slide10 n2 rocking nl - epflcivil 706 – n2, rocking, check nltha edce-epfl-enac-sgc 2016 -2-...

CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -1-

EDCE: Civil and Environmental Engineering CIVIL 706 - Advanced Earthquake Engineering

N2 method and Rocking behaviour NL Time History Analysis check


Content

•  Equal displacement rule

•  Target displacement, EC 8 part 1, Annex B

•  Approximate graphical assessment

•  N2 displacement demand prediction accuracy

•  Rocking (S-model and Griffith model)

•  Fast assessment for cantilever out-of-plane

•  Amplification with floor response spectrum


Equal displacement rule

•  Favorable effect, behaviour factor q •  Displacement ductility = behaviour factor value •  Avoid brittle failures (shear failures)

displacement

elastic

xe = xel / q xp = xel

Fy = Fel / q

Fel

plastic

force

Empirical rule: equal displacements


Equal displacement rule, check NLTHA •  164 recorded earthquakes, methodology


Equal displacement rule, check NLTHA •  164 recorded earthquakes, results


EC 8 P1, Annex B : Target displacement •  Long period, equal displacement rule


EC 8 P1, Annex B : Target displacement •  Short period, relative larger displacement


Determination of target displacement •  Alternative procedure used in USA, NZ, etc.

0.0

1.0

2.0

3.0

4.0

5.0

0.00 0.02 0.04 0.06 0.08 0.10

Sd [m]

S a [

m/s

2 ]

Performance point Sd = 35 mm

0.0

1.0

2.0

3.0

4.0

5.0

0.00 0.05 0.10 0.15 0.20 Sd

Sa

Damping according to « Substitute structure approach » ( βéquivalent , ksecant )

F

x

β=5%

β=10%

β=20%


EC 8 P1, Annex B : Target displacement •  Short period, used assumption (Tlim=TC)


EC 8 P1, Annex B : Target displacement •  Short period, relative larger displacement


EC 8 P1, Annex B : Target displacement •  Short period, graphical approximation


Parameters for EC8 (SIA 261) soil classes

classe description S TB[s] TC[s] TD[s]

A massifs rocheux 1,00 0,15 0,4 2,0

B gravier ou sable très compact 1,20 0,15 0,5 2,0

C compact ou semi-compact 1,15 0,20 0,6 2,0

D lâche à semi-compact 1,35 0,20 0,8 2,0

E couche C ou D sur A ou B 1,40 0,15 0,5 2,0

F sensible, organique - - - -


Elastic response spectra in EC 8 (SIA 261)

0

1

2

3

4

0.01 0.1 1 10période T[s]

Se/

a gd

classe de sol A

classe de sol B

classe de sol C

classe de sol D

classe de sol E


Displacement demand prediction accuracy •  Sets of 12 recorded TH, Abrahamson matching

10-2 10-1 100 1010

1

2

3

4

5Sa

[m/s

2]SIA 261 soil class A, before modification

10-2 10-1 100 1010

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class A, after modification

10-2 10-1 100 1010

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class B, before modification

10-2 10-1 100 1010

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class B, after modification

10-2 10-1 100 101

period [s]

0

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class C, before modification

10-2 10-1 100 101

period [s]

0

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class C, after modification


0 0.1 0.2 0.3 0.4 0.5 0.60

0.005

0.01

0.015di

spl.

dem

and

[m]

12 modified ESMD-earthquakes for soil class A, R=2

Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.60

0.005

0.01

0.015

disp

l. de

man

d [m

]

Takeda model (alpha=0.4, beta=0.0), R=3

Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6period [s]

0

0.005

0.01

0.015

disp

l. de

man

d [m

]

strength reduction factor R=4

Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6period [s]

0

0.005

0.01

0.015

disp

l. de

man

d [m

]


Prop1

Prop2

EC8

Displacement demand prediction accuracy


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.005

0.01

0.015

0.02

0.025

0.03di

spl.

dem

and

[m]

12 modified ESMD-earthquakes for soil class B, R=2

Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.005

0.01

0.015

0.02

0.025

0.03

disp

l. de

man

d [m

]


Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7period [s]

0

0.005

0.01

0.015

0.02

0.025

0.03

disp

l. de

man

d [m

]


Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7period [s]

0

0.005

0.01

0.015

0.02

0.025

0.03

disp

l. de

man

d [m

]


Prop1

Prop2

EC8



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04di

spl.

dem

and

[m]

12 modified ESMD-earthquakes for soil class C, R=2

Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

disp

l. de

man

d [m

]


Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8period [s]

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

disp

l. de

man

d [m

]


Prop1

Prop2

EC8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8period [s]

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

disp

l. de

man

d [m

]


Prop1

Prop2

EC8



0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08di

spl.

dem

and

[m]

12 modified ESMD-earthquakes for soil class D, R=2

Prop1

Prop2

EC8

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

disp

l. de

man

d [m

]


Prop1

Prop2EC8

0 0.2 0.4 0.6 0.8 1period [s]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

disp

l. de

man

d [m

]


Prop1

Prop2

EC8

0 0.2 0.4 0.6 0.8 1period [s]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

disp

l. de

man

d [m

]


Prop1

Prop2

EC8



Site effect

Earthquake =

source * propagation * site

Site effect due to the reflexions of seismic waves in the "soft » soil deposits


Microzonation : H/V measurements


Microzonation : SASW measurements


Microzonation : Yverdon

0

1

2

3

4

0.01 0.1 1 10période T[s]

Se [

m/s

2 ]

Yverdon, S1

Yverdon, S2

Yverdon, S3

Yverdon, S4


Microzonation : Yverdon, response spectra

0

1

2

3

4

0.01 0.1 1 10période T[s]

Se [

m/s

2 ]Yverdon, S1

Yverdon, S2

Yverdon, S3

Yverdon, S4


•  ADRS format

0

1

2

3

4

0 0.02 0.04 0.06 0.08 0.1

Sd [m]

Se [

m/s

2 ]

soil class A, zone 2

soil class E, zone 2

Yverdon, Zone S1

Yverdon, Zone S2

Yverdon, Zone S3

Yverdon, Zone S4



•  Zone S1 : are usual procedures valid ?

0

1

2

3

4

0 0.01 0.02 0.03 0.04 0.05 0.06

Sd [m]

Se [

m/s

2 ]

soil class A, zone 2

Yverdon, Zone S1



Conclusions – N2 method

•  Equal displacement rule from Tlim=TC

•  Target displacement in Part 1, Annex B

•  Approximate graphical assessment

•  Accuracy (R=3 ✔, R<3 !, R>3 ")

•  Procedure also valid for special response spectra from microzonation studies


Rocking behaviour, check NLTHA •  164 recorded earthquakes, methodology

duct

ility

dem

and

frequency

non-linear behavior

164 recorded earthquakes

acce

lera

tion

time

disp

lace

men

t

time

non-linear dynamic response

12 MDOF

MDOF

13 f09 Rfo

rce

displacement

4 hysteretic models

SDOF

disp

lace

men

t

time

non-linear dynamic response

statistical

analysis

statistical

analysis

4 R


Rocking behaviour, check NLTHA •  Hysteretic models

force

S-modelforce

displacement displacement

modified Takeda-modelelastoplastic (EP-model)

Flag-model


Rocking behaviour, check NLTHA •  164 recorded earthquakes, results

initial frequency [Hz]

disp

lace

men

t duc

tility

dem

and

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

123456789

10S-model

initial frequency [Hz]di

spla

cem

ent d

uctil

ity d

eman

d

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

123456789

10modified Takeda-model

R = 4

R = 3.5

R = 3

R = 2.5

R = 2


Rocking behaviour, check NLTHA •  164 recorded earthquakes, results

frequency [Hz]

disp

lace

men

t duc

tility

dem

and

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

123456789

10S-model

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00123456789

10Flag-model

frequency [Hz]di

spla

cem

ent d

uctil

ity d

eman

d R = 4

R = 3.5

R = 3

R = 2.5

R = 2


Rocking behaviour, check NLTHA •  Slightly modified R – µΔ relationships

frequency [Hz]

disp

lace

men

t duc

tility

dem

and

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

5

6

7S-model

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

7Flag-model

frequency [Hz]di

spla

cem

ent d

uctil

ity d

eman

d R = 4

R = 3.5

R = 3

R = 2.5

R = 2

= 4 /3 ·R - 1 /3 = 3 /2 ·R - 1 /2


Masonry out-of-plane behaviour •  Griffith : rigid body behaviour UNREINFORCED MASONRY WALLS 837

pivot

2/3 h

∆e = 2/3 t

F0

Mgt/2 pivots

pivotInertia force distribution

F0/2Mg/2

R=F0/2-Mgt/2h

h/6

∆e = 2/3 t

F=0

∆e = 0 ∆e = 0

F=0

Inertia forcedistribution

R’=F0/2+Mgt/2h

R’=F0

F0/2

(a) Parapet Wall at incipient Rockingand Point of Instability

(b) Simply-Supported Wall at Incipient Rockingand Point of Instability

Figure 3. Inertia forces and reactions on rigid URM walls.

