Geometric optimization of the crashworthiness performance of 3D-printed PLA-GF tubes with square-to-circular transitions via squircle geometry

Abstract

This study explores the crashworthiness performance of 3D-printed tubes composed of polylactic acid reinforced with glass fiber (PLA-GF), with particular emphasis on the geometric transition from square to circular cross-sections through the use of arc fillets. Tubes with profiles ranging from sharp-cornered squares to fully circular shapes, including intermediate transitional designs, were subjected to quasi-static axial compression testing to evaluate their structural performance. During testing, the load-displacement responses were systematically recorded, and the corresponding failure modes were carefully examined. The crashworthiness analysis focused on evaluating key indicators, including initial peak force ( $F_{ip})$ , total absorbed energy ( $E_T$ ), average force ( $F_{avg}$ ), crash force efficiency (CFE), and specific energy absorption (SEA). Additionally, to identify the most effective geometry, a multi-attribute decision-making framework based on the Complex Proportional Assessment (COPRAS) method was employed. The results revealed that the circular configuration achieved substantial enhancements compared with the square tube, including a 37% increase in $F_{avg}$ , a 41% increase in $E_T$ , a 33% improvement in CFE, and a 70% increase in SEA. Additionally, the SR25 structure exhibited notable improvements over the square tube, particularly in SEA (+40%) and CFE (+24%), highlighting the benefits of intermediate geometric transitions. The COPRAS evaluation further identified the circular tube as the most favorable configuration, attributed to its well-balanced across all evaluated indicators. This work provides clear experimental evidence that controlled geometric transitions between square and circular profiles can significantly affect the crashworthiness of 3D-printed PLA-GF structures.

Introduction

Crashworthiness plays an essential role in the design and development of structures subjected to crash forces, particularly within critical industries such as automotive, aerospace, and defense. It fundamentally refers to a structure’s capacity to effectively manage the absorption and dissipation of kinetic energy during collisions, thereby reducing the magnitude of forces transmitted to occupants, sensitive components, or essential systems.^1,2 This capability not only protects human life but also preserves structural integrity and functionality after crash events. For automotive manufacturers, achieving high crashworthiness performance is vital to observe with stringent safety regulations imposed by governments and international agencies worldwide. Compliance with these regulations is mandatory for obtaining vehicle certifications required before market release in different regions.^3,4 Prominent regulatory bodies, such as the National Highway Traffic Safety Administration (NHTSA) in the United States and the European New Car Assessment Programme (Euro NCAP), have established comprehensive crash testing protocols. These protocols carefully evaluate vehicle safety performance through multiple standardized tests, including frontal, side, and rollover impacts. The resulting safety ratings not only ensure public safety but also heavily influence consumer preferences and purchasing decisions, thereby impacting a manufacturer’s reputation and market competitiveness.^5,6

In the design of the crashworthy components, the careful selection of materials and structural geometries is paramount to efficiently managing and dissipating impact energy during collisions.^1,7,8 The core objective is to achieve controlled and predictable deformation of the vehicle structure, which absorbs the crash energy while minimizing the transmission of harmful forces to the occupants, thereby enhancing passenger safety.^9–12 Thin-walled tubes have been widely employed as primary energy-absorbing structures due to their exceptional energy absorption-to-weight ratios and highly predictable, stable deformation behaviors under impact.^13–16 Their slender walls enable efficient dissipation of kinetic energy through controlled buckling and folding mechanisms, making them ideal for use in crashworthiness applications where both lightweight design and reliable energy management are critical. ¹⁷ A range of thin-walled geometry tubes have been thoroughly studied in recent decades. Nia and Hamedani ¹⁸ examined how thin-walled tubes of various cross-sectional shapes (circular, square, rectangular, hexagonal, triangular, pyramidal, and conical) deform and absorb energy when subjected to quasi-static axial force. Pyramidal and conical tubes provided more steady load-displacement behavior, making them more appropriate for impact applications, although circular tubes demonstrated the largest energy absorption.

Chen and Masuda ¹⁹ investigated the crash behavior of hexagonal thin-walled tubes with internal partition plates under axial compression using finite element analysis. Their results showed that, during the crash process, folds form along the entire tube length and collapse simultaneously. Under this condition, the compressive force does not significantly decrease because the force contributed by the central region remains stable even as folds develop on the outer walls. Consequently, to minimize fluctuations in the compression force and enhance the average crash force, the introduction of corner features, particularly those formed by the intersection of three plates, was identified as an effective geometric modification for thin-walled tubes. Xie et al. ²⁰ claim that filleting the corners of square tubes significantly reduces peak forces and enhances energy absorption. As fillet radius increases, both specific energy absorption and crash force efficiency improve, especially in thicker-walled tubes. Shojaeefard et al. ²¹ confirmed that combined (square-to-circular) sections have a lower initial peak force than pure square or circular sections, and their energy absorption is significantly higher than square sections and comparable to circular ones. Istiyanto et al. ²² combined the simulations and experiments to examine how varying corner radius (0-3 mm) in square thin-walled steel structures affects crashworthiness. Results show that increasing the radius reduces peak force and energy absorption, with simulation errors under 3%.

