LIFE ASSESSMENT OF STEAM TURBINE …ommi.co.uk/PDF/Articles/166.pdf · Life assessment of steam turbine components based on probabilistic procedures OMMI, (Volume 5, Issue 2) August

Life assessment of steam turbine components based on probabilistic procedures www.ommi.co.uk OMMI, (Volume 5, Issue 2) August 2008

© Siemens Power Generation 1

LIFE ASSESSMENT OF STEAM TURBINE COMPONENTS

BASED ON PROBABILISTIC PROCEDURES

*Albert Bagaviev *Siemens Power Generation, Mülheim/Ruhr, Germany

Now with E-ON Anlagenservice GmbH, Gelsenkirchen, Germany [email protected]

Albert Bagaviev, Dr.-Ing., Head of Computational Engineering Department at E-ON Anlagenservice GmbH, 45896 Gelsenkirchen, Germany His main professional activity (last 10 years) is in the area of computational life time assessment of rotating and stationary power plant components including high temperature damage and non-linear fracture mechanics applications in the frames of deterministic and probabilistic approaches.

Abstract The most important factors in the operation of large steam turbines are the long service life endurance and high availability. Operational safety is ensured by regular inspections in combination with mechanical calculations. In this context, risk-based turbine component life assessment approaches - which balance out safety and economic requirements and take into account variability and uncertainty of the material data, component geometry, applied loads etc. - have become more and more important in recent years. This paper demonstrates the application of computational methods in the framework of probabilistic fracture mechanics for failure assessment in the typically heavy-loaded components of stream turbines. Keywords: steam turbine, probabilistic failure assessment, fracture mechanics, creep, fatigue, crack growth

1. Introduction The mechanical integrity of the turbine components is a matter of great practical importance for both economic and safety reasons. This integrity must be assured by design, materials, manufacturing procedure, inspection and operational modes. Many turbine components originally designed approximately 30 years ago for base-load operation are now being commercially pressed for the flexible operation of cyclic duty for economic reasons. As a result, a higher flexibility comes at the price of increased risk. Computational life-assessment techniques are able to quantify the penalty in terms of reduced turbine operational life resulting from the changed operating mode. Generally, both the component design and remaining life analysis, performed either by the deterministic or probabilistic method, are used to fulfil the condition shown in Fig.1 and to define how far the system or structure can withstand the applied load.

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MaterialResistanceMaterial

ResistanceLoad and

Crack DrivingForce

Load andCrack Driving

Force

Static Elastic-PlasticProperties

Static Elastic-PlasticProperties

Creep and RelaxationProperties

Creep and RelaxationProperties

High and Low Cycle FatigueProperties

High and Low Cycle FatigueProperties

Fracture MechanicsProperties

Fracture MechanicsProperties

Applied Stress and StrainApplied Stress and Strain

Component GeometryComponent Geometry

Operational CyclingOperational Cycling

Crack GeometryCrack Geometry

Fig. 1: "Demand (Load) – Capacity (Material Resistance) " approach Design safety is usually ensured in the deterministic approach by specifying a safety margin as NN SR , s. Fig. 2, whereby the nominal resistance RN is usually two or three standard deviations below the mean value. The load is multiplied by a load factor accounting for uncertainties in geometry and/or maximum load and temperature conditions. The resulting safety factors of traditional deterministic approaches (based on the state-of-the-art technique related to the design time) do not provide a means to quantify expected structural reliability and can lead to unbalanced design wherein some components are over-designed.

µS µR

kS·σS

kR·σR

Prob

abili

ty d

ensi

ty

SN RN

Fig. 2: Design safety approach




This can result in the situation shown in Fig. 3. In such cases the first design component life can be extended by applying reliability methods and modern computational methods like the Finite Element Method (FEM) [7] for assessment of the component geometry and applied loads, to further approach the actual component limits extending from the values predicted by the classical methods.

DesignExtension of design life(NDE,operational data)

Real component life to failure(Material data scatter, load uncertainty)

Operationalhours

Failure rateof component

Fig. 3: Component life

Therefore, it is essential to identify excessively high safety factors in component dimensions, allowable crack sizes or component inspection intervals. The failure assessment methods based on statistical methods making use of modern techniques including First Order Reliability Methods, FORM, and Monte-Carlo simulations with adaptive and importance sampling, provide an alternative to the traditional engineering approach by using arbitrary safety factors to ensure desired levels of reliability and safety. This work presents the application of probabilistic fracture mechanics analysis techniques to the failure assessment of the rotational and stationary steam turbine components.




