Crashworthiness of multi-cell circumferentially corrugated square tubes with cosine and triangular configurations

International Journal of Mechanical Sciences 165 (2020) 105205

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Crashworthiness of multi-cell circumferentially corrugated square tubes with cosine and triangular configurations Zhixiang Li a,b,c, Wen Ma a,b,c, Ping Xu a,b,c,d, Shuguang Yao a,b,c,∗ a

Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha, 410075, China Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Central South University, Changsha, 410075, China c National & Local Joint Engineering Research Center of Safety Technology for Rail Vehicle, Central South University, Changsha, 410075, China d State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410075, China b

a r t i c l e

i n f o

Keywords: Crashworthiness Multi-cell configuration Circumferentially corrugated square tubes Multiple loading conditions Complex proportional assessment Optimization

a b s t r a c t Multi-cell and multi-corner configurations are the most common methods to improve the crashworthiness performance of thin-walled tubes. However, the crashworthiness of a thin-walled tube combining these two configurations has not yet been evaluated. This paper proposed a series of multi-cell circumferentially corrugated square tubes (CCSTs) in which multi-corner configurations were achieved by introducing cosine and triangular corrugations. The crashworthiness of these multi-cell CCSTs were investigated under multiple loading angles. The complex proportional assessment (COPRAS) method was used to select the optimal structures. The results showed that the cosine and triangular multi-cell CCSTs with crisscross ribs, namely, C1 and T1 in this paper, exhibited the best performance. Then, two different multi-objective optimization methods were used to determine the optimal geometric parameters of C1 and T1. The results showed that one optimization method mainly balanced the crashworthiness performance at different loading angles but did not improve the performance, whereas the other optimization method significantly improved the crashworthiness of the tubes.

1. Introduction Thin-walled tubes are widely used as energy absorbing structures in industrial fields [1–4]. In recent years, the requirements for safety performance have increased. These increased safety requirements have led to the common circular and square tubes being unable to meet the requirements of energy absorption. Therefore, some new designs have been proposed to improve energy absorption. The existing literature shows that multi-cell and multi-corner configurations are the two most effective methods to enhance the crashworthiness performance of thinwalled structures [2,5,6]. Multi-cell designs can be achieved by adding inner ribs into simple single tubes. Zhang [7] investigated the crushing performance of multicell circular tubes with different inner ribs and first proposed a theoretical model based on the constituent element method to predict the crushing force. Zhang [7] also resolved some problems found in multi-cell square tubes, including the influence of geometric compatibility among elements, the type of trigger, the structural parameters and topology changes on the crush resistance of the structure. Tran [8] numerically and theoretically studied the crashworthiness of multi-cell triangular tubes and performed an optimization procedure, through which an op-

timum configuration was determined. Tang et al. [9] proposed a type of cylindrical multi-cell column that exhibited improved energy absorption efficiency. After comparison, they also found that the cylindrical multicell columns performed better than square multi-cell columns in terms of energy absorption. Nia [10] compared the crashworthiness performance of multi-cell thin-walled tubes with triangular, square, hexagonal and octagonal sections. This comparison showed that the energy absorption capacity of multi-cell configurations is obviously greater than that of simple single tubes and that hexagonal and octagonal multi-cell tubes have the greatest energy absorption efficiency (SEA). Wang et al. [5,6] studied the effect of cell numbers on the crushing behaviour of multi-cell square tubes. They found that the half-wavelength caused by different cell numbers had a vital influence on the mechanical properties of the tubes. Sun et al. [11–14] imposed functionally graded thickness on multi-cell tubes to provide improved performance. The studies mentioned above all show that the multi-cell configurations can significantly promote the deformation stability and crashworthiness performance of thin-walled tubes. Multi-corner configurations can be achieved in many ways. Polygon setting is the most common way to increase number of corners. Different polygonal thin-walled tubes have been designed as energy absorbers, and their energy absorption has also been shown to be efficient

∗ Corresponding author at: Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha, 410075, China E-mail address: [email protected] (S. Yao).

https://doi.org/10.1016/j.ijmecsci.2019.105205 Received 13 July 2019; Received in revised form 27 September 2019; Accepted 30 September 2019 Available online 1 October 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

Z. Li, W. Ma and P. Xu et al.

[10,15–21]. On the basis of polygonal sections, the number of corners can be further increased by pushing the existing corners inward along diagonal lines. Sun [22] used this method on a square tube and obtained a so-called “crisscross tube”. The “crisscross tube” performed better in energy absorption efficiency than a square tube. Tang [23,24] repeated this method on a square tube and produced a column with a concave profile, which had more corners than the “crisscross tube” obtained by Sun. In addition, some mathematical expressions can also be used to realize multi-corner tubes. For example, Wu [25] used a Fourier series expansion to control the cross-section and obtained a multi-cell corner tube, which also exhibited good crashworthiness performance. Deng et al. [26,27] introduced a sinusoidal function into the cross-section of the circular tube and achieved a lateral corrugated tube. Their results showed that 6 corrugations was the best for the energy absorption. They noted that the corners in the sinusoidal lateral corrugated tube were all curved corners; when replacing these corners as straight corners, the authors obtained a star-shaped tube [28–31]. The effects of the material properties, processing method, and geometric parameters on the crashworthiness of the star-shaped tube were emphasized. Li [2,32] introduced a cosine expression into the configuration of the square tube and constructed a multi-corrugation (corner) square tube, whose energy absorption was much better than that of the simple square tube. However, multi-corner configurations also bring unstable deformation to thin-walled tubes. Most of the multi-corner tubes mentioned above achieve progressive deformations by adding triggers. However, in practical applications, how to add these triggers and where they should be added are still quite difficult problems. It is already known that the multi-cell configuration can provide improved deformation stability. Therefore, combining the multi-cell and multi-corner configurations to construct a cross-sectional configuration of tubes may be a good idea to improve the energy absorption performance and stability of thinwalled tubes. As far as we know, thin-walled tubes combining multi-cell and multi-corner configurations have not been investigated by any researchers and have not been emphasized in any literature. Furthermore, most studies on the crashworthiness of thin-walled tubes have focused on axial crushing. However, in an actual impact accident, the impact angle is often uncertain. Therefore, increasing attention has been given to oblique crushing. For instance, Isaac [33] investigated the structural response and crashworthiness performance of a hexagonal thin-walled grooved tube subjected to oblique impact loading conditions. Yao [34] studied the crashworthiness of circular tubes with functionally graded thickness under multiple loading angles. Qiu [35] comprehensively investigated the crashing behaviours of different multi-cell hexagonal cross-sectional columns under axial and oblique loads. Fang [36] analysed the effect of cell number and oblique loads on the crashing behaviours of thin-walled tubes. Pirmohammad [19] compared the crashworthiness of triangular, square, hexagonal and circular tubes under multiple loading angles. Tarlochan [20] described a computationally aided design process to investigate thin-walled structures under different loading conditions. Rad [37] first studied a thin-walled tube under a 3D oblique load. The above studies show that the oblique crushing performance is very important for thin-walled tubes. To fully understand the performance of the structure, different loading conditions must be considered. In addition, the mentioned literature on oblique crushing also provides some useful methods to help us better study the structure presented in this paper. In this paper, we proposed a series of multi-cell circumferentially corrugated square tubes (CCSTs) with different cell configurations and cell numbers. The CCSTs can be used as energy absorption components in automobiles and rail vehicles. These tubes are actually a combination of multi-cell and multi-corner configurations, wherein the multi-corner configuration was achieved by introducing cosine and triangular corrugations into the cross-section of the tube. The crashworthiness of the multi-cell CCSTs were investigated under multiple loading conditions. Using the complex proportional assessment (COPRAS) method, the optimal structures were selected. Then, optimizations for the selected struc-

