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Retaining structures
M. di PriscoDepartment of Civil and Environmental Engineering,Politecnico di Milano
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CONCRETE RETAINING (cantilever) WALL
Surcharge
Load condition
23SLS
SLU=
δ
δ
%15.0: =s
rebar ρ
MC (2010) classification of SFRC:
Class “2b” - Vf = 25kg/m3
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Advantages of R/C sheet piles over steel:
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Sheet piles
b Jansze et al. 2005
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V u
[ k N ]
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No splitting cracks
occurred!
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General framework
Ground slope
Varenna
Caslino
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General framework
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July 2003
Ground slope stabilization
The slope in Caslino d’Erba (Lecco, Italy)
Safety factor= 1.1
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General framework
CASLINO
Nailing
Anchored bulkhead
New design solution
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Sheltering Structures
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Description of the retaining structure
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2003 2004 2005 2006
2007 2008 2009 2010
Preliminary
investigations
Soil investigations
Element design Element
positioning
Plate
installation
monitoring (full equipment)
Chronology
Stability
checks
Design and casting elements
Monitoring (partial equipment )
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Construction phases: preliminary operations on site
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Construction phases: slope scaling
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Construction phases: safety operations
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Construction phases: drainage setting-up
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Construction phases: testing on plates
L t f
L t b L f r e e
L c o ns t r a i nt
L e
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Construction phases: relief, template positioning and anchor fixing
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Construction phases: template formwork
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Construction phases: ground anchor boring and grouting
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Construction phases: casting of bedding bottom layer
C i h l h dli
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Construction phases: panel handling
Construction phases: anchor plate positioning
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Construction phases: anchor plate positioning
Construction phases: final view
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Construction phases: final view
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New design aspects
Main Concepts
• to strenghten the structure “ground slope”• to use precast elements to accelerate the in-situ operations
• to apply FRC materials to benefit of local thoughness
• to use such construction as an environmental structurecarefully monitored to check the reliability of both the
structure and the monitoring exp. techniques• to calibrate NDT, by perfectly knowing the mechanical
characteristics of the materials
N d i t
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New design aspects
EDL
EUL
Ne design aspects
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2D Jambuapproach
New design aspects
0.5
0.75
1
1.25
1.5Safety
factor
0 1 2 3 4 5 6 7 8 9 10 11 12 13Step
Failure 1
Failure 2
Failure 2: 1.4Failure 1: 1.4
Actual anchor load values
after 10 days from post-tensioning
Anchor loads at the onset of post-tensioning
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constituent quantity [kg/ m3]
Cement (CEM I
52.5R)400
Coarse aggregate A 569 (da < )
Coarse aggregate B 403 (da < )
Sieved sand 676 (da
< )
Calcareous filler 96 (da < )
Steel fibres 60/0.8 50
Superplasticizer/
cement ratio 2.2 %
Water/binder ratio 0.39
Material characterization: mix design
or pane s: wor a ty
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p y
suitable tests
SFRC for panels: mechanical tests
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0 1000 2000 3000
COD [µm]
0
4
8
12
N
[ M P a ]
Mean
SFRC for panels: mechanical tests
fibre content
50 kg/m3
df = 0.8 mm
Lf = 60 mm
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SFRC Classification
C60–S5–12–XC4–F4–DH1–DS1
UNI 11039
4a
f R1k / f Lk ≈ 0,94 > 0.4
f R3k / f R1k ≈ 0,70 > 0.5
MC 2010 (EN14651)
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0 1000 2000 3000COD [µm]
0
4
8
12
N
[ M P a ]
Mean
0 1200 2400 3600COD [µm]
0
4
8
12
N
[ M P a ]
fctf = 6.09 MPa [s = 0.84]
f1tf = 6.10 MPa [s = 0.77]
fFtf* = 2.57 MPa [s = 0.79]
P P
l l l
L=3.5 l
h + a = l
a
h
Material characterization: workability suitable tests
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Material characterization: workability suitable tests
Test 1 Test 2 Test 3 Cast Slump T50Ratio Scatter Ratio Scatter Ratio Scatter [mm] [sec]
Sector [kg/m3] [%] [kg/m3] [%] [kg/m3] [%] 1 625 8.8
1 40.01 -20.24 60.33 -7.73 44.43 -25.3 2 618 10.6
2 54.48 8.61 63.36 -3.10 69.33 16.56 3 560 12.0
3 54.16 7.98 68.12 4.19 53.62 -9.85 4 666 5.4
4 47.25 -5.79 61.15 -6.48 64.66 8.71 5 580 12.6
5 45.34 -9.62 64.12 -1.93 57.69 -3.01 6 714 6.2
6 50.35 0.37 76.00 16.23 61.94 4.13 7 685 6.8
7 46.58 -7.13 70.63 8.02 62.41 4.93 8 686 6.2
8 57.56 14-76 66.95 2.39 62.87 5.69 9 658 8.1
9 55.71 11.06 57.82 -11.57 58.37 -1.87 10 641 10.5
Mean 50.16 65.39 59.48 Mean 643 8.7
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Structural designMain doubts
hardening
behaviour, lcs ?
which advantages
with respect to plainconcrete ?
adequate safety
coefficient ?
which ductility?
