Novel Design and Optimization of Aircraft Stiffened Panels for Improved Critical Buckling Load Resistance

Abstract

This study proposes two novel stiffened panel configurations, designated X and X-30, manufactured from the conventional aerospace alloy Al 2024-T3 to enhance the critical buckling resistance under in-plane compression. Their performance was evaluated against traditional T-, I-, L-, and Omega-type stiffeners, as well as a newly introduced Y-panel found in the literature. Initial results show that both proposed designs achieve 80–200% higher buckling capacity than conventional panels, with only a 4.54% difference between the X and X-30 configurations. A weight-constrained optimization was then conducted using a Box–Behnken design of experiments combined with a multi-objective genetic algorithm in Ansys DesignXplorer. After correcting inconsistencies in the initial optimization ranges, the prediction error in the optimized buckling values was reduced to 4%. The optimized X panel attained the highest critical buckling load of 2920 kN, followed by the X-30 panel with 2744.3 kN, corresponding to a 114–258% improvement over traditional stiffener geometries. A Pearson correlation matrix further suggested that, for all the stiffened panels except Omega, the base plate showed a strong correlation with the critical buckling load, typically ranging from 0.83 to 0.99. In contrast, for the X-30 panel, the lower base part also showed a strong correlation of 0.93.

1. Introduction

Buckling is a standard structural failure mode in aircraft components subjected to compressive loading, particularly in stiffened panels that form the fuselage and wings. Such failure can pose a serious challenge to aircraft safety by reducing the ability to withstand compressive loads. Therefore, it is crucial to evaluate the critical buckling load of the stiffened panels to ensure the aircraft’s overall strength and stability and to prevent sudden failure during flight. At the same time, minimizing fuel consumption requires that these structures remain as lightweight as possible. This balance can be achieved by optimizing the critical buckling load (N_cr) and ultimate collapse loads of the stiffened panels through a detailed understanding of their geometric design and dimensions under compressive loading and various boundary conditions.

By definition, critical buckling occurs when a slender structural element, such as a panel or column, becomes unstable under compressive forces. This instability occurs when the applied load exceeds a specific limit, causing the structure to bend abruptly in a direction perpendicular to the load, significantly reducing its ability to carry weight [1,2]. For instance, for an omega-stringer-stiffened panel, critical buckling occurs when the structure deforms out of plane, leading to a substantial decrease in its load-carrying ability [3,4]. In stiffened panels, such as those used in aircraft, this behavior is vital for establishing the structural limits before collapse [5]. In contrast, when critical buckling marks the first point of instability, post-buckling analysis is used to assess structural performance beyond it. It provides insights into deflection (the deformation after buckling) and residual strength (the additional load that the structure can sustain after buckling) [6]. Understanding this post-buckling behavior is crucial for evaluating the remaining load-carrying capacity and collapse strength of stiffened panels under axial compressive loads [7]. Torres et al. [8] suggests that including post-buckling behavior in refining the design can minimize the risk of failure and improve damage tolerance in composite stiffened panels.

Structural components of an aircraft are subjected to various levels of dominant stress types, such as bending, shear, tension, and compression. However, some parts are especially vulnerable to buckling under compressive loads. For instance, the fuselage is highly susceptible to buckling, especially near joints and cutouts, such as windows and doors. The wing structure, including spars and ribs, may buckle due to aerodynamic forces. Tail structures and stabilizers are at risk of buckling due to aerodynamic pressure during turbulent conditions [9]. Schilling and Mittelstedt [10] noted that stiffened panels, such as those in the Airbus A350 fuselage, are particularly prone to local buckling under pressure. Buckling of the fuselage occurs due to the combination of internal pressure and axial compressive loads, leading to large out-of-plane deformations [11]. The wing spar is another essential component that carries the load in an aircraft’s wing structure, which is particularly vulnerable to buckling due to the high aerodynamic forces it experiences [12]. The wing skin, particularly the upper panels, is prone to buckling due to compressive aerodynamic loads during flight. The upper skin panel undergoes significant out-of-plane deformations, leading to localized buckling. These deformations, often referred to as “skin depressions,” can compromise the wing’s structural integrity and aerodynamic performance [13]. Additionally, if the ribs are not properly stiffened or contain imperfections, they can buckle more easily under pressure [14].

For stiffened panels, the critical buckling load (Pcr) and ultimate collapse load are typically determined by numerical analysis and experimental testing, as closed-form solutions are generally unavailable. Researchers have used these approaches to observe various aspects of buckling behavior in structures, as summarized in

Table 1. Summary of the works related to stiffened panels and buckling.

