Reliability and sensitivity assessment of laminated composite plates with high-dimensional uncertainty variables using active learning-based ensemble metamodels

2.1 Maximum stress criterion

The maximum stress criterion is one of the primary failure criteria used to assess the strength of composite materials. This criterion is based on the concept that failure occurs when a material exceeds its maximum stress-carrying capacity in one of its directions. In other words, if any of the principal stresses applied to a composite material exceeds its corresponding strength, failure occurs.

To apply this criterion, the principal stresses computed within the structure are compared with the material’s maximum strengths in each direction. Failure happens when one of the following inequalities is not satisfied:

where σ ₁ , σ ₂, and σ ₃ represent the principal components of normal stresses, and σ ₄ , σ ₅, and σ ₆ represent the principal components of shear stress, while X _t, Y _t, Z _t, X _c, Y _c, and Z _c correspond to the ultimate tensile and compressive strengths of the lamina in the 1, 2, and 3 directions, and R, S, and T denote the ultimate shear strengths of the lamina in the 2–3, 1–3, and 1–2 planes, respectively.

For a 2D in-plane stress state (σ ₃ = 0, σ ₄ = 0, σ ₅ = 0), which is often the case for laminates, the maximum stress failure criterion is expressed as follows:

2.2 Tsai-Hill criterion

The Tsai-Hill failure criterion is based on the concept of distortion energy [50]. It is an adaptation of the Von Mises criterion, initially designed for isotropic materials, but it has been specifically tailored for anisotropic materials like composites. This failure criterion assesses the strength of a composite material by considering the intricate interactions between stresses and strains that occur in an anisotropic material.

The failure condition according to the Tsai-Hill criterion is defined as follows:

where the values of X, Y, and Z represent X _t, Y _t, Z _t or X _c, Y _c, and Z _c depending on the sign of σ ₁, σ ₂, and σ ₃.

For a 2D in-plane stress state (σ ₃ = 0, σ ₄ = 0, σ ₅ = 0), the Tsai-Hill failure criterion is expressed as follows:

2.3 Tsai-Wu criterion

The Tsai-Wu criterion is a widely used failure model for assessing the safety of composite materials. It is based on Beltrami’s theory of failure by total strain energy [51]. According to this criterion, a lamina or composite material is considered to fail if the following inequality is not satisfied:

where

For a 2D in-plane stress state (σ ₃ = 0, σ ₄ = 0, σ ₅ = 0), the Tsai-Wu failure criterion is expressed as follows:

3.1 Failure probability

The failure probability of structures is defined through the LSF G(x). This probability is defined as the percentage of the region in which the LSF G(x) is less than zero. It is calculated using the following integral:

where f x represents the joint density of the input random variables.

The failure probability can be expressed as follows:

where E represents the expectation function with respect to f x , and I Df is the indicator function, which is defined as follows:

The integral in Eq. (8) is complex to solve directly. For this reason, the MCS is employed to estimate this integral. This method involves generating a population, denoted as P, which is then used to simulate the system N times. The number of failure events is counted, and therefore, the probability of failure can be estimated by the following expression:

Furthermore, to obtain a reliable degree of precision for this result, it is necessary to calculate the coefficient of variation (CoV) of P ˆ f , defined by the following relationship:

In most cases, a precision level of CoV < 0.05 is commonly employed, as indicated by researchers [52,53].

3.2 Local sensitivity of failure probability

The primary objective of local sensitivity analysis of failure probability is to measure the local impacts of the distribution parameters of input random variables on the failure probability. It has been employed in the context of probability by the researcher [48]. To achieve this, the gradient of the failure probability with respect to these variables is evaluated:

where θ i represents the parameter of the probability distribution function of the ith random variable.

Furthermore, to ensure a fair comparison of the sensitivities of all random variables in the studied system, it is necessary to standardize these sensitivities with respect to the probability of failure, making them dimensionless. A normalization method proposed by Wu and Mohanty [54] is expressed as follows:

where σ i represents the standard deviation of the probability distribution function of the ith random variable.