In favour of safety
no redundancy effect was taken into
account !
Experimental programme
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Not Post-tensioned beams
30
3 0
Total: 15 beams (6 typologies) 3 0
30
R0
2 x
S0
3 x
p p g
Post-tensioned beams
3 0
30
3 0
30
S1
3 x 3 x
3 0
30
P1
2 x 2 x
P2 3 0
30
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Material characterization: bending strength
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f If[MPa]
f peak[MPa]
f Ft[MPa]
f eq(0-0.6)[MPa]
f eq(0.6-3)[MPa]
σFt1[MPa]
σFt2[MPa]
D0 D1
R0A - 3.850 3.465 - - - - - -
R0B - 4.509 4.058 - - - - - -
P1A - 3.716 3.344 - - - - - -
Average - 4.025 3.622 - - - - - -Variation
Coefficient [%]- 10.56 10.55 - - - - - -
S0A 3.899 6.181 3.509 5.559 2.879 2.501 0.189 1.425 0.512
S0B 4.350 6.805 3.915 6.063 3.890 2.728 0.581 1.391 0.649
S0C 4.736 6.644 4.263 5.600 4.014 2.520 0.747 1.187 0.732
S1A 4.334 7.023 3.900 6.216 4.545 2.797 0.874 1.437 0.733
S1B 4.231 4.982 3.808 4.623 2.496 2.081 0.208 1.092 0.553
S1C 4.195 6.279 3.776 5.602 3.661 2.521 0.570 1.314 0.651
S2A 4.575 6.526 4.118 5.825 3.672 2.621 0.526 1.263 0.638
S2B 5.003 5.571 4.503 4.920 2.409 2.214 0.097 0.983 0.506S2C 4.131 5.269 3.718 4.650 2.809 2.092 0.358 1.120 0.604
Average 4.384 6.142 3.945 5.451 3.375 2.453 0.461 1.246 0.620
Variation
Coefficient [%]7.69 11.61 7.68 10.79 22.19 - - - -
Material characterization: elastic modulus and time hardening
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0 0.02 0.04 0.06 0.08 0.1
ε [%]
0
5
10
15
20
25σ [MPa]
cilindro S1C-5
4 cylinders
cylinder ∆σ [MPa] ∆ε [-] E [MPa]
R0A-3 11.87 4.044E-04 29350
S0A-1 13.75 4.573E-04 30069S1C-5 14.40 4.741E-04 30375
S2A-5 13.43 4.188E-04 32069
Mean value 13.36 4.386E-04 30466 (3.8 %)
0 7 14 21 28tempo [gg]
0
20
40
60
80
Rcm [MPa]
evoluzione resistenza a compressione
cls tradizionale
cls fibrorinforzato
Experimental tests: post-tensioning
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step(1) (2) (3) (4) (5)
1 -5.16 -1.02 3.03 -1.11 -3.13
2 -2.12 -2.12 -2.12 -2.12 -2.12
3 -3.23 0.91 -3.14 -7.28 -5.25
4 -4.24 -4.24 -4.24 -4.24 -4.24
5 -7.37 -3.23 -3.23 -7.37 -7.37
6 -12.53 -4.25 -0.20 -8.48 -10.50
7 -9.49 -5.35 -5.35 -9.49 -9.49
8 -10.60 -2.32 -6.37 -14.65 -12.62
9 -11.61 -7.47 -7.47 -11.61 -11.61
10 -14.74 -6.46 -6.46 -14.74 -14.74
σ [MPa]
B C
(2) (3)
A D
(1) (4)(5)
E
f cm 60 MPa
k 1.48 -
s 3.6 MPa
f ck 54.67 MPa
f ctk 4.34 MPa
Strength at 7 days
(control type B)
Experimental tests: demoulding and lifting
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k·g
F
Fv
Fh
α
x
kNm f W M ctm,fl R 3.17 =⋅=
kNm x M M S 9.1)(max ==
S R M M >
Experimental tests: demoulding and lifting
f cm,1 = 34 Mpa
β = β(h)= 1.23
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Experimental tests: test set-up
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Prova di flessione a quattro punti
P/2 P/2L/4 L L L L/4
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L 1 L 2 L 3
L 4
L 7L 6 L 8
Vert A1 Vert A2 Vert B1 Vert C1 Vert D1 Vert E2 Vert E1
A B C D E
lato 1
Experimental tests: test instrumental equipment
Experimental tests: test instrumental equipment
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L 1 L 2 L 3
L 4
L 7L 6 L 8
Vert A1 Vert A2 Vert B1 Vert C1 Vert D1 Vert E2 Vert E1
A B C D E
lato 1
Experimental tests: test instrumental equipment
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Vert A1 Vert A2 Vert B1 Vert C1 Vert D1 Vert E2 Vert E1
L 7
L 9
L 1
L 3
L 5
L 6
A B C D E
lato 1
Experimental tests: reference grid
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Digital Image Correlation
Experimental tests: reference grid
Experimental tests: post-tensioned beams
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0 30 60 90 120
δ [mm]
0
50
100
150
200
250
P [kN]
S1A
S1B
S1C
0 20 40 60 80 100δ [mm]
0
100
200
300P [kN]
P2A
P2B
0 20 40 60 80 100
δ [mm]
0
100
200
300
P [kN]
S2A
S2B
S2C
SFRC
P/C
0 30 60 90 120δ [mm]
0
50
100
150
200
250P [kN]
P1A
P1B
4 strands
5 strands
Experimental tests: not post-tensioned beams
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0 0.004 0.008 0.012 0.016 0.02
θ [1/m]
0
5
10
15
20
25M [kNm]
S0A
S0A sx
S0B
S0C
Carichi pre-esistenti (mezzeria):Momento da peso proprio: 2.46kNmMomento da ripartitore: 1.1kNm
0 4 8 12δ [mm]
0
10
20
30
40
50P [kN]
S0A
S0B
S0C
Carichi pre-esistenti (oltre al peso proprio)Carico del ripartitore: 2.2kN
0 0.04 0.08 0.12
θ [1/m]
0
40
80
120
160
200M [kNm]
R0A
R0B
R0B dx
0 40 80 120δ [mm]
0
100
200
300
400P [kN]
R0A
R0B
SFRC
R/C
Midspan loads MG
MQ
Load device = 2.2 kN
Experimental tests: comparisons
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Obseration!
