dst bk - penny & diana ppt.pptx

PENGARUH PENAHAN LATERAL ELAS-TOMERIC RUBBER

TERHADAP RESPONS SEISMIK MENARA MASJID DENGAN PENAMBAHAN

BASEMEN

Penny Dwiadhiputri 1206218064

DISCRETE SHEAR TRIANGULAR – BATOZ KATILI

A new triangular element with 3 nodes and 3 DOF per node based on the Reissner/Mindlin plate theory. The transverse shear effect are represented using moment equilibrium and constitutive

equations. The formulation based on DKMT element and specially developed to solve problem in DST-BL

element which not passed the patch test for thick plate. That’s why Batoz and Katili represent Bergan’s free formulation of constant bending moment plus

incompatible energy orthogonal higher order bending modes.

FORMULATION OF DST-BK ELEMENT

Geometry of the Triangular Element

Vertical displacement using linear approximation

Rotation displacement using quadratic incompatible approximation

𝑤=∑𝑖=1

3

𝑁 𝑖𝑤 𝑖

𝛽𝑥=∑𝑖=1

3

𝑁 𝑖 𝛽𝑥𝑖+∑𝑘=4

6

𝑃𝑘𝐶𝑘 𝛼𝑘 𝛽 𝑦=∑𝑖=1

3

𝑁 𝑖 𝛽𝑦 𝑖+∑𝑘=4

6

𝑃𝑘 𝑆𝑘𝛼𝑘

⟨ 𝑃𝑘 ,𝜂 ⟩=4 ⟨−(𝜉− 13 )(𝜉− 13 ) ( 𝜆−𝜂 )⟩

The incompatible shape function for rotation

𝜆=1−𝜉 −𝜂

⟨ 𝑃𝑘 , 𝜉 ⟩=4 ⟨ ( 𝜆−𝜉 )(𝜂− 13 )−(𝜂− 13 )⟩

Π𝑏𝑒=12 ⟨𝑢𝑛 ⟩ [𝑘𝑐 ] {𝑢𝑛 }+ 12 ⟨𝛼𝑛 ⟩ [𝑘𝛼 ] {𝛼𝑛}

[𝑘𝑐 ]= 𝐴𝑒 [ 𝐵𝑐 ]𝑇 [ 𝐻𝑏 ] [ 𝐵𝑐 ] [𝑘𝛼 ]=∫𝐴𝑒

❑

[ 𝐵𝛼 ]𝑇 [ 𝐻𝑏 ] [ 𝐵𝛼 ] 𝑑𝐴

[ 𝐻𝑏 ]=𝐷 [ 1 𝑣 0𝑣 1 0

0 0 1−𝑣2 ]

𝐷=𝐸 h3

12(1−𝑣2)

The bending stiffness matrix

𝐴𝑒=12 (𝑥21 𝑦31−𝑥31 𝑦21 )

Π𝑏𝑒=12 ⟨𝑢𝑛 ⟩ [𝑘𝑐 ] {𝑢𝑛 }+ 12 ⟨𝛼𝑛 ⟩ [𝑘𝛼 ] {𝛼𝑛}

The bending stiffness matrix

[ 𝐵𝑐 ]= 12 𝐴𝑒 [0 𝑦23 0 0 𝑦31 0 0 𝑦12 0

0 0 𝑥32 0 0 𝑥13 0 0 𝑥210 𝑥32 𝑦23 0 𝑥13 𝑦 31 0 𝑥21 𝑦12][ 𝐵𝛼 ]= 1

2𝐴𝑒 ¿

[𝑘𝑐 ]= 𝐴𝑒 [ 𝐵𝑐 ]𝑇 [ 𝐻𝑏 ] [ 𝐵𝑐 ] [𝑘𝛼 ]=∫𝐴𝑒

❑

[ 𝐵𝛼 ]𝑇 [ 𝐻𝑏 ] [ 𝐵𝛼 ] 𝑑𝐴

Moment Equilibrium Eq. Linear Constitutive Eq. Kinematical Eq. of Curvature & Shear Strain

Π 𝑠𝑒=12 ⟨𝛼𝑛 ⟩ [𝑘𝑠 𝛼 ] {𝛼𝑛 }

[𝑘𝑠 𝛼 ]=𝐴𝑒 [ 𝐵𝑠𝛼 ]𝑇 [ 𝐻𝑠 ] [ 𝐵𝑠 𝛼 ]

