QUICK UPDATE: Demi just had her paper published ‘Assessing the Stability of Unreinforced Masonry Arches and Vaults: A Comparison of Analytical and Numerical Strategies’, in the Journal of Architectural Heritage. You can find it here
This post is second in a series covering different assessment methods for stability of masonry structures. Part 1 covered classical and equilibrium methods; this post covers suitable numerical modeling techniques as well as different examples of physical modeling for masonry stability.
4. Numerical modeling
Several methods of numerical modeling for masonry structures exist, as demonstrated by the flowchart in Fig. 10.
As the first level of Fig. 10 suggests, numerical modeling of masonry structures can be divided into four main categories: macro-modeling, homogenized modeling, simplified micro-modeling, and detailed micro-modeling. Asteris et al.  provide discussions, summarized below with some additions where noted, on the differences between these modeling approaches. Fig. 11 also depicts the different numerical modeling approaches. In this section, macro-modeling and simplified micro-modeling are the focus.
4.1 Macro-modeling: masonry as a one-phase material
The macro-modeling approach models both bricks and mortar (or all bricks, in the case of dry masonry) as a homogeneous continuum as in Fig. 11(b). As the subsets under macro-modeling in Fig. 10 suggest, these numerical models are typically finite element models.
Finite element analysis (FEA) is used widely today for continuum finite element modeling many reinforced concrete and steel structures. However, like elastic theory, FEA is inappropriate for the modeling of masonry structures due to the different behavior of masonry materials.
Block  compares the continuum FEA of a masonry arch to a thrust line analysis to demonstrate the most apparent inadequacy of FEA in assessing masonry structures: the continuum modeling approach assumes that masonry can carry tension and that the arch acts as a continuous unit when in reality it is made of separate elements. In Fig. 12, arches of two different thickness ratios are examined using FEA (left) and thrust line analysis (right). FEA predicts that the two arches behave similarly – both arches have similar stress distributions, and failure is predicted in neither arch. However, thrust lines drawn through each arch demonstrates that the thinner arch forms 4 hinges (top right arch: the thrust line exits the section 4 times) and thus has an unstable geometry under self-weight. The decision to use continuum FEA for masonry analysis is analogous to using elastic theory to assess masonry: both FEA and elastic theory make incorrect assumptions about masonry that render the methods inappropriate for masonry applications.
As a whole, though macro-modeling may be appealing for use in large-scale masonry structures due to continuum smearing of brick element properties and joint properties, this same assumption makes the modeling approach precisely inadequate for capturing the true detailed behavior of masonry structures.
4.2 Simplified micro-modeling: masonry as a two-phase material
As Fig. 10 suggests, the second approach of numerical modeling of masonry structures, simplified micro-modeling, transitions gradually from finite to discrete element modeling. Fig. 11(c) provides a visual representation of this approach: each brick is modeled as the true brick geometry but expanded to include joint thickness (if any). The two elements considered are thus that of the brick element and the brick-brick interface element, the latter of which takes its stiffness properties from the true joint stiffness. Because of the reduction in computational expense from macro-modeling, simplified micro-modeling is often more applicable for a wider range of structures.
The incorrect assumptions of using finite element analysis was discussed earlier, but some researchers have used finite element modeling (FEM) while bypassing the assumption of continuity by modeling both each masonry element and joint interface-element through discontinuous finite element modeling. In section 9 of , Roca et al. discuss a few works that use this modeling method. Asteris et al.  comment that these discontinuum methods of macro-modeling actually give results in good agreement with experimental results.
Originally conceived for the modeling of jointed rock, discrete element modeling is used to model masonry as a collection of distinct blocks interacting along their boundaries. Cundall and Hart 1989  provide the following conditions for discrete element modeling:
- Finite displacements and rotations of discrete bodies are allowed
- New contacts between blocks are automatically recognized and updated as the calculation progresses
Asteris et al.  describe the different types of discrete element modeling methods that exist for masonry structures; these methods differ by the different ways that they model 1) behavior of the discontinuities and 2) behavior of the continuous material. This section will focus on the distinct element method (abbreviated here as DEM, but not to be confused with “discrete element modeling”), which Asteris et al.  conclude is most suitable for simulating the response of discontinuous media under static or dynamic loading. Note that this strength was precisely described in Section 3.5 as the shortfall of limit state analysis,.
