Comparison of Different Criteria and Analytical Models for the Analysis of Composite Cylinders Assisted by Online Software

Abstract

Due to their higher strength-to-weight ratio and ability to operate in harsh environments, the usage of fiber-reinforced cylindrical shells experienced a significant increase in the past decades. The key novelty of this study lies in implementing dual analytical approaches to address the complex failure mechanisms and stress distributions in composites. Two distinct theoretical solutions were investigated, membrane theory and Mindlin–Reissner theory, for failure prediction in filament-wound structures, while uniquely providing a platform for easy comparison of theoretical approaches. Experimental data from different setups, materials, and winding angles were collected in the literature and compared using the developed online MECH-Gcomp software. Failure analysis was also carried out by applying five different failure criteria well-established for composite materials. The results from the Mindlin–Reissner theory showed 46.9% deviation and those for the membrane theory 36.2% deviation, considering more than 120 cases. Sobol sensitivity analysis identified pressure (P), transverse tensile strength, winding angle, and radius as the most influential parameters regarding the index of failure of composite cylinders.

1. Introduction

Unlike traditional isotropic materials, fiber-reinforced composites exhibit complex stress distributions and failure mechanisms due to their anisotropic nature and multi-layered configuration [1]. Predicting the failure of such structures is a challenging task that has been studied for decades, starting from adapting well-consolidated criteria already applied to isotropic materials [2,3,4]. Newer analytical models were developed considering extra features, key mechanical aspects, and other characteristics of composite materials, e.g., by computing distinguished failure modes for fiber fracture and matrix cracking [5,6,7], by incorporating material nonlinearities, especially regarding shear strains [8,9], improving synergistic relations between stresses [10], differentiating a lamina strength from its in situ strength [11], and accounting for post-failure behavior [12] and fiber kinking [13]. However, advanced criteria commonly require more parameters, leading to more material characterization and experimental tests. In addition, micromechanical-based models assisted by numerical methods, such as the finite element method (FEM), are also applied for composite failure prediction [14,15]. These models can achieve higher accuracy by incorporating interface properties, delamination, damage mechanics, nonlinear material models, fiber waviness, and accounting for fiber packing arrangements [16,17,18].

Composite pressure vessels have gained significant attention in recent decades due to their superior strength-to-weight ratio, corrosion resistance, and adaptability to harsh environments, making them a preferred choice over traditional metallic counterparts [19,20]. These vessels are widely used in aerospace, automotive, energy storage, and chemical industries, particularly for applications such as hydrogen storage, gas transport, and pressurized fluid containment [21,22]. The filament winding (FW) process is the most common manufacturing method for these vessels, allowing precise fiber orientation to optimize mechanical performance [23]. However, given the complex behavior of this class of material and due to the presence of both hoop, axial, and radial stresses in pressure vessels, predicting its failure remains challenging [2,22,24]. In The World-Wide Failure Exercise I (WWFEI) [25] and World-Wide Failure Exercise II (WWFEII) [26], several models, with distinct approaches, assumptions, and degrees of complexity, were tested regarding their capabilities on predicting failure in composite laminates submitted to 2D (WWFEI) and 3D (WWFEII, 14 years later) loading conditions. Even with accurate input from both matrix and fiber individual properties and lamina properties, approximately one third of the predictions yielded errors above 50% in WWFEI and WWFEII.

The analysis of composite pressure vessels has been historically based on Classical Lamination Theory (CLT), which assumes Kirchhoff–Love hypotheses and neglects transverse shear effects [27,28]. While CLT is effective for thin laminates, its limitations arise in thicker structures where shear deformations become significant. To address this, the First-Order Shear Deformation Theory (FSDT) introduces a constant transverse shear strain but requires correction factors for accuracy [29,30]. Higher-order shear deformation theories (HSDTs), such as Reddy’s Third-Order Theory (TSDT) [31,32], eliminate correction factors and provide more accurate shear strain distributions. Other approaches, like Touratier’s theory [33], use trigonometric and hyperbolic functions to enhance precision in thick. The Carrera Unified Formulation (CUF) further extends modeling capabilities by enabling flexible displacement field representations [34]. However, the increasing complexity in the analysis with advanced theories, sometimes demanding additional composite characterization, makes them more useful for engineering applications with a strict degree of structural responsibility and less so for more ordinary cases or general educational purposes.

