SEISMIC PROTECTIVE SYSTEMS: PASSIVE ENERGY DISSIPATION

Transcription

1 SEISMIC PROTECTIVE SYSTEMS: PASSIVE ENERGY DISSIPATION Presented & Developed by: Michael D. Symans, PhD Rensselaer Polytechnic Institute Initially Developed by: Finley A. Charney, PE, PhD Virginia Polytechnic Institute & State University Passive Energy Dissipation

2 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

3 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

4 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

5 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

6 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

7 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

8 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 contentsrelated damage Passive Energy Dissipation

9 Pseudo-Spectral Pseudoacceleration, Acceleration, g g % Damping 10% 20% 30% 40% Effect of Added Damping (Viscous Damper) T=.50 T= Spectral Displacement, Inches T=1.5 T=2.0 - Decreased Displacement - Decreased Shear Force T=3.0 T=4.0 Passive Energy Dissipation

10 Pseudo-Spectral Pseudoacceleration, Acceleration, g g % Damping 10% 20% 30% 40% Effect of Added Stiffness (Added Bracing) T=.50 T= Spectral Displacement, Inches T=1.5 T=2.0 - Decreased Displacement - Increased Shear Force T=3.0 T=4.0 Passive Energy Dissipation

11 Effect of Added Damping and Stiffness (ADAS System) Pseudo-Spectral Pseudoacceleration, Acceleration, g g % Damping 10% 20% 30% 40% T=.50 T= Spectral Displacement, Inches T=1.5 T=2.0 - Decreased Displacement - Decreased Shear (possibly) T=3.0 T=4.0 Passive Energy Dissipation

12 Pseudo-Spectral Pseudoacceleration, Acceleration, g g Effect of Reduced Stiffness (Seismic Isolation) 5% Damping 10% 20% 30% 40% T=.50 T=1.0 - Decreased Shear Force - Increased Displacement Spectral Displacement, Inches T=1.5 T=2.0 T=3.0 T=4.0 Passive Energy Dissipation

13 Pseudo-Spectral Pseudoacceleration, Acceleration, g g Effect of Reduced Stiffness (Seismic Isolation with Dampers) % Damping 10% 20% 30% 40% T=.50 T=1.0 - Decreased Shear - Increased Displacement Spectral Displacement, Inches T=1.5 T=2.0 T=3.0 T=4.0 Passive Energy Dissipation

14 Effect of Damping and Yield Strength on Deformation Demand 1.4 Peak Displacement (in) Yield Strength (kips) Damping Ratio (%) Passive Energy Dissipation

15 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

16 Distinction Between Natural and Added Damping Natural (Inherent) Damping ξ is a structural property, dependent on system mass, stiffness, and inherent energy dissipation mechanisms ξ NATURAL = 0.5 to 7.0% C Added Damping ξ is a structural property, dependent on system mass, stiffness, and the added damping coefficient C ξ ADDED =10 to 30% Passive Energy Dissipation

17 1981/1982 US-JAPAN PROJECT Response of Bare Frame Before and After Adding Ballast Model Weight Bare Model 18 kips Loaded Model 105 kips Passive Energy Dissipation

18 1981/1982 US-JAPAN PROJECT Change in Damping and Frequency with Accumulated Damage FREQUENCY, HZ DAMPING, % CRITICAL PEAK GROUND ACCELERATION, %G Passive Energy Dissipation

19 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

20 Reduction in Seismic Damage Energy Balance: Hysteretic Energy E = E + E + ( E + E ) + I S K DI DA E H Inherent Damping Added Damping Damage Index: DI () t = u u max ult + ρ E H F u Source: Park and Ang (1985) y ( t) ult DI Collapse Damage State Passive Energy Dissipation

21 Duration-Dependent Damage Index DI () t = u u max ult + ρ E H F u y ( t) ult Source: Park and Ang (1985) u max u ult ρ = maximum displacement = monotonic ultimate displacement = calibration factor 1.0 DI E H = hysteretic energy dissipated Fy = monotonic yield force 0.0 Collapse Damage State Passive Energy Dissipation

22 % Damping Absorbed Energy, inch-kips Energy (kip-inch) KINETIC + STRAIN DAMPING HYSTERETIC Damping Reduces Hysteretic Energy Dissipation Demand Time, Seconds Absorbed Energy, Inch-Kips Energy (kip-inch) KINETIC + STRAIN 20% Damping DAMPING Time, Seconds HYSTERETIC Passive Energy Dissipation

23 Effect of Damping and Yield Strength on Hysteretic Energy 140 Hysteretic Energy (kip-in) Yield Strength (kips) Damping Ratio (%) Passive Energy Dissipation

24 Energy and Damage Histories, 5% Damping Hysteretic Viscous + Hysteretic Total E I = 260 Energy, In-Kips E DI +E DA E H Damage Index Time, Seconds Max=0.55 Analysis performed on NONLIN Passive Energy Dissipation

25 Energy and Damage Histories, 20% Damping Hysteretic Viscous + Hysteretic Total E I = 210 Energy, In-Kips E DI +E DA E H Damage Index Max= Time, Seconds Passive Energy Dissipation

26 Reduction in Damage with Increased Damping % Damping 20% Damping 40% Damping 60% Damping 0.50 Damage Index Time, Seconds Passive Energy Dissipation

