Sensitivity analysis methodology to identify the critical material properties that affect the open hole strength of composites

Aeronautical industries are concerned about generating efficient and cost-effective statistical design allowables of composite laminate responses. Within this framework of uncertainty quantification and management (UQ & M), accounting for the various uncertainties is crucial to calculate the design allowables which, in turn, adds confidence to the design and certification process. In this context, as a precursor to UQ & M, we establish a global sensitivity analysis framework to identify the material properties that influence the laminate responses. In this study, an open hole tensile (OHT) test configuration is considered. In a first step, sensitivity analysis based on an analytical model is performed using Sobol, FAST, and Morris methods, whereas in a second step, sensitivity analysis based on a detailed 3D finite element model (FEM) incorporating damage is performed using Morris. The paper compares different sensitivity approaches in terms of computational cost and results. Further, different case studies (varied notched hole sizes, width-to-diameter ratios, laminate stacking sequences, etc.) are performed to study how the sensitive parameters vary for different configurations. The study identified the Morris method, along with the selected hyper parameters, as being suitable for FEM based sensitivity analysis due to its low computational cost. Five material properties (out of 23) were identified as influencing the open hole laminate tensile strength. Using the FE simulations performed for the Morris analysis, a first estimation of the design allowables (A and B basis values) on the OHT strength is also presented. The results from this paper contribute to the next step of UQ&M, where only the identified sensitive material properties can be considered for detailed experimental characterization or for building meta models to calculate the design allowables. A framework like this, complements experimental campaign to reduce the associated costs and knock-down factors, thus shortening the certification process.


