NUREG/CR-6645 

BNL-NUREG-52576 

Reevaluation of 

Regulatory Guidance on 

Modal Response Combination 

Methods for Seismic Response 

Spectrum Analysis 

Brookhaven National Laboratory 

U.S. Nuclear Regulatory Commission 

Office of Nuclear Regulatory Research 

Washington, DC 20555-0001
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'4
NUREG/CR-6645 

BNL-NUREG-52576 

Reevaluation of 

Regulatory Guidance on 

Modal Response Combination 

Methods for Seismic Response 

Spectrum Analysis 

Manuscript Completed: September 1999 

Date Published: December 1999 

Prepared by 

R. Morante, Y. Wang 

Brookhaven National Laboratory 

Upton, NY 11973-5000 

R. M. Kenneally, NRC Project Manager 

Prepared for 

Division of Engineering Technology 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission 

Washington, DC 20555-0001 

NRC Job Code W6521
ABSTRACT

Regulatory Guide 1.92, "Combining Modal Responses and Spatial Components in Seismic Response 

Analysis," was last revised in 1976. The objectives of this project were to re-evaluate the current 

regulatory guidance for combining modal responses in response spectrum analysis; evaluate recent 

technical developments; and recommend revisions to the regulatory guidance. In addition, Standard 

Review Plan Section 3.7.2, "Seismic System Analysis," was reviewed to identify related sections 

which may need to be revised. The objectives were addressed through a literature review of past 

studies, supplemented by analysis of a piping system model previously utilized in NUREG/CR

5627, "Alternate Modal Combination Methods in Response Spectrum Analysis." 

This project evaluated (1) methods for separation of the in-phase and out-of-phase modal response 

components; (2) methods for combination of the out-of-phase modal response components; (3) the 

contribution of"missing mass"; and (4) the combination of the three elements of response to produce 

the total response. Numerical results from response spectrum analyses were compared to 

corresponding time history analysis results to assess the accuracy of the various combination 

methods tested. 

During the course of the project, several insights relating to potential improvements in the 

methodology for seismic analysis were identified and documented. These include (1) improvements 

in correlation between mode superposition time history and direct integration time history; (2) use 

of response spectrum generation single degree of freedom oscillator responses to define the 

frequency above which modal responses are in-phase with the input time history; and (3) evaluation 

of the effects of potential differences in mass distribution used in static and dynamic analyses of a 

piping system. 

The conclusions of the project are (1) several of the modal response combination methods evaluated 

should be included as acceptable methods in the next revision to Regulatory Guide 1.92; (2) deletion 

of several of the modal response combination methods currently in the Regulatory Guide should be 

considered; (3) the Regulatory Guide should be revised to specifically define acceptable procedures 

for constructing the total response spectrum analysis solution; and (4) several variations of modal 

response combination methods, which were examined to try to improve accuracy, showed promising 

results and warrant further study. 

This project did not specifically analyze the modal response combination methods applied to highly 

flexible, low frequency systems. A separate evaluation for these systems should be considered.

Hii NUREG/CR-6645
CONTENTS 

Page 

ABSTRACT ................................................................ iii 

FIGURES ................................................................. vii 

TAB LES ................................................................... vii 

EXECUTIVE SUMMARY .................................................... ix 

ACKNOWLEDGMENTS ..................................................... xi 

ABBREVIATIONS ......................................................... xiii 

1 INTRODUCTION ................... 1 

2 DESCRIPTION OF MODAL RESPONSE COMBINATION METHODS ....... 5 

2.1 Combination of Out-of-Phase Modal Response Components ................ 7 

2.1.1 Square Root of the Sum of the Squares (SRSS) Method .............. 7 

2.1.2 NRC Grouping Method ....................................... 8 

2.1.3 NRC Ten Percent Method ..................................... 9 

2.1.4 Rosenblueth's Double Sum Combination (DSC) .................. 11 

2.1.5 NRC Double Sum Combination (NRC-DSC) ..................... 12 

2.1.6 Der Kiureghian's Complete Quadratic Combination (CQC) ......... 12 

2.1.7 ASCE Standard 4 Recommended Methods ....................... 12 

2.2 Separation of Modal Responses into Out-of-Phase Components and 

In-Phase Components ............................................. 13 

2.2.1 Lindley-Yow Method ........................................ 13 

2.2.2 Hadjian Method ............................................ 15 

2.2.3 Gupta Method ............................................. 16 

2.2.4 ASCE Standard 4 Recommended Methods ....................... 18 

2.3 Contribution of H-igh Frequency Modes ............................... 18 

2.3.1 Missing Mass Method ....................................... 18 

2.3.2 Static ZPA Method ......................................... 20 

2.4 Complete Solution for Response Spectrum Analysis ..................... 20 

2.4.1 M ethod I ................................................. 20 

2.4.2 M ethod 2 ................................................. 21 

2.4.3 Method 3 ......................................... 22

¥ NUREG/CR-6645
CONTENTS (continued) 

Page 

3 QUANTITATIVE STUDIES OF MODAL RESPONSE COMBINATION 

M ETH ODS ........................................................... 23 

3.1 BM3 Piping Analysis Model ........................................ 23 

3.2 Description of Analyses Performed ................................... 24 

3.2.1 Baseline Time History Analysis ................................ 24 

3.2.2 Response Spectrum Analysis .................................. 24 

3.3 Discussion of Numerical Results ..................................... 26 

3.3.1 Out-of-Phase Modal Response Combination ...................... 26 

3.3.2 Complete Solution .......................................... 26 

3.4 Summary of Results ............................................... 27 

4 SUPPLEMENTARY STUDY OF ADDITIONAL MODAL RESPONSE 

COMBINATION METHODS .................................... 51 

4.1 Basis for Study of Additional Modal Response Combination Methods ....... 51 

4.2 Description of Additional Modal Response Combination Methods .......... 52 

4.2.1 Grouping Methods 1 and 2 .................................. 52 

4.2.2 Gupta Method - Variations to Value off2 ........................ 55 

4.3 Discussion of Numerical Results for Additional Modal Response Combination 

M ethods ........................................................ 56 

4.4 Summary of Results ............................................... 58 

5 CONCLUSIONS AND RECOMMENDATIONS ............................ 65 

5.1 Technical Conclusions ............................................. 65 

5.2 Recommendation for Revision to Regulatory Guide 1.92 and Standard Review 

Plan 3.7.2 ....................................................... 66 

5.3 Recommendations for Additional Evaluation of Modal Response Combination 

M ethods ........................................................ 67 

6 REFERENCES ........................................................ 69 

APPENDIX A: Description of BM3 Piping Model ........................... A-I 

APPENDIX B: Time History and Response Spectrum Input for BM3 Analysis ...... B-1 

APPENDIX C: Potential Differences Between Dynamic and Static Mass Distributions 

for the Same Computer Model ................................ C-1 

APPENDIX D: Tabulation of Mode Correlation Coefficients for the Double Sum 

Combination (DSC) and Complete Quadratic Combination (CQC) 

M ethods ................................................ D-1

NUREG/CR-6645 Ai
CONTENTS (continued)

Page

APPENDIX E: 

APPENDIX F: 

APPENDIX G: 

APPENDIX H: 

APPENDIX I:

3-1 

3-2 

3-3

Comparison of Time History Solution Methods - Mode Superposition 

vs. Direct Integration ....................................... E-l 

Use of Response Spectrum Generation SDOF Oscillator Responses to 

Establish the Threshold Frequency for In-Phase Modal Responses ... F-i 

Evaluation of Lindley-Yow Method by Professor A.S. Veletsos .... G-i 

Independent Analysis of BM3 Piping Model by Professors 

A.K. Gupta and Abhinav Gupta .............................. H-i 

Calculation of Missing Mass Contribution to Total Response ....... I-i 

List of Figures

BM 3 Piping M odel ..................................................... 47 

BM3 Input Spectrum (1%) ................................................ 48 

BM3 Input Spectrum (5%) ................................................ 49

List of Tables 

3-1 BM3 Model Modal Frequencies ........................................... 29 

3-2 BM3 Model-8 Mode Combinations-Time History vs. Response Spectrum 

Support Reactions ...................................................... 30 

3-3 BM3 Model-8 Mode Combination-Time History vs. Response Spectrum 

Pipe End M oments ...................................................... 31 

3-4 BM3 Model-14 Mode Combination-Time History vs. Response Spectrum 

Support Reactions ............................................. 32 

3-5 BM3 Model-14 Mode Combination-Time History vs. Response Spectrum 

Pipe End Moments ....................................................... 33 

3-6 BM3 Model-31 Mode Combination-Time History vs. Response Spectrum 

Support Reactions ............................................. 34 

3-7 BM3 Model-31 Mode Combination-Time History vs. Response Spectrum 

Pipe End M oments ...................................................... 35 

3-8 BM3 Model-Modal Time History vs. Method I-Response Spectrum 

(14 Modes Plus Missing Mass)-Support Reactions ............................. 36 

3-9 BM3 Model-Modal Time History vs. Method I-Response Spectrum 

(14 Modes Plus Missing Mass)-Pipe End Moments ............................ 37

Vii NUREG/CR-6645
CONTENTS (continued) 

Page 

3-10 BM3 Model-Modal Time History vs. Method 2-Response Spectrum 

(14 Modes Using Lindley-Yow Approach Plus Missing Mass)

Support Reactions ............................................. 38 

3-11 BM3 Model-Modal Time History vs. Method 2-Response Spectrum 

(14 Modes Using Lindley-Yow Approach Plus Missing Mass)

Pipe End M oments .............................................. , ....... 39 

3-12 BM3 Model-Modal Time History vs. Method 2-Response Spectrum 

(31 Modes Using Lindley-Yow Approach Plus Missing Mass)

Support Reactions ...................................................... 40 

3-13 BM3 Model-Modal Time History vs. Method 2-Response Spectrum 

(31 Modes Using Lindley-Yow Approach Plus Missing Mass)

Pipe End M oments ...................................................... 41 

3-14 BM3 Model-Modal Time History vs. Method 3-Response Spectrum 

(14 Modes Using Modified Spectrum Approach Plus Static ZPA)

Support Reactions ...................................................... 42 

3-15 BM3 Model-Modal Time History vs. Method 3-Response Spectrum 

(14 Modes Using Modified Spectrum Approach Plus Static ZPA)

Pipe End M oments ...................................................... 43 

3-16 BM3 Model-Modal Time History vs. Method 3-Response Spectrum 

(14 Modes Using Modified Spectrum Approach Plus Static ZPA)

Support Reactions-5% Damping ..... ...................................... 44 

3-17 BM3 Model-Modal Time History vs. Method 3-Response Spectrum 

(14 Modes Using Modified Spectrum Approach Plus Static ZPA)

Pipe End Moments-5% Damping .......................................... 45 

3-18 Equivalence of R,: Lindley-Yow Plus Missing Mass vs. Static ZPA ............ 46 

4-1 BM3 Model-14 Mode Combination-Time History vs. Response Spectrum

Support Reactions Comparison Among Different Methods ...................... 60 

4-2 BM3 Model-14 Mode Combination-Time History vs. Response Spectrum

Pipe End Moments Comparison Among Different Methods ...................... 61 

4-3 BM3 Model-14 Modes Plus Missing Mass Combination-Time History vs. 

Response Spectrum - Support Reactions Comparison Among Different 

M ethods .............................................................. 62 

4-4 BM3 Model-14 Modes Plus Missing Mass Combination-Time History vs. 

Response Spectrum-Pipe End Moments Comparison Among Different 

M ethods ................................................... 63 

NUREGICR-6645 viii
EXECUTIVE SUMMARY

Regulatory Guide 1.92 "Combining Modal Responses and Spatial Components in Seismic 

Response Analysis" was last revised in 1976. The objective of this project were to re-evaluate 

the current regulatory guidance for combining modal responses in response spectrum analysis; 

evaluate recent technical developments; and recommend revisions to the regulatory guidance. In 

addition, Standard Review Plan (SRP) Section 3.7.2, "Seismic System Analysis," was reviewed 

to identify related sections which may need to be revised. The objectives were addressed through 

a literature review of past studies, supplemented by analysis of a piping system model previously 

utilized in NUREG/CR-5627, "Alternate Modal Combination Methods in Response Spectrum 

Analysis". 

The project evaluated (1) methods for separation of the in-phase and out-of-phase modal 

response components; (2) methods for combination of the out-of-phase modal response 

components; (3) the contribution of "missing mass"; and (4) the combination of the three 

elements of response to produce the total response. Numerical results from response spectrum 

analyses were compared to corresponding time history analysis results to assess the accuracy of 

the various combination methods tested. 

The methods selected for evaluation were those which have been subjected to the greatest level 

of prior review and assessment. For separation of the in-phase and the out-of-phase modal 

response components, the methods proposed by Lindley-Yow, Hadjian and Gupta were 

evaluated. For combination of the out-of-phase modal response components, the Square Root of 

the Sum of the Squares (SRSS), NRC Grouping, NRC Ten Percent, Rosenblueth's Double Sum 

Combination (DSC), the NRC DSC, and Der Kiureghian's Complete Quadratic Combination 

(CQC) methods were evaluated. For treatment of the "missing mass" contribution, the method 

of Kennedy was evaluated. Response spectrum analyses were conducted by combining elements 

of the above methods to construct complete response spectrum analysis solutions. 

Baseline time history solutions, utilizing both mode superposition and direct integration methods, 

were also generated. Excellent correlation between the two time history methods was obtained 

by adding the missing mass contribution to the mode superposition solution. Missing mass 

proved to be significant even though the mode superposition solution included 31 modes, up to 

70 Hz. 

Based on the literature review and the numerical results generated in this project, it is concluded 

that Rosenblueth's DSC and Der Kiureghian's CQC methods for combining out-of-phase modal 

response components, coupled with the Lindley-Yow or Gupta method for separation of the in

phase and the out-of-phase modal response components, and inclusion of Kennedy's missing 

mass contribution to account for modes with frequencies above that corresponding to the Zero 

Period Acceleration (ZPA) of the response spectrum, constitute the best methodologies currently 

available for response spectrum analysis.

ix NUREG/CR-6645
EXECUTIVE SUMMARY

The NRC Grouping, NRC Ten Percent and NRC-DSC methods for combining out-of-phase 

modal response components produced more conservative but less accurate results. Removal of 

these methods from the Regulatory Guide should be considered, because absolute summation of 

closely spaced modal responses has no documented technical basis. SRSS remains applicable in 

the absence of closely spaced modes. 

Separation of modal responses into in-phase and out-of-phase components for modes with 

frequencies below the ZPA frequency of the response spectrum produced more accurate results 

than commonly applied past methods, in which all modes below the ZPA frequency were 

considered to be out-of-phase. It is important to note that at low frequency (<frequency of the 

peak spectral acceleration) modal responses should be treated as out-of-phase. A limitation of 

the Lindley-Yow formulation is that low frequency modal responses are separated into out-of

phase and in-phase components; consequently, when significant low frequency modal responses 

exist, the Lindley-Yow formulation must be appropriately modified. The Gupta formulation 

correctly assumes low frequency modes are out-of-phase. This project did not specifically 

analyze modal response combination methods applied to systems with significant low frequency 

response. 

At the request of the Project Peer Review Panel, an extensive investigation of individual modal 

response contributions to both the response spectrum and time history solutions was conducted 

for selected support reactions, in order to explain significant differences in prediction. Once the 

phenomenon was understood, several variations of the initial modal response combination 

methods were defined and additional response spectrum analyses were conducted. Improvement 

in accuracy was achieved. As part of this additional analytical effort, a methodology was defined 

to utilize the SDOF oscillator responses from the response spectrum generation analysis to 

establish the frequency above which modal responses are in-phase with the input time history. 

This technique should be applicable to all response spectra. 

During the course of the project, static analysis results for the mass times the ZPA were 

generated utilizing system mass distributions developed by both the dynamic analysis option and 

the static analysis option of the same computer code (based on SAP V). Significant differences 

in solution were noted, which are attributed to differences in the treatment of mass. In the 

dynamic analysis model, mass is apportioned to node points; in the static analysis model, mass is 

treated as distributed along the element length. While this situation may not exist in all piping 

analysis codes, it is important to verify that consistent static analysis results are obtained for both 

dynamic and static mass distributions. Lack of consistency usually indicates that the element 

breakup in the piping model is not sufficiently refined to accurately predict support reactions and 

internal bending moments.

NUREG/CR-6645 X
ACKNOWLEDGMENTS

The authors wish to thank Mr. Wallace E. Norris, Mr. Roger M. Kenneally and Dr. Nilesh C. 

Chokshi of the U.S. Nuclear Regulatory Commission for their guidance and support in carrying 

out this project The authors also wish to thank the Peer Review Panel - Dr. Robert P. Kennedy 

and Professor A.S. Veletsos - for their valuable input and recommendations during the conduct 

of this project and preparation of this report. The authors wish to acknowledge Dr. Paul Bezler 

of Brookhaven National Laboratory for his technical assistance and advice in the early stage of 

this program. 

Following completion of the initial draft of this report, Professors A.K. Gupta and Abhinav 

Gupta of North Carolina State University volunteered to review and comment upon the technical 

content. Their technical comments and independent analytical results are greatly appreciated, 

and have been incorporated into the final report. 

Special thanks to Ms. Susan Signorelli, Ms. Barbara J. Roland, and Ms. Linda Thompson of 

Brookhaven National Laboratory for typing and formatting this manuscript.

Ai NUREG/CR-6645
ABBREVIATIONS

ASCE American Society of Civil Engineers 

BNL Brookhaven National Laboratory 

CQC Complete Quadratic Combination 

DSC Double Sum Combination 

MODF Multiple Degree of Freedom 

NRC Nuclear Regulatory Commission 

RB-DSC Rosenblueth's Double Sum Combination 

RG Regulatory Guide 

RSA Response Spectrum Analysis 

SDOF Single Degree of Freedom 

SRP Standard Review Plan 

SRSS Square Root of the Sum of the Squares 

ZPA Zero Period Acceleration

xiii NUREG/CR-6645
1 INTRODUCTION

General Design Criterion 2, "Design Basis for Protection Against Natural Phenomena" of 

Appendix A, "General Criteria for Nuclear Power Plants" to 1OCFR Part 50, "Licensing of 

Production and Utilization Facilities" requires that nuclear power plant structures, systems, and 

components which are important to safety be designed to withstand the effects of earthquakes 

without loss of capability to perform their safety functions. In addition, Paragraph (a)(1) of 

Section VI of Appendix A, "Seismic and Geologic Siting Criteria for Nuclear Power Plants" to 

1OCFR Part 100, "Reactor Site Criteria" identifies the use of a suitable dynamic analysis as one 

method of ensuring that structures, systems, and components can withstand seismic loads. 

Related requirements for new license applicants (after January 10, 1997) are specified in 

Paragraph (a) (1) of Section IV of Appendix S, "Earthquake Engineering Criteria for Nuclear 

Power Plants" to 10 CFR Part 50. 

The United States Nuclear Regulatory Commission (NRC) issues Regulatory Guides (RG) which 

describe methods acceptable to the NRC staff for satisfying regulations. One such guide is RG 

1.92, "Combining Modal Responses and Spatial Components in Seismic Response Analysis," 

(Reference 1). This guide was last revised in 1976, prior to a number of significant technical 

developments for combining modal responses. 

The 1989 revision to Standard Review Plan (SRP) Section 3.7.2, "Seismic System Analysis," 

(Reference 2) recognized a number of recent technical developments by reference, and stated that 

their application to nuclear power plant seismic analysis is subject to review on a case-by-case 

basis. Also incorporated into SRP Section 3.7.2 as Appendix A was a procedure developed by 

Kennedy (Reference 3) to address high frequency mode effects. 

The objective of this project was to evaluate these recent developments for modal response 

combination, through a literature review and analytical effort, and to provide recommendations 

for revision of RG 1.92 and SRP Section 3.7.2 which reflect the current state of technology for 

combining modal responses in seismic response spectrum analysis.. 

The design of structures, systems, and components for seismic loads is complicated by the 

uncertainty about the future seismic event. For seismic analysis where an accurate record of the 

seismic input exists, time history analysis is the best approach to mathematical prediction of 

seismic structural response. In the seismic design process, two primary approaches are available 

to account for the uncertainty in the seismic ground motion: perform a number of time history 

analyses utilizing an appropriate set of acceleration records or perform response spectrum 

analysis utilizing a bounding peak acceleration vs. frequency spectrum. Either approach, 

properly implemented, should provide a conservative basis for seismic design. 

A significant feature of response spectrum analysis is that only the maximum dynamic response 

is predicted. In time history analysis, the response vs. time is predicted. The mathematical 

simplicity of response spectrum analysis is achieved by calculating the independent peak modal

I NUREG/CR-6645
1 INTRODUCTION

responses and then applying a rule for combining these, in order to predict the peak dynamic 

response. The input response spectrum defines the acceleration to be applied to each natural 

mode of vibration of the structure, depending on its.modal frequency. However, the spectrum 

provides no information about the time phasing of excitation of these modes. 

It is evident that the modal response combination method used to predict the peak dynamic 

response is a critical element of the response spectrum analysis method. Many researchers have 

studied this specifically for seismic analysis, and a number of increasingly sophisticated methods 

have been developed. In 1984, Kennedy (Reference 3) reviewed alternate methods for modal 

response combination for the NRC and provided recommendations for revision to regulatory 

guidance. Gupta (Reference 4, Chapter 3) provides an excellent review of these developments 

up to the late 1980's, including his own modal response combination method. In 1994, Mertens 

(Reference 5) presented a refinement of previously developed methods for combining amplified 

modal responses. 

This present investigation does not address the issue of seismic design methodology - time 

history analysis vs. response spectrum analysis. It is focused on comparison and evaluation of 

different modal response combination methods for use in response spectrum analysis. Because 

the objective of response spectrum analysis is to predict, with reasonable accuracy, the peak 

dynamic response to a time varying acceleration input, comparison to time history solutions is 

the primary method employed to evaluate the applicability and limitations of the various modal 

response combination methods. 

The initial phase of this program focused on review of the technical literature and selection of 

candidate modal response combination methods for more detailed evaluation. Acceptable 

methods in RG 1.92 were also included to provide a comparison to more recent technical 

developments. References 3 and 4 provided an excellent starting point. In addition, prior 

numerical studies conducted by Brookhaven National Laboratory (References 6, 7, 8) on 

response spectrum analysis of piping systems provided a quantitative database for alternate 

modal response combination methods. Industry standards such as ASCE Standard 4 (References 

9, 10) were also reviewed. The work of Maison et al (Reference 11) provided a numerical study 

of different modal response combination methods applied to a building structure. A significant 

body of additional reference material was included in the literature review. 

The methods selected for evaluation were those which have been subjected to the greatest level 

of prior review and assessment. The evaluation addressed (1) methods for combination of the 

out-of-phase modal response components; (2) methods for separation of the in-phase and out-of

phase modal response components; (3) the contribution of high frequency modal responses; and 

(4) combination of the three elements of response to produce the total response. The term "in

phase" denotes the response component that is in-phase with the time varying input acceleration; 

the term "out-of-phase" denotes the response component that is not in-phase with the time

NUREGICR-6645 2
1 INTRODUCTION

varying input acceleration. Gupta (Reference 4) refers to these as the "rigid" or "pseudo-static" 

response and the "damped periodic" response, respectively. 

It is important to note that individual modal responses which are each "out-of-phase" with 

respect to the time varying input acceleration may be nearly in-phase with each other. This is 

commonly referred to as the "closely spaced modes" issue, because modes close in frequency are 

likely to respond nearly in-phase with each other when excited by the same time varying input 

acceleration. This is addressed by methods for combination of the "out-of-phase" modal 

responses. 

Modes with frequencies higher than the frequency (/rpA) at which the spectral acceleration returns 

to the Zero Period Acceleration (ZPA) respond pseudo-statically to the time varying input 

acceleration and, therefore, are in-phase with each other. The contribution of these modes to the 

total response is most accurately and efficiently treated by static analysis of the "missing mass" 

(i.e., system mass not participating in the modes with frequencies belowfzpA) multiplied by the 

ZPA. Modes with frequencies in the amplified region of the response spectrum (f<fzpj 

generally have both an in-phase component and an out-of-phase component. The in-phase modal 

response components and the missing mass contribution are combined algebraically, to produce 

the total in-phase response component. 

For combination of the out-of-phase modal response components, Square Root of the Sum of the 

Squares (SRSS), NRC Grouping, NRC Ten Percent, Rosenblueth's DSC (Reference 12), NRC

DSC, and Der Kiureghian's CQC (Reference 13) methods were evaluated. For separation of the 

in-phase and the out-of-phase modal response components, the methods proposed by Lindley and 

Yow (Reference 14), Hadjian (Reference 15), and Gupta (Reference 4) were evaluated. For 

treatment of the missing mass contribution, the method of Kennedy (Reference 3) was evaluated.

3 NUREGICR-6645
2 DESCRIPTION OF MODAL RESPONSE COMBINATION METHODS 

To lay the groundwork for the ensuing discussions of modal response combination methods, it is 

necessary to define the different regions of a typical seismic response spectrum and key 

frequencies which divide these regions. The major application of seismic response spectrum 

analysis in the nuclear industry is for systems and components attached to building structures. 

A building-filtered in-structure response spectrum depicting spectral acceleration vs. frequency is 

the typical form of seismic input for such analyses. This type of spectrum usually exhibits a 

pronounced peak at the fundamental frequency of the building/soil dynamic system. An 

idealized in-structure response spectrum is shown below; the spectral regions and key 

frequencies are indicated.

low-frequency -<- 1k- mid-frequency -+I

(Predominately out-of

phase response)

-- * high-frequency

(Transition from out-of- (In-phase, 

phase to in-phase static res 

response) 

I 

I 

i 

I P

pseudo

ponse)

fp fc? fzPA Frequency (Hz)

fsp = frequency at which the peak spectral acceleration is reached; typically the fundamental 

frequency of the building/soil system 

fzpA = frequency at which the spectral acceleration returns to the zero period acceleration 

(ZPA) 

fnp = frequency above which the SDOF modal responses are considered to be in-phase with 

the time varying input acceleration used to generate the response spectrum

NUREGICR-6645

0 

4) 

U U 

C.) 

4) 

rJ2

5
2 COMBINATION METHODS

The high frequency region of the spectrum (>fzr) is characterized by no amplification of the 

peak acceleration of the input time history. A SDOF oscillator having a frequency >fZlA is 

accelerated in-phase and with the same acceleration magnitude as the applied acceleration, at 

each instant in time. A system or component with fundamental frequency >fzpA is correctly 

analyzed as a static problem subject to a loading equal to its mass times the ZPA. The system or 

component is said to respond "pseudo-statically." This concept can be extended to the high 

frequency (>fzrA) modal responses of multi-modal systems or components. The mass not 

participating in the amplified modal responses (i.e., "missing mass") multiplied by the ZPA is 

applied in a static analysis, to obtain the response contribution from all modes with frequencies 

>fZPA

In the low-frequency region of the spectrum (<fsp), the modal responses of SDOF oscillators are 

not in-phase with the applied acceleration time history, and generally are not in-phase with each 

other. These are designated "out-of-phase" modal responses. Since a response spectrum 

provides only peak acceleration vs. frequency, with no phasing information, the out-of-phase 

peak modal responses for a multi-modal structural system requires a rule or methodology for 

combination. Based on the assumption that the peak modal responses are randomly phased, the 

square root of the sum of the squares (SRSS) method was developed and adopted. Modifications 

to the SRSS method were subsequently developed, in order to account for potential phase 

correlation when modal frequencies are numerically close (i.e., "closely spaced modes"). 

In the mid-frequency region (fsP tofzp,), it has been postulated that the peak SDOF oscillator 

modal responses consist of two distinct and separable elements. The first element is the out-of

phase response component and the second element is the in-phase response component It is 

further postulated that there is a continuous transition from out-of-phase response to in-phase 

response. IffF <fzPA can be defined, then the mid-frequency region can be further divided into 

two sub-regions:fsp <f<f~p andfp : < fr f<ZA. 

It is noted that past practice in the nuclear power industry has been to assume that individual 

modal responses in the mid-frequency region (fsp <f<fzp) are out-of-phase, and that the 

combination methods applicable to the low-frequency region are also applicable to the mid

frequency region. 

Three elements are needed to define a suitable methodology for the mid-frequency region: 

1) A definition forfn,. 

2) A method for separating the in-phase and out-of-phase components of individual peak 

modal responses. 

3) A phase relationship for combining the total out-of-phase response component with the 

total in-phase response component.

NUREG/CR-6645 6
2 -'COMBINATION METHODS

Methods for modal response combination are described in the following sections, where the 

following notation is used: 

Sa, = Spectral Acceleration for mode i 

R, = Response of mode i 

ai = In-phase response ratio for mode i 

Rrx = In-phase response component for mode i 

Rpi = Out-of-phase response component for mode i 

Rr = Total in-phase response component from all modes 

Rp = Total out-of-phase response component from all modes 

Rt = Total combined response from all modes 

CAk = Modal response correlation coefficient between modesj and k. 

2.1 Combination of Out-of-Phase Modal Response Components 

In generalized form, all of the out-of-phase modal response combination methods can be 

represented by a single equation: 

Rp E E Cj Rpj RPk (Eqn. 2-1) 

j ,-1 k-1 kl 

The mode correlation coefficients Cjk are uniquely defined for each method. 

2.1.1 Square Root of the Sum of the Squares (SRSS) Method 

At the foundation of all methods for combining uncorrelated modal responses is the SRSS 

method. All of the methods for combination of the out-of-phase response components are 

equivalent to SRSS if there are no "closely spaced" modes. 

In this case, 

CAk = 1.0 forj k 

(Eqn. 2 - 2) 

CAk = 0.0 forj*k

7 NUREG/CR-6645
2 COMBINATION METHODS

and Equation 2-1 reduces to: 

Rp=[• Rp 2 (Eqn. 2-3) 

2.1.2 NRC Grouping Method 

The NRC Grouping Method (Reference 1) is the most commonly applied method of accounting 

for closely spaced modes in the nuclear power industry. The system modal responses are' 

grouped and summed absolutely before performing SRSS combination of the groups. To 

illustrate the method, consider a system with 10 modes, with responses Rp, through Rp1 0, and 

associated frequenciesf 1 throughf1 0 having the following distribution: 

A > 1.1f 

A,f A < 1.1f 

fA > 1.1" 

fA < 1.1f 

fA > l.f 

f > L.1f 

fA 9 A0 < 1.1lf 

The modal responses are grouped such that the lowest and highest frequency modes in a group 

are within 10% and no mode is in more than one group. Using the distribution above, the 

following groups are created: 

GR, = Rp1 

GR2 = IRp2I + IRP31 + IRP41 

GR3 = I Rp5I + I Rp61 

GR4 = Rp 7

NUREG/CR-6645 8
2 COMBINATION METHODS

GR5 = JRp8I + IRp9I + IRp1oI

Rp = [ GpI] (Eqn. 2 - 4)

In this illustration, n = 5. 

The major criticism of the NRC Grouping Method is the use of absolute summation within each 

group. If modal responses are assumed to be correlated because they have closely spaced 

frequencies, then summation should be algebraic within each group. The bias toward 

conservatism in the NRC Grouping Method is somewhat contradictory to the basic premise for 

grouping. 

When expressed in terms of the mode correlation coefficients Ci k, the NRC Grouping Method 

may be defined as follows:

forj =k

for j* k, not in same group 

for j #k, in the same group, Rpj and Rpk 

have same sign 

for j* k, in the same group, Rpj and Rpk 

have opposite signs

(Eqn. 2 - 5)

In implementing the NRC Grouping Method, the approach presented in the illustration is more 

straightforward. 

2.1.3 NRC Ten Percent Method 

The NRC Ten Percent Method (Reference 1) is a generally more conservative variation of the 

NRC Grouping Method. Closely spaced modes are defined as modes with frequencies within 

10% of each other and absolute summation of the closely spaced modal responses is specified. 

The difference is that modal responses are not grouped.

NUREG/CR-6645

Cik = 1.0 

Cjk - o.o 

CJk = 1.0 

Cjk = -1.0

9
2 COMBINATION METHODS

In terms of the mode correlation coefficients, CJk, the NRC Ten Percent Method is defined as 

follows:

for j =k

for j* k, andfj andfk separated by > 10% of the 

lower frequency 

for j * k, andfj andfk separated by g 10% of the 

lower frequency; Rpj and RpI same sign 

for j * k, fj andfk separated by : 10% of the 

lower frequency; Rpj and Rpk opposite signs

(Eqn. 2 - 6)

The definition of Cj k is analogous to that for the NRC Grouping Method, except that grouping is 

not performed. 

As an illustration of the difference between these two methods, assume three modal responses 

Rpl, Rp2, Rp3 with frequenciesf,, f 2 ,f 3 and a frequency distribution defined as follows: 

1 f .11A

By the NRC Grouping Method,

[2 1 

[i-1

where

GR1 = IRp1 I + I P 21

GR2 = Rp3

or

Rp = RKI Rpi" + 2 1Rp,,* Rp l

NUREG/CR-6645

Cjk = 1.0 

Cjk = 0.0 

Cjk = 1.0 

Cjk = -1.0

10
2 COMBINATION METHODS

By the NRC Ten Percent Method, 

Rp = Rp + 21Rp, * RP2I+21RP2 * RP3I 

The NRC Ten Percent Method has an additional contribution to Rp becausef 3 < 1.1 f 2 . The 

NRC Ten percent Method will always produce results > the NRC Grouping Method. 

2.1.4 Rosenblueth's Double Sum Combination (DSC) 

Rosenblueth (Reference 12) provided the first significant mathematical approach to evaluation of 

modal correlation for seismic response spectrum analysis. It is based on the application of 

random vibration theory, utilizing a finite duration of white noise to represent seismic loading. A 

formula for calculation of the coefficients Cjk as a function of the modal circular frequencies (tj, 

wak), modal damping ratios (Ipj, P•), and the time duration of strong earthquake motion (tD) was 

derived. Using the form of the equation from Reference 1, 

I 

1+ J--+ 

Pi• Caj + Pk (Ok, 

(Eqn. 2 - 7) 

where ()() = I6)l- P2)] 

13)= 13() + 2 

'D 0'() 

Appendix D tabulates numerical values of Cj, for the DSC Method as a function of frequency, 

frequency ratio, and strong motion duration time for constant modal damping of 1%, 2%, 5% and 

10%. The effect of tD is most significant at 1% damping and low frequency. For 5% and 10% 

damping, tD = 10 sec. and 1000 sec. produced similar values for Cjk regardless of frequency. The 

most significant result is that Cjk is highly dependent on the damping ratio. For 2%, 5% and 10% 

damping, Cjk = 0.2, 0.5 and 0.8 respectively, at a frequency ratio of 0.9 (modal frequencies within 

10%). In comparison, the definition of closely-spaced modes used in the NRC Grouping and Ten 

Percent Methods are not damping-dependent

I1I NUREG/CR-6645
2 COMBINATION METHODS

2.1.5 NRC Double Sum Combination (NRC-DSC) 

The NRC-DSC Method (Reference 1) is an adaptation of Rosenblueth's method, described in 

Section 2.1.4. The coefficients Cjk are defined by Equation 2-7. A conservative modification, 

consistent with the NRC Grouping and Ten Percent methods, is that the product Cjk Rpj Rpk is 

always taken as positive. In Rosenblueth's method, the product may be either positive or 

negative, depending on the signs of Rpj and Rpk. Consequently, NRC-DSC will always produce 

results > Rosenblueth's method. 

2.1.6 Der Kiureghian's Complete Quadratic Combination (CQC) 

Der Kiureghian (Reference 13) presents a methodology similar to Rosenblueth's Double Sum 

Combination (Reference 12) for evaluation of modal correlation for seismic response spectrum 

analysis. It is also based on application of random vibration theory, but utilizes an infinite 

duration of white noise to represent seismic loading. The following formula for calculation of 

the coefficients Cjk as a function of modal circular frequencies and modal damping ratios was 

derived: 

CA 8 (pjPkCoj(ok)' * (pj(.j + Pk( 3 )k) * (t.j2-k 

Jk =(2_ h + 4Pj[k~jtk(c2 + ,)k) + 4( ' + p0)2cagk2 (Eqn.2-8) 

While the form of Equation 2-8 differs significantly from Equation 2-7, the two equations 

produce equivalent results if tD is assumed very large in Equation 2-7. This is shown in 

Appendix D, where Cjk is tabulated for DSC with t. = 1000 sec. and for CQC. 

2.1.7 ASCE Standard 4 Recommended Methods 

For combination of out-of-phase modal response components, ASCE Standard 4 (Reference 9) 

specifies the DSC Method (Equation 3200-16). The NRC methods and CQC are also recognized 

in the, commentary. 

Draft ASCE Standard 4 (Reference 10) specifies a modified DSC Method (Equation 3200-19) or 

the CQC Method (Equation 3200-22) as an alternative. The commentary to the Draft ASCE 

Standard (Reference 10) indicates that Equation 3200-19 produces correlation coefficients 

"which are practically the same" as Equation 3200-22. Although not indicated, the modified 

DSC Method presented in Equation 3200-19 was developed by Gupta (Reference 4). 

Because they essentially duplicate the DSC and CQC methods, the ASCE methods will not be 

referenced in the numerical evaluation presented in Chapter 3. However, the numerical results 

for DSC and CQC are applicable.

