Availableonlineatwww.sciencedirect.com . ScienceDirect c000e5 Part B: engineering ELSEVIER Composites: Part B 39(2008)694-703 www.elsevier.com/locate/compositesb Probabilistic analysis of a SiC/Sic ceramic matrix composite turbine vane Pappu L.N. Murthy a,, Noel N Nemeth, David N. Brewer Subodh Mital c US Army Research Laboratory, National Aeronautics and Space Administration, Glenn Research Center, Clereland, OH 44135, United States niversity of Toledo, Toledo, OH 43606, United States Received 15 December 2006: received in revised form 22 May 2007: accepted 29 May 2007 Available online 28 June 2007 Abstract To demonstrate the advanced composite materials technology under development within the Nasa Ultra-Efficient Engine Technol- ogy (UEET) Program, it was planned to fabricate, test, and analyze a turbine vane made entirely of silicon carbide- fiber-reinforced sil- con carbide matrix composite( SiC/SiC CMC)material. The objective was to utilize a five-harness satin weave melt-infiltrated (MI)SiC Sic composite material to design and fabricate a stator vane that can endure 1000 h of engine service conditions. The vane was designed to withstand a maximum temperature of 1315C(2400F)within the substrate and the hot surface temperature of 1482C(2700F) with the aid of an environmental/thermal barrier coating(EBC/TBC)system. Furthermore, the vane was designed such that the expected maximum stresses to be encountered were kept within the proportional limit strength of the material. Any violation of this design requirement was considered as the failure. This paper presents results of a probabilistic analysis and reliability assessment of the vane robability of failure to meet the design requirements was computed using the probabilistic analysis methods embedded in the nessus software. In the analysis, material properties, strength, and pressure loading were considered as random variables. The variations in properties and strength were based on the actual experimental data. In the present analysis, the pressure loads were considered normally distributed with a nominal variation. a temperature profile on the vane was obtained by performing a computational fluid dynan (CFD)analysis and was assumed to be deterministic. The results suggest that for the current vane design, the chance of not meet design requirements is about 1.6% Published by elsevier Ltd Keywords: Probabilistic analysis; CMC vane: Cumulative distribution function; Probability density function; Scatter: Weibull distribution; Strength; Proportional limit; Design requirements; Ceramic matrix composite 1. Introduction amount of cooling, which reduces the turbine inlet temper- atures, thereby reducing the thermal efficiency. CMCs have Advanced high-temperature ceramic matrix composites desirable properties such as lighter weight and higher ther- (CMCs) have been recognized as viable candidate materials mal stability compared to the conventional metallic materi- for propulsion system components. Use of these advanced als. Hence one can surmise that CMCs can perform well at materials will lead to increase in thermal efficiency and a much higher temperatures thereby increasing the engine reduction in NOx emissions. These objectives can be efficiency. Furthermore, higher combustion temperatures accomplished mainly by raising the turbine inlet tempera- have the beneficial effect of lowering the NOx emissions ture and nating cooling of the turbine blades, vanes, Research under NASas High-Speed Research Enabling rs.Conventional materials require a large Propulsion Materials(HSR/EPM) program led to the emergence of the Sylramic(Dow Corning Corporation, Midland, MI)SiC fiber with chemical-vapor-infiltrated E-mail address: Pappu. L Murthy(@nasa. gov(P L.N. Murthy CVI-SiC/melt-infiltrated (MI)-Sic matrix 5-harness 1359-8368S- see front matter Published by Elsevier Ltd. doi:10.1016/j.compositesb.2007.05.006
Probabilistic analysis of a SiC/SiC ceramic matrix composite turbine vane Pappu L.N. Murthy a,*, Noel N. Nemeth a , David N. Brewer b , Subodh Mital c a National Aeronautics and Space Administration, Glenn Research Center, Cleveland, OH 44135, United States b US Army Research Laboratory, National Aeronautics and Space Administration, Glenn Research Center, Cleveland, OH 44135, United States c University of Toledo, Toledo, OH 43606, United States Received 15 December 2006; received in revised form 22 May 2007; accepted 29 May 2007 Available online 28 June 2007 Abstract To demonstrate the advanced composite materials technology under development within the NASA Ultra-Efficient Engine Technology (UEET) Program, it was planned to fabricate, test, and analyze a turbine vane made entirely of silicon carbide-fiber-reinforced silicon carbide matrix composite (SiC/SiC CMC) material. The objective was to utilize a five-harness satin weave melt-infiltrated (MI) SiC/ SiC composite material to design and fabricate a stator vane that can endure 1000 h of engine service conditions. The vane was