# On the Folding Mechanics of Square Columns with Double-Surfaced Gradients

Authors: Junxian Zhou; Ruixian Qin; Bingzhi Chen
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The implementation of graded thicknesses in structural designs is an efficient strategy for improving the energy absorption performance of thin-walled columns. In this study, the mechanical responses of square columns with double-surfaced gradients under axial crushing forces were analytically and numerically investigated. A mathematical expression for the mean crushing force on a novel tube design is derived according to the super folding element method by introducing two novel parameters: the average thickness of the efficient energy absorbing region (EEAR) and global average thickness. The analytical predictions show good agreement with the simulation results that were obtained using the nonlinear finite element program ANSYS/LS-DYNA. The results demonstrate that a greater coefficient of average thickness of the EEAR versus global average thickness improves the energy absorption efficiency. However, increasing this coefficient largely depends on the sectional material distribution law. To explore the effects of this law on crushing responses, three different material distribution types were investigated. A significant improvement in specific energy absorption (approximately 30%) was obtained for a double-surfaced column with an optimized material distribution type compared to that of a square column with a uniform thickness.


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## Body

## 1. Introduction

Based on a greater emphasis on the passive safety of transportation, which includes the security of passengers and vital equipment, the performance requirements of energy absorption devices have increased significantly. An energy absorption device typically consists of a thin-walled structure with a steady and foreseeable collapse mode, high manufacturability, and moderate cost [1, 2].Additionally, the implementation of lightweight designs is important for energy conservation. Therefore, novel structures with high specific energy absorption (SEA) are required. The energy absorption of a thin-walled tube is closely related to its section configuration and crushing mode. Accordingly, research on the crushing behaviors of metal tubes with various cross-sectional patterns has received significant attention. Alexander [3] utilized experimental data to develop an approximate mathematical model of the concertina crushing mode to estimate the mechanical responses of cylindrical tubes subjected to quasi-static axial loads. However, depending on the diameter-to-wall-thickness ratio (D/t) and length-to-diameter ratio (L/D), a cylindrical column may be crushed not only in the concertina mode, but also in diamond and mixed modes [4], which were not considered in the theoretical model [3]. Many researchers have investigated the concertina and diamond modes [4–7] and developed improved theoretical models for predicting the folding patterns and mean crushing loads of cylindrical tubes.Similarly, crushing modes of square tubes are closely related to the cross-sectional width-to-wall-thickness ratio (B/t) and length-to-cross-sectional-width ratio (L/B); such relationships are summarized in a graph proposed by Abramowicz and Jones [8], which was constructed based on several experiments. Through theoretical research, Wierzbicki and Abramowicz [9] modeled the basic folding mechanism of a square tube with a quasi-inextensional collapse mode and referred to their findings as the super folding element theory. In this theory, a quasi-inextensional basic folding element was established. This element consists of trapezoidal, horizontal cylindrical, conical, and toroidal surfaces. Three energy-dissipating approaches—folding along stationary plastic hinge lines, rolling at inclined plastic hinge lines, and quasi-inextensional deformation in the toroidal area—were considered for the element. The inclined hinge lines and toroidal surface were essentially confined to the corner regions. Therefore, it is believed that the corner regions contribute significantly to energy absorption.Consequently, a strategy that increases the number of corners (polygonal columns) [10–13] has been adopted to improve the energy absorption efficiency. Yamashita et al. [11] found that the buckling mode and dynamic performance of polygonal tubes are affected by the angles between neighboring flanges. Additionally, angles in the range of 90–120° are the most reasonable angles for achieving the highest mean load. However, designs based on polygonal cross sections have a finite amount of energy absorption improvement because the additional side walls lead to corner angles exceeding the reasonable range (90–120°).The problem of increasing the number of corners while maintaining corner angles in the range of 90–120° can be solved through the use of tubes with novel cross sections, known as a non-convex multi-corner (NCMC) columns, which were developed by Tang and Liu et al. [14, 15]. Compared to traditional rectangular structures, a significant improvement in SEA can be obtained by using these novel columns. Similarly, an improvement in SEA can also be achieved by using another multi-corner structure, known as a star-shaped column, which was proposed by Fan et al. [16, 17].Based on previous studies [14–17], a potential deficiency of NCMC and star-shaped tubes has been identified. Specifically, the crushing mode will deform into a noncompact mode as the tubes become thicker. Inspired by the structure of bamboo, Liu et al. [15] inserted perforated plates into NCMC tubes to counteract this deficiency. Additionally, Liu et al. [17] placed two polygonal shells on the inside and outside of a star-shaped column to avoid irregular folding patterns.More recently, some researchers put their attention on graded structures. Baykasoglu and Cetin studied crashworthiness of circular tubes with gradient thicknesses [18]. The crushing resistances of circular tubes with axially graded thickness were then explored by Xu et al. [19]. Graded thicknesses in the longitudinal direction were also extended to square [20] and multicell columns [21]; and corresponding performance optimization was presented. In addition, Zhang et al. investigated crushing resistances of non-uniform-thickness conical tubes [22] and bending collapse behaviors of square tubes with variable thickness [23]; they also summarized state of the art of energy absorption of graded structures and materials [24]. These studies have proved that it is a significant approach to improve the energy absorption performance by changing the material distribution of thin-walled structures. In addition, with advancements in material rolling technology, metal plates with graded thickness could be manufactured without difficulty or high costs. Accordingly, such tubes can be manufactured using tailor-rolled-blank technology (TRB), which is also used to produce lightweight vehicle components efficiently [25, 26].Similarly, Zhang et al. [27] improved the energy absorption performance by optimizing the cross-sectional material distribution of square tubes. This approach is similar to the strategy of increasing the number of corners. It considers the plastic energy to be dissipated centrally in the corner regions, as reported in [9]. Therefore, this approach concentrates more material in the corner regions and improves the energy absorption performance of square tubes.Research [27] shows two material distribution methods for the cross sections of square tubes, namely, the single surface gradient (SSG) and double-surfaced gradient (DSG) methods; collapse modes and crashworthiness were experimentally and numerically investigated under axial loading. Compared to a traditional square tube, a maximum improvement of 27% in terms of specific energy absorption was obtained by the DSG square tube.However, the energy absorption mechanism of the DSG tube is still unknown and the influence of cross-sectional material distribution on energy absorption is unclear. In this study, the crushing mechanism of a DSG square tube was theoretically analyzed based on the super folding element method to predict energy dissipation of the tube. A mathematical model was developed for the mean crushing force. Additionally, simulations were conducted to validate the accuracy of the theoretical predictions and investigate the influence of key variables of the mathematical model on the mechanical responses using the nonlinear finite element software ANSYS/LS-DYNA. Finally, the law of cross-sectional material distribution, which can effectively improve energy absorption, was investigated.


## 2. Material Distribution Types

Zhang et al. [27] proposed two material distribution types for square tubes, namely, the SSG and DSG, as shown in Figure 1. This figure illustrates that the cross-sectional shapes are defined by three parameters: the thickness of the middle regions (t1), corner thickness (t2), and cross-sectional width (B). The authors tested numerous specimens under quasi-static axial impact conditions using two types of gradient thicknesses. These tests were conducted to investigate the effects of the two types (SSG and DSG) on folding patterns and crashworthiness performance. According to [27], despite maintaining consistent values for all the specimens (B was maintained at 36 mm,t1ranged from 0.6 to 1.2, andt2ranged from 1.2 to 1.8 for all tubes), there were still significant differences between the mechanical responses of the SSG and DSG tubes.Figure 1
Material distributions in the various types of square columns.First, the folding wavelength and lobes of the SSG square tubes gradually increased with a thickening of the corner regions. Consequently, with the maximum thickness gradient, irregular deformations occurred in the thinnest regions of the SSG square column. For the DSG square columns, amplification of the folding lobes and folding wavelength was not noticeable and irregular deformation did not occur, even with the same thickness gradient as that of the SSG tube mentioned previously. Second, based on the longer wavelength, the force-displacement curves of the SSG square columns fluctuated significantly more than those of the DSG square columns and the distances between neighboring peaks in the curves were much greater than those of the DSG structures. Finally, compared to a traditional square column, the mean crushing load of the DSG square column with the maximum graded thickness increased by 35.5%, which is greater than the increase provided by the SSG square tube with the same graded thickness.To sum up, this study focused on the theoretical analysis of the folding mechanism and energy absorption enhancement of DSG tubes.


## 3. Theoretical Analysis of a Square Column with a DSG

An analytical solution for the mechanical responses of square tubes under axial progressive loading was derived by Wierzbicki and Abramowicz through the introduction of the coefficient of effective crushing distance [9].According to their study, a modified expression for the mean static impact forcePm can be defined as shown in (1)Pm=13.06σ0B1/3t5/3,where σ0 is the equivalent flow stress and B and t are the sectional width and wall thickness, respectively.The expression for the half folding wavelengthH is (2)H=1.27B2/3t1/3.Considering strain hardening effects, the equivalent material flow stress with power law hardening behavior can be calculated as follows [28]:(3)σ0=σyσu1+n,where σy denotes the initial yield strength and σu andnare the ultimate strength and power law exponent, respectively.


### 3.1. Basic Collapse Elements

As shown in Figure2, two basic folding elements for the theoretical analysis of rectangular tube crushing responses are considered in the super folding element method: Type I, which was studied by Wierzbicki and Abramowicz [9], and Type II, which was investigated in [29–31]. In these studies, the collapse mechanisms of the two basic folding elements were accurately described using a rigorous kinematic approach. Additionally, the energy dissipation of the two elements was precisely analyzed using four energy dissipating contributors: the horizontal plastic hinge line, inclined plastic hinge line, toroidal surface, and conical surface. Therefore, these two basic folding elements can be used to predict the impact responses of columns with different cross-sectional shapes [9, 14, 15, 32, 33].Figure 2
Simplified collapse mechanisms of the two folding elements: (a) Type I and (b) Type II; and (c) deformation areas of the Type I element.
(a) (b) (c)The energy dissipation of Type I, which is illustrated in Figure2(a), can be estimated using the following equation [9].(4)Etype1=M016I1Hbt+2πC+4I3H2b,where I1 and I3 are 0.58 and 1.11 (for square tubes), respectively,Cdenotes the flange length of a corner, andb denotes the smaller radius of the toroidal surface [9]. Similarly, the energy absorption formula for the Type II folding element, which is illustrated in Figure 2(b), was deduced by Abramowicz and John [29] and defined as follows:(5)Etype2=M08H2t+2πC+4H.This energy dissipation expression was modified by introducing an improved deformation mechanism [30] and rewritten as follows: (6)Etype2=M02πH2t+2πC+πH.