The use of software plays a crucial role in modern education, enhancing learning experiences across various disciplines. These tools provide interactive and engaging ways to grasp complex concepts, offering simulations, data visualization, and automated feedback that traditional methods cannot [35,36]. They also foster self-paced learning, allowing students to explore topics at their own speed while reinforcing their understanding through hands-on practice. From virtual laboratories to AI-powered tutors, educational software bridges the gap between theory and practice, making learning more accessible and effective [37]. Moreover, software tools support educators by streamlining assessments, personalizing content, and improving collaboration among students, ultimately enhancing the overall learning environment. For undergraduate students, software tools are particularly valuable in helping them transition from foundational knowledge to more applied learning [38].

For engineers, proficiency in industry-standard software is essential, as modern engineering relies heavily on digital tools for design, analysis, and manufacturing. In the field of composite materials, some book authors [1,39,40] developed their own software tool to provide an interactive learning experience that reinforces theoretical knowledge through practical application. By integrating graphical resources and pre-implemented equations, the tools allow students and researchers to visualize complex mechanical behavior, such as stress distribution, deformation, and material responses, which are often challenging to grasp with plain text only. Moreover, a dedicated software tool allows students to verify their knowledge by readily testing different scenarios and solving problems without manually performing extensive calculations. This not only saves time but also helps them focus on conceptual understanding rather than tedious calculations.

In this context, this study uses an online software tool for the analysis of composite structures. The software MECH-Gcomp (www.ufrgs.br/mechg, accessed on 22 February 2025) includes modules for micromechanics, ply mechanics, laminates, and sandwich structures, and more recently a new module dedicated to cylinders and pressure vessels was implemented, which was used to investigate the accuracy of classical failure theories (maximum stress, maximum strain, Azzi-Tsai, Tsai-Wu, and Hashin) for such structures. Failure data from the literature, with different loading conditions, materials, and experimental setups, are applied for comparison purposes. Stresses in individual laminae are computed from the loading conditions using two different approaches, either based on the CLT, where radial stresses are neglected [28], or focusing on thick-wall cylinders, accounting for radial stresses [29]. Both approaches were incorporated into the MECH-Gcomp software, along with the abovementioned failure criteria (FC). Additionally, a global sensitivity analysis was carried out to quantify the influence of input parameters on the structural response of composite cylinders, specifically evaluating the index of failure (IF).

The key novelty of this study lies in implementing dual analytical approaches to address the complex failure mechanisms and stress distributions in composites. Two distinct theoretical solutions were investigated—membrane theory and Mindlin–Reissner theory—for failure prediction in filament-wound structures. In addition, a new software module for composite cylinders and pressure vessels was built into the MECH-Gcomp platform to enable systematic evaluation of classical failure criteria under different loading conditions. Considering the wide range of materials, geometries, and load cases analyzed, the module also offers valuable insights into the reliability of the models. Moreover, a global sensitivity analysis was carried out to identify the critical parameters influencing the strength of those structures, providing meaningful guidance for design optimization.

2. Materials and Methods

2.1. Modeling Composite Cylinders and Pressure Vessels Using MECH-Gcomp

MECH-Gcomp is an online software tool developed using Django [41], a Python framework (Python 2.7) that structures web development efficiently by separating creation steps. Key benefits include seamless front-end and back-end integration, flexible data handling, and a structured development process [42,43]. It enables the creation of multiple pages within a website, with each sector treated as a separate module, making maintenance and expansion easier. Following Django’s Model-View-Template (MVT) structure, a variant of Model-View-Controller (MVC), the software ensures clear separation of concerns. The Model handles data storage and management, the View processes user requests and manages logic, and the Template defines the user interface by integrating static elements with processed data. This architecture enhances maintainability and scalability.

