter,again except for the hysteretic damper. For high values of the damping parameter both the friction andthe hysteretic damper increase floor accelerations, by up to 60%.

FIGURE 5-18 FLOOR ACCELERATIONS

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20

Damping Parameter

Acc

eler

atio

n Ra

tio


0.000.20

0.400.600.801.00

1.201.40

0 5 10 15 20

Damping Parameter

Acc

eler

atio

n Ra

tio


0.00

0.50

1.00

1.50

2.00

0 5 10 15 20

Damping Parameter

Acc

eler

atio

n Ra

tio


5.8 EQUIVALENT VISCOUS DAMPING

The effects of supplemental damping are often expressed as equivalent viscous damping, for example in UBCand FEMA-273. The results presented above can be used to approximate the effect of the dampers byequilibrating the response to an equivalent viscous damping value in the structure without dampers.

The procedure used was:


1. Evaluate each building with no supplemental dampers for increasing viscous damping, from 0% to 60%.Drifts were calculated for each variation, as shown in Figure 5-7.

2. For each damper variation, determine the amount of viscous damping which provides a similar maximumdrift to that obtained from the particular analysis.

This approach is only valid for structures responding at the same period. For the hysteretic and frictiondampers, and to a lesser extent the visco-elastic dampers, the devices add stiffness and change the frequencyof response. Therefore, the damping calculations are only an approximation for these types.

Figure 5-15 shows the values calculated for the uniform damper distributions for the El Centro earthquake.These can be correlated with the reduction factors in Appendix A for this earthquake (Figures A-1, A-4, A-7and A-10) for the uniform distribution.

Note that the analysis of the structures with devices has an inherent 5% viscous damping in addition to thedamping provided by the devices. Therefore, equivalent damping less than 5% implies a negative effect of thedamping devices.

• The hysteretic damper produces equivalent viscous damping slightly above the 5% assumed for the“base” structures for low coefficients for the 3 story structure but otherwise less than 5%. For the 5 storystructure they provide less than 5% for all parameters. It is only for the 10 story building that thesedevices provide more than 5% damping consistently, up to a maximum of 35% damping. Comparingthese to Figure A-1, the lowest damping (0% for the 3 story structure) is where the drifts were 130% ofthe drifts with no devices. The 35% damping for the 10 story structure corresponded to drifts about one-half those of the structure without devices.

• The friction damper followed a similar pattern to the hysteretic damper except that it was more effectivefor the 3 story structure, providing up to 25% damping for high coefficients. For the 10 story buildingthe friction dampers provided 55% damping for high damping coefficients, the maximum of any type.See Figure A-4 for drift reductions associated with this damping.

• The viscous dampers provided more than 5% damping for all building types, with maximum values of50% for the 3 story building, 40% for the 5 story building and 25% for the 10 story building. See alsoFigure A-7.

• The visco-elastic dampers provided similar levels of damping to the viscous dampers, a range of 25% to50%. See also Figure A-10.


FIGURE 5-19 EQUIVALENT DAMPING (EL CENTRO)

0%

10%

20%

30%

40%

50%

60%

0 5 10 15 20

Damping Parameter

Equ

ival

ent D

ampi

ng

H U 3 StoryF U 3 StoryV U 3 StoryVE U 3 Story

0%

10%

20%

30%

40%

50%

0 5 10 15 20

Damping Parameter

Equ

ival

ent D

ampi

ng

S U 5 StoryF U 5 StoryV U 5 StoryVE U 5 Story

c

0%

10%

20%

30%

40%

50%

60%

70%

0 5 10 15 20

Damping Parameter

Equ

ival

ent D

ampi

ng

H U 10 StoryF U 10 StoryV U 10 StoryVE U 10 Story

c

5.9 OPTIMUM DEVICES

The analyses of these three buildings illustrate the complexity of the response of yielding structures withdamping devices. To assist in interpreting results, the results have been processed to obtain the “best” 12devices in terms of response quantities which may be important depending on one of three commonobjectives of adding dampers:

a) Minimising drifts. For concrete frames this is often the most important parameter as it relates almostdirectly to ductility demands on frame elements.

b) Minimising the shear resisted by the frame. For columns which are deficient in shear, a commonoccurrence in older frame structures, this may be a retrofit objective.

c) Minimising floor accelerations. This will be a retrofit objective when loads on components or contents aretoo high.

Tables 5-5 to 5-8 summarise the devices which best meet these objectives for the 3, 5 and 10 story buildingsrespectively. Each table lists the 12 devices which best met these objectives, in terms of ratio of drift, frameshear or acceleration relative to the response without any dampers. As for the previous results, these ratios arebased on the maximum values from the three earthquakes.


3 Story Building

For the 3 story building, Table 5-5, the drifts can be reduced by a factor of 2.5 to 3 by using viscous or visco-elastic dampers with a high coefficient, C, of at least 6000. These drift reductions are associated with areduction in the frame shear force of from 10% to 15%. The optimum viscous devices for drift reduce flooraccelerations more than the visco-elastic dampers. The uniform damper distribution is generally the mosteffective in reducing drifts.

To reduce the frame shear, the most effective devices are viscous and visco-elastic devices, as for the driftratios, but in this case the reverse triangular distribution is the most effective. The frame shear forcereductions are much less than drift reductions, with a maximum reduction of 20%. Devices which reduce theshear ratios are also effective in reducing drifts and floor accelerations.

Reductions in floor accelerations are optimised by using viscous or visco-elastic devices with a low dampingcoefficient. The maximum reductions in accelerations, by 30%, are achieved using the triangular distribution.The devices which are optimum for floor accelerations are not particularly effective in reducing either drifts orshear forces.

5 Story Building

The optimum devices for the 5 story building, Table 5-6, generally follow similar trends to the 3 story buildingalthough the optimum devices are less effective in controlling drifts but more effective in reducing shears andfloor accelerations.

The drifts were reduced by a factor of 2 with high coefficient viscous or visco-elastic dampers in either auniform or reverse triangular distribution. Dampers which were effective in reducing drifts were also effectivein reducing shears and accelerations.

The visco-elastic dampers with a high coefficient and a reverse triangular distribution were most effective inreducing shear forces, by up to 34%. The optimum dampers for shear generally also appeared as optimum fordrift.

The maximum reductions in floor accelerations were achieved with a relatively low slip force friction damper,which reduced values by 35%. Viscous dampers were also effective. Generally, the triangular distribution ofdampers was most effective for accelerations. The optimum dampers in this group were not very effective inreducing drifts or shear forces.

10 Story Building

The optimum results for the 10 story building, Table 5-7, are dominated by friction dampers, rather than theviscous or visco-elastic devices which were best for the two lower buildings.

High slip force friction dampers with a uniform distribution can reduce drifts by up to 53%. These dampersalso reduce frame shear forces but increase floor accelerations, by up to 74%. Visco-elastic dampers with a


high damping coefficient are less effective than the friction dampers but do not increase the flooraccelerations to the same extent as the friction dampers.

The dampers which are most effective at controlling drift are generally also most effective at controlling frameshears, producing a maximum reduction of 29% in this parameter. The reverse triangular distributions aremore effective than the uniform distribution, unlike for drifts where the uniform distribution was optimum.

The friction devices with a much lower slip force can reduce accelerations by up to 19%. Moderate values ofthe slip force can reduce drifts and shear forces as well as accelerations.

5.10 SUMMARY OF PERFORMANCE

For the 3 story building the optimum damper types are either viscous or visco-elastic dampers with a dampingcoefficient of 7000 KN-sec/m or higher in either a uniform or reverse triangular distribution. The formerwill minimise the drifts, the latter the frame shear force. If floor accelerations are important then the dampingcoefficient should be reduced to less than 2500 KN-sec/m but the effectiveness in reducing drifts and shearswill be much less.

The dampers which are most effective for the 3 story building are also optimum for the 5 story building. Forthis building, friction dampers with a low slip force are the most effective in controlling floor accelerations ifless effectiveness in reducing drifts and shears is acceptable.

Friction dampers are the most effective for the 10 story building. Dampers with a high slip force are mosteffective for reducing drifts and shears but will increase floor accelerations. Reducing the slip force willreduce floor accelerations at the cost of effectiveness in reducing for drifts and shears.

There was a general trend in that the uniform distributions of dampers was best at controlling drifts, thereverse triangular distribution best at controlling frame shears and the triangular distribution was mosteffective in reducing floor accelerations.


TABLE 5-5 OPTIMUM DEVICES FOR 3 STORY BUILDING

DeviceType

DampingCoefficient

Distribution DriftRatio

ShearRatio

AccelerationRatio

Minimum Drift RatiosV 8500 U 0.33 0.82 0.77V 8000 U 0.36 0.85 0.94

VE 9000 U 0.36 0.83 1.05VE 8500 U 0.37 0.85 1.01V 7000 U 0.38 0.85 0.92V 7500 U 0.39 0.85 0.94

VE 8000 U 0.39 0.84 1.02V 6500 U 0.39 0.85 0.89V 9500 T 0.40 0.90 0.77V 10000 R 0.40 0.82 0.87

VE 7000 U 0.40 0.84 0.99V 6000 U 0.40 0.85 0.88

Minimum Shear RatiosVE 10000 R 0.41 0.79 0.92VE 9500 R 0.42 0.80 0.91VE 8500 R 0.44 0.80 0.89VE 9000 R 0.43 0.80 0.90VE 8000 R 0.45 0.81 0.88V 10000 R 0.40 0.82 0.87

VE 7000 R 0.48 0.82 0.87VE 7500 R 0.46 0.82 0.88V 9500 R 0.41 0.82 0.86V 8500 U 0.33 0.82 0.77V 8500 R 0.43 0.82 0.85V 9000 R 0.42 0.82 0.86

Minimum Acceleration RatiosV 2000 T 0.87 0.92 0.70

VE 2000 T 0.78 0.91 0.72V 1500 U 0.82 0.90 0.72V 2500 T 0.82 0.92 0.73V 2000 U 0.73 0.89 0.73V 1500 T 0.91 0.92 0.73

VE 1500 U 0.71 0.89 0.74VE 1500 T 0.83 0.92 0.74V 3000 T 0.76 0.92 0.74

VE 2500 T 0.73 0.91 0.74VE 2000 U 0.68 0.89 0.75V 3500 R 0.70 0.88 0.75



DeviceType

DampingCoefficient


ShearRatio

AccelerationRatio

Minimum Drift RatiosVE 7500 U 0.49 0.73 0.76VE 10000 R 0.50 0.66 0.77V 10000 R 0.51 0.74 0.75V 7500 U 0.51 0.78 0.76

VE 9500 R 0.52 0.68 0.76V 9500 R 0.52 0.76 0.75

VE 9000 R 0.54 0.70 0.76V 9000 R 0.54 0.78 0.75

VE 7000 U 0.54 0.75 0.77V 6000 U 0.54 0.85 0.74

VE 6000 U 0.55 0.80 0.74V 8500 R 0.56 0.80 0.75

Minimum Shear RatiosVE 10000 R 0.50 0.66 0.77VE 9500 R 0.52 0.68 0.76VE 9000 R 0.54 0.70 0.76VE 8500 R 0.56 0.72 0.76VE 7500 U 0.49 0.73 0.76VE 8000 R 0.58 0.74 0.76V 10000 R 0.51 0.74 0.75

