Abstract
Purpose
This study aims to understand how the facesheet size, orientation and core size influence the analytical failure mechanism mode of glass fibre reinforced polymer (GFRP)/polyvinyl chloride (PVC) sandwich structures subjected to three-point bending. The purpose of this study was to develop failure-mode map of GFRP/PVC sandwich structures. Sandwich structures with different facesheet and core thicknesses were used to develop the failure map.
Design/methodology/approach
The sandwich structure and facesheet were fabricated using a vacuum-assisted resin infusion method with core sizes of 10, 15 and 20 mm and facesheet thicknesses of 1.5 and 3 mm and were arranged in three different orientations: angle-ply, cross-ply and quasi-isotropic. The key failure modes that occur in sandwich structures were used to predict possible failures in the developed material. Analytical equations were used in MATLAB for each observed failure mode. The probable failure modes, namely, face yielding, core shear and indentation equations, were used to construct the failure maps and were compared with the experimental data.
Findings
The boundary of the two failure modes shifts with changes in the facesheet and core thicknesses. The theoretical stiffness of sandwich panels was higher than the experimental stiffness. Based on strength-to-weight ratio, specimens E10-4, A15-8 and E20-8 exhibited the best optimum values owing to their shorter distance to the boundary lines.
Originality/value
In this study, a failure map was used to predict the possible failure modes for different GFRP facesheet orientations and thicknesses and PVC core thickness sandwich structures. Little is known about the prediction of the failure modes of unidirectional GFRP arranged in different orientations and thicknesses and PVC core thicknesses for sandwich structures. Few studies have used failure mode maps with unidirectional GFRP oriented in angle-ply, cross-ply and quasi-isotropic directions as a facesheet for sandwich structures compared to bidirectional mats. This study can serve as a guide for the correct selection of materials during the design process of sandwich structures.
Keywords
Citation
Kosgey, E.C., Kanny, K. and Mwangi, F.M. (2024), "Failure mode maps for glass fibre-reinforced polymer/polyvinyl chloride foam-cored sandwich structures", World Journal of Engineering, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/WJE-03-2024-0147
Publisher
:Emerald Publishing Limited
Copyright © 2024, Edwin Cheruiyot Kosgey, Krishnan Kanny and Festus Maina Mwangi.
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial & non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
A sandwich panel consists of a low-density core placed between two stiff and thin facesheets. The role of the core is to reinforce the facesheets during the out-of-plane properties, provide strength against indentation and buckling and resist shear forces (Mirsalehi et al., 2024). Sandwich structures are widely applied in engineering fields such as marine, automotive and civil construction (Li et al., 2023; Xie et al., 2024; Palomba et al., 2022; Mater et al., 2024; Prabhu et al., 2023a; Prabhu et al., 2023b; Prabhu et al., 2024) because of their excellent response to bending forces. Several core types are employed in sandwich structures, categorized as corrugated cores, honeycomb cores, foam cores, lattice cores and origami cores, where they exhibit different failure modes (Ma et al., 2021; Feng et al., 2020).
Foam cores, such as polyvinyl chloride (PVC), are often used because of their low weight and high energy absorption (Imielińska et al., 2008). Foam-cored sandwich structures exhibit failure modes, namely, face yield, face wrinkling, core shear, debonding and indentation (Valenza et al., 2010; Liu et al., 2022). The occurrence of these failure modes in bending can be predicted using the failure mode maps. Failure maps play an essential role in the design of sandwich panels, owing to the probable failure modes that provide a benchmark for designing lightweight structures (Xie et al., 2024; Zhang et al., 2022). The goal in using failure map is to ascertain the correlation between the failure mode and design configuration for particular loading parameters, facesheets and core properties (Triantafillou and Gibson, 1987). When two failure modes co-occur, the failure equations are equated to the two modes that provide the boundary equation (Triantafillou and Gibson, 1987; Kotzem et al., 2021). This boundary equation is then used to plot a graph using axes to represent the design configuration for each failure mode (Triantafillou and Gibson, 1987; Xie et al., 2024).
