Probabilistic buckling assessment and reliability of fiber-metal laminate, composite and aluminum cylindrical panels under compression with load and fabrication uncertainties

The objective of this article is to study the buckling behavior and reliability of fiber metal laminated (FML), composite and aluminum cylindrical panels under uniaxial compression taking into account fabrication and loading uncertainties. A 3D finite element modeling with ANSYS software has been implemented for this purpose. The panels are discretized using shell elements and the eigenvalue buckling analysis is conducted for the prediction of elastic buckling. The influences of load and fabrication uncertainties on the buckling load factor are studied with probabilistic analyses and the reliability of the panels is calculated. It is found that the thickness of aluminum layers is the most significant uncertain variable for the critical buckling load factor of FML panels, whereas the fiber misalignment angle of their composite layers is insignificant. As the metal volume fraction decreases, the sensitivity of the elastic buckling load factor to variations of the axial load distribution is reduced whereas its sensitivity to variations of the thickness of composite layers is increased. Consequently, the metal volume fraction is an important design parameter for the uncertain buckling behavior of the panels.


Introduction
Fiber-metal laminates (FMLs) are widely applied for the manufacture of lightweight thin-walled structures in the aerospace industry.The outstanding mechanical performance of FMLs, such as high fatigue resistance, excellent impact resistance, high fracture toughness and low density, makes them very appealing for aerospace applications [1].The buckling of lightweight cylindrical structures is a problem of major practical importance, since it is a primary cause of failure and, as a result, an accurate prediction of their buckling behavior is required for their design [2][3][4][5].
The buckling strength of a component fitted in an engineering structure is influenced by various factors such as its structural details and the type of loading.However, there is variability in the production procedure of actual industrial applications due to the presence of imperfections which cannot be eliminated.As a result, the actual dimensions of the structure deviate from its nominal dimensions [6,7].Furthermore, the curing of composite materials, which is a basic step of their manufacturing process, has technical difficulties which cause deviations from the desired structural details.Typical uncertainties observed due to the manufacturing process of composite materials are the fiber misalignment and the variation of ply thickness [8].Uncertainties characterize also the loading of cylindrical panels [9], since the ideal uniform distribution of the axial load is not always achieved in real structures.The aforementioned uncertainties must be taken into account in order to predict the elastic buckling behavior and reliability of FML panels.
The scientific research concerning the probabilistic mechanical behavior and reliability of advanced materials and structures is intensive.Kalantari et al. [10] performed multi-objective optimization and robust design of multi-directional carbon/glass fiber-reinforced hybrid composites when different sources of uncertainties are considered.In the study of Chamis [8], a methodology is developed to computationally assess the probabilistic composite behavior due to uncertainties in the constituent properties, in the fabrication process and in structural variables.Babich et al. [11] considered an approximate probabilistic model of accumulation of microdefects in a material under repeated loading, which makes it possible to define theoretical parameters of the fatigue failure.
Wang et al. [12] introduced a reliability-based optimization framework to design filament-wound cylindrical shells with variable angle tow, considering the uncertainty in the winding angle.In the paper of Kumar et al. [13] the probabilistic buckling behavior of sandwich panels considering random system parameters using a radial basis function is presented.Alibrandi et al. [14] proposed an efficient procedure for the reliability analysis of frame structures with respect to the buckling limit state taking into account that the elastic parameters are uncertain and modeled as random variables.Alfano and Bisagni [9] developed a probabilistic approach to the buckling behavior of composite and sandwich cylindrical shells in order to achieve reliability based knockdown factor.
This article presents a probabilistic analysis of FML, composite and aluminum cylindrical panels under axial compression, in order to assess their elastic buckling behavior and reliability.The effects of three sources of uncertainties are studied, namely ply thickness, fiber misalignment and axial load distribution.To the authors' knowledge, studies investigating the effect of these uncertainties on the buckling behavior of panels composed of GLARE (GLAss-fiber REinforced aluminum laminate) have not been published elsewhere.Furthermore, the reliability of such panels under compressive loading which may cause their buckling failure has not been previously calculated in the published literature.GLARE belongs to the broad category of FMLs and has been applied in several primary and secondary aerospace structures so far [1,15].

