limit load calculation model of ductile failure of defective pipe and pressure vessel orynyak i.v....

LIMIT LOAD CALCULATION LIMIT LOAD CALCULATION MODEL OF DUCTILE MODEL OF DUCTILE

FAILURE OF DEFECTIVE FAILURE OF DEFECTIVE PIPE AND PRESSURE PIPE AND PRESSURE

VESSELVESSEL

Orynyak I.V.Orynyak I.V. G.S. Pisarenko Institute for Problems of G.S. Pisarenko Institute for Problems of Strength, National Academy of Sciences Strength, National Academy of Sciences of Ukraine, Kyiv, Ukraineof Ukraine, Kyiv, Ukraine

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1th Hungarian-Ukrainian Joint Conference on

Safety-Reliability and Risk Engineering Plants and Components

11,12 April 2006, Miskolc, Hungary

ICI

I

KK

aK

a

K ICCR

a

0lim

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2W2a

Application of Limit Load in Fracture Application of Limit Load in Fracture MechanicsMechanics

Thus another restriction (criterion) is needed Thus another restriction (criterion) is needed

rS 1rS

1rK

1

1

M

rK

DT

BT

1O

u

y

Wa

CR 1][

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Examples of Limit Load Calculation Examples of Limit Load Calculation

PP

RoRo

RiRi

RtRt

PRR

P

RPR

rdrd

ii

o

orir

r

rr

if ][ ][or ln][][

boundary 0)( ;)(

criterion ][

mequilibriu 0

NNNN

MM MM

t

2N

M

ttt

tN

NtN ][

Mt

t

u

u

1][][

2

N

N

M

Mabs

4/][][ 2tM tN ][][

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The Existing Formulas for Defected Pipes The Existing Formulas for Defected Pipes

Battelle’s local 1Battelle’s local 1)(/)1(1),(

Mca

t

RpLL

f

LL

)(

1),(

11

Mt

Rp GLL

f

LL

)(/157.0exp11),(1 atRc

t

aca

t

RpLL

u

LL

5.02 )61.11()( M

c – defect halflength

l – defect halfwidth

a – defect depth = 1-a/t - dimensionless

ligament thickness

= c / Rt - dimensionless

length1 = c / Ra - dimensionless length

c – defect halflength

l – defect halfwidth

a – defect depth = 1-a/t - dimensionless

ligament thickness

= c / Rt - dimensionless

length1 = c / Ra - dimensionless length

Battelle’s local 2Battelle’s local 2

Nuclear Electric globalNuclear Electric global

- strength reduction cofficient - strength reduction cofficient

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Strength Characteristic of MaterialStrength Characteristic of Material

Illustrating the viability of the UTS as a failure criterion for plastic collapse in a steel line pipe Illustrating the viability of the UTS as a failure criterion for plastic collapse in a steel line pipe

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Analytical Modeling of Ductile Failure Analytical Modeling of Ductile Failure

Static limit load theorem (lower bound) of the theory of plasticity:

an elastic–plastic structure will not collapse under monotone loads if

a) it is in static equilibrium and

b) the yield function is nowhere violated

Static limit load theorem (lower bound) of the theory of plasticity:

an elastic–plastic structure will not collapse under monotone loads if

a) it is in static equilibrium and

b) the yield function is nowhere violated Corollaries of the theorem :

1. Accounting for any additional bond in the body can only increase the limit load

2. Decreasing the volume of the body cannot increase the limit load

3. The solution which gives maximal value of limit load is more appropriate

Corollaries of the theorem :

1. Accounting for any additional bond in the body can only increase the limit load

2. Decreasing the volume of the body cannot increase the limit load

3. The solution which gives maximal value of limit load is more appropriate

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AXIAL SURFACE CRACK IN A CYLINDERAXIAL SURFACE CRACK IN A CYLINDER

1. Assumptions:

a) shear force is equal to zero

b) shearing moments is equal to zero

c) axial force is proportional to the net-section

1. Assumptions:

a) shear force is equal to zero

b) shearing moments is equal to zero

c) axial force is proportional to the net-section

0L0xM

)(xtN u

2. Equations of equilibrium 2. Equations of equilibrium

xxx Q

dxdM

dxdQ

R

Np

3. The Gist of the models3. The Gist of the models

The The presence of an axial flaw causes imbalance between the presence of an axial flaw causes imbalance between the circumferential stress and the internal pressure which must circumferential stress and the internal pressure which must be balanced by the increment in the transverse forces to be balanced by the increment in the transverse forces to maintain equilibrium. maintain equilibrium. TThe transverse forces, he transverse forces, in turn, in turn, induce induce bending moments. The cylinder passes into the limit state bending moments. The cylinder passes into the limit state when the bending moments reach a critical value when the bending moments reach a critical value corresponding to the chosen limit conditioncorresponding to the chosen limit condition

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.

