SEISMIC PROTECTIVE SYSTEMS: PASSIVE ENERGY ......Pseudo-Spectral Acceleration, g - Increased...
Transcript of SEISMIC PROTECTIVE SYSTEMS: PASSIVE ENERGY ......Pseudo-Spectral Acceleration, g - Increased...
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
Passive Energy Dissipation 15 – 6 - 2Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 3Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 4Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 5Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 6Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 7Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 8Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 9Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 10Instructional Material Complementing FEMA 451, Design Examples
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 Stiffness(Added Bracing)
- Decreased Displacement- Increased Shear Force
Pseu
do-S
pect
ral A
ccel
erat
ion,
g
Passive Energy Dissipation 15 – 6 - 11Instructional Material Complementing FEMA 451, Design Examples
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 and Stiffness(ADAS System)
- Decreased Displacement- Decreased Shear (possibly)
Pseu
do-S
pect
ral A
ccel
erat
ion,
g
Passive Energy Dissipation 15 – 6 - 12Instructional Material Complementing FEMA 451, Design Examples
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 Reduced Stiffness(Seismic Isolation)
- Decreased Shear Force- Increased Displacement
Pseu
do-S
pect
ral A
ccel
erat
ion,
g
Passive Energy Dissipation 15 – 6 - 13Instructional Material Complementing FEMA 451, Design Examples
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 Reduced Stiffness(Seismic Isolation with Dampers)
- Decreased Shear- Increased Displacement
Pseu
do-S
pect
ral A
ccel
erat
ion,
g
Passive Energy Dissipation 15 – 6 - 14Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 15Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 16Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 17Instructional Material Complementing FEMA 451, Design Examples
1981/1982 US-JAPAN PROJECTResponse of Bare Frame Before and After Adding Ballast
Model Weight
Bare Model 18 kipsLoaded Model 105 kips
Passive Energy Dissipation 15 – 6 - 18Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 19Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 20Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 21Instructional Material Complementing FEMA 451, Design Examples
( ) ( )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
Passive Energy Dissipation 15 – 6 - 22Instructional Material Complementing FEMA 451, Design Examples
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
)
Passive Energy Dissipation 15 – 6 - 23Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 24Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 25Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 26Instructional Material Complementing FEMA 451, Design Examples
0.00
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0.30
0.40
0.50
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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
Passive Energy Dissipation 15 – 6 - 27Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 28Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 29Instructional Material Complementing FEMA 451, Design Examples
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,UnbondedBrace Dampers, and Friction Dampers
• Concept of Equivalent Viscous Damping• Modeling Considerations for Structures
with Passive Damping Systems
Passive Energy Dissipation 15 – 6 - 30Instructional Material Complementing FEMA 451, Design Examples
Cross-Section of Viscous Fluid Damper
Source: Taylor Devices, Inc.
Passive Energy Dissipation 15 – 6 - 31Instructional Material Complementing FEMA 451, Design Examples
Possible Damper Placement Within Structure
AugmentedBracing
Passive Energy Dissipation 15 – 6 - 32Instructional Material Complementing FEMA 451, Design Examples
Chevron Brace and Viscous Damper
Passive Energy Dissipation 15 – 6 - 33Instructional Material Complementing FEMA 451, Design Examples
Diagonally Braced Damping System
θ
Passive Energy Dissipation 15 – 6 - 34Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 35Instructional Material Complementing FEMA 451, Design Examples
Fluid Damper within Diagonal Brace
San Francisco State Office Building
San Francisco, CA
Huntington TowerBoston, MA
Passive Energy Dissipation 15 – 6 - 36Instructional Material Complementing FEMA 451, Design Examples
Toggle Brace Damping System
θ1
θ2)cos(
sin
21
2
1 θθθ+
==UUAF D
U1
UD
Passive Energy Dissipation 15 – 6 - 37Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 38Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 39Instructional Material Complementing FEMA 451, Design Examples
)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
Passive Energy Dissipation 15 – 6 - 40Instructional Material Complementing FEMA 451, Design Examples
( ) ( )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
