NASA Technical Memorandum 10209 1 

Probabilistic Structural Analysis Methods 

of Hot Engine Structures 

C.C. Chamis and D.A. Hopkins 

Lewis Research Center 

Cleveland, Ohio 

Prepared for the 

34th International Gas Turbine and Aeroengine Congress and Exposition 

sponsored by the American Society of Mechanical Engineers 

Toronto, Canada, June 4-8, 1989 

4 HA SA-Ttl- l 0 20 9 1 ) 

BHALYSIS NETHODS OP BOT EIGIIE STIt0CTI)RES 

(NASA. Lewis Research Center) 17 pCSCL 20K 

PRO 3 ABIL IS41 IC STRU CT UR Ai, N89-28030 

Unclas 

G3/39 0224994 
PROBABILISTIC STRUCTURAL ANALYSIS METHODS OF HOT ENGINE STRUCTURES 

C.C. Chamis and D.A. Hopkins 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135 

SUMMARY 

Development of probabilistic structural analysis methodology for hot 

engine structures is a major activity at NASA Lewis Research Center. 

sists of three program elements: (1) composite load spectra methods, (2) probabilistic structural analysis methods, and (3) probabilistic analysis and m structural reliability application. Recent progress includes: (1) quantifica03 etion of the effects of uncertainties for several variables on high pressure W I fuel turbopump (HPFT) turbine blade temperature, pressure, and torque of the 

space shuttle main engine (SSME), (2) the evaluation of the cumulative distribution function for various structural response variables based on assumed 

uncertainties in primitive structural variables, and (3) evaluation of the 

failure probability. Collectively, the results demonstrate that the structural 

durability of hot engine structural components can be effectively evaluated in 

a formal probabilisticlreliability framework. 

It conINTRODUCTION 

It is becoming increasingly evident that deter-ministic structural analysis 

methods will not be sufficient to properly design critical components in hot 

engine structures. These structural components are subjected to a variety of 

complex, and severe cyclic loading conditions including high temperatures and 

high temperature gradients. Most of these are quantifiable only as best engineering estimates. These complex loading conditions subject the material to 

coupled nonlinear behavior which depends on stress, temperature, and time. 

Coupled nonlinear material behavior is nonuniform, is very difficult to determine experimentally, and perhaps impossible to describe deterministically. In 

addition, hot rotating structural components are relatively small. Fabrication 

tolerances on these components, which in essence are small thickness variations, can have significant effects on the component structural response. Fabrication tolerances by their very nature are statistical. Furthermore the 

attachment of components in the structural system generally differs by some 

indeterminant degree from that assumed for designing the component. In summary, all four fundamental aspects: (1) loading conditions, (2) material 

behavior, (3) geometric configuration, and (4) supports - on which structural 

analyses are based, are of a random nature. The direct way to formally 

account for all these uncertain aspects is to develop probabilistic structural 

analysis methods where all participating variables are described by appropriate probability functions. 

NASA Lewis Research Center is currently developing probabilistic structural analysis methods for select engine structural components (fig. 1). 

Briefly, the deterministic, three-dimensional, inelastic analysis methodology 

developed under the Hot Section Technology (HOST) and Research and Technology 
Base Programs is being augmented to accommodate the complex probabilistic 

loading spectra, the thermoviscoplastic material behavior, and the materia 

degradation associated with the environment of aerospace propulsion system 

structural components. The goal of the methodology is to address the prob 

depicted schematically in figure 2. 

The development of probabilistic structural analysis methodology cons 

em 

sts 

of the following program.elements (ref. 1): (1) composite load spectra models, 

(2) computational probabilistic structural analysis methods, and (3) probabilistic analysis and structural reliability application. The development of the 

probabilistic structural analysis methodology is a joint program of NASA Lewis 

in-house and sponsored research. The objective of the proposed paper is to 

illustrate recent progress on the application of this methodology to determine 

the reliability of turbine blade components of rocket propulsion systems. 

progress of specific elements of the program are described in papers presented 

in several recent conferences (refs. 2 to 8). Recent activities have focused 

on extending the methods to include the combined uncertainties in several factors on the structural response (fig. 3). The objective of the present paper 

is to briefly describe recent progress in three program elements: composite 

load spectra models, probabilistic finite element structural analysis, and 

probabilistic strength degradation modeling. Progress is described in terms 

of fundamental concepts, computer code development, and representative numerical results. 