The front end is built using JavaScript [44], HTML, and CSS. JavaScript, the most widely used web language, enables large applications with simple scripts and clear structure but has debugging and security challenges [45,46]. It manages user interactions, while HTML and CSS define and enhance interface elements. Users navigate through different software modules, input data, generate plots and reports, and perform interactive actions. Saved material properties are linked to user accounts and managed using MySQL [47] (MySQL 1.3.9), a fast, reliable, and widely compatible database system [48,49]. Once data are collected, Python [50] handles back-end computations, including stress–strain analysis, safety factors, and failure envelopes. While Python is slower than C or C++, its open-source libraries (e.g., NumPy and SciPy), offer high readability and efficiency, making it widely adopted in research [51,52]. Python performs back-end computations, handling stress analysis, failure envelopes, and other mechanical evaluations. Though slower than C or C++, Python scientific libraries have made it widely adopted in research [53].

The software has been developed by students of the Composite and Nanocomposite Group (GCOMP) of the Federal University of Rio Grande do Sul (UFRGS) since 2012. Although most books on mechanics of composites focus on continuous reinforcements [1,39,54], micromechanics of composites reinforced with hybrid fibers, short fibers, particles, and nanofillers are included first in the software. Other modules were included later for ply mechanics and sandwich structures, and, more recently, cylinders and pressure vessels [55,56].

To analyze cylinders and pressure vessels, the first step consists in determining the properties of the laminae to be used in the vessel. This could be performed using micromechanics by informing the values of the properties of the individual constituents (fiber, matrix, and their volume fraction) or directly inserting them in the “Laminae” page. The next step is to define the material, thickness, and fiber orientation angle of each ply, along with the stacking sequence, which is performed in the “Laminate” page.

The “Pressure Vessels” module, depicted in Figure 1, is used to select the laminate in the “Material” section and to inform the internal radius of the structure in the “Geometry” section. It is also necessary to select between “thin walls” or “thick walls”, since the software applies distinct algorithms for each option, bearing in mind that some authors [28] consider thin shell geometries those with the R/t ratio higher than 10, where R is the radius taken at the mid-surface and t is the wall thickness, whereas other authors [57] adopt a more conservative approach, classifying as thin walls R/t ratios above 20.

The algorithm implemented in MECH-Gcomp for thin walls is based on the study by Li et al. [28], where the Classic Laminate Theory (CLT) [58], based on Kirchhoff–Love assumptions, is applied to retrieve strains and curvatures of each ply individually. The authors coupled internal pressure (P), external pressure (−P), torque (T), axial loads (F), hygrothermal loads (Part 1), bending moments (Part 2), and shear forces (Part 3). After computing the strains from each of the three parts, the superposition law is applied to compute the total strain. Good agreement in comparison with FEM models was reported. The applied hypotheses include sections to remain plane and perpendicular to the neutral axis, constant through-thickness stresses, negligible shear from Parts 1 and 2, linear elasticity, and negligible out-of-plane stresses [28,59].

For thick walls, the algorithm is based on the Fronk’s model [29], where Mindlin–Reissner shell theory is applied, introducing a first-order shear term into the modeling, allowing for transverse shear deformation. Therefore, the constraints of constant through-thickness stress and negligible radial stresses, used in the previous model, are disregarded [59]. However, this model requires axisymmetric loads, so only the loads depicted in Part 1 can be applied (i.e., internal pressure, external pressure, axial, and hygrothermal loads). Besides, using the full constitutive matrix for engineering constants, instead of the compacted one applied by Li’s model, the assumption of a transversally isotropic material stands, i.e., among the three mutually perpendicular planes of symmetry, there must also be an axis of symmetry (e.g., the axis in the fiber direction). In practical terms, the constitutive matrix is built using the following considerations [39]:

$$E_2=E_3;G_{12}=G_{13};{\nu }_{12}={\nu }_{13};{\nu }_{23}={\displaystyle \frac{E_3}{G_{23}}}-1$$

(1)

where E, G, and ν are the elastic modulus, shear modulus, and Poisson’s ratio, respectively. The axis 1 is parallel to the fiber direction, while 2 and 3 are transverse to the fiber in-plane and out-of-plane, respectively.

