Failure Initiation Analysis of a PRSEUS BWB Wing Subjected to Structural Damage

Abstract

In this study, a finite element approach was used to analyze the PRSEUS-based undamaged wing structure of a civil aircraft with a blended-wing-body configuration. The displacement, stress, and strain distribution of the PRSEUS wing structure were studied under an aerodynamic load with three different values of the factor of safety. This was used as a reference to study the response of the same wing configuration, first with a single stringer, where failure was initiated at the fourth loading value, while the second loading condition was sufficient to initiate failure in the triple-stringer damage wing. In addition, damage to rib components was investigated, and it was shown that damage to a single rib and double rib did not impose significant risks to the structural integrity of the wing structure, and the results have shown that the values of displacement, stress, and strain do not differ much from those of the undamaged wing, even as the length of the rib damage is increased.

1. Introduction

Multiple degrees of action must be taken to mitigate the effects of human activity on the climate. The aviation sector has already begun to make significant strides in the direction of more environmentally friendly aviation [1] and the deeper incorporation of composites [2,3]; some of these composites are eco-friendly due to their bio-based composition [4], but generally, total emissions, including operation and use, exhibit a much more environmentally friendly trend for composites as a result of emissions saved via decreased fuel consumption. The structural integrity of the airplane is significantly impacted by composites, and the economic viability of aviation is also significantly impacted [5,6]. By adopting lighter yet more durable materials, manufacturers may build larger, more passenger-capable planes that are also lighter and require less fuel to fly. The ability of composites to be layered with the fibers running in various directions is a useful characteristic. As a result, engineers can create structures with unconventional features.

The Blended-Wing-Body (BWB) concept displayed remarkable performance compared to conventional design [7,8], and different configurations of BWB design may opt for further optimized performances [9]. The BWB concept is an example of a future design that might displace the conventional airplane [10,11,12,13]. The idea of an amphibious aircraft having all of the characteristics of a flying wing is not new. The Tube-and-Wings (T & W) type of cylinder fuselage and flying wing aircraft are combined in the BWB plane. The main objective of this design is to house all components of the aircraft within a single supporting structure. The fuel is in the wings, and the cockpit and passengers are in the middle. Depending on the configurations, the landing gear and the luggage compartment are either in the middle or in the transition zone. In a BWB configuration, there is no longer a distinction between the wings and the fuselage, which should result in up to 30% less fuel consumption than an airplane with the same dimensions and technology [7]. It also offers additional advantages including lessened noise emissions and lighter takeoff weight. This is a desirable answer to the economic and environmental problems we are currently facing.

One of the most ambitious initiatives seeking to lessen the negative effects of aviation on the environment is the Environmentally Responsible Aviation (ERA) project [14,15,16]. The ERA project was launched by NASA to investigate potential aircraft layouts. The main premise of the project is to provide new, robust, and lightweight structures that consume less fuel and produce fewer pollutants. The main structural idea behind the ERA project’s development of the next-generation airframe technology is called PRSEUS [17,18]. The PRSEUS concept displayed great potential in meeting the unique loading requirements of the pressurized cabin of the BWB structure [19,20]; this is mainly achieved by exploiting the orthotropic nature of carbon fibers, leading to breakthrough damage arrest capabilities through the suppression of delamination between skin and flanges [21,22,23]. According to this concept, the majority of the loads that bend the wings are supported by the frame members, while the majority of the loads that bend the fuselage are supported by the stringers. The PRSEUS concept is made up of foam core, pultruded rods, and dry carbon epoxy components, as illustrated in Figure 1. After being put together into a stiffened panel, the resin is infused into it, and it is then set for out-of-autoclave curing. Due to the near complete elimination of fasteners, through-thickness stitching also creates a structure that is lightweight and has outstanding crack-arresting qualities [24,25,26,27]. The excellent mechanical qualities of the PRSEUS structure enable it to overcome some of the problems caused by the unusual BWB structure [23,28]. For instance, it shows decreased pressurization resistance in the structure.

In this study, a Finite Element (FE) approach was proposed to explore three main aspects. First, an FE method that employs CATIA, HyperMesh, and Abaqus [29] was used to conduct the preprocessing and phase of the numerical model to study the response of an undamaged BWB wing structure under aerodynamic load conditions, since damage evaluations for composite panels may call for expensive tools and time-consuming methods. In order to verify the presence of failure mechanisms in the undamaged BWB wing, pressurization loads were applied by applying different safety factors (1.5, 2, 2.5, and 3). Second, data from the undamaged wing analysis were used as a reference in analyzing the response of a BWB wing containing stringer damage, for purpose of assessing the risk imposed by cutting through single or triple stringer components. Third, the same steps were followed to examine the effect of damaging single or double rib components on the structural integrity of the BWB wing. The major objective of this paper is to demonstrate that the PRSEUS design is conducive to maintaining the structural integrity of a damaged BWB wing while simultaneously depending on a robust and simplified FEM.

2. Introduction of Finite Element Analysis and Hashin Failure Criteria for Composite Structures

Assessment of failure in composites may call for pricey equipment and labor-intensive procedures. The two most frequently used approaches to evaluate composite damage situations in engineering applications where analysis speed and efficiency are crucial are analytical models and commercially accessible finite element material models.

