NUREG/CR--5627

TI91 002468

II" ' 'lii I __ iii llll III [ ' _ ................... I IIIII'I I I ] IIII

,Alternate Modal CombinationMethods in ResponseSpectrum Analysis

-- " I ............... I II . II I II III I I Hill_ .............

Manuscript Completed: September 1990Date Published: October 1990

Prepared by :_ _ _ e -- _-P. Bezler, J. R. Curreri, Y. K. Wang, A. K. Gupta _. _ ,- = ..

u=.'_

Brookhaveni.:ationalLaboratory _5g_N _ '_ = _Upton, NY 11973 _ _' _ to

Prepared for _ _ °_ _a _ rd _"_°_Divisionof Engineering _ _ _ _ _ _ _ _ =_

U.S.washington,OfficeOfNuclearNuclear RegulatoryResearChDcNlt_egu,aorY20555Commission _ _'_!il__"_. " "__i 'i:'2_:_.____:,__"=__-_ _NRC FIN A3955 ,__ _ ._-_.__._,

lr:

" .... ,_ " '_' Irll1'' ' qll'¢_, ', " '" "'lll_"

ABSTRACT

In piping analyses using the response spectrum method Square

Root of the Sum of the Squares (SRSS) with clustering between

closely spaced modes is the combination procedure most commonly

used to combine between the modal response components. This

procedure is simple to apply and normally yields conservative

estimates of the time history results. The purpose of this study

is to investigate alternate methods to combine between the modal

response components. These methods are mathematically based to

properly account for the combination between rigid and flexiblemodal responses as well as closely spaced modes. The methods are

those advanced by Gupta, Hadjian and Lindley-Yow to address rigidresponse modes and the Double Sum Combination (DSC) method and the

Complete Quadratic Combination (CQC) method to account for closely

spaced mod-s. A direct comparison between these methods as well

as the SRSS procedure is made by using them to predict the response

of six piping systems. The results provided by each method are

compared to the corresponding time history estimates of results as

well as to each other. The degree of conservatism associated witheach method is characterized.

EXECU_LT_VE SUMMARX

The response spectrum method is most commonly used to estimatethe seismic response of piping systems. In this method a response

quantity is computed for each mode, each direction of excitationand each excitation input. The total response is then formed by

summing over all modes, inputs and directions of excitation_ Sincein this method all time phasing information between response

components is lost, no "exact" way to combine the response

components to form the total response exists.

To assure acceptable design, the Nuclear Regulatory Commission

(NRC) has defined explicit rules to perfo_n these combinations.

Specifically the NRC accepts square-root-sum-of-the-squares (SRSS)combination, corrected to account for closely spaced modes, for the

summation of modal response components and SRSS combination for thesummation of the directional components. However, although it has

procedures extant, the NRC recognizes the need to evaluate new andalternate methods and, in this study, has supported the evaluationof modal combination methods that address frequency dependent

correlation effects and methods purported to more correctly treat

closely spaced modes.

Three methods to address frequency dependent correlation

effects and three methods to treat closely spaced modes are

evaluated. For frequency effects, the methoc0s advanced by A.K.

Gupta, A.H. Hadjian and D.W. Lindley-J.R. Yow are considered. For

closely spaced modes, the Rosenblueth Double Sum (DSC) method, the

Complete Quadratic (CQC) method and the SRSS (i.e., no correction

for close spacing) are considered. Additionally, for comparison,

an estimate of response based on the NRC accepted procedure ismade.

The adequacy or impact of these alternate methods are estab-

lished by the comparison of the estimates of response developed

using these methods to time history estimates of the same response

quantities, with the time history estimates being accepted as thetrue response. Response estimates were made for six different

piping systems. Two of these systems were the RHR and AFW systemsfrom Zion Nuclear Power Plant, and for these, thirty three

different seismic excitation records yielding thirty three sets of

response estimates were developed. These data give a statistic

basis to the study. The remaining four problems ranged from simp±e

to complex, involved the consideration of only one seismicexcitation each and were used to round out the problem set

Ali evaluations were performed at BNL. In all evaluations

only uniform support excitation in the X coordinate direction anda cut off frequency approximation of i00 Hz is used. For the time

history evaluations, the modal auperposition method is used. For

' the response spectrum solutions, the modal method is used and the

v

excitation spectra are the unbroadened spectra derived from the

corresponding excitation time history records for the same levelof damping.

The study results are summarized in both tabular and figureformat. They show that all of the methods provide estimates of

response that range from small underestimates to large

overestimates of the time history response. Regarding the methodsto address frequency correlation effects, the response estimatesdeveloped using these methods provide results that exhibit smaller

mean error and dispersion than the method currently accepted byNRC. Regarding the methods to treat closely spaced modes, thestudy results were inconclusive. It is concluded that the use of

either the Gupta, Hadjian or Lindley-Yow methods to address

frequency correlation effects in conjunction with either the DSC

or CQC methods to treat closely spaced modes are desirable

alternates to the methods currently accepted by the NRC and their

use is recommended. Application of these methods will provide

estimates of response that closely approach the time history

response and never exhibit the gross conservatism typically evidentwith application of the accepted procedure.

vi

TABLE OF CONTENTS

PAGE

ABSTRACT iii• • • • • • • • • • • • • • • • • • • • o • • • _ • • • • • • • • w, • • • e, o • • • • • • • • • • • •

EXECUTIVE SUMMARY .......................................... v

i. INTRODUCTION .......................................... 1

2. DESCRIPTION OF CANDIDATE METHODS ...................... 3

2.1 Frequency_ DeDendent_______o=_rrelation Effects .......... 3

2 1 1 Gupta Method _ 4• • _ an, • • • • • • • • • • • • • • • • • @ • • o • • _, • • • •

2 1 2 Lindley-Yow Method 7• • • • B • • • • • • • • • u • • • • • • • • • • •

2. I. 3 H_adj ian Method ............................ 8

2. I. 4 M issinq Mass_IEffect-Residual Rig_Response ................................. 8

2 2 Closely-Spaced Modes 9• ..... - __0 • • • • • • • • • • o • • • • • • • • • • • • • • • • •

2.2.1 Square-Root-Sum-of-Squares (SRSSl Method° . i02.2.2 Rosenblueth Double Sum Combination (DSC)

2.2.3 C__,o__mpletev__,_ _,,,_Method .................................... 12

3 PROBLEM SET DESCRIPTION 13

STUDY DESCRIPTION 21

5. STUDY RESULTS ......................................... 27

5.1 RHR and AFW - Deciree of Exceedance Results ....... 29

5°2 BNL Problems - Degree of Exceedance Result______s...... 29

6. DISCUSSION OF RESULTS ................................. 45

7. CONCLUSIONS ........................................... 48

8 . REFERENCES ............................................ 50

APPENDIX I A-i• • ° • • • • • ° • • • • • ,s o • • o ° • • ° • • • • • • • • ° • ,t- _ ° • • • • • • o e

APPENDIX II AA-i• • e • • • • • • • • • ° • ° o • e • ° • ° • • • • • • • • • • • • • • ° ° • ° • •

vii

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LIST OF FIGURES

FIGURE

1 Typical Design Spectra ........................ 6

2 RHR Piping Model 14o_oooooo_ooooooeoooeoooOeoo,oo

3 AFW Piping Model ............................... 15

4 Z Bend Finite Element Model ................... 17

5 BMI and BM2 Piping Model ...................... 18

6 BM3 Piping Model .............................. 19

7 Input Excitation Records, RHR Problem ......... 23

8 Input Excitation Records, AFW Problem ......... 24

9 Input Excitation Records, Z Bend and BMIProblems ...... ................................ 25

i0 Input Excitation Records, BM2 and BM3 Problems 26

II RHR Problem Statistical Results ............... 39

12 AFW Problem Statistical Results ............... 40

13 Z Bend Problem Distribution of Results ........ 41

14 BM1 Problem Distribution of Results ........... 42

15 BM2 Problem Distribution of Results ........... 43

16 BM3 Problem Distribution of Results ............ 44

i

viii

LIST OF TABLES_

T__ABLES PAGE

1 Model Parameters ............................. 20

2 Representative Table of Results .............. 28

3a RHR Support Force Mean Value Degree ofExceedance Results ........................... 31

