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. 2024 Jan 17;2024:8867496. doi: 10.1155/2024/8867496

An Optimum Design of a Subsonic Aircraft Wing due to the Aerodynamic Loading

Ibtisam J Ismeal 1, Mehmet Bakirci 1, Muhsin J Jweeg 2,✉
PMCID: PMC10807939  PMID: 38268746

Abstract

Aircraft designers are mainly interested in finding the level of pressure, stresses, and deformations of the parts of the aircraft wing. In many aviation accidents, the failure of the wing is the main cause of disasters, as it is considered the main surface that generates the necessary lift for the aircraft in addition to its other functions in controlling the transverse stability. In this work, a numerical study was performed to obtain the optimum wing structural design parameters for high strength and minimum weight for the L-39 A/C wing. The wing was modeled as a honeycomb with different thicknesses using the software SOLIDWORKS 2020. The pressure distribution was predicted using the FLUENT 2022 R1 package. Having obtained the aerodynamic pressure, the deformations and stresses were obtained using the ANSYS program. The results were compared with other researchers using other models, such as using ribs and stringers in the interior structure of the wing. The current results were found to be reliable and acceptable from the design point of view of the high stiffness-to-weight ratio.

1. Introduction

The design of the internal structure of the aircraft wing mainly depends on the use of ribs and stringers, and the wing structure is divided into many cells which results in a low stiffness-to-weight ratio. The new aeroplane design trends are focused on the use of lightweight sections which are capable of supporting the payload and aerodynamic loading, such as the honeycomb structures. The authors of [1–3] presented an experimental and computational study of the bending behaviors regarding honeycomb sandwich panels with different shapes of the cores (in other words, circular, hexagonal, and square) and 2 types of facings: one is aluminum and the other is composite. In comparison with the other core shapes, the square honeycomb core had the highest load, which increased due to the increase in facing thickness, and aluminum skin facing had a larger value of the load when compared to composite skin facing.

Crushing behavior related to a honeycomb sandwich structure was investigated via experiments. In addition, a numerical model for capturing some specific deformation and failure features in the process of crushing was developed using experimental data to ensure its validity. A three-point bending test was performed on an Al honeycomb sandwich panel [4]. The strength of the sandwich structures under bending loads with a variety of face materials was studied theoretically. Titanium and aluminum may be used as face materials. It has been discovered that titanium alloy has superior sandwich construction qualities [5–7].

The bending stress of a glass fiber-reinforced plastic (GFRP) sandwich construction was investigated for a lightweight vehicle. There have been 3 different adhesives used to adhere to the face and core. The research demonstrated that a lightweight chassis vehicle's honeycomb sandwich panel design with three adhesives might withstand considerable bending stress [8–10]. The researchers provided an experimental investigation on honeycomb sandwich panel compression properties in relation to different design parameters such as cell size, foil thickness, and size of the sample (i.e., width, length, and height dimensions) regarding the honeycomb structure. Sandwich samples were constructed by bonding Al honeycomb cores and 1 mm thick CFRP laminate faces, and compression tests were performed on them. It could be seen that when the foil thickness and core height decrease, the yield stress increases [11, 12].

The impact of the honeycomb thickness on the vibration response of sandwich panels was studied using experiments with various boundary conditions. Free vibration analysis was performed on different support conditions. It was revealed that the impact of core height on the basic natural frequency of the honeycomb sandwich panels is considerable. As the height of the core rises, so does frequency [13–15].

Finite and experimental element analytic methods have been used to look at the behavior of aluminum honeycomb structures under the low-speed impacts. The ASTM D7766 standard was utilized for conducting low-velocity impact tests on the honeycomb structures that were created. The impact force was investigated as a function of cell width and height. The maximal impact force values in the honeycomb composite constructions have been found to grow as cell width and height decrease. The experimental and finite element approach results are roughly 85 percent in agreement [16–18].

The goal of the current work is to design and analyze a lightweight L-39 aircraft wing using a honeycomb structure which can resist aerodynamic loading. A comparison study will be performed by using wing design with ribs and stringers.