2.1. Finite Element Analysis

Engineers started processing increasingly complex structures, and a potent tool called the Finite Element Method (FEM) was developed in the aerospace industry [30,31]. The Finite Element Method (FEM) is a numerical technique used to address specific physics issues. It is a technique that enables the determination of an approximative solution over a spatial domain, i.e., the calculation of a field (of scalars, vectors, or tensors) that relates to specific equations and predetermined conditions. The aim of discretization, the first stage in the finite element approach, is to subdivide or separate the continuous complicated structure into elements. Consequently, the structure should be modeled using the proper finite elements. The elements’ quantity, kind, size, and layout are decided. Mesh refinement for solution convergence typically determines the accuracy of a finite element analysis because a finer mesh chooses to reduce the magnitude of discontinuities.

2.2. Hashin Failure Criteria

The majority of the failure criteria are phenomenological, indicating that they cannot be deduced through a micromechanical inspection. The primary goal of the failure criteria is to assess the mechanical resistance that the materials have generated, which corresponds to an irreversible degradation that is either obtained by actual material rupture or at the limit of the elastic domain (e.g., microrupture in the matrix, rupture of the fibers, fiber–matrix decohesion). The Hashin [32,33] criteria were initially designed for unidirectional composites, and later, woven composites were included in the above generalization. Four distinct failure modes are distinguished by Hashin, one of the interactive failure criteria (i.e., fiber tensile failure, matrix tensile failure, fiber compressive failure, and matrix compressive failure). The criteria are expressed as follows:

Fiber tension (σ₁₁ ≥ 0):

$$F_{ft}={\left(\frac{{\mathsf{\sigma }}_{11}}{X^T}\right)}^2+\alpha {\left(\frac{{\mathsf{\sigma }}_{12}}{S^L}\right)}^2$$

(1)

Fiber compression (σ₁₁ $<$ 0):

$$F_{fc}={\left(\frac{{\mathsf{\sigma }}_{11}}{X^C}\right)}^2$$

(2)

Matrix tension (σ₂₂ ≥ 0):

$$F_{mt}={\left(\frac{{\mathsf{\sigma }}_{22}}{Y^T}\right)}^2+{\left(\frac{{\mathsf{\sigma }}_{12}}{S^L}\right)}^2=1$$

(3)

Matrix compression (σ₂₂ $<$ 0):

$$F_{mc}={\left(\frac{{\mathsf{\sigma }}_{22}}{2S^T}\right)}^2+\left[{\left(\frac{Y^C}{2S^T}\right)}^2-1\right]\frac{{\mathsf{\sigma }}_{22}}{Y^C}+{\left(\frac{{\mathsf{\sigma }}_{12}}{S^L}\right)}^2=1$$

(4)

where σ_ij represents stress components; $X^T$ and $X^C$ denote tensile and compressive strengths in the fiber direction, respectively; $Y^T$ and $Y^C$ express tensile and compressive strengths in the matrix direction, respectively; longitudinal and transverse shear strengths are expressed by $S^L$ and $S^T$, respectively. In addition, $\alpha$ determines the shear stress contribution for the fiber damage initiation criterion in tension.

In general, fibers carry the axial load and are substantially stiffer than the matrix. Therefore, the fiber tensile failure mode is much more important because it is followed by a strong energy release and may potentially lead to the failure of nearby fibers and matrix. When the loading carried by the fibers exceeds a critical value, fiber breakage occurs suddenly. Overall, fiber failure frequently causes the entire structure to fail, whereas the effect of delamination is not taken into account. The adopted failure criteria influence the accuracy of the analysis [34], and while there are many commonly used criteria, one can still modify and construct a failure criterion best on stress components [35,36,37]. Abaqus/CAE provides direct input for Hashin failure criteria for fiber-reinforced composites. The Hashin-based criterion incorporates a progressive damage model in which the stiffness (constitutive matrix) of a damaged location is reduced by considering previously calculated failure mode indexes. In addition, the progressive damage model takes into account the fracture energy by employing stress–displacement curves, but this is outside the scope of this paper. It is noteworthy that the stress components must be checked for reasons of evaluating the strength of the composite structural components [38].

3. Numerical Modeling and Meshing of the BWB Wing Structure

An airplane needs to be able to lift its own weight as well as that of the pilot, passengers, cargo, and fuel in order to be able to fly. Wings produce the most lift, which is what makes an airplane fly. The wing is one of an aircraft’s most important technological components. Its main responsibility is to increase the weight of the aircraft in the air in accordance with the speed and pitch of the aircraft. It also acts as a fuel tank, supports the engines, and controls the ailerons of the aircraft. The top skin, lower skin, and leading edge skin are the three sections that make up the skins of the wing, which enclose the structure of the wing. Stiffeners are placed on these skins to provide rigidity and prevent buckling. Spars are the primary structural elements of a wing. They normally extend from the fuselage toward the tip of the wing parallel to the lateral axis of the aircraft and are connected to the fuselage by wing fittings, plain beams, or a truss. The ribs primarily retain the shape of the wing and provide resistance to the wing in torsion, and they prevent the skins and stiffeners from buckling.

3.1. Geometrical Modeling of the BWB Wing

Geometries acquired from CATIA were used in this study to model the wing assembly, which is illustrated in Figure 2. The surfaces of each component need to be identified. The greatest distance, which is measured at the center of the root section, is 1823.11 mm. The distance between the top skin component and the lower skin component is 1614.54 mm at the rear section of the root of the wing box and 1572.33 mm at the front section.

