Optimal structural design of helical springs with Ludwik-type elastic–plastic materials

Motivated by the limitations of idealized power-law assumptions in spring design, this work revisits the optimization of compressive helical springs using a more realistic Ludwik-type elastic–perfect plastic material model. Unlike earlier approaches, we explicitly incorporate the pitch angle in computing the total wire length, improving geometric accuracy. A unified root-solving algorithm is introduced to handle the Karush–Kuhn–Tucker conditions efficiently, eliminating the need for case-by-case treatment. The proposed design is benchmarked against the DIN standard, which is often overlooked in analytical studies. To ensure practical relevance, finite element simulations are performed in COMSOL and show good agreement with theoretical predictions. The combination of refined geometry, nonlinear mechanics, and comparative validation provides a more robust optimization framework that bridges theoretical modeling with engineering practice. We believe this approach offers new insight into spring design for advanced structural materials.