2. Honeycomb Structures

With regard to sandwich structures, honeycomb cores are available in a range of materials, including paper and card for applications needing low strength and stiffness and low loads (like interior doors for homes) and high strength and stiffness, incredibly lightweight sections for aviation structures, are available for honeycomb cores used in sandwich structures. Honeycombs can be formed into composite structures which are both flat and curved without needing a lot of mechanical force or heat [19–21]. A honeycomb's typical shape is shown in Figure 1. The cells could be hexagonal, triangular, or square. Examples of honeycomb include glass fiber-reinforced plastic, aluminum, and honeycomb made of kraft paper. Aluminum and carbon honeycomb cores were used in this experiment. Figure 1 shows a simplified honeycomb.

Figure 1.

Figure 1

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Schematic of a typical sandwich structure.

Honeycomb characteristics are anisotropic, meaning that they differ between out-plane and in-plane strengths and stiffnesses. The walls of the cells first bend, and deformation is linear elastic in a case where a honeycomb is squeezed in plane, that is, when stress acts orthogonal to the cell axis; plane X1 X2 is depicted in Figure 2.

Figure 2.

Figure 2

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Definitions of parameters for a honeycomb cell [11]. (a) One honeycomb cell. (b) Multihoneycomb cell.

Composite sandwich structures, on the other hand, represent a modeling problem due to the fact that their core region is made up of several cells with complex geometries.

The isotropic beam analysis method in [22–25] is the most widely used model of unit cells for determining effective characteristic determination. The author makes an assumption that the honeycomb deformations' linear elastic response and the consequent core characteristics are solely dependent on the bending of the core cell walls. The stretching and shearing of cell walls were studied as additional deformation modes [26–28]. FEM has been used in numerous previous publications for estimating effective material characteristics regarding honeycomb architectures. The core of multifunctional sandwich composites could then be sized using analytical equations in the design process to account for anticipated energy needs as well as desired service loads. Figure 3 provides an illustration of the homogenization approach used in this study.

Figure 3.

Figure 3

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Overview of the homogenization methodology sought in the current investigation.

In Figure 3, the superscript in E11, E12, and E13 refers to the layer number and the subscript refers to the principal direction. Eff is the effective modulus, and Ecore is the core modulus.

2.1. Material Properties of the Sandwich Panels

The material properties used in this analysis for carbon fibers and Al 7075-T6 materials are listed in Table 1. A 3D deformable shell geometry is used to model the skins and the honeycomb core. The material properties for aluminum (Al 7075-T6) are as follows: elastic modulus = 72 000 N/mm2, Poisson's ratio = 0.3, shear modulus = 26900 N/mm2, tensile strength = 570 N/mm2, yield strength = 505 N/mm2, mass density = 2810 kg/m3, thermal expansion coefficient = 2.36E.05 1/K, thermal conductivity = 130 W/(m·K), and specific heat = 960 J/(kg·K).

Table 1.

Work plan.

Displacements and von Mises stresses
  1 cell 6 cells 9 cells
Mach no. M = 0.4 M = 0.4 M = 0.4
Skin thickness (mm) 2 3 4 2 3 4 2 3 4
Core thickness (mm) 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6
Mach no. M = 0.6 M = 0.6 M = 0.6
Skin thickness (mm) 2     3     4     2     3     4     2     3     4    
Core thickness (mm) 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6
Mach no. M = 0.8 M = 0.8 M = 0.8
Skin thickness (mm) 2 3 4 2 3 4 2 3 4
Core thickness (mm) 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6
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Material properties of carbon fiber laminates are E = 140 GPa, density = 1760 kg/m3, and Poisson's ratio = 0.22.

The work methodology is shown as block diagram in Figure 4, which can be summarized as follows: (1) Three internal structure configurations were suggested: 1 cell, 6 cells, and 9 cells. (2) For each case in (1), three different air speeds were chosen 0.4, 0.6, and 0.8 Mach numbers. (3) The aerodynamic pressure distribution on the wing for each case in (2) is obtained using the FLUENT program. (4) For each case in (1), the skin thicknesses of 2, 3, and 4 mm are used. (5) For each skin thickness in (4), three core thicknesses of 2, 4, and 6 mm are used. (6) The displacement and stresses are obtained for each tested case in the block diagram using the ANSYS package.

Figure 4.

Figure 4

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Contour of the total deformation at one cell skin 2 core 2 (Mach 0.4).

