1. SAE TECHNICAL
2001-01-3044PAPER SERIES
Body-in-White Weight Reduction via Probabilistic
Modeling of Manufacturing Variations
Andreas Vlahinos
Principal, Advanced Engineering Solutions, LLC
Jason Weber
Staff Technical Specialist, Ford Motor Company
Reprinted From: Proceedings of the 2001 SAE International Body
Engineering Conference on CD-ROM (IBEC2001CD)
International Body Engineering
Conference and Exhibition
Detroit, Michigan
October 16-18, 2001
400 Commonwealth Drive, Warrendale, PA 15096-0001 U.S.A. Tel: (724) 776-4841 Fax: (724) 776-5760

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3. 2001-01-3044
Body-in-White Weight Reduction via Probabilistic
Modeling of Manufacturing Variations
Copyright © 2001 Society of Automotive Engineers, Inc.
ABSTRACT
A design is robust when it is not sensitive to variations in
noise parameters such as manufacturing tolerances,
material properties, environmental temperature,
humidity, etc. In recent years several robust design
concepts have been introduced in an effort to obtain
optimum designs and minimize the variation in the
product characteristics [1,2]. In this study, a probabilistic
design analysis was performed in order to develop a
robust design with the mean value of the resulting stress
at target, and minimum standard deviation. The
methodology for implementing robust design used in this
research effort is summarized in a reusable workflow
diagram.
Andreas Vlahinos
Principal, Advanced Engineering Solutions, LLC
Jason Weber
Staff Technical Specialist, Ford Motor Company
terms of probability distribution functions. Monte Carlo
and response surface sampling techniques are
implemented in determining the response distribution.
Six sigma design criteria are established to size the
component and compare this design to the one
developed using the traditional nominal value criteria.
The Parametric Deterministic FEA Model
For this study, the battery tray (FE model shown in figure
1) was selected. The tray is made from a composite
material SMC and is supporting the battery (FE model
shown in figure 2).
INTRODUCTION
Currently, to account for manufacturing variations, auto
body designs are based on the nominal or worst case
scenario values, which leads to over-designed
components. If the scatter in material properties,
thickness and dimensions is accounted for in the finite
element analysis stress prediction, it is expected that
lighter designs will be produced. This type of stochastic
approach can be used to investigate the robustness and
sensitivity of a proposed solution and to minimize the
risk of failing corporate and consumer tests. In addition,
it can potentially reduce the cost by allowing more
variation in components that are not critical to
performance requirements.
In this research effort, probabilistic modeling of
manufacturing variations for a structural auto body
component (battery tray of an SUV) is performed to
determine the sensitivity and the response distribution
(stress, stiffness, fatigue life) due to the scatter of the
random variables. The scatter of the modulus of
elasticity and the thickness and loading are defined in
Figure 1. FE Model Of The Battery Tray
The battery is modeled with an elastic isotropic material
of uniform density appropriately adjusted to produce the
battery weight. The battery is supported by the tray with
a set of springs at appropriate locations. The parametric
model contains 2105 shell, 324 solid and 48 spring
elements.