Introduction
Aeronautical industries are constantly faced with the challenge of accounting for the various uncertainties related to the response of composite materials. Uncertainties arising from material properties, manufacturing processes, test setups, data reduction methods, loading conditions, specimen geometry, laminate stacking sequences, manufacturing defects etc., can significantly vary composite structural performance. Hence, uncertainty quantification and management (UQ&M) methods 1,2 are used to furnish the variability associated to each input parameter and provide the design allowables, thereby adding confidence and robustness to the design and certification process.
The standard approach of obtaining the design allowables is through experimental testing of 18 (termed as B18 reduced sampling) or 30 (termed as B30 robust sampling) specimens from three or five batches, respectively, to obtain the B basis design allowable. 3 This process is quite extensive and expensive as it covers the whole manufacturing and testing costs of the specimens. In addition, modifying the laminate layup, the geometry or the loading configuration requires the entire process being repeated. With the development of high-fidelity numerical models that can predict with good accuracy, industries have started to complement experimental tests with virtual tests to reduce the recurring and non-recurring costs, leading to greater freedom in the design space and more efficient designs.
The failure strength of a composite laminate at the mesoscale depends on the material properties of the fiber-matrix system, the laminate configuration, specimen geometry, etc. Material properties account for the elastic, strength, and fracture properties of the inter-and intralaminar material system. These numerous properties are characterized using extensive laboratory tests to obtain the material card and are then used in finite element (FE) simulations to predict the response of the composite laminates. FE simulations account for the deterministic mean values of the material properties but do not account for the variability in these material properties. However, in reality, composite laminate response can be significantly influenced by the variability of these material properties. In this context, it is important to account for their statistical distribution obtained from the experimental tests, which indeed demands extensive experimental test campaigns. Hence, it is beneficial to understand which influential properties are the ones that affect the laminate's response.
Global sensitivity analysis (GSAs) is multi-disciplinary and is applied across various fields to estimate the sensitive parameters that influence a target output response. 4 In the context of composite laminates and associated material properties, integrating a sensitivity analysis into a composite laminate test campaign helps to screen the influential and non-influential material properties. Once the sensitive parameters have been identified, the experimental tests to obtain these corresponding material properties can be performed with greater care, ensuring that the associated uncertainty is minimal. For example, for an interlaminar fracture toughness test, switching from a linear elastic fracture mechanics (LEFMs) based data reduction to a J-integral based toughness measurement facilitates to avoid the high uncertainties associated with the measurement of the crack length during the test. 5,6 Further, within the framework of uncertainty quantification, these selected key properties can be well characterized by increasing the number of specimens to 18 or 30 (B18 and/or B30 sampling as recommended in the Composites Materials Handbook CMH-17 3 ) to obtain a better statistical distribution. In addition, identifying the sensitive material properties can help towards more efficient FE based optimization methods and/or generate meta models for machine learning, where only the key parameters need to be considered to optimize the output response. These numerous benefits substantially support the importance of performing sensitivity analysis within the context of composite materials, with the aim of improving the design of composite structures.
Notched strength of composite laminates is an important parameter in the aeronautical industry as it can be the limiting factor in the design of composite structures. Openhole strength testing of laminated composites is a good practical example and is often employed as a baseline test to understand a material's structural performance. 7 This test is a characteristic test that has an interaction between geometric and material properties. Further, the test replicates the scenario of fastener holes and cut-outs which are commonly seen in composite structural components.
Moreover, the open-hole strength and the failure modes are strongly influenced by the different specimen configurations (notched hole size, width-to-diameter ratios etc.), 7 and hence makes it challenging and an optimum test to validate a sensitivity analysis case study.
Numerous researchers have investigated the open-hole tensile response of a composite laminate through experimental, 8 numerical, 9 and analytical methods. 10 For a generic quasi-isotropic laminate, delamination and in-plane fibre failure are two damage modes that lead to final failure. Hallet et al., 11 however, reported complex interactions between the matrix cracks, fibre splits and delaminations during the damage process. The failure stress and damage mechanisms vary significantly with different configurations such as notched hole size, 12 layup configuration, 13 width-todiameter ratios, sub-laminate scaling 7 etc. Advancement in the FE numerical models ensured the good predictability of not only the failure strength, but also the interaction between the damage modes. These models are crucial for GSA and UQM approaches, despite the hindrance of their heavy computational costs.
GSA has been widely used in different types of applications 14 and various review papers have analyzed the different sensitivity methods and the feasibility in terms of predictions and/or computational time. 4,15 The choice of a sensitivity method depends on various factors such as the: computational cost of a single model evaluation, number of input variables, complexity of the model, allotted time for the project, etc. 4 GSA is generally broken down into the following methods based on regression, screening, variance and/or meta model. For instance, variance-based methods decompose the variance of the model output for every input, provides quantitative measures but with the drawback of high computational cost. The screening-based method on the other hand is optimum for models demanding high computational cost since these methods require less model evaluations, but their drawback is that they provide only the qualitative measures to rank factors and not quantify the effects of different factors on the output. Vallmajó et al. 16 performed a local sensitivity analysis (LSAs) on the OHT strength using an analytical model and provided a UQM framework to obtain a B-basis design allowable for notched composite parts. Cózar et al. 17 performed an LSA on the projected delamination area of the impact scenario and on the compression after impact strength of composite laminates using FE models. Further, only the sensitive parameters identified were considered to build a response surface for calculating the design allowables using UQM approach. Thapa et al. 2 performed uncertainty quantification and a GSA (using polynomial chaos expansion based sobol indices) utilizing a progressive failure analysis to obtain the key parameters for the tensile, compressive and shear response of a laminate with a circular cutout. To the authors' knowledge, GSA using high-fidelity FE models considering progressive damage for composite response prediction has not been previously reported because of the complexity of the models and the computational time required.
The main objective of this paper is to establish an economical and efficient global sensitivity analysis framework in regards to the open hole tensile response of composite laminates. This work demonstrates the material properties that can influence the open hole strength of a notched composite laminate. In a first step, an analytical framework based on OHT strength prediction is combined with different global sensitivity analysis methods to identify the key material and geometrical properties. Three sensitivity analysis methods (Sobol, FAST and Morris) methods were studied and compared for a baseline configuration using an analytical model that has nine input parameters. In the next step, using a detailed 3D FE numerical model that also accounts for inter-and intra-laminar progressive damage, a sensitivity analysis using the Morris method was performed to estimate the influential material properties, out of a total of 23 properties accounted for, the ones that influenced material properties. In addition, different case studies were performed where we varied the number of ply clusters, notched hole sizes, width-to-diameter ratios and laminate stacking configurations were varied to study how the influential material properties vary for different cases and to also validate the GSA approach. Finally, a first estimation of the design allowables on the OHT strength is also presented.