Past 

COMPOSITE LOAD SPECTRA 

The fundamental assumption for developing composite load spectra is that 

each individual load condition is the probabilistic time synthesis of four 

primitive parts: (1) steady state, (2) periodic, (3) random, and (4) spike. 

Each of these parts, except random, is described by a deterministic portion 

and a probabilistic perturbation about this deterministic portion as depicted 

schematically in figure 4. One justification for describing each loading 

condition in terms of primitive parts is that experts, over the years, have 

developed good judgments of the ranges of the perturbations about nominal 

(deterministic) conditions. The objective of the Composite Load Spectra program is to formalize the fundamental assumption in a computer code using: (1) 

available data from various rocket engines, (2) probability theory, and (3) a 

dedicated expert system. 

A schematic diagram of the composite load spectra (CLS) computer code is 

shown in figure 5. 

different engir;e factors on the SSME high pressure fuel turbopump turbine 

blade are Snown in figures 6 and 7. Figure 6 depicts the nominal temperatures, while figure 7 indicates the temperature changes due to hot gas seal 

geometry and respective perturbations indicated in the figure caption. for 

example, the greatest temperature change due to gas seal geometry (fig. 7) is 

53.3 "F. The combined contributions of this and other factors (not shown 

here) is 87.6 "F dhich is in addition to the greatest nominal temperature of 

;860 OF. 

this is not the case, because at these levels, small temperature changes have 

dramatic effects on durability and cooling requirements. 

Representative results obtained for the perturbations of 

Although at first glance a change of 87.6 "F may seem insignificant, 

2 
Another representative example is shown in figure 8, where the effect of 

comparable factors on the torque of the high pressure oxidizer turbopump 

(HPOTP) are plotted as a function of time, for the start-up portion of a mission cycle. 

Similar plots can be obtained for pressures or any other loading condition 

(ref. 5). 

different factors for each different load condition. 

These bounds are substantially wider at some times then at others. 

The current CLS code permits the simultaneous perturbation of 47 

PROBABILISTIC FINITE ELEMEN; STRUCTURAL ANALYSIS 

The fundamental assumption for developing probabilistic finite element 

structural analysis (PFEM) is that the uncertainties in each primitive structural variable can be represented by an assumed probabilistic distribution. 

Primitive structural variables are those which are used to describe a structure such as: (1) stiffness, (2) strength, (3) thickness and tolerance, (4) 

spatial location, (5) attachment, and (6) various nonlinear dependencies (temperature, stress, time, etc.). A schematic of the probabilistic distributions 

for some primitive variables is shown in figure 9. Subsequently, the uncertainties in the load conditions (described by the composite load spectra) and 

the uncertainties in the primitive structural variables are computationally 

simulated by performing probabilistic structural analyses to determine the random structural response of a specific SSME structural component. The structural response is generally described in terms of usual quantities such as 

displacement, frequencies, buck1 ing loads, and structural fracture toughness. 

The integration is illustrated in figure 10. 

It is instructive to compare component development by the traditional 

engineering approach and component evaluation using PFEM. 

summarized in table I. The former approach relies on physical experiments for 

quantification and requires that the material, fabrication process, and test 

methods be available. The latter approach is entirely computational and 

requires the integration of available structural analysis methods with available probability theory. The former approach has the advantage of demonstrating a specific technology while the latter has the advantage of assessing 

undeveloped, but with high payoff potential, candidate technologies. In addition, the former approach requires a large number of experiments to verify a 

design, while in the latter a design can be verified with strategically 

selected few experiments. 

The parallelism is 

PFEM has been formalized and integrated into a computer code identified 

as numerical evaluation of stochastic structures under stress (NESSUS). A 

schematic diagram of NESSUS is shown in figure 11. The user interacts with 

NESSUS through a dedicated expert system schematically shown in figure 12 

(ref. 8). Representative results obtained using NESSUS are shown in figure 13. 

The distributions assumed for the primitive variables are listed in the table 

in figure 13. Both the individual and the combined effects of the primitive 

variable uncertainties on the combined stress (Von Mises) are shown in figure 13 in terms of cumulative distribution functions (CDF). The information 

generated during the PFEM can be used to establish the confidence level. A 

sample result is shown in figure 14 for one blade location. 

