6 Quantification of Prediction Bounds Caused by Model Form Uncertainty 61 Fig. 6.6 Main-effect ANOVA of predicted frequencies (contact stiffness FE model) hand, with four through-thickness elements greatly increases the time-to-solution. With these considerations, a discretization with three elements through the thickness is selected for the portal frame [14]. The resulting, nominal FE model features a total of 105,400 nodes and 75,534 finite elements. A modal analysis performed to extract the low-frequency mode shapes is solved in 1 min on a typical PC workstation, which is fast enough to perform a few hundred runs for the parameter studies presented next. This discretization, with three through-thickness elements (top beam, side columns, brackets, base plate), is identical for all runs performed. 6.5.3 Parameter Study of the Contact Stiffness Representation Spring coefficients of the contact stiffness representation are investigated next. The first step is to decide ranges over which these values must be varied. It is observed that, for values lower than 10C10 N/m, the order of the modes changes. This is an indication that the contact surface condition deviates from the experimental condition and, therefore, the spring coefficients used in the FE model are not appropriate. On the other hand, the resonant frequencies asymptote to constant values when the spring coefficients exceed 10C15 N/m. It indicates that the jointed connection is essentially “rigid” and further increasing the spring stiffness values has no effect. These two observations set the lower and upper bounds of stiffness coefficients to the range (10C10 N/m; 10C15 N/m). The parameter study is pursued using this range of values. To keep the study to a manageable size, all of the bolts of the bracket-to-vertical column and bracket-to-top beam connections are grouped together and assigned the same stiffness values in the normal, shear and tangential directions. These coefficients are denoted by knn (normal), kss (shear) and ktt (tangential). Similarly, the 10–24 socket cap screws of the bracket-to-base plate connections are grouped together and assigned the same stiffness values knn, kss and ktt. This parameterization of the model gives six spring coefficients where each is varied within the range (10C10 N/m; 10C15 N/m). A two-level, full-factorial design-of-computer-experiments is defined to propagate the ranges of spring stiffness values through the model. Predictions are used to observe the effect on the first six resonant frequencies of changing one spring at a time. This analysis is similar to the main-effect analysis applied to measured resonant frequencies in Sect. 6.4.4. The ANOVA results of Fig. 6.6 clearly indicate that one of the bolt shear parameters (“bolt kss”) and the base plate normal stiffness (“base knn”) are the two most influential parameters for the six resonant frequencies. They must be considered to quantify the prediction uncertainty. Observing that the bolt shear stiffness coefficient is one of the most influential parameters of the FE model matches the intuition gained from experimental campaigns, where it is observed that most of the target modes are sensitive to shearing effects of the jointed connections. The ANOVA also indicates that the other four stiffness coefficients can be eliminated from further consideration since varying them from the lower to upper bounds does not significantly change the frequency predictions. These four parameters are kept constant and equal to mid-range values (3.16 10C12 N/m, midpoint on a log10-scale).
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