3b RHR Support Force Stan. Dev. Degree ofExceedance Results ........................... 32

3c RHR Pipe End Moment Mean Value Degree ofExceedance Results ........................... 33

3d RHR Pipe End Moment Stan. Dev. Degree ofExceedance Results ........................... 34

4a AFW Support Force Mean Value Degree ofExceedance Results ........................... 35

4b AFW Support Force Stand. Dev. Degree ofExceedance Results ......... • ................. 36

4c AFW Pipe End Moment Mean Value Degree ofExceedance Results ........................... 37

4d AFW Pipe End Moment Stand. Dev. Degree ofExceedance Results ........................... 38

ix

i. INTRODUCTION

The dynamic response of piping systems is most commonly

estimated using the response spectrum method. In this method a

response quantity is computed for each mode, each direction of

excitation and each input, if multiple inputs are considered. The

total response is then formed by summing over all modes, inputs and

directions of excitation using combination rules designed to

provide estimates of system response suitable for design qualifi-

cation. Although the United States Nuclear Regulatory Commission

(USNRC) has defined explicit rules to perform these combinations,

new and alternate combination methods, which potentially provide

better estimates of system response, have been advanced [6, 7, 8].

In 1984 the USNRC Piping Review Committee completed a

comprehensive evaluation and review of the Seismic Design Require-

ments for Nuclear Power Plant Piping. The committee's endorsements

and recommendations to improve the dynamic load design of piping

are presented in NUREG-1061 [i]_ Among the items recommended for

additional study were: an assessment of the Independent SupportMotion (ISM) method when combined with Code Case N-411 damping, an

investigation of methods which address frequency dependentcorrelation effects between modes and an evaluation of more

rigorous methods to address closely spaced modes. The assessmentof the ISM method when combined with Code Case N-411 damping was

completed by BNL and reported in NUREG/CR-5105 [2]. The investi-

gation of methods to address frequency dependent correlation

effects and to accommodate closely spaced modes are the subject of

this report.

Specifically, three methods to address frequency dependent

correlation effects and three methods to treat closely spaced modes

are evaluated in this study. For frequency effects, the methods

advanced by A.K. Gupta, A.H. Hadjian and D.W. Lindley - J R. Yow

are considered. For closely spaced modes, the Rosenblueth Double

Sum Combination (DSC) method, the Complete Quadratic Combination(CQC) method and the conventional square-root-sum-of-squares (SRSS)method are considered.

The adequacy or impact of these alternate methods are

established by the comparison of response spectrum estimates of

response developed using these methods to time history estimates

of the same response quantities. It is noted that the response

spectrum method is a design method and is not expected to give

results that are in total agreement with the time history results.Just as a design response spectrum represents an average of several

response spectra based on several ground motions that among

themselves exhibit considerable scatter, the response spectrum

method results are likely to compare with the corresponding time

history results only approximately. However, on the average, the

error between the response spectrum and the time history results

are likely to be quite small, tending to zero as the number of

1

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ground motions considered in the analysis are increased. This willbe true only when the modal combination methods used in theanalysis are rational and applied correctly.

A description of the candidate methods, the study problem setand the study approach are given in Sections 2, 3 and 4. A summarydescription of the study results are given in Section 5 with acomplete listing of all study results included as Appendix I. Thediscussion of results is provided in Section 6 with a summary ofthe study conclusions presented in Section 7. Appendix II presentsthe results of a separate investigation which support the studyconclusions.

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2. DESCRIPTION OF CANDIDATE METHODS

When the response spectrum method is used to compute inertialresponse, estimates of response quantities are developed for eachmode, each direction of excitation and each input, if multipleinputs are considered. The total response is then formed bysumming over all modal component, spatial component and excitationcomponent responses. Unfortunately, since in the response spectrummethod all time phasing infol_ation between response components islost, ne "exact" way to form the sums exists. For this reason theNuclear Regulatory Commission (NRC) has specified summing pro-cedures which it deems acceptable for the design of nuclear powerplant piping. These include square-root-sum-of-the-squares (SRSS)combination over spatial components, SRSS combination, correctedto consider the interaction of closely spaced modes, over modalcomponents and absolute combination over excitation components[3,4]. These methods are considered to provide conservativeestimates of the response.

To improve the dynamic load design of piping, the NRC PipingReview Committee recommended that investigations be undertaken toevaluate modal combination methods that properly treat frequencydependent correlation effects and the interaction between closelyspaced modes. The recommendations are consistent with therecommendations for change presented by R.P. Kennedy in NUREG/CR-1161 [5] and recognize that, realistic and technically sound rulesto address these issues have become available, the accepted SRSSmethod does not address the frequency dependent correlation issueand closely spaced modes are addressed in a simplistic and possibleoverly conservative fashion. This study was performed to examinethe recommendations.

2.1 Frequency Dependent Correlation Effects

Three methods to address frequency dependent correlationeffects have been evaluated. These are methods advanced byA.K. Gupta [6], A.H. Hadjian [7] and D.W. Lindley - J.R. Yow [8].

In each of these methods the modal responses are consideredto consist of two components, a rigid component, RR_, and a dampedperiodic component, RP_, where i is the mode number. All the rigidcomponents are considered to be perfectly correlated with the inputground motion and are thus perfectly correlated with each other_The total rigid response, RR, is then appropriately given by thealgebraic sum of all the modal components of rigid response. Thatis:

n

RR = _ RR_i=l

where n is the total number of modes.

The damped periodic components, on the other hand, are

considered to exhibit the variable degree of correlation normallyassociated with modal responses. Their sum is provided by any of

the candidate modal combination rules which properly consider theinteraction of modes as a function of frequency. That is the total

damped periodic response, RP, is given by:

n n

Rp = ( X RpjRP )j=l k=l

where Cjk are modal coupling factors, the expression for whichvaries with the combination rule. In this study the DSC, CQC and

SRSS combination rules are used to form this sum, as describedbelow.

In forming the above sums, it should be realized that in the

response spectrum method, the response in mode i (R_), and its

rigid and damped periodic components (RR_ and RP_), all have theirunique signs, and those signs are retained throughout the analysis.

The uniqueness comes from the fact that the participation factor

7_ is related to the sign of ,the modal vector (4)- Thus, the

product 7_(4_), to which all responses RL are related, has unique

signs. Some computer programs incorrectly put an absolute sign on

The three methods evaluated differ in the specific definition

of the modal components of rigid response and damped periodic

response and in the method used to sum between the total rigid bodyand total damped periodic responses to form the total response.

A description of each method, as implemented in this study, isprovided below. A comparison of the methods, with a discussion of

the implication of the differences, is provided in the position

paper [9] prepared by R.P. Kennedy for the USNRC Viping ReviewCommittee.

2.1.1 Gupta Method

In this method [6] the modal responses are considered to be

segregated into three categories according to their associated

frequencies. These categories or regimes are differentiated bythree frequencies:

fl - a frequency below which a modal response is assumed

to be wholly a damped periodic component (RR_ = 0)

f2 - a frequency above which a modal response is assumed

to be wholly a rigid component (RP_ = 0)

fr - the frequency at which the spectral acceleration of

the excitation returns to the zero period acceler-

ation, ZPA (S, = ZPA) o

Specifically:= sa=J (2 sv

f2 = (fl + 2 fr)/3

where S. _ and Sv mx are the maximum spectral acceleration andvelocity of the input. These parameters are illustrated in Figure1 which shows a typical design spectra in tripartite formate. Usingthese frequencies modal coefficients, _, are defined such that:

mL = (log (fJfl))/log(fjfl) , fl _ f_ S fs

_ = l.o, f_ > fs

The above equations for the coefficient s L are based on theaverage values from several earthquake ground motions. Theequation of _ between fl and f2 is idealized as a straight line ona semi-log chart. Just as the response spectra from several groundmotions used to obtain the design spectrum show considerablescatter, the distribution of _ and the values of fl and f2 showscatter from earthquake to earthquake. The idealized semi-logstraight line is, however, a rational simplification of a complexproblem.

