Geometrical Optimization of TopHat Structure Subject to Axial Low Velocity Impact Load Using Numerical Simulation
Hung Anh Ly^{*}, Hiep Hung Nguyen, Thinh ThaiQuang
Department of Aerospace Engineering, Faculty of Transportation Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam
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To cite this article:
Hung Anh Ly, Hiep Hung Nguyen, Thinh ThaiQuang. Geometrical Optimization of TopHat Structure Subject to Axial Low Velocity Impact Load Using Numerical Simulation. International Journal of Mechanical Engineering and Applications. Special Issue: Transportation Engineering Technology — part Ⅱ. Vol. 3, No. 31, 2015, pp. 4048. doi: 10.11648/j.ijmea.s.2015030301.17
Abstract: Crashworthiness is one of the most important criteria in vehicle design. A crashworthy design will reduce the injury risk to the occupants and ensure their safety. In structure design, the energy absorption and dispersion capacity are typical characteristics of crashworthy structure. This research continues the previous studies, focuses on analyzing the behavior of tophat and doublehat thinwalled sections subjected to axial load. Due to limitations on the experimental conditions, this paper focuses on analyzing the behaviors of tophat and doublehat thinwalled sections by theoretical analysis and finite element method. Two main objectives are setting up finite element models to simulate tophat and doublehat thinwalled structures in order that the results are consistent with the theoretical predict; and using the results of these models to optimize a tophat column subject to mean crushing force and sectional bending stiffness constraints by the "Twostep RSMEnumeration" algorithm. An approximate theoretical solution for a tophat column with different in thickness of hatsection and closing back plate is also developed and applied to the optimization problem.
Keywords: Crashworthiness, Impact, Optimization, Tophat
1. Introduction
Crashworthiness is the ability to protect the passengers in case of collision. For example, a helicopter with crashworthy design can guarantee the life of pilots even in cases where the aircraft crashed; or a car impacts at high speed, but the driver’s safety is assured thanks to the airbag, seatbelts, and the structures which absorb and disperse impact energy. In structure design, crashworthiness is related to the energy absorption capacity of the structure. Normally, the thinwalled structures made of steel are used for this task. The impact energy is absorbed on the folding wave of the thinwalled structure. To make the design process easier, behaviors of these structures have been studied for many years [1] [2] [3]. This research continues the previous studies, focuses on analyzing the behavior of tophat and doublehat thinwalled sections subjected to axial load.
In this paper, FE models of tophat structure is set up and comparing with the theoretical analysis developed by M.D. White et al. [4] [5] and Q. Wang et al. [6]. Then, the models are used as the data for optimizing a tophat column subject to mean crushing force and sectional bending stiffness constraints. The "Twostep Response Surface Methodology (RSM)  Enumeration" algorithm introduced by Y. Xiang et al. [7] is used in cases of the thickness of hatsection and closing back plate are the same and different . The results will be compared and evaluated.
2. Theoretical Review
2.1. Behavior of TopHat Section
2.1.1. The Theoretical Analysis of M.D. White et al. and Q. Wang et al
Superfolding element model of W. Abramowicz and T. Wierzbicki (as shown in Figure 1) applies to isotropic materials with properties that do not change over time, solid and perfectly plastic; the simple solution is
(1)
where is the thickness of section, is the rolling radius, is the perimeter of a superfolding element, is the length of a folding wave, is the effective crushing distance, and is the energy equivalent flow stress in the region of plastic flow.
By dividing hatsection into four "L" shape superfolding elements, M. D. White et al. [4] [5] were able to apply the model of W. Abramowicz and T. Wierzbicki to their analytical solution for a tophat section with strain hardening materials, which give
(2)
(3)
The general form of the relation between the dynamic crushing force and the mean static crushing force is
(4)
The average strain rate during axial crushing of an asymmetric superfolding element which was estimated by W. Abramowicz and T. Wierzbicki [8] is
(5)
where is the mean velocity, is the impact velocity, and ( is the final length of a folding wave [9]). The mean dynamic crushing force for a strain hardening, strain rate sensitive tophat section is
(6)
In a different way, Q. Wang et al. [6] made the corrections for the theoretical analysis of tophat section developed by M.D. White et al. [4] [5] is
(7)
The same method as the static theoretical prediction was used to obtain the results of parameters and by minimizing Eq. (7) respect to and , and equaled to zero.