The transverse stiffness matrixTo express shear energy,

[ 𝐵𝑠𝛼 ]=[ 𝐻 𝑠 ]−1 [ 𝐻𝑏 ] [𝑇 2 ] [𝑇 𝛼 ]

[ 𝐻 𝑠 ]=kGh [1 00 1]

[𝑇 2 ]= 1(2 𝐴𝑒 )2 [ [𝑡2 ] 0

0 [𝑡 2 ] ][𝑡 2 ]=[ 𝑦31

2 𝑦212 2 𝑦 31𝑦21

𝑥312 𝑥21

2 2 𝑥31𝑥21− 𝑦31𝑥31 − 𝑦21𝑥21 𝑦 31𝑥21+𝑦 21𝑥31]

[𝑇 𝛼 ]=[−8𝐶4 0 00 0 −8𝐶6

−4𝐶 4 4𝐶5 −4𝐶6

−8𝑆4 0 00 0 −8𝑆6

−4𝑆4 4𝑆5 −8 4]

Discrete constraints

𝛾 𝑠𝑘=𝛾𝑠𝑘= (𝑤, 𝑠 )𝑘+ 𝛽𝑠𝑘;𝑘=4,5,6

𝛾𝑠𝑘= ⟨𝐶𝑘 𝑆𝑘 ⟩ {𝛾𝑠𝑘 }=⟨𝐶𝑘 𝑆𝑘 ⟩ [𝐵𝑠𝛼 ] {𝛼𝑛}𝛽𝑠𝑘=𝐶𝑘 𝛽𝑥 𝑘+𝑆𝑘 𝛽 𝑦 𝑘

k i j m p

4 1 2 5 65 2 3 6 46 3 1 4 5

𝑳𝒌

𝟐 𝜶𝒌+𝑳𝒌

𝟔 (𝑪𝒎𝑪𝒌+𝑺𝒎𝑺𝒌 )𝜶𝒎+𝑳𝒌

𝟔 (𝑪𝒑𝑪𝒌+𝑺𝒑𝑺𝒌 )𝜶𝒑−𝑳𝒌 ⟨𝑪𝒌 𝑺𝒌 ⟩ [𝑩𝒔𝜶 ] {𝜶𝒏 }=𝒘 𝒊−𝒘 𝒋−𝟏𝟐 𝒙 𝒋 𝒊 (𝜷𝒙𝒊+𝜷𝒙𝒋 )−

𝟏𝟐 𝒚 𝒋𝒊 (𝜷𝒚 𝒊+𝜷𝒚 𝒋 )

Assuming a linear variation of

{𝛼𝑛}=[ 𝐴𝑛 ] {𝑢𝑛 }

[ 𝐴𝑛 ]=[ 𝐴𝛼 ]−1 [ 𝐴𝑤 ]

[ 𝐴𝛼 ]= [ 𝐴𝛼 𝑏 ]− [ 𝐴𝛼 𝑠 ]

[ 𝐴𝛼 𝑏 ]=[ 𝐿 ] [𝐶 ] [ 𝐴𝛼 𝑠 ]=[𝐿4 𝐶4 𝐿4𝑆4

𝐿5𝐶5 𝐿5𝑆5

𝐿6𝐶6 𝐿6𝐶6] [𝐵𝑠𝛼 ] ; (𝐿𝑘 𝐶𝑘=𝑥 𝑗𝑖 ;𝐿𝑘 𝑆𝑘=𝑦 𝑗𝑖 )

[ 𝐿 ]=23 [𝐿4 0 00 𝐿5 00 0 𝐿6 ] [𝐶 ]=3

4 [ 1 13 (𝐶 4𝐶5+𝑆4𝑆5 ) 1

3 (𝐶4𝐶6+𝑆4 𝑆6 )

13 (𝐶4𝐶5+𝑆4𝑆5 ) 1 1

3 (𝐶5𝐶6+𝑆5𝑆6 )

13 (𝐶4 𝐶6+𝑆4𝑆6 ) 1

3 (𝐶5𝐶6+𝑆5𝑆6 ) 1 ]