designed to withstand a maximum temperature of 1315 C (2400 F) within the substrate and the hot surface temperature of 1482 C (2700 F) with the aid of an environmental/thermal barrier coating (EBC/TBC) system. Furthermore, the vane was designed such that the expected maximum stresses to be encountered were kept within the proportional limit strength of the material. Any violation of this design requirement was considered as the failure. This paper presents results of a probabilistic analysis and reliability assessment of the vane. Probability of failure to meet the design requirements was computed using the probabilistic analysis methods embedded in the NESSUS software. In the analysis, material properties, strength, and pressure loading were considered as random variables. The variations in properties and strength were based on the actual experimental data. In the present analysis, the pressure loads were considered normally distributed with a nominal variation. A temperature profile on the vane was obtained by performing a computational fluid dynamics (CFD) analysis and was assumed to be deterministic. The results suggest that for the current vane design, the chance of not meeting design requirements is about 1.6%. Published by Elsevier Ltd. Keywords: Probabilistic analysis; CMC vane; Cumulative distribution function; Probability density function; Scatter; Weibull distribution; Strength; Proportional limit; Design requirements; Ceramic matrix composite 1. Introduction Advanced high-temperature ceramic matrix composites (CMCs) have been recognized as viable candidate materials for propulsion system components. Use of these advanced materials will lead to increase in thermal efficiency and a reduction in NOx emissions. These objectives can be accomplished mainly by raising the turbine inlet temperature and eliminating cooling of the turbine blades, vanes, and combustors. Conventional materials require a large amount of cooling, which reduces the turbine inlet temperatures, thereby reducing the thermal efficiency. CMCs have desirable properties such as lighter weight and higher thermal stability compared to the conventional metallic materials. Hence one can surmise that CMCs can perform well at much higher temperatures thereby increasing the engine efficiency. Furthermore, higher combustion temperatures have the beneficial effect of lowering the NOx emissions. Research under NASA’s High-Speed Research Enabling Propulsion Materials (HSR/EPM) program led to the emergence of the Sylramic (Dow Corning Corporation, Midland, MI) SiC fiber with chemical-vapor-infiltrated (CVI)–SiC/melt-infiltrated (MI)–SiC matrix 5-harness 1359-8368/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.compositesb.2007.05.006 * Corresponding author. E-mail address: Pappu.L.Murthy@nasa.gov (P.L.N. Murthy). www.elsevier.com/locate/compositesb Available online at www.sciencedirect.com Composites: Part B 39 (2008) 694–703
P LN. Murthy et al./ Composites: Part B 39(2008)694-703 satin weave CMC material as one of the most promising assessment. For the purpose of the present study, it is con candidates for propulsion system components (Refs. sidered that any violation of a design requirement is a fail- 2]). Successful demonstration of the new CMC technol- ure. In addition, cumulative probability distribution ogy for propulsion system components is one of the goals functions of critical stresses and the sensitivities of various of NASA's UEET Program. Under this program, the ther- random variables are computed. The proportional limit mal capability of the material has been raised to 1315 and strength distributions are computed from the experi- (2400 F). This material is sought for combustor liners mental coupon data and Weibull statistics. Measured dis and turbine vanes, which see gas temperatures in excess tributions of thermal properties and pressure loads are of 1650C(3000F). Furthermore, the hot side is coated not available at this point. These variables are considered with an environmental/thermal barrier coating (EBC/ normally distributed with nominal coefficients of variation TBC) system that is stable up to about 1482C(2700 F) (COVs) in the current analysis. The vane thickness is con- (Ref. [3]. To demonstrate the new CMC technology, it sidered deterministic in these analyses. It is worth noting was planned to fabricate, test, and analyze a turbine stator that Federal Aviation Administration regulations require vane made entirely of the MI SiC/SiC composite materia that for commercial aircrafts, the probability of failure developed under NASA,s UEET Program. The turbine sta- should range from a high value of 10-3 for minor failure tor vane was to be fabricated utilizing this CMC material conditions to an extremely low value of 10-' for cata- and tested in a high-pressure burner rig at NASA Glenn strophic failure conditions. Additionally, NASA space mis- Research Center. This rig is capable of simulating the sions are striving for a catastrophic