3. Detailed Case Study

The L-39 A/W wing weight is 6500 kg, with the profile NACA 64A 012. The span is 9.12 m, the gross area is 18.8 m2, the quarter chord line sweepback angle is 1°45, the leading edge sweepback angle is 6°26, the taper ratio is 0.475, the aspect ratio is 4.4, the mean aerodynamic chord is 2.15 m, the tip chord is 1.33 m, the root chord is 2.8 m, and the geometric shape is trapezoidal.

The calculations are based on the angle of attack 12° because of the maximum pressure around the A/C wing model as shown in Table 1. First, a 12° angle of attack is adopted because at this attitude, the maximum deformations and stresses are developed in the wing structure. The pressure developed at this position is found using the FLUENT package which is exported directly to the ANSYS package (fluid-structure interaction) which is considered an exact modeling of the wing structure under aerodynamic modeling. This was used throughout the work presented here.

3.1. Wing One Cell

The determination of displacements and stress using different structural modeling methods is presented here. Finite element modeling was employed as explained above to predict the pressure distribution to be applied on the wing structure, which corresponds to the real case to give the true picture of deformation and stresses on the wing skin.

The effects of the design parameters of the wing structure are discussed as follows:

  1. Effects of skin thickness

  2. Effects of the number of cells

  3. Effects of core thickness (invariably of skin thickness)

3.1.1. Effects of Skin Thickness

Using a skin thickness of 2 mm and a core thickness of 2 mm, the maximum deformation of 154.28 mm and the resulted von Mises of 210.17 MPa were obtained, as shown in Figures 4 and 5.

Figure 5.

Figure 5

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Contour of the equivalent stress at one cell skin 2 core 2 (Mach 0.4).

Increasing a skin thickness to 3 mm, the deformation becomes 106.41 mm and the resulted von Mises becomes 143.64 MPa, as shown in Figures 6 and 7. Table 2 shows the results of using different thicknesses (invariably of core). Note that in Table 2, some of the points in bold failed because the value of the von Mises stress exceeded a yield strength of 505 N/(mm)2. The least mass was 200.21 kg and the largest mass was 421.71 kg using material Al 7075-T6. Table 2 shows that the change in the total von Mises stress was largest (793.92 MPa) at a skin thickness of 2 mm, a core thickness of 4 mm, and Mach 0.8 (failed), while the lowest stress (108.41 MPa) was obtained at a skin thickness of 4 mm, a core thickness of 6 mm, and Mach 0.4. It was shown that the largest equivalent strain of 0.0075414 mm/mm was obtained in the model with a skin thickness of 3 mm and a core thickness of 2 mm at Mach 0.8, while the least deviation of 0.0010246 mm/mm was obtained in the model with a skin thickness of 2 mm and a core thickness of 6 mm at Mach 0.8.

Figure 6.

Figure 6

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Contour of the total deformation at one cell skin 3 core 2 (Mach 0.4).

Figure 7.

Figure 7

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Contour of the equivalent stress at one cell skin 3 core 2 (Mach 0.4).

Table 2.

Wing one cell at angle of attack 12° (2 cells).

Mach no. Skin thickness (mm) Core thickness (mm) Max deformation (mm) Equivalent stress (MPa) Equivalent strain (mm/mm) Mass (kg)
0.4 2 2 154.28 210.17 0.003177 200.21
3 2 106.41 143.64 20404 258.8
4 2 81.681 110.74 0.001557 317.96
2 4 146.39 217.08 0.003156 253.46
3 4 101.88 140.54 0.002052 311.5
4 4 78.785 108.41 0.001577 369.96
2 6 138.16 195.19 0.002785 306.56
3 6 98.262 137.65 0.001964 363.82
4 6 76.48 104.37 0.001484 421.71
0.6 2 2 338.84 473.87 0.007152 200.1
3 2 233.76 324.6 0.004612 258.8
4 2 179.45 253.16 0.003528 317.96
2 4 321.44 487.8 0.007093 253.46
3 4 223.74 316 0.004617 311.5
4 4 173.04 244.05 0.003549 369.96
2 6 308.4 515.52 0.007238 306.56
3 6 215.75 309.67 0.004423 363.82
4 6 167.95 235.52 0.003338 421.71
0.8 2 2 547.06 774.53 0.011689 200.1
3 2 377.45 530.72 0.007541 258.8
4 2 289.77 413.8 0.005768 317.96
2 4 518.87 793.92 0.011543 253.46
3 4 361.19 514.4 0.007548 311.5
4 4 279.36 398.98 0.005801 369.96
2 6 489.42 718.16 0.010246 306.56
3 6 348.26 503.78 0.007197 363.82
4 6 271.12 358.11 0.005433 421.71
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3.1.2. Effects of Core Thickness

Using a skin thickness of 2 mm and a core thickness of 2 mm, the maximum deformation of 338.84 mm and the resulted von Mises of 473.87 MPa were obtained, as shown in Table 3.

