COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Computational Design of Beam Sections under Impact Loading
S. J. Hou 1*, Q. Li 2, S. Y. Long 1, X. J. Yang 1
1
2
College of Mechanics and Aerospace, Hunan University, Changsha, Hunan, 410082 China
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006,
Australia
Email: [email protected], [email protected], [email protected]
Abstract: The paper sought for the optimal shapes of the partly tapered and totally tapered thin-walled cylindrical
beam cross-section under a crashworthiness criterion based on the explicit Finite Element (FE) simulation by using
response surface method (RSM) together with nonlinear constrained optimization programming of Matlab. The
effect of selection of quartic polynomial basis functions on the optimization results is investigated. In the
optimization procedure, the strain energy absorption per unit weight is defined as the objective function to reflect the
crashworthiness criterion. The optimal results of cross-sectional shapes of the two kinds of tapered beam were given
together with the traditional cylindrical beam, and the comparisons in terms of the crash energy absorption and weight
efficiency between the tapered cylindrical beam and the conditional cylindrical beam were also presented.
Key words: response surface method, optimization, crashworthiness, nonlinear finite element analysis, tapered
cylindrical
INTRODUCTION
The energy-absorbing behavior and weight efficiency of automotive front end structure are very important because they
are directly related to the passenger injury criteria [1]. Crashworthiness design is of special interest in the automotive
industry and in the transportation safety field to ensure the vehicle structural integrity and more importantly the occupant
safety in the even of the crash [2-4].
Several literatures can be found about the crashworthiness optimization of the thin-walled structures [5]. Conventional
cylindrical tube of uniform cross-section is optimization by using optimization methods, Finite Element (FE) simulations
and approximation methods [6, 7]. In the research, the emphasis is laid on the methodological discussion of the
crashworthiness design and not on the shape optimization of the beam cross-section. In 2002, Lee et al optimized only the
cylindrical tube of uniform cross-section by using response surface method based on stochastic process [8]. Meanwhile,
Chiandussi et al and Avalle et al optimized the tubular partly tapered thin-walled structure by the minimization of a load
uniformity parameter respectively in 2002 [9, 10]. Relative merits of square single-cell, double-cell, triple-cell and
foam-filled thin-walled structures in energy absorption were presented by W. Chen et al in 2001 [11]. In 2002, H. S. Kim
proposed and investigated a new type of multi-cell with four square elements at the corner [12]. In 2004, Lanzi et al
defined the total structural weight as the objective function to optimize the shape of conical absorbers with elliptical
cross-sections by using the response surfaces coupled with Genetic Algorithms to perform both constrained single- and
multi-objective optimizations [13]. However, partly and totally tapered cylindrical sections of thin-walled beam
structure have never been size-optimized and shape-optimized in the terms of crash energy absorption and weight
efficiency. Especially, the comparisons between the tapered and conventional cylindrical beam in the terms of crash
energy absorption and weight efficiency presented in this study have almost never been seen and studied in correlative
literatures.
In this study, optimal cross-sectional sizes of tapered and conventional cylindrical thin-walled beams were sought for the
front end part of higher crash energy absorption and weight efficiency of car body structure based on the explicit finite
element code. The Response Surface Method (RSM) is used to form the response surface of the Specific Energy
Absorption (SEA) vs. design variables. Compared with the conventional cylindrical beam also showed here, tapered
⎯ 476 ⎯
beams especially totally tapered cylindrical show dramatic improvements in energy-absorption characters.
Commercially available nonlinear dynamic explicit finite element code LS-DYNA3D was used throughout this study as
the analysis engine.
RESPONSE SURFACE METHOD [14]
Response surface method is considered appropriate in design optimization of complex mechanics problems like
contact-impact. In this approach, an approximate function of mechanical responses is assumed a priori in terms of basic
function as,
y% ( x ) = ∑ Nj=1 a j ϕ j ( x )
(1)
where N represents number of terms of basis functions ϕ j ( x ) .
