The document describes a new index called U** that can analyze load paths in vehicle body structures under distributed loading conditions, such as those that reproduce natural vibration modes. U** is introduced as it overcomes limitations of the existing U* index, which only applies to concentrated loads. The paper applies the U** analysis to a vehicle body model to identify load paths under the first vertical bending vibration mode. Key load paths are determined and interpreted based on the U** distribution. Shape optimization targets are also identified from consistency checks between U** and U* load path analyses under different loading scenarios.

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2009-01-0770
Load Path Analysis of Vehicle Body Structures under Eigenmode Deformation
of Bending Vibration
Yasuhisa Okano, Takuya Matsunaga and Shinichi Maruyama
Nissan Motor Co., Ltd.
Masashi Hanazato and Kunihiro Takahashi
Keio University
Copyright © 2009 SAE International
ABSTRACT index is introduced that is also applicable to distributed
loads through the use of complementary energy. This
The load path U* analysis is an effective tool for new index makes it possible to analyze load paths under
investigating the load paths in body structures. In the distributed load conditions that reproduce the natural
present study, a new index U** is introduced to modes of the vehicle body. The U** index was applied to
investigate structures under distributed loading. The new study measures for improving the low-order natural
parameter U** is a complementary concept of U*. frequencies of the body framework. Examples of
Although the conventional index U* cannot be applied to improvement measures devised for this purpose are
cases of distributed loading conditions, the new index presented along with an interpretation of the shape
U** can be applied to those cases. optimization results thus obtained in terms of the
concept of load paths.
This paper describes the application of a load path U**
analysis to improve efficiently the first eigenvalue of the THEORY OF CONVENTIONAL U* INDEX[1,2]
vertical bending mode in a vehicle body structure model.
It also explains how target parts for shape optimization U* INDEX FOR EXPRESSING LOAD TRANSFER
are interpreted on the basis of a load path U** analysis
when a load is applied to reproduce the first vertical A brief explanation is given here of the theory of the
bending mode. conventional U* index based on previous studies. An
elastic structure is modeled in terms of internal stiffness
INTRODUCTION in Fig. 1(a). Point A in the figure is the loading point,
point B is the support point and point C is any arbitrary
Any study of vehicle body structures today must point. The relationship between the load and
consider measures for achieving high stiffness, high displacement at the three points is given by Eq. (1),
strength and weight savings, among other attributes. A using internal stiffness expressed by subscripted K.
critical issue in satisfying these various performance Subscripted p is the load and d is a column vector
requirements simultaneously is to identify accurately the expressing displacement.
transfer of force through the body structures. As a new
method for accomplishing that, the U* index has been ­ pA ½ ª K AA K AB K AC º ­d A ½
° ° «K ° °
K BC » ®d B ¾
proposed for expressing the concept of load paths in
terms of internal stiffness[1,2]. However, the ® pB ¾ « BA
K BB
» (1)
conventional U* index has the drawback of being °p ° « K CA K CC » °d C °
applicable only to a concentrated load condition and not ¯ C¿ ¬ K CB ¼¯ ¿
to a distributed load condition. In this study, a new U**
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The energy U stored in the structure in Fig. 1(a) is given
by Eq. (2). A
U**
1 1
U pA ˜ d A K AAd A K AC d C

3. ˜ d A (2)
A
2 2 B
This expression takes into account that point B is fixed. 0 1 S2
s/ l
Tensor and vector notation are used for the product (a) Uniformity
notation. Next, the energy U' stored in the structure is
Curvature of U**
S1
found when point C is fixed and the same displacement 1
dA as in (a) is applied, as shown in Fig. 1(b). Using the
ratio between U and U' thus obtained, U* is defined as B B
shown in Eq. (3). A (c) Consistency of
load path
1 1 0 s/l 1
§U c · § 2U ·
U* { 1 ¨ ¸ ¨1
¨ (K d ) ˜ d ¸
¸
(b) Continuity
©U ¹ © AC C A ¹
(3)
Fig. 3 Three indexes of desirable structures
As is indicated in this expression, U* is a value that is
dependent on KAC, which expresses the strength of the The notation s is the distance along the path and l is the
coupling between the loading point and the arbitrary total length of the path. The index of consistency in Fig.
