SO7-00 " 

f"y 

NASA Technical Memorandum 102324 

Probabilistic Structural Analysis of Aerospace 

Components Using NESSUS 

Michael C. Shiao and Vinod K. Nagpal 

Sverdrup Technology, Inc. 

NASA Lewis Research Center Group 

Cleveland, Ohio 

and 

Christos C. Chamis 

National Aeronautics and Space Administration

OR1!' Lewis Research Center 

Cleveland, Ohio 

Presented at the 

29th Structures, Structural Dynamics and Materials Conference 

cosponsored by the AIAA, ASME, ASCE, AHS, and ACS 

Williamsburg, Virginia, April 18-20, 1988 

STRUCTURAL t90-22823 

ANALYSIS OF AFRnSPACE COMPUNINTS USING J. 

NESSUS (NASA) 10 D CSCI 20K

UnC1aS 

G3/39 0286260
OR'

TMtS 

C..UR 

PROBABILISTIC STRUCTURAL ANALYSIS OF AEROSPACE

COMPONENTS USING NESSUS 

Michael C. Shiao* and Vinod K. Nagpalt

Sverdrup Technology, Inc.

NASA Lewis Research Center Group

Cleveland, Ohio 44135 

and 

Christos C. Chamis

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135 

Summary 

Probabilistic structural analysis of a space shuttle main 

engine (SSME) turbopump blade is performed by using the 

computer code NESSUS (Numerical Evaluation of Stochastic 

Structures Under Stress). The goal of the analysis is to derive 

probabilistic characteristics of blade response for given 

probabilistic descriptions of uncertainties in blade geometry, 

material properties, temperature, and pressure distributions. 

Probability densities are calculated for critical blade responses. 

Risk assessment is performed assuming different failure models. 

Introduction 

Probabilistic structural analysis assesses the effects of 

uncertain or random input parameters on structural response. 

Unlike the traditional deterministic analysis, in which 

geometry, material properties, and loading are assumed 

known, probabilistic structural analysis treats these inputs as 

samples from probability distributions. This approachmakes 

it possible to assess the effects of fluctuating loads, variable 

material properties and uncertain geometry on structural 

response. In the deterministic approach, these uncertainties 

are not quantified and are acknowledged by an empirical safety 

Research Engineer. 

tDepartment Director. 

Senior Aerospace Scientist, Structures Division.

factor. Probabilistic structural analysis provides a more 

realistic and systematic way to evaluate structural performance 

and durability. 

The NESSUS computer code (ref. 1), developed under the 

research program, Probabilistic Structural Analysis Methods 

for Select Space Propulsion System Components (PSAM) 

(ref. 2), combines finite element methods with probabilistic 

load and geometry input descriptions and structural reliability 

algorithms to provide an integrated approach to probabilistic 

structural analysis. 

Structure of NESSUS 

NESSUS consists of the following three major modules: 

NESSUS/PRE, NESSUS/FEM, and NESSUS/FPI. 

NESSUS/PRE is a preprocessor used for the preparation 

of the statistical input needed to perform the probabilistic finite 

element analysis. It allows the user to describe a spatial domain 

defined by a set of discrete points, typically corresponding to 

the nodal points of a finite element mesh. One or more random 

fields may then be specified over this spatial domain by 

defining the mean value and standard deviation of the random 

variable at each point, together with an appropriate form of 

correlation. Correlated random variables are decomposed into 

a set of uncorrelated vectors by an eigenvalue transformation 

procedure. For strong correlation problems, the number of 

dominant random variables in the set of uncorrelated vectors 

will be much less than that of weakly correlated random 

variables. The computational time required for the analysis
Z "2 

Lu 

Lu Do 

EjD

a 2 

0 

0 

-J

a point probability estimate. The cumulative distribution 

function can be obtained numerically by running FPI at several 

response values. One alternative for generating the distribution 

function for any given response is to conduct a Monte Carlo 

simulation study. However, in general, direct simulation is 

very costly. NESSUS/FPI provides a method which not only 

produces an accurate probability distribution, but also requires 

less computing time than Monte Carlo simulation, especially 

in the low probability region. 

E 1 E0 E2

YOUNG'S MODULUS 

Figure I.—Response function. 

DISPLACEMENT 

Figure 2.—Modified Newton iteration. 

will also decrease due to the reduction of the number of 

uncorrelated random variables. 