4. variable. It was also assumed that the modulus of
elasticity exhibits a Gaussian distribution with a mean
value of 5723 N/mm2 and a standard deviation 570
N/mm2.
Figure 2. FE Model of the Battery and Tray Assembly
Propability D ensity of Thickness
0 .7
0 .6
0 .5
Density
0 .4
Probability
0 .3
0 .2
0 .1
0
3 42
Thickness (mm)
Propability of Th ickness (Input V ariable t)
1
0.9
0.8
0.7
0.6
Probability
0.5
0.4
0.3
0.2
0.1
0
2 3 4 51
Thickne ss ( mm)
It is assumed that the tray is rigidly fixed at the support
locations. The input parameters of the FEA model can
Figure 3. Probability Distribution of Thickness t (input
variable)
be any dimension, material property and loading. For
this study three parameters were considered: the wall
thickness (t), the modulus of elasticity (E) and the
vertical loading (q). We call these model parameters
and for any set of values of the three model parameters
a solution of the FEA model can produce two model
output variables: the maximum Von Mises stress (max e)
and the maximum equivalent strain (max e).
The Probabilistic FEA Model
Uncertainty in the input parameters of the FEA model
can be introduced by assuming certain randomness in
the input parameters. In this study, it was assumed that
the thickness (t), and the modulus of elasticity (E) are
-4
P ropability Density of Modulus of Elasticity x 10
3 .5
3
2 .5
Density
2
Probability
1 .5
1
0 .5
0
4000 5000 6000 7000 8 0003 000
2
Modulus of Elasticity (N/mm )
P ropability of Inpu t Va riable E
1
0.9
0.8
0.7
0.6
Probability
0.5
0.4
0.3
0.2
0.1
0
4000 6000 8000 100002000
2
Modulus of Elastic ity ( N/mm )
characterized by a Gaussian distribution and that the
vertical loading (q) is characterized by a lognormal
distribution. These assumptions are based on historical
Figure 4. Probability Distribution of Modulus of
Elasticity E (input variable)
data. The distribution parameters (mean values and
standard deviations) can be specified to define a set of
random values for the model parameters. The mean
value of the thickness was considered as a controllable
parameter and it was declared as an optimization
design variable. The rest of the distribution parameters
(mean values of E & q) and the standard deviation of t, E
and q were considered uncontrollable or noise
parameters. Figures 3, 4 and 5 show the probability
distributions and the probability of the input variables,
namely, thickness, modulus of elasticity and vertical
loading. The ANSYS3 probabilistic Design System was
used to generate the values from the distribution
parameters. A set of 100 points from these distributions
was used to perform FEA analysis on the tray. It was
-5
Propability Density of Vertical Lo adin g x 10
3 .5
3
2 .5
Density
2
Probability
1 .5
1
0 .5
0
2 4 6 8 100
Vertical Loading (mm/sec
2
) x 104
P ropability of Inpu t Va riable V
1
0.9
0.8
0.7
0.6
Probability
0.5
0.4
0.3
0.2
0.1
0
4000 6000 8000 100002000
V ertical Loading ( mm/sec
2
)
assumed that the thickness exhibits a Gaussian
distribution with a mean value of 3.0 mm and a standard
deviation of 0.3 mm. In the optimization model, the mean
value of the thickness was an unknown design
Figure 5. Probability Distribution of Vertical
Loading q (input variable)

5. The lognormal distribution with a mean of 58842 m/sec2
and standard deviation of 12000 m/sec2 was considered
for the vertical loading.
The following four sampling techniques were used to
combine the input variables and produce a set of output
variables max e and max e:
1. Direct Monte Carlo Sampling
2. Latin Hypercube Sampling
3. Central Composite Design with response
surface
4. Box-Behnken Matrix Design with response
surface
Using one of the four sampling techniques, the
probabilistic model determines a set of designs (unique
values of the model input parameters), uses the
parametric FEA model and computes a set of output
variables. By post-processing the output variables, the
probabilistic model computes the distribution parameters
of the output variables.
Direct Monte Carlo Sampling
Figure 6 shows a scatter plot of the input variables t and
E using the direct Monte Carlo sampling technique.
Figure 7 and 8 show the effect (and scatter) of the
thickness variation on the max equivalent strain and
stress respectively. Figures 9 and 10 show the
histogram of the max equivalent strain and stress
respectively. The mean (average) value of the max Von
Mises stress is 40.80 N/mm2 and the standard deviation
is 19.57 N/mm2. The mean (average) value of the max
strain is 1.0545846e-002 and the standard deviation is
6.1042144e-003.
Scatter plot of Input Variables t and E with Direct Monte Carlo Sampling
9000
8000
(N/mm)
7000
2ModulusofElasticity
5000
6000
4000
3000
2 2.5 3 3.5 4 4.51.5
Thickness (mm)
Scatter plot of Thickness and max Strain with Direct Monte Carlo Sampling
0.04
0.035
0.03
0.025
Strain
0.02
max
0.015
0.01
0.005
0
2 2.5 3 3.5 4 4.51.5
Thickness (mm)
Figure 7. Scatter Plot of the Input Variables t and
max strain using Direct Monte Carlo Sampling
Scatter plot of Thickness and max Stress with Direct Monte Carlo Sampling
140
120
(N/mm)
100
2MisesStress
80
maxVon
60
40
20
0
2 2.5 3 3.5 4 4.51.5
Thickness (mm)
Figure 8. Scatter Plot of the Input Variables t and max
stress using Direct Monte Carlo Sampling
Histogram of max Strain with Direct Monte Carlo Sampling
20
18
16
14
Frequency
12
10
Relative
8
6
4
2
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
max Strain
Figure 9. Histogram of max strain using Direct
Figure 6. Scatter Plot of the Input Variables t and E Monte Carlo Sampling
using Direct Monte Carlo Sampling