-separated performance level
- different initial stiffnesses
- ductility
42
374
207
101
0 0.2 0.4 0.6 0.8δ [mm]
0
4
8
12
16P [kN]
R0B
S1C
S0A
0 30 60 90 120δ [mm]
0
100
200
300
400
P [kN]
R0B
S1C
S0A
Experimental tests: comparisons
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0 30 60 90 120
0
100
200
300
P [kN] P2
S2
0 30 60 90 120
stroke [mm]
0
100
200
300
P [kN] P1
S1
"ensioncontrib#tion
Com$ression
contrib#tion
P1B (4 strands - PC)
S1B (4 strands) S!C)
120 kN
100 kNdesig
n
Experimental results: observation
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" Contri#uto delle fi#re a tra$ione
" Buona ripeti#ilit% del comportamento degli elementi strutturali
No cracking of ducts!
#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I
03Sizes
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Concrete
Ecm 30460 Mpa
fcm 58 MPa
fIF 5.94 MPa
Post-tensioning steel
fptk 1860 Mpa
Fpk 1650 MPa
F 0.6! Inc"
#n 139 mm$
Ep 195000 MPa
#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I#%#&I'I P)E&IMI%#)E *E& +)#,ICCI *I ,)#I
Come si / ginti allo stio el pro2lema flesso-torsionale
P&I,EC%IC *I MI%
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#nc"or stran stretc"ing ;s . )eference noeisplacement
CASE 1: Indefinite Plastic hinges
CASE 2: Plastic hinge with finite ductility
perimental In;estigationIn;estigationIn;estigationIn;estigation ? 4assani e +an=iroli
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@AB
@AB #. assani '. +an=iroli Il pro2lema torsionale in elementi fi2rorinfor=atiD ,eorie e
sperimenta=ione . Master t"esis $010-$011
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Monitoring of retaining structure
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Load vs. strand sliding Which load during
ground anchor post-
tensioning?
Meteo conditions
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Rainfall
Temperature
Temperature evolution
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winter summer
Long term monitoring of load cells
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EDL
EDR
EUL
EUR
Load cell single strand
Monitoring of retaining structure
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LEFE assessment
X
Y
Z
Concluding remarks
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• a new retaining structure continuously monitored via GPRS isnow also ready to be used as a lab for NDT
• a synergic action between University and local and regionaladministration gives rise to a new environmental lab on slope
stability used by Polo Regionale di Lecco
• new concept design and advanced materials have been used to
design a new retaining structure solution
• main aims: to check via GPRS the structure in the time, to investigateon the monitoring control of protection structure database, to validate
Eurocode 7 approach for special structures
Possible use combining plates and geomembranes for
deep strenghthening + skin retrofitting
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&HPE)# *I ',#I&I#I%E *I % PE%*I&HPE)# *I ',#I&I#I%E *I % PE%*I&HPE)# *I ',#I&I#I%E *I % PE%*I&HPE)# *I ',#I&I#I%E *I % PE%*I
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'itoD'itoD'itoD'itoD
Comne i Caslino HEr2a @CB
Parete i materiale morenico eterogeneo
#lte==a complessi;a $5 m parete
s2;erticale i 16 m.
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Calcestr==o fi2rorinfor=ato 'F)C
Post-tensione a ca;i non aerenti
*imensioni contente.