Authors	Year	Research Methodology	Research Object	Main Findings
Quinn et al. [5]	2009	Experimental, FEM	Prismatic sub-stiffened panel	The use of longitudinal sub-stiffeners successfully increased the initial buckling load by 87.2% and the ultimate collapse load by 17.7% for a panel of equivalent mass.
Mittelstedt and Beerhorst [15]	2009	Closed-form analytical method, FEM	Composite panel with omega stringer	The closed-form analytical technique for the critical buckling load of composite plates was found to agree with the exact numerical result. The result demonstrated that the increased stringer height or web angle increases the buckling load.
Houston et al. [16]	2017	Classical plate theory, FEM	Stiffened panel with Buckling Containment Features (BCF).	The research developed design charts that identify the minimum-mass design for a stiffened panel, which occurs at the transition between global and local buckling when Buckling Containment Features (BCF) are used.
Layachi et al. [17]	2017	Topology optimization, FEM	Grid sub-stiffened panel	The application of topology optimization, by introducing elliptic patterns into the stiffeners, successfully reduced the panel mass by 1.5% to 2% while maintaining a nearly identical critical buckling load (~222 kN). The optimized design also promoted a more favorable local buckling mode with a higher number of half-waves.
Chen et al. [18]	2019	FEM, Particle swarm optimization	Sub-stiffened panel	Three optimized sub-stiffened panel designs showed that the panel with a straight sub-stiffener was the most effective, achieving a critical buckling load of 241.0 kN—more than three times the load of the conventional panel (73.5 kN). It also increased the ultimate collapse load by 46%. The grid-stiffened panel was found to have the lowest sensitivity to manufacturing defects.
Ye et al. [19]	2020	Genetic algorithm, FEM.	Composite stiffened panel with sub stiffeners	Optimizing composite sub-stiffened panels increased the buckling strength by 122.66% while mass is constrained. However, the appearance of sub-stiffeners also made the panel’s ultimate collapse load more sensitive to the skin-stiffener interface properties.
Alhajahmad and Mittelstedt [20]	2021	Genetic algorithm, FEM	Curvilinear grid-stiffened panels	A new design framework was introduced that integrates a genetic algorithm with Python and the Abaqus interface. Weight was reduced up to 30% compared to the orthogrid and 18% compared to the angle grid design.
Chandra et al. [21]	2021	FEM	Curved stiffened panel (riveted or FSW, CSF Panel)	Curved rivet stiffened panel showed marginally small buckling load than the curved FSW stiffened panel. The curved stiffened panel with a small stringer showed a higher buckling load, while the closed hat stringer had better buckling strength than the open hat and z stiffener.
Prabowo et al. [7]	2022	FEM	Stiffened panel	The study showed that increasing the plate thickness increased the ultimate collapse load significantly, with a 5 mm plate showing 65.7% and 20.61% higher strength than 3 mm and 4 mm plates, and increasing the longitudinal stringer thickness moderately improved strength by up to 11.8%. In contrast, changes in sub-stiffener thickness had a negligible effect.
Chagraoui et al. [22]	2023	FEM	Sub stiffened panel with omega stringer	The introduction and optimization of sub-stiffeners of omega stringers into a T-stiffened composite panel resulted in a 242.63% increase in the primary buckling load without adding any mass. Furthermore, the cohesive interface properties were found to significantly change the deformation pattern of post-buckling, ultimate collapse load and the initiation and propagation of various failure modes.
Zhang et al. [23]	2024	FEM, Experimental, Analytical	Composite stiffened panel	This review concludes that analyzing composite stiffened panels requires integrating experimental, analytical, and numerical methods, as each approach offers unique strengths and faces distinct limitations. Future advances hinge on effectively combining these methodologies.
Sarwoko et al. [1]	2024	FEM	Stiffened panel	The adoption of a Y-shaped stiffener geometry significantly increased the critical buckling load and the ultimate collapse load compared to an I-type stiffener of the same volume. Again, reducing the distance between stiffeners and adding transverse stiffeners significantly enhanced buckling resistance and energy absorption.
Rayhan et al. [24]	2025	ML, FEM	Additively manufactured lattice stiffened panel	The Simple Cubic (SC) lattice panel showed higher buckling resistance at equivalent relative densities, showing up to 77% higher critical buckling load than Face Centered Cube (FCC) and Body Centered Cube (BCC) panels. Among all the machine learning models, polynomial regression achieved the highest prediction accuracy with a near-perfect R2 score of 0.9999 and the lowest error (MSE = 0.0001).
Stamatelos and Labeas [25]	2025	FEM, Analytical Macro Modeling Optimization	Composite stiffened panel, aircraft wing (Airbus A330)	The result showed a framework that reduced highly detailed models to plate-level size without compromising accuracy. Provides less than 5% deviation in predicting analytical buckling load. Sizing of the composite wing box using the proposed model was completed within minutes.
Zhang et al. [2]	2025	FEA, PNN. NSGA-III, TOPSIS	Composite stiffened panel	The result demonstrated that the Parallel Neural Network (PNN) accurately predicts the buckling load of the stiffened panel (R2 = 0.968, MAPE = 6.7%). The NSGA-III algorithm performs better at finding high-quality Pareto solutions than NSGA-II. Achieved a 22.18% reduction of mass and a buckling load increase of 0.11% compared to the original design.

. Quinn et al. [5] showed that prismatic sub-stiffening demonstrably enhances the initial plate buckling performance of integrally machined aluminum alloy stiffened panels, achieving up to an 89% gain over equivalent mass designs. Mittelstedt and Beerhorst [15] developed a closed-form analytical solution to determine the critical buckling modes and loads of compressed composite plates braced by longitudinal omega-stringers. Houston et al. [16] employed initial design rules for predicting the buckling behavior of stiffened panel elements. Layachi et al. [17] analyzed the stability performance of grid-stiffened panels under mechanical loading for use in the aerospace sector. Chen et al. [18] demonstrated that introducing sub-stiffeners significantly improves the critical buckling loads and post-buckling ultimate strength of stiffened panels under uniaxial loading. Ye et al. [19] investigated that optimized sub-stiffeners significantly improve the critical buckling load of composite stiffened panels and influence their post-buckling deformation and failure mechanisms. Alhajahmad and Mittelstedt [20] optimized tow-steered composite fuselage panels stiffened with curvilinear grids for minimum weight, subject to buckling constraints, demonstrating significant weight reduction and favorable buckling patterns compared to traditional stiffening designs. Chandra et al. [21] investigated the buckling characteristics of curved stiffened fuselage panels, showing that friction stir-welded panels exhibited slightly lower buckling loads than riveted panels, and that smaller stringer pitch and close hat stringer shapes enhanced the critical buckling load. Prabowo et al. [7] showed that the critical buckling load (Pcr) and ultimate collapse response of stiffened panels, demonstrating that increased plate thickness and specific material types significantly enhance strength. In contrast, sub-stiffener thickness has a negligible impact. Chagraoui et al. [22] introduced an optimization framework for omega sub-stiffened composite panels, demonstrating that these sub-stiffeners significantly improve initial buckling performance and alter post-buckling deformation paths and ultimate failure loads. Sarwoko et al. [1] showed that adopting a Y-shaped stiffener geometry, incorporating transverse stiffeners and reducing the distance between stiffeners, enhances the buckling resistance and collapse performance of stiffened aircraft panel structures. Zhang et al. [23] synthesized a decade of experimental, analytical, and numerical analysis on the critical buckling and post-buckling behaviors of composite panels. Rayhan et al. [24] found that simple cubic lattice unit cells provided superior buckling loads compared with face-centered cubic (FCC) and body-centered cubic (BCC) structures across a range of relative densities. Stamatelos and Labeas [25] presented a macro-modelling framework for the accurate and rapid prediction of buckling loads in large-scale composite panels, incorporating a bending stiffness matrix that accounts for essential eccentricities and coupling. Zhang et al. [2] proposed a new multi-objective optimization technique for composite panels, facilitating optimal designs that inherently influence their buckling and post-buckling performance.