The distance between the upper and lower skin components at the wing’s tip, on the other hand, is shortest at the front part, measuring 190.81 mm, and greatest at the rear section, measuring 198.58 mm., while the distance at the center is only as far as 227.32 mm. The wing box is 7318.18 mm wide at the base and 1241.78 mm wide at the tip. To maintain the profile of the wing, a total of 39 ribs were used. The distance between the ribs ranges from a minimum of 524.95 mm to a maximum of 752.88 mm between the first two ribs at the root. The rib spacing in the majority of the construction ranges from 665.75 mm to 710.23 mm. The pultruded rods were represented by a total of 52 line components in the model. The stiffener rods are separated on average by 235.6 mm. Additionally, it should be noted that the average distance between the rods and skin-related components is 49.9 mm. In addition to the design requirements linked to the various materials discussed in the previous section, the maximum vertical displacement of the wing must also be considered, as large displacements can result in significant aerodynamic losses and vibration issues. Hence, the maximum displacement of the wing is determined as 10% of the wing’s maximum length; this case, the maximum wing length is approximately 24 × 10³ mm.

3.2. Finite Element Model of BWB Wing

HyperMesh and Abaqus both offer the import and export of CAD data in the leading industry formats. HyperMesh also has strong tools for cleaning up imported geometry, making it possible to make meshes of the highest quality quickly. In general, the accuracy of a finite element analysis is dictated by mesh refinement for solution convergence, as a finer mesh reduces the amplitude of discontinuities. Fast mesh production of outstanding quality is one of the primary advantages of HyperMesh. As a result, the leading and trailing edge components were meshes in HyperMesh, whereas the wing box components were meshes in Abaqus. This is due to the need for additional geometrical modification, more specifically, modeling cut damages, and the fact that it is easier to edit the mesh after the geometry change in Abaqus. Wing meshing was performed using line beams, B31 elements were employed to discretize the pultruded rods, while general-purpose mesh elements S4R and S3 were used for the composite parts of the wing. B31 is a 2-node beam element with linear interpolation formulations in three-dimensional space. This element allows for transverse shear deformation. S4R and S3 are general-purpose elements that provide reliable and precise solutions for thin and thick shell problems under all loading circumstances. The small-strain shell elements use a Mindlin–Reissner type of flexural theory that includes transverse shear [29]. The modeling of the undamaged wing used a total of 241,911 elements, as illustrated in Figure 3.

3.3. Material Properties

This concept comprises foam core, pultruded rods, and dry carbon epoxy components; the lamina has seven plies stacked in the sequence [45, −45, 0, 90, 0, −45, 45]_T, and it is 1.32 mm thick. The fiber proportions are 44.9, 44.9, and 12.2 at 0°, 45°, and 90°, respectively. The material properties of the components are presented in

Table 1. Material properties of the composite components of the wing structure.

Engineering Constants	Young’s Moduli (MPa)	E1 = 67,154.93 E2 = 33,542.99 E3 = 34,956.42
	Poisson’s Ratio	v12 = v13 = 0.4 v23 = 0.095
	Shear Moduli (MPa)	G12 = G13 = 16,340.57 G23 = 5515.81
	Density t/mm3	ρ = 1.60 × −9
Strength Properties	Fiber Tensile and Compressive Strength (MPa)	XT = 724.64 XC = 546.06
	Matrix Tensile and Compressive Strength (MPa)	YT = 320.61 YC = 264.31
	Longitudinal and Transverse Shear Strength (MPa)	SL = ST = 206.15

and

Table 2. Material properties of noncomposite components of the wing structure.

Material	E, MPa	ν	ρ, t/mm3
Foam Core Rohacell 110WF	144.79	0.32	9.99 × −11
Rod Toray T800/3900-2B	126,932.48	0.3	1.60 × −9

.

3.4. Boundary Conditions and Loading

An airplane experiences air loads as well as the gravitational attraction of the aircraft’s weight while it is in flight. The pressure that is evenly distributed throughout the surface of the aircraft creates the aerodynamic loads. These weights fluctuate in size and location depending on the style of flight, such as cruising, maneuvers, and wind gusts. The amount and distribution of lift over the wingspan are determined by the airfoil’s shape and the wing’s angle of attack. In Abaqus, the wing end that is connected to the aircraft’s body is subject to the boundary condition of zero degrees of freedom. This is because the wing is firmly fastened to the airplane, whether by welding, riveting, or another technique. At the ends of each stringer and along each line that makes up the wing’s root edge, the model has zero degrees of freedom. The static analysis of the civil aircraft design manual’s instructions was adhered to in this investigation. Additionally, the aerodynamic forces applied to the wing were calculated using Computational Fluid Dynamics (CFD) analysis. The load distribution on the upper and lower skin components of the wing in cruising conditions was computed by CFD, and it was found to equal 0.00785 MPa on the upper skin component and 0.00057 MPa on the lower skin components when the Factor of Safety (F.o.S.) equals 1. In this study, the F.o.S. was first set to equal the standard value of 1.5, and it was increased to 2, 2.5, and lastly 3,

Table 3. Values of pressure load at different values of factor of safety.

Factor of Safety	Pressure Load on the Upper Skin Component (MPa)	Pressure Load on the Lower Skin Component (MPa)
1.5	0.01125	0.00085
2	0.01516	0.00114
2.5	0.01895	0.001425
3	0.02277	0.00171

demonstrates the equivalent pressure load at these factors. Limit loads (the maximum loads to be anticipated in service) and ultimate loads are used to specify strength requirements (limit loads multiplied by prescribed factors of safety). The structure must be capable of supporting limit loads without deforming permanently. Under any load up to and including the limit load, the deformation may not impede safe operation. Static tests conducted to ultimate load must account for the ultimate deflections and deformations generated by the stress, as implied by the code of federal regulations (14 CFR 25.305(b)). In addition, a surface traction load that has a magnitude of 0.0043 MPa is distributed over roughly two-thirds of the lower skin component. Figure 4 is an illustration of the pressure load and root boundary conditions of the wing.