*escri=ione el sito e aspetti i interesse
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1.1.1.1. #%#&I'I#%#&I'I#%#&I'I#%#&I'I P)E&IMI%#)EP)E&IMI%#)EP)E&IMI%#)EP)E&IMI%#)E elelelel graticciograticciograticciograticcio iiii tra;itra;itra;itra;i
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I%*ICE
P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- I& P)&EM# I%+E+%E)I',ICI& P)&EM# I%+E+%E)I',ICI& P)&EM# I%+E+%E)I',ICI& P)&EM# I%+E+%E)I',IC
*#& M*E&I%E *E& +)#,ICCI #& M*E&I%E *E& ,)#E *I*#& M*E&I%E *E& +)#,ICCI #& M*E&I%E *E& ,)#E *I*#& M*E&I%E *E& +)#,ICCI #& M*E&I%E *E& ,)#E *I*#& M*E&I%E *E& +)#,ICCI #& M*E&I%E *E& ,)#E *I
05
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&IMI,I *E& M*E&I%E&IMI,I *E& M*E&I%E&IMI,I *E& M*E&I%E&IMI,I *E& M*E&I%E
•#ssen=a ca;i i precompressione
•Impossi2ile tenere in contoellHintera=ione M-,
•%ecessitJ i inagarenmericamente la risposta
torsionale con e sen=a laprecompressione
#'PE,,I#'PE,,I#'PE,,I#'PE,,I *#*#*#*# 'I&PP#)E'I&PP#)E'I&PP#)E'I&PP#)E
•Intro=ione non linearitJ el
materiale
•Moella=ione piK raffinataella ner;atra
•Intro=ione ei ca;i i post-
tensione nella moella=ione
•alta=ione ella resisten=atorsionale in presen=a i
a=ione flettente
)*)*)*)*
P)'ECI%E *E& ) *I ,E'IP)'ECI%E *E& ) *I ,E'IP)'ECI%E *E& ) *I ,E'IP)'ECI%E *E& ) *I ,E'I
)eali==a=ione i n moello nmerico in grao i pre;eere il)eali==a=ione i n moello nmerico in grao i pre;eere il)eali==a=ione i n moello nmerico in grao i pre;eere il)eali==a=ione i n moello nmerico in grao i pre;eere il
comportamento ellacomportamento ellacomportamento ellacomportamento ella
tra;e i 2oro a torsione 7 torsione e precompressione 7 flessotra;e i 2oro a torsione 7 torsione e precompressione 7 flessotra;e i 2oro a torsione 7 torsione e precompressione 7 flessotra;e i 2oro a torsione 7 torsione e precompressione 7 flesso----
torsionetorsionetorsionetorsione
02
06P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,I
'F)C @'teel fi2re reinforce concreteB'F)C @'teel fi2re reinforce concreteB'F)C @'teel fi2re reinforce concreteB'F)C @'teel fi2re reinforce concreteB
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CMP)E''I%ECMP)E''I%ECMP)E''I%ECMP)E''I%ED
cr;a i 'argin L ramosoftening
,)#I%E,)#I%E,)#I%E,)#I%ED
'oftening ilineare i
Prisco et al. @AB
@AB M. i Prisco &. Ferrara M. Colom2o M. Mari -n t"e ientification of 'F)C constitti;e la innia>ial tension. )I&EM p22lication. $006
feN1D )esisten=a post-fessrati;a
'&EfeN$D )esisten=a post-fessrati;a '&
I legami costitti;i tili==ati
FI)EDFI)EDFI)EDFI)ED
Fi2re ncinate
#cciaio a 2asso contento
i car2onio
&O60 mm F=08 mmantitJD 50 kg7m.
02
0GP&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,I
M*E&& C',I,,I ,I&I#,M*E&& C',I,,I ,I&I#,M*E&& C',I,,I ,I&I#,M*E&& C',I,,I ,I&I#,
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'oftareD ##' 6.10##' 6.10##' 6.10##' 6.10
Moello costitti;oD Concrete *amage Plasticit MoelConcrete *amage Plasticit MoelConcrete *amage Plasticit MoelConcrete *amage Plasticit Moel
Moello a anno 2asato slla plasticitaH
*e meccanismi i rottraD cracking a tra=ione e crs"ing a
compressione.
&egame non associatoD istin=ione tra sperficie i sner;amento @FFFFB e
poten=iale plastico @++++B.
Poten=iale plastico @fn=.