3 
The curves in figure 14 can be used in a number of ways such as: (1) all 

the blades tested in the assumed conditions will have a mean combined stress 

between 57.9 and 62.1 ksi 90 percent of the time and (2) the combined stress 

in all the blades tested (under the assumed conditions) will range from 42 to 

83 ksi. This indicates that a wide scatter in the combined stress is probable. Two implications that follow are: (1) assessing the durabilityllife of 

the blades using only material uncertainties will not be sufficient, and (2) 

obtaining wide scatter in measured stress/strain magnitudes does not necessarily indicate test procedure difficulties. 

All the NESSUS results presented herein were obtained using about 1000 

perturbations for each case studied. These perturbations are relatively small 

in number compared to direct Monte Carlo simulation which will normally require 

20 000 and greater simulations to generate the same curves (refs. 6 and 7). 

The reduced (but with comparable accuracy) number of simulations is a result 

of a NESSUS feature which utilizes the fast probability integration method 

(ref. 6) to select subsequent perturbations in a self-adaptive manner. 

PROBABILISTIC ANALYSIS FOR STRENGTH DEGRADATION 

The fundamental assumption for developing probabilistic analysis methods 

for strength degradation is that the uncertainties in the primitive variables 

affecting strength can be described by assumed distributions. Two different 

models are selected to demonstrate the concept. The models express the number 

of mechanical load cycles to failure. One of the models is based on linear 

elastic fracture mechanics and the other on a strength degradation model 

recently studied at NASA Lewis (ref. 9). The models with their respective 

primitive variables are summarized in table 11. 

Both of these models are used to predict the number of cycles to failure 

in a material typical for SSME components. The input for the fracture mechanics model is summarized in table 111. The CDF obtained from this input is 

shown in figure 15. The input for the strength degradation model is summarized in table IV and the corresponding CDF is shown in figure 16. Both CDF’s 

exhibit wide ranges for the probable number of cycles to failure. The linear 

fracture mechanics model shows a mean of 10 000 cycles while the strength 

degradation model shows a mean of 10 million cycles. Based on this comparison 

the linear fracture mechanics model penalizes the material by three decades. 

It is important to note the differences between the two models: (1) the linear 

fracture mechanics model assumes the existence of a crack-like defect and then 

evaluates the number of cycles required to grow this defect to a critical size 

for imminent rapid propagation to fracture; (2) the strength degradation model 

does not presume the existence of defects and, therefore, includes both defect 

initiation and propagation resulting in greater number of cycles; (3) the linear fracture mechanics model has five primitive variables while the strength 

degradation model has 13 (assuming that the greater the number of primitive 

variables in the model the more inclusive the representation of the physics in 

the model, then the strength degradation model will be more accurate); and 

(4) The linear fracture mechanics model requires determination of c, m, and 

ai by specialized and often complex test methods while the strength degradation model uses available room temperature material properties. The computer 

code to perform both of these simulations is described in reference 9. 

4 
CONCLUSIONS 

The development or probabilistic structural analysis methods for hot 

engine structures continues. Recent progress on its application to select 

SSME components includes: (1) the effects of the uncertainties of several factors on blade temperatures, pressures, and torque, (2) the evaluation of the 

cumulative distribution function of structural response variables based on 

assumed uncertainties in the structural primitive variables, (3) evaluation of 

failure probability, and (4) life assessment in terms of cumulative distribution function using linear fracture mechanics and strength degradation models. 

Three different computer codes are being developed in parallel: (1) Composite 

Load Spectra (CLS) for the probabilistic description of SSME component loads, 

(2) NESSUS, for the probabilistic structural analysis of select SSME structural 

components, and (3) a life durability code for the assessment of the fatigue 

cycles to failure of structural components in SSME mission environments. Collectively, the results obtained to date demonstrate that the structural durability of SSME critical components can be evaluated using the probabilistic 

methodology under development. 

REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

Chamis, C.C., 1986, "Probabilistic Structural Analysis Methods for Space 

Propulsion System Components," NASA TM-88861. 

Chamis, C.C. and Hopkins, D.A., Session Chairs, 1987, "Session IV, 

Structural Dynamics," Structural Integrity and Durability of Reusable 

Space Propulsion Systems, NASA CP-2471, pp. 117-200. 

Liu, W.K., Besterfield, G.H., and Belytschko, T., 1987, "A Probabilistic 

Hu-Washizu Variational Principle," 28th Structures, Structural Dynamics 

and Materials Conference, Part 1, AIAA, New York, pp. 252-259. 