VE 7000 U 0.54 0.75 0.77VE 7500 R 0.60 0.76 0.76V 9500 R 0.52 0.76 0.75

VE 6500 U 0.56 0.77 0.76V 7500 U 0.51 0.78 0.76

Minimum Acceleration RatiosF 175 T 0.90 0.99 0.65V 7000 T 0.68 0.92 0.68F 125 U 0.89 0.97 0.68F 200 T 0.90 0.99 0.68V 6500 T 0.69 0.93 0.68V 6000 T 0.70 0.93 0.68V 5500 T 0.71 0.93 0.68V 5000 T 0.72 0.93 0.69

VE 6500 T 0.73 0.95 0.69VE 6000 T 0.75 0.95 0.69VE 5500 T 0.76 0.95 0.69F 150 T 0.89 0.98 0.69



DeviceType

DampingCoefficient


ShearRatio

AccelerationRatio

Minimum Drift RatiosF 1000 U 0.47 0.75 1.48F 950 U 0.48 0.75 1.74F 1000 R 0.48 0.71 1.24F 900 U 0.48 0.75 1.51F 850 U 0.48 0.78 1.45F 800 U 0.49 0.81 1.43F 950 R 0.51 0.72 1.20F 750 U 0.53 0.83 1.61

VE 10000 R 0.53 0.89 0.89F 900 R 0.54 0.74 1.19

VE 9500 R 0.54 0.89 0.89VE 9000 R 0.55 0.90 0.89

Minimum Shear RatiosF 1000 R 0.48 0.71 1.24F 950 R 0.51 0.72 1.20F 900 R 0.54 0.74 1.19F 1000 U 0.47 0.75 1.48F 950 U 0.48 0.75 1.74F 850 R 0.57 0.75 1.16F 900 U 0.48 0.75 1.51F 800 R 0.61 0.77 1.14F 850 U 0.48 0.78 1.45F 750 R 0.63 0.80 1.10F 800 U 0.49 0.81 1.43F 700 R 0.66 0.82 1.08

Minimum Acceleration RatiosF 250 T 0.99 0.93 0.81F 200 T 1.06 0.96 0.81F 300 R 0.89 0.88 0.81

VE 3000 U 0.74 0.91 0.81F 250 U 0.86 0.88 0.82F 150 U 0.98 0.93 0.82F 150 T 1.10 0.95 0.82F 250 R 0.93 0.92 0.84F 400 T 0.91 0.90 0.85F 100 U 1.06 0.95 0.85V 7000 T 0.89 0.91 0.86V 6500 T 0.89 0.91 0.86


6666 PRACTICAL DEVICE PROPERTIESPRACTICAL DEVICE PROPERTIESPRACTICAL DEVICE PROPERTIESPRACTICAL DEVICE PROPERTIES

The time history analyses considered a range of devices without consideration of just how practical thedamping parameters were. There are of course limits to the devices which can be used, based on bothpractical and economic bounds.

6.1 HYSTERETIC DEVICES

The hysteretic devices are generally metal yielding, such as steel under axial or shear loads. Other materialssuch as lead may be used. Table 6-1 lists the steel area required to provide the range of yield forces used inthese studies. These areas are based on a steel strength of 250 MPa. As the yield displacement is proportionalto steel strength, and the lower the yield displacement the more efficient the damper, the lowest practical steelstrength will provide optimum performance.

TABLE 6-1 STEEL AREA FOR HYSTERETIC DAMPERS ACTING AS BRACES

YieldForce(KN)

SteelArea

(mm2)

Size ofSteelFlat

(mm x mm)50 200 10 x 20100 400 10 x 40150 600 10 x 60200 800 10 x 80250 1000 20 x 50300 1200 20 x 60350 1400 20 x 70400 1600 20 x 80450 1800 20 x 90500 2000 25 x 80550 2200 25 x 88600 2400 25 x 96650 2600 25 x 104700 2800 25 x 112750 3000 25 x 120800 3200 25 x 128850 3400 25 x 136


YieldForce(KN)

SteelArea

(mm2)

Size ofSteelFlat

(mm x mm)900 3600 25 x 144950 3800 25 x 1521000 4000 25 x 160

The steel areas listed in Table 6-1 are within practical limits for most structures. If configured as a bracespecial techniques will be required to ensure that the damper can function in both tension and compression.This is typically achieved by enclosing the damper in a concrete filled tube.

Most damper configurations will provide a vertical component of force as well as horizontal and this verticalload will generally be resisted by existing frame columns. The ability of existing columns to resist added axialload may form an upper limit on the yield force which can be used.

Figure 6-1 plots the maximum displacements in the hysteretic dampers for the three buildings for eachdamping yield force. The values plotted are for the uniform distribution but the displacements are similar forthe other two distributions evaluated. The displacements are within a band of 60 mm to 100 mm for allbuildings and yield forces. The yield displacement for these dampers is approximately 11 mm and so thedisplacement ductility demands range from 5.5 to 9.

FIGURE 6-1 HYSTERETIC DAMPER DISPLACEMENT

0

20

40

60

80

100

120

0 200 400 600 800 1000 1200DAMPER YIELD FORCE (KN)

DA

MPE

R D

ISPL

ACE

ME

NT

(mm

)

3 Story U Displacement5 Story U Displacement10 Story U Displacement

The sizes of other type of devices can be calculated based on the material properties. For example, the shearyield strength of lead is about 10.5 MPa. A yield force of 100 KN would require a shear area of 9524 mm2, ora circular section of diameter of 110 mm. Available literature from Robinson Seismic Ltd lists devices forforces from 1 KN to 1000 KN at displacements up to 100 mm so these could be used for the hystereticdampers used for this study.


Lead extrusion dampers (LEDs) provide yield forces up to 1000 KN and displacements up to 1000 mm andso could provide the requirements of these devices. The hysteresis loop of an LED is essentially rectangularso would probably match the friction device properties closer than the hysteretic damper, depending on theflexibility of the components used to attach it to the structure.

6.2 FRICTION DEVICES

The friction devices considered for this study had a slip force range similar to the yield force range used forthe hysteretic devices, from 15 KN to 1000 KN. Figure 6-2 plots the displacements associated with these slipforces for the three buildings.

Unlike the hysteretic dampers, the friction dampers show a trend of decreasing displacements as the slip forceincreases. For low slip forces the displacements are in the same range as for the hysteretic devices, 60 mm to100 mm, but for high slip forces the displacements are about one-half, 30 mm to 50 mm. This is a function ofthe greater effectiveness of friction devices in reducing drifts, compared to the hysteretic dampers.

FIGURE 6-2 FRICTION DAMPER DISPLACEMENT

0102030405060708090

100

0 200 400 600 800 1000 1200DAMPER SLIP FORCE (KN)

DA

MPE

R D

ISPL

ACE

ME

NT

(mm

)


The slip force could be provided by a variety of means. For example, 8 x ½” (12.7 mm) A325 bolts can beused in a steel-brass slip bolted connection to provide a slip force of 270 KN. Many installed frictiondampers (for example, the proprietary Pall dampers from Canada) provide slip forces in the range of 300 KNto 600 KN and so the range of slip forces used in these studies are practical.

As for the hysteretic dampers, the friction dampers will usually apply additional forces to the existingstructure. This may form an upper limit to the slip force.

As for all types, the friction dampers are mobilised by interstory drifts and so need to be connected from floorto floor. Depending on the configuration used to achieve this, these dampers may act more as a hystereticdamper if there is significant displacement in the connecting members. The hysteretic damper has beendefined as a system with a yield displacement of 11 mm and the friction damper as a device with a slipdisplacement of 1.1 mm. Analytically, the two systems are identical apart from the amplitude of the yield


displacement. If the friction damper were mounted in a brace which had an extension of 11 mm at the pointof slip then the results would be the same as those for the hysteretic damper.

6.3 VISCOUS DAMPERS

The force in the viscous dampers is equal to the damping coefficient times the velocity. Figure 6-3 plots thepeak velocities for each damping coefficient and building type. There are two general trends in the velocityplots:

1. Velocity reduces with increasing damping coefficient. This is because the increased damping reducesdisplacements without significantly changing the frequency of response and so the velocity reducesproportionately.

2. The velocity is highest for the 3 story building and reduces as the height of the building increases. Thisvelocity is a function of the ratio of period to drift and this trend would not necessarily apply for allbuilding types.

These trends determine the function of damping force versus damping coefficient for the three buildings, asshown in Figure 6-4. The damping force is consistently higher for the 3 story building than the 5 story whichis itself higher than the 10 story building. Although the damping force increases with damping coefficient theincrease is not proportional because the velocity is reducing. For example, if the damping coefficient in the 3story building is increased by 100%, from 4000 to 8000, the damping force increases by only 60%.

FIGURE 6-3 VISCOUS DAMPER VELOCITY

0.0000.050

0.1000.150

0.2000.250

0.3000.350

0.400

0 2000 4000 6000 8000 10000DAMPING COEFFICIENT (KN-sec/m)

DA

MPE

R V

ELO

CITY

(KN

/m)

3 Story U Velocity5 Story U Velocity10 Story U Velocity


FIGURE 6-4 VISCOUS DAMPER FORCE

0200400600800

100012001400160018002000


DA

MPI

NG

FO

RCE

(KN

) 3 Story U Force5 Story U Force10 Story U Force

There is not a lot of published data on the size of viscous dampers to provide a specified damping force.

The type of viscous dampers which are most common, although not for buildings, are automobile shockabsorbers. The performance of these is non-symmetrical for positive and negative directions of loading withforces termed “bump” and “rebound”. These dampers would need to be used in pairs to providesymmetrical damping forces. The damping coefficient, C, is in the range of 1 to 35 KN-sec/m for automobileuse, with the upper value for large truck shock absorbers.

Test dampers quoted in the literature with a stroke of ± 51 mm were 280 mm long and weighed 10 N. Thesedampers had a damping coefficient of 15 KN-sec/m, which is in the same range as automobile shockabsorbers.

It is apparent that a large number of these small dampers would be required to provide damping coefficientsof the range used here (1000 to 8000 KN-sec/m would require 50 to 400 dampers).

The Taylor Devices, Inc. web site quotes damper forces in the range of 100 KN to 6000 KN with anexponent of 0.3 to 1.0 (all analyses in this study assumed an exponent of 1.0). The upper force level is muchhigher than the peak value of 1800 KN for these buildings and so it does seem that practical devices may beavailable. However, details of the velocity required to achieve these force levels are not provided. Exampleprojects list maximum damping forces up to 5600 KN at displacements of ± 52 mm.

There are quoted prices on the Internet for VDDs of $7,000 for a 150 kip (675 KN) device and $13,000 for a300 kip (1350 KN) device (prices in US dollars). Figures 6-5 and 6-6 illustrate low capacity and high capacityTaylor Devices viscous dampers.

The high capacity dampers are large and would be difficult to fit within the constraints of most buildingstructures. Multiple smaller dampers would probably be used if high damping forces were required.


