Topology Optimization of Elastoplastic Structure Based on Shakedown Strength

The traditional approach to structural lightweight optimization design, which is based on the elastic limit rule, often results in a structure that exhibits either weight redundancy or strength redundancy to some extent. This study introduces a novel integration of shakedown analysis with structural topology optimization, departing from the conventional elastic limit rule. Shakedown analysis identifies a non-failure external load region beyond the elastic limit but below the plastic limit, independent of loading history. The proposed method, for the first time, accounts for the influence of self-equilibrium residual stress at the element level, redefining effective and ineffective elements in topology optimization. Shakedown total stress replaces elastic equivalent stress, offering a comprehensive measure. Utilizing Melan’s lower bound theorem, a gradient-based topology optimization framework for shakedown analysis is developed, ensuring structures stay within the elastic-plastic range, preventing excessive plastic deformation. The approach, employing the moving asymptotes method after adjoint sensitivity analysis of shakedown total stress, is applied to a three-dimensional L-shaped bracket. Even with a remarkable 50% reduction in weight, the maximum total shakedown stress of the bracket reveals that it only increases by a modest 17.20% from its initial value. Moreover, compared to traditional topology optimization methods based on either elastic stress or stiffness, the proposed method based on total shakedown stress leads to a higher shakedown limit. Specifically, the configuration designed using the total shakedown stress exhibited increases of 2.01% and 9.82% in the shakedown limit compared to those obtained using stiffness and equivalent elastic stress, respectively. This suggests that the proposed method can effectively balance the trade-off between shakedown strength and structural stiffness, achieving a 2.01\% rise in shakedown strength with only a 2.24% compromise in structural stiffness. These findings highlight the method's effectiveness and potential, emphasizing the benefit of redefining effective and ineffective elements using shakedown stress in topology optimization.