DEM uses an explicit approach to determine equations of motion for discrete bodies in increments of time. Discontinuities are modeled as deformable contacts; that is, joints hold a finite measurable stiffness. The bodies themselves may be rigid or deformable .
Cundall  and Lemos et al.  developed code for two-dimensional application discrete element method as the program UDEC. Cundall has since developed a three- dimensional version, 3DEC, in  and .
Another discrete element modeling method similar to the distinct element method is discontinuous deformation analysis (DDA). While this method is not considered in this thesis, differences between DEM and DDA are detailed in section 3.2.2(b) of Asteris et al. 
4.3 Other modeling approaches
For more details on detailed micro-modeling, the reader is directed to section 3.3 in Asteris et al. . Ultimately, however, Asteris et al. comment that the more detailed results of detailed micro-modeling come at the expense of computational intensity. Roca et al. also touch on homogenized modeling, described as “midway between micro- modeling and macro-modeling,” in section 11 of .
4.4 Numerical modeling: discussion
This section outlined the different numerical modeling approaches available for the analysis of masonry structures, focusing on macro-modeling and simplified micro-modeling. It was shown that the continuum methods of macro-modeling, namely FEM, are not adequate for masonry analysis due to incorrect assumptions. Of the simplified micro-modeling approaches, the discrete element method has been accepted as one of the most powerful methods for modeling masonry under static and dynamic loading conditions across time-steps.
5. Experimental and physical modeling
Physical scale modeling has served as representative of the predicted behavior of masonry structures, an important point of comparison during the development of the above analytical and numerical methods. For example, DeJong  makes use of a shake table to simulate the mechanism developed in an arch under earthquake loading; his analytical methods predict the same result. Ochsendorf similarly used experimental results of scale buttresses and arches under settlement to verify his analytical methods in several sections of his dissertation.  Van Mele et al.  use a sophisticated and systematic setup to create scale models to simulate the behavior of a groin vault under applied displacements, comparing the results against 3DEC results.
Summary: Limit State Analysis vs. DEM
Heyman’s contributions [1, 9, 21, 27] have solidified plastic theory as the theory by which to assess masonry structures (as opposed to elastic theory.) Limit state analysis at present is considered a reliable engineering method of assessing masonry stability. Its limitations in 2D have prompted more recent work in extending the principles to three dimensions, primarily with the equilibrium approach in 3D; the failure mechanisms are too complex to predict for spatial structures, making the upper limit difficult to investigate. Though some authors consider it overly conservative, the slicing technique is a reliable method for assessing spatial structures using 2D equilibrium.
Distinct element modeling (hereon abbreviated as DEM) has been established as a numerical modeling technique with the best potential for modeling the behavior of masonry structures. While limit analysis is limited to assessing the stability of masonry structures, DEM can give the same information and more, as summarized by Fig. 13.
Asteris et al.  provide a few more nuances to the comparison of attributes:
- DEM not only allows for large displacements and rotations of discrete units, but also complete detachment.
- DEM is able to update existing contacts and recognize new contacts automatically.
- However, contact detection and deformable bodies can make DEM computationally demanding.
- Because DEM models contacts, contact parameters are needed when assessing existing structures. These contact parameters are not always able to be determined. Limit analysis, on the other hand, does not require input material properties.
- Limit analysis is generally quicker and more intuitive, allowing for easier discussion and understanding of alternative solutions.
- However, for structures with many blocks and complex geometries, limit analysis can get more computationally tedious.