failure rate of 10 engine service environment. The test was to demonstrate that the vane can successfully withstand the harsh engine 2. Vane subelement fabrication environment conditions for up to 1000 h In order to make sure that the vane lasts the duration successfully, it was Vane subelements were fabricated from a silicon designed such that the maximum expected stresses are bide- fiber-reinforced SiC/SiC composite and were coated within the proportional limit stress of the CMC material. with an environmental barrier coating(EBC). In order to It should be noted that the life of Sic/Sic materials sub- address realistic critical design features of a turbine airfoil, jected to stresses that exceed the proportional limit is often the vane subelement cross section was derived from an everely limited due to the harsh environmental attack of existing production aircraft engine vane. A unique woven the fibers via matrix cracl cloth configuration was used to provide a sharp trailing Due to the brittle nature of the CMC constituents, the edge with continuous fiber-reinforcement (Ref [5D. Fabri properties of CMCs show considerable scatter Reproduc- cation of vanes with a sharp trailing edge was considered to ibility is a major issue and a concern. Examination of the be one of the more challenging features for fabricating a MI SiC/SiC stress-strain behavior indicated a substantial ceramic composite vane. The vanes were densified through amount of scatter in the first matrix cracking strength(pro- the CVI/slurry cast/silicon MI process. Both nondestruc portional limit)as well as the ultimate strength(Ref. [4). tive and metallographic examinations revealed that the Furthermore, variations and uncertainties are usually pres- quality of the final as-fabricated composite vanes was con- ent in geometry, thermal properties, and loading conditions sistent with that typically obtained for the same composite as well. Vane designs based solely upon the mean values for material fabricated into flat panels. One consisted of a thin- the material properties, geometrical variables, and loads walled (1. 5 mm) shell with a continuously reinforced sharp may be unconservative and may lead to unexpected prema- trailing edge. a vane subelement manufactured in this ture failures. These failures are due to a violation of the study is shown in Fig. 1. Each vane had a constant cross design constraint: the maximum stresses in critical locations section over a height of 50 mm, with a trailing edge radius have exceeded the proportional limit. Thus the uncertainties, of 0.26 mm, a leading edge radius of 3. 1 mm, and a cord which add concerns regarding the reliability of the vane per- length of 50 mm, as shown in Fig. 1. All vanes were man formance under the service conditions, need to be quantified. ufactured with the CvI/slurry-cast/MI SiC/SiC material Current research effort is primarily directed towards system using Sylramic SiC fiber-reinforcing cloth assessment of the reliability of an all-CMC turbine stator A fiber architecture was developed to address the fabri vane subjected to engine service conditions. Given the scat- cation challenges presented at the vane trailing edge as well ter in material properties, uncertain loading conditions, as provide a fiber architecture in the remaining regions of and geometrical variations, a probabilistic analysis of the the vane that had been well characterized and successfully vane is performed in order to quantify the risk of not meet- demonstrated in other CMC turbine engine components ing the design requirements. Since the stresses in the vane The fiber tows forming the trailing edge section are inter- depend upon the pressure as well as the temperature gradi- locked, thereby enhancing the through-the-thickness ents through-the-thickness, material thickness variations strength capability of the composite material. The sharp will have significant effect on the stresses. Variations in trailing edge is then naturally formed within the vertex of the temperature profile(caused by the variation in the the Y-shaped cloth. This avoids sharply bent fiber filaments gas temperature) and in material thickness and their effect and its ass ssociated strength reduction. The vane test pro-