Passive Energy Dissipation 15 – 6 - 41Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 42Instructional Material Complementing FEMA 451, Design Examples
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
Excitation Frequency (Hz)
Passive Energy Dissipation 15 – 6 - 43Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 44Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 45Instructional Material Complementing FEMA 451, Design Examples
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=
Passive Energy Dissipation 15 – 6 - 46Instructional Material Complementing FEMA 451, Design Examples
Actual Hysteretic Behavior of Fluid Damper
Seismic Loading
Harmonic Loading
Source:Constantinou and Symans (1992)
Passive Energy Dissipation 15 – 6 - 47Instructional Material Complementing FEMA 451, Design Examples
-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=
Passive Energy Dissipation 15 – 6 - 48Instructional Material Complementing FEMA 451, Design Examples
-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=
Passive Energy Dissipation 15 – 6 - 49Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 50Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 51Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 52Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 53Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 54Instructional Material Complementing FEMA 451, Design Examples
-800
-600
-400
-200
0
200
400
600
800
-15 -10 -5 0 5 10 15
Displacement, Inches
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
Passive Energy Dissipation 15 – 6 - 55Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 56Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 57Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 58Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 59Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 60Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 61Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 62Instructional Material Complementing FEMA 451, Design Examples
( ) ( ) ωγγτ /tGtG)t( &′′+′=
Apply Fourier Transform:
( ) ( ) 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
Passive Energy Dissipation 15 – 6 - 63Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 64Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 65Instructional Material Complementing FEMA 451, Design Examples
Actual Hysteretic Behavior of Viscoelastic Damper
Seismic Loading
Harmonic Loading
Passive Energy Dissipation 15 – 6 - 66Instructional Material Complementing FEMA 451, Design Examples
Advantages of Viscoelastic Dampers
• High reliability
• May be able to use linear analysis
• Somewhat lower cost
Passive Energy Dissipation 15 – 6 - 67Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 68Instructional Material Complementing FEMA 451, Design Examples
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 Damping Systems
Passive Energy Dissipation 15 – 6 - 69Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 70Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 71Instructional Material Complementing FEMA 451, Design Examples
Kelvin Model (Continued)
Apply Fourier Transform:
( )[ ] 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
Passive Energy Dissipation 15 – 6 - 73Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 74Instructional Material Complementing FEMA 451, Design Examples
Maxwell Model (Continued)
DD KC=λRelaxation Time:
CD
)(C ω
ω
KD
)(K S ω( ) ( )tuCtPKC)t(P d
D
D && =+
Apply Fourier Transform:
( ) ( )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=
Passive Energy Dissipation 15 – 6 - 75Instructional Material Complementing FEMA 451, Design Examples
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)
Excitation Frequency (Hz)
KD
Passive Energy Dissipation 15 – 6 - 76Instructional Material Complementing FEMA 451, Design Examples
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
Excitation Frequency (Hz)
CD
Passive Energy Dissipation 15 – 6 - 77Instructional Material Complementing FEMA 451, Design Examples
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
+=
Passive Energy Dissipation 15 – 6 - 78Instructional Material Complementing FEMA 451, Design Examples
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,UnbondedBrace Dampers, and Friction Dampers
• Concept of Equivalent Viscous Damping• Modeling Considerations for Structures
with Passive Damping Systems
Passive Energy Dissipation 15 – 6 - 79Instructional Material Complementing FEMA 451, Design Examples
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 =
Passive Energy Dissipation 15 – 6 - 80Instructional Material Complementing FEMA 451, Design Examples
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 Damping Systems
Passive Energy Dissipation 15 – 6 - 81Instructional Material Complementing FEMA 451, Design Examples
Steel Plate Dampers(Added Damping and Stiffness System - ADAS)
L
a
L
t
b
V
V
Passive Energy Dissipation 15 – 6 - 82Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 83Instructional Material Complementing FEMA 451, Design Examples
ADAS Device(Tsai et al. 1993)
Experimental Response (Static)(Source: Tsai et al. 1993)
