This post is first in a series covering different assessment methods for stability of masonry structures. This post covers classical and equilibrium methods; Part 2 covers suitable numerical modeling techniques as well as different examples of physical modeling for masonry stability.
The persistence of some of the oldest structures in the world in masonry has demonstrated the high potential for masonry structures to last through various conditions over long periods of time. Masonry’s compressive strength is extraordinarily high – it is estimated that a stone pillar would have to be 2 kilometers tall in order to fail by crushing.  As a result, in contrast to materials such as concrete and steel that make up most of present-day structures, the limit state of masonry is often dictated by its geometry and not its material properties.
Research into the stability of masonry structures is valuable for two main reasons. Firstly, this research enables us to understand and preserve the structures of the past. Many structures of rich cultural heritage are made of masonry, but their stability is challenged by environmental and anthropogenic threats, such as earthquakes or terrorist attacks. [2–6] The second reason is forward-looking. In some areas of the world, masonry materials are abundant and are thus the most economic choice of building material. An understanding of stability in masonry structures can make possible design tools for materially efficient structures.
Examples of masonry structures are given below. Philadelphia City Hall (1901) is the world’s tallest masonry structure at 167 meters height. [A] The King’s College Chapel (1515) in Cambridge, UK is not even a fifth of the height of Philadelphia City Hall, but the complex geometry of its fan vaults make it a compelling study of masonry stability. [B] Finally, the Armadillo Vault (2016) is a prime example of how an understanding of masonry stability can inform efficient design today. [C]
Methods and theories of structural analysis for masonry structures
The structural analysis of masonry arches and structures have preoccupied countless scientists since the 17th century. In this post, studies on 1. Classical methods and 2. Limit state analysis (including equilibrium analysis and kinematic analysis) are presented. A future post will explore 3. Numerical modeling and discuss existing studies that use each method to assess masonry structures. A more comprehensive overview of studies on each analysis method can be found in [7–9].
1. Classical methods: hanging models and graphical methods
One of the earliest theories of determining equilibrium in masonry structures dates back to Robert Hooke in 1676, who asserted the following principle (now known as Hooke’s law of inversion):
Ut pendet continuum flexile, sic stabit continuum rigidum.
As hangs the flexible line, so but inverted will stand the rigid arch. 
Hooke intuited that a hanging chain in pure tension could inform the shape of a structure in pure compression. More specifically, rigidizing and inverting the shape of a hanging chain and would give the load path of an arch in pure compression (for example, see Fig. 1, left). The compressive material analogous to the tensile chain would be masonry elements.
While Hooke did not decipher the equation for the catenary himself, in 1698, Gregory independently derived the equation and concluded that “when an arch of any other figure is supported, it is because in its thickness some catenaria is included” . As will be discussed later, this claim is remarkably close to the theorem for the structural behavior of masonry structures that is widely accepted today.
Roca et al.  also note that throughout the 18th century and into the early 20th century, funicular models inspired the design and assessment of masonry bridges and buildings. For example, in 1748, Poleni first applied Gregory’s concept by demonstrating an equilibrium of Saint Peter’s cracked dome as the inverted catenary (Fig. 1, right). Architect Gaudì took this modeling method into the third dimension through his hanging chain models that informed the design of Colonia Güell near Barcelona (Fig. 2).
Meanwhile, in 18th-century France, others such as Coulomb took a mechanical approach to assessing stability. Rather than simply checking the stability through geometric fitting, Coulomb developed mathematical foundations for describing different collapse modes of the masonry, locating unfavorable positions of hinges or rupture .
Hooke’s initial insight was further developed with graphic statics, one of the first equilibrium analysis methods for structural systems. This graphical method first emerged in the late 16th century through Simon Stevin’s parallelogram rule  and was formalized by Culmann  (Fig. 3).
As a graphical method of equilibrium analysis, graphic statics emphasizes the relationship between form and force. In the case of the equilibrium analysis of masonry structures, a funicular polygon of the self-weight of each masonry block can be constructed to conceptualize a thrust line. The thrust line is a theoretical line illustrating one possible path of compressive forces in a structure, representing static equilibrium. Just as a hanging chain with a given load can have an infinite number of catenary shapes due to the variable horizontal reaction forces at the ends of the chain, a masonry arch of a given loading thus has an infinite number of thrust lines depending on the horizontal reactions at its supports . These thrust lines can be used to assess the stability of a structure.
