CDNJ04820ENC_001

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Proceedings

of the

First In te rn at ion al Con ference

on

S T R U C T U R A L

M E C H A N I C S

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Conference Organ isa t ion by :

Commission of the European Communities, Brussels

Bundesanstalt für Materialprüfung (BAM), Berl in, Germany

in cooperation with:

The Franklin Institute Research Laboratories (FIRL), Philadelphia, Pa.

Inst i tut für Kerntechnik (IKT), Technische Universität Berl in

Instytut Podstawowych Problemów Techniki ( IPPT),

Polska Akademia Nauk. Warsaw

Inst i tut für Stat ik und Dynamik (ISD), Technische Universität Stuttgart

Nuklear-Ingenieur-Service GmbH (NIS), Hanau

Nuclear Uti l i t ies Service (NUS), Rockvi l le, Md.

Ingenieurunternehmen für speziel le Stat ik, Dynamik und Konstrukt ion (SDK),

Lörrach

Sponsoring Societ ies :

American Concrete Inst i tute (ACI)

The American Society of Mechanical Engineers (ASME),

Nuclear Engineering Division

The American Nuclear Society (ANS)

Atomic Energy Society of Japan (AESJ)

The British Nuclear Energy Society (BNES)

Kerntechnische Gesellschaft (KTG),

incorporated in the Deutsches Atomforum e.V.

Schweizerische Vereinigung für Atomenergie (SVA)

Vereinigung der Grosskesselbetreiber e.V. (VGB)

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Proceedings

of the

First International Conference on

S T R U C T U R A L

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PREFACE

The purpose of the First International Conference on STRUCTURAL

MECHANICS IN REACTOR TECHNOLOGY was to bring together

engineers and scientists who are actively engaged in solving structural

mechanics problems in the field of reactor techno logy and fundam entalists

in the general field of engineering mechanics to present and discuss

applied and fundamental papers on structural mechanics problems

in reactor technology.

The meeting of more than 800 reactor technologists and engineering

mechanicians from 33 countries all over the world has brought together

a wealth of information and inspiration for the benefit of both reactor

technology and structural mechanics science.

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CONTENTS

Pages

Session K 1 Survey Lectures : Earthqua ke R esponse Analysis

and Aseism ic D esign

Chairmen :

R.J. SCAVUZZO,

Departmen t of Mech anical En gineering, Rensselaer Polytechnic

Institute of Connecticut, Hartford G raduate Center East Windsor

Hill Connecticut, U.S.A.

H. SHIBATA,

Institute of Industrial Science, University of Tokyo, Tokyo, Japan

K 1/1 * Earthqu ake Response Analysis of Rea ctor Str uc ture s

N.M.

NEWMARK,

Departmen t of Civil Engineering, U niversity of Il linois, Urbana,

Illinois, U.S.A.

Discussion

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VI

Pages

K 1 /4 * Problèm es de séismes : techn iques ut i lisées pour les

réacteurs nucléaires en France 11

D. COSTES et al..

Départeme nt des Etudes de Piles, C .E.A., Centre d Etudes

Nucléaires de Saclay, Gif-sur-Yvette, France

Discussion 12

Session K 2 Aseism ic Design of Nuclear Pow er Plant Stru ctur es

Chairmen :

J . M . BIGGS,

Department of Civil Engineering, Massachusetts Institute of

Technology, Cambridge, Massachusetts, U.S.A.

N.M. NEWMARK,

Departmen t of Civil Engineering, U niversity of Il linois, Urbana,

Illinois, U.S.A.

K 2 /1

*

The Earthqu ake Response Analysis for a BW R Nuclear

Po w er Plant Using Reco rded Da ta 15

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VII

Pages

K 2/5 Aseism ic Design of Nu clear Rea ctor Bui ld ing - Stress

Analysis and St i f fness Evaluat ion of the Ent ire Bui ld ing

by the F in i te E lement M eth od

93

Y. TSUSHIMA, Y. HAYAMIZU, K. NISHIYAMA,

Takenaka Kom uten Co. Ltd., Technical Research L aboratory,

Tokyo, Japan

Discussion

108

K 2/6 Aseismic Design for Japan Exp erime ntal Fast Rea ctor

(Joyo) 109

K. AKINO, M. KATO,

The Japan Atomic Power Company, Tokyo, Japan

Discussion 122

K 2 /7 Berechnung der Erdbebensch wingungen von S t ruk turen

m i t d e r F in i t e - E le m e n t - M e t h o d e - M e c h a n is c h e M o d e l l e

von Kernkra f twerken mi t E inbauten

123

K. MARGUERRE, M. SCHALK, H. WÖLFEL,

Institut für Mecha nik, Te chnische Hochschule Darmstadt,

Darmstadt, Germany

Discussion 139

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VIII

Pages

K 3 /4 Ana lys is o f So i l -S t ru c tu re In te rac t ion E f fec ts under

Se ismic Exc i ta t ion

195

C.J. COSTANTINO,

School of Engineering, The City College of the City University

of New

York

New

York

U.S.A.

K 3 /5 So i l -Fo unda t ion In terac t ion o f Reactor S t ruc ture s Sub jec t

to Seism ic Exc itat ion 211

T.H. LEE, D.A. WESLEY,

Gulf General Atomic, San Diego, California, U.S.A.

Discussion 232

K 3 /6 Dyna mic Ca lcu la t ions Us ing a F ram ew ork Ana log y to

Pred ict th e Seism ic Response of a Nuc lear Reactor 235

D.A. JOBSON,

United Kingdom Atomic Energy Authority, Reactor Group, Risley,

Warrington, United Kingdom

Discussion 256

K 3 /7 Param etr ic Ana lys is o f So i l -S t ruc ture In te rac t ion fo r a

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IX

Pages

K 4 /3 Paper deleted

K 4 /4 Ase ismic Des ign o f Asy mm etr ic S t ruc tu res and the

Equipment Conta ined

299

CH. CHEN,

Gilbert Associates Inc., Reading, Pennsylvania, U.S.A.

Discussion 317

K 4 /5 Paper deleted

K 4 /6 Dynam ic Ana lys is o f V i ta l P ip ing Systems Sub jec ted to

S e is m ic M o t io n 319

CH. CHEN,


Discussion 328

K 4/ 7 Seism ic Response Spe ctra for Equ ipm ent Design in

Nuc lear Power P lan ts

329

J . M . BIGGS,

Department of Civil Engineering, Massachusetts Institute of

Technology, Cambridge, Massachusetts, U.S.A.

Discussion 343

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Topical Grouping of the Proceedings

of the First Internat ional Conference on

Structura l Mecanics in Reactor Technology

V o l . 1 . S U R V E Y O F T H E C O N F E R E N C E : R E A CT O R T E C H N O L O G Y

Prefaces : Opening Address

Topical Scope of the Conference

Part A. Ge neral Lectures

On the Dissemination of Scient i f ic Information

Power Reactor Development Strategies

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XII

V o l .

3 . R E A CT O R C O M P O N E N T S

Part E. Shock and Vib rat i on Analysis of Reac tor Co m pon ents

E 1 Therm al Sho ck, Pressure Pulse, and Impac t Response

Analysis

E 2 Dyna mics of Fast Reactor Excursion and Co ntainm ent

E 3 Fuel Rod Vibra tions in Parallel Flow

E 4 Reactor Com ponen t Vibrat ions

Part F . Str uc tur al Analysis of, Core Su pp ort and Co olant

Ci rcu i t S t ruc tures

F 1 Structura l Ana lysis of Reactor Core Sup port Structures

F 2 Structural Analysis of Miscel laneous Reactor Comp onents

F 3 Structural Analysis of Coolant Circuit Com ponents — I

F 4 Structural Analysis of Coolant Circuit Com ponents — II

F 5 Structural Analysis of Coolant Circuit Com ponents —

III

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XIII

V o l .

6 . R E A CT O R P L A N T S T R U C T U R E S A N D C O N T A I N M E N T

Part J . Ana lysis o f She l l S t ructu res; Co nta inm ent

J 1 Analysis of Thin -Sh ell Structures — I

J 2 Analysis of Thin -Sh ell Structures — II

J 3 Con tainm ent of Powe r Reactor Plants — I

J 4 Co ntainm ent of Pow er Reactor Plants — II

Part K. Seismic Response Analysis of Nuclea r Po we r Plant

Systems

Κ 1 * Survey Lectures: Earthquake Response Ana lysis and

Aseismic Design

Κ 2 Aseism ic Design of Nuclear Pow er Plant Structures

Κ 3 Seismic Load ing and Interac tion Effects

Κ 4 Aseism ic Design of Nuclear Powe r Plant Piping and

Equipment

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K

1/1*

EARTHQUAKE RESPONSE ANALYSIS OF REACTOR STRUCTURES

N . M .

N E W M A R K ,

Depa rtment of Civil En gineering,

University of Illinois, Urban a, Illinois, U.S.A.

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DISCUSSION

Q

R . S C H N E I D E R , G e r m a n y

Y o ur r e s e a r c h is m a i n l y b a s e d on a c c e l e r a t i o n r e c o r d by E l - C e n t r o - E a r t h q u a k e

(ma g n i tu d e M Ä 7 .0 ) . In C e n t r a l E u ro p e t h e m a x im u m ma g n i tu d e i s a b o u t M = 6 -6 .5 . W h a t

a b o u t t h e s i m i l a r i t y b e t w e e n g r o u n d an d d e s i g n s p e c t r a b e t w e e n a M s: 6 . 0 - e a r t h q u a k e a n d

th e 7.0 ma g n i tu d e e a r th q u a k e ? C a n o n e u s e th e d e s ig n s p e c t r a c i t e d in y o u r l e c tu r e a l s o for

C e n t r a l E u r o p e m a x i m u m e a r t h q u a k e s j u s t by m u l t i p l y i n g th e s p e c t r a by d i m i n i s h i n g f a c t o r s

N. M. NEW MA RK, U. S. A.

F i r s t , my r e s e a r c h i s n ot b a s e d ma in ly o n th e E l -C e n t ro 1 94 0 r e c o rd , b ut o n t h e

r e c o rd s o f ma n y o th e r e a r th q u a k e s a n d of m o t io n s d u e t o b l a s t i n g a n d imp a c t a s w e l l . T h e

g e n e r a l c o n c l u s i o n s a r e a p p l i c a b l e , r e g a r d l e s s of e a r t h q u a k e m a g n i t u d e . I w o u ld r e c o m m e n d

tha t a g round ve loc i ty o f

1

5 c m /s e c o r ó i n . / s e c b e u s e d a s a m in im u m d e s ig n v a lu e , a n d

th i s c o r r e s p o n d s t o a b o u t 0. 1 g a c c e l e r a t i o n . T h e u s e of a n y th in g l e s s w o u ld in my o p in io n

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- 3 -

Q

H. SATO, Japan

W e h a v e n ot b e e n s u c c e s s f u l t o k no w a c c e l e r a t i o n a nd d i s p l a c e m e n t s i m u l t a n e o u s

ly b y i n s t ru m e n t a t i o n . D i s p l a c e m e n t m o t io n mig h t b e l a rg e r t h a n th a t w e k n ow a t t h e m o m e n t

w h e n i t c o n t a in s l o n g e r p e r io d b ey o n d th e c a p a b i l i t y of t h e a c c e l e ro m e t e r . W o u ld y o u p l e a s e

m a k e an y c o m m e n t o n t h e a c c u r a c y of t h e p r o p o s e d m a x i m u m v a l u e of d i s p l a c e m e n t ?

. N. M. NEW MA RK, U. S. A.

A

In g e n e r a l l a r g e d i s p l a c e m e n t s a r e n ot c o n t r o l l i n g f a c t o r s i n t h e d e s i g n of n u c l e a r

p o w e r p l a n t f a c i l i t i e s a n d c o mp o n e n t s e x c e p t p o s s ib ly fo r v e ry l o n g p e r io d e l e me n t s , w i th

p e r io d s l o n g e r t h a n 5 s e c o n d s . H o w e v e r , e v e n fo r t h e s e , o n e c a n u s e t h e m a x im u m g ro u n d

v e lo c i ty a s a m e a s u re t o o b t a in a c o n s e rv a t iv e e s t i m a t e of t h e r e s p o n s e . In t h e c a s e of f a u l t

mo t io n s a t t h e s u r f a c e n e a r t h e e p i c e n t e r , v a lu e s h a v e b e e n r e c o r d e d a s mu c h a s 20 ft f o r

v e r y l a r g e e a r t h q u a k e s , b u t t h e g r o u n d v e l o c i t y a s s o c i a t e d w i t h t h e m i s v e r y m u c h l e s s t h a n

th e m a x i m u m g ro u n d v e lo c i ty . H e n c e it i s n o t a s e r io u s m a t t e r t h a t w e d o n o t h a v e go o d r e c

o r d s o f m a x i m u m t r a n s i e n t g r o u n d d i s p l a c e m e n t i f w e h a v e r e c o r d s o f m a x i m u m g r o u n d v e

l o c i t y o n e c a n c i p h e r t h e s e v a l u e s f r o m m e a s u r e m e n t s o f m a x i m u m g r o u n d a c c e l e r a t i o n .

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K 1/2*

EVALUATION OF THE REQUIREMENTS OF NUCLEAR SYSTEMS

FOR ACCOMODATING SEISMIC EFFECTS

T.W. PICKEL, Jr.,

Reactor Division,

Oak Ridge National Laboratory, Oak Ridge, Tennessee, U.S.A.

A logical approach to the evaluation of nuclear .system requirements for ac

comodating seismic effects, is presented in this paper.. This approach· in

volves selection of the type of analysis best suited to the circumstances of

a specific nuclear system, selection of a method of solution for the equa

tions resulting from the analysis method selected, development of a suitable

mathematical model wherein the component parts of the system are adequately

defined, and definition of the interactions of forces among these components

of the system.

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- 6

DISCUSSION

Q

M. S. RA O, India

1. W h a t w o u ld h a p p e n w h e n th e e p i c e n t r e of t h e e a r th q u a k e c o in c id e s w i th t h e r e a c t o r s i t e ?

2 .

In In d i a t h e r e w a s a n e a r t h q u a k e a t K oy n a d a m S i t e a b o u t 5 y e a r s a g o w h ic h w a s s u s p e c t e

to be due to the impo undin g of a la r ge m a ss of wa te r . Som e of the nu c le a r s ta t io ns in Ind ia

a r e b e in g p l a n n e d n e a r l a r g e h y d ro p o w e r s t a t i o n s t o o p e ra t e a s b a s e lo a d p l a n t s . T h i s q u e s t

h a s p a r t i c u l a r r e l e v a n c e to t h i s t h i n k in g .

Λ M. B E N D E R , U . S . A .

1. Th i s i s so un l ik e ly tha t it shou ld be ign ore d fo r eva lua t ion pu rp os es .

2 .

T h e t r i g g e r i n g of e a r th q u a k e s is n o t w e l l u n d e r s to o d bu t s u r f a c e e f f e c ts f ro m w a te r p r e s

s u re a r e u n l ik e ly t o h a v e s ig n i f i c a n c e . H ig h p r e s s u re i n j e c t e d i n to t h e s u b s t ru c t u r e mig h t b e

of s ign i f icance as ind ica ted by e f fec ts o f in jec t ion wel l s in Colorado .

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K 1/3*

PHILOSOPHY AND PRACTICE OF THE ASEISMIC DESIGN

OF NUCLEAR POWER PLANTS :

SUM MA RY OF THE GUIDELINES IN JAP AN

T. HISADA,

Kajima Institute of Construction Techno logy, K ajima Corpo ration, Tokyo,

K. AKINO,

The Japan Atomic Pow er Co., Tokyo,

T. IWATA,

The Kansai Electric Power Co.,

O. KAWAGUCHI,

Power and Nuclear Fuel Development Corporation,

K. OMATSUZAWA,

The Tokyo Electric Power Co., Nuclear Power Department, Tokyo,

H. SATO,

Institute of Industrial Science, University of Tokyo , Tokyo,

Y. SONOBE,

Chiba Institute of Techno logy,

H. TAJIMI,

Nihon University, Japan

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DISCUSSION

Q

H . B A R N E R T , G e r m a n y

I s t h e r e a n y a c t i v i t y i n t h e f i e ld of f o r e c a s t o f e a r th q u a k e s i n J a p a n ? W o rk o n

th e s e q u e s t i o n s mig h t b e im p o r t a n t if d e s ig n c r i t e r i a t u rn o u t t o b e t o o h a rd , f o r e x a m p le

fo r t h e s h u td o w n s y s t e m.

K . O M A T S U Z A W A , J a p a n

Y e s . A n o rg a n i s a t i o n h a s b e e n e s t a b l i s h e d to s tu d y h ow to fo r e c a s t t h e e a r th q u a k e

T h e a c t i v i t y o f t h i s o rg a n i s a t i o n i s a s f o l l o w s :

1. A c c u r a t e m e a s u r e m e n t of l e v e l of g r o u n d s u r f a c e i n a s h o r t e r p e r i o d .

2 .

M e a s u r e m e n t of m i c r o - e a r t h q u a k e n e a r t he a r e a w h e r e l a r g e e a r t h q u a k e i s e x p e c te d in

c o m p a r a t i v e l y n e a r f u t u r e .

3.

A c c u r a t e m e a s u r e m e n t of h o r i z o n t a l m o v e m e n t of t h e g r o u n d b y l a s e r b e a m .

A t p r e s e n t , h o w e v e r , i t i s n o t p r a c t i c a l t o fo r e c a s t t h e l a r g e e a r th q u a k e b e fo re a fe w min

u t e s o r s e c o n d s fo r s h u t t i n g do w n th e r e a c to r , a l t h o u g h u n d e r c o n s id e ra t i o n i n o th e r f i eld

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o

L . E S T E V A , M e x ic o

Y ou me n t io n e d s o m e n u m b e r s b y w h ic h y o u mu l t i p ly th e l a t e r a l f o r c e c o e f f i c i e n t s

of o rd in a ry s t ru c t u r e s i n o r d e r t o o b t a in t h e l a t e r a l f o r c e c o e f f i c i e n t s a p p l i c a b l e t o t h e

v a r i o u s t y p e s of s t r u c t u r a l e l e m e n t s of n u c l e a r r e a c t o r s . O n t h e o t h e r h a n d , y o u a l s o s h o w e d

c h a r t s of g r o u n d v e l o c i t i e s a n d a c c e l e r a t i o n s f or g iv e n r e t u r n p e r i o d s . H a v e t h o s e c o e f f i c ie n t s

b e e n d e r iv e d f ro m th e me n t io n e d c h a r t ? If s o , i n w h a t m a n n e r ? W h a t i s t h e q u a n t i t a t i v e

ju s t i f i c a t i o n of t h o s e n u m b e r s ?

K . OMATSUZAWA, Japan

A

In t h e d e s ig n of A c l a s s i t e m s , l a t e r a l f o r c e s a r e d e t e r m in e d b y b o th s t a t i c an d

d y n a m i c a n a l y s i s . A nd t h e m e m b e r s of s t r u c t u r e s a r e a l s o d e t e r m i n e d b y t h e l a r g e r f o r c e s .

S t a t i c s e i s m ic c o e f f i c i e n t i s b a s e d on th e b u i ld in g s t a n d a rd l a w of J a p a n , a n d th e d y n a mic

a n a l y s i s i s m o r e s c i e n t i f i c a l l y d e t e r m i n e d b y t h e l o c a l s e i s m i c i t y .

So, t h e e a r t h q u a k e f o r c e s i n a lo w s e i s m i c i t y a r e a , b e i n g d e t e r m i n e d b y t h e s t a t i c s e i s m i c

c o e f f i c i e n t ( f a c to r of 3 ) , a r e c o n t ro l l i n g t h e d e s ig n , a n d o n th e c o n t r a ry , i n a h ig h s e i s m ic i ty

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10 -

K . O M A T S U Z A W A , J a p a n

T h e d a mp in g f a c to r s u s e d fo r o u r d e s ig n p u rp o s e s o f r e a c to r f a c i l i t i e s a r e a s

follows :

R e i n f o r c e d c o n c r e t e 5

W e l d e d s t e e l s t r u c t u r e 1

R i v e t e d o r b o l t e d s t e e l s t r u c t u r e 2

V i t a l p ip in g s y s t e m 0 . 5

C o n t r o l r o d d r i v e m e c h a n i s m 3 . 5

F u e l a s s e m b l i e s i n w a t e r 7 . 0

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K 1/4*

PROBLÈMES DE SÉISMES : TECHNIQUES UTILISÉES

POUR LES RÉACTEURS NUCLÉAIRES EN FRANCE

D.

COSTES et al.,

Département des Etudes de Piles,

C.E.A.,Centre

d Etudes Nucléaires de Saclay, Gif-sur-Yvette, France

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- 12 -

D I S C U S S I O N

Q

R . S C H N E ID E R , G e rm a n y

P o u r v o s p ro j e t s e n F ra n c e e s t - c e q u e v o u s a v e z u t i l i s é p o u r l a d é t e rm i n a t i o n

d e v os s p e c t r e s s e u l e m e n t d es c h o c s a m é r i c a i n s c o m m e " E l - C e n t r o " ? Q u e l le s a c c é l é r a t i o n s

a v e z - v o u s c h o i s i ?

D.

C O S T E S , F r a n c e

L e s é i s m e a l g é b r i q u e a é t é d é t e r m i n é e s s e n t i e l l e m e n t d ' a p r è s d e s s é i s m e s a m é -

r i c a i n s , p a r c e q ue l e s e n r e g i s t r e m e n t s f r a n ç a i s c o r r e s p o n d a i e n t s e u l e m e n t à d e s m i c r o -

s é i s m e s , b e a u c o u p m o i n s r i c h e s en f r é q u e n c e s d i v e r s e s q u 'u n g r a n d s é i s m e .

L e s i n t e n s i t é s n o m i n a l e s c h o i s i e s p o u r l e s r é a c t e u r s f r a n ç a i s so n t de 7 o u 8 s e l o n l e s r é g i o n

l e s s é i s m e s m a j o r é s s ' e n d é d u i s e n t .

D.

L U T O S C H , G e r m a n y

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13

J ' a i m o n t r é l 'a d a p t a ti o n a u s é i s m e d ' E l - C e n t r o . P o u r un e c o l l e c t io n i m p o r t a n t e de s é i s m e s ,

o n p e u t t r o u v e r u n e f r é q u e n c e fo n d a me n ta l e mo y e n n e u n p e u mo d i f i é e .

2.

L ' a p p l i c a t i o n du s é i s m e a l g é b r i q u e à u n e c o l l e c t i o n d e r é s o n a t e u r s m o n t r e l e t e m p s a u

b ou t d u q u e l o n o b t i e n t l a r é p o n s e m a x i m a l e d e c h a q u e m o d e , c a r a c t é r i s é c o m m e u n r é s o n a

t e u r .

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K 2/1*

THE EARTHQUAKE RESPONSE ANALYSIS

FOR A BWR NUCLEAR POWER PLANT USING RECORDED DATA

K. MUTO,

Muto Institute of

Structural

Mechanics, inc., Tokyo,

K. OMATSUZAWA,

The Tokyo

lectric

Power Company, Inc.,

Nuclear Power Department, Tokyo, Japan

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16 -

DISCUSSION

Q

J . P . L A F A I L L E , B e l g i u m

W h e n s tu d y in g th e r e s p o n s e o f a b u i ld in g w i th a 2 -d ime n s io n a l mo d e l i t i s imp o s

s ib l e t o d e t e c t a n y r e s p o n s e o f t h e b u i ld in g in a d i r e c t i o n d i f f e r e n t f ro m th e e x c i t a t i o n . T h i s

coupl ing e f fec t would occur i f the exc i ta t ion d id no t occur in a p r inc ipa l d i rec t ion of ine r t ia .

W e re t h e r e c h e c k s ma d e to v e r i fy t h a t t h e r e w e re n o t s u c h e f f e c t s ?

K . MUTO, Japan

The ea r th qu ak e in May 1970 had grou nd mo t ion in bo th NS and EW di r ec t io ns , an

c o u p le d e f f e c ts w o u ld h a v e o c c u r r e d i n t h e b u i ld in g . O u r me a s u r e m e n t w a s ma d e in N S d i r e

t ion on ly be ca us e of econ om y. An a ly s is was then l imi ted in th is d i re c t i on . But the fa i r ly

a c c u r a t e c o i n c i d e n c e of t h e o r y a nd m e a s u r e m e n t w a s t a k e n .

I n f u t u r e , f or a t h r e e - d i m e n s i o n a l a p p r o a c h , a l ot of i n s t r u m e n t s a nd m e a s u r e m e n t s w o u l d

b e d e s i r a b l e a s w e l l a s r e l e v a n t a n a l y s i s .

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- 17

Q

A. H. HA DJ IA N, U. S. Λ.

V i s c o u s d a mp in g i s a e q u iv a l e n c e a n d g e n e ra l l y d e t e rmin e d b y t e s t s o f a c tu a l

How ar e then the dam ping coe f f ic ien ts

m a t r i x m a d e p r o p o r t i o n a l to t h e s t i f fn e s s m a t r i x ?

s t ru c t u r e s . H ow a r e th e n th e d a m p in g c o e f f i c i e n t s 'V . d e t e rm in e d a n d w h y w a s th e d a m p in g

K. MUTO, Japan

A

T h e d a mp in g c o e f f i c i e n t V of t h e v ib r a t i o n e l e m e n t i s c o m p u te d f ro m b o th t h e

n a tu r a l p e r io d (T ) a n d th e d a m p in g f a c to r ( h) w h ic h a r e u s u a l ly me a s u r e d b y th e v ib r a t i o n

t e s t . In c a s e of o n e m a s s s y s t e m , t h e V ma y be e x p r e s s e d a s fo l l o w s :

V = T h / K

As for the defini t io n of V , re fe r the equ atio n (1):

{ F l i =

[ B ] .

| v )

.

+

[ v B ] .

W i

in which

|Fj = e x t e rn a l f o r c e v e c to r

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K 2/2

STRONG MOTION EARTHQUAKES AND THEIR EFFECTS

ON NUCLEAR POWER PLANTS

R.B.

M A T T H I E S E N ,

Schoo l of Engineering and Applied Science,

University of California, Los Angeles, California,

C.B. SMITH,

Norm an Engineering Co., Los Angeles, California, U.S.A.

ABSTRACT

For the past four years, UCLA has been studying the effects of earth

quakes on nuclear power plants. Structural vibrators, hydraulic rams , and ex

plosive blasts have been used to excite reactor structures and equipment.

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nia (see Matthiesen and Smith [1 ], [2], [3] ; Bleiweis, Hart, and Smith [4];

Ibanez,

Matthiesen, Smith, and Wang [5] ). The tests have used structural vi

brators,

hydraulic

rams,

and explosive blasts to excite structures and equip

ment.

The results of these tests have been used to develop mathematical models

which are considered valid for the level of response in the tests when they

reproduce the experimental data. The mathematical models are then used to

predict the response of the system to various digitized earthquake records.

The experimental tests have provided new information concerning the dynamic

properties of the large structures and components used in nuclear power plants.

Data have been obtained on reactor containment buildings, stacks, water towers,

steam generators, pressure vessels, cores, primary coolant pumps, pressurizers,

and other items. In the few cases where information has been available, we

have compared experimentally determined parameters (natural frequencies, damp

ing,

effective masses, and mode shapes) with theoretical studies published by

others.

A number of techniques have been used to excite the structures tested.

These will be listed in order of increasing force capability. Ambient vibra

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- 21 -

of response approaching strong motion earthquakes. In one method we use a hy

draulic ram to produce a large static displacement of equipment such as a

steam generator or primary coolant pipe. When the displacing force is sud

denly removed, the equipment undergoes large amplitude free vibrations.

Another technique makes use of explosive charges placed in the soil adja

cent to

the

reactor containment building. The explosives have been placed in

bore holes located at distances from 100 to 1000 feet from the containment

building (Figure 2). Tests have been performed with varying quantities of

high explosive, ranging from one pound up to 2000 pounds (Figure 3 ) . Plans

are being made for additional tests using 25,000 pounds of explosive. The re

sults of theSie tests indicate that explosive blasts are a useful tool for dy

namic testing. Excellent agreement has been obtained in comparisons of forced

vibration test data and blast data.

We use the same type of instrumentation for recording the response due to

the blasts. The duration of the blast excitation is shorter, so in addition

to the strip chart recorders we employ FM magnetic tape recorders (Figure 4 ) .

The electronic records can be processed automatically, first using a subrou

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- 22 -

The second deficiency in the simulation methods is that the frequency

spectrum of the exciting force is different from strong motion earthquakes.

The ambient methods use "broad band noise" to excite the system. The forced

vibration tests excite the system with energy at a single frequency which is

incrementally varied over the range of interest. The impulse tests excite the

system with a combination of high frequency motion and motion at the natural

frequency of the excited component. The blast test produces a "fairly narrow

band" (0-100 Hz) excitation, but the duration of ground motion is short com

pared to earthquakes and the dominant frequencies in the spectrum are gener

ally higher than the dominant frequencies in an earthquake.

The third deficiency concerns the level of excitation used in the simula

tion tests. Typically, the equipment and structural response we measured in

the structural vibration tests is one-hundredth to one-thousandth of the value

that would result from a strong motion earthquake. We have observed in our

work that reactor structures respond differently under ambient vibrations

(one-millionth to one-ten thousandth of strong motion earthquake motion) than

during the forced vibration tests. Even over the limited range of forces pos

sible with the structural vibrators, we have observed significant departures

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- 23 -

Test results analyzed to date have given positive evidence that these

nonlinear effects occur at higher levels of excitation. Significant changes

have been observed; these can be correlated with the effective stiffness and

with the effective damping of the system examined.

In the above definitions we refer to the total system, so that harden

ing and softening refer to a combined effect of both stiffness and damping.

Most of the results we have observed fall in the softening system category.

3. USE OF BLAST TESTS

To illustrate the use of explosive blast testing, we shall outline one

such test and describe some of the pertinent results which have been obtained.

In this test, explosives were detonated in bore holes that were typically'

20 m deep. The distance and depth varied slightly from test to test. The re

actor containment building, pressure vessel, piping, core, and steam'generator

were instrumented with accelerometers. Other accelerometers were placed on

the soil away from the building. In addition, two three-component bore hole

seismometers were located at the bottom of borings placed between the blasts

and the containment building.

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- ?4 -

eral modes of vibration simultaneously, careful placement of accelerometers is

a necessity. The resulting records must be Fourier analyzed if modal coupling

is present.

ReactOT equipment and piping responses were also obtained during the

blast tests. The data are be in, used to obtain the dynamic parameters of each

equipment item, and also to sec how the parameters vary with the level of ex-

citatioT'..

Figure 5 is a plot of the response of the top of a steair, generator to a

series oí' blast tests. Xote that the initial response during the blast (the

forced vibrations ) occurs at

u

high frequency. After approximately one

second, the forced vibrations end and the steam generator undergoes free vi

brations at its natural frequency. In the largest tests, the lateral acceler

ations of the steam generator exceeded the design seismic loading.

The blast data can yield information regarding modal frequencies, mode

shapes,

and modal damping. IVc are studying each piece of equipment as well as

the primary coolant piping. When possible, blast test da'ta are compared to

other data and calculated values.

We are also examining how the equipment parameters change at higher force

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- 25 -

4.

COMPARISON WITH EARTHQUAKE RESPONSE

Only a limited number of nuclear reactors in the United States have

strong motion seismographs and until February 1971 none of these had been

subjected to anything other than small earthquakes. In California, the UCLA

research reactor is equipped with a USCGS strong motion instrument. In

addi

tion, the reactor structure is instrumented with accelerometers which are re

corded when the strong motion instrument trips.

Also in California the San Onofre Nuclear Generating Station has a Tele-

dyne MTS-100 system with sensors on the containment structure basement, the

steam generator, and the pressurizer. The station has an AR-240 strong mo

tion accelerograph, as well as several passive peak recording seismometers .

Several records have been obtained at San Onofre, including the earth

quakes of 7 August 1966, 8 April 1968 (Borrego

Mountain),

12 September 1970

(Lytle Creek) and most recently 9 February 1971 (San

Fernando).

Copies of records from the recent earthquakes were made available to the

authors by the Southern California Edison Company. These were the records

from 12 September 1970, when an earthquake near Cajon Pass triggered the

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- 26 -

undergoing large displacements. Further study is needed to assess the in

fluence of the connecting piping on the steam generator response before a

definite conclusion can be reached.

Shortly after the September earthquake data had been analyzed, Southern

California experienced the 9 February 1971 San Fernando earthquake. In terms

of damage and destruction, this M = 6.5 earthquake was the worst one in Cali

fornia since the 1933 Long Beach earthquake. Strong motion accelerograph

records were obtained at the San Onofre Nuclear Generating Station, at the

UCLA reactor, and at numerous structures in Los Angeles.

Figure 11 shows the San Onofre steam generator record. Peak accelera

tions of approximately ±0.2g were recorded at the top of the steam generator.

It is interesting to note that the frequency in the time trace where the

largest amplitude vibrations occurred is approximately 2.9 Hz. From the work

of Ibanez, et. al. [S], this is seen to correspond not to the steam generator

but to the interaction of the steam generator with the pump. Additional anal

ysis of the San Onofre records is underway.

At UCLA the strong motion instrument tripped during the February earth

quake and again on four subsequent aftershocks during the day of the earth

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- 27 -

REFERENCES

[1] MATTHIESEN, R.B., SMITH, C.B., A Simulation of Earthquake Effects on the

UCLA Reactor Using Structural Vibrators, UCLA Department of Engineering

Report NEL-105 (October 1966).

[2] SMITH, C.B., MATTHIESEN, R.B., Forced Vibration Tests of the Experi

mental Gas-Cooled Reactor (EGCR), UCLA Engineering Report #69-42 (August

1969).

[3] MATTHIESEN, R.B., SMITH, C.B., Forced Vibration Tests of the Carolinas-

Virginia Tube Reactor (CVTR), UCLA Engineering Report #69-8 (February

1969).

[4] BLEIWEIS, P., HART, G.C., SMITH, C.B., Enrico Fermi Nuclear Power Plant

Dynamic Response During Blasting, ANS Transactions 1 3, 1, pp. 231-232

(June 1970).

[5] IBANEZ, P., MATTHIESEN, R.B., SMITH, C.B., WANG, G.S.C. , San Onofre

Nuclear Generating Station Vibration Tests, UCLA-ENG-7037 (August

1970).

[6] SCHMITT, R.C., Evaluation and Comparison of Structural Dynamics Inves

tigation of the Carolinas Virginia Tube Reactor Containment, Report

»IN-1372,

Idaho Nuclear Corporation (May 1970).

[7] Personal communication, B.J. MORRILL (USCGS) to R.B. MATTHIESEN, November

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28 -

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- 29

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- 30 -

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- 31 -

37.4

37.2

37.0

> -

o

36.8

σ

ÜJ

or

u .

36.6

364

o bio.«

D bla«

O ^ - O -

—

D

~~-.

□

t tsst - west

t tett - north

°\

\

\

\

\

IO SO 100

A CCELERA TION ( thousandths o f a g )

4 0 0

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32

3 . 0

2 9

2.8

N

f

>-

g 2.7

UI

3

UI

2.6

2.5

{

ï

I

%

Π earthquake

otlon tests

o f 12 sept 70

D

0.2 OS 1.0 5.0 IO 5 0 100 20 0

DISPLACEMENT (thousandths o f an Inch)

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-

33

— * '-**■·· ¡■u

r Ä y v - * ^ > V - * y v V ^ y y ^

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- 34 -

DISCUSSION

0

N. J. M. R EE S, U. Κ.

D id y o u c o n s i d e r t h e e f fe c t of a i r b l a s t o n y o u r s t r u c tu r e s f ro m y o u r b u r i e d

e x p lo s io n s s in c e f ro m y o u r s l i d e s t h e y a p p e a re d t o h a v e a l l b e e n v e n t e d o n e s ?

C. B. SM ITH , U. S. A.

W e d id n o t m e a s u re t h e a i r b l a s t c o n t r i b u t io n . O n a g u a rd s h a c k a b o u t 1 0 0 ' f r o m

th e b l a s t s , n o w in d o w s w e re b ro k e n d u r in g a n y of t h e t e s t s .

Q

J . D . S T E V E N S O N , U . S . A .

D id y o u t ry t o d e t e r m in e t h e s o u r c e of n o n - l i n e a r e f f e c t s ? S l ip p a g e of s u p p o r t s

v s .

c h a n g e in s t r e s s l e v e l in c o m p o n e n t s .

A

C. B. SM ITH , U. S. A.

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K 2/3

EARTHQUAKE CALCU LATIONS : THEIR IMPORTA NCE

W I T H RESPECT TO AREAS OF AVERAGE

AN D LOW SEISMIC AC TIV ITY A ND THE A PPLICATION

OF COMPUTER ORIENTED METHODS

A.E. HUBER, P .O . SCHILDKNECHT,

SDK Ingenieurun ternehme n für spezielle Statik, Dyna mik und Konstruktion

GmbH,

Lörrach, Germany

ABSTRACT

In the past design criteria for earthquake resistant conventional structures

in areas of low and medium seismic activities have been semi-empirical. Even

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- 36 -

1.0 Dynamische Berechnung von Konstruktionen

1.1 Berechnung des Bewegungsablaufs (Time History Modal Analysis)

Wir betrachten eine linear elastische Konstruktion mit i Freiheitsgraden,

die mit i Massen behaftet sind.

Das verallgemeinerte Eigenwertsproblem eines solchen Systems und dessen

Lösung, nämlich die Eigenwerte und Eigenvektoren, werden hier als bekannt

vorausgesetzt. Auf dieses Problem wird im letzten Teil eingegangen.

Betrachten wir zuerst eine statische Belastung des Systems. Diese lässt

sich nach den Eigenvektoren entwickeln. Die Deformation, welche dem norma-

lisierten Eigenvektor A. entspricht, erhält man durch eine statische

3

im

f ι

Lastgruppe L. , die die Massenkräfte eines frei schwingenden Systems

ersetzt.

L. = w

2

— A. = Masse χ Beschleunigung (1)

im m g im

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' I U I ( 4 )

37 -

zu

\#>\ - TL.