Methodology
Open-hole tension: Analytical model Furtado et al. 10 proposed an efficient analytical framework that can predict the notched strength of a multi-directional composite laminate using only three ply-based material properties. The framework integrates a finite fracture mechanics model, 18 invariant based trace theory and master ply concept to estimate the stiffness and strength, 19,20 and an analytical fracture mechanics model to estimate the laminate fracture toughness. 21 The proposed analytical model only requires three material properties (the longitudinal Young's modulus, E 11 , longitudinal strength, XT, and the fracture toughness in the fiber direction, G°), in addition to the laminate stacking sequence and the specimen geometrical properties (width of the specimen, W, and the diameter of the hole, D) as the input. Furtado et al. 10 compared the predicted notched strength (both tension and compression load cases) from the analytical model with the experimental values for different laminates and material configurations and obtained excellent correlations. Interested readers are referred to 10,18 for more details on the formulation of the model.

Open-hole tension: Numerical model
We used a mesoscale FE model to simulate the open hole tensile response, where Abaqus/Explicit 3D solid elements (C3D8R) were used to model the plies. The interfaces were modeled using Abaqus finite thickness cohesive elements (COH3D8). 22 Progressive intra-ply damage was accounted for with the continuum damage model proposed by Maimí et al. [23][24][25] The damage activation functions are based on LARC04 failure criteria. The predictive capability of the model has been demonstrated for various loading cases such as OHT, 8,26 double edged notched tension, 8 impact and compression after impact. 25,[27][28][29] The intralaminar model was implemented in a VUMAT user-written subroutine. An intralaminar element was deleted once the fiber damage variable (d 1 ) reached 1. Further details of the model can be found in the works from Maimí et al. 23,24 The simulations were performed in Abaqus using a dynamic explicit solver without using mass scaling. A uniform in-plane element size of 0.5 mm was used for all the elements (both inter and intralaminar) throughout the study. Each ply was modeled as a solid layer with one element through the ply thickness, while the thickness of the cohesive elements was set to 0.01 mm as recommended in the Abaqus documentation. 22 Tensile load was introduced by applying a pre-defined displacement at one of the specimen edges while keeping the other edge constrained. The pre and post processing of the models were automated using python scripts and the final failure (OHT load) is defined as the maximum load in the force-displacement curve from the simulation.

Sensitivity analysis
In this work, we used three sensitivity analysis methods: Sobol, 30-32 FAST (Fourier Amplitude Sensitivity Test) 33 and MOAT (Morris One-At-a-Time). 34,35 Sobol and FAST are the two most commonly used variance-based methods. These methods quantify the output variance for every input parameter and also calculate the interaction effects among the different input variables. The Sobol method provides the Sobol indices (SIs) such as the first, second, and total order sensitivity indices as a quantitative measure of the SA. The first order SI represents the main effect of each input variable on the output response, while the second order SI measures the effect due to the second-order interaction between two input variables. The total SI (introduced by Homma et al. 36 ) denotes the effect due to an input variable and also its interaction with all the remaining input variables. The FAST method, originally developed by Cukier et al., 33 also computes the first and the total (developed in an improved version by Saltelli 37 as eFAST) order effects. Variance-based methods are efficient for complex and nonlinear models but have a drawback of high computational cost.
MOAT is a screening-based one-at-a-time method where each parameter, x i , is varied along a grid size of Δ i to create a trajectory in the defined design space. For a model with n input parameters, one such trajectory contains n variations and (n + 1) model evaluations. In each model evaluation, only one input parameter is varied at a time. The elementary effect of the ith parameter can be calculated using where EE is the ratio of the change in the model output to the input parameter variation. Each input variable is varied across selected p levels in the grid space. The MOAT method uses hyper parameters r (denotes the number of repetitions/trajectories) and p (number of levels in the design space grid). The improved Morris sampling technique (proposed by Campolongo et al. 35 ) provides a better scan of the input design space by finding trajectories with the maximum dispersion in the input space. The Morris method provides two sensitivity measures: μ*, which provides the overall influence of the input parameter on the output, and σ, which assesses the non-linear or interaction effect of that input parameter with the rest of the parameters. A Morris plot of μ* versus σ qualitatively screens the various input parameters into: (a) insensitive (low μ* and low σ), (b) with high influence on output but with less interaction/non-linear response (high μ* and low σ), (c) with high influence on output as well as high interaction with other parameters (high μ* and high σ) and (d) with less influence on the output but high interaction with other parameters (low μ* and high σ). Note that the Morris sensitivity measures cannot be compared quantitatively between two different Morris plots. We used the open source Python library SALIB 38 to create the input matrices and perform the sensitivity analysis.
In the whole process of GSA, the first step is to create sampling for a selected sensitivity analysis method. The user defines the number of input parameters, their bounds, sampling method, and the hyper-parameters depending on the GSA method. Once the sampling is performed with input variations, then each row of the obtained test matrix corresponds to one model evaluation (either using analytical or FE simulations). The whole input matrix is run to obtain the output response (OHT strength in this case). Along with the input and output matrix, GSA sensitivity analysis is performed to identify the sensitive parameters depending on the corresponding GSA method.

