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.

iii

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

iv

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.

v

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

vi

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

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.

vii

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.

Table of Contents

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

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

1.3 Significance of Research........................................................................................... 4

1.4 Dissertation Organization ......................................................................................... 5

1.5 Prior Publications Contributing to this Dissertation ................................................. 6

2.1 Introduction ............................................................................................................... 8

2.2 Structural behavior of concrete shells during earthquakes ....................................... 9

2.2.1 Case study: response spectrum analysis of reinforced concrete shell ................. 10

2.2.2 Parametric study on key parameters affecting seismic behavior ........................ 15

2.3 Masonry shells in seismic areas .............................................................................. 20

2.4 Form Finding .......................................................................................................... 29

2.5 Conclusion and research gap .................................................................................. 33

3.1 Introduction ............................................................................................................. 36

3.1.1 Form finding of arches ........................................................................................ 38

3.1.2 Review of analysis techniques for masonry arches under earthquake loading ... 39

3.2 Form finding Methodology ..................................................................................... 42

3.2.1 Thrust lines due to combination of gravity and horizontal acceleration ............. 42

viii

3.2.2 Form finding procedure ....................................................................................... 46

3.2.3 Application example ........................................................................................... 51

3.2.4 Validation through kinematic limit state analysis ............................................... 54

3.3 Results ..................................................................................................................... 57

3.3.1 Support size variations for an arch with rise-to-span ratio of 1/2 ....................... 57

3.3.2 Different design acceleration .............................................................................. 59

3.3.3 Different arch rise-to-span ratios......................................................................... 60

3.4 Discussion ............................................................................................................... 61

3.4.1 Support size variations for r/s equal to 1/2 ......................................................... 61

3.4.2 Design accelerations ............................................................................................ 63

3.4.3 Rise-to-span ratios ............................................................................................... 64

3.5 Applications and limitations ................................................................................... 64

3.6 Conclusion .............................................................................................................. 69

4.1 Introduction ............................................................................................................. 71

4.1.1 Scope and outline ................................................................................................ 72

4.2 Form finding under earthquake loading .................................................................. 74

4.2.1 Form finding method for corrugated shells under earthquake loading ............... 74

4.2.2 Application to masonry shells ............................................................................. 79

4.2.3 Non-linear pushover ............................................................................................ 83

4.3 Results ..................................................................................................................... 86

4.3.1 Pushover curves................................................................................................... 87

4.3.2 Collapse mechanism and flow of forces ............................................................. 90

4.4 Discussion ............................................................................................................... 93

4.4.1 Evaluation of form finding method ..................................................................... 93

ix

4.4.2 Influence of shape on flow of forces and collapse mechanism ........................... 96

4.4.3 Design opportunities ........................................................................................... 98

4.5 Conclusion ............................................................................................................ 101

5.1 Introduction ........................................................................................................... 103

5.1.1 Advantages of double-layer shells .................................................................... 104

5.2 Form Finding Methodology .................................................................................. 107

5.2.1 Extension from thrust lines to nets .................................................................... 107

5.2.2 Implementation of form finding method ........................................................... 109

5.3 Results of the Form Finding Process .................................................................... 116

5.3.1 Square plan ........................................................................................................ 116

5.3.2 Different plan geometries .................................................................................. 118

5.4 Form manipulations to align crowns..................................................................... 119

5.5 Conclusions ........................................................................................................... 123

6.1 Solutions to research questions ............................................................................. 124

6.2 Recommendations for Future Research ................................................................ 129

6.2.1 Single-layer uniform thickness shells from 3D approach ................................. 129

6.2.2 Construction aspects .......................................................................................... 130

6.2.3 Large scale testing and dynamic loads .............................................................. 135

6.2.4 Gridshells .......................................................................................................... 136

6.2.5 Applications for historic structures ................................................................... 137

1

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 Mw 1994 Northridge

earthquake, which was responsible for 57 deaths and economic losses of $46 billion [4].

Year Mw Fatalities Location

2017 7.3 and 7.1 1,000 Iran and Mexico

2016 7.8 676 Ecuador

2015 7.8 8,898 Nepal

2014 6.2 729 Wenping, China

2013 7.7 825 Pakistan

2012 6.7 113 Philippines

2011 9 20,896 Honshu, Japan

2010 7 316,000 Haiti

2009 7.5 1117 Sumatra, Indonesia

2008 7.9 87,587 Sichuan, China

2007 8 514 Pisco, Central Peru

2006 6.3 5,749 Java, Indonesia

2005 7.6 80,361 Pakistan

2004 9.1 227,898 Northern Sumatra

Table 1.1 - Earthquakes and fatalities worldwide since 2004.

2

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

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,

3

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/m

2 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.

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 earthquake-

prone 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. Identify the key parameters that affect the seismic behavior of continuous shell structures.

4

ii. Implement efficient first-order equilibrium analysis techniques to analyze arches under

seismic loading.

iii. Develop and validate a form finding approach for arches subjected to in-plane earthquake

loading.

iv. Establish and validate a form finding methodology for corrugated shells subjected to

seismic loading.

v. Devise a form finding methodology for double-layer shells that can withstand earthquake

loading.

vi. 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

5

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.

