THE MODELING OF THE THE MODELING OF THE LIMIT STATE OF DUCTILE LIMIT STATE OF DUCTILE

THICK-WALLED PIPES THICK-WALLED PIPES WITH AXIAL SURFACE WITH AXIAL SURFACE

DEFECTSDEFECTS

Orynyak I.V., Ageyev S.M.Orynyak I.V., Ageyev S.M.

G.S. Pisarenko Institute for Problems G.S. Pisarenko Institute for Problems ofof Strength, Kiev, UkraineStrength, Kiev, Ukraine

PlanPlan1.1. Existing models for pipes with defects.Existing models for pipes with defects.2.2. The problems of modeling of the thick-The problems of modeling of the thick-

walled pipes. walled pipes. 3.3. The proposed analytical model for The proposed analytical model for

thick-walled pipes. thick-walled pipes. 4.4. The theoretical analysis of the results. The theoretical analysis of the results. 5.5. The comparison with experimental The comparison with experimental

data.data.6.6. Discussion.Discussion.7.7. Application to a repair technology.Application to a repair technology.

Existing models for pipes with Existing models for pipes with defectsdefects

C – crack half-lengthℓ – crack half-width a – crack depth

- «local» formula of the Battelle Memorial Institute

- dimensionless ligament thickness

- dimensionless crack length

- dimensionless limit pressure or strength reduction coefficient

- «global» formula of the Battelle Memorial Institute

- «global» formula Staat

- «local» formula Staat

- formula DNV

The problems of modeling of The problems of modeling of the thick-walled pipesthe thick-walled pipes 1. The choice of limit 1. The choice of limit

characteristic.characteristic.2. The choice of criterion of 2. The choice of criterion of ductile failure. ductile failure. 3. The irregularity of defect’s 3. The irregularity of defect’s form. form. 4. Interaction of closely 4. Interaction of closely situated defects. situated defects. 5. The taking into account the 5. The taking into account the wall thickness. wall thickness. 6. External/internal defects. 6. External/internal defects.

The choice of limit The choice of limit characteristiccharacteristic

The choice of criterion of ductile failure The choice of criterion of ductile failure ((for for unflawed thick-walled pipesunflawed thick-walled pipes).).

№mm mm

Specimen orientation MPa

MPa MPa MPa MPa

1 88,9 4,0 longitudinal 336 486 42,7-47,0 45,8 52,9

2 88,9 8,8 longitudinal 324 457 94,2-100,6 100,8 116,4

3 88,9 22,2 longitudinal 288 438 307,1 303,1 350,0

4 101,6 10,0 longitudinal 284 408 97,5 89,9 103,3

5 101,6 10,0 transverse 390 457 100,2 115,7

6 139,7 12,5 longitudinal 266 400 73,5-76,0 78,9 91,1

7 139,7 12,5 transverse 338 432 85,2 98,4

8 88,9 4,0 longitudinal 512 642 57,9-61,8 60,5 69,93

9 88,9 8,8 longitudinal 506 634 135,4-170,7 139,9 161,5

10 88,9 22,2 longitudinal 473 614 416,9-421,8 424,9 490,6

11 101,6 10,0 longitudinal 689 740 183-175 162,2 187,3

12 101,6 10,0 transverse 717 759 166,4 192,1

13 139,7 12,5 longitudinal 648 702 152,0 138,4 159,8

14 139,7 12,5 transverse 668 719 141,8 163,7

The proposed analytical model for thick-The proposed analytical model for thick-walled pipes.walled pipes.

Analytical model of the Institute for problems Analytical model of the Institute for problems of strength (for thin-walled pipes)of strength (for thin-walled pipes)

- the equation of forces equilibrium in the radial direction

- the circumferential force;х – axial coordinate;

- transverse force

dimensionless limit pressure for the thin-walledpipe with axial surface defect

- limit condition

- external

- internal

External/internal defects

the limit bending moment

The taking into account the wall The taking into account the wall thicknessthickness

- local equation of equilibrium for thin-walled pipe

- the solution of differential equation

The pipe with external defect The pipe with internal defect

The theoretical analyze of the resultsThe theoretical analyze of the resultsThe dimensionless limit pressure versus dimensionless crack length for the model of thick-walled pipes with external/internal defects.

