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Guidelines for Performing

Probabilistic Analyses of BoilerPressure Parts

Technical Report

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EPRI Project ManagerR. Tilley

EPRI 3412 Hillview Avenue, Palo Alto, California 94304 PO Box 10412, Palo Alto, California 94303 USA800.313.3774 650.855.2121 [email protected] www.epri.com

Guidelines for PerformingProbabilistic Analyses of Boiler

Pressure Parts

1000311

Final Report, December 2000


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CITATIONS

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Engineering Mechanics Technology4340 Stevens Creek Blvd., Suite 166San Jose, CA 95129

Principal InvestigatorD. Harris

This report describes research sponsored by EPRI.

The report is a corporate document that should be cited in the literature in the following manner:

Guidelines for Performing Probabilistic Analyses of Boiler Pressure Parts, EPRI, Palo Alto,CA: 2000. 1000311.


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v

REPORT SUMMARY

Using probabilistic methodologies for life assessment of boiler components provides a morerealistic basis for managing the inspection, maintenance, repair, and replacement actions forthose components. Such a realistic basis also couples with economic parameters to allow utilitiesto make better overall decisions in their efforts to reduce operating and maintenance costs. In thecompetitive environment for power generation, more accurate assessments of risks and benefitsmust be incorporated into utility decision making. Probabilistic techniques are a proven way torefine the basis for such decisions. This document reviews some life prediction methodologiesand discusses relevant statistical principles. It provides guidelines on the generation and use ofsuch results in maintenance and inspection planning.

Background

In the past, maintenance decisions and corresponding expenditures could be based onengineering analyses using approximate models for component damage mechanisms and addingconservative safety factors to account for both model and data inaccuracies. However, because ofthe emerging competitive environment and more financially oriented management, utilities arefinding that such conservative approaches are non-optimum in balancing costs and benefits.Decisions need to be justified from an economic point of view that better incorporates risks ofequipment failure. Probabilistic techniques have been used in other industries to provide such a

risk-based bridge to economic decision making. Recent EPRI analytical models such as theBoiler Life Evaluation and Simulation System (BLESS) incorporate options to allowprobabilistic analyses. Utilities can now use these options to improve decisions on boilercomponent inspections, maintenance, repair, and replacement.

Objectives

To review probabilistic methodologies for use in boiler component life management

To provide guidelines on the generation and use of such results in maintenance andinspection planning

Approach

Using a probabilistic approach, the probability of failure within a certain time range can beestimated, rather than providing a deterministic failure time. The deterministic result uses asingle set of input variables and is used with a safety factor to account for inaccuracies in themodel and its input. Probabilistic results are generated by running a series of analyses in whichkey input parameters are varied to reflect the actual variation occurring in the population ofsimilar components. These results are generally expressed as failure probabilities (or failurerates) versus time. The time for remedial action can be keyed to the time at which failure rate


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vi

becomes excessive. This document concentrates on boiler pressure parts, but much of thediscussion is readily applicable to other components that degrade due to material aging.

The probabilistic approach reviewed in this document is based on an underlying mechanisticmodel of lifetime. Representative lifetime models are reviewed with a focus on boiler pressureparts. Special attention is paid to probabilities associated with crack initiation and growth, which

are leading causes of material degradation and component failure.

Statistical background information is provided in the area of probabilistic structural analysis. Thedevelopment of probabilistic models of component lifetimes is also discussed.

Results

When using probabilistic methodologies for component life management, the following pointsneed to be kept in mind:

Lifetime models are available

Scatter and uncertainties in inputs to the models usually preclude accurate deterministic

results

Probabilistic lifetime models can be obtained by quantifying the scatter and uncertainty andincorporating them into the underlying deterministic lifetime model

Numerical procedures for generation of failure probabilities are available, and numericalresults can usually be obtained using a personal computer

Probabilistic results can be used in analyses of expected future operating costs, which are ofgreat use in component life management

EPRI PerspectiveAs utilities come under increased pressure to reduce costs and extend the lifetime of plant

components, interest has increased in procedures for rationally planning inspection andmaintenance. EPRI and other organizations have facilitated the use of risk-principles to prioritizemaintenance actions. Building on software tools such as the BLESS code and processes such asthose developed for extending intervals between turbine maintenance outages (TURBO-X), thisreport provides guidance for the use of probabilistic approaches in managing boiler componentlife. This information can then be incorporated into the component cost-benefit models tooptimize overall costs.

KeywordsProbabilistic analysisBoiler components

Life assessment


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EPRI Licensed Material

vii

CONTENTS

1INTRODUCTION.................................................................................................................. 1-1

2REVIEW OF DETERMINISTIC LIFE PREDICTION PROCEDURES FOR BOILERPRESSURE PARTS ............................................................................................................... 2-1

2.1 Crack Initiation........................................................................................................... 2-1

2.1.1 Fatigue Crack Initiation ......................................................................................... 2-1

2.1.2 Creep Crack Initiation ........................................................................................... 2-3

2.1.3 Creep/Fatigue Crack Initiation............................................................................... 2-6

2.1.4 Oxide Notching ..................................................................................................... 2-7

2.2 Crack Growth ............................................................................................................ 2-8

2.2.1 Crack Tip Stress Fields......................................................................................... 2-8

2.2.2 Crack Driving Force Solutions............................................................................. 2-10

2.2.3 Calculation of Critical Crack Sizes ...................................................................... 2-13

2.2.4 Fatigue Crack Growth......................................................................................... 2-13

2.2.5 Creep Crack Growth ........................................................................................... 2-14

2.2.6 Creep/Fatigue Crack Growth .............................................................................. 2-15

2.3 Simple Example Problems....................................................................................... 2-18

2.3.1 Fatigue of a Crack in a Large Plate..................................................................... 2-18

2.3.2 Creep Damage in a Thinning Tube ..................................................................... 2-19

3SOME STATISTICAL BACKGROUND INFORMATION...................................................... 3-1

3.1 Probability Density Functions .................................................................................... 3-1

3.2 Fitting Distributions.................................................................................................... 3-4

3.3 Combinations of Random Variables ........................................................................ 3-11

3.3.1 Analytical Methods.............................................................................................. 3-11

3.3.2 Monte Carlo Simulation Principles ................................................................... 3-12

3.3.3 Monte Carlo Confidence Intervals .................................................................... 3-16

3.3.4 Monte Carlo Simulation Importance Sampling ................................................. 3-19


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3.3.5 First Order Reliability Methods Basics.............................................................. 3-21

3.3.6 First Order Reliability Methods General ........................................................... 3-27

4DEVELOPMENT OF PROBABILISTIC MODELS FROM DETERMINISTIC BASICS.......... 4-1

4.1 Discussion................................................................................................................. 4-14.2 Simple Example Problems......................................................................................... 4-4

4.2.1 Fatigue Crack Growth in a Large Plate ................................................................. 4-4

4.2.2 Creep Damage in a Thinning Tube ..................................................................... 4-11

4.2.3 Hazard Rates...................................................................................................... 4-15

4.3 Inspection Detection Probabilities............................................................................ 4-18

5EXAMPLE OF A PROBABILISTIC ANALYSIS ................................................................... 5-1

5.1 Gathering the Necessary Information ........................................................................ 5-1

5.1.1 Component Geometry and Material ...................................................................... 5-1

5.1.2 Operating Conditions ............................................................................................ 5-3

5.2 Performing the Analysis............................................................................................. 5-6

5.3 Combining Data....................................................................................................... 5-12

6USE OF PROBABILISTIC RESULTS.................................................................................. 6-1

6.1 Target Hazard Rates ................................................................................................. 6-1

6.2 Economic Models ...................................................................................................... 6-5

7CONCLUDING REMARKS .................................................................................................. 7-1

ADETAILS OF BLESS EXAMPLE ........................................................................................A-1

BREFERENCES....................................................................................................................B-1


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LIST OF FIGURES

Figure 2-1 Strain Life Data for A106B Carbon Steel in Air at 550F (290C) With MedianCurve Fit [From Keisler 95] .............................................................................................. 2-2