27 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

28 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

29 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

30 Cross-Section of Viscous Fluid Damper Source: Taylor Devices, Inc. Passive Energy Dissipation

31 Possible Damper Placement Within Structure Augmented Bracing Passive Energy Dissipation

32 Chevron Brace and Viscous Damper Passive Energy Dissipation

33 Diagonally Braced Damping System θ Passive Energy Dissipation

34 Fluid Dampers within Inverted Chevron Brace Pacific 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

35 Fluid Damper within Diagonal Brace San Francisco State Office Building San Francisco, CA Huntington Tower Boston, MA Passive Energy Dissipation

36 Toggle Brace Damping System U 1 AF = U U D = sinθ 2 cos( θ θ 2 ) U D θ 2 θ 1 Passive Energy Dissipation

37 Toggle Brace Deployment Huntington Tower, Boston, MA - New 38-story steel-framed building direct-acting and toggle-brace dampers kn (292 kips), +/- 101 mm (+/- 4 in.) - Dampers suppress wind-induced vibration Passive Energy Dissipation

38 FORCE, KIPS u( Harmonic Behavior of Fluid Damper Loading Frequency Pt ( ) = Psin( ω t)cos( δ ) + Pcos( ω t)sin( δ ) t ) = u 0 sin( ωt ) Imposed Motion Total Force TIME, SECONDS ELASTIC FORCE DAMPING FORCE TOTAL FORCE Phase Angle (Lag) Note: Damping force 90 o out-of-phase with elastic force. δ ω Passive Energy Dissipation

39 K S P( t ) = K S u( t ) + Cu( & P0 P0 = cos( δ ) K L = sin( δ ) u u C = t ) K L ω δ = sin 0 0 Storage Stiffness Loss Stiffness Damping Coeff. Phase Angle u o 1 * K P Z P 0 Total Force, P P Z K S P o PZ = K Luo = Po E = πp u πp u D Z o = sin( δ ) sin( δ ) o o Damper Displacement, u Passive Energy Dissipation

40 Frequency-Domain Force-Displacement Relation Apply Fourier Transform: P( ( t) + Cu( t) P( t ) = K S u & ( ω ) K iω u( ω ) ω ω ) = K u / P( ω ) Complex Stiffness: ( ω ) K * = P = P u S + L [ K ik ] u( ω ) ( ω ) ( ω ) S + ( ) * ω = K ( ω ) u( ω ) ( * R K ) = K ( * ) Note: and S I K = K L L I K S δ K ( ω ) ( ω ) R Compact Force-Displ. Relation for Viscoelastic Dampers * K L ( ω ) Passive Energy Dissipation

41 Dependence of Storage Stiffness on Frequency for Typical Single-Ended Fluid Damper Storage Stiffness (lb/in) Storage Stiffness, lbs/inch Cutoff Frequency Excitation Frequency, Hz. (Hz) Passive Energy Dissipation

42 Dependence of Damping Coefficient on Frequency for Typical Single-Ended Fluid Damper Damping Coefficient, C, lb-sec/in Cutoff Frequency Excitation Frequency, Hz(Hz) Passive Energy Dissipation

43 120 Dependence of Phase Angle on Frequency for Typical Single-Ended Fluid Damper Phase Angle (degrees) δ δ Frequency (Hz) δ Passive Energy Dissipation

44 Dependence of Damping Coefficient on Temperature for Typical Fluid Damper Damping Coefficient, lb-sec/in Damping Constant, k-sec/in Temperature (Degrees C) Temperature, Degrees C Passive Energy Dissipation

45 Behavior of Fluid Damper with Zero Storage Stiffness o K P( t ) = C u& = L u& K S = 0 δ = 90 u o ω Damper Force, P P o E = πp d o u o Damper Displacement, u Passive Energy Dissipation

46 Actual Hysteretic Behavior of Fluid Damper Harmonic Loading Seismic Loading Source: Constantinou and Symans (1992) Passive Energy Dissipation

47 Force-Velocity Behavior of Viscous Fluid Damper Exp Damper Force, Kips P (t) = C u & α sgn(u) & Damper Deformational Velocity, in/sec Passive Energy Dissipation

48 120 Nonlinear Fluid Dampers P (t) = C u & α sgn( u & ) α = α = 10. Force, Kips Displacement, Inches Passive Energy Dissipation

49 Energy Dissipated Per Cycle for Linear and Nonlinear Viscous Fluid Dampers Linear Damper: E D = π P o u o Hysteretic Energy Factor Nonlinear Damper: E = λ P D o u o λ = 4 2 α 2 α Γ Γ(2 + α) Γ = Gamma Function Passive Energy Dissipation

50 Relationship Between λ and α for Viscous Fluid Damper 4.0 AF = λ/π = 1.24 Hysteretic Energy Factor λ AF = 1.11 AF = Damper Velocity Exponent α Passive Energy Dissipation

51 Relationship Between Nonlinear and Linear Damping Coefficient for Equal Energy Dissipation Per Cycle C C π NL = ( u ω ) 1 α o λ L Note: Ratio is frequency- and displacement-dependent and is therefore meaningful only for steady-state harmonic response. Passive Energy Dissipation

52 Ratio of Nonlinear Damping Coefficient to Linear Damping Coefficient (For a Given Loading Frequency) Loading Loading Frequency Freqency = 1 Hz = (6.28 Hz rad/sec) 12 Ratio of CNL to CL Max Disp = 1.0 Max Disp = 2.0 Max Disp = Velocity Exponent Passive Energy Dissipation