This sensitivity can be estimated using the MCS method:

If the input variables are independent, the joint density can be decomposed into f x = ∏ i = 1 n f x i . Thus, this sensitivity can be expressed as follows:

According to Wu [48], two types of sensitivity can be defined, the first one being sensitivity of the probability of failure with respect to the mean μ, and the second one with respect to the standard deviation σ:

4.1 LSF incorporating failure criteria

In the context of reliability analysis theory, failure is always described using an LSF, denoted as G(X). This function establishes a relationship between the limit state of the structure and various factors, including applied loads, material properties, and other relevant parameters. In this research, to ensure the reliability and safety of composites, and since none of the failure criteria guarantee the most conservative results in all conditions according to Martinez and Bishay [55], it is assumed that failure occurs when one of the three failure criteria is satisfied. To this end, an LSF based on the failure criteria presented in the previous section has been defined as follows:

where X = {x ₁ , x ₂,…, x _n } represents the vector of input random variables, which typically include loads and material properties, and the failure occurs when the LSF has a negative value, G(X) ≤ 0.

4.2 Construction of a metamodel combining three metamodels

To construct a consolidated average metamodel from multiple metamodels, the commonly used method is the metamodel ensemble approach, which has been recently incorporated into structural reliability analysis by researchers [56,57]. This approach involves training multiple metamodels using the same sample points and then combining them to obtain a more accurate prediction than what each individual model could provide separately. The combination of these metamodels is achieved by determining the weights associated with each of them.

In the literature, there are two primary strategies for determining these weights. First, there is global weighting, where the weights w _i are constant for each metamodel ŷ _i . This includes methods such as weight calculation based on root-mean-square error (RMSE) [58], Bayesian model averaging [59], or determining weights using an optimization problem [42]. Second, there is local weighting, where the weights depend on input values X. An example of this is variance-based local weighting [60].

In this article, the method of weight calculation based on RMSE [58] is employed. This method suggests the calculation of the average metamodel using the following formula:

where Ŷ _ens(x) represents the global prediction of the EM at given x, ŷ _i represents the output of the ith meta-model at the specific point x, M stands for the total number of surrogates employed in the EM, and w _i corresponds to the weight assigned to the ith meta-model.

Moreover, this approach enables the detection of regions where substantial prediction errors may occur by computing the variance V Y ˆ ens ( x ) of the EM, defined as follows:

The weights are computed using a heuristic formulation that employs a “leave-one-out” cross-validation strategy to determine the RMSE. The weight calculation is as follows:

where Y ^(k) represents the real response at a given point x ^(k), Ŷ ^(k) represents the predicted response obtained from the ith surrogate model trained using all points of the design of experiments (DoEs) except the pair (x ^(k), y ^(k)), and N _DoE is the total number of DoE, and α and β are two parameters that need to be defined. In this article, the same parameter values as those used by Goel et al. [58] have been applied, α = 0.05 and β = −1.

4.3 U _EM learning function

The accuracy of determining the failure probability largely depends on the precise classification of points near the system’s state limit, where Y(x) = 0. To ensure this precision, Echard et al. [53] implemented the U-learning function, which was developed in the context of the AK-MCS algorithm. This function identifies points that may potentially cross the predicted separator of the metamodel Ŷ(x) = 0 incorrectly.

In this article, an adaptation of this LSF has been introduced. It involves the introduction of a minimum distance between the studied point and the points already present in the design of experiments. This adaptation aims to achieve a balanced distribution of all samples in the space, preventing excessive point concentration in a single area and ensuring a uniform distribution. This learning function is defined as follows.

where Ŷ _ens(x) represents the global prediction of the EM at given x, V Y ˆ ens ( x ) is the variance of the metamodel in x, and d min ( x ) is the minimal distance, calculated using the Euclidean norm, between the point x and the nearest point in the DoE.

4.4 Stopping criterion for the algorithm

The stopping criterion adopted in this method is based on the evolution of the failure probability values over the last five iterations and is proposed as follows:

where i represents the value of the last iteration and P f i − k − 1 and P f i − k represent the failure probabilities for iterations (i−k−1) and (i−k), respectively.

4.5 Calculation of failure probability and its sensitivity

The U _EM learning function is computed for all points in the population. The point with the lowest value is then incorporated into the DoE. The best next point, denoted as X*, is then determined by:

At each iterative step, X* is identified and incorporated into the DoE. Subsequently, metamodels are refined using this new DoE, and the failure probability is calculated at each iteration according to the following expression:

Then, the local sensitivity of the failure probability is calculated according to the following expression:

5.1 ANN

ANNs, a subset of machine learning, form the cornerstone of deep learning algorithms. They draw their name and structure from the human brain, aiming to simulate how the human brain processes information. These networks consist of layers of nodes, including an input layer, one or more hidden layers, and an output layer. Each node is interconnected with others and is assigned an associated weight (W _i ), as depicted in Figure 1.