6. Histogram of max Stress with Direct Monte Carlo Sampling
18
16
14
12
Frequency
10
Relative
8
6
4
2
0
20 40 60 80 100 120 1400
max Von Mises Stress (N/mm
2
)
Figure 10. Histogram of max stress using Direct
Monte Carlo Sampling
Latin Hypercube Sampling
Figure 11 shows a scatter plot of the input variables t
and E using the Latin Hyper Cube Sampling technique.
Figures 12 and 13 show the effect (and scatter) of the
thickness variation on the max equivalent strain and
stress respectively. Figures 14 and 15 show the
histogram of the max equivalent strain and stress
respectively. The mean (average) value of the max Von
Mises stress is 40.18 N/mm2 and the standard deviation
is 19.40 N/mm2. The mean (average) value of the max
strain is 1.0358670e-002 and the standard deviation is
5.9027582e-003.
Scatter plot of Input Variables t and E with Latin Hypercube Sampling
9000
8000
(N/mm)
7000
2ofElasticity
6000
5000
Modulus
4000
3000
2000
1 1.5 2 2.5 3 3.5 4 4.5 5
Thickness (mm)
Figure 11. Scatter Plot of the Input Variables t and E
using Latin Hypercube Sampling
Scatter plot of Thickness and max Strain with Latin Hypercube Sampling
0.04
0.035
0.03
0.025
Strain
0.02
max
0.015
0.01
0.005
0
1 1.5 2 2.5 3 3.5 4 4.5 5
Thickness (mm)
Figure 12. Scatter Plot of the Input Variables t and
max strain using Latin Hypercube Sampling
Scatter plot of Thickness and max Stress with Latin Hypercube Sampling
160
140
Stress(N/mm)
120
2
100
VonMises
80
60
max
40
20
0
1 1.5 2 2.5 3 3.5 4 4.5 5
Thickness (mm)
Figure 13. Scatter Plot of the Input Variables t and max
stress using Latin Hypercube Sampling
Histogram of max Strain with Latin Hypercube Sampling
25
20
RelativeFrequency
15
10
5
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040
max Strain
Figure 14. Histogram of max strain using
Latin Hypercube Sampling

7. Histogram of max Stress with Latin Hypercube Sampling
25
20
RelativeFrequency
15
10
5
0
20 40 60 80 100 120 140 1600
max Von Mises Stress (N/mm
2
)
Figure 15. Histogram of max stress using
Latin Hyper Cube Sampling
Central Composite Design
Figures 16 and 17 show the histogram of the max
equivalent strain and stress respectively. The mean
(average) value of the max Von Mises stress is 39.98
N/mm2 and the standard deviation is 22.08 N/mm2. The
mean (average) value of the max strain is 1.3305926e-
002 and the standard deviation is 4.8547397e-003.
Histogram of max Strain with Central Composite Design
0.035
0.03
0.025
Frequency
0.02
Relative
0.015
0.01
0.005
0
-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03-0.01
max Strain
Figure 16. Histogram of max strain using the
Central Composite Design
Box-Behnken Matrix Design
Figures 18 and 19 show the histograms of the max
equivalent strain and stress respectively. The mean
(average) value of the max Von Mises stress is 40.95
N/mm2 and the standard deviation is 24.60 N/mm2. The
mean (average) value of the max strain is 1.6345224e-
002 and the standard deviation is 5.8322964e-003.
Histogram of max Stress with Central Composite Design
0.07
0.06
0.05
Frequency
0.04
Relative
0.03
0.02
0.01
0
50 100 150 200 2500
max Von Mises Stress (N/mm
2
)
Figure 17. Histogram of max stress using the
Central Composite Design
Histogram of max Strain with Box-Behnken Matrix Design
0.035
0.03
0.025
Frequency
0.02
Relative
0.015
0.01
0.005
0
-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-0.01
max Strain
Figure 18. Histogram of max strain using the
Box-Behneken Matrix Design.
Designing for Six Sigma quality with
Probabilistic Design and Optimization
Factors like geometric dimensions (mean value of
thickness) of a part can be controlled by designers in a
typical automotive design. Uncontrollable or noise
factors such as manufacturing imperfections (standard
deviation of the thickness), environmental variables
(loading), product deterioration (material properties) are
sources of variations whose effects cannot be
eliminated. The goal of a robust design is to reduce a
product's variation by reducing the sensitivity of the
product to the sources of variation rather than by
controlling these sources. An effort was made to reduce
response variation by selecting appropriate settings for
controllable parameters to dampen the effects of hard-
to-control noise variables. The methodology for
implementing robust design used in this research effort
is summarized in a workflow diagram shown in Figure
20.

