Safety factors for designing the unstiffened shell structure are subdivided into a factor covering structural and material uncertainties (section 7.4.1) and a factor covering the model uncertainty (section 7.4.2).
7.4.1 Safety factor covering material and structural uncertainties
The safety factor covering material and structural uncertainties that has been modelled in the Monte Carlo simulation can now be calibrated according to a chosen target reliability level. For demonstration purposes, the -value is chosen to be 3.09 and 2.33, corresponding to a structural reliability level of 0.999 and 0.990, respectively (on the basis of [Esn96]). Assuming normal distributions, the allowable buckling load at the
1 Set Z1
2 Sets Z1+Z2
design point can then be computed for each Monte Carlo simulation by rewriting
The partial safety factor is then given through [Fab02]:
%
x50
xD
R
(7-3)
where xD is the design point related to the -value and x50% is the characteristic value.
In the case of stiffness and geometrical properties the characteristic value is commonly taken as the 50 % quantile [EN 02] .
The resulting safety factors are given in Table 7-5.
Table 7-5 Safety factor covering material and structural uncertainties
Design stage
For a reliability level R0.999, the value of 0.91 reflects the uncertainty at the preliminary design stage. With the availability of measurements from set Z1, this uncertainty re-duces to 0.93, resulting from the lower scatter of the buckling load under the given scatter of input variables. Due to only small differences in the input data derived form set Z1 and updated with set Z2, the updated computation does not lead to a change in safety factor. The safety factor when considering the Monte Carlo analysis using aver-aged measurement data of set Z1 and set Z2 (denoted as Frequentist inTable 7-5) is also 0.93. At a reliability level of R0.990, safety factors are show slightly increased values with identical tendencies.
7.4.2 Safety factor covering model uncertainties
The model uncertainty describes the degree of uncertainty related to deviation be-tween model inference and the test result. This deviation can be caused by a lack of
information about input parameters, by negligence of relevant parameters or by the ability of the model to properly describe the physical effect.
A Bayesian model is set up to compute a safety factor describing the above mentioned deviation. This factor can then be updated in the light of new information available.
A multiplicative error model is employed, written as e q
Y (7-4)
q describes the simulated data, e is the error term and Y is the vector of test results.
Assuming lognormal distributions for q and e, the posterior distribution becomes also lognormal distributed and can be computed as
with being the estimator for the mean of the error term. Prior and likelihood func-tions are written as follows:
)
For the predesign stage, the mean error term is modelled using an uniformative prior via a uniform distribution ranging from 0.5 to 1.5. As a first guess, the knockdown factor given by the NASA SP could also be used. However, since this knockdown factor is associated with another computational method, an uniformative prior is chosen.
At the preliminary design stage, the safety factor is updated two times, with results from set Z1 and set Z2, respectively (Table 7-6). Thus, a safety factor covering the un-certainties related to the usage of nominal data is developed.
The posterior probability density function for is computed using Equation (7-6) and (7-4), whereas the integral in the numerator does not have to be solved according to the MAP rule (section 3.4.5). An uninformative uniform prior is chosen for the first update. The resulting posterior distributions are depicted in Figure 7-7 and the safety factor is computed as the 50 % quantile of the estimated mean model error [Fab02].
The most probable stochastic moments of are read from the joint posterior distribu-tion funcdistribu-tion (Figure 7-7). From the resulting lognormal posterior distribudistribu-tion funcdistribu-tion
the safety factor is computed as the 5 % quantile to account for the uncertainty of the model error.
Figure 7-7 Probability density function of µθand σθ
Lognormal distribution functions with characteristics of the test data and the data of the MC inference (Table 7-4, p.111) are assumed.
In the absence of any other information, the prior mean error term in Equation (7-4) for the predesign stage is modelled using an uninformative prior via a uniform distribu-tion ranging from 0.5 to 1.5. Evaluating the probability density funcdistribu-tion gives a most probable value for the mean and standard deviation of the error term. From these, the 50 % and 5 % quantile are derived.
The computation is first run with respect to test results of set Z1 and then updated with respect to test results of set Z2. Each subsequent analysis uses the previous distri-bution function for the error term as the prior distridistri-bution.
At the preliminary design stage the resulting safety factor covers the model uncertain-ties related to the usage of nominal data. Since the test results of the second cylinder set Z2 showed slightly lower results compared to set Z1, the updated multiplicative safety factor is lower (0.87 as compared to 0.94 for the 5 % quantile).
For the specific design phase this effect can also be observed for the updating process but is less severe since the model inference is in high agreement with test results.
The updated safety factor at the 5 % quantile for the Bayes model is identical to the Frequentist model that gives a factor of 0.94.
Table 7-6 Safety factor m covering model uncertainty
design stage
information
source prior
update
Z1 Z2
5% 50% 5% 50%
predesign preliminary uniform 0.94 0.95 0.87 0.92
specific design
Z1Z2-Bayes lognormal 0.95 1.00 0.94 0.99
Z1Z2-Freq - 0.94 1.00