A survey is presented of recently published papers in the field of optimum structural design of plates, largely with respect to the minimum-weight design of plates subject to such constraints as fundamental frequency maximization. It is shown that, due to the availability of powerful computers, the trend in optimum plate design is away from methods tailored to specific geometry and loads and toward methods that can be easily programmed for any kind of plate, such as finite element methods. A corresponding shift is seen in optimization from variational techniques to numerical optimization algorithms. Among the topics covered are fully stressed design and optimality criteria, mathematical programming, smooth and ribbed designs, design against plastic collapse, buckling constraints, and vibration constraints.

Optimum Structural Design with Plate Bending Elements—A Survey

Structure optimization at airframe level is mainly focused on sizing design variables detailing the thin-walled properties of aircraft structures. Typical design variables are cross sectional dimensions for 1D and 2D elements with an additional complexity brought by composite materials with their directional and multi-layer aspects. Even if the scope of these design variables is clear and well understood, the vision of the structure behaviour is multi-criteria and encompasses various fidelity levels. Its design requires several stages from the future project to the detailed definition of structural parts using various analysis tools from different disciplines. These several stages require adequate structural optimization processes to offer the best response with the right level of details to answer questions being sought at each maturity level of the design. A review of methods & tools developed and applied at AIRBUS to deliver automated sizing for aircraft structures along their development will be presented. Ways forward and major stakes for the future will be discussed.

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Numerical Optimization applied to structure sizing at AIRBUS: A multi-step process

Buckling sensitivity analysis and optimal design of thin laminated structures

Optimum design of composite wing structures by a refined optimality criterion

An examination of available shell finite elements suitable for buckling analysis of thin walled airframe structures leads to the selection of a simple, accurate, design-oriented element, which is, then, used with slight modifications to obtain explicit, closed form equations for the stiffness and geometric stiffness matrices. In turn, these equations are used to derive explicit expressions for the analytic sensitivities of stiffness and geometric stiffness matrices with respect to shell shape design variables. With analytic shape sensitivities of structural matrices and corresponding buckling eigenvalues at hand, the resulting new computer capability makes it possible to construct buckling constraint approximations for Approximation-Concepts based structural synthesis, as well as to examine sources of numerical noise which might appear when parametric studies or finite difference sensitivities are carried out using existing FE codes. The simplicity of the shell elements used and the elimination of the need to carry out numerical integration, lead to computational savings, especially when repetitive analyses have to be carried out during shape design optimization of typical airframes.The new capability is aimed at capturing both local and global modes of buckling failure with the same FE model. Subcomponent interaction during buckling can, thus, be taken into account during shape optimization of wing and fuselage structures. Numerical tests involving plates, thin walled channel sections and a complete wing box of a typical fighter airplane demonstrate the effectiveness and accuracy of the new design-oriented capability.

Finite Element-Based Analytic Shape Sensitivities of Local and Global Airframe Buckling Constraints

A rational multilevel approach for minimum weight structural design of truss and wing structures including local and system buckling constraints is presented. Overall proportioning of the structure is achieved at the system level subject to strength, displacement and system buckling constraints, while the detailed component designs are carried out separately at the component level satisfying local buckling constraints. Total structural weight is taken to be the objective function at the system level while employing the change in the equivalent system stiffness of the component as the component level objective function. Finite element analysis is used to predict static response while system buckling behavior is handled by incorporating a geometric stiffness matrix capability. Buckling load factors and the corresponding mode shapes are obtained by solving the eigenvalue problem associated with the assembled elastic stiffness and geometric stiffness matrices for the structural system. At the component level various local buckling failure modes are guarded against using semi-empirical formulas. Mathematical programming techniques are employed at both the system and component level.

L. A. Schmit

Papers

A multilevel approach for minimum weight structural design including local and system buckling constraints