### 3.2. Energy Dissipation of Type I Folding Element with a DSG

This study focused on the quasi-inextensional mode, which consists of four Type I elements subjected to a single wavelength crushing. In accordance with the super folding element theory [9], the collapse mechanism of the Type I folding element is illustrated in Figure 2(c). In a complete collapse, energy dissipation is derived from three main contributors [9]: the horizontal plastic hinge lines (see Figure 2(c), area 2), toroidal surface (see Figure 2(c), area 1), and inclined plastic hinge lines (see Figure 2(c), area 3).Considering a Type I folding element with a uniform thickness, the dissipated energy at the horizontal plastic hinge lines can be expressed as follows [9]:(7)E2=2πM0C,where M0 is the plastic limit moment (M0=1/4σ0t2) andC represents the flange length around the corner. Therefore, obtaining an accurate expression for the plastic limit moment of a DSG Type I element with varying thickness is relatively complicated.In this study, the bending energy dissipated by a DSG plate with a nonuniform thickness was approximated using an integration method. This ensured that the theoretical analysis was efficient and the final mean load expression had a concise form. Subsequently, as shown in Figure3, a two-dimensional coordinate system was established on a DSG plate with a flange length ofC/2. The vertical y-axis represents the thickness and the horizontal x-axis represents the flange length. As illustrated in Figure 3, the origin of the coordinate system is located at the end of the plate midline.t(x) is a continuous thickness function over the length interval [0, C/2] and can be fitted according to the geometric configuration of the DSG plate. Accordingly, the bending energy of an infinitesimal plate with a width ofdx can be expressed as follows:(8)dE=12σ0πt2ξdx,where ξ is an arbitrary point within the interval [x, x+dx]. Furthermore, the bending energy of a DSG folding element with a flange length ofC can be calculated as follows: (9)E2DSG=2∫0C/212σ0π2tx2dx=2Cσ0πt2ζ,where ζ denotes a point within the interval [0, C/2]. To facilitate theoretical analysis, a new parameter, namely, the global average thickness (ta), was introduced. This parameter can be expressed as follows: (10)ta≈2tζ.Using (10), (9) can then be simplified to produce (11) as follows:(11)E2DSG≈12σ0πCta2.Based on the uncomplicated nature of the thickness functions for DSG square tubes, the parameter ta can be calculated easily.Figure 3
Cross section of a DSG square tube.It is worth noting that when the tube length and width are constant, the change in the average thickness determines the mass of the tube.The remaining energy absorbed by the Type I folding element stems from two efficient contributors [9]: the inclined plastic hinge lines and toroidal surface. The region participating in the energy absorption by these contributors (i.e., the efficient energy absorbing region (EEAR)) is largely focused near the corners, as shown by the blue triangles in Figures 4(a) and 4(b). The energy dissipated by the EEAR in a Type I folding element with constant thickness can be obtained by integrating the two contributors as follows [9]: (12)E1+E3=M016HbtI1+4H2bI3.Figure 4
(a) Geometry of the Type I basic folding element, (b) triangular simplified regions representing the EEAR, and (c) cross section of the Type I element with a DSG.
(a) (b) (c)For the DSG Type I element, deriving a rigorous expression for the energy dissipated by the two contributors is relatively difficult. Therefore, the method used for handling the horizontal plastic hinge lines in the DSG section was also adopted for this purpose. The energy contribution of the inclined plastic hinge lines and toroidal surface in the DSG Type I element can be calculated using the following equations.(13)E1DSG=∫0xe4σ02tx2Hb/2txI1dxxe=4σ02tη2Hb2tηI1≈4σ0te2HbteI1,(14)E3DSG=∫0xeσ02tx2H2/bI3dxxe=σ02tη2H2bI3≈σ0te2H2bI3,(15)2tη≈te,where xe denotes the width of the EEAR, as shown in Figures 4(a)–4(c),t(x) is the thickness function, and η denotes a point within the interval [0, xe]. The parameter te is the average thickness of the EEAR, which can be calculated as follows:(16)te=2∫0xetxdxxe.Therefore, the primary challenge in solving forte is determining the width of the EEAR (i.e., xe, where xe is equal to the base-side length of the blue triangles shown in Figure 4(a)). In this figure, the initial geometry of the folding element is defined by the height2H, total widthC, and angle π-2φ0 between two neighboring plates. The geometry in the current configuration can be described by the angle of rotation of the side panels α or the crushing distance 2Hδ, where δ denotes the crushing distance coefficient. The other angle γ is related to φ0 and α as follows [9]: (17)γ=tan-1⁡tan⁡φ0sin⁡α,φ0=π4.Based on the geometric characteristics of the Type I folding element, the parameterxe can be calculated using the following equations.(18)α=cos-1⁡H1-δH=cos-1⁡1-δ,(19)xe=sin⁡αH=εH;ε=1-1-δ2,where ε is the width coefficient of the EEAR. Therefore, for a Type I element that experiences complete compression with a crushing distance coefficient of δ=1, xe=H. However, based on the investigation conducted by Wierzbicki and Abramowicz [9], the actual crushing distance coefficient δ is approximately 0.7–0.75. Therefore, theoretically, the width coefficient of the EEAR ε is approximately 0.95–0.97. In the following analysis, the two coefficients δ and ε were set to 0.70 and 0.95, respectively.


### 3.3. Mean Crushing Force

As mentioned above, the energy dissipation of a square column stems from three main contributors (horizontal plastic hinge lines, inclined plastic hinge lines, and the toroidal surface). This assumption is also appropriate for the DSG square column. Therefore, equating the external work to the internal energy absorption of the DSG structure in one wavelength period yields the following equation.(20)2HδPm=4E1DSG+E2DSG+E3DSG=14σ064I1te2bHte+16I3te2H2b+8ta2πC.Considering thatLC is the section perimeter, (20) can be simplified to produce (21) as follows:(21)Pm1/4σ0=1δte232I1bte+8I3Hb+ta2πLCH.Minimizing (21) with respect to the unknown parametersH andb yields the following equations.(22)∂Pm∂H=0:8I3te31b/te=πta2LCH/te2(23)∂Pm∂b=0:4I1I3bte2=HteThe two significant parametersb andH can be obtained by solving (22) and (23) as follows:(24)b=Ateta2LCte31/3,(25)H=Bteta2LCte32/3,where(26)A=πI3128I121/3,(27)B=4I1I3πI3128I122/3.Through the substitution of (24) and (25) into (21), the mean crushing force can be obtained as follows:(28)Pm1/4σ0=1δ32AI1te2ta2LCte31/3+πLCta2Btete3ta2LC2/3+8I3Bte2Ata2LCte31/3.Equation (28) can then be simplified to produce the following equation.(29)Pm=1δPsσ0teta2LC1/3,where (30)Ps=24πI1I31281/3.For square tubes withI1=0.58 and I3=1.11[9], the parameter Ps is equal to 6.021.


## 3.1. Basic Collapse Elements

As shown in Figure2, two basic folding elements for the theoretical analysis of rectangular tube crushing responses are considered in the super folding element method: Type I, which was studied by Wierzbicki and Abramowicz [9], and Type II, which was investigated in [29–31]. In these studies, the collapse mechanisms of the two basic folding elements were accurately described using a rigorous kinematic approach. Additionally, the energy dissipation of the two elements was precisely analyzed using four energy dissipating contributors: the horizontal plastic hinge line, inclined plastic hinge line, toroidal surface, and conical surface. Therefore, these two basic folding elements can be used to predict the impact responses of columns with different cross-sectional shapes [9, 14, 15, 32, 33].Figure 2
Simplified collapse mechanisms of the two folding elements: (a) Type I and (b) Type II; and (c) deformation areas of the Type I element.
(a) (b) (c)The energy dissipation of Type I, which is illustrated in Figure2(a), can be estimated using the following equation [9].(4)Etype1=M016I1Hbt+2πC+4I3H2b,where I1 and I3 are 0.58 and 1.11 (for square tubes), respectively,Cdenotes the flange length of a corner, andb denotes the smaller radius of the toroidal surface [9]. Similarly, the energy absorption formula for the Type II folding element, which is illustrated in Figure 2(b), was deduced by Abramowicz and John [29] and defined as follows:(5)Etype2=M08H2t+2πC+4H.This energy dissipation expression was modified by introducing an improved deformation mechanism [30] and rewritten as follows: (6)Etype2=M02πH2t+2πC+πH.


## 3.2. Energy Dissipation of Type I Folding Element with a DSG

This study focused on the quasi-inextensional mode, which consists of four Type I elements subjected to a single wavelength crushing. In accordance with the super folding element theory [9], the collapse mechanism of the Type I folding element is illustrated in Figure 2(c). In a complete collapse, energy dissipation is derived from three main contributors [9]: the horizontal plastic hinge lines (see Figure 2(c), area 2), toroidal surface (see Figure 2(c), area 1), and inclined plastic hinge lines (see Figure 2(c), area 3).Considering a Type I folding element with a uniform thickness, the dissipated energy at the horizontal plastic hinge lines can be expressed as follows [9]:(7)E2=2πM0C,where M0 is the plastic limit moment (M0=1/4σ0t2) andC represents the flange length around the corner. Therefore, obtaining an accurate expression for the plastic limit moment of a DSG Type I element with varying thickness is relatively complicated.In this study, the bending energy dissipated by a DSG plate with a nonuniform thickness was approximated using an integration method. This ensured that the theoretical analysis was efficient and the final mean load expression had a concise form. Subsequently, as shown in Figure3, a two-dimensional coordinate system was established on a DSG plate with a flange length ofC/2. The vertical y-axis represents the thickness and the horizontal x-axis represents the flange length. As illustrated in Figure 3, the origin of the coordinate system is located at the end of the plate midline.t(x) is a continuous thickness function over the length interval [0, C/2] and can be fitted according to the geometric configuration of the DSG plate. Accordingly, the bending energy of an infinitesimal plate with a width ofdx can be expressed as follows:(8)dE=12σ0πt2ξdx,where ξ is an arbitrary point within the interval [x, x+dx]. Furthermore, the bending energy of a DSG folding element with a flange length ofC can be calculated as follows: (9)E2DSG=2∫0C/212σ0π2tx2dx=2Cσ0πt2ζ,where ζ denotes a point within the interval [0, C/2]. To facilitate theoretical analysis, a new parameter, namely, the global average thickness (ta), was introduced. This parameter can be expressed as follows: (10)ta≈2tζ.Using (10), (9) can then be simplified to produce (11) as follows:(11)E2DSG≈12σ0πCta2.Based on the uncomplicated nature of the thickness functions for DSG square tubes, the parameter ta can be calculated easily.Figure 3
Cross section of a DSG square tube.It is worth noting that when the tube length and width are constant, the change in the average thickness determines the mass of the tube.The remaining energy absorbed by the Type I folding element stems from two efficient contributors [9]: the inclined plastic hinge lines and toroidal surface. The region participating in the energy absorption by these contributors (i.e., the efficient energy absorbing region (EEAR)) is largely focused near the corners, as shown by the blue triangles in Figures 4(a) and 4(b). The energy dissipated by the EEAR in a Type I folding element with constant thickness can be obtained by integrating the two contributors as follows [9]: (12)E1+E3=M016HbtI1+4H2bI3.Figure 4
(a) Geometry of the Type I basic folding element, (b) triangular simplified regions representing the EEAR, and (c) cross section of the Type I element with a DSG.
(a) (b) (c)For the DSG Type I element, deriving a rigorous expression for the energy dissipated by the two contributors is relatively difficult. Therefore, the method used for handling the horizontal plastic hinge lines in the DSG section was also adopted for this purpose. The energy contribution of the inclined plastic hinge lines and toroidal surface in the DSG Type I element can be calculated using the following equations.(13)E1DSG=∫0xe4σ02tx2Hb/2txI1dxxe=4σ02tη2Hb2tηI1≈4σ0te2HbteI1,(14)E3DSG=∫0xeσ02tx2H2/bI3dxxe=σ02tη2H2bI3≈σ0te2H2bI3,(15)2tη≈te,where xe denotes the width of the EEAR, as shown in Figures 4(a)–4(c),t(x) is the thickness function, and η denotes a point within the interval [0, xe]. The parameter te is the average thickness of the EEAR, which can be calculated as follows:(16)te=2∫0xetxdxxe.Therefore, the primary challenge in solving forte is determining the width of the EEAR (i.e., xe, where xe is equal to the base-side length of the blue triangles shown in Figure 4(a)). In this figure, the initial geometry of the folding element is defined by the height2H, total widthC, and angle π-2φ0 between two neighboring plates. The geometry in the current configuration can be described by the angle of rotation of the side panels α or the crushing distance 2Hδ, where δ denotes the crushing distance coefficient. The other angle γ is related to φ0 and α as follows [9]: (17)γ=tan-1⁡tan⁡φ0sin⁡α,φ0=π4.Based on the geometric characteristics of the Type I folding element, the parameterxe can be calculated using the following equations.(18)α=cos-1⁡H1-δH=cos-1⁡1-δ,(19)xe=sin⁡αH=εH;ε=1-1-δ2,where ε is the width coefficient of the EEAR. Therefore, for a Type I element that experiences complete compression with a crushing distance coefficient of δ=1, xe=H. However, based on the investigation conducted by Wierzbicki and Abramowicz [9], the actual crushing distance coefficient δ is approximately 0.7–0.75. Therefore, theoretically, the width coefficient of the EEAR ε is approximately 0.95–0.97. In the following analysis, the two coefficients δ and ε were set to 0.70 and 0.95, respectively.