In general, the selection of basic function should ensure accurate enough to achieve fast convergence [6]. This study
adopts a set of quartic polynomial functions as basis functions listed as
1, x1 , x2 , L , xn , x12 , x1 x2 , L , x1 xn , L , xn2 , x13 , x12 x2 , L , x12 xn , x1 x22 , L , x1 xn2 ,L , xn3 ,
x14 , x13 x2 , L , x13 xn , x12 x22 , L , x12 xn2 , L , x1 x23 , L , x1 xn3 ,L , xn4
(2)
During the analyses of FE results of this study, relative error is defined as
RE =
y% ( x ) − y ( x )
y( x)
(3)
where y% ( x ) represents approximation solution of RSM and y ( x ) represents the numerical solution by FE analysis.
DEFINITION OF CRASHWORTHINESS OPTIMIZATION PROBLEM
In finite element framework, the total strain energy can be computed as
E total = ∫ A(ε )dV
(4)
V
where A(ε ) represents the density of the structural total strain energy.
The strain energy absorbed per unit structural weight is defined as the Specific Energy Absorption (SEA: with the unit of
kJ/kg) shown in Eq. (5).
SEA =
Total Energy Absorbed Etotal
Total Structural Weight
(5)
Therefore, as a result, the optimization problem is mathematically formulated as
⎧Maximize: y = SEA( x )
⎨
x L ≤ x ≤ xU
⎩ s.t.
(6)
where x = ( x1 x2 L xk ) represents the vector of k design variables in the cross-sectional size, x L = ( x1L x2L L xkL )
represents the vector of lower limits of k design variables, and xU = ( x1U x2U L xkU ) represents the vector of upper
limits of k design variables.
FINITE ELEMENT MODELING
The configurations of the model are illustrated in Fig. 1. The structures considered in this study are the cylindrical
thin-walled beam with partly tapered (Fig. 1(b)), totally tapered (Fig. 1(c)) and traditional circular (Fig. 1(d))
cross-sections. The dimensions d1 and c of the tapered end are chosen as design variables. For the partly tapered
configuration, the variable interval of the two design parameters are 30 mm ≤ d1 ≤ 70mm and 50mm ≤ c ≤ 130 mm .
The length L of the thin-walled beam is a constant of 400 mm for all the three kinds of cross-sections. Nevertheless,
when c = L = 400 mm, the configuration of Fig. 1(c) is obtained, and when d1 = d 2 = 80 mm together with c = 0 , Fig.
⎯ 477 ⎯
1(d) is obtained. The loading condition for the three cases is that the thin-walled beam structures impact a rigid wall
with the 10 m/s of an initial velocity and 20 ms of a crash duration time. The weight of the lumped mass at the free end
of the beam is 500 kg.
Figure 1: The configuration of the computational model with loading condition
The thin-walled structures are made of the aluminum alloy AA6061-T4 with the material mechanical properties of density
ρ = 2.7 × 103 kg/m 3 , Young’s modulus E = 70.0 GPa , Poisson’s ratio μ = 0.28 , yield stress σ s = 110.3 MPa and
Tangent elastic modulus Et = 450 MPa which is used to be the strain hardening plastic modulus during the definition of
bilinear kinetic hardening material model for the thin shell element. As the aluminum is insensitive to the strain rate
effect, this effect is neglected in the finite element computation.
In establishment and analysis of the finite element model, commercial finite element software ANSYS is used to
establish the geometrical model and to complete the finite element model building. In this study, the 4-node shell
element with both bending and membrane capabilities introduced by Belytschko et al [15] is used to simulate the
thin-walled beam structures. Both in-plane and normal loads are permitted. This kind of element has 12 degrees of
freedom at each node and in this study it corresponds to an isotropic bilinear kinetic hardening material model. Actual
calculations and the definition of contact are performed on the dynamic non-linear explicit finite element code
LS-DYNA3D which is also used to be the post-processor for the visualization and data acquisition of total strain energy.
Response surface method is adopted to deal with the data acquired and then optimal design values are obtained by using
constrained nonlinear multivariable optimization function contained by the technical computing language MATLAB.
