Optimisation of Fibre-Reinforced Hybrid Composites Under Combined Loading

Abstract

Fibre-reinforced hybrid composites offer an effective balance between strength, weight, and cost by combining multiple fibre types within a single matrix. This study focuses on optimising the design of carbon/glass fibre-reinforced hybrid composites under combined bending and torsional loading using finite element analysis (FEA) and response surface methodology. Twelve different layup configurations, including sandwich and non-sandwich hybrid designs, were analysed to identify the optimal ply angles and fibre volume fractions that maximise failure load while minimising material cost and density. The results reveal that sandwich-type layups, such as [C₃G]_S, [C₂G₂]_S, and [CG₃]_S, demonstrate superior strength-to-weight performance, achieving failure loads exceeding 300 N. The study also confirms that optimal ply angles range from 12° to 30°, depending on the layup configuration, and that increasing the carbon fibre volume fraction generally enhances failure load, though an optimal balance with glass fibres must be maintained. The findings provide valuable design guidelines for engineers seeking to tailor hybrid composites for aerospace, automotive, and structural applications. Future work should focus on experimental validation and extending the analysis to additional loading conditions, such as impact and fatigue, to further improve the robustness of hybrid composite structures.

1. Introduction

Fibre-reinforced hybrid composites are sophisticated materials that integrate multiple types of fibres within a single matrix. This integration leverages the strengths of each fibre type while mitigating their individual weaknesses. Consequently, these materials are extensively utilised across various industries due to their enhanced properties and customised performance [1]. In the aerospace industry, hybrid composites are used in aircraft components for their high strength-to-weight ratio and durability. The automotive sector employs hybrid composites in vehicle parts to reduce weight and improve fuel efficiency. In construction, hybrid composites are incorporated into building materials to enhance structural performance and longevity. Additionally, hybrid composites are used in sports equipment such as bicycles, tennis rackets, and helmets.

Composite structures, e.g., aircraft wings, are usually subjected to combined bending and torsional loading. Wang et al. [2] established a mathematical model for the buckling and post-buckling of the wing box under different torsion-bending ratios. Mukherjee et al. [3] studied the static and stability behaviours of thin-walled composite beams with variable stiffness under flexural–torsional coupling conditions. Ren et al. [4] studied the static behaviour of stiffened steel tubes under combined compression–bending–torsion loading. Patuelli et al. [5] presented a finite element method that is capable of predicting the static and dynamic behaviour of beam structures with bending–torsion coupling.

One of the advantages of carbon/glass fibre-reinforced hybrid composites is the enhancement of flexural strength, known as the positive hybrid effect [6]. Dong and Davies [7] studied the flexural properties of carbon- and glass fibre-reinforced epoxy hybrid composites and concluded that for unidirectional composites, positive hybrid effects are observed when glass/epoxy laminas are placed on the compressive face, whereas negative hybrid effects dominate when they are placed on the tensile face. For sandwich-type hybrid composites, carbon/epoxy laminas should form the skin, and glass/epoxy laminas should constitute the core. Under tensile/compressive loading, a positive hybrid effect of +7.4% was reported for the failure strain in tension for interply carbon/glass fibre-reinforced hybrid composites, while a negative hybrid effect of −6.4% was observed in compression. For intraply carbon/glass fibre-reinforced hybrid composites, the best synergistic effects of +17.8% and +39.6% were reported in tensile and compressive strengths, respectively [8].

Since the behaviours of hybrid composites under combined loadings are complex, numerical methods, e.g., finite element analysis (FEA), are often needed [9]. Optimisation is achieved with the aid of numerical simulation. Moazed et al. [10] defined structural efficiency metrics and performance indices for the integrated selection of the design parameters that govern the static and dynamic performance of laminated thin-walled composite beams. The design charts and simplified approach were then used as a tool for selecting laminated composite beam(s) with the required stiffness and mass. Important parameters in the design of laminated composite beams include the number of layers, the thickness of the laminate, fibre angle orientation, materials of construction, and cross-sectional shape. Monte et al. [11] optimised the composite specimen with the objective of minimising the maximum displacement for a given load by considering the orientation of the layers as the design variable. In a similar approach, Infante et al. [9] optimised hybrid carbon/glass fibre-reinforced composites under combined loading of bending and torsion.

