Keywords: composite structures, postbuckling, perturbation methods, global-local methods.

Most of the real life structural designs are complex assemblies of thousands of components. An engineering approach is to use an idealized simulation model of the components. In the design process, a coarse finite element model of the whole structure is used to determine the loads in each component and the components are designed independently. This multilevel approach or global-local approach [1,2] works well only if a linear static behavior is considered because to predict the load paths for nonlinear behaviour we require a very fine finite element mesh, which is computationally expensive. Apart from this, the design optimization procedure necessitates several analyses of the complete model. This significantly increases the computation cost.

The most important step in structural design is to predict the load paths in the structure because based on these internal loads the structural subcomponents are designed. A highly idealized and simplified global model is used to predict the load paths in the structure. For example at the preliminary stage of the design of fuselage or wing structure, each panel between the stiffeners or ribs or spars are modeled via a single element to avoid the computational overhead associated with full scale finite element analysis. The aim of the global analysis is to give the internal load distribution. Based on these internal loads, the subcomponents are designed against strain, stability or buckling, and manufacturing constraints. This works well in the linear static behavior. In practice, however, it is customary to allow the secondary load carrying components (like skin panels of fuselage) to go beyond their critical load where the linear analysis is no longer true. So, a linear global finite element analysis is not accurate enough to predict the load paths where some or all of the subcomponents have buckled. Murphy et al. [3] proposed a nonlinear idealization for axially loaded structures only. The strategy is based on representing a subsection with a single one dimensional nonlinear spring element in the global model. A detailed finite element analysis of the subsections is performed to generate the spring data. The proposed methodology cannot be extended very easily to biaxially loaded structures or structures under bending loading.

This paper focuses on developing a general methodology to predict the approximate load path of a loaded composite structure with some or all buckled components. After buckling, though the stiffness of the structure reduces considerably, the load response is essentially linear. This strategy is based on taking advantage of the linear postbuckled load response by quantifying the reduced stiffness of the structural subcomponent and using it in linear global finite element analysis. Since the linear analysis is inexpensive compared to full scale nonlinear analysis, the methodology promises considerable time saving.

The nonlinear equilbrium equations are derived usingt a von Kármán plate model. The total potential of the plate is derived in terms of edge displacements. Since at the bifurcation point, the derivative does not exist a perturbation technique is applied to derive the postbuckled stiffness expressions. The load path prediction scheme consists of three steps. First, determine the buckling factor of all the structural components. Second, reduce the applied loading linearly so that only the component having the lowest buckling load buckles. Perform a linear analysis with all the components having prebuckling stiffness. Next, apply an incremental load that will buckle the component, which was second lowest in terms of buckling capacity. Perform another linear analysis under the incremental load with the prebuckling stiffness of the buckled component(or s) replaced by postbuckled stiffness. Continue repeating the second step unless the total applied load is reached. Third, add the response obtained in all the linear analyses performed in step 2 to give the final load paths. Three different examples are presented to demonstrate the procedure. The first example is a simply supported flat composite panel subjected to different edge displacement compression loading. We observe that the present approach gives good correspondence with the full nonlinear analysis in predicting the load paths provided they are not close to zero. The second example is a two panel arrangement with different thickness subjected to end loadings. The result indicates that the predicted load paths are in good agreement with full nonlinear analysis. The third example is a four panel composite wing structure subjected to bending load. Here, the top skin panels are under compression due to bending load and known to undergo local buckling. For the loading case solved, we observe that the proposed methodology can very well be extended to such complex loading cases. In this case, two of the root panels in top skins have buckled and the load paths predicted in the postbuckled regime are in good agreement with the full nonlinear analysis.

1

L.A. Schmit, R.K. Ramanathan, "Multilevel Approach to Minimum Weight Design including Buckling Constraints", AIAA Journal, 16(2), 97-104, 1978. doi:10.2514/3.60867

2

J. Sobieszczanski, B.B. James, A. Dovi, "Structural Optimization by Multilevel Decomposition", AIAA Journal, 28(3), 1175-1182, 1985. doi:10.2514/3.9165

3

A. Murphy, M. Price, A. Gibson, C.G. Armstrong, "Efficient Nonlinear Idealisations of Aircraft Fuselage Panels in Compression", Finite Elements in Analysis and Design, 40, 87-96, 2004. doi:10.1016/j.finel.2003.11.009

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