use of load depend ent ritz vectors in modal …
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The 14th
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USE O
1 Ph.D 2 Disting
ABSTRACT
The performawidespread uresponse. Mobut it fails in is to present tdynamic forccalculated ressystems whervectors, in cinclusion of choosing ade
KEYWORDS
1. INTROD
Evaluation ofanalyzed usinhowever, it isa simple methstructures (mdistribution owith consider
In order to oattractive feaprocedures win the modal inertia force expansion aresystems. A sbuilding is pr
The accuracythe conditionapproximate evaluation ofbuildings, thebuildings to Chintanapakd
orld Confere17, 2008, Beij
OF LOAD
D. Student of guished Profes
T:
ance of a struse in perforodal Pushoversome cases othe use of Lo
ce; instead ofsponse of irrere higher mo
case of stiffehigher modequate number
S: Modal Pus
DUCTION
f structural reng either lineas very time cohod has been
mainly regulaof load, goodrable higher-m
overcome thistures, for inst
with invariant expansion odistribution
e then combistep-by-step sresented in Re
y of MPA muns under whicprocedure, M
f “regular” bue next step ia degree o
dee [2] have
ence on Earthijing, China
D DEPEND
Hossein K
f EQ. Eng., Islssor, Earthqu
ComplEmail: hk
ructural systermance basedr Analysis (M
of the irregulaoad Dependenf commonly uegular systemode effects car lower storie effects witr of required
shover Analys
esponse is a kar or nonlineaonsuming andn preferred in ar structures)d estimates womode contribu
s limitation, Mtance, it retaiforce distribuf the effectivfor each mo
ined by the Csummary of eferences [2]
ust be evaluatech it is appli
MPA has been uildings. Becas to determinf accuracy wshown that:
hquake Engi
DENT RITAN
Kayhani1 and
lamic Azad Uake Engineerex; Islamic A
kayhani@gma
m can be estd seismic desMPA) has beear structural snt Ritz vectorused eigen-m
ms when limitannot be ignoies, increase hout really cRitz vectors f
sis, Ritz vecto
key concept inar methods. N
d may not be practice. Alth, due to limould not be pution to respo
MPA proceduins the conceputions. In the ve earthquakeode. These “mCQC rule to o
the MPA proand [3].
ed for a wideicable for sei shown to beause vertical ine whether owhich is con“The MPA p
ineering
1
TZ VECTNALYSIS
d Mohsen G
University, Sciring and Struczad Universit
ail.com, mohs
timated usingsign due to ien suggested ystems (i.e. srs (LDR) whi
mode shape inted number ofored. The num
the accuracycomputing thfor vertically
ors, LDR, No
n performancNonlinear dynsuitable for ehough it may
mitations in ipromised for onse.
ure has been ptual simpliciMPA procedu
e forces is detmodal” demaobtain an estiocedure to e
e range of strusmic evaluati accurate enoirregularities
or not the MPnsidered suffprocedure is
TORS IN M
Ghafory-Ash
ience & Reseactural Departty, Tehran, Irasen@ashtiany
g a non-lineaits simplicityto increase th
stiffer lower sich takes into n the MPA in f modes is tomerical resuly of force rehem). There regular or irr
nlinear Static
ce based seismnamic analysiveryday engi provide goodits fundamenall types of s
proposed byity and compure, the seismtermined by nands due to imate of the stimate the s
uctural systemion of structu
ough in estimsignificantly PA can estimficient for prless accurate
MODAL P
htiany2
arch Complextment of Scienan y.com
ar static analyy in estimatinhe accuracy otories). The oaccount the order to imp
o be considerelts have indicesponse signare also som
regular structu
c Analysis, St
mic engineeriis is considerneering practd approximat
ntal concepts structures esp
y Chopra [3] putational easmic demand du
nonlinear stathe first fewtotal seismic seismic dema
ms and grounures. By studating seismicinfluence the
mate seismic ractical applie relative to t
PUSHOVE
x,Tehran, Irannce and Resea
ysis which hang inelastic sof Pushover objective of thspatial distrib
prove the acced, especiallycated that usinificantly (becme suggestionures considere
tiffness Irregu
ing. Structurered as ‘exact’ tice; the NSP tion for some
like the firpecially for st
which offerse of current pue to individu
atic analysis uw terms of th
demand for ands for a m
nd motions todying the biac demands fore seismic demdemands of ication? Chothe reference
ER
narch
as found structural analysis; his paper bution of curacy of y for stiff ing LDR cause of ns about ed.