*rcker PragerB
'p. i sner;amento
&egge i scorrimento
Φ⋅−+Φ⋅⋅= tan)tan( pq G t
22
0σ ε
⋅
⋅∂
∂= λ
σ
Gδε
ij
pl
ij
03
1
1=−⋅−⋅+⋅⋅−⋅
−
= )())((maxmax plc pl
pq F ε σ σ γ σ ε β α
α
)/(
/
12
1
00
00
−⋅
−=
cb
cb
σ σ
σ σ α
( ) ( )α α ε σ
ε σ β +−−⋅= 11
)(
)( pl
tt
pl
cc
12
13
−⋅
−⋅=
c
c
K
K )(γ
QConcrete *amage Plasticit MoelR
02
#&I*#I%E *EI &E+#MI C',I,,II#&I*#I%E *EI &E+#MI C',I,,II#&I*#I%E *EI &E+#MI C',I,,II#&I*#I%E *EI &E+#MI C',I,,II
08P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,I
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fIFm feN1m feN$m fcm Ecm
MPa MPa MPa MPa MPa
4.68 5.$0 3.G3 G498 $5094
Pro;a a tra=ione monoassiale s pro;ino intagliatoConfrontoConfrontoConfrontoConfronto sperimentalisperimentalisperimentalisperimentali - *o=io $008
#%+& i *I,#%##%+& i *I,#%##%+& i *I,#%##%+& i *I,#%# ME':ME':ME':ME': )E+)I#I%E)E+)I#I%E)E+)I#I%E)E+)I#I%E
I'C'#I'C'#I'C'#I'C'#
02
#&I*#I%E *EI &E+#MI C',I,,II#&I*#I%E *EI &E+#MI C',I,,II#&I*#I%E *EI &E+#MI C',I,,II#&I*#I%E *EI &E+#MI C',I,,II
09P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,I
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fIFm feN1m feN$m fcm Ecm
MPa MPa MPa MPa MPa
5.94 8.39 4.8G 69.00 39$G0
Pro;a a tra=ione iniretta ? flessione s 4 pntiConfrontoConfrontoConfrontoConfronto sperimentalisperimentalisperimentalisperimentali - *o=io $008
ME':ME':ME':ME': E&EME%,IE&EME%,IE&EME%,IE&EME%,I
#*)#,ICI#*)#,ICI#*)#,ICI#*)#,ICI
)E+)I#I%E)E+)I#I%E)E+)I#I%E)E+)I#I%E
I'C'#I'C'#I'C'#I'C'#
13P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #
,)'I%E,)'I%E,)'I%E,)'I%E
M*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%E
M ll
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Moello M*E&& +EME,)ICM*E&& +EME,)ICM*E&& +EME,)ICM*E&& +EME,)IC ???? Moello 4
#%E&& )I+I*
O
Cernieratorsionale
F)C:E,,# )I+I*#O
)itegno torsionale
I%C&I
C%,#,, D
ire=ione normale D istacco e assen=a i compenetra=ione
ire=ione tangen=iale D
anello → µO0.45 @acciaio - calcestr==oB
forc"etta → µO0.04 @calcestr==o - teflonB
14P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #
,)'I%E,)'I%E,)'I%E,)'I%E
M*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%E
)i lt ti i i M ll
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)isltati nmerici Moello
,ra;e ,picco
k%m S
'perim 31.03 -
33.39 G.6
5.0 e-03
2.0 e-03
8.0 e-04
)I'&,#,I %ME)ICI)I'&,#,I %ME)ICI)I'&,#,I %ME)ICI)I'&,#,I %ME)ICI ---- Moello 4
'P)#
, 'T
',,
PE M#T principali -
picco
aro fessrati;o
sperimentale
15P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&
P)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%E
EFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&EEFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&EEFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&EEFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&E
Formla=ione teorica e risltati sperimentali
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' $50 > 1850
mm
'c"ema 'tatico'c"ema 'tatico'c"ema 'tatico'c"ema 'tatico
16P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&
P)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%E
EFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&EEFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&EEFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&EEFFE,, *E& P)ECMP)E''I%E '& C#P#CI,#H ,)'I%#&E
Moello C e ;alia=ione ei moelli 7C
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Moello C e ;alia=ione ei moelli 7C
)I'&,#,I #&I*#I%E)I'&,#,I #&I*#I%E)I'&,#,I #&I*#I%E)I'&,#,I #&I*#I%E ???? Moello
M*E&& CM*E&& CM*E&& CM*E&& C
Moello %mericoMoello %mericoMoello %mericoMoello %merico'c"ema'c"ema'c"ema'c"ema
'tatico'tatico'tatico'tatico
----0.50.50.50.5----$ 0.9$ 0.9$ 0.9$ 0.9
ffff'P'P'P'P
----0.50.50.50.5----$Wf$Wf$Wf$Wf'P'P'P'P
#&I*#I%E#&I*#I%E#&I*#I%E#&I*#I%E ---- Moello
C
M*E&& CM*E&& CM*E&& CM*E&& C
)isltati nmerici per la ner;atra )I1
1GP&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&
P)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%E
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M*E&& CM*E&& CM*E&& CM*E&& C ????
)I41
PEM#T principaliθO 9.5 mra
PEM#T principali
θO 15 mra
PEM#T principali
θO 30 mra
,%M 33.39 k%m
,pr %M 55.10 k%m
,pr ," 5G.90 k%m
Pra ,orsione @ris. nmericoB
,orsione LPrecompressione @ris.