53 Ratio of Nonlinear Damping Constant to Linear Damping Constant (For a Given Maximum Displacement) Maximum Displacement = 1 Ratio of CNL to CL Velocity Exponent Load Freq = 6.28 Hz Load Freq = 3.14 Hz Load Freq = 2.09 Hz 1/3 Hz (2.09 rad/s) 1/2 Hz (3.14 rad/s) 2 Hz (6.28 rad/s) Passive Energy Dissipation

54 Example of Linear vs Nonlinear Damping Alpha = 1.0 Alpha = 0.5 Force, Kips Frequency = 1 Hz (6.28 rad/s) Max. Disp. = 10.0 in. λ = 3.58 C Linear = 10.0 k-sec/in C Nonlinear = 69.5 k-sec 0.5 /in Displacement, Inches Passive Energy Dissipation

55 Recommendations Related to Nonlinear Viscous Dampers Do NOT attempt to linearize the problem when nonlinear viscous dampers are used. Perform the analysis with discrete nonlinear viscous dampers. Do NOT attempt to calculate effective damping in terms of a damping ratio (ξ) when using nonlinear viscous dampers. DO NOT attempt to use a free vibration analysis to determine equivalent viscous damping when nonlinear viscous dampers are used. Passive Energy Dissipation

56 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

57 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

58 Viscoelastic Dampers L A P(t) Viscoelastic Material A W u(t) P(t) Developed in the 1960 s for Wind Applications h h Section A-A Passive Energy Dissipation

59 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

60 Harmonic Behavior of Viscoelastic Damper u( t ) = u 0 sin( ωt ) Imposed Motion Total Force Loading Frequency Pt ( ) = Psin( ω t)cos( δ ) + Pcos( ω t)sin( δ ) 0 0 Phase Angle (Lag) STRESS, PSI 800 ELASTIC FORCE 600 DAMPING FORCE 400 TOTAL FORCE TIME, SECONDS Passive Energy Dissipation

61 K S G' A = KL h P(t) = KSu(t) + Cu(t) & = G'' A h K L C = ω δ = sin 1 τ Z τ 0 Storage Stiffness Loss Stiffness Damping Coeff. Phase Angle G ( ω ) G = Storage Modulus γ η = = tan ( δ ) 0 G ( ω ) G = Loss Modulus * Loss Factor G Shear Stress τ Z G τ 0 τ Z = G γ o = τ o sin ( δ ) () t + G γ () t ω τ ( t ) = G γ & / E D Shear Strain = πτ γ Ah = πτ γ Ah sin δ = πg γ Z o o o 2 ( ) V Passive Energy Dissipation o

62 Frequency-Domain Stress-Strain Relation Apply Fourier Transform: ( t) + G γ ( t) ω τ ( t ) = G γ & / ( ω ) + G iω γ ( ω ) ω τ ( ω ) = G γ / [ G + ig ] γ ( ω ) τ ( ω ) = τ ( ω ) = G + Complex Shear Modulus: ( ω ) ( ω ) τ G * ( ω ) = = G + γ τ ( ) * ω = G ( ω ) γ ( ω ) [ 1 iη ] γ ( ω ) [ 1 iη ] I δ * G G ( ω ) ( ω ) G ( ω ) R Compact Stress-Strain Relation for Viscoelastic Materials Passive Energy Dissipation

63 Dependence of Storage and Loss Moduli on Temperature and Frequency for Typical Viscoelastic Damper Storage or Loss Modulus (MPa) Increasing Temperature Frequency (Hz) Passive Energy Dissipation

64 Dependence of Loss Factor on Temperature and Frequency for Typical Viscoelastic Damper 1.5 Increasing Temperature Loss Factor Frequency (Hz) Passive Energy Dissipation

65 Actual Hysteretic Behavior of Viscoelastic Damper Harmonic Loading Seismic Loading Passive Energy Dissipation

66 Advantages of Viscoelastic Dampers High reliability May be able to use linear analysis Somewhat lower cost Passive Energy Dissipation

67 Disadvantages of Viscoelastic Dampers Strong Temperature Dependence Lower Force and Displacement Capacity Not Force Limited 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

68 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

69 Modeling Viscous Dampers: Simple Dashpot C D P(t) u(t) Newtonian Dashpot P( t ) = C D u & ( t ) Useful For : Fluid Dampers with Zero Storage Stiffness This Model Ignores Temperature Dependence Passive Energy Dissipation

70 Modeling Linear Viscous/Viscoelastic Dampers: Kelvin Model C D Newtonian Dashpot P(t) Hookean Spring K D P( Useful For : Viscoelastic Dampers and Fluid Dampers with Storage Stiffness and Weak Frequency Dependence. This Model Ignores Temperature Dependence t ) = K D u u(t) ( t ) + C u& ( t ) D Passive Energy Dissipation

71 Kelvin Model (Continued) P( t ) = K D u ( t ) + C u& ( t ) d Apply Fourier Transform: P( ω ) = [ K iω C ] u ( ω ) D + d K S ( ω ) Complex Stiffness: K Storage Stiffness: S * [ ( )] * K K D K ( ω ) = R ω = Loss Stiffness: ( ω ) = K + D iω C C ( ω ) d [ ( )] * K ω C ω K ( ω ) = I = L Damping Coefficient: ( ω ) D K C ( ω ) = L = C D ω K D C D ω Passive Energy Dissipation