In summary, the study of critical buckling and the optimization of lightweight panels capable of carrying higher loads remain essential research topics in aeronautical and aerospace engineering. Although extensive research has been conducted over the past few decades, the field continues to evolve due to its wide-ranging applications in structural mechanics. The development of novel stiffened panel configurations, their optimization, and subsequent validation for practical implementation remain at the forefront of current research. Moreover, with the advancement of manufacturing technologies such as 3D printing, the rapid realization and testing of innovative structural designs have become increasingly feasible. In the present study, two novel designs of aircraft stiffened panels are proposed and were systematically compared with conventional configurations currently used in aircraft structures. Furthermore, optimization was performed to enhance the structural stability of the panels and investigate the performance of the newly proposed panels with conventional designs.

2. Statistical Methods

2.1. Design of Experiments

Design of experiments (DOE) is used to generate the design points within the defined design space for conducting experiments. The responses obtained at these points are then used to construct the response surface equation. Moreover, DOE enables the development of the response surface equation using fewer design points, improving efficiency. It is an advanced statistical approach that goes beyond passive observation by actively adjusting input parameters to accelerate the search for an optimal solution, outpacing conventional methods. DOE creates a balanced set of experiments around a central point by adjusting all factors simultaneously, allowing identification of the objective function’s gradient for optimization. This approach enhances efficiency and robustness, with final testing limited to validating simulation results. Widely used in process development, DOE improves performance and resilience to external variations. To ensure reproducibility and adaptability across different production contexts, the methodology must be symmetrically structured and organized into sequential steps [26].

2.2. Box–Behnken Experimental Design

Box–Behnken design (BBD) is a statistical optimization methodology developed in 1960 that uses three-level incomplete factorial designs to efficiently study the relationships between multiple input variables and output variables [27]. For three factors, BBD makes experimental points at the center and midpoints of a cube’s edges, all located on a sphere’s surface [28]. This spherical arrangement provides rotatability, meaning prediction accuracy remains consistent across the entire experimental domain, which enhances the reliability and strength of the model predictions.

BBD operates at three levels for each factor: usually represented as −1 for the low level, 0 for the central or middle level, and +1 for the high level. These three levels allow the design to represent nonlinear relationships between variables and their responses, which is critical for realistic modeling of complex systems. The central level (0) serves dual purposes: it enables estimation of curvature in the response surface (quadratic effects), and it provides replication at the design center to evaluate experimental error and model adequacy. The mathematical determination of the experimental runs required is expressed as

N = 2k (k − 1) + C₀

(1)

where N denotes the overall number of experimental trials, k denotes the count of factors being evaluated, and C₀ denotes the number of central point replications [28].

The fundamental analytical tool in BBD is the second-order polynomial regression model, which mathematically represents the connection between independent variables (factors) and dependent variables (responses). The general form of this model is expressed as

$$y={\beta }_0+{\sum }_{i=1}^P {\beta }_i{x}_i+{\sum }_{i=1}^P{\sum }_{j=1}^P{\beta }_{ij}{x}_i{x}_j+{\sum }_{i=1}^P{\sum }_{j=1}^P{\sum }_{k=1}^P{\beta }_{ijk}{x}_i{x}_j{x}_k$$

(2)

Here, ${ \beta }_0$ represents the intercept when all factors are at the central level, ${ \beta }_i$ represents how the changing of i affects y when the other factors are constant, ${ \beta }_{ij}$ represents the two-way interaction between the factors i and j, and ${ \beta }_{ijk}$ is the three-way interaction among the factors i, j, and k [29].

The second-order polynomial model in Box–Behnken design represents three types of effects: main effects (individual factor influence), quadratic effects (nonlinear relationships), and interaction effects (how factor combinations jointly influence responses). This is essential for stiffened panel optimization where design parameters interact nonlinearly to affect structural performance. Once regression coefficients are determined, a three-dimensional response surface is generated that graphically shows how responses vary across the design space, allowing researchers to identify optimal conditions, robust performance regions, and factor interactions while communicating results clearly to stakeholders.

BBD offers substantial advantages for stiffened panel optimization by conclusively reducing experimental attempts such as requiring only 15 runs compared to 81 for full-factorial designs—an 82% reduction while avoiding impractical extreme factor combinations through its “missing corners” characteristic that focuses resources on practical conditions [27,28]. BBD shows superior statistical efficiency for three to four factors with optimal spherical point arrangement, achieving R² values exceeding 0.90–0.99 that represent 90–99% of experimental variance and ensure reliable model predictions across the entire design space [27]. Unlike sequential one-factor-at-a-time approaches, BBD enables simultaneous exploration of multiple variables and their interactions, allowing researchers to develop models for competing objectives (such as maximizing stiffness while minimizing weight) and to recognize optimal solutions balancing conflicting engineering goals through explicit capture of interaction effects that reveal how one factor’s influence depends on another’s level [29].

2.3. Mathematical Background of Response Surface Methodology

Response Surface Methodology (RSM) is employed to analyze relationships when the input variables are quantitative. In this context, the variables x₁ and x₂ are used to maximize the process yield Y. In short, these variables significantly affect the yield of the process, as outlined below:

$$\mathrm{Y} = f({\mathrm{x}}_1, {\mathrm{x}}_2) + \epsilon$$

(3)

Indeed, Response Surface Methodology (RSM) is highly impactful for both developing and analyzing systems, with the primary goal of optimizing the response influenced by various factors. RSM plays a crucial role in the development, design, and implementation of new engineering studies. Its widespread use spans across industrial, natural, social, and engineering disciplines. Given its broad range of applications, researchers have taken great interest in exploring the origins and development of RSM. Numerous studies have applied RSM to manufacturing and engineering problems. For instance, it has been used to analytically model surface roughness and cutting force components, showing that feed rate and the hardness of the specimen strongly impact both [30].

