ML20118A748
| ML20118A748 | |
| Person / Time | |
|---|---|
| Site: | Cooper  |
| Issue date: | 04/17/1968 |
| From: | EARTH SCIENCES ASSOCIATES |
| To: | |
| Shared Package | |
| ML20118A746 | List: |
| References | |
| NUDOCS 9209250275 | |
| Download: ML20118A748 (55) | |
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EARTIlOUAKU AN EYSIS OF TI1E REACTon BUILDING COOPER NUCLEAR GENURATING STATION o
for ..
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BURNS AND ROE, Inc.
-. .
- Engincors and contractors
- by -
EARTH SCIENCES
A Telodyne Company
- 171 North Santa Anita Avenue
. Pasadena / California
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17 April 3 968 ,
9209250275 920921 PDR ADOCK 05000298 ,
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' 5' TAULl OF CONTENTS y .
1 J
. Title Page INTRODUCTION.
1 DESCRIPTION CP TIIE DUILDING 1 1
, DEFINITION OF TIIE EARTIIQUARE ,
s- '2 MATIIEMATICAL MODEL; 3 l .
. METilOD OF ANALYSIS .
4 1 .
RESULTS .
6 DRYWELL .- 7 a .
SUPPRESSION CIIAMBER' '.
8'
- LIST OF REFERENCES 9 .
l APPENDIX A Figuros and Results 10 APPENDIX B Responso Spectra at Reactor 11 Building Floors APPENDIX C Mathematical Theory 12 e
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EART1100A13: ANAINS3S OP TH1: lumCTOR BUII.DIMC 1
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COOPMR HUC).P.AH GENERATING STAT.1ON INTRODUCTION _ _ _ . -
Cooper Nuc1cai Generating Station is located In Nemahn 1
County, Nebra'.ka on the West bnuk of the kissouri, River. Tho l '
soil conditions and the scismicity of the arca necessit5tc
. that class I structures and certain key components in the Gon, crating Station be analyzed for'the affcet of carthqua.hc
.f.rccc. ,
i This report describes the analysis of the reactor building
- and drywell for the efiv'en carthquake forces. The analysis
- ' I covers displacements, accelerations, moments and shears in the "reac' t or building, the shears and moments in the drywell, and ,
the response . spectra for the floors of' the reactor building,
,_ DESCRIPTION OF THE BUILDING -
. The reactor building is a rectangular reinforced concrete structure topped by a braced stccl frame. This steel structure houscs the refuelling floor and the ovc$rhead crane. The trans--
verse (East-West) section, and the longitudinal (North-South) section are shown in Figurcs 1 and 2, respectively.*
The building is founde'd on a concrete raat, the top of the mat being at elevation 859'-9". The natura3 grade of the soil
{ is at clcvati,on 903'-0". The refuelling' floor' is at elevation 1001'-0", and the top of the building is at clovation 1054'-0". , ,
At.its ba'sc the building is 141'-6" squarc. This dimension
. reduces in hhc East-West direction.to 105'-9". '
The reactor pressure vesse) is supported ot} a concretc .
pedestal and is surrounded by a concreto and stool biological
? .
- Figures are 3ocated at the end of this text, in Appendix A.
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shic3d, A stec) drywell containn the pressure vensel and itn f' The suppression chataber in a 3nrgo steel torus r.u ppor t system.
located in the baco of the bui). ding. The torus han-a horiv.on-=
l tal major diameter of 103 '-9" and a . minor internal diameter of 28'-9". .
, , del'INITION OF TilR P.hRT110UAKE "
The Preliminary Sufoty Analysis Rcpor't (Reference 1) for enoper Nuclear Station sets forth the critoria for the " Maximum
- Pro.bable Design Earthquake" and the " Hypothetical Maximum j
Desito Barthquahe". Tabic 1 summarines the recommended maximum '
palues of ground acceleration. ,
.. -~_ . . . . .-
^ '
- Table 1. VALUES OF MAXIMUM GROUND ACCELERATION.
Desinn Earthouake Maximum Hypothetical
~
Probable Maximum e
0.10g 0.20g Horiz'ontal Component vertical Component 0.05g 0.10g
! It was recommended that the maximum valros of Table 1 be used for the ground accoloration at both the rock' surf ace and the base of the structure.