72 Kelvin Model (Continued) C D K D P(t) u(t) Kelvin Model C ( ω) = K ( ω ) = C D S K D P(t) u(t) Equivalent Kelvin Model Passive Energy Dissipation

73 Modeling Linear Viscous/Viscoelastic Dampers: Maxwell Model P(t) C D K D u(t) Newtonian Dashpot; Hookean Spring P( P& = () t C u& () t Useful For : Viscoelastic Dampers and Fluid Dampers with Strong Frequency Dependence. This Model Ignores Temperature Dependence t ) + C K D D D Passive Energy Dissipation

74 Passive Energy Dissipation Maxwell Model (Continued) = C D K D λ Relaxation Time: C D ) ( C ω ω K D ) ( K S ω () () t u C t P K C ) t ( P d D D & & = + Apply Fourier Transform: ( ) ( ) ω ω ω ω ω u C i P K C i ) ( P d D D = + Complex Stiffness: 2 2 D D * 1 C i 1 C ) ( K ω λ ω ω λ λ ω ω = ( ) [ ] D * S 1 K K ) ( K ω λ ω λ ω ω + = = R Storage Stiffness: Loss Stiffness: Damping Coefficient: ( ) 2 2 D S 1 C K ) C( ω λ ω ω ω + = = ( ) [ ] 2 2 D * L 1 C K ) ( K ω λ ω ω ω + = = I

75 Maxwell Model Parameters from Experimental Testing of Fluid Viscous Damper Storage Stiffness, lbs/inch (lb/in) K D Excitation Frequency, Hz. (Hz) Passive Energy Dissipation

76 Maxwell Model Parameters from Experimental Testing of Fluid Viscous Damper Damping Coefficient, lb-sec/in C D Excitation Frequency (Hz) Passive Energy Dissipation

77 Maxwell Model (Continued) Note: K C( ω) S P(t) C D K D u(t) = ( ω ) = C D 2 1+ λω K D λ ω 1 + λ ω 2 2 P(t) u(t) - If K D is very large, λ is very small, K S is small and C = C D - If C D is very small, λ is very small, K S is small and C = C D - If K D is very small, λ is very large, C is small and K S = K D. Maxwell Model Equivalent Kelvin Model C D K D Passive Energy Dissipation

78 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

79 Effect of Linkage Flexibility on Damper Effectiveness Because the damper is always in series with the linkage, the damper-brace assembly acts like a Maxwell model. K θ AE, Effective 2 cos L Brace = 2 θ Hence, the effectiveness of the damper is reduced. The degree of lost effectiveness is a function of the structural properties and the loading frequency. Passive Energy Dissipation

80 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

81 Steel Plate Dampers (Added Damping and Stiffness System - ADAS) V t V b L a L Passive Energy Dissipation

82 Implementation of ADAS System Wells Fargo Bank, San Francisco, CA - Seismic Retrofit of Two- Story Nonductile Concrete Frame; Constructed in Dampers Within Chevron Bracing Installed in Yield Force Per Damper: 150 kips Passive Energy Dissipation

83 Hysteretic Behavior of ADAS Device ADAS Device (Tsai et al. 1993) Experimental Response (Static) (Source: Tsai et al. 1993) Passive Energy Dissipation

84 Ideal Hysteretic Behavior of ADAS Damper (SAP2000 and ETABS Implementation) Initial Stiffness Secondary Stiffness Ratio F y F = β kd + ( 1 β ) F Z y Force, F Z& = k F y ( ) α Z D& 1 if D& Z > 0 D& otherwise Yield Sharpness Displacement, D. Z is a Path Dependency Parameter Passive Energy Dissipation

85 Parameters of Mathematical Model of ADAS Damper t V b V L k = n F ( 2 + a / b) y = nf L 3 y bt 4L 3 EI b a L n f I y b = = = Number of plates Yield force of each plate Second moment of area of each plate at b (i.e, at top of plate) Passive Energy Dissipation

86 Unbonded Brace Damper Steel Brace (yielding core) (coated with debonding chemicals) Stiff Shell Prevents Buckling of Core Concrete Passive Energy Dissipation

87 Implementation of Unbonded Brace Damper Source: ASCE Civil Engineering Magazine, March Plant and Environmental Sciences Replacement Facility - New Three-Story Building on UC Davis Campus - First Building in USA to Use Unbonded Brace Damper Unbonded Braced Frames with Diagonal or Chevron Brace Installation - Cost of Dampers = 0.5% of Building Cost Passive Energy Dissipation

88 Hysteretic Behavior of Unbonded Brace Damper Passive Energy Dissipation

89 Testing of Unbonded Brace Damper Testing Performed at UC Berkeley Typical Hysteresis Loops from Cyclic Testing Passive Energy Dissipation

90 Advantages of ADAS System and Unbonded Brace Damper Force-Limited Easy to construct Relatively Inexpensive Adds both Damping and Stiffness Passive Energy Dissipation

91 Disadvantages of ADAS System and Unbonded Brace Damper Must be Replaced after Major Earthquake Highly Nonlinear Behavior Adds Stiffness to System Undesirable Residual Deformations Possible Passive Energy Dissipation

