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A strain tensor method for three-dimensional Michell structures

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Abstract

In the design of discrete structures such as trusses and frames, important quantitative goals such as minimal weight or minimal compliance often dominate. Many numerical techniques exist to address these needs. However, an analytical approach exists to meet similar goals, which was initiated by Michell (1904) and has been mostly used for two-dimensional structures so far. This paper develops a method to extend the existing mainly two-dimensional approach to apply to three-dimensional structures. It will be referred as the Michell strain tensor method (MSTM). First, the proof that MSTM is consistent with the existing theory in two dimensions is provided. Second, two- and three-dimensional known solutions will be replicated based on MSTM. Finally, MSTM will be used to solve new three-dimensional cases.

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Notes

  1. This definition relates to truss-like structures. As recently shown in Sigmund et al. (2016), other kind of structures, such as plate- or shell-like structures with varying thicknesses, may be more optimal in terms of stiffness or compliance than their truss-like counterparts.

  2. These rules, and MSTM by derivation, are valid in T-regions only, defined in Sokół and Rozvany (2012) or Lewińsky and Sokół (2014) as regions where Michell structures have orthogonal tension and compression elements.

References

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Benjamin P. Jacot & Caitlin T. Mueller

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  1. Benjamin P. Jacot
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  2. Caitlin T. Mueller
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Correspondence to Benjamin P. Jacot.

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Jacot, B.P., Mueller, C.T. A strain tensor method for three-dimensional Michell structures. Struct Multidisc Optim 55, 1819–1829 (2017). https://doi.org/10.1007/s00158-016-1622-z

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