Skip to main content
Log in

Minimum thickness of elliptical masonry arches

  • Published:
Acta Mechanica Aims and scope Submit manuscript


In this paper, we compute the minimum thickness and the location of the imminent intrados hinge of symmetric elliptical masonry arches when subjected to their weight. While this problem (Couplet’s problem) was solved rigorously for semicircular arches more than a century ago, no results have been available for elliptical arches. Motivated from the recent growing interest in identifying the limit equilibrium states of historic structures, this paper first computes two neighboring physically admissible thrust-lines which can just be located in elliptical arches by adopting either a polar or a cartesian coordinate system. These two distinguishable, physically admissible thrust-lines are neighboring thrust-lines to Hooke’s catenary which is not a physically admissible thrust-line as has been shown recently. Accordingly, the paper shows that the answer for the minimum thickness of symmetric elliptical masonry arches is not unique and that it depends on the coordinate system adopted and the associated stereotomy exercised. This result is confirmed by developing a variational formulation after selecting the appropriate directions of the rupture that initiates at the intrados hinge. The paper concludes that Hooke’s limiting catenary, although not a physically admissible thrust-line, offers a conservative value for the minimum thickness in most practical configurations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Hooke, R.: A Description of Helioscopes, and Some Other Instruments. London (1675)

  2. Heyman J.: The Masonry Arch. Ellis Horwood, Chichester (1982)

    Google Scholar 

  3. Heyman J.: Structural Analysis: A Historical Approach. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  4. O’Dwyer D.: Funicular analysis of masonry vaults. Compt. Struct. 73, 187–197 (1999). doi:10.1016/S0045-7949(98)00279-X

    Article  MATH  Google Scholar 

  5. Block P., DeJong M., Ochsendorf J.: As hangs the flexible line: equilibrium of masonry arches. Nexus Netw. J. 8(2), 13–24 (2006). doi:10.1007/s00004-006-0015-9

    Article  MathSciNet  MATH  Google Scholar 

  6. Gregory D.: Catenaria. Phil. Trans. 1695–1697(19), 637–652 (1697). doi:10.1098/rstl.1695.0114

    Google Scholar 

  7. Heyman J.: The safety of masonry arches. Int. J. Mech. Sci. 11, 363–385 (1969)

    Article  Google Scholar 

  8. Poleni G.: Memorie Istoriche Della Gran Cupola Del Tempio Vaticano. Stamperia del Seminario, Padova (1743)

    Google Scholar 

  9. Barlow P.W.: Treatise on the Strength of Materials. Lockwood, London (1867)

    Google Scholar 

  10. Roca P., Cervera M., Gariup G., Pela L.: Structural analysis of masonry historical constructions. Classical and advanced approaches. Arch. Comput. Methods Eng. 17, 299–325 (2010). doi:10.1007/s11831-010-9046-1

    Article  MATH  Google Scholar 

  11. Makris, N., Alexakis, H.: From Hooke’s “Hanging Chain” and Milankovitch’s “Druckkurven” to a variational formulation: The Adventure of the Thrust-line of Masonry Arches. Report series in EEAM 2012-02, September 2012, University of Patras, Greece (2012)

  12. Makris, N., Alexakis, H.: The effect of stereotomy on the shape of the thrust-line and the minimum thickness of semicircular masonry arches. Arch. Appl. Mech., tentatively accepted (2013)

  13. Alexakis, H.: Limit State Analysis and Earthquake Resistance of Masonry Arches. PhD thesis (in Greek), February 2013, Department of Civil Engineering, University of Patras, Greece (2013)

  14. Couplet, P.: De la poussée des voûtes, Histoire de l’Académie Royale des Sciences, pp. 79–117, 117–141. Académie royale des sciences, Paris (1729, 1730)

  15. Foce, F.: On the safety of the masonry arch. Different formulations from the history of structural mechanics. In: Huerta, S. (ed.) Essays in the History of Theory of Structures, pp. 117–142. Instituto Juan de Herrera, Madrid (2005)

  16. Albuerne, A., Huerta, S.: Coulomb’s theory of arches in Spain ca. 1800: the manuscript of Joaquín Monasterio. In: Chen, B., Wei, J. (eds). Proceedings 6th International Conference on Arch Bridges (ARCH’10), pp. 354–362. College of Civil Engineering, Fuzhou University, Fuzhou, China (2010)

  17. Milankovitch, M.: Beitrag zur Theorie der Druckkurven. Dissertation zur Erlangung der Doktorwürde, K.K. technische Hochschule, Vienna (1904)

  18. Milankovitch M.: Theorie der Druckkurven. Zeitschrift für Mathematik und Physik 55, 1–27 (1907)

    Google Scholar 

  19. Ochsendorf, J.: Collapse of Masonry Structures. PhD thesis, Department of Engineering, University of Cambridge, Cambridge, UK (2002)

  20. Heyman, J.: La Coupe des Pierres. In: Proceedings 3rd International Congress on Construction History, vol. 2, pp 807–812. Neunplus1, Berlin (2009)

  21. Moseley, H.: The Mechanical Principles of Engineering and Architecture. London (1843)

  22. Foce F.: Milankovitch’s Theorie der Druckkurven: Good mechanics for masonry architecture. Nexus Netw. J. 9(2), 185–210 (2007). doi:10.1007/s00004-007-0039-9

    Article  MATH  Google Scholar 

  23. Coulomb, C. A.: Essai sur une application des regles des maximis et minimis à à quelques problémes de statique relativs a l’arquitecture. In: Mémoires de mathematique et de physique présentés à à l’académie royal des sciences per divers savants et lus dans ses assemblées. 1, París, pp. 343–382 (1773)

  24. De Rubeis, A.: On the definition of the geometrical safety factor of masonry arches. In: Sinopoli, A. (ed.) Arch Bridges, pp. 119–122. A.A. Balkema, Rotterdam (1998)

  25. Bićanić N., Stirling C., Pearce C.J.: Discontinuous modeling of masonry bridges. Comput. Mech. 31, 60–68 (2003). doi:10.1007/s00466-002-0393-0

    Article  MATH  Google Scholar 

  26. Lamé M.G., Clapeyron E.: Mémoire sur la stabilité des voûtes. Annales des mines 8, 789–836 (1823)

    Google Scholar 

  27. Timoshenko S.P.: History of Strength of Materials. McGraw-Hill Book Company, Inc., New York (1953)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nicos Makris.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alexakis, H., Makris, N. Minimum thickness of elliptical masonry arches. Acta Mech 224, 2977–2991 (2013).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: