Table Of ContentResearch for Development
Gabriele Milani
Vasilis Sarhosis Editors
From Corbel
Arches to Double
Curvature Vaults
Analysis, Conservation and Restoration
of Architectural Heritage Masonry
Structures
Research for Development
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Gabriele Milani Vasilis Sarhosis
Editors
From Corbel Arches
to Double Curvature Vaults
Analysis, Conservation and Restoration
of Architectural Heritage Masonry Structures
Editors
Gabriele Milani Vasilis Sarhosis
Department of Architecture, Built University of Leeds
Environment and Construction Engineering School of Civil Engineering
(ABC) Leeds, UK
Politecnico di Milano
Milan, Italy
ISSN 2198-7300 ISSN 2198-7319 (electronic)
Research for Development
ISBN 978-3-031-12872-1 ISBN 978-3-031-12873-8 (eBook)
https://doi.org/10.1007/978-3-031-12873-8
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
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Contents
Introduction ...................................................... 1
Gabriele Milani and Vasilis Sarhosis
Historical Review of Masonry Arches and Vaults. Trends
in Historic Preservation and the Relevance of Understanding
Curved Masonry Structures ........................................ 21
Stefano Della Torre and Lorenzo Cantini
Survey and Monitoring Methods for Masonry Arches and Vaults ...... 39
Yu Yuan, Gabriel Stockdale, and Gabriele Milani
Digital Setting Out Techniques for Tile Vaults Without Formwork ..... 61
Julián García, Carlos Martín, Roberto Molinos, and Joaquín Grau
Experimental Behaviour of Masonry: Static and Dynamic
Behaviour of Arches and Vaults ..................................... 79
Nicoletta Bianchini, Angelo Gaetani, Nuno Mendes,
and Paulo B. Lourenço
Revisiting Classic Methods for the Equilibrium Analysis
of Masonry Arches and Domes ...................................... 123
Danila Aita
Rigid Block Limit Analysis of Masonry Arches with Associated
and Non-associated Slides .......................................... 169
Yiwei Hua and Gabriele Milani
NURBS-Based Limit Analysis of Masonry Vaults ..................... 205
Nicola Grillanda, Andrea Chiozzi, and Gabriele Milani
Discrete Element Modelling of Masonry Arch Bridges, Arches
and Vaults ........................................................ 233
José V. Lemos, Francesca Gobbin, Tamás Forgács, and Vasilis Sarhosis
v
vi Contents
Complex Simplicity—Design of Innovative Sustainable Thin-Shell
Masonry Structures ............................................... 257
Michael H. Ramage, Ana Gatóo, and M. Wesam Al Asali
Introduction
Gabriele Milani and Vasilis Sarhosis
Structures of architectural heritage have inevitably suffered damage with time, and
this is particularly true for vaults and domes. Earthquakes, soil settlements, material
degradation and lack of maintenance are the main reasons for that. With Sustainable
Development Goal 11 (SDG 11), countries have pledged to “make cities and human
settlements inclusive, safe, resilient and sustainable”. Within this goal, Target 11.4
aims to “strengthen efforts to protect and safeguard the world’s cultural heritage”.
In Europe, inside cathedrals, church buildings, castles, towers and palaces the role
played by curved masonry and wood structures, in the form of arches, vaults and
domes are among the most well recognized. Arches are also very common and of
major importance in Roman and Medieval bridges, as well as in aqueducts. The evolu-
tion from trilithons to arches, passing through corbel vaults was a Roman intuition;
the further refinement of the idea, achieved in the Middle Age and Renaissance, to
conceive structures resistant thanks to their geometry lead in Europe to build double
curvature structures and domes with impressive artistic value. There is an imperative
need to ensure that these outstanding structures continue to be an integral part of
local life given the increasing level of support and expertise required to maintain
them at a time when the limited resources in communities are becoming ever more
stretched.
The present book focuses on arches. In architecture, an arch is a structural element
with a curved shape that rests in equilibrium on springing and typically covers
moderate spans. The utilization of curved geometries for some masonry elements is
a direct intuitive consequence of the fact that, as observed by Di Pasquale (1992)“in
masonry the form of the resistant structure depends on loads”, Fig. 1.
B
G. Milani ( )
Department of Architecture, Built Environment and Construction Engineering, Polytechnic of
Milan, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
e-mail: [email protected]
V. Sarhosis
School of Civil Engineering, University of Leeds, Leeds LS2 9JT, UK
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 1
G. Milani and V. Sarhosis (eds.), From Corbel Arches to Double Curvature Vaults,
Research for Development, https://doi.org/10.1007/978-3-031-12873-8_1
2 G.MilaniandV.Sarhosis
Fig. 1 In masonry, the form of the resistant structure depends on loads, as observed by Di
Pasquale (1992)
-a -b
Fig. 2 Difference between corbel arch (a Corbel arch at Mayan ruins in Coba, Mexico) and true
arch (b Titus tunnel bridge, Hatay, Turkey)
In a masonry arch, there is something that goes beyond the mechanical features of
the masonry material and invokes concepts related to the geometry, i.e. to the method
through which the blocks are assembled into a structural element. Specializing such
general definition to arches, such concept is immediately clear when corbel arches
are compared with true ones, Fig. 2.
Figure 3 highlights the distinction between masonry regarded as a material and as
a method from a structural analysis standpoint. Masonry as a material begins with the
definition of the material and texture and the behavior is dominated by the strength
and stiffness. Geometry plays a secondary role. Masonry as a method begins with
and is controlled by geometry and form. Materials and textures are secondary.
Introduction 3
Distinction between method and material
Masonry as method Masonry as material
Behavior independent from Behavior dependent on material
material and texture
Load carrying capacity controlled Load carrying capacity controlled
by geometry by strength and stiffness
Materials and textures provide Geometry provides secondary
secondary effects to capacity effects to capacity
Advantageous for framing Advantageous for in-and out-of-
systems plane loaded walls
Fig. 3 Distinction between masonry as “method” and masonry as “material”
Before going to discuss the role played by the geometry on the stability of masonry
arches, it is therefore necessary to analyze “masonry” as material. As highlighted by
Huerta (2001), three are the main features to consider, namely: (i) Heterogeneity;
(ii) Almost no resistance to tensile and good compressive strength; (iii) High friction
coefficient. As far as the heterogeneity is concerned, masonry is a heterogeneous
composite material constituted by the assemblage of blocks and mortar. According
to experimental evidence, it exhibits a non-isotropic behavior both in the elastic range
and at collapse (Page 1981; Dhanasekar et al. 1985; Milani et al. 2006a), see Fig. 4.
In the case of vaults, the texture can be completely different in the different parts of
the same vault, Fig. 5, a feature that adds complexity to an already difficult problem
consequent to the curved shape of the surface.
Going to consider the second factor, i.e. the inability of masonry to withstand
tensile stresses, it should be pointed out that masonry tensile strength depends
