Structural Optimization and Stability

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AbstractIn many mechanical and aerodynamic systems structural strength, stability and lightness are of primaryimportance. The stability of a structure with a specified support and external loading can be characterized byan eigenvalue problem whose solution determines the non-trivial equilibrium configurations of the system.The maximum external load that can be imposed on the structure is determined by the least eigenvalue of thesystem. The problem of designing an optimal structure that can support the largest load is more involved. It requiresthe determination of structural parameters, e.g., the variable cross-sectional area of the structure, whichmaximize the least eigenvalue of the system. An important example is due to Lagrange, who attempted todetermine the shape of the strongest column with a specific length and mass. This apparently simple problemremained unsolved for more than one hundred and fifty years. Its analytical solution is still surrounded bycontroversy. The main thrust of this research is that the standard optimization procedure is by no means the best way toaddress the structural optimization problem involved. Depending on the failure criterion, there is a certainrelation between the structural mode of failure and the physical parameters of the optimal structure. In theLagrange problem the maximum bending stress in the ideal column is constant along its length. This conditionimplies that the square of the fundamental buckling mode is proportional to the cube of the cross-sectional areafor the optimal column. This relation was vital in determining the exact analytical solution to the problem. Italso plays a major role in the new methodology proposed here, which consists of two stages:(a) Determining the relation between the mode-shape of failure and the physical parameters of the optimalstructure. This relation leads to an eigenvalue problem where elements of the stiffness matrix are knownfunctions of an eigenvector of the system, and(b) Solving the inverse eigenvalue problem of determining the unknown parameters in the stiffness matrixsubject to the eigenvector constraint.The project has practical engineering applications in new and emerging technologies related to fabricationof microstructures and their integration in new generation of mechanical components. Another importantapplication of the proposed research is in design of safe, reliable, and lightweight aerospace structuresThe proposed activity will also contribute to the curriculum development of Machine Design Lab andStress Analysis. The students will take part in a design competition involving optimal design of structuresThe project will support disadvantaged minority students by (a) directly involving them in the research,and (b) stimulating their interest in advancing themselves to pursue graduate study