Keywords: composite material, reliability, optimization, finite elements.

Due to their high stiffness and low mass, composite materials take more and more importance in the design of plane structures (e.g. wings, tails). Their extension to the design of fully composite fuselage is under development.

The great increase in computer performance contributes to the generalization and to the intensive use of optimization methods and their application to composite structures.

However these optimization methods lead to deterministic optimum designs, and don't include the variability of the design parameters. These configurations ignoring the different uncertainties can result in unreliable designs. The origin of uncertainties can be for example geometrical, modeling, simulation or manufacturing.

Several authors recently addressed these uncertainties. Moreover the reliability studies taking into account the uncertainties were often coupled with optimization strategies, leading to the Reliability Based Design Optimization methods (RBDO). Many studies use the reliability index and FORM/SORM approaches to evaluate the failure probability. The necessity of working in two different spaces (physical space and normalized Gaussian space) is the main drawback of these method. Numerous computations are needed to converge to the optimized design. In our paper we combine a reliability analysis with an optimization one, using the response surface approximation. This method is applied to a composite cylinder, with longitudinal and transversal stiffeners (a portion of fuselage). The skin and the stiffeners are both made of composite material. The first buckling load is considered as the dimensioning criteria.

Firstly we perform a sensitivity analysis with some parameters (e.g. material parameters, geometrical parameters). The number, position and geometry of the longitudinal stiffeners appear to be the more influential parameters and are considered in the further calculations. The continuous parameters are supposed to be uncertain with a normal distribution. In order to be consistent with the manufacturing process and to limit the number of parameters, we define four zones in the fuselage. The stiffeners have the same profile in all the zones.

The objective of the optimization problem is to set the failure probability lower than an acceptable value . The repartition of the longitudinal stiffeners, given by the number of stiffeners in each region is the main parameters. The algorithm proposed uses the inertial properties of the omega profiles of the longitudinal stiffeners. A maximization of the first critical load will automatically lead to minimize a geometrical parameter (which increases the inertia, and therefore the stiffness) in the buckled zones. The main interest of this propriety is that is a continuous parameter which allow us to easily use the response surface methodology.

The algorithm consists of three steps:

1. Building of the response surface for varying between and
2. Determination of the failure probability by a statistical exploration of the response surface. The influence of the parameters is also estimated by ANOVA analysis.
3. Variation of the number of stiffeners in function of the value and influence of the at the optimum point of the response surface.

These three steps are run until the convergence criterion is satisfied.

All the optimization and statistical analysis are computed on the response surface obtained by a Latin Hypercube Sampling and quadratic approximation. The Monte-Carlo simulation and the optimization are immediate as the approximate function is analytically defined.

The methodology proposed provides good results. the first critical load is improved by almost 20%, when the mass increases by only 2.2%. The failure probability of the optimized structure is acceptable (0.125%) and the influence of each zone of the cylinder section is established.

The work developed in this paper is a practical way to reach a reliable objective for the buckling behavior of such structures without enormous calculation cost nor increase of the mass.

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