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**Response Spectrum Analysis Enhancements**

GTStrudl Version 30 Response Spectrum Analysis Enhancements Related To NRC Regulatory Guide 1.92, Revision 2 COMBINING MODAL RESPONSES AND SPATIAL COMPONENTS in SEISMIC RESPONSE ANALYSIS Michael H. Swanger, Ph.D. Georgia Tech CASE Center GTSUG 2008 June 23-26, 2008 Las Vegas, NV

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**Topics Background GTStrudl Enhancements, Version 30 Example**

NRC Reg Guide 1.92, Rev 1 Positions Response Spectrum Characteristics Response Spectrum Solution Strategy NRC Reg Guide 1.92, Rev 2 Positions Response spectrum Solution Strategy GTStrudl Enhancements, Version 30 The RESPONSE SPECTRUM LOAD/ MODE FACTORS Command The ALGEBRAIC Mode Combination Total Response Example NRC Reg Guide 1.92 Rev 1 vs Rev 2

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Background 3

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Background

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**Background NRC Reg Guide 1.92, Rev 1 Positions**

Response Spectrum Characteristics All modes are assumed to be out-of-phase with the ground acceleration and out-of-phase with each other Acceleration All modes having frequencies ≤ some arbitrary cutoff frequency are deemed “significant” for inclusion in the response spectrum analysis Frequency Note: 1976, the date of Reg 1.92, Rev 1, was prior to many of the significant developments in response spectrum analysis that we take for granted today!

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**Background NRC Reg Guide 1.92, Rev 1 Positions**

Response Spectrum Solution Strategy For each ground motion direction, k = 1, 2, 3, the modal maximum responses from all “significant” modes, having no time and phase characteristics, are combined according to a statistical rule, such as SRSS. The total response is computed from the SRSS of the combined modal responses in each ground motion direction

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**Background NRC Reg Guide 1.92, Rev 1 Positions**

Response Spectrum Solution Strategy If frequencies are not closely spaced§: SRSS Mode Combination Method § two consecutive modes are defined as closely spaced if their frequencies differ from each other by no more than 10 percent of the lower frequency

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**Background NRC Reg Guide 1.92, Rev 1 Positions**

Response Spectrum Solution Strategy If frequencies are closely spaced: NRC Grouping Method NRC Ten Percent Method NRC Double Sum Method td = duration of earthquake

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Characteristics Frequency Low Frequency Out-of-Phase Response Mid Frequency Transition from Out-of-Phase to In-Phase Response High Frequency In-Phase Rigid Static F1 = frequency at which peak spectral acceleration is observed F2 = frequency above which the SDOF (modal) oscillators are in-phase with the transient acceleration input used to generate the spectrum and in phase with each other FZPA = frequency at which the spectral acceleration returns to the zero period acceleration; maximum base acceleration of transient acceleration input used to generate the spectrum

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Characteristics Frequency Low Frequency Out-of-Phase Response Mid Frequency Transition from Out-of-Phase to In-Phase Response High Frequency In-Phase Rigid Static fi ≤ F1 Maximum response from periodic or transient response in the modal frequency fi. Maximum modal (oscillator) responses are out-of-phase with one another. 10

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Characteristics Mid Frequency Transition from Out-of-Phase to In-Phase Response Low Frequency Out-of-Phase Response High Frequency In-Phase Rigid Static Response Frequency fi ≥ F2 Maximum response from steady state response. The maximum modal responses are in phase with one another. 11

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Characteristics Mid Frequency Transition from Out-of-Phase to In-Phase Response Low Frequency Out-of-Phase Response High Frequency In-Phase Rigid Static Response Frequency F1 < fi < F2 Response is part periodic and part rigid. Maximum modal responses transition from out-of-phase to in phase. 12

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Solution Strategy For each mode i, in each ground motion direction k, the response is separated into a periodic part and a rigid part: The periodic modal response portions are combined using a double sum rule:

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Solution Strategy The rigid modal responses are combined algebraically, including the residual rigid contribution from the missing mass: The total response in each ground motion direction is computed from the SRSS of the modal combinations of the periodic and rigid responses:

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Solution Strategy Finally, the complete response is computed by performing the SRSS on the total responses in the three ground motion directions: A rule is also acceptable for combination of the spatial response components