nmericoB
5.0 e-03
2.0 e-03
8.0 e-04
1.4 e-02
8.3 e-03
2.4 e-03
1.6 e-03
8.2 e-04
2.7 e-04
amento capacitJ torsionale @60SB
per effetto
ella precompressione
M*E&I%E *EI C#I *I P',M*E&I%E *EI C#I *I P',M*E&I%E *EI C#I *I P',M*E&I%E *EI C#I *I P',----,E%'I%E,E%'I%E,E%'I%E,E%'I%E ---- M*E&& *M*E&& *M*E&& *M*E&& *
)isltati nmerici e confronto con il caso i pressione niforme
alle estremitJ
18P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&M*E&I%E *E&
P)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%E
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M*E&& *M*E&& *M*E&& *M*E&& * ???? )I41
ca;i i
post-
tensione
alle estremitJ
Moello CMoello CMoello CMoello C ???? %o%o%o%o
Ca;iCa;iCa;iCa;i
Moello *Moello *Moello *Moello * ---- Ca;iCa;iCa;iCa;i
Ca;i i post-
tensione
Elementi 3*
Precompressione ∆, s ca;o
Precompressione Pressione
niforme
M*E&& CM*E&& CM*E&& CM*E&& C ???? )I41
*MI%I *I I%,E)#I%E F&E''I%E*MI%I *I I%,E)#I%E F&E''I%E*MI%I *I I%,E)#I%E F&E''I%E*MI%I *I I%,E)#I%E F&E''I%E ---- ,)'I%E,)'I%E,)'I%E,)'I%E
*omini i intera=ione presenti in letteratra
19P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- I%,E)#I%E F&E''I%EI%,E)#I%E F&E''I%EI%,E)#I%E F&E''I%EI%,E)#I%E F&E''I%E----,)'I%E,)'I%E,)'I%E,)'I%E
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C&' 2iancoC&' 2iancoC&' 2iancoC&' 2ianco @1B@1B@1B@1B 'F)C'F)C'F)C'F)C @$B@$B@$B@$B 'F)C'F)C'F)C'F)C
precompressoprecompressoprecompressoprecompresso @3B@3B@3B@3B
@1B )epot on torsion in strctral concrete. ,ec"nical report #CI Committee 4457#'CE.@$B M. #. Mansr. ening-torsion interaction for concrete 2eams reinforce it" steel fi2res.
Maga=ine of Concrete )esearc"198$.@3B ). %araanan an #. '.
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*imensioniD 85 > 1G8 > $400 mm
Precompressione ca;o
Carico ;erticale applicato per raggingere il momento flettente ric"iesto
Ceimento rota=ionale anelli rigii
#&I*#I%E M*E&& *#&I*#I%E M*E&& *#&I*#I%E M*E&& *#&I*#I%E M*E&& * ? %araanan et al. @AB
@AB ). %araanan an #. '.
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@AB ). %araanan an #. '.
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#&I*#I%E M*E&& *#&I*#I%E M*E&& *#&I*#I%E M*E&& *#&I*#I%E M*E&& * ? eforma=ioni plastic"e massime principalieforma=ioni plastic"e massime principalieforma=ioni plastic"e massime principalieforma=ioni plastic"e massime principali alpicco elle cr;e
MO0 MO0.$5M
MO0.50M MO0.G5M
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,,,,
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,,,,
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,,,,
',,',,',,',,
5.0 e-032.0 e-03
8.0 e-04
5.0 e-03
2.0 e-03
8.0 e-04
5.0 e-033.0 e-038.0 e-04
2.0 e-021.0 e-024.0 e-03
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Moelli geometrici e nomenclatre
$3P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
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M*E&I%E *E& %E)#,)# )I1M*E&I%E *E& %E)#,)# )I1M*E&I%E *E& %E)#,)# )I1M*E&I%E *E& %E)#,)# )I1
+eometria ei e moelli *ettagli el moello
Mes" $5mm
*ettagli el moello
Mes" 50mm
M*E&& *.0 @1$00 mmB M*E&& *.1 @3000 mmB
%ME%C,)# e C#)#,,E)I',IC:E M*E&&I *.0. e *.1.%ME%C,)# e C#)#,,E)I',IC:E M*E&&I *.0. e *.1.%ME%C,)# e C#)#,,E)I',IC:E M*E&&I *.0. e *.1.%ME%C,)# e C#)#,,E)I',IC:E M*E&&I *.0. e *.1.
mesh cavi n°El.. tipo El. nodi
D.0.25U 25x25 non
aderenti 17168 C3D8 21057
D.0.50U 50x50 non
aderenti 9664 C3D8 12093
D.0.25B 25x25 aderenti 17168 C3D8 21057
mesh cavi n°El.. tipo El. nodi
D.1.25U 25x25 non
aderenti 67959 C3D8 75121
D.1.50U 50x50 non
aderenti 42127 C3D8 48510
D.1.25B 25x25 aderenti 67959 C3D8 71521
M*E&& *.0
M*E&& *.1
#%#&I'I '&,E PE) %E)#,)# )I1#%#&I'I '&,E PE) %E)#,)# )I1#%#&I'I '&,E PE) %E)#,)# )I1#%#&I'I '&,E PE) %E)#,)# )I1
P)# ,)'I%E ? F&E''I%E e ,)'I%E
$4P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
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•Pra torsionePra torsionePra torsionePra torsione
•Intera=ione flessione e torsione per i;ersiIntera=ione flessione e torsione per i;ersiIntera=ione flessione e torsione per i;ersiIntera=ione flessione e torsione per i;ersili;elli i caricoDli;elli i caricoDli;elli i caricoDli;elli i caricoD
•MO0.$5MO0.$5MO0.$5MO0.$5 MMMM
•MO0.50MO0.50MO0.50MO0.50 MMMM
•MO0.G5MO0.G5MO0.G5MO0.G5 MMMM
#%#&I'I %ME)IC:E '&,ED#%#&I'I %ME)IC:E '&,ED#%#&I'I %ME)IC:E '&,ED#%#&I'I %ME)IC:E '&,ED
'c"ema statico'c"ema statico'c"ema statico'c"ema statico
fctm feN1m feN$m fcm Ecm
MPa MPa MPa MPa MPa
3.90 5.G4 3.59 G$.03 30466
ProprietJ meie el materialeProprietJ meie el materialeProprietJ meie el materialeProprietJ meie el materiale
)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1 ???? M*E&& *.0M*E&& *.0M*E&& *.0M*E&& *.0
Moello *.0. ? Pra ,orsione
$5P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
,)#E 1$00 mm,)#E 1$00 mm,)#E 1$00 mm,)#E 1$00 mm
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PE ma> principal -θO
8 mra
PE ma> principal -θO
10 mra
PE ma> principal -θO
14 mra
Tpicco[kNm]
%
D.0.25U 57.06 -
D.0.50U 58.75 2.9
D.0.25B 57.51 0.7
assi scarti S rispetto al moello corrett
****....0000....$5$5$5$5
risltati con ca;i aerenti atteni2ili
,)#E 1$00 mm,)#E 1$00 mm,)#E 1$00 mm,)#E 1$00 mm
3.0 e-031.5 e-034.0 e-04
1.2 e-026.0 e-032.0 e-03
4.0 e-042.0 e-048.0 e-05
)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1 ???? M*E&& *.0.M*E&& *.0.M*E&& *.0.M*E&& *.0.
