Practical Plastic Layout Optimization of Trusses Incorporating Stability Considerations

Frequently, the topologies generated using conventional discrete structural layout optimization methods may be shown to be in unstable equilibrium with the specified applied loading. Instability may occur due to the lack of lateral bracing at a node lying within a line of parallel compression members (or ‘chain’), or at the end of a compression member. In practice, this would typically lead either to complete failure of a framework or to a failure which is localised to a small section of the overall structure. Previously proposed methods to overcome this problem have tended to suffer from either a lack of rigour in identifying unstable solutions, a failure to properly consider all means of ensuring the stability of a framework, or are likely to be very computationally expensive. This paper discusses the drawbacks of current methods and develops a robust and efficient alternative method capable of identifying optimum solutions which do not contain unstable nodes. The method involves the application of nominal lateral force load cases at unstable nodes in a plastic linear programming problem formulation, hence requiring the introduction of bracing members or the consideration of alternative, stable, layouts. The magnitude of these disturbing forces is calculated such that it complies with current British and European codes of practice. Additionally, the problem of including Euler buckling considerations in linear, plastic optimization is discussed. It is shown that the relationship between axial force and minimum required available section size for certain structural sections is quasi-linear and that linear approximations may be used to determine suitable stress when sizing compression members. Furthermore, an iterative stress-adaptation method may be used to ensure that members are adequately proportioned against Euler buckling and slenderness limits. Examples of stable solutions identified by this approach are presented, along with proposed future developments to increase the practicality and efficiency of the method. In particular, a benchmark problem is examined and is shown to have a significantly lower optimum volume when using the proposed method than that achieved using an alternate algorithm.