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September 21, 2011
E.H. Baker1, C. James2, S.F. Duffy2,1, and R.L. Bratton3
Reliability Prediction Methodologies: Comparison of the Proposed ASME code for Nuclear Grade Graphite to the Standard Practice Protocol for Advanced Ceramics
1Connecticut Reserve Technologies, Inc.2Cleveland State University3Idaho National Laboratory
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ASME code now includes sections to characterize the brittle material behavior of nuclear grade graphite for reliability prediction of graphite components. Brittle material characterization and component reliability have been thoroughly studied by the ceramics community for 35+ years. The purpose of this study was to compare the section of the ASME code delineating graphite reliability to the standard practices utilized by the ceramic community.
Limitations:
• Graphite exhibits nonlinear elasticity with inelastic behavior. In this study the material was treated as purely linear elastic.
• Time dependent behavior was not included as it pertains to graphite.
• The study assumes that the flaws always originated within the volume of the material instead of either on a surface or on an edge.
INTRODUCTION
33
This section in the ASME code specifies the use of the maximum deformation energy theory (Beltrami) to convert a multiaxial stress state into an equivalent stress that is used in the reliability calculation. This will be referred to as the ASME method throughout. The equivalent stress is calculated as follows:
with:
when the i principal stress is positive (tensile), and
when the i principal stress is negative (compressive)
Where Rtc is the ratio of the mean compressive to mean tensile strength for the specific grade of graphite, and from the Material Data Sheet (GB-2200) ν = 0.15.
ASME Code Section GB-3213: Graphite
44
Eq. 3.1
The ASME code includes a simplified assessment of the reliability of a graphite component according to the following:
• Material characterization based on the lower confidence bound (95%) of the two-parameter Weibull distribution. The parameter estimates are calculated by the method of least squares.
• Specimen failure data must be ranked for the least squares method. The code specifies the following rank formula:
• Although the simplified assessment utilizes an FEA to determine various peak stress values, the FEA is not utilized for the actual reliability calculation in the ASME method. An equivalent stress value is obtained from averaging stress values across a component ligament from all pertinent “Design Loadings,” i.e., multiple load cases are used to compute an equivalent stress.
• Calculate (II-3300) the design allowable stress for various probabilities of failure (matching the service level loading). The design allowable stress is then compared to the component maximum equivalent stress.
SIMPLIFIED ASSESSMENT
55
Summary of the Ceramic Reliability Protocol• Utilizes Weibull material parameters estimated from specimen fracture
strength tests to characterize ceramic material behavior• A range of test specimen geometries and boundary conditions are
utilized. • Specimens that are as similar in geometry and loads to the final
component design are considered optimal.• Fractographic analysis of failed specimens is recommended in order to
censor the failure data according to critical flaw initiation location.• The ASTM parameter estimation standard includes calculation of two-
parameter maximum likelihood (MLE2) unbiased parameters as well as the confidence bounds on the MLE2 parameters. The ceramic protocol requires the use of the MLE2 biased material parameter estimates for the component reliability prediction.
66
• The reliability calculation includes a stress-volume integration calculation throughout a component.
• The reliability calculation depends on the choice of material fracture criterion.
• The principle of independent action (PIA) fracture criterion and various forms of the Batdorf fracture criteria methodologies are typically applied.
• The PIA fracture criterion specifies that compressive principal stresses are to be treated as zero and therefore do not contribute in the reliability calculation.
77
A Comparison of Reliability ProtocolsMaterial Characterization
ASME Graphite CeramicMean strengths obtained in tension and compression. Weibull distribution parameters in tension and compression are assumed to be the same.
Weibull parameters pooled from as many specimen geometries as possible. Encouraged to use specimen geometries as similar to the final component as possible.
Fractographic analysis of failed specimens is not available; therefore, the failure data is treated as uncensored and critical flaws are assumed to originate within the volume of the material.
Perform fractographic analysis to censor the failure data by critical flaw initiation location.
Ranking Formulation: Pf = i / (n+1) Ranking Formulation: Pf = (i - 0.5) / n
Specimen-specific (volume dependent)
Full assessment: Specimen-specific, biased MLE three-parameter Weibull estimates
Simple assessment: lower confidence bound, specimen-specific, LIN two-parameter Weibull estimates
Material-specific (volume independent)
Material-specific two-parameter biased MLE Weibull estimates
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Component Reliability
ASME Graphite CeramicFull assessment: FEA of component required
Simple assessment: FEA of component only used for stress analysis, not reliability prediction
FEA of component is often required
ASME Fracture criteria is based on the maximum deformation energy criterion (Beltrami) equivalent stress.
