seismic protective systems: passive energy dissipation major

Passive Energy Dissipation 15 – 6 - 1Instructional Material Complementing FEMA 451, Design Examples

SEISMIC PROTECTIVE SYSTEMS:PASSIVE ENERGY DISSIPATION

Presented & Developed by:Michael D. Symans, PhDRensselaer Polytechnic Institute

Initially Developed by:Finley A. Charney, PE, PhDVirginia Polytechnic Institute & State University


Major Objectives

• Illustrate why use of passive energy dissipation systems may be beneficial

• Provide overview of types of energy dissipation systems available

• Describe behavior, modeling, and analysis of structures with energy dissipation systems

• Review developing building code requirements


Outline: Part I

• Objectives of Advanced Technology Systems and Effects on Seismic Response

• Distinction Between Natural and Added Damping

• Energy Distribution and Damage Reduction• Classification of Passive Energy Dissipation

Systems


Outline: Part II• Velocity-Dependent Damping Systems:

Fluid Dampers and Viscoelastic Dampers• Models for Velocity-Dependent Dampers• Effects of Linkage Flexibility• Displacement-Dependent Damping

Systems: Steel Plate Dampers, Unbonded Brace Dampers, and Friction Dampers

• Concept of Equivalent Viscous Damping• Modeling Considerations for Structures

with Passive Energy Dissipation Systems


Outline: Part III

• Seismic Analysis of MDOF Structures with Passive Energy Dissipation Systems

• Representations of Damping• Examples: Application of Modal Strain

Energy Method and Non-Classical Damping Analysis

• Summary of MDOF Analysis Procedures


Outline: Part IV

• MDOF Solution Using Complex Modal Analysis

• Example: Damped Mode Shapes and Frequencies

• An Unexpected Effect of Passive Damping• Modeling Dampers in Computer Software• Guidelines and Code-Related Documents

for Passive Energy Dissipation Systems


Outline: Part I




Systems


Objectives of Energy Dissipation and Seismic Isolation Systems

• Enhance performance of structures at all hazard levels by:

Minimizing interruption of use of facility (e.g., Immediate Occupancy Performance Level)

Reducing damaging deformations in structural and nonstructural components

Reducing acceleration response to minimize contents-related damage


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20

Spectral Displacement, Inches

Pseu

doac

cele

ratio

n, g

T=.50 T=1.0

T=1.5

T=2.0

T=3.0

T=4.0

5% Damping

10%

20%

30%40%

Effect of Added Damping(Viscous Damper)

- Decreased Displacement- Decreased Shear Force

Pseu

do-S

pect

ral A

ccel

erat

ion,

g


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20


Pseu

doac

cele

ratio

n, g

T=.50 T=1.0

T=1.5

T=2.0

T=3.0

T=4.0

5% Damping

10%

20%

30%40%

Effect of Added Stiffness(Added Bracing)

- Decreased Displacement- Increased Shear Force

Pseu

do-S

pect

ral A

ccel

erat

ion,

g


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20


Pseu

doac

cele

ratio

n, g

T=.50 T=1.0

T=1.5

T=2.0

T=3.0

T=4.0

5% Damping

10%

20%

30%40%

Effect of Added Damping and Stiffness(ADAS System)

- Decreased Displacement- Decreased Shear (possibly)

Pseu

do-S

pect

ral A

ccel

erat

ion,

g


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20


Pseu

doac

cele

ratio

n, g

T=.50 T=1.0

T=1.5

T=2.0

T=3.0

T=4.0

5% Damping

10%

20%

30%40%

Effect of Reduced Stiffness(Seismic Isolation)

- Decreased Shear Force- Increased Displacement

Pseu

do-S

pect

ral A

ccel

erat

ion,

g


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20


Pseu

doac

cele

ratio

n, g

T=.50 T=1.0

T=1.5

T=2.0

T=3.0

T=4.0

5% Damping

10%

20%

30%40%

Effect of Reduced Stiffness(Seismic Isolation with Dampers)

- Decreased Shear- Increased Displacement

Pseu

do-S

pect

ral A

ccel

erat

ion,

g


Effect of Damping and Yield Strength on Deformation Demand

0

0.2

0.4

0.6

0.8

1

1.2

1.4

5 10 15 20 25 30

Damping Ratio (%)

Peak

Dis

plac

emen

t (in

)

102030

Yield Strength(kips)


Outline: Part I




Systems


Natural (Inherent) Damping

Added Damping

ξ

%0.7to5.0NATURAL =ξ

ξ is a structural property, dependent onsystem mass, stiffness, and the added damping coefficient C

%30to10ADDED =ξ

C

Distinction Between Natural and Added Damping

is a structural property, dependent onsystem mass, stiffness, and inherentenergy dissipation mechanisms


1981/1982 US-JAPAN PROJECTResponse of Bare Frame Before and After Adding Ballast

Model Weight

Bare Model 18 kipsLoaded Model 105 kips


0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30 35 40

PEAK GROUND ACCELERATION, %G

FREQUENCY, HZ

DAMPING, %CRITICAL

1981/1982 US-JAPAN PROJECTChange in Damping and Frequency with Accumulated Damage


Outline: Part I• Objectives of Advanced Technology Systems

and Effects on Seismic Response• Distinction Between Natural and Added

Damping• Energy Distribution and Damage Reduction• Classification of Passive Energy Dissipation

Systems


Energy Balance:

HDADIKSI EEEEEE ++++= )(

Damage Index:

( ) ( )ulty

H

ult

max

uFtE

uutDI ρ+=

Added DampingInherent Damping

Reduction in Seismic DamageHysteretic Energy

Source: Park and Ang (1985)

DI1.0

DamageState

0.0 Collapse


( ) ( )ulty

H

ult

max

uFtE

uutDI ρ+=

Duration-Dependent Damage Index

DI1.0

DamageState

0.0

Source: Park and Ang (1985)

= maximum displacement

= monotonic ultimate displacement

= calibration factor

= hysteretic energy dissipated

= monotonic yield force

maxu

ultuρ

HE

yF Collapse


0

20

40

60

80

100

120

140

160

180

200

0 4 8 12 16 20 24 28 32 36 40 44 48

Time, Seconds

Abs

orbe

d En

ergy

, Inc

h-K

ips

DAMPING

HYSTERETIC

KINETIC +STRAIN

0

20

40

60

80

100

120

140

160

180

200

0 4 8 12 16 20 24 28 32 36 40 44 48

Time, Seconds

Abs

orbe

d En

ergy

, inc

h-ki

ps

HYSTERETIC

DAMPING

KINETIC +STRAIN

10% Damping

20% Damping

Damping ReducesHysteretic EnergyDissipation Demand

Ene

rgy

(kip

-inch

)

Ene

rgy

(kip

-inch

)


Effect of Damping and Yield Strength on Hysteretic Energy

0

20

40

60

80

100

120

140

5 10 15 20 25 30

Damping Ratio (%)

Hys

tere

tic E

nerg

y (k

ip-in

)

102030

Yield Strength(kips)


Energy and Damage Histories, 5% Damping

0

50

100

150

200

250

300

0.00 5.00 10.00 15.00 20.00

Ener

gy, I

n-K

ips

HystereticViscous + HystereticTotal

0.00

0.20

0.40

0.60

0.80

1.00

0.00 5.00 10.00 15.00 20.00

Time, Seconds

Dam

age

Inde

x

EDI+EDA

EH

EI = 260

Analysisperformedon NONLIN

Max=0.55


0

50

100

150

200

250

300

0.00 5.00 10.00 15.00 20.00

Ener

gy, I

n-K

ips

HystereticViscous + HystereticTotal

0.00

0.20

0.40

0.60

0.80

1.00

0.00 5.00 10.00 15.00 20.00

Time, Seconds

Dam

age

Inde

x

Energy and Damage Histories, 20% Damping

EH

EI = 210

Max=0.30

EDI+EDA


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.00 5.00 10.00 15.00 20.00

Time, Seconds

Dam

age

Inde

x

5% Damping 20% Damping40% Damping 60% Damping

Reduction in Damage with Increased Damping


Outline: Part I




Systems


Classification of Passive Energy Dissipation Systems

Velocity-Dependent Systems• Viscous fluid or viscoelastic solid dampers• May or may not add stiffness to structure

Displacement-Dependent Systems• Metallic yielding or friction dampers• Always adds stiffness to structure

Other• Re-centering devices (shape-memory alloys, etc.)• Vibration absorbers (tuned mass dampers)




Systems: Steel Plate Dampers,UnbondedBrace Dampers, and Friction Dampers


with Passive Damping Systems


Cross-Section of Viscous Fluid Damper

Source: Taylor Devices, Inc.


