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Batura A . S . , Orynyak I.V.

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ENGINEERING METHODS FOR STRESS INTENSITY FACTOR CALCULATION FOR 2-D AND 3-D BODIES WITH CRACKS

BaturaA.S., Orynyak I.V.

Pisarenko’ Institute for Problems of Strength , Kyiv, Ukraine

National Academy of Sciences of Ukraine

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Weight Function Method for plane bodies

- weight function,

- the law of stress distribution,

G – geometry parameters.

- asymptotical (singular) part of WF,

- correction (regular) part of WF.

Then for any specified stress law

(for example

) obtain

where

and

doesn’t depend upon geometry.

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Weight Function Method for plane bodies

In particular, for a plane body with an edge crack

The main idea of Weight Function Methods:

If we have the SIF solution for one particular loading we can obtain the SIF solution for any other law of loading.

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Application of WFM for a pipes

Crack compliance method

(modification of Cheng & Finnie approach)

In the circular pipe additional force N and moment M appear. Angle and displacement discontinuity can be expressed in the next form:

where YN, YM – are the dimensionless SIF, induced by M and N as in the plane body,

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Application of WFM for a pipes

Crack compliance method

(modification of Cheng & Finnie approach)

- caused by loading,

- caused by force,

- caused by moment.

Obtain result SIF :

SIF is smaller than in the case of straight plane !

Using equilibrium equations for a ring and initial parameter method, get the expression for a dimensionless SIF decrease from the case of straight plane (Y0):

where - dimensionless pressure.

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Application of WFM for a pipes

Crack compliance method

(modification of Cheng & Finnie approach)

Result plots

Conclusion:

Advanced SIF formula for pipes was obtained. The feature of the SIF decreasing at rising of the pressure was found.

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Weight Function Method for 3-D bodies

(1)

(2)

- elliptical crack,

- for semi-

elliptical crack,

- for quarter-elliptical crack

-correction part

- asymptotical part for elliptical crack.

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loading dependent

geometry dependent

If

is known we obtain

and can calculate

for any law of loading.

where

- is a known SIF for any law of loading.

Weight Function Method for 3-D bodies

Similarly to the 2-D case,

So

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Check of the PWFM accuracy for

semi-elliptic cracks

SIF along crack front (angle), homogeneous loading

90

0

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Dependence SIF from ratio a/l

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Dependence SIF from ratio a/l

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The solution: approximation of the stress law with function of the next type: ,

calculation of the SIF array for each stress function

. Approximate SIF function can be build as linear combination of precalculated .

Weight Function Method for 3-D bodies.

Simplified (speed up) approach.

The problem: triple integral (square and contour) with singularity at the edge high computation cost (especially for repeating – fatigue, stress-corrosion,… – calculations) !!!

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Weight Function Method for 3-D bodies.

Simplified (speed up) approach.

Polynomial example

The expression for dimensionless SIF functions :

Weight Function Method for 3-D bodies.

Simplified (speed up) approach.

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Polynomial example

For

simple expressions for IijA,C(α) were obtained :

Semi-elliptical crack on the inner surface of the cylinder.

Application of the peveloped methods:

Software “ReactorA”

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- This program is intended for calculation of reactor pressure vessel residual life and safety margin with respect to brittle fracture.

- User sets loading and temperature fields in the different moments of time. Then material fracture toughness, embrittlement parameters are also set by user.

- Residual life is calculated deterministically and probabilistically (MASTER CURVE approach) for various points of crack front

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ReactorAadvantages

- The sizes of stress and temperature fields' aren't bounded
- Number of time moments is bounded only by the memory size
- Cladding is taken into account
- Welding seam and heat-affected area are taken into account
- Deterioration is taken into account not only as shift of the material fracture toughness function but also as its inclination
- Original feature of the software is using of the author variant of the weight function method. It allows to set loading on the crack surface in the form of table.

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3. Residual Life calculation of the NPP pressure vessel using fracture mechanics methods

Input Data

1) Stress field for time

Table arbitrary size

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Input Data

2) Temperature field for time

Table arbitrary size

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weld seam

heat-affected zone

basematerial

cladding

crack

base material

cladding

crack

Input Data

3) Crack types

a)Axialwith weld seam

b)circumferential

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4) The basic material characteristics

1. Arctangents

2. Exponent

3. User (pointed) function

Common shape of the crack growth resistance function is

for user function Atakes from coordinates of first point

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5) Shift and inclination conceptions

1.Shift

2.Inclination

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6) Dependence of shift on radiation

a)Analytical form

b)Table form

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Results

Scenario – Break of the Steam Generator Collector WWER-1000 operated at full power

It is given: - stress field,

- temperature field,

= 1000, 2000, 2800, 3000, 3160, 3600, 4000 sec - time points

Axial crack. Half-lengthl -40 мм.,

deptha - 50 мм.

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a)Dependences of the calculated and critical SIF from temperature for time = 3000 sec

SIFfor base material

--//-- forweld seam

Critical SIFfor base material

--//-- for weld seam

--//-- forheat-affected area

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b)History of the dependences calculated SIF fromtemperature forsome points and all times intervals andcritical SIF

T

historyforbasic material

--//-- for weld seam

criticalSIFforbasic material

--//-- forweld seam

--//-- forheat-affected area

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c)Table of the calculated temperature margin

for all points of crack front and time points

fields for chosen history points

minimal margin

margin for time points

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d) Figure of the calculated margin

calculated temperature margin

shift of the temperature by user table

shift of the temperature by analytical model

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Calculated temperature margin

Results for other crack geometries

New geometry for axial crack

Half lengthl - 60мм

Depth a - 40 мм

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Calculated temperature margin

New geometry for axial crack

Half lengthl - 40мм

Depth a - 60 мм

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calculated temperature margin

New geometry for circumferential crack

Half lengthl - 60мм

Depth a - 30 мм

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Implementation MASTER CURVE

Conception

1. Failure probability calculation for structural element

2. Failure probability calculation forcrack

3. Calculation parameters

Pf = 63,2% Кmin = 20 В0 = 25 мм b = 4

4. In addition

Кmin, K0(Т), В0, b - arbitrarily

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Result for main scenario

Time point t4= 3000 sec

Axial crackhalf lengthl -40 мм.,

deptha - 50 мм.

For timeDT =0 failure probability equal 1.07*10-05

SIF dependences on angle

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Dependences of logarithm probability on DT

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Probability density forDT = 50

Application of the developed methods:

Software “WFM”

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- This program is intended for SIFcalculation for different (1-D and 2-D) types of cracks and for endurance estimation with using different fatigue and stress-corrosion laws.

- User sets “maximum”, “minimum” and “corrosion” loading fields.

- SIF, grow of the crack dimensions in time and endurance are calculated. “Until specified depth” or “until specified count of cycles” modes are presented.

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1. Damages

2. Cracks

WFM: implemented types of damages and cracks

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WFM: example of result window

- Input and output data can be exchanged with clipboard.

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CONCLUSION

1. Efficient methods of stress intensity factor (SIF)

calculation are developed.

2. The computer software which reflected all modern requirements for brittle strength analysis of Reactor Pressure Vessel is created.