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Solution Strategy Computation of rigid response factor αki ; The Gupta Method: Mid Frequency Transition from Out-of-Phase to In-Phase Response Low Frequency Out-of-Phase Response High Frequency In-Phase Rigid Static Response Frequency

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Solution Strategy Periodic responses are combined using a double sum rule: εij computed according to the following methods: SRSS Method NRC Double Sum Method (Rosenbleuth correlation coefficient) CQC method (Der Kiureghian’s correlation coefficient) 17

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Solution Strategy Computation of the Residual Rigid Response for all fi ≥ FZPA by the Missing Mass Method: The Missing Mass Method is quite accurate and is most important for adequately capturing the high-frequency response near supports

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**Background NRC Reg Guide 1.92, Rev 2 Positions**

Response Spectrum Solution Strategy Note: Under Rev 2, the response spectrum solution also may be performed according to Reg 1.92, Rev 1 provided that the residual rigid response due to the missing mass is included 19

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**GTStrudl Enhancements, Version 30**

RESPONSE SPECTRUM LOAD/MODE FACTORS Command Syntax Purpose: To compute α and (1 – α2)1/2 for each active mode for the defined response spectrum load 20

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**GTStrudl Enhancements, Version 30**

RESPONSE SPECTRUM LOAD/MODE FACTORS Command Example UNITS CYCLES SECONDS RESPONSE SPECTRUM LOAD ‘100R’ SUPPORT ACCELERATION TRANSLATION X FILE ‘ELC-RS’ MODE FACTORS COMPUTE RIGID RESPONSE FZPA 40.0 END RESPONSE SPECTRUM LOAD RESPONSE SPECTRUM LOAD ‘100P’ MODE FACTORS COMPUTE PERIODIC RESPONSE FZPA 40.0 Note: FZPA is specified (FZPA 40.0); therefore: F1 = Samax/(2πSvmax) F2 = (F1 + 2FZPA)/3

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**GTStrudl Enhancements, Version 30**

The ALGEBRAIC Mode Combination

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**GTStrudl Enhancements, Version 30**

The ALGEBRAIC Mode Combination Example LOAD LIST ‘100R’ $ Rigid RS Components COMPUTE RESPONSE SPECTRUM DISPLACEMENTS MODE COMBINATION ALGEBRAIC COMPUTE RESPONSE SPECTRUM FORCES MODE COMBINATION ALBEGRAIC CREATE PSEUDO STATIC LOAD ‘PS100R’ FROM ALGEBRAIC OF LOAD ‘100R’ . LOAD LIST ‘100P’ $ Periodic RS Components COMPUTE RESPONSE SPECTRUM DISPLACEMENTS MODE COMBINATION CQC COMPUTE RESPONSE SPECTRUM FORCES MODE COMBINATION CQC CREATE PSEUDO STATIC LOAD ‘PS100P’ FROM CQC OF LOAD ‘100P’

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**GTStrudl Enhancements, Version 30**

Total Rigid, Directional, and Solution Response Example $* ** $* ** Total Rigid Response UNITS CYCLES SECONDS FORM MISSING MASS LOAD ‘100M’ FROM RESPONSE SPECTRUM LOAD ‘100R’ – CUTOFF FREQUENCY 40.0 . STIFFNESS ANALYSIS CREATE LOAD COMBINATION ‘100RTOT’ SPECS ‘PS100R’ 1.0 ‘100M’ 1.0 $* ** Total Directional Response CREATE LOAD COMBINATION ‘100TOT’ TYPE RMS SPECS ‘PS100P’ 1.0 – ‘100RTOT’ 1.0 $* ** Total Solution Response CREATE LOAD COMBINATION ‘EQTOT’ TYPE RMS SPECS - ‘100TOT’ 1.0 ‘200TOT’ 1.0 ‘300TOT’ 1.0

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**Example 1 Columns: W14X53 Beams (Global X): W18X35**

12’) 10’) 10’) Columns: W14X53 Beams (Global X): W18X35 Beams (Global Z): W18X50 210 Joints, 474 Members Additional Mass: 1 kip, all joints, Global X and Z Seismic Loading: El Centro RS, Global X and Z

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**El Centro Response Spectrum**