92 Friction Dampers: Slotted-Bolted Damper Pall Friction Damper Passive Energy Dissipation

93 Sumitomo Friction Damper (Sumitomo Metal Industries, Japan) Passive Energy Dissipation

94 Pall Cross-Bracing Friction Damper Interior of Webster Library at Concordia University, Montreal, Canada Passive Energy Dissipation

95 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 Cross-Bracing Friction Dampers Installed in Dampers Exposed for Aesthetics Passive Energy Dissipation

96 Hysteretic Behavior of Slotted-Bolted Friction Damper Passive Energy Dissipation

97 Ideal Hysteretic Behavior of Friction Damper μn Normal Force F N ut D = μ & () ut &( ) Force, Kips 0 u 0 u 0 Coefficient of Friction μn F D Alternatively, = Nμ sgn [ u& ( t) ] Displacement, Inches 1 sgn(x) -1 x Passive Energy Dissipation

98 Advantages of Friction Dampers Force-Limited Easy to construct Relatively Inexpensive Passive Energy Dissipation

99 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

100 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

101 Equivalent Viscous Damping: Damping System with Inelastic or Friction Behavior F D EH E H E S u D E s based on secant stiffness ξ H = 4 E π H E S Note: Computed damping ratio is displacement-dependent Passive Energy Dissipation

102 Effect of Inelastic System Post-Yielding Stiffness on Equivalent Viscous Damping 0.6 F D Equivalent Hysteretic Damping Ratio α = 0 α = 0.3 α ξ H K u y αk μu y u 2( μ 1)(1 α) = πμ(1 + μα α) Ductility Demand μ Note: May be Modified (κ) for Other (less Robust) Hysteretic Behavior Passive Energy Dissipation

103 Equivalent Viscous Damping: Equivalent System with Linear Viscous Damper C F D ED E S u D C = 2m ωξ H E s and ω are based on Secant Stiffness of Inelastic System Passive Energy Dissipation

104 Equivalent Viscous Damping: Caution! It is not possible, on a device level, to replace a displacement-dependent device (e.g. a Friction Damper) with a velocity-dependent device (e.g. a Fluid Damper). Some simplified procedures allow such replacement on a structural level, wherein a smeared equivalent viscous damping ratio is found for the whole structure. This approach is marginally useful for preliminary design, and should not be used under any circumstances for final design. Passive Energy Dissipation

105 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

106 Modeling Considerations for Structures with Passive Energy Dissipation Devices Damping is almost always nonclassical (Damping matrix is not proportional to stiffness and/or mass) For seismic applications, system response is usually partially inelastic For seismic applications, viscous damper behavior is typically nonlinear (velocity exponents in the range of 0.5 to 0.8) Conclusion: This is a NONLINEAR analysis problem! Passive Energy Dissipation

107 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

108 Seismic Analysis of Structures with Passive Energy Dissipation Systems Yes (implies viscous or viscoelastic behavior) Linear Behavior? Classical Damping? Yes No No Nonlinear Response-History Analysis Complex Modal Response Spectrum Analysis or (Complex Modal) Response-History Analysis Modal Response Spectrum Analysis or (Modal) Response-History Analysis Passive Energy Dissipation

109 Seismic Analysis of MDOF Structures with Passive Energy Dissipation Systems Mv( & t ) + C I v( & t ) + C A v( & t ) + F S ( t ) = MRv&& g ( t ) Inherent Damping: Linear Added Viscous Damping: Linear or Nonlinear CA f ( ω) Restoring Force: (May include Added Devices) Linear or Nonlinear Passive Energy Dissipation

110 MDOF Solution Techniques Mv( & t ) + C I v( & t ) + C A v( & t ) + F S ( t ) = MRv&& g ( t ) Explicit integration of fully coupled equations: Treat C I as Rayleigh damping and model C A explicitly. Use Newmark solver (with iteration) to solve full set of coupled equations. System may be linear or nonlinear. Passive Energy Dissipation

111 MDOF Solution Techniques Mv( & t ) + C I v( & t ) + C A v( & t ) + F S ( t ) = MRv&& g ( t ) Fast Nonlinear Analysis: Treat C I as modal damping and model C A explicitly. Move C A (and any other nonlinear terms) to right-hand side. Left-hand side may be uncoupled by Ritz Vectors. Iterate on unbalanced right-hand side forces. System may be linear or nonlinear. Passive Energy Dissipation

112 Mv( & t Mv( & t ) + ) + C C I I Fast Nonlinear Analysis v( & t v( & t ) + ) + C K A E v( & t Move Added Damper Forces and Nonlinear Forces to RHS: v( t ) + K E v( t ) = MRv&& ) + g ( t F H ) ( t F ) H = ( t MRv&& ) C A g ( t Nonlinear Restoring Force v( & t ) ) Transform Coordinates: M ~ & y( t Linear Terms Apply Transformation: v( t) = Φy( t) C ~ K ~ T ) + y( & t ) + y( t ) = Φ MRv&& ( t ) Φ I E Orthogonal basis of Ritz vectors: Number of vectors << N g Nonlinear Terms T F H ( t ) C ~ A y( & t ) Uncoupled Coupled Passive Energy Dissipation

113 Fast Nonlinear Analysis FAST SLOW Relative Solution Speed: FNA/Full Integration Nonlinear DOF / Total DOF 1.0 Passive Energy Dissipation

114 MDOF Solution Techniques Mv( & t ) + C I v( & t ) + C A v( & t ) + F S ( t ) = MRv&& g ( t ) Explicit integration or response spectrum analysis of first few uncoupled modal equations: Treat C I as modal damping or Rayleigh damping Use Modal Strain Energy method to represent C A as modal damping ratios. System must be linear. Applicable only to viscous (or viscoelastic) damping systems. Passive Energy Dissipation

115 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

116 Modal Damping Ratios M v& + Cv& + Kv = MRv& g v = Φy & y i + 2ξ ω y& + ω i i i 2 i y i = Γ v& i g Specify modal damping values directly Passive Energy Dissipation

117 Passive Energy Dissipation M M 2 M C n 1 i i T i i T i i i = = φ φ φ φ ω ξ Modal Superposition Damping Artificial Coupling Skyhook Note: There is no need to develop C explicitly.

