Five different models are set up to compute the resistance of the measured cylinders against buckling and to compare results against the buckling loads from the tests. The five models differ in the number of parameters considered (Table 6-5).
Table 6-5 Parameters considered in different models
Model number
Geometric imperfection
Load imperfec-tion
Layup as built
Bedding of
cylin-ders
1 x x - -
2 x x x -
3 x - - -
4 - x - -
5 x x - x
The computational results for all 11 cylinders tested are depicted in Figure 6-12 as computed buckling load over tested buckling load. In case of agreement, the results would lie on the blue diagonal line representing a linear resistance model with zero error associated to it.
Model 3 (no consideration of load imperfections) computes similar results for all cylin-ders, indicating that the considered geometric imperfections have only minor influence on the scatter of the results. Model 4, which neglects geometric imperfections but incorporates load imperfections, shows a high degree of scatter with certain offset from the reference diagonal. Models 2 and 4 scatter around the diagonal, where for model 4 the numerical analysis significantly exceeds the test result for several cylin-ders.Models 1 and 2 show similar behaviour in terms of scatter around the diagonal.
Figure 6-12 Model uncertainty
According to EN 1990 [EN 02], Annex D, the model uncertainty of each type of model can now be further quantified by analysing the mean value correction factor, denoted b, and the variance of the error term, V. The factor b can be interpreted as the least squares best-fit to the slope of the resistance model, given by the following relation between experimental resistance rei and theoretical resistance rti:
n
i ti n
i ti
eir r
r b
1 2 1
(6-1)
The error term V is given through:
1exp 2
s
V (6-2)
Refer to EN 1990 [EN 02], Annex D for further details.
The results are summarised in Table 6-6. The variation of the error terms does not vary largely between the models. With values between 3.88 % for model 3 and 4.87 % for model 5, the variance associated with the model uncertainty is in the range of the measured buckling load scatter which showed a coefficient of variance of 3.96 %.
The analysis shows that for the cylinders investigated, the highest loss in resistance prediction accuracy occurs for model 3 and is due to the negligence of load imperfec-tions occurring during the cylinder tests. Here, the mean value correction for the re-sistance is 15 % lower than for model 1. In comparison, the influence of geometric imperfections, illustrated by the difference of results between model 4 and 1, is small
with only 3 %. The consideration of the layup changes the model bias by only 1 % and reduces the variance of the error term slightly. Highest b-value is achieved for model 5 using a detailed modelling approach for the cylinder mounting (section 6.1.5).
Applying this method to the test results of DEGENHARDT ET AL. [Deg10] and computations made by BROGGI [Bro11] it is found that the introduction of thickness imperfections leads only to minor reduction of the model bias. HÜHNE [Hüh05] reached the maximum b-factor when including geometric imperfections but also high variance of the error term, indicating that this model might also be improved by consideration of additional parameters. A comparison of these results with non-linear computations for the per-fect geometry shows that the influence of the geometric imperper-fections measured has a much larger influence than for the cylinders investigated within this study. This is ex-pected due to the lower radius to thickness ratio as well as the lower imperfections associated with the cylinders. EN 1990 [EN 02] offers a transparent way of quantifying and comparing these influences for different studies, whereas the results always have to be assessed in the context of the geometries and manufacturing methods used.
Table 6-6Model uncertainty characterisation
Schillo [Sch15]
Degenhardt [Deg10], Broggi
[Bro11] Hühne [Hüh05]
11 nominally identical cylinders 10 nominally iden-tical cylinders
Within this work, the mean shear force appearing at buckling that is measured, causing additional moments, is about 5 % of the mean buckling load but making up for 15 % of model bias. To make statements about prediction models for unstiffened cylinders, it is very important to consider this parameter.
6.3 Summary and discussion
Predictions for the resistance of all cylinders tested are made, using four different FE-models with a varying number of parameters considered. While the consideration of geometrical imperfections only increased the model prediction agreement by 3 %, the largest contribution comes from the introduction of load imperfections which made up for 15 % of the model bias. The comparably small influence of the geometric imperfec-tions is remarkable since there is commonly a much higher load reduction associated with it (refer to section 2.2.1). The NASA SP predicts a higher susceptibility towards geometric imperfections with increasing R/t ratio (Figure 2-5, page 14). With a nominal R/t ratio of 147, the cylinders considered are situated at the far left of the NASA SP design curve but a significant knockdown would still be expected (=0.52, refer to section 2.4.1). In addition to the expected lower influence the main reason could be associated with the quasi-isotropic layup and the manufacturing process, leading to decreased residual stresses, low tolerances of the geometry and long-waved imperfec-tion modes rather than short-waved. Hence, a general assessment of the sensitivity of unstiffened cylinders towards geometric and load imperfections should include a wider range of R/t values and different manufacturing methods.
The mean vector of transverse load imperfections occurring during test was measured to be around 3 kN, corresponding to approximately 5 % of the axial load applied and contributing by 15 % to the model bias. No data is available to compare this amplitude with other test setups. It is considered to be higher than other test facilities that are purpose-built for uniaxial compression tests of unstiffened cylinders. Since small load imperfections can have large influence on the buckling load, these forces should be measured during test.
Despite the fact that no material inherent flaws and the simplification of rigidly strained cylinder edges are considered, the prediction agreement for all models con-sidering load imperfections was very high. However, further research should investi-gate the sensitivity of the models with respect to the thickness variation that has only been included deterministically in these models, as well as an appropriate representa-tion of the cylinder stiffness from coupon tests.
7 Relia ilit ased ali ratio of safet fa tors
The following sections describe the Bayesian approach to derive and update calibrated safety factors for the structural and model uncertainty. Parts of this section are also treated in [Sch17].