Classical, refined and component-wise analysis of reinforced-shell structures

This paper compares early and very recent approaches to the static analysis of reinforced-shell wing structures. Early approaches were those based on the pure semimonocoque theory along with the beam assumptions of the Euler Bernoulli and Timoshenko type. The recent approaches are based on a hierarchical, one-dimensional formulation. These are obtained by adopting various polynomial expansions of the displacement field above the cross-section of the structure according to the unified formulation which was recently proposed by the first author. Two classes were developed in the unified formulation framework. In the first class, Taylor expansion models were developed by exploiting N-order Taylor-like polynomials; classical beam theories (Euler-Bernoulli and Timoshenko) were obtained as special cases of Taylor expansion. In the second class, Lagrange expansion models were built by means of four- and nine-point Lagrange-type polynomials over the cross-section of the wing. The component-wise approach was obtained by using different four- and nine-point Langrangian descriptions for different wing components including panels, ribs, spar caps, stringers, and transverse ribs. The finite element method was used to develop numerical applications in the weak form. Finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not formally depend on the order and the expansion. A number of typical aeronautical structures were analyzed, and semimonocoque results were compared to classical (Euler-Bernoulli and Timoshenko), refined (Taylor expansion), and component-wise (Lagrange expansion) models. Stress and displacement fields of simple statically determinate, redundant, and open-section wing-box structures were analyzed. Finite element models by a commercial software that make use of solid and shell elements were used for comparison purposes. Results have highlighted the enhanced capabilities of the present refined and component-wise formulations. The present component-wise approach appears to be the natural tool to analyze wing structures because it leads to results that can only be obtained by the use of three-dimensional elasticity (solid) elements whose costs are at least one order of magnitude higher than component-wise cases. Component-wise models in conjunction with finite elements could be seen as a modern way of analyzing reinforced-shell structures by removing classical assumptions of constant shear in the spar webs and panels.