4. Limit condition: 4. Limit condition:

010

1xx

xappl MxMM

5. Second boundary of limit area, x = x1

5. Second boundary of limit area, x = x1

where for crack-like where for crack-like defectdefect

4

5.002

1t

xMxxM u

01

0

x

LL dxxNRp LL

LL Accx

1

)(1

taA /1 000 for rectangular formfor rectangular form

for semi-elliptical formfor semi-elliptical form 5.020 )/(1 cxaxa )4/1(4/0 A

for parabolic formfor parabolic form 20 )/(1 cxaxa 3/13/20 A

0axa

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6. The maximum value of the applied moment : 6. The maximum value of the applied moment :

1

0 0

/x x

appl dxdxRxNpM

7. The analytical formulas 7. The analytical formulas

0

200

2

121

121

LLfor rectangular formfor rectangular form

for semi-elliptical formfor semi-elliptical form 3/411

3/4132/3)1(11

02

02

02

LL

0

200

2

11

19/8)1(11

LLfor parabolic formfor parabolic form

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t

x=0 x=c x1 x

x

x [M(0)]

[M(x1)]

Qx(x)

Mx(x)

Illustration of the stages of solution for the rectangular defect

Illustration of the stages of solution for the rectangular defect

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0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

5.0

3

LL2

LL

1

LL

The crack shape influence on the dimensionless limit pressure:

1 - for a rectangular crack,

2 - for a semi-elliptical crack,

3 - for a parabolic crack

The crack shape influence on the dimensionless limit pressure:

1 - for a rectangular crack,

2 - for a semi-elliptical crack,

3 - for a parabolic crack

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IVIIIIII 1 25.0LL

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

1

LL

5.0

IVIIIII

I

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

25.0 5.0

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

1 2

The limit pressures for rectangular axial cracks as given by The limit pressures for rectangular axial cracks as given by the different formulas: (I) “local” formula (1the different formulas: (I) “local” formula (1aa), (II) “local” ), (II) “local” formula (1formula (1bb), (III) “global” formula (2), (IV) – our solution ), (III) “global” formula (2), (IV) – our solution (13(13aa).).

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N 2c,mm

R,mm

t,mm

tn,

mm kg/mm2

kg/mm2

1 360 381 9.5 3.79 58.64 37.41 0.638 0.620 0.634

2 212 381 9.5 3.78 59.36 40.78 0.687 0.689 0.684

3 73 381 9.5 3.78 58.64 48.94 0.835 0.885 0.875

4 358 381 9.5 5.69 57.44 25.03 0.436 0.432 0.445

5 207 381 9.5 5.69 57.44 28.83 0.502 0.505 0.516

6 71 381 9.5 5.69 54.14 43.45 0.803 0.799 0.801

7 357 381 9.5 7.72 56.46 10.97 0.194 0.229 0.239

8 210 381 9.5 7.72 57.16 16.17 0.282 0.309 0.320

9 72 381 9.5 7.72 56.46 39.09 0.692 0.701 0.694

10 209 381 9.5 8.51 57.44 8.58 0.149 0.236 0.243

u tRP /exp еxp theor LL

Comparison with Battelle’s experimental results for Comparison with Battelle’s experimental results for defected pipesdefected pipes

t

x

Qx(x)

L2 L2 k2 А В

нt

нсt

z z

t

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THE TREATMENT OF THE MULTIPLE DEFECTSTHE TREATMENT OF THE MULTIPLE DEFECTS

1

,

121

121

2

2

kzlRt

zLRt

zL

0.34

0.38

0.42

0.46

0.5

0 0.1 0.2 0.3 0.4 0.5

The strength reduction coefficient is

determined from

The strength reduction coefficient is

determined from

The strength reduction coefficient vs. the distance between adjacent defects

The strength reduction coefficient vs. the distance between adjacent defects

Lk

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INFINITE DENT IN A THIN-WALLED PIPEINFINITE DENT IN A THIN-WALLED PIPE

1. The form considered 1. The form considered

B

O

R

A

R

w

R

lRRW cos2

WW << << RR – dent depth – dent depth

R

l

– – jump in the angle of the jump in the angle of the tangent to the pipe surfacetangent to the pipe surface

2. Equations of equilibrium 2. Equations of equilibrium

ds

dMQ

Q

ds

dN

ds

dQNp

;0

)( ;