It was also noted in  that there are lack of customized packages for limit state analysis, increasing the time to prepare the model. This limitation can be seen in the overview of limit state analysis programs, where masonry structures to be assessed had to be built within each software. Because masonry’s failure is dictated by geometry, there is a missed opportunity by making thrust line analysis rather inaccessible to architectural designers.
Finally, many have used experimental scale models as a basis of comparison for analytical and numerical methods in masonry assessment.
 J. Heyman, “The stone skeleton,” International Journal of Solids and Structures, vol. 2, pp. 249–279.
 P. Roca, M. Cervera, and G. Gariup, “Structural analysis of masonry historical constructions. Classical and advanced approaches,” Archives of Computational Methods in Engineering, vol. 17, pp. 299–325, 2010.
 J. Heyman, “The masonry arch,” in Structural Analysis: A Historical Approach, ch. 5, Cambridge: Cambridge University Press, 1998.
 J. Heyman, The Masonry Arch. West Sussex, England: Ellis Horwood Limited, 1982.
 J. A. Ochsendorf, The Collapse of Masonry Structures. PhD thesis, Department of Engineering, University of Cambridge, Cambridge, 2002.
 J. Heyman, The Stone Skeleton. New York: Press Syndicate of the University of Cambridge, 1995.
 P. Block, Thrust Network Analysis: Exploring Three-dimensional Equilibrium. PhD thesis, Department of Architecture, MIT, Cambridge, MA, 2009.
 M. Rippmann and P. Block, “Funicular shell design exploration,” in Proceedings of the 33rd Annual Conference of the ACADIA (Waterloo/Buffalo/Nottingham, Canada), September 2013.
 P. G. Asteris, V. Plevris, V. Sarhosis, L. Papaloizou, A. Mohebkhah, P. Komodromos, and J. V. Lemos, “Numerical modeling of historic masonry structures,” in Handbook of Research on Seismic Assessment and Rehabilitation of Historic Structures, ch. 7, pp. 213–256, Hershey, PA: Engineering Science Reference, 2015.
 P. A. Cundall and R. D. Hart, “Numerical modelling of discontinua,” in Proceedings of the 1st US Conference on Discrete Element Methods, CSM Press, 1989.
 Itasca Consulting Group, Minneapolis, MN, 3DEC version 5.0 Theory and Background, 1988.
 P. A. Cundall, “UDEC – a generalized distinct element program for modelling jointed rock,” tech. rep., Report PCAR-1-80, Peter Cundall Associates Report, European Research Office, U.S. Army, Contract DAJA37-79-C-0548, 1980.
 J. V. Lemos, R. D. Hart, and P. A. Cundall, “A generalized distinct element pro- gram for modeling jointed rock mass (a keynote lecture),” in Proceedings of the International Symposium on Fundamentals of Rock Joints (Bjorkliden, Sweden, September 1985), (Lulea, Sweden), pp. 335–343, Centek Publishers, 1985.
 P. A. Cundall, “Formulation of a three-dimensional distinct element model – Part I: A scheme to detect and represent contacts in a system composed of many poly- hedral blocks,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 25, pp. 107–116, 1988.
 R. Hart, P. A. Cundall, and J. V. Lemos, “Formulation of a three-dimensional distinct element model – Part II: Mechanical calculations for motion and interaction of a system composed of many polyhedral blocks,” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 25, pp. 117–126, 1988.
 M. J. DeJong, Seismic Assessment for Masonry Structures. PhD thesis, Department of Architecture, MIT, Cambridge, MA, 2009.
 T. Van Mele, J. McInerney, M. DeJong, and P. Block, “Physical and computational discrete modeling of masonry vault collapse,” in Proceedings of the 8th International Conference on Structural Analysis of Historical Constructions (Wroclaw, Poland), October 2012.
 P. A. Cundall and R. D. Hart, “Numerical modeling of discontinua,” Engineering Computations, vol. 9, no. 2, pp. 101–113, 1992.
Author: Demi Fang ’17
Adapted and excerpted from the author’s senior-thesis-in-progress. See also: Part 1