Limit Analysis of Masonry Arch Bridges Using Discontinuity Layout Optimization

  • Matthew GilbertEmail author
  • John Valentino
  • Colin Smith
  • Tom Pritchard
Conference paper
Part of the Structural Integrity book series (STIN, volume 11)


Limit analysis provides a simple yet powerful and effective means of verifying the safety of masonry arch bridges. However, a disadvantage of traditional rigid block limit analysis methods is that any soil fill surrounding the arch barrel in a bridge cannot readily be modelled directly, since the locations of potential failure planes in the soil are not known in advance. Also, it will seldom be feasible to model every individual unit in the masonry parts of a bridge with a rigid block. In this contribution discontinuity layout optimization (DLO) is used to extend the scope of the traditional rigid block analysis method, enabling direct modelling of both masonry and soil elements. DLO involves discretizing a solid body using nodes, and then interconnecting pairs of nodes with potential failure plane discontinuities. Optimization can then be used to identify the critical subset of discontinuities at failure which defines the geometry of the critical failure mechanism. In this contribution DLO is applied to both soil and masonry problems before being applied to a number of representative masonry arch bridge problems.


Masonry arch bridges Limit analysis Masonry Discontinuity layout optimization Rigid block analysis 



The financial support of Network Rail is gratefully acknowledged. The authors also acknowledge the contributions of Maxime Gueguin, who programmed the homogenized masonry model described herein.


  1. Amodio, S., Gilbert, M., Smith, C.C.: Modelling masonry arch bridges containing internal spandrel walls. In: Proceedings of the 9th Arch Bridges Conference, Porto (2019)Google Scholar
  2. Cavicchi, A., Gambarotta, L.: Lower bound limit analysis of masonry bridges including arch–fill interaction. Eng. Struct. 29(11), 3002–3014 (2007)CrossRefGoogle Scholar
  3. De Buhan, P., De Felice, G.: A homogenization approach to the ultimate strength of brick masonry. J. Mech. Phys. Solids 45(7), 1085–1104 (1997)MathSciNetCrossRefGoogle Scholar
  4. Gilbert, M., Melbourne, C.: Rigid-block analysis of masonry structures. Struct. Eng. 72(21), 356–361 (1994)Google Scholar
  5. Ferris, M.C., Tin-Loi, F.: Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints. Int. J. Mech. Sci. 43(1), 209–224 (2001)CrossRefGoogle Scholar
  6. Gilbert, M., Casapulla, C., Ahmed, H.M.: Limit analysis of masonry block structures with non-associative frictional joints using linear programming. Comput. Struct. 82, 873–887 (2006)CrossRefGoogle Scholar
  7. Gilbert, M., Smith, C.C., Pritchard, T.J.: Masonry arch analysis using discontinuity layout optimization. Proc. Inst. Civ. Eng. Eng. Comput. Mech. 163, 155–166 (2010)Google Scholar
  8. LimitState: LimitState:GEO Manual, Version 3.5 edition. Accessed 15 Apr 2019 (2019)
  9. Livesley, R.K.: Limit analysis of structures formed from rigid blocks. Int. J. Numer. Methods Eng. 12, 1853–1871 (1978)CrossRefGoogle Scholar
  10. Makrodimopoulos, A., Martin, C.M.: Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int. J. Numer. Meth. Eng. 66(4), 604–634 (2006)CrossRefGoogle Scholar
  11. Melbourne, C., Gilbert, M.: The behaviour of multi-ring brickwork arch bridges. Struct. Eng. 73(3), 39–47 (1995)Google Scholar
  12. Smith, C.C., Gilbert, M.: Application of discontinuity layout optimization to plane plasticity problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 463(2086), 2461–2484 (2007)MathSciNetCrossRefGoogle Scholar
  13. Valentino, J., Gilbert, M., Gueguin, M., Smith, C.C.: Limit analysis of masonry structures using discontinuity layout optimization and homogenization. Int. J. Numer. Methods Eng. Prep. (2019)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Matthew Gilbert
    • 1
    Email author
  • John Valentino
    • 1
  • Colin Smith
    • 1
  • Tom Pritchard
    • 2
  1. 1.Department of Civil and Structural EngineeringUniversity of SheffieldSheffieldUK
  2. 2.LimitState Ltd.SheffieldUK

Personalised recommendations