satin weave CMC material as one of the most promising candidates for propulsion system components (Refs. [1,2]). Successful demonstration of the new CMC technology for propulsion system components is one of the goals of NASA’s UEET Program. Under this program, the thermal capability of the material has been raised to 1315 C (2400 F). This material is sought for combustor liners and turbine vanes, which see gas temperatures in excess of 1650 C (3000 F). Furthermore, the hot side is coated with an environmental/thermal barrier coating (EBC/ TBC) system that is stable up to about 1482 C (2700 F) (Ref. [3]). To demonstrate the new CMC technology, it was planned to fabricate, test, and analyze a turbine stator vane made entirely of the MI SiC/SiC composite material developed under NASA’s UEET Program. The turbine stator vane was to be fabricated utilizing this CMC material and tested in a high-pressure burner rig at NASA Glenn Research Center. This rig is capable of simulating the engine service environment. The test was to demonstrate that the vane can successfully withstand the harsh engine environment conditions for up to 1000 h. In order to make sure that the vane lasts the duration successfully, it was designed such that the maximum expected stresses are within the proportional limit stress of the CMC material. It should be noted that the life of SiC/SiC materials subjected to stresses that exceed the proportional limit is often severely limited due to the harsh environmental attack of the fibers via matrix cracks. Due to the brittle nature of the CMC constituents, the properties of CMCs show considerable scatter. Reproducibility is a major issue and a concern. Examination of the MI SiC/SiC stress-strain behavior indicated a substantial amount of scatter in the first matrix cracking strength (proportional limit) as well as the ultimate strength (Ref. [4]). Furthermore, variations and uncertainties are usually present in geometry, thermal properties, and loading conditions as well. Vane designs based solely upon the mean values for the material properties, geometrical variables, and loads may be unconservative and may lead to unexpected premature failures. These failures are due to a violation of the design constraint: the maximum stresses in critical locations have exceeded the proportional limit. Thus the uncertainties, which add concerns regarding the reliability of the vane performance under the service conditions, need to be quantified. Current research effort is primarily directed towards assessment of the reliability of an all-CMC turbine stator vane subjected to engine service conditions. Given the scatter in material properties, uncertain loading conditions, and geometrical variations, a probabilistic analysis of the vane is performed in order to quantify the risk of not meeting the design requirements. Since the stresses in the vane depend upon the pressure as well as the temperature gradients through-the-thickness, material thickness variations will have significant effect on the stresses. Variations in the temperature profile (caused by the variation in the gas temperature) and in material thickness and their effect on the vane reliability should be considered in the risk assessment. For the purpose of the present study, it is considered that any violation of a design requirement is a failure. In addition, cumulative probability distribution functions of critical stresses and the sensitivities of various random variables are computed. The proportional limit and strength distributions are computed from the experimental coupon data and Weibull statistics. Measured distributions of thermal properties and pressure loads are not available at this point. These variables are considered normally distributed with nominal coefficients of variation (COVs) in the current analysis. The vane thickness is considered deterministic in these analyses. It is worth noting that Federal Aviation Administration regulations require that for commercial aircrafts, the probability of failure should range from a high value of 103 for minor failure conditions to an extremely low value of 109 for catastrophic failure conditions. Additionally, NASA space missions are striving for a catastrophic failure rate of 106 . 2. Vane subelement fabrication Vane subelements were fabricated from a silicon carbide-fiber-reinforced SiC/SiC composite and were coated with an environmental barrier coating (EBC). In order to address realistic critical design features of a turbine airfoil, the vane subelement cross section was derived from an existing production aircraft engine vane. A unique woven cloth configuration was used to provide a sharp trailing edge with continuous fiber-reinforcement (Ref. [5]). Fabrication of vanes with a sharp trailing edge was considered to be one of the more challenging features for fabricating a ceramic composite vane. The vanes were densified through the CVI/slurry cast/silicon MI process. Both nondestructive and metallographic examinations revealed that the quality of the final as-fabricated composite vanes was consistent with that typically obtained for the same composite material fabricated into flat panels. One consisted of a thinwalled (1.5 mm) shell with a continuously reinforced sharp trailing edge. A vane subelement manufactured in this study is shown in Fig. 1. Each vane had a constant cross section over a height of 50 mm, with a trailing edge radius of 0.26 mm, a leading edge radius of 3.1 mm, and a cord length of 50 mm, as shown in Fig. 1. All vanes were manufactured with the CVI/slurry-cast/MI SiC/SiC material system using Sylramic SiC fiber-reinforcing cloth. A fiber architecture was developed to address the fabrication challenges presented at the vane trailing edge as well as provide a fiber architecture in the remaining regions of the vane that had been well characterized and successfully demonstrated in other CMC turbine engine components. The fiber tows forming the trailing edge section are interlocked, thereby enhancing the through-the-thickness strength capability of the composite material. The sharp trailing edge is then naturally formed within the vertex of the Y-shaped cloth. This avoids sharply bent fiber filaments and its associated strength reduction. The vane test program with test rig description, test conditions, and vane test P.L.N. Murthy et al. / Composites: Part B 39 (2008) 694–703 695