FIGURE 6-5 TAYLOR DEVICES 225 KN VISCOUS DAMPERS

FIGURE 6-6 TAYLOR DEVICES 5850 KN AND 9000 KN VISCOUS DAMPERS


6.4 VISCO-ELASTIC DEVICES

The maximum forces in the visco-elastic devices, plotted in Figure 6-7, follow a generally similar pattern tothose for the viscous dampers but the forces are higher. This is because the force due to the elastic stiffnessof the damper is added to the viscous force.

The required thickness of the visco-elastic devices is defined by the maximum displacement in the device,plotted in Figure 6-8. Displacements ranged from a high of almost 90 mm to a low of 30 mm. Displacementswere smallest for the lowest building and decreased with increasing damping coefficient.

Visco-elastic dampers are generally designed for peak shear strains of 150% (DBE) to 250% (MCE) and so therequired thickness for this range of displacements would be from 12 mm (30 mm at 250% strain) to 60 mm(90 mm at 150% strain).

Typical loss modulus properties of visco-elastic dampers for this frequency and strain level would be aboutG”/ω = 0.1 MPa-sec (Figure 3-22). The damping coefficient is calculated as C = G”Ab/ωt. Assuming athickness of 30 mm, the required damper area for a damping coefficient of unity, C=1 KN-sec/m, is

223 3000003.0

101.0030.01

)/"(mmm

xx

GCtAb ====

ω

The dampers are typically installed as pads with maximum dimensions of 200 mm x 200 mm with two padsper unit. Each unit provides an area of 80,000 mm2 and so the damping coefficient provided per unit is80,000/300 = 267 KN-sec/m. The most effective coefficients, C = 5000 to 10,000, would require from 18 to36 units per floor.

If the material were used as a wall damper, bonded between plates, C=10,000 would require a total area of 3square metres which could be provided within a wall panel, especially if multiple plates were used.

The elastic stiffness component of the visco-elastic damper associated with this damping coefficient can becalculated using the shear modulus of 0.2 MPa associated with the loss modulus of 0.1 MPa. For an areasufficient to provide C = 10,000 the elastic stiffness K = 20,000 KN/m. This is equivalent to a steel bracewith an area of 878 mm2. At 30 mm displacement, the elastic force would be 600 KN.


FIGURE 6-7 VISCO-ELASTIC DAMPER FORCE

0

500

1000

1500

2000

2500


DA

MPI

NG

FO

RCE

(KN

) 3 Story U Force5 Story U Force10 Story U Force

FIGURE 6-8 VISCO-ELASTIC DAMPER DISPLACEMENT

0102030405060708090

100


DA

MPE

R D

ISPL

ACE

ME

NT

(mm

)



7777 DAMPING DESIGN PROCEDURESDAMPING DESIGN PROCEDURESDAMPING DESIGN PROCEDURESDAMPING DESIGN PROCEDURES

7.1 APPLICABLE CODES

The two codes which have the most comprehensive provisions for the implementation of damping and energydissipation devices are FEMA 273 and the SEAOC “Blue Book”. The latter provisions are used as asupplement to the Uniform Building Code (UBC) and so are generally applicable to new buildings.

Most applications for which supplemental damping is considered will be for seismic upgrading of existingbuildings for which the FEMA 273 procedures are appropriate and so these notes reference procedures in thisdocument.

The FEMA Guidelines have provisions on general requirements, modelling of devices, analysis procedures,detailed system requirements, design and construction review and required tests of devices. The Commentaryto the Guidelines also provides an example design using linear viscous dampers.

These notes do not duplicate the material in FEMA or SEAOC and it is recommended that you consult thesesources before starting a project using supplemental dampers.

7.2 SECTION OF DEVICE TYPE AND PROPERTIES

The studies done so far do not provide a clear preference for the selection of device. For the 3 and 5 storystructures the viscous and visco-elastic devices provided the best performance. For the 10 story structure thefriction dampers were better. For a given device type, the optimum distribution depended on whether theobjective was to reduce drifts, structure shears or floor accelerations.

Although future developments may make the selection of device type and properties explicit in a designprocedure, at this stage these guidelines can only provide some general statements which may assist inselecting a range specific types and properties. Evaluations will then need to be performed to decide betweendevices and to refine the properties of the particular device or devices.

The points which can influence the selection of devices include:

Type of Building

Flexible buildings are inherently more suited to supplemental damping than stiff buildings. In general, themore flexible the building the lower the amount of damping that has to be added to gain reductions inresponse. Although there are no hard and fast rules, buildings suited for dampers will almost always be


moment frames. Shear wall and braced frame buildings will not usually be suitable as the in-structuredeformations will be too small to generate sufficiently high damping forces.

In theory, it may be possible to modify existing structural elements of stiff buildings to incorporate damping.For example, add dampers into existing bracing or separate walls between floors and add dampers. Thesewould require a large design and evaluation effort and would only be considered for projects which justifiedthis level of effort.

Costs

Almost every project will be driven by retrofit costs. In terms of first cost, the cheapest devices are hystereticyielding, followed by friction dampers then visco-elastic and finally the most expensive are viscous dampers.Of course, the price-performance ratio is more important than the absolute cost and there is no simplehierarchy for this. There is no point in using hysteretic dampers because they have the lowest first cost whenthey do not provide any benefits.

However, if several types of devices can produce benefits then they will generally be ranked by the costs.Visco-elastic and viscous devices can usually provide similar benefits and on a cost basis visco-elastic would befavoured. However, there are generally a much larger number of visco-elastic devices than viscous devices soconnection costs may reverse this order. As for so many factors involved with in-structure damping, there isno simple answer.

Availability may also affect costs depending on the location of the project. For example, large capacity viscousdampers are available in the U.S. but may be more expensive, and have a long delivery time, in other countries.

Strength of Existing Building

For existing structures, there will be usually be constraints on the maximum damping parameters imposed bythe strength of the existing building. The vertical component of damper forces will generally be accumulateddown existing columns in the building. This will limit the maximum yield or slip force for the hysteretic anddamping devices.

You will generally have evaluated the building without dampers to have arrived at the point where youconsider supplemental damping. This evaluation will provide an estimate of the extra load which can beadded to columns without causing failure. For hysteretic and friction dampers the magnitude of the extra loadcan be calculated from simple statics using the yield/slip force accumulated over all levels. These forces maybe offset partly by a reduction in overall response. However, for the most common case where the damperreaction is resisted by internal columns the axial load will not be reduced by the response reduction.

Structural Form

The form of the existing building may lend itself to particular damping device types. Some damping types areconcentrated in a few locations (yielding braces), other can be distributed over a wide area (visco-elastic).Small visco-elastic dampers provide relatively small forces per unit and so may be considered in locations suchas non-structural walls or to connect concrete cladding panels to floors. Installation of damping walls may notalways be practical as they restrict the internal layout of the building.


Deficiencies and Extent of Improvement Required

Probably the most important factors affecting device selection are the type of deficiency and the magnitude ofthe deficiency. Some dampers are effective at reducing drift but less effective at reducing story shear, althoughthis seems to happen more in theory than practice. A theoretical comparison of friction and viscous damperswould suggest that the former would reduce drifts effectively because of the added stiffness but be lesseffective in reducing force because the damper forces would be in phase with the structure. However, thetime history analyses tended to show that dampers effective at reducing drift also reduced forces on theexisting structure. Nevertheless, although the best damper type for drift was also usually the same type forforces, the optimum value of the damping parameter was not the same (see Tables 5-5 to 5-7).

Derivation of Optimum Devices from Time History Results

Table 7-1 shows the damper types and property ranges which can reduce drifts by at least 15% compared tothe structure without dampers. Table 7-2 lists similar device ranges which can reduce drift by at least 30%.The tables list values for the three distributions included in the time history evaluation.

TABLE 7-1 DAMPER PROPERTIES TO REDUCE DRIFT > 15%

Damper Distribution 3 Story 5 Story 10 StoryHysteretic U

TR

50-7575-125

200-500

400+Friction U

TR

165+ 225+475+250+

300+550+350+

Viscous UTR

1500+2500+2000+

1000+1000+1000+

3500+7500+4500+

Visco-Elastic UTR

1000+1000+1000+

500+1000+500+

2000+4000+2000+


TABLE 7-2 DAMPER PROPERTIES TO REDUCE DRIFT > 30%

Damper Distribution 3 Story 5 Story 10 StoryHysteretic U

TR

Friction UTR

255+ 350+

475+

550+

650+Viscous U

TR

2500+4000+3500+

3000+6000+4500+

5500+

7000+Visco-Elastic U

TR

2000+3500+2500+

4000+8000+5000+

4000+

5000+

All structures have a constant floor weight of 1800 KN/floor and so the values in the tables can be related tothe damping force as a proportion of the floor weight.

1. For the hysteretic and friction dampers, the optimum damper yield/slip forces range from 50 KN to 650KN, which is equivalent to 3% to 35% of the floor weight.

2. For the viscous and visco-elastic dampers the average velocity can be taken as 0.25 m/sec (Figure 6-3).Damping coefficients of 1000 to 8000 KN-s/m correspond to forces of approximately 300 KN to 1800KN (Figure 6-4). These damping forces are in a range of 16% to 100% of the floor weight.

Tables 7-1 and 7-2, plus the results presented previously, clearly demonstrate that the optimum damper typeand weight are not a factor solely of structure type and floor mass, else the results would be similar for thethree buildings considered.

The other factors which presumably influence the effectiveness of dampers are the dynamic characteristics,that is, the periods and mode shapes. For the three buildings considered, the period increased with increasingbuilding height, with elastic periods increasing from 0.79 seconds for the 3 story building, 1.56 seconds for the5 story building and 2.86 seconds for the 10 story building. Approximately, the period ratios are 1:2:4 for thethree buildings. The three buildings have generally similar mode shapes with the effective mass in mode 1reducing slightly from 91% in the 3 story building to 82% in the 10 story building.

The maximum drifts in the structures without dampers increase with increasing period. The 3 story peak driftis 1.5%, the 5 story peak drift is 2.0% and the 10 story peak drift is 2.4% (Table 5-4). These are the maximumvalues from the three earthquakes but the same trend occurred for individual earthquakes.

The best that can be extracted from the evaluation to date, in terms of providing a design procedure, is toassume that the differences in device performance are related to differences in the periods of the structure,which also corresponds to differences in drifts.


Based on this, a few general “rules” can be extracted to assist in selecting dampers, but these should be treatedwith some caution, recognising that they are extracted from a very limited data set:

• Hysteretic dampers are relatively inefficient for all the buildings considered here. They do providemoderate drift reductions for buildings with periods greater than 1.5 seconds or drifts greater than 2%.Note that a hysteretic damper with an increased elastic stiffness may be categorised as a friction damper interms of these results. If hysteretic dampers are used, they require a yield strength of at least 3% of thestory weight, with the minimum required yield level increasing with period to at least 10% for long periodbuildings (greater than 2.5 seconds).

Where hysteretic dampers are effective, the uniform distribution tends to be best although in somesituations either the triangular or reverse triangular distribution may be effective. The results suggest thatthe results from this type of damper are very sensitive to both the structural and the damper properties.

