form finding of arches and shell structures subjected ...

FORM FINDING OF ARCHES AND SHELL STRUCTURES SUBJECTED TO SEISMIC LOADING

Tim L. Michiels

A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

Adviser: Sigrid Adriaenssens June 2018)

© Copyright by Tim L. Michiels, 2018. All rights reserved.

Abstract The overall geometry of arches and shell structures plays an essential role in their capacity to withstand earthquakes, and shells in particular resist seismic loading extremely well because of their lightweight nature and high geometric stiffness. Seismic loading, however, is rarely considered when initially determining the form of arches and shells, although this could significantly improve their design’s material-efficiency and seismic performance. Therefore, in this dissertation the first computational methodologies are presented that generate shapes for arch and shell structures designed to sustain self-weight and seismic loading in a material-efficient way. The research focuses on arches and shells that withstand the applied loads through compressive internal loading, making the developed forms suitable for construction in materials with substantial compressive strength such as unreinforced concrete, stone, earth, ice or masonry. Additionally, the resulting geometries are scalable as long as the compressive strength of the material is not exceeded. A form finding algorithm for arches with varying thickness is presented that relies on a methodological application of a series of geometric manipulations of a thrust line, generated under combined gravity and horizontal loading. This algorithm is subsequently extended to find material-efficient forms for thin shells that can similarly withstand these loads. This is accomplished by corrugating the supporting edges of the shell so that a compressive load path can form within the depth of the supports. It is demonstrated that the obtained shapes are superior to non-form-found geometries in their material use and their horizontal pushover capacity through a kinematic limit state analysis for arches and by using a non-linear pushover analysis for corrugated shells. Additionally, a second method for shells is developed that expands the 2D thrust line concepts to a 3D hanging net model approach obtained in a dynamic relaxation solver. This method accounts for self-weight combined with seismic loads acting in any horizontal direction. The approach yields single-layer shells with varying thickness or lighter double-layer interconnected thin shells. As the first of their kind, the conceptual form finding approaches presented in this dissertation will facilitate the design of material-efficient and safe arches and shell structures in seismic zones.

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Acknowledgements This dissertation would not have seen its final form without the support, advice and help of many. First and foremost, I would like to thank my advisor, Prof. Sigrid Adriaenssens for guiding me through this 4-year journey. I truly appreciated the freedom she gave me to pursue my interests and ideas but am equally thankful for our many in-depth conversations on my research and shell structures. I am also grateful to Professors Maria Garlock and Branko Glisic from Princeton’s CEE Department for all of their feedback and help along the way. Maria Garlock served on my PhD committee and generously took the time to review this manuscript. She also introduced me to the concept of structural art and inspired me to analyze the seismic behavior of Félix Candela shells. Branko Glisic made himself available anytime I walked into his office seeking help on the structural challenge of the day. Additionally, I would like to thank Professor John Ochsendorf for serving on my committee and for being so generous with his ideas and critical reflections. His enthusiasm for and encouragement to pursue the study of vaulted structures and historic buildings using graphic statics were instrumental to this dissertation. I am also grateful to Professor Matthew DeJong, for hosting me at Cambridge University and for helping me set up and interpret the validation framework provided in Chapter 4. Our numerous discussions greatly enlightened my understanding of shell behavior during earthquakes. Similarly, I am indebted to Professor Juan José Jorquera-Lucerga from the Universidad Politécnica de Cartagena. Our long conversations during his research stay at Princeton formed the basis behind the method presented in Chapter 5. A great number of other people contributed to my Princeton experience. My Form Finding Lab colleagues Victor Charpentier and Olek Niewiarowski, as well as my fellow-CEE graduate students Tracy Huyn, Kasparas Spokas, Becca Napolitano and Isabel Morris all helped me in their own way. Lionel Du Peloux generously shared his dynamic relaxation code during his research stay at the Form Finding Lab, which significantly sped up the implementation of Chapter 5. I furthermore had the opportunity to work with a group of outstanding people on a variety of projects. Eric Teitelbaum, Laura Salazar, Amber Lin and Prof. Forrest Meggers helped me design and build a large rammed earth installation. Prof. Stefano Gabriele and Giulia Tomasello from the University of Roma Tre welcomed me in Rome and shared their ideas about the funicularity of shells structures (and pasta). It was also a pleasure to work with a set of talented undergraduate students such as Lu Lu, Russel Archer, Aaron Katz and Demi Fang. None of this, however, would have been possible without the support and love from my former colleagues, friends and family. My mentor at the Getty Conservation Institute, Claudia Cancino, encouraged me to pursue a PhD and gave me the opportunity to continue to work on the GCI’s Seismic Retrofitting Project. This allowed me to stay in touch with preservation practice and provided me access to an additional group of extraordinary international scholars. My friends in

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Belgium, although physically far away, always seemed close by. Their repeated visits and our cherished moments back home, made distance feel irrelevant. I am also grateful to my American family, Alexa and Michael, for welcoming me so warmly and generously, time-after-time. A special word of thanks goes out to my parents. They selflessly encouraged me to pursue my aspirations, even though it took me far away from them. Their emphasis on the importance of education helps me to this day, and I cannot thank them enough for all their unwavering support in everything I do. Finally, I want to thank Niki for her love and unparalleled support. During our years together, she did not only transform me into a better English writer, but also into a better person. Words to thank her will always fall short.

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Table of Contents 1.1

Motivation and Background ............................................................................................. 1

1.2

Research Objectives ......................................................................................................... 3

1.3

Significance of Research .................................................................................................. 3

1.4

Dissertation Organization ................................................................................................. 4

1.5

Prior Publications Contributing to this Dissertation ........................................................ 4

2.1

Introduction ...................................................................................................................... 6

2.2

Structural behavior of concrete shells during earthquakes............................................... 6

2.2.1 2.2.2

Case study: response spectrum analysis of reinforced concrete shell ................................... 7 Parametric study on key parameters affecting seismic behavior ........................................ 11

2.3

Masonry shells in seismic areas ..................................................................................... 14

2.4

Form Finding .................................................................................................................. 19

2.5

Conclusion and research gap .......................................................................................... 23

3.1

Introduction .................................................................................................................... 25

3.1.1 3.1.2

3.2

Form finding of arches ........................................................................................................ 26 Review of analysis techniques for masonry arches under earthquake loading ................... 27

Form finding Methodology ............................................................................................ 29

3.2.1 3.2.2 3.2.3

Thrust lines due to combination of gravity and horizontal acceleration ............................. 29 Form finding procedure ...................................................................................................... 32 Application example ........................................................................................................... 35

3.2.4

Validation through kinematic limit state analysis ............................................................... 37

3.3

Results ............................................................................................................................ 40

3.3.1

Support size variations for an arch with rise-to-span ratio of 1/2 ....................................... 40

3.3.2

Different design acceleration .............................................................................................. 41

3.3.3

Different arch rise-to-span ratios ........................................................................................ 42

3.4

Discussion ...................................................................................................................... 42

3.4.1

Support size variations for r/s equal to 1/2 ......................................................................... 42

3.4.2 3.4.3

Design accelerations ........................................................................................................... 43 Rise-to-span ratios............................................................................................................... 44

3.5

Applications and limitations........................................................................................... 44

3.6

Conclusion...................................................................................................................... 48

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4.1

Introduction .................................................................................................................... 49

4.1.1

4.2

Form finding under earthquake loading ......................................................................... 51

4.2.1 4.2.2

4.3

Form finding method for corrugated shells under earthquake loading ............................... 51 Application to masonry shells ............................................................................................. 55

Validation method: non-linear pushover analysis .......................................................... 57

4.3.1

4.4

Scope and outline ................................................................................................................ 50

Non-linear pushover ............................................................................................................ 58

Results ............................................................................................................................ 60

4.4.1

Pushover curves .................................................................................................................. 60

4.4.2

Collapse mechanism and flow of forces ............................................................................. 62

4.5

Discussion ...................................................................................................................... 64

4.5.1 4.5.2 4.5.3

Evaluation of form finding method ..................................................................................... 64 Influence of shape on flow of forces and collapse mechanism ........................................... 66 Design opportunities and limitations .................................................................................. 68

4.6

Conclusion...................................................................................................................... 70

5.1

Introduction .................................................................................................................... 71

5.1.1

5.2

Form Finding Methodology ........................................................................................... 73

5.2.1 5.2.2

5.3

Advantages of double-layer shells ...................................................................................... 71 Extension from thrust lines to nets ...................................................................................... 73 Implementation of form finding method ............................................................................. 75

Results of the Form Finding Process.............................................................................. 80

5.3.1 5.3.2

Square plan.......................................................................................................................... 80 Different plan geometries.................................................................................................... 82

5.4

Form manipulations to align crowns .............................................................................. 83

5.5

Conclusions .................................................................................................................... 85

6.1

Solutions to research questions ...................................................................................... 86

6.2

Recommendations for Future Research ......................................................................... 89

6.2.1 6.2.2 6.2.3

Single-layer uniform thickness shells from 3D approach ................................................... 89 Construction aspects ........................................................................................................... 89 Large scale testing and dynamic loads ................................................................................ 93

6.2.4 6.2.5

Gridshells ............................................................................................................................ 94 Applications for historic structures ..................................................................................... 94

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Introduction 1.1 Motivation and Background Since 2004, earthquakes alone have caused the deaths of more than 752,000 people worldwide (see table 1.1 [1]–[3]). Many of these casualties can be attributed to building failures. In response to such cataclysmic natural disasters, over the past several years there has been a surge in attention paid to the disaster preparedness and the resilience of buildings. For example, the United States is becoming increasingly aware of the vulnerabilities of its infrastructure. California, for example, has invested significant resources in preparation for a major (about 7.9 Mw) earthquake expected to be induced by the San Andreas fault ever since the 6.7 M w 1994 Northridge earthquake, which was responsible for 57 deaths and estimated economic losses of $46 billion [4]. Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004

Mw 7.3 and 7.1 7.8 7.8 6.2 7.7 6.7 9 7 7.5 7.9 8 6.3 7.6 9.1

Fatalities 1,000 676 8,898 729 825 113 20,896 316,000 1117 87,587 514 5,749 80,361 227,898

Location Iran and Mexico Ecuador Nepal Wenping, China Pakistan Philippines Honshu, Japan Haiti Sumatra, Indonesia Sichuan, China Pisco, Central Peru Java, Indonesia Pakistan Northern Sumatra

Table 1.1 – Selected earthquakes and fatalities worldwide since 2004. Many regions in the United States, however, are barely prepared for a similar seismic event, such as the Pacific Northwest [5] where the Cascadia subduction zone is anticipated to induce a cataclysmic 9.0 Mw earthquake, which according to FEMA will cause approximately a million buildings to collapse or be severely damaged [3]. Due to the worldwide threat of imminent earthquakes, the further development of seismic-resistant structures which are safe, cost-effective and environmentally friendly is crucial. The presented research focuses on shell structures to

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address this challenge because shells have proven to be safe shelters during natural disasters such as earthquakes, hurricanes and tornados [6]–[8]. For example, the reinforced concrete shells of Félix Candela withstood the 1985 Mexico City earthquake unharmed, while buildings in the surrounding neighborhoods sustained significant damage [7]. To the best of the author’s knowledge, collapse of concrete shells or masonry shells constructed in the past century have not been reported. Over the past two decades both design research and the construction of shell structures has been spurred by the development of a series of computational tools for parametric design, form finding and structural optimization [9]. These tools have a particularly large impact on shell structures specifically, as the overall behavior of their surfaces is mostly influenced by their global form [10]. Furthermore, advances in digital fabrication techniques using robots and drones are beginning to be employed for shell construction, as illustrated by a variety of innovative prototype shells that have been successfully built in recent years using these new methods and tools [11]–[15]. Moreover, traditional construction techniques have also been revisited during the past decade to build a handful of ground-breaking new shell structures that have a remarkably low environmental impact. Prominent examples are the Mapungubwe Interpretive Center (Rich, Ramage and ODB, South-Africa, 2008) and the Droneport project (Foster+Partners, BRG and ODB, Italy, 2016) at the 2016 Venice Biennale, both constructed from thin earthen tiles employing the traditional thin-tile vaulting technique [16]. This technique has the advantage of requiring minimal formwork and the resulting thin-tile masonry shells have an exceptionally low embodied carbon content (as low as 60 kgCO2e/m2 compared to around 440 kgco2e/m2 for traditional floor and roof structures) [17]. Despite the ever-looming threat that earthquakes pose to the built environment, the empirically observed good behavior of shells in seismic areas, the advances in computational design and analysis techniques, and the innovation and rediscovery of construction methods for sustainable shells, limited research has been conducted to study the effect of earthquake loading on the structural behavior of shells. Therefore, the presented research introduces a set of computational strategies for the generation of forms for arches and shells in seismic areas.

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1.2 Research Objectives The overall research goal of this thesis is to present a series of novel computational methodologies that generate material-efficient shapes for arch and shell structures in earthquakeprone regions. The scope is limited to arches and shells that resist the external applied load predominantly through compressive internal loading, thus making the developed forms suitable for construction in materials with substantial compressive strengths such as unreinforced concrete, stone, ice or masonry, including earthen blocks. This overall goal is achieved by addressing the following specific research objectives: i. ii. iii. iv. v. vi.

Identify the key parameters that affect the seismic behavior of continuous shell structures. Implement efficient first-order equilibrium analysis techniques to analyze arches under seismic loading. Develop and validate a form finding approach for arches subjected to in-plane earthquake loading. Establish and validate a form finding methodology for corrugated shells subjected to seismic loading. Devise a form finding methodology for double-layer shells that can withstand earthquake loading. Establish future research directions and applications of the developed form finding approaches.

1.3 Significance of Research Various form finding techniques for arches and shell structures have been developed, but the application of these techniques has typically been limited to design under vertical (often gravitational) loading assuming that the self-weight of the shell is more important than the induced live-loads (such as wind and snow loading) [9]. This assumption may not hold for seismic loads, which are proportional to the mass of structures. To the best of the author’s knowledge, no form finding approaches have been developed for either arches or shell structures that take seismic loading into account to determine the overall geometry of these structures during the initial steps of the design process. Therefore, the methods presented in this dissertation are the first of its kind and can facilitate the construction of efficient and safe arches and shell structures in seismic zones. Furthermore, because all presented form finding techniques are material-independent, they can be applied to construct buildings from a range of materials. As the form finding approaches are conducted under the assumption that the compressive strength of these materials will exceed the compressive stresses that the structures experience by an order of magnitude, the resulting shapes can be scaled. A final contribution of this dissertation is the provision of design guidelines for arches and corrugated shells under seismic loading.

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1.4 Dissertation Organization This dissertation is organized as follows. In Chapter 2 an introduction into the behavior of shells during earthquakes is presented through a case study, a literature review and a parametric study of reinforced concrete shells that evaluates the effect of shell shape on the vibrational properties, stresses and deformations under seismic loading. A review of the behavior of masonry shells during earthquakes is also provided, as well as an overview of the state-of-the-art of form finding approaches. In Chapter 3 analysis techniques for arches without tensile capacity are detailed and thrust line concepts are employed to perform form finding of arches subjected to in-plane seismic loading. Through a parametric study, design guidelines for arches in seismic areas are also provided. In Chapter 4 the form finding methodology for arches is elaborated upon and transformed into a method that can be used for the design of corrugated shell structures. The method is validated in this chapter using non-linear finite element analysis, and design guidelines for corrugated shells are provided. In Chapter 5, the 2-dimensional form finding approach presented in Chapter 3 for arches is extended to three dimensions by employing hanging net models to obtain geometries for a variety of double-layer shells that can withstand seismic loading through compression load paths. In Chapter 6, the main conclusions of this dissertation are compiled and areas of future work for the design of shell structures in seismic areas are identified.

1.5 Prior Publications Contributing to this Dissertation Portions of this dissertation have appeared in a set of peer-reviewed journal and conference papers. These sections have been edited for continuity and are included with permission of the respective publishers. Peer-reviewed journal papers: i.

ii.

iii. iv.

T. Michiels and S. Adriaenssens, “Identification of key design parameters for earthquake resistance of reinforced concrete shell structures,” Eng. Struct., vol. 153, pp. 411–420, 15 2017. T. Michiels, R. Napolitano, S. Adriaenssens, and B. Glisic, “Comparison of thrust line analysis, limit state analysis and distinct element modeling to predict the collapse load and collapse mechanism of a rammed earth arch,” Eng. Struct., vol. 148, pp. 145–156, Spring 2017. T. Michiels and S. Adriaenssens, “Form finding algorithm for masonry arches subjected to in-plane earthquake loading,” Comput. Struct., vol. 195, pp. 85–98, 15 2018. T. Michiels, S. Adriaenssens, and M. DeJong, “Form finding of corrugated shell structures for seismic design and validation using non-linear pushover analysis.,” Eng. Struct., Under review at time of writing.

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T. Michiels, S. Adriaenssens, and J. J. Jorquera-Lucerga, “Parametric study of masonry shells form-found for seismic loading,” J. Int. Assoc. Shell Spat. Struct., vol. 58 (4), pp. 267– 275, 2017.

v.

Conference papers: i.

ii.

iii.

T. Michiels, S. Adriaenssens, and L. Rhode-Barbarigos, “Size optimization of a cylindrical thin shell subjected to 1992 Landers earthquake,” in Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam. Future Visions. 17 - 20 August 2015, Amsterdam, The Netherlands, 2015. T. Michiels, M. Garlock, and S. Adriaenssens, “Seismic assessment of Félix Candela’s concrete shells and their behavior during the 1985 Mexico City earthquake. A case study on the church of our lady of the miraculous medal,” in Structural Analysis of Historical Constructions: Anamnesis, diagnosis, therapy, controls - Proceedings of the 10th International Conference on Structural Analysis of Historical Constructions, SAHC 2016, 2016, pp. 1544–1550. T. Michiels, M. DeJong, and S. Adriaenssens, “The optimal form of corrugated shells designed to withstand earthquakes,” in Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2018, Boston. Creativity in Structural Design. 16 20 July 2018, Boston, U.S.A., 2018 (accepted).

The author of this dissertation was the first and primary author of the listed publications. The coauthors for these works served primarily as advisers and editors.

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Research Context and Literature Review Sections of Chapter 2 are based on following publications: T. Michiels and S. Adriaenssens, “Identification of key design parameters for earthquake resistance of reinforced concrete shell structures,” Eng. Struct., vol. 153, pp. 411–420, 15 2017. T. Michiels, M. Garlock, and S. Adriaenssens, “Seismic assessment of Félix Candela’s concrete shells and their behavior during the 1985 Mexico City earthquake. A case study on the church of our lady of the miraculous medal,” in Structural Analysis of Historical Constructions: Anamnesis, diagnosis, therapy, controls - Proceedings of the 10th International Conference on Structural Analysis of Historical Constructions, SAHC 2016, 2016, pp. 1544–1550.

2.1 Introduction In this chapter the research context for this dissertation is provided through a literature review and a set of numerical analyses. The key factors that affect the behavior of shells during earthquakes are also identified. As earlier research on shells in earthquake areas has focused predominantly on reinforced concrete shells, these structures are addressed first. Subsequently, the past behavior of a set of masonry shell structures during earthquakes is investigated through a literature review, providing examples of masonry shells that behaved well during seismic events, and highlighting the causes of damage for shells that collapsed. Finally, a state-of-the art of form finding techniques for shells (applicable to any shells built from a material that works in compression) is presented. In the subsequent Chapters 3, 4 and 5, additional literature review is provided to complement the initial information presented in this chapter.

2.2 Structural behavior of concrete shells during earthquakes In the past decades, large span thin shell reinforced concrete roof structures have shown their potential to resist extreme loading during natural disasters. For example, the Miami Marine Stadium (Miami, USA, 1967), a reinforced concrete hyperbolic paraboloid (hypar) folded shell, withstood the Category 5 Hurricane Andrew (1992) without being damaged [18] and a residential reinforced concrete dome (Pensacola Beach, USA, 2002) survived the Category 5 Hurricane Ivan (2004) while several buildings on its block collapsed or suffered irreparable damage. Similarly, thin reinforced concrete shells have withstood the effects of earthquakes. For example, the thin reinforced concrete shell structures designed and constructed by Félix Candela (Mexico City, 1950s and 1960s) withstood the 8.0 Mw earthquake that shook Mexico City in 1985 without reported damage. Taking a closer look at one of these Candela shells, the

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Church of Our Miraculous Medal (Mexico City, Mexico, 1953-1955), provides insights into the key parameters that affect how such a shell responds to an earthquake (see section 2.2.1). As this case study shows that the high eigenfrequencies of the shell have an important impact on the its behavior during earthquakes, the effect of shell form on these eigenfrequencies is addressed through a parametric analysis (see section 2.2.2), which sheds light on the importance of the lightweight nature and geometric stiffness of shells during seismic events.

2.2.1 Case study: response spectrum analysis of reinforced concrete shell The geometry of the Church of Our Miraculous Medal consists of four identical bays of hypars, each spanning 21 m. These structural shells have a thickness of 4 cm and are positioned adjacent to another set of hypars forming the apse with a height of 14.95 m [19] (see Figure 1 and 2).

Figure 1 - Exterior (left) and interior (right) of the Church of Our Miraculous Medal (Mexico City, Mexico, 1953-1955) (photos by Bruce White) [19]. The presented response spectrum analysis builds upon a finite element model constructed by [20] and similarly considers two adjacent bays. Geometrically linear elastic behavior for the structure is assumed for the analysis. The geometry is composed of a repeating set of half-bays with each half-bay supported by 3 supports (see Figure 2) [20]. Each of the supports is considered to be pinned, as it is assumed that no bending moments can be transferred to either the columns or the foundations. The thickness of the shell is modeled as 4 cm with a thickened edge on the top of 14 cm. Density of the concrete is assumed to be 2400 kg/m3. Conservative values for the properties of the concrete are taken similar to the ones used in [20]: a maximum compressive strength of 14 MPa and a maximum tensile strength of 1.5 MPa, to account for possible weaknesses in the old, manually applied concrete [19]. The Young’s modulus of concrete is 23.6 GPa [19]. It should be noted that steel reinforcement was present in the entire structure, thus even if stresses were to exceed this tensile strength there would be extra capacity provided by the steel. The concrete beams placed around the triangular edges of the bays and the façade were not taken into account for this analysis which is again a safe assumption as the beams are expected to reduce stress concentrations around the edges.

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Figure 2 – Dimensions of a half-bay with supports indicated by triangles (adapted from Thrall, Garlock and Billington) [21]. A mesh convergence study showed that a mesh with 40 shell elements (first-order quads) along the 6.8 m width of the bay was sufficient for stress convergence. The aspect ratio of the shell elements was kept to around 1 in the rest of the shell. Altair HyperMesh in combination with the OptiStruct solver was used to perform the analysis [22]. Stress results and results from the modal analysis were validated using the results provided in [19] and by running a similar model in SAP2000. The principal stresses under gravity loading in the middle and top surface are provided for reference in Figure 3. The stresses in the bottom surface are even lower and are therefore not displayed. The minimal principal stresses (compression) in the bottom surface remain under 2 MPa and are of little interest given the assumed compressive strength of the concrete of 14.5 MPA.