OPTIMIZATION DESIGN PROCESSES
For the maximum crash energy absorption and weight efficiency, a same optimization process is employed for all the
optimization examples. By using commercial finite element analysis software ANSYS and non-linear explicit finite
element code LS-DYNA3D, the specific energy absorptions of the thin-wall beam at the chosen design points were
obtained and used to establish the response surface or response curve of SEA vs. design variables. By using MATLAB
program codes and internal MATLAB function of non-linear constrained optimization, the figures of response surfaces
and the optimal design parameters during the design intervals are proposed. Relative fitting error of response surface is
also obtained in the plot software Origin. Meanwhile, the unitary and sectional deformations of the three cases are
respectively illustrated for the optimal design parameter of each case.
Case 1: Partly Tapered Cylindrical Cross-section The configuration of partly tapered cross-section is shown in Fig.
1(b). 40 design points were used to the establishment of response surface of SEA vs. d1 and c . From the SEA FE
results and by utilizing the formulae of response surface method, MATLAB code was assembled in order to obtained
the quartic surface fitting, and the Response Surface (RS) of SEA vs. d1 and c together with the relative error of RS
fitting at design points was illustrated in Fig. 2.
By using constrained nonlinear multivariable optimization function of MATLAB, the optimal SEA of RS shown in Fig.
2(a) is 13.7529kJ/kg at d1 = 60.76mm, c = 106.08, where the total weight of the thin-walled structure is 0.789 kg. In the
Fig. 2(a), the variation tendency of SEA vs. d1 and c is also illustrated for the cylindrical thin-walled beam with
tapered end. Fig. 3 shows example of the deformed mesh at the time of 20ms with the optimal design variables, from
which it can be seen that the structure developed stable and progressive folding deformation patterns.
⎯ 478 ⎯
Figure 2: The RS of SEA vs d1 and c together with the Relative Error of the RS fitting at design points
Figure 3: The unitary of partly tapered cross-sectional configuration
Case 2: Totally Tapered Cross-section The configuration of totally tapered cross-section is shown in Fig. 1(c). Here,
c = L = 400 mm, and then there is only one design d1 . 5 design points were chosen and quadratic basis function was
utilized. From the SEA FE results and by utilizing the formulae of response surface method, MATLAB code was
assembled in order to obtain the quadratic curve fitting, and the Response Curve (RC) of SEA vs. d1 together with the
fitting relative error at design points was shown in Fig. 4.
Figure 4: The RC of SEA vs d1 together with the Relative Error of the RC fitting at design points
⎯ 479 ⎯
By using constrained nonlinear multivariable optimization function of MATLAB, the optimal SEA of RC shown in Fig.
3(a) is 18.537kJ/kg at d1 = 30mm, where the total weight of the thin-walled structure is 0.561 kg. In the Fig. 4(a), the
variation tendency of SEA vs d1 is also illustrated for the cylindrical thin-walled beam with tapered end. Fig. 5 shows
example of the deformed mesh at the time of 20ms with the optimal design variables, from which it can be seen that the
structure developed stable and progressive folding deformation patterns.
Figure 5: The unitary deformation of the totally tapered cross-sectional configuration
Case 3: Traditional Circular Cross-section In order to comparison between the traditional cylindrical hollow beam
and the tapered cylindrical hollow beam in terms of crash energy-absorption and weight efficiency, the case of
conventional circular cross-section is listed in Fig. 1(d). Just let the design variables d1 = d 2 =80mm and c = 0 mm,
the regular thin-walled cylindrical beam can be obtained. Because the design variables here are constants, optimization
process isn’t needed. The SEA is 11.2104kJ/kg and the structural total weight is 0.816kg for this kind of configuration.
Fig. 6 shows example of the deformed mesh during the time of 20ms with the above given design variables, from
which it can be seen that the structure developed stable and progressive folding deformation patterns.