Possible Damper Placement Within Structure

AugmentedBracing


Chevron Brace and Viscous Damper


Diagonally Braced Damping System

θ


Fluid Dampers within Inverted Chevron BracePacific Bell North Area Operation Center (911 Emergency Center)

Sacramento, California(3-Story Steel-Framed Building Constructed in 1995)

62 Dampers: 30 Kip Capacity, +/-2 in. Stroke


Fluid Damper within Diagonal Brace

San Francisco State Office Building

San Francisco, CA

Huntington TowerBoston, MA


Toggle Brace Damping System

θ1

θ2)cos(

sin

21

2

1 θθθ+

==UUAF D

U1

UD


Toggle Brace Deployment

Huntington Tower, Boston, MA- New 38-story steel-framed building- 100 direct-acting and toggle-brace dampers - 1300 kN (292 kips), +/- 101 mm (+/- 4 in.)- Dampers suppress wind-induced vibration


Harmonic Behavior of Fluid Damper

Note: Damping force 90o out-of-phase with elastic force.

P t P t P t( ) sin( ) cos( ) cos( )sin( )= +0 0ω δ ω δ

Loading Frequency

PhaseAngle(Lag)

-1500

-1000

-500

0

500

1000

1500

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

TIME, SECONDS

FOR

CE,

KIP

S

ELASTIC FORCEDAMPING FORCETOTAL FORCE

)tsin(u)t(u ω= 0 Imposed Motion

Total Force

ωδ


)cos(uPK

0

0S δ= )sin(

uPK

0

0L δ=

)t(uC)t(uK)t(P S &+=

ωLKC = ⎟⎟

⎠

⎞⎜⎜⎝

⎛= −

0

1sinPPZδ

Storage Stiffness Loss Stiffness Damping Coeff. Phase Angleuo

Damper Displacement, u

PoPZ KS

Tota

l For

ce, P

( )δsinPuKP ooLZ ==( )δππ sinuPuPE oooZD ==

*K


( ) ( )tuCtuK)t(P S &+=

Frequency-Domain Force-Displacement Relation

Apply Fourier Transform:

( ) ( ) ωωωωω /uiKuK)(P LS +=

[ ] ( )ωω uiKK)(P LS +=

Complex Stiffness:

( ) ( )( )ωωω

uPK * =

Compact Force-Displ. Relationfor Viscoelastic Dampers( ) ( ) ( )ωωω uKP *=

ℜ

ℑ( )ω*K

( )ωSK

( )ωLKδ

( ) S* KK =ℜNote: ( ) L

* KK =ℑand


0

2000

4000

6000

8000

10000

0 5 10 15 20 25

Excitation Frequency, Hz.

Stor

age

Stiff

ness

, lbs

/inch

CutoffFrequency

Stor

age

Stiff

ness

(lb/

in)

Excitation Frequency (Hz)

Dependence of Storage Stiffness on Frequencyfor Typical “Single-Ended” Fluid Damper


Dependence of Damping Coefficient on Frequencyfor Typical “Single-Ended” Fluid Damper

0

20

40

60

80

100

120

140

0 5 10 15 20 25

Excitation Frequency, Hz

Dam

ping

Coe

ffici

ent C

, lb-

sec/

inD

ampi

ng C

oeffi

cien

t, lb

-sec

/in

CutoffFrequency



Dependence of Phase Angle on Frequency for Typical “Single-Ended” Fluid Damper

Frequency (Hz)

Phas

e A

ngle

(deg

rees

)

0

40

80

120

0 5 10 15 20 25

20

60

100

δ

δ

δ


0

20

40

60

80

100

120

0 10 20 30 40 50

Temperature, Degrees C

Dam

ping

Con

stan

t, k-

sec/

in

Dependence of Damping Coefficient on Temperature for Typical Fluid Damper

Dam

ping

Coe

ffici

ent,

lb-s

ec/in

Temperature (Degrees C)


Behavior of Fluid Damper with Zero Storage Stiffness

Damper Displacement, u

Dam

per F

orce

, P

Po

uo

uKuC)t(P L &&ω

==oS 900K =⇒= δ

ood uPE π=


Actual Hysteretic Behavior of Fluid Damper

Seismic Loading

Harmonic Loading

Source:Constantinou and Symans (1992)


-300

-200

-100

0

100

200

300

-6.00 -4.00 -2.00 0.00 2.00 4.00 6.00

Damper Deformational Velocity, in/sec

Dam

per F

orce

, Kip

s

0.2 0.4 0.6 0.8 1.0

Force-Velocity Behavior of Viscous Fluid DamperExp

)usgn(uC)t(P &&α=


-120

-80

-40

0

40

80

120

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

Displacement, Inches

Forc

e, K

ips

α = 01.

α = 10.

Nonlinear Fluid Dampers)usgn(uC)t(P &&

α=


Linear Damper: ooD uPE π=

Nonlinear Damper: ooD uPE λ=

)2(2

124

2

α

α

λ α

+Γ

⎟⎠⎞

⎜⎝⎛ +Γ

×=

Energy Dissipated Per Cycle for Linear and Nonlinear Viscous Fluid Dampers

Γ = Gamma Function

Hysteretic Energy Factor


Relationship Between λ and αfor Viscous Fluid Damper

3.0

3.2

3.4

3.6

3.8

4.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Damper Velocity Exponent α

Hys

tere

tic E

nerg

y Fa

ctor

λ

AF = λ/π = 1.24

AF = 1.11

AF = 1.0


Relationship Between Nonlinear and Linear DampingCoefficient for Equal Energy Dissipation Per Cycle

αωλπ −= 1

oL

NL )u(CC

Note: Ratio is frequency- and displacement-dependentand is therefore meaningful only for steady-stateharmonic response.


Loading Freqency = 6.28 Hz

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2

Velocity Exponent

Rat

io o

f CN

L to

CL Max Disp = 1.0

Max Disp = 2.0Max Disp = 3.0

Ratio of Nonlinear Damping Coefficient to Linear DampingCoefficient (For a Given Loading Frequency)

Loading Frequency = 1 Hz (6.28 rad/sec)


Maximum Displacement = 1

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2Velocity Exponent

Rat

io o

f CN

L to

CL

Load Freq = 6.28 HzLoad Freq = 3.14 HzLoad Freq = 2.09 Hz

Ratio of Nonlinear Damping Constant to Linear Damping Constant (For a Given Maximum Displacement)

1/3 Hz (2.09 rad/s)1/2 Hz (3.14 rad/s)2 Hz (6.28 rad/s)


-800

-600

-400

-200

0

200

400

600

800

-15 -10 -5 0 5 10 15


Forc

e, K

ips

Alpha = 1.0Alpha = 0.5

Example of Linear vs Nonlinear Damping

Frequency = 1 Hz (6.28 rad/s)Max. Disp. = 10.0 in.

λ = 3.58

CLinear = 10.0 k-sec/inCNonlinear = 69.5 k-sec0.5/in0.5


Recommendations Related to Nonlinear Viscous Dampers

• Do NOT attempt to linearize the problem when nonlinearviscous dampers are used. Perform the analysis withdiscrete nonlinear viscous dampers.

• Do NOT attempt to calculate effective damping in termsof a damping ratio (ξ) when using nonlinear viscous dampers.

• DO NOT attempt to use a free vibration analysis to determine equivalent viscous damping when nonlinearviscous dampers are used.