PIA fracture criterion calculated on the principal stresses; Batdorf fracture criteria calculated on the equivalent stress based on the various Batdorf methodologies.
Group integration point volumes based on their equivalent stress values. No groupings
Adjust the value of the Weibull threshold parameter based on the maximum equivalent stress in the component.
No adjustment
Reliability a function of all integration points with an equivalent stress > the adjusted threshold; compressive stresses contribute to the equivalent stress
Typically PIA methodology excludes compressive stresses
Volume groupings must be larger than the cube of 10 times the maximum grain size
Mesh refinement is encouraged for accurate reliability prediction
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Ceramic Protocol: PIA Reliability ModelA number of reliability models have been presented in the literature that assess component reliability based on multiaxial states of stress. One of the two most popular for modeling reliability of components is the principle of independent action (PIA) model. Here the component probability of failure is expressed in the following generic format
where is identified as a failure function per unit volume. For the simplest form of PIA takes the form
dV 1 = P f exp
0,, 3210
3
0
2
0
1
mmm
1010
In the failure function 1 , 2 and 3 are the three principal stresses at a given point. The parameter 0 is the Weibull material scale parameter.
The PIA model is the probabilistic equivalent to the deterministic maximum stress failure theory. The PIA model has been widely applied in brittle material design. However, the PIA model does not specify the nature of the defect causing failure. In essence the model is phenomenological, which does not imply that the model is not useful. The simplicity of phenomenological models can often times be a strength, not a weakness.
This is a noninteractive reliability model in that the failure modes do not interact with each other to cause failure. The failure mode associated with the simplest form of PIA is a tensile failure mode. One can easily introduce a compressive failure mode. Take
3,2,1321
kk
1111
For each value of k the corresponding term in is computed based on whether the principle stress is tensile or compressive, i.e.,
The subscript “c” denotes compression and the subscript “t ” denotes tension. With an noninteractive failure model one can easily allow separate and distinct failure modes to arise. Here one mode is for tension and one is for compression.
The approach also allows for separate and distinct Weibull distribution parameters for either the tension or compression failure mode. The ASME method does not. One conducts tensile tests and utilizes the same Weibull distribution parameters for the compression failure mode in the ASME method.
0
0
k
m
t
kk
t
0
0
k
m
c
kk
c
1212
Three-Parameter Weibull DistributionScale Parameter• The second parameter is commonly referred to as the scale parameter,
or when working with specimen failure data the "specimen-specific" (volume dependent) characteristic strength parameter. It is this second parameter that is then transformed to the "material-specific" (volume independent) parameter also known as the material scale parameter.
• Most of the following discussion deals with the specimen-specific Weibull distribution parameters.
• The scale parameter in the two-parameter Weibull distribution is the stress value for which the Weibull parameters estimate a probability of failure of 63.21%. In the linear-linear Weibull plot this would be the zero intercept where Y = LN(LN(1/(1-Pf))).
• What occurs with this second parameter, the scale parameter, in the three-parameter Weibull distribution?
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Three-Parameter Weibull Distribution• Mathematically defined as
• Weibull distribution parameters are often estimated from the following form of the three-parameter reliability equation:
• Let's perform the subtraction in the numerator and shift our σ failure stresses by the threshold value:
• where σ* = σ - γ.
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• The following substitution β = λ - γ is necessary, in order to find the 63.21% scale parameter of the three-parameter Weibull distribution. An alternative three-parameter Weibull distribution utilizing λ would then be:
• A similar re-characterization of the three-parameter Weibull distribution may be found in ASME II-3200 equations 17 and 18.
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Slopeα, 3‐parameterm, 2‐parameter
Scaleβ, 3‐parameter
Scalere‐characterizedλ, 3‐parameterσθ, 2‐parameter
Thresholdγ, 3‐parameter
Blue DiamondsMLE2B 4.257 15.546MLE3 2.706 10.227 5.052
MLE3 (re‐char) 2.706 15.279 5.052Red Squares
MLE2B 2.706 10.227
The Sigma data set is the original data set where both the MLE2 and MLE3 estimations have been performed and plotted. Sigma* is the transformed data set to which only the MLE2 estimation have been performed and plotted. This is to visually determine the format of the three-parameter Weibull distribution that will allow for a direct comparison between the scale parameter from the two-parameter distribution and the scale parameter from the three-parameter distribution
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• The re-characterization of the three-parameter Weibull distribution yields an estimate of the scale parameter, λ, which may be directly compared to the two-parameter scale parameter, σθ. These values are highlighted in shaded green in the above table. When the three-parameter Weibull distribution is characterized, the scale parameter, β, may only be directly compared to the two-parameter scale parameter of the shifted failure stresses and in fact they are equivalent. These values are shown in the purple shade in the above table.
