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Abstract

This study explores the crashworthiness performance of square structures inspired by the cross-sectional geometry of a seahorse skeleton, made from polyethylene terephthalate glycol reinforced with carbon fiber (PETG-CF). Three key parameters were studied, including inner shape, inner diameter, and rib thickness, each varied across four levels. Quasi-static axial compression tests were conducted to assess the crashing performance of the specimens. Detailed failure histories were documented, and data on crash load, absorbed energy, and displacement responses were recorded. To evaluate crashworthiness, several indicators were analyzed, containing the initial peak force ( $F_{ip}$ ), the energy absorption (U), the mean force ( $F_{m}$ ), the specific energy absorption (SEA), and the crash force efficiency (CFE). Analysis of variance (ANOVA) was employed to quantify the percentage contribution of each studied parameter to the crashworthiness indicators. An integrated multi-attribute decision-making (MADM) approach, merging the analytic hierarchy process (AHP) and the technique for order of preference by similarity to ideal solution (TOPSIS), was utilized to identify the optimal structure for improved crashworthiness performance. The ANOVA results reveal that inner shape predominantly influences $F_{m}$ (53.31%) and U (53.45%), inner diameter has the greatest effect on SEA (38.33%), and rib thickness contributes most significantly to $F_{ip}$ (50.46%) and CFE (63.51%). Furthermore, the AHP-TOPSIS results indicated that the P30/T5 configuration exhibited the best overall crashworthiness performance, achieving $F_{ip}$ = 22.20 kN, U = 1105.22 J, $F_{m}$ = 19.74 kN, SEA = 19.08 J/g, and CFE = 0.889. This superior performance is attributed to the well-balanced combination of the crashworthiness indicators.

Keywords

Energy absorption composite filament light-weight structures failure examination decision making

Introduction

Ensuring occupant safety during vehicular collisions remains a fundamental objective in automotive engineering, driving continuous research and innovation in the field of crashworthiness. Crashworthiness denotes to the ability of a material or structural system to protect vehicle occupants during collision events by efficiently absorbing and dispersing the kinetic energy generated upon crash.^1–3 This concept is central to passive safety design and plays a fundamental role in minimizing injuries and fatalities during accidents. As the automotive industry increasingly focuses on sustainability, the importance of crashworthiness has grown, not only in traditional internal combustion vehicles but also in electric and lightweight vehicles where different materials and design strategies are employed.^4,5 Improving crashworthiness contains optimizing structural geometries, integrating energy-absorbing materials, and innovating with bio-inspired or multifunctional designs that balance safety and performance. Consequently, advancing crashworthiness is vital to developing safer, lighter, and more efficient vehicles aligned with the global push toward sustainable and intelligent transportation systems.^6,7

A wide range of material classes can be utilized in the development of crashworthy structures, each bringing different advantages tailored to specific applications and loading environments. Traditionally, metals like steel and aluminum have been the primary choices for energy-absorbing components, largely due to their well-characterized mechanical behavior and predictable plastic deformation under loading.^8,9 These metals are renowned for their ability to dissipate substantial amounts of kinetic energy through ductile failure mechanisms, offering an effective balance of strength, toughness, and energy absorption, key attributes for ensuring occupant protection during crash events.^10,11 In recent years, however, the focus has expanded beyond conventional metals to include advanced polymer-based materials, which present compelling advantages in terms of weight reduction and energy efficiency. Polymers are commonly classified into two groups: thermosets and thermoplastics.^12,13 Thermosetting, which exhibit high stiffness and strength due to their cross-linked molecular architecture, have been extensively studied for crashworthiness applications.^14,15 However, growing interest has been directed toward thermoplastic materials, driven by their favorable processing characteristics, lower manufacturing costs, recyclability, and suitability for mass production.¹⁶

To harness the combined advantages of both thermosetting and thermoplastic matrices, researchers have increasingly focused on fiber-reinforced thermoplastics.^17–19 By embedding high-performance fibers, such as carbon or glass, into thermoplastic matrices, it is possible to achieve hybrid material systems that exhibit the desirable mechanical strength and rigidity of thermosets, alongside the toughness and manufacturability of thermoplastics.^20,21 This approach addresses the growing demand for lightweight, crashworthy, and sustainable materials that can be efficiently produced and tailored for energy-absorbing structures. Among these, polyethylene terephthalate glycol-modified reinforced with carbon fiber (PETG-CF) has emerged as a notable example, demonstrating excellent printability, high specific strength, and superior energy absorption under crash loading.^22,23 This synergy between matrix and reinforcement opens new opportunities for designing next-generation crashworthy components. In this regard, Sivakumar et al.²⁴ concentrated on creating thin-walled PETG composite tubes reinforced with carbon fiber (CF) that have octagonal corrugated lattice patterns on their lateral surfaces. Important fused filament fabrication (FFF) parameters, including line width, infill density, printing speed, nozzle temperature, and layer height, were optimized in the study. The PETG-CF composite tubes with the octagonal lattice designs were tested for experimental results, such as compressive strength and dimensional length error. The findings showed that 0.1 mm layer height, 22°C nozzle temperature, 20 mm/s printing speed, 0.1 mm line width, and 100% infill density were the ideal FFF parameters for optimizing compressive strength. The optimized 3D-printed samples exhibited better compressive strength and minimal dimensional deviation, making them suitable for safety-critical automotive applications.

Moreover, using both experimental and numerical methods, Mallek et al.²⁵ examined the dynamic behavior of honeycomb sandwich panels with auxetic and hexagonal cellular cores under low-velocity impacts. Panels composed of PETG/CF and virgin PETG. Significant differences in Young’s modulus and tensile yield strength were found dependent on the orientation when the tensile properties of these materials were assessed across several printing orientations. Impact energy and damage patterns were then evaluated using low-velocity drop-weight impact experiments conducted at 1.5 and 2.5 m/s. The findings showed that the hexagonal structure was more susceptible to impact forces, but the auxetic honeycomb structure showed better impact resistance. Furthermore, the impact resistance of the panels was improved by the addition of carbon fiber reinforcing. Awd Allah et al.²⁶ looked into how printing factors affected the crashworthiness of PETG-CF square tubes. Three primary printing factors that varied over four levels were the focus of the study: layer height, infill pattern structure, and infill density. The quasi-static axial compression test was used to evaluate the specimens’ structural performance. The study used the complex proportional assessment (COPRAS) method to determine the ideal arrangement. With a honeycomb infill pattern, 30% infill density, and a 0.2 mm layer height, the HC-30-0.20 design had the best overall crashworthiness performance, according to the COPRAS analysis.

Alongside significant advancements in materials and manufacturing technologies, a diverse range of cross-sectional geometries is frequently employed in the design of crashworthy structures to maximize energy absorption and maintain structural integrity during impact events.^27–29 These geometries include traditional circular sections,^10,30 various polygonal shapes,^31,32 as well as more complex multi-cell configurations,³³ each offering distinct advantages in terms of deformation behavior and crash performance. Recently, there was a growing emphasis on the development of lightweight, energy-efficient structural designs that incorporate innovative cross-sectional geometries, often inspired by biological systems. These bio-inspired forms are being actively investigated as a means to address the increasingly severe performance requirements related to mechanical efficiency, crash safety, and environmental sustainability.^32,34 Nature-optimized structures, improved through evolutionary processes, often exhibit superior strength-to-weight ratios and mechanically efficient designs, making them ideal for crashworthy components.^35,36 A particularly compelling example is the seahorse tail cross-section, which has attracted significant interest due to its unique articulated morphology. This design combines flexibility and robustness, allowing controlled deformation under load, an attribute highly desirable for impact mitigation and energy absorption applications.³⁷

The literature review conducted in this study identifies a significant gap in experimental research regarding the crash performance of the seahorse tail cross-section from 3D-printed composite material. In response to this research gap, the current study aims to experimentally assess the crashworthiness characteristics of square structures inspired by the cross-sectional geometry of a seahorse skeleton, fabricated from polyethylene terephthalate glycol reinforced with carbon fiber (PETG-CF) using fused deposition modeling (FDM) technology. These 3D-printed structures were subjected to quasi-static axial compression loading. These parameters include inner shape geometry (square, circular, polygon, and hexagonal patterns), inner diameter (25, 30, 35, and 40 mm), and rib thickness (4, 5, 6, and 7 mm). The best structure for enhanced crashing performance was then found using a hybrid decision-making framework that combined the analytic hierarchy process (AHP) with the technique for order preference by similarity to ideal solution (TOPSIS).