1 ~ >

, 1 J

I m i l im J l i

M i t H i l f e d e r O r t h o g o n a l i t ä t s b e d i n g u n g e n k a n n g e z e i g t w e r d e n , d a s s

„(D 1

B '"

= - J - ^ ^ Ρ Α. (5 )

m w „ f—r- ι im

m ι = 1

v e r g i . I l l S . 9 4

Betrachten wir eine Lastgruppe P., bei der an der Stelle i=j die Last

P. = P. angreift und alle übrigen Lasten P. = O (für i ï j] sind, so

erhält man aus (5)

Ρ = -4 P. Α. (6)

Zur eindeutigen Kennzeichnung der Lastgruppe genügt dabei der Index (j)

anstelle von (1).

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38 -

'¿Ρ

1

(t) + w

2

D

1

* (t) = w

2

f (t)

m m m m j

(10)

Berücksichtigt man viskose Dämpfung, so ist rP' (t) die Lösung der

Gleichung

"cP

1

(t) + 2 b

m

íPJ (t) + w

2

LP

1

(t) = w

2

f. (t)

m m m m m m j

(11)

mit b als Dämpfungsfaktor.

Die Lösung der Gleichung (11) lautet:

t

rpl (t) - V f ,^ „"

b

i

\R

b

2

m m

f. (t) e~

b

™

(t

~-' . sin \/w

m

-b^ . (t-t)dt

'-0

(12)

Wenn wir die Belastungen in allen Punkten j=l...je betrachten, erhält man

durch Summation über j aus Gleichung (8) und (9):

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- 39 -

Mit dieser Belastungsfunktion und mit b = c w„ erhält man aus Gl . (13):

3

m m m

m=me j = j e W.

q, = -

t>

A.

~z>~

6., p. -1 A. . y (15)

4

i ¡£j- im ifj- jk *} g ]m »m

dabei ist:

t

y = — \ ζ (t) e "

c

m

w

m

( t

" £ ' sin w (t-t)dt (16)

m w l —

m — —

m

r

1

to

Man kann also den Bewegungsablauf (time history) für eine beliebige Grund-

beschleunigung berechnen.

In der Praxis interessieren in Bezug auf die Festigkeitsanforderungen nur

die Maximalwerte der Verschiebungen bzw. der dynamischen Belastungen.

(Um die Übersichtlichkeit zu wahren, werden diese Betrachtungen auf die

Verschiebungen beschränkt. Sie lassen sich analog für dynamische Be-

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1.2 Berechnung mittels Response Spektren (Response Spectrum Modal Analysis)

Betrachtet man ein gedämpftes System mit einem Freiheitsgrad, das einer

Grundbeschleunigung unterworfen wird, so erhält man die Relativverschie-

bung zu: (vergi. 13] , S. 85)

t

y (t, w, c) = i- \ ζ (t) e"

c W m ( t

~ y sin w

m

(t-t)dt (17)

m

'T

ζ = Grundbeschleuniqunq eines Erdbebens

w = Eigenfrequenz des gedämpften Systems, für die Praxis

kann sie durch die ungedämpfte Eigenfrequenz ersetzt

werden.

y = Relativverschiebung

c = Dämpfungsfaktor

Es sei hier vermerkt, dass diese Relativverschiebung gleich dem in Gl.(16)

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- 41 -

i m y

2

=

■=

k y

2

und my = k . y

erhält man mit w

2

= —

m

y = w y = w y

und somit: ¿ S = S = w S

d

Durch diese Verknüpfung lassen sich alle drei Response Spektren in einem

Diagramm mit logarithmischer Teilung darstellen, (siehe Abb. 1)

Zusammenfassend kann man folgende Definition treffen:

Das Response Spektrum für ein vorgegebenes Erdbeben ist ein Diagramm, das

die Veränderung des maximalen Ansprechens (max. Verschiebung, Geschwin-

digkeit,

Beschleunigung) eines Einmassensystems mit der Eigenfrequenz

zeigt,

wenn es einer dem gegebenen Erdbeben entsprechenden Grundbeschleu-

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- 42 -

In Abb . 2 ist ein Entwurfs -Spek trum d arge stel lt, das durch Normierung und

Mittelung aus vier verschiedenen Accelerogrammen gewonnen wurde.

Dieses Res ponse Spektru m zeigt geglättete Kurve n, die eine Reihe von

Accelerogrammen abdecken.

2.0 Möglichkeiten zur Bestimmung von seismischen Eingabeparametern für die

dynamische Berechnung von Konstruktionen

2.1 Allgemeines

Die folgenden Absch nitte sollen kurz die wesent liche n Verfahren sk izzie

ren,

mit denen man aus den Ergebnissen einer seismologischen Untersuchung

die seismischen Eingabeparameter für eine dynamische Berechnung von Kon

struktionen bestimmen kann. Entsprechend den vorangegangenen Ausführungen

sind also für den Standort charakt eristi sche Time Histori es oder Resp onse

Spektre n zu besti mmen . Aus einer seismolog ischen Untersu chung kö nnen u.a.

folgende Angaben für den betreffenden Standort zur Verfügung stehen :

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- 43 -

Für das Problem der Verstärkung von Erdbeben durch überlagernde, weichere

Baugrundschichten sei auf entsprechende Literatur verwiesen. Es soll hier

jedoch der Vollständigkeit halber erwähnt werden, dass die lokalen Bau

grundverhältnisse das dynamische Verhalten der Konstruktionen sehr wesent

lich beeinflussen können.

2.2 Maximale Grundbeschleunigung a, maximale Grundgeschwindigkeit v

und maximale Grundverschiebung d

Sofern genügend Seismogramme mit entsprechendem Auflösungsvermögen

exi

stieren, können a, v und d daraus sehr genau ermittelt werden. In diesem

Fall wird man allerdings die später erwähnten, exakteren Methoden wählen.

2.21 Intensität I und maximale Grundverschiebung d vorgegeben

Wir wenden uns nun dem Fall zu, dass lediglich die Intensität I und die

maximale Grundverschiebung d vorliegen. (Auf die Bedeutung einer Angabe

von d für mitteleuropäische Verhältnisse wird später eingegangen; für

amerikanische Verhältnisse kann man auf diese Angabe verzichten.)

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- 44 -

Solche Schwächungsgesetze sind z.B. bei L. Esteva 141 zu finden und

haben die Form:

a = C, e °

2 1

(R +

C3)'

C

^

< 2 0 a )

ν = Ki e

1

(R + K

2

e

K3l

)

K

'·

(20b)

K,C = Konstanten

I = Intensität

R = Abstand des Hypozentrums

Ein entsprechendes Schwächungsgesetz kann man für die maximale Grundver-

schiebung d formulieren.

Die Konstanten sind im wesentlichen abhängig von lokalen Bodenbedingun-

gen,

von der Art der geologischen Formation, die von den Schockwellen

passiert werden, von den Schockmechanismen etc. Durch lokale seismische

Aufzeichnungen und/oder durch geeignete Wahl von Aufzeichnungen in Ge-

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-

45 -

2.4 Bestimmung von Response Spektren aus a, ν und d

Aus einem Grundspektrum lassen sich Response Spektren entwickeln. Empiri-

sche Untersuchungen haben ergeben, dass man Response Spektren einfach

durch Multiplikation von a, v und d des Grundspektrums abschätzen kann,

(vergi. 191).

Dieser Zusammenhang ist durch die empirisch festgestellte Tatsache be-

gründet,

dass das Response Spektrum für

20-25%

Dämpfur.gsrate näherungs-

weise mit dem Grundspektrum zusammenfällt (vergi. 1141).

Ausgehend von dieser empirischen Feststellung und von einem vorgegebenen

Grundspektrum kann man diese Faktoren mit Hilfe stochastischer Bewegungs-

modelle auf mathematisch-physikalischem Wege bestimmen. Auf stochastische

Bewegungsmodelle wird später noch eingegangen.

Ein Satz solcher Multiplikationsfaktoren für elastisches Materialverhal-

ten und verschiedene Dämpfungsraten ist beispielsweise in 1101 zu finden,

(vergi.

Abb. 3 ) . Diese Faktoren basieren auf dem El Centro Erdbeben von

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- 46 -

2.6 Bewegungsablauf (Time Histories)

Ausgeh end von einem (oder mehrere n) aufgezeich neten Acceler ogram m, das

als charakteristisch für die Umgebung des Standortes angesehen werden

k a n n , wird dieses Acce lerog ramm entsprechen d der vorgege benen Intensität

oder der maxima len Grundbesch leunigu ng norm iert. Um individuellen Fluktu-

ationen Rechnung zu tragen, kann man weitere Accelerogramme durch Ampli-

tudenmod ulation und zeitliche Verzerru ng entwic keln .

Aus diesen Accelerogrammen kann man einerseits ein als Berechnungsgrund-

lage dienendes Response Spektrum berechnen, andererseits können sie als

direk te Eingabewer te für die dynami sche Berechnung d iene n. Die Erzeugung

von künstlichen Accelerogrammen mit Hilfe stochastischer Bewegungsmodelle

wird im folgenden er wähn t.

2.7 Stochastische Bewegungsmodelle

Einsch lägige Unt ersuch ungen haben ergebe n, dass sich Erdbeben mitte ls

stocha stische r Bewegu ngsmod elle sehr gut repräsenti eren lass en,

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-

47

-

Die konstante Spektraldichte

So

bedeutet

für

praktisc he Anwend ungen ledig-

lich eine Normierungskonstante.

Es

bestehen unter anderem folgende Mö g-

lichkeiten,

So zu

bestim men:

1) durch Normierung

auf

eine vorgegeben e Intensität

2) durch Normierung

auf

eine vorgegebene ma ximale G run dbe -

schleunigung

bzw.

Verschiebung

3) durch eine derartige Normierung, dass

ein

vorgegebenes

Grundspektrum

mit dem

ermittel ten R esponse Spe ktrum

für

ca. 25% der

kritischen Dämpfung

im

Mittel möglichst

gut

übereinstimmt.

Für

die

relative Spektraldi chte stehen sehr anpassungsfäh ige

und

theore-

tisch fundierte Näherungsformeln

zur

Verfügu ng (vergi.

13) S.

3 3 9 ) ,

z.B.:

S. (ω)

=

- ^

(21)

0 - ã

ι τ 4ζ

2

ug

/ g

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Die einhüllende Intensitätsfunktion der in Betracht zu ziehenden Accelero

gramme kann für einen bestimmten Standort aus (wenigen) aufgezeichneten

Accelerogrammen abgeschätzt werden. Ihr Einfluss auf das Response Spektrum

ist für in der Praxis vorkommende Dämpfungsraten relativ gering, so dass

eine Näherung dieser Funktion bereits gute Ergebnisse liefert.

2.8 Anmerkungen

In den vorangegangenen Abschnitten wurden die wesentlichen Möglichkeiten

zur Festlegung von Entwurfsparametern (Accelerogramme, Response Spektren)

schematisch aufgezeigt. Diese Möglichkeiten können selbstverständlich auf

verschiedene Weise variiert und kombiniert werden. Zu Kontrollzwecken wird

man in der Praxis verschiedene, weitgehend voneinander unabhängige Wege

beschreiten.

3.0 Möglichkeiten zur Bestimmung von seismischen Eingabeparametern in

Gebieten mit mittlerer und geringer seismischer Aktivität

3.1 Allgemeines

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- 49 -

Aus den vorhan denen Aufzeich nunge n kann man jedoch im wesen tlich en folgen-

d e ,

verwertbare Informationen sammeln:

1) Häufigkeit von Erdbeben

2) maxima le Grundversc hiebun g d

3) Intensität der Erdbeben I

4) Lage der Erdbebenherde

5) Pauscha le Anga ben über die vorhe rrsch enden Frequen zen

Aus diesen Angaben lassen sich wei ter e, wich tige Daten a bschä tzen.

3.2 Maxi male Grundb eschl eunigu ng a, maxim ale Grun dgesc hwin digk eit ν

und maximale Grundverschiebung d

Mit den in 2.21 genannt en Bezie hungen k ann man a und ν aufgrund d er für

den Standort ermittelten Intensität abschätzen. Eine Bestimmung der Kon-

stanten von Schwächungsgesetzen für a, ν und d entsprechend 2.22 ist damit

ebenfa lls möglich . Man sollte sich aber bewuss t sein , dass diese Sch wä-

chungskonstanten aufgrund der oben genannten Abschätzungen für v und a

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- 50 -

wird (und sekundär durch die Eigenschaften der zu berechnenden Konstruk

tion)

,

kann man annehmen, dass sich diese Faktoren mit dem relativen

Frequenzgehalt nicht wesentlich ändern. Es scheint damit gerechtfertigt,

die in 2.4 zitierten numerischen Werte der Multiplikationsfaktoren zumin

dest für eine Abschätzung des Response Spektrums zu übernehmen (vergi.

Abb. 3 ) . Eine Überprüfung dieser Multiplikationsfaktoren für lokale Gege

benheiten aufgrund weniger, registrierter Accelerogramme ist im Bereich

des vorhandenen Auflösungsvermögens der Seismogramme möglich und sollte

vorgenommen werden. Eine vollständige Überprüfung kann man mit wenigen,

zukünftigen Aufzeichnungen von höherem Auflösungsvermögen erreichen.

3.5 Direkte Bestimmung von Response Spektren aus Seismogrammen

und Time Histories

Diese Möglichkeiten scheiden für den Frequenzbereich, der das Auflösungs

vermögen der Seismogramme übersteigt, praktisch aus. Dieser Frequenzbe

reich ist allerdings für die mitteleuropäischen Erdbebengebiete von Be

deutung. Es ist deshalb anzuregen, die Seismographen in Zukunft so auszu

legen,

dass dieser Weg beschritten werden kann.

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- 51 -

4.0 Vergleich von mitteleuropäischen Response Spektren mit

amerikanischen Response Spektren und Minimalwerten

In Abb. 4 sind charakteristische Grundspektren und Response Spektren von

2%

kritischer Dämpfung für amerikanische und mitteleuropäische Verhält

nisse aufgetragen. Das angegebene Grundspektrum für mitteleuropäische

Verhältnisse entspricht etwa einer Intensität VI (Mercalli-Sieberg).

Der Frequenzgehalt entspricht in Mitteleuropa registrierten Werten.

Zusätzlich sind in Abb. 4 die für die Auslegung von Kernkraftwerken

ent

sprechend 10 empfohlenen minimalen Grund-und Response-Spektren gezeigt,

die auch in Gebieten, in denen Erdbeben unwahrscheinlich sind, angesetzt

werden sollten.

Aus dem unterschiedlichen Frequenzgehalt ist ersichtlich, dass sich die

Messungen in Mitteleuropa nur auf kleinere Erdbeben beziehen, die in

relativ grosser Entfernung vom Erdbebenherd registriert wurden. Da diese

schwachen Erdbebenintensitäten nicht charakteristisch für stärkere Erd

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7.0 Zusammenfassung

Den vorhergehenden Ausführungen ist zu entnehmen, dass die Lücke von

spärliche n seismischen Ausgan gsdate n für dynamisc he Berechnungen von

Konstruktionen einerseits mit Hilfe von Abschätzungen teilweise ge-

schlossen werde n kan n, andere rseits bieten die ma themat isch-p hysika lisch

fundierten, stochastischen Berechnungsmethoden eine echte Altern ative,

diese Lücke vollständig zu umgehen.

An Hand von Response Spektr en wurde gez eig t, dass quasi -stati sche Berech-

nung en, wie sie beispi elswei se nach entspre chende n Normen vor geschlag en

werden, sowohl eine Überschätzung als auch eine starke Unterschätzung

der seismischen Wirkun gen in bestimmt en Frequenzbere ichen e rgebe n.

Dynamisc he Berechnu ngen können also hel fen, einerse its das hohe Sicher-

heitsb edürfni s bei Kernk raftwe rksan lagen in realistische r Weise zu be -

fried igen, andere rseits können sie eine unwirtsc haftlic he Dimensionierun g

verhindern. Insbesondere ist es in vielen Fällen möglich, aufgrund dyna-

mischer Analysen Massnahmen zu ergreifen, die ein Ansprechen der Kon-

struktionen auf seismische Erschütterungen enorm reduzieren oder über-

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- 53 -

The available solution techniques for eigenvalue problems can be classi

fied into Transformation Methods and Iterative Methods. Among the trans

formation methods the Jacobi method and the Givens method are the most

wellknown. The Jacobi method performs orthogonalization on the original

matrix to achieve a diagonal matrix with the eigenvalues of the matrix on

the diagonal. The Givens method, similar to the Jacobi method, yields a

tridiagonal matrix by transformation. The advantage is that the complete

eigensolution is performed with the transformation methods. A big

dis

advantage is that the solution is accomplished by operating on the system

matrices, i.e. inversion of the matrix using the Jacobi method. The facts

clearly make the transformation methods less attractive than the widely

used iterative methods, particularly the conjugate gradient method.

The conjugate gradient method is the widely used iterative method. This

scheme to minimize a function was first developed by Hestenes and Stiefel

( 115)

) .

Bradbury and Fletcher ( 116) ) developed from that the Rayleigh

Quotient Solution. Also Prato (1171) presents an application of the con

jugate gradient method as a solution technique for static, vibration and

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- 54 -

The Rayleigh quo tie nt impli es a nume rato r whic h is twice the strain energy

and a deno mina tor whic h is twice the max imu m kinetic ener gy of the stru c

ture.

The Rayleigh quotient can therefore be accomplished by taking the

sum of the potential and the kinetic energies of the individual elements.

Equati on (2) can therefore be rewritten as:

Σ. 6. k. 6.

R (Δ) = i-i (3)

Σ δ.

Τ

m. δ.

where k = element stiffness matrix

m = element mass matrix

i

= decomposed generalized displac ement

r = number of discrete elements

The initially estimated vector Δ

0

is changed in each iter ation cy cle to

ward s the eigenvector Δ . Theor etical ly, the solution is achieved with the

conjugate g radien t method after η cycl es, where η is the number of system

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- 55 -

P

i + 1

= VR (Δ.

+ 1

) (9)

. _ P j

+

i

T p

i+i

ßi =

p ~ 7 ~ <

1 0 )

1 p

i

« i + i

=

-

p

i

+

i

+ β

Λ

(11)

From any approx imati on to a local minim um Δ., a search is mad e along a

dire ctio n q. to find a bett er app roxi mati on Δ. .. The step length a. in

the q. dire ctio n must now be chosen such tha t the function is min imi zed in

^ 1

that direction.

Rewriti ng equati on (2) using the expr essi on of equ atio n (4) yiel ds

(Δ.+α q ) Κ(Δ + α q )

R (Δ.+α.q.) =

1 1 1

=

ί

1

(12)

( Δ . + a . q . )

1

M ( A

i

+ a

i

q

i

)

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- 56 -

Since both roots often appear positive it is suggested to actually calcu

late for both values of a. the Rayleigh quotient. This is done without

any additional computer time since the parameters of (1 5), (16) and (17)

were already available and valid for both a.'s. Then the a. associated

1 1

with the lower of the two Rayleigh quotients was chosen to estimate a new

vector Δ. ,.

The second eigenvalue and its associated eigenvector can therefore be

determined by posing a new minimization problem.

R (Δ

2

)

Τ

Δ

2

ΚΔ

2

~Τ

Δ

2

ΜΔ

2

is a minimum subject to the orthogonality condition

Δ? Μ Δ Ι = 0

In this restricted subspace the Rayleigh quotient has a unique minimum

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- 57 -

9.0 Seismische Berechnung eines Reaktorcontainments mit Einbauten

Abb. 7 zeigt das System, das aus Containment, Reaktorstützung,

Druckbehälter, biologischem Schild und Zwischendecke besteht.

In Abb. 8 ist das dynamische Modell dargestellt.

Die Berechnung basiert auf dem Response Spektrum für mitteleuropäische

Verhältnisse (Abb. 4 ) .

Die ersten vier normalisierten Eigenvektoren und die zugehörigen Eigen

frequenzen sind aus Abb. 9 zu ersehen.

Abb. 10 zeigt die Beschleunigungen für jede einzelne Frequenz.

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Literature

1 NORRIS, C.H., HANSEN, R.J. etc., "Structural Design for Dynamic Loads",

McGraw Hill, New York, Toronto, London (1959)

2 ZUDANS, Ζ., FISHMAN, H.M., REDDY, G.V.R., CHOW, T.Y., "Technical Report:

Lums, Manual for the Dynamic Response of Lumped Mass Systems Program",

The Franklin Institute Research Laboratories

3 WIEGEL, R.L., "Earthquake Engineering", Prentice-Hall, Inc., Englewood

Cliffs., N.Y. (1970)

4 HANSEN, R.J., "Seismic Design for Nuclear Power Plants", The M.I.T.

Press (1970)

5 "Nuclear Reactors and Earthquakes", TID-70 24, USAEC (1963)

6 HILLER, W., ROTHE, J.P., SCHNEIDER, G., "The Rhinegraben Progress

Report 1969"

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- 59 -

14 ESTEVA, L., ROSENBLUETH, Ε., "Espectros de Temblores a Distancia

Moderados y Grandes", Boletin, Sociedad Mexicana de Ingenieria Sismica,

V. 2, No. 1, March 1964

15 HESTENES, M.R., STIEFEL, E., "Methods of Conjugate Gradients for Solving

Linear Systems", Journal of Research of the National Bureau of Standards

Vol. 49, No. 6, Research Paper 2379, December 1952

16 BRADBURY, W.W., FLETCHER, R., "New Iterative Methods for Solution of

the Eigenproblem", Numerische Mathematik 9, pp. 259-267, 1966

17 PRATO, C A . , "Plate and Shallow Analysis by Conjugate Gradients",

Ford Foundation Research Report, R 69-53, MIT, September 1968

18 FOX, R.L., KAPOOR, M.P., "A Minimization Method for the Solution of the

Eigenproblem Arising in Structural Dynamics", Case Western Reserve

University, Cleveland, Ohio, September 1968

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60 -

0j05 0.1 Q2

Frequency, cps

Response spectrum

El Centro earth qua ke 19¿0

from [10]

Fig.

1

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f i l

Relat ive Values Of Spectrum Ampl i f icat ion Factors

Percent Of Critical

Damping

0

0,5

Am pl i f ica t ion Factor For

Displacement

2,5

2,2

Velocity

£.0

Acce lera t ion

6,4

5,8

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ί·2

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S3 -

O

O

O

o

in

o

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- 64 -

8

o

S

in

S

Osi

o

o

o

LT)

o

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FIG.7 REACTOR CONTAINM ENT

WITH INTERIOR STRUKTURES FIG.8 DYN AM IC MODEL

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M0DE1 (3.78CPS)

MODE2(7.60CPSI

MOOE 3I13.68CPS) MODE4 (1856CPS1

FIG.9 MODE SHAPE (NORMALIZED)

MODE 1(3.78 CPS)

— r t

w

~

MODE3(13,68CPS]

MODE 5 11990 CPSl

MODE 6 (28.80 CPS)

FIG.10 ACCELERATION IN g

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- 67 -

DISCUSSION

-^ K . ZILCH , Germany-

S o me c o mme n t s o n t h e i n p u t d a t a o f a s e i s mic d e s ig n .

M a y I e m p h a s i z e t h a t t he r e s u l t s o f o u r a s e i s m ic c a l c u l a t i o n c a n o n ly b e a s e x a c t a s t h e

inpu t . Tha t me an s , we have to t r y to ge t good inpu t da ta as we l l a s ma ke p ro gr es s in the

c a l c u l a t i o n me th o d s . In a r e a s of l ow s e i s m ic i ty th e g r e a t p ro b l e m i s t h e l a c k of s a t i s f a c t o ry

s e i s m i c d a t a . T h e r e f o r e , a l l a v a i l a b l e m a t h e m a t i c a l a n d e n g i n e e r i n g t o o l s sh o u l d b e u s e d to

u t i l ize the da ta g iven . In th is con te x t . I wan t to re fe r to we l l known pr ob ab i l i s t ic an a ly s is o f

s e i s m i c d a t a , f o r e x a m p l e r e l a t i o n s h i p s b e t w e e n t h e e x p e c t e d r e t u r n p e r i o d s o f e a r t h q u a k e s

and the magni tude (1 ) , (2 ) .

F o r e x a m p le : l og N = a -b M

N i s t h e m e a n y e a r l y n u m b e r o f e a r t h q u a k e m a g n i t u d e s g r e a t e r t h a n M .

S u c h me th o d s g iv e a t l e a s t s o me h in t s f o r a d e s ig n , a n d o n ly i n fo rma t io n o n mo re t h a n o n e

s p e c i f i e d d e s i g n e a r t h q u a k e e n a b l e s t h e e n g i n e e r s t o m a k e a r e a l r i s k a n a l y s i s r e s u l t i n g i n

d e f in i t i v e n u mb e r s o f r e l i a b i l i t y a n d to a v o id t h a t s o me g iv e n d e f in i t i o n s r e ma in s u b j e c t i v e

a n d o p e n to i n d iv id u a l i n t e rp r e t a t i o n .

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- 68 -

s h e l l w e re c o mp u te d a c c o rd in g to s h e l l t h e o ry . T h e d i f f e r e n c e b e tw e e n s h e l l t h e o ry an d b e a m

th e o ry fo r t h e r e a c to r s u p p o r t a n d th e r e a c to r s h i e ld is n e g l ig ib l e .

o

H . R I E K E R T , G e r m a n y

A r e t h e c o m m e n t s o n t h e a d v a n t a g e of c o n j u g a t e g r a d i e n t m e t h o d s o v e r t r a n s f o r m

m e t h o d s b a s e d o n t h e o r e t i c a l a s p e c t s o r o n p r a c t i c a l c o m p a r i s o n s . C o n j u g a te g r a d i e n t m e t h

o d s c o n v e rg e t h e o re t i c a l l y i n n s t e p s , b u t u s u a l ly n o t i n p r a c t i c e . S o i t w o u ld b e o f i n t e r e s t

t o k n o w , w h e t h e r t r a n s f o r m m e t h o d s a s f o r i n s t a n c e t h e Q R - m e t h o d o r t h e H o u s e h o l d e r -

me th o d c o u ld n o t b e a p p l i e d h e re w h e re a d v a n ta g e c o u ld b e t a k e n o f t h e s y mme t ry o f t h e ma

t r i x p r o b l e m .

P .

O . S C H I L D K N E C H T , G e r m a n y

A s I h a v e p o in t e d o u t , t h e c o n v e r g e n c e o f t h e c o n ju g a t e g r a d i e n t m e th o d d e p e n d s

to a h ig h d e g re e on th e a s s u m p t i o n of t h e s t a r t i n g v e c to r . S in c e w e a r e a b l e t o ma k e r e a s o n

a b le a s s u mp t io n s fo r t h i s v e c to r a s l o n g a s w e d e a l w i th i d e a l i z e d tw o -d ime n s io n a l d y n a mic

p r o b l e m s , t h e c o n j u g a t e g r a d i e n t m e t h o d ( a s a n e n e r g y m e t h o d ) i s f a v o r a b l e f o r l a r g e s y s t e m

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K 2/4

ASEISMIC DESIGN OF STRUCTURES W I T H NUCLEAR REACTORS -

METHOD OF EARTHQUAKE RESPONSE ANALYSIS

FOR COMPOSITE STRUCTURES EVALUATED

FOR DAMPING EFFICIENCIES

BY MA TER IAL A N D STRUCTURE TYPE

Y. TSUSHIMA, J . JIDO,

Takenaka Komuten Co. Ltd.,

Technical Research Laboratory, Tokyo, Japan

ABSTRACT

The purpose of this paper Is to summarize analytical procedures which

have been employed in the evaluation of dynamic properties and dynamic

response values for earthquake motions putting emphasis on estimating the

damping capacities of a special structure like a nuclear power plant which

is made of various materials such as concrete, steel, special alloys, etc.,

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- 70 -

Especially, Reactor Building has heavy weight in comparison with others.

Response values of Reactor itself, which is the most important part of

this structure, are affected by the dynamical properties of Reactor

Building.

(c) Reactor Building being spatially constructed of many walls which react

with the external force acting in various directions makes calculation

of stiffness of the building difficult.

(d) Reactor Building being massive and having a short period, the dynamic

properties (natural period, damping capacity) are.affected by the

interaction between foundation and ground. There is theoretically no

difference in dynamic properties at time of earthquake motion between a

structure like this and normal structures but it must be analyzed by the

best method which can be considered for its dynamic properties.

(e) Reactor Building has heavy weight and rigid stiffness of which response

values are quite large. On the other hand, Nuclear Reactor has light

weight and less rigid stiffness of which response values are very small.

Thus the difference in these two structures causes several problems at

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- 71 -

Dry Well

Dry Well Is a typical thin shell structure of which dynamic proper

ties must be calculated considering movements in bending and shearing,

which is similar to movement of free end of cantilever with change in shape

of shell. Stiffness of this is also calculated using F.E.M..

Other Structures

Stiffnesses of other structures like Reactor Pressure Vessel and

Shield Wall are calculated using the bending shear deflection theory.

Details about the stiffness calculation of structures are explained in

the paper, "Aseismic Design of Nuclear Reactor Building — Stress Analysis

and Stiffness Evaluation of the Entire Building by the Finite Element

Method," [1],

3. DAMPING PROPERTIES OF STRUCTURES

Generally, the damping properties of structures are usually assumed as

follows :

(a) viscous damping due to viscous properties of materials molecules

(b) hysteretic damping due to imperfect elasticity of members of

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- 72 -

expressing the stiffness of ground by complex numbers, an experiment was made

expressing the stiffnesses of structure and ground by complex numbers

(K R + 1 K T ) and evaluating the damping ratios of the structure as hysteretic

damping.

If the stiffness matrices are expressed by complex numbers, the equation

of single freedom motion for the dynamic analysis of the structures can be

expressed as follows:

Mi +

( K R

+

1 K T ) X

= -Mx

0

(1)

where M :

mass,

χ : acceleration of earthquake motion.

Details of the equation of motion will be explained elsewhere.

Imaginary Parts of Stiffness Matricest^J

Therefore, in Eq. (1) , the solution of equation may also be assumed as

follows :

. -huit iüjt , ~

\

χ = A

e

e (2)

Substituting Eq. (2) into Eq. (1 ), the following equation is obtained.

,, K R + ίΚτ , .

(-ηω + 1 ω )

2

+

-ü-jj i = 0

(i)

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- 73 -

hj = h u

0

j

2

A¡j

2

(10b)

Eliminating the sign ω

0

< from the above two equations, the damping ratio of

J-th mode can be expressed by the following equation:

hj = (-1 + /l+»Jh

2

)/2h (11)

The damping ratio (hj) is constant by hypothesis and assuming the damping

ratio (h) to be small enough compared with 1, It can be concluded that the

damping ratio of each mode is equal to the damping ratio established by

material and structural type.

Complex Stiffness Matrices of Multi-Degrees of Freedom

In calculating the complex stiffness matrices of a composite structure

such as one containing a nuclear reactor, at first, the entire structure is

divided into a number of groups (G) by damping ratios expressed by the sign

G

H

and the complex stiffness matrices (KR +

1 K T ) Q

in the local coordinate

system for each group are prepared. The Individual complex stiffness matrix

contributing to each group is calculated by Eq. (12) from the individual

real stiffness matrix

(QKR)

which is calculated by F.E.M. and other methods.

[ KR + IKIJG = (1 + 12

G

H)[

G

K

R

J (12)

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- 74 -

In preceding studies, .there have been two well known techniques for

calculation of stiffnesses: one of them deals with the ground statically and

the other deals with the ground dynamically being based on the application

of the wave propagation theory. Aiming for practical uses in dynamic

analyses of structures, the coefficient of subgrade.reaction, from which the

stiffnesses can be calculated, Is defined as a linear relation between stress

and strain of soil and expressed analytically as a function of stress of

ground, the shape of a foundation and its area. In addition to effect of

spring, the interaction also contains the effect which is known as dissipa-

tion of energy and by which the effect of spring is decreased and effect of

damping resultingly grows. For the'purpose of estimating the effect of

damping capacity, the stiffness must be calculated considering dynamic

properties of the interaction.

This problem of dynamic properties of interaction has been solved using

the experimental results of forced vibration tests by means of exciter.

This has been also analytically and numerically expressed based on the theo-

retical displacement of foundation on an elastic semi-infinite caused by a

harmonic force as expressed by H. Tajiml and T. Kobori-R. Minai.

7/25/2019 CDNJ04820ENC_001


7 Γ )

As the other method for calculation of damping capacity without theoreti

cal calculations, some values of damping ratio are determined by engineering

Judgment and using this result the stiffness and damping capacity are commonly

calculated.

Because of lack of sufficient space to describe in detail the theory of

complex stiffness expressed as a function of exciting frequencies, only the

numerical results calculated by Tajiml's theory will be introduced. The

abovementioned Investigations made by Tajiml and Koborl - Minai, described

the interaction between foundation and ground on the underside of a founda

tion,

but theories regarding effect of surrounding ground on the lateral

sides have not yet been established.

Therefore, for practical purposes the resistance effect of surrounding

ground must be commonly estimated by means of coefficient of subgrade reaction

and some values of damping ratio are determined by engineering judgment and

stiffness and damping capacity must be calculated using these damping ratios.

5. EQUATION OF MOTION

The structures are idealized by the lumped mass-spring system shown in

76

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[KJ

"rr

+

i^sr k

S

s

kr.? + ík

i^rs Κ

Γ 2

T

i*

r

2 —

K

ri * l*ri

r,

+ iki

+

ik¿

s

k

S2

+ ik

S 2

— k

sl

+ ik¿i — k

rn

+

ik¿

n

k

2

s + i

k

2

r

k

2s + i

k

¿s k

22

+

i k¿

2

™

k

2i

+

i

k

2i —

k

2n + l

k

2

n

k

ir

+

i

k

'ir

k

is + i

k

is

k

l2

+

i

k

i2 —

k

ii

+

i

k

ii

k

ln

+

l

k

in

K

n r

,

Τ ΐΚ,,η *^n<z * l^rit; *^η·5 ** ì^n-ì ^ril * ^ r

—

kr

(20)

nr

T

i -nr

K

ns

T

i

K

ns

K

n2

T

i

K

n2 ~

K

ni l

K

ni ~

K

nn i

K

nn

/■Ηβη ri η

k

rr+i

k

rr = k

R

+ik

R

+/ (k

s

(y ) +

3

(y ) ) (y-H

BE

)

2

) dy+ Σ Σ

(k

r2rl

tk'

2I

,,

)

•J

o

r,=2

r,=2

(H

r 2

-H

B E

)(H

r l

-H

B E

)

fHBn

HBn η η

k r s + i k f s = / ( k

s

( y ) + i k ¿ ( y ) ( y - H

B E

) ) dy+ Σ Σ ( k

r 2

r i

+

ik

r

2π

*

r, = 2 r,=2

kss

+

i

k

ss

(Η

Γ 2

-Η

Β Ε

)

fHBE η η

■ /(kg(y)+

1

k

s

(y)) dy+ Σ Σ (k

r2

ri

+

ikr2ri)

J

°

r

2

=2 r,=2

(21)

(22)

(23)

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- 77 -

sign (χ) expresses the complex eigenvectors. Substituting Eq. (30) and the

_1

equation {x) = [MJ

2

(Y ) into Eq. (29), Eq. (29) is changed as follows:

_1 _1

[KJ[MJ

2

{Y) = A

2

[MJ[MJ

2

{Y) '·'■ (3D

1

~ 2

Premultiplying both sides by [MJ :

_1 _1 _1 _1

[MJ

2

[KJ[MJ

2

(Y) - X

2

[MJ

2

[MJ[MJ

2

{Y)

or in other form [KJ{ï} = x

2

{Y) (32)

here [KJ is symmetric and [KJ is also symétrie since

_1 _1 _1 _1

[KJT = ([MJ

2

[KJ[MJ

2

T

= [MJ

2

[ KJ

T

[ MJ

2

= [KJ

now in order to obtain the solution, Eq. (32) must be analyzed as follows:

The method of analyzing Eq. (32) consists of two parts;t5J

(a) 1st step: The given matrix [KJ is reduced to almost triangular

(Hessenberg) form [Hj by elementary similarity transformations.

It follows that

[TJ

_1

[KJ[TJ = [HJ (33)

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Orthogonality Property of Complex Matrix ^ J

Matrix [MJ and [KJ are respectively real and complex symmetric matrix,

let Xi and Xj be eigenvectors and let (x

1

) and {χj) be complex eigenvectors

corresponding respectively to Xi and Xj. Here 1

/

j

Then [Kj(xi) = Xi[Mj{xi) and [ Kj Uj l = Xj[MJ(xj) (39)

If the first equation in premultiplied by

(x-|)'

r

and the second by

{xj)

T

,

the following are respectively obtained:

{Xj}

T

[KJ{Xi) = Xi(Xj}

T

[MJ{xi) (40)

and {Xi)

T

[KJ{Xj) = Xj(Xi)T[MJ(Xj) (Hi)

Now, if transpose of each side of Eq. (40) is taken, remembering that [KJ

and [MJ are [KJ

T

= [KJ and [MJ

T

= [MJ, the following equation is obtained:

(Xi)T[Kj(xj) = Xi{Xi)

T

[Mj{Xj) (12)

Finally, substractlng Eq. (Il) from Eq. (42), the following equation is

obtained.

(Xi-Xj){x

1

)

T

[MJ(x

j

) = 0 (43)

Therefore, since X^

f*

X j ,

by hypothesis, it follows that

ÍXijTtMjíXj) = 0 1

fi

J (44)

On the other hand, since X^ = x., by hypothesis, it follows that

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- 79 -

Substituting Eqs. (16) and (47) into Eq. (17)

m m

Σ [Mj(x,)q¡ + Σ [ Kjtxjjq, = -x

0

Σ [Mjíx^JS, (49)

J=i j=i j=i

J J

Then premultlplying by the transpose of an arbitrary modal vector Xi

T

which

is not the same as the j-th mode and taking advantage of the orthogonality

properties,

a single uncoupled equation of motion for j-th mode is obtained.

q, + X

2

q] = -BjXo (50)

or q\

0

+ X,

2

= -x„ (51)

where q j

0

= Qj/ßj

For arbitrary loading earthquake motions, the solution of each modal

response Eq. (51) can be performed by the Duhamel Integral.

When the modal responses consisting of accelerations, velocities and

displacements for all significant modes have been determined at any time

"t",

the response values of mass points at this time are then obtained by

Eqs.

(52), (53) and (51).

m . m .