point. With the U* index that expresses the quantified 3(c) requires that the two load paths denoted as S1 and
strength of the coupling between these two points, it is S2 be consistent. The former path is for loading at point
possible to identify the condition of load transfer through A with point B fixed and the latter path is for loading at
the structure. point B with point A fixed. In other words, consistency
signifies that the places where U* is high from the
LOAD PATHS direction of the loading point A, i.e., the places stiffly
coupled with the loading point, have the possibility of
A load path U* distribution like that shown in Fig. 2 can improving stiffness if they are also simultaneously
be expressed by calculating U* successively at each coupled stiffly with the support point. The reader is
point in the structure. It is assumed that the applied load referred to reference [1] for a more detailed description
is transferred through the places where there is a strong of these three indexes.
coupling between the loading point and the support point.
Accordingly, the ridgeline that is formed when INTRODUCTION OF A NEW U** INDEX
generating contours can be defined as the load path S in
a U* distribution[1,2]. On the basis of U*, the transfer of DIFFERENCES WITH THE CONVENTIONAL U* INDEX
force in a structure can be identified globally and also
examined locally along the load path. The U* index is applicable in the case of a concentrated
load, but it cannot be used to treat a distributed load. In
THREE INDEXES OF STRUCTURAL DESIGN this study, the U** index described here was introduced
in order to treat cases involving a distributed load.
The three indexes of uniformity, continuity and Although the U** index has already been discussed
consistency, as proposed in reference [1], are applied as elsewhere[3], a detailed explanation of the index is given
the criteria for evaluating a structure in a load path below.
analysis based on U*. According to these indexes, a
structure is deemed desirable when the solid lines in Fig. CONCEPT OF THE U** INDEX[3]
3(a), (b) and (c) are in the direction indicated by the
arrows [1] . The U* index always applies the same displacement to
the loading point A, whereas the U** index always
applies an identical load to point A. In calculating the U*
pA , d A pA' , d A A index, a forced displacement is applied to the loading
A A point. This gives rise to a contradiction in that a point is
first fixed and then the point is displaced. Because the
S U** index applies a load to the loading point, it does not
C C cause a contradiction like what occurs in calculating the
U* index. For that reason, the U** index is also
B B B applicable to a distributed load condition.
0 U* 1
(a) (b)
In addition, U* is defined in terms of the work done by
Fig. 1 Internal stiffness Fig. 2 Load path the load because the displacement is the variable. In
contrast, U** is defined in terms of the complementary
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pA ,d A p A , d A' distribution was calculated using customized
A A MSC.Nastran by DMAP. As the first step, a natural
frequency analysis was conducted using a body-in-white
(BIW) model of a rear-wheel-drive passenger vehicle
C C with the four shock absorbers supported by a simple
structure as shown in Fig. 5. The modes obtained in this
B frequency analysis at all of the body nodes were then
B
applied to the body as forced displacements, and the
(a) (b) resultant reaction forces were found. The reaction forces
Fig. 4 Internal compliance thus obtained were applied as the inputs to reproduce
the first-order vertical bending mode.
work done by the load because the applied load is the
variable. Since a linearly elastic structure is being U** VALUE CALCULATION AND IDENTIFICATION OF
treated here, work and complementary work have the LOAD PATHS
same values. However, the respective equations differ
as noted below because different variables are adopted. Figure 6 shows the U** distribution of the force
inputs for reproducing the first-order vertical
Figure 4 shows the same elastic structure as that in Fig. bending mode. It is seen in the figure that the U**
1. What is different from Fig. 1 is that a forced load pA is value decreases toward the support point from the
applied instead of the forced displacement dA. point of its maximum value at the rear of the body.
The major load paths identified are shown in Fig. 7
The relationship between the displacement and load is and are explained below.
given by Eq. (4), using internal compliance expressed by
subscripted C. Load path (1): This path transfers the load from the
rear of the body through the rear side member and
the top of the rear wheel housing to the rear shock
­d A ½ ªC AA C AC º ­ p A ½ absorber.
® ¾ «C ® ¾
¯d C ¿ ¬ CA C CC » ¯ pC ¿
¼ (4)
Load path (2): This path transfers the load from the
Because the displacement at the support point B is zero, rear of the body through the rear side member and
the elements pertaining to point B are omitted here. The the bottom of the rear wheel housing to the rear
complementary energy Ucom stored in the structure in Fig. shock absorber.
4(a) is given by Eq. (5).