## 3.3. Mean Crushing Force

As mentioned above, the energy dissipation of a square column stems from three main contributors (horizontal plastic hinge lines, inclined plastic hinge lines, and the toroidal surface). This assumption is also appropriate for the DSG square column. Therefore, equating the external work to the internal energy absorption of the DSG structure in one wavelength period yields the following equation.(20)2HδPm=4E1DSG+E2DSG+E3DSG=14σ064I1te2bHte+16I3te2H2b+8ta2πC.Considering thatLC is the section perimeter, (20) can be simplified to produce (21) as follows:(21)Pm1/4σ0=1δte232I1bte+8I3Hb+ta2πLCH.Minimizing (21) with respect to the unknown parametersH andb yields the following equations.(22)∂Pm∂H=0:8I3te31b/te=πta2LCH/te2(23)∂Pm∂b=0:4I1I3bte2=HteThe two significant parametersb andH can be obtained by solving (22) and (23) as follows:(24)b=Ateta2LCte31/3,(25)H=Bteta2LCte32/3,where(26)A=πI3128I121/3,(27)B=4I1I3πI3128I122/3.Through the substitution of (24) and (25) into (21), the mean crushing force can be obtained as follows:(28)Pm1/4σ0=1δ32AI1te2ta2LCte31/3+πLCta2Btete3ta2LC2/3+8I3Bte2Ata2LCte31/3.Equation (28) can then be simplified to produce the following equation.(29)Pm=1δPsσ0teta2LC1/3,where (30)Ps=24πI1I31281/3.For square tubes withI1=0.58 and I3=1.11[9], the parameter Ps is equal to 6.021.


## 4. Numerical Simulations

The columns considered in this study were DSG square tubes and numerical simulations were conducted for the columns using the nonlinear finite element software ANSYS/LS-DYNA to simulate crushing. The boundary conditions for the numerical simulations are illustrated in Figure5. As shown in this figure, the bottom of the tube was clamped, and the top was crushed by a massive rigid wall with a constant velocity of 10 m/s in the axial direction. Two contacts were considered in the model. Automatic single surface contact was introduced to simulate the self-contact of the column and automatic node-to-surface contact was applied between the column and rigid wall. Static and dynamic friction were included in these contacts with the same coefficient values of 0.2. Identical materials and geometry dimensions with a lengthL =320 mm and cross-sectional widthB= 90 mm were used for all finite element models. To ensure a reasonable deformation mode and lower initial peak force, triggers with a depth of 0.6 mm were installed on the two parallel plates at a distanceH below the crushed end of the columns, as illustrated in Figure 6.Figure 5
Boundary conditions for the specimens in the numerical simulations.Figure 6
FE model of a DSG square tube with imperfections.


### 4.1. Numerical Models

Figure7 presents the cross sections of the DSG square tubes with two types of material distributions. In order to intuitively reflect the effect of material distribution type on the energy absorption performance, linear thickness functions with obvious characteristics were adopted by the two types of tubes (Type 1 and Type 2). The main parameters of the sectional geometry are the maximum thickness t1, minimum thickness t2, and sectional widthB. The difference between the two distribution types was the width of the uniform thickness area. In Figure 7, the uniform thickness area and gradient thickness area are denoted by pink and gray, respectively. There was almost no area with a uniform thickness in the Type 1 tubes, except for the joints between adjacent plates. In contrast, the Type 2 tubes had a large uniform thickness area, which was characterized by the width we. To achieve higher energy absorption efficiency compared to the Type 1 DSG tubes, a strategy for increasing the amount of material in the EEAR was adopted for the Type 2 tubes. Therefore, the width of the uniform thickness area we was set equal to the width of the EEAR xe (i.e., we=xe=0.95H).Figure 7
Two types of material distributions.The columns were modeled using solid elements. Mesh convergence analysis was conducted to determine a sufficient mesh density to obtain an accurate mechanical response. Simulations of the two representative columns with different numbers of elements along their thicknesses were performed to determine the optimal number of elements, as shown in Figure8(a). The optimal element size was also determined, as shown in Figure 8(b). As a result, three elements along the thickness direction with a mesh size of 1 mm along the width and length directions were implemented to ensure sufficient mesh density and achieve accurate results.Figure 8
Force convergence: (a) different numbers of elements along the thickness and (b) element size.
(a) (b)


### 4.2. Material Characterization

The material used in the simulations was the aluminum extrusion alloy AA6060 T4, which has the following properties [28]: Young’s modulus E = 68.21 GPa, Poisson’s ratio μ = 0.3, initial yield strength σy = 80 MPa, ultimate strength σu = 173 MPa, and the power law exponentn=0.23. The engineering tensile-stress strain curve of the material is shown in Figure 9.Figure 9
Engineering stress-strain curve of AA6060 T4.


### 4.3. Numerical Results

The geometric parameters of the specimens are listed in Table1. All specimens were divided into three groups: A, B, and C. Columns in each group have the same global average thickness and a constant mass, and global average thicknesses for groups A, B, and C are 3.0 mm, 3.2 mm, and 3.4 mm, respectively. A concise numbering method was used for the columns. For instance, A-TRA denotes a traditional group A column with a constant thickness of 3.0 mm, and Type 1-A-1 denotes a DSG square tube with a Type 1 material distribution type. The final number indicates that the magnitude of the thickness gradient is equal to one. All finite element models were computed using the ANSYS/LS-DYNA software. Correspondingly, the crushing processes of all specimens are illustrated in Figure 10, which shows the folding shapes of the columns at displacements of 20 and 230 mm.Table 1
Dimensions and physical properties of the specimens.
Specimen t 1 (mm) t 2 (mm) w e t e (mm) t a (mm) Mass (g) A-TRA 3.00 3.00 0 3.00 3.00 935.3 Type 1-A-1 3.20 2.80 0 3.08 3.00 935.3 Type 1-A-2 3.40 2.60 0 3.16 3.00 935.3 Type 1-A-3 3.60 2.40 0 3.24 3.00 935.3 B-TRA 3.20 3.20 0 3.20 3.20 997.6 Type 1-B-1 3.40 3.00 0 3.28 3.20 997.6 Type 1-B-2 3.60 2.80 0 3.36 3.20 997.6 Type 1-B-3 3.80 2.60 0 3.44 3.20 997.6 C-TRA 3.40 3.40 0 3.40 3.40 1060.5 Type 1-C-1 3.60 3.20 0 3.48 3.40 1060.5 Type 1-C-2 3.80 3.00 0 3.56 3.40 1060.5 Type 1-C-3 4.00 2.80 0 3.64 3.40 1060.5 Type 2-A-1 3.20 2.19 0.95H 3.20 3.00 935.3 Type 2-A-2 3.40 1.38 0.95H 3.40 3.00 935.3 Type 2-B-1 3.40 2.36 0.95H 3.40 3.20 997.6 Type 2-B-2 3.60 1.59 0.95H 3.60 3.20 997.6 Type 2-C-1 3.60 2.53 0.95H 3.60 3.40 1060.5 Type 2-C-2 3.80 1.72 0.95H 3.80 3.40 1060.5Figure 10
Deformation patterns of the columns.Based on the theoretical analysis in Section3, two parameters are essential for predicting the mean load of the DSG tube theoretically: the global average thickness of the tube ta, which can be easily obtained based on the geometric configuration, and average thickness of the EEAR te, which can be determined for a known half wavelengthH using (25). However, calculatingH using (16) and (19) requires a known value of te, meaning the calculation process is circular.Based on the impact responses obtained from the numerical simulations, the relationship between the half wavelength and thickness gradient of the Type 1 DSG tubes was plotted as the half wavelength versus the maximum thickness (t1) curves, as shown in Figure 11. The half wavelength increased slightly or remained unchanged with an increase in the thickness gradient. This trend was observed for all three average thickness conditions (Figures 11(a)–11(c)). Furthermore, the theoretical predictions (red lines in Figure 10) for the traditional square tubes agree well with the simulation results for DSG tubes with the same average thicknesses as the traditional square tubes, regardless of the thickness gradient. Therefore, the te value of a DSG tube can be calculated using the theoretical half wavelength of a traditional square tube with the same average thickness as the DSG tube.Figure 11
Half wavelength maximum thickness (t1) curves of Type 1 DSG tubes in groups: (a) A, (b) B, and (c) C.