Material selection is another critical factor for crashworthy structures that requires a delicate balance among multiple factors, including energy absorption capacity, weight reduction, manufacturability, and cost-effectiveness.^23,24 Traditionally, steels and aluminum have been extensively utilized due to their excellent ability to undergo controlled plastic deformation during impact events. These metals effectively dissipate kinetic energy, reducing occupant injury risk by lowering peak force transmission. ²⁵ Recently, polymeric materials have gained increasing attention in automotive and aerospace industries because of their unique combination of high specific strength, stiffness, and remarkable energy absorption capabilities.^26–28 Polymers present significant advantages in applications where weight savings are critical to improving fuel efficiency and overall vehicle dynamics without compromising structural integrity.^29–31 Their inherently lighter mass compared to metals leads to reduced vehicle weight, which contributes not only to fuel economy but also to lower emissions and enhanced handling performance. Polymers used in crashworthiness applications are broadly categorized into two primary classes: thermoset polymers and thermoplastic polymers. Thermosets, in particular, have emerged as a preferred choice for many crashworthy components due to their high mechanical strength, durability, and resistance to elevated temperatures, making them well-suited for demanding impact conditions.^32,33 Moreover, thermoplastic polymers offer the significant benefit of high recyclability, making them a more environmentally sustainable choice compared to many alternative materials. This environmental advantage has encouraged increased research into utilizing thermoplastic polymers for energy-absorbing components.^34–36 Although there are multiple techniques available for fabricating thermoplastic structures, the traditional methods often require several complex steps and tend to incur higher manufacturing costs. However, with recent technological advancements, 3D printing has emerged as a more efficient and cost-effective method for producing intricate cellular structures. Among the various additive manufacturing (AM) technologies currently being explored and implemented, Fused Deposition Modeling (FDM) is one of the most widely used approaches.^37–40

Recent advances in AM have revolutionized the design of crashworthy structures by enabling the fabrication of complex, customized geometries that were previously difficult or impossible to produce using traditional manufacturing methods. In particular, AM allows the integration of high-performance polymers reinforced with fibers, combining lightweight design with enhanced mechanical strength and energy absorption capabilities. These capabilities provide exceptional opportunities to optimize structural performance for impact resistance while reducing material usage and enabling tailored, application-specific solutions.^41–43

Building upon the literature reviewed above, square and circular cross-sections are two of the most widely utilized geometries, and each has unique benefits and drawbacks. In modular systems, square tubes are easier to align and package and usually have a higher starting stiffness. Sharp corners in square profiles, however, frequently result in localized stress concentrations that weaken deformation stability and cause early local buckling. ⁴⁴ Conversely, circular tubes offer a more consistent distribution of stress during crashing, which results in stable axisymmetric folding and reduced peak crash force. ⁴⁵ Despite extensive research on square and circular tubes, no experimental study, to the authors’ knowledge, has investigated the geometric transition between these profiles, particularly for 3D-printed thin-walled structures made from fiber-reinforced thermoplastic polymers. To address this gap, the present study examines the axial crash behavior of 3D-printed polylactic acid reinforced with glass fiber (PLA-GF) tubes incorporating arc fillets. A range of geometries, including square, circular, and intermediate squircle-type transitional profiles, was systematically evaluated to determine their effect on crashworthiness. Key indicators, such as initial peak force ( $F_{ip})$ , total absorbed energy (E _T ), average force ( $F_{\mathit{a}\mathit{v}\mathit{g}}$ ), crash force efficiency (CFE), and specific energy absorption (SEA), were quantified through quasi-static compression testing. Additionally, the Complex Proportional Assessment (COPRAS) method was applied to rank the tube configurations based on the experimental data.

Methodology

Material

In this study, PLA-GF filament sourced from eSUN Co. Ltd (China) was used to fabricate tubes with cross-sectional transitions from square to circular via varying fillet radii. PLA-GF is a composite of conventional PLA reinforced with short glass fibers, which significantly enhances its mechanical, thermal, and dimensional properties, making it suitable for functional and semi-structural additive manufacturing applications.^46,47 While neat PLA is biodegradable, easy to print, and dimensionally accurate, it suffers from low toughness, brittleness, and limited thermal resistance. ⁴⁸ Incorporating glass fibers addresses these limitations, producing a composite with higher tensile strength, stiffness, impact resistance, and durability under compressive, tensile, and flexural loads. ⁴⁹ PLA-GF also improves dimensional stability by reducing shrinkage and warpage during printing, ensuring precise geometries critical for interlocking components and energy-absorbing structures. ⁵⁰ Its thermal performance is similarly enhanced; whereas neat PLA has a heat deflection temperature of 55–60°C, PLA-GF maintains mechanical integrity at temperatures up to 150°C, extending its applicability to moderately heated components such as automotive fixtures and mechanical housings.^51,52 From a processing perspective, PLA-GF retains the printability of conventional PLA and is compatible with standard FDM 3D printers, though hardened steel or ruby-tipped nozzles are recommended due to the abrasive nature of glass fibers. ⁵³ Environmentally, PLA-GF remains partially biodegradable, offering a more sustainable alternative to fully synthetic composites such as carbon fiber-reinforced ABS or PETG.^54,55 Table 1 provides an overview of PLA-GF’s mechanical and physical characteristics as reported by the manufacturer.