International Journal of Mechanical Sciences 165 (2020) 105205

ture were carried out in two different ways to find the optimal geometric configurations. 2. Finite element (FE) modelling 2.1. Problem description In this study, the crashworthiness of multi-cell CCSTs was analysed. Two kinds of corrugated profiles, namely, cosine and triangular, were considered, as shown in Fig. 1(a) and (b), respectively. The red side in the cosine cross-section is controlled with a cosine expression: [ ( ) ] 𝐿 𝐿 𝐿 5 𝑦 = 𝐴 cos 𝜋𝑥 + 0 , 𝑥 ∈ − 0 , 0 (1) 𝐿0 2 2 2 where A represents the amplitude of the cosine function and L0 is the nominal side length. Then, by performing a circular array, the three other sides are obtained. Each corrugated side is distributed along the side of a square, and the four corrugated sides constitute the crosssection of a CCST with a cosine profile. When utilizing the same method on a triangular corrugated side, the cross-section of a CCST with triangular profile will be obtained, as shown in Fig. 1(b). The multi-cell configuration is generated by adding inner ribs into the CCST. In this study, six different inner ribs are taken into consideration. Therefore, 6 multi-cell cosine CCSTs and 6 multi-cell triangular CCSTs were obtained, which were named C1, C2 …… C6 and T1, T2 …… T6, respectively, as shown in Fig. 2. In the initial design, we set L0 = 40 mm and A = 3 mm for all tubes. Moreover, the tube length was set to 100 mm. The crashworthiness performance of the multi-cell CCSTs under multiple loading angles was analysed. The test tube was clamped on a fixed plate, and a moving plate axially or obliquely loaded the tube to make it deform (Fig. 3). The studies on oblique loading for square tubes in existing literature define the oblique loads with a single angle in a 2D system, and the loads are always along a side of the square section. Considering that in a real-world loading case, energy absorbing components are subject to oblique loads at various uncertain angles, we adopted two loading patterns when considering the oblique loadings. Pattern 1 is loaded along a side with 𝛼 > 0°, whereas pattern 2 is loaded along a corner with 𝛽 > 0°, as shown in Fig. 3(b) and (c), respectively. 2.2. Crashworthiness criteria Three criteria were used to evaluate the crashworthiness performance of the multi-cell CCSTs. The criteria are the SEA, the maximum force (Fmax ) and the crushing force efficiency (CFE). These three criteria are defined hereafter. 𝐸𝐴 𝑆𝐸 𝐴 = (2) 𝑚 where EA and m are the energy absorption and mass, respectively. The SEA represents the energy absorption efficiency of the absorber; a high SEA is always preferred. Fmax is the maximum force in the crushing process. A high Fmax always brings large injuries to passengers, so the smaller Fmax is, the better the absorber. 𝐶𝐹 𝐸 =

𝐹𝑚𝑒𝑎𝑛 𝐹max

(3)

where Fmean is the mean crushing force. An ideal energy absorber always requires a high CFE. 2.3. FE model and validation In this study, the FE software LS-DYNA was used to investigate the crashworthiness performance of the multi-cell CCSTs. The FE model consisted of three parts: a fixed plate, a moving plate and a test tube, as

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 1. Sketch of the cross-sectional profiles of CCSTs: (a) cosine profile and (b) triangular profile.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Multi-cell CCSTs. Fig. 3. Sketch of FE models of multi-cell CCST under different loading conditions: (a) axial loading, (b) loading along a side and (c) loading along a corner.

shown in Fig. 3. The fixed plate and moving plate are modelled with Mat_rigid because these objects should not exhibit any deformation during the crushing process. The test tube was modelled with Mat_24. The true stress-strain curves of the tube material obtained from the material test are shown in Fig. 4. The material had the following properties: density 𝜌 = 2700 kg/m3 , Young’s modulus E = 69.79 GPa, yield stress

𝜎 y = 54.00 MPa, ultimate stress 𝜎 u = 163.46 MPa and Poisson’s ratio 𝜈 = 0.33. The test tube was modelled with the Belytschko-Tsay shell element with one centre integration point and five through-thickness integration points. By performing a mesh convergence analysis, the element size of 1 × 1 mm was found to be able to ensure the accuracy of the

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 4. True stress-strain curves of the multi-cell CCSTs. Table 1 Comparison of the crashworthiness indicators between the test and the FE simulation.

Test FE Absolute error (%)

Fmax (kN)

Fmean (kN)

EA (kJ)

SEA (kJ/kg)

CFE

43.34 44.01 1.55

32.62 31.14 4.54

1.96 1.87 4.59

21.57 20.60 4.50

0.75 0.71 5.33

numerical results. The “AUTOMATIC_SURFACE_TO_SURFACE” contact algorithm was used to account for the contacts between the two plates and the test tube. Moreover, the “AUTOMATIC_SINGLE_SURFACE” was defined to simulate the self-contact of the test tube. The coefficients of friction for both types of contacts were set to 0.15 [35,37]. In addition, on the premise that the kinetic energy does not exceed 5% of the internal energy throughout the simulation, the loading velocity was chosen as 3 m/s to improve the computational efficiency. Here, we carried out an axial crushing test on tube T2 to validate the FE model. A comparison of the deformation processes and the forcedisplacement curves between the test and the FE analysis are shown in Fig. 5(a) and (b), respectively. The FE results agreed well with the testing results. In addition, some crashworthiness criteria were also compared and are listed in Table 1. The maximum absolute error was 5.33%, which was acceptable. Above all, the FE model had good accuracy and can be used for further analysis. 3. Crashworthiness performance under multiple loading angles 3.1. Deformation modes The deformation modes of the cosine multi-cell CCST under different loading angles are shown in Fig. 6. For the cosine multi-cell CCSTs, most developed a progressive deformation mode when loaded axially, 𝛼 = 10° and 𝛽 = 10°, and deformed in global bending under other loading conditions except for C4. The results show that C4 exhibited unordered deformation under axial loading and bending under oblique loading, which will deteriorate its energy absorption. The deformation modes of the triangular multi-cell CCSTs under different loading angles are shown in Fig. 7. The deformation mode distribution of the triangular CCSTs was basically the same as that of the cosine CCSTs. Similarly, T4 also showed unideal deformation. However, there were also some differences, such as T6 exhibiting nonprogressive deformation under 𝛼 = 10° and 𝛽 = 10° loadings.