Figure 3 - Maximum principal stresses in the top and middle face of two bays under gravity loading (tension is positive).

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2.2.1.a Modal Analysis A modal analysis was performed to understand the vibrational properties of the double bay. In order to obtain a mass participation of about 90% in all directions (respectively 91% east-west, 88% north-south and 85% vertical), it was necessary to take into account 250 eigenmodes. Considering this high number of modes is also required to ensure convergence of stresses for the response spectrum analysis and is recommended for shells with double curvature [23]. All modes were determined taking into account preloading from self-weight, although its effect was negligible (0.3% for the first mode). The 1st eigenmode, with a frequency of 3.09 Hz, has a modal mass contribution of 42% in the north-south direction of the church (see Figure 2 for orientation), the 2nd most important mode in the north-south direction is the 13th mode with a mass contribution of 10% (6.53 Hz). In the east-west direction, the 5th (5.18 Hz) and 14th (6.96 Hz) modes are most important, with modal mass contributions of 27% and 10% respectively. It is noteworthy that these frequencies are high, especially compared to frame structures, but are not surprising for shell structures as frequency (𝜔) is related to stiffness (k) and mass (m): 𝜔 = √𝑘/𝑚. The shells of the church are very stiff due to the anticlastic double curvature in the hypars. Additionally, they have very low mass due to their thinness (4 cm). 2.2.1.b Response Spectrum Analysis Comparing these fundamental frequencies to the response spectrum measured during the 1985 earthquake indicates the reasons why the shells did not suffer any significant damage. The highest acceleration response due to the earthquake was in the frequency range of 0.4 to 0.7 Hz, far below the frequencies of the bays (3.09 Hz and up, see Figure 4). Therefore, dynamic amplification did not occur in the shells. Still the principal stresses (see Figure 5) due to the ground motion increased relative to the ones shown in Figure 3 as demonstrated by the results of the linear response spectrum analysis (taking into account 250 modes to get 90% mass participation). The earthquake excitation was applied in the east-west direction, identically to the real event and 5% modal damping was applied. The earthquake excitation induces additional tensile stresses, that nonetheless remain under the rupture stress of 1.5 MPa almost everywhere. Overall, the principal tensile stresses are higher than those due to gravity loading (which rarely exceed 0.7 MPa) but remain under 1 MPa almost everywhere throughout the shells. The maximum tensile strength of the concrete is exceeded locally around the supports and edges (reaching 2.2 MPa), where stress concentrations are found in the model. However, it can be assumed that the stiffening beams around the edges significantly reduce the stress concentrations in these areas. It can therefore be concluded that the shells behaved entirely elastically during the 1985 earthquake. This observation explains why no cracks were noticed in the shells after the event.

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Acceleration response spectrum (cm/s2)

1200

Response Spectrum Mexico City Earthquake (EW-direction)

900 600 300

All eigenfrequencies higher than 3.09 Hz

0 0.0

0.5

1.0

1.5

2.0

2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

Figure 4 - Acceleration Response Spectrum of the 1985 Michoacán earthquake as measured in the Lake Zone of Mexico City.

Figure 5 - Maximum principal stresses in the top surface and minimum principle stresses in the middle surface of two bays due to 1985 Michoacán earthquake loading. While no comprehensive post-disaster survey of Candela’s buildings has been completed after the recent 7.1 Mw 2017 Central Mexico earthquake, none of the Candela shells have been reported to have collapsed. The hypars of Los Manantiales restaurant, however, did sustain damage as extremely severe ground motions ravaged the Xochimilco neighborhood and caused failure around the concrete footings as well as one major crack in the shell surface (see Figure 6) [24], [25]. The structure’s stability was not jeopardized though. An in depth-study to explain and replicate the failure mechanism is out of the scope of this research, as it would require specific earthquake data and a damage survey, which were unavailable at the time of writing. Nevertheless, from photographs (see Figure 6) it can be deduced that it is likely that the crack in the shell formed due to a localized exceedance of tensile capacity of the concrete, which may be attributed to the movement and failure of the support footings. Despite the crack, the shell remained stable and did not suffer irreparable damage.

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Figure 6 – Damage at Candela’s Los Manantiales restaurant after the 2017 Central Mexico Earthquake. Left: crack in one the hypars. Right: close-up of damaged concrete foundation (images courtesy of Juan Ignacio Del Cueto).

2.2.2 Parametric study on key parameters affecting seismic behavior The case study of Candela’s Church of Our Miraculous Medal indicates that its shells did not experience dynamic amplification due to their lightness and high stiffness, which ensure high eigenfrequencies. The subsoil in Mexico City amplifies only a narrow range of lower frequencies (0.4 to 0.7 Hz in 1985) [26], [27], which allowed the studied shells to behave elastically during the 1985 Mexico City earthquake. To investigate if lightness and geometric stiffness are key parameters for other shell structures as well, a literature review and parametric study is conducted on singly-curved barrel vaults and synclastic doubly-curved shells with square plan. A code-based design input spectrum (obtained from the A.S.C.E. for downtown Los Angeles [28]), as well as the spectrum of an exceptionally strong earthquake with a strong vertical component (1992 Landers Earthquake [29]) have been employed as detailed in section 2.2.2.b. 2.2.2.a Literature review on behavior of concrete shells during earthquakes To the best of the author’s knowledge, no thin reinforced concrete shell structure, other than Los Manantiales [24], has ever been reported to have sustained damage due to an earthquake. Typically, the doubly-curved geometry of the shell surface allows for high structural efficiency as shells tend to carry forces predominantly through membrane action, which enables them to be very thin and thus have a low mass per unit area. Therefore, the induced forces through dynamic actions such as earthquakes are relatively low because these forces are directly proportional to the mass of the shell. The overall shape of a shell structure is the most important factor of the

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common design parameters, such as support conditions, type of material, thickness and overall shape, that determine if a shell will be adequate in terms of safety, stability and stiffness [10]. The form of roof shell structures, however, is typically shaped to perform optimally under gravity loading [9]. Additionally, reinforced concrete roof shells are often designed to carry the loads to the foundations through compressive membrane action (carrying the forces through inplane action), avoiding tensile stresses due to bending and torsional moments [21]. Nonetheless, shells could sustain structural damage due to the tensile stresses generated by the unanticipated bending and torsional moments caused by earthquakes resulting in cracks as observed in the Los Manantiales hypar shells. In practice, the design of shells in earthquake-prone regions is carried out differently than typical self-weight focused shell design. Instead of concentrating on gravity loading, much more attention is paid to the resistance to large bending moments [30]. While both the earthquake resistance capacity of thin reinforced concrete roof shells and the potential danger of tensile stresses in these shells under earthquake loading have been documented in literature [30], few studies have characterized the structural behavior of thin reinforced concrete roof shells when subjected to earthquake-induced dynamic loads [23]. Most studies of reinforced concrete shells and seismic action are limited to hemispherical domes or hyperboloid shells [6], [31]–[34]. Reinforced concrete cooling towers in particular, which are hyperboloid shells of revolution supported on columns, have been analyzed under earthquake loading by several researchers. Numerical simulations [35], [36] and experimental tests on scale models [37] have shown that severe damage and even collapse of reinforced concrete cooling towers can occur during very strong earthquake loading (PGA exceeding 1.735 g during the experimental tests on scale models). However, in practice, no instances of reinforced concrete cooling towers have been reported to have collapsed or experienced major structural damage within the shell surface due to actual seismic events. Furthermore, several studies reported that damage to these reinforced concrete hyperboloid towers would occur first in the columns supporting the shell [37], [38]. Plastic hinges in the shell of revolution itself, would only occur after significant damage in these supporting columns [35]. Other studies also identified the columns supporting the shell as the most vulnerable elements and thus directed their focus mainly on the soil-pile interaction and its effect on the shell [39], [39]. None of these studies specifically focused on the behavior of the reinforced concrete hyperboloid shell itself. General knowledge about reinforced concrete shells and roof shells in particular, such as the underlying reasons for shells’ earthquake resistance, or how key design parameters influence the vibration properties of shell structures, are thus not well-documented. Additionally, recent optimization studies performed on roof shell structures obtain the ideal membrane stress state under a variety of load conditions (such as self-weight, service load, snow load and wind load), but do not take into account dynamic loading of earthquakes [40]–[43]. The optimization of the ideal membrane stress state without taking into account seismic action can thus lead to a decrease in the redundancy of the shell by changing shape and/or thickness to account for one of

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the optimization criteria, but might thereby increase the susceptibility to collapse under dynamic loading [30]. 2.2.2.b High fundamental frequencies, low dynamic amplification To address this lack of general understanding of key contributing factors to the seismic behavior of shells, a parametric study on barrel vaults and doubly-curved cylindrical shells is presented. In this study the performance of singly-curved and doubly-curved shells with different spans was compared and contrasted under the code-based A.S.C.E. design input spectrum and the 1992 Landers earthquake (see [29]). The study focused on shallow (rise-to-span ratios of 0.02 to 0.35) shells and analyzed how change in shape and span affect the fundamental frequencies as well as deformations and stresses under the considered seismic loads. The study showed that 8 cm thick shells with concrete compressive strength of 30 MPa and small to medium spans (from 5 m up to 15 m, in increments of 5 m) exhibited elastic behavior when subjected to either the horizontal or vertical components of the considered ground motions, as the permissible compressive (30 MPa) and tensile (3 MPa) strengths were never exceeded under the considered earthquakes. Also, shells with larger spans, for example 20 m, were unlikely to experience cracks caused by excessive tensile stresses due to the horizontal components of the considered seismic actions. This structural behavior could be attributed to the high geometric stiffness of these shells (which increases with their curvature [44]) in combination with their lightweight nature. These characteristics ensured that the shells have high fundamental frequencies (see Figure 7), and thus their structural modes were only minimally excited by the examined earthquakes, which mainly amplified vibrations at lower frequencies (below 2 Hz) [29]. It was also shown that doubly-curved shells performed far better than singly-curved shells as their geometric stiffness is higher. The fundamental frequencies of doubly-curved shells were therefore at least 3 times larger than those for singly-curved shells for the same height and material volume (for example, 12 Hz vs. 3 Hz for respectively doubly- and singly-curved 20 m span shells, see Figure 7), and their response to both the Landers earthquake and the A.S.C.E.-7 prescribed earthquake in terms of stresses and deformations was superior. As long as the rise of the doubly-curved shell was greater than 0.14 times the span (for a 20 m span shell), the principal compressive stresses did not exceed 6 MPa (20% of the ultimate compressive strength). The principal tensile stresses did not exceed 2.88 MPa (under the 3 MPa limit), but only reached such high values around the supports and were significantly lower throughout most of the shell [29]. While this initial research demonstrates that the earthquake resistance of reinforced concrete roof shells can strongly be influenced by the shells’ shape and span, it should be emphasized that proper shape alone might not be sufficient during certain cataclysmic seismic events. In particular, near-field seismic events with strong vertical components or earthquakes with high frequency content could still cause structural damage. In the case of high frequency content,

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dynamic amplification is more likely to occur, while strong vertical components will induce greater bending moments because the shell is subjected to large out-of-plane action. In locations where these types of seismic events can be expected, other seismic protection measures beyond shell shape should be considered. Base isolation of the supports, for example, might be a particularly suitable seismic protection measure as reinforced concrete shells typically transfer loads to the ground through a limited amount of supports.

Figure 7 - Evolution of fundamental frequency for reinforced concrete shells (singly-curved and doubly-curved) with constant thickness of 8 cm and square plan but changing span.

2.3 Masonry shells in seismic areas The construction of modern thin concrete shells started in the 1910s, but became prevalent only in the 1930s [45]. Until then, masonry vaults in stone, fired bricks, tiles and adobe (mudbrick) were commonly used building materials for continuous shells. These masonry vaulted structures have also empirically shown their potential to resist earthquakes as illustrated by many enduring monuments scattered throughout seismic regions, such as Turkey, Mexico, Italy, Peru and Iran as described below. While the collapse of reinforced concrete shells due to earthquakes has not been reported to the author’s knowledge, a limited number of masonry shells have failed or sustained significant damage during earthquakes [46]–[48], demonstrating a need for further examination of their collapse mechanisms under seismic action. Collapses can typically be related to the introduction of excessive tensile stresses due to thrusts of the curved shapes or to the failure of the supporting masonry structures. For example, the initial dome of the Hagia Sophia (Istanbul, Turkey, 537) collapsed twice due to major earthquakes in 553 and 557 [49]. These failures were due to the lack of proper buttressing of the main dome by the lateral arches and two semi-domes, which were unable to account for the outward thrust of the main dome [50], [51]. The dome’s shape was thus modified to reduce 14

its outward thrust by increasing the height of the dome and inserting a series of windows at the base, resulting in the reconstructed dome shape from 562, which has essentially remained the same until today (see Figure 8). A quarter of the main dome of the Hagia Sophia collapsed again though after a severe earthquake in 986. This time the repairs addressed the issue of the excessive thrust by adding two enormous corner buttresses on the north and south sides of the dome (see Figure 8). After yet another partial collapse during an earthquake in 1346, the buttressing was perfected [45] and an additional tension ring was added in 1847 [49]. Despite the damage the Hagia Sophia suffered during its early history, given the number of earthquakes the structure has endured ever since (one about every 100 years), it is considered an example of the great potential of domes to be earthquake-resistant [45]. The surviving brick masonry dome rests on a square base, and is spherical in shape with a span of 31 m, an angle of embrace of 140° and a thickness of about 0.95 m. Its fundamental frequency measured in-situ is 1.84 Hz [52]. Domes, such as the Hagia Sophia’s, are intrinsically more stable during earthquakes than arches or vaults as several load paths in (doubly-curved) domes exist to transfer the compressive thrusts to the supports [48]. As seen for the Hagia Sophia, it is crucial for these domes to be properly supported or buttressed so the thrusts can be carried off in compression to the substructure. Other examples of domes that have stood the test of time and have also been analyzed by researchers for their good seismic behavior include the unreinforced concrete spherical dome of the Pantheon (Rome , Italy, 126 – span 42.5 m), which has extremely thick walls that can accommodate the thrust of the dome; the brick double-layer masonry octagonal dome of Santa Maria Del Fiore (Florence, Italy, 1436, Brunelleschi – span 45.5 m), where additional tensile strength is provided through wooden and iron chains and an interlocking pattern of the masonry; and the double-layer brick masonry dome of St. Peter (Rome, Italy, 1590, Michelangelo and Giacomo della Porta – span 41.5 m) which is situated on top of a buttressing cross that can counteract the thrusts exerted by the dome on the supporting pillars [48]. semi-domes

buttresses

Figure 8 – Cross-section (east-west) of the Hagia Sophia in its current form. The dome is buttressed by two semi-domes (one on the east side and one on the west side) and two massive buttresses on both the north as the south sides (Image in the public domain). 15

While domes have performed relatively well during earthquakes, vaults, especially Gothic ones, have been identified as more vulnerable to earthquake damage [48]. Gothic vaults are typically relatively thin and have weaknesses due to the imperfect connections between ribs and webs, but the main danger in seismic events arises from their support conditions. Gothic vaults are often supported on slender and deformable columns making the vaults very vulnerable to support movements, which is only partly mitigated by the infill of the abutments above these columns. Therefore, if the supports of vaults move or fail altogether, the vaults crack or collapse. Such collapses have been widely recorded. For example, the masonry vaults of a set of churches in Coastal Peru, built in the 16th and 17th centuries, copying the construction details from largely aseismic Spain, collapsed during earthquakes in the 17th and 18th centuries because they lacked proper buttressing or ring beams to resist the additional horizontal thrust induced by seismic action [53]. A better documented example is the collapse of two brick masonry vaults of the Basilica of St. Francis of Assisi (Assisi, Italy, 1253) due to the 1997 Mw 6.1 Umbria and Marche earthquake while the other vaults of the structure survived (see Figure 9). While the structure had sustained 10 previous earthquakes, the 1997 earthquake caused the destruction of the cross-rib vaults close to the façade and the transept, and induced a series of cracks all over the other vaults [54]. The reason for the partial collapse of these vaults has been attributed to the excessive volume of non-cohesive backfill which exerted unacceptable pressures on the webs of the vaults, thereby reducing their curvature. Numerical simulations employing pushover analysis showed that under an equivalent horizontal acceleration of 0.2 the vaults were expected to collapse. The vaults near the façade collapsed due to a progressive loss of curvature of the ribs and the collapse of these ribs in turn led to the destruction of the vaults [54]. Another, arguably more plausible, explanation for this collapse is that interventions to the roof structure disproportionately stiffened the structure, redirecting excessive forces from the roof to the vaults [55].

Figure 9 – Collapsed vault “Volta del Cimabue” at the Basilica of St. Francis of Assisi after the 1997 earthquake (CC BY-SA 3.0). Another site where a significant number of vaulted structures collapsed while others survived was at the mud-brick city of Arg-e-Bam in Iran during the 2003 6.3 Mw Bam earthquake, which

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caused horizontal and near-fault vertical accelerations. A structural analysis of a set of barrel vaults revealed that under static loads, the walls supporting some of these vaults were already subjected to tilt and shear as no thrust line could be fit within the masonry [47], [56]. The horizontal loading induced by the earthquake exacerbated this state, forcing the thrust line further out of the masonry section, thus creating tensile cracking resulting in hinges that contributed to collapse [47]. Because many buttressing elements also failed due to lack of cohesion caused by termite damage [57], there was no way for the vaults to exert thrusts without collapsing. Other authors have reported that sets of barrel vaults performed well during the earthquake, as long as the thrust of the horizontal seismic movement could be contained within the masonry which in certain cases was facilitated by thick supporting pillars, and infill between the arches above these pillars (see Figure 10) [58], [59].

Figure 10 – Surviving barrel vaults after the Arg-e-Bam earthquake (Image T. Mahdi). While multiple vaults collapsed, surviving vaulted roofs (both singly-curved barrel vaults and doubly-curved pavilion vaults) were found all over the earthquake zone. Overall, six factors were reported to have contributed to the survival or collapse of these vaults. The first (1) and most prominent factor was the stability of the supporting structure: if the supporting walls or piers collapsed (due to out-of-plane action or a lack of tying elements), the vaults they supported logically collapsed as well [58], [59]. A second (2) factor was the thickness of the vaults: thicker vaults performed better than their thinner counterparts [58] as they had a larger section that could work in compression to encompass the generated thrust lines. A third (3) factor was the additional weight on top of the vaults: vaults were commonly covered with non-cohesive material as traditionally earthen water-proofing layers were cumulatively added on a yearly basis (up to an additional thickness of 50 cm) [59]. These layers added weight and thus increased the thrust of the vaults but did not contribute to the structural thickness. Soil conditions (4) also were reported to have an impact: vaulted structures constructed on rocky subsoil performed better [58]. Finally, also the quality of workmanship (5) and level of prior deterioration (6) played a role [57], [59]. It is also noteworthy that most of the recently constructed brick masonry domes

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and vaults in the city of Bam performed satisfactorily during the earthquake [58], [59]. Their successful behavior is attributed to the good connection of the shells to the supporting walls, typically employing steel-reinforced ring beams to contain the thrusts [58], [59]. Similar good behavior of well-supported masonry vaults of adobe and brick has been recorded during the 2017 Mw 8.2 Chiapas earthquake and the 2017 Mw 6.1 and 2018 Mw 7.2 Oaxaca earthquakes. A series of vaults constructed in the affected region around Oaxaca sustained all three earthquakes without any damage, while significant damage to other nearby buildings was recorded [60]. The shape of the vaults varied from doubly-curved pavilion vaults (in adobe and brick masonry) with a plan size of up to 7 m by 7 m and a rise of 2.1 m, to less conventional doubly-curved vault shapes that were supported on edges defined by a Gaussian curve (see Figure 11) [60], [61]. All vaults had a thickness of 12 to 15 cm [61]. As the outward thrust of the vaults was limited due to the vaults’ thinness, supports of small concrete ring beams or steelreinforced masonry proved sufficient.

Figure 11 – Pavilion vault (left) and Gaussian vault (right) that sustained three earthquakes in quick succession (Images Ramón Aguirre Morales). The aforementioned examples, and especially the failures in Assisi and Bam, illustrate that seismic action may induce tensile stresses in unreinforced masonry which can lead to cracking and collapse. Additionally, they showed that proper supports or buttressing that can resist the increased outward thrusts due to seismic action are of vital importance. It is also noteworthy that in the case of heavy masonry shells, such as in Bam, these outward thrusts will be significantly larger than those for a thin concrete or masonry shell. For example, an 8 cm thick concrete shell will exert an outward thrust onto its supports that is about 5 times lower than the thrust exerted by a 50 cm thick masonry structure (like in Bam) of similar shape (assuming a concrete density of 2400 kg/m3 and a masonry density of 2000 kg/m3). In contrast, the tensile strength of (unreinforced) concrete (around 3 MPa) can be an order of magnitude larger than that of masonry (around 0.35 MPa for tile vaulting). For masonry shells it is therefore of great

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importance to guarantee that they can behave in compression when exposed to seismic action, and that the supports are able to withstand the increased outward thrusts. If not, the tensile capacity of the shells will be exceeded almost instantly, which will lead to cracking and possibly collapse. An interesting solution to mitigate this problem of low tensile strength has been presented in the 2017 Rwanda Cricket Stadium. The stadium consists of three catenary vaults (the largest span is 16 m) and was constructed in a moderately seismic region. To increase the tensile capacity of the thin-tile vaulting, the masonry was reinforced with geomesh between the layers of tiles, with the goal of enhancing the structure’s seismic performance through increased tensile capacity of the masonry [62].

Figure 12 – The Rwanda Cricket Stadium (2017) constructed using the tile vaulting technique with geogrid reinforcement to provide additional ductility during seismic events (Image Light Earth Designs).

2.4 Form Finding As the overall form of shells plays a fundamental role in their structural behavior, shell researchers have dedicated significant efforts to develop methods that yield shell surface geometries that exhibit mostly membrane behavior under the design load. The process in which a set of parameters is controlled to generate geometries for that design loading is termed form finding [9]. The design loading for form finding is typically self-weight for masonry or concrete shells, while the geometry is the unknown in the form finding process. Shell designers have resorted to physical form finding methods to aid design, often hanging models that simulate gravity, such as the ones used by Gaudí and Isler and for Frei Otto’s Mannheim Multihalle (Germany, 1974) [63]. Alternatively, researchers have employed computational models to numerically simulate such hanging models, or to guide shell design through a set of numerically controlled parameters [9].