Methodology

Material

In this work, polyethylene terephthalate glycol filament reinforced with 20 wt% carbon fiber (PETG-CF20) was sourced from LYNX Co. Ltd (Egypt). To ensure material consistency throughout the experiments, all PETG-CF filament spools used were sourced from the same production batch. PETG, an engineering-grade polyester copolymer, is extensively used in additive manufacturing due to its superior properties compared to conventional thermoplastics.³⁸ It provides excellent impact resistance, toughness, chemical stability, and dimensional accuracy, along with low moisture absorption, which helps preserve the integrity of printed components under varying environmental conditions.³⁹ The addition of carbon fibers (CF) to the PETG matrix further enhances its performance by improving stiffness, tensile strength, and thermal stability while reducing shrinkage and minimizing warping during printing. This reinforcement also offers greater resistance to creep and deformation under mechanical loads.⁴⁰ These attributes make PETG-CF an ideal choice for high-performance applications that require a combination of lightweight design, mechanical strength, and reliability. Furthermore, its ease of processing and versatility allow for widespread use in various industrial sectors for both prototyping and end-use production.⁴¹ Table 1 summarizes the physical and mechanical properties of PETG-CF as provided by the manufacturer. It should be noted that the carbon fibers in PETG-CF are short, discontinuous, and uniformly dispersed within the PETG matrix, as specified by the manufacturer. Their orientation is primarily random within the filament and further influenced by the deposition path during the fused deposition modeling (FDM) process. Consequently, the fibers remain embedded within individual extruded filament strands and do not form continuous connections between structural elements.

Table 1.

PETG-CF properties.

Property	PETG-CF	Unit
Density	1.17	g/cm³
Melting point	210 $\pm$ 5	$℃$
Elongation at break	5.6	%
Tensile strength	41	MPa
Flexural modulus	1973	N/mm²
Flexural strength	65	MPa
Impact strength	28	KJ/M²
Filament diameter	1.75 $\pm$ 0.05	mm
Printing temperature	240–280	$℃$

Design parameters

Selecting appropriate parameters is an important aspect of any study, as it directly influences both the methodology and the reliability of the results. In the present work, particular emphasis was placed on identifying and selecting key design parameters that significantly affect structural performance under crash conditions. Three primary design parameters were considered, and each parameter was evaluated across four distinct levels. As noted by Deleo and Feraboli⁴² and Browne et al.,⁴³ there is no standardized dimension or shape for crashworthy structures. This variability highlights the significance of tailoring design parameters to meet the specific goals and constraints of each study. In this study, the inner shape, inner diameter, and rib thickness of a bio-inspired cross section were examined to achieve an optimal balance between structural integrity, lightweight design, and mechanical performance, ensuring high efficiency. Figure 1 presents the actual morphology of the seahorse skeleton along with a schematic representation of its cross-section, highlighting the studied design parameters. To present the design variables in a clear and structured manner, Table 2 provides a detailed summary of each selected parameter and its four distinct levels.

Figure 1.

(a) Actual morphology of the seahorse skeleton³⁷ and (b) schematic representation of the studied cross-sectional and design parameters.

Table 2.

Studied parameters.

Parameters/level	Level #1	Level #2	Level #3	Level #4
Inner shape	Circular (c)	Square (SQ)	Polygon (P)	Hexagonal (H)
Infill diameter, mm	25	30	35	40
Rib thickness, mm	4	5	6	7

Specimen preparation

The Prusa® i3 MK3, a widely used profitable 3D printer with a print volume of 250 mm × 211 mm × 218 mm, was utilized to fabricate PETG-CF tubes. This printer employs FDM recognized for its cost-efficiency, ease of use, and capability to produce complex geometries with high precision. During the process, filament is fed continuously into a heated extruder nozzle, where it is melted to the appropriate extrusion temperature. Once liquefied, the material is deposited layer by layer onto the build platform according to a predefined toolpath. As each layer cools and solidifies, the structure gradually takes shape based on the CAD model specifications. The fabrication process is controlled through slicing software, which defines key printing parameters. These parameters are essential for achieving dimensional accuracy, surface finish quality, and mechanical integrity of the final printed components.

A critical aspect of the FDM process is the precise control of layer height in the z-direction, which determines the thickness of each deposited layer. In this study, the layer height was set to 0.2 mm, which, as reported by Awd Allah et al.,²⁶ provides good energy absorption performance. Maintaining this layer height ensures consistent and uniform deposition, contributing to the formation of a solid and stable structure. Additionally, choosing the right printing speed is essential, as it affects the dimensional accuracy, layer adhesion, and mechanical properties of the printed structures. While higher speeds can reduce print time, they may lead to weaker layer bonding, increased surface roughness, and geometric inaccuracies due to inconsistent material flow. On the other hand, slower speeds improve print quality by enhancing layer precision and interlayer adhesion, but they significantly increase fabrication time, which may not be suitable for large-scale or time-sensitive projects. To ensure that the 3D-printed PETG-CF tubes met both structural integrity and time efficiency standards, 100 mm/s was selected as the optimal balance. This speed allowed for consistent extrusion, reducing printing defects while maintaining an efficient fabrication process.

To achieve optimal printing conditions and ensure proper layer adhesion, both the printing bed and nozzle temperature were controlled throughout the process.⁴⁴ The nozzle temperature was set to 275°C, which is within the ideal range for melting PETG-CF filament. This setting is essential as it ensures the filament reaches the optimal extrusion consistency, allowing it to flow smoothly through the nozzle without causing blockages. Consistently maintaining this nozzle temperature is key to achieving precise layer deposition and strong interlayer bonding, directly influencing the mechanical properties and dimensional accuracy of the final printed component. Additionally, the printing bed temperature was set to 90°C to facilitate optimal adhesion between the initial layers and the heated surface. This temperature is important for preventing warping, a common issue in 3D printing, by ensuring that the first layers adhere firmly to the print bed as they cool. This approach helps create a stable foundation for the subsequent layers, ultimately improving the structural integrity of the printed component. Another key factor affecting the success of the 3D printing process is the ambient temperature around the printer. The ambient temperature was kept at approximately 25°C. Significant fluctuations in temperature can lead to issues such as warping, delamination, and poor adhesion between layers, as the material may cool too quickly or unevenly, disrupting the printing process.^16,45

Honeycomb infill structures with a 20% density and a raster angle of +45°/−45° were used in this work to balance material usage, structural integrity, and energy absorption. The 20% infill provides sufficient stiffness and strength while minimizing material consumption, and the +45°/−45° raster orientation enhances interlayer bonding and distributes stresses more evenly under load. Inspired by the hexagonal geometry of beehives, this design maximizes structural integrity while minimizing material usage, making it widely employed in structural, automotive, and aerospace applications where lightweight, stiff, and crashworthy components are required.^46–48 The 3D-printed tubes were accurately designed with specific and precise dimensions to ensure consistency in the testing process. Each tube was considered with a height of 80 mm and outer side measurements of 60 mm by 60 mm. The design of these test samples was created using SolidWorks®. Once the design was finalized, it was exported as Stereo Lithography (STL) files. These files were then imported into the 3D printing software. Following the importation of the STL files into the 3D printing software. Table 3 provides a detailed overview of each manufactured tube and the studied parameters, summarizing their key specifications.

Table 3.

Key specifications and parameters of the 3D-printed tubes.

Specimen code	Inner shape	Inner dimeter, mm	Rib thickness, mm	Mass, g
SQ30/T5	Square (SQ)	30	5	87.260
C30/T5	Circle (C)	30	5	80.066
P30/T5	Polygon (P)	30	5	82.745
H30/T5	Hexagonal (H)	30	5	83.296
C25/T5	Circle	25	5	79.834
C35/T5	Circle	35	5	81.168
C40/T5	Circle	40	5	82.735
C30/T4	Circle	30	4	79.793
C30/T6	Circle	30	6	80.124
C30/T7	Circle	30	7	79.013

Note. For square, hexagonal, and octagonal shapes, the outer edge was taken to be 30 mm, which is the same as a circle.