(χ) = Σ ßj q

J o

(xj) (52) (χ) = Σ ßj q

J o

( Xj } (53)

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Calculation of Dynamic Properties

The stiffness matrix of each group and stiffnesses of ground were indi

vidually calculated as follows:

(a) The stiffness matrices (real part) of Reactor Building (G-l) and Dry

Well (G-2) were calculated by F.E.M. and those of Shield Wall (G-3),

Truss

(G-4),

Skirt

(G-5),

Reactor Pressure Vessel (G-6) and Stabilizer

(G-7) were calculated by the bending shear deflection theory.

(b) The complex stiffness matrix of each group was calculated by Eq. (12)

putting the damping ratio

(QH)

of each group shown in Table IV into

this equation. The damping ratio of each group can be estimated from

the damping ratios shown in Table III considering the material and

structural type of each group.

(c) The complex stiffness of ground was calculated in two ways.

o Case A : The real part of complex stiffness was approximately calcu

lated by the method based on Tajimi's theory and the imaginary part of

this was done putting the damping ratios into Eq. (1 2). The damping

ratios used in this calculation are shown in Table III.

o Case Β : The complex stiffness was theoretically calculated by the

- 81 -

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approximate fundamental natural circular frequency by hypotheses, and that

the stiffness of ground is constant as a result.

Table VI shows the comparison of natural periods, and damping ratios

among three cases of whole structures.

Fig.

6 shows the first three participation...functions of Reactor Building.

Fig. 7, Fig. 8 and Fig. 9 show those of the entire structure for case 3.

Calculation of Dynamic Response Values for Earthquake Motions..

1 — ^ —

The dynamic response values were calculated for case 2 and case 3 to

make -clear the relationship between the damping ratio and the response.

(a) El Centro I9I0 N-S component (max. acceleration = 300 gal)

(b) Taft 1952 Ε-W component (max. acceleration = 300 gal)

Fig. 10 and Fig. 11 show the max. values of displacement and overturn

ing moment for case 3 respectively. Fig. 12 shows the comparison between

case 2 and case 3 for the overturning moment to the Taft 1952 Ε-W component.

Fig. 12 shows that the values of case 3 are smaller by about IO? than

those of case 2.

9. CONCLUSIONS

82

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TABLE I OASEA :

STIFFNESS

of

GROUND

(Kí(txk) Κκ( t<n/rnd))

KM

K S I

K R

STIFFNESS

Ζ 3 4 Χ 1 θ ' + ; Ζ 5 4 Χ 1 θ '

1 0 * Χ 1 0

5

+· 3 .06X10*

2.44X10

l l

+ i Z 4 4 X 1 0

TABLE Π OASEB :

STIFFNESS of GROUND

fKe(t/d») K R ( torrad)»

K B ?

K a t

K»

S T I F F N E S S

Z S 4 X 1 0 ' + ι 23 4 X 1 0

1

S 2 4 X 1 0

5

+ .

W 8 X 1 0 *

Z 4 2 X 1 0

U

+ i 2 5 0 X 1 0

U

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- 83 -

TABLE IV DAMPING RATIOS

of

GROUPS

GROUP

NAME

G - 0

G - 1

G - 2

G - 3

G - 4

G - 5

S T R U C T V R E N A M E S

of

NVCLËAR POAQSR PLANI

F O U N D A T I O N

R E A C T O R B U I L D I N G

D R Y WE L L

S H I E L D W A L L

T R A S S

S K I R T

R E A C T O R P R E S S U R E

Wfä

skçwn

T A B .

m

0.0 5

0.01

aos

0.0 1

0.01

84

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TABLE \J DOMPARISON TABLE :

PERIODS and DAMPING RATIOS for THE ENTIRE STRUCTURE

MODE

AS.

1

2

3

4

CASE

1

PERIOD

( S E C )

0.1927

O r0 3 2

0.0850

0.0799

DAMPING

RATIO

( A )

Ü.0498

O0 49 5

0035?

aa ι οι

CASE

2

PERIOD

( S E C )

0 2 4 8 8

0 1 2 2 3

O 0 8 5 2

O 0 8 3 5

DAMPING

RATIO

( A )

0 0 4 9 8

O0498

O0364

O049 1

CASE 3

PERIOD

( S E C )

0 2 4 2 1

0 1 19 1

O0851

O0812

DAMPING

RATIO

( A )

0 0 7 9 9

O0934

O0362

0 1 2 3 2

85 -

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Xn

nRWP AXn

86 -

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1 £ ί 2 0 0 0 _ _

s-:»

llL

ι

ν

;.- l> ι.· 1/ .M M V * _ > L V _ M ¿ | ,

G1 REACTOR BUILD ING

G 2 DRY W E L L

G 3 S H I E L D W A L L

G6 REACTOR PRE SSU RE

V E S S E L

8 7

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BASE GROUNO

F I G .

4

- K v ^ - K a - i K a

2

Ksi ·

i Ksi

g-.,g

t

-ΛΛΛ-

EQUIVALENT FOUNDATION-GROUND SYSTEM

U) : NATURAL CIRC ULA R

FREQUENCY

UJi:

1ST

MODE NATURAL

CIRCULAR FREOENCY

FKÎUJ)

. Fk(.uj).i

F

Ctw)

88 -

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A

Φ&

2.0

f

G - 1

JJg TN|2I . .' 1 » .■

\x

ι

„.

-τ—

Φ

G/-2

IO ·

Ρ—t—

P A R T I C I P A T I O N F U N C T I O N

1

FREQUENCY

« ι 2 0 .670» I0

0 . 1 0 8 » ' O

3

FREQUENCY 0 .260»1θ ' 0 .207«1θ'

P E R I O D

( S E C ) 0 . 2 42

D A M P I N G

R A T I O 0 . 0 7 9 9

?

:

G-6

* -4-

- i — Φ

G - 4

16

- 89

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ΐο

fl

ίο

P A R T I C I P A T I O N F U N C T I O N

: G - I

6

FREQUENCY"? O.W5*»tf 0.395» '0

F U E Q U E N C » O . 7 M »I O ' O J 6 T > O

PERIOD ISEO

O.OaS

M M P i í . 0 « A T i C 0 0 3 6 ?

1

G -

6

. . .

I

I

-+-■

y ¿

s

G-2 I ,

"o

h

ι φ ιι

>

ι.'

^^V

G - 5

0 - 3

1

90

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ì 4

2

3

B - l

G-1

MAXIMU M OVERTURNING MOMENT

. ■ El CENTRO 1940 NS

■ l A F I 19S2 EW

DESIGN

MAX. AC C . - » O G « .

10 »

10" tm

MOMENT

- 91 -

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D I S C U S S I O N

Q

T. H. L E E , U. S. A.

In t h e s o i l - s t r u c tu r e i n t e r a c t i o n , t h e r a d i a t i o n d a mp in g i s a f u n c t io n o í t h e e x

c i t i n g f r e q u e n c y . In y o u r T a b le I II , c o n s t a n t v a lu e s w e re g iv e n fo r r a d i a t i o n d a m p in g . A r e

th e s e v a lu e s t h e a v e ra g e v a lu e s o v e r t h e f r e q u e n c y r a n g e o r t h e ma x imu m v a lu e s ?

J . J ID O , J a p a n

T a b le III i n my p a p e r s h o w s th e d a m p in g r a t i o s to t h i s p a r t i c u l a r s t r u c tu r e i n

J a p a n . I s u p p o s e th e o t h e r s t r u c t u r e s w o u l d h a v e d i f f er e n t d a m p i n g r a t i o s f r o m t h i s e x a m p l e .

o

K. UCHIDA, Japan

Y ou u s e th e a b s o lu t e v a lu e s a s th e e x p r e s s io n of d i s p l a c e m e n t s i n y o u r p a p e r . I

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K 2/5

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ASEISMIC DESIGN OF NUCLEAR REACTOR BUILDING

STRESS AN AL YSIS A N D STIFFNESS EV AL UA TIO N

OF THE ENTIRE BUILD ING

BY THE FINITE ELEMENT MET HO D

Y. TSUSHIMA, Y. HAYAM1ZU, K. NISHIYAMA,

Takenaka Komuten Co. Ltd.,

Technical Research Laboratory, Tokyo, Japan

ABSTRACT

The purpose of this paper is to evaluate the spatial characteristics of

stress and stiffness of a nuclear reactor building having a complex wall

arrangement, a normal tendency of nuclear reactor buildings, by the finite

- 94 -

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(3) share ratios of each wall and shell for entire external force

(Ί) stress distribution of each wall and shell

In order to solve these problems, In this paper it Is attempted to

analyze stress distribution and stiffness evaluation of the entire structure

by using the finite element method and its application. This analysis method

can evaluate the spatial characteristics and this is the new method of the

authors.

In this method, eacli wall and shell is treated individually as an

assembly of flat elements, and Is reduced to a small order stiffness matrix

which has the vector of nodal displacements on floor level and on vertical

edge surface at some intervals. Subsequently, these small order stiffness

matrices are superposed considering the actual condition of the entire

structure. This superposed stiffness matrix represents a stiffness matrix

of the entire structure. Giving appropriate forces to the entire stiffness

matrix, each nodal displacement mentioned above is computed by Gauss Reduc

tion or other method. Stress analysis of each wall and shell can be com

puted from these nodal displacements.

2.

95

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using the finite element method and are reduced to the smaller order stiff-

ness matrices with the nodal displacements, 2 ^ 5 in lateral direction "u"

and, 6 ï. 13 In vertical direction "v" is shown in Fig. 1 (C ).

Let [KA],

[ K

B

] ,

[ KC ], [KD] and [χΕ] be the five stiffness matrices of

each structural component, and [K

A

] = [ K

E

], [K

B

] = [ K

D

] by assuming the

symmetry of structural model on the vertical center plane at right angle to

u direction. Then the stiffness matrix of the entire structural model

becomes as follows:

[ K

S

] = [KA] + [KB] + [

K

C] + [KD] + [KE] (1)

In which [Kg] is the stiffness matrix of the entire structure. The relation-

ships between the external forces and the displacements become respectively

[[ΚΑ] + [κΕ]](ν

Α

> = [ K A E ] (

V A ) >

• = [[KB] + [KD]]

VA

BD

11'

BD

21'

BD

Tf

B D

K

12'

BD

K

22'

BD

K

BD

K

13

BD

2 3

„BD

'

u

V

A

V

( ? )

(3)

- 96 -

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and the stiffness matrix of Eq. (6) represents the stiffness matrix [Kg] of

Eq. (1) .

Hence,

when the external forces of Eq. (6) are known, the lateral

dis

placements of each floor (u) and the vertical nodal displacements ( v

Ä

) , (Vg)

can be decided solving this equation, and the stress distribution of struc

tural components can be computed easily by using these solved displacements.

It is needless to say, in the stress analysis, these displacements are given

at the specific nodal points 2·*·13 as the boundary condition.

Then,

multiplying the stiffness matrices of the structural components

by these displacements mentioned above the results obtained are the external

forces acting on these structural components. By computing the ratios of

these acting external forces to the total external forces acting on the

entire structure, those ratios are the share ratios of lateral external force

when the deformation of floor slab is ignored.

Next, if (q^l and (qg)are null vectors, eliminating the vertical nodal

displacements fv^} and (v g) , the stiffness matrix of Eq. (6) is reduced to

the stiffness matrix concerned with (p) and (u) , i.e. it becomes as follows:

- 97

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Now the blocK diagram showing the computational procedure of this

method is shown in Fig. 2.

3. EXAMPLE OF ANALYSIS

3-1. Outline of Building

This model is of a nuclear reactor building of BWR type having power

generating capacity of 500 MW In Japan.

The plan of ground floor and the section A-Α in X direction is shown in

Fig. 3(a) and (b). This building has four stories having 15.5Ί m total

height above ground level, and two stories having 16.70 m total depth below

ground level and it is supported by a stiff and deep shale layer. This

building is completely square in plan at the lower part having a length on

one side of 63.00 m, while the upper part is also considerably symmetrical

and thus the influence of tortion may be'negligible. The structural compo

nents of the building are made almost all of reinforced concrete except for

the steel roof truss. Major items of specification of concrete are as

follows

:

- 98 -

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The spatial arrangement of walls and shell which will be effective as

resistant structural components for external forces is shown in Fig. Ί. The

structural components (walls and shell) numbers are 32 for the X direction

and 29 for the

ï direction. The thicknesses and the divided mesh condition

of'

a typical wall and shell In these components are shown in Fig. 5 (a) and

(b).

These figures show the number of the floors.at the left hand and the

number of the selected nodal points to satisfy the vertical nodal displace-

ments on the edge surface of structural components. AI30, a "t" at the

right hand and inside of these figures shows thickness (mm). Of course,

in these structural components, the most numbers of nodal points, are in the

flask type shell as shown In Fig. 5(b), and it is needed to solve the about

3000 order simultaneous equations for a half part of this shell.

After superposing all the stiffness matrices of the structural components

reduced to the matrix order of the lateral displacements and the selected

vertical nodal displacements, the stiffness matrix of the entire structure

considering spatial characteristics is obtained, and In this case the Btiff-

ness matrix is of the order of I80 χ 180 square. Of these ordere, 13 are the

lateral displacements and the remaining orders are the vertical nodal

dis-

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shear theory.

c) Stress Distribution of Walls

The principal stresses and reactive forces distributions of W2 , WC and

WH are shown in Fig. 8 (a ), (b) and (c) , respectively.

Fig.

8 (a) shows the principal stress and reactive force distribution

which are parallel to the direction of lateral force. In this figure, it

will be seen very clearly that the tensions flow from the right side of the

top to the left side of the base and compressions.flow from the left side of

the top to the right side of the base. It will be seen also that the stress

values near middle stories are larger than those near top and bottom stories

since this wall has a narrow width above the vicinity of middle stories. The

stress distribution around the opening is considerably disturbed and the

values are large in comparison with those of other parts. On the other hand,

the reactive force distribution under the base slab shows a nearly triangular

distribution except the reactive force of the compressive edge. This reac

tive force of the compressive edge becomes very large because, of considera

tion of a condition completely fixed under the base slab. This distribution

- 100 -

can be performed In a short time. But in the bending-shear theory, the

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entire structure is replaced by a single bar which is subjected to bending

and shear stresses, and walls at right angles to the direction of lateral

force, subwalls and a flask type shell which has the characteristics itself

are treated in the same way as the normal resisting walls. Therefore, the

results of analysis using the bending-shear theory may tend to overvalue

stiffness of the entire structure for the actual condition.

While the method explained In this paper has some unsolved problems re

maining at present, by using this method, stiffness of the entire structure

can be obtained considering the spatial characteristics of wall arrangement.

These stiffnesses are compared with the periods and the participation

functions obtained by performing eigenvalue analyses which employ the same

weight distribution. This weight distribution is shown in Table II. The

eieenvalue analysis Is performed by using the Jacobi's method.

The results of analysis are shown in Table II and Fig. 10 for two stiff

nesses obtained by using the bending-shear theory and the method explained in

this paper. Table II shows both periods from the first order to the seventh

order for the X and Y directions. Fig. 10 shows both participation functions

- 101 -

bending-shear theory In stiffness evaluation is apparent as shown in Fig. 10

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and Table II.

From the results analyzed only for the specific structural model treated

in this paper, which may not necessarily represent the general tendency, it

can be seen that the use of this method makes it highly possible to carry out

a reasonable structural design.

ACKNOWLEDGMENT

The authors wish to thank Dr. H. Tajimi, Professor of Ninon University,

Mr.

F. Horle, Chief Research Officer of Odaka Laboratory, and Mr. I. Funahashl,

Research Manager of Takenaka

.Technical.

Research Laboratory, for their helpful

discussions and suggestions, and .the cooperation of structural engineers

engaged in the design of the nuclear reactor building mentioned in this paper.

REFERENCES

[I] H.C. Martin: An Introduction to Matrix Methods of Structural

Analysis. McGraw-Hill, I966

102

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( B E G I N )

103

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d>"

Φ-

®~

© -

© -

© -

© -

© -

y W2

^ - W H /[ ,—WC

YJ-WSl j l / J / l

Λ

| \ ^ l · Jwf\

1T|—i—

\

J Å \ \

-fl—

Juiz]

DIRECTION

of

LATERALEXTERNAL

FORCE

CD-

104

A

7/25/2019 CDNJ04820ENC_001


L

3

s

s

(5

ΰ

5?n

b

I S S

51S

J)

12.6

I l i

-

i t i

®

s

Ifi.

S 17

1 «

3

223

15(1

©

s

s

7â S

a i

E

ÎO

îs

22.'

£

\

1

> ?

©

«*

1 * —

Ί 21f,

A ' -K

c

r»

®

FIG. 6 SHARE RATIOS of

LATERAL EXTERNAL FORCES

(PERCENT)

105 -

7/25/2019 CDNJ04820ENC_001


M 5 « ι » ω ι ι

- 106

7/25/2019 CDNJ04820ENC_001


so tggi^AwT)

- 107 -

7/25/2019 CDNJ04820ENC_001


\ N

vS.

7ni

t , h

V

\ /

l i

M

1 i

' 1

i

I ι /

y

f i

h

( ist

lì

- 1 0 - 2 0 - 1 0 0 10 20 30

X-DIRECTION

By Evaluation of Bonding Shear Theory

By Evaluation of MMhod in This Paptr

Y- DIRECTION

FIG. 10

COMPARISON of PARTICIPATION FUNCTIONS in

X.and

Y DIRECTIONS

ï OH

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Q

A

D I S C U S S I O N

H . W O L F E L , G e r m a n y

H ow d o y o u c a l c u l a t e t h e d y n a m ic l o a d s c a u s e d by e a r th q u a k e ?

Y. HAYAM1ZU, Japan

G e n e ra l l y , w e o b t a in t h e d y n a mic l o a d s by t h e fo l lo w in g m e th o d :

1.

W e c a l c u l a t e t h e d y n a mic l o a d s by t h e s t a n d a rd c o d e i n J a p a n .

2.

B y u s in g a n a n a ly t i c a l m o d e l e v a lu a t e d b y th e b e n d in g - s h e a r t h e o ry a n d th e o th e r m e th o d

w e p e r f o r m t h e d y n a m i c r e s p o n s e a n a l y s i s , a n d f r o m t h e r e s u l t of t h i s r e s p o n s e a n a l y s i s ,

w e o b t a i n t h e m a x i m u m s h e a r f o r c e s a t e a c h f lo o r l e v e l . W e a s s u m e th a t t h e s e s h e a r f o r c e s

a r e t h e d y n a mic l o a d s .

K 2/6

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ASEISMIC DESIGN

FOR JAPAN EXPERIMENTAL FAST REACTOR (JOYO)

Κ. AKINO, M. KATO,

The Japan Atomic P ower Company, Tokyo, Japan

ABSTRACT

This paper explains the aseismic design of Japan Experimental Fast Reactor (50

MWt)

called "JEFR" or Japanese nickname "JOYO" which is being constructed at Oarai site in

I bararli Prefecture, along the shore of the Pacific Ocean.

Even though the aseismic design of JOYO Is being progressed now in detail, fundamental

- 110 -

Association in April, 1970, and was edited by a special committee, explains the above philoso

7/25/2019 CDNJ04820ENC_001


phy and method, and a member of the committee will introduce this Technical Guidelines at

the Conference. Therefore, this paper does not touch upon such general philosophy, criteria

and method regarding the aseismic design of nuclear facilities, and it refers only to special topics

on the aseismic design appeared in the project of JOYO.

2.

DESIGN EARTHQUA KE

2.1.

Special Site Condition

In the case of nuclear power plants which are being constructed or planned in Japan,

sound rock layers for bearing heavy reactor building sufficiently to withstand strong earth

quakes are searched in the course of site selections. In the case of JOYO, even though it does

not generate electric power, its size and weight, structural c om plex ity, construction cost and

safety requirements are comparable w ith those of commercial nuclear power plants, and a

subsoil profile of JOYO's site shows very deep sand layers up to 162 m below the ground

surface. However, since the bottom of reactor building foundation was located 32 m deep

fro m the ground surface, this project presented us w ith a new problem how the design earth

quake be selected considering an effect of very thick sand stratum between the bottom of

building and the base rock.

I l l

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( 1 )

Deep sand stratum of 22 m below the ground surface can be regarded as a vibra tory base

as well as the base rock of shale, because both records represented a constant velocity

spectrum of ground movements as shown in Fig. 1.

(2) Predominant periods of the ground are 0.1 5,0 .5 and 1.1 sec on the surface, and 0.5 and

1.1 sec at the elevation of 22 m below the surface as shown in Fig. 2. Those periods mean

that 0.15 sec is the natura l pe riod o f the over burd en an d 1.1 sec and 0.5 sec can be regard-

ed as the natural periods o f who le sand strata corresponding to the first and second mode

vibrations, respectively.

Therefore, -2 2 m layer can be chosen as the vibratory base instead of -1 65 m layer, bu t

to perform conservative calculation the latter elevation was defined as the vib ratory base.

2.4. Amplification of Sand Stratum

In order to evaluate the amount of amplification of ground movement due to the exist-

ence of deep sand strata, a theoretical calculation and actual observation were carried out.

In the theoretical calculation by means of the theory for multilayer reflections, the

following matters were considered:

( 1 ) Kanai's report [4 ] was referred to ,

(2) The base rock located -1 65 m be low the surface was regarded as the vibrato ry base,

- 112 -

2.5. Selection of

Earthquake

Waves

As apparent fro m the previous investigations, tw o differe nt tendencies should be taken

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into consideration for selecting the design earthquake waves, the one has the spectral peak at

a range of 0.3 - 0.4 sec periods and the othe r peak at 1.0 - 1.1 sec. Lockin g for many records

obtained by Strong M otion Accelograms and after exam ining the m, the follow ing two waves

were selected and their values of the maximum accelerations were decided respectively to

normalize them for the purpose of designing JOYO:

El Centra NS, 1940, Maximum Acceleration = 150 gals

Akita Record EW, 1964 (obtained at Niigata Earthquake on building of A kita Prefec

tura Government), Maximum A cceleration — 100 gals

By the way, Fig. 4 represents the response spectra fo r 5% of critic al dam ping of the

above two design earthquakes.

3. PLANT LAYOUT

An original conceptual layout of buildings indicated that the reactor building together

with the containment vessel was one individual structure, and several other buildings, in which

many A class facilities were supported, were arranged around the reactor building. However,

the bottom of the reactor building foundation is located at -32 m below the ground surface

since an elevation of the operating floor has to be coincided with the ground surface for con

- 113 -

duce rocking and swaying v ibration modes can generate some amplifications in a broad range

7/25/2019 CDNJ04820ENC_001


instead of selecting plural input waves.

There are several references presented by Ta jimi 16) , Timoshen ko et al. | 7 ] , Tor ium i

[81 etc. to account the spring constants, but those formulas gave different results, namely,

It can be said that wo rking o ut th e spring constants is an uncertain pro blem , therefore setting

a certain range for the spring constant is an advisable technique.

The calculated numbers of ΙΚβ and Ks obtained from the above formulas and averages

of these numbers were regarded as corresponding to the case of the hardest soil condition,

and one half of th e above averaged numbers were regarded as correspo nding t o the othe r case

of the softest soil condition. Duplicate response calculations applying to El Centra Earth

quake wave fo r bot h soil cond itions w ere perfo rm ed, and the designs of a ll A class item s have

been required to satisfy the both cases.

4.3.

Calculated Results

Calculated results of the response analyses were shown in Fig . 6A fo r the fir st m odel

and in Fig. 6B for the second model. With respect to the first model, the dynamic response

analysis gave an insignificant result for the design of the buildings and containment vessel,

compared with the distribu tion of respondent acceleration and the static requirement wh ich

was defined as the seismic coefficient represented by the step-wise full l ines in Fig. 6A. How

114 -

F U E L A S S E M B L IE S A N D C O N T R O L R O D S

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5 . 1 .

Capabi l i ty of Scramming

The m os t i m po r t an t p ro b l em i n t he ase ism i c des ign o f JO YO was whe t he r o r no t t he

sa f e t y rods can be i nse r t ed i n t o t he co re whe n a des t ruc t i ve ea r t hqu ake exc i t es t he reac t o r

bu i l d i ng . D yna m i c behav i o rs o f key i t em s (see F i gs. 8A a nd 8B) re l a t i ng t o t h i s p rob l em w i l l

be as f o l l o ws :

(1 ) Th e reac tor vesse l moves togeth er w i th the con crete s t r uct ure ,

(2 ) The co re ba r re l and co re cove r s t ruc t u re , wh i c h are ve r t i ca l can t i l eve rs , m ove indepe nd

en t l y and som e re l a t i ve d i sp l acem en t be t ween t he t op o f ba r re l and t he bo t t om o f cove r

s t ruc t u re occu r i n sod i um coo l an t , and

(3) Th e hexag onal fue l and b lank et assembl ies lean on the core barre l ow ing to the ex is tence

o f c l ea rances be t ween t he assem b l ies , and t h i s am oun t o f de f o rm a t i ons is c r i t i ca l .

5 . 2 . D e s i g n M o d i f i c a t i o n

in the or ig ina l des ign, there were no pads a long the outer sur face of hexagonal assem

b l i es cons i de r i ng bow i ng de f o rm a t i on , cha rg i ng and d i scha rg i ng , and an accum u l a t i on o f 3 . 2

m m c l ea rance be t ween each assem b ly i n t i m a t ed m uc h de f o rm a t i o n o f t he assem b ly co l um ns .

I n o rde r t o es t i m a t e t he am oun t o f t he above de f o rm a t i on , num erous t heo re t i ca l ca l cu l a t i ons

- 115 -

As an exam p l e , t he f l oo r response spec t ra f o r t he 4 t h f l oo r wh i ch co r respon d t o an e leva -

7/25/2019 CDNJ04820ENC_001


t i o n suspend i ng t he reac t o r vessel and f o r 1% o f c r i t i ca l dam p i ng a re show n i n F i g . 9 . C om pu t e r

ca lc u la t io n dr ew the curves A for the hardest so i l and Β for the sof te st , and the th i rd C was the

a r t i f i c i a l des i gn cu rve . I n m ak i ng t he cu rve C t he f o l l ow i ng , we re t aken i n t o cons i de ra t i on :

(1) H i l l I cove rs t he e l as t i c v i b ra t i on m ode o f t he reac t o r bu i l d i ng wh i ch appears i n t he cu rve

B,

(2 ) ¡1 cove rs t he roc k i ng v i b ra t i on m ode wh i c h is a f f ec t ed m a i n l y by chang i ng the sp r i ng

cons t an t s o f so i l ,

(3 )

III

was d raw n judg i ng f rom t he response due t o the Ak i t a wave ,

(4 ) The l e f t f oo t co r respo nds t o t he m ax i m um response acce l e ra t i on o f t he bu i l d i ng a t the

same e levat ion, and

15) The r i gh t f oo t co r responds t o t he m ax i m u m response d i sp l acem en t o f t he bu i l d i ng due t o

E l Cen t ra wave .

7 . S O D I U M C O O L A N T P I P IN G

LM FB R p i p i ng des i gn needs a pecu l i a r de l i be ra t i on ow i ng t o i t s h i gh t em pera t u re , t h i n

p i pe th i ckness , doub l e -wa l l ed p r i m ary sys t em and ase ism i c supp or t s . F i r s t o f a l l , a w i n d i n g p i p -

ing arrangement was made to reduce thermal expansion s t ress as low as poss ib le for the main

- 116 -

9 .

C O N C L U S I O N

Co ns t ru c t i on w or k o f t he c on t a i nm en t vessel is be i ng ca r r i ed ou t and eng i nee r i ng de t a i l

7/25/2019 CDNJ04820ENC_001


des i gn o f m any com ponen t s a re a l so p roceed i ng now . Som e ou t com es wh i ch have been so l ved

o r con c l ude d up t o t h i s da t e i n t he p repa ra t i on and des i gn s tages a re m en t i o ned be f o re . We have

a respo ns i b i l i t y f o r f i na l i z i ng t he des ign and cons t ruc t i on o f JO YO , and we do no t kn ow w ha t

new bo t he rsom e p rob l em s i n t he ase i sm i c des i gn m ay a r i se i n f u t u re , bu t we shou l d f i nd app ro

pr ia te so lu t ions on a l l such cases to the best o f our knowledge.

10. A C K N O W L E D G E M E N T

Thanks a re due t o t he Power Reac t o r and Nuc l ea r Fue l Deve l opm en t Corpo ra t i on f o r g i v

i ng us pa r t i c i pa t i on i n t he p ro j ec t and f o r pe rm i t t i ng ou r p resen t a t i on o f t he paper i n t he

con f e rence .

Ap pre c i a t i o n is exp ressed t o eng i nee rs and resea rch m em be rs o f t he con t ra c t o r and venders

for the i r ass is tances and cooperat ions in the analyses and ca lcu la t ions and to Mr. T . Uchida for

h i s k i n d f u l subm i t t a l o f use f u l da ta f o r ea r t hqua ke obse rva t i ons a t Toka i and Oara i pe r f o rm ed

by Japan A t om i c Energy Research I ns t i t u t e .

R E F E R E N C E S

( 1 1 K . K a n a i e t a l . : " E x p e c t a n c y o f t h e m a x i m u m v e l o c i t y a m p l i t u d e o f e a r t h q u a k e m o t i o n s

1 1 7

7/25/2019 CDNJ04820ENC_001


JUT» A* ( .1. U l k

Fig. 1. Re ponse accélération spectra of earthquake

·■■> observed simultaneou sly around JMTR and JPD R.

I

Olla·

¿ I l l a · - ·

I ιΛ Λ/

1

'

'~>\

I'

/ . .s ~

Ι

y>

- 118

7/25/2019 CDNJ04820ENC_001


nini ,

A H M

n*

IMU

I M .

S Ì N T I M

HUI

Ml

I

n m · . A R K A M ta SK I < >M

» A H Í

(OUf.lM. SVSTKM

l

IH

M

(ï?)

ΠI

-;;.v"

ΙΠ

I >Ofr,v,M'.r

\\)ì

1

19

7/25/2019 CDNJ04820ENC_001


I I

: 33 ■JJ\Jj

¡

tACTOa III II Ms ,,

UI IU AL T BLI LDI NC

VESSEL

IEL

INTI»KALS

SOCL

ASE o r so r r rsT so« ,

s mixe s or sou.

Fig- 6A . Resul ts of earthquake response analys is for the f i rs t mo del , or fo r bui ldings

and containment vessel.

,.

V

■

i

- 120

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P R I M A R

1

I Ί Ι Μ A M P I I *

m τι

KT

121 -

^D ISPLAC EM EN T ι™;

7/25/2019 CDNJ04820ENC_001


· *

" — PERIOD

U M I

Fig. 9. Typical floor response spectra for reactor buildin

- 122 -

7/25/2019 CDNJ04820ENC_001


DISCUSSION

Q

C. B. SMITH, U. S.A.

In the vibration tests of graphite shielding:

a) How was the experimental structure tested ?

b) Did your measurements include observations of impact between individual graphite

blocks ?

c) In your opinion is impact likely to be a significant factor after the graphite has undergo

radiation damage ?

l

Ú

J^ K. A K IN O , J a p a n

a) It is the largest shaking table in the world, its owner is Desaster Prevention Center,

Japan Government, at Tsukuba in Ibaràgi Prefecture. The maximum loading capacity is

500 metric tons, and its control is carried out by displacement of either sinusoidal or ran

dom vibrations.

K 2/7

7/25/2019 CDNJ04820ENC_001


BERECHNUNG DER ERDBEBENSCHWINCUNCEN VON STRUKTUREN

MIT DER FINITE ELEMENT METHODE — MECHANISCHE

MODELLE VON KERNKRAFTWERKEN MIT EINBAUTEN

K MARGUERRE, M SCHALK H. WÖLFEL,

Institut für Mechanik,

Technische Hochschule Darmstadt,

Darmstadt, Germany

Prof.

K. KLOTTER zum 70 Geburststag gewidmet

ABSTRACT

For the analysis of seismic vibrations, complex structures are usually idea

lized by lumped parameter models. With the finite element method however,

- 124

1. Rechenverfahren allgemein

1

7/25/2019 CDNJ04820ENC_001


Am geeignetsten für die Schwingungsberechnung von

kompliz.

ert^n

Sti'ukturen ist das Verfahren der finiten Elemente, a) weil sich wa-

ndt Strukturen praktisch beliebiger Topologie erfassen lassen, b)

•weil bei guten Ansitzen die Genauigkeit bei gleichen Aufwand we.-.·.·n;

lieh iijher ist als z, B. bei der Abbildung auf ein System nit

dis-

kreten 'lassen (lumped parameter

model).

Als "Koordinaten" kön-en

Verschiebung- oder Kraftgrößen gewählt werden; wir verwenden hier

Verschiebungen und Drehungen,

kurz:

Verrückungen, Die

Matrixsch.·.

;1:

weise ist für ein computer-orientlertes Rechenverfahren besonders

zweckmäßig.

ì·i A

u

£

G

¿

y

ìl£.

n

_^£

r

_Schwlngungs g_le i chun¿

Der Zusammenhang zwischen dem Gesant-Vektor <fj,(t) der glob.'.j

einander unabhängigen) Verrückungen einer schwingenden Struk

aen zugehörigen globalen Kräften p

JO

<t) wird hergestellt dure

t u r ι.n>

1; ι C c

- 125 -

sammenhang zwischen den lokalen und den globalen Koordinaten, mit

Hilfe einer

7/25/2019 CDNJ04820ENC_001


Inzidenzmatrix I„ ;

dann zeigt sich, daß

L · -

transponiert - auch für die Erfüllung der

statischen Kcmpatlbllltätsbedingungen sorgt, mit dem Ergebnis [

S ] :

S - ΣΙ/S/L . (2)

ί

E , l í í , r i ¿ n _ t m _ a f c r ; l ^ e r i

Die Steifigkeitsmatrizen S^ der Elemente stellen den Zusammenhang

her zwischen den Rand-Verrückungen Γ und den Randkrüften ƒ :

V r'

r

r ·

Bei kontinuierlichen Elementen sind die Glieder s.. dieser Matrizen

1. a, transzendente Funktionen. Da die s,,. nicht nur unbequem sind,

sondern überdies in die Lösung der Gl (1) numerische Schwierigkelten

- 126 -

i. 3

¿osung_der_Schwiíigungsg^leichung

Der Weg zur Lösung der Gleichung (3) führt über die Eigenwerte und

7/25/2019 CDNJ04820ENC_001


Eigenvektoren des schwingenden Gebildes. Bei p = 0 geht (3) mit den

Ansatz <j(t) =

φ-e

über in das allgemeine Eigenwertproblem

(C -ω

ι

·Μ)·ψ

= 0

y

CO

für dessen Lösung heute ausgezeichnete numerische Verfahren zur Ver-

fügung stehen; man findet die Eigenwerte <A und die zugehörigen

Eigenvektoren A

t

.

Für p(t) i O baut man nun die Unbekannten <J(t) auf aus den 1.1-

genvektoren. Ist φ die Ν χ Ν - Katrix der Eigenvektoren, so ersetzt

man cj durch einen Vektor

η

vermöge der Transformation

<Jf(t)

= φ-tjU)

= Z f - n J < )

. (5)

Dank d e r O r t h o g o n a l i t ä t s r e l a t i o n e n

frC-φ^Ο

, ψ; · /1 ·ψ

κ

-O

jar

L*K ('I ')

- 127 -

selen. Dann entkoppelt die Operation (5) auch den Danpfungsanteil,

und man erhält für n eine Gleichung vom Typus

7/25/2019 CDNJ04820ENC_001


> r i u . - r j

L

* r

L L

- j

L

+ C

t t

- y _ = p i l i ) , (7)

die man mit Hilfe des Duhamelschen Integrals

A - é . ( t

_

T )

o

103t, das die Bewegung aus der Ruhe herau3 (Anfangsbedingungen

y(0) =

li

(0) = 0 ) beschreibt.

Erdbeben

Wie modifizieren sich die allgemeinen Überlegungen in dem besonderen

Fall der (Erdbeben-)Fußpunkterregung? Ausgangspunkt 1st die Gl (1 " ),

Schreiben wir für die an allen Fußpunkten gleiche Erregung ζ (statt

q ) , so lautet sie

- 128 -

von der Wirklichkeit im Detail zu sehr ab, und dann geht man zwecK-

mäßig einen Umweg: man rechnet aus q» durch Multiplikation mit den

örtlichen Massen iriy eine "dynamische Belastung" aus und bestimmt

7/25/2019 CDNJ04820ENC_001


dann statisch die zugehörigen Schnittkräfte mit Hilfe eines Modells,

das an den interessierenden Punkten die Wirklichkeit genauer be-

schreibt.

Soweit die "time history analysis". Nun aber weist beim Erdbeben die

erregende Funktion p(t) sc schnelle Vorzeichenwechsel auf, daß man,

um q(t) numerisch zu bestimmen,zu einer ungemein feinen Untertei-

lung der t-Achse gezwungen 1st, d. h. zu einem erheblichen Rechen-

zeit - Aufwand. Man beschränkt sich daher meistens darauf, die

Ant-

wort der Struktur auf ein Spektrum zu bestimmen (response spectrum

modal

analysis),

das a) für ein bestimmtes Erdbebengebiet allgemein-

gültiger 1st als ^r gend eine'gemessene p(t)-Kurve und b) sehr viel

weniger Aufwand erfordert.

Da das Spektrum-Verfahren aus der Erdbebenliteratur geläufig ist,

wollen wir nur eine Bemerkung machen zu der Frage der Resultiercndcn-

bildung. Was man ausrechnet ist für Jede Eigenform der Beitrag r>,

- 129 -

Stelle nicht eingehen auf die Wechselwirkung Gebäude-Boden, weil für

die Standorte in unserem Raum bisher zu wenig verlässliche Aussagen

7/25/2019 CDNJ04820ENC_001


über die Boden-Kennwerte vorliegen.]

3.1 Grundsätzliches zur_Modellabbildung

Die Wahl des Modells hängt ab von dem, was man sucht. Handelt es

sich um lokale Zerstörungswirkungen (Flugzeugabsturz), so ist das

Schwingungsverhalten an dieser Stelle — Wellenausbreitung — wichtig.

Anders beim Erdbeben; dort sind gefährlich (rufen Resonanzvergröße-

rungen hervor) nur Frequenzen unter 10, notfalls 15 Hz, also sind

nur die untersten Elgenfrequer.zen der KKW-Gebäude wichtig, d. h.