When comparing results from a single time history with thosefrom the corresponding response spectrum, it would be moreappropriate to calculate the mL-values from the time historyresponses numerically. The values of _-calculated from the aboveformula will in general not agree with the numerically calculatedvalues for a single time history. However, in the present study,the _-values from the formula were used based on the values of thekey frequencies (fl, f2) estimated from the appropriate equationsabove. This procedure would give a poorer agreement between theresponse spectrum and the time history response quantities thanwould the procedure where the numerically calculated _-values areused in the response spectrum modal combination.

The rigid and periodic response components are then given by:

RR_ = R_

RP_ = R_ (i -_L2) h

where R_ is the predicted modal response corresponding to frequencyf_ The total rigid and periodic responses are then:

n

RR=_ RR_i=_.

5

=

n n

RP = _ _ Cjk RPj RP kj=l k=l

where the Cjk are the combination method specific, modal couplingcoefficients, as defined in the following sectlons, and n is thetotal number of modes Deing considered. Finally the total

response, R, is formed by combining the total rigid and periodic

responses. In this method these two components of response areconsidered to be uncorrelated. Therefore:

R = (RR 2 + RP 2)h

2. i. 2 LiIL_!_e_Y-Yow Method

This method [8] follows the Gupta method except in thedefinition of the modal coefficients. These are given by:

ol = ZPA/SaL and 0.0 _< _ _< 1.0

where ZPA is the zero period acceleration and Sa_ is the spectralacceleration for the mode i. The modal and total components of

rigid and damped periodic response are then given by:

RR_ = R_

RP L = R_ (I - _2)h

n

i=l

n n

RP = >_ _ Cjk RPj RP kj=l j=l

In this method the two components are again considered to be

uncorrelated and the total response is therefore:

R - (RR 2 + RP 2)_

In the low frequency range (f_ < 2 Hz) the value of _ given

by ZPA/S,_ continues to have a significant value that is physicallynot rational. If a structural or mechanical system has significant

modes in the low frequency range, a careful evaluation of this

method of calculating _ would be desirable. All the problems used

in the present study have fundamental frequencies that are greaterthan 2 Hz and therefore do not run into this type of difficulty.

2. i. 3 HadjiaD Method

This method [7] can also be cast in the same format as theGupta method. The definition of the modal coefficients isidentical to the definition used in the Lindley-Yow method:

_ = ZPA/S,_ and 0.0 _< _ _< 1.0

With this definition of _L t:!is method would exhibit the same typeof difficulty with problems having significant modes in the lowfrequency range as the Lindley-Yow method.

The modal and total rigid and periodic response components arerespectively:

RR_ = Rj_i

RP_ = R_ (i - _)

n

RR=_ RR Li=l

n n

RP = _ _ Cjk RPj RP kj=l k=l

The above definition of corresponding rigid and damped periodicmodal response components is based on the assumption that thesecomponents are additive or in phase• To be consistent with thisassumption, the total response is given by:

R = . IRPI

2• I. 4 _ISS_0_q/ Mass Effect-Residual Bic_id Response

_ The right hand side of the equation of motion of a multi-degree of freedom system is given by -[M]{Ub)us, in which [M] is themass matrix of the system, {Ub) is the influence vector repre-senting the static response of the system when subjected to a unit

displacement in the direction of the input motion, and ug is _heinput acceleration history. Assuming there are N-total degrees of

: freedom in the system, there will be N-normal modes {_)• Thecorresponding participation factors are 7L = {4_}T[M]{Ub) • Thefollowing identity can be easily established [6]_

N

i=l

8

ilml,,l_q_, , , rl ,,,, I,iqlI,iqlll_ r, ,1,,i, ',11 ,, ,' I_r,,

The right hand side can thus be written as:

N

i=l

When only n-modes of vibration are included, n < N, the "missing"

part of the right hand side can be written as"

N,#

- [MI {Ub- Z 7_ 4_) U, or- [M] {Ubo)U,.i=n+l

The term [MI {Ubo) has been called the "missing mass". If all the

significant terms of the missing mass vector are negligibly small

as compared 'to the total mass vector [M] {Ub), the missing masseffect would also be negligible and can be ignored. Such is the

case in tall buildings and the missing mass effect is notconsidered.

In many mechanical systems, including piping systems, the

missing mass can be significant for practical number of modes (n,

n < N) included in the analysis. Assuming that the frequency ofthe last mode included in the analysis is greater than or equal tc

the rigid frequency fr, the missing mass would only generate rigid

response. This response is called the "residual rigid response. '_

The relative displacement vector for the residual rigid response

is {Uo) = - {K) -I [M] {Ubo} U, in the time history method or simply[K]- [M] {Ubo) ZPA in the response spectrum method. If the

corresponding response is Ro the above equations for the total

rigid response (in all the three methods) should be modified to:

n

RR = RO + Z RR_i=l

Since the concepts of missing mass and the residual rigid

response are well established in the scientific literature [6, 12],and since the effect is common to the three combination methods

° (Gupta, Hadjian, Lindley-Yow) , it was not considered in the present

study. Both the time history and the response spectrum analyses

were performed without considering the residual rigid response.

However, in an actual design problemr the residual rigid response

must always be considered, and included when significant.z

2.2 Closely-S_paced Modes

Numerous methods to perform the combir_ation of peak modalresponses are available. The methods deemed acceptable tc the NRC

are described in Regulatory Guide 1.92 [4]. _hese are the Grouping

Method, the Ten Percent Method and the Double Sum Method. Tl_e

Grouping and Ten Percent Methods are typically used by industry and

: 9

=

=

these rules are essentially SRSS combination corrected withabsolute combination between modes with associated naturalfrequencies spaced within I0 percent of each other. These are

simple rules that account for the weak correlation cr couplingbetween modes that have widely different natural fre£_encies,widely spaced mc;des, and the strong correlation or coupling _etweenclosely spaced modes. The Double Sum Method is a more sophisti-cated method that is a conservative variant of the Rosenblueth

Double Sum Combination Method considered in this study.

Other methods that are available are more sophisticated andare founded in random vibration theory. These methods, althoughmore complex to apply, are considered to more correctly define thecoupling between modes. _ concise description and comparison ofavailable methods is p_e_ented in the above referenced positionpaper prepared by R.P. Kennedy for the USNRC.

In this study_ the combination between peak modal responsesis performed u_ing three alterDate combination rules. These

include SRSS combination with no correction for closely spacedmodes, the Rosenblueth Double Sum Combination rule and the CompleteQuadratic Combination rule. A brief description of each of thesemethods is provided below.

2.2.1 S__quare-Root-Sum-of-___S_quares {SRSS) Method

If the peak modal responses are assumed to be randomly phasedthey should exhibit weak correlation. In that instance, combi-nation of the peak modal response by the simple SRSS combinationrule is appropriate_ Recalling that in this study the rigid bodycomponent of the peak modal responses are being treated separately,the combination rule under discussion is used to combine only thedamped periodic components of the peak modal responses.

The generic form of the equation for the combination of thedamped periodic components is:

n n

RP = ( E E Cjk RPj RP_)%j=l k=l

where the Cjk are the modal coupling coefficients. For _imple SRSScombination

C_k = 1.0 for j=k

Cjk = 0.0 for j_k

and the combination can be expressed as:

I0

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n

RP = ( _ (RpL) 2)

j=l

In the NRC accepted Grouping and Ten i_cent Methods, thecombination rule is identical to the abeve with the following

addition being m_de to the sum:

n n

7 7j=l k=l

where the summation is over all j and k modes whose frequencies are

closely spaced. The definition of closeness is slightly differentfor the two methods but is essentially all modes within ten percent

of each other.

In this study a base line response spectrum estimate is

derived using the Reg. Guide 1.92 grouping method and used to gaugethe results of the candidate methods. It is referred to as the

Reg. Guide 1.92 solution.