Figure 1

ANNs architecture.

Each neuron transforms the input into an output using an activation function, which has the role of introducing a nonlinearity in the output of a neuron, and in this study, we use the sigmoid function as activation function, which is defined as follows:

At the level of each neuron, a weighted set of inputs from the previous layer is summed, and the activation function is applied to this sum as follows:

where, w ij l represents the weight of the connection between neuron j in layer l − 1 and neuron i in layer l , a j l − 1 is the activation of neuron j in layer l − 1, and b i l the bias term in neuron i in layer l .

Considering that the sigmoid function is sensitive in the interval [−1,1], we proceed to normalize all inputs and outputs (targets) as described by Rafiq et al. [61]:

For inputs:

where S is the normalized value of the variable X and X _min and X _max are the minimum and maximum values of the variables.

For targets:

where out is the normalized value of the targets T, and T _min and T _max are the minimum and maximum values, respectively, of the targets.

In this study, an ANN was employed, featuring three hidden layers, each comprising 20 nodes. To determine the problem’s weight values, the Levenberg-Marquardt (LM) algorithm implemented in MATLAB was utilized to address the nonlinear least squares problem.

5.2 SVR

SVR is an evolution of the SVM that expands its capabilities. It introduces the concept of an ε-insensitive tube, as illustrated in Figure 2. Within this tube, deviations from the target output are allowed without any penalties, while outside the tube, penalties are applied to deviations. SVR is fundamentally a machine learning algorithm [36] that relies on a loss function L _ε called e-insensitive function. This function is designed to handle cases where we are less concerned with small deviations within the ε tube but want to penalize larger deviations outside it, making SVR particularly useful for robust regression tasks. This loss function L _ε is defined as follows:

Figure 2

Curve of the SVR approximation function with the slack variables.

5.3 Kriging metamodel

The Kriging model, a nonlinear interpolation meta-model developed for Geostatistics by Matheron [63], also known as Gaussian process, is a statistical interpolation method employed to predict or estimate unknown values within a given space by utilizing observations made at various points within that space. This approach is based on establishing a stochastic random field that simulates the behavior of the limit function. Subsequently, the technique known as the best linear unbiased predictor (BLUP) is utilized to optimally estimate the value of this model at a specific point. The Kriging function is written as a realization of a random function as described by [64].

where F ( x ) = [ { f 1 ( x ) } , … , { f p ( x ) } ] represents the basic functions and β = { β 1 , … , β p } is the vector of regression coefficients, and in this article, ordinary Kriging is chosen, which means that F ( x ) is a vector of ones and the product F ( x ) T β = β is a scalar.

Z(x) is a stationary Gaussian process that has an unknown form, zero mean and the following covariance functions between two points of space:

where σ is the variance of the process and σ 2 R θ ( x i , x j ) is the autocorrelation function between the points x i and x j .

In this type of Kriging, the Gaussian process is stationary, which means that the autocorrelation function R depends only on the difference between the points and on a set of hyperparameters θ ∈ R + n , and this correlation model can be formulated as follows:

where i, j = 1,., n, with n representing the number of random variables, and N ₀ denoting the number of points in the design of experiments.

The identification of the hyperparameters β , σ , and θ can be determined using the maximum likelihood method defined as follows:

where R ( θ ) = R θ ( x i , x j ) is the matrix of correlation between each pair of points of the design of experiments, and it is a symmetric correlation matrix and 1 is the vector of size n and filled with 1.

Then, since maximizing this likelihood is equivalent to minimizing its opposite natural logarithm, the first-order optimality conditions for this likelihood logarithm is used to determine the two estimations of the parameters β and σ ².

These two equations lead to:

However, the two parameters in Eqs. (55) and (56) depend on the correlation parameter θ, so it is first necessary to obtain it using maximum likelihood estimation:

By expanding the optimization problem of the Eq. (57) and eliminating the constant terms, the likelihood function is reduced to:

After the determination of the parameter θ, the best linear unbiased predictor (BLUP) is used to determine G ˆ ( x ) the estimation of the response G(x) and to determine σ G ˆ 2 ( x ) the variance of G ˆ ( x ) at the point x.

where r ( x ) = { R θ ˆ ( x , x i ) } i = 1 , … , n

where u ( x ) = 1 t R ( θ ) − 1 r ( x ) − 1 and 1 is the vector of size n and filled with 1.