Fig. 4 illustrates thrust line analysis of a masonry arch of arbitrary geometry under self-weight. In the figure, (a) shows the thrust line through the arch and the theoretical inverted chain under self-weight of each voussoir, or masonry block. (b) shows the funicular polygon, or the forces and reactions in equilibrium. (c) and (d) isolate the local forces experienced by the shaded block, demonstrating the relationship between form and force that is apparent through graphical analysis.
2. Elastic theory and analysis
Though masonry structures typically behave inelastically, several studies have used elastic theory to assess masonry structures.
For example, in 1858, Rankine  developed the middle third rule, which stated that in order for a structure to be stable and take compression forces, its thrust line must pass through the middle third of the structure (Fig. 5). The theory behind the middle third rule relies on the assumptions that the masonry cannot take tension and that its behavior is linear-elastic. As illustrated in Fig. 6, under these assumptions, the stress distribution along the section of a voussoir is completely in compression as long as the force lies within the middle third of the voussoir.
However, many structures throughout history seem to be much thinner than dictated by this rule. As it turns out, the assumptions of elasticity and zero tensile strength can ultimately cause the middle third rule to overestimate the required thickness . As will be discussed later, this middle third rule can generally be relaxed to a “middle half” rule.
Linear elastic theory is commonly applicable in the analysis of steel and concrete structures. However, the material of masonry structures have very different properties from steel and concrete. Masonry’s low tensile capacity results in a complex nonlinear response at low or moderate stress levels; an elastic approach is thus inappropriate.  Furthermore, since masonry structures are constituted by the accumulation of blocks rather than a continuous material, assuming an elastic continuum overlooks the blocky interactions (such as displacements and rotations of individual blocks) involved in the structure’s stability. These blocky interactions, rather than elastic material deformations, are responsible for the deformations that arise in masonry structures. 
Most generally agree that linear elastic analysis is not useful for estimating the ultimate response or assessing strength of masonry structures. Despite the conviction of Heyman’s work in plastic analysis in the 1950s, elastic analysis of masonry structures continued as an “academic” exercise: in 1960, Robert Mark completed the first elastic analysis of a Gothic building using photoelasticity to determine stress states in French cathedrals. This work inspired more elastic analysis of masonry structures through numerical modeling in the following decades, particularly appealing due to the lower computational cost of assuming elasticity.  In some of these cases, Roca comments that “the limitations of the method were counterbalanced by the very large expertise and deep insight of the analysts.”  Finally, Meli and Pena  demonstrate that the elastic approach may have some potential in providing preliminary information for the study of masonry structures.
Ultimately, however, the inaccurate assumptions of elastic theory for the assessment of masonry structures outweigh the convenient advantages of using it in most applications.
3. Plastic theory: limit state analysis
Heyman was the first to firmly establish plastic theory as the appropriate mode of masonry structure assessment in his seminal essay titled “The Stone Skeleton” , though he based some of his ideas upon Kooharian’s work  from a decade prior. A fuller picture of the principles limit state analysis synthesizes work from both Heyman and Kooharian and is summarized well by Gilbert , which is paraphrased here.
Before stating the theorems of limit state analysis, the conditions to be satisfied for plastic analysis, as determined by Horne , are:
- The equilibrium condition: calculated internal actions should represent a state of equilibrium between internal and external loads. (If an energy method is being used, the compatibility condition should be satisfied instead.)
- The mechanism condition: sufficient releases should be made to transform the structure into a mechanism.
- The yield condition: The stresses in the material do not surpass shear, crushing, or tensile strength limits.
The limit state may be defined as the point or state beyond which a structure will collapse. Put more rigorously, let λ represent the load factor multiplied by the applied load on a structure. The structure will fail when λ reaches the failure load factor λp; determining a limit state thus determines the collapse load of a structure.
The above conditions allow for the following fundamental theorems of plastic analysis, as stated by Horne :
- Static theorem (lower bound theorem): if a load factor is applied such that the equilibrium and yield conditions are satisfied, then the load factor gives a lower bound solution to the limit state. The failure load factor is the greatest static load factor.
λ = λl ≥ λp
λp = max(λl)
- Kinematic theorem (upper bound theorem): if a load factor is applied such that the mechanism condition is satisfied and the work done by the applied load equals the work done in plastic energy dissipation, then the load factor gives an upper bound solution to the limit state. The failure load factor is the least kinematic load factor.
λ = λu ≤ λp
λp = min(λu)
- Uniqueness theorem: if a load factor is applied such that all three conditions of equilibrium, mechanism, and yield are satisfied, then the load factor gives the limit state.