2.2.2 Rosenblueth D_um Combination CDSC) Met_hg__

This method [I0] is theoretically based in random vibration

theory. As noted above, a variant of it, designated the double sum

method, is one method deemed acceptable by the NRC staff for

combining peak modal responses. For the DSC method the combinationrule is :

n n

Rp = ( 7 X cek RPk)j=l k=l

with the coupling coefficients defined as:

cjk = (1 + ((_'j -_'k)/(/_'., _"j + /_'k _))2) -_

where

_'j = _j + 2/t_j

and _j and _j are the modal frequency and the damping ratio for thej th mode respectively and t is the duration of the strong motion

segment of the seismic excitation. The coupling coefficient is

based on representing the earthquake by a finite segment (duration

ii

t) of the white noise. The value of duration must be specified for

this method. In this study uniform damping was assumed (all EL are

identical) and the duration was estimated from the Husid plot [13]for each seismic excitation record.

In the NRC accepted version of the procedure, the combinationis given by:

n

= (Z Z I Jlj=l k=l

with the coupling coefficients as defined above. As can be seen

the accepted version is identical to the Rosenblueth Method except

for the use of the absolute summation within the radical. As such,

the NRC Double Sum method can be assumed to provide responseestimates similar teo, but more conservative than the DSC method.

The NRC Double Sum method was not evaluated in this study.

2.2.3 Complete Quadr___ati___cCombination (CQC) Method

This method [Ii] is also theoretically based in randomvibration theory. It differs from the DSC method in the definitionof the coupling coefficients. That is:

n n

RP = ( X Z cek RPk)j=l k=l

and

Cjk = -

where _j and _j are the natural frequency and damping ratiorespectively for the j th mode. The coupling coefficient is based

on representing the earthquake by white noise of infinite duration.

It can be shown that when the two modes have identical dampingvalues, and t = _ is substituted in the Rosenblueth, equation thecoefficients developed in the CQC or DSC methods are identical.

Although it is not obvious from the equations, the CQC method

treats the effect of modal damping more correctly than the DSC

method. It can be numerically shown that the effect of different

damping for each mode on the coupling coefficient is significant,

that the CQC equation accounts for it, and the Rosenblueth equationdoes not. A modified Rosenblueth equation that considers the

effect of different damping values (and the finite duration) isgiven in reference 6.

12

3. PROBLEM SET DESCRIPTION

The problem set consisted of the same six piping systems used

in the BNL evaluations of the Independent Support Motion (ISM)

method. The systems are identified as; the RHR piping model, the

AFW piping model, the Z-bend piping system model, and the BMI, BM2

and BM3 piping system models. Each of these system models are

fully described in the ISM reports, NUREG/CR-3811 [14] and

NUREG/CR-5105 [2]. A brief description will be provided here.

The RHR and AFW piping systems are housed in the containmentstructure of the Zion Nuclear Power Plant in Illinois. The

containment structure consists of the containment shell and a

separate concrete internal structure supporting a four-loop PWR

nuclear steam supply system. Details pertaining to this system

were made available to BNL by LLNL who analyzed it as part of the

Zion SSMRP studies performed by Lawrence Livermore Laboratory for

NRC [15]. The Zion RHR and AFW models were also used as problemsin the LLNL studies of PVRC damping, peak broadening [16] and

response margins [17,18].

The RHR model, shown in Figure 2, consists of a 12 inch, Sch.

40 and Sch. 160, pipe line header running from a wall anchor at the

internal structure of the containment building to an anchor in the

AFT complex and an 8-inch, Sch. 40, branch line from the Refueling

Water Storage Tank (RWST) nozzle to the 12-inch pipe. The pipe

material for the entire system is stainless steel. The line is

rated at a design temperature of 400°F and a design pressure of 75

psi. One isolation valve exists at the upper end of the model,

near the containment wall. The system exhibits eighteen natural

frequencies in the range from 4 Hz to 34 Hz with closely spaced

modes appearing above 12 Hz.

The AFW model, shown in Figure 3, consists of a 16-inch, Sch.

120, main feedwater (MFW) _ine running from a steam generator

nozzle to a containment penetration, and a 3-inch, Sch. 160,

Auxiliary Feedwater (AFW) branch line running from the 16-inch MRW

line to a containment penetration. The entire line, rated for a

design temperature of 400°F and a design pressure of 200 psi, is

carbon steel. The spring supports and snubbers are shown in the

figure. This system exhibited 37 natural frequencies in the rangefrom 3 Hz to 34 Hz indicating that it had significant lower

frequency content.

As noted above, both the RHR and AFW piping systems wereevaluated in the Zion SSMRP studies. In those studies statistical

methods were used to develop a data base of seismic excitation sets

suitable for these problems. In the ISM study the response of

these problems were determined for thirty three different sets of

seismic excitation from that data set. In this evaluation,

thirtythree, lD (X direction), uniform support motion, excitation

sets were extracted form the ISM input data set. Solutions

13

INTERNAL ,Yz_ XAU_UARY-, L-TURBINE

12"@REFUELING

, STORAGETANK

8"@ IB, H

21 " LEGEND:

ANCHOR,SPRINGSUPPORT

,77ANCHOR

Figure 2- RHR Piping Model

14

_'_li i .... , , , ,,11 _ ,Li ,,,di J iIJ ,!' ,

corresponding to each of these thirtythree inputs and 2% uniform

damping were developed for each problem providing some statisticalbasis to the study results.

The Z bend is a simple, planar configuration of 4-inch, Sch.

40 steel pipe consisting of two elbows and three straight lengths,as depicted in Figure 4. This system simulates a system used in

laboratory tests to determine the seismic margins inherent in

piping systems. It is a stiff system exhibiting i0 naturalfrequencies in the range from 9 Hz to 160 Hz. For the tests the

pipe was supported from and excited by three hyd_caulic actuators

designed to drive the pipe uniformly. In this evaluation the test

was simulated by imposing uniform excitation of the three supports

in the X coordinate direction and using 2% uniform damping.

The BMI and BM2 evaluations both involve the piping model

shown in Figure 5. The piping consists of 2-inch and 6-inch, Sth.

40, carbon steel pipe, with a configuration corresponding to a line

from a cooling water system in a nuclear power plant. It exhibited

27 natural frequencies ranging from 5 Hz to 9 Hz. In BMI the

piping was considered to be part of a PWR system and the inputs tothe model were derived from a 3D analysis of a PWR containment and

internal structure. For BM2 the piping was assumed to be part ofa BWR system. The inputs for this problem were derived from a lD

stick model of a BWR structure, In this evaluation the response

of the piping system was determined considering a uniform exci-tation of all supports in the X coordinate direction that was first

consistent with the inputs for the PWR model, BMI, and then

consistent with the inputs for the BMR model, BM2. In both cases

1% uniform damping was used.

The BM3 model, Figure 6, is a portion of a piping system from

the HFBR Test Reactor at BNL° It consists of 3 inch, 4 inch and

8-inch, Sch. 40, stainless steel pipe. It exhibited 32 natural

frequencies ranging from 3 Hz to 84 Hz. The inputs for this model

were derived from a two dimensional frame model with soil springsof the reactor building internal structures. For this evaluation,the response of the piping was determined for uniform excitation

of all supports in the X coordinate direction. The excitation was

consistent with the inputs derived with the frame model and with1% uniform damping.

A summary of model parameters are presented in Table i.

16

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20

4. STUDY DESCRIPTION

In this study, the adequacy and impact of alternate methods

to combine between modal components are investigated. To assure

that the impact of these methods and the differences between them

is not obscured by other aspects of the calculational procedure,

certain simplifications to the normal r_sponse spectrum calcula-

tional mode were made. Specifically, in both the response spectrum

and time history analyses, only uniform excitation of all supports

in one coordinate direction, the X direction, was considered.

Further, in the response spectrum calculations, the input spectra

was derived directly from the time history record used in the

corresponding time history evaluation, with no broadening of that

spectra being performed. Finally, the same level Of uniform

damping was used in both the time history and response spectrumevaluations.

By contrast, in a conventional response spectrum evaluation,

broadened spectra would be defined for each coordinate direction,

with each of these spectra being an envelope of the spectra for all

supports in the system. In the conventional time history evalu-

ation each support would be excited by its individual excitation.

These excitations would be similar but not necessary identical.

As mentioned, uniform levels of damping were used in all

response spectrum and time history evaluations. For the RHR, AFW

and Z bend problems, the level of damping used was 2% of critical

damping. For the BMi, BM2 and BM3 problems, the level of damping

was 1% of critical damping.