1 1
U com d A ˜ pA C AA pA

5. ˜ pA
2 2 (5)
Next, the complimentary energy U'com stored in the
structure is found when the same load pA is applied with
point C fixed, as shown in Fig. 4(b). Using the ratio
between Ucom and U'com thus obtained, U** is defined as Fig. 5 First-order vertical bending mode
expressed in Eq. (6).
1
U' d C ˜ C CC d C
U * * { 1 com
U com 2U (6)
LOAD PATHS IN U** INDEX
Under a distributed load condition, the ridgeline
from the point of maximum U** to the support point max value
is defined as the load path.
APPLICATION OF U** ANALYSIS TO VEHICLE
BODY STRUCTURES
ANALYSIS CONDITIONS min U** max
In order to identify the load paths of the first-order
vertical bending mode, a U** analysis was conducted
Fig. 6 U** distribution
under input conditions reproducing this mode. U**
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Load path (3): This path transfers the load from the
rear of the body through the rear side member and
the rear spring attachment point to the shock
absorber.
In the graph in Fig. 8, the distance s from the point
of the maximum U** value has been
nondimensionalized by the total load path length l
and is shown along the horizontal axis of the graph,
with the value of U** shown along the vertical axis.
The results indicate that all three load paths are
desirable in terms of uniformity and continuity.
EXAMINATION OF LOAD PATH CONSISTENCY
The consistency of the load paths was then
min U* max
examined. Figure 9 shows the U* distribution when
forced displacements were applied to the four shock
absorbers, with the rear body supported where the Fig. 9 U* distribution
maximum U** value was distributed in relation to the
forces applied to reproduce the first-order vertical
bending mode. The sign of the forced
displacements applied to the four shock absorbers
was reversed from positive to negative with respect
to the displacement at the rear body location of the Path
maximum U** distribution in the first-order vertical
bending mode. The major load paths identified are Path
shown in Fig. 10 and are explained below.
Path
Path
Fig. 10 Major load paths
Path
Load path (1): This path transfers the load from the
rear shock absorber through the rear wheel housing
and the rear portion of the rear side member to the
rear of the body.
Path
Load path (2): This path transfers the load from the
rear shock absorber through the rear wheel housing
and the front part of the rear side member to the
rear seat cross member.
Fig. 7 Major load paths
Load path (3): This path transfers the load from the
1.0 rear shock absorber through the rear wheel housing,
Path the front part of the rear side member and the side
0.8 Path sill to the rear seat cross member.
0.6
Path
In the graph in Fig. 11, the distance s from the loading
U**
point has been nondimensionalized by the total load
0.4 path length l and is shown along the horizontal axis, with
the value of U* shown along the vertical axis. The results
0.2
suggest that load paths (2) and (3) are more desirable
0.0
than load path (1) with respect to uniformity and
continuity. Load path (1) in Fig. 10 and load path (2) in
0 0.2 0.4 0.6 0.8 1
Fig. 7 are more desirable in terms of consistency.
s/l However, it is also seen that there are no load paths in
Fig. 8 Uniformity and continuity of load paths Fig. 7 corresponding to load paths (2) and (3) in Fig. 10.
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1.0 COMPARISON WITH ANALYSIS RESULTS FOR
IMPROVEMENT OF NATURAL VIBRATION
0.8
FREQUENCY BY SHAPE OPTIMIZATION
0.6
EXPLANATION OF MODEL
U*
0.4 Path
Path The vehicle body structure was investigated using
0.2
Path the traction method in the OPTISHAPE-TS[5]
program for shape optimization, with the aim of
0.0 improving the natural frequencies of the body
0 0.2 0.4 0.6 0.8 1 framework efficiently. The initial model used in the
s/l shape optimization exercise is shown in Fig. 13.
Fig. 11 Uniformity and continuity of load paths This model was created by applying solid elements
30 mm in thickness to the rear floor panel and the
INVESTIGATION OF MEASURES FOR wheel housing of the BIW model of a rear-wheel-
IMPROVING NATURAL FREQUENCY OF FIRST- drive passenger vehicle mentioned earlier. The solid
ORDER VERTICAL BENDING MODE elements were coupled to these body parts at
shared nodes.