4. Static Analysis of an Undamaged BWB Wing

4.1. Static Analysis Approach for the Undamaged BWB Wing Structure

The preprocessing step of the analysis was established using three different programs (i.e., CATIA V5, HyperMesh, and Abaqus), and the general schematic is presented in Figure 5. Using this module, materials should be created and assigned to each part. The properties stated in Table 1 and Table 2 were the guidelines by which the materials were initially designed. If the material of the rods is a homogeneous solid, the piece of the part should then be identified using the proper tool. A generic static step was created and given a 5 s time limit. Since linear static analysis is not time-dependent, this time period can be characterized as random. Additionally, the field output requests were established.

4.2. Results and Failure Onset Assessment of Undamaged BWB Wing

After the static linear analysis was performed, the displacement results were inspected in the visualization module.

Table 4. Results of out-of-plane displacement, stress, strain, and fiber tensile failure criterion of undamaged BWB wing at different values of F.o.S.

Factor of Safety	1.5	3
Out-of-Plane Displacement (mm)
Von Mises Stress (MPa)
Maximum In-Plane Principal Strain
Hashin Fiber Tensile Failure Criterion

demonstrates that the out-of-plane displacement is 2480 mm at the standard factor of safety 1.5, 3474 mm at the factor of safety 2, 4436 mm at the factor of safety 2.5, and 5407 mm when the factor of safety is 3. Along the wing, the vertical displacement was distributed uniformly, and the displacement decreased as it approached the root of the wing, and a significant area of the wing showed essentially no vertical displacement because the root was restricted in movement along all axes and rotated around the axis. The lower part of the wing showed the most von Mises stress, specifically outside the wing box at the leading edge component. This region of the wing where the front edge and the front spar come in contact bears large shear stresses. The distribution of von Mises stress is substantially higher at the component that makes up the trailing edge; that region of the assembly has a geometrical discontinuity, as the trailing edge sweep angle sharpens. Moreover, the value of the largest stress does not respond linearly to the increase in the pressure load. The largest in-plane principal strain is measured at an intersection between two rib components closest to the upper section of the root. The distribution of the lower wing section reveals that the smallest values of in-plane principal strain coincide with the region of the lower wing where the longitudinal strain is compressive. Nonetheless, since the lower section is subjected to tensile stresses, a larger portion of it displays tensile longitudinal strains, while compressive longitudinal strains are spread over a larger section of the upper side of the wing. Finally, it can be determined from the values of the fiber tensile failure criterion that the pressure loads utilized in this study are not sufficient to invoke the failure of the structure. The fiber tensile failure criterion is indicative of a structural failure in this study, and since the value has not reached 1, the failure is not initiated.

5. Static Analysis of BWB Wing Containing Damage Cutting through Stringers

When under stress, the stringers that are attached to the boundary where the wing skin is affixed offer support and keep the skin from buckling. The stiffeners also support axial loads generated by bending moments in the wing. Between the spars, there is space for stringers. This will make the skin more resistant to buckling when sheared. It is therefore of great interest to examine the effects of cutting through the stringers of the wing, in addition to the effect of the damage length on the structural integrity.

5.1. Modeling of Stringer Damage in Damaged BWB Wing

To create a cut, it was necessary to take into consideration the effects of the notch’s size and orientation. As a result, four damaged wing models were constructed, two of which had the damage cut oriented perpendicular to the direction of the stringers. One of these models had damage that went through one stringer component (Figure 6), and the other had damage that went through three stringers (Figure 7).

5.2. Results and Failure Onset Assessment of BWB Wing Containing Single-Stringer and Triple-Stringer Damage

After the static linear analysis was finished, the deformation data were looked at in the visualization module. As can be seen from

Table 5. Results of out-of-plane displacement of single-stringer and triple-stringer damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Stringer Damage	Triple-Stringer Damage
1.5
3

, the pressure load applied to the upper and lower skin components of the wing box caused the largest vertical displacement at the tip of the wing, measuring 2496, 3496, 4464, and 5440 mm when the factor of safety equals 1.5 2, 2.5, and 3, respectively, in the wing assembly with a single damaged stringer, while these measurements were 2496, 3496, 4464, and 5441 mm in the triple-stringer damage wing; these values appear to be equivalent. The vertical displacement is uniformly distributed along the length of the wing and decreases as it approaches the wing’s root. As the sections of the wing closer to the root are inspected, it is observed that the rate at which the displacement reduces slows and that a sizable portion of the wing exhibits virtually little vertical displacement because the root was constrained in its movement along all axes. It can be seen that the displacement significantly exceeds the value corresponding to 10% of the wingspan as the factor of safety is increased. When F.o.S. is 1.5, the maximum vertical displacement negligibly exceeds the imposed displacement restrictions.