Example 1 El Centro Response Spectrum UNITS FEET CYCLES SECONDS CREATE RESPONSE SPECTRUM ACCELERATION - LINEAR VS FREQUENCY LINEAR FILE 'ELC-RS' FREQUENCY RANGE FROM TO AT DAMPING RATIOS 0.05 USE ACCELERATION TIME HISTORY FILES 'ELCENTRO' INTEGRATE USING DUHAMEL DIVISOR END OF CREATE RESPONSE SPECTRUM FZPA F1 = 1.9 HZ F2 = 27.3 HZ 26

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**Example 1 Revision 1 Revision 2 27 UNITS INCHES KIPS**

DEAD LOAD 'DLX' DIR X ALL MEMBERS DEAD LOAD 'DLZ' DIR Z ALL MEMBERS INERTIA OF JOINTS FROM LOAD 'DLX' SAME DOFS INERTIA OF JOINTS FROM LOAD 'DLZ' SAME DOFS INERTIA OF JOINTS WEIGHT EXISTING TRANSL X 1.0 Z 1.0 UNITS CYCLES SECONDS EIGENVALUE PARAMETERS SOLVE USING GTSEL FREQUENCY SPECS 0.0 TO 40.0 PRINT MAX END DYNAMIC ANALYSIS EIGENVALUE UNITS INCHES KIPS DEAD LOAD 'DLX' DIR X ALL MEMBERS DEAD LOAD 'DLZ' DIR Z ALL MEMBERS INERTIA OF JOINTS FROM LOAD 'DLX' SAME DOFS INERTIA OF JOINTS FROM LOAD 'DLZ' SAME DOFS INERTIA OF JOINTS WEIGHT EXISTING TRANSL X 1.0 Z 1.0 UNITS CYCLES SECONDS EIGENVALUE PARAMETERS SOLVE USING GTSEL FREQUENCY SPECS 0.0 TO 40.0 PRINT MAX END DYNAMIC ANALYSIS EIGENVALUE 27

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**Example 1 Revision 1 Revision 2 28 $* ****

$* ** Define response spectrum loads for response in the $* ** global X and Z directions RESPONSE SPECTRUM LOAD 100 SUPPORT ACCELERATION TRANSLATION X FILE 'ELC-RS' END RESPONSE SPECTRUM LOAD RESPONSE SPECTRUM LOAD 300 TRANSLATION Z FILE 'ELC-RS' UNITS INCHES KIPS CYCLES SEC DAMPING RATIOS PERFORM RESPONSE SPECTRUM ANALYSIS $* ** $* ** Define response spectrum loads for rigid response in $* ** the global X and Z directions RESPONSE SPECTRUM LOAD ‘100R' SUPPORT ACCELERATION TRANSLATION X FILE 'ELC-RS' MODE FACTORS COMPUTE RIGID RESPONSE FZPA 40.0 END RESPONSE SPECTRUM LOAD RESPONSE SPECTRUM LOAD ‘300R' TRANSLATION Z FILE 'ELC-RS' $* ** Define response spectrum loads for periodic response $* ** in the global X and Z directions RESPONSE SPECTRUM LOAD ‘100P' MODE FACTORS COMPUTE PERIODIC RESPONSE FZPA 40.0 RESPONSE SPECTRUM LOAD ‘300P' UNITS INCHES KIPS CYCLES SEC DAMPING RATIOS PERFORM RESPONSE SPECTRUM ANALYSIS LOAD LIST ‘100R' ‘300P' PRINT DYNAMIC LOAD DATA 28

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**Example 1 Revision 2 29 { 790} > PRINT DYNAMIC LOAD DATA .**

LOADING - 100R STATUS - ACTIVE RIGID Response Modal Scaling (NRC Guide 1.92, Rev. 2, Combination Method A) =========================================================================== F1 = F2 = FZPA = MODE FACTOR MODE FACTOR MODE FACTOR MODE FACTOR MODE FACTOR MODE FACTOR E E LOADING - 100P STATUS - ACTIVE PERIODIC Response Modal Scaling (NRC Guide 1.92, Rev. 2, Combination Method A) ============================================================================== F1 = F2 = FZPA = MODE FACTOR MODE FACTOR MODE FACTOR MODE FACTOR MODE FACTOR MODE FACTOR . E E E E E E+00 29