)(

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3. Applied moment determination3. Applied moment determination

QpRN

1

10

,0

,1

0

Q

dSPR

PQ

tRdsdsPR

PM uappl

2)(1

0 0

1

0

4. Limit condition 4. Limit condition

01 1

0

MMM appl

22

14

t

MM uBA

wherewhere

22 12 t

R

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5. Analytical results 5. Analytical results

242

2

1

t

R

t

R

t

Rp

u

LL 24 1 Rt

l

R

W

t

R

R

W

t

R

21arccos1

21arccos 24

2

2

tW

tW

1

2

tWtpRM /6/

What is the characteristic dent dimension? What is the characteristic dent dimension?

- Elastic solution- Elastic solution

Wt

61-“Elastic-“Elastic

” ”

Wt

tW 21

lim

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FINITE DENT IN A THIN-WALLED PIPEFINITE DENT IN A THIN-WALLED PIPE

The base for analysis is thatThe base for analysis is that

Finite length dent finite length Finite length dent finite length crackcrack

1.1. Rectangular dent ifRectangular dent if

if if

0WxW cx

0)( xW cx

121

1212

2

LL

2. “Parabolic” form2. “Parabolic” form if if

5.02

0

5.020

5.020

)/(112

)/(12)/(1)(

)(cxa

cxacxa

t

xWx

cx

0

200

2

,11,19/8),1(11

LL

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AXIALLY SYMMETRIC DEFECT AXIALLY SYMMETRIC DEFECT

1. Complete analogy with the “crack” solution 1. Complete analogy with the “crack” solution

2. The only difference is the limit moments : 2. The only difference is the limit moments :

4

5.0022 t

cxMxM u 4

5.02

1

txxM u

3. Limit condition: 3. Limit condition:

010

1xx

xappl MxMM oror 0

0 xx

c

appl McMM

4. The solution for the limit pressure: 4. The solution for the limit pressure:

22

22

22

/5.0

14)1(

14)1(

minALL

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FINITE LENGTH FINITE WIDTH 3D DEFECT FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT)(SLOT)1. Differential equation of equilibrium 1. Differential equation of equilibrium

QRd

dMQ

dx

dM

Rd

dQ

dx

dQ

R

Np

xx

x

;

2. “Slot” solution is between the “crack” and “axially sym.” solutions

2. “Slot” solution is between the “crack” and “axially sym.” solutions Mechanism of transferring the applied moments from Mechanism of transferring the applied moments from sections to sections where sections to sections where

0

0 Rl /0

xc

c x

constR

tA

constR

tA

Rd

dQu

u

0

0

0

0 0

101

00

0)( 1 Q

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FINITE LENGTH FINITE WIDTH 3D DEFECT FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT)(SLOT)3. Limit force and moments in circumferential direction 3. Limit force and moments in circumferential direction

cxtN u 0 1

4/0 220 tMM u 4/)1( 22

1 tM u anandd

4. Limit conditions in circumferential direction 4. Limit conditions in circumferential direction

ptimization parameterptimization parameter

R

tlAM u

appl 22

00

0

)/1(

2 102

00

1 AAR

tlAM u

appl

5. Limit conditions in axial direction 5. Limit conditions in axial direction

122

1

122

1

1122

1

/5.0

14)1(

14)1(

min

ALL

01 1 A

2

222

2

121

121

C

LL12 1 A

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FINITE LENGTH FINITE WIDTH 3D DEFECT FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT)(SLOT)

N t R

(inch)

(t-a) 2c

(inch)

2l Fracture pressure lbf/inch2

(inch) (inch) (inch) Exp Theor Our model

1 0.125 2.74 0.062 1.502 0.375 1530 1279 1305

2 0.128 2.74 0.062 1.498 1.498 1910 1280 1200

3 0.127 2.74 0.062 1.502 0.124 1630 1282 1350

4 0.122 2.74 ----- ----- ------ 2300 1914 1914

5 0.112 2.76 0.092 1.584 0.398 2150 1656 1610

6 0.112 2.70 0.033 1.504 0.394 1050 616 800

7 0.124 2.59 0.029 0.478 0.157 2100 1126 1720

8 0.123 2.59 0.023 0.241 0.380 2275 1943 1470

9 0.123 2.59 0.034 2.348 0.404 950 620 720

10 0.248 2.62 0.037 2.106 0.387 1700 701 1380

11 0.246 1.62 0.061 0.542 0.385 3300 3235 3530

Comparison of the calculated and experimental values of the Comparison of the calculated and experimental values of the fracture pressure for aluminum pipes with 3D defectsfracture pressure for aluminum pipes with 3D defects