+) vgl. den Fußpunkt-erregten Einmassenschwinger, dessen Gleichung

lautet ra χ + c y = o

das Modell muß diese Frequenzen liefern. Die Feinstruktur einzelner

Teile zu berücksichtigen würde einen ungerechtfertigten Aufwand be-

deuten und kann überdies numerisch gefährlich sein. Beim Explosions-

stoß kommt es auf die Ausdehnung an: handelt es sich um einen Rohr-

130

f) Bei Doppelsymmetrie sind beide Schwlngungsrichtungen voneinan-

der und von der Drehung des Gesaratgebäudes um die Hochachse

unabhängig. Beim Balken trennen sich die beiden Biegungen und

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die Torsion,

Sind eine oder mehrere dieser Voraussetzungen nur näherung3weise

erfüllt,

so liefert die Betrachtung des ebenen (oder eindimen-

sionalen) Modells nur ein — vielfach allerdings ausreichendes -

Näherungsergebnis.

Oft auch sind bei einem Gebäude die Voraussetzungen für eine Ver-

einfachung der Struktur nicht in allen Gebäudeteilen gleichermaßen

erfüllt.

Es kann dann notwendig werden, das Gebäude oder einzelne

Teile für die verschiedenen Schwingungsrichtungen auf Jeweils andere

Modelle abzubilden.

3-_3_KKW-Einbauten

Gebäude und Einbauten führen gekoppelte Schwingungen aus, müssen

also als Gesamtsystem betrachtet werden. Sind Jedoch die Massen

- 131 -

Flg.

3 zeigt das Reaktorgebäude (und gleichzeitig das Schwingungs

modell) eines Siedewasserreaktors (BWR). Auch hier ist wieder die

Abbildung auf einen Balken möglich; da es sich Jedoch um einen un

7/25/2019 CDNJ04820ENC_001


symmetrischen Querschnitt handelt, müssen die gekoppelten Biege-

Torsionsschwingungen berechnet werden. Fig.

Ί zeigt zwei der

räum

lichen Eigenformen eines BWR von etwa 800 MW. Die tiefste Eigenfre

quenz liegt bei 3,6 Hz, Fig.

Ίο) und d ) zeigen den Verschlebungs-

und Beschleunigungsverlauf dieses

Gebäudes für das auf lm/sec nor

mierte und geglättete Beschleunigungsspektrum des El Centro Bebens

von

19Ί0 mit 7Ϊ

Dämpfung.

In Flg. 5 1st der Schnitt durch einen Siedewasserreaktor (800 MW)

gezeigt und in Flg. 6 das zugehörige Schwingungsmodell mit den zu

den 3 tiefsten Biegefrequenzen gehörigen Eigenformen, ferner in Flg.7

der Verschiebungs- und Beschleunigungsverlauf. Das Schwingungsmodell

1st hier eine aus mehreren Balken zusammengesetzte Struktur.

In einem Maschinenhaus sind die Steifigkeiten i. A. aufgelöst, Schei

ben sind kaum vorhanden, so daß als Schwingungsmodelle nur räumliche

(oder mehrere ebene) Modelle in Betracht kommen. Fig. 8 zeigt das

- 132

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Literatur

[lj Ahorner, L., H. Murawski und G. Schneider

Die Verbreitung von schadensverursachenden Erdbeben auf

dem Gebiet der Bundesrepublik Deutschland.

Zeitschrift für Geophysik, Band 36, 1970

[2] Hansen, R.J.

Seismic Design for Nuclear Power Plants.

M.Ι.Τ, Press, Cambridge, Mass. und London 1970

[3] Hurty, W.C . und M.F. Rubinstein

Dynamics of Structures.

Prentice-Hall, Nev/ Jersey

196Ί

13 3 -

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¿fe

irâkr

134 -

v^w

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räumliches

Schwingungsmodell

Fig 2 Sys tem der

Einbauten (PWR)

- 135 -

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Jm f

2

=

5,5Hz

= 16 Hz

- 136 -

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o C r -

- c

-c

0-

W

Γ

—

— )

Λ Λ

f, =3,6 Hz

»3=11Hz

SCH

13 7

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138

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139

DISCUSSION

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Q Y. HAYAMIZU, Jap an

1. By which me thod do you ana lyz e the e ige nva lue o f the ma tr ix ?

2 .

Up to how man y deg re es o f f ree dom can you ana lyze the e igenv a lue o f the m at r i x

fi^ H. W O L F E L , G e r m a n y

1. W e u s e t h e H o u s e h o ld e r Q R a lg o r i t h m .

2.

O u r e x a m p le s h a d a b o u t 50 d e g re e s of f r e e d o m.

Λ

Κ. OMATSUZAW A, Ja pa n

Could you te l l us the na tura l f requency of the tu rb ine founda t ion ? And how much

i s t h e ma x imu m re l a t i v e d i s p l a c e me n t t o t h e t u rb in e b u i ld in g ?

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K 3/1

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STATISTICAL TREATMENT OF SEISMIC LOADINC

ON REACTOR BUILDINGS AND EQUIPMENTS

A. AMIN, A.H.-S. ANG,

Department

of

Civil Engineering,

University

of

Illinois, Urbana, Illinois, U.S.A.

142 -

1 . INTRODUCTION

Random vibration concepts h ave been used in seismic analysts since the early

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fifties;

see for example G oodma n, Ros enblu eth, and Newma rk [1] . The proc edur e of taking

the square-root of the sum of squares of the maximu m modal respo nses, for exam ple, ¡s

based on certain approximat ions derived from the theory of random vibrati on. This leads

to reasonable approxima tions for regular and symmetrical buildings for which only the

lateral deflections and the story drifts are the quantitie s of primary interest. Howev er,

in the seismic analysis of nuclear reactor facilities, the systems are quite complex,

and response quantit ies other than displacements are also of signi fican ce. Moreov er, in

certain soil dynamic evaluations, the number of exceedances beyond a high stress level

is also of in tere st. Beca use of the compl exity of the syste ms involved in a reactor

facility, a direct random vibration analysis, therefore, appears to be potentially useful

and desirable.

In this paper the availabl e random vibration concepts that ar e of practical

signific ance for reactor facilities are summa rized, and specific numerical results

obtained therefrom are compared with those from a normalized set of recorded accelerograms.

This comparison demonstrates the validity of using a direct random vibration approach in

143

í n wh i ch v . ( t ) i s the r a te o f up- c ross ing a t l eve l χ = b and i s ob ta ined f r om the j o i n t

b

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v

b

( t )

=

/ *

f

x X <

b

' * , t ) d x

( 3 i

Assuming that

the

earthquake response

îs a

Gaussian Process,

the

design response

level

b

corresponding to an exceedance probability p (t.) can be evaluated numerically

from the above equations. Also if the response is assumed to be a stationary process,

the evaluation

of b can be

obtained

in

closed form

as

follows:

b = α σ

(k)

χ

where,

t J ° ' M O

0

« [2 £n ( Α - V )Y

n

<5>

q = - £n(l - ρ } (6)

144

vibration. Pertinent equations of relevance to the subsequent presentation can be

summarized as follows:

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G.(oj)

= u

2

G (ω) (7)

σ = ƒ G (lo)du) (8)

σ?

= ƒ

G-(ω)du

(9)

¡ η which G (ω) is the power spectral density of the response X(t) which is related to the

input earthquake spectral density G..(iij) according to eq. (10)

y

G (ω) = Η (ω) Η"(ω) G.. (D) (10)

Χ X X y

whe re Η is the compi ex frequency response function and - denotes the compi ex conjugate.

σ-

1

n

which

145

ι ι

B

x

k

v<YV \i

(,6)

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and,

do (17)

do (18)

σ

κι

=

ü

k¡t

tn

ƒ

-OÛ

co

ƒ

-00

G., (ω)

z

k

(o)z

Ä

(o)

IÚ

2

G ·.

(ω)

z

k

(o)z

¿

(o)

For most power spectral density functions, G » ( ω ) , developed to describe eart h-

quakes, the Int egrals In e q s. (17) and (18) can be readily evaluat ed by the method of

residues and using compl ex arith metic featur es available In most compu te rs. Finally,

the double summation

In

e q s.

(15) and (16) may

sometimes

be

appr oximate d, resp ectively,

by

1

n

Β

L· 0

- 146 -

O l y m p i a ( Α / Ι 3 / Ί 9 ) , an d T a f t ( 7 / 2 1 / 5 2 ) . T h e r e c o r d s w e r e n o r m a l i z e d t o h a v e t h e sam e

a r e a u n d e r t h e u n da m p e d p s e u d o - v e l o c i t y c u r v e ( V - ωχ ) f o r Τ - 0 . 1 t o 2 . 5 s e c o n d .

m

T he f o l l o w i n g e x p r e s s i o n i s us e d f o r t h e s p e c t r a l d e n s i t y o f t h e g r o u n d

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a c c e l e r a t i o n ,

G _ ( u ) - G , ( u ) G

2

( o ) ( 2 1 )

1 ♦ Ί β

2

( ^ - )

2

I Q).

G . (u j ) - χ-* —, =■ S (2 2 )

' [1 - ( £ - ) ¥ ♦ Ίβ

2

<^

2

°

ω. ι u i .

0.650 +

2.2Í» (H-)

2

+ 1.63 (—)*

ω, ω,

ε

2

( ω )

ΓΤΤ ω

2

(23)

[ι - (ÜL)

2

]

2

+ 2.2Ί

(<±-)

2

ω

2

ω

2

- Ι -1 2 -* ΐ

i n w h i c h ω, = 1 5 -5 s e c , 0 . - 0 .6 * 4 2 , ω„ » 1 5 * 7 s e c , S - .0 0 5 2 f t s e c . F i g u re 2

I 1 2 o

14V

respectively, the median and the response level corresponding to a 1035 exceedance

probability. These are presented In Columns 5 through 8 In Table I, and are given

in terms of the average and second highest values of Columns 3 and Ί, respectively.

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The results summarized In Table I indicate that the approximation of

eq. (19) may be poor even for canti levered beam-like structures; see for example the

response y in Column 5, versus the corresponding result In Column 6 obtained using

eq. (15) for the variance. The errors caused by the approximation of eq. (19) become

more pronounced for structures 2 and 3, as evidenced in Table I; these are structures

having modal frequencies close to each other. Since this property is not uncommon in the

higher mode responses of complex systems, the evaluation of the variances through

eqs. (15) and (16) appears necessary when considering complex structures.

Columns 6 and 8 of Table I demonstrate the validity of the proposed random

vibration approach. Specifically, these show that the assumptions of a G aussian response

and Poisson occurrence of level-crossings produce a reliable means for estimating maximum

earthquake responses In MDF-systerns.

Ί. RESP ONSE OF SECONDARY SYSTEMS

148

\

+ 2 ß

2 V k

+

Vk

=

" ík %

+ 2β

1

ω

1

A

( 2 5 )

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\

+ 2 β

ι

ω

ι Λ

+

» u \ " - *

( t )

<

26

>

where ω.. , y

k

= mod al f re q u e n c y a nd p a r t i c i p a t i o n f a c t o r , r e s p e c t i v e l y , f o r th e p r i ma r y

mode k; <l>Aj) " am pl i tu de of mode k at f l o o r j ; and β , β = damping values fo r the

pr ima ry and secondary sys tems , r es pe c t i v e l y , and ω = f r equency o f the secondary sys tem.

The complex f reque ncy response o f the equ ipmen t , determ ined f rom eqs . (24)

th rough (26) , i s

Η

χ

(ω) = - Σ V k

( j ) a

k

(u l )

[<V

2

(<»)z

lk

(tü)r' (27)

k = l

In which z-

k

is give n by eq . (14) and

a (ω) - 1 + 2i B, (£ - ) (28)

-

149 -

b u l l d i n g . H o w e v e r ,

the

f l o o r - s p e c t r a c an n o t

be

dir e c t l y u s e d

for

e q u i p m e n t s , s u ch

as

p i p i n g s y s t e m s , t h a t are c o n n e c t e d to s e v e r a l fl o o r s . A r e s p o n s e s p e c t r u m a p p r o a c h for

p i p i n g s y s t e m s is de s cr ibe d in Ref. [5]; ¡t is me n t i o n e d t h a t a r an do m v ibr a t i o n a p p r o ach

t o the an al y s is of p i p i n g s y s t e m s w o u l d be p r e f e r a b l e . S u ch an a p p r o ach s h o u l d al s o

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I m p r o v e

the

accuracy

of

r e s u l t s

for MDF

s y s t e m s r e s t i n g

on a

s in gl e fl o o r .

No

c o n c e p t u a l

difficu l t y e xis t s

for the

a p p l i c a t i o n

of

r an do m v ibr a t i o n p r i n ci p l e s

to MDF

s e co n dar y

s y s t e m s al o n g the l ine s p r o p o s e d h e r e i n . Ho w e v e r , addi t io n al s t u die s are n e e de d to

simplify the fo u r - f o l d s u mma t io n in v o l v e d in c a l c u l a t i n g all the co u p l e d t e r m s In the

e x p r e s s i o n for σ two s u mmat io n s for the building modes and two for the e q u i p m e n t ) .

This

i

tern

Is

c u r r e n t l y u n d e r s t u d y .

5. CONCLUSIONS

The fundamental periods and damping of most re ac tor f a c i l i t i e s a re such tha t

the seismic responses of these structures can be trea ted as a st a tion a ry random process.

Pra ct ic al ly fe asi bl e procedures for the an al ysi s.o f rea ctor f a c i l i ti e s and equipments

are ava il ab le from random vi b ra ti on theory. The use of random vi b ra ti on leads to re sul ts

that are In agreement with those obta ine d from the d ir e c t in te g ra tion of a normal ized

set of recorded accel erograms. This shows that the proposed s ta t i s ti c a l approach can be

150

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T A B L E I - C O M P A R I S O N OF R E S U L T S : R A N D O M V I B R A T I O N VS. I N T E G R A T I O N

O F A C C E L E R O G R A M S

(B - .05 In all

m o d e s )

R e s p o n s e

Q u a n t i t i e s

(1)

4

Uj χ 10

(u

?

- u

6

) χ IO

1

*

( u

1 0

- u

3

) χ IO

11

Values

Range

(2)

from Records

2nd

A v e .

(3)

(a) Structure 1

291-477

208-409

98-158

383

283

122

Highest

(4)

,T - 2 Sec

467

v

322

144

b

. 5 0

( 3 0 )

AVE.

ï

(5)

1.01

.95

1.00

Ï ; Σ

(6)

1.00

•

9 5

.94

b

. i o

2 n d H

Γ

( 7 )

.99

.99

■ 9 7

( 3 0 )

I g h e s t

(8)

• 98

.99

.9Ί

151

x ( t )

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FIG. I FIRST PASSAGE PROBLEM

0.8

0.7

I I

0.625

m

ι

0 .682 m

0,738 m

0.795 m

1

1.05 k

I

1.10k

ι

1.15k1

ï

agi

t i

i

2

CMi

e L

T

¿L

y ^ u

+ yg

π

2EZZZ3ZZEJ

T k»x

kx

'c.M. ,. x, Cflj

yï

=

«Jj+y

g

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0.852 m

0,909 m

0,966 m

1,022m

1.079 m

1,136 m '

Struc

1.20k

'

1.25k

ι

1.30 k

■

1.35k

l

1,40

k

'

1.45

k

ι

1.50k

fure

viii/r/ritimtiim

Pion

k

V7

I

r «

t^ w

m, J

Structure 2

taammmazax

i=5

¡ = 4

¡=3

¡=2

¡=l

Typical Pion

ç.

l

k i V e

Ü

* i ' W '

v-e

V i

m, J

m, J

m, J

m, J

m, J

Structure 3

FIG. 3 STRUC TURES USED IN THE COMPARITIVE STUDIES

153

iitiiiiiiiizzm

^P~*

—Equipmiipment

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Γ

■rmitiY/ittiirm Floo r j of

Primary System

•nir»»»»»» »»»»?»»

FIG.

4 SDF EQUIPMENT MOUNTED ON A PRIMA RY

SYSTEM

lOOOi 1 r

154 -

D I S C U S S I O N

N. N. KUL KARN I, Ind ia

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Q

In c a s e of p r e s s u r e t u b e t y p e of r e a c t o r s f u e ll in g m a c h i n e s a l s o o p e r a t e d u r i n g

o p e r a t i o n f o r o n - l o a d i n g f u e ll i ng . If a n e a r t h q u a k e o c c u r s d u r i n g o p e r a t i o n M C A c a n r e s u l

I t i s n e c e s s a r y t o e s t a b l i s h a c r i t e r i o n f o r t h e d e s i g n o f f u e l l i n g m a c h i n e s . C a n t h e a u t h o r s

p r e s e n t s o m e s t a t i s t i c a l d a t a f o r s u c h a c a s e ?

J . M. DO YL E, U. S. A.

I t s e e ms th a t i f y o u c o n s id e r t h e fu e l l i n g ma c h in e s a s a n e q u ip me n t i t e m, t h e

methods ou t l ined in our paper cou ld be used to ob ta in the f loor mot ion a t the loca t ion of the

m a c h in e . T h e r e f o r e , t h e i n fo r m a t io n y o u w a n t w o u ld n e e d to b e c a l c u l a t e d fo r e a c h in d iv id u a

c a s e . I t w o u ld d e p e n d , o f c o u r s e , o n t h e d e s i g n b a s i s e a r th q u a k e , a n d th e d y n a m ic p r o p e r t i e

of t h e p r i m a r y s t r u c t u r e .

K 3/2

THE RESPONSE SPECTRUM ANALYSIS

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TO AN ARTIFICIAL EARTHQUAKE

WITH TWO GROUND PREDOMINANT PERIODS

H. SATO,

Institute of

Industrial

Science,

University

of Tokyo, Tokyo, Japan

ABSTRACT

Analysis of the response spectrum of structure system simulated by one-degree-of-freedom

to an artificial earthquake; stationary random vibration with two ground predominant period,

is made. This is investigated as the extensive study for the case of single predominant pe

riod. Then for the estimation of maximum of the artificial earthquake and response wave form

- 156 -

the system parameters as ground predominant period, natural period and damping ratio of struc

ture to the spectrum, shich is easily masked for the spectrum to actual earthquake record be

cause of its complexity.

Response spectrum is originally plotted by taking maximum of time history of the re

7/25/2019 CDNJ04820ENC_001


sponse.

As for the acceleration response it also can be represented by taking the ratio of

maximum between input earthquake and the response, response spectrum of accleration amplifi

cation factor. According to the analysis that the maximum is in proportion to the standard

deviation and random vibration corresponding to earthquake has the characteristic of band

limitted white noise filtered through one-degree-of-freedom system the natural period of

which is equal to predominant period of ground, the response spectrum obtained through the

simulation agrees well by covering a number of spectra to earthquake records in sense of an

envelope [2]. The author has made an investigation that he applied the probability density

function of extreme by Rice [3] in order to find maximum, that is, where the function is

small enough was assumed the maximum [4]. The results were similar to the case that the stand

ard deviation was adopted for the maximum.

These analyses were all performed by the simulated earthquake with sinßle predominant

period in spite that the spectrum to earthquake has several peaks. Then this paper studies

the statistical analysis to simulated earthquake with two ground predominant period which is

H,(s)=

2 u j j i h j i s * L i j

2u)j2hj2S-«o \

- 157 -

(D

s

1

*2ui

ìx

h

ì

i SM u

ì

\

s ^ o o ^ h ^ s + ü i j l

I f λ =0 i n e q . ( l ) , i t i s e q u a l t o t h a t o f s i n g l e p r e d o m i n a n t p e r i o d s y s t e m o n w h i c h a n u m b er o f

7/25/2019 CDNJ04820ENC_001


s t u d i e s h a v e b e e n c a r r i e d o u t . ω , i = 2ir/Tj j ,

1I1J2 = 2T T/ TJ 2,

h j ι and h j

2

a r e th e p r e d o m i n a n t c i r c u -

l a r f r e q u e n c y a n d t h e e q u i v a l e n t d a m p i n g r a t i o o f t h e g r o u n d m o d e l . T j ι and T

3

2 a r e t h e g r o u n d

p r e d o m i n a n t p e r i o d .

The p r o b a b i l i t y d e n s i t y f u n c t i o n o f e x t r e m e p ( y ) f o r a t i m e f u n c t i o n a l r an do m p r o c e s s

I ( t ) w i t h G a u s s i a n d i s t r i b u t i o n i s g i v e n a s f o l l o w s b y R i c e [ 3 ] ,

1

/UU- l i

Io I» I

2

y

2

l2

p ( y )

=

- _ _ — e x p{ - y

2

} *

— — 3 -

y e x p ( ) { l - e r f — y } ( 2 )

• Ί π / Ϊ 7 ΰ 2 r . I0 I 1 .- I 2 ) 2 / Î 7 L . 2 / 2 ( I o U - 1 2 )

w h e r e y = I ( t ) / / T

0

(3)

a n d

I » T | f | l l t s ) |

2

k ¿ω , i

2

=

I

i

5

i

7

j ° ° | s H ( s ) |

2

k d

M

, ^ ¿ ^ ^ ( s l l ' k to (4 )

H ( s ) = H , ( s ) H

i

( s ) (5 )

l l ( s ) = H

1

, ( s ) H

>

( s ) l i

j

( s ) (6 )

F o r t h e ra nd om v i b r a t i o n c o r r e s p o n d i n g t o e a r t h q u a k e a nd f o r t h a t of r e s p o n s e o f s t r u c t u r e

s y s t e m t o i t , H ( s) i s g i v e n a s e q . ( 5 ) a nd e q . ( 6 ) . T h e se a r e s u b s t i t u t e d i n t o e q . ( 4 ) a nd e q . ( 2 )

- 158 -

nant periods of the ground.

Even if T|2 becomes longer than that of the examples, the tendency does not change. Fig.l

(b) shows another example of the combination of two predominant periods. In this case the

longer predominant period exists at five times as much as the short one, however, the sensiti

vity decreaseing the amplification factor at original short predominant period and increasing

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that at another long predominant period is much smaller than the case aforementioned.

These results make it obvious that the maximum value of a response spectrum occurs when

the natural period of the structure model coincides with the predominant period of the simu

lated earthquake containing single component, in other words even if the natural period of the

structure is equal to either of multi predominant period of the ground, the amplification fac

tor is not larger than that for single predominant period.

Fig.2 (a) and (b) ahow the displacement response spectra by the statistical computation.

The parameters used for these correspond to those in Fig.l (a) and (b) respectively. These

figures explain that the appearance of longer predominant period simply increases the

dis

placement response in longer period than longer predominant period. This phenomenon is really

observed about the response spectrum for violent earthquake as Niigata {June 16, 1964).

In Fig.3 the analytical results and those by actual earthquake records such as El Centro

(NS,

May 18, 1940) and Taft (NS, July 21, 19S2) are compared. λ=0.9 are taken for the analysis

- 159 -

i s t he s imples t expres s ion of the e l a s t o -p l a s t i c deformat ion sys t em. The equa t ion of mot ion

for th e system can be w ri t t e n as

mx=-cx-f-ma t)

f=ky, i=y : f< |F | 9)

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f=F x-y)/|x-y|: f>|P|

where

m: mass , c:

damping consta nt

o f th e

structure model,

k:

spring constant

o f

the structure

model, x:

relative displa cement

o f

mas s

t o t h e

ground,

y:

relative displacement

o f t o p o f

spring

to the

ground,

F:

yield force

a n d

a t ) :

t h e

ground accelera tion.

Th e

syst em can

be r e -

presented by a block diag ram s how n i n

Fig.8.

Laplace transform o f input t o nonlinea r element

Z s)

can be

given

a s

follows,

Z s)= — — —

-H s))

10)

KS + iu

2

+2ü)¡,h

k

iOs-nü 2 d

k

h

b

+ic)

eq. l)

i s

used

a s H s ) f or t h e

case

o f t wo

predominant period

o f

ground.

u

b

a n d h

h

ar e

natu-

ral circular frequency

a n d

damping ratio structure model

f or

linear behavi our,

an d κ is

equi-

valent linear gain

f o r t h e

nonlinear element.

Ü and X in

Fig.8 mean relative velocity

a n d

d i s -

placement respectively. This block diagram shows that dis placement o f t h e system i s obtaine d

as output o f the op en loop through a n integral. This suggests that response o f displa cement i s

- 160 -

5. EXAMPLE OF CALCULATED RESPONSE SPECTRUM FOR THE NONLINEAR SYSTEM

Fig.9 is example of displacement response spectrum. ß=°° coincides with linear response.

The parameters shown are same with those found in case of réponse spectrum of linear accelera-

tion amplification factor. In this figure predominant periods exist in short period, so that

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details of difference hardly seen. This will be shown later. General characteristics appearing

according to nonlinearity do not change in comparison with the spectrum for single predomiant

period. These characteristics are that the stronger the nonlinearity is, that is, the smaller

β is, the larger the displacement response is in short period. For example β=0.3 implies yield

seismicity of 0.09 for the case of the maximum input acceleration 0.3g. As β increases the re-

sponse spectrum with nonlinear characteristic approaches the linear response spectrum. λ«0.9

is the parameter which made good agreement with acceleration response spectrum to actual re-

cords in linear system.

Next the velocity response spectrum is payed attention to. Some tendency as in single

pre

dominant period, which in short period the spectrum for nonlinear system becomes larger than

that for linear system and in. longer period this characteristic reverses, is found. Fig.10

shows an example of velocity response spectrum using same parameters as Fig.9. Taking that ve-

locity response does not show permanent excursion as displacement response into consideration,

- 161 -

behaviour of machine structure system if it is appended to building structure system.

It is made obvious that although there is some differences as for the response spectrum

between the analysis by the artificial and actual earthquake, the characteristic of the spec

trum for the latter can be explained applying the analysis of equi linearization for the non

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linear element.

6. EXPECTATION AND VARIANCE OF THE ACCELERATION AMPLIFICATION FACTOR

The analysis by the statistical computation makes it possible to predict seismic force

applied to structure system during earthquake by knowing the system parameters as ground pre

dominant period, natural period and damping ratio of structure system [10]. However these pa

rameters are given as design value, the realization of these usually differs from the estima

tion.

According to observation the predominant period appearing in earthquake recorded at an

observatory point moves around as is seen in Fig.12 after Kanai [11]. This can be said that

appearance of predominant period possesses a probabilistic characteristic. Prediction of na

tural period and damping ratio also have same sort of probabilistic characteristic as the

ground predominant period. Then if probability density function is fitted for realization of

the system parameters, the response spectrum given by the statistical analysis as Fig.l can be

- 162 -

E[A]=3.40 σ „=0.163 (21)

for log-normal probability density function with ov =o

Tb

=0.10. Taking the damping ratio as a

stochastic variable,

E[A] = 3.08 σ„ =

0.346

(22)

are provided for

0^=0.004

and same σ

τ

and o

T b

of normal distribution. Since the integral is

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carried out by Simpson method for eq.(21) and by Monte Carlo method for

eq.(22),

direct com

parison is difficult. However the general tendency that E[A] diminishes and σ increases does

not vary.

Although eq.(21) and eq.(22) are obtained for h

b

=0.07 and the spectrum of the single pre

dominant period, Fig.IS shows an effect of two ground predominant period. In Fig. 15 only the

natural period of structure system is provided the probabilistic characteristic, and the nat

ural period is varied keeping ratio of the standard deviation to the natural period constant.

Fig.15 (a) is as for two ground predominant period and Fig.15 (b) is as for single predominant

period. The behaviour of mean and 30* width show that they keep constant for the former and

they are almost same tendency as the originally estimated spectrum for the latter. Zigzag

curve depends on using Monte Carlo method. However, the result means that once the ground pre

dominant period appears more than one at close period each other, the predicted amplification

factor should be constant irrespective of the change of natural period. This is considered im

portant from practical viewpoint in estimating seismic forces.

- Ì63 -

REFERENCES

[1] HOUSNER, G;» ., MARTEL, R .R .an d ALFORD, J.L., "Spectrum analysis o f strong mot ion eart h-

quakes".

Bull. Seism. Soc. Am., 43- 2, 1953- 4.

[2]* TAJIMI, H., "Basic th eories on aseismic design of structures'.', Rep . I nst . Ind. Sci.,

7/25/2019 CDNJ04820ENC_001


Univ. of Tok yo, 8-4, 1959-3.

[3] RICE, S.O., "Mathmatical t heory of random noise ", Bell Syst . Tech. J., 23, 1944, 24,

194S.

[4]* SATO, H., "A study on aseismic design of machine st ruct ure ", Rep . Inst. Ind. Sci., Univ.

of Toky o, 15-1, 1965-11.

[5] SATO, H., "Response of structure system t o a model earthquake motion with two ground pr e-

dominant p erio ds", J. Inst. Ind. Sci., Univ. of Tok yo, 21-11, 1969-11.

[6]* SATO, H., "Response of nonlinear structure system to a model eathquake motion with two

ground pr edominant pe riod", Proc. JSME, 700- 17, 1970-10.

[7] KANAI, K., "Semi-empirical formula for the seismic characterist ic of ground", Bull . ERI,

Univ. of Tokyo, 35, 1957- 1.

[8] NEWTON Jr., G.C., et al .. Analytical design of linear feedback cont rol s, John Wiley, 1957.

[9] SAWARAGI, Y., "A survey on st atistical st udy of nonlinear cont rol sys t ems". Trans■ Fac■

of Eng., Univ. of Kyot o, 14, 1958-9.

[10]*

SATO, H., "A study on confidence limits o f characterist ics of resp onse sp ectr um". Proc.

3rd Japan Earthq. Eng. Symp., 1970-11.

164

«κ/α,

4 . 0 '

λ

• ο

λ

Χ 0.6

Ο 0.2 α 0.8

ψ,

-0.015

ft =3.0

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3.0

2 . 0

1.0

Δ 0.4 Ο- 1.2

h , , ' 0 4 Tg,= 0 . 2 0 . e c

h g ;

0.3 Tg .-0.5 jec

h b - 0 . 0 7

O 0.2 0.4 0.6 0.8 1.0 1.2 '»

b

s e c

165

Χ 0.01 cm/gol

5.0

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3.0

1.0

g / , ' Ο 015

ΊΊ

'

10

hg,· 04

Tg,» 0.20

sec

h » ' 0 . 0 7

/¿¿^

X *

^ ' -

yS

^ V

hg,-o.3 y'sf^s*'

s*·

T,,.

0.5sec S,K''s'"'.-■''

/A S.- - ^κ

fy/s^s^

„

/ # X

^ ^ · °

gs^

°

°

2

Δ 0.4

_ ^ « - * '

„- *

λ

X 0.6

α

o.a

■ O 1.2

0 2

0.4

0.8

1.2 Ttsec

«,/o_

4.0

166

O Τ,; 0 .2 sec 1 ,, · o .5 sec Λ- 0 . 8~ Ι .Ο

._ . Τ, ,-0 .2sec Tg= 0.4sec λ=0.75~Ι.Ο

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3.0

2.0

1.0

• E L

C E N T R O

î 0,· 0.015 (ώ,= 3.0

h,»0.4 h,_0.3 h_=0.07

O 0.2 0.4 0.6 0.8 1.0 T

65

ec

Fig. 3

Comparison of the acceleration response spectra by the analysis with those by actual

167

Λ =1 .0 ^ =0.015

T„=0.2s

Γι Tn=0.5s

A =3.0

h„=0.A

hj¿=0.3

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Λ -»- ANALYSIS

-O -

EL CENTRO

- 168

ί

Α Β

F /

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A

/ /

0 / /

x

Β

F ig . 7 Scheme d e s c r ip t i o n o f c h a r a c t e r i s t i c o f t h e e l a s to -p l a s t i c de fo rma t ie

1 6 )

4ΰΰ

" f e w

^ ° ·

2 5

T«=0.5s

Ρ

■VO.3

X P

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0.3

0.2

0.1 f

0.0

o

0.3 ·

α 0.6

Δ Ι Ο - /

j t ^ ' / ^ S j t '

o o / /

s f

>■

/

\

J

¿ ^ ^

Tb

S

00 1.0 2.0 3.0 4.0 5.0

Fig.9 Comparison

of the

displacement response spe ctra

as for

linear

and

nonlinear system

- 170

80.0

a CENTRO

· LINEAR

- « — NONLINEAR

(0 3 g)

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40.0

20.0

0.0

h.=0

07

' / , . « - - ' ' λ 0,9 . i=02s , v=ass

- «- fl=oe, - « - fl.1.5

~ 4 -

Λ. = 0

. l,,=0.2s

β--αα

—Ο -

Α =0

.

V=0.5s

.

fl.oo

Tes

0.0 0.2

0.4 0.6

0.8

1.0

Fig.11 Comparison

of the

velocity response spectrum

of the

nonlinear system

by the

analysi

with that

by

actual earthquake record

- 171

iVlO-

8-

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6-

4-

2-

0

O

0.2 0.4

I ■ 1

0.6

0.8

Tg sec

Fig.13 A relation betwee n the ground pre dominant period in earthq uake observed at a p lace

and its occurren ce frequency (after Kanai fll])

172

V 0 2

VO.

5

s ly 0.4 l y 0.3

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λ«0.9 1ν0.07

E (A)

EÍAh3t>E(A)-3.-

Ο. 0 .2 0. 4 0.6 0.8 1.0 1.2 1.4

γ

Μ

hg-0.4 h» 0-07, (R/TÛ · 0 2

- 173 -

DISCUSSION

__ C h . C HE N , U .S . A .

Q

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In one of the s l id es the re la t io nsh ip between ground predom inant p er iod s and the

occ ure nce frequency i s shown. D oes th is re la t io nsh ip apply to a l l ground con dit ion s , ir re

spec t ive o f com petent rock or so f t so i l ?

H. SATO, Japan

Δ

N o,

i t i s a datum mea su red at an ob serv ato ry . I wi sh I cou ld have th is k ind o f

data for var io us cond it ions from the eng i ne er 's v iew po int .

_ G. S C HN E I D E R, Ge r m a n y

^ What is you r opin ion about Prof. Kanai'β me thod to find out natu ral or predo min ant

p e r io d s o f m ic r o s e i s m ic s ig n a l s d e p en d in g o n g e o lo g i c e n v ir o n m e n t a n d t o u s e t h e s e r e s u l t s

for predic t ing predominant per iod s in earthquake s igna ls ? S ince mo st rea cto rs a re or

174

o t h e r s h a v e i n t r o d u c e d c o r r e c t i o n s in o r d e r t o a c c o u n t f o r fi n it e d u r a t i o n an d n o n - s t a t i o n a r i t

F o r t h e c a s e y ou s tu d i e d , w h a t w a s t h e a d v a n ta g e of w o rk in g w i th a tw o - m a s s s y s t e m ?

H ow d id y o u o b t a in t h e p r o p e r t i e s o f t h e tw o - m a s s s y s t e m f ro m th o s e of t h e c o n t in u o u s s y s

t e m ? H ow d o y o u o b t a in a m p l i f i c a t i o n fu n c t io n s for r e s p o n s e s p e c t r a r a th e r t h a n fo r F o u r i e

7/25/2019 CDNJ04820ENC_001


s p e c t r a ?

H . S A T O , J a p a n

I c a n t a k e t h e a d v a n ta g e b y th i s r a th e r s im p le a n a l y s i s t o ma k e th e e ff e ct o f

a n o t h e r g r o u n d p r e d o m i n a n t p e r i o d t o t h e r e s p o n s e s p e c t r u m o b v i o u s . O n c e w e k n o w t h e

e f fe c t, w e c a n c o m p o s e t he m u l t i - d e g r e e - o f - f r e e d o m s y s t e m a s fa r a s t h e n u m b e r of d e g r e e

i s c o n s id e re d . I m a d e u s e o f t h e p ro b a b i l i t y d e n s i t y fu n c t io n o f e x t r e m e s b y S . O . R IC E . T h e

m a x i m u m i s r e p r e s e n t e d b y t h e p o in t w h e r e t h e p r o b a b i l i t y d e n s i t y fu n c ti o n b e c o m e s s m a l l

e n o u g h . T h i s i s m a d e fo r t h e r a n d o m p ro c e s s c o r r e s p o n d in g to t h e e a r th q u a k e mo t io n a n d

th e r e s p o n s e to i t. T h e r a t i o s for b o th e s t im a t io n s a r e t a k e n a s t h e a m p l i f i c a t i o n f a c to r .

K 3/3

THE INFLUENCE OF SEISMIC PULSE TIME

ON STRUCTURE-FOUNDATION INTERACTION

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R.J . SCAVUZZO,

Department of Mechanical Engineering,

Rensselaer Polytechnic Institute, Hartford Grad uate Center, East Windso r Hill, Conn ecticut,

R.R. LITTLE,

Department of Mechanical Engineering,

The University of Toledo, Toledo, Ohio, U.S.A.

ABSTRACT

The influence of the duration of seismic ground motions on inertia forces

of a power plant is investigated considering ground-structure interaction

effects.

This study is based on the ground accelerations measured during the

176

T h e s t r u c t u r a l m o d e l u s e d in t h i s s t u d y is a s i m p l i f i e d r e p r e s e n t a t i o n

o f a n u c l e a r p o w e r p l a n t p r e v i o u s l y e m p l o y e d . T h i s d y n a m i c m o d e l c o n s i s t s

o f t h r e e m a s s e s , a b a s e m a s s , a c o n t a i n me n t v e s s e l m a s s a nd an i n t e r n a l

s t r u c t u r e m a s s ( F ig ur e 1 ) . F i xe d - b a s e fr e q u e n c i e s o f t h e c o n t a i n me n t v e s s e l

7/25/2019 CDNJ04820ENC_001


ma s s an d i n t e r n a l s t r u c t u r e m a s s ar e a p p r o x i m a t e l y 4 .0 c p s an d 5.0 c p s ,

r e s p e c t i v e l y . T h e n a t u r a l fr e q u e n c y o f t h e co n t a i n me n t v e s s e l is c l o s e t o

m e a s u r e d v a l u e s o f t h e f u n d a m e n t a l mo d e o f t h e E G CR b y M a t t h i e s e n a n d S mit h ,

[7].

NOMENCLATURE

a

A

b

c

E

F ( t )

f ( t )

D i l a t a t i o n ( Ρ) w a v e v e l o c i t y

A r e a o f t h e s t r u c t u r e b a s e

S h e a r ( S) w a v e v e l o c i t y

H a l f t h e b a s e w i d t h

Y o u n g ' s m o d u l u s

L a t e r a l f o r c e at t h e b as e o f a s t r u c t u r e

Su rface s h e ar s t r e s s w h e n |x| < c

177 -

Making use of the coordinate system introduced in Figure 2, the boundary

conditions used in this solution are:

o - 0 , y - 0

y

ff(t) , |x|<c,

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Τ χ

Μ o . |x|»c.

