The effect of stereotomy on the shape of the thrust-line and the minimum thickness of semicircular masonry arches

Abstract

More than a century ago, the Serbian engineer and astronomer Milutin Milankovitch presented a remarkable formulation for the thrust-line of arches that do not sustain tension, and by taking radial cuts and a polar coordinate system, he published for the first time the correct and complete solution for the theoretical minimum thickness, t, of a monolithic semicircular arch with radius R. This paper shows that Milankovitch’s solution, t/R = 0.1075, is not unique and that it depends on the stereotomy exercised. The adoption of vertical cuts which are associated with a cartesian coordinate system yields a neighboring thrust-line and a different, slightly higher value for the minimum thickness (t/R = 0.1095) than the value computed by Milankovitch. This result has been obtained in this paper with a geometric and a variational formulation. The Milankovitch minimum thrust-line derived with radial stereotomy and our minimum thrust-line derived with vertical stereotomy are two distinguishable, physically admissible thrust-lines which do not coincide with R. Hooke’s catenary that meets the extrados of the arch at the three extreme points. Furthermore, the paper shows that the catenary (the “hanging chain”) is not a physically admissible minimum thrust-line of the semicircular arch, although it is a neighboring line to the aforementioned physically admissible thrust-lines. The minimum thickness of a semicircular arch that is needed to accommodate the catenary curve is t/R = 0.1117—a value that is even higher than the enhanced minimum thickness t/R = 0.1095 computed in this paper after adopting a cartesian coordinate system; therefore, it works toward the safety of the arch.

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References

  1. 1

    Huerta S.: Galileo was wrong: the geometrical design of masonry arches. Nexus Netw. J. 8, 25–52 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

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

  3. 3

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

    Google Scholar 

  4. 4

    O’Dwyer D.: Funicular analysis of masonry vaults. Comput. Struct. 73, 187–197 (1999)

    Article  MATH  Google Scholar 

  5. 5

    Block P., DeJong M., Ochsendorf J.: As hangs the flexible line: equilibrium of masonry arches. Nexus Netw. J. 8, 13–24 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

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

  7. 7

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

  8. 8

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

    Google Scholar 

  9. 9

    Couplet, P.: De la poussée des voûtes, Histoire de l’Académie Royale des Sciences, Paris (1729, 1730)

  10. 10

    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)

  11. 11

    Albuerne, A., Huerta, S.: Coulomb’s theory of arches in Spain ca. 1800: the manuscript of Joaquín Monasterio. In: Chen, B., Wei, J. (ed.) Proc. 6th Int. Conf. on Arch Bridges (ARCH’10), Fuzhou, China, pp. 354–362 (2010)

  12. 12

    Heyman, J.: La Coupe des Pierres. In: Kurrer, K.E., Lorenz, W., Wetzk, V. (eds.) Proc. 3rd Int. Cong. on Construction History, Neunplus1, Berlin, vol. 2, pp. 807–812 (2009)

  13. 13

    Loo Y.C., Yang Y.: Cracking and failure analysis of masonry arch bridges. J. Struct. Eng. 117, 1641–1659 (1991)

    Article  Google Scholar 

  14. 14

    Molins C., Roca P.: Capacity of masonry arches and spatial frames. J. Struct. Eng. 124, 653–663 (1998)

    Article  Google Scholar 

  15. 15

    Thavalingam A., Bicanic N., Robinson J.I., Ponniah D.A.: Computational framework for discontinuous modelling of masonry arch bridges. Comput. Struct. 79, 1821–1830 (2001)

    Article  Google Scholar 

  16. 16

    Lourenço P.B.: Computation on historic masonry structures. Progr. Struct. Eng. Mater. 4, 301–319 (2002)

    Article  Google Scholar 

  17. 17

    Milani E., Milani G., Tralli A.: Limit analysis of masonry vaults by means of curved shell finite elements and homogenization. Int. J. Solids Struct. 45, 5258–5288 (2008)

    Article  MATH  Google Scholar 

  18. 18

    Varma M., Jangid R.S., Ghosh S.: Thrust line using linear elastic finite element analysis for masonry structures. Adv. Mater. Res. Vols. 133(134), 503–508 (2010)

    Article  Google Scholar 

  19. 19

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

    Article  Google Scholar 

  20. 20

    Foce F.: Milankovitch’s Theorie der Druckkurven: good mechanics for masonry architecture. Nexus Netw. J. 9, 185–210 (2007)

    Article  MATH  Google Scholar 

  21. 21

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

    Google Scholar 

  22. 22

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

    Google Scholar 

  23. 23

    AutoCAD LT v.2010 design software, Copyright 2009 Autodesk, Inc

  24. 24

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

  25. 25

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

    Google Scholar 

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Correspondence to Nicos Makris.

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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 83, 1511–1533 (2013). https://doi.org/10.1007/s00419-013-0763-4

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Keywords

  • Stone arches
  • Limit equilibrium analysis
  • Line of resistance
  • Catenary curve
  • Variational formulation