The NG9W component of the 21 July 1952 earthquake recorded '
at Taft, California was specified as an appropriate accelcrogram for the Cooper site. Since the NG9N component,had a recorded maximum acccleration of 0.157 gravity the accelerogram amplitude was multilp3ied respectively by 0.100/0.157 and 0.200/0.157 to represent the horizontal components of the Maxiuum Probabic and Hypothetical Maximum design carthquakes.
Accolcration response spectra arc shown in Figures 3 and 4 for cach design carthquake. Figure 3 is the response spec-trum for the Maximum Probab3 c Denign Earthquaho, eniculated *
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- ~' rr gir+w.m,__
with damping net to 5 percent of critical. This dumping valuo l
"Mhen to repreuent energy lonnes in the reinforced concrete building when excited at the Maximum "robable intennity. The s.kooth responne curve of Figure 3 reprocents the Maximum Prob-
- 1 able Hesign Spectrum. It has been drawn as an ap poximato average of the peaks and valleyu contained within the Taft - .
response spectrum. 'SimilarJy, Figurc 4,is the response spec- ,
trtlm for the Hypothetical Maximum Design Earthquake. The damp-ing value of 7 percent represents energy lossos in the reinforced ,
concreto building when excited at the Hypothetical Maximum in-tensity., The Hypothetical Maximum Design Spectrum,is represented '
.by the smooth curvo.of Figurc 4. ,
i Figurc 5 is an additional plot of the responso spectrum for the Maximum Probable Design Earthquake calculated for 5', ,
percent damping. 'he T tripartite log graph pe,rmits simultaneous display of' acceleration, velocity, and displacement responsc.
MATIIRMATICAL MODEL
- 4 The reactor building and drywell are represented mathe-maticully by means of a finito number of lumped masses joined
, together with flexible splines to repr'esent the weight and .
, stiffness of the ,structuro (Figurc 6) . The drywell was c'on-sidered as a series of lumped masses connected rigidly to' the i building at clovations 961',-10-3/4"'and 885'-3-1/8".
The
- . mr4ss'es representin'g the. drywell and the exterior wal) are si'own in-Figurc 6 as non-coincident with the reactor building.
This is donc for pictoral clarity. In the' analysis all masses are considered to be coaxial. Thus, 'the rotations and dis-placements are identical for the building and drywell:at eleva-tions 961'-10-3/4" and 885'-3-1/8", and, the exter;3r wall and ,
building coincide at clovations 903 '-6" and 854 '-9".
1 At each' node, e.r mass point, the mas'ses were assigned i translational and rotational inertia and were permitted trans-lational and rotab5onal displacements. .
The proporties of the structure in termn of the mathe-
. 3- . . ,.
J ., s .s g, co.l w.idel were provided by Burns and Roc, Inc. (l?cforcncco q C,d 'go.) Those datn are tabulated in Figurc 7 for the re-p*h pai.uilding in the East-West direction, in Figurc 0 for the roMfor 1(uilding in the North-South direction, and in Figurc 9
{p r de arywell and exterior. wall below grade. .'-o mass at coob { L ar level includes the weight of the concretc floor and th< Cr[butary we'ight of the walls and equipment be' tween adjoin-inty f: pors. The mass of the drywell and its appendages wero ,
pref ortioned to cach of the nodal mass points representing the d ryed.?. . The stiffness characteristics of the building between t:ht ] gaped masses were determined from calculations of th'e area -
t.Y. - sucnt-of-inertia of the concretc walls between the floors.
Simi.br section propert'y calculations were made for the drywell.
pho time-depende'nt accelerations .cf the design carthquakes are zxssumed to occur at the base of the model (clevation 854 '-9") . -
In nc-r.panding to horizontal acceleration at ,the base, the build-ing ter.fis to rock on ' the foundation. A rotational spring n, = 9. 55:410' kip-ft/ rad is taken to represent' rocking inter-2 : : .:m 'between building and' foundation. ' Further, the dynamic 4 soll' .;cessur'c due to the exterior wall, rocking against the b Ackip2E around the building is modeled by a linear spring 5 ki'p/ft attached at midheight of the exterior Q = 6 4 x 10 vJd ) .. The soil spring values were pr'ovided by Burns 'and Roc, ,
ing . j hused upon recommendations of Reference 3. ,
MCT4C@JlPANALYSM, Cc response of the reactor buil'ciing )o the carthquake hecc(crograms'was calculated on a time-dependent basis using the mode' superposition method. The characteristic free vibra-t'lon chkpes '(modes) were used us principal coordinates per-mitting the independent evaluation of each coordinate responso.