the loading angles of 𝛼 and 𝛽 were the same, the force-displacement characteristics of the two conditions were also basically the same. In the beginning of deformation, the force of axial loading increased sharply to the initial peak value, and the corresponding displacement was approximately 5 mm (Fig. 8(a)). However, the increase rate in the initial deformation decreased significantly with increasing oblique loading angle, which led to an increase in the displacement corresponding to the initial peak force (Fig. 8(b)–(h)). Moreover, under axial (𝛼 = 𝛽 = 0°) and oblique loading of 10° (𝛼 = 10° and 𝛽 = 10°), the forces fluctuated after the initial peak force because the progressive folds were formed in deformation (Fig. 8(a), (b) and (e)), whereas the forces in oblique loading angles of 20° and 30° dropped only after the initial peak forces because the tubes deformed with bending in this stage (Fig. 8(c), (d), (g) and (h)). There were also two exceptions, namely, C4 and T4. Since their deformations were obviously different from those of other tubes, their forces exhibited different characteristics in each loading angle. The T6 also showed some difference in the oblique angle of 10° (Fig. 8(b) and (e)) because it did not deform with a progressive mode (Fig. 7). In addition, the difference in forces between the cosine and triangular CCSTs with the same inner ribs was very small, which illustrates that the crashworthiness of these tubes is basically the same. 3.3. Energy absorption Fig. 9 shows the values of Fmax , SEA and CFE under different loading angles of the considered CCSTs. Fig. 9(a) and (b) shows that the Fmax and SEA basically had the same changing trend with respect to the loading angle or inner rib shape. For each tube, both the maximum Fmax and SEA appeared when the loading angle was 𝛼 = 𝛽 = 0°, and the Fmax and SEA decreased obviously with increasing 𝛼 or 𝛽. In addition, when a tube was loaded with oblique angle 𝛼 or the same angle 𝛽, the Fmax and SEA only had a small difference. Comparing the cosine and triangular CCSTs with the same inner ribs, the triangular CCSTs had smaller Fmax and higher SEA when the loading angles were 0° and 10°, except for C4 and T4. When the loading angles were 20° and 30°, it was not obvious which type of CCST had higher or lower Fmax and SEA. These findings illustrated that the crashworthiness of the triangular CCSTs was better than that of the cosine CCSTs to some extent. Moreover, the Fmax and SEA increased with increasing cell number (or number of ribs). For example, when the cross-sectional profile changed from T1 to T3, under axial loading, the Fmax increased from 33.52 to 57.80 kN, and the SEA increased from 18.23 to 22.54 kJ/kg; under other oblique angles, both Fmax and SEA also increased to a certain extent. The CFE of all tubes is plotted in Fig. 9(c). The change trend of CFE with respect to the loading angle of all tubes is the same: CFE increased when the loading angle increased from 0° to 10°, then decreased from 10° to 20°, and then increased from 20° to 30° The maximum CFE occurred when the loading angle was 10° (𝛼 = 10° or 𝛽 = 10°) for every tube, and the minimum CFE of most tubes appeared when the loading angle was 20° (𝛼 = 20° or 𝛽 = 20°). In addition, for the cosine and triangular CCSTs with the same inner ribs, the two CFEs were almost equal. 4. Selection of the optimal multi-cell CCST The crashworthiness study in the present work was very complex because it involved many cross-sectional tubes, multiple loading angles and multiple crashworthiness evaluation criteria. Therefore, it was difficult to select the best among the considered tubes. However, it was actually a multicriteria decision-making (MCDM) problem. There are many methods to solve MCDM problems. Herein, we adopted the COPRAS method [38] to select the optimal structure because it is efficient and simple and has been widely used.

3.2. Force-displacement characteristics 4.1. COPRAS method The force-displacement curves of all tubes under different loading angles are shown in Fig. 8. The results showed that for each tube, when

The procedure of the COPRAS method is described hereafter [38].

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 5. Validation of the FE model: (a) deformation comparison and (b) force-displacement curve comparison.

Fig. 6. Deformation of the cosine CCSTs under different loading angles.

Step 1: Use the criteria values of the alternatives to generate an initial decision matrix X. ⎡ 𝑥11 ⎢ [ ] 𝑥 X= 𝑥𝑖𝑗 𝑚𝑛 = ⎢ 21 ⎢⋯ ⎢𝑥 ⎣ 𝑚1

𝑥12 𝑥22 ⋯ 𝑥𝑚2

⋯

𝑥1𝑛 ⎤ ⎥ 𝑥2𝑛 ⎥ ⋯⎥ 𝑥𝑚𝑛 ⎥⎦

(4)

where xij represents the value of the jth criteria of the ith alternative and m and n denote the number alternatives and criteria, respectively. Step 2: Generate a normalized decision matrix R. As the units of different criteria are different, the order of magnitude of their values are also different. To make them comparable, the values should be normalized. 𝑥𝑖𝑗 [ ] 𝐑 = 𝑟𝑖𝑗 𝑚𝑛 = ∑𝑚 (5) 𝑖=1 𝑥𝑖𝑗 Step 3: Determine the weight of each criterion.

The importance of different criteria may be different for the object to be evaluated. The weights are used to consider the importance of different criteria. The COPRAS method gives a way to determine the weights, which is described as follows:

1) Compare any two conditions at a time, and the total set of comparisons is N = n(n-1)/2. In the present study, the overall comparison set is 3 (3∗ (3−1)/2), as shown in Table 2. 2) Give a score Sij to the two criteria being selected from the comparison sets based on their importance. Specifically, score Sij = 3 for the most important criterion, whereas Sij = 1 for the other one. If the two criteria being compared have the same importance, assign Sij = 2 for both criteria. In the present study, three criteria, namely, Fmax , SEA and CFE, were considered, and their scores are listed in Table 2.