Nagpal, V.K., Rubinstein, R., and Chamis, C.C., 1987, "Probabilistic 

Structural Analysis to Quantify Uncertainties Associated with Turbopump 

Blades," 28th Structures, Structural Dynamics and Materials Conference, 

Part 1, AIAA, New York, pp. 268-273. 

Spencer, B.F. and Hopkins, D.A., 1988, "An Expert System for Probabilistic 

Description of Loads on Space Propulsion System Structural Components," 

29th Structures, Structural Dynamics and Materials Conference, Part 3, 

AIAA, New fork, pp. 1262-1266. 

Wu, Y.T. and Burnside, O.H., 1988, "Validation of the NESSUS Probabilistic 

Finite Element Analysis Computer Program," 29th Structures, Structural 

Dynamics and Materials Conference, Part 3, AIAA, New fork, pp. 1267-1274. 

Shiao, M.C., Nagpal, V.K., and Chamis, C.C., 1988, "Probabilistic 

Structural Analysis of Aerospace Components Using NESSUS," 29th 

Structures, Structural Dynamics and Materials Conference, Part 3, AIAA, 

New 'fork, pp. 1275-1282. 

5 
8. Millwater, H., Palmer, K., and Fink, P., 1988, "NESSUS/Expert - An Expert 

System for Probabilistic Structural Analysis Methods," 29th Structures, 

Structural Dynamics and Materials Conference, Part 3, AIAA, New York, 

pp. 1283-1288. 

Traditional component development 

Parameter identification 

Characterization - database 

Component fabrication 

Component testing 

Component test results database 

Stati sti cal inferences 

Re1 i abi 1 i ty/conf i dence 1 evel 

9. Boyce, L. and Chamis, C.C., 1988, "Probabilistic Constitutive 

Relationships for Cyclic Material Strength Models," 29th Structures, 

Structural Dynamics and Materials Conference, Part 3, AIAA, New York, 

pp. 1299-1306. 

PSAM component evaluation 

Parameter identification 

Probabilistic distributions 

Component mechanistic model 

Component anal ysi s 

Component analysis results database 

Statistical inference 

Re1 i abi 1 i ty/conf i dence 1 evel 

Physi cal 

TABLE 11. - PROBABILISTIC ANALYSIS FOR MATERIAL STRENGTH DEGRADATION AND FRACTURE MODELS 

Computational 

fatigue crack growth model 

where C, M, ho and a, are random variables 1 

Strength reduction model 

1 

s 

m 

where NMF, NMO, S, So, TF, T, To, SF, 0, (1 n, in and q are random variables 0' 

6 
TABLE 111. - PRIMITIVE VARIABLE DISTRIBUTIONS FOR THE FATIGUE CRACK GROWTH MODEL 

L 

Vari ab1 e 

Material property, C 

Materi a1 property, M 

A1 ternati ng range 

Initial crack size, aI 

Final crack size, aF 

Component/crack shape 

stress, ha 

parameter, Y 

Vari ab1 e type 

Random 

Random 

Random 

Random 

Determi n i s t i c 

Deterministic 

Di stri bution type 

Lognormal 

No rnial 

Lognormal 

Log n o rina 1 

N/A 

N /A 

Mean 

8.66~10-~' in. /cycl e 

6.48~10~' i 

cycle/ksi in. ?7i 

90 ksi 

118~10-~ in. 

7.87~10-~ in. 

1 .o 

Standard deviation 

0.866~10-~~ (10 percent) 

3.24~10-~ (5 percent) 

9 (10 percent) 

17.7~10-~ (15 percent) 

N/A 

N/A 

TABLE IV. - PRIMITIVE VARIABLE DISTRIBUTIONS FOR THE STRENGTH REDUCTION MODEL 

I Variable 

Melting temperature, TF 

Ultimate tensile strength, SF 

Log ot final cycle, NMF 

Reference temperature, TO 

Residual compressive stress, 00 

Log of reference cycle, N 

Reference fatigue strengtr: SO 

Current temperature, T 

Current mean stress, a 

Current fatigue strength, S 

Temperature exponent, N 

Stress exponent, M 

Cycle exponent. Q 

Normal 

Lognormal 

Log normal 

Normal 

L ogn o rma 1 

Lognormal 

Lognormal 

Normal 

Lognormal 

Log normal 

Normal 

rlormal 

tlo rma 1 

2732 OF 

130 ksi 

68 OF 

-2.9 ksi 

7 

72.6 ksi 

1562 OF 

21.8 ksi 

36.3 ksi 

0.5 

0.5 

0.5 

a 

Standard deviation 

I 82.0 (3 percent) 

