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
P LN. Murthy et al./ Composites: Part B 39(2008)694-703 Proportional limit strength, MPa Fig. 7. Weibull plot of CMC in-plane measured proportional limit strength 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. estimated parameters show the uncertainty range as to the two variables (one relating to strength or resistance R, true values of the parameters, which in this case is between and the other to the loads, S, of the structure) as shown 9.2 and 5.3 for y and 187.0 and 168.7 for B. The in-plane in Fig 8. Both R and S are random in nature and that ran- proportional limit Weibull parameter estimates were also domness is characterized by their respective probability based on 24 measurements which, as noted before, yielded density functions. The nominal(deterministic) values, RN reasonably narrow confidence bounds. In summary, both and SN, used in a safety-factor-based approach, are also the in-plane modulus and proportional limit data fit the shown in Fig. 8. In a deterministic approach, the design two-parameter Weibull distribution reasonably well. The safety is assured by requiring that the rn be greater than degree of scatter was considerably larger in the proportional SN with a specified safety margin. A safe design requires limit data than the in-plane modulus data. The sample size that RN is greater than Sn at all times. The intent of this of 24 for each measured quantity provided an adequate approach and other similar deterministic approaches can degree of confidence that the estimated parameters were be understood by considering the area of the overlap of representative of their true values. The degree of scatter in two probability density functions(shaded region), which the data in both Figs. 6 and 7 highlight the necessity of using provides a measure of the probability of failure. This over- probabilistic methodology in any risk assessment of a com- lap depends upon ponent to meet its design requirements. 1. The relative positions of the curves represented by the 6. Probabilistic analysis approach mean values (us and uR)of the two variables S and R As the distance between the two curves increases. the The need to account for uncertainties in engineering area of the overlap(probability of failure)decreases design has long been recognized. No structure can be guar- anteed to be absolutely safe because of the unpredictability of the loads, uncertainties in the in situ material properties, the use of simplified assumptions in the analysis(which include limitations of the numerical methods used), and human factors(errors and omissions). Nevertheless, the probability of failure is usually required to be shown to application. Therefore, the estimation of failure probability and risk assessment of a structure becomes an important task for the design/ analysis engineers Design requirements dictat structure satisfies arious criteria of safety, serviceability and durability under the action of anticipated loads during its useful life- time. That is, the structural strength or resistance should be Fig 8. Failure probability evaluation. SN and RN are nominal values of greater than the effects caused by the action of loads. a stress and resistance(strength), respectively. The ss) and fdr)are implified model (referred to as R-S model)consists of corresponding failure probability density functions
estimated parameters show the uncertainty range as to the true values of the parameters, which in this case is between 9.2 and 5.3 for c and 187.0 and 168.7 for b. The in-plane proportional limit Weibull parameter estimates were also based on 24 measurements which, as noted before, yielded reasonably narrow confidence bounds. In summary, both the in-plane modulus and proportional limit data fit the two-parameter Weibull distribution reasonably well. The degree of scatter was considerably larger in the proportional limit data than the in-plane modulus data. The sample size of 24 for each measured quantity provided an adequate degree of confidence that the estimated parameters were representative of their true values. The degree of scatter in the data in both Figs. 6 and 7 highlight the necessity of using probabilistic methodology in any risk assessment of a component to meet its design requirements. 