A similar expression, Equation (4), also derived using standard modal analysis procedures,is used to de!ne the e"ective displacement (#e).

#e =!n

i=1 mi!2i

!ni=1 mi!i

(4)

It can be shown from Equation (4) that

#e = 2=3#t (for a parapet wall) and (5a)

#e = 2=3#m (for simply-supported wall) (5b)

where #t and #m are the top of wall and mid-height wall displacements, respectively.Note that both Equations (3) and (5) are based on the assumption of a triangular-shaped

relative displacement pro!le. This can be justi!ed for a rocking wall where the displacementsdue to rocking far exceed the imposed support displacements. The accuracy of this assumptionhas been veri!ed with shaking table tests and THA as described in Reference [12]. Thus, theresultant inertia force is applied at two-thirds of the height of a parapet wall, and one-third ofthe upper half of the simply supported wall measured from its mid-point (Figures 3(a) and3(b)).

Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:833–850


Griffith model : rigid body behaviour •  Force – displacement relationships 838 K. DOHERTY ET AL.

Forc

e

∆ f=2/3 t

F0=M egt/h F0= 4M egt/hKo

KoF0= 4(1+ψ)M egt/h

loadbearing

Non-loadbearing

Ψ=overburden weight/(Mg/2)

Displacement

Forc

e

Displacement

−∆ f −∆ f∆ f=2/3 t

(a) Parapet Wall (b) Simply Supported Wall

Figure 4. Force–displacement relationships of rigid URM walls.

3. MODELLING OF CRACKED UNREINFORCED MASONRY WALLSAS RIGID BLOCKS

The spring force function F(!e) can be obtained by determining the total horizontal reaction(or base shear) at di"erent displacements using basic principles of static equilibrium. Forexample, the overturning equilibrium of a parapet wall about the pivot point at the base ofthe wall can be used to determine F(!e).For a parapet wall at the point of incipient rocking (i.e. !e = 0+ or alternatively !t = 0+),

moment equilibrium leads to (refer Figure 3(a)) the expression:

Mgt=2=F0(2=3)h (6a)

Solving for F0 (F at !e = 0+) and substituting Equation (3) into Equation (6a) gives

F0 =Me(gt=h) (6b)

For a parapet wall at the point of instability (!e = 2=3t or alternatively !t = t), the force Frequired for static equilibrium of the wall is given by

F =0 (6c)

Therefore, the F(!e) function for a parapet wall can be constructed in accordance withEquations (6b) and (6c) as shown in Figure 4(a).Similarly, moment equilibrium can also be used to determine F(!e) at the point of incipient

rocking (!e = 0+) for a wall simply supported at the top and bottom. By considering momentequilibrium of the upper half of a simply supported wall (of height = h=2 and mass=M=2)about the pivot point in the cracked cross-section at the mid-height of the wall leads to

(Mg=2)t=2=R(h=2)− (F0=2)(h=6) (7a)



Griffith model : rigid body behaviour •  Tri-linear hysteretic model 842 K. DOHERTY ET AL.

Forc

e

Rigid body(bi-linear model)

Experimentalnon-linear

Tri-linear model

F0

∆1 ∆2 ∆fDisplacement

Note : Only the positivedisplacement range isshown

Figure 6. Force–displacement relationship of deformable URM walls.

value of ! for parapet walls to be in the order of 3 per cent using this technique. The viscousdamping factor can also be calculated from dynamic equilibrium as the net di!erence betweenthe experimentally determined inertia force and the restoring force (according to the recordedacceleration and displacement, respectively) at any instant of time during the rocking response.Subsequent free-vibration experiments carried out on a range of simply supported walls [12]indicated that damping ratios were of a similar order. This critical damping ratio can betranslated into a viscous damping factor using the following equation to carry out non-linearTHA:

C = 2!!Me = 4"f!Me (10)

where ! is the angular velocity of the linearized system. Further details considering thefrequency dependence (and hence amplitude dependence) is provided in Reference [12].