Advantages of Fluid Dampers

• High reliability• High force and displacement capacity• Force Limited when velocity exponent < 1.0• Available through several manufacturers• No added stiffness at lower frequencies• Damping force (possibly) out of phase with

structure elastic forces• Moderate temperature dependency• May be able to use linear analysis


Disadvantages of Fluid Dampers

• Somewhat higher cost• Not force limited (particularly when exponent = 1.0)• Necessity for nonlinear analysis in most practical

cases (as it has been shown that it is generally not possible to add enough damping to eliminate all inelastic response)


Viscoelastic Dampers

Section A-A

h

h

W

Developed in the 1960’sfor Wind Applications

Viscoelastic Material

L

A

A

u(t)

P(t)P(t)


Implementation of Viscoelastic Dampers

Building 116, US Naval Supply Facility, San Diego, CA- Seismic Retrofit of 3-Story

Nonductile RC Building- 64 Dampers Within Chevron Bracing Installed in 1996


Harmonic Behavior of Viscoelastic Damper

-800

-600-400

-2000

200

400600

800

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

TIME, SECONDS

STR

ESS,

PSI

ELASTIC FORCEDAMPING FORCETOTAL FORCE

P t P t P t( ) sin( ) cos( ) cos( )sin( )= +0 0ω δ ω δ

Loading Frequency

PhaseAngle(Lag)

)tsin(u)t(u ω= 0 Imposed Motion

Total Force


SG' AK

h=

LG'' AK

h=

SP( t ) K u(t ) Cu(t )= + &

LKC =ω ⎟⎟

⎠

⎞⎜⎜⎝

⎛= −

0

1sinττδ Z

Storage Stiffness Loss Stiffness Damping Coeff. Phase Angle

γ 0

Shear Strain

She

ar S

tress

G’τ0τZ

G’ = Storage ModulusG’’ = Loss Modulus

( ) ( ) ωγγτ /tGtG)t( &′′+′=

Loss Factor

( )( ) ( )δωωη tan

GG

=′′′

=

( ) VGsinAhAhE 2ooooZD γπδγπτγπτ ′′===

( )δτγτ sinG ooZ =′′=

*G


( ) ( ) ωγγτ /tGtG)t( &′′+′=


( ) ( ) ωωγωωγωτ /iGG)( ′′+′=

[ ] ( )ωγωτ GiG)( ′′+′=

[ ] ( )ωγηωτ i1G)( +′=

Complex Shear Modulus:

( ) ( )( ) [ ]ηωγωτω i1GG* +′==

Compact Stress-Strain Relationfor Viscoelastic Materials

( ) ( ) ( )ωγωωτ *G=

ℜ

ℑ( )ω*G

( )ωG ′

( )ωG ′′δ

Frequency-Domain Stress-Strain Relation


Dependence of Storage and Loss Moduli on Temperature and Frequency for Typical Viscoelastic Damper

Frequency (Hz)

Stor

age

or L

oss

Mod

ulus

(MPa

)

0

1

2

3

0 1 2 3 4 5

Increasing Temperature


Dependence of Loss Factor on Temperature and Frequency for Typical Viscoelastic Damper

Frequency (Hz)

Loss

Fac

tor

0

0.5

1.0

1.5

0 1 2 3 4 5

Increasing Temperature


Actual Hysteretic Behavior of Viscoelastic Damper

Seismic Loading

Harmonic Loading


Advantages of Viscoelastic Dampers

• High reliability

• May be able to use linear analysis

• Somewhat lower cost


Disadvantages of Viscoelastic Dampers

• Strong Temperature Dependence

• Lower Force and Displacement Capacity

• Not Force Limited• Necessity for nonlinear analysis in mostpractical cases (as it has been shown that it isgenerally not possible to add enough damping to eliminate all inelastic response)


Outline: Part II

• Velocity-Dependent Damping Systems:Fluid Dampers and Viscoelastic Dampers

• Models for Velocity-Dependent Dampers• Effects of Linkage Flexibility• Displacement-Dependent Damping





Modeling Viscous Dampers:Simple Dashpot

Useful For :Fluid Dampers with Zero Storage Stiffness

This Model Ignores Temperature Dependence

( )tuC)t(P D &=

u(t)

P(t)DC

Newtonian Dashpot


Useful For :Viscoelastic Dampers and Fluid Dampers with Storage Stiffness and Weak Frequency Dependence.

This Model Ignores Temperature Dependence

Modeling Linear Viscous/Viscoelastic Dampers: Kelvin Model

DC

( ) ( )tuCtuK)t(P DD &+=

Newtonian Dashpot

DKu(t)

P(t)

Hookean Spring


Kelvin Model (Continued)


( )[ ] D*

S KK)(K =ℜ= ωωStorage Stiffness:

Loss Stiffness:CD

)(C ω

ω

KD

)(K S ω

( ) ( )tuCtuK)t(P dD &+=

[ ] ( )ωωω uCiK)(P dD +=

Complex Stiffness:

dD* CiK)(K ωω +=

( )[ ] ωωω D*

L CK)(K =ℑ=Damping Coefficient:

( )D

L CK)(C ==ω

ωωPassive Energy Dissipation 15 – 6 - 72Instructional Material Complementing FEMA 451, Design Examples

Kelvin Model (Continued)

Equivalent Kelvin Modelu(t)

P(t)DC)(C =ω

DS K)(K =ω

Kelvin Model

DK u(t)

P(t)DC


Useful For :Viscoelastic Dampers and Fluid Dampers with StrongFrequency Dependence. This Model Ignores Temperature Dependence

Modeling Linear Viscous/ViscoelasticDampers: Maxwell Model

CD KD u(t)

P(t)

( ) ( )tuCtPKC)t(P D

D

D && =+Newtonian Dashpot;Hookean Spring


Maxwell Model (Continued)

DD KC=λRelaxation Time:

CD

)(C ω

ω

KD

)(K S ω( ) ( )tuCtPKC)t(P d

D

D && =+


( ) ( )ωωωωω uCiPKCi)(P d

D

D =+

Complex Stiffness:

22D

22

2D*

1Ci

1C)(K

ωλω

ωλωλω

++

+=

( )[ ] 22

22D*

S 1KK)(K

ωλωλωω

+=ℜ=

Storage Stiffness:

Loss Stiffness:Damping Coefficient:

( )22

DS

1CK)(C

ωλωωω

+==( )[ ] 22

D*L 1

CK)(Kωλ

ωωω+

=ℑ=


Maxwell Model Parameters from Experimental Testing of Fluid Viscous Damper

0

2000

4000

6000

8000

10000

0 5 10 15 20 25

Excitation Frequency, Hz.

Stor

age

Stiff

ness

, lbs

/inch

Stor

age

Stiff

ness

(lb/

in)


KD


Maxwell Model Parameters from Experimental Testing of Fluid Viscous Damper

0

20

40

60

80

100

120

140

0 5 10 15 20 25

Dam

ping

Coe

ffici

ent,

lb-s

ec/in


CD


Note: - If KD is very large, λ is very small, KS is small and C = CD

- If CD is very small, λ is very small, KS is small and C = CD

- If KD is very small, λ is very large, C is smalland KS = KD. KD

CD

Maxwell Model (Continued)

CD KD u(t)

P(t)Maxwell Model

Equivalent Kelvin Modelu(t)

P(t)

22D

1C)(C

ωλω

+=

22

22D

S 1K)(K

ωλωλω

+=




Systems: Steel Plate Dampers,UnbondedBrace Dampers, and Friction Dampers




Because the damper is alwaysin series with the linkage, thedamper-brace assembly actslike a Maxwell model.

Hence, the effectiveness of the damper is reduced. The degree of lost effectiveness is a function of the structural properties and the loading frequency.

Effect of Linkage Flexibility on Damper Effectiveness

θ

θ2, cos2

LAEK EffectiveBrace =








Steel Plate Dampers(Added Damping and Stiffness System - ADAS)

L

a

L

t

b

V

V


Implementation of ADAS System

Wells Fargo Bank,San Francisco, CA- Seismic Retrofit of Two-Story Nonductile Concrete Frame; Constructed in 1967

- 7 Dampers Within Chevron Bracing Installed in 1992

- Yield Force Per Damper: 150 kips


ADAS Device(Tsai et al. 1993)

Experimental Response (Static)(Source: Tsai et al. 1993)

Hysteretic Behavior of ADAS Device


F k D F Zy= + −β β( )1

( )⎪⎩

⎪⎨⎧ >⋅−=

α

otherwiseif 01 ZD

DZD

FkZ

y

&

&

&&

Ideal Hysteretic Behavior of ADAS Damper

Initial Stiffness Secondary StiffnessRatio

Yield Sharpness

Z is a Path Dependency Parameter

(SAP2000 and ETABS Implementation)

Displacement, D.