2. Integrity assessment methodology The deterministic failure assessment models form the basis of the probabilistic approach. Some basic features of deterministic analysis are briefly outlined below.

2.1. Deterministic approach

Steam turbine components are among the most highly loaded and stressed power plant components. These components are usually continually operated in a temperature range where the creep mechanisms under steady state stress are activated. Additionally, the cycling loading occurs as a result of transient operations such as turbine start-up, shutdown, load change, etc.. These thermal stresses are mainly responsible for low cycle fatigue (LCF) of materials. The creep and fatigue accumulated over operational time are two principal degradation mechanisms which eventually lead to crack initiation and growth, in critical high temperature turbine components. It can be quantified using a linear cumulative law:

( ) ∫∆

⋅+∆⋅=statesteadyt

0

nI dt*)C(aKC

dNda & (1)

The crack propagation under cyclic loading conditions by a fatigue mechanism can be expressed in terms of stress intensity range ∆KI by the Paris-Erdogan equation:

( )nIKC

dNda

∆⋅= , (2)

where C and n are the material dependent parameters. The creep crack growth &a can be determined experimentally and described by the creep fracture mechanics parameter C* in the form of:

( )φ∗⋅= CDdtda , (3)

where D and φ are the material dependent parameters. In order to estimate creep crack growth during dwell periods (under the steady state operational conditions), the parameter C* is evaluated using reference stress method:

2

ref

Icrefref

KC

σ

⋅ε⋅σ=∗ & (4)

The evaluation method of the reference stress, σref , will be discussed below in greater detail. The turbine component integrity evaluation uses the widely used Failure Assessment Diagrams (FAD techniques) (see, e.g. R6 and SINTAP methodologies). This approach enables integrity analysis of the components with defects due to brittle fracture and plastic collapse at the same time. The FAD locus divides the area of the dimensionless co-ordinate frame (Sr , Kr) into “safe” and “unsafe” regions (see Fig. 4).




0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Sr

Kr

UNSAFE REGION

SAFE REGION

Kr = FADSSY(Sr)Kr = FADSSY(Sr)

Kr = FAD(Sr)Kr = FAD(Sr)SSY approachapplication area

SSY approachapplication area

Fig. 4: FAD Diagrams

Once the brittle fracture parameter Ic

Ir K

KK = and the plastic collapse parameter

YrS

σσ

= for the case under investigation are calculated, the position of the evaluation

point (Sr , Kr) relative to the FAD locus is obtained. Taking into account the main failure mechanism, three limit state (failure) functions are used:

• The design of steam turbine rotors and shrunk-on disks (nuclear units) neglects, in a wide range of operational speeds, the occurrence of plastic collapse. Therefore the reduced failure assessment is predominantly based on linear elastic fracture mechanics (LEFM) with small plasticity effect correction at the crack tip (small scale yielding), (line FADSSY, Fig. 4). Thus, the resulting limit state function has the form:

)geometry,,,a,(KK)geometry,,,a,,K( YIIcYIc σξσ−=σξσΨ . (5) • Low pressure rotors and modern shrunk-on disks are made of materials with

high ductility. This leads to the necessity of applying the limit state function based on the J-Integral concept, (without the plastic collapse load criterion):

)geometry,,,,a(JJ)geometry,,,a,,J( YIcYIc σσξ−=σξσΨ (6)




• Stationary steam turbine components like cylinder casings, valve casings and piping have a common feature: they have a finite geometry (wall thickness) with respect to the system “crack – remaining ligament”, so that the concept of plastic collapse can be applied to characterise the remaining through-the-wall plastification. Limit state function is based on the failure assessment diagram (depending on the chosen R6 Option for the case under investigation), or e.g.:

π

π−=σξσΨ

r

rrYIc

S2

secln8

SK),,a,,K( (7)

2.2. Probabilistic approach Denote, by )x,...,x( N1 , an N-dimensional random vector, characterising within the framework of the underlying model, the uncertainties in the geometry, material properties, applied loads and existing/assumed defects which can be modelled as random variables. The probability of failure is defined as

∫≤Ψ

=0)x,...,x(

N1N1f

N1

dx...dx)x,...,x(fP (8)

with )x,...,x( N1Ψ being the limit state function (failure surface), and )x,...,x(f N1 being the joint probability density function of )x,...,x( N1 . A number of commonly used approximation methods are available for performing the integration of the above multidimensional integral in probabilistic uncertainty analysis.