All evaluations were perfon_ed at BNL. The time history

evaluations for the RHR and AFW problems were developed using the

SMACS computer code with all supports excited uniformly in the x

coordinate direction with a time history record that corresponded

to the time history record used to excite the first support group

in the earlier ISM studies [14]. The response spectrum evaluations

for the RHR and AFW problems and both the time history and response

spectrum evaluations for the Z bend, BMI, BM2 and BM3 problems were

all performed using the computer code PSAFE2. Again for the Z

bend, BM1, BM2 and BM3 problems the excitation corresponded to that

used to excite the first group for each problem in the ISM studies.

In all response spectrum evaluations a cut off frequency of I00 Hz

was used. For the time history evaluations, the modal super-

position method of solution was used considering the same modes

used in the response spectrum solutions. This was done to assure

consistency between the time history and response spectrumsolutions.

ii[nthe Z bend, BMI, BM2 and BM3 models, there are significant

modes beyond the i00 Hz cut-off frequency used in the present

study. For design purposes, the calculations for these models

should have included the effect of missing mass - the residual

21

rigid response (section 2.1.4). In this study it was not necessarybecause the same modes were considered in both the response

spectrum and the time history methods. However if a correction for

missing mass had been made in both the time history and responsespectrum results developed in the study, the agreement between

these solutions would have improved.

The inputs used in the evaluations are depicted in Figures 7

through i0. Figure 7 shows the support point acceleration time

history record and the corresponding response spectra for the RHR

problem. Two sets of records are shown on this figure corres-

ponding to the response spectra exhibiting the minimum and maximum

acceleration peaks. The minimum peak acceleration is 0.74G

occurring for record 19 while the maximum peak acceleration is

2.81G occurring for record 25. Similar data for the AFW problem

are shown in Figure 8. For that problem the minimum of 0.82G

occurs for record i0 and the maximum of 2.81G again occurs in

record 25. The support point acceleration time history record and

corresponding response spectra for the Z bend are shown on the left

side of Figure 9, while those for the BMI problem are shown on the

right side of Figure 9_ The input records for the BM2 and BM3

problems are shown on the left and right side, respectively, ofFigure 10.

As will be noted when the results are reviewed, response

spectrum estimates of response were developed based on the Gupta,

Hadjian and Lindley-Yow methods to address frequency effects and

the DSC, CQC and SRSS methods to combine over modes. In addition,

response spectrum estimates of response based on the use of the

Regulatory Guide i.92 grouping method, were also developed. These

were developed to permit a comparison of the new procedures to a

currently accepted procedure. In reviewing the results based onRegulatory Guide i_92, it must be remembered that the results were

developed using unbroadened spectra.

Again as will be noted when the results are reviewed, the

specific candidate procedures are not evaluated separately. That

is, there are no evaluations for the Gupta method only or the DSC

method only. Instead evaluations are made for each of the methods

to address frequency correlation effects when coupled with each of

the methods for treating closely' spaced modes. Specifically, three

response spectrum evaluations involving the Gupta method were

performed for each time history record. Gupta coupled with DSC,

Gupta coupled with CQC and Gupta coupled with SRSS. For each time

history evaluation, then, there were nine response spectrum

evaluations involving the candidate methods and an additional

response spectrum evaluation involving the Regulatory Guide

procedure. The result tables depict ten columns of response

spectrum results corresponding to these evaluations.

22

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5. STUDY RESULTS

The basic results of the study are presented as tables in

Appendix I. The tables list the maximum time history results aswell as the corresponding response spectrum results for each

candidate method and the Regulatory Guide Grouping procedure.

Separate tables are included for support force results and for pipeend moment results. As an example of these tables, the support

force results for the RHR model and earthquake No. 3 are presentedin Table 2.

The first column indicates the element number for which the

data applies while the second column lists a force code identifier

(i corresponds to a support force, 6, 12 or 18 corresponds to a

pipe end moment). The maximum value from the time history recordof the response (force or moment) due to 'the particular earthquakeis listed in the third column of the tables. Subsequent columns

summarize the response spectrum results list¢_ according to the

particular methodology used to address frequency correlationeffects, i.e., Gupta, Hadjian and Lindley-Yow. Three columns ofvalues are shown under each of these methods corresponding to the

method used to address closely spaced modes, i.e., DSC, CQC and

SRSS. A last column shows the results that were obtained using the

SRSS with clustering in accordance with Regulatory Guide 1.92. At

the end of each column of response spectrum results, two more

entries, corresponding to the mean value of the data in the columnand the standard deviation of that data, are provided.

The response spectrum results presented in the tables are not

actually the response quantities but rather the percentage by which

the respective response spectrum (RS) estimate of a response

quantity exceeded the time history (TH) estimate of that quantity.This percentage, termed the Degree of Exceedance is defined as:

Degree of Exceedance = i00 (RS-TH)/TH

If this value is equal to zero then the response spectrum and time

history estimates are the same. If the value is negative, the

response spectrum estimate is less than the time history estimate.

The data in these tables were obtained at selected nodes and

elements for each piping system. For the Z bend, BMI, BM2 and BM3

problems this included all nodes and elements. For the RHR and AFW

problems, only a selection corresponding to points where peakstresses would be expected to occur, such as at elbows reducers and

tees, were included. Since the data in these tables should be peakvalues, comparisons due to various excitations and to the resultsof different combination methods should be significant.

27

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28

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5.1 _HR and AFW - Deqree of Exceedance Result__s

For each of these problems, sixty-six tables, corresponding

to the thirty-three seismic events considered for each problem, are

included in Appendix I. These results were processed to obtain

statistical properties. Table 3a shows the mean value of the

support force Degree of Exceedance results for the RHR problem.

Table 3b lists the corresponding standard deviations of the support

force Degree of Exceedance results while Tables 3c and 3d show themean and standard deviation of the end moment Degree of Exceedance

results for this problem. Tables 4a through 4d show the same typeof information for the mean and standard deviation results for the

AFW problem. The format of these tables is identical to that for

the single seismic events, Table 2, with the column listing the

time history values being deleted and the entries being thestatistical estimates.

The mean value results for the RHR and AFW problems are also

shown graphically in Figures iI and 12 respectively. On each of

these figures the support force results are shown on the upper plotwhile the end moment results are shown on the lower plot. Each

point on each plot corresponds to a tabulated data point. All the

data corresponding to a specific candidate method is shown in a

column under a heading designating the method used to address

frequency correlation effects and above a subheading designatingthe method used to treat closely spaced modes. The last column,

with the subheading REG 1.92, depicts the estimates of results

developed using the Regulatory Guide Grouping procedure.

These plots show the total range and dispersion of the dataas a function of location since, for a candidate method, every

plotted point corresponds to a different node or element. Thedistribution of results, and in particular the occurrence of

equivalent results, is clearly depicted in the figures, with pointscorresponding to equivalent results being plotted adjacent to eachother on a horizontal line. As an example, referring to the upper

plot of Figure 12, six of the mean values of the support force

degree of exceedance parameter, for the AFW problem, were essen-

tially null for the Gupta method coupled with either the DSC, CQCor SRSS methods.

5.2 BNL Problems - Deqree of _xceedance Results

_o tables are included in Appendix I for the Z bend, BMI, BM2

and the BM3 problems. One table presents the support force Degreeof Exceedance results while the other table presents the end moment

Degree of Exceedance results for each problem. The results are

also shown graphically in Figures 13 through 16. These figures

have the same format as Figures Ii and 12 with the difference being

that in these figures the actual data for a single seismic event

are shown. Results with magnitudes approaching zero are however

not depicted in these figures. The elimination of these small

29

responses was based on the considerations that first, they do notcontrol design, and second, that they could introduce misleadinglylarge Degree of Exceedance values into the data plots. Inordin-ately large Degree of Exceedance values could occur since as the

time history estimate of a response approached zero, division byzero could produce large values even for small differences betweenthe response spectrum and time history estimates. It was felt thatthe inclusion of these large magnitude Degree of Exceedanceestimates would cause a misleading increase in the dispersion ofthe data shown on the figures.