Figure 6: The unitary deformation of the traditional circular cross-sectional configuration
RESULTS COMPARISON BETWEEN REGULAR HEXAGON AND SQUARE CROSS-SECTION
Partly and totally tapered cylindrical thin-walled beam were optimized, and the conventional circular cross-section
hollow beam was also analyzed. In this section, the performances of tapered cross-sectional configurations are
compared with traditional cross-sectional configuration in terms of crash energy absorption and weight efficiency so
that optimal cross-section profile of the three cases can be chosen.
The optimal design variables and specific energy absorptions of approximation and finite element analyses are given in
Table 1 with accompanying relative error of fitting results. Meanwhile, the curves of SEA vs. time for the three cases at
optimal design variables at the time 20ms are shown together in Fig. 7.
⎯ 480 ⎯
It is clearly seen from the Table 1 and Fig. 7 that optimization results of the two tapered cases, especially the totally
tapered case, outperform the traditional circular case in the crash energy absorption and weight efficiency. For the
optimal design variables, the energy absorption per unit of the tapered cases is clearly more that that of the conventional
configuration. Of the three cases, the totally tapered cylindrical thin-walled beam is the optimal profile, the weight
efficiency of which obviously exceeds that of the other two cases, and the structural total weight of the totally tapered
case is less than the conventional case. The comparisons between the tapered cases and the conventional circular case
and between the partly and totally tapered cases in terms of increase rates of SEA and structural total weight of optimal
design variables are given in Table 2-4. From Table 2-4, it can be known that tapered configurations exceed
conventional configuration in the energy absorption per unit no matter and totally tapered configuration outperform the
other two cases. In addition, the total weight of optimal design profile of the totally tapered one is 31.25% lower than
the conventional one and 28.90% lower than the partly tapered one. From Table 4 and 5, it can be known that compared
with the other two beam configurations of optimal parameters, the totally tapered case predominates up to 68.82% and
40.21% in SEA respectively.
Table 1 Optimization results of the three cases
Optimal Design
Total
Variables
Weight
(mm)
(kg)
Partly Tapered
d1=60.76 ,
c=106.08
0.789
Totally Tapered
d1=30 , c=400
0.561
Traditional Circular
d1=80 , c=0
0.816
Type of
Cross-section
Approx. SEA
by RSM
(kJ/kg)
SEA by FE
Analysis
Relative
Error
(kJ/kg)
(%)
13.7529
13.4980
1.89
18.5376
18.9257
−2.05
11.2104
Figure 7: Curves of SEA vs time for the three cases with optimal design variables
Table 2 Increase rate of partly tapered than traditional circular case
SEA (kJ/kg)
Optimal Weight (kg)
Partly Tapered
13.4980
0.789
Traditional Circular
11.2104
0.816
Increase Rate
20.41%
-3.31%
⎯ 481 ⎯
Table 3 Increase rate of totally tapered than traditional circular case
Totally Tapered
Traditional Circular
Increase Rate
SEA (kJ/kg)
18.9257
11.2104
68.82%
Optimal Weight (kg)
0.561
0.816
−31.25%
Table 4 Increase rate of totally tapered than partly tapered case
Totally Tapered
Partly Tapered
Increase Rate
(a) Partly tapered
SEA (kJ/kg)
18.9257
13.4980
40.21%
(b) Totally tapered
Optimal Weight (kg)
0.561
0.789
−28.90%
(c) Traditional Circular
Figure 8: The Von Mises stress and stable progressive deformation of the thin-walled
hollow beam of the three cases of the optimal design variables
Fig. 8 shows the Von Mises Stresses of the three optimal configurations at the time of 20ms, which were obtained by
using finite element analysis results of LS-DYNA3D.
From Fig. 8, one can know that the stable progressive folding deformation and the uniform stress distribution of the
partly tapered case (shown in Fig. 8(a)) distinctly exceed those of the other cases (shown in Figs. 8(b) and 8(c)).
SUMMARY
Partly and totally tapered cylindrical cross-section sizes of thin-walled beam structures are optimized and investigated
with the objective function of specific energy absorption (SEA) in this study for the first time, together with the
comparisons between the conventional circular cross-sectional hollow beam and the tapered configurations in terms of
crash energy absorption and weight efficiency.