Problem definition and case studies
The OHT baseline case selected is a quasi-isotropic laminate ([45/90/À45/0] 3s ) with 24 plies of 0.131 mm per ply. The nominal laminate thickness is 3.12 mm. The hole diameter is selected as 2 mm with the in-plane dimensions: 12 mm (width) and 24 mm (length), and maintaining a width-to-diameter (W/D) ratio of 6.
In the first step, global sensitivity analysis is performed on the OHT response predicted by the analytical model. 10 Table 1 presents the nine input properties (seven material and two geometric properties) considered for the analytical model based sensitivity analysis. The selected material is IM7/8552 and the mean (μ) and standard deviation (σ) values were obtained from. 39 The material properties follow a normal distribution and are varied within their 99.7% confidence interval (±3 standard deviations from the mean). Sensitivity analyses are performed with the three different GSA methods explained above, with nine input parameters and the OHT strength as the output response.
In the next step, we move from analytical-based GSA to FEM-based GSA to incorporate all the material properties associated to the OHT test. Table 2 presents all 23 material properties considered in the FEM model that are the input parameters that account for the elastic, strength and fracture properties of both the fiber and matrix of the material system. Morris SA is used in this step due to the expensive computational time associated with the variance-based SA. Literature recommends the effectiveness of MOAT to give satisfactory predictions with a reduced number of sample runs, which applies in the case of FE-based SA. 4 The first objective is to select the Morris hyper-parameters (number of trajectories r and number of grid levels p) and then use these to study other configurations.
We set up different case studies associated with the OHT response to study the change in the sensitive material properties, compared to the baseline case explained above. The different case studies studied here are the effect of: (a) ply clustering, (b) notched hole size, (c) width-to-diameter ratios and (d) laminate stacking configurations, on the sensitive material properties that affect the OHT strength. Table 3 presents the different laminates employed to study the effect blocked plies have. Laminate Lx1, Lx2 and Lx4 contain 1, two and four blocked plies, respectively, and are of the same laminate thickness. Table 4 details specimens with different hole diameters to study the effect of notched hole size. Specimen width and length and the hole diameter are varied by keeping the W/D ratio as a constant value of 6. Table 5 introduces different specimens with varying W/D ratios. WD-9, WD-6, WD-3 and WD-1.5 have W/D ratios of 9, 6, three and 1.5., respectively, which is obtained by increasing the diameter of the hole and keeping the specimen width and length constant. Table 6 presents laminates with different stacking sequence configurations such as quasi isotropic, a hard and a soft laminate. The percentage of plies associated with each laminate is also mentioned in the table.

