NASA SP 8007 is a guideline for design of unstiffened shells under different loading conditions. The analytical solution for computation of the buckling load is based on a simplification of the classic theory suggested by BATDORF [Bat47] and an extension for orthotropic shells by JONES [Jon68]. All possible imperfections or uncertainties are covered through a global knockdown factor.
The basis of this knockdown factor is the test data mentioned earlier that was summa-rised and compared by WEINGARTEN [Wei65b] [Wei65a]. The test results he retrieves from literature are reproduced in Figure 2-5. On the horizontal axis the R/t ratio of the cylinders is depicted while on the vertical axis the ratio of computed buckling load to test result is shown.
Figure 2-5 Knock-down factor curve used for NASA SP [Wei65a]
The curve shown in Figure 2-5 is the basis of the safety concept used for the NASA SP-8007 guideline for design of unstiffened shells under axial compression ([Wei65b], [Pet68]).
As outlined in section 2.1, the test data cannot be analysed statistically since different boundary conditions, materials and test set ups have been used. Hence, an equation to determine a conservative knockdown factor depending on the R/t ratio was devel-oped that lies below all test data analysed and is computed as
For orthotropic shells the exponent is modified using
* 4 parame-ters. Using the buckle aspect ratio
R m
nL
(2-11)
with m being the number of buckle half waves in axial direction and n being the num-ber of buckle waves in circumferential direction, the buckling load for orthotropic shells is given as
Note the appearance of square in the numerator.
The recommendations of the NASA SP-8007 do also apply for cylinders made of fibre reinforced plastics. Notably, the previously mentioned influence of the coupling terms of the laminate stiffness matrix, further investigated by GEIER ET AL [Gei02], cannot be captured.
Also, none of the cylinders depicted in Figure 2-5 is made of FRP. Although some CFRP cylinders are later tested and the buckling loads are compared to the design loads given by the NASA SP, it is only concluded that the approach leads to very conservative
results (see e.g. [Deg10]). There is however not enough statistical meaningful data to determine the real inherent safety of the structure designed using this approach.
2.4.2 Single perturbation load approach (SPLA)
Based on findings by ESSLINGER [Ess70] that the collapse is initiated at a single buckle, HÜHNE [Hüh05] develops a design concept called single perturbation load approach (SPLA) which incorporates the usage of a transverse point load on the outer cylinder surface using finite element analysis. The idea is that this point load triggers the first buckle and hence allows for the computation of conservative load estimation. Hühne performs a parameter study concerning the amplitude of the transverse load and finds that above a certain load level, the computed buckling load does not further decrease.
The point load is applied during an initial load step within the finite element analysis.
During the second load step, the axial compression load is applied. The design load is then computed via a non-linear FEA. Hence, a global knockdown factor with respect to a perfect cylinder can be found without further knowledge about manufacturing uncer-tainties.
In some cases the computed design load exceeds the buckling load observed in exper-iments [Hüh05]. The SPLA is hence further investigated within the framework of the EU-project DESICOS (New Robust Design Guidelines for Imperfection Sensitive Compo-site Launcher Structures, www.desicos.eu). It is found to be conservative in terms of covering possible geometric imperfections [Kri12b].
2.4.3 Probabilistic approach
Probabilistic approaches for the design of unstiffened shells were first proposed by BOLOTIN [Bol62] and later picked up by others [Fra69], [Roo69].
ELISHAKOFF AND ARBOCZ analysed cylindrical shells by considering the Fourier coefficients used to approximate the geometric imperfections as random variables within an ana-lytical framework [Eli82], [Eli87]. However, no statistical data base for comparison with real cylinders was available. In order to validate the procedure, 30 beer cans are meas-ured and tested at the TU Delft [Arb79]. A comparison of the resulting probability density functions shows no good agreement and it is concluded that not all uncertain-ties are accurately represented in the model.
SCHENK AND SCHUELLER [Sch03] consider random geometric imperfections of 7 copper electro-plated, isotropic cylindrical shells referred to in [Arb79] and generate artificial geometric imperfections by applying a Karhunen-Loeve expansion to generate eigen-functions with uncorrelated random variables [Sch01]. The standard deviation of the
resulting simulation compares reasonably well with the standard deviation of the test results.
CHAMIS AND ABUMERI [Cha05] perform Monte Carlo simulations treating ply thickness, fibre volume ratio and fibre longitudinal modulus as random variables. No geometric imperfections are considered. Static and dynamic analyses are compared but no com-parison to test results is possible.
DEGENHARDT ET AL. [Deg10] carry out Monte Carlo simulations and treat material parame-ters probabilistically. The resulting probability density function does not match the test result.
BROGGI ET AL.[Bro11] use the measurements taken by DEGENHARDT to perform new Mon-te Carlo simulations. Here, the focus is on generating artificial, representative geomet-ric imperfection patterns by using random fields. Evolutionary power spectra are used to capture the statistical properties of the random field. The method leads to good agreements for the torsional load case. For axial compression it is found that not all uncertainties are captured in the model.
KRIEGESMANN [Kri10] performed Monte Carlo analysis for the cylinders tested by HÜHNE
[Hüh08] and considers geometric imperfections as well as boundary imperfections as random variables. The probabilistically derived lower bounds were conservative with respect to the test results but less conservative than the NASA SP. Due to the small sample size, the results could not be validated. Later, KRIEGESMANN applies an extension of the semi-analytic approach suggested by ELISHAKOFF AND ARBOCZ [Eli82], [Eli87] to the set of shells tested by DEGENHARDT ET AL. [Deg10]. Good approximations of the probabil-ity densprobabil-ity function of the test results are found for both the semi-analytic as well as the Monte Carlo simulation.
BIAGI AND DEL MEDICO [Bia08] derive reliability based knock-down factors by statistically analysing so- alled e ui ale t i pe fe tio a plitudes . The p o edu e o sists of computing the collapse load of the imperfect shell for different imperfection ampli-tudes with an in-house code, which includes a characteristic imperfection shape that is used to derive equivalent imperfection amplitude corresponding to the collapse load measured during the experiment. These equivalent imperfection amplitudes are com-puted for cylinder tests found in literature and lead to a statistical distribution that is used as input for a Monte Carlo simulation. From these simulations, a knockdown factor can be computed in accordance with a target reliability level. The method relies on using one characteristic imperfection shape for all cylinders considered.