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 7. Deformation of the triangular CCSTs under different loading angles.

Table 2 Individual weights of different criteria (number of comparison sets N = (3−1)/2 = 3). SelectionAssigned score of each comparison set Set 2 Set 3 criteria Set 1 Fmax SEA CFE

2 2

3 1

Total Σ

3 1

Wj

wj

5 5 2 G = 12

5/12 = 0.4167 5/12 = 0.4167 2/12 = 0.1667 1

Step 6: Select the minimal value of S-i : 𝑆−𝑖

min

= min 𝑆−𝑖

(11)

Step 7: Compute the relative priority (Q). The relative priority of the ith alternative Qi can be achieved as follows: ∑ 𝑆− min 𝑚 𝑖=1 𝑆−𝑖 (12) 𝑄𝑖 = 𝑆+𝑖 + ) ∑𝑚 ( 𝑆−𝑖 𝑖=1 𝑆− min ∕𝑆−𝑖 Step 8: Determine the quantitative utility U. The quantitative utility of the ith alternative is calculated as follows:

3) Sum up the total scores of the jth criterion from all comparison sets according to the following formula: ∑𝑚 𝑆𝑖𝑗 = 𝑊𝑗 (6) 𝑖=1

4) Compute the weight of the jth criterion wj as follows: 𝑤𝑗 =

𝑊𝑗 𝐺

where G is the total score of all criteria. Step 4: Obtain the weighted normalized decision matrix D. [ ] 𝐃 = 𝑑𝑖𝑗 𝑚𝑛 = 𝑟𝑖𝑗 × 𝑤𝑗

(7)

(8)

where dij is the weighted normalized value of the jth criterion for the ith alternative. Step 5: Achieve the total beneficial and non-beneficial attributes for each alternative: ∑𝑛 𝑆+𝑖 = 𝑦+𝑖𝑗 (9) 𝑗=1

𝑆−𝑖 =

∑𝑛

𝑦 𝑗=1 −𝑖𝑗

(10)

where S + i and S-i are the total beneficial and non-beneficial attributes for the ith alternative.

𝑈𝑖 =

𝑄𝑖 × 100% 𝑄max

(13)

According to the values of Ui , we can obtain the ranking of the performance priorities of all alternatives. The best alternative has a quantitative utility value of 100%. 4.2. Results Here, we separately selected the optimal design of cosine and triangular multi-cell CCSTs using the COPRAS method. According to Steps 1 and 2, the initial and normalized decision matrixes of the cosine and triangular multi-cell CCSTs were obtained based on the numerical results, as shown in Tables A1–A4. To further calculate the weighted normalized decision matrix, we cannot directly use the weights obtained in Table 2 because there are multiple loading angles; therefore, the weights should be further recomputed. We assigned a weight wa to each angle according to their importance, as listed in Table 3. Then, the final weights of each criterion corresponding to each loading angle were obtained as wf = wa × wj . The final weights are shown in Table 4. Then, the nominalized weighted decision matrixes of the cosine and triangular multi-cell CCSTs were obtained and are displayed in Tables A5 and A6, respectively.

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 8. Force-displacement curves: (a) 𝛼 = 𝛽 = 0°, (b) 𝛼 = 10°, (c) 𝛼 = 20°, (d) 𝛼 = 30°, (e) 𝛽 = 10°, (f) 𝛽 = 20° and (g) 𝛽 = 30°

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 9. Crashworthiness indicators of the cosine and triangular CCSTs under different loading angles: (a) Fmax , (b) SEA and (c) CFE.

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International Journal of Mechanical Sciences 165 (2020) 105205

Table 3 Weights wa of different loading angles. Case 0 A= 10° 𝛼 = 𝛽 = 20° 𝛼 = 𝛽 = 30° 𝛼 = 𝛽 = Total

𝛽 = 0° 10° 10° 20° 20° 30° 30°

wa

Total

0.4 0.15 0.15 0.1 0.1 0.05 0.05 1

0.4 0.3 0.2 0.1 1

Table 4 Final individual weights wf of different criteria at different loading angles. Case

wf _Fmax

wf _SEA

wf _CFE

𝛼 = 𝛽 = 0° 𝛼 = 10° 𝛼 = 20° 𝛼 = 30° 𝛽 = 10° 𝛽 = 20° 𝛽 = 30° Total

0.4 × 0.4167 = 0.1667 0.15×0.4167 = 0.0625 0.1 × 0.4167 = 0.0417 0.05 × 0.4167 = 0.0208 0.0625 0.0417 0.0208 0.04167

0.1667 0.0625 0.0417 0.0208 0.0625 0.0417 0.0208 0.04167

0.0667 0.0250 0.0167 0.0083 0.0250 0.0167 0.0083 0.01667

Table 5 Results of COPRAS of the cosine multi-cell CCSTs. Section

C1

C2

C3

C4

C5

C6

S+ SQi Ui Rank

0.0911 0.0517 0.1793 100% 1

0.1055 0.0680 0.1726 96.26% 2

0.1072 0.0891 0.1584 88.35% 5

0.0777 0.0493 0.1702 94.95% 4

0.1033 0.0673 0.1711 95.42% 3

0.1069 0.0914 0.1568 87.46% 6

Table 6 Results of COPRAS of the triangular multi-cell CCSTs. Section

T1

T2

T3

T4

T5

T6

S+ SQi Ui Rank

0.0938 0.0505 0.1834 100% 1

0.1061 0.0689 0.1718 93.66% 2

0.1077 0.0900 0.1580 86.14% 5

0.0724 0.0470 0.1686 91.92% 4

0.1032 0.0692 0.1687 91.97% 3

0.0999 0.0912 0.1495 81.52% 6

Finally, the results of COPRAS of the cosine and triangular multicell CCSTs were achieved, as shown in Tables 5 and 6, respectively. By accessing the comprehensive performance of the CCSTs under multiple loading angles, we found that the optimal configurations of the cosine and triangular CCST are C1 and T1, respectively. The two tubes had the same inner ribs, namely, a crisscross shape. Both tubes had 4 cells, which was the minimum value in all considered tubes. Therefore, increasing the cell number does not necessarily improve the crashworthiness performance of thin-walled tubes when multiple loading angles are considered comprehensively. 5. Crashworthiness optimization of the selected structure In the last section, C1 and T1 were determined as the optimal designs of the cosine and triangular CCSTs, respectively. Therefore, in this section, multi-objective optimizations were conducted to find the best geometric configurations for C1 and T1. 5.1. Definition of the optimization problem In the last section, the parameters Fmax , SEA and CFE were considered as the crashworthiness criteria to select the optimal structure. Thus,

Fig. 10. Process of optimization Ⅰ.