The results of a static analysis performed on the wing components point to a maximum stress of 930.5, 1339, 1734, and 2133 MPa in the model with the smaller cut at the factor of safety equal to 1.5, 2, 2.5, and 3, respectively. In all the models containing single-stringer damage, the maximum stress is located outside the wing box. Specifically, the maximum stress is found in the lower section of the leading edge component, at a location that is fairly close to the root section. In parts near the contact with the wing box assembly, the front edge component and the trailing edge component both experience a noticeable increase in stress intensity, according to a preliminary analysis of the stress distribution in the assembly. The relatively acute geometrical change in the trailing edge component and the modeled cut damage are reasons for the relatively high stresses in that region. In contrast, the maximum von Mises stress in the model containing triple-stringer damage is 934.6 MPa at the standard factor of safety 1.5, 1345 MPa when the factor of safety is 2, 1741 MPa when the factor of safety equals 2.5, and 2142 MPa when the factor is 3. The distribution slightly differs in the vicinity of the damage when it is larger. It can be seen that larger stresses concentrate at the tips while smaller stress extends to larger regions adjacent and parallel to the cut; these locations are denoted by A and A′ in Figure 8. And the results of von Mises stress of single-stringer and triple-stringer damage BWB wings at different values of F.o.S. are shown in

Table 6. Results of von Mises stress of single-stringer and triple-stringer damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Stringer Damage	Triple-Stringer Damage
1.5
3

.

Results of the maximum in-plane principal strain of single-stringer and triple-stringer damage BWB wings at different values of F.o.S. are presented in

Table 7. Results of the maximum in-plane principal strain of single-stringer and triple-stringer damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Stringer Damage	Triple-Stringer Damage
1.5
3

. The maximum in-plane strain in the wing assembly was found to occur at the point of contact with the rib component at the top section of the trailing edge, with a high value of 0.00809 at the standard factor of safety 1.5, 0.01156 at the factor of safety 2, 0.01492 at the factor of safety 2.5, and 0.01830 at the factor of safety 3 in the model containing the smaller cut. These values respectively correspond to 0.00820, 0.01159, 0.01496, and 0.01835 in the triple-stringer damage model. The existence of the stringers and ribs had a substantial impact on the strain distribution, which disrupts the smooth transition due to discontinuities. Examining the lower skin panels suggests that when the cut is larger, the intense strain which is concentrated around the tips spreads onto a larger region, and the smaller strains are spread on a larger region over the lower skin panel.

Results of Hashin fiber tensile failure criterion of single-stringer and triple-stringer damage BWB wings at different values of F.o.S. are shown in

Table 8. Results of Hashin fiber tensile failure criterion of single-stringer and triple-stringer damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Stringer Damage	Triple-Stringer Damage
1.5
3

. The fiber tensile failure criterion equals 0.299 for the single-stringer damage model and 0.554 for the triple-stringer damage model at the standard factor of safety 1.5. When the factor of safety is 2, the fiber tensile failure criterion equals 0.575 in the model with single-stringer damage and 1.066 in the triple-stringer damage model. This criterion equals 0.927 and 1.719 in the single-stringer and triple-stringer damage models, respectively, at the factor of safety 2.5. Finally, when the factor of safety is 3, this criterion equals 1.368 in the single-stringer damage model and 2.534 in the triple-stringer model. The maximum value was measured at the tips of the cut in all the models. It is noticed that the large values of fiber tensile failure criterion are distributed over a relatively wider section in the single-stringer damage models, containing front sections of the lower wing section and also spreading towards the root of the wing box. On the other hand, when the cut is larger, the large values are principally constrained in the regions closest to the tips of the cut.

5.3. Comparison of Results between Undamaged BWB Wing and Wing Containing Stringer Damage

Generally, the distribution of the out-of-plane displacement in the damaged models almost matches that of the pristine wing models.

Table 9. Comparison of out-of-plane displacement in the undamaged wing and wing containing stringer damage.

	Maximum Out-of-Plane Displacement (mm)
Factor of Safety	Undamaged Wing	Single-Stringer Damage	Ratio of the Values	Triple-Stringer Damage	Ratio of the Values
1.5	2480	2496	1.006	2496	1.006
2	3474	3496	1.006	3496	1.006
2.5	4436	4464	1.006	4464	1.006
3	5407	5440	1.006	5441	1.006

suggests that the ratio is constant at a value that equals 1.006, and the size of the cut does not have a noticeable effect on the change in the out-of-plane displacement, as the only difference is displayed when the factor of safety is 3, and the difference is small that it can be considered negligible.

On the other hand,

Table 10. Comparison of von Mises stress in the undamaged wing and wing containing stringer damage.

		Von Mises Stress (MPa)
Factor of Safety	Component	Undamaged Wing	Single-Stringer Damage	Ratio of the Values	Triple-Stringer Damage	Ratio of the Values
1.5	Wing Assembly	873.3	930.5	1.065	934.6	1.070
	Lower Skin Component	311.0	393.0	1.253	501.2	1.611
2	Wing Assembly	1256	1339	1.066	1345	1.070
	Lower Skin Component	443.8	544.9	1.250	695.2	1.566
2.5	Wing Assembly	1627	1734	1.065	1741	1.070
	Lower Skin Component	572.4	692.0	1.208	882.9	1.542
3	Wing Assembly	2001	2133	1.065	2142	1.070
	Lower Skin Component	702.1	840.3	1.196	1072	1.526

demonstrates that the response of the von Mises stress distribution is clearly affected by the size of the cut. The ratio of change in the single-stringer damage model is 1.065 and 1.066 under different loads, while it is slightly larger in the triple-stringer damage models. However, analyzing the lower skin panel clarifies the effect of the cut, as this ratio changes from 1.253 in the small cut model to 1.611 in the larger cut model when the factor of safety is 1.5. The tips of the cut witness a significant rise in longitudinal stresses and shear stresses. In addition, it can be seen that as the factor of safety increases, the ratios decrease.