Forc

e, F

Fy


Parameters of Mathematical Modelof ADAS Damper

L

a

L

t

b

V

V

( )3

b

LEIb/a2nk +

=

L4btnf

F3

yy =

=n Number of plates

=yf Yield force of each plate

=bI Second moment of area of each plate at b (i.e, at top of plate)


Unbonded Brace Damper

Stiff Shell PreventsBuckling of Core

Steel Brace (yielding core)(coated with debonding chemicals)

Concrete


Implementation of Unbonded Brace Damper

Plant and Environmental Sciences Replacement Facility

- New Three-Story Building on UC Davis Campus

- First Building in USA to Use Unbonded Brace Damper

- 132 Unbonded Braced Frames with Diagonal or Chevron Brace Installation

- Cost of Dampers = 0.5% of Building Cost

Source: ASCE Civil Engineering Magazine, March 2000.


Hysteretic Behavior of Unbonded Brace Damper


Testing of Unbonded Brace Damper

Testing Performedat UC Berkeley

Typical Hysteresis Loops from

Cyclic Testing


Advantages of ADAS Systemand Unbonded Brace Damper

•Force-Limited

•Easy to construct

•Relatively Inexpensive

•Adds both “Damping” and Stiffness


Disadvantages of ADAS Systemand Unbonded Brace Damper

• Must be Replaced after Major Earthquake

• Highly Nonlinear Behavior

• Adds Stiffness to System

• Undesirable Residual Deformations Possible


Friction Dampers: Slotted-Bolted Damper

Pall Friction Damper


Sumitomo Friction Damper(Sumitomo Metal Industries, Japan)


Pall Cross-Bracing Friction Damper

Interior of Webster Library at Concordia University, Montreal,

Canada


Implementation of Pall Friction Damper

McConnel Library at Concordia University, Montreal, Canada- Two Interconnected Buildings of 6 and 10 Stories

- RC Frames with Flat Slabs- 143 Cross-Bracing Friction

Dampers Installed in 1987- 60 Dampers Exposed for Aesthetics


Hysteretic Behavior of Slotted-Bolted Friction Damper


-60

-40

-20

0

20

40

60

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2


Forc

e, K

ips

μN

− μN

u0−u0

F N u tu tD = μ&( )&( )

Normal Force

Coefficient of Friction

Ideal Hysteretic Behavior of Friction Damper

( )[ ]tusgnNFD &μ=Alternatively,

1

-1

sgn(x)

x


Advantages of Friction Dampers

• Force-Limited

• Easy to construct

• Relatively Inexpensive


Disadvantages of Friction Dampers

• May be Difficult to Maintain over Time

• Highly Nonlinear Behavior

• Adds Large Initial Stiffness to System

• Undesirable Residual Deformations Possible








uDEH

ES

EH

ES

S

HH E

Eπ

ξ4

=

Es based on secant stiffness

FD

Note: Computed damping ratio is displacement-dependent

Equivalent Viscous Damping:Damping System with Inelastic or Friction Behavior


Effect of Inelastic System Post-Yielding Stiffness on Equivalent Viscous Damping

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9Ductility Demand

Equi

vale

nt H

yste

retic

Dam

ping

Rat

io

0.000.050.100.150.200.250.30

α

μ

K

αK

uy μuy

)1()1)(1(2

αμαπμαμξ

−+−−

=H

Note: May be Modified (κ) for Other (less Robust)Hysteretic Behavior

α = 0

α = 0.3

FD

u


uDED

ES

Es and ω are based on Secant Stiffness of Inelastic System

Equivalent Viscous Damping:“Equivalent” System with Linear Viscous Damper

C

HmC ωξ2=

FD


• It is not possible, on a device level, to “replace” adisplacement-dependent device (e.g. a Friction Damper)with a velocity-dependent device (e.g. a Fluid Damper).

• Some simplified procedures allow such replacement ona structural level, wherein a “smeared” equivalent viscousdamping ratio is found for the whole structure. Thisapproach is marginally useful for preliminary design, andshould not be used under any circumstances for final design.

Equivalent Viscous Damping:Caution!








Modeling Considerations for Structures with Passive Energy Dissipation Devices

• Damping is almost always nonclassical(Damping matrix is not proportional to stiffnessand/or mass)

• For seismic applications, system responseis usually partially inelastic

• For seismic applications, viscous damper behavioris typically nonlinear (velocity exponents in therange of 0.5 to 0.8)

Conclusion: This is a NONLINEAR analysis problem!


Outline: Part III






Seismic Analysis of Structures with Passive Energy Dissipation Systems

Modal Response Spectrum Analysisor

(Modal) Response-History Analysis

Nonlinear Response-HistoryAnalysis

Complex Modal Response Spectrum Analysis

or(Complex Modal) Response-History

Analysis

LinearBehavior?

ClassicalDamping?

Yes

Yes(implies viscousor viscoelasticbehavior)

No

No


)t(vMR)t(F)t(vC)t(vC)t(vM gSAI &&&&&& −=+++

Seismic Analysis of MDOF Structureswith Passive Energy Dissipation Systems

Inherent Damping:Linear

Added Viscous Damping:Linear or Nonlinear

Restoring Force:(May include Added Devices)Linear or Nonlinear( )AC f≠ ω



MDOF Solution Techniques

Explicit integration of fully coupled equations:

• Treat CI as Rayleigh damping and model CAexplicitly.

• Use Newmark solver (with iteration) to solve fullset of coupled equations.

System may be linear or nonlinear.



MDOF Solution Techniques

Fast Nonlinear Analysis:Treat CI as modal damping and model CAexplicitly. Move CA (and any other nonlinearterms) to right-hand side. Left-hand side maybe uncoupled by Ritz Vectors. Iterate onunbalanced right-hand side forces.

System may be linear or nonlinear.


Fast Nonlinear Analysis)t(vMR)t(F)t(vK)t(vC)t(vC)t(vM gHEAI &&&&&& −=++++

)()( tytv Φ=

)t(vC)t(F)t(vMR)t(vK)t(vC)t(vM AHgEI &&&&&& −−−=++

Transform Coordinates:Orthogonal basis of Ritz vectors:Number of vectors << N

)t(yC~)t(F)t(vMR)t(yK~)t(yC~)t(yM~ AHT

gT

EI &&&&&& −−−=++ ΦΦ

Uncoupled Coupled

Move Added Damper Forces and Nonlinear Forces to RHS:

Apply Transformation:

Nonlinear TermsLinear Terms

Nonlinear Restoring Force


Rel

ativ

e So

lutio

n Sp

eed:

FNA

/Ful

l Int

egra

tion

Nonlinear DOF / Total DOF

1.0

1.0

FAST SLOWFast Nonlinear Analysis

0.0

10.0


)t(vMR)t(F)t(vC)t(vC)t(vM gSAI &&&&&& −=+++MDOF Solution Techniques

• Treat CI as modal damping or Rayleigh damping

• Use Modal Strain Energy method to represent CAas modal damping ratios.

System must be linear.Applicable only to viscous (or viscoelastic)damping systems.

Explicit integration or response spectrumanalysis of first few uncoupled modal equations:


Outline: Part III






Modal Damping Ratios

gvMRKvvCvM &&&&& −=++

yv Φ=

giiiiiii vyyy &&&&& Γ=++ 22 ωωξ

Specify modal damping values directly


MM

2MC

n

1ii

Ti

iTi

ii

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∑

=

φφφφ

ωξ

Modal Superposition Damping

Artificial Coupling

Skyhook

Note: There is no need to develop C explicitly.


KMCR βα +=

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30

Frequency, radians/second

Dam

ping

Rat

io

MASSSTIFFNESSCOMBINED

Rayleigh Proportional Damping

Skyhook


φ 2,1

φ 2,2

φ 2,3

φ 2,4

φ 2,1 - φ 2,2

φ 2,2 - φ 2,3

φ 2,3 - φ 2,4

φ 2,4

FloorDisplacement

DamperDeformationDeformation of Structure

in its Second Mode

Derivation of Modal Strain Energy Method


21k,ik,ikikstory,i,D )(CE −−= φφπω

iTi

2ii

Tii,S M

21K

21E φφωφφ ==

i,S

storiesn

1kkstory,i,D

i E4

E

πξ

∑==



21

1

2

N stories

k i ,k i ,kk

i Ti i i

C ( )

M

−=

φ − φξ =

ω φ φ

∑


2 2

T Ti A i i A i

i T *i i i i i

C CM m

φ φ φ φξ = =

ω φ φ ω

Note: IF CA is diagonalized by Φ,THEN

ii

ii m

cω

ξ *

*

2=


Modal Strain Energy Damping Ratio

ii

iATi

i mC

ωφφξ *2

=

Note: φ is the Undamped Mode Shape

The Modal Strain Energy Method is approximate ifthe structure is non-classically damped since theundamped and damped mode shapes are different.