NSGA-II (Non-dominated Sorting Genetic Algorithm II) is an evolutionary algorithm for solving multi-objective optimisation problems by ranking solutions using non-dominated sorting and preserving diversity with a crowding distance metric. It efficiently evolves a population toward a well-distributed approximation of the Pareto front. When NSGA-II and FEA are coupled, because for each run in NSGA-II, the FEA model needs to be updated, multi-objective optimisation is preventively time-consuming and infeasible for practical use [12]. Dong [12] developed several design rules for both unidirectional and multi-directional hybrid composites. These rules prove to be a better and more efficient approach to optimisation with minimum cost and weight as the objective functions when compared to NSGA-II optimisation.

In this study, a carbon/glass fibre-reinforced hybrid composite is optimised to minimise density and cost while meeting a required failure load under combined bending and torsional loading. Various layup configurations are identified, and for each, the optimal fibre orientation is determined. The fibre volume fractions of the carbon/epoxy and glass/epoxy plies are systematically varied to compute failure loads using an FEA-based model. A response surface is then developed to quantify the influence of fibre volume fractions on failure load. Using these response surfaces, potential layups that satisfy the failure load requirement are identified. Finally, the costs and densities of the selected configurations are evaluated, and a Pareto front is constructed.

2. Methodology

2.1. Materials

In this study, the hybrid composites are an interlayer structured consisting of an epoxy matrix being reinforced by carbon and glass fibres. Each layer is reinforced by either high-strength carbon fibre or E-glass fibre. The properties of the fibres and epoxy resin are shown in

Table 1. Constituent material properties.

Material	Elastic Modulus (GPa)	Tensile Strength (MPa)	Density (g/cm3)	Cost ($/Litre)
Epoxy	3.1	69.6	1.09	26.2
High-strength carbon fibre	230	4900	1.8	151.2
E-glass fibre	72	3450	2.58	10.8

, based on data from the literature [13]. The properties of the composites are derived using Hashin’s Composite Cylindrical Model (CCM) [14].

Interfacial adhesion between the fibre and epoxy binder is implicitly represented through the input material properties, which are derived from experimental data in the literature. The higher interfacial shear strength and wettability of carbon fibres compared to glass fibres are reflected in the assigned orthotropic strength values, consistent with previous modelling approaches.

In this study, symmetric angle-ply stacking sequences are chosen because they are much stronger and tougher than the quasi-isotropic laminates. Also, they do not have problems with weak transverse layers [15].

2.2. FEA-Based Model

The model consists of an 8-ply laminate that is subject to combined bending and torsional loading. An AutoCAD 2023 schematic of the geometry and loading conditions is shown in Figure 1. As can be seen in Figure 1, the composite plate contains 8 plies which are 0.25 mm each, totalling a laminate thickness of 2 mm. The length and the width of the composite specimen are 115 and 65 mm, respectively. A predefined downwards displacement is applied to the composite plate via a cylindrical loading nose, which is located at the Cartesian coordinates (100, 50), with respect to the origin indicated in Figure 1.

The composite layup is defined in Ansys ACP. For each type of ply, i.e., carbon/epoxy and glass/epoxy plies, the orthotropic material properties and all strength components are defined. When these failure modes are assumed, the composite is modelled as a surface body. With reference to Figure 1, two edges, x = 0 and y = 0, are constrained, and a displacement of 30 mm in the z-axis is applied to the loading nose. Simulation is performed in Ansys Mechanical, similar to the approach in a previous study [14].

The layups for the carbon/glass fibre-reinforced hybrid composite are defined to be left to right, corresponding to ply 8 to ply 1, and C and G denote carbon/epoxy and glass epoxy, respectively. For example, layup [C₇G] denotes that ply 1 is glass/epoxy and plies 2–8 are carbon/epoxy. In this layup, the glass/epoxy ply is located on the bottom surface of the specimen, thus being compressed.

For a composite laminate in flexure, it has been found that failure mostly initiates at the compressive side [6]. The compressive behaviour of the composite is complex, with several possible failure modes: pure compression, delamination/shear, and microbuckling or kinking [16]. In this study, the Lo–Chim model [17] is used to predict the compressive failure in microbuckling or kinking.