NUREGICR-6645 12
-2 ' •COMBINATION METHODS

2.2 Separation of Modal Responses into Out-of-Phase Components and 

In-Phase Components 

Three methods have received considerable prior review and evaluation: Lindley-Yow (Reference 

14), Hadjian (Reference 15), and Gupta (Reference 4). It should be noted that the mathematical 

statement of each method is not restricted to the mid-frequency range (fs, <f<fzPA) of the 

response spectrum. However, as discussed at the beginning of this chapter, it is in the mid

frequency range that the separation of individual peak modal responses into out-of-phase and in

phase modal response components is applicable. Forf < fsj, modal responses should be 

considered out-of-phase and combined by the methods presented in Section 2.1. Forf >fzpA, 

modal responses are in-phase and are most accurately accounted for by the method of Kennedy 

(Reference 3). 

The similarities and differences, as well as the limitations, of the three methods are described in 

the following sections. 

2.2.1 Lindley-Yow Method 

In its most general form, the Lindley-Yow method (Reference 14) may be defined by the 

following equations:

0 ' ai . 1.0

Rri = Ri * ai

Rpi = Ri * - af

Rr= Rri 

i.l

Rp = l

I V2

(Eqn. 2 - 9) 

(Eqn. 2 - 10) 

(Eqn. 2 - 11) 

(Eqn. 2 - 12) 

(Eqn. 2 - 13) 

Rt = ^Rr (Eqn. 2 - 14) 2 + Rp 2

where the Cjk'sare defined by one of the methods for combining the out-of-phase modal response 

components described in Section 2.1.

NUREG/CR-6645

ai = ZPA/Sai

k Cjk Rpj Rpk k-l

13
2 COMBINATION METHODS

From these mathematical relationships, the following characteristics of the Lindley-Yow method 

are observed: 

0 aj -)- 1.0 asfi -"fzA (Sa1 = ZPA). Consequently,kfl =fzpA in the Lindley-Yow method. 

0 The in-phase component of modal response for every mode has an associated 

acceleration equal to the ZPA. 

* The out-of-phase component of an individual peak modal response has an associated 

modified spectral acceleration given by 

Sai = [Sai2 - ZPA 2 (Eqn. 2- 15) 

* 1.f = (RpT + Rr•j"; which infers that the in-phase and out-of-phase response 

components of an individual peak modal response are uncorrelated and, therefore, 

combine by SRSS. 

a All in-phase modal response components (RrT) are summed algebraically to obtain Rr. 

0 All-out-of-phase modal response components (Rp,) are combined by a suitable method 

(as described in Section 2.1) to obtain Rp. 

0 The total response, Rt, is obtained by SRSS combination of Rr and Rp; i.e., Rr and Rp 

are uncorrelated. 

0 a, attains its minimum value atf =fsp, but increases forfi <fsp until it attains a value of 

1.0 when Sa, = ZPA in the low frequency region of the spectrum. Values of ai > 1.0 

have no meaning because (1 - a)j' becomes imaginary. 

An obvious limitation of the Lindley-Yow method is in the low frequency range (f<fsp) of the 

response spectrum. There is no physical basis for assuming that low frequency modal responses 

become increasingly in-phase with the input acceleration time-history, which is an outcome if the 

Lindley-Yow method is applied to low frequency modal responses. Modal responses in the low 

frequency range are generally out-of-phase with the input acceleration time history. Therefore, 

the Lindley-Yow method is applicable to structural systems which do not have significant modal 

responses withfi <fsp. Lindley and Yow (Reference 14) do not address this limitation. For the 

sample problems presented in Reference 14, the lowest system frequency is greater thanfsp of the 

applied response spectrum. Therefore, the results reported in Reference 14 are not affected by 

this limitation. Circumventing this limitation in the Lindley-Yow method is straightforward: 

apply it only to those modes withf, >fsp and set a, = 0 forf, <fsp.

NUREGICR-6645 14
2 COMBINATION METHODS

Professor A.S. Veletsos, one of the project peer reviewers, provided an independent evaluation of 

the Lindley-Yow method and its limitation, as part of his review of the project results and 

conclusions. His evaluation is included as Appendix G to this report. 

For a structural system with fundamental frequency >fsP, the Lindley-Yow method lends itself to a 

relatively straightforward physical interpretation. In the limit, if all modes are retained in the 

solution, the total mass participation is unity. Applying the Lindley-Yow method is equivalent to 

performing a static analysis of the system loaded by total mass times the ZPA, and performing the 

response spectrum analysis for amplified modesf <fzpA using modified spectral accelerations, Sai 

given by Equation 2-15. The total dynamic response is then obtained by SRSS combination. 

The Lindley-Yow method automatically provides for algebraic combination of modal responses 

abovefzpA, because aj = 1.0, Rpi = 0, and Rri = Rý. However, to completely account for the modal 

response abovefzpA, all system modes of vibration need to be included in the analysis. This 

contribution is most accurately and efficiently calculated by use of the missing mass method 

discussed in Section 2.3. Therefore, while in theory, the Lindley-Yow method includes the in

phase contribution from modes abovefzpA, its practical application is for modal responses below 

fzA, coupled with the missing mass method for modal contributions abovefzpA. It is noted that 

the Lindley-Yow/missing mass approach will produce identical results for any modal analysis 

cutoff frequency ->fzxA. 

2.2.2 Hadjian Method 

The Hadjian Method (Reference 15) is similar in formulation to the Lindley-Yow method, with 

two notable differences: 

"* Equation 2-1i is replaced by 

Rp, Ri *I( - 0ai )(Eqn. 2 -16) 

"* Equation 2-14 is replaced by 

Rt = Rp +I Rr (Eqn. 2-17) 

"* The modified spectral acceleration is given by 

Sai = Sai - ZPA (Eqn. 2 - 18)

is NUREGICR-6645
2 COMBINATION METHODS

The Hadjian method has the same limitation as the Lindley-Yow method in the low frequency 

range, because the definition of a, is identical. However, the Hadjian Method possesses internal 

contradictions with respect to the assumed phase relationships between in-phase and out-of

phase response components. Combining Equations 2-10 and 2-16 yields 

Ri = Rpi + Rri (Eqn. 2-19) 

This implies that the in-phase and out-of-phase response components for each mode are in-phase 

with each other. However, all Rri's are in-phase and summed algebraically, per Equation 2-12, to 

obtain Rr. Therefore, it would follow that all Rpi's are also in-phase and should be summed 

algebraically to obtain Rp. This contradicts Equation 2-13, in which the Rpi's are assumed to be 

predominantly out-of-phase. Kennedy (Reference 3) previously identified this contradiction. On 

this basis, the Hadjian method is not recommended and is not included in the numerical studies 

presented in Chapter 3. 

2.2.3 Gupta Method 

The Gupta Method (Reference 4) is identical in form to the Lindley-Yow method. The one very 

significant difference is the definition of ac. Equations 2-10 through 2-14 remain the same. In 

the Gupta method, ai is an explicit function of frequency. The original basis for definition of ai 

is semi-empirical, derived from numerical studies using actual ground motion records. A best fit 

equation, which defines cc1 as a continuous function of frequency, was developed from the results 

of the numerical studies. 

Two spectrum-dependent frequencies (1,f2) are first defined as follows: 

A - (Eqn. 2-20) 2x Svma 

where Sa. and Sv, are the maximum spectral acceleration and velocity, respectively. 

f2 = (f, + 2 fzPA )/3 (Eqn. 2-21)

NUREG/CR-6645 16
2 COMBINATION METHODS

Gupta's definition of ai is given by: 

ai = 0 forfi sf, 

ai=' In il (f for A . :• A2• (Eqn. 2-22) 

ai = 1.0 for A >f 2 

For a sharply peaked, in-structure response spectrum, 

A-=fsp 

because Sv. = Max (SaE / co) = Sam. / casp 

Substitution into Equation 2-20 yields 

A = -(0 =fsP 

The corresponding definition off2 yields 

S= (fsp + 2fzm) / 3 

For a sharply peaked, in-structure response spectrum, the Gupta method has the following 

characteristics: 

"* Forf <Sfsp, ai = 0. 

Consequently, all modal responses withf <fsP are treated as out-of-phase. The limitation 

in the Lindley-Yow definition of ai forf ..fsp does not apply to Gupta's method. 

" Forf 2 <5f£<fzpA, ai = 1.0 

Consequently, all modal responses withfi 2:f 2 are treated as in-phase. This infers thatf, = 

A2 in the Gupta method. 

" Only modal responses withfsp <f, <f2 are separated into out-of-phase and in-phase 

response components.

17 NUREG/CR-6645
2 COMBINATION METHODS

The potential limitations of the Gupta method lie in the semi-empirical basis for definition of ac 

as a function off. The range of applicability is difficult to assess without a comprehensive 

numerical study using ground and in-structure acceleration records. In Reference 4, Gupta 

indicates that ai can be numerically evaluated if the time history used to generate the response 

spectrum is known. It is implied without stating that numerical evaluation of ac is more accurate 

than the semi-empirical definition of ai given by Equation 2-22. 

The overall structure of the Gupta method is superior to the Lindley-Yow method because there 

is no limitation for modal responses withf, <fsp. In addition, any value offi<fzpA can be 

accommodated by settingf 2 =fu, in lieu of Equation 2-21. 

For the initial numerical studies described in Chapter 3, the Lindley-Yow method was selected to 

evaluate the importance of separating modal responses into out-of-phase and in-phase response 

components. For the follow-up numerical studies described in Chapter 4, the Gupta method was 

selected in order to evaluate the influence offjp on the response spectrum solution. This was 

accomplished by selecting three different numerical values forf2. 

2.2.4 ASCE Standard 4 Recommended Methods 

For separation of in-phase and out-of-phase response components, ASCE Standard 4 (Reference 

9) recognizes the Lindley-Yow, Hadjian, and Gupta Methods in the commentary. 

Draft ASCE Standard 4 (Reference 10) specifies separation of the in-phase and out-of-phase 

response components consistent with Gupta's method (Eqns. 3200-18, 3200-20, and 3200-21) 

except thatf2 =f (defined as the "cutoff frequency or ZPA frequency") is substituted for 

Equation 2-21. The frequencyfr is not clearly defined, but is_<fzpA. The Lindley-Yow and 

Hadjian methods are recognized in the commentary to Reference 10. 

2.3 Contribution of High Frequency Modes 

2.3.1 Missing Mass Method 

The "Missing Mass" Method is a convenient, computationally efficient and accurate method to 

(1) account for the contribution of all modes with frequencies above the frequency (fzpA) at which 

the response spectrum returns to the Zero Period Acceleration (ZPA) and (2) account for the 

contribution to support reactions of mass which is apportioned to system support points. It 

constitutes the total effect of all system mass which does not participate in (i.e., "missing" from) 

the modes with frequencies belowfzpA. The system response to the missing mass is calculated by 

performing a static analysis for applied loads equal to the missing mass multiplied by the 

spectrum ZPA. This method is mathematically rigorous and is considered the only acceptable

NUREG/CR-6645 Is
'COMBINATION i METHODS

method to account for high frequency modal contributions (f fzpA) and mass apportioned to 

system support points. 

Kennedy (Reference 3) documented this method and recommended that it be included in 

Regulatory Guidance. The 1989 revision to the SRP Section 3.7.2, "Seismic System Analysis," 

(Reference 2) incorporated Kennedy's recommendation as Appendix A. The mathematical 

details are presented in both References 2 and 3. For completeness, the mathematical 

formulation is included as Appendix I to this report. The guideline provided in References 2 and 

3, that the missing mass contribution needs to be considered only if the fraction of missing mass 

at any degree of freedom exceeds 0.1,should be eliminated. This guideline does not consider the 

total mass which is missing, which in the limit could be 10%. In a static analysis this represents 

a 10% reduction in the applied load. The missing mass contribution should be calculated in all 

response spectrum analyses, because its potential effect on support reactions is difficult to judge 

based on the fraction of missing mass. This calculation has been automated in a number of 

piping analysis codes and does not represent a significant computational effort. 

The missing mass contribution to the response spectrum analysis solution represents response 

which is completely in-phase with the time varying acceleration input and can be scaled to the 

instantaneous acceleration to obtain its contribution at any specific point in time. This 

characteristic is not important in response spectrum analysis because only peak response is 

predicted. In this case, the ZPA is used to generate the missing mass loading. However, the 

importance of the missing mass contribution is not limited to response spectrum analysis only. 

Mode superposition time history analysis is most accurately and efficiently performed by a 

procedure similar to that employed in response spectrum analysis (Reference 4). Only modes 

which vibrate at frequencies below fzA need to be included in the transient mode superposition 

solution. The missing mass contribution, scaled to the instantaneous acceleration, is then 

algebraically summed with the transient solution at the corresponding time to obtain the total 

solution. This method is more rigorous and accurate than including additional modes in the 

transient mode superposition solution. Even if additional modes are included, it is still necessary 

to calculate the missing mass for the excluded, higher frequency modes and system support 

points. This is quantitatively demonstrated in Appendix E of this report. 

Use of the Missing Mass method for calculating the contribution of high frequency modes is 

recommended in Draft ASCE Standard 4 (Reference 10) for both response spectrum analysis 

(Eqn. 3200-8) and mode superposition time history analysis (Eqn. 3200-5). In Reference 10, this 

is referred to as the "residual rigid response due to the missing mass."

19 NUREG/CR-6645
2 COMBINATION METHODS

2.3.2 Static ZPA Method 

The Lindley-Yow Method (Reference 14) defines the acceleration of the in-phase response 

component of all modes to be the ZPA of the response spectrum. As discussed in Section 2.2.1, 

the algebraic summation of the in-phase response components for all modes (Rr) is equivalent to 

the static response for a load equal to the total mass times the ZPA. When using the 

Lindley-Yow method, an alternate approach to including the contribution of high frequency (f> 

fzp ) modes is to calculate Rr directly by the Static ZPA method. This eliminates the need for 

calculation of the missing mass, since it is automatically included in the static analysis of total 

mass times ZPA. The out-of-phase response component (Rp) is calculated in accordance with 

the Lindley-Yow method. 

During the course of numerically verifying the equivalence between Rr calculated by Lindley

Yow plus Missing Mass approach and Rr calculated by the Static ZPA Method, a significant 

result was obtained which led to a supplementary study of differences in mass distribution 

defined in a dynamic analysis model and in a static analysis model. Using a BNL version of 

SAP V adapted for piping analysis, correlation was not initially achieved. Further investigation 

identified the source of the discrepancy to be the different treatments of the piping system mass. 

Calculation of Rr by the Lindley-Yow plus Missing Mass approach utilized the dynamic analysis 

model, in which the distributed piping system mass is replaced by discrete masses at the nodes of 

the model. Calculation of Rr by the Static ZPA Method initially utilized a static analysis model, 

in which the mass remains distributed along the pipe elements. For the BM3 model, the 

differences in mass distribution led to significant discrepancies in the support reactions predicted 

by the two analyses. When the discrete mass distribution from the dynamic analysis model was 

utilized in the static ZPA method, all discrepancies disappeared and equivalence between the two 

approaches to calculate Rr was verified. This issue is further evaluated in Appendix C; 

guidelines for ensuring that the model discretization is sufficient to accurately represent the 

distributed mass are also provided. 

2.4 Complete Solution for Response Spectrum Analysis 

For the numerical studies presented in Chapter 3, three approaches are used to define the 

complete response spectrum analysis solution. For simplicity, these have been designated 

Meth6d 1, Method 2, and Method 3, and are defined below. The coefficients Cjk are defined by 

one of the out-of-phase combination methods (Section 2.1.1 through 2.1.6). In the numerical 

studies, all six methods were tested in conjunction with Methods 1, 2, and 3. 

2.4.1 Method 1 

Method 1 represents the common method applied to response spectrum analysis since the 1980's. 

Amplified modal responses (f<fzpA) are combined by SRSS with a correction for closely spaced 

modal frequencies. The contribution of unamplified modal responses (f>fz,) is calculated by

NUREG/CR-6645 20
2 COMBINATION METHODS

the missing mass method of Section 2.3.1. These two components are then combined by SRSS 

to produce the total solution. Mathematically, this is represented by 

Rp = [E E: CjkRjR

n = no. of modes below fzpA (Eqn. 2-23)

Rr = Rmissing mass 

Rt = JRp2 +Rr 2 

2.4.2 Method 2 

Method 2 introduces the concept of in-phase and out-of-phase modal response components for 

the amplified modes (f<fzm)- Mathematically, the complete solution is represented by 

Rpi = Ri * (1I - i) 

Rri = Ri * ai

Rp Rp njý =j .ojR] 

n = no. of modes below fz,,A

Rr = Rr 

i=l

(Eqn. 2-24)

+ Rmissing mass

Rt = IRp 2 +Rr 2 

The method recommended in Draft ASCE Standard 4 (Reference 10) for obtaining the complete 

response spectrum analysis solution (Eqns. 3200-17 and 3200-18) is essentially equivalent to 

Method 2. 

Method 2 is equally applicable to both the Lindley-Yow (Section 2.2.1) and the Gupta (Section 

2.2.3) methods. Only the definition of ai changes. For the initial numerical study described in

21 2NUREG/CR-6645
2 COMBINATION METHODS

Chapter 3, the Lindley-Yow method was selected for implementation. For the follow-up 

numerical study described in Chapter 4, the Gupta method was implemented. 

2.4.3 Method 3 

Method 3 is a variation of Method 2, which utilizes the Static ZPA Method of Section 2.3.2 to 

calculate Rr. Mathematically, the complete solution is represented by 

Rpi = * (I - i) 

Rp = Cjk RPj RpkI 

n = no. of modes below fzpA (Eqn. 2-25) 

Rr = Ra 0 ZPA 

Rt = Rp2+ Rr2 

Method 3 is compatible only with the Lindley-Yow method, because calculation of Rr by the 

Static ZPA Method is based on the Lindley-Yow definition for a, per Equation 2-9.

NUREG/CR-6645 22
3 QUANTITATIVE STUDIES OF MODAL RESPONSE 

COMBINATION METHODS 

In addition to the qualitative review presented in Section 2, a series of analyses were conducted 

to provide a quantitative basis for recommending revisions to regulatory guidance on modal 

combination methods. A piping system model previously utilized in the NUREG/CR-5627 study 

conducted by BNL (Reference 6) was selected for analysis. This model is designated BM3. 

Detailed descriptions of the BM3 model and the seismic loading are presented in Appendices A 

and B respectively. A brief description of the model and loading is provided in Section 3.1 

3.1 BM3 Piping Analysis Model 

Figure 3-1 shows a schematic of the BM3 piping model, including node, element, support, and 

pipe size information. Table 3-1 provides a list of the first 31 natural frequencies (up to 70 Hz) 

for the BM3 model. Appendix A also includes a printout of the computer model, the nodal mass 

distribution, and the mode participation factors. Figures 3-2 and 3-3 show the 1% and 5% 

damping, unbroadened acceleration response spectra of the input time history. The 1% response 

spectrum was used as x-direction input for all response spectrum analysis (RSA) conducted, with 

one exception. One analysis using the 5% spectrum was conducted in order to assess the effect 

of damping on the behavior of the modal response combination methods. 

The input response spectrum is typical of an in-structure response spectrum which results from 

filtering and amplification of the seismic ground motion through the building structure/soil 

system. It exhibits a sharp, highly amplified peak at the fundamental frequency of the 

structure/soil system. Piping systems attached to building structures are usually subjected to this 

type of seismic input, instead of ground motion associated with broad-banded response spectra. 

For this study, the unbroadened spectrum was used to provide a direct comparison to time history 

analysis results. The input time history of acceleration used to generate the response spectra and 

to perform the time history analyses is provided in Appendix B. 

For the BM3 input response spectrafzpA = 16.5 Hz and the ZPA = 0.54 g. The BM3 model has 

14 modes below fzpA. These modes are associated with the amplified modal responses. From 

Table 3-1, the following modes are closely spaced, based on a 10% criterion: modes 3 and 4; 

modes 5 and 6; modes 6 and 7; modes 8, 9, and 10; modes 9, 10 and 11; modes 11 and 12. The 

number of closely spaced modes was judged to be adequate to assess differences between the six 

(6) methods tested for combination of the out-of-phase modal responses. For development of 

complete RSA solutions utilizing Methods 1, 2, and 3 defined in Section 2.4, only the first 14 

modes (f<fzpA) are needed.

23 NUREG/CR-6645
3 QUANTITATIVE STUDIES

3.2 Description of Analyses Performed 

3.2.1 Baseline Time History Analysis 

To provide a baseline for assessing the accuracy of RSA methods, time history analysis of BM3 

was conducted using the acceleration input tabulated in Appendix B. Both mode superposition 

and direct integration solutions were obtained. Appendix E contains a discussion of the 

comparison between the two time history solutions and the tabulated results for support reactions 

and pipe end moments, for 1% and 5% damping. 

In the NUREG/CR-5627 study (Reference 6), a mode superposition time history analysis had 

been performed using the first 31 modes of the BM3 model (f• = 70 Hz). In this study, a direct 

integration analysis was performed and compared to the previously obtained solution. 

Significant discrepancies for a number of support reactions prompted additional investigation of 

the time history solutions. It was determined that the missing mass contribution to the mode 

superposition solution was significant for these support reactions. Inclusion of the missing mass 

in the mode superposition solution resolved the discrepancies. The tabulated results in Appendix 

E show the mode superposition solution with and without the missing mass contribution, as well 

as comparison to the direct integration solution. 

The excellent correlation obtained between the mode superposition and the direct integration 

solutions demonstrates that constant modal damping can be reasonably approximated in the 

direct integration method by careful selection of the target frequencies for the calculation of the 

damping coefficients a and P3. Guidelines for performing both mode superposition and direct 

integration analysis are provided in Appendix E. 

For comparison to complete RSA results, the mode superposition solution with the missing mass 

contribution was used as the baseline. There would be negligible differences in the comparison 

if the direct integration solution was used as the baseline. The selection was made because 

several comparisons to partial modal summations are included in the quantitative studies; 

baselines for these partial modal solutions can only be developed by the mode superposition 

method. 

3.2.2 Response Spectrum Analysis 

A series of eight (8) different comparisons between time history and RSA solutions were 

generated. These results are presented in Tables 3-2 through 3-17, for support reactions and pipe 

end moments. Pipe end moments are defined as (Mx2 + My2 + Mz2 •". In each Table, 

predictions using all six of the out-of-phase response combination methods defined in 

Section 2.1 are presented. Actual numerical values are shown for the baseline time history

NUREG/CR-6645 24
3 QUANTITATIVE STUDIES

solution. The RSA results are shown as the ratio of the RSA numerical value to the time history 

numerical value. This form of presentation was chosen to facilitate easy recognition of 

overprediction and underprediction by the RSA methods. A ratio of 1.0 indicates exact 

agreement; a ratio greater than 1.0 indicates RSA overprediction; a ratio less than 1.0 indicates 

RSA underprediction. The mean value and standard deviation of the ratios are shown at the 

bottom of the table for each RSA method, as an indication of the overall correlation and scatter, 

compared to the time history solution. In calculating the mean and standard deviation of the 

ratios, all output locations were weighted equally, regardless of their response magnitude. 

The eight (8) comparative analyses are 

1) 8 Mode Time History vs. 8 Mode RSA Combination, Rpi = 

2) 14 Mode Time History vs. 14 Mode RSA Combination, Rpi = R, 

3) 31 Mode Time History vs. 31 Mode RSA Combination, Rpi = • 

4) Complete Time History vs. Method 1, n = 14 

5) Complete Time History vs. Method 2 (Lindley-Yow), n=14 

6) Complete Time History vs. Method 2 (Lindley-Yow), n=31 

7) Complete Time History vs. Method 3, n = 14 

8) Complete Time History vs. Method 3, n = 14 (5% damping) 

Analysis No.'s 1, 2, and 3 were conducted to study the ability of the six (6) out-of-phase 

response combination methods to match the time history results as the number of included modes 

is increased. For these three analyses, the modal responses, R1 , are assumed to be out-of-phase; 

i.e., Rp2 = Rý. This method of RSA was used in the early applications of seismic analysis in the 

nuclear power industry. 

Analysis No.'s 4, 5, and 7 were conducted to evaluate the ability of Methods 1, 2, and 3 to 

match the complete time history solution. Analysis No. 8 tested the sensitivity of the results to a 

change in damping from 1% to 5%. Analysis No. 6 was included to verify that Method 2 

produces the same results when additional modes abovefzpA are included in the analysis. 

The equivalency between Rr calculated by Method 2 and Rr calculated by Method 3 was also 

verified by a separate numerical study. Table 3-18 compares the results of four (4) procedures to

25 NUREG/CR-6645
3 QUANTITATIVE STUDIES

obtain Rr, including the Static ZPA Method. In applications of the Static ZPA Method, the 

dynamic lumped mass representation is used. See the discussion in Section 2.3.2 and 

Appendix C. 

3.3 Discussion of Numerical Results 

3.3.1 Out-of-Phase Modal Response Combination 

Tables 3-2 through 3-7 present the 8, 14, and 31 mode RSA comparisons to time history results. 

The 8 mode RSA shows the best correlation to time history. This is reasonable because the 

highly amplified responses are predominately out-of-phase with the acceleration input. The 14 

mode RSA shows poorer correlation to time history. The six (6) additional modal responses, up 

tofzpA, would be expected to have a significant in-phase component, which is not appropriately 

treated by the out-of-phase modal response combination methods. The 31 mode RSA shows the 

poorest correlation to time history. The additional 17 modes withf>fzpA are completely in

phase with each other. The out-of-phase modal combination methods are inappropriate. 

Among the combination methods, the RB-DSC and CQC methods consistently show the best 

correlation to the time history solution. The NRC Grouping, Ten percent and DSC methods 

produce the most conservative predictions for locations where the time history results are 

exceeded. However they produce very similar predictions for locations where the time history 

results are either underpredicted or are reasonably matched. For the BM3 analysis, SRSS 

performed better overall than the three NRC methods, but worse than the RB-DSC and CQC 

methods. The increased conservatism introduced by the three NRC methods does not appear to 

be justified, because there is only a minimal corresponding improvement when RSA 

underpredicts the time history results. 

The small difference in predictions between the RB-DSC and CQC methods is attributed to the 

definition of t16fi 5 seconds for the RB-DSC method. Consequently, the mode correlation 

coefficients for RB-DSC are somewhat larger than the CQC coefficients. See Appendix D. 

3.3.2 Complete Solution 

Tables 3-8 through 3-17 present comparisons of RSA Methods 1, 2, and 3 to the complete time 

history solution. Method I represents the common RSA methodology utilized since the 1980's. 

Method 1 produces consistently inferior results when compared to Methods 2 and 3. This is 

attributed to the lack of consideration of the in-phase response component for the amplified 

modal responses. Both overpredictions and underpredictions are more extreme, which is 

consistent with the treatment of in-phase response as out-of-phase response.

NUREG/CR-6645 26
3'i QUANTITATIVE STUDIES

Methods 2 and 3 produce essentially identical results, which are in reasonable agreement with 

the time history solution. The best correlation with the least scatter is achieved with the RB-DSC 

and CQC methods. 

The largest overpredictions occur at support reactions of small magnitude, relative to the largest 

support reactions. Differences among the mode combination methods are also most extreme at 

these support reactions. This behavior is most likely attributed to the absence of a dominant 

modal contribution to the total support reaction. Without a dominant modal contribution, the 

limited capacity of response spectrum analysis methods to recreate the actual phase relationships 

between modal responses is more likely to result in less accurate predictions. A typical example 

is Node 1, MX. 

Comparison of Tables 3-10 and 3-11 to Tables 3-12 and 3-13 demonstrates the insensitivity of 

the Lindley-Yow Method (Method 2) to the number of modes included in the RSA, provided all 

modes belowfp. are included. Table 3-18 provides a comparison of four (4) approaches to 

calculate the total in-phase response component, Rr. The equivalence between Rr for Method 2 

and Rr for Method 3 is evident. 

Results for the 5% damping input spectrum are presented in Tables 3-16 and 3-17, utilizing 

Method 3. The greater accuracy of the RB-DSC and CQC combination methods is more evident 

at 5% damping than at 1% damping. At 5% damping, the coefficients Cik are numerically larger 

than at 1% damping (see Appendix D). Consequently, differences between the combination 

methods are more pronounced. The overall level of conservatism for RSA methods is higher at 

5% damping than at 1%; however, the scatter is also significantly larger. The NRC methods and 

SRSS exhibit the largest increases in conservatism and scatter. 

3.4 Summary of Results 

Based on the quantitative studies described above, the following observations are made: 

"* Including modes abovef. in the out-of-phase modal response combination is always 

incorrect and cannot be justified. The contribution of modes abovef.. is accurately 

calculated by the "missing mass" procedure and it is considered mandatory to include this 

contribution. 

" Assuming amplified modal responses (f<fv) have only out-of-phase components can lead to 

significant error in the RSA prediction. Method 1, which is based on this assumption, is 

significantly less accurate than Methods 2 and 3, which incorporate separation of modal 

responses into in-phase and out-of-phase response components.

27 NUREG/CR-6645
3 QUANTITATIVE STUDIES

"* Of the six methods tested for combination of the out-of-phase response components, RB

DSC and CQC consistently produce the most accurate results, and are essentially equivalent 

to each other. Increased accuracy is most notable in the 5% damping case. SRSS compared 

fairly well to RB-DSC and CQC for 1% damping, but deviated significantly for 5% damping. 

"• Method 2, using the Lindley-Yow formulation, and Method 3 are equivalent; they provide 

two approaches to achieve the same solution. Method 3 involves less computation because 

the total in-phase response component is calculated in a single step.

NUREG/CR-6645 28
3 QUANTITATIVE STUDIES

Table 3-1 

BM3 Model Modal Frequencies

e f Mode Number (radians/sec) (cycles/sec) 

1 18.3 2.91 

2 27.6 4.39 

3 34.7 5.52 

4 35.8 5.70 

5 43.8 6.98 

6 46.1 7.34 

7 49.5 7.88 

8 64.7 10.30 

9 69.5 11.06 

10 70.6 11.23 

11 72.2 11.50 

12 78.1 12.43 

13 87.2 13.88 

14 101.3 16.12 

15 113.3 18.04 

16 117.0 18.62 

17 122.7 19.52 

18 122.9 19.56 

19 137.1 21.82 

20 139.4 22.18 

21 143.6 22.86 

22 163.1 25.95 

23 245.2 39.02 

24 248.3 39.52 

25 258.9 41.21 

26 288.5 45.91 

27 296.9 47.25 

28 330.7 52.64 

29 372.5 59.29 

30 440.4 70.09 

31 444.2 70.70

29 NUREG/CR-6645
3 QUANTITATIVE STUDIES

Forces in lbs 

Moments in in-lbs

Table 3-2 

BM3 Model 

8 Mode Combination 

Time History vs. Response Spectrum 

Support Reactions

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE (Tr) VALUE SRSS JGROUPING TEN PER. NRC-DSC [ R-DSC CQC 

1 Fx 3.34 0.87 0.87 0.88 0.88 0.87 0.87 

1 F, 2.91 1.07 1.07 1.08 LOS L07 L07 

1 Fz 0.93 1.50 153 1.54 1.53 1.49 L50 

1 Mx 32.25 0.83 0.91 0.93 0.87 0.81 0.81 

1 my 724.63 0.91 0.91 0.91 0.91 0.91 0.91 

1 Mi 18736 0.90 0.90 0.91 0.91 0.90 0.90 

4 Fx 25.04 0.88 0.88 0.88 0.88 0.88 0.88 

4 Fz 11.26 L44 1.46 L47 1.46 L43 L43 

7 1y 13.30 L12 1.13 L19 1.14 .11 L12 

11 iY 13.43 1.37 1.40 1.64 1.42 1.34 1.35 

11 Fz 70.88 0.85 0.85 0.85 0.85 0.S 0.85 

15 Fx 484.98 0.75 0.76 0.76 0.76 0.76 0.75 

17 F, 25.01 1.90 1.92 2.32 1.97 1.85 1.87 

17 F1 36.05 1.52 1.53 L75 1.56 I.55 1.54 

36 F, 8170 L00 L01 L02 L01 1.00 1.00 

36 Fz 36.61 1.35 L51 L78 L44 1.43 1.40 

38 Ft 41.85 L19 1.21 1.23 1.20 L1 L19 

38 1y 49.23 L02 1.03 1.03 L03 L02 L02 

38 Fz 12.26 1.39 1.39 1.40 1.39 1.39 1.39 

38 Mx 602.71 2.14 2.22 2.29 2.18 2.15 2.14 

38 my 883.44 1.33 1.33 1.34 1.33 1.32 1.33 

38 Mz 3486.10 1.02 1.03 L03 1.03 1.02 1.02 

23 Fx 137.41 1.16 1.16 1.16 L16 1.16 1.16 

23 Fy 69.96 L10 1.14 1.23 1.13 L10 1.10 

31 Fx 6.33 L00 1.21 1.23 105 0.95 0.97 

31 F, 16.51 L17 L25 1.28 120 L15 L16 

31 Fz 18.29 L66 1.95 L97 L74 1.5 1.61 

31 Mx 1486.40 L58 1.94 1.94 1.67 1.48 I-.5 

31 My 129.92 1.45 1.76 1.81 1.5 L37 L41 

31 Mz 446.70 1.22 1.53 1-56 1.31 L14 1.17 

Mean Value of 30 Comp. 1.22 1.29 1.35 1.25 1.21 1.21 

Standard Dev. of 30 Comp. 0.33 0.38 0.43 0.35 0.32 0.33.