Sensitivity analysis results: Analytical model
In this section, we compare the sensitivity results from three different SA methods (Sobol, FAST and Morris) based on the analytical model for the baseline laminate. First, we study the minimum number of runs required to have reliable sensitivity indices for Sobol and FAST methods using the analytical model. Then, we discuss the identified sensitive parameters obtained using the analytical model and a comparison between the three different GSA methods. The total number of runs/model evaluations for Sobol is determined by N = (n + 2)r, if second order interaction effects are neglected. Different replication times r are studied to understand the evolution of the SI. Figure 1(a) and (b) present the evolution of the total SI provided by Sobol and FAST methods, respectively, of the nine input parameters, for increasing number of total runs (N) obtained from the analytical model. Figure 1(a) also presents a 95% confidence interval (using bootstrap re-sampling) on the Sobol total sensitivity index for all the input parameters. Compared to Sobol, FAST GSA showed stable sensitivity indices at lower values of newton. Hence, we selected N = 8000 for Sobol and N = 5000 for FAST, where the estimators of the statistics stabilized, for the comparative analysis. Figure 2 presents the Morris plots for different hyper parameters: different p levels (8, 16 and 32) and r repetitions (5, 10, 20, 40, 80 and 100). Recall that the Morris sample size is determined by N = (n + 1)r, and to determine the fewest sample runs to obtain effective Morris screening, it is important to inspect the results for different values of the hyper parameters. Increasing r leads to higher numbers of runs/model evaluations (N = 50 for r = 5 to N = 1000 for r = 100), whereas increasing p denotes a finer discretization of the grid levels in the input domain. Overall, the results from the Morris plots for different p and r values show good consistency (see Figure 2). The most sensitive parameter, XT, and the second most important, G°, (according to the μ* values) are identified in all the cases, hence we select p = 8 and r = 40 (since the variation in the results is negligible among all the cases) as hyper parameters to compare against the other two GSA methods. Note that the Morris sensitivity measures cannot be compared quantitatively between the different Morris plots. Figure 3(a) and (b) present the comparison of the first order and total order SI of the nine input parameters from the Sobol and FAST GSA methods, respectively, for the selected sample sizes, as discussed above. Figure 4(a) presents the Morris μ* versus σ plot for the selected hyper parameters p = 8 and r = 40. Figure 4(b) shows absolute mean values (μ*) with the 95% confidence interval (obtained through bootstrap re-sampling) on the μ* for all nine input parameters. All three SA methods provided the same results (comparing Figures 3 and 4) in terms of identifying the sensitive and insensitive parameters. Fiber longitudinal tensile strength (XT) was identified as the most sensitive parameter. All three methods demonstrated that XT has a main effect on the OHT strength and also presents high interactions (high total order SI and high σ in Morris) with the other input parameters. The second sensitive parameter in ranking is the fiber tensile fracture toughness (G°). The third sensitive parameter is the diameter (D) of the specimen hole (when D is varied within its tolerances: 2 ± 0.04 mm), but the effect is seen to be very low. All the other parameters are considered comparatively insensitive towards the OHT strength of the laminate. Even though the agreement between the GSA methods is the same, the total number of model evaluations varies critically for Morris compared to other two methods: 8000, 5000 and 400 total runs for Sobol, FAST and Morris, respectively. Since the GSA was based on an analytical model, the high number of model evaluations are still feasible due to the low computational effort required per model evaluation. Nevertheless, this can be crucial when moving to FE-based high-fidelity numerical models that also incorporate progressive damage.