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Figure 13 – Physical form finding models under gravity loading. Hanging model by Heinz Isler (left) and hanging model for the Mannheim Multihalle (right) (images Heinz Isler Archive and Institute for Lightweight Structures and Conceptual Design (ILEK), University of Stuttgart). A range of such numerical shell form finding approaches have been developed. They can be grouped into two categories: geometric stiffness methods and dynamic equilibrium methods [64]. Geometric stiffness methods, such as the Force Density Method (FDM) [65] or Thrust Network Analysis (TNA) [66] are material-independent and typically prescribe either force densities (FDM) or forces (TNA) to generate geometry. Both FDM as TNA have been used extensively in practice. FDM has been employed primarily to design tensioned roofs, but has also been used for timber grid shell roofs such as the Mannheim Multihalle (which as mentioned earlier was also aided by a physical model) and the Solemar Therme (Geier + Geier and Linkwitz, Germany, 1987) [67]. TNA, in contrast, has predominantly been employed to find the shape of continuous shells designed to work mostly in compression in materials such as stone, (tile) masonry or concrete. Examples (see Figure 14) respectively include the Armadillo Vault (Block Research Group, Italy, 2016), the Droneport prototype (Foster+Partners, Block Research Group and ODB, Italy, 2016) and the NEST HiLo concrete shell roof prototype (Block Research Group, Zurich, 2017) [15], [68], [69]. The form finding process in TNA is broken up into two steps. First, the equilibrium of horizontal thrusts is solved. Then, the heights of the nodes are computed based on the boundary conditions, the external vertical loads and the obtained horizontal equilibrium [70]. This decoupling allows for the form finding by working with a reciprocal form diagram (the planar projection of the thrust network) and force diagram. By manipulating the force diagram, the designer shapes both the flow of forces as well as the shape of the shell. In the initial formulation of the method, the form finding process only works for a set of vertical design loads [66]. The method has been reformulated, however, to serve as an analysis tool in a seismic context [71], but this reformulation eliminates the decoupling between horizontal and vertical forces, and thus the method’s direct applicability to form finding is lost as detailed further in Chapter 4.

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Figure 14 – Armadillo stone vault (left) and compressed earthen blocks Droneport prototype shell (right) as displayed during the 2016 Venice architectural biennale. Both shells were formfound using thrust network analysis. Dynamic equilibrium methods, such as Dynamic Relaxation (DR) [72] and Particle Spring systems (PS) [73], sometimes referred to as hanging cloth models [74] (see Figure 15), solve a dynamic equilibrium problem to obtain a static equilibrium solution [64]. In these dynamic equilibrium methods, the geometry can be altered by adjusting mass as well as spring lengths and stiffnesses. Examples of application include the grid shells of the British Museum (Foster+Partners, Buro Happold and Williams, England, 1994) in which DR was used to distribute a triangular grid pattern on a mathematically predefined surface geometry [75] and the Dutch Maritime Museum (Ney & Partners, The Netherlands, 2011) [76] (see Figure 16). These dynamic equilibrium methods have the advantage that they can find equilibrium solutions under any type of static loading, including horizontal loading.

Figure 15 - Digital reproduction obtained with dynamic relaxation solver of Isler's hanging net displayed in Figure 13.

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Figure 16 - The grid shell covering the courtyard of the Dutch Maritime Museum was formfound using dynamic relaxation. All the discussed methods (FDM, TNA, DR and PS) have predominantly been used to find shell shapes that act under an ideal membrane stress state under gravity loading [9]. However, to account for the loading that arises from earthquakes, where horizontal loads (which could act in any directions) inevitably cause significant bending stresses [30], both gravity and horizontal seismic loads must be considered in the form finding process.

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2.5 Conclusion and research gap This chapter provided the research context for this dissertation by looking into the behavior of concrete and masonry shells during earthquakes, and by summarizing the state-of-the-art for the form finding of shells. The section on concrete shells (2.2) provides a case study of a structure composed of concrete hypars in Mexico City and explains that it survived the 1985 earthquake there undamaged because its shells behaved elastically as stress levels remained under the ultimate tensile and compressive strength limits during the seismic event. This good seismic performance is a direct result of the conceptual design of the shell. Because the structure is lightweight and stiff through its doubly-curved hypar shape, its eigenfrequencies (which are a function of mass and stiffness) are high (3.09 Hz and greater). Because the earthquake amplified mostly low frequencies (0.40.7 Hz), dynamic amplification did not occur and thus the induced base shear in the lightweight structure remained low. This trend that lightness and stiffness of shell structures are essential to their good behavior during earthquakes was also confirmed by a more general parametric study of singly- and doubly-curved cylindrical shells with square plan under two other earthquake spectra. Reinforced concrete shells span large distances with minimal materials and thus intrinsically tend to be lightweight. As the base shear induced by seismic actions is proportional to the mass of the structure, the induced forces are lower than those in heavier construction alternatives. Shells, because of their curved geometry, have an inherently large geometric stiffness. The parametric study of reinforced concrete shells showed that doubly-curved shells are stiffer than singly-curved shells (assuming material, mass and span to be constant) and thus have fundamental frequencies that are up to 3 times higher than their singly-curved counterparts (for example 12 Hz vs. 3 Hz respectively for the doubly- and singly-curved cylindrical reinforced concrete shells with a 20 m span and an 8 cm thickness displayed in Figure 7). Indeed, the combination of low mass and high stiffness through shape has a direct effect on the vibrational properties of shells because they ensure high eigenfrequencies. As earthquakes in general mostly excite lower frequencies (typically under 2 Hz), this means in practice that dynamic amplification due to resonance is unlikely to occur for reinforced concrete shell structures (especially for doubly-curved ones). Furthermore, as compressive stresses remain low (about 20% or lower than the ultimate compressive strength throughout the parametric study), the risk of buckling is low as well. If, however, the tensile strength of the material were to be exceeded in several locations of the shell, cracks and thus plastic hinges would be introduced. Such cracking could theoretically lead to significant damage and collapse by causing stiffness reduction of the shells and thus possibly a drop in eigenfrequencies, making the shells more vulnerable to resonance, but this has not yet been observed in practice. It is therefore important to design shell shapes so that they will behave predominantly in compression also under seismic loading.

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While to the best of the author’s knowledge there is no empirical evidence of collapse of reinforced concrete shells due to a seismic event, in the case of masonry shells a number of collapses have been reported. Masonry shells typically have negligible tensile capacity and are therefore more susceptible to the formation of tensile cracks, making it more likely for them to collapse due to the formation of a set of hinges caused through seismic action. The failure of masonry vaults during earthquakes has been linked in literature to a combination of factors. One recurring cause is improper form of the shell (the initial dome of the Hagia Sophia for example), which is often linked to a second cause of insufficient buttressing or support to account for the increased outward thrust of the shell due to earthquakes (this occurred in the vaults of Arg-eBam for example). A third recurring reason for damage or collapse of masonry vaults is the additional superimposed weight on top of these shells that generates excessive outward thrust during seismic events (this happened in the Basilica in Assisi for example). Nonetheless, many examples of masonry shells that survived repeated seismic action undamaged, such as the reconstructed Hagia Sophia dome or a set of thin vaults in Oaxaca (Mexico), have been presented. Overall, the literature on concrete and masonry shells, as well as the results produced in the case study and parametric study presented in this chapter, establish that it is vital that shells remain in compression during a seismic event and that tensile stresses are avoided or remain low to avert cracking. To accomplish this, the shell needs to be properly supported so that outward thrusts can be carried off to the foundations and the shell needs to have a proper form that guarantees a compression load path under seismic loading. The overall form of shells is thus one of the major factors governing seismic resilience. While several methods to obtain efficient forms for shell structures have been developed, none of them have been adapted to obtain shapes that take into account seismic loads in the conceptual design phase. Chapters 4 and 5 address this research gap and provide two different form finding methodologies to incorporate seismic loading in shell design from the start. Chapter 3 will provide a form finding methodology for arches subjected to seismic loading, as this will form the basis for the ensuing shell form finding presented in Chapters 4 and 5.

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Form finding of Arches under Seismic Loading Large sections of this chapter have been published as the following journal article, but edits have been made for continuity within this dissertation. T. Michiels and S. Adriaenssens, “Form finding algorithm for masonry arches subjected to inplane earthquake loading,” Comput. Struct., vol. 195, pp. 85–98, 15 2018.

3.1 Introduction Before pursuing the form finding of shells in Chapters 4 and 5, this chapter will present a form finding methodology for arches under seismic loading. Arches can be considered as the 2dimensional simplification of certain shells (such as barrel vaults and domes that are continuously supported on their edges). This simplification is typically conservative, as it eliminates a possible load path in the 3rd dimension (such as the hoop stresses for a dome, for example). If the cross section of the arch is stable without relying on the load carrying capacity in the 3rd dimension, the corresponding shell will a fortiori be stable as well. The shape of arches has been the subject of extensive research ever since Robert Hooke published his findings that “the true mathematical and mechanical form of arches” is the inverse of a hanging chain [77]. Most ensuing research (see section 3.1.1), however, has focused on the analysis of existing masonry arches, rather than on how to shape these types of arches appropriately for specific loading conditions. In particular, literature has not yet addressed the question of finding the appropriate form for masonry arches under earthquake loading. Therefore, this chapter presents the first form finding algorithm that allows for the design of arches subjected to self-weight and in-plane horizontal loading. New material-efficient arch shapes are obtained by considering both horizontal and gravitational acceleration in the form finding process. By interpreting the obtained forms, insights into the influence of form on the earthquake resistance of the arches are presented. First, the relevant literature on form finding and analysis techniques of arches is presented (Sections 3.1.1 and 3.1.2) , followed by a description of the form finding procedure (Section 3.2. The form finding procedure relies on thrust line analysis, which is a first-order equilibrium method that represents a lower-bound approach to the analytic problem of arch stability under gravity and horizontal loading. Subsequently, kinematic limit state analysis, another first-order equilibrium analysis, which presents an upper-bound approach to the same problem is used for

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validation. Through a methodological application of a series of geometric manipulations of the thrust line in the form finding process, shapes are obtained that can resist the design acceleration by guaranteeing a compression-only load path (Section 3.2.2). Forms are obtained for horizontal accelerations of 0.15, 0.3 and 0.45 g, as well as for arches of different rise-to-span ratios (1/2, 1/4 and 1/8), see Section 0. The obtained shapes require up to 65% less material than circular arches with constant thickness that are designed to withstand the same horizontal acceleration and selfweight, regardless of acceleration magnitude. The findings of this research will thus allow for the more material-efficient design of masonry arches in seismic areas.

3.1.1 Form finding of arches To the best of the author’s knowledge, the finding of appropriate shapes for masonry arches under horizontal loading has not been reported in literature to date, whereas the finding of appropriate shapes under gravity loading has been extensively researched. After Hooke’s seminal work, a panoply of authors expanded on his findings, focusing on analyses techniques that could help shape arches. De La Hire’s work on thrust lines [78] and Coulomb’s studies on hinge formation and sliding in arches [79] were both milestones that helped pave the way for the extensive treatises on masonry bridge design in the 18th and 19th centuries. For an extensive overview of the evolution of arch design the reader is referred to [80] and [81]. All of this research, however, was focused on the design and construction of arches subjected to gravity loading. Even after renewed interest in masonry arches was spurred by Heyman’s work in the 1960s [46], [82], the research focus remained on how to analyze and optimize these arches under vertical loading. Indeed, several authors tackled the problem of the optimal arch under vertical loads through analytical and numerical approaches several authors on the analysis of arches under earthquake loading, to inform form finding procedures for arches that can better resist such loads. An overview of this body of work is given in Section 3.1.2.[83]–[85]. Peng carried out an interesting study employing limit state analysis in combination with a genetic algorithm to find the form of arches [86] and shape optimization tools were used to design concrete [87] and steel arches in the context of bridge design. Another approach was developed to obtain the forms of spatial leaning arches, relying on the use of the thrust lines to create funicular forms [88]. Nevertheless, none of these recent studies address the form finding of arches when considering gravity and horizontal loading. Uzman et al. presented a method to optimize the design of parabolic and circular arches with varying cross sections under an array of loads, but their method does not allow to account for several load cases [89]. Therefore, it does not cope with the horizontal forces that can occur in both in-plane directions during a seismic event. This chapter thus relies on the extensive work of several authors on the analysis of arches under earthquake loading, to inform form finding procedures for arches that can better resist such loads. An overview of this body of work is given in Section 3.1.2.

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3.1.2 Review of analysis techniques for masonry arches under earthquake loading The analysis of masonry arches under earthquake loading, in contrast to their form finding, has been approached through a variety of methods. One method, thrust line analysis, is an equilibrium method that hinges on Hooke’s hanging chain analogy. It relies on three assumptions that are generally accepted for masonry: 1) masonry has no tensile strength; 2) sliding between blocks does not occur; and 3) the compressive stresses remain low compared to the material strength of the masonry so that crushing does not occur. If these assumptions hold, Heyman’s safe theorem states that an arch will be stable if a thrust line can be found that fits within the geometry of the arch under the considered loading [82]. While only static loads are taken into account in this analysis method, DeJong et al. and Huerta have simulated equivalent earthquake loading by imposing a horizontal load on arches by tilting them [90], [91]. This approach induces a horizontal acceleration, but also reduces the compressive stresses under gravity. However, as crushing is assumed not to occur, these reduced compressive stresses can be ignored [79], and thus an arch will be stable if a thrust line can be found that fits within the masonry under the combination of the considered horizontal acceleration and gravity. Thrust line concepts have been further expanded to 3D-networks in the thrust network analysis, which allows for the structural design and analysis of arches and shells under vertical loading [66], [70]. Relying on the same principles, other authors have developed alternate continuum approaches to perform lower bound equilibrium analysis looking at thrust surfaces in vaults and shells [92]. One of the strengths of the thrust network analysis method, however, is that it uses a dual approach, visualizing the forces in the eventual form using graphic statics by employing force network polygons. This method was reformulated and extended neglecting this duality with graphic statics to be able to cope with horizontal forces, for example due to earthquakes [71] and a sample application to arches was provided. Overall, the thrust line analysis approach is a lower bound (static) solution to the problem of stability: every thrust line that can be found within the masonry arch, represents one possible equilibrium solution between the internal and external loads [93]. Therefore, it is necessary to determine the maximum acceleration under which a thrust line can fit within the arch’s geometry, which can be done through linear programming [94]. This thrust line also provides a qualitative idea about the expected collapse mechanism of the arch, as the locations where the thrust line is closest to the boundary are the locations where hinges for this mechanism are expected to form. Another equilibrium method that replaces the earthquake with an equivalent horizontal acceleration is based on the kinematic limit analysis of rigid blocks [95] and builds upon the same three assumptions as thrust line analysis. It furthermore relies on the observation that unreinforced masonry arches fail through a four-hinge mechanism under base motion [96]. It determines the critical collapse mechanism and associated acceleration needed to activate this mechanism through a series of virtual work calculations [97]. Kinematic limit analysis has been

27

automated and applied to a set of different arch geometries with constant thickness and has been validated experimentally and numerically by several authors [98]–[100]. In this method, the acceleration required to activate every possible collapse mechanism is determined. The lowest acceleration that triggers a collapse mechanism corresponds to the critical acceleration and determines the capacity of the arch. This kinematic limit state approach is therefore an upper bound solution to the same problem solved by thrust line analysis [93]. Upper bound limit state analysis has also been expanded to analyze three-dimensional vaults [101] and has been applied in the context of seismic loading [102]. Both thrust line analysis and limit state analysis replace the dynamic earthquake loading by an equivalent horizontal load and are essentially stability analyses. They can predict the minimum acceleration required to onset the formation of the critical mechanism of the arch [103], but cannot account for what happens once the mechanism is activated under base motion. Once the critical mechanism is activated, the blocks that have formed will start rocking back and forth. The question of whether the arch collapses or not, becomes a problem of rigid body dynamics. The dynamic response of arches through these continuous cycles of rocking can then be determined analytically or numerically by solving the equation of motion for a single-degree-offreedom (SDOF) system [103], [104]. However, if the acceleration to onset the mechanism is not exceeded, the rocking will not occur. Therefore, both thrust line analysis and limit state analysis yield conservative solutions. Other approaches used to assess the behavior of masonry arches during earthquakes include nonlinear finite element modelling (FEM), and distinct element modeling (DEM). Despite the widespread use of FEM for structural analysis, the application of FEM to masonry remains a convoluted task. Because of its discontinuous nature, masonry cannot be modeled as an elastic continuum [90] and therefore, computationally expensive non-linear analyses are necessary. These non-linear finite element models have nevertheless been applied successfully to masonry arch bridges [105] and seismic analyses were performed through non-linear static pushover simulations [106], [107] and non-linear dynamic analysis, which provided similar results [108]. Non-linear FEM requires high expertise from its operators but has the advantage of being able to capture three-dimensional effects. DEM can similarly capture these three-dimensional effects and inherently incorporates the heterogeneous nature of masonry. This method relies on finitedifferences principles to characterize the interaction of discontinuous blocks. Blocks are allowed to rotate and displace, deform and form new contacts and movements are traced at each time-step allowing the user to understand the collapse of the arch over time [90], [109]. This method has been applied to arches under seismic action [99], [103], [104] and its results were found to correspond well with data obtained through analytical limit state analysis [99], [104] and experimental tests [90]. The convoluted nature of non-linear FEM and DEM, and their high computational demands are in stark contrast to the straightforward analytical solutions that can be obtained employing thrust

28

line and limit state analysis under an equivalent horizontal load. This observation makes these latter methods (thrust line and limit state analysis) more suitable for the design and form finding of masonry arches under earthquake loading, especially as these methods have been shown to be accurate and conservative [90], [94].

3.2 Form finding Methodology 3.2.1 Thrust lines due to combination of gravity and horizontal acceleration The form finding method presented in this chapter relies on thrust line analysis performed under combined vertical and horizontal loading. The method does not, however, rely on simulating the horizontal acceleration by tilting the arch, as suggested in earlier work [90], [91] (see Section 1.2). Instead, an algorithm is used to compute compression-only equilibrium conditions under the combination of gravity loading and a horizontal acceleration. If the resultants of the stresses under this loading on each voussoir of the arch fall within the intrados and extrados of the arch, it is guaranteed that at least one equilibrium condition exists in which the arch can carry the resulting loads through a compression-only load path according to Heyman’s safe theorem [82]. The formulation of the equilibrium problem to find the thrust lines that fit within the masonry arch under horizontal loading is based on the mathematical formulation first provided by O’Dwyer for his force networks for vertical external loads [110]. In the method, presented in this chapter, the effect of the horizontal acceleration is added and the problem is solved iteratively (as similarly detailed in [71] for 3D networks). The arch is discretized into n voussoirs and the thrust line therefore similarly connects a series of n nodes, one for each voussoir with horizontal coordinate 𝑥𝑖 and vertical coordinate 𝑧𝑖 . To ensure equilibrium, each node is located on the line of action of the external forces that act at the centroid of each voussoir Ci with (i : 1..n ) (see Figure 17). The line of action is thus determined by the magnitude of the weight of each block 𝐷𝑖 and the force due to the equivalent horizontal acceleration 𝑎, which is proportional to the weight (𝑎 ∙ 𝐷𝑖 ). The internal forces that act on the nodes are denoted as 𝐹𝑖 (see Figure 17). The locations of the nodes are constrained as each node needs to fit between the points of the intrados 𝐼𝑖 (𝑋𝑖,𝐼 , 𝑍𝑖,𝐼 ) and the extrados 𝐸𝑖 (𝑋𝑖,𝐸 , 𝑍𝑖,𝐸 ), which can be expressed based on the height (𝑧𝑖 ) of the node: 𝑍𝑖,𝐼 ≤ 𝑧𝑖 ≤ 𝑍𝑖,𝐸 ∀𝑧𝑖

(3.1)

For a stable arch, equilibrium is guaranteed at each node. This equilibrium can be expressed as a function of the vertical and horizontal coordinates of the node (𝑧𝑖 𝑎𝑛𝑑 𝑥𝑖 ), the applied external loading 𝐷𝑖 , the acceleration a, and the horizontal components of the forces interconnecting these nodes (𝐹𝑖,𝐻 ) (see Figure 17).

29

Figure 17 – Force equilibrium in nodes of the thrust line. The weights of the voussoirs (𝐷𝑖 ) and the acceleration (a) are known. From the equation of vertical equilibrium, it follows that (with 𝐹𝑖,𝑉 being the vertical component of 𝐹𝑖 ): 𝐷𝑖 = 𝐹𝑖,𝑉 + 𝐹𝑖+1,𝑉

(3.2)

𝑧𝑖 − 𝑧𝑖−1 𝑧𝑖+1 − 𝑧𝑖 + 𝐹𝑖+1,𝐻 ∗ 𝑥𝑖 − 𝑥𝑖−1 𝑥𝑖+1 − 𝑥𝑖

(3.3)

Which can be rewritten as: 𝐷𝑖 = 𝐹𝑖,𝐻 ∗

The equation of horizontal equilibrium yields: 𝐹𝑖+1,𝐻 = 𝐹𝑖,𝐻 + 𝑎 ∗ 𝐷𝑖

(3.4)

Substituting (3) in (4) gives: 𝐷𝑖 = 𝐹𝑖,𝐻 ∗

𝑧𝑖 − 𝑧𝑖−1 𝑧𝑖+1 − 𝑧𝑖 + (𝐹𝑖,𝐻 + 𝑎 ∗ 𝐷𝑖 ) ∗ 𝑥𝑖 − 𝑥𝑖−1 𝑥𝑖+1 − 𝑥𝑖

(3.5)

Which can be rewritten in function of 𝑧𝑖 . 𝐷𝑖 = (

𝐹𝑖,𝐻 𝐹𝑖,𝐻 + 𝑎 ∗ 𝐷𝑖 𝐹𝑖,𝐻 𝐹𝑖,𝐻 + 𝑎 ∗ 𝐷𝑖 − )𝑧𝑖 − 𝑧𝑖−1 + 𝑧 𝑥𝑖 − 𝑥𝑖−1 𝑥𝑖+1 − 𝑥𝑖 𝑥𝑖 − 𝑥𝑖−1 𝑥𝑖+1 − 𝑥𝑖 𝑖+1

(3.6)

Assuming a value for the horizontal and vertical component of the reaction 𝐹𝑖,𝐻 and 𝐹𝑖,𝑉 at one location (for example 𝐹1,𝐻 and 𝐹1,𝑉 located at the left support), allows for the computation of all forces at each voussoir using the equations 3.3 and 3.4. Self-weights of the voussoirs 𝐷𝑖 , as well 30

as the horizontal acceleration are known and therefore, it is possible to obtain a set of solutions, resulting in the 𝑥𝑖 and 𝑧𝑖 coordinates of each node of the thrust line using (eq. 3.6). Each solution set that also satisfies the constraint that each node needs to fit between the intrados and extrados (expressed in eq. 3.1), fulfills the safe theorem [46] and will thus be able to sustain the horizontal design acceleration a. To obtain such an equilibrium solution, the values of 𝐹𝑖,𝐻 and 𝐹𝑖,𝑉 need to be adjusted iteratively. A good starting value for 𝐹1,𝑉 is 50% of the total self-weight of the arch, while for the horizontal component 𝐹1,𝐻 a first approximation of 15% of the self-weight is advised (for accelerations acting from left to right, as in Figure 17). Expression 3.6 can be visualized and implemented through graphic statics, permitting a visualized real-time evaluation of the thrust line within the masonry arch, which furthermore provides a qualitative indication of the expected collapse mechanism if the horizontal design acceleration were surpassed [94], [111]. The graphic construction starts by formulating a load line, based on the vertical loads (𝐷𝑖 ) and horizontal loads (𝑎 ∙ 𝐷𝑖 ) (see Figure 18). Subsequently, the location of a pole in the force diagram is picked, which is equivalent to fixing the two unknowns 𝐹1,𝐻 and 𝐹1,𝑉 which enables the determination of all values 𝐹𝑖 and thus also 𝐹𝑖,𝐻 , 𝐹𝑖,𝑉 (see eq. 3.3 and eq. 3.4), 𝑥𝑖 and 𝑧𝑖 (see eq. 3.6). The force diagram represents the relation between 𝐷𝑖 , 𝑎, and 𝐹𝑖 , as can be seen in Figure 18. The corresponding thrust line can be drawn based on the force diagram, which should fit within the geometry to obtain a valid equilibrium condition. If this is not the case, another equilibrium condition should be sought by picking a different pole iteratively.