these three parameters were taken as objectives to define the optimization problem. In addition, the amplitude A and wall thickness t were regarded as variables to find the best configurations of C1 and T1. The global response surface method (GRSM) was utilized to solve the optimization problems because of its high accuracy and efficiency [39]. An ideal structure showing superior crashworthiness performance always has lower Fmax and higher SEA and CFE. Therefore, the optimization problem was expressed as follows: ⎧min (𝐹 , −𝑆𝐸𝐴, −𝐶𝐹 𝐸 ) max ⎪ (14) ⎨𝑠𝑡 0 𝑚𝑚 ≤ 𝐴 ≤ 5 𝑚𝑚 ⎪ 0.5 𝑚𝑚 ≤ 𝑡 ≤ 2 𝑚𝑚 ⎩ However, in the present study, the optimization problem was complex because 7 different loading conditions were considered. Here, we adopted two methods to carry out the optimization process to confirm that the optimization results can satisfy the complex loading conditions. We named the two optimization processes optimization Ⅰ and optimization Ⅱ, which are shown in Fig. 10 and Fig. 11, respectively. In optimization Ⅰ, the optimizations were conducted separately under the 7 different loading angles. Therefore, there were 7 Pareto solutions. For each Pareto solution, we used the technique for order of preference by similarity to ideal solution (TOPSIS) [37] to select an optimal point (optimal design). Then, the crashworthiness of the 7 optimal designs under multiple loading angles were compared with the COPRAS method, and a superior optimal point was selected as the final design. In optimization Ⅰ, 7 optimizations were conducted with respect to 7 loading angles, and the 7 optimizations were considered to have the same importance. In optimization Ⅱ, the importance of the 7 loading angles was considered different. Therefore, weights were used to take into account the difference. The weighs are the same as those used in Section 4, as shown in Table 4. The design of experiments (DOE) responses were processed into dimensionless form first using the following equation: 𝑟𝑑 =

𝑟 − 𝑟𝑙 𝑟𝑢 − 𝑟𝑙

(15)

where rd represents the dimensionless response and ru and rl denote the upper and lower limits of the initial response, respectively. The dimensionless responses of the 7 loading angles were summed to construct a new DOE, and then the DOE was used to construct the

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International Journal of Mechanical Sciences 165 (2020) 105205

In the present study, the HyperKriging method was adopted to construct the approximate model because of its good accuracy for highly nonlinear problems. Two parameters, namely, the coefficient of determination (R2 ) and the relative average absolute error (RAAE), were adopted to assess the accuracy of the approximate models. These two parameters are defined as follows: 𝑛 ( ∑

𝑅2 = 1 −

𝑖=1 𝑛 ∑

(

𝑖=1 1 𝑛

𝑦𝑖 − 𝑦̄𝑖

)2

)2 𝑦𝑖 − 𝑦̄

(16)

𝑛 ( ) ∑ |𝑦 − 𝑦̄ | 𝑖| | 𝑖

𝑖=1

𝑅𝐴𝐴𝐸 = √

1 𝑛

𝑛 ( ∑ 𝑖=1

) 𝑦𝑖 − 𝑦̄

(17)

where n is the number of sample points, yi denotes the values of the ith point, 𝑦̄𝑖 is the fit predictions at the same points, and 𝑦̄ is the mean of the input values. R2 is a measure of how well the fit can reproduce known data points. The value of R2 typically varies between 0 and 1. The higher the value of R2 , the better the quality of the fit. In practice, a value of R2 above 0.92 is often very good. RAAE indicates the ratio of the average absolute error to the standard deviation. A low ratio is more desirable because this indicates that the variance in the fit predicted value is dominated by the actual variance in the data and not by modelling error. 5.3. Optimization results 5.3.1. Optimization Ⅰ Table 9 shows the values of R2 and RAAE under different conditions in optimization I. It is clear that all the approximate models of C1 and T1 in optimization I had good accuracy. The Pareto solutions of C1 and T1 under different loading angles are plotted in Fig. 12 and Fig. 13, respectively. Using the TOPSIS method, an optimal point of each loading angle was selected, and their geometric parameters for C1 and T1 are listed in Tables 10 and 11, respectively. After comparing the 7 designs with the COPRAS method, the superior optimal points with geometric parameters A = 1.85 mm and t = 0.57 mm for C1 and A = 2.03 mm and t = 0.50 mm for T1 were obtained as the final design, as shown in Tables 10 and 11.

Fig. 11. Process of optimization Ⅱ. Table 7 Sampling matrix of the full-factorial DOE. Variable

Level

Value

A t

6 4

0, 1, 2, 3, 4, 5 0.5, 1, 1.5, 2

Table 8 Validation matrix generated with Latin hypercube DOE. No.

1

2

3

4

5

6

7

8

A t

0.62 1.06

3.08 1.57

1.33 1.27

3.75 1.12

0.81 0.69

3.58 1.81

2.5 1.8

4.64 0.77

approximate model. Therefore, in optimization Ⅱ, only one Pareto solution, in which the results of the 7 loading angles were all considered, was obtained. Finally, the TOPSIS method was also used here to find an optimal design. 5.2. Approximate models An approximate model is always used in optimization problems because it can reduce the cost and time of calculations while ensuring the accuracy of the optimization results. To achieve the approximate models, we should first perform the DOE to obtain adequate data. Fullfactorial DOE was used here because it can generate uniform sample points. The levels of variables A and t were set to 6 and 4, respectively; therefore, a total of 24 design samples were generated. The values of each level of variables are listed in Table 7. In addition, another design matrix, which was generated randomly with the Latin hypercube DOE method, was used to validate the accuracy of the approximate model, as shown in Table 8.

5.3.2. Optimization Ⅱ Table 12 displays the values of R2 and RAAE of the approximate models in optimization Ⅱ. The results showed that both models had good accuracy. The Pareto solutions of C1 and C2 in optimization Ⅱ are shown in Fig. 14(a) and (b), respectively. Note that in this optimization, the values of the three objectives were weighted dimensionless values rather than the true values. The upper limits of the three objects were basically equal to their respective weights; therefore, Fmax u = SEAu = 0.4167 and CFEu = 0.1667, as shown in Table 2. Nevertheless, the optimization can still help us find an optimal design when considering the importance of different loading angles. The geometric parameters of the optimal designs for C1 and T1 selected with TOPSIS are listed in Table 13. 5.3.3. Comparison of the two optimizations The results of the two optimizations and the initial design were compared for C1 and T1, as shown in Figs. 15 and 16, respectively. The optimization results of C1 and C2 are somewhat similar, so we take C1 as an example hereafter. Optimization Ⅰ was conducted for 7 loading angles, and the optimization result for a loading angle of 𝛼 = 20° was selected as the optimal one. It is an already well-known fact that Fmax and SEA are conflicting. To reduce the Fmax , a certain SEA must be sacrificed. This trend can