Outline: Part III






m = 1.0 k-sec2/in.

m = 1.5

m = 1.5

m = 1.5

m = 1.5

k = 200 k/in.

k = 250

k = 300

k = 350

k = 400

c = 10.0 k-sec/in.

c = 12.5

c = 15.0

c = 17.5

c = 20.0

Example: Application of Modal Strain Energy MethodProportional Damping


Frequency Damping(rad/sec) Ratio, ξ

4.54 0.11312.1 0.30218.5 0.46223.0 0.57527.6 0.690

Modal Damping Ratios from Modal Strain EnergyMethod for Proportional Damping Distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30

Frequency, rad/sec

Dam

ping

Rat

io

Proportional


m = 1.0 k-sec2/in.

m = 1.5

m = 1.5

m = 1.5

m = 1.5

k = 200 k/in.

k = 250

k = 300

k = 350

k = 400

c = 10 k-sec/in.

c = 10

c = 10

c = 20

c = 30

Nearly Proportional Damping

Example: Application of Modal Strain Energy Method



4.54 0.12312.1 0.31818.5 0.45523.0 0.55727.6 0.702

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30

Frequency, rad/sec

Dam

ping

Rat

io

Proportional

Nearly Proportional

Modal Damping Ratios from Modal Strain EnergyMethod for Nearly Proportional Damping Distribution


m = 1.0 k-sec2/in.

m = 1.5

m = 1.5

m = 1.5

m = 1.5

k = 200 k/in.

k = 250

k = 300

k = 350

k = 400

c = 20 k-sec/in

c = 30

Nonproportional Damping

Example: Application of Modal Strain Energy Method


Frequency Damping(rad/sec) Ratio. ξ

4.54 0.08912.1 0.14418.5 0.13423.0 0.19427.6 0.514

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30

Frequency, rad/sec

Dam

ping

Rat

io

ProportionalNearly ProportionalNonproportional

Modal Damping Ratios from Modal Strain EnergyMethod for Nonproportional Damping Distribution


MM

MCn

ii

Ti

iTi

ii

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∑

=1

2 φφφφ

ωξ

Modal Superposition Damping

Artificial Coupling

Skyhook

Modal Superposition Damping can be used to construct the damping matrix from the modal damping ratios obtained via the Modal Strain Energy Method


⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−−

−

=

53751700051753201500

0015527512000512522010000010010

.....

......

..

C

Comparison of Actual Damping Matrix and Damping Matrix Obtained from MSE Damping Ratios

Actual Damping Matrix

Modal Superposition DampingMatrix Using MSE Damping Ratios

Proportional Damping

c = 10.0 k-sec/in.

c = 12.5

c = 15.0

c = 17.5

c = 20.0

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−−

−

=

5.375.170005.175.320.1500

00.155.275.120005.125.220.100000.100.10

AC


c = 10.0 k-sec/in.

c = 10.0

c = 10.0

c = 20.0

c = 30.0

Nearly Proportional Damping

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−

−−−−

−

=

0.500.200000.200.300.1000

00.100.200.100000.100.200.100000.100.10

AC

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−−−

−−−−−−−−−−−−

=

53781773104220010081713311516902280

7310115327212166042201690212022669010022801660669010

..........

.....

.....

.....

C





c = 20.0 k-sec/in.

c = 30.0

Nonproportional Damping

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

=

0.500.200000.200.20000

000000000000000

AC

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−

−−−−−

−−

=

92011521607206601159219107220981

2169104139254560072722925278962

066009814560962653

..........

..........

.....

C





Example: Seismic Analysis of a Structurewith Nonproportional Damping

• Discrete Damping vs Rayleigh Damping• Discrete Damping: Rigid vs Flexible Braces


4.54 0.08912.1 0.14418.5 0.13423.0 0.19427.6 0.514

MSE Results

c = 20.0 k-sec/in.

c = 30.0Damping ratios in modes 1 and 4 usedto construct Rayleigh damping matrix.


Computing Rayleigh Damping Proportionality Factors (Using NONLIN Pro)


0

2

4

6

8

10

12

14

0.0 5.0 10.0 15.0 20.0

First Mode Damping, % Critical

Pea

k R

oof D

ispl

acem

ent,

Inch

es

Discrete DampersRayleigh Damping

Example: Discrete (Stiff Braces) vs Rayleigh Damping

c

1.5c

Very Stiff Braces


0

200

400

600

800

1000

1200

1400

1600

0.0 5.0 10.0 15.0 20.0


Pea

k B

ase

She

ar, K

ips

Discrete DampersRayleigh Damping: InertialRayleigh Damping: Columns

Example: Discrete (Stiff Braces) vs Rayleigh Damping

Very Stiff Braces

c

1.5c


0

2

4

6

8

10

12

14

0.0 5.0 10.0 15.0 20.0


Pea

k R

oof D

ispl

acem

ent,

Inch

es

Very Stiff BracesReasonably Stiff Braces

Example: Effect of Brace Stiffness(Discrete Damping Model)

c

1.5c


0

200

400

600

800

1000

1200

1400

1600

0.0 5.0 10.0 15.0 20.0


Pea

k In

ertia

l For

ce, K

ips

Very Stiff BracesReasonably Stiff Braces

Example: Effect of Brace Stiffness (Discrete Damping Model)

Pea

k B

ase

She

ar, K

ips

(from

Iner

tial F

orce

s)

c

1.5c


0

2

4

6

8

10

12

14

0.0 5.0 10.0 15.0 20.0


Pea

k R

oof D

ispl

acem

ent,

Inch

es

Discrete Model with Flexible BracesRayleigh Damping

Example: Discrete (Flexible Braces) vs Rayleigh Damping

c

1.5c


0

200

400

600

800

1000

1200

1400

1600

0.0 5.0 10.0 15.0 20.0


Peak

Iner

tial F

orce

r, Ki

ps

Discrete Model with Flexible Braces

Rayleigh Damping

Pea

k B

ase

She

ar, K

ips

(from

Iner

tial F

orce

s)

Example: Discrete (Flexible Braces) vs Rayleigh Damping

c

1.5c


-10 0 0-8 0 0-6 0 0-4 0 0-2 0 0

02 0 04 0 06 0 08 0 0

10 0 0

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20T im e , S e co n d s

Shea

r, K

ips

C o lu m n sB rac e s (D a m p e r) STIFF BRACES

-1 0 0 0-8 0 0-6 0 0-4 0 0-2 0 0

02 0 04 0 06 0 08 0 0

1 0 0 0

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0T im e , S e c o n d s

Shea

r, K

ips

C o lu m n sB ra c e s (D a m p e r) FLEX BRACES

Example: Effect of Brace Stiffness on Peak Story Shear Forces


Outline: Part III






Summary: MDOF Analysis Procedures(Linear Systems and Linear Dampers)

• Use discrete damper elements and explicitly include thesedampers in the system damping matrix. Perform responsehistory analysis of full system. Preferred.

• Use discrete damper elements to estimate modal dampingratios and use these damping ratios in modal response history or modal response spectrum analysis. Dangerous.

• Use discrete damper elements to estimate modal dampingratios and use these damping ratios in a response historyanalysis which incorporates Rayleigh proportionaldamping. Dangerous.


Summary: MDOF Analysis Procedures(Linear Systems with Nonlinear Dampers)

• Use discrete damper elements and explicitly include thesedampers in the system damping matrix. Perform responsehistory analysis of full system. Preferred.

• Replace nonlinear dampers with “equivalent energy”based linear dampers, and then use these equivalentdampers in the system damping matrix. Perform responsehistory analysis of full system. Very Dangerous.

• Replace nonlinear dampers with “equivalent energy”based linear dampers, use modal strain energy approachto estimate modal damping ratios, and then performresponse spectrum or modal response history analysis.Very Dangerous.