By using response surface method (RSM) and MATLAB codes, total strain energy datum obtained by finite element
analysis of LS-DYNA3D are dealt with so as to address the fitting surface or curve of SEA vs. design variables. The
expressions of SEA vs. design variables are obtained by compiling the MATLAB code and then optimal design
parameters are found for the design cases by using the nonlinear constrained optimization package of MATLAB.
Meanwhile, the unitary folding deformations are also proposed respectively for all the cases.
Through the comparison between the conventional case and the two tapered cases with the optimal design variables in
terms of crash energy absorption and weight efficiency, one can know that the tapered configurations, especially the
totally tapered cross-sectional thin-walled beam outperforms the conventional circular cross-sectional thin-walled beam
and that the totally tapered one distinctly exceed the partly tapered one. The Von Mises stress distributions together with
progressive folding deformations in the same figure are illustrated together for the three cases of the optimal design
variables, from which one can also draw the conclusions that in terms of stress distribution and folding deformation, the
optimal partly tapered case exceed the other two profiles of thin-walled beam structures.
⎯ 482 ⎯
Acknowledgement
The financial supports from National 973 Scientific & Technological Innovation Project (2004CB719402) and National
Natural Science Foundation of China (10372029) are gratefully acknowledged. Moreover, thanks a lot for Prof. Qing Li’s
instruction and help.
REFERENCES
1.
Mahmood H, Aouadi F. Characterization of frontal crash pulses. AMD-Vol.246/BED-Vol.49, Crashworthiness,
Occupant Protection and Biomechanics in Transportation Systems, ASME, 2000; 15-22.
2.
Belingardi G, Gugliotta A, Vadori R. Fragmentation of composite material plates submitted to impact loading:
comparison between numerical and experimental results. Key Engineering Materials, 1998; 144: 75-88.
3.
Ramakrishna S, Hamada H. Energy absorption characteristics of crash worthy structural composite materials.
Key Engineering Materials, 1998; 141-143: 585-620.
4.
Wang B, Zheng J. Dynamic Buckling of a column with alhesive bonds. Key Engineering Materials, 2000;
177-180: 703-708.
5.
Li Q, Steven GP, Querin OM. Optimization of thin shell structures subjected to thermal loading. Structural
Engineering and Mechanics, 1999; 7(4): 401-412.
6.
Kurtaran H, Eskandarian A, Marzougui D. Crashworthiness design optimization using successive response
surface approximations. Computational Mechanics, 2002; 29: 409-421.
7.
Li W, Li Q, Steven GP, Xie YM. An evolutionary shape optimization procedure for contact problems in
mechanical designs. Journal of Mechanical Engineering Science, 2003; 217: 435-446.
8.
Lee SH, Kim HY, Oh IS. Cylindrical tube optimization using response surface method based on stochastic
process. Journal of Materials Processing Technology, 2002; 130-131: 490-496.
9.
Avalle M, Chiandussi G, Belingardi G. Design optimization by response surface methodology: application to
crashworthiness design of vehicle structures. Structural and Multidisciplinary Optimization, 2002; 24: 325-332.
10. Chiandussi G, Avalle M. Maximisation of the crushing performance of a tubular device by shape optimization.
Computers & Structures, 2002; 80: 2425-2432.
11. Chen WG, Wierzbicki T. Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy
absorption. Thin-Walled Structures, 2001; 39: 287-306.
12. Kim HS. New extruded multi-cell aluminum profile for maximum crash energy absorption and weight efficiency.
Thin-Walled Structures, 2002; 40: 311-327.
13. Lanzi L, Castelletti LML, Anghileri M. Multi-objective optimization of composite absorber shape under
crashworthiness requirements. Composite Structure, 2004; 65: 433-441.
14. Myers RH, Montgomery DC. Response Surface Methodology, first edition. John Wiley & Sons Ltd., New York,
American, 1995.
15. Belytschko T, Lin JI, Tsay CC. Explicit algorithms for the nonlinear dynamics of shells. Computer Methods in
Applied Mechanics and Engineering, 1984; 42: 225-251.
⎯ 483 ⎯