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International Journal of Mechanical Sciences 165 (2020) 105205

Table 9 Accuracy of the approximate models in optimization Ⅰ. Parameter R

2

RAAE

C1 T1 C1 T1

𝛼 = 𝛽 = 0°

𝛼 = 10°

𝛼 = 20°

𝛼 = 30°

𝛽 = 10°

𝛽 = 20°

𝛽 = 30°

0.9989 0.9987 0.0262 0.0271

0.9959 0.9243 0.0507 0.1618

0.9902 0.9968 0.0720 0.0401

0.9921 0.9968 0.0549 0.0435

0.9682 0.9733 0.1210 0.1189

0.9400 0.9544 0.1751 0.1460

0.9925 0.9902 0.0721 0.0737

Table 10 Geometric parameters and results of COPRAS of the selected points of C1. No.

1 (𝛼 = 𝛽 = 0°)

2 (𝛼 = 10°)

3 (𝛼 = 20°)

4 (𝛼 = 30°)

5 (𝛽 = 10°)

6 (𝛽 = 20°)

7 (𝛽 = 30°)

A (mm) t (mm) S+ SQi Ui Rank

1.87 0.75 0.0151 0.0130 0.0249 92.37% 4

2.01 0.70 0.0146 0.0118 0.0248 92.11% 5

1.85 0.57 0.0129 0.0084 0.0269 100.00% 1

2.05 0.70 0.0145 0.0119 0.0259 96.00% 2

2.47 0.74 0.0146 0.0131 0.0246 91.52% 6

1.16 0.67 0.0131 0.0102 0.0244 90.70% 7

1.69 0.69 0.0142 0.0113 0.0256 94.95% 3

Fig. 12. Pareto solutions of the selected CCST under different loading angles: (a) 𝛼 = 𝛽 = 0°, (b) 𝛼 = 10°, (c) 𝛼 = 20°, (d) 𝛼 = 30°, (e) 𝛽 = 10°, (f) 𝛽 = 20° and (g) 𝛽 = 30°

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 13. Pareto solutions of the selected CCST under different loading angles: (a) 𝛼 = 𝛽 = 0°, (b) 𝛼 = 10°, (c) 𝛼 = 20°, (d) 𝛼 = 30°, (e) 𝛽 = 10°, (f) 𝛽 = 20° and (g) 𝛽 = 30° Table 11 Geometric parameters and results of COPRAS of the selected points of T1. No.

1 (𝛼 = 𝛽 = 0°)

2 (𝛼 = 10°)

3 (𝛼 = 20°)

4 (𝛼 = 30°)

5 (𝛽 = 10°)

6 (𝛽 = 20°)

7 (𝛽 = 30°)

A (mm) t (mm) S+ SQi Ui Rank

2.19 0.68 0.0145 0.0119 0.0248 84.10% 3

2.45 0.68 0.0146 0.0121 0.0245 83.06% 5

2.90 0.62 0.0138 0.0107 0.0246 83.33% 4

2.24 0.68 0.0145 0.0119 0.0242 82.08% 6

2.38 0.70 0.0148 0.0126 0.0242 81.94% 7

2.03 0.50 0.0124 0.0070 0.0295 100.00% 1

2.05 0.72 0.0148 0.0128 0.0267 90.38% 2

also be found in Fig. 15(a) and (b). In addition, since the importance of different criteria was considered the same, the overall performance can only be improved by increasing CFE. As the optimal result came from the optimization in loading angle of 𝛼 = 20°, the CFE in 𝛼 = 20° (𝛽 = 20° for T1) was improved most, as shown in Fig. 15(c). In addition, as the importance of the crashworthiness performance of different load-

ing angles was considered the same, optimization Ⅰ will reduce the difference in the criterion of different loading angles, especially for Fmax and SEA. Therefore, the ranges of Fmax and SEA of C1 in optimization Ⅰ were 11.35 kN and 8.19 kJ/kg, respectively, which were substantially smaller than those in the initial design (26.11 kN and 11.53 kJ/kg, respectively). The ranges of Fmax and SEA of T1 were 8.54 kN and 7.65 kJ/kg, respec-

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International Journal of Mechanical Sciences 165 (2020) 105205

Fig. 14. Pareto solutions of optimization Ⅱ: (a) C1 and (b) T1.

Fig. 15. Comparison of the two optimization results of C1: (a) Fmax , (b) SEA and (c) CFE.

Fig. 16. Comparison of the two optimization results of T1: (a) Fmax , (b) SEA and (c) CFE.

Table 12 Accuracy of the approximate models in optimization Ⅱ.

Table 13 Geometric parameters of the selected optimal design in optimization Ⅱ.

Parameter

C1

T1

Parameter

C1

T1

R2 RAAE

0.9969 0.0372

0.9988 0.0226

A t

1.99 0.81

0.78

2.47

Z. Li, W. Ma and P. Xu et al.

tively, which were also much smaller than those in the initial design (24.04 kN and 13.39 kJ/kg, respectively). In optimization Ⅱ, the importance of the three criteria was considered in the form of weights. Because the weight of CFE was much smaller than that of Fmax and SEA, the main work of optimization Ⅱ is maximizing SEA and minimizing Fmax while having a relatively smaller effect on CFE. Figs. 15 and 16 showed that optimization Ⅱ greatly reduced the Fmax while ensuring that the SEA decreased slightly. In addition, the CFE of optimization Ⅱ changed smaller than the CFE of optimization Ⅰ in comparison with the initial design. Obviously, optimization Ⅰ balanced the crashworthiness performance at different loading angles but did not improve the performance. In contrast, optimization Ⅱ significantly improved the crashworthiness of the tubes. Therefore, the method adopted in optimization II was more reasonable. 6. Conclusion In this paper, the crashworthiness performance of a series of multicell CCSTs under multiple loading angles was studied numerically. Three criteria, namely, Fmax , SEA and CFE, were used to assess the performance of these tubes, and the optimal structures for the cosine and triangular CCSTs were selected with the COPRAS method. Then, the optimizations with two methods were conducted to find the best configurations of the selected structures. Some important conclusions were drawn and are listed hereafter: (1) Most of the considered multi-cell CCSTs deformed progressively when the loading was axial or 10°, whereas these CCSTs deformed unstably when the loading angles were 20° and 30° For each tube, the force level decreased with increasing loading angle, and the greatest CFE appeared when the loading angle was 10° (𝛼 = 10° or 𝛽 = 10°). (2) The optimal tubes selected with the COPRAS method were C1 and T1 for the cosine and triangular CCSTs, respectively. Both C1 and T1 had 4 cells, which was the minimum value in the considered CCSTs. This finding indicated that when considering multiple crashworthiness criteria and multiple loading angles, increasing the number of cells is not necessarily beneficial. (3) Optimization Ⅰ balanced the crashworthiness performance at different loading angles but did not improve the performance. In contrast, optimization Ⅱ significantly improved the crashworthiness of the tubes. For the multi-objective crashworthiness optimization of thin-walled tubes, optimization Ⅱ was more reasonable than optimization Ⅰ.

