International Journal of Mechanical Engineering and Applications
Volume 3, Issue 3-1, June 2015, Pages: 40-48

Geometrical Optimization of Top-Hat Structure Subject to Axial Low Velocity Impact Load Using Numerical Simulation

Hung Anh Ly*, Hiep Hung Nguyen, Thinh Thai-Quang

Department of Aerospace Engineering, Faculty of Transportation Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam

Email address:

(H. A. Ly)
(H. H. Nguyen)
(T. Thai-Quang)

To cite this article:

Hung Anh Ly, Hiep Hung Nguyen, Thinh Thai-Quang. Geometrical Optimization of Top-Hat Structure Subject to Axial Low Velocity Impact Load Using Numerical Simulation. International Journal of Mechanical Engineering and Applications. Special Issue: Transportation Engineering Technology — part . Vol. 3, No. 3-1, 2015, pp. 40-48. 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 top-hat and double-hat thin-walled sections subjected to axial load. Due to limitations on the experimental conditions, this paper focuses on analyzing the behaviors of top-hat and double-hat thin-walled sections by theoretical analysis and finite element method. Two main objectives are setting up finite element models to simulate top-hat and double-hat thin-walled structures in order that the results are consistent with the theoretical predict; and using the results of these models to optimize a top-hat column subject to mean crushing force and sectional bending stiffness constraints by the "Two-step RSM-Enumeration" algorithm. An approximate theoretical solution for a top-hat column with different in thickness of hat-section and closing back plate is also developed and applied to the optimization problem.

Keywords: Crashworthiness, Impact, Optimization, Top-hat


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 thin-walled structures made of steel are used for this task. The impact energy is absorbed on the folding wave of the thin-walled 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 top-hat and double-hat thin-walled sections subjected to axial load.

In this paper, FE models of top-hat 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 top-hat column subject to mean crushing force and sectional bending stiffness constraints. The "Two-step Response Surface Methodology (RSM) - Enumeration" algorithm introduced by Y. Xiang et al. [7] is used in cases of the thickness of hat-section and closing back plate are the same  and different . The results will be compared and evaluated.

2. Theoretical Review

2.1. Behavior of Top-Hat Section

2.1.1. The Theoretical Analysis of M.D. White et al. and Q. Wang et al

Figure 1. (a). Cross-section of a top-hat column. (b) Four asymmetric elements forming a collapse profile [2-5].

Super-folding 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 super-folding 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 hat-section into four "L" shape super-folding 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 top-hat 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 super-folding 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 top-hat section is

(6)

In a different way, Q. Wang et al. [6] made the corrections for the theoretical analysis of top-hat 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 Top-Hat Section with Different in Thickness of Hat-Section and Closing Back Plate

Figure 2. Cross-section of new top-hat column.

An approximate analytical solution for a new top-hat section with different in the thickness of hat-section  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 top-hat section is the sum of energy dissipated on hat-section  and the closing back plate

The energy absorbed by hat-section is equal to the total dissipated energy on four super-folding 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 hat-section 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 quasi-static 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 top-hat section can be written as

(13)

Figure 3. (a) Cross-section of top-hat column and (b) distribution of spot-weld [7].

with  and  are the root of the set of equations  and .

2.2. Optimization for Top-Hat Section

Y. Xiang et al. [7] introduced the "Two-step RSM-Enumeration" algorithm to optimize the mass of a top-hat 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 cross-sectional area,  is the geometry design variables vector with  and  are the lower and upper boundary respectively,  is the number of spot-welds 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 "Two-step RSM-Enumeration" optimization method consists of two steps. In the first step, the optimal cross-sectional area with the optimized  will be found based on the data of FE top-hat column models with a large number of spot-welds or complete weld. In the second step, the FE top-hat model with the optimal cross-sectional geometry  is tested with the increasing number of spot-weld  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 top-hat section, which is the same with Y. Xiang’ case. The second optimization problem  takes care of the case of new top-hat section (different in thickness).