Hysteretic Behavior of ADAS Device
Passive Energy Dissipation 15 – 6 - 84Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 85Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 86Instructional Material Complementing FEMA 451, Design Examples
Unbonded Brace Damper
Stiff Shell PreventsBuckling of Core
Steel Brace (yielding core)(coated with debonding chemicals)
Concrete
Passive Energy Dissipation 15 – 6 - 87Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 88Instructional Material Complementing FEMA 451, Design Examples
Hysteretic Behavior of Unbonded Brace Damper
Passive Energy Dissipation 15 – 6 - 89Instructional Material Complementing FEMA 451, Design Examples
Testing of Unbonded Brace Damper
Testing Performedat UC Berkeley
Typical Hysteresis Loops from
Cyclic Testing
Passive Energy Dissipation 15 – 6 - 90Instructional Material Complementing FEMA 451, Design Examples
Advantages of ADAS Systemand Unbonded Brace Damper
•Force-Limited
•Easy to construct
•Relatively Inexpensive
•Adds both “Damping” and Stiffness
Passive Energy Dissipation 15 – 6 - 91Instructional Material Complementing FEMA 451, Design Examples
Disadvantages of ADAS Systemand Unbonded Brace Damper
• Must be Replaced after Major Earthquake
• Highly Nonlinear Behavior
• Adds Stiffness to System
• Undesirable Residual Deformations Possible
Passive Energy Dissipation 15 – 6 - 92Instructional Material Complementing FEMA 451, Design Examples
Friction Dampers: Slotted-Bolted Damper
Pall Friction Damper
Passive Energy Dissipation 15 – 6 - 93Instructional Material Complementing FEMA 451, Design Examples
Sumitomo Friction Damper(Sumitomo Metal Industries, Japan)
Passive Energy Dissipation 15 – 6 - 94Instructional Material Complementing FEMA 451, Design Examples
Pall Cross-Bracing Friction Damper
Interior of Webster Library at Concordia University, Montreal,
Canada
Passive Energy Dissipation 15 – 6 - 95Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 96Instructional Material Complementing FEMA 451, Design Examples
Hysteretic Behavior of Slotted-Bolted Friction Damper
Passive Energy Dissipation 15 – 6 - 97Instructional Material Complementing FEMA 451, Design Examples
-60
-40
-20
0
20
40
60
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Displacement, Inches
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
Passive Energy Dissipation 15 – 6 - 98Instructional Material Complementing FEMA 451, Design Examples
Advantages of Friction Dampers
• Force-Limited
• Easy to construct
• Relatively Inexpensive
Passive Energy Dissipation 15 – 6 - 99Instructional Material Complementing FEMA 451, Design Examples
Disadvantages of Friction Dampers
• May be Difficult to Maintain over Time
• Highly Nonlinear Behavior
• Adds Large Initial Stiffness to System
• Undesirable Residual Deformations Possible
Passive Energy Dissipation 15 – 6 - 100Instructional Material Complementing FEMA 451, Design Examples
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 Damping Systems
Passive Energy Dissipation 15 – 6 - 101Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 102Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 103Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 104Instructional Material Complementing FEMA 451, Design Examples
• 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!
Passive Energy Dissipation 15 – 6 - 105Instructional Material Complementing FEMA 451, Design Examples
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 Damping Systems
Passive Energy Dissipation 15 – 6 - 106Instructional Material Complementing FEMA 451, Design Examples
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!
Passive Energy Dissipation 15 – 6 - 107Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 108Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 109Instructional Material Complementing FEMA 451, Design Examples
)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
Passive Energy Dissipation 15 – 6 - 110Instructional Material Complementing FEMA 451, Design Examples
)t(vMR)t(F)t(vC)t(vC)t(vM gSAI &&&&&& −=+++
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.
Passive Energy Dissipation 15 – 6 - 111Instructional Material Complementing FEMA 451, Design Examples
)t(vMR)t(F)t(vC)t(vC)t(vM gSAI &&&&&& −=+++
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.
Passive Energy Dissipation 15 – 6 - 112Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 113Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 114Instructional Material Complementing FEMA 451, Design Examples
)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:
Passive Energy Dissipation 15 – 6 - 115Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 116Instructional Material Complementing FEMA 451, Design Examples
Modal Damping Ratios
gvMRKvvCvM &&&&& −=++
yv Φ=
giiiiiii vyyy &&&&& Γ=++ 22 ωωξ
Specify modal damping values directly
Passive Energy Dissipation 15 – 6 - 117Instructional Material Complementing FEMA 451, Design Examples
MM
2MC
n
1ii
Ti
iTi
ii
⎥⎥⎦
⎤
⎢⎢⎣
⎡= ∑
=
φφφφ
ωξ
Modal Superposition Damping
Artificial Coupling
Skyhook
Note: There is no need to develop C explicitly.