Table 1.

Mechanical and physical characteristics of PLA-GF Filament.

Property	Unit	Value
Filament diameter	mm	1.75 ± 0.05
Density	g/cm3	1.31
Melting point	°C	155–185
Tensile strength	MPa	59.27
Elongation at break	%	7.99
Flexural strength	MPa	85.01
Flexural modulus	MPa	4414.89
Impact strength (notched)	kJ/m2	10.16
Print speed	mm/s	40–100
Printing temperature	°C	190–230

Specimen preparation

PLA-GF tubes were produced using a Prusa® i3 MK3 3D printer (build volume 250 × 211 × 218 mm) via FDM. In this process, filament is continuously fed into a heated nozzle, melted, and deposited layer by layer following a predetermined toolpath to gradually form the CAD geometry. Slicing software controlled critical printing parameters, including nozzle temperature, bed temperature, layer thickness, print speed, and deposition path, ensuring mechanical integrity, dimensional accuracy, and surface quality. Precise control of layer height in the z-direction ensured uniform, stable layers, contributing to the overall structural stability of the specimens.

The printing speed in this work was set at 80 mm/s, a moderate speed balancing part quality and manufacturing efficiency. Higher speeds reduce build time but can cause geometric errors, surface roughness, and poor interlayer bonding, while lower speeds improve precision but increase fabrication time. The selected speed ensures consistent extrusion of PLA-GF, minimizing defects such as warping, delamination, or under-extrusion, while meeting project time and experimental requirements. Nozzle and bed temperatures were set to 220°C and 75°C, respectively, to ensure proper filament viscosity, smooth extrusion, strong layer bonding, and bed adhesion. The ambient temperature was maintained at 25°C to prevent uneven cooling, warping, or delamination, thereby improving print quality and reliability. ⁵⁶ The gyroid infill pattern was chosen for its complex, triply periodic minimal surface (TPMS) architecture, inspired by biological structures like butterfly wings. ⁵⁷ This geometry provides an efficient balance of strength, weight, and energy absorption, uniformly distributing stress and enhancing structural stability. Its excellent mechanical performance, material efficiency, and environmental benefits, through reduced material usage, make it a preferred infill strategy in high-performance, lightweight 3D-printed components.^58–60 All specimens were printed under identical machine and environmental conditions, including temperature, humidity, layer thickness, and infill settings, to ensure consistency across samples.

To systematically investigate the effect of geometric transitions on energy absorption behavior, a series of specimens with varying cross-sectional profiles were designed. As illustrated in Figure 1, the four sharp corners of the square cross-section were replaced with arc fillets of uniform radius, R. When R = 0, the cross-section remains a perfect square, whereas increasing R progressively transforms the geometry toward a circle, achieving a fully circular profile when R equals half the tube width. The fillet radii (5-30 mm) were chosen to provide a controlled transition from square to circular profiles while ensuring manufacturability and structural integrity, and wall thickness values were selected based on preliminary empirical trials. As noted by Deleo and Feraboli ⁶¹ and Browne et al., ⁶² there is no standardized dimension or shape for crashworthy structures, highlighting the importance of tailoring design parameters to meet specific study goals. The mass of the fabricated specimens is listed in Table 2, whereas their detailed geometry and dimensional specifications are illustrated in Figure 1.

Figure 1.

Representation for shape and dimensions of printed tubes.

Table 2.

Specimen dimensions used in the experimental test.

Specimen code	Square	SR5	SR10	SR15	SR20	SR25	Circle
Mass, g	44.73	43.53	41.92	40.27	38.94	37.55	34.95

Note. Square denotes the sharp-cornered square tube, Circle represents the fully circular tube, and the intermediate specimens are labeled SR5, SR10, SR15, SR20, and SR25, where the number indicates the fillet radius in millimeters used to transition from the square to circular cross-section.

Quasi-static compression testing

PLA-GF 3D-printed tubes were tested for quasi-static axial compression using a standard testing machine (Type Jinan WDW, China) with a 100 kN capacity. The crosshead speed of 10 mm/min was chosen in compliance with established procedures from earlier investigations on the crashworthiness of comparable tubular structures,^24,63 providing a controlled quasi-static loading environment suitable for analyzing progressive folding, deformation behavior, and energy absorption characteristics. As mentioned by Awd Allah et al. ¹³ and Abd El Aal et al., ⁶⁴ there is no universally accepted standard for quasi-static compression testing; nonetheless, such investigations are essential for understanding material response under slow, controlled loading, which approximates scenarios involving gradual impacts or sustained static forces.^65,66

During testing, the printed tubes were carefully positioned between two parallel steel plates to ensure uniform load distribution and controlled testing conditions during quasi-static compression. Load-displacement data were continuously recorded using an automated system, enabling real-time monitoring of the tubes’ deformation and capturing critical information on energy absorption, structural response, and failure modes. Tests were also video-recorded to provide qualitative observations of structural changes. For each design, three identical specimens were tested to ensure repeatability and statistical reliability. These procedures allowed for a comprehensive assessment of the tubes’ crashworthiness, from quantitative load-displacement metrics to key performance indicators derived from the experimental data.