Interestingly, it can be seen from

Table 11. Comparison of maximum in-plane principal strain in the undamaged wing and wing containing stringer damage.

		Maximum In-Plane Principal Strain
Factor of Safety	Component	Undamaged Wing	Single-Stringer Damage	Ratio of the Values	Triple-Stringer Damage	Ratio of the Values
1.5	Wing Assembly	0.0081	0.0080	0.987	0.0082	1.012
	Lower Skin Component	0.0050	0.0067	1.340	0.0082	1.640
2	Wing Assembly	0.0116	0.0115	0.991	0.0116	1.000
	Lower Skin Component	0.0072	0.0092	1.277	0.0113	1.569
2.5	Wing Assembly	0.0150	0.0149	0.993	0.0149	0.993
	Lower Skin Component	0.0093	0.0117	1.258	0.0144	1.548
3	Wing Assembly	0.0183	0.0183	1.000	0.0183	1.000
	Lower Skin Component	0.0114	0.0142	1.245	0.0175	1.535

that the maximum in-plane principal strain appears to have either stayed constant or decreased with a negligibly small difference in the overall assemblies. The effect of the cut is clarified once again when the lower skin panels are examined separately, and similar to the stress, the larger cut has the largest ratio, which equals 1.640, while the smaller cut has a ratio of 1.340 when the factor of safety equals 1.5, and these ratios decrease as the factor of safety is increased.

Overall, from Equation (1), it can be determined that the values of longitudinal stress and shear stress are the governing stresses for the onset of fiber tensile failure and thus, in this particular case, the onset of failure in the assembly.

Table 12. Maximum values of stress components in the wing models containing stringer damage.

Stress	Stringer Damage	Maximum Tensile Stress (Maximum Compressive Stress), MPa
		F.o.S. 1.5	F.o.S. 2	F.o.S. 2.5	F.o.S. 3
S11	Single-stringer	765.4 (−716.7)	1097 (−1028)	1418 (−1330)	1741 (−1634)
	Triple-stringer	767.5 (−719.0)	1100 (−1031)	1421 (−1334)	1745 (−1639)
S22	Single-stringer	255.4 (−311.1)	364.9 (−444.9)	470.8 (−574.4)	577.7 (−705.0)
	Triple-stringer	265.2 (−310.5)	365.9 (−443.9)	472.1 (−573.0)	579.2 (−703.3)
S12	Single-stringer	111.6 (−116.6)	153.9 (−166.8)	194.8 (−215.4)	236.1 (−264.4)
	Triple-stringer	112.7 (−116.8)	155.4 (−167.1)	196.7 (−215.8)	238.3 (−264.9)

displays the maximum values of stress components in the damaged wing models, while

Table 13. Comparison of Hashin fiber tensile failure in the undamaged wing and wing containing stringer damage.

Factor of Safety	Undamaged Wing	Single-Stringer Damage	Ratio of the Values	Triple-Stringer Damage	Ratio of the Values
1.5	0.195	0.299	1.533	0.554	2.84
2	0.392	0.575	1.466	1.066	2.71
2.5	0.653	0.927	1.419	1.719	2.632
3	0.983	1.368	1.391	2.534	2.577

displays the values of fiber tensile failure criterion in the models; it can be seen that tensile failure is not initiated in the pristine wing assembly even as the factor of safety is increased. In the case of single-stringer damage, the value is less than 1 in the first two analyses when the factor of safety is 1.5, 2, and 2.5, while it is greater than 1 when the factor equals 3, suggesting that at some point between the factors 2 and 3, the load is large enough for the failure to be initiated when the cut is small. On the other hand, the failure in the triple-stringer damage model is initiated when the factor of safety equals 2, and as was mentioned, the significantly larger longitudinal and shear stresses around the tips of the cut lead to this failure onset. Similarly, the ratios decrease as the factor of safety increases, suggesting that the stiffness of the model does not fully change linearly with the increase in pressure load.

The stringers that are attached to the boundary where the wing skin is attached provide support and prevent the skin from buckling when it is subjected to stress. Hashin failure criteria show that the planar stresses in the lower skin, whether longitudinal or shear stresses, become almost twice as intense when three stringers are damaged than when a single stringer is damaged.

6. Static Analysis of BWB Wing Containing Damage Cutting through Ribs

The role of the wing rib is crucial because it gives the wing its aerodynamic shape, distributes the load, and redistributes shear forces. To allow fuel or machinery to pass through it, the ribs are sliced. Wing ribs with cutouts are lighter and more load-resistant. An analysis of stress and strain conditions was conducted to assess the criticality of damaging rib components.

6.1. Modeling of Rib Damage in Damaged BWB Wing

The effect of cutting through rib components on the mechanical behavior of a BWB wing assembly was explored through two different models. The models evidently contain damage cuts of different lengths. The models contain a cut modeled in a direction parallel to the stringer; the first of these cuts is through a single rib component (Figure 9), while the last one contains damage that cuts through two rib components (Figure 10).