118 Rayleigh Proportional Damping C R = α M + β K Skyhook Damping Ratio MASS STIFFNESS COMBINED Frequency, radians/second Passive Energy Dissipation

119 Derivation of Modal Strain Energy Method Deformation of Structure in its Second Mode Floor Displacement φ 2,1 φ 2,2 φ 2,3 φ 2,4 Damper Deformation φ 2,1 - φ 2,2 φ 2,2 - φ 2,3 φ 2,3 - φ 2,4 φ 2,4 Passive Energy Dissipation

120 Derivation of Modal Strain Energy Method E 2 D,i,story k = πωick( φi,k φi,k 1 ) E S,i = 1 T φ i Kφi = ωi φ 2 T i Mφ i ξ i = n stories k= 1 E 4πE D,i,story k S,i Passive Energy Dissipation

121 Derivation of Modal Strain Energy Method ξ = i Nstories k = 1 C( φ φ ) k i,k i,k 1 2ωφ M φ T i i i 2 T T φ C φ φ C φ ξ i = = T * 2ω φ Mφ 2m ω i A i i A i i i i i i Note: IF C A is diagonalized by Φ, THEN c * ξ i i = 2m * i ω i Passive Energy Dissipation

122 Modal Strain Energy Damping Ratio φ C φi ω T i A ξ i = * 2mi i Note: φ is the Undamped Mode Shape The Modal Strain Energy Method is approximate if the structure is non-classically damped since the undamped and damped mode shapes are different. Passive Energy Dissipation

123 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

124 Example: Application of Modal Strain Energy Method m = 1.0 k-sec 2 /in. Proportional Damping m = 1.5 k = 200 k/in. c = 10.0 k-sec/in. m = 1.5 k = 250 c = 12.5 m = 1.5 k = 300 c = 15.0 m = 1.5 k = 350 k = 400 c = 17.5 c = 20.0 Passive Energy Dissipation

125 Modal Damping Ratios from Modal Strain Energy Method for Proportional Damping Distribution 0.8 Frequency Damping (rad/sec) Ratio, ξ Damping Ratio Proportional Frequency, rad/sec Passive Energy Dissipation

126 Example: Application of Modal Strain Energy Method m = 1.0 k-sec 2 /in. Nearly Proportional Damping m = 1.5 k = 200 k/in. c = 10 k-sec/in. m = 1.5 k = 250 c = 10 m = 1.5 m = 1.5 k = 300 k = 350 k = 400 c = 10 c = 20 c = 30 Passive Energy Dissipation

127 Modal Damping Ratios from Modal Strain Energy Method for Nearly Proportional Damping Distribution 0.8 Frequency Damping (rad/sec) Ratio, ξ Damping Ratio Proportional Nearly Proportional Frequency, rad/sec Passive Energy Dissipation

128 Example: Application of Modal Strain Energy Method m = 1.0 k-sec 2 /in. Nonproportional Damping m = 1.5 k = 200 k/in. m = 1.5 k = 250 m = 1.5 k = 300 m = 1.5 k = 350 c = 20 k-sec/in k = 400 c = 30 Passive Energy Dissipation

129 Modal Damping Ratios from Modal Strain Energy Method for Nonproportional Damping Distribution 0.8 Frequency Damping (rad/sec) Ratio. ξ Damping Ratio Proportional Nearly Proportional Nonproportional Frequency, rad/sec Passive Energy Dissipation

130 Modal Superposition Damping C Skyhook n 2ξ iω i T = M φi φi M T i=1 φi Mφ i Artificial Coupling 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

131 Comparison of Actual Damping Matrix and Damping Matrix Obtained from MSE Damping Ratios Proportional Damping c = 10.0 k-sec/in. c = 12.5 C A = Actual Damping Matrix c = 15.0 Modal Superposition Damping Matrix Using MSE Damping Ratios c = 17.5 c = 20.0 C = Passive Energy Dissipation

132 Comparison of Actual Damping Matrix and Damping Matrix Obtained from MSE Damping Ratios Nearly Proportional Damping c = 10.0 k-sec/in. c = 10.0 C A = Actual Damping Matrix c = 10.0 Modal Superposition Damping Matrix Using MSE Damping Ratios c = 20.0 c = 30.0 C = Passive Energy Dissipation

133 Comparison of Actual Damping Matrix and Damping Matrix Obtained from MSE Damping Ratios Nonproportional Damping Actual Damping Matrix C A = c = 20.0 k-sec/in. c = 30.0 C Modal Superposition Damping Matrix Using MSE Damping Ratios = Passive Energy Dissipation

