Keywords: postbuckling, composite stiffened panels, skin buckling, global buckling, strip elements, strength criterion of Puck.

It is well-known that the load carrying capacity of stringer stiffened composite panels is not exhausted when the local skin buckling load is reached. Typically, the skin buckling load can be exceeded significantly before collapse occurs. Therefore, fast and accurate methods to analyse the postbuckling behaviour are required for the design of such panels.

In general, commercial finite element tools enable an accurate simulation of the postbuckling behaviour, but the computational effort is very high as a detailed idealisation of the structure must be used. To reduce the computational effort and to provide a method that is also applicable in early design stages, an approach for the investigation of the postbuckling behaviour of stringer stiffened composite panels is developed. This approach is based on the analytical solution of an appropriate plate resp. shell theory [1] and enables the analysis of the behaviour of flat as well as of curved panels.

The main idea of the approach is to reduce the number of degrees of freedom considerably compared to a finite element idealisation by the use of strip elements, i.e. the panels are only discretised in the circumferential direction, while no discretisation is required in the longitudinal direction (direction of stiffeners).

For each element of the structure the element stiffness matrix is determined in the following way: Considering large displacements, an appropriate shell resp. plate theory is used to set up the system of partial differential equations for curved resp. flat elements. To describe the buckling and postbuckling behaviour of the panels in the longitudinal direction, trigonometric functions are applied [2]. This leads to a reduction of the system of partial differential equations to a system of ordinary differential equations. The solution of the differential equations in the circumferential direction is found by numerical integration. Finally, the resulting system of algebraic equations is transformed in order to obtain the element stiffness matrix. Knowing the element stiffness matrices, the total stiffness matrix is found by the application of the direct stiffness method.

Based on the determination of the total stiffness matrix, the same methods as in finite element codes are used in order to carry out linearised buckling analyses (computation of skin buckling and global buckling load) as well as nonlinear analyses of the postbuckling behaviour. In particular, both, a Newton iteration as well as an arc-length procedure [3] are implemented to perform the computation of the structural response in the postbuckling region. Based on the results of the postbuckling analysis, the stress distribution in the structure is computed according to the classical lamination theory and it is checked whether material failure occurs in the postbuckling region using the strength criterion of Puck [4].

In this paper details of the underlying theory regarding the linearised buckling analysis as well as the postbuckling analysis are given. Furthermore, the approach is applied to a flat stiffened composite panel [5] and the results are compared to corresponding finite element solutions using the code NASTRAN. For both, the buckling loads obtained by the linearised buckling analysis and the stress distribution in the postbuckling region a good agreement with the finite element results is found. The most critical areas in the structure are observed by the strength criterion of Puck and it is found that the panel is able to work in the postbuckling region up to global buckling without material failure.

1

Rittweger, A., "Statik, Stabiliät und Eigenschwingungen anisotroper Rotationsschalen beliebigen Meridians mit der Übertragungsmatrizenmethode", Phd thesis, Lehrstuhl und Institut für Leichtbau RWTH Aachen, 1982.

2

Riks, E., "Buckling and Postbuckling Analysis of Stiffened Panels in Wing Box Structures", International Journal of Solids and Structures, 37, 6795-6824, 2000. doi:10.1016/S0020-7683(99)00315-7

3

Riks, E., "An Incremental Approach to the Solution of Snapping and Buckling Problems", International Journal of Solids and Structures, 15, 529-551, 1979. doi:10.1016/0020-7683(79)90081-7

4

Puck, A., Schürmann, H., "Failure Analysis of FRP Laminates by Means of Physically Based Phenomenological Models - Part A", Composites Science and Technology, 58, 1045-1067, 1998. doi:10.1016/S0266-3538(96)00140-6

5

Lanzi, L., Bisagni, C., "Postbuckling Experimental Tests and Numerical Analyses on Composite Stiffened Panels", Proceedings of 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, Norfolk (USA), AIAA-2003-1795, 2003.

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