RSM works very well with accurate and good measurements. So, researchers analyze all the variables and prioritize significant variables and exclude the insignificant ones. Moreover, Hill and Hunter categorized response surface analysis into four distinct phases, such as the plan of statistical experiment, estimation of response surface coefficients, validation of the model whether it fits or not, and assessment of the result [31].

Response Surface Methodology (RSM) is both a computational and scientific tool used to model and analyze scenarios involving multiple factors that influence a desired response, to maximize the outcome [32]. In many RSM problems, the exact relationship between the target and the input variables is initially unknown [33]. To address this, RSM begins by estimating the response variable y in relation to a set of independent variables x. In most cases, a low-order polynomial is applied to different regions of the response surface. The first-order model, which serves as a functional approximation, is effectively represented using these independent variables and is written as follows:

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\cdots + {\mathrm{\beta }}_{\mathrm{k}}{x}_k+ \epsilon$$

(4)

When the system exhibits curvature, a second-order or higher-degree polynomial is used to accurately model the response surface [32]. Here, Equation (4) is the linear form of Equation (3), where the $\epsilon$ indicates the random error.

The parameters of approximation polynomials are measured using the least-squares method. This allows for response surface analysis to be carried out on the corresponding surface. Additionally, the analysis of the fitted surface is compared to that of the true system to determine whether the fitted surface provides a good representation of the true response [34].

In turn, the model variables help determine an appropriate experimental setup for efficient acquisition of data. As a result, the fitted surface is regarded as the appropriate response surface. Without a doubt, RSM is a powerful tool for selecting the best process parameters of a system or for locating a region within the factor space that meets the desired performance criteria [35]. Similarly, the simultaneous evaluation of multiple outcomes begins with developing suitable response surface models for each output. Then it optimizes a set of conditions that ensure all responses remain within acceptable limits, particularly at the lower bound [36].

2.4. Genetic Algorithm (GA)

A Genetic Algorithm (GA) is an optimization method based on the foundation of natural selection. It seeks the best operating conditions for solution of a problem by simulating the mechanism of inheritance, selection, crossover, and mutation to improve a set of candidate solutions iteratively. Genetic algorithms are frequently used to solve massive combinatorial optimization problems due to their ability to run in parallel and with excellent global search capacity [37].

The process of using a genetic algorithm is as follows:

(i)

Design Encoding: The initial and most critical step is encoding the design variables (the stiffener geometry and layout) into a digital format, often referred to as a chromosome. This sequence represents the genetic blueprint for a single candidate solution (a stiffened plate design). While binary coding is the foundational method, real-valued encoding is frequently used in complex engineering applications to directly represent the continuous dimensions of the stiffeners, such as height, thickness, and spacing.

(ii)

Population Initialization: A diverse, randomized initial population of stiffened plate designs is generated to begin the search. This step ensures that the algorithm explores many different designs, which helps prevent it from getting stuck on a poor solution early on, prevents premature convergence to a local optimum, and ensures a comprehensive search of the entire design space.

(iii)

Fitness Evaluation: Each individual in the population is assessed based on its performance against the design objective. In the context of plate stability, the critical buckling load is calculated by Finite Element Analysis (FEA) and serves as the fitness function. A higher critical buckling load indicates a fitter individual, resulting in a higher likelihood of selection for reproduction.

(iv)

Selection Operator: According to the calculation of the fitness value of the previous step, the individuals that are demonstrating superior fitness (i.e., those with a higher critical buckling load) are chosen to become “parents.” The goal is to preferentially select the “best” genes for inheritance by the next generation. Common selection strategies, such as the betting wheel (or roulette wheel) method, are used to ensure that while top performers are favored, less-fit individuals are retained with a low probability, maintaining genetic diversity.

(v)

Crossover Operator: Crossover is the primary mechanism for generating novel designs. It involves recombining the genetic code of two parent solutions to create two new offspring. This operation mimics biological reproduction by exchanging design parameters (genes) at one or more points along the encoded sequence. As illustrated in Figure 1, this mixing allows successful characteristics from two different designs to be rapidly integrated into a potentially superior new design.

(vi)

Mutation Operator: Following crossover, the mutation operator introduces random, slight modifications to the genes of the offspring. This step is crucial for maintaining genetic diversity and preventing premature convergence. Mutation ensures that the algorithm can explore areas of the design space that might not be reachable through simple crossover of existing population members.

(vii)

Termination and Optimal Solution: The evolutionary cycle, selection, crossover, and mutation are repeated over many generations. This iterative process drives the population toward increasingly higher fitness levels. The algorithm terminates when a predefined stop condition is satisfied. The stiffened plate design corresponding to the maximum attained fitness represents the optimal solution.

GA is particularly effective for complex engineering problems, like stiffened panel buckling optimization, where multiple design variables, such as the thickness of the panel, stiffener height, and stiffener thickness interact in highly nonlinear ways, making traditional optimization methods less efficient. A Multi-Objective Genetic Algorithm (MOGA) is essentially an extension of a standard Genetic Algorithm (GA). To achieve a balanced trade-off between multiple performance criteria of the specimen under buckling conditions, a Multi-Objective Genetic Algorithm (MOGA) is employed. This method allows the continuous optimization of conflicting objectives—such as maximizing the critical buckling load while minimizing material usage or weight.

3. Theoretical Concept of Plate Buckling

3.1. Kirchhoff’s Plate Theory

In the 1850s, Kirchhoff proposed the plate theory, which is mainly used to model the bending of thin plates under compressive loading. The idea is that the deformation of a plate can be described by a set of differential equations that predict its behavior under various conditions, such as bending, stretching, or compression.