#### 4.3.1. Verification of Theoretical Predictions

Based on the inertia and strain rate effects of the tested materials, the mean loads under dynamic loading were larger than the theoretical results predicted by (29), which were derived for quasi-static compression conditions. The dynamic amplification coefficient, which was proposed by Hanssen et al. [34] based on an experimental study of the dynamic and quasi-static responses of thin-walled columns, can be introduced to fix this discrepancy. The amplification of the aluminum is dominated by inertial effects because it is insensitive to the strain rate. Langseth and Hopperstad [35] proposed a range of 1.3–1.6 for the coefficients of AA6060 T4 tubes with no triggers, while a lower value was proposed for tubes with triggers. In this study, to deal with the triggers, a value of 1.15 was adopted to account for the theoretical prediction of a specimen with an imperfection. Therefore, (29) was modified to create (31), which can be used to predict the crushing resistance of DSG tubes under dynamic compression. (31)Pm=1.15×6.0211δσ0teta2LC1/3.Using the Type 2-A-1 sample as an example, it was found thatte=3.2 mm and ta=3.0 mm. By substituting these values into (31), the theoretical solution for the mean load can be expressed as(32)Pm=1.15×6.021×10.7×106×3.232×3601/3=49.65×103N.Similarly, a theoretical solution for each column was obtained and combined with the simulation results listed in Table2. The numerical mean crushing force F-n is defined as follows: (33)F-n=∫0δnLFXdxδnL,where δn is the effective crushing distance coefficient of the numerical solution and F(X) is the crushing force.Table 2
Theoretical solutions and numerical results of impact responses.
Numerical simulation Eq. (31) (δ=0.70 and Db=1.15) Eq. (31) (δ=δn and D=1.15) F - n (kN) F i a (kN) F m a x (kN) CFE (%) SEA (kJ/kg) P m (kN) Error (%) δ n P m (kN) Error (%) A-TRA 45.21 117.22 117.22 38.57 11.14 46.54 +2.9 0.72 45.25 +0.1 Type 1-A-1 48.09 116.40 116.40 41.31 11.52 47.78 −0.6 0.70 47.78 −0.6 Type 1-A-2 49.33 115.87 115.87 42.57 11.81 49.02 −0.6 0.70 49.02 −0.6 Type 1-A-3 52.38 115.16 115.16 45.48 12.18 50.26 −4.0 0.68 51.73 −1.2 B-TRA 50.66 128.72 128.72 39.35 11.53 51.83 +2.3 0.71 51.10 +0.9 Type 1-B-1 53.33 128.50 128.50 41.50 11.97 53.12 −0.4 0.70 53.12 −0.4 Type 1-B-2 55.55 128.31 128.31 43.29 12.29 54.42 −2.0 0.69 55.20 −0.6 Type 1-B-3 58.03 127.77 127.77 45.41 12.65 55.72 −4.0 0.68 57.35 −1.1 C-TRA 57.26 138.16 138.16 41.44 12.09 57.34 +0.1 0.70 57.34 +0.1 Type 1-C-1 59.64 138.57 138.57 43.04 12.60 58.68 −1.6 0.70 58.68 −1.6 Type 1-C-2 62.55 138.30 138.30 45.23 13.02 60.03 −4.0 0.69 60.90 −2.6 Type 1-C-3 64.75 138.19 138.19 46.85 13.29 61.38 −5.2 0.68 63.18 −2.4 Type 2-A-1 48.44 117.20 117.20 41.33 11.60 49.65 +2.4 0.70 49.65 +2.4 Type 2-A-2 53.67 116.05 116.05 46.25 13.22 52.75 −1.7 0.72 51.28 −4.4 Type 2-B-1 54.42 127.93 127.93 42.53 12.21 55.07 +1.2 0.70 55.07 +1.2 Type 2-B-2 62.72 127.14 127.14 49.33 13.88 58.31 −7.0 0.69 59.15 −5.7 Type 2-C-1 61.23 137.88 137.88 44.40 12.93 60.71 −0.8 0.70 60.71 −0.8 Type 2-C-2 66.01 136.70 136.70 48.28 13.94 64.08 −2.9 0.70 64.08 −2.9 F i   a= initial peak force. D   b= dynamic amplification coefficient.The SEA was calculated as follows:(34)SEA=Eintm=F-n·δnLm,where Eint andm are the dissipated energy and mass, respectively. The crushing force efficiency (CFE) is an important parameter for evaluating the load consistency of an energy absorbing device. A higher value indicates excellent energy absorbing stability. The CFE is defined as follows:(35)CFE=F-nFmax.The results listed in Table2 reveal that the theoretical mean crushing forces agree well with the simulated values. Furthermore, there is good agreement between the simulation results and modified theoretical mean crushing forces, which were calculated using the effective crushing distance coefficient derived from the numerical simulations. Therefore, for AA6060 T4 DSG square tubes and the geometric configurations described above, a dynamic enhancing coefficient of 1.15 adequately accounts for imperfections in the tubes.In general, all the modified theoretical solutions for the Type 1 DSG tubes were closer to the simulation values compared to the Type 2 DSG tubes. The maximum error of Type 1 DSG tubes was approximately −2.6% (for Type 1-C-2). For the two Type 2 DSG tubes (Type 2-A-2 and Type 2-B-2), the errors were −4.4% and −5.7%, respectively. This is largely attributed to the irregular deformation modes of the two tubes, as shown in Figure10.As illustrated in Figure10, all initial folds developed at the tops of the tubes because imperfections were introduced in the top regions. All Type 1 DSG tubes folded in the symmetric quasi-inextensional mode, from which the expression for the mean load was derived above. However, the folding patterns of the two Type 2 DSG tubes (Type 2-A-2 and Type 2-B-2) were noticeably different from those of the other specimens. Therefore, the derived mean load expression was not appropriate for these tubes. Additionally, a notable characteristic of the irregular deformation mode was vertical irregular folds, as shown by the blue dashed boxes in Figure 10. The occurrence of this irregular folding is mainly attributed to the long folding wavelength. Additionally, the spacing of the two opposite plates was significantly reduced during the inward movement of the irregular fold (see Type 2-A-2 and Type 2-B-2). Simultaneously, the other two opposing plates bulged outward in the middle regions, which were relatively thinner. These thinner middle regions are characterized by a minimum thickness value t2, which is relatively small, as indicated by the dashed circle in Figure 10. Accordingly, the relatively long folding wavelength and smaller value of t2 lead to atactic folding. For example, although Type 2-A-2 had a similar global average thickness (ta), and maximum thickness (t1) compared to Type 1-A-2, there was a significant reduction in the t2 value of Type 2-A-2 compared to that of Type 1-A-2. Therefore, an irregular fold occurred in the Type 2-A-2 specimen but not in the Type 1-A-1 specimen. The difference between the deformation modes of the Type 1-B-2 and Type 2-B-2 specimens can be explained similarly. Additionally, the t2 values of the other four Type 2 DSG tubes (Type 2-A-1, Type 2-B-1, Type 2-C-1, and Type 2-C-2) did not decrease sufficiently or trigger the irregular mode.


#### 4.3.2. Influences of the Two Main Parameters (te and ta)

According to the theoretical analysis presented in Section3, the crushing resistances of the DSG square tubes with consistent materials, cross-sectional widths, and boundary conditions were primarily influenced by two crucial parameters: the average thickness of the EEAR te and global average thickness ta. In this section, the effects of these two parameters on the crushing responses will be discussed.Figures12 and 13 present the force-displacement curves of the grouped Type 1 and Type 2 DSG tubes, respectively. For the Type 1 tubes with ta=3mm (see Figure 12(a)), although the thickness gradients of the Type 1-A-1, Type 1-A-2, and Type 1-A-3 specimens increased, curve undulations of those tubes are similar to those of the traditional tube, namely, A-TRA. Additionally, the initial peak forces of the four tubes are nearly identical because they are strongly linked to the global average thickness. These trends were also observed in other groups of Type 1 and Type 2 DSG tubes. However, the curve fluctuations, peak numbers, and peak stages in the curves for the Type 2-A-2 and Type 2-B-2 tubes were significantly different from those of the other tubes (see Figures 13(a) and 13(b)), which can be attributed to their irregular folding patterns.Figure 12
Crushing load responses of Type 1 DSG tubes for three average thickness conditions: (a)ta=3.0mm, (b) ta=3.2mm, and (c) ta=3.4mm.
(a) (b) (c)Figure 13
Crushing load responses of Type 2 DSG tubes for three average thickness conditions: (a)ta=3.0mm, (b) ta=3.2mm, and (c) ta=3.4mm.
(a) (b) (c)According to (31), the mean load increases with an increasing value of te, while the other parameters remain unchanged. The extent of this increase is closely related to the coefficient of te/ta. This can also be proven based on the simulation results in Table 2. The maximum increase in the mean load of the Type 1 DSG tubes was 13.1% compared to the traditional C-TRA tube, which was determined using the Type 1-C-3 tube. For the Type 2 DSG tubes, the maximum increase in the mean load was 23.8% compared to the B-TRA tube, which was determined using the Type 2-B-2 tube. The reason why the Type 2 DSG tubes excelled at mean load enhancement is that the EEAR of Type 2 tube can contain more material compared to Type 1, when ta is constant. Therefore, Type 2 is more efficient at increasing the coefficient of te/ta. This coefficient will be discussed in detail later.In addition to the parameterte, the global average thickness ta is another essential variable related to the mean load. Its influence on mechanical responses is discussed below. Several important phenomena can be observed in the curves in Figures 12 and 13 and corresponding data in Table 2. First, nearly identical initial peak forces were obtained for the tubes with a constant value of ta, regardless of the material distribution type and thickness gradient. Second, the initial peak force increases with an increasing global average thickness, which is also independent of the material distribution type and thickness gradient. As illustrated in Figures 12(a)–12(c), the initial peak force of the Type 1 DSG tubes increased from approximately 116 to 138 KN with an increase in ta from 3.0 to 3.4 mm. This trend can also be observed in the response curves (Figures 13(a)–13(c)) of the Type 2 DSG tubes. Finally, the ideal energy absorber has a high mean load and low initial peak force. Therefore, the best DSG square tube should have a large coefficient of te/ta. Again, this coefficient proved to be an important factor affecting crashworthiness. However, such a DSG square tube may experience atactic deformation, as demonstrated by the Type 2-A-2 and Type 2-B-2 specimens. As mentioned above, the occurrence of irregular deformation is strongly linked to the minimum thickness t2, which is closely related to the material distribution.