6. Probabilistic analysis approach The need to account for uncertainties in engineering design has long been recognized. No structure can be guaranteed to be absolutely safe because of the unpredictability of the loads, uncertainties in the in situ material properties, the use of simplified assumptions in the analysis (which include limitations of the numerical methods used), and human factors (errors and omissions). Nevertheless, the probability of failure is usually required to be shown to be within a specified acceptable range for each specific application. Therefore, the estimation of failure probability and risk assessment of a structure becomes an important task for the design/analysis engineers. Design requirements dictate that a structure satisfies various criteria of safety, serviceability and durability under the action of anticipated loads during its useful lifetime. That is, the structural strength or resistance should be greater than the effects caused by the action of loads. A simplified model (referred to as R–S model) consists of two variables (one relating to strength or resistance, R, and the other to the loads, S, of the structure) as shown in Fig. 8. Both R and S are random in nature and that randomness is characterized by their respective probability density functions. The nominal (deterministic) values, RN and SN, used in a safety-factor-based approach, are also shown in Fig. 8. In a deterministic approach, the design safety is assured by requiring that the RN be greater than SN with a specified safety margin. A safe design requires that RN is greater than SN at all times. The intent of this approach and other similar deterministic approaches can be understood by considering the area of the overlap of two probability density functions (shaded region), which provides a measure of the probability of failure. This overlap depends upon 1. The relative positions of the curves represented by the mean values (lS and lR) of the two variables S and R. As the distance between the two curves increases, the area of the overlap (probability of failure) decreases. Fig. 7. Weibull plot of CMC in-plane measured proportional limit strength 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. Fig. 8. Failure probability evaluation. SN and RN are nominal values of stress and resistance (strength), respectively. The fS(s) and fR(r) are corresponding failure probability density functions. P.L.N. Murthy et al. / Composites: Part B 39 (2008) 694–703 699
P L N. Murthy et al. Composites: Part B 39(2008)694-70 2. The dispersion of the two curves is characterized by the integration is performed over the failure region Q2 where standard deviations (os and GR) of the two variables S g0]. rent study the FPI approach is implemented as illustrated It should be noted that since the cumulative distribution in the following procedure function(CDF) of z at Zo equals the probability that [g of the CDF can be computed by varying the Zo and (1)A set of input random variables are identified, and the computing the point probability corresponding probabilistic distributions are selected The probability of failure, p, is given by the integral For a given set of random variables, a deterministic Pp= -s(X, x,x-)dXYidX ax finite element analysis is run using the ANSYS finite element code(Ref. [lID. The response results are col- lected from the finite element analysis output. in which x(X1, X2.. X,) is the joint probability density (2) The above process is repeated a number of times to function for the random variables X1,,... Xn, and the generate a table of response variable values that analysis using ANSYS output options Random input variables Advanced first-order (AFORM)using fast probability integr Sensitivities of the input Response cumulative distribution random variables function(CDF)(stress at critical location Fig 9. Probabilistic analysis flow diagram. ANSYS (ANSYS, Inc, Canonburg, PA)is finite element code
2. The dispersion of the two curves is characterized by the standard deviations (rS and rR) of the two variables S and R. For narrower curves, the area of overlap and thus the probabilities of failure are smaller. 3. The shape of the curves that are represented by the probability density functions. To achieve a safe design, the design variables must be chosen such that the area of overlap is minimized. The basic concept of the classical theory of structural reliability and risk-based design starts with the identification of relevant load and resistance parameters, called basic variables or sometimes referred to as random variables Xi (such as loads, material properties, and so forth) and the functional relationship between the response variable Z (e.g., stress at a point, deflection, frequency, etc.) and the basic random variables. Mathematically, it can be described as Z ¼ ZðX1; X2; X3; ... XnÞ ð2Þ A limit state function (sometimes referred to as performance function) is defined as g ¼ ZðXÞ Z0 ð3Þ where Z0 is a particular value of Z. A limit state function can be an implicit or explicit function of random variables and is divided in such a way that g(X) = 0 is a boundary between the failure region [g 6 0] and safe region [g > 0]. It should be noted that since the cumulative distribution function (CDF) of Z at Z0 equals the probability that [g 6 0], the CDF can be computed by varying the Z0 and computing the point probability. The