GTStrudl Damping Models for Dynamic Analysis Michael H. Swanger, Ph.D. CASE Center Georgia Tech

Topics Background Simple Modal Damping Weighted Average Composite Modal Damping Rayleigh Proportional Damping The Viscous Damper Element General Composite Modal Damping New Damping Models/Functions

Background

Simple Modal Damping DAMPING RATIOS 0.05 5 0.03 7 0.02 10 TRANSIENT LOAD ‘TH1’ SUPPORT ACCELERATION TRANSLATION X FILE ‘ELCENTRO’ INTEGRATE FROM 0.0 TO 55.0 AT 0.01 END LOAD LIST ‘TH1’ PERFORM TRANSIENT ANALYSIS RESPONSE SPECTRUM LOAD ‘RS1’ TRANSLATION X 1.0 FILE ‘RSCurve1’ LOAD LIST ‘RS1’ PERFORM RESPONSE SPECTRUM ANALYSIS

Simple Modal Damping Frequency-dependent Damping Ratios Per NRC Reg. 1 Simple Modal Damping Frequency-dependent Damping Ratios Per NRC Reg. 1.61, March 2007

Simple Modal Damping STORE RESPONSE SPECTRUM … Design Response Spectrum per FEMA 356

Simple Modal Damping UNITS IN LBS CYC SEC StdMASS CREATE RESPONSE SPECTRUM ACCELERATION LINEAR VS FREQUENCY LINEAR FILE 'RS1‘ FREQUENCY RANGE FROM 0.10000 TO 40.00000 AT 0.10000 DAMPING RATIOS 0.01 0.06 0.15 USE ACCELERATION TIME HISTORY FILES 'ELCENTRO' INTEGRATE USING DUHAMEL DIVISOR 20.00000 END OF CREATE RESPONSE SPECTRUM

Weighted Average Composite Modal Damping

Weighted Average Composite Modal Damping Recommended Modal Damping Values per NRC Reg. 1.61, March 2007

Weighted Average Composite Modal Damping UNITS INCHES KIPS JOINT RELEASES GROUP 'support' KFX 5.0 DFX 0.04 INERTIA OF JOINTS WEIGHT EXISTING TRANSL ALL 0.5 DAMPING 0.04 CONSTANTS MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.04 MEMBERS GROUP 'beams' MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.04 MEMBERS GROUP 'columns‘ eigenvalue analysis DYNAMIC PARAMETERS RESPONSE DAMPING STIFFNESS 1.0 MASS 0.0 END COMPUTE MODAL DAMPING RATIOS AVERAGE

Weighted Average Composite Modal Damping Uniform Damping, Damping Ratios: All Components 4% TYPE PLANE FRAME XY MATERIAL STEEL OUTPUT LONG NAME UNITS INCHES KIPS JOINT COORDINATES $ Name X coord Y coord $ -------- ----------------- ----------------- 1 X 0.00000 Y 0.00000 2 X 120.00000 Y 0.00000 . 18 X 240.00000 Y 360.00000 19 X 360.00000 Y 360.00000 20 X 480.00000 Y 360.00000 DEFINE GROUP 'support' ADD JOINTS 1 to 5 STATUS SUPPORT JOINT GROUP 'support' $* ** $* ** Modal damping ratio for spring support $* ** stiffnesses JOINT RELEASES GROUP 'support' KFX 5.0 DFX 0.04