Passive Energy Dissipation 15 – 6 - 118Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 119Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 120Instructional Material Complementing FEMA 451, Design Examples
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
∑==
Derivation of Modal Strain Energy Method
Passive Energy Dissipation 15 – 6 - 121Instructional Material Complementing FEMA 451, Design Examples
21
1
2
N stories
k i ,k i ,kk
i Ti i i
C ( )
M
−=
φ − φξ =
ω φ φ
∑
Derivation of Modal Strain Energy Method
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=
Passive Energy Dissipation 15 – 6 - 122Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 123Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 124Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 125Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 126Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 127Instructional Material Complementing FEMA 451, Design Examples
Frequency Damping(rad/sec) Ratio, ξ
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
Passive Energy Dissipation 15 – 6 - 128Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 129Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 130Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 131Instructional Material Complementing FEMA 451, Design Examples
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−
−−−−
−
=
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
Passive Energy Dissipation 15 – 6 - 132Instructional Material Complementing FEMA 451, Design Examples
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
Comparison of Actual Damping Matrix and Damping Matrix Obtained from MSE Damping Ratios
Actual Damping Matrix
Modal Superposition DampingMatrix Using MSE Damping Ratios
Passive Energy Dissipation 15 – 6 - 133Instructional Material Complementing FEMA 451, Design Examples
c = 20.0 k-sec/in.
c = 30.0
Nonproportional Damping
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
=
0.500.200000.200.20000
000000000000000
AC
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−−
−−−−−
−−
=
92011521607206601159219107220981
2169104139254560072722925278962
066009814560962653
..........
..........
.....
C
Comparison of Actual Damping Matrix and Damping Matrix Obtained from MSE Damping Ratios
Modal Superposition DampingMatrix Using MSE Damping Ratios
Actual Damping Matrix
Passive Energy Dissipation 15 – 6 - 134Instructional Material Complementing FEMA 451, Design Examples
Example: Seismic Analysis of a Structurewith Nonproportional Damping
• Discrete Damping vs Rayleigh Damping• Discrete Damping: Rigid vs Flexible Braces
Frequency Damping(rad/sec) Ratio, ξ
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.
Passive Energy Dissipation 15 – 6 - 135Instructional Material Complementing FEMA 451, Design Examples
Computing Rayleigh Damping Proportionality Factors (Using NONLIN Pro)
Passive Energy Dissipation 15 – 6 - 136Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 137Instructional Material Complementing FEMA 451, Design Examples
0
200
400
600
800
1000
1200
1400
1600
0.0 5.0 10.0 15.0 20.0
First Mode Damping, % Critical
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
Passive Energy Dissipation 15 – 6 - 138Instructional Material Complementing FEMA 451, Design Examples
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
Very Stiff BracesReasonably Stiff Braces
Example: Effect of Brace Stiffness(Discrete Damping Model)
c
1.5c
Passive Energy Dissipation 15 – 6 - 139Instructional Material Complementing FEMA 451, Design Examples
0
200
400
600
800
1000
1200
1400
1600
0.0 5.0 10.0 15.0 20.0
First Mode Damping, % Critical
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
Passive Energy Dissipation 15 – 6 - 140Instructional Material Complementing FEMA 451, Design Examples
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 Model with Flexible BracesRayleigh Damping
Example: Discrete (Flexible Braces) vs Rayleigh Damping
c
1.5c
Passive Energy Dissipation 15 – 6 - 141Instructional Material Complementing FEMA 451, Design Examples
0
200
400
600
800
1000
1200
1400
1600
0.0 5.0 10.0 15.0 20.0
First Mode Damping, % Critical
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
Passive Energy Dissipation 15 – 6 - 142Instructional Material Complementing FEMA 451, Design Examples
-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
Passive Energy Dissipation 15 – 6 - 143Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 144Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 145Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 146Instructional Material Complementing FEMA 451, Design Examples
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!