NESSUS/FEM is a finite element code used for structural 

analysis and parameter sensitivity evaluation. It generates a 

database containing all the response information corresponding 

to a small variation of each independent random variable. The 

algorithm used in NESSUS/FPI requires an explicit response 

function in terms of uncorrelated random variable in order to 

perform a reliability analysis. In complicated structural analysis 

problems, response can only be available implicitly through 

a finite element model. To overcome this difficulty, the response 

function is estimated numerically. As an example in figure 1, 

the response (dependent variable) to be considered is the 

displacement D and the uncertainty involved is the modulus 

E (independent variable). NESSUS/FEM calculates the 

displacements D0 , D1 , and D2 corresponding to the modulus 

at E0 , E1 , E2 . Displacements D1 and D2 are computed by a 

method akin to the modified Newton nonlinear algorithm 

illustrated in figure 2. 

NESSUS/FPI implements an advanced reliability method 

known as the Fast Probability Integrator (refs. 3 and 4). This 

module extracts data from the database generated by 

NESSUS/FEM to develop a response or performance function 

in terms of uncorrelated random variables. The probabilistic 

structural response is calculated from the performance model. 

For a given response value, the probability of exceedence at 

this value is estimated by a reliability method, which treats 

the problem as a constrained minimization. This step is called

Concept of Probabilistic Structural 

Analysis by NESSUS 

Defining the loading, material properties, geometry, and 

boundary conditions as random variables, the probabilistic 

structural analysis is then performed with NESSUS. The 

structure can be considered as a filter to those random 

variables. The response variables, which can be displacements, 

strains and stresses at one or several locations, or natural 

frequencies, are the output of the filter and will be described 

by a cumulative distribution function (CDF) or a probability 

density function (PDF). The concept is illustrated in figure 3. 

Turbine Blade Study by NESSUS 

Probabilistic structural analysis requires that the uncertainties 

involved be recognized and quantified. An SSME turbopump 

blade as shown in figure 4 is subjected to complex mechanical 

f A STRUCTURALI .- I J

E I MODEL H 

X Y 

(a) Input PDF (load, material, geometry). 

(b) Output PDF (displacement, strain, stress).

Figure 3.—Concept of probabilistic structural analysis. 

STOCHASTIC 

THE RMOMECI-1AN CAL 

LOADS 

J BLADE 

STOCHASTIC 

MATERIAL 

PROPERTIES

/

UNCERTAIN

L 

\c SHAPE 

-.

FIRTREE 

Figure 4.—A typical turbine blade. 
PRESSURE,

PSI 

519 

466 

413 ..J 

:::. 

254 

201 

148 

95 

42 

TEMPERATURE. 

1836 

1826 

1817 

1807 

1798 

1788 

1778 

1769 

1759 

1750 

and thermal loads (ref. 5). The dominant loads on the turbine 

blade are centrifugal force, thermal load, and the differential 

pressure across the airfoil. Centrifugal force is induced by the 

blade rotation. Since it is difficult to maintain constant 

rotational speed, the centrifugal force is defined as a random 

variable. Random thermal load is due to combustion 

irregularities that cause a random temperature distribution in 

the blade. Differential pressure is also random because of 

pressure fluctuations. Uncertainties in the blade geometry exist 

as a result of the manufacturing process. The stochastic 

material properties are caused by nonuniformities in the

material resulting from the fabrication. Uncertainty in the 

temperature distribution causes an additional level of material 

property uncertainty. 

An SSME turbopump blade is modelled by 40 four-node shell 

elements with 55 nodal points. In this study, seven random 

fields are considered as listed in table I. Random variables 

for coordinate and temperature are defined at each grid point. 

Those for modulus and lateral pressure are defined at the 

element level. The mean values of coordinates random 

variables are the same as the deterministic coordinates. Mean 

temperature and pressure are shown in figures 5 and 6. It is 

TABLE 1.—RANDOM INPUT DATA

Random fields Number of 

dependent 

random 

variables

Mean Standard 

deviation 

(or COV)

Correlation 

length, 

in.

Number of 

independent 

random 

variables 

X-coordinate 55 Deterministic 0.01 in. 5.0 13 

coordinate 

Y-coordinate 55 Deterministic .01 in. 5.0 13 

coordinate 

Z-coordinate 55 Deterministic .01 in. 5.0 13 

coordinate 

Temperature 55 Steady state 60 F 3.0 22 

temperature 

Modulus 40 23 x 106 psi .10 (COV) 3.0 16 

Pressure 36 Steady state .20 (COV) 0.0 36 

pressure 

Rotation 1 40 000 rpm .01 (COV) ---

Figure 5.—Steady state temperature distribution. Figure 6.—Steady state pressure distribution.

3 
.125 4 

.115 

.105 

k

.045 

035 (d) 

(a) Mean (pounds per square inch). (b) Standard deviation (pounds per square inch). 