• The initial peak force ( $F_{ip}$ ) represents the maximum force detected at the end of the pre-crash stage, is directly obtained from the load-displacement curve. This value is critically important, as it is closely associated with the potential severity of passenger injury in crash scenarios.^67,68

• Total absorbed energy ( $E_T$ ) is defined as the total energy absorbed during the crash process. It stands for the area below the load-displacement curve.

$E_T={\int }_0^{{\mathrm{l}}_{\max }}\hspace{0.17em}\mathrm{F}\left(\mathrm{l}\right)\hspace{1em}\text{dl}$

(1)

where F(l), ${\mathrm{l}}_{\max }$ are the instantaneous crash force and overall crash displacement.

• • The average force ( $F_{avg}$ ) is the average force sustained over the total crash distance. Throughout the deformation process, $F_{avg}$ should stay as high as feasible for efficient energy absorption.^69,70

${\mathrm{F}}_{\text{avg}}=\frac{{\int }_0^{{\mathrm{l}}_{\max }}\hspace{0.17em}\mathrm{F}\left(\mathrm{l}\right)\hspace{1em}\text{dl}\hspace{0.17em}}{{\mathrm{l}}_{\max }}$

(2)

• • Crashing force efficiency (CFE) is defined as the ratio of ${\mathrm{F}}_{\text{avg}}$ to $F_{ip}$ . A high CFE indicates steady and effective energy absorption, contributing to reduced occupant injury and improved crash performance. ⁷¹

$\text{CFE}=\frac{{\mathrm{F}}_{\text{avg}}}{{\mathrm{F}}_{\mathrm{ip}}}$

(3)

• • Specific energy absorbed (SEA) quantifies the energy absorbed per unit mass of the specimen, serving as a key indicator for lightweight design. ⁷²

$\text{SEA}=\frac{E_T}{{\hspace{0.17em}\mathrm{m}}_{\mathrm{c}}}$

(4)

where the mass of the crashed portion, ${\hspace{0.17em}\mathrm{m}}_{\mathrm{c}}$ , is calculated as:

${\hspace{0.17em}\mathrm{m}}_{\mathrm{c}}=\left(\frac{\mathrm{M}}{\mathrm{L}}\right){\mathrm{l}}_{\max }$

(5)

where, $\mathrm{M}\hspace{0.17em}\text{and}\hspace{0.17em}L$ are the original mass and length of the specimen, respectively.

Decision making

When multiple options are available, Multi-Attribute Decision-Making (MADM) procedures are frequently used to determine the best combination of alternatives. The Complex Proportional Assessment (COPRAS) method was selected for this study because it is straightforward, efficient, and particularly suitable for situations involving several conflicting criteria, such as assessing crash performance across different design parameters. ⁷³ COPRAS compares each alternative’s performance relative to the others, providing a clear and systematic ranking that facilitates identification of the combination of factors resulting in the best overall performance. By evaluating multiple design parameters simultaneously, COPRAS offers a comprehensive framework for decision-making. A notable strength of the method is its ability to handle both qualitative and quantitative data, enabling holistic assessment without requiring complex mathematical models and producing clear, interpretable results. For a detailed explanation of COPRAS, refer to 74,75, which outline the procedure for calculating and interpreting the results. Below is a description of how COPRAS is used:

Step 1: Create the initial decision matrix X.

$\mathrm{X}={\left[{\mathrm{x}}_{\text{ij}}\right]}_{\text{mn}}=\left[\begin{array}{lllll}{\mathrm{x}}_{11}& \dots & {\mathrm{x}}_{1\mathrm{j}}& \dots & {\mathrm{x}}_{1\mathrm{n}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ {\mathrm{x}}_{\mathrm{i}1}& \dots & {\mathrm{x}}_{\text{ij}}& \dots & {\mathrm{x}}_{\text{in}}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ {\mathrm{x}}_{\mathrm{m}1}& \dots & {\mathrm{x}}_{\text{mj}}& \dots & {\mathrm{x}}_{\text{mn}}\end{array}\right];\mathrm{i}=1,\dots ,\mathrm{m},\mathrm{j}=1,\dots ,\mathrm{n}$

(6)

where n and m stand for the number of indicators and competitors, respectively. The performance of the ${\mathrm{i}}_{\text{th}}$ tube in relation to the ${\mathrm{j}}_{\text{th}}$ indication is thus shown by ${\mathrm{x}}_{ij}$ .