8. Histogram of max Stress with Box-Behnken Matrix Design
0.07
0.06
0.05
Frequency
0.04
Relative
0.03
0.02
0.01
0
50 100 150 200 2500
max Von Mises Stress (N/mm
2
)
Figure 19. Histogram of max stress using the
Box-Behneken Matrix Design
The objective is to select automatically the mean value
of the geometric design variables that minimize variation
and produce a design that meets a target value. To
automate this, we can set up an optimization loop that
uses as design variables the mean values of thickness.
For a given set of mean values (using the probabilistic
model) this approach can produce the mean and
standard deviation of the response. In this case the
mean max Von Mises stress and its standard deviation.
The next step is to compute the value of the
con = mean (max e) + 3 * std dev of (max e),
compare it to a target value and select the mean values
of the design variables that minimize the standard
deviation subject to con < Target
The problem can be expressed in mathematical terms
as:
Select the mean values of the model design variables
within a given range:
t min < mean ti < t max
that minimize the standard deviation of the response
minimize [std. dev of (max e)]
subject to the constraint:
[mean (max e) + 3 * std dev of (max e)] < Target
The ANSYS script files to implement this process are
available upon request.
CONCLUSIONS
The example presented demonstrates the
advantage of using an automated probabilistic
design process.

The sampling technique has a negligible effect
on the mean values of the response and a small
effect on the standard deviation of the response.

For this component and number of design
variables, the Box-Behneken Matrix design
technique is recommended since it produces the
same quality of results as the other techniques,
but with the minimum number of FEA runs.

The above conclusion may not be valid if a large
number of design variables forms a highly non-
linear problem. In that case, the Latin Hypercube
and Box-Behneken Matrix should be used to
validate the "goodness" of the Box-Behneken
Matrix technique.
With the probabilistic design and optimization
approach, engineers are enabled to identify
better designs that meet the performance
objectives and are less sensitive to
manufacturing variations.

FEA software tools have incorporated
probabilistic design and allow distributed
computing that enables the implementation of
this technology.

By incorporating the physical scatter into the
model, the risk of failing legal or consumer tests
can be minimized
ACKNOWLEDGMENTS
This research effort was partially funded by the
Department of Energy. The authors would like to
express their appreciation to Bob Kost of DOE, Terry
Penney and Keith Wipke of NREL and Eleni Beyko,
N. Purushotthaman and Qutub Khaja of Ford Motor
Company.

9. Figure 20 Reusable workflow template that determines the mean values of design variables that produce robust design

10. REFERENCES
1. S. Schmidt, R. Launsby: Understanding Industrial
Designed Experiments, 4th edition, Air Academy
Press,2000.
2. M. S. Phadke: Quality Engineering Using Robust
Design, Prentice Hall, 1989.
3. ANSYS Inc: Probabilistic Design Techniques.
4. S. Kelkar, S. Patil, J. Hua Zhou, U. Naik: Systematic
Design of Automotive Body Structures Joints Using
CAE and DOE, International Body Engineering
Conference, Detroit, September 1993.
5. Stefan Reh, Personal communications ANSYS Inc.
CONTACT
Andreas Vlahinos, Ph.D. email: andreas@aes.nu