Figure 18 – Circular arch with thrust line due to gravity (blue) and thrust line due to combined gravity and horizontal loading of 0.3 g (green) (shown left) and corresponding force diagrams for gravity (shown center) and gravity and horizontal loading (shown right).

31

3.2.2 Form finding procedure While the method described in section 3.2.1, can be used to assess the ability of an arch to withstand an earthquake coming from one direction, a procedure needs to be established to find the shape of arches that can sustain in-plane earthquake action in either direction. This procedure relies on performing thrust line analyses in combination with a set of iterative geometric manipulations that build on the graphic statics approach detailed in section 3.2.1. The procedure, which produces symmetric shapes, is scripted within the parametric modeling plugin Grasshopper for the CAD software Rhino3D [112]. The problem of finding an arch that can sustain a design peak ground acceleration (a) and has a predefined rise-to-span ratio (r/s) is considered. Because of the assumption of no-crushing, the problem is scalable and the actual span is of no relevance [46]. The form finding process starts by picking an arbitrary arch (for example, a catenary arch) that has the same span and rise as the targeted rise and span. After dividing this initial arch into a set of voussoirs with weight 𝐷𝑖 , a thrust line is calculated under the combination of gravity and horizontal acceleration, as detailed in section 3.2.1. As the initially defined arch may not be able to accommodate a fitting thrust line, the constraint expressed in (eq. 3.1) can be released and instead, any thrust line that fits closely to the desired shape is a good starting point, if its start and end fit within the masonry (eq. 3.7) (see green curve in Figure 19a). 𝑍1,𝐼 ≤ 𝑧1 ≤ 𝑍1,𝐸 𝑎𝑛𝑑 𝑍𝑛,𝐼 ≤ 𝑧𝑛 ≤ 𝑍𝑛,𝐸

(3.7)

This thrust line will be adjusted iteratively by changing the location of the pole (thus changing 𝐹1,𝐻 and 𝐹1,𝑉 ). Convergence can be improved by restricting the height of the initial thrust line so that its highest point does not exceed the highest point of the extrados (see Figure 19a, green curve). (3.8) 𝑀𝑎𝑥 (𝑧𝑖 ) ≤ 𝑀𝑎𝑥 (𝑍𝑖,𝐸 ) Subsequently, the thrust line is mirrored along the axis of symmetry of the original arch shape (as the earthquake can hit in either direction) to obtain a second thrust line to account for the other potential direction of the earthquake (see Figure 19a, red curve). The mirrored thrust line is then moved horizontally and the distance between the start and end points of both thrust lines will define the support thickness (see Figure 19b).

32

Figure 19 – A: Initial thrust line for a catenary arch under combined horizontal and gravity loading in green and mirrored thrust line in red. B: the initial and mirrored thrust line are moved horizontally so that they define the support thickness. Each thrust line is then offset by a distance (for example 20% of the initial catenary thickness) towards the top and bottom of the initial and mirrored thrust line, leading to four curves of which the outline always encompasses the initial and mirrored thrust lines (see Figure 20a). Then, the envelope of these four offset curves is taken (see Figure 20b). This envelope defines the new shape of the arch, which is again divided into n set of voussoirs, with the new extrados 𝐸𝑖′ (𝑋𝑖,𝐸 , 𝑍𝑖,𝐸′ ) , the new intrados 𝐼𝑖′ (𝑋𝑖,𝐼′ , 𝑍𝑖,𝐼′ ) and weights 𝐷𝑖′ . As the choice of support size affects the span, the initially obtained thrust line is adjusted by moving the original pole in the graphic procedure so that the new arch satisfies the desired r/s ratio: 𝑋𝑛,𝐼′ − 𝑋1,𝐼′ = 𝑠

(3.9)

𝑍𝑚𝑎𝑥,𝐼′ − 𝑍1,𝐼′ = 𝑟

(3.10)

Figure 20 – A: offset of the initial thrust lines. B: New intrados and extrados after taking the envelope of the curves of the offset curves. The initial thrust lines are contained within the geometry.

33

Figure 21 – Flow chart detailing the different steps in the form finding process.

34

Once a satisfactory shape is obtained iteratively for the chosen support size, the capacity of the new arch shape needs to be verified under horizontal acceleration and self-weight. If needed, the offset thickness of the arch can be increased or decreased to obtain exactly the desired acceleration capacity. Determining this capacity is performed through the thrust line analysis approach outlined in Section 3.2.1, which is now carried out with the new weights 𝐷𝑖′ based on the new intrados and extrados. The entire form finding approach is visualized in the flow chart shown in Figure 21. A variety of arches that satisfy the stability criterion for the design acceleration can be obtained using the provided methodology (see Section 0). From a design perspective, the presented approach allows the designer to manipulate the arch’s shape in real time by moving the pole to obtain efficient arches that can sustain the design earthquake, while influencing rise, span and support thickness.

3.2.3 Application example The form finding methodology, presented in Section 3.2.2, is applied to an arch with rise-to-span ratio (r/s) equal to 1/2. The design acceleration is chosen as 0.3 g, which corresponds to a moderate-to-high earthquake depending on a range of factors (for reference, the maximum ground acceleration ag with probability of exceedance equal to 10% in 50 years in the most seismic areas of Italy is defined as larger than 0.25 g [113]). While the entire form finding process can be carried out dimensionless, the span for this specific example is fixed at 8 m. The start point of the form finding process is thus a catenary arch with a span of 8 m and a rise of 4 m, so its r/s is identical to the one of the desired final arch. The catenary arch is assigned a thickness of 0.15 m (see Figure 22a). The resulting arch is unable to withstand a 0.3 g horizontal acceleration as no thrust line can be found that fits under the combined loading due to gravity and horizontal acceleration. A thrust line is thus chosen that fits within the geometry at the extrados and intrados at the supports, and does not exceed the highest point of the extrados of the catenary arch (eq. 5 and 6), see Figure 22a. Next the thrust line is mirrored along the line of symmetry and moved horizontally to define the thickness of the support (see Figure 22b). For masonry arches, it is more desirable to have thick supports rather than a thick crown (see section 3.1). The support size can be explored in real time by the designer by adjusting the pole location in the graphic statics framework, and is chosen here as 1.15 m, which corresponds to a supportthickness-to-span ratio of 0.14 (see Figure 22b) and is the most efficient solution in terms of material use (as will be shown in Section 3.1). The initial and mirrored thrust lines are then offset 3 cm (20% of the initial arch’s thickness) towards the top and the bottom (see Figure 22c) and the envelope of the resulting four offset curves is taken (Figure 22d). The pole location is then adjusted to obtain the exact desired r/s of 1/2. Finally, a new thrust line analysis is carried out to show that the obtained shape withstands the design acceleration of 0.3 g (blue curve Figure 22d).

35

Figure 22 – Step-by-step example of the form finding process for an arch with rise to span ratio of 1/2 under a horizontal acceleration of 0.3 g. A: initial catenary and form finding thrust line under gravity and horizontal loading of 0.3 g (shown in green). B: two thrust lines, one is mirrored with respect to the axis of symmetry and moved to obtain a support size of 1.15m. C: mirrored thrust line (green) and offset lines (red). D: envelope curves(red) forming the new intrados and extrados of new shape and new thrust line under gravity and 0.3 g (blue). The form finding algorithm converges after 7 iteration steps on the pole location (affecting the support reactions) for this arch. Table 2.1 provides an overview of the initial assumptions and starting values, and reports for each iteration the shape-determining values discussed in the flow chart of Figure 21 (in bold). The start point of the iteration is governed by the initial catenary arch and the initially imposed equilibrium condition for the thrust line. By assigning the left horizontal reaction F1,H as 15% of the weight of the arch, and the left vertical reaction as F1,V as half the weight of the arch, the location of the pole of the force diagram is determined. The horizontal and vertical reaction at the right support (Fn,H and Fn,V) are reported for completeness. The reaction forces determine the first location of the pole. As the support size and the initial value of the offset distance d have been chosen as well, the span and rise follow from the construction outlined in Section 3.2.2. The capacity (acap) of the resulting arch (see Section 3.2.1) is only half the desired capacity (0.15 g) based on these starting values. During the first iteration, the location of the pole is shifted so that the thrust line fits within the thickness of the initial catenary and the thrust line’s height does not exceed the desired rise (see initial check in flow chart shown in Figure 21). The result is an arch with a capacity of 0.28 g (6.67% under the desired value) which approximates the 0.3 g design acceleration. However, the span after 1 iteration is 5% under the target (see table 2.1). Therefore, during iterations 2 and 3, the span and rise are corrected, leading to an arch with a satisfactory rise and span, but with a capacity of only 0.25 g. Thus, the offset distance d needs to be adjusted iteratively (see the last check in the flow chart) to adapt the capacity. After iteration 4, this leads to an arch with a capacity of 0.32 g and

36

almost the desired span and rise. Subsequently, in iterations 5, 6 and 7 the span, rise and distance d are further refined, so that the span, rise and capacity differ less than 1% from the target values. Figure 23 displays a plot of the convergence towards the target values of span, rise and capacity. Target arch

Iteration

Start

1

2

3

4

5

6

7

acceleration (acap)

0.3 g

F1,V / FV,total

0.151

0.091

0.092

0.114

0.114

0.114

0.114

0.114

span (s)

8m

Fn,V / FV,total

0.451

0.391

0.392

0.414

0.414

0.414

0.414

0.414

Rise (r)

4m

F1,H / FV,total

0.499

0.434

0.458

0.458

0.458

0.460

0.460

0.459

support size

1.1 m

Fn,H / FV,total

0.501

0.566

0.542

0.542

0.542

0.540

0.540

0.541

Starting Catenary catenary span

8.4 m

span (m) rise (m)

8.89 4.10

7.62 4.01

8.01 4.35

8.00 4.00

7.97 3.97

8.01 4.00

8.01 4.01

8.00 4.00

catenary rise

4m

catenary thickness

0.2 m

d (mm)

15

15

15

15

30

30

27

27

acap (x*g)

0.15

0.28

0.25

0.25

0.32

0.32

0.30

0.30

Difference from target values 0.0% 0.1% 0.9%

Table 2.1 – Iteration steps to obtain form-found arch with support size 1.1 m for an arch of 8 m rise and 4 m span.

Figure 23 – Convergence of span, rise and capacity (acap) towards the target values.

3.2.4 Validation through kinematic limit state analysis Additionally, an extra validation of the horizontal capacity of each form-found arch is carried out by performing a limit state analysis. This method has the additional benefits that it provides an upper bound approach in contrast to the lower bound already used for the form finding process itself. Furthermore, it identifies the expected collapse mechanism. The approach has been presented in literature to compute the seismic capacity of masonry arches of constant thickness [99], [100], however, in this chapter its methodology is generalized to free form arches with varying thickness. First, a failure mechanism is predefined by imposing four hinges at the intersection of voussoirs (A, B, C and D from left to right) (see Figure 24a). Voussoirs between hinges are combined to form a segment, so that the arch is divided into three segments (AB, BC and CD). Next, the acceleration that is required to onset this pre-imposed

37

mechanism is computed based on a virtual work calculation. This calculation is repeated for all possible locations of hinges A, B, C and D, after which the critical failure mechanism is identified as the one with the lowest onset acceleration. The critical failure mechanism for the arch obtained in Section 3.2.3 is displayed in Figure 24 as an example. The virtual work calculation is performed as follows: each of the four hinges A, B, C and D can form at the intrados or extrados, dividing the arch into three segments, AB, BC, and CD. As the arch is in equilibrium before the onset of collapse, the sum of the work of all the forces in the system must equal zero in the instant right before a mechanism is formed, based on the principal of virtual work. Thus, the sum of the work done by the gravity acceleration and the work done by the horizontal acceleration over all three segments is equal zero: ∑ 𝑗{𝐴𝐵,𝐵𝐶,𝐶𝐷}

𝑀𝑗 𝑔𝑣𝑗 + ∑

𝑗{𝐴𝐵,𝐵𝐶,𝐶𝐷}

𝑀𝑗 𝑎𝑢𝑗 = 0

(3.11)

where Mj, 𝑗 {𝐴𝐵, 𝐵𝐶, 𝐶𝐷} is the mass of each of the three segments, and 𝑣𝑗 and 𝑢𝑗 are respectively the vertical and horizontal component of the virtual displacement at the centroid of each segment. g is the gravity acceleration and a is the horizontal acceleration. As detailed in [99], [100] and [94], the collapse of the arch is a one degree-of-freedom mechanism, and thus the virtual displacements 𝑣𝑗 and 𝑢𝑗 can be expressed in function of the rotation of segment BC around the instantaneous center (I, see Figure 24). After applying a small rotation 𝛾 to the central block BC, the relative rotations of segments AB and CD (𝛿 and 𝜃 respectively) can be computed as follows: 𝛿=γ

𝑥𝐼 −𝑥𝐵 𝑥𝐵 −𝑥𝐴

(3.12)

𝜃=γ

𝑥𝐶 −𝑥𝐼 𝑥𝐷 −𝑥𝐶

(3.13)

Then the vertical (𝑣AB, 𝑣BC and 𝑣CD) and horizontal components (𝑢AB, 𝑢BC and 𝑢CD) of virtual displacements at the locations of centroids of segments AB, BC and CD can be calculated as well: 𝑣𝐴𝐵 = 𝛿 |𝑥𝐴 −𝑥𝐴𝐵 | and 𝑢𝐴𝐵 = 𝛿 |𝑦𝐴𝐵 −𝑦𝐴 |

(3.14)

𝑣𝐵𝐶 = 𝛾 |𝑥𝐼 −𝑥𝐵𝐶 | and 𝑢𝐵𝐶 = 𝛾 |𝑦𝐼 −𝑦𝐵𝐶 |

(3.15)

𝑣𝐶𝐷 = 𝜃 |𝑥𝐶𝐷 −𝑥𝐷 | and 𝑢𝐶𝐷 = 𝜃 |𝑦𝐶𝐷 −𝑦𝐷 |

(3.16)

where 𝑥AB, 𝑥BC and 𝑥CD are the x-coordinates and 𝑦AB, 𝑦BC and 𝑦CD are the y-coordinates of the centroids of the segments AB, BC and CD, respectively.

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Solving for 𝑎 yields expression 3.17, in which the rotation 𝛾 cancels out after inserting eqs. (3.12 - 3.16). 𝑎=−

∑𝑗{𝐴𝐵,𝐵𝐶,𝐶𝐷} 𝑀𝑗 𝑢𝑗 = − 𝑀𝐴𝐵 𝑢𝐴𝐵 ∑𝑗{𝐴𝐵,𝐵𝐶,𝐶𝐷} 𝑀𝑗 𝑔𝑣𝑗

(3.17)

Application of equation 3.17 for every possible location of A, B, C and D allows for the identification of the minimum horizontal acceleration that causes collapse, as well as the associated collapse mechanism of the arch. For the form-found arch obtained in Section 3.2.3 with a span of 8 m and a rise of 4 m for an acceleration of 0.3 g, the capacity is confirmed as 0.3 g for the critical failure mechanism (shown in Figure 24). The limit state analysis was carried out for all arches obtained through the form finding process and confirmed that all these form-found arches can sustain the same horizontal acceleration as predicted through thrust line analysis.

Figure 24 – A: Location of hinges (A, B, C, D) and forces acting on each segment (AB, BC and CD) for limit state analysis. B: Collapse mechanism activated by acceleration of 0.3g.

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3.3 Results 3.3.1 Support size variations for an arch with rise-to-span ratio of 1/2 The final shape of the arch (see Figure 25a) obtained in the example in Section 3.2.3 is just one of the many design solutions that can be obtained as a function of the designer’s decisions along the form finding process. Selecting the support size is the most important parameter that influences the form-found arch shape. As the obtained form-found shapes are scalable and independent of span, a variety of shapes can be created for different support-size-to-span ratios. In Figure 26 the support-thickness over-span ratio versus material use is given for different formfound arches with a r/s of 1/2 that can withstand a 0.3 g acceleration. The X-axis represents the support size relative to the span. Because the analysis is carried out in 2D, the material efficiency can be expressed in terms of relative area. The Y-axis therefore represents the ratio of the area of the form-found shape and the (minimum) area of the circular arch (see Figure 25b) with constant thickness and identical r/s ratio that can withstand the same 0.3 g acceleration and self-weight. The minimum thickness required for this circular arch to withstand an acceleration of 0.3 g is determined by performing a kinematic limit state analysis as described in section 3.2.4 for increasing thickness. (For reference: the minimum thickness for this circular arch with r/s equal to 1/2 corresponds to a thickness-over-radius (t/R) ratio of 0.206, which is almost double the minimum thickness required for an arch that only needs to withstand gravity loads (t/R=1.075 [100]) . The green dotted line in Figure 26 represents the ratio between the minimum required area of a catenary arch with constant thickness able to sustain the same acceleration (see Figure 25c), divided by the area of the equivalent circular arch. Thus, all the form-found solutions on the black curve below the red dotted line (area ratios lower than 1) in Figure 26 perform better than the circular arch with constant thickness. All solutions on the black curve below the green line additionally perform better than the catenary arch with constant thickness in terms of required material.

Figure 25 – All three shapes can withstand the same horizontal acceleration of 0.3 g and have identical rise-to-span ratio (1/2). A: form-found arch. B: circular arch. C: catenary arch.

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Figure 26 – Relative area of form-found shapes (some of which are displayed in A to E) compared to the area of the circular arch with constant thickness that can withstand the same 0.3 g horizontal acceleration for a r/s ratio of 1/2.

3.3.2 Different design acceleration The form finding method presented in Section 3.2.2, is also employed to obtain a series of shapes with r/s equal to ½ designed to withstand a range of design accelerations corresponding to a range of earthquake magnitudes. Horizontal accelerations range from small (0.15 g – orange curve) to moderate (0.3 g – black curve) and extreme (0.45 g – blue curve) [43]. Plots of a series of solutions for different support sizes relative to span for these horizontal acceleration magnitudes are presented in Figure 27. The area of each obtained shape is divided by the area of the circular arch that can withstand the same design acceleration. The shape of the most material efficient arch for each acceleration is also provided, as well as the ratio of minimum area for the catenary arch divided by the one for the circular arch for comparison as in Figure 26 (horizontal dotted lines).

Figure 27 - Relative area of form-found shapes compared to the area of the circular arch for accelerations of 0.15 g, 0.3 g and 0.45 g and a constant r/s ratio of 1/2.

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3.3.3 Different arch rise-to-span ratios The analyses in Sections 3.3.1 and 3.3.2. are carried out for an arch rise to span ratio of 1/2. In Figure 28 form-found arch shapes under 0.3 g acceleration for r/s = 1/2 and shallow (r/s=1/4) and very shallow arches (r/s=1/8) [106] are plotted as a function of support size to span ratio versus material efficiency. Material efficiency is expressed as the area of the form-found arch divided by the area of the corresponding circular arch with constant thickness with the same r/s that can withstand the same acceleration. The horizontal dotted lines represent the ratio of the areas of the corresponding catenary and circular arches, thus if an arch shape has a lower area ratio, it uses less material than both the catenary and circular arches to sustain the same horizontal acceleration.

Figure 28 - Relative area of form-found shapes of r/s ratios 1/8, 1/4 and 1/2 compared to the area of the circular arch with same r/s ratio that can withstand a horizontal acceleration of 0.3g.

3.4 Discussion 3.4.1 Support size variations for r/s equal to 1/2 Figure 25 illustrates how the form finding approach presented in this chapter has the capability to greatly reduce the amount of material needed to design and construct material efficient masonry arch shapes. In comparison with more traditional circular and catenary arch shapes of constant thickness and with the same rise-to-span ratio and subjected to the same horizontal acceleration of 0.3 g, the form-found arch requires 68% and 41% less material respectively. The location of most material reduction is around the nexus of the arch, which is much thinner and leads to a lower overall height than for the circular or the catenary arch. Both the circular and the catenary arches need to be so much thicker because they need to accommodate the asymmetrical thrust line due to the horizontal load (see green thrust line in Figure 29). This leads to an inefficient

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distribution of material if the constant thickness is maintained. The form-found arch, in contrast, only employs material in the area that is required to fit such asymmetric thrust line. The material reduction not only has the benefit of decreasing the self-weight of the arch, but also of lowering the horizontal thrusts exerted on the supports. While the shape displayed in Figure 25 corresponds to the most materially efficient solution, it is shown in Figure 26 that many more form-found shapes perform better in terms of material efficiency than circular or catenary arches. Looking at very thin and very thick supports provides further understanding of the arch shapes. In the case of very thin supports (the minimum thickness can be formulated in terms of a maximum stress criterion [94]), the location of the thrust line is fixed at the supports (see Figure 26a), resulting in undesirably thick arch sections, requiring the same amount of material as a circular arch. From a structural perspective, the “realistic” thickest supports occur when the highest point of the form finding thrust line is located in the middle of the crown voussoir (corresponding to a support-to-span ratio of 0.2). This solution results in the arch shape tapering gradually from thick supports to a thin central section. Beyond this point, the choice of an even larger support size would only unnecessarily further increase the overall thickness. This formfound arch shape with thick supports (support-to-span ratio of 0.2) requires 47% of the amount of material needed for the circular arch, but several solutions that require even less material can be found by choosing a slightly less thick support size. The series of arch forms based on different support sizes indeed shows that support-to-span ratios between 0.125 and 0.165 yield the lowest material use for r/s =1/2 and horizontal acceleration 0.3 g. As can be seen in Figure 26, any arch shape obtained through the method, presented in this chapter, is more material efficient than a circular arch, while all obtained arches with a support-thickness-to-span ratio greater than 0.07 perform better than the corresponding catenary arch under an earthquake of 0.3 g.