[ 𝐴𝑤 ]=12 [ 2 𝑥12 𝑦12 −2 𝑥12 𝑦12 0 0 00 0 0 2 𝑥23 𝑥23 −2 𝑥23 𝑦23−2 𝑥31 𝑦 31 0 0 0 2 𝑥31 𝑦31 ]

Discrete constraints

[𝑘𝑏 ]=[𝑘𝑐 ]+[ 𝐴𝑛 ]𝑇 [𝑘𝛼 ] [ 𝐴𝑛 ]

Bending stiffness matrix

[𝑘𝑠 ]=[ 𝐴𝑛 ]𝑇 [𝑘𝑠𝛼 ] [ 𝐴𝑛 ]

Shear stiffness matrix

[𝑘 ]= [𝑘𝑏 ]+ [𝑘𝑠 ][𝑘 ]= [𝑘𝑐 ]+ [ 𝐴𝑛 ]𝑇 ( [𝑘𝛼 ]+ [𝑘𝑠𝛼 ] ) [ 𝐴𝑛 ]

DSTBK Stiffness Matrix

Π𝑏𝑒=12 ⟨𝑢𝑛 ⟩ [𝑘𝑏 ] {𝑢𝑛 }

Π 𝑠𝑒=12 ⟨𝑢𝑛 ⟩ [𝑘𝑠 ] {𝑢𝑛 }

NUMERICAL RESULT

Twisted PlateMorley Acute Plate

Razzaque Skew Plate Circular Plate

Square PlateDistributed Load

ClampedMaillage Variation (A & B)

L/h Variation (20, 0, 5, 4)

Simple SupportedMaillage Variation (A & B)


Concentrated LoadClamped

Maillage Variation (A & B)


Simple SupportedMaillage Variation (A & B)


Squared Plate

DataE 10.92fz 1

L/h 1000L 1000h 1v 0.3

Df 1

Boundary ConditionsSimple supported (w = βs = 0)Clamped (w = βs = βn = 0)

Symmetric Conditionsβx = 0 at BC & βy = 0 at CD


Razzaque Skew Plate Circular PlateSquared Plate

Maillage A Maillage BN x N DST BK IDKT DKT DST BK IDKT DKT2 x 2 1.452 1.464 1.547 1.133 1.109 1.2144 x 4 1.309 1.321 1.347 1.229 1.23 1.2586 x 6 1.282 1.291 1.303 1.249 1.249 1.2628 x 8 1.276 --- 1.286 1.256 --- 1.264Ref. 1.265

Distributed Load - Clamped

0 2 4 6 80.8000.8500.9000.9501.0001.0501.1001.1501.2001.250 Wc Distributed Load - Clamped

DST BK (A)IDKT (A)DKT (A)DST BK (B)IDKT (B)DKT (B)

N

Wc/

Wc

ref


Concentrated Load - Clamped

1 2 3 4 5 6 7 80.850

0.900

0.950

1.000

1.050

1.100

1.150

Wc Concentrated Load - Clamped


N

Wc/

Wc

ref




Distributed Load – Simple SupportedMaillage A Maillage B

N x N DST BK IDKT DKT DST BK IDKT DKT2 x 2 11.582 11.405 11.688 11.664 12.01 12.8384 x 4 11.631 11.549 11.635 11.576 11.733 126 x 6 11.623 11.578 11.621 11.582 11.666 11.88 x 8 11.617 --- 11.613 11.588 --- 11.721Ref. 11.601

Concentrated Load – Simple Supported

1 2 3 4 5 6 7 80.800

0.850

0.900

0.950

1.000

1.050

Wc Distributed Load - Simple Supported


N

Wc/

Wc

ref

1 2 3 4 5 6 7 80.9200.9400.9600.9801.0001.0201.0401.0601.0801.1001.120

Wc Concentrated Load - Simple Supported


NW

c/W

c re

f



Distributed Load – Clamped Concentrated Load – Clamped

L/h = 20 L/h = 10 L/h = 5 L/h = 4DST BK 1.334 1.505 2.158 2.639DST BL 1.352 1.548 2.252 2.751