λ = λp
At the limit state, the structure is about to collapse; both a statically and kinematically admissible collapsing mechanism exists at the limit state.  Gilbert  provides a graphical depiction of the relationship between theorems in Fig. 7.
For the limit state analysis of masonry structures, Heyman 1966  essentially provides the lower bound theorem, while Kooharian 1952  establishes the upper bound theorem. Both theorems are detailed below.
3.1 The lower bound: the safe theorem
While Horne  established general conditions for plastic analysis, Heyman  builds on these conditions to establish the assumptions for plastic analysis of masonry structures in particular:
- No sliding failure: the masonry blocks are rigid and have infinite friction.
- No crushing failure: it has been shown extensively that even the largest stresses experienced by masonry structures are orders of magnitude smaller than the com- pressive strength of masonry material.
- No tensile strength: masonry materials are extremely weak in tension, so this assumption is conservative to a reasonable degree.
In the safe theorem, Heyman states that for a structure under a given loading, as long as one equilibrium solution, or thrust line, can be found within the structure’s cross-section, the structure is stable – the structure will never collapse under that loading. This assessment method is here referred to as equilibrium analysis.
Unlike elastic analysis, the safe theorem appropriately overlooks the structure’s behavior in the presence of small settlements or cracks – these microscopic deformations do not change the ability of the thrust line to fit within the section. (For an in-depth discussion, see Chapter 2: The Plastic Theorems in .)
Of the infinite number of thrust lines in a structure for a given loading, a structure may still have an infinite number of thrust lines that lie within the structure, but the structure’s safety may be assessed by two of these thrusts: its extreme thrusts. Fig. 8 illustrates the maximum and minimum thrusts in a semi-circular arch. (b) demonstrates how the inward movement of supports leads to the minimum thrust state, and (c) demonstrates how the outward movement of supports leads to the maximum thrust state. The funicular polygon (Fig. 8, right) illustrates how the maximum thrust is associated with the largest possible horizontal reactions, and the minimum thrust with the least possible horizontal reactions. The minimum thrust is the thrust line that fits within the structure with the least magnitude of horizontal reaction, representing the minimum amount of horizontal reaction force that the structure can provide. Analogously, the maximum thrust is the thrust line that fits within the structure with the highest magnitude of horizontal reaction, representing the maximum amount of horizontal reaction force that the structure can provide. 
Once the thrust line meets the edge of the masonry, a plastic hinge develops at the intersection of the thrust line and masonry boundary. Three hinges may be tolerated for a stable arch, but the development of a fourth hinge turns the structure into a mechanism. Note that by the safe theorem, the structure’s geometry determines the magnitude of horizontal reaction that the structure can resist.
Heyman  also introduces a way to quantify how safe the structure is. For any given structure, the structure’s depth can be reduced by some factor as far as possible such that the reduced structure is just able to contain one thrust line. This reduction factor is called the geometrical factor of safety. Fig. 9 illustrates the method of determining the geometrical factor of safety of a thrust line in a semi-circular arch.
Another way of stating the middle third rule of elastic theory is dictating that an arch has a geometrical factor of safety of at least 3. Heyman  comments that when assuming plastic theory for the masonry arch, the middle third rule may be relaxed to the “middle half rule” – that is, a geometrical factor of safety of at least 2 is accepted as a safe practical factor for the masonry arch.
The geometrical factor of safety has since been extended in various ways; for example, Harvey  has proposed consideration of the “thrust zone” which considers material crushing strength.
In summary, the safe theorem and the geometrical factor of safety take into account the tendency of masonry to fail not due to inadequate strength but through instability and inadequate geometry. Because the safe theorem is a lower bound theorem, it is particularly efficient in assessing the safety of existing structures; stable structures will give a load factor below the failure load factor (λl < λp). The highest load factor gives the failure load factor (max(λl) = λp).
3.2 The upper bound: kinematic limit analysis
It was discussed how thrust line analysis can predict where plastic hinges will form on a structure under applied loading. If enough (4 or more) hinges are developed, the structure becomes a mechanism and fails.
Kooharian’s “unsafe theorem”  takes a different approach to observing the failure of a structure. The theorem states that “collapse will occur (or will have occurred previously) if a kinematically admissible collapse state can be found.” This “kinematically admissible collapse state” is achieved when the work done by external loads is at least as large as that done by the internal forces. The failure factor load can thus be determined by finding the minimum load factor for which work done by external loads is equal to that done by internal forces for a kinetic mechanism.