3.4.2 Design accelerations The study using different horizontal design accelerations (see section 3.3.2), shows that the presented approach remains efficient regardless of whether the design acceleration is large (0.45 g) or low (0.15 g). The required material of the form-found arches (r/s 1/2) can be reduced by as much as 65% compared to circular arches with constant thickness regardless of the magnitude of the horizontal acceleration as shown in Figure 27. The results of the design acceleration study confirm that the support size should be large rather than small to obtain the most efficient arches for all considered accelerations. Greater horizontal accelerations (0.3 g and 0.45 g) require thicker supports relative to the span as the thrust line due to horizontal accelerations becomes more asymmetrical (see Figure 29). The overall shape for a different design acceleration changes very little, except for the size of the supports. For arches with a r/s of 1/2, designed for an acceleration of 0.45 g, it is recommended that the support is between 0.19 and 0.26 times the span, while for accelerations of 0.3 and 0.15 g the recommended values are between 0.12 and 0. 17 times the span, and between 0.05 and 0.1 times the span respectively.

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Figure 29 – Form-found shapes for arches and the thrust line used for their form finding under accelerations of 0.15 g (green), 0.3 g (blue) and 0.45 g (red).

3.4.3 Rise-to-span ratios The form finding method is very effective for arches with a large r/s ratio as demonstrated by the results in Sections 3.1 and 3.2. The larger the rise relative to the span, the more vulnerable an arch will be to the horizontal components of earthquake loading, and the more effective the form finding technique presented here, will be in achieving weight reduction and this material efficiency. The results shown in Figure 28, demonstrate that the method still has significant benefits for shallow arches as well. For example, for form-found arches with a r/s of 1/4, 43% and 33% less material is needed compared respectively to a circular and catenary arch of constant thickness that can withstand the same acceleration of 0.3 g. Even for very shallow arches (r/s = 1/8), which are intrinsically much less likely to fail due to horizontal accelerations due to their nearly flat geometry, the required material can still be reduced by 21% compared to a circular arch (and 19% compared to a catenary arch) of constant thickness.

3.5 Applications and limitations The form finding approach, presented in this chapter, allows for the generation of a variety of shapes, with different r/s ratios that can sustain the horizontal design acceleration. While there is a minimum material form-found shape solution for each r/s and design acceleration, one might want to veer away from this solution for a variety of reasons. For example, while the most efficient shape from the perspective of material use follows a similar pattern regardless of design acceleration or r/s ratio, this solution might not be the easiest to construct. The pattern in the shapes that requires the least material can be observed in Figure 26, Figure 27 and Figure 28 for a predefined r/s and a. The thickest sections for all these arches are the supports. From the supports, the arch tapers to very thin sections on either side of the arch, roughly situated at a

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quarter and three-quarters of the arch’s length. These locations represent the area where the initial thrust line (used for form finding), and its mirrored equivalent intersect, resulting in the thinnest zones. The thrust lines also intersect at the arch’s crown and thus the arch becomes thin in that region. These thickness variations might not be desirable for construction purposes as it can be beneficial to have a constant thickness over part of or over the full arch. This constant thickness can be accomplished by increasing the support thickness or the overall offset of the initial thrust line, both parameters which can be controlled in real time in the form finding process. The shapes obtained with the presented approach are theoretical and not necessarily practical forms; these forms should be rationalized from a construction perspective to be market viable. After rationalizing the forms, construction of an arch with all individually carved voussoirs can be avoided by constructing the arch using thin-tile masonry, allowing for varying thickness along the arch and construction without the use of formwork [114]. Another possible construction approach could be to fully digitize the manufacturing process and mill the blocks to size, as was done for the Armadillo Vault (Venice, 2016, designers ODB engineering and Block Research Group) [15], [115]. Additionally, because the form finding process relies solely on geometry, the obtained shapes can also be applied to any other material that predominantly works in compression, including concrete, masonry and earth. The shape of arches that are thin at the central section and become thicker at the supports is very similar to those found in some gothic arch and vault structures [116]. These arches are often of constant thickness around the central vault, but the area between the arch springing and the wall is often filled with rubble, to contain the horizontal thrust. A similar approach could be taken for the construction of new arches, where the shape and amount of rubble could be informed by the presented form finding process. In that scenario, an arch with constant thickness could be built that is subsequently made thicker around the supports based on the support size obtained through the form finding process. The form finding process has been applied to find the shape of arches under self-weight. Nonetheless, the methodology can also be used to find better shapes for arches that are subjected to additional permanent loads. This is accomplished by adding the permanent load (vertical or horizontal) to the appropriate voussoir in the force diagram, and then carrying out the same form finding process as described in section 3.2.2 The effect of infill, for example, be can analyzed by increasing the mass of the voussoir below the infill and adjusting the center of mass accordingly. Additionally, other permanent loads, such as those carried off to the arch by columns, can be incorporated following that methodology. Figure 30 provides an example of how the method can be used to obtain the form of an arch in such a scenario. The obtained shape is the result of applying a large load (8 times the weight of the central two voussoirs) at the center of the arch, simulating additional weight due to the presence of a column. The considered rise-to-span ratio is 1:2 and the design acceleration is 0.3 g. The resulting arch is a more pointed version of the one presented in Figure 29 due to the additional central weight.

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Figure 30 – Form-found arch obtained under a horizontal acceleration of 0.3 g but with an additional vertical load simulating a column acting on the center of the arch. Furthermore, the presented form finding approach can also inform the design of vaults and shells in seismic areas. For example, the outline of the obtained shapes could be used as the sections of a 3D undulating vault as suggested in [117], resembling the Fosfato Thomas horizontal silo (Eladio Dieste, 1965, Montevideo) [118]. The vault surface can then be built with constant thickness. The 3D vault shape can be generated by changing the undulation depth of the supports based on the shape of the 2D form-found arch. In such a 3D generated vault surface, a thrust line can be fitted and thus a series of compression only load paths are guaranteed in several 2D sections under equivalent horizontal loading (see Figure 31). These 3D surfaces based on the 2D arches have been form-found relying on geometric properties only, hence they can be adapted also for the construction of shells, made from any material with compressive strength. A detailed explanation on how to manipulate the shapes of such corrugated 3D vaults, which requires an adaptation of the method for arches, is presented in detail in Chapter 4.

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Figure 31 – Example of how 2D-arch with varying section, could be adapted to 3D-undulating vault fitting at least one thrust line per section under horizontal acceleration (green). The strengths of the presented form finding approach are the scalability of the shapes and the comprehensible graphic interpretation of the results. However, the method relies on a series of assumptions and simplifications. The first assumption of unlimited compressive strength is valid in most scenarios as stresses in arches tend to remain well below the maximum compressive strengths of the material. If the arches become very thin or the crushing strength of the material is very low though, it should be verified that the forces at each voussoir do not exceed the compressive material strength. As the forces at each voussoir are an output of the algorithm through the force diagram, these forces can easily be compared to the material strength and the thickness can be adjusted accordingly. The second assumption is that sliding between blocks does not occur. When the thrust line is close to being perpendicular to the interfaces between the voussoirs this assumption is reasonable. In the case of very large accelerations, however, and especially if the joints between voussoirs run horizontally instead of perpendicularly to the intrados, sliding might be an issue which needs to be considered. This verification could be done by incorporating a finite friction criterion, as has been proposed for vault and arch analyses [119], [120]. Finally, the presented approach assumes that a rocking mechanism cannot form in the arch, thus neglecting the beneficial damping and energy dissipation during an earthquake event.

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3.6 Conclusion In this chapter, a form finding method to obtain efficient shapes of masonry arches in earthquake-prone areas is presented. It builds on thrust line analysis which can be used for a first-order earthquake assessment of masonry arches. Therefore, the technique has been adapted into a form finding process that achieves material efficient shapes for masonry arches subjected to horizontal in-plane accelerations and self-weight. The form finding procedure ensures that a thrust line fits into the arch geometry regardless of the in-plane earthquake direction, by obtaining a suitable shape through a set of geometric manipulations on an initial thrust line. During the form finding process the support size can be manipulated to obtain a variety of shapes. All results of this form finding process are validated by a kinematic limit state analysis, which also provides the collapse mechanism of the obtained shapes. It was shown that the amount of material needed for earthquake-resistant arches obtained through this form finding process considering horizontal loading and self-weight, was significantly less (65% for r/s =1/2) than the material needed for circular and catenary arches with constant thickness, regardless of the magnitude of the horizontal acceleration. It was also demonstrated that the thickness of the supports plays a crucial role in the seismic capacity of masonry arches and that with increasing design acceleration comes increased support size. Guidelines for the support size as a function of the earthquake magnitude were also provided. A comparison of the effect of different r/s ratios confirmed that arches with a greater r/s ratio are more vulnerable to horizontal accelerations, and thus they can benefit most from being shaped by the presented method. Nevertheless, the shape of shallow arches (r/s =1/4) can still be significantly improved (43% reduction in material use compared to circular arches with constant thickness) by adopting the developed form finding method and even very shallow arches (r/s = 1/8) which are less prone to earthquake action can benefit (20% compared to circular arches with constant thickness). The generation of more efficient forms for masonry arches under horizontal and gravity loads can lead to arch material and foundation cost savings in the construction of new masonry 2D arches and 3D undulated vaults in seismic zones. The presented approach can also inform the rebuilding of arches that collapsed during previous earthquakes. Insights regarding the necessary support thicknesses can be used to verify and even retrofit existing arches by strategically adding rubble around the supports. Because the method is the first one to explicitly incorporate the horizontal forces induced by seismic action in the form finding design phase, its further exploration might allow for a whole new realm of safe and efficient forms for structural surfaces made from masonry for earthquake-prone areas.

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Form Finding of Corrugated Shells Subjected to Seismic Loading This chapter is largely based on following journal publication, which was under review at the time of writing: T. Michiels, S. Adriaenssens, and M. DeJong, “Form finding of corrugated shell structures for seismic design and validation using non-linear pushover analysis.,” Eng. Struct., Under Review.

4.1 Introduction As indicated in Chapter 2, the geometry of shell structures plays an essential role in their capacity to withstand earthquakes. Seismic loading, however, is rarely considered when determining the overall geometry of shells. Therefore, this chapter presents a novel form finding methodology for the conceptual seismic design of corrugated shells. The method, which builds on the form finding algorithm for arches developed in Chapter 3, ensures that a compression load path exists to carry lateral earthquake accelerations by deriving shell geometries from a series of funicular polygons obtained through a graphic statics procedure for combined gravity and horizontal loads. While the method can be applied to any material that resists compressive stresses, it is employed in this chapter to find the shapes of corrugated thintile masonry shells, as these shells show promise as a sustainable and durable construction solution in a resource-constrained context. Non-linear pushover analysis is then used to quantify lateral capacity and evaluate form finding results in terms of material efficiency to resist lateral loads. The analysis also provides insights regarding the collapse mechanisms and flow of forces. It is demonstrated that the lateral capacity before cracking in the corrugated shell shapes is up to 79% higher than the capacity of a non-form-found reference shell shape considering identical material use. All form-found shells were observed to fail through a similar collapse mechanism which is defined by four crack zones. The location of these crack zones can be manipulated through the form finding process and the locations where reinforcement could be most efficiently introduced can be identified. Finally, the flow of forces within the form-found shells is used to propose alternative designs that provide additional openings in the shell surface while maintaining similar seismic capacity. Thus, the chapter provides a new approach for the conceptual design of safe corrugated shell structures in earthquake prone areas.

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4.1.1 Scope and outline The focus of this chapter is the design of corrugated shells that cover a rectangular area. Corrugations increase the effective thickness of the shell cross section, thereby increasing the bending capacity of the cross-section and the effective depth wherein load could be resisted in compression [121]. Numerous corrugated shells have been constructed, including iconic elegant designs by Eladio Dieste in Uruguay [118], [122] (see Figure 32). While it has been noted that corrugations in shells contribute to their overall stability [123] and that the double curvature increases their lateral stiffness limiting deformation due to the horizontal thrusts [118], the optimization of these corrugations to increase seismic capacity has not yet been addressed. The method presented in this chapter permits the shaping of corrugated shells so that they can withstand a specific equivalent horizontal design acceleration, while transmitting the bulk of the internal forces to the supports in compression. The proposed method can thus be applied to construct shells from a variety of materials that perform well under compressive forces. This chapter is constructed as follows. First the proposed method is presented (Section 4.2.1) and applied (Section 4.2.2) to thin-tile masonry shell design. The validation methodology (Section 4.3.1) for the masonry vaults is then described, accompanied by a brief literature review on analysis techniques for masonry shells. The results (Section 0) of the non-linear pushover analysis are then presented and discussed (Section 4.5), followed by concluding remarks (Section 0).

Figure 32 – Cadyl horizontal silo designed by Eladio Dieste in Young, Uruguay (1978) (Image Vicente Del Amo Hernández).

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4.2 Form finding under earthquake loading 4.2.1 Form finding method for corrugated shells under earthquake loading As indicated in Section 2.4, to date no form finding methodology has been presented that takes into account seismic loading from the start of the design process for shells. In Chapter 3, however, a methodology was presented to obtain the shapes of arches with varying thickness subjected to in-plane seismic loading. This form finding process for arches consists of a methodological application of a series of geometric manipulations of an asymmetrical thrust line, which is obtained under gravity loading and a predefined horizontal acceleration. This results in arch shapes that can resist the design acceleration by guaranteeing a compression-only load path. The form finding methodology presented in this chapter extends and adapts this method from arches to shells. Instead of providing arches with varying thickness (see Chapter 3) or doublelayered shells which will be presented in Chapter 5, this new method yields single-layer thin shells with uniform thickness. The method can be used to find shells of any size that work predominantly in compression under combined gravity and horizontal acceleration, and with a wide range of rise-to-span ratios. Before starting the form finding process, a series of parameters are defined, including the desired span (s) and rise (r), the thickness (t) of the shell and the horizontal peak ground acceleration (a) that the shell should be able to withstand. Additionally, a corrugation depth (d) and corrugation period (p) need to be chosen for the edges. It is suggested that initially the corrugation depth be defined using the support thickness-to-span ratios provided for arches in [124]. Subsequently, the corrugation depth, along with the other parameters, could then be iterated as part of the design process. To find the corrugation depth, the initial geometric and acceleration parameters are used to find an arch (in the x-z plane, see Figure 33a) enveloping a 2D-load path to carry the equivalent seismic forces in compression to the supports [125]. First, a funicular polygon needs to be found under the combined loading of gravity acceleration (in the negative z-direction) and the design acceleration (acting in the positive x direction) for a catenary arch with rise, span and thickness identical to the ones of the desired shell. This arch is discretized into n voussoirs and therefore the funicular polygon connects a series of n nodes, one for each voussoir with horizontal coordinate 𝑥𝑖 and vertical coordinate 𝑧𝑖 , and weight 𝐷𝑖 (𝑖 = 1. . 𝑛). The funicular polygon obtained for this arch (see Figure 33a) should have a rise (𝑟𝑓𝑝 ) equal to the rise of the desired shell (r) (see eq. 4.1) and a span (𝑠𝑓𝑝 ) equal to the sum of the desired shell span (s) and the desired corrugation depth (d) (see eq. 4.2). 𝑟𝑓𝑝 = 𝑟 51

(4.1)

𝑠𝑓𝑝 = 𝑠 + 𝑑

(4.2)

This can be rewritten as a function of 𝑥 and 𝑧 as in equations 4.3 and 4.4: max(𝑧𝑖 ) − 𝑧1 = 𝑟

(4.3)

𝑥𝑛 − 𝑥1 = 𝑠 + 𝑑

(4.4)

Finding a funicular polygon with the prescribed rise (𝑟𝑓𝑝 ) and span (𝑠𝑓𝑝 ), can be achieved through a real-time iterative graphics statics procedure by moving the pole of the force diagram as detailed in [94], [111], or a solution can be calculated numerically by solving a series of equations expressing horizontal and vertical equilibrium at each node of the funicular polygon. Based on the equilibrium of vertical and horizontal forces (eqs. 4.5 and 4.6), the location of each node of the funicular polygon can be written as a function of 𝐷𝑖 (eq. 4.7), where 𝐹𝑖,𝑉 and 𝐹𝑖,𝐻 are respectively the vertical and horizontal component of the force acting on voussoir i, and a is the horizontal acceleration. The detailed and illustrated derivation of these equations have been provided in Chapter 3.

𝐷𝑖 = (

𝐷𝑖 = 𝐹𝑖,𝑉 + 𝐹𝑖+1,𝑉

(4.5)

𝐹𝑖+1,𝐻 = 𝐹𝑖,𝐻 + 𝑎 ∗ 𝐷𝑖

(4.6)

𝐹𝑖,𝐻 𝐹𝑖,𝐻 + 𝑎 ∗ 𝐷𝑖 𝐹𝑖,𝐻 𝐹𝑖,𝐻 + 𝑎 ∗ 𝐷𝑖 − )𝑧𝑖 − 𝑧𝑖−1 + 𝑧 𝑥𝑖 − 𝑥𝑖−1 𝑥𝑖+1 − 𝑥𝑖 𝑥𝑖 − 𝑥𝑖−1 𝑥𝑖+1 − 𝑥𝑖 𝑖+1

(4.7)

Equations 4.5, 4.6 and 4.7 express all possible equilibrium solutions representing an infinite number of funicular polygons. By limiting the length of the funicular polygons, through the constraints provided by equations 4.3 and 4.4, a single solution can be obtained as presented in Figure 33a. The procedure is also detailed in the flowchart presented in Figure 34. Subsequently, this asymmetrical funicular polygon is mirrored along the vertical z-axis and moved horizontally (along the x-axis, see Figure 33b), so that the distance between the start point of the funicular polygon and the endpoint of the mirrored funicular polygon equals the corrugation depth d, thereby ensuring that the final shell will have its desired span (s). The mirrored and moved funicular polygons presented in Figure 33b provide the outline of the section view of the final shell. Next, the shell is corrugated in the third dimension. This is accomplished by moving the mirrored funicular polygon along the y-axis to generate a corrugation period, see Figure 33c. The guiding curves for the final shell are then obtained by copying the initial and mirrored funicular polygons along the y-axis for a desired number of periods (see Figure 33d). Fitting a NURBS surface through all these funicular polygons generates the middle surface of the shell (see Figure 33e). Finally, the middle surface is offset by

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half the thickness in both directions to obtain the final shell (see Figure 33f). The entire form finding procedure is visualized in the flowchart provided in Figure 34.

Figure 33 – Step by step form finding process. .

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Figure 34 – Step by step form finding process.

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4.2.2 Application to masonry shells The presented form finding method can be applied to design shells made from a variety of materials. Masonry shells are chosen because they can be built using local materials and local labor, thus requiring about 7 times less embodied carbon than typical roof construction from other materials [17]. The surge in interest in new masonry vaulting is exemplified by recent innovative projects such as the Mapungubwe National Park Interpretive Center (Peter Rich, South-Africa, 2009) [16] and the Droneport project exhibited at the 2016 Venice Biennale [68], [126]. These projects use traditional thin-tile vaulting while the form of the latter was obtained through state-of-the-art numerical form finding. These recent masonry shells have been built in areas with low-seismicity. A world map overlaying seismicity and locations which have a tradition in either earthen or vaulted masonry construction shows that many regions have the potential to adopt (earthen) masonry vaulting, but are at significant seismic risk (see Figure 35) [125]. Furthermore, the obtained shell shapes lend themselves well to the thin-tile construction method. The span (s) of the form-found masonry shell is chosen to be 8 m. The thickness is 12 cm which is practical for thin-tile construction of such span [126]. The rise (r) of the shell is determined as 4 m, so that the rise-to-span ratio of the form-found shell equals the rise-to-span ratio (1/2) of the Cadyl Horizontal Silo shown in Figure 32. This shell, henceforth referred to as the Dieste shell will be used as a point of comparison for the shapes that will be obtained through the form finding process. The shape of the Dieste shell is defined by a series of two alternating catenary arches that form the guides for the doubly-curved surface. Additionally, the performance of the form-found shell will also be compared to an 8 m span catenary vault (shaped by extruding a catenary along a straight line), which does not have an edge corrugation.

Figure 35 - World map overlaying earthquake hazard (red is greatest hazard; white is smallest) and locations with either a tradition of earth construction or masonry vault construction (dark overlay).

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The design acceleration (a) was set at 0.3 g (for reference, the maximum ground acceleration ag with probability of exceedance equal to 10% in 50 years in the most seismically active areas of Italy is prescribed as larger than 0.25 g [113]). These four parameters will be kept constant for all the obtained shells (except for one additional form finding example which will be carried out for an acceleration of 0.45 g). Finally, the corrugation depth d, which corresponds to the distance between the end points of the initial and the mirrored funicular polygon in the x-direction (see Figure 33b and Figure 33c), and the period of the corrugation p, which equals twice the distance between the initial and the mirrored funicular polygon in the y-direction (see Figure 33c and Figure 33d), must be defined. Both parameters are design choices that will affect the eventual capacity and failure mechanism of the shell, as will be further discussed in Section 4.4. Five form-found shells (FF1 to FF5) with varying corrugation depths and periods are generated and are compared to the shape of the Dieste shell. The corrugation depth (0.8 m) and period (2.4 m) of the first shape (FF1) are chosen to maximally improve earthquake capacity, while keeping the additional material use low (3% increase). The second shell (FF2) has the same corrugation depth (0.4 m) and equal corrugation period (1.5 m) as the Dieste shell. The third shell (FF3) has the same period (1.5 m), but has a depth of 0.9 m, so that the additional material use is limited to 10%. The fourth shell (FF4) has the same depth (0.4 m) as the Dieste shell, but its period is increased to 2.6 m to limit the additional material use to 5%. Finally, the fifth shell (FF5), has a depth of 1.1 m, which corresponds to the ideal depth [124] for minimal material use for an arch with a span of 8 m, a rise of 4 m and a period of 6 m so that the same volume is used for this shell as was for Dieste’s. Figure 36 shows a plan, a section and a perspective view of each shell. An overview table compiling corrugation depths and periods, and material use is provided in section 4.4 together with the capacity results of the pushover analysis.

Figure 36 - Perspective, section and plan view of the form-found shells and the Dieste shell.