Problem delineation
A cylindrical GLARE panel is subjected to uniaxial compression.The panel is thin and consists of alternating thin aluminum layers (0.2 -0.5 mm) and unidirectional (UD) S2-glass/epoxy composite plies.Figure 1 illustrates the stacking sequence a GLARE 2A-3/2-0.458panel in combination with the applied axial loading.The three aluminum layers are located externally and in the middle of the depicted GLARE, while the four composite plies are inserted in between.The dimensions of the panel are given as follows: b=0.75 m (width), R=0.55 m (radius of curvature), L=1 m (length), φ=78.1 o (angle of the panel) and t=1.875 mm (thickness).It is noted that t is very small with respect to the other dimensions and the panel is shallow.
As it is depicted in figure 1, the panel is statically compressed with a uniaxial loading N z along its rounded boundaries.The loading N z is constant for the deterministic analysis and varies linearly along the edges for the probabilistic analysis.The classical simply supported boundary conditions are considered in this study according to reference [16].Specifically, the following displacements of the panel are constrained: radial displacements along the four edges, circumferential displacements along the curved edges AB and CD, axial displacements along the straight edges BC and AD.Taking into account the applied increasing loading and the support type of the panel, it will deform in the axial direction and it will buckle for a critical value of N z .The critical deterministic buckling load N det , corresponding to constant axial load distribution, is calculated numerically with ANSYS FEM software [17] and the classical eigenvalue buckling analysis.
The probabilistic elastic buckling behavior of GLARE, aluminum and composite panels (having the aforementioned L, b, R, t values) is investigated in combination with the Probabilistic Design System (PDS) of ANSYS software [17].Details and a flow-chart [18] concerning the procedures for a probabilistic analysis with ANSYS PDS are given in references [17,18].The considered random variables and the application of N z in the FEM models are analyzed in section 4 of the present study.A suitable definition of reliability is introduced and the reliability of the panels is calculated.

FEM modeling procedure
In this work a 3D FEM procedure is carried out in combination with the classical eigenvalue buckling analysis in order to calculate the critical buckling load factor of the axially compressed panels.The examined panels have been simulated numerically using SHELL 181 elements, considering that this type of elements is appropriate for modeling thin laminate shell structures [17].The idealization of the behavior of the two material systems is achieved using linear elastic material models with isotropic (aluminum) and orthotropic (composite) material properties.
It is noted that in order to verify the convergence of the numerical results, FE models have been constructed with varying mesh density and in each case the results were practically invariant between the fine and the very fine meshed models.Furthermore, the stress levels of each layer of the laminate corresponding to the critical buckling load factor are always checked, in order to ensure that the elastic buckling limits are not exceeded.
The probabilistic FEM analysis is implemented to the succeeding GLARE grades: GLARE 2A The abbreviations Alum and 0 o c stand for the aluminum (2024-T3) and composite plies, respectively.The probabilistic analysis is also implemented to the symmetric UD glass-epoxy composite laminate ([0] 15 ) and to aluminum panels.The 0 o orientation angle of the fibers is parallel with the longitudinal axis of the cylindrical panel (z-axis of the cylindrical coordinate system depicted in figure 1.a).It is noted that the 0 o fiber orientation angle has been chosen for all analyzed materials in order to study separately the impact of variations of their metal volume fraction (MVF) on the probabilistic buckling response of the panels.
The last number of the abovementioned nomenclature of GLARE grades indicates the nominal thickness of each aluminum layer in mm.Each composite ply of the examined UD composite and GLARE panels has a nominal thickness of 0.125 mm and consists of S2-glass UD fiber prepreg.The material properties of the UD composite and 2024-T3 aluminum can be found in reference [19].
The numerical procedure and the appropriateness of the applied boundary conditions of this work have been validated as it is described in reference [18].A very good convergence between the theoretical and the FEM results has been found in reference [18] for all examined material systems.