• Friction dampers can be effective for the full range of buildings considered here with the effectivenessgenerally increasing with increasing slip force. Moderate drift reductions can be achieved with slip forcesranging from 9% of the story weight for medium period structures to 16% for long period structures.Drift reductions greater than 30% require increases in these slip forces to a minimum of 14% (3 story) to30% (10 story) of the story weight.

The uniform distribution generally is most reliable in providing drift reductions although for somebuildings the reverse triangular distribution is also effective.

• The viscous dampers can be effective for all structures. To achieve moderate drift reductions, at least15%, the 5 story building requires a smaller damping coefficient than either the 3 or the 10 story building.For high reductions, 30% or more, the required damping coefficient increases with the period of thebuilding. High damping coefficients are required so as to provide damping forces of at least 16% of thefloor weight (C=1000) but in some cases as high as 70% of the floor weight (C=5500).

As for the other damping devices, the uniform distribution generally requires a smaller coefficient toprovide a given drift reduction than the other two distributions considered.

• Visco-elastic dampers generally require a smaller coefficient than the viscous dampers to provide the samelevel of drift reduction. The exception is for dampers to provide drift reductions of greater than 30% inthe 5 story building, where a coefficient one-third higher than the viscous damper is required.

For moderate drift reductions the reverse triangular distribution of visco-elastic dampers is as effective asthe uniform distributions. As the former requires less total damping capacity it would be a more cost-effective solution in this situation.

As the characteristics of the structure are an integral factor in the performance of a structure with addeddampers there is no guarantee that these trends will apply to other buildings with similar periods. Until webetter define performance, it would be advisable to also investigate devices which fall outside theserecommendations.

The results used to develop Tables 7-1 and 7-2 are extracted from the detailed time history results inAppendix A. Drift results were used for this process and different ranges would apply for frame force or


floor acceleration ratios. If these latter quantities are important, use the tables in the Appendix to assesswhich devices and properties best achieve the aims.

7.3 DEVICE DESIGN

The first step in damper design is to develop a configuration for installing the dampers. Generally, multipledevices will be used at each level to provide redundancy. Both FEMA and SEAOC require that a higherreliability/redundancy factor be applied to calculated actions if there are less than four devices per story ineach direction and so you should aim to use at least this number and try to locate them symmetrically aboutthe centre of stiffness.

All dampers dissipate energy by deformations imposed by inter-story drifts. The devices connect successivefloor levels of the building such that horizontal motions cause deformations, either displacements orvelocities, in the device. As discussed earlier, the configurations most often used are dampers installed ininclined braces or dampers installed between the tops of disconnected walls and the floors above. The greaterthe angle the brace makes with the horizontal the less efficient the former type of configuration will be.

A key for efficient performance of all devices is the elastic stiffness of the device supports and connections.Any inter-story drift which is taken out as deformations in the supports or connections reduces theeffectiveness of the device and so all components other than the devices themselves should be as stiff aspossible. This will maximise the relative displacement and velocity between the ends of the damper.

Many damping devices are proprietary, patented items and design and supply is provided by the manufacturerto achieve specified performance requirements. The design engineer should develop these performancerequirements based on the results of the evaluation of performance described later in this chapter. Thespecifications will include such items as yield or slip force, damping coefficients and maximum displacementsand/or velocities.

The exceptions to the use of proprietary devices will be some hysteretic and friction devices. These can bedesigned and detailed using the relevant material codes – they will almost always be steel. As for proprietaryitems, they need to be designed to meet the performance requirements developed for the project. If youintend to design a device yourself, read the literature relating to the device type carefully as there are aspectswhich affect performance which may not be apparent. For example, a friction damper comprised of hightension bolts clamping a steel-to-steel interface will have a severely reduced slip force under successive cycles.Steel on brass is much better – see Bibliography for sources of information.

All devices obviously need to be connected to the structure. Connection design will need to be based oncapacity design principles to ensure that all connections remain elastic and they should comply with thedetailed system requirements in FEMA.

Both FEMA and SEAOC require prototype tests of devices. These tests are generally to levels ofdisplacement and/or force above the maximum values obtained from the design. Device and connectiondesign must take account of this and so will need a minimum level of over-strength, in addition to anyredundancy factors as noted above.


7.4 EVALUATION OF PERFORMANCE

There is a hierarchy of four levels of structural analysis appropriate for the evaluation of existing buildings(FEMA). Each higher level procedure provides a more accurate model of the actual performance of abuilding subjected to earthquake loads, but requires greater effort in terms of data preparation time andcomputational effort.

1. The Linear Static Procedure (LSP) is suitable only for regular buildings, which respond primarily withinthe elastic range. This procedure represents the earthquake loads as an equivalent set of static loads and isthe basic method for most seismic design codes.

2. The Linear Dynamic Procedure (LDP) is able to model irregular buildings but is also suitable mainly forbuildings which respond primarily within the elastic range. This is the response spectrum method ofanalysis, also defined in most seismic codes.

3. The Non-linear Static Procedure (NSP) can evaluate buildings loaded beyond the elastic range but isunable to fully capture the dynamics of response, especially higher mode effects. This is often termed a“Pushover Analysis” and has been developed primarily for the evaluation of existing buildings.

4. The Non-linear Dynamic Procedure (NDP) is the most complete form of analysis, modelling bothdynamic effects and inelastic response. However, it is sensitive to modelling and ground motionassumptions.

The SEAOC recommendations for passive energy dissipation systems permit the LSP for a restricted range ofbuildings (regular buildings of 5 stories or less) and require NDP for all other buildings. The NSP is notreferenced in SEAOC.

The FEMA Guidelines permit the linear procedures (LSP and LDP) only if the framing system exclusive ofthe energy dissipation devices remains essentially linearly elastic after the effects of added damping areconsidered. The Guidelines also impose other restrictions on regularity and device types which limit the useof the linear procedures.

FEMA allows the nonlinear procedures (NSP and NDP) to be used to implement passive energy dissipationdevices without restriction. As for all buildings evaluated using the FEMA Guidelines, there are impedimentsto using the NDP in that more comprehensive knowledge of the structure is required than for other methodsand the analysis and design is required to be subject to review by an independent third-party professionalengineer. This latter provision can have cost and schedule implications for the design process.

Most buildings for which damping devices are being considered will fall outside the limitations for the linearprocedures and so the options evaluated are the NSP and NDP. The NSP can be adapted for thedisplacement dependent devices but it is difficult to use a static procedure for the velocity dependent devicesas the response is so specific to the dynamic response of the building.


7.4.1 NSP FOR DISPLACEMENT DEPENDENT DEVICES

The NSP (pushover) procedure for displacement dependent devices is the same as for structures withoutdevices. A target displacement is calculated and components evaluated against acceptance criteria for theforces and deformations at this target displacement.

If FEMA Method 1 is used the benefit of adding the devices is provided by the increase in building stiffnessand the reduction in target displacement associated with the reduction in effective period. No direct accountis taken of the added damping provided by the energy dissipation devices.

The alternative NSP procedure in the FEMA Commentary, Method 2, is based on the ATC-40 requirements.This method is based on calculating the target displacement as the point where the spectral capacity curve andthe demand design curve intersect. Once the target displacement is obtained the method follows the sameevaluation procedure as Method 1.

Method 2 incorporates damping by establishing a demand design curve be reducing the 5% dampedacceleration spectrum to allow for the equivalent damping due to inelastic action in the seismic framing systemand the added damping provided by the energy dissipation devices. As the damping is displacementdependent, the solution for the target displacement is iterative.

This second method is complex to apply when energy dissipation devices are used as the hysteresis loop areaof the building without devices must be calculated and then a second analysis used to calculate the area of thehysteresis loops of the individual devices.

FEMA does not express a preference for either method although the fact that Method 1 is in the Guidelinesand Method 2 only in the Commentary seems to imply greater acceptance of Method 1. Given that Method 1is much simpler to apply for displacement dependent devices than Method 2, and that both methods haveconsiderable uncertainties, there seems no reason to use the second method.

Our HCG spreadsheets for NSP evaluation include both methods but Method 2 as implemented does not yethave a procedure for adding the device damping. Method 1 can be used as-is for displacement dependentdevices, Method 2 can be used as-is but will be conservative. It is difficult to justify the effort required toimplement Method 2 fully for the reasons above.

7.4.2 NSP FOR VELOCITY DEPENDENT DEVICES

As for the displacement dependent devices, FEMA also provides the use of both NSP (pushover) methodsfor velocity dependent devices.

For the implementation of Method 1, the target displacement is reduced to take account of the dampingadded by the velocity-dependent energy dissipation devices. The damping is calculated based on the ratio ofthe work done by the devices to the maximum strain energy in the frame.

Unlike for displacement dependent devices, Method 1 is iterative for velocity-dependent devices. Method 2applied to velocity dependent devices is similar to that for displacement dependent devices as the calculationof the work done by the devices in implicit to this method.


Implementation for both methods is complicated by the need to assess maximum actions at three stages:maximum drift, maximum velocity, and maximum acceleration. This requires calculations for each individualmode of response and later combination of results by SRSS.

The HCG analysis tools have not been upgraded to implement the NSP for velocity dependent dampingdevices and probably will not be. Both methods entail extremely complex procedures to attempt to representa dynamic phenomenon within a static framework and the procedures do not appear to be well validated. It isrecommended that the NDP be used for velocity dependent dampers pending a further assessment of themerits and practicalities of the NSP.

7.4.3 NDP FOR ALL DEVICES

Although FEMA does not require the NDP (time history analysis) for any devices, it does permit this methodto be used in all cases, subjected to the same requirements as for all evaluations using this method of analysis.

The HCG Performance Based Design procedures use the same input and output spreadsheets for the NSPand the NDP so there is minimal extra data preparation for the NDP. The processing is much simpler for theNSP as the actions and deformations are extracted directly, without the need for assessing individual modes asis required for the NSP.

The negatives of using the NDP are the requirements for comprehensive building knowledge and peer reviewnoted above. However, generally the quality of the results obtained compared to the NSP will favour theNDP.

As exception is for displacement dependent devices, where the FEMA Method 1 NSP can be applied withoutmodification. For this type of device, the calculations of record will generally use the NSP. However, weshould still perform a NDP evaluation for these structures wherever possible so that we can continue to assessthe quality of results obtained.

7.5 EXAMPLE 10 STORY BUILDING

As an example of the application of the NSP method of evaluation, the 10 story building frame example isused, first with no devices and then with 400 KN yield force hysteretic dampers (HD 400) and with 800 KNslip force friction dampers (FD 800). As the NSP is not recommended for velocity-dependent devices, theviscous and visco-elastic devices were not included in this example.

For each configuration, the roof displacements were calculated using the NDP and the NSP Methods 1 and 2as defined in FEMA-273.

The evaluation of these building was for the same motions used for the time history analyses, which wasequivalent to an NZS4203 Intermediate Soil for Z=1.2 (or UBC Zone 4, Soil Type C with no near faultamplification). Figure 7-1 shows the 5% damped spectra from these two codes. The time histories werescaled using the UBC procedure to be compatible with these spectra (see Section 5.3). These time historiescomply with the FEMA-273 requirements for the NDP.