6.5 15 percent) 

.8 (10 percent) 

2.0 (3 percent) 

-0.145 (5 percent) 

0.7 (10 percent) 

3.6 (5 percent) 

46.7 (3 percent) 

1.1 (5 percent) 

1.8 (5 percent) 

0.15 (3 percent) 

0.15 (3 percent) 

0. I5 i 7 percent) 

7 
HIGH PRESSURE TURBOPUMP BLADE 

LOX POSTS 

MAIN COMBUSTION CHAMBER 

FIGURE 1. - SELECT SSME COMPONENTS 

INCLUDED IN THE PROBABLILISTIC 

STRUCTURAL ANALYSIS METHODS 

DEVELOPMENT PROGRAM. 

ORIGINAL PAGE 

BLACK AIL'D WHITE PHCTOQRAPH 

a 
PROBLEM: CONFIDENCE 

PROB [ 0 DIFFICULT ANALYSIS 

P R 0 B L E M 

NATURAL FREQUENCY < 

OF BLADE, 0 

VAR I ABLES 

0 GEOMETRY 

0 STIFFNESS CDF 

MASS OF 0 

0 APPROXIMATE METHOD 

DEFINED 

7) I' 

1 '1 0 GIVES INTERVAL ESTIMATES 

0 USES NESSUS DATA BASE 90% CONFIDENCE BOUNDS 

ANSWER - PROBABILITY 

- VARIABILITY 

ESTIMATE 

I 

0 

/' 

FIGURE 2. - PROBABILISTIC STRUCTURAL ANALYSIS DEFINITION, 

PLANNED ACCWLISHRENT STATUS 

DEVELOP PROBABILISTIC STRUCTURAL 0 FORMULATED PROBABILISTIC FINITE 

ANALYSIS RETHODS/CWUTER CODES ELEMENT METHODS 

0 DEVELOPED SECOND GENERATION FOR QUANTIFYING THE INDIVIDUAL 

COMPUTER CODE - OPERATIONAL AT 

LEWIS. UNCERTAINTIES INCLUDED: 

AND COMBINED EFFECTS OF UNCERTAlNTlES ON THE STRUCTURAL DUR- 

- LOADS ABILITY OF SW COMPONENTS 

- GEOMETRY 

- MATERIAL 

- SPATIAL POSITION 

- ATTACHMENT 

COMBINED PROBABILITY OF OCCURRENCE 

SIGNIFICANCE 

THE COMBINED PROBABILITY OF MAGNITUDE OF STRUCTURAL DURABILITY 

VARIABLES CAN BE DETERMINED 

20 40 60 80 100 

BLADE STRESS, KSI 

FIGURE 3. - PROBABILISTIC STRUCTURAL ANALYSIS METHODS (PSAM). 

9 
r- rEXPERT 

SYSTEM 

a 

c z STEADY STATE PERIODIC u 

r 

0 SOPHISTICATED PROBABILISTIC METHODS 

DISCRETE PROBABILITY DISTRIBUTION 

MONTE CARLO 

BARRIER CROSSING 

DRIVER (USER 

INTERFACE 

MODULE) 

- DATA 

BASE 

RULE - 

BASE 

TIME CD-88-3216J 

FIGURE 4. - EACH LOAD CONDITION IS THE PROBABILISTIC TIME 

SYNTHESIS OF FOUR PRIMITIVE PARTS: COMPOSITE LOAD SPECTRA. 

COMPOS I TE 

OUTPUT LOAD 

SPECTRA 

COMPOSITE LOAD SPECTRA EXPERT SYSTEM LOAD EXPERT SYSTEM DESIGN PHILOSOPHY 

SCALING COEFFICIENTS 

NOMINAL ENGINE COEFFICIENTS 

DUTY CYCLE LOAD PROFILES 

ENGINE CONFIGURATION AND GEOMETRY DATA 

REQUIREMENT 

I RULE-BASE PRODUCTION SYSTEM 

IF-THEN RULES 

SIMPLE INFERENCE SCHEME 

10 
1468. 