ularity
es can be method, which is types of
rst mode tructures
s several pushover ual terms using the he modal
inelastic multistory
o identify as of this r seismic
mands on irregular
opra and e regular
TO f
m T
ft 2
Itlodlmia
Imf
ufani 3 Ti
A
The 14th
WoOctober 12-1
frame in estisoft-and-weakmodes it is su
The objectivestiffer lower frames determthe use of LD
2. PROPOSEIGENV
In 1982 Wilstook into accless computator LDR (Loadistribution oload vector,mode-accelerimpossible toanalogous to
− Static
− Pseud
− LDR
In each of themanner by firfirst vector costatic solutionusually consiforce or dispaccurate dispnoted that thein the continu
3. NUMER
To evaluate tin this researstories. Both AISC-89 (AS
orld Confere17, 2008, Beij
imating the sk lower half;
usceptible to l
e of the propostories for the
mined by MPADR vectors in
SED METHOVECTOR
son et al. [8] count the effetional time th
ad Dependentof excitation){ }( )f s can
ration procedo generate vecpseudo static
c correction
do static vect
R vectors
e first two sumrst term or inorresponds ton. This is theidered, and implacement resplacement rese first pseudouation of the p
RICAL ANAL
the effectivenrch: 1) Regul
type of systSD method)
ence on Earthijing, China
eismic dema; stiff, stronglack of accura
osed method ie estimation o
PA procedure wMPA procedu
OD: MPA U
introduced aects of spatialhan use of exat Ritz) vectors, the first Ritbe compute
dure). Since ctors which a
c vector used i
{ ( )}U t =∑or
{ ( )}r
iU t
=
= ∑
{ } [ ]sU K=
{ ( )}r
i
U t=
= ∑
mmation metn a static mano static solutioe major benemplementing sponse calcusponse do noto static vectorpaper, only th
LYSIS
ness of using ar moment reems were defor PGA=0.3
hquake Engi
nds of frame, or stiff-andacy for the ca
is to improve of the seismicwith the “exaure can impro
USING LDR O
a new dynaml distribution
act eigenvectos. Consider thtz vector whid as: 1{ }X =
this vector ware not excitein the “mode-
1
{ } ( )r
i ii
y tϕ=
+∑
1{ } ( )
r
i iy tϕ=
+∑1] { ( )}f s−
*
2
{ } ( )r
i iX y t=∑
thods the full nner by the seon and additioefit of using
LDR vectorlation (displat guarantee ar, { }sU , is idhe results for L
LDR or { sUesisting framesigned accor3g, soil type
ineering
2
es with strong-strong lower
ases in which
the accuracyc demands. Cact” nonlinearove the accura
OR PSUEDO
mic analysis mof the dynam
ors known as he input forcech is the stat
1[ ] { }K f− (whwill be useded by the assu-acceleration”
1[ ] ({ (K f s−+
1([ ] [ rK K−+ −
1
1( [ ]T
XX M X
+
loading vecto
econd term, wonal vectors rLDR vectorss ensures theacement respaccurate forcedentical to theLDR has been
}s vectors; twe systems andding to Irani
e III and Res
g or stiff-andr half”. As thhigher-mode
y of MPA for tComparison or response hisacy of MPA f
O STATIC V
method basedmic loading aWYD Ritz vee as { ( , )}F s ttic response ohich is knowd for extractiumed loading” based respo
)} { ( )})rs f s−
1] ){ ( )}r f s g−
*11/2
1
( ))
y tX
or has been awhich is not threpresent dynas in MPA appe consideratioponse converge responses de starting vecn presented.