Passive Energy Dissipation 15 – 6 - 147Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 148Instructional Material Complementing FEMA 451, Design Examples
)t(vMR)t(F)t(vC)t(vC)t(vM gSAI &&&&&& −=+++
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.
Passive Energy Dissipation 15 – 6 - 149Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 150Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 151Instructional Material Complementing FEMA 451, Design Examples
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:
Passive Energy Dissipation 15 – 6 - 152Instructional Material Complementing FEMA 451, Design Examples
⎥⎦
⎤⎢⎣
⎡
ΦΦΛΦΦΛ
=*
**
P
Extracting Damped Mode Shapes
Damped Mode Shapes
Passive Energy Dissipation 15 – 6 - 153Instructional Material Complementing FEMA 451, Design Examples
Damped Mode Shapes
Real
Imaginary
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
++++
=
44
33
22
11
ibaibaibaiba
φ
U1
U2
U3
U4
Passive Energy Dissipation 15 – 6 - 154Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 155Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 156Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 157Instructional Material Complementing FEMA 451, Design Examples
-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
Example: Damped Mode Shapes and FrequenciesSystem with Non-Classical Damping
Passive Energy Dissipation 15 – 6 - 158Instructional Material Complementing FEMA 451, Design Examples
-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
Passive Energy Dissipation 15 – 6 - 159Instructional Material Complementing FEMA 451, Design Examples
-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
Example: Damped Mode Shapes and FrequenciesSystem with Non-Classical Damping
Passive Energy Dissipation 15 – 6 - 160Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 161Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 162Instructional Material Complementing FEMA 451, Design Examples
Toggle Brace Deployment
Huntington Tower
Passive Energy Dissipation 15 – 6 - 163Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 165Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 166Instructional Material Complementing FEMA 451, Design Examples
Computed Damping Ratios for System With A = 20
A20
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
ioDamping EnergyModal Strain EnergyLog DecrementComplex Eignevalues
Damping Coefficient (k-sec/in)
Energy RatioModal Strain EnergyLog DecrementDamped Eigenproblem
Passive Energy Dissipation 15 – 6 - 167Instructional Material Complementing FEMA 451, Design Examples
Computed Damping Ratios for System With A = 30
A30
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
Energy RatioModal Strain EnergyLog DecrementDamped Eigenproblem
Passive Energy Dissipation 15 – 6 - 168Instructional Material Complementing FEMA 451, Design Examples
Computed Damping Ratios for System With A = 50
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
Damping EnergyModal Strain EnergyLog DecrementComplex Eigenvalues
Damping Coefficient (k-sec/in)
Energy RatioModal Strain EnergyLog Decrement
Damped Eigenproblem
Passive Energy Dissipation 15 – 6 - 169Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 170Instructional Material Complementing FEMA 451, Design Examples
-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
Passive Energy Dissipation 15 – 6 - 171Instructional Material Complementing FEMA 451, Design Examples
-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
Passive Energy Dissipation 15 – 6 - 172Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 173Instructional Material Complementing FEMA 451, Design Examples
• 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)
Passive Energy Dissipation 15 – 6 - 174Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 175Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 176Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 177Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 178Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 179Instructional Material Complementing FEMA 451, Design Examples
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.