probability of failure, pf, is given by the integral pf ¼ Z ... Z X fX ðX1; X2; ... XnÞdX1dX2dXn ð4Þ in which fX(X1,X2 ... Xn) is the joint probability density function for the random variables X1,X2 ... Xn, and the integration is performed over the failure region X where g 6 0 (Ref. [9]). If the random variables are statistically independent, then the joint probability density function can be replaced by individual density functions. This integral can be computed by the standard Monte Carlo procedure which is rather straightforward (Ref. [9]). However, depending upon the number of random variables involved and the level of Pf (usually very small), this must be repeated thousands of times to accurately build the response variable’s stochastic characteristics. Although the method is inherently simple, the large number of output sets that must be generated to build an accurate cumulative distribution function of the output variable becomes its obvious disadvantage. Furthermore, if the deterministic computation of the response is complicated (e.g., need for a large nonlinear finite element analysis), the computational costs could become prohibitive. Thus the need for more efficient approaches to perform such tedious tasks in a timely and cost-effective manner to routinely assess the reliability of a design cannot be overemphasized. For over two decades, NASA Glenn Research Center has been involved in developing efficient probabilistic analysis tools for aerospace applications. As a result of this effort, a collection of methods called fast probability integration (NESSUS/FPI) techniques were developed to solve a large class of engineering problems (Ref. [10]). In the current study the FPI approach is implemented as illustrated in the following procedure: (1) A set of input random variables are identified, and the corresponding probabilistic distributions are selected. For a given set of random variables, a deterministic finite element analysis is run using the ANSYS finite element code (Ref. [11]). The response results are collected from the finite element analysis output. (2) The above process is repeated a number of times to generate a table of response variable values that Fig. 9. Probabilistic analysis flow diagram. ANSYS (ANSYS, Inc., Canonburg, PA) is finite element code. 700 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 correspond to the perturbed values of the selected modulus and proportional limit statistics are input random variables based on observed scatter(measured data)and (3) The FPI analysis then uses the previously generated the Weibull parameter analysis described earlier table to compute the CDF and corresponding sensi- The probabilistic analysis, using the advanced tivities of the first-order reliability method(AFORM), is done in two steps. The first step involves the computa In addition to the CDF of the response, the FPI tech tion of the probabilistic characteristics (mean nique provides additional information regarding the sensi value and standard deviation) of the hoop stress tivity of the response to the random variables as shown at the critical location. Fig. 10a shows such schematically in the flow diagram of Fig. 9. The magnitude CDF with its predicted statistical characteristics of the sensitivity factor provides a way to rank the random Sensitivity factors for the hoop stress are shown variables that have the most influence on the uncertainty of in Fig. 10b. The Youngs modulus has the highest the response variable. This helps the user to prioritize the effect on the variability of hoop stress among the data collection resources. Also, by controlling scatter in random variables selected. The second step the more significant variables, the reliability can be involves risk quantification by employing the Improved basic r-s reliability model, where S represents the computed stress at the critical location, and 7. Results and discussion R is the resistance(strength/proportional limit) based on the observed scatter and the weibull As mentioned in the Probabilistic Analysis section, two parameter analysis described earlier cases were evaluated for the risk assessment of the vane the advanced first-order reliability methods, the design as described below: probability of failure is computed as 0.00994 (approximately 10 failures out of 1000 trials). Sen- Case I: Young's modulus, Poisson's ratio the coefficient sitivity information from this analysis indicates of thermal expansion for computing the variabl that scatter in strength virtually controls the reli ity in the stress, and the proportional limit of ability of the vane design. It is worth mentioning he material were considered random variables again that since the vere originally designed Other parameters were assumed to be determinis- to assure that under the high-pressure burner rig tic for