calculul static neliniar
TRANSCRIPT
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Calculul static neliniarP100-1/2006
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Calcul inelastic – De ce?
Calcul elastic:
FD ≤ Fcap – verificare bazata pe forte
F
FD
Fcap
Calcul inelastic:
D ≤ cap – verificare bazata pe
deplasare
F
D cap
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Modele pentru calcul neliniar
plastic hinge
plastic hingeelastic beam
M
M
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Calcul seismic
Calcul dinamic neliniar:
- ofera deformatii maxime (ductile)
- forte maxime (fragile)
Avantaje:
• consecventa comportare “reala”
Dezavantaje:
• complicat
• actiunea seismica (accelerograme)
• dupa cedare – rezultate nefolositoare
• volum de calcul (verificare a calculului)
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Calcul static neliniar
D
F
D
mecanism
F
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Cerinta de deplasare?
Spectre de deplasare inelastice
Probleme:
• Spectre – SGLD
• Conversie NGLD - SGLD
cy= .1
cy= .15
cy= .2
cy= .25
cy= 10
Deplasare
Perioada(s)
0.0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4
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Echivalare NGLD-SGLD
M {U.. (t) } + C{U
. (t) } + {F(t) } M{1} ug
.. (t)
{U (t)} = {} D(t)
)t(uM
)t(F)t(D2)t(D g
M = mk ; D* = DL
M
*
*
;
T* = 2 M
K
*
*; F*(t) =
Ft()
M* = {}T M {}; L* = {}T M{1}
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Relatiile intre marimile NGLD si SGLD echivalent
pentru deplasari D = *M
*LD*
pentru forte F = F*
pentru factorii capacitatii de rezistenta cy = F
Mgy
= c*y
pentru factorul de ductilitate = *
daca forma este a unui vector propriu T = T*
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2 4 6 8 10 12 14 16
p+8e
p+8e sdof
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• Consecventa matematica
• SGLD aproximeaza perioada MGLD prin metoda Rayleigh
• Aproximeaza deplasarea la varf - Acuratete satisfacatoare
Observatii:
• Structuri cu rigiditate si rezistenta uniforme pe verticala – aportul modurilor superioare este mic
• Erori mai mari la evaluarea fortei taietoare de baza in raport cu deplasarea la varf
• Eficienta mai mare in cazul cutremurelor cu banda ingusta
• Metoda de echivalare este aceeiasi ca in cazul Eurocode 8 (difera numai modul de prezentare)
• Problema principala: vectorul de forma al deplasarilor si implicit al distributiei fortei laterale
Echivalare NGLD-SGLD
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5 10D (cm)
F (kN)
700
350
Curba push-over NGLD
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5 10D (cm)
F (kN)
700
350
Biliniarizare
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F (kN)
3.5 7
D* = Dt*M*/L*cm)
F* = F/(kN)
1000
500
Dt (cm)
Transformare NGLD-SGLD
700
350
5 10
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Cerinta de deplasare
Se cunoaste:
• T* si cy*
Spectre de deplasare inelastice – D*
Se determina D = D* L*/M*
cy= .1
cy= .15
cy= .2
cy= .25
cy= 10
Deplasare
Perioada(s)
0.0
0.1
0.2
0.3
0.4
0.5
0 1 2 3 4
T*
D*
Alternativ (eliminarea determinarii spectrelor):
Se determina D = D* L*/M*
Solutia nu este valabila decat pentru constructiile noi - rezistenta conform P100-1/2006
)T(cS)T(SD DeD*
2
2
T
)T(S)T(S eDe
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Verificare
capacitate > cerinta
• deplasari relative
• eforturi pentru cedarile fragile
• rotiri plastice
• mecanism
• factori de suprarezistenta
u/ i
D
Verificari:
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Avantajele calculului static neliniar:
• Simplitate
• Modelul scade in complexitate (comportare histeretica)
• Volumul de calcul scade considerabil
• Un singur calcul (medierea cerintelor prin spectru)
• Se poate aplica si pentru elemente fragile
• Modele tridimensionale
• Elimina problema alegerii accelerogramelor
Dezavantaje:
• Limitat la structuri cu rigiditate si rezistenta uniforme pe verticala (amplificari dinamice)
• Nu este recomandat structurilor inalte (aportul modurilor superioare este important)
Calcul static sau dinamic neliniar?