Figure 2-2 Creep Rupture Data for 1 Cr Mo With Curve Fit [From Grunloh 92](1 ksi=6.895MPa) ............................................................................................................ 2-4

Figure 2-3 Creep/Fatigue Damage Plane Showing Combinations Corresponding toCrack Initiation................................................................................................................. 2-6

Figure 2-4 Crack-Like Defect Initiated by Oxide Notching........................................................ 2-7

Figure 2-5 Depiction of Procedure for Determination of Oxide Thickness for a Time at TloFollowed by a Time at Thi................................................................................................. 2-8

Figure 2-6 Coordinate System Near a Crack Tip..................................................................... 2-9

Figure 2-7 Through-Crack of Length 2a in a Large Plate Subject to Stress . ....................... 2-11

Figure 2-8 Single Edge-Cracked Strip in Tension With J-Solution ......................................... 2-12

Figure 2-9 Fatigue Crack Growth as a Function of the Cyclic Stress Intensity Factor for2 Cr 1 Mo at Various Temperatures [Drawn From Viswanathan 89]........................... 2-14

Figure 2-10 Creep and Creep/Fatigue Crack Growth Data and Fits. Left Figure Is forConstant Load and Right Figure Is for Cyclic Load With Various Hold Times[From Grunloh 92].......................................................................................................... 2-17

Figure 2-11 Half-Crack Length as a Function of the Number of Cycles to 20 ksi(137.9 MPa) for Example Fatigue Problem.................................................................... 2-19

Figure 2-12 Time to Failure as a Function of the Wall-Thinning Rate for the CreepRupture Example Problem (1 Mil/Year=25.4 m/yr)....................................................... 2-21

Figure 3-1 Plot of Data of Table 3-2 on Log-Linear Scales ...................................................... 3-8

Figure 3-2 Lognormal Probability Plot of Data of Table 3-2 ..................................................... 3-8

Figure 3-3 Normal Probability Plot of Data of Table 3-2........................................................... 3-9

Figure 3-4 Cumulative Probability of the Sum of Two Lognormals as Computed byNumerical Integration and Monte Carlo With 20 and 500 Trials ..................................... 3-15

Figure 3-5 Values of Factors in Table 3-5 Divided by the Number of Failures ....................... 3-19

Figure 3-6 Cumulative Probability of the Sum of Two Lognormals as Computed byNumerical Integration and Monte Carlo Simulation With 20 Trials, With and WithoutImportance Sampling..................................................................................................... 3-21

Figure 3-7 Pictorial Representation of Joint Density Function in Unit Variate SpaceShowing Failure Curve and Most Probable Failure Point (MPFP).................................. 3-22

Figure 3-8 Plot of the Performance Function in Reduced Variate Space for the ExampleProblem of Two Lognormals for z=2. The Origin Is at the Upper Right Corner, andthe Most Probable Failure Point Is Indicated.................................................................. 3-24


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Figure 3-9 Cumulative Probability of the Sum of Two Lognormals as Computed by theFirst Order Reliability Method and Numerical Integration ............................................... 3-25

Figure 3-10 Example of a Performance Function With the Vector Normal to the Axis of

One of the Variables Showing the Insensitivity of to That Variable.............................. 3-26

Figure 3-11 The Direction Cosines of x and y for the Example Problem of the Sum of

Two Lognormals ............................................................................................................ 3-27Figure 3-12 Diagrammatic Representation of a Procedure for Finding Most Probable

Failure Point .................................................................................................................. 3-31

Figure 4-1 Probabilistic Treatment of Strain-Life Data for A106B Carbon Steel in Air at550F (288C) Showing Various Quantiles of the Keisler Curve Fit [From Keisler 95]...... 4-3

Figure 4-2 Cumulative Distribution of Critical Crack Size for the Fatigue Crack GrowthExample Problem (104Trials) .......................................................................................... 4-6

Figure 4-3 Lognormal Probability Plot of the Failure Probability as a Function of theNumber of Cycles for the Fatigue Example Problem (104Trials)...................................... 4-7

Figure 4-4 Plot on Lognormal Scales of the Distribution of Cycles to Failure for the

Example Fatigue Crack Growth Problem and the Same Problem With the Mean andStandard Deviation Divided by Two (106Trials) ............................................................... 4-7

Figure 4-5 Cumulative Failure Probabilities in the Lower Probability Region of theFatigue Crack Growth Example Problem With Two Distributions of the FractureToughness (10

6Trials)..................................................................................................... 4-8

Figure 4-6 Cumulative Failure Probabilities in the Lower Probability Region for theFatigue Crack Growth Example Problem. Solid Line is Monte Carlo With 106Trials,Points are Results From Rackwitz-Fiessler...................................................................... 4-9

Figure 4-7 Cumulative Failure Probabilities in the Lower Probability Region for theFatigue Crack Growth Problem as Obtained Using Importance Sampling With 1000Trials With Different Shifts in Parameters of Input Random Variables. Points are

From Rackwitz-Fiessler. ................................................................................................ 4-10Figure 4-8 Direction Cosines for the Fatigue Crack Growth Example Problem as a

Function of the Number of Cycles to Failure .................................................................. 4-11

Figure 4-9 Creep Rupture Data for 1 Cr 1/2 Mo as Obtained From Van Echo 66 WithLeast Squares Linear Fit (Stress in ksi, T

ain Degrees Rankine, t

Rin Hours).................. 4-13

Figure 4-10 Cumulative Distribution of the Random Variable A Used to Describe theScatter in the Larson-Miller Data for 1 Cr 1/2 Mo Steel .............................................. 4-13

Figure 4-11 Results of Example Problem of Creep Damage in a Thinning Tube asObtained by Monte Carlo Simulation With 10

4Trials ...................................................... 4-15

Figure 4-12 Hazard Function vs. Cycles for the Fatigue Crack Growth Example Problem..... 4-17

Figure 4-13 Failure Rate as a Function of Time for the Thinning Problem Only.Histogram Results Are for the Smallest 2,000 Failure Times in 106Trials, the Line Isfor the Closed Form Result. ........................................................................................... 4-18

Figure 4-14 Nondetection Probability as a Function of Crack Depth Divided by PlateThickness for Fatigue Cracks in Ferritic Steel for Three Qualities of UltrasonicInspection ...................................................................................................................... 4-21

Figure 4-15 Failure Probability as a Function of Time for the Inspection ExampleProblem......................................................................................................................... 4-22


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Figure 4-16 Comparison of Fatigue Example Problem of Infinite Plate With FiniteThickness Results With No Inspection Using PRAISE Code.......................................... 4-23

Figure 5-1 Schematic Representation of Typical Header With Illustration of SegmentConsidered in Analysis .................................................................................................... 5-2

Figure 5-2 Summary of Header Geometry Analyzed (Length Dimensions in Inches)............... 5-3

Figure 5-3 Log-Linear Plot of Probability of Leak as a Function of Time for OxideNotching Initiation and Creep Fatigue Crack Growth in Header Example Problem(2,000 Trials) ................................................................................................................... 5-7

Figure 5-4 Lognormal Probability Plot of BLESS Results for the Header ExampleProblem (Probability in Percent) ...................................................................................... 5-7

Figure 5-5 Lognormal Hazard Function for the Header Example Problem............................... 5-9

Figure 5-6 Comparison of Hazard as a Function of Time as Obtained From the BLESSOutput Data and the Fitted Lognormal Relation ............................................................. 5-10

Figure 5-7 Results of Header Example Problem With Varying Shifts and Number ofTrials. The Solid Line is the Curve Fit Based on the Estimated LognormalDistribution With No Shifts (the Line of Figure 5-6). ...................................................... 5-12

Figure 6-1 Log-Log Plot of Frequency Severity Data for Boiler Components FromTable 6-2 ......................................................................................................................... 6-4


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LIST OF TABLES

Table 2-1 Table of the Function h1(,n) in the J-Solution for an Edge Cracked Plate in

Tension for Plane Strain [From Shih 84] ........................................................................ 2-12