Passive Energy Dissipation 15 – 6 - 180Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 181Instructional Material Complementing FEMA 451, Design Examples
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
Passive Energy Dissipation 15 – 6 - 182Instructional Material Complementing FEMA 451, Design Examples
EnergyDissipationDevice (EDD)
Energy Dissipation Nomenclature
EnergyDissipationAssembly (EDA)
1993 - Tentative General Requirements for the Design and Construction of Structures Incorporating Discrete Passive
Energy Dissipation Devices (2 of 3)
Passive Energy Dissipation 15 – 6 - 183Instructional Material Complementing FEMA 451, Design Examples
- 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
1993 - Tentative General Requirements for the Design and Construction of Structures Incorporating Discrete Passive
Energy Dissipation Devices (3 of 3)
Passive Energy Dissipation 15 – 6 - 184Instructional Material Complementing FEMA 451, Design Examples
- 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)
Passive Energy Dissipation 15 – 6 - 185Instructional Material Complementing FEMA 451, Design Examples
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
1994 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (2 of 4)
Part 1 – Provisions & Part 2 – Commentary (FEMA 222A & 223A)
Passive Energy Dissipation 15 – 6 - 186Instructional Material Complementing FEMA 451, Design Examples
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
1994 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (3 of 4)
Part 1 – Provisions & Part 2 – Commentary (FEMA 222A & 223A)
Passive Energy Dissipation 15 – 6 - 187Instructional Material Complementing FEMA 451, Design Examples
- 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
1994 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (4 of 4)
Part 1 – Provisions & Part 2 – Commentary (FEMA 222A & 223A)
Passive Energy Dissipation 15 – 6 - 188Instructional Material Complementing FEMA 451, Design Examples
- 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)
Passive Energy Dissipation 15 – 6 - 189Instructional Material Complementing FEMA 451, Design Examples
- 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
Passive Energy Dissipation 15 – 6 - 190Instructional Material Complementing FEMA 451, Design Examples
-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)
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) (2 of 9)
Coefficient Method
Capacity Spectrum Method
Passive Energy Dissipation 15 – 6 - 191Instructional Material Complementing FEMA 451, Design Examples
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) (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
Passive Energy Dissipation 15 – 6 - 192Instructional Material Complementing FEMA 451, Design Examples
Mathematical Modeling of Displacement-Dependent Devices
Eq. (9-20)Force in Device
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) (4 of 9)
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
Passive Energy Dissipation 15 – 6 - 193Instructional Material Complementing FEMA 451, Design Examples
Mathematical Modeling of Solid Viscoelastic Devices
Eq. (9-22)Force in Device
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) (5 of 9)
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
Passive Energy Dissipation 15 – 6 - 194Instructional Material Complementing FEMA 451, Design Examples
Mathematical Modeling of Fluid Viscoelastic and Fluid Viscous Devices
Maxwell Model
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) (6 of 9)
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 ′
Passive Energy Dissipation 15 – 6 - 195Instructional Material Complementing FEMA 451, Design Examples
Pushover Analysis for Structures with EDD’s (Part of NSP)
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) (7 of 9)
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.
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) 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
Passive Energy Dissipation 15 – 6 - 197Instructional Material Complementing FEMA 451, Design Examples
Design Process for Velocity-Dependent Dampers using NSP (2)
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) (9 of 9)
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
Passive Energy Dissipation 15 – 6 - 198Instructional Material Complementing FEMA 451, Design Examples
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)
Passive Energy Dissipation 15 – 6 - 199Instructional Material Complementing FEMA 451, Design Examples
- 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.
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (1 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)
Passive Energy Dissipation 15 – 6 - 200Instructional Material Complementing FEMA 451, Design Examples
- 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.
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (2 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)
Passive Energy Dissipation 15 – 6 - 201Instructional Material Complementing FEMA 451, Design Examples
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.
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (3 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)
Passive Energy Dissipation 15 – 6 - 202Instructional Material Complementing FEMA 451, Design Examples
Effective Damping Ratio (used to determine factors, B, that reduce structure response)
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (4 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)
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
Passive Energy Dissipation 15 – 6 - 203Instructional Material Complementing FEMA 451, Design Examples
Equivalent Viscous Damping from EDS
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (5 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)
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
Passive Energy Dissipation 15 – 6 - 204Instructional Material Complementing FEMA 451, Design Examples
⎭⎬⎫
⎩⎨⎧
=+
V75.0;B
VmaxVIV
min
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (6 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)
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
Passive Energy Dissipation 15 – 6 - 205Instructional Material Complementing FEMA 451, Design Examples
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
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (7 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)
Passive Energy Dissipation 15 – 6 - 206Instructional Material Complementing FEMA 451, Design Examples
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 Viscous Damping
Decreased DisplacementDecreased Shear Force (can not take full advantage of)
2000 - NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures (8 of 8)
Part 1 – Provisions & Part 2 – Commentary (FEMA 368 & 369)