• Step 2: Convert matrix X into matrix R, which is zero-dimensional.

$R={\left[r_{ij}\right]}_{mn}=\frac{x_{ij}}{{\displaystyle \sum }_{i=1}^m{x}_{ij}\hspace{0.17em}}$

(7)

where the normalized ${\mathrm{j}}_{\text{th}}$ indicator for ${\mathrm{i}}_{\text{th}}$ design is denoted by ${\mathrm{r}}_{ij}$ .

• Step 3: Compute the weightage of each attribute separately ( ${\mathrm{w}}_j)$ .

${\mathrm{W}}_j={\displaystyle \sum }_{i=1}^{\mathrm{N}}{\mathrm{w}}_{ij}$

(8)

${\mathrm{w}}_j=\frac{{\mathrm{W}}_j}{\mathrm{H}}=\frac{{\mathrm{W}}_j}{{\displaystyle \sum }_{\mathrm{j}=1}^N{\mathrm{W}}_j\hspace{0.17em}},\mathrm{N}=\left(\mathrm{n}\hspace{0.17em}\left(\mathrm{n}-1\right)/2\right)$

(9)

• Step 4: Multiply the matrix $R$ by the weight of each consistent indication to obtain matrix $D$ , which includes the weightage of each indicator.

$D={\left[y_{ij}\right]}_{mn}=r_{ij}\hspace{0.5em}x\hspace{0.17em}{\hspace{0.17em}w}_j$

(10)

where ${\mathrm{y}}_{ij}$ is the weighted normalized ${\mathrm{j}}_{\text{th}}$ indicator for ${\mathrm{i}}_{\text{th}}$ design.

• Step 5: In matrix D, positive and negative signals are represented by the symbols $+{\mathrm{y}}_{ij}$ and $-{\mathrm{y}}_{ij}$ , respectively. A credible competitor in design is one with a higher positive indicator value and a lower negative indicator value. The sum of the positive ( ${\mathrm{S}}_{+\mathrm{i}}$ ) and negative ( ${\mathrm{S}}_{-\mathrm{i}}$ ) crash indicators for each design choice would therefore be obtained using Equations (11) and (12).

${\mathrm{S}}_{+i}={\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}+{\mathrm{y}}_{ij}\hspace{1em}i=1,\dots ,\mathrm{m}$

(11)

${\mathrm{S}}_{-i}={\displaystyle \sum }_{j=1}^n-{\mathrm{y}}_{ij}\hspace{1em}i=1,\dots ,m$

(12)

• Step 6: Using Equations (13) and (14), describe the relative importance ( ${\mathrm{Q}}_{\mathrm{i}}$ ) and the quantitative value ( ${\mathrm{U}}_i$ ) for individually design attribute.

${\mathrm{Q}}_i={\mathrm{S}}_{+i}+\frac{{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{m}}{\hspace{0.5em}\mathrm{S}}_{-\mathrm{i}}\hspace{0.17em}}{S_{-i}\hspace{0.17em}{\displaystyle \sum }_{i=1}^m\hspace{0.17em}\frac{1}{S_{-i}\hspace{0.17em}}}$

(13)

${\mathrm{U}}_{\mathrm{i}}=\frac{{\mathrm{Q}}_{\mathrm{i}}}{{\mathrm{Q}}_{\max }}$

(14)

Results and discussion

Crash response and deformation history

Figures 2–8 present the axial load-displacement, energy-displacement, and failure history of the designed specimens under quasi-static compression. These data provide insight into how the geometric transition from square to circular profiles influences the tubes’ load-carrying capacity and energy absorption behavior. As shown in Figures 2–8(a), the crash load increases sharply to reach the initial peak force, followed by fluctuations around the average force during the post-peak phase. These variations are attributed to progressive, localized damage mechanisms, including fiber breakage, interfacial debonding, delamination, and matrix cracking. A secondary increase in load is observed as the tubes approach the compaction stage, where mechanical resistance is dominated by densified material and direct platen contact rather than the intact composite, consistent with observations reported by Abdewi et al. ⁷⁶ It was also observed that the energy increased nonlinearly with displacement for all manufactured tubes as shown in Figures 2–8 (a). This nonlinear behavior highlights the complex interplay between geometry, material properties, and structural stability.

Figure 2.

(a) Load-displacement, energy absorption-displacement curves, (b) top view of crashed specimen, and (c) deformation history of square tube.

Figure 3.

(a) Load-displacement, energy absorption-displacement curves, (b) top view of crashed specimen, and (c) deformation history of SR5 tube.

Figure 4.

(a) Load-displacement, energy absorption-displacement curves, (b) top view of crashed specimen, and (c) deformation history of SR10 tube.

Figure 5.

(a) Load-displacement, energy absorption-displacement curves, (b) top view of crashed specimen, and (c) deformation history of SR15 tube.

Figure 6.

(a) Load-displacement, energy absorption-displacement curves, (b) top view of crashed specimen, and (c) deformation history of SR20 tube.

