Issue 78

A. Gallo et alii, Fracture and Structural Integrity, 78 (2026) 489-503; DOI: 10.3221/IGF-ESIS.78.28

the imposed constraints ensure admissible and physically meaningful values. A rigorous identification would require additional experimental investigation of damage evolution, which is beyond the scope of the present optimization study.

S HAPE OPTIMIZATION OF A METAL HOLLOW PLATE IN BENDING

T

he third case study investigates shape optimization of a hollow plate subjected to bending conditions and aims to assess the consistency and broad applicability of the proposed optimization algorithm to structural optimization problems. Unlike previous case studies, the optimization variables describe geometry and position of the hole, thus influencing the structural layout and load-carrying capability of the system. FEM analyses were carried out using ANSYS coupled with MATLAB, as previously described. The domain was discretized using four-node quadrilateral plane stress elements, ensuring an adequate balance between numerical accuracy and computational efficiency. The adopted mesh density, shown in Fig. 6 and consisting of 26454 elements and 29819 nodes, was satisfactory to accurately describe stress concentration effects due to the presence of the hole. Boundary conditions were defined to reproduce the bending behaviour of the plate through a self-balancing force system, whereas appropriate displacements constraints prevented rigid body motion (Fig. 6).

Figure 6 : Finite element mesh and boundary conditions of the hollow plate. The plate length and width are fixed and respectively equal to ൌ 1200 and ൌ 800 , while other geometrical entities and mechanical linear elastic properties of steel are reported in Tab. 3 alongside the optimized values. All boundary conditions were consistently updated during the optimization process to account for changes in the hole diameter and position, ensuring meaningful comparison among different design configurations. The objective was to minimize the plate mass while an equality constraint was introduced to preserve the maximum von Mises stress to initial geometry reference value. The structural mass ഥ of hollow plate can be expressed in terms of material density , plate thickness s , and area A : ഥ ൌ | ௦ୀଵ ൌ ሺ െ ଶ ሻൌ ሺ ሻ (26) The equality constraint is expressed imposing equal to zero the difference between computed current configuration, ௙௩௢௠௟௟௢௪௜௡௚ , and reference, ௥௩௘௠௙௘௥௘௡௖௘ , stress values of maximum von Mises stress: ௙௩௢௠௟௟௢௪௜௡௚ ൌ ௥௩௘௠௙௘௥௘௡௖௘ → ∆ ൌ ௙௩௢௠௟௟௢௪௜௡௚ െ ௥௩௘௠௙௘௥௘௡௖௘ ൌ 0 (27) By introducing the Lagrange multiplier λ , the constrained optimization problem was solved by minimization of Lagrange function ℒ , similarly to the pseudo-objective function introduced in the previous paragraph [7,11,14,18]: ℒ ൌ ഥ െ ൌ ൫ , ௙ , ℎ ௙ , ൯ (28)

501