Membrane theory offers computational efficiency and simplicity, making it ideal for preliminary designs and educational purposes where thin-walled structures are involved. However, it neglects transverse shear deformation and through-thickness stresses, which can lead to underestimation of deformations and stresses in thicker laminates [28]. In contrast, Mindlin–Reissner theory accounts for shear deformation and provides more accurate results for thick-walled vessels, but at the cost of increased computational complexity and the requirement for additional material parameters [29]. The theory is also limited to axisymmetric loading conditions. For practical engineering applications, membrane theory often suffices for thin vessels (radius-to-thickness ratio > 10), while Mindlin–Reissner becomes necessary for accurate analysis of thicker structures or when shear effects are significant [29].

The structural analysis of pressure vessels must consider the boundary conditions, specifically whether the component has closed or open ends. Although stress calculations are performed exclusively on the cylindrical walls and not on the domes, closed-end conditions must be accounted for, as internal pressure induces additional axial stresses.

The analysis takes into account section properties, namely, total thickness, cross-sectional area, linear density, moment of inertia, radius of gyration, longitudinal elastic modulus, stiffnesses, and the change in both inner and outer radii. Then, it evaluates stresses and strains under applied loads, determining the IF for the critical ply based on the selected failure criterion.

Additional results of local and global stresses and strains on both bottom and top coordinates of each ply can also be shown. The local coordinate system adopted to display stresses (σ) and strains (ε) considers the fiber in direction 1, with directions 2 and 3 transverse to it, in-plane and out-of-plane, respectively, as shown in Figure 2. For the global coordinate system, the axis of the cylinder/vessel is x, the hoop direction is θ, and the radial direction is r. These results can be plotted as a function of the radial coordinate (where 0 mm is the center of the radius) for easier readability.

2.2. Validation of Results

Experimental data from the literature for composite vessels/cylinders manufactured by FW and loaded until failure were adopted as reference for the validation. The material and geometrical inputs applied by the authors were used in MECH-Gcomp, where both global and local stresses were retrieved using the two algorithms mentioned above. From the developed stress fields, the failure predictions were performed according to five different FC implemented in the software: maximum stress, maximum strain, Tsai-Wu [3], Azzi-Tsai [4], and Hashin [5]. The computed IF values were compared with experimental data from studies described below:

• Al-Khalil [60] analyzed glass fiber-reinforced polymer (GFRP) cylinders submitted to biaxial testing, incorporating internal pressure and axial compression. Different ratios of pressure and compression were applied, enabling the plotting of a failure envelope for their cylinders.
• Gargiulo et al. [2] manufactured 60 cylinders in carbon fiber-reinforced polymer (CFRP) with three different winding angles (WAs). They were tested under axial load, both internal and external pressure, and hybrid loading conditions. To simulate actual applications, the ends were closed during the tests.
• Guess [61] investigated FW cylinders using quasi-isotropic stacking sequences for CFRP. A combination of axial load and internal pressure was applied. Lamina properties were measured by the author using a different resin but the same reinforcing fiber. Strength properties not informed by the authors were taken from Tsai and Hahn [3].
• Madrid et al. [19] applied internal pressure in FW cylinders with open ends. Therefore, no axial loads were applied. Six different WAs were tested, and their failure modes were compared using Digital Image Correlation. The properties of the CFRP were taken from Scherer [62].
• Swanson et al. [20] investigated the mechanical behavior of CFRP cylinders using hoop layers (WA = 88.86°) submitted to axial and torsion loads. Longitudinal strengths and engineering constants were not informed. Thus, CFRP properties from Tsai and Hahn [3] were used for the missing data.