this analysis. Table 2 shows the probabilis test conditions the maximum stresses will be tic characteristics of these random variables the below the proportional limit stress, a failure would mean that the vane design fails to meet this criterion Table 2 Parameters for probabilistic analyses Case il: In the second case. two additional variables Property Distributio namely, internal cooling air pressure and external value deviation type aerodynamic pressure on the vane-were added as In-plane modulus, GPa 181.513.8 Weibull random variables. Since at this time. no measured 0.170.009 Normal data on pressures are available, a hypothetical Coeffecient of thermal expansion 0.23 study was conducted to evaluate the effect of cer in-plane),10-°/C tain loading parameters. The internal cooling air portional limit, MPa 166.0 Weibull pressure is assumed to be normally distributed a1.00 0. 0.75 0.50 套050 §025 000 93.1 96.3 103.5 oIsson's Stress. MPa Fig. 10. Probabilistic analysis results for three of the random variables for Sic/Sic turbine stator blade design, case I Stress mean value is 97.4 MPa and standard deviation, 1.4 MPa(a) Cumulative distribution function( CDF) of hoop stress (b) Sensitivity factors
correspond to the perturbed values of the selected input random variables. (3) The FPI analysis then uses the previously generated table to compute the CDF and corresponding sensitivities of the response. In addition to the CDF of the response, the FPI technique provides additional information regarding the sensitivity of the response to the random variables as shown schematically in the flow diagram of Fig. 9. The magnitude of the sensitivity factor provides a way to rank the random variables that have the most influence on the uncertainty of the response variable. This helps the user to prioritize the data collection resources. Also, by controlling scatter in the more significant variables, the reliability can be improved. 7. Results and discussion As mentioned in the Probabilistic Analysis section, two cases were evaluated for the risk assessment of the vane design as described below: Case I: Young’s modulus, Poisson’s ratio, the coefficient of thermal expansion for computing the variability in the stress, and the proportional limit of the material were considered random variables. Other parameters were assumed to be deterministic for this analysis. Table 2 shows the probabilistic characteristics of these random variables. The modulus and proportional limit statistics are based on observed scatter (measured data) and the Weibull parameter analysis described earlier. The probabilistic analysis, using the advanced first-order reliability method (AFORM), is done in two steps. The first step involves the computation of the probabilistic characteristics (mean value and standard deviation) of the hoop stress at the critical location. Fig. 10a shows such a CDF with its predicted statistical characteristics. Sensitivity factors for the hoop stress are shown in Fig. 10b. The Young’s modulus has the highest effect on the variability of hoop stress among the random variables selected. The second step involves risk quantification by employing the basic R–S reliability model, where S represents the computed stress at the critical location, and R is the resistance (strength/proportional limit) based on the observed scatter and the Weibull parameter analysis described earlier. By using the advanced first-order reliability methods, the probability of failure is computed as 0.00994 (approximately 10 failures out of 1000 trials). Sensitivity information from this analysis indicates that scatter in strength virtually controls the reliability of the vane design. It is worth mentioning again that since the vanes were originally designed to assure that under the high-pressure burner rig test conditions, the maximum stresses will be below the proportional limit stress, a failure would mean that the vane design fails to meet this design criterion. Case II: In the second case, two additional variables – namely, internal cooling air pressure and external aerodynamic pressure on the vane – were added as random variables. Since at this time, no measured data on pressures are available, a hypothetical study was conducted to evaluate the effect of certain loading parameters. The internal cooling air pressure is assumed to be normally distributed Table 2 Parameters for probabilistic analyses Property Mean value Std. deviation Distribution type In-plane modulus, GPa 181.5 13.8 Weibull Poisson’s ratio 0.17 0.009 Normal Coeffecient of thermal expansion (in-plane), 106 /C 4.5 0.23 Normal Proportional limit, MPa 166.0 27.2 Weibull Fig. 10. Probabilistic analysis results for three of the random variables for Sic/Sic turbine stator blade design, case I. Stress mean value is 97.4 MPa and standard deviation, 1.4 MPa. (a) Cumulative distribution function (CDF) of hoop stress. (b) Sensitivity factors. P.L.N. Murthy et al. / Composites: Part B 39 (2008) 694–703 701