• As the threshold ranges from 0 ≤ γ < σmin the Weibull modulus α must take into account the increased scatter in the data. With β = f(α) and λ = β + γ then as γ increases, α decreases (more scatter in the data) and β decreases (since it is a function of α) leaving λ lower than σθ.
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The figure to the right shows the effect of increasing the threshold. Note that αand β are plotted according to the left-hand vertical axis while λ is plotted according to the right-hand vertical axis. The vertical line indicates the optimum threshold parameter for three-parameter MLE for this particular data set.
Modulus () changes as a function of threshold ()
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Threshold ParameterThe previous figure lends itself well as a discussion point regarding the ASME code adjustment to the threshold parameter. According to the ASME code, when the maximum equivalent stress in the component is less than the scale parameter, the threshold parameter is reduced in the following manner:
What is the net effect on the reliability prediction?Consider the complete range of potential threshold values from zero (i.e., the two-parameter distribution) up to the MLE3 estimated. From a goodness-of-fit standpoint a threshold anywhere between those bounds would theoretically have a goodness-of-fit somewhere between the goodness-of-fit of both bounds.
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MLE2B MLE3MLE3 ADJ MLE3f
slope 31.678 6.73325 6.73325 13.0226
scale 19.265219.2274
119.2274
119.2505
5
threshold 0 14.9499 11.21 11.21
R2 93.0% 95.2% 9.9% 94.3%
However, the ASME code does not include the dependence of the slope and scatter parameters on the threshold value. As indicated in the figure to the right as the threshold varies from zero to the MLE3 optimized value (and beyond) the slope and scatter adjust themselves accordingly. Therefore, any adjustment to the threshold parameter must be accompanied by adjustments to the slope and scatter.
ASME
ASME
2020
Weibull Distribution Ranking Scheme• When estimating Weibull distribution parameters by the linear
regression technique, it is necessary to assign a reliability rank to each of the failure strength data.
• The estimated parameters from the linear regression technique are being influenced by the choice of ranking formulation.
• Various ranking formulations exist in the literature. A proposed ASTM standard practice stipulates the use of the following ranking formula due to the minimum bias associated with this estimator
• In section II-3100 of the proposed ASME code uses
2121
The figure to the left depicts a data set from (Barnett, 1966). Here the underlying distribution was assumed to be a two-parameter Weibull distribution. The effects of the two probability of failure estimators are shown.
The figure to the right depicts the same data with the assumption that the the underlying distribution is a three-parameter Weibull distribution.
2222
Failure Envelopes• The Weibull parameters along with Rtc were set to the same values for the PIA model with compression and the ASME method.
• In Quadrant I the PIA model with compression uses the tensile Weibull material scale parameters. In contrast the ASME would use the tensile Weibull characteristic strength parameter.
• In Quadrant IV, the PIA model with compression uses the compressive Weibull material scale parameter. The ASME method use the tensile Weibull characteristic strength parameter after reducing the compressive principal stresses by Rtc
• The failure envelopes do not match (see figure). Neither would reliability surfaces.
2323
Axially Loaded Pressurized Tube
DimensionsInner Diameter 60.4 mmOuter Diameter 71.6 mmGage Length 45.7 mm
Linear Elastic Material PropertiesYoung's Modulus 11508 MPaPoisson's Ratio 0.150
LOAD CASE AXIAL HOOP
1 12 02 -48 03 0 124 11 85 -35 7
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Material ExamplesTensile Specimens• Specimen strength data was provided for 190 tensile specimens. • The provided data included the diameter, maximum load, extensometer
displacement, maximum strain, and the tensile strength of each specimen.
• The tensile strength was calculated as load divided by the cross-sectional area of the gauge section. The gauge length was 63.5 mm and the average diameter of the 190 specimens was 15.90 mm.