The material and geometrical data applied in each comparison are mentioned in Appendix A. The engineering constant G₂₃ was not provided by any author, so it was assumed equal to G₁₂. Since the authors reported only the total thickness, ply thicknesses were considered the same in each case. Since ultimate strains were not reported, the maximum strain criterion used values directly calculated from ultimate strengths, assuming linear elasticity [39]. The out-of-plane shear strength S₂₃, required by the Hashin criterion, was also not informed, and the following approximation was adopted [12]:

$$S_{23}=0.5\times Y_C$$

(2)

where Y_C is the transverse compression strength. The interaction term F₁₂ in the Tsai-Wu criterion was computed according to [63]:

$$F_{12}={\displaystyle \frac{ - 1}{2X_T{X}_C}}$$

(3)

where X_T and X_C are the longitudinal tensile and compressive strengths, respectively.

2.3. Sensitivity Analysis

A global sensitivity analysis was conducted to quantify the influence of the input parameters on the structural response of composite cylinders, based on the evaluation of the IF from the maximum stress criterion. This allows identifying the most impactful parameters, enabling dimensional reduction and structural optimization without compromising accuracy. This also enhances computational efficiency and simulation reliability, ensuring that variations in critical parameters are properly controlled. Furthermore, identification of parameters with minimal influence enables model simplification, reducing complexity while maintaining response accuracy. This analysis ensures a more efficient design aligned with safety requirements for composite cylinders or pressure vessels.

The stress calculations in the laminate layers were performed analytically using the thin-walled theory to determine the stress distribution across the composite layers. The sensitivity analysis methodology is based on the Sobol indices [64], which decompose the variance of the response function into individual and interactive contributions of the input parameters. These indices are obtained through Monte Carlo simulations, where the total variance of the response is distributed among the input variables via high-dimensional integrations.

The first-order Sobol indices (S_i) represent the fraction of variance due to each variable independently, while the total-order indices (S_Ti) account for both direct effects and interactions between variables. The S_i captures the direct influence of a single input variable, X_i, on the output IF, being defined as follows:

$$S_i={\displaystyle \frac{V_i}{V\left(Y\right)}}$$

(4)

where $V_i$ is the partial variance caused by X_i, and $V\left(Y\right)$ is the total variance of the output response. A high value of S_i indicates that the input X_i has a significant impact on the IF.

In contrast, the S_Ti accounts for both the direct effect of X_i and its interaction with other input variables, being given by the following equation:

$$S_{Ti}=1 - {\displaystyle \frac{V_{ - i}}{V\left(Y\right)}}$$

(5)

where $V_{-i}$ is the variance excluding the input X_i. The S_Ti reveals the overall importance of X_i in influencing the IF, including any synergistic effects with other parameters. This methodology allows for a comprehensive understanding of how each input parameter and its interactions influence the IF of the composite laminate structure.

Thirteen input variables were considered, each varying within a predefined range to represent real operating conditions. The longitudinal and transverse elastic moduli, E₁ and E₂, varied within [50, 125] GPa and [3.0, 10.0] GPa, respectively. The shear modulus G₁₂ ranged from [2.0, 8.0] GPa. The tensile and compressive strengths along the fiber direction, X_T and X_C, varied within [800, 2300] MPa and [200, 1500] MPa, respectively, whereas the transverse tensile strength, Y_T, and shear strength, S, varied between [10, 40] MPa and [20, 110] MPa, respectively. The ply thickness t varied within [0.2, 0.4] mm, and WA varied within [12°, 88°]. The internal pressure, P, ranged from [0, 100] MPa, the axial load F from [0, 100] kN, and the torque, T, from [0, 1.5] kNm. The radius of the cylinder, r, varied within [10, 100] mm. The number of plies was kept constant at 14.

The sampling process was carried out using the Monte Carlo technique with n_samples = 100, leading to a total of 2800 simulations, following the expression (2d + 2) × n_samples, where d = 13 represents the number of input variables.

