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<p>This paper applies well-established shape optimization techniques to obtain undercut forms for shaft shoulders subjected to axial, bending, torsion, and combined loads aiming at minimization stress concentration factors. The design variables, based on DIN 509 standard forms, were used to model the undercut as an elliptical shape with displacement and rotation. Inequality constraints were imposed on the design variables to guarantee that the optimal form obtained is manufacturable. The optimization problem was solved using ANSYS®, applying a hybrid optimization method, and the boundary value problem associated with the stress field was obtained by the finite element method with three-dimensional modelling of the shaft. The results showed that the geometry of the optimal undercut shapes indicated a dependence on the type of load applied, and the proposed scheme was successful in specifying optimal design variables that can be manufactured. Comparisons between the optimization undercut shapes obtained&#xa0;—&#xa0;with both the simple fillet and a form chosen from DIN 509 standard&#xa0;—&#xa0;showed reductions, respectively, of up to 49% and 40% in stress concentration factors. Additionally, a tendency of stress gradients smoothing along the undercut’s path leads to a quasi-uniform stress field. Above all, the findings of this work are relevant for design of shafts in real applications, as the optimal forms achieved perform better in terms of stress concentration factors than the forms provided in the current DIN 509 standard.</p>

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Shape optimization applied in minimizing stress concentration factors: a case study for manufacturable shaft design

  • 期刊论文

  • 发表日期：

    2025年1月11日

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  • João Luiz do Vale,
  • Jéderson da Silva

摘要

This paper applies well-established shape optimization techniques to obtain undercut forms for shaft shoulders subjected to axial, bending, torsion, and combined loads aiming at minimization stress concentration factors. The design variables, based on DIN 509 standard forms, were used to model the undercut as an elliptical shape with displacement and rotation. Inequality constraints were imposed on the design variables to guarantee that the optimal form obtained is manufacturable. The optimization problem was solved using ANSYS®, applying a hybrid optimization method, and the boundary value problem associated with the stress field was obtained by the finite element method with three-dimensional modelling of the shaft. The results showed that the geometry of the optimal undercut shapes indicated a dependence on the type of load applied, and the proposed scheme was successful in specifying optimal design variables that can be manufactured. Comparisons between the optimization undercut shapes obtained — with both the simple fillet and a form chosen from DIN 509 standard — showed reductions, respectively, of up to 49% and 40% in stress concentration factors. Additionally, a tendency of stress gradients smoothing along the undercut’s path leads to a quasi-uniform stress field. Above all, the findings of this work are relevant for design of shafts in real applications, as the optimal forms achieved perform better in terms of stress concentration factors than the forms provided in the current DIN 509 standard.