Buckling prediction of cylindrical shells by vibration considering stochastic modelling, combined loading and industrial-scale validation

Kurzfassung

The research summarised in this thesis focuses on advancing the state of the art regarding a non-destructive method to predict the buckling load of various structures using the change in their frequency response as a function of the load level applied, commonly known as the Vibration-Correlation Technique (VCT). The structures of choice for this research were cylindrical shells, for which VCT is particularly appealing, given their high sensitivity to imperfections which often leads to large discrepancies between their theoretical and experimental buckling loads. Owing to the recent developments regarding this technique and its low adoption outside the academic environment, the research was focused on key areas that are equally relevant to both academic and industrial environments. First, the significance of the load ratio and the number of measurements over the buckling load estimation provided by VCT was studied for multiple cylindrical shells, given the existent lack of agreement over these aspects of the experimental test setup. In an initial step, these aspects were studied using numerical data from the finite element models of 10 nominally identical cylindrical shells. Then, in a second step, a similar study was performed on experimental data gathered from the existent literature, which confirmed the trends observed in the study based on numerical data. The main conclusions of this investigation were that the most important test parameter is the load level at which the frequency measurement is taken and that the number of measurements is mostly relevant in decreasing the possible errors given by unexpected frequency deviations between the multiple measurements. The VCT predictions of the structures studied had a negligible variation with respect to the prediction using all available measurements, regardless of the number of measurements taken into account, as long as the ratio between the highest load level considered and the buckling load was at least 70%. Conversely, having at least 9 measurements proved enough in ensuring a negligible variation of the predictions with respect to the one using all available measurements in presence of frequency measurement deviations within ±0.5Hz. Furthermore, it was also noticed that there are cases in which the VCT predictions decrease, rather than increase, with increasing the load level taken into account. While this behaviour is relatively easy to identify when performing VCT prediction while gradually increasing the maximum load level considered, the general conservative nature of the method in these cases may no longer be ensured. Second, the sensitivity of VCT in presence of various magnitudes of shape and loading imperfections was also assessed. For this study, the FE model of a cylindrical shell with a large knock-down factor, related to its shape imperfection, and for which VCT gave robust experimental buckling load predictions was chosen as the starting point. Then, multiple magnitudes of its measured out-of-plane and of a realistic in-plane imperfection pattern were applied to the cylinder to study the reliability of VCT to provide robust buckling load predictions in presence of such imperfections. It was found that the VCT predictions are generally insensitive to the aforementioned imperfections, provided that the maximum load ratio accounted was in line with the one found to be sufficient in the first study. However, it was also found that there might be several shape imperfections for which VCT may be unreliable, primarily due to the VCT predictions decreasing with increasing the load ratio taken into account, aspect also highlighted in the first study. Furthermore, in the event of local buckling occurring, VCT proved unreliable in estimating the load level at which these events occur. Similarly, the VCT estimations of the global buckling load using the measurements before local buckling occurred were also relatively poor. Nevertheless, predicting the global buckling using VCT may be possible when taking into account exclusively the measurements beyond the point at which local buckling occurred. In these scenarios, the established VCT procedure often provided unconservative predictions, reason why using an empirical VCT procedure and taking shape imperfections into account in the buckling analysis is recommended, as the estimations provided in this case are generally more conservative. Third, the robustness of VCT for cylindrical shells subjected to combined loading was assessed. This numerical study used the finite element models of 6 cylindrical shells, in which additional bending, shear or torsion were applied, be it individually, or all together, besides the main compressive load. The study revealed that VCT can provide reliable estimations of the axial loads at which cylinders buckle under the aforementioned combined loading scenarios, provided that the amount of bending, shear, and/or torsion are relatively low. Moreover, whether the compression load is applied sequentially, or simultaneously with the additional ones was only relevant regarding the amount of the latter beyond which VCT failed to provide robust predictions. In this context, VCT provided better estimations when compression was applied after the aforementioned additional