The effects of actual variations, also called uncertainties, in geometry and material properties on the structural response of a space shuttle main engine turbopump blade are evaluated. A normal distribution was assumed to represent the uncertainties statistically. Uncertainties were assumed to be totally random, partially correlated, and fully correlated. The magnitude of these uncertainties were represented in terms of mean and variance. Blade responses, recorded in terms of displacements, natural frequencies, and maximum stress, was evaluated and plotted in the form of probabilistic distributions under combined uncertainties. These distributions provide an estimate of the range of magnitudes of the response and probability of occurrence of a given response. Most importantly, these distributions provide the information needed to estimate quantitatively the risk in a structural design

Nagpal, Vinod K.

English

NASA Technical Reports Server

N88-22401
ADVANCED PROBABILISTIC METHODS FOR QUANTIFYING THE EFFECTS OF
VARIOUS UNCERTAINTIES IN STRUCTURAL RESPONSE*
Vinod K. Nagpal
Sverdrup Technology, Inc.
(Lewis Research Center Group)
Lewis Research Center
A probabilistic structural analysis of a space shuttle main engine (SSME)
turbopump blade has been underway at the Lewis Research Center for the last
3 years. The immediate objectives of this study are to evaluate the effects
of actual variations, also called uncertainties, in geometry and material prop-
erties on the structural response of the turbopump blades. In this study a
normal distribution has been assumed to represent the uncertainties statisti-
cally. Uncertainties were assumed to be totally random, partially correlated,
and fully correlated. The magnitudes of these uncertainties have been repre-
sented in terms of mean and variance. These two quantities were selected such
that the absolute magnitudes of uncertainty at any point were either less than
or equal to 10 percent of their original values.
Many studies have demonstrated that probabilistic analysis methods for compo-
nents under random loading are more reliable than deterministic approaches.
Probabilistic methods have been predominantly used in fatigue, fracture mechan-
ics, and structural reliability analyses under ramdon vibrations (e.g., stud-
ies reported by Dover, 1979; Yang, 1981; Wirshing, 1981; Kawamoto, 1982; and
Huang and Nagpal, 1984). Particularly in fatigue, improvements in estimating
fatigue life ranged up to several hundred percent in comparison with a widely
used deterministic approach developed by Miner (1945). Because of the poten-
tial, the application of probabilistic methods has been accepted in the other
fields of engineering in recent years. These fields include input loading, fi-
nite elements, and metallurgy. A probabilistic approach to develop a compos-
ite loading for SSME components is underway at the Battelle Laboratories under
the technical guidance of R. Kurth (1985) and has been discussed by Shinozuka
and Lin (1981). Other studies using probabilistic finite-element methods have
been briefly reviewed by DasGupta (1986). A variational approach for develop-
ing the probabilistic finite elements is under development by Belytschko and
Liu (1985). The usefulness and importance of the probabilistic approach, espe-
cially for turbopump blades, has been summarized by Chamis (1986).
Blade response, recorded in terms of displacements, natural frequencies, and
maximum stress, was evaluated and plotted in the form of probabilistic distri-
butions under combined uncertainties. These distributions provide an estimate
of the range of magnitudes of the response and probability of occurrence of a
given response. Most importantly, these distributions provide the information
needed to estimate quantitatively the risk in a structural design.
*Work performed for the Structural Mechanics Branch under contract NAS3.
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A wide range of response, such as a maximum stress distribution, implies that
small natural uncertainties in the geometry and material properties can cause a
large variation in the response. Consequently, designs based on deterministic
methods will not be as safe as probability-based designs. Higher stresses,
which have low but certain probability of occurrence, are not included in the
deterministic designs. In most cases available experimental results are
insufficient to provide an entire range of stresses and their probabilities of
occurrence. Limited experimental results as well as deterministic methods pro-
vide estimates of the mean response but do not cover the scatter. Consequent-
ly, the risk in design even after using the factor of safety may or may not be
within acceptable limits.
The results of the study indicate that an uncertainty up to i0 percent in the
material properties had no significant effect on the response. However, uncer-
tainty up to i0 percent in geometry, only through the thickness, has signifi-
cant effect on the structural response. Based on the results, a number of
probabilistic models has also been developed using regression analysis. These
models predict the response for given uncertainties in the geometry and mate-
rial property parameters. The models for uncertainties in temperature com-
bined with those in geometry and material properties are under development.