(2)

ο = τ = 0

y xy

y =

By employing Laplace and Fourier transforms the solution for the displace-

ment at the origin caused by a shear force which varies arbitrarily with

time in the interval - c < χ < c is obtained (eq. (3)). In order to simplify

the inversion of these transforms Poisson's ratio is made equal to 1/4.

For this case, a = /3 b where a and b are the Ρ and S wave velocities,

respectively.

178

It should be noted that for t < —, there is no contribution from the

second integral in eq. (3). In the interval — < t < § the function,

bt c a b

Im g ( — ) has one form and for t > c- the term has a second form. A singularity

occurs when t = — where v is the Rayleigh wave velocity. However, the

second integral of eq. (3) is bounded in the Cauchy sense.

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Interaction Equations

For an N-mass structure subjected to an arbitrary lateral foundation

acceleration, the horizontal loads at the foundation can be expressed in

the following form using the notation of O Hara and Cunniff [8],

F(t)

5

Λ

Γ

8

(l) sin u.(t-t)dT + M ü(t)

J o

(6)

where the positive direction of F(t) is assumed to be the same as the shear

stress f(t) which acts on the surface in the interval - c < χ < c, M. are

the effective modal masses and u(t) is the lateral displacement at the

179 -

In eq. (8) the properties of the structure are defined in terms of the

base mass M , the effective modal masses M., the circular natural frequencies,

ω,, the base half width c. The properties of the ground are specified by

the shear modulus, u , and the shear wave velocity b. Because Poisson s

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ratio has been made equal to 1/4 and a value of 100 lb/ft was used for the

weight density of the soil in all calculations, the stiffness of the ground

is completely specified by the shear wave velocity. Values of 500 ft/sec.,

1000 f t / s e c , 2000-ft/sec. and 4000 ft/sec. were employed in these parametric

studies.

4. DISCUSSION

The solutions to twenty-eight problems were studied in the evaluation

of the effects of time on soil-structure interaction. The input parameters

specified in these problems are tabulated on Table I. In addition, the

acceleration response ratio, which is defined as the acceleration response

determined from the foundation acceleration, ü (t) , (eq. (8)), divided by

the acceleration response of the input acceleration, ü (t ), at the fixed-

- 180 -

where Ζ is either the input acceleration, ü (t), or the foundation acceler-

ation,

ü(t ). The spectrum response is the maximum value of

S(UJ,

t) , for any

time, t. Spectrum responses of the two mode problems (Cases 1 to 24) are

presented on Tables II and III. The time, t, at which the integral of eq.

(9) is a maximum is also listed on these tables. By comparing the results of

problems which are similar except for the length of the input motion, the

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effect of time on foundation-structure interaction can be evaluated.

Results of the studies on the El Centro input are the easiest to evaluate

Peak accelerations of this motion of 0.25 g's or more occur between 1.7

seconds and 4.89 seconds. After five seconds, the magnitude of input accel-

eration motion decreases. Peak values of the spectrum response integral

based on this input motion varies from 4.95 seconds to 10.42 seconds and

depends upon the specified response frequency, ω. On Figures 9 and 10, the

integral, S(ii),t),

is plotted as a function of time for the El Centro earth-

quake motion. The magnitude of S(ui,t) is large for the fifteen seconds

considered because there is no damping associated with the integral of eq.

(9).

However, the peak acceleration responses of the calculated foundation

motion occurs between 2.2 and 3.24 seconds. On Figures 11 and 12, S(io,t)

- 181 -

tendency for the integral of the foundation motion, S(u>,t), to increase with

time because of radiation damping. As a result, if the portion of the Taft

earthquake between three and eleven seconds are used as input, the maximum

seismic forces acting on the structure can be evaluated considering inter

action effects.

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Using the one-mode two-mass system described above, input motions from

five seconds to thirty seconds in duration are considered (Cases 25 to 28).

As seen on Table I, the acceleration response ratio is unaltered after the

first ten seconds of motion are considered. The peak response occurs at

9.9 seconds which is similar to the two-mode model. The increase in the

acceleration response ratio associated with these cases is caused by the

reduction in total weight of the idealized structure.

In the preceding discussion, the spectrum response at the fixed-base

frequencies of the idealized structure was considered. These frequencies

were considered because structure inertia loads are determined at these

values.

However, the response of light weight structures without structural

damping attached to the base mass, Μ , can.be determined from the spectrum

- 182 -

maximum inertia forces than for the heavy structural components. At some

frequencies,

the acceleration responses increased with time.

6. ACKNOWLEDGEMENT

The authors are grateful to Dr. Paul C. Jennings of the California

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Institute of Technology for providing the earthquake motions used as input in

this study and to the United States Atomic Energy Commission for supporting

the work presented in this paper through USAEC Contract AT-(40-1)-3822.

BIBLIOGRAPHY

TABLE I

TABULATION OF PROBLEMS STUDIED

Case

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

M

0

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

Structure

m

l

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

m

2

.310

.310

.310

.310

.310

.310

.310

.310

.310

.310

.310

.310

.310

.310

.310

.310

Properties

£

1

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

£

2

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

Ground

A

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

υ

Proper

Ρ

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

100

ties

b

500

500

500

1000

1000

1000

2000

2000

2000

4000

4000

4000

500

500

500

1000

Free Field Earthquake

Input, Ü (t)

1940 N-S EL Centro

1940

N-S EL

Centro

1940 N-S EL Centro

1940

N-S EL

Centro

1940 N-S EL Centro

1940 N-S EL Centro

1940

N-S EL

Centro

1940

N-S EL

Centro

1940

N-S EL

Centro

1940 N-S EL Centro

1940

N-S

EL

Centro

1940

N-S EL

Centro

1952 N21E Taft

1952 N21E Taft

1952

N21E

Taft

1952 N21E Taft

Duration

of

Seismic

First

First

First

First

First

First

First

First

First

First

First

First

First

First

First

First

5

10

15

5

10

15

5

10

15

5

10

15

5

10

15

5

Input

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

Ac ce le ation

Ratio*

.201,

.188,

.188,

.339,

.317,

.317,

.588,

.550,

.550,

1.091,

1.020,

1.020,

.279,

.137,

.177,

.406,

.200

.182

.169

.274

.249

.231

.406

.369

.342

.794

.723

.670

.125

.126

.156

.200

Response

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17

18

19

20

21

22

23

24

25

26

27

28

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.4

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.475

.310

.310

.310

.310

.310

.310

.310

.310

0

0

0

0

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.06

4.95

4.95

4.95

4.95

4.95

4.95

4.95

4.95

-

-

-

-

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

15400

100

100

100

100

100

100

100

100

100

100

100

100

1000

1000

2000

2000

2000

4000

4000

4000

1000

1000

1000

1000

1952

N21E

Taft

1952

N21E

Taft

1952 N21E Taft

1952 N21E Taft

1952

N21E

Taft

1952

N21E

Taft

1952 N21E Taft

1952

N21E

Taft

1952 N21E Taft

1952

N21E

Taft

1952 N21E Taft

1952 N21E Taft

First

First

First

First

First

First

First

First

First

First

First

First

10

15

5

10

15

5

10

15

5

10

15

30

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

sec.

ccc.

sec.

.260,

.260,

.651

.404,

.404

.949,

.707,

.707,

.188

.251

.354

.305

.364

.691

.534

.648

495

288

288

288

1

m

2

£ ,

2 6

base mass» lb-sec /ft. χ 10

containment-vessel mass, lb/ sec / ft. χ 10

internal-st ructure mass, lb-sec /f t . χ 10

fixed-base containment-vessel frequency, cps

fixed-base internal-structure frequency, cps

A - base area of st ructure- ft .

ν - Poisson s rat io

3

ρ - ground density, lb/ft.

b - soil shear wave velocity, ft/sec.

Ratio of base motion acceleration spectrum response to free-field acceleration spectrum response at the

fixed-base structure freqquencies

(f.,f_).

184

TABLE II

COMPARISON OF ACCELERATION SPECTRA FOR THE

TWO MODE MODEL SUBJECT TO

THE EL CENTRO EARTHQUAKE MOTION

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S O I L C O N D I T I O N

F r e e - F i e l d M o t i o n

b = 9 0 0 f t / s e c

b = 1 0 0 0 f t / i c e

b = 2 0 0 0 f t / s e c

b = 4 0 0 0 f t / s e c

F r e e - F i e l d M o t i o n

b = 5 0 0 f t / s e c

b = 1 0 0 0 f t / a e c

b = 2 0 0 0 f t / s e c

b = 4 0 0 0 f t / « e c

'n

c p s

4 . 0 6

4 . 0 6

4 . 0 6

4. 06

4 . 0 6

4 .

9 5

4 .

9 5

4 . 9 5

4 . 9 5

4. 95

D U R A T I O N O F I N P U T M O T I O N

5 S E C

1 . 5 4 @ 5 . 0 0 . e c

0 . 3 1 @ 2 . 2 0 » e c

0 . 5 2 @ 2 . 1 9 « e c

0 . 9 0 @ 2 . 5 8 » e c

1 . 6 8 @ 2 . 5 6 » e c

1 .

86 (8 4 . 95 »ec

0 . 3 7 @ 2 . 7 2 a e c

0 . 5 1 @ 2 . 7 2 » e c

0 . 7 6 @ 3 . 2 4 » e c

1 . 4 8 @ 3 . 2 4 a e c

1 0 S E C

1 . 6 5 @ 9 . 6 7 » e c

0 . 3 1 @ 2 . 2 0 » e c

0 . 5 2 @ 2 . 1 9 « e c

0 . 9 0 @ 2 . 5 8 » e c

1 . 68 @ 2 . 56 ae c

2 . 0 5 @ 6 . 9 8 » e c

0 . 37 @ 2 . 72 »e c

0 . 51 @ 2 . 72 »e c

0. 7 6 (S> 3. 2 4 a e c

1 . 4 8 @ 3 . 2 4 a e c

15

1 . 6 5 * ?

0. 31 £

0 . 5 2 é

0 . 90 @

1 . 6 8 @

2 . 2 1 t ?

0 . 37 i t

0. 51 @

0. 76 @

1 . 4 8 @

S E C

9 . 6 7 a e c

2 . 2 0 a e c

2 . 1 9 a e

2 . Μ

Μ ^

Σ . 5 6 » e

1 0 . 4 2 a r

2 .

7 2 a e

2 . 7 2 a e

3 . 2 4 a e

3 . 2 4 a e c

- 185 -

Containment Vessel

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Internal Structure

Base

186

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- 187

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, , l i l

Aluiu iP

||mrt 1

h

ι iff

11

r

li I

Ι

I

l l l l i l 1

iffl I l i | Ji p i l

m

' I l ι 1

188

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- 189

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"'

Ili

n.^uti/lIMJl

_*HWI

fi 1

'

' ' ■

.1

ï

iE

f i » | 1

ÌÌkUiL·

¡Imff

1

J J U J I L

flyjijv

rti'llljM/

1

γγψΓίψ

Π Μ Ε . SECONDS

190 -

1

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u i

τ τη

·>,

έ \ Λ

m

14

•

tå

SS

lÜI

siika

Piff

1fr

■ 6 P

- η

' · i l ·

.U-I

«

ι

ι >

t

5 t 7 , g 10 u u

l

> >» 1* Ib

191

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I j ι

¡A

fil i H

I P P I I I I I '1

?

' W I M '

k

il

ι -

192

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-AV

ΛΙΛ M f l

iiÉlt

li

III

u l

ll

ί ι

l

o i: u

u

193 -

DISCUSSION

0

H. SHIBATA, Japan

1.

W e g e n e ra l l y c o n s id e r t h e e f fe c t of h o r i z o n ta l mo t io n . B u t t h e e f fe c t of r o t a t i o n a l mo t io n

o f g ro u n d s u r f a c e s h o u ld b e s t ro n g e r t h a n th a t o f t h e h o r i z o n ta l mo t io n .

7/25/2019 CDNJ04820ENC_001


2.

T h e n o t c h e s i n F ig . 15 mig h t b e v e r y s ig n i f i c a n t f o r th e d e s ig n . B u t t o e s t im a t e t h e e ig e n -

f re qu en c ie s in an ac cu ra cy wi th in 10% is ve ry d i f f icu l t , so the v iew po in t of the ma rg in of

sa fe ty , the e f fec t of peak s should be a l so co ns id er ed in av er ag e . The n app ly ing the e f fec t of

suc h redu c t io n fo r the des ign ma y be l im i ted . How do you co ns id er the cas e of app ly ing th i s

effect to the design ?

yfi R . J . SCA VU ZZO , U .S .A .

1.

In t h e a n a ly s i s w h ic h i s p r e s e n t e d , r o t a t i o n a l mo t io n w a s n o t c o n s id e re d . H o w e v e r , in

R ef . ( 4) t h e e f f ec t o f r o t a t i o n w a s c o n s id e re d , g ro u n d m o t io n s w e re b a s e d o n th e f i n i t e e l e m e n t

a n a l y s i s o f I s e n b e r g (R ef . ( 6) ). T h e s e r e s u l t s s h o w e d t h a t a c c e l e r a t i o n s a s s o c i a t e d w i th r o c k

-

194

R . J . S C A V U Z Z O , U . S . A .

A

No my conclusion would not change. The reason for this statement is that the

first portion of the earthquake which does not have high accelerations would not affect peak

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structural loads significantly which will occur during the violent portion of the ground moti

Thus, the first 10 seconds of motion could be neglected.

Q

M . B E N D E R , U . S .

A.

1.

Would you suggest quantitative criteria for differentiating between light and heavy mas

structures for the purpose of analysis and suggest the most appropriate approach to give a

conservative design for intermediate mass elements of the structure ?

2.

With respect to uncertainties in the soil stiffness characteristics and variabilities in

structural response attributable to unknown physical properties, how would you use your

analysis methods as a design tool ?

K 3/4

ANALYSIS OF SOIL-STRUCTURE INTERACTION EFFECTS

UNDER SEISMIC EXCITATION

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C.J. COSTANTINO,

School of Engineering,

The City College of the C ity University of N ew York, New York, U.S.A.

ABSTRACT

This report describes a numerical technique to treat the complete

dynamic soil-structure interaction problem. The structure embedded within

the free-field soil system is represented by its rigid body and elastic free-

free modes, while the soil is treated by the finite element method including

nonlinear material properties. The application of the developed computer

- 196 -

configuration of interest is shown in Figure 1 and consists of a general

structure embedded within a soil/rock foundation made up of an arbitrary

number of material layers, each layer possessing its own, generally non

linear,

constitutive law. To this system, the seismic motion history is

applied in the form of either displacement, velocity or acceleration motion

records. This immediately brings to the fore an important aspect of the

problem, namely, how should the input motions be applied to the system.

7/25/2019 CDNJ04820ENC_001


Seismic motion records are generally surface records and in fact what is

needed are records with depth at a given site. This problem, however, is

currently unanswerable and is considered as external to the soil-structure

interaction problem of interest herein.

The wave propagation from the input location into the free-field

soil system can be treated by finite element methods of analysis (see

Costantino [3], [4] ) including the effects of nonlinear properties of the

soil.

The computer code developed for this problem is termed the SLAM Code

for identification. The finite element approach has been taken in this

development to allow the user a general flexibility in treating problems of

rather complex geometry (material layering, structural inclusions, complex

- 197 -

relationship.

b) The structure is represented by its rigid body modes together with its

lower free-free elastic modes.

c) Potential separation and sliding between the structure and the free-field

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can be treated by means of a special element (of zero thickness) placed

between the structure and the soil.

It should be mentioned that if nonlinear behavior of the structure

is to be considered, then the use of the finite element mesh through the

structure must be used, as previously discussed.

2. FREE-FIELD ANALYSIS

To treat the free-field wave propagation problem, the soil/rock

material is divided into small elements, these elements being connected to

each other at their vertices. The types of elements used depend upon the

particular problem of interest. For three dimensional problems, tetrahe

drons,

cubes, etc. can be used. For the two dimensional problem for which

- 198 -

where M-, is the total nodal mass composed of the mass contributions from

each adjacent element, ( F,^ F

W M

) are the horizontal and vertical forces

R K

applied to the nodes (if any) and

(

F«.»,

f

utt

)

are the node resisting forces

developed by the distortions of the surrounding elements, the summation

being taken over all of the surrounding elements. Clearly, a displacement

7/25/2019 CDNJ04820ENC_001


field causing only rigid body motions of the elements will develop no

resisting forces at the nodes. The details for computing the node resist-

ing forces from the element distortion are presented by Costantino

\_4~\.

Combining the equations for all the nodes, a set of second order

equations are developed for the entire mesh which can be written symboli-

cally as

i ^ \ x 4- K x ■ F

A

♦ F ( 2 )

where M is a diagonal mass matrix, χ is a displacement vector con-

sisting of the horizontal and vertical displacements of the nodes, Κ is

the usual banded system elastic stiffness matrix and F is the vector of

- 199 -

a) elastic material, either isotropic or anisotropic,

b) linear compressible fluid,

c) elastic plastic material satisfying the Mises yield criterion with

arbi-

trary strain hardening effects,

d) elastic plastic material satisfying the Coulomb-Mohr yield criterion,

e) a nonlinear material law which contains a stiffening effect under hydro-

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static pressure as well as a plastic dissipation under deviatoric

strains to account for compaction effects in soils.

The last three of this list are the only nonlinear laws currently

available in the code, and have been included in an attempt to at least

crudely approximate some known responses of soil/rock materials. Cuite

apparently, none of these models are completely adequate but until further

advances in the state of the art occur, only such approximations are avail-

able for applications to earth media.

4 . S O I L - S T R U C T U R E I N T E R A C T I O N

The treatment of the interaction between the structure and soil

- 200 -

M/ x

t

+

F* « -Ρ (6)

where Ρ is the vector of interaction forces developed between the nodes

and the structure. With these interaction forces, the corresponding modal

loads applied to the structure are then

T

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Q

s

« F P (7)

where the superscript indicates the transpose of the matrix. Substituting

eq. (6) and (7) into eq. (1), the equations of motion for the structure

become

•VÍ, ♦ M»

=

-

r

K

(8)

where M_ is a nondiagonal mass matrix including the inertial coupling

between the structure and the free-field, and is defined by

M j · M

s

«■ p

T

^ F (9)

201

differences only near the stress front. In all such computations, displace

ment calculations show good correlation with available solutions while

stress calculations contain these typical oscillations. This is true also

for nonlinear material problems.

The other curves of Fig. 3 indicate the computed solution at the

7/25/2019 CDNJ04820ENC_001


same time but with a rigid mass included in the mesh. The first curve is

for the case where the mass of the inclusion equals that of a rectangular

element while the second curve is for the case where the mass of the inclu

sion is ten times that of an element.

5. SEPARATION BETWEEN SOIL AND STRUCTURE

In treating this separation problem, it is desirable to use a tech

nique which does not deviate from the method of analysis outlined above. To

accomplish this objective, a new finite element model was developed. For

the two dimensional problem (planar

motion),

a rectangular element is used

which has a finite dimension in one direction and a zero dimension in the

normal direction. The properties of this element are determined by using

- 202

-

which the input displacement motion is compared with the horizontal motion

in the middle of the mesh (2250 feet from the left most boundary). As can

be noted, the total motion response at the downstream location is a replica

of the input motion up until a time of approximately 2.1 seconds. Beyond

this time, the motion response is modified due to the reflections trans

mitted from the downstream or right most boundary of the mesh. It is quite

clear that the mesh must be long enough so that within the response time of

7/25/2019 CDNJ04820ENC_001


interest no fictitious boundary reflections will be encountered. Nontrans-

mitting or "quiet" boundary considerations can be used to help reduce such

reflection effects but these will not be considered herein.

Shock spectra for both the input motion and for the downstream

motion were computed and are shown in Fig. 7. As can be noted, the spectra

for the computed motions lies below that for the input at all frequencies.

At the low frequency ranges (below 10 cps) the differences are due to the

shorter record length of the computed motions as well as the boundary

reflections. At the high frequency end of the spectra, the differences are

due to the characteristics of the mesh used. An approximate relationship

for the highest frequencies in a given mesh is

- 203 -

shock spectra intensity factors were computed for the horizontal and verti

cal spectra of the upstream, center and downstream points on the soil-

structure interface. These intensity factors were defined by Miller and

Costantino

\,2]

as

2>„ J l i « . U f

s

v ' f, I * — U T

(ID

7/25/2019 CDNJ04820ENC_001


( · ·

\ ' ] \ ί

1 ««τ

where Sp = displacement intensity factor, Sv = velocity intensity factor, and

5„ = acceleration intensity factor, f is the frequency of the linear oscil

lator

(cps),

and Τ is the corresponding period. The parameter ΙΖ,π_

χ

|

i s

the value of the peak displacement of the shock spectra associated with a

given frequency, )Z I the corresponding peak pseudo-velocity and \ϊ |

the corresponding peak acceleration. These factors are measures of the in

tensity of the motion at the low, mid and high frequency ranges. For the

relatively short record lengths used in this problem, the displacement in

- 204 -

ACKNOWLEDGEMENT

The development of the SLAM Code was supported in part by the

National Science Foundation under Grant No. GK 3214 with The City College

of New York.

REFERENCES

7/25/2019 CDNJ04820ENC_001


1 BARON, M.L., The response of a cylindrical shell to a trans

verse shock wave , Proc, of the Second U.S. National Congress

of

Appi.

Mechanics, ASME

(1955).

2 MILLER, C.A. and COSTANTINO, C.J., Structure-foundation

interaction of a nuclear power plant with a seismic disturbance;

1

Nuclear Eng, and Design, 14, 332 (1970).

3 COSTANTINO, C.J., Finite element approach to stress wave

problems , J. Eng. Mech. Division, ASCE, 93, 153

(1967).

4 COSTANTINO, C.J., Two dimensional wave propagation through

nonlinear media , Journal of Computational Physics, 4, 147

(1969)

- 205

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SEISMIC

f

INPUT

J

—

MOTIONS ι

- 206 -

U J Í — I

Η -Ι 1 1 1 1 1 1 1 1

ι 11 1

M

m 111 ι I

1

I

R e c ta n g u l a r / V- Rigid Inclusio n

Element Mesh

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No Inclusion

M a s s R a t i o « I

Moss Ratio

= 10

A n a l y t i c S o l u t i o n at a give

Λ-

- 207 -

4 5 0 0 '

"Λ

• C G .

7/25/2019 CDNJ04820ENC_001


^w tyr 7Π7 Wr W7 wT W 7~ * w

Input Motion

Histories

Rol ler Supports

Along Base

jm I tm ih imi Tffr ττπ ίητ

__Reetangulor

E le me n t s (5 0 * 5 0 ' )

2 0

F i g . 5 E l e m e n t M esh u s e d f o r S o i l - S t r u c t u r e S y s t em

208

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20 9

„

■Μ ν,

X

\ B A C /

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ν

\ \

\Λ .

^~ ι

■

.Ι5Π

Ksy

■νϊ·ί/ Γ?Λ

■ ƒ W 1 / V. S V P o i n t Β, H o r i z o n t a l

Point Ar U '

Ho π I D η tol \

Ν

- ν — P o i n t C, H o r i z o n t a l

,

\

■ . ^ ^ - F r « t - F I « l d \

N,

v

\

H o r i z on ta l

^v

7/25/2019 CDNJ04820ENC_001


K 3/5

SOIL-FOUNDATION INTERACTION

OF REACTOR STRUCTURES SUBJECT TO SEISMIC EXCITATION

7/25/2019 CDNJ04820ENC_001


T.H. LEE, D.A. WESLEY,

Gulf General Atomic, San Diego, California, U.S.A.

ABSTRACT

A theoretical Investigation has been conducted to study the soil-structure dynamic inter

action effects on the seismic response of reactor structures. The analysis was made by consid

ering

a

linear, damped, unsymmetric, three-dimensional flexible structu re coupled with

an

- 212 -

s p o n s o r e d s t u d y . T he y c o n s ¡ d e r e d t h e d y n a m ic i n t e r r e l a t i o n s h i p b e t we e n on e l a s t i c h a l f - s p a c e

a n d a c o n v e n t i o n a l

N-mass

s t r u c t u r e i n a l a t e r a l t r a n s l a t I o n a l m o t i o n . No d a m p in g i n t h e

s t r u c t u r e c a n b e c o n s i d e r e d t o e x i s t i n t h e i r f o r m u l a t í o n . T a j i m i [ 6 ] d i s c u s s e d Pa rm e l e e ' s

p a p e r o n t h e h i g h e r m ode a s p e c t s a n d d e r i v e d t h e e q u a t i o n s f o r s t e a d y - s t a t e r e s p o n s e o f an

N-mass

s t r u c t u r e w i t h l a t e r a l t r a n s l a t i o n a nd r o c k i n g d e g re e s o f f r e e d o m . T a j i m i ' s p r i m a r y

e m p h a s is w as to d e m o n s t r a t e t h e f o r m u l a t i o n f o r an

N-mass

b u i l d i n g m o d el t h r o u g h t h e u s e o f

a moda

1

t r a n s f o r m â t ï o n t e c h n i q u e , a n d n o n u m e r i c a l r e s u i t s w e r e p r e s e n t e d . I n t h e s e ¡ n v e s t i -

g a t i o n s u t i l i z i n g a n e l a s t i c h a l f - s p a c e , t h e g eo m et r y a nd m o t i o n o f t h e s y s t e m a r e h i g h l y

7/25/2019 CDNJ04820ENC_001


i d e a l i z e d s o t h a t t h e p r o b l e m s n e c e s s a r i l y w e r e t r e a t e d i n a r a t h e r r e s t r i c t e d m a n n e r . H e n c e

t h e t e c h n i q u e s a r e n o t a d e q u a t e f o r g e n e r a l a n a l y s i s o f c o m p l e x u n s y m m e t r i c s t r u c t u r e s s u c h a

n u c l e a r p o w e r p l a n t s .

I n t h e p r e s e n t w o r k , a m o re g e n e r a a p p r o a c h t o t h e p r o b l e m i s p r e s e n t e d b y c o n s i d e r i n g

a l i n e a r , d a mp e d, u ns ym me t r i e , t h r e e - d i m e n s i o n a l f l e x i b l e s t r u c t u r e c o u p l e d w i t h a n e l a s t i c ,

h o m o g e n e o u s , Ì s o t r o p i c h a l f - s p a c e . T he s t r u c t u r e i s s i m u l a t e d b y a d i s c r e t e s y s t e m w h i c h c a n

h a v e , í n a d d i t i o n t o i t s m od al c o o r d i n a t e s , s i x r i g i d - b o d y d e g re e s o f fr e e d o m . A d d i t i o n a l c o

s i d e r a t i o n s , su c h as t h r o u g h - s o i l c o u p l i n g b e tw e e n t r a n s l a t i o n a nd r o c k i n g , h av e b ee n i n c l u d e

Í n t h e a n a l y s i s . The s e i s m i c e x c i t a t i o n i s d e f i n e d by a f r e e - f i e l d d i s p l a c e m e n t c o l um n m a t r i

wh i eh is a

1

lo ^/ ed t o h a ve t h r e e t r a n s l a t i o n a l a nd th r e e r o t a t i o n a l c o m p o n e n t s , e a c h h a v i n g a

p r e s c r i b e d t im e h i s t o r y . T he t h r e e - d i m e n s i o n a l i n t e r a c t i o n e q u a t i o n s w e re f o r m u l a t e d f r o m t h

- 213 -

For a rigid base, it has been shown ¡n [7] that

_B 1

B O O

___ 1 _B

n n A

Β ·ο _ Β , . . . .

n

_» ,

0

v

Τ = -=- m u. u . + ·=■ Ι . ,Ω .Ω . + m e.

..u.Çl.r. _

ι , J , k =

1,2,3/

( 2 ;

2 1 1 2 I J I J

i j k i j k

'

J

where u. and Ω, ar e, respectively , the components of u°and Ω, e.., is the permutati on symbol[8],

β ' J R — — 1 JK

m Ís t he total mass of the ba se , r. are compon ents of the posi tion ve ctor of the mass cen ter

B

of the base mat, and I., are the inertia tensors of the base with respect to the x. coord i nate

IJ

p

ι

system defined as

7/25/2019 CDNJ04820ENC_001


l

* ¡

=

f

B

p ( 6

i j

r

k

r

k -

r

i

r

j

) d T

(3)

τ

w h e r e ρ ¡ s t h e m as s d e n s i t y , 6 . . i s t h e K r o n e c k e r d e l t a , r, a r e t h e c o m p o n e n t s o f t h e p o s i t i o n

1

J

κ

p

v e c t o r o f a p a r t i c l e w i t h o r i g i n a t 0 , a nd t h e i n t e g r a l i s o v e r t h e v o lu m e o f t h e b a s e , τ .

L e t t he r e m a i n d e r o f t h e s t r u c t u r e ( s u p e r s t r u c t u r e ) be r e p r e s e n t e d b y a l um p e d- m a ss m o d e l .

T he k i n e t i c e n e r g y Τ may t h e n b e e x p r e s s e d as a f i n i t e sum c o n s i s t i n g o f t h e f o l l o w i n g Ν t e rm s

T

S

= i m °

n

ù = ù

5

U , n = 1.2. . .N)

( Ί )

¿ Ε,η χ. η

w h e r e m i s a d i a g o n a l i n e r t i a m a t r i x o f t h e d i s c r e t e m a s s e s , a n d { u $ } i s a n N - c o m p o n e n t d i s -

- 2 1 4

w h e r e

pr

PS

n

5B

-

-

_

D

Λ Λ

in Up nr

t n I q n t q p

D

m.

A . B y

ts

(

i n i r np ps

( p . r - 1 ,2 , . . . 6 )

( q . p . s . t - 1 , 2 , . . . L )

( 9 a )

O b )

U , n - I . 2 . . . . N ) ( 9 c )

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T h e s t r a i n e n e r g y o f a l um p e d -m a s s s t r u c t u r e m ay b e w r i t t e n a s

U - - i - k u u (1 0)

2 qs q 5

o r , i n t er m s o f t h e g e n e r a l i z e d c o o r d i n a t e s , a s

w h e r e

U

■

Ì

K

p t V t

0 , )

K

S

- k Y t

p t qs qp s t

E q u a t io n s ( 2 ) , ( 8 ) , an d ( I I ) w i l l b e u se d i n c o n j u n c t i o n w i t h t h e v i r t u a l w o rk e x p r e s s i o n

- 215 -

• S SB'"R

K q + C q + K q - - H U (16a)

n m m n m m n m m n r r

M

B

Ü

R

+ M

S

Ü

R

+ M

S B

q - Q

M

(n.m - 1,2,...L) (16b)

p r r p r r p n η p ( p , r -

1,2,...6)

α

where H can be put ín partitioned form as

(17)

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H-Stì

Q

with all the submatrices being 3X3. The elements in I have already been defined in Eq. 3) and

M

B B

-«,.m

B

(18)

U 'J

u

BC B B

do*

M., - - e. .. r.m (19)

ík ιjk j

I n E q . ( 1 6 a ) , C i s k no w n a s t h e g e n e r a l i z e d d a m o in q m a t r i x o f t h e s u p e r s t r u c t u r e , an d i t

nm

may b e d e r i v e d fr o m th e d i s s i p a t i o n t e r m in t h e L a g r a n g e ' s e q u a t i o n s . T h e p r i m e d e n o t e s t h e

t r a n s D o s e o f a m a t r i x .

2 . 2 S o l u t i o n o f t h e D yn am ic E q u a t io n s

and the matr ix D is define d as

nm

21 6

D = ω

2

Τ "

]

( 25 )

nm nm

T he r e s p o n s e o f t h e s t r u c t u r e - b a s e s y s t e m i s th u s c o u p l e d w i t h t h e d i s p l a c e m e n t s o f t h e e l a s t

h a l f - s p a c e .

I n E q . ( 2 * 0 , t h e m a t r i x Κ ( ϊ ω ) i s t h e d y n a m ic s t i f f n e s s m a t r i x o f t h e h a l f - s p a c e m ed iu m

I t s e l e m e n t s a r e c o m p l e x f u n c t i o n s t a k e n f r o m t h e s o l u t i o n s o f t h e d y n a m ic r e s p o n s e o f e l a s t i

s e m i - i n f i n i t e s o l i d s u n d er h a r m o n i c s u r f a c e l o a d i n g o r h a r m o n i c i n c i d e n t w a v e s . T he i m a g i n a r

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M

p a r t o f a n e l e m e n t i n Κ m a t r i x a c c o u n t s f o r t h e e n e r g y d i s s i p a t i o n d ue t o r a d i a t i o n o f t h e

w av e ( r a d i a t i o n d a m p i n g ) .

I n v i e w o f E q . ( 1 5 ) , E q . ( 2 3 ) r e d u c e s t o

L

2

*

1

+ K

M

)

u

1

= - ω

2

Μ

Σ

D

G

(26)

\ p r p ry r pr r

w h e r e

H

1

- H

B

+ M

S

♦ M

S B

D M

S B

'

pr pr pr pn nm mr

Equation (26) represents a system of six algebraic equations with complex coefficients.

r

When the free-field excitation com pon ent s, ü , are prescri bed, Eq. (26) can be solved for the

-Ι

Γ

- 217 -

With the excepti on of certain physical phenomena such as faulting, the time history of an

earthq uake distur bance is sufficient ly well-behav ed to guarantee the existe nce of a Fourier

transform. Let

{ ü

G

(

u

) } = / { ü

G

( t ) } . ' » » dt

(35)

be the Fourier transform column matrix of the ground accele ration s so that the response trans

forms of the absolute accelera tions of the structural masses are

S

5 G

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| u

u

) }

-

[ H ¡ U ) ] { Ü U > ) }

(36)

Upon inverting Eq. (3 6) , the time history of the struct ural response is obta ined as

{ ü

5

( t ) (

= I

r [ H

S

( l

M

) ] { ü

G

(

u

) } e

1

' * ,

(37)

2.3 Through-Soil Coupling Effects

The medium stiffness matrix K

M

in the interaction equations represents the influence of

the foundation fle xibili ty. The element s on its diagonal are the most significant ones since

the off-diagonal elements account for the stiffness coupl ing effects through the medium. The

dynamic stiffness prope rties of a semi-i nfinite elas tic solid under forced vibration s have been

- 218 -

I n F l g . 2 . T h e s e c u r v e s a r e i n c l o s e a g r e e m e n t w i t h t h o s e g i v e n e a r l i e r b y P a r m e l e e [ I ] . T h

c o m p u t a t i o n w o r k a s s o c i a t e d w i t h t h e F o u r i e r s y n t h e s i s m e th o d w as d o ne w i t h t h e a i d o f t h e f a

F o u r i e r t r a n s f o r m a l g o r i t h m . V e r i f i c a t i o n o f t i m e - h i s t o r y r e s u l t s w as a l s o made b y c o m pa ri ng

t h e r e s p o n s e s c o m p u te d f o r r i g i d g r o u n d c a s e s w i t h t h e v a l u e s g i v e n b y o t h e r c o m p u t e r p r o g r a m

U s i n g c o n v e n t i o n a l m e t h o d s , t h e f i x e d - b a s e n a t u r a l f r e q u e n c i e s , mode s h a p e s , an d m o da l

c h a r a c t e r i s t i c s o f t h e PCRV w e r e o b t a i n e d . T h e t e rm s " t w o - m o d e P CR V" a s u s e d h e r e r e f e r s t o

t h e i d e a l i z e d PCRV s t r u c t u r e h a v i n g tw o f i x e d - b a s e n a t u r a l f r e q u e n c i e s . Some e x p l a n a t i o n i s

n e c e s s a r y c o n c e r n i n g t h e g e o m e t r y an d c o o r d i n a t e s o f t h e m o d e ls u se d i n t h e e x a m p l e p r o b l e m s .

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T h e s u p e r s t r u c t u r e i n M o d el I ( F i g . 3 a ) i s r e p r e s e n t e d by N m a ss es i n s e r i e s , ea c h h a v i n

o n e d e g r e e o f f r e e d o m r e l a t i v e t o t h e r i g i d - b o d y d i s p l a c e m e n t s o f t h e s y s t e m . T h e e a r t h q u a k e

m o t i o n I n t h i s m od el w as a ss um e d t o h a v e o n l y a s i n g l e c o m p o n e n t , l a t e r a l t r a n s l a t i o n , an d t h

b a s e o f t h e m o d el i s a l l o w e d t o t r a n s l a t e a n d r o t a t e ( r o c k i n g ) a b o u t th e X 3 - a x i s , w h i c h i s

p e r p e n d i c u l a r t o th e d i r e c t i o n o f t h e g ro u n d m o t i o n . T h e r o t a t i o n s o f t h e lu mp ed s t r u c t u r a l

m a ss es r e l a t i v e t o t h e x . f r a m e w e r e i g n o r e d . W hen t h e r o c k i n g d e g r e e o f f r e e d o m o f t h e b a s e

i s o m i t t e d . M o d el I r e d u c e s t o t h e s y s t e m s t u d i e d b y S c a v u z z o , e t a l . [ Ί ] .

M o d e l I I ( F i g . 3b ) r e p r e s e n t s a n i d e a l i z e d u n s y m m e t r i c s y s t e m w i t h an e c c e n t r i c m ass ( o r

a p p e n d a g e ) w i t h m u l 11 c o m p o n e n t g r o u n d d i s p l a c e m e n t i n p u t . T h e s e i s m i c m o t i o n w a s a s su m e d t o

b e a h o r i z o n t a l m o t i o n c o m b in e d w i t h v e r t i c a l g r o u n d m o v e m en t . T h e tw o d i s p l a c e m e n t c o m p o n en

c a n h a v e a p r e s c r i b e d p h a s e r e l a t i o n . T h e t o p m a s s , m i . I s s u p p o r t e d by a " b e a m - t y p e " m em ber

- 219 -

The ground f l e x i b i l i t y is represented by the parameter V which Is the shear wave ve lo c-

it y o f the ha lf-s pa ce medium de fine d as

V

s

- /¡Tp" (38)

where μ and ρ are the modulus o f r ig id it y and the mass de ns ity o f the hal f-s pa ce medium,

r e s p e c t i v e l y . T he c o n d i t i o n ν - ™ is th e l i m i t i n g ca se w h er e th e g ro u n d I s r i g i d . Th e

resu l ts ob tained with r i g id ground represent the responses o f the system with o ut co nside r ing

t h e I n t e r a c t i o n e f f e c t s .