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FINITE LENGTH FINITE WIDTH 3D DEFECT FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT)(SLOT)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

25.0 5.0

1 2

5.0

3.0

1.0

5.0

3.0

1.0

5.0

3.0

1.0

5.0

3.0

1.0

)/(21 tRl

)/(21 tRc

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THROUGH CRACKTHROUGH CRACK

1. Equilibrium differential equations 1. Equilibrium differential equations

2. Boundary region2. Boundary region

3

2

1

c 1c

x

x

Q

R

Q

R

NP x

0

x

L

R

N

0

x

N

R

L x

x

MQ x

x

R

MQ

On the boundaries On the boundaries

0 ,0 ,0 1 QLcx

0 ,0 ,0 QLc

On the boundaries On the boundaries

ccLL

LL

11

3. Redistribution axial stresses

3. Redistribution axial stresses

cxN 0 при ,0 1 при , cxctN u

1

2122

11

22

0 0 //2)1(

0 /1)(

cxccxcxkk

cxckxtxN ux

210 )( txN ux

32

2122

1111

221 0

0 //2)1(

0 /1)(

cxccxcxkk

cxcxktxN ux

)1)(1( k )1)(1(1 k

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Redistribution axial stresses (epures) Redistribution axial stresses (epures)

c 1ctu

tu

tu tu

x

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4. Redistribution tangent stresses 4. Redistribution tangent stresses

12111

2

0 0 )(12

0 /2),(

cxcc/x/cRk

cxcRxktxL u

2121111

21 0

0 )(12

0 /2),(

cxcc/x/cRk

cxcRxktxL u

32

2121111

2112

0 0 )(12

0 /2),(

cxkkcc/x/cR

cxkkcxRtxL u

Determination Determination

5.0max tL u )4/(1 kcR


Redistribution tangent stresses (epures)

Redistribution tangent stresses (epures)

3

2

1

c 1c x

const

constx

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5. Redistribution circumference stresses 5. Redistribution circumference stresses

0 )(

0 )()(

212

1

cx

cxtN u

0 /-22

0 2

0 /

312

2122

1222

122

1

2122

112

1

12

1

cRk/cRψk/cRψkR/cψk

/cRψkR/cψk

cRk

LLLL /)1(1 12


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3

2

1

c 1c x

1

Redistribution circumference stresses (epures)

Redistribution circumference stresses (epures)


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6. Limit state in the longitudinal direction 6. Limit state in the longitudinal direction

x

Q

R

Q

R

NP x

R

t

R

Qu

)(

Limit equilibrium equation of the moments for the determination of Limit equilibrium equation of the moments for the determination of )(

11

)()1(

)1()(2

2

212

LL

LLLL

LL

LLLL

7. Limit state in the circumferential direction

7. Limit state in the circumferential direction

)1(4

)( 21

00

t

ddRM

0)( 0)( 0

M

dQ MM


Limit equilibrium equation of the momentsLimit equilibrium equation of the moments

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8. Comparison with the Battelle’s formula8. Comparison with the Battelle’s formula

5.02 )61.11(

BattelleBattelle 00 0.150.15 0.30.3 0.50.5 0.70.7

0.250.25 0.9360.936 0.9560.956 0.9550.955 0.9620.962 0.9620.962 0.9590.959

0.50.5 0.8140.814 0.8460.846 0.8520.852 0.8590.859 0.8750.875 0.8740.874

0.750.75 0.7020.702 0.7130.713 0.720.72 0.7290.729 0.7370.737 0.7130.713

11 0.5950.595 0.5890.589 0.5890.589 0.6010.601 0.6110.611 0.5960.596

1.51.5 0.4360.436 0.3730.373 0.4180.418 0.4190.419 0.420.42 0.3970.397

22 0.3480.348 0.3030.303 0.3080.308 0.2990.299 0.2910.291 0.2860.286

33 0.2240.224 0.1870.187 0.1920.192 0.1760.176 0.1660.166 0.1550.155

tN ux ,


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THROUGH CRACK - experimentTHROUGH CRACK - experiment