DKT 1.286 1.286 1.286 1.286Yuan & Miller 1.329 1.513 2.203 2.700


DKT 5.64 5.64 5.64 5.64Yuan & Miller 6.257 8.222 15.952 21.694

Distributed Load – Simple Supported Concentrated Load – Simple Supported


DKT 4.063 4.063 4.063 4.063Salermo & Goldberg 4.108 4.242 4.908 5.179


DKT 11.613 11.613 11.613 11.613Yuan & Miller 12.026 13.305 18.446 22.305

Square Plate Concentrated Load (thickness=1)

Number of ElementIf NxN=2x2 , so m=2If NxN=4x4 , so m=4If NxN=6x6 , so m=6If NxN=8x8 , so m=8

Maillage AIf maillage B, so

ClampedIf simply supported, so

Concentrated load

Thickness

Square Plate Distributed Load (thickness=1)

Number of ElementIf NxN=2x2 , so m=2If NxN=4x4 , so m=4If NxN=6x6 , so m=6If NxN=8x8 , so m=8

Maillage AIf maillage B, so

ClampedIf simply supported, so

Thickness

Distributed Load

Square Plate with thickness variationThicknessIf L/h=20, so thickness data = 50 If L/h=10, so thickness data = 100 If L/h=5, so thickness data = 200 If L/h=4, so thickness data = 250

Element 2x2 Element 4x4

Element 6x6Element 8x8

Square PlateMaillage A


Element 6x6Element 8x8

Square PlateMaillage B

Circular Plates Twisted PlateMorley Acute Plate


Circular Plates under Uniform LoadingClamped Hasil dari JurnalClamped R/h 50

NELT Wc Mcr E6 9843.3 2.57 64548.0

24 9854.2 2.23 65322.896 9791.6 2.09 64350.3

240 9777.6 2.06 64115.2exact 9783.5 2.03 64091

0 50 100 150 200 250 3000.99400.9960

0.99801.00001.00201.0040

1.00601.0080

Wc/WcExact Clamped R/h=50

NELT

Wc/

WcE

xact

0 50 100 150 200 250 3000.0000

0.20000.40000.60000.80001.0000

1.20001.4000

Mc/McExact Clamped R/h=50

NELT

Mc/

McE

xact

0 50 100 150 200 250 3000.9900

0.99501.00001.00501.01001.0150

1.02001.0250

E/Eexact Clamped R/h=50

NELT

E/Ee

xact

Clamped R/h 50 NELT Wc Mcr E

6 9843 2.56 6454624 9855.1 2.25 6532996 9802.5 2.09 64456

240 9789.5 2.05 64204Exact 9783.5 2.03 64091




NELT Wc Mcr E6 11.343 2.41 76.032

24 11.547 2.16 80.77696 11.542 2.07 81.178

240 11.539 2.05 81.230exact 11.551 2.03 81.025


6 11.342 2.31 76.0324 11.548 2.16 80.78296 11.554 2.07 81.302

240 11.551 2.04 81.371Exact 11.551 2.03 81.025

0 50 100 150 200 250 3000.9700

0.9750

0.9800

0.9850

0.9900

0.9950

1.0000

1.0050


NELT

Wc/

WcE

xact

0 50 100 150 200 250 3000.9000

0.9500

1.0000

1.0500

1.1000

1.1500

1.2000

1.2500


NELT

Mc/

McE

xact

0 50 100 150 200 250 3000.9000

0.9200

0.9400

0.9600

0.9800

1.0000

1.0200


NELT

E/Ee

xact




NELT Wc Mcr E6 1.289 2.30 9.233

24 1.335 2.15 10.63896 1.339 2.07 10.970

240 1.339 2.05 11.032exact 1.339 2.03 11.103


6 1.289 2.3 9.23324 1.335 2.14 10.63896 1.339 2.07 10.985

240 1.339 2.04 11.056Exact 1.339 2.03 11.103

0 50 100 150 200 250 3000.9400

0.9500

0.9600

0.9700

0.9800

0.9900

1.0000

1.0100


NELT

Wc/

WcE

xact

0 50 100 150 200 250 3000.9500

1.0000

1.0500

1.1000

1.1500


NELT

Mc/

McE

xact

0 50 100 150 200 250 3000.7500

0.8000

0.8500

0.9000

0.9500

1.0000

1.0500


NELT

E/Ee

xact



Circular Plates under Uniform LoadingSimply Supported

Hasil dari JurnalSS R/h 50NELT Wc Mcr E

6 37392 5.43 28772224 39233 5.29 34008396 39633 5.20 353622

240 39700 5.18 356102exact 39831 5.16 359088

0 50 100 150 200 250 3000.9000

0.9200

0.9400

0.9600

0.9800

1.0000

1.0200

Wc/WcExact Simply Supported R/h=50

NELT

Wc/

WcE

xact

0 50 100 150 200 250 3000.97000.98000.99001.00001.01001.02001.03001.04001.05001.0600