6.2. Results and Failure Onset Assessment of BWB Wing Containing Single-Rib and Double-Rib Damage

As illustrated by

Table 14. Results of out-of-plane displacement of single-rib and double-rib damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Rib Damage	Double-Rib Damage
1.5
3

, the displacement data were examined in the visualization module following the completion of the static linear analysis. As expected, when the factor of safety is equal to 1.5, 2, 2.5, and 3 in the case of the wing assembly with a single damaged rib, the pressure load applied to the upper and lower skin components of the wing box caused the largest vertical displacement at the tip of the wing, measuring 2473, 3464, 4424, and 5392 mm, respectively. The corresponding values in the double-rib damage are 2474, 3645, 4425, and 5394 mm. Along the length of the wing, the vertical displacement is evenly distributed and decreases as it approaches the wing’s root. The largest displacement occurs outside the wing box in all the analyzed models. Similar to the wing models with stringer damage, it can be observed that as the safety factor increases, the displacement greatly exceeds the value equivalent to 10% of the wing span.

According to the findings displayed in

Table 15. Results of von Mises stress of single-rib and double-rib damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Rib Damage	Double-Rib Damage
1.5
3

, the model with the smaller cut would experience maximum stresses of 930.4, 1339, 1734, and 2133 MPa in the single-rib damage models and 931.0, 1340, 1735, and 2134 MPa in the double-rib damage models at factors of safety equal to 2, 2.5, and 3, respectively. The largest stress in every model with damage is outside the wing box in the leading edge component. A preliminary investigation of the stress distribution in the assembly shows that the front edge component and the trailing edge component both experience a substantial increase in stress intensity in parts close to contact with the wing box assembly. On the other hand, the tips of the cut display particularly smaller stress than the surrounding sections of the lower skin, and the largest stress in the lower skin does not occur at the vicinity of the cut; it is denoted by B and B′ in Figure 11. Overall, the stress distribution in all the components of the wing is fairly similar when the cut is small and when it is large.

The point of contact between the upper skin and the rib components near the root of the wing was found to have the largest in-plane strain, with high values of 0.0080 at the standard factor of safety 1.5, 0.01156 at the factor of safety 2, 0.01492 at the factor of safety 2.5, and 0.01830 at the factor of safety 3 in the model with the smaller cut, as illustrated in

Table 16. Results of the maximum in-plane principal strain of single-rib and double-rib damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Rib Damage	Double-Rib Damage
1.5
3

. The double-rib damage model converts these numbers to 0.0080, 0.01156, 0.01492, and 0.01830, respectively. It can be noticed that the strain values and distribution are the same even if the cut is larger. The strain distribution was significantly affected by the presence of the stringers and ribs, which causes the smooth transition to be disrupted by discontinuities; this disruption is not more noticeable in the upper section, which is subjected to compression. On the other hand, the lower section of the wing has a larger in-plane principal strain distributed over a larger section.

Table 17. Results of Hashin fiber tensile failure criterion of single-rib and double-rib damage BWB wings at different values of F.o.S.

Factor of Safety	Single-Rib Damage	Double-Rib Damage
1.5
3

demonstrates that the fiber tensile failure criterion equals 0.1917 and 0.1918 in the single-rib and double-rib damage models, respectively, at the standard factor of safety 1.5. The criterion is 0.390 in the model with single-rib damage and 0.391 in the model with double-rib damage when the factor of safety is 2. At a factor of safety of 2.5, this criterion is equal to 0.650 and 0.651 in the single-rib and double-rib damage models, respectively. The criterion equals 0.983 in the single-rib damage model and 0.983 in the double-rib damage model when the factor of safety is set to 3. These maximum values of the fiber tensile failure criterion were not measured at the vicinity of the cut, but rather at a region close to the root of the wing, coinciding with the largest longitudinal axial strain in the lower skin panel, coinciding with the locations denoted as B and B’ in Figure 11.

6.3. Comparison of Results between Undamaged BWB Wing and Wing Containing Rib Damage

Table 18. Comparison of out-of-plane displacement in the undamaged wing and wing containing rib damage.

	Maximum Out-of-Plane Displacement (mm)
Factor of Safety	Undamaged Wing	Single-Rib Damage	The Ratio of the Values	Double-Rib Damage	Ratio of the Values
1.5	2480	2473	0.997	2474	0.997
2	3474	3464	0.997	3465	0.997
2.5	4436	4424	0.997	4425	0.997
3	5407	5392	0.997	5394	0.997

demonstrates that the out-of-plane displacement decreases when the model contains single-rib damage, and even though it does increase once this rib cut is larger, this increase is negligibly small. These displacement values are smaller than those obtained from the undamaged BWB wing analysis, as the ratio is seemingly constant at a value of 0.997 throughout the analyses.

Table 19. Comparison of von Mises stress in the undamaged wing and wing containing rib damage.

		Von Mises Stress (MPa)
Factor of Safety	Component	Undamaged Wing	Single-Rib Damage	Ratio of the Values	Double-Rib Damage	Ratio of the Values
1.5	Wing Assembly	873.3	930.4	1.065	931.0	1.066
	Lower Skin Component	311.0	310.4	0.998	310.5	0.998
2	Wing Assembly	1256	1339	1.066	1340	1.067
	Lower Skin Component	443.8	442.9	0.997	443.0	0.998
2.5	Wing Assembly	1627	1734	1.065	1735	1.066
	Lower Skin Component	572.4	571.2	0.997	571.3	0.998
3	Wing Assembly	2001	2133	1.065	2134	1.066
	Lower Skin Component	702.1	700.6	0.997	700.7	0.998

, on the other hand, shows that the size of the rib cut does not have a noticeable impact on the von Mises stress. In the single-rib damage model, the ratio of change is 1.065 and 1.066 under the three loading factors; this ratio slightly increases in the models containing double-rib damage. Analysis of the lower skin panel, however, reveals the stress over the planar components is small compared with the pristine wing models. Further examination of lateral components, such as the ribs and the spars, reveals otherwise; there is a significant increase in von Mises stress, especially over the spars.