1368. 

1260. 

1168. 

1868. 

1885. 

980. 

888. 

781. 

680. 

588. 

X SEERDY STRTE TEPPERRTCRE 399.H 

8 

FIGURE 6. - COMPOSITE LOAD SPECTRA PREDICTED STEADY STATE TEMPERATURE. 

X -.02071 IEWEPLITURE PERTUPB9iIO% CziTl ?LE i3 HOT 3GS MLTR TEW5RRTJRES ONE STUNGRRO DEUIRTiON OF ,860 P 

FIGURE 7. - TEMPERATURE PERTURBATION DATA DUE TO HOT GAS SEAL 

GEOMETRY DELTA TEMPERATURES. ONE STANDARD DEVIATION. 

11 
v) P 

L 

v) 4 

s 

c 

W 

W 

a 

8 

1 .o 

.8 

.6 

0 RAN 

.4 

.2 

0 1 2 3 4 5 

TIME. SEC 

FIGURE 8. - TRANSIENT MODEL HPOTP TORQUE. 

Y V z 

Y 

E 

E 

3 

0 

U 0 

Y 0 

A STIFFNESS 

ILL SPATIAL LOCATION 

STRENGTH TOLERANCE ar/\ ATTACHMENT TEMPERATURE DEPENDENCE 

MAGNITUDE 

CD-BB-32181 

FIGURE 9. - PROBABILISTIC STRUCTURAL ANALYSIS: THE UNCERTAINTIES IN EACH STRUCTURAL PRIMITIVE VARIABLE CAN BE 

DESCRIBED BY ASSUPIED CORRESPONDING PROBABILISTIC DISTRIBUTIONS. 

LOCAL: AT EACH REGIDN BOUNDED BY NODAL LINES 

Mil = Ivhl'IDlhldv 

CI~= Ivt~~l~tflt~lt~cldv 

Kil= ivt~dTtD1tBldv 

THE UNCERTAINTIES IN THE STRUCTURAL 

RESPONSE VARIABLES (u, W, u, 6, G) ARE 

DETERMINED FROM THE RESULTS OF THE 

MULTIPLE FINITE ELEMENT ANALYSES CD-UU-32187 

FIGURE 10. - UNCERTAINTIES IN THE LOAD CONDITIONS AND STRUCTURAL 

PRIMITIVE VARIABLES ARE INTEGRATED BY PERFORMING MULTIPLE FINITE 

ELEKNT ANALYSES. 

12 
FINITE ELEMENT LIBRARY PERTURBAT ION VARIABLES : DATA FLOW THROUGH THE 

MAIN PROCESSORS IN NESSUS 

DATABASE 

PERTURBATION PROCESSOR 

! 

(1) TOPOLOGY 

NODAL COORDINATES 

NODAL SHELL THICKNESS 

NODAL SHELL OR BEAM NORMALS PLANE STRESS PLANE STRAIN AXISYWTRIC 

(I I ) GEONETRIC PROPERN 

THICKNESS OF PLAIN STRESS 

ELEMENTS 

3D BRICK 3D SHELL LINEAR BEAM ELASTIC BEAM SECTION PROPERTIES 

e 

e 

e 

(111) MATERIAL PROPERTY 

ELASTICITY MODULUS ALL ELEMENTS INTEGRATED NUMERICALLY 

LINEAR LAGRANGIAN INTERPOLATING POISSON'S RATIO 

SELECTIVELY REDUCED INTEGRATION MATERIAL DENSITY 

(IV) BOUNDARY CONDITIONS 

FUNCTIONS THERMAL EXPANSION COEFFICIENl 

BASE SPRING STIFFNESSES 

(V) LOADS 

FAST PROBABILITY 

INTEGRATION PROCESSOR 

PROBABILITY DENSITY 

AND CUMULATIVE 

NODAL FORCE VECTORS 

ELEENT PRESSURES AND EDGE 

NODAL TEMPERATURES 

TRACTIONS 

FIGURE 11. - PROBABILISTIC FINITE ELEMENT COMPUTER CODE NESSUS. 

FIGURE 12. - NESSUS DEDICATED EXPERT SYSTEM. 