Summary: MDOF Analysis Procedures(Nonlinear Systems with Nonlinear Dampers)

• Use discrete damper elements and explicitly include thesedampers in the system damping matrix. Explicitly modelinelastic behavior in superstructure. Perform response historyanalysis of full system. Preferred.

• Replace nonlinear dampers with “equivalent energy”based linear dampers and use modal strain energy approachto estimate modal damping ratios. Use pushover analysisto represent inelastic behavior in superstructure. Usecapacity-demand spectrum approach to estimate systemdeformations. Do This at Your Own Risk!


Outline: Part IV• MDOF Solution Using Complex Modal

Analysis• Example: Damped Mode Shapes and

Frequencies• An Unexpected Effect of Passive Damping• Modeling Dampers in Computer Software• Guidelines and Code-Related Documents




MDOF Solution for Non-Classically Damped Structures Using Complex Modal Analysis

Modal Analysis using Damped Mode Shapes:

• Treat CI as modal damping and model CAexplicitly.

• Solve by modal superposition using damped(complex) mode shapes and frequencies.

System (dampers and structure) must be linear.


Damped Eigenproblem

State Vector:

0)t(Kv)t(vC)t(vM A =++ &&&

Linear Structure

EOM for Damped Free Vibration

⎭⎬⎫

⎩⎨⎧

=vv

Z&

HZZ =&State-SpaceTransformation:

⎥⎦

⎤⎢⎣

⎡ −−=

−−

0

11

IKMCM

H AState Matrix:

Assume CI is negligible


Solution of Damped Eigenproblem

P HPΛ =

*

λ⎡ ⎤Λ = ⎢ ⎥λ⎣ ⎦

ntn nZ P eλ=Assume Harmonic Response for n-th mode:

n n nP HPλ =Substitute Response intoState Space Equation:

Damped Eigenproblem for n-th Mode

Damped Eigenproblem for All Modes

Eigenvalue Matrix:(* = complex conjugate)

* *

*P⎡ ⎤Φλ Φ λ

= ⎢ ⎥Φ Φ⎣ ⎦Eigenvector Matrix:

[ ]ndiagλ = λ


Complex Eigenvaluefor Mode n:

Modal Damping Ratio:

21n n n n niλ = −ξ ω ± ω − ξ

Extracting Modal Damping and Frequencyfrom Complex Eigenvalues

( )n

nn λ

λξ ℜ−=

nn λω =Modal Frequency:

Analogous toRoots of Characteristic Equation for SDOFDamped Free VibrationProblem

1i −=Note:


⎥⎦

⎤⎢⎣

⎡

ΦΦΛΦΦΛ

=*

**

P

Extracting Damped Mode Shapes

Damped Mode Shapes


Damped Mode Shapes

Real

Imaginary

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

++++

=

44

33

22

11

ibaibaibaiba

φ

U1

U2

U3

U4


Outline: Part IV• MDOF Solution Using Complex Modal

Analysis• Example: Damped Mode Shapes and

Frequencies• An Unexpected Effect of Passive Damping• Modeling Dampers in Computer Software• Guidelines and Code-Related Documents



m = 1.0 k-sec2/in.

m = 1.5

m = 1.5

m = 1.5

m = 1.5

k = 200 k/in.

k = 250

k = 300

k = 350

k = 400

c = 20 k-sec/in

c = 30

Example: Damped Mode Shapes and FrequenciesNonproportional Damping


4.5412.118.423.027.6

0.0890.1440.1340.1940.516

12345

4.5812.318.924.025.1

0.0890.1410.0640.0270.770

12345

Frequency(rad/sec)

DampingRatio

Frequency(rad/sec)

DampingRatio

Using UNDAMPEDMODE SHAPES

Using DAMPEDMODE SHAPES*

Example: Damped Mode Shapes and FrequenciesSystem with Non-Classical Damping

*Table is for model withVERY STIFF braces.

Obtained from MSE Method Significant Differences in Higher Mode Damping Ratios


-1.2

-0.8

-0.4

0

0.4

0.8

1.2

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

Imaginary Component of Mode Shape

Rea

l Com

pone

nt o

f Mod

e S

hape

Level 1Level 2Level 3Level 4Level 5

ω = 4.58ξ = 0.089

Mode = 1



-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Time, Seconds

Mod

al A

mpl

itude Level 5

Level 4Level 3Level 3Level 1

0

1

2

3

4

5

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00Modal Displacement

Stor

y Le

vel

1 2

2

3 4

4 1 3


-1.2

-0.8

-0.4

0

0.4

0.8

1.2

-1.2 -0.8 -0.4 0 0.4 0.8 1.2

Real Component of Mode Shape

Imag

inar

y C

ompo

nent

of M

ode

Shap

e

Level 5Level 4Level 3Level 2Level 1

ω = 12.34ξ = 0.141

Mode = 2



Outline: Part IV






Huntington Tower- 111 Huntington Ave, Boston, MA- New 38-story steel-framed building- 100 Direct-acting and toggle-brace dampers - 1300 kN (292 kips), +/- 101 mm (+/- 4 in.)- Dampers suppress wind vibration

The larger the dampingcoefficient C, the smaller

the damping ratio ξ.

Why?Note: Occurs for toggle-braced systems only.

An Unexpected Effect of Passive Damping


Toggle Brace Deployment

Huntington Tower


Example: Toggle Brace Damping System

U1

U2

U3

360

150

159 65

Units: inchesPassive Energy Dissipation 15 – 6 - 164Instructional Material Complementing FEMA 451, Design Examples

Methods of Analysis Used to Determine Damping Ratio

•Energy Ratios for Steady-State Harmonic Loading: ξ = ED/4πES

•Modal Strain Energy

•Free Vibration Log Decrement

•Damped EigenproblemC =10 to 40 k-sec/in (increments of 10)A =10 to 100 in2 (increments of 10)


Computed Damping Ratios for System With A = 10

A10

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 10 20 30 40 50

Damping Constant (k-sec/in)

Dam

ping

Rat

io

Damping EnergyModal Strain EnergyLog DecrementComplex Eigenvalues

Damping Coefficient (k-sec/in)

Energy RatioModal Strain EnergyLog DecrementDamped Eigenproblem



A20

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 10 20 30 40 50


Dam

ping

Rat

ioDamping EnergyModal Strain EnergyLog DecrementComplex Eignevalues





A30

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 10 20 30 40 50


Dam

ping

Rat

io







A50

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 10 20 30 40 50Damping Constant (k-sec/in)

Dam

ping

Rat

io



Energy RatioModal Strain EnergyLog Decrement

Damped Eigenproblem


Why Does Damping Ratio Reduce for LowBrace Area/Damping Coefficient Ratios?

U1

U2

U3

UD

Displacement in Damper is Out-of-Phasewith Displacement at DOF 1


-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0 5 10 15 20 25 30 35 40 45 50

T ime, Seconds

Dis

plac

emen

t, in

.

Device Lateral

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

0 5 10 15 20 25 30 35 40 45 50

T ime, sec

Dis

plac

emen

t, in

Device Lateral

A=10C=40

A=50C=40

Phase Difference Between Damper Displacement and Frame Displacement

A/C=0.25

A/C=1.25

Damper

Damper Frame

Frame


-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

REAL

IMA

GIN

AR

Y

3

1

2

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

REAL

IMA

GIN

AR

Y

1

2

3-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

REAL

IMA

GIN

AR

Y 2

1

3

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

REAL

IMA

GIN

AR

Y 2

1

3

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

REAL

IMA

GIN

AR

Y

1

2

3

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

REAL

IMA

GIN

AR

Y

1

2

3

C=0 C=10 C=20

C=30 C=40 C=50

Damped Mode Shapes for System With A=20 in2

1

2

3

U1

U2

U3


Interim Summary Related to Modeling and Analysis (1)

• Viscously damped systems are very effective inreducing damaging deformations in structures.

• With minor exceptions, viscously damped systemsare non-classical, and must be modeled explicitlyusing dynamic time history analysis.

• Avoid the use of the Modal Strain Energy method(it may provide unconservative results)


• Damped mode shapes provide phase angle information that is essential in assessing andimproving the efficiency of viscously dampedsystems. This is particularly true for linkagesystems (e.g. toggle-braced systems).

• If damped eigenproblem analysis procedures arenot available, use overlayed response history plotsof damper displacement and interstory displacementto assess damper efficiency. (This would be required for nonlinear viscously damped systems.)