Weighted Average Composite Modal Damping $* ** $* ** Eigenvalue analysis INERTIA OF JOINTS LUMPED INERTIA OF JOINTS WEIGHT EXISTING TRANSL ALL 0.5 EIGENVALUE PARAMETERS SOLVE USING GTLANCZOS NUMBER OF MODES 20 PRINT MAX END DYNAMIC ANALYSIS EIGENVALUE $* ** Compute weighted average composite modal $* ** damping ratios DYNAMIC PARAMETERS RESPONSE DAMPING STIFFNESS 1.0 COMPUTE MODAL DAMPING RATIOS AVERAGE $* ** Response spectrum analsis UNITS INCHES CYCLES CREATE RESPONSE SPECTRUM ACCELERATION LIN VS FREQUENCY LIN FILE 'RS' FREQUENCY RANGE FROM 1.00000 TO 40.00000 AT 1.00000 DAMPING RATIOS 0.04 USE ACCELERATION TIME HISTORY FILES 'ELCENTRO' INTEGRATE USING DUHAMEL DIVISOR 20.00000 END OF CREATE RESPONSE SPECTRUM MEMBER INCIDENCES $ Name Start joint End joint $ -------- -------- -------- $ Columns 1 1 6 2 6 11 3 11 16 4 2 7 . 23 14 15 24 16 17 25 17 18 26 18 19 27 19 20 DEFINE GROUP 'columns' ADD MEMBERS 1 TO 15 DEFINE GROUP 'beams' ADD MEMBERS 16 TO 27 MEMBER PROPERTIES GROUP 'beams' T 'WBEAM9' 'W18x35' GROUP 'columns' T 'WCOLUMN9' 'W14x53' $* ** $* ** Damping ratios for member structural $* ** stiffness damping CONSTANTS MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.04 - MEMBERS GROUP 'beams' GROUP 'columns'

Weighted Average Composite Modal Damping RESPONSE SPECTRUM LOAD 'RS1' SUPPORT ACCELERATION TRANSLATION X 1.0 FILE 'RS' END PERFORM RESPONSE SPECTRUM ANALYSIS COMPUTE RESPONSE SPECTRUM DISPLACEMENTS OUTPUT MODAL CONTRIBUTIONS ON LIST RESPONSE SPECTRUM DISPL JOINT 16 $* ** $* ** Mode superposition transient analysis TRANSIENT LOAD ‘TH1’ TRANSLATION X FILE ‘ELCENTRO’ INTEGRATE FROM 0.0 TO 10.0 AT 0.01 PERFORM TRANSIENT ANALYSIS LIST TRANSIENT MAX DISPL JOINT 16 RS Analysis, 4% Uniform TH Analysis, 4% Uniform TH Analysis, No Damping

Weighted Average Composite Modal Damping Damping Ratios (non-uniform): Beams 5%, Columns 3%, Foundation 10% TYPE PLANE FRAME XY MATERIAL STEEL OUTPUT LONG NAME UNITS INCHES KIPS JOINT COORDINATES $ Name X coord Y coord $ -------- ----------------- ----------------- 1 X 0.00000 Y 0.00000 2 X 120.00000 Y 0.00000 . 18 X 240.00000 Y 360.00000 19 X 360.00000 Y 360.00000 20 X 480.00000 Y 360.00000 DEFINE GROUP 'support' ADD JOINTS 1 to 5 STATUS SUPPORT JOINT GROUP 'support' $* ** $* ** Modal damping ratio for spring support $* ** stiffnesses JOINT RELEASES GROUP 'support' KFX 5.0 DFX 0.10

Weighted Average Composite Modal Damping $* ** $* ** Eigenvalue analysis INERTIA OF JOINTS LUMPED INERTIA OF JOINTS WEIGHT EXISTING TRANSL ALL 0.5 EIGENVALUE PARAMETERS SOLVE USING GTLANCZOS NUMBER OF MODES 20 PRINT MAX END DYNAMIC ANALYSIS EIGENVALUE $* ** Compute weighted average composite modal $* ** damping ratios DYNAMIC PARAMETERS RESPONSE DAMPING STIFFNESS 1.0 COMPUTE MODAL DAMPING RATIOS AVERAGE $* ** Response spectrum analsis UNITS INCHES CYCLES CREATE RESPONSE SPECTRUM ACCELERATION LIN VS FREQUENCY LIN FILE 'RS' FREQUENCY RANGE FROM 1.00000 TO 40.00000 AT 1.00000 DAMPING RATIOS 0.01 0.07 0.15 USE ACCELERATION TIME HISTORY FILES 'ELCENTRO' INTEGRATE USING DUHAMEL DIVISOR 20.00000 END OF CREATE RESPONSE SPECTRUM MEMBER INCIDENCES $ Name Start joint End joint $ -------- -------- -------- $ Columns 1 1 6 2 6 11 3 11 16 4 2 7 . 23 14 15 24 16 17 25 17 18 26 18 19 27 19 20 DEFINE GROUP 'columns' ADD MEMBERS 1 TO 15 DEFINE GROUP 'beams' ADD MEMBERS 16 TO 27 MEMBER PROPERTIES GROUP 'beams' T 'WBEAM9' 'W18x35' GROUP 'columns' T 'WCOLUMN9' 'W14x53' $* ** $* ** Damping ratios for member structural $* ** stiffness damping CONSTANTS MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.05 - MEMBERS GROUP 'beams' MODAL DAMPING PROPORTIONAL TO STIFFNESS 0.03 - GROUP 'columns'