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For three-d im ensio nal an al ys is , the model Is assumed to have a c i rc u la r base wi th radius

r . th e s te a d y -s ta te r e s u l ts s u p p l ie d by B y c r o f t [1 1 , 1 2 ] fo r a th r e e -d im e n s i o n a l e l a s t i c

h a l f -s p a c e w ere u t i l i z e d . B y c r o f t ' s r e s u l ts a re v a l i d w i th i n th e f re q u e nc y ra ng e o f i ii r

0

/V <

1.5. However, the extens io n o f a hal f -sp ace analy sis to h igher frequency fac to rs has subse-

qu ently been made by Olen [15] and by Awo jobi [1 7 ].

Figures 4 and 5 present the response amplitudes of the 1100 MW(e) PCRV simulated by

Model I . Both the single-mo de (so ft-mo unted ) and two-mode (hard-mo unted) cases were co ns ide red .

T he se re s u l ts c l e a r l y I n d i c a te th a t a l a r g e n u c le a r po w er s ta t i o n w i l l b e ha ve d i f fe r e n t l y f ro m

the c o n v e n t io n a l h i g h - r i s e b u i l d i n g s d u r i n g e a r th q u a k e s . T he I n t e r a c t i o n e f f e c ts te n d to

reduce the response o f a l ow, heavy stru ctu re due to the presence o f re la t i v e ly larg e amounts

- 2 2 0 -

^ . 2 T i m e - H i s t o r y R e s po n se

T he r e s p o n se o f a n i n t e r a c t i o n s y s te m u n d e r a n a r b i t r a r y t i m e - h i s t o r y i n p u t w as d e t e r m i n

f o r an e x a m p l e p r o b l e m b a s e d o n M o d el I I I w h i c h w as s u b j e c t e d t o a g r o u n d t r a n s l a t i o n a l e x c i -

t a t i o n i n t h e X

3

d i r e c t i o n . T he t i m e h i s t o r y o f t h e N -S c o m p on e n t o f th e E l C e n t r o , C a l i f o r n

e a r t h q u a k e ( Ma y 1 9 ^ 0 ) w a s u s e d a s i n p u t a n d a m o d a l d a m p i n g f a c t o r o f . 0 5 w a s u s e d f o r a l l t h

s t r u c t u r a l m o d e s . W he n t h e g r o u n d i s t r e a t e d a s a d e f o r m a b l e m e d i u m , t h e b a s e d i s p l a c e m e n t

v e c t o r h as s i x n o n - z e r o c o m p o n e n t s , a n d t h e s y s t e m r e s p o n d s w i t h t h r e e - d i m e n s i o n a l m o t i o n t o

s e i s m i c e x c i t a t i o n . T im e h i s t o r i e s o f th e s i x c o m po n en ts o f { u l ( t ) } a r e sh ow n i n F i g . 8 f o r

7/25/2019 CDNJ04820ENC_001


t h e c a s e w i t h t h e s t r u c t u r e o n f i r m s o i l ( Vg = 1 00 0 F P S ) . A s e x p e c t e d , t h e X

3

( N o r t h - S o u t h )

b a se t r a n s l a t i o n a n d r o c k i n g a b o u t t h e X _ - a x i s a r e s e en t o h a v e m uc h l a r g e r m a g n i t u d e s t h a n

t h e o t h e r f o u r c o m p o n e n t s .

F i g u r e 9 s ho ws t h e c o m p a r i s o n o f t h e t o t a l b a s e a c c e l e r a t i o n s i n X

3

d i r e c t i o n w i t h t h e

a p p l i e d f r e e - f i e l d e x c i t a t i o n i n p u t . W hen t he r e a c t o r i s o n s o i l , t h e m a xi mu m v a l u e s f o r t h e

b a se a c c e l e r a t i o n t e n d t o be r e d u c e d b y t h e i n t e r a c t i o n e f f e c t s . F o r a r o c k f o u n d a t i o n

m e d i u m ,

t h e p ea k b a se a c c e l e r a t i o n s d o n o t d i f f e r a p p r e c i a b l y f r o m t he f r e e - f i e l d m axim um s

w h en d a m p in g i s p r e s e n t i n t h e s t r u c t u r e . T h i s i s m e n t i o n e d b e c a u s e t h e b a s e a c c e l e r a t i o n s

c a n b e s ï grvî f i c a n t l y a m p l i f i e d b y t h e i n t e r a c t i o n e f f e c t s i f t h e s t r u c t u r e i s u n d a mp ed { s e e ,

f o r e x a m p l e , t h e r e s u l t s i n S c a v u z z o , e t a l . [ 1 8 ] ) .

221

REFERENCES

[1] P AR MELEE, R., Buildi ng-Fo undati on Interaction Effe cts , J ournal of the Engineer ing

Mechanics Division, ASC E, EM2, April 1967, pp. 131- 151.

[2] PAR MELEE, R., PER ELMAN, D., LEE, S., KEER, L., Seismic Respon se of Structur e-Found ation

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System s, Journal of the Engineering Mechanics Divisio n, ASC E, Vol . 9Ί , N o. EM6,

December I960, pp. 1295 -1315.

[3] LUCO, J., Dynamic Interaction of a Shear Wall with the Soi l, Journal of the Enginee ring

Mechanics Division, ASC E, EM2, April 1969, pp. 333-3Ί 6.

[Ί] SCAVUZZ O, R., BAILEY, J., RAFTOPOULOS, D., Lateral Structural-Foundation Interaction of

Nuclear Power Plants with Large Base Masse s, USAEC Contract No. AT-(Ί 0-1)-3 822, Tech.

Report No. 3., September 1969.

[5] DUKE, C. M., et al. , Strong Earthq uake Motion and Site Co ndit ions : Holly wood, UCLA

Department of Engineering, June 1969.

[6] TAJ IMI, H., Discussion of R ef. 2., Journal of the Engineering Me chanics Divis ion, ASC E,

EM6, December 1967, pp. 29Ί -298.

222 -

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223 -

A « p l i t u d e t

Single-Mode

COUVENTIΟΝΑΙ

BU

Ι LD

ΙHC

Modal Cvrplitg Fac to r ( - .05

P o i s s o n ' s R.tlo . - 0

TRAÍS LITIOK-BPCIÍIIIO

7/25/2019 CDNJ04820ENC_001


(a) Amplitude of Top nasi

3 .0

l i t i ,

frequency

It 10.0 .

^

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(a) Model I

Conventional N-Mass Building

(b) Model II

Unsymmetric System

(c) Model III

Idealized Three-Dimensional

Four-Mode Structure

225

-

SINGLE-MODE PCU

Modal Oanping Factor - .0Í.

Poisson's Ral¡o - 0

Slruci.irp ΓΐΛΐυΓΛι frri iu rnc y

·- · V

r

- 200

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fa) Αηρ I i . ¿de (.1 Top M.n,

- 226

ΙΓ»Λ»1*|Ιon-tockIng

i . l lu« Hjf, · . Γ . . ■ ' . . . r-fl

i\:f\

ι u i

po?

:ooo

'.'30

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> < i \

Λ;/\ \

/ .<_>'- -

_«._;>.

*

^ S

227 -

15

(A) TOTAL VERTICAL DISPLACEMENT OF APP ENDAG E MASS

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b -

(B) TOTAL HORIZ ONTAL DISPLACEMEN T OF APP ENDAG E MASS

v

IN

PHASE WITH

u (v -

I

u )

V

S

F T / 5 E C

g

9 9 3 g'

- 228

V

s

- 1500

FT/SEC

MODAL DAMPING

FACTOR

FOR

STRUCTURE

ζ. = 0.02

■ THROUGH-SO IL COUPLING

NEGLECTED

THROUGH-SOIL COUPLING

INCLUDED

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East-West Translat ion

8-

4 -

-

O-

U-

-8 -

X CT

- - » - ' - ΐ ν / \ / ί

Vertical Translation

4

2

^

- 4

*iO~

3

-

' ^ P

V

\ Α Λ _ Λ A /

/ v /

v v

V U

North-South Rocking

8h

4

< C

7

xO

Torsion

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1

r-

North-South Translat ion

¿ ■ i ■ 4 ■ j, ■

TIME .SECOND

8

_

4

¿O

- 4

-8

East-West Rocking

"S

3 4 5-

T I M E , SECOND

- 230

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- 231 -

OB

0.4

0

-0.4

■ j

Λ Λ Λ

Λ - Λ

Λ / Λ

v

s

= °°

Fps

Μ / U V V

W

Ι

/ \ /

^

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-0.8

V ι

/ V

~ΜΑΧ. 0.908g

V =2 00 0 FPS

V

S

- 2 3 :

DISCUSSION

Q

A. HA DJI AN , U. S. Α.

C o n t r a r y t o F ig . 5 a n d m o re i n l i n e w i th T a j im i ' s t h in k in g , B i e l e k ' s ( 1 97 1 ) P h . D .

T h e s i s a t C a l t e c h s h o w s th a t t h e 2 n d mo d e of a tw o m a s s m o d e l of a c o n ta in m e n t s t r u c t u r e

d o e s n o t s h o w a n y v a r i a t i o n o f t h e f r e q u e n c y a s c o mp a re d to a f i x e d b a s e d mo d e l . T h i s w a s

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p a r a m e t r i c a l l y s h o w n t o b e t r u e f o r a w i d e v a r i a t i o n o f s o i l p r o p e r t i e s .

T. H. L E E , U. S. A.

T h e in t e r a c t i o n e f f e c t s o n h ig h e r mo d e s , i n my o p in io n , w i l l d e p e n d o n th e s y s t e m

p a r a m e te r s , a t t h i s s t a g e I d o n ' t k n ow w h a t c a s e B ie l e k h a s s tu d i e d an d w h a t a p p ro a c h h e h a

u s e d . I w i l l b e g l a d t o h a v e a c o p y of h i s t h e s i s f o r f u r t h e r s t u d i e s .

H . SATO, Japan

233

T. H. L E E , U. S . A.

Δ

Prof.

Taj im i 's op in ion was on the ana lys is o f a ca nt i lev er- typ e s t ruc ture . I t i s

qui te poss i b le that for th is part icu lar type o f sy s t em , in ter act io n e f fec ts on the h igher m od es

are tru ly neg l ig ib le . The con s ide rat io n of the f irs t few m od es for analy s is o f h i gh -r i se b u i ld

ings appears to be just i f iab le .

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Q

K. AKINO, Japan

W ith r e s p e c t t o f l e x ib l e s t r u c t u r e s , e s p e c ia l l y t a l l b u i ld in g s , we h a v e e x p e r im e n

tal data in Japan for thei r natural pe rio ds . If we sup po se that the f ir st p erio d is unity , the

seco nd i s appr ox im ate ly 1 /3 and the th ird i s app rox im ate ly 1 /5 , and thos e num bers cor resp on d

to natura l per io ds o f shea r mode v ibrat ion mod es o f a can t i lev er beam . Th ere for e , i t can be

sa id that v ibrat ion m od es o f ta l l bu i ld ings are independent o f the so i l in tera ct ion . How do you

th ink wheth er Jap anes e have to rec on s id er the in f luence o f s tru ctu re- gro un d intera ct ion

upon the res po nse o f h igher m od es , as you po inted out in your paper , ( the secon d pa ragraph

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K 3/6

DYNAMIC CALCULATIONS USING A FRAMEWORK ANALOGY

TO PREDICT THE SEISMIC RESPONSE

OF A NUCLEAR REACTOR

D.A. JOBSON,

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United Kingdom Atomic Energy Authority,

Reactor Group, Risley, Warrington, United Kingdom

ABSTRACT

The feasibility of predicting the response of a nuclear reactor system to a spectrally

defined earthquake is established in the context of a particular example. The associated

dynamic calculations were carried out on FRAMES, which is a UXAEA Reactor Group program for

- 236 -

(iii) Response of the core, including the support plates and the steel restraint

structure.

2.

SOIL/STRUCTURE MODES

The ground strata consisted of silty sand and clay, resting on soft rock. Beneath this

was a thin layer of sand and then clay to a very great depth; the reactor raft was founded

on the rock.

The application of spectral analysis first requires that the relevant natural frequencies

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and modal shapes of the system be found. If soil/structure interaction is ignored, there is

no ambiguity as to what is meant by the "system". A relatively flexible building founded on

solid bed rock extends only as far as the latter. The above method is obviously inadequate

for situations where the raft and the biological shield are both massive and stiff, whereas

the ground is relatively deformable. The boundaries of the "stress bulb" on which the reactor

sits were based on the dead-weight stresses Induced in the ground. These limits were chosen

as the line beyond which the vertical stress felt by the soil was less than 10Í of the mean

vertical stress under the foundation and indicated that most of the enclosed soil volume was

clay.

237

Σ

Γ * Ρ Χ

=

_________________

V^x*

+

V

+

Φ

ζ

2

) ' Y

S

r

m( <

Px

2

+ V

+

^

Γ

„

_

V m i - 2 i _ 2 χ _ 2 \ '

1

)

The summation is taken over all the r masses, each of which has component displacements

φ , tri and φ respectively in that mode, relative to the reference axes Ox, Oy and Oz .

Neglect of the relatively small masses of the steel and graphite features of the system in

this calculation involved negligible error. The maximum displacements of the raft were

obtained from u

0

and the relevant Γ, together with the modal amplitude (q_) of the component

considered. The spectral amplitudes (u

0

Γ φ_) of the raft so obtained are given in Table I.

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The tabulated values are spectral and not absolute displacements. Interpreted physically,

they are the peak displacements of the raft, as seen from moving ground that is substantially

uninfluenced by the presence of the reactor. As the methods described above are novel, they

were compared with a conventional analysis of soil-structure interaction, using the method of

Whitman [5]. Although there was a remarkably good agreement of frequencies, the modal shapes,

see Figs 5 and 6, are much more complicated than could be derived from the Whitman analysis

alone,

and hence the responses are different.

3. DIAGRID, VESSEL AND DUCT SYSTEM

- 238 -

movement however, such as that seen at the next critical (3*39 Hz, see Fig. 8 ) . Although

there were uncertainties about the behaviour of the heat exchangers, subsequent computer runs

showed that the core movements were not in general sensitive to their response. Only an

unlikely synchronism between the natural frequency of a heat exchanger and one of the modes

of the main system could substantially modify the top duct movement and its interaction with

the main modes of the vessel. The dynamic modelling and characteristics of the core, together

with its associated restraint structure will be considered in the next section.

4. GR APHITE CORE STACK AND RESTRAINT STRUCTURE

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Figure 9 shows the graphite core construction in which, at the assembly stage, each

brick is separated in plan from its neighbour by a small gap (exaggerated for clarity in the

figure),

the individual columns of bricks being stabilized by keys to those surrounding it.

A layered model was used to simulate this structure, each node of the latter being

supported vertically by columns having a stiffness derived from tilting tests on a model core

stack. The octagonal bricks are arrayed on a square pitch with clearance between each, and

an interlinking system of mutually perpendicular keys. The further interstitial bricks,

which are loosely fitted between them and are similarly interlinked by sliding keys, thus

give rise to what is equivalent to a cross-braced lattice pattern. Reference has already

- 239 -

or to unit moment. It was concluded that in this frequency range, the interactive effect, or

receptance of the core on the diagrid would be substantially the same as that for a rigid

body. An equivalent dynamical system was thus devised, consisting of two lumped masses, lying

on the centre-line of the core.

The forced responses further showed that there was negligible straining of the cross-

section of the core due to excitation by alternating forces and moments at the base of the

core in the range 0-5 Hz. Core distortion could therefore be represented in this frequency

range simply by flexing of its centre-line and led to the use of a flexible "dumb-bell"

for computations in this range. Such a simple model could not adequately represent the core

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at higher frequencies, where its behaviour became increasingly complex. In the 3-5 Hz range

it was difficult to decide whether a rudimentary modelling of the core, with a fairly complete

representation of the vessel and associated circuits, would yield better results than a

sophisticated modelling of the supported core, with a crude modelling of the vessel/duct and

heat exchanger system. It was judged that the latter would certainly be more appropriate

above 5 Hz and it was finally decided to analyse the 3-5 Hz range by a combination of both

models.

5. RESPONSE OF CORE/VESSEL COMPLEX

- 240 -

In an alternative representation, particularly aimed at computation of the higher modes

of the core (above 5 H z ) , the core and restraint structure were modelled in full detail.

This allowed only a crude simulation of the vessel and attached ducts. Lumped masses were

used to represent each of the latter, based on the total mass of the vessel, plus a nominal

allowance for the attached ducts.

It followed that such a model would give the same fundamental frequency in each of the

two principal planes, and this came out at 3*09 Hz . It corresponds to 2·95 Hz (longitudinal)

and 3·77 Hz (transverse) obtained from the previous model. A further limitation was that no

other frequencies were found in this range to correspond to the different ways in which the

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ducts could participate in core motion.

Further significant core modes were found at 5*9 Hz and 13·3 Hz, see Fig.

1 0 ( a ) ,

(b) and

( c ) . Computation of the modal participation factors, again ignoring any reactive effect on

the foundations, led to the results given in Table IV. Negligible distortion of the cross-

section of the core was found for the frequency range covered by Tables II, III and IV. This

was attributed not only to the effectiveness of the lattice layers in preserving their shape,

when account is also taken of the stability of the graphite columns and of the restraint

structure, but also to the absence of any excitation of the 'breathing' modes of the core

found at

b'U,

9·Α and 1Λ-2 Hz (see Fig.

1 1 ( a ) ,

(b) and (c)).

- 241 -

di = dj =

ψ -

Χι u. = u (5)

λι a a

The modal participations for the higher (structural) frequency require however that

account be taken not only of the relatively large movement of the small structural mass m

2

at

this frequency, but also the relatively small movement of the much larger foundation mass mi.

An approximate analysis shows Xi to be very nearly, -XjiT^/mi. It follows that the modal

participation factor at the higher frequency is virtually zero since IhX =- 0. Physically it

implies that the higher mode is such that the mass-centre of the combined system remains

virtually stationary. These conditions appear to be well satisfied for the reactor considered

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and require that the participation factors for the structural modes be reduced by taking into

account the reactive effect on the foundation. Using m, φ and Μ, Φ to denote the 'structural

and "foundation" masses respectively, the modified modal participation factor corresponding

to Γ

χ

is:

»ιφ + ΣΜΦ

Ρ ι _ x x /¿Λ

'χ ~ Σ/ηφ

2

+ ΣΜΦ

2

κ

'

Since Φ

2

« φ

2

the effect on the denominator of ΣΜΦ

2

is virtually negligible. It follows

that the Reactive Factor (r) to be applied to Γ

χ

is approximately:

- 242 -

Extensiv-? use was made of FRAMES, which is a powerful general-purpose program for the

dynamic analysis of skeletal structures. This was deployed in conjunction with grid-framework

methods, which enable elastic continua to be represented by equivalent lattices.

Modal analyses were carried out by means of an eigenvalue/eigenvector sub-routine, which

-vas based on the use of a matrix deflation method that determined each frequency in turn from

the lowest value upwards. Dynamic interaction between sub-systems was accounted for by

noddling the dynamic flexibilities of the attached system at each interface.

Deformable ground implied that the natural modes of the raft/structure needed to include

7/25/2019 CDNJ04820ENC_001


that part of the sub-soil which participated significantly in the motion. In assessing the

latter account also had to be taken of the fact that the ground properties depended on the

associated bearing pressure. Definition of the boundary for modal analysis is to some extent

arbitrary but is net judged to be critical. The various layers of ground within the support-

ing stress bulb were represented by an equivalent elastic framework and its distributed mar.'-

was lumped on the nodes of the latter. The boundary of the framework war- anchored by equiva-

lent springs having stiffnesses based on a finite element modelling oí the surrounding ground.

Subsequent analysis of the forced response of the structural features mounted on the raft

showed that the sideways movement was amplified so far as the core was concerned by associated

243 -

R REICHS

[1] JOlìSOll, D. A. and LITHERLAND, J. R., vibration analysis by computer: a user's guide tc

programs for the natural and forced oscillations of skeletal structures , TRG Report

1919(R),

(1969)

[2] HUDSON, D. E., 'Response spectrum techniques in engineering seismology , Proceedings c

;

1956 World Conference. Earthquake Engineering, Earthquake Engineering Research Institut'-,

1956

[3] JOBSON, D. Α., Lattice analogies for plane elastic problems . TRG Report

1339(R),

Part 2, (1966)

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[4·] BIGGS, J.

VI.,

introduction to structural dynamics . McGraw

Hill,

New York, (1969)

[5] WHITMAN, R. V., Seismic design for nuclear power plants . MIT Press, (1970)

[6J HRENNIKOFF, A,, Sulution of problems of elasticity by the framewerk method , J. App.

Medi..

ASME, pp A169-175, December 19Λ1

[7J JOBSON, D. Α., Grid Analogies for the elastic bending of plates . TRG Report 13<VO(R),

Part 2, (1967)

[8] JOBSON, D. Α., The representation of elastic solids by space lattices . TRG Report

V J U R ,

(1967)

- 244 -

TABLE I

RESPONSE OF FOUNDATION

Mode

Longitudinal

direction

2

Frequency

/(Hz)

0-46

Maximum spectral

displacement of top

of slab/(in.)

1.75

Maximum

rotation

/(rdn)

0-001C

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3

6

7

9

Transverse

direction

0·50

0·76

0-88

0-97

0·15

0-10

0-01

0-01

0-0006

0

0

0

0-0011

24 5

T AB LE I I I

INTERMEDIATE FREQUENCY RANGE UNDIMINISHED RESPONSE

Mode

Longitudinal axis

Frequency/(Hz)

2-95

3-61

4-03

Lateral acceleration of core/(g)

base

0-0118

0-0015

0-0058

top

0-0688

0-0102

0-0162

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Transverse axis

3-77

0-0012

0-0065

TABLE IV

HIGH FREQUENCY RANGE UNDIMINISHED RESPONSE

Lateral acceleration of core/(g)

246

G O A P H i T t C O R E

P R E S S U R E V E S S F

L

H F A T E X C H A N G E R S

S E C O N D A R Y B I O L O G I C A

S H I E L D

O U T L E T D U C T

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E X P A N S I O N J O . N T S

D I A G R I D G R I D

P R I M A R Y B I O L O G I C A L

S H I E L D

J. _ I N L E T D U C T

S U P P O R T t E E T i P A D S

24

7

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24 8 -

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^r *-^—

94 Sb

/ Λ

ι»·

\

\

trr

(kA

ritr

i

-f^.-

~i

7/25/2019 CDNJ04820ENC_001


\

u I h

\

■ψ:-

L OUGIT UDIU&.L 5E C T IOU

iyjf2QE.2_

0_K>J_ ç^.._

I 2.

LOUGITUDIrJ^L ¿ECTiOÌJ

Longi tudinal Modes due to Soi l -S t ruc ture In te rac t ion

■1

3

Sn

t i—

» Ί

ι

I

<L4

3 ^

bl

/

/

1

φ

1

1

39

2 »

<J

/

/

/

/

ta

%t

4 o

* 7

/

(··»

b l

5 2 ,

4)

Ï O

.'-.

„ ■ ·

l i .

1

«

'Λ

IM

1

1

J

4 -

.

7 3

M -

»

1 2 .

M

I

11

i'

-Ι

Γι

i

(74

bS

-.

3 2

7/25/2019 CDNJ04820ENC_001


\

3—

7 «

'9

* V

TS

•20

IO

Λ

W

^

24

M oot i

M t S c . »

11

MODE.

Z.

b»

2

CB.OSS 5ECTIQU

FIGURE 6

CB.OSS 5ECTOU

T r a n s v e r s e Mo de s d u e t o S o i l - S t r u c t u r e I n t e r a c t i o n

251 -

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t o

CJI

IO

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FIGURE 8 - .Vcdal Shape cf Ve ss el/ Du ct Sy st e- at 3-39 Hz

253

7/25/2019 CDNJ04820ENC_001


, ϋ — u i

" /

" *

■ « M -

1

\

'\

\

V?

Vs.

\

\ „

r ^ T ^

10

· · / ,

's

/

ri

1

',,

/

7/25/2019 CDNJ04820ENC_001


(a) 3 C") Hi

(c) 13 3 Hi

i -I G 'J R E 1 0 - A s y r r m e t r i c C o r e M o d e s

H

1

N t*

1

v

1

\

" X \ \

<-. « -*

V

\^

sS

^^

M

/

/

'Χ

N

ƒ

"

I

(o) 6 17 Hi

/

IO

n

it

/

I

4ft

14

«J

•1

1

\

1

w

fel

\

•1 '

1 1

\

H 1

»

\

1)

sîV.

I η ν

11

— -

——_ '

_

, · » '

,. Á

'X

.o

<

1

tl

u

ι

«

M

ι

i

II

\

/

(c) 14-2 Hi

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FIGURE 11 - Symmetric Core Modes

256

DISCUSSION

Q

K. AKINO, Japan

W e c a r r i e d ou t v ib r a t i o n t e s t f o r t h e g r a p h i t e s h i e ld in g s t ru c t u r e , a n d l e s t r e s u l

s h o w u s t h a t t h e g r a p h i t e p i l e -u p s t ru c tu r e d o e s n o t h a v e e ig e n v a lu e s .

In y o u r p a p e r , y o u p ro p o s e d th e t r a m e w o r k a n a lo g y in c lu d in g th e g r a p h i t e c o re s t ru c t u r e n u

you s ta t ed n a t ur a l pe r iod of the s t ru c t ur e to be t> cp s . Do you have any exp er im en ta l ev ide nce

c o n c e rn in g th a t f i g u re ?

7/25/2019 CDNJ04820ENC_001


. W. T. LAW TO N, U. K.

A

T h e g r a p h i t e c o r e w a s s i m u l a t e d by a 3 - d i m e n s i o n a l a r r a y ot d a m p e d m a s s e s

a n d s p r in g s ; t he s t i f f n e s s e s in t h e h o r i z o n ta l p l a n e w e re c a l c u l a t e d t o r e f l e c t t h e f r e e d o m to

p a r t i c u l a r h o r i z o n t a l m o v e m e n t s p e r m i t t e d b y t h e k e y ed c o n s t r u c t i o n , w h i le t he v e r t i c a l

s p r i n g s w e r e d e t e r m i n e d f r o m e x p e r i m e n t a l m e a s u r e m e n t of t h e l a t e r a l s ti f fn e s s of a c o l u m

K

3/7

PARAMETRIC ANALYSIS OF SOIL-STRUCTURE INTERACTION

FOR A REACTOR BUILDING

R.V. WHITMAN,

J.T.

CHRISTIAN,

J.M.

BIGGS,

Department

of

Civil Engineering,

Massachu setts Institute of Techno logy, Cambridge, Massachu setts, U .S.A.

7/25/2019 CDNJ04820ENC_001


ABSTRACT

A reinforced concrete reactor building

to be

located over deep soil

deposits is analyzed to determine the effects of varying flexibility of

the soil and variable damping. The soil structure interaction is

- 258 -

by stabilizer springs and by a drywell floor

seal,

which is treated as a

spring in the analysis. The entire edifice is supported on a circular base

mat.

The structure is of reinforced concrete, except for the reactor vessel

and its biological shield.

The ground response spectrum for the design basis earthquake is shown

in Fig. 2. This is based on a peak ground acceleration of 0.2 g. Only

horizontal ground motions are considered in this paper.

MATHEMATICAL MODEL

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The building together with its contents is treated as one integrated

dynamic system. The dynamic model is composed of 60 nodes, at which mass

is lumped, and these are connected by structural segments and springs as

illustrated in Fig. 3.

The soil and foundation is accounted for by translation and rocking

springs at node 60. The rotation of all horizontal planes is assumed to be

identical because vertical deformations in the walls of the exterior build

ing and the primary containment are negligible. The circular base mat is

- 259 -

due to foundation effects alone. An analysis was carried out for the case

of a rigid foundation by setting both foundation springs to large values.

The frequencies of the first five modes of the structure on a rigid

foundation are listed in Table 2. Fig. 4 shows the shapes of the first

three modes.

The first mode involves the outer cylinder only. The second and third

modes involve response of all the structure except the cylinder. In the

second mode all parts of the structure are in phase, but the reactor in

ternals (mass 14) are out of phase in the third mode.

When the damping ratio is 7* in all modes , the responses tabulated in

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Table 3 occur. It is clear that the most important modes are the first and

second. The cylinder responds most strongly in the first mode, and the rest

of the structure responds in the second and third modes. This is predict

able from the mode shapes.

The spectral accelerations and participation factors are also shown in

Table 2. Because of the shape of the response spectrum, the first three

modes have the same spectral accelerations of 0.38 g. The higher modes

have decreasing participation factors and decreasing spectral accelerations.

- 260 -

that case.

A prediction of the fundamental frequency of the combined system can be

made by the Dunkerley-Southwell approximation (Jacobsen and Ayre, 1958):

Λ

>-

Λ ♦ Λ

f f

RF RS

where f „ and f

R

_ are the fundamental frequencies of the cases with a rigid

foundation and with a rigid structure. Application of this rule to the re-

sults of Table 4 gives values of 1.18, 0.92, and

0.706

Hz for the cases of

hard, medium, and soft foundations, respectively.

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RESPONSE OF COMBINED STRUCTURE AND FOUNDATION

The response of the combined structure and soil system with 7% damping

in each mode was calculated with the computer model with the results in-

dicated in Tables 5 through 9. In addition to the runs summarized in

these Tables, computer runs were made in which the stiffness of the drywell

- 261 -

turai deformation becomes less important. Modes 3 and 4 are primarily struc

tural modes, mode 4 being almost entirely internal.

Thus,

modes 3 and 4

of the combined case look very much like modes 2 and 3 of the rigid founda

tion case.

A further understanding of the soil-structure interaction is obtained

by examining the proportion of energy distributed among structural deform

ation,

swaying, and rocking. This is tabulated for the first two modes in

Table 6, which shows that as the foundation becomes softer there is less

energy in the structural deformation. In the first mode the decreased

structural energy comes from an increase in rocking energy. In the second

mode it comes from both rocking and swaying but primarily from swaying. In

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all cases there is a significant contribution from both soil and structure,

but the structural part of the response decreases with decreasing foundation

stiffness.

The effects of all these factors on accelerations at various points

in the structure are seen in Table 7. The pattern of modal domination is as

expected from the previous paragraphs. The behavior of the cylinder and

roof is dominated by the first mode, and accelerations decrease for the

- 262 -

D = - i - Σ D . D.

η . ni i

η

where D represents damping in the n'th mode

E represents the total strain energy in the n'th mode

E .represents the strain energy in part i for the n'th mode

D. represents the damping for part i.

In the present case, the system consists of three parts:

1. The superstructure including the building, the containment, and

the reactor (D. = 4%)

2.

The soil rocking spring (D, = 5*)

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3. The soil swaying spring (D3 = 25*)

To compute the modal strain energy in the three parts it is convenient

first to compute the total kinetic energy in the entire system, which must

also be the total strain energy in the mode, E . The strain energy in the

two soil springs is easily computed from their modal displacements. The

energy in the structure is then the total minus that in the two springs.

The results of this calculation are shown in Table 10 . To illustrate

- 263 -

than it was in the case of uniform 7% damping, but the same general trends

are still evident. All accelerations are smaller than they were for uniform

7%

damping. With the uniform damping of 7% the acceleration of the founda

tion mat is greater than the peak ground acceleration, whereas with variable

damping it is, in most cases, approximately equal to the ground acceleration.

The latter result, which seems more reasonable (Biggs and Whitman, 1970),

is the result of larger damping in the second mode.

The forces and moments at selected points are tabulated in Tables 8 and

9. Again the pattern of modal dominance is more complicated, but the same

general trends are observed as in the case of uniform damping. The magni

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tudes of all forces and moments tabulated are reduced by variable damping.

The decreases are least for the outer structure (dominated by the first mode)

and greatest for points whose response is dominated by the second mode.

CONCLUSIONS - COUPLED FOUNDATION AND STRUCTURE

The analysis of the combined system of soil and structure leads to the

following conclusions applicable to this specific case:

264

CONCLUSIONS - GENERAL

The study illustrates how the interaction of structure and soil may

affect the response of a reactor building. It also shows how a detailed

examination of modal response can reveal patterns in the soil-structure

interaction.

The response of the model with weighted damping is significantly less

than that with uniform damping, especially those portions where the response

is strongly affected by foundation swaying. This is because a large portion

of the energy of the dominant first and second modes is in the soil rocking

7/25/2019 CDNJ04820ENC_001


and swaying springs. Results obtained using nominal uniform damping in all

modes may be conservative for the internal portions of a reactor building and

for equipment mounted on the foundation and internal structure.

- 265 -

TABLE 1

FOUNDATION SPRING CONSTANTS

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FOUNDATION TYPE SPRING CONSTANTS

SWAYING (kg)

ROCKING (k

R

)

HARD

1.2 X 10

6

K/ft. 7.50 X 10

9

K-ft/Radian

6 9

- 266

TABLE 3

RESPONSE OF RIGID FOUNDATION CASE

ACCELERATIONS

LOCATION

FOUNDATION

ROOF (PT.43)

TOP OF

SHIELD (PT.22)

ACCELI

0

0

0

:RATION

20g

60g

39g

DOMINANT

MODE

—

1

2

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BOTTOM

(b)

OF VESSEL (PT.9)

LOCATION

BASE (PT.59)

BOTTOM

BOTTOM

OF SUPPORT (PT.21)

OF SHIELD (PT.26)

SHEARS

0

AND MOMENTS

SHEAR

306

3.97

3.63

X

X

X

io

2

10

2

10

2

K

κ

κ

1XJM.

MODE

1

3

2

31g

MOMENT

438.

3.29

2.70

X

X

X

10

4

io

4

io

4

K-

K-

K-

■ft

■ft

-ft

2

DOM.

MODE

1

3

TABLE 4

RESPONSE OF RIGID STRUCTURE CASE

FOUNDATION

TYPE

UNCOUPLED

FREQUENCIES (Hz)

ROCKING SWAYING

COUPLED

FREQUENCIES (Hz)

f

l

¡2

ENERGY DISTRIBUTIONS

MODE 1 MODE 2

ROCKING SWAYING ROCKING SWAYING

HARD 1.57 2.25 1.36 3.82

71%

29%

28% 72%

MEDIUM

1.11 1.84 1.00 3.02

81% 19% 21% 79%

SOFT 0.81 1.45 0.74 2.14

81% 19% 19% 81%

Note: Uncoupled frequencies refer to one degree of freedom systems for rocking only and swaying only.

Coupled frequencies refer to system with two degrees of freedom where rocking and swaying occur

simultaneously.

TABLE 5

DESC RIPTIO N OF SOIL-STRU CTUR E INTERACTION RUNS

MODAL RESPONSE 7% DAMPING

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FREQUENCY (Hz) SPECTRAL ACCELERATION PAR TICIPA TION FACTOR

1 2 3 4 1 2 3 4 1 2 3UN NO.

FOUNDATION

TYPE

4

5

6

HARD

MEDIUM

SOFT

1 . 2 4

0 . 9 6

0 . 7 4

3 . 4 6

2 . 8 6

2 . 2 5

4 . 1 1

3 . 8 8

3 . 8 1

4 . 6 6

4 . 6 4

4 . 6 2

. 3 0 3

. 2 3 4

. 1 7 9

. 3 8

. 3 8

. 3 8

. 3 8

. 3 8

. 3 8

. 3 8

. 3 8

. 3 8

1 . 6 7

1 . 6 2

1 . 5 9

1 . 7 9

. 7 9

. 8 2

- 2 . 3 1

- . 3 9

- . 0 6 ■

. 7 2

. 0 2

- . 0 8

2 6 8

TABLE 6

ENERGY RATIOS

RUN

NO.

MODE

NO.

STRUCTURE

23.6%

ENERGY RATIOS

SWAYING

20.1%

ROCKING

56. 3%

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30.4%

13.5%

5.8%

7.7%

1.8%

63.5%

17.5%

79.3%

16.0%

82.4%

6.

1%

69.0%

14.9%

76.3%

15.8%

RUN

NO.

4

5

6

4,

3.

2,

Dominant

Modes

BASE

(PT.59)

SHEAR MOMENT

,07xl0

4

K

40

91

1

4.85xl0

6

K-l

3.70

2.80

1

6,

6

5

TABLE 8

FORCES AND MOMENTS

BOTTOM OF

SUPPORT (PT.21)

SHEAR MOMENT

,2xl0

2

K

. 0

.7

2

3.92xl0

4

K-l

2.70

2.15

2

2

1,

0.

BOTTOM

SHIELD

SHEAR

.83xl0

2

K

,37

9 8

2

OF

(PT.8)

MOMENT

2.74xl0

4

K-l

1.48

1.07

2,1

BOTTOM OF

SKIRT (PT.8)

RUN NO. SHEAR MOMENT

BOTTOM OF

DRYWELL (PT.42)

SHEAR MOMENT

BOTTOM OF

CYLINDER (PT. 58)

SHEAR MOMENT

4

5

6

7.75xl0

2

K

3.49

2.34

1.28xl0

4

K-l

0.52

0.35

10.5xl0

3

K

7.4

6.0

7.84xl0

5

K-l

4.88

3.65

2.89xl0

4

K

2.14

1.61

3.76xl0

6

K-l

2.75

2.03

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Dominant

Modes 2,1

2,1 1,2

270 -

RUN NO.

TABLE 9

FORCES IN CONNECTIONS

SPRING

VESSEL TO SHIELD (3-22) SHIELD TO CYLINDER (22-29)

4

5

6

4.74 X 10 ' K

2.21

1.50

13.6 X 10 K

7.08

4.98

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DOMINANT

MODES 1,2

1,2

TABLE 10

WEIGHTED MODAL DAMPING

RUN NO. DAMPING IN

MODE:

RUN MODAL SPECTR AL ACCELERA TION (g )

NO. 1 2 3 4

4 . 2 7 9 . 2 5 6 . 3 9 1

. 6 1 9

5 . 2 2 1

. 2 4 0

. 6 0 4 . 6 3 2

6 . 1 7 0 . 2 4 0 . 6 2 8 . 6 3 2

DOMINANT MODES

TABLE 11

ACCELERATIONS FOR WEIGHTED DAMPING

ACCELERATION (g) AT:

TOP OF TOP OF

FOUNDATION ROOF SHIELD VESSEL

(PT.43) (PT.22) (PT. 9)

.179

.195

.200

2

. 504

. 396

. 2 0 0

1

. 3 8 2

. 229

. 3 1 0

1 , 2 , 3 i n

1 i n 5 &

4

6

. 334

. 1 9 2

. 146

2 in 4

1 i n 5 & 6

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TABLE 12

FORCES AND MOMENTS FOR WEIGHTED DAMPING

RUN

NO.