Interim Summary Related to Modeling and Analysis (2)


Outline: Part IV






Computer Software Analysis Capabilities

Linear Viscous Fluid DampersNonlinear Viscous Fluid DampersViscoelastic DampersADAS Type SystemsUnbonded Brace SystemsFriction SystemsGeneral System Yielding

SAP2000;ETABS DRAIN

RAMPerform

YesYesYesYesYes

YesNOYesYesYes

YesYes*YesYesYes

Yes Yes YesPending Yes Yes

*Piecewise Linear


Use a Type-1 truss bar element with stiffness proportional damping:

LAEK = KC β=

For dampers with low stiffness:Set A = L, E = 0.01 andβ = CDamper/0.01

Modeling Linear Viscous Dampers in DRAIN

ij

kj k

LDamper

Result:01.0K = DamperCC =

uCuKuCF Damper &&& === β


Modeling Linear Viscous Dampers in DRAIN

i j km

Dampers may be similarly modeled usingthe zero-length “Type-4”connection element.

Nodes j and m have the same coordinates


CD KD

i k

Modeling Viscous/Viscoelastic DampersUsing the SAP2000 NLLINK Element

The damper is modeled as a Maxwell Element consisting of a linear or nonlinear dashpot in series with a linear spring.

To model a linear viscous dashpot, KD must be set to a large value, but not too large or convergence will not be achieved. To achieve this, it is recommended that the relaxation time (λ = CD/KD) be an order of magnitude less than the loading time step Δt. For example, let KD = 100CD/Δt. Sensitivity to KD should be checked.

SAP2000 often has difficulty converging when nonlinear dampersare used and the velocity exponent is less than 0.4.


Modeling ADAS, Unbonded Brace, and Friction Dampers using the SAP2000 NLLINK Element

ZFkDF y)1( ββ −+=

( )⎪⎩

⎪⎨⎧ >−=

otherwiseD0ZDifZ1D

FkZ

y &

&&&

α

For bilinear behavior, use α of approximately 50. Larger values canproduce strange results.

k

βk

Fy

α=50α=4α=2

F

DNote: Z is an internal hysteretic variable with magnitude less than or equal to unity. The yield surface is associated with a magnitude of unity.


Outline: Part IV






1993 - Tentative General Requirements for the Design and Construction of Structures Incorporating Discrete Passive

Energy Dissipation Devices (1 of 3)

- Draft version developed by Energy Dissipation Working Group (EDWG) of Base Isolation Subcommittee of Seismology Committee of SEAONC (Not reviewed/approved by SEAOC; used as basis for 1994 NEHRP Provisions)

- Philosophy: For Design Basis Earthquake (10/50), confine inelastic behavior to energy dissipation devices (EDD); gravity load resisting system to remain elastic

- Established terminology and nomenclature for energy dissipation systems (EDS)

- Classified systems as rate-independent or rate-dependent (included metallic, friction, viscoelastic, and viscous dampers)

- Required at least two vertical lines of dampers in each principal direction of building; dampers to be continuous from the base of the building

- Prescribed analysis and testing procedures


EnergyDissipationDevice (EDD)

Energy Dissipation Nomenclature

EnergyDissipationAssembly (EDA)




- Elastic structures with rate-dependent devices: Linear dynamic procedures (response spectrum or response history analysis)

- Inelastic structures or structures with rate-independent devices: Nonlinear dynamic response history analysis

- Prototype tests on full-size specimens (not required if previous tests performed and documented by ICBO)

- General acceptability criteria for energy dissipation systems:- Remain stable at design displacements- Provide non-decreasing resistance with increasing displacement

(for rate-independent systems)- Exhibit no degradation under repeated cyclic load at design displ.- Have quantifiable engineering parameters

- Independent engineering review panel required to oversee design and testing




- Includes Appendix to Chapter 2 entitled: Passive Energy Dissipation Systems

- Material is based on:- 1993 draft SEAONC EDWG document- Proceedings of ATC 17-1 Seminar on Seismic Isolation, Passive

Energy Dissipation, and Active Control (March 1993)- Special issue of Earthquake Spectra (August 1993)

- Applicable to wide range of EDD’s; therefore requires EDD performance verificationvia prototype testing

- Performance objective identical to conventional structural system (i.e., life-safety for design EQ)

- At least two EDD per story in each principal direction, distributed continuouslyfrom base to top of building unless adequate performance (drift limits satisfied and member curvature capacities not exceeded) with incomplete verticaldistribution can be demonstrated

- Members that transmit damper forces to foundation designed to remain elastic

1994 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (1 of 4)

Part 1 – Provisions & Part 2 – Commentary (FEMA 222A & 223A)


WBCBVV Smin ==

V = Minimum base shear for design of structure without EDSB = Reduction factor to account for energy dissipation provided by EDS(based on combined, inherent plus added damping, damping ratio)

Vmin = Minimum base shear for design of structure with EDS [Use for linear static (ELF) or linear dynamic (Modal) analysis]

Analysis/Design Procedure for Linear Viscous Energy Dissipation Systems

Note: After publication, it was recognized that this procedure may not be appropriate since it allows reduction in forces due to bothinelastic structural response (R-factor) and added damping(B-factor). For yielding structures, added damping will not reduce forces.0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35

Combined Damping Ratio (%)

Red

uctio

n Fa

ctor

, B

Non

linea

r Dyn

amic

Ana

lysi

s




Analysis/Design Procedure for EDD’s other than Linear Viscous Dampers

−+

−+

+

+=

ΔΔDDD FF

k eff

22D

S

Dneqneq 2

TWW4

Wm2m2cΔππ

ωξω ===

Eq. (C2A.3.2.1a)Effective Device Stiffness at Design Displacement

Eq. (2A.3.2.1)Equivalent Device Damping Coefficient

Forc

e

Deformation

+Δ

−Δ+

DF−

DF

Slope = effDk

2) Performance Verification: Nonlinear response history analysis

EDD Behavior

1) Preliminary Design: Linear dynamic modal analysis using effective stiffness and damping coefficient of energy dissipation devices. Use B-factor toreduce modal base shears.

Area = DWSE4WD

strcombined πξξ ∑+=

Eq. (C2A.3.2.1c)Combined Equivalent Damping Ratio




- For nonlinear response-history analysis, mathematical modeling should account for:- Plan and vertical spatial distribution of EDD’s- Dependence of EDD’s on loading frequency, temperature, sustained loads,

nonlinearities, and bilateral loads

- Prototype Tests on at least two full-size EDD’s(unless prior testing has been documented)

- 200 fully reversed cycles corresponding to wind forces- 50 fully reversed cycles corresponding to design earthquake- 10 fully reversed cycles corresponding to maximum capable earthquake

- Acceptability criteria from prototype testing of EDD’s:- Hysteresis loops have non-negative incremental force-carrying capacities

(for rate-independent systems only)- Exhibit limited effective stiffness degradation under repeated cyclic load- Exhibit limited degradation in energy loss per cycle under repeated cyclic load- Have quantifiable engineering parameters- Remain stable at design displacements




- Includes an appendix to Chapter 13 entitled:Passive Energy Dissipation

- The appendix in the 1994 NEHRP Provisions was deleted since it was deemed to be insufficient for designand regulation. It was replaced with 3 paragraphs thatprovide very general guidance on passive energydissipation systems.

