Analytic shape sensitivities and approximations of local and global airframe buckling constraints

Abstract

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 the 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. Sub-component interaction during buckling can, thus, be taken into account during shape optimization of wing and fuselage structures. Numerical tests involving isotropic and laminated 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. Also the reduced order eigensystem which takes modeshapes at the reference design variable as the basis vectors for the pertubed design is derived and compared to the Rayleighy Quotient approximation.