Failure mode maps are developed based on the core and facesheet properties. Several studies have been conducted to elaborate on the failure maps. Yuan et al. (2022) studied the failure behaviour of double-layer aluminium of 0.5–3 mm facesheet thickness and 5–25 mm core thickness aluminium. The authors reported that, for a high facesheet thickness, the critical failure mode of the double-layered sandwich panel was core shear, while the low ratio of core thickness to span length led to face yield failure. Xie et al. (2024) studied the failure modes of curved sandwich beams with a 25 mm core thickness and 2.4 mm facesheet thickness. The authors reported that the parametric analysis of panel with cross-ply fibre angle layup had the highest stiffness 0.38 kN/mm while angle-ply fibre layup had the lowest stiffness at 0.31 kN/mm with local indentation failure.
Ávila (2007) studied the failure mode of functionally graded polystyrene cores with different densities and thicknesses. The authors reported that yielding occurred regardless of the core layer stacking sequence. Valenza et al. (2010) developed a failure mode map for glass fibre reinforced polymer (GFRP)/PVC sandwich panels with various span lengths and facesheet thicknesses. The developed theoretical model predicted the experimental failure modes of types I and II core shear and facesheet failure in the tension mode. Steeves and Fleck (2004b) and Steeves and Fleck (2004a) studied the collapse mechanism of sandwich panels of PVC/GFRP and identified three dominant failure mechanisms: face yielding, core shear and indentation. The authors optimized the geometry by constructing a failure map and deduced the minimum weight as a function of the structural load index. Vitale et al. (2017) studied the mechanical behaviour and failure mode of natural fibre honeycomb and PVC core reinforced by jute and glass fibre facesheet sandwich panels under three-point bending. The authors reported that the failure map indicated different dominant failure modes: core crushing, face wrinkling, core shear and debonding.
From the discussed literature, information still needs to be included in the influence of unidirectional glass fibre orientations when combined with different facesheet and core thicknesses to predict the failure mode using a failure map. This study aims to understand how the facesheet size, orientation and core size influence the failure mechanism mode of GFRP/PVC sandwich structures subjected to three-point bending. The present failure modes of GFRP as a facesheet are available in the literature. However, little information is available about predicting the failure modes of unidirectional GFRP arranged in different orientations and thicknesses on one hand and PVC core thicknesses for sandwich structures on the other. Few studies have used failure mode maps with unidirectional GFRP oriented in angle-ply, cross-ply and quasi-isotropic directions as a facesheet for sandwich structures compared to bidirectional mats. Information on the predicted failure modes in sandwich structures serves as a guideline for academicians, engineers and designers during the development and optimization of materials for structures using GFRP and PVC, such as boats and wind turbine blades.
2. Analytical modelling of three-point bending behaviour of sandwich panel
The failure load of the sandwich panel under a quasi-static flexural test was predicted based on the mechanical properties of the core and facesheet. The assumption is that the facesheet behaves linearly and fails according to the classical beam theory (Mostafa et al., 2014). Consider a simply supported sandwich panel in three-point bending mode with a uniform cylindrical roller, shown in Figure 1. The central section of the sandwich panel moves by deflection u because of the applied load P in the loading roller. Let L be the sandwich panel separation distance between the two supports, b the width of the beam, H the overhang at each end, tf the face thickness and tc the core thickness. The important mechanical properties of the core are the Young’s modulus Ec, the compression strength σc, shear strength τc and shear modulus Gc. The mechanical properties of the facesheet include yield strength σf and the axial Young’s modulus Ef (Steeves and Fleck, 2004b). The load of the different failure modes can be calculated using the mechanics of the materials formula. The likely failure modes for foam-cored sandwich structures are face yielding, core shear failure, indentation, debonding and face wrinkling (Mostafa et al., 2014).