2.2.1. Mean value estimation method (MVM)

MVM typically involves construction of the Taylor expansion of the limit state function Ψ(x1,...,xN) around the mean values ix of random uncorrelated variables ( N,...,1i = ). The first and second moments of resulting approximation are then used to calculate the failure probability [2]:

( ) ( )( )Nj

N

1i

N

1jii

ji

N12

ii

N

1i i

N1N1N1 xxxx

xx)x,...,x(

!21xx

x)x,...,x()x,...,x()x,...,x( −−

∂∂Ψ∂

+−∂

Ψ∂+Ψ≈Ψ ∑∑∑

= ==

The expectation of this expression or the first moment is:

[ ] ( )( )Nj

N

1i

N

1jii

ji

N12

N1N1N1 xxxxxx

)x,...,x(!2

1)x,...,x()x,...,x(E)x,...,x( −−∂∂

Ψ∂+Ψ≈Ψ=Ψ ∑∑

= =

The same procedure applied to the expression for the second moment or variance yields:

[ ] [ ] [ ]∑∑∑= ==

∂Ψ∂

∂Ψ∂

+

∂Ψ∂

≈ΨN

1i

N

1j j

N1

i

N1ji

2

i

N1N

1iiN1 x

)x,...,x(x

)x,...,x(x,xCov2x

)x,...,x(xV)x,...,x(V

If the limit state function Ψ(x1,...,xN) is normal distribution, one can evaluate the failure probability as:




( ) ( ) ( )[ ]( )[ ]

( )[ ]( )[ ]

[ ][ ]

,)x,...,x(V

)x,...,x(EP

x,,xVx,,xE0

x,,xVx,,xEx,,x

Pr0x,,xPr

N1

N1f

21

21

21

212121

Ψ

Ψ−Φ≈

Ψ

Ψ−<

Ψ

Ψ−Ψ=<Ψ

K

K

K

KKK

where Φ(•) stands for the standard normal cumulative density function.

2.2.2. First-order reliability methods (FORM)

FORM can be thought of as an extension of the mean value estimation method. The first-order approximation of the limit state function (Taylor expansion around the design point) is considered at the closest surface point (most probable point (MPP) or design point) to the origin of the standard Gaussian space. First, the original random variables )x,...,x( N1 are transformed into independent standard Gaussian variables )u,...,u( N1 . The original limit state function is then mapped onto the new limit state function 0)u,...,u( N1U =Ψ . Secondly, the point on the surface (in standard Gaussian space) with the shortest distance to the origin is determined. This point is referred to as design point and the distance from the design point to the origin is called reliability index, β. The probability of failure Pf is then approximated by:- ( )β−Φ=<Ψ 0)u,...,u(Pr N1U

2.2.3. Monte Carlo simulation method

To evaluate the multidimensional integral

∫≤Ψ

=0)x,...,x(

N1N1f

N1

dx...dx)x,...,x(fP ,

n realizations of the random vector of model variables ( ) ( )( ) ( ) ( )( )nN

n1

1N

11 x,...,x,,x,...,x K are

generated. Corresponding output samples ( ) ( ) ( ) ( ) ( ) ( ) )x,...,x(,),x,...,x( nN

n1

n1N

11

1 ΨΨ K are counted as follows:

[ ]∑=

≤ΨΩ≈n

1iN1f 0)x,...,x(

n1P

where




[ ] ≤Ψ

=≤ΨΩotherwise0

,0)x,...,x(if10)x,...,x( N1

N1

The approximation approaches the exact value of failure probability when the number of trials n grows to infinity. An improvement of the crude Monte Carlo method is the Monte Carlo simulation with the importance sampling. Using FORM, the design or most probable point can be obtained and the samples are taken around the “important “ MPP with importance-sampling density.

2.2.4. Random Variables

In this section, the employed probability distributions are presented. In the application examples, variables are assumed to be uncorrelated.