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Z BEND PIPE END MOMENT RESPONSE

Figure 13 - Z Bend Problem Distribution of Results

41

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42

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Figure Ii - RHR Problem Statistical Results

'39

W GUPTA HAD J IAN LINDLE Y' YOWOZ

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-50 t .a . = ........ j , j . _ J t _ JD$C CQC SRSS DSC CQC 8RSS DSC CQC SRSS REG

1.92

BM2 SUPPORT FORCE RESPONSE

¢150 _ ....,' " , _' ', , , -" ," , " - _-" j

LU0 GUPTA HADJIAN LINDLEY-YOWZ<

100 ....LU

XLLJ o _ •• • • •

LI..O 50- . . • : : :LU : : : : : :LM e o • • • •

oa i! ii ;= :" :". .. ... ::: :" :. :. ;.....

Ua 0 [ ,-,,,,.:'"............,!'=.... ,:.:.i_,. ;,B. "" " iii....2.. i"................ l!_'.'.':-i,,,," ,--::.._:':"':= " " :" .,-::..d'V"....... '........_ .._::=="_='= :, :: ..

i:°" i:'":. :- :

-50 l .-_ _ t _____. ! j t --! ' J .... 'DSC CQC SRSS DSC CQC 8RSS DSC CQC SRSS REG

1.92

BM2 PIPE END MCMENT RESPONSE

Figure 15 - BM2 Problem Distribution of Results

43

..... , , ' v t ' 'i...... _ r ......... I I

_oo

100 - GUPTA HADJIAN LINDLEY-YOW -ILl

Z

aW 50 ....LU

. . :" :. :.x ...... :,. "..... :.. :.. :,, •w i _. i.Lt. : : :eo •

0 ..... ;..:. :.0 _ .......',.e t._ ;.... :.... :.... "_j _. iu _ , ,, .", , --- AA ..... ,,ou. mu ev ,og'ee ' "" - _ --' o¢

"' '- ! '" F i.,.I_ ."" •:" "'"• ;. • ;.. t.. I(9 • • " :. :. :.LU " " : " " '

• • • ® • • •

50• . . : : :• m • • • •

DSC CQC SRSS DSC CQC 8RSS DSC CQC SRSS REG

1.92BM3 SUPPORT FORCE RESPONSE

...... I ", ...... I , "'u' l' i u..... i

UJ GUPTA HADJIAN LINDLEY-YOW ..O 100Z<

el ee

nii ;8UJ. °

X 50 : : : . :UJ : : : " •• • • eo q_o • • |

_Lm .. -0 .. , • _ ..

O .... !:.. .... .:. ...........

l'.: i'.: i...... "" "" 'LU ... .. l" I"w .. ' : " :o'.::....'::".:'.:: ..

0 _ F,'." l_i:::." ............ . s. ::: ..": ::: •

LM qp e. • •

- 50 _._-L...__..L..___ I I l n I. t ........ I

DSC CQC'SRSS DSC CQC SRSS DSC CQC SRSS REG

1.92BM3 PIPE END MOMENT RESPONSE

Figure ]6 - BM3 Problem Distribution of Results

44

6. DISCUSSION OF RESULTS

The result_ of the study provide a body of data that can beused to assess the relative merit of each candidate method and the

currently accepted method. An ideal method would have small meanerror and a small standard deviation about the mean error.

With the exception of the detailed results for each earthquake

event for the RHR and AFW problems, all the pertinent study results

are depicted in Figures _ Ii through 16. Since the RHR and AFWresults are a statistical set, Figures Ii and 12 can be considered

to depict the overall trends to be expected. Figures 13 through

16, on the other hand, show case specific examples of results.

Referring first to Figures Ii and 12, and accepting the time

history estimate of response as the true response, the followingobservations can be made:

I. Each of the candidate methods can provide underestimates of

time history response. The magnitude of the underestimates

appears least for the Lindley-Yow procedure and comparable for

the Gupta and Hadjian procedures.

2. The currently accepted procedure (REG 1.92) also providesunderestimates of time history response. The magnitude of

these underestimates are comparable to those exhibited by thecandidate methods.

3. The dispersion of the mean value results for the candidatemethods is not large. Most results are clustered near the

time history response (Degree of Exceedance = 0) alid

essentially all results are bounded between Degree of

Exceedance values of ± 25%. The dispersion of results appears

least for the Lindley-Yow procedure and comparable for the

Gupta and Hadjian procedures.

4. The REG 1.92 procedure shows a greater dispersion of results: than the candidate methods. The results range from Degree of

Exceedance values of -25% to Degree of Exceedance values

exceeding 150%, with most results exhibiting positive Degreeof Exceedance values.

5. Essentially no variation of results as a function of a method

used to combine over periodic modal responses (DSC, CQC and

SRSS) are evident.

Observations 3 through 5 can be substantiated by reference to

Tables 3a, 3c, 4a and 4c. The last rows on these tables list the

average and standard deviation of the mean value results for theRHR and AFW problems° As can be seen, the means of the mean valueresults for the candidate methods are very comparable and show

values between -4 and +8. The corresponding standard deviation

45

values are of the order of i0 indicating the results are alltightly clustered. The corresponding values for the REG 1.92procedure are four to five times larger indicating a higher meanand greater dispersion for this method. Referring again to themean value listings for the candidate methods only slight differ-ences are evident between the DSC, CQC and SRSS methods.

Referring next to Figures 13 through 16, the followingobservations can be made:

I. Greater variability between the results for the candidatemethods in a given problem, and between the results for thecandidate methods between problems, is evident. On the whole,the results developed with the Lindley-Yo w method show themost consistency over the problem set.

2. All methods provide underestimates of time history response.For these single seismic event evaluations, underestimates aslarge as 75% on the Degree of Exceedance scale are evident.

3. The REG 1.92 procedure shows the greatest dispersion ofresults, with some response estimates exceeding the timehistory response by a fa(tor of four.

4. Essentially no variation of results attributable to the DSC,CQC or SRSS methods is evident.

All these observations are consistent with those noted above for

the RHR and AFW problems.

The magnitude of the error associated with the Gupta methodcan be reduced if a greater effort is given to the development ofthe modal coefficients (_). These coefficients, which quantifythe correlation between each modal response and the input excita-tion, can be computed numerically from the input time historyrecord. The formula for the coefficients given in section 2.1.1and used in tnis study provide idealized estimates of the coeffi-cients that may not give representative results in individualcases. The formula was used in this study because it was antici-pated that it would be used in most applications of the Guptamethod.

The insensitivity of the results to the method used to combineover the periodic modal responses, or the apparent lack of anyimpact of closely spaced modes, leads to either of the followingconclusions. Either, the spacing of modes has no impact onresults, or, the problem set was deficient in such a way that thiseffect was not revealed. Since a large body of evidence exists tosupport the contention that closely spaced modes impact on responseresults, the second conclusion is felt to be the case in thisstudy.

46

To support this conclusion, an evaluation of the problem set

was conducted. In particular, the frequency of occurrence of

closely spaced modes within the problem set, their modal order, the

magnitude and correspondence of their associated modal participa-

tion factors, their frequency magnitude relative to the input

forcing spectra and the correspondence between the modal participa-tion factors and the input excitation (X) direction, were investi-

gated. Characteristics that would assure large impact include

closely spaced modes at a low frequency, modes that exhibit largemodal participation factors in the X direction, and clustered

frequencies occurring in the amplified region of the input spectra.

The following observations were made:

i) For the RHR problem, the frequency of the first set of closely

spaced modes occurs at a frequency corresponding to the ZPA

or non-amplified region of the input. The associated modalparticipation factors are dominant in the Y and Z directions

respectively which are not consistent with each other or the

input excitation. A null impact should be expected.

ii) For the AFW problem, closely spaced modes occur at a low

frequency order corresponding to a frequency in the amplified

region of the spectra. Associated modal participation factors

are X and Z respectively, one corresponding to the input

direction but the other not. Small impact should be expected.

iii) For the Z bend, their are no clustered modes and consequen-

tially a null impact should be expected.

iv) For the BMI, BM2 and BM3 problems, clustering occurs for

higher order modes at frequencies corresponding to the ZPA

region of the input and with associated modal participation

factors that do not correspond with each other or the input.

A null impact should be expected.