(c) Coefficient of variation. (d) Probability of failure. 

Figure 7.—Effective stress on pressure side. 

4****^, ---Mm

(b) 

(C) 

(a)

102 

91 

80 

48 

37 

26

assumed that the random fields are independent of each other. 

However, the correlation coefficients between random 

variables in the same field are defined as 

p(X, X) = exp(—sX,IS) (1) 

where Xi and X1 are the ith and jth random variables in the 

X-coordinate random field; AXjj is the distance between X1 

and X. S is the correlation length. The larger the correlation length, the greater the correlation between the random 

variables in random field. When S is extremely large, the

random field can be represented by a single random variable. 

When S approaches zero, the random variables are 

uncorrelated. In each correlated random field, the dependent 

random variables are functions of independent random 

variables through a decomposition procedure. The blade 

responses being analyzed are the effective stresses on both 

pressure and suction sides and displacements in the X- and Zdirections. The mean, standard deviation, and coefficient of 

variation of effective stress on the pressure side are shown 

on figures 7(a) to (c). The largest mean effective stress is 

102 ksi at the lower corner of the leading edge. The coefficient 

4.69 U 

4.22 

3.74 

3.27 

2.80 

2.32 

1 85 

1.38 

0.90 

0.43 

090 

.08 

.07 

.06 

.05 

04 

03 

02

4
(a)

691 

62 

55 

47 - 

40 

4_ (b)

5.14 

4.62 

4.10 

3 58 

3 07 

2.55 

2.03 

151 

0.99 

0.47 

of variation associated with this critical stress is 3 percent. 

Based on this information, a blade risk analysis is performed. 

The failure mode is defined as the event where stress is greater 

than material strength. The probability of failure is calculated by 

Pj(i) 

= 

f5 (X) FR.(X) dX (2) 

where f, is the probability density function of effective stress 

at the ith node computed by NESSUS/FPI. FR, is the 

cumulative distribution function of strength at the ith node, 

which is normally distributed with mean equal to 120 ksi and

standard deviation equal to 12 ksi. The failure probability at 

each nodal point is obtained and shown in figure 7(d). As 

expected, the location of highest failure probability is the lower 

left corner (blade root, leading edge). A similar study was 

performed for the effective stress on the suction side as shown 

in figure 8. Since the stress is much lower on the suction 

surface than that on the pressure surface, the failure probability 

is not as critical. For structures like blades, the displacements 

must be tightly controlled. The maximum deflection cannot 

exceed its allowable magnitude for clearance considerations. 

The mean and standard deviation of blade displacement in X-

(c)

.281 

.26 

.23 

20 

17 

15 

12 

09 

06 

.04 (d) 

(a) Mean (pounds per square inch). (b) Standard deviation (pounds per square inch). 

(c) Coefficient of variation. (d) Probability of failure. 

Figure 8.—Effective stress on suction side.

6.3x10 

5.6 - 

4.9 

:: u 

2.8

5
and Z-directions are shown in figures 9 and 10. Thus, it can 

be decided whether the deflection control is acceptable or not. 

NESSUS/FPI also computes the sensitivity of the probability 

distribution for any response to the parameters of the independent random variables. This information, quantified by 

sensitivity factors, can be used to define the roles of the random 

variables in the reliability analysis. Sensitivity factors indicate 

which random variables are crucial and require more attention 

during the manufacturing process and in the environmental 

(loading, temperature) conditions. The sensitivity factors for 

the effective stress at the airfoil leading edge root are obtained. 

The five most influential random variables are listed in table II.

As expected, the temperature random field has insignificant 

effect on the stress. The Z-coordinate random field is a crucial 

one. The rotational speed which induces centrifugal forces is 

another important random variable. The uncertainty of the 

effective stress at root will be minimized significantly if the 

uncertainties of those five random variables are reduced. This 

information leads the designer to improve the design and 

reliability of the blade in a most efficient way. A similar 

analysis has been done for tip displacement in the X-direction. 

The five most influential random variables are shown in table 

III. In this case, the temperature has a dominant effect. The 

modulus should also be included in the reliability model. 

14 7x103 

ITO 1iII

98 

- 81

5.8x1O 

-

- 5.2 

4.5 

:: 

Ask - 

(b) 

(a) Mean. (b) Standard deviation.

Figure 9.—Displacement in X-direction (inches).
12x10 3 U 

- 

-28 60 I 

tTI1

-39 G7

-61 40 

L - :: 

(a) -13 (b) 0 j 

(a) Mean. (b) Standard deviation. 