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4.3 Validation method: non-linear pushover analysis The methods used to analyze masonry shells under seismic loading in literature can be divided into three categories: equilibrium methods, distinct element modeling (DEM) and finite element modeling (FEM). Equilibrium approaches to the problem include both upper and lower bound limit state solutions. Upper bound solutions such as kinematic limit state analysis have been applied to gain understanding of vaults under a series of load conditions, including seismic loads [101], [102], [127]. Lower bound limit analysis to analyze the stability of masonry has also been presented [92] and thrust network analysis has been reformulated to be able to account for horizontal loading due to seismic action [71]. As the form finding algorithm is based on equilibrium methods, these methods are not used for validation. Another approach to analyze masonry vaults under earthquake loading is Discrete Element Modeling (DEM), which incorporates the heterogeneous nature of masonry blocks by modeling the interaction between discontinuous blocks relying on finite-difference principles [90], [109]. The method has been applied to arches and vaults under seismic action [99], [103], [104] and its results were found to correspond well with data obtained through analytical limit state analysis [90], [99]. The application of DEM requires the input of initial cracking patterns or the definition of discrete blocks. Thus, it is less suitable for thin-tile shells which are formed from multiple layers of tiles in alternating directions, creating an integrated continuum with no obvious joint planes through the shell thickness. The seismic capacity of the form-found shapes presented in this study is therefore determined through non-linear pushover analysis using finite element modeling (FEM). The thin-tile masonry shells are analyzed as a homogeneous material because they are composed of several overlapping layers of tiles with changing orientation and intermediate layers of mortar. While an idea of when initial cracking occurs could be obtained using an elastic model, non-linear analyses are necessary to capture the ultimate capacity. Further, non-linear softening due to cracking must be included due to the brittle nature of masonry; bilinear elastoplastic models are computationally less expensive but can significantly overestimate post-cracking capacity. These non-linear FEM models have been applied successfully to a wide variety of masonry structures [128], [129] and seismic analyses are typically performed through non-linear static pushover simulations or non-linear dynamic analysis (NDA) [129]–[131]. NDA considers a wider range of dynamic parameters and can therefore accurately predict load and displacement capacity, as well as collapse mechanisms but is also very computationally expensive [132]. Because massproportional non-linear pushover analysis has been shown to yield similar load capacity estimations and collapse mechanisms as NDA for masonry structures, and is significantly more computationally efficient [132], it is used in this research to assess load capacities and collapse mechanisms for the form-found shells. Additionally, an extensive and detailed parametric study using non-linear pushover analysis employing non-linear FEM has been carried out before to

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successfully assess the seismic capacity of a thin-tile masonry vault [126], and this study will thus be used as the basis for the analysis methodology in this paper.

4.3.1 Non-linear pushover The structural capacity assessment of the shells was performed in the FEM software DIANA following the non-linear pushover analysis procedure outlined in Eurocode 8 [133]. Under constant gravity, horizontal loads proportional to the mass of the structure are applied incrementally, approximating the inertial forces experienced during seismic action [126]. Such mass-proportional pushover analysis has been shown to be an accurate method for the seismic assessment of masonry structural systems and has been deemed more accurate to estimate load capacities and failure mechanisms than mode-proportional pushover approaches for masonry structures [132]. Additionally, the obtained shells are very stiff (fundamental frequencies ranging between 4.28 Hz and 6.85 Hz, and second mode frequencies above 10 Hz) and thus higher mode effects are not expected to be significant for these shells with 8 m spans. Nonetheless, all pushover approaches are known to underestimate the displacement capacity of masonry in comparison to non-linear dynamic time history analysis (NDA). Therefore, load capacities will be used predominantly for the seismic assessment presented. While the absolute values of these load capacities should still be used with caution, they do provide clear information on the relative performance the shells and thus pushover analysis can accurately be employed to demonstrate the benefit of the proposed method. In the considered mass-proportional pushover implementation, the shell structure yields incrementally with the increase of the horizontal loading, and thus the stiffness reduces with each load increment [134]. The damage in the shell also develops with increasing load and is tracked sequentially. The loading conditions are prescribed using the arc-length method with indirect displacement control within DIANA [135], which provides solution increments as input without the need to predefine displacements. The loading conditions are prescribed using the arc-length method with indirect displacement control within DIANA [135], which provides solution increments as input without the need to predefine displacements. This incremental-iterative method allows for the capturing of the snapthrough and snap-back behavior by simultaneously constraining the norm of the incremental displacements and the size of the increments [135]. Instead of focusing on global non-linear behavior, the indirect displacement control sequentially considers part of the displacements and thus allows for a better capturing of local collapse mechanisms. Iterations are performed using the Regular Newton-Raphson method, where at each iteration the stiffness matrix (relating force and displacement vectors) is evaluated based on the last known or predicted situation. The masonry behavior is simulated employing the Total Strain Rotating Crack material model (TSRC). Stresses in this model are described as a function of strain and follow a smeared

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cracking approach based on fracture energy [136], [137]. The constitutive relations are evaluated in the principal directions of the strain vector, which also determines the orientation of the cracks [131], which are free to rotate throughout the solution. Tensile stresses are assumed to diminish linearly after the maximum tensile strength (ft) is reached and will occur perpendicular to the direction of the major principal stresses [138] (see Figure 37a). Softening behavior is governed by fracture energy (Gf), which corresponds to the area under the curve of the tension side of the stress-strain diagram. The fracture energy is kept constant and is dependent on the element length to produce results independent of mesh size. Meanwhile, in compression elastoplastic behavior is assumed with a constant post-crushing compressive strength (fc). The compressive and tensile strength of the materials, as well as the Poisson's ratio and the density are presented in table 4.1 [126]. For the fracture energy, a relatively low value of 0.015 N/mm is used, in line with the values assumed in the reference study for thin-tile masonry [126], and similar to the values used for adobe masonry, which is more brittle than thin-tile masonry [131] . Because the primary goal of the study is a comparison between the seismic capacity of the overall shell geometries, the material properties are kept constant. This allows for direct comparison between shapes, assuming that altering the material properties would have a negligible effect on the relative capacity of the different shells.

Figure 37 - Stress strain relations in tension and compression characterizing masonry. Property

Value

Compressive Tensile strength strength coefficient Poisson Density Fracture energy

8.5 MPa 0.35 MPa 0.2 1900 kg/m3 0.015 N/mm

Table 4.1 – Material properties. DIANA’s CT30S element is used for all elements. The CT30S Element is a quadratic six-node triangular isoparametric curved shell element with uniform thickness [135]. A six-point

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integration scheme is employed over the area of the element, while a seven-point integration scheme is used in the thickness direction, as in [126]. A mesh convergence study identified that elements with a mesh size of approximately 200 mm were satisfactory. The capacity of the shells is evaluated by plotting the displacement in the top node of the shell versus the base shear, while documenting the applied horizontal accelerations at the moments of the first crack, the peak capacity and the final converged load step. The latter provides information on the shell’s relative ductility, though this can be very sensitive to convergence difficulties. The analysis of the shells is performed on one bay of each form. Symmetry boundary conditions (y-displacement equals zero) are applied along the nodes on the edges in the x-z plane. All nodes along the corrugated edges of the base are pinned. The pushover force is always applied along the negative x-axis; applying the force in the opposite direction yields the same capacities due to symmetry. The pushover capacity of the shell in the perpendicular direction, along the direction of the corrugated edge (y-axis) is significantly higher than the capacity in the x-direction as the consecutive bays are supporting one another. The shell bays that buttress one another thus act jointly as a large diaphragm, transferring the horizontal load through in-plane compressive forces to the supports. Because pushover along the corrugated edge is thus not governing the design, only the capacity in the weakest direction (x) is reported in the results section.

4.4 Results 4.4.1 Pushover curves Pushover curves are obtained for the five form-found shells for an acceleration of 0.3 g, as well as for the Dieste shell and an extruded catenary vault. Table 4.2 provides an overview of the corrugation depths and periods, and the additional volume that each shell uses compared to the Dieste shell. The table also lists the peak ground accelerations (PGA) at which the first crack forms for each shell, as well as the ultimate (peak) horizontal acceleration capacity before collapse. To compare the efficiency of the forms regardless of the increases in material use, PGA normalized by volume is also reported. It is observed that the first crack in the shells occur at absolute accelerations ranging from 0.28 g to 0.38 g, while the form finding process is devised to provide a capacity of 0.3 g (see Figure 38). The Dieste shell, in contrast, cracks at an acceleration of 0.2 g, while a catenary vault shows first cracks at 0.07 g. The masonry shells presented in this study do not fail due to the formation of one crack, but rather through the formation of a collapse mechanism determined by crack zones. It is thus observed that the first crack occurs at 68% of the peak capacity on average. All the shells designed to withstand 0.3 accelerations have an ultimate capacity that reaches at least 0.44 g.

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Normalized Hor. Span = 8m Depth Period Additional PGA (% of g) PGA (% of g) Stiffness Rise = 4m D (m) P (m) Volume (%) 1st crack Peak 1st crack Peak (%g/mm) 0.20 0.29 0.20 0.29 1.31 Dieste 0.4 1.5 0.36 0.50 0.35 0.48 2.35 FF1 0.8 2.4 3% 0.28 0.44 0.24 0.38 2.13 FF2 0.4 1.5 15% 0.38 0.56 0.35 0.51 2.61 FF3 0.9 1.5 10% 0.32 0.47 0.30 0.45 2.17 FF4 0.4 2.6 5% 0.30 0.46 0.30 0.46 1.17 FF5 1.1 6 0.0% 0.07 0.10 0.07 0.10 0.08 Catenary -8%

Fund. Freq. (Hz.) 3.46 6.54 6.41 6.85 6.47 4.28 1.27

Table 4.2- Overview of shapes with their corrugation depth and period, additional volume compared to the Dieste shell and peak ground acceleration capacities. Horizontal stiffness and fundamental frequency of each shell are reported as well. All form-found shells behave significantly better than the Dieste shell. Accounting for the volume of materials used (normalized PGA), the first cracks appear on average at a normalized acceleration of 0.31 g, a 58% increase in capacity from the Dieste shell. The ultimate capacity of the shells is similarly 59% higher on average. While all form-found shapes have a considerably higher capacity, significant differences can be observed between the efficiency of the shells. For example, shell FF1 has a normalized capacity of 0.35 g (first crack) and normalized peak capacity of 0.48 g, which is respectively 79% and 69% higher than that of the Dieste shell. FF2, however, which used 15% more material than the Dieste shell, only performs 25% (normalized 1st crack) and 32% (peak) better and is thus a considerably less efficient solution.

Figure 38 – Absolute (left) and normalized for volume (right) pushover capacity without cracking (light gray) and after cracking (dark grey) for different shapes. Figure 39 represents the pushover curves for the form-found shells, as well as for the Dieste shell. All data points correspond to converged solutions. The pushover curve for each shape is split into two parts where the first part (in lighter color and marked with a cross) represents the zone in which the shell is uncracked, while the second part of the curve captures the behavior with cracks (in darker color with a horizontal marker).

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Figure 39 – Pushover curves for form-found shells and Dieste shell.

4.4.2 Collapse mechanism and flow of forces The evolution of cracks as the lateral force increases demonstrates how the collapse mechanisms develop. For example, Figure 40 shows the evolution of the cracks that form the final collapse mechanism for FF1. Cracks in the top and bottom surface are plotted together for each load step. The first cracks (see Figure 40a) occur at the left support at the bottom surface of the shell. With increasing loads, these cracks continue to develop along the edge of the bay at the bottom of the shell forming crack zone A (see Figure 40b). Subsequently, cracks start to form at the top surface on the right side of the shell at the center of the bay, creating crack zone E (see Figure 40c). Once cracks appear on both left and right sides of the shell (zones A and E respectively), increasing the load causes these cracks to open further (see Figure 40d). Additionally, cracks start forming about 1 m to the right of the left support at the top surface, creating crack zone B. Simultaneously, some small cracks also appear at the bottom surface around mid-span (zone C) (see Figure 40e). Failure occurs through a mechanism formed by 4 crack zones, which in the case of FF1 does not include the crack zone C at mid-span. While the cracks in zone A (bottom surface) and zones B and E (top surface) continue to open wider, the fourth crack zone (D) is formed about one meter to the left of crack zone E, roughly symmetrically to crack zone B. The cracks in zone D occur at the bottom surface. Figure 41 provides an overview of the collapse mechanisms of all five form-found shapes and of the Dieste shell. All shells fail through a mechanism that is governed by 4 crack zones. However, the sequence and location of the crack zones is dependent on geometry. For example, the Dieste shell fails in a similar fashion to FF1 through the ABDE mechanism. FF2, FF3 and FF4 fail through the activation of crack zones A and E around the edges, but additionally through crack zone C at the bottom surface at mid-span, in combination with a fourth hinge (either C or B) at roughly the quarter-span. FF5 fails through the ABDE mechanism again. The type of collapse mechanism is linked to the depth of corrugation, where small corrugation depths at the base lead to a mechanism that involves a hinge at midspan (hinge C). Larger corrugation depths at the base lead to the ABDE-mechanism without the hinge at midspan.

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Figure 40 – Evolution of collapse mechanism. Cracks in zones A, C and D are in the bottom surface. Cracks in zones B and E are at the top surface.

Figure 41 - Section view and perspective of corresponding collapse mechanisms.

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The flow of forces is visualized by the principal stresses at the horizontal design load of 0.3 g for FF1. Maximum and minimum principal stresses (Pmax and Pmin) are provided for top and bottom surfaces (see Figure 42). Positive values indicate tension and negative values denote compression. The stress plots show that a primary compression load path, carrying most of the internal forces occurs at the center of the bay (when pushing from right to left). The compressive load path starts at the bottom surface at the inner most section of the corrugation on the right side (see Figure 42d, I). The greatest compressive forces are then carried near the top surface, along the right middle section of the right side of the bay (see Figure 42c, II) The compression load path then shifts back towards the inside surface (see Figure 42d, III) on the left side of the bay. Compressive stresses are finally largest at the left support around the top surface (see Figure 42d, IV). These 4 zones correspond to the cracking areas of the failure mechanism (zones ABDE, Figure 40f), though the cracks form at the opposite extreme fibers of the cross sections at those locations, as expected. While tensile stresses do arise in both the top and bottom surfaces under an acceleration of 0.3 g, they do not exceed the ultimate tensile strength of the material (0.35 MPa).

Figure 42 - Principal maximum and minimum stresses in the top and bottom surfaces of FF1 under an acceleration of 0.3 g.

4.5 Discussion 4.5.1 Evaluation of form finding method The objective of the form finding process is to produce shell shapes that can withstand design peak ground accelerations without cracking. The form finding procedure meets this objective as all shells obtained for a PGA of 0.3 reach this desired capacity or exceed it without cracking. The only exception is FF2 whose capacity (0.28 g) is only 6% under the target value. Additionally, the form finding procedure was also tested for higher accelerations and was again found to 64

produce shells that match the design acceleration. For example, applying the form finding technique to design a shell (FF1b) that withstands an earthquake of 0.45 g with a corrugation depth equal to 1.9 m (determined based on the 2D form finding process [124]) and a corrugation period identical to FF1 (p = 2.4 m), the first cracks were found to occur at an acceleration of 0.45 g. Figure 43 shows FF1 next to the shell form-found for 0.45 g (FF1b). The larger capacity to withstand horizontal loading is obtained through the form of the initial funicular polygon in 2D (see the thicker black line in the sections at the top) and the greater corrugation depth.

Figure 43 – Section and plan views (center) of FF1 form-found for 0.3 g (left) and shell FF1b form-found for 0.45 g (right), as well as perspective views. The analysis of form-found arches in earlier research has shown that the thickness of the arch’s supports is a dominant factor for the lateral load capacity, and the most materially efficient solutions are the ones with thicker supports (ranging from 0.9 m to 1.4 m for an 8 m span arch) [124]. Similarly, the shells with greater corrugation depths perform better than their counterparts with smaller corrugation depths. For example, FF3 has a corrugation depth of 0.9 m, versus 0.4 m for FF2, while both shells have the same corrugation period (p = 1.5 m). FF3 cracks at a horizontal acceleration of 0.38 g, while FF2 cracks at 0.28 g due to its smaller depth. Moreover, FF3 uses less material than FF2, because the small corrugation depth of FF2 combined with a small corrugation period, generates an overly pronounced top corrugation, i.e. the depth of the corrugation at approximately the third span is much larger than at the base (see figure 2). If this top corrugation is smoothed by introducing a larger corrugation period (p =2.6 m), as in FF4 (d = 0.4 m), the material use drops by 10% relative to Dieste’s shell, and the lateral capacity increases to 0.32 g. Thus, while a small corrugation period might be beneficial for construction purposes (see also Section 5.3), too small a corrugation period can have a negative effect on material efficiency and horizontal load capacity. The potential of the form finding procedure to improve the pushover capacity of shells is exemplified by the increased capacity of the form-found shapes compared to the Dieste shell, 65

while accounting for volume (normalized PGA). Dieste’s shell cracks at 0.2 g, while the other shells (except for the underperforming FF2) reach a capacity that is at least 50% higher (0.3 g). FF5 was form-found to use the same volume of material as the Dieste shell and therefore its PGA does not require normalization, and it performed exactly 50% better. The best performing shells accounting for volume (FF1 and FF3 at 0.35 g), have a capacity which is 79% higher than the Dieste shell. These higher capacities can again be attributed to the form of the initial funicular polygons. The Dieste shell is based on a series of regular catenaries, the ideal shape for structures under gravity. The form-found shapes in contrast, are based on funicular polygons that take into account both vertical and horizontal acceleration. The pushover curves also provide information about the relative lateral stiffness of the different shells. The horizontal stiffness values of FF1, FF2, FF3 and FF4 are largely similar. The Dieste shell, however, is about half as stiff, similar to the horizontal stiffness of FF5. FF5 is significantly less stiff than the other form-found shells because it has a much higher corrugation period (p=6 m versus p= 1.5 to 2.6 m), resulting in significantly less curvature. The observation that stiffness can be adapted by adjusting corrugation period without resulting in a significant loss of overall capacity can be a useful insight to shell design. It should also be noted that the formation of the first crack does not mark the ultimate pushover capacity of the shells. It is only after the development of a large amount of additional cracks in concentrated cracking zones that a four-hinge mechanism is formed, leading to collapse. While the first cracks occur around the 0.3 g design PGA for all form-found shapes, this design capacity is on average just 59% of the maximum capacity, which ranges from 0.44 g (FF2) to 0.56 g (FF3). However, this ultimate capacity is largely dependent on the value of the fracture energy used in the study. Because the presented results have been obtained using a relatively low, and thus conservative, value for the fracture energy (0.015 N/mm) these ultimate capacities are potentially underestimated. Repeating the study with a higher fracture energy of 0.05 N/mm, which is more realistic, the ultimate capacity of the shells increases by 29% on average. The overall behavior of the shells observed in the pushover curves and the collapse mechanisms remain the same regardless of fracture energy. The actual value of the ultimate capacity of the shells is of limited interest for this study, as the primary goal is to perform a comparative analysis between the shells and assess the effectiveness of the form finding method.

4.5.2 Influence of shape on flow of forces and collapse mechanism The flow of forces when pushing the structure from right to left can be best understood by looking at one bay at the design acceleration of 0.3 g. The stress plots presented earlier in Figure 42 for FF1, indicate that the compressive stresses are largest (ranging from -0.5 MPa to -0.1 MPa) in the central zone of the bay as would be expected by the design based on the guiding funicular polygon. The largest tensile stresses occur around the edges separating the different bays, and most prominently along these edges by the left support at the bottom surface. This

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corresponds to the location of the first cracks. The principal compressive stress vectors in the top surface are additionally plotted in Figure 44. The figure shows that the forces experienced at the edges are carried off towards the center of the bay. This central section carries the bulk of the compression forces. This compression load path corresponds neatly with how a 2D thrust line would form in an arch. The load path starts at the innermost point of the support at the left (zone I in Figure 44), which is similar to the hinge that would form around the intrados in the 2D case of an arch subjected to earthquake loading [96]. Then, the load path moves towards the top of the middle-right of the bay (zone II), and the bottom of the middle left of the bay (zone III), just as the second and third hinges would form around the extrados and intrados of an arch respectively. Finally, the largest compressive stresses are found at the top surface on the far left (zone IV), corresponding to the hinge at the extrados. Overall compressive stresses remain very low at a maximum of -0.489 MPa, which is 5.7% of the ultimate compressive strength.

Figure 44 – Principal compressive stresses (N/m2) for FF1 in the top surface for a horizontal acceleration of 0.3 g. The collapse mechanisms displayed in Figure 41 can directly be linked to this flow of forces and the shapes of the shells, and even to the arches formed by the enveloping section. All shapes fail through a 4-hinge mechanism of which two hinges are invariably hinge A (extrados on the left) and hinge E (intrados on the right). The location of the hinges in between varies, as there are 3 places where the two additional crack zones form. The crack zones correspond to the areas where the shell is only singly-curved, and not doubly-curved like most of the shell surface. The locations of these singly-curved sections vary but can be easily discerned in the section cuts provided in Figure 41. The shell is singly-curved where the two funicular polygons cross, which happens three times for each form. The funicular polygons cross once on the left side (around quarter-span) and once symmetrically on the right side, with the location dependent on the corrugation depth. The other crossing of the funicular polygons is always exactly in the center of the section. Because it is known that the shells will fail through a mechanism formed around these three zones and at the two zones at the base, designers can opt to improve the shells ductility efficiently by controlling these locations in the form finding process or by adding reinforcement if desired. 67

4.5.3 Design opportunities and limitations The stress plots provided in Figure 42 and Figure 44 show that the inner and outer most sections of the corrugated edge supports are most important for the development of a compression load path. Thus, it is possible to remove less vital sections around the supports to create openings in the side of the shell, as shown in Figure 45. This would reduce material required and could provide further functionality to the design (e.g. allow entry along the length of the shell or allow light to enter the shell if it is a roof structure). The shell in Figure 45 is again form-found for 0.3 g, has a corrugation depth of 1.1 m and a period of 3 m, and uses 1.8% less material than the Dieste shell. Applying the same analysis methodology shows that the shell retains a pushover capacity of 0.3 g (normalized) when the first crack forms (at an absolute PGA of 0.29 g). The shell collapses at a pushover force equivalent to 0.59 g (normalized) through an A-B-D-E collapse mechanism. Thus, removing sections of shell material is possible, while not compromising capacity.