FIGURE 7-1 5% DAMPED SPECTRUM FOR EVALUATION

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

PERIOD (Seconds)

ACC

ELER

ATI

ON

(g)

UBC Ca 0.40 Cv 0.56

NS4203 Intermediate Z = 1.2

7.5.1 NDP RESPONSE

Table 7-3 lists the peak roof displacements for the three configurations for each of the three earthquakes. Inaccordance with the requirements for the NDP, the target displacement is the maximum of the three values,the final column in Table 7-3. The peak displacements with the HD 400 and FD 800 dampers are respectively63% and 35% of the value for the structure without any devices.

Figure 7-2 plots the profiles of displacements, drifts and floor accelerations. Each profile is the envelope fromthe three earthquakes. Drifts were calculated from the instantaneous displacement profiles at every time step,rather than just from the envelopes as was done for the time history analyses. The maximum drifts with theHD 400 and FD 800 dampers are respectively 75% and 44% of the value for the structure without anydevices. (Referring to Appendix A, the drift ratios recorded for these two damper configurations are 79% and49% respectively. This is an indication of the difference caused by using the more approximate measure ofpeak drifts.)

The peak drifts are reduced by a smaller amount than the peak roof displacements. The drift plots in Figure7-2 show that this is because the dampers reduce the drifts in the top stories by a proportionately greateramount than drifts in the lower stories. However, the drifts in the lower stories are numerically highest and sothe reduction in peak drift is less than indicated by the reduction in displacement.

Figure 7-2 also plots the acceleration profiles, which illustrate how a damper which has a positive effect ondrifts can have a negative effect on floor accelerations. The accelerations for the structure with no devices andwith the HD 400 damper are generally similar. The FD 800 device increases these floor accelerations by over40%.


TABLE 7-3 EXAMPLE DEVICES IN 10 STORY BUILDING

Configuration Earthquake 1Roof

Displacement(mm)

Earthquake 2Roof

Displacement(mm)

Earthquake 3Roof

Displacement(mm)

MaximumRoof

Displacement(mm)

No DevicesHysteretic 400 KN UFriction 800 KN U

586334169

495367193

377323206

586367206

FIGURE 7-2 NDP RESPONSE DETAILS

05

10152025303540

0 100 200 300 400 500 600 700

DISPLACEMENT (mm)

ELE

VA

TIO

N (m

)

NO DEVICEHD 400FD 800

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00%

1

3

5

7

9

STO

RY

DRIFT (mm/mm)

FD 800HD 400NO DEVICE

05

10152025303540

0.000 0.200 0.400 0.600 0.800

ACCELERATION (g)

ELE

VA

TIO

N (m

)

NO DEVICEHD 400FD 800


7.5.2 NSP RESPONSE

The pushover curves were generated for each configuration using the NSP procedure specified by FEMA-273. Figure 7-3 plots these curves. These curves reflect the characteristics of each damper type:

• The HD 400 damper increases the frame force linearly up to a yield displacement of about 100 mm, atwhich point the force is 400 KN higher than for the frame without dampers. This difference is equal tothe yield force of the damper. This indicates that the damper stiffness is about the same as that of thestructure.

• The FD 800 damper has a first yield point at about 30 mm, when the dampers slip. At this point thedamper force is 800 KN and the frame force about 150 KN. The load increases from this point, with atotal force equal to a constant value of 800 KN greater than for the structure with no devices.

FIGURE 7-3 PUSHOVER CURVES

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600

DISPLACEMENT (mm)

PUSH

OV

ER

FORC

E (K

N)

No DeviceHD 400FD 800

The target displacements were calculated for the structure without devices and for the two damperconfigurations using both Method 1 and Method 2 as defined by FEMA-273. As discussed above, Method 2was not a complete implementation as it did not include the damping due to devices.

Figures 7-4 and 7-5 plot the performance point for the configuration with the FD 800 devices for Method 1and Method 2 respectively. These plots illustrate the differences between the two methods; Method 1calculates the target displacement using a relationship between an elastic oscillator and the correspondinginelastic oscillator. No direct account is taken of the energy dissipated. Method 2 reduces the 5% dampedspectrum to account for damping to produce a Reduced Demand Spectrum. The target displacement is thenthe intersection of the pushover curve with the Reduced Demand Spectrum.


A full implementation of Method 2 would require that the Reduced Demand Spectrum be reduced furtherdepending on the hysteresis of the damping devices.

FIGURE 7-4 NSP TARGET DISPLACEMENTS METHOD 1

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 50 100 150 200 250 300 350 400Roof Displacement (mm)

Base

She

ar (F

ract

ion

of W

eigh

t)

Pushover Curve

Bilinear Curve

Performance Point

Capacity Demand Spectrum: FEMA273 Procedure, X- Direction: Fraction of full load: 1.00Performance Point: 382

FIGURE 7-5 NSP TARGET DISPLACEMENTS METHOD 2

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400Spectral Displacement (mm)

Spec

tral A

ccele

ratio

n (g

)

Pushover CurveBilinear CurveElastic Demand SpectrumReduced Demand SpectrumPerformance Point

Capacity Demand Spectrum: ATC40 Procedure, X- Direction: Fraction of full load: 1.00Performance Point: 433


Table 7-4 compares the roof displacements from the NDP with the equivalent value calculated from the NSPMethod 1 and Method 2. These results are used to assess the relationship between the NDP and NSP forabsolute displacements and for quantifying the effects of dampers, using results in Table 7-4:

• The roof displacements, listed in the first section of Table 7-4, show that the FEMA Method 1consistently over-estimates displacements relative to the NDP. Method 2 consistently producesdisplacements which are higher again than Method 1.

• The second section of Table 7-4 normalises the displacements to the NDP values. This shows that theover-estimation of displacement by the NSP methods is more pronounced with added damping. Method1 is quite close to the NDP for no devices, only 8% higher, but is 31% and 24% higher for the HD 400and FD 800 devices respectively. Method 2 shows the same trend but with even higher numbers. For theFD 800, Method 2 produces displacements two times as high as the NDP.

• The final section of Table 7-4 normalises the displacements to the value for no devices, to see whether theNSP procedures incorporate the reductions due to damping to the same extent as the NDP. The NDPshows damped displacements of 63% and 35% of the No Devices value for the HD 400 and FD 800respectively. The NSP Method 1 provides equivalent values of 75% and 40% and Method 2 85% and56%.

These results show that, compared to the NDP, the NSP (1) over-estimates maximum displacements and (2)under-estimates the effect of damping on displacements. These effects are more pronounced for Method 2than Method 1, although as noted Method 2 has not been implemented to include the added damping.

TABLE 7-4 COMPARISON OF NDP AND NSP RESULTS

NDP NSPMethod 1

FEMA

NSPMethod 2ATC-40

Roof Displacement (mm) No Devices Hysteretic 400 KN U Friction 800 KN U

586367206

636480255

731623(1)

412(1)

Normalised to NDP Value No Devices Hysteretic 400 KN U Friction 800 KN U

1.001.001.00

1.081.311.24

1.251.70(1)

2.00(1)

Normalised to No Device Value No Devices Hysteretic 400 KN U Friction 800 KN U

1.000.630.35

1.000.750.40

1.000.85(1)

0.56(1)

NOTE :

[1] The NSP Method 2 as implemented does not include damping from the devices and so these results will havedisplacements over-estimated.


Of course, there is no guarantee that the NDP is the more accurate of the two procedures. However, the twoprocedures use the same structural models and material properties and so these sources of uncertainties oftenquoted for the NDP do not exist in this comparison. The only difference is the manner in which the seismicloads are applied and the response to the earthquake is calculated. The NDP applies the ground accelerationsdirectly to impose inertia loads on the structure and incorporates the hysteretic energy dissipation through thesection hysteresis. The NSP uses indirect specification of the seismic inertia loads through the responsespectrum and incorporates hysteretic energy dissipation by using equivalent non-linear spectra (Method 1) orequivalent viscous damping to reduce the spectrum (Method 2).

As the NDP is explicit and avoids the approximations inherent in the NSP the reliability of the results shouldbe higher for this procedure. This is particularly so for such a simple structure as this example which has onlyflexural members and a limited number of plastic hinges. The conclusions reached in this section are based onthe assumption that the NDP results are the most accurate and form the benchmark by which results from theother procedures are evaluated.

7.6 DESIGN RECOMMENDATIONS

The studies performed as part of the development of these guidelines have confirmed that the design of addeddampers is complex because of the interaction of the dampers with the existing structure. It is not possible todevelop an explicit design procedure at this stage and the best that can be provided are some pointers to helpdevelop a project specific design:

• Use the FEMA-273 provisions for detailed design and evaluation requirements. The SEAOC Blue Bookcan also be used to obtain further information on design requirements.

• Quantify the deficiencies you want to remedy with dampers, which will generally be one or more ofexcessive drifts (and associated element deformations), excessive element shear forces and/or excessivefloor accelerations.

• Selection of device type is a function of a number of factors, discussed earlier in this chapter. Theseinclude the type of building, the retrofit budget, and the type and magnitude of deficiencies.

• Optimum device parameters, such as yield force, slip force and damping coefficient are also a function ofthese factors. The time history results give some guidance as to the types which seemed best suited tospecific building types. However, you should evaluate a wide range of properties for a specific project.

• The distribution of dampers over the height of the building seems to depend on the type of deficiencyyou are trying to remedy.

• Hysteretic and friction dampers function more as structural elements than dampers in that the response ismore a function of the stiffness they add than the energy dissipation. The energy dissipation is a functionof the elastic stiffness – the higher the initial stiffness, the more effective the device as a damper.


• Hysteretic and friction dampers can be designed as for any other type of structural strengthening element,ignoring the energy dissipation function. This is a conservative approach with the advantage of simplicity.

• All damper types can be evaluated using a non-linear time history analysis (NDP) and the hysteretic andfriction types can also be evaluated using non-linear pushover analysis (NSP). FEMA-273 provides forthe NSP to be used to evaluate viscous and visco-elastic devices. However, the procedure is complex andis not recommended.

• The limited analyses performed to date suggest that the NSP analysis produces conservative resultscompared to the NDP. At this time, the NDP is recommended for all projects with the NSP used as acheck on results.


8888 SUMMARYSUMMARYSUMMARYSUMMARY

8.1 IN-STRUCTURE DAMPING AND ENERGY DISSIPATION

Earthquake mitigation strategies, of which in-structure damping is one, attempt to reduce the demand on astructure, rather than the more usual approach of adding capacity. The three general classifications of seismicmitigation hardware are Seismic Isolation, Passive Energy Dissipation and Active Control. These guidelines arerestricted to the range of devices within the Passive Energy Dissipation classification.

Although seismic isolation is a subset of the general field of passive energy dissipation, in-structure dampingvaries from isolation in two major respects:

1. In-structure damping is distributed up the height of the building rather than concentrated at one plane.

2. Most of the effectiveness of isolation is the period shift effect, lengthening the period of response, with alesser effect from damping. In-structure damping has a minor effect on period and in fact often shortensthe period if anything. Response reductions rely entirely on energy dissipation.

From an engineering mechanics viewpoint, a fundamental difference is that an isolation system acts in serieswith the structure whereas in-structure damping acts in parallel with the structure. An isolation systemabsorbs energy and filters the motion before it passes into the structural system. For a structure with in-structure damping, all energy passes into the combined system which then dissipates this energy depending onthe characteristics of each of the components (structural system and devices). This requires that the dampingbe tuned to the structure for optimum performance, a more complex design problem than isolation.