wo types of sd 2) Irregularan National B
sponse reduct
d-strong first his method cocontribution
the vertically of the seismicstory analysisfor the case of
VECTOR IN
d on Rayleighand yielded mectors (Wilso
{ ( )}. ( )f s g t=
of the time inwn as pseuon of other
g. The obtaineonse spectrum
) ( )g t
( )g t
accounted for he case for LDamic contribuproach whereon of higher-mges faster tha
due to higher tor of LDR v
structural systr frame systeBuilding codtion factor R
story; soft, wonsiders the cannot be neg
irregular fram demands in
s response, shf irregular stru
STEAD OF
h-Ritz methodmuch better ron, Yuan and D
({ }( )f s is thndependent podo static vevectors it w
ed response fm approach [5]
either in the DR vectors; sution neglectee first few mmode participan force respmodes). It sh
vector approa
tems were coems with stiffde which is siR=6. The reg
weak, or first few glected.
mes with irregular
hows that uctures.
d, which results in Dickens) he spatial ortion of ector in
would be format is ].
dynamic since the ed by the
modes are pation in ponse so hould be ch. Thus
onsidered fer lower imilar to ular and
TO i
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Caw Ftcrp
The 14th
WoOctober 12-1
irregular fram
For the input0.7g (to ensucontribution o
Considering tand pseudo swith nonlinea
Figure 3 showthe evaluatiocalculating drequired in Mproperties are
T1=0.98 sec T2=0.361 T3=0.211
orld Confere17, 2008, Beij
mes and their
Figure 1 Regu
ground motiure the nonlinof higher mod
Fig
the effect of htatic vector foar time history
ws sample mn of two app
damping in eaMPA approache in the veloci
T1=1.49 sec T2=0.554 T3=0.323
ence on Earthijing, China
respective pe
ular and Irregula
ion, a set of nnear responsedes has been u
gure 2 Spectral
higher modesfor such irreguy analysis wh
odal propertieproaches (LDach mode forh. It is importity sensitive r
T1=1.91sec T2=0.71 T3=0.42
hquake Engi
eriods for the
ar frame models
near-field groue) with impuused.
accelerations for
s on responseular systems hich is assume
es for the 8-sDR vectors an
r the analysistant to considregion or not.
ineering
3
first 3 modes
that have been u
und motionslsive characte
r selected groun
e for the case (also for regued to be exact
story irregularnd eigenvectos of SDOF (S
der rational da Different sys
T1=0.T2=0.2T3=0.
T1=2.35 secT2=0.888 T3=0.531
are shown in
used in the appli
recorded on ser which can
d motions and m
of stiffer lowular systems ft response) ha
r frame whichors). RayleighSingle Degreamping ratiosstems have be
62 sec 205 099
T1=0T2=0T3=0
c
n Figure 1.
ication of MPA w
soft soil (typen cause consi
mean spectrum
wer stories, thfor measuringave been exam
h have been oh damping ratee of freedoms since it is noeen coded acc
0.87 sec 0.311 0.183
T1=1T2=0T3=0
with LDR
e 3) scaled toderable respo
e use of LDRg the accuracymined.
obtained and tio has been
m) systems wot known whecording to the
1.23 sec 0.442 0.261
T1=T2=0T3=0
o PGA of onse and
R vectors y, in line
used for used for
which are ether the e number
1.6 sec 0.558 0.334
TO oeton
Fcb
f 3 Ie
The 14th
WoOctober 12-1
of stories, reexample 4sIrthat since speof each compnumber of LD
Figure
Figure 4(a) scan be seen thbetween 3rd estiffer part offirst few mod
3.1. Story Sh
In the case eigen-modes
orld Confere17, 2008, Beij
egularity, typer-e(2) represeectral content puted vector DR vectors ca
Figure 3 S
4 Comparison o
hows the diffhere is no sig
eigenvector anf the structuredes.
hear and Mom
of shear respshould be co
ence on Earthijing, China
e of vectors nt the resultsof the first vedepends on t
an significant
Sample Modal pr
of Eigenvectors a
fference betwegnificant partind 3rd LDR ve and as a res
ments
ponse of irreonsidered for
hquake Engi
(LDR or eigs for a 4 storyector in LDR the requestedly reduce the
roperties for 8 st
(a) and LDR vector
een First LDRcipation of loector (in a se
sult their cont
egular structur 16-story irr
ineering
4
genvector) any Irregular-frvector appronumber of Lcomputation
tory irregular fra
rs (a) First LDRv
R vector or Power stories int of three).It itribution to th
ure, to satisfregular frame
d number of ame, using 2
oach is spreadLDR vectors. nal effort need
ame for eigen-mo
(b) vector vs. 1st eig
Pseudo static n either case. is apparent thhe final respo
fy 90% of me but in case
f modes that eigen-modes
d among all thAppropriate
ded for MPA.