Quasi-static axial test

A standard testing machine (Model Jinan WDW, China) with a 100 kN capacity was used to accomplish quasi-static axial compression tests on the 3D-printed tubes. Following established protocols from previous studies on the crashworthiness of similar tubular structures, the crosshead speed was set at 10 mm/min.^10,49,50 To ensure consistent testing conditions, the printed tubes were carefully aligned between two identical, parallel steel plates before the compression tests began. This alignment was essential for ensuring uniform load application across the entire cross-sectional area of the tubes, minimizing the risk of irregularities or distortions that could affect the accuracy of the results. Additionally, an automated data collection system was employed to continuously capture load-displacement data during the test. The collected data was essential for evaluating the crashworthiness and structural integrity of the 3D-printed tubes under quasi-static loading conditions. In addition to the load-displacement data, the deformation and collision events of the test specimens were closely monitored and documented through real-time video recording. This enabled a visual capture of the tubes as they underwent compression. For each compression test, three identical samples were exposed to the similar conditions, and the results from these trials were averaged to improve the reliability and statistical validity of the findings. This repetition was essential to ensure that the results were not skewed by inconsistencies or outliers. From the load-displacement curves obtained during the quasi-static compression tests, several key crashworthiness indicators were derived, including.

• Initial peak force ( $F_{ip})$ is got straightforward from the crashing load-displacement plot.⁵¹

• Energy absorption (U) indicates the whole energy absorbed through the crashing procedure. It is precisely defined as:

U = \int_{0}^{l_{\max}} F (l) dl

(1)

where,

F (l) and l_{\max}

are the instantaneous, respectively, crashing load and total crash displacement.

• Mean force ( $F_{m})$ is offered:

F_{m} = \frac{U}{l_{\max}}

(2)

• Specific energy absorption (SEA) is calculated as the energy absorption divided by the crashed mass:

SEA = \frac{U}{m_{c}}

(3)

m_{c} = (\frac{M}{L}) l_{\max}

(4)

where,

\frac{M}{L}

is the mass of the energy absorber before testing divided by length.

• Crashing force efficiency (CFE) is the ratio of $F_{m}$ to $F_{ip}$ ⁵²:

CFE = \frac{F_{m}}{F_{ip}}

(5)

Decision making

The analytic hierarchy process (AHP) is an influential decision-making tool capable of effectively handling both tangible and intangible attributes by incorporating the subjective judgments of individuals involved in the decision-making process.^53,54 This structured approach allows for a complete evaluation of multiple criteria, ensuring a more informed and rational decision. However, in certain scenarios, the method can become increasingly complex, particularly when dealing with many attributes and alternatives. As the number of pairwise comparisons grows, the process may become overwhelming and difficult to manage.^55,56 The technique for order preference by similarity to ideal solution (TOPSIS) is particularly effective in handling tangible attributes and efficiently evaluating multiple alternatives.⁵⁷ This technique to decision-making is popular and practical since it may rank alternatives according to how close they are to the ideal and negative-ideal solutions. However, one of the challenges of TOPSIS is the need for a robust method to define the relative importance of various attributes in relation to the overall objective.^58,59 The AHP serves as a complementary tool by providing a structured and systematic procedure for assigning appropriate weights to these attributes, thereby enhancing the effectiveness of the TOPSIS method.^60,61 To leverage the strengths of both methods, a combined multi-attribute decision-making (MADM) approach integrating TOPSIS and AHP is implemented to determine the most suitable structure. Using AHP to establish the relative importance of attributes and TOPSIS to rank the alternatives built on their closeness to the ideal solution, this hybrid approach enhances decision-making efficiency and accuracy.^62,63

Step 1

The TOPSIS method starts by constructing a decision matrix $D$ that includes $n$ criteria (attributes) and $m$ alternatives. The evaluation and ranking of the options according to several criteria is based on this matrix. The decision matrix is represented as follows:

D = [\begin{array}{l} x_{11} & x_{12} & x_{13} & \dots & x_{1 n} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2 n} \\ x_{31} & x_{32} & x_{33} & \dots & x_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x_{m 1} & x_{m 2} & x_{m 3} & \dots & x_{m n} \end{array}]; i = 1, \dots, m, j = 1, \dots, n

(6)

In this case, $x_{i j}$ represents the $i - t h$ alternative’s performance value in relation to the $j - t h$ characteristic.

Step 2

Construct the normalized decision matrix to guarantee that all attribute values are dimensionless and comparable. The following formula is used to obtain the normalized decision matrix $R$ :

r_{i j} = \frac{x_{i j}}{\sqrt{\sum_{i = 1}^{m} {x_{i j}}^{2}}}

(7)

The resulting normalized decision matrix is expressed as:

R = [\begin{array}{l} r_{11} & r_{12} & r_{13} & \dots & r_{1 n} \\ r_{21} & r_{22} & r_{23} & \dots & r_{2 n} \\ r_{31} & r_{32} & r_{33} & \dots & r_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ r_{m 1} & r_{m 2} & r_{m 3} & \dots & r_{m n} \end{array}]

(8)

Step 3

Assign weights according to each attribute’s significance to ascertain the relative value of the various attributes in relation to the overall goal. These weights are obtained using the AHP’s pairwise comparison matrix. Using the Saaty⁶⁴ nine-point preference scale, which has two different sets of numbers, the pairwise comparison matrix is produced. The first set, (1, 3, 5, 7, and 9), indicates the gradual priority of one criterion relative to another, where the values represent the degree of importance as one move from left to right on the scale. For example, a 1 means the two criteria are equally important, a 3 indicates a slight preference, and a nine signifies a very strong preference. The second set, (2, 4, 6, and 8), contains intermediate values used to represent preferences that fall between the corresponding values of the first set. These are employed when there is a need to refine the level of importance between criteria. Using these values, the pairwise comparison matrix is created, and consistency is guaranteed by the reflexivity principle. The scaling system planned by Saaty⁶⁴ is given in Table 4. The pair-wise comparison matrix $A$ , which is an $n x n$ matrix, is represented as follows:

A = [\begin{array}{l} 1 & a_{12} & a_{13} & \dots & a_{1 n} \\ \frac{1}{a_{12}} & 1 & a_{23} & \dots & a_{2 n} \\ \frac{1}{a_{13}} & \frac{1}{a_{23}} & 1 & \dots & a_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{1}{a_{1 n}} & \frac{1}{a_{2 n}} & \frac{1}{a_{3 n}} & \dots & 1 \end{array}]

(9)

In the pair-wise comparison matrix $A$ , the diagonal elements represent self-comparison, meaning each attribute is compared to itself. Since an attribute is equally important when compared to itself, the diagonal elements are all set to 1, i.e., $a_{i j} = 1$ when $i = j$ , for all $i, j = 1, 2, \dots, n$ . The values on either side of the diagonal reflect the relative significance of attribute $i$ compared to attribute $j$ . Specifically, if $a_{i j}$ represents the relative importance of attribute $i$ compared to attribute $j$ , then the reciprocal relationship holds: $a_{i j} = 1 / a_{i j}$ for $i \neq j$ , ensuring that the matrix is a positive reciprocal square matrix. This reciprocal property ensures consistency within the matrix, as the strength of the relative importance of one attribute compared to another is mirrored on the opposite side of the diagonal. The normalization of the geometric mean (NGM) method is employed to assess the relative importance of the attributes under consideration. In this approach, let $w_{i}$ represent the weight or importance degree of the $i - t h$ attribute. This weight reflects the influence of each attribute to the general decision-making procedure and is determined based on the normalized geometric mean of the attribute values. Through this method, the relative significance of each attribute is quantified, enabling more informed decisions when comparing and evaluating the attributes in a multi-criteria decision-making framework.

w_{i} = {(\prod_{j = 1}^{n} a_{i j})}^{\frac{1}{n}} / \sum_{i = 1}^{n} {(\prod_{j = 1}^{n} a_{i j})}^{\frac{1}{n}} i, j = 1, 2, \dots, n

(10)