Consider the buckling of a thin rectangular plate subjected to biaxial compression load. The governing equation for this scenario, based on Kirchhoff’s or classical thin plate theory (CPT), is given in equation [39]:

$$D\left({\displaystyle \frac{{\partial }^{4}w}{\partial x^4}}+2{\displaystyle \frac{{\partial }^{4}w}{\partial x^2\partial y^2}}+{\displaystyle \frac{{\partial }^{4}w}{\partial y^4}}\right)+N_x{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}+N_y{\displaystyle \frac{{\partial }^{2}w}{\partial y^2}}=0$$

(5)

Here, we represent the transverse deflection, and D is the flexural rigidity of the plate. The flexural rigidity can be expressed as follows:

$$D={\displaystyle \frac{E{h}^3}{12\left(1-v^2\right)}}$$

(6)

where h is the plate’s thickness, E is Young’s Modulus, and v is Poisson’s ratio. For two boundary conditions, and for each edge that is elastically restrained, the equations are as follows:

$$D{\displaystyle \frac{{\partial }^{2}w}{\partial n^2}}+Dv{\displaystyle \frac{{\partial }^{2}w}{\partial s^2}}\mp {\displaystyle \frac{K_r^m}{L}}{\displaystyle \frac{\partial w}{\partial n}}=0$$

(7)

The additional boundary equation is the following:

$$N{\displaystyle \frac{\partial w}{\partial n}}+D{\displaystyle \frac{{\partial }^{3}w}{\partial n^3}}+\left(2-v\right)D{\displaystyle \frac{{\partial }^{3}w}{{\partial }^2\partial n}}\pm K_l^{m}w=0$$

(8)

Here, N is the normal coordinate measured outward from the boundary of the plate, ${\displaystyle \frac{\partial w}{\partial n}}$ denotes the slope of the plate deflection along the normal edge, ${\displaystyle \frac{{\partial }^{3}w}{\partial n^3}}$ denotes the rate of change of bending moment [40,41].

3.2. Plate Buckling

Buckling occurs in thin rectangular plates under compression when the applied load reaches a specific critical value. For simply supported rectangular plates on all sides, the critical buckling load is given by [39,42] the following:

$$N= {\displaystyle \frac{K_C{\pi }^{2}D}{b^2}}$$

(9)

In Equation (9) above, D can be calculated using Equation (6). Here, a and b represent the lengths of the unloaded and loaded sides of the plate, respectively. k_c is the buckling coefficient, which varies based on the aspect ratio a/b and the boundary conditions (support type) along the plate edges.

3.3. Critical Buckling Load Validation Against Plate Theory

Table 2. Validation of critical buckling load for rectangular plates with different boundary conditions.

No.	Case	Analytical Result (kN)	Computational Result (kN)	$\mathbf{Error}=\frac{\mathbf{A}\mathbf{n}\mathbf{a}\mathbf{l}\mathbf{y}\mathbf{t}\mathbf{i}\mathbf{c}\mathbf{a}\mathbf{l} - \mathbf{C}\mathbf{o}\mathbf{m}\mathbf{p}\mathbf{u}\mathbf{t}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{a}\mathbf{l}}{\mathbf{A}\mathbf{n}\mathbf{a}\mathbf{l}\mathbf{y}\mathbf{t}\mathbf{i}\mathbf{c}\mathbf{a}\mathbf{l}} \mathbf{\ast } 100\mathbf{\%}$
1	All edges are simply supported	2.067	2.067	0%
2	Loading edges are fixed support, and unloaded edges are simply supported	2.288	2.279	0.39%
3	Loading edges are simply supported, and unloaded edges are fixed supported	3.798	3.647	3.97%
4	All edges fixed support	3.987	3.804	4.58%

compares the analytical and computational critical buckling loads of a plate structure. The analytical results of the critical buckling load were obtained by determining the values of buckling coefficients [42] in Equation (9). The percentage error indicates a strong correlation between the experimental and numerical outcomes and validates the FE method for stiffened panel analysis.

4. Materials and Methods

The geometry used in this simulation refers to the exact shape and arrangement of the components that make up the stiffened and sub-stiffened panel, which were validated in Section 3.3 and were mainly guided by the literature [5]. The simulation incorporated the dimensions of the stiffened panel as reported in the experimental data: 590 mm in length, 50 mm in width, a stiffener height of 28 mm, and a plate thickness of 2.2 mm. The structural geometry considered consists of a plate and three attached stiffeners, as illustrated in Figure 2, which labels each element. The design modification of the stiffened panels is achieved by altering the type of stiffeners, as further discussed in Section 4.3.

4.1. Material Model

In this stiffened panel, Aluminum Alloy 2024-T3 is considered for the critical buckling load analysis which was also utilized to validate the experimental work earlier. The mechanical properties of the material are given in

Table 3. Material properties of the stiffened panel based on information in [43].

Property	Value
Density (kg/m3)	2780
Young’s modulus (GPa)	73.4
Poisson’s ratio (-)	0.33
Yield Strength (MPa)	315
Ultimate Strength (MPa)	550

[43].

To ensure the validity of the FE code and the present work for an aircraft stiffened panel, the experimental outcomes of a stiffened and sub-stiffened panel reported in the literature [5] are compared, and the results are presented in

Table 4. Comparison between computational and experimental results for both panels.

Panel Category	Loading Type	Experimental Value [5] kN	Computational Value, kN	$\mathbf{Error}=\frac{\mathbf{E}\mathbf{x}\mathbf{p}\mathbf{e}\mathbf{r}\mathbf{i}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{a}\mathbf{l} - \mathbf{C}\mathbf{o}\mathbf{m}\mathbf{p}\mathbf{u}\mathbf{t}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{a}\mathbf{l}}{\mathbf{E}\mathbf{x}\mathbf{p}\mathbf{e}\mathbf{r}\mathbf{i}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{a}\mathbf{l}} \mathbf{\ast } 100\mathbf{\%}$
Stiffener	Critical buckling load	74.5	70.6	5.23
Sub Stiffener	Critical buckling load	140.2	147.89	5.48
Stiffener	Ultimate collapse load	216.6	218.3	0.77
Sub Stiffener	Ultimate collapse load	255	247.96	2.76

. The post-buckling load versus displacement curve is shown in Figure 3. It is important to note that the current work is based on critical buckling load only. However, the non-linear ultimate collapse load is also calculated to represent the capability of the FE code in capturing both linear and non-linear buckling behavior.