4. MODELLING OF CRACKED UNREINFORCED MASONRY WALLSAS DEFORMABLE (SEMI-RIGID) BLOCKS

The bilinear force–displacement relationship described in the previous section is based onthe assumption that URM walls behave essentially as rigid bodies which rock about pivotpoints positioned at cracks. It has been con"rmed by experimental static push-over tests thatthe individual blocks of the URM wall can deform signi"cantly when subjected to highpre-compression. This results in: (i) pivot points possessing "nite dimensions (rather thanbeing in"nitesimally small) so that the resistance to rocking is associated with a lever armsigni"cantly less than half the wall thickness (as for a rigid wall) and (ii) the wall possessing"nite lateral sti!ness (rather than being rigid) prior to incipient rocking. Importantly, thethreshold resistance to rocking is reduced signi"cantly from the original level associated witha rigid wall, to a ‘force plateau’ as shown in Figure 6. It can be further seen from Figure 6that the F–# relationship observed during the experiment deviates signi"cantly from thisbilinear relationship and assumes a curvilinear pro"le. This is largely due to the non-linear


UNREINFORCED MASONRY WALLS 843

Table II. Empirically derived trilinear F–! de"ning displacements.

State of degradation at cracked joint !1=!f !2=!f

New 6% 28%Moderate 13% 40%Severe 20% 50%

deformations that occur in the mortar joint. However, there is relatively little deviation fromthe original bilinear model at large displacements.This curvilinear pro"le can be idealized by a trilinear model that is de"ned by three dis-

placement parameters: !e;1, !e;2 and !e;max and the force parameter F0 (refer Figure 6). Toconstruct the trilinear model, the bilinear model is "rst constructed in accordance with F0and K0. The amplitude of the force plateau is, therefore, controlled by the ratio !2=!f . Fordisplacements in the range exceeding !2, the trilinear and the bilinear models coincide. Fordisplacements between !1 and !2, the force is constant. The initial slope of the trilinearmodel is governed by the force amplitude of the plateau and the value of !1.The ratios !1=!f and !2=!f are related to the material properties and the state of degra-

dation of the mortar joints at the pivot points. Data recorded during many quasi-static anddynamic tests of 14 simply supported walls suggests nominal values for the ratios of !1=!fand !2=!f for walls in ‘new’, ‘moderately degraded’ and ‘severely degraded’ condition asshown in Table II. The interpretation of the ‘moderately degraded’ and ‘severely degraded’conditions are highly subjective. From the experimental tests, the e#ective width of the mortarin the cracked bedjoint for walls classi"ed as severely degraded was approximately 90 percent of the original width. Moderately degraded walls had e#ective bedjoint widths that wereessentially equal to their original widths. However, the exposed vertical faces of the mortarjoints had rounded due to some rocking having taken place. Full details of these tests aregiven in Reference [13]. This trilinear F–! relationship proved to be e#ective for the wallstested in this study over the full range of degradation.The traditional method of selecting a secant sti#ness for use with a substitute structure rep-

resentation of a multi-degree-of-freedom system is not straightforward for non-ductile systemssuch as URM. One method commonly used is to adopt the secant sti#ness from the system’snon-linear force–displacement curve corresponding to the point of maximum (permissible)displacement. For ductile systems, this is often associated with a point on the post-peak soft-ening section of the non-linear force–displacement curve where the force has reduced to somefraction (75–80 per cent is common) of the peak force value. In this study, and for masonryin general, it was not simple to de"ne this point due to material strength variability and a lackof de"nitive yield and=or softening points. However, it was observed that the sti#ness corre-sponding to a line going through the point on the trilinear force–displacement curve where!=!2 as shown in Figure 7 was reasonably consistent with this notion.The e#ective secant sti#ness, Ks-e# , for the semi-rigid wall obtained in this manner can be

expressed mathematically in generic terms as

Ks-e# = K0!

1− 1!2=!1

"

(11)



Out-of-Plane : rigid body behaviourSeveral stability conditions Static conditions: rocking onset

italian code Overturning : Betbeder-Matibet

agd ≤α ⋅ g

a0* = PGA 2 ≤α ⋅ g

V ≈agd10

≤12⋅α ⋅ g ⋅ r

⇒ agd ≤ 5 ⋅twhw⋅ g ⋅ r


Casestudy,defini.onsCantilever non bearing wall Dimensions: hw = 3 m (3rd storey)

tw = 0.1 m - 0.4 m Seismic: Z2

CO I soil CSC

Building: T1 = 0.3 s

T1 = 1.2 s T1 = 1.0 s T1 = 1.67 s

844 K. DOHERTY ET AL.

∆f

F0

∆1

(∆2-∆f)Ko

Ks-eff= Ko (∆2-∆f) / ∆2

∆2

Note : Only the positivedisplacement range isshown

Figure 7. E!ective secant sti!ness (Ks-e! ) of semi-rigid walls.

where K0 is de"ned as shown in Figures 5 and 6 and values for #2=#1 are given byTable II. The e!ective undamped natural frequency, fs-e! , for the equivalent SDOF system isaccordingly given by the following equation:

fs-e! =!