Mesh refinement is performed to ensure the accuracy and convergence of results in finite element analysis (FEA).

2.3. Optimal Ply Angle

In this study, 12 possible layups, including 1 full carbon layup ([C₈]), 1 full glass layup ([G₈]), and 10 hybrid layups, are considered. For the hybrid layups, there are 3 sandwich layups ([C₃G]_S, [C₂G₂]_S, and [CG₃]_S) and 7 non-sandwich layups ([C₇G], [C₆G₂], [C₅G₃], [C₄G₄], [C₃G₅], [C₂G₆], and [CG₇]).

For each layup, the ply angle ([(±θ)₂]_S) for achieving the maximum failure load needs to be determined. As an example, the [C₄G₄] laminates of various ply angles are studied. The fibre volume fractions, V_fc and V_fg, are 0.3 and 0.5, respectively. The deformation is shown in Figure 2. It is shown that significant twist occurs for a ply angle of 0°.

For comparison, two quasi-isotropic stacking sequences, [0/90/±45]_S and [0/±45/90]_S, coupled with the aforementioned 12 layups, are simulated.

The maximum failure criteria for layers 1 and 8 are shown in Figure 3 and Figure 4, respectively. It is shown that when the ply angle is 0° or 26°, layer 1 has higher maximum failure criteria than layer 8, suggesting that failure will occur at layer 1, which is in compression. When the ply angle is 45° or 90°, layer 8 has higher maximum failure criteria than layer 1, suggesting that failure will occur at layer 8, which is in tension.

2.4. Response Surface

The failure load vs. ply angle for [C₄G₄] is shown in Figure 5, from which it is shown that the optimal ply angle is θ = 26°. After the ply angle is set to 26°, both fibre volume fractions, i.e., V_fc and V_fg, are varied between 0.3 and 0.65 at 5 levels, and the corresponding failure load is recorded. The resulting failure loads are shown in

Table 2. Failure loads for layup [C₄G₄].

Vfc	Vfg	Failure Load (N)
0.3	0.3	192.42
0.3	0.3875	224.74
0.3	0.475	259.41
0.3	0.5625	263.20
0.3	0.65	239.94
0.3875	0.3	199.54
0.3875	0.3875	229.20
0.3875	0.475	269.80
0.3875	0.5625	276.70
0.3875	0.65	252.60
0.475	0.3	208.30
0.475	0.3875	241.82
0.475	0.475	274.85
0.475	0.5625	286.89
0.475	0.65	264.04
0.5625	0.3	217.38
0.5625	0.3875	250.04
0.5625	0.475	284.53
0.5625	0.5625	300.51
0.5625	0.65	274.68
0.65	0.3	229.22
0.65	0.3875	259.01
0.65	0.475	296.71
0.65	0.5625	308.35
0.65	0.65	285.03

and can be plotted as a contour plot, as shown in Figure 6. It can be seen from Figure 6 that the failure load depends on both V_fc and V_fg. The minimum failure load occurs when V_fc is 0.3 and V_fg is 0.3, and the maximum failure load occurs when V_fc is 0.65 and V_fg is 0.5625. It is also shown that when V_fg is greater than 0.5625, the failure load decreases with increasing V_fg.

2.5. Determination of Candidates

For a given required failure load, solutions for each identified layup are obtained by varying the fibre volume fractions of carbon/epoxy and glass/epoxy plies. Since multiple solutions exist, this study sets V_fc to several predefined values, e.g., 0.3, and V_fg is adjusted based on the contour plots so that the target failure load is just achieved. Once the fibre volume fractions are established, the material composition is defined based on the layup, and the density and cost are calculated using the property data in Table 1.

3. Results and Discussion

3.1. Optimal Ply Angles

The optimal ply angles for all layups are shown in

Table 3. Optimal ply angles for achieving the highest failure loads.

Layup	Ply Angle (°)
[C8]	16
[C7G]	30
[C6G2]	26
[C5G3]	25
[C4G4]	26
[C3G5]	26
[C2G6]	28
[CG7]	15
[G8]	21
[C3G]S	13
[C2G2]S	12
[CG3]S	12

. The optimal ply angles range from 12° to 30°.