odes and LDR v
envector (b) 3rd
vector and fiFigure 4(b) s
hat LDR vectoonse can be ca
mass participof using LD
have been us). It should b
he basis vectorselection of
vectors
vector comparis
irst eigenvectshows the comors have activaptured by us
pation requireDR vectors, 3
used; For be noted rs, shape required
son
tor. As it mparison vated the sing only
ement, 8 3 vectors
TO we
otb
Fnev
The 14th
WoOctober 12-1
would be sueigenvector wshear of 16 sonly cause 10the higher mobut fewer vec
Figure 6 showno noticeableeigenvectors.vectors than superiority of
orld Confere17, 2008, Beij
ufficient. Figuwith respect totory irregular0% error. Thiodes. In case ctors would b
Figure 5 Com
Table 1
Story
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
ws the similare benefit due As the struceigen-modes,f LDR vector
ence on Earthijing, China
ure 5 and Tao nonlinear tir structure aftis trend can bof the regula
e needed to s
mparison of stor
MPA error perc
4 Story-IrreguEigen LD
- -- -- -- -- -- -- -- -- -- -- -- -
-5.14 7.6-1.76 0.9-25.55 -8.-45.27 -26
r results for tto the fact th
ctures get tall, still there iss is noticeabl
hquake Engi
able 1 show ime history anter using 4 ei
be seen for stiar structures uatisfy 90% pa
ry shear between
centage in Story
ular 8 Story-DR Eigen - - - - - - - - - - - - - - - - - 11.85 - 2.16 - 4.19 - 5.86 67 1.70 93 -17.96 .75 -31.24 6.34 -43.31
he story momhat LDR vecler, more accus no significane.
ineering
5
the advantanalysis (NLTHigen-modes wiffer part of stuse of LDR varticipation of
n NLRHA and M
Shear response
-Irregular 12LDR E
- - - - - -1- -7- -- 1
10.24 46.99 84.70 15.43 64.27 --8.91 -1-19.16 -3-31.02 -4
ments. Using ctors for theseurate results nt difference.
age of MPA HA) for vario
will have 40%tructures becavectors does nf mass.
MPA using eigenv
using LDR vect
2 Story-IrregularEigen LDR
- - - - - - - -
14.55 -22.137.88 -7.291.57 -1.831.74 1.134.95 5.358.01 8.880.00 10.416.76 7.106.52 0.0319.09 -5.5231.54 -13.8941.70 -23.69
LDR vectors e kinds of strcan be obtain. In the case
analysis witous types of f
% error while ause these panot cause sign
vectors and LDR
tors or eigenvect
r 16 Story-Irr Eigen
-4.48 2.47 1.57 2.76
3.97 5.79 6.84 9.12 11.08 9.72 1.63 -4.64
-14.55 -24.86
-35.42 -43.67
for low-rise ructures rapidned by using of irregular f
th LDR vectframes. The fiusing 3 LDR
arts are not exnificant impr
R vectors
tors
regular LDR
-14.16 1.58 2.38 0.45 3.06 8.14 10.38 11.71 11.56 10.01 6.55 7.12 9.49 7.56 -0.26 -9.59
regular structdly converge less number
frames consid
ors over first story R vectors xcited till rovement
tures has to exact of LDR
dered the
TO 3 F
toThritpabv
The 14th
WoOctober 12-1
3.2. Drift Re
Figure 7 and
the height of of regular struTable 3 showheight of strresponse of important. Usthe obtained parameters, haccurate resubenefits of invectors algori
orld Confere17, 2008, Beij
esponse
Table 2 show
structure incructures there
ws the accuracructure. Howa structure, se of LDR ve
results (exchitherto studiults, even witncluding highithm.