! Good . ' *:uracy was obtained by using the first six (G) modes of vi'. : : . : :,n . The method of analysis is in accordance with the pt iocj ples defined in Refcrences 5, 6, 7, and 8.
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. .Th e mathematical procedure is described comp]ctc]y in Ap-pendix C. In the modal coordinato system, the response of the z.eructure to the r.pplica loadu is actormined at each interva)
~
of time. By transforming back to the original coordinato system the displacements, bending moments and shears at'cach of tho
- nodal' points can be calcu]ated. The 3ateral and rotational ,
accolcrations are a' Iso calculated.
A macro flow diagram for the computer .piogram' used for
.this study is shown in Figurc 10. -
The use of an historical accol,crogram for the design carth- ,
. quake poses some mathematic'al problems. '
4
---It. cnn be -shown h.t at -cart h qua k es exc i te oscillators o'f different periods to a ' greater or lesser degrec depending on I the component motions' of the carthquake. This phenomenon'id, demonstrated most clearly on the response spectrum (Figure 3).
4 i This spectrum is for the Taft carthquake recorded on 21 July
- 1952 in the N697 direction, normaiized to .1 gravity (damping
= 5% critical. For design purposes the peaks a'd n valleg's are smoothed out. This is accomplished by constructing a curve to bound the peaks and a similar curve-to, bound the valleys. The design spectrum is then considered as a smooth curve drawn mid-way between the maximum' and minimum curves. .
For each of ths first six periods, participation factors ,
- are computed. The participation factor is the ratio of the ,
response to the, actual spectrum of an oscillator of given period to that of the smoothed spectrum. These participation factors are shown in Table 2. -
4
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8 g
- 4 4 8 e 4 4
m 8 9
9
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3
.F Modal Participation FacCors i
., Mode E-W N-S-
. 1 .757
- 757 2 .764 -
.893 1.
' 3 *1.
4 .96
.96
~~
5 1. 1.
-~~
6 1.04 1.04 l
- f l -
Table 2 4
I "RESULTS .
3 2
- - Analyscs were carried out for both the North-South and -
East-West directions using the design cr rthquake with a maximum
- accolc* ration of .1 gravity. The response in the East-West'di-rection gave higher values for moments and shears in the reactor
- building and so the analysis for the maximum, hypothetical carth- '
quake was carried out for t'he East-Wcst direction only. ;
All analyses were performed for the,cmpty drywell condition.
The first six modes of vibration ~for the East-West dirce-tion are shown in. Figure 11._ The -associated periods .are given in 5Able 3. ~
- q i .
' ' Period, Sec. .
E-W Modo N-S .
.433 1 .438 4
,- 2 .139 .161 - .
3 .081 .078 -
- ~j
< . . g 4
4 .073 ' .073' ,
. 5 . .054 .055 i
4 '
-6 .052 .052- ;
Tabic 3
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yhc fundamental period is .433 secondu, representing a rel-
- c. Live 3y stiff structure. The fundamental mode is dominated by roci:ing 6n the soil foundation. ,
For cach of the three-analyscs the following data woro obtained: ,
- 1. Maximum' rolativo displaccment *
- 2. Maximum total acceleration
- . 3. Maximum rotational accele, ratio 6
- 4. Maximum moments .
- 5. Maximum s, hears, , ,
This data was plotted graphically'for the f]oors of'the I
reactor bu'ilding and is shown in Figures 12,-13, 14, 15,,and 16.
. .From. the accelerations computed for cach 'nodul point the l l *
. ,. lateral ac.cclcrations at each floor level at each interval of time were punched on cards. These punched cards were then veed to, compute the respo6se' spectra for single degree of freedom.
oscillators damped at .5 percent and 1. percent of critical .
for cach *of the floors for both the maximum pro! able and the l
hypothetical maximum earthquakes. The plots of the spoetra are given in Appendix B. -
DRWELL
! A coupled analysis /
was performed for the reactor building and the drywell with a damping-coefficicnt.of 2. = 5% critical.