Moello *.0. ? F&E''I%E e ,)'I%E
$6P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
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M O 0.50 M
M O 0.$5 M
*istri2=ione PE massime
principali
θ O 5 mra
MMMMffff,,,,piccopiccopiccopicco
----1.6S1.6S1.6S1.6S
----4.$S4.$S4.$S4.$S
3.0 e-031.5 e-038.0 e-04
3.0 e-03
1.5 e-038.0 e-04
)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1 ???? M*E&& *.1M*E&& *.1M*E&& *.1M*E&& *.1
Moello *.1. ? Pra ,orsione
$GP&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
,)#E 3000 mm,)#E 3000 mm,)#E 3000 mm,)#E 3000 mm
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PE ma> principali - θO 8 mra
PE ma> principali - θO 10 mra
PE ma> principali - θO 15 mra
Tpicco[kNm]
%
D.1.25U 55.30 -
D.1.50U 56.53 2.2
D.1.25B 55.21 0.2
2.5 e-04
1.0 e-04
8.0 e-05
2.0 e-031.0 e-03
3.0 e-04
5.0 e-032.0 e-03
8.0 e-04
*eforma=ioni plastic"e PE massime
)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1)I'&,#,I %ME)ICI PE) %E)#,)# )I1 ???? M*E&& *.1.M*E&& *.1.M*E&& *.1.M*E&& *.1.
Moello *.1. ? F&E''I%E e ,)'I%E
$8P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
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M = 0.25 MU
M = 0.50 MU
M = 0.75 MU
p
principali
θ O 13 mra
MMMMffff,,,,piccopiccopiccopicco
----1.9S1.9S1.9S1.9S
----4.8S4.8S4.8S4.8S
----11.4S11.4S11.4S11.4S
1.0 e-02
5.0 e-031.0 e-03
5.0 e-032.0 e-038.0 e-04
5.0 e-032.0 e-038.0 e-04
I%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%E
Effetto ella lng"e==a sllHintera=ione flesso-torsionale
$9P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
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,ra;e Corta,ra;e Corta,ra;e Corta,ra;e Corta ???? 1$00 mm1$00 mm1$00 mm1$00 mm
,ra;e &nga,ra;e &nga,ra;e &nga,ra;e &nga ???? 3000 mm3000 mm3000 mm3000 mm
----4.0S4.0S4.0S4.0S----4.3S4.3S4.3S4.3S
----4.5S4.5S4.5S4.5S
I%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%E
)isltati nmerici e omini i intera=ione
30P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
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,)#E & O 1$00,)#E & O 1$00,)#E & O 1$00,)#E & O 1$00
mmmmmmmm,)#E & O 3000,)#E & O 3000,)#E & O 3000,)#E & O 3000
mmmmmmmm
C%C&'I%IC%C&'I%IC%C&'I%IC%C&'I%I
31P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- C%C&'I%IC%C&'I%IC%C&'I%IC%C&'I%I
P)#P)#P)#P)# ,)'I%E,)'I%E,)'I%E,)'I%E DDDD
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•Moello nmerico &imitata so;rastima ella capacitJ torsionale
sperimentale
•Corrisponen=a el Naro fessrati;o
•Moello nel complesso affia2ile.
,)'I%E,)'I%E,)'I%E,)'I%E eeee P)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%E DDDD
•ona corrisponen=a con i ;alori teorici @'ke ening ,"eorB
•)isltati confronta2ili tra Moello C @assen=a i ca;iB e Moello* @ca;i 3*B.