2.1 Stiffness
The overall deflection u at the central location of a simply supported sandwich panel loaded in three-point bending is given as the sum of the deflections caused by the bending of the facesheets and the shear of the core (Steeves and Fleck, 2004b):
2.2 Strength
In sandwich structures, the primary failure modes include face yield, face wrinkling, core shear and indentation (Steeves and Fleck, 2004b; Allen, 2013). Facesheet failure occurs when the axial compressive/tensile stress of facesheet exceeds its maximum strength. The failure of the sandwich structure resulted from compression failure in the upper facesheet. From the classical theory of sandwich panels, the axial strain is assumed to exhibit linear variation over the cross-section of the panel, as in the case of the Euler–Bernoulli theory (Mostafa et al., 2014). Failure by indentation is more challenging and is dealt with as a particular case (Steeves and Fleck, 2004b). For the four failure modes, the relevant predicted failure loads are as follows:
2.2.1 Face yielding or micro buckling
Face yielding occurs when the critical stress of the facesheets reaches the yield stress of the facesheet material (Zhang et al., 2022). The facesheet is considered to be elastic perfectly plastic, and face yield occurs only when the facesheets reach total plastic deformation. Shear stress in the facesheets are negligible compared with the normal stresses σf; hence, the equivalent stress is equal to the normal stress (Triantafillou and Gibson, 1987). The threshold load just before onset of failure of a sandwich beam is given by Steeves and Fleck (2004b):
2.2.2 Core shear failure
The bending moment on a sandwich structure is mainly carried by front and back facesheets, while the transverse shear force is mainly carried by core (Zhang et al., 2020). Core shear occurs when the thickness of the facesheet is relatively small compared to that of the core. Thus, core shear failure is assumed to occur at a uniform shear strength with negligible additional strength of the facesheets (Steeves and Fleck, 2004a). The threshold loads necessary to cause core shear damage is given by:
2.2.3 Face wrinkling
In facesheet wrinkling mode, it is assumed that the top facesheet and the upper surface of the foam core undergo the same deformation in the y-direction when wrinkling occurs in the top facesheet. The top facesheet buckles in a wrinkle-like manner when the normal stress σf in the facesheet reaches the critical stress (Omachi et al., 2020). Thus, the face wrinkling is described as localized unstable elasticity of the facesheet contributed by short wavelength elastic buckling of the upper facesheet that does not match the wavelength of the underlying elastic core (Steeves and Fleck, 2004a). The face-wrinkling threshold load is given by:
2.2.4 Indentation
Indentation refers to a specific local mode of failure that is often observed in sandwich structures with thicker cores and thinner facesheets. This form of failure occurs jointly with plastic deformation and core yield, marking a notable advancement in material stress transfer (Jin et al., 2023). The indentation of sandwich structures occurs by the plastic yield of the core with facesheets deforming either elastically or plastically, as discussed in detail by Steeves and Fleck (2004a). The assumption is that the facesheet maintains an elastic state during the indentation scenario, and the core assumes a rigid, perfectly plastic solid. Hence, the minimum load is given by:
3. Experiment
3.1 Materials and fabrication
3.1.1 Materials
The sandwich structures used in this study comprised a PVC polymeric foam core and unidirectional (UD) glass fibres of 500 gsm. Core thicknesses of 10, 15 and 20 mm with a density of 80 kg/m3 were used. The fibres were arranged to form different stacking sequences and orientations, keeping ply numbers as four and eight layers to ensure a uniform facesheet thickness. The technical material properties of the core obtained from the manufacturer are listed in Table 1. The chemicals used were LR30 epoxy resin and LH30 slow hardener mixed at a ratio of 100:25 by weight for the resin to hardener. LR30 epoxy resin and LH30 hardener have low viscosity and are suitable for the vacuum infusion process. This mixture was designed to exhibit excellent toughness and mechanical properties after post curing. All materials and chemicals used were purchased from AMT composites, Durban.