Probabilistic analysis
As already mentioned, uncertainties concerning fabrication and load distribution are considered in this study.We define the thickness of aluminum (TA) and the thickness of UD glass-epoxy layers (TC) along with the fiber orientation angle (A) in the composite plies as random input variables (RVs).The lowest eigenvalue of the eigenvalue buckling analysis, which is the critical buckling load factor (NC), is defined as the random output parameter.
The simulation of the uncertain uniaxial loading is implemented using a line pressure load N z , as depicted in figure 1. N z is always equal to 1 N/m at points B and C. N z is equal to P1 at points A and D, which is defined as a RV.The axial load N z varies linearly along the curved edges AB and CD.A triangular distribution has been chosen for P1 [17] with two cases of loading range.In case I, the variation of P1 is within ±10% from the most likely value (1 N/m) and in case II the variation is ±20% from the most likely value.Consequently, in case I the minimum value is P1=0.9N/m and the maximum value is P1=1.1 N/m.In case II the minimum value is P1=0.8N/m and the maximum value is P1=1.2N/m.It is noted that NC=N det when P1=1 N/m.
It is considered that the three remaining RVs vary according to the normal distribution.It is very usual to assume that these fabrication details are normally distributed [6-8, 10, 17].The tolerance of the thickness of aluminum layers is obtained from industrial data sheets referring to 2024-T3 aluminum [20], whereas the tolerance of the thickness of UD S2-glass/epoxy composite plies is found in reference [10].The tolerance of angle A is ±3 o [10].The normal distribution of the RVs is defined as it is described in [18].
As mentioned above (section 2) the ANSYS PDS was utilized in order to carry out the probabilistic analysis.The Monte Carlo simulation method was chosen coupled with the Latin Hypercube Sampling technique.The convergence of the probabilistic results is accomplished by executing a large number of simulation cycles for each case and is always validated taking into account the ancillary ANSYS suggestions.
In our study we define the reliability R of the panels as follows:  = ( ≤   ) (1) The probability P of equation ( 1) can be easily calculated using the cumulative distribution function (cdf) F NC (N) of NC which is defined by:   () = ( ≤ ) = ∫   ()  −∞ (2) where f NC is the probability density function of NC.
In figure 2 the definition of R is presented graphically.