P L.N. Murthy et al. Composites: Part B 39(2008)694-70 mm 50 mm 50 mm Fig. 1. SiC/SiC CMC turbine stator vane 3. Vane analysis Prior to testing, computational fluid dynamics(CFD) and finite element analyses were performed to predict the temperature and stress conditions present in the vane dur- ing rig testing(Ref. [6]. Analyses were performed for a pre- liminary vane design that did not include trailing edge cooling. The CFD analysis for a cascade of blades was per formed using a two-dimensional Euler (i.e, inviscid flow) equation solver. Local pressure and velocity results were used to determine heat transfer coefficients for the vane exterior surface. Calculated in-plane tensile stress values ranged from 27 MPa in the axial direction to a maximum transverse or"hoop"stress of 105 MPa. The predicted interlaminar tensile(ILT) stresses were found to be rather high, although in a very small area. This vane has some through-the-thickness reinforcement (because of the Fig. 2. Vanes in holder prior to testing SiC/Sic test vane is in center with unique geometry at the critical location as explained ear metallic vanes on either sid lier) that is likely to provide a higher ILT strength tha found in a two-dimensional flat specimen. Therefore, even configuration is described in reference [5]. An interesting though ILT stresses were high, because of the small region point to note is that the CMc vane was surrounded by a and the additional reinforcement provided, they were not metallic vane on either side to help establish close to real- considered to be a major design issue. The finite element istic flow around the SiC/SiC test specimen( Fig. 2) model of the vane with the boundary conditions is shown 1.52 axially 21.8 Nodes held Fig 3. Finite element model and boundary conditions for vane. (a) Finite element mesh details and vane geometry. All dimensions are in millimeters.(b) Boundary conditions for finite element model
configuration is described in reference [5]. An interesting point to note is that the CMC vane was surrounded by a metallic vane on either side to help establish close to realistic flow around the SiC/SiC test specimen (Fig. 2). 3. Vane analysis Prior to testing, computational fluid dynamics (CFD) and finite element analyses were performed to predict the temperature and stress conditions present in the vane during rig testing (Ref. [6]). Analyses were performed for a preliminary vane design that did not include trailing edge cooling. The CFD analysis for a cascade of blades was performed using a two-dimensional Euler (i.e., inviscid flow) equation solver. Local pressure and velocity results were used to determine heat transfer coefficients for the vane exterior surface. Calculated in-plane tensile stress values ranged from 27 MPa in the axial direction to a maximum transverse or ‘‘hoop’’ stress of 105 MPa. The predicted interlaminar tensile (ILT) stresses were found to be rather high, although in a very small area. This vane has some through-the-thickness reinforcement (because of the unique geometry at the critical location as explained earlier) that is likely to provide a higher ILT strength than found in a two-dimensional flat specimen. Therefore, even though ILT stresses were high, because of the small region and the additional reinforcement provided, they were not considered to be a major design issue. The finite element model of the vane with the boundary conditions is shown Fig. 1. SiC/SiC CMC turbine stator vane. Fig. 2. Vanes in holder prior to testing. SiC/SiC test vane is in center with metallic vanes on either side. Fig. 3. Finite element model and boundary conditions for vane. (a) Finite element mesh details and vane geometry. All dimensions are in millimeters. (b) Boundary conditions for finite element model. 696 P.L.N. Murthy et al. / Composites: Part B 39 (2008) 694–703