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**Response Spectrum Loadings 100R and 100P**

Example 1 Revision 2 Response Spectrum Loadings 100R and 100P Mode # X mass % Freq (HZ) α (1-α2)1/2 Total % Active % (Modes having X mass participation ≥ 0.05% listed) F2 = HZ F1 = 1.86 HZ 30

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**Example 1 Revision 1 Revision 2 31 $* ****

$* ** Compute modal and combined modal results LOAD LIST COMPUTE RESPONSE SPECTRUM DISPL MODE COMBINATION CQC COMPUTE RESPONSE SPECTRUM FORCES MODE COMBINATION CQC COMPUTE RESPONSE SPECTRUM REACTIONS MODE COMBINATION CQC CREATE PSEUDO STATIC LOAD 'PS100' FROM CQC OF LOAD ‘100’ CREATE PSEUDO STATIC LOAD 'PS300' FROM CQC OF LOAD ‘300’ $* ** $* ** Compute rigid modal and combined rigid modal results LOAD LIST ‘100R’ ‘300R’ COMPUTE RESPONSE SPECTRUM DISPL MODE COMBINATION ALG COMPUTE RESPONSE SPECTRUM FORCES MODE COMBINATION ALG COMPUTE RESPONSE SPECTRUM REACTIONS MODE COMBINATION ALG CREATE PSEUDO STATIC LOAD ‘PS100R’ FROM ALG OF LOAD ‘100R' CREATE PSEUDO STATIC LOAD ‘PS300R’ FROM ALG OF LOAD ‘300R' $* ** Compute Periodic modal and combined periodic modal $* ** results LOAD LIST ‘100P’ ‘100P’ COMPUTE RESPONSE SPECTRUM DISPL MODE COMBINATION CQC COMPUTE RESPONSE SPECTRUM FORCES MODE COMBINATION CQC COMPUTE RESPONSE SPECTRUM REACTIONS MODE COMBINATION CQC CREATE PSEUDO STATIC LOAD ‘PS100P’ FROM CQC OF LOAD ‘100P’ CREATE PSEUDO STATIC LOAD ‘PS300P’ FROM CQC OF LOAD ‘300P’ 31

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**Example 1 Revision 1 Revision 2 $* ****

$* ** Compute total combined modal results, including missing $* ** mass,in the global X and Z directions FORM MISSING MASS LOAD ‘100M’ FROM RESPONSE SPECTRUM LOAD 100 - DAMPING RATIO 0.05 CUTOFF FREQUENCY 28.77 FORM MISSING MASS LOAD ‘300M’ FROM RESPONSE SPECTRUM LOAD 300 - LOAD LIST ‘100M’ ‘300M’ STIFFN ANALYSIS GTSES $* ** Compute total response in the global X direction LOAD LIST ALL CREATE LOAD COMBINATION ‘100TOT’ TYPE RMS - SPECS ‘PS100’ 1.0 ‘100M’ 1.0 $* ** Compute total response in the global Z direction CREATE LOAD COMBINATION ‘300TOT’ TYPE RMS - SPECS ‘PS300’ 1.0 ‘300M’ 1.0 $* ** Compute total solution CREATE LOAD COMBINATION 'EQTOT' TYPE RMS - SPECS ‘100TOT’ 1.0 ‘300TOT’ 1.0 $* ** $* ** Compute total combined rigid results, including missing $* ** mass, in the global X and Z directions FORM MISSING MASS LOAD ‘100M’ FROM RESPONSE SPECTRUM LOAD ‘100P’ - DAMPING RATIO 0.05 CUTOFF FREQUENCY 28.77 FORM MISSING MASS LOAD ‘300M’ FROM RESPONSE SPECTRUM LOAD ‘300P’ - LOAD LIST ‘100M’ ‘300M’ STIFFN ANALYSIS GTSES CREATE LOAD COMBINATION ‘100RTOT’ SPECS ‘PS100R’ 1.0 ‘100M’ 1.0 CREATE LOAD COMBINATION ‘300RTOT’ SPECS ‘PS300R’ 1.0 ‘300M’ 1.0 $* ** Compute total response in the global X direction LOAD LIST ALL CREATE LOAD COMBINATION ‘100TOT’ TYPE RMS - SPECS ‘100RTOT’ 1.0 ‘PS100P’ 1.0 $* ** Compute total response in the global Z direction CREATE LOAD COMBINATION ‘300TOT’ TYPE RMS - SPECS ‘300RTOT’ 1.0 ‘PS300P’ 1.0 $* ** Compute total solution CREATE LOAD COMBINATION ‘EQTOT’ TYPE RMS - SPECS ‘300TOT 1.0 ‘300TOT’ 1.0

