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Masonry Arches: Thrust Line and Strength: History
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Contributor: Danila Aita

The concept of ‘thrust line’ has its roots in historical contributions that laid the foundation for the development of modern tools useful for the limit analysis of masonry arches and vaults. This concept can be traced back to two different, but related, notions: ‘line of resistance’ and ‘line of pressure’. In this paper, both historical contributions and recent developments of the thrust line method will be discussed with a focus on those formulations which take into account the material’s properties, in particular compressive strength.

  • masonry arch
  • limit analysis
  • thrust line
  • compressive strength
  • line of pressure
  • line of resistance
This paper aims to examine the historical roots of the thrust line concept, developed in the context of 19th-century studies on the static analysis of masonry arches, focusing on those contributions capable of taking into account the masonry strength, placing them as a basis for possible modern developments.

Limit Analysis of Masonry Structures

To better understand how this topic can be framed from a theoretical point of view, it may be necessary to first review the approaches related to the structural analysis of unreinforced masonry structures, a crucial problem in contemporary structural mechanics due to the complex mechanical behaviour of masonry, a heterogeneous material with low tensile strength and good compressive strength [1][2][3][4].
As is well known, different approaches are used to address this matter. In the recent scientific literature, discrete element (DEM) [5][6][7] and finite element methods (FEM) are available as powerful tools to capture the mechanical response of a masonry structure under gravitational or seismic loads by taking into account the blocks arrangement, the material properties, the features of the interfaces, and the presence of reinforcements, sometimes with reference to specific case studies or to interpret experimental tests [8][9][10]. For a comprehensive discussion of modelling strategies for masonry characterization and analysis of unreinforced masonry structures, the interested reader is referred to [11][12][13].
However, as observed among others by [12], when modelling the structural behaviour of masonry using refined formulations, several mechanical parameters must be considered, sometimes affected by uncertainties or difficult to determine. Furthermore, these methods can be computationally demanding. For this reason, simple approaches based on the static and kinematic theorem of limit analysis are still considered valuable in engineering practice today for rapidly assessing masonry structures, providing useful information for conservation purposes [14]. The objective of classical limit analysis is to assess structural safety and/or determine the maximum load-bearing capacity of a structure. Starting from the fundamental contribution by Kooharian [15], Jacques Heyman [16][17][18] transferred the philosophy of plastic theory from the steel to the stone skeleton, studying the mechanics of masonry structures in the theoretical background of limit analysis. Re-evaluating pre-elastic contributions [19], he assumed simplifying hypotheses on the masonry material (zero tensile strength, infinite compressive strength, no sliding between the blocks). Considering these assumptions, ref. [20] offers an overview of the theoretical aspects of limit analysis applied to masonry structures, in particular arches, vaults, and domes. Heyman’s no-tension model has been widely and successfully used to examine the stability of masonry systems by exploiting both the static and kinematic theorems of limit analysis: on the one hand, his well-known safe theorem made it possible to re-evaluate the idea of thrust line and to leverage graphical statics to assess the equilibrium of masonry arches and vaults using both classical [21][22] and computational approaches [23][24]; on the other, kinematic analysis was applicable to examine possible collapse modes of both vaulted structures and masonry buildings under gravitational or seismic loads [25][26][27][28].
Although Heyman limit analysis [16][17][18] has proven to be a reliable technique for the structural analysis of masonry structures, a number of scholars (see, for example [29]) observe that such a simplified approach does not take into account important factors that should be considered to avoid overestimating the load-bearing capacity of an unreinforced masonry structure, such as the possibility of sliding between the masonry blocks, and masonry crushing.
Given the complexity of the theoretical background of studies on the structural response of masonry structures, only briefly outlined above, and observing that limit analysis plays a crucial role, this paper seeks to draw the readers’ attention to an important topic: the relationship between the thrust line method, framed in the static theorem of limit analysis, and the introduction of material strength in the structural analysis of masonry arches.
The subject offers the opportunity to explore the intriguing historical context in which the concept of thrust line emerged, between limit analysis and elastic analysis, a framework with theoretical inconsistencies, as will be described in Section 1.2. Considering a limited compressive strength allows one to overcome the simplifications due to the classical Heyman’s approach, which reduce the analysis of masonry arch structures to an essentially geometric problem [21]. Although it is recognized that taking into account material characteristics, like strength, is not always necessary to evaluate a masonry structure’s stability [21][24], it can be significant when it comes to conservation aspects, such as damage [30][31][32] and reinforcement techniques [33]. The influence of a limited strength on the structural response of masonry arches and bridges is also confirmed by experimental investigations, as described by [31]. Considering strength is essential to examine the effects of material properties on the collapse behaviour of masonry structures [32][34] and to perform rigorous structural assessment of large masonry vaults or domes; see, for example, the large-scale masonry arch bridge studied by [35] and the Global Vipassana Pagoda in Mumbai [36][37].
Investigating the relationship between thrust line and compressive strength offers the opportunity to critically examine the so-called Méry method, which is still widely used in engineering practice, highlighting the current misinterpretation of the original approach proposed by the author [38].
Finally, state-of-the-art reviews on the main historical contributions based on the thrust line approach [39][40] focus on the equilibrium conditions and mathematical developments of the method, while an analysis on the relationship between thrust line and strength in a historical perspective is missing. This paper aims to contribute in this direction.

This entry is adapted from the peer-reviewed paper 10.3390/encyclopedia6030057

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