P LN. Murthy et al./ Composites: Part B 39(2008)694-703 in Fig. 3. In this study, analysis is focused primarily on the hoop stress since measured data is readily available for the in-plane modulus and proportional limit strength. Based upon this data, the statistics pertaining to strength and modulus could be established. Such statistical data is not available for ILT tensile strength 4. Probabilistic analysis As mentioned earlier, the measured MI SiC/Sic mate- rial strength and modulus showed a substantial amount of scatter. The deterministic analyses, specifically near the 105115125135145155165175185195205215225 trailing edge region, indicated that the most critical stresses are in-plane hoop stresses. These stresses are affected pri Fig. 5. Frequency histogram of proportional limit strength of MI SiC/SiC marily by the in-plane stiffness of the material as well as 27.2 MP. points. Mean value is 166 MPa and standard deviation, for 24 dat by the loading conditions and need to be compared to the stress allowables (i.e, proportional limit in this case) The in-plane Youngs modulus and proportional limit lus statistics are based upon the measured data Probabilistic-distribution-related parameters are strength of the UEET material data taken for 24 samples is shown in the form of histograms in Figs. 4 and 5, respec ssumed for the remaining variables. All other per tively. It is evident that these two measured material prop- tinent parameters (e.g, material thickness, gas erties show considerable scatter. It is possible that a temperature, or other loading parameters)are correlation exists between these two variables. However, Case Il: In addition to the variables considered in case I ulus and proportional limit strength have been assumed to two other parameters related to the loading condi tions--internal pressure of the cooling air and the be independent random variables. Design/ analysis solely external aerodynamic pressure on the vanewere based upon mean values for these properties, therefore, considered as random variables with assumed dis. might lead to unexpected failures during the rig testin tributions and nominal values for the means and due to the wide scatter range. Consequently, it was decided COVS perform a probabilistic (risk) analysis to quantify the probability of vane performance not meeting the design requirement, which is referred to as failure (i.e, the hoop 5. Estimation of weibull parameters stress exceeding the proportional limit). Two cases for probabilistic analysis were evaluated: The stochastic behavior of the MI SiC/SiC in-plane Youngs Modulus and proportional limit at 1200C Case I: Only the material Young,'s modulus, Poisson's(2200F)were characterized from experimental data using ratio, coefficient of thermal expansion and propor- the two-parameter Weibull distribution [7]. This informa- tional limit are considered as random variables. tion was subsequently included in the probabilistic analy Among these variables the strength and modu- sis. The two-parameter Weibull distribution is expressed as 0.30 P=1 0.24 where P is the probability of occurrence, a is the particular value of data for which probability is to be calculated, B is 80.18 the Weibull characteristic value -the value where the prob. ability of occurrence is 63. 21%-and y is the Weibull mod- ulus which measures the degree of dispersion or scatter in 0.12 the data. For the composite proportional limit - which could be regarded as a strength measurement -a is a value of strength while B is the characteristic strength. Both a and B have units of stress. The Weibull modulus y is dir less. As y increases the amount of dispersion decreases. 145155165175186195205215225 Typical values describing monolithic ceramic strength dispersion range from about 5 to more than 30. Ceramic for 24 data points. mean value is i815 Gpa and standard deviation. is interpreted as a probability of failure when the distribu 13.8 GPa tion is used to describe strength. Likewise, characterizing
in Fig. 3. In this study, analysis is focused primarily on the hoop stress since measured data is readily available for the in-plane modulus and proportional limit strength. Based upon this data, the statistics pertaining to strength and modulus could be established. Such statistical data is not available for ILT tensile strength. 4. Probabilistic analysis As mentioned earlier, the measured MI SiC/SiC material strength and modulus showed a substantial amount of scatter. The deterministic analyses, specifically near the trailing edge region, indicated that the most critical stresses are in-plane hoop stresses. These stresses are affected primarily by the in-plane stiffness of the material as well as by the loading conditions and need to be compared to the stress allowables (i.e., proportional limit in this case). The in-plane Young’s modulus and proportional limit strength of the UEET material data taken for 24 samples is shown in the form of histograms in Figs. 4 and 5, respectively. It is evident that these two measured material properties show considerable scatter. It is possible that a correlation exists between these two variables. However, for simplicity and lack of measured data, the Young’s modulus and proportional limit strength have been assumed to be independent random variables. Design/analysis solely based upon mean values for these properties, therefore, might lead to unexpected failures during the rig testing due to the wide scatter