Figure 7.

(a) Load-displacement, energy absorption-displacement curves, (b) top view of crashed specimen, and (c) deformation history of SR25 tube.

Figure 8.

(a) Load-displacement and energy absorption-displacement curves, (b) deformation history, and (c) top view of the crashed circular tube.

The damage processes were exposed by analyzing the specimen photos taken during and after the test as shown in Figures 2–8 (b) and (c). As observed in Figures 2–6 (b) and (c), the specimens with square and small fillet radius corners exhibit clear signs of weak interlayer adhesion, evidenced by wall fragmentation and the outward separation of broken pieces. Additionally, the presence of internal diagonal fissures and inward buckling patterns suggests that local buckling dominated the initial stage of collapse. This localized instability ultimately led to catastrophic tearing at the corners regions known for their high stress concentrations.^77,78 On the other hand, the photographs of the SR25 and circular specimens (Figures 7 and 8 (b) and (c)) reveal that the tubes initially underwent uniform axial shortening. As compression progressed, localized buckling emerged along the tube walls, accompanied by outward expansion at the mid-height region. The upper and lower portions exhibited pronounced delamination along the vertical fiber direction, reflecting the progressive failure of fiber-matrix interfaces. Throughout the deformation, matrix cracking and fiber breakage were observed, while delamination propagated both radially and longitudinally, enabling gradual energy dissipation.

Crashworthiness indicators

Initial peak and average crash forces ( $F_{ip}$ , $F_{avg}$ )

The initial peak force is a critical indicator of the initial shock transmitted during a crash, with lower values generally being more desirable as they reduce the initial impact load on occupants or connected structures. ¹³ Figure 9 compares the performance of square, fillet-radius-modified square (SR5-SR25), and circular tube specimens in terms of $F_{ip}$ and $F_{avg}$ . Among the tested specimens, the SR5 and SR15 configurations exhibit the highest $F_{ip}$ of 9.52 and 9.01 kN, respectively, indicating the greatest initial stiffness or resistance to collapse; however, this may result in a less desirable, harsh initial impact. However, these same specimens show lower average force with values of 3.10 kN and 3.17 kN, suggesting less stable progressive deformation and potential for catastrophic failure after the $F_{ip}$ is reached. These results agree with Istiyanto et al. ²² who demonstrate that tubes with smaller corner radius exhibit reduced $F_{avg}$ due to both geometric transition from circular to square and corner-thickness thinning during forming. In contrast, the Square and SR25 specimens show the lowest $F_{ip}$ with 6.78 and 6.42 kN, respectively, suggesting they are more effective at minimizing the initial shock. The Circle specimen displays the highest $F_{avg}$ of 5.92 kN, highlighting its efficiency in absorbing energy over the entire crash distance. Other specimens, such as Square and SR20, show relatively high average forces (approximately 4.5-5.5 kN), second only to the Circle. The results indicate that, although tubes with small fillet radii can withstand higher peak forces at the onset of compression, rounded or circular geometries provide superior overall crashworthiness by sustaining higher average forces over the deformation process and minimizing abrupt failure events. The results strongly align with prior findings by Shojaeefard et al. ²¹ and Vishwanath et al. ⁷⁹

Figure 9.

$F_{ip}$ and $F_{avg}$ of the designed tubes.

Total absorbed energy ( $E_T$ )

Figure 10 presents the total absorbed energy of various designed tube specimens subjected to quasi-static axial compression test. The Square specimen demonstrates moderate $E_T$ (266.49 J), which is likely attributed to localized buckling and early collapse initiation at the sharp corners. Rounded-square specimens with small fillet radius (SR5, SR10, SR15) exhibit lower $E_T$ performance, at values of 224.97, 248.20, and 235.05 J, respectively. This suggests that minor geometric modifications may introduce instability without significantly enhancing deformation mechanisms. However, as the corner radius increases further (SR20 and SR25), $E_T$ shows a marked improvement, reaching to 327 and 333.28 J respectively. The Circular specimen absorbs the highest energy (375.51 J), confirming its superior crashworthiness. These observations are consistent with Mamalis et al., ⁸⁰ who demonstrated that circular profiles promote symmetric and ductile collapse modes that result in higher energy dissipation compared to polygonal cross-sections. Similarly, Alghamdi ⁸¹ and Abramowicz and Jones ⁷⁷ highlighted that stress concentrations at corners in square tubes can lead to brittle or unstable collapse, while smooth geometries support progressive folding and plastic hinge formation.

Figure 10.

$E_T$ of the designed tubes.

From a mechanical standpoint, energy absorption during axial collapse primarily involves plastic bending and frictional work between folding lobes. Longer, more stable deformation strokes and a more even distribution of stresses are achievable by circular and rounded profiles. Furthermore, the observed trend also supports the work of Hanssen et al., ⁸² who showed through simulation that circular tubes maintain consistent deformation modes across different loading conditions. Unlike polygonal sections which exhibit variable failure modes depending on the corner geometry and loading axis. ⁷⁷ Overall, the results emphasize the importance of optimizing cross-sectional geometry, particularly corner radius, as a design parameter in crashworthy structures. The transition from square to circular profiles leads to a more ductile and energy-efficient collapse, making rounded and circular tubes ideal for applications such as automotive crash boxes, aerospace structures, and railway crash energy absorbers.