International Journal of Mechanical Sciences 165 (2020) 105205

Table A1 Initial decision matrix of the cosine CCSTs. Case

C1

C2

C3

C4

C5

C6

𝛼 = 𝛽 = 0° Fmax (kN) SEA (kJ/kg) CFE 𝛼 = 10° Fmax (kN) SEA (kJ/kg) CFE 𝛼 = 20° Fmax (kN) SEA (kJ/kg) CFE 𝛼 = 30° Fmax (kN) SEA (kJ/kg) CFE 𝛽 = 10° Fmax (kN) SEA (kJ/kg) CFE 𝛽 = 20° Fmax (kN) SEA (kJ/kg) CFE 𝛽 = 30° Fmax (kN) SEA (kJ/kg) CFE

35.44 16.62 0.63 22.10 13.26 0.80 16.19 7.08 0.58 9.33 5.09 0.73 23.08 13.78 0.80 15.68 7.02 0.60 10.72 5.33 0.66

44.46 20.39 0.70 31.64 16.78 0.81 20.29 7.30 0.55 12.90 5.72 0.68 30.93 16.99 0.84 21.39 8.76 0.63 14.23 6.37 0.68

58.40 21.57 0.72 41.48 17.83 0.84 26.25 7.16 0.53 16.28 5.48 0.65 44.27 18.15 0.80 26.43 8.03 0.59 17.08 5.87 0.67

36.01 15.52 0.61 20.75 10.14 0.70 15.80 6.91 0.62 8.62 4.53 0.75 20.04 7.67 0.55 12.87 5.04 0.56 10.08 4.19 0.59

43.90 19.97 0.71 30.82 16.12 0.81 20.49 7.81 0.59 14.33 5.89 0.64 31.68 16.77 0.82 20.01 7.78 0.60 13.18 5.52 0.65

60.63 21.77 0.74 42.45 17.69 0.86 27.92 7.60 0.56 17.27 5.64 0.67 46.51 17.49 0.77 24.79 7.46 0.62 15.92 5.09 0.66

Table A2 Initial decision matrix of the triangular CCSTs. Case

T1

T2

T3

T4

T5

T6

𝛼 = 𝛽 = 0° Fmax (kN) SEA (kJ/kg) CFE 𝛼 = 10° Fmax (kN) SEA (kJ/kg) CFE 𝛼 = 20° Fmax (kN) SEA (kJ/kg) CFE 𝛼 = 30° Fmax (kN) SEA (kJ/kg) CFE 𝛽 = 10° Fmax (kN) SEA (kJ/kg) CFE 𝛽 = 20° Fmax (kN) SEA (kJ/kg) CFE 𝛽 = 30° Fmax (kN) SEA (kJ/kg) CFE

33.52 18.23 0.73 24.52 14.57 0.78 14.07 7.05 0.66 9.46 4.84 0.67 23.43 14.82 0.83 15.45 7.53 0.64 10.55 5.41 0.67

43.69 21.60 0.75 35.22 17.80 0.76 20.20 7.75 0.58 12.83 5.84 0.69 32.82 18.18 0.83 21.06 9.03 0.65 14.05 6.47 0.69

57.80 22.54 0.75 44.56 18.94 0.82 26.07 7.53 0.55 16.64 5.63 0.65 44.31 19.40 0.84 28.05 8.40 0.58 17.55 6.05 0.66

33.82 13.97 0.58 17.72 9.69 0.77 17.41 7.20 0.58 8.55 4.65 0.76 19.21 8.37 0.61 12.90 4.87 0.53 9.62 4.01 0.59

45.23 20.85 0.71 32.22 17.28 0.82 20.73 8.09 0.60 13.92 6.16 0.68 33.35 17.81 0.82 21.47 8.13 0.58 13.62 5.91 0.66

59.00 21.48 0.74 44.73 15.19 0.69 28.06 7.67 0.56 17.68 5.77 0.66 44.62 16.81 0.77 26.58 7.64 0.58 16.85 5.37 0.65

Table A3 Normalized decision matrix of the cosine CCSTs.

Declaration of Competing Interest We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgements The present work was supported by the National Key Research Project of China (grant number 2016YFB1200404-04), the Fundamental Research Funds for the Central Universities of Central South University (grant number 2019zzts269), the National Natural Science Foundation of China (grant number 51675537), National Key Research Project of China and the Project of the State Key Laboratory of High-Performance Complex Manufacturing (grant number ZZYJKT2018-09). Appendix 1 Table A1–Table A6.

Case

C1

C2

C3

C4

C5

C6

𝛼 = 𝛽 = 0° Fmax SEA CFE 𝛼 = 10° Fmax SEA CFE 𝛼 = 20° Fmax SEA CFE 𝛼 = 30° Fmax SEA CFE 𝛽 = 10° Fmax SEA CFE 𝛽 = 20° Fmax SEA CFE 𝛽 = 30° Fmax SEA CFE

0.1271 0.1435 0.1533 0.1168 0.1444 0.1660 0.1275 0.1614 0.1691 0.1185 0.1573 0.1772 0.1174 0.1517 0.1747 0.1294 0.1592 0.1667 0.1320 0.1647 0.1688

0.1594 0.1760 0.1703 0.1672 0.1827 0.1680 0.1598 0.1664 0.1603 0.1639 0.1768 0.1650 0.1574 0.1870 0.1834 0.1765 0.1987 0.1750 0.1752 0.1968 0.1739

0.2094 0.1862 0.1752 0.2192 0.1942 0.1743 0.2068 0.1632 0.1545 0.2068 0.1694 0.1578 0.2253 0.1998 0.1747 0.2181 0.1821 0.1639 0.2103 0.1813 0.1714

0.1291 0.1340 0.1484 0.1096 0.1104 0.1452 0.1245 0.1575 0.1808 0.1095 0.1400 0.1820 0.1020 0.0844 0.1201 0.1062 0.1143 0.1556 0.1241 0.1294 0.1509