3.1.2 Fabrication
The specimens were fabricated using LR30 epoxy resin and LH30 slow hardener using the vacuum-assisted resin infusion method (VARIM). First, the weight of the fibres was measured using a digital weighing scale. Then, the required LR30 resin and LH30 slow hardener were calculated using a ratio 100:25 by weight of the fibres. The fibres were then laid in the mould with the core ready for infusion. Finally, LR30 and LH30 were mechanically stirred in a glass beaker and placed at the suction point of the infusion assembly. The suction point was clamped when all the mixtures were used and left for 24 h before demoulding. The specimens for the tensile test were made of GFRP only, whereas the sandwich structure for the three-point bend (3PB) included the foam core. The production process of the sandwich structures is shown in Figure 2.
The manufactured sandwich samples were post-processed in an oven, with the maximum temperature maintained at 80°C for 4 h. The samples were produced as panels and cut to the required sizes using a band saw with a diamond blade, as per the ASTM standard for the respective tests. In the current study, samples containing cross-ply (0/90), angle-ply (+45/−45) and quasi-isotropic glass-fibre-oriented facesheet and PVC foam cores of various thicknesses were marked for identification with notations and thickness sizes. For example, A20-4 means “A” for angle-ply orientation, “20” for core size (mm), “4” to indicate 4 layers of plies and 8 to indicate 8 layers of plies. A similar notation was “C” for cross-ply and E represents quasi-isotropic orientation.
3.2 Tensile test
Tensile properties were conducted for the facesheet only. These properties were used during the formulation of the boundary lines as they help in the prediction of the possible failure modes. The tensile tests of the fibre composite facesheet were carried out following (ASTMD3039/D3039M, 2000) standard. The ends of the specimen were carefully clamped at the gripping area to avoid slippage that can result in premature failure. The specimens were tested at test rate of 1 mm/min in tension using MTS Criterion, Model 43, an electronic universal testing machine installed with a load cell capacity of 30 kN. The test rate of 1 mm/min was chosen after testing other rates following ASTM D3039 standard where constant strain rate in gage section has to occur between 1 and 10 min. The tensile stresses and strains were calculated based on the load and displacement data obtained from the raw data. Table 2 shows tensile test data (yield strength and Young’s modulus) for GFRP facesheet.
3.3 Three-point bending experiments
A three-point bending test was performed in universal testing machine MTS Model 43 equipped with 30 kN load cell at test rate of 3 mm/min as per ASTMC393/C393M (2020). Similar standard was used by Xie et al. (2024) for three-point bending test of sandwich panels. The lengths of the sandwich specimens were 200 mm, width of 60 mm and span length was maintained at 150 mm. The support and loading bars were steel cylinders with diameters of 25 mm. The data were captured using the data acquisition system software, which records the load against the corresponding displacement. The peak of the load–displacement curve was considered as the critical load failure point.
4. Failure maps construction
Numerous factors influence the failure load and failure mechanisms of sandwich panels as discussed in the analytical section. The failure mode is determined by the properties of the facesheet and core materials, such as orientation of the facesheet, thicknesses of the facesheet and core, and span length. The failure map mechanism was constructed following the methodologies described in (Steeves and Fleck, 2004b; Słonina et al., 2023). Failure mode maps were developed using MATLAB (MathWorks, 2023b). The sandwich beam failure load depends on the properties of the facesheet (Ef, tf), core thickness (Ec, tc) and span length L (mm). The failure mode maps are represented by a graph that shows the possible failure modes. Boundaries were used to represent zoning for the different failure modes. The critical load on the sandwich panel is considered as a function of the material properties and geometric parameters of the material. To obtain boundaries for various failures for stresses in facing and core, equations (4)(5), (5)(7) and (4)(7) were compared side by side and corresponding relations were obtained as shown in equations (8)–(10).
For core shear versus face yield (CS/FY):
5. Results and discussion
5.1 Failure mechanisms
The constructed failure maps for panels with facesheets of various thicknesses, ply-orientations and core thicknesses, based on the analytical parameters elaborated in Section 2. The failure maps do not indicate the failure load of the sandwich panel; rather, they indicate the possible failure mode that will occur.