Table 3.

Wing one cell at angle of attack 12° (6 cells).

Mach no. Skin thickness (mm) Core thickness (mm) Max deformation (mm) von Mises stress (MPa) Equivalent strain (mm/mm) Mass (kg)
0.4 2 2 154.28 210.17 0.003177 200.2
2 4 146.39 217.08 0.003156 253.5
2 6 138.16 195.19 0.002785 306.6
3 2 106.41 143.64 0.00204 258.8
3 4 101.88 140.54 0.002052 311.5
3 6 98.262 137.65 0.001964 363.8
4 2 81.681 110.74 0.001557 318
4 4 78.785 108.41 0.001577 370
4 6 76.48 104.37 0.001484 421.7
0.6 2 2 338.84 473.87 0.00715 200.2
2 4 321.44 487.8 0.00709 253.5
2 6 308.4 515.52 0.007238 306.6
3 2 233.76 324.6 0.00461 258.8
3 4 223.74 316 0.00462 311.5
3 6 215.75 309.67 0.00442 363.8
4 2 179.45 253.16 0.00353 318
4 4 173.04 244.05 0.00355 370
4 6 167.95 235.52 0.00334 421.7
0.8 2 2 547.06 774.53 0.011689 200.21
2 4 518.87 793.92 0.011543 253.46
2 6 489.42 718.16 0.010246 306.56
3 2 377.45 530.72 0.007541 258.8
3 4 361.19 514.4 0.007548 311.5
3 6 348.26 503.78 0.007197 363.82
4 2 289.77 413.8 0.005768 317.96
4 4 279.36 398.98 0.005801 369.96
4 6 271.12 358.11 0.005433 421.71
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Note that in Table 3, some of the points in bold failed because the value of the von Mises stress exceeded a yield strength of 505 N/mm2. The least mass was 200.21 kg, and the largest mass was 421.71 kg. Table 3 shows that the change in the total von Mises stress was largest (793.92 MPa) at a skin thickness of 2 mm, a core of thickness 4 mm, and Mach 0.8 (failed), while the lowest stress (108.41 MPa) was obtained at a skin thickness of 4 mm, a core thickness of 6 mm, and Mach 0.4. Table 3 also shows that the largest equivalent strain of 0.0075414 mm/mm was obtained in the model with a skin thickness of 3 mm and a core thickness of 2 mm at Mach 0.8, while the least deviation of 0.0010246 mm/mm was obtained in the model with a skin thickness of 2 mm and a core thickness of 6 mm at Mach 0.8. Some cases failed due to the maximum stresses developed compared to the yield stress at the used material (Al 7075-T6) when a stress ratio is less than 1. They are labelled in bold. Even the masses are low compared to the mass used in the A/C design.

3.2. Wing 6 Cells

Finite element modeling was used as stated previously to predict the pressure distribution to be applied on the wing structure. This corresponds to the real case to give the true picture of deformation and stresses on the wing skin. The effects of the design parameters of the wing structure are discussed as follows:

  1. Effects of skin thickness.

  2. Effects of the number of cells

  3. Effects of core thickness (invariably of skin thickness)

3.2.1. Effects of Skin Thickness

Using a skin thickness of 2 mm and a core thickness of 2 mm, the maximum deformation of 149.83 mm and the resulted von Mises of 208.83 MPa were obtained, as shown in Figures 8–11. Increasing the skin thickness to 3 mm, the deformation becomes 104.24 mm and the resulted von Mises becomes 141.84 MPa, as shown in Figures 12 and 13.

Figure 8.

Figure 8

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Contour of the total deformation at 6 cell skin 2 core 2 (Mach 0.4).

Figure 9.

Figure 9

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Contour of the equivalent stress at 6 cell skin 2 core 2 (Mach 0.4).

Figure 10.