3.2. Failure Loads

Failure in hybrid laminates under combined bending and torsional loading initiates on the compressive side, typically through microbuckling and kinking, consistent with our earlier studies [6,7]. Delamination may also occur due to interlaminar shear stresses induced by torsion. Similar interlaminar effects under combined loading have also been reported for basalt FRP tendons in seawater environments [18], reinforcing the importance of considering shear-driven damage in hybrid composite design. Sandwich-type layups such as [C₃G]_S, [C₂G₂]_S, and [CG₃]_S exhibit superior strength-to-weight ratios because carbon plies on the outer surfaces provide high flexural stiffness, while glass plies in the core reduce weight and cost. This structural arrangement effectively redistributes stresses and delays compressive failure.

The failure loads of symmetric angle ply and quasi-isotropic stacking sequences for V_fc = 0.3 and V_fg = 0.5 are presented in

Table 4. Failure loads of symmetric angle ply and quasi-isotropic stacking sequences for V_fc = 0.3 and V_fg = 0.5.

	Failure Load (N)
Layup	Angle Ply	[0/90/±45]S	[0/±45/90]S
[C8]	306.84	189.60	225.71
[C7G]	293.67	219.99	260.80
[C6G2]	273.85	227.89	232.39
[C5G3]	273.75	209.12	233.15
[C4G4]	274.11	208.45	235.00
[C3G5]	277.99	208.33	234.97
[C2G6]	281.57	200.68	237.46
[CG7]	256.73	200.84	228.59
[G8]	269.53	188.85	212.39
[C3G]S	314.50	190.36	223.45
[C2G2]S	291.43	171.23	227.43
[CG3]S	270.57	175.74	211.27

and graphically shown in Figure 7. It is shown that symmetric angle ply stacking sequences yield the highest failure loads. For those two quasi-isotropic stacking sequences, [0/±45/90]_S has higher failure loads than [0/90/±45]_S.

The contour plot varies with the layup. The contour plot for [C₂G₆] is shown in Figure 8. It is shown that the minimum failure load occurs when V_fc is 0.3 and V_fg is 0.3, and the maximum failure load occurs when V_fc is 0.65 and V_fg is 0.5625. It is shown that the failure load increases with V_fg, and V_fc has little effect on the failure load. Similar contour patterns are found for layups [CG₇], [C₃G₅], [C₄G₄], and [C₅G₃].

The contour plot for [C₆G₂] is shown in Figure 9. It is shown that the minimum failure load occurs when V_fc is 0.3 and V_fg is 0.3, and the maximum failure load occurs when V_fc is 0.65 and V_fg is 0.475. It is also shown that when V_fg is greater than 0.475, the failure load decreases with increasing V_fg.

The contour plot for [C₇G] is shown in Figure 10. It is shown that the minimum failure load occurs when V_fc is 0.3 and V_fg is 0.65, and the maximum failure load occurs when V_fc is 0.65 and V_fg is 0.3875. It is also shown that when V_fg is greater than 0.475, the failure load decreases with increasing V_fg.

More complex contour shapes are seen for sandwich-type layups. As an example, the contour plot for [C₂G₂]_S is shown in Figure 11. In general, the maximum failure load occurs when both V_fc and V_fg are 0.65 and increases with the number of carbon/epoxy plies.

3.3. Candidates

Two target failure loads, 250 and 300 N, are considered. When the target failure load is 250 N, the solutions are given as dots, as shown in Figure 12. The lower bound, represented by the red line, forms the Pareto front, and the optimal candidates are given in

Table 5. Optimal candidates for a failure load of 250 N.