2 This served the purpose of. determining the feedback effects of the*drysiell forces into the reactor bpilding. _However, in tho' l
case of the drywell a' damping coefficient of,),-= 1% critical.
ii more realistic.
In order.to got a good approximation to the forces acting on the drywell, 'for which it is to be designed, a subset of calculations was performed. This was donc by estimating what
] part of the, loads on the drywell- can be attributed to the forced ,
- l displacement from the reactor building and what.part can bc attributed to the inertia) forbes of the drywell itself. At .
the period of vibration of the drywell, of 073 seconds,
- , . hm ,. ., ,, . c,- , . . - . , ..n-...
4
- the ratio of the respons.c of a single degree of freedom oscil-
,Litc$r to the design enrthquake at W, critical dumping to the
- rospcense a't 5% critien1 damping is 1.30
- 1.00. ,
(Reference 2.)
Therefore, the loads due to inertial' forces were incronced by
- this ratio and new loads on the drywell were obtained. The design shears and moments are plotted in Figurc 17.
i SUPPRESSION Cl!NMER ,
The suppression chamber is a stebl torus with a major diameter of 101'09". The rocking of the building on the soil below, when the building is subjected to carthqunho forces, ,
produces eiertical accelerations.over and above the vertical seismic accclerations. The maxi. mum rotational acceleration ht the base of the building in the East-West direction for .the maximum probabic earthquak'e is .052 radians /sec2 and for the -- -
hypothetical maximum is 083 radians /sec*. These represent l
accelcrations at a radius of 50.87' of 8.2 and 13.0 percent of
!. g respectively.
't - '
The suppression chamber must be designed for vertical
. accelerations of I
a) 8.2 + 5.0 = 13.2 percent of g for the Maximum
\ Probable Dosign Earthquake ,
b) 13. 0 + 10. 0= 23. 0 percent of g for the IIypothetical Maximtm Design Earthquake e
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k . g g
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,,w v,,- ....,_,,.4,. , , _, , , ,, ,,
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- 1. P.S.A'.R., Volume III, Appendb: A,,Section III, p. 11.
- 2. P.S. A.R. , Volume III, Appendix A, Figurc X-1.
- 3. Woodward-Clyde-Shcrard, "Progrc5s Report XI", Revised January 17, 1968. ,
- 4. N. M. Newhark, " Design Criterin for Nuclear Reactors Sub '
jected to Earthquake Hazards", Urbana, Illinois, 25 May'
! 19G7.
l 5. R. W. Clough, :'Use of Modern Computers in Structural Analys'is", Journal of the Structural Division 'of the
~
- ASCE, 1958. .
- 6. R. W. Clough, "Earthq'uake Analysis by Response Spectrum Superposition", Bulletin of The Scismological Society of ~
America, Vol. 52, No. 3, July 1962.
- 7. M. Ru'ocnstein, "M*'rix domputer Analysis of Structures",
Prentice IIall, 1966.
B. G. I,. Rogers, "An Introduction to the Dynamics of Framed -
Structurcs", John Wiley, 1959. .
- 9. Burns and Roc, " Computations of Masses and noments of lnertia for Reactor Building", January 31, 1968. ,
- 10. Burns and Roc, " Computations of Arcas a..d Second Moments of Arca for Reactor Building", January 31,'1960'.
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APPENDIX C ,
Mathemat.ical Theory .
y . . * ,
e g e e
k e
9 6
4 g e
e 4
e
- 4., \
4 4
4 4 .
.i. .
6 g 4 e
e e O'
% e ,
4 g e
y -- '
y
i .
DYnhtU C hith1NGili i
) .
l
- ', k'hcpurposeofthischnpteristodoncribethemethod f l
used to enrry out the detailed clautic dynntnic nnnlyuis of the I 1
reactor building.