2.1.2. An Approximate Analytical Solution for a TopHat Section with Different in Thickness of HatSection and Closing Back Plate
An approximate analytical solution for a new tophat section with different in the thickness of hatsection and the thickness of closing back plate as shown in Figure 2 can be found by the same Q. Wang’s procedure.
The total dissipated energy of tophat section is the sum of energy dissipated on hatsection and the closing back plate
The energy absorbed by hatsection is equal to the total dissipated energy on four superfolding elements with the average perimeter . Eq. (1) can be rewritten as
(8)
By using Eqs. (4), (5) and (8), the mean dynamic crushing load of hatsection is
(9)
Considering the closing back plate, the energy absorbed in crushing process given by M.D. White et al. [4] is
(10)
where is the width of the closing back plate. The mean crushing load of the plate for the quasistatic axial crushing force is
(11)
By using Eqs. (4), (5) and (11), the mean dynamic crushing force of the closing back plate is
(12)
We assume that the different between and in Eqs. (9) and (12) can be neglected. Therefore, the total mean dynamic crushing force of the new tophat section can be written as
(13)
with and are the root of the set of equations and .
2.2. Optimization for TopHat Section
Y. Xiang et al. [7] introduced the "Twostep RSMEnumeration" algorithm to optimize the mass of a tophat column (as shown in Figure 3) subject to mean crushing force and sectional stiffness constraints . The optimization problem can be written as
(14)
where is the crosssectional area, is the geometry design variables vector with and are the lower and upper boundary respectively, is the number of spotwelds on one side, and is the smallest allowed values of and . With the pure finite element approaching, the mean crushing force equation is determined by the RSM method from data of FE models. The "Twostep RSMEnumeration" optimization method consists of two steps. In the first step, the optimal crosssectional area with the optimized will be found based on the data of FE tophat column models with a large number of spotwelds or complete weld. In the second step, the FE tophat model with the optimal crosssectional geometry is tested with the increasing number of spotweld until reaches the desired value.
In this paper, two cases of optimization problem will be carried out and compared. The first optimization problem takes care of the case of normal tophat section, which is the same with Y. Xiang’ case. The second optimization problem takes care of the case of new tophat section (different in thickness).
The crosssectional area and the bending stiffness given by Y. Xiang et al. [7] for the case of normal tophat section are
(15)
And
(16)
In the case of different in thickness , equation of crosssectional area and the bending stiffness are
(17)
(18)
Where
3. Finite Element Model
The BelytschkoTsay 4node shell elements with 5 integration points is used to simulate column wall with finer mesh size . In this study, the wall column material is mild steel RSt37 which was used by S.P. Santosa et al. [10] with mechanical properties: Young’s modulus , initial yield stress , ultimate stress , Poisson’s ratio , density , and the power law exponent . The empirical CowperSymonds uniaxial constitutive equation constants and . The material model used to simulate mild steel is piecewise linear plasticity. The true stress – effective plastic strain curve of RSt37 steel was calculated from the engineering stressstrain curve of Santosa and was given in Table 1. The nodes in the lowest cross section of the column are clamped.
Mild steel RSt37  
Effective plastic strain (%)  True plastic stress (MPa) 
0.0  251 
2.0  270 
3.9  309 
5.8  339 
7.7  358 
9.6  375 
11.4  386 
13.2  398 
The indenter is modeled solid elements with Young’s modulus and Poisson’s ratio . The contact between the indenter and the column is nodes to surface. The contact used for the column wall is single surface to avoid interpenetration of folds generated during axial collapse. The indenter is only permitted to displace in axis with the initial velocity .