Table 3-1 Characteristics of Some Commonly Encountered Probability Functions Usedto Describe Scatter and Uncertainty ................................................................................ 3-3

Table 3-2 Values of Cffor Fatigue Crack Growth Data ............................................................ 3-7

Table 3-3 Information on Parameters of Distribution of CfData of Table 3-2 ......................... 3-10

Table 3-4 Results of Monte Carlo Example With 20 Trials..................................................... 3-14

Table 3-5 Summary of Some of the Statistics from the Monte Carlo Trials ............................ 3-15Table 3-6 Confidence Upper Bounds on N

trp

ffor Various Numbers of Failures ...................... 3-18

Table 3-7 Coordinates and Direction Cosines of the MPFP for Example Problem Withz = 2 .............................................................................................................................. 3-25

Table 3-8 Intermediate Steps in Iterative Procedure for Finding the MPFP for theExample Problem with z=3 ............................................................................................ 3-32

Table 3-9 Intermediate Steps in Iterative Procedure for Finding the MPFP for theExample Problem With z=2 ........................................................................................... 3-33

Table 4-1 Random Variables for the Fatigue Crack Growth Example Problem........................ 4-5

Table 4-2 Steps in Estimating the Hazard Function for the Fatigue Crack Growth

Example Problem .......................................................................................................... 4-16Table 4-3 Parameters of the Equation Describing the Non-Detection Probability .................. 4-20

Table 5-1 Summary of Time-Variation of Operating Conditions ............................................... 5-5

Table 5-2 Summary of Initiation Time for Various Operating Scenarios................................... 5-8

Table 6-1 Examples of Hazards of Common Activities as Measured by Fatality Rate ............. 6-2

Table 6-2 Partial List of Boiler Component Failure Rate and Consequences........................... 6-3

Table 6-3 Calculation of Expected Cost of Continuing Operation of Example Header forAnother 20 Years............................................................................................................. 6-7

Table 6-4 Calculation of Expected Cost of Replacing Header Now and Then Operatingfor Another 20 Years........................................................................................................ 6-8


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1-1

1INTRODUCTION

Deregulation of the electric utility industry has lead to increased competition in the generation ofelectrical power, which has led to increasing need for systematic means of inspection andmaintenance planning. Power plant components are subject to aging due to a variety ofmechanisms and must occasionally be replaced or repaired. It is not economical to replace themearlier than necessary, nor is it economical to run them until they cause an unscheduled outage orsafety problem. Hence, it is desired to define an optimum time range for remedial action.

The purpose of this document is to review probabilistic methodologies for use in component life

management and to provide guidelines on the generation and use of such results in maintenanceand inspection planning. The points to be made are that:

Lifetime models are available

There are scatter and uncertainties in inputs to the models that usually preclude accuratedeterministic results

Probabilistic lifetime models can be obtained by quantifying the scatter and uncertainty andincorporating them into the underlying deterministic lifetime model

Numerical procedures for generation of failure probabilities are available, and personalcomputers are becoming so fast and cheap that generation of numerical results is usually not

a problem The probabilistic results generated (failure probability as a function of time) are of direct use

in analyses of expected future operating costs, which are of great use in component lifemanagement

This document reviews some life prediction methodologies, followed by a discussion of relevantstatistical principles. Examples of probabilistic analyses are provided, including the analysis of aheader using the EPRI BLESS software. The use of the failure probability results in arun/retirement decision is demonstrated. All of this information is available elsewhere, but not ina single convenient document. Guidance on exercising the resulting probabilistic models is givenand interpretation of the results is discussed.

Due to uncertainties and inherent randomness in parameters that determine the component life,the precise time of failure can rarely be accurately defined. To account for inherent inaccuracies,conservative safety factors are often applied to deterministic results. Use of probabilisticapproaches provides a more useful way of accounting for analysis uncertainties. Using aprobabilistic approach, the probability of failure within a certain time range can be estimated,rather than providing a deterministic failure time. The availability of probabilistic informationcan be viewed as a plus, because this can lead directly to application of risk-based concepts in


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Introduction

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run/replace/retire decisions. Risk is conventionally defined as the product of the probability offailure and the consequences of failure. Hence, one component of risk (the failure probability) isa direct outcome of probabilistic analyses. The time for remedial action can be keyed to the timeat which failure rate becomes excessive excessive being based on level of risk. Includingconsequences in the process allows attention to be focused on the items of importance. Itemswith a high failure rate but low consequences do not require the level of attention that wouldoccur if only failure rate was used in the decision process.

Another advantage of using failure probability is that it can be used to estimate the futureexpected cost of failure, which can be an important component of expected future operating cost.These expected costs can be incorporated into financial models to optimize equipment lifemanagement over an entire group of plants. These costs are expressed in terms that are readilycommunicated to utility management.

This document concentrates on boiler pressure parts, but much of the discussion is readilyapplicable to other components that degrade due to material aging.


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2-1

2REVIEW OF DETERMINISTIC LIFE PREDICTION

PROCEDURES FOR BOILER PRESSURE PARTS

The probabilistic approach reviewed in this document is based on an underlying mechanisticmodel of lifetime. Hence, such models are fundamental to the approach, and this section willreview some representative lifetime models.

Boiler pressure parts are subject to material degradation by a wide variety of mechanisms. Boilerpressure parts are considered to be headers (superheater, reheater, and economizer), tubing(superheater and reheater), and pipes. Viswanathan 89 provides a comprehensive summary ofdegradation mechanisms and life prediction approaches, with his Chapter 5 being devoted toboiler components.

Material degradation in boiler pressure parts is mostly due to creep and/or fatigue. This caninvolve crack initiation and/or crack growth. Corrosion, pitting, oxidation and erosion can alsobe problems. Oxidation and fire-side corrosion can be troublesome in tubing, but will not becovered here. Viswanathan 89 and 92 provide information on these topics. Crack initiation inheaders by oxide notching is an important degradation mechanism, and the model used inBLESS [Grunloh 92] is discussed and considered in an example problem. The life predictionmethodologies for creep and fatigue are quite different. Also, the procedures to be employed forcrack initiation are quite different than for crack growth. The crack growth methodologies

discussed here are based on fracture mechanics and are applicable when the lifetime is controlledby the behavior of a single (or a few) dominant cracks.

2.1 Crack Initiation

Crack initiation can occur due to fatigue, stress corrosion cracking, creep, oxide notching or acombination of these. Stress corrosion will not be discussed here. Fatigue crack initiation occursdue to cyclic loading and may occur in the absence of time-dependent material response. Incontrast to this, creep crack initiation occurs due to time spent in the stress and temperature rangewhere time-dependent material response (creep) is important. Creep generally is not a problem inmetals at temperature less than about 1/3 to 1/2 of the absolute melting temperature. In the steels

used in electrical power generating plants, this is about 800F (427C). Oxide notching is aproblem that is aggravated by load cycling.

2.1.1 Fatigue Crack Initiation

The initiation of fatigue cracks is due to cyclic loading and can occur at temperatures well belowthe range where creep is important. This is the degradation mechanism most familiar to


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Review of Deterministic Life Prediction Procedures for Boiler Pressure Parts

2-2

engineers. The cyclic lifetime is measured in the laboratory as a function of the cyclic stress (orstrain) level, and expressed as an S-N curve. The data is most often in terms of the cycles tofailure, rather than cycles to initiation. Figure 2-1 provides an example, which is from Keisler 95

and is for A106 carbon steel in air at 550F (290C). The amount of scatter in the data is usuallylarge, and the desirability of a probabilistic approach is immediately apparent. The data in

Figure 2-1 are actually the cycles for a 25% load drop in the test, which correspondsapproximately to a 3 mm deep crack. The data is in terms of the strain amplitude (one-half thepeak-to-peak value).

The following functional form is often used to represent fatigue data:

abBN A= + Eq. 2-1

The line in Figure 2-1 is a plot of the best fit obtained for this data by Keisler and Chopra[Keisler 95], which corresponds toA=0.11,B=27.47, and b=0.534.