• Fracture toughness

The fracture toughness for the casing steels follows the two or three parameter Weibull distribution with the density taken in general form:

( )

η

γ−−

η

γ−ηα

=−α

Ic1

IcIc

Kexp

KKf

with position parameter, γ, shape parameter, α, and scale parameter, η. The fracture toughness of the rotor steel is assumed to follow a lognormal density distribution:

( ) ( )

⋅

−−

π⋅⋅= 2

K

2KIc

KIcIc

Ic

Ic

Ics2

m)Kln(exp

2sK1Kf

• Yield strength

For the yield strength values a distribution with a lognormal density is assumed:

( ) ( )

⋅

−σ−

π⋅⋅σ=σ

σ

σ

σ2

2Y

YY

Y

Y

Ys2

m)ln(exp

2s1f

with a coefficient of variation of 0.12.

• Crack growth rate

The fatigue and creep crack growth data are subject to considerable scatter. Fatigue crack growth rates are correlated by a Paris-Erdogan-type relationship of the form:




( )nIKC

dNda

∆⋅= (10)

The scatter of the parameter C is fitted by a lognormal distribution. The n parameter is assumed to be constant. Steady state creep crack growth is modelled by a relationship of the form

( )φ∗⋅= CDdtda , (11)

approximating the D parameter as lognormal variable and φ as a constant.

• Crack initiation probability

Based on the field data on the number of indications found during inspections of number of different rotor zones at different locations, the crack initiation probability qi can be estimated as a binomially distributed variable. For the components with indications found during the inspection, the crack initiation probability is set to be 1.

• Crack size and shape probability

The crack size determination is based on the Flat-Bottom Hole (FBH) inspection methodology of crack detection by non-destructive examination technique. The FBH diameter is multiplied by a correction factor followed by reshaping of FBH into elliptically shaped crack with the area corresponding to the area of the FBH with the corrected diameter. The crack shape description based on the crack depth to crack length ratio is assumed to be distributed normally with the mean of 0.4 and the standard deviation of 0.1.

• Applied Load

For the description of the applied load randomness, a normal distribution with a coefficient of variation of 0.1 is chosen. The mean value is calculated using on the results of the two- or three-dimensional Finite Element (FE) modelling.

3. Application examples

3.1. Steam intermediate pressure turbine rotor

3.1.1 Reliability based design example

As an example, an intermediate pressure rotor of non-Siemens PG design is considered. The rotor was manufactured in 1967 from 1% Cr Steel for the operational conditions with 535°C and 40 bar of (reheated) steam temperature and steam pressure respectively.




During the ultrasonic non-destructive examination of the rotor, some indications of the FBH diameter of 5 mm were found at the rotor axis in the steam inlet area. To analyze the operational limits for the future operations consisting of the allowable operational cycle number and allowable operational hours, the integrity analysis of the rotor was performed using the probabilistic approach. The transient and steady state operational conditions are controlled by the thermocouples which are installed in the stationary turbine components. Based on this information the thermal and mechanical boundary conditions needed for the computational purposes can be obtained (see Fig 5).

Steam temperatureSteam temperature

Componentsurface temperature

Componentsurface temperature

Componentaveraged temperature

Componentaveraged temperature

Fig. 5: Example of the measured steam temperature start-up time dependence The results of recent re-calculation of the rotor for the designed conditions are shown in Figs. 6 and 7. The temperature distribution in the rotor under steady state operational conditions shows that the inlet part is operated in a temperature range where the creep mechanisms under steady state stress are activated.

Fig. 6: Steady state temperature distribution




The original design (1967) was based on the standard, at the time, practice to assess safety by a comparison of applied load and material resistance (for both time dependent and time independent cases). Fig. 7 shows the tangential and the effective (von Mises) stress distribution under steady state operational conditions after approximately 300,000 operational hours.

Fig. 7: Distribution of the tangential and von Mises stress after 300,000 OH

• Applied load

The applied load can be summarized as operational cycling consisting of steady state operational conditions and transient operational conditions (turbine start-ups and shutdowns, load changes), (Fig. 8).