Four hexahedron solid elements are used to simulate a spotweld. Contact spotweld is also used between the surface nodes and the spotweld elements. The material model used to simulate the spotweld is spotweld with the same mechanical properties of mild steel RSt37. In fact, after welding, the spotweld area has different mechanical properties from the original properties of material. However, these differences are ignored in this study due to the limit of experiment.
The boundary conditions are shown in Figure 4.
4. Results
4.1. The First Optimization Problem
Consider a tophat column as shown in Figure 3. We want to find the crosssectional area such that the weight is minimized subject to mean crushing force and sectional stiffness constraints with smallest allowed values and . The column in length is under axial crushing by an indenter. Velocity of the indenter is . The crushing displacement is . The column made of mild steel RSt37. The geometry design variables vector with the limit sizes are and in . The data for optimizing is given in Table 2. The initial values of and are and , respectively.
Table 2. Design matrix of .
No.  Error  Error  
1  49.75  40.25  0.77  11.00  11.804  11.442  3.16%  11.725  0.67% 
2  58.50  41.25  1.55  14.50  39.277  38.191  2.84%  40.042  1.91% 
3  50.00  42.75  2.57  31.25  94.389  93.538  0.91%  99.351  4.99% 
4  47.25  43.50  2.76  38.75  107.877  107.783  0.09%  114.432  5.73% 
5  77.75  44.50  0.89  28.25  16.774  16.628  0.88%  16.804  0.18% 
6  59.00  45.75  1.20  35.00  25.578  27.187  5.92%  27.842  8.13% 
7  72.25  46.75  2.07  10.50  64.137  63.335  1.27%  66.965  4.22% 
8  60.75  47.75  0.84  15.75  14.250  14.087  1.15%  14.350  0.70% 
9  52.75  48.75  0.63  27.00  8.497  8.985  5.43%  9.005  5.64% 
10  51.50  50.00  2.33  20.50  79.236  77.529  2.20%  82.312  3.74% 
11  74.50  50.50  1.09  37.00  22.690  24.038  5.61%  24.385  6.95% 
12  57.00  51.00  2.47  12.50  87.045  83.796  3.88%  89.439  2.68% 
13  48.75  52.50  2.98  36.50  124.445  123.709  0.60%  131.563  5.41% 
14  42.00  53.25  2.21  14.75  74.415  68.369  8.84%  72.837  2.17% 
15  79.75  54.25  2.02  35.50  65.473  67.787  3.41%  70.454  7.07% 
16  43.75  56.00  2.55  22.00  92.824  90.333  2.76%  96.228  3.54% 
17  65.50  56.00  2.40  24.25  85.985  85.638  0.41%  90.357  4.84% 
18  44.75  58.00  2.82  33.25  114.664  111.905  2.47%  118.884  3.55% 
19  42.75  58.25  2.30  18.50  76.296  75.186  1.48%  79.890  4.50% 
20  70.75  59.50  1.00  34.75  19.716  20.845  5.41%  21.066  6.41% 
21  76.00  60.25  1.06  25.25  21.739  22.532  3.52%  22.892  5.04% 
22  45.75  61.50  1.82  22.50  53.427  52.337  2.08%  54.848  2.59% 
23  75.50  62.00  1.62  26.00  43.767  45.882  4.61%  47.411  7.69% 
24  62.50  63.75  1.17  19.25  25.940  25.630  1.21%  26.285  1.31% 
25  78.00  64.75  2.17  12.25  76.103  72.066  5.60%  75.773  0.44% 
26  63.75  65.25  0.54  29.75  6.813  7.341  7.20%  7.251  6.03% 
27  40.25  67.00  2.91  17.25  124.542  112.127  11.07%  120.169  3.64% 
28  56.25  67.25  1.43  13.75  35.982  34.933  3.00%  36.259  0.76% 
29  55.00  68.75  2.63  27.75  103.062  101.268  1.77%  107.014  3.69% 