2-220
PROBABILISTICSTRUCTURALNALYSISESTIMATESOFRESPONSE
The probabilistic approach provides the scatter in the response and also pro-
vides the probability of occurrence. This information helps assess the risk
quantitatively for all levels of response. Contrarily, the deterministic
response provides an estimate of meanvalue and does not cover the scatter.
Consequently, the risk even after using a safety factor mayor maynot be
within the acceptable range.
- 10% MEAN + 10%o_ 5600 6050 6550
GEOMETRY //_--r.-7-,_,-,....._._"__ FIRSTVIBRATION
f'_ FREQUENCY
RESPONSE.__,,,.4
/ _ _x'- _VARIATION -'
'- I L,
- 10°/o MEAN + 10% FINITE ELEMENT 70 105 140 ksi
MATRIAL PROPERTIES MODEL MAXIMUM STRESS
5050
FIRSTVIBRATION
FREQUENCY
105
MAXIMUM STRESS
STATISTICAL VARIATIONS
(UNCERTAINTIES)
PROBABILISTIC DETERMINISTIC RESPONSE
RESPONSE (MEAN RESPONSE)
CD-88-32631
2-221
OBJECTIVE
The analysis was performed on the airfoil of a typical SSMEturbine blade.
Nominal dimensions of the airfoil are i to i_ inches long, about I inch wide,
and a _ inch thick. Generally, these blades are madeof anisotropic materi-
als. The numberof blades per turbine stage is about 65. The other parts of
the blade are the platform, shank, and fir tree.
QUANTIFY UNCERTAINTIES
ASSOCIATED WITH
PROBABILISTIC RESPONSE
• GEOMETRY
UNCORRELATED
CORRELATED
• MATERIAL PROPERTIES
• LOADING
• TEMPERATURE
AIRFOIL
_- PLATFORM
/
A TYPICAL TURBINE BLADE
CD-88-32632
2-222
PROBABILISTIC STRUCTURAL RESPONSE
The probabilistic structural response of the blade was recorded in terms of
first three natural frequencies, root stress, which is also the maximum
stress, and blade tip displacements. The root stress is the stress at the
junction of the airfoil and the platform. Since the blade airfoil is consid-
ered fixed at this junction for this analysis, stress at that location becomes
the maximum. The tip displacements were estimated in three directions. Two
displacements were in the plane of the airfoil, and one was perpendicular to
the plane of the airfoil.
, NATURALFREQUENCIES
FIRST
SECOND
THIRD
. I_lf'_T I&IIAVIliJllU_ eTOi'_
II_Vl _IllrI_l Ill V Ill I V I I II_VV
• TiP DISPLACEMENTS
CD-88-32633
2-223
METHODOLOGY
The developed methodology is generic and can be used to analyze other
structural componentswhich may or maynot be related to this problem. This
chart describes the major steps used for performing this analysis. The first
step is the generation of randomnumbers. Correlated or uncorrelated random
numbersare used for perturbation of the geometry, spatial location, and/or ma-
terial properties. Uncorrelated numbers indicate no bearing of the perturba-
tion of one point on any other point on the airfoil.
The second step is modeling the blade with a finite-element model. Eighty tri-
angular shell elements with 55 nodes were used to model the airfoil.
The third step involves setting up of the design to run simulations. Full fac-
torial design was used because it is considered more economical than other
available approaches.
The fourth step involved analyzing the response and developing its probabilis-
tic distribution and models to predict the response for given uncertainties.
• SELECTED RANDOM DISTRIBUTION TO SIMULATE UNCERTAINTIES IN GEOMETRY,
SPATIAL LOCATION, AND MATERIAL PROPERTIES
NORMAL
• MODELED BLADE WITH FINITE-ELEMENT MODEL
80 ELEMENTS
55 NODES
• USED NUMERICAL EXPERIMENT DESIGN TO PERFORM SIMULATION
FULL FACTORIAL DESIGN
• ANALYZED RESPONSE
PROBABILISTIC MODELS
PROBABILISTIC DISTRIBUTIONS
STATISTICAL TESTS
CD-88-32634
2-224
SAMPLERANDOMNESSU EDIN PERTURBATIONS
Both uncorrelated and correlated randomnumbers are generated by the comput-
er. They can be generated using any distribution and with preselected statis-
tical properties such as mean, standard deviation, etc. In the present study
a normal distribution with selected meanand variance was used. The random
numbers have the property that they can be multiplied by any constant without
losing their randomness. A plot of randomnumbers is shown.