Weighted Average Composite Modal Damping RESPONSE SPECTRUM LOAD 'RS1' SUPPORT ACCELERATION TRANSLATION X 1.0 FILE 'RS' END PERFORM RESPONSE SPECTRUM ANALYSIS COMPUTE RESPONSE SPECTRUM DISPLACEMENTS OUTPUT MODAL CONTRIBUTIONS ON LIST RESPONSE SPECTRUM DISPL JOINT 16 $* ** $* ** Mode superposition transient analysis TRANSIENT LOAD ‘TH1’ TRANSLATION X FILE ‘ELCENTRO’ INTEGRATE FROM 0.0 TO 10.0 AT 0.01 PERFORM TRANSIENT ANALYSIS LIST TRANSIENT MAX DISPL JOINT 16 WtAvgCMD, Non-Uniform, KFX = 50 k WtAvgCMD, Non-Uniform, KFX = 1 k

Rayleigh Proportional Damping

Rayleigh Proportional Damping Classical Damping = 5% at ω = 3.16 Hz and 100 Hz DAMPING PROPORTIONAL TO STIFFNESS 1.5271E-4 MASS 1.9238 Non-classical Damping at ω =3.16 Hz and 100 Hz Beams and columns 5% Supports 15% CONSTANTS RAYLEIGH DAMPING PROPORTIONAL TO STIFFNESS 1.5271E-4 - MEMBERS GROUP LIST 'beams‘ ‘columns’ RAYLEIGH DAMPING PROPORTIONAL TO MASS 1.9238 MEMBERS 6 TO 20 GROUP ‘ beams’ RAYLEIGH DAMPING PROPORTIONAL TO MASS 5.7714 MEMBERS 1 TO 5 UNITS INCHES KIPS JOINT RELEASES 1 TO 5 KFX 5.0 DFX 4.5813E-4

Rayleigh Proportional Damping Classical Damping TYPE PLANE FRAME XY MATERIAL STEEL OUTPUT LONG NAME UNITS INCHES KIPS JOINT COORDINATES $ Name X coord Y coord $ -------- ----------------- ----------------- 1 X 0.00000 Y 0.00000 2 X 120.00000 Y 0.00000 . 18 X 240.00000 Y 360.00000 19 X 360.00000 Y 360.00000 20 X 480.00000 Y 360.00000 DEFINE GROUP 'support' ADD JOINTS 1 to 5 STATUS SUPPORT JOINT GROUP 'support‘ JOINT RELEASES GROUP 'support' KFX 5.0

Rayleigh Proportional Damping Classical Damping MEMBER INCIDENCES $ Name Start joint End joint $ -------- -------- -------- $ Columns 1 1 6 2 6 11 3 11 16 4 2 7 . 23 14 15 24 16 17 25 17 18 26 18 19 27 19 20 DEFINE GROUP 'columns' ADD MEMBERS 1 TO 15 DEFINE GROUP 'beams' ADD MEMBERS 16 TO 27 MEMBER PROPERTIES GROUP 'beams' T 'WBEAM9' 'W18x35' GROUP 'columns' T 'WCOLUMN9' 'W14x53‘ INERTIA OF JOINTS LUMPED INERTIA OF JOINTS WEIGHT EXISTING TRANSL ALL 0.5 $* ** $* ** Rayleigh proportional damping for $* ** structural damping at 0.05 for $* ** w = 19.84 rad/sec (3.16 Hz) and $* ** 635 rad/sec (100 Hz) DAMPING PROPORTIONAL TO STIFFNESS 1.5271E-4 MASS 1.9238 $* ** $* ** Perform direct integration time history $* ** analysis UNITS INCHES CYCLES TRANSIENT LOAD 'T1' SUPPORT ACCELERATION TRANSLATION X FILE ‘ELCENTRO’ INTEGRATE FROM 0.0 TO 10.0 AT 0.01 END DYNAMIC ANALYSIS PHYSICAL NEWMARK BETA 0.25 LIST TRANSIENT MAX DISPL JOINTS 1 16 Rayleigh Damping, Classical, Physical Analysis No Rayleigh Damping, Physical Analysis Modal TH Analysis, 5% Uniform Damping