3. Results and Discussion

The outputs from the software, along with the loading conditions applied, are depicted in Appendix B for the five papers mentioned above. Since the authors applied different experimental setups, different materials, WA, and loading conditions, they are separately discussed at first.

Figure 3 and Figure 4 show the IF values for Al-Khalil et al. [60] comparisons using thin and thick walls, respectively. A very good agreement (13% deviation) was achieved by the FC when only internal pressure was applied. This good agreement can be attributed to the predominance of hoop stresses, which are well captured by the FC under internal pressure loading. A high WA resulted in negligible stresses in direction 2 (transverse to the fibers), as the fibers were primarily aligned to resist the hoop stresses. In contrast, shear stresses played a significant role due to the material’s inherently low shear strength, highlighting the importance of considering shear effects in failure predictions.

The criteria that considered the synergistic effect between axial tension and shear (Azzi-Tsai and Hashin) overestimated the failure, while the criteria that neglected this synergy (maximum strain and maximum stress) underestimated it. When axial compression was applied combined with internal pressure, σ₂ became relevant. The average deviation increased to 24%, and only maximum stress and Hashin could keep a good accuracy. Given the high R/t ratio (around 40), predictions for thin walls and thick walls were very close. This is because, for such high R/t ratios, the stress distribution across the wall thickness becomes nearly uniform, reducing the influence of wall thickness on failure predictions.

The authors reported a reasonable agreement in comparison to the Tsai-Wu criterion, which has shown the worst performance among the five criteria investigated here. The radial stress computed through Fronk’s model [29] was significantly lower (around 50%) than the approximation of σ_r = −P adopted in [60]. This discrepancy highlights the limitations of simplified assumptions in capturing the true stress state, particularly in thin-walled structures where radial stresses are often neglected. The use of more accurate models, such as Fronk’s, can provide better insights into stress distribution and improve the reliability of failure predictions.

In Gargiulo et al. [2], none of the five criteria could achieve a reasonable agreement. In some cases, Azzi-Tsai and Tsai-Wu yielded negative IFs (shown as 0.00 in the results). The authors [2] stated that the use of nonlinear models was imperative to correct them, citing the nonlinear model for shear behavior proposed by Tsai and Hahn [3]. In this study, the predictions of the five criteria underestimated the IF values, yielding an average deviation, for all load cases and criteria, of 57.4%, where the best predictions were obtained by maximum stress and maximum strain, both with 42%. The authors tested three different WAs (35°, 55°, and 75°), all of them showing similar deviations, despite the different stress fields. This suggests that the failure mechanisms were not significantly influenced by the fiber orientation alone but were instead dominated by other factors, such as the type of loading and material nonlinearities. The primary challenge arose in load cases involving external pressure, where the deviations were significantly higher compared to other loading conditions. External pressure introduces a complex stress state, including significant radial and hoop stresses, which are not adequately captured by linear failure criteria. This complexity is further intensified by the material nonlinear response to compression and shear, which linear models fail to represent accurately.

As for the data reported by Guess [61], only the cylinders manufactured from Thornel 75 S could be accurately predicted, with maximum strain criteria yielding 9% deviation. The high level of accuracy suggests that the maximum strain criterion is particularly well-suited for predicting failure in Thornel 75 S composites, possibly due to its ability to effectively capture the strain limits of the material under those conditions. However, the Azzi-Tsai criterion was an exception, generating spurious IF values for the cases involving axial loading only. This inconsistency highlights a significant limitation of criteria that rely heavily on stress interactions, particularly when the stress state is dominated by a single loading component, such as pure axial loading.