Similar to the distribution of von Mises stress, it can be seen from

Table 20. Comparison of maximum in-plane principal strain in the undamaged wing and wing containing rib damage.

		Maximum In-Plane Principal Strain
Factor of Safety	Component	Undamaged Wing	Single-Rib Damage	Ratio of the Values	Double-Rib Damage	Ratio of the Values
1.5	Wing Assembly	0.0081	0.0080	0.987	0.0080	0.987
	Lower Skin Component	0.0050	0.0050	1.000	0.0050	1.000
2	Wing Assembly	0.0116	0.0115	0.991	0.0115	0.991
	Lower Skin Component	0.0072	0.0072	1.000	0.0072	1.000
2.5	Wing Assembly	0.0150	0.0149	0.993	0.0149	0.993
	Lower Skin Component	0.0093	0.0093	1.000	0.0093	1.000
3	Wing Assembly	0.0183	0.0180	0.983	0.0183	1.000
	Lower Skin Component	0.0114	0.0114	1.000	0.0114	1.000

that the maximum in-plane main strain decreases or does not change across all assemblies. When the lower skin panels are examined separately, the effect of the cut length is not noticeable as the increase is only seen at the fifth decimal number. The spars, on the other hand, demonstrate an increase in longitudinal and lateral strains and thus the maximum in-plane principal strain.

Overall, it can be concluded from

Table 21. Maximum values of stress components in the wing models containing rib damage.

Stress	Rib Damage	Maximum Tensile Stress (Maximum Compressive Stress), MPa
		F.o.S. 1.5	F.o.S. 2	F.o.S. 2.5	F.o.S. 3
S11	Single-rib	766.5 (−716.9)	1097 (−1029)	1418 (−1330)	1741 (−1635)
	Double-rib	765.7 (−717.2)	1097 (−1029)	1418 (−1331)	1741 (−1635)
S22	Single-rib	255.8 (−309.5)	365.3 (−442.6)	471.4 (−571.4)	578.4 (−701.3)
	Double-rib	265.5 (−309.2)	366.4 (−442.1)	472.7 (−570.8)	580.0 (−700.6)
S12	Single-rib	112.2 (−116.6)	154.7 (−166.8)	195.8 (−215.4)	237.3 (−264.4)
	Double-rib	112.3 (−116.6)	154.8 (−166.8)	195.9 (−215.4)	237.4 (−264.5)

and

Table 22. Comparison of Hashin fiber tensile failure criterion in the undamaged wing and wing containing rib damage.

Factor of Safety	Undamaged Wing	Single-Rib Damage	Ratio of the Values	Double-Rib Damage	Ratio of the Values
1.5	0.195	0.191	0.979	0.191	0.979
2	0.392	0.390	0.994	0.391	0.997
2.5	0.653	0.650	0.995	0.651	0.996
3	0.983	0.983	1.000	0.983	1.000

that the pressure loads utilized in this study are not sufficient to initiate structure failure in the wing models which contain rib damage. In fact, it is interesting to note that the fiber failure criterion decreases as the cut is modeled; it does indeed slightly increase in the double-rib damage models, yet the increase is small. The effect of the cut on the wing structural integrity is substantial on the level of the spar components, which displays a substantial increase in the value of the fiber tensile failure criterion.

Damaged single or double rib components have no effect on the BWB wing’s structural integrity. Since the ribs assist in transferring stress from the wings to the spars, it is likely that further ribs will need to be damaged before a visible difference in structural integrity can be observed.

7. Conclusions

The preprocessing stage was either carried out on Abaqus in the analysis of the single- and triple-stringer PRSEUS panel or carried out on CATIA, HyperMesh, and Abaqus all at once in the analysis of the BWB wing models. The current study explored a finite element method through the use of processing and postprocessing features provided by Abaqus. The behavior of a BWB wing structure subjected to various damage scenarios and under an increase in the factor of safety has been studied using modeling techniques, which has resulted in substantial time and resource savings. The following is a summary of the conclusions drawn from the results of the different models examined:

(1) The loads acting on the skin are transferred and distributed among the frames by means of stringers. It was shown that the load distribution was disrupted when the damage on the lower area of the wing cut through the above stiffening components, and the lower skin component displayed more stresses and strains in comparison to the other components. However, the PRSEUS concept, which was used for the BWB wing structure, is helpful in maintaining the integrity of the wing structure when a single stringer is damaged until the factor of safety is too large for the damaged structure. The triple-stringer damage is not maintainable, and it is large enough for the structural failure to be initiated as early as when the factor of safety equals 2.

(2) The integrity of the BWB wing structure was barely impacted by the damage to rib components. It is probable that additional ribs will need to be damaged before the deterioration in the structural integrity of the wing can be seen because the ribs help carry the force acting on the wing to the spars.

(3) The decrease in the ratio values as the pressure load increases necessitates an investigation of the probability of a nonlinear response of stiffness to the load.

One of the key characteristics unique to the PRSEUS structural concept is the impeccable damage arrest capabilities. Therefore, future studies should analyze the damage progress in the stringer damage models, and a nonlinear analysis can be taken into account to analyze the various responses that would be shown and compare them to the findings of the linear static analysis from this study. Nevertheless, the adopted FEM establishes the groundwork for a reliable modeling process to study the response of a BWB wing under aerodynamic loads.