Rayleigh Proportional Damping Non-classical Damping TYPE PLANE FRAME XY MATERIAL STEEL OUTPUT LONG NAME UNITS INCHES KIPS JOINT COORDINATES $ Name X coord Y coord $ -------- ----------------- ----------------- 1 X 0.00000 Y 0.00000 2 X 120.00000 Y 0.00000 . 18 X 240.00000 Y 360.00000 19 X 360.00000 Y 360.00000 20 X 480.00000 Y 360.00000 DEFINE GROUP 'support' ADD JOINTS 1 to 5 STATUS SUPPORT JOINT GROUP 'support‘ $* ** $* ** Rayleigh damping factor for spring support $* ** stiffnesses corresponding to 15% damping in $* ** the support for w = 19.84 rad/sec (3.16 Hz) and $* ** 635 rad/sec (100 Hz) JOINT RELEASES GROUP 'support' KFX 5.0 DFX 4.5813E-4

Rayleigh Proportional Damping Non-classical Damping MEMBER INCIDENCES $ Name Start joint End joint $ -------- -------- -------- $ Columns 1 1 6 2 6 11 3 11 16 4 2 7 . 23 14 15 24 16 17 25 17 18 26 18 19 27 19 20 DEFINE GROUP 'columns' ADD MEMBERS 1 TO 15 DEFINE GROUP 'beams' ADD MEMBERS 16 TO 27 MEMBER PROPERTIES GROUP 'beams' T 'WBEAM9' 'W18x35' GROUP 'columns' T 'WCOLUMN9' 'W14x53‘ INERTIA OF JOINTS LUMPED INERTIA OF JOINTS WEIGHT EXISTING TRANSL ALL 0.5 $* ** $* ** Rayleigh damping factors for joint and member $* ** inertias corresponding to 15% damping in the $* ** support and 5% elsewhere for w = 19.84 rad/sec $* ** (3.16 Hz) and 635 rad/sec (100 Hz) 1 to 5 TRANSL ALL 0.5 DAMPING 5.7714 6 TO 20 TRANSL ALL 0.5 DAMPING 1.9238 CONSTANTS RAYLEIGH DAMPING PROPORTIONAL TO MASS 5.7714 - MEMBERS 1 TO 5 RAYLEIGH DAMPING PROPORTIONAL TO MASS 1.9238 - MEMBERS 6 TO 15 GROUP 'beams' $* ** $* ** Rayleigh damping for member stiffnesses at 5% $* ** for w = 19.84 rad/sec (3.16 Hz) and $* ** 635 rad/sec (100 Hz) RAYLEIGH DAMPING PROPORTIONAL TO STIFFNESS 1.5271E-4 - GROUP LIST 'beams' 'columns'

Rayleigh Proportional Damping Non-classical Damping $* ** $* ** Perform direct integration time history $* ** analysis UNITS INCHES CYCLES TRANSIENT LOAD 'TH1' SUPPORT ACCELERATION TRANSLATION X FILE 'ELCENTRO' INTEGRATE FROM 0.0 TO 10.0 AT 0.01 END DYNAMIC ANALYSIS PHYSICAL NEWMARK BETA 0.25 LIST TRANS MAX DISPL JOINT 1 16 Rayleigh Damping, Non-Classical, Physical Analysis Rayleigh Damping, Classical, Physical Analysis

