**2.3. Probabilistic research**

4 Advances in Computational Stability Analysis

**Figure 2.** Perturbation load mechanism

**Figure 3.** Single perturbation load approach (SPLA)

0

5

10

15

**N1**

**F1**

**Buckling load N (kN)**

**Buckling load F**

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**F0**

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**N0 line (a)**

**2.2. Single-perturbation-load approach** 

Hühne [1] proposed an approach based on a single buckle as the worst imperfection mode leading directly to the load carrying capacity of a cylinder. Figure 2 explains its mechanism; the lateral perturbation load P is disturbing the otherwise unloaded shell, and the axial compression load F is applied until buckling. This is repeated with a series of different perturbation loads, starting with the undisturbed shell and the respective buckling load F0. In Figure 3 buckling loads F depending on the perturbation loads P are depicted. The figure shows that the buckling load belonging to a perturbation load larger than a minimum value P1 is almost constant. A further increase of the pertubation load has no significant change on the buckling any more. The buckling load F1 is considered to be the design buckling load. This concept promises to improve the knock-down factors and allows designing any CFRP cylinder by means of one calculation under axial compression and a single-perturbation load. Within a DLR-ESA study, this approach was confirmed analytically and experimentally, cf. [2]. However, there is still the need for a multitude of further studies.

> 0 2 4 6 810 **Perturbation load P (N)**

**Perturbation load P (F)**

**line (c)**

**P1 = Minimum single perturbation load**

**P1**

**line (b)**

In general, tests or analysis results are sensitive to certain parameters as boundary conditions or imperfections. Probabilistic methods are a possibility to assess the quality of results. The stochastic simulation with Monte Carlo (e.g. [17]) allows the statistical description of the sensitivity of the structural behaviour. It starts with a nominal model and makes copies of it whereas certain parameters are varied randomly. The random numbers, however, follow a given statistical distribution. Each generated model is slightly different, as in reality.

Recently, probabilistic simulations found the way into all industrial fields. In automotive engineering they are successfully applied in crash or safety (e.g. [18]). Klein et al. [20] applied the probabilistic approach to structural factors of safety in aerospace. Sickinger and Herbeck [21] investigated the deployable CFRP booms for a solar propelled sail of a spacecraft using the Monte Carlo method.

Velds [22] performed deterministic and probabilistic investigations on isotropic cylindrical shells applying finite element buckling analyses and showed the possibility to improve the knock-down factors. However, setting-up of a probabilistic design approach still suffers by a lack of knowledge due to the incomplete base of material properties, geometric deviations, etc..

Arbocz and Hilburger [23] published a probability-based analysis method for predicting buckling loads of axially compressed composite cylinders. This method, which is based on the Monte Carlo method and first-order second-moment method, can be used to form the basis for a design approach and shell analysis that includes the effects of initial geometric imperfections on the buckling load of the shell. This promising approach yields less conservative knock-down factors than those used presently by industry.