#### 4.3.3. Influence of Material Distribution Type

For a DSG tube with a constant global average thicknessta, the minimum thickness t2 decreases with an increasing maximum thickness t1. The t2 values of Type 2 DSG tubes decrease more rapidly based on their material distribution characteristics. Therefore, each Type 2 DSG tube has a smaller t2 value compared to the Type 1 DSG tube with the same t1 and ta values, as indicated by the physical properties listed in Table 1. Based on the reduction of t2 for the Type 2-A-2 and Type 2-B-2 tubes beyond a certain limit, irregular folds occurred. There was a relatively small reduction in the t2 values of the Type 1-A-2 and Type 1-B-2 tubes, which deformed via regular deformation modes.As mentioned above, in the theoretical expression (31), the load efficiency of a DSG column increases with an increase in te, while the other parameters remain unchanged. Therefore, the purpose of changing the material distribution is to increase the value of te. Of the two types of material distributions, Type 2 is more efficient than Type 1 for increasing te. As shown in Table 1, the te value of each Type 2 DSG column was larger than that of the Type 1 DSG column with the same ta and t1 values. For example, the te value of Type 2-A-2 increased to 3.4 mm when t1 increased to 3.4 mm. However, for Type 1-A-2, te only increased to 3.16 mm. Therefore, the mean load and SEA of the Type 2-A-2 tube are higher than those of the Type 1-A-2 tube, which was proven by the theoretical prediction and simulation results. The efficiency of the Type 2 material distribution for promoting te mainly stems from the fact that the EEAR of the Type 2 DSG tubes holds more material compared to that of the Type 1 DSG tubes.Finally, from the perspective of crashworthiness improvement, the main concerns are the load efficiency and regular folding patterns that a given tube can produce. Therefore, a material distribution type that ensures a fixedt2 value and will not trigger irregular folding while allowing the EEAR to hold as much material as possible is desirable.


## 4.1. Numerical Models

Figure7 presents the cross sections of the DSG square tubes with two types of material distributions. In order to intuitively reflect the effect of material distribution type on the energy absorption performance, linear thickness functions with obvious characteristics were adopted by the two types of tubes (Type 1 and Type 2). The main parameters of the sectional geometry are the maximum thickness t1, minimum thickness t2, and sectional widthB. The difference between the two distribution types was the width of the uniform thickness area. In Figure 7, the uniform thickness area and gradient thickness area are denoted by pink and gray, respectively. There was almost no area with a uniform thickness in the Type 1 tubes, except for the joints between adjacent plates. In contrast, the Type 2 tubes had a large uniform thickness area, which was characterized by the width we. To achieve higher energy absorption efficiency compared to the Type 1 DSG tubes, a strategy for increasing the amount of material in the EEAR was adopted for the Type 2 tubes. Therefore, the width of the uniform thickness area we was set equal to the width of the EEAR xe (i.e., we=xe=0.95H).Figure 7
Two types of material distributions.The columns were modeled using solid elements. Mesh convergence analysis was conducted to determine a sufficient mesh density to obtain an accurate mechanical response. Simulations of the two representative columns with different numbers of elements along their thicknesses were performed to determine the optimal number of elements, as shown in Figure8(a). The optimal element size was also determined, as shown in Figure 8(b). As a result, three elements along the thickness direction with a mesh size of 1 mm along the width and length directions were implemented to ensure sufficient mesh density and achieve accurate results.Figure 8
Force convergence: (a) different numbers of elements along the thickness and (b) element size.
(a) (b)


## 4.2. Material Characterization

The material used in the simulations was the aluminum extrusion alloy AA6060 T4, which has the following properties [28]: Young’s modulus E = 68.21 GPa, Poisson’s ratio μ = 0.3, initial yield strength σy = 80 MPa, ultimate strength σu = 173 MPa, and the power law exponentn=0.23. The engineering tensile-stress strain curve of the material is shown in Figure 9.Figure 9
Engineering stress-strain curve of AA6060 T4.


## 4.3. Numerical Results

The geometric parameters of the specimens are listed in Table1. All specimens were divided into three groups: A, B, and C. Columns in each group have the same global average thickness and a constant mass, and global average thicknesses for groups A, B, and C are 3.0 mm, 3.2 mm, and 3.4 mm, respectively. A concise numbering method was used for the columns. For instance, A-TRA denotes a traditional group A column with a constant thickness of 3.0 mm, and Type 1-A-1 denotes a DSG square tube with a Type 1 material distribution type. The final number indicates that the magnitude of the thickness gradient is equal to one. All finite element models were computed using the ANSYS/LS-DYNA software. Correspondingly, the crushing processes of all specimens are illustrated in Figure 10, which shows the folding shapes of the columns at displacements of 20 and 230 mm.Table 1
Dimensions and physical properties of the specimens.
Specimen t 1 (mm) t 2 (mm) w e t e (mm) t a (mm) Mass (g) A-TRA 3.00 3.00 0 3.00 3.00 935.3 Type 1-A-1 3.20 2.80 0 3.08 3.00 935.3 Type 1-A-2 3.40 2.60 0 3.16 3.00 935.3 Type 1-A-3 3.60 2.40 0 3.24 3.00 935.3 B-TRA 3.20 3.20 0 3.20 3.20 997.6 Type 1-B-1 3.40 3.00 0 3.28 3.20 997.6 Type 1-B-2 3.60 2.80 0 3.36 3.20 997.6 Type 1-B-3 3.80 2.60 0 3.44 3.20 997.6 C-TRA 3.40 3.40 0 3.40 3.40 1060.5 Type 1-C-1 3.60 3.20 0 3.48 3.40 1060.5 Type 1-C-2 3.80 3.00 0 3.56 3.40 1060.5 Type 1-C-3 4.00 2.80 0 3.64 3.40 1060.5 Type 2-A-1 3.20 2.19 0.95H 3.20 3.00 935.3 Type 2-A-2 3.40 1.38 0.95H 3.40 3.00 935.3 Type 2-B-1 3.40 2.36 0.95H 3.40 3.20 997.6 Type 2-B-2 3.60 1.59 0.95H 3.60 3.20 997.6 Type 2-C-1 3.60 2.53 0.95H 3.60 3.40 1060.5 Type 2-C-2 3.80 1.72 0.95H 3.80 3.40 1060.5Figure 10
Deformation patterns of the columns.Based on the theoretical analysis in Section3, two parameters are essential for predicting the mean load of the DSG tube theoretically: the global average thickness of the tube ta, which can be easily obtained based on the geometric configuration, and average thickness of the EEAR te, which can be determined for a known half wavelengthH using (25). However, calculatingH using (16) and (19) requires a known value of te, meaning the calculation process is circular.Based on the impact responses obtained from the numerical simulations, the relationship between the half wavelength and thickness gradient of the Type 1 DSG tubes was plotted as the half wavelength versus the maximum thickness (t1) curves, as shown in Figure 11. The half wavelength increased slightly or remained unchanged with an increase in the thickness gradient. This trend was observed for all three average thickness conditions (Figures 11(a)–11(c)). Furthermore, the theoretical predictions (red lines in Figure 10) for the traditional square tubes agree well with the simulation results for DSG tubes with the same average thicknesses as the traditional square tubes, regardless of the thickness gradient. Therefore, the te value of a DSG tube can be calculated using the theoretical half wavelength of a traditional square tube with the same average thickness as the DSG tube.Figure 11
Half wavelength maximum thickness (t1) curves of Type 1 DSG tubes in groups: (a) A, (b) B, and (c) C.


### 4.3.1. Verification of Theoretical Predictions

Based on the inertia and strain rate effects of the tested materials, the mean loads under dynamic loading were larger than the theoretical results predicted by (29), which were derived for quasi-static compression conditions. The dynamic amplification coefficient, which was proposed by Hanssen et al. [34] based on an experimental study of the dynamic and quasi-static responses of thin-walled columns, can be introduced to fix this discrepancy. The amplification of the aluminum is dominated by inertial effects because it is insensitive to the strain rate. Langseth and Hopperstad [35] proposed a range of 1.3–1.6 for the coefficients of AA6060 T4 tubes with no triggers, while a lower value was proposed for tubes with triggers. In this study, to deal with the triggers, a value of 1.15 was adopted to account for the theoretical prediction of a specimen with an imperfection. Therefore, (29) was modified to create (31), which can be used to predict the crushing resistance of DSG tubes under dynamic compression. (31)Pm=1.15×6.0211δσ0teta2LC1/3.Using the Type 2-A-1 sample as an example, it was found thatte=3.2 mm and ta=3.0 mm. By substituting these values into (31), the theoretical solution for the mean load can be expressed as(32)Pm=1.15×6.021×10.7×106×3.232×3601/3=49.65×103N.Similarly, a theoretical solution for each column was obtained and combined with the simulation results listed in Table2. The numerical mean crushing force F-n is defined as follows: (33)F-n=∫0δnLFXdxδnL,where δn is the effective crushing distance coefficient of the numerical solution and F(X) is the crushing force.Table 2
Theoretical solutions and numerical results of impact responses.
Numerical simulation Eq. (31) (δ=0.70 and Db=1.15) Eq. (31) (δ=δn and D=1.15) F - n (kN) F i a (kN) F m a x (kN) CFE (%) SEA (kJ/kg) P m (kN) Error (%) δ n P m (kN) Error (%) A-TRA 45.21 117.22 117.22 38.57 11.14 46.54 +2.9 0.72 45.25 +0.1 Type 1-A-1 48.09 116.40 116.40 41.31 11.52 47.78 −0.6 0.70 47.78 −0.6 Type 1-A-2 49.33 115.87 115.87 42.57 11.81 49.02 −0.6 0.70 49.02 −0.6 Type 1-A-3 52.38 115.16 115.16 45.48 12.18 50.26 −4.0 0.68 51.73 −1.2 B-TRA 50.66 128.72 128.72 39.35 11.53 51.83 +2.3 0.71 51.10 +0.9 Type 1-B-1 53.33 128.50 128.50 41.50 11.97 53.12 −0.4 0.70 53.12 −0.4 Type 1-B-2 55.55 128.31 128.31 43.29 12.29 54.42 −2.0 0.69 55.20 −0.6 Type 1-B-3 58.03 127.77 127.77 45.41 12.65 55.72 −4.0 0.68 57.35 −1.1 C-TRA 57.26 138.16 138.16 41.44 12.09 57.34 +0.1 0.70 57.34 +0.1 Type 1-C-1 59.64 138.57 138.57 43.04 12.60 58.68 −1.6 0.70 58.68 −1.6 Type 1-C-2 62.55 138.30 138.30 45.23 13.02 60.03 −4.0 0.69 60.90 −2.6 Type 1-C-3 64.75 138.19 138.19 46.85 13.29 61.38 −5.2 0.68 63.18 −2.4 Type 2-A-1 48.44 117.20 117.20 41.33 11.60 49.65 +2.4 0.70 49.65 +2.4 Type 2-A-2 53.67 116.05 116.05 46.25 13.22 52.75 −1.7 0.72 51.28 −4.4 Type 2-B-1 54.42 127.93 127.93 42.53 12.21 55.07 +1.2 0.70 55.07 +1.2 Type 2-B-2 62.72 127.14 127.14 49.33 13.88 58.31 −7.0 0.69 59.15 −5.7 Type 2-C-1 61.23 137.88 137.88 44.40 12.93 60.71 −0.8 0.70 60.71 −0.8 Type 2-C-2 66.01 136.70 136.70 48.28 13.94 64.08 −2.9 0.70 64.08 −2.9 F i   a= initial peak force. D   b= dynamic amplification coefficient.The SEA was calculated as follows:(34)SEA=Eintm=F-n·δnLm,where Eint andm are the dissipated energy and mass, respectively. The crushing force efficiency (CFE) is an important parameter for evaluating the load consistency of an energy absorbing device. A higher value indicates excellent energy absorbing stability. The CFE is defined as follows:(35)CFE=F-nFmax.The results listed in Table2 reveal that the theoretical mean crushing forces agree well with the simulated values. Furthermore, there is good agreement between the simulation results and modified theoretical mean crushing forces, which were calculated using the effective crushing distance coefficient derived from the numerical simulations. Therefore, for AA6060 T4 DSG square tubes and the geometric configurations described above, a dynamic enhancing coefficient of 1.15 adequately accounts for imperfections in the tubes.In general, all the modified theoretical solutions for the Type 1 DSG tubes were closer to the simulation values compared to the Type 2 DSG tubes. The maximum error of Type 1 DSG tubes was approximately −2.6% (for Type 1-C-2). For the two Type 2 DSG tubes (Type 2-A-2 and Type 2-B-2), the errors were −4.4% and −5.7%, respectively. This is largely attributed to the irregular deformation modes of the two tubes, as shown in Figure10.As illustrated in Figure10, all initial folds developed at the tops of the tubes because imperfections were introduced in the top regions. All Type 1 DSG tubes folded in the symmetric quasi-inextensional mode, from which the expression for the mean load was derived above. However, the folding patterns of the two Type 2 DSG tubes (Type 2-A-2 and Type 2-B-2) were noticeably different from those of the other specimens. Therefore, the derived mean load expression was not appropriate for these tubes. Additionally, a notable characteristic of the irregular deformation mode was vertical irregular folds, as shown by the blue dashed boxes in Figure 10. The occurrence of this irregular folding is mainly attributed to the long folding wavelength. Additionally, the spacing of the two opposite plates was significantly reduced during the inward movement of the irregular fold (see Type 2-A-2 and Type 2-B-2). Simultaneously, the other two opposing plates bulged outward in the middle regions, which were relatively thinner. These thinner middle regions are characterized by a minimum thickness value t2, which is relatively small, as indicated by the dashed circle in Figure 10. Accordingly, the relatively long folding wavelength and smaller value of t2 lead to atactic folding. For example, although Type 2-A-2 had a similar global average thickness (ta), and maximum thickness (t1) compared to Type 1-A-2, there was a significant reduction in the t2 value of Type 2-A-2 compared to that of Type 1-A-2. Therefore, an irregular fold occurred in the Type 2-A-2 specimen but not in the Type 1-A-1 specimen. The difference between the deformation modes of the Type 1-B-2 and Type 2-B-2 specimens can be explained similarly. Additionally, the t2 values of the other four Type 2 DSG tubes (Type 2-A-1, Type 2-B-1, Type 2-C-1, and Type 2-C-2) did not decrease sufficiently or trigger the irregular mode.