NN R, R, мммм t, t, мммм 22l ,l ,мммм в ,в ,МПаМПаPPexp exp R/ tR/ t

МПаМПаPPcal cal R/R/ tt

МПаМПа

11 381381 9,559,55 222,25222,25 537,9537,9 191,7191,7 187,47187,47

22 381381 9,639,63 222,25222,25 537,9537,9 194,5194,5 188,23188,23

33 381381 9,149,14 222,25222,25 553,1553,1 190,3190,3 190,98190,98

44 381381 9,149,14 222,25222,25 553,1553,1 190,3190,3 190,98190,98

55 381381 9,229,22 114,3114,3 533,1533,1 322,8322,8 342,96342,96

66 381381 9,329,32 25,425,4 533,1533,1 486,9486,9 522,06522,06

77 381381 9,379,37 25,425,4 533,1533,1 481,4481,4 522,20522,20

88 381381 9,509,50 83,8283,82 560,7560,7 384,8384,8 432,56432,56

99 381381 9,409,40 83,8283,82 560,7560,7 387,6387,6 432,23432,23

1010 381381 9,969,96 224,0224,0 563,4563,4 206,9206,9 202,86202,86

1111 381381 9,539,53 177,8177,8 532,4532,4 216,6216,6 233,40233,40

1212 381381 9,539,53 177,8177,8 534,4534,4 227,6227,6 234,28234,28

1313 381381 9,539,53 177,8177,8 495,2495,2 226,2226,2 217,10217,10

1414 381381 9,539,53 177,8177,8 495,2495,2 223,4223,4 217,10217,10

1515 381381 9,539,53 137,2137,2 518,6518,6 293,8293,8 292,22292,22

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THROUGH CRACK - experimentTHROUGH CRACK - experiment

NN R, ммR, мм t, ммt, мм 2l ,мм2l ,мм в ,МПав ,МПаPexp R/ tPexp R/ t

МПаМПа

Pcal R/ tPcal R/ tМПаМПа

1616 381381 9,539,53 162,6162,6 551,7551,7 267,6267,6 262,56262,56

1717 381381 9,539,53 162,6162,6 862,1862,1 309,0309,0 410,29410,29

1818 381381 9,539,53 162,6162,6 583,4583,4 205,5205,5 277,65277,65

1919 381381 9,149,14 137,7137,7 648,3648,3 297,2297,2 355,49355,49

2020 381381 9,149,14 138,9138,9 648,3648,3 285,5285,5 352,28352,28

2121 381381 9,149,14 139,7139,7 688,3688,3 304,1304,1 376,20376,20

2222 381381 9,179,17 152,4152,4 589,0589,0 269,6269,6 298,93298,93

2323 381381 9,179,17 152,4152,4 583,4583,4 272,3272,3 296,08296,08

2424 381381 9,179,17 152,4152,4 580,0580,0 257,9257,9 294,36294,36

2525 381381 9,129,12 152,4152,4 641,4641,4 266,1266,1 325,50325,50

2626 381381 8,338,33 381381 587,6587,6 126,2126,2 94,9294,92

2727 381381 8,338,33 508,0508,0 587,6587,6 100,7100,7 64,6964,69

2828 381381 8,338,33 222,25222,25 587,6587,6 217,9217,9 184,54184,54

2929 381381 8,338,33 177,8177,8 558,6558,6 222,0222,0 233,30233,30

3030 381381 8,338,33 279,4279,4 557,2557,2 182,7182,7 120,66120,66

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Leak before breakLeak before break

crsurfacethrough ,)(

cr

crcrthrough

121

1212

2

Precondition of the leak before break phenomenon

Precondition of the leak before break phenomenon

- net thickness section

The boundary between leak and break is obtainedThe boundary between leak and break is obtained

11,0, surfacethroughsurface

Then

cr

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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4

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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4

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CONCUSIONSCONCUSIONS

1.1. The limit load used in the criteria formulation of The limit load used in the criteria formulation of fracture mechanics originated from the theory of fracture mechanics originated from the theory of plasticity and should be treated by appropriate plasticity and should be treated by appropriate methods. methods.

2.2. The theoretical models of ductile failure of The theoretical models of ductile failure of defected bodies provides understanding of the defected bodies provides understanding of the ductile failure mechanisms, establishes the ductile failure mechanisms, establishes the dimensionless parameters that have the most dimensionless parameters that have the most influence on the limit load.influence on the limit load.

3.3. Theoretical models can be used for choosing the Theoretical models can be used for choosing the analytical pattern in constructing empirical analytical pattern in constructing empirical formulas and the checkpoints when performing formulas and the checkpoints when performing the FEM calculations.the FEM calculations.

limit load calculation model of ductile failure of defective pipe and pressure vessel orynyak i.v....

Documents

limit loaddecreasing

limit state

dimensionless limit

maximal value of limit

limit loadthe solution

boundary of limit area

appropriateaxial surface

cylinder illustration