I0
-I0
-2°O
I I
100 200
SEOUENTIAL NUMBERS
I
300
CD-88-36235
'}_995
FRACTORIAL DESIGN FOR THREE VARIABLES
A factorial design is an economical method for performing real experiments or
computational simulations. Unlike a parametric approach, the influence of
individual variables and their interactions can be estimated discretely.
Another advantage of the factorial design is that the range of the variables
can be extended by amending the design scientifically.
There are two limitations with the factorial design: it can be used only for
the independent variables, and the results of the study are only applicable
within the range studied.
A representation of the factorial design for three variables is shown in the
attached figure. The solid circles at the corners or in the center of the box
represent the experiment or simulation points. A range of variables is defined
as the difference between its two extreme levels.
(LEVEL 1)1
(LEVEL !
VARIABLE 2
(LEVEL 1)2
(LEVEL 1)3 VARIABLE 1
VARIABLE 3
(LEVEL 2)3 2)1
CD-88-32636
2-226
RANDOM COORDINATE PERTURBATION OF SSME BLADE 
A geometric perturbation is done by perturbing the nodal coordinates in all 
three dimensions. The figure shows an exaggeration of perturbation of nodal 
coordinates in the x direction only. Perturbation in all three directions 
is difficult to show in graphical form. The geometric perturbation in this 
study is intended to simulate realistic geometric variation due to manufactur- 
ing tolerances and the errors in finite-element modeling. The variations in 
geometry at any point were limited to less than 10 percent. 
Perturbations were created using random numbers. Uncorrelated and partially 
correlated random numbers were used to simulate truly uncorrelated or patterned 
variations in the geometry. 
X 
Y J 
X-COMPONENT, 
in. 
.010 
.005 
0 
- .005 
- .010 
CD-00-32637 
RANDOM YOUNG'S MODULUS DISTRIBUTIONS IN SSME BLADE 
Material properties perturbation is done by perturbing the properties of each 
finite element. The attached figure is a demonstration of the variation of mo- 
dulus of elasticity for a blade that is made of an isotopic material. For a 
blade made of anisotropic material the entire material property matrix is per- 
turbed at the same time. Material properties, like geometry, were perturbed 
using uncorrelated and partially correlated random numbers. 
These perturbations are to represent realistic variation in the material prop- 
erties. The maximum variation at any location was limited to 10 percent. The 
"10 percent'' number was selected based on expert opinion. 
Y 
X 
J 
YOUNG'S 
MODULUS, 
psi 
21 
CD-88-32638 
PROBABILISTIC MODELS
Probabilistic models were developed using regression analysis. The general
form of the model is represented at the top of the following figure. The left
side of the model represents dependent variables, which can be any variable in
the response. The right side of the model consists of means and standard devi-
ations of uncertainties and their coefficients. Over 20 models were developed.
For demonstration purposes coefficients of only one of the models are plotted.
This model is for estimating the variation first natural frequency due to com-
bined effect of uncertainties in geometric and material properties.
MODEL
DEPENDENT
VARIABLE
A
Y = (CONST) + (COEF)lo 1 +... + (COEF)6oC33 +... + (COEF)21o'3oC33
WHERE
a i STANDARD DEVIATION OF PERTURBATION IN i DIRECTION
oCjj STANDARD DEVIATION OF PERTURBATIONS OF MATERIAL PROPERTY MATRIX
ELEMENT Cjj
MAGNITUDE
OF
COEFFICIENTS
6185
m
CONST
320
COEF1 V//,/J COEF3 COEF4 COEF5 COEF6
1600 CD-88-32639
2-229
INFLUENCE OF MATERIAL PERTURBATIONS ON NATURAL-FREQUENCY DISTRIBUTION
The response variables were evaluated for both uncorrelated and correlated ran-
dom variations in geometry and material properties. The variation in the natu-
ral frequencies for partially correlated variations in material properties was
slightly wider than for uncorrelated variations. However, the increment in
variations was not significant. The response for variation in geometry alone
and combined variation in geometry and material properties was studied.