134 Example: Seismic Analysis of a Structure with Nonproportional Damping Discrete Damping vs Rayleigh Damping Discrete Damping: Rigid vs Flexible Braces c = 20.0 k-sec/in. c = 30.0 MSE Results Frequency Damping (rad/sec) Ratio, ξ Damping ratios in modes 1 and 4 used to construct Rayleigh damping matrix. Passive Energy Dissipation

135 Computing Rayleigh Damping Proportionality Factors (Using NONLIN Pro) Passive Energy Dissipation

136 Example: Discrete (Stiff Braces) vs Rayleigh Damping 14 Peak Roof Displacement, Inches Discrete Dampers Rayleigh Damping First Mode Damping, % Critical Very Stiff Braces c 1.5c Passive Energy Dissipation

137 Example: Discrete (Stiff Braces) vs Rayleigh Damping Peak Base Shear, Kips Discrete Dampers Rayleigh Damping: Inertial Rayleigh Damping: Columns First Mode Damping, % Critical Very Stiff Braces c 1.5c Passive Energy Dissipation

138 14 Example: Effect of Brace Stiffness (Discrete Damping Model) Peak Roof Displacement, Inches Very Stiff Braces Reasonably Stiff Braces c 1.5c First Mode Damping, % Critical Passive Energy Dissipation

139 Example: Effect of Brace Stiffness (Discrete Damping Model) Peak Base Peak Shear, Inertial Kips Force, (from Kips Inertial Forces) Very Stiff Braces Reasonably Stiff Braces First Mode Damping, % Critical c 1.5c Passive Energy Dissipation

140 Example: Discrete (Flexible Braces) vs Rayleigh Damping 14 Peak Roof Displacement, Inches Discrete Model with Flexible Braces Rayleigh Damping First Mode Damping, % Critical c 1.5c Passive Energy Dissipation

141 Example: Discrete (Flexible Braces) vs Rayleigh Damping Peak Base Shear, Peak Inertial Kips (from Forcer, Inertial Kips Forces) Discrete Model with Flexible Braces 200 Rayleigh Damping 0 c c 20.0 First Mode Damping, % Critical Passive Energy Dissipation

142 Shear, Kips Example: Effect of Brace Stiffness on Peak Story Shear Forces Columns Braces (Damper) STIFF BRACES Time, Seconds Shear, Kips Colum ns Braces (Dam per) FLEX BRACES Tim e, Seconds Passive Energy Dissipation

143 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

144 Summary: MDOF Analysis Procedures (Linear Systems and Linear Dampers) Use discrete damper elements and explicitly include these dampers in the system damping matrix. Perform response history analysis of full system. Preferred. Use discrete damper elements to estimate modal damping ratios and use these damping ratios in modal response history or modal response spectrum analysis. Dangerous. Use discrete damper elements to estimate modal damping ratios and use these damping ratios in a response history analysis which incorporates Rayleigh proportional damping. Dangerous. Passive Energy Dissipation

145 Summary: MDOF Analysis Procedures (Linear Systems with Nonlinear Dampers) Use discrete damper elements and explicitly include these dampers in the system damping matrix. Perform response history analysis of full system. Preferred. Replace nonlinear dampers with equivalent energy based linear dampers, and then use these equivalent dampers in the system damping matrix. Perform response history analysis of full system. Very Dangerous. Replace nonlinear dampers with equivalent energy based linear dampers, use modal strain energy approach to estimate modal damping ratios, and then perform response spectrum or modal response history analysis. Very Dangerous. Passive Energy Dissipation

146 Summary: MDOF Analysis Procedures (Nonlinear Systems with Nonlinear Dampers) Use discrete damper elements and explicitly include these dampers in the system damping matrix. Explicitly model inelastic behavior in superstructure. Perform response history analysis of full system. Preferred. Replace nonlinear dampers with equivalent energy based linear dampers and use modal strain energy approach to estimate modal damping ratios. Use pushover analysis to represent inelastic behavior in superstructure. Use capacity-demand spectrum approach to estimate system deformations. Do This at Your Own Risk! Passive Energy Dissipation

147 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

148 MDOF Solution for Non-Classically Damped Structures Using Complex Modal Analysis Mv( & t ) + C I v( & t ) + C A v( & t ) + F S ( t ) MRv&& Modal Analysis using Damped Mode Shapes: = g ( t ) Treat C I as modal damping and model C A explicitly. Solve by modal superposition using damped (complex) mode shapes and frequencies. System (dampers and structure) must be linear. Passive Energy Dissipation

149 Damped Eigenproblem Mv( && t ) + C v( & t ) + Kv( t ) = A 0 EOM for Damped Free Vibration Assume C I is negligible Linear Structure State Vector: Z State-Space Transformation: State Matrix: v& = v H = Z & = HZ M 1 I C A M 0 1 K Passive Energy Dissipation

150 Solution of Damped Eigenproblem Assume Harmonic Response for n-th mode: Z n = Pe λ n n t Substitute Response into State Space Equation: P λ = HP n n n Damped Eigenproblem for n-th Mode PΛ = HP Damped Eigenproblem for All Modes Eigenvalue Matrix: (* = complex conjugate) λ Λ= * λ λ = diag [ λ ] n Eigenvector Matrix: * * P Φλ Φ λ = * Φ Φ Passive Energy Dissipation