Sensitivity analysis results: FEM model
In this section, we move from analytical based models to FE numerical models to account for the progressive damage evolution and thereby obtaining more reliable strength predictions. In GSA-based FE models, we account for all the material properties (as provided in Table 2), but this can lead to high computational costs per model evaluation.
Hence, exploiting the conclusions of the previous study, we incorporate the Morris method with the FE model, due to its capability to screen for insensitive parameters at a relatively low cost. We use the same baseline OHT configuration as explained in the previous section. All the material properties that are inputs to the OHT FE model are considered for the GSA, while the geometrical variables (D and W/D) are not included in the FE-based GSA.  Similar to the approach with the analytical model, first we performed the Morris sensitivity analysis for different hyper parameters to find the optimum p and r values. Figure 5 presents the Morris plots for the different hyper parameters: different p levels (8, 16 and 32) and r repetitions (5, 10, 20, and 30). We considered 21 material properties that constitute the elastic, strength and fracture of intra and interlaminar material systems. Note that GYT and GSL are excluded from the list of input material properties for the current sensitivity analysis as these parameters are normally assumed in the literature as: GYT = G Ic (the matrix tensile fracture toughness is assumed as the mode I interlaminar fracture toughness) and GSL ¼ G IIc (the matrix shear fracture toughness is assumed as mode II interlaminar fracture toughness) due to the complexity to experimentally characterize them. Each Morris plot in Figure 5 corresponds to newton simulations that depend on the value of r, ranging from N = 110 (r = 5) to N = 660 (r = 30). The results are seen to be stable at r = 20, and when r is increased to 30, the results remain consistent qualitatively. The effect of p was not found to be as significant as compared to r, and in addition, the computational cost is decided by r. Hence, we selected p = 16 and r = 20 as the hyper parameters. Figure 6(a) presents the Morris μ* versus σ plot for the selected hyper parameters p = 16 and r = 20 for the FEMbased GSA for 21 material properties. Figure 6(b) shows the μ* values for all the material properties along with the 95% confidence interval on the μ* values. From Figure 6, all 21 material properties are placed diagonally in the plot, with five being placed in the region of high overall effect on output and high interaction with the other parameters. The remaining parameters that lie in the region with low values of μ* and σ are considered insensitive. Fiber longitudinal tensile strength (XT) and fiber tensile fracture toughness (GXT) are two of the five sensitive parameters that influence the OHT strength. Both these parameters are characteristic fiber tensile properties that play a crucial role in a tensile test. Analytical model based GSA also deemed these two parameters as the most sensitive. In the framework of cohesive law, XT defines the onset of failure and GXT is the area under the stresscrack opening curve. Apart from these two anticipated properties, the cohesive law parameters are observed as key properties that affect the OHT strength. fXT and fGT  are the ratios of XT and GXT, respectively, in the first branch of longitudinal tensile cohesive law that define the shape of the cohesive law. This is in line with the findings from Maimí et al., 40 where they concluded that the first part of the cohesive law plays a major role in the strength of the OHT specimens.

Morris analysis results: Different case studies
OHT strength is strongly influenced by the test configurations such as the specimen geometry and the layups. 7 Hence, in addition to the baseline OHT configuration studied above, we employ the same sensitivity analysis approach to the other case studies. We used the selected Morris hyper parameters identified in the previous section along with all the 23 material properties (Note that GYT and GSL are included in this GSA due to several laminate configurations studied in this section being different from the baseline) detailed in Table 2. With the above-defined values, the sensitivity analysis of each configuration requires 480 simulations (n = 23, r = 20 and p = 16).

Clustering of plies
The first case focuses on laminates with different levels of clustered plies. Figure 7(a)-(c) present the Morris sensitivity plots for laminates with one (Lx1), two (Lx2) and four (Lx4) plies clustered. Figure 7(d) shows the OHT strength from 480 model evaluations (each point represents OHT strength obtained from the i th simulation) for the different cases studied for ply clustering. For Lx1, the same five  properties (as found with the baseline configuration) are observed to be sensitive. But with increased ply blocking, mode II interlaminar fracture toughness ðG IIc Þ shows a higher interaction (σ) with the other parameters and also a higher total effect (μ*) on OHT strength. These results are in line with the literature, as when plies are clustered, delamination becomes a dominant failure mode, 12 and hence G IIc is seen to have a greater effect on the OHT strength as this parameter is associated with the delamination propagation. Blocked plies are prone to have early induced transverse matrix cracks at the vicinity of the hole and free edge, which then grow into delaminations at the interfaces, and hence have lower OHT strength compared to unblocked plies (Figure 7(d)).