Regarding the data from Madrid et al. [19], the observations are consistent with those for Gargiulo [2], with all five criteria significantly underestimating the IF. In this case, the low number of layers may also corroborate such results by inserting higher stress concentrations [65]. Unlike the other studies analyzed here, Madrid et al. [19] tested five samples per configuration, mitigating the dispersion effect. However, even with different failure modes, none of the WA yielded accurate predictions since the lower WA triggered failures associated with σ₂, higher angles with τ₁₂, and the 90° configuration with σ₁. Since X_T is significantly higher than Y_T for CFRP materials, experiments in unidirectional plates point to a rapid decrease in strength as the load varies with fiber orientation, and this decrement in strength is predicted by most FC [3,66,67]. The same behavior is expected for cylinders submitted to internal pressure only, with no axial load, when fibers are wound close to 90°. However, different values for optimal WA have been observed experimentally [19,68], and this behavior was not followed by the FC tested here. For the 90° configuration, the values for σ₁ at maximum pressure were around 470 MPa, with negligible values for σ₂ and τ₁₂. Given that X_T = 1374 MPa, it is unlikely that any FC could predict such failure.

As for the experimental data of Swanson et al. [20], only Tsai-Wu and maximum stress showed reasonable agreement, with the other criteria yielding deviations above 30%. The direct torque in a cylinder with only hoop layers made the shear stress higher than in the previous cases, and the general trend was overestimating the IF instead of underestimating it. The accuracy of the FC increased as the torque decreased and the axial load increased for both axial tension and compression. It is important to highlight that X_T and X_C were not provided by those authors, but obtained from a third study [3].

Based on the analysis of the deviations from those five authors, it may be implied that there is room for improvement in the failure prediction of FW cylindrical structures and that accuracy is related to the loading conditions. Since all components analyzed had an R/t ratio above 20, no significant difference between thin and thick wall models was observed or expected according to the ratio specified by Ventsel and Krauthammer [57]. Inaccurate material properties lead to poor predictions in some cases. For the Tsai-Wu and Azzi-Tsai criteria, for instance, X_T and Y_T are considered, even for load cases where σ₁ < 0 and σ₂ < 0.

Additionally, most of the analyzed data refers to a single sample per configuration, bringing repeatability concerns due to the inherent complexities of the experimental setup for pressurizing composite cylinders. Variations in material properties, manufacturing inconsistencies, and the sensitivity of failure mechanisms to small deviations in fiber orientation contribute to this dispersion. Anyway, the software estimates are a reliable approximation of the experimental data under the studied conditions. The average deviations observed were 36.2% for the membrane theory and 46.9% for Mindlin–Reissner. These values are consistent with expected uncertainties in composite failure predictions [26] and highlight the need for refined material characterization and failure criteria to further improve accuracy.

Figure 5 presents the results of the global sensitivity analysis based on the Sobol index, used to quantify the influence of input parameters on the response of the composite cylinders based on the IF computed using the maximum stress criterion. The figure displays both the first-order Sobol index (S_i) and the total-order index (S_Ti), allowing the assessment of individual effects and interactions among variables. An S_i value close to 0 means little influence on the IF, while an S_i close to 1 indicates a strong influence. Similarly, an S_Ti close to 0 means that the parameter has little impact even when considering interactions.

Among the analyzed parameters, P, Y_T, WA, and r exhibited the highest S_i values, indicating a direct and significant impact on the IF. In contrast, E₁, X_T, X_C, T, and E₂ have low S_i values, meaning they exert little direct influence on failure. In particular, X_T and X_C show low sensitivity because, within the analyzed parameter ranges, the maximum stress criterion almost always activates failure in transverse tension rather than the fiber direction. Consequently, variation in these strengths has a limited effect on IF.

Regarding S_Ti, P and r are the most influential parameters, with values greater than 0.4. This indicates that their influence on failure is significantly enhanced when accounting for interactions with other parameters. The WA still shows high S_Ti, emphasizing its dominant role in the structural response. The G₁₂ and S, with moderate S_i values, exhibit relatively higher S_Ti values, suggesting greater influence through interactions. The t follows a similar trend, with an intermediate S_Ti. Moreover, X_T, X_C, E₁, T, and E₂ display low S_Ti values, confirming a limited impact even when interactions are considered.