To ensure the consistency and validity of the pairwise comparison matrix in the evaluation procedure, a consistency check is conducted. The consistency check helps determine whether the relative significance values assigned to the attributes maintain a reasonable level of transitivity. If inconsistencies are detected beyond an acceptable threshold, adjustments may be necessary to refine the comparisons, ensuring a more robust and justifiable decision-making process. Let $C$ represent an $n -$ dimensional column vector that encapsulates the summation of the weighted values corresponding to the significance degrees of the attributes. Mathematically, this can be stated as:

C = {[C_{i}]}_{n x 1} = A W^{T}, i = 1, 2, \dots, n

(11)

where

A W^{T} = [\begin{array}{l} 1 & a_{12} & a_{13} & \dots & a_{1 n} \\ \frac{1}{a_{12}} & 1 & a_{23} & \dots & a_{2 n} \\ \frac{1}{a_{13}} & \frac{1}{a_{23}} & 1 & \dots & a_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{1}{a_{1 n}} & \frac{1}{a_{2 n}} & \frac{1}{a_{3 n}} & \dots & 1 \end{array}] [w_{1}, w_{2}, w_{3}, \dots, w_{n}] = [\begin{array}{l} c_{1} \\ \begin{array}{l} c_{2} \\ \begin{array}{l} c_{3} \\ \begin{array}{l} \dots \\ c_{n} \end{array} \end{array} \end{array} \end{array}]

(12)

The vector $C V = {[c v_{i}]}_{n x 1}$ can be used to represent the constancy values for the attributes, with each member $c v_{i}$ being unique as follows:

c v_{i} = \frac{c_{i}}{w_{i}}, i = 1, 2, \dots, n

(13)

To minimize inconsistencies in the pairwise comparison matrix, Saaty⁶⁴ proposed using the extreme eigenvalue $λ_{\max}$ to assess the effectiveness of judgments. The extreme eigenvalue serves as a key indicator in the reliability assessment of the comparison matrix. It can be determined using the following formula:

λ_{\max} = \frac{\sum_{i = 1}^{n} c v_{i}}{n}, i = 1, 2, \dots, n

(14)

with the extreme eigenvalue

λ_{\max}

, the consistency index (CI) can be considered using the following formula:

C I = \frac{λ_{\max} - n}{n - 1}

(15)

If the consistency index (CI) is equal to zero ( $C I = 0$ ), it indicates that the pairwise comparison matrix is perfectly consistent. In other words, the judgments made in the comparison process are completely logical and transitive. Remarkably, the closer the extreme eigenvalue $λ_{\max}$ is to the total number of attributes $n$ , the higher the consistency of the evaluation. Any deviation from $λ_{\max} = n$ suggests some degree of inconsistency in the pairwise comparisons. To additional assess the acceptability of the inconsistency, the consistency ratio (CR) is used as a guideline. The CR is intended as follow:

C R = \frac{C I}{R I}

(16)

Here, $R I$ signifies the average random index, which is determined based on the order of the pairwise comparison matrices. If the $C R$ falls under the threshold of 0.10, the assessment of attribute position is deemed reliable and acceptable.

Table 4.

Scaling system.

Scale value	Description	Interpretation
1	Equal	Both criteria hold the same level of importance
3	Moderate	One criterion is moderately more important than the other
5	Strong	One criterion is significantly more important than the other
7	Very strong	One criterion is strongly dominant in importance
9	Extreme	One criterion is extremely more important than the other
2	Middle (1 and 3)	Falls between equal and moderate importance
4	Middle (3 and 5)	Falls between moderate and strong importance
6	Middle (5 and 7)	Falls between strong and very strong importance
8	Middle (7 and 9)	Falls between very strong and extreme importance

Step 4

The weighted normalized matrix $v_{i j}$ must then be constructed. The weight $w_{i}$ of the associated attribute, which was established by the pairwise comparison of attributes in the preceding step, is multiplied by each column of the normalized decision matrix $r_{i j}$ . Every normalized value $r_{i j}$ is modified to reflect its weighted importance based on the weight $w_{i}$ , which represents the relative importance of each attribute. The weighted normalized matrix $v_{i j}$ is considered using the formula:

v_{i j} = r_{i j} x w_{i}

(17)

The resulting weighted normalized matrix $V$ is expressed as:

V = [\begin{array}{l} v_{11} & v_{12} & v_{13} & \dots & v_{1 n} \\ v_{21} & v_{22} & v_{23} & \dots & v_{2 n} \\ v_{31} & v_{32} & v_{33} & \dots & v_{3 n} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ v_{m 1} & v_{m 2} & v_{m 3} & \dots & v_{m n} \end{array}]

(18)

Step 5

The ‘negative-ideal’ (worst) and ‘ideal’ (best) solutions are found in this step. The degree to which each alternative resembles the ideal and negative-ideal solutions will be assessed using these solutions, which show the greatest and worst outcomes for each attribute. The ‘ideal’ solution reduces the downsides for all cost characteristics and maximizes the benefits for all beneficial attributes, making it the finest feasible set of attribute values. Conversely, the ‘negative-ideal' answer is the poorest conceivable combination of numbers, indicating the worst performance in every aspect. The following formula is used to obtain the ideal and negative-ideal solutions:

v^{+} = {(\sum_{i}^{\max} v_{i j} / j ϵ J), (\sum_{i}^{\min} v_{i j} / j ϵ J^{'}) / i = 1, 2, \dots, m} = {v_{1}^{+}, v_{2}^{+}, v_{3}^{+}, \dots, v_{n}^{+}}

(19)

v^{-} = {(\sum_{i}^{\min} v_{i j} / j ϵ J), (\sum_{i}^{\max} v_{i j} / j ϵ J^{'}) / i = 1, 2, \dots, m} = {v_{1}^{-}, v_{2}^{-}, v_{3}^{-}, \dots, v_{n}^{-}}

(20)

where

J = (j = 1, 2, \dots, n) / j

represent the set of indices corresponding to the beneficial attributes. These are the attributes where higher values are preferable, meaning they contribute positively to the overall evaluation of the alternatives. While

J^{'}

(j = 1, 2, \dots, n) / j

represent the set of indices corresponding to the non-beneficial attributes (or cost attributes). These are the attributes where lower values are preferable, meaning they represent undesirable characteristics that should be minimized in the decision-making procedure.

Step 6

The parting distances between each option and the ideal and negative-ideal solutions are computed in this stage to evaluate each alternative’s proximity to these benchmark solutions.

• Parting from the ideal solution: The separation of each alternative $i$ from the ideal solution $S^{+}$ is computed using the Euclidean distance formula. The separation is given by:

S_{i}^{+} = \sqrt{\sum_{j = 1}^{n} {(v_{i j} - v_{j}^{+})}^{2}}, i = 1, 2, \dots, m

(21)

• Parting from the negative-ideal solution: Correspondingly, the separation of each alternative $i$ from the negative-ideal solution $S^{-}$ is assessed as:

S_{i}^{-} = \sqrt{\sum_{j = 1}^{n} {(v_{i j} - v_{j}^{-})}^{2}}, i = 1, 2, \dots, m

(22)

Step 7

This stage involves calculating each alternative’s relative proximity to the optimal answer. $S_{i}^{-}$ , the separation from the negative-ideal solution, is divided by the total separation, which is the sum of the separations from the ideal and negative-ideal solutions, to achieve this. The following formula provides the relative closeness $C_{i}$ for each choice $i$ :

C_{i} = \frac{S_{i}^{-}}{S_{i}^{+} + S_{i}^{-}}, i = 1, 2, \dots, m

(23)

C_{i}

will have a value between 0 and 1: An alternative that is nearer the optimal answer is indicated by a

C_{i}

value near 1. When

C_{i}

is near zero, it means that the alternative is nearer the negative-ideal solution.

Step 8

Once you have calculated the $C_{i}$ values for each alternative, the next step is to rank the alternatives based on these values.