Mc/McExact Simply Supported R/h=50

NELT

Mc/

McE

xact

0 50 100 150 200 250 3000.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

E/Eexact Simply Supported R/h=50

NELT

E/Ee

xact

SS R/h 50 NELT Wc Mcr E

6 37391 5.43 28771024 39234 5.28 34009096 39680 5.2 354240

240 39795 5.17 357530Exact 39831 5.16 359088



Circular Plates under Uniform LoadingSS R/h 5

NELT Wc Mcr E6 38.888 5.27 299.19

24 40.926 5.22 355.5496 41.432 5.18 371.09

240 41.557 5.16 374.7Exact 41.599 5.16 374.49

SS R/h 5NELT Wc Mcr E

6 38.889 5.28 299.2024 40.926 5.22 355.5396 41.386 5.18 370.48

240 41.469 5.17 373.29exact 41.599 5.16 374.49

0 50 100 150 200 250 3000.9

0.92

0.94

0.96

0.98

1

1.02


NELT

Wc/

WcE

xact

0 50 100 150 200 250 3000.99

0.995

1

1.005

1.01

1.015

1.02

1.025


NELT

Mc/

McE

xact

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2


NELT

E/Ee

xact

Simply Supported

Hasil dari Jurnal



Circular Plates under Uniform Loading

Mcr=(Mx1+Mx2)/2

SS R/h 2NELT Wc Mcr E

6 3.052 5.16 23.51624 3.215 5.20 28.22396 3.249 5.18 29.486

240 3.254 5.17 29.725exact 3.262 5.16 29.983

SS R/h 2 NELT Wc Mcr E

6 3.052 5.16 23.51524 3.215 5.2 28.22396 3.252 5.18 29.531

240 3.259 5.16 29.847Exact 3.262 5.16 29.983

0 50 100 150 200 250 3000.9000

0.9200

0.9400

0.9600

0.9800

1.0000

1.0200


NELT

Wc/

WcE

xact

0 50 100 150 200 250 3000.9960

0.9980

1.0000

1.0020

1.0040

1.0060

1.0080

1.0100


NELT

Mc/

McE

xact

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2


NELT

E/Ee

xact

Simply Supported

Hasil dari Jurnal



Circular Clamped R/h = 50NELT = 6, so n = 1If NELT = 24, so ubah n = 2 dan boundaryIf NELT = 96, so ubah n = 4If NELT = 216, so ubah n = 6

Boundary untuk NELT = 24



Thickness





Thickness

Circular Simply supportedBoundary untuk NELT = 6

If NELT = 24, boundary :



NELT=6

NELT=96

NELT=24

NELT=216

Circular

Razzaque Skew PlateWc

N 2x2

N 4x4

N 6x6

N 8x8

MyN 2x2

N 4x4

N 6x6

N 8x8



DataE 1085v 0.31h 0.1L 100

Df 0.10fz 1

Boundary ConditionsAB & CD: w = βx = 0



Wc MyN x N DST BK DKT Q4γ DST BK DKT Q4γ2 x 2 6.620 6.389 3.976 94.07 89.53 37.94 x 4 7.556 7.524 6.737 94.47 95.9 77.66 x 6 7.760 7.739 7.371 95.20 95.86 87.18 x 8 7.838 7.819 7.61 95.57 95.93 90.9

12 x 12 7.896 7.896 7.785 95.87 95.98 93.7Ref. 7.945 95.89

Displacement and Moment at Center

0 2 4 6 8 10 120.400

0.500

0.600

0.700

0.800

0.900

1.000

1.100Dispacement at Center, Wc

DST BKDKTQ4y

N

Wc/

Wc

ref

0 2 4 6 8 10 120.200

0.400

0.600

0.800

1.000

1.200Moment at Center, My

DST BKDKTQ4y

NM

y/M

y re

f

Razzaque

NxN = 2x2, so n = 2If NxN = 4x4, so n = 4If NxN = 6x6, so n = 6If NxN = 8x8, so n = 8If NxN = 12x12, so n = 12