Layup	Failure Load (N)	Vfc	Vfg	Density (g/cm3)	Cost ($/L)
[C]8	306.84	0.3	-	1.303	63.70
[C3G]S	314.50	0.3	0.3	1.362	53.17
[C3G]S	314.40	0.3	0.36	1.384	52.94
[C3G]S	314.42	0.3	0.42	1.406	52.71
[C6G2]	251.87	0.3	0.44	1.414	52.63
[C2G2]S	282.72	0.3	0.3	1.420	42.64
[C2G2]S	268.11	0.3	0.36	1.465	42.18
[C2G2]S	270.68	0.3	0.42	1.509	41.72
[CG3]S	250.51	0.52	0.31	1.529	38.87
[CG3]S	255.14	0.5	0.34	1.559	37.90
[CG3]S	251.87	0.45	0.35	1.561	36.22
[C3G5]	251.21	0.3	0.44	1.580	36.03
[CG3]S	251.39	0.42	0.38	1.589	34.94
[CG3]S	251.25	0.4	0.39	1.597	34.20
[CG3]S	250.35	0.37	0.4	1.603	33.14
[CG3]S	250.12	0.32	0.41	1.605	31.46
[CG3]S	251.84	0.3	0.42	1.613	30.72
[CG7]	253.77	0.3	0.39	1.625	25.63
[G]8	250.25	-	0.44	1.746	19.42

. It is shown from Table 5 that sandwich layups, including [C₃G]_S, [C₂G₂]_S, and [CG₃]_S, are dominant, with the exception of three non-sandwich layups: [C₆G₂], [C₃G₅], and [CG₇]. Full carbon and full glass layups are also optimal candidates.

When the target failure load is 300 N, the solutions are given as dots, as shown in Figure 13. The lower bound, represented by the red line, forms the Pareto front, and the optimal candidates are given in

Table 6. Optimal candidates for a failure load of 300 N.

Layup	Failure Load (N)	Vfc	Vfg	Density (g/cm3)	Cost ($/L)
[C]8	306.84	0.3	-	1.303	63.70
[C7G]	301.46	0.34	0.3	1.357	62.81
[C3G]S	314.50	0.3	0.3	1.362	53.17
[C3G]S	314.40	0.3	0.36	1.384	52.94
[C3G]S	314.42	0.3	0.42	1.406	52.71
[C2G2]S	305.27	0.35	0.3	1.438	45.77
[CG3]S	300.37	0.58	0.43	1.673	39.36
[CG3]S	300.95	0.54	0.47	1.711	37.65
[CG3]S	302.62	0.51	0.49	1.728	36.48
[C3G5]	300.56	0.47	0.51	1.743	35.00
[CG3]S	301.54	0.44	0.53	1.760	33.83
[CG3]S	302.11	0.4	0.56	1.787	32.23
[CG3]S	302.52	0.36	0.58	1.802	30.75
[CG3]S	301.77	0.34	0.64	1.866	29.43

. Similar to failure load 250 N, it can be seen that sandwich layups, including [C₃G]_S, [C₂G₂]_S, and [CG₃]_S, are dominant. In addition, two non-sandwich layups, [C₇G] and [C₃G₅], and a full carbon layup are optimal candidates. A full glass layup cannot achieve a failure load of 300 N.

4. Conclusions

This study has optimised the design of carbon/glass fibre-reinforced hybrid composites under combined bending and torsional loading using finite element analysis (FEA) and response surface methodology. The main conclusions are as follows:

• Optimal ply orientation: The best-performing laminates have ply angles ranging from 12° to 30°, depending on the layup configuration.
• Superior layup types: Sandwich-type layups such as [C₃G]_S, [C₂G₂]_S, and [CG₃]_S show the highest strength-to-weight ratios and cost-effectiveness compared to full-carbon or full-glass laminates.
• Fibre volume fraction effects: The failure load is highly sensitive to both carbon (V_fc) and glass (V_fg) fibre volume fractions. The optimum region is V_fc = 0.3–0.65 and V_fg < 0.5625 to prevent strength reduction.
• Strength performance: The best-performing hybrid layups achieved failure loads exceeding 300 N while maintaining relatively low density and cost.
• Failure mechanisms: Failure tends to initiate on the compressive side due to microbuckling/kinking, with sandwich configurations mitigating these effects through load redistribution.
• Design implications: The proposed prediction formulas and contour maps provide practical guidelines for selecting ply orientation and fibre ratios during preliminary design.
• Future work: Experimental validation, inclusion of interlaminar debonding effects, and extension to other loading modes (impact and fatigue) are recommended to further enhance the robustness of hybrid composite designs.