The failure mode maps constructed from the geometrical dimensions of the materials are illustrated in Figures 3 and 4, showing the simultaneous curves for core sizes of 10, 15 and 20 mm. From Figures 3 and 4, it can be observed that the face yield and indentation occupy the largest area of the map, whereas core shear occurs only in a limited area. A possible reason for this behaviour is that the breaking loads of the face yield and indentation are functions of the material properties of the facesheet. Furthermore, the PVC foam core has good resistance to shear compared to the GFRP facesheet, which contributes to the small area of the core shear failure. The face yield is dominant in regions with high values of tc/L and tf/L. This scenario can be explained by considering that, for this geometrical configuration, the indentation of the upper facesheet increases, which prevents premature indentation failure. Moreover, stress is applied onto the weaker areas of the sandwich panels, which causes the axial stress to exceed the tensile strength of the facesheet.
A three-point bending test was conducted with different combinations of the facesheet and core at a constant span length, and failure modes were identified. Figure 5 shows the failure modes obtained from the three-point bend experiment conducted on the GFRP/PVC sandwich panel. The dominant failure modes of indentation, face yield and core shear, which were analysed in Section 2, are visible. Specimen E10-4 exhibited face yield failure in the experiment, which was well predicted by the failure map shown in Figure 3. This result was replicated well in all other facesheet orientations with similar core sizes and facesheet thicknesses. Specimens C20-4 and E20-8 exhibited indentation failure. The thickness of the core contributes to indentation, as the loading roller leads to the densification of the PVC foam cells, thereby creating a local indent at that point.
5.2 Load carrying capacity
Table 3 shows a summary of the experimental results, where Pmax is the maximum load, and Dexp and Dthe are the experimental and theoretical elastic bending stiffnesses, respectively. Dexp was obtained from the experimental data, as follows:
5.3 Optimization of structural variables
Table 4 shows the distances of the structural variables to the intersection point. The distance from the location of the structural variables to the intersection point was calculated using the formula used to calculate the distance between two points in a plane. Specimens near the boundary line possess optimum structural variables because the failure modes in the core and facesheet occur simultaneously, which indicates that no excess material is utilized in the sandwich structure. Therefore, the intersection points of the boundary lines are most important (Hao et al., 2020). The obtained distance was used to compute the level of optimization of the strength-to-weight ratio. The strength-to-weight ratio of the sandwich panel was calculated using equation (12) (Hao et al., 2020):
Where Wt is the strength-to-weight ratio,
Taking A10-4 as an example, Pmax = 1,427.8 N, b = 0.06 m, L = 0.15 m,
Figure 6 shows a concise pattern in which the strength-to-weight ratio of the sandwich structure is inversely proportional to the distance from the point of the structural variables to the intersection of the failure boundary line. Specimens E10-4, A15-8 and E20-8 had the best strength-to-weight ratio and shortest distance to the boundary line. On the other hand, specimens C10-8 and A20-8 had lower strength-to-weight ratios and larger distances to boundary lines.
6. Conclusion
This study focusses on the analytical failure modes of GFRP/PVC sandwich panels under a three-point bending load. The failure mode map was developed to predict the failures in sandwich panels for the relationship between the facesheet thickness, core thickness, and span length between the loading supports. The facesheet thickness of 1.5 and 3 mm, and core thickness of 10, 15 and 20 mm were used for design. From the experiments used for validation, three failure modes were dominant: face yield, indentation and core shear. For the maximum load carrying capacity, the theoretical stiffness was higher than experimental stiffness by 42.23%. These failure modes are expressed in a failure mode map on the graph of tc/L versus tf/L. The information from the failure map was used to compute the strength-to-weight ratio of the sandwich composite to optimize the structural parameters and ensure minimal wastage of the materials. The strength-to-weight ratio varied from 20,739.15 to 91,395.81 N m/kg, whereas distance to intersection varied from 0.0558 to 0.1248. It was observed that specimens with the shorter distance to boundary lines gave the best results. The analytical model developed showed results consistent with the experimental data, thus indicating a reliable tool to predict failure mechanisms in sandwich structures.