P L.N. Murthy et al. Composites: Part B 39(2008)694-70 0.80 0.60 ess, MPA of thermal rati Fig. ll. Probabilistic analysis results for five of the random variables for SiC/SiC turbine stator vane design, case Il. (a) Cumulative distribution function (CDF)of hoop stress Stress mean value is 96.5 MPa and standard deviation, 14.7 MPa.(b) Sensitivity factors b0.6 Failures Proportional 10/1000 limit strength ≥0.0 -0.4 case1:Pr=0.00994(-1% case2:Pr=00162(-1.6%) Case 2 Fig. 12. Risk analysis results summary for SiC/SiC turbine stator vane design, Pr is failure probability with a mean value of 862 kPa(125 y) and a coef- technique were used to perform a formal reliability assess- ficient of variation(COV)of 0.04 (i. e, a standard ment of an all-ceramic matrix composite(CMC) turbine deviation of 35 kPa(5 psi)). The external aerody stator vane. Two cases were evaluated -one with three ran- namic pressure is also assumed to be normally dis- dom variables for stress(all related to material properties tributed with a mean value of 552 kPa(80 y )and and the other with five random variables for stress(with a COV of 0.08. Fig. lla shows a CDF of hoop the addition of load-related parameters ). Results show that stress at the critical location. Fig. 1lb shows the the failure probability is 10/1000 and 16/1000 for the two ranking of the sensitivity factors(at a probability cases, respectively, for not meeting the design requirements level of 0.999). As expected, the loading parame- under the high-pressure burner rig test conditions. Results ters, namely the internal and external pressures, also showed that load-related parameters have a more sig- e the dominant ones that control the scatter in nificant effect on the uncertainty in hoop stress than mate- hoop stress. The stress distribution is used to rial- or geometry-related parameters. Reliability of the quantify risk by using the standard R-S model. vane design can be controlled primarily by reducing the Accordingly, the probability of failure is com- scatter in the proportional limit of the vane material. How puted as 0.0162(-1.6% or 16 failures out of ever, as the CMc materials continue to develop and fabri- 1000 trials). In this case, stress uncertainty also cation techniques for complex CMC structural shapes are contributes to this failure probability. The higher perfected, these failure rates will reduce. In any case, such failure probability is due to an increase in uncer- methodologies provide a quantifiable way to asses the risk tainty of hoop stress, which, in turn, occurs and provide a quantifiable tool to tailor specific designs for as random variables. A summary of the nes a given reliability because of the inclusion of loading parameter assessment for both cases is shown in Fig. 12 References [1 DiCarlo JA. CMC Material Development Status. Paper presented at Concluding remark the NASA Ultra Efficient Engine Technology (UEET) Technology Forum. itar restricted. 2002. Available from the nasa center for The ANSYS finite element code and the probabilistic [2] Murthy Pappu LN, Mital Subodh K, DiCarlo James ACharacter analysis methods in the FPI(fast probability integration) izing the Properties
with a mean value of 862 kPa (125 w) and a coef- ficient of variation (COV) of 0.04 (i.e., a standard deviation of 35 kPa (5 psi)). The external aerodynamic pressure is also assumed to be normally distributed with a mean value of 552 kPa (80 w) and a COV of 0.08. Fig. 11a shows a CDF of hoop stress at the critical location. Fig. 11b shows the ranking of the sensitivity factors (at a probability level of 0.999). As expected, the loading parameters, namely the internal and external pressures, are the dominant ones that control the scatter in hoop stress. The stress distribution is used to quantify risk by using the standard R–S model. Accordingly, the probability of failure is computed as 0.0162(1.6% or 16 failures out of 1000 trials). In this case, stress uncertainty also contributes to this failure probability. The higher failure probability is due to an increase in uncertainty of hoop stress, which, in turn, occurs because of the inclusion of loading parameters as random variables. A summary of the risk assessment for both cases is shown in Fig. 12. 