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**Example 1 Revision 1 Revision 2**

{ 804} > LOAD LIST 'PS100' '100M' '100TOT' { 805} > OUTPUT BY MEMBER { 806} > LIST REACTION JOINT 7 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL PS 100M 100TOT Revision 2 { 848} > LOAD LIST 'PS100P' 'PS100R' '100M' '100RTOT' '100TOT' { 849} > OUTPUT BY MEMBER { 850} > LIST REACTION JOINT 7 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL PS100R 100M 100RTOT PS100P 100TOT

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**Example 1 Revision 1 Revision 2**

{ 808} > LOAD LIST '100TOT' '300TOT' 'EQTOT' { 809} > OUTPUT BY MEMBER { 810} > LIST REACT JOINT 7 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL 100TOT 300TOT EQTOT Revision 2 { 852} > LOAD LIST '100TOT' '300TOT' 'EQTOT' { 853} > OUTPUT BY MEMBER { 854} > LIST REACT JOINT 7 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL 100TOT 300TOT EQTOT

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**Example 2 Material Concrete Columns: 18”x18”**

50.0 FT 10’) 10’) 10’) Material Concrete Columns: 18”x18” Floor and Wall Panel Thicknesses: 12” 2520 Joints, 342 Members, 2670 Plate FEs

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**Response Spectrum Loadings 100R and 100P**

Example 2 Revision 2 Response Spectrum Loadings 100R and 100P Mode # X mass % Freq (HZ) α (1-α2)1/2 . . (Total X mass particpation, modes 1-24 = 0.06%!) Total % (f ≤ 40 HZ) Active % (mass participation ≥ 0.001%) F2 = HZ F1 = 1.86 HZ 36

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**Example 2 Revision 1 Revision 2 37 37**

{ 804} > LOAD LIST 'PS100' '100M' '100TOT' { 805} > OUTPUT BY MEMBER { 806} > LIST REACTION JOINT 21 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL PS 100M 100TOT Revision 2 { 848} > LOAD LIST 'PS100P' 'PS100R' '100M' '100RTOT' '100TOT' { 849} > OUTPUT BY MEMBER { 850} > LIST REACTION JOINT 21 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL PS100R 100M 100RTOT PS100P 100TOT 37 37

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**Example 2 Revision 1 Revision 2 38**

{ 808} > LOAD LIST '100TOT' '300TOT' 'EQTOT' { 809} > OUTPUT BY MEMBER { 810} > LIST REACT JOINT 21 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL 100TOT 300TOT EQTOT Revision 2 { 852} > LOAD LIST '100TOT' '300TOT' 'EQTOT' { 853} > OUTPUT BY MEMBER { 854} > LIST REACT JOINT 7 ACTIVE UNITS INCH KIP CYC DEGF SEC RESULTANT JOINT LOADS SUPPORTS JOINT LOADING / FORCE // MOMENT / X FORCE Y FORCE Z FORCE X MOMENT Y MOMENT Z MOMENT GLOBAL 100TOT 300TOT EQTOT 38

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Concluding Remarks The Rev 2 response spectrum solution methodology appears to be a reasonably rational way to incorporate more recent knowledge about periodic and rigid response characteristics. The effect of the Rev 2 rigid response modifications may increase or decrease the magnitude of response predictions, depending on where the modal frequencies are distributed on the response spectrum curves with respect to F1, F2, and FZPA. The more concise way in which rigid response is treated in the Rev 2 solution may reign in the trend toward higher and higher cutoff frequencies. The Rev 2 solution does require additional dynamic loading conditions, longer compute times, and more results data to manage. Are differences in results worth the extra effort?

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**Concluding Remarks Practical Issues:**

It may take a very large number of modes to encompass all frequencies ≤ FZPA . Computer resources are still finite! No specified role for mass participation percentage under RG 1.92.

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