Figure 45 – Form-found shell with cutouts along the base corrugation. The generalized conceptual design approach presented in this paper yields forms that withstand seismic loading, relying predominantly on the compressive capacity of the shell. As compressive stresses in the structures under seismic forces remain an order of magnitude less than the maximum compressive strengths of the material (for example 17 times under the ultimate compressive strength for FF1), failure due to exceedance of compressive strength will not occur. Thus, the examined shapes are scalable up to the point that this ultimate compressive strength criterion is violated, or the frequency of the structure reduces to the point that higher mode effects become significant. The presented methodology is also material-independent and can therefore be applied to construct shells from any type of material that is able to resist compression. The forms obtained through the form finding process can be evaluated rapidly and effectively using the presented non-linear pushover analysis. Nonetheless, a comprehensive design would incorporate further seismic analyses that consider the vibrational properties of the shell and multi-directional nature of earthquakes in greater depth, which could be accomplished with nonlinear dynamic time history analysis (NDA). Such an analysis would be expected to provide a

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more accurate estimation of the ultimate displacement capacity of the masonry for regular structures [139] or simple structures with single curvature [104], however, for complicated threedimensional shells with double curvature, the ability of FEM modelling to accurately predict cracking and subsequent collapse under dynamic loading is arguable at best. Nonetheless, the pushover analyses presented in this paper provide clear information on the relative capacity of the shells and the relative improvement in performance is therefore clear. While the pushover might not predict the actual capacity with precision, the relative benefit of the proposed method is demonstrated. Further, NDA requires a site-specific input spectrum and because its practical application is difficult due to the intrinsic complexities and high computational demand [140], it is typically suggested to be used when detailed vulnerability assessment is necessary [141] and structural performance levels of damage or displacements are clearly defined [132], [142], which is not the case in this conceptual design research phase. Finally, it is also important to note that the form finding methodology is limited to a 2D-section and thus yields geometries that account for horizontal accelerations that act in the span direction (x). Therefore, it is vital to provide a series of repeating bays in the direction perpendicular to that of the design acceleration (y), so that stability is provided through bays that support one another, ensuring that collapse is indeed governed by the span direction. Even with sufficient repeating bays, detailed analysis should be conducted to ensure local failure does not occur at the end bays due to loading in the corrugation direction.

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4.6 Conclusion In this chapter a novel form finding approach for vaulted shell structures located in seismic regions is presented and a parametric study on a variety of masonry shells to evaluate the method’s effectiveness is performed. The presented form finding method relies on the geometric manipulations of a funicular polygon, which is obtained under the combined loads due to gravity and horizontal accelerations. By mirroring and translating this polygon, a shell shape is obtained with corrugated edges. By controlling the period and depth of these corrugated edges, a variety of geometries can be obtained, independent of material properties. Several design geometries were generated, and their capacities evaluated using non-linear pushover analysis. The analyses confirm that the form finding methodology is effective in generating shell shapes that can withstand the prescribed design acceleration. Furthermore, the form-found shapes performed significantly better than a similar shell shape based on Eladio Dieste’s Silo in Young (Uruguay), which was not form-found for horizontal loading. Using the same amount of material, the capacity before cracking of the best shape is 79% higher than the capacity of the Dieste shell. The ultimate capacity of all form-found shells is on average 59% higher than the capacity at which the first cracking occurs, using conservative material properties. The results also indicate that the corrugation depth and corrugation period of the obtained shapes can be manipulated to tune the shells to certain desired behaviors. For example, the corrugation period can be altered to modify the shells’ lateral stiffness. Furthermore, the corrugation also affects the collapse mechanism. All the investigated shells fail through a crack zone at both supports, and one additional crack zone in the flattest area around the left or right quarter-span. The fourth crack will form either symmetrically around the other-quarter span, or alternatively at mid-span. This advance knowledge of the expected failure mechanism can be beneficial to make educated choices on where to use reinforcement – if at all. The efficient behavior under horizontal loading of the obtained shells can be best understood by looking at the flow of compressive stresses through one bay. The design geometry provides a load path to transfer the horizontal seismic loads to the supports predominantly in compression. This also means that certain areas of the corrugated supports are not essential for seismic capacity, and thus material in these regions can be removed while still maintaining earthquake resistance. This opens a new realm of design possibilities. In conclusion, in this chapter a novel design methodology is presented to shape shell structures that will act predominantly in compression under earthquake loading. While particularly well-suited to be built using low-embodied carbon thin-tile masonry, the shapes can also be used to construct shells from other materials that work well in compression. This study can thus contribute to the construction of more efficient, safe and sustainable shell structures in earthquake areas.

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3D Form Finding Approach for Shells Subjected to Seismic Loading Sections of this chapter have been adapted from the following journal article: T. Michiels, S. Adriaenssens, and J. J. Jorquera-Lucerga, “Parametric study of masonry shells form-found for seismic loading,” J. Int. Assoc. Shell Spat. Struct., vol. 58 (4), pp. 267–275, 2017.

5.1 Introduction In this chapter a form finding approach is presented that allows for the shape generation of single-layer shells with varying thickness and of double-layer shells with uniform thickness in seismic areas. Through a parametric study, this method is illustrated for a wide variety of boundary conditions and yields a set of shell geometries including asymmetrical ones in contrast to those provided in Chapter 4. The form finding approach examined here assumes that the shells will be constructed from a material that resists limited to no tensile stresses (such as masonry, unreinforced concrete, earth, stone or ice, as assumed in the previous chapters). Therefore, for shells subjected to both vertical gravity and horizontal seismic loading, a compression-only load path (in 2D often referred to as a thrust line) should be present within the thickness of the shell to avoid collapse mechanisms. Through the application of an inversed hanging net model subjected to lateral loading in a dynamic relaxation solver, shell forms are generated for which it can be ensured that such a load path exists. To illustrate this methodology, a variety of shapes are generated based on a set of parameters including boundary conditions and net stiffness. The shapes discussed in this chapter are the second novel set of compression-only double-layer shells reported in literature (after those presented in Chaper 4), whose forms are successful and efficient in withstanding combined gravity and seismic loading that can act in any possible direction.

5.1.1 Advantages of double-layer shells As indicated in Chapter 2, shell shapes have never been designed specifically for construction in seismic areas, and thus their form finding processes have not had to account for lateral loads induced by seismic actions. Earthquakes, however, can be expected to have an important impact

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on the behavior and thus the shape of these shell structures, as their large horizontal forces can be expected to induce large bending moments that cannot be accommodated in thin shells made from low-tensile strength materials. Despite these challenges, shell structures (for example, the thin-tile masonry considered in Chapter 4), still hold great potential as low-cost and environmentally-friendly shelter in earthquake areas. In general, they tend to behave well during earthquakes due to their high stiffness because of their double curvature and low mass. This combination often results in high fundamental frequencies limiting dynamic amplification (see Chapter 2). To account for the large horizontal loads and anticipated bending moments, a form finding methodology is proposed that considers an equivalent horizontal earthquake load, which can act in any direction, from the start of the form finding process. The method combines funicular shapes that can accommodate a compression-only load path also under a large horizontal load, while not relying on any tension capacity of the material. Certain shapes that result from the form finding process, however, require a great structural depth, which could lead to thick and thus possibly heavy shells. To avoid heavy shells, while still providing sufficient structural depth, it is suggested to implement the obtained forms as interconnected double-layer thin shells. This way, ample effective thickness is provided so that a compressive thrust surface can form over a wide depth of the structure, while maintaining the lightweight nature and associated advantages of the shells. Double-layer shells have been a reliable structural form for almost 2000 years. They can be traced back to the Roman Empire, where they were employed, for example, in a barrel vault at the cult complex in Argos (Greece, 1st century BC) [143]. Many more domes and shells have been constructed with double-layered masonry ever since, including emblematic ones such as the dome of the Florence Cathedral (Italy, 1436, Brunelleschi) and Saint Peter’s Basilica in Rome (Italy, 1590, Michelangelo and Giacomo della Porta) [144]. These examples consist of a shell that is structurally connected to a second shell skin, providing more depth and thus a larger section in which compressive load paths can develop. Other more recent examples of doublelayer vaults include the former textile factory Vapor Aymerich, Amat I Jover in Terassa (Spain, 1909, Lluís Muncunill) constructed in thin-tile masonry, the early 20th century thin-tile domes by the Guastavino Company such as the double dome of St. Paul’s Chapel at Columbia University (New York City, USA, 1907, R. Guastavino Jr.) and the dome of the National Museum of Natural History (Washington DC, USA, 1909, R. Guastavino Jr.) (see Figure 46a and Figure 46b), and a series of double-layer Mexican style vaulting in Oaxaca (Mexico, 2015, Ramón Aguirre Morales) [61]. The latter consists of a double-layer shell for thermal insulation (see Figure 46c).

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b

a

c

Figure 46 - a: Double dome of St. Paul’s Chapel at Columbia University, New York City, USA, 1907 (The Guastavino Company, Avery Library of Columbia University) b: Double dome of the National Museum of Natural History in Washington DC, USA, 1909 (The Guastavino Company, Avery Library of Columbia University). c: Double-layer Mexican style vault in Oacaxa, Mexico, 2015 (Designer and image: Ramón Aguirre Morales).

5.2

Form finding methodology

5.2.1 Extension from thrust lines to nets The proposed method builds on the no-tensile model developed by Heyman, which relies on three commonly accepted principles [82] which are also assumed in the form finding of arches and corrugated shells (see Chapters 3 and 4): (1) compressive stresses remain low compared to the compressive strength of the material, so crushing does not occur; (2) the tensile strength of the material is negligible and can therefore be considered zero; and finally, (3) sliding does not 73

occur, which implies infinite friction between blocks (in the case of masonry). When these assumptions hold, a structure is self-supporting as long as one equilibrium compressive load path network (or thrust network) fits within the geometry of the masonry under its own weight [66], [145]. The theorem can be expanded to a combination of gravity loads and horizontal loads and has been used to find the shapes for masonry arches under earthquake loading as detailed in Chapter 3. The dynamic loading induced by earthquakes can be approximated by an equivalent horizontal load due to an acceleration which is proportional to the gravity load, the peak ground acceleration (PGA) of an earthquake. In this chapter, an equivalent horizontal load of 0.3 g is considered which corresponds to a moderate-to-high earthquake depending on a range of factors as explained in Chapters 3 and 4. If the aforementioned assumptions hold, a structure will stand under an equivalent horizontal load, if a compressive-only load path exists within the structure under such a load. This concept can be visualized most easily for an arch. Figure 47 shows a semicircular arch and two thrust lines that correspond to equilibrium states under different loading conditions. The blue thrust line, which fits into the arch geometry, represents one of many equilibrium states under the gravity acceleration (g). An analogy for this thrust line is a hanging chain under gravity loads, which is inverted [77]. The green thrust line represents one of the equilibrium states under the combination of self-weight and an additional horizontal force due to an acceleration of 0.3 times the gravity load. This thrust line also fits within the geometry, thus it can be concluded that the circular arch shown in Figure 47 will be able to survive in-plane accelerations of at least 0.3 g.

Figure 47 - This cylindrical arch is stable under gravity loading, and under combined gravity and in-plane acceleration of 0.3 g as thrust lines fit within the geometry under both load cases. The 2D idea of an inverted hanging chain fitting into the geometry, which was used extensively in Chapters 3 and 4, can be expanded to an inverted hanging net in 3D. If such a funicular net can fit within the shell’s geometry under the combination of self-weight and horizontal acceleration, the shell will be stable. Because the direction of the acceleration induced by an earthquake is unknown, at least one such net should fit within the shell geometry for each possible acceleration direction and orientation.

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5.2.2 Implementation of form finding method 5.2.2.a Nets under combined gravity loading and horizontal acceleration The form finding methodology starts with identifying a series of funicular nets under combined gravity and horizontal accelerations, which is accomplished by using cable elements in a network subjected to loading and solving in a dynamic relaxation routine. The basis of the dynamic relaxation method adopted is to trace step-by-step the motion of each node of the cable net for small time increments until, due to artificial kinetic damping, the net comes to rest in static equilibrium [72], [146]. The form finding process may be started from a flat mesh or an arbitrary geometry, with the motion caused by applying external load at the nodal masses. The routine is implemented in the solver Marsipulami [147], [148] for Grasshopper in the CAD software Rhinoceros [112]. For more details about the dynamic relaxation method the reader is referred to [72]. In this implementation, the shell surface is represented digitally by discretizing the surface into a network of linear elastic springs connected at n nodes with free rotation and initial coordinates Xi, Yi, Zi (with i = 1..n). Gravity and earthquake accelerations (a) are applied to each node, generating loads. These loads are expressed as a vector (Vj, ki ) at each node (index i) and the magnitude of the components of the load vector is proportional to a load multiplier (Lki) assigned to each node. This load multiplier (Lki) is proportional to the mass of the shell around the node and thus updated iteratively (index k) during the form finding process as the shell thicknesses changes. The load vector applied at each node is furthermore dependent on the acceleration direction (index j). During the first iteration, the initial load magnitude equals 1, as it is assumed that the loads of the square grid are distributed equally to the nodes (see eq. 5.1 and Figure 48, step 1). 𝐿𝑘=1 = 1 𝑓𝑜𝑟 𝑖 = 1. . 𝑛 𝑖

(5.1)

Assuming the first direction of the design acceleration is the positive x-direction (j=1), the load vector V j,ki is written as a function of a and Lki as in equation 5.2. 𝑉𝑖

𝑗=1,𝑘

= 𝐿𝑘𝑖 [𝑎, 0, 1]

(5.2)

In the first iteration uniform loads are applied, thus this can be expressed as in equation 5.3. 𝑉𝑖

𝑗=1,𝑘=1

[𝑎, 0, 1] = 1 [𝑎, 0, 1] = 𝐿𝑘=1 𝑖

(5.3)

The system behaves like a cable net that stands in pure compression under a combined loading of gravity and horizontal acceleration, as it is the inverse of a net that hangs in pure tension under such loading. Interactive computation of the shell form starts from a flat mesh of springs (at nodes n), supports (any 3 or more nodes that are pinned) and loads (V j,ki). The elastic stiffness of

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the net (ES) is furthermore specified and updated iteratively so that the rise of the shell matches the desired rise (r). The result of the converged dynamic relaxation process for earthquake loading in the positive x-direction (j=1), assuming the 4 corners of the grid act as pinned supports, is the geometry of the net displayed in Figure 48, step 2. This dynamic relaxation process is repeated with the horizontal acceleration acting in different directions, for example negative x-direction leading to a second net (j=2) displayed in Figure 48, step 3. In the current implementation, 4 directions are considered, each with positive and negative orientations for the horizontal loads (positive and negative x and y, and these loads rotated over 45 degrees, see Figure 48, step 4). Additionally, a load case representing gravity loading without any horizontal acceleration is added to these 8 load cases as well, thus amounting to a total of j=9 load cases. The 9 corresponding load vectors (V j=1..9 , ki ) are provided in equations 5.4 to 5.11. 𝑉𝑖1,𝑘 = 𝐿𝑘𝑖 [𝑎, 0, 1]

(5.4)

𝑉𝑖2,𝑘 = 𝐿𝑘𝑖 [−𝑎, 0, 1]

(5.5)

𝑉𝑖3,𝑘 = 𝐿𝑘𝑖 [0, 𝑎, 1]

(5.5)

𝑉𝑖4,𝑘 = 𝐿𝑘𝑖 [0, −𝑎, 1]

(5.6)

𝑉𝑖5,𝑘 = 𝐿𝑘𝑖 [

𝑎

√2 √2

𝑉𝑖6,𝑘 = 𝐿𝑘𝑖 [− 𝑉𝑖7,𝑘 = 𝐿𝑘𝑖 [

𝑎

,

𝑎

,

, 1]

𝑎

√2 √2

, 1]

(5.7) (5.8)

−𝑎 −𝑎 , , 1] √2 √2

(5.9)

𝑎 −𝑎 , , 1] √2 √2

(5.10)

𝑉𝑖8,𝑘 = 𝐿𝑘𝑖 [

𝑉𝑖9,𝑘 = 𝐿𝑘𝑖 [0, 0, 1]

(5.11)

In the first iteration (k=1), these load vectors all have equal magnitude (because 𝐿1𝑖 =1) as illustrated in Figure 48, step 4. Applying the j load vectors and superimposing the j resulting nets leads to Figure 48, step 5. Subsequently, the envelope of all these nets is taken (Figure 48, step 6) retaining only a top and bottom shell surface. This is accomplished numerically by defining n vertical lines through the n points of the starting grid (Xi , Yi , Zi) and finding the intersection points of these lines with the j nets, resulting in n series of j intersection points with coordinates

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𝑘,𝑗

(Xi , Yi , 𝑍𝑇,𝑖 ) for each iteration k. Retaining the n top nodes (𝑁𝑇𝑜𝑝,𝑖 ) that correspond to the intersection points with the highest z-coordinate𝑠 𝑍𝑇𝑜𝑝,𝑖 , and similarly the n bottom nodes (𝑁𝐵𝑜𝑡𝑡𝑜𝑚,𝑖 ) that correspond to the intersection points with the lowest z-coordinate 𝑍𝐵𝑜𝑡,𝑖 of the surfaces as in equations 5.12 and 5.13, allows one to reconstruct the enveloping top and bottom surface displayed in Figure 48, step 6: 𝑘,𝑗

(5.12)

𝑘,𝑗

(5.13)

𝐹𝑜𝑟 𝑖 = 1. . 𝑛:

𝑍𝑇𝑜𝑝,𝑖 = max(𝑍𝑇,𝑖 𝑤𝑖𝑡ℎ 𝑗: 1. .9)

𝐹𝑜𝑟 𝑖 = 1. . 𝑛:

𝑍𝐵𝑜𝑡,𝑖 = min(𝑍𝑇,𝑖 𝑤𝑖𝑡ℎ 𝑗: 1. .9)

The rise (r) of the shell can be influenced by changing the stiffness of the cable elements so that the highest top node 𝑁𝑇𝑜𝑝, 𝑚 (with index i=m and height 𝑍𝑇𝑜𝑝,𝑚 ) is at the desired rise from the starting flat grid, by satisfying the constraint provided in equation 5.14: 𝑍𝑇𝑜𝑝,𝑚 − 𝑍𝑚 = 𝑟

(5.14)

Figure 48 - Form finding process step by step for a shell with square plan simply supported on the four corners. 5.2.2.b Adjustments for non-uniform weight distribution The form finding method thus far assumed that the loads at every node of the grid are equal in magnitude and therefore, the mass of the shell is uniformly distributed. For thin, double-layer shells with a form of lightweight connectors, this assumption is fair and the form finding process 77

does not require further iterations. Due to the setup of the form finding process, it is guaranteed that a funicular net can be found that fits within the top and bottom surfaces for each of the design loads. Thus, a compressive load path will exist during any of the design loads and the shell will be stable under said loading. However, if one considers the bottom and top layers as the boundaries of a continuous shell with varying thickness (or the connectors between the double-layer shells have significant weight), the initial assumption of uniform load distribution needs to be updated. The load vector multiplier 𝐿𝑘𝑖 in such a situation is not equal for all nodes but is a function of the local thickness of the shell. Therefore, 𝐿𝑘𝑖 is updated based on the magnitudes of the distance between the bottom and top layers (see Figure 49, step 2 and eq. 5.15). 𝐿𝑘𝑖 = 𝑍𝑇𝑜𝑝,𝑖 − 𝑍𝐵𝑜𝑡,𝑖

(5.15)

This update leads to new load vectors (eqs. 5.4 to 5.11) and thus new nets and updated shapes for each iteration until the thicknesses (and 𝐿𝑘𝑖 ) have stabilized. Depending on the chosen parameters, the load vector multipliers 𝐿𝑘𝑖 typically converge after 2 or 3 iterations. After these iterations, a funicular net can again be found (by also adjusting the stiffness parameter of the net) so that it fits between the top and bottom layers (see Figure 49, step 3), and therefore the shell will stand under the design horizontal load. A detailed overview of this method is presented in the flow chart provided in Figure 50.

Figure 49 - A funicular net (dark blue in step 3) can be found that fits between top and bottom layer, using the updated loads based on the distances between top and bottom layers.

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Figure 50 – Flowchart of form finding process to obtain top and bottom layer of double-layer shells.

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5.3 Results of the form finding process 5.3.1 Square plan In Figure 51, 15 variations of shell forms obtained through this method for a horizontal acceleration of 0.3 g are shown. The base grid was a square of 8 m by 8 m with each side divided in 10 equal length segments resulting in 121 nodes. Because of the assumption of zero-tensile strength with unlimited compressive strength, the obtained shapes are scalable (for as long as the compressive stresses in the shell do not exceed the ultimate compressive strength of the material). A wide variety of shell shapes can therefore be obtained starting from a square planar grid. The factors determining the final form are the support conditions and net stiffness. Only pinned supports (marked in red) are used, assuming the shells cannot transfer bending moments to the foundation. Once the base grid and support conditions are defined only one parameter needs to be determined to obtain a final shape. For example, the rise of the shell can be chosen as being half its span, and the stiffness of the cable net is adjusted iteratively to obtain such rise (as was done for the top left shape of Figure 51). The actual value of this fictitious axial elastic stiffness parameter of the cable element is of little relevance as the study results in compression-only shapes independent of material properties. Figure 51 demonstrates that the resulting shells can be rather thick in comparison to their span. Thick shells, however, are not very desirable as they require more material, and thus increase the mass, which has a double negative impact during seismic action. Heavier shells will inevitably experience larger base shear, as these forces are directly proportional to the mass. Furthermore, the eigenfrequencies of heavy shells will drop (i.e. periods will become longer), making them more susceptible to dynamic amplification. Thus, it is proposed to construct two thin shells, which are interconnected intermittently by diaphragms. With this appraoch the shells’ depth is guaranteed, but their mass remains low. It should further be noted that the proposed shapes in Figure 51 are illustrations of the outcomes of the form finding process and their usefulness for construction strongly depends on the chosen construction material. For example, masonry shells ideally have line supports, rather than point supports (as in the shape obtained in Figure 49). Line supports allow for more compressive load paths and are less susceptible to movement or settlement. If a point support of such a masonry structure moves relative to the other supports, important bending moments will arise in these stiff shells, which requires the shells to have some tensile capacity. These shapes are still relevant though, as point supports can be suitable for reinforced concrete structures (which have some tensile capacity) because they can be more easily base-isolated in highly seismic areas.

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Figure 51 - Top and bottom layers of shells that can accommodate a funicular net under a horizontal acceleration of 0.3 g. Red crosses are the locations of pinned supports.

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Comparing the top left shell (with 4 point supports), and bottom right shell (4 curved line supports) in Figure 51 also shows that using curved line supports effectively reduces the shell thickness at the top, which is an additional benefit for most shells. Similarly, shells with linear supports, like the middle ones in the 1st and 2nd rows in Figure 51, result in thin shells with shapes corresponding to the cross-sections obtained through a 2D-form finding process for arches in Chapter 3. Overall, Figure 51 provides a starting point for the discussion of more appropriate shapes for double-layer thin shells in earthquake areas.

5.3.2 Different plan geometries There are two main differences between the method presented in Section 5.2 and the method presented in Chapter 4 to find the shapes of corrugated shells. One major distinction is that the shells obtained in Chapter 4 are single-layer with constant thickness in contrast to the resulting single-layer shells of varying thickness or double-layer uniform thickness shells of this chapter. The other primary difference is that the method presented in Chapter 4 is only able to take into account horizontal accelerations that act along one axis, requiring supporting bays in the direction perpendicular to the design acceleration’s direction, so that stability is provided through diaphragm action. This limits the possibilities of plan sizes or necessitates some creative workarounds such as buttressing. In contrast, the major benefit of using the 3D-nets presented in this chapter is that forms can be obtained that do not require repeating bays or specially designed buttresses. This method allows for the design of asymmetrical shell forms, as illustrated in Figure 51. Moreover, the 3D form finding method can be used to obtain shapes for irregular plans, as exemplified in Figure 52 for a shape form-found for an acceleration of 0.3g.