SHEAR

3.61x10 K

BASE

(PT.59)

MOMENT

4.47x10 K-l

BOTTOM OF SUPPORT (PT.21) BOTTOM OF SHIELD (PT.26)

SHEAR MOMENT SHEAR MOMENT

4.6x10 Κ

3.10xl0

4

K-l 2.36xl0

2

K

2.24x10 K-l

2.91 "

2.37 "

3.47 "

2.65 "

4.2 "

3.8 "

2.16

1.68

1.16

0.78

1.25

0.88

Dominant

Modes 1

1,2

1,2

1,2,3

1,2,3

RUN NO. BOTTOM OF SKIRT (PT.8)

SHEAR MOMENT

6.62x10 Κ 1.12x10 K-l

BOTTOM OF DRYWELL (PT.42) BOTTOM OF CYLINDER (PT.58)

SHEAR MOMENT SHEAR MOMENT

8.2x10 Κ

6.41xl0

5

K-l 2.66xl0

4

K

3.42x10 K-l

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3.22

2.12

Dominant

Modes 1,2,3

0.52

0.34

1,2,3

5.8 "

4.6 "

1,2

4.20

3.14

1,2

2.01

1.52

2.57

1.92

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α05 007 0.1

02 0.3 05 0.7 1.0

U N D A M P E D P E R I O D I S K . )

Figure I: IDEA LIZA TION OF STRUCTURE

Figure 2·. GROUND RESPONSE SPECTRUM-DESIGN BASIS EARTHQUAKE

f r # q : 2 3 3 C pi

I r t q .

4 . 2 5 c p i

I r t q . ' 4 . 6 7 c p t

.-I

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Figure 3 : THE DYNAMIC MODEL

MODE 3

Figure 4 : MODE SH AP ES - RIGID FOUNDATION CASE

- 275 -

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Iraq.>3.46 cp»

Β

276 -

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277 -

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( r eq . = 2 . 25 Cps

- 278 -

DISCUSSION

K. MAR.GUERRE, Germany

T h e s o i l i s r e p re s e n t e d b y s p r in g s a n d d a mp in g . W h a t a b o u t t h e ma s s o f t h e s o i l

J. M. BIG GS . U. S. A.

A n e f f e c t i v e s o i l ma s s w a s a d d e d to t h e b a s e ma t - b o th fo r t r a n s l a t i o n a n d ro c k

in g . T h e v a lu e s u s e d w e r e t h o s e p r e v io u s ly d e r iv e d b y D r . W h i tm a n .

Q

7/25/2019 CDNJ04820ENC_001


Q

G . K L E IN , G e rma n y

In t h i s c o u n t ry t h e r e i s a s t r o n g d i s c u s s i o n a b o u t d a m p in g . A r e y o u r f i g u re s :

C o n c r e t e s t r u c t u r e 4

S o i l s w a y in g 2 5 %

S o i l r o c k in g 5 %

- 279

s e a l is r u b b e r , p e r h a p s t h e s p r in g c o n s t a n t s h o u ld b e t a k e n a s z e r o . I n t h e c a s e of a b e l l o w s

se a l , the sp r in g i s ve ry s t if f and has a s ign i f ican t e f fec t on the fo r ce s in the pe de s ta l and

dry wel l .

Q

D.

L U N T O S C H , G e r m a n y

I s n ' t i t a r a t h e r p o o r r e p r e s e n t a t i o n m o d e l l i n g a c y l i n d r i c a l a n d c o n i c a l s h e l l b y

only one lumped mass fo r each r ing ? What about the accuracy of the mode l ?

7/25/2019 CDNJ04820ENC_001


J. M. BIG GS , U. S. A.

O v a l l i n g o f a c y l in d r i c a l s h e l l d o e s n o t o c c u r u n d e r e a r th q u a k e lo a d in g b e c a u s e

i t i s a n t i - s y m m e t r i c a l . L o c a l b e n d in g m i g h t b e s i g n i f ic a n t if l a r g e c o n c e n t r a t e d m a s s e s w e r e

a t t a c h e d to t h e s h e l l . H o w e v e r , t h i s d o e s n o t u s u a l ly o c c u r a n d th e r e f o r e I t h in k u s e of a

s in g l e n o d e fo r t h e c o mp le t e r i n g i s s a t i s f a c to ry .

7/25/2019 CDNJ04820ENC_001


K 4/1*

DEVELOPMENT OF ASEISMIC DESIGN OF PIPINGS

VESSELS AND EQUIPMENTS IN NUCLEAR FACILITIES

H. SHIBATA, A. WATARI, H. SATO, T. SHIGETA,

Institute of Industrial Science, University of Tokyo , Tokyo ,

A. OKUMURA,

Faculty of Science and E ngineering, Waseda University,

S. FUJII, M. IGUCHI,

Faculty of Engineering, University of Tokyo, Tokyo

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This report involves the development of aseismic design procedures of pipings,

vessels and equipments in Japan. These mechanical structures show their va

282

DISCUSSION

- ^ G. KLEIN, Germ any

Do you use di f f erent damping factors for con s ider ing :

1. De s ign base earthquake ?

2. Maximum potent ia l earthquake ?

H. SHIBATA, Japan

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We de sign only for des ig n earthqu ake . But for the an aly s is of hypo thet ical

earthquake usua l ly we use the sam e value . In som e ca se s , we use another approach to chec

the ma rgin of the safety , for exa mp le , e las to- pla s t i c an aly s i s .

K 4/2

SEISMIC DESIGN COEFFICIENTS OF EQUIPMENT

IN NUCLEAR POWER PLANTS

C.-W.

LIN,

Westinghouse lectric Corporation,

Pow er Systems, Pittsburgh,

Pennsylvania,

U.S.A.

7/25/2019 CDNJ04820ENC_001


ABSTRACT

Seismic coefficients

for the

design

of

equipment

in

nuclear power plants have been

shown to be proportional to the single degree response spectrum. Using the two degree

response spectra, constructed according to a numerical time integration study for the El-

Centro 1940 N-S earthquake obtained by previous investigators, equipment seismic design

- 284 -

Using the same model, this paper illustrates how the results presented by Penzien and

Chopra [ 5], in obtaining two degree of freedom response spectra, can be easily converted to

useful information for different site conditions when either the local ground power spectral

density or the single degree response spectrum is known.

2.

DYNAMIC RESPONSE OF A TWO DEGREE OF FREEDOM SYSTEM

Using the fact that the mass of equipment is usually small compared with the supporting

structure, the interaction effect will be more from the structure to the equipment than vice

versa.

Consequently, Penzien and Chopra [5] suggest a separate two degree of freedom system

for each of the N normal modes of the building without the equipment in analyzing this inter-

action effect. Figure 1 shows this system, in which subscripts n and a Indicate the quanti-

7/25/2019 CDNJ04820ENC_001


ties derived from the n th building mode without the equipment and from the equipment, re-

spectively. M is the

mass,

K the spring constant, C the damping factor, X the displacement

relative to the support, and U (t) is the support motion, with

N N

U ft) - ( E m . θ, /

Ζ

ra. θ

2

) U (t) - α U (t) (1)

1-1

i i n

i e l

i i n

8 " 8

where θ Is the dimensionless η th building mode shape quantity for i th floor, with m

- 2 8 5 -

t -ω ζ ( t - ζ ) . .

Χ + a U = α o ƒ s i n o ( t -

ç) e U (ç ) d (5)

n n g n n n g

ς

After rearranging eq. (4) by grouping Χ + α U to one side of the equation and then elimin

ating this using eq. (5 ), the resulting equation can be substituted into eq. (2) and then

solved for X - X . The net result is :

a η

et ω t

-ω*ξ*

(t - λ)

(Χ - Χ ) - - - ¡V

1

I

sino* (t - A)e

a a

(6)

a n ω a

a o

λ -ω Ε (λ - ζ) dçdX

7/25/2019 CDNJ04820ENC_001


where:

ω* - ω (Β + 1 )

1 / 2

, ξ* - ξ (β + 1 )

1 / 2

a a a * a a a

3. VARIANCE OF (Χ - Χ )

a η

Since design earthquakes are usually very strong and have long duration, it can be de-

duced from Caughey, Stumpf, and Bycroft [7], [ 8], that each of such earthquakes forms a

- 286 -

as obtained in the process will have a sharp peak at o - ω . Hence, when making contour

integration with respect to o, the main contribution to the integral will come from the

region around ω - ω . Using the same analogy as Caughey and Stumpf [7] , which originated

from the Laplace's method of evaluating integrals, eq. (8) may be very closely approximated

b y :

α o t t

o

2

( t ) « G (o ) ( - W o

2

ƒ ƒ s i n o * ( t - λ ) s i n o * ( t ' - λ ' ) ( 1 0 )

η ω η a a

a o o

- ω * ζ * ( t + t ' - λ - λ ' ) «

e

< ' | ζ ( ω ) |

2 { C 0 S w ( X

"

Χ Ί

- ω ς λ '

-e [coSüíA cosu) λ + ζ βΐηω λ cosuX *

η η η

7/25/2019 CDNJ04820ENC_001


[COSCDX COSLÚ

η

-ω ζ (λ + λ ' )

+ ζ sin to λ ' cosLoX + — βΐηωλ simo λ*Ί + e

η η ω η

η

- ω

2

+ ω

2

[1 + ζ s imo (λ + λ ' ) + — - β ΐηω λ s in to λ ' ] } d io ) dXdX'

- 287 -

single degree resp onse spe ctrum S (in ,f, ) , is in p rop ort ion to the sq uare root of its power

spectrum density. Therefore,

S* (ω ,Ç )

σ * ( ω , ο , ζ , ï , β , α ) " _

a

.

" ,". ο ( ο , ω , ζ , ζ , β , α ) ( 14)

max η a η a'

a

η S (ο ,ζ ) max η a η' a a' η

a η η

Eq. (14) has

a

direct application when the t wo degree of freedom resp onse spectra obtained

by

Penzien and Chopr a [5] for the El-Centro 1940 earth quake are used. Using t heir not ation

with:

C (ω , ο , ζ , ξ , β , α ) Ξ Ι (Χ - Χ ) ο

2

/ Ι (15)

an η* a' ^η '

s

a ' a η ' a η a g

1

max

represent ing the seismic coefficient based on El-Centro earthquake, eq. (14) can be writt en

7/25/2019 CDNJ04820ENC_001


ΰ ( ο , ο , ξ , ζ , β , α )

an n a' n a a' η

S* (ω , ξ )

' C (ο , o .

Ç

, ξ . β , α )

(16)

where

(<"_·

ξ

)

an

η η

- 288 -

e) Multiply each C by its proper building participation factor

a

and the normalized

mode shape θ and take the root-mean-squared sum, i.e.,

C.* - [ Σ (α θ< C

* ) 2 ]

1/2

ia η in an

η

f) Design the equipment to resist a maximum horizontal seismic force of:

F - C, *W

a ia a

where W is the weight of the equipment.

5. DISCUSSION AND CONCLUDING R EMARKS

For a given set of building and equipment parameters, such as ο , ο , ζ , ζ , β and

7/25/2019 CDNJ04820ENC_001


η a η a &

α , the present study shows that the ratio of the seismic coefficients for two different

design earthquakes is in proportion to the ratio of the spectral accelerations. Using this

conclusion, together with the two degree response spectra obtained by Penzien and Chopra [5],

based on a numerical time integration study for El-Centro 1940 N-S earthquake, equipment

seismic design coefficients for other earthquakes can be easily obtained.

289 -

REFERENCES

[1] ALFORD, J. L., HOUSNER, G. W., and MARTEL, R. R., "Spectrum Analyses of Strong-Motion

Earthquakes," Earthquake Research Laboratory, California Institute of Technology,

August 1951.

[2] CLOUGH, R. W.

,

"Earthquake Analysis by Response Spectrum Superposition," Bulletin of

the Seismologicai Society of America, Vol. 5 2, No. 3.

[3] JOHN A. BLUME 6. ASSOCIATES, ENGINEERS, "Summary of Current Seismic Design Criteria for

Nuclear Facilities," San Francisco, California, September 1967.

[4] KEITH, J., "Seismic Design of Critical Equipment in Nuclear Reactor Plants," John A.

7/25/2019 CDNJ04820ENC_001


Blume & Associates, Engineers, 1968.

[5] PENZIEN, JOSEPH and CHOPRA, ANIL K., "Earthquake Response of Appendage on a Multi-

Story Building," Proceedings of the Third World Conference on Earthquake Engineering,

Vol. II, 1965, New Zealand, pp. 476-487.

[6] SEXTON, H. JOSEPH, KEITH, EDWARD J., Discussion of the above paper, Proceedings of the

Third Wcrld Conference on Earthquake Engineering, Vol. II, 1965, New Zealand,

290 -

APPENDIX II - NOTATION

The following symbols are used in t his p aper :

C - damping factor of the equipment .

C - damping factor of the nth building mode.

K » spring constant of the equipment supp ort.

K ■ sp ring constant of the generalized nt h building mass.

M » mass of the equipment .

M - generalized mass of the nth building mode.

7/25/2019 CDNJ04820ENC_001


W

α

equipment weight.

S . S * ■ spectral accelerations,

a' a

m - mass of the building at ith floor.

G " power spectral density.

291 -

TABLE 1

PARAMETERS FOR TWO DEGREE

RESPONSE SPECTRA FIGS. 4-12

CURVE NO.

1

2

3

4

5

2π

Τ α -

Up

0.20

0.40

βο

0.002

0.010

0.025

0.002

0.010

7/25/2019 CDNJ04820ENC_001


6

7

8

9

1

0.60

0.80

0.025

0.002

0.010

0.025

0.002

FIXED

REFERENCE

U s «

- E -

C„

X n l t )

X

0

( l ]

ΠΙ1

m

¿>¿JHy l4,

MwSwwwuyMsWA&ú ^

FIG.

1

TWO DEGREE OF FREEDOM SYSTEM

7/25/2019 CDNJ04820ENC_001


- S

a

(«n , in ), EL-CENTRO

SO

| .

n |

(

n

) , * 7 . l TIMES

1940 EARTHQU AKE EMPIIFIED TYPICAL EARTHQUAKE

293 -

16.0

14.0

12.0

So im i, £n)

Sa (un, £n)

10.0

7/25/2019 CDNJ04820ENC_001


- 294

Can 2

»Ί 1

1

2

1 t

Ι ι

/

V/

4

7

\ /

5

. \ / \ 'Ü

Λ / /

\Y

S

V"»

y y - -7 ¿ -u L

a

n ■ 1.0

ín ■ 0.05

i

0

= 0.00

^&» *\.

/ / / /

7/25/2019 CDNJ04820ENC_001


///λ

0.2

0.4 0.6

T„

0.8

1.0

FIG.5 SEISMIC COEFFICIENTS

295 -

-an 2

β ' , 1

1

11

3

4

\ 1

5

Á

M 1 /

6

J

7.

\ I

e

x\v '

W / π

w ¿& f*&

n ' 10

£„ « 0.02

f

0

- 0.02

7/25/2019 CDNJ04820ENC_001


¡I

' z^^fc^,"

'—A..s s Λ

0.2 0.4 0.6 0.8 1.0

T„


296

1 >

//

//y

//yi

/ /r/ /

4

7

^ ¡ é ^

S A X ^ ¿ ^ ^

W¿k

" n = ' 0 0

£

n

=

0.05

ίο = °

0 5

Ï Ï S L

1

' >' ι ' >'

0.2 0.4 0.6

Tn

0.8 1.0

7/25/2019 CDNJ04820ENC_001



- 297

1

¿yi

\ y*

7

^ ^ i *"

10

"n * 1.0

£„ = o.io

í o ■= 0.02

J l

" 1 2 ^ * ^ ·

^?4¿ieéáé

0.2

0.4 0.6

Tn

0.8 1.0

7/25/2019 CDNJ04820ENC_001


FIG. 11 SEISMIC COEFFICIENTS

2 9 8

D I S C U S S I O N

Q

P .

M I T T E R B A C H E R , S w i t z e r l a n d

D o y o u c o n s id e r t h e c a s e o f a h o r i z o n ta l t u b e o r t a n k w i th e n d - c lo s u re a n d n ot

c o mp le t e ly f i l l e d w i th w a te r u n d e r t h e i n f lu e n c e o f a n e a r th q u a k e ?

J . D . S T E V E N S O N , U . S . A .

F o r d e s i g n p u rp o s e s t h e s im p l i f i e d m e th o d p re s e n t e d i n T ID 7 0 2 4 , c h a p te r 6 ,

e a r t h q u a k e d e s i g n of n u c l e a r f a c i l i t i e s i s n o r m a l l y u s e d . T h i s p r o c e d u r e d o e s c o n s i d e r s l o s h

ef fec t .

7/25/2019 CDNJ04820ENC_001


0

H . W O L F E L , G e r m a n y

I t h in k y o u r me th o d is v e ry c o n s e rv a t i v e . D id y o u c o m p a re y o u r r e s u l t s w i th a

t ime h i s to ry a n a ly s i s a n d c a n y o u g iv e u s a n e s t ima t io n o f t h e f a i l u r e ?

K 4/4

ASEISMIC DESIGN OF ASYMMETRIC STRUCTURES

AND THE EQUIPMENT CONTAINED

CH. CHEN,


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ABSTRACT

The dynamic structural response Is a function of these basic assumptions regarding

seismic input, soil structure interaction, structural and dynamic properties of

- 300 -

centers of different floors are not on the same vertical

axes.

Hence the translational

displacements and torsional displacement are coupled. The results indicate that

the coupled displacements have a significant effect on the equipment design.

2.

SEISMIC INPUT

The intensity of shaking and the frequency of occurrence of a possible

future earthquake at a nuclear power plant site should be first estimated by the

seismologist and the geologist. Engineers interested in this topic can consult

to references [5] through [9]· The recommendation from the analysis of the local

geology and seismology will then be translated into some forcing function or input

at the base of the structure. Due to the nondeterrainistic nature of earthquake

records,

there are vorks dealing with random input [10 - l6]. Unfortunately, owing

to the scarcity of strong motion acceleregrams, the statistical properties of the

1

7/25/2019 CDNJ04820ENC_001


random motion can not be determined precisely [9> l *]* Hence there are efforts

devoted to the analysis of least favorable response [IT , l8] . But the disadvantage

of this approach is that the design may be too pressimistic to be practical even

for the least favorable response in the local sense only [17]. The other alternative,

though not necessarily a better way due to the uncertainty about the nature of a

- 301 -

simplicity in its application. However the different damping values between soil

and structure may cause difficulty in the analysis. An intuitive formula for combined

damping value was suggested by Professor Biggs as mentioned in reference [27].

Another pitfall of this method is that the basic assumption of no separation between

base and soil in the derivation of spring constants may sometimes be violated in

the analysis. Other more sophisticated schemes such as the finite element method

or the lumped parameter method are sometimes used to model the

soil.

Related works

in this area can be found in reference [28]. These methods can also he used to

predict the soil amplification or the influence of local soil properties, if the

soil model is extended to the bedrock. But these methods again suffer from the

difference in damping values between soil and structure. The linear elastic half

space theory was also applied to predict the influence of structure on the free

field motion [29, 30, 31] and to predict the structure response on flexible foundation

7/25/2019 CDNJ04820ENC_001


[32]. Generally speaking, the effect of interaction is small for structures built

on competent rock. As to the soil amplification effect or the influence of local

soil properties, both the shear beam model and finite element method have been used

[ 2 8 ] .

- 302 -

the shear walls. Winokur and Glück

lk2]

analyzed statically the asymmetric multistory

structures by combining the stiffness matrix of each individual stiffening element

into the overall stiffness matrix. Glück [1*3] analyzed the asymmetric structure

by the continuous method, and was discussed by Nynhoven ii Adams, Biswas & Tso

5. EXAMPLE

The linear elastic dynamic analysis of the coupled auxiliary structures

of a nuclear power plant is presented here as an example. The characteristics of

these structures are quite different from those conventional structures mentioned

before. First of all, due to the requirement of biological shielding from the radio

active emission caused by hypothetical accident, the vails and floors are exceedingly

thick. Secondly, as a result of the equipment layout, interior walls are discontinuous

from floor to floor. Thirdly, several structures are coupled together.

7/25/2019 CDNJ04820ENC_001


To reveal the torsional effect about the vertical axis of an asymmetric

structure, the floor is assumed to behave like a diaphragm rigid in its own plane

and with three degrees of freedom, two translational, and one torsional [38, ^ i ,

k6 ¡. Due to the Stubby proportion of the actual structure, the flexibility of the

where m, m are the lumped mass and the

T, is the transformation matrix defined

303 -

moment of inertia respectively, and

(Ό

where C , C are the coordinates of the center of gravity of the i floor. The

transformed individual matrices can be combined into the overall matrix easily.

The equations of motion of the system are

[M] {D} + [C] {D} + [K] {D}

{F}

(5)

where mass matrix [M] is composed of the subraatrices M, on the diagonal. [C] is

7/25/2019 CDNJ04820ENC_001


the damping matrix, [K] is the stiffness matrix and {D} is the relative displacement

vector. The overhead dot indicates time derivative. The force vector { F } i3 defined

(F) = - I T ] ' [M

2

] {G}

(6)

- 304 -

where

{Dj} = [U] {D} (12)

and

[Kj] =

([U]')"

1

[Κ] [ U]

- 1

(13)

Let the normalized eigenvectors of [Κ ] "be [ V]. Premultiplying both sides of eg.

t l

(il) by [V] and making use of the orthonormal property of the eigenvectors, we

h av e

t v ] '

(Dj.) + iv ]* n y Iv] IV ] ' {Dj} = {0} (11,)

7/25/2019 CDNJ04820ENC_001


{

V

+ [

V

{D

N

}

=

{0} (15 )

w h er e

- 305 -

and the ground acce le ra t ion vec tor ÍG) i s rep laced by

{0} = G f ( t ) {e} (22)

o

and G is the maximum ground acceleration, f(t) is the time function of ground motion,

and {e} is the earthquake directional vector.

If one wants to adopt time history of strong motion earthquake as input,

then eg. (20) can be used directly to solve for the modal response. If one wants

to use the design spectrum method, then the solution of eg. (20) with zero displacement

and velocity as initial conditions and with small damping 12] is

D, = i - Í V G

ε

- ^

( ΐ _ τ )

f(-r) Sin ω, (t-x)dx (23)

J

i

J

°

J

°

¡

7/25/2019 CDNJ04820ENC_001


Defing the spectrum value as 120]

S (ω) =

ω[ί*0 f(T)

e

ß uJ ( t _ x )

Sin

U

(t-T)dx]max

(Sk)

Jo °

306

ACKNOWLEDGMENT

Thanks axe due to Dr. G. J. Patterson, research engineer of Gilbert Associates,

Inc., for some stimulating discussion, and to Mr. M. Plica, project structural

engineer of the same company, for preparing the mat hematic model in the example.

7/25/2019 CDNJ04820ENC_001


- 307 -

REFERENCES

1. Biggs, J.M. Introduction to Structural Dynamics McGraw Hill 196k.

2. Hurty, W.C. , M.F. Rubinstein, Dynamics of Structures, Prentice Hall I96I1.

3. Lin, ï. K. Probabilistic Theory of Structural Dynamics, McGraw Hill

I967.

1*. Blume, J.Α., N.M. Newmark, L.H. Corning, Design of Multistory Reinforced Concrete

Buildings for Earthquake Motions, Portland Cement Association 1961.

5. U.S.A.E.C. (Division of Technical

Information),

Nuclear Reactor and Earthquake,

TID-702I», August 1963.

6. U.S.A.E.C. (Division of Technical

Information),

Summary of Current Seismic Design

Practice for Nuclear Reactor Facilities, TID-25021, September 1967·

7.

HouBner, G. Vf.. "Engineering Estimates of Ground Shaking and Maximum Earthquake

Magnitude", U^" World Earthquake Engineering Conference, Vol. I, Santiago, Chile

7/25/2019 CDNJ04820ENC_001


1969.

Θ. Lomenick, T.F. and NSIC Staff, Earthquake and Nuclear Power Plant Design, Nuclear

Safety Information Center, Oak Ridge National Laboratory, 0RNL-NSIC-28, July 1970.

9. Housner, G.W. , "Strong Ground Motion," "Earthquake Engineering", Edited by R. L.

Wiegel, Prentice-Hall, Inc. 1970.

- 308 -

21. Newmark, N. M., W. J.

Hall,

"Seismic Design Criteria for Nuclear Reactor

Facilities," Fourth_ World Conference on Earthquake Engineering, Vol. II, Chile,

1969.

22.

Cornell, C.A. , "Design Seismic Input," Seismic Design for Nuclear Power Plants,

Edited by R. J. Hansen, The M.I.T. Press, 1970.

23. Housner, G.W., "Design Spectrum," Earthquake Engineering, Edited by R. L. Wiegel,

Prentice-Hall, 1970.

2U,

Jennings, P.C., G.W. Housner, N.C.

Tsai,

Simulated Earthquake Motions, Earthquake

Engineering Research Laboratory, California Institute of Technology, 1968.

25. Barkan, D.D., Dynamics of Base and Foundations, Translated from Russian by L.

Drashenska, McGraw-Hill, I962.

26.

Whitman, R.V., F.E. Richart, "Design Procedures for Dynamically Loaded Foundations,"

ASCE, Soil Mechanics and Foundations Division, Nov. 1967·

27.

Whitman, R.V. , "Soil Structure Interaction," Seismic Design for Nuclear Power

Plants, Edited by R.J. Hansen, The M.I.T. Press, 1970.

7/25/2019 CDNJ04820ENC_001


28. Werner, S.D., A Study of Earthquake Input Motions for Seismic Design, prepare for

U.S.A.E.C. By Agbabian Jacobsen Associates, R-691^-925, June 1970.

29. Parmelee, R.A. , "Building - Foundation Interaction Effects," ASCE Engineering

Mechanics Division, April 1967·

- 309 -

Ui. Manning, Τ.Α., Jr., The Analysis of Tier Buildings with Shear Walls, Ph. D.

Dissertation, Stanford University, April 1970.

»2. Winokur, Α., J. Glück, "lateral Loads in Asymmetric Multistory Structure",

ASCE. Structural Division, March 1968.

Ί3. Glück, J., Lateral-Load Analysis of Asymmetric Structures , ASCE, Structural

Division, February 1970.

ΊΙ*.

Wynhoven, J.H.

,

P.F. Ada ms, J.K. Biswas, W.K. Iso, Discussion of "Lateral-Load

Analysis of Asymmetric Multistory Structures", ASCE, Structural Division,

November 1970.

•»5. Bergstrom, R.N.

,

S.L. Chu, R.J. Small, "Dynamic Analysis of Nuclear Power Plants

for Seismic Loading", Presented at the ASCE Annual Meeting, Chicago, Illinois,

October 1969.

U6.

Plica, Μ., C. Chen, Dynamic Analysis of the Auxiliary Structures, Gilbert

Associates, Inc., Reading, Pennsylvania, Report No. 17^ 8, January 1971.

7/25/2019 CDNJ04820ENC_001


hj. Ayre, H.S ., "Interconnection of Translational and Torsional Vibrations in

Buildings",

Bulletin of the Seismic Society of America, 2 8, 1938.

hü . Biggs, J.M., J.M. Roesset, Seismic Analysis of Equipment Mounted on a Massive

Structure , Seismic Design for Nuclear Power Plants, Edited by R. J. Hansen, The

M.I.T.

Press, 1970.

- 3 1 0 -

TABLE I - MODAL FREQUE NC IES

f ICENYA4.UtS

0.1*4047910

0.11 1*1 »6 JO

O . U 4 » » M

0.10*4** 4*0

0.41*111*10

0.142**1 M D

0.2TT0I161D

0.151»47740

0.4·«791410

0.9*79 401*0

0.IS470I71D

0 . · » · 09«7 0

O l

O l

OS

O l

O l

0 *

0 4

0 *

0 »

04

0 *

0 *

P R E O u e w c i i i - o i D / s t c

1 4 . 1 0 1

1 7 . 4 4 *

2 4 . 4 4 6

2 4 . 4 2 4

1 0 . 1 1 0

1 7 . 7 7 1

9 2 . 4 ) 2

9 4 . 0 2 0

4*.«8 1

1 4 . 4 1 2

« 2 . 1 1 «

« 1 . 1 7 0

F R f o u e K i t i - c r s

2 . 2 * 1

t . a o *

1 .476

6 . 1 ) ·

6 . 1 1 1

6 . 0 1 1

• . » » t

4 . 3 Ϊ 1

1 1 . 1 1 *

1 2 . 2 0 0

1 4 . 7 1 1

1 4 . 1 2 *

n a i o o s - s i c

0 . 4 4 1

0 . 1 1 6

O.ÏSI

0 . 1 1 6

0.20T

0 . 1 6 6

Ü . l l *

0 . 1 0 1

0.040

0.0»i

0 . 0 6 1

0*0*1

7/25/2019 CDNJ04820ENC_001


0.10140*420

ο . ι ι » · » ι ο * ο

0.1217*9*10

0.1*4400000

0.1M44114O

0.217792*40

0 1

0 1

0 1

OS

OS

O l

1 0 0 . 7 0 1

1 0 1 . 0 0 «

1 1 0 . 9 4 7

1 1 4 . 0 1 7

1 1 4 . 9 4 9

1 4 7 . 9 4 9

1 4 . 0 2 1

1 7 . 2 0 1

1 1 . 4 4 1

1 » . 4 » 9

2 1 . 1 1 9

1 1 . 4 * 4

0.067

0 . 0 1 1

0.037

0.036

0.066

0.0«»

DYNAMIC R ESPONSE IN MODE I FOR 0.27 G EARTHQUAKE AT THE ORIG IN OF THE GLOBAL CO ORDIN ATES

MODE SHAPE

3.19717D-05

7.5615 JD-06

9.28002D-07

9.576830-05

1.05763D-05

2.698390-06

1.772440-02

7 . 5 8 1 8 6 0 - 0 *

5.441440-04

9.70*720-05

1.08*370-05

2 . 7 3 2 1 8 0 - 0 6

3.881310-06

3.*56610-06

9.*8686D-08

1.3027*0-05

9.087700-06

1.576930-07

2 . * 8 8 3 7 U - 0 5

1.666620-05

1.871**0-07

* . 1 5 0 * 7 0 - 0 5

* . 1 8 7 2 5 D - 0 5

6.080660-07

3.99*13D-06

* . 7 B 6 2 5 D - 0 6

'5.108920-08

1.25662D-05

8.793690-06

2 . 1 0 2 9 8 0 - 0 8

1.285*0D-05

1.2821SO-05

8.3933*0-08

3.888770-02

1.083560-01

7 . 6 6 2 9 6 0 - 0 *

4 . 2 3 5 4 5 D - 0 2

1.387230-01

8.396810-04

4.565360-02

1.681*90-01

9.09755D-04

OPKT IN X QUAKE

C M , R A D

3.502*3D-05

8.28353D-06

1.016610-08

1.0*9120-0*

1.158610-05

2 . 9 5 6 0 2 0 - 0 8

1.9*1670-02

- 8 . 3 0 5 7 6 0 - 0 *

5.960980-06

1.06313D-0«

1.187910-05

2.9930*0-08

* . 2 5 1 8 9 0 - 0 6

3.7866*0-06

1.03926D-09

l.*27l2D-05

9.95537D-06

1.727*90-09

2 . 7 2 5 9 5 D - 0 5

1.8257*0-05

2.05012D-09

* . 5 * 6 7 5 0 - 0 5

* . 5 8 7 0 * 0 - 0 5

6.661230-09

* . 3 7 5 * 8 0 - 0 6

5.2*3230-06

- 5 . 5 9 6 7 1 D - 1 0

1.376600-05

9.633300-06

- 2 . 3 0 3 7 7 0 - 1 0

1.«.00130-05

1.404570-05

- 9 . 1 9 * 7 2 D - 1 0

- * . 2 6 0 0 6 0 - 0 2

1.187010-01

- 8 . 3 9 * 6 0 0 - 0 6

- * . 6 3 9 8 * D - 0 2

1.519680-01

- 9 . 1 9 8 5 2 0 - 0 6

- 5 . 0 0 1 2 6 0 - 0 2

1.8*20*0-01

- 9 . 9 6 6 1 6 0 - 0 6

OP N T I N y auAKE

C M , R A O

9 . 0 3 * 6 0 0 - 0 *

2 . 1 3 6 7 S D - 0 *

2.622360-07

2.706230-03

2 . 9 8 8 6 6 0 - 0 *

T.62512D-0T

5.008570-01

- 2 . 1 * 2 * 9 0 - 0 2

1.537650-0«

2.7*2370-03

3 . 0 6 * 2 3 0 - 0 *

7.72061D-07

1 . 0 9 6 7 9 0 - 0 *

9.76T72O-05

2.680800-08

3.681280-0«

2.568010-0«

« . « 5 6 0 9 0 - 0 8

7.031660-0«

« . 7 0 9 5 « D - 0 «

5.28B3«D-08

1.172840-03

1.1832*0-03

1.718280-07

1.12866D-0«

1.352500-0«

- l . « « 3 6 B D - 0 8

3.55096D-0«

2.«8«930-0«

- 5 . 9 * 2 6 2 0 - 0 9

3 . 6 3 2 3 0 D - 0 *

3 . 6 2 3 1 1 0 - 0 *

- 2 . 3 7 1 8 0 D - 0 B

-1.09889D 00

3.061930 0 0

- 2 . 1 6 9 4 1 D - 0 «

- 1 . 1 9 6 8 6 0 0 0

3.9200*0

0 0

- 2 . 3 7 2 7 8 D - 0 *

-1.29008D 00

«.751570 00

- 2 . 5 7 0 7 9 D - 0 «

AC C I N X OUAKE

N . R A O / S E C S O

6.971500-05

1.6«B82D-05

2.023530-06

2 . 0 8 8 2 5 0 - 0 *

2 . 3 0 6 1 9 D - 0 5

5.88389

0-06

3.86484O-02

- 1 . 6 5 3 2 « 0 - 0 3

1.186520-03

2.1161*0-0«

2.36450D-05

5.957570-06

β.463290-06

7.537210-06

2.068630-07

2.8*06*0-05

1.98159D-05

3.*3BS2D-07

5.«259«0-05

3.63*090-05

« . 0 8 0 7 2 0 - 0 7

9.050190-05

9.130380-05

1.325900-06

8.709280-06

1.043650-05

- 1 . 1 1 4 0 1 0 - 0 7

2.74008D-05

1.917*80-05

- « . 5 8 5 5 9 0 - 0 8

2.802850-05

2.79576O-05

- 1 . 8 3 0 1 9 0 - 0 7

- 8 . « 7 9 5 5 D - 0 2

2.362720-01

- 1 . 6 7 0 9 2 0 - 0 3

- 9 . 2 3 5 4 8 0 - 0 2

3.024880-01

- 1 . 8 3 0 9 « 0 - 0 3

- 9 . 9 5 4 8 7 D - 0 2

3.666530-01

- 1 . 9 8 3 7 4 0 - 0 3

AC C I N r OUAKE

N . R A O / S E C S Q

1.798310-03

« . 2 5 3 1 6 0 - 0 «

5.2197*D-05

5.386680-03

5.948850-04

1.517760-0«

9.969430-01

- « . 2 6 * 5 7 0 - 0 2

3.060650-02

5.*5862D-03

6.099280-04

1.536770-0«

2.18312U-0«

l.9«*2*D-04

5.336070-06

7.327500-0«

5.111560-04

8.8697«0-06

I.399630-03

1.374220-04

1.0526

30-05

2.3 14520-03

2.355200-03

3.420190-05

2.246580-04

2.69212U-0«

- 2 . 8 7 3 6 2 0 - 0 6

7.068100-0«

4.946 190-04

- 1 . 1 8 2 8 6 D - 0 6

7.230010-0«

7 . 2 1 1 7 2 0 - 0 *

- 4 . 7 2 1 0 0 D - 0 6

- 2 . 1 8 7 3 2 0 0 0

6.09*690

0 0

- « . 3 1 0 1 8 0 - 0 2

- 2 . 3 8 2 3 1 0 0 0

7.802730 0 0

- « . 7 2 2 9 5 D - 0 2

-2.567880 00

9.457890

0 0

- 5 . 1 1 7 1 0 0 - 0 2

β

y-i

H

I

w

M

o

co

M

H

0

c

S

O

7/25/2019 CDNJ04820ENC_001


SPECTRUM RESPONSE ·

PARTICIPATION FACTOR

PARTICIPATION FACTOR

0.5322

IN X OUAKE -

IN Y OUAKE ·

0.4181

10.7847

DYNAMIC RESPONSE IN MODE 2 FOR 0.27 G EARTHQUAKE AT THE ORI GIN OF THE GLOBAL COOR DINATES

MOOE SHAPE

1.856670-05

2.26575D-06

2.735100-08

4.604680-05

4.03470D-06

1.360480-07

7.944970-03

2.126130-03

-4.29429D-05

4.71816D-05

4.101440-06

1.522400-07

1.99599D-06

9.4179«D-07

1.091«10-08

6.920T1D-06

2.32523D-06

-2.659150-08

1.36808D-05

«-12361D-06

-1.10580D-07

2.21312D-05

8.062510-06

-1.391430-07

1.707790-06

6.B6426D-07

-9.8725BO-09

5.916670-06

1.47659D-06

-2.705820-08

5.710510-06

1.099080-06

-2.11576D-08

2.348210-02

2.149350-02

-3.76814D-04

5.914290-02

2.435370-02

-4.391350-04

9.56988D-02

2.708280-02

-4.99324O-04

OPMT IN X QUAKE

CM,RAO

3.B0075D-04

4.63816D-05

5.59897D-09

9.*261«D-04

B.25935D-05

2.785000-08

1.626400-01

4.352350-02

-8.790740-06

9.65B43D-04

8.39597D-05

3.116460-08

4.085940-05

1.92792D-05

2.23420D-09

1.416720-04

4.759920-05

-5.44349D-09

2.800560-04

8.441340-05

-2.26365D-08

4.530430-04

1.650460-04

-2.84836D-08

3.49597D-05

1.40517D-05

-2.020990-09

1.211190-04

3.022690-05

-5.539020-09

1.16B980-04

2.249900-05

-4.33111D-09

4.806980-01

4.39989D-01

-7.71367D-05

1.210700 00

4.98538D-01

-8.989440-05

1.959030

00

5.544060-01

- 1 . 0 2 2 1 5 D - 0 4

OPMT IN Y QUAKE

C M , R A D

3 . « 6 7 5 2 0 - 0 8

« . 2 3 1 5 1 D - 0 9

5 . 1 0 8 0 8 0 - 1 3

8 . 5 9 9 7 0 0 - 0 8

7 . 5 3 5 2 1 0 - 0 9

2 . 5 « 0 B 3 D - 1 2

1.48380D-05

3 . 9 7 0 7 6 0 - 0 6

- 8 . 0 2 0 0 1 0 - 1 0

8 . 8 U 6 3 D - 0 8

7 . 6 5 9 8 6 0 - 0 9

2 . 8 4 3 2 3 0 - 1 2

3 . 7 2 7 7 1 0 - 0 9

1.75889D-09

2 . 0 3 8 3 2 0 - 1 3

1.29251D-0B

4 . 3 4 2 6 0 0 - 0 9

- ♦ . 9 6 6 2 3 0 - 1 3

2 . 5 5 5 0 2 0 - 0 8

7 . 7 0 1 2 4 0 - 0 9

- 2 . 0 6 5 1 8 0 - 1 2

4 . 1 3 3 2 3 D - 0 B

1.505750-08

- 2 . 5 9 8 6 3 D - 1 2

3 . 1 8 9 4 6 0 - 0 9

1.28197D-09

- 1 . B 4 3 8 0 0 - 1 3

1.105000-08

2 . 7 5 7 6 7 0 - 0 9

- 5 . 0 5 3 3 B 0 - 1 3

1.066490-08

2 . 0 5 2 6 4 D - 0 9

- 3 . 9 5 1 3 8 D - 1 3

4 . 3 8 5 5 2 0 - 0 5

4 . 0 1 4 1 3 0 - 0 5

- 7 . 0 3 7 3 8 0 - 0 9

1.104550-0«

4 . 5 4 8 2 9 0 - 0 5

- 8 . 2 0 1 2 9 0 - 0 9

1.78727D-04

5 . 0 5 7 9 8 0 - 0 5

-9.32537D-09

ACC IN X OUAKE

M.RAO/SECSQ

1.183930-03

1.4*«780-0«

1.7**070-06

2.936220-03

2.572770-0*

8.6752*0-06

S.066190-01

1.35575D-01

-2.738300-03

3.008580-03

2.615330-0«

9.70772D-06

1.272760-0«

6.005440-05

6.95950D-07

«.«13060-0«

l.«8271O-0«

-1.6956*0-06

8.72369D-04

2.62946D-04

-7.05122D-06

1.411220-03

5.1*ll*D-0*

-8.872600-06

1.088990-0*

*.377070-05

-6.295360-07

3.77283D-04

9.*15610-05

-1.725390-06

3.6*1360-0*

7.0083BD-05

-1.3*913D-06

l.*97360 00

1.370560 00

-2.402790-02

3.77131D

00

1.552940 00

-2.800190-02

6.10234D

00

1.726960 00

- 3 . 1 8 3 9 9 D - 0 2

ACC IN Y QUAKE

M . R A D / S E C S Q

1.080120-07

1.318110-08

1.591160-10

2 . 6 7 8 7 9 0 - 0 7

2 . 3 * 7 2 0 0 - 0 8

7 . 9 1 * 6 3 0 - 1 0

* . 6 2 2 0 1 D - 0 5

1.236680-05

- 2 . * 9 8 2 2 D - 0 7

2 . 7 * * 8 1 D - 0 7

2 . 3 8 6 0 3 0 - 0 8

8 . 8 5 6 6 0 0 - 1 0

1.16117D-08

5 . * 7 8 9 2 D - 0 9

6 . 3 * 9 3 2 0 - 1 1

* . 0 2 6 1 5 0 - 0 8

1.35271D-08

- 1 . 5 * 6 9 7 0 - 1 0

7 . 9 5 8 8 * D - 0 8

2 . 3 9 8 9 2 0 - 0 8

- 6 . * 3 3 0 0 D - 1 0

1.287*90-07

♦ . 6 9 0 3 9 0 - 0 8

- 8 . 0 9 * 6 9 0 - 1 0

9 . 9 3 5 U D - 0 9

3 . 9 9 3 3 1 0 - 0 9

- 5 . 7 * 3 4 1 D - U

3 . 4 * 2 0 * 0 - 0 8

8 . 5 9 0 1 0 0 - 0 9

-1.57*12D-10

3.322110-08

6.39392D-09

-1.23085D-10

I.366080-0*

1.25039D-0*

-2.19213D-06

3.**066D-0*

l.*16780-0*

-2.55*690-06

5.56731D-0*

1.575550-0*

-2.90*83D-06

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SPECTRUM RESPONSE «

PARTICIPATION FACTOR IN

PARTICIPATION FACTOR IN

62*9

OUAKE ·

CUAKE -

10.4121

0.0009

313

TABLE IV - SRSS OF ALL THE MODES

SASS OF THE DYNAMIC RESPONSE OF ALL 42 KODES FC« 0.77 C EARTHQUAKE AT THE CENTERS OF HASSES

1 5 5

n-

1

1

2

3.

