Kinematic stability is an often overlooked, but crucial, aspect when mathematical optimization models are developed for truss topology design and sizing optimization (TTDSO) problems. In this paper, we propose a novel mixed integer linear optimization (MILO) model for the TTDSO problem with discrete cross-sectional areas and Euler buckling constraints. Random perturbations of external forces are used to obtain kinematically stable structures. We prove that, by considering appropriate perturbed external forces, the resulting structure is kinematically stable with probability one. Furthermore, we show that necessary conditions for kinematic stability can be used to speed up the solution of discrete TTDSO problems. Using the proposed TTDSO model, the MILO solver provides optimal or near optimal solutions for trusses with up to 990 bars.

当为桁架拓扑设计和尺寸优化（TTDSO）问题开发数学优化模型时，运动稳定性是一个经常被忽视但至关重要的方面。在本文中，我们针对具有离散横截面积和Euler屈曲约束的TTDSO问题，提出了一种新颖的混合整数线性优化（MILO）模型。外力的随机扰动用于获得运动学稳定的结构。我们证明，通过考虑适当的扰动外力，所得结构在运动学上是稳定的，概率为1。此外，我们表明运动学稳定性的必要条件可以用来加快离散TTDSO问题的解决速度。使用建议的TTDSO模型，MILO求解器可为高达990巴的桁架提供最佳或接近最佳的解决方案。