30  61.50  69.25  1.26  23.50  30.473  29.634  2.83%  30.387  0.28% 
31  55.50  70.50  2.73  39.50  111.631  111.986  0.32%  117.855  5.28% 
32  73.50  71.50  1.94  32.25  61.310  63.768  3.85%  66.105  7.25% 
33  53.75  72.25  1.66  30.50  44.993  47.604  5.49%  49.268  8.68% 
34  68.75  73.75  1.70  29.50  48.667  50.601  3.82%  52.256  6.87% 
35  64.50  74.25  0.71  24.25  11.650  11.563  0.76%  11.553  0.84% 
36  66.50  75.25  1.78  17.50  53.418  52.623  1.51%  54.750  2.43% 
37  69.50  76.50  1.49  20.75  39.997  39.790  0.52%  40.992  2.43% 
38  46.50  77.50  1.31  16.50  32.350  30.521  5.99%  31.513  2.66% 
39  71.50  79.00  1.97  32.75  63.299  65.996  4.09%  68.371  7.42% 
40  67.75  79.75  0.59  38.25  9.120  8.941  2.01%  8.797  3.67% 
The equation of mean crushing force which formed by RSM algorithm can be written as
(19)
The results in first step of are shown in Table 3. Values of mean crushing force predicted by RSM method and mean crushing force from FE model are approximate.
Table 3. The optimal crosssectional dimensions of .






 
RSM  FE  
40  51  1.80  15  50.011  50.298  4.0905  435.6 
Figure 5. The second step results of .
RSM (from FE results)  RSM (from results of Q. Wang’s Eqs.)  Directly from M.D. White’s Eqs.  
 51  50  51 
 40  40  40 
 1.80  1.85  1.78 
 15  15  15 
 435.6  444.0  430.8 
4.2. The Second Optimization Problem
In this section, the "Twostep RSMEnumeration" algorithm is used again to optimize the mass of a new tophat column subject to ’s constraints. To reduce the calculation time, Eq. (13) is used to predict the mean crushing force, then correct them to fit with the simulation results. The FE results of 21 tophat models with different in thickness of hatsection and closing back plate are given in column of Table 5. The models are in length, and crushing distance is . Impact velocity is . As shown in Table 5, the difference between result of mean crushing force from approximate theoretical prediction and simulating result is significant when the thickness of hatsection or the thickness of closing back plate is much smaller than the perimeter (in that case, ). In addition, when the difference between and is significant, the deviation of folding wave of hatsection and the closing back plate becomes more apparent, especially in the case . Observing the deformation of models No. 10, No. 24 and No. 51 in Figure 6, which have the small ratio, we can see the buckling phenomenon occurs at the closing back plate. This may be the reason for the decline of in FE models.
Table 5. Design matrix of .
No. 








1  60.50  40.00  1.37  0.89  38.25  31.628  30.004  30.004 
2  47.75  41.00  1.59  2.37  21.00  44.344  45.019  45.019 
3  44.50  42.00  2.05  1.41  10.00  54.908  58.783  58.783 
4  70.50  42.25  0.57  2.42  31.00  14.398  14.416  
5  67.00  42.75  2.37  1.28  19.25  74.225  75.816  75.816 
6  58.25  43.50  2.67  2.75  37.75  104.205  105.855  
7  68.25  44.25  0.94  0.51  13.50  15.689  14.501  14.501 
8  58.75  45.00  2.83  1.55  19.75  99.248  97.802  
9  41.75  45.75  2.28  1.50  23.75  69.384  68.828  
10  77.25  46.25  2.61  1.03  13.75  86.272  83.498  83.498 
