Keywords: structural optimization, size optimization, self-weight, beam structure, inexact restoration, nonlinear programming.

This paper investigates the size optimization of beam structures under self-weight loading. The study formulates a gradient-based optimization framework incorporating first-order sensitivity analysis of both the objective function and constraints. A key challenge arises from the nonlinear dependence of critical internal forces on cross-sectional geometry. Two formulations of the ultimate limit state strength constraints are compared: a bounded format, which directly limits internal forces, and a relative format, which expresses constraints as ratios of internal forces to resistances. The paper provides full analytical derivatives for both formats, emphasizing their linearization properties. The proposed methodology is tested on both statically determinate and indeterminate structures, enabling a systematic evaluation of constraint behavior, convergence robustness, and computational efficiency. Results show that while the bounded format offers better constraint control for large sections, it may cause infeasibility near small designs. Conversely, the relative format improves robustness across a wider design space. The findings highlight important trade-offs between formulation styles and provide guidance for selecting constraint formats in self-weight-dominated structural optimization problems.

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