Results and discussions

Inner shape effect

Figure 2 presents the force-displacement and absorbed energy curves as functions of the crashing displacement for PETG-CF bio-inspired cross-section featuring various inner shapes, including square, circular, polygonal, and hexagonal designs, during axial quasi-static compression tests. Initially, the tubes in the pre-crash zone are more complex, dividing into two distinct linear regions separated by a small, curved transition zone. This curved region indicates a gradual shift in the deformation mechanism, potentially signifying an initial localized collapse before the structure fully engages in a more uniform load-bearing response. A comparable trend was reported by Abd El Aal et al.⁶⁵ for PLA 3D-printed tubes. In this phase the structure experiences minimal deformation, and the load progressively increases with displacement. The load continues to rise until reaching $F_{i p}$ , the point marking the maximum load-bearing capacity before significant structural damage occurs. After reaching the $F_{i p}$ , the tubes transition into the post-crashing phase, where the load starts to oscillate around the $F_{m}$ . These oscillations reflect the material’s plastic deformation behavior, demonstrating its continued ability to absorb energy while undergoing further deformation. A similar trend was reported by Awd Allah et al.³² Following this phase, the specimens enter the compaction stage, where the applied load steadily increases. During this stage, the machine’s table plays a critical role by supporting the entire tube and enduring the rapidly rising stresses caused by the ongoing material deformation. As compression continues, the tube material undergoes significant densification, resulting in a sharp increase in the applied load. This compaction effect signifies the final stage of energy absorption, where the material reaches its densification limit, and further deformation becomes increasingly challenging. At this point, the material’s capacity to absorb additional energy through deformation diminishes, and the load rises as the remaining uncompressed sections are fully compacted. This behavior aligns with the findings of Awd Allah et al.⁶⁶ and Abdewi et al.⁶⁷ who observed similar trends in their studies on the crashworthiness performance of various structures.

Figure 2.

Force and absorbed energy against displacement curves for PETG-CF tubes with different inner shape.

The energy versus displacement graphs presented in Figure 2 offer a more understanding of the material’s behavior during compression tests. In the initial elastic deformation phase, the material primarily absorbs energy in the form of elastic strain energy. This phase is characterized by a linear increase in energy absorption, as the material deforms under the applied load but returns to its original shape upon load removal, indicating a reversible deformation mechanism. As the applied stress continues to rise, the material enters the plastic deformation phase, where irreversible structural changes occur. During this stage, the energy absorption rate increases significantly, reflecting the material’s transition from elastic behavior to a more complex plastic flow regime. This phase marks a critical shift in the material’s response, where permanent deformation becomes dominant. The nonlinearity of the energy absorption curve highlights that the material’s stress response is not a simple linear process but instead involves multiple stages, containing yielding, plastic flow, and the early stages of fracture. These observations indicate that the material’s failure mechanism is complex, driven by a combination of elastic and plastic behaviors. Furthermore, Figure 3 delivers a detailed summary of the crash histories for each of the tested tubes, highlighting the proceedings that occurred and the structural changes that happened during the compression testing. The figure clearly validates that the printed PETG-CF tubes established cracks, indicating the start of structural breakdown. As the load sustained to be applied, the cracks progressively expanded, and the material began to deform more severely. Following the formation of these cracks, an obvious increase in wrinkling and folding was observed along the surface of the tube. This deformation intensified as the tube struggled to absorb further compressive forces, leading to a more pronounced collapse. The increase in wrinkling and folding signifies the material’s struggle to withstand the applied stress.

Figure 3.

Crashing histories for PETG-CF tubes with different inner shape.

Figure 4 offers a comparison of the crashworthiness indicators for the PETG-CF tubes, each featuring a distinct inner shape. As underscored by Awd Allah et al.⁶⁸ minimizing the $F_{ip}$ is important for ensuring optimal occupant protection in the event of a crash. Additionally, the data presented in Figure 4(a) clearly demonstrate that the highest documented $F_{ip}$ was observed in the H30/T5 tube, with a value of 23.24 kN. This value is notably higher than the forces measured for the other tested configurations. Specifically, the $F_{ip}$ of H30/T5 is 1.019 times greater than that of SQ30/T5, 1.037 times higher than C30/T5, and 1.047 times greater than P30/T5. Furthermore, when evaluating the $F_{m}$ , the P30/T5 configuration recorded the highest value at 19.74 kN, significantly outperforming the other configurations. Specifically, this value is 1.064, 1.069, and 1.027 times higher than those of SQ30/T5, C30/T5, and H30/T5, respectively.

Figure 4.

Crashworthiness indicators for PETG-CF tubes with different inner shape.

Regarding U, Figure 4(b) clearly demonstrates that the P30/T5 and C30/T5 specimens recorded the maximum and minimum values, measuring 1105.22 J and 1034.05 J, respectively. The significant disparity in energy absorption between these specimens underscores the superior performance of the P30/T5 configuration. Specifically, the energy absorption of P30/T5 was 1.064, 1.069, and 1.027 times greater than that of SQ30/T5, C30/T5, and H30/T5, respectively. With the highest SEA of 19.08 J/g, as shown in Figure 4(b), the P30/T5 specimen outperformed the other configurations in terms of energy absorption relative to mass. In contrast, the SQ30/T5 structure noted the lowest SEA value at 17.00 J/g. Notably, the SEA of P30/T5 was 1.122 times higher than that of SQ30/T5, and 1.034 times higher than both C30/T5 and H30/T5. This performance of P30/T5 can be attributed to its optimized geometric design, which likely enhances the material’s ability to absorb and dissipate energy more efficiently, resulting in a more favorable energy-to-mass ratio compared to the other structures.

As noted by Alshahrani et al.⁶⁹ the CFE is an important parameter for evaluating a material’s or structure’s ability to effectively manage and distribute forces during impact events. Figure 4(c) demonstrates the significant influence of inner shape on the CFE and its subsequent impact on structural performance under compressive loading. With a CFE value of 0.889, the P30/T5 configuration exhibited the highest efficiency among the evaluated specimens. This suggests that P30/T5 effectively distributes impact forces better than the other specimens. Specifically, the CFE of P30/T5 exceeds that of SQ30/T5, C30/T5, and H30/T5 by factors of, respectively, 1.093, 1.079, and 1.075.

The crashworthiness indicators for the four different inner tube shapes are slightly different, suggesting that the overall structural response is governed by factors that outweigh the specific geometry of the inner ribs. This is primarily due to the dominant role of the outer tube, which serves as the main load-bearing and energy-absorbing component. Its structural contribution likely overshadows the minor variations introduced by the inner configurations. Moreover, the outer tube constrains the inner section, forcing all geometries to adopt a comparable and stabilized deformation mode during crashing.

Inner diameter effect

Figure 5 demonstrates the effect of inner diameter on both crash force and energy absorption as functions of displacement. During the pre-crash phase, the specimens with inner diameters of 25 mm, 35 mm, and 40 mm demonstrated a distinctly linear response followed by a small, curved zone, as shown in Figure 5. As the loading continues and displacement increases, the material transitions beyond the elastic phase, entering a plastic deformation stage where energy absorption mechanisms become more complex. The distinct linear response observed in the early phase underscores the critical role of inner diameter in shaping the material’s elastic properties and its ability to resist deformation before yielding. After reaching the $F_{ip}$ , the tubes enter the post-crashing phase, marked by load oscillations around the $F_{m}$ . These fluctuations reflect plastic deformation, where the material continues to absorb energy through controlled structural changes. In the subsequent compaction phase, the load steadily rises as the material densifies. Figure 5 also displays the energy-displacement curves for structures with different inner diameters. In the initial deformation stage, the curves exhibit a predominantly linear relationship between energy and displacement, indicating an elastic response of the materials. However, as displacement increases, the curves deviate from linearity, signing a transition to a nonlinear deformation phase. This shift suggests that the material behavior becomes more complex and less predictable, possibly due to factors like plastic deformation, damage accumulation, or strain localization. The observed nonlinear response underscores the intricate mechanical properties of the tested materials under compression.^70,71

Figure 5.

Force and absorbed energy against displacement curves for PETG-CF tubes with different inner diameter.

The crash histories of the PETG-CF tested specimens are exposed in detail in Figure 6, which additional underscores how inner diameter affects the failure processes of these printed specimens. Remarkably, the failure process is initiated by the formation of a transverse crack at the bottom of the tube, which marks the onset of localized stress concentration. This initial crack is likely attributed to the accumulation of compressive forces at the tube’s base, where stress intensifies as the structure resists deformation. As the applied load continues to increase, the crack propagates further across the tube’s circumference, compromising its structural stability.

Figure 6.

Crashing histories for PETG-CF tubes with different inner diameter.