Figure 2-1Strain Life Data for A106B Carbon Steel in Air at 550F (290C) With Median Curve Fit[From Keisler 95]


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2-3

In cases where the cyclic stress level varies during the lifetime, such as is usually the case due todifferent loads, the cycles to failure is generally computed by use of Miners rule. The damage

per cycle of strain of amplitude is taken to be equal to 1/N(), so the damage for n cycles of

amplitude is n/N(). The total damage is then the sum for each of the contributors, and failure istaken to occur when the damage totals to unity. For instance, if a given time period consists of

nicycles of strain amplitude i, and there areLdifferent amplitudes of cycling, then the fatiguedamage for this time period is

DN

f

i

ii

L

==

( )1

Eq. 2-2

The value ofN(i)is obtained from Equation 2-1. The number of time periods to failure is the

number of time periods to reachD=1. Failure can be final failure of a specimen, the presenceof a crack, or, in the particular case of the Keisler data, a crack of about 3 mm in size.

Fracture mechanics principles can be used to then grow the crack once it initiates. Theaccumulated fatigue damage does not change the stresses in a complex body, except for thepresence of the crack.

2.1.2 Creep Crack Initiation

Creep cracks can initiate after a period of time under steady loading. The lifetimes of uniaxialtensile specimens are typically measured for a range of temperatures as a function of the appliedstress. The rupture lifetimes are often then plotted as function of a so-called Larson-Millerparameter, which is defined as

LMP T C tA R= +[ log( )] Eq. 2-3

TAis the absolute temperature. Cis evaluated as part of the curve fitting procedure. Figure 2-2 is

a typical plot of creep rupture data for 1 Cr Mo steel. A plot of the best fit is also shown, buta considerable amount of scatter is again observed, and the usefulness of a probabilistic approachis apparent.


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2-4

Figure 2-2Creep Rupture Data for 1 Cr Mo With Curve Fit [From Grunloh 92] (1 ksi=6.895MPa)

The following expression is the curve fit shown in Figure 2-2, with Cin Equation 2-3

equal to 20:

LMP= 42869 5146 956 2[log ] [log ] Eq. 2-4

The log is to the base 10, the temperature is in degrees Rankine (1R=5/9K), the stress is inthousand pounds per square inch (ksi), and the rupture time is in hours. Equations 2-3 and 2-4can be considered to provide a function that gives the rupture time as a function of stress and

temperature, tR(,T).

The time to rupture for varying stress conditions is often evaluated by the creep counterpart of

Miners rule, which is called Robinsons rule. For a set of times tispent at a stress

iand

temperature Ti, the creep damage is evaluated by use of the expression

Dt

t Tc

i

R i ii

L

==

( , )1

Eq. 2-5


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2-5

Failure is considered to occur when the creep damage reaches unity. Failure is the rupture of alaboratory specimen, or, in a larger component, can be considered to be the initiation of a creepcrack.

An alternative procedure for considering creep damage has been suggested that is usuallyassociated with the names of Kachanov and Rabotnov. Skrzypek and Hetnarski [Skryzpek 93]provide a recent summary. This is the continuum creep damage mechanics methodology, whichhas the advantage of being easily expanded to complex geometries and stress/temperaturehistories. Finite element procedures can easily be implemented. Unlike fatigue damage, creepstraining and associated damage can result in appreciable redistribution of stresses relative toinitial elastic conditions. Such factors are readily treated by continuum creep damage mechanics.The discussion here is limited to simple stress conditions. The creep damage enters into therelation between the creep strain rate and the stress and temperature. In the simplest case ofuniaxial tension, the creep rate is often expressed by a so-called Norton relation. The followingform contains a term to account for temperature variations and describes the minimum strain rate(which is also known as the steady state or secondary creep rate).

/

=

AeQ T n

Eq. 2-6

The parameter Qis the activation energy for creep divided by the gas constant. Tis the absolutetemperature. Creep damage can be included in the stress-strain rate relation in the following way

/

=

LNM

OQP

Ae Q Tn

1Eq. 2-7

The term is the creep damage, which accumulates according to the relation

d

dt t TR

= + 1

1 1( ) ( , )( ) Eq. 2-8

The only additional material constant is , which can be evaluated from data on the tertiary creep

characteristics of the material (the increase in strain rate that occurs as failure is approached).

Failure occurs when =1. For constant stress and temperature conditions this corresponds tofailure at t

Rfor that stress and temperature. For stress and temperature that vary in a known

fashion, Equation 2-8 can be used to evaluate the time to failure by separating variables and

integrating. The initial damage is 0 and failure occurs when =1. This leads directly to thefollowing relation

dtt t TR

tR

[ ( ), ]=z 10 Eq. 2-9

This is the counterpart of Robinsons rule (Equation 2-5) expressed as an integral rather than asum. An example of the use of continuum creep damage mechanics to a simple problem isprovided in Section 2.3.2.


24/140



2-6

In some instances, primary creep can be important, and is often included as another term in thecreep ratestress relation. Typically, the following relation is employed:

( )( )

/ /( )

/( )

/

= ++

+ +

+Be p

p tAeQ T

pm

p p

Q T n11

1 1

1Eq. 2-10

No creep damage is included in this expression. The first term is the primary creep strain rate,and the second term is the secondary creep strain rate.B, m,p, Q,A,and nare material constantsthat are obtained from curve fits to creep straintime test data. They are considered to beindependent of temperature, at least over a limited range of temperature.

There are ways to include both of these terms in one expression [Stouffer 96] and include creepdamage at the same time. Such representations are much more convenient to include in finiteelement computations for life prediction of complex geometries. Such representations are beyondthe scope of these guidelines.

2.1.3 Creep/Fatigue Crack Initiation

Creep/fatigue crack initiation is based on the fatigue and creep damage expressed byEquations 2-2 and 2-5, respectively. It is tempting to assume that failure (crack initiation) willoccur when the sum of the creep and fatigue damage is equal to unity. However, it has beenexperimentally observed that there is some interaction between the damage mechanisms, andfailure is considered when the creep and fatigue damage conditions fall outside a line on a creep-fatigue damage plot. Figure 2-3 schematically shows the usual treatment.

Figure 2-3Creep/Fatigue Damage Plane Showing Combinations Corresponding to Crack Initiation


25/140



2-7

2.1.4 Oxide Notching

Crack-like defects can be initiated by the growth and cracking of oxide layers. Figure 2-4 is aphotomicrograph of a crack-like defect that has initiated due to repeated cracking of the oxidescale during start-stop cycles.

Figure 2-4Crack-Like Defect Initiated by Oxide Notching

This initiation mechanism is considered in a deterministic fashion in the BLESS software[Grunloh 92, Harris 93]. A corresponding probabilistic treatment is not available. Section 4.2.1of Grunloh 92 provides the details of the oxide notching model in BLESS. In this case, thegrowth of the steam-side oxide under constant temperature conditions is expressed as

h C e t oxC T C

= 12 3/

Eq. 2-11

The values of C1 C

3are taken to be deterministically defined. The temperature is in absolute

degrees. Simple procedures for evaluation of the oxide thickness when temperature varies are

given by Grunloh 92 and are depicted in Figure 2-5 for a time t1at T

1and t

2at T

2. The oxide

thickness is taken to continuously increase as long as the adjacent metal temperature is greaterthan T

lo-ox. If the metal temperature decreases below T

lo-ox, the oxide is assumed to crack, and the

crack depth is incremented by an amount equal to the increment in the oxide thickness since thelast time it cracked. The oxide thickness is then rezeroed and grown during subsequent timesabove T

lo-ox. Once the oxide notch crack depth reaches a specified depth, it is considered to be an

initiated crack that then grows by fracture mechanics principles, as discussed in Section 2.2. The

value of Tlo-oxis 700F (371C) in BLESS and the depth of notching at which fracture mechanicsprinciples takes over is 0.030 inches (0.76 mm).


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2-8

Figure 2-5

Depiction of Procedure for Determination of Oxide Thickness for a Time at TloFollowed bya Time at Thi

2.2 Crack Growth

The initiation of a crack most often does not mean that the component has reached the end of itsuseful life. Fracture mechanics procedures can be used to estimate the remaining time to growthe crack to the point where it will pose a significant risk to continued operation. Similarly,cracks may initially be present in a component, and fracture mechanics is again called for.