Figure 52 – Shell shape with asymmetric plan form-found for 0.3 g horizontal acceleration.

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5.4 Form manipulations to align crowns The parametric study provided in Chapter 3 demonstrated that for arches subjected to in-plane seismic loading it is beneficial to have thin crowns and thick supports. Similarly, in Chapter 4 it was shown that shell shapes that ensure greater depth at the supports performed well under seismic loading. The presented form finding approach in Section 5.2, however, generates shells with thick crowns and thin supports, which may be suboptimal when constructing certain types of shells using low-tensile materials. To accommodate the aligning of crowns rather than supports, the form finding method is reformulated, incorporating elements of the approach presented in Chapter 4 for corrugated shells into the approach outlined in Section 5.2. The essence of the method remains unchanged, as a 3D inverted hanging chain model solved in a dynamic relaxation approach is still used to find shell surfaces that withstand combined gravity and horizontal accelerations in any direction. However, before taking the envelopes of the superimposed surfaces obtained through different load cases, the surfaces are moved relative to one another to generate support thickness. The form finding process starts in a similar fashion with a flat grid of nodes, some of which are assigned as supports (see Figure 53, step 1). Subsequently, uniform loads are assumed at all nodes due to the gravity acceleration and an equivalent horizontal acceleration (for example, in the positive x-direction as in Figure 53, step 2). This horizontal acceleration is applied in the same 4 directions (and each time with positive and negative orientations) as above, and corresponding surfaces are found using 3D hanging net models obtained through a dynamic relaxation solver for each load direction. Then the method is slightly adapted as follows. The load vectors can be grouped into pairs along acceleration directions (where each vector in the pair has identical direction but opposite orientation). For example, Figure 53, steps 3 and 4 display the load vectors and resulting surfaces for a first pair of load combinations in which the horizontal acceleration acts in the positive (green) and negative (red) x-direction. 3 additional pairs are obtained by rotating this initial pair 45 degrees at a time, resulting in surfaces that are funicular for gravity and horizontal acceleration in the positive and negative y-direction and along the diagonals through the xy-quadrants (see eqs. 5.7 to 5.11). Each net is then moved by a distance ds (depth of the support) along the direction of the corresponding applied horizontal acceleration but in the opposite orientation (see Figure 53, step 5 and Figure 54 for a close-up). The magnitude of this depth of support ds is chosen so that the crowns of the nets align. As a first estimate for ds the optimal support thickness for an arch with similar rise-to-span ratio and form-found for the same acceleration can be used (this data is provided in Chapter 3, Section 3.3.1). The other surfaces obtained for accelerations in the other directions are moved by the same distance and again along the direction with opposite orientation of their respective design horizontal accelerations (see Figure 53, steps 6 and 7). This

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results in a collection of surfaces as displayed in Figure 53, step 7. Taking the envelope of these curves allows the generation of a top and bottom surface as explained in the previous section (see Figure 53, step 8). The result is a shell with supports that have greater depth and a thin crown. The rise of the shell as well as the size of the supports and the associated crown thickness can be manipulated through changing the elastic stiffness (ES) and depth of support ds in the form finding process. Finally, Figure 53, step 9 demonstrates that the initial net fits within the envelope of the new shell, thus guaranteeing a compressive load path for the assumed loads, which can again be updated iteratively using the process described in Section 5.2.2.b.

Figure 53 – Form finding process for double-layered shells that allows for aligned crowns and thicker supports.

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Figure 54 – Example of moving a structural surface form-found for acceleration in positive xdirection for a distance ds in negative x-direction.

5.5 Conclusions A form finding approach for shell structures that takes self-weight and earthquake loading into account from the start of design process was presented in this chapter. Rather than relying on 2D thrust lines, as in Chapters 3 and 4, the method relies on 3D nets obtained using a spring network model solved using dynamic relaxation. By taking the envelope of a set of funicular surfaces under both gravity and horizontal accelerations in multiple directions, shapes are obtained that can accommodate a compressive load path under the design earthquake load. As the method relies on 3D nets, the approach can be applied to find shapes for continuous shells with a variety of plans, including asymmetrical ones. The shells can furthermore be constructed from any type of material that can withstand compression stresses. The presented method is used to generate a variety of scalable shell shapes that can resist an acceleration of 0.3 g in any horizontal direction. Because most shell shapes would require significant thickness, adding undesirable additional mass, it is suggested to construct the shells out of two interconnected layers of thin shells. Furthermore, as it can be beneficial for shell structures made of low-tensile materials to have thick supports and thin crowns, a methodology to remodel the shells shifting structural depth from crowns to supports is presented as well. The forms obtained in this chapter are the first instances reported in literature of compressiononly shells form-found for earthquake loading, considering all possible orientations for seismic loading. Furthermore, this chapter pioneers the idea of generating structural depth for shells in earthquake areas by building double-layer interconnected shells.

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Conclusions and Further Research In this chapter the main conclusions of this dissertation are summarized, and it is shown how the research objectives presented in Chapter 1 have been accomplished. In addition, recommendations for further research for the form finding of shells subjected to seismic loading are discussed.

6.1 Solutions to research questions This dissertation presents the first research that addresses the form finding of arches and shell structures while accounting for seismic loading from the start of the design process. Three new methods are presented to find material-efficient and safe arch and shells subjected to earthquake loading. The first method focuses on arches (Chapter 3), the second on corrugated shells (Chapter 4), and the third on single-layer shells with varying thickness and double-layer shells with uniform thickness (Chapter 5). Initial research on the behavior of shell structures in earthquake areas (Chapter 2) demonstrated that shells have intrinsically good properties to withstand earthquakes. Shells tend to be lightweight, and thus the induced forces they experience due to seismic actions, which are proportional to mass, remain low. Moreover, shells usually have a high geometric stiffness, as they are typically doubly- or singly- curved surfaces. This combination of low mass and high stiffness leads to beneficial vibrational properties by ensuring high eigenfrequencies (short periods). Because of these high frequencies, shells are less at risk for dynamic amplification of low-frequency content seismic events. Empirical evidence supports the position that shells have intrinsically good properties to withstand earthquakes. Indeed, while several examples of survival and good behavior of reinforced concrete shells have been reported in literature, none of these shells have been known to collapse due to seismic action. This does not mean that all shells will automatically be earthquake-proof. Repairable damage was observed for example, in the Los Manantiales reinforced concrete shell in Mexico City after the 2017 earthquake there, likely because of a local exceedance of the tensile capacity of the shell due to support movements. Furthermore, several examples of collapses of masonry shells, which have lower tensile capacity than their reinforced concrete counterparts, have been reported. These collapses could typically be attributed to excess weight on top of the shell, failure of the support system, or improper shell form.

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Form finding for shells under earthquake loads had not been performed before this dissertation though, and most conceptual design of shell form focused predominantly on static loads such as self-weight. To incorporate seismic loading into the form finding process to create a rapid and efficient first-order design approach, the earthquake design load was simplified from a dynamic multi-directional load, to an equivalent horizontal load, thus neglecting dynamic rocking behavior and causing the problem to be quasi-static. The multi-directional character of earthquakes was another factor that needed to be accounted for, which was accomplished by sequentially applying the equivalent horizontal load in different directions, and subsequently guaranteeing that a compression-only load path existed under every combination of gravity and the equivalent horizontal accelerations. For arches (see Chapter 3), the form finding implementation relied on thrust line analysis, an equilibrium approach rooted in graphic statics. Through a methodological application of a set of geometric manipulations of a thrust line, forms were obtained for arches with varying thickness that could resist in-plane design accelerations by guaranteeing a compression-only load path under combined gravitational and horizontal loading. During this form finding process, the size of the supports of the arches could be manipulated to produce a variety of shapes. A parametric study showed that the most material-efficient arch shapes had thick supports and thin crowns and that these newly obtained shapes could significantly reduce the amount of material needed to create earthquake-resistant arches compared to more commonly used circular or catenary arches, regardless of the considered horizontal acceleration. Furthermore, the results of a parametric study for arches with different rise-to-span ratios subjected to a variety of accelerations, allowed for the formulation of design guidelines for arches in seismic areas. Because the work was carried out independent of scale and material properties, it can be applied to construct arches of different dimensions in a variety of materials that can withstand compressive forces. This method created for arches was further developed into a form finding approach for 3D corrugated shells covering a rectangular plan (see Chapter 4). The resulting shells consist of repeating bays that can withstand horizontal accelerations acting in either direction, while relying on diaphragm action provided by the adjoining bays in the perpendicular direction. The method ensures that a compression load path exists to carry forces induced by lateral accelerations by deriving shell geometries from a series of funicular polygons which are again obtained through a graphic statics procedure for combined gravity and horizontal accelerations, as was the case for arches. Rather than relying on varying thickness to provide a thick section that can accommodate a compression load path, the shells have uniform thickness but are doubly-curved instead. This additional curvature is provided in the direction perpendicular to the in-plane arched section to produce structural depth and thus effective thickness, without the need of additional material use. The effectiveness of this methodology was demonstrated by comparing the pushover capacity of the form-found shells to the capacity of a similar non-form-found shell using non-linear pushover analyses. These analyses showed that the lateral capacity before cracking of the corrugated shell shapes was up to 79% higher than the capacity of a non-form-found reference

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shell shape considering identical material use. Moreover, the non-linear pushover analysis identified the collapse mechanisms of the shells, which could in turn be manipulated through the form finding process. This approach also allowed guidelines for the most efficient locations for placing reinforcement in the shells. Additionally, by analyzing the flow of forces within the form-found shells, it was possible to propose alternative designs with large openings in the shell surface while maintaining similar seismic capacity. The last form finding methodology presented in this dissertation (see Chapter 5) replaces the thrust line approaches of Chapters 3 and 4 with a 3D approach. This is done by performing a set of manipulations to structural 3D nets that are shaped under gravity and horizontal accelerations using spring networks solved employing dynamic relaxation. While previous methods had the limitation that accelerations could only be accounted for in two directions, the new method is able to combine nets form-found for as many acceleration directions as desired. Through a parametric study in which support conditions and net stiffness were varied, a series of new shapes of shells that can withstand a seismic event with a PGA of 0.3 g were presented. As Chapters 3 and 4 indicated that providing sufficient support depth for low-tensile capacity shells leads to more efficient material use for shells designed under seismic loads, an additional form manipulation approach to generate thin crowns and thick supports was presented as well. The resulting shapes of the form finding processes presented in Chapter 5 were achieved by building shells with varying thicknesses. To keep the mass of these shells low and thereby preserve their beneficial vibrational properties, however, it is suggested to construct them using double-layer, interconnected, thin shells. This way the necessary structural depth is provided, while maintaining the lightweight nature of the shells. Finally, all the methods presented in this dissertation yield forms that withstand seismic loading, relying solely on the compressive capacity of the shells. As compressive stresses in the structures under seismic forces remain an order of magnitude under the maximum compressive strengths of the material (for example 17 times under the ultimate compressive strength of the low strength masonry in Chapter 4), failure due to exceedance of compressive strength will not occur. Thus, for as long as this ultimate compressive strength criterion is not violated, all the examined shapes are scalable. The presented methodologies are also material-independent and can therefore be applied to construct shells from any type of material that is able to resist compression, such as concrete, stone, masonry, earth or ice. The application of these form finding methods can thus have a significant impact by allowing the construction of safe and efficient shells in earthquakeprone regions around the world.

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6.2 Recommendations for future research There are many opportunities to make further contributions to the design of shell structures in seismic areas. This section describes potential areas for research exploration that extend the work presented in this dissertation.

6.2.1 Single-layer uniform thickness shells from 3D approach The shell geometries that result from the form finding process in Chapter 5 are double-layered. Constructing double-layer shells is a structurally efficient solution that could offer opportunities, such as a concealed space for thermal insulation [61], but may also require more materials than single-layer shells. Therefore, it is worth investigating if the double-layered shells could be replaced by single-layer ones based on the 3D form finding process in Chapter 5. A similar method to the one successfully employed to turn the arches with varying thicknesses in Chapter 3 into shells of uniform thickness in Chapter 4 could be devised. This was accomplished in Chapter 4 by corrugating the edges of the shell within the boundaries of the support thickness of the arch. This process could be replicated by fitting a single-layer shell surface through the crown of the double-layer shells, and by curving its supports so that its structural depth matches the thickness of supports of the form-found shells. Another option worth investigating is a hybrid between a double- and a single-layer shell, in which the crown of the shell consists of a singlelayer thin shell, while the supports split apart into two shells. As such, the single layer shell is buttressed and a load path under seismic loads is provided.

6.2.2 Construction aspects 6.2.2.a Additive manufacturing: 3D-printing and use of UAVs The obtained forms in this dissertation are scalable and material-independent, and therefore they do not account for construction techniques. However, a major challenge for the presented shells with irregular geometries will be to build them accurately and efficiently. This construction could be accomplished using traditional techniques (such as thin-tile vaulting, reinforced concrete, or masonry construction relying on formwork), but the rapidly evolving field of shell construction using digital fabrication methods seems especially deserving of further research. Indeed, additive manufacturing (such as 3D printing [149]–[151] and unmanned aerial vehicles (UAVs)- based construction [152], [153]) largely eliminates the fabrication issues, such as the need to construct extensive temporary formwork or the reliance on skilled labor, associated with crafting complex geometries. Therefore, it is worth exploring how the form-found shapes provided in this dissertation can be cost-efficiently built using these techniques. 3D printing of structural components, either in polymers or concrete is increasingly becoming an established fabrication method, as exemplified by 3D printed buildings around the world [151].

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The technique offers a solution to fabricating structural pieces, or possibly entire shells, in nonstandardized forms such as the ones developed in this dissertation, without adding significant building costs because of the non-planar nature of the components. For example, the arch shapes with varying thicknesses obtained in Chapter 3 could be divided into segments (voussoirs) that could be 3D printed and then assembled on-site. Interlocking joints could be placed at the edges of the voussoirs, so that no mortar would be required during the construction process. Furthermore, voussoirs could largely be hollow with structurally connecting ribs to reduce the required amount of material. Arches could possibly even be printed in one piece (an example of a 3D printed, not form-found arch, is presented in Figure 55) and transported as a unit if the size of the 3D printer permits.

Figure 55 - Example of a 3D-printed arch (image DUS Architects via techcircuit.net). The size of 3D printers, however, is one of the main limitations to printing entire arches or shells as the machine needs to be larger than the manufactured structure. As indicated, this could be solved by compartmentalizing the printed structure. For the corrugated shells presented in Chapter 4, this could be accomplished by printing each bay individually after which the bays could be stacked for easier transportation. If the span of the bay is too long to fit within the printer, the bay could be split into two cantilevering sides which are connected at the crown, or an approach with limited temporary formwork or supports could be taken. An example of 3Dprinted scale models of the arch and corrugated shells is provided in Figure 56.

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Figure 56 – 3D-printed scale models of form-found arch and corrugated shells. The same approaches for 3D printing could be used to construct double-layer shells. Segments of the interconnected layers could be made from multi-layer building blocks with varying thicknesses, employing constructive systems that resemble the terra cotta segmental arches that were used in the early steel frame construction of the late 19th century [154]. Efficient distribution of the interconnecting diaphragms between top and bottom layers, and effective compartmentalization of elements, would require significant further study though.

Figure 57 - Segmental tile arch. A similar concept could be employed to construct double-layer shells by varying the size of the prefabricated blocks. Another construction approach that circumvents the size limitation imposed by 3D printers which is worth investigating, would be to use UAVs (often referred to as drones) to perform the additive manufacturing process. Instead of employing large elements to reduce assembly costs, many small elements could be assembled on the construction site through an automated process. Currently UAVs are limited in the amount of construction material they can carry per flight though [152] and thus small, lightweight building materials are advised. This approach could have significant advantages, as building materials such as soil tiles could be fabricated on-site, and once the shell supports are in place, construction could be performed semi-automatically. This might be particularly advantageous in situations when roads have been destroyed due to an earthquake and thus the transportation of materials to the construction site is hampered. The corrugated shells from (lightweight) thin-tile vaults proposed in Chapter 4, could lend themselves as a good case study for UAV-based shell construction. Because of the shape of these 91

shells, the amount of formwork could be limited, and the construction of the shell surface could be carried out almost entirely by UAVs. All the presented shapes (form-found for 0.3 g) in Chapter 4 could be built using two identical and reusable planar formwork arches. These guiding arches in the shape of funicular polygons are also the base-curves for the different NURBS surfaces. Thus, the arches could be repositioned alternating every half period (as with the polygons) and tiles could be cantilevered out from one arch to the next, before moving on to the following bay while reusing the formwork (which would need to be lowered or folded to move it). Once the arch formwork is put in place, the tiles of the arch could be flown in and placed using a programmed UAV, while another UAV pours the quick-setting mortar. Once the arches are in place, the rest of the surface can be completed by a UAV cantilevering the tiles following the prescribed geometry of the shell. Additionally, initial research on shells printed using UAVs has called for shell geometries that are designed not only to take into account self-weight, but also other horizontal loads [152] and their associated larger bending moments. As self-weight needs to be kept as low as possible for additive manufacturing purposes, other loads (such as wind, or snow) will become more important leading to a more complex preemptive design [152]. Because the methods developed in this dissertation take these horizontal loads into account from the start of the design process, they can be readily adapted for this preemptive design as they can generate shapes that account for horizontal wind loading and/or (asymmetric) snow loading. 6.2.2.b Interconnection of double-layer shells To build the double-layer shells proposed in Chapter 5, a series of additional constructional challenges will require further investigation. A main concern is the structural collaboration between the top and bottom shells, which is essential to guarantee that a compression-only thrust surface can develop over the entire thickness of the structure, despite the discrete connections. Additionally, differential movement between the shells needs to be avoided. Thus, the design of connections that can transfer shear forces between both shells is an important task that remains to be tackled. Possible solutions to interconnect the shells are to incorporate vertical masonry diaphragms, or another form of ties. Additionally, the voids between the shells could be filled with insulating material, as was done in the Mexican double shells (see Chapter 5) to generate a shell-insulation-shell sandwich. The effectiveness of these connections to transfer loads from one shell to the other needs to be analyzed in detail, which it is suggested should be done through non-linear finite element analysis validated by tests on scale models, see Section 6.2.3. 6.2.2.c Shape optimization for construction constraints Different construction methods come with varying constraints, some of which may dominate the design process. Therefore, research into incorporating additional constraints into the presented form finding methodologies can lead to forms that can be constructed more efficiently.

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One example of an important constraint that can be considered from the start, is the maximum allowable thrust exerted by the shell or arch on the support system (most likely underground anchors, steel ties or a concrete ring beam) during an earthquake. As highlighted in Chapter 2, shells and arches whose supports are prone to movements or failure due to the additional horizontal thrusts induced by seismic action, are likely to collapse or sustain severe cracking. Thus, preventing this important failure mode by limiting the thrusts will enhance the safety of the shells during seismic events. In all presented form finding methods, the thrusts are controlled directly through the formulation of form-defining thrust lines or nets. The resulting shape can thus easily be tailored by placing a cap on the maximum value of the forces that are transmitted to the foundations, which will automatically limit the resulting shell geometries. Curvature is another very important factor that governs the behavior of the proposed shell structures. In Chapter 4, for example, it was demonstrated that corrugated shells are most likely to crack at the locations of single curvature, while structural depth through double curvature was shown to increase the pushover capacity of the considered shells. Nevertheless, it may be necessary to prescribe a limit on the maximum allowable curvature to ensure constructability. For example, there is a practical limit to the local curvature of a surface when constructing shells using thin-tile vaulting, which is governed by the size of the tiles.

6.2.3 Large scale testing and dynamic loads Form finding is only the first step in the conceptual design process, and further validations of the structural capacity of the final shapes are always necessary. Typically, in this design stage it is useful to have the option to cycle through a large number of design variations relatively quickly. Therefore, it makes sense to simplify the complex seismic loads in this initial stage to avoid time-consuming computations. In this dissertation, a first-order approach was used to simulate seismic loading, the results of which were subsequently validated using the more sophisticated (but still quasi-static) non-linear pushover analysis (for corrugated shells). To get a comprehensive picture of how the designed shells behave under full dynamic loading, further analysis is suggested. The results of such an analysis could then be used to finetune and enhance the form finding methodologies and could furthermore be employed to develop detailed design guidelines for arches and shells constructed in seismic areas. Because of the variability and complexity of earthquake loading, it is recommended to conduct such further analysis through a combination of shake table testing on scale models, as well as numerical modeling. The type of numerical modeling depends strongly on the chosen building material. If a construction technique using voussoirs or discrete blocks is used, distinct element modeling is suggested. For materials which behave as a homogenous unit (such as concrete or thin-tile vaulting) non-linear pushover analyses as well as non-linear time history analyses are recommended.

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6.2.4 Gridshells Gridshells account for a significant portion of the large span roof shells that are constructed nowadays. The form finding approaches presented in this thesis are formulated for continuous shells though. All provided shell shapes and form finding methodologies could be adapted to find the initial shape of grid shells as well. Furthermore, studying the design of grid shells under earthquake loading opens a series of additional design opportunities. For example, the connections between the elements of the gridshells could be designed to influence the vibrational properties of the shell.

6.2.5 Applications for historic structures A large portion of the vaults and arches in the current built environment are part of historic structures. Some of the methods implemented in this thesis could be employed to analyze and retrofit these historic buildings. For example, the kinematic limit state analysis used for the arch validation, as well as the thrust line approach under combined gravity and seismic loads, could be employed to perform first-order seismic analysis of constructed vaults and arches. The guidelines for support thicknesses developed in Chapter 3 could additionally be used to retrofit arches by increasing the masonry sections above the supports. Furthermore, the TLA and LSA methods presented in Chapter 3 have been validated experimentally under gravity loading, also taking into account the limits on compressive strength of the material [94]. The presented expansion of the method to include horizontal acceleration could be further verified and fine-tuned using tilting tests, which reproduce equivalent horizontal accelerations. The major advantage of these simplified techniques is that they provide a quick, first-order, conservative estimation of the seismic capacity of arches and vaults. Additionally, a thrust line is an easily interpreted and intuitive graphical representation of the flow of force, in contrast to the range of intricate stress-, displacement- and crack-plots that result from finite element analyses or discrete element analysis. Therefore, they have great didactic value and can enhance the communication between preservation professionals when making intervention decisions. Conducting further study on the application of first-order methods to assess seismic capacity of arches and shells can thus have a significant impact on the conservation of monuments. In that regard, it would also be of interest to see if the dynamic relaxation methods used to find the shapes of 3D shells can be expanded to study the stability of vaults under seismic loading.

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