To build the Kriging model in this article and to solve the global optimization problem of Eq. (58), which is complex and cannot be solved analytically, an optimization algorithm is developed in MATLAB to solve this problem.

6.1 Stage 1: convergence analysis and numerical validation

This stage holds paramount significance within our methodology, aiming to validate the FE model and determine the optimal mesh to use. The first substage involves validating the obtained results by comparing them with analytically calculated results. Subsequently, the second substage aims to determine the optimal mesh size. It is well known that a finer mesh yields more precise results, albeit often at the cost of longer computational time. Therefore, a convergence study becomes necessary, employing various mesh sizes to ascertain the optimal size that provides both adequate accuracy and reasonable computation time.

Moreover, in the absence of precise analytical results, it is conceivable to assess the model’s validation and convergence by comparing it to other established finite element models based on different theories or concepts. Alternatively, running it within a different finite element software is also a viable approach.

6.2 Stage 2: reliability analysis

The objective of this stage is to achieve a precise estimation of failure. It includes the following steps:

1. Generate the Monte Carlo population S using Latin hypercube sampling (LHS) technique.

2. Select from S a number N ₀ (e.g., N ₀ = 12) of points X = {X ₁,…, X N 0 }, and calculate the corresponding system responses for the initial N ₀ points, Y = {Y ₁ ,…, Y N 0 };

3. Build the three metamodels Kriging–ANN–SVR using {X, Y} and determine the metamodel mean Y ˆ ens ( x ) and its variance V Y ˆ ens ( x )

4. Estimate the failure probability P f using the equation:

   (61) P f ≈ ∑ i = 1 N I Y ˆ ens ( x ) ≤ 0 ( x i ) N .

5. Evaluate the stopping criterion in Eq. (46). If the condition is met, proceed to the next stage; otherwise, find the best point X* from all points in S to add it to the DoE, by minimizing the learning function U _EM. Then calculate the response Y(X*). Next, update the X = { X ∪ X * } , Y = { Y ∪ G ( X * ) } and return to step (3).

6. Calculate the coefficient of variation of P f using Eq. (11) and check if CoV < 0.05 proceed to stage 3; otherwise, expand the population by adding S _new to S, and return to step (1).

6.3 Stage 3: reliability sensitivity analysis

Once the failure probability calculation algorithm is completed, the sensitivity of the failure probability with respect to a given parameter θ _i is estimated using the following expression:

7.1 Techniques and applications utilized

7.1.1 Tools and software

The simulations were conducted using the ANSYS APDL software with the SHELL181 finite element, which provides six degrees of freedom at each node, as illustrated in Figure 4. This software was used to model the laminates. Furthermore, the necessary codes for performing reliability analysis and sensitivity analysis were developed within the MATLAB environment. A coupling between the two software tools was established.

Figure 4

Quadrilateral four-node shell element in ANSYS software (SHELL181).

This element is based on the FSDT, which assumes that the transverse deformation through the thickness is constant and also takes into account the transverse shear displacement. The parameter settings and assumptions used for this element in ANSYS are as follows:

1. KEYOPT(1) = 0: This parameter is set to include both membrane forces and moments in the calculation of the element stiffness.

2. KEYOPT(3) = 0: It is set to zero to utilize reduced integration, which reduces computational time while maintaining acceptable accuracy.

3. KEYOPT(5) = 1: This option selects the standard formulation of the shell under the FSDT theory.

4. KEYOPT(8) = 3: By setting this parameter to 3, results are computed at the top, bottom, and mid-plane of each layer, allowing us to subsequently assess failure criteria at three points for each layer, aiming to ensure more reliable results.

5. KEYOPT(9) = 0: No subroutine is used to provide initial thickness, meaning that thickness data is directly defined within the model.

6. KEYOPT(10) = 0: By default, the normal stress is assumed to be zero.

7. KEYOPT(11) = 0: The standard orientation axes of the element are used by default.

7.2 Convergence analysis and validation of the FE model

A FE validation study was conducted to confirm the accuracy of the model. Furthermore, a convergence study was carried out by varying the mesh size (2 × 2), (4 × 4), (6 × 6), (8 × 8), and (10 × 10) to determine the optimal mesh, allowing for precise analytical results while minimizing resource utilization and computation time. Precise deflection values for the plates were extracted from the research of Reddy and Pandey [65,66]. The properties of the epoxy graphite material T300/5208 and the applied force for this validation are presented in Table 1.

