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Structural and Multidisciplinary Optimization

, Volume 60, Issue 5, pp 1957–1966 | Cite as

Catenary arch of finite thickness as the optimal arch shape

  • Dimitrije NikolićEmail author
Research Paper
  • 103 Downloads

Abstract

Based on the analogy with the load path of a homogeneous hanging chain, catenary has been for over three centuries considered to be an ideal masonry arch’s shape. In the present research, after the thrust line theory, complete insight into the equilibrium analysis of catenary arch of finite thickness under its own weight is provided. It is concluded that, in accordance with the common criterion—the existence of only normal (axial) forces—the ideality has previously been unjustifiably attached to such an arch, since there are shear forces present as well. However, with respect to so-called Couplet-Heyman’s assumptions (unilateral rigid no-tension material), it is shown that any catenary arch is stable under its own weight.

Keywords

Catenary arch Thrust line analysis Masonry arch Minimum thickness Optimal shape 

Notes

Acknowledgements

I would like to thank Professor Ivica Bošnjak for providing valuable remarks and help regarding thrust line analysis carried out in this research. The paper was done within the Project No. TR36042 supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia.

Compliance with Ethical Standards

Conflict of interest

The author declare that there is no conflict of interest.

References

  1. Alexakis H, Makris N (2013) Minimum thickness of elliptical masonry arches. Acta Mech 224(12):2977–2991MathSciNetCrossRefGoogle Scholar
  2. Alexakis H, Makris N (2015) Limit equilibrium analysis of masonry arches. Arch Appl Mech 85(9):1363–1381CrossRefGoogle Scholar
  3. Bélidor BFd (1729) La science des ingénieurs dans la conduite des travaux de fortification et d’architecture civile. ParisGoogle Scholar
  4. Benvenuto E (1991) An introduction to the history of structural mechanics. Springer, New YorkCrossRefGoogle Scholar
  5. Bernoulli J (1690) Analysis problematis antehac propositi, de inventione lineæ descensus a corpore gravi percurrendæ uniformiter, sic ut temporibus æqualibus æquales altitudines emetiantur: & alterius cujusdam Problematis Propositio. Acta Eruditorum, 217– 219Google Scholar
  6. Bernoulli J (1704) Meditationes, annotationes, animadversiones theol. et philos. a me concinnatae et collectae ab anno 1677-1704Google Scholar
  7. Bossut C (1774) Recherches sur l’Èquilibre des voûtes. Historie de l’Académie Royale des Sciences 1778:534–566Google Scholar
  8. Cocchetti G, Colasante G, Rizzi E (2012) On the analysis of minimum thickness in circular masonry arches. Appl Mech Rev 64(5):27 pagesGoogle Scholar
  9. Como M (2017) Statics of historic masonry constructions. Springer, BerlinCrossRefGoogle Scholar
  10. Coulomb CA (1773) Essai sur une application des régles des maximis & minimis à quelques problémes de statique, relativs a larquitecture. Mémoires de Mathématique et de Physique, présentés à l’Académie Royal des Sciences per divers Savans, & lûs dans ses Assemblées 1776:343–382Google Scholar
  11. Couplet P (1730) Seconde partie de l’examen de la poussee des voûtes. Histoire de l’Académie Royale des Sciences 1732:117–141Google Scholar
  12. Cowan HJ (1981) Some observations on the structural design of masonry arches and domes before the age of structural mechanics. Archit Sci Rev 24(4):98–102CrossRefGoogle Scholar
  13. Foce F (2007) Milankovitch’s Theorie der Druckkurven: good mechanics for masonry architecture. Nexus Netw J 9(2):185–210CrossRefGoogle Scholar
  14. Gáspár O, Sipos AA, Sajtos I (2018) Effect of stereotomy on the lower bound value of minimum thickness of semi-circular masonry arches. Int J Archit Herit 12(6):899–921CrossRefGoogle Scholar