range. Consequently, it was decided to perform a probabilistic (risk) analysis to quantify the probability of vane performance not meeting the design requirement, which is referred to as failure (i.e., the hoop stress exceeding the proportional limit). Two cases for probabilistic analysis were evaluated: Case I: Only the material Young’s modulus, Poisson’s ratio, coefficient of thermal expansion and proportional limit are considered as random variables. Among these variables the strength and modulus statistics are based upon the measured data. Probabilistic-distribution-related parameters are assumed for the remaining variables. All other pertinent parameters (e.g., material thickness, gas temperature, or other loading parameters) are considered deterministic in this evaluation. Case II: In addition to the variables considered in case I, two other parameters related to the loading conditions—internal pressure of the cooling air and the external aerodynamic pressure on the vane—were considered as random variables with assumed distributions and nominal values for the means and COVs. 5. Estimation of Weibull parameters The stochastic behavior of the MI SiC/SiC in-plane Young’s Modulus and proportional limit at 1200 C (2200 F) were characterized from experimental data using the two-parameter Weibull distribution [7]. This information was subsequently included in the probabilistic analysis. The two-parameter Weibull distribution is expressed as P ¼ 1 exp a b c ð1Þ where P is the probability of occurrence, a is the particular value of data for which probability is to be calculated, b is the Weibull characteristic value – the value where the probability of occurrence is 63.21% – and c is the Weibull modulus which measures the degree of dispersion or scatter in the data. For the composite proportional limit – which could be regarded as a strength measurement – a is a value of strength while b is the characteristic strength. Both a and b have units of stress. The Weibull modulus c is dimensionless. As c increases the amount of dispersion decreases. Typical values describing monolithic ceramic strength dispersion range from about 5 to more than 30. Ceramic composites likely fall within this same range. P in Eq. (1) is interpreted as a probability of failure when the distribution is used to describe strength. Likewise, characterizing Fig. 4. Frequency histogram of in-plane Young’s modulus of MI SiC/SiC for 24 data points. Mean value is 181.5 GPa and standard deviation, 13.8 GPa. Fig. 5. Frequency histogram of proportional limit strength of MI SiC/SiC for 24 data points. Mean value is 166 MPa and standard deviation, 27.2 MPa. P.L.N. Murthy et al. / Composites: Part B 39 (2008) 694–703 697
P L N. Murthy et al. Composites: Part B 39(2008)694-70 the in-plane Youngs modulus using the Weibull distribu- data would yield improved goodness-of-fit scores, however tion a is the value ofy oung's modulus, B is the character- there are too few data points to conclude with any reason- istic value, and o and b both have units of stress able certainty that the underlying distribution has a three- Results from the measurements of the in-plane elastic parameter Weibull behavior (Ref. [7D). Also provided in modulus for 24 specimens are shown in Fig. 6 in the form Table I are the 90% confidence bounds(the 5 and 95% val of a Weibull plot. Table I lists the values of the Weibull ues)on the Weibull parameters. The relative spread in the parameters estimated from this data set. This information values is a function of the number of data points used in was obtained using the CARES/LIFE code, and these pro- the estimation and the Weibull modulus , Note that there cedures are described in reference [8]. The Weibull line that is always more relative spread in y than in B. In this case was best-fit to the data and corresponds to the parameters in 24 specimens were used in the estimation, which yielded rea- Table I is also shown in Fig. 6. The parameters were sonably narrow confidence bounds. A standard rule-of obtained using the maximum likelihood estimation method thumb is that 30 or more specimens are desirable to obtain (Ref. [8]. From Fig. 6 it can be seen that there is significant sufficiently narrow confidence bounds scatter in Youngs modulus where the data have a mean of Results from the measurements of the in-plane propor 81.5 GPa, a standard deviation of 13.8 GPa, and a Cov tional limit for 24 specimens and the best-fit Weibull line of 7.6%. For the Weibull distribution, the scatter is described obtained from maximum likelihood analysis are shown in with the Weibull modulus ,, which has a value of 14. 1. The Fig. 7. Table I lists the values of these estimated parameters. data visually shows a good fit to the two-parameter Weibull There is a significant scatter in the data, which have a mean distribution, and this is confirmed with Kolmogorov-Smir- of 166.0 MPa, a standard deviation of 27.2 MPa, and a nov(K-S)(Ref. [8) and Anderson-Darling(A-D)(Ref. [8D COV of 16.4%. The Weibull modulus y has an estimated goodness-of-fit significance levels of 49 and 82%, respec- value of 7. 