Figure 6 Comp
Figure 7 Co
ence on Earthijing, China
w the results f
reases, the usis no importacy of the prop
wever, it shoubut in some
ectors for regucept for the ted, for irreguth smaller nuher mode re
parison of story
omparison of drif
hquake Engi
for drift ratio
e of LDR vecant differenceposed method
uld be noted irregularitie
ular frames, dtime of calcuular frames eumber of LDRsponses (via
moment betwee
ft ratio between
ineering
6
( i
ii
h1−Δ−Δ ) o
ctors will leade in using eithd for the “Ovthat higher
s (like the odoes not haveulations and
especially for R vectors usstatic correc
en NLRHA and M
NLRHA and MP
of the irregula
d to better esther LDR vectoverall Drift Inmodes have
one consideree any superior
number of taller frames
ed for the anction concept
MPA using eigen
PA using eigenv
ar structures.
timates of exaors vs. eigenvndex ( /TOP HΔ
e little influeed here) theyrity over the uvectors requs LDR vectornalysis. This t) in the star
nvectors and LD
vectors and LDR
It can be see
act values. In vectors. )” where H ince on disply may becomuse of eigenv
uired). But likrs will produis mainly du
rting vector f
DR vectors
R vectors
en that as
the case
s overall lacement me more vectors in ke other
uce more ue to the for LDR
TO
F R
The 14th
WoOctober 12-1
From the abo
Regular struc
• Best partic
• For mcalcu
orld Confere17, 2008, Beij
Ta
Story
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Tab
Story
16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
ove mentioned
ctures:
results are cipation. most of low rulation. As th
ence on Earthijing, China
able 2 MPA erro
4 Story-IrreguEigen LD
- - - - - - - - - - - -
-13.99 -10-18.98 -17-22.42 -7-45.41 -26
ble 3 MPA erro
4 Story-IrreguEigen LD
- -- -- -- -- -- -- -- -- -- -- -- -
-16.62 -16-18.90 -17-28.84 -13-45.32 -27
d results, the f
obtained usi
rise to mediuhe height of s
hquake Engi
or percentage in
ular 8 StoryDR Eigen - - - - - - - - - - - - - - - - - 8.12 - -14.14- 2.15 - 16.29
0.43 9.23 7.93 -47.657.49 -26.626.92 -38.07
r percentage in d
ular 8 Story-DR Eigen - - - - - - - - - - - - - - - - - 3.80 - 3.66 - 7.25 - 9.00 .01 0.83 .41 -41.57 .15 -32.09 .01 -38.51
following obs
ing first set
um rise regulstructure incr
ineering
7
drift ratio using
-Irregular 12LDR E
- - - - - -1- -1- -1- -1
9.21 8-11.76 12.11 12
16.13 910.48 -3-43.46 -4-18.82 -1-27.43 -3
drift index using
-Irregular 12LDR E
- - - - - 3- 4- 7- 1
3.76 13.50 17.25 89.71 -2.65 -2
-35.98 -1-22.97 -2-27.53 -3
servation can
of LDR vec
lar models ( Hreases, neglec
LDR vectors or
2 Story-Irregularigen LDR - - - - - - - -
8.55 -21.586.00 -15.453.88 -13.80
11.39 -11.558.97 9.15 5.66 16.062.58 12.739.49 9.66 39.74 -37.284.37 2.12 6.99 -6.39
32.77 -17.84
g LDR vectors or
2 Story-IrregularEigen LDR
- - - - - - - -
3.50 3.514.81 4.837.70 7.691.37 11.406.12 16.192.60 12.708.01 8.251.05 -0.27
26.94 -21.5616.42 -6.7623.86 -11.3032.97 -17.64
be made for
ctors with m
3≤BH ) fircting upper m
r eigenvectors
r 16 Story-Irre Eigen L
-9.08 --32.01 -3-21.42 -2-23.45 -2-5.80 -8.92 117.39 114.46 18.61 1.91 2.77 -2.06 -4.30 -9.48 -17.16 -32.40 -
r eigenvectors
r 16 Story-Irr Eigen
2.71 2.56 6.72 8.29
10.90 12.15 12.13 9.76 7.10 4.26 -4.79 -8.06
-12.39 -16.56
-22.58 -32.28
regular and ir
more than 90
rst mode is dmodes may ca
egular LDR 12.66 31.86 20.71 24.19 -6.08 10.08 19.27 15.71 9.12 2.39 6.77 5.84 4.53 5.27 5.28 -2.47
regular LDR 2.95 2.81 6.97 8.61 11.31 12.58 12.50 10.21 7.96 6.72 5.22 5.23 4.50 4.56 3.95 -1.48
rregular struc
0% cumulativ
ominant for rause serious
ctures:
ve mass
response errors in
TO
I
4 T
ubum
R
The 14th
WoOctober 12-1
estimsuper
• Use oslight
• For thlead first f
Irregular stru
• Best massof veeigen
• Use eigencorretheir
• In thfirst s
4. CONCL
The paper prstructures wituse of LDR vbesides accuruse of exact meaningful dselecting the