loads (sequential load introduction), such that the change in the frequency response reflected a change in the compression load alone. Last, it was also shown that VCT can provide good estimations on the compression load level at which cylinders buckle when bending, shear and torsion are applied together besides compression, also provided that the magnitudes of the former are small. Last, the results of an experimental campaign covering aspects from all previous investigations conducted was shown to validate the results found. In this experimental campaign VCT was applied to a real-scale launcher structure, tested under two load cases in which uneven compression was applied. These load cases had significant bending components, given that they were defined such that buckling was precisely initiated on two key areas on the structure. During both of these tests, the vibration modes considered to provide reliable predictions were identified in-situ. Taking into account the maximum load ratio of a valid vibration measurement during the first load case of 86%, the relative error of VCT buckling load prediction was -5%. Conversely, the relative error of the VCT buckling load prediction for the second load case was -15%, as the load ratio of the last valid vibration measurement was lower than for the first load case, at 77%. Nevertheless, given the approximately linear relation between the VCT predictions and the maximum load level taken into account, an extrapolation of the VCT buckling load prediction corresponding to a 86% load ratio was made for the second load case, which yielded a relative error comparable to the one achieved for the first load case, namely 6%. Another positive outcome was that the preliminary numerical investigation performed in preparing the test provided valuable insight confirmed by experiment. Some of the key aspects accurately identified through numerical analyses were the frequency window in which the first vibration modes would occur, the span of the circumferential waves of the vibration modes and the expected VCT prediction error as a function of the maximum load level used. In addition to this, a fair estimation on the frequency drop as the load applied increased was also provided by the preliminary numerical analysis. The research summarised in this thesis focuses on advancing the state of the art regarding a non-destructive method to predict the buckling load of various structures using the change in their frequency response as a function of the load level applied, commonly known as the Vibration-Correlation Technique (VCT). The structures of choice for this research were cylindrical shells, for which VCT is particularly appealing, given their high sensitivity to imperfections which often leads to large discrepancies between their theoretical and experimental buckling loads. Owing to the recent developments regarding this technique and its low adoption outside the academic environment, the research was focused on key areas that are equally relevant to both academic and industrial environments. First, the significance of the load ratio and the number of measurements over the buckling load estimation provided by VCT was studied for multiple cylindrical shells, given the existent lack of agreement over these aspects of the experimental test setup. In an initial step, these aspects were studied using numerical data from the finite element models of 10 nominally identical cylindrical shells. Then, in a second step, a similar study was performed on experimental data gathered from the existent literature, which confirmed the trends observed in the study based on numerical data. The main conclusions of this investigation were that the most important test parameter is the load level at which the frequency measurement is taken and that the number of measurements is mostly relevant in decreasing the possible errors given by unexpected frequency deviations between the multiple measurements. The VCT predictions of the structures studied had a negligible variation with respect to the prediction using all available measurements, regardless of the number of measurements taken into account, as long as the ratio between the highest load level considered and the buckling load was at least 70%. Conversely, having at least 9 measurements proved enough in ensuring a negligible variation of the predictions with respect to the one using all available measurements in presence of frequency measurement deviations within ±0.5Hz. Furthermore, it was also noticed that there are cases in which the VCT predictions decrease, rather than increase, with increasing the load level taken into account. While this behaviour is relatively easy to identify when performing VCT prediction while gradually increasing the maximum load level considered, the general conservative nature of the method in these cases may no longer be ensured. Second, the sensitivity of VCT in presence of various magnitudes of shape and loading imperfections was also assessed. For this study, the FE model of a cylindrical shell with a large knock-down factor, related to its shape imperfection, and for which VCT gave robust experimental buckling load predictions was chosen as the starting point. Then, multiple magnitudes of its measured out-of-plane and of a realistic in-plane imperfection pattern were applied to the cylinder to study the reliability of VCT to provide robust buckling load