NUREGICR-6645

1 % Damping

30
3 QUANTITATIVE STUDIES Table 3-3 

BM3 Model 

8 Mode Combination 

Time History vs. Response Spectrum (M1M2

7

2+~ 2)" Pipe End Moments 

Moments in in-lbs 1 % Damping 

ELEM. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES 

NO. ( VALUE I SRSS J GROUPING TEN PER. NRC-DSC RB-DSC CQC 

1 727.97 0.90 0.90 0.90 0.90 0.90 0.90 

2 720.75 0.39 0.39 0.89 0.89 0.$9 0.39 

3 31.89 0.91 0.91 0.92 0.92 0.91 0.91 

4 710.23 0.98 0.98 0.98 0.98 0.98 0.98 

5 631.26 L03 1.03 1.04 L03 1.03 L03 

6 1999.70 0.95 0.96 0.96 0.96 0.95 0.95 

7 1326.20 0.96 0.96 0.96 0.96 0.96 0.96 

a 1049.10 1.01 1.02 1.06 1.02 1.00 1.00 

9 1769.90 L1.0 01 1.01 1.01 1.01 1.01 

10 1675.30 100 L01 L03 1.01 1.00 1.0 

11 3931.90 0.36 0.86 0.86 0.86 0.86 0.86 

12 7620.70 0.87 0.87 0.8 0.87 0.87 0.87 

13 6299.40 0.90 0.90 0.91 0.90 0.90 0.90 

14 5414.80 0.71 0.71 0.73 0.71 0.71 0.71 

15 4056.20 0.75 0.75 0.81 0.76 0.76 0.75 

16 3257.00 0.77 0.78 0.86 0.79 0.78 0.78 

17 2532.00 L72 1.73 2.10 L78 1.71 L72 

13 2696.00 L03 1.04 111 O5 1.03 L03 

19 1937.60 L16 L17 1.31 L19 L16 L16 

20 7915.90 1.12 L13 L15 L13 L13 1.13 

21 1836.00 1.23 1.26 1.27 1.24 1.22 1.22 

22 2268.30 1.07 L10 LI LOS 1.06 1.07 

23 2151.00 1.0 L10 L11 1.09 1.07 1.07 

24 1751.20 1.10 L13 1.14 1.11 1.09 1.09 

25 53S.77 1.17 1.45 L48 1.25 L09 1.12 

26 49.58 1.26 1.5 1.57 1.34 L19 1.2 

27 327.90 1.97 2.20 2.26 2.04 1.91 1.94 

28 440.31 16 1.92 1.95 1.89 1.85 1.85 

29 456.61 L75 L78 1.80 L77 1.75 1.75 

30 1522.40 1.58 1.95 1.95 1.68 1.49 L53 

31 8396.90 L19 1.21 1.22 1.20 1.19 119 

32 8240.10 1.21 1.23 1.24 1.22 1.21 1.21 

33 7816.90 1.21 1.22 1.24 1.21 1.21 1.21 

34 6420.40 L08 1.09 1.09 1.08 L08 L08 

35 6909.70 L06 1.06 L07 L06 L06 L06 

36 1887.30 1.26 1.27 1.29 1.27 1.26 1.26 

37 3598.70 .10 L11 L12 .11 L10 1.10 

Mean Value of 37 Coip. 1.13 1.17 1.20 1.14 __LI2 L12 

Standard Dev. of 37 Comp. 0.30 0.35 0.37 0.32 0.29 0.29

31 NUREGICR-6045
3 QUANTITATIVE STUDIES

Forces in Ibs 

Moments in in-lbs

Table 3-4 

BM3 Model 

14 Mode Combination 

Time History vs. Response Spectrum 

Support Reactions

NODE REAC TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE (TI) VALUE I SRSS GROUPING TEN PER. I NRC-DSC RB-DSC CQC 

1 Ft 3.96 0.79 0.83 0.90 0.84 0.79 0.79 

1 F, 3-51 1.55 1.65 2.11 1.75 L41 L44 

1 Fz 2.57 2.91 3.00 4.10 3.29 2.51 2.63 

1 Mx 43.23 6.37 6.57 9.05 7.22 5.45 5.71 

1 my 794.03 0.87 0.87 0.91 0.88 0.86 0.86 

1 MA 215.80 0.97 1.01 1.19 LOS 0.93 0.94 

4 Fx 29.24 0.83 0.84 0.92 0.86 0.81 0.82 

4 Fz 27.64 2.72 2.80 3.83 3.07 2.34 2.45 

7 y 13.87 L12 1.20 1.28 1.18 .10 L10 

11 FY 13.25 .1.48 1.55 1.85 1.60 L43 L44 

11 Fz 93.01 0.86 0.87 LO0 0.91 0.82 0.83 

15 FX 706.21 0.62 0.63 0.65 0.63 0.63 0.62 

17 1y 25.56 1.91 1.95 2.36 1.99 L85 1.88 

17 Fz 62.29 1.27 1.37 1.48 1.31 1.28 1.28 

36 Fy 46.89 L93 2.16 2.27 2.03 1.91 L92 

36 Fz 68.17 L15 1.37 L47 1.21 L16 L16 

38 Fx 121.91 0.98 L09 1.20 L02 1.02 1.00 

38 F, 43.98 1.20 1.22 1.25 1.21 1.20 1.20 

38 Fz 42.65 0.90 1.05 1.14 0.96 0.96 0.94 

38 Mx 713.03 L97 2.27 2.41 2.09 1.95 L96 

38 MY 2783.00 0.93 1.07 1L16 0.99 0.98 0.97 

38 Mx 3117.80 1.19 1.21 1.24 1.20 L19 119 

23 F, 159.21 1.35 1.43 1.52 1.39 1.39 138 

23 Fy 28.00 3.76 4.72 5.18 4.12 3.68 3.72 

31 F1 6.99 1.45 1.70 1.85 1.56 1.46 1.46 

31 1Y 13.95 L78 1.96 2.05 1.85 1.80 1.79 

31 Fz 1.0 2.04 2.43 2.46 2.15 L95 1.98 

31 Mx 1102.30 2.21 2.72 2.75 2.35 2.09 2.14 

31 MY 2.59.45 L09 1.32 1.41 L17 1.11 LII 

31 M 1 608.00 1.31 1.59 1.70 L41 1.31 1.32 

Mean Value of 30 Cr•np, L65 L382 2.091 781 1.58 1.60 

Standard Dev. of 30 Comp, 1.14 1.24 1.67 1.30 0.97 L02

NUREGICR-6645

1 % Damping

32
3 QUANTITATIVE STUDIES 

Table 3-5 

BM3 Model 

14 Mode Combination 

Time History vs. Response Spectrum 

Moments in in-lbs (M 2

+My+MK2) Pipe End Moments I % Damping 

ELEM. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALU 

NO. (FiT) VALUE SRSS j GROUPING J TEN PER. J NRC-DSC f RB-DSC J CQC 

1 776.55 0.92 0.93 1.00 035 0.90 0.91 

2 759.36 0.91 0.91 0.97 0.93 0.89 0.90 

3 968.28 1.22 1"25 1.55 1.32 L12 1LIS 

4 773.63 1.06 L07 1.20 1.10 1.02 L03 

5 701.53 1.36 1.39 1.71 1.47 1.26 1.29 

6 2207.60 0.93 0.94 0.99 0.95 0.91 0.92 

7 1948.20 0.94 0.94 0.98 0.95 0.93 0.93 

$ 837.44 2.75 2.86 3.77 3.08 2.43 2.52 

9 1357.40 2.12 2.18 2.73 2.31 1.93 L99 

10 2183.80 0.96 0.96 L09 LO 0.92 0.93 

11 4173.00 0.82 6.83 0.84 0.83 0.82 0.82 

12 8283.10 0.81 0.81 0.82 0.82 0.81 0.81 

13 6579.90 0.87 0.87 0.88 0.87 0.87 0.87 

14 13174.00 0.59 0.60 0.63 0.61 0.60 0.60 

is 2294.00 1.50 155 L72 1.55 154 153 

16 1693.10 1.99 2.09 2.37 2.09 2.06 2.04 

17 2456.50 1.84 191 2.29 1.92 1.84 1.34 

18 2377.20 1.23 1.26 1.35 L25 1.22 1.22 

19 1635.50 L45 1.49 L67 L49 1.44 L45 

20 7586.00 1.25 1.32 1.36 1.27 1.25 1.25 

21 1616.20 L49 1.61 L66 1.54 L49 1.49 

22 1977.10 1.51 1.60 1.70 1.56 L54 1.53 

23 1900.20 1.51 1.61 1.70 1.56 1.54 1.53 

24 1607.70 1.5 1.64 L73 1.57 155 1.54 

25 265.36 3.01 3.74 4.00 3.29 2.89 294 

26 271.57 2.87 3.53 3.79 3.13 2.78 2.82 

27 348.19 2.11 2.49 2.63 2.26 2.07 2.09 

28 552.58 1.65 1.75 1.79 1.69 L65 1.65 

29 581.00 1.56 1.64 1.67 1.59 156 1.56 

30 1102.30 2.34 2.87 2.92 2.49 2.22 2.27 

31 7586.80 1.61 L75 L84 1.67 L65 1.64 

32 9093.40 1.41 154 1.64 1.46 L44 1.43 

33 11281.00 1.15 1.28 1.36 1.20 LI8 1.17 

34 6061.80 1.21 1.22 1.24 1.22 L21 1.21 

35 6634.10 1.22 1.28 L32 1.24 1.23 1L23 

36 2451.60 1.18 1.30 1.37 1.22 1.20 119 

37 4009.70 1.18 1.27 1.33 1.21 1.20 19 

Mean Value of 37 Comp. 1.46 1.S8 172 L 53 1.44 L45 

Standard Dev. of 37 Comp. 0.58 0.73 0.84 0.65 0.55 0.56

33 NUREG/CR-6645
3 QUANTITATIVE STUDIES

Forces in Ibs 

Moments in in-lbs

Table 3-6 

BM3 Model 

31 Mode Combination 

Time History vs. Response Spectrum 

Support Reactions

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE (TI) VALUE SRSS GROUPING TEN PER. NRC-DSC RB-DSC CQC 

1 FX &40 0.82 0.83 0.M5 0.83 0.82 0.82 

1 , 10.14 2.28 2.34 2.39 2.34 2.31 2.31 

1 Fz L60 4.82 5.01 6.75 5.43 4.19 4.37 

1 MN 49.98 5.63 5.85 7.95 6.36 4.85 5.07 

1 my 776.73 0.89 0.89 0.93 0.90 0.88 0.88 

1 MA 193.46 2.72 2.79 2.87 2.80 2.73 2.74 

4 Fx 68.75 0.66 0.67 0.68 0.67 0.66 0.66 

4 Fz 19.55 3.92 4.07 5.49 4.42 3.40 3.55 

7 1Y 13.31 1.19 1.28 1.36 1.25 L17 1.17 

11 F, 13.31 1.51 L60 1.88 L64 L43 1.44 

11 Pa 81.14 1.00 LOO L15 L04 0.95 0.96 

15 Fx 713.32 0.61 0.62 0.64 0.62 0.62 0.62 

17 F, 25.61 1.90 1.95 2.35 L98 1.85 1.87 

17 Fz 64.26 1.24 1.33 1.45 1.28 1.25 1.25 

36 1y 46.31 1.96 2.19 2.30 2.06 1.94 L95 

36 FZ 48.5 2.07 2.33 2.44 2.15 2.08 2.08 

38 Fx 128.79 0.95 L06 L16 1.00 1.00 0.93 

38 F 43.2 1.21 1.24 1.26 1.23 1.21 1.21 

38 ]z 31.40 1.26 1.47 1.58 1.34 1.33 1.31 

38 Afx 720.04 1.95 2.25 2.39 2.07 1.93 L94 

38 My 2142.00 1.23 L42 1.54 1.30 1.30 1.28 

38 Mz 3088.90 1.20 1.22 1.25 1.22 1.20 1.20 

23 Fx 259.22 0.88 0.93 0.99 0.91 0.91 0.90 

23 1 26.04 4.07 5.1 5.60 4.47 3.93 4.01 

31 FP 23.24 0.84 0.88 0.91 0.86 0.84 0.84 

31 14.21 1.75 1.92 2.02 L82 1.77 1.76 

31 Fz 16.07 L97 2.34 2.38 2.08 1.88 L92 

31 MA 1136.90 2.14 2.64 2.67 2.28 2.02 2.07 

31 My 608.43 0.83 0.89 0.91 0.85 0.83 0.83 

31 M 1 1767.90 0.81 0.87 0.89 0.83 0.81 0.81 

Mean Valur of 30 Comp. J 1.81 1 L97 2.23 1.93 L74 L76 

Standard Dev. of 30 Comp. 1.27 1.38 1.84 L44 LI0 .L5

NUREG/CR-6645

1 % Damping

34
3 QUANTITATIVE STUDIES 

Table 3-7 

BM3 Model 

31 Mode Combination 

Time History vs. Response Spectrum SPipe End Moments 

Moments in in-lbs (M 2 +M2+ 21 % Damping 

ELEM. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES 

NO. (TH) VALUE SRSS GROUPING TEN PER. j NRC-DSC RB-DSC CQC 

1 23.39 0.91 0.92 0.98 0.93 0.89 0.90 

2 318.58 0.87 0.87 0.92 0.88 0.8 0.86 

3 1412.20 0.96 0.98 L16 1.02 0.90 0.92 

4 762.56 LOS L09 1.23 L13 1.04 1.05 

5 598.91 L66 1.69 2.05 1.78 1.54 1.57 

6 2134.40 0.97 0.97 1.02 0.99 0.95 0.96 

7 1876.40 0.98 0.98 1.01 0.99 0.97 0.97 

8 852.96 2.70 2.82 3.71 3.03 2.39 2.48 

9 1415.50 2.03 2.09 2.62 2.22 1.86 1.91 

10 1945.30 1.07 1.08 1.23 1.12 L03 L04 

11 3826.10 0.90 0.91 0.92 0.91 0.90 0.90 

12 8440.50 0.80 0.80 0.81 0.80 0.80 t.o 

13 6741.60 0.85 0.85 0.86 0.85 .85 0.8.5 

14 1324L00 0.59 0.60 0.63 0.60 0.60 0.60 

15 2290.80 1.50 L55 1.72 1.5 1.54 153 

16 1698.70 1.99 2.09 2.36 2.08 2.05 2.03 

17 2455.40 L14 1.91 2.29 1.92 1.84 1.84 

18 2360.10 1.24 1.27 1.36 1.26 L23 1.23 

19 1623.90 1.46 1.50 1.68 15 L46 1.46 

20 7539.20 1.26 1.33 1.36 1.28 1.26 1.26 

21 1595.60 1.5 1.65 1.70 1.57 L53 1.53 

22 2042.80 1.47 1-56 1.65 1.2 L50 1.49 

23 1800.80 1.61 1.71 L81 L66 L63 L63 

24 1522.10 1.66 1.78 1.89 1.72 1.69 L68 

25 926.28 1.23 1.38 L44 1.29 1.20 1.21 

26 665.57 1.55 1.76 L85 1.63 1.52 1.5 

27 399.00 1.97 2.28 2.40 2.09 1.93 1.94 

28 742.38 1.34 1.40 L44 1.37 134 L34 

29 824.73 1.26 1.32 1.34 1.29 1.27 1.27 

30 1989.00 L44 1.71 L73 L52 1.38 1.40 

31 7437.70 L65 1.79 1.88 L70 68 .L67 

32 8931.50 L43 157 1.67 L49 1.47 1.46 

33 1142L00 L14 1.27 L35 1.19 L17 16 

34 6096.60 1.20 1.21 1.24 1.21 1.21 1.21 

35 6633.90 1.22 1.28 1.32 1.24 1.23 1.23 

36 1942.80 1.51 L67 1.76 1.57 54 1.53 

37 3538.70 1.35 1.45 1.51 1.39 1.37 1.36 

Mean Value of 37 Comp. 136 1.43 .56 L41 1.34 1.35 

Standard Dev. of 37 Comp. 0.42 0.46 0.60 0.47 0.39 0.40

35 NUREG/CR-6645
3 QUANTITATIVE STUDIES

Forces in lbs 

Moments in in-lbs

Table 3-8 

BDM3 Model 

Modal Time History vs. Method 1 

Response Spectrum (14 modes plus missing mass) 

Support Reactions

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE (TH)VALUE SRSS IGROUPING TEN PER. I NRC-DSC I RB-DSC I CQC 

1 Fx 43.71 .10 1.10 1.10 1.10 1.10 .10 

1 F,/ 4.36 1.26 1.34 1.70 L41 LIS L18 

1 Fz L60 4.74 4.87 6.63 5.33 4.10 4.27 

1 Mi 49.88 5.55 5.72 7.86 6.28 4.76 4.99 

1 my 776.40 0.89 0.89 0.93 0.90 0.88 0.88 

1 M5 278.42 0.78 0.82 0.95 0.84 0.75 0.75 

4 Fx 116.79 0.80 0.80 0.81 0.80 0.80 0.80 

4 z 20.01 3.77 3.89 5.30 4.26 3.26 3.41 

7 Fy 13.27 1.20 1.29 1.37 1.27 1.18 L18 

11 F. 13.31 1.48 1.55 1.84 1.59 1.42 L44 

11 Fz 81.34 0.99 L00 1.15 1.03 0.95 0.96 

15 Fx 731.47 0.60 0.61 0.63 0.61 0.61 0.61 

17 FY 25.60 1.91 1.95 2.35 1.98 1.85 L88 

17 Fz 65.36 1.22 1.30 L41 1.25 1.23 1.22 

36 Fy 46.69 L93 2.17 2.28 2.03 1.92 L92 

36 Fz 42.12 2.02 2.36 2.51 2.11 2.04 2.03 

38 Fx 732.18 0.89 0.89 0.89 0.89 0.89 0.89 

38 F, 43.44 1.21 1.24 1.26 1.23 1.21 1.21 

38 FZ 29.95 1.40 L60 L71 L47 1.47 L45 

38 Mx 719.05 1.95 2.25 2.39 2.07 1.93 L94 

38 My 2084.97 1.31 L50 L62 1.38 1.37 1.36 

38 Mi 3085.87 1.20 1.22 1.25 1.21 1.20 1.20 

23 FX 259.59 L02 LOS L10 L03 L03 L02 

23 1y 26.08 4.04 5.06 5.56 4.43 3.96 399 

31 Fx 55.05 0.92 0.92 0.93 0.92 0.92 0.92 

31 F, 14.17 L75 1.92 2.02 1.82 L77 L76 

31 Fz 16.08 1.97 2.34 2.38 2.08 L88 1.91 

31 Mx 1137.30 2.15 2.64 166. 2.28 2.02 2.07 

31 My 612.38 0.74 0.80 0.83 0.76 0.75 0.75 

31 Mi 1773.20 0.79 0.85 0.88 0.82 0.79 0.80 

Mean Value of 30 Comp. 1.72 1.86 214 1.84 164 1.66 

IStandard Dev. of 30 Camnp, 1.23:[ 1.34 1.81 I 1.41 106 L 1.10

NUREGICR-6645

1% Damping

36
Table 3-9 

BM3 Model

3 QUANTITATIVE STUDIES

Modal Time History vs. Method 1 

Response Spectrum (14 modes plus missing mass) 

Moments in in-lbs (M.2+My2+M2) Pipe End Moments

1 % Damping

ELEM. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. (TET) VALUE SRSS GROUPING TEN PER. J NRC-DSC RB-DSC CQC 

1 790.95 0.91 0.92 0.99 0.93 0.89 0.90 

2 836.77 0.87 0.88 0.93 0.89 0.86 0.87 

3 1429.28 0.96 0.98 .15 1.01 0.91 G.92 

4 763.87 1.07 1.09 1.22 1.12 105 1.06 

5 597.98 1.65 1.68 2.04 1.77 1.5 1.56 

6 2133.18 0.97 0.98 1.03 0.99 0.96 0.96 

7 1877.81 0.98 0.98 1.01 0.99 0.97 0.97 

8 952.74 2.70 2.82 3.70 3.03 2.39 2.47 

9 1415.21 2.04 2.09 2.61 2.22 L5 1.90 

10 1946.24 08 .L09 123 L12 L04 L05 

11 3829.46 090 0.91 0.92 0.91 0.90 0.90 

12 8428.48 0.80 0.80 0.80 0.80 0.79 0.79 

13 6741.03 0.86 0.86 0.87 0.86 0.85 0.85 

14 13219.91 0.59 G60 0.63 0.61 0.60 0.60 

15 2293.75 1.50 1.55 L72 1.55 i.5 I.53 

16 1696.61 1.99 2.09 2.37 2.08 2.06 2.03 

17 2455.62 1.84 1.90 2.29 1.92 1.84 1.84 

18 2363.76 123 1.27 1.36 L26 123 123 

19 1625.77 1.46 150 1.68 L50 146 1.46 

20 7544.81 1.26 133 1.37 1.28 1.26 1.26 

21 1590.16 1.52 L64 1.70 1.56 1.5 1.52 

22 2046.41 L46 L.56 L64 1.51 1.49 L49 

23 1807.75 L61 L71 L81 L66 L64 L63 

24 151&31 L63 1.76 L85 L69 1.67 L65 

25 922.51 1.26 L41 1.47 1.32 1.24 1.25 

26 677.35 1.51 L72 L8O L60 L48 150 

27 390.71 1.99 2.31 2.43 2.11 1.95 L96 

28 73742 1.33 L40 L44 1.36 1.33 1.33 

29 830.76 1.22 1.27 1.29 1.24 1.23 1.22 

30 1994.56 1.42 L70 L72 1.50 1.37 1.39 

31 7428.79 1.65 L78 1.88 L71 1.68 L67 

32 8785.32 1.46 L60 L69 .52 1.49 L48 

33 10970.06 LIS L32 L40 1.23 1.21 1.20 

34 5993.82 1.22 L24 1.26 1.24 123 1.23 

35 678822 1.20 1.25 1.29 1.22 121 120 

36 1874.84 1.62 1.77 1.86 L67 1.64 L63 

37 3501.06 1.37 1.48 1.54 1.41 1.40 L39 

Mean Value of 37 Comp. 136 1.44 1.57 1.42 1.34 1.35 

Standard Dev. of 37 Comp. 0.42 0.46 0.60 0.47 0.39 0.40

37 NUREG/CR-6645
3 QUANTITATIVE STUDIES 

Table 3-10 

BM3 Model 

Modal Time History vs. Method 2 

Response Spectrum (14 modes using Lindley-Yow approach plus missing mass) 

Forces in lbs Support Reactions 1 % Damping 

Moments in in-lbs 

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE ___ VALUE __ SRSS GROUPING TEN PER. NRC-DSC RB-DSC CQC 

1 Fx 43.71 L06 1.06 1.06 1.06 1.06 1.06 

1 1Y 4.36 0.85 0.87 0.9s 0.89 0.81 0.82 

1 Fz L60 2.21 2.28 3.02 2.47 1.94 2.02 

1 Mx 49.88 2.50 2.60 3.52 2.82 2.15 2.25 

1 my 776.40 0.89 0.89 0.90 0.89 0.89 0.89 

1 Mi 278.42 0.83 0.84 0.87 0.85 0.83 0.83 

4 Fx 116.79 0.90 0.90 0.91 0.91 0.90 0.90 

4 FZ 20.01 1.81 1.87 2.45 2.02 1.61 L66 

7 Fy 13.27 1.05 1.07 LI0 L07 1.04 1.05 

11 Fy 13.31 1.14 .17 1.29 1.19 1.13 L13 

11 1z 81.34 0.87 0.87 0.91 0.88 0.86 0.86 

15 Fx 731.47 0.81 0.81 0.81 0.81 0.81 0.81 

17 F, 25.60 1.42 1.44 L62 L46 1.39 1.40 

17 Fz 65.36 0.97 0.99 1.02 0.98 0.97 0.97 

36 F,/ 46.69 1.35 1.44 1.47 1.39 1.35 1.35 

36 Ps 42.12 1.28 1.42 L49 1.32 1.30 1.29 

38 Fx 732.18 L05 1.05 L05 1.05 L05 L05 

38 1y 43.44 1.10 1.11 1.12 1.11 LI0 L10 

38 Ps 29.95 1.27 1.31 1.34 1.28 1.28 1.28 

38 Mx 719.05 1.25 138 L43 1.30 L24 1.24 

38 MY 2084.97 1.27 130 1.33 1.28 1.28 1.28 

38 Mz 3085.87 1.10 L10 1.11 LI0 1.10 1.10 

23 FX 259.59 1.32 1.32 1.33 1.32 1.32 1.32 

23 F, 26.08 2.14 2.57 2.80 2.31 2.09 2.11 

31 Fx 55.05 L02 1.03 1.03 L02 1.02 102 

31 F,/ 14.17 1.26 1.33 1.36 1.29 1.26 1.26 

31 Ps 16.08 1.43 1.64 L65 L48 L38 L40 

31 Afx 1137.30 1.49 1.79 1.80 1.57 L41 L44 

31 MY 612.38 0.99 LOO L01 0.99 0.99 0.99 

31 Ms 1773.20 100 1.00 L01 LO0 LO0 LO0 

Mean Value of 30 Comp. 1.25 1.31 L43 130 1.22 1.23 

Standard Dev. of 30 Comp. 0.42 0.48 0.67 0.49 0.35 0.37

NUREGICR-6645 38
Table 31 3 QUANTITATIVE STUDIES 

BM3 Model 

Modal Time History vs. Method 2 

Response Spectrum (14 modes using Lindley-Yow approach plus missing mass) 

Pipe End Moments

Moments in in-lbs ( +M. 2 +M2

2)12 1 % Damping

ELEML TIME HISTORY RESPONSE SPCTRUM (EXPRESSED AS RATIOS OF THE TH VALUES 

NO. (TH) VALUE SRSS I GROUPINGJ TEN PER. NRC-DSC j RB-DSC CQC 

1 790.95 0.90 0.90 0.91 0.90 0.99 0.89 

2 336.77 0.90 0.90 0.91 0.90 0.89 0.39 

3 1429.28 0.91 0.91 0.95 0.92 0.90 0.90 

4 763.87 0.98 0.98 1.02 0.99 0.97 0.98 

5 597.98 1.20 1.21 1.32 1.24 1.17 18 

6 2133.18 0.94 0.94 0.95 0.94 0.92 0.94 

7 1877.81 0.95 0.95 0.96 0.95 0.94 0.94 

8 352.74 1.59 L64 1.96 L72 1.49 1.52 

9 1415.21 1.42 1.44 1.60 L47 1.37 1.39 

10 1946.24 0.97 0.97 1.01 0.98 0.96 0.96 

11 3829.46 0.91 0.91 0.91 0.91 0.91 0.91 

12 8428.48 0.86 0.86 0.86 0.86 0.86 0.86 

13 6741.03 0.89 0.89 0.90 0.39 0.89 0.89 

14 13219.91 0.31 0.81 0.81 0.81 0.81 0.81 

15 2293.75 1.20 1.22 1.28 1.22 1.21 1.21 

16 1696.61 1.42 1.46 1.56 1.46 L44 1.44 

17 2455.62 1.38 1.40 1.57 1.41 1.38 1.38 

18 2363.76 1.09 1.10 1.13 .10 L09 1.09 

19 1625.77 1.22 1.23 1.31 1.24 .22 1.22 

20 7544.31 LI1 1.12 1.14 1.11 1.11 1.11 

21 1590.16 L30 1.33 1.36 1.31 1.30 1.30 

22 2046.41 1.49 L.50 1.52 1.49 1.49 1.49 

23 1807.75 1.38 1.41 1.44 1.39 1.38 1.38 

24 1518.31 1.42 1.45 1.48 1.44 1.43 1.43 

25 922.51 1.11 LI 1.20 L13 1.10 1.11 

26 677.35 1.20 1.29 1.31 1.23 .18 L9 

27 390.71 1.56 L68 L73 L60 1.53 1.55 

28 737.42 1.24 L26 1.27 1.25 1.24 1.24 

29 830.76 1.18 1.20 1.20 L19 LI LI 

30 1994.56 1.26 1.39 139 1.29 1.23 1.25 

31 7428.79 1.58 L61 1.63 1.60 1.58 1.58 

32 8785.32 L46 1.49 1.52 1.47 1.46 1.46 

33 10970.06 1L26 1.30 1.32 1.27 1.27 1.27 

34 5993.82 L13 1.14 1.14 L14 L14 1.14 

35 6788.22 1.23 1.24 1.25 1.23 1.23 1.23 

36 1874.34 1.31 1.35 1.38 1.33 1.32 1.32 

37 3501.06 1.25 1.28 1.29 L27 1.25 1.25 

Mean Value of 37 Comp. 19 1.21 1.26 L21 _ 1.15 L19 

Standard Dev. of 37 Comp. 0.22 0.24 0.28 0.24 0.22 0.22

39 NUREGICR-6645
3 QUANTITATIVE STUDIES 

Table 3-12 

BM3 Model 

Modal Time History vs. Method 2 

Response Spectrum (31 modes using Lindley-Yow approach plus missing mass) 

Forces in lbs Support Reactions 1 % Damping 

Moments in in-lbs 

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE (IIr VALUE SRSS [GROUPING TEN PER. NRC-DSC RB-DSC 

1 Fx 43.71 1.06 L06 L06 1.06 L06 L06 

1 F, 4.36 0.91 0.94 L05 0.96 0.88 0.89 

1 F1 1.60 2.22 2.29 3.03 2.48 L95 2.03 

1 Mi 49.88 2.51 2.61 3.52 2.83 2.16 2.26 

1 my 776.40 0.89 0.89 0.90 0.89 0.89 0.89 

1 MA 278.42 0.86 0.87 0.89 0.87 0.85 0.85 

4 Fz 116.79 0.91 0.91 0.91 0.91 0.91 0.91 

4 Fz 20.01 1.82 1.87 2.46 2.02 L61 L67 

7 iy 13.27 LOS 1.07 1.10 1.07 L04 L05 

11 F, 13.31 L14 1.18 1.29 L19 L13 113 

11 Fz 81.34 0.87 0.87 0.91 0.88 0.86 0.86 

15 Fx 731.47 0.81 0.81 0.81 0.81 0.81 0.81 

17 Fy 25.60 L42 L44 L62 1.46 1.39 L40 

17 Fz 65.36 0.97 0.99 1.02 0.938 0.97 0.97 

36 Fy 46.69 1.35 1.44 1.47 1.39 1.35 1.35 

36 Fz 42.12 1.29 1.42 L.50 1.33 1.31 1.30 

38 Fx 732.18 1.05 L05 L05 L05 1.05 L05 

38 F,. 43.44 1.10 .11 L12 .11 L10 LI0 

38 Fz 29.95 1.27 1.31 1.34 1.28 1.28 1.28 

38 Mi 719.05 1.25 1.38 L43 1.30 1.24 1.24 

38 MY 2084.97 L27 1.30 1.33 1.2 .238 1.28 

38 Mi 308587 LI0 L10 L1 1.0 .Li L10 

23 Fx 259.59 1.32 1.32 1.33 1.32 1.32 1.32 

23 1 26.08 2.14 2.57 2.80 2.31 2.09 2.11 

31 Fx 55.05 .L03 1.03 1.03 L03 L03 L03 

31 Fy 14.17 1.26 1.33 1.36 1.29 1.26 1.26 

31 ft 16.08 1.43 1.64 1.65 L48 1.38 L40 

31 MN 1137-30 1.49 1.79 1.80 1.57 1.41 L44 

31 M 612.38 0.99 L01 L01 L00 0.99 0.99 

31 M 1 1773.20 1.00 1.01 1L02 L00 1.00 L00 

Mean Value of 30 Comp. 1.26 1.32 L43 L31 1.22 L23 

Standard Dev. of 30 Comp. 0.42 0.48 0.67 0.49 U.S.5 0.37

NUREG/CR-6645 40
Table 3 QUANTITATIVE STUDIES 3-13 

BM3 Model 

Modal Time History vs. Method 2 

Response Spectrum (31 modes using Lindley-Yow approach plus missing mass) 

Pipe End Moments 

Moments in in-lbs (Mx2+My2+M)1 2 1 % Damping 

ELEM. TIME HISTORY RESPONSE PCTRUM (EXPRESSED AS RATIOS OF THE VALUES 

NO. (TH) VALUE SRSS (GROUPING TEN PER. NRC-DSC RB-DSC CQC 

1 790.95 0.90 &.90 0.91 0.0 U9 0.90 

2 836.77 0.90 0.90 0.91 0.90 0.89 0.89 

3 1429.28 0.91 0.92 0.96 0.93 0.90 6.90 

4 763.37 0.98 0.98 1.02 099 0-97 6.98 

5 597.98 1.20 1.21 1.3 1.24 .17 LI8 

6 2133.18 0.94 0.94 0.95 0.94 0.94 0.94 

7 1877.31 0.95 0.95 0.96 0.95 0.94 0.94 

3 852.74 1.59 1.64 L96 1.72 L49 1.5 

9 1415.21 L42 L44 1.60 1.47 1.37 1.39 

10 1946.24 0.97 0.97 1.01 0.98 0.96 0.96 

11 3829.46 0.91 0.91 0.91 0.91 0.91 0.91 

12 8428.48 0.36 0.86 0.86 0.36 0.36 0.36 

13 674L03 0.39 0.89 0.90 0.89 0.89 0.39 

14 13219.91 0.81 0.31 0.81 0.31 0.81 0.31 

is 2293.75 1.20 L22 1.28 .22 1.21 1.21 

16 1696.61 1.42 L46 1.56 L46 1.44 1.44 

17 2455.62 1.38 L40 1.57 L41 1.38 1.38 

18 2363.76 1.09 Li 1.13 .10 L09 L09 

19 1625.77 .22 1.23 1.31 1.24 1.22 1.22 

20 7544.3i .11 L12 L14 LI1 L11 L11 

21 1590.16 1.30 1.33 1.36 1.31 1.30 1.30 

22 2046.41 1.49 .50 152 1.50 L49 1.49 

23 1807.75 1.38 1.41 L44 1.39 1.38 138 

24 1518.31 L42 L45 L48 1.44 L43 L43 

25 922.51 L12 L19 1.21 LIS L11 L12 

26 677.35 1.21 1.30 1.32 1.24 L19 1.20 

27 390.71 1.56 L70 L73 L60 I.53 1.55 

28 737.42 1.24 1.26 1.27 1.25 1.24 1.24 

29 830.76 L19 1.20 1.21 L19 1.19 1.19 

30 1994.56 1.27 139 L40 1.30 1.23 1.25 

31 7428.79 1.58 L61 L63 L60 1.58 1.58 

32 8785.32 1.46 1.49 1.52 L47 L46 1.46 

33 10970.06 1.26 1.30 1.32 1.27 1.27 1.27 

34 5993.32 1.13 L14 L14 L14 L14 L14 

35 6788.22 1.23 1.24 1.25 1.23 1.23 1.23 

36 1874.84 1.31 1.35 139 1.33 1.32 1.32 

37 3501.06 1.25 1.28 1.29 L27 1.25 1.25 

Mean Value of 37 Comp. 1.19 1.22 1.26 1.21 1.18 1.19 

Standard Dev. of 37 Comp. 0.22 0.24 0.28 0.24 0.22 0.22

41 NUREG/CR-6645
3 QUANTITATIVE STUDIES 

Tablc 3-14 

BM3 Model 

Modal Time History vs. Method 3 

Response Spectrum (14 modes using modified spectrum approach plus static ZPA) 

Forces in lbs Support Reactions 1 % Damping 

Moments in in-lbs 

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE (TH)VALUE SRSS GROUPINGI TEN PER. NRC-DSC RB-DSC C C 

1 Fx 43.71 1.06 1.06 1.06 L06 L06 1.06 

1 FY 4.36 0.85 0.87 0.98 0.89 0.81 0.82 

1 Fz 1.60 2.21 2.28 3.01 2.47 1.94 2.02 

1 Mx 49.88 2.50 2.60 3.52 2.82 2.16 225 

1 my 776.40 0.89 0.89 0.90 0.89 0.89 0.89 

1 Mz 278.42 0.83 0.84 0.87 0.85 0.83 0.83 

4 Fx 116.79 0.90 0.90 0.91 0.91 0.90 0.90 

4 FZ 20.01 1.81 1.87 2.45 2.01 L61 1.66 

7 F, 13.27 1.05 L07 LI.0 1.07 1.04 L05 

11 FL 13.31 L14 .17 1.29 18 L12 13 

11 F 81.34 0.87 0.8&7 0.91 0.88 0.86 0.86 

15 Fx 731.47 0.81 0.81 0.81 0.81 0.81 0.81 

17 Fy 25.60 L42 1.43 1.61 L45 1.39 L40 

17 FZ 65.36 0.97 0.99 1.02 0.98 0.97 0.97 

36 FY 46.69 1.35 1.43 1.47 1.38 1.34 1.34 

36 Fz 42.12 1.28 1.41 1.49 132 1.30 1.29 

38 Fx 732.18 1.05 Los L0S 05 105 .L05 

38 1y 43.44 1.10 11 L12 1.11 LIO L10 

38 1z 29.95 1.27 1.31 1.34 1.28 1.28 1.28 

38 Mx 719.05 L24 1.37 L42 1.29 1.23 1.23 

38 MY 2084.97 1.27 1.30 1.33 1.28 1.28 1.28 

38 NU 3085.87 LIO 1.11 1.11 LIO L10 LIO 

23 F_ 259.59 1.32 1.32 1.33 1.32 1.32 1.32 

23 26.08 2.12 2.56 2.78 2.29 2.07 2.09 

31 F1 55.05 1.02 1.03 L03 L02 1.02 L02 

31 F1 14.17 1.26 1.33 1.36 1.28 1.26 1.26 

31 Pa 16.08 1.42 L63 1.64 L48 1.37 1.39 

31 Mx 1137.30 1.50 1.79 1.80 1.57 1.41 L44 

31 MY .612.38 0.99 1.00 L01 0.99 0.99 0.99 

31 M1 1773.20 LOO 1.00 1.01 1.00 LO0 1.00 

SteanVadu, of 30 ComP. 12 1.31 1.42 L304 L22 0.23 

Standard Dev. of 30 Comp. 0.42 0.48 0.67 0.49 0.35 10.37

NUREG/CR-6645 42
3 QUANTITATIVE STUDIES

Table 3-15 

BM3 Model 

Modal Time History vs. Method " 

Response Spectrum (14 modes using modified spectrum approach plus static ZPA) 

Moments in in-lbs (M.+M+M) 1

' Pipe End MomentsDam )in

ELEM. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. (H) VALUE SRSS J GROUPING I TEN PER. NRC-DSC j RB-DSC CQC 

1 790.95 0.90 0.90 0.91 0.90 0.89 0.89 

2 836.77 0.90 0.90 0.91 0.90 0.89 0.89 

3 1429.28 0.91 0.91 0.95 0.92 0.90 0.90 

4 763.87 0.98 0.98 1.02 0.99 0.97 0.98 

S 597.98 1.20 1.21 1.31 1.24 1.17 L17 

6 2133.18 0.94 0.94 0.95 0.94 0.92 0.92 

7 1877.81 0.95 0.95 0.96 0.95 0.94 0.95 

8 852.74 1.59 L64 1.96 L72 1.49 1.52 

9 1415.21 1.42 1.45 1.61 1.49 1.37 1.39 

10 1946.24 0.97 0.97 L01 0.98 0.96 0.96 

11 3829.46 0.91 0.91 0.92 0.91 0.91 0.91 

12 8428.48 0.86 0.86 0.86 0.86 0.86 U.6 

13 6741.03 0.89 0.89 0.90 0.89 0.89 0.89 

14 13219.91 0.81 0.81 0.81 0.1 0.81 0.81 

15 2293.75 1.20 1.22 1.28 1.22 1.21 1.21 

16 1696.61 1.42 1.46 1.56 1.46 1.44 L44 

17 2455.62 1.38 1.40 1.57 L41 1.38 1.38 

18 2363.76 1.09 1.10 L13 L10 1.09 L09 

19 1625.77 1.22 1.23 1.30 1.24 1.22 I.2 

20 7544.81 1.11 L12 L14 11 L11 L11 

21 1590.16 1.30 1.33 1.34 1.31 1.29 1.30 

22 2046.41 1.48 1.50 1.52 1.49 1.49 1.49 

23 1807.75 1.38 1.41 L43 1.39 1.38 1.38 

24 1518.31 1.42 1.45 1.48 L43 1.42 1.42 

25 922.51 LI1 LI 1.20 L13 10 L10 

26 67735 1.20 1.29 1.31 1.23 1.18 L19 

27 390.71 1.55 L68 L73 1.60 1.52 i15 

28 737.42 1.24 1.26 1.27 1.25 1.24 1.24 

29 830.76 1.18 L19 120 L19 1.18 1.18 

30 1994.56 1.26 1.39 1.40 1.29 1.23 1.25 

31. 7428.79 1.58 1.61 L63 1.60 1.8 1.58 

32 8785.32 1.46 1.49 L51 1.47 L46 L46 

33 10970.06 1.26 1.30 .32 1.27 1.27 1.27 

34 5993.82 1.13 1.14 L14 L14 1.14 114 

35 6788.22 1.23 1.24 1.25 1.23 1.23 1.23 

36 1874.84 1.31 1.35 1.39 1.33 L32 1.32 

37 350L06 1.25 L28 1.30 1.27 1.25 L25 

Mean Value of 37 Comp. 1.19 1.22 1.26 1.21 LI8 19 

Standard Dev. of 37 Comp 0.22 0.24 0•U 0.24 0.22 0.22

43 NUREGICR-6645
3 QUANTITATIVE STUDIES 

Table 3-16 

BM3 Model 

Modal Time History vs. Method 3 

Response Spectrum (14 modes using modified spectrum approach plus static ZPA)