11  49.00  47.00  2.42  0.94  32.00  77.091  72.316  
12  60.00  47.50  0.82  0.81  23.00  13.857  12.939  
13  51.75  48.25  1.94  1.85  22.50  56.905  57.285  
14  62.25  49.00  1.68  2.48  33.75  53.259  53.613  
15  71.25  49.25  2.54  1.30  25.75  86.499  83.495  83.495 
16  55.75  50.25  1.11  2.98  27.00  32.718  34.606  
17  57.25  51.00  1.00  1.52  13.00  20.573  20.507  20.507 
18  40.25  51.75  0.91  2.90  15.25  22.771  26.087  
19  65.75  52.50  0.62  2.07  17.00  13.475  13.705  
20  64.50  53.00  2.32  1.94  28.75  79.019  77.932  
21  53.50  53.75  1.99  2.50  20.75  63.403  66.726  
22  72.75  54.50  1.77  2.16  23.25  54.295  54.311  
23  46.25  55.25  1.26  1.44  39.75  30.684  29.394  
24  75.50  55.50  2.56  0.71  28.00  87.603  82.300  82.300 
25  75.25  56.50  2.94  0.86  12.25  106.298  99.918  
26  73.50  57.00  1.80  1.76  18.00  52.483  51.818  
27  63.25  58.00  1.53  1.83  30.50  42.944  42.033  
28  56.50  58.25  1.20  1.36  28.25  27.750  26.687  
29  78.50  59.25  2.74  0.99  36.25  101.911  92.038  92.038 
30  71.75  59.50  1.87  2.61  15.75  60.290  60.456  60.456 
31  47.00  60.75  2.17  1.69  14.75  65.890  67.575  
32  73.00  61.00  0.87  2.19  35.50  22.847  22.432  
33  43.25  61.50  1.45  2.65  22.00  41.541  44.789  
34  63.50  62.50  1.64  1.16  35.75  44.877  42.175  
35  50.25  63.25  1.57  2.42  24.25  46.171  48.059  
36  64.75  63.50  2.40  1.65  24.75  82.194  82.826  82.826 
37  61.75  64.00  1.41  2.72  26.00  43.055  44.653  
38  61.00  65.25  2.96  1.99  16.50  113.316  116.791  116.791 
39  74.50  65.75  2.22  1.18  25.25  71.934  67.991  
40  50.75  66.50  1.03  2.94  34.00  30.624  31.925  
41  42.00  67.00  2.49  1.21  29.50  84.205  80.288  80.288 
42  52.50  67.50  1.90  2.29  34.50  62.239  62.547  
43  69.25  68.50  0.67  2.57  10.50  16.135  17.202  
44  54.75  69.00  2.67  0.75  30.50  95.038  88.040  
45  41.00  69.50  2.78  0.79  12.00  95.344  91.090  91.090 
46  54.25  70.00  2.85  2.23  11.25  106.045  113.448  
47  48.25  71.25  2.16  2.12  30.00  72.733  73.433  
48  76.50  72.00  1.08  1.88  18.50  26.777  27.805  27.805 
49  79.00  72.00  0.51  0.64  32.75  7.314  6.680  
50  58.00  72.75  0.79  1.73  36.75  17.980  17.418  
51  66.50  73.75  0.72  0.58  32.25  11.727  10.844  10.844 
52  77.50  74.50  1.47  2.29  17.25  43.169  43.821  
53  53.25  74.75  1.25  2.02  18.00  31.819  33.217  33.217 
54  44.00  75.50  2.89  2.67  37.25  119.596  122.987  
55  69.25  76.50  1.16  1.04  33.50  26.670  24.902  
56  45.00  77.25  1.75  2.85  26.75  57.156  59.366  59.366 
57  50.00  77.50  1.32  1.09  39.00  32.378  30.413  
58  46.00  78.25  2.12  2.80  14.25  70.941  79.738  
59  79.50  78.75  0.64  1.61  39.50  14.156  13.445  
60  67.50  80.00  2.04  0.58  20.25  61.578  50.201  50.201 
61  40.00  57.25  1.91  0.50  15.00  49.593  44.057  44.057 
In this study, for simplicity, we assume that the decrease of the mean crushing force is just affected by the ratio. The linear relationship between mean force error and the shown in Figure 7 can be written as
(20)
by linear least squares algorithm from the results of FE models.
Figure 7. Mean force error vs. ratio.