3.1.2 Failure assessment example

Deterministic model

Using the estimate for the C* based on the reference stress σref [1], and assuming the secondary creep in the form of Norton-Bailey relationship

2nmref

cref tA ⋅σ⋅=ε& , (12)

the following expression for C* can be obtained: 2I

n2mref KtAC ⋅⋅σ⋅= −∗ (13)

For the embedded elliptical indication under the far field stress σref, the stress intensity factor for the rotor axis location can be assumed to take a simplified form:




⋅π

⋅σ=

caQ

aK refI (14)

The following relationship is used for C*:

⋅π

⋅⋅σ⋅=∗

caQ

atAC 2nmref . (15)

Finally,

( ) ∫∆

⋅

⋅σ⋅π⋅+∆⋅=

statesteady 21

t

0

nmref

nI dt

caQ

atAKCdNda , (16)

Here the stress intensity range for LCF crack growth involves the transient operational conditions such as turbine start-up and shutdown (Fig. 8).

Rotor surfacestress

Rotor axisstress

Steam temperature

Rotor surfacetemperature

Rotor axistemperature

Time

Tem

pera

ture

Com

pres

sive

stre

ss

Time

Tens

ilest

ress

Fig. 8: Typical steam turbine load cycle: major load change temperatures and stresses

For the component regions, where the low cycle fatigue is the dominant crack growth mechanism, the procedure of critical crack depth and length calculation can be simplified as follows:

( )

( )nIc

nIa

KCdNdc

KCdNda

∆⋅=

∆⋅= ⇒

2n

ac

dcda

= ⇒ ( )

n22

2n1

02n1

02n1

acaac++++

−+= (17)




The critical crack depth acrit can be found by solving the implicit equation

( )

⋅π

⋅σ=

critcrit

crit

critrefIc

aca

Q

aK , (18)

where KIc is the fracture toughness of the material. The critical crack length ccrit is obtained automatically. The procedure can be applied straightforwardly for the cases with the non-uniform, (with spatial gradient), far field stress distributions.

Calculation of σref

The procedure of the σref calculation consists of the two steps. First step involves the simple stress linearization procedure. Its goal is to express the far field stress distribution Σ(x), obtained from Finite Element computation, as a combination of the reference membrane stress σm and reference bending stress σb components needed to calculate the stress intensity factors.

Evaluation path

Stre

ss

x1 x2

Σ(x) FEM-calculated

Α⋅x+B

Fig. 9: On the linearisation procedure

Using two equations expressing the equivalence of the forces and moments over the evaluation path between x1 and x2 (Fig. 9):




( ) ( )

( ) ( )∫∫

∫∫

⋅+⋅⋅=⋅Σ⋅

⋅+⋅=⋅Σ

2

1

2

1

2

1

2

1

x

x

x

x

x

x

x

x

dxBxAxdxxx

dxBxAdxx

, (19)

the following formulas for σm and σb can be obtained:

( ) ( )

m1refb

2222

refm

bxA

dxxxx

1dxBxAxx

1

σ−+⋅=σ

⋅Σ⋅−

=⋅+⋅⋅−

=σ ∫∫ (20)

On the second step, one evaluates the length L of the evaluation path between x1 and x2, since an excessively long evaluation path can lead to the non-conservative stress intensity factor value.

Using the singular stress distribution field at the front of the crack tip ( )ax2

KI

−⋅π⋅ ,

the intersection point with the far filed stress can be easily found (Fig. 10).

a L

Far field stressDistribution (FEM)

Singular (crack tip)stress distribution

Evaluation path length

Stre

ss

Fig. 10: On the reference stress definition




Solving two implicit equations (with two unknowns K and L) expressing the equivalence of the forces and moments over the evaluation path L, the stress intensity factor K and the so called “influence length” L can be easily calculated:

( )( )∫∫ ⋅Σ=

−⋅π⋅

L

0

L

a

dxxdxax2

K (21a)

( )( )∫∫ ⋅⋅Σ=

−⋅π⋅

⋅ L

0

L

a

dxxxdxax2

xK (21b)

Instead of K as a unknown, the handbook stress intensity factor can be used, in this case only the ”influence length” L is to be calculated. To give parameter L physical meaning we note that the crack tip “feels” the influence of the far stress field distribution between 0 and L only. The linearisation path is to be limited by the length L. The application of the method for the embedded cracks is straightforward.

