Structural analysis of box beams using symbolic manipulation technique

Abstract

The aeroelastic analysis of aircraft wings requires an accurate determination of the influence coefficients. In the past, energy methods have been commonly used to analyze box-type structures and the results have been found to agree well with the experiments. However, when analysis of large wing-type structures is desired, it becomes necessary to automate the energy method. In this article, a method has been developed based on symbolic manipulation as an automated technique to find solutions to box-type structures. Various manipulations required for the energy method have been automatically implemented in a computer program with solutions available at each stage in a symbolic form. The numerical results for several example problems have been compared with alternate theoretical as well as experimental results. Good agreement has been noted in all the cases considered in this article. ATRIX methods have long been used for the structural analysis of complex structures. With the availability of high-speed computers, it is now possible to use the matrix- based finite-element method in the analysis and design of very complex structures. In particular, the finite-element method is being extensively used as an analysis tool to verify test data on many practical engineering applications. Also, as a result of enormous effort in this area, several efficient finite-element codes have been developed to meet the ever-increasing de- mand for accurate theoretical modeling, analysis, and results. A review of the finite-element method in connection with the analysis of highly indeterminate complex structures is pre- sented in Ref. 1. For wing-type structures, conventional methods of analysis using the simple bending theory of beams do not yield good results. To overcome this difficulty, the energy method has been commonly used in the past.2 This energy method is based primarily on equilibrium considerations with the use of elastic strain energy in the structure for deriving the displacements. In the case of highly indeterminate structures such as aircraft wings, the equilibrium conditions alone are not sufficient to solve for the displacements. Therefore, the condition that the actual displacements and the stress distribution correspond to the minimum of strain energy is used in addition to the equi- librium conditions to get the displacement and to further eval- uate the stresses. A review of the literature indicates that in addition to the energy method Klein's method has been used for the analysis of aircraft wing structures. This method is based on the so- lution of a set of equations consisting of equilibrium and force- displacement relations of all the elements of the wing. This method was developed by Klein 3 and was used to analyze wings subjected to bending-type loadings. In the original de- velopment of the method for semimonocoque structures, it was assumed that the wing structure was composed of uni- form, nontapered axial elements (spar, rib, and web chords) taking only axial loads and constant thickness sheets (cover