A review of the data set shows that only for the AFW problem

was there any indication of an influence of closely spaced modes.This is consistent with the observations above and confirms the

conclusion that the problem set was deficient in characteristics

that make an investigation of alternate methods to address closelyspaced modes viable. Unfortunately, in order to make full use of

existing data, the problem set was prescribed at the outset of the

study precluding any evaluation or optimization of it for the

purpose of this study.

h

47

7. CONCLUSIONS

The Gupta, Hadjian and Lindley-Yow methods are shown toaddress frequency correlation effects rationally. All of thesemethods provide results that exhibit smaller mean error andstandard deviation than the procedure currently accepted by theNRC. Regarding alternate methods to combine mode:_ with closelyspaced frequencies the study results were inconclusive. Howeverthere is adequate information in the literature showing tnatclosely spaced modes can have a significant impact on results.Both the Rosenblueth Double Sum Method (DSC) and the CompleteQuadratic Sum Method (CQC) are considered to address the phenomenonmore correctly than the simple procedure currently accepted. Thisis clearly demonstrated in the study performed by Maison, Neuss andKasai [19], a description of which is presented in Appendix II.Therefore the use of the Gupta, Hadjian or Lindley-Yow methods toaddress frequency effects in conjunction with either the DSC or CQCmethods for modes with closely spaced frequencies should beconsidered desirable alternates to the methods currently acceptedby the NRC.

It is _mportant that any modal analysis technique, includingthe response spectrum and the time history method, account for theresidual rigid response. The residual rigid response was omittedfrom the present calculations because it did not directly affectthe comparison between the time history and the response spectrummethod. Also discussed earlier in the report, in the responsespectrum method and in the combination of modal responses, it isnecessary to retain the unique signs of modal responses (RL). Forexample, this uniqueness will be destroyed if an absolute sign isapplied to the participation factor as it is done in some computerprograms.

As noted in the Introduction, the response spectrum method isa design method and is not expected to give results that are intotal agreement with the time history results. On the average, theerrors between the response spectrum and the time history resultsare likely to be quite small, tending to zero as the number ofground motions considered in the analysis is increased. This willbe true only when the modal combination methods are rational andapplied correctly. Understanding of this concept is important whenreviewing the results from the present study. There are manyresponse spectrum calculated results that are less than (and othersthat are more than) those from the corresponding time historyresults. That does n_t make the response spectrum method unconser-vative (or conservative). For design purposes, a rationallyformulated response spectrum analysis is superior to a single timehistory analysis.

To put things in perspective, there are many other sources ofuncertainty and conservatism in the _eismic design of structures.These are, for example, a low probability design earthquake event

48

--l

C

and the corresponding definition of parameters (such as peak ground

acceleration), use of a mean plus one standard deviation spectral

shape (84o1% probability of nonexceedance), peak broadening offloor response spectra, etc. All of the above assure that the

likelihoed of exceeding the calculated seismic stresses during an

actual seismic event is extremely small. None of them assure thatthe calculated stresses will never be exceeded.

49

8. REFERENCES

"Report of the U Si. U ..... Nuclear Regulatory Commission, . .

Nuclear Regulatory Commission Piping Review Committee,"NUREG-1061, April 1985.

2. Bezler, P., Wang, Y.K., Reich, M., "Response Margins Investi-

gation of Piping Dynamic Analyses Using the Independent

Support Motion Method and PVRC Damping," NUREG/CR-5105, March1988.

3. U.S. Nuclear Regulatory Commission, "Dynamic Testing and

Analysis of Systems, Components and Equipment," StandardReview Plan, NUREG-0800, Section 3.9.2.

. "Combining Model Responses4 U.S. Nuclear Regulatory Commission,

and Spatial Components in Seismic Response Analysis,"

Regulatory Guide 1.92, Rev. _, February 1976.

. "Recommended Revisions to5 Lawrence Livermore Laboratory,

Nuclear Regulatory Commission Seismic Design Criteria,"

NUREG/CR-II61, June 1979.

6. Gupta, A.K., Response Spectrum Method, Blackwell, 1990.

7. Hadjian, A.H., "Seismic Response of Structures by the Response

Spectrum Method," Nuclear Engineering and Design, Vol. 66, No.

2, August, 1981.

8. Lindley, D.W., and Yow, J.R., "Modal Response Summation for

Seismic Qualification," Proceedings 2nd ASCE Conference on

Civil Engineering and Nuclear Power, Knoxville, Tennessee,

September 1980.

• " SMA9 Eennedy, R.P., _'Position Paper on Response Combinations,12211.02-R2-0, March 1984.

i0. Rosenblueth, E., and Elorduy, J., "Responses of Linear Systems

to Certain Transient Disturbances," Proceedings 4th World

Conference on Earthquake Engineering, Santiago, Chile, 1969.

• "A Replace-Ii Wilson, E.L., Der Kiureghian, A. and Bayo, E.P.,

ment for the SRSS Method in Seismic Analysis," Earthquake

Engineering and Structural Dynamics, Vol. 9, 1981, pp. 187-192.

• "Seismic Analysis of12 American Society of Civil Engineers,

Safety Related Nuclear Structures and Commentary on Standard

for Seismic Analysis of Safety Related Nuclear Structures,"

ASCE Standard 4-86, September 1986.

5O

mm

"An Evaluation of Scaling Methods13. Nau, J.M. and Hall, W.J.,

for Earthquake Response Spectra," Structural Research SeriesNo. 499, Department of Civil Engineering, University of

Illinois at Urbana-Champaign, Urbana, Illinois, May 1982.

14. Subudhi, M., Bezler, P., Wang, Y.K. and Alforque, R.,

"Alternate Procedures for the Seismic Analysis of Multiple

Supported Piping Systems_' NUREG/CR-3811, August 1984.

15. Bohn, M. et al., "Application of the SSMRP Methodology to the

Seismic Risk at the Zion Nuclear Power Plant," NUREG/CR-3429,

January 1984.

• "Impact of Changes in Damping and16. Chuang, T.Y., et al ,

Spectrum Peak Broadening on the Seismic Response of Piping

Systems," NUREG/CR-3526, March 1984.

17. Johnson, J.J., et al., "Response Margins of the Dynamic

Analysis of Piping Systems," NUREG/CR-3996, September 1984.

18. Johnson, J.J. and Benda, B.J., "Quantification of Margins in

Piping System Seismic Response Methodologies and Damping,"

NUREG/CR-5073, February 1988.

"The Comparative19. Maison, B.F., Neuss, C.F., and Kasai, K.,

Performance of Seismic Response Spectrum Combination Rules in" Earthquake Engineering and StructuralBuilding Analysis,

Dynamics, Vol. 11, 1983, pp. 623-647.

51

APPENDIX I

A-i

t

APPENDIX I

Deqree of Exceeda, nce _e,su!t Tables

Page

o Table Descriptors A-IeoooooooJomoooooooooeooeoaeeeleoeme

o RHR Model A-2ooooeooaooeooooomeoooooogooooeeoooooeoooooe

o AFW Model A-68oooooolooee_ot_eQ_to_oooeeoo,ooeeooooeeoeoe

Z Bend Model ' A 134O _ oooooooooooooooooooloooeoeoooeoooeoeoeee

o BMI Model A-137oo_oooooooooeeoooooeoooooeooooooooooeoeeeoo

o BM2 Model A- 140ooooeoooseoeooeomooooooaeoeo_l_ooooooooeoe

o BM3 Model A-143, IoÙoooooooooloeooeoooooooeoeoooeoooeoo.ooeoe

A-ii

=

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-_ A-4

TABLE DESCRIPTIONS

Abbreviations and Symbols Description

TH Time History Data

RS Response Spectrum Estimate ofResult

Degree of Exceedance Degree of Exceedance =IO0(RS-TH)/TH

DSC Rosenblueth Double Sum Method

for combining modal responses.

CQC Complete Quadratic CombinationMethod for Combining Modal

Responses.

SRSS Square root of Sum of SquareMethod for Combining Modal

Responses.

REG 1.92 NRC Accepted Grouping Method for

Combining Modal Responses.

Gupta Gupta Method to account forfrequency dependent correlationeffects.