This study can be extended to failure mode maps using quasi-static indentation and dynamic tests such as low-velocity impact. Currently, the authors are working on analytical and experimental research on failure modes of impact energy propagation for GFRP/PVC sandwich panels. Furthermore, with disruptive technology, sandwich structures can be fabricated using 3D printing in continuous and discontinuous methods, using materials with a density equivalent to PVC for comparison.
Figures
Figure 1
Geometrical parameters of a sandwich panel as loaded in three-point bending test
Figure 2
Vacuum infusion process
Figure 3
Failure mode maps for sandwich structures with 1.5 mm facesheet and different core thicknesses
Figure 4
Failure mode maps for a 3 mm facesheet depicting different core thicknesses
Figure 5
Failure modes
Figure 6
Relationship between strength-to-weight ratio of sandwich structure and distance from point of structural variables to intersection point
Mechanical properties of PVC foam core
| Density | Compressive strength | Compressive modulus | Shear strength | Shear modulus | Tensile strength | Tensile modulus |
|---|---|---|---|---|---|---|
| ASTM D1622 | ASTM D1621-10 | ASTM D1621-10 | ASTM C273 | ASTM C273 | ASTM D1623 | ASTM D1623 |
| 80 kg/m3 | 1.60 MPa | 74 MPa | 1.20 MPa | 30 MPa | 2.74 MPa | 176 MPa |
Table courtesy of Kosgey et al. (2024), https://creativecommons.org/licenses/by/3.0/
Tensile properties of GFRP facesheet
| Structural variables | Young’s modulus, Ef (GPa) | Yield strength, σf (MPa) |
|---|---|---|
| A4 | 2.62 | 77.99 |
| C4 | 9.33 | 396.36 |
| E4 | 7.07 | 197.43 |
| A8 | 2.84 | 122.42 |
| C8 | 17.60 | 296.59 |
| E8 | 10.32 | 338.93 |
Table courtesy of Kosgey et al. (2024), https://creativecommons.org/licenses/by/3.0/
Geometrical parameters of sandwich panels
| Structural variables | L (mm) | tf (mm) | tc (mm) | tf /L | tc /L | Pmax (kN) | Pfy (kN) |
Pcs (kN) |
Pin (kN) |
Dexp (N/mm) | Pthe (N/mm) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| A10-4 | 150 | 1.5 | 10 | 0.01 | 0.067 | 1.43 | 2.15 | 1.656 | 20.46 | 159.53 | 171.69 |
| C10-4 | 150 | 1.5 | 10 | 0.01 | 0.067 | 1.67 | 10.94 | 1.656 | 31.26 | 231.97 | 355.26 |
| E10-4 | 150 | 1.5 | 10 | 0.01 | 0.067 | 1.44 | 5.45 | 1.656 | 28.49 | 286.28 | 312.08 |
| A10-8 | 150 | 3.0 | 10 | 0.02 | 0.067 | 1.52 | 7.64 | 1.872 | 21.90 | 214.40 | 356.91 |
| C10-8 | 150 | 3.0 | 10 | 0.02 | 0.067 | 3.10 | 18.51 | 1.872 | 40.23 | 301.94 | 671.13 |
| E10-8 | 150 | 3.0 | 10 | 0.02 | 0.067 | 1.83 | 21.15 | 1.872 | 33.67 | 354.22 | 598.62 |
| A15-4 | 150 | 1.5 | 15 | 0.01 | 0.1 | 1.18 | 3.08 | 2.376 | 34.62 | 384.11 | 317.54 |