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Specimen Tensile Length Radius Gage Vol.Analysis Volume 63.5 7.95 12608.3
NORMAL DISTR. Count Mean Std Dev Skewness Kurtosis190 20.1825 2.12377 -0.708277 0.755841
WEIBULL DISTR. Rank Form (i - 0.5) / nTWO PARAMETER M Sig Not Char StrLIN2 Biased 11.4833 48.0032 21.0947 MLE2 Biased 11.6198 47.535 21.0917 MLE2 Unbiased 11.5349 47.821 21.092 90% Conf Lower 10.5062 20.8642 90% Conf Upper 12.6716 21.3235
THREE PARAMETER M Sig Not Threshold LIN3 Biased 11.4833 48.0032 6.33902E-9 MLE3 Biased 11.6198 47.535 5.76742E-6
THREE PARAMETER M Char Str Threshold LIN3 Biased 11.4833 21.0947 6.33902E-9 MLE3 Biased 11.6198 21.0917 5.76742E-6
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Material ExamplesCompressive Specimens• Specimen strength data was provided for 192 compressive specimens. • The provided data included the diameter, maximum load, extensometer
displacement, maximum strain, and the compressive strength of each specimen.
• The compressive strength was calculated as load divided by the cross-sectional area of the gauge section. The gauge length was 50.8 mm and the average diameter of the 192 specimens was 25.40 mm.
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Strength Compression Tension Rtc
Mean 81.1782 20.1825 4.02Weibull Characteristic 83.0618 21.0917 3.94Weibull Material Scale 130.2290 47.5350 2.74* MLE2B
Specimen Compressive Length Radius Gage Vol.Analysis Volume 50.8 12.7 25740.7
NORMAL DISTR. Count Mean Std Dev Skewness Kurtosis192 81.1782 4.09847 -0.389528 -0.0744632
WEIBULL DISTR. Rank Form (i - 0.5) / nTWO PARAMETER M Sig Not Char StrLIN2 Biased 25.2514 124.008 82.9431 MLE2 Biased 22.5831 130.229 83.0618 MLE2 Unbiased 22.4192 130.659 83.0624 90% Conf Lower 20.4372 82.5996 90% Conf Upper 24.6272 83.5265
THREE PARAMETER M Sig Not Threshold LIN3 Biased 5.90421 125.165 60.4047 MLE3 Biased 5.84008 125.472 60.7609
THREE PARAMETER M Char Str Threshold LIN3 Biased 5.90424 22.4106 60.4046 MLE3 Biased 5.84008 22.0453 60.7609
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Summary - Material Examples
2929
Material Example: JAERI IG-110• Tensile, compressive and bending failure data for IG-110 were
extracted from JAERI-M-92-009. • Weibull material parameters were estimated from the failure data sets. • Failure stress measurements for an axially loaded pressurized tube
were extracted from JAERI-Research-96-016. • Reliability predictions based on the estimated Weibull parameters were
made for the tube and compared to the measured results.
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Strength Compression Tension Rtc
Normal Mean 69.323 21.180 3.27Weibull Characteristic 70.495 21.931 3.21Weibull Material Scale 83.239 32.748 2.54* MLE2B
ASME CARES Inputs PWC CARES Inputsm σθ Poisson Rtc m σ0 Rtc
15.807 21.930 0.15 3.27 Tensile 15.8072 32.7475 2.54Compressive 35.0639 83.2391
The failure strength data generated in tension and compression discussed previously are superimposed in the plot to the left along with bend bar data that was not used here
This data was also used to compute the Rtc values for both the PIA and ASME methods
3131
The parameters estimated in the previous overhead were used to predict the 5%, 50% and 95% percentiles of reliability against multi-axial data from Battiste et al (2010).
Predictions (green for PIA and orange for ASME) were made using the modified ASME method, but volume grouping was not utilized.
Note that different tube geometry and FEA meshes were used in each figure.
3232
Summary• Clarifications needed for proper implementation of a reliability protocol
have been highlighted. • Comparisons to the standard practice for ceramic reliability have been
made. • The scale parameter, threshold parameter, and data ranking
formulation of the Weibull distribution have been discussed. • Reliability failure envelops were presented.• Weibull parameters from measured data have been utilized to predict
component reliability for the ASME and the PWC methods. • The final example showed the somewhat non-conservative prediction
of the ASME method compared to a relatively more conservative prediction of the PWC method. A more complete comparison utilizing this example would include further mesh refinement for the PWC method and inclusion of the volume groupings for the ASME method.
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