  15. Gregory D (1697) Catenaria. Phil Trans 19:637–652Google Scholar
  16. Heyman J (1972) Coulombs memoir on statics: an essay in the history of civil engineering. Cambridge University Press, CambridgezbMATHGoogle Scholar
  17. Heyman J (1997) The stone skeleton: structural engineering of masonry architecture. Cambridge University Press, CambridgeGoogle Scholar
  18. Hooke R (1676) A description of helioscopes and some other instruments. John Martyn, LondonGoogle Scholar
  19. Huerta S (2001) Mechanics of masonry vaults: the equilibrium approach. In: Lourenço PB, Roca P (eds) Historical constructions, pp 47–69Google Scholar
  20. Huerta S (2006) Structural design in the work of Gaudí. Archit Sci Rev 49(4):327–339CrossRefGoogle Scholar
  21. Huerta S (2008) The analysis of masonry architecture: a historical approach. Archit Sci Rev 51(4):297–328CrossRefGoogle Scholar
  22. Kurrer KE (2018) The history of the theory of structures: searching for equilibrium, 2nd edn. Wilhelm Ernst & Sohn Verlag für Architectur und technische Wissenschaften GmbH & Co. KGGoogle Scholar
  23. Lluis i Ginovart J, Coll-Pla S, Costa-Jover A, López Piquer M (2017a) Hookes chain theory and the construction of catenary arches in Spain. Int J Archit Herit 11(5):703–716zbMATHGoogle Scholar
  24. Lluis i Ginovart J, Costa-Jover A, Coll-Pla S, López Piquer M (2017b) Layout of catenary arches in the Spanish enlightenment and modernism. Nexus Netw J 19(1):85–99CrossRefGoogle Scholar
  25. Makris N, Alexakis H (2013) The effect of stereotomy on the shape of the thrust-line and the minimum thickness of semicircular masonry arches. Arch Appl Mech 83(10):1511–1533CrossRefGoogle Scholar
  26. Méry E (1840) Sur l’équilibre des voûtes en berceau. Annales des Ponts et Chaussées 1(1):50–70Google Scholar
  27. Milanković M (1907) Theorie der Druckkurven. Zeitschrift für Mathematik und Physik 55:1–27zbMATHGoogle Scholar
  28. Moseley H (1843) The mechanical principles of engineering and architecture. Longman Brown, Green and Longmans, LondonGoogle Scholar
  29. Nikolić D (2017) Thrust line analysis and the minimum thickness of pointed masonry arches. Acta Mech 228(6):2219–2236MathSciNetCrossRefGoogle Scholar
  30. Osserman R (2010) How the Gateway arch got its shape. Nexus Netw J 12(2):167–189CrossRefGoogle Scholar
  31. Poleni G (1748) Memorie istoriche della gran cupola del Tempio Vaticano. Stamperia del Seminario, PadovaGoogle Scholar
  32. Poncelet JV (1852) Examen critique et historique des principales théories ou solutions concernant l’équilibre des voûtes. Comptes rendus 35:577–587Google Scholar
  33. Radelet-de Grave P (2003) Essays on the history of mechanics. Springer Basel AG, chap The use of a particular form of the parallelogram law of forces for the building of vaults (1650–1750), pp 135–163CrossRefGoogle Scholar
  34. Romano A, Ochsendorf JA (2010) The mechanics of Gothic masonry arches. Int J Archit Herit 4(1):59–82CrossRefGoogle Scholar
  35. Salimbeni L (1787) Degli archi e delle volte. Dionigi Ramanzini, VeronaGoogle Scholar
  36. Stirling J (1717) Lineæ tertii ordinis Neutonianæ. OxfordGoogle Scholar
  37. Truesdell C (1960) The rational mechanics of flexible or elastic bodies 1638–1788. Introduction to Leonhardi Euleri Opera Omnia, 2nd series vol XI. Zürich, Orell FüssliCrossRefGoogle Scholar
  38. Ware S (1809) A treatise of the properties of arches, and their abutment piers. Longman Hurst, Rees and Orme, LondonGoogle Scholar
  39. Young T (1824) Supplement to the fourth, fifth and sixth editions of the Encyclopaedia Britannica, vol 2. Archibald Constable, Edinburgh. Bridge, pp 497–520Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Technical Sciences, Department of ArchitectureUniversity of Novi SadNovi SadSerbia

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