Figure 10.—Displacement in Z-direction (inches). 

TABLE 11.—FIVE MOST INFLUENTIAL INDEPENDENT

RANDOM VARIABLES TO THE EFFECTIVE

STRESS AT ROOT 

Random fields nth Independent Sensitivity Ranking 

random variable factor 

in each field 

Z-coordinate I 0.204 4 

Z-coordinate 2 .610 I 

Z-coordinate 5 .273 3 

Modulus 12 .188 5 

Rotation speed 1 .326 2 

TABLE III.—FIVE MOST INFLUENTIAL INDEPENDENT

RANDOM VARIABLES TO THE TIP DISPLACEMENT

(X-COMPONENT)

Random fields nth Independent Sensitivity Ranking 

random variable factor 

in each field 

Z-coordinate 2 0.160 4 

Z-coordinate 2 .145 5 

Temperature I .855 1 

Temperature 3 .223 3 

Modulus 1 .345 2

Conclusion 

A space shuttle main engine (SSME) turbopump blade is 

analyzed probabilistically by using NESSUS (Numerical 

Evaluation of Stochastic Structures Under Stress) to account 

for the uncertainties in the blade geometry, material properties, 

temperature, and loading conditions. Probability distributions 

of structural responses such as displacements and stresses are 

obtained. Risk assessment is performed based on different 

failure criteria with known probability distributions of response 

variables. Sensitivity of any response to each random variable 

is also calculated. A redesign is necessary when the blade does 

not meet the safety requirements. Sensitivity factors provide 

a guide to modify the design in an efficient way for the 

reliability improvement. 

Lewis Research Center 

National Aeronautics and Space Administration 

Cleveland, Ohio, November 14, 1989

7 
References 

1. "Probabilistic Structural Analysis Methods for Select Propulsion 

System Structural Components (PSAM)." Vols. 1-3. NASA 

CR-185125-VOL-1,-2,-3, 1989. 

2. Chamis, CC.. "Probabilistic Structural Analysis Methods for Space 

Propulsion System Components," Probabilistic Engineering Mechanics, 

Vol. 2, June 1987. pp. 100-110 (also, NASA TM-88861). 

3. Lind, NC., Krenk, D., and Madsen. HO., Methods of Structural Safety. 

Prentice-Hall, New York, 1985.

4. Wu, Y.T., "Demonstration of a New, Fast Probability Integration Method 

for Reliability Analysis," Advances in Aerospace Structural Analysis, 

O.H. Burnside and C.H. Parr, eds., ASME, New York, 1985, 

pp. 63-73. 

5. Newell, iF., Kurth, RE., and Ho, H., "Composite Load Spectra for 

Select Space Propulsion Structural Components," R1/RD86- 123 AR-I, 

Rockwell International Corp., Canoga Park, CA, 1987, NASA 

CR- 179496.
NASA National Aeronautics aric Report Documentation Page 

Space Admnstraton 

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

NASA TM-102324 

4. Title and Subtitle 5. Report Date 

Probabilistic Structural Analysis of Aerospace Components Using NESSUS

6. Performing Organization Code 

7. Author(s) 8. Performing Organization Report No. 

Michael C. Shiao, Vinod K. Nagpal, and Christos C. Chamis E-5020 

10. Work Unit No. 

553-13-00 

9. Performing Organization Name and Address

11. Contract or Grant No. 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3191 13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address Technical Memorandum 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001

14. Sponsoring Agency Code 

15. Supplementary Notes 

Presented at the 29th Structures, Structural Dynamics and Materials Conference cosponsored by the AIAA, 

ASME, ASCE, AHS, and ACS, Williamsburg, Virginia, April 18-20, 1988. Michael C. Shiao and Vinod K. 

Nagpal, Sverdrup Technology, Inc., Lewis Research Center Group, Brook Park, Ohio 44142; Christos C. Chamis, 

NASA Lewis Research Center. 

16. Abstract 

Probabilistic structural analysis of a space shuttle main engine (SSME) turbopump blade is performed by using 

the computer code NESSUS (Numerical Evaluation of Stochastic Structures Under Stress). The goal of the analysis is 

to derive probabilistic characteristics of blade response for given probabilistic descriptions of uncertainties in 

blade geometry, material properties, temperature, and pressure distributions. Probability densities are calculated 

for critical blade responses. Risk assessment is performed assuming different failure models. 

17. Key Words (Suggested by Author(s)) 18. Distribution Statement 

Probabilistic; Risk; Failure; Reliability; Correlation; Unclassified - Unlimited 

Random; Sensitivity Subject Category 39 

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

Unclassified Unclassified II A03

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