### 4.3.2. Influences of the Two Main Parameters (te and ta)

According to the theoretical analysis presented in Section3, the crushing resistances of the DSG square tubes with consistent materials, cross-sectional widths, and boundary conditions were primarily influenced by two crucial parameters: the average thickness of the EEAR te and global average thickness ta. In this section, the effects of these two parameters on the crushing responses will be discussed.Figures12 and 13 present the force-displacement curves of the grouped Type 1 and Type 2 DSG tubes, respectively. For the Type 1 tubes with ta=3mm (see Figure 12(a)), although the thickness gradients of the Type 1-A-1, Type 1-A-2, and Type 1-A-3 specimens increased, curve undulations of those tubes are similar to those of the traditional tube, namely, A-TRA. Additionally, the initial peak forces of the four tubes are nearly identical because they are strongly linked to the global average thickness. These trends were also observed in other groups of Type 1 and Type 2 DSG tubes. However, the curve fluctuations, peak numbers, and peak stages in the curves for the Type 2-A-2 and Type 2-B-2 tubes were significantly different from those of the other tubes (see Figures 13(a) and 13(b)), which can be attributed to their irregular folding patterns.Figure 12
Crushing load responses of Type 1 DSG tubes for three average thickness conditions: (a)ta=3.0mm, (b) ta=3.2mm, and (c) ta=3.4mm.
(a) (b) (c)Figure 13
Crushing load responses of Type 2 DSG tubes for three average thickness conditions: (a)ta=3.0mm, (b) ta=3.2mm, and (c) ta=3.4mm.
(a) (b) (c)According to (31), the mean load increases with an increasing value of te, while the other parameters remain unchanged. The extent of this increase is closely related to the coefficient of te/ta. This can also be proven based on the simulation results in Table 2. The maximum increase in the mean load of the Type 1 DSG tubes was 13.1% compared to the traditional C-TRA tube, which was determined using the Type 1-C-3 tube. For the Type 2 DSG tubes, the maximum increase in the mean load was 23.8% compared to the B-TRA tube, which was determined using the Type 2-B-2 tube. The reason why the Type 2 DSG tubes excelled at mean load enhancement is that the EEAR of Type 2 tube can contain more material compared to Type 1, when ta is constant. Therefore, Type 2 is more efficient at increasing the coefficient of te/ta. This coefficient will be discussed in detail later.In addition to the parameterte, the global average thickness ta is another essential variable related to the mean load. Its influence on mechanical responses is discussed below. Several important phenomena can be observed in the curves in Figures 12 and 13 and corresponding data in Table 2. First, nearly identical initial peak forces were obtained for the tubes with a constant value of ta, regardless of the material distribution type and thickness gradient. Second, the initial peak force increases with an increasing global average thickness, which is also independent of the material distribution type and thickness gradient. As illustrated in Figures 12(a)–12(c), the initial peak force of the Type 1 DSG tubes increased from approximately 116 to 138 KN with an increase in ta from 3.0 to 3.4 mm. This trend can also be observed in the response curves (Figures 13(a)–13(c)) of the Type 2 DSG tubes. Finally, the ideal energy absorber has a high mean load and low initial peak force. Therefore, the best DSG square tube should have a large coefficient of te/ta. Again, this coefficient proved to be an important factor affecting crashworthiness. However, such a DSG square tube may experience atactic deformation, as demonstrated by the Type 2-A-2 and Type 2-B-2 specimens. As mentioned above, the occurrence of irregular deformation is strongly linked to the minimum thickness t2, which is closely related to the material distribution.


### 4.3.3. Influence of Material Distribution Type

For a DSG tube with a constant global average thicknessta, the minimum thickness t2 decreases with an increasing maximum thickness t1. The t2 values of Type 2 DSG tubes decrease more rapidly based on their material distribution characteristics. Therefore, each Type 2 DSG tube has a smaller t2 value compared to the Type 1 DSG tube with the same t1 and ta values, as indicated by the physical properties listed in Table 1. Based on the reduction of t2 for the Type 2-A-2 and Type 2-B-2 tubes beyond a certain limit, irregular folds occurred. There was a relatively small reduction in the t2 values of the Type 1-A-2 and Type 1-B-2 tubes, which deformed via regular deformation modes.As mentioned above, in the theoretical expression (31), the load efficiency of a DSG column increases with an increase in te, while the other parameters remain unchanged. Therefore, the purpose of changing the material distribution is to increase the value of te. Of the two types of material distributions, Type 2 is more efficient than Type 1 for increasing te. As shown in Table 1, the te value of each Type 2 DSG column was larger than that of the Type 1 DSG column with the same ta and t1 values. For example, the te value of Type 2-A-2 increased to 3.4 mm when t1 increased to 3.4 mm. However, for Type 1-A-2, te only increased to 3.16 mm. Therefore, the mean load and SEA of the Type 2-A-2 tube are higher than those of the Type 1-A-2 tube, which was proven by the theoretical prediction and simulation results. The efficiency of the Type 2 material distribution for promoting te mainly stems from the fact that the EEAR of the Type 2 DSG tubes holds more material compared to that of the Type 1 DSG tubes.Finally, from the perspective of crashworthiness improvement, the main concerns are the load efficiency and regular folding patterns that a given tube can produce. Therefore, a material distribution type that ensures a fixedt2 value and will not trigger irregular folding while allowing the EEAR to hold as much material as possible is desirable.