4.2. Mesh Study & Boundary Condition

Meshing is an essential process for discretizing a continuous structure into thousands of small, finite elements, enabling software like ANSYS to solve complex equations and replicate real-world behavior numerically. The computational value of the stiffened panel was mentioned for different mesh sizes and compared to the reference value. Among all these mesh sizes, the lowest error percentage was observed for a 10 mm mesh, as shown in

Table 5. Mesh study of stiffened panel.

No.	Mesh Size (mm)	Computational Value (kN)	Reference Value [5] (kN)	Error, %
1	10	72.4	74.5	2.85
2	9	68.7	74.5	7.7
3	8	72.3	74.5	3
4	7	71.7	74.5	3.8
5	6	71.6	74.5	3.8

.

The linear buckling analysis assumes structural linearity up to a critical-load threshold that initiates instability. In this analysis, the model is simply supported on all sides, and the displacements on the top and bottom sides are unrestricted in the y-axis, while being fixed in the x- and z-axes. Rotation on the top and bottom sides is fixed along the x- and z-axes and unrestricted along the y-axis. A uniformly distributed load of 10⁵ N is applied to both the top and bottom edges. At the plate’s center, displacement is constrained in the x- and y-directions and permitted along the z-axis. Figure 4 illustrates the detailed boundary setup.

4.3. Scenario Variation

This study explored different scenarios by analyzing physical parameter variations that affect the structural performance of the model. Physical variation refers to the change in the type of stiffeners that are attached to the plate. The volume and density of the stiffeners were treated as dependent variables.

The aim of examining different stiffener types is to assess their impact on the performance of stiffened panels under loading conditions. Seven stiffener types—Type I, Type L, Type T, Type Y, Type X, Type X (with 30 mm fillet), and Type Omega (as illustrated in Figure 5)—were evaluated. The width and length of the stiffened panels were set to 720 mm and 1200 mm, respectively. To ensure consistency, the volume of each stiffener was maintained close to 8.64 × 10⁶ mm³ (approximately 24.019 kg). The dimensions for these stiffener types are detailed in

Table 6. Codes of stiffener type variations.

Stiffener Type	Code
I	Type-I
L	Type-L
T	Type-T
Y	Type-Y
X	Type-X (proposed)
X-30	Type-X with 30 mm fillet (proposed)
Omega	Type-Omega

and

Table 7. Dimensions of various stiffener types.

Type	Geometry	Dimension (mm)	Type	Geometry	Dimension (mm)
I		A = 150 mm B = 8 mm	T	A = 75 mm B = 6 mm C = 100 mm D = 8 mm
L		A = 75 mm B = 6 mm C = 100 mm D = 8 mm	Y	A = 37.5 mm B = 8 mm C = 50 mm D = 6 mm E = 45 degree F = 6 mm G = 80.90 mm
X		A = 96 mm B = 80.90 mm C = 2.87 mm D = 2.87 mm E = 78 mm F = 2.87 mm	X-30	A = 96 mm B = 80.90 mm C = 30 mm D = 26.44 mm E = 2.885 mm F = 2.885 mm G = 78 mm
Omega		A = 30 mm B = 127.5 mm C = 30 mm D = 6.675 mm E = 65 degree

.

5. Optimization Framework

To enhance the structural efficiency of plate models, an optimization framework was developed to maximize the critical buckling load under mass and volume constraints. The design space is defined by the following geometric parameters: stiffener height, stiffener thickness, stiffener width, and base thickness. By systematically exploring these variables, the objective was to identify the most efficient configuration that offers superior buckling resistance without exceeding weight limitations. The applied optimization methodology is outlined in the following flowchart.

The optimization process followed the framework shown in Figure 6. Initially, design variables such as stiffener height, thickness, width, and base thickness are defined within their feasible ranges along with material properties. A Box–Behnken Design of Experiments (DOE) is employed to generate samples, and a genetic aggregation response surface model is constructed to predict buckling performance. A multi-objective genetic algorithm is then applied to maximize the critical buckling load under mass and volume constraints. The optimized solutions are then validated using finite element analysis; if the prediction error exceeds 4%, the variable ranges are refined to ±5% of the candidate solution and re-optimized until convergence is achieved.

The input parameters and their respective ranges for each stiffener configuration are summarized in

Table 8. Initial range of different stiffener panels.

SN	Stiffener Type	Input Parameters	Range
1.	I-Stiffener Panel	Base Plate = 5 mm Stiffener Thickness = 8 mm	3 mm to 7 mm 6 mm to 10 mm
2.	L-Stiffener Panel	Base Plate = 5 mm Stiffener body = 6.85 mm	3 mm to 7 mm 4 mm to 8 mm
3.	T-Stiffener Panel	Base Plate = 5 mm Vertical Part = 6 mm Upper Part = 8 mm	3 mm to 7 mm 4 mm to 8 mm 6 mm to 10 mm
4.	Y-Stiffener Panel	Base Plate = 5 mm Lower Part = 6 mm Vertical Part = 6 mm Upper Part = 8 mm	3 mm to 7 mm 4 mm to 8 mm 4 mm to 8 mm 6 mm to 10 mm
5.	X-Stiffener	Base Plate = 5 mm Lower Part = 2.87 mm Upper Part = 2.87 mm X = 2.87 mm	3 mm to 7 mm 2 mm to 4 mm 2 mm to 4 mm 2 mm to 4 mm
6.	X-Stiffener with 30 mm Fillet	Base Plate = 5 mm Lower Part = 2.885 mm Middle Part = 2.885 mm Upper Part = 2.885 mm Middle part X = 2.885 mm	3 mm to 7 mm 2 mm to 4 mm 2 mm to 4 mm 2 mm to 4 mm 2 mm to 4 mm
7.	Omega-Stiffener Panel	Base Plate = 5 mm Stiffener thickness = 6.675 mm	3 mm to 7 mm 5 mm to 9 mm

. For consistency, the base plate thickness is kept fixed at 5 mm across all models, while the stiffener dimensions such as height, thickness, and width are varied. The design ranges are chosen as ±2 mm from the initial reference dimensions to ensure a practical and manufacturable domain. This approach allowed systematic exploration of seven stiffener types, including I, L, T, Y, X, X with 30 mm fillet, and Omega panels, for evaluating their buckling performance.

