Orynyak I.V., Radchenko S.A.

IPS NASUIPS NASU

Pisarenko’ Institute for Problems of Strength , Kyiv, Ukraine National Academy of Sciences of Ukraine 

Pisarenko’ Institute for Problems of Strength , Kyiv, Ukraine National Academy of Sciences of Ukraine 

STRESS STATE CALCULATION OF COMPLEX PIPING SYSTEMS AT STATIC AND DYNAMIC LOADINGS

STRESS STATE CALCULATION OF COMPLEX PIPING SYSTEMS AT STATIC AND DYNAMIC LOADINGS

1th Hungarian-Ukrainian Joint Conference on

Safety-Reliability and Risk Engineering Plants and Components

11,12 April 2006, Miskolc, Hungary

IPS NASUIPS NASU

3D model of pipeline system of RBMK Chernobyl NPP3D model of pipeline system of RBMK Chernobyl NPP

Problem statementProblem statement

Calculation

designis executed according to currently standards

doesn’t require high accuracy

uses high safety factors

uses old design solution

fitness for service

requires high accuracy

requires determination of

local distribution of stress: for calculation of defects for application of systems of

monitoring for application of strength

criteria

IPS NASUIPS NASUProblems of calculation pipe bend

A

C

B

D

A

C

B

D

zKzK

1d

0d

B

O

R

BR

BtR2

- parameter of flexibility - parameter of flexibility

- parameter of curvature

EI

sMK

ds

d

bendpipeforf

pipesraightforK

1,

1

1. Pipe bend as BEAM element in a piping 1. Pipe bend as BEAM element in a piping

2. The local stress-displacement fields of a SHELL 2. The local stress-displacement fields of a SHELL

• Saint Venant’s task;

• Geometrical non-linearity - internal pressure;

• End-effect; -3

-2

-1

0

1

2

3

0 30 60 90 120 150 180

end-effect

Saint Venant'ssolution

z

x

k

2

IPS NASUIPS NASU

Software “InfoPipeMaster” - Software “InfoPipeMaster” - information and calculating programinformation and calculating program

1. Gathering and storage the information about pipeline:

2. Calculation of the stress strain state:

static calculation static calculation

computer portrait, schemas, drawings and photos;

databases Materials, Soils, Objects and Defects;

results of technical inspection;

results of calculation

3. Comparative analysis of the results of the inspections executed during the different period of time .

4. Preparation of reports about a condition of object according to results of observation

dynamic analysis dynamic analysis

calculation of defects calculation of defects

IPS NASUIPS NASUDatabases structureDatabases structure

«Object location»

«Geometry» «Calculation results»«Loads»• object type

• dimensions

• schemas, photos, drawings

• object label

• coordinates

• pressure

• temperature

• distribution forces

• weight

• displacements and angles

• forces and moments

• stresses

«Defect location»

«Geometry» «Calculation results»«Loads»• defect type

• dimensions• defect label

• element number

• coordinates

• pressure

• axial force

• bending moments

• stress intensity factor

• reference stress

• safety factor

DB «Objects»

DB «Defects»

IPS NASUIPS NASU The features of calculation modules The features of calculation modules • Pipelines are considered as beam structure where characteristics of cross sections are determined from theory of shells

• The calculation method is based on a method of initial parameters

• The continuity of the solution at transition from dynamics to a statics is provided

• Analytical solution for pipe bend are used

• Large library of elements with taking into account environment influence

• Accounting nonlinear supports and soil characteristics

• Convenient system for building and editing

• Abilities for data export – import with other software

IPS NASUIPS NASU

Equilibrium equationsEquilibrium equations

000

B

K

dBdK yx

yzxy mQ

BK

dB

dK

00

zyz mQ

dBdK

0

zz q

dBdQ

0

y

допy q

BN

dB

dQ

00

xy

доп

qB

Q

dBdN

00

000 2

21EtBPR

EFBN

EIKK

dBdθ доп

zin

z

GIK

B

θ

dBd xy

200

EI

KK

BdB

d yout

y 00

The equations for displacementsThe equations for displacements

0

2

0 2GIBRKK

dBd x

кр

yz w

zy

Bu

dB

d

00

w

122

21

0

2

00 EIBRKK

EFsN

EhRP

TBdB

du zinдоп

Tyw

The equations for curved beamThe equations for curved beam

The equations for angles of cross-section center The equations for angles of cross-section center

IPS NASUIPS NASUAnalytical solution for axial displacementAnalytical solution for axial displacement

.sin

221

4sincossin

4sincos

2sin

2sincos

4sin5cos8cos8

4cos54sinsin9

2sin3cos2

2sincos22

cos12

sin2

21

cos1sincos

0

2

0

2

000

220

220

000

2

000

020

00

0

00

00

0

00

00

EtPR

TBBq

BqQNEFB

BqBq

BNBQ

BmtBPR

FBN

K

IK

EIBK

Buu

y

xyдоп

xy

допy

z

доп

inz

in

zyw

IPS NASUIPS NASU Static calculation. Pipe bend - SHELLStatic calculation. Pipe bend - SHELL

r

R

O

B

O1

t

x

y

z

v u

w

Equilibrium equations:Equilibrium equations:

PS

NQS

SQRSR

Nxx

sin11

0cos11

SNL

SR

QSN

RSx

0sin1 2

xx Q

NLS

RS

0cos1

xx M

MSM

RSQ

01 2

x

xxM

MSRS

SQ

HN

HNx

12H

L

HM

HM x

21 H

M x

Physical equations:Physical equations:

IPS NASUIPS NASU

Swvu

S

sincos1

Rwv

R

1

v

SSuu

R1cos1

RS

wv

S

wvw

S

cossinsincos1

22

2

2

22

2

2

1

R

ww

R

w

S

wRSSR

vSS

uuR 2

2 cos22sin11cos1

- strains- strains

- curvatures- curvatures

Geometric equations:Geometric equations:

Solution methodsSolution methods

Complete - 8-th order Simplified - 4-th order- semianalytical - semianalytical

- analytical

2

z

x

k

zk

2

Stress distribution on outside surface for in-plane bending of 900 bend having rigid flanges

Stress distribution on outside surface for in-plane bending of 900 bend having rigid flanges

Example: bend radius ; inside radius ; thickness . Example: bend radius ; inside radius ; thickness .

Circumferential stress factorCircumferential stress factorLongitudinal stress factorLongitudinal stress factor-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 30 60 90 120 150 180

analytical solutionexperimentnumerical solution 4numerical solution 8ADINAP

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 30 60 90 120 150 180

analytical solutionexperimentnumerical solution 4numerical solution 8ADINAP

mmB 3750 mmR 125 mmt 5.12

IPS NASUIPS NASU

Flexibility factor for in-plane bending of 900 bend having rigid flanges

Flexibility factor for in-plane bending of 900 bend having rigid flanges

mmt 5.12mmR 125

0

2

4

6

8

10

12

200 250 300 350 400 450 500

Saint Venant's solutionour resultsexperimentADINAP

K

0B

IPS NASUIPS NASUThe calculation of multi-branched 3D piping

Pump station view

IPS NASUIPS NASUPump station calculation model

Table of element

Calculation results

displacements

angles

forcesmoment

s

geometry

IPS NASUIPS NASU Dynamic analysisDynamic analysis

1. Determination of own frequencies and forms of vibrations of the piping. 2. Calculation forced vibrations of system.3. Restoration of value of external force by the measured displacements of the piping.

Tasks:Tasks:

Problems:Problems:1. Choice of the optimal method of the solution of statically indefinite beam system. 2. Determination of a local flexibility of curved beam by dynamic loading.Solution methods:Solution methods:1. Dynamic stiffness method for a case of harmonious loading. 2. A method of dynamic analysis for a case of non-stationary loading.

Advantages:Advantages:1. Accurate analytical solutions are used. 2. The continuity of the solution at transition from dynamics to a statics is provided.

IPS NASUIPS NASU

Harmonious vibrations. Harmonious vibrations. Dynamic stiffness methodDynamic stiffness method

1. Straight beam.1. Straight beam.

02

4

4

yz

yW

EI

F

dx

Wd z

y

dx

dW

z

zz

EI

K

dx

d

y

z Qdx

dK

- the equations of movement at cross vibrations:- the equations of movement at cross vibrations:

inertial member - frequency of vibrations- frequency of vibrations

x

y

dx

X0X1

01 XKX

stiffness matrix

IPS NASUIPS NASU

- the constraint equation :- the constraint equation :

xkYkEI

QxkY

kEI

KxkY

kxkYWW y

yz

y

yyz

z

yy

z

yyy 433221000

0

xkYWkxkYkEI

QxkY

kEI

KxkY yyyy

yz

yy

yz

zyzz 43221 0

00

0

xkYEIkxkYEIkWxkYk

QxkYKK yzyzyzyyy

y

yyzz 43

221 00

0

0

xkYkKxkYEIkxkYEIkWxkYQQ yyzyzyzyzyyyyy 432

23

1 0000

zy EI

Fk

24

2. Curved beam.2. Curved beam.

1

1

2

2

3

3

4

4 5

1

;,1,e

izb

iz

;sincos,1, ieii

eiy

biy

;sincos ,1 ie

iyiei

bi

;sincos,1 ieii

eiy

biy, UWW

;1eiz,

biz, WW

.sincos1 ieiy,i

ei

bi WUU

The transition matrixThe transition matrix

Harmonious vibrations. Harmonious vibrations. Dynamic stiffness methodDynamic stiffness method