In doncribing the scismic annlyuls procedurc, matrix notation and simpic matrix nigebra operntions arc cmployed. This approach has been adopted beenuso matrices pro-vido the moct effective means of expressing the operations to d be performed and also beenuco they lend to officient digital i computer programs. ,
J J PORMUIATION OF T!!E ANALYSIS :
Genornl. The application of clectronic digital computcrs to evalunting the dynamic responso of complex structures has mado possible eniculations which were unmanageable by manual means. "
/
.The thco'ry of structurni dynamics has long been understood. '
- llowever,'significant evaluations of. practical structures're-sponding to arbitrary dynamic loads owe their existence to tlye
.rclatively recent advent of digital computers.
in most practical casos, the structure is rcpinred by a lumped paramotor :idealization. Typic' ally, the mass which is distributed throughout the structure, is assumed to be concen-This allows the pattern trated at a finite number of points.
of. structural displacements to be described by a limited' number of dispinecment coordinates.' Then, the deformation-resistance churnctorintics of th'c structure are determined for relative The advantage of,this' l'
displacement of the mass coordinntes. ,
lumped paramotor idenlization is that the equilibrium of the '
system may'be expressed by a finite number of ordinary differ- ..
ential equation,s; wherens, a partini differen'tini equation is required to doccribe the actual distributed mass system.
The system of ordinary differential equations are solved
.by th,c modo-superposition motlyod. The churnetcristic free vibration shapes are used as princip]c coordinatos to which i ,
9
- . - . - - - , , , - - . ...--n .-m- - - . , . - ,, -- - .
' o .
i
) *
- i the cy' stem j a transformed, thus uncoupling the rouponnn of the l zyutem int.o coordinates whi.ch may be evaluated independently .
of cach other. The principio ndvantage of the mode-superposition method is that the renponse of an cinqtic system is generally exproar.cd adequately within the first ,fow modou of vibration.
Thus, good accurney mny be obtained by including only a few ', ,
principic coordinates in an annlysis.
Equntion of Equilibrium. The structuro is represented by n All of the building ,
j lumpod mass system as shown in rigurc C-1. ,
l and equipment mass distributed betucen mid-hcigh*t of successive ~
Uncoupled linear l
stories in as conecntrated at cach floor level.
l springs are used to represent substructure' rocking and sliding '
respectively within the foundation' material. Thc. essential ,
deformations of 'this system nre define'd by the two displacemen'ts'
- of each lump'a mass, u n andwy. ,
Assuming a viscous (velocity proportional) form of damping, .
the equilibrium of a discretc ma'ss system at time t is expressed i . .
by the following matrix equation: * . .
7 -. -
-r 4 "u " " = 0 (1) y ", + c + K .
na w -
w w .
- st - - - .s t u t .
t
, where u a trbnslational displacement of the system relative to .
! is base .
u a total dispinecmont of the system .
'~
w -= rotational disp 1ncement of the system ,
- = differentintion with respect to time H" = translational mass ' '
M = mass moment-of-inertin c = damping matrix .
. K = stiffness matrix Each lumped mass in free t'o trans1nto and rotate, thus, the I matrix Eg. l' is 2 (N+1) in hizc. ' Thoro arc (N+1-degr'cas- '
of-freedom for both translation and. rotation. -
' ),,g . -.
b i : .
e
?
.. c)N., . .
- ht .
i,
()l.I-l -
- ,,bl~ 11 - 2 -
.u* 3.. V> o' u
f<.C ^y cWn
()n I .
4
/
4
.. )3 .
's *
.. , 43 . ()2 " -
' - . 42 - - u 4.9 s
()1 4,
~~~~ 0 .
s . - -
q 'g
., We,e .
+ = y, i i a
. _ _ _ __ ._ b. ... .
., oo -
...,., y, 5 '
,v g
l '
u .
g , -
,f . y, x :> . .
4 .
i . .
J. * .
? '
0 C
e Figure C-1. !!ORIZONTAL MODEL ,
e G
G 4
4 .
4
. L' e
e w- - - - - - - , , - . . - g -
_w~g., . - - ,, --e-g-m--,, -y-.-., -- .,.--,r--wv.-c, , -,-,-.---y-,
. l
. . \
Decuuno the ground anotion of fcets nre caused by n vibra- ")
tofy inotion of the supporto, it is necesunry to utudy the I
' condition of equilibritun for such n enne in ordur to c): press -
ground motion as an offective load. The total translationni ,
dinpinecmont u of a coordinato is the sum of the ground inotion u9 plus the relativo displacement u (F3 9uro C-3) . Rotation w of cach mass is not directly excited by transintion of the supports.
u +uau - .