0.1574 0.1724 0.1727 0.1629 0.1756 0.1680 0.1614 0.1781 0.1720 0.1820 0.1821 0.1553 0.1612 0.1846 0.1790 0.1651 0.1765 0.1667 0.1623 0.1705 0.1662

0.2174 0.1879 0.1800 0.2243 0.1927 0.1784 0.2199 0.1733 0.1633 0.2194 0.1743 0.1626 0.2367 0.1925 0.1681 0.2046 0.1692 0.1722 0.1960 0.1572 0.1688

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International Journal of Mechanical Sciences 165 (2020) 105205

References

Table A4 Normalized decision matrix of the triangular CCSTs. Case

T1

T2

T3

T4

T5

T6

𝛼 = 𝛽 = 0° Fmax SEA CFE 𝛼 = 10° Fmax SEA CFE 𝛼 = 20° Fmax SEA CFE 𝛼 = 30° Fmax SEA CFE 𝛽 = 10° Fmax SEA CFE 𝛽 = 20° Fmax SEA CFE 𝛽 = 30° Fmax SEA CFE

0.1228 0.1536 0.1714 0.1232 0.1559 0.1681 0.1112 0.1557 0.1870 0.1196 0.1472 0.1630 0.1185 0.1554 0.1766 0.1231 0.1651 0.1798 0.1283 0.1629 0.1709

0.1600 0.1820 0.1761 0.1770 0.1904 0.1638 0.1596 0.1711 0.1643 0.1622 0.1776 0.1679 0.1660 0.1906 0.1766 0.1678 0.1980 0.1826 0.1708 0.1948 0.1760

0.2117 0.1899 0.1761 0.2240 0.2026 0.1767 0.2060 0.1663 0.1558 0.2104 0.1712 0.1582 0.2241 0.2034 0.1787 0.2235 0.1842 0.1629 0.2134 0.1821 0.1684

0.1239 0.1177 0.1362 0.0891 0.1037 0.1659 0.1376 0.1590 0.1643 0.1081 0.1414 0.1849 0.0971 0.0877 0.1298 0.1028 0.1068 0.1489 0.1170 0.1207 0.1505

0.1656 0.1757 0.1667 0.1619 0.1849 0.1767 0.1638 0.1786 0.1700 0.1760 0.1873 0.1655 0.1687 0.1867 0.1745 0.1711 0.1783 0.1629 0.1656 0.1779 0.1684

0.2161 0.1810 0.1737 0.2248 0.1625 0.1487 0.2217 0.1694 0.1586 0.2236 0.1754 0.1606 0.2256 0.1762 0.1638 0.2118 0.1675 0.1629 0.2049 0.1616 0.1658

Table A5 Weighted normalized decision matrix of the cosine CCSTs. Section

C1

C2

C3

C4

C5

C6

𝛼 = 𝛽 = 0° Fmax SEA CFE 𝛼 = 10° Fmax SEA CFE 𝛼 = 20° Fmax SEA CFE 𝛼 = 30° Fmax SEA CFE 𝛽 = 10° Fmax SEA CFE 𝛽 = 20° Fmax SEA CFE 𝛽 = 30° Fmax SEA CFE

0.0212 0.0239 0.0102 0.0073 0.009 0.0042 0.0053 0.0067 0.0028 0.0025 0.0033 0.0015 0.0073 0.0095 0.0044 0.0054 0.0066 0.0042 0.0027 0.0034 0.0014

0.0266 0.0293 0.0114 0.0105 0.0114 0.0042 0.0067 0.0069 0.0027 0.0034 0.0037 0.0014 0.0098 0.0117 0.0046 0.0074 0.0083 0.0044 0.0036 0.0041 0.0014

0.0349 0.031 0.0117 0.0137 0.0121 0.0044 0.0086 0.0068 0.0026 0.0043 0.0035 0.0013 0.0141 0.0125 0.0044 0.0091 0.0076 0.0041 0.0044 0.0038 0.0014

0.0215 0.0223 0.0099 0.0069 0.0069 0.0036 0.0052 0.0066 0.003 0.0023 0.0029 0.0015 0.0064 0.0053 0.003 0.0044 0.0048 0.0039 0.0026 0.0027 0.0013

0.0262 0.0287 0.0115 0.0102 0.011 0.0042 0.0067 0.0074 0.0029 0.0038 0.0038 0.0013 0.0101 0.0115 0.0045 0.0069 0.0074 0.0042 0.0034 0.0035 0.0014

0.0362 0.0313 0.012 0.014 0.012 0.0045 0.0092 0.0072 0.0027 0.0046 0.0036 0.0013 0.0148 0.012 0.0042 0.0085 0.0071 0.0043 0.0041 0.0033 0.0014

Table A6 Weighted normalized decision matrix of the triangular CCSTs. Section

T1

T2

T3

T4

T5

T6

𝛼 = 𝛽 = 0° Fmax SEA CFE 𝛼 = 10° Fmax SEA CFE 𝛼 = 20° Fmax SEA CFE 𝛼 = 30° Fmax SEA CFE 𝛽 = 10° Fmax SEA CFE 𝛽 = 20° Fmax SEA CFE 𝛽 = 30° Fmax SEA CFE

0.0205 0.0256 0.0114 0.0077 0.0097 0.0042 0.0046 0.0065 0.0031 0.0025 0.0031 0.0014 0.0074 0.0097 0.0044 0.0051 0.0069 0.0030 0.0027 0.0034 0.0014

0.0267 0.0303 0.0117 0.0111 0.0119 0.0041 0.0067 0.0071 0.0027 0.0034 0.0037 0.0014 0.0104 0.0119 0.0044 0.0070 0.0083 0.0030 0.0036 0.0041 0.0015

0.0353 0.0317 0.0117 0.0140 0.0127 0.0044 0.0086 0.0069 0.0026 0.0044 0.0036 0.0013 0.0140 0.0127 0.0045 0.0093 0.0077 0.0027 0.0044 0.0038 0.0014

0.0207 0.0196 0.0091 0.0056 0.0065 0.0041 0.0057 0.0066 0.0027 0.0022 0.0029 0.0015 0.0061 0.0055 0.0032 0.0043 0.0045 0.0025 0.0024 0.0025 0.0012

0.0276 0.0293 0.0111 0.0101 0.0116 0.0044 0.0068 0.0074 0.0028 0.0037 0.0039 0.0014 0.0105 0.0117 0.0044 0.0071 0.0074 0.0027 0.0034 0.0037 0.0014

0.0360 0.0302 0.0116 0.0141 0.0102 0.0037 0.0092 0.0071 0.0026 0.0047 0.0036 0.0013 0.0141 0.0110 0.0041 0.0088 0.0070 0.0027 0.0043 0.0034 0.0014

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