Forces in lbs Support Reactions

Moments in in-lbs

5 % Damping

NODE REAC TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE M(T) VALUE SRSS GROUPING TEN PER. NRC-DSC I RB-DSC CQC 

1 Fx 44.22 L05 L05 LOS 1.05 LOS LOS 

1 ... 3.86 0.76 0.79 0.94 0.94 0.65 0.65 

1 Fz 0.97 3.35 3.43 4.61 4.56 L70 L74 

1 Mx 18.89 6.03 6.20 8.57 8.41 2.41 2.49 

1 my 694.80 0.97 0.97 0.98 0.98 0.96 0.96 

1 MN 240.22 0.86 0.87 0.90 0.93 0.86 0.86 

4 Fx 113.70 0.92 0.92 0.92 0.92 0.92 0.92 

4 Fz 13.85 2.43 2.47 3.29 3.25 1.34 1.36 

7 FL. 7.95 1.21 1.25 1.29 1.39 L13 14 

11 F,/ 7.13 1.55 1.61 1.80 1.94 L40 1.41 

11 FIZ 7L89 0.95 0.95 0.99 L00 0.92 0.92 

15 Fx 690.97 0.84 0.84 0.84 0.85 0.85 0.85 

17 F, 20.14 L55 1.57 1.73 L78 1.42 L43 

17 Fz 60.05 0.98 1.00 1.04 LOS 0.99 0.99 

36 F,. 38.70 1.38 1.46 1.50 1.58 1.29 1.30 

36 Fz 42.33 L19 1.27 1.32 L35 1.19 1.20 

38 Fi 740.68 1.04 L04 1.04 L04 1.04 L04 

38 F, 38.18 1.20 1.20 1.21 1.22 L19 L19 

38 Fz 29.85 1.23 1.26 1.30 131 1.30 1.30 

38 Afx 465.42 L49 L63 L72 L85 L35 L35 

38 My 2080.83 1.24 1.27- 1.29 1.30 1.29 1.29 

38 Mz 2713.20 L19 1.20 1.20 1.22 L19 119 

23 Fl 270.50 125 1.26 1.26 1.27 1.26 1.26 

23 F, 10.58 3.83 4.76 5.33 .88 2.69 2.76 

31 Fx 54.46 L04 L04 L04 L04 L04 L04 

31 1. 9.76 L72 1.77 L81 L88 L73 L74 

31 Fz 8.66 2.12 2.35 2.38 2.49 189 1.90 

31 Mx 560.28 2.21 2.59 2.61 2.69 10 1.82 

31 My 582.28 1.01 1.02 1.02 1.03 L.01 L01 

31 M1 1685.20 1.03 1.04 1.04 1.05 L03 1.03 

Mean Value of 30 Comp. 1.59 1.67 1.87 1.91 1.30 1.31 

Standard Dev. of 30 Comp. 1.10 1.21 1.65 1.67 &045 0.47

NUREG/CR-6645 44
Table 3-17 

BM3 Model

3 QUANTITATIVE STUDIES

Modal Time History vs. Method 3 

Response Spectrum (14 modes using modified spectrum approach plus static ZPA) 

. . . Pipe End Moments

Moments in in-lbs M 1 +My'+M')' 5 % Damping

ELEM. TIME HISTORY RESPONSE SPECRUM (EXPRESSED AS RATIOS OF THE TH VALUES 

NO. (TI) VALUE SRSS GROUPING I TEN PER. J NRC-DSC RB-DSC CQC 

1 713.05 0.96 0.96 0.97 0.98 0.95 0.95 

2 765.02 0.96 0.96 0.97 0.97 0.95 0.95 

3 1323.65 0.95 0.95 0.99 0.99 0.92 0.92 

4 684.42 1.02 1.02 1.05 1.06 1.00 LOO 

5 58.66 1.25 1.26 1.38 1.39 L14 14 

6 1893.29 1.00 LOO 1.01 L02 0.99 0.99 

7 1664.64 1.01 L0i 1.02 L03 1.01 1.01 

8 764.93 1.58 1.62 1.96 1.98 1.18 L19 

9 1319.86 1.35 1.36 1.53 L54 1.18 .18 

10 1714.12 LO0 Leo L04 LOS 0.97 0.97 

11 3507.84 0.96 0.96 0.97 0.97 0.96 0.96 

12 7650.43 0.92 0.92 0.92 0.93 0.92 0.92 

13 6075.21 0.95 0.95 0.96 0.96 0.95 0.95 

14 12487.74 0.83 0.83 0.84 0.85 0.84 0.84 

15 2030.00 1.26 1.28 1.33 1.35 1.30 1.30 

16 1057.35 2.08 2.12 2.25 2.31 2.17 2.18 

17 1495.69 2.01 2.03 2.23 2.26 1.96 1.97 

18 2027.05 1.20 1.21 .23 1.26 L19 L19 

19 1370.50 1.34 1.35 L40 1.43 1.32 1.32 

20 "612.74 L21 .23 1.24 L25 L21 121 

21 1337.38 1.44 L47 1.48 1.5 1.42 L43 

22 1803.08 1.63 L64 1.66 68 164 .L64 

23 1352.91 1.75 1.78 1.81 1.84 1.78 L78 

24 1229.48 L67 L70 L73 1.75 1.70 1.70 

25 958.32 1.02 1.05 1.06 1.08 L1O Leo 

26 689.97 L09 L14 1.16 1.19 L04 1.05 

27 322.88 L67 L77 1.81 1.89 1.60 L60 

28 624.60 1.39 1.40 1.41 1.44 1.38 1.38 

29 737.73 1.29 129 1.30 1.32 1.29 129 

30 1784.26 1.24 2 1.33 1.34 .17 L18 

31 7194.98 1.56 1.58 1.60 1.62 1.58 1.58 

32 8768.03 1.40 L42 1.44 1.46 1.42 L42 

33 10912.71 1.22 1.24 1.26 1.27 1.24 1.24 

34 5338.54 1.22 .22 1.23 1.24 1.22 1.22 

35 6348.71 127 1.28 1.28 1.29 1.27 1.27 

36 175.84 1.32 1.35 1.37 1.40 1.34 1.34 

37 3263.83 129 1.31 1.32 1.34 1.30 1.30 

Mean Value 0f37 Cornp. L28 1.30 1.34 1.36 1.26 126 

Standard Dev. of 37 Cornp. 0.30 0.32 0.35 0.36 0.31 0.31

45 4 NUREGICR-6645
I

Note 1: Dynamic model mass disf•ibutlon used In static ZPA calculation

Table 3-18 

Equivalence of 1R 

Lindley-Yow Plus Missing Mass vs. Static ZPA 

Node Reac. Eqn. 2-12 Missing Mass Eqn. 2-12 Missing Mass Eqn. 2-12 Missing Mass Total R, Total P, Total R, Stati ZPA 

No. Type n-8 n > 8 n-14 n;>,14 n=31 n'>31 n-8 n=14 nm31 (Note 1) 

1 Fx 1.40 -47.60 1.95 -48.20 6.47 -52.70 -46.20 -46.25 -46.23 -46.21 

1 Fy 0.42 0.24 1.35 -0.71 -11.90 12.50 0.66 0.64 0.60 0.66 

1 Fz 0.05 0.22 1.30 -1.04 0.27 0.00 0.27 0.26 0.27 0.26 

1 Mx 23.30 -12.80 -18.10 28.50 10.60 -0.27 10.50 10.40 10.33 10.53 

1 My -290.00 -38.90 .347.00 18.50 -329.00 0.30 -328.90 -328.50 -328.70 -329.25 

1 "Mz 64.40 94.20 102.00 56.60 67.50 90.60 158.60 158.60 158.10 158.43 

4 Fx -10.10 -92.70 -13.60 -89.20 -53.90 -48.90 -102.80 -102.80 -102.80 -102.75 

4 Fz -1.19 -4.29 -12.90 7.43 .4.95 -0.48 -5.48 -5.47 -5.43 -5.49 

7 Fy -1.07 1.05 -3.70 3.69 -0.31 0.30 -0.02 -0.01 -0.01 -0.01 

11 Fy -2.68 1.50 -0.80 -0.40 -1.20 0.00 -1.18 -1.20 -1.20 -1.17 

11 Fz 30.40 7.26 49.40 -11.80 37.40 0.20 37.66 37.60 37.60 37.65 

15 Fx -236.00 -238.00 -448.00 -25.60 -456.00 -18.40 -474.00 -473.60 -474.40 -474.26 

17 Fy -2.17 5.97 3.72 0.08 3.81 -0.01 3.80 3.80 3.80 3.78 

17 Fz 16.80 -52.00 -32.10 -3.05 -33.90 -1.15 -35.20 -35.15 -35.05 -35.17 

36 Fy 45.00 -36.10 9.33 -0.41 7.65 1.27 8.90 8.92 8.92 8.92 

36 Fz 26.60 5.19 65.30 -33.60 42.20 -10.60 31.79 31.70 31.60 31.68 

38 Fx 9.05 -776.00 -137.00 -630.00 -146.00 -621.00 -766.95 -767.00 -767.00 -767.29 

38 Fy -16.90 5.67 -11.70 0.55 -11.30 0.08 -11.23 -11.15 -11.22 -11.24 

38 Fz 1.16 -32.30 -48.10 16.90 -33.20 2.02 -31.14 -31.20 -31.18 -31.17 

38 Mx 400.00 -590.00 -183.00 -5.94 -190.00 0.99 -190.00 -188.94 -189.01 -189.06 

38 My 114.00 -2320.00 -3130.00 928.00 -2280.00 79.30 -2206.00 -2202.00 -2200.70 -2202.83 

38 Mz 1190.00 -392.00 837.00 -33.00 806.00 -3.17 798.00 804.00 802.83 804.58 

23 Fx 34.40 -338.00 -153.00 -152.00 -304.00 -0.54 -303.60 -305.00 -304.54 -304.11 

23 Fy 55.40 -56.00 1.71 -2.39 -0.74 0.05 -0.60 -0.68 -0.69 -0.68 

31 Fx 4.04 -60.50 -7.24 -49.30 -23.30 -33.20 -56.46 -56.54 -56.50 -56.51 

31 Fy -5.69 5.42 -1.19 0.97 -0.08 -0.15 -0.27 -0.22 -0.23 -0.25 

31 Fz 1230 -10.10 -0.05 2.29 2.21 0.03 2.20 2.24 2.24 2.24 

31 Mx 1040.00 -818.00j 78.00 138.00 215.00 1.31 222.00 216.00 216.31 219.27 

31 M -74.30 655.00 231.00 350.00 577.00 4.16 580.70 581.00 581.16 580.84 

31 Mz -289.00 1990.00 549.00 1150.00 1700.00 5.43 1701.00 1699.00 1705.43 1702.48

0I
BM3 Piping Model

M %'J ), ANCHOR. -~SPRING SUPPORT 

"o NODE NUMBER o ELEi•• T NUMBER 

8" 'f 

7a% Figure 3-1 cn -9b 

Ln

LEGEND*..
8.00 

7.00 

6.00 

5.00 

-' 4.00 

3.00 

2.00 

1.00 

0.00+

0.10 1.00 10.00 100.00 

Freq. (lz) 

Figure 3-2 BM3 Input Spectrum (1%)
3.00

2.50 

2.00 

S1.50 

1.00 

0.50 

0.00 , 

0.10 1.00 10.00 100.00 

Freq. (hz) 

h Figure 3-3 BM3 Input Spectrum (5%) 

U'
4 SUPPLEMENTARY STUDY OF ADDITIONAL MODAL 

RESPONSE COMBINATION METHODS 

4.1 Basis for Study of Additional Modal Response Combination Methods 

Although the agreement between the Response Spectrum Analysis (RSA) results and the Time 

History Solution is generally acceptable from an engineering perspective, the M. support reaction 

at Node 1 provides an excellent basis for a detailed investigation of the effects of RSA 

approximations on the prediction of peak response. It is noted that the magnitude of the reaction 

is relatively small when compared to the maximum moment reaction (-2%), and the RSA results 

exceed the Time History Solution by a factor of 2 or greater. Therefore, the inaccuracy of RSA 

in this case is not significant from a design standpoint. However, the specific elements involved 

in generating this significant overprediction were deemed worthy of a more detailed 

investigation. 

It is important to recall thatfzpA = 16.5 Hz was initially chosen based on the response spectrum, 

and that the first fourteen (14) natural modes of vibration for BM3 have frequencies below 16.5 

Hz. Modal contributions above 16.5 Hz are accounted for by the missing mass contribution. 

Therefore, the evaluation initially concentrated on the combination of the first 14 modes and 

comparison to the 14 mode time history solution. 

A numerical study was performed to examine the individual modal contributions of the first 

fourteen (14) modes in both the time history solution and the RSA solution. For Node 1, M. an 

interesting peculiarity of the modal contributions to the total predicted response is that two 

modes, Mode 9 (11.06Hz) and Mode 11 (11.50Hz), dominate the response. The responses are 

opposite in sign, and close in magnitude (-154 vs. 219), with identical times of peak response in 

the Time History Solution. This type of modal behavior poses a difficult challenge to the RSA 

methodologies employed in the initial study phase. 

Examination of Tables 3-2 and 3-4 for Node 1, MK shows the deterioration in accuracy between 

the 8 mode combination and the 14 mode combination. The SRSS combination of modal 

responses produces a gross overprediction. The DSC and CQC modal combination approaches 

should improve the situation. However, for 1% damping,fA 10HI-z, and frequency ratio = 0.95, 

the mode correlation coefficient is 0.15-0.23 (see Appendix D). This provides only a minor 

correction. The largest overpredictions are generated by the NRC Grouping, Ten Percent and 

DSC Methods, in which closely spaced modes are combined by absolute sum. The complete 

solution utilizing the Lindley-Yow assumption (see Tables 3-10, 3-12, or 3-14) provides 

improvement because significant fractions of the modal responses for modes 9 and 11 are 

included in the algebraic (in-phase) summation, and therefore tend to cancel each other. 

However, the fractions of the mode 9 and 11 responses included in the out-of-phase response 

combination are still significant enough to introduce inaccuracy on the conservative side.

51 NUREG/CR-6645
4 SUPPLEMENTARY STUDY

4.2 Description of Additional Modal Response Combination Methods 

The behavior of the RSA solution for the Node 1, MK reaction led to a recommendation by the 

Project Peer Reviewers that a sensitivity study be conducted using alternate modal response 

combination methods, which might produce more accurate results. Two alternate methods were 

selected based on the specific behavior at Node 1, MK. The Gupta Method was also selected for 

inclusion in this sensitivity study. In the Gupta Methodf2 defines the frequency above which the 

modal responses are assumed to be perfectly correlated and consequently combined algebraically. 

Three different assumptions for frequencyf 2 were tested. 

The mathematical formulations for the additional modal response combination methods are 

presented below. The two new methods, developed in conjunction with the project Peer 

Reviewers, are designated GROUP-1 and GROUP-2. Applications of Gupta's Method are 

designated GUP-0, GUP-1 and GUP-2. For comparison, the Lindley-Yow CQC results are also 

included in the tabulated results under the heading LINDLEY. Terms used in the following 

sections were previously defined in Chapter 2. 

4.2.1 Grouping Methods 1 and 2 

Grouping Methods consist of two similar procedures, designated GROUP-1 and GROUP-2. 

GROUP-1 was proposed by Dr. R.P. Kennedy following his peer review of the initial numerical 

results. GROUP-2 is a variation of GROUP-1 in which the mode 8 through mode 14 responses 

are not split into out-of-phase and in-phase response components. 

Group-1 

The 14 modes (/5 <fzpA) are first divided into their in-phase and out-of-phase response 

components according to the Lindley-Yow method. This has been previously described in 

Section 2.2.1. 

The out-of-phase modal response components are combined in accordance with the following 

formula: 

Rp = jRp2 + Rp2 +(Rp 3 + RP4)2 +(Rp5 + Rp6)

2 + RP7 (RPS +RP9 + RPIO + RP11 + R4n12 RP33 ÷ Rp1 2

NUREGICR-6645 52
4 SUPPLEMENTARY STUDY

The in-phase modal response components are algebraically summed: 

14 

Rr =E Rri 

i=i 

The total response for the first 14 modes is: 

Rt = FRp2 +Rr2 

which assumes that the in-phase and out-of-phase modal response components Rr and Rp, are 

uncorrelated. 

Group-2 

The first 7 modes Q7 < 9.8Hz) are divided by the Lindley-Yow method. The out-of-phase 

response components are combined in accordance with the following formula: 

Rp = IRp2 + Rp2 + (RP3 +RP) + (Rp5 +Rp6)2 + Rp2 

which is identical to GROUP-I for the first 7 modes. The in-phase response components for the 

first 7 modes are algebraically summed with the responses for modes 8 through 14, which are 

treated as completely in-phase with the input time history: 

7 14 

Rr E Rr, + E R, 

iMl i-8 

The total response for the first 14 modes is again given by: 

RP =-Rp2 +Rr 2 

GROUP-1 and GROUP-2 differ only in their treatment of modes 8 through 14. GROUP-1 and 

GROUP-2 both assume that all modes above 9.8 Hz (3.5 times the frequency of the spectral 

peak) are in-phase and combine algebraically. However, GROUP-1 maintains the Lindley-Yow 

split for all modes withf<fzpA.

53 NUREG/CR-6645
4 SUPPLEMENTARY STUDY

A detailed evaluation of the 1% damping, response spectrum generation calculation concluded 

that SDOF oscillators with frequencies above 9.5 Hz respond in-phase with the input time 

history. This is documented in Appendix F. The Appendix F procedure can be applied to any 

response spectrum, in order to pinpoint the frequency (designatedfp) above which SDOF 

oscillators respond in-phase with the input time history. Knowledge offxp for a given response 

spectrum provides a definitive separation between out-of-phase modal responses for which a 

combination rule is needed and the in-phase modal responses for which algebraic combination is 

appropriate. 

A significant characteristic of GROUP-I and GROUP-2 is that 100% correlation is assumed 

between closely spaced modes. The modal correlation coefficients are assumed to be either zero 

or one, similar to the NRC Grouping Method. However, algebraic combination is employed 

instead of the absolute sum combination used in the NRC Grouping Method. Grouping of modes 

in GROUP-I and GROUP-2 is problem dependent; it cannot be viewed as a generally applicable 

alternative for the treatment of closely spaced modes. 

The primary basis for the grouping is the shape of the BM3 input response spectrum. The single 

well defined, narrowly banded peak in the response spectrum at 2.8 Hz is the result of strong 

filtering of the ground motion through the building structure/soil. This spectral shape is 

encountered frequently in the analysis of distributed systems (e.g., piping) attached to building 

structure. The strong filtering produces a dynamic input to the distributed system which more 

closely resembles a single-frequency excitation rather than a broad spectral input characteristic of 

seismic ground motion. The CQC and DSC mode combination formulations for closely spaced 

modes are rooted in random vibration theory, which is more appropriate to earthquake ground 

motion. The phasing of closely spaced modal responses for strongly filtered input motion may 

be more analogous to the response of MDOF systems to a single-frequency input motion. The 

basis for the grouping of modes in GROUP-i and GROUP-2 can be defined as follows: 

* The peak of the response spectrum occurs at 2.8 Hz 

• Modes with frequencies less than 1.75 times the peak (1.75 x 2.8 Hz = 4.9 Hz) 

have uncorrelated out-of-phase response components 

Modes with frequencies greater than 3.5 times the peak (3.5 x 2.8 Hz = 9.8 Hz) 

respond in-phase 

Between 1.75 and 3.5 times the peak frequency, out-of-phase modal response 

components are grouped and combined algebraically if they are closely spaced 

(i.e., within 10% of the lowest frequency mode comprising a group). This implies 

they are perfectly correlated. 

The selection of 1.75 and 3.5 times the peak frequency for division of the frequency band is 

subject to further study and refinement.

NUREGICR-6645 54
4 SUPPLEMENTARY STUDY

4.2.2 Gupta Method - Variations to Value offA 

The Gupta Method was previously described in Section 2.2.3. The total in-phase response and 

total out-of-phase response are obtained by: 

14 

Rr= Rri 

i=l 

r14 14 1/ 

Rp= C11RpjRPk/2 

j=1 k=1 j 

Cjk is defined by Eq. 3200-19 of ASCE Draft Standard 4 (Reference 10): 

wherefj fk are the natural frequency and P1 j k < are damping ratio for the jth and kth mode 

respectively. This equation is essentially equivalent to the CQC method. See Section 2.1.7. 

The total response for the first fourteen (14) modes is obtained by SRSS combination of Rp and 

Rr: 

Rt = /Rp2 +Rr 2 

The Gupta Method is well documented (Reference 4) and has been recommended by the ASCE 

(Reference 10). In the initial phase of this study, it was decided not to include the Gupta Method 

because it uses semi-empirical relationships developed from earthquake ground motion records, 

which may not be applicable to building-filtered seismic motion. However, the follow-up 

sensitivity study provided an opportunity to include several variations of the Gupta Method. The 

key parameter varied wasf 2, which defines the frequency above which modal responses are 

combined algebraically (i.e., a = 1.0). GUP-0 assumesf 2 equals 3 times the frequency at the 

spectral peak (3 x 2.8 Hz = 8.4 Hz). GUP-I uses Gupta's definition off 2 from Reference 4 (see 

Section 2.2.3). This yields a numerical valuef 2 = 11.9 Hz. GUP-2 uses the definition off2 =fzPA; 

therefore,f 2 = 16.5 Hz. The selection off2 has a significant impact on the predicted response.

55 NUREG/CR-6645
4 SUPPLEMENTARY STUDY

4.3 Discussion of Numerical Results for Additional Modal Response 

Combination Methods 

The results of the mode response combination sensitivity study are tabulated in Table 4-1 and 

4-2, without the missing mass contribution above 14 modes, and Table 4-3 and 4-4, with the 

missing mass contribution above 14 modes. The missing mass contribution is algebraically 

summed with the in-phase component of response from the first 14 modes before combination by 

SRSS with the out-of-phase component of response. The last two rows of the tables provide the 

mean value of the response ratio and the corresponding standard deviation. 

All of the alternate combination methods except GUP-2 produced more accurate results than 

Lindley-Yow/CQC. GUP-2 produced the most conservatives results on average, and had the 

largest standard deviation. This is noteworthy because GUP-2 and Lindley-Yow/CQC use 

essentially equivalent definitions for Cjk, and settingf 2 =fzpA in the Gupta Method is equivalent to 

the Lindley-Yow definition offn. The only real difference is the numerical value of ac as a 

function of frequency, in the rangefsp to fzpA. 

GROUP-i, GROUP-2, and GUP-0 produced results in agreement with each other at most 

locations. The mean of the ratios and standard deviation of the ratios are the best of the methods 

tested. It is important to note that the onset of completely in-phase response is assumed to occur 

between modes 7 and 8, in all three methods (see Table 3-1). GUP-l, withf 2 = 11.9 Hz, assumes 

completely in-phase response occurs between modes 11 and 12 (see Table 3-1). GUP-1 

corresponds to Gupta's Method as presented in Reference 4 and described in Section 2.2.3. For 

the BM3 model and seismic input, GUP-1 produces superior correlation to the time history 

solution, compared to Lindley-Yow/CQC. 

The trend toward improvement in correlation to the time history solution with decreasingfp(fo) 

indicates that appropriate estimation of fn, <fzpA is a significant aspect of RSA. One method to 

identifyfp is provided in Appendix F. 

The largest discrepancies among the different methods occur for support reactions which are 

"orthogonal" to the x-directional load input: Fy, F, M.. These reactions are typically smaller in 

magnitude than the primary reactions Fx, MK, K. For the primary reactions, there is very little 

scatter in the results from one method to another. The 14 mode solution and the complete 

solution exhibit similar correlation with the corresponding time history solutions. For the 

"orthogonal" reactions, considerable scatter between methods is exhibited for both the 14 mode 

and complete solutions. 

Examining the results for Node 1, M., which initially instigated the follow-up sensitivity study, it 

is evident that the RSA prediction is extremely sensitive to the specific modal response 

combination method employed. Of particular surprise was the deterioration in accuracy

NUREGICR-6645 56
4 SUPPLEMENTARY STUDY

exhibited by GROUP-I, GROUP-2, and GUP-0 between the 14 mode solution and the complete 

solution. Further investigation of the time history solution for Node 1, MK revealed that the time 

of maximum response shifts as successive modal contributions are added. At the end of the 14 

mode summation, the maximum response occurs at 13.88 seconds. From Table E-l, it is noted 

that the complete solution for Node 1, MN maximizes at 6.68 seconds. 

The missing mass contribution above 14 modes occurs at the time of and is scaled to the peak 

acceleration of the input time history: -.54g at 7.35 seconds. The reaction at Node 1, M. due to 

missing mass above 14 modes is of the same order as the 14 mode time history summation: 28.54 

vs. 43.23. However, they occur at 13.88 seconds and 7.35 seconds respectively, while the 

complete time history solution maximizes at 6.68 seconds. It is obvious that the phase 

relationship between these two components of the time history solution is very important in 

arriving at the peak response of the complete solution. This element is lacking in the RSA 

method regardless of the specific mode combination method employed. 

As indicated by the scatter of predictions by RSA in the initial study phase and in the follow-up 

sensitivity study for Node 1, M., there will be cases where RSA cannot produce a reasonably 

accurate prediction of the peak response. However, it is important to note that this appears to be 

a problem for the "orthogonal" response quantities (i.e., Fy, F,, K for an x-direction input). A 

typical design RSA would include three directions of input motion, with the major F., F. and M. 

contributions coming from y and z direction input. In the SRSS combination of directional 

responses, errors in Fy, F, and M. due to x-direction input should have only a minor effect on the 

total predicted response due to three directional loading. 

An investigation of the modal response combination methods for three directions of input motion 

was not considered as part of this study, because the introduction of additional variables would 

have complicated the evaluation of the combination methods. This would be a logical and very 

useful extension of the results reported herein. 

Appendix H presents independent results generated by Professors A.K. Gupta and Abhinav 

Gupta of North Carolina State University for three directional loading of the BM3 model. These 

results verify the supposition stated above. Large errors in "orthogonal" output quantities (e.g., 

Node 1, Mi) are not significant in the combined response to three directional loading. 

Appendix H also documents a proposed alternate approach to determinef 2 (f'n) for use in the 

Gupta Method. The proposed method is qualitative and requires some judgment, but has an 

advantage over the quantitative method proposed in Appendix F. For the Appendix F approach, 

response spectra generation analyses are needed to determinefn, as a function of damping ratio. 

The method proposed in Appendix H requires only the response spectra plots for different 

damping values. A single value for f 2, independent of damping, is estimated from the 

superimposed plots, and is defined as follows: "The key frequencyf 2 is the minimum frequency

57 NUREG/CR-6645
4 SUPPLEMENTARY STUDY

at which spectra at various damping values converge." It appears that the Appendix H method 

would require unbroadened spectra for the determination of f 2

For the BM3 x-direction seismic input, f2 = 6.0 Hz is estimated in Appendix H. The method of 

Appendix F, identified 

f 1, = 9.5 Hz @ 1% damping 

f• = 7.5 Hz @ 5% damping 

for the BM3 seismic input. 

Since accurate definition offu, is essential to improving the correlation between RSA solutions 

and time history solutions, the methods proposed in Appendix F and Appendix H warrant 

additional study and refinement. The Appendix F approach is judged to provide an upper bound 

tofa,; the Appendix H method, based on its application to the current problem, may provide a 

lower bound to fn,. It is noted that for 1% damping, GUP-0 (f2 = 8.4 Hz) provided a better match 

to the time history, on average, than the solution presented in Appendix H (f2 = 6.0 Hz). See 

Table 4-3 and Appendix H Table 3. 

4.4 Summary of Results 

Based on the supplementary quantitative study described above, the following observations are 

made: 

"* For highly-filtered, in-structure response spectra, improvement in accuracy can be attained by 

better definition of the onset of completely in-phase modal response (ft,). 

" Algebraic grouping of closely-spaced modes may be more applicable to highly-filtered 

response spectra than the RB-DSC and CQC methods. 

" Large inaccuracies in prediction of small magnitude responses, due to one direction of 

seismic input, will likely be insignificant in the total RSA solution for combined three 

direction of seismic input. 

"* Candidate methodologies for definition of fa. have been proposed; these warrant further 

investigation and refinement. 

" The response of MDOF systems to a single frequency input may provide a mathematical 

basis for combining out-of-phase modal response components for highly-filtered, in-structure 

response spectra. Additional study in this area is warranted.

NUREG/CR-6645 58
4 ý SUPPLEMENTARY STUDY

The flexibility inherent in Gupta's method, to specify any value of fnp (through definition of 

f 2 ) is an advantage over the Lindley-Yow method. The sensitivity of the results to change in 

f2 highlights the importance of improving the definition of fk.

59 NUREG/CR-6645
4 SUPPLEMENTARY STUDY 

Table 4-1 

BM3 Model 

14 Mode Combination 

Time History vs. Response Spectrum 

Support Reactions Comparison Among Different Methods 

Forces in lbs 

Moments in in-lbs

1 % Damping

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. TYPE (TI-I) VALUE GROUP-1 GROUP-2 GUP-0 I GUP-1 GUP-2 LINDLEY 

1 Ft 3.96 0.84 0.85 0.85 0.82 0.81 0.85 

1 Fy 3.51 0.96 0.96 0.94 0.97 L15 LOS 

1 Fz 2.57 0.75 0.76 0.77 0.96 171 L35 

1 M, 43.23 0.81 0.72 0.74 1.49 3.51 2.63 

1 my 794.03 0.87 0.88 0.88 0.87 0.87 0.88 

1 Mz 215.80 0.87 0.88 0.87 0.86 0.88 0.90 

4 Fx 29.24 0.83 0.83 0.84 0.82 0.82 0.84 

4 Fz 27.64 0.74 0.75 0.75 0.92 1.60 1.28 

7 F,/ 13.87 1.04 1.04 1.00 L04 1.06 1.04 

11 Fy 13.25 1.15 1.15 L01 LI 1.27 L13 

11 Fz 93.01 0.80 0.81 0.82 0.80 0.80 0.83 

15 Fx 706.21 0.81 0.82 0.84 0.78 0.73 0.80 

17 Fy 25.56 L42 L42 127 1.53 L65 L41 

17 FZ 62.29 0.99 0.96 0.90 L01 11 0.99 

36 F. 46.89 1.15 1.13 L07 1.26 151 L35 

36 Fz 68.17 1.13 1.13 1.16 1.11 L13 L16 

38 Fx 12191 1.33 1.32 1.32 1.34 1.26 1.25 

38 Fy 43.98 108 1.08 L07 L10 L14 L09 

38 Fz 42.65 132 131 1.32 132 1.23 124 

38 MX 713.03 LO, 0.86 0.77 L07 1.43 1.26 

38 My 2783.00 1.32 1.32 132 1.33 124 1.25 

38 Mk 3117.80 L08 1.08 L07 L10 L13 L09 

23 Fx 159.21 1.46 1.43 1.43 1.50 147 1.39 

23 Fy 28.00 1.01 0.97 0.69 1.43 2.44 197 

31 FX 6.99 1.30 126 126 1.41 L46 1.33 

31 Fy 13.95 1.24 L16 LI 127 L42 129 

31 FZ 15.50 L17 1.17 126 1.52 1.68 L45 

31 PAX 1102.30 L10 1.10 124 160 1.80 1.49 

31 My 259.45 1.13 1.09 L06 LI L18 1.09 

31 Mz 608.00 .15 1.11 1.10 1.27 1.31 1.17 

Mean Value of 30 Camp. L06 1.04 1.03 1.16 L36 I M 

Standard D'v. of 30 Comp. 0.20 0.20 0.21 0.25 0.54 0.36

NTUREG/CR-6645 60
4 SUPPLEMENTARY STUDY

Table 4-2 

BM3 Model 

14 Mode Combination 

Time History vs. Response Spectrum 

Pipe End Moments Comparison Among Different Methods 

Moments in in-lbs (M.2 +My2+MK2)

1 /

2

1 % Damping

ELEM. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUES) 

NO. (TI) VALUE GROUP-1 GROUP-2 GUP-0 GUP-1 I GUP-2 j LINDLEY 

1 77.55 088 0.89 0.89 0.88 0.9 0.90 

2 759.36 0.89 0.89 0.89 0.88 0.88 0.89 

3 968.28 0.87 0.87 0.88 0.88 0.98 0.94 

4 773.63 0.94 0.95 0.95 0.94 0.97 0.97 

5 701. C6.97 0.97 0.97 0.98 1.10 1.04 

6 2207.60 6.90 0.90 0.91 0.90 0.90 0.91 

7 1948.20 0.91 0.92 0.92 0.92 0.92 0.92 

8 837.44 1.22 1.20 1.17 1.28 1.79 1.56 

9 1357.40 1.28 1.27 1.27 1.31 1.57 1.44 

10 2183.80 0.89 0.90 0.90 0.89 0.90 0.91 

11 4173.00 0.87 0.88 0.88 0.86 0.84 0.87 

12 8283.10 0.86 0.87 0.87 0.85 0.84 0.86 

13 6579.90 0.89 0.90 0.90 0.89 0.88 0.89 

14 13174.00 0.81 0.82 0.84 0.78 0.72 0.80 

15 2294.00 1.24 L15 L14 1.22 1.33 1.22 

16 1693.10 1.50 1.35 1.29 1.49 L69 1.45 

17 2456.50 1.39 1.37 1.26 1.49 1.61 138 

18 2377.20 LOS L08 L06 LIO 1.15 L09 

19 1635.50 1.22 1.21 LI 1.26 1.33 1.22 

20 7586.00 1.10 L09 L08 1.12 1.17 .11 

21 1616.20 130 1.29 128 1.34 1.40 1.32 

22 1977.10 1.48 1.45 L44 1.52 1.54 L45 

23 1900.20 1.49 1.46 L45 1.53 1.54 L45 

24 1607.70 1.54 15 1.51 1.5W 1.58 1.50 

25 265.36 0.90 0.88 1.20 1.64 2.16 L77 

26 271.57 1.11 LO 1.28 1.69 2.14 L77 

27 348.19 1.29 1.25 128 1.52 1.73 152 

28 552.58 1.30 1.24 1.23 1.34 1.43 1.33 

29 581.00 1.26 1.19 LI8 1.28 1.36 1.27 

30 1102.30 1.30 1.29 1.41 1.77 1.96 L64 

31 7586.80 1.63 L59 1.5 1.66 1.66 L59 

32 9093.40 1.49 1.47 1.45 L51 L50 1.45 

33 11281.00 1.30 1.29 1.28 1.32 1.29 1.27 

34 6061.80 1.15 L14 L14 L17 1.19 L5 

35 6634.10 1.25 1.24 1.24 1.27 1.26 1.23 

36 2451.60 1.24 1.23 1.23 1.26 1.25 1.23 

37 4009.70 1.26 1.25 1.25 127 126 1.23 

Mean Value of 37 Comp. 1.16 1.14 1.15 1.23 1.32 1.23 

Standard Dev. of 37 Comp, 0.23 0.22 0.21 0.29 0.38 0.28

61 NUREG/CR-6645
4 SUPPLEMENTARY STUDY 

Table 4-3 

BM3 Model 

14 Modes Plus Missing Mass Combination 

Time History vs. Response Spectrum 

Support Reactions Comparison Among Different Methods 

Forces in lbs 

Moments in in-lbs

1 % Damping

NODE REAC. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TH VALUFS 

NO. TYPE ("tH) VALUE GROUP-i GROUP-2 GUP-G GUP-1 I GUP-2 LINDLEY 

1 Fa 43.71 1.06 L06 L06 L06 L07 L06 

1 Fy 4.36 0.72 0.72 0.71 0.74 0.90 0.82 

1 Fz L60 0.91 0.83 0.84 1.22 2.63 2.03 

1 Mx 49.88 0.64 0.38 0.39 LI0 3.00 2.26 

1 MY 776.40 0.88 0.89 0.89 0.88 0.88 0.89 

1 MA 278.42 0.81 0.81 0.81 0.79 0.79 0.83 

4 FX 116.79 0.90 0.90 0.90 0.89 0.87 0.90 

4 FZ 20.01 0.33 0.80 0.81 1.OS 2.14 .L67 

7 1• 13.27 L05 1.05 1.01 1.06 L09 L0O 

11 Fy 13.31 L14 L14 L02 L18 L27 L13 

11 FZ 31.34 0.83 0.84 0.84 0.83 0.86 0.86 

15 Fx 73L47 0.$1 0.82 0.84 0.78 0.73 0.80 

17 Fy 25.60 1.42 L42 1.27 1.52 L65 1.41 

17 Fz 6.36 0.97 0.94 0.88 LO L08 0.97 

36 1y 46.69 15 L14 1.08 126 1.52 1.35 

36 FIZ 42.12 L23 122 L20 121 1.39 1.30 

38 Fx 732.18 L05 1.07 1.07 L07 L04 1.05 

38 1• 43.44 1.09 1.09 LOS L11 L15 .10 

38 PZ 29.95 L42 1.33 1.34 1.36 129 1.28 

38 MA 719.05 1.00 0.85 0.76 1.06 1.42 1.25 

38 MY 2084.97 L41 1.34 1.34 136 1.30 L28, 

38 M? 3085.87 L09 1.08 LOS 1.11 15 LI0 

23 FX 259.59 1.35 1.38 1.33 L42 L36 1.33 

23 1y 26.08 LOS L02 0.72 1.54 2.62 2.11 

31 Fx 55.05 L03 L04 1.04 LOS L04 L03 

31 Fy 14.17 1.21 1.14 1.16 L26 L40 1.27 

31 Fz 16.08 1.14 1.15 124 L48 L62 1.40 

31 Mx 1137.30 1.08 1.08 1.23 1.56 L74 1.45 

31 MY 612.38 0.99 L02 L00 1.04 L00 0.98 

31 Mz 1773.20 0.99 L01 L00 1.04 1.02 0.99 

Mean Value of 30 Comp = L04 L02 .L00 L14 .L37 1.23 

Standird De'. of 30 Comp. 0.20 0.22 0.22 OM 0._6 0.37

NU•.EG/CR-6645 62
4 SUPPLEMENTARY STUDY

BM3 Model 

14 Modes Plus Missing Mass Combination 

Time History vs. Response Spectrum 

Pipe End Moments Comparison Among Different Methods 

Moments in in-lbs (M.2+My 2+M42)t 1 % Damping

ELEM. TIME HISTORY RESPONSE SPECTRUM (EXPRESSED AS RATIOS OF THE TI S 

NO. (Tl) VALUE GROUP-1 GROUP-2 J GUP-0 [ GUP-1 GUP-2 LINDLEY 

1 790.95 0.88 0.88 0.89 0.87 0.88 0.89 

2 836.77 0.8S 0.89 0.89 0.87 0.87 0.89 

3 1429.28 0.86 0.86 0.86 0.85 0.88 0.89 

4 763.87 0.95 0.95 0.95 0.95 0.98 0.97 

5 597.98 L09 1.09 1.09 1.13 128 L18 

6 2133.18 0.92 0.92 0.92 0.92 0.93 0.93 

7 1877.81 0.93 0.94 0.94 0.94 0.95 0.94 

8 8..74 120 1.18 L15 1.25 176 153 

9 1415.21 1.24 1.23 1.23 127 151 1.39 

10 1946.24 0.93 0.94 0.94 0.93 0.96 6.96 

11 3829.46 0.90 0.91 0.91 0.90 0.89 0.91 

12 8428.48 0.86 0.86 0.87 0.85 0.83 0.86 

13 6741.03 0.88 0.89 0.89 0.88 0.87 0.88 

14 13219.91 0.81 0.2 0.84 0.78 0.72 0.80 

15 2293.75 1.24 L15 1.14 1.22 1.33 122 

16 1696.61 150 1.35 1.29 1.49 L69 L45 

17 2455.62 1.39 1.37 1.26 1.49 L61 1.38 

1s 2363.76 1.09 LOS 1.07 1.11 L15 L09 

19 1625.77 1.22 1.21 1.18 1.26 133 1.23 

20 7544.81 11 L10 L09 1.13 L17 L12 

21 1590.16 128 1.27 125 1.33 1.39 130 

22 2046.41 L152 1.51 1.50 158 157 149 

23 1807.75 143 137 1.36 L45 1.49 1.39 

24 1518.31 1.47 1.42 1.41 1.49 152 L43 

25 922.51 1.03 1.02 1.06 Los L14 L12 

26 677.35 1.06 L03 LOS L13 126 L19 

27 390.71 1.37 .33 1.37 L55 171 155 

28 737.42 1.22 121 1.20 128 131 1.24 

29 830.76 1.18 .17 L16 122 1.24 L19 

30 1994.56 1.12 L14 L17 1.32 1.38 125 

31 7428.79 L63 l59 1.66 .L67 1.59 

32 8785.32 1.51 1.48 1.47 L53 1.52 1.47 

33 10970.06 1.31 1.30 1.29 133 1.30 1.8 

34 5993.82 1.14 L13 1.12 L16 LI 1.14 

35 6788.22 1.25 1.25 1.24 127 1.26 1.23 

36 1874.84 134 128 1.27 1.33 1.38 1.32 

37 3501.06 129 1.26 1.25 1.29 1.30 1.26 

Mean Value of 37 Com. L16 1.15 1.14 1.19 1.25 1.19 

Standard Dev. of 37 Comp. 0.23 0.21 0.20 0.24 0.29 0.22

NUREG/CR-6645

Table 4-4

63
5 CONCLUSIONS AND RECOMMENDATIONS

5.1 Technical Conclusions 

Based on the results of this evaluation the following technical conclusions have been developed: 

1) Rosenblueth's DSC method and Der Kiureghian's CQC method for combining the out

of-phase modal response components (Rpi) represent a technical improvement over SRSS 

and the current methods accepted in Regulatory Guide 1.92 for treatment of closely 

spaced modes. 