2.2.1 Crack Tip Stress Fields

Fracture mechanics principles are most often based on the analysis of the stresses near a cracktip. The stresses depend on the stress strain relation of the material, which, for uniaxial tension,can often be expressed as

=FHG

IKJD

n

Eq. 2-12

When n=1 andD=E, this is the familiar Hookes law of linear elasticity, and is the elasticstrain. When n 1, then this can represent nonlinear elastic behavior which is the same asplasticity as long as no unloading occurs. The strain is then the plastic strain. This is theRamberg-Osgood representation of plasticity. If the strain is a rate, rather than a strain directly,then this can represent the secondary creep relation of Equation 2-6. This is readily applied toprimary creep also. Hence, Equation 2-12 can be used to describe a variety of material responses.


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2-9

Figure 2-6 shows the coordinate system near a crack tip.

Figure 2-6Coordinate System Near a Crack Tip

The deformation field near a crack tip in a homogeneous isotropic body whose stress-strainrelation is given by Equation 2-12 is characterized by the so-called Hutchinson-Rice-Rosengrensingularity and is given as [Kanninen 85, Kumar 81, Anderson 95]

),(~

),(~

),(~

1

11

1

1

1

nurDI

Ju

nrDI

J

nrI

JD

in

n

n

n

i

ij

n

n

n

ij

ij

n

n

n

ij

++

+

+

=

=

=

Eq. 2-13

where ~ , ~ ij ij and ~ui are dimensionless tabulated functions [Shih 83] andIn is a dimensionlessconstant [Anderson 95, Kanninen 87] that depends on nand whether the conditions are planestress or plane strain. These equations show that i) the stresses and strains are large as rapproaches zero, ii) the deformation field (for a given n) always has the same spatial variation,and iii) the magnitude of the field (for a givenDandn) is controlled by the single parameterJ.Dimensional considerations require thatJhas the units ofDr, which is (stress)x(length) or (F/L).Jis a measure of the crack driving force. The parameterJ is Rices J-integral, whichis the valueof the strain energy release rate with respect to crack area (joules/m

2, in-lb/in

2, etc.). Specific

examples ofJ solutions are given in Section 2.2.2. The case of linear elasticity is when ninEquation 2-12 is equal to 1. This case is of particular interest, and Equations 2-13 can be writtenexplicitly as follows:

x

y

xy

K

r

K

r

K

r

=

= +

=

2 21

2

3

2

2 21

2

3

2

2 2 2

3

2

cos ( sin sin )

cos ( sin sin )

sin cos cos

Eq. 2-14


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2-10

As expected, the stresses are controlled by a single parameter, which is denoted as K and iscalled the stress intensity factor. KandJ are related to one another by the expression

J

K

E

K

E

=

R

S

||

T

||

2

2 21

plane stress

plane strain( )

Eq. 2-15

Equation 2-14 shows that Khas the units of (stress)x(length)1/2

. (Eis Youngs modulus andisPoissons ratio). Specific examples of Ksolutions are discussed in Section 2.2.2.

If the strain in Equation 2-12 is replaced by a strain rate, the stress-strain rate relation is as givenin Equation 2-6. Equations 2-13 still describe the stress and strain rate field near the crack tip,

and the field is controlled by a single parameter which is the rate analog of theJ-integral, whichis denoted as C*. If primary creep is also included, as in Equation 2-10, then C*is applicable tothe secondary creep and there is another parameter to account for the primary creep. This

parameter is referred to as Ch* , and is the parameter controlling the crack tip stress field when

primary creep is dominant.

2.2.2 Crack Driving Force Solutions

Equation 2-13 shows that the stress-strain-displacement field near a crack tip is controlled by asingle parameterJ. As Equation 2-14 shows, if n=1, then the parameter Kcontrols the field, butK

is related toJ

by Equation 2-15. The magnitude and type of loading, as well as the geometry ofthe cracked body, have an influence on the crack tip fields, and this influence enters into theexpression forJor K, which are referred to here as the crack driving forces. From dimensionalconsiderations, the stress intensity factor Khas dimensions of (stress x square root of length). Forthe linear problems to which Kis applicable, Kmust vary linearly with the applied loads. For a

through-crack in a large plate, such as is shown in Figure 2-7, K must be proportional to a ,because ais the only length available.


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2-11

Figure 2-7

Through-Crack of Length 2a in a Large Plate Subject to Stress .

The proportionality constant turns out to be 1/2, as obtained from the limiting case of theelasticity solution for stresses near the tip of an elliptical hole in a plate. In general, stressintensity factor solutions can be written as

K a F= geometry)( Eq. 2-16

There are numerous such K-solutions available for a wide range of crack configurations andloadings. Tada, Paris and Irwin [Tada 00] is an example of such a compendium.

If the crack driving force is expressed in terms ofJ, which has units of in-lb/in2or Joules/m2, thecrack driving force can be written as

J a G geometry n= ( , ) Eq. 2-17

For the simple problem of Figure 2-7, and n=1, G=. If the material is creeping, thenJisreplaced by C* and is replaced by .J-solutions are tabulated in Kanninen 87, Anderson 95and Kumar 81. As an example of aJ-solution, consider the edge-cracked strip in tension shown

in Figure 2-8. The expression forJis given in the figure. The function h1(,n)has been

determined by finite element computations and tabulated [Shih 84]. Table 2-1 is the table forplane strain.


30/140



2-12

J

D

ah nn

n n n

=

+

+

11

11

( , )

( ) ( )

= a h/

=

+

1 2 2

1

2

=R

S|

T|

1455. plane strain

1.072 plane stress

Figure 2-8Single Edge-Cracked Strip in Tension With J-Solution

Table 2-1

Table of the Function h1(,n) in the J-Solution for an Edge Cracked Plate in Tension forPlane Strain [From Shih 84]

a/h n=1 2 3 5 7 10 13 16 20

0.125 5.01 7.17 9.09 12.7 16.3 21.7 27.3 34.1 45.2

0.250 4.42 5.20 5.16 4.54 3.87 3.02 2.38 1.90 1.48

0.375 3.97 3.48 2.88 1.92 1.28 0.704 0.396 0.225 0.111

0.500 3.45 2.62 2.02 1.22 0.754 0.373 0.188 0.0952 0.0391

0.625 2.89 2.16 1.70 1.11 0.744 0.420 0.243 0.142 0.0710

0.750 2.38 1.86 1.55 1.13 0.858 0.585 0.409 0.290 0.186

0.875 1.93 1.62 1.43 1.18 1.00 0.812 0.672 0.563 0.452


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2-13

A variety of other geometries have been analyzed with solutions analogous to the one shown inFigure 2-8. See for instance, Kanninen 87, Kumar 81, and Anderson 95. Many more crack caseshave been analyzed for the linear problem, because the procedures involved (usually finiteelements) are more straightforward and linear superposition is applicable. Tada 00 is an exampleof a compendium of stress intensity solutions.

2.2.3 Calculation of Critical Crack Sizes

Since the stresses and strains surrounding a crack tip are controlled by the value of theJ-integral,a reasonable failure criterion is that failure occurs when the applied value ofJequals somecritical value,J

Ic. The value ofJ

Icis obtained in a test. This criterion is often valid and has been

widely used. There are many complications, however, including the influence of non-singularterms on the stresses and strains, as well as well as increasing resistance of the material to crackgrowth as the crack extends. These complications will not be considered here. Anderson 95provides details. In the case of conditions where the body remains substantially elastic, thefailure criterion can be expressed in terms of the stress intensity factor, with a critical value being

denoted as KIc. The critical value ofJor Kis usually called the fracture toughness.