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.

6

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. 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.

ii. 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.

iii. 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.

iv. 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.

v. 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.

7

Conference papers:

i. 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.

ii. 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.

iii. 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 co-

authors for these works served primarily as advisers and editors.

8

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

9

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

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.

10

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 Figure 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

11

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.

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

12

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).

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

13

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.

14

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

0

300

600

900

1200

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Acc

eler

atio

n r

esp

on

se

spec

tru

m (

cm/s

2 )

Frequency (Hz)

Response Spectrum Mexico City Earthquake (EW-direction)

All eigenfrequencies higher

than 3.09 Hz

15

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.

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

16

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

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 in-

plane 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

17

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

18

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

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

19

(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].

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.

20

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,

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.

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

21

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 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

22

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].

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).

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

semi-domes

buttresses

23

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 was 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].

24

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 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], [55]. 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 [56], 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) [57], [58].

25

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 [57], [58]. A second (2) factor was the thickness of the vaults: thicker

vaults performed better than their thinner counterparts [57] 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) [58]. 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

[57]. Finally, also the quality of workmanship (5) and level of prior deterioration (6) played a

26

role [56], [58]. It is also noteworthy that most of the recently constructed brick masonry domes

and vaults in the city of Bam performed satisfactorily during the earthquake [57], [58]. 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 [57], [58].

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 [59]. 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) [59], [60]. All vaults had a thickness of 12 to 15 cm [60]. As the outward

thrust of the vaults was limited due to the vaults’ thinness, supports of small concrete ring beams

or steel-reinforced masonry proved sufficient.

Figure 11 – Pavilion vault (left) and Gaussian vault (right) that sustained three earthquakes

in quick succession (Images Ramón Aguirre Morales).

27

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, this outward thrust 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

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 [61].

28

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).

29

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)

[62]. Alternatively, researchers have employed computational models to numerically simulate

such hanging models, or to guide shell design through numerically controlled parameters [9].

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 [63].

30

Geometric stiffness methods, such as the Force Density Method (FDM) [64] or Thrust

Network Analysis (TNA) [65] 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) [66]. 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], [67], [68]. 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 [69]. 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 [65]. The method has been reformulated, however, to serve as an

analysis tool in a seismic context [70], 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.

31

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 form-found using thrust network analysis.

Dynamic equilibrium methods, such as Dynamic Relaxation (DR) [71] and Particle Spring

systems (PS) [72], sometimes referred to as hanging cloth models [73] (see Figure 15), solve a

dynamic equilibrium problem to obtain a static equilibrium solution [63]. 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 [74] and the

Dutch Maritime Museum (Ney & Partners, The Netherlands, 2011) [75] (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.

32

Figure 15 - Digital reproduction obtained with dynamic relaxation solver of Isler's hanging

net displayed in Figure 13.

Figure 16 - The grid shell covering the courtyard of the Dutch Maritime Museum was form-

found 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.

33

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 (0.4-0.7 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

34

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.

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

35

increased outward thrust of the shell due to earthquakes (this occurred in the vaults of Arg-e-

Bam 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.

36

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 in-plane 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 2-

dimensional 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 [76]. 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

37

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

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 3.3. 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 self-weight, regardless of acceleration magnitude. The findings of this research will thus

allow for the more material-efficient design of masonry arches in seismic areas.

38

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 [77] and Coulomb’s studies on hinge

formation and sliding in arches [78] 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 [79] and [80]. 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], [81], 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 [82]–[84]. Peng carried out an interesting

study employing limit state analysis in combination with a genetic algorithm to find the form of

arches [85] and shape optimization tools were used to design concrete [86] 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 [87]. 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 [88]. 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

39

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.

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 [81]. 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 [89], [90]. 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 [65], [69]. 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 [91]. One of the

40

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 [70] 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 [92]. 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

[93]. 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 [94] 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 [95]. It

determines the critical collapse mechanism and associated acceleration needed to activate this

mechanism through a series of virtual work calculations [96]. Kinematic limit analysis has been

automated and applied to a set of different arch geometries with constant thickness and has been

validated experimentally and numerically by several authors [97]–[99]. 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

41

bound solution to the same problem solved by thrust line analysis [92]. Upper bound limit state

analysis has also been expanded to analyze three-dimensional vaults [100] and has been applied

in the context of seismic loading [101].

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

[102], 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-

of-freedom (SDOF) system [102], [103]. 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

non-linear 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 [89] and therefore, computationally expensive non-linear analyses are necessary.

These non-linear finite element models have nevertheless been applied successfully to masonry

arch bridges [104] and seismic analyses were performed through non-linear static pushover

simulations [105], [106] and non-linear dynamic analysis, which provided similar results [107].

Non-linear FEM requires high expertise from its operators but has the advantage of being able to

42

capture three-dimensional effects. DEM can similarly capture these three-dimensional effects

and inherently incorporates the heterogeneous nature of masonry. This method relies on finite-

differences 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 [89], [108]. This method has

been applied to arches under seismic action [98], [102], [103] and its results were found to

correspond well with data obtained through analytical limit state analysis [98], [103] and

experimental tests [89].

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 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 [89], [93].

3.2 Form finding me