2) Separation of the out-of-phase (Rp1) and the in-phase (Rr.) modal response components 

for amplified modes (f<fzpA), in accordance with the Lindley-Yow method and the Gupta 

method (see Method 2, Section 2.4.2), provides a more accurate RSA prediction than the 

traditionally applied method, which assumes out-of-phase response (Rpi = I) for 

amplified modes (see Method 1, Section 2.4.1). 

3) The contribution from high frequency modes (f> fzmA) is accurately calculated by the 

Missing Mass method and should be included in all RSA. The Static ZPA method is a 

convenient way to account for the high frequency modes when implementing the Lindley

Yow method (see Method 3, Section 2.4.3).. 

4) Specific structural systems with significant participation of low frequency modes 

(f<fv pJ were not analyzed. The Lindley-Yow method is not suitable for analysis of 

low frequency systems because of limitations imposed by the definition of the in-phase 

response component. The Gupta method is not restricted because low frequency modes 

are correctly treated as out-of-phase (i.e., Rpi = R•). 

5) Numerical evaluation of the highly filtered, in-structure response spectrum applied to the 

BM3 model indicates that the onset of totally in-phase response (Rri = R.) occurs at a 

frequency (fn,) significantly belowfzpA. The value offn, can be determined for any 

response spectrum by the method delineated in Appendix F. This method is potentially 

applicable to definingf 2 for the Gupta Method (i.e.,f 2 =fr). (Note: In Appendix H, Gupta 

also proposes a new approach to define f 2). 

6) Excellent correlation between mode superposition and direct integration time history 

analysis is achievable for constant modal damping. The missing mass contribution to the 

mode superposition solution must be included, regardless of the number of modes 

retained in the solution. Appendix E provides guidelines for each time history analysis 

method, to improve correlation.

65 NUREG/CR-6645
5 CONCLUSIONS AND RECOMMENDATIONS

7) The discrete, nodal mass distribution in a dynamic analysis model should be sufficiently 

refined to produce a static solution in close agreement with a static solution based on 

continuously distributed mass. Appendix C provides a discussion and guidelines for 

ensuring an adequate dynamic mass distribution. 

8) Broad-banded, ground motion spectral input is more consistent with the technical bases 

for Rosenblueth's DSC and Der Kiureghian's CQC Methods for modal combination of 

out-of-phase response components. However, the quantitative results in Chapter 3 

indicate their superiority to currently accepted methods, when applied to a highly-filtered, 

in-structure response spectrum. 

9) For highly filtered, in-structure response spectra, improvements to the RSA combination 

methods evaluated in Chapter 3 appear achievable, based on the supplementary study 

results presented in Chapter 4. A mathematically based grouping method for highly 

filtered in-structure response spectrum input and refinement of Gupta's Method, based on 

new approaches to definef 2 for such spectral input, warrant further evaluation. 

10) The BM3 model can potentially serve as a benchmark problem for evaluating alternate 

methodologies for RSA. Sufficient model, loading, and time history response data is 

provided in the report. 

5.2 Recommendations for Revision to Regulatory Guide 1.92 and Standard 

Review Plan 3.7.2 

Based on the technical conclusions delineated in Section 5.1, the following revisions to 

regulatory guidance are recommended: 

1) The regulatory guidance on modal response combination for RSA should be consolidated 

in one document. The most logical candidate is R.G. 1.92. This would then be 

referenced by SRP 3.7.2. 

2) Acceptable methodologies for developing the complete RSA solution should be defined 

in R.G. 1.92. 

3) For out-of-phase response combination, Rosenblueth's DSC and Der Kiureghian's CQC 

methods should be identified in R.G.1.92 as acceptable methods. SRSS is acceptable in 

the absence of closely spaced modes. The 10% definition for closely spaced modes is 

applicable only at low damping ratios (r2%). For higher damping ratios, closely spaced 

modes should be defined by 5 times the damping ratio (e.g., 25% for 5% damping; 50% 

for 10% damping). See Section 2.1.4.

NUREG/CR-6645 66
5 CONCLUSIONS MND RECOMMENDATIONS

The NRC Ten Percent and NRC-DSC methods should be deleted from R.G. 1.92. 

Because of its extensive prior use, the NRC Grouping method is a candidate for retention 

in R.G. 1.92, even though absolute summation of closely spaced modes has no technical 

basis. The maximum damping ratio should be limited to 2%, because the 10% definition 

of closely spaced modes is not applicable at higher damping levels. 

4) For separation of out-of-phase and in-phase components for the amplified modes, the 

methods of Lindley-Yow and Gupta should be identified in R.G. 1.92 as acceptable 

methods. R.G. 1.92 should include a cautionary note that the Lindley-Yow method is not 

suitable for analysis of systems with significant low-frequency response Of<f,• e.J 

5) For the complete RSA solution, Method 2 (Lindley-Yow or Gupta formulation) and 

Method 3 should be identified in R.G. 1.92 as acceptable methods, subject to the 

limitations of the Lindley-Yow method cited in (4) above. Method 1 should be strongly 

discouraged or identified as unacceptable. 

6) The missing mass formulation documented in Appendix I should be included in R.G. 

1.92, in order to provide a complete definition for Method 2. The contribution of high 

frequency modes (ffzP•) must be included in all RSA, without exception. Appendix A 

to SRP 3.7.2 should be deleted. 

7) R.G. 1.92 should address the importance of the missing mass contribution in mode 

superposition time history analysis and define an acceptable procedure, analogous to the 

approach used in Method 2 for RSA. Only modes withf<fzA participate in the modal 

solution; the missing mass contribution, scaled to the instantaneous input acceleration, is 

treated as an additional mode in the algebraic summation of modal responses at each time 

step. 

8) SRP 3.7.2 should contain a discussion of dynamic model vs. static model mass 

distributions, and provide guidelines for ensuring an adequate discrete mass distribution 

in the dynamic model. Appendix C provides a framework for revision of SRP 3.7.2. 

5.3 Recommendations for Additional Evaluation of Modal Response 

Combination Methods 

During the course of this study, a number of areas worthy of additional evaluation were 

identified. The following recommendations are not restricted to regulatory guidance issues, but 

also address perceived opportunities for improvement of RSA methodology. 

1) Numerical study of the RSA methods should be expanded to include additional problems 

and three directions of seismic input, in order to quantify the level of correlation

67 NUREG/CR-6645
5 CONCLUSIONS AND RECOMMENDATIONS

achievable in design-type calculations. 

2) The methodologies developed in Appendix F and Appendix H for definition offn, should 

be tested for a wide range of spectral shapes and, if proven, should be incorporated as a 

integral element of the RSA process. 

3) A modal response combination method rigorously based on MDOF response to a single 

frequency input should be developed and tested for highly-filtered, in-structure response 

spectra. 

4) A detailed study of modal response combination methods for systems with significant 

low-frequency response (f< f spectral peak) should be conducted. 

5) The present project did not address independent support motion (ISM) of multiply

supported systems. The applicability of the results and conclusions of this project to 

improving ISM analysis methods should be investigated.

NUREG/CR-6645 68
6 REFERENCES

1. U.S. Nuclear Regulatory Commission, "Combining Modal Responses and Spatial 

Components in Seismic Response Analysis," Regulatory Guide 1.92, Revision 1, February 

1976. 

2. U.S. Nuclear Regulatory Commission Standard Review Plan for the Review of Safety 

Analysis Reports for Nuclear Power Plants, NUREG-0800, Section 3.7.2, "Seismic System 

Analysis," Revision 2, August 1989. 

3. Kennedy, R.P., "Position Paper on Response Combinations," Report No. SMA 12211.02-R2

0, March 1984 (Published in Report of the U.S. Nuclear Regulatory Commission Piping 

Review Committee: Evaluation of Other Dynamic Loads and Load Combinations, NUREG

1061, Vol. 4, pp. B-43 to B-95, April 1985). 

4. Gupta, A.K., "Response Spectrum Method in Seismic Analysis and Design of Structures," 

CRC Press, Inc., 1993. 

5. Mertens, P.G., "A New Algebraic White-Noise Modal Combination Rule - GAC(A)," 

Nuclear Engineering and Design 147(1994) 299-309. 

6. Bezler, P., Curreri, J.R., Wang, Y.K., and Gupta, A.K, "Alternate Modal Combination 

Methods in Response Spectrum Analysis," NUREG/CR-5627, BNL, October 1990. 

7. Wang, Y.K., Subudhi, M., and Bezler, P., "Investigation of the Conservatism Associated 

with Different Combinations Between Primary and Secondary Piping Responses," 

NUREG/CR-3086, BNL, January 1983. 

8. Bezler, P., Hartzman, M., and Reich, M., "Piping Benchmark Problems Dynamic Analysis 

Uniform Support Motion Response Spectrum Method," NUREG/CR-1677, Vol.1, BNL, 

August 1980. 

9. American Society of Civil Engineers, "Seismic Analysis of Safety Related Nuclear 

Structures and Commentary on Standard for Seismic Analysis of Safety Related Nuclear 

Structures (4-86), September 1986. 

10. American Society of Civil Engineers, "Seismic Analysis of Safety Related Nuclear 

Structures and Commentary," ASCE 4, Revision, Draft, April 1995. 

11. Maison, B.F., Neuss, C.F., and Kassaj, K., "Earthquake Engineering and Structural 

Dynamics, Vol. II, p. 623-647; "The Comparative Performance of Seismic Response 

Spectrum Combination Rules in Building Analysis," John Wiley & Sons, 1983.

69 NUREG/CR-6645
6 REFERENCES

12. Rosenblueth, E., and Elorduy, J., "Responses of Linear Systems to Certain Transient 

Disturbances," Proceedings of the Fourth World Conference on Earthquake Engineering, 

Santiago, Chile, 1969. 

13. Der Kiureghian, A. "A Response Spectrum Method for Random Vibrations," University of 

California at Berkeley, June 1980. 

14. Lindley, D.W., and Yow, T.R., "Modal Response Summation for Seismic Qualification," 

Proceedings of the Second ASCE Conference on Civil Engineering and Nuclear Power, Vol. 

VI, Paper 8-2, Knoxville, TN, September 1980. 

15. Hadjian, A.H., "Seismic Response of Structures by the Response Spectrum Method," 

Nuclear Engineering and Design, Vol. 66, no. 2, pp. 179-201, August 1981.

NUREG/CR-6645 70
APPENDIX A 

DESCRIPTION OF BM3 PIPING MODEL

A-1 ANUREG/CR-6645
z* 2BM3 Piping Model 

rm 

,, 

EN 

E 

6 

LEGEND:

X ANCHOR. -- f SPRING SUPPORT 

NODE NUMBER o ELID&V~ NUMBER

8" 0
449 

450 

1 MODEL 3-- -UNIFORM 1 t TIME HISTORY ANALYSIS--. tha.x a11 dl. inp 

CONTROL INFORMATION 

NUMBER OF NODAL POINTS a 38" 

NUMBER OF ELE8MWT TYPES * 2 

NUMBER OF STATIC LOAD CASES u 0 

NUM1ER OF DYNAMIC LOWAD CASES I 1 

NUMBER OF ANCHOR MVMT CASES m 0 

NUMBER OF FREQUEIES a 31 

SOw1uTIOc MODE (MODEX) - 0 

ST" 0, EXECuTION 

.1, DATA CHECK 

SS CALCMUlTION FLAG a 0 

EQ.0 vO 

EQ.1 YES 

ASME CODE EVALTION FIAG a 0 

Et.1 CLASS I PIPING 

EQ.2 CLASS2 OR CLASS 3 PIPING 

ACCELERATION DUE TO GRAVITY P386.4 

BANDWIDTH MINIMIZATION FLAG I 1 EQ.0N 

ARAW Y NODE NUMBERING FLAG w 0 EQ.0 NO0 

I EQ.1 YES 

NU41ER OF SUPPORT GROUPS • I 

FLAG FOR NODAL COORD. INPUT UNITS, 0 

0 CONSISTENT UNIT 

W EQI FEET 76 INCHES

LIST OF ANALYSIS TO BE PERFORMED 

LAD CASE DISK FILE ANALYSIS TYPE 

1 0 UNIFORM TIME HISTORY ANALYSIS 

I NO=AL POINT INPUT DATA 

0VNDE BOUNDARY CONDITION CODES NODAL POINT COORDINATES 

NUMBERX Y Z XX TY ZZ X Y 

1 0 0 0 0 0 0 .000 .000 

2 0 0 0 0 0 0 15.000 .000 

3 0 0 0 0 0 0 19.500 -4.500 

4 0 0 0 0 0 0 19.500 -180.000 

5 0 0 0 0 0 0 19.500 -199.500 

6 0 0 0 0 0 0 19.500 -204.000 

7 0 0 0 0 0 0 19.500 -204.000 

8 0 0 0 0 0 0 24.000 -204.000 

9 0 0 0 0 0 0 96.000 -204.000 

10 0 0 0 0 0 0 254.000 -204.000 

S11 0 0 0 0 0 0 333.000 -204.000 

12 0 0 0 0 0 0 411.000 -204.000 

13 0 0 0 0 0 0 483.000 -204.000 

14 0 0 0 0 0 0 487.500 -204.000 

15 0 0 0 0 0 0 487.500 -204.000 

16 0 0 0 0 0 0 487.500 -204.000

z .000 

.000 

.000 

.000 

.000 

4.500 

139.500 

144.000 

144.000 

144.000 

144.000 

144.000 

144.000 

148.500 

192.000 

235.500

T 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000
17 0 0 0 0 0 0 

18 0 0 0 0 0 0 

19 0 0 0 0 0 0 

20 0 0 0 0 0 0 

21 0 0 0 0 0 0 

22 0 0 0 0 0 0 

23 0 0 0 0 0 0 

24 0 0 0 0 0 0 

25 0 0 0 0 0 0 

26 0 0 0 0 0 0 

27 0 0 0 0 0 0 

28 0 0 0 0 0 0 

29 0 0 0 0 0 0 

30 0 0 0 0 0 0 

31 0 0 0 0 0 0 

32 0 0 0 0 0 0 

33 0 0 0 0 0 0 

34 0 0 0 0 0 0 

35 0 0 0 0 0 0 

36 0 0 0 0 0 0 

37 0 0 0 0 0 0 

38 0 0 0 0 0 0 

1KQUATIOtI NUMWE~ 

1 1 2 3 4 5 6 

2 7 8 9 10 11 12 

3 13 14 15 16 17 18 

4 19 20 21 22 23 24 

S 25 26 27 28 29 30 

6 31 32 33 34 35 36 

>1 7 37 38 39 40 41 42 

8 43 44 45 46 47 48 

9 49 50 51 52 53 54 

10 55 56 57 58 59 60 

11 61 62 63 64 65 66 

12 67 68 69 70 71 72 

13 73 74 75 "76 77 78 

14 79 80 81 82 83 84 

15 85 86 87 88 89 90 

16 91 92 93 94 95 96 

17 97 98 99 100 101 102 

18 103 104 105 106 107 108 

19 109 110 111 112 113 114 

20 115 116 117 118 119 120 

21 121 122 123 124 125 126 

22 127 128 129 130 131 132 

23 133 134 135 136 137 138 

24 139 140 141 142 143 144 

25 145 146 147 148 149 150 

26 151 152 153 154 155 156 

27 157 158 159 160 161 162 

28 163 164 165 166 167 168 

29 169 170 171 172 173 174 

* 30 175 176 177 178 179 180 

31 181 182 103 184 185 186 

32 187 188 189 190 191 192 

33 193 194 195 196 197 198 

34 199 200 201 202 203 204 

35 205 206 207 208 209 210 

36 211 212 213 214 215 216

492.000 -204.000 

575.000 -204.000 

723.000 -204.000 

727.500 -208.500 

727.500 -264.000 

727.500 -264.000 

727.500 -264.000 

733.500 -264.000 

753.500 -264.000 

845.500 -264.000 

851.500 -264.000 

851.500 -264.000 

851.500 -264.000 

851.500 -270.000 

851.500 -360.000 

727.500 -264.000 

727.500 -264.000 

727.500 -264.000 

727.500 -264.000 

739.500 -264.000 

847.500 -264.000 

955.500 -264.000

240.000 

240.000 

240.000 

240.000 

240.000 

205.000 

190.000 

184.000 

184.000 

184.000 

178.000 

160.000 

142.000 

136.000 

136.000 

255 .000 

270.000 

306.000 

414.000 

426.000 

426.000 

426.000

.000 .000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 .000
37 217 218 219 220 221 

38 223 224 225 226 227 

1S PR ING ELEMENTS

ELEN TYPE 

NUMBER OF ELEMENTS

1 

30

ELMENT IOAD CASE MUJLTIPLIERS

CASE (A) .0000CASE (B) .0000

ELEMNT NODE SUPPORT CODE CODE DIRECTION COSINES WRT GLOBAL AXES 

NtU•ER (N) GROUP KD KR X- Y- z1 

1 

1 

1 

1 

1 4 

4 

7 

11 11 

15 

17 

1'7 

36 

36 

38 

38 

38 

38 

38 

38 

23 

23 

31 

31 

31 

31 

31 

31 ELEMENT

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 1 

1 

2 

2 

2 2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

I

1 

1 

1 

0 

0 

0 

1 1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

0 

0 0 

1 

1 

1 

1 

1 

0 

0 

0 

NPUT

0 

0 

0 

1 

1 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 1 

1 

1 

0 

0 0 

0 

0 

1 

1 

1 

DATA

1.000 

.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

.000 

.000 

.000 

1.000 

.000 

.000 

.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

.000

.000 

1.000 

.000 

.000 

1.000 

.000 

.000 

.000 

1.000 

1.000 

.000 

.000 

1.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

1.000 

.000 

.000 

1.000 

.000

.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

1.000 

.000 

.000 

1.000 

.000 

.000 

1.000 

.000' 

.000 

.000 

.000 

1.000 

.000 

.000 

1.000

SPECIFIED 

DISPLACEMENT 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000

NTROL INFORMATION

NUMBER OF PIPE ELEMENTS 

NUMBER OF MATERIAL SETS

* 37 

- 1

222 

228

CASE (C) .0000CASE (D) 

.0000

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

PH

,r 

IP I 

,0 

U't

SPECIFIED 

ROTATION 

.000" 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000 

.000

SPRING 

RATE 

1.0000+11 

1.0000+11 

1.0000E+ll 

1.0000E+20 

1.0000E+20 

1.0000E+20 

1.0000+E08 

1.0000+08 

1.0000E+08 

1.0000R+08 

1.0000+E05 

1.00008+05 

1.0000+E08 

1.0000E+05 

1.0000E+08 

1.0000+E05 

1.0000E+11 

1.0000E+11 

1.0000E+11 

1.0000E+20 

1.0000E+20 

1.OOOOE+20 

1.0000E+05 

1.0000+E08 

1.0000+11 

1.0000E+11 

1.0000+R11 

1.0000E+20 

1.0000E+20 

1.0000E+20
MAXIIMUM NUMBER OF MATERIAL 

TEMPERATURE nIN POINTS a 1 

NUMBER OF SECTION PROPERTY SETS a 3 

NUMBER OF BRAC4HI POINT NODES m 0 

MAXIMUM NUMBER OF TANGENTS 

C00 0TO A BRANCH POINT - 3 

FLAG FOR NEGLECTING AXIAL 

DEFORMATIONS IN BDED ELEMENTS a 0 

(EQ.1, NEGLECT) 

IMATERIAL PROPERTY TABLES

OMATERIAL NUMBER n NUMBER OF 

TEMPERATURE POINTS a IDENTIFICATION ( 

POINT 

NUMBER TEMPERATURE 

1 .00 

1 iSECTION PRO! 

SECTION OUTSIDE 

NUMBER DIAMETER 

1 3.500 

2 4.500 

3 8.625

1.) 

1)

)

YOUNG'S 

MlODULUS 

2.900E+07 

SE R T Y 

WALL 

IICIOESS 

.2160 

.2370 

.3220

POISSON' S 

RATIO 

.300 

TABLE 

SHAPE FACTOR 

FOR SHEAR 

.0000 

.0000 

.0000

THERMAL 

EXPANSION 

6.440E-06 

WREIGHIT/ 

UNIT LENGTH 

8.98009-01 

1.35808+00 

4.1870E+00

UNITL•G= D E S C R I PT I ON 

2 .3240E-03 

3.5145E-03 

1.0836E-02

ELEMENT LOAD 

X-DIRECTION GRAVITY 

Y-DIRECTION GRAVITY 

Z-DIRECTION GRAVITY 

THERMAL DISTORTION 

PRESSURE DISTORTION 

IPIPE ELEMENT 

ELEMENT ELEMENT NODE 

NUMBER TYPE -I

C'ASE MULTIPLIERS 

CASE A CASE B CASE C 

.000 .000 .000 

.000 .000 .000 

.000 .000 .000 

.000 .000 .000 

.000 .000 .000

IN 

NODE .J

PUT 

MATL. 

NUMBER

1 TANGENT 1 2 

2 BEND 2 3

1 2.

DATA 

SECTION REFERENCE 

NUMBER TEMPERATURE 

(BEND 

RADIUS) 

1 50.00 

1 50.00 

4.500)

CASE D 

.000 

.000 

.000 

.000 

.000

DESIGN 

PRESSURE 

(THIRD 

POINT) 

200.00 

200.00 

(TI)

PEAK 

PRESSURE 

(X3 

ORDINATE) 

.00 

.00 

19.500)

TEST END CODES NODE INPUT' 

PRESSURE END-I END-J INCRENENT TAG (Y3- (Z3- (BEND ORDINATE) ORDINATE) DEGREE) 

.00 0 0 1 II 

.00 0 0 IC .000) ( .000) ( 90.0001)

In
3 

4 

5

TANGENT 

TANGENT 

BEND

6 TANGENT 

7 BEND 

8 TANGENT 

9 TANGENT 

10 TANGENT 

11 TANGENT 

12 TANGENT 

13 BEND 

14 TANGENT 

15 TANGENT 

16 BEND 

17 TANGENT 

18 TANGENT 

19 BEND

20 

21 

22 

23 

24 

25 

26 

27 

28 

29

TANGENT 

TANGENT 

TANGENT 

BEND 

TANGENT 

TANGENT 

BEND 

TANGENT 

TANGENT 

BEND

30 TANGENT 

31 TANGENT 

32 TANGENT 

IPIPE ELEME

ELEMENT 

NUMBER 

33 

34 

35 

36

3 4 1 1 50.00 4 5 1 1 50.00 

5 6 1 1 50.00 

4.500) 

6 7 1 1 5o.00 

.7 8 1 1 50. 00 

4.500) 

8 9 1 1 5o.00 

9 10 1 1 50.00 10 11 1 1 50.00 11 12 1 1 50.00 12 13 1 1 50.00 

13 14 1 1 50.00 

4.500) 

14 15 1 1 50.00 

15 16 1 1 50.00 

16 17 1 1 50.00 

4.500) 

17 18 1 1 50.00 

18 19 1 1 50.00 19 20 1 1 50.00 

4.500) 

20 21 1 1 50.00 21 22 1 2 50.00. 

22 23 1 2 50.00 23 24 1 2 50.00 

6.000) 

24 25 1 2 50.00 25 26 1 2 50.00 

26 27 1 2 50.00 

6.000) 

27 28 1 2 50.00 28 29 1 2 50.00 

29 30 1 2 50.00 

6.000) 

30 31 1 2 50.00 

21 32 1 3 50.00 

32 33 1 3 50.00 

NT INPUT DATA

ELEMENT NODE NODE 

TYPE -I -

TANGENT 

TANGENT 

BEND 

TANGENT

33 34 

34 35 

35 36

MATL. SECTION 

NUMBER NUMBER

1 

1 

1

3 

3 

3

36 37 1

REFERENCE 

TEMPERATURE 

(BEND 

RADIUS) 

50.00 

50.00 

50.00 

12.000)

3 50.00

200.00 

200.00 

200.00 

(TV) 

200.00 

200.00 

(TI) 

200.00 

200.00 

200.00 

200.00 

200.00 

200.00 

(TI) ( 

200.00 

200.00 

200.00 

(TI) 

200.00 

200.00 

200.00 

(TI) 

200.00 

200.00 

200.00 

200.00 

(TI) 

200.00 

200.00 

200.00 

(TV) ( 

200.00 

200.00 

200.00 

(TI) ( 

200.00 

200.00 

200.00 

DESIGN 

PRESSURE 

(THIRD 

POINT) 

200.00 

200.00 

200.00 

(TI)

.00 

.00 

.00 

-19.500) 

.00 

.00 

19.500) 

.00 

.00 

.00 

.00 

.00 

.00 

487.500) 

.00 

.00 

.00 

487.500) 

.00 

.00 

.00 

727.500) 

.00 

.00 

.00 

.00 

727.500) 

.00 

.00 

.00 

851.500)0 

.00 

.00 

.00 

851.500) 

.00 

.00 

.00 

PEAK 

PRESSURE 

(X3

ORDINATE) 

.00 

.00 

.00 

727.500)

200.00 .00

.60 0 0 

.00 0 0 

.00 0 0 

-204.000) ( .000) 

.00 0 0 

.00 0 0 

-204.000) ( 144.000) 

.00 0 0 

.00 0 0 

.00 0 0 

.00 0 0 

.00 0 0 

.00 0 0 

-204.000) ( 144.000) 

.00 0 0 

.00 0 0 .. 00 0 0 

-204.000) ( 240.000) 

.00 0 0 

.00 0 0 

.00 0 0 .- 204.000)( 240.000) 

.00 0 0 

.00 0 0 

.00 0 0 

.00 0 0 

-264.000) ( 184.000) 

.00 0 0 

.00 0 0 

.00 0 0 

-264.000) ( 184.000) 

.00 0 0 

.00 0 0 

.00 0 0 

-264.000) ( 136.000) 

.00 0 0 

.00 0 0 

.00 0 0 

TEST END CODES 

PRESSURE END-I END-J 

(Y3- (Z3

ORDINATE) ORDINATE) 

.00 0 0 

.00 0 0 

.00 0 0 

-264.000).( 426.000)

.00 0 0 1I

1 

1 

90.0001) 

1 

90.0001) 

1 

1 

1 

1 

1 

90.0001) 

1 

1 

90.0001) 

1 

1 

90.0001) 

1 

1 

1 

(90.0001) 

1 

1 

(90.0001) 

1 

1 

(90.0001) 

1 

1 

1 

NODE 

INCREMENT (BEND 

DEGREE) 

1 

1

II 

II 

IC 

II 

IC 

II 

II 

II I1 

II 

ic 

II 

II 

IC 

II 

II 

IC 

II 

II II 

IC 

II 

II 

IC 

II 

II 

IC 

II 

II 

IIJ 

TAG 

II 

II 

IC

PA ci

1,,
T 37 TANGENT 37 38 1 3 50.00 200.00 .00 .00 0 0 

1

1BANDWIDTH MINIMI ATION 

Mfl4BmD (BALDWIDIvIm cON L PARAMETER) 

- I 

0EQUATIONNZ4W5ERS AFTER BMDWIDTH MINIMIZATION 

0NODE SEQUENCE X Y Z XX YY ZZ 

1 38 223 224 225 226 227 228 

2 37 217 218 219 220 221 222 

3 36 211 212 213 214 215 216 

4 35 205 206 207 208 209 210 

5 34 199 200 201 202 203 204 

6 33 193 194 195 196 197 198 

7 32 187 188 189 190 191 192 

8 31 181 182 183 184 185 186 

9 30 175 176 177 178 179 180 

10 29 169 170 171 172 173 174 

11 28 163 164 165 166 167 168 

12 27 157 158 159 160 161 162 

13 . 26 151 152 153 154 155 156 

14 25 145 146 147 148 149 150 

15 24 139 140 141 142 143 144 

16 23 133 134 135 136 137 138 

17 22 127 128 129 130 131 132 

18 21 121 122 123 124 125 126 

19 20 115 116 117 118 119 120 

20 19 109 110 111 112 113 114 

21 18 103 104 105 106 107 108 

22 17 97 98 99 100 101 102 

23 15 85 86 87 88 89 90 

24 13 73 74 75 76 77 78 

25 11 61 62 63 64 65 66 

26 9 49 50 51 52 53 54 

27 7 37 38 39 40 41 42 

28 5 25 26 .27 28 29 30 

29 3 13 14 15 16 17 18 

30 2 7 8 9 10 11 12 

31 1 1 2 3 4 5 6 

32 16 91 92 93 94 95 96. 

33 14 79 80 81 82 83 84 

34 12 67 68 69 70 71 72 

35 10 55 56 57 58 59 60 

36 8 43 44 45 46 47 48 

37 6 31. 32 33 34 35 36 

38 4 19 20 21 22 23 24 

ORWADWWfID PRIOR 7 RESEUNCING: 72 

BAJDVWW1II APTJ R Cfl4G u 18 

1EQUATION P ARAMETERS 

"WrAL NMAT., OF EQATIQONS - 228 

BI WID'M - 18 

hUMSER 0 EOUATIONS IN A BLOCK w 228 

NWBER O DBLOCK• 1 

INODAL LOADS (STATI C) OR

NODE LOAD X-AXIS 

NM43ER CASE FORCE 

IDYNAMIC ANALYSIS

MASSES (DYNAMIC)

Ii

Y-AXIS 

FORCH

Z-AXIS 

FORCE

X-AXIS 

MOMENT

Y-AXIS MOMENT Z-AXIS MOMENT
Nodal Masses of the Model BM3

Unit:lbs/'m/sec**2

Node No. Mass 

1- 0.017 

2 0.026 

3 0.212 

4 0.227 

5 0.031 

6 0.165 

7 0.165 

8 0.092 

9 0.267 

10 0.275 

11 0.182 

12 0.174 

13 0.092 

14 0.059 

15 0.101 

16 0.059 

17 0.105 

18 .0.268 

19 0.18 

20 0.073 

21 0.207 

22 0.088 

23 0.043 

24 0.052 

25 0.197 

26 0.178 

27 0.048 

28 0.063 

29 0.048 

30 0.175 

31 0.158 

32 0.163 

33 0.276 

34 0.78 

35 0.687 

36 0.687 

37 1.17 

38 0.585

A-9 NUREG/CR-6645
MODE 

NUMBER

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31

CIRCULAR 

FREQUENCY 

(RAD/SEC) 

1.8268E+01 

2.7565E+01 

3.4657E+01 

3.5840E+01 

4.3841E+01 

4.6141E+01 

4.9509E+01 

6.4727E+01 

6.9466E+01 

7.0578E+01 

7.2241E+01 

7.8104E+01 

8.7205E+01 

1.0130E+02 

1.1333E+02 

1.1700E+02 

1.2265E+02 

1.2291E+02 

1.3707E+02 

1.3935E+02 

1.4361E+02 

1.6305E+02 

2.4517E+02 

2.4831E+02 

2.5890E+02 

2.8845E+02 

2.9688E+02 

3.3072E+02 

3.7253E+02 

4.4039E+02 

4.4421E+02

NUREG/CR-6645

FREQUENCY 

(CYCLES/SEC) 

2.9074E+00 

4.3871E+00 

5.5158E+00 

5.7041E+00 

6.9775E+00 

7.3436E+00 

7.8796E+00 

1.0302E+01 

1.1056E+01 

1.1233E+01 

1.1498E+01 

1.2431E+01 

1.3879E+01 

1.6122E+01 

1.8038E+01 

1.8621E+01 

1.9521E+01 

1.9561E+01

2.1815E+01 

2.2179E+01 

2.2857E+01 

2.5950E+01 

3.9020E+01 

3.9521E+01 

4.1205E+01 

4.5908E+01 

4.7249E+0i 

5.2636E+01 

5.9290E+01 

7.0090E+01 

7.0699E+01

PERIOD 

(SEC) 

3.4395E-01 

2.2794E-01 

1.8130E-01 

1.7531E-01 

1. 4332E-01 

1.3617E-01 

1. 2691E-01 

9.7071E-02 

9.0450E-02 

8.9025E-02 

8.6975E-02 

8.0446E-02 

7.2050E-02 

6.2026E-02 

5.5439E-02 

5.3702E-02 

5.1227E-02 

5.1122E-02 

4.5840E-02 

4.5088E-02 

4.3750E-02 

3.8536E-02. 

2.5628E-02 

2.5303E-02 

2.4269E-02 

2.1783E-02 

2.1164E-02 

1.8999E-02 

1.6866E-02 

1.4267E-02 

1. 4144E-02

A-10
IMODAL PARTICIPATION FACTORS

MODE 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

i2 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27

28 52.636 -3.3621E-01

FREQ (CPS) 

2.907 

4.387 

5.516 

5.704 

6.977 

7.344 

7.880 

10.302 

11.056 

11.233 

11.498 

12.431 

13.879 

16.122 

18.038 

18.621 

19.521 

19.561 

21.815 

22.179 

22.857 

25.950 

39.020 

39.521 

41.205 

45.908 

47.249

X-DIRECTION 

-2.3473E-02 

-5.5673E-01 

-4.7719E-02 

-9.9555E-03 

-1.5342E-01 

5.5364E-01 

-3.7119E-01 

-4.0272E-01 

5.5995E-01 

9.2828E-02 

-1.4351E+00 

4.2956E-01 

-3.2298E-01 

9.7048E-02 

-1.0796E-01 

-4.2217E-01 

-3.2449E-02 

-1.4330E-01 

-3.8758E-01 

-3.6788E-03 

-6.9162E-02 

1.6498E-01 

1.2766E-01 

-2.4041E-02 

-2.0186E-02 

4.7943E-01 

-8.3761E-02

NUREG/CR-6645

Y-DIRECTION 

2.7695E-01 

9.0279E-02 

-9.4682E-01 

-2.1351E-01 

-4.5827E-01 

3.9813E-01 

3.9334E-01 

1.1940E+00 

1,8119E-01 

-1.4703E-01 

-1.9653E-01 

-2.4824E-01 

-1.5196E-01 

4.1157E-01 

4.0371E-02 

-4.0211E-02 

-2.8297E-01 

8.6254E-02 

1.1586E-01 

-9.8705E-01 

8.0846E-02 

-1.8727E-02 

7.3949E-01 

-3.3916E-01 

3.3298E-01 

-1.6665E-02 

-2.6544E-01 

4.7051E-02

Z-DIRECTION 

3.5437E-02 

-1.0926E-01 

-1.9203E-01 

6.8419E-01 

7.4423E-01 

-3.5022E-01 

4.5914E-01 

-1.1466E-02 

-7.0772E-02 

-2.7472E-04 

-3.4271E-01 

-4.5713E-01 

-5.3436E-02 

2.9944E-01 

4.3020E-01 

2.9074E-01 

-2.1127E-01 

:-6.1027E-01 

-5.3784E-02 

7.5035E-02 

1.1833E+00 

1.6100E+00 

-2.9101E-02 

-6.4757E-03 

-1.9897E-02 

-6.2062E-02 

2.6515E-02 

1.1705E-01

A-11
59.290 -2.5127E-03 

70.090 4.5397E-02 

70.699 -5.3524E-01

2.4614E-04 

1.7356E-03 

-1.2430E-02

-5.0636E-01 

-6.2325E-01 

-6.2982E-02

NUREGACR-6645

29 

30 

31

A-12
APPENDIX B

TIME HISTORY AND RESPONSE SPECTRUM INPUT 

FOR 

BM3 ANALYSIS

B-1 NUREG/CR-6645
8.00 

7.00 

6.00 

5.00 

S4.00 

N 

3.00 

2.00

1.00

0.00 .. . . :.. . . . . . 