2.2.4 Fatigue Crack Growth

Since the stress-strain field near the tip of a crack is controlled by a single parameter, it isreasonable to presume that the rate of growth of a crack in a body subject to cyclic loading(da/dN)is controlled by the cyclic value of the crack tip stress parameter. For linearly elastic

bodies, the cyclic parameter is K, which is equal to Kmax

Kmin

. This has been observed to be thecase for a wide variety of metals and conditions, and the following relation is often found toprovide a good fit to data

da

dNC K

f

m= Eq. 2-18

This is the so-called Paris relation and is a suitable representation under a wide variety ofconditions. At extremes of crack growth rates, such as very slow (~10-3in/cycle), the crack growth behavior can deviate from this relation, and more complex

representations are appropriate. The Forman relation is widely used in such instances; see forinstance Henkener 93.

Figure 2-9 is an example of fatigue crack growth data. The material is 2 Cr 1 Mo steel atvarious temperatures. The figure is drawn from Viswanathan 89. The room temperature fit is also

shown, and the dashed line is a least squares curve fit to the 1100F (593C) data. Both of thelines in Figure 2-9 are of the form of Equation 2-18, that is, the Paris relation. This figure shows

that the fatigue crack growth rate is not a strong function of temperature, with data for 700F(371C) being comparable to the room temperature line. The 1100F (593C) data isconsiderably above the data for the lower temperatures, however. The amount of scatter in the

data is seen to be quite large, especially for the 1100F (593C) data. This suggests the use of aprobabilistic treatment. The values of C

fand mfor the lines in Figure 2-9 are as follows:


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2-14

Cf

m

Room temperature 1.41x10-11

3.85

1100F (593C) 8.07x10-10 2.95

The values of Cfare applicable when Kis in ksi-in

1/2(1 ksi-in

1/2=1.099 MPa-m

1/2) and da/dNis in

inches per cycle.

Figure 2-9

Fatigue Crack Growth as a Function of the Cyclic Stress Intensity Factor for 2 Cr 1 Mo atVarious Temperatures [Drawn From Viswanathan 89]

2.2.5 Creep Crack Growth

The stresses near a crack tip in a body that is undergoing secondary creep according to Equation2-6 are controlled by the parameter C*, which is the time analog of theJ-integral. Hence, itwould be reasonable for the creep crack growth rate (da/dt) to be controlled by the value of C*.This has been observed to be the case, but many complicating factors arise. The primaryrestriction is that the body must be fully in the steady-state creep range; elastic and primary creepstrain rates must be negligible. Even if the material exhibits no primary creep, there is still an

elastic response that must be considered. Under the case of secondary creep dominating,Equation 2-19 has been found to provide a good representation of data

da

dtC C

c

q= * Eq. 2-19


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2-15

When the strain rates consist of elastic, primary and secondary creep rates, the situation becomesmore complex. A variety of procedures have been suggested, such as described in Riedel 87 ,Saxena 98, and Bloom and Malito [Bloom 92]. The approach of Bloom is to consider a time-dependent crack driving force that has terms corresponding to elastic, primary creep, andsecondary creep. The crack driving force is referred to as C

t(t)and is given as

C t CK

E n t

n p

n pC

t

C

t

p mp m

h

p p

( ) *( )

( )

( )( )

*

/[( )( )]( )( )

=

+L

NM

O

QP

+ ++ +

FHG IKJ

+

+

2 1 12 2 1

2

1 1

1

1

1

1

1 1

1

+

+

*/( + )

Eq. 2-20

The first line is the elastic transient that occurs on initial loading, the second line is the secondarycreep, and the third line is the steady-state creep. As time becomes very large, the third linedominates.

The parameter C*his the primary creep analog of the steady state parameter C*. It is obtainable

from the J-solution by replacing (1/Dn) with

B p e Q Tp

( ) //( )

11 1

+ +

The creep crack growth rate is then considered to be related to Ct(t) by use of Equation 2-19 withC* replaced by C

t(t).This provides the relation

da

dtC C t

c t

q= [ ( )] Eq. 2-21

2.2.6 Creep/Fatigue Crack Growth

When cyclic loading occurs at temperatures and cycling rates where creep is important, theincrement of crack growth per cycle has been found to be related to the average value of C

t(t)

during the cycle plus a fatigue contribution. The growth per cycle is given by the expression

a C K C C t fm

c t

q

hf| ,cycle ave= + Eq. 2-22

The average value of Ctis given by


34/140



2-16

C C t dt t tt

t

o

h

,ave= z ( ) Eq. 2-23

In this expression, this the duration of the time at load and t

ois a small time, such as the rise time

of the loading. Figure 2-10 provides an example of creep crack growth and creep/fatigue crackgrowth data. The data is for 1 Cr Mo steel 1000F (538C). The left-hand part of the figureis for creep crack growth (i.e. no load cycling), and the right-hand part of the figure is for cyclicloading with various hold times. The line in the figure is best fit to the data and corresponds toC

c= 0.0246 and q= 0.825 in Equation 2-21. (The value of C

cis for C

t(ave)in kips/inch-hour and

crack growth rates in inches per hour, 1 kip/inch-hour = 1.75x105J/m

2-hr). There is a

considerable amount of scatter observed in Figure 2-10, which suggests the usefulness of aprobabilistic approach.


35/140



2-17

Figure 2-10Creep and Creep/Fatigue Crack Growth Data and Fits. Left Figure Is for Constant Load andRight Figure Is for Cyclic Load With Various Hold Times [From Grunloh 92].


36/140



2-18

2.3 Simple Example Problems

Two simple problems are presented to serve as demonstration of the procedures involved in lifeprediction. The problems in this section are deterministic. Their probabilistic counterparts areincluded in Section 4.2.

2.3.1 Fatigue of a Crack in a Large Plate

Consider a through crack in a large plate, such as shown in Figure 2-7. The initial half-crack

length is ao,

and the plate is subject to a stress that cycles between 0 and . Hence the cyclicstress intensity factor is given by the expression

K a= Eq. 2-24

The fatigue crack growth relation is the Paris relation of Equation 2-18. Failure occurs when themaximum applied stress intensity factor is equal to the critical value, K

Ic. The critical crack size,

ac, is

aK

c

Ic= LNM

OQP

12

Eq. 2-25]

A differential equation for the crack length as a function of the number of fatigue cycles is

obtained by inserting the relation for Kinto the Paris relation for crack growth rate.

da

dNC K C a C a

f

m

f

m

f

m m m= = = / /2 2

This equation can be solved by separating variables and integrating, thereby providing thefollowing end result for the cycles to failure,N

f, for a given initial crack size a

o.

NC a a

f

f

m m

o

m

c

mm=

L

NM

O

QP

1 1 12

22 2 2 2 2 / ( ) / ( )/

Eq. 2-26

As an example of the above relations, Figure 2-11 is a plot of avsNfor a of 20 ksi(137.9 MPa), an initial crack half-length of 0.050 inches (1.27 mm), and C

fand mfor the room

temperature line in Figure 2-9.

The results of Figure 2-11 are fairly typical of fatigue crack growth problems with initial cracksthat are quite small; not much happens for a long period, but once the crack starts to growappreciably, it quickly becomes long. Also, the cycles to failure are not strongly influenced bythe critical crack size if the initial size is small. If the critical crack size is larger than about4 inches (100 mm), then the cycles to failure is nearly 1,300,000, independent of critical cracksize. This is a consequence of the nearly vertical slope of the line in Figure 2-11 as aexceedsabout 2 inches (50 mm).


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2-19

Figure 2-11Half-Crack Length as a Function of the Number of Cycles to 20 ksi (137.9 MPa) for Example

Fatigue Problem

In most practical situations, a closed form expression can not be obtained for the crack size as afunction of the number of cycles. Stress intensity factor solutions for realistic problems are morecomplex than in this example, so the integration can not be performed. Another complicatingfactor is that the fatigue crack growth relation is usually more complex than the Paris relation.The above example demonstrates the principles involved, but realistic problems usually requirenumerical procedures for computation of crack sizes and lifetimes.