4, which indicates considerably more scatter than tively. The A-D test is more sensitive to the tails of the dis- the in-plane modulus data. The K-S and A-D goodness-of- tribution, thus the interpretation of the percentages is that a fit significance levels were 87 and 74%, respectively, which better fit is achieved towards the tails than the central por- indicates a good fit across the entire range of data to the tions. Fitting a three-parameter Weibull distribution to the estimated parameters. The 90% confidence bounds on the 5 -1 190 Yound’ s modulus,GPa Fig. 6. Weibull plot of CMC in-plane Young,s modulus for 24 specimen measurements (also shown is best-fit line through data). For any x, P is the robability that the value is less than or equal to x. Table l for in-plane elastic modulus and proportional limit strength Property Weibull 90% confidence Weibull characteristic 90% confidence K-S goodness-of-fit A-D goodness-of-fit statistic modulus y bounds on y value B(MPa bounds on B statistic(and (and significance level %) 189.1×103 1946×103 0.43 9. 183.8×103(49%) Strength 24 7.4 0 0.51 168.7 (87%) Also shown are confidence bounds amete nd goodness-of-fit statistics
the in-plane Young’s modulus using the Weibull distribution, a is the value of Young’s modulus, b is the characteristic value, and a and b both have units of stress. Results from the measurements of the in-plane elastic modulus for 24 specimens are shown in Fig. 6 in the form of a Weibull plot. Table 1 lists the values of the Weibull parameters estimated from this data set. This information was obtained using the CARES/LIFE code, and these procedures are described in reference [8]. The Weibull line that was best-fit to the data and corresponds to the parameters in Table 1 is also shown in Fig. 6. The parameters were obtained using the maximum likelihood estimation method (Ref. [8]). From Fig. 6 it can be seen that there is significant scatter in Young’s modulus where the data have a mean of 181.5 GPa, a standard deviation of 13.8 GPa, and a COV of 7.6%. For the Weibull distribution, the scatter is described with the Weibull modulus c, which has a value of 14.1. The data visually shows a good fit to the two-parameter Weibull distribution, and this is confirmed with Kolmogorov–Smirnov (K–S) (Ref. [8]) and Anderson–Darling (A–D) (Ref. [8]) goodness-of-fit significance levels of 49 and 82%, respectively. The A–D test is more sensitive to the tails of the distribution, thus the interpretation of the percentages is that a better fit is achieved towards the tails than the central portions. Fitting a three-parameter Weibull distribution to the data would yield improved goodness-of-fit scores, however there are too few data points to conclude with any reasonable certainty that the underlying distribution has a threeparameter Weibull behavior (Ref. [7]). Also provided in Table 1 are the 90% confidence bounds (the 5 and 95% values) on the Weibull parameters. The relative spread in the values is a function of the number of data points used in the estimation and the Weibull modulus c. Note that there is always more relative spread in c than in b. In this case 24 specimens were used in the estimation, which yielded reasonably narrow confidence bounds. A standard rule-of thumb is that 30 or more specimens are desirable to obtain sufficiently narrow confidence bounds. Results from the measurements of the in-plane proportional limit for 24 specimens and the best-fit Weibull line obtained from maximum likelihood analysis are shown in Fig. 7. Table 1 lists the values of these estimated parameters. There is a significant scatter in the data, which have a mean of 166.0 MPa, a standard deviation of 27.2 MPa, and a COV of 16.4%. The Weibull modulus c has an estimated value of 7.4, which indicates considerably more scatter than the in-plane modulus data. The K–S and A–D goodness-of- fit significance levels were 87 and 74%, respectively, which indicates a good fit across the entire range of data to the estimated parameters. The 90% confidence bounds on the Fig. 6. Weibull plot of CMC in-plane Young’s modulus for 24 specimen measurements (also shown is best-fit line through data). For any x,P is the probability that the value is less than or equal to x. Table 1 Weibull parameters for in-plane elastic modulus and proportional limit strength Property Sample size Weibull modulus c 90% confidence bounds on c Weibull characteristic value b (MPa) 90% confidence bounds on b K–S goodness-of-fit statistic (and significance level %) A–D goodness-of-fit statistic (and significance level %) Modulus 24 14.1 17.7 189.1 · 103 194.6 · 103 0.17 0.43 9.9 183.8 · 103 (49%) (82%) Strength 24 7.4 9.2 177.5 187.0 0.13 0.51 5.3 168.7 (87%) (74%) Also shown are confidence bounds on parameters and goodness-of-fit statistics. 698 P.L.N. Murthy et al. / Composites: Part B 39 (2008) 694–703