6. Results and Discussions

6.1. Initial Critical Buckling Load Analysis

The stiffened panel structures used in the study incorporated seven different stiffener types: Type-I, Type-L, Type-T, Type-Y, Type-X, Type-X with 30 mm fillet, and Type-Omega. Each type ensured a consistent volume of 8.64 × 10⁶ mm³ (approximately 24.019 kg) across all variants. The initial results before optimization of these panels are shown in Figure 7. According to the illustration, panels reinforced with Type-X stiffeners exhibited the highest critical buckling load, reaching 2403.5 kN, substantially higher than the other configurations. Type-X (with 30 mm fillet) stiffener ranked second with a value of 2333.3 kN, followed by Type-Y at 1769.50 kN, Type-Omega at 1356.20 kN, Type-L stiffeners at 887.91 kN, and Type-I stiffeners at 813.08 kN. The Type-T stiffener showed the lowest buckling load of 777.85 kN. It means that the X stiffeners increase the critical buckling load by approximately 209%, 200%, 174.23%, 79.5%, 37.6%, and 4.75%, when compared to the T-, I-, L-, Omega-, Y-, and X-Stiffeners with 30 mm fillet, respectively. The lowest performance was observed in panels with Type-T stiffeners, which recorded a critical buckling load of 777.85 kN, more than one-third of that of the Type-X configuration.

Figure 8 illustrates the first mode buckling contours for each stiffener type. These results reveal that local buckling occurred primarily in the plate portion of the structure across all configurations. The X-type stiffener showed the most pronounced buckling waveforms, suggesting a higher buckling coefficient and, consequently, a greater critical buckling load. This was followed by the Type-X with a 30 mm fillet stiffener. Meanwhile, the Y-, L-, I-, and T-type stiffeners exhibited similar buckling patterns; however, the I-type configuration also displayed signs of local buckling within the stiffener itself. In addition, the Omega stiffener showed a buckling pattern that corresponds to a global buckling pattern.

6.2. Optimization Results

The optimization process was carried out using the Box–Behnken method, a robust Design of Experiments (DOE) technique. This method enabled an experiment of multiple input variables, such as base plate dimensions, stiffener configurations, and other structural parameters, to generate five candidate points for each type of stiffener. The percentage of error was calculated with the help of the analytical value and the computational value. For each stiffener type, the deviation between the analytical and computational buckling loads was calculated. This was done using the following formula:

$$\mathrm{Error} = {\displaystyle \frac{\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}-\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}{\mathrm{A}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}} \times 100\%}$$

6.2.1. Results of First Optimization with All Candidate Points and Error Percentage

The main objective of the first optimization is to minimize the error and improve the accuracy of the computational predictions, particularly for stiffener types with high deviations. As a sample, all the candidate points of the I-Stiffener are shown as a sample in

Table 9. First optimized result of I-Stiffener (A sample data).

Using the Box–Behnken Method
Input Variables	CP-1	CP-2	CP-3	CP-4	CP-5
Base plate, mm	5.064	5.029	5.1557	5.0187	5.0187
Stiffener, mm	7.8959	7.9537	7.7471	7.9641	7.9641
Mass, kg	24.017	24.019	24.013	24.01	24.01
Buckling Load (Analytical), kN	814.94	814.52	812.18	813.27	813.27
Buckling Load (Computational), kN	813.79	813.67	812.82	812.59	812.59
Deviation	0.14%	0.10%	0.71%	0.83%	0.83%

CP = Candidate Point.

. The results from the first optimization are presented in Table 9 and

Table 10. First optimized result of L-Stiffener, T-Stiffener, Y-Stiffener, X-Stiffener, X30-Stiffener, and Omega-Stiffener.

Using the Box–Behnken Method (CP1)
Input Variables	L-Stiffener	T-Stiffener	Y-Stiffener	X-Stiffener	X30-Stiffener	Omega-Stiffener
Base plate, mm	6.2662	6.2725	6.1269	5.6001	6.1852	4.9638
Stiffener, mm	5.103	-	-	-	-	-
Vertical Part, mm	-	4.3173	4.1912	-	-	-
Horizontal_T, mm	-	6.1383	-	-	-	-
Lower Part, mm	-	-	4.1316	2.2168	1.3596	-
Middle Part, mm	-	-	-	-	1.7556	-
Upper Part, mm	-	-	8.307	3.7319	4.8671	-
X, mm	-	-	-	2.154	1.354	-
Omega, mm	-	-	-	-	-	6.7551
Mass, kg	23.988	23.994	24.019	24.011	23.968	24.016
Buckling Load (Analytical), kN	1334.5	1278.6	2581.9	2795.4	2715.9	1364.5
Buckling Load (Computational), kN	1205.2	1206.2	2559.5	2677.6	2630.4	1364.4
Deviation	9.68%	5.66%	0.86%	4.21%	3.14%	0.01%

for each stiffener configuration. Among all five candidate points, the first candidate point, named CP-1, shows the least deviation. All the values of CP-1 of Type-L, Type-T, Type-Y, Type-X, Type-X with 30 mm fillet, and Type-Omega are shown in Table 10. The deviations were more than 4% at all of those five candidate points, and those stiffeners were optimized again. Types such as Type-L, Type-T, Type-X, and Type-X with 30 mm fillet showed a deviation of more than 4%. Especially the Type-L Stiffener showed the highest deviation of all. Although the first candidate point of Type-X with 30 mm fillet did not show a deviation of more than 4% as shown in Table 10, it was optimized again, as the deviation was more than 4% in the third candidate point (CP-3).

6.2.2. Results of the Second Optimization with All Candidate Points and Error Percentage

In the second phase, the focus was on the stiffener types that exhibited an error percentage greater than 4%, such as the L, T, X, and X-30 stiffeners shown in

Table 11. Second optimized result of L-Stiffener, T-Stiffener, X-Stiffener, and X-Stiffener with 30 mm fillet.