Figure 10

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Contour of the equivalent stress at 6 cell skin 3 core 2 (Mach 0.4).

Figure 11.

Figure 11

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Contour of the total deformation at 6 cell skin 3 core 2 (Mach 0.4).

Figure 12.

Figure 12

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Contour of the total deformation at 6 cell skin 2 core 2 (Mach 0.6).

Figure 13.

Figure 13

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Contour of the equivalent stress at 6 cell skin 2 core 2 (Mach 0.6).

Table 4 shows the results of using different thicknesses (invariably of core). Note that in the table, some of the points in bold failed because the value of the von Mises stress exceeded a yield strength of 505 N/(mm)2. The least mass was 210.46 kg and the largest mass was 451.26 kg using material Al 7075-T6. Table 5 shows that the change in the total von Mises stress was largest (765.07 MPa) at a skin thickness of 2 mm, a core thickness of 2 mm, and Mach 0.8 (failed), while the lowest stress (100.77 MPa) was obtained at a skin thickness of 4 mm, a core thickness of 6 mm, and Mach 0.4. This table also shows that the largest equivalent strain of 0.0011286 mm/mm occurs in the model with a skin thickness of 4 mm and a core thickness of 6 mm at Mach 0.8, while the least deviation is 0.001417 mm/mm obtained in the model with a skin thickness of 2 mm and a core thickness of 6 mm at Mach 0.4.

Table 4.

Wing 6 cells at the angle of attack 12°.

Mach no. Skin thickness (mm) Core thickness (mm) Max deformation (mm) von Mises stress (MPa) Equivalent strain (mm/mm) Mass (kg)
0.4 2 2 149.83 208.83 0.00308 210.46
3 2 104.24 141.84 0.002064 268.93
4 2 80.478 113.23 0.001575 328.02
2 4 138.66 203.51 0.002887 273.77
3 4 98.149 136.93 0.001962 331.6
4 4 76.558 104.97 0.001509 389.87
2 6 129.37 196.4 0.002781 336.7
3 6 93.145 130.62 0.001839 393.69
4 6 73.405 100.77 0.001417 451.26
0.6 2 2 329.04 467.96 0.0069023 210.46
3 2 228.96 321.24 0.0046238 268.93
4 2 176.79 260.86 0.003631 328.02
2 4 304.44 456.06 0.0064775 273.77
3 4 215.53 307.49 0.0044095 331.6
4 4 168.13 237.86 0.003391 389.87
2 6 283.99 441.37 0.0065513 336.7
3 6 204.5 294.99 0.004155 393.69
4 6 161.18 227.4 0.003184 451.26
0.8 2 2 531.3 765.07 0.011286 210.46
3 2 369.72 526.43 0.007557 268.93
4 2 285.5 426.58 0.005938 328.02
2 4 491.52 746.52 0.010595 273.77
3 4 347.99 504.07 0.00723 331.6
4 4 271.47 389.03 0.005558 389.87
2 6 458.47 724.37 0.010264 336.7
3 6 33.07 480.62 0.006771 393.69
4 6 260.24 372.02 0.00522 451.26
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Table 5.

The results of using different thicknesses (invariably of core).

Mach no. Skin thickness (mm) Core thickness (mm) Max deformation (mm) Equivalent stress (MPa) Equivalent strain (mm/mm) Mass (kg)
0.4 2 2 149.83 208.83 0.00308 210.46
2 4 138.66 203.51 0.002887 273.77
2 6 129.37 196.4 0.002781 336.7
3 2 104.24 141.84 0.002064 268.93
3 4 98.149 136.93 0.001962 331.6
3 6 93.145 130.62 0.001839 393.69
4 2 80.478 113.23 0.001575 328.02
4 4 76.558 104.97 0.001509 389.87
4 6 73.405 100.77 0.001417 451.26
0.6 2 2 329.04 467.96 0.0069023 210.46
2 4 304.44 456.06 0.0064775 273.77
2 6 283.99 441.37 0.0065513 336.7
3 2 228.96 321.24 0.0046238 268.93
3 4 215.53 307.49 0.0044095 331.6
3 6 204.5 294.99 0.004155 393.69
4 2 176.79 260.86 0.003631 328.02
4 4 168.13 237.86 0.003391 389.87
4 6 161.18 227.4 0.003184 451.26
0.8 2 2 531.3 765.07 0.011286 210.46
2 4 491.52 746.52 0.010595 273.77
2 6 458.47 724.37 0.010264 336.7
3 2 369.72 526.43 0.007557 268.93
3 4 347.99 504.07 0.00723 331.6
3 6 330.17 480.62 0.006771 393.69
4 2 285.5 426.58 0.005938 328.02
4 4 271.47 389.03 0.005558 389.87
4 6 260.24 372.02 0.00522 451.26
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3.2.2. Effects of Core Thickness