## 4.3.1. Verification of Theoretical Predictions

Based on the inertia and strain rate effects of the tested materials, the mean loads under dynamic loading were larger than the theoretical results predicted by (29), which were derived for quasi-static compression conditions. The dynamic amplification coefficient, which was proposed by Hanssen et al. [34] based on an experimental study of the dynamic and quasi-static responses of thin-walled columns, can be introduced to fix this discrepancy. The amplification of the aluminum is dominated by inertial effects because it is insensitive to the strain rate. Langseth and Hopperstad [35] proposed a range of 1.3–1.6 for the coefficients of AA6060 T4 tubes with no triggers, while a lower value was proposed for tubes with triggers. In this study, to deal with the triggers, a value of 1.15 was adopted to account for the theoretical prediction of a specimen with an imperfection. Therefore, (29) was modified to create (31), which can be used to predict the crushing resistance of DSG tubes under dynamic compression. (31)Pm=1.15×6.0211δσ0teta2LC1/3.Using the Type 2-A-1 sample as an example, it was found thatte=3.2 mm and ta=3.0 mm. By substituting these values into (31), the theoretical solution for the mean load can be expressed as(32)Pm=1.15×6.021×10.7×106×3.232×3601/3=49.65×103N.Similarly, a theoretical solution for each column was obtained and combined with the simulation results listed in Table2. The numerical mean crushing force F-n is defined as follows: (33)F-n=∫0δnLFXdxδnL,where δn is the effective crushing distance coefficient of the numerical solution and F(X) is the crushing force.Table 2
Theoretical solutions and numerical results of impact responses.
Numerical simulation Eq. (31) (δ=0.70 and Db=1.15) Eq. (31) (δ=δn and D=1.15) F - n (kN) F i a (kN) F m a x (kN) CFE (%) SEA (kJ/kg) P m (kN) Error (%) δ n P m (kN) Error (%) A-TRA 45.21 117.22 117.22 38.57 11.14 46.54 +2.9 0.72 45.25 +0.1 Type 1-A-1 48.09 116.40 116.40 41.31 11.52 47.78 −0.6 0.70 47.78 −0.6 Type 1-A-2 49.33 115.87 115.87 42.57 11.81 49.02 −0.6 0.70 49.02 −0.6 Type 1-A-3 52.38 115.16 115.16 45.48 12.18 50.26 −4.0 0.68 51.73 −1.2 B-TRA 50.66 128.72 128.72 39.35 11.53 51.83 +2.3 0.71 51.10 +0.9 Type 1-B-1 53.33 128.50 128.50 41.50 11.97 53.12 −0.4 0.70 53.12 −0.4 Type 1-B-2 55.55 128.31 128.31 43.29 12.29 54.42 −2.0 0.69 55.20 −0.6 Type 1-B-3 58.03 127.77 127.77 45.41 12.65 55.72 −4.0 0.68 57.35 −1.1 C-TRA 57.26 138.16 138.16 41.44 12.09 57.34 +0.1 0.70 57.34 +0.1 Type 1-C-1 59.64 138.57 138.57 43.04 12.60 58.68 −1.6 0.70 58.68 −1.6 Type 1-C-2 62.55 138.30 138.30 45.23 13.02 60.03 −4.0 0.69 60.90 −2.6 Type 1-C-3 64.75 138.19 138.19 46.85 13.29 61.38 −5.2 0.68 63.18 −2.4 Type 2-A-1 48.44 117.20 117.20 41.33 11.60 49.65 +2.4 0.70 49.65 +2.4 Type 2-A-2 53.67 116.05 116.05 46.25 13.22 52.75 −1.7 0.72 51.28 −4.4 Type 2-B-1 54.42 127.93 127.93 42.53 12.21 55.07 +1.2 0.70 55.07 +1.2 Type 2-B-2 62.72 127.14 127.14 49.33 13.88 58.31 −7.0 0.69 59.15 −5.7 Type 2-C-1 61.23 137.88 137.88 44.40 12.93 60.71 −0.8 0.70 60.71 −0.8 Type 2-C-2 66.01 136.70 136.70 48.28 13.94 64.08 −2.9 0.70 64.08 −2.9 F i   a= initial peak force. D   b= dynamic amplification coefficient.The SEA was calculated as follows:(34)SEA=Eintm=F-n·δnLm,where Eint andm are the dissipated energy and mass, respectively. The crushing force efficiency (CFE) is an important parameter for evaluating the load consistency of an energy absorbing device. A higher value indicates excellent energy absorbing stability. The CFE is defined as follows:(35)CFE=F-nFmax.The results listed in Table2 reveal that the theoretical mean crushing forces agree well with the simulated values. Furthermore, there is good agreement between the simulation results and modified theoretical mean crushing forces, which were calculated using the effective crushing distance coefficient derived from the numerical simulations. Therefore, for AA6060 T4 DSG square tubes and the geometric configurations described above, a dynamic enhancing coefficient of 1.15 adequately accounts for imperfections in the tubes.In general, all the modified theoretical solutions for the Type 1 DSG tubes were closer to the simulation values compared to the Type 2 DSG tubes. The maximum error of Type 1 DSG tubes was approximately −2.6% (for Type 1-C-2). For the two Type 2 DSG tubes (Type 2-A-2 and Type 2-B-2), the errors were −4.4% and −5.7%, respectively. This is largely attributed to the irregular deformation modes of the two tubes, as shown in Figure10.As illustrated in Figure10, all initial folds developed at the tops of the tubes because imperfections were introduced in the top regions. All Type 1 DSG tubes folded in the symmetric quasi-inextensional mode, from which the expression for the mean load was derived above. However, the folding patterns of the two Type 2 DSG tubes (Type 2-A-2 and Type 2-B-2) were noticeably different from those of the other specimens. Therefore, the derived mean load expression was not appropriate for these tubes. Additionally, a notable characteristic of the irregular deformation mode was vertical irregular folds, as shown by the blue dashed boxes in Figure 10. The occurrence of this irregular folding is mainly attributed to the long folding wavelength. Additionally, the spacing of the two opposite plates was significantly reduced during the inward movement of the irregular fold (see Type 2-A-2 and Type 2-B-2). Simultaneously, the other two opposing plates bulged outward in the middle regions, which were relatively thinner. These thinner middle regions are characterized by a minimum thickness value t2, which is relatively small, as indicated by the dashed circle in Figure 10. Accordingly, the relatively long folding wavelength and smaller value of t2 lead to atactic folding. For example, although Type 2-A-2 had a similar global average thickness (ta), and maximum thickness (t1) compared to Type 1-A-2, there was a significant reduction in the t2 value of Type 2-A-2 compared to that of Type 1-A-2. Therefore, an irregular fold occurred in the Type 2-A-2 specimen but not in the Type 1-A-1 specimen. The difference between the deformation modes of the Type 1-B-2 and Type 2-B-2 specimens can be explained similarly. Additionally, the t2 values of the other four Type 2 DSG tubes (Type 2-A-1, Type 2-B-1, Type 2-C-1, and Type 2-C-2) did not decrease sufficiently or trigger the irregular mode.


## 4.3.2. Influences of the Two Main Parameters (te and ta)

According to the theoretical analysis presented in Section3, the crushing resistances of the DSG square tubes with consistent materials, cross-sectional widths, and boundary conditions were primarily influenced by two crucial parameters: the average thickness of the EEAR te and global average thickness ta. In this section, the effects of these two parameters on the crushing responses will be discussed.Figures12 and 13 present the force-displacement curves of the grouped Type 1 and Type 2 DSG tubes, respectively. For the Type 1 tubes with ta=3mm (see Figure 12(a)), although the thickness gradients of the Type 1-A-1, Type 1-A-2, and Type 1-A-3 specimens increased, curve undulations of those tubes are similar to those of the traditional tube, namely, A-TRA. Additionally, the initial peak forces of the four tubes are nearly identical because they are strongly linked to the global average thickness. These trends were also observed in other groups of Type 1 and Type 2 DSG tubes. However, the curve fluctuations, peak numbers, and peak stages in the curves for the Type 2-A-2 and Type 2-B-2 tubes were significantly different from those of the other tubes (see Figures 13(a) and 13(b)), which can be attributed to their irregular folding patterns.Figure 12
Crushing load responses of Type 1 DSG tubes for three average thickness conditions: (a)ta=3.0mm, (b) ta=3.2mm, and (c) ta=3.4mm.
(a) (b) (c)Figure 13
Crushing load responses of Type 2 DSG tubes for three average thickness conditions: (a)ta=3.0mm, (b) ta=3.2mm, and (c) ta=3.4mm.
(a) (b) (c)According to (31), the mean load increases with an increasing value of te, while the other parameters remain unchanged. The extent of this increase is closely related to the coefficient of te/ta. This can also be proven based on the simulation results in Table 2. The maximum increase in the mean load of the Type 1 DSG tubes was 13.1% compared to the traditional C-TRA tube, which was determined using the Type 1-C-3 tube. For the Type 2 DSG tubes, the maximum increase in the mean load was 23.8% compared to the B-TRA tube, which was determined using the Type 2-B-2 tube. The reason why the Type 2 DSG tubes excelled at mean load enhancement is that the EEAR of Type 2 tube can contain more material compared to Type 1, when ta is constant. Therefore, Type 2 is more efficient at increasing the coefficient of te/ta. This coefficient will be discussed in detail later.In addition to the parameterte, the global average thickness ta is another essential variable related to the mean load. Its influence on mechanical responses is discussed below. Several important phenomena can be observed in the curves in Figures 12 and 13 and corresponding data in Table 2. First, nearly identical initial peak forces were obtained for the tubes with a constant value of ta, regardless of the material distribution type and thickness gradient. Second, the initial peak force increases with an increasing global average thickness, which is also independent of the material distribution type and thickness gradient. As illustrated in Figures 12(a)–12(c), the initial peak force of the Type 1 DSG tubes increased from approximately 116 to 138 KN with an increase in ta from 3.0 to 3.4 mm. This trend can also be observed in the response curves (Figures 13(a)–13(c)) of the Type 2 DSG tubes. Finally, the ideal energy absorber has a high mean load and low initial peak force. Therefore, the best DSG square tube should have a large coefficient of te/ta. Again, this coefficient proved to be an important factor affecting crashworthiness. However, such a DSG square tube may experience atactic deformation, as demonstrated by the Type 2-A-2 and Type 2-B-2 specimens. As mentioned above, the occurrence of irregular deformation is strongly linked to the minimum thickness t2, which is closely related to the material distribution.


## 4.3.3. Influence of Material Distribution Type

For a DSG tube with a constant global average thicknessta, the minimum thickness t2 decreases with an increasing maximum thickness t1. The t2 values of Type 2 DSG tubes decrease more rapidly based on their material distribution characteristics. Therefore, each Type 2 DSG tube has a smaller t2 value compared to the Type 1 DSG tube with the same t1 and ta values, as indicated by the physical properties listed in Table 1. Based on the reduction of t2 for the Type 2-A-2 and Type 2-B-2 tubes beyond a certain limit, irregular folds occurred. There was a relatively small reduction in the t2 values of the Type 1-A-2 and Type 1-B-2 tubes, which deformed via regular deformation modes.As mentioned above, in the theoretical expression (31), the load efficiency of a DSG column increases with an increase in te, while the other parameters remain unchanged. Therefore, the purpose of changing the material distribution is to increase the value of te. Of the two types of material distributions, Type 2 is more efficient than Type 1 for increasing te. As shown in Table 1, the te value of each Type 2 DSG column was larger than that of the Type 1 DSG column with the same ta and t1 values. For example, the te value of Type 2-A-2 increased to 3.4 mm when t1 increased to 3.4 mm. However, for Type 1-A-2, te only increased to 3.16 mm. Therefore, the mean load and SEA of the Type 2-A-2 tube are higher than those of the Type 1-A-2 tube, which was proven by the theoretical prediction and simulation results. The efficiency of the Type 2 material distribution for promoting te mainly stems from the fact that the EEAR of the Type 2 DSG tubes holds more material compared to that of the Type 1 DSG tubes.Finally, from the perspective of crashworthiness improvement, the main concerns are the load efficiency and regular folding patterns that a given tube can produce. Therefore, a material distribution type that ensures a fixedt2 value and will not trigger irregular folding while allowing the EEAR to hold as much material as possible is desirable.


## 5. Modification of Material Distributions

### 5.1. Improvement of the Type 2 Material Distribution

As discussed above, for a Type 2 DSG tube with a constant global average thickness, concentrating material within the EEAR will lead to a significant decrease in the minimum thickness (t2), which limits the promotion of the average thickness of the EEAR (te). Therefore, a modified material distribution type should maximize the value of te under such limitations. Based on this principle, the Type 2 material distribution was modified to create a new distribution.The geometric configuration of this new material distribution type (Type 3) is illustrated in Figure14. As shown in the figure, the non-EEAR was given a uniform thickness equal to the minimum thickness t2, which ensures that the EEAR contains the most material for a fixed value of t2. Additionally, the EEAR has a uniform thickness area with a width of we and gradient thickness area with a width of wg, as shown in Figure 14. In other words, the width of the EEAR xe is equal to the sum of we and wg. The crushing response improvements provided by the Type 3 material distribution are discussed below.Figure 14
Material distribution of Type 3 tubes.


