# Topology Optimisation in Structural Steel Design for Additive Manufacturing
## 5 Topology Optimisation in Structural Design of Steel Elements and Joints
### Buckling and Local Instability Phenomena  
Member buckling phenomena has been addressed in Discrete Optimisation for a long time. However, within the Optimisation of Continua, multiple global and local, real and fictional, buckling modes hampered progress after precursory works undertaken in Instituto Superior Tecnico [437]. Some notorious problems have been found in the appearance of buckling associated with low-density regions, high local stresses, which add to the repetition of modes, convergence problems, and the need for an extensive computational capacity for dealing with so many modes [137; 388; 438].  
Recent advances already allow accounting for buckling in TO endeavours in a computationally feasible manner [438; 439; 440; 441; 442], mitigating some of the aforementioned problems.  
Figure 11: Examples of topologically optimised and additively manufactured metallic connections for civil engineering structures. Left to right and top to down: (**a**) Non-TO and TO joint specimens by ARUP [355]; (**b**) several joints designed in the University of Leeds [139]; (**c**) wire and arc additive manufacturing (WAAM) printed bolted joint in the Technical University of Darmstadt [422] (reproduced under the Creative Commons Attribution-NonCommercial-NoDerivs License); (**d**) WAAM printed node in the Technical University of Darmstadt [422] (reproduced under the Creative Commons Attribution-NonCommercial-NoDerivs License); (**e**) joint specimens topologically optimized with the bi-directional evolutionary structural optimisation (BESO) method for axial loads and bending in the RMIT University [427].  
Though, those methods rely on linearized buckling, which is generally regarded as an inadequate simplification for many engineering problems [137; 443; 444] and therefore, face significant opposition from many researchers. Adding to such a discussion, a recent research paper using non-linear pre-buckling analyses in the context of microstructural design [445] found that under certain loading conditions, linear and non-linear buckling analyses yield similar results. Considering the knowledge gathered by structural engineers in linear and non-linear buckling over time, it is expected that such a conclusion may be proven valid for several other cases.  
On the other hand, advances have been made in the last few years on the non-linear buckling of topologically optimized continua [446; 447]. Such remarkable achievements employ non-incremental analyses and recursive design.  
Despite the mentioned problems, under which buckling phenomena can only be regarded as a mostly unsolved problem in the TO of continua, some practical applications found simpler or more sophisticated strategies for modelling instabilities in optimisation methods. That is the case of [PERSON] work on local instability in optimised aluminium cross-sections [341].  
Regarding Discrete Optimisation, methods for considering buckling also evolved. Among a recent contribution, one can highlight the work by [PERSON] and [PERSON] on trusses [397; 448], [PERSON] et al. suggested a practical approach for TO in tensegrity structures [449; 450], as well as the research by [PERSON] et al. [451] on methods for mitigating member instability in reticulated structures.