Using a skin thickness of 2 mm and a core thickness of 2 mm, the maximum deformation is 149.83 mm and the resulted von Mises is 208.83 MPa. Increasing a skin thickness of 3 mm, the deformation is 104.24 mm and the resulted von Mises is 141.84 MPa, as shown in Figures 12–15.

Figure 14.

Figure 14

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Contour of the equivalent stress at 6 cell skin 3 core 2 (Mach 0.6).

Figure 15.

Figure 15

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Contour of the total deformation at 6 cell skin 3 core 2 (Mach 0.6).

Table 5 shows the results of using different thicknesses (invariably of core). Note that in the table, some of the points in bold failed because the value of the von Mises stress exceeded yield strength = 505 N/(mm)2. The least mass was 210.46 kg and the largest mass was 451.26 kg using material Al 7075-T6. Table 5 also shows that the change in the total von Mises stress was largest (765.07 MPa) at a skin thickness of 2 mm, a core thickness of 2 mm, and Mach 0.8 (failed), while the lowest stress (100.77 MPa) was obtained at a skin thickness of 4 mm, a core thickness of 6 mm, and Mach 0.4. This table also shows that the largest equivalent strain of 0.0011286 mm/mm occurs in the model with a skin thickness of 4 mm and a core thickness of 6 mm at Mach 0.8, while the least deviation of 0.001417 mm/mm was obtained in the model with a skin thickness of 2 mm and a core thickness of 6 mm at Mach 0.4.

The summary of the results which are extracted from the contours is shown in Table 5. Some cases failed due to the maximum stresses developed compared to the yield stress at the used material (Al 7075-T6) when a stress ratio is less than 1. They are labelled in bold.

4. Comparison Study

The obtained results using the honeycomb structures are compared with the results obtained by using the ribs and stringers as shown in Table 6.

Table 6.

Comparison between the current work and that reported by Gargan [29].

Current work
Skin thickness (mm) Core thickness (mm) Max deformation (mm) von Mises stress (MPa) Stress ratio Mass (kg)
2 2 338.84 473.87 1.066 200.21
2 4 321.44 487.8 1.035 253.46
2 6 308.4 515.52 0.979 306.56
3 2 233.76 324.6 1.556 258.8
3 4 223.74 316 1.598 311.5
3 6 215.75 309.67 1.631 363.82
4 2 179.45 253.16 1.995 317.96
4 4 173.04 244.05 2.064 369.96
4 6 167.95 235.52 2.144 421.71
Mr. Gargan [29]
Skin thickness (mm) Displacement (m) von Mises stress (MPa) Mass (kg)
15 0.056 460 414
2 0.047 361 440
25 0.041 298 467
3 0.037 254 493
3.5 0.032 221 520
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Table 6 shows a comparison study between the suggested modeling and the work achieved by Gargan [29].

5. Conclusions

From this work, the following points can be concluded:

  1. The effectiveness of using honeycomb in reducing stress is greater than the effectiveness of using the ribs and stringers.

  2. Using 7075-T6 aluminum alloy for both the skin and core gives the least weight of the wing structure with a largest weight (421.71 kg) using one cell and a minimum weight using 6 cells (200.2 kg). The maximum wing mass was 451.26 kg and the minimum mass was 210.46 kg which are considered to be lightweight wings in any case compared to the wings with ribs and stringers with a minimum mass of 414 kg.

  3. The stiffness of the wings using the honeycomb structures is greater than that using the ribs and stringers, which results in less deformation, which in turn increases the flutter and the divergence speeds and gives a greater chance to avoid failure under the same aerodynamic modeling.

  4. Finally, the finite element modeling of a three-dimensional wing and using the idea of cells as a stiffening element were found to give good results and to determine the areas of most stress and acceptable deformation levels.

Acknowledgments

The research was supported by the authors.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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