Sample histograms of variations in the first three natural frequencies are
shown.
UNCORRELATED PARTIALLY CORRELATED
100 f_-]-_ 100
50 50
0 I 0
5.400 5.575 5.750 5.20 5.65 6.10
FIRST
100
0
7.50 7.95 8.40
100
0
7.2 7.9 8.6
SECOND
100
0
13 13.5 14
100
0
12.6 13.6 14.6
THIRD
NATURAL FREQUENCIES, kHz CD-88-32640
2-230
• METHODOLOGY TO QUANTIFY UNCERTAINTIES 1N THE DESCRIPTORS OF SSME BLADES
HAS BEEN DEVELOPED AND APPLIED.
• METHODOLOGY IS GENERIC AND EXTENDABLETO ANY STRUCTURAL COMPONENT AND
ALL ASPECTS OF STRUCTURAL ANALYSIS AND DESIGN.
• GEOMETRIC UNCERTAINTIES SHOWED SIGNIFICANT EFFECT ON COMPONENT
STRUCTURAL RESPONSE.
• MATERIAL PROPERTIES UNCERTAINTIES AND THEIR INTERACTIONS HAVE
INSIGNIFICANT EFFECTS ON COMPONENT STRUCTURAL RESPONSE.
• RANGE OF VARIATION IN STRUCTURAL RESPONSE QUANTIFIED AND EXPLICIT
PROBABILISTIC MODELS HAVE BEEN DEVELOPED.
• INFLUENCE OF UNCORRELATED AND PARTIALLY CORRELATED VARIATIONS IN
GEOMETRY AND MATERIAL PROPERTIES ON THE RESPONSE HAVE BEEN STUDIED.
CD-88-32641
2-231
REFERENCES
Belytschko, T., and Liu, W.K., 1985, "Probabilistic Finite Elements: Varia-
tional Theory," Structural Integrity and Durability of Reusable Space Pro-
pulsion Systems, NASA Report CP-2381.
Box, G.E.P., Hunter, W.G., and Hunter, J.S., 1982, An Introduction to Design_
Data Analysis_ and Model Building, John Wiley and Sons.
Chamis, Christos C., 1986, "Probabilistic Structural Analysis Methods for Space
Propulsion System Components," presented in 3rd Space System Technology
AIAA Conference, San Diego, California.
DasGupta, G., 1986, "Literature Review on Probabilistic Structural Analysis
and Stochastic Finite Element Methods," Probabilistic Structural Analysis
Methods for Select Space Propulsion System Components, Southwest Research
Institute, Annual Report.
Dover, W.D., 1979, "Fatigue Crack Growth in Tubular Welded Connections," Sec-
ond International Conference on Behavior of Offshore Structures, held at
Imperial College, London, England.
Huang, T.C., and Nagpal, Vinod K., 1984, "Probabilistic Factors, Experimental
Design and Statistical and Variance Analyses for Fatigue Under Random
Vibrations," edited by T.C. Huang and P.D. Spanos, New Orleans.
Kawamoto, James, et al., 1982, "An Assessment of Uncertainties in Fatigue
Analysis of Steel Jacket Offshore Platform," Applied Ocean Research,
Vol. 4, No. I.
Kurth, R., 1985; "Composite Load Spectra For Select Space Propulsion Structural
Components, Structural Integrity and Durability of Reusable Space Propul-
sion Systems," NASA Report CP-2381.
Lin, Y.K., 1981, "Random Vibrations of Civil Engineering Structures," Proceed-
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St. Louis, Missouri.
Miner, M.A., 1945, "Cumulative Damage in Fatigue," Journal of Applied Mechan-
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Shirozuka, M., and Tan, R., 1981, "Probabilistic Load Combinations and Crossing
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neering, ASCE, St. Louis, Missouri.
Wirshing, Paul H., 1981, "Fatigue Reliability Analysis in Offshore Structures,"
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Yang, J.N., 1981, "Reliability Analysis of Aircraft Structures," Proceedings
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St Louis, Missouri.
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Advanced probabilistic methods for quantifying the effects of various uncertainties in structural response

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