Crashing force efficiency (CFE)

CFE quantifies the consistency of load distribution during the crash process, plays a fundamental role in the performance of energy-absorbing structures. ⁸³ Values approaching 100% indicate that the structure maintains a nearly constant force near the peak, minimizing sudden deceleration and reducing the risk of structural instability or occupant injury. ⁸⁰ Figure 11 illustrates the CFE of the designed tubes. Among the specimens, the Circular tube exhibited the highest CFE of 84.86%, highlighting its superior ability to evenly distribute impact forces and effectively absorb crash energy. The sustained, progressive collapse behavior of circular profiles further enhances their crashworthiness, a finding consistent with the observations reported by Alghamdi. ⁸¹ While the Square specimen performed moderately (63.85% CFE), which is consistent with research by Jin and Altenhof, ⁸⁴ who found square tubes to be structurally viable but less efficient than circular ones. Moreover, the SR25 specimen showed excellent performance, with a CFE of 79.38%, suggesting that a higher structural reinforcement ratio greatly improves energy dissipation. This is consistent with the findings of Xie et al., ²⁰ who discovered that the CFE of specimens with a larger fillet radius and circle cross-sections is noticeably higher than that of square ones. In contrast, SR5 and SR15 showed the lowest CFE with 32.57 and 35.17%, respectively, followed by SR10 and SR20 with values of 49.42 and 57.35%, respectively.

Figure 11.

CFE of the designed tubes.

Specific energy absorption (SEA)

Figure 12 illustrates the SEA of different specimen geometries under quasi-static axial compression test. The results in Figure 12 make it abundantly evident that the SEA rises when the cross-sectional geometry transitions from sharp-edged forms, like squares and low-radius (SR5, SR10, SR15) specimens, to more rounded configurations (SR20, SR25), with the circular specimen marking the top value (13.15 J/g). This trend aligns with the findings of Abramowicz and Jones, ⁷⁷ who reported that circular tubes outperform square ones in terms of energy absorption. Similarly, Palanivelu et al. ⁸⁵ founded that circular cross-sections enhance CFE and optimize SEA, making them perfect for lightweight energy-absorbing applications. Overall, the experimental results show good agreement with accepted crashworthiness principles, confirming that geometric optimization, particularly by using circular or high-SR designs, is a very successful method for maximizing specific energy absorption in structural elements meant to reduce impacts.

Figure 12.

SEA of the designed tubes.

Optimum design

The main COPRAS decision matrix, presented in Table 3, includes data from seven uniquely designed specimens, each evaluated for its crashworthiness performance. Five critical crashworthiness indicators were selected for this process: $F_{ip}$ , $E_T$ , $F_{avg}$ , CFE, and SEA. Consequently, a crash indicator matrix (R) was developed, as shown in Table 4. To ensure an unbiased evaluation, each indicator was initially assigned an equal weight of 0.20, reflecting their comparable contribution to crashworthiness. This approach prevents any single criterion from dominating the ranking and provides a balanced assessment of structural performance, as detailed in Table 5. Next, the matrix ( $D$ ), displayed in Table 6, was computed to assess each specimen’s overall crashworthiness performance. During this stage, the indicators were categorized based on their desirability: the $F_{ip}$ was identified as a non-beneficial criterion, where a lower value is preferable to minimize the initial impact force transmitted to occupants. ¹⁵ Conversely, the remaining indicators ( $E_T$ , $F_{avg}$ , CFE, and SEA) were considered beneficial, with higher values indicating improved energy absorption, force distribution, and overall crash performance. Following this classification and application of the assigned weights, the final COPRAS evaluation involved calculating the $Q_i$ and $U_i$ values for each specimen, as summarized in Table 7. These values reflect the overall performance score of each configuration based on the combined effect of the crash indicators. The ranking results indicate that the circular specimen, followed closely by the SR25 specimen, outperformed all other designs, achieving the highest overall score and demonstrating superior crashworthiness characteristics. In contrast, the SR5 specimen ranked lowest, reflecting comparatively poor performance across the evaluated indicators. To verify the robustness of the assigned weights and the consistency of the resulting rankings, a sensitivity analysis was conducted by varying each indicator weight by ± 0.1. The analysis confirmed that the ranking of top-performing specimens, particularly the Circle configuration, remained consistent despite these substantial variations in weighting, demonstrating the reliability and stability of the COPRAS-based evaluation.

Table 3.

Matrix (X).

