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On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight

Discussion on the Michell class concept

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Abstract

The details of the Theorem of Michell are examined. Some remarks are made about the quantity of structure and the Maxwell number of a framework and about the Michell optimality criterion. The two-loads bending problem is examined, comparing solutions derived from another one found by Sokół and Lewiński (Struct Multidisc Optim 42:835–853, 2010) with others obtained from simulated annealing (SA) search. The collected data suggest that the Michell theorem is a sufficient test for a framework to be optimal, but maybe no necessary. As a consequence, there could exist problems for which the theorem is useless.

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Notes

  1. We find minor mistakes in the Table 1 of Sokółand Lewiński (2010): x D/d and y D/d must be x D/L and y D/L accordingly with the proportions in the Fig. 20 of the paper. The SA search confirms the authors’ layout and cost, although the best result obtained by SA was of \({\cal{Q}}/(4PL) =0.953814 \), with an error of up to 1.2% respect to the analytical result of the authors.

  2. The complete geometry and connectivity definition of the SA solutions—and others benchmarks structures— can be downloaded from the page http://habitat.aq.upm.es/gi/mve/dt/.

  3. \({\cal{Q}}\) is reduced by \(\alpha=\sqrt{({\cal{Q}}^{\perp}-\Delta {\cal{Q}}^{\perp})/{\cal{Q}}^{\|}}\), where \({\cal{Q}}^{\perp}\) accounts for the horizontal part of \({\cal{Q}} = |{q}_{i}|\ell_{i}\), as \({\cal{Q}}^{\|}\) does for the vertical one, being \(\Delta {\cal{Q}}^{\perp}\) the horizontal part cancelled removing members. Note that this reductions is obtained with new y coordinates: y new = αy.

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Acknowledgments

The writers are indebted with Prof. Jesús Ortiz (Madrid) for his enlightening in the subtle concepts under the field T. They also thank Blanca Estevan, María Cifuentes, Carlos Vázquez, Pilar Vázquez and Elena Moreno, who made a carefully reading and reviewing of the English expression of their ideas. The present paper was prepared within the one-year staying of Vázquez at CIMNE (Universidad Politécnica de Barcelona) under the surveillance of Dr. Eugenio Oñate, financed by the Universidad Politecnica de Madrid.

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Authors and Affiliations

  1. CIMNE—International Center for Numerical Methods in Engineering, Edificio C1, Campus Norte UPC, C/Gran Capitán s/n., 08034, Barcelona, España

    Mariano Vázquez Espí

  2. Departamento de Estructuras de Edificación, Universidad Politécnica de Madrid, Escuela Técnica Superior de Arquitectura, Avda. Juan de Herrera, 4., 28040, Madrid, España

    Jaime Cervera Bravo

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  1. Mariano Vázquez Espí
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  2. Jaime Cervera Bravo
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Correspondence to Mariano Vázquez Espí.

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Vázquez Espí, M., Cervera Bravo, J. On the solution of the three forces problem and its application in optimal designing of a class of symmetric plane frameworks of least weight. Struct Multidisc Optim 44, 723–727 (2011). https://doi.org/10.1007/s00158-011-0702-3

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