13 
RANDOM VARIABLE DISTRIBUTION TABLE 

YX SSME 

BLADE 

- 

:ASE 

- 

1 

2 

3 

4 

5 

6 

7 

8 

9 

RANDOM 

VARIABLE, 

RV 

X - COOR (N)b 

Y - COOR (N) 

2 - COOR (N) 

T - TEMP (N) 

E - YOUNG'S 

NODULUS (N) 

L - LATERAL 

PRESSURE (N) 

F - CONC. 

LOAD (N) 

R - ROTATION 

SPEED (N) 

C - COMBINE 

aCOV OF RV 

0.005 (STDJ 

,005 (STD) 

.005 (STD) 

60 OF (STD) 

0.10 

0.20 

0.20 

0.02 

CDF OF VON MISES STRESS DUE TO SINGLE/ 

CWINED RANDOM VARIABLE PERTURBATION 

2 COORDINATE 

B V ':ZF 1 , 

Eo 

6 

L 

? Y COORDINATE 

- 

PI EO "EL 

a 

X COORDINATE 5 1.0 

V 

0 60 120 

LATERAL 

PRESSURE 

EL 

EL 

0 EL60 120 

YOUNG'S 

MODULUS 

TEMPERATURE 

ROTATION 

SPEED 

EL 

ti 

0 EL 60 120 

CONCENTRATED 

COMBINED CASE 

COMBINED STRESS AT ROOT. KSI 

aCOV: COEFFICIENT OF VARIATION (STD/MEAN) 

hN) NORMAL DISTRIBUTION 

%TD: STANDARD DEVIATION 

FIGURE 13. - PROBABILISTIC STRUCTURAL ANALYSIS USING NESSUS. 

1 .o 

L 

IE: .5 

E! 

2 

a 

Dz I0 

40 000 62 500 85 OOO 

COMB I NED STRESS 

FIGURE 14. - 90% CONFIDENCE BOUND RESULT FROM 

UNCERTAINTY IN THE INPUT PARAMETERS. 

14 
1 .o 

E .8 

8 

0 

Y V 4 

U 

CL .6 

CL 0 LL 

0 

3.0 3.4 3.8 4.2 4.6 5.0 

LOG OF CYCLES 

FIGURE 15. - SIMULATED FAILURE PROBABILITY USING A FATIGUE 

CRACK GRWH MODEL. 

CL 0 LL 

> 

t 

1.0 - 

.8 - 

.6 - 

.4 - 

.2 - 

0 I 

5 6 7 8 9 

LOG OF CYCLES 

FIGURE 16. - SIMULATED FAILURE PROBABILITY USING 

A STRENGTH REDUCTION RODEL. 

15 
National Aeronautics and Space Administration 

1. Report No. 

NASA TM-102091 

Report Documentation Page 

2. Government Accession No. 3. Recipient's Catalog No. 

4. Title and Subtitle 

Probabilistic Structural Analysis Methods of Hot Engine Structures 

5. Report Date 

6. Performing Organization Code 

C.C Chamis and D.A. Hopkins 

7. Author@) 

I E-4855 

8. Performing Organization Report No. 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

10. Work Unit No. 

553- 13-00 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

114. Sponsoring Agency Code 

I 

5. Supplementary Notes 

Prepared for the 34th International Gas Turbine and Aeroengine Congress and Exposition sponsored by the 

American Society of Mechanical Engineers, Toronto, Canada, June 4-8, 1989. 

6. Abstract 

Development of probabilistic structural analysis methods for hot engine structures is a major activity at Lewis 

Research Center. It consists of three program elements: (1) composite load spectra methodology, (2) probabilistic 

structural analysis methodology, and (3) probabilistic structural analysis application. Recent progress includes: 

(1) quantification of the effects of uncertainties for several variables on high pressure fuel turbopump (HPFT) turbine blade temperature, pressure, and torque of the space shuttle main engine (SSME), (2) the evaluation of the 

cumulative distribution function for various structural response variables based on assumed uncertainties in primitive 

structural variables, and (3) evaluation of the failure probability. Collectively, the results demonstrate that the 

structural durability of hot engine structural components can be effectively evaluated in a formal probabilistic/ 

reliability framework. 

Structural analysis; Finite elements; Strength degradation; 

Fracture mechanics; Failure probability; Structural 

durability; Fatigue life; Probability distributions 

Unclassified -Unlimited 

Subject Category 39 

7. Key Words (Suggested by Author@)) 

9. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of pages 22. Price' 

Unclassified Unclassified I A03 

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