(2) ilifferentiating tuico with respect to time, ,.
u 9
+U=Y .-
9 (3) l .
cxpresses total translational acceleration'ii as the sum of .,
ground acceleration U g
,and relativo neccleration 'ti.
p.
If the vibratory rnotion of the support is expressed. ns a .
I ground accciaration, its effectivo load enn be derf ved by com-binir.g Eqs. 3 and 1. . ,
- - - ,~ - ., .. _ .
U "
I + C ~
+ X ,
=- u9 (t) (4)
- 11" U O w 0 t ,s t
. A 's , , , , , ,. ,
Eq. 4 inay be exr.:e ssed symbolically as 4
Md + C0'+ KU = P cff (5) >
l The grourid acccieration acting on the structure produces nn '
effcet equiva1cnt to a dynamic load applic'd to cach translationni l rnass . The negative sign indicates that the force acts in a direction opposing tho~neceloration U. .
STIFFl1F.SS 14ATI TX The proceding discunsion shows that the esucntial charactor-istics of the r.tructure arc exprcased by means of three tantriccs,
[11], (C), and [K), reprencnting its mass distribution, viscous ,
. v15-
Of thouc, the 0ninping, and ntiffncos propertien, roupectively.
ansu
, man o inntrix in n simpic ' listing, in diagonni form, of th On the other hand, the ,
concentrated at cach mnus 1x) int.
- stiffncss matrix may be complex and r.cquire a considerabic nmount of computation. The most convpnient appronch to the stiffncss analysis is by matrix methods. . .
Story Stiffncus. The ~
stiffncss of an in'dividual story clement connecting mnus points reprocents the forces developed in that ,
clomont by unit displaccmonts at its ends. For n typical cicmont, as shown in Figurc C-2, the end forces in terms of cnd -
/
dispinecments are given by .
' M- ha hb t n
, (g)
. - .a ,
1 Mb k h3 tb *
,I ,
b - - .,
j.
r .
or symbolically ,
(7)
{S)' a [k].[r] l then shear deformations are included, . .
2E,JI, ( 1-8 }
2EI (2+8 ?,
k.a =
L. N1.+2 p / hb*L N1+2p/ .
_ ~
8 in which p = 6EI/L AG' A = the effective shear aren and none St:if fness.
- The bas.c stiffncss may be used to model rocking j '
and sliding at the foundation. - The rigid mass of Figure C-3
.i Linear rocking and trans-j
- reprosoni;s the total substructurc mass. . .
Intional resistant springs represent the restarin,g forces withio tbc foundation materin1.
Utilizing the notation of Figurc C-3, the base stiffncss is expressed directly as - .
6 1
,-1s- . .
8
.e n
.s g
- g. _ ._ __...._ _
. e f
- chord f
0, /r it
'.- - - . I centerline . .
4- , , .
, j. . . .
o J. f I .
j .
. _ . 3 ,
f .-
1 l'iguro C-2. TYPICAL STORY ELE!1E11T
. t
'e 4 .
- - - - - . _ _ . _ . - - _ . _ _ _ _ - . _ _ _ _ _ _ - _ QRQQQ QQQQf f y
- . .. . ~ - . -__._ _ - . - - _ _ - . - - - . - - - . .. . .
e l .I 1 . l t I 4 a a ,
- l N
xO oo /.
/ WA._. .
+ h > f ,
.m _ i-
-[-
7 I / L.s oof
] yq '
, ' L_ .
g c.)] i
/ ,/- -
,/ , .
,/ _
/ .
- r d I
<0 . E- i . .
~
9
. t, -
Figure C-3. DASE STIFPNESS ...
/ .
a k - g g
- f*
T.
. .e g k .
- 1
.. i
~ "I 0' 'k o - o ' i ~'
o n (o}* O o -h ) -
- 0 0
{ or nymbolienlly , j .. [s] = [k] [r] . Tho-total. Stiffncos of the
, Total Stiffncss of the System.
