1. 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

2. 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

3. Background 3

4. Background

5. 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!

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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:

14. 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:

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. GTStrudl Enhancements, Version 30
The ALGEBRAIC Mode Combination

23. 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’

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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?

40. 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.