An investigation was made of the three indexes that a
desirable structure should satisfy, as described in ANALYSIS CONDITIONS
section before. Because the results in Fig. 8 indicated
that the load paths were desirable with respect to In the shape optimization exercise, maximization of
uniformity and continuity, only consistency was the natural frequency of the first-order vertical
examined here. From the results in Figs. 6 and 9, it was bending mode was set as the objective functional
assumed that the consistency of the load paths would be under the constraint that the volume of the solid
improved by providing reinforcements connecting the elements, which were the target of the optimal
places having large U** and U* values, respectively. For design, should be reduced from the present level.
example, providing stiffness elements connecting the
rear shock absorbers and the front and rear ends of the SHAPE OPTIMIZATION RESULTS
rear side members would presumably be effective for
that purpose. These reinforcements are respectively The shape optimization results are shown in Fig. 14.
denoted as and in Fig. 12. The resultant Reinforcements and for connecting the rear
improvement in the natural frequency of the first-order shock absorbers and the front and rear ends of the
vertical bending mode is shown in Table 1. rear side members are newly represented in the
results. With regard to the solid elements of the
spare tire pan and the rear floor panel, the thickness
of these parts was reduced from the original
specification. However, it is clear that applying
improvement measures to these parts has little
effect on improving the natural frequency of the first-
order vertical bending mode.
Fig. 12 Reinforced model
Table 1 Effects of reinforcement
Specification of Change in natural frequency of
model first-order vertical bending mode
Original model 0 Hz
Reinforced model +4.2 Hz
Fig. 13 Initial model
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(3) A comparison of the shape optimization results
and the results of the structural changes based
on U** confirmed that both sets of results
displayed the same tendencies. In addition, it
was found that the shape optimization results
were explainable in terms of the improved
consistency of the load paths.
REFERENCES
[1] Takahashi, K., Conditions for desirable
Fig. 14 Calculated optimal model structures based on a concept of load transfer
courses, Proceedings of the First International
Table 2 Results of optimization Structural Engineering and Construction
Conference, 2001, pp. 699-702.
Specification of Change in natural frequency of
model first-order vertical bending mode [2] Sakurai, T., et al., Load Path Optimization and
Original model 0 Hz U* Structural Analysis for Passenger Car
Compartments under Frontal Collision,
Calculated optimal +3.5 Hz Preprints of International Body Engineering
model Conference, 2003, pp. 181-186.
[3] Wang, Enyang., Zhang, Xiaoguang., Takahashi,
Table 2 shows the change in the natural frequency of K., Load Path U* Analysis of Square Pipe
the first-order vertical bending mode between the initial under Collision, Society of Automotive
model before shape optimization and the shape Engineers of Japan Spring Conference, 2007,
optimized model. Compared with the initial model, the No. 17-07.
natural frequency of the first-order vertical bending mode
of the shape optimized model increased by 3.5 Hz. [4] Sakurai, T., Takahashi, K., Kawakami, H., Abe,
M., Reduction of Calculation Time for Load
INTERPRETATION OF SHAPE OPTIMIZATION Path U* Analysis of Structures, Journal of Solid
RESULTS Mechanics and Materials Engineering, 2007,
Vol. 1, No. 11, pp. 1322-1330.
Good agreement was seen between the reinforcements
provided to improve the consistency of the load paths [5] http://www.quint.co.jp/eng/pro/ots/index.htm
(Fig. 12) and the shape optimization results presented in
Fig. 14. This agreement can be interpreted as indicating
that the shape optimization results in Fig. 14 show the CONTACT
improved consistency of the load paths. It is clear from
the results in Fig. 7 that the spare tire pan and rear floor Yasuhisa Okano
panel did not serve as load paths. That can be E-mail: yasu-okano@mail.nissan.co.jp
understood as the reason why the solid elements Phone:+81-50-2029-1327
provided at these locations became thinner in the shape Address:1-1, Morinosatoaoyama, Atsugi-shi, Kanagawa,
optimization results in Fig. 14. 243-0123, Japan
CONCLUSIONS
In this study, a body-in-white (BIW) model was
subjected to a load path analysis and shape
optimization, with the aim of improving the low-order
natural frequencies of the body framework. The
following results were obtained.
(1) A new U** index was introduced to treat the low-
frequency dynamic problems of the BIW.
(2) The structural changes made on the basis of the
results calculated for U** concerning the vertical
bending mode of the BIW showed that dynamic
stiffness can be improved efficiently.
Author:Gilligan-SID:13330-GUID:23758887-59.66.116.210