151 Extracting Modal Damping and Frequency from Complex Eigenvalues Complex Eigenvalue for Mode n: λ = ξ ω ± iω 1 ξ 2 n n n n n Analogous to Roots of Characteristic Equation for SDOF Damped Free Vibration Problem Modal Frequency: ω n = λ n Modal Damping Ratio: ξ n = R ( λ ) λ n n Note: i = 1 Passive Energy Dissipation

152 Extracting Damped Mode Shapes P ΦΛ Φ * Φ Λ = * Φ * Damped Mode Shapes Passive Energy Dissipation

153 Damped Mode Shapes U 1 Imaginary U 2 U 3 U 4 φ = a a a a ib ib ib ib Real Passive Energy Dissipation

154 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

155 Example: Damped Mode Shapes and Frequencies m = 1.0 k-sec 2 /in. Nonproportional Damping m = 1.5 k = 200 k/in. m = 1.5 k = 250 m = 1.5 k = 300 m = 1.5 k = 350 k = 400 c = 20 k-sec/in c = 30 Passive Energy Dissipation

156 Example: Damped Mode Shapes and Frequencies System with Non-Classical Damping Using UNDAMPED MODE SHAPES Using DAMPED MODE SHAPES* Frequency (rad/sec) Damping Ratio Frequency (rad/sec) Damping Ratio Obtained from MSE Method *Table is for model with VERY STIFF braces. Significant Differences in Higher Mode Damping Ratios Passive Energy Dissipation

157 Example: Damped Mode Shapes and Frequencies System with Non-Classical Damping Real Component of Mode Shape Level 1 Level 2 Level 3 Level 4 Level 5 Mode = 1 ω = 4.58 ξ = Imaginary Component of Mode Shape Passive Energy Dissipation

158 Modal Amplitude Level 5 Level 4 Level 3 Level 3 Level Time, Seconds Story Level Modal Displacement Passive Energy Dissipation

159 Example: Damped Mode Shapes and Frequencies System with Non-Classical Damping Imaginary Component of Mode Shape Level 5 Level 4 Level 3 Level 2 Level 1 Mode = 2 ω = ξ = Real Component of Mode Shape Passive Energy Dissipation

160 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

161 An Unexpected Effect of Passive Damping The larger the damping coefficient C, the smaller the damping ratio ξ. Why? Note: Occurs for toggle-braced systems only. Huntington Tower Huntington Ave, Boston, MA - New 38-story steel-framed building Direct-acting and toggle-brace dampers kn (292 kips), +/- 101 mm (+/- 4 in.) - Dampers suppress wind vibration Passive Energy Dissipation

162 Toggle Brace Deployment Huntington Tower Passive Energy Dissipation

163 Example: Toggle Brace Damping System 360 U 1 U U Units: inches Passive Energy Dissipation

164 Methods of Analysis Used to Determine Damping Ratio Energy Ratios for Steady-State Harmonic Loading: ξ = E D /4πE S Modal Strain Energy Free Vibration Log Decrement Damped Eigenproblem C =10 to 40 k-sec/in (increments of 10) A =10 to 100 in 2 (increments of 10) Passive Energy Dissipation

165 Computed Damping Ratios for System With A = 10 A10 Damping Ratio Damping Energy Ratio Energy Modal Strain Energy Log Decrement Complex Damped Eigenproblem Eigenvalues Damping Coefficient Constant (k-sec/in) Passive Energy Dissipation

166 Computed Damping Ratios for System With A = A20 Damping Ratio Damping Energy Ratio Energy Modal Strain Energy Log Decrement Complex Damped Eigenproblem Eignevalues Damping Coefficient Constant (k-sec/in) Passive Energy Dissipation

167 Computed Damping Ratios for System With A = 30 A30 Damping Ratio Damping Energy Ratio Energy Modal Strain Energy Log Decrement Complex Damped Eigenproblem Eigenvalues Damping Damping Coefficient Constant (k-sec/in) Passive Energy Dissipation

168 Computed Damping Ratios for System With A = A50 Damping Ratio Damping Energy Ratio Energy Modal Strain Energy Log Decrement Complex Damped Eigenproblem Eigenvalues Damping Damping Coefficient Constant (k-sec/in) (k-sec/in) Passive Energy Dissipation

169 Why Does Damping Ratio Reduce for Low Brace Area/Damping Coefficient Ratios? U 1 U 3 U 2 U D Displacement in Damper is Out-of-Phase with Displacement at DOF 1 Passive Energy Dissipation

170 Phase Difference Between Damper Displacement and Frame Displacement Displacement, in Device Damper Lateral Frame A=10 C=40 A/C=0.25 Time, Seconds Displacement, in Damper Device Lateral Frame A=50 C=40 A/C= Time, sec Passive Energy Dissipation

171 Damped Mode Shapes for System With A=20 in C=0 0.4 C= C=20 IMAGINARY IMAGINARY IMAGINARY REAL REAL REAL C= C= C=50 IMAGINARY IMAGINARY IMAGINARY REAL U 3 U U 2 REAL REAL Passive Energy Dissipation

172 Interim Summary Related to Modeling and Analysis (1) Viscously damped systems are very effective in reducing damaging deformations in structures. With minor exceptions, viscously damped systems are non-classical, and must be modeled explicitly using dynamic time history analysis. Avoid the use of the Modal Strain Energy method (it may provide unconservative results) Passive Energy Dissipation