A.

D PKT

. θ

i

f

2

'.i 2

θ I

κ l

a «

o ;

Κ

0 '

IH Ι QUAKE OF HT IN V OUAKE

C M . R A D

. 4 3 2 7 2 E - 0 2

- 4 7 2 * 2 E - 0 2

■ 7 4 1 0 3 E - 0 6

. 4 T M T I - 0 2

. 9 6 0 9 6 E - 0 2

. 2 0 4 9 4 E - 0 9

. 9 3 7 3 0 C - 0 I

. 4 2 0 1 6 E - 0 1

. 9 9 7 4 6 E - 0 4

„ 4 9 9 M C - 0 2

. 1 0 M 2 E - 0 2

. 1 4 ) 1 1 7 E - 0 9

. 6 1 2 9 3 E - 0 3

• 3 4 4 2 * 1 - 0 3

C M . R A D

. . 4 7 3 1 0 E - 0 3

. 7 5 4 0 4 Ε - 0 2

. 4 3 0 S O E - 0 «

. 1 9 0 9 4 E - 0 2

E.9*997E-02

r . e i 7 9 3 E - 0 6

Í . 7 9 6 9 1 E - 0 1

Γ . Α 6 4 6 2 Ε - 0 1

I . O 2 3 6 9 E - 0 4

F . 9 3 4 6 9 E - 0 2

. 5 1 9 9 6 E - 0 1

Γ . 7 Β 9 1 9 Ε - 0 9

Γ . 4 3 2 7 2 Ε - 0 3

■ 3 9 4 9 2 Ε - 0 2

ACC IM I QUAKE

M . K A D / S f C S t )

1.82T23E 0 0

1.477&1E

0 0

0 . 1 3 4 3 6 E - 0 2

3 . 0 8 4 3 3 E 0 0

2 . 9 1 4 8 3 E 0 0

1.085 82E-01

5 . 4 0 0 6 4 E 0 0

3 . T 2 6 8 3 E 0 0

2 . 1 8 7 7 8 E - 0 1

7 . 9 1 4 0 1 E 0 0

9 . 8 1 4 8 3 E 0 0

9 . 4 7 9 1 0 E - 0 1

1.47108E 0 0

8 . 8 9 2 9 0 E - 0 1

ACC IN T OUAKE

H . * A O / S E C S Q

8 . T 0 & 2 9 E - 0 1

1.T9T10E 0 0

4 . 9 1 6 2 4 E - 0 2

1.43862E 0 0

2 . 9 9 9 3 3 E 0 0

T . 6 B 9 6 8 E - 0 2

3 . 9 0 3 1 1 E 0 0

6 . 3 2 1 9 2 E 0 0

1.22092E-01

4 . 0 0 9 8 7 E 0 0

1.98430E 0 1

• ■ S 4 0 8 9 E - 0 1

T . 9 4 1 3 2 E - 0 1

1.80269E 0 0

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Y

6

l

8.

. I 7 1 8 4 E - 0 *

. 9 7 1 B 9 E - 0 2

. 2 1 4 3 4 E - 0 2

. 8 9 4 3 6 E - 0 *

. 0 9 4 9 6 C - 0 2

. 3 9 7 9 B E - 0 3

. U I M H »

. 2 B . 2 4 E - 0 2

. 9 7 B 4 1 E - 0 2

• 2 9 3 4 2 E - 0 3

» . 2 9 8 6 3 Ε - 0 6

. 9 4 0 2 9 E - 0 2

I . U I 2 3 E - 0 2

» . 9 8 4 4 7 Ε - 0 *

. 9 S 1 2 7 E - 0 2

U 1 7 1 Z 2 E - 0 2

) . 8 7 0 4 6 E - 0 A

I . T 5 1 1 6 E - 0 2

I . 4 1 4 T 6 E - 0 2

. 2 7 8 0 7 Ε - 0 9

3 . 3 5 9 9 5 E - 0 2

2 . 4 7 8 4 6 E 0 0

1.50T22E 0 0

6 . 8 9 4 9 2 E - 0 2

3 . 8 9 4 0 A E 0 0

1.07974F 0 0

9 . B 0 5 3 & E - 0 2

4 . 7 4 4 9 1 E 0 0

2 . 3 3 6 9 9 E 0 0

1.47026E-01

3 . 6 4 7 T 3 E - 0 2

1.87T40E 0 0

3 . 4 0 7 4 3 E 0 0

7 . 9 8 4 9 8 E - 0 2

2 . 7 7 7 4 0 E 0 0

3 . 3 6 7 4 3 E 0 0

1.08639E-01

3 . 4 4 4 2 7 E 0 0

3 . 6 1 7 7 0 : 0 0

1.40448E-01

3 1 4

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315

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316

-

õ

4

Ζ

O

υ

υ

<

DI»

ν EQUIPA

\STRUCT

P H R A G M NO. 8

ΛΕΝΤ DAMPING 0 5%

URAL DAMPING 5°/o

1 Υ QUAKE X RESPONSE

\ 0 27 G EARTHQUAKE

\

~ ^

\

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.01 .02 04 .06 08.10 .20

40 .60 80 1.0

- 317 -

DISCUSSION

0

P .

C. RI ZZ O, U. S. A.

If y o u u s e d d e s i g n s p e c t r a a p p r o a c h f o r s t r u c t u r e , w h a t m e t h o d w a s u s e d t o

g e n e ra t e f l o o r r e s p o n s e s p e c t r a ? I f B ig g s ' me th o d is u s e d , f o r w h a t p l a n t s ( fo r e ig n a n d

U. S . ) has i t been us ed s uc ce ss f u l l y ?

. Ch. CH EN , U. S. A.

R e s p o n s e s p e c t r u m m e t h o d ( B i g g s ' m e t h o d ) w a s u s e d to g e n e r a t e f l o o r r e s p o n s e

c u rv e s . I n t h e U . S. t h i s m e th o d w a s a p p l i e d s u c c e s s f u l ly i n a n u c l e a r p o w e r p l a n t i n P e n n

s y lv a n ia (I d o n ' t w a n t t o me n t io n t h e n a m e of t h e p l a n t w i th o u t o u r c l i e n t ' s p e rm i s s io n ) . T h i s

me th o d w a s a l s o a p p l i e d t o a j o b i n J a p a n ,

7/25/2019 CDNJ04820ENC_001


Q

H . S A T O , J a p a n

I would l ike to as k a qu es t ion abo ut F i gs . 3 and 4 . I t se em s to m e tha t the sa m e

- 318

a n d s o m e t i m e s h i g h e r v a l u e .

Q

K. AKINO, Japan

I have two qu es t io ns wh ich a r e re la te d . The f i r s t i s , in the U . S . A . who p rov ide s

a c o n c e p t u a l d e s i g n of s t r u c t u r a l l a y o u t ; i n y o u r c a s e , W e s t i n g h o u s e o r G i l b e r t ?

Th e sec on d i s , if you a re g iven the s t ru c t u ra l layou t shown in F ig . 1 in you r pa pe r , you have

t o c a r r y o ut u n r e l i a b l e c o m p l i c a t e d d y n a m i c a n a l y s i s i n c l u d in g t o r s i o n a l v i b r a t i o n m o d e .

H o w e v e r , a s t r u c t u r a l e n g i n e e r c a n p r o v i d e a b e t t e r b a l a n c e d an d m o r e s t a b l e s t r u c t u r a l

layout than you have now. Which i s the be t te r way ? E i ther you ca r ry ou t the ana lys is as you

d o o r s t r u c t u r a l e n g i n e e r s p r o v i d e a m o r e a d e q u a t e s t r u c t u r a l la y o ut f or th e a s e i s m i c d e s i g n

B y th e w a y , d o y o u k n o w a n id e a l p ro p o s a l p r e p a re d b y K a i s e r E n g in e e r s w h ic h i s a x i s y m n ic

7/25/2019 CDNJ04820ENC_001


r i c a l c i r c u l a r r e a c t o r a n d a u x i l i a r y b u i l d i n g s .

. Ch. CH EN , U. S.A .

A

K 4/6

DYNAMIC ANALYSIS OF VITAL PIPING SYSTEMS SUBJECTED

TO SEISMIC MOTION

CH. CHEN,

Gilbert Associates Inc., Reading , Penn sylvania, U.S.A.

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ABSTRACT

The linear dynamic analysis of the three dimensional piping system of a nuclear

power plant is based on a lumped parameter model. Both time history input and design

- 320 -

damping matrix [C] are both symmetric, {x} is the relative displacement vector. The

dots over the variables indicate time derivatives, {y} is the input acceleration vector

at the support. The piping response is affected by the mathematical model used, and

some modelling consideration vas discussed by Harrington and Vorus [6j.

Either the time history of the support acceleration or the floor design

spectrum can be used as input. In view of the fact that hundreds of piping systems are

analyzed dynamically in a typical FWR plant, and that the time history analysis is time

consuming, it is more practical to use floor design spectrum as input. Of course time

history analysis has the advantage of providing resposes as a function of time on

condition that the mathematical model is correct and that the assumed material properties

and time history input are exact.

If the time history is used as input, eq. (l) can be solved either by direct

time integration [7» Ö, 93» or by superposition of normal modes. With either the time

history method by superposition of normal modes or the design spectrum method, the

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eigenvalues and eigenvectors of the system have to be solved first. They are obtained

by solving the free vibrational equations.

1

- 321 -

where we made use of the orthonormal conditions

[φ]

Τ

[ Μ] [φ] = [ Y ] , (6)

[ φ]

Τ

[Κ] [φ] - f ω ? ] , (Τ)

and the proportional damping relations

[C] - Ç[M] + α

[Κ].

(8)

The i component of eqs. (5) is

τ^

+ (C + cui )

f\

±

+ ω^η

1

-

-ί Φ

1

)

Τ

[M] {y} (9)

1

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where {φ.} ie the transpose of the i column of [φ ]. If we define the percentage of

critical damping β as

ß -

ζ

* °""i

- 322

ing.rJl - '

here we assumed small dampin g,^1 - β -*■ 1. Let the acceleration response spectrum

value be [18, 19]

Sa(u) = ω y [/e ^ - ^ f (

T

)Simo(t-t) di]

o o max.

(lo)

Then the maximum modal response is

γ. Sa

(n

i

)max = — J ^ . (

IT

)

i

l h e m ax im um d i s p l a ce m en t s a t e ac h d e g r ee o f fr eed o m i n i m od e a r e

( x . ) = ( η . ) { φ . } ( 1 3 )

ι max ι max ι

If one is interested in the maximum equivalent forces applied at each degree of freedom

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the maximum absolute modal acceleration will be derived first. It is [.'O]

( r l

i

+ Y

i

y

o

f ( t ) )

m a x

=

Y

i

S a

( U , )

■¿2'λ

frequency modes, and then take SRSS with the rest of the modes. Since some modes are

insignificant in comparison with others, it is desirable to choose the contributing

ones.

Following the derivation, we can see that the modal acceleration as defined in eq. (19)

is a natural basis for modal selection.

The final maximum stress at a point is then compared with the allowable one.

In case of overstress, perturbation technique can be applied to choose the design

changes [22]. A general approach without analysis is to put rigid restraints, e.g.

snubbers,

at location of maximum deflection; hopefully this can drive the fundamental

frequency toward the higher frequency side of the peak area of the design spectrum. Of

course this does not promise to be on economical redesign.

3. COMPARISON OF STIFFNESS AND FLEXIBILITY MATRIX METHOD

For stiffness matrix method, the overall stiffness matrix is obtained by

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combining the individual branch stiffness matrices as done in displacement finite

element method [23]. This matrix includes elements corresponding to branch points which

are not assigned as mass points. These unwanted elements can be eliminated by condensa-

tion scheme as follows. Let the overall stiffness matrix be partitioned as

- 324

The translations and rotations of branch points not assigned as mass points are obtained

{

V "

-

[ K

j j

r l

[ R

j i

] {

V

(2 T)

The total displacement vector is

(x) = # ) (28)

X

J

Applying these displacements {χ} to individual branch stiffness matrix, the internal

stresses and support reaction can be calculated accordingly.

If one wants to use flexibility matrix method, the flexibility matrix [A] in

eqs. (3) can be obtained either by taking the inverse of [K] in eq.

(2k)

or by applying

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unit load method. When unit load method is used, we will solve the set of simultaneous

equations

[Κ] (x) - IJ (29)

325 -

obtain the amplification curves and to combine the modal responses. With the time

history method, special care should be exercised to obtain the proper time history and

to perform parametric study. Due to the abrupt changes of the unsmoothed response

spectrum obtained from the actual strong motion earthquake records, the general trend

is to use simulated earthquake [28] as input such that the unsmoothed response spectrum

derived from it will simulate closely the design spectrum.

For primary coolant loop of a PWR plant, the mass is not small comparing with

the supporting structure. The response will usually be overestimated if the floor

design spectrum is used as input. Under this case, the loop and the structure can be

combined into one model and analyzed using ground design spectrum as input [29]. The

other alternative is to perform component mode analysis [30].

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ACKN0WLEGEMENT

- 326 -

REFERENCES

1 . C r a w f o r d , L . , " P i p i n g u n d e r D y na m ic L o a d i n g " , J

;

_ o f t h e A m eri c a i ; So c i e t y ol Naval

E n g i n e e r s , I n c . , M ay 1 9 5 6 .

2 .

U . Ξ . Α . E . C . ( D i v i s i o n o f T e c h n i c a l I n f o r m a t i o n ) " Su mm ary o f C u r r e n t S e i s m i c P e s i f: ;

P r a c t i c e f o r N u c l e a r R e a c t o r F a c i l i t i e s " , T I D - 2 5 0 2 1 , S e p te m b e r 1 9 6 7 .

3 .

B e r k o w i t z , L . , " S e i s m i c A n a l y s i s o f P r i m a r y P i p i n g S y s t em f o r N u c l e a r G e n e r a t i n g

S t a t i o n s " , R e a c t o r a n d F u e l P r o c e s s i n g T e c h n o l o g y , V o l . 1 2 , N o. L , M a rc h 1 9 o 9 ·

U. L i n , C . W ., " S e i s m i c A n a l y s i s o f P i p i n g S y s t e m " , N u c l e a r E n g i n e e r i n g a nd D e s i g n ,

V o l . 1 1 , N o . 2 , M ar ch 1 9 7 0 .

5 . A r c h e r , J . S . , " C o n s i s t e n t Mass M a t r i x f o r D i s t r i b u t e d M ass S y s t e m " , ASCK, S t r u c t u r a l

D i v i s i o n , A u g u s t 1 9 ^ 3 ■

6 . H a r r i n g t o n , R . L . , Vo ru s , W . S . , "D yn am i c S h oc k A n a l y s i s of S h i p b o a r d E q u i p m e n t " ,

p r e s e n t e d a t t h e M e e t i n g o f t h e H am p to n R o ad s S e c t i o n o f t h e S o c i e t y o f N a v a l

A r c h i t e c t u r e s a nd M a r in e E n g i n e e r s , I 9 6 6 ,

7 · W i l s o n , E . L . , R .W . C l o u g h , " Dy na m ic R e s p o n se b y S t e p - b y - S t e p M a t r i x A n a l y s i s " ,

7/25/2019 CDNJ04820ENC_001


S ym po si um o n t h e Use o f C o m p u t e r e i n C i v i l E n g i n e e r i n g , P o r t u g a l , 1 9 o 2 .

8 . C h a n , S . P . , H . L . C o x , W .A . B e n f i e l d , " T r a n s i e n t A n a l y s i s o f F o r c e d V i b r a t i o n s oV

C om pl ex S t r u c t u r a l - M e c h a n i c a l S y s t e m s " , J

■

Roy . A e ro . Soc . , 66 iy6,? .

- 327 -

22. Higney, J.T., Application of Perturbation Techniques to the Navy's Dynamic Design

Analysis Method , The Shock and Vibration Bulletin, Naval Research Laboratory,

December 19^9·

23· Ziekiewicz, O . C , The Finite Element Method in Structural and ^Continuum Mechanics ,

McGraw

Hill,

I967.

2h

. ASME Codes and Standards, Interpretations of the Code for Pressure Piping,

Mechanical Engineering, November 1970.

25. Hovanessian, S.A., L.A. Pipes, Digital Computer Methods in Engineering , McGraw

Hill 1969.

26.

Ketter, R.L., S.P. Prawel, Jr., Modern Methods of Engineering Computation , McGraw

Hill 1969.

27.

Biggs, J.M., Roesset, J.M., Seismic Analysis of Equipment Mounted on a Massive

Structure , Seismic Design for Nuclear Power Plants , edited by R.J. Hansen, the

M.I.T. Press. 1970.

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28. Jennings, P.C., G.W. Housner, U.C. Tsai, Simulated Earthquake Motions , Earthquake

Engineering Research Laboratory, California Institute of Technology, 1968.

29. Chen, C , J. David, Dynamic Analyses of Vital Piping Systems Subjected to Seismic

Motion , GAI Report No. 1729, Gilbert Associates, Inc., Reading, Pa. 1970.

328

DISCUSSION

Q

K. AKINO, Japan

Do you calcu late s t re s s e s of p iping sys tem s due to earthquake loading in your

code through e i ther forc es or mom ents ?

Is the re com pa tibi l i ty of your com pu ter co de with piping co de , or USAS B31 . 1 and B31 . 7

which inc lude f l ex ibi l i ty facto rs and s t re s s ind ices ?

Ch. CHE N, U. S. A.

In B31. 1 code the s t r es se s a re calcula ted by mo m ent s . The piping progr am co m

pl ies wi th a l l the requir em ents spec i f i ed in the code.

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K 4/7

SEISMIC RESPONSE SPECTRA FOR EQUIPMENT DESIGN

IN NUCLEAR POWER PLANTS

J.M. BIGGS,

Depa rtment of Civil Enginee ring,

Massa chusetts Institute of Techno logy, Ca mbridge , Massa chusetts, U.S.A.

ABSTRACT

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Floor response spectra for the seismic analysis of equipment are generated by a very

simple, generalized method based

on the

ground response spectrum

and the

results

of a

response spectrum analysis

of the

supporting structure.

In

this method

the

effects

of the

structure's modes

are

computed separately

and

then combined

by an

empirical procedure.

As

- 330 -

motion. It is limited to the case of uncoupled systems, i.e., cases in which the mass of the

equipment is relatively small and does not affect the overall response of the structure. It

would not, for example, apply to the reactor vessel in a reactor building because that item

has appreciable mass and should be included as part of the dynamic model for analysis of the

structure. However, the vast majority of equipment and piping has relatively small mass and

may be considered uncoupled. Because of the large number of pieces of equipment in a power

plant, it is neither practical nor desirable to include them in the model of the complete

building.

A method similar to that presented here was introduced by the author in 1968. [1] The

procedures and numerical functions recommended here represent an updating and improvement of

the original method based upon additional studies of equipment-structure interaction in re

sponse to recorded earthquake ground motion.

To illustrate the nature of the problem and application of the proposed method, consider

a typical BWR reactor building. The dynamic model to be used for analysis is shown in Fig. 1.

This is a lumped-parameter model with nodes located on the exterior concrete building, the

7/25/2019 CDNJ04820ENC_001


concrete containment or drywell, the sacrificial shield and reactor vessel, and the concrete

pedestal supporting the vessel. The exterior building is connected to the interior struc

tures only through the foundation mat, but the drywell, shield and vessel are interconnected

- 331 -

either the building or the equipment.

BASIC CONCEPTS

The maximum acceleration response of the equipment may be considered to be an amplifica

tion of either (1) the ground response spectrum or (2) the peak acceleration of the structure

at the point where the equipment is attached. These two approaches are complimentary since

the first 1s more accurate for long equipment natural periods and the second is more accurate

for short periods. Therefore, the proposed method uses both approaches, each in the range

where it is the more accurate. The amplifications factors have been determined empirically.

As will be shown below, the factors are essentially a function of only the ratio of equip

ment to structure periods (T /T ) and the amount of damping in each. This fact makes possible

a very simple computational procedure.

The maximum equipment response is determined for each mode of the structure. The total

equipment response is then taken to be the root-mean-square of the responses due to the struc

7/25/2019 CDNJ04820ENC_001


tural modes.

Before the method is described in detail, it will be helpful to identify two limiting

cases of equipment response: (1) Very flexible equipment relative to the structure

- 332 -

vide the two amplification factors for a range of period ratio. Although the model is a two-

degree system, it should be noted that this does not imply that the structure has only one

degree of freedom. Instead, the lower mass and its supporting spring represents any one of

the uncoupled normal modes of the structure.

These four particular earthquakes were chosen because they are typical of strong motion

records, and also because they have different frequency content, i.e., the maximum responses

occur in different frequency ranges.

Since the results are not completely independent of the actual value of the periods,

analyses were made for various values of Τ . The points plotted in Figs. 2 and 3 each repre

sent the maximum amplification factor computed for values of T

r

ranging from 0.05 to 2.50

sees.

However, this does not result in excessive conservatism as may be seen by inspection

of Fig. 5, which shows the variation of peak amplification factor (at Τ /Τ = 1) with Τ .

Each point plotted is the maximum of the responses due to the four earthq uake records. The

value actually used for the peak amplification (10.4) is only slightly unconservative at cer

tain values of Τ . It is somewhat conservative for Τ > 1.0, but such periods do not usually

s s

r

'

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occur in nuclear power plant structures. Plots for other period ratios show similar results

and it is concluded that ignoring the effect of Τ does not produce significant error.

The amplification curves adopted (shown in Figs. 2 and 3) are generally upper bounds for

-

3 3 3

-

r e s p o n s e s p e c t r u m :

1 . D e t e r m i n e A , t h e e q u i p m e n t r e s p o n s e a c c e l e r a t i o n a s i f i t w e r e s u p p o r t e d o n t h e

g r o u n d . T h i s is o b t a i n e d by r e a d i n g t h e g r o u n d r e s p o n s e s p e c t r u m f o r t h e e q u i p m e n t

p e r i o d a n d d a m p i n g .

2 .

F o r e a c h s i g n i f i c a n t m o d e o f t he s t r u c t u r e :

( A ) I f - T

e

/ T

s n

< 0 . 9

A

C o m p u t e A = A ( j — ) , w h e r e t h e r a t i o in p a r e n t h e s i s i s o b t a i n e d f r o m F i g . 2.

s n

( B ) I f T

e

/ T

s n

> 0 . 5

Γ φ A

C o m p u t e

A =

s

"

s n

·

A

( ñ ~ ) > w h e r e t h e r a t i o

in

p a r e n t h e s i s i s g i v e n

by

n e eg

F i g . 3 , a n d

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T

i

< 1 . 2 5 ,

C

n e

= 1

- 334 -

In step 2 (Β) the multiplier Γ. Φ- appears because this is a measure of the effect of

mode η on the equipment. For example, if r = 0, mode η does not participate in the seismic

response. If φ = 0, mode η produces no motion at the equipment support. In either case,

no equipment acceleration is associated with mode n. C is an empirical correction factor

which ensures the correct result when Τ is very large. If Τ 1s much larger than any of the

structural periods, A./A„ = 1 for all modes and A„ must equal A . When the modes are com-

e eg e ^ eg

bined in Step 4, the construction of C ensures this result. The range 1.25 - 2.25 was se

lected to provide spectra consistent with computed responses to actual earthquake records.

The combination of modal effects by root-mean-square in Step 4 is consistent with the

method most commonly used for analysis of structures based on response spectra. Any other

method of modal combination could also have been used, but it should be consistent with that

used for the structure. Thus, when all Τ /T values are small, A will be equal to the pre

dicted maximum acceleration of the supporting structure, as it should be.

The computations required by this procedure are extremely simple and can even be execu

ted by hand. When a computer is used the calculations are almost trivial.

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VERIFICATION OF METHOD

The proposed method is intended to provide a floor response spectrum which is an enve

- 335 -

very similar to that predicted by the proposed method. It should be re-emphas1zed that agree

ment between the two curves being compared in each plot is not expected since the proposed

general method 1s Intended to be an envelope of all possible seismic inputs.

The Parkfield input is Included in Fig. 7 because that record was not one of those used

1n developing the amplification curves (Figs. 2 and 3). This serves to prove that the pro

posed amplification curves are Indeed general and not dependent on the detailed nature of the

seismic motion.

The comparisons in F1gs. 8 and 9 are derived from another BWR reactor building which is

similar but not identical to that shown in Fig. 1. The Taft and El Centro earthquakes have

been normalized to

0.08g.

In this case the first four modes of the structure contribute sig

nificantly to the floor response spectrum. The periods of these modes are 0.28, 0.19, 0.17

and 0.14 sees. The response in this case is therefore quite complicated, but even so, the

general method produces a very reasonable result which is at all points more conservative

than the t1me-h1story results.

As a result of these and many other such comparisons which have been made, it may be

7/25/2019 CDNJ04820ENC_001


concluded that the proposed method produces conservative, yet reasonable, results throughout

the range of equipment period.

- 336 -

ground motion records. This is true whether the motions are actual earthquake records or

artificial motions mathematically derived from a ground response spectrum. In either case

one cannot be sure that the selected records are conservative for the particular multi-degree

structure supporting the equipment. On the other hand the proposed method is intended to be

an envelope of all probable seismic inputs.

It is hoped that the method presented provides a more realistic approach to the critical

problem of seismic design of equipment in nuclear power plants.

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3 3 7 -

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- 338

_ I 1 I I

FIGURE 3: AM PLI FIC AT ION OF STRUCTURE'S o

MOTION

S t ruc t ura l Da mping ' . 0 4

E q u i p m e n t D a m p i n g ».005

o Tof f

• El Centro

+ H e l e n a

χ Golden Gate

t

/'

/ i

o

1

i k

I 1

—

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—

^^ ^

+

+

—

33 9

1 1 Γ

. Jk / a l ue Used

M a x i m u m o f 4 E a r t h q u a k e s

T o f t

E l Centro

H e l e n a

Golden Gate

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FIGU RE 5: PEAK AM PLIFICATION (è *I .O ) VS STRUCTURE PERIOD

Is

J L

340 -

i -

<

or

ω

in

ζ

O

α . 2-

(n

UJ

FIGURE 7 : FLOOR RESPONSE SPECTRUM POINT 4 6 RESPONSE

TO PARKFIELD (. 20 g )

G e n e r a l i z e d M e t h o d

Time History Analysis

S t r u c t u r a l D am pin g » 0 4

E q u i p m e n t D a m p i n g « . 0 0 5

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<

2

3 4 1

FIGURE 9 : FLOOR RESPONSE SPECTRUM POINT 3 RE SPO NS E TO

TAFT ( .0

8 g

)

< 3 -

Z

O

χ

<

2

G e n e r a l i z e d M e t h o d

T i m e - H i s t o r y A n a l y s i s

S t r u c t u r a l D a m p i n g = . 0 4

E q u i p m e n t D a m pi ng = . 0 0 5

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342

: ::

: : ; : ·

M :

■ ■ ■

;

:

:;.|8i

i

■

.... ,

R H ■

:

- - ■

■

: ■

:

:

·..

:

A

y.

.

Ï

■

:::::

. :;.

. :

¡

1

TTÉTI

■ : . - .

M a

:

:

l i a .:

F I G U R E

1 1

l i t

1

•L

■ ' . .

fili

|

ΐ .

¡ f

I

1

::::

.B

5

n

S

Ef

f

C .

F

:□

fl

j :

L a

Ρ Ρ Γ '

PQit

_ .

V

L

;

v . | ■

r"

1 |

4T

Γ

R

|

K

u .

JQ

ί

a

F

TF

31

Ì

11

l i i

I f

' : |

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¡ -

í tsl

1 1

lh|

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A R I

ϋ | 3 Ρ .

¡J

D

S

I F

ψ

Ef

M l

Ei

IR

Ν

Th

J

Q l

ί

Ri

[

■

ÍF

Π"

çt,

D C

| Ë I

1

...

— ■

....

—

~

r

.

7/25/2019 CDNJ04820ENC_001


' t , .

:

rr

■

ti

b

'i'I

:

■

:

.:

: ; · ■ :

1

:

-

;:

:

: ■ :

: '

:· -

l

;-

:

■

1

■ j ■ '

- 343

DISCUSSION

Q

R . J . S C A V U Z Z O , U . S . A .

In t h e c o m p a r i s o n of t h e t i m e h i s t o r y a n a l y s e s w i th t h e s p e c t r u m a n a l y s e s , i s

t h e s a m e m a th e m a t i c a l mo d e l u s e d o f t h e p o w e r p l a n t ? If t h e m o d e l i s t h e s a m e w o u ld n ' t

y o u e x p e c t t o o b t a in t h e a g re e me n t s h o w n ?

. J. M. BIG GS, U. S. A.

A

T h e s a m e d y n a m ic mo d e l i s u s e d in b o th c a s e s . H o w e v e r , t h e r e s u l t i n g f l o o r

r e s p o n s e s p e c t r u m w o u l d b e d i f f e r e n t f o r t w o r e a s o n s :

1. t h e a mp l i f i c a t i o n f a c to r mig h t n o t b e c o r r e c t f o r t h a t p a r t i c u l a r e a r th q u a k e a n d s t r u c tu r a l

p e r i o d , a n d

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2.

t h e u s e of r o o t - m e a n - s q u a r e f o r c o m b i n a t i o n of t h e m o d a l e f fe c ts m i g h t be i n e r r o r . N e v e r

t h e l e s s , t h e r e s u l t s i n d i c a t e t h a t t h e s e e r r o r s a r e n o t s e r i o u s .

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SALES OFFICES

The Of f ice for O ff ic ia i Pub l icat ions sel ls a l l doc um ents pub l ished by the C om mission

the European Communi t ies a t the addresses and a t the pr ice g iven be low. Whe

ordering, specify c lear ly the exact reference and the t i t le of the document.

G R E A T B R I T A I N A N D T H E C O M M O N W E A L T H

H.M. Stationery Office

P.O. Box 569

L o n d o n S E 1

U N I T E D S T A T E S O F A M E R I C A

European Comm unity Information Service

2100 M St reet . N.W.

Sui te 707

I TALY

Libreria dello Stato

Piazza G. Verdi 10

00198 Rom a — Te l . (6 ) 85 09

C CP 1/2640

Agencies :

00187 Rom a — V i a de l T r i t one 61 / A e 61 / B

00187 Rom a — V i a XX Se t t em bre (Pa l azzo M i n i s t e

del le f inanze)

20121 Mi lano — Gal ler ia V i t tor io Emanuele 3

80121 Napo l i — V i a Ch i a i a 5

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Wash i ng t on , D . C . 20 037 50129 F i renze — V i a Cavour 46 / R

16121 Genova — V i a X I I O t t ob re 172

4 0 1 2 5 B o l o g n a — S t r a d a M a g g i o r e 2 3 / A

R E A C T O R T E C H N O L O G Y

Solutions for Special

Problems

I

DESIGN CONDIVONS and OPERATIONAL LIMITATIONS

S

η

g

β

;¡

■

η

S

I

fc

Β

R E A C T O R C O R E : N U C L E A R C O M P O N E N T S

(P ar t ic i« .

M a t r ix ) , P e l le» . C ladd ing , Cap» ; F u e l - ,

M odera t o r - . Re f lec t o r - , and Con t ro l -E lem ent s

R E A C T O R C O R E : S T R U C T U R A L C O M P O N E N T S

fuel Element Assemblies

S pac er , Hangen , S hroud* :

Core S uppor t and Gr id S t ruc t u re»

P R I M A R Y C O O L A N T C I R C U I T S T R U C T U R E S

K ip ing ,

J unc t ions . B e l lows ; ,

Primary Heat Exchangers;

Special Pumps. Circulators, etc.

RE A CTOR V E S S E LS

Calandria Vessels:

Steel Pressure Vessels.

Prpstressed Con crete Pressure Vessels

R A D I A T I O N S H I E L D S

Reactor Therm al Shields; ,

Reactor Biological Shields:

S haded Fue l E lem ent Cas k s

■ 1 R E A C T O R C O N T A I N M E N T

S 1 Mechanical Safeguarding Barnen;

f i 1 Sted Shells.

5

8

S

Prestressed Concrete Shells

■ GRI DS and FRA M E S ,

S LA B S end P LA TE S

3-d ntension?

C O N T I N U A

PRACTICAL EXPERIENCE

M E C H A N I C A L / T H E R M A L

B O U N D A R Y & S O U R C E

C O N D I T I O N S

s t a t ionary , t r ans ien t

c y c l i c , dy nam ic

THERMO

AND FLUID-

DYNAMICS

S T R U C T U R A LM E C H A N I C S

*

%

*&

S· j * ·

* > "

SAFFTV

ANO RELIABILITY

ANALYSIS

t

I

I

( T H E R M O I -

E L A S T I C I T Y

I.THERMO)-

P L A S T I C I T Y

( T H E R M O l -

" V I S C O E L A S T I C I T Y

F R A C T U R E '

N U C L E A R M A T E R I A L S

Metals

S T R U C T U R A L M A T E R I A L S

Metals

Ceramics

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E N G I N E E R I N G

ΟΝΞ028Κ)ΓΝαθ

Documents

CDNJ04820ENC_001