Element 12x12

Razzaque



DataE 10.92v 0.3h 0.1L 100fz 1Df 0.001

Morley Acute PlateVariation

N = 4,8,16,32,64Maillage A

L/h =1000

L/h =100

Maillage B

L/h =1000

L/h =100

Boundary ConditionsABCD: w = 0



WcN Maillage A Maillage B4 39896000 460830008 37827000 43223000

16 38499000 4246600032 39113000 4203100064 39518000 41729000

exact 40800000

WcN Maillage A Maillage B4 39709 461458 37561 43294

16 38190 4257632 39107 4225164 40282 42169

exact 40800

0 8 16 24 32 40 48 56 640.80000.85000.90000.95001.00001.05001.10001.1500

Wc - L/h = 1000

Maillage AMaillage B

N

Wc/

Wc

exac

t

0 8 16 24 32 40 48 56 640.8000

0.8500

0.9000

0.9500

1.0000

1.0500

1.1000

1.1500Wc - L/h = 100

Maillage AMaillage B

NW

c /W

c ex

act

Displacement at Center, L/h = 1000 Displacement at Center, L/h = 100

Morley 4x4

NxN = 4x4, so n = 4

Boundary element 4x4

Morley 8x8 NxN = 8x8, so n = 8


Morley 16x16 NxN =16x16, so n = 16


Morley 32x32 NxN =32x32, so n = 32




Morley

Element 64x64



Twisted PlateWc

N = 2

N = 4

N = 6

N = 8

N = 16

N = 32

W3N = 2

N = 4

N = 6

N = 8

N = 16

N = 32

DataE 15.6L 10v 0.3h 2.5



Wc W3N x N DST BK DST BL DKT DST BK DST BL DKT2 x 2 1.894 1.71 1.6 7.740 6.84 6.44 x 4 1.927 1.826 1.6 7.773 7.46 6.46 x 6 1.967 1.955 1.6 7.902 8.108 6.48 x 8 2.073 2.236 1.6 8.297 8.583 6.4

12 x 12 2.145 2.331 1.6 8.583 8.935 6.4Ref. 1.6 6.4

Displacement at Center and Nodal 3

0 2 4 6 8 10 120.800

0.900

1.000

1.100

1.200

1.300

1.400

1.500Displacement at the Center, Wc

DST BKDST BLDKT

N

Wc/

Wc

ref

0 2 4 6 8 10 120.800

0.900

1.000

1.100

1.200

1.300

1.400

1.500Displacement at nodal 3, W3

DST BKDST BLDKT

NW

3/W

3 re

f

Twisting element 2x2NxN = 2x2, so n = 2If NxN = 4x4, so n = 4If NxN = 8x8, so n = 8If NxN = 16x16, so n = 16If NxN = 32x32, so n = 32


Twisting element 4x4

NxN = 4x4, so n=4



NxN = 6x6, so n=6



NxN = 8x8, so n=8



NxN =16x16, so n=16



NxN =32x32, so n=32


Boundary condition:w = 0

Constant Curvature Past Test

DataE 1000v 0.3k 5/6

• Batoz, J.L. dan Katili, I. (1992). On a Simple Triangular Reissner/Mindlin Plate Element Based on Incompatible Modes and Discrete Constraints. International Journal for Numerical Methods in Engineering, 35, 1603-1632.

• Katili, I., Formulation et évaluation de nouveaux éléments finis pour l'analyse linéaire des plaques et coques de forme quelconque, Université de Technologie de Compiègne, France, 1993.

• Katili, I. (1993). A New Discrete Kirchhoff-Mindlin Element Based on Mindlin-Reissner Plate Theory and Assumed Shear Strain Fields-Part I: An Extended DKT Element for Thick-Plate Bending Analysis. International Journal for Numerical Methods in Engineering, 36, 1859-1883.

• Katili, I., Metode Elemen Hingga untuk Pelat Lentur, Penerbit Universitas Indonesia (UI-Press), Jakarta, 2004.

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