The crashworthiness indicators for the tested tubes with varying inner diameters are presented in Figure 7. The C30/T5 specimen stands out for having the highest values for $F_{ip}$ , while the $F_{m}$ value reaches its peak for C25/T5 at 22.42 kN and 18.86 kN, respectively, as illustrated in Figure 7(a). Specifically, the $F_{ip}$ value for C30/T5 was 1.049, 1.100, and 1.196 times greater than those of the C25/T5, C35/T5, and C40/T5 specimens, respectively. Similarly, the $F_{m}$ for C25/T5 was 1.021, 1.104, and 1.119 times higher than the values observed for C30/T5, C35/T5, and C40/T5 specimens.

Figure 7.

Crashworthiness indicators for PETG-CF tubes with different inner diameter.

Figure 7(b) illustrates the U for the bio-inspired cross-sectional PETG-CF tubes. The C40/T5 and C25/T5 specimens exhibited the lowest and highest energy absorption performance, respectively. The C40/T5 specimen recorded the lowest U value of 943.74 J, while the C25/T5 specimen accomplished the highest U value of 1056.00 J. This noteworthy difference in energy absorption highlights the critical role of the inner diameter in influencing the material’s energy dissipation capability. Specifically, the C25/T5 specimen demonstrated energy absorption values that were 1.021, 1.104, and 1.119 times higher than those of the C30/T5, C35/T5, and C40/T5 specimens, respectively. This improved energy absorption in the C25/T5 specimen can be attributed to its optimal structural design, where the smaller inner diameter likely allows for better force distribution and greater resistance to deformation under impact. The more compact structure in the C25/T5 tube may help it absorb and dissipate energy more effectively, preventing premature failure and leading to higher overall energy absorption performance. Conversely, the larger inner diameter of the C40/T5 specimen may result in reduced structural integrity and less efficient energy dissipation, hence its lower energy absorption value. With the highest SEA value of 18.90 J/g, the C25/T5 structure clearly outperformed the C40/T5 structure, which documented the lowest SEA of 16.30 J/g, as shown in Figure 7(b). This significant difference in SEA underscores the influence of structural design on energy absorption relative to mass. Additionally, when compared to the SEA values of the C30/T5, C35/T5, and C40/T5 tubes, the SEA of the C25/T5 specimen was 1.024, 1.123, and 1.160 times higher, respectively.

The CFE values for the PETG-CF tested structures with different inner diameter are shown in Figure 7(c). The results indicate that the C40/T5 specimen excelled in handling and distributing impact forces, achieving the highest CFE value of 0.899. In contrast, the C30/T5 specimen had the lowest CFE at 0.824. Additionally, the CFE of the C40/T5 specimen was 1.019, 1.092, and 1.073 times higher than those of the C25/T5, C30/T5, and C35/T5 specimens, respectively.

Rib thickness effect

The influence of rib thickness on the crash performance of the tested tubes is illustrated in Figure 8. A clear linear trend is observed during the initial loading phase, followed by a nonlinear trend until the specimens approach their respective peak loads. Following this pre-crash region, the load-displacement curves exhibit oscillatory behavior in the post-crash zone, attributed to the progressive collapse and fracture mechanisms of the PETG-CF tube walls (as shown in Figure 9). These fluctuations persist until the onset of the compaction phase, during which the structures undergo more substantial deformation and absorb a greater amount of energy. Furthermore, the energy absorption versus displacement curve in Figure 8 follows a nonlinear trend, indicating that energy absorption becomes more complex as displacement increases.

Figure 8.

Force and absorbed energy against displacement curves for PETG-CF tubes with different rib thickness.

Figure 9.

Crashing histories for PETG-CF tubes with different rib thickness.

Figure 10(a) illustrates the impact of rib thickness on crashworthiness indicators, showing that the C30/T5 tube recorded the highest $F_{ip}$ value, reaching a peak of 22.42 kN. In contrast, the C30/T4 specimen displayed the lowest $F_{ip}$ at 17.56 kN. Furthermore, the $F_{ip}$ value of the C30/T5 tube was 1.277, 1.009, and 1.059 times higher than that of the C25/T5, C35/T5, and C40/T5 tubes, respectively. Additionally, the C30/T5 tube demonstrated an improvement in $F_{m}$ , surpassing the C25/T5, C35/T5, and C40/T5 tubes by factors of 1.133, 1.067, and 1.017, respectively.

Figure 10.

Crashworthiness indicators for PETG-CF tubes with different rib thickness.

As shown in Figure 10(b), the U of the PETG-CF tubes vary with rib thickness. The C30/T5 specimen recorded the highest U value of 1034.05 J, demonstrating superior impact energy dissipation. In contrast, the C30/T4 tube exhibited the lowest U at 912.48 J. Consequently, the U of the C30/T5 specimen was 1.133, 1.067, and 1.017 times higher than that of the C30/T4, C30/T6, and C30/T7 tubes, respectively. This emphasizes the significant influence of rib thickness on the energy absorption capacity of the structure. As shown in Figure 10(b), the C30/T5 and C30/T4 tubes exhibited the highest and lowest SEA values in this set, with measurements of 18.45 and 16.34 J/g, respectively. Additionally, the SEA of the C30/T5 tube was 1.129, 1.068, and 1.004 times higher than that of the C30/T4, C30/T6, and C30/T7 tubes, respectively.

Additionally, the C30/T4 specimen recorded the highest CFE among the tubes examined, as shown in Figure 10(c). This indicates that the C30/T4 structure is more efficient at management and dispensing impact forces compared to the other specimens. In contrast, the C30/T6 tube exhibited the lowest CFE, measuring 0.779. The CFE of the C30/T4 tube was 1.127, 1.191, and 1.082 times higher than that of the C30/T5, C30/T6, and C30/T7 tubes, respectively.

Failure analysis

One failure mode observed during the axial compression of the tubes involves distinct stages of damage initiation, propagation, and energy absorption. The failure sequence begins with the formation of longitudinal cracks, typically originating at the bottom, center, or top of the tubes where stress is concentrated. As the material exceeds its elastic limits, these cracks form while the structure absorbs the initial impact. Subsequently, circumferential wrinkles develop in the tube walls due to continued compaction and plastic deformation. The localized damage from the cracks weakens the tube’s hoop integrity, causing buckling and folding. The appearance of these wrinkles marks the transition from localized damage to widespread structural failure, eventually leading to a rapid, catastrophic collapse. This sequence exemplifies a high energy-absorbing failure mode, classified in the literature, notably by Hull,⁷² as the ‘fragmentation’ mode of progressive crashing. This mode is considered ideal for crashworthiness as it provides stable energy absorption through a controlled crush zone.

Scatter in the test results

Table 5 presents the statistical distribution of the experimental results, expressed in terms of the coefficient of variation (CV). The CV serves as a measure of data dispersion relative to the mean, thereby providing insight into the consistency and reliability of the results. Several factors, such as specimen preparation, handling, storage conditions, and experimental design, can introduce variability into the results. The highest CV observed was 9.97% for

F_{m}

in the case of the P30/T5 configuration. All CV values remain below 10%, which is generally considered acceptable in experimental research, reflecting high repeatability and significant accuracy of the tests.⁷³

Table 5.

Summery for the obtained data from the quasi-static test.