Specimen code	Attributes
	${\mathrm{F}}_{\text{ip}}$ , kN	${\mathrm{F}}_{\text{avg}}$ , kN	CFE, %	${\mathrm{E}}_{\mathrm{T}}$ , J	SEA, J/g
Square	6.78	4.33	63.85	266.49	7.75
SR5	9.52	3.1	32.57	224.97	5.71
SR10	8.18	4.04	49.42	248.2	7.71
SR15	9.01	3.17	35.17	235.05	6.45
SR20	8.89	5.1	57.35	327	10.49
SR25	6.42	5.09	79.38	333.28	10.86
Circle	6.98	5.92	84.86	375.51	13.15

Table 4.

Matrix (R).

Specimen code	Attributes
	${\mathrm{F}}_{\text{ip}}$	${\mathrm{F}}_{\text{avg}}$	CFE	${\mathrm{E}}_{\mathrm{T}}$	SEA
Square	0.12154894	0.14081301	0.15859414	0.13254912	0.12475853
SR5	0.17067049	0.10081301	0.08089916	0.11189754	0.09191887
SR10	0.14664754	0.13138211	0.12275211	0.12345188	0.12411462
SR15	0.16152743	0.10308943	0.08735718	0.11691122	0.10383129
SR20	0.15937612	0.16585366	0.14244908	0.16264611	0.16886671
SR25	0.11509502	0.16552846	0.19716841	0.16576971	0.17482292
Circle	0.12513446	0.19252033	0.21077993	0.18677443	0.21168706

Table 5.

Distinct weighting for each indicator.

	$N=\frac{n\left(n-1\right)}{2}=\frac{5\left(5-1\right)}{2}=10$
	${\mathrm{F}}_{\text{ip}}$	${\mathrm{F}}_{\text{avg}}$	CFE	${\mathrm{E}}_{\mathrm{T}}$	SEA
$W_{ij}$	2	2	-	-	-
	2	-	2	-	-
	2	-	-	2	-
	2	-	-	-	2
	-	2	2	-	-
	-	2	-	2	-
	-	2	-	-	2
	-	-	2	2	-
	-	-	2	-	2
	-	-	-	2	2
$W_j$	8	8	8	8	8
G	40	-	-	-	-
$w_j$	0.20	0.20	0.20	0.20	0.20

Table 6.

Matrix (D).

Specimen code	Attributes
	${\mathrm{F}}_{\text{ip}}$	${\mathrm{F}}_{\text{avg}}$	CFE	${\mathrm{E}}_{\mathrm{T}}$	SEA
Square	0.0243098	0.028163	0.031719	0.02651	0.024952
SR5	0.0341341	0.020163	0.01618	0.02238	0.018384
SR10	0.0293295	0.026276	0.02455	0.02469	0.024823
SR15	0.0323055	0.020618	0.017471	0.023382	0.020766
SR20	0.0318752	0.033171	0.02849	0.032529	0.033773
SR25	0.023019	0.033106	0.039434	0.033154	0.034965
Circle	0.0250269	0.038504	0.042156	0.037355	0.042337

Table 7.

COPRAS results.

Specimen code	$Q_i$	$U_i$	Rank
Square	0.14421818	0.7500	4
SR5	0.10051895	0.5228	7
SR10	0.12758880	0.6635	5
SR15	0.10697634	0.5563	6
SR20	0.15303555	0.7959	3
SR25	0.17537659	0.9121	2
Circle	0.19228559	1.0000	1

Conclusions

This study investigated the crashworthiness performance of 3D-printed PLA-GF tubes with geometric transitions from square to circular cross-sections using arc fillets. Quasi-static axial compression tests, coupled with COPRAS-based multi-criteria evaluation, revealed that the tube geometry plays a pivotal role in governing energy absorption, force distribution, and overall crash performance. Among the tested configurations, the circular tube exhibited the highest overall performance, with the average force ( ${\mathrm{F}}_{\text{avg}}$ ) increasing by 36.7%, total absorbed energy ( ${\mathrm{E}}_{\mathrm{T}}$ ) by 40.9%, crash force efficiency (CFE) by 33.0%, and specific energy absorption (SEA) by 69.6% relative to the square tube. The SR25 transitional profile also demonstrated significant enhancements, particularly in SEA (+40.2%) and CFE (+24.3%), highlighting the effectiveness of intermediate geometric transitions in optimizing crash performance. In contrast, the SR5 configuration recorded the highest initial peak force ( ${\mathrm{F}}_{\text{ip}}$ = 9.52 kN), indicative of stress concentration effects at sharp corners. These quantitative findings provide strong evidence that controlled geometric transitions can substantially improve energy dissipation and structural stability.

While this study provides valuable insights into the quasi-static crashworthiness of 3D-printed PLA-GF tubes with varying geometric transitions, it is limited to low strain-rate testing and does not capture dynamic crash behavior. Future work should include high strain-rate experiments to evaluate performance under realistic crash conditions. Additionally, investigating hybrid fiber-matrix composites, alternative infill patterns, or multi-material 3D printing could further enhance energy absorption and structural integrity. These approaches will help extend the applicability of the findings to lightweight crashworthy components in automotive and aerospace applications.