Probabilistic model

The plastic collapse stress for the rotors is assumed to be a 0.2% yield stress offset. Then the limit (failure) function is given by:

σσ

⋅π

⋅

π⋅

σσ

=−1

Y

YIc

I

2cosln8

KK (22)

The operational cycling regime is assessed deterministically and consists of: turbine cold start-up, dwell period, turbine shutdown (Fig. 8). During the dwell period one hot start-up is considered in addition. The corresponding crack growth is:

( ) ( )[ ]∑=

∆+=N

1ii0 t,Naat,Na

with

( )[ ] [ ]( ) ∫∆+

⋅

⋅

σ⋅π⋅+∆⋅=∆statesteadyi

i

21

tt

t

i

i

in

mref

niIi dt

caQ

atAKCt,Na

To achieve an independence of the limit state function on its explicit form, all variables with non-standard distributions (Table 1) are to be transformed into variables with the standard normal distribution via comparison of the cumulative forms of the distributions (in the case of single and independent variables):




( ) ( )( )( )uFx

uxF1 Φ=

Φ=−

where x is the variable under consideration and u is standard normally distributed variable with mean value 0 and standard deviation 1. The following distributions were assumed in the probabilistic analysis1:

Table 1: Basic variables

Fracture toughness

Yield stress

Initial crack size

Fatigue CG parameter C

Creep CG parameter D

Stress Stress range

Lognormal Lognormal Exponential Lognormal Lognormal Normal Normal

3.1.3 Target probability example

The damage probability is usually determined from the combined probabilities of turbine failure P1, missile striking a structure containing components critical to safety P2, and penetration or significant damage occurring in the structure and in the component P3 [3]:

3214 PPPP ⋅⋅= The analysis is limited only to assessment of P1, whereby the P1 involves in original sense of [2] the probability of the penetration of the surrounding turbine inner and outer casing. Based on [2], the target probability rate can be assumed:

141 yr10P −− ⋅=

The value of the cumulative failure probability over 10 turbine operating years amounts to Pf = 10-3 This corresponds to the reliability index β = 3.1 related to an event with severe consequences of failure [1].

1 Due to the absence of the detailed material data information regarding the properties degradation, some conservative estimates of the probability distributions were made.




3.1.4 Calculation result

Fig. 11 shows the results of the calculations performed using FORM and Monte-Carlo Simulations. The Monte-Carlo simulation results were used as an “exact” value of the failure probability to control the results obtained using FORM technique.

1,0E-06

1,0E-05

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

0 2 4 6 8 10

Operational Years

Failu

re P

roba

bilit

y

FORM

MC

10-4 yr-1

Fig. 11: Results of Monte-Carlo simulations and calculated with FORM, compared

with the target probability rate The plot can give guidance on the rational selection of the next inspection time; i.e., the next inspection is to be done after approximately 7 operational years (with 640 cold and hot start-ups) corresponding with the acceptable value of the failure probability (target probability) 7.4·10-4.

3.2. Steam turbine valve casing

As an example of the stationary turbine component, an intermediate pressure valve of non-Siemens PG design is considered. The valve casing was manufactured in 1967 from 1% Cr Steel for the operational conditions with 550°C and 50 bar of (reheated) steam temperature and steam pressure respectively. During the inspection, the indication with the length of 32 mm was found (Fig. 12) at the inner surface of the valve wall. The crack depth of 12 mm was measured as well. The reason of the crack detected was the old welding zone locally repaired (probably manufacture welding) with the degraded material properties of the heat affected zone (HAZ), which was opened as a result of the transient operation.




The experience shows, that the cast steel casings contains material defects from the manufacturing process itself due to the complex design and the cross-sectional transitions areas and ribs, i.e. those stress concentration locations, which are exposed to transient stresses. Two options were considered: to repair the casing by welding (time consuming) or, alternatively, to release the casing with the existing crack without any repair for some operational time. To analyze the operational limits for the future operations, the integrity assessment of the crack was performed in the framework of the probabilistic approach.

Fig. 12: Surface crack found during inspection

The transient stresses and steady state operational conditions are controlled by the thermocouples which are installed directly in the valve wall.