predictions in presence of such imperfections. It was found that the VCT predictions are generally insensitive to the aforementioned imperfections, provided that the maximum load ratio accounted was in line with the one found to be sufficient in the first study. However, it was also found that there might be several shape imperfections for which VCT may be unreliable, primarily due to the VCT predictions decreasing with increasing the load ratio taken into account, aspect also highlighted in the first study. Furthermore, in the event of local buckling occurring, VCT proved unreliable in estimating the load level at which these events occur. Similarly, the VCT estimations of the global buckling load using the measurements before local buckling occurred were also relatively poor. Nevertheless, predicting the global buckling using VCT may be possible when taking into account exclusively the measurements beyond the point at which local buckling occurred. In these scenarios, the established VCT procedure often provided unconservative predictions, reason why using an empirical VCT procedure and taking shape imperfections into account in the buckling analysis is recommended, as the estimations provided in this case are generally more conservative. Third, the robustness of VCT for cylindrical shells subjected to combined loading was assessed. This numerical study used the finite element models of 6 cylindrical shells, in which additional bending, shear or torsion were applied, be it individually, or all together, besides the main compressive load. The study revealed that VCT can provide reliable estimations of the axial loads at which cylinders buckle under the aforementioned combined loading scenarios, provided that the amount of bending, shear, and/or torsion are relatively low. Moreover, whether the compression load is applied sequentially, or simultaneously with the additional ones was only relevant regarding the amount of the latter beyond which VCT failed to provide robust predictions. In this context, VCT provided better estimations when compression was applied after the aforementioned additional loads (sequential load introduction), such that the change in the frequency response reflected a change in the compression load alone. Last, it was also shown that VCT can provide good estimations on the compression load level at which cylinders buckle when bending, shear and torsion are applied together besides compression, also provided that the magnitudes of the former are small. Last, the results of an experimental campaign covering aspects from all previous investigations conducted was shown to validate the results found. In this experimental campaign VCT was applied to a real-scale launcher structure, tested under two load cases in which uneven compression was applied. These load cases had significant bending components, given that they were defined such that buckling was precisely initiated on two key areas on the structure. During both of these tests, the vibration modes considered to provide reliable predictions were identified in-situ. Taking into account the maximum load ratio of a valid vibration measurement during the first load case of 86%, the relative error of VCT buckling load prediction was -5%. Conversely, the relative error of the VCT buckling load prediction for the second load case was -15%, as the load ratio of the last valid vibration measurement was lower than for the first load case, at 77%. Nevertheless, given the approximately linear relation between the VCT predictions and the maximum load level taken into account, an extrapolation of the VCT buckling load prediction corresponding to a 86% load ratio was made for the second load case, which yielded a relative error comparable to the one achieved for the first load case, namely 6%. Another positive outcome was that the preliminary numerical investigation performed in preparing the test provided valuable insight confirmed by experiment. Some of the key aspects accurately identified through numerical analyses were the frequency window in which the first vibration modes would occur, the span of the circumferential waves of the vibration modes and the expected VCT prediction error as a function of the maximum load level used. In addition to this, a fair estimation on the frequency drop as the load applied increased was also provided by the preliminary numerical analysis. Given the reliable in-situ VCT buckling load predictions and in that the post-test investigations a single vibration mode that provided a better prediction than the in-situ ones and only for the second load case, VCT can be considered a robust nondestructive method to determine the axial load level at which cylinders buckle. Furthermore, this was shown to hold true for a real-scale structure subjected to load cases with significant bending components, displaying thus the versatility of the method and its retained robustness when applied to structure sizes relevant to industrial environments. This feat is even more impressive when considering the tests important time restrictions and the occurrence of unexpected events, which for the first load case caused loosing track of the majority of vibration modes monitored beyond the 6th load step, out of the 9 initially defined, and for the second load case caused loosing track of all previously tracked vibration modes beyond the 7th load step, out of the 8 initially defined