Hadjian Hadjian Method to account forfrequency dependent correlationeffects.

Lindley-Yow Lindley-Yow Method to accountfor frequency dependentcorrelation effects.

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APPENDIX II

AA-±

A comparison of the Rosenblueth Double Sum Combination (DSC)

method, the Complete Quadratic Combination (CQC) or Der Kiureghiandouble sum method, SRSS and the absolute sum combination rules was

made by Maison, Neuss and Kasai [19]. They analyzed the fifteen

story steel moment resisting frame structure of the University of

California Medical Center Health Sciences East Building located inSan Francisco. Two building models were formulated. The first wasthe "regular" building in which the centers of stiffness and mass

were coincident. The second w_ an irregular building with mass

offset from the stiffness center of the building. The regular

building did not have interaction between modes with closely spacedfrequencies. Therefore, as one would expect the two double sum

methods and the SRSS rules gave comparable results, which were also

very close to the time history results for the regular building;

the absolute sum rule overestimated the response values signifi-

cantly. In the irregular building, the modes in the two orthogonal

directions became coupled leading to interacting modes with closelyspaced frequencies. Three ground motions were used: San Fernando

(Pacoima Dam, SOOE, 1971), Imperial Valley (El Centro, SOOE, 1940),and Sar, Fernando (Orion Blvd., NOOW, 1971) . The double sum

calculations were performed using the modal correlation coefficient

from the Rosenblueth-Elorduy equation (DSC), and from the Der

_j Kiureghian equation (CQC). In the former, the effective durations was taken to be i0 seconds.

A statistical summary of errors is given in Table i. Theearthquake motion was applied in the east and west direction The

-_ response in the north-south direction, and the rotational torquei response was generated due to the eccentricity between the mass and

the stiffness centers. The parallel east-west responsive values

_ form the two double sum calculations are comparable; the SRSSvalues have relatively higher errors; the errors from the absolute

' sum calculations are the highest. Similar conclusions can be made

- about the torsional responsep except that the absolute sum values

=_- now have much higher errors. All the combination rules have the

highest errors in the orthogonal north-south response. The double

sum method using the Rosenblueth-Elorduy modal correlationcoefficient gives the best resu].ts. The results from the SRSS and

the absolute sum combination rules are literally unacceptable. Theorthogonal north-south response values foz_ the San Fernando-

: Pacoina Dam excitation are shown in Figures i, 2 and 3. The orderof accuracy between different combination rules observed from the

figure is the same as that concluded on the basis of Table 1

_- In this example, the response results developed in the time

history and the two double s'_m response spectrum solutions are less

than the results developed in the JRSS response spectrum solution.

This occurred because in reality the significant modes have

opposite signs in the response computation. For another problem,the opposite could be the case when the significant modes have the

same sign in the response computation.4

AA-I

Table I. Error in Response Spectrum Results with Respect to timeHistory Results.

__. ........... ,,, . lllll ii .................. ,ll iii i i,i ,,i,lb

% Error in Resultsi i i,|l i i i i _ _ iii,. --.._-_,__

Double Sum SRSS Abs. S_m

Response Description Rosenblueth- Der

Quantity Elord_Ysc) Kiureghian............... (COC) ._ .............. . .

Paral!,,,91 ,IE-W) Response

Deflection Average Error 7 6 18 27Maximum Error 19 17 26 67

Shear Average Error 8 8 22 49Maximum Error 20 ig 35 122

Overturning Average Error 6 7 25 39Moment Maximum Error 18 18 34 120

:: " Ji ii : :--__ .... - m, .i ii iiii ii

Orthogonal (N-S)I_esponse

Deflection Average Error 18 32 251 491Maximum Error 33 67 350 800

bi,_ar Average Error 17 24 217 528Maximum Error 31 55 307 661

Overturning Average Error 16 25 218 520Moment Maximum Error 25 51 299 658

i ...... i i i i,1., i lull I " :7

Torsional Response

Torques Average Error 9 7 13 137Maximum Error 27 26 40 288

Story Deflection (inch)

Figure 1 Comparison of Modal Combination Rules

Story Deflections

Ai-3

'.,llr_,r_',,,,,, 'lll"'',illrill' '_lllIl'l'lll'rll lprrl,,,,,,i,,i,,,,,. 111'iiiil,, i!!,i,,IIIGI,,II.ii, ,,_,,111[1,,,_i,..,G,I' , ,li,l,,,ilNI' iii ,r,nll'll,,",, ,ii-,i;illill_',t_,,, ,r , %1' 'TIIm,'l_' ,'" '_'rll!lrlFl"'rl!_tl' ' I'?,r,,rllqlli'"'_,"'_irl'll!II"_"" "iii ..... 1,_' rl Illlll'IIl" ",Iii' "Ill IIII '_' .... ,_' 'i',' ,l,i'H

Story Shear (kips)

Figure 2 Comparison of Modal Combination Rules

Story Shears

AA-4

Story Overturning Moment (x 10 e kip-inch)

Figure 3 COmparison of Modal Combination Rules

Story Overturning Moments

AA-5

IHII IIIIII II IIIIII iiiiii I I I III

NRC _ORM 335 U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER(2.89J (A=llgn_l by NRC, Add Vol., Supp., Rev,,NRCM 1102, lind Addendum Numborl, ii in¥.)

no1,_2o2 BIBLIOGRAPHIC DATA SHEETrsee,n,,,,ctto,_s0,,,hereverse) NUREG /CR- 5627

............ ..... _NL-NUREG- 522572. TITLE AND SUBTITLE

Alternate Modal Combination Methods in Response 3, DATEREPORTPUBLISHED

Spectrum Analysis _,ONT. I _E,nOctober 1990

4. FINOR GPANT NUMBER

A-39555. AUTHOR(S, 6. TYPE OF REPORT "'

P. Bezler, J.R. Curreri, Y.K. War,g, A.K. Gupta Technical Report

7. PERIOD COVERED H,,cl.s,veDJtL,,I

October 1, 1988 -September 30, 1990

8. PER FORMING ORGAN IZAI ION .- NAME AND ADDR ESS (li NRC. p,ov,de D,v,sion. Oft,ce or Region, U.S Nuclear Regulatory Commission, and mallmg addre==,"it contractor, proviclename snd mailing addteJs./

Brookhaven National Laboratory

Upton, NY I1973

, i

9. SPONSOR ING ORGANIZATION -- NAME AND ADDR ESS (lt NRC. type "Same a=aborts". ,t contractor, p,ovide NRC D,v,s,on. O/lice or Region, U.,_. Nuclear Regulator), Commission,and tr,..1: ..... _,_,o,, '

Division of EngineeringOffice of Nuclear Regulatory Research

U.S. Nuclear Regulatory Commission_ DC 20555, Washi nat(_n : , , ,

10. SUPPLEMENTARY NOTES

11. ABSTRACT (200 wo,a_ or It,is_

This technical report was prepared to document the results of an

investigation of alternate methods to combine between modal response

components when using the response spectrum method of analysis. These

methods are mathematically based to properly account for the combinationbetween rigid and flexible modal responses as well as closely spaced

modes. The methods are those advanced by Gupta, Hadjian and Lindley-

Yow to address rigid response modes and the Double Sum Combination (DSC)method and the C_mplete Quadratic Combination (CQC) method to accountfor closely spaced modes. A direct comparison between these methods as

well as the SRSS procedure is made by using them to predict the responseof six piping systems. The results provided by each method are

compared to the corresponding time history estimates of results as wellas to each other. The degree of conservatism associated with each_ethod is characterized.

12. K E Y WO RDS/DESC R '.PTOR S fL,st words or phrases that will a_sist re*earcher; in locating the report. ) 13. AVAI LABI I.ITY STATEMENT

Unlimited14. SECURITY CLASSIFICATION

Nuclear Power Plants -- Pipes; Pipes -- Seismic Effects; (This P.tge)

Mathematical Models; Reactor Safety; UnclassifiedPipes -- Response Functions; (T_i,.0po,.Reactor Components -- Seismic Effects; UnclassifiedFrequency Analysis ; _5NUMBEROFPAGES

16. PRICE

"' i iiii i i i '1 ii.u -- -

NRC FORM 335 !2-891