The response reductions from in-structure damping are much less dramatic than from isolation. Isolation canreduce structural forces and deformation by a factor of from 4 to 6. In-structure damping generally providesreductions by factors of 1.5 to 2 at best. However, it is less intrusive than isolation and cheaper to install.

Almost by definition, buildings not suitable for base isolation are the best candidates for in-structure damping.It is most effective on flexible buildings with slender lateral load systems and is also suitable for soft soil sites.The suitability of flexible buildings arises from the fact that in-structure damping is activated by inter-storymovement, either velocity or displacement. The greater the movement the greater the damping which givesrise to a paradox in that the aim of the damping is to reduce the movements which give rise to the damping.


8.2 DAMPER TYPES AND PROPERTIES

There are four main categories of device:

1. Yielding metal devices, such as steel cantilevers, yielding braces and lead extrusion dampers. The force isdisplacement dependent and energy dissipation is through hysteretic yielding.

2. Friction devices, such as brake pads clamped with bolts at brace intersections. As for the yielding metal,the force is displacement dependent and energy dissipation is through a frictional hysteresis.

3. Viscous dampers, usually fluid forced through an orifice. The force is velocity dependent and energydissipation is by the fluid viscosity.

4. Visco-elastic dampers, usually a solid copolymer such as the product developed by 3M which was basicallythick Scotch tape bonded between steel plates. These materials have an elastic stiffness, with adisplacement dependent force, as well as a viscous component which produces a velocity dependent force.Some visco-elastic devices are liquid. Damping is through the material viscosity.

There are other more exotic passive devices such as shape memory alloys but these guidelines are restricted tothese four types.

The velocity dependent dampers provide damping forces which are out of phase with the displacements andso these forces are not directly additive to the structure forces. This makes the velocity dependent dampermore efficient, in theory, than the displacement dependent devices. In practice, although the velocity anddisplacements are out of phase, there is some degree of coupling between the two sets of forces, especially fornon-linear dampers or if the structural system yields.

Practical dampers may be configured to yield in bending, shear or axially. The dampers are configured suchthat displacements or velocities are imparted to the devices by inter-story movements.

Dampers may be configured as diagonal braces or placed horizontally from the top of a partial height wall toan adjacent column. They can also be configured to connect the top of a wall panel to the soffit of the girderof the floor above. The wall panel is a cantilever from the wall below, with a gap between the top of the walland the floor above. As an alternative to a wall panel, the dampers can be mounted on a steel frame.Proposals have been made to use the cladding panels of a building to mount shear or flexural dampers butthere is no record of this being implemented.

8.3 DAMPING DECAY

One procedure for quantifying the damping provided by devices is by duplicating analytically a physicalmethod of measuring damping, the snap-back test, which is to release a structure from a deformed positionand measure the decay in displacements over successive cycles.

Decay analyses were performed on a 10 story yielding frame structure with a range of devices. This identifiedthe following characteristics:


• Structural yielding (beams and columns) had only a very slight effect on damping as measured by decay.This is because the structure immediately unloads to its elastic state over one-half cycle and then vibratesas for the non-yielding model.

• The hysteretic dampers provided increased damping, about 8%, for the first cycle but in subsequent cyclesthe damping reduced to that for the base structure with the elastic stiffness of the dampers. This isbecause the dampers did not cycle plastically after the initial release.

• The friction dampers produced a similar response to the hysteretic dampers but with much higherdamping in the initial cycle, over 30%.

• The viscous dampers produced relatively constant damping, from 8% to 18% for the properties includedin this study. The damping did not increase linearly with the damping coefficient; increasing the dampingcoefficient by a factor of 5 increased damping by a factor of 2.2.

• The visco-elastic dampers provided almost constant damping but with some decrease with decreasingamplitude because of the stiffening effect of the elastic component. These devices seemed to provideapproximately as much damping as a viscous damper with the same coefficient. For example, C = 2500produced 10½% damping for the visco-elastic device, compared to 10% for the C = 2000 viscous deviceand 12% for the C = 3000 viscous device.

At first examination, these results appear to indicate much better performance from viscous devices (VD andVE) than hysteretic devices (HD and FD) in that the damping for the latter only applies for the first cycle.However, this more likely identifies problems with quantifying damping using this procedure rather thannecessarily ineffectiveness of the devices. The intention of using supplemental dampers for seismic protectionis generally to reduce the peak amplitude of response and the HD and FD dampers may be effective in this.

8.4 TIME HISTORY ANALYSIS

Three prototype buildings were studied using the time history method of analysis. The buildings wereconcrete frames with heights of 3, 5 and 10 stories respectively. The buildings were designed for a low seismiczone and the performance was evaluated with varying devices, and device distributions, for earthquake recordscorresponding to a high seismic zone. The aim of the study was to determine which devices andconfigurations could improve the performance so as to be satisfactory for the higher load.

Results were very building specific. For the 3 story building the optimum damper types were either viscous orvisco-elastic dampers with a damping coefficient of 7000 KN-sec/m or higher in either a uniform or reversetriangular distribution. The former will minimise the drifts, the latter the frame shear force. For optimumeffect on floor accelerations the damping coefficient needed to be reduced to less then 2500 KN-sec/m butthe effectiveness in reducing drifts and shears was much less at this value.

The dampers which are most effective for the 3 story building were also optimum for the 5 story building.For this building, friction dampers with a low slip force were the most effective in controlling flooraccelerations but were less effective in reducing drifts and shears.


Friction dampers were the most effective for the 10 story building. Dampers with a high slip force were mosteffective for reducing drifts and shears but increased floor accelerations. Reducing the slip force reduced flooraccelerations at the cost of effectiveness in reducing for drifts and shears.

There was a general trend in that the uniform distributions of dampers was best at controlling drifts, thereverse triangular distribution (highest capacity dampers at the base) were best at controlling frame shears andthe triangular distribution (highest capacity dampers at the roof) was most effective in reducing flooraccelerations.

8.5 DESIGN PROCEDURES

The studies performed as part of the development of these guidelines have confirmed that the design of addeddampers is complex because of the interaction of the dampers with the existing structure. It is not possible todevelop an explicit design procedure at this stage and the best that can be provided are some pointers to helpdevelop a project specific design:

• Use the FEMA-273 provisions for detailed design and evaluation requirements. The SEAOC Blue Bookcan also be used to obtain further information on design requirements.

• Quantify the deficiencies you want to remedy with dampers, which will generally be one or more ofexcessive drifts (and associated element deformations), excessive element shear forces and/or excessivefloor accelerations.

• Selection of device type is a function of a number of factors. These include the type of building, theretrofit budget, and the type and magnitude of deficiencies.

• Optimum device parameters, such as yield force, slip force and damping coefficient are also a function ofthese factors. The time history results give some guidance as to the types which seemed best suited tospecific building types. However, you should evaluate a wide range of properties for a specific project.

• The distribution of dampers over the height of the building seems to depend on the type of deficiencyyou are trying to remedy.

• Hysteretic and friction dampers function more as structural elements than dampers in that the response ismore a function of the stiffness they add than the energy dissipation. The energy dissipation is a functionof the elastic stiffness – the higher the initial stiffness, the more effective the device as a damper.

• Hysteretic and friction dampers can be designed as any other type of structural strengthening element,ignoring the energy dissipation function. This is a conservative approach with the advantage of simplicity.

• All damper types can be evaluated using a non-linear time history analysis (NDP) and the hysteretic andfriction types can also be evaluated using non-linear pushover analysis (NSP). FEMA-273 provides forthe NSP to be used to evaluate viscous and visco-elastic devices. However, the procedure is complex andis not recommended.


• The limited analyses performed to date suggest that the NSP analysis produces conservative resultscompared to the NDP. At this time, the NDP is recommended for all projects with the NSP used as acheck on results.


8.6 RECOMMENDATIONS

The design of in-structure damping is difficult and it is only suitable for a restricted range of buildings.Unfortunately, this range is not well defined and so a lot of effort may be expended simply to prove that abuilding is not suited to added damping. These guidelines are intended to eventually ensure that we filter outunsuitable projects before we expend all this effort.

The more efficient types of damper, at least in theory, are the most expensive – fluid viscous dampers.Hysteretic dampers tend to merge with structural elements and for some types if is difficult to differentiatebetween a structural brace and a damper.


9999 BIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHYBIBLIOGRAPHY

The FEMA Guidelines have an extensive reference list. Good sources are also the ATC17-1 Seminar Notesand papers from the World Conferences in Earthquake Engineering.

[1]. Uniform Building Code Appendix Division III Earthquake Regulations for Seismic-Isolated Structures, UBC,American Association of Building Officials, Whittier, CA, 1994.

[2]. NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA-273, Federal EmergencyManagement Agency, Washington D.C. October, 1997

[3]. Recommended Lateral Force Requirements and Commentary, 7th Edition, Structural Engineers Associationof California, 1999.

[4]. Proceeding of Seminar on Seismic Isolation, Passive energy Dissipation and Active Control, ATC 17-1, AppliedTechnology Council, San Francisco, CA, 1993.

[5]. Proceeding of Seminar on Base Isolation and Passive energy Dissipation, ATC 17, Applied TechnologyCouncil, San Francisco, CA, 1986.

[6]. DRAIN 2D - A General Purpose Computer Program for Dynamic Analysis of Inelastic Plane Structures, A EKanaan and G H Powell, Report No. EERC 73-6 and 73-22, University of California, Berkeley(revised September 1973 and August 1975).

[7]. ANSR II - Analysis of Nonlinear Structural Response, Users Manual, D P Mondkar and G H Powell,Report No. UCB/EERC - 79/17, Earthquake Engineering Research Centre, University ofCalifornia, Berkeley 1979.

[8]. ETABS Three Dimensional Analysis of Building Systems - Users Manual, A Habibullah, Computers andStructures, Inc. Berkeley, CA 1986.

[9]. Base Isolation of Structures - Design Guidelines, Holmes Consulting Group, Revision 0, July, 2001.

[10]. Performance Based Evaluation of Buildings – Non-Linear Pushover and Time History Analysis, HolmesConsulting Group, Revision 5, November, 2000.