1
- structuro, X, in developed by npplying the standard dispincc~ >
ment rathod' of matrix unnlysis. f ' T (9) !
.K n A hh -
~
J The cicment'stiffncss matrix, k, is composed of the 2 x 2 i i matrices in Eq. G and 0 arranged down the diagonni ,- ._. f' ._ Na b , , kb ka N .
- ~ ~, ,
i . ha b .-
~
bh a N l i .k e .N * , - . 110) I . ha h b k b k'a 1 i - k, 0 l O k 0 0 l
~
Element _ displacements, r, are expressed in terms of. mass point i dispinecments, R, by the displaccmont transformation A. . Symbolica11y, ,
. - {r) = [A] (~R):.
(11) Tabic-1 lists-the completo' displacement trannformation,-A, for: l the structure of Pigurc C-1. Note that the rocking-and trans-l Intional springs k, and k 6 develop resisting forces directly-l proportion ~nl -to u and w - r e spec t.ively . -Thun, these two springs i .. . _. _o . o h a... ... -ve clement dinplacements which arc identienl: to the mann point dinp1hecments. I l
. :.l.h '
i: I I ._ ,v .; s ,- - 1
- ! ,i! ,3 ,
. (-
' >23 ,I I- 1 I-), -1/4 3 1/4 g .
Ii
?
I' 1 uN ~.' OND
.:.1/4 5 1/45 *
- 1 '
~
l, IC N -1
-1/43 _t 1/4N-1 .
I- !"N-2 14 l, .,. 1 ' ; ? i .cN-1 _1/29N-1 . 1/4 N- 1 - I b . 1
, .. I -4 i u
- 1
- r .
*g 2
;- (
.>.=
. 1 u,
- . A I .
. [
I- 1 u _yf4 174 2 o 2 I
! t 2
!- a l
<-- -- - -?'
y
; y f.2
. _;f4 2
1742 , x o- . . j . 1/411 1 w c l -1/4 1 - x..:.
.a- .
1/4 1 1 1 w ' -1/4 1 r-2
-_ Of _-,_
i--------- - 1 l . 0 :- a I
- i. l w
- 0 I 1 2 l- N 9 / . - . - . .
w,~ i- . o
% / l (r) = [a]'(R}
Displacementi Trr.neformation.
- ' Table 1. ,
4 -
. . , -._ _ . . , ~ . . , . . . .._ ~- ._,
. t
4 a DYtmMIC lil*S1'ONSM hiode Shupon und PrectuencI6r,1 , In ny. 5, the term on the right hand ulde reprencuts the . > offective carthqunho forces tending to excito motions of the structure. The rouponse to thiu cxcitation is' rcprononted by the displacement vector 11 !!owever, in order to eniculato f j this rouponse, it is convenient first to express the dynamic ) properties of the system in terms of 'its frco vibration mode 4 shapes ar.d . frequencies. This involves the solution of the . following equation for its chnrnctoristic valuen: .
^1
{ - [x)[ t ()n ~
*$ I"3I+i} n : :
) - where
- th . .
i l e a frequ n y fn m dc . n '
# { { g} n ~ ~ " 0" 'h"E f "th mode The solution is a standard mathematieni operation which ,
may be solved-on digital computers by standard library programs; therefore, it need not be discussed at any further length here. { ' It should be noted that the modo shapes satisfy the ortho - sonnlity conditions: - (ty)[ [MJ (6g) ,
= 0' mfn l
(t i) oj [x] [t i ), , , In addition, tho modo shapes are normalized so that [M) (4g ), (6g) = 1 m=n' Therefore, the following reintionships apply: , [f in [M3-[f in 1* EI ' (3 3) IK3 If i (14) a
' Ef in) 3 (of) ,
. - la-
.-,,r..---y., -
..,,_-..-v..--. ,.,...,y ,.-,,_mn,-...,,,,%,y ,,t ., - ,, , , , . , -. - y - - , . . - . , 4w.,
+
I:ormal coordinaten: ,
- ,, The next step is to c>
- preuu thu horj v.outui or vertical j motionc of the dynamletilly nxcited structure in terms of the ,!