Notched hole size effect
We considered different specimen configurations where the diameter of the hole is increased but the W/D ratio of the specimen is kept as six and the same stacking sequence as the baseline laminate is maintained. Figure 8(a)-(c) present the Morris sensitivity analysis results of specimens with hole sizes of D = 0.5, D = 2 and D = 6 mm, respectively. With the same W/D ratio, the former refers to a small specimen and the latter to a larger specimen. Figure 8(d) shows the OHT strength from 480 simulations for the different notch size specimens.
For the smallest specimen, only XT and G IIc are the sensitive parameters, with all the other parameters being completely insensitive. With increased hole diameters or larger specimens, fiber fracture toughness (GXT) and the cohesive law parameters (fXT and fGT) move to the right of the plot.
In the literature, this is explained as the size effect of the hole, where in smaller specimens, the fracture process zone (FPZ) extends to the edge of the specimen in comparison to the larger specimens where the FPZ is confined to the vicinity of the hole. 39 In fact, for the smallest specimen (D-0.5), in fact, the FPZ is larger than the specimen width and the average stress at the fracture plane tends to the unnotched strength of the laminate. Hence only XT is found dominant and none of the cohesive law parameters are found in the sensitive region (Figure 8(a)). For the other two cases (D-2 and D-6), FPZ is developed and thus cohesive law parameters are also sensitive. Compared to a smaller specimen, the largest specimen has a smaller FPZ in terms of specimen width and hence the average stress acting on the fracture plane is lower. This is why the OHT strength is higher for D-2 when compared to D-6, as seen in Figure 8(d).

Different width to diameter ratios
We studied specimens with different W/D ratios by varying the hole diameter but keeping the width constant in all cases.    Table 5 for the definition of the different width to diameter ratios of the specimens).
The layup is the same as that of the baseline laminate. Four configurations were studied where the specimen with the highest W/D ratio refers to a smaller hole and vice versa. Figure 9(a)-(d) present the Morris sensitivity results of specimens with W/D ratios 9, 6, 3 and 1.5, respectively. In all the cases, the same five sensitive parameters identified for the baseline configuration are repeated here. For a high W/D ratio specimen, the edge of the hole is very close to the specimen's free edge and hence the damage is more brittle. 7 Hallet et al. 13 reported that for a high W/D ratio specimen, failure can be more pull-out dominant than delamination. In fact, this is seen in the sensitivity analysis results where G IIc is comparatively less significant for WD-9 (Figure 9(a)) compared to other smaller W/D ratios.

Laminate stacking configuration
As a last case, we studied three different laminate stacking configurations ranging from soft (42% of plies oriented in 90°) to quasi-isotropic to hard laminates (42% of plies oriented in 0°). Figure 10 The conclusions from the above case studies demonstrate that the proposed GSA analysis identified the same dependencies which are already reported in the literature. Nevertheless, this adds confidence to the proposed FEbased GSA framework which can be applied to other test configurations with less understanding.
A first rough estimation of the design allowables In this section, we exploit the numerous simulations performed for the Morris analysis (see Figure 8(d)) to obtain a first estimation of the design allowables on the OHT strengths for different configurations. Figure 11 shows the OHT strength variations for the two configurations of hole size effect (D = 2 and 6 mm) studied in 6.2. Figure 11 provides a box plot and mean of the 480 OHT strengths   Table 2), and the A and the B basis design values obtained using 480 simulations following the CMH-17 approach. 3 The deterministic numerical OHT strengths for both configurations (543 and 445 MPa for D = 2 and 6 mm, respectively) are predicted well when compared with the experimental data (555 and 438 MPa for D = 2 and 6 mm, respectively), with a maximum error of 2%.
The most widely used design allowables A and B basis values are defined as 95% of the lower one-sided confidence bound of the first and 10th percentiles of the population, respectively. 3 For the estimation of design allowables following CMH-17 approach, firstly the distribution that best fits the data (480 OHT strengths) needs to be studied. For the configuration D = 2 mm, the OHT strengths did not follow the Weibull, normal or log-normal distribution and so a non-parametric procedure was used to estimate the A and B basis values. For D = 6 mm, the OHT strengths followed the normal distribution, and subsequently the basis values were calculated following the CMH-17 tabular data for normal distribution. For more details on these procedures, the reader is referred to the CMH-17. 3 A (465 and 364 MPa for D = 2 and 6 mm, respectively) and B (485 and 395 MPa for D = 2 and 6 mm, respectively) basis design allowables estimated are around 16% and 11% lower than the experimental OHT strengths, hence leading to conservative fail safe design of aeronautic parts. Note that this is only a first estimation of the design allowables with the sole intention to exploit the various FE results from Morris GSA.