Second Optimization Using the Box–Behnken Method (CP1)
Input Variables	L-Stiffener	T-Stiffener	X-Stiffener	X-30 Stiffener
Base plate, mm	6.2788	6.4595	5.728	6.2753
Stiffener, mm	5.1035	-	-	-
Vertical Part, mm		4.1123	-	-
Horizontal_T, mm	-	5.8432	-	-
Lower Part, mm	-	-	2.1133	1.2945
Middle Part, mm	-	-	-	1.7534
Upper Part, mm	-	-	3.5431	4.6269
X, mm	-		2.1437	1.4189
Omega, mm	-		-	-
Mass, kg	24.019	24.017	24.015	24.018
Buckling Load (Analytical), kN	1348.7	1277.2	3035.6	2793.5
Buckling Load (Computational), kN	1354.8	1270.5	2920	2744.3
Deviation	0.45%	0.52%	3.80%	1.76%

. For these designs, the optimization process was repeated, but with a more refined approach. The range of the input parameters was reduced by ±5% around the CP-1 (candidate point 1) values from the first optimization. This adjustment was done to narrow the possible design configurations and reduce the error in computational predictions. The second optimization resulted in a new set of candidate points, where the error percentages were reduced significantly. For instance, the L-Stiffener, which initially had errors of 9.68%, was optimized to a much lower error, bringing the design closer to the analytical results. Similarly, for other stiffener types like the T, X, and X with 30 mm fillet, where the deviations were initially higher, the refined optimization process helped improve the accuracy of the computational buckling loads shown in Table 11.

After the optimization (Figure 9), all types of stiffeners exhibited a notable enhancement in their load-carrying capacity. The X-Stiffener continued to outperform other stiffeners with an increased buckling load of 2920 kN, representing an improvement of 19.9% over its pre-optimized counterpart. The X-Stiffener also showed an almost identical percentage gain of 18.06% over its previous buckling load. The Y-, L-, and T-type configurations recorded the most pronounced improvements of 44.64%, 52.58%, and 63.33%, respectively, indicating their high sensitivity to geometric optimization. However, the Omega- and I-type stiffeners showed relatively minor improvements of 0.97% and 0.12%, respectively.

6.2.3. Correlation Analysis of Stiffener Geometry and Buckling Load

Figure 10 presents the Pearson correlation coefficient heatmap, which illustrates the input parameters and the corresponding critical buckling load for different stiffener configurations. Here, the color gradient represents the degree of the correlation, where the darker shades represent a stronger impact, and the lighter shades represent a weaker impact or negligible relationships. From the figure, it can be seen that the base plate parameter exhibits a strong positive correlation, consistently, exceeding 0.9 in most cases. This suggests that the base plate plays a dominant role in the overall buckling load of the stiffened panels. Though the Omega-Stiffener showed comparatively less impact than the others. The stiffener body shows a moderate correlation with certain configurations, such as I- and L-Stiffeners, indicating that this variable contributes to the buckling load appreciably but not as decisively as the base plate. The lower part parameter demonstrates a high correlation of 0.93 with the X-30 Stiffener, suggesting that this variable has a significant influence on the buckling load of the X-30 Stiffener. Similarly, a strong correlation of 0.86 is found between the Omega body and the Omega-Stiffener. On the other hand, other input parameters, such as the Vertical Part, Horizontal_T, Middle Part, and Upper Part, exhibit a very low correlation and no correlation (represented as 0 in Figure 10), indicating the limited influence and no influence, respectively, on the buckling behavior.

7. Conclusions

This present work investigated the linear and nonlinear buckling analysis of seven different stiffened panel structures, where four of these stiffened panels are used regularly in aircraft. Recently, the Y-Stiffener was introduced to improve structural performance. In the present study, two new designs, such as the X-Stiffener and X-Stiffener with 30 mm fillet, were proposed and compared with the existing design. Based on the critical buckling load and the ultimate panel collapse load, the panels were then optimized. The work was carried out using Ansys Mechanical after a brief experimental and theoretical validation was obtained. The results obtained led to the following conclusions:

• In this paper, we proposed two novel stiffened panel configurations (X and X with 30 mm fillet) that are compared with the traditional stiffened panels (I-, L-, T-, Omega-, and Y-Stiffener). Among the initial design cases, the X-stiffened panel achieved the maximum critical buckling load of 2403.5 kN, indicating superior structural stability compared to the other stiffener configurations. Conversely, the T-stiffened panel recorded the minimum buckling load of 777.85 kN. Quantitatively, the X-stiffened design improves the buckling capacity by 37.6% over the Y-stiffened panel and by 209% relative to the T-stiffened panel, highlighting its enhanced stiffness efficiency and load-carrying performance.
• These initial designs were optimized to enhance the critical buckling load, and the results were validated with the candidate points generated by Box–Behnken Design, and the error percentage was calculated. If the error percentage was found to be more than 4% in any of the stiffened panels, then those panels were optimized again.
• After the optimization, the critical buckling load of each stiffened panel was enhanced significantly. The X-Stiffener performed best with a critical buckling load of 2920 kN and the I stiffened panel showed the lowest critical buckling load of 813.79 kN which indicates that the X-stiffened panel increases the critical buckling load by 258.81% compared to the I stiffened panel and 19.9% over its pre-optimized counterpart.
• The Pearson correlation matrix suggests that base plate thickness has a strong influence on the critical buckling load of both the X and X-30 stiffened panels, with 0.97 and 0.98 for the panels, respectively. However, for the X-30 type, the lower part of the stiffener on the panel also shows a strong correlation of 0.93.

Future research could begin by exploring post-buckling optimization methods to achieve an optimal balance between strength, weight, damage tolerance, and manufacturability in advanced stiffened panel designs. Additionally, comprehensive analysis under different boundary conditions, such as varying load cases, support configurations, and constraint scenarios should be conducted to better understand the structural behavior and validate the optimization framework across diverse applications. Furthermore, experimental studies are essential to verify the numerical predictions and to assess the real-world performance of optimized stiffened panel configurations, thereby bridging the gap between computational models and practical implementation.