2.3.2 Creep Damage in a Thinning Tube

An example of creep rupture with wall thinning is presented as an example of creep lifeprediction. This example problem could be representative of a superheater/reheater tube, inwhich case the stresses are dominated by pressure and easily estimated. Consider a tube withconstant internal pressurep, mean radiusR,and a thickness that decreases with time according to

h t h t o( )= Eq. 2-27

The stress strain rate relation is given by Equation 2-7 and the damage kinetics by Equation 2-8.Although the stress that controls creep rupture can be a combination of the principal stress,equivalent stress, and the hydrostatic stress, for this example consider the maximum principalstress to be governing. This is the hoop stress due to the pressure, which is given by

( )( )

tpR

h t

pR

h to

= =

Eq. 2-28

Consider temperature to be fixed. The above equation can be used along with Equation 2-9 toobtain the time to failure. In general, numerical integration is necessary. As a simple example, ifthe curve fit in Figure 2-2 is taken to be a straight line, rather than the second order relation of


38/140



2-20

Equation 2-4, then a closed form relation for the rupture time can be obtained. The following is agood linear representation of the data of Figure 2-2 for stresses less than 20 ksi (137.0 MPa)

log ( ) . .= = A B LMP x LMP6 615 1538 10 4 Eq. 2-29

Using the definition ofLMPfrom Equation 2-3, this can be rearranged to give the followingexpression for the rupture time

t TR

C A BT

BT( , )

/( )

/

=

+101

Eq. 2-30

Use the following definitions:

=

=

== =

= = =

+

+

1

10

101

/

/

/

/

/

BT

C

t hpR

h

tC

R

C A BT

T

o

o

C

C A BT

o

BT

R

o

= time for wall to thin to zero

hoop stress at initial thickness

time for creep rupture at intial stress

The time to rupture with wall thinning and creep damage is then obtained by using Equation 2-30for the rupture time and Equation 2-28 for the stress in conjunction with Equation 2-9. Using theabove definitions and grinding through the algebra leads to the following relation for the rupturetime t

R:

t

t t

t

R

TC

T

=

+ L

NM

O

QP

11

1 1

1 1

( )

/( )

Eq. 2-31

As an example, consider a 2.125 inch (54 mm) diameter 1 Cr 1/2 Mo tube with a wallthickness of 0.4 inches (10.2 mm) operating at 1,000F (1,460R = 811.1K) and 2400 psi(16.55 MPa) pressure. Using the above definitions, the following values are obtained

= 1/BT= 1/(1460x1.538x10-4) = 4.453

CR= 10

-C+A/(BT)= 10

-20+6.615/(1460x0.0001538)= 2.88x10

9

o= 2.4x1.0625/0.4 = 6.38 ksi (44.0 MPa)

tC= 7.51x10

5hours = 85.7 years


39/140



2-21

Figure 2-12 provides a plot of the failure time for various wall thinning rates . It is seen thatthere is a strong interaction between the creep and thinning degradation mechanisms, because thefailure time is much smaller than if only one mechanism is acting.

Figure 2-12Time to Failure as a Function of the Wall-Thinning Rate for the Creep Rupture ExampleProblem (1 Mil/Year=25.4 m/yr)

In most practical situations, a closed form expression for the creep lifetime can not be obtained,because of more complex stress and temperature histories and more complex geometries. Thestresses in the above example are statically determined, so the stress analysis is simple. In fact,creep strain and damage can result in large changes in stress in complex bodies, and detailedlifetime evaluations often require finite element computations. This simple example problemserves to show the principles involved.

The above discussion provides examples of deterministic lifetime models. Although such modelsare available, results obtained in their application to real plant components are subject to manysources of uncertainty, including scatter in material properties, uncertainty in service conditions(pressure, temperature, etc.), and derivation of model constants from test data. Probabilisticmodels account for these uncertainties and scatter, and are discussed in Section 4, but first somestatistical background information is provided in Section 3.


40/140


41/140


3-1

3SOME STATISTICAL BACKGROUND INFORMATION

Some background information on statistics is provided in areas of particular interest inprobabilistic structural analysis. No attempt is made to be comprehensive, and those familiarwith statistics can proceed directly to Section 4. The following basic references are suggested foradditional information [Ang 75, Ang 84, Ayyub 97, Hahn 67, Wolstenholme 99]

3.1 Probability Density Functions

For a continuous random variable,x, the probability ofxfalling within a range of values isdescribed by the probability density function,p(x).Mathematically, this is expressed as

probability that falls between and =x x x dx p x dx+ ( ) Eq. 3-1

Various characteristics of random variables are of use, the most common ones being the mean(or average) and the standard deviation. If a set ofxdata is available, the mean is given by theexpression

xN

xi

i

N

=

=

11

Eq. 3-2

and the standard deviation, commonly denoted as , is

2 2

1

2 2

1

1 2

1

1

1

1=

=

L

NMM

O

QPP

= =

N x x N x N xii

N

i

i

N

( )

/

Eq. 3-3

The square of the standard deviation is called the variance. Sometime theN-1in the denominator

in Equation 3-3 is replaced byN. IfN-1is used, then the value of is called unbiased; ifN=1

then is said to be biased. IfNis large, it doesnt make much difference. The values of the mean

and variance can be obtained from the probability density function (pdf) by use of the following

x xp x dx

x x p x dx

=

=

zz

( )

( ) ( )2 2Eq. 3-4


42/140


Some Statistical Background Information

3-2

The coefficient of variation, cov, is equal to the standard deviation divided by the mean, and isoften referred to.

The probability thatxis less than some value is often also of interest. This is referred to as thecumulative distribution function and is denoted as P(x).The cumulative distribution function isobtained from the probability density function by the equation

P x p y dy

x

( ) ( )=z Eq. 3-5

Since probabilities are always between 0 and 1, the maximum value of P(x)is 1 andEquation 3-5 implies that

p x dx( ) =

z 1The probability thatxis greater than some value is also of interest. This is known by variousnames, usually the complementary cumulative distribution, and sometimes the reliability orsurvivor function. The complementary cumulative distribution is equal to one minus thecumulative distribution.

The median of a random variable is also often of interest. This is the value that is exceeded witha 50-50 chance. Mathematically, this is expressed as

1

2

1

250

50

= =z p x dx P x

x

( ) ( )or Eq. 3-6

Another item of interest is the failure rate. This is the probability of failure betweenxandx+dxgiven that failure has not already occurred. This is also called the hazard function, h(x), and isrelated to the pdf and cumulative by the expression

h xp x

P x( )

( )

( )=

1Eq. 3-7

There are many different probability density functions. Just about any function that integrates tounity can be used as a pdf. The type of distribution to be used in a given situation can be selectedbased on fits to data, theoretical considerations, convenience, or personal taste. The most

compelling reason is fits to data, if sufficient data is available. The most commonly useddistribution is the normal, or Gaussian, distribution. Any random variable that is the sum ofmany other random variables is normally distributed, regardless of the distributions of theindividual variables. This is a consequence of the central limit theorem. Table 3-1 summarizesseveral of the distributions most often encountered in probabilistic lifetime analysis. Theirusefulness here is in describing the scatter or uncertainty in the variables entering into thelifetime model.


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Some

Table 3-1Characteristics of Some Commonly Encountered Probability Functions Used to Describe Scatter and U

Name

ProbabilityDensity

Function

CumulativeDistribution

Function

Range ofRandomVariable Mean

StandardDeviation

uniform 1

b

x a

b

a, a + b a

b+

2

b

12

sim

exponential 1

e x /

1 e x/0, sim

fun

normal orGaussian 1

2

2

22

e

x x

( ) 1

21

2+

FHG

IKJ

L

NM

O

QPerf

x x

, x moerf

lognormal1

2

502

2

xe

x xL

NM

OQP

ln( / ) 1

21

2

50+ F

HGIKJ

L

NMM

O

QPP

erfln( / )x x

0, x e5022 / x e e

502 2 2 log

the

Weibull c

b

x

be

c

x b cFHG

IKJ

1

( / ) 1 e x b

c( / ) 0, b

c 1

1+

FHG

IKJ

see note 2 a fsim

c=c=2

1. x50is equal to exp[mean of ln(x)], is the standard deviation of ln(x)

2. standard deviation of a Weibull variate is bc c

21

11 2

+FHG

IKJ

+FHG

IKJ

LNM

OQP

RS|

T|UV|

W|, is the gamma function, which is w

instance, Abramowitz 64).


