An Improved Strain Energy-Based Optimality Criteria Method for Weight-Optimal Sizing of Hyperstatic Structures Under Frequency, Buckling, or Displacement Constraints

This paper presents a rigorously derived optimality criteria (OC)-based sizing method for efficiently designing minimum-weight linear statically indeterminate (hyperstatic) frame structures subject to frequency, buckling, or multiple displacement constraints. By expressing the constraints in terms of elastic strain energy, the constrained problem is transformed into an unconstrained form using a Lagrange multiplier formulation. The first derivative test applied to the Lagrangian is used to derive novel OC, which generalizes previous approaches relying on the heuristic assumption of uniform strain energy density (SED) distribution across structural members, strictly applicable to statically determinate structures. Further, an iterative computationally efficient sizing algorithm is developed from the OC, whose accuracy and applicability are demonstrated in three case studies: a portal frame with a frequency constraint, a reticulated roof structure with a buckling constraint, and a cantilevered truss with multiple displacement constraints. Numerical results demonstrate the effectiveness of the proposed OC optimal sizing method and algorithm in converging to the global optimum compared to uniform SED-based OC approaches, which yield suboptimal heavier designs, and to global metaheuristic numerical optimization techniques, which are computationally much costlier.