### 5.2. Crushing Response Comparison

In theory, for two DSG tubes with the same cross-sectional width and global average thickness, a higher energy absorption efficiency is achieved by the tube with a largerte value. The te value for a Type 1 or Type 3 DSG tube can be obtained easily by using the relationships between its geometric parameters, as shown in Table 3. Additionally, it is necessary to determine if a modified Type 2 DSG tube (i.e., Type 3 DSG tube) is more efficient than a Type 1 DSG tube in terms of energy absorption for the same geometric constraints, namely, cross-sectional width (B), global average thickness (ta), and minimum thickness (t2). The geometric constraints ofB= 90 mm, ta= 3.2 mm, and t2 = 2.0 mm were used for the two types of DSG tubes. The maximum te values are listed in Table 4. The Type 1 DSG tube with geometric constraints was labelled Type 1-B-MAX. Similarly, considering the same geometric constraints, three Type 3 DSG tubes with wg+we=xe were established, where the wg values were set to 0.25xe, 0.5xe, and 0.75xe to investigate the effects of this parameter. These three tubes were labelled Type 3-B-MAX1, Type 3-B-MAX2, and Type 3-B-MAX3, respectively. All the dimensions of the Type 1 and three Type 3 tubes are listed in Table 4.Table 3
Relationships between geometric parameters.
Gradient thickness type Width of the EEAR Widths of the gradient thickness area and uniform thickness area Maximum thickness Average thickness of the EEAR Type 1 x e = 0.95 H w g = B 2; t 1 = 2 t a - t 2 t e = t 2 - t 1 x e B + t 1 w e = 0 Type 3 x e = 0.95 H w g + w e = x e t 1 = B t a - t 2 + t 2 x e + t 2 w e x e + w e t e = B ( t a - t 2 ) 2 x e + t 2Table 4
Dimensions and crushing responses of two types of specimens with dimensional constraints.
Specimen Dimension Simulation result Constraint Mass (g) t a (mm) t 2 (mm) t 1 (mm) t e (mm) w g w e P m (kN) CFE (%) SEA (kJ/kg) B-TRA 997.6 3.20 3.20 3.20 3.20 0 B / 2 50.66 39.35 11.53 Type 1-B-MAX 997.6 3.20 2.00 4.40 3.71 B / 2 0 62.35 48.71 13.60 Type 3-B-MAX1 997.6 3.20 2.00 4.37 4.10 0.25xe 0.75xe 67.10 54.80 15.06 Type 3-B-MAX2 997.6 3.20 2.00 4.73 4.10 0.50xe 0.50xe 67.87 55.42 14.80 Type 3-B-MAX3 997.6 3.20 2.00 5.22 4.10 0.75xe 0.25xe 68.53 55.87 14.72Numerical simulations of the four tubes under the boundary conditions mentioned above were carried out. The force-crushing distance curves and deformation modes are presented in Figures15(a) and 15(b), respectively. First, as shown in Figure 15(b), all four tubes were crushed in progressive and regular deformation modes. The folding lobes of the three Type 3 DSG tubes were slightly larger than those of the Type 1 DSG tube. Second, the simulation results listed in Table 4 show that, compared to the traditional square tube, the improvements in terms of CFE and SEA for the Type 1-B-MAX tube were 23.7% and 17.9%, respectively. Greater improvements in these two indexes were obtained by the Type 3 DSG tubes, with values of 41.9% and 30.6%, respectively. Additionally, for the three Type 3 DSG tubes with wg+we=xe, the mean load increased slightly with an increasing value of wg, while the SEA decreased slightly. This decrease is related to the effective crushing distance, which decreases with an increasing value of wg, as shown in the response curves in Figure 15(a). Therefore, although wg has a slight effect on the crushing responses, from the perspective of energy absorption efficiency, a smaller wg value is required to guarantee a gentle thickness gradient change. Finally, under the same geometric constraints with constant values of ta and t2, a higher value of te was achieved by the Type 3 DSG tubes compared to the Type 1 DSG tube, which led to a higher energy absorption efficiency. Therefore, the key to improving the crushing performance of a DSG tube is an appropriate material distribution type.Figure 15
Crushing responses of specimens with dimensional constraints: (a) force-crushing distance curves and (b) folding patterns.


## 5.1. Improvement of the Type 2 Material Distribution

As discussed above, for a Type 2 DSG tube with a constant global average thickness, concentrating material within the EEAR will lead to a significant decrease in the minimum thickness (t2), which limits the promotion of the average thickness of the EEAR (te). Therefore, a modified material distribution type should maximize the value of te under such limitations. Based on this principle, the Type 2 material distribution was modified to create a new distribution.The geometric configuration of this new material distribution type (Type 3) is illustrated in Figure14. As shown in the figure, the non-EEAR was given a uniform thickness equal to the minimum thickness t2, which ensures that the EEAR contains the most material for a fixed value of t2. Additionally, the EEAR has a uniform thickness area with a width of we and gradient thickness area with a width of wg, as shown in Figure 14. In other words, the width of the EEAR xe is equal to the sum of we and wg. The crushing response improvements provided by the Type 3 material distribution are discussed below.Figure 14
Material distribution of Type 3 tubes.


## 5.2. Crushing Response Comparison

In theory, for two DSG tubes with the same cross-sectional width and global average thickness, a higher energy absorption efficiency is achieved by the tube with a largerte value. The te value for a Type 1 or Type 3 DSG tube can be obtained easily by using the relationships between its geometric parameters, as shown in Table 3. Additionally, it is necessary to determine if a modified Type 2 DSG tube (i.e., Type 3 DSG tube) is more efficient than a Type 1 DSG tube in terms of energy absorption for the same geometric constraints, namely, cross-sectional width (B), global average thickness (ta), and minimum thickness (t2). The geometric constraints ofB= 90 mm, ta= 3.2 mm, and t2 = 2.0 mm were used for the two types of DSG tubes. The maximum te values are listed in Table 4. The Type 1 DSG tube with geometric constraints was labelled Type 1-B-MAX. Similarly, considering the same geometric constraints, three Type 3 DSG tubes with wg+we=xe were established, where the wg values were set to 0.25xe, 0.5xe, and 0.75xe to investigate the effects of this parameter. These three tubes were labelled Type 3-B-MAX1, Type 3-B-MAX2, and Type 3-B-MAX3, respectively. All the dimensions of the Type 1 and three Type 3 tubes are listed in Table 4.Table 3
Relationships between geometric parameters.
Gradient thickness type Width of the EEAR Widths of the gradient thickness area and uniform thickness area Maximum thickness Average thickness of the EEAR Type 1 x e = 0.95 H w g = B 2; t 1 = 2 t a - t 2 t e = t 2 - t 1 x e B + t 1 w e = 0 Type 3 x e = 0.95 H w g + w e = x e t 1 = B t a - t 2 + t 2 x e + t 2 w e x e + w e t e = B ( t a - t 2 ) 2 x e + t 2Table 4
Dimensions and crushing responses of two types of specimens with dimensional constraints.
Specimen Dimension Simulation result Constraint Mass (g) t a (mm) t 2 (mm) t 1 (mm) t e (mm) w g w e P m (kN) CFE (%) SEA (kJ/kg) B-TRA 997.6 3.20 3.20 3.20 3.20 0 B / 2 50.66 39.35 11.53 Type 1-B-MAX 997.6 3.20 2.00 4.40 3.71 B / 2 0 62.35 48.71 13.60 Type 3-B-MAX1 997.6 3.20 2.00 4.37 4.10 0.25xe 0.75xe 67.10 54.80 15.06 Type 3-B-MAX2 997.6 3.20 2.00 4.73 4.10 0.50xe 0.50xe 67.87 55.42 14.80 Type 3-B-MAX3 997.6 3.20 2.00 5.22 4.10 0.75xe 0.25xe 68.53 55.87 14.72Numerical simulations of the four tubes under the boundary conditions mentioned above were carried out. The force-crushing distance curves and deformation modes are presented in Figures15(a) and 15(b), respectively. First, as shown in Figure 15(b), all four tubes were crushed in progressive and regular deformation modes. The folding lobes of the three Type 3 DSG tubes were slightly larger than those of the Type 1 DSG tube. Second, the simulation results listed in Table 4 show that, compared to the traditional square tube, the improvements in terms of CFE and SEA for the Type 1-B-MAX tube were 23.7% and 17.9%, respectively. Greater improvements in these two indexes were obtained by the Type 3 DSG tubes, with values of 41.9% and 30.6%, respectively. Additionally, for the three Type 3 DSG tubes with wg+we=xe, the mean load increased slightly with an increasing value of wg, while the SEA decreased slightly. This decrease is related to the effective crushing distance, which decreases with an increasing value of wg, as shown in the response curves in Figure 15(a). Therefore, although wg has a slight effect on the crushing responses, from the perspective of energy absorption efficiency, a smaller wg value is required to guarantee a gentle thickness gradient change. Finally, under the same geometric constraints with constant values of ta and t2, a higher value of te was achieved by the Type 3 DSG tubes compared to the Type 1 DSG tube, which led to a higher energy absorption efficiency. Therefore, the key to improving the crushing performance of a DSG tube is an appropriate material distribution type.Figure 15
Crushing responses of specimens with dimensional constraints: (a) force-crushing distance curves and (b) folding patterns.


## 6. Conclusion

DSG square tubes were constructed by concentrating additional material in the corner regions by changing the cross-sectional material distributions. The energy dissipating performance of the DSG square structures under an axial dynamic load was investigated both theoretically and numerically.A mathematical model for the mean crushing force of the DSG square columns was developed. Two key parameters, namely, the global average thickness (ta) and average thickness of the EEAR (te), were proposed. The analytical prediction results were in good agreement with the simulation results. The influences of two key parameters on the impact responses of the columns were discussed.For the same length, section width, and global average thickness of columns, a largerte value led to a greater mean crushing load. Similarly, when the other parameters remained unchanged, a smaller ta value led to a lower initial peak force for the columns. Therefore, a larger coefficient of te versus ta results in a column with excellent energy absorption efficiency and better crashworthiness performance.To investigate the influence of the cross-sectional material distribution law on increasing the coefficient ofte/ta, two material distribution types (Type 1 and Type 2) were presented, and their characteristics were studied. First, Type 2 was more efficient than Type 1 for promoting te/ta. The te/ta of each Type 2 tube is larger than the Type 1 tube with the same thickness gradient as the Type 2 tube. In addition, a thinner thickness of the middle of each edge was found in cases with a large thickness gradient of Type 2 tubes, not Type 1 tubes. Thus, Type 2 tubes more easily deform in irregular modes. Therefore, a modified material distribution type (Type 3) was established, which maintained the advantages of Type 2 and remedied its shortcomings.An excellent performance of energy absorption efficiency was achieved by Type 3 tubes, and they were not deformed in the irregular deformation mode that occurred on Type 2 tubes. The SEA values of Type 3 tubes with dimensional constraints increased by 30.6% compared to those of traditional square tubes, which is higher than those of the other two types. Therefore, the Type 3 is superior in the three types.In summary, as a novel design, the DSG square tube is a superior structure that can meet increasingly stringent requirements for energy absorption efficiency. Although more extensive investigations must be carried out, this study presents a advancement in the comprehensive understanding of the folding mechanisms of square tubes with graded thicknesses, which is advantageous for the design improvement of such structures in terms of deriving more efficient energy absorbing devices.



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*Source: 4706563-2019-05-09.xml*