5,0

5,0The comparison of analytical models for thin-walled pipe:

and for the thick-walled pipe:

The comparison with experimental data (Staat’s data)

№

1 3,105 0,8 0,865 0,85 0,851 0,014 -0,001 0,851 0,826 0 -0,025

2 1,769 0,5 0,681 0,639 0,744 -0,063 -0,105 0,704 0,598 -0,04 -0,146

3 3,656 0,5 0,566 0,549 0,583 -0,017 -0,034 0,605 0,533 0,022 -0,05

4 9,63 0,5 0,52 0,518 0,502 0,018 0,016 0,541 0,497 0,039 -0,005

5 4,009 0,35 0,408 0,393 0,417 -0,009 -0,024 0,473 0,377 0,096 -0,04

№

1 3,642 0,48 0,564 0,546 0,633 -0,069 -0,087 0,585 0,476 -0,048 -0,157

2 6,549 0,48 0,525 0,518 0,506 0,019 0,012 0,551 0,451 0,045 -0,055

3 15,018

0,49 0,52 0,519 0,491 0,029 0,028 0,536 0,445 0,045 -0,046

4 1,044 0,284 0,693 0,622 0,705 -0,012 -0,083 0,689 0,424 -0,016 -0,281

5 2,456 0,284 0,433 0,396 0,477 -0,044 -0,081 0,495 0,309 0,018 -0,168

6 3,783 0,31 0,389 0,371 0,424 -0,035 -0,053 0,45 0,306 0,026 -0,118

7 6,69 0,284 0,327 0,321 0,35 -0,023 -0,029 0,378 0,263 0,028 -0,087

8 6,775 0,3 0,338 0,332 0,311 0,027 0,021 0,387 0,274 0,076 -0,037

9 15,159

0,284 0,311 0,31 0,302 0,009 0,008 0,339 0,251 0,037 -0,051

10 15,244

0,24 0,263 0,262 0,281 -0,018 -0,019 0,294 0,209 0,013 -0,072

11 1,411 0,091 0,415 0,337 0,584 -0,169 -0,247 0,527 0,133 -0,057 -0,451

12 1,411 0,034 0,362 0,249 0,574 -0,212 -0,295 0,502 0,051 -0,072 -0,523

inte

rnal

exte

rnal

1. The comparison our models with experimental 1. The comparison our models with experimental data for the pipe with internal defect.data for the pipe with internal defect.

- “external” formula- “external” formula

- “internal” formula- “internal” formula

3. The comparison our models with Staat’s “global” formula.3. The comparison our models with Staat’s “global” formula.

2. The comparison our models with Staat’s “local” formula.2. The comparison our models with Staat’s “local” formula.

internal external

internal external

Discussion

1 2

№

11 1,186 0,091 0,492 (0,415

)

0,404 (0,337

)

0,584

12 1,172 0,034 0,448 (0,362

)

0,355 (0,279

)

0,574

external (Staat)

5,0

3,0

Influence of the form of the defects

Application

0,4

0,5

0,6

0,7

0,8

0,9

1

0 1 2 3 4 5 6

dimensionless crack length

stre

ngth

red

uctio

n co

effi

cien

t

- the pipe without defect

- the pipe with defect

- the pipe with sleeve

- experiment

the pipe with defect

the pipe with sleeve

5,0

mm

mm

9

7,25,0

- pipe’s geometry- defect’s geometry

- the added thick pipe’s wall as a result of used sleeve (equal 8 MPa)

1.

2.

3. Numerical analyze and simplify analytical model

4.

5.