A-1

AAAA TIME HISTORY RESULTSTIME HISTORY RESULTSTIME HISTORY RESULTSTIME HISTORY RESULTS

TABLE A-1 RESPONSE RATIOS FOR TIME HISTORY ANALYSES

UniformDistribution

TriangularDistribution

ReverseTriangular

DistributionNumber

ofStories

DamperType

Coeff DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

3 H 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.003 H 15 1.09 0.98 0.99 1.06 0.99 0.99 1.05 0.99 0.993 H 30 1.15 0.98 0.98 1.11 0.98 0.98 1.10 0.98 0.983 H 45 1.21 0.99 1.01 1.16 0.99 0.99 1.14 0.98 0.983 H 60 1.24 0.99 1.02 1.20 0.99 1.01 1.17 0.98 0.983 H 75 1.27 1.00 1.04 1.23 1.00 1.03 1.20 0.98 0.993 H 90 1.28 0.99 1.04 1.26 1.00 1.04 1.21 0.98 1.003 H 105 1.26 0.98 1.05 1.28 1.00 1.06 1.22 0.98 1.003 H 120 1.26 0.98 1.05 1.30 0.99 1.06 1.24 0.98 1.003 H 135 1.26 0.99 1.06 1.31 0.99 1.06 1.24 0.99 1.013 H 150 1.27 0.99 1.06 1.31 0.99 1.07 1.24 0.97 1.013 H 165 1.28 0.99 1.06 1.30 0.99 1.07 1.24 0.97 1.013 H 180 1.28 0.98 1.05 1.30 0.99 1.07 1.23 0.97 1.023 H 195 1.28 0.98 1.05 1.30 0.99 1.07 1.22 0.97 1.023 H 210 1.26 0.98 1.06 1.30 0.99 1.06 1.22 0.98 1.013 H 225 1.23 0.98 1.05 1.31 0.99 1.06 1.22 0.98 1.023 H 240 1.20 0.98 1.04 1.30 0.99 1.04 1.22 0.98 1.023 H 255 1.17 0.99 1.04 1.30 1.00 1.03 1.22 0.98 1.033 H 270 1.14 0.99 1.04 1.31 1.00 1.03 1.22 0.98 1.033 H 285 1.11 1.00 1.03 1.31 1.01 1.02 1.21 0.98 1.043 H 300 1.09 1.00 1.02 1.31 1.01 1.01 1.21 0.98 1.04

3 F 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.003 F 15 1.04 0.97 0.93 1.04 0.97 0.93 1.02 0.98 0.963 F 30 1.06 0.95 0.92 1.07 0.96 0.92 1.03 0.97 0.933 F 45 1.08 0.94 0.90 1.09 0.95 0.91 1.03 0.96 0.933 F 60 1.08 0.94 0.89 1.11 0.95 0.90 1.04 0.95 0.933 F 75 1.08 0.95 0.87 1.12 0.95 0.89 1.04 0.95 0.923 F 90 1.07 0.95 0.85 1.12 0.96 0.88 1.04 0.94 0.913 F 105 1.04 0.95 0.84 1.12 0.96 0.88 1.04 0.94 0.913 F 120 1.00 0.95 0.83 1.11 0.96 0.88 1.03 0.93 0.90

A-2

UniformDistribution


ReverseTriangular

DistributionNumber

ofStories

DamperType

Coeff DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

3 F 135 0.94 0.96 0.83 1.10 0.96 0.89 1.03 0.93 0.893 F 150 0.86 0.96 0.82 1.08 0.96 0.89 1.01 0.93 0.873 F 165 0.84 0.96 0.80 1.06 0.97 0.87 1.01 0.93 0.863 F 180 0.83 0.96 0.78 1.03 0.97 0.85 1.00 0.92 0.853 F 195 0.82 0.95 0.79 1.00 0.98 0.84 0.99 0.93 0.833 F 210 0.79 0.95 0.82 0.98 0.98 0.82 0.96 0.93 0.823 F 225 0.75 0.95 0.85 0.94 0.98 0.81 0.92 0.93 0.813 F 240 0.72 0.94 0.88 0.90 0.98 0.81 0.87 0.93 0.803 F 255 0.68 0.94 0.91 0.85 0.98 0.81 0.86 0.93 0.793 F 270 0.64 0.93 0.93 0.82 0.98 0.82 0.86 0.93 0.783 F 285 0.61 0.93 0.95 0.83 0.98 0.85 0.86 0.93 0.783 F 300 0.57 0.93 0.97 0.83 0.99 0.87 0.85 0.93 0.77

3 V 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.003 V 500 0.94 0.93 0.83 0.97 0.94 0.84 0.94 0.95 0.913 V 1000 0.87 0.91 0.76 0.95 0.92 0.77 0.90 0.92 0.873 V 1500 0.82 0.90 0.72 0.91 0.92 0.73 0.86 0.91 0.833 V 2000 0.73 0.89 0.73 0.87 0.92 0.70 0.84 0.90 0.803 V 2500 0.67 0.89 0.77 0.82 0.92 0.73 0.81 0.89 0.783 V 3000 0.61 0.88 0.80 0.76 0.92 0.74 0.76 0.88 0.763 V 3500 0.55 0.87 0.82 0.71 0.92 0.76 0.70 0.88 0.753 V 4000 0.51 0.87 0.83 0.68 0.91 0.77 0.66 0.87 0.773 V 4500 0.47 0.87 0.84 0.66 0.92 0.78 0.63 0.87 0.783 V 5000 0.44 0.85 0.86 0.62 0.92 0.79 0.59 0.86 0.783 V 5500 0.42 0.85 0.87 0.59 0.92 0.81 0.55 0.85 0.803 V 6000 0.40 0.85 0.88 0.58 0.91 0.81 0.51 0.85 0.813 V 6500 0.39 0.85 0.89 0.56 0.91 0.82 0.46 0.84 0.833 V 7000 0.38 0.85 0.92 0.54 0.91 0.83 0.44 0.84 0.853 V 7500 0.39 0.85 0.94 0.52 0.91 0.84 0.43 0.84 0.853 V 8000 0.36 0.85 0.94 0.51 0.91 0.86 0.44 0.83 0.843 V 8500 0.33 0.82 0.77 0.50 0.91 0.91 0.43 0.82 0.853 V 9000 0.47 0.92 0.90 0.42 0.82 0.863 V 9500 0.40 0.90 0.77 0.41 0.82 0.863 V 10000 0.40 0.82 0.87

3 VE 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.003 VE 500 0.88 0.93 0.84 0.94 0.94 0.85 0.90 0.95 0.913 VE 1000 0.79 0.90 0.77 0.89 0.92 0.78 0.83 0.92 0.883 VE 1500 0.71 0.89 0.74 0.83 0.92 0.74 0.78 0.90 0.843 VE 2000 0.68 0.89 0.75 0.78 0.91 0.72 0.73 0.88 0.823 VE 2500 0.64 0.88 0.79 0.73 0.91 0.74 0.69 0.87 0.803 VE 3000 0.59 0.88 0.81 0.71 0.91 0.76 0.68 0.86 0.78

A-3

UniformDistribution


ReverseTriangular

DistributionNumber

ofStories

DamperType

Coeff DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

3 VE 3500 0.53 0.87 0.85 0.69 0.92 0.78 0.66 0.87 0.783 VE 4000 0.49 0.87 0.86 0.66 0.92 0.81 0.63 0.85 0.773 VE 4500 0.47 0.86 0.88 0.64 0.92 0.81 0.61 0.85 0.803 VE 5000 0.45 0.85 0.90 0.61 0.91 0.83 0.57 0.84 0.823 VE 5500 0.43 0.85 0.91 0.58 0.91 0.83 0.52 0.84 0.833 VE 6000 0.42 0.85 0.93 0.56 0.91 0.85 0.50 0.83 0.853 VE 6500 0.41 0.85 0.96 0.55 0.90 0.86 0.49 0.83 0.863 VE 7000 0.40 0.84 0.99 0.54 0.90 0.88 0.48 0.82 0.873 VE 7500 0.40 0.84 1.01 0.52 0.90 0.88 0.46 0.82 0.883 VE 8000 0.39 0.84 1.02 0.50 0.90 0.89 0.45 0.81 0.883 VE 8500 0.37 0.85 1.01 0.49 0.90 0.92 0.44 0.80 0.893 VE 9000 0.36 0.83 1.05 0.49 0.91 0.95 0.43 0.80 0.903 VE 9500 0.44 0.89 0.91 0.42 0.80 0.913 VE 10000 0.41 0.79 0.92


5 F 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.005 F 25 0.90 1.00 1.01 0.96 1.01 1.00 0.93 1.00 1.005 F 50 0.86 0.99 0.97 0.90 1.01 0.99 0.83 0.99 1.025 F 75 0.88 0.97 0.88 0.86 1.01 0.94 0.85 0.98 1.015 F 100 0.89 0.97 0.77 0.88 1.00 0.85 0.87 0.97 1.00

A-4

UniformDistribution


ReverseTriangular

DistributionNumber

ofStories

DamperType

Coeff DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

5 F 125 0.89 0.97 0.68 0.89 0.98 0.78 0.88 0.95 0.955 F 150 0.88 0.96 0.70 0.89 0.98 0.69 0.89 0.95 0.915 F 175 0.88 0.94 0.73 0.90 0.99 0.65 0.88 0.94 0.855 F 200 0.86 0.93 0.75 0.90 0.99 0.68 0.87 0.94 0.795 F 225 0.85 0.91 0.74 0.91 0.99 0.71 0.86 0.94 0.745 F 250 0.83 0.89 0.76 0.89 0.98 0.73 0.85 0.93 0.725 F 275 0.80 0.88 0.78 0.89 0.98 0.73 0.85 0.92 0.735 F 300 0.77 0.89 0.77 0.89 0.96 0.77 0.83 0.91 0.765 F 325 0.74 0.89 0.77 0.88 0.95 0.79 0.82 0.90 0.795 F 350 0.70 0.89 0.79 0.86 0.94 0.80 0.80 0.88 0.825 F 375 0.68 0.88 0.85 0.84 0.94 0.79 0.79 0.88 0.865 F 400 0.66 0.87 0.87 0.86 0.93 0.79 0.77 0.87 0.845 F 425 0.64 0.86 0.83 0.87 0.93 0.77 0.75 0.87 0.815 F 450 0.61 0.86 0.93 0.87 0.94 0.77 0.72 0.86 0.795 F 475 0.59 0.86 0.98 0.85 0.94 0.78 0.69 0.85 0.785 F 500 0.58 0.85 1.03 0.83 0.94 0.77 0.66 0.84 0.82

5 V 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.005 V 500 0.84 0.97 0.88 0.92 1.00 0.91 0.90 0.98 0.935 V 1000 0.79 0.94 0.81 0.82 0.98 0.84 0.81 0.95 0.895 V 1500 0.76 0.94 0.76 0.79 0.97 0.79 0.78 0.93 0.855 V 2000 0.75 0.93 0.73 0.78 0.95 0.76 0.77 0.93 0.815 V 2500 0.73 0.92 0.72 0.77 0.95 0.73 0.76 0.92 0.795 V 3000 0.70 0.91 0.71 0.76 0.95 0.72 0.74 0.92 0.775 V 3500 0.68 0.90 0.71 0.75 0.94 0.71 0.73 0.91 0.765 V 4000 0.66 0.90 0.71 0.74 0.94 0.70 0.71 0.90 0.755 V 4500 0.63 0.89 0.71 0.73 0.94 0.69 0.69 0.89 0.755 V 5000 0.61 0.88 0.71 0.72 0.93 0.69 0.68 0.89 0.745 V 5500 0.58 0.86 0.72 0.71 0.93 0.68 0.66 0.88 0.745 V 6000 0.54 0.85 0.74 0.70 0.93 0.68 0.64 0.87 0.745 V 6500 0.61 0.84 0.74 0.69 0.93 0.68 0.62 0.86 0.745 V 7000 0.60 0.82 0.83 0.68 0.92 0.68 0.61 0.85 0.745 V 7500 0.51 0.78 0.76 0.69 0.92 0.69 0.59 0.84 0.745 V 8000 0.65 0.93 0.72 0.57 0.82 0.745 V 8500 0.67 0.93 0.70 0.56 0.80 0.755 V 9000 0.54 0.78 0.755 V 9500 0.52 0.76 0.755 V 10000 0.51 0.74 0.75