Results and discussion
Figure 3 shows a characteristic diagram of how sensitive the critical buckling load factor is with respect to the variability of the RVs for the GLARE 2A-3/2-0.458cylindrical panel under axial compression (case I).For the considered RVs, it is demonstrated from figure 3 that the critical buckling load factor is insensitive to variability of the fiber misalignment angle.However, NC is quite sensitive to variability of any of the remaining RVs.It is obvious that the sensitivity of TA is considerably higher than the absolute value of the sensitivity of P1.Similarly, the absolute value of the sensitivity of P1 is clearly higher than the sensitivity of TC.The descending order of the sensitivities presented herein is also valid for all analyzed GLARE materials for both cases of load distribution, except for the case I of GLARE 5, where the sensitivity of TC is higher than the absolute value of the sensitivity of P1.
The Spearman rank-order correlation coefficients (r s ) between each one of the RVs and NC have been calculated and are presented in tables 1 and 2, for all analyzed panels.It is noted that the r s values are equal to the corresponding probabilistic sensitivities and vary according to: -1 ≤ r s ≤ 1 [21].It is seen that the correlation coefficients of P1 in table 2 (case II) are higher (as absolute values) than the correlation coefficients of P1 in table 1 (case I).This is reasonable, since the variation of P1 values is higher in case II and as a result there is a stronger correlation between P1 and NC in this case.It is also observed that the r s values of TA and TC of table 2 are lower than the respective r s values of table 1, due to the aforementioned stronger correlation between P1 and NC in case II.
It is observed from tables 1 and 2, where the analyzed materials are presented in accordance with the descending order of their MVF, that the correlation coefficients of P1 and TC are reduced (as absolute values) and increased, respectively, as the MVF decreases.Since all composite layers have the same nominal thickness with the same tolerance, this is attributed to the fact that the number of UD glass-epoxy layers increases with the decrease of the MVF.As a result, the combined uncertainty of the thicknesses of composite layers becomes more significant as the number of these layers increases.Simultaneously, the uncertainty of P1 becomes less significant for the critical buckling load factor.The relation between the r s values of TC and P1 becomes obvious for the UD glass-epoxy panels: the thickness uncertainty of each one of the 15 composite layers is superimposed making TC a more significant input variable for NC in comparison with P1, for both cases of load distribution.
However, it is shown from tables 1 and 2 that, with respect to the different GLARE materials, the r s values of TA are not substantially affected from the MVF reduction and this is explained as follows.Contrary to the number of UD glass-epoxy layers which is variable, the number of aluminum layers is constant (three) for the GLARE panels of this study.Furthermore, the tolerances of all aluminum layers of the GLARE panels in this study are the same regardless of their different thickness [20].This is why the r s values of TA for GLARE materials do not vary considerably for both cases of axial load distribution in tables 1 and 2.
The effect of the number of layers and their thickness uncertainty is also observed from the difference of the correlation coefficients of TA between aluminum and GLARE panels.It is shown, from tables 1 and 2, that the r s values of TA for the aluminum panel (one layer) are substantially lower than the corresponding values for the GLARE panels (three layers).The correlation coefficients of A in tables 1, 2 are close to zero, verifying the aforementioned insignificance of the uncertainty in the fiber misalignment angles.
From our probabilistic analysis, we have observed that as the number of simulation loops becomes large, the mean value of the calculated critical buckling load factor tends to be equal to the deterministic critical buckling load: NC mean ≈ N det .
Figure 4 illustrates the cdf curves of NC of the analyzed panels for both cases of axial load distribution (N z in N/m).Such curves are a primary review tool for the probabilistic buckling design of structural components.It is shown that the cdf curve of the aluminum panel for case I (continuous curve) has a considerable deviation from the corresponding curve for case II (dashed curve).Although this observation is valid for all materials, it is shown that as we move from the right to the left of figure 4 (as the MVF is reduced) the deviation between the two loading cases decreases and, finally, the curves for UD glass-epoxy laminate (case I and II) are almost identical.The aforementioned fact that the sensitivity of P1 reduces as the MVF decreases contributes to this behavior.In table 3 the reliabilities of the analyzed panels are given for cases I and II.The deviation described in the previous paragraph between the two loading cases affects the reliability of the panels defined in figure 2. However, it is seen from table 3 that the influence of the considered variations of P1 on the R value of each material is small.It is observed from table 3 that with the considered uncertainties all panels have a rather low reliability.According to table 3, the UD glass-epoxy panel is the most reliable, but it has a much lower deterministic buckling load than the GLARE and aluminum panels.

Conclusions
The elastic buckling of simply supported cylindrical panels consisting of different materials and subjected to uniaxial compression with fabrication and load uncertainties is investigated using probabilistic FEM analysis.Three different GLARE grades are analyzed along with a UD glass-epoxy composite and monolithic 2024-T3 aluminum.The panels have identical dimensions.

Figure 1 .
Figure 1.(a) Dimensions and (b) fine mesh (top view) of the considered GLARE panel under linearly varying axial compression.

Figure 2 .
Figure 2. Graphical definition of reliability R.

Figure 3 .
Figure 3. Case I sensitivities of the critical buckling load factor (NC) to the RVs for the GLARE 2A panel.

Figure 4 .
Figure 4. Buckling load factor cdf curves of the analyzed cylindrical panels under axial compression for cases I and II.

Table 3 .
Reliability (R) values of the simply supported cylindrical panels under axial compression.

Table 1 .
Case I r s values between RVs and NC for the analyzed simply supported cylindrical panels under axial compression.

Table 2 .
Case II r s values between RVs and NC for the analyzed simply supported cylindrical panels under axial compression.