Ochsendorf  uses this procedure to specifically study the effect of lateral acceleration on the collapse mechanisms of a masonry arch. Gilbert  introduces rigid-block analysis for masonry structures, which models the displacements and rotations of each block and uses linear programming to determine the minimum of the upper bound; i.e., the minimum applied load for a mechanism to equate external and internal work.
3.3 Limit state analysis of complex spatial structures
Although the thrust line method behind the safe theorem is limited to 2D analysis, there have been various approaches to extending these principles to 3D structures; as noted in section 5.2 of , extending these principles to 3D is an active field of research today. These methods largely make use of the equilibrium approach rather than the kinematic approach. The kinematic approach is useful when the exact critical failure mechanism is known; this mechanism is more difficult to pinpoint in 3D structures than in 2D arch analysis, especially when structures or loadings are asymmetric. 
The slicing technique, which analyzes 2D sections in order to assess 3D structures, dates back to Poleni  but was also used by Heyman  and Block . These methods have been found to be reliable but conservative [30, 33], prompting others to explore three-dimensional equilibrium as an alternative.
These works include O’Dwyer , which applies Heyman’s limit analysis to vaults and domes; Block , which establishes thrust network analysis to generate compression- only surfaces under consideration of gravity loading only; and Andreu et al. [35, 36], which study the use of cable net solutions as the 3D extension of the thrust line in Heyman’s safe theorem.
3.4 Existing tools and software for limit state analysis
Block et al.  details interactive web applets for equilibrium analysis of two-dimensional masonry arches. The interactivity encourages the user to develop structural intuition for changes in the behavior of masonry arches and funicular polygon under different input conditions. However, at the time of the writing of this section, the web applets, located at http://web.mit.edu/masonry/interactiveThrust/examples.html, were no longer accessible due to new security settings across the Internet that block Java applets.
Harvey’s zone of thrust methodology  for the lower bound was implemented in the Obvis software ArchieM.  This software provides interactive equilibrium analyses of masonry arched structures, targeted for practicing engineers. Users must work with the software’s custom “.brg” file type, making it hard to transfer models from other modeling programs in and out of the software.
The LimitState:RING software is based on Gilbert’s rigid block analysis and is currently used by several engineering firms.  The software thoroughly models masonry bridges under many different conditions that may be specified by the user. Like ArchieM, the user should expect to work exclusively within the software rather than being able to import and export models from different programs.
The lack of transferability with other programs is noted as undesirable because it poses a disadvantage in terms of practicality: the extra step of exporting and importing files for use in different programs presents an extra cumbersome step in the design workflow. It has been noted that thrust line analysis closely engages form with force; this relationship is not fully utilized if the limit state analysis program is not closely linked with the structural design program.
All of the above tools assess two-dimensional masonry arch structures. In the third dimension, Block Research Group has developed RhinoVAULT, a plug-in for the widely- used architectural modeling program Rhino 3D. [39, 40] Based on Block’s thrust network analysis, RhinoVAULT allows users to design compression-only shells. While it has been shown that thrust network analysis can assess historic masonry structures (see ), the RhinoVAULT tool itself focuses on the design rather than the assessment of masonry structures.
There is yet to exist a tool in either two or three dimensions that performs simple thrust line analysis in a file type easily transferrable to other programs.
3.5 Plastic theory and limit state analysis: discussion
The limit state analysis of masonry structures through the upper and lower bound theorems was outlined in this section. These analysis methods rely firmly upon plastic theory, which is more suitable to assess the stability of masonry structures: unlike elastic theory, plastic theory accounts for the fact that masonry failure is dictated primarily by geometry rather than strength.
The power of geometry dictating masonry failure provides motivation for architectural designers to explore limit state analysis as a way of relating form and forces. There exist two extreme applications in the existing tools available for limit state analysis: isolated engineering design of masonry arches, and purely creative design of compression-only structures. There exists some untapped potential in developing principles from limit state analysis that may inform but not entirely dictate the forms created by a designer.
While limit state analysis is accepted as a powerful engineering tool for masonry stability assessment, many recognize its shortfalls, put best by Roca et al. : it is unable to describe the response and damage for moderate load levels that do not necessary lead to a limit condition. This limitation can be addressed through numerical modeling, which will be discussed in the next post of this series.
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Author: Demi Fang ’17
Excerpted and adapted from the author’s senior-thesis-in-progress.