| C15-4 | 150 | 1.5 | 15 | 0.01 | 0.1 | 1.38 | 15.70 | 2.376 | 52.88 | 398.38 | 572.83 |
| E15-4 | 150 | 1.5 | 15 | 0.01 | 0.1 | 1.20 | 7.82 | 2.376 | 48.20 | 285.71 | 517.83 |
| A15-8 | 150 | 3.0 | 15 | 0.02 | 0.1 | 1.82 | 10.58 | 2.592 | 36.61 | 298.26 | 562.61 |
| C15-8 | 150 | 3.0 | 15 | 0.02 | 0.1 | 2.87 | 25.63 | 2.592 | 67.29 | 297.77 | 909.50 |
| E15-8 | 150 | 3.0 | 15 | 0.02 | 0.1 | 1.30 | 29.28 | 2.592 | 56.30 | 238.39 | 838.67 |
| A20-4 | 150 | 1.5 | 20 | 0.01 | 0.133 | 1.50 | 4.02 | 3.096 | 50.42 | 518.03 | 489.46 |
| C20-4 | 150 | 1.5 | 20 | 0.01 | 0.133 | 1.79 | 20.45 | 3.096 | 77.01 | 663.19 | 799.73 |
| E20-4 | 150 | 1.5 | 20 | 0.01 | 0.133 | 1.71 | 10.19 | 3.096 | 70.19 | 301.55 | 736.78 |
| A20-8 | 150 | 3.0 | 20 | 0.02 | 0.133 | 1.68 | 13.52 | 3.312 | 52.97 | 536.44 | 781.49 |
| C20-8 | 150 | 3.0 | 20 | 0.02 | 0.133 | 3.04 | 32.74 | 3.312 | 97.31 | 631.27 | 1148.90 |
Table by authors
Summary of the distances from structural variables to intersection points
| Structural variables | tf/L | tc/L | Pmax (kN) | Density (kg/m3) |
Strength-to-weight ratio (N m/kg) | Distance to intersection |
|---|---|---|---|---|---|---|
| A10-4 | 0.01 | 0.067 | 1.43 | 543.48 | 74,493.67 | 0.0558 |
| C10-4 | 0.01 | 0.067 | 1.67 | 547.10 | 86,506.99 | 0.0595 |
| E10-4 | 0.01 | 0.067 | 1.44 | 586.96 | 69,632.44 | 0.0593 |
| A10-8 | 0.02 | 0.067 | 1.52 | 730.77 | 46,093.07 | 0.0572 |
| C10-8 | 0.02 | 0.067 | 3.10 | 753.21 | 91,395.81 | 0.0632 |
| E10-8 | 0.02 | 0.067 | 1.83 | 833.33 | 48,717.35 | 0.0618 |
| A15-4 | 0.01 | 0.1 | 1.18 | 478.32 | 70,135.60 | 0.0856 |
| C15-4 | 0.01 | 0.1 | 1.38 | 488.87 | 80,181.82 | 0.0898 |
| E15-4 | 0.01 | 0.1 | 1.20 | 451.92 | 75,380.96 | 0.0923 |
| A15-8 | 0.02 | 0.1 | 1.82 | 663.99 | 31,687.99 | 0.0866 |
| C15-8 | 0.02 | 0.1 | 2.87 | 626.74 | 52,956.27 | 0.0937 |
| E15-8 | 0.02 | 0.1 | 1.30 | 681.14 | 22,030.40 | 0.0922 |
| A20-4 | 0.01 | 0.133 | 1.50 | 363.66 | 33,397.17 | 0.1160 |
| C20-4 | 0.01 | 0.133 | 1.79 | 413.25 | 35,153.16 | 0.1211 |
| E20-4 | 0.01 | 0.133 | 1.71 | 393.86 | 35,233.92 | 0.1210 |
| A20-8 | 0.02 | 0.133 | 1.6764 | 573.01 | 20,739.15 | 0.1217 |
| C20-8 | 0.02 | 0.133 | 3.0393 | 609.88 | 35,326.84 | 0.1248 |
| E20-8 | 0.02 | 0.133 | 2.6203 | 601.54 | 30,878.92 | 0.1231 |
Table by authors
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Acknowledgements
The authors acknowledge Durban University of Technology research office for postgraduate support finances.
Availability of data and materials: The data sets used during the current study are available from the corresponding author upon reasonable request.
Competing interests: The authors declare that they have no competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.