REFERENC
1. BuildBuild
2. ChintUsingEarth
3. Chopfor bu
4. ChopEd., P
5. GhafAcceVol.
6. WilsoStruc
7. WilsoSupe
orld Confere17, 2008, Beij
mation of the rriority over eiof LDR vectotly underestimhe case of lowto better resufew LDR vec
uctures:
response esti participationectors requesnvectors, so itof LDR vec
n-modes, sincection concepteffects can bee case of usiset of LDR ve
LUSION
roposed an Mth stiffer low
vectors is supracy, are the meigenvectors
differences in number of re
CES
ding Seismicdings, FEMA-tanapakdee, Cg Vertically hquake Enginpra AK, Goeluildings.” EQ
pra AK. (200Prentice Hall,fory-Ashtianyeleration App13, No. 2 andon, E. L., (20ctures, Inc. Beon, E.L., rposition of R
ence on Earthijing, China
response paraigenvectors (eors (because omated results w rise to med
ults. Since, atctors (dependi
imates are obn. One shouldsted, as this t is important ctors providece the startingt. Consideringe compensateing single groectors with m
MPA proceduwer stories. Fr
erior to use omost importan
for this specusing either
quired Ritz v
c Safety Cou-273 .Federal C. and Chopr
“Regular” neering Reseal RK. (2002)
Q. Eng. and St1). Dynamics, Englewood
y, M. (1989). proach"; Iranid 3. 002), Three Derkeley, Calif
Yuan, M.WRitz Vectors",
hquake Engi
ameters. For except for theof their naturefor higher sto
dium rise regut the end of Ling on the siz
btained using d keep in min
number incto choose ads better estimg vector in Lg the effect oed by the firstound motion,
more than 90%
ure using LDrom the numeof exact eigenvnt factors in scial kind of irset of vector
vectors is a rul
uncil. (1997)Emergency M
ra, A.K. (200and Irregula
arch Center, U“A modal pu
tructural Dyns of StructureCliffs, New J"Seismic Res
ian Journal o
Dimensional fornia, USA. W., Dicken, EQ. Eng. an
ineering
8
this group ofe number of ve) usually leadories compariular models,
LDR vectors ae of the syste
the first set ond that the shcreases shapeequate numbemates of respLDR vector gf higher (rigidt vector of LD, results are m
% cumulative
DR vectors. Uerical results vectors basedselecting an arregularity. Frs. It should ble of thumb a
). NEHRP GManagement 3). “Evaluatioar Generic University of ushover analynamics; 31(3)es: Theory andJersey. sponse of Sixof Science an
Static and Dy
ns, J.M.nd Structural D
f models usinvectors requireds to over estiing to use of eusing more thalgorithm an
em) will conv
of LDR vectoapes of LDRes of first feer of LDR veponse calculgeneration algd) modes to th
DR. more scatteremass particip
Use of the Lit can be see
d modal propeapproach, use or the case obe kept in miand it needs m
Guidelines foAgency: Wason of The MoFrames” ReCalifornia, Bysis procedur: 561-582. d Application
x Correlated End Technology
ynamic Analy
(1982). "DDynamics, vo
ng LDR vectoed in some caimated resultseigenvectors. han 2 or 3 LDeigenvalue perge to the ex
ors with less R vectors are dew vectors wctors. ation in comgorithm is eqhe response o
ed. Best resulpation.
DR vectors, en that for thierties. As the of LDR vect
of regular struind that the ru
more investiga
or the Seismshington, DC.odal Pushoveeport No.PEE
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