1997 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures

Part 1 – Provisions & Part 2 – Commentary (FEMA 302 & 303)


- Chapter 9 entitled: Seismic Isolation and Energy Dissipation(Developed by New Technologies Team under ATC Project 33)

- Performance-based document- Rehabilitation objectives based on desired performance levels for selected hazard levels

- Global Structural Performance Levels- Operational (OP)- Immediate Occupancy (IO)- Life-Safety (LS)- Collapse Prevention (CP)

- Hazard levels- Basic Safety Earthquake 1 (BSE-1): 10/50 event- Basic Safety Earthquake 2 (BSE-2): 2/50 event (Maximum Considered EQ - MCE)

- Rehabilitation Objectives- Limited Objectives (less than BSO) - Basic Safety Objective (BSO): LS for BSE-1 and CP for BSE-2- Enhanced Objectives (more than BSO)

1997 - NEHRP Guidelines for the Seismic Rehabilitation of Buildings (FEMA 273)1997 - NEHRP Commentary on the Guidelines for the Seismic Rehabilitation of

Buildings (FEMA 274) (1 of 9)

Most ApplicablePerformance Levels

ApplicableRehabilitationObjectives


-Simplified vs. Systematic Rehabilitation- Simplified: For simple structures in areas of low to moderate seismicity- Systematic: Considers all elements needed to attain rehabilitation objective

- Systematic Rehabilitation methods of analysis:- Linear static procedure (LSP)- Linear dynamic procedure (LDP)- Nonlinear static procedure (NSP)- Nonlinear dynamic procedure (NDP)



Coefficient Method

Capacity Spectrum Method



Buildings (FEMA 274) (3 of 9)• Basic Principles:

– Dampers should be spatially distributed (at each story and on each side of building)

– Redundancy (at least two dampers along the same line of action; design forces for

dampers and damper framing system are reduced as damper redundancy is increased)

– For BSE-2, dampers and their connections designed to avoid failure (i.e, not weak link)

– Members that transmit damper forces to foundation designed to remain elastic

• Classification of EDD’s– Displacement-dependent– Velocity-dependent– Other (e.g., shape memory alloys and fluid restoring force/damping

dampers)

Manufacturing quality control program should be established along with prototype testing programs and independent panel review of system design and testing program


Mathematical Modeling of Displacement-Dependent Devices

Eq. (9-20)Force in Device



DkF eff=

−+

−+

+

+=

DD

FFkeff

Eq. (9-21)Effective Stiffnessof Device

−D

−F

Displacement, D

Forc

e, F

+D

+F

Slope = effk

2aveeff

Deff Dk

W21π

β =Eq. (9-39)Equivalent ViscousDamping Ratio ofDevice

Area = DW


Mathematical Modeling of Solid Viscoelastic Devices

Eq. (9-22)Force in Device



DCDkF eff&+=

KDD

FFkeff ′=

+

+=

−+

−+Eq. (9-23)Effective Stiffnessof Device

Forc

e

Deformation

+D

−D+F

−F

Slope = effk

EDD Behavior

Area = DW

Storage Stiffness

12ave1

D KD

WCωπω

′′==

Eq. (9-24)Damping Coefficientof Device

Loss Stiffness

Average Peak Displ. Circular frequency of mode 1


Mathematical Modeling of Fluid Viscoelastic and Fluid Viscous Devices

Maxwell Model



DCFF && =+ λ

Fluid Viscoelastic Devices:

Eq. (9-25)Linear or Nonlinear Dashpot Model( )DsgnDCF 0

&& α=

Fluid Viscous Devices:

Caution: Only use fluid viscous device model if = 0 for frequenciesbetween 0.5 f1 and 2.0 f1; Otherwise, use fluid viscoelastic device model.

K ′


Pushover Analysis for Structures with EDD’s (Part of NSP)



No dampers

With ADAS Dampers

Bas

e Sh

ear

Roof Displacement

Base Shear

Roof Displ.

No dampers

With FrictionDampers

Bas

e Sh

ear

Roof Displacement

No dampers

With Viscoelastic Dampers

Bas

e Sh

ear

Roof Displacement

No dampers

With Viscous Dampers

Bas

e Sh

ear

Roof Displacement

Performance point without dampersPerformance point with dampers

Reduced Displacement Reduced DamagePassive Energy Dissipation 15 – 6 - 196Instructional Material Complementing FEMA 451, Design Examples

Design Process for Velocity-Dependent Dampers using NSPSteps1) Estimate Target Displacement (performance point)2) Calculate Effective Damping Ratio and Secant Stiffness of building with dampers

at Target Displacement3) Use Effective Damping and Secant Stiffness to calculate revised Target Displacement4) Compare Target Displacement from Steps 1 and 4.

If within tolerance, stop. Otherwise, return to Step 1.


Buildings (FEMA 274) 8 of 9)

k

jj

eff W4

W

πββ

∑+=

Effective damping ratio of building with dampers at Target Displ.;j = index over devices

∑=i

iik F21W δ Maximum strain energy in building with dampers at Target Displ.;

i = index over floor levels


Design Process for Velocity-Dependent Dampers using NSP (2)



2rjj

S

2

j CT

2W δπ=

Work done by j-th damper with buildingsubjected to Target Displacement(assumes harmonic motion with amplitude equal toTarget Displacement and frequency correspondingto Secant Stiffness at Target Displacement)

∑∑

+=

i

2ii

j

2rjj

2jS

eff m4

cosCT

φπ

φθββ

Alternate expression for Effective Damping Ratiothat uses modal amplitudes of first mode shape

Checking Building Component Behavior (Forces and Deformations)

For velocity-dependent dampers, must check component behavior at three stages:1) Maximum Displacement2) Maximum Velocity3) Maximum Acceleration


2000 – Prestandard and Commentary for the Seismic Rehabilitation of Buildings (FEMA 356)

• Prestandard version of 1997 NEHRP Guidelines and Commentary for the Seismic Rehabilitation of Buildings (FEMA 273 & 274)

• Prepared by ASCE for FEMA

• Prestandard = Document has been accepted for use as the start of the formal standard development process (i.e., it is an initial draft for a consensus standard)


- Appendix to Chapter 13 entitled Structures with Damping Systems(completely revised/updated version of 1994 and 1997 Provisions; Brief commentary provided)

- Intention:- Apply to all energy dissipation systems (EDS)- Provide design criteria compatible with conventional

and enhanced seismic performance- Distinguish between design of members that are part of EDS and members that are independent of EDS.

-The seismic force resisting system must comply with the requirements for the system’s Seismic Design Category, except that the damping system may be used to meet drift limits.

No reduction in detailing is thereby allowed,even if analysis shows that the damping systemis capable of producing significant reductions inductility demand or damage.




- Members that transmit damper forces to foundation designed to remain elastic

- Prototype tests on at least two full-size EDD’s(reduced-scale tests permitted for velocity-dependent dampers)

- Production testing of dampers prior to installation.

- Independent engineering panel for review of design and testing programs

- Residual mode concept introduced for linear static analysis. This mode, which is in addition to the fundamental mode, is used to account for the combined effects of higher modes. Higher mode interstory-velocities can be significant and thus are important for velocity-dependent dampers.




Methods of Analysis:

• Linear Static (Equivalent Lateral Force*) - OK for Preliminary Design

• Linear Dynamic (Modal Response Spectrum*)- OK for Preliminary Design

• Nonlinear Static (Pushover*)- May Produce Significant Errors

• Nonlinear Dynamic (Response History)- Required if S1 > 0.6 g and may be used in all other cases

*The Provisions allow final design using these procedures, butonly under restricted circumstances.




Effective Damping Ratio (used to determine factors, B, that reduce structure response)



HVmIm βμβββ ++=

Hysteretic Damping Due to Post-Yield Behavior in Structure

Equivalent Viscous Damping ofEDS in the m-th Mode

Inherent Damping Due to Pre-Yield EnergyDissipation of Structure (βI = 5% or less unless higher values can be justified)

Effective Damping Ratio in m-th mode of vibration


Equivalent Viscous Damping from EDS



m

jmj

Vm W4

W

πβ

∑= Equivalent Viscous Damping in m-th mode

(due to EDS)

Maximum Elastic Strain Energy of structurein m-th mode∑=

iimimm F

21W δ

HVmIm βμβββ ++=

μ Adjustment factor that accounts for dominance ofpost-yielding inelastic hysteretic energy dissipation


⎭⎬⎫

⎩⎨⎧

=+

V75.0;B

VmaxVIV

min



Base Shear Force

Minimum base shear fordesign of seismic forceresisting system

To protect against damper system malfunction, maximum reductionin base shear over a conventional structure is 25%

Minimum base shear for designof structure without EDS

Spectral reduction factor based on the sum ofviscous and inherent damping


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 5 10 15 20 25 30

Total Effective Damping Ratio, %

Spec

tral R

educ

tion

Fact

or

Maximum AddedDamping WRT Minimum Base Shear= 14 - 5 = 9%

33.1VV75.0Vmin =≥

Maximum base shearreduction factor




0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20


Pseu

doac

cele

ratio

n, g

T=.50 T=1.0

T=1.5

T=2.0

T=3.0

T=4.0

5% Damping

10%

20%

30%40%

Effect of Added Viscous Damping

Decreased DisplacementDecreased Shear Force (can not take full advantage of)