0.10 1.00 10.00 100.00 

Freq. (hz) 

BM3 Input Spectrum (1%)
3.00 C

2.50 

2.00 

to ,• 1.50 

1.00 

0.50 

0.00 

0.10 1.00 10.00 100.00 

Freq. (hz) 

% BlDM3 Input Spectrum (5%) gn
CA 

K - -- - -

0IC1C 

Bc IE ITRYRCR
-264.0 184.0 

-264.0 136.0 

-264.0 426.0

1i0.0 

1 1 14950 20 0.001

TZXZ HZSTORY ANALYSXS 

0.01

1

1500 1.0 C

.000 .025 

.060 .703 

.110 2.707 

.160 4.481 

.210 7.839 

.260 7.898 

.310 7.422 

.360 -3.220 

.410 -19.350 

.460 -22.220 

.510 -8.580 

.560 18.610 

.610 35.460 

.660 31.170 

.710 29.340 

.760 6.5S6 

.810 1.531 

.860 -5.220 

.910 18.613 

.960 31.190 

1.010 33.949 

1.060 30.180 

1.110 57.912 

1.160 75.680 

1.210 6.520 

1.260 -398.830 

1.310-143.250 

1.360 -68.300 

1.410 52.250 

1.460 105.300 

1.510 45.060 

1.560 -31.120 

1.610 -66.070 

1.660 -29.300

PEUATIONAL Z"VEL 

.020 .063

.070 

.120 

.170 

.220 

.270 

.320 

.370 

.420 

.470 

.520 

.570 

.620 

.670 

.720 

.770 

.820 

.370 

.920 

.970 

1.020 

1.070 

1.120 

1.170 

1.220

1.0060 

3.036 

5.053 

8.272 

7.763 

6.549 

-6.119 

-21.900 

-20.680 

-4.5S0 

25.200 

33.590 

31.120 

26.646 

3.236 

-. 130 

-3 . 630 

23. 618 

32.500 

32.890 

33.240 

64.040 

70.900 

-16 .180

1.270-113.940 

1.320-137.700 

1.370 -45.000 

1.420 72.600 

1.470 100.650 

1.S20 27.540 

1.570 -41.830 

1.620 -63.940 

1.670 -20.800

.030 .127 

.080 1.488 

.130 3.341 

.180 5.720 

.230 8.433 

.280 7.730 

.330 4.9324 

.380, -8.900 

.430 -23.220 

.480 -18.410 

.530 .050 

.580 30.370 

.630 31.710 

.680 31.290 

.730 22.460 

.780 1.835 

.830 -2.250 

.880 .330 

.330 26.386 

.980 33.750 

1.030 31.460 

1.080 38.420 

1.130 69.5S0 

1.180 60.900 

1.230 -33.133 

1.280-126.200 

1.330-126.150 

1.380 -21.051 

1.430 88.450 

1.480 31.100 

1.530 10.580 

1.580 -51.170 

1.630 -58.020 

1.680 -14.5320

.040 

.090 

.140 

.190 

.240 

.290 

.340 

.390 

.440 

.490 

.540 

.590 

.640 

.690 

.740 

.790 

.840 

.890 

.940 

.990 

1.040 

1.090 

1.140 

1.190 

1.240

.247 1.933 

3.653 

6.460 

8.349 

7.780 

2.590 

-12.090 

-23.510 

-15.490 

5.380 

33.830 

31.080 

31.390 

17.180 

1.780 

-3.930 

6.127 

28.850 

34.540 

30.040 

44.810 

73.800 

46.110 

-61.080

1.290-135.900 1.340-109.900 

1.330 3.690 

1.440 99.200 

1.490 77.830 

1.540 -5.064 

1.590 -5$.310 

1.640 -49.480 

1.690 -9.130

.050 .436 

.100 2.343 

.150 4.016 

.200 7.193 

.250 8.119 

.300 7.760 

.350 -. 200 

.400 -15.814 

.450 -23.130 

.500 -12.170 

.SS0 11.680 

.600 35.640 

.650. 31.166 

.700 30.820 

.750 11.540 

.800 2.030 

.850 -5.020 

.900 12.550 

.950 30.020 

1.000 34.560 

1.050 29.340 

1.100 51.475 

1.1S0 76.150 

1.200 27.670 

1.250 -81.110 

1.300-142.200 

1.350 -*0.300 

1.400 28.670 

1.450 104.800 

1.S00 62.140 

1.5S0 -18.940 

1.600 -64.200 

1.650 -39.460 

1.700 .3.370

BUREG/CR-6645

24T 24 

25T 25 

263 26 

6.0 

27T 27 

28T 28 

293 29 

6.0 

30T 30 

31T 21 

32T 32 

33T 33 

34T 34 

35B 35 

12.0 

36T 36 

37T - 37

25 

26 

27 

TI 

28 

29 

30 

TI 

31 

32 

33 

34 

35 

36 

TI 

37 

38

1 1 

1 

851 

1 

1 

1 

851 

1 

1 

1 

1 

1 

1 

72"7 

1 

1

2 

2 

2 

.S 

2 

2 

2 

.5 

2 

3 

3 

3 

3 

3 

.5 

3 

3

I

B-5
1.710 2.640 

1.760 13.490 

1.310 5.210 

1.860 -37.870 

1.910 -44.430 

1.960 -20.220 

2.010 22.690 

2.060 50.900 

2.110 46.500 

2.160 29.530 

2.210 14.300 

2.260 14.350 

2.310 -10.127 

2.360 -40.270 

2.410 -55.050 

2.460 -20.100 

2.510 25.070 

2.560 40.210 

2.610 26.130 

2.660 2.293 

2.710 -19.818 

2.760 -49.700 

2.310 -43.630 

2.860 14.222 

2.910 74.050 

2.960 90.090 

3.010 26.590 

3.060 -30.694 

3.110 -63.370 

3.160 -55.090 

3.210 14.938 

3.260 77.000 

3.310 38.030 

3.360 -61.870 

3.410-132.440 

3.460-101.280 

3.510 -34.416 

3.560 31.600 

3.610 51.340 

3.660 58.620 

3.710 25.204 

3.760 -33.270 

3.810 -17.920 

3.860 44.630 

3.910 47.570 

3.960 -17.930 

4.010 -90.240 

4.060-115.210 

4.110 -89.450 

4.160 -7.650 

4.210 95.900 

4.260 148.210 

4.310 126.400 

4.360 46.740" 

4.410 -59.390 

4.460-140.460 

4.510-143.250 

4.560 -53.190 

4.610 66.470 

4.660 120.700

1.720 7.910 

1.770 12.720 

1.820 -1.060 

1.870 -44.660 

1.920 -40.640 

1.970 -14.660 

2.020 32.910 

2.070 50.890 

2.120 43.980 

2.170 25S.20 

2.220 13.650 

2.270 11.800 

2.320 -16.677 

2.370 -45.040 

2.420 -51.130 

2.470 -9.890 

2.520 31.250 

2.S70 38.300 

2.620 22.040 

2.670 -2.550 

2.720 -25,091 

2.770 -52.860 

2.820 -35.990 

2.870 27.998 

2.920 81.260 

2.970 83.490 

3.020 12.040 

3.070 -39.100 

3.120 -653550 

3.170 -46.680 

3.220 31.850 

3.270 79.050 

3.320 20.270 

3.370 -80.710 

3.420-135.540 

3.470 -88.960 

3.520 -19.460 

3.570 38.390 

3.620 53.910 

3.670 36.650 

3.720 11.500 

3.770 -36.140 

3.820 -6.674 

3.070 52.060 

3.920 38.150 

3.970 -33.551 

4.020 -99.270 

4.070-114.590 

4.120 -76.210 

4.170 12.124 

4.220 113.500 

4.270 147.310 

4.320 117.530 

4.370 24.450 

4.420 -70.560 

4.470-150.200 

4.520-131.860 

4.570 -25.880 

4.620 86.900 

4.670 115.300

1.730 11.6932 

1.780 12.340 

1.830 -9.275 

1.380 -48.440 

1.930 -36.040 

1.980 -7.S30 

2.030 41.160 

2.080 50.370 

2.130 40.770 

2.180 21.754 

2.230 13,840 

2.280 7.500 

2.330 -23.257 

2.380 -49500 

2.430 -44.950 

2.480 .560 

2.530 36.340 

2.510 35.700 

2.630 17.430 

2.680 -7.200 

2.730 -31.340 

2.780 -53.330 

2.030 -25.760 

2.880 41.354 

2.930 86.860 

2.980 72.430 

3.030 -. 760 

3.080 -46.480 

3.130 -65.650 

3.180 -34.860 

3.230 47.100 

3.280 75.700 

3.330 .830 

3.330 -97.000 

3.430-132.490 

3.480 -76.420 

3.530 -4.590 

3.530 42.380 

3.630 56.150 

3.680 52.200 

3.730 -3.020 

3.700 -35.800 

3.330 6.614 

3.880 55.970 

3.930 26.060 

3.980 -49.341 

4.030-105.970 

4.080-111.900 

4.130 -60.730 

4.180 33.040 

4.230 128.010 

4.280 143.810 

4.330 105.240 

4.380 3.020 

4.430 -96.210 

4.480-155.300 

4.530-116.610 

4.580 -6.244 

4.630 103.150 

4.680 105.710

1.740 13.840 

1.790 11.590 

1.840 -18.870 

1.890 -49.200 

1.940 -30.680 

1.990 1.474 

2.040 46.760 

2.090 49.540 

2.140 36.980 

2.190 18.100 

2.240 14.633 

2.290 2.070 

2.340 -29.545 

2.390 .53.310 

2.440 -37.650 

2.490 10.110 

2.540 39.640 

2.590 32.830 

2.640 12.430 

2.690 -11.440 

2.740 -38.050 

2.790 -S2.120 

2.840 -13.340 

2.890 53.810 

2.940 90.700 

2.990 58.100 

3.040 -11.720 

3.090 -53.110 

3.140 -63.900 

3.190 -19-850 

3.240 60.000 

3.290 67.100 

3.340 -19.900 

3.390-111.450 

3.440-124.440 

3.490 -63.210 

3.540 5.570 

3.590 46.040 

3.640 57.860 

3.690 45.550 

3.740 -16.340 

3.790 -32.590 

3.840 20.760 

3.890 56.520 

3.940 12.130 

3.990 -64.670 

4.040.110.800 

4.090-106.970 

4.140 -43.930 

4.190 54.610 

4.240 138.790 

4.290 138.900 

4.340 88.790 

4.390 -177.985 

4.440-112.440 

4.430-155.600 

4.540 -97.660 

4.590 13.100 

4.640 114.700 

4.690 93.570

1.750 14.260 

1.800 9.390 

1.850 -28.840 

1.900 -47.540 

1.950 -25.310 

2.000 11.810 

2.050 49.810 

2.100 48.330 

2.150 33.160 

2.200 15.660 

2.250 15.120 

2.300 -3.854 

2.350 -35.200 

2.400 -SS.620 

2,450 -29.410 

2.500 18.160 

2.550 40.830 

2,600 29.700 

2.650 7.233 

2.700 -15.455 

2.750 -44.430 

2.800 -48.890 

2.850 .310 

2.900 64.380 

2.950 92.200. 

3,000 42.290 

3,050 -21.500 

3.100 -59.020 

3.150 -60.550 

3.200 -2.720 

3.250 70.250 

3.300 54.100 

3.350 -41.140 

3.400-123.830 

3.450-113.420 

3.500 -49-130 

3.550 21.940 

3.600 43.730 

3.650 58.800 

3.700 36.660 

3.750 -26.690 

3.800 -26.660 

3.850 33.940 

3.900 53.760 

3.950 -2.690 

4.000 -78.600 

4.050-113.900 

4.100 -99.680 

4.150 -26.260 

4.200 75.930 

4.250 145.480 

4.300 133.100 

4.350 68.820 

4.400 -38.970 

4.450-127.390 

4.S00-151.300 

4.550 -76.090 

4.600 42.890 

4.650 120.700 

4.700 80.350

B-6 NUREG/CR-6645
A

4.710 66.900 

4.760 -12.130 

4.810 -77.800 

4.860 -48.520 

4.910 S.828 

4.960 27.212 

5.010 31.767 

5.060 54.406 

5.110 S.SSO 

5.160 96.100 

5.210 26.340 

5.260 -59.950 

5.310-125.050 

5.360-100.000 

5.410 -19.920 

5.460 73.590 

5.510 69.310 

5.560 6.090 

5.610 -60.550 

5.660 -47.990 

5.710 16.663 

5.760 68.200 

5.810 56.900 

5.360 20.530 

5.910 -12.073 

5.960 -39.620 

6.010 -23.420 

6.060 27.790 

6.110 40.050 

6.160 26.630 

6.210 -15.060 

6.260 -23.310 

6.310 -13.379 

6.360 -8.950 

6.410 28.230 

6.460 42.950 

6.510 15.470 

6.560 -47.530 

6.610 -90.970 

6.660 -87.140 

6.710 -29.630 

6.760 49.190 

6.810 81.660 

6.860 70.890 

6.910 12.040 

6.960 -48.116 

7.010 -87.850 

7.060 -85.640 

7.110 19.780 

7.160 147.900 

7.210 154.150 

7.260 -9.040 

7.310-166.450 

7.360-199.400 

7.410 -78.470 

7.460 71.130 

7.510 148.400 

7.560 108.300 

7.610 -13.927 

7.660-125.100

4.720 53.190 

4.770 -30.350 

4.320 -79.400 

4.870 -34.956 

4.920 11.809 

4.970 29.890 

5.020 33.100 

5.070 60.560 

5.120 92.980 

5.170 87.260 

5'.220 10.090 

5.270 -78.380 

5.320-126.600 

5.370 -88.120 

5.420 1.700 

S.470 80.700 

5.520 60.290 

5.570 -9.037 

5.620 -66.000 

5.670 -34.340 

5.720 27.730" 

5.770 72.050 

5.820 48.100 

5.870 14.330 

5.920 -19.580 

5.970 -41.330 

6.020 -13.170 

6.070 33.310 

6.120 40.120 

6.170 19.130 

6.220 -19.150 

6.270 -23.000 

6.320 -11.870 

6.370 -S.970 

6.420 36.780 

6.470 3i.750 

6.520 6.080 

6.570 -61.120 

6.620 -93.460 

6.670 -80.840 

6.720 -12.126 

6.770 59.850 

6.820 83.550 

6.870 62.720 

6.920 -1.990 

6.970 -57.360 

7.020 -94.380 

7.070 -71.000 

7.120 46.806 

7.170 163.750 

7.220 131.300 

7.270 -45.377 

7.320-186.400 

7.370-184.800 

7.420 -47.140 

7.470 94.400 

7.520 150.100 

7.570 89.500 

7.620 -41.410 

7.670-134.350

4.730 38.710 

4.780 -47.780 

4.830 -77.130 

4.880 -22.100 

4.930 16.510 

4.980 31.140 

5.030 36.332 

5.080 66.810 

5.130 99.230 

5.180 75.150 

S.230 -6.380 

5.280 -94.900 

5.330-124.140 

5.380 -74.640 

S.430 23.380 

5.480 82.600 

5.530 48.950 

5.580 -23.540 

5.630 -67.350 

5.680 -19.900 

5.730 30.400 

5.780 72.950 

5.830 39.740 

5.830 8.700 

5.930 -26.450 

5.980 -40.840 

6.030 -1.350 

6.080 36.530 

6.130 39.100 

6.180 10.267 

6.230 -21.340 

6.280 -21.259 

6.330 -10.620 

6.380 -. 301 

6.430 42.760 

6.480 34.000 

6.530 -5.390 

6.580 -72.690 

6.630'-94.480 

6.680 -71.600 

6.730 5.7S0 

6.780 67.540 

6.830 33.470 

6.880 52.170 

6.930 -15.190 

6.980 -65.737 

7.030 -99.600 

7.080 -51.800 

7.130 74.450 

7.180 172.600 

7.230 101.150 

7.280 -79.830 

7.330-200.200 

7.380-163.900 

7.430 -15.687 

7.480 114.000 

7.530 146.250 

7.580 67.410 

7.630 -67.080 

7.680-139.100

4.740 4.790 

4.040 

4.090 

4.940 

4.990 

S.040 

S.090 

5.140 

S.190 

5.240

22.900 -62.290 

-70.940 

-10.920 

20.270 

31.300 

41.640 

72.399 

102.200 

60.370 

-23.250

5.290-108.610 5.340-113.560 

5.390 -58.930 

5.440 43.670 

5.490 30.S00 

S.540 35.770 

5.590 -37.540 

5.640 -65.170 

5.690 -6.430 

5.740 51.120 

5.790 70.580 

5.640 32.480 

5.890 2.164 

5.940 -32.030 

5.990 -37.730 

6.040 9.490 

6.090 38.280 

6.140 36.540 

6.190 .742 

6.240 -22.740 

6.290 -18.869 

6.340 -10.140 

6.390 8.060 

6.440 45.750 

6.490 23.850 

6.540 -13.610 

6.590 -81.420 

6.640 -93.640 

6.690 -59.680 

6.740 22.550 

6.790 73.550 

6.840 11.160 

6.390 39.700 

6.940 -27.070 

6.990 -73.330 

7.040-100.100 

7.090 -29.710 

7.140 101.430 

7.190 174,250 

7.240 65.960 

7.290-112.040 

7.340-207.200 

7.390-138.200 

7.440 15.267 

7.490 129.900 

7.540 137.200 

7.590 42.110 

7.640 -90.270 

7.690-137.250

4.750 5.790 4.100 -72.300 

4.850 -61.060 

4.900 -1.644 

4.950 23.770 

5.000 31.340 

5.050 48.030 

S.100 78.840 

5.150 101.160 

S.200 43.860 

5.250 -41.154 

S.300-118.910 

5.350-110.380 

S.400 -40.500 

5.450 60.900 

5.$00 75.990 

S.SSO 21.310 

5.600 -50.S30 

5.650 -58.730 

S.700 5.590 

5.75O 61.130 

5.100 64.910 

5.850 26.270 

5.900 -4.749 

5.950 -36.390 

6.000 -31.030 

6.050 19.650 

6.100 39.370 

6.150 32.410 

6.200 -83230 

6.250 -23.650 

6.300 -16.310 

6.350 -9.950 

6.400 18.110 

6.450 45.640 

6.500 22.870 

6.550 -32.937 

6.600 -87.220 

6.650 -91.160 

6.700 -45.690 

6.750 37.410 

6.300 78.250 

6.350 76.910 

6.900 26.120 

6.950 -37.960 

7.000 -30.510 

7.050 -95.350 

7.100 -5.780 

7.150 126.350 

7.200 163.300 

7.250 28.530 

7.300-141.300 

7.350-206.900 

7.400-109.200 

7.450 44.560 

7.500 141.S00 

7.550 124.300 

7.600 14.440 

7.650-110.000 

7.700-129.600

B-7 BKUREG/CR-6645
7.710-117.100 

7.760 -8.010 

7.810 96.280 

7.860 139.200 

7.910 92.010 

7.960 -2.S20 

8.010 -92.750 

8.060-104.900 

3.110 -48.260 

8.160 32.890 

1.210 77.420 

8.260 107.540 

8.310 82.870 

8.360 25.950 

8.410 -44.040 

8.460 -96.990 

3.510-114.970 

8.560 -74.040 

8.610 28.380 

8.660 121.100 

8.710 107.250 

8.760 -17.340 

8.010-121.150 

8.860-136.300 

8.910 -32.320 

3.960 105.970 

f.010 173.050 

9.060 128.100 

9.110 -2.990 

9.160-104.600 

9.210-109.750 

3.260 -32.300 

9.310 54.980 

9.360 82.530 

9.410 60.990 

9.460 1.330 

9.510 -43.220 

9.560 -35.140 

9.610 10.790 

9.660 37.550 

9.710 34.030 

9.760 10.240 

9.810 -31.990 

9.860 -56.130 

9.910 -52.211 

9.960 -41.607 

10.010 -38.660 

10.060 -30.160 

10.110 34.520 

10.160 52.400 

10.210 -30.710 

10.260-131.000 

10.310-128.900 

10.360 -36.700 

10.410 54.940 

10.460 78.150 

10.510 68.080 

10.560 49.720 

10.610 -3.090 

10.660 -22.430

7.720-100.820 

7.770 15.471 

7.820 111.410 

7.870 136.400 

7.920 75.840 

7.970 -21.820 

8.020-103.800 

8.070 -96.650 

8.120 -33.402 

8.170 46.180 

8.220 83.650 

8.270 108.040 

8.320 72.570 

8.370 12.300 

8.420 -56.790 

3.470-103.440 

3.520-112.S00 

8.570 -57.180 

8.620 51.310 

8.670 128.200 

8.720 839.000 

8.770 -45.240 

8.820-131.800 

8.870-123.250 

8.920 -3.390 

8.970 127.360 

9.020 174.100 

9.070 105.600 

9.120 -30.840 

9.170-113.S00 

9.220 -99.700 

9.270 -11.8350 

9.320 63.940 

9.370 83.690 

9.420 51.460 

9.470 -11.480 

9.520 -46.510 

9.570 -26.160 

3.620 18.210 

9.670 38.360 

9.720 32.680 

9.770 1.470 

9.820 -38.050 

9.870 -58.110 

9.920 -49.047 

9.970 -40.300 

10.020 -39.390 

10.070 -19.540 

10.120 44.600 

10.170 43.200 

10.220 -55.300 

10.270-139.300 

10.320-115.600 

10.370 -14.033 

10.420 65.430 

10.470 76.330 

10.520 66.548 

10.570 41.880 

10.620 -12.810 

10.670 -21.220

7.730 -80.770 7.780 37.660 

7.830 123.450 

7.880 129.500 

7.930 57.720 

7.980 -40.510 

3.030-110.140 

8.080 -87.100 

8.130 -17.496 

8.180 56.360 

8.230 90.380 

8.280 105.310 

8.330 61.660 

8.380 -2.220 

8.430 -68.300 

8.480-103.550 

8.530-107.260 

8.580 -37.870 

8.630 74.260 

3.680 130.400 

8.730 66.080 

8.780 -69.470 

8.830-139.700 

8.880-105.500 

8.930 26.080 

8.980 144.900 

9.030 170.450 

9.080 80.610 

9.130 -55.700 

9.180-118.700 

9.230 -86.700 

9.280 7.790 

3.330 70.120 

9.380 81.760 

3.430 40.670 

9.480 -22.580 

9.530 -47.720 

9.580 15.3980 

9.630 25.000 

9.680 37.530 

9.730 29.760 

3.780 -8.120 

3.830 -43.630 

3.880 -538.775 

9.930 -46.472 

9.930 -39.340 

10.030 -40.220 

10.080 -6.290 

10.130 52.230 

10.180 30.860 

10.230 -79.890 

10.280-144.53,0 

10.330 -99.380 

10.380 7.322 

10.430 73.060 

10.480 74.550 

10.530 64.570 

10.530 32.073 

10.630 -13.320 

10.680 -19.420

7.740 -57.460 7.790 58.690 

7.840 132.300 

7.890 119.250 

7.940 37.920 

7.930 -53.170 

8.040-112.400 

8.090 -75.870 

8.140 -. 406 

8.190 64.200 

8.240 97.560 

8.290 99.760 

8.340 50.380 

8.390 -16.790 

3.440 -79.670 

8.490-112.400 

8.540 -39.150 

8.590 -16.8350 

8.640 93.720 

3.690 127.700 

8.740 39.720 

8.790 -90.020 

8.840-144.100 

3.890 -84.000 

8.940 54.630 

8.990 158.450 

9.040 161.400 

9.090 53.730 

9.140 -76.180 

9.190-119.850 

9.240 -70.720 

9.290 26.030 

9.340 74.320 

9.390 76.760 

9.440 28.463 

9.490 -31.490 

9.540 -46.320 

9.590 -6.030 

9.640 30.620 

9.690 36.150 

9.740 24.720 

5.790 -17.364 

9.840 -48.640 

9.890 -57.850 

3.940 -44.559 

9.930 -38.671 

10.040 -39.990 

10.090 8.012 

10.140 57.070 

10.190 14.480 

10.240-101.080 

10.230-144.400 

10.340 -80.500 

10.390 26.120 

10.440 77.580 

10.490 72.620 

10.540 60.990 

10.530 20.610 

10.640 -22.380 

10.690 -17.070

7.750 -32.620 7.800 78.4S0 

7.850 137.600 

7.900 106.520 

7.950 17.550 

8.000 -77.270 

8.050-110.570 

8.100 -62.600 

8.150 16.910 

8.200 71.036 

8.250 103.810 

8.300 92.030 

8.350 38.570 

8.400 -30.740 

8.450 -83.080 

8.500-114.750 

8.550 -88.030 

8.600 5.270 

8.650 109.450 

8.700 120.100 

8.750 11.180 

8.800-107.190 

8.850-143.450 

8.900 -59.430 

8.950 81.450. 

9.000 167.700 

9.050 147.100 

9.100 25.690 

3.150 -92.250 

3.200-116.600 

9.250 -52.310 

9.300 42.200 

3.350 79.170 

9.400 69.500 

9.450 15.039 

9.500 -33.230 

9.550 -42.020 

3.600 2.830 

9.650 34.830 

9.700 35.000 

9.750 17.930 

9.800 -25.280 

3.850 -52.920 

9.900 -55.430 

9.950 -43.041 

10.000 -38.390 

10.050 -37.020 

10.100 22.020 

10.150 57.350 

10.200 -6.460 

10.250-118.280 

10.300-138.900 

10.350 .53.300 

10.400 41.880 

10.450 79.020 

10.500 70.240 

10.550 55.970 

10.600 a.510 

10.650 -22.930 

10.700 -13.870

B-8 UPUREG/CR-6645
S.'

10.710 -9.230 

10.760 35.050 

10.810 50.340 

10.860 47.700 

10.910 -2.800 

10.960 -80.335 

11.010-109.170 

11.060 -67.750 

11.110 32.260 

11.160 93.000 

11.210 72.700 

11.260 24.209 

11.310 1.110 

11.360 4.400 

11.410 20.259 

11.460 28.160 

11.510 25.360 

11.560 26.130 

11.610 2.620 

11.660 -. 617 

11.710 12.390 

11.760 7.362 

11.610 -36.880 

11.860 -62.050 

11.910 -67.710 

11.960 -46.440 

12.010 -3.320 

12.090 102.520 

12.110 129.250 

12.160 23.730 

12.210-114.550 

12.260-134.700 

12.310 -57.470 

12.360 23.170 

12.410 94.600 

12.460 132.200 

12.510 72.050 

12.560 -88.900 

12.610-167.950 

12.660-104.590 

12.710 7.563 

12.760 75.900 

12.610 81.370 

12.860 61.730 

12.910 -. 938 

12.960 -54.490 

13.010 -55.390 

13.060 1.220 

13.110 44.500 

13.160 56.100 

13.210 31.340 

13.260 -21.780 

13.310 -47.160 

13.360 -46.800 

13.410 -16.857 

13.460 13.000 

13.510 30.130 

13.560 28.560 

13.610 3.535 

13.660 -7.823

10.720 -2.640 10.730 6.040 

10.770 41.830 10.760 46.702 

10.820 50.408 10.130 50.610 

10.870 43.270 10.280 35.840 

10.920 -18.566 10.930 -34.600 

10.970 -91.960 10.980 -99.770 

11.020-108.390 11.030-104.140 

11.070 -49.800 11.080 -30.170 

11.120 49.860 11.130 64.890 

11.170 94.820 11.180 92.830 

11.220 62.990 11.230 52.770 

11.270 16.200 11.280 9.850 

11.320 .640 11.330 .620 

11.370 7.500 11.380 10.880 

11.420 21.710 11.430 23.080 

11.470 27.938 11.480 26.599 

11.520 26.670 11.530 28.040 

11.570 22.440 11.580 17.770 

11.620 -. 720 11.630 -2.350 

11.670 .920 11.680 3.420 

11.720 15.210-11.730 16.890 

11.770 -.919 11.780 -10.882 

11.820 -42.439 11.830"-47.303 

11.870 -66.320 11.880 -69.500 

11.920 -63.690 11.930 -58.970 

11.970 -42.270 11.980 -36.400 

12.020 15.710 12.030 37.772 

12.070 118.400 12.080 129.100 

12.120 117.700 12.130 100.550 

12.170 -7.277 12.160 -36.660 

12.220-130.000 12.230-139.700 

12.270-123.630 12.280-109.630 

12.320 -39.980 12.330 -23.450 

12.370 38.750 12.380 54.421 

12.420 105.630 12.430 115.870 

12.470 131.100 12.480 125.800 

12.520 42.340 12.530 9.520 

12.570-116.450 12.530-139.620 

12.620-165.200 12.630-156.800 

12.670 -62.260 12.660 -58.S30 

12.720 25.470 12.730 41.240 

12.770 62.240 12.780 85.580 

12.620 77.950 12.630 74.920 

12.670 52.410 12.380 40.580 

12.920 -14.149 12.930 -26.120 

12.970 -61.040 12.980 -64.720 

13.020 -45.610 13.030 -34.020 

13.070 12.043 13.080 22.360 

13.120 49.380 13.130 53.370 

13.170 53.440 13.180 49.390 

13.220 22.296 13.230 11.442 

13.270 -29.890 13.200 -36.501 

13.320 -49.350 13.330 -50.840 

13.370 -42.S00 13.380 -36.330 

13.420 -9.740 13.430 -2.940 

13.470 16.894 13.460 19.980 

13.520 33.240 13.530 34.370 

13.570 23.711 13.530 18.285 

13.620 .530 13.630 -1.720 

13.670 -6.370 13.680 -8.390

10.740 16.;190 10.790 49.470 

10.840 50.520 

10.890 25.300 

10.940 -50.370 

10.990-104.600 

11.040 -95.750 

11.090 -9.260 

11.140 77.510 

11.190 87.940 

11.240 42.840 

11.290 5.360 

11.340 .990 

11.390 14.340 

11.440 24.980 

11.490 25.182 

11.540 28.850 

11.590 12.500 

11.640 -2.390 

11.690 6.430 

11.740 16.320 

11.790 -20.966 

11.840 -52.194 

11.890 -70.930 

11.940 -54.260 

11.990 -28.420 

12.040 60.600 

12.090 134.500 

12.140 78.700 

12.190 -68.190 

12.240-143.600 

12.290 -93.380 

12.340 -7.610 

12.390 69.210 

12.440 124.100 

12.490 114.650 

12.540 -24.570 

12.590-156.300 

12.640-142.900 

12.690 -34.850 

12.740 55.000 

12.790 66.060 

12.840 72.070 

12.890 27.110 

12.940 -36.660 

12.990 -65.070 

13.040 -21.900 

13.090 31.390 

13.140 56.070 

13.190 44.480 

13.240 -. 454 

13.290 -41.440 

13.340 -50.970 

13.390 -30.440 

13.440 3.030 

13.490 23.010 

13.540 34.550 

13.590 12.739 

13.640 -3.700 

13.690 -6.510

10.750 26.350 

10.800 50.360 

10.850 49.790 

10.900 12.090 

10.950 -65.840 

11.000-107.640 

11.050 -83.370 

11.100 12.120 

11.150 87.160 

11.200 81.100 

11.250 33.290 

11.300 2.550 

11.350 2.150 

11.400 17.730 

11.450 26.995 

11.S00 24.690 

11.550 28.360 

11.600 7.220 

11.650 -1.610 

11.700 9.400 

11.750 13.120 

11.800 -29.822 

11.850 -57.220 

11.900 -70.340 

11.950 -50.090 

12.000 -17.840 

12.050 82.640 

12.100 134.700 

12.150 52.740 

12.200 -93.600 

12.250-141.700 

12.300 -75.610 

12.350 7.642 

12.400 82.510 

12.450 129.750 

12.S00 96.600 

12.550 -57.940 

12.600-165.400 

12.650-124.950 

12.700 -12.586 

12.750 66.710 

12.800 $4.360 

12.850 68.120 

12.900 12.990 

12.55O -46.100 

13.000 -62.030 

13.050 -10.038 

13.100 38.670 

13.150 57.010 

13.200 38.620 

13.250 -11.920 

13.300 -44.720 

13.350 -49.600 

13.400 -23.784 

13.450 8.275 

13.500 26.500 

13.550 32.320 

13.600 7.634 

13.650 -5.810 

13.700 -3.800

U-E G/CR-6645 B;-9
13.710 -. 966 

13.760 6.950 

13.810 -4.710 

13.860 -11.340 

13.910 -. 145 

13.960 5.302 

14.010 -5.539 

14.060 -25.570 

14.110 -14.900 

14.160 13.130 

14.210 33.650 

14.260 23.190 

14.310 6.485 

14.360 -23.480 

14.410 -36.650 

14.460 -33.660 

14.510 .- 3.950 

14.560 26.370 

14.610 19.740 

14.660 -13.744 

14.710 -50.430 

14.760 -49.020 

14.310 -18.410 

14.860 22.800 

14. 910 39.330 

14.960 26.350 

3 

3

13.720 

13.770 

13.820 

13.870 

13.320 

13.970 

14.020 

14.070 

14.120 

14.170 

14.220 

14.270 

14.320 

14.370 

14.420 

14.470 

14.520 

14.570 

14.620 

14.670 

14.720 

14.770 

14.820 

14.870 

14.920 

14.370

1.530 

5.750 

-7.390 

-9.560 

2.271 

4.430 

-9.860 

-26.500 

-9.890 

19.140 

33.560 

26.900 

-. 483 

-26.700 

-33.370 

-29.350 

3.510 

28.290 

14.520 

-22.525 

-53.190 

-45.030 

-9.6539 

28.240 

39.300 

20.500

13.730 

13.780 

13.830 

13.880 

13.930 

13.980 

14.030 

14.080 

14.130 

14.180 

14.230 

14.280 

14.330 

14.380 

14.430 

14.480 

14.530 

14.530 

14.630 

14.680 

14.730 

14.780 

14.830 

14.880 

14.930 

14.980

3.890 

3.714 

-9.770 

-7.475 

4.110 

2.820 

-14.591 

-25.680 

-4.550 

24.620 

32.700 

23.390 

-7.492 

-29.506 

-39.130 

-24.120 

10.750 

23.400 

3.550 

-31.200 

-54.160 

-39.870 

-. 613 

32.530 

37.950 

14.210

13.740 5.338 

13.790 1.142 

13.840 -11.531 

13.890 -5.222 

13.940 5.130 

13.990 .SSO 

14.040 -19.160 

14.090 -23.220 

14.140 1.170 

14.190 29.140 

14.240 31.680 

14.290 10.700 

14.340 -13.966 

14.390 -32.120 

14.440 -38.730 

14.490 -18.070 

14.540 17.290 

14.590 26.850 

14.640 1.920 

14.690 -39.120 

14.740 -53.630 

14.790 -33.640 

14.840 8.164 

14.930 35.820 

14.940 35.310 

14.990 7.653

13.750 7.000 

13.800 -1.760 

13.850 -12.200 

13.900 -2.745 

13.950 5.458 

14.000 -2.179 

14.050 -23.000 

14.100 -19.470 

14.150 7.146 

14.200 32.248 

14.250 30.620 

14.300 12.990 

14.350 -19.350 

14.400 -34.310 

14.450 -36.870 

14.500 -11.270 

14.550 22.640 

14.600 23.910 

14.650 -5.490 

14.700 -45.690 

14.750 -51.870 

14.800 -26.470 

14.850 16.090 

14.900 33.130 

14.950 31.400.

NUREG/CR-6645 B-10
APPENDIX C

POTENTIAL DIFFERENCES BETWEEN DYNAMIC AND STATIC 

MASS DISTRIBUTIONS FOR THE SAME COMPUTER MODEL

C-1 NUPXG,/ECR-6645
The analysis of nuclear power plant structures, systems and components often requires that static 

analysis solutions and dynamic analysis solutions for the same computer model be combined to 

produce the total stress state, for comparison to a design allowable stress. One such combination is 

deadweight plus seismic load. To produce accurate load combination results, it is important that the 

mass distribution for the static analysis and the dynamic analysis be comparable and sufficiently 

refined. During the course of the present study (see Section 2.3.2), convergence of the Lindley-Yow 

plus missing mass in-phase response component to the static analysis solution for total mass times 

ZPA was tested for the BM3 piping model used in the numerical studies. The first attempt to 

correlate these two solutions was unsuccessful, because the piping computer code, based on SAP V, 

treats the distributed mass of the piping model differently in static analyses and dynamic analyses. 