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Exemplu de calcul
Capacitatile de rezistenta:- rezistente medii ale materialelor Doua ipoteze privind distributia fortelor:• Ipoteza I – conform modului 1 pe fiecare directie: moment seismic maxim• Ipoteza II – aceleratie constanta (distributia urmareste distributia maselor): momente si forte taietoare maxime la baza
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Determinarea cerintei de deplasare
)T(cS)T(SD DeD*
T SDe(T) c D* D
ip.1
dir.X 1.115 0.2041 1.331473 0.271754 0.352979
dir.Y 1.046 0.179621 1.470089 0.264059 0.345776
ip.2
dir.X 1.115 0.2041 1.331473 0.271754 0.271754
dir.Y 1.046 0.179621 1.470089 0.264059 0.264059
D = D* L*/M*
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Forta –Deplasare – Ip. I
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Forta –Deplasare – Ip. II
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• Prima articulatie plastica ~ 3500 kN
• Forta seismica de calcul Fb = 2184 kN
• Suprarezistenta 3500/2184 ~ 1.6:
• rezistentelor medii
• conditii de conformare, procente minime
• Verificarea factorului u/ i ~ 1.3 (nu implica subdimensionare)
• Forta taietoare de baza maxima ~ 4400 kN
• Considerand raportul rezistente medii / de calcul ~ 1.35
• Suprarezistenta 4400/(2184*1.35) ~ 1.5
Structura proiectata conform P100/92
• Suprarezistenta mai mare:
• Dimensiuni mai generoase (verificare la deplasare restrictive)
• Armare la procente minime
Suprarezistenta structurii
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Mecanismul de plastificare - X
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Mecanismul de plastificare - Y
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Verificarea deplasarilor relative
caz Ip I-x Ip I-y Ip II-x Ip II-y
etaj drx /h dry /h drx /h dry /h
9 0.001564 0.001301 0.000827 0.000592
8 0.003910 0.003491 0.001432 0.000942
7 0.007603 0.006549 0.002676 0.001631
6 0.011578 0.009753 0.005356 0.003649
5 0.015489 0.013341 0.009379 0.007475
4 0.018404 0.016275 0.013690 0.011992
3 0.019936 0.018252 0.017507 0.016189
2 0.019186 0.017912 0.019270 0.018906
1 0.016459 0.015471 0.016479 0.015785
dra= 0.02
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Determinarea rotirilor capabile:
• Relatii de evaluare directa – experimente (Metoda A)
• Relatii care se bazeaza:
• pe evaluarea analitica a capacitatii de deformatie sectionala (u)
• legile constitutive ale betonului si armaturii
• formule empirice de determinare a lungimii plastice conventionale a articulatiei plastice (metoda B)
• P100-1/2006: Metoda B (orientativ)
• Eurocode 8 – part 3 (Evaluarea si consolidarea cladirilor)
• Metoda A
• Metoda B
• Preluare in propunerea de normativ pentru evaluarea structurilor de beton armat
Verificarea rotirilor plastice capabile
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coeficient privind tipul elementului (stalp, perete)h - este înălţimea secţiunii transversaleLv = M/V braţul de forfecare în secţiunea de capăt - forţa axială adimensionalizata’- coeficienţii de armare a zonei comprimate, respectiv întinsefc şi fyw - rezistenţele betonului la compresiune şi ale oţelului din etrieri (MPa), - factorul de eficienţă al confinăriixcoeficientul de armare transversală
Eurocode 8 –part 3 – Metoda A
c
ywx f
f
Vcum h
Lf
25
4
35,02,0
3,0'
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Eurocode 8 - part 3 – Metoda B
v
plplyu
el
plum L
LL
5,01
1
85,0
7.31c
ywsxccc f
fff
1512 f
fccccc
cc
ywsxcu f
f 5,0004,0
)(
)(15,02,0
30 MPaf
MPafdh
LL
c
yblvpl
• model de beton confinat
• Lungimea articulatiei plastice
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Verificarea rotirilor plastice
plmax plcap plmax plcap plmax plcapABA
CAZ DE INCARCARE
ARTICULATII PLASTICE INGRINZI
ARTICULATII PLASTICE IN STALPI
moment pozitiv moment negativ
AA BB AA BB AA BB
Ip. I-x 0.01962 0.0434 0.02563 0.0190 0.0270 0.0251 0.01134 0.04253 0.0289
Ip. II-x 0.01885 0.04694 0.02470 0.0183 0.0253 0.0251 0.01305 0.03599 0.0256
Ip. I-y 0.02184 0.0368 0.02575 0.0220 0.0266 0.0228 0.01176 0.01683 0.0251
Ip. II-y 0.02395 0.03519 0.02293 0.0243 0.0277 0.0227 0.01454 0.01987 0.0280
plmax pl
max plmax
plcappl
cap plcap
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Eurocode 8 – Proiectarea bazata pe deplasare – Spectru ADRS
T
Sa
TCTB TD
Sd
Sa
TD
TB TC
T TC
Sd
TD TB
ad ST
S2
2
4
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1.5
3
4
2
Sd
Sa
Spectrele ce exprima cerinta
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5 10Dt (cm)
F (kN)
1000
500
NGLD – Curba F-D
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5 10Dt (cm)
F (kN)
1000
500
Biliniarizare
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F (kN)
3.5 7D* = Dt/cm)
F* = F/(kN)
700
350
Dt (cm)
NGLD-SGLD
1000
500
5 10
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Sa = F*/m*(% g)
3.5 7D* = Dt/cm)
0.35
0.175
F* = F/(kN)
Diagrama de capacitate
700
350
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Sd
Sa
ductilitate tinta
Perioada necesara
Proiectarea baza pe deplasare (determinare q)
Deplasare tinta
Rezistenta necesara
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Precautie pentru Romania
N2 Spectra Comparison
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.5 1 1.5 2
T(s)
PGA=0.4g
PGA=0.2g
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Va multumesc pentru atentie
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Example
0.0
0.5
1.0
1.5
0 5 10 15 20
Sd = D* (cm)
ag=0.60g
ag=0.15g
ag=0.30g
T*=0.79s
=2.9
=1.5
1.14
0.39
4.4 6.1 8.9 17.7
Sa (g)
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1.5
3
4
2
Dy*
Say
Target Displacement
T*
Sde
Sae
Sd =
Sa =
Sa
Sd
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Sde
Sae
Sd Dy*
Say
Sa
Reduction Factor
Dd*
Sad
*yd DS
yeayae FFSSR
R
dyadays FFSSR
qRRRSSFFR ssadaede