Abstract
Transmissible loads are external loads defined by their line of action, with actual points of load application chosen as part of the topology optimization process. Although for problems where the optimal structure is a funicular, transmissible loads can be viewed as surface loads, in other cases such loads are free to be applied to internal parts of the structure. There are two main transmissible load formulations described in the literature: a rigid bar (constrained displacement) formulation or, less commonly, a migrating load (equilibrium) formulation. Here, we employ a simple Mohr’s circle analysis to show that the rigid bar formulation will only produce correct structural forms in certain specific circumstances. Numerical examples are used to demonstrate (and explain) the incorrect topologies produced when the rigid bar formulation is applied in other situations. A new analytical solution is also presented for a uniformly loaded cantilever structure. Finally, we invoke duality principles to elucidate the source of the discrepancy between the two formulations, considering both discrete truss and continuum topology optimization formulations.
Highlights
In classical structural layout and topology optimization formulations the locations of external loads to be carried by the structure are specified in advance
An alternative ‘migrating loads’ approach to layout optimization with transmissible loads has been developed by Gilbert et al (2005) and Darwich et al (2010) as an extension of the classical plastic, linear programming (LP), based ground structure formulation (after Dorn et al (1964))
Since the optimal structure includes T-type regions, the end result shown in Fig. 5a is erroneous; this appears to comprise a central funicular R-type region with geometry defined by θPR− ∈ [−π/4, π/4], spanning between two T-type regions in which the structural members are aligned at ±π/4 to the line of action of the transmissible load, and artificially braced by the presence of the zero-cost vertical rigid bars
Summary
In classical structural layout and topology optimization formulations the locations of external loads to be carried by the structure are specified in advance. An alternative ‘migrating loads’ approach to layout optimization with transmissible loads has been developed by Gilbert et al (2005) and Darwich et al (2010) as an extension of the classical plastic, linear programming (LP), based ground structure formulation (after Dorn et al (1964)) In their formulation, additional LP variables were introduced to represent potential loads applied at each node present along the line of action of every given transmissible load. The resulting insight is shown to apply to the continuous topology optimization formulation of Fuchs and Moses (2000)
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