Specimen code	Data	$F_{ip}$ , kN	U, J	$F_{m}$ , kN	SEA, J/g	CFE
SQ30/T5	Avg	22.80	1038.66	18.55	17.00	0.814
	SD	1.70	43.74	1.16	1.15	0.078
	CV, %	7.47	4.21	6.25	6.75	9.58
C30/T5	Avg	22.42	1034.05	18.47	18.45	0.824
	SD	2.01	98.37	1.51	1.81	0.053
	CV, %	8.97	9.51	8.17	9.79	6.45
P30/T5	Avg	22.20	1105.22	19.74	19.08	0.889
	SD	1.71	98.97	1.97	1.78	0.064
	CV, %	7.69	8.95	9.97	9.31	7.24
H30/T5	Avg	23.24	1075.99	19.21	18.45	0.827
	SD	1.70	59.03	1.67	1.23	0.077
	CV, %	7.32	5.49	8.67	6.67	9.31
C25/T5	Avg	21.38	1056.00	18.86	18.90	0.882
	SD	1.51	53.60	1.70	1.87	0.057
	CV, %	7.06	5.08	9.03	9.88	6.50
C35/T5	Avg	20.38	956.42	17.08	16.83	0.838
	SD	1.61	63.94	1.28	1.53	0.078
	CV, %	7.88	6.68	7.47	9.11	9.28
C40/T5	Avg	18.74	943.74	16.85	16.30	0.899
	SD	1.45	91.78	1.16	1.10	0.084
	CV, %	7.76	9.72	6.89	6.73	9.31
C30/T4	Avg	17.56	912.48	16.29	16.34	0.928
	SD	1.14	88.10	1.09	1.17	0.048
	CV, %	6.49	9.66	6.68	7.18	5.17
C30/T6	Avg	22.22	969.17	17.31	17.28	0.779
	SD	2.15	89.99	1.69	1.71	0.062
	CV, %	9.67	9.28	9.78	9.89	7.95
C30/T7	Avg	21.17	1016.87	18.16	18.39	0.858
	SD	1.76	77.65	1.59	0.99	0.063
	CV, %	8.33	7.64	8.77	5.38	7.39

Analysis of variance (ANOVA)

The ANOVA is a powerful statistical tool used to quantify the contribution of each design parameter to the crashworthiness indicators. Table 6 summarizes the ANOVA results for all considered indicators. It is evident from the analysis that different parameters dominate the performance depending on the specific metric: rib thickness has the most significant influence on both the

F_{ip}

(50.46%) and CFE (63.51%); inner shape primarily affects the

F_{m}

(53.31%) and U (53.45%); and inner diameter is the dominant factor controlling the SEA (38.33%).

Table 6.

ANOVA results.

	Degree of freedom	Seq SS	Adj SS	Adj MS	Contribution, %
$F_{ip}$ (kN)
Inner shape	3	10.652	0.625	0.208	35.21
Inner diameter	3	4.336	7.361	2.454	14.33
Rib thickness	3	15.268	15.268	5.089	50.46
Total	9	30.256	-	-	100
$F_{m}$ (kN)
Inner shape	3	6.036	1.075	0.358	53.31
Inner diameter	3	2.423	2.992	0.998	21.40
Rib thickness	3	2.863	2.863	0.955	25.29
Total	9	11.322	-	-	100.00
U (J)
Inner shape	3	18929.644	3380.885	1126.962	53.45
Inner diameter	3	7568.389	9335.845	3111.948	21.37
Rib thickness	3	8917.537	8917.537	2972.512	25.18
Total	9	35415.571	-	-	100.00
SEA (J/g)
Inner shape	3	3.241	2.331	0.777	31.84
Inner diameter	3	3.902	4.694	1.565	38.33
Rib thickness	3	3.036	3.036	1.012	29.83
Total	9	10.178	-	-	100.00
CFE
Inner shape	3	0.003682	0.003493	0.001164	19.76
Inner diameter	3	0.003119	0.003783	0.001261	16.74
Rib thickness	3	0.011835	0.011835	0.003945	63.51
Total	9	0.018636	-	-	100.00

Interaction between process parameters

This section provides a detailed examination of the interactions between the design parameters, each of which was tested at four distinct levels. To illustrate these interactions, Figure 11 presents the interaction plots for the crashworthiness indicators, based on the mean values of the collected data. Interaction plots are a widely used statistical tool that help visualize how the effect of one parameter changes depending on the level of another. By analyzing these plots, researchers can determine whether a parameter’s influence is consistent across different levels of other parameters or if more complex interactions exist. Such analyses are particularly valuable for system optimization and understanding the combined effects of multiple design variables on responses, such as crashworthiness in this study.⁷⁴

Figure 11.

Interaction plots for (a) $F_{ip}$ , (b) $F_{m}$ , (c) $U$ , (d) SEA, and (e) CFE (data means).

Figure 11 reveals the complex and interdependent effects of inner shape, inner diameter, and rib thickness on key crashworthiness metrics. An analysis of the plots for $F_{ip}$ , $F_{m}$ , $U$ , and SEA shows a consistent trend: the polygon and hexagonal inner shapes significantly outperform the circle and square configurations. The optimal combination for maximizing energy absorption is a polygon shape with a rib thickness of five and an inner diameter of 30. However, the plot for CFE introduces a critical design trade-off. While the polygon shape excels at absorbing energy, the circle shape, the worst performer in the other metrics, demonstrates the highest efficiency and stability during collapse, particularly at a rib thickness of 5. A key insight from these graphs is the presence of strong interactions between the design parameters. An interaction occurs when the effect of one factor depends on the level of another. Visually, this is represented by lines that are not parallel. For instance, in the energy absorption plot, the effect of changing the rib thickness from four to five is an increase for the polygon shape (red line), but a much smaller increase for the circle shape (blue line). This means the benefit of a thicker rib is not constant; it is highly dependent on the chosen inner shape. This concept is important because it shows that optimizing one variable at a time is ineffective. The peak performance is not achieved by finding the “best” shape, “best” diameter, and “best” thickness independently, but by identifying the specific combination of these factors that works best together.

Selection of the best structure

To systematically rank and determine the ideal structure, this study uses a hybrid decision-making methodology that integrates the AHP and TOPSIS techniques. The AHP technique is utilized to control the relative significance of the selection criteria, which include 10 distinct tubes. Consequently, the TOPSIS technique is useful to evaluate and rank these variants based on their performance concerning five key criteria:

F_{i p}

F_{m}

U

, SEA, and CFE. In this context, a user-defined criterion can be introduced to evaluate the performance of the material under quasi-static axial loading. Table 7 presents the initial decision matrix

D

, which serves as the foundation for the TOPSIS method. Using equation (7), the initial decision matrix is normalized as exposed in Table 8. The relative importance of each performance criterion is then determined using the AHP process and the scale shown in Table 9. The order

U > CFE > F_{ip} > F_{m} > SEA

is an example of a user-defined criterion. The AHP pairwise comparison matrix is shown in Table 10. The normalized weights assigned to each attribute was 0.12957, 0.06364, 0.51004, 0.03292, and 0.26383, respectively, for

F_{ip}

F_{m}

, U, SEA, and CFE. These weights represent the relative significance of the numerous attributes under evaluation, with the normalization process ensuring that the sum of all weights equals one.

Table 7.

Initial decision matrix D.

Specimen code	Attributes
Specimen code	$F_{ip}$ , kN	$F_{m}$ , kN	U, J	SEA, J/g	CFE
SQ30/T5	22.80	18.55	1038.66	17.00	0.814
C30/T5	22.42	18.47	1034.05	18.45	0.824
P30/T5	22.20	19.74	1105.22	19.08	0.889
H30/T5	23.24	19.21	1075.99	18.45	0.827
C25/T5	21.38	18.86	1056.00	18.90	0.882
C35/T5	20.38	17.08	956.42	16.83	0.838
C40/T5	18.74	16.85	943.74	16.30	0.899
C30/T4	17.56	16.29	912.48	16.34	0.928
C30/T6	22.22	17.31	969.17	17.28	0.779
C30/T7	21.17	18.16	1016.87	18.39	0.858

Table 8.

Normalized decision matrix R.

Specimen code	Attributes
Specimen code	$F_{ip}$	$F_{m}$	U	SEA	CFE
SQ30/T5	0.33878	0.32439	0.32436	0.30320	0.30110
C30/T5	0.33313	0.32299	0.32292	0.32906	0.30480
P30/T5	0.32987	0.34520	0.34515	0.34029	0.32884
H30/T5	0.34532	0.33593	0.33602	0.32906	0.30591
C25/T5	0.31768	0.32981	0.32978	0.33708	0.32626
C35/T5	0.30282	0.29868	0.29868	0.30016	0.30998
C40/T5	0.27845	0.29466	0.29472	0.29071	0.33254
C30/T4	0.26092	0.28487	0.28496	0.29142	0.34327
C30/T6	0.33016	0.30270	0.30266	0.30819	0.28816
C30/T7	0.31456	0.31757	0.31756	0.32799	0.31738

Table 9.

Considered parameters for MADM to choose the optimum structure.

Indicators	Notation	Significance
Initial peak force	$F_{ip}$	$↓$ (decrease)
Mean crash load	$F_{m}$	$↑$ (increase)
Energy absorption	U	$↑$ (increase)
Specific energy absorption	SEA	$↑$ (increase)
Crash force efficiency	CFE	$↑$ (increase)
Criteria	-	$U > CFE > F_{ip} > F_{m} > SEA$

Table 10.