Ks-avg=Me

2!(12)

The experimentally observed ‘resonance’ frequency for each of the test walls was found toagree well with estimates given by Equation (12) using e!ective secant sti!ness values asde"ned above. (Note: using the approach of Section 3 where the e!ective sti!ness was takenas the slope of the line going through the centroid of the area under the force–displacementcurve gives similar results.)

5. DISPLACEMENT DEMAND PREDICTION BY SUBSTITUTE STRUCTUREIDEALIZATION

The DB analysis methodology provides a rational means for determining seismic design ac-tions as an alternative to the more traditional ‘quasi-static’ force-based approach. In the DBmethod, the dynamic lateral displacement capacity of a structure, subjected to an excitationis determined based on a comparison of the displacement demand imposed on the structureduring a seismic event with a pre-determined critical displacement capacity. The ‘substitutestructure’ methodology proposed by Shibata and Sozen [14] was adopted to simplify highlynon-linear systems into a linearized DB procedure. An elastic SDOF oscillator is selected withlinear properties that characterize those of the real non-linear structure. The e!ectiveness ofthe linearized DB procedure is reliant on the assumption that both the ‘substitute structure’and real system will reach the same critical displacement under the same excitation.It was observed from the parametric studies using Guassian Pulses (described in Section 3)

that incipient instability most likely occurs as a consequence of the large displacement am-pli"cations associated with resonance of the wall. The e!ective resonant frequency, fe! , isrelated to a particular e!ective secant sti!ness. It appears that the displacement demand ofURM walls arising from rocking can be predicted using the linearized DB analysis procedure



Griffith model : rigid body behaviour •  Sets of 12 recorded TH, methodology


Abrahamson matching for 12 TH – EC8/SIA

10-2 10-1 100 1010

1

2

3

4

5Sa

[m/s

2]SIA 261 soil class A, before modification

10-2 10-1 100 1010

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class A, after modification

10-2 10-1 100 1010

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class B, before modification

10-2 10-1 100 1010

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class B, after modification

10-2 10-1 100 101

period [s]

0

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class C, before modification

10-2 10-1 100 101

period [s]

0

1

2

3

4

5

Sa [m

/s2]

SIA 261 soil class C, after modification


Non linear time history analysis •  Examples


Non linear time history analysis •  Determination using incremental analysis


Non linear time history analysis •  Ultimate increasing value : results assessment


Non linear time history analysis

tw[m]

tw / Hw

[-]α g

[m/s2]2 α g[m/s2]

NL (mean 12 TH > tw/2)ζ = 1% - 3% - 5% [m/s2]

NL (mean 12 TH > tw)ζ = 1% - 3% - 5% [m/s2]

0.10 0.033 0.3 0.7 0.29 – 0.38 – 0.43 0.35 – 0.51 – 0.610.12 0.040 0.4 0.8 0.35 – 0.46 – 0.52 0.40 – 0.61 – 0.730.15 0.050 0.5 1.0 0.44 – 0.57 – 0.65 0.50 – 0.73 – 0.910.18 0.060 0.6 1.2 0.53 – 0.68 – 0.78 0.60 – 0.93 – 1.090.20 0.067 0.7 1.3 0.58 – 0.76 – 0.86 0.67 – 1.02 – 1.210.25 0.083 0.8 1.6 0.73 – 0.94 – 1.08 0.81 – 1.27 – 1.510.30 0.100 1.0 2.0 0.87 – 1.13 – 1.29 1.00 – 1.46 – 1.810.40 0.133 1.3 2.6 1.16 – 1.51 – 1.72 1.33 – 1.95 – 2.42

•  Aggregate results for different wall thicknesses


Amplification with floor response spectra •  Methodology


•  Limit case : stiff building

Amplification with floor response spectra


•  Limit case : flexible building



T1 = 1.2 s T1 = 1.67 s



Conclusions – Rocking (S and Griffith)

•  Slight modifications of R – µΔ relationships for S-model and flag-model

•  Onset rocking condition on the safe side for cantilever walls with rigid body behaviour

•  Approach also valid for walls in upper stories by taking into account adequate floor response spectra

slide10 n2 rocking nl - epflcivil 706 – n2, rocking, check nltha edce-epfl-enac-sgc 2016 -2-...

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