I%,E)#I%EI%,E)#I%EI%,E)#I%EI%,E)#I%E F&E''I%EF&E''I%EF&E''I%EF&E''I%E ????,)'I%E,)'I%E,)'I%E,)'I%E conconconcon precompressioneprecompressioneprecompressioneprecompressioneDDDD
•Corrisponen=a el Naro fessrati;o sperimentale @aletteratraB
•*ecremento ella resisten=a torsionale per alti ;alori i momentoflettente
•)isltati nmerici confronta2ili tra moella=ione ca;i aerenti enon aerenti
•Inflen=a ella lng"e==a ella tra;e slla capacitJ torsionale
'I&PPI F,)I'I&PPI F,)I'I&PPI F,)I'I&PPI F,)I
3$P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E ---- 'I&PPI F,)I'I&PPI F,)I'I&PPI F,)I'I&PPI F,)I
,)'I%E,)'I%E,)'I%E,)'I%E EEEE P)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%E
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,)'I%E,)'I%E,)'I%E,)'I%E EEEE P)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EP)ECMP)E''I%EDDDD
•Campagna sperimentale s tra;i precompresse cosX a caratteri==arecompitamente il comportamento i na singola ner;atra el
graticcio
I%,E)#I%EI%,E)#I%EI%,E)#I%EI%,E)#I%E F&E''I%EF&E''I%EF&E''I%EF&E''I%E ????,)'I%E,)'I%E,)'I%E,)'I%E conconconcon precompressioneprecompressioneprecompressioneprecompressioneDDDD
•#nalisi nmerica a collasso consierano lHeffettoellHintera=ione flesso- torsionale
•Campagna sperimentale s tra;i precompresse soggette a flessionee torsione
•Pro;a sperimentale a collasso ella strttra graticcio.
+)#IE PE)+)#IE PE)+)#IE PE)+)#IE PE)
&H#,,E%I%E&H#,,E%I%E&H#,,E%I%E&H#,,E%I%E
P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #
,)'I%E,)'I%E,)'I%E,)'I%E
M*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%EM*E&I%E P)# ,)'I%E
Confronto Moello # e Moello
C%F)%, %ME)ICC%F)%, %ME)ICC%F)%, %ME)ICC%F)%, %ME)IC ---- Moell0 # e
Moello 4
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Moello 4
M*E&& #M*E&& #M*E&& #M*E&& #
incolo i Copling
@#2aNsB
M*E&& M*E&& M*E&& M*E&&
incoli
moellati
,ra;e ,piccok%m
S
'perim 31.03 -
33.39 G.6
# 3$.69 5.3
incolo iincolo iincolo iincolo i #2aNs#2aNs#2aNs#2aNs
@#B@#B@#B@#B
incoli moellatiincoli moellatiincoli moellatiincoli moellati
@B@B@B@B
Incompati2ile conprecompressione
Maggiore controllo ei
;incoli
Moello piK realistico
P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #M*E&I%E CMP),#ME%, #
,)'I%E,)'I%E,)'I%E,)'I%E
,)#E C#&E,,I E '))E%,I%,)#E C#&E,,I E '))E%,I%,)#E C#&E,,I E '))E%,I%,)#E C#&E,,I E '))E%,I%
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02
P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,I
'F)C @'teel fi2re reinforce concreteB'F)C @'teel fi2re reinforce concreteB'F)C @'teel fi2re reinforce concreteB'F)C @'teel fi2re reinforce concreteB
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02
P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,I
*#%% # CMP)E''I%E*#%% # CMP)E''I%E*#%% # CMP)E''I%E*#%% # CMP)E''I%E ???? #2aNs 6.10#2aNs 6.10#2aNs 6.10#2aNs 6.10
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02
P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- M*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,IM*E&I%E #* E&EME%,I FI%I,I
Poten=iale plastico @fn=.
* k P B
Φ⋅−+Φ⋅⋅= tan)tan( pq G t
22
0σ ε
M*E&& C',I,,I ,I&I#,M*E&& C',I,,I ,I&I#,M*E&& C',I,,I ,I&I#,M*E&& C',I,,I ,I&I#,
QConcrete *amage Plasticit MoelR
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*rcker PragerB
'p. i sner;amento @&ee Fen;esB
&egge i scorrimento⋅
⋅∂
∂= λ
σ
Gδε
ij
pl
ij
031
1=−⋅−⋅+⋅⋅−⋅
−= )())((
maxmax plc pl pq F ε σ σ γ σ ε β α
α
)/(
/
12
1
00
00
−⋅
−=
cb
cb
σ σ
σ σ α
( ) ( )α α ε σ
ε σ β +−−⋅= 11
)(
)( pl
tt
pl
cc
12
13
−⋅
−⋅=
c
c
K
K )(γ
Parametri i inptD
ElasticitJD
EEEE ν.
Plasticità:
Φ = angolo i ilatan=a
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,ra;e &O3000mm,ra;e &O3000mm,ra;e &O3000mm,ra;e &O3000mm
'for=i minimi principali in corrisponen=a i momento
flettente massimo
I%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%EI%,E)#I%E F&E''I%E ,)'I%E
Effetto ella lng"e==a sllHintera=ione flesso-torsionale
P&I,EC%IC *I MI%
'& CMP),#ME%, # ,)'I%E *I ,)#I FI))I%F)#,E---- %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1 %E)#,)# )I1
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,ra;e &O1$00mm,ra;e &O1$00mm,ra;e &O1$00mm,ra;e &O1$00mm
'for=i minimi principali in corrisponen=a i momento
flettente massimo
,ra;e &O3000mm,ra;e &O3000mm,ra;e &O3000mm,ra;e &O3000mm
'for=i minimi principali in corrisponen=a i momento
flettente massimo
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