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3-4

A function that is often encountered is the so-called error function, erf(x). It is useful because itconveniently expresses the cumulative normal distribution, as shown in Table 3-1. The errorfunction is defined as

erf ( )x e dyyx

z2 2

0Eq. 3-8

The complementary error function is often also of use. It is one minus the error function, and isdenoted as erfc(x). That is

erfc erf ( ) ( )x x= 1 Eq. 3-9

The error function is related to the cumulative unit normal variate, which is the cumulativedistribution for a normally distributed variate with zero mean and a standard deviation of unity.The standard normal variate is often tabulated in statistics texts, and is denoted as P(x) or ( )x .

Using the ( )x notation, the relationship to the error function is

( )xx

= +LNM

OQP

1

21

2erf Eq. 3-10

The error function is widely tabulated, either as erf(x) or ( )x . Abramowitz and Stegun

[Abramowitz 64] provide convenient tables and curve fits. They also provide curve fits forinverse error functions.

3.2 Fitting Distributions

A key part of developing probabilistic lifetime models is the definition of the distributions of therandom input variables. This is most often done by use of data. Equations 3-2 and 3-3 can beused to compute the mean and standard deviation from the data, and Table 3-1 then used to getthe constants in the distribution functions. The constants, such as band cfor the Weibulldistribution, are often referred to as the parameters of the distribution. Another way to evaluatethe parameters of the distribution that entails the distribution type is the method of maximumlikelihood. This method has some theoretical advantages but is often more difficult toimplement. The parameters are evaluated by finding their values that will maximize thelikelihood, which is expressed as

L p xii

N

( ,parameters) parameters= b g1 Eq. 3-11

For instance, if a Weibull distribution is assumed, the expression for the likelihood would be

L b cc

b

x

bei

c

x b

i

N

ic

( , ) ( / )= F

HG I

KJ

=

1

1

Eq. 3-12


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3-5

The values of band cthat maximize L would be selected. It is most often not easy to solveEquation 3-12, and numerical methods must be employed.

Just knowing the parameters is not sufficient to define the distributionthe distribution typemust also be known, such as lognormal or Weibull. If maximum likelihood is used, then thedistribution type has already been assumed, but it is still not known how well the data has beenrepresented. A good way to assess the fit to the distribution type is to plot the distribution on astandard paper, such as normal probability paper. Such paper is available for selected distributiontypes, including normal, lognormal, and Weibull, but this paper can be constructed byappropriate transformations. The data, sayNvalues ofx, can be used to obtain the cumulativedistribution, P(x), by first sorting the data in ascending order. Alternatively, a histogram of thedata can be constructed. Denote this sorted list asNvalues of s. The values of Pare thenestimated by a relation such as

Pi

Ni = Eq. 3-13

Many relations similar to Equation 3-13 have been suggested as better approximations for P.One widely used relation is

P

i

Ni

1

2Eq. 3-14

and this will be used herein. Once the values of si(sortedx

i) are known, a P

i s

iplot is made on

paper whose scales are transformed in such a manner that the cumulative distribution will plot asa straight line if the data is of the distribution type that the paper was constructed for. The

simplest example is for an exponential distribution. In this case, P(x)=1-e-x/. Rearranging and

taking logarithms, this can be written as

xP

=

FHG

IKJ

ln1

1

so that if ln[1/(1-P)] is plotted versusx, andxis exponentially distributed, then the data will plot

as a straight line with slope related to the parameter . Similar transformations can be made forother distribution types. For the lognormal distribution, the cumulative distribution can bewritten as

ln( ) ln( ) ( )x x P= 50

2 2 erfc-1 Eq. 3-15


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3-6

In this equation, erfc-1(x) is the inverse complementary error function, that is if erfc()=then

erfc-1()=. Abramowitz and Stegun contain convenient curve fits to the inverse complementary

unit normal variate (see page 933 of Abramowitz 64), which can be used to define a curve fit tothe inverse complementary error function. Hence, if erfc

-1(2P) is plotted versusxon a log scale,

the data will plot as a straight line if it is lognormally distributed. The counterpart ofEquation 3-15 for a normal variate is

x x P= 2 2erfc-1 ( ) Eq. 3-16

As an example, consider the 1100F (593C) fatigue crack growth data in Figure 2-9. There are25 data points. The linear least squares curve fit to the data gave a C

fand m(in Equation 2-18) of

8.07x10-10

and 2.95, respectively. In order to characterize the scatter in the 1100F (593C) data,fix mat 2.95 and evaluate C

ffor each of the 25 data points in Figure 2-9. This provides the data

in Table 3-2. In this table, Pis evaluated by use of Equation 3-14 and Y=erfc-1(2P).


47/140



3-7

Table 3-2Values of C

ffor Fatigue Crack Growth Data

(values of Cfare for crack growth in inches per cycle and K in ksi-in

1/2)

i 1010Cf

Sorted

1010Cf P

X=ln(Cf)

Sorted Y

1 7.0350 6.2580 .0200 -21.1900 1.4540

2 7.7870 6.2920 .0600 -21.1900 1.1000

3 6.2580 6.5250 .1000 -21.1500 .9066

4 6.2920 7.0350 .1400 -21.0700 .7640

5 6.5250 7.1660 .1800 -21.0600 .6472

6 8.1490 7.2100 .2200 -21.0500 .5458

7 7.5300 7.2870 .2600 -21.0400 .4547

8 7.4740 7.4740 .3000 -21.0100 .3707

9 9.3170 7.5300 .3400 -21.0100 .2917

10 9.3730 7.5690 .3800 -21.0000 .2163

11 10.6000 7.6880 .4200 -20.9900 .1435

12 9.5870 7.7140 .4600 -20.9800 .0724

13 8.0210 7.7840 .5000 -20.9700 -.0022

14 7.1660 7.7870 .5400 -20.9700 -.0724

15 7.6880 8.0210 .5800 -20.9400 -.1435

16 7.2100 8.1490 .6200 -20.9300 -.2163

17 7.7140 8.1970 .6600 -20.9200 -.2917

18 9.9450 8.5400 .7000 -20.8800 -.3707

19 10.3200 9.3170 .7400 -20.7900 -.4547

20 10.7000 9.3730 .7800 -20.7900 -.5458

21 7.7840 9.5870 .8200 -20.7700 -.6472

22 7.2870 9.9450 .8600 -20.7300 -.7640

23 7.5690 10.3200 .9000 -202.6900 -.9066

24 8.5400 10.6000 .9400 -20.6600 -1.1000

258.1970 10.7000 .9800 -20.6600 -1.4540

The data in Table 3-2 is plotted on log-linear scales in Figure 3-1. Figure 3-2 is the same plot onlognormal probability scale, and Figure 3-3 is the corresponding normal probability scale plot.The solid line in these figures is the linear least squares fit to the corresponding X-Y data. BothFigures 3-2 and 3-3 provide a good fit to the data. The straight lines shown in these figures arethe result of linear least squares curve fits on the transformed scales of the figures. The


48/140



3-8

lognormal fit (Figure 3-2) appears to be somewhat better than the normal fit (Figure 3-3). Whichof these is better can be judged by the goodness of fit, which is an entire topic in itself, see, forinstance, DAgostino 86. Such goodness-of-fit tests are restricted to the range of data, and theinterest in probabilistic structural models is often in extrapolations beyond the data.

Figure 3-1Plot of Data of Table 3-2 on Log-Linear Scales

Figure 3-2Lognormal Probability Plot of Data of Table 3-2


49/140



3-9

Figure 3-3Normal Probability Plot of Data of Table 3-2

Another way to see how well the data is being fitted within the range is the coefficient of linearcorrelation, which can be evaluated from the data by the expression

=

R

S

|

T|

U

V

|

W|

=

= =

( ) ( )

( ) ( )

/

x x y y

x x y y

i i

i

N

i

i

N

i

i

N

2 2

1

2

1

Documents

000000000001000311-Guidelines for Performing Probabilistic Analyses of Boiler Pressure Parts