These results highlight the nonlinear and interdependent nature of the structural response of composite cylinders, with high interaction effects observed for r, WA, and P. The analysis also highlights the crucial role of Y_T, as failure is predominantly governed by transverse tensile strength rather than fiber-direction failure. These findings suggest that dimensional reduction may be achieved by disregarding parameters with low S_Ti, optimizing computational efforts without significant accuracy loss. Furthermore, design optimization should prioritize P, WA, r, and Y_T, as these parameters most strongly affect failure.

Figure 6 presents the correlation matrix between the input parameters and the IF, providing deeper statistical insight into the relationships between material properties, geometric parameters, and structural failure. The correlation coefficients help identify which variables have the most significant impact on failure. A key observation is the correlation between P and IF, which is 0.45, indicating a moderate positive relationship. Thus, as expected, increasing P leads to a proportional rise in IF, meaning higher internal loads contribute significantly to structural failure. The Y_T and IF show a negative correlation of −0.30, suggesting that an increase in Y_T helps reduce the IF. This aligns with the fact that failure in composite cylinders is often governed by transverse tensile stresses, and higher transverse tensile strength delays failure.

The parameter t has a correlation of −0.14 with IF, indicating a weak negative relationship. While not as significant as other parameters, this suggests that increasing thickness slightly reduces the IF by enhancing structural stiffness and load-bearing ability. However, the relatively low correlation value implies that thickness alone is not a dominant factor in failure. The R/t ratio and IF exhibit a correlation of 0.39, highlighting its role in influencing structural behavior. A higher R/t typically means a thinner-walled structure, which correlates with an increased IF, emphasizing the need to carefully design this ratio to optimize strength and weight.

The WA shows a weak negative correlation of −0.21 with IF, indicating that variation in fiber orientation has some influence on failure, but not as dominantly as pressure or transverse tensile strength. This suggests that while WA affects structural performance, its impact on failure is not as direct as those of P or R/t. Regarding material properties, the X_T has a correlation of −0.10 with IF, indicating that increasing X_T slightly reduces the IF. This supports the idea that failure is not primarily driven by fiber-directional loading but rather by transverse and shear stresses. Finally, E₁ and IF show a correlation close to zero, confirming that the longitudinal fiber modulus has little direct influence on failure.

4. Conclusions

A module dedicated to the failure analysis of cylinders and pressure vessels of composite materials was implemented in online software. It allows for a quick and easy way to compute the section properties, stiffness, and stresses and strain in all layers based on two analytical models. Tools for the graphical visualization of stresses and strains were also implemented, and the software applies different failure criteria from the literature to evaluate the index of failure of the structure.

The comparison between the software predictions and five experimental campaigns revealed that the accuracy of failure criteria (Azzi-Tsai, Hashin, Tsai-Wu, maximum strain, and maximum stress) varies significantly depending on the loading scenario. While some criteria showed good agreement in specific cases, others demonstrated limitations, reinforcing the importance of using multiple failure models for a more comprehensive analysis. Analysis of over 120 cases shows reasonable approximation, although still influenced by the inherent variability in material property and composite testing.

The global sensitivity analysis using the Sobol index highlighted key parameters influencing failure of the composite cylindrical sections, with pressure, transverse tensile strength, winding angle, and radius of the cylinder showing significant impacts. In contrast, parameters like longitudinal tensile strength and elastic moduli showed minimal influence. The analysis also revealed that failure is mainly governed by transverse tensile strength. Correlation results confirmed the positive correlation between pressure and IF and the negative correlation between transverse tensile strength and IF, further emphasizing their critical role. These findings suggest that dimensional reduction is feasible by neglecting parameters with small influence, optimizing computational efforts without sacrificing accuracy.

Therefore, numerical models are required for more accurate predictions of the mechanical behavior of composite pressure vessels/cylinders. Even so, the MECH-Gcomp software offers a readily available analytical approach for teaching purposes, parameter investigation, qualitative analysis, and preliminary design stages. Its robust analytical framework, user-friendly interface, and ability to handle various loading cases and failure mechanisms make it a trustworthy resource for academic and industrial professionals interested in the design of composite cylinders and pressure vessels.