The Viscous Damper Element

The Viscous Damper Element MEMBER PROPERTIES GROUP 'beams' T 'WBEAM9' 'W18x35' GROUP 'columns' T 'WCOLUMN9' 'W14x53‘ INERTIA OF JOINTS LUMPED INERTIA OF JOINTS WEIGHT EXISTING TRANSL ALL 0.5 $* ** $* ** Rayleigh proportional damping for $* ** structural damping at 0.05 for $* ** w = 19.84 rad/sec (3.16 Hz) and $* ** 635 rad/sec (100 Hz) DAMPING PROPORTIONAL TO STIFFNESS 1.5271E-4 - MASS 1.9238 $* ** Viscous damper elements at supports DAMPING ELEMENT DATA 'DE1' INC 1 GLOBAL CTX 1.E4 'DE2' INC 2 GLOBAL CTX 1.E4 'DE3' INC 3 GLOBAL CTX 1.E4 'DE4' INC 4 GLOBAL CTX 1.E4 'DE5' INC 5 GLOBAL CTX 1.E4 END

The Viscous Damper Element $* ** $* ** Perform direct integration time history $* ** analysis UNITS INCHES CYCLES TRANSIENT LOAD 'TH1' SUPPORT ACCELERATION TRANSLATION X FILE 'ELCENTRO' INTEGRATE FROM 0.0 TO 10.0 AT 0.01 END DYNAMIC ANALYSIS PHYSICAL NEWMARK BETA 0.25 LIST TRANS MAX DISPL JOINT 16 Viscous Damping, CTX = 1.e7 lb Viscous Damping, CTX = 100 lb

General Composite Modal Damping

General Composite Modal Damping . DEFINE GROUP 'support' ADD JOINTS 1 to 5 STATUS SUPPORT JOINT GROUP 'support' JOINT RELEASES GROUP 'support' KFX 5.0 MEMBER INCIDENCES $ Name Start joint End joint $ -------- -------- -------- $ Columns 1 1 6 2 6 11 MEMBER PROPERTIES GROUP 'beams' T 'WBEAM9' 'W18x35' GROUP 'columns' T 'WCOLUMN9' 'W14x53' $* ** $* ** Define damping properties for modal $* ** damping ratio computation: $* ** Rayleigh damping factors for structural $* ** damping at 0.05 for w = 19.4 rad/sec (3.16 Hz) $* ** and 635 rad/sec (100 Hz) DAMPING PROPORTIONAL TO STIFFN 1.5271E-4 - MASS 1.9238 $* ** Viscous damper elements at supports DAMPING ELEMENT DATA 'DE1' INC 1 GLOBAL CTX 1.E4 'DE2' INC 2 GLOBAL CTX 1.E4 'DE3' INC 3 GLOBAL CTX 1.E4 'DE4' INC 4 GLOBAL CTX 1.E4 'DE5' INC 5 GLOBAL CTX 1.E4 END PRINT DAMPING ELEMENT DATA $* ** Eigenvalue analysis INERTIA OF JOINTS LUMPED INERTIA OF JOINTS WEIGHT EXISTING TRANSL ALL 0.5 EIGENVALUE PARAMETERS SOLVE USING GTLANCZOS NUMBER OF MODES 20 PRINT MAX DYNAMIC ANALYSIS EIGENVALUE

General Composite Modal Damping $* ** $* ** Compute modal damping ratios assuming $* ** classical proportional damping COMPUTE MODAL DAMPING RATIOS PROPORTIONAL INCLUDE COUPLING ALL $* ** Mode superposition transient analysis UNITS INCHES CYCLES TRANSIENT LOAD 'TH1' SUPPORT ACCELERATION TRANSLATION X FILE 'ELCENTRO' INTEGRATE FROM 0.0 TO 10.0 AT 0.01 END PERFORM TRANSIENT ANALYSIS LIST TRANSIENT MAX DISPL JOINT 16 DYNAMIC ANALYSIS PHYSICAL NEWMARK BETA 0.25 LIST TRANS MAX DISPL JOINT 16 General Composite modal damping, No CTX General Composite modal damping, CTX = 1.e7 lb General Composite modal damping, CTX = 100 lb

General Composite Modal Damping DAMPING PROPORTIONAL TO STIFFN 1.5271E-4 MASS 1.9238 COMPUTE MODAL DAMPING RATIOS PROPORTIONAL

New Damping Models/Functions Direct Computation of “Rayleigh” Damping Ratios Caughey Damping

New Damping Models/Functions Superposition of Modal Damping Matrices