Prospects of an efficient FEM-based morris GSA for uncertainty quantification and management
In the response of composite laminates, it is critical to capture the interaction of damage modes in order to have a good prediction on the failure strength of the laminate. Hence, FEM simulation with constitutive damage models integrated with GSA is seen as one of the best options. The drawback of the computational time of FEM simulations can be tackled by using Morris GSA, which provides reliable qualitative sensitive analysis results as demonstrated in this paper. The selected Morris hyper-parameters, despite depending on the nonlinearity of the model, can be used as a reference for the GSA of other loading cases of composite laminates. Within the framework of UQM, the conclusions of the GSA are paramount. In the next step, the five identified sensitive parameters can be experimentally characterized by testing more specimens. The following three characterization tests can be performed in the laboratory using a greater number of specimens (18 specimens as per CMH-17) 3 for a better statistical distribution: Longitudinal tensile test for XT (ASTM D3039 M), 41 Double edged notched tensile test 42 or compact tension test 43 for GXT, fXT and fGT, C-ELS (ISO 15114) 44 or ENF (ASTM D7905) 45 mode II interlaminar test for G IIc . Further, to propagate the uncertainties, machine learning can be used to predict statistical design allowables, as performed by Furtado et al. 46 In this step, the insensitive material properties identified from GSA can be kept constant while varying only the five identified sensitive properties to have efficient meta-models for the UQM. The design allowables estimated through this methodology will be compared with the A and B basis values obtained in A first rough estimation of the design allowables.

Conclusions
Aeronautical industries are trying to quantify the various uncertainties associated with composite materials responses. Within the framework of uncertainty quantification and management (UQ&M), the industries aim to obtain statistical design allowables for different composite tests to quantify the confidence on the design and certification processes and structural safety. In this aspect, global sensitivity analysis (GSAs) on the response of composite structures is a significant precursor to UQ&M. In this paper, we demonstrate an efficient integration of GSA methods with analytical or finite element numerical models accounting for progressive damage models to estimate the sensitive material properties that affect the open hole tensile strength of a composite laminate. In the first step, analytical model based GSA was performed on the baseline configuration and different GSA methods were compared to study their prediction and computational efficiency. In a latter step, the analytical model was replaced by FE high-fidelity models and we proposed a FEM-based Morris sensitivity analysis framework to estimate the sensitive parameters, despite the Morris approach being a qualitative analysis but associated with a relatively low cost. We identified five sensitive material properties, properties (out of 23), that influence the open hole tensile strength of the laminated composites, i.e., the longitudinal tensile strength of the unidirectional ply, the first part of the tensile fiber cohesive law and the mode II interlaminar fracture toughness. Further, the framework was investigated with different case studies (ply clusters, notched hole size, laminate configuration, etc.) to evaluate the efficiency when predicting the sensitive parameters. Finally, a first rough estimation of the design allowables was performed exploiting the OHT strengths results obtained from the Morris sensitivity analysis. The next step would be to define a UQ&M analysis framework considering the results from the GSA analysis that will help to reduce the dimensionality of the problem.
The proposed FEM-based GSA framework contributes towards an efficient UQ&M, where only the sensitive material properties identified can be considered for a detailed experimental characterization or to construct metamodels for machine learning applications to estimate design allowables. In a nutshell, complementing experimental tests with virtual tests to obtain the design allowables help to move towards a safer and efficient designs with reduced number of tests which, in turn, provides more freedom to the design space, reduced costs and knock-down factors.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.