8. Concluding remarks The ANSYS finite element code and the probabilistic analysis methods in the FPI (fast probability integration) technique were used to perform a formal reliability assessment of an all-ceramic matrix composite (CMC) turbine stator vane. Two cases were evaluated – one with three random variables for stress (all related to material properties) and the other with five random variables for stress (with the addition of load-related parameters). Results show that the failure probability is 10/1000 and 16/1000 for the two cases, respectively, for not meeting the design requirements under the high-pressure burner rig test conditions. Results also showed that load-related parameters have a more significant effect on the uncertainty in hoop stress than material- or geometry-related parameters. Reliability of the vane design can be controlled primarily by reducing the scatter in the proportional limit of the vane material. However, as the CMC materials continue to develop and fabrication techniques for complex CMC structural shapes are perfected, these failure rates will reduce. In any case, such methodologies provide a quantifiable way to asses the risk and provide a quantifiable tool to tailor specific designs for a given reliability. References [1] DiCarlo JA. CMC Material Development Status. Paper presented at the NASA Ultra Efficient Engine Technology (UEET) Technology Forum, ITAR restricted, 2002. Available from the NASA Center for Aerospace Information. [2] Murthy Pappu LN, Mital Subodh K, DiCarlo James A. Characterizing the Properties of a Woven SiC/SiC Composite Using Fig. 11. Probabilistic analysis results for five of the random variables for SiC/SiC turbine stator vane design, case II. (a) Cumulative distribution function (CDF) of hoop stress. Stress mean value is 96.5 MPa and standard deviation, 14.7 MPa. (b) Sensitivity factors. Fig. 12. Risk analysis results summary for SiC/SiC turbine stator vane design, Pf is failure probability. 702 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 W-CEMCAN Computer Code. NASA/TM-1999-209173, 1999 conference on composites. materials and structures, ITAR p: //gltrs grc nasa. gov/reports/1999/TM-1999-209173 pdf restricted, 2003. Available from the NASA Center for Aerospace 3] Lee kN. Current status of environmental barrier coatings for Si- Information based ceramics. Surf Coat Tech 2000: 133-134: 1-7 [7 Weibull w. A statistical distribution function of wide applicability. J [4 Calomino AM. Mechanical behavior and characterization of 1316C Appl Mech Trans ASME 1951; 18(3): 293-7 in situ BN coated MI/SiC/SiC composite. Paper Presented at the [8] Nemeth Noel N et al. CARES/LIFE ceramics analysis NASA Ultra Efficient Engine Technology (UEET) Technology liability evaluation of structures life prediction program. NASA/ Forum. ITAR restricted. 2002. Available from the NASA Center Tm?2003-106316,2003.http://gltrs.grcnasa.gov/reports/2003/tm for Aerospace Information. 2003-106316pdf Verrilli MJ. CMC Vane subelement fabrication and testing in a gas [9] Nowak As, AndrzejS, Collins KR Reliability of structures. McGraw turbine environment. Paper presented at the NASA Ultra Efficient Hill Science/Engineering/Math: 2000. ngine Technology (UEET) Technology Forum, ITAR restricted. [10] NESSUS(Numerical Evaluation of Stochastic Structures Under 003. Available from the NASA Center for Aerospace Information. Stress). Final NASA Code, New Manuals, ver. 6.2: 1995. [6 Thomas DJ. Srivastava R. Analysis of CMC Vanes subjected to [11] ANSYS, Ver. 7.1, Canonsburg, PA: ANSYS, Inc. 2003. gas turbine environment testing. In: Proceedings of the 27th annual
W-CEMCAN Computer Code. NASA/TM-1999-209173, 1999. http://gltrs.grc.nasa.gov/reports/1999/TM-1999-209173.pdf. [3] Lee KN. Current status of environmental barrier coatings for Sibased ceramics. Surf Coat Tech 2000;133–134:1–7. [4] Calomino AM. Mechanical behavior and characterization of 1316 C in situ BN coated MI/SiC/SiC composite. Paper Presented at the NASA Ultra Efficient Engine Technology (UEET) Technology Forum, ITAR restricted, 2002. Available from the NASA Center for Aerospace Information. [5] Verrilli MJ. CMC Vane subelement fabrication and testing in a gas turbine environment. Paper presented at the NASA Ultra Efficient Engine Technology (UEET) Technology Forum, ITAR restricted, 2003. Available from the NASA Center for Aerospace Information. [6] Thomas DJ. Srivastava R. Analysis of CMC Vanes subjected to gas turbine environment testing. In: Proceedings of the 27th annual conference on composites, materials and structures, ITAR restricted, 2003. Available from the NASA Center for Aerospace Information. [7] Weibull W. A statistical distribution function of wide applicability. J Appl Mech Trans ASME 1951;18(3):293–7. [8] Nemeth Noel N et al. CARES/LIFE ceramics analysis and reliability evaluation of structures life prediction program. NASA/ TM?2003-106316, 2003. http://gltrs.grc.nasa.gov/reports/2003/TM- 2003-106316.pdf. [9] Nowak AS, Andrzej S, Collins KR. Reliability of structures. McGrawHill Science/Engineering/Math; 2000. [10] NESSUS (Numerical Evaluation of Stochastic Structures Under Stress). Final NASA Code, New Manuals, ver. 6.2: 1995. [11] ANSYS, Ver. 7.1, Canonsburg, PA: ANSYS, Inc. 2003. P.L.N. Murthy et al. / Composites: Part B 39 (2008) 694–703 703