The cross-sectional area  and the bending stiffness  given by Y. Xiang et al. [7] for the case of normal top-hat section  are

(15)

And

(16)

In the case of different in thickness , equation of cross-sectional area  and the bending stiffness  are

(17)

(18)

Where

3. Finite Element Model

The Belytschko-Tsay 4-node 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 Cowper-Symonds 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 stress-strain curve of Santosa and was given in Table 1. The nodes in the lowest cross section of the column are clamped.

Table 1. True stress – Effective plastic strain data of mild steel RSt37.

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 spot-weld. Contact spotweld is also used between the surface nodes and the spot-weld elements. The material model used to simulate the spot-weld is spotweld with the same mechanical properties of mild steel RSt37. In fact, after welding, the spot-weld 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.

Figure 4. Boundary conditions.

4. Results

4.1. The First Optimization Problem

Consider a top-hat column as shown in Figure 3. We want to find the cross-sectional 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 cross-sectional dimensions of .

        RSM FE    
40 51 1.80 15 50.011 50.298 4.0905 435.6

In the second step of optimization procedure, the FE model with cross-sectional dimensions obtained in the first step is test again with the number of spot-weld n=2,3,… until reaches the expected mean crushing force. As shown in Figure 5, the optimal is top-hat column with 11 spot-welds on each side. Finally, we can conclude that the top-hat column, which has values of cross-sectional dimensions a,b,t and f are 40,51,1.8 and 15 mm respectively and 11 spot-welds on each flanges, is the optimum results of (SO)_1. Note that in the first step of optimization procedure, if we use the results of mean crushing force from the theoretical solution of Q. Wang et al. [6] to find the equation of P_m by RSM method, or use Eqs. (2) and (6) of M.D. White et al. [4] [5] directly, we can have results in Table 4. These values are so close. Hence, if the optimization problem does not need high precision, we can use theoretical equations to save the computation time.

Figure 5. The second step results of .

Table 4. Optimization results by three different methods.

  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

Figure 6. Deformation of top-hat columns No. 10B, No. 24B, No. 51B.

In this section, the "Two-step RSM-Enumeration" algorithm is used again to optimize the mass of a new top-hat 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 top-hat models with different in thickness of hat-section 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 hat-section  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 hat-section 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 cross-sectional 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 cross-sectional area  is smaller than the older  about .

In second step of , as shown in Figure 8, the top-hat column with 12 spot-welds or more on each flange will satisfy the requirement about mean crushing force.

Figure 8. The second step results of.

Therefore, the top-hat column, which has values of cross-sectional dimensions  and  are     and   respectively and 12 spot-welds on each flanges, is the optimum results of .  mass of the optimal top-hat 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 cross-sectional area (a little more better).

5. Conclusions

In this paper, the theoretical predictions for behavior of top-hat thin-walled structure are used as a basis for checking the correctness of the model. Results of the investigation of spot-weld pitch show that when the spot-weld pitch decreases, the mean crushing force will be increased to the saturation value which approximates to the analytical result. The value of spot-weld pitch that mark the start of the saturation region is equaled to . In optimization problem for a top-hat 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 "Two-step RSM-Enumeration" algorithm can help to save significant computation time if the optimization problem does not require high accuracy. Allowing the different in thickness of hat-section and closing back plate in optimization problem can help improve the energy absorption capacity of top-hat 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 top-hat thin-walled section with different in thickness of hat-section 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/SEED-Net, JICA for CRA Program.

Nomenclature

length of a folding wave

width of a top-hat section

depth of a top-hat section

sectional bending stiffness

Cowper-Symonds coefficients

Young’s modulus

internal energy absorbed by a super-folding element in asymmetric folding mode

energy absorbed by the closing back plate

energy absorbed by the hat-section

total energy absorbed

width of flange

second moment of area

perimeter of a top-hat section

mean crushing force

rolling radius of toroidal surface

thickness of section

effective crushing distance

flow stress of material

Reference

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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 thin-walled structures. His expertise is structural analysis using finite element method.

Thinh Thai-Quang 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.

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