Research 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 Series Editors Emilio Bartezzaghi, Milan, Italy Giampio Bracchi, Milan, Italy Adalberto Del Bo, Politecnico di Milano, Milan, Italy Ferran Sagarra Trias, Department of Urbanism and Regional Planning, Universitat Politècnica de Catalunya, Barcelona, Barcelona, Spain Francesco Stellacci, Supramolecular NanoMaterials and Interfaces Laboratory (SuNMiL), Institute of Materials, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Vaud, Switzerland Enrico Zio, Politecnico di Milano, Milan, Italy Ecole Centrale Paris, Paris, France The series Research for Development serves as a vehicle for the presentation and dissemination of complex research and multidisciplinary projects. 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THE SERIES IS INDEXED IN SCOPUS · 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 Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland 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