Normalized pair-wise comparison matrix.

	$F_{ip}$	$F_{m}$	U	SEA	CFE
$F_{ip}$	1	3	1/5	5	1/3
$F_{m}$	1/3	1	1/7	3	1/5
U	5	7	1	9	3
SEA	1/5	1/3	1/9	1	1/7
CFE	3	5	1/3	7	1

The calculated value of

λ_{\max}

is 5.238, with a corresponding

C R

of 0.053. This

C R

value is notably below the commonly accepted threshold of 0.10, which is considered the maximum allowable limit for ensuring adequate consistency in decision-making processes. According to Kou et al.⁷⁵ and Temesi,⁷⁶ a

C R

below this threshold signifies that the judgments within the pairwise comparison matrix are highly consistent and reliable. This indicates minimal contradictions or inconsistencies in the provided judgments, ensuring the decision-making process remains objective. Consequently, the pairwise comparison matrix can be regarded as free from significant errors or biases, confirming that the resulting conclusions are based on sound and rational judgment. After that, the weighted normalized matrix (

v_{i j}

) must be calculated. As indicated in Table 11 this entails applying the appropriate weights for each criterion, which were established in the earlier phases of the decision-making procedure, to the normalized values from the decision matrix. Then, using equations (19) and (20) respectively, the ideal (best) and negative-ideal (worst) solutions are determined and exposed in Table 12.

Table 11.

Weighted normalized decision matrix V.

Specimen code	Attributes
Specimen code	$F_{ip}$	$F_{m}$	U	SEA	CFE
SQ30/T5	0.04390	0.02064	0.16544	0.00998	0.07944
C30/T5	0.04317	0.02055	0.16470	0.01083	0.08042
P30/T5	0.04274	0.02197	0.17604	0.01120	0.08676
H30/T5	0.04474	0.02138	0.17138	0.01083	0.08071
C25/T5	0.04116	0.02099	0.16820	0.01110	0.08608
C35/T5	0.03924	0.01901	0.15234	0.00988	0.08178
C40/T5	0.03608	0.01875	0.15032	0.00957	0.08774
C30/T4	0.03381	0.01813	0.14534	0.00959	0.09057
C30/T6	0.04278	0.01926	0.15437	0.01014	0.07603
C30/T7	0.04076	0.02021	0.16197	0.01080	0.08374

Table 12.

Positive and negative ideal solution.

Attributes	$F_{ip}$	$F_{m}$	U	SEA	CFE
Positive ( $v^{+}$ )	0.03381	0.02197	0.17604	0.01120	0.09057
Negative ( $v^{-}$ )	0.04474	0.01813	0.14534	0.00957	0.07603

These solutions serve as benchmarks against which the performance of each alternative is assessed. The ideal solution represents the best possible outcome for each criterion, while the negative-ideal solution represents the worst possible outcome. The distance between each alternative and the ideal and negative-ideal solutions is then calculated using the separation measurements. These measures are important in determining the relative proximity of each alternative to the best and worst solutions. The parting from the ideal solution is calculated using equation (21), which represents the Euclidean distance among an alternative and the ideal solution. Similarly, the separation from the negative-ideal solution is calculated using equation (22), which also uses the Euclidean distance but in relation to the negative-ideal solution. The separation measures are presented in Table 13. The qualified closeness of each alternative to the ‘ideal’ solution is calculated using equation (23). The final ranking, based on the AHP and TOPSIS, identifies the optimal structure, as shown in Table 13. Through the application of AHP and the TOPSIS methods, P30/T5 emerged as the top-performing alternative in the assessment. This material was found to successfully meet all the user-defined performance, fulfilling each criterion in the specified order of influence.

Table 13.

Final ranking informed by AHP and TOPSIS.

Specimen code	$S_{i}^{+}$	$S_{i}^{-}$	$C_{i}$	Ranking
SQ30/T5	0.01847	0.02056	0.52678	5
C30/T5	0.01792	0.02010	0.52870	4
P30/T5	0.00971	0.03285	0.77184	1
H30/T5	0.01546	0.02669	0.63327	3
C25/T5	0.01169	0.02544	0.68510	2
C35/T5	0.02606	0.01065	0.29006	9
C40/T5	0.02622	0.01541	0.37009	7
C30/T4	0.03098	0.01819	0.37000	8
C30/T6	0.02775	0.00933	0.25159	10
C30/T7	0.01721	0.01891	0.52351	6

Conclusions

This study investigates the crash behavior and deformation characteristics of square structures inspired by the cross-sectional geometry of a seahorse skeleton, fabricated from PETG-CF material. Key design parameters, inner shape, inner diameter, and rib thickness, were examined. The fabricated tubes underwent quasi-static axial compression tests, and crashworthiness was evaluated using the initial peak crash force ( $F_{ip}$ ), total absorbed energy (U), mean force ( $F_{m}$ ), specific energy absorption (SEA), and crash force efficiency (CFE). ANOVA was conducted to evaluate the contribution percentage of each investigated parameter to the crashworthiness indicators. A hybrid AHP-TOPSIS method was employed to rank the tested configurations, leading to the following conclusions:

Among the analyzed inner shapes, the polygon design exhibited superior performance across key crashworthiness indicators, including U, $F_{m}$ , SEA, and CFE, with 1105.22 J, 19.74 kN, 19.08 J/g, and 0.889, respectively. Conversely, the hexagonal shape recorded the highest $F_{ip}$ with a value of 23.24 kN. Additionally, the smallest inner diameter exhibited the best performance in terms of U, $F_{m}$ , and SEA, with values of 1056 J, 18.86 kN, and 18.90 J/g, respectively. In contrast, the 30 mm inner diameter achieved the highest $F_{ip}$ = 22.42 kN, while the 40 mm inner diameter attained the maximum CFE = 0.899. With respect to rib thickness, the 5 mm rib thickness demonstrated superior performance in key crashworthiness indicators, yielding higher values of 22.42 kN, 1034.05 J, 18.47 kN, and 18.45 J/g, respectively, for $F_{ip}$ , U, $F_{m}$ , and SEA. In contrast, the 7 mm rib thickness achieved the highest CFE with a value of 0.858. Additionally, the tubes failed progressively, with initial longitudinal cracks followed by wall wrinkling, ultimately leading to catastrophic collapse. ANOVA results demonstrate that the influence of design parameters on crashworthiness performance varies across different indicators. Rib thickness showed the highest contribution to $F_{ip}$ (50.46%) and CFE (63.51%), inner shape was dominant for $F_{m}$ (53.31%) and U (53.45%), while inner diameter had the greatest impact on SEA (38.33%). The P30/T5 configuration was identified as the optimal design based on the AHP-TOPSIS analysis, outperforming all other tested configurations across the evaluated crashworthiness parameters.

Footnotes

Acknowledgements

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/390/46.

Author contributions

Mahmoud M. Awd Allah (Conceptualization; Formal analysis Investigation; Methodology; Validation; Writing – original draft; Writing – review & editing). Mahmoud F. Abd El-Halim (Formal analysis; Methodology; Visualization; Writing – original draft; Writing – review & editing). Marwa Abd El-baky (Conceptualization; Funding acquisition; Resources; Supervision; Validation; Writing – original draft; Writing – review & editing).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Deanship of Research and Graduate Studies at King Khalid University through Large Research Project under grant number RGP2/390/46.

ORCID iDs

Mahmoud M Awd Allah

Marwa A Abd El-baky

Data Availability Statement

This statement indicates that the corresponding author is willing to provide access to the datasets generated and/or analyzed during the study upon reasonable request.*

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Experimental crashworthiness evaluation of seahorse-inspired PETG-CF square tubes under axial compression loading

Abstract

Keywords

Introduction

Methodology

Material

Design parameters

Specimen preparation

Quasi-static axial test

Decision making

Results and discussions

Inner shape effect

Inner diameter effect

Rib thickness effect

Failure analysis

Scatter in the test results

Analysis of variance (ANOVA)

Interaction between process parameters

Selection of the best structure

Conclusions

Footnotes

Acknowledgements

Author contributions

Declaration of conflicting interests

Funding

ORCID iDs

Data Availability Statement

References