I amplitudes of tha various modcu of vibration, rather than in l terms of the displacementn of. the individuni macnca. Thin 3 transformation of coordinaten is c):prc'soed nu follows for - i horiv.ontal dicpinecments: ' { fui) a Lt in.) [>:n) : . (r) . X th -
'v!horo n
= amplitude of the n ciode of vibrut4on.
F.u.r. t.h. ermore , l I
~
".)
- bDin)
.. n} - (10)
I i) " bDin) n) ' .
'* (17)
TJ,2,c, qubstitution of Eqs. 15, 16, and 17 into Eq._5'yicids- . . Ef))[t in){ n)t + [C)[t in 3{ n) t + bE E Din] X[ n)t l ,
" [Pcff)t , (10) '
1 T)}i3 gguntion is nod promultiplied by [tin T *
.[D IIN bIin) n)t
- Ifin T[C][t in 3{. n)t
,d *
'f.in }. E b 'in nt in Ecff)t *
' ( 19)'
R.o.r '.nost structuren the exact form of the . dumping matrix
.in unknown. hiso, in most caccc its affect on the vibration modo, shapes of the structure is small; therefore, an assumption as to, the form of thic matrix is justi f f *.ibic. 'Specifically it 1
39 ynnumed that the damping matrix 3 n oc)cct.cd no thnt. ort.h- . 99 ann)ity of the damping forces in mnJntnined: 4 _~
..m., ,..,,ne., ,. , . , y - ,..
r,,
* ~
(6),i[c][h1,,a 1 o m/n
. nnd so that .
T ( ti )n (C] ( 6g ), a ng m=n
- in which a n a 2),wn ' ""U n
" E'"9 " '" "
modo. -
'th er ciore , the following equation l's satisfied: .
[ fin) ] C ] [tg ,) = [an ) 0) . wht::re [an ) is a diagonal snutrix. This implies that the damping in :"uncouplod" i wi'th respect to the modo shapes. The form of
- Eq.19 may be simplified somcwhat by d$ fining the q'lantity-
- 1 .. -
[P3nt" IE in 3 [Pcff)t .
. (21) and b'y substituting it along with Eqc. 13, 14, and 20 into '
Equation 19. .
' @3 [xn ]e + [a n3I*.n 3t + f5*n 3IXn }t = (Pn )t .
(22).
- Equation 22 represents a set of equatio.nc of the form
+ aXt+*X'"P
'N ~ '
t dr rewriting in ter.ms of the damping ratio t + $* t** t " Ek . -( )
. AnGlvsin of the Modal Response:
- nnsed on the assumption that the accoloration N V"#I"" '
t
. linear 3y within cach time interval '6t, the solut' ion of Eq. 23 procedes as follows: '
y 1. Initial Ca.iculations - q .. . X D-ta=o*P$=o~ '* t=o ~5 X t=o. * (248) f P = (1 + 6 t) w -l Y m*) '
,, (24b) %.owew
e
/
- 2. 'The respunne X e ja now determined at dineroto U me pojntn from the fol3owing not. of recurnlon equationn M
L u k (Ptt - 2\ma - w"b) 1 (2Gu) n+N2 3! t p,t u (25b) (25c).
*l X { n b + I tefXt . ,
whtsv - (2Ga) , a**X g 1 l'fNg 1 6 v
" t 1, t i t9't.-t3t + Ati 3 y t-6 t e,
(2Gb) Ift.t in s.k.11, bot h ui t.h renpcet to the period of vibra-t (<:a of t he- 9 :vi n i. vide, and also wit.h renpoc$t to the "porloda'. of tjet- elt uund O!.t'I l l.ili ots , thjD procedure wilJ give good .
..... oi . icy i n L t.i f ina l ret.u l tr. . -
.1...i ..n ..r ::t : ur t ura1 til np,1;teemeni n and Forcen t j Yh. bi r i v.out al , d100) ace:.a nt.c, velocitics ar$d. accelerationc
.i t . .. : . ri.i . . are datormined by the f ollowing coordinate trancfor-n.i t i on. . .pa t.ionu: ,
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, (b1}e n)t - (27D) i j (Uit l I Jn) n)t
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[ the ground. ,, When the relptive mann displacements havo'been Octormined, t.be elantic forces asnociated with 1.hoco deformations may bc - obtained directly by promultiplying by the stif fncus matrix '
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