The dynamic analysis treats the distributed piping mass as lumped masses at the node points, based 

on adjacent pipe element lengths. The static analysis treats the distributed piping mass times 

acceleration as a distributed load along the length of the pipe element. The static analysis procedure 

is the more accurate representation for calculation of stresses and reactions at supports. However, 

for the dynamic analysis solution, a finite number of mass degrees of freedom are defined by assigning 

the mass to the locations of the node points. 

In dynamic analysis, the definition of node points is primarily driven by the physical characteristics 

of the piping system (support points, branch points, and in-line components) and by the dynamic 

behavior which is to be predicted (frequency range and mode shapes). A third criterion for definition 

of node points should also be included: sufficient refinement for accurate prediction of stresses and 

support reactions. 

Tables C-I and C-2 present a comparison between two static analysis solutions, for BM3 support 

reactions and pipe end moments, respectively. The column labeled "static/dyn" are the results for the 

dynamic lumped mass representation multiplied by ZPA. The column labeled "static/sta" are the 

results for the static distributed mass representation multiplied by ZPA. Both analyses utilized the 

same computer model and computer code. 

Examination of Table C-1 indicates that reaction forces F., Fy , 4 are in reasonable agreement; 

however, moment reactions at the fixed support points (nodes 1, 31, 38) have significant differences 

(>10%). Table C-2 indicates that the pipe end moments for elements in the vicinity of the fixed 

support points also have significant differences (>10%). Two effects contribute to poor correlation 

of the moment. In the dynamic analysis, mass apportioned to the fixed supports (nodes 1, 31, 38) 

do not produce any moment effect. Also, concentrated loads (i.e., lumped masses) do not generally 

produce the same moments as a distributed load. Both of these effects can be minimized in the 

dynamic analysis by defining of a sufficient number of node points in the vicinity of fixed supports. 

During development of the mathematical model, a sensitivity study should be conducted to ensure 

that the node point distribution is sufficient to produce an accurate static solution. Successive 

refinement of the node point distribution should be performed until there is reasonable correlation 

between results generated with the dynamic lumped mass representation and the static distributed 

mass representation.

NUREG/CR-6645 C-2
Some existing computer codes for piping analysis may be less susceptible to this condition because 

corrections have been built into the code. It is important for the analyst to understand the treatment 

of mass and ensure that an adequate lumped mass distribution has been defined for the dynamic 

analysis.

C-3 H REG/CR-6645
Support Reactions of Static Analyses due to Different Mass 

Distributions

Forces in lbs 1 % Damping

Moments in in-lbs

NODE REAC. ZPA ZPA 

NO. TYPE static/dyn static/sta 

I Fx -46.21 -50.50 

I Fy 0.66 0.66 

1 Fz 0.26 0.01 

I Mx 10.53 32.80 

1 my -329.25 -478.00 

1 Mz 158.43 -823.00 

4 Fx -102.75 -98.00 

4 Fz -5.49 -4.26 

7 Fy -0.01 -0.85 

11 Fy -1.17 -0.12 

11 Fz 37.65 36.00 

15 Fx -474.26 -475.00 

17 Fy 3.78 3.84 

17 Fz -35.17 -34.40 

36 Fv 8.92 8.70 

36 Fz 31.68 34.80 

38 Fx -767.29 -770.00 

38 FY -11.24 -11.20 

38 Fz -31.17 -35.60 

38 Mx -189.06 -195.00 

38 My -2202.83 -2520.00 

38 Mz 804.58 799.00 

23 Fx -304.11 -297.00 

23 Fy -0.68 -0.15 

31 Fx -56.51 -60.60

31 

31 

31 

31 

31

I Fz 

Mx 

Mi

-0.25 

2.241 

219.27 

580.84 

1702.48

3.49 

344.00 

502.00 

2200.00

Table C-I

NUREG/CR-6645 C-4
Pipe End Moments of Static Analyses due to Different Mass 

Distribution 

Resultant Moments in in-lbs 1% Damping

ELEM. ZPA ZPA 

NO. static/dyn static/sta 

1 357.75 961.37 

2 450.75 808.88 

3 995.58 1241.14 

4 327.00 526.13 

5 241.67 303.25 

6 868.02 627.91 

7 725.62 486.36 

8 354.59 179.07 

9 502.09 498.37 

10 912.98 836.15 

11 1617.18 1642.18 

12 3951.97 3927.76 

13 2956.60 2925.35 

14 8613.31 8664.29 

15 475.77 503.56 

16 467.76 445.32 

17 310.22 302.07 

18 706.10 755.46 

19 471.56 519.70 

20 2028.91 2009.32 

21 580.58 524.32 

22 1971.49 1987.20 

23 1167.59 1191.47 

24 1123.54 1119.43 

25 922.09 788.22 

26 657.25 536.09 

27 340.76 257.71 

28 600.98 502.43 

29 710.13 578.43 

30 181215 2279.14 

31 7523.23 7288.41 

32 9107.41 8828.37 

33 10884.84 10502.09 

34 2199.42 2703.06 

* 35 4815.86 5424.33 

36 1247.94 1402.92 

37 2352.78 2649.47 

Table C-2

C-5 5UREG/CR-6645
APPENDIX D

TABULATION OF MODE CORRELATION COEFFICIENTS FOR THE 

DOUBLE SUM COMBINATION (DSC) AND THE COMPLETE 

QUADRATIC COMBINATION (CQC) METHODS

D-1 DUJREG/CR-6645
CA 

9, N

J% DAMPING 

MODE CORRELATION COEFFICIENTS 

f1/f2 DSC (T 

- 10 see) DSC (T = 20 sec) DSC (T '1000 se) ndCQC 

2Hz 1.0 1.0 1.0 1.0 

.95 .54 .36 .15 

.91 .22 .12 .04 

.87 .11 .06 .02 

5-z 1.0 1.0 1.0 1.0 

.95 .32 .24 .15 

.91 .10 .07 .04 

.87 .05 .03 .02 

10Hz 1.0 1.0 1.0 1.0 

.95 .23 .19 .15 

.91 .07 .06 .04 

.87 .03 .03 .02 

20Hz 1.0 1.0 1.0 1.0 

.95 .19 .17 .15 

.91 .06 .05 .04 

.87 .03 .02 .02 

30Hz 1.0 1.0 1.0 1.0 

.95 .18 .16 .15 

.91 .05 .05 .04 

.87 .02 .02 .02
2% DAN-TINGi

MODE CORRELATION COEFFICIENTS 

f ,, _ /f2 DSC (T - 10 sec) DSC (T = 20 sec) DSC (T 1000 sec) And CQC 

2 Hz 1.0 1.0 1.0 1.0 

".95 .69 .58 .41 

.91 .35 .25 .15 

.87 .19 .13 .07 

.83 .12 .08 .04 

5 Hz 1.0 1.0 1.0 1.0 

.95 .55 .49 .41 

.91 .23 .19 .15 

.87 .12 .09 .07 

.83 .07 .06 .04 

10Hz 1.0 1.0 1.0 1.0 

.95 .49 .45 .41 

.91 .19 .17 .15 

.87 .09 .08 .07 

.83 .06 .05 .04 

20 Hz 1.0 1.0 1.0 1.0 

.95 .45 .43 .41 

.91 .17 .16 .15 

.87 .08 .08 .07 

.83 .05 .05 .04 

30 Hz 1.0 1.0 1.0 1.0 

.95 .44 .43 .41 

.91 .16 .16 .15 

.87 .08 .08 .07 

.83 .05 .05 .04

In 

I, 4U' a.M
z = 

C, 

0

'F

MODE CORRELATION COEFFICIENTS 

Sf/f DSC (T 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) an-d CQC 

2Hz 1.0 1.0 1.0 1.0 

.95 .88 .86 .82 

.91 .65 .60 .53 

.87 .45 .39 .33 

.83 .32 .27 .22 

.79 .23 .19 .15 

.75 .17 .14 .11 

5HIz 1.0 1.0 1.0 1.0 

.95 .85 .83 .82 

.91 .58 .56 .53 

.87 .38 .36 .33 

.83 .26 .24 .22 

.79 .18 .17 .15 

.75 .13 .12 .11 

10Hz 1.0 1.0 1.0 1.0 

.95 .83 .83 .82 

.91 .56 .54 .53 

.87 .36 .34 .33 

.83 .24 .23 .22 

.79 .17 .16 .15 

.75 .12 .12 .11
% CDAM INT( 

MODE CORRELATION COEFFICIENTS (CONT'D)

IT

Sf/f2 DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC 

20 Hz 1.0 1.0 1.0 1.0 

.95 .83 .82 .82 

.91 .54 .53 .53 

.87 .34 .34 .33 

.83 .23 .22 .22 

.79 .16 .16 .15 

.75 .12 .11 .11 

301Hz 1.0 1.0 1.0 1.0 

.95 .82 .82 .82 

.91 .54 .53 .53 

.87 .34 .34 .33 

.83 .23 .22 .22 

.79 .16 .15 .15 

1 .75 .12 .11 .11

i 

CA 

.1 

01 Ab 

(n
CA 

rln

10% DAMPING 

MODE CORRELATION COEFFICIENTS 

Sf/f DSC (T" 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC 

2Hz 1.0 1.0 1.0 1.0 

.95 .96 .95 .95 

.91 .86 .84 .81 

.87 .72 .70 .66 

.83 .60 .56 .52 

.79 .48 .45 .41 

.75 .39 .36 .32 

.72 .32 .30 .26 

.68 .27 .24 .21 

.65 .22 .20 .17 

.62 .19 .17 .14 

.59 .16 .15 .12 

.57 .14 .13 .10 

.54 .12 .11 .09 

.51 .11 .10 .08

'P
10% DAMPING 

MODE CORRELATION COEFFICIENTS (CONT'D)

C'A 

c.'

f _ _/f2 DSC (T = 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC 

5Hz 1.0 1.0 1.0 1.0 

.95 .95 .95 .95 

.91 .83 .83 .81 

.87 .69 .68 .66 

.83 .56 .54 .52 

.79 .45 .43 .41 

.75 .36 .35 .32 

.72 .29 .28 .26 

.68 .24 .23 .21 

.65 .20 .19 .17 

.62 .17 .16 .14 

.59 .15 .14 .12 

.57 .13 .12 .10 

.54 .11 .11 .09 

.51 .10 .09 .08
10% DAMPING 

MODE CORRELATION COEFFICIENTS (CONT'D)

A /f2 DSC (T - 10 sec) DSC (T = 20 sec) DSC (T = 1000 sec) and CQC 

10 Hz 1.0 1.0 1.0 1.0 

.95 .95 .95 .95 

.91 .83 .82 .81 

.87 .68 .67 .66 

.83 .54 .54 .52 

.79 .43 .43 .41 

.75 .35 .34 .32 

.72 .28 .28 .26 

.68 .23 .23 .21 

.65 .19 .19 .17 

.62 .16 .16 .14 

.59 .14 .14 .12 

.57 .12 .12 .10 

.54 .11 .10 .09 

.51 .09 .09 .08

I? CA1

Go
10% DAMPING 

MODE CORRELATION COEFFICIENTS (CONT'D)

cm 

c"

f /f2 DSC (T - 10 sec) DSC (T = 20 sec) DSC (T 1000 sec) and CQC 

20Hz 1.0 1.0 1.0 1.0 

.95 .95 .95 .95 

.91 .82 .82 .81 

.87 .67 .67 .66 

.83 .54 .53 .52 

.79 .43 .42 .41 

.75 .34 .34 .32 

.72 .28 .27 .26 

.68 .23. .22 .21 

.65 .19 .19 .17 

.62 .16 .16 .14 

.59 .14 .14 .12 

.57 .12 .12 .10 

.54 .11 .10 .09 

.51 .09 .09 .08
10% DAMPING 

MODE CORRELATION COEFFICIENTS (CONT'D)

I 

C, 

00

fl/f DSC (T 

- 10 sec) DSC (T = 20 see) DSC (T = 1000 sec) and CQC 

30 Hz 1.0 1.0 1.0 1.0 

.95 .95. .95 .95 

.91 .82 .82 .81 

.87 .67 .67 .66 

.83 .53 .53 .52 

.79 .42 .42 .41 

.75 .34 .34 .32 

.72 .28 .27 .26 

.68 .23 .22 .21 

.65 .19 .19 .17 

.62 .16 .16 .14 

.59 .14 .14 .12 

.57 .12 .12 .10 

.54 .10 .10 .09 

.51 .09 .09 .08
APPENDIX E 

COMPARISON OF TIME HISTORY SOLUTION METHODS: 

MODE SUPERPOSITION VS. DIRECT INTEGRATION

E- I EUREG/CR-6645
Comparison of Mode Superposition vs. Direct Integration Time History 

In mode superposition time history analysis, the residual or "missing!" mass effect must be 

calculated and algebraically summed with the modal responses. To utilize modal time history analysis 

efficiently, only modal responses with frequencies up to the ZPA frequency need to be calculated; the 

"missing mass" not participating in these modes should be treated as an additional mode, with its peak 

response calculated from a static analysis of the system for ZPA times "missing mass" (as defined in 

Appendix I of report). The time variation for the missing mass mode is obtained by scaling this peak 

response to the input time history. Clearly, knowledge of the response spectrum is very useful when 

mode superposition time history is being performed. 

The value of the response spectrum is perhaps even more significant when a direct integration 

time history analysis is being conducted. The spectrum provides useful information for (1) the 

definition of the integration time step and (2) the development of a - 0 damping coefficients to 

simulate constant modal damping. The integration time step must be sufficiently small to capture 

responses which oscillate at all frequencies below the ZPA frequency. In addition, knowing the 

frequency range of spectral amplification can help in the selection of target frequencies to obtain a 

best fit a -1 combination for approximating constant modal damping. 

For the direct integration time history analyses of BM3, a and P3 were determined by 

specifying the target modal damping at the frequency of the fundamental mode (2.9 Hz) of the BM3 

model and at an intermediate frequency between this frequency andfzA (16.5 Hz). A frequency of 

11.5 Hz was selected, in order to achieve the best fit over the 2.9 Hz to 16.5 Hz range of interest. 

This resulted in a minimum damping equal to 80% of the target modal damping, which occurs at 

- 5.7 Hz, and a maximum damping equal to 130% of the target modal damping, which occurs at 16.5 

Hz. Sensitivity to variations in the damping value diminishes asfpA is approached; consequently, 

somewhat higher effective modal damping in the high frequency end of the amplified range of the 

spectrum has relatively negligible effect on the total dynamic solution. 

In the case of the BM3 model and loading, comparison between the direct integration solution 

and the mode superposition plus missing mass solution indicates close agreement for both 1% and 

5% damping. See Tables E-1 through E-4. The missing mass contribution to the mode superposition 

solution is significant. Without accounting for this contribution, one might conclude that direct 

integration is somewhat inaccurate when compared to mode superposition because of its inherent 

inability to represent constant modal damping. Table E-5 shows the missing mass, representing the 

residual after 31 modes (> 70 Hz). By including the missing mass contribution, excellent correlation 

was obtained between the complete mode superposition solution and the direct integration solution.

NUREG/CR-6645 E-2
Based on this comparative study, it is plausible that criticism of direct integration methods 

may be rooted in three potential sources for differences between direct integration and mode 

superposition solutions: 

1) integration time step too large in the direct integration analysis; 

2) poor definition of a - P coefficients in the direct integration analysis; 

3) failure to include the "missing mass" contribution in the mode 

superposition analysis.

E-3 E- UREG/CR-6645
Table E-1 

Comparison of Peak Support Reaction Forces of BM3 

Due to Different Approaches

Forces in lbs 

Moments in in-lbs 1 % Damping

NODE REAC. MODAL TIME MODAL(31m) TIME DIRECr TIME 

NO. TYPE (31 modes) at sec.j Plus MM.i at secI NTEGRATIONI at sec. 

1 FX 8.40 7.34 43.71 7.36 43.63 7.35 

1 Fy 10.14 7.34 4.36 4.56 3.35 4.69 

1 Fz 1.60 7.48 1.60 7.48 1.59 7.35 

1 Mx 49.98 6.68 49.88 6.68 45.14 6.68 

1 My 776.73 7.36 776.40 7.36 778.86 7.35 

1 Mz 193.46 4.54 278.42 7.36 279.M5 7X3 

4 FX 68.75 7.34 116.79 7.361 117.78 7.35 

4 Fz 19.55 7.34 20.01 7.34 21.03 7.35 

7 Fy 13.31 12.90 13.27 12.90 13.78 12.89 

11 Fy 13.31 12.72 13.31 12.72 13.74 12.73 

11 Fz 81.14 7.36 81.34 7.36 82.49 7.35 

15 FX 713.32 7.36 731.47 7.36 740.29 7.35 

17 Fy 25.61 11.00 25.60 11.00 27.45 11.01 

17 Fz 64.26 7.34 65.36 7.34 66.32 7.35 

36 Fy 46.31 4.80 46.69 4.80 47.26 4.80 

36 FZ 48.85 7.18 42.12 3.88 41.80 4.46 

38 Fx 128.79 4.48 732.18 7.36 7.36.41 7.35 

38 Fy 43.52 7.36 43.44 7.36 42.51 7.36 

38 Fz 31A0 4.48 29.95 4.48 29.49 4.48 

38 Mx 720.04 7.34 719.05 7.34 719.65 7.33 

38 My 2142.00 4.48 2084.97 4.48 2055.80 4.48 

38 Mi 3088.90 7.36 3085.87 7.36 3018.90 7.36 

23 Fx 259.22 4.48 259.59 4.48 265.71 4.47 

23 Fy 26.04 7.38 26.08 738 24.60 7.38 

31 Fx 23.24 7.34 5S.05 7.34 55.78 7.35 

31 Fy 14.21 10.56 14.17 10.56 14.20 10.56 

31 Fz 16.07 11.84 16.08 11.84 16.90 11.83 

31 Mx 1136.90 11.84 1137.30 11.84 1173.80 11.84 

31 My 608.43 7.34 612.38 7.34 620.26 7-3 

31 MB 1767.90 7.34 1773.20 7.34 1776.60 7.35

HUREG/CR-6645 E-4
Table E-2 

Comparison of Peak Pipe Resultant Moments of BM3 

Due to Different Approaches

Moments in in-lbs 1 % Damnlnin

ELEAL MODAL TIME MODAL (31r) TIME DIMECT TIME 

No. (31 modes)! at Sec.I PIus MAuL at sec INTEGRATIONI at sec. 

1 823.40 7.36 790-95 7.36 791.02 735 

2 818M 736 836.77 736 83S.40 735 

3 1412.22 7.36 1429.28 736 1441.59 7.35 

4 762.56 7.36 763.87 7.36 764.19 7.35 

5 598.91 7.36 597.98 7.36 60254 7.35 

6 2134.36 7.36 2133.18 7.36 2141LO9 7.35 

7 1876.38 7.36 1877.81 7.36 1280.06 7.35 

8 852.96 7.38 852.74 7.38 83243 4.65 

9 1415.6 7.38 1415.21 7.38 1438.67 4.66 

10 1945.32 7.36 1946.24 7.36 1967.00 7.35 

11 3826.15 7.36 3829.46 7.36 3804.39 7.35 

12 8440.47 7.36 842.48 7.36 8437.39 7.35 

13 6741.62 7.36 6741.03 7.36 6738.07 7.35 

14 13241.93 7.36 13219-91 7.36 13373.34 7.35 

15 2290.84 7.36 2293.75 7.36 2364.57 4.67 

16 1698.69 10.56 1696.61 10.56 1697.84 10.56 

17 2455A7 10.66 2455.62 10.66 2492.35 10.66 

i1 2360.13 7.36 2363.76 7.36 2349.31 735 

19 1623.90 7.36 1625.77 7.36 1623.68 7.35 

20 7539.21 7.36 7544.81 7.36 7484.45 735 

21 1595.61 7.36 1590.16 7.36 1564.96 7.35 

22 2042.84 10.56 2046.1 10.56 2036M8 10.56 

23 1800.75 10.56 1807.75 10.56 1802.73 10.56 

24 1522.11 10.O6 151831 10.56 1514.11 1O.56 

25 926.28 7.4 922.51 7.4 91S.35 734 

26 665.57 7.34 677.35 7.34 674.46 7.34 

27 399.00 7.34 390.71 7.34 38738 7.34 

28 742.39 7.34 737.42 7.34 734.60 7.34 

29 824.73 7.34 830.76 7.34 825.51 7.34 

30 1988.96 7.34 1994.56 7.34 2042.35 735 

31 7437.70 10.56 7428.79 1O.56 7377.43 10.56 

32 893145 7.34 8785.32 7.34 8703.23 7.35 

33 11421.16 7.34 10970.06 7.34 10866.98 7-34 

34 6096.62 7.36 5993.82 736 5966.82 735 

35 6633.91 7.36 6788.22 7.36 6757.74 735 

36 1942.86 7.36 1874.84 7.36 1892.65 735 

37 3538.741 736 3501.06 736 3487.86 735

E-5 NUREG/CR-6645
Table E-3 

Comparison of Peak Support Reaction Forces of BM3 

Due to Different Approaches

Forces in lbs 

Moments in in-lbs %°/ Th)nimrno

NODE REAC. MODAL TIME MODAL (31m) TIME DIRECT TIME 

NO. "TYPE (31 modes) at sec. Plus M.M.I at see INTEGRATIONI at see. 

1 Fx 7.87 7.36 44.22 7.36 44.19 7.35 

1 Fy 11.33 7.34 3.86 7.26 2.24 7.39 

1 Fz 0.97 7.50 0.97 7.50 1.02 7.49 

1 Mx 19.11 9.04 18.89 9.04 18.84 4.51 

1 MY 695.11 7.36 694.80 7.36 703.87 7.36 

1 Mz 156.85 7.38 240.22 7.36 239.67 7.37 

4 F'X 65.32 7.36 113.70 7.36 113.82 7.35 

4 Fz 13.38 7.36 13.85 7.36 14.07 7.36 

7 Fy 8.01 7.60 7.95 7.60 8.26 7.61 

11 Fy 7.13 12.74 7.13 12.74 7.54 12.73 

11 Fz 71.69 7.36 71.89 7.36 72.60 7.36 

15 Fx 672.79 7.36 690.97 7.36 692.19 7.36 

17 Fy 20.15 7.38 20.14 7.38 20.26 7.38 

17 Fz 58.91 7.36 60.05 7.36 60.27 7.36 

36 Fy 37. 7.38 38.70 7.38 39.84 7.37 

36 Fz 50.25 12.60 42.33 12.60 41.88 12.61 

38 FX 136.14 7.34 740.68 7.36 744.53 7.35 

38 Fy 38.25 7.38 3818 7.38 39.69 7.37 

38 Fz 31.43 7.18 29.85 7.18 29.61 7.19 

38 Mx 466.39 7.36 465.42 7.36 481.87 7.49 

38 MY 2143.30 7.18 2080.83 7.18 2065.30 7.19 

38 Mz 2715.60 7.38 2713.20 7.38 2820.50 7.37 

23 Fx 270.09 12.60 270.50 12.60 268.35 12.60 

23 y 10.54 7.38 10.58 7.38 10.97 6.57 

31 Fx 22.57 7.34 54.46 7.36 55.10 7.35 

31 Fy 9.62 7.38 9.76 7.38 10.16 7.50 

31 Fz 8.66 7. 8.66 7.28 9.02 7.28 

31 Aft 559. 12.60 560.28 12.60 598.88 7.29 

31 My 578.29 7.34 582.28 7.34 586.45 7.35 

31 Mz 1679.80 7.34 1685.20 7.34 1685.60 735

UPREG/CR-6645 E-6
Table E-4 

Comparison of Peak Pipe Resultant Moments of BM3 

Due to Different Approaches

Moments in in-lbs 5 % Damping

ELEM, MODAL TIME MODAL (31m) TM DIRECT TIM 

No. (31 modes) atmsec. Plus M.MAL at sec INTEGRATION at see. 

1 745.97 7.36 713.05 736 72144 7.36 

2 746.03 7.36 765.02 736 77240 7.36 

3 1306.36 736 1323.65 7.36 1326.79 7.36 

4 683.45 7.36 684A2 7-36 693.18 736 

5 529.22 736 528.66 736 53801 7.37 

6 1894.42 7.36 1893.29 736 1919g7 7.37 

7 1663.22 736 1664.64 7.36 1695.52 7.37 

8 765.19 7.36 764S3 7.36 78LLS 7.37 

9 1320.17 736 1319S6 738 1362.63 7.37 

10 1713.20 736 1714.12 736 1730.62 7.36 

11 3504.53 7.36 3507.84 7.36 3557.59 7.37 

12 766241 736 7658.43 736 7733.3 736 

13 6075.86 7.36 6075.21 7.36 6149.10 7.36 

14 12509.76 7.36 12487.74 736 12525.94 736 

15 2027.62 7U3 2030.00 7.3 2061.56 7.37 

16 1057.72 7.26 107.35 7.26 1032.69 7.26 

17 149687 7.38 1495.69 73S L523.29 7.37 

i1 2023.46 736 2027.05 736 2107.67 7.37 

19. 1368.94 7.38 13'0.50 7.38 1433.54 7.37 

20 6607.12 736 6612.74 7.36 6886.72 7.37 

21 1340.38 738 133738 7.38 1397.33 7.37 

"22 1789.34 736 1803-08 7.36 182239 735 

23 133929 736 1352.91 7.36 1368.9 736 

24 1236.93 7.36 1229A4 736 1232.76 7.36 

25 962.11 7.34 958.32 7.34 95248 7.34 

26 677.97 7.34 689.97 7.34 67,08 7.34 

27 328.88 12.6 322.88 12.6 320.59 12.6 

28 630.00 7.36 624.60 7.36 635.75 7.35 

29 731.43 7.36 737.73 7.36 749.77 735 

30 1777.89 7.34 1784.6 7.34 1798.77 7.35 

31 722L12 7.36 719438 736 7240.74 7.35 

32 8935.01 7.34 9768.03 7.34 9747.80 735 

33 11395.76 7.34 10912.71 7.34 1091338 7.35 

34 5486.76 736 5338.54 736 5519.24 7.37 

35 6167.02 7.36 634&.71 7.36 6373 7.36 

36 1M519 7,36 17554 7.36 1763.67 7W3 

37 3308.12 7.36 3263.3 7.36 3278.70 73

E-7 7NUREG/CR-6645
Table B-5 

Nodal Masses Participating In 31 Mode Combination 

(bs-sec"** m) 

Note: Negative missing mass indicates 31 mode mass exceeds total mass. 

Node No. Total Mass Participating Mass of 31 modes Missing Mass after 31 modes 

1 0.017 0.000 0.017 

2 0.026 0.000 0.026 

3 0.212 0.001 0.211 

4 0.227 0.000 0.227 

5 0.031 0.024 0.007 

6 0.165 0.166 -0.001 

7 0.165 0.168 -0.003 

8 0.092 0.093 -0.001 

9 0.267 0.270 -0.003 

10 0.275 0.277 -0.002 

11 0.182 0.183 -0.001 

12 0.174 0.174 0.000 

13 0.092 0.092 0.000 

14 0.059 0.053 0.006 

15 0.101 0.008 0.093 

16 0.059 0.055 0.004 

17 0.105 0.106 -0.001 

18 0.268 0.272 -0.004 

19 0.180 0.183 -0.003 

20 0.073 0.073 0.000 

21 0.207 0.170 0.037 

22 0.088 0.066 0.022 

23 0.043 0.038 0.005 

24 0.052 0.053 -0.001 

25 0.197 0.202 -0.005 

26 0.178 0.189 -0.011 

27 0.048 0.053 -0.005 

28 0.063 0.072 -0.009 

29 0.048 0.051 -0.003 

30 0.175 0.165 0.010 

31 0.158 0.000 0.158 

32 0.163 0.150 0.013 

33 0.276 0.280 -0.004 

34 0.780 0.881 -0.101 

35 0.687 0.090 0.597 

36 0.687 0.008 0.679 

37 1.170 0.007 1.163 

38 0.585 0.000 0.585

NUREG/CR-6645 E[-8
APPENDIX F

USE OF RESPONSE SPECTRUM GENERATION SDOF 

OSCILLATOR RESPONSES TO ESTABLISH THE THRESHOLD 

FREQUENCY FOR IN-PHASE MODAL RESPONSE

F-1 K-PnJ-EG/CR-6645
Methodology for Identification of the Lowest-Frequency SDOF Oscillator which 

Responds In-phase with the Input Time History during Response Spectrum 

Generation 

A key element in the utilization of the Response Spectrum Method to predict peak 

dynamic response due to a seismic time history input is establishing appropriate phase 

relationships between the individual peak modal responses, so that the result of modal 

combination provides a reasonable estimate of the peak dynamic response. 

During the generation of a response spectrum from a ground or in-structure time history 

record, the complete time history of each SDOF oscillator response is calculated and 

processed to identify the peak response. This peak response becomes a single point on the 

response spectrum plot. Associated with each SDOF oscillator peak response is the time 

of occurrence and direction of the peak response. This information is typically not 

retained since it is not needed to generate the response spectrum. However, valuable 

conclusions can be derived from this information by comparison to the time and direction 

of the peak acceleration in the time history record. 

The lowest SDOF oscillator frequency (f.) for which the time and direction of peak 

response coincides with the time and direction of the peak of the input time history 

represents the onset of in-phase response with the input, provided all higher-frequency 

SDOF oscillator responses exhibit the same behavior, Le., for f > f4,, all SDOF 

oscillator peak responses occur at the same time and in the same direction as the peak of 

the input time history. To further verify that in-phase response exists, a comparison of the 

crossings of the acceleration equal to zero datum between the input time history and 

SDOF oscillator time history response should be performed for SDOF oscillator 

frequencies in the vicinity of f. 

Once fl, is established by this procedure, which can be fully automated and made a part 

of the response spectrum generation algorithm, it is straightforward to specify in the 

modal combination methodology that all structural modes with frequency > f4, are to be 

combined algebraically. Application of this approach to the BM3 time history record for 

1% and 5% damping produced the following results: 

1% 41 

1% damping 9.5 HZ 16.5 HZ 

5% damping 7.5 HZ 16.5 HZ 

Algebraic combination of structural modal responses for f > f,,. is accomplished by the 

"missing mass" procedure using the ZPA. In the frequency range f, to f., individual

NUREG/CR-6645 i:-2
structural modal responses are algebraically combined with the."missing mass" 

contribution. Consequently, the response of all modes with f > f,, are combined 

algebraically. 

Numerical Results 

Numerical data for the 1% and 5% damping cases are provided: the response spectrum 

plots and tables showing the peak spectral acceleration and associated time of occurrence 

for the frequency range 0.2 to 34 Hz. In-phase response was verified by comparing the 

crossings of the acceleration equal to zero datum, as discussed above.

F-3 NUREG/CR-6645
4b, 

8.00) 

7.00 

6.00 

5.00 

4.00 

3.00 

2.00 

1.00 

0.00 

0.10 1.00 10.00 100.00 

Freq. (hz) 

DM3 Input Spedtnm (1%)
Occurence Time of Spectral Accelerations of the BM3 Input Spectrum

1% Damping

Freq (hz) Peak Accel (g) O 0cc. at sec. Freq (hz) Peak Accel (g) 0cc. at sec. 

0.20 0.06 I 6.64 4.80 1.36 7.34 

0.30 0.13 12.28 5.00 1.31 12.76 

0.40 0.13 a 8.49 5.25 1.69 12.62 

0.50 0.20 12.43 5.50 1.27 12.61 

0.60 0.35 I 10.49 5.75 1.04 12.59 

0.70 0.39 11.04 1 6.00 0.76 12.59 

0.80 0.37 10.96 6.25 0.76 7.32 

0.90 0.41 12.40 6.50 0.69 7.32 

1.00 0.76 9.2 6.75 0.70 12.60 

1.10 0.64 -42.09 7.00 0.74 7.19 

1.20 0.59 12.82 7.25 0.70 7.17 

1.30 0.91 12.27 7.50 0.67 7.34 

1.40 1.03 5.94 7.75 0.66 7.31 

1.50 1.46 10.12 8.00 0.61 7.34 

1.60 0.95 a 12.41 8.50 0.75 7.31 

1.70 0.91 4.45 9.00 0.60 I 12.59 

1.80 I .1.61 4.67 9.50 0.69 7.34 

1.90 T. 1.92 4.62 10.00 0.61 7.33 

2.00 F 1.57 8.82 10.50 0.70 7.35 

2.10 1.18 7.68 1 1.00 0.59 7.32 

2.20 2.65 5.47 11.50 0.61 7.33 

2.30 2.85 7.14 12.00 0.56 7.33 

2.40 3.26 7.68 12.50 0.59 7.34 

2.50 4.47 9.17 13.00 0.59 7.34 

2,60 4.75 12.92 13.50 -0.59 7.34 

2.70 5.29 11.31 14.00 0.58 7.34 

2.80 7.44 9.41 14.50 0.59 7.33 

2.90 4.27 8.27 15.00 0.58 7.33 

3.00 4.61. 8.26 16.00 0.55 7.33 

3.15 4.13 9.90 17.00 0.56 7.33 

3.30 3.96 12.43 18.00 0.55 7.33 

3.45 4.05 9.19 20.00 0.55 7.34 

3.60 2.44 7.52 22.00 0.55 7.34 

3.80 2.09 7.84 25.00 0.54 7.34 

4.00 2.29 7.51 28.00 0.54 7.33 

4.20 1.52. 12.25 31.00 0.54 7.34 

4.40 1.34 7.35 34.00 0.54 • 7.33 

4.60 1.37 7.35 __

1:-5 - NUREG/CR-6645
In 3.00

2.50 

2.00 

S 1.50

1.00 

0.50 

0.00 

0.10 1.00 10.00 

Freq. (hz) 

UM3 Input Spectrum (5%)

100.00
Occurence Time of Spectral Accelerations of the BM3 Input Spectrum 

5% Damping

Freg (hz) Peak Acce! (g) Occ. at sec. F Peak Accel (g) Occ. at se. 

0.20 0.05 6.57 4.80- 1.16 7.34 

0.30 0.10 12.22 5.00 1.03 7.34 

0.40 0.10 12.82 5.25 1.07 12.62 

0.50 0.14 1 12.43 5.50 0.99 . 12.61 

0.60 0.24 10.47 5.75 0.88 12.60 

0.70 0.28 10.99 6.00 0.75 7.32 

0.80 0.29 10.93 6.25 0.72 7.32 

"0.90 0.27 11.79 6.50 0.68 7.32 

1.00 0.38 9.19 6.75 0.64 7.32 

1.10 0.45 12.07 -7.00 0.63 7.18 

1.20 0.49 12.02 7.25 0.62 7.18 

1.30 0.48 I 12.26 7.50 0.62 7.33 

1.40 0.59 5.91 7.75 0.61 7.33 

1.50 0.71 4.45 8.00 0.60 7.33 

1.60 0.70 . 1 4.43 8.50 0.58 7.32 

1.70 0.83 4.43 9.00 0.54 7.35 

1.80 L 1.13 4.65 9.50 0.58 7.35 

1.90 1.27 4.62 10.00 0.59 7.35 

2.00 1.12 4.60 10.50 0.60 7.34 

2.10 1.15 4.60 11.00 0.59 7.34 

2.20 1.51 5.45 11.50 0.59 7.34 

2.30 1.69 5.42- 12.00 0.58 7.34 

240 1.66 5.39? 12.50 0.57 7.34 

2.50 2.11 9.15 13.00 0.58 7.34 

2.60 2.57 9.12 13.50 0.58 7.34 

2.70 2.83 9.28 14.00 0.58 7.34 

2.80 2.74 8.31 14.50 0.58 7.33 

2.90 2.79 8.11 15.00 0.57 7.33 

3.00 2.70 7.91 16.00 0.56 7.33 

3.15 2.28 7.89 17.00 0.55 7.33 

3.30 1.95 7.88 18.00 0.55 7.33 

3.45 1.90 7.86 20.00 0.55 7.34 

3.60 1.68 7.53 22.00 0.55 7.34 

C CA 734

2t Rf 1%Q

4.00 1.42 7.51 1 o28.00 0.54

4_20

4_40

1.33

1.31

7.36

7.35

44 I I~.I- 1.25 ___ 460 _ 125

7.351

0.54

7.33 

7.34 

7.33

FiUREG/CR-6645

4 •0

iF-7

4.2 1.33w

7..•7, za.w V.,•

,j 1.V

.•*t.uv
APPENDIX G 

EVALUATION OF LINDLEY-YOW METHOD 

by 

Professor A. Veletsos 

Rice University

G-1 NUREG/CR-6645
APPENDIX G

Fundamental to the Lindley-Yow method is the premise that the response of a system may be 

expressed as the sum of two components: 

1. An "in-phase" component, the instantaneous value of which is equal to the corresponding, 

static effect of the forcing function; and 

2. A series of "out-of-phase" components, the characteristics of which depend on the charac

teristics both of the forcing function and of the system itself. 

The maximum value of the total response is then determined by SRSS combination of the 

numerical values of the component maxima. 

For the base-excited systems examined in the report, the in-phase component at any time t is 

proportional to the ground acceleration, y(t), and the out-of-phase component for a given mode of 

vibration is proportional to the instantaneous pseudo-acceleration, A(t), of a single-degree-of

freedom system having the natural frequency of that mode. The maximum values ofy(t) and A(t) 

will henceforth be identified as y. and A., respectively. 

