2 nd International Conference on Engineering Optimization September 6 - 9, 2010, Lisbon, Portugal 1 Buckling optimization of composite stiffened panels: some important issues Claude Fleury 1 , Michaël Bruyneel 2 , Benoît Colson 2 , Alain Remouchamps 2 1 Aerospace & Mechanical Department, University of Liège, Liège, Belgium, [email protected] 2 SAMTECH s.a., Liège Science Park, Angleur, Belgium, [email protected] Abstract In this paper, the buckling optimization of thin-walled stiffened composite panels is studied. Some important issues are recalled, and an efficient solution procedure is discussed. It is shown how to properly represent the buckling behavior with the finite element method. Several approximations of the Sequential Convex Programming method (SCP) are compared on an industrial test case consisting in the optimization of a curved stiffened composite panel submitted to compression and shear. Keywords: buckling, post-buckling, stiffened panels, composite, SCP 1. Introduction When compression and shear are present in a structure, it must be designed to withstand buckling [1]. Despite a great deal of effort over the last decades dedicated to this topic [2,3,4], handling buckling optimisation for industrial applications is still an issue. Oscillations usually appear during the iterative process of minimizing the mass for buckling loads larger than a prescribed value, leading to a slow convergence process or even worst no convergence at all [5]. Mode switching [6,7], multiple eigen-values [8] and local or global influence of certain modes make the problem more complicated. Moreover, the reliability of a linear buckling analysis based on the eigen-values is questionable for structures capable of withstanding large displacements observed in the post-buckling range, or assuming a limit point in the equilibrium path. To simulate such behaviors and approach reality, a non linear analysis is needed, which requires a specific continuation method [9] for identifying the collapse (limit) load of the structure. Lighter and safer composite structures may be obtained by simulating buckling, post-buckling and collapse. Solving such problems remains however challenging. Proposals for an efficient solution to this problem are relatively new, since buckling, post-buckling and collapse optimizations have been of interest to researchers quite recently [10-16]. The purpose of this paper is twofold. On one hand, attention will be focused on how to properly represent the buckling behavior of composite stiffened panels (local/global, linear/nonlinear). On the other hand, algorithms suitable to optimization will be discussed. The Sequential Convex Programming approach [17-29] is adopted. The various concepts and issues discussed here are supported through an illustrative example. However they are still valid for real life problems such as the optimization of wings, fuselages or even the full airplane. Constraints of the optimization problem are formulated in the form of buckling and collapse reserve factors: the buckling reserve factors results from a linear finite-element analysis, and the collapse reserve factor is computed by a nonlinear finite element simulation. The structural analyses are conducted with the SAMCEF finite element code [30]. The BOSS Quattro optimization toolbox [31] is used to set up and solve the optimization problem. 2. Structural stability analysis 2.1. Structural analysis Classically the buckling load factors are obtained by solving an eigen-values problem around a linearized configuration. In the formalism of the finite elements, it comes that: ( 0 = - j j Φ S K λ m j ,..., 1 = (1) where K and S are the stiffness and initial stress stiffness matrices, respectively. λ j and Φ j are the buckling load factors and buckling modes, respectively. The reliability of a linear buckling analysis is questionable for structures capable to withstand large displacements observed in the post-buckling range, or assuming a limit point in the equilibrium path. To simulate such behaviours and approach reality, a non linear analysis is needed. This requires a specific continuation method [9] for the identification of the collapse (limit) load of the structure. In this case, a system of non linear equations is solved, together with an additional constraint imposing to stay on the equilibrium path, even if it is unstable:
Buckling optimization of composite stiffened panels: some important issues
Jun 29, 2023
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