5 VE 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.005 VE 500 0.82 0.97 0.89 0.86 0.99 0.91 0.85 0.97 0.945 VE 1000 0.78 0.95 0.81 0.80 0.98 0.84 0.80 0.94 0.90

A-5

UniformDistribution


ReverseTriangular

DistributionNumber

ofStories

DamperType

Coeff DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

5 VE 1500 0.78 0.95 0.76 0.79 0.97 0.79 0.77 0.93 0.865 VE 2000 0.79 0.92 0.74 0.78 0.96 0.76 0.76 0.93 0.835 VE 2500 0.77 0.92 0.73 0.79 0.96 0.73 0.76 0.92 0.805 VE 3000 0.74 0.91 0.72 0.81 0.96 0.72 0.75 0.89 0.785 VE 3500 0.72 0.90 0.72 0.81 0.95 0.71 0.74 0.89 0.775 VE 4000 0.69 0.88 0.72 0.80 0.95 0.70 0.73 0.87 0.765 VE 4500 0.65 0.87 0.72 0.79 0.95 0.69 0.71 0.86 0.765 VE 5000 0.62 0.85 0.73 0.78 0.95 0.69 0.69 0.85 0.755 VE 5500 0.59 0.83 0.73 0.76 0.95 0.69 0.67 0.84 0.755 VE 6000 0.55 0.80 0.74 0.75 0.95 0.69 0.65 0.82 0.755 VE 6500 0.56 0.77 0.76 0.73 0.95 0.69 0.64 0.81 0.755 VE 7000 0.54 0.75 0.77 0.72 0.94 0.69 0.62 0.78 0.755 VE 7500 0.49 0.73 0.76 0.71 0.94 0.74 0.60 0.76 0.765 VE 8000 0.67 0.94 0.71 0.58 0.74 0.765 VE 8500 0.68 0.93 0.72 0.56 0.72 0.765 VE 9000 0.54 0.70 0.765 VE 9500 0.52 0.68 0.765 VE 10000 0.50 0.66 0.77


10 F 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

A-6

UniformDistribution


ReverseTriangular

DistributionNumber

ofStories

DamperType

Coeff DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

10 F 50 1.11 0.98 0.95 1.10 0.99 0.98 1.07 0.99 0.9810 F 100 1.06 0.95 0.85 1.14 0.97 0.90 1.08 0.98 0.9510 F 150 0.98 0.93 0.82 1.10 0.95 0.82 1.06 0.94 0.9010 F 200 0.91 0.88 0.89 1.06 0.96 0.81 0.99 0.93 0.8710 F 250 0.86 0.88 0.82 0.99 0.93 0.81 0.93 0.92 0.8410 F 300 0.81 0.89 0.91 0.96 0.92 0.92 0.89 0.88 0.8110 F 350 0.78 0.90 0.96 0.93 0.91 0.94 0.84 0.87 0.8610 F 400 0.76 0.88 1.04 0.91 0.90 0.85 0.81 0.87 0.9410 F 450 0.74 0.88 1.01 0.88 0.91 0.93 0.78 0.87 0.9710 F 500 0.72 0.88 1.13 0.86 0.92 1.02 0.76 0.86 1.0010 F 550 0.69 0.87 1.44 0.85 0.93 1.01 0.73 0.85 1.0310 F 600 0.66 0.86 1.41 0.85 0.94 1.01 0.71 0.85 1.0710 F 650 0.62 0.86 1.40 0.86 0.95 1.01 0.68 0.83 1.0710 F 700 0.58 0.85 1.47 0.86 0.97 1.02 0.66 0.82 1.0810 F 750 0.53 0.83 1.61 0.86 0.97 1.09 0.63 0.80 1.1010 F 800 0.49 0.81 1.43 0.86 0.97 1.13 0.61 0.77 1.1410 F 850 0.48 0.78 1.45 0.86 0.97 1.40 0.57 0.75 1.1610 F 900 0.48 0.75 1.51 0.85 0.97 1.17 0.54 0.74 1.1910 F 950 0.48 0.75 1.74 0.85 0.96 1.27 0.51 0.72 1.2010 F 1000 0.47 0.75 1.48 0.83 0.96 1.38 0.48 0.71 1.24

10 V 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.0010 V 500 1.07 0.98 0.96 1.08 0.99 0.97 1.04 0.99 0.9810 V 1000 1.07 0.97 0.92 1.11 0.98 0.95 1.04 0.98 0.9610 V 1500 1.04 0.95 0.89 1.11 0.97 0.92 1.03 0.97 0.9310 V 2000 1.00 0.93 0.88 1.10 0.96 0.91 1.01 0.95 0.9110 V 2500 0.93 0.92 0.87 1.09 0.95 0.89 0.98 0.93 0.9010 V 3000 0.89 0.91 0.87 1.07 0.95 0.89 0.95 0.92 0.8910 V 3500 0.85 0.90 0.87 1.04 0.94 0.88 0.90 0.91 0.8810 V 4000 0.81 0.90 0.87 1.01 0.94 0.88 0.87 0.90 0.8810 V 4500 0.78 0.90 0.87 0.97 0.94 0.87 0.84 0.90 0.8710 V 5000 0.74 0.91 0.87 0.95 0.93 0.87 0.81 0.90 0.8710 V 5500 0.69 0.92 0.87 0.93 0.93 0.87 0.78 0.90 0.8710 V 6000 0.66 0.92 0.88 0.91 0.92 0.86 0.75 0.90 0.8710 V 6500 0.89 0.91 0.86 0.73 0.90 0.8710 V 7000 0.89 0.91 0.86 0.70 0.90 0.8710 V 7500 0.82 0.91 0.89 0.68 0.90 0.8710 V 8000 0.66 0.89 0.8710 V 8500 0.63 0.89 0.8710 V 9000 0.62 0.88 0.8710 V 9500 0.60 0.88 0.8710 V 10000 0.58 0.87 0.87

A-7

UniformDistribution


ReverseTriangular

DistributionNumber

ofStories

DamperType

Coeff DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

DriftRatio

ShearRatio

AccelRatio

10 VE 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.0010 VE 500 0.99 1.00 0.97 1.03 1.00 0.98 0.98 1.00 0.9910 VE 1000 0.93 0.99 0.93 1.01 1.00 0.95 0.93 0.99 0.9710 VE 1500 0.86 0.97 0.90 0.99 0.99 0.93 0.89 0.99 0.9410 VE 2000 0.82 0.94 0.89 0.95 0.99 0.91 0.85 0.97 0.9210 VE 2500 0.78 0.92 0.88 0.91 0.98 0.90 0.82 0.95 0.9010 VE 3000 0.74 0.91 0.81 0.88 0.97 0.89 0.79 0.93 0.9010 VE 3500 0.71 0.92 0.88 0.86 0.97 0.88 0.76 0.90 0.8910 VE 4000 0.68 0.93 0.88 0.83 0.96 0.88 0.73 0.89 0.8910 VE 4500 0.66 0.93 0.88 0.80 0.95 0.88 0.71 0.90 0.8810 VE 5000 0.65 0.93 0.88 0.78 0.95 0.87 0.69 0.90 0.8810 VE 5500 0.64 0.92 0.88 0.76 0.95 0.87 0.66 0.91 0.8810 VE 6000 0.62 0.94 0.92 0.76 0.95 0.87 0.64 0.91 0.8810 VE 6500 0.75 0.94 0.87 0.63 0.91 0.8810 VE 7000 0.75 0.94 0.87 0.61 0.91 0.8810 VE 7500 0.75 0.95 0.87 0.59 0.91 0.8810 VE 8000 0.58 0.90 0.8810 VE 8500 0.56 0.90 0.8810 VE 9000 0.55 0.90 0.8910 VE 9500 0.54 0.89 0.8910 VE 10000 0.53 0.89 0.89

A-8

FIGURE A-1 EFFECTIVENESS OF HYSTERETIC DAMPERS EQ1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 50 100 150 200 250 300

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift

H U 3 Story DriftH T 3 Story DriftH R 3 Story Drift

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 100 200 300 400 500Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 200 400 600 800 1000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


A-9


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 50 100 150 200 250 300

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 100 200 300 400 500

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 200 400 600 800 1000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


A-10


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 50 100 150 200 250 300

Damping Parameter

Equ

ival

ent D

ampi

ng


0.00

0.20

0.40

0.60

0.801.00

1.20

1.40

1.60

1.80

0 100 200 300 400 500

Damping Parameter

Equ

ivale

nt D

ampi

ng


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 200 400 600 800 1000

Damping Parameter

Equ

ivale

nt D

ampi

ng


A-11

FIGURE A-4 EFFECTIVENESS OF FRICTION DAMPERS EQ1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 50 100 150 200 250 300

Damping Parameter

Drift /5%DampedDrift

F U 3 Story DriftF T 3 Story DriftF R 3 Story Drift

0.00

0.20

0.40

0.60

0.80

1.00

1.20


Drift /5%DampedDrift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 200 400 600 800 1000

Damping Parameter

Drift /5%DampedDrift F U 10 Story Drift

F T 10 Story DriftF R 10 Story Drift

A-12


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 50 100 150 200 250 300

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 100 200 300 400 500

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 200 400 600 800 1000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


A-13


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 50 100 150 200 250 300

Damping Parameter

Equ

ival

ent D

ampi

ng


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 100 200 300 400 500

Damping Parameter

Equ

ivale

nt D

ampi

ng


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 200 400 600 800 1000

Damping Parameter

Equ

ivale

nt D

ampi

ng


A-14

FIGURE A-7 EFFECTIVENESS OF VISCOUS DAMPERS EQ1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift

V U 3 Story DriftV T 3 Story DriftV R 3 Story Drift

0.00

0.20

0.40

0.60

0.80

1.00

1.20


Drif

t / 5

% D

ampe

d D

rift


0.000.100.200.300.400.500.600.700.800.901.00

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


A-15


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


A-16


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Equ

ival

ent D

ampi

ngV U 3 Story DriftV T 3 Story DriftV R 3 Story Drift

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Equ

ivale

nt D

ampi

ng


0

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Equ

ivale

nt D

ampi

ng


A-17

FIGURE A-10 EFFECTIVENESS OF VISCO-ELASTIC DAMPERS EQ1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift

VE U 3 Story DriftVE T 3 Story DriftVE R 3 Story Drift

0.00

0.20

0.40

0.60

0.80

1.00

1.20


Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


A-18


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Drif

t / 5

% D

ampe

d D

rift


A-19


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Equ

ival

ent D

ampi

ngVE U 3 Story DriftVE T 3 Story DriftVE R 3 Story Drift

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Equ

ivale

nt D

ampi

ng


0

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2000 4000 6000 8000 10000

Damping Parameter

Equ

ivale

nt D

ampi

ng


In-structure Damping and Energy Dissapation

Documents

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soft soil

seaoc blue

engineering

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applied sine

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