IPS NASUIPS NASU

zy

x

w

EI

M

xzz

yz Q

x

M

2

2

t

wFq

x

Q yy

y

acceleration

iy

iy

iy

iy

iy

iy A

wwwwlt

w

22

21

2

2

2

;24384

124192

1696

1248

124

14

12

41

4

2

31

30

2

221

20

2

31

02

41

0

xkA

lk

EI

xqxlk

EI

xQxlk

EI

xKlxkx

xkww i

yy

iy

izi

ziy

iy

;6384

166144

1248

112

13

12

41

3

2

31

02

31

20

2

2210

2

31

0

xkA

lk

EI

xqxkw

xlk

EI

xQxlk

EI

xKlxk iy

yiy

iy

izi

ziz

;2384

124296

116

12

02

41

2

2

20

02

20

02

31

02

2210 xm

Alkxqlxmxm

wxlk

xQxlk

EI

KK i

yyi

ziy

iy

izi

z

02

41

2

210

20

020

02

31

0 3841

82481 mA

lkxq

xlkEIKlxmxm

wxlk

QQ iyy

izi

ziy

iy

iy

23844882 2

4

2

30

2

20

20

20 l

AEI

lq

EI

lQ

EIlKlw i

yy

iy

iz

iz

iy

EI

xq

EI

xQ

EI

xKxww y

iy

izi

ziy

iy 2462

430

20

00

Non-stationary vibrations. Non-stationary vibrations. Method of dynamic analysisMethod of dynamic analysis

- the equations of movement at cross vibrations:- the equations of movement at cross vibrations:

- the constraint equation :- the constraint equation :

IPS NASUIPS NASUExamples – dynamic stiffness methodExamples – dynamic stiffness method

P

A B

l

W

х

thtglPl

M

820 4

42

2

1

EI

lF

tPtP cos0

-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300

M(l /2)

frequency, rad/s

Е = 2∙106 МPа; G = 8∙105 МPа; = 0.3; = 8000 kg/m3; l = 5 m; R = 0.1 m; h = 0.005 m.

1. The diagram of the bending moment in a middle point of articulate beam

srad136

1.25

2. The restoration of value of exciting force by known displacement

P0, H

-4,E+07

-3,E+07

-2,E+07

-1,E+07

0,E+00

1,E+07

2,E+07

3,E+07

4,E+07

5,E+07

6,E+07

0 500 1000 1500

frequency , rad/s

1

10 P10 W

2. The restoration of value of exciting force by known displacement

1. The diagram of the bending moment in a middle point of articulate beam

IPS NASUIPS NASU

1. The frequency of vibrations is lower than first own frequency

-1,4E-06

-1,0E-06

-6,0E-07

-2,0E-07

2,0E-07

6,0E-07

1,0E-06

1,4E-06

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3 1,4

numerical solution theoretical solution

w , m

t , s

ttP 68sin

Examples – method of dynamic analysisExamples – method of dynamic analysis

IPS NASUIPS NASU

2. The frequency of vibrations is higher than first own frequency ttP 300sin

-7,E-07

-3,E-07

1,E-07

5,E-07

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

numerical solution theoretical solution

w, m

t , s

Examples – method of dynamic analysisExamples – method of dynamic analysis

IPS NASUIPS NASUThe determination of dynamic flexibility

of pipe bendThe determination of dynamic flexibility

of pipe bend- flexibility factor by harmonious vibrations

B

BABBABAK

2

10607241167211672 2

B

BCBABBCBA

B

BCBAK

iiii

4

151060721632212144

4

32212144 2

2

22

22 )1( A

0

2

4

6

8

10

0 50 100 150 200

K

B

2

EhR

B 2

242 )1(12

BhR2

- flexibility factor by non-stationary vibrations

IPS NASUIPS NASUExampleExample

200

600

1000

1400

1800

2200

2600

1,2s 2,2s 3.2s 1,1s

mode of vibrations (m, n)

fre

qu

en

cy ,

Hz

experiment

FEM

K=1

Saint Venant's solution

our results

L. Salley and J. Pan. A study of the modal characteristics of curved pipes // Applied Acoustics. – 2002. – V.63. – pp. 189-202.

Е = 2.07∙106 МPа; = 0.3; = 8000 kg/m3; R = 0.0806 m; h = 0.00711 m;В = 0.457 m; l=0.2 m;

l

l

Rh

B

IPS NASUIPS NASUThe calculation of defects The calculation of defects

cracks

corrosion

defects of form

IPS NASUIPS NASUPressure calculation modelPressure calculation model

altitude of the pipelineoperating pressure

IPS NASUIPS NASUStrength analysis of defectsStrength analysis of defects

dangerous zone

conditionally dangerous zone

safety zone

defect state

IPS NASUIPS NASUApplication of software “InfoPipeMaster” Application of software “InfoPipeMaster”

Chernobyl NPP

Zaporozhye NPP

South-Ukrainian NPP

Ammoniac pipeline «Togliatti-Odessa»

System oil pipelines JSC«Ukrtransnafta»

System gas pipelines DE«Ukrtransgas»