Since then, we have the column in Table 5. is equaled to if exist, or is calculated by the relationship shown in Eq. (20). The equation of mean crushing force which formed by RSM algorithm from values of can be written as
(21)
Table 6. Optimal crosssectional dimensions of .







 
RSM  FE  
40  55.75  2.05  0.5  15  50.016  50.211  4.0048  407.075 
The results in first step of are given in Table 6. The new crosssectional area is smaller than the older about .
In second step of , as shown in Figure 8, the tophat column with 12 spotwelds or more on each flange will satisfy the requirement about mean crushing force.
Figure 8. The second step results of.
Therefore, the tophat column, which has values of crosssectional dimensions and are and respectively and 12 spotwelds on each flanges, is the optimum results of . mass of the optimal tophat column of has been reduced. Note that the equation of mean crushing force Eq. (23) used in the first step of is based on data from the approximate theoretical solution corrected by FE models. Therefore, this optimal result of may not be the most optimal result. That mean the most optimal result can coincide with this result or can give the smaller crosssectional area (a little more better).
5. Conclusions
In this paper, the theoretical predictions for behavior of tophat thinwalled structure are used as a basis for checking the correctness of the model. Results of the investigation of spotweld pitch show that when the spotweld pitch decreases, the mean crushing force will be increased to the saturation value which approximates to the analytical result. The value of spotweld pitch that mark the start of the saturation region is equaled to . In optimization problem for a tophat column, using the results of mean crushing force from the theoretical solution of Q. Wang et al. to find the equation of by RSM method, or use Eqs. (2) and (6) of M.D. White et al. directly in the first step of "Twostep RSMEnumeration" algorithm can help to save significant computation time if the optimization problem does not require high accuracy. Allowing the different in thickness of hatsection and closing back plate in optimization problem can help improve the energy absorption capacity of tophat structures in case the weight is no change, mean that the weight can be reduce (about for the problem in this paper). The theoretical analysis for tophat thinwalled section with different in thickness of hatsection and closing back plate is quite suitable in the case of deformation of the flanges and the closing back plate is similar. Results of this theory should be calibrated with experimental results.
Acknowledgements
The research for this paper was financially supported by AUN/SEEDNet, JICA for CRA Program.
Nomenclature
 length of a folding wave 
 width of a tophat section 
 depth of a tophat section 
 sectional bending stiffness 
 CowperSymonds coefficients 
 Young’s modulus 
 internal energy absorbed by a superfolding element in asymmetric folding mode 
 energy absorbed by the closing back plate 
 energy absorbed by the hatsection 
 total energy absorbed 
 width of flange 
 second moment of area 
 perimeter of a tophat section 
 mean crushing force 
 rolling radius of toroidal surface 
 thickness of section 
 effective crushing distance 
 flow stress of material 
Reference
Biography
Hung Anh Ly is a lecturer in the Department of Aerospace Engineering – Faculty of Transport Engineering at Ho Chi Minh City University of Technology (HCMUT). He received his BEng in Aerospace Engineering from HCMUT in 2005, his MEng in Aeronautics and Astronautics Engineering from Bandung Institute of Technology  Indonesia (ITB) in 2007 and his DEng in Mechanical and Control Engineering from Tokyo Institute of Technology  Japan (Tokyo Tech) in 2012. He is a member of the New Car Assessment Program for Southeast Asia (ASEAN NCAP). His main research interests include strength of structure analysis, impact energy absorbing structures and materials. 
Hiep Hung Nguyen received the Bachelor of Engineering in Aerospace of Engineering with Second Class Honours from Ho Chi Minh City University of Technology (HCMUT), Vietnam in April, 2015. He spent two years studying in the field of impact of thinwalled structures. His expertise is structural analysis using finite element method. 
Thinh ThaiQuang received the Bachelor of Engineering in Aerospace of Engineering with First Class Honours from Ho Chi Minh City University of Technology (HCMUT), Vietnam in April, 2015. His research interests include the areas of structural impact, finite element methods, and structural mechanics. 