Fig. 13: Steady state temperature distribution

Fig. 13 shows the three-dimensional geometry model and the temperature distribution at the steady state operational conditions. The casing material is operated in a temperature range where the creep mechanisms, under steady state stressing, are activated. The stress level (max principal stress component) at the start of the service life and relaxed after 300000 operational hours are shown in Fig. 14.




elastic

after 300,000 OH

Fig. 14 Elastic (left) and relaxed after 300,000 OH (right) distribution of the max

principal stress

3.2.1 Failure assessment example

The failure assessment procedure discussed previously can be applied to the integrity assessment of the valve casing. Due to the finite wall thickness (compared to the embedded crack in the rotor), the stress gradient has to be considered. Therefore, the stress intensity factor for a semi-elliptical surface crack in a finite width plate under bending and tension [4,5] will be used:

( )6507.1bmI

ca4673.11

FaHK

⋅+

⋅⋅π⋅σ⋅+σ= , (23)

where F and H are the geometric correction factors. This stress intensity factor solution was used to derive the stress evaluation path length, linearisation procedure for the σref expression and the reference membrane stress σm and reference bending stress σb components similarly to the case of the rotor procedure considered in the previous sections.




The following distributions were assumed in the probabilistic analysis2:

Table 2: Basic variables

Fracture toughness

Yield stress

Initial crack size

Fatigue CG parameter C

Creep CG parameter D

Stress Stress range

Two-parametric Weibull

Lognormal Exponential Lognormal Lognormal Normal Normal

3.2.2 Target probability example

Based on the statistics of pressure vessel failures [6], the target probability rate can be assumed to be 10-5 per vessel year. The value of the cumulative failure probability over 10 operating years amounts to Pf = 10-4. This corresponds to the reliability index β = 3.7, related to an event with severe consequences of failure [2].

3.2.3 Calculation results

Fig. 15 shows the results of the calculations performed using FORM, and Monte-Carlo simulations as an “exact” solution.

2 Due to the absence of the detailed material data information regarding the properties degradation, some conservative estimates of the probability distributions were made.




1,0E-06

1,0E-05

1,0E-04

1,0E-03

0 1 2 3 4 5 6

Operational years

Failu

re p

roba

bilit

y

FORM

MC

10-5 yr-1

Fig. 15: Results of Monte-Carlo simulations and calculated with FORM, compared with the

target probability rate.

The intersection point shows, that the allowable (target ) failure probability value is met after approximately 5 operational years, including 330 operational cycles of cold and hot starts-ups and shutdowns. The corresponding assumed, value of the failure probability (target probability) is 5·10-5.

4. Summary In this paper, the integrity assessments of steam turbine components – rotor and valve casing – within the framework of probabilistic fracture mechanics methodology are presented. The analysis is performed using both First Order Reliability Method and Monte-Carlo simulation. The probabilistic procedure provides the means of incorporating uncertainty and material property scatter into analysis, with “release for further operation”/”retire”/”replace” decision based on an acceptable level of reliability. The acceptable level of failure probability used in the analysis is to be set separately, and can only be established by authorized personnel. The simple analytical stress linearisation and stress evaluation path length procedures for the stress distributions computed numerically (e.g. with Finite Element Method) are presented. Within these procedures, the membrane and bending stress components as well as the reference stress can be evaluated to calculate the stress intensity factors.




References

[1] Webster, G.A.; Nikbin, K.M.; Chorlton, M.R.; Celard, N.J.C.; Ober, M.: A comparison of high temperature defect assessment methods, Mater High Temp, 15, 1998

[2] General principles on reliability for structures, International standard ISO 2394, Second edition, 1998-06-01

[3] Bush, S.H.: Probability of damage to nuclear components due to turbine failure, Nuclear Safety, 14(3), 1973

[4] Newman, J.C.; Raju, I.S.: An empirical stress intensity factor equation for the surface crack, Engng Fract Mech 15, 1981

[5] Fett, T.; Munz, D.: Stress intensity factors and weight functions, Advances in fracture series, Computational Mechanics Publications, 1997

[6] Bush, S.H.: Statistics of pressure vessel and piping failures, Journal of Pressure Vessel Technology, 110, 1988

[7] ABAQUS/Standard, Version 6.3.1, HKS


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LIFE ASSESSMENT OF STEAM TURBINE …ommi.co.uk/PDF/Articles/166.pdf · Life assessment of steam turbine components based on probabilistic procedures OMMI, (Volume 5, Issue 2) August