The maximum response obtained by this approach is clearly larger than the maximum static 

response, and in recognition of this fact, the method is said to apply only to system frequencies for 

which the amplification factor of response, AF, defined as the ratio of the maximum dynamic to 

maximum static response, is equal to or greater than unity. This factor, which is effectively the 

ratio Ao/9, is, of course, the reciprocal of the factor a used in the report. The relatively high 

natural frequency for which AF = 1 is denoted in the following by the symbols 

For system frequenciesf•_fh, the out-of-phase component of response is zero, and the in-phase 

component is indeed equal to the static response. This is true both of single-degree-of-freedom 

systems and multi-degree-of-freedom systems with fundamental natural frequencies greater than 

fh. As duly noted in the report, a static analysis of the system in this case, or an analysis that 

considers only the in-phase component of response will yield the correct solution. 

As the natural frequency of the system decreases belowfh, it is well known that the AF increases 

and, after reaching a peak, which depending on the degree of periodicity of the forcing function 

may be quite high, it decreases, returning to the unit value, and then attains even lower values. Let 

f, be the natural frequency of the relatively low-frequency system for which AF returns to unity. 

According to the Lindley-Yow method, the response history of a single-degree-of-freedom system 

with such a frequency will be identical to that of the corresponding static response. Expressed 

differently, the temporal variation of the pseudo-acceleration A(t) will be the same in this case as 

that of the input acceleration y(t). This conclusion is simply unacceptable, as precise analyses will 

reveal that the response of such a low-frequency system has no resemblance to the static response, 

but is dominated instead by oscillations of a frequency equal or close to the natural frequency of 

system.

NUREGICR-6645 G-2
APPENDIX G

Since the characteristics of the solution cannot be expected to change abruptly with a change in the 

natural frequency of the system, the Lindley-Yow method should also be inapplicable over a range 

of natural frequencies higher than but close tof. By contrast, the method should give quite 

satisfactory results over a range of natural frequencies close to but lower thanfh. Furthermore, the 

higher the amount of damping of the system, the lower will be, in the latter case, the contribution 

of the out-of-phase component of response compared to that of the in-phase component, and 

hence, the wider will be expected to be the range of applicability of the Lindley-Yow method. 

For broad-banded response spectra of the type associated with the 1940 El Centro earthquake 

record, which Lindley and Yow utilized, the frequencyf, may indeed be significantly lower than 

the fimdamental natural frequency of the secondary systems that are of interest in the present 

study. However, the base-input motion for such a system is typically dominated by oscillations of 

a frequency equal to or close to the fundamental natural frequency of the primary system; the 

associated response spectrum is sharply peaked; and the fundamental natural frequency of the 

secondary system may well be close tof, particularly when the primary system is comparatively 

stiff. It follows that the limitations of the Lindley-Yow method in the neighborhood of the low

frequency limitf, are not of academic interest only. 

I believe that these matters should be emphasized in the description of method, as I interpret the 

objective of Chapter II of the report to be to provide not merely a description of the various 

methods, but also some evaluation of them. Incidentally, Gupta's method is superior in this 

respect, and this should also be duly acknowledged. 

And now, to what I regard to be a more important point. When defined by the 

displacements [ u(t) ) of the points of mass concentration relative to the moving base, the response 

of an n-degree-of-freedom system can be expressed in the form 

n 

{u(t)} = c, (1) 

j=1 

Where { 4•j = the jth mode of the vibration of the system; cj = the participation factor for that 

mode; and AOt) = the instantaneous pseudo-acceleration of a single-degree-of-freedom system 

with a natural frequency equal to the jth natural frequency of the given system. 

If Af) in Eq. 1 is expressed as 

A(= (t (t) + AAj(t) (2) 

that is, as the sum of the ground acceleration and an incremental pseudo-acceleration, AAft), Eq. I 

can be rewritten as

G-3 NUREG/CR-6645
APPENDIX G

n 

{u(t)} = C I{ ,}[.O (t) +AAj(t)] (3) 

i-I 

or as 

u(t)} = { u8S(t) } : c 1 {4j }AAj(t) (4) 

where {us (t)) represents the instantaneous values of the static displacements, and the series 

expression defines the corresponding dynamic increments. 

In the Lindley-Yow method, the maximum value of the incremental pseudo-acceleration for a 

given mode, denoted in the following simply as A4., is given by 

AA. A.2 - Y.2) (5) 

This is not an exact definition of AAo The exact maximum is given by the peak value of the 

difference of A(t) andy(t), and generally occurs at a time different from that for which A(t) attains 

its maximum value. 

An indication of the difference between the exact value of A40 and that determined from Eq. 5 is 

provided in Table I and Fig. 1, which presents the relevant response spectra for undamped systems 

subjected to a relatively simple ground motion. The acceleration trace of the ground motion 

considered consists of a sequence of three half-sine pulses of alternating- signs and durations ti, 

2t,, and ti, as shown in the inset diagram. The velocity and displacement traces of the motion are 

full-cycle and half-cycle pulses, respectively. The results are displayed in dimensionless form, 

with the pseudo-accelerations normalized with respect to the maximum input acceleration, and the 

abscissa representing the product of the natural frequency of the system and the duration of the 

middle acceleration pulse. 

At high values of the frequency parameter 2fl1, the two sets of results in Fig. 1 both tend to a 

limiting value of zero. However, at the lower frequency values, they differ significantly to the left 

of the highest spectral peak. The very large differences in the results obtained in the low

frequency range raise serious questions about the applicability of either procedure in this range. 

Neither procedure is logical for very-low-frequency systems. While I believe that it is possible to 

modify the procedure to yield satisfactory results in this frequency range as well, this extension is

NUREGICR-6645 G-4
APPENDIX G

clearly beyond the scope of the present project. In this case, working with the deformation and 

pseudo-velocity values of the response spectra holds the secret, in my view, to improvement. 

Table 1. Pseudo-acceleration values for undamped single-degree-of-freedom systems 

obtained by different procedures. Systems excited by a half-cycle displacement 

pulse. 

(\ 2 AA. 

2fto 

0.5 2.70 3.090 

0.8 3.77 3.894 

1.0 2.85 2.038 

1.5 2.67 2.400 

2.0 2.21 2.160 

2.5 1.27 0.967 

3.0 1.12 0.500

G-5 NUREG/CR-6645
APPENDIX G

Exact AA 

Y.

5 

4ý 

.2 

M 

z 

0 

£ 0

SI 0., U l I 0.5 1

Fig. 1 Approximate and exact response spectra for incremental pseudo-acceleration of 

single-degree-of-freedom, undamped systems subjected to indicated base motion.

NUREGICR-6645

I 

I 

I 

I 

I 

I 

I

OL 

0.: 2ftl

| I I I S 10

G-6

2•, 

- !YT

I
APPENDIX H

INDEPENDENT ANALYSIS OF BM3 PIPING MODEL 

by 

Professors A.K. Gupta and Abhinav Gupta 

North Carolina State University

H-1 NUREG/CR-6645
APPENDIX H

Proposed Definition of f for Gupta's Method 

Unlike the RG 1.60 spectrum, the actual earthquake spectra do not have a sharp transition from the 

amplified to constant (ZPA) acceleration regions. To reduce the arbitrariness and in accordance 

with the Draft ASCE 4, we propose that the key frequencyf= be defined as follows: "The key 

frequencyf 2 is the minimum frequency at which spectra at various damping values converge." 

Although, there would still be some room for variable interpretation by different people, this 

definition is much more precise than the definition off, at which the spectral acceleration becomes 

equal to the ZPA. (In the Draft ASCE definition off2 andf, have not been appropriately separated, 

f, is assumed to be equal tof 2. This observation is truein the RG 1.60 spectra, but not in the 

spectrum at hand.) 

For the input motion considered in this study, the above definition givesf 2 approximately equal to 

6.0 Hz., as shown in the attached Fig. 1. We evaluated the values for rigid response coefficient, a, 

numerically for the input time history considered in this study and compared it with those 

evaluated using the definition of a given in section 2.2.3 of the report andf2 = 6.0 Hz. As shown 

in the attached Fig. 2, the proposed definition off2 gives good estimation of a. 

Independent Analysis of the BM3 Piping Model 

We modeled the piping system BM3 on PIPESTRESS and performed a time history analysis using 

14 modes and the missing mass effect. Table 1 compares the frequencies calculated by us (using 

PIPESTRESS) with those given in the report (using SAP). The two programs have different bend 

elements. Yet, the frequencies from the two sets of analyses are practically identical. Table 2 

compares the time history results for x-direction input motion calculated by us with those given in 

the report. As shown in this table, the results from two sets of analyses are close to each other 

except for the reaction FY at node 1 for which the difference is 24%. Large difference in this 

reaction force may occur due to its insignificant value for the x direction input motion considered 

and effect of different bend stiffness. 

Comparison of Gupta ff = 6.0 HzM and Lindlev-Yow Methods to 

Time History Results for Three Directions of Input Motion 

We performed response spectrum analyses using Lindley-Yow method and Gupta method withf2 

= 6.0 Hz. and using numerically calculated values of rigid response coefficient a. Tables 3 

through 6 give the results for Lindley-Yow and Gupta methods and Tables 7 through 10 give the 

corresponding results for numerically calculated values of a. 

We performed the time history and response spectrum analyses for all three directions of input 

motion. For vertical direction, the input motion was considered to be 2/3 of that in the horizontal

NUREGICR-6645 H-2
APPENDIX H

direction. Table 6 compares the time history results with the response spectrum results for 

Lindley-Yow and Gupta method withf 2 = 6.0 Hz. The response values in this table are obtained 

by combining individual responses for input motions in x, y and z directions using SRSS rule. As 

shown in this table, the mean and standard deviation of responses from the two methods of 

combining modal responses are almost identical. Further, these mean and standard deviation 

values are practically the same as those calculated using numerically calculated value of a and 

given in Table 10. 

Response tables 4-1 and 4-2 (end of Chapter 4 in the report) compare time history and response 

spectrum results for all the support reactions when input is applied in x-direction only. It is 

desirable to compare either only those reactions that get significant contribution from x-direction 

input analysis or all the reactions after combining responses from all the three (x, y and z) input 

motion analyses. 

To illustrate the above, let us consider reactions MX at node 1, FY at node 23, and FX, FY and FZ 

at node 31. Table 11 gives the values of these reactions calculated from the time history and 

response spectrum analyses for each of the three directions of input motion and also the responses 

due to the three inputs combined by SRSS rule. As seen in this table, these reactions get most of 

their response from excitation in a particular direction. For excitation in other two directions, their 

values are insignificant. The insignificant values can have large differences in the time history and 

response spectrum results. However, the results from the two analyses are close to each other for 

significant response values and for the combined response.

H-3 NUREG/CR-6645
APPENDIX H

Frequency. Hz

Figure 1: Input Response Spectra (12 = 6.0 Hz)

a

Frequency. Hz

Figure 2: Variation of Rigid Response Coefficient with Frequency

NUREG/CR-6645

IL7 

ts 

I CC 

9 

M. 

zn

H-4
APPENDIX H

Table 1: Frequencies (Hz) of BM3 Piping System 

Mode SAP PWESTRESS Percent Difference 

S P (P-S)*100/S 

1 2.9074 2.9074 0.00 

2 4.3871 4.3871 0.00 

3 5.5158 5.5158 0.00 

4 5.7041 5.7042 0.00 

5 6.97T75 6.9776 0.00 

6 7.3436 7.3434 0.00 

7 7.8796 7.8794 0.00 

8 10.302 10.3013 -0.01 

9 11.056 11.0558 0.00 

10 11.233 11.2329 0.00 

11 11.498 11.4974 -0.01 

12 12.431 12.4307 0.00 

13 13.879 13.8790 0.00 

14 16.122 16.1230 0.01

H-5 NUREG/CR-6645
APPENDIX H

Table 2: Comparison of Timb History Results from SAP and PIPESTRESS, X Direction Input 

NODE REAC.I SAP PIPESTRESS Ratio 

NO. TYPE (S) (P) P/S 

1 FX 43.71 45.18 1.03 

1 FY. 4.36 3.32 0.76 

1 FZ 1.6 1.65 1.03 

1 MX 49.88 49.42 0.99 

1 MY 776.4 783.27 1.01 

1 MZ 278.42 273.95 0.98 

4 FX 116.79 119.88 1.03 

4 FZ 20.01 21.61 1.08 

7 FY 13.27 13.75 1.04 

11 FY 13.31 13.58 1.02 

11 FZ 81.34 83.02 1.02 

15 FX 731.47 743.50 1.02 

17 FY 25.6 26.89 1.05 

17 FZ 65.36 66.74 1.02 

36 FY 46.69 48.92 1.05 

36 FZ 42.12 41.82 0.99 

38 FX 732.18 760.46 1.04 

38 FY 43.44 43.35 1.00 

38 FZ 29.95 29.93 1.00 

38 MX 719.05 727.98 1.01 

38 MY 2084.97 2086.88 1.00 

38 MZ 3085.87 3078.87 1.00 

23 FX 259.59 266.49 1.03 

23 FY 26.08 25.94 0.99 

31 FX 55.05 57.23 1.04 

31 FY 14.17 14.17 1.00 

31 FZ 16.08 16.61 1.03 

31 MX 1137.3 1157.15 1.02 

31 MY 612.38 622.11 1.02 

31 MZ 1773.2 1794.08 1.01 

Mean Value of 30 Comp. 1.01 

Standard Dev. of 30 Comp. 0.05

F = Force (lbs) SM = Moment (in-lbs)

NUREG/CR-6645 H-6
APPENDIX H

Table 3: Comparison of Time History and Response Spectrum Results, X Direction Input 

NODE REAC. Time-History Gupta Method Ratio Lindley-Yow Ratio 

NO. TYPE (TH) (GUP) GUP/TH (LY) LY/TH 

1 FX 45.18 45.43 1.01 46.21 1.02 

1 FY 3.32 3.08 0.93 3.58 1.08 

1 FZ 1.65 1.34 0.81 3.24 1.97 

1 MX 49.42 1 17.06 0.35 112.68 2.28 

1 MY 783.27 707.04 0.90 687.72 0.88 

1 MZ 273.95 241.32 0.88 229.08 0.84 

4 FX 119.88 109.30 0.91 104.60 0.87 

4 FZ 21.61 16.22 0.75 33.36 1.54 

7 FY 13.75 12.25 0.89 13.84 1.01 

11 FY 13.58 11.16 0.82 15.00 1.10 

11 FZ 83.02 71.77 0.86 69.80 0.84 

15 FX 743.50 657.80 0.88 586.10 0.79 

17 FY 26.89 25.36 0.94 35.81 1.33 

17 FZ 66.74 55.83 0.84 63.26 0.95 

36 FY 48.92 45.12 0.92 63.09 1.29 

36 FZ 41.82 45.80 1.10 54.48 1.30 

38 FX 760.46 776.20 1.02 768.90 1.01 

38 FY 43.35 44.52 1.03 47.82 1.10 

38 FZ 29.93 37.11 1.24 38.32 1.28 

38 MX 727.98 507.96 0.70 901.20 1.24 

38 MY 2086.88 2587.20 1.24 2672.40 1.28 

38 MZ 3078.87 3157.20 1.03 3384.00 1.10 

23 FX 266.49 326.10 1.22 343.30 1.29 

23 FY 25.94 10.60 0.41 55.43 2.14 

31 FX 57.23 56.06 0.98 56.71 0.99 

31 FY 14.17 14.19 1.00 17.90 1.26 

31 FZ 16.61 13.95 0.84 22.35 1.35 

31 MX 1157.15 809.88 0.70 1636.80 1.41 

31 MY 622.11 576.24 0.93 601.44 0.97 

31 MZ 1794.08 1670.40 0.93 1759.20 0.98 

Mean Value of 30 Comp. 0.90 1.22 

Standard Dev. of 30 Comp. 0.19 0.36

F - Force (Ibs) 

Gupta Method, with fi= 2 .

8 Hz and f2=6.0 Hz

M = Moment (in-lbs)

NUREG/CR-6645

I i

H-7
APPENDIX H

Table 4: Comparison of Time History and Response Spectrum Results, Y Direction Input 

NODE REAC. Time-History Gupta Method Ratio Lindley-Yow Ratio 

NO. TYPE (TH) (GUP) GUP/TH (LY) LY/TH 

1 FX 16.92 17.94 1.06 17.81 1.05 

1 FY 110.20 96.04 0.87 98.51 0.89 

1 FZ 3.99 3.43 0.86 4.25 1.07 

.1 MX 165.51 143.52 0.87 176.40 1.07 

1 MY 138.03 148.56 1.08 158.28 1.15 

1 MZ 1864.83 1497.60 0.80 1807.20 0.97 

4 FX 24.46 22.70 0.93 26.93 1.10 

4 FZ 21.55 15.96 0.74 32.19 1.49 

7 FY 127.13 150.60 1.18 138.60 1.09 

11 FY 199.14 202.70 1.02 189.50 0.95 

11 FZ 24.15 17.93 0.74 34.97 1.45 

15 FX 41.97 36.29 0.86 59.33 1.41 

17 FY 97.96 102.70 1.05 97.60 1.00 

17 FZ 13.05 7.83 0.60 16.25 1.24 

36 FY 375.52 376.10 1.00 353.70 0.94 

36 FZ 17.77 14.70 0.83 27.19 1.53 

38 FX 11.12 7.18 0.65 14.65 1.32 

38 FY 197.70 179.20 0.91 180.80 0.91 

38 FZ 3.11 2.27 0.73 5.03 1.62 

38 MX 2707.97 3094.80 1.14 2817.60 1.04 

38 MY 211.07 153.72 0.73 336.48 1.59 

38 MZ 6093.94 5581.20 0.92 5698.80 0.94 

23 FX 20.51 21.07 1.03 26.87 1.31 

23 FY 250.13 261.00 1.04 235.20 0.94 

31 FX 3.62 1.31 0.36 6.39 1.76 

31 FY 92.25 93.03 1.01 89.26 0.97 

31 FZ 16.68 1.83 0.11 32.13 1.93 

31 MX 2346.41 1747.20 0.74 2940.00 1.25 

31 MY 327.44 314.64 0.96 325.08 0.99 

31 MZ 673.64 568.44 0.84 752.52 1.12 

Mean Value of 30 Comp. 0.86 1.20 

Standard Dev. of 30 Comp. 0.22 0.27

F = Force (lbs) 

* Gupta' Method, with fj=2.8 Hz and f2=6.0 Hz 

NUREG/CR-6645 H-8

M = Moment (in-lbs)
APPENDIX H

Table 5: Comparison of Tilne History and Response Spectrum Results, Z Direction Input 

NODE REAC. Time-History Gupta Method Ratio Lindley-Yow Ratio 

NO. TYPE (TH) (GUP) GUP/TH (LY) LY/TH 

1 FX 5.46 5.70 1.04 5.23 0.96 

1 FY 16.42 16.80 1.02 14.23 0.87 

1 FZ 36.40 33.29 0.91 39.89 1.10 

I MX 710.77 721.20 1.01 627.00 0.88 

1 MY 462.39 344.40 0.74 652.44 1.41 

1 MZ 508.10 520.80 1.02 430.20 0.85 

4 FX 40.01 38.07 0.95 34.89 0.87 

4 FZ 263.24 275.30 1.05 237.30 0.90 

7 FY 28.26 29.06 1.03 32.37 1.15 

11 FY 37.30 26.19 . 0.70 46.82 1.26 

11 FZ 211.19 207.50 0.98 185.30 0.88 

15 FX 69.48 62.23 0.90 89.16 1.28 

17 FY 19.51 15.80 0.81 23.08 1.18 

17 FZ 123.97 111.70 0.90 133.80 1.08 

36 FY 29.18 30.50 1.05 30.17 1.03 

36 FZ 882.28 882.90 1.00 856.30 0.97 

38 FX 16.45 10.33 0.63 24.16 1.47 

38 FY 8.21 8.03 0.98 11.29 1.38 

38 FZ 265.01 251.70 0.95 255.20 0.96 

38 Mx 578.94 612.36 1.06 706.92 1.22 

38 MY 8476.37 7932.00 0.94 8162.40 0.96 

38 MZ 580.75 567.96 0.98 794.52 1.37 

23 FX 44.83 37.37 0.83 44.20 0.99 

23 FY 23.43 12.55 0.54 36.68 1.57 

31 FX 16.34 15.08 0.92 17.43 1.07 

31 FY 15.81 2.92 0.18 24.89 1.57 

31 FZ 150.11 156.00 1.04 146.80 0.98 

31 MX 10204.29 10142.40 0.99 10062.00 0.99 

31 MY 666.02 620.64 0.93 708.36 1.06 

31 MZ 1876.89 1759.20 0.94 19,69.20 1.05 

Mean Value of 30 Comp. 0.90 1.11 

Standard Dev. of 30 Comp. 0.18 1 0.21

F = Force (Ibs) 

* Gupta Method, with fi=2.8 Hz and f 2=6.0 Hz

M = Moment (in-lbs)

H-9 NUREG/CR-6645
APPENDIX H

Table 6: Comparison of Time History and Response Spectrum Results, Combined Responses for 

All Three Input Motions 

NODE REAC. Time-History Gupta Method Ratio Lindley-Yow Ratio 

NO. TYPE (TH) (GUP) GUP/TH (LY) LY/TH 

1 FX 48.55 49.18 1.01 49.80 1.03 

1 FY 111.47 97.55 0.88 99.60 0.89 

1 FZ 36.66 33.49 0.91 40.25 1.10 

1 MX 731.46 735.54 1.01 661.02 0.90 

1 MY 919.98 800.37 0.87 961.09 1.04 

I MZ 1952.13 1603.83 0.82 1871.77 0.96 

4 FX 128.73 117.95 0.92 113.51 0.88 

4 FZ 265.00 276.24 1.04 241.79 0.91 

7 FY 130.96 153.87 1.17 143.00 1.09 

11 FY 203.06 204.69 1.01 195.77 0.96 

11 FZ 228.20 220.29 0.97 201.07 0.88 

15 FX 747.92 661.73 0.88 595.80 0.80 

17 FY 103.44 106.96 1.03 106.49 1.03 

17 FZ 141.39 125.12 0.88 148.89 1.05 

36 FY 379.81 380.02 1.00 360.55 0.95 

36 FZ 883.45 884.21 1.00 858.46 0.97 

38 FX 760.72 776.30 1.02 769.42 1.01 

38 FY 202.57 184.82 0.91 187.36 0.92 

38 FZ 266.71 254.43 0.95 258.11 0.97 

38 MX 2863.26 3195.43 1.12 3041.51 1.06 

38 MY 8732.04 8344.69 0.96 8595.33 0.98 

38 MZ 6852.21 6437.41 0.94 6675.26 0.97 

23 FX 271.01 328.91 1.21 347.18 1.28 

23 FY 252.56 261.52 1.04 244.41 0.97 

31 FX 59.62 58.07 0.97 59.67 1.00 

31 FY 94.66 94.15 0.99 94.38 1.00 

31 FZ 151.94 156.63 1.03 151.93 1.00 

31 MX 10534.33 10323.61 0.98 10609.74 1.01 

31 MY 968.42 903.46 0.93 984.47 1.02 

31 MZ 2682.39 2491.61 0.93 2745.69 1.02 

Mean Value of 30 Comp. 0.98 0.99 

Standard Dev. of 30 Comp. 0.08 0.09

-. F = Force (Ibs) 

* Gupta Method, with ft=2.8 Hz and f2=6.0 Hz

M = Moment (in-lbs)

NUREG/CR-6645 H- 10
APPENDIX H

Table 7: Comparison of Time •Iistory and Response Spectrum Results with Numerically Calculated 

Values of Rigid Response Coefficients, a, X Direction Input 

NODE REAC. Time-History Response Spectrum Ratio 

NO. TYPE (TH) (RS) RS/TH 

1 FX 45.18 44.86 0.99 

1 FY 3.32 3.45 1.04 

1 FZ - .65 1.84 1.12 

1 MX 49.42 52.86 1.07 

I MY 783.27 707.04 0.90 

1 MZ 273.95 273.84 1.00 

4 FX 119.88 110.70 0.92 

4 FZ 21.61 21.31 0.99 

7 FY 13.75 12.71 0.92 

11 FY 13.58 11.63 0.86 

11 FZ 83.02 73.18 0.88 

15 FX 743.50 659.90 0.89 

17 FY 26.89 28.15 1.05 

17 FZ 66.74 59.28 0.89 

36 FY 48.92 48.74 1.00 

36 FZ 41.82 44.23 1.06 

38 FX 760.46 771.20 1.01 

38 FY 43.35 44.31 1.02 

38 FZ 29.93 35.63 1.19 

38 MX 727.98 639.96 0.88 

38 MY 2086.88 2486.40 1.19 

38 MZ 3078.87 3141.60 1.02 

23 FX 266.49 313.30 1.18 

23 FY 25.94 28.42 1.10 

31 FX 57.23 56.05 0.98 

31 FY 14.17 14.26 1.01 

31 FZ 16.61 15.69 0.94 

31 MX 1157.15 1065.24 0.92 

31 MY 622.11 585.60 0.94 

31 MZ 1794.08 1694.40 0.94 

Mean Value of 30 Comp. 1.00 

Standard Dev. of 30 Comp. 0.09

F = Force (lbs) M = Moment (in-lbs)

H-1f NUREG/CR-6645
APPENDIX H

Table 8: Comparison of Timl History and Response Spectrum Results with Numerically Calculated 

Values of Rigid Response Coefficients, a, Y Direction Input 

NODE REAC. Time-History Response Spectrum Ratio 

NO. TYPE (TH) (RS) RS/TH 

1 FX 16.92 15.21 0.90 

1 FY 110.20 104.60 0.95 

1 FZ 13.99 3.84 0.96 

1 MX 165.51 181.76 0.98 

1 MY 138.03 131.76 0.95 

1 MZ 1864.83 1872.00 1.00 

4 FX 24.46 22.42 0.92 

4 FZ 21.55 24.38 1.13 

7 FY 127.13 120.80 0.95 

11 FY 199.14 212.50 1.07 

11 FZ 24.15 26.60 1.10 

15 FX 41.97 41.55 0.99 

17 FY 97.96 116.20 1.19 

17 FZ 13.05 11.23 0.86 

38 FY 375.52 366.20 0.98 

36 FZ 17.77 18.73 1.05 

38 FX 11.12 9.55 0.86 

38 FY 197.70 178.50 0.90 

38 FZ 3.11 3.17 1.02 

38 MX 2707.97 2954.40 1.09 

38 MY 211.07 213.12 1.01 

38 MZ 6093.94 5530.80 0.91 

23 FX 20.51 22.15 1.08 

23- FY 250.13 266.80 1.07 

31 FX 3.62 3.70 1.02 

31 FY 92.25 91.40 0.99 

31 FZ 16.68 18.05 1.08 

31 MX 2346.41 2187.60 0.93 

31 MY 327.44 320.88 0.98 

31 MZ 673.64 629.88 0.94 

Mean Value of 30 Comp. 1.00 

Standard Dev. of 30 Comp. 0.08

F = Force (Ibs) M = Moment (in-lbs)

NUREGICR-6645 H- 12
APPENDIX H

Table 9: Comparison of Time Ifistory and Response Spectrum Results with Numerically Calculated 

Values of Rigid Response Coefficients, a, Z Direction Input 

NODE REAC. Time-History Response Spectrum Ratio 

NO. TYPE (TH) (RS) RS/TH 

1 FX 5.46 4.82 0.88 

1 FY 16.42 14.54 0.89 

1 FZ 36.40 34.94 0.96 

1 Mx 710.77 703.08 0.99 

1 MY 462.39 426.48 0.92 

1 MZ 508.10 429.24 0.84 

4 FX 40.01 36.14 "0.90 

4 FZ 263.24 265.60 1.01 

7 FY 28.26 26.79 0.95 

11 FY 37.30 37.08 0.99 

11 FZ 211.19 200.10 0.95 

15 FX 69.48 64.63 0.93 

17 FY 19.51 16.83 0.86 

17 FZ 123.97 119.40 0.96 

36 FY 29.18 31.71 1.09 

36 FZ 882.28 876.40 0.99 

38 FX 16.45 14.91 0.91 

38 FY 8.21 8.53 1.04 

38 FZ 265.01 252.50 0.95 

38 MX 578.94 642.00 1.11 

38 MY 8476.37 7981.20 0.94 

38 MZ 580.75 601.92 1.04 

23 FX 44.83 39.56 0.88 

23 FY 23A3 23.76 1.01 

31 FX 16.34 15.81 0.97 

31 FY 15.81 13.81 0.87 

31 FZ 150.11 153.70 1.02 

31 MX 10204.29 10140.00 0.99 

31 MY 666.02 648.00 0.97 

31 MZ 1876.89 1825.20 0.97 

Mean Value of 30 Comp. 0.96 

Standard Dev. of 30 Comp. 0.06

F = Force (Ibs) M = Moment (in-lbs)

NUREG/CR-6645

. 1.

H- 13
APPENDIX H

Table 10: Comparison of Timq History and Response Spectrum Results with Numerically Calcu

lated Values of Rigid Response Coefficients, a, Combined Responses for All Three Input Motions 

NODE REAC. Time-History Response Spectrum Ratio 

NO. TYPE (TH) (RS) RS/TH 

1 FX 48.55 47.61 0.98 

1 FY 111.47 105.66 0.95 

I FZ 36.66 35.20 0.96 

1 MX 731.46 723.38 0.99 

1 MY 919.98 836.15 0.91 

1 MZ 1952.13 1940.00 0.99 

4 FX 128.73 118.59 0.92 

4 FZ 265.00 267.57 1.01 

7 FY 130.96 124.39 0.95 

11 FY 203.06 216.02 1.08 

11 FZ 228.20 214.72 0.94 

15 FX 747.92 664.36 0.89 

17 FY 103.44 120.74 1.17 

17 FZ 141.39 133.78 0.95 

36 FY 379.81 370.79 0.98 

36 FZ 883.45 877.72 0.99 

38 FX 760.72 771.40 1.01 

38 FY 202.57 184.11 0.91 

38 FZ 266.71 255.02 0.96 

38 MX 2863.26 3090.34 1.08 

38 MY 8732.04 8362.25 0.96 

38 MZ 6852.21 6389.19 0.93 

23 FX 271.01 316.56 1.17 

23 FY 252.56 269.36 1.07 

31 FX 59.62 58.35 0.98 

31 FY 94.66 93.53 0.99 

31 FZ 151.94 155.55 1.02 

31 MX 10534.33 10427.84 0.99 

31 MY 968.42 930.48 0.96 

31 MZ 2682.39 2568.87 0.96 

Mean Value of 30 Comp. 0.99 

Standard Dev. of 30 Comp. 0.07

F = Force (ibs) M = Moment (in-lbs)

NUREGICR-6645 H- 14
APRENDIX H 

Table 11: Comparison of Time History and Response Spectrum Results for Some Specific Reactions 

NODE REAC. X-Direction Y-Direction Z-Direction Combined 

NO. TYPE TH RS TH RS TH RS TH RS 

I MX 49.42 17.06 165.51 143.52 710.77 721.20 731.46 735.54 

23 FY 25.94 10.60 250.13 261.00 23.43 12.55 252.56 261.52 

31 FX 57.23 56.06 3.62 1.31 16.34 15.08 59.62 58.07 

31 FY 14.17 14.19 92.25 93.03 15.81 2.92 94.66 94.15 

31 FZ 16.61 13.95 16.68 1.83 150.11 156.00 151.94 156.63

F = Force (Ibs) M = Moment (in-lbs) 

TH - Time-History 

RS - Rsponse Spectrum using Gupta Method with fi=2.8 Hz and f2--6.0 Hz

NUREG/CR-6645

t

H- 15
APPENDIX I

CALCULATION OF MISSING MASS CONTRIBUTION 

TO 

TOTAL RESPONSE

1-1 NUREGICR-6645
APPENDIX I

Mathematical descriptions of the "missing mass" contribution to total response are contained in 

report References 3, 4, and 10. Reference 3 presents a step-by-step, mechanistic approach. 

Reference 4 presents a more complete mathematical description, which provides additional 

insight Reference 10 essentially incorporates the mathematical description of Reference 4. It is 

recommended that Reference 4, Section 3.4 be reviewed to attain an understanding of the 

procedure. 

The following steps can be utilized to calculate the response contribution of all system modes of 

vibration with frequencies equal to or greater than f•. 

Each direction of earthquake input motion must be considered separately. 

Step 1: Determine the modal responses only for those modes that have natural frequencies less 

than that at which the spectral acceleration approximately returns to the ZPA (fJ). 

Step 2: For each degree of freedom (DOF) included in the dynamic analysis, determine the 

fraction of DOF mass included in the summation of all of the modes included in Step 1. 

This fraction d, for each DOF i is given by: 

N 

di =2 Cnj kfi (1) 

n=1 

where, 

n is the mode number (1, 2, ..., N) 

N is the total number of modes included in Step 1 

is the eigenvector value for mode n and DOF i 

is the direction of input motion 

c•,i is the participation factor for mode n in the jth direction: 

- {Cn}T[m]{n} (2) 

where 8

5j is the Kronecker delta, which is one if DOF i is in the direction of the 

earthquake input motion and zero if DOF i is a rotation or not in the direction of the 

earthquake input motion. This assumes that the three orthogonal directions of earthquake 

input motion are coincident with the DOF directions.

NUREG/CR-6645 1-2
APPENDIX I

Next, determine the fraction of DOF mass not included in'the summation of these modes: 

ej = di- a ij (3) 

Step 3: Higher modes can be assumed to respond in phase with the ZPA and, thus, with each 

other; hence, these modes are combined algebraically, which is equivalent to pseudo 

static response to the inertial forces from these higher modes excited at the ZPA. The 

pseudo static inertial forces associated with the summation of all higher modes for each 

DOF i are given by: 

Pi =ZPA xMi xej (4) 

where Pi is the force or moment to be applied at DOF i. 

MK is the mass or mass moment of inertia associated with DOF i. 

The structure is then statically analyzed for this set of pseudo static inertial forces applied 

to all of the degrees of freedom to determine the maximum responses associated with 

high-frequency modes not included in Step 1. 

This procedure requires the computation of individual modal responses only for lower-frequency 

modes. Thus, the more difficult higher-frequency modes need not be determined. The procedure 

ensures inclusion of all modes of the structural model and proper representation of DOF masses.

1-3 SNUREGICR-6645
NRC FORM 335 U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER 

(2-89) (Assigned by NRC, Add Vol.. Supp., Rev.. 

NRCM 1102. and Addendun Nmbers. If any.) 3201,3202 

BIBUOGRAPHIC DATA SHEET NUREGICR-6645 

(See insn,=n, on ft t,,ne) BNL-NUREG-52576 

2. TITLE AND SUBTITLE 

3. DATE REPORT PUBLISHED 

Reevaluation of Regulatory Guidance on Modal Response Combination Method for MONTH YEAR 

Seismic Response Spectrum Analysis December 1999 

4. FIN OR GRANT NUMBER 

W-6521 

AUTHOPRS) & TYPE OF REPORT 

R. Morante, Y. Wang 

7. PERIOD COVERED (Indusive Dates) 

S. PERFORMING ORGANIZATION - NAME AND ADDRESS INARC. provide Dhfsion, Oice or Region, U.S. NuclearRegulatory Commission, and maikng address W onf'actoi, provide name and imaTng adft=ss.) 

Department of Advanced Technology 

Brookhaven National Laboratory/Bldg. 130/P.O. Box 5000 

Upton, NY 11973-5000 

9. SPONSORING ORGANIZATION -NAME AND ADDRESS ('ANRC, &e Same as aboveiW'oontraclor, provide NRCAM , Offi'e orRegion, U.S. NucearRegtdatoryComatssion, and mafn addres.) 

Division of Engineering Technology 

Office of Nuclear Regulatory Research 

U.S. Nuclear Regulatory Commission 

Washington, DC 20555-0001 

10. SUPPLEMENTARY NOTES 

R.M. Kenneafly, Project Manager

11. ABSTRACT (20 wo dsortess)

Regulatory Guide 1.92, *Combining Modal Responses and Spatial Components in Seismic Response Analysis,' was last revised in 

1976. The objectives of this project were to re-evaluate the current regulatory guidance for combining modal responses in 

response spectrum analysis; evaluate recent technical developments; and recommend revisions to the regulatory guidance. In 

addition, Standard Review Plan Section 3.7.2, *Seismic System Analysis," was reviewed to Identify related sections which may need 

to be revised. The objectives were addressed through a literature review of past studies, supplemented by analysis of a piping 

system model previously utilized in NUREG/CR-5627. "Alternate Modal Combination Methods in Response Spectrum Analysis.' 

This project evaluated (1) methods for separation of the in-phase and out-of-phase modal response components; (2) methods for 

combination of the out-of-phase modal response components; (3) the contribution of 'missing mass'; and (4) the combination of the 

three elements of response to produce the total response. Numerical results from response spectrum analyses were compared to 

corresponding time history analysis results to assess the accuracy of the various combination methods tested. 

During the course of the project, several insights relating to potential improvements in the methodology for seismic analysis were 

identified and documented. These include (1) Improvements in correlation between mode superposition time history and direct 

integration time history; (2) Ose of response spectrum generation single degree of freedom oscillator responses to define the 

frequency above which modal responses are in-phase with the input time history; and (3) evaluation of the effects of potential 

differences in mass distribution used in static and dynamic analyses of a piping system.

12 KEY WORDSIDESCRIPTORS (List words orphrses eaerinlocatngtherepo) inhat alassist 13. AVA&ABILIIY STATEMENT 

Seismic effects UNLIMITED 

Regulatory Guides 14. SECURITY CLASSIFICATION 

Frequency analysis (This Page) 

Frequency measurement UNCLASSIFIED 

Systems analysis (Ms Report) 

Spectral response 

UNCLASSIFIED 

I. NUMBER OF PAGES 

16. PRICE

=,

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