The results, presented in Table 2, indicate that the maximum deflection values obtained from numerical simulations of the plate are nearly equal to the exact solutions. This confirms the validation of the FE-based model. Furthermore, in the analysis of results convergence based on mesh size, the choice was made to use the (8 × 8) mesh for all subsequent steps. This selection is justified by its ability to provide sufficiently accurate results compared to the exact solution while exhibiting minimal differences compared to the next mesh size, which is (10 × 10).

7.3 Example 1: Composite plate under Uniformly distributed transverse load

The first example involves a laminated composite plate consisting of 4 layers with an antisymmetric angular arrangement [−45°, 45°, −45°, 45°] made of graphite/epoxy T300/5208, as depicted in Figure 5. This laminate is simply supported at its edges and subjected to a uniformly distributed load over its surface. Dimensional characteristics such as length, width, and thickness, as well as the properties of the graphite/epoxy material (elasticity modules in different directions, Poisson’s ratio in direction 12, maximum strengths in different directions), and the applied surface force are all considered as random variables in this example. The probability distributions of these variables are presented in Table 3.

Figure 5

Composite plate under uniformly distributed transverse load.

7.3.1 Reliability analysis of the composite plate

The reliability results for the laminate are presented in Table 4. The exact probability was computed using the MCS method with 100,000 simulations. It is noteworthy that the proposed method yielded results very close to the exact value, with an estimation error of only 2.27% compared to the MCS method. Furthermore, it is important to highlight that the proposed method required only 102 simulations to achieve this probability, which is significantly fewer than the simulations needed by the AK-MCS method.

Table 4

Results of reliability analysis for the antisymmetric angle-ply composite [−45°, 45°, −45°, 45°]

Method	Number of calls	P f	Error (%)
MCS	100,000	0.009131	—
AK-MCS [53]	12 + 159 = 171	0.009596	5.09
Proposed method	12 + 90 = 102	0.009339	2.27

Figure 6 illustrates the evolution curve of the estimated failure probability using the proposed method and the AK-MCS method. It can be observed that the proposed method rapidly converged to the exact solution starting from the 86th iteration, and the algorithm did not require many simulations to meet the stopping criterion, as it reached it after only about 16 additional simulations. In contrast, the AK-MCS method required a much higher number of simulations to meet the same stopping criterion.

Figure 6

Evolution of probability for example 1.

Figure 7 presents the evolution of weighting coefficients associated with the three metamodels used in the proposed method. Overall, it can be observed that the Kriging metamodel dominates in the initial and final segments of the curve, while in the central part of the curve, covering approximately one-third, the SVR metamodel takes precedence at certain points. On the other hand, the ANNs metamodel consistently maintains low weights in comparison to the other two, except in 3 out of the 102 iterations where it outperforms the other two metamodels.

Figure 7

Evolution of weight contribution of metamodels for example 1.

7.3.2 Sensitivity analysis of the composite plate

Figures 8 and 9, respectively, depict the sensitivity of failure probability regarding the mean and standard deviation of variables. It is clearly observed that the thickness t of the laminate layers has a significant impact on the failure probability. Furthermore, given its positive sensitivity to the mean, it can be affirmed that it positively contributes to the increase in the failure probability. Variables such as Young’s moduli E ₁₁ and E ₂₂, as well as the maximum strengths Y _T /Z _T and Y _C /Z _C, length L, width l, and force P exhibit almost the same level of impact. Among these, E ₂₂, L, l, and P significantly contribute to the failure probability, while E ₁₁, Y _T /Z _T, and Y _C /Z _C have a reducing effect on the failure probability. As for the remaining variables, their impact on the failure probability is negligible.

Figure 8

Sensitivity of failure probability to the mean of variables for example 1.

Figure 9

Sensitivity of failure probability to the standard deviation of variables for example 1.

To enhance reliability and reduce the probability of failure of this composite plate, it is imperative to take appropriate measures. Based on the results of our sensitivity analysis, it is evident that enhancing the robustness of critical variables is essential. Specifically, by prioritizing the optimization of the thickness t of the laminate layers and selecting high-quality composite materials characterized by high Young’s modulus E ₁₁ and optimal maximum strengths (Y _T /Z _T and Y _C /Z _C), while ensuring that dimensions (length and width) align with the required specifications, it becomes feasible to significantly diminish the probability of failure. This proactive approach will bolster the overall resilience of the structure and promote a more dependable performance of the composite plate.

7.4 Laminates with hole under in-plane loading

The second example is a laminated composite plate comprising 4 layers, arranged in a symmetric cross-ply configuration [0°, 90°, 90°, 0°], made of graphite/epoxy T300/5208, and featuring a circular hole at the center of the plate. This laminate is clamped on one side and simply supported on the other, while being subjected to a linear tensile load on that side, as illustrated in Figure 10.

Figure 10

Composite laminates with a hole under in-plane loading.

In the context of this example, given the absence of an analytical solution to validate the FE model, the structure was modeled using two types of predefined FEs in the ANSYS software, namely, shell181 and shell281. A comparison of the displacement results for the node located at the center of the edge under a uniformly distributed load revealed consistency between the two approaches.

Regarding meshing, a specific strategy was employed, with an increase in density near the central hole and a decrease in other areas. To assess mesh convergence, various mesh sizes were utilized, gradually decreasing. The optimal mesh size was chosen by observing that the lateral displacement results of the specified node showed no significant improvement for smaller meshes, while maintaining stability for the larger mesh.

In this example, the maximum strengths in different directions are considered constant and are presented in Table 1. The variables in this problem are the length, the width, the thickness of each layer, the radius of the hole, the fiber orientation for each layer, the elastic modulus in different directions, Poisson’s ratio in direction 12, and the applied linear force. The probability distributions of these variables are presented in Table 5.

7.4.1 Reliability analysis of the composite plate

The reliability results for the laminate are presented in Table 6. The exact probability was computed using the MCS method with 50,000 simulations. It is observed in the table that the proposed method yielded an acceptable result with an error of 9.14%, and it achieved this with only 154 simulations. In contrast, the AK-MCS method required a larger number of simulations to produce a result with a greater error than the proposed method.

Table 6

Results of reliability analysis for the symmetric cross-ply [0°, 90°, 90°, 0°]

Method	Number of calls	P f	Error (%)
MCS	50,000	0.0339	—
AK-MCS [53]	12 + 241 = 253	0.0396	16.81
Proposed method	12 + 142 = 154	0.0308	9.14

The evolution of failure probability as a function of the number of simulations is presented in Figure 11, illustrating the trajectory of the estimated failure probability using both our proposed method and the AK-MCS method. It is remarkable that our method converged to the exact solution by the 154th iteration, while the AK-MCS method required 253 simulations to reach the same stopping criterion. Therefore, it is encouraging to note that our method required a relatively modest number of simulations, especially considering the large number of random variables involved, totaling 18 variables.

Figure 11

Evolution of probability for example 2.

Figure 12 shows the evolution of the weighting coefficients of the three meta-models used in the proposed method. In this example, the Kriging meta-model does not dominate as much as in the previous example. In fact, it can be observed that the three metamodels almost equally contributed throughout the process. This shows the advantage of this approach, which combines the strengths of each meta-model by assigning them different weights based on their accuracy.

Figure 12

Evolution of weight contribution of metamodels for example 2.

7.4.2 Sensitivity analysis of the composite plate

Figures 13 and 14, respectively, illustrate the sensitivity of failure probability concerning the mean and standard deviation of variables. It is evident that the mechanical properties of the laminates have a negligible impact on the probability of failure. However, regarding the thickness of the laminates, only the thicknesses t ₂ and t ₃, corresponding to laminates with fibers oriented at 90°, have moderate and negative effects on the probability of failure. Concerning the dimensions of the laminate, the length L of the laminate has no effect, while the width and hole diameter have a significant impact on the probability of failure, with negative and positive effects, respectively.

Figure 13

Sensitivity of failure probability to the mean of variables for example 1.

Figure 14

Sensitivity of failure probability to the standard deviation of variables for example 2.

As for the fiber orientation angles, they have no effect on the probability of failure when their mean changes. However, for variations in their standard deviations, angles θ ₂ and θ ₃ have an impact on the probability of failure. Finally, just like in the first example, the applied force has a substantial impact as a contributor to the probability of failure.

According to the results of our sensitivity analysis, to enhance the reliability and reduce the probability of failure of this composite plate, it is essential to decrease the hole